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arXiv:physics/0102080v1 [physics.atom-ph] 23 Feb 2001Resonant ddµFormation in Condensed Deuterium Andrzej Adamczak∗ Institute of Nuclear Physics, Radzikowskiego 152, PL-31-3 42 Krak´ ow, Poland Mark P. Faifman† Russian Scientiﬁc Center, Kurchatov Institute, RU-123182 Moscow, Russia (Dated: February 2, 2008) Abstract The rate of ddµ muonic molecule resonant formation in dµatom collision with a condensed deuterium target is expressed in terms of a single-particle response function. In particular, ddµ formation in solid deuterium at low pressures is considered . Numerical calculations of the rate in the case of fcc polycrystalline deuterium at 3 K have been per formed using the isotropic Debye model of solid. It is shown that the energy-dependent ddµ formation rates in the solid diﬀer strongly from those obtained for D 2gaseous targets, even at high dµkinetic energies. Monte Carlo neutron spectra from ddfusion in ddµmolecules have been obtained for solid targets with diﬀerent concentrations of ortho- and para-deuterium. The recent experimental results performed in low pressure solid targets (statistical mixture of ortho -D2and para-D 2) are explained by the presence of strong recoilless resonance peaks in the vicini ty of 2 meV and very slow deceleration ofdµatoms below 10 meV. A good agreement between the calculated a nd experimental spectra is achieved when a broadening of D 2rovibrational levels in solid deuterium is taken into accou nt. It has been shown that resonant ddµformation with simultaneous phonon creation in solid gives only about 10% contribution to the fusion neutron yield. The neutron time spectra calculated for pure ortho-D 2and para-D 2targets are very similar. A practically constant value of th e mean ddµ formation rate, observed for diﬀerent experimental condit ions, is ascribed to the fact that all the recent measurements have been performed at temperatures T/lessorsimilar19 K, much lower than the target Debye temperature Θ D≈110 K. In result, the formation rate, obtained in the limit T/ΘD≪1, depends weakly on the temperature. PACS numbers: 34.10+x, 36.10.Dr ∗Electronic address: andrzej.adamczak@ifj.edu.pl †Electronic address: faifman@imp.kiae.ru 1I. INTRODUCTION Theoretical study of resonant formation of the muonic molec uleddµin condensed deu- terium targets is the main subject of this paper. The resonan tddµformation, ﬁrst observed by Dzhelepov and co-workers [1], is a key process of muon cata lyzed fusion ( µCF) in deu- terium (see e.g. reviews [2, 3]). A muonic deuterium atom dµis created when a negative muonµ−is captured into an atomic orbital in a deuterium target. Aft erdµdeexcitation to the 1Sstate and slowing down, the ddµmolecule can be formed in dµatom collision with one of the D 2target molecules. The resonant formation is possible due to presence of a loosely bound state of ddµ, characterized by the rotational number J= 1 and vibra- tional number v= 1, with binding energy \|εJv=11\| ≈1.97 eV. This energy, according to the Vesman mechanism [4], is completely transferred to exci ted rovibrational states of the molecular complex [( ddµ)dee]. The scheme of calculation of ddµformation rate in gaseous deuterium has been developed for many years [5, 6, 7, 8], and h as lead to a good agreement with the experiments performed in gaseous targets [9, 10]. O n the other hand, this theory, when directly applied to solid deuterium targets, leads to s trong disagreement with the ex- perimental results [11, 12, 13]. Therefore, it is necessary to calculate the ddµformation rate with solid state eﬀects taken into account, which is the main purpose of this paper. Our calculations are based on the theoretical results (tran sition matrix elements, reso- nance energies) obtained in the case of ddµformation in a single D 2molecule. In Sec. II the main formulas used for this case are brieﬂy reported. A ge neral formula for the energy- dependentddµformation rate in a D 2condensed target is derived in Sec. III, using the Van Hove formalism of the single-particle response function [1 4]. This formula is then applied (Sec. IV) for harmonic solid targets, in particular for a cub ic Bravais lattice. A phonon expansion of the response function is used to study phonon co ntributions to the resonant formation. Numerical results for 3 K zero pressure frozen de uterium targets (TRIUMF ex- perimental conditions [11, 13]), with the fcc polycrystall ine structure, are shown in Sec. V. The formation rates have been calculated assuming the isotr opic Debye model of the solid and the values of Debye temperature and lattice constant obs erved in neutron scattering experiments. The calculated rates of resonant ddµformation and back decay have been used for Monte Carlo simulations of ddfusion neutron and proton time spectra. Since the initial di stributions of 1Smuonic atom energy contain contributions from hot dµ’s (∼1 eV) [15, 16], inﬂuence of slow deceleration of dµatoms below 10 meV [17] on these time spectra is investigated in Sec. VI. The simulations take into account the processes o f incoherent and coherent dµ atom scattering in solid deuterium. In particular, the Brag g scattering, phonon scattering, and rovibrational transitions in D 2molecules are included. We consider a dependence of the resonant formation rate and time spectra on broadening o f the rovibrational D 2energy levels, due to the binding of the molecules in the lattice [18 ]. Since it has been predicted in Refs. [19, 20, 21] that strong ddµformation takes place only in solid para-D 2, study of this process in pure ortho-D 2and para-D 2targets is another aim of this work. The neutron spectra calculated for these two so lids are discussed in Sec. VI. 2II. RESONANT FORMATION IN A FREE MOLECULE First we consider resonant formation of the ddµmolecule in the following reaction (dµ)F+ (D 2)I νiKi→/bracketleftbig (ddµ)Jv Sdee/bracketrightbig νfKf, (1) where D 2is a free deuterium molecule in the initial rovibrational st ate (νiKi) and the total nuclear spin I. The muonic atom dµhas total spin Fand CMS kinetic energy ε. The complex [( ddµ)dee] is created in the rovibrational state ( νfKf) and the molecular ion ddµ, which plays the role of a heavy nucleus of the complex, has to tal spin S. The rate λSF νiKi,νfKfof the process above depends on the elastic width ΓSF νfKf,νiKiof [(ddµ)dee] complex decay [22, 23, 24, 25] in reactions ΓSF νfKf,νiKi− − − − − − → (dµ)F+ (D 2)I νiKi/bracketleftbig (ddµ)Jv Sdee/bracketrightbig νfKf − → ˜λfstabilization processes,(2) where ˜λfis the total rate of the stabilization processes, i.e. deexc itation and nuclear fusion inddµ ddµ→  µ+t+p+ 4.0 MeV µ+3He +n+ 3.3 MeV µ3He +n+ 3.3 MeV.(3) When fusion takes place, the muon is generally released and c an again begin the µCF cycle. However, sometimes the muon is captured into an atomic orbit al of helium (sticking), which stops further reactions. The value of ΓSF νfKf,νiKiis given in atomic units ( e=/planckover2pi1=me= 1) by the formula ΓSF νfKf,νiKi= 2πAif/integraldisplayd3k (2π)3\|Vif(ε)\|2δ(εif−ε), (4) whereVif(ε) is a transition matrix element and εifis a resonance energy deﬁned in Ref. [8]. The factor Aifis due to averaging over initial and summing over ﬁnal projec tions of spins and angular momenta of the system. Vector kis the momentum of relative dµand D 2 motion ε=k2/2M, (5) andMis the reduced mass of the system. Integration of Eq. (4) over kleads to ΓSF νfKf,νiKi=Mkif πAif\|Vif(εif)\|2, k if=k(εif). (6) Since ΓSF νfKf,νiKiand˜λfare much lower ( ∼10−3meV) than ε, Vesman’s model can be applied and the energy-dependent resonant formation rate has the Di rac delta function proﬁle λSF νiKi,νfKf(ε) = 2πNB if/vextendsingle/vextendsingleVif(ε)/vextendsingle/vextendsingle2δ(ε−εif). (7) 3whereNis the density of deuterium nuclei in the target. According t o Ref. [8] the coeﬃcients AifandBifin the above equations are equal to Aif= 4WSFξ(Ki)2Ki+ 1 2Kf+ 1, Bif= 2WSF2S+ 1 2F+ 1,(8) where WSF= (2F+ 1)/braceleftbigg1 21F 1S1/bracerightbigg2 , ξ(Ki) =/braceleftBigg 2 3forKi= 0, 1 3forKi= 1,(9) and the curly brackets stand for the Wigner 6 jsymbol. In formula (8) the usual Boltzmann factor describing the population of rotational states in a g as target is omitted because we calculate the formation rate separately for each initial ro tational state. If the muonic atoms in a gas have a steady kinetic energy distribution f(ε,T) at target temperature T, Eq. (7) can be averaged over the atom motion leading to a mean resonan t rateλSF νiKi,νfKf(T). III. RESONANT FORMATION IN A CONDENSED TARGET Since a muonic deuterium atom can be approximately treated a s a small neutron-like particle, methods used for description of neutron scatteri ng and absorption in condensed matter are applicable in the case ddµformation in dense deuterium targets. Below we adapt the method developed by Lamb [26], and then generalized by Si ngwi and Sj¨ olander [27] using the Van Hove formalism of the single-particle respons e function Si[14], for calculation of the resonant ddµformation rates. A Hamiltonian Htotof a system, consisting of a dµatom in the 1 Sstate and a heavy condensed D 2target, can be written down as follows Htot=1 2Mdµ∇2 Rdµ+Hdµ(r1) +HD2(̺1) +V(r1,̺1,̺2) +H, (10) whereMdµis thedµmass and Rdµdenotes the position of dµcenter of mass in the coor- dinate frame connected with the target (see Fig. 1). Operato rHdµis the Hamiltonian of a freedµatom,r1isdµinternal vector; HD2denotes the internal Hamiltonian of a free D 2 molecule. It is assumed that ddµformation takes place in collision with the l-th D 2target molecule. The position of its mass center in the target frame is denoted by Rl;̺1is a vector connecting deuterons inside this molecule. Function Vstands for the potential of the dµ–D2 interaction [8], leading to ddµresonant formation. Vector ̺2connects the dµand D 2centers of mass. We neglect contributions to the potential Vfrom the molecules other than the l-th molecule because we assume here that distances between diﬀe rent molecules in the target are much greater than the D 2size. The kinetic energy εof thedµatom and its momentum k in the target frame are connected by the relation ε=k2/2Mdµ. (11) 4The Hamiltonian Hof a pure D 2target, corresponding to the initial target energy E0, has the form H=/summationdisplay j1 2Mmol∇2 Rj+/summationdisplay j/summationdisplay j′/negationslash=jUjj′, (12) whereRjis the position of j-th molecule center of mass in the target frame (Fig. 2), Ujj′de- notes interaction between the j-th andj′-th molecule, and Mmolis the mass of a single target molecule. The coordinate part Ψ totof the initial wave function of the system can be written as a product Ψtot=ψ1S dµ(r1)ψνiKi D2(̺1) exp(ik·Rdµ)\|0/angb∇acket∇ight, (13) where \|0/angb∇acket∇ightstands for the initial wave function of the condensed D 2target, corresponding the total energy E0. Eigenfunctions of the operators HdµandHD2are denoted by ψ1S dµandψνiKi D2, respectively. Using the relation Rdµ=Rl+̺2, the wave function Ψ tottakes the form Ψtot=ψ1S dµ(r1)ψνiKi D2(̺1) exp(ik·̺2) exp(ik·Rl)\|0/angb∇acket∇ight, (14) which is similar to that used in the case of ddµformation on a single D 2, except the factor exp(ik·Rl)\|0/angb∇acket∇ight. This factor depends only on positions of mass centers of the target molecules. After formation of [( ddµ)dee] complex, the total Hamiltonian of the system is well ap- proximated by the operator H′ tot Htot≈H′ tot=Hddµ(r,R) +HC(̺) +V(̺,r,R) +/tildewideH, (15) whereHddµis an internal Hamiltonian of ddµmolecular ion, vectors randRare its Jacobi coordinates. Relative motion of ddµanddin the complex is described by a Hamiltonian HC which depends on the respective internal vector ̺. The ﬁnal Hamiltonian /tildewideHof the target, with the eigenfunction \|/tildewiden/angb∇acket∇ightand energy eigenvalue /tildewideEn, is expressed by the formula /tildewideH=1 2MC∇2 Rl+/summationdisplay j/negationslash=l1 2Mmol∇2 Rj+/summationdisplay j/summationdisplay j′/negationslash=jUjj′ =−/parenleftbigg 1−Mmol MC/parenrightbigg1 2Mmol∇2 Rl+H= ∆H+H,(16) whereMCis the mass of the complex. The respective coordinate part Ψ′ totof the total ﬁnal wave function of the system is Ψ′ tot=ψJv ddµ(r,R)ψνfKf C(̺)\|/tildewiden/angb∇acket∇ight. (17) whereψJv ddµandψνfKf Cdenote eigenfunctions of the Hamiltonians HddµandHC, respectively. The energy-dependent resonant ddµformation rate λSF νiKi,νfKf(ε) in the condensed target, for the initial \|0/angb∇acket∇ightand ﬁnal \|/tildewiden/angb∇acket∇ighttarget states and a ﬁxed dµtotal spinF, is calculated using the formula λSF νiKi,νfKf(ε) = 2πNB if\|Ai0,fn\|2δ(ε−εif+E0−/tildewideEn), (18) 5with the resonance condition ε+E0=εif+/tildewideEn, (19) taking into account the initial and ﬁnal energy of the target . The resonant energy for a free D2is denoted by εifand the transition matrix element is given by Ai0,fn=/angb∇acketleftΨ′ tot\|V\|Ψtot/angb∇acket∇ight. (20) Using Eqs. (14) and (17) the matrix element (20) can be writte n as a product Ai0,fn=/angb∇acketleft/tildewiden\|exp(ik·Rl)\|0/angb∇acket∇ightVif(ε) (21) whereVif(ε) is the transition matrix element calculated for a single D 2molecule [8]. The rate (18) can be additionally averaged over a distribution ρn0of the initial target states at a given temperature Tand summed over the ﬁnal target states, which leads to λSF νiKi,νfKf(ε) = 2πNB if\|Vif(ε)\|2/summationdisplay n,n0ρn0\|/angb∇acketleft/tildewiden\|exp(ik·Rl\|0/angb∇acket∇ight\|2 ×δ(ε−εif+E0−/tildewideEn).(22) FactorBif, deﬁned by Eqs. (8), is due to the averaging over the initial p rojections and summation over the ﬁnal projections of spin and rovibration al quantum numbers. This factor takes also into account a symmetrization of the total wave function of dµ+D2system over three deuterium nuclei. Now we introduce a time variable tto eliminate the δfunction in the equation above and then we involve time-dependent operators, which is familia r in scattering theory (see, e.g., Refs [28, 29]). Using the Fourier expansion of the δfunction δ(ε−εif+E0−/tildewideEn) =1 2π/integraldisplay∞ −∞dtexp/parenleftbig /parenleftbig/parenleftbig −it(ε−εif+E0−/tildewideEn)/parenrightbig /parenrightbig/parenrightbig (23) one has λSF νiKi,νfKf(ε) =NBif\|Vif\|2/integraldisplay∞ −∞dtexp/parenleftbig /parenleftbig/parenleftbig −it(ε−εif)/parenrightbig /parenrightbig/parenrightbig/summationdisplay n,n0ρn0 × /angb∇acketleft0\|exp(−ik·Rl)\|/tildewiden/angb∇acket∇ight/angb∇acketleft/tildewiden\|exp(it/tildewideEn) exp(ik·Rl) exp(−itE0)\|0/angb∇acket∇ight. (24) Assuming that the perturbation operator ∆ His well-approximated by its mean value ∆H≈ /angb∇acketleft0\|∆H\|0/angb∇acket∇ight ≡∆εif=−(1−Mmol/MC)ET<0, (25) which is valid when the target relaxation time is much smalle r than theddµlifetime of the order of 10−9s, the matrix element in Eq. (24) can be expressed as /angb∇acketleft/tildewiden\|exp(it/tildewideEn) exp(ik·Rl) exp(−itE0)\|0/angb∇acket∇ight =/angb∇acketleft/tildewiden\|exp/parenleftbig /parenleftbig/parenleftbig it(H+ ∆H)/parenrightbig /parenrightbig/parenrightbig exp(ik·Rl) exp(−itH)\|0/angb∇acket∇ight ≈ /angb∇acketleft/tildewiden\|exp(it∆εif) exp(itH) exp(ik·Rl) exp(−itH)\|0/angb∇acket∇ight =/angb∇acketleft/tildewiden\|exp(it∆εif) exp/parenleftbig /parenleftbig/parenleftbig ik·Rl(t)/parenrightbig /parenrightbig/parenrightbig \|0/angb∇acket∇ight,(26) 6where Rl(t) denotes the Heisenberg operator and ETin formula (25) is the mean kinetic energy of the target molecule at temperature T. Using the identity/summationtext n\|/tildewiden/angb∇acket∇ight/angb∇acketleft/tildewiden\|= 1 in Eq. (24) we obtain λSF νiKi,νfKf(ε) =NBif\|Vif(ε)\|2/integraldisplay∞ −∞dtexp/parenleftbig /parenleftbig/parenleftbig −it(ε−ε′ if)/parenrightbig /parenrightbig/parenrightbig ×/angbracketleftbig exp/parenleftbig /parenleftbig/parenleftbig −ik·Rl(0)/parenrightbig /parenrightbig/parenrightbig exp/parenleftbig /parenleftbig/parenleftbig ik·Rl(t)/parenrightbig /parenrightbig/parenrightbig/angbracketrightbig T,(27) where/angb∇acketleft···/angb∇acket∇ight Tdenotes both the quantum mechanical and the statistical ave raging at temper- atureT, andε′ ifbeing the resonance energy ε′ if=εif+ ∆εif, (28) shifted by ∆ εif<0. Note that such a resonant energy shift was neglected in pap ers [26, 27], where absorption of neutrons and γ-rays by heavy nuclei were considered. An estimation of the shift in the case of γemission from a nucleus bound in a solid, similar to Eq. (25) w as given in Ref. [30]. A self pair correlation function Gs(r,t) is deﬁned by the following equation [14] /angbracketleftbig exp/parenleftbig /parenleftbig/parenleftbig −ik·Rl(0)/parenrightbig /parenrightbig/parenrightbig exp/parenleftbig /parenleftbig/parenleftbig ik·Rl(t)/parenrightbig /parenrightbig/parenrightbig/angbracketrightbig T=/integraldisplay d3rGs(r,t) exp(ik·r), (29) and the single-particle response function Si(κ,ω) is given by the formula Si(κ,ω) =1 2π/integraldisplay d3rdtG s(r,t) exp/parenleftbig /parenleftbig/parenleftbig i(κ·r−ωt)/parenrightbig /parenrightbig/parenrightbig . (30) Thus, by virtue of Eqs. (27) and (30), the resonant formation rate in a condensed target can by expressed in terms of the response function λSF νiKi,νfKf(ε) = 2πNB if\|Vif(ε)\|2Si(κ,ω), (31) where the momentum transfer κand energy transfer ωto the target are deﬁned as follows κ=k, ω =ε−ε′ if. (32) The advantage of the Van Hove method is that all properties of the target, for given momen- tum and energy transfers, are contained in the factor Si(k,ω). It is possible to rigorously calculate Siin the case of a perfect gas and in the case of a harmonic solid. However, a liquid target or a dense gas target is a diﬃcult problem to solve. Proceeding as above one can obtain a similar formula for ΓSF′ νfKf,νiKiin a condensed target (in general, dµspinF′after back decay can be diﬀerent from dµspinFbefore the formation) ΓSF′ νfKf,νiKi= 2πAif/integraldisplayd3k (2π)3\|Vif(ε)\|2/tildewideSi(κ,ω′), ω′= ˜ε′ if−ε, ˜ε′ if=εif+ ∆˜εif,(33) /tildewideSiis the response function calculated for the state \|/tildewiden/angb∇acket∇ightand ∆˜εif≡ /angb∇acketleft/tildewiden\|∆H\|/tildewiden/angb∇acket∇ight=−(MC/Mmol−1)/tildewideET, (34) where /tildewideETdenotes the mean kinetic energy of the complex bound in the ta rget. 7IV. RESONANT FORMATION IN A HARMONIC SOLID It has been shown by Van Hove [14] that the self correlation fu nction in the case of a gas or a solid with cubic symmetry takes the general form Gs(r,t) =/parenleftbigg /parenleftbigg/parenleftbiggMmol 2πγ(t)/parenrightbigg /parenrightbigg/parenrightbigg3/2 exp/parenleftbigg /parenleftbigg/parenleftbigg −Mmol 2γ(t)r2/parenrightbigg /parenrightbigg/parenrightbigg . (35) For a cubic Bravais lattice, in which each atom is at a center o f inversion symmetry, γ(t) is given by the formula γ(t) =/integraldisplay∞ −∞dwZ(w) wnB(w) exp(−iwt), (36) whereZ(w) is the normalized vibrational density of states such that /integraldisplay∞ 0dwZ(w) = 1, Z(w) = 0 for w>w max, Z(−w)≡Z(w),(37) nB(w) is the Bose factor nB(w) = [exp(βw)−1]−1, β= (kBT)−1. (38) and the Boltzmann constant is denoted by kB. The response function (30), after substitution of Eqs. (35) , (36) and integration over r, can be written as follows Si(κ,ω) =1 2πexp/parenleftbigg /parenleftbigg/parenleftbigg −κ2 2Mmolγ(∞)/parenrightbigg /parenrightbigg/parenrightbigg ×/integraldisplay∞ −∞dtexp(−iωt) exp/parenleftbigg /parenleftbigg/parenleftbiggκ2 2Mmol[γ(∞)−γ(t)]/parenrightbigg /parenrightbigg/parenrightbigg ,(39) γ(∞) denotes the limit of γ(t) att→ ∞. This formula can be expanded in a power series of the momentum transfer κ, which leads to Si(κ,ω) = exp( −2W)/bracketleftBigg δ(ω) +∞/summationdisplay n=1gn(ω,T)(2W)n n!/bracketrightBigg , (40) where 2Wis the Debye-Waller factor, familiar in the theory of neutro n scattering, 2W=κ2 2Mmolγ(∞) =κ2 2Mmol/integraldisplay∞ 0dwZ(w) wcoth/parenleftbig1 2βw/parenrightbig , (41) and the functions gnare given by g1(w,T) =1 γ(∞)Z(w) w[nB(w) + 1], gn(w,T) =/integraldisplay∞ −∞dw′g1(w−w′,T)gn−1(w′,T), /integraldisplay∞ −∞dwg n(w) = 1.(42) 8In the case of a cubic crystal structure 2 Wcan also be expressed as 2W=1 3/angb∇acketleft0\|u2\|0/angb∇acket∇ightκ2, (43) where uis the displacement of a molecule from its lattice site. Subs titution of Eq. (40) to Eq. (31) leads to the following formation rate λSF νiKi,νfKf(ε) = 2πNB if\|Vif(ε)\|2exp(−2W)/bracketleftBigg δ(ω) +∞/summationdisplay n=1gn(ω,T)(2W)n n!/bracketrightBigg , (44) The ﬁrst term in expansion (44) represents a sharp peak descr ibing theδproﬁle recoilless formation. The next terms give broad distributions corresp onding to subsequent multi- phonon processes. In particular, the term with n= 1 describes formation connected with creation or annihilation of one phonon. If 2W≪1 we deal with so-called strong binding [26] where only the fe w lowest terms in the above expansion are important. On the other hand, in the l imit 2W≫1 (weak-binding) many multi-phonon terms give comparable contributions to ( 44). Therefore, for suﬃciently largeκ2it is convenient to use the impulse approximation in which γ(t) is replaced by its value neart= 0 γ(t)≈γ(0) +it−2 3ET. (45) This leads to the asymptotic formula for Si Si(κ,ω) =1 ∆√πexp/parenleftbigg /parenleftbigg/parenleftbigg −/parenleftbiggω− R ∆/parenrightbigg2/parenrightbigg /parenrightbigg/parenrightbigg , (46) where ∆ = 2/radicalBig 2 3ETR,R=κ2 2Mmol. (47) The mean kinetic energy ETof a molecule in the solid, which also determines the resonan ce energy shift (25), is equal to ET=3 2/integraldisplay∞ 0dwZ(w)w/bracketleftbig nB(w) +1 2/bracketrightbig . (48) The energy ETcontains a contribution from the zero-point vibrations and it approaches 3kBT/2 only at high temperatures T≫wmax/kB. Function (46) is a Gaussian with re- sponse centered at the recoil energy R. Therefore in the weak binding region the resonant formation rate takes the Doppler form obtained by Bethe and P laczek1for resonant ab- sorption of neutrons in gas targets [31]. However, the reson ance width (47) in the solid at temperature Tis diﬀerent from the Doppler width in a Maxwellian gas ∆ gas= 2√kBTR unless the temperature is suﬃciently high. This phenomenon was pointed out by Lamb in his paper [26] concerning resonant neutron absorption in so lid crystals. By virtue of the equations above one can introduce for the solid an eﬀective t emperature Teﬀ Teﬀ=2 3ET/kB. (49) 1In fact, formula (46) is the limit of the Bethe formula in the c ase of a very narrow natural resonance width Γ →0. 9V. RESONANT FORMATION IN FROZEN DEUTERIUM The following considerations concern the solid deuterium c rystals used in the TRIUMF experiments [32, 33], though the results presented below ca n be applied to targets obtained in similar conditions [12, 34]. At TRIUMF thin solid deuteri um layers have been formed by rapid freezing of gaseous D 2on gold foils at T= 3 K and zero pressure. According to Ref. [35] such deuterium layers have the face-centered cubic (fcc) po lycrystalline structure. Since the distance between the neighboring molecules is a few times gr eater than the diameter of a D2molecule and the Van der Waals force that binds the solid is we ak, one can neglect perturbations of the resonant formation potential Vdue to these neighbors. The deuterium crystals at zero pressure are quantum molecul ar crystals. The amplitude of zero-point vibration at 3 K equals 15% of the nearest neigh bor distance. A single-particle potential in this case is not harmonic and the standard latti ce dynamics leads to imaginary phonon frequencies. However, the standard dynamics can be a pplied after a renormalization of the interaction potential, taking into account the short -range pair correlations between movement of the neighbors [35]. In result, the theoretical c alculations [36] of the phonon dispersion relations give a good agreement with the neutron scattering experiments [37] and the Debye model for solid deuterium can be used as a good appro ximation of the phonon energy distribution Z(w) =/braceleftBigg 3w2/w3 Difw≤wD, 0 if w>wD,(50) with the Debye energy wD=kBΘDand Debye temperature Θ Dtaken from the neutron experiments. For T= 3 K we use the Debye model of an isotropic solid with Θ D= 108 K corresponding to the maximal phonon energy wD= 9.3 meV. Thus, we are dealing with the limitT/ΘD≪1 where γ(∞) =3 2w−1 D,ET=9 16wD≈5.2 meV, T eﬀ=3 8ΘD≈40 K, (51) are very good approximations of Eqs. (41), (48) and (49). The Debye-Waller factor and mean kinetic energy ETat lowest temperatures are determined by contributions fro m the zero-point D 2vibration in the lattice, and therefore these quantities do not tend to zero at T→0. The zero-point energy is not accessible energy, but its eﬀ ects are always present. The values of the resonance energies depend on initial and ﬁn al rovibrational quantum numbers of the system. In solid hydrogens at low pressures th ese quantum numbers remain good quantum numbers, but excited energy levels broaden int o energy bands (rotons and vibrons) due to coupling between neighboring molecules [18 ]. The calculations presented in the literature concern pure solid H 2, HD and D 2targets and only lowest quantum numbers. The problem of a heavier impurity, such as ( ddµ)dcomplex in D 2, has not been considered yet. However, knowing that the width of the rotational bands can reach about 1 meV [18], a possible inﬂuence of this eﬀect on the calculated formatio n rates and fusion neutron time spectra is discussed in the next section. At low temperatures all D 2molecules are in the ground vibrational state νi= 0 andddµ is formed via the excitation of the complex to the state νf= 7. Unless a catalyst is applied, rapidly frozen deuterium is a mixture of ortho-D 2(Ki= 0) and para-D 2(Ki= 1). In the TRIUMF experiments gaseous deuterium was pumped through a h ot palladium ﬁlter before freezing. Therefore the solid target was a statistical mixt ure (2:1) of the ortho- and para- states (Ki=stat). Since the para-ortho relaxation without a catalyst is ver y slow (0.06%/h) 10in solid deuterium [38], the population of these states is no t changed during experiments of a few days. The lowest resonance energies εifandε′ if, for ﬁxedνi,νfand diﬀerent values of F,Ki, SandKfare shown in Table I [10]. A few of them have negative values, w hich means that to satisfy the resonance condition ε=εifan energy excess in the dµ+D2system should be transferred to external degrees of freedom. This is possibl e in dense targets, where energy of neighboring molecules can be increased. Such an eﬀect, du e to triple collisions in gas targets, has been ﬁrstly discussed in Ref. [39]. In a solid, t he energy excess is lost through incoherent phonon creation. According to (25), (28), and (5 1), in the considered 3 K solid deuterium all resonant energies ε′ ifare shifted by ∆ εif≈ −1.81 meV. One can see that all resonances for F=1 2are placed at higher energies, which is caused by dµhyperﬁne splitting ∆Ehfs= 48.5 meV. All resonance energies ε′ if/lessorsimilarwD≈10 meV are connected with formation from the upper spin state F=3 2ofdµ. However, only resonances corresponding to the dipole transitions Ki= 0→Kf= 1 andKi= 1→Kf= 0,2 can give a signiﬁcant contribution to the formation rate at lowest energies. Othe r transition matrix elements described in Ref. [40] tend to zero when ε→0 (see Figs. 3 and 4 obtained for Ki= 0 and Ki= 1 ). The low energy rates ( ε/lessorsimilarwD) are calculated using formula (44) with a few most sig- niﬁcant terms of the response function expansion (40) taken into account. Fig. 5 shows the function Si(κ,ε−ε′ if) corresponding to the two dipole transitions in para-D 2. The sub- threshold resonance, with ε′ if≈ −9.0 meV, gives contributions to the formation rate only through the phonon creation processes. For ε′ if≈1.6 meV, the non-phonon process is pos- sible and it is represented by a vertical line. Diﬀerent peak s in this ﬁgure describe processes connected with diﬀerent numbers of created phonons. In part icular, one-phonon processes, which are proportional to Z(w) with the characteristic Debye cutoﬀ, can be clearly distin - guished. Since the n-phonon term in (40) is proportional to κ2n, theddµformation rate tends to zero at ε→0. Note that the phonon annihilation gives negligible contr ibution to the rate at very low target temperatures T≪ΘD. In order to compare the calculated formation rates with expe riments the summed rates λF Ki(ε) are introduced λF Ki(ε) =/summationdisplay Kf,SλSF νiKiνfKf, ν i= 0, ν f= 7. (52) In Fig. 6 the formation rates λF Ki(ε) in the solid ortho-D 2and para-D 2are shown for F=3 2. In the case of resonances satisfying the condition ε′ if≤wDwe have 2W < 1 and the expansion (44) is used. The two strong peaks represent the re coilless formation process, without phonon excitations. The delta function proﬁle of ev ery peak is shown as a rectangle with a height equal to the formation rate strength divided by the total decay width ( ≈ 0.8×10−3meV). The strength deﬁned as the value of the factor standing beforeδ(ω) in the expansion (44), is equal to 0.1061 eV ·µs−1for the resonance Ki= 0→Kf= 1 in solid ortho-D 2. The transition Ki= 1→Kf= 2 in para-D 2gives 0.07544 eV ·µs−1as the resonance strength. Higher resonance energies involve many multi-phonon terms and therefore we use the asymptotic form (46) of Siforε′ if>wD. All formation rates presented in the ﬁgures are normalized to the liquid hydrogen density N0= 4.25×1022atoms/cm3. Though in Monte Carlo simulations, involving energy-depen dent rates of diﬀerent pro- cesses, the “absolute” formation rates λF Ki(ε) should be used, it is convenient to introduce an 11eﬀective formation rate ¯λF Ki(ε) which leads to the nuclear ddfusion in [(ddµ)dee] complex. Back decay of the complex to the dµ+D2system, characterized by the quantum numbers K′ i andF′, strongly inﬂuences the fusion process because the back-de cay rates are comparable with the eﬀective fusion rate ¯λf≈374µs−1[7]. Since in a solid target rotational deexci- tation of the asymmetric complex is much faster than back dec ay and fusion, it is assumed that back decay takes place only from the state Kf= 0. The eﬀective formation rate is then deﬁned by the following formula ¯λF Ki(ε) =/summationdisplay Kf,SλSF νiKiνfKf(ε)Pfus S, ν i= 0, ν f= 7, (53) where the fusion fraction Pfus Sis given by Pfus S=¯λf ΓS,ΓS=¯λf+/summationdisplay F′ΓSF′,ΓSF′=/summationdisplay K′ i,Kf=0ΓSF′ νfKf,νiK′ i. (54) Since the frequency of lattice vibrations ( ∼wD//planckover2pi1∼107µs−1) is many orders of mag- nitude greater than the back-decay and fusion rates, energe tic phonons created during theddµformation process are dissipated. At 3 K the number of phonon s with energies w/greaterorsimilarkBT≈0.26 meV is strongly suppressed by the Bose factor nB(w). Therefore back decay with phonon annihilation at T≪ΘDis negligible. In particular, the phonon channel of decay of ddµ, formed from dµstateF=3 2due to the subthreshold resonances, is closed because this would require an annihilation of a phonon with e nergy of a few meV. In this case back decay is connected with the spin-ﬂip transition to F′=1 2. Since the corresponding en- ergy release of a few tens of meV is much greater than the Debye energy (∆Ehfs≫wD), the ddµdecay rate is dominated by contributions from simultaneous phonon creation processes. After integration of formula (33) over direction of vector kone obtains ΓSF′ νfKf,νiKi=Aif π/integraldisplay∞ 0dkk2\|Vif(ε)\|2/tildewideSi(k2,ω′), (55) and then substitution of expansion (40) and integration of t he recoilless term lead to ΓSF′ νfKf,νiKi=Aif π/bracketleftBigg M/tildewidekif\|Vif(˜ε′ if)\|2exp(−2/tildewiderWif) +∞/summationdisplay n=1/integraldisplay∞ 0dkk2\|Vif(ε)\|2exp(−2/tildewiderW)gn(ω′,T)(2/tildewiderW)n n!/bracketrightBigg ,(56) where 2/tildewiderW=k2 2MCγ(∞),2/tildewiderWif= 2/tildewiderW(/tildewidekif),/tildewidekif=/radicalBig 2M˜ε′ if. (57) It is assumed in the formula above that the phonon energy spec trum of solid deuterium containing [( ddµ)dee] is similar to that of a pure deuterium lattice. The problem o f lattice dynamics of a quantum solid deuterium crystal containing a s mall admixture of a heavier isotope has not been considered yet in literature, at least t o the knowledge of the authors. However, this approximation is reasonable since the Debye t emperatures of solid hydrogen 12and deuterium at 3 K are very similar [35], independently of t he mass diﬀerence of these isotopes. Therefore it is assumed that during the ddµlifetime the mean kinetic energy /tildewideETof the complex reaches the energy ETcharacterizing a pure deuterium solid. Thus the resonance energy shift (34) is approximated by ∆˜εif≈ −(MC/Mmol−1)ET≈ −2.77 meV, (58) which gives ˜ ε′ if=εif−2.77 meV. The eﬀective formation rates in 3 K solid deuterium for F=3 2are shown in Fig. 7. The phonon part of the rates below a few meV is about two orders of magnitude lower than the average rate of 2.7 µs−1derived from the experiment [11, 13]. This means that atε≪wDthe phonon contribution to the total resonant formation rat e is even smaller than the non resonant ddµformation rate λnr≈0.44µs−1[9], and that the estimation of the phonon contribution given in Ref. [20] is strongly overesti mated. Therefore, the experimental results can only be explained by resonant ddµformation at energies ε/greaterorsimilar1 meV, where the rate exceeds signiﬁcantly the value of 1 µs−1. A cusp at 0.3 meV in para-D 2is due to the formation with simultaneous one-phonon creation, co nnected with the subthreshold resonanceKi= 1→Kf= 0. This implies a signiﬁcant diﬀerence between the resonan t formation in ortho-D 2and para-D 2below 1 meV. However, this diﬀerence is diﬃcult to measure because of a broad distribution of dµenergy. Note that a similar subthreshold phonon eﬀect in the case of resonant dtµformation in solid deuterium has been discussed in Ref [41]. In the solid target the fusion fraction Pfus S≈0.3 and the total resonance width ΓS≈ 0.8×10−3meV for both S=1 2andS=3 2. The back-decay rate ΓSF′fromS=1 2toF′=1 2 equals about 843 µs−1. DecayS=1 2→F′=1 2is impossible. In the case of S=3 2we have obtained ΓSF′≈281µs−1forF′=1 2and ΓSF′≈610µs−1forF′=3 2. Phonon creation processes give dominant contributions to the back-decay ra tes, e.g., the non-phonon part of ΓSF′, given by the ﬁrst term of expansion (56), equals 169 µs−1. Therefore the dµenergy spectrum, after back decay in the solid, is not discrete. In Fig. 8 the eﬀective rates in solid deuterium for F=1 2are presented. For the sake of comparison the formation rate for 3 K ortho-D 2gas is also plotted. The “gas” curve has been calculated using the asymptotic formula (46) for SiwithTeﬀ= 3 K.This ﬁgure shows that in a real solid deuterium target the rates are smea red much more than in a gas target with the same temperature, because of the zero-point vibrations. Therefore even at relatively high dµenergies of some 0.1 eV one should not neglect the solid eﬀect s and use the formation rates calculated for a 3 K Maxwellian gas. VI. MONTE CARLO CALCULATIONS The calculated energy-dependent ddµformation rates have been applied in our Monte Carlo simulations of µCF in 3 K solid deuterium targets. The ﬁnal dµenergy distribution after back decay, including simultaneous phonon creation p rocesses, has been determined through a numerical integration of Eq. (56). The calculated distribution is shown in Fig. 9 forS=1 2,Kf= 0 andF′=1 2. The rotational transitions to K′ i= 0,1,2 with no phonon creation are seen as the delta peaks. The continuous energy s pectrum describes phonon creation contribution to dµenergy. Note that, opposite to ddµformation rates, this phonon 13contribution (for a given rotational transition peak) exte nds towards lower energies. The averagedµenergy after ddµback decay equals about 30 meV, for the presented spectrum. Theddfusion neutron and proton spectra depend on the time evoluti on ofdµenergy. This energy is determined by diﬀerential cross sections of d iﬀerent scattering processes of dµ atoms in a given solid target, including elastic scattering , rovibrational transitions, spin-ﬂip reactions and phonon processes. The scattering cross secti ons in a solid are calculated using the Van Hove method. Some results of such calculations for dµatoms in fcc solid deuterium have been presented and discussed in Ref. [42]. The incohere nt processes, such as spin-ﬂip or rovibrational transitions, are described by the self pair c orrelation function Gs(r,t) deﬁned by Eq. (29). The Bragg scattering and coherent phonon scatte ring are connected with a pair correlation function G(r,t) [14]. In Fig. 10 is shown the total cross section for dµ(F=3 2) scattering in the statistical mixture of 3 K solid ortho-D 2and para-D 2. Bragg scattering, with the Bragg cutoﬀ at εB= 1.1 meV, and incoherent elastic scattering do not change dµenergy because of the very large mass of the considered solid target. Below 1.7 meV the dµatom is eﬀectively acceler- ated, mainly due to the rotational deexcitation of para-D 2molecules [21, 42]. This transition is enabled by muon exchange between deuterons in dµ+D2scattering. The curve “0 →1” in Fig. 10, describing the rotational deexcitation, includ es contributions from simultaneous incoherent phonon processes. This cross section at ε= 2.5 meV equals 0.22 ×10−20cm2, which is about three times less (taking into account the stat istical factor of 1/3 for K= 1 states) than the estimation given in paper [21]. Phonon anni hilation is a much weaker dµ acceleration mechanism than the rotational deexcitation. Since the coherent amplitude for dµelastic scattering on a single D 2molecule is greater by two orders of magnitude than the incoherent amplitude, th e coherent processes involving conservation of momentum dominate low energy dµscattering in solid deuterium. It is especially important below a few meV, where the shapes of coh erent and incoherent cross sections diﬀer strongly. The small phonon creation cross se ction below 1.1 meV, leading to dµenergy loss, is due to the incoherent amplitude. Coherent ph onon creation is impossible belowεB. This limit is obtained in the case of coherent one-phonon cr eation process, for the total momentum conservation involving the smallest (no n-zero) inverse lattice vector τ, which also ﬁxes the position of the ﬁrst peak of the Bragg scat tering atεB= 1.1 meV. For τ=0one-phonon creation is possible only if the dµvelocity is not lower than the sound velocity in the crystal, which is well-known in neutron phys ics. According to Ref. [38] the mean sound velocity in solid deuterium equals about 1.2 ×105cm/s and this corresponds todµenergy of 15 meV. Therefore, neglecting the inverse lattice contribution to the one- phonon creation cross section in Ref. [21] leads to the sever e underestimation of dµslowing down at lowest energies and subsequent overestimation of dµkinetic energy. Above 1.7 meV phonon creation already prevails over all acce leration processes. However, the eﬀective deceleration rate below wDis strongly suppressed by the dominating Bragg elastic scattering. At energies above some 10 meV subsequen t rotational and then vibrational excitations of D 2molecules become important and they provide a very fast mech anism of dµdeceleration at higher energies. The total cross section for dµ(F=3 2) scattering in a pure 3 K ortho-D 2target (see Fig. 11) is quite similar to that shown in Fig. 10. A signiﬁcan t diﬀerence is the lack of rotational deexcitation. Therefore phonon annihilation i s the only, and weak, acceleration mechanism. It dominates the inelastic cross section below 1 .4 meV. Fig. 12 presents the time evolution of average dµ(F=3 2) atom energy εavg, obtained from 14our Monte Carlo calculations. It is assumed that the target i s inﬁnite and that dµatoms have initially a Maxwellian energy distribution with a mean energy of 1 eV. A statistical initial population of dµtotal spin is used and the theoretical non-resonant part of t he total spin-ﬂip rate λ3 2,1 2is multiplied by a single scaling factor of 0.4, in order to ke ep agreement with the experimental values [10, 43] of the spin-ﬂip rate. T he calculations have been performed for ortho-D 2, para-D 2and their statistical mixture (stat). One can see that dµmean energy of 10 meV is reached already after 5 ns. Then, belo w the Debye energy, deceleration become very slow. The lowest value of εavgis determined by the intersection point of the cross sections of the acceleration processes an d phonon creation process. In the case of a statistical mixture εavg≈1.7 meV, for K= 0 we have εavg≈1.4 meV. Finally, for pure para-D 2, with a contribution to the total cross section from the rota tional transition K= 1→0 three times greater than that shown in Fig. 10, εavg≈2.2 meV. Thus, dµatoms are never thermalized and their energy is signiﬁcantly grea ter than 1 meV. For para-D 2the mean energy is always greater than the energy of the lowest re sonance peak ε′ if= 1.6 meV. However, even if εavgis smaller than ε′ if, a signiﬁcant part of dµatoms has energy ε≥ε′ if because of a large admixture of hot dµatoms att= 0 [15, 16] and slow deceleration below 10 meV. Since at energies of a few meV the lowest delta peaks are domin ant in the resonant formation, their contributions to the mean eﬀective format ion rate are shown in Fig. 13 for gas and solid deuterium (stat) targets, assuming steady Max well distributions of dµ(F=3 2) energy, with diﬀerent εavg. The maximum average rate of about 6 µs−1in the solid is due to the resonance energy shift of −1.8 meV. The experimental result of 3 µs−1can be explained because εavgis greater than 1 meV. However, in order to obtain large fusio n neutron and proton yields through resonant ddµformation, the width ΓSof the resonance peaks in solid can not be too narrow. The peak resonant rates o f a few 104µs−1have been obtained assuming the discrete values of the rovibrational D2energies in solid deuterium and ΓS∼10−3meV. These resonant rates are many orders of magnitude great er than the inelastic scattering rate ∼10µs−1. In such a case dµatoms are very quickly (compared to dµ(F=3 2) lifetime) removed from the regions of resonance peaks and t he contribution of the recoilless resonances to the neutron yield is negligible. T he Monte Carlo simulations have shown that the neutron yield from the phonon part of the reson ant rates gives only some 10% of the yield observed in the experiments. In result, the calc ulated time spectra, obtained for the small ΓS, are dominated by weak non-resonant ddµformation, which disagrees with the experimental data. Therefore, we have investigated inﬂ uence of a broadening of the non-phonon resonant peaks, due to the presence of molecular rovibrational bands in solid, discussed in Ref. [18]. Since in the literature there is no in formation concerning the proﬁle of such bands, we have assumed a rectangular shape of the reso nance peaks. The resonance strengths have been ﬁxed and their widths have been changed i n the limits 0.001–1 meV. It turns out that good Monte Carlo results are obtained for ΓS≈0.5 meV, which is consistent with the rotational bandwidths of about 1 meV reported in Ref . [18]. This gives the resonant formation rate of 294 µs−1for the recoilless peak in ortho-D 2, and respectively 214 µs−1in para-D 2. In Fig. 14 one sees the resonant formation rate at lowest ene rgies for ΓS= 0.5 meV and for the statistical mixture of ortho- and para-states. A lso shown is the Monte Carlo distribution of dµ(F=3 2) energy, calculated for times t= 10 ns and t= 30 ns. The Maxwell distribution of initial dµenergy, with εavg= 1 eV, has been assumed. Two minima in thedµenergy distribution appear quickly at the positions of the r esonance peaks since the respective ddµformation rates are comparable with the total inelastic sca ttering rate of 15about 30µs−1. Theddfusion neutron spectrum, calculated assuming the same init ialdµenergy and resonance proﬁles, is shown in Fig. 15. A 3.2 ×10−6concentration of nitrogen is included in order to ﬁt the TRIUMF target conditions. The solid line plot ted in this ﬁgure has been cal- culated using the steady-state kinetics model with the eﬀec tive formation rate ¯λ3/2 stat= 3µs−1 and total spin-ﬂip rate λ3 2,1 2= 36µs−1taken from the ﬁts to the experimental data [13]. The slope of the spectrum at t/lessorsimilar80 ns is determined by the rates ¯λ3/2 stat,λ3 2,1 2, anddµscattering rate which also changes the population of dµ(F=3 2) atoms in the vicinity of the resonant peaks. The steady-state kinetics model does not include the process ofdµdeceleration. Therefore, ﬁts using this model could entangle the decelera tion rate with the formation and spin-ﬂip rates. The mean formation rate, calculated direct ly in the Monte Carlo runs, is a function of time, and it stays at the level of 1–3 µs−1. The spectrum slope at large times t/greaterorsimilar100 ns, when dµ(F=3 2) atoms practically disappear, are due to the nonresonant ddµ formation from F=1 2and to the muon transfer to nitrogen contamination. The shape of the time spectra practically does not change whe n the mean energy εavgof the initial single Maxwell distribution varies in the limit s 0.01–1 eV. On the other hand, the spectra change strongly if a signiﬁcant part of dµatoms att= 0 has energy smaller than the energy of the lowest resonant peak, which can be observed using a more complicated (e.g. two-Maxwell distribution). Assuming that ΓSis greater than 0.5 meV we obtain results which begin to diﬀer signiﬁcantly from the analytical curve calculated with the experimental parameters. In particular, the ratios of neutron yields fro m the short and large times begin to disagree. Fits of the calculated spectra to the experimen tal data would enable a better determining of ΓSand a shape of the initial dµenergy. However, this is not the purpose of this work. A qualitative comparison of Monte Carlo spectr a with the experimental data has already been performed in article [13]. In this case good ﬁts were not obtained since at that time the resonant ddµformation rates in solid D 2anddµscattering rates including coherent eﬀects in the solid were not yet available. Our calculations show that strong resonant ddµformation takes place both in ortho-D 2 and para-D 2. There are certain diﬀerences between the neutron time spec tra from these targets (see Fig. 16), caused by the diﬀerent positions and s trengths of the lowest resonance peaks. Also dµslowing down process diﬀers slightly in the two cases. The ne utron yield at larger times is smaller for ortho-D 2since in this case the resonance peak is placed at higher energy of 2.3 meV. Therefore, dµatoms are removed faster from the peak compared to the situation in para-D 2, where the resonance is observed at 1.6 meV. A greater mean dµenergy in para-D 2(cf. Fig. 12) leads also to a stronger overlap of the resonanc e peak and dµenergy distribution at t/greaterorsimilar20 ns. However, the diﬀerences between the spectra can be cle arly seen only in high-statistics experiments. VII. CONCLUSIONS The methods used for description of resonant neutron and γ-ray absorption in condensed matter have been directly applied for calculation of resona ntddµformation and back-decay rates in condensed deuterium targets. These rates are expre ssed in terms of the Van Hove single-particle function, which depends on properties of a given target. In particular, we have derived the analytical formulas for the rate in the case of resonant ddµformation in a harmonic solid deuterium. The calculations show great diﬀ erences between resonant ddµ 16formation in 3 K solid deuterium and in 3 K D 2gas. In solid, the formation at a few meV, which determines the experimental results, is dominated by presence of the strong recoilless resonant peaks. On the other hand, the formation with simult aneous phonon creation is important above the Debye energy. The resonance proﬁles in t he solid at higher energies are similar to that in D 2gas, but with the eﬀective temperature equal to 40 K. This tem perature is determined by the energy of zero-point vibration of D 2molecule in the lattice. Phonon creation is always important in the case of ddµback decay because it is connected with energy release of a few tens meV, which is much greater than th e Debye energy. A condition T/ΘD≪1 is fulﬁlled for any solid deuterium target at low pressure. There- fore, the parameters determining solid state eﬀects (Debye -Waller factor, mean energy of D 2 vibration in solid) weakly depend on target temperature T. They are expressed in terms of the Debye energy wDwhich does not signiﬁcantly change with the varying solid te mpera- tureT. In result, the resonant ddµformation rates in solid deuterium for diﬀerent Tare very similar and one may expect that the average formation rates, derived from measurements performed at diﬀerent temperatures, will also be very close . This is conﬁrmed by the results of experiments carried out at TRIUMF and at JINR. The structure of a solid deuterium target depends on its temp erature and history. Targets maintained at T/greaterorsimilar4 K have the hcp structure [35]. Though our calculations have been performed for fcc crystals, the obtained results are also go od approximations of the resonant rates in hcp polycrystals since the Debye temperature and ne arest neighbor distance are similar for these two lattices. In general, the formulas der ived in this paper can be used in a wide range of target temperature and density, with appropr iate experimental values of the Debye temperature and lattice constant taken into account. The Monte Carlo calculations show that dµdeceleration below the Debye energy is very slow and that mean energy of dµ(F=3 2) atom is always signiﬁcantly greater than 1 meV. The energy distribution of dµ’s during their lifetime is very broad (at least a few meV), th erefore a strong overlap of this distribution and lowest resonance p eaks takes place, leading to a large meanddµformation rate in solid deuterium. However, explanation of the experiments is possible only if the broadening of rovibrational molecular levels in solid is taken into account. We obtained reasonable results assuming that the strengths of the recoilless resonant peaks are constant and that the rotational bands increase the reso nance peak width to 0.5 meV. Note that, according to Ref. [18], high pressures lead to a gr eater broadening and even to a mixing of rotational states. This could complicate a compa rison of theory and high-pressure experiments. The phonon part of the resonant rate give only a bout 10% contribution to the calculated neutron time spectra. Theddfusion neutron spectra calculated for ortho-D 2and para-D 2solid targets are quite similar. Small diﬀerences between the spectra are due to the diﬀerent energies and strengths of the lowest resonant peaks, and to a slightly hig her meandµenergy in para-D 2. These diﬀerences can be clearly seen only in high-statistic s experiments. Our calculations do not conﬁrm a lack of strong resonant ddµformation in solid ortho-D 2, predicted in the papers [20, 21]. In order to verify the theory it is necessary to perform measurements in pure ortho-D 2and para-D 2solid targets under the same conditions. Acknowledgments We wish to thank L. I. Ponomarev for stimulationg discussion s. We are grateful to G. M. Marshall for a critical reading of the manuscript. This work was supported in part 17through Grant INTAS 97-11032. [1] V. P. Dzhelepov et al., Zh. Eksp. Teor. Fiz. 50, 1235 (1966) [Sov. Phys. JETP 23, 820 (1966)]. [2] W. H. Breunlich et al., Annu. Rev. Nucl. Part. Sci. 39, 311 (1989). [3] L. I. Ponomarev, Contemp. Phys. 31, 219 (1990). [4] E. A. Vesman, Zh. Eksp. Teor. Fiz. Pisma 5, 113 (1967) [Sov. Phys. JETP Letters 5, 91 (1967)]. [5] L. I. Ponomarev and M. P. Faifman, Zh. Eksp. Teor. Fiz. 71, 1689 (1976) [Sov. Phys. JETP 44, 886 (1976)]. [6] L. I. Menshikov and M. P. Faifman, Yad. Fiz. 43, 650 (1986) [Sov. J. Nucl. Phys. 43, 414 (1986)]. [7] L. I. Menshikov et al., Zh. Eksp. Teor. Fiz. 92, 1173 (1987) [Sov. Phys. JETP 65, 656 (1987)]. [8] M. P. Faifman, L. I. Menshikov, and T. A. Strizh, Muon Cata lyzed Fusion 4, 1 (1989). [9] A. Scrinzi et al., Phys. Rev. A47, 4691 (1993). [10] C. Petitjean et al., Hyp. Interact. 118, 127 (1999). [11] P. E. Knowles et al., Hyp. 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Adamczak, Hyp. Interact. 119, 23 (1999). [43] N. I. Voropaev et al., Hyp. Interact. 118, 135 (1999). FIG. 1: System of coordinates used for the calculation of res onant formation of the com- plex [( ddµ)dee] in a condensed deuterium target. FIG. 2: Position of impinging dµatom with respect to the condensed target. 19FIG. 3: Transition matrix elements \|Vif(ε)\|2forKi= 0 and Kf= 0,1,2 versus dµenergy FIG. 4: Transition matrix elements \|Vif(ε)\|2forKi= 1 and Kf= 0,1,2 versus dµenergy 20FIG. 5: Response function Si(κ,ε−ε′ if) (in arbitrary units) for the para-D 2crystal at 3 K. The dashed line is obtained for the subthreshold resonance ε′ if≈ −9.0 meV, the solid line corresponds toε′ if≈1.6 meV. The vertical line represents the rigid lattice term δ(ε−ε′ if) exp(−2W). FIG. 6: Formation rate λF Ki(ε) forF=3 2in 3 K ortho-D 2(solid line) and para-D 2(dashed line). The labels “1 →2” and “0 →1” denote the rotational transition Ki→Kfcorresponding to the lowest non-phonon processes. 21FIG. 7: Eﬀective formation rate ¯λF Ki(ε) forF=3 2in 3 K solid ortho-D 2and para-D 2. The labels “1→2” and “0 →1” denote the rotational transition Ki→Kfcorresponding to the lowest non-phonon processes. FIG. 8: Eﬀective formation rate ¯λF Ki(ε) for F=1 2in 3 K solid ortho-D 2and para-D 2. The label “gas” denotes the curve obtained for 3 K gaseous deuter ium (Ki= 0), using the asymptotic formula (46) for the response function SiwithTeﬀ= 3 K. 22FIG. 9: Distribution of ﬁnal dµenergy after ddµback decay from S=1 2,Kf= 0 to F′=1 2, K′ i= 0,1,2. The three peaks describe the rotational transitions with out a simultaneous phonon excitation. FIG. 10: Total cross section for dµ(F=3 2) scattering in statistical mixture of solid ortho-D 2 and para-D 2. The label “1 →0” denotes the rotational deexcitation K= 0→1 of a target D 2 molecule. The curves “ −phonon” and “+phonon” stand for dµscattering with phonon annihilation and creation, respectively. The Bragg cross section is calc ulated for the fcc polycrystalline lattice. 23FIG. 11: Cross section for dµ(F=3 2) scattering in solid ortho-D 2. The labels are identical to those in Fig. 10. FIG. 12: Calculated time evolution of average dµenergy εavgforF=3 2in 3 K solid ortho-D 2, para-D 2, and their statistical mixture (stat). A Maxwell distribut ion of dµinitial energy, with mean energy of 1 eV, has been assumed. 24FIG. 13: The eﬀective resonant ddµformation rate as a function of mean CMS energy εavgof dµ(F=3 2) atom for gas and solid deuterium targets. A steady Maxwell d istribution of dµenergy is assumed for a given εavg. The contributions from the two lowest resonant peaks to the formation rate are taken into account. FIG. 14: Resonant ddµformation rate for F=3 2in the statistical mixture of ortho-D 2and para-D 2 for the resonance peak width ΓS= 0.5 meV. Monte Carlo distribution of dµenergy at t=10 ns andt=30 ns after the muon stop is plotted (in arbitrary units). 25FIG. 15: The Monte Carlo fusion neutron spectrum for the stat istical mixture of 3 K solid ortho-D 2 and para-D 2(solid line). The dashed line represents the spectrum obtai ned using an analytical steady state kinetics model with ¯λ3/2 stat= 3µs−1. The initial dµenergy is given by a Maxwell distribution with mean energy of 1 eV. The width ΓSof the non-phonon resonances is ﬁxed at 0.5 meV. A 3.2 ×10−6concentration of nitrogen is included. FIG. 16: Calculated neutron spectra from 3 K solid ortho-D 2and para-D 2. The Maxwell distri- bution of initial dµenergy with εavg= 1 eV and ΓS=0.5 meV have been assumed for the both targets. 26TABLE I: The lowest resonance energies of ddµ formation in dµscattering from single D 2 molecule ( εif) and from 3 K solid deuterium target ( ε′ if). These energies are given in the re- spective CMS systems. εif(meV) ε′ if(meV) F K iKfS −7.218 −9.0283 21 01 2 −3.667 −5.4773 21 11 2 0.5368 −1.2723 20 01 2 3.422 1.6123 21 21 2 4.088 2.2793 20 11 2 11.18 9.3683 20 21 2 42.10 40.301 21 01 2 45.66 43.851 21 11 2 49.86 48.051 20 01 2 52.74 50.941 21 21 2 53.41 51.601 20 11 2 60.50 58.691 20 21 2 27
http://arXiv.org/abs/physics/0102082 physics/0102082 ppp hhh yyysssiiicccsss///000111000222000888222 Problems with “GR without SR: A gravitational -domain description of first -order Doppler effects” (gr-qc/9807084) Eric Baird February 2001 Einstein’s goal of producing an advanced gravitational model that was independent of special relativity’s “non -gravitational” derivations has arguably still not been achieved. The author produced a paper in 1998 outlining a possible method of attack on the problem based on shift equivalence principles (“Doppler mass shift”). Some problems with this paper have since come to light. We list some further developments, some papers not cited in gr -qc/9807084, some experimental results missed by the author, and identify a problem with the paper’s approach to aberration effects. 1. INTRODUCTION In April 1950, Einstein publish ed an article in Scientific American stating his opinion that a full gravitational model should not depend on special relativity as a foundation [1] – while physics within flat spacetime is certainly very convenient [2], the simplifying assumption of iner tial mass without gravitational mass is not a natural feature of more advanced gravitational models. Paper gr -qc/9807084 by this author “GR without SR” [3] described a paradigm intended to avoid special relativity’s “flat” geometry, with velocity - dependen t curvature between relatively -moving masses acting as a regulating mechanism for local lightspeed constancy (with conventional Doppler shifts appearing as gravitational -domain effects). The most suprising aspects of this alternative model were i) that it seemed to allow a different “relativistic” shift equation to that used by special relativity, and ii) that this altered relationship did not seem to generate the usual incompatibilities between classical and quantum models (“black hole information paradox” [4]), and still seemed to be compatible with all current experimental data. Since the first version of the paper was uploaded to the LANL archive, some problems have come to light, namely 1. the paper’s accidental reversal of some aberration relationships a nd effects, and 2. the omission of published experimental evidence supporting special relativity’s Doppler relationships. In this supplementary paper we document these problems, and also list some additional work that has been produced or become known to th e author since September 1998. 2. ABERRATION PROBLEMS A basic derivation of relativistic aberration effects for shift equations (1), (2) and (3) (as identified in [3] section 5.1) has shown that special relativ ity’s aberration formula [5], with its forward -tilted rays is a “general” relativistic result [6], and does not depend on the assumption of flat spacetime or on special relativity’s particular choice of Doppler equations. The paper therefore seems to be i n error when it relates a rearward -observed redshift on a receding object to an apparent rearward concentration of fieldlines in a receding object – while the author’s calculated fieldline density -change ratios are probably correct, the actual relationship seems to be inversely proportional rather than directly proportional (it is easy to make this sort of accidental inversion when working with an “unexplored” model, e.g. Newton [7]). The correct argument would seem to be the one presented elsewhere in the paper for the inverse relationship between the object’s perception of environmental Doppler effects and of environmental fieldline density – the “moving” object sees more environmental mass ahead than behind, but the proposed Doppler mass shift effect act s to enhance the gravitational influence of aft (redshifted) masses and lessen the attraction of the forward (blueshifted) material – the Doppler effect acts against the calculated effect of aberration on fieldline density [8]. A recent piece by Carlip [9] has come to a broadly similar conclusion regarding the equal magnitude of displacement and aberration effects, with velocity components acting to make any supposed gravitational -field aberration effects undetectable. Problems with “GR without SR: A gravitational -domain description of first -order Doppler effects” Eric Baird 2/4 http://arXiv.org/abs/physics/0102082 physics/0102082 ppphhhyyysssiiicccsss///000111000222000888222 3. MISSED EXPERIMENTAL RESULTS 3.1 “Transv erse” tests using non-transverse measurements From “GR without SR” [3] section 9.1: “ While it may seem improbable that we have not yet been able to verify that the SR shift prediction (2) is more accurate than the earlier equation (1), the author has so far been unable to find any direct evidence favoring (2)” In fact, a number of experiments that are listed as tests of special relativity’s “transverse” redshift predictions are actually based on analysis of non- transverse data (e.g. Ives -Stilwell [10]). These experiments had been missed because of their usual classification as “transverse” tests [11]. At least two of the experiments listed in MacArthur’s review piece [11], [10][12] do lend themselves to reanalysis, and do appear to give a very good match to the predictions of (2) rather than (1), which seems to settle the issue in favour of special relativity. However, th ese experiments were carried out at a time when it seems to have been widely accepted that lab -transverse redshift effects could only be generated by (2) (“... no change in frequency. ” [13] “classically one would not expect a frequency shift from a source t hat moves by right angles.” [14]), suggesting the possibility that an experimenter looking for either a null result or a Lorentz redshift might attribute an “inexplicable” double -strength redshift [15] to a combination of (2) and some additional redshifting effect in the apparatus (such as mirror recoil). Observations made at 90° LAB Eqn (1) Eqn (2) Eqn (3) “bad” textbook predictions [13] null result Lorentz redshift null result Corrected predictions [15] null result Lorentz redshift Double Lorentz redshift Since these experiments seem to be the only ones to date that give unequivocal support for special relativity’s choice of shift equation, it would be helpful if similar future experiments [16] could be designed to test for possible agreement with (1), as well as with (2) and (3) [17]. 4. ADDITIONAL REFERENCES AND FURTHER WORK 4.1 Black hole information paradox This paradox (Susskind review article [4]) has also been discussed in detail by Preskill, Danielson and Schiffer [18][19], and does not appear to apply to a physics based on (1). Unruh’s work on indirect radiation through signal horizons in non -SR models [20][21] described “sonic horizon” radiation as being an analogue of Hawking radiation. Visser has since presented this effect as a full Hawking radiation effect [22][23][24]. Visser’s paper appeared in the LANL archive after research for “GR without SR” had been completed. 4.2 Aberr ation and angle -dependent shifts The angle -changes and angle -dependent frequency -shifts associated with hypothetical relativistic models based on (1), (2) and (3) have now been calculated from general principles, without presupposing flat spacetime [6]. In this exercise, all three calculations generate the same aberration formula as special relativity [5], and generates the result that for any given laboratory angle, (2) predicts wavelengths that are Lorentz redshifted compared to (3), but (1) predicts wavelengths that are doubly Lorentz -shifted compared to (3) [6][15]. Lorentz relationships are often presented as being unique to (2) (but see Visser [25]). 4.3 E=mc2 An exact derivation of the E=mc2 result from (1) for a pair of plane -waves aligned with the moving object’s path has now been given and discussed in [26]. The relationshi ps given in [6] also let us apply this result to pairs of plane -waves emitted at any other angle. 4.4 Transverse effects The Lorentz -squared redshift predictions for a lab-transverse detector for (1) are general an d apply to relativistic and non -relativistic calculations ( see: Lodge’s 1893 “ spurious or apparent Doppler ” prediction [27]). Special relativity’s predictions are the root product of the predictions made for (1) and (3) not just for the non-transverse case [28], but also for any other angle defined in a given frame [29]. Problems with “GR without SR: A gravitational -domain description of first -order Doppler effects” Eric Baird 3/4 http://arXiv.org/abs/physics/0102082 physics/0102082 ppphhhyyysssiiicccsss///000111000222000888222 4.5 Time dilation without acceleration Textbooks tend to suggest that muon pathlengths are only explainable using special relativity [30]. The mathematics indicates otherwise – for a muon created at the edge of the Earth’s atmosphere with a given rest mass, rest frame lifetime and momentum, the “new” calculated penetration depth under special relativity is the same as the older Newtonian prediction [15]. Lab-transverse redshifts (“aberration redshifts”) already feature in a range of models that do not include physical time -dilation effects [15]. 4.6 Acceleration effects Acceleration of an object towards the obse rver introduces non -linear behaviour that is not compatible with flat -space approximations [31] (see also “acceleration radiation” and Bekenstein/Hawking radiation under more standard theory). A full gravitational description of the “twins" problem can run into similar problems under general relativity [32][33]. 5. WORK BY OTHER AUTHORS Matt Visser and W.G. Unruh have derived purely classical indirect radiation effects in transonic fluid flows, and identified these effects as Hawking radiation effects. This work would also seem to apply to indirect radiation through the r=2M surface in systems of physics based around shift equation (1). Steve Carlip has produced a study of the aberration -gravity problem, and related the lack of additional gravitational aberrati on effects to the existence of gravitational velocity components [9]. S. Dinowitz has described a model that sounds similar in concept to the Doppler mass shift idea [34]. Wolfgang Rindler has described how ge neral relativity could (in theory) have been developed in the Nineteenth Century, independently of special relativity [35]. 6. CONCLUSIONS The sort of model described in [3] calls on several obscure areas of physi cs theory that have not yet been fully explored, because of complicating non -linearities or incompatibilities with special relativity. Progress in at least some of these areas is now being stimulated by work on the black hole information paradox. Experime nts such as Ives/Stilwell indicate that Nature’s shift laws do obey (2) rather than (1), apparently invalidating this approach and making the existence of a possible non -SR solution to the information paradox irrelevant. However, these experiments were pr obably designed to differentiate between the “null shift” and “redshift” predictions of (2) and (3). If experimenters were not aware of the (largely undocumented) double Lorentz redshift predictions associated with (1), it is still conceivable that further experiments, designed to differentiate between (1) and (2), may still tip the balance of evidence towards the non -SR equation. Radial Doppler frequency -changes and ruler -changes freq’ / freq = (c-v) / c … (1) freq’ / freq = ( )( )vcvc + −/ … (2) freq’ / freq = c / (c+v) … (3) ( perceived ruler -lengths alter by the same ratio as perceived frequency ) Lab-transverse Doppler frequency - changes and ruler -changes freq’ / freq = 2 2/ 1 cv− … (t1) freq’ / freq = 2 2/ 1 cv− … (t2) freq’ / freq = 1 … (t3) ( … these predictions apply to observations at a “90°” angle, defined in the laboratory observer’s frame ) Problems with “GR without SR: A gravitational -domain description of first -order Doppler effects” Eric Baird 4/4 http://arXiv.org/abs/physics/0102082 physics/0102082 ppphhhyyysssiiicccsss///000111000222000888222 REFERENCES [1] A. Einstein, "On the Generalized Theory of Gravitation" Sci. Am. 182 (4) 13 -17 (April 1950). [2] Misner, Thorne and Wheeler (“MTW”) Gravitation (Freeman, NY, 1971), section 6. [3] Erk “GR without SR: A gravitational -domain description of first-order Doppler effects” arXiv reference: gr-qc/9807084 [4] Leonard Susskind, "Black Holes and the Information Paradox," Sc i. Am. 276 (4) p.40 -45 (April 1997). [5] A. Einstein, "On the Electrodynamics of Moving Bodies" section 7 (1905), translated in The Principle of Relativity (Dover, NY, 1952) pp.35 -65. [6] Eric Baird, “Relativistic angle -changes and frequency - changes” arXiv reference: physics/0010006 [7] Eric Baird, “Newton’s aether model” arXiv reference: physics/0011003 [8] This “environmental aberration” issue probably deserves to be described in a separate paper. [9] S. Carlip, “Aberration and the Speed of Grav ity” Phys. Lett. A 267 81-87 (2000). [10] Herbert E. Ives and G.R. Stilwell, “An experimental study of the rate of a moving clock” J. Opt. Soc. Am 28 215 -226 (1938). [11] D.W. MacArthur, “Special relativity: Understanding experimental tests and formulatio ns” Phys. Rev. A 33 1-5 (1986). [12] Hirsch I. Mandelberg and Louis Witten, “Experimental verification of the relativistic Doppler effect” J. Opt. Soc. Am. 52 529 -536 (1962). [13] W.G.V. Rosser, An Introduction to the Theory of Relativity (Butterworths, Lon don, 1964) section 4.4.7 pp.160. [14] Richard A. Mould, Basic Relativity , (Springer -Verlag, NY, 1994) pp.80. [15] Eric Baird, “Transverse redshifts without special relativity” arXiv reference: physics/0010074 [16] Some of the key frequency -shift relation ships will be identified and discussed in a further paper. [17] R. Klein, R. Grieser et al, “Measurement of the transverse Doppler shift using a stored relativistic 7Li+ ion beam” Z. Phys. A 342 455 -461 (1992). [18] John Preskill, “Do Black Holes Destroy Inf ormation?” published in International Symposium on Black Holes, Membranes, Wormholes and Superstrings January 16 -18, 1992 (World Scientific, Singapore, 1993) eds. Sunny Kalara and D.V. Nanopoulos pp.22 -39. [19] Ulf H. Danielson and Marcelo Schiffer “Quantu m mechanics, common sense, and the black hole information paradox” Phys. Rev. D 48 4779 -4784 (1993). [20] W.G. Unruh, “Experimental Black -Hole Evaporation?” Phys. Rev. Letts. 46 1351 -1353 (1981). [21] W.G.Unruh, “Sonic analogue of black holes and the ef fects of high frequencies on black hole evaporation,” Phys.Rev. D 51 2827 -2838 (1995). [22] Matt Visser, “Hawking Radiation without Black Hole Entropy” Phys. Rev. Letts. 80 3436 -3439 (1998). [23] Matt Visser, “Acoustic black holes: horizons, ergospheres and Hawking radiation” Class. Quantum Grav. 15 1767 -1791 (1998) [24] Matt Visser, “Acoustic black holes” arXiv reference: gr-qc/9901047 [25] Matt Visser, “Acoustic propagation in fluids: an unexpected example of Lorentzian geometry” arXiv reference: gr-qc/9311028 [26] Eric Baird, “Two exact derivations of the mass/energy relationship, E=mc2” arXiv reference: physics/0009062 [27] Oliver Lodge, “Aberration Problems,” Phil.Trans.Roy.Soc. (1893) sections 56 -57. [28] T.M. Kalotas and A.R. Lee, “A ‘two -line’ derivation of the relativistic longitudinal Doppler formula” Am. J. Phys 58 187 -188 (1990). [29] This relationship is sometimes hidden by defining angles differently under the different models being compared [30] Clifford M. Will, Was Einstein Right?: Putting General Relativity to the Test (Oxford University Press, Oxford, 1988), Appendix pp. 245 -257. [31] Eric Baird, “Warp drives, wavefronts and superluminality” arXiv reference: physics/9904019 [32] C.B Leffert and T.M. Donahue, “Clock Paradox and the Physics of Discontinuous Gravitational Fields” Am. J. Phys. 26 514 -523 (1958). [33] C. Møller, “Motion of free particles in discontinuous gravitational fields” Am. J. Phys. 27 491 -493 (1959). [34] S. Dinowitz, “Field Distortion Theory,” Physics Essays 9 393- (1996). [35] Wolfgang Rindler “General Relativity before special relativity: An unconventional overview of relativity theory” Am. J. Phys. 62 887 -893 (1994). additional papers to appear during 2001: http://arXiv.org/find/gr -qc,physics/1/au:+baird/0/ 1/0/all/0/1
arXiv:physics/0102083v1 [physics.ins-det] 26 Feb 2001Compact vacuum phototriodes for operation in strong magnetic ﬁeld M.N.Achasov,1B.A.Baryshev, K.I.Beloborodov, A.V.Bozhenok, S.V.Burdin, V.B.Golubev, E.E.Pyata, S.I.Serednyakov, Z.I.Stepanenko, Yu.A.Tikhonov, P.V.Vo robyov G.I.Budker Institute of Nuclear Physics, Siberian Branch o f the Russian Academy of Sciences, Novosibirsk, 630090, Russia Abstract The results of tests of 1” vacuum phototriodes in a magnetic ﬁ eld up to 4.5 T are presented. It was found that output amplitude decreases by a bout 6% per tesla in the magnetic ﬁeld range from 2.0 to 4.0 T. For devices with an a node mesh pitch of 16µm, the output amplitude at 4.0 T is 30% lower than that at zero ﬁ eld. 1 Introduction Scintillation calorimeters which are an important part of a ll modern elemen- tary particle detectors are often located in harsh environm ent of strong mag- netic ﬁeld and high radiation ﬂuxes. Photodetectors having to operate in such conditions have to satisfy special requirements. A photodetector type satisfying these requirements is the v acuum phototriode (VPT), which is a single-dynode photomultiplier tube with p roximity focusing of photoelectrons. This makes it possible to operate in a str ong axial magnetic ﬁeld. Use of radiation-hard glass for VPT manufacturing mak es it also tolerant to high doses of ionizing radiation. VPTs are already widely used in calorime- try, for example in the detectors DELPHI [1,2] and OPAL [3] at LEP. They are also planned for use in the end-cap PbWO calorimeter of the CM S detector for LHC [4]. Another type of photodetector for calorimetry, the avalanc he photodiode (APD), shows also excellent performance in strong magnetic ﬁeld [5 ]. APDs have how- 1E-mail: achasov@inp.nsk.su, FAX: +7(383-2)34-21-63 Preprint submitted to Elsevier Preprintever some drawbacks, such as a small active area and direct co unting of charged particles which can make VPTs preferable in some cases. Several years ago VPTs with the capability to work in magneti c ﬁelds up to 2.5 T [6] were developed in BINP; about 3500 devices of this ki nd operate in the electromagnetic calorimeters of the SND [7], CMD-2 [8 ] and KEDR [9] detectors. In 1999 a new low cost VPT was developed for ope ration in strong magnetic ﬁeld. In this paper the results of tests of th e new devices in a magnetic ﬁelds up to 4.5 T are presented, and its performance compared with that of similar photodetectors described in [10,11]. 2 Phototriode parameters and experimental set-up The VPTs were manufactured using conventional bulb technol ogy described elsewhere in [6]. All VPT electrodes are connected to pins on its base. The semitransparent bialkali photocathode was formed on the in ner surface of a window glass S52-1 or S52-2 with transparency of more than 96 % in the wave- length range from 300 to 900 nm. The quantum eﬃciency of the ph otocathode in the maximum of spectral sensitivity measured with calibr ated light source was about 20%. The VPT performance in magnetic ﬁeld depends on the anode mes h pitch. The smaller the pitch, the larger fraction of secondary emis sion electrons from the dynode is collected on the anode, but on the other hand a sm aller part of accelerated photoelectrons reach the dynode. Prototypes w ith the anode mesh pitch sof 250, 100, 50 and 16 µm were manufactured. The devices height is 40 mm, tube diameter is 25 mm, the photocathode spectral sens itive region is from 360 to 600 nm, maximum of photocathode spectral sensiti vity is λmax= 420 nm, total photocathode sensitivity is 95 µA/lm, typical quantum eﬃciency atλmaxis about 20%, dark current is less than 1 nA, gain without magn etic ﬁeld is 15 for s= 50µm and 10 for s= 16µm, anode mesh transparency is 60% for s= 50µm and 52% for s= 16µm. The layout of the test system used to check the operation in hi gh magnetic ﬁelds is shown in Fig.1. Measurements were performed using a charge sensitive preampliﬁer with a sensitivity of 0.7 V/pC and a shaper with i ntegration and diﬀerentiation time constants of 2 µs. Green LED with wavelength of about 520 nm was used as a source of light signals. A magnetic ﬁeld wi th a strength up to 4 .5T was produced by a superconducting solenoid. The VPT axis c ould be tilted by up to 30 degrees with respect to the magnetic ﬁeld direction. The dependences of output signal on photocathode and dynode voltages at zero magnetic ﬁeld are shown in Figs.2,3. For further measur ements the pho- 2tocathode and dynode voltages were ﬁxed to Uc=−1000V and Ud=−200V respectively. After absorption of high dose of radiation the input glass wi ndow may darken thus decreasing the VPT sensitivity. The dependence of tran sparency of VPT windows made of S52-1 and S52-2 glass on the radiation dose is shown in Fig.4. The radiation harder S52-2 glass was chosen as a mater ial for the VPT window. 3 VPT performance in magnetic ﬁeld The dependence of VPT output amplitude on the magnetic ﬁeld s trength was measured both illuminating the entire photocathode area, a nd its central part (10mm in diameter). Fig.5 shows the dependences of the outpu t signal on magnetic ﬁeld for VPTs with diﬀerent anode mesh pitches in ca se of illumina- tion of entire photocathode. The amplitude drop at B= 4.0T varies from 70% for tubes with s= 250 µm mesh to 30% for s= 16µm. The dependence for s= 50µm and s= 16µm with illumination of the central part of photocathode is shown in Fig.6. The output signal decreases by about 6% per tesla in a range of ﬁeld from 2.0 to 4.0T. The diﬀerence in amplitude drops for illumination of the full photocathode area and of its central part can be expl ained by eﬀec- tive cut-oﬀ of the peripheral area of the photocathode in axi al magnetic ﬁelds. Photoelectrons from this area, propagating along the magne tic ﬁeld, cannot reach the dynode which due to manufacture constraints has sm aller diameter. The dependence of the output amplitude on α, the angle between the magnetic ﬁeld and the tube axis, is shown in Fig.7. The initial amplitu de increase by ∼ 15% with αcan be attributed to the increase of the secondary electron e mission coeﬃcient on the dynode for larger impact angles of the photo electrons. At larger tilt angles another eﬀect, the decrease of anode mesh transparency for photoelectrons, apparently becomes dominant. The amplitu de dependence for mesh pitch s= 50µm and α= 30◦on the magnetic ﬁeld strength is shown in Fig.8. The tests demonstrate that the VPTs with 16 µm anode mesh are the best for operation in strong magnetic ﬁeld. The output signal dec reases by less than 30% at 4.0 T, for a angles between the tube axis and the ﬁel d up to 40◦. Recently, the results of the RIE VPTs (diameter of 25 mm) tes ts in a magnetic ﬁeld were presented in Ref.[11]. The output amplit ude of the device making use of a ﬁne mesh with 100 lines per mm decreases by abou t 40% at 4T and α= 0◦. As reported in Ref.[10], in commercial Hamamatsu 25 mm vacuum phototetrodes the output signal amplitude decrease by about 70% in the same conditions. 34 Conclusions We describe the development of prototypes of compact vacuum phototriodes with quantum eﬃciency of ∼20% and gain 10 ÷15 for operation in strong magnetic ﬁeld. Their performance in the ﬁelds up to 4.5 T was t ested. It was found that the decrease of output amplitude is about 6% per te sla in the magnetic ﬁeld range from 2.0 to 4.0 T. For VPT with anode mesh p itch of 16µm the output amplitude at 4.0 T is 30% less than that without ma gnetic ﬁeld. References [1] P.Abreu et al., Nucl. Instr. and Meth., A378 (1996), 57. [2] P.Checcina et al., Nucl. Instr. and Meth., A248 (1986), 3 17. [3] M.Akrawy et al., Nucl. Instr. and Meth., A290 (1990), 76. [4] J.P.Ernenwein, Nucl. Phys. — Proceedings Supplements, B78 (1999), 186. [5] J.Marler et al., Nucl. Instr. and Meth., A449 (2000), 311 . [6] P.M.Beschastnov et al., Nucl Instr and Meth. A342 (1994) , 477. [7] M.N.Achasov et al., Nucl. Instr. and Meth. A449 (2000), 1 25. [8] R.R.Akhmetshin, et al. Nucl. Instr. and Meth., A379 (199 6), 509. [9] V.M.Aulchenko, et al. Nucl. Instr. and Meth., A379 (1996 ), 502. [10] M.Bonesini et al., Nucl. Instr. and Meth., A387 (1997), 60. [11] N.A.Bajanov et al., Nucl. Instr. and Meth., A442 (2000) , 146. 4/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1SHAPER ADCUd UcANODE DYNODE PHOTOTRIODE αBPHOTOCATHODE LEDPREAMPLIFIER CALIBRATIONOPTICAL FIBER Fig. 1. Test system layout 5Uc, VPulse height (a.u.) 0.20.40.60.81 0 200 400 600 800 1000 Fig. 2. The VPT output signal as a function of photocathode vo ltageUc. The dynode voltage is ﬁxed to Ud=−200V. The kink at Uc=−200 V reﬂects the transition from photodiode to a phototriode operation mode of the devic e. Ud, VPulse height (a.u.) 0.20.40.60.81 400 600 800 1000 Fig. 3. The VPT output signal as a function of dynode voltage Ud. The photocathode voltage is ﬁxed to Uc=−1000V 6Fig. 4. Transparency of S52-1 and S52-2 glasses, 1 mm thick, a s a function of absorbed radiation dose for diﬀerent wavelength. 7Relative amplitude 0.20.40.60.81 0 1 2 3 4 5 B, T Fig. 5. Relative output amplitude as a function of magnetic ﬁ eld at α= 0◦for VPTs with anode mesh spacing 16 µm (•), 50µm (/squaresolid), 100µm (/trianglesolid), 250µm (/triangledownsld). Full photocathode illumination. Relative amplitude 0.50.60.70.80.911.1 B, T 0 1 2 3 4 5 Fig. 6. Relative output amplitude as a function of magnetic ﬁ eld at α= 0◦for VPTs with anode mesh spacing 16 µm (•), 50µm (/squaresolid). Illumination of the central part of photocathode. 8Fig. 7. Relative output amplitude as a function of tilt angle αin 4T magnetic ﬁeld for VPTs with anode mesh spacing 16 µm (•), 100 µm (/trianglesolid) in case of full photocathode illumination. The line through ( ∗) corresponds to VPT with mesh spacing 16 µm, when only the central part of photocathode was illuminated. Relative amplitude 0.20.40.60.81 0 1 2 3 4 5 0 1 2 3 4 5 B, T Fig. 8. Relative output amplitude as a function of magnetic ﬁ eld at α= 30◦for VPT with anode mesh spacing 50 µm and full photocathode illumination ( /squaresolid) or illumination of the central part only ( ∗). 9
arXiv:physics/0102084v1 [physics.class-ph] 26 Feb 2001True energy-momentum tensors are unique. Electrodynamics spin tensor is not zero R. I. Khrapko1 In the electrodynamics the variational principle results i n a pair of the canonical tensors: the energy- momentum tensor Tcikand the spin tensor Υcikj(upsilon) Tcik=−∂iAl·Fkl+gikFpqFpq/4,Υcikj=−2A[iFk]j. However, they contradict experience. It is obvious in view o f a asymmetry of the energy- momentum tensor, and it is checked up directly. For example, in a constant uniform magnetic ﬁeldBx=B,B y=Bz= 0, Fyz=Fyz=−B, A y=Bz/2, A z=−By/2, the energy-momentum tensor gives zero value of a ﬁeld pressu re across ﬁeld lines: Tyy=Tzz= 0, what is wrong. The canonical energy-momentum tensor does no t satisfy the conservation equation, and its divergence is equal to a wrong expression −∂iAk·jk, but not to −Fikjk: ∂kTcik=−∂iAk·jk . The true energy-momentum tensor in the electrodynamics is t he Minkowski tensor. Tik=−FilFk l+gikFlmFlm/4. Only this tensor satisﬁes experiments. Only this tensor loc alizes energy-momentum. But, as appear, a true spin tensor in electrodynamics is unknown, Υikj= ? True deﬁnitions of energy-momentum and spin tensors do not a dmit any arbitrariness because the trasvections of the tensors and an 3-element dVkare observable quantities: an inﬁnitesimal 4-momentum dPiand 4-spin dSik: dPi=TikdVk, dSik= ΥikjdVj The well-known attempt to correct the canonical energy-mom entum tensor by subtraction the Rosenfeld’s pair, (T Rik,Υ Rikj) = (∂jΥc{ikj}/2,Υcikj),Υ{ikj}= Υikj−Υkji+ Υjik, from the canonical pair of tensors does not lead to the Minkow ski tensor merely because the Rosen- feld’s identity is obviously wrong: Tcik−T Rik=Tcik−∂jΥc{ikj}/2 =Tik−Aijk/negationslash=Tik. 1Moscow Aviation Institute, 4 Volokolamskoe Shosse, 125871 , Moscow, Russia. E-mail: tahir@k804.mainet.msk.su Subject Khrapko 1But the subtraction leads to an elimination of the electrody namics spin tensor: Υcikj−Υcikj= 0. Meanwhile, the subtraction is inadmissible on principle be cause an addition of any construction, including, for example, ˜Tik=∂lψikl, ψi(kl)= 0, to an energy-momentum tensor of concrete matter corresponds to a local change of matter. And what is more, the addition of ∂lψiklcan change the total 4-momentum and angular 4-momentum of a system. For exa mple, it is easy to express the energy-momentum tensor of an uniform ball of radius Rin the form ˜Tik=∂lψikl: ψ00α=−ψ0α0=ǫxα/3 (r<R ), ψ00α=−ψ0α0=ǫR3xα/3r3(r>R ) give ˜T00=−ǫ(r<R ),˜T00= 0 (r>R ), α= 1,2,3. Obviously, an addition of this construction changes the tot al 4-momentum. Nevertheless, the authors of all textbooks on the theory of ﬁ elds repeat the same mistake. As an example I quote here from Landau and Lifshitz [1] using our no tations. “It is necessary to point out that the deﬁnition of the (energ y-momentum) tensor Tikis not unique. in fact, if Tikis deﬁned by Tk i=q,i·∂Λ ∂q,k−δk iΛ, (32.3) then any other tensor of the form Tik+∂ ∂xlψikl, ψijk=−ψikj, (32.7) will also satisfy equation ∂Tik/∂xk= 0, (32.4) since we have identically ∂2ψikl/∂xk∂xl= 0. The total four-momentum of the system does not change, since... we can write /integraldisplay∂ψikl ∂xldVk=1 2/integraldisplay/parenleftBigg dVk∂ψikl ∂xl−dVl∂ψikl ∂xk/parenrightBigg =1 2/contintegraldisplay ψikldakl, were the integral on the right side of the equation is extende d over the (ordinary) surface which ‘bond’ the hypersurface over which the integration on the le ft is taken. This surface is clearly located at inﬁnity in the three-dimensional space, and since neithe r ﬁeld nor particles are present at inﬁnity this integral is zero. Thus the fore-momentum of the system i s, as it must be, a uniquely determined quantity.” But it seems to be incorrect./contintegraltextψikldakl= 0,only ifψikldecreases on inﬁnity rather quickly. I present a three-dimensional analogy concerning an electri c currentIand its magnetic ﬁeld Hαβ: /integraldisplay ∂βHαβdaα=1 2/contintegraldisplay Hαβdlαβ=/integraldisplay jαdaα=I/negationslash= 0 Varying the electrodynamics action integral with respect t o a metric tensor in the Minkowski space, one can obtain the Minkowski energy-momentum tensor which is called the metric energy- momentum tensor. But, in our opinion, it is a happy accident, because it is impossible to obtain, for 2example, a spin tensor by varying an action integral with res pect to a torsion or contortion tensor. It cannot be done either for electromagnetic ﬁeld or for a ﬁel d which obviously has spin in the Minkowski space. It is a consequence of the fact, that the tor sion and contortion tensors are equal to zero in Minkowski space and even in a Riemann space (unlike gik). As appear, the variational principle is not capable to give a true spin tensor in a Rieman n space. So we are not sure, that varying an action integral in U4space (with a torsion) one can obtain true energy-momentum and spin tensors as it is aﬃrmed, for ex ample, in [2]. In any case it is obvious that a metric (symmetric) and a canonical energy-momentum t ensors will be essentially diﬀerent. In electrodynamics the elimination of spin tensor leads to a strange opinion that a circularly polarized plane wave with inﬁnite extent can have no angular momentum [3], that only a quasiplane wave of ﬁnite transverse extend carries an angular momentum whose direction is along the direction of propagation. This angular momentum is provided by an oute r region of the wave within which the amplitudes of the electric Eand magnetic Bﬁelds are decreasing. These ﬁelds have components parallel to wave vector there, and the energy ﬂow has compone nts perpendicular to the wave vector. “This angular momentum is the spin of the wave” [4]. Within an inner region the EandBﬁelds are perpendicular to the wave vector, and the energy-momentum ﬂ ow is parallel to the wave vector [5]. But let us suppose now that a circularly polarized beam is abs orbed by a round ﬂat target which is divided concentrically into outer and inner parts. Accor ding to the previous reasoning, the inner part of the target will not perceive a torque. Nevertheless R . Feynman [6] clearly showed how a circularly polarized plane wave transfers a torque to an abs orbing medium. What is true? And if R. Feynman is right, how one can express the torque in terms of po ndermotive forces? From our point of view, classical electrodynamics is not com plete. The task is to discover the nonzero spin tensor of electromagnetic ﬁeld. NOTE The subject matter of this paper had been rejected by the foll owing journals: Amer. J. Phys., Journal of Experimental & Theoretical Physics, Theoretica l and Mathematical Physics, Physics - Uspekhi. The subject matter of this paper has been published in Abstracts of 10-th Russian Gravitation Conference (Vladim ir, 1999), p. 47. http://www.mai.ru/projects/mai works/index.htm “Spin density of electromagnetic waves”. REFERENCES 1. L. D. Landau, E. M. Lifshitz. The Classical Theory of Field s (Pergamon, New York, 1975), 4th ed., p. 78. 2. F. W. Hehl, et al.. “General relativity with spin and torsion: Foundation and p rospects” Rev. Mod. Phys., 48, 393. (1976). 3. W. Heitler. The Quantum Theory of Radiation (Clarendon, O xford, 1954), p. 401. 4. H. C. Ohanian. “What is spin?” Amer. J. Phys. 54, 500, (1986). 5. J. D. Jackson. Classical Electrodynamics (John Wiley, Ne w York, 1962), p. 201. 6. R. P. Feynman et al.. The Feynman Lectures On Physics (Addi son-Wesley, London, 1965), v. 3, p. 17-10. 3
arXiv:physics/0102085v1 [physics.atom-ph] 26 Feb 20012s Hyperﬁne Structure in Hydrogen Atom and Helium-3 Ion Savely G. Karshenboim1,2⋆ 1D. I. Mendeleev Institute for Metrology, 198005 St. Petersb urg, Russia 2Max-Planck-Institut f¨ ur Quantenoptik, 85748 Garching, G ermany Abstract. The usefulness of study of hyperﬁne splitting in the hydroge n atom is limited on a level of 10 ppm by our knowledge of the proton stru cture. One way to go beyond 10 ppm is to study a speciﬁc diﬀerence of the hyperﬁn e structure intervals 8∆ν2−∆ν1. Nuclear eﬀects for are not important this diﬀerence and it i s of use to study higher-order QED corrections. 1 Introduction The hyperﬁne splitting of the ground state of the hydrogen at om has been for a while one of the most precisely known physical quantities, h owever, its use for tests of QED theory is limited by a lack of our knowledge of the proton structure. The theoretical uncertainty due to that is on a level of 10 ppm . To go farther with theory we need to eliminate the inﬂuence of the nucleus. A few ways have been used (see e. g. [1]): •to remove the proton from the hydrogen atom and to study a two- body system, which is like hydrogen, but without any nuclear stru cture, namely: muonium [2] or positronium [3]; •to compare the hyperﬁne structure intervals of the 1 sand 2sstates (this work); •to measure the hyperﬁne splitting in muonic hydrogen and to c ompare it with the one in a normal hydrogen atom (a status report on the n= 2 muonic hydrogen project is presented in Ref. [4]; compariso n of the 1sand 2shfs and possibility to measure 1 shfs is considered in Ref. [5]. Recently there has been considerable progress in measureme nt and calculation of the hyperﬁne splitting of the ground state and the 2 s1/2state in the hydrogen atom. The 2 s1/2hyperﬁne splitting in hydrogen was determined to be [6] ∆ν2(H) = 177 556 .785(29) kHz , (1) While less accurate than the classic determination of the gr ound state hyperﬁne splitting, the combination of 1s and 2s hfsintervals D21(H) = 8∆ν2−∆ν1. (2) ⋆E-mail: sek@mpq.mpg.de2 Savely G. Karshenboim which is determined in the hydrogen atom [6] as D21(H) = 48.528(232) kHz , has more implications for tests of bound state QED because th ere is signiﬁcantly less dependence on the poorly understood proton structure c ontributions. Specif- ically, the theoretical uncertainty for the ground state fr om the proton structure is about 10 kHz, while the uncertainty for the combination is estimated to be few Hz. On the theoretical front, there has been considerable progr ess in the calcula- tion of the ground state hyperﬁne splitting. Taken together with earlier calcula- tions ofD21[14,15], which were possible because of cancellations of a n umber of large terms, one can now give quite accurate values for ∆ν1and∆ν2, We collect in Tables 1 and 2 along with the hydrogen results, the known ex perimental and theoretical results for the deuterium atom and the3He+ion. The helium results [7] ∆ν1(3He+) = 8665 649 .867(10) kHz (3) and [8] ∆ν2(3He+) = 1083 354 .969(30) kHz (4) lead to the most accurate value for the diﬀerence D21(3He+) = 1 189.979(71) kHz . (5) Table 1. Comparison of the QED part of the theory to the experiment for hydrogen and deuterium atoms and for the3He+ion. The results are presented in kHz Atom Experiment QED theory for D21 D21(exp) Refs.: 2s/1s Old New H 48.528(232) [6]/[9] 48.943 48.969(2) H 49.13(40) [10]/[9] D 11.16(16) [11]/[12] 11.307 11.3132(4) 3He+1189.979(71) [8]/[7] 1 189.795 1191.126(40) 3He+1 190.1(16) [13]/[7] 2 Theory We consider a hydrogen-like system with a nucleus of charge Z, massM, spinI, and magnetic moment µ. The basic scale of the hyperﬁne splitting is then given2s Hyperﬁne Structure in Hydrogen and Helium 3 by the Fermi formula, EF=8 3Z3α2Ryd\|µ\| µB2I+ 1 2I/parenleftbiggM m+M/parenrightbigg3 . (6) Here we take the ﬁne structure constant αderived from g-2 value of electron α−1= 137.035 999 58(52). In addition we use a value the Rydberg constan t of Ryd=cRy= 3.289 841 931 ·1012kHz. We present the hyperﬁne structure as a sum ∆νn=∆νn(QED) +∆νn(nuclear structure) . (7) 2.1 Non-recoil limit First we consider the external-ﬁeld approximation. For a po int-like nucleus, they can be compactly parameterized by the equation ∆νn(N-R) =EF n3/bracketleftbigg Bn+α πD(2) n(Zα) +/parenleftBigα π/parenrightBig2 D(4) n(Zα) +.../bracketrightbigg . Here, with γ=/radicalbig 1−(Zα)2, [22] the Breit relativistic contribution is B1=1 γ(2γ−1)≃1 +3 2(Zα)2+17 8(Zα)4+... (8) and B2=2/parenleftbig 2(1 +γ) +/radicalbig 2(1 +γ)/parenrightbig (1 +γ)2γ(4γ2−1)≃1 +17 8(Zα)2+449 128(Zα)4+... (9) and the functions D2r n(Zα) represent rloop radiative corrections. In the limit Zα= 0 they reduce to the power series expansion of the electron g−2 factor, and the diﬀerence is refered to a binding correction. For the ground state, D(2) 1=1 2+π(Zα)/parenleftbigg ln(2) −5 2/parenrightbigg + (Zα)2/bracketleftBigg −8 3ln2(Zα) +/parenleftbigg16 3ln(2) −281 180/parenrightbigg ln(Zα) +GSE 1(Zα) +GVP 1(Zα)/bracketrightBigg (10) and for the excited state D(2) 2=1 2+π(Zα)/parenleftbigg ln(2) −5 2/parenrightbigg + (Zα)2/bracketleftBigg −8 3ln2(Zα) +/parenleftbigg32 3ln(2) −1541 180/parenrightbigg ln(Zα) +GSE 2(Zα) +GVP 2(Zα)/bracketrightBigg .(11)4 Savely G. Karshenboim At present the functions GSEhave been determined numerically at Z= 1 and Z= 2 [16], GSE 1(Z= 1) = 16.079(15) (12) and GSE 1(Z= 2) = 15.29(9) (13) whileGV P 1is known analytically [17]: GVP 1=−8 15ln(2) +34 225+π(Zα)/bracketleftbigg −13 24lnZα 2+539 288/bracketrightbigg +... (14) To present results for the 2 sstate, we can use the results of Ref. [14] for D21, which however include terms only up to order α(Zα)2EF. After we recalculated some integrals from paper [14] the result is GSE 2=GSE 1+/parenleftbigg −7 +16 3ln(2)/parenrightbigg ln(Zα)−5.221233(3) + O/parenleftbig π(Zα)/parenrightbig (15) and GVP 2=GVP 1−7 10+8 15ln(2) + O/parenleftbig π(Zα)/parenrightbig . (16) Continuing to the two-loop corrections, all terms known to d ate are state- independent, so we give only the ground state result [18,19, 20], D(4) 1=a(4) e+ 0.7718(4)π(Zα)−4 3(Zα)2ln2(Zα). Non-leading terms, including single powers of ln( Zα) and constants, are both state-dependent, but unknown. When the nucleus is not point-like, the leading correction i s known as the Zemach correction, ∆νn(Zemach) =8EF πn3(Zαm)/integraldisplaydp p2/parenleftBig GE(p)/tildewideGM(p)−1/parenrightBig . (17) Inaccuracy arisen from uncertainties in the form factors GEand/tildewideGM, which are both normalized to unity at zero momentum, and from the lack o f knowledge of polarization eﬀects, is large as about 10 ppm or 4 ppm respect ively, but those leading terms are state-independent and do not contribute i nto the the diﬀerence D12. For atoms with nuclear structure the following result was fo und [15] ∆D21(Rec) = (Zα)2m M/braceleftBigg −9 8+/bracketleftbigg −7 32+ln(2) 2/bracketrightbigg/parenleftbigg 1−1 x/parenrightbigg −/bracketleftbigg145 128−7 8ln(2)/bracketrightbigg x/bracerightBigg , (18) wheregM/Zm p=x= (µ/µB)(M/m)(1/ZI). It does not depend on the nuclear structure eﬀects such a distribution of the nuclear charge and mag- netic moment. Contrary, the leading recoil term for the ∆νn(which has order (Zα)(m/M)ln(M/m) [21] is essentially nuclear-structure dependent.2s Hyperﬁne Structure in Hydrogen and Helium 5 3 Present status of D21theory 3.1 Old theory and recent progress The Breit, Zwanziger and Sternheim corrections [22,14,15] lead to a result D21=EF(Zα)2×/braceleftBigg/bracketleftbigg5 8+177 128(Zα)2/bracketrightbigg +α π/bracketleftbigg/parenleftbigg −7 +16 3ln(2)/parenrightbigg ln(Zα)−5.37(6)/bracketrightbigg +α π/bracketleftbigg −7 10+8 15ln(2)/bracketrightbigg +m M/bracketleftbigg −9 8+/parenleftbigg −7 32+ln(2) 2/parenrightbigg /parenleftbigg 1−1 x/parenrightbigg −/parenleftbigg145 128−7 8ln(2)/parenrightbigg x/bracketrightbigg/bracerightBigg . (19) Some progress was achieved before we started our work. In par ticular, we need to mention two results: •Integrals, used for in Ref. [14], were evaluated later by P. M ohr1with higher accuracy and the constant was found to be -5.2212 instead of - 5.37(6). The theoretical prediction based on Eq. (19) but with a correcte d value of the contstant is Table 1 as “old theory”. •Some nuclear-structure- and state-dependent corrections were found [23] for arbitrarynS. 3.2 Our results The similar diﬀerence has been under investigation also for the Lamb shift and a number of useful auxiliary expressions have been found for c alculating the state dependent corrections to the Lamb shift [24]. Let us mention that an improvement in the accuracy and new res ult on higher nhfs can be expected with progress in optical measurements an d we present here a progress also for higher n, deﬁningDn1=n3∆νn−∆ν1. •We have reproduced the logarithmic part of the self energy co ntribution and found for arbitrary ns ∆Dn1=α π(Zα)2EFln(Zα)/parenleftbigg −8 3/parenrightbigg ×/bracketleftbigg 2/parenleftbigg −ln(n) +n−1 n+ψ(n)−ψ(1)/parenrightbigg −n2−1 2n2/bracketrightbigg .(20) 1Unpublished. The result is quoted accordingly to Ref. [8].6 Savely G. Karshenboim The calculation is based on a result in Ref. [24] for the singe logarithmic correction due to the one-loop self energy and one-loop vacu um polarization. •We have reproduced the vacuum polarization contribution an d found that for arbitrary ns ∆Dn1=α π(Zα)2EF/parenleftbigg −4 15/parenrightbigg ×/bracketleftbigg 2/parenleftbigg −ln(n) +n−1 n+ψ(n)−ψ(1)/parenrightbigg −n2−1 2n2/bracketrightbigg .(21) •Integrals used by Zwanziger [14] have been recalculated and the constant was found to be -5.221233(3). •We also found two higher-order logarithmic corrections ∆Dn1=α2 π2(Zα)2EFln(Zα)/parenleftbigg −4 3/parenrightbigg ×/bracketleftbigg 2/parenleftbigg −ln(n) +n−1 n+ψ(n)−ψ(1)/parenrightbigg −n2−1 2n2/bracketrightbigg (22) and ∆Dn1=α πm M(Zα)2EFln(Zα)/parenleftbigg16 3/parenrightbigg ×/bracketleftbigg 2/parenleftbigg −ln(n) +n−1 n−ψ(n) +ψ(1)/parenrightbigg −n2−1 2n2/bracketrightbigg .(23) •We found two higher-order non-logarithmic corrections ∆DSE n1=α(Zα)3EF/braceleftBigg/bracketleftbigg139 16−4 ln(2)/bracketrightbigg ×/bracketleftbigg −ln(n) +n−1 n+ψ(n)−ψ(1)/bracketrightbigg +/bracketleftbigg ln(2) −13 4/bracketrightbigg ×/bracketleftbigg ψ(n+ 1)−ψ(2)−ln(n)−(n−1)(n+ 9) 4n2/bracketrightbigg/bracerightBigg (24) and ∆DVP n1=α(Zα)3EF×/braceleftBigg/bracketleftbigg5 24/bracketrightbigg /bracketleftbigg −ln(n) +n−1 n+ψ(n)−ψ(1)/bracketrightbigg2s Hyperﬁne Structure in Hydrogen and Helium 7 +/bracketleftbigg3 4/bracketrightbigg ×/bracketleftbigg ψ(n+ 1)−ψ(2)−ln(n)−(n−1)(n+ 9) 4n2/bracketrightbigg/bracerightBigg .(25) •We have also found a term proportional to the magnetic radius . To the best of our knowledge that is the ﬁrst contribution, which is prop ortional to the magnetic radius and on the level of the experimental accurac y. The complete nuclear-structure correction is ∆DNucl n1=−(Zα)2/bracketleftbigg ψ(n+ 1)−ψ(2)−ln(n)−(n−1)(n+ 9) 4n2/bracketrightbigg ×∆ν1(Zemach + polarizability) +4 3(Zα)2/bracketleftbigg ψ(n)−ψ(1)−ln(n) +n−1 n−/parenleftbiggRM RE/parenrightbigg2n2−1 4n2/bracketrightbigg/parenleftBig mRE/parenrightBig2 EF. (26) 4 Present status To calculate the corrections presented in the previous sect ions, we have used an eﬀective non-relativistic theory. In particular we have studied two kinds of terms. One is a result of the second order perturbation theor y with two δ(r)- like potentials, evaluated in the leading non-relativisti c approximation, while the other is due to a more accurate calculation of a single del ta-like poten- tial. Both kinds contribute into the state-independent lea ding logarithmic cor- rections (α2(Zα)2ln2(Zα),α(Zα)2(m/M)ln2(Zα), andα(Zα)3ln(Zα)) and to next-to-leading state-dependent terms ( α2(Zα)2ln(Zα),α(Zα)2(m/M)ln(Zα), andα(Zα)3). The crucial question is if we found all corrections in thes e orders. Rederiving a leading term in order α(Zα)2ln(Zα) within our technics, we can easily incorporate the anomalous magnetic moment of the ele ctron in the cal- culation and restore the nuclear mass dependence. Since we r eproduce the well- known result for the α(Zα)2ln(Zα) term, we consider that as a conﬁrmation of two other logarithmic corrections found by us. In the case of α(Zα)3it might be a contribution of an eﬀective operator, proportional to ( ∆/m)δ(r). That can give no logarithmic corrections, but leads to a state-depen dent constant. We are now studying this possibility. Summarizing all corrections, the ﬁnal QED result is found to be: DQED 21=EF(Zα)2×/braceleftBigg/bracketleftbigg5 8+177 128(Zα)2/bracketrightbigg +α π/bracketleftbigg/parenleftbigg −7 +16 3ln(2)/parenrightbigg ln(Zα)−5.221233(3)/bracketrightbigg8 Savely G. Karshenboim +α π/bracketleftbigg −7 10+8 15ln(2)/bracketrightbigg +m M/bracketleftbigg −9 8+/bracketleftbigg −7 32+ln(2) 2/bracketrightbigg /parenleftbigg 1−1 x/parenrightbigg −/bracketleftbigg145 128−7 8ln(2)/bracketrightbigg x/bracketrightbigg/bracerightbigg +α2 2π2/parenleftbigg −7 +16 3ln(2)/parenrightbigg ln(Zα) −α π2m M/parenleftbigg −7 +16 3ln(2)/parenrightbigg ln(Zα) +α(Zα)/braceleftBigg/bracketleftbigg139 16−4 ln(2) +5 24/bracketrightbigg /bracketleftbigg3 2−ln(2)/bracketrightbigg +/bracketleftbigg13 4−ln(2) −3 4/bracketrightbigg /bracketleftbigg ln(2) +3 16/bracketrightbigg/bracerightBigg . (27) Numerical results (in kHz) for hydrogen and deuterium atoms and the helium- 3 ion are collected in Table 2. One can see that the new correct ions essentially shift the theoretical predictions. A comparison of the QED p redictions (in kHz) against the experiments is summarized in Table 1. We take the values of the fundamental constants (like e. g. the ﬁne structure constan tα) from the recent adjustment (see Ref. [25]). Table 2. QED contributions to the D21in hydrogen, deuterium and helium-3 ion. The results are presented in kHz Contribution H D3He+ (Zα)2EF 47.2275 10.8835 1152.9723 +α(Zα)2EF(SE) 1.9360 0.4461 37.4412 +α(Zα)2EF(VP) -0.0580 -0.0134 -1.4148 + (Zα)2m MEF -0.1629 -0.0094 0.7966 +α2(Zα)2EF 0.0033(16) 0.0008(4) 0.070(35) +α(Zα)2m MEF -0.0031(15) -0.0004(2) -0.022(11) +α(Zα)3EF(SE) 0.0282 0.0065 1.3794 +α(Zα)3EF(VP) -0.0019 -0.0005 -0.0967 An important point is that the diﬀerence is sensitive to 4th o rder corrections and so is competitive with the muonium hfsas a test of the QED. The diﬀerence between the QED part of the theory and the experiment is an ind ication of higher-order corrections due to the QED and the nuclear stru cture, which have to be studied in detail. In particular, we have to mention tha t while we expect that we have a complete result on logarithmic corrections an d on the vacuum-2s Hyperﬁne Structure in Hydrogen and Helium 9 polarization part of the α(Zα)3term we anticipate more contributions in the orderα(Zα)3due to the self-energy. A complete study of this term oﬀers a possibility to determine the magnetic radius of the proton, deuteron and helion- 3. Acknowledgments The author would like to thank Mike Prior, Dan Kleppner and es pecially Eric Hessels for stimulating discussions. An early part of this w ork was done during my short but fruitful stay at University of Notre Dame and I am very grateful to Jonathan Sapirstein for his hospitality, stimulating disc ussions and participation in the early stage of this project. The work was supported in p art by RFBR grant 00-02-16718, NATO grant CRG 960003 and by Russian Stat e Program “Fundamental Metrology”. References 1. S. G. Karshenboim: invited talk at ICAP 2000, to be publish ed, e-print hep- ph/0007278 2. K. Jungmann: in Hydrogen atom: Precision Physics of Simple Atomic System , ed. S. G. Karshenboim et al., Springer-Verlag, 2001, to be publi shed 3. R. Conti et al: inHydrogen atom: Precision Physics of Simple Atomic System , ed. S. G. Karshenboim et al., Springer-Verlag, 2001, to be publi shed 4. R. Pohl et al: inHydrogen atom: Precision Physics of Simple Atomic System , ed. S. G. Karshenboim et al., Springer-Verlag, 2001, to be publi shed 5. K. Jungmann, V. G. Ivanov and S. G. Karshenboim: in Hydrogen atom: Precision Physics of Simple Atomic System , ed. S. G. Karshenboim et al., Springer-Verlag, 2001, to be published 6. N. E. Rothery and E. A. Hessels: presented at Hydrogen atom II conference (unpublished) 7. H. A. Schluessler, E. N. 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Kinoshita and M. Nio: Phys. Rev. Lett. 72, 3803 (1994) 19. M. I. Eides and V. A. Shelyuto: Pis’ma ZhETF 61, 465 (1995), JETP Lett. 61, 478 (1995); Phys. Rev. A 52, 954 (1995)10 Savely G. Karshenboim 20. S. G. Karshenboim: ZhETF 103, 1105 (1993) /in Russian/; JETP 76(1993) 541; Z. Phys. D 36, 11 (1996) 21. R. Arnowitt: Phys. Rev. 92, 1002 (1953); A. Newcomb and E. E. Salpeter: Phys. Rev.97, 1146 (1955) 22. G. Breit: Phys. Rev. 35, 1477 (1930) 23. S. G. Karshenboim: Phys. Lett. A 225, 97 (1997) 24. S. G. Karshenboim: Z. Phys. D 39, 109 (1997) 25. P. J. Mohr and B. N. Taylor: in Hydrogen atom: Precision Physics of Simple Atomic System , ed. S. G. Karshenboim et al., Springer-Verlag, 2001, to be p ub- lished
arXiv:physics/0102086v1 [physics.gen-ph] 26 Feb 2001NEW HIERARCHIC THEORY OF CONDENSED MATTER AND ITS COMPUTERIZED APPLICATION TO WATER & ICE Alex Kaivarainen H2o@karelia.ru URL: http://www.karelia.ru/˜alexk This work contains review of original quantum Hierarchic th eory of con- densed matter, general for liquids and solids and its numero us branches. Com- puter program (copyright, 1997, Kaivarainen), based on new theory, was used for comprehensive simulations of water and ice physical pro perties. Condensed matter is considered as gas of 3D standing waves (c ollective ex- citations) of diﬀerent nature: thermal de Broglie waves (wa ves B), IR photons and thermal phonons. Quantitative interrelation between m icroscopic, meso- scopic (as intermediate) and macroscopic properties of con densed matter are demonstrated. New theories of total internal energy, inclu ding contributions of kinetic and potential energy, heat capacity, surface tensi on, vapor pressure, ther- mal conductivity, viscosity and self-diﬀusion are describ ed. Hierarchic theory of osmotic pressure, based on new state equation, new theori es of light refrac- tion, Brillouin light scattering and M¨ ossbauer eﬀect are p resented also in article and compared with available experimental data for water and ice. Lot of hid- den parameters, inaccessible for experiment, describing t he dynamic and spatial properties of 24 quantum collective excitations of matter, can be calculated also, as demonstrated on examples of water and ice. Total number of physical parameters of liquids and solids in wide T-interval, including that of phase transitions, to be possible to evalu ate using CAMP computer program, is about 300. The agreement between theoretical and available experimen tal results is very good. The evidence of high-T mesoscopic molecular Bose cond ensation (BC) in water and ice in form of coherent clusters is obtained. The new mecha- nisms of the 1st and 2nd order phase transitions, related to s uch clusters forma- tion/melting, their assembly/disassembly and symmetry ch ange is proposed. Theory uniﬁes dynamics and thermodynamics on microscopic, mesoscopic and macroscopic scales in terms of quantum physics. The idea of new optoa- coustic device: Comprehensive Analyzer of Matter Properti es (CAMP) with huge informational possibilities, based on computer progr am, elaborated and its multisided applications are described. This work may be considered as a theoretical part of MANUEL to CAMP - program. The computer pr ogram (CAMP) is applicable for any condensed matter, if primary fo ur experimen- tal parameters are known in the same T-interval. It may be ord ered from the author. The number of articles, devoted to diﬀerent aspects and poss ibilities of new theory see at the electronic journal ”Archive of Los-Alamos ”: http://arXiv.org/ﬁnd/cond- mat,physics/1/au:+kaivarainen A/0/1/past/0/1 1CONTENTS: 1. INTRODUCTION 2. THE NEW NOTIONS AND DEFINITIONS, INTRODUCED IN HIERARCHIC THEORY OF MATTER 3. THE MAIN STATEMENTS AND BASIC FORMULAE OF HIERARCHIC MODEL 3.1. Parameters of individual de Broglie waves (waves B) 3.2. Parameters of de Broglie waves of molecules in com- position of condensed matter 3.3. Phase velocities of standing de Broglie waves, form- ing new types of quasiparticles 3.4. Concentration of quasiparticles, introduced in hi- erarchic model of condensed matter 4 . HIERARCHIC THERMODYNAMICS 4.1. The internal energy of matter as a hierarchical sys- tem of collective excitations 4.2. The contributions of kinetic and potential energy to the total internal energy 4.3. Some useful parameters of condensed matter 5 QUANTITATIVE VERIFICATION OF HIERARCHIC THEORY ON EXAMPLES OF ICE AND WATER 5.1. Discussion of theoretical temperature dependencies 5.2. Explanation of nonmonotonic temperature anoma- lies in aqueous systems 5.3. Physiological temperature and the least action prin- ciple 5.4. Mechanism of phase transitions in terms of the Hierarchic theory 5.5. The energy of quasiparticles discreet states. Acti- vation energy of water dynamics 5.6. The life-time of quasiparticles and frequencies of their excitation 6. INTERRELATION BETWEEN MESOSCOPIC AND MACROSCOPIC PROPERTIES OF MATTER 26.1. The state equation for real gas 6.2. New state equation for condensed matter 6.3. Vapor pressure 6.4. Surface tension 6.5. Mesoscopic theory of thermal conductivity 6.6. Mesoscopic theory of viscosity for liquids and solids 6.7. Brownian diﬀusion 6.8. Self-diﬀusion in liquids and solids 7. OSMOSIS AND SOLVENT ACTIVITY: CONVENTIONAL AND HIERARCHIC MODELS 8. NEW APPROACH TO THEORY OF LIGHT REFRACTION 8.1. Refraction in gas 8.2. Light refraction in liquids and solids 9. BRILLOUIN LIGHT SCATTERING 9.1. Traditional approach 9.2. Fine structure of scattering 9.3. Mesoscopic approach 9.4. Quantitative veriﬁcation of hierarchic theory of Brillouin scattering 10. HIERARCHIC THEORY OF M ¨OSSBAUER EFFECT 10.1. General background 10.2. Probability of elastic eﬀects 10.3. Doppler broadening in spectra nuclear gamma- resonance (NGR) 10.4. Acceleration and forces, related to thermal dy- namics of molecules and ions. Vibro-Gravitational interaction 11. ENTROPY-INFORMATIONAL CONTENT OF MATTER. SLOW RELAXATION, MACROSCOPIC OSCILLATIONS. EFFECTS OF MAGNETIC FIELD 11.1. Theoretical background 11.2. The entropy - information content of matter as a hierarchic system 11.3. Experimentally revealed macroscopic oscillations 11.4. Phenomena in water and aqueous systems, induced by magnetic ﬁeld 311.5 Coherent radio-frequency oscillations in water, re- vealed by C. Smith 11.6. Inﬂuence of weak magnetic ﬁeld on the properties of solid bodies 11.7. Possible mechanism of perturbations of nonmag- netic materials under magnetic treatment GENERAL CONCLUSION REFERENCES 41. INTRODUCTION A quantum and quantitative theory of liquid state, as well as a general theory of condensed matter, was absent till now. This fundam ental problem is crucial for diﬀerent brunches of science and technology. The existing solid states theories did not allow to extrapolate them successfu lly to liquids. Widely used molecular dynamics method is based on classical approach and corresponding computer simulations. It cannot be consider ed as a general one. The understanding of hierarchic organization of matter and developing of gen- eral theory needs a mesoscopic bridge between microscopic a nd macroscopic physics, between liquids and solids. The biggest part of molecules of solids and liquids did not fo llow classical Maxwell-Boltzmann distribution. This means that only quan tum approach is valid for elaboration of general theory of condensed matter . Our theoretical study of water and aqueous systems was initiated in 1986. It was stimulated by necessity to explain the nontrivi al phenomena, obtained by diﬀerent physical methods in our investigation s of water-protein solutions. For example, the temperature anomalies in water physical proper- ties, correlating with changes in large scale protein dynam ics were found in our group by specially elaborated experimental approaches (Ka ivarainen, 1985). It becomes evident, that the water clusters and water hierarch ical cooperative properties are dominating factors in self-organization, f unction and evolution of biosystems. The living organisms are strongly dependent on water proper- ties, representing about 70% of the body mass. On the other ha nd, due to its numerous anomalies, water is an ideal system for testing a new theory of condensed matter. If the theory works well with respect to wa ter and ice, it is very probable, that it is valid for other liquids, glasses or crystals as well. For this reason we have made the quantitative veriﬁcation of our hierarchic concept (Kaivarainen, 1989, 1992, 1995, 1996, 2000) on examples of w ater and ice. Our theory considers two main types of molecular heat motion :translational (tr)andlibrational (lb) anharmonic oscillations, which are characterized by cer- tain distributions in three- dimensional (3D) impulse spac e. The most probable impulse or momentum (p) determine the most probable de Broglie wave (wave B) length ( λB=h/p=vph/νB) and phase velocity ( vph). Conformational in- tramolecular dynamics is taken into account indirectly, as far it has an inﬂuence on the intermolecular dynamics and parameters of waves B in c ondensed matter. Solids and liquids are considered as a hierarchical system o f collective excita- tions - metastable quasiparticles of the four new type: eﬀectons, transitons, convertons and deformons, strongly interrelated with each other. When the length of standing waves B of molecules exceed the di stances be- tween them, then the coherent molecular clusters may appear as a result of high temperature molecular Bose-condensation (BC). The possib ility of BC in liquids and solids at the ambient temperatures is one of the most impo rtant results of our model, conﬁrmed by computer simulations. Such BC is meso scopic one, in contrast to macroscopic BC, responsible for superﬂuidity a nd superconductiv- ity. The value of the standing wave B length, which determine the edges of the primary eﬀecton (tr or lb) in selected directions (1,2,3) ma y be considered as a mesoscopic parameter of order . The primary transitons andconvertons have common features with coherent dissipative structures introduced by Chatzidimitrov-Dreisman and Br¨ andas in 51988. Such structures were predicted on the background of co mplex scaling method and Prigogin theory of star- unitary transformation s. Estimated from principle of uncertainty, the minimum boson ’s ”degrees of freedom”( nmin) in these spontaneous coherent structures are equal to: nmin≥τ(2πkBT/h) (1.1) where τis relaxation time of coherent-dissipative structures. If, for example, τ≃10−12c, corresponding to excitation of a molecular sys- tem by infrared photon, then at T= 300 Kone get nmin≃250. It means that at least 250 degrees of freedom, i.e. [250 /6] molecules, able to translations and librations act coherently and produce a photon absorption/emission phe- nomenon. The traditional consideration of an oscillating i ndividual molecule as a source of photons is replaced by the notion of a correlation pattern in such a model. The interaction between atoms and molecules in condensed ma tter is much stronger and thermal mobility/impulse much lesser, than in gas phase. It means that the temperature of Bose condensation can be much higher in solids and liquids than in the gas phase. The lesser is interaction between molecules or atoms the lower temperature is necessary for initiation o f Bose condensation. This is conﬁrmed in 1995 by Ketterle’s group in MIT and later b y few other groups, showing the Bose-Einstein condensation in ga s of neutral atoms, like sodium (MIT), rubidium (JILA) and lithium (Rice Univer sity) at very low temperatures, less than one Kelvin. However, at this temper atures the number of atoms in the primary eﬀectons (Bose condensate) was about 20,000 and the dimensions were almost macroscopic: about 15 micrometers. For comparison, the number of water molecules in primary lib rational eﬀec- ton (coherent cluster), resulting from mesoscopic BC at fre ezing point 2730K is only 280 and the edge length about 20 ˚A (see Fig. 7). OurHierarchic theory of matter unites and extends strongly two earlier existing most general models of solid state (Ashkroft and Me rmin, 1976): a) the Einstein model of condensed matter as a system of indep endent quan- tum oscillators; b) the Debye model, taking into account only collective phen omena - phonons in a solid body as in continuous medium. Among earlier models of liquid state the model of ﬂickering c lusters by Frank and Wen (1957) is closest of all to our model. In our days the qu antum ﬁeld theoretical approach to description of biosystems with som e ideas, close to our ones has been developed intensively by Umesawa’s group (Ume zawa, et. al., 1982; Umezawa, 1993) and Italian group (Del Giudice, et al., 1983; 1988, 1989). Arani et al. introduced at 1998 the notion of Coherence Domai ns (CD), where molecules are orchestrated by the internal electroma gnetic waves (IR photons) of matter [13]. This idea is close to our notion of co llective excita- tions of condensed matter in the volumes of 3D translational and librational IR photons, termed primary electromagnetic deformons (see ne xt section). The new physical ideas required a new terminology . It is a reason why one can feel certain discomfort at the beginning of readi ng this work. To 6facilitate this process, we present below a description of a new quasiparticles, notions and terms, introduced in our Hierarchic Theory of ma tter (see Table 1). Most of notions and properties, presented below are not p ostulated, but a results of our computer simulations. 2. THE NEW EXCITATIONS, INTRODUCED IN HIERARCHIC THEORY OF MATTER AND THEIR PROPERTIES The Most Probable (primary) de Broglie Wave (wave B) The main dynamics of particles in condensed matter (liquid o r solid) repre- sents thermal anharmonic oscillations of two types: translational (tr) and libra- tional (lb). The corresponding length of the most probable wave B of molecule or atom of condensed matter can be estimated by two following ways: [λ1,2,3=h/mv1,2,3 gr=v1,2,3 ph/ν1,2,3 B]tr,lb (2.1) where the most probable impulse (momentum): p1,2,3=mv1,2,3 gris equal to product of the particle mass (m) and most probable group velo city (v1,2,3 gr). We prefer to use term impulse , instead momentum, for the end not to confuse the latter notion with momentum of impulse, deﬁned as: mv1,2,3 grλ1,2,3=h (2.1a) The length of wave B could be evaluated, as the ratio of Plank c onstant to im- pulse and as the ration of most probable phase velocity ( v1,2,3 ph)tr,lbto most prob- able frequency ( ν1,2,3 B)tr,lb. The indices (1,2,3) correspond to selected directions of motion in 3D space, related to the main axes of the molecule s symmetry and their tensor of polarizability. In the case when molecular m otion is anisotropic one, we have: λ1/ne}ationslash=λ2/ne}ationslash=λ3 (2.2) It is demonstrated in our work, using Virial theorem (see eqs . 4.10 - 4.14), that due to anharmonicity of oscillations - the most probabl e kinetic energy of molecules ( Tkin)tr, lbis lesser than potential one ( V)tr, lb: (mv2/2)<(kT/2) where most probable (mean) velocity of particle of matter is equal to corre- sponding group velocity ( v=vgr). Consequently, the most probable 3D wave B length is big enough: /parenleftbiggV0 N0/parenrightbigg1/3 < λ1,2,3> h/(mkBT)1/2(2.2a) It is a condition of mesoscopic molecular Bose condensation (BC). The Most Probable (Primary) Eﬀectons (tr and lb) 7A new type of quasiparticles (excitations), introduced as 3 D superposition of three most probable pairs of standing waves B of molecules, a re termed primary eﬀectons . The shape of primary eﬀectons in a general case can be approx imated by parallelepiped, with the length of edges determined by 3 m ost probable standing waves B. The volume of primary eﬀectons is equal to: Vef= (9/4π)λ1λ2λ3. (2.3) The number of molecules or atoms forming eﬀectons is: nm= (Vef)/(V0/N0), (2.4) where V0andN0are molar volume and Avogadro number, correspondingly. Thenmincreases, when temperature is decreasing and may be about h undreds in liquids or even thousands in solids, as shown in our work. In liquids, primary eﬀectons may be registered as a clusters and in solids as domains or microcrystalline. The thermal oscillations in the volume of corresponding eﬀe ctons are syn- chronized. It means the coherence of the most probable wave B of molecules and their wave functions. We consider the primary eﬀectons as a result of partial Bose condensation of molecules of condensed matter. Primary eﬀec- tons correspond to the main state of Bose-condensate with th e packing number np= 0, i.e. with the resulting impulse equal to zero. Primary eﬀectons (librational in liquids and librational a nd translational in solids), as a coherent clusters, represent self-organizat ion of condensed matter on mesoscopic level, like I. Prigogin dissipative structur es. However, revealed in our work mesoscopic high - T Bose condensation, is a quantum p henomenon. It is important to note, that the coherent oscillations of mole cules in the volume of the eﬀectons can not be considered as phonons (acoustic waves) , because they are not accompanied by ﬂuctuation of density in contrast to s econdary deformons (see below). ”Acoustic” (a) and ”Optical” (b) States of Primary Eﬀectons The ”acoustic” a-state of the eﬀectons is such a dynamic state when molecules or other particles composing the eﬀectons, oscillate in the same phase (i.e. with- out changing the distance between them). The ”optic” b-state of the eﬀectons is such dynamic state when particles oscillate in the counte rphase manner (i.e. with periodical change of the distance between particles). This state of primary eﬀectons has a common features with Fr¨ olich’s mode. It is assumed in our model, that kinetic energies of ”acousti c” (a) and ”opti- cal” (b) modes are equal [ Ta kin=Tb kin] in contrast to potential energies [ Va< Vb]. It means that the most probable impulses in ( a) and ( b) states and, conse- quently, the wave B length and spatial dimensions of the eﬀectons in the both states are equal ([λ1,2,3]a= [λ1,2,3]b).The energy of intermolecular interaction (Van der Waals, Coulomb, hydrogen bonds etc.) i na-state are big- ger than that in b-state. Consequently, the molecular polarizability in a-state also is bigger than in b-state. It means that dielectric properties of matter may change as a result of shift of the ( a⇔b)1,2,3 tr,lbequilibrium of the eﬀectons. 8Primary Transitons (tr and lb) Primary transitons represent intermediate transition states between ( a) and (b) modes of primary eﬀectons - translational and librational . Primary tran- sitons (tr and lb) - are radiating or absorbing IR photons cor responding to translational and librational bands in oscillatory spectr a. Such quantum tran- sitions are not accompanied by the ﬂuctuation of density but with the change of polarizability and dipole moment of molecules only. The v olumes of primary transitons and primary eﬀectons coincides (see Table 1). Primary Electromagnetic and Primary Acoustic Deformons (t r, lb) Electromagnetic primary deformons are a new type of quasipa rticles (excita- tions) representing a 3 Dsuperposition of three standing electromagnetic waves. The IR photons (tr, lb) are radiated and absorbed as a result o f (a⇔b)1,2,3 tr,lb transitions of primary eﬀectons, i.e. corresponding prima ry transitons. Elec- tromagnetic deformons appear as a result of superposition o f 3 standing IR photons, penetrating in matter in diﬀerent selected direct ions (1,2,3). We as- sume, that each of 3 pairs of counter-phase photons form a sta nding wave in the volume of condensed matter. The linear dimension of each of three edges of primary deform on is deter- mined by the wave length of three standing IR photons, superp osing in the same space volume: λ1,2,3= [(n˜ν)−1]1,2,3 tr,lb(2.5) where: nis the refraction index and (˜ ν)tr,lb- the wave number of transla- tional or librational band. These quasiparticles as the big gest ones, are respon- sible for the long-range (distant) space-time correlation in liquids and solids. In the case when ( b→a)tr,lbtransitions of primary eﬀectons are accompa- nied by big ﬂuctuation of density (like cavitational ﬂuctua tion in liquid or defect formation in solid), they may be followed by emission of phon ons instead of pho- tons. It happens, when primary eﬀectons are involved in the v olume of macro- and supereﬀectons (see below). Primary acoustic deformons may originate or annihilate in such a way. However, the probability of collec tive spontaneous emission of photons during ( b→a)tr,lbtransition of primary eﬀectons is much higher than that of phonons, related to similar transition o f macroeﬀectons (see below), as it leads from our theory. The coherent electromagnetic radiation as a result of self- correlation of many dipole moments in composition of coherent cluster, lik e primary eﬀectons, containing N≫1 molecules is already known as superradiance (Dicke, 1954). The time of collective transition in the case of superradian ce is less than that of isolated molecule and intensity of superradiance (I∼N·hν/τ∼N2)is much bigger than that from the same number of independent mol ecules (I∼ N·hν/T1∼N).The ( b→a) transition time of the primary eﬀectons has the reverse dependence on the N(τ∼1/N).The relaxation time for isolated atoms or molecules ( T1) is independent on N. The main energy is radiated in the direction of most elongated volume, i.e. ends of tubes. 9Secondary eﬀectons (tr and lb) In contrast to primary eﬀectons, this type of quasiparticle s isconventional . They are the result of averaging of the frequencies and energ ies of the ”acoustic” (a) and ”optical” ( b) states of eﬀectons with packing numbers nP>0, having the resulting impulse more than zero. For averaging the ener gies of such states the Bose-Einstein distribution was used under the conditio n when T < T 0(T0 is temperature of degeneration and, simultaneously, tempe rature of ﬁrst order phase transition). Under this condition the chemical poten tial:µ= 0 and distribution has a form of Plank equation. Secondary eﬀectons (tr and lb) In contrast to primary eﬀectons, the ”acoustic” ( a) and ”optical” ( b) states of secondary (mean) eﬀectons are the result of averaging the energies of the eﬀectons with packing numbers nP>0, having the resulting impulse diﬀerent from zero. For this averaging the Bose-Einstein distributi on was used under the condition: T < T 0(T0is temperature of degeneration for mesoscopic Bose condensation (BC), equal to temperature of ﬁrst order p hase transition). Under this condition it is assumed, that the chemical potent ial:µ≃0 and Bose-Einstein distribution has a form of Plank equation [4] . Secondary transitons (tr and lb) Secondary transitons, like primary ones are intermediate t ransition state be- tween ( a) and ( ¯b) states of secondary eﬀecton - translational and libration al. As well as secondary eﬀectons, these quasiparticles are condi tional, i.e. a result of averaging. It is assumed that the volumes of secondary trans itons and secondary eﬀectons coincide. The ( ¯ a⇔¯b)tr,lbtransition states of secondary eﬀectons, in contrast to that of primary eﬀectons, are accompanied by the ﬂuctuation of density. Secondary transitons are responsible for radiati on and absorption of phonons. Secondary ”acoustic” deformons (tr and lb) This type of quasiparticles is also conditional as a result o f 3D superposi- tion of averaged thermal phonons. These conventional phono ns originate and annihilate in a process of ( ¯ a⇔¯b)1,2,3thermoactivated transitions of secondary conventional eﬀectons. These states correspond to transla tional and librational transitons. Convertons (tr⇔lb) These important excitations are introduced in our model as a reversible transitions between translational and librational primar y eﬀectons. The (acon) convertons correspond to transitions between the ( atr⇋alb) states of these eﬀectons and (bcon) convertons - to that between their ( btr⇋blb) states. As 10far as the dimensions of translational primary eﬀectons are much less than libra- tional ones, the convertons could be considered as [dissociation ⇋association] of the primary librational eﬀectons (coherent clusters). Bot h of convertons, ( acon) and (bcon), are accompanied by density ﬂuctuation, inducing phonons with cor- responding frequency in the surrounding medium. All kinds o f Convertons may be termed ’ﬂickering’ clusters. The ca- and cb- deformons, induced by convertons Three-dimensional (3D) superposition of phonons, irradiated by t wo types of convertons, acon andbcon, represents in our model the acoustic ca- and cb-deformons. They have properties similar to that of secondary deformons, discussed above. The c-Macrotransitons (Macroconvertons) and c-Macrodefo rmons Simultaneous excitation of the aconandbcontypes of convertons in the vol- ume of primary librational eﬀectons leads to origination of big ﬂuctuations, like cavitational ones, termed c-Macrotransitons or Macroconvertons. In turn, cor- responding density ﬂuctuations induce in surrounding medi um high frequency thermal phonons. The 3D-superposition these standing phon ons forms c- Macrode- formons. Macroeﬀectons (tr and lb) Macroeﬀectons (A and B) are collective simultaneous excitations of the pri- mary and secondary eﬀectons in the [ A∼(a,¯a)]tr,lband [B∼(b,¯b)]tr,lbstates in the volume of primary electromagnetic translational and librational defor- mons, respectively. This correlation of primary and second ary states results in signiﬁcant deviations from thermal equilibrium. The A and B states of macroef- fectons (tr and lb) may be considered as the most probable vol ume-orchestrated (correlated) thermal ﬂuctuations of condensed matter. Macrodeformons or Macrotransitons (tr and lb) This type of conventional quasiparticles is considered in o ur model as an intermediate transition state of macroeﬀectons. The ( A→B)tr,lband (B→ A)tr,lbtransitions are represented by the coherent transitions of primary and sec- ondary eﬀectons in the volume of primary electromagnetic deformons - transla- tional and librational. The ( A→B)tr,lbtransition of macroeﬀecton is accompa- nied by simultaneous absorption of 3 pairs of photons and tha t of phonons in the form of electromagnetic deformons. If ( B→A)tr,lbtransition occurs without emission of photons, then all the energy of the excited B-sta te is transmitted to the energy of ﬂuctuation of density and entropy of Macroeﬀec ton as an isolated mesosystem. It is a dissipative process: transition from th e more ordered struc- ture of matter to the less one, termed Macrodeformons. The bi g ﬂuctuations of density during ( A⇔B)tr,lbtransitions of macroeﬀectons, i.e. macrodeformons are responsible for the Rayleigh central component in Brill ouin spectra of light 11scattering [15]. Translational and librational macrodefo rmons are also related to the corresponding types of viscosity and self-diﬀusion [ 16]. The volumes of macrotransitons, equal to that of macrodeformons (tr or lb) and macroeﬀec- tons, coincide with that of trorlb primary electromagnetic deformons, correspondingly. Supereﬀectons This mixed type of conventional quasiparticles is composed of translational and librational macroeﬀectons correlated in space and time in the volumes of superimposed electromagnetic primary deformons (transla tional and librational - simultaneously). Like macroeﬀectons, supereﬀectons may exist in the ground (A∗ S) and excited ( B∗ S) states representing strong deviations from thermal equi- librium state. Superdeformons or Supertransitons This collective excitations have the lowest probability as compared to other quasiparticles of our model. Like macrodeformons, superde formons represent the intermediate ( A∗ S⇔B∗ S) transition state of supereﬀectons. In the course of these transitions the translational and librational mac roeﬀectons undergo simultaneous [(A⇔B)trand(A⇔B)lb] transitions The (A∗ S→B∗ S) transition of supereﬀecton may be accompanied by the absor p- tion of two electromagnetic deformons - translational and l ibrational simultane- ously. The reverse ( B∗ S→A∗ S) relaxation may occur without photon radiation. In this case the big cavitational ﬂuctuation originates. Such a process plays an important role in the processes of sublimation, evaporat ion and boiling. The equilibrium dissociation constant of the reaction: H2O⇋H++HO−(2.6) should be related with equilibrium constant of supertransi tons: KB∗ S⇋A∗ S. The A∗ S→B∗ Scavitational ﬂuctuation of supereﬀectons can be accompani ed by the activation of reversible dissociation of small fraction of water molecules. In contrast to primary and secondary transitons and deformo ns, the notions of [macro- and supertransitons] and [macro- and superde- formons] coincide. Such types of transitons and deformons represent the dy- namic processes in the same volumes of corresponding primar y electromagnetic deformons. Considering the transitions of all types of translational deformons (primary, secondary and macrodeformons), one must keep in mind that th elibrational type of modes remains the same. And vice versa, in case of libr ational defor- mons, translational modes remain unchanged. Only the reali zation of a con- vertons and supereﬀectons are accompanied by the interconv ersions between the translational and librational modes, between translation al and librational eﬀec- tons. 12Interrelation Between Quasiparticles Forming Solids and L iquids Our model includes 24types of quasiparticles (Table. 1):  4 -Eﬀectons 4 -Transitons 4 -Deformons translational and librational ,including primary and secondary(I)  2 -Convertons 2−C-deformons 1−Mc-transiton 1−Mc-deformon the set of interconvertions between translational and librational primary eﬀectons(II) /bracketleftbigg2 -Macroeﬀectons 2 -Macrodeformons/bracketrightbiggtranslational and librational (spatially separated)(III) /bracketleftbigg1 -Supereﬀectons 1 -Superdeformons/bracketrightbiggtranslational ⇋librational (superposition of trandlbeﬀectons and deformons in the same volume)(IV) Each next level in the hierarchy of quasiparticles (I - IV) introduced in our model, is based on uniﬁcation of the properties of the pre vious ones. All of these quasiparticles are constructed on the same physica l principles. Part of them is a result of 3D - superposition of diﬀerent types of sta nding waves: de Broglie waves, IR electromagnetic photons and phonons. Such a system in equilibrium state can be handled as a gas of quasiparticles. As far each of the eﬀecton’s types: [ tr] and [ lb], macroeﬀec- tons [tr+lb] and supereﬀectons [ tr/lb] have two states (acoustic and optic) the total number of excitations, as one can calculate from the ta ble above, is equal to: Nex=31 This classiﬁcation reﬂects the duality of matter and ﬁeld and represent their self-organization and interplay on mesoscopic and macrosc opic levels. Our hierarchical system includes a gradual transition from theOrder (pri- mary eﬀectons, transitons and deformons) to Disorder (macro- and superdefor- mons). It is important, however, that in accordance with the model proposed, this thermal Disorder is ”organized” by hierarchical superposition of deﬁnite types of the ordered quantum excitations. It means that the ﬁ nal dynamics condensed matter only ”looks” as chaotic one. Our approach m akes it possible to take into account the Hidden Order of Condensed Matter in f orm mesoscopic Bose condensate and its dynamics for better understanding o f Disorder. The long-distance correlation between quasiparticles is d etermined mainly by the biggest ones - an electromagnetic primary deformons ,involving in its volume a huge number of primary and secondary eﬀectons. The v olume of primary deformons [tr and lb] could be subdivided on two equa l parts, within 13the nodes of 3D standing IR electromagnetic waves. The big nu mber of the eﬀectons in each of these parts is equal also. The dynamics eﬀ ectons is correlated in such a way, that when one half of their quantity in the volum e of big primary deformon undergo ( a→b)tr,lbtransitions, the other half of the eﬀectons undergo the opposite ( b→a)tr,lbtransition. These processes may compensate each other due to exchange of IR photons and phonons in equilibrium cond itions. The increasing or decreasing in the concentration of primar y deformons is directly related to the shift of ( a⇔b)tr,lbequilibrium of the primary eﬀectons leftward or rightward, respectively. This shift, in turn, l eads also to correspond- ing changes in the energies and concentrations of secondary eﬀectons, deformons and, consequently, to that of super- and macro-deformons. It means the ex- isting of feedback reaction between subsystems of the eﬀect ons and deformons, necessary for long-range self-organization in macroscopic volumes of condensed matter. Table 1. Schematic representation of the 18 types of quasipar- ticles of condensed matter as a hierarchical dynamic system , based on the eﬀectons, transitons and deformons. Total number of quasi- particles , introduced in Hierarchic concept is 24. Six collective ex- citations, related to convertons - interconversions between primary 14librational and translational eﬀectons and their derivati ves are not represented here for the end of simplicity. The situation is possible when spontaneous oscillations be tween the sub- systems of eﬀectons and deformons are not accompanied by the change in the total internal energy due to compensation eﬀect . In such a way a long-period macroscopic oscillations in liquids, revealed experiment ally (Chernicov, 1990a, 1990b), could be explained. Such kind of phenomena, related to equilib- rium shift of two subsystems, could be responsible for long r elaxation (memory) of water containing systems after diﬀerent pertur bations (like magnetic treatment, ultra high dilution, etc.) . The instability of macrosystem arises from competition between discrete quantum andaveraged thermal equilibrium types of energy distributions of coherent molecular clus- ters, as it leads from our theoretical calculations. The total internal energy of substance is determined by the c ontributions of all types of quasiparticles with due regard for their own ene rgy, concentration and probability of excitation. It leads from our simulation s, that contributions ofsuper- andmacro eﬀectons and corresponding super- andmacro deformons as well as polyeﬀectons and coherent superclusters to the internal energy of matter normally are small, due to their low probability of excitati on, big volume and, consequently, low concentration. Polyeﬀectons and superclusters are the result of primary eﬀ ectons assembly (one-dimensional, two- or three-dimensional), stabilize d by Josephson’s junc- tions. The sizes of primary eﬀectons (translational and libration al) determine the mesoscopic scale of the condensed matter organization. Dom ains, nods, crys- tallites, observed in solid bodies, liquid crystals, polym ers and biopolymers are the consequence of primary eﬀectons and their association. 3. THE MAIN STATEMENTS AND BASIC FORMULAE OF HIERARCHIC MODEL As far the acoustic ( a) and optical ( b) thermal coherent modes of molecules in composition of elementary cells or bigger clusters of the condensed matter are anharmonic, the quantum a⇔btransitions (beats) with absorption and radiation of phonons or photons can exist. The number of acoustic and optical modes is the same and equal to three (Kaivarainen, 1995), if oscillation of all p-atoms in the basis are coherent in both optical and acoustic dynamic states. Remnant modes are degenerated. The states of system, minimizing the uncertainty relation, when: [∆p·∆x∼/planckover2pi1and∆x=L∼/planckover2pi1/∆p]1,2,3(3.1) are quantum coherent states . 15A system of the eﬀectons could be considered as a partially de generate Bose- gas. The degree of the degenerateness is proportional to the number of molecules in the volume of primary eﬀectons. Degeneration in liquids g rows up at lowering temperature and make a jump up as a result of (liquid →solid) phase transition as it leads from our theory and computer calculations. It is known from the Bose-Einstein theory of condensation, d eveloped by London (1938), that if the degeneration factor: λ= exp( µ/kT) (3.2) is close to λ≃1 at a low chemical potential value: µ≪kT (3.2a) then the contribution of bosons with the resulting impulse Pef≃0 (like primary eﬀectons) cannot be neglected, when calculating in ternal energy. We assume in our theory that for all types of primary and secon dary eﬀectons of condensed matter (solids and liquids), the condition (3. 2a) is valid. Partial Bose-Einstein condensation leads to the coherence of the waves B of molecules and atoms forming primary eﬀectons in the both: ac oustic (a) and optic (b) states. Primary eﬀectons are described with wave f unctions coherent in the volume of an eﬀecton. In non ideal Bose-gas, despite the partial Bose-condensati on, the quasiparti- cles exist with nonzero impulse, termed as secondary eﬀectons. These eﬀectons obeys the Bose-Einstein statistics. The sizes of primary eﬀectons determine the mesoscopic scal e of the con- densed matter organization. According to our model, the domains, nods, crystallites, and clusters observed in solid bodies and in l iquid crys- tals, polymers and biopolymers - can be a consequence of prim ary translational or librational eﬀectons. Stabilization of molecules, atoms or ions in composition of coherent clus- ters (eﬀectons) and correlation between diﬀerent eﬀectons could be provided by distant Van der Waals interaction and new Resonant Vibro-Gravitational Interaction, introduced in our theory (see Section 10.4). It leads from quantitative consequences of mesoscopic concept, that [gas →liquid] phase transition is related with appearance the con ditions for par- tial Bose-condensation, when the primary librational eﬀec tons, containing more than one molecule emerge (Kaivarainen, 1995, 1996). At the s ame time it means the beginning of degeneration when the chemical potential µ→0. At this con- dition wave B length, corresponding to librations, starts t o exceed the mean distances between molecules in the liquid phase. It means that the temperature, at which the phase transition [gas→liquid] occurs, coincides with the temperature of partial Bose- con densation ( Tc),[i.e. primary librational eﬀectons formation] and degeneration temperature ( T0). The changes of quasiparticles volume and shape in three dimensional (3D) space are related to corresponding changes in the impul se space. The total macroscopic Bose-condensation, in accordance wi th our model, responds to conditions, when [ a⇔b] equilibrium of primary eﬀectons strongly shifts to the main (a)- state and (b)- state becomes thermall y inaccessible. 16At the same time the wave B length tends to macroscopic value. For quantum systems at temperature (T) higher than degeneration temper ature T0(T > T 0), when chemical potential ( µi=∂Gi/∂ni)<0 has a negative value, the mean number of Bose-particles ( ni) in state (i) is determined by the Bose- Einstein distribution: ni={exp[(ǫi−µi)/kT]−1}−1(3.3) where ǫiis the energy of the particle in state ( i). For ”normal” condensed matter ǫi≫µ≪kT. The Bose-Einstein statistics, in contrast to the Maxwell- B oltzmann statis- tics, is applied to the indistinguishable Bose- particles w ith zero or integer spin values. The Fermi-Dirac distribution is valid for systems o f indistinguishable particles with a semi-integer spin obeying the Pauli princi ple. In the case of condensed matter at the temperature: 0< T < [T0∼=Tc] N∗particles of Bose condensate have a zero impulse (Ashkroft, Mermin, 1976): N∗≃N[1−(T/T0)3/2] (3.4) where N is the total number of particles in a system. 3.1. Parameters of individual de Broglie waves (waves B) The known de Broglie relation expressing Wave-Particle Dua lity, has a simple form: − →p=/planckover2pi1− →k=h/− →λB − →p=/planckover2pi1/− →LB=m− →vgr where /vectork= 2π//vectorλ= 1//vectorLBis the wave number of wave B with length /vectorλ= 2π/vectorLB, /vector p is the impulse (momentum) of particle with mass ( m) and group velocity ( vgr),/planckover2pi1=h/2πis the Plank constant. Each particle can be represented as wave packet with group ve locity: vgr=/parenleftbiggdωb dk/parenrightbigg 0 and phase velocity: vph=ωb k(3.4a) where: ωBis the angle frequency of wave B determining the total energy of the waveB: (EB=/planckover2pi1ωB). 17Total energy is equal to the sum of kinetic ( Tk) and potential ( VB) energies and is related to particle’s mass and product of phase and gro up velocities (vgrvph) as follows (Grawford, 1973): EB=/planckover2pi1ωB=Tk+VB=(/planckover2pi1k)2 2m+VB=mvgrvph (3.4b) where (m) is particle mass; (c) is light velocity. From 3.4a and 3.4b it is possible to get an important relation between phase and group velocities of wave B and its kinetic, potential and total energy: vph vgr=Tk+VB 2Tk=EB 2Tk(3.4c) 3.2. Parameters of de Broglie waves of molecules in composit ion of condensed matter The formulae given below allow to calculate the frequencies of the corre- sponding primary waves B in the directions 1,2,3 in aandbstates of primary eﬀectons (translational and librational) (Kaivarainen, 1 989, 1995, 1996): /bracketleftbig νa 1,2,3/bracketrightbig tr,lb=/bracketleftBigg ν1,2,3 p exp(hν1,2,3 p/kT)−1/bracketrightBigg tr,lb(3.5) /bracketleftbig νb 1,2,3/bracketrightbig tr,lb=/bracketleftbig νa 1,2,3+ν1,2,3 p/bracketrightbig tr,lb(3.6) The most probable frequencies of photons/bracketleftbig ν1,2,3 p/bracketrightbig tr,lbare related to the wave numbers of the maxima of corresponding bands (tr and lib)/bracketleftbig ˜ν1,2,3 p/bracketrightbig tr,lbin os- cillatory spectra: /bracketleftbig ν1,2,3 p/bracketrightbig tr,lb=c/bracketleftbig ˜ν1,2,3 p/bracketrightbig tr,lb(3.7) where (c) is light velocity. For water the most probable freq uencies of photons, corresponding to ( a⇔b)trtransitions of primary translational eﬀectons are determined by maxima with the wave numbers: /tildewideν(1) p= 60cm−1;/tildewideν(2) p=/tildewideν(3) p= 190cm−1. The band ˜ ν(1) p= ˜ν(2) p= ˜ν(3) p= 700 cm−1corresponds to the ( a⇔b)lb transitions of primary librational eﬀectons. The degenera teness of frequencies characterizes the isotropy of the given mobility type for mo lecules. The distribution (3.5) coincides with the Plank formula, fo r the case when frequency of a quantum oscillator is equal to the frequency o f photon and: νp=npνp (3.8) 18where ¯ np= [exp( hνp/kT−1)]−1is the mean number of photons with the fre- quency νp.. The transition a→bmeans that ¯npincreases by one νb=νa+νp=nνp+νp=νp(n+ 1) (3.9) The derivation of the formula (3.5) is based upon the assumpt ion that ( a⇔ b)1,2,3transitions are analogous to the beats in a system of two weak ly inter- acting quantum oscillators. In such a case the frequency ( ν1,2,3 p) of photons is equal to the diﬀerence between the frequencies of waves B forming a primary eﬀecton s in (b) and ( a) states as a frequency of quantum beats (Grawford, 1973): /bracketleftBig ν1,2,3 p=νb 1,2,3−νa 1,2,3= ∆ν1,2,3 B/bracketrightBig tr,lb(3.10) where ∆ ν1,2,3 Bis the most probable diﬀerence between frequencies of waves B in the marked directions (1,2,3). The ratio of concentrations for waves B in aandbstates ( na B/nb B) at such consideration is equal to the ratio of wave B periods Ta,bor the inverse ratio of wave B frequencies in these states: (Ta/Tb)1,2,3= (νb/νa)1,2,3. At the same time, the ratio of concentrations is determined w ith the Boltzmann distribution. So, the formula is true: /parenleftbiggna B nb B/parenrightbigg 1,2,3=/parenleftbiggνb νa/parenrightbigg 1,2,3= exp/parenleftBigg hν1,2,3 B kT/parenrightBigg = exp/parenleftBigg hν1,2,3 p kT/parenrightBigg (3.11) Substituting the eq.(3.10) into (3.11) we derive the eq.(3. 5), allowing to ﬁnd (νa 1,2,3)tr,lband (νb 1,2,3)tr,lbfrom the data of oscillation spectroscopy at every temperature. The energies of the corresponding three waves B(Ea 1,2,3andEb 1,2,3) and that of the primary eﬀectons as 3D standing waves with energies ( Ea efandEb ef) in aandbstates are equal to: /bracketleftbig Ea 1,2,3=hνa 1,2,3/bracketrightbig tr,lb;/bracketleftbig Ea ef=h(νa 1+νa 2+νa 3/bracketrightbig tr,lb(3.12) /bracketleftbigEb 1,2,3=hνb 1,2,3/bracketrightbig tr,lb;/bracketleftbig Eb ef=h(νb 1+νb 2+νb 3/bracketrightbig tr,lb(3.13) In our model energies of quasiparticles in each state are thu s determined only by the three selected coherent modes in directions (1,2,3). All remnant degrees of freedom: (3 n−3), where nis the number of molecules forming eﬀectons or deformons, are degenerated due to their coherence. 19The mean packing numbers for ¯ aand¯bstates are thereby expressed with the formula (1.27), and the mean energies ( ¯Ea 1,2,3=h¯νa 1,2,3and¯Eb 1,2,3=h¯νb 1,2,3) - with Bose-Einstein distribution (1.21;1.28), coincident with the Plank formula at chemical potential µ= 0. Finally, the averaged Hamiltonians of ( a,¯ a) and ( b,¯b) states of the system containing primary and secondary eﬀectons (translational and librational) have such a form: /bracketleftbig¯Ha 1,2,3=Ea 1,2,3+¯Ea 1,2,3=hνa 1,2,3+h¯νa 1,2,3/bracketrightbig tr,lb(3.14) /bracketleftbig¯Hb 1,2,3=Eb 1,2,3+¯Eb 1,2,3=hνb 1,2,3+h¯νb 1,2,3/bracketrightbig tr,lb(3.15) where /bracketleftBigg ¯νa 1,2,3=νa 1,2,3/bracketleftbigexp(hνa 1,2,3)/kT−1/bracketrightbig=¯va ph ¯λ1,2,3 a/bracketrightBigg tr,lb(3.16) /bracketleftBigg ¯νb 1,2,3=νb 1,2,3/bracketleftbigexp(hνb 1,2,3)/kT−1/bracketrightbig=¯vb ph ¯λ1,2,3 b/bracketrightBigg tr,lb(3.17) ¯νa 1,2,3and ¯νb 1,2,3are the mean frequency values of each of three types of co- herent waves B forming eﬀectons in ( ¯ a) and ( ¯b) states; ¯ va phand ¯vb phare the corresponding phase velocities. The resulting Hamiltonian for photons, which form the primary deformons andphonons forming secondary deformons , are determined with the term-wise subtraction of the formula (3.14) from the formula (3.15): \|∆¯H1,2,3\|tr,lb=h\|νb 1,2,3−νa 1,2,3\|tr,lb+h\|¯νb 1,2,3−¯νa 1,2,3\|tr,lb= =h(ν1,2,3 p)tr,lb+h(ν1,2,3 ph)tr,lb (3.18) where the frequencies of six IR photons, propagating in dire ctions ( ±1,±2,±3) and composing the primary deformons in the interceptions ar e equal to: (ν1,2,3 p)tr,lb=\|νb 1,2,3−νa 1,2,3\|tr,lb= (c/λ1,2,3 p·n)tr,lb (3.19) where: [c] and [ n] are the light velocity and refraction index of matter; λ1,2,3 ph are the wavelengths of photons in directions (1,2,3); and: (ν1,2,3 ph)tr,lb=\|νb 1,2,3−νa 1,2,3\|tr,lb= (vs/λ1,2,3 ph)tr,lb (3.20) are the frequencies of six phonons (translational and libra tional) in the directions 20(±1,±2,±3), forming secondary acoustic deformons; vsis the sound speed; ¯λ1,2,3 phare the wavelengths of phonons in three selected directions . The corresponding energies of photons and phonons are: E1,2,3 p=hν1,2,3 p;E1,2,3 ph=h¯ν1,2,3 ph(3.21) The formulae for the wave B lengths of primary and secondary e ﬀectons are derived from (3.5) and (3.16): λ1,2,3 a)tr,lb=λ1,2,3 b=va p/νa 1,2,3= = (va p/ν1,2,3 p)/bracketleftbigexp(hν1,2,3 p)/kT−1/bracketrightbig tr,lb(3.22) ¯λ1,2,3 a)tr,lb=¯λ1,2,3 b= ¯va ph/¯νa 1,2,3= = (¯va ph/¯ν1,2,3 ph)/bracketleftBig exp(h¯ν1,2,3 ph)/kT−1/bracketrightBig tr,lb(3.23) The wavelengths of photons and phonons forming the primary andsecondary deformons can be determined as follows (λ1,2,3 p)tr,lb= (c/nν1,2,3 p)tr,lb= 1/(/tildewideν)1,2,3 tr,lb where: (˜ ν)1,2,3 tr,lbare wave numbers of corresponding bands in the oscillatory spectra of condensed matter. (¯λ1,2,3 ph)tr,lb= (¯vs/¯ν1,2,3 ph)tr,lb For calculations according to the formulae (2.59) and (2.60 ) it is necessary to ﬁnd a way to calculate the resulting phase velocities of wave s B forming primary and secondary eﬀectons ( va phand ¯va ph). 3.3. Phase velocities of standing de Broglie waves, forming new types of quasiparticles In crystals three phonons with diﬀerent phase velocities ca n propagate in thedirection set by the longitudinal wave normal. In a general c ase, two quasi- transversal waves ”fast” ( vf ⊥) and ”slow” ( vs ⊥) and one quasi-longitudinal ( v/bardbl) wave propagate (Ashkroft and Mermin, 1976). The propagation of transversal acoustic waves is known to be accompa- nied by smaller deformations of the lattice than that of longitudinal waves, when they are caused by external impulses. The thermal phonons, sponta- neously originating and annihilating under conditions of h eat equilibrium may be accompanied by even smaller perturbations of the structu re and could be considered as transversal phonons. Therefore, we assume, that in the absence of external impuls es in solid state: vf ⊥≈vs ⊥=v1,2,3 phand the resulting thermal phonons velocity is determined as: vres s= (v(1) ⊥v(2) ⊥v(3) ⊥)1/3=vph (3.24) 21In liquids the resulting sound speed has an isotropic value: vliq s=vph. According to our model, the resulting velocity of elastic wa ves in condensed me- dia is related to the phase velocities of primary and seconda ry eﬀectons in both (acoustic and optic) states and that of deformons (translat ional and librational) in the following way: /bracketleftbigvs=fava ph+fbvb ph+fdvd ph/bracketrightbig tr,lb(3.25) /bracketleftbig¯vs=¯fa¯va ph+¯fb¯vb ph+¯fd¯vd ph/bracketrightbig tr,lb(3.26) where: va ph, vb ph,¯va ph,¯vb phare phase velocities of the most probable and mean eﬀectons in the ”acoustic” and ”optic” states; and vd ph=vd ph=vs are phase velocities of primary and secondary acoustic defo rmons, equal to phonons velocity. Nevertheless, ( a→b)tr,lbor (b→a)tr,lbtransitions of primary eﬀectons are mainly related with absorption or emission of photons, the r ate of such process (relaxation time) is limited by the rate of changing the mode of oscillations in (a) and ( b) state, i.e. by sound velocity ( vs=vph). The phonons [absorp- tion/radiation] during these transitions could accompani ed the like processes in composition of macrodeformons; fa=Pa Pa+Pb+Pd;fb=Pb Pa+Pb+Pd;fd=Pd Pa+Pb+Pd(3.27) and fa=¯Pa ¯Pa+¯Pb+¯Pd;fb=¯Pb ¯Pa+¯Pb+¯Pd;fd=¯Pd ¯Pa+¯Pb+¯Pd(3.28) are the probabilities of corresponding states of the primar y (f) and secondary quasiparticles; Pa, Pb, Pdand¯Pa,¯Pb,¯Pd- relative probabilities of excitation (thermoaccessibilities) of the primary and secondary eﬀec tons and deformons (see eqs. 4.10, 4.11, 4.18, 4.19, 4.25 and 4.26). Using eq. (3.4c) it is possible to express the phase velociti es inband¯bstates of eﬀectons ( vb phand ¯vb ph) via ( va phand ¯va ph) in the following way: /bracketleftBigg vb ph vbgr/bracketrightBigg tr,lb=/bracketleftbiggEb tot 2Tb k/bracketrightbigg tr,lb=/bracketleftbigghνb res m(vbgr)2/bracketrightbigg tr,lb(3.29) From this equation, we obtain for the most probable phase velocity in (b) state: 22(vb ph)tr,lib=/bracketleftbig λres phνres b/bracketrightbig tr,lib=/bracketleftbigg (va ph)νb res νares/bracketrightbigg tr,lb(3.30) We keep in mind that according to our model vb gr=va grand ¯vb gr= ¯va gr, i.e. the group velocities of both states are equal. Likewise for the mean phase velocity in ¯b-state of eﬀectons we have: (¯vb ph)tr,lb=/bracketleftbigg/parenleftbig¯va ph/parenrightbig¯νres b ¯νresa/bracketrightbigg tr,lb(3.31) where in (3.30): /bracketleftbiggνb res= (νb 1νb 2νb 3)1/3 νa res= (νa 1νa 2νa 3)1/3/bracketrightbigg tr,lb(3.32) are the resulting frequencies of the most probable (primary ) eﬀectons in band astates. They can be calculated using the eqs. (3.5 and 3.6); f requencies; and in (3.31): /bracketleftBig νb res= (νb 1νb 2νb 3)1/3/bracketrightBig tr,lb(3.33) /bracketleftBig νa res= (νa 1νa 2νa 3)1/3/bracketrightBig tr,lb(3.34) are the resulting frequencies of the mean eﬀectons in ¯band¯ astates. They can be estimated according to eqs. (3.17 and 3.16). Using eqs. (3.25 and 3.30), we ﬁnd the formulas for the resulting phase velocities of the primary translational and librational eﬀectons in ( a) state: /parenleftbig va ph/parenrightbig tr,lb= vs(1−fd) fa 1 +Pb Pa/parenleftBig νbres νares/parenrightBig  tr,lb(3.35) Similarly, for the resulting phase velocity of secondary eﬀ ectons in (a) state we get from (3.26) and (3.31): /parenleftbig ¯va ph/parenrightbig tr,lb= vs(1−¯fd) ¯fa 1 +¯Pb¯Pa/parenleftBig ¯νbres ¯νares/parenrightBig  tr,lb(3.36) As will be shown below, it is necessary to know va phand ¯va phto determine theconcentration of the primary and secondary eﬀectons. When the values of resulting phase velocities in aand¯ astates of eﬀectons are known, then from eqs. (3.30) and (3.31) it is easy to express resulting phase v elocities in band¯b states of translational and librational eﬀectons. 233.4. Concentrations of quasiparticles, introduced in Hier archic model of condensed matter It has been shown by Rayleigh that the concentration of the st anding waves of any type with wave lengths within the range: λtoλ+dλis equal to: nλdλ=4πdλ λ4(3.37) or, expressing wave lengths via their frequencies and phase velocities λ=vph/ν we obtain: nνdν= 4πν2dν v2 ph(3.38) For calculation the concentration of standing waves within the frequency range from zero to the deﬁnite characteristic frequency, for exam ple, to the most probable ( νa) or mean (¯ νa) frequency of wave B, then eq. (3.38) should be integrated: na=4π v3 phνa/integraldisplay 0ν2dν=4 3π/parenleftbiggνa vph/parenrightbigg3 =4 3π1 λ3a(3.39) Jeans has shown that each standing wave formed by photons or p honons can be polarized twice. Taking into account this fact the concen trations of standing photons and standing phonons in the all three directions (1, 2,3) are equal to: n1,2,3 p=8 3π/parenleftBigνp 1,2,3 c1,2,3/n/parenrightBig3 ¯n1,2,3 ph=8 3π/parenleftbigg ¯νph 1,2,3 v1,2,3 ph/parenrightbigg3(3.40a,b) where: [c] and [n] are the light speed in vacuum and refractio n index of matter; vph=vs- velocity of thermal phonons, equal to sound velocity. The standing waves B of atoms and molecules have only one line ar polariza- tion in directions (1,2,3). Therefore, their concentratio ns are described by an equation of type (3.39). According to our model (see Introduction), superposition o f each of three diﬀerently oriented (1,2,3) standing waves B forms quasi-p articles which we have termed eﬀectons . They are divided into the most probable (primary) (with zer o resulting impulse) and mean (secondary) eﬀectons. Quasipa rticles, formed by 3D superposition of standing photons and phonons, originat ing in the course of (a⇔b) and (¯ a⇔¯b) transitions of the primary and secondary eﬀectons, respectively, are termed primary and secondary deformons (Table 1). Eﬀectons and deformons are the result of thermal translations (tr) and libra- tions (lb) of molecules in directions (1,2,3). These quasiparticles a re generally approximated by a parallelepiped with symmetry axes (1,2,3 ). 24As far three coherent standing waves of corresponding nature take part in the construction of each eﬀecton , it means that the concentration of such quasi- particles must be three times lower than the concentration o f standing waves expressed by eq. (3.39). The coherence of molecules in the vo lume of the ef- fectons and deformons due to partial Bose-condensation is t he most important feature of our model, which leads to degeneration of waves B o f these molecules. Finally, we obtain the concentration of primary eﬀectons, p rimary transitons and convertons: /parenleftbignef/parenrightbig tr,lb=4 9π/parenleftBigg νa res va ph/parenrightBigg3 tr,lb=nt=nc (3.41) where νa res= (νa 1νa 2νa 3)1/3 tr,lb(3.42) is the resulting frequency of a-state of the primary eﬀecton; νa 1, νa 2, νa 3are the most probable frequencies of waves B in a-state in directions (1,2,3), which are calculated according to formula (3 .5);va ph- the resulting phase velocity of eﬀectons in a-state, which corresponds to eq. (3.35 ). Theconcentration of secondary (mean) eﬀectons and secondary t ransitons is expressed in the same way as eq. (3.41): (¯nef)tr,lb=4 9π/parenleftBigg ¯νa res ¯va ph/parenrightBigg3 tr,lb=nt (3.43) where phase velocity ¯va phcorresponds to eq. (3.36); νa res= (¯νa 1¯νa 2¯νa 3)1/3(3.44) - the resulting frequency of mean waves B in ¯ a-state. The mean values ¯ νa 1,2,3 are found by the formula (3.16). Maximum concentrations of the most probable and mean eﬀectons ( nmax ef) and (¯nmax ef), as well as corresponding concentrations of transitons ( nmax t) and (¯nmax t) follow from the requirement that it should not be higher tha n the con- centration of atoms. If a molecule or elementary cell consists of [ q]atoms , which have their own degrees of freedom and corresponding impulses, then nmax ef=nmax t=nmax ef=nmax t=qN0 V0 The concentration of the electromagnetic primary deformon sfrom eq. (3.40): /parenleftbig nd/parenrightbig tr,lb=8 9π/parenleftbiggνres d c/n/parenrightbigg3 tr,lb(3.46) 25where ( c) and ( n) are light speed and refraction index of matter; /parenleftbigνres d/parenrightbig tr,lb=/parenleftBig ν(1) pν(2) pν(3) p/parenrightBig1/3 tr,lb(3.47) - the resulting frequency of primary deformons, where /parenleftbigν1,2,3 p/parenrightbig tr,lb=c/parenleftbig˜ν1,2,3 p/parenrightbig tr,lb(3.48) are the most probable frequencies of photons with double pol arization, related to translations and librations; c - the speed of light; ˜ νp- the wave numbers, which may be found from oscillatory spectra of matter. Theconcentration of acoustic secondary deformons derived from eq. (3.40) is: /parenleftbig ¯nd/parenrightbig tr,lb=8 9π/parenleftbigg¯νres d vs/parenrightbigg3 tr,lb(3.49) where vsis the sound velocity; and /parenleftbig¯νres d/parenrightbig tr,lb=/parenleftBig ¯ν(1) ph¯ν(2) ph¯ν(3) ph/parenrightBig1/3 tr,lb(3.50) is the resulting frequency of secondary deformons (translational and librational); in this formula: /parenleftBig ¯ν1,2,3 ph/parenrightBig tr,lb=/vextendsingle/vextendsingle¯νa−¯νb/vextendsingle/vextendsingle1,2,3 tr,lb(3.51) are the frequencies of secondary phonons in directions (1,2 ,3), calculated from (3.16) and (3.17). Since the primary and secondary deformons are the results of transitions (a⇔band ¯a⇔¯b)tr,lbof the primary and secondary eﬀectons, respectively, then the maximum concentration of eﬀectons, transitons and deformons must coincide: nmax d=nmax d=nmax ef=nmax t=nmax ef=nmax t=qN0 V0(3.52) 4. HIERARCHIC THERMODYNAMICS 4.1. The internal energy of matter as a hierarchical system of collective excitations 26The quantum theory of crystal heat capacity leads to the foll owing equation for the density of thermal internal energy (Ashkroft, Mermi n, 1976): ǫ=1 Vi/summationtextEiexp(−Ei/kT) i/summationtextexp(−Ei/kT)(4.1) where V - the crystal volume; Ei- the energy of the i-stationary state. According to our Hierarchic theory, the internal energy of m atter is deter- mined by the concentration ( ni) of each type of quasiparticles, probabilities of excitation of each of their states ( Pi) and the energies of corresponding states (Ei). The condensed matter is considered as an ”ideal gas” of 3D s tanding waves of diﬀerent types (quasiparticles and collective exc itations). However, the dynamic equilibrium between types of quasiparticles is very sensitive to the external and internal perturbations. The total partition function - the sum of the relative probab ilities of excita- tion of all states of quasiparticles is equal to: Z=/summationdisplay tr,lb  /parenleftBig Pa ef+Pb ef+Pd/parenrightBig + +/parenleftBig ¯Pa ef+¯Pb ef+¯Pd/parenrightBig + +/bracketleftbig/parenleftbig PA M+PB M/parenrightbig +PM D/bracketrightbig   tr,lb+ + (Pac+Pbc+PcMd) +/parenleftbig PA S+PB S+Ps D∗/parenrightbig (4.2) Here we take into account that the probabilities of excitati on of primary and secondary transitons and deformons are the same ( Pd=Pt;¯Pd=¯Pt) and related to the same processes: (a⇔b)tr,lb and (¯ a⇔¯b)tr,lbtransitions. The analogous situation is with probabilities of a, b and cM convertons and corresponding acoustic deformons excitations: Pac, PbcandPcMd=PcMt. So it is a reason for taking them into account in the partition fu nction only ones. The ﬁnal formula for the total internal energy of ( Utot) of one mole of matter leading from mesoscopic model, considering the system of 3D standing waves as an ideal gas is: Utot=V01 Z/summationdisplay tr,lb/braceleftbigg/bracketleftbigg nef/parenleftbigPa efEa ef+Pb efEb ef+PtEt/parenrightbig +ndPdEd/bracketrightbigg + +/bracketleftbig ¯nef/parenleftbig¯Pa ef¯Ea ef+¯Pb ef¯Eb ef+¯Pt¯Et/parenrightbig + ¯nd¯Pd¯Ed/bracketrightbig + +/bracketleftbig nM/parenleftbigPA MEA M+PB MEB M/parenrightbig +nDPD MED M/bracketrightbig tr,lb+ 27+V01 Z/bracketleftbig ncon/parenleftbigPacEac+PbcEbc+PcMtEcMt/parenrightbig + +(ncdaPacEac+ncdbPbcEbc+ncMdPcMdEcMd)/bracketrightbig + +V01 Zns/bracketleftbig /parenleftbig PA∗ SEA∗ S+PB∗ SEB∗ S/parenrightbig +nD∗PD∗ SED∗ S/bracketrightbig (4.3) where all types the eﬀecton’s contributions in total intern al energy correspond to: Uef=V01 Z/summationdisplay tr,lb/bracketleftbignef/parenleftbigPa efEa ef+Pb efEb ef/parenrightbig + +¯nef/parenleftbig¯Pa ef¯Ea ef+¯Pb ef¯Eb ef/parenrightbig +nM/parenleftbigPA MEA M+PB MEB M/parenrightbig /bracketrightbig tr,lb+ +V01 Zns/parenleftbig PA∗ sEA∗ s+PB∗ sPB∗ s/parenrightbig (4.4) all types of deformons contribution in Utotis: Ud=V01 Z/summationtext tr.lb/parenleftbig ndPdEd+ ¯nd¯Pd¯Ed+nMPD MED M/parenrightbig tr,lb+ +V01 ZnsPD∗ SED∗ S(4.5) and contribution, related to [ lb/tr] convertons: Ucon=V01 Z/bracketleftbig ncon/parenleftbigPacEac+PbcEbc+PcMtEcMt/parenrightbig + +(ncdaPacEac+ncdbPbcEbc+ncMdPcMtEcMd)/bracketrightbig (4.5a) Contributions of all types of transitons ( Ut) also can be easily calculated. The intramolecular conﬁgurational dynamics of molecules i s automatically taken into account in our approach as it has an inﬂuence on the intermolecular dynamics, dimensions, and on concentration of quasipartic les as well as on the energy of excitation of their states. These dynamics aﬀects the positions of the absorption bands in oscillatory spectra and values of sound velocity, that we use for calculation of internal energy. The remnant small contribution of intramolecular dynamics to Utotis related to oscillation energy corresponding to fundamental molecu lar modes ( νi p). It may be estimated using Plank distribution: Uin=N0i/summationdisplay 1h¯νi p=N0i/summationdisplay 1hνi p/bracketleftbig exp/parenleftbighνi p/kT/parenrightbig −1/bracketrightbig−1(4.5b) where ( i) is the number of internal degrees of freedom. i= 3q−6 for nonlinear molecules; i= 3q−5 for linear molecules qis the number of atoms forming a molecule. It has been shown by our computer simulations for the case of w ater and ice thatUin≪Utot. It should be general condition for any condensed matter. 28Let us consider now the meaning of the variables in formulae ( 4.2 -4.5), necessary for the internal energy calculations: V0is the molar volume; nef,¯nefare the concentrations of primary (eq. 3.41) and secondary ( eq. 3.42) eﬀectons; Ea ef, Eb efare the energies of the primary eﬀectons in aandb states: /bracketleftbigEa ef= 3hνa ef/bracketrightbig tr,lb(4.6) /bracketleftbigEb ef= 3hνb ef/bracketrightbig tr,lb, (4.7) where /bracketleftbigνa ef=1 3/parenleftbigνa 1+νa 2+νa 3/parenrightbig/bracketrightbig tr,lb(4.8) /bracketleftbigνb ef=1 3/parenleftbig νb 1+νb 2+νb 3/parenrightbig/bracketrightbig tr,lb(4.9) are the characteristic frequencies of the primary eﬀectons in the ( a) and ( b) - states; νa 1,2,3, νb 1,2,3are determined according to formulas (3.5 and 3.6); Pa ef, Pb ef- the relative probabilities of excitation (thermoaccessi bilities) of eﬀectons in ( a) and ( b) states [2-4] introduced as:  Pa ef= exp −/vextendsingle/vextendsingle/vextendsingleEa ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingleνa ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.10)  Pb ef= exp −/vextendsingle/vextendsingle/vextendsingleEa ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingleνb ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.11) where E0= 3kT= 3hν0 (4.12) is the equilibrium energy of all types of quasiparticles det ermined by the tem- perature of matter (T): ν0=kT h(4.13) is the equilibrium frequency. 29¯Ea ef,¯Eb efare the characteristic energies of secondary eﬀectons in ¯ aand¯b states: /bracketleftbig¯Ea ef= 3h¯νa ef/bracketrightbig tr,lb(4.14) /bracketleftbig¯Eb ef= 3h¯νb ef/bracketrightbig tr,lb, (4.15) where /bracketleftbig¯νa ef=1 3/parenleftbig¯νa 1+ ¯νa 2+ ¯νa 3/parenrightbig/bracketrightbig tr,lb(4.16) /bracketleftbig¯νb ef=1 3/parenleftbig ¯νb 1+ ¯νb 2+ ¯νb 3/parenrightbig/bracketrightbig tr,lb(4.17) are the characteristic frequencies of mean eﬀectons in ¯ aand¯bstates; ¯ νa 1,2,3,¯νb 1,2,3 determined according to formulae (3.16 and 3.17). ¯Pa ef,¯Pb efare the relative probabilities of excitation (thermoacces sibilities) of mean eﬀectons in ¯ aand¯bstates (Kaivarainen, 1989a) introduced as:  ¯Pa ef= exp −/vextendsingle/vextendsingle/vextendsingle¯Ea ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingle¯νa ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.18)  ¯Pb ef= exp −/vextendsingle/vextendsingle/vextendsingle¯Ea ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingle¯νb ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.19) Parameters of deformons (primary and secondary) [tr and lb] : nd,¯ndare the concentrations of primary (eq. 3.46) and secondary ( eq. 3.49) deformons; Ed,¯Edare the characteristic energies of the primary andsecondary defor- mons,equal to energies of primary and secondary transitons: /bracketleftbigEd= 3hνres d=Et/bracketrightbig tr,lb(4.20) /bracketleftbig¯Ed= 3h¯νres d=¯Et/bracketrightbig tr,lb(4.20) where: characteristic frequencies of the primary and secon dary deformons are equal to: /bracketleftBig νres d=1 3/parenleftBig ν(1) p+ν(2) p+ν(3) p/parenrightBig /bracketrightBig tr,lb(4.22) 30/bracketleftBig ¯νres d=1 3/parenleftBig ¯ν(1) ph+ ¯ν(2) ph+ ¯ν(3) ph/parenrightBig /bracketrightBig tr,lb(4.23) The frequencies of the primary photons are calculated from t he experimental data of oscillatory spectra using (3.48). The frequencies of secondary phonons are calculated as: /parenleftBig ν1,2,3 ph/parenrightBig tr,lb=\|νa−νb\|1,2,3 tr,lb(4.24) where ν1,2,3 aandν1,2,3 bare founded in accordance with (3.16) and (3.17). Pdand ¯Pdare the relative probabilities of excitation of primary and secondary deformons in medium, surrounding eﬀectons, intr oduced as the prob- abilities of intermediate transition states: (a⇔b)tr,lband (¯ a⇔¯b)tr,lb: /parenleftbigPd=Pa ef·Pb ef/parenrightbig tr,lb(4.25) /parenleftbig¯Pd=¯Pa ef·¯Pb ef/parenrightbig tr,lb(4.26) Parameters of transitons [tr and lb] (nt)tr,lband (¯nt)tr,lbare concentrations of primary and secondary transitons, equal to concentration of primary (3.41) and secondary (3.4 3) eﬀectons: (nt=nef)tr,lb; (nt=nef)tr,lb (4.27) (Ptand¯Pt)tr,lbare the relative probabilities of excitation of primary and sec- ondary transitons, equal to that of primary and secondary de formons: (Pt=Pd)tr,lb; (¯Pt=¯Pd)tr,lb (Etand¯Et)tr,lbare the energies of primary and secondary transitons: /bracketleftBig Et=Ed=h(ν(1) p+ν(2) p+ν(3) p)/bracketrightBig tr,lb(4.28) /braceleftBig ¯Et=¯Ed= 3h/bracketleftbig\|¯νa ef−ν0\|+\|¯νb ef−ν0\|/bracketrightbig1,2,3/bracerightBig tr,lb(4.29) Primary and secondary deformons in contrast to transitons, represent the quasi- elastic mechanism of the eﬀectons interaction via medium. 31Parameters of macroeﬀectons [tr and lb] (nM=nd)tr,lbare the concentrations of macroeﬀectons equal to that of primary deformons (3.46); (EA MandEB M)tr,lbare the energies of A and B states of macroeﬀectons; ( νA M andνB M)tr,lbare corresponding frequencies, deﬁned as: /bracketleftbig EA M= 3hνA M=−kTlnPA M/bracketrightbig tr,lb(4.29a) /bracketleftbig EB M= 3hνB M=−kTlnPB M/bracketrightbig tr,lb(4.29b) where /bracketleftbig PA M=Pa·Pa/bracketrightbig tr,lb(4.29c) and /bracketleftbig PB M=Pb·Pb/bracketrightbig tr,lb(4.29d) are the relative probabilities of excitation of A and B state s of macroeﬀectons. Parameters of macrodeformons [tr and lb] (nD M)tr,lbis the concentration of macrodeformons equal to that of macr oeﬀec- tons (macrotransitons) corresponding to concentration of corresponding primary deformons: see eq.(3.46); (PD M)tr,lb= (PA M·PB M)tr,lb (4.29e) are the probabilities of macrodeformons excitation; (EM D)tr,lb=−kTln(PD M)tr,lb= 3h(νD M)tr,lb (4.29f) are the energies of macrodeformons; Parameters of convertons and related excitations The frequency and energy of a-convertons and b- convertons: νac=\|(νa ef)lb−(νa ef)tr\|;Eac= 3hνac νbc=\|(νb ef)lb−(νb ef)tr\|;Ebc= 3hνbc (4.30) where: characteristic frequencies ( νa ef)lband (νa ef)trcorrespond to (4.8). where characteristic frequencies ( νb ef)lband (νb ef)trcorrespond to (4.9). Probabilities of (a) and (b) convertons, equal to that of cor responding acous- tic c-deformons excitations: /parenleftbiggPac= (Pa ef)tr·(Pa ef)lb Pbc= (Pb ef)tr·(Pb ef)lb/parenrightbigg (4.30a) 32Probability and energy of c - Macrotransitons (Macroconvertons) excitation [simultaneous excitation of (a) and (b) con- vertons ],equal to that of c- Macrodeformons is: PcMd=Pac·Pbc;EcMt=EcMd=−kT·lnPcMd (4.30b) The characteristic frequency of cM-transitons and cM-defo rmons is: νcMt=νcMd=EcMd/3h The concentrations of (a), (b)-convertons ( ncon) andc-Macrotransitons ( ncMd) are equal to that of primary eﬀectons ( nef). The concentrations of acoustic deformons, excited by conve rtons The concentrations of ca-deformons andcb-deformons, representing 3D stand- ing phonons, excited by a-convertons and by b-convertons correspondingly are: /parenleftbign/parenrightbig cad,cbd=8 9π/parenleftbiggνac,bc vs/parenrightbigg3 (4.30c) where [ vs] is the sound velocity and νac= (νa ef)lb−(νa ef)tr, ν bc= (νb ef)lb−(νb ef)tr (4.30d) are characteristic frequencies of a- and b-convertons, equal to the diﬀerence between characteristic frequencies of primary librationa l and translational ef- fectons (see eqs.4.8 and 4.9) in aandbstates correspondingly. The concentration of cM-deformons , excited by cM-transitons (or Macro- convertons) is equal to: ncMd=8 9π/parenleftbiggνcMd vs/parenrightbigg3 (4.30e) where: νcMdis characteristic frequency of c-Macrodeformons, equal to that of c-Macrotransitons (Macroconvertons) . The maximum concentration of all convertons-related excit ations is also lim- ited by concentration of molecules Parameters of supereﬀectons: (nS=nd)lbis the concentration of supereﬀectons, equal to that of prim ary librational deformons (3.46); PA∗ S;PB∗ Sare the relative probabilities of excitation of A∗andB∗: PA∗ S= (PA M)tr·(PA M)lbPB∗ S= (PB M)tr·(PB M)lb (4.30f) andEA∗ S;EB∗ Sare the energies of A and B states of supereﬀectons from (3.27) and (3.28); EA∗ S=−kT·lnPA∗ S EB∗ S=−kT·lnPB∗ S 33Parameters of superdeformons: nD∗is the concentration of superdeformons, equal to that of sup ereﬀectons; PD∗ S= (PD M)tr(PD M)lb (4.30g) is the relative probability of superdeformons; ED∗ Sis the energy of superdeformons, deﬁned as: ED∗ S=−kTlnPD∗ S (4.30h) Substituting the parameters of quasiparticles, calculate d in this way into eqs. (4.2 and 4.3), we obtain the total internal energy of one mole of matter in solid or liquid phase. For water and ice the theoretical re sults coincide with experimental one fairly well (see Fig. 2). It is important that our equations are the same for solid and l iq- uid states. The diﬀerence in experimental parameters, such as molar vol ume, sound velocity, refraction index, positions of translatio nal and librational bands determines the diﬀerence of internal energy and of more than 100 another pa- rameters of any state of condensed matter, which can be calcu lated using eq. (4.3). It is important to stress that our concept is general for soli ds and liquids, for crystals, glasses and amorphous matter. 4.2. The contributions of kinetic and potential energy to the total internal energy The total internal energy of matter ( Utot) is equal to the sum of total kinetic (Ttot) and total potential ( Vtot) energy: Utot=Ttot+Vtot The kinetic energy of wave B(TB) of one molecule may be expressed using its total energy ( EB), mass of molecule (m), and its phase velocity as wave B (vph): TB=mv2 gr 2=E2 B 2mv2 ph(4.31) The total mass ( Mi) of 3D standing waves B forming eﬀectons, transitons and deformons of diﬀerent types are proportional to number of mo lecules in the volume of corresponding quasiparticle ( Vi= 1/ni): Mi=1/ni V0/N0m (4.32) the limiting condition for minimum mass of quasiparticle is : 34Mmin i=m (4.33) Consequently the kinetic energy of each coherent eﬀectons i s equal to /bracketleftBigg Ti kin=E2 i 2Miv2 ph/bracketrightBigg (4.34) where: E iis a total energy of given quasiparticle. The kinetic energy of coherent primary and secondary deform ons and tran- sitons we express analogously to eq. (4.34), but instead of t he phase velocity of waves B we use the light speed and resulting sound velocity vres(eq.3.24), respectively: /bracketleftbigg Ti kin=E2 i 2Mic2/bracketrightbigg dand/bracketleftbigg Ti kin=E2 i 2Mi(vress)2/bracketrightbigg d(4.35) The kinetic energies of [ tr/lb] convertons: /bracketleftbigg Ti kin=(Ei/3)2 2Mi(vress)2/bracketrightbigg =/bracketleftbigg Ti kin=E2 i 6Mi(vress)2/bracketrightbigg con(4.35a) According to our model, the kinetic energies of the eﬀectons inaandband also in the ¯ aand¯bstates are equal. Using (4.34) and (4.35) we obtain from eq.( 4.3) the total thermal kinetic energy for 1 mole of matter: Ttot=V01 Z/summationdisplay tr,lb   nef/summationtext(Ea)2 1,2,3 2Mef(va ph)2∗/parenleftbig Pa ef+Pb ef/parenrightbig + ¯nef/summationtext/parenleftbig¯Ea/parenrightbig2 1,2,3 2Mef(va ph)2∗/parenleftbig¯Pa ef+¯Pb ef/parenrightbig + + nt/summationtext(Et)2 1,2,3 2Mt(vress)2Pd+ ¯nt/summationtext/parenleftbig¯Et/parenrightbig2 1,2,3 2¯Mt(vress)2¯Pd + nd/summationtext(Ed)2 1,2,3 2Mdc2Pd+ ¯nd/summationtext/parenleftbig¯Ed/parenrightbig2 1,2,3 2Md(vress)2¯Pd + +/bracketleftBigg nM/parenleftbigEA M/parenrightbig2 6MM(vA ph)2∗/parenleftbigPA M+PB M/parenrightbig +nD/parenleftbigED/parenrightbig2 6MD(vress)2PM D/bracketrightBigg/bracerightBigg tr,lb+ +V0ncon Z/bracketleftBigg/parenleftbig Eac/parenrightbig2 6Mc(vress)2Pac+/parenleftbig Ebc/parenrightbig2 6Mc(vress)2Pbc+/parenleftbig EcMd/parenrightbig2 6Mc(vress)2PcMd/bracketrightBigg + V01 Z/bracketleftBigg ncda/parenleftbig Eac/parenrightbig2 6Mc(vress)2Pac+ncdb/parenleftbig Ebc/parenrightbig2 6Mc(vress)2Pbc+ncMd/parenleftbig EcMd/parenrightbig2 6Mc(vress)2PcMd/bracketrightBigg + +V01 Z/bracketleftBigg nS/parenleftbig EA∗ S/parenrightbig2 6MS/parenleftbigvA∗ ph/parenrightbig2∗/parenleftBig PA∗ S+PB∗ S/parenrightBig +nS(ED∗)2 6MS(vress)2PD∗ S/bracketrightBigg (4.36) 35where the eﬀective phase velocity of A-state of macroeﬀecto ns is introduced as: /bracketleftBigg 1 vA ph=1 va ph+1 ¯va ph/bracketrightBigg tr,lb→/bracketleftBigg vA ph=va ph·¯va ph va ph+ ¯va ph/bracketrightBigg tr,lb(4.37) and the eﬀective phase velocity of supereﬀecton in A∗-state: vA∗ ph=(vA ph)tr·(vA ph)lb (vA ph)tr+ (vA ph)lb(4.38) Total potential energy is deﬁned by the diﬀerence between to tal internal (eq. 4.3) and total kinetic energy (eq. 4.36): Vtot=Utot−Ttot(4.39) Consequently, we can separately calculate the kinetic and p otential energy con- tributions to the total thermal internal energy of matter, u sing four experimental parameters, obtained at the same temperature and pressure: 1)density or molar volume; 2)sound velocity; 3)refraction index and 4)positions of translational and librational bands in oscillatory spectrum of condensed matter. It is important to stress that the same equations are valid fo r liquids and solids. The contributions of all individual types of quasiparticle s in thermodynamics as well as a lot of characteristics of these quasiparticles a lso may be calculated, using hierarchic theory. 4.3. Some useful parameters of condensed matter The total Structural Factor can be calculated as a ratio of the kinetic to the total energy of matter: SF=Ttot/Utot(4.40) The structural factors, related to contributions of transl ations (SFtr) and to librations (SFlb) could be calculated separately as: SFtr =Ttr/Utotand SFlb =Tlb/Utot(4.41) 36Dynamic parameters of quasiparticles, introduced in Hiera rchic theory The frequency of c- Macrotransitons or Macroconvertons exc itation, repre- senting [dissociation/association] of primary libration al eﬀectons - ”ﬂickering clusters ”as a result of interconversions between primary [lb] and [tr ] eﬀectons is: FcM=1 τMcPMc/Z (4.42) where: PMc=PacPbcis a probability of Macroconvertons excitation; Zis a total partition function (see eq.4.2); the life-time of Macroconvertons is: τMc= (τacτbc)1/2(4.43) The cycle-period of (ac) and (bc) convertons are determined by the sum of life-times of intermediate states of primary translationa l and librational eﬀec- tons: τac= (τa)tr+ (τa)lb; τbc= (τb)tr+ (τb)lb;(4.44) The life-times of primary and secondary eﬀectons (lb and tr) ina- and b- states are the reciprocal values of corresponding state fre quencies: [τa= 1/νa;τa= 1/νa]tr,lb; (4.45) [τb= 1/νb;τb= 1/νb]tr,lb (4.45a) [(νa) and ( νb)]tr,lbcorrespond to eqs. 4.8 and 4.9; [(νa) and ( νb)]tr,lbcould be calculated using eqs.4.16; 4.17. The frequency of (ac) and (bc) convertons excitation [lb/tr ]: Fac=1 τacPac/Z (4.46) Fbc=1 τbcPbc/Z (4.47) where: PacandPbcare probabilities of corresponding convertons excitation s (see eq.4.29a). The frequency of Supereﬀectons and Superdeformons (bigges t ﬂuctuations) excitation: 37FSD=1 (τA∗+τB∗+τD∗)PD∗ S/Z (4.48) It is dependent on cycle-period of Supereﬀectons: τSD=τA∗+τB∗+τD∗ and probability of Superdeformons activation ( PD∗ S),like the limiting stage of this cycle. The averaged life-times of Supereﬀectons in A∗andB∗state are dependent on similar states of translational and librational macroeﬀ ectons : τA∗= [(τA)tr(τA)lb] = [(τaτa)tr(τaτa)lb]1/2(4.49) and that in B state: τB∗= [(τB)tr(τB)lb] = [(τbτb)tr(τbτb)lb]1/2(4.50) The life-time of Superdeformons excitation It is determined by frequency of beats between A∗and B∗states of Supereﬀectons as: τD∗= 1/\|(1/τA∗)−(1/τB∗)\| (4.51) The frequency of A⇋Bcycle excitations of translational and librational macroeﬀectons is deﬁned in a similar way: /bracketleftbigg FM=1 (τA+τB+τD)PD M/Z/bracketrightbigg tr,lb(4.52) where: (τA)tr,lb= [(τaτa)tr,lb]1/2(4.53) and (τB)tr,lb= [(τbτb)tr,lb]1/2(4.54) (τD)tr,lb= 1/\|(1/τA)−(1/τB)\|tr,lb(4.55) The frequency of primary translational eﬀectons (a⇋b)tr transitions: 38Ftr=1/Z (τa+τb+τt)tr(Pd)tr (4.56) where: ( Pd)tris a probability of primary translational deformons excita tion; [τa;τb]trare the life-times of (a) and (b) states of primary translational ef- fectons (eq.4.45). The frequency of primary librational eﬀectons as ( a⇋b)lbcycles excitations: Flb=1/Z (τa+τb+τt)lb(Pd)lb (4.57) where: ( Pd)lbis a probability of primary librational deformons excitati on;τaand τbare the life-times of (a) and (b) states of primary libration al eﬀectons deﬁned as (4.45). The life-time of primary transitons (tr and lb) as a result of quantum beats between (a) and (b) states of primary eﬀectons could be intro duced as: [τt=\|1/τa−1/τb\|−1]tr,lb (4.58) The fraction of molecules (Fr) in each selected type of excit ation (quasiparticle): Fr(i) =P(i)/Z (4.59) where: P(i) is thermoaccessibility (relative probability) of given e xcitation andZis total partition function (4.2). The concentration of molecules in each selected type of exci tation: Nm(i) =Fr(i)(NA/V0) = [P(i)/Z](NA/V0) (4.60) where: NAandV0are the Avogadro number and molar volume of matter. The concentration of each type of independent excitations (quasiparticles) N(i) =Fr(i)n(i) = [P(i)/Z]n(i) (4.61) where: n(i) is a concentration of given type (i) of quasipart icles;Fr(i) is a fraction of corresponding type of quasiparticles. 39The average distance between centers of i-type of randomly distributed quasiparticles: d(i) = 1/[N(i)]1/3= 1/[(P(i)/Z)·n(i)]1/3(4.62) The ratio of average distance between centers of quasiparti cles to their linear dimension [l= 1/n(i)1/3]: rat(i) = 1/[(P(i)/Z)]1/3(4.63) The number of molecules in the edge of primary translational and primary librational eﬀectons: κtr=/parenleftbig Vtr ef/vm/parenrightbig1/3=/bracketleftbig (1/ntr ef)/(V0/NA)/bracketrightbig1/3(4.63a) κlb=/parenleftbig Vlb ef/vm/parenrightbig1/3=/bracketleftbig (1/nlb ef)/(V0/NA)/bracketrightbig1/3(4.63b) where: (1 /ntr,lb ef) is the volume of primary translational or librational eﬀec - tons; ( V0/NA) is the volume, occupied by one molecule in condensed matter . A lot of other parameters, characterizing diﬀerent physica l properties of condensed matter are also possible to calculate, using Hier archic theory and our computer program elaborated, as will be shown in the next chapters. 5. QUANTITATIVE VERIFICATION OF HIERARCHIC THEORY ON EXAMPLES OF ICE AND WATER All the calculations, based on Hierarchic theory, were perf ormed on the personal computers. The special software: ”Comprehensive analyzer of matter properties” [copyright 1997, Kaivarainen] was worked out. This program allows to evaluate more than three hundred parameters of any conden sed matter if the following basic experimental data are available in the temp erature interval of interest: 1. Positions of translational and librational bands in IR sp ectra; 2. Sound velocity; 3. Molar volume; 4. Refraction index. The basic experimental parameters for ice: The wave numbers (˜ νtr), corresponding to positions of translational and librational bands in oscillatory IR spectra were taken from book of Eisenberg and Kauzmann (1969). Wave numbers for ice at 0oCare: 40/parenleftBig ˜ν(1) ph/parenrightBig tr= 60cm−1; /parenleftBig ˜ν(2) ph/parenrightBig tr= 160 cm−1; /parenleftBig ˜ν(3) ph/parenrightBig tr= 229 cm−1 Accordingly to our model, the IR photons with corresponding frequencies are irradiated and absorbed a result of ( a⇔b) primary translational deformons in ice. Temperature shifts of these bands positions are close t o zero: ∂/parenleftBig ˜ν1,2,3 ph/parenrightBig tr/∂T≈0 Wave numbers of librational IR bands, corresponding to absorption of photons, related to ( a⇔b)1,2,3 lbtransitions of primary librational eﬀectons of ice are: /parenleftBig ˜ν(1) ph/parenrightBig lb=/parenleftBig ˜ν(2) ph/parenrightBig lb=/parenleftBig ˜ν(3) ph/parenrightBig lb≈795cm−1. The equality of wave numbers for three directions (1,2,3) indicate the spatial isotropy of the librations of H2Omolecules. In this case deformons and eﬀectons have a cube geometry. In general case they have a shape of para llelepiped (like quasiparticles of translational type) with each of three ribs, corresponding to most probable de Broglie wave length in selected direction. The temperature shift of the position of the librational ban d maximum for ice is: ∂/parenleftBig ˜ν1,2,3 ph/parenrightBig lb/∂T≈ −0.2cm−1/C0 The resulting thermal phonons velocity in ice, responsible for secondary acoustic deformons, is taken as equal to the transverse sound velocit y (Johri and Roberts, 1990): vres s= 1.99·105cm/s This velocity and molar ice volume ( V0) are almost independent on temperature (Eisenberg, 1969): V0= 19.6cm3/M≃const The basic experimental parameters for Water The wave numbers of translational bands in IR spectrum, corr esponding to quantum transitions of primary translational eﬀectons bet weenacoustic (a) and 41optical (b) states with absorption or emission of photons, forming electromag - netic 3D translational deformons at00Care (Eisenberg, 1969): /parenleftBig ˜ν(1) ph/parenrightBig tr= 60cm−1;/parenleftBig ˜ν(2) ph/parenrightBig tr≈/parenleftBig ˜ν(3) ph/parenrightBig tr≈199cm−1 with temperature shifts: ∂/parenleftBig ˜ν(1) ph/parenrightBig tr/∂T= 0; ∂/parenleftBig ˜ν(2,3) ph/parenrightBig tr/∂T=−0.2cm−1/C0 The primary librational deformons of water at 00Care characterized by follow- ing degenerated wave numbers of librational bands in it IR sp ectrum: /parenleftBig ˜ν(1) ph/parenrightBig lb≈/parenleftBig ˜ν(2) ph/parenrightBig lb≈/parenleftBig ˜ν(3) ph/parenrightBig lb= 700 cm−1 with temperature shift: ∂/parenleftBig ˜ν1,2,3 ph/parenrightBig lb/∂T=−0.7cm−1/C0 Wave numbers are related to the frequencies ( ν) of corresponding transitions via light velocity as: ν=c˜ν The dependence of sound velocity (vs)in water on temperature within the temperature range 0 −1000Cis expressed by the polynomial (Fine and Millero, 1973): vs= 1402 .385 + 5 .03522 t−58.3087·10−3t2+ + 345 .3·10−6t3− −1645.13·10−9t4+ 3.9625·10−9t5(m/s). The temperature dependence of molar volume ( V0) ofwater within the same temperature range can be calculated using the polynomial (K ell, 1975; Kikoin, 1976): V0= 18000 /[(999,83952+ 16 .945176 t− −7.98704·10−3t2− −4.6170461 ·10−5t3+ 1.0556302 ·10−7t4− −2.8054253 ·10−10t5)/ /(1 + 1 .687985 ·10−2t)] (cm3/M) Therefraction index for ice was taken as an independent of temperature ( nice= 1.35) and that for water as a variable, depending on temperatur e in accordance with experimental data, presented by Frontas’ev and Schrei ber (1966). The refraction index for water at 200C is approximately: 42nH2O≃1.33 The temperature dependences of diﬀerent parameters for ice and water, com- puted using the formulas of our mesoscopic theory, are prese nted in Figs.(1-4). It is only a small part of available information. In principl e, it is possible to cal- culate about 200 diﬀerent parameters for liquid and solid st ate of any condensed matter [3]. 5.1. Discussion of theoretical temperature dependences an d comparison with experimental data It will be shown below that our hierarchic theory makes it pos sible to cal- culate unprecedented big amount of parameters for liquids a nd solids. Those of them that where measured experimentally and taken from li terature are in excellent correspondence with theory. Fig. 1 .(a, b, c).Temperature dependences of the resulting ther- moaccessibility ( Z) (eq.4.2) and contributions related to primary and secondary eﬀectons and deformons for ice (a,b) and water (c). 43The resulting thermoaccessibility minimum (Fig. 1a) for ic e (Z) corresponds to the temperature of -1700C. The interval from -198 to -1730C is known indeed as anomalies one due to the fact that the heat equilibrium of i ce establishes very slowly in the above range (Maeno, 1988). This fact can be expl ained by the less probable ice structure (minimum value of partition functio n Z ) near −1700C. For the other hand, experimental anomaly, related with maximum heat capacity ( Cp), also is observed near the same temperature. It can be expla ined, if we present heat capacity as: Cp=∂ ∂T(1 ZU∗) =−1 Z2∂Z ∂TU∗+1 Z∂U∗ ∂T One can see, that heat capacity is maximal, when ( ∂Z/∂T ) = 0 and Z is min- imal. It is a condition of Z(T) extremum, just leading from ou r theory at -1700C (Fig.1a). In liquid water the temperature dependences of Z and its comp onents are linear. The thermoaccessibility of mean secondary eﬀecton s in water decreases, while that of primary eﬀectons increases with temperature, just like in ice (Fig. 1 b,c). Fig. 2 . (a,b). Temperature dependences of the total internal energy ( Utot) and diﬀerent contributions for ice (a) and water (b) (eqs. 4.3 - 4.5). Following contributions to Utotare presented: (Uef+¯Uef) is the contribution of primary and secondary eﬀec- tons; ( Ud+¯Ud) is the contribution of primary and secondary de- formons; ( Uef+Ud) is the contribution of primary eﬀectons and deformons; (¯Uef+¯Ud) is the contribution of secondary eﬀectons and defor- mons. It leads from our calculations, that contributions of macro - and supereﬀec- tons to the total internal energy and that of macro- and super deformons, as 44well as all types of convertons, are much smaller than those o f primary and secondary eﬀectons and deformons. On lowering down the temperature the total internal energy o f ice (Fig. 2a) and its components decreases with temperature coming close r to absolute zero. The same parameters for water are decreasing almost linearl y within the interval (100−0)0C(Fig. 2b). In computer calculations, the values of Cp(t) can be determined by diﬀeren- tiating Utotnumerically at any of temperature interval. It follows from Fig. 2a that the mean value of heat capacity fo r ice in the interval from -75 to 0oCis equal to: ¯Cice p=∆Utot ∆T≈39J/M K = 9.3 cal/M K For water within the whole range ∆ T= 1000C, the change in the internal energy is: ∆ U= 17−9.7 = 7.3kJ/M (Fig.2b). This corresponds to mean value of heat capacity of water: Cwater p =∆Utot ∆T= 73J/M K = 17.5cal/M K These results of our theory agree well with the experimental mean values Cp= 18 Cal /M K for water and Cp= 9cal/M K for ice. Mesoscopic molecular Bose condensation at physiological temperature: possible or not? The possibility of existence of mesoscopic (intermediate b etween microscopic and macroscopic) Bose condensation in form of coherent clus ters in condensed matter at the ambient temperature was rejected for a long tim e. The reason of such shortcoming was a wrong primary assumption, that the thermal oscillations of atoms and molecules in condensed matter are theharmonic ones (see for example: Beck and Eccles, 1992). The condition of harmonic oscillations means that the averaged kinetic ( Tk) and potential ( V) energy of molecules are equal to each other and linearly dependent on t emperature (T). This condition leads from Virial theorem (Clausius, 1870) f or the case of classical systems: Tk=V=1 2kT (5.1) The averaged kinetic energy of the oscillating particle may be expressed via its averaged impulse ( p) and mass ( m): Tk=p2/2m (5.1a) The most probable wave B length ( λB) of such particle, based on assumption (5.1), is : λB=h/p=h/(mkT)1/2(5.2) 45It leads from this formula that around the melting point of ic e:T= 273 K the value of λBis less than 1 ˚A and much less than the approximate distance between centers of molecules ( l˜ 3˚A) in ice and water: λB< l (5.2a) This result leads to wrong conclusion that water and ice are c lassical systems, where Bose condensation (BC) is impossible. The sa me wrong conclusion, based on 5.1 and 5.2 follows for any conden sed matter at T around its melting point. The BC is possible only at conditions, when the wave B of parti cles is equal or bigger, than the average distance between their c enters (l): λB≥l= (V0/N0)1/3(5.2b) In contrast to low-temperature macroscopic BC, accompanied supercon- ductivity and superﬂuidity, the condition of mesoscopic high temperature BC may be expressed as: L > λ B> l (5.2c) where Lis a macroscopic parameter, comparable with dimensions of t he whole sample. Condition of partial or mesoscopic BC (5.2c) is general for a ny liq- uids and solids as conﬁrmed by our theory and computer simula tions for water and ice. Correct comparisons of ratio between average kinetic and po tential energy of matter and applying to Virial theorem may give a rig ht answer to question: is this system classical or quantum ? It leads from our theoretical dependencies, presented at Fi g. 3a, bthat the total kinetic energy of water ( Tkin) is approximately 30 times less than the potential energy ( Vp) at the same temperatures. In the case of ice, they diﬀer even more: ( Tkin/V)<1/100. The resulting Tkinof water increases almost twice over the range (0 −1000C) : from 313 to 585 J/M. However, the change of the total internal energy ( Utot=Tkin+Vp) is determined mainly by the change in potential energy Vp(t) of ice and water. 46Fig. 3 . (a,b). Temperature dependences of the kinetic ( Tkin) and potential ( Vp) energy for the ice (a) and water (b), calculated, using eqs.(4.36), .(4.39). . Note that Utot=Tkin+Vpand was calculated from eq.(4.3). We can analyze the above ratio between total kinetic and pote ntial energies in terms of the Viral theorem worked out by Clausius (Clausius, 1870; see also Prokhorov, 1988). It is important to note, that this theorem is valid for both: classical and quantum-mechanical systems. This famous theorem for a system of any kind of particles syst em - relates the averaged kinetic ¯Tk(/vector v) =/summationtext imiv2 i/2 and potential ¯V(r) energies in the form: 2¯Tk(/vector v) =/summationdisplay imiv2 i=/summationdisplay i/vector ri∂V/∂/vector r i (5.2d) The potential energy V(r) is a homogeneous n-order function like: V(r)∼rn(5.3) where the value of power ( n) is equal to ratio of doubled average kinetic energy to average potential energy: n=2Tk(/vector v) V(r)(5.3a) For example, for a harmonic oscillator: n= 2 and ¯Tk=¯V. For Coulomb interaction: n=−1 and ¯T=−¯V /2. For water our calculation of TkandVgives: nw∼1/15 and for ice: nice∼ 1/50. It follows from (5.1) that in water and ice the dependence of potential energy on distance (r) is very weak: Vw(r)∼r(1/15);Vice∼r(1/50)(5.4) These results can be considered as indication of distant int eractions in water and ice, as an associative cooperative systems. We get here a strong evidence that water and ice can not be con- sidered as a classical systems, following condition (5.1). It is important also to note, that the direct interrelation e xists between the inﬁnitive spatial scale of Bose condensation, determin ed by wave B length: λ= (h/p)→ ∞ (eq.5.2) and condition of nonlocality as independence of potential on distance at Tk→0;p→0;n→0: V(r)→const (5.4a) This result is true not only for real condensed matter system s, but also for systems of virtual particles, forming the vacuu m (see: http://arXiv.org/abs/physics/0003001). 475.2. Explanation of temperature anomalies, nonmonotonic T-deviations in aqueous systems Hierarchic theory is the ﬁrst one enable to predict and give a clear explana- tion to deviations of temperature dependencies of some phys ical parameters of water from monotonic ones. It clarify also the interrelation between these deviations (transitions) and corresponding temperature anomalies in properties of bios ystems, such as large- scale dynamics of proteins, the enzymes activity, dynamic e quilibrium of [assembly- disassembly] of microtubules and actin ﬁlaments, etc. Fig. 4 .(a) : The temperature dependencies of the number of H2Omolecules in the volume of primary librational eﬀecton ( nlb M)ef,left axis) and the number of H2Oper length of this eﬀecton edge ( κ, right axis); (b): the temperature dependence of the water pr i- mary librational eﬀecton (approximated by cube) edge lengt h [llib ef= κ(V0/N0)1/3]. The number of H2Omolecules within the primary libration eﬀectons of water, which could be approximated by a cube, decreases fr omnM= 280 at 00tonM≃3 at 1000(Fig. 4a). It should be noted that at physiological temperatures (35 −400) such quasiparticles contain nearly 40 water molecules. This number is close to that of water molecules that can be enc losed in the open interdomain protein cavities judging from X-ray data. The ﬂickering of these clusters, i.e. their ( dissociation ⇋association ) due to [ lb⇔tr] conversions in accordance with our model is directly related to the large -scale dynamics of proteins. It is important that the linear dimensions of such water clus ters (11 ˚A) at physiological temperature are close to dimensions of prote in domains (Fig. 4b). Such spatial correlation indicate that the properties of wa ter ex- erted a strong inﬂuence on the evolution of biopolymers, nam ely, their dimensions and dynamic properties due to ”ﬂickering” of inter- subunit water clusters. We assume here that integer and half-integer values of numbe r of water molecules per eﬀecton’s edge [ κ] (Fig. 4a) reﬂect the conditions of increased and 48decreased stabilities of water structure correspondingly . It is apparently related to the stability of primary librational eﬀectons as coopera tive and coherent water clusters. Nonmonotonic behavior of water properties with temperatur e is widely known and well conﬁrmed experimental fact (Drost-Hansen, 1976, 1 992; Clegg and Drost-Hansen, 1991; Etzler, 1991; Roberts and Wang, 1993; R oberts and Wang, 1993; Roberts, et al., 1993, 1994; Wang et al., 1994). We can explain this interesting and important for biologica l func- tions phenomenon because of competition between two factor s: quan- tum and structural ones in stability of primary librational eﬀectons. The quantum factor such as wave B length, determining the value of the eﬀecton edge: /bracketleftBig lef=κ(V0/N0)1/3˜λB/bracketrightBig lb(5.5) decreases monotonously with temperature increasing. The structural fac- toris a sensitive parameter depending on the H2Oeﬀective length: lH2O= (V0/N0)1/3andtheir number [κ] in the eﬀecton’s edge, approximated by cube. We suggest that when ( lef) corresponds to integer number of H2O, i.e. [κ= (lef/lH2O) = 2,3,4,5,6...]lb(5.6) thecompetition between quantum and structural factors is minimum and pri- mary librational eﬀectons are most stable. On the other hand , when ( lef/lH2O)lb is half-integer, the librational eﬀectons are less stable ( thecompetition is maxi- mum). In the latter case ( a⇔b)lbequilibrium of the eﬀectons must be shifted rightward - to less stable state of these coherent water clus ters. Consequently, the probability of dissociation of librational eﬀectons to a number of much smaller translational eﬀecton, i.e. probability of [lb/tr ] convertons increases and concentration of primary librational eﬀectons decreas es. Experimentally the nonmonotonic change of this probability with temperatu re could be reg- istered by dielectric permittivity, refraction index meas urements and by that of average density of water. The refraction index change sho uld lead to corre- sponding variations of surface tension, vapor pressure, vi scosity, self-diﬀusion in accordance to our hierarchic theory (Kaivarainen, 1995, 20 00). In accordance to our model the density of liquid water in comp o- sition of librational eﬀectons is lower than the average in t he bulk water. In the former case all hydrogen bonds of molecules are satura ted like in ideal ice in contrast to latter one. We can see from Fig.4a that the number of water molecules in primary lb eﬀecton edge (κ) is integer near the following temperatures: 60(κ= 6); 170(κ= 5); 320(κ= 4); 490(κ= 3); 770(κ= 2) (5.7) These temperatures coincide very well with the maximums of relaxation time in pure water and with dielectric response anomalies (Roberts , et al., 1993; 1994; Wang, et al., 1994). The special temperatures predicted by o ur theory are close also to chemical kinetic (Aksnes, Asaad, 1989; Aksnes, Libn au, 1991), refrac- tometry (Frontas’ev, Schreiber 1966) and IR (Prochorov, 19 91) data. Small 49discrepancy may result from the high sensitivity of water to any kind of pertur- bation, guest-eﬀects and additional polarization of water molecules, induced by high frequency visible photons. Even such low concentratio ns of inorganic ions ester and NaOH as used by Aksnes and Libnau (1991) may change w ater proper- ties. The increase of H2Opolarizability under the eﬀect of light also may lead to enhancement of water clusters stability and to correspondi ng high-temperature shift of nonmonotonic changes of water properties. The semi integer numbers of [ κ] for pure water correspond to temperatures: 00(κ= 6.5); 120(κ= 5.5); 240(κ= 4.5); 400(κ= 3.5); (5.7a) 620(κ= 2.5); 990(κ= 1.5) The conditions (5.7a) characterize the less stable water st ructure than con- ditions (5.7). The ﬁrst order phase transitions - freezing a t 00and boiling at 1000of water almost exactly correspond to κ= 6.5 and κ= 1.5. This fact is important for understanding the mechanism of ﬁrst order pha se transitions. The temperature anomalies of colloid water-containing sys tems, discovered by Drost-Hansen (1976) and studied by Etzler and coauthors (1987; 1991) occurred near 14-160; 29-320; 44-460and 59-620C. At these temperatures the extrema of viscosity, disjoining pressure and molar exc ess entropy of water between quartz plates even with a separation 300-500 ˚A has been observed. These temperatures are close to predicted by our theory for b ulk water anoma- lies, corresponding to integer values of [ κ] (see 5.7). Some deviations can be a result of interfacial water perturbations, induced by coll oid particles and plates. It is a ﬁrst theory which looks to be able to predict and explai n the existence of Drost-Hansen temperatures. The dimensions, concentration and stability of water clust ers (primary li- brational eﬀectons) in the volume of vicinal water should be bigger than that in bulk water due to their less mobility and to longer waves B l ength. Interesting ideas, concerning the role of water clusters in biosystems were developed in works of John Watterson (1988a,b). It was revealed in our laboratory (Kaivarainen, 1985; Kaiva rainen et al., 1993) that nonmonotonic changes of water near Drost-Hansen temperatures are accompanied by in-phase change of diﬀerent protein large-s cale dynamics, re- lated to their functioning. The further investigations of l ike phenomena are very important for understanding the molecular mechanisms of th ermoadaptation of living organisms. 5.3. Physiological temperature and the least action princi ple The Fig.5 a, bshows the resulting contributions to the total kinetic ener gy of water of two main subsystems: eﬀectons and deformons. The minimum of deformons contribution at 430is close to the physiological temperatures for warm-blooded animals. 50Fig. 5. Temperature dependences of two resulting contributions - eﬀectons ( Tef kin) and deformons ( Td kin) of all types to the total kinetic energy of water. The minima at temperature dependences of diﬀerent contribu tions to the total kinetic energy of water at Fig.5 correspond to the best imple mentation of the least action principle in the form of Mopertui-Lagrange. In such a form, this principle is valid for the conservative h olonom systems, where limitations exist for the displacements of the particles of this sys- tem, rather than the magnitudes of their velocities. It states that among all the kinematically possible displacements of a system fr om one conﬁguration to another, without changing total system energy, such a dis placements are most probable for which the action (W) is least: ∆ W= 0. Here ∆ is the symbol of total variation in coordinates, velocities and time. The action is a fundamental physical parameter which has the dimension of the product of energy and time characterizing the dynamics o f a system. According to Hamilton, the action: S=t/integraldisplay t0Ldt (5.8) is expressed through the Lagrange function: L=Tkin−V, (5.9) where T kinand V are the kinetic and potential energies of a system or a su b- system. According to Lagrange, the action (W) can be expressed as: W=t/integraldisplay t02Tkindt (5.10) 51We can assume that at the same integration limit the minimum v alue of the action ∆W ≃0 corresponds to the minimum value of T kin.. Then it can be said that at temperature about 430the subsystems of deformons is most stable (see Fig. 6). This means that the equilibrium between the acousti c and optic states of primary and secondary eﬀectons should be most stable at th is temperature. 5.4. Mechanism of the 1st and 2nd order phase transitions in terms of the hierarchic theory The abrupt increase of the total internal energy (U) as a resu lt of ice melting (Fig. 6a), equal to 6 .27kJ/M , calculated from our theory is close to the experi- mental data (6 kJ/M ) (Eisenberg, 1969). The resulting thermoaccessibility (Z) during [ice →water] transition decreases abruptly, while potential and kinetic energies increase (Fig. 6b). Fig. 6. The total internal energy ( U=Tkin+Vp) change during ice-water phase transition and change of the resulting ther moacces- sibility (Z) - (a); changes in kinetic ( Tkin) and potential ( Vp) energies (b) as a result of the same transition. It is important that at the melting point H2Omolecules number in a primary translational eﬀecton (ntr M)efdecreases from 1 to ≃0.4 (Fig. 7a). It means that the volume of this quasiparticle type gets smaller than the v olume occupied by H2Omolecule. According to our model, under such conditions the individua l water molecules get the independent translation mobility. The number of water molecules forming a primary libration eﬀecton decreases abruptly from about 3000 to 280, as a result of melting. The number of H2Oin the secondary librational eﬀecton decreases correspondingly from ∼1.25 to 0.5 (Fig. 7b). Fig. 8 a, bcontains more detailed information on changes in primary li bra- tional eﬀecton parameters in the course of ice melting. The theoretical dependences obtained allow us to give a clea r interpretation of the ﬁrst order phase transitions. The condition of meltin g atT=Tcris realized in the course of heating when the number of molecule s in the volume of primary translational eﬀectons nMdecreases: ntr M≥1(T≤Tcr)Tc→ntr M≤1(T≥Tcr) (5.11) 52Number of molecules ntr,lb Min primary translational and librational eﬀectons may be calculated using (4.63 a,b): ntr,lb M=/parenleftBig Vtr,lb ef/vm/parenrightBig =/bracketleftBig (1/ntr,lb ef)/(V0/NA)/bracketrightBig1/3 (5.11a) where: (1 /ntr,lb ef) is the volume of primary translational or librational eﬀec - ton; (V0/NA) is the volume, occupied by one molecule in condensed matter . Fig. 7. Changes of the number of H2Omolecules forming pri- mary ( ntr M)efand secondary (¯ ntr M)eftranslational eﬀectons during ice-water phase transition (a). Changes in the number of H2O molecules forming primary ( nlb M)efand secondary (¯ nlb M)eflibrational eﬀectons (b) as a result of phase transitions. The process of boiling, i.e. [liquid →gas] transition, as seen from Fig. 7a, is also determined by condition (5.11), but at this case it is realized for primary librational eﬀectons. This means that [gas →liquid] transition is related to origination (condensa- tion) of the primary librational eﬀectons which contain more than one molecule of substance. In a liquid as compared to gas, the quantity of rotational deg rees of freedom is decreased due to librational coherent eﬀectons formatio n, but the number of translational degrees of freedom remains the same. The translational degrees of freedom, in turn, also decreas es, however, during [liquid →solid] phase transition, when the wave B length of molecules corre- sponding to their translations begins to exceed the mean dis tances between the centers of molecules (Fig. 7a). This process is accompanied by partial Bose- condensation of translational waves B and by the formation o f coherent primary translational eﬀectons, including more than one molecule. The size of librational eﬀectons grows up abruptly during [ water →ice] transition. 53Fig. 8. Changes of the number of H2Omolecules forming a primary librational eﬀecton ( nlb M)ef, the number of H2Omolecules (κ) in the edge of this eﬀecton (a) and the length of the eﬀecton edge: llb ef=κ(V0/N0)1/3(b) during the ice-water phase transition. The enlarged primary librational eﬀectons and librational polyeﬀectons, orig- inating from the eﬀectons ”side-to-side” assembly due to Jo sephson’ junctions, may serve as a centers of crystallization, necessary for [ liquid →solid] transi- tion. We assume here that probability of mesoscopic Bose con densation (BC) of molecules, involved in translations, as a condition of [l iquid-solid] transition, increases in vicinity of crystallization centers, stimula ted by interfacial eﬀects. The process of polyeﬀectons formation in very pure water is s low due to rela- tively low probability of their collision, necessary for pr imary eﬀectons assembly. Such mechanism may be responsible for getting the supercool ed water, i.e. liq- uid water, existing few degrees below 00C.The presence of impurities in form of colloid particles in water - stimulates the enlargement o f librational eﬀectons and their assembly, increasing in such a way the temperature of [water →ice] phase transition and making it closer to 00C.The opposite sharp transition of pure [ ice→water ],however, occur always at 00Cat normal pressure of 1 atm. We may explain this phenomenon because the uniﬁed syste m of primary [translational + librational BC] of ice is more cooperative than the system of only librational BC of water. Consequently, the reaction of ice to temperature is more sensitive, than that of water and phase transition [ ice→water ] is more sharp, than [ water →ice]. In contrast to ﬁrst order phase transitions, the 2nd order ph ase transitions are not related to the abrupt change of primary eﬀectons volu me and concentration, but only to their stability, related to their (a⇋b)1,2,3 tr,lb equilibrium shift, symmetry changes and polymerization. Such phe- nomena may be a result of a gradual [temperature/pressure] - dependent de- crease in the diﬀerence between the energy of aiandbistates of one of three standing waves B, forming primary eﬀectons/bracketleftbig hνp=h/parenleftbig νb−νa/parenrightbig/bracketrightbigi tr,lb.Such eﬀect, registered by IR spectroscopy, is known as a soft mode low-frequency shift: 54/bracketleftbig hνp=h/parenleftbig νb−νa/parenrightbig /bracketrightbigi tr,lb→0 at/bracketleftbig λTc b=λTca/bracketrightbigi tr,lb>/parenleftbigV0/N0/parenrightbig1/3/bracerightBigg (5.12) The non-monotonic changes of sound velocity and the low-fre quency shift of translational and librational bands in oscillatory spec tra, according to our theory, should be followed by jump of heat capacity, compres sibility and coef- ﬁcient of thermal expansion. The parameters of elementary c ells,depending on geometry, stability and dynamics of primary eﬀectons are changing also. All these predictions of our theory are in accordance with exper imental data. Consequently, theory propose a new clear mechanism of 1st an d 2nd order phase transitions. The number of molecules in the volume of p rimary eﬀec- tons (5.11a) may be considered as a parameter of order for 1st order phase transition. The value of the constant of ( a⇋b)1,2,3 tr,lbequilibrium K(a⇋b)1,2,3 tr,lb= [a]/[b]1,2,3 tr,lb= [hνa/hνb]1,2,3 tr,lb may serve as the parameter of order for 2nd order phase transition. The critical values of both parameters of order are close to o ne 5.5. The energy of quasiparticle discrete states. Activation energy of dynamics in water Over the entire temperature range for water and ice, excludi ng conditions of 2nd order phase transitions, the energies of ”acoustic” a-states of primary eﬀectons (translational and librational ones) are lower th an the energies of ”op- tic”b-states (Fig.9). The energy of an ideal eﬀecton (3RT) has the intermediate values. 55Fig. 9. Temperature dependences for the energy of primary eﬀectons in ”acoustic” ( a) and ”optical” ( b) states and that for the energy of a harmonic 3D oscillator (the ideal thermal eﬀecto n:E0= 3RT) for water and ice calculated according to the formulae (4.6 , 4.7 and 4.12): a) for primary translational eﬀectons of wate r ina andbstates; b) for primary librational eﬀectons of water in aandb states; c) for primary translational eﬀectons of ice in aandbstates; d) for primary librational eﬀectons of ice in aandbstates. According to the eq.(4.10 and 4.11) the thermoaccessibility of (a) and ( b) states is determined by the absolute value of the diﬀerence: \|Ea ef−3kT\|tr,lb;\|Eb ef−3kT\|tr,lb. where E0= 3kT= 3hν0is energy of an ideal eﬀecton. The ( a⇔b) transitions (quantum beats) can be considered as autoosci l- lations of quasiparticles around the thermal equilibrium s tate ( E0), which is quantum - mechanically prohibited. In terms of synergetics , the primary eﬀec- tons are the medium active elements. (b→a) transitions are related to origination of photons, and ele ctromagnetic deformons, while the reverse ones ( a→b) correspond to absorption of them, i.e. annihilation of deformons. 56The nonequilibrium conditions in the subsystems of eﬀecton s and deformons can be induced by the competition between discrete quantum a nd continuous heat energy distributions of diﬀerent quasiparticles. Som etimes these nonequi- librium conditions could lead to macroscopic long-period o scillations in con- densed matter. The temperature dependences of the excitation (or ﬂuctuati on) energies for translational and librational macroeﬀectons in A(a,¯a) and B(b,¯b) states: (ǫA M)tr,lb; (ǫB M)tr,lband that for macrodeformons ( ǫM D)tr,lband superdeformons (ǫs D∗), for water (a,b) and ice (c,d) can be calculated according t o formulas (5.8, 5.9 and 5 .10). E0= 3RTis the energy of ideal quasiparticle, corresponding to thermal equilibrium energy. The knowledge of the excitation energies of macrodeformons is important for calculation the viscosity and coeﬃcient of self-diﬀusi on (see sections 6.6 and 6.8). The A and B states of macro- and supereﬀectons represent the s igniﬁcant de- viations from thermal equilibrium. The transitions betwee n these states termed: macro- and superdeformons represent the strong ﬂuctuation s of polarizabilities and, consequently, of refraction index and dielectric perm eability. The excitation energies of A and B states of macroeﬀectons ar e determined as: (ǫA M)tr,lb=−RTln(Pa efPa ef)tr,lb=−RTln(PA M)tr,lb (5.13) (ǫB M)tr,lb=−RTln(Pb efPb ef)tr,lb=−RTln(PB M)tr,lb (5.14) where Pa efand¯Pa efare the thermoaccessibilities of the ( a)−eq.(4.10) and (¯ a)− eq.(4.18) - states of the primary and secondary eﬀectons, correspo ndingly; Pb ef and¯Pb efare the thermoaccessibilities of ( b)−eq.(4.11) and ( ¯b)−eq.(4.19) states. The activation energy for superdeformons is: ǫs D∗=−RTln(Ps D) =−RT[ln(PM D)tr+ ln(PM D)lb] = (5.15) = (ǫM D)tr+ (ǫM D)l The value ( ǫM D)tr≈11.7kJ/M ≈2.8 kcal/M characterizes the activation energy fortranslational self-diﬀusion of water molecules , and ( ǫM D)lb≈31kJ/M ≈7.4 kcal/M - the activation energy for librational self-diﬀusi on of H2O. The latter valueis close to the energy of the hydrogen bond in water (Eisenberg, 1969). On the other hand,the biggest ﬂuctuations-superdeformons are responsible for the process of cavitational ﬂuctuations in liquids and the emergency of defects in solids. They determine vapor pre ssure and sublimation, as it will be shown in our work. 57Fig. 10. Temperature dependences of the oscillation frequen- cies in ( a) and ( b) state of primary eﬀectons - translational and librational for water (a) and ice (b), calculated from (Fig. 9). The relative distribution of frequencies on Fig.10 is the sa me as of energies on Fig. 9. The values of these frequencies reﬂect the minimum life-times of corresponding states. The real life-time is dependent also on probability of ”jump” from this state to another one and on probability of st ates excitation. 5.6. The life-time of quasiparticles and frequencies of the ir excitations The set of formula, describing the dynamic properties of qua siparticles, in- troduced in mesoscopic theory was presented earlier. For the case of ( a⇔b)1,2,3transitions of primary and secondary eﬀectons (tr and lb), their life-times in (a) and (b) states are the reciprocal val ue of corresponding frequencies: [ τa= 1/νaandτb= 1/νb]1,2,3 tr,lb. These parameters and the resulting ones could be calculated from eqs.(2.27; 2 .28) for primary eﬀectons and (2.54; 2.55) for secondary ones. The results of calculations, using eq.(4.56 and 4.57) for fr equency of excita- tions of primary tr and lb eﬀectons are plotted on Fig. 11a,b. The frequencies of Macroconvertons and Superdeformons wer e calculated using eqs.(4.42 and 4.48). 58Fig. 11. (a) - Frequency of primary [tr] eﬀectons excitations, calculated from eq.(4.56); (b) - Frequency of primary [lb] eﬀectons excitations, calcu lated from eq.(4.57); (c) - Frequency of [ lb/tr] Macroconvertons (ﬂickering clusters) excitations, calculated from eq.(4.42); (d) - Frequency of Superdeformons excitations, calculated from eq.(4.48). At the temperature interval (0 - 100)0Cthe frequencies of translational and librational macrodeformons (tr and lb) are in the interval o f (1.3−2.8)·109s−1and(0.2−13)·106s−1(5.16) correspondingly. The frequencies of (ac) and (bc) converto ns could be deﬁned also using our software and formulae, presented at the end of Section IV. The frequency of primary translational eﬀectons [ a⇔b] transitions at 200C, calculated from eq.(4.56) is ν∼7·1010(1/s) (5.17) It corresponds to electromagnetic wave length in water with refraction index (n= 1.33) of: λ= (cn)/ν∼6mm (5.18) For the other hand, there are a lot of evidence, that irradiat ion of very diﬀerent biological systems with such coherent electromagnetic ﬁel d exert great inﬂuences on their properties (Grundler and Keilman, 1983). 59Between the dynamics/function of proteins, membranes, etc . and dynamics of their aqueous environment the strong interrela tion exists. The frequency of macroconvertons, representing big densit y ﬂuctuation in the volume of primary librational eﬀecton at 370C is about 107(1/s) (Fig 11c), the frequency of librational macrodeformons at the same tem perature is about 106s−1,i.e. coincides with frequency of large-scale protein cavities pulsa- tions between open and closed to water states (see Fig.11). This conﬁrm our hypothesis that the clusterphilic interaction is respo nsible for stabilization of the proteins cavities open state and that transition from the open state to the closed one is induced by coherent water cluster dissociatio n. The frequency of Superdeformons excitation (Fig.11d) is mu ch lower: νs∼(104−105)s−1(5.19) Superdeformons are responsible for cavitational ﬂuctuati ons in liquids and orig- ination of defects in solids. Dissociation of oligomeric pr oteins, like hemoglobin or disassembly of actin and microtubules could be also relat ed with such big ﬂuctuations. Superdeformons could stimulate also the reve rsible dissociation of water molecules, which determines the pH value H2O⇋HO−+H+(5.20) Recombination of HO−andH+may be accompanied by emission of UV and visible photons. Corresponding radiation could be resp onsible for fraction of so-called biophotons. The parameters, characterizing an average spatial distrib ution of primary lb and tr eﬀectons in the bulk water are presented on the next Fig.12. 60Fig. 12. Theoretical temperature dependencies of: (a) - the space between centers of primary [lb] eﬀectons (cal cu- lated in accordance to eq.4.62); (b) - the ratio of space between primary [lb] eﬀectons to thei r length (calculated, using eq.4.63); (c) - the space between centers of primary [tr] eﬀectons (in a c- cordance to eq.4.62); (d) - the ratio of space between primary [tr] eﬀectons to thei r length (eq.4.63). One can see from the Fig.12 that the dimensions of primary tra nslational eﬀectons are much smaller and concentration much higher tha n that of primary librational eﬀectons. We have to keep in mind that these are t he averaged spatial distributions of collective excitations. The form ation of polyeﬀectons - coherent clusters of lb (in liquids) and tr (in solids) prima ry eﬀectons, interacting side-by-side due to Josephson eﬀect is possible also. Fig. 13. Temperature dependences for the concentrations of pri- mary eﬀectons (translational and librational) in ( a) and ( b) states: (Na ef)tr,lb,(Nb ef)tr,lbfor water ( aandb); the similar dependencies for ice ( candd). Concentrations of quasiparticles were calculated from eqs.:(Na ef)tr,lb= (nefPa ef/Z)tr,lb; (Nb ef)tr,lb= (nefPb ef/Z)tr,lb 61These dependences can be considered as the quasiparticles distribution functions. To get such information using conventional tools, i.e. by me ans of x-ray or neutron scattering methods is very complicated task. How ever, even in this case the ﬁnal information about properties of collective ex citations will not be so comprehensive as it leads from our theory. The results, presented above, conﬁrms the correctness of ou r model for liquids and solids, as a hierarchic system of 3D standing waves of diﬀerent nature. It will be demonstrated below that applica tion of Hierarchic theory could be useful for elucidation and quant itative analysis of very diﬀerent physical properties. 6. INTERRELATION BETWEEN MESOSCOPIC AND MACROSCOPIC PARAMETERS OF MATTER 6.1. The state equation for real gas The Clapeyrone-Mendeleyev equation sets the relationship between pressure (P), volume ( V) and temperature ( T) values for the ideal gas containing N0 molecules (one mole): PV=N0kT=RT (6.1) In the real gases interactions between the molecules and the ir sizes should be taken into account. It can be achieved by entering the corres ponding amend- ments into the left part, to the right or to the both parts of eq . (1). It was Van der Waals who choosed the ﬁrst way more than a hundre d years ago and derived the equation: /parenleftBig P+a V2/parenrightBig/parenleftbig V−b/parenrightbig =RT (6.2) where the attraction forces are accounted for by the amendin g term ( a/V2), while the repulsion forces and the eﬀects of the excluded vol ume accounted for the term (b). Equation (2) correctly describes changes in P,V and T relate d to liquid-gas transitions on the qualitative level. However, the quantit ative analysis by means of (2) is approximate and needs the ﬁtting parameters. The pa rameters (a) and (b) are not constant for the given substance and depend on tem perature. Hence, the Van der Waals equation is only some approximation descri bing the state of a real gas. We propose a way to modify the right part of eq.(1), substitut ing it for the part of the kinetic energy (T) of 1 mole of the substance (eq.4 .31 in [1, 2]) in real gas phase formed only by secondary eﬀectons and deformo ns with nonzero impulse, aﬀecting the pressure: 62PV=2 3¯Tkin=2 3V01 Z/summationdisplay tr,lb/bracketleftBigg ¯nef/summationtext3 1/parenleftbig¯Ea 1,2,3/parenrightbig2 2m/parenleftbig va ph/parenrightbig2/parenleftbig¯Pa ef+¯Pb ef/parenrightbig + + ¯nd/summationtext3 1/parenleftBig ¯E1,2,3 d/parenrightBig2 2m(vs)2¯Pd  tr,lb(6.3) The contribution to pressure caused by primary quasipartic les as Bose-condensate with the zero resulting impulse is equal to zero also. It is assumed when using such approach that for real gases the model of a system of weakly interacted oscillator pairs is valid. The validity of such an approach for water is conﬁrmed by available experimental data indicating the presence of dimers, trimers and larger H2Oclusters in the water vapor (Eisenberg and Kauzmann, 1975). Water vapor has an intensive band in oscillatory spectra at ˜ ν= 200 cm−1. Possibly, it is this band that characterizes the frequencie s of quantum beats between ”acoustic” (a) and ”optic” (b) translational oscil lations in pairs of molecules and small clusters. The frequencies of libration al collective modes in vapor are absent. The energies of primary gas quasiparticles ( hνaandhνb) can be calculated on the basis of the formulae used for a liquid (see section... ). However, to calculate the energies of secondary quasiparti cles in (¯ a) and (¯b) states the Bose-Einstein distribution must be used for th e case when the temperature is higher than the Bose-condensation temperat ure (T > T 0) and the chemical potential is not equal to zero ( µ <0). According to this distribution: /braceleftbigg ¯Ea=h¯νa=hνa exp(hνa−µ kT)−1/bracerightbigg tr,lb /braceleftBigg ¯Eb=h¯νb=hνb exp/parenleftBig hνb−µ kT/parenrightBig −1/bracerightBigg tr,lb(6.4) The kinetic energies of eﬀectons (¯ a)tr,lband (¯b)tr,lbstates are equal, only the potential energies diﬀer as in the case of condensed matter. All other parameters in basic equation (6.3) can be calculat ed as previously described. 6.2. New state equation for condensed matter Using our eq.(4.3 from [1,2]) for the total internal energy o f condensed matter (Utot), we can present state equation in a more general form than (3 ). For this end we introduce the notions of internal pressure (Pin), including all type of interactions between particles of matter and excluded molar volume (Vexc): 63Vexc=4 3πα∗N0=V0/parenleftbiggn2−1 n2/parenrightbigg (6.5) where α∗is the acting polarizability of molecules in condensed matt er (see section...); N0is Avogadro number, and V0is molar volume. The general state equation can be expressed in the following form: PtotVfr= (Pext+Pin)(V0−Vexc) =Uef (6.6) where: Uef=Utot(1+V/Tt kin) =U2 tot/Tkinis the eﬀective internal energy and: (1 +V/Tkin) =Utot/Tkin=S−1 is the reciprocal value of the total structural factor ( eq.2.46a of[1]);Ptot= Pext+Pinis total pressure, PextandPinare external and internal pressures; Vfr=V0−Vexc=V0/n2(see eq.5) is a free molar volume; Utot=V+Tkinis the total internal energy, V and Tkinare total potential and kinetic energies of one mole of matter. For the limit case of ideal gas, when Pin= 0;Vexc= 0; and the potential energy V= 0, we get from (6) the Clapeyrone - Mendeleyev equation (see 1): PextV0=Tkin=RT One can use equation of state (6) for estimation of sum of all types of internal matter interactions , which determines the internal pressure Pin: Pin=Uef Vfr−Pext=n2U2 tot V0Tkin−Pext (6.7) where: the molar free volume: Vfr=V0−Vexc=V0/n2; and the eﬀective total energy: Uef=U2 tot/Tkin=Utot/S. For solids and most of liquids with a good approximation: Pin≫[Pext∼1 atm. = 105Pa]. Then from (7) we have: Pin∼=n2Utot V0S=n2 V0·Utot/parenleftbigg 1 +V Tkin/parenrightbigg (6.8) where S=Tkin/Utotis a total structural factor; Tkinand V are total kinetic and potential energies, respectively. For example for 1 mole of water under standard conditions we o btain: Vexc= 8.4cm3;Vfr= 9.6cm3;V0=Vexc+Vfr= 18cm3; Pin∼=380000 atm. = 3 .8·1010Pa(1 atm. =105Pa). The parameters such as sound velocity, molar volume, and the positions of translational and librational bands in oscillatory spec tra that determine Uef(4.3) depend on external pressure and temperature. 64The results of computer calculations of Pin(eq.7) for ice and water are presented on Fig. 14 a,b. Polarizability and, consequently, free volume ( Vfr) and Pinin (6.6) depend on energy of external electromagnetic ﬁelds. Fig. 14. (a) Theoretical temperature dependence of internal pressure ( Pin) in ice including the point of [ice ⇔water] phase tran- sition; (b) Theoretical temperature dependence of interna l pressure (Pin) in water. Computer calculations were performed using eq. (6.7). The minima of Pin(T) for ice at −1400and−500Cin accordance with eq.(9) correspond to the most stable structure of this matter, rela ted to temperature transition. In water some kind of transition appears at 350C, near physiological temperature. There may exist conditions when the derivatives of internal pressure P inare equal to zero: (a) :/parenleftbigg∂Pin ∂Pext/parenrightbigg T= 0 and ( b) :/parenleftbigg∂Pin ∂T/parenrightbigg Pext= 0 (6.9) This condition corresponds to the minima of potential energy, i.e. to the most stable structure of given matter. In a general case there may be a few metastable states when conditions (6.9) are fulﬁlled. Equation of state (6.7) may be useful for the study of mechani cal properties of condensed matter and their change under diﬀerent inﬂuenc es. Diﬀerentiation of (6.6) by external pressure gives us at T = const: Vfr+∂Pfr ∂Pext(Pex+Pin) +Vfr∂Pin ∂Pext=∂Pef ∂Pext(6.10) Dividing the left and right part of (6.10) by free volume Vfrwe obtain: /parenleftbigg∂Pin ∂Pext/parenrightbigg T=/parenleftbigg∂Pef ∂Pext/parenrightbigg T−/bracketleftbig 1 +βT(Pext+Pin)/bracketrightbig T(6.11) 65where: βT=−(∂Vfr/∂Pext)/Vfris isothermal compressibility. From (6.9) and (6.11) we derive condition for the maximum stability of matter structure: /parenleftbigg∂Pef ∂Pext/parenrightbigg T= 1 + β0 TPopt tot (6.12) where: Popt tot=Pext+Popt inis the ”optimum” total pressure. The derivative of (6.6) by temperature gives us at Pext=const: Ptot/parenleftbigg∂Vfr ∂T/parenrightbigg Pext+Vfr/parenleftbigg∂Pin ∂T/parenrightbigg Pext=/parenleftbigg∂Uef ∂T/parenrightbigg Pext=CV (6.13) where /parenleftbigg∂Vfr ∂T/parenrightbigg Pext=/parenleftbigg∂V0 ∂T/parenrightbigg Pext−4 3πN0/parenleftbigg∂α∗ ∂T/parenrightbigg Pext(6.14) and/parenleftbigg∂Vtot ∂T/parenrightbigg Pext=∂Pin ∂T(6.14a) From our mesoscopic theory of refraction index (see section ..) the acting polar- izability α∗is: α∗=/parenleftBig n2−1 n2/parenrightBig 4 3πN0 V0(6.15) When condition (6.9b) is fulﬁlled, we obtain for optimum int ernal pressure (Popt in) from (6.13): Popt in=CV//parenleftbigg∂Vfr ∂T/parenrightbigg Pext−Pext (6.16) or Popt in=C Vfrγ−Pext, (6.17) where γ= (∂Vfr/∂T)/Vfr (6.18) is the thermal expansion coeﬃcient; Vfris the total free volume in 1 mole of condensed matter: Vfr=V0−Vexc=V0/n2(6.19) 66It is taken into account in (6.13) and (6.19) that (∂Vexc/∂T)∼=0 (6.20) because, as has been shown by our computer simulations, ∂α∗/∂T∼=0 Dividing the left and right parts of (6.13) by PtotVfr=Uef, we obtain for the heat expansion coeﬃcient: γ=CV Uef−1 Ptot/parenleftbigg∂Pin ∂T/parenrightbigg Pext(6.21) Under metastable states, when condition (6.9 b) is fulﬁlled , γ0=CV/Uef (6.22) Putting (6.8) into (6.12), we obtain for isothermal compres sibility of metastable states corresponding to (6.9a) following formula: β0 T=V0Tkin n2U2 tot/parenleftbigg∂Uef ∂Pext−1/parenrightbigg (6.23) It seems that our equation of state (6.7) may be used to study d iﬀerent types of external inﬂuences (pressure, temperature, elect romagnetic radiation, deformation, etc.) on the thermodynamic and mechanic prope rties of solids and liquids. 6.3. Vapor pressure When a liquid is incubated long enough in a closed vessel at co nstant tem- perature, then an equilibrium between the liquid and vapor i s attained. At this moment, the number of molecules evaporated and conde nsed back to liquid is equal. The same is true of the process of sublimat ion. There is still no satisfactory quantitative theory for vapor pressure calcula- tion. We can suggest such a theory using our notion of superdeformons , represent- ing the biggest thermal ﬂuctuations (see Table 1 and Introdu ction). The basic idea is that the external equilibrium vapor pressure is rela ted to internal one (PS in) with coeﬃcient determined by the probability of cavitatio nal ﬂuctuations (superdeformons) in the surface layer of liquids or solids. In other words due to excitation of superdeformons with prob ability ( PS D), the internal pressure ( PS in) in surface layers, determined by the total contribu- tions of all intramolecular interactions turns to external one - vapor pressure (PV). It is something like a compressed spring energy realizati on due to trigger switching oﬀ. 67For taking into account the diﬀerence between the surface an d bulk internal pressure ( Pin) we introduce the semi-empirical surface pressure factor ( qS) as: PS in=qSPin−Pext=qS·n2Utot V0S−Pext (6.24) where: P incorresponds to eq.(7);S=Tkin/Utotis a total structure factor. The value of surface factor ( qS) for liquid and solid states is not the same: qS liq< qS sol (6.25) Fig. 15. a) Theoretical ( −) and experimental ( ··) temperature dependences of vapor pressure ( Pvap) for ice (a) and water (b) includ- ing phase transition region. Computer calculations were pe rformed using eq. (6.26). Multiplying (6.24) to probability of superdeformons excit ation we obtain for vapor pressure, resulting from evaporation or sublimat ion, the following formulae: Pvap=PS in·PS D=/parenleftbigg qSn2U2 tot V0Tkin−Pext/parenrightbigg ·exp/parenleftbigg −ES D kT/parenrightbigg (6.26) where: PS D= exp/parenleftbigg −ES D kT/parenrightbigg (6.27) is a probability of superdeformons excitation (see eqs. 3.3 7, 3.32 and 3.33). 68We can assume, that the diﬀerence in the surface and bulk inte rnal pressure is determined mainly by diﬀerence in total internal energy ( Utot) but not in kinetic one ( Tk). Then a pressure surface factor could be presented as: qS=γ2= (Uin/Utot)2 where: γ=US tot/Utotis the surface energy factor , reﬂecting the ratio of surface and bulk total energy. Theoretical calculated temperature dependences of vapor p ressure, described by (6.26) coincide very well with experimental ones for wate r atqS liq= 3.1 (γl= 1.76) and for ice at qS sol= 18 ( γs= 4.24) (Fig. 15). The almost ﬁve-times diﬀerence between qS solandqS liqmeans that the surface properties of ice diﬀer from bulkones much more than for liquid water. The surface factors qS liqandqS solshould be considered as a ﬁt pa- rameters. The qS=γ2is the only one ﬁt parameter that was used in our hierarchic mesoscopic theory. Its calculation from t he known vapor pressure or surface tension can give an important info rmation itself. 6.4. Surface tension The resulting surface tension is introduced in our mesoscop ic model as a sum: σ= (σtr+σlb) (6.28) where: σtrandσlbare translational and librational contributions to surfac e tension. Each of these components can be expressed using our mesoscopic state equation (6.7), taking into account the diﬀerence between s urface and bulk total energies ( qS), introduced in previous section: σtr,=1 1 π(V lbef)2/3 tr,lb/bracketleftbiggqSPtot(PefVef)tr,lb−Ptot(PefVef)tr,lb (Pef+Pt)tr+ (Pef+Pt)lb+ (Pcon+PcMt)/bracketrightbigg (6.29) where ( Vef)tr,lbare volumes of primary tr and lib eﬀectons, related to their concentration ( nef)tr,lbas: (Vef)tr,lb= (1/nef)tr,lb; rtr,lb=1 π(Vef)2/3 tr,lb is an eﬀective radius of the primary translational and libra tional eﬀectons, local- ized on the surface of condensed matter; qSis the surface factor, equal to that used in eq.(6.24-6.26); [ Ptot=Pin+Pext] is a total pressure, corresponding to eq.(6.6); ( Pef)tr,lbis a total probability of primary eﬀecton excitations in the (a) and (b) states: 69(Pef)tr= (Pa ef+Pb ef)tr (Pef)lb= (Pa ef+Pb ef)lb (Pt)trand (Pt)lbin (29) are the probabilities of corresponding transiton ex cita- tion; Pcon=Pac+Pbcis the sum of probabilities of [ a] and [ b]convertons; PcMt= PacPbcis a probability of Macroconvertons excitation (see Introd uction). The eq. (6.29) contains the ratio: (Vef/V2/3 ef)tr,lb=ltr,lb (6.30) where: ltr= (1/nef)1/3 trandllb= (1/nef)1/3 libare the length of the ribs of the primary translational and librational eﬀectons, approxim ated by cube. Using (6.30) and (6.29) the resulting surface tension (6.28 ) can be presented as: σ=σtr+σlb=πPtot(qS−1)·/bracketleftbig (Pef)trltr+ (Pef)llb/bracketrightbig (Pef+Pt)tr+ (Pef+Pt)lb+ (Pcon+PcMt)(6.31) where translational component of surface tension is: σtr=πPtot(qs−1)(Pef)trltr (Pef+Pt)tr+ (Pef+Pt)lb+ (Pcon+PcMt)(6.32) and librational component of σis: σlb=πPtot(qS−1)(Pef)lbllb (Pef+Pt)lb+ (Pef+Pt)lb+ (Pcon+PcMt)(6.33) Under the boiling condition when qS→1 as a result of ( US tot→Utot), then σtr, σlbandσtends to zero. The maximum depth of the surface layer, which determines the σlbis equal to the length of edge of cube ( llb), that approximates the shape of primary librational eﬀectons. It decreases from about 20 ˚A at 00Ctill about 2.5 ˚A at 1000C(see Fig. 4b). Monotonic decrease of ( llb)with temperature could be accompanied by nonmonotonic change of probabilities of [lb/tr] convertons and macroconvertons excitations (see c omments to Fig 4a ). Consequently, the temperature dependence of surface tensi on on temperature can display anomalies at deﬁnite temperatures. This conseq uence of our theory is conﬁrmed experimentally (Adamson, 1982; Drost-Hansen a nd Lin Singleton, 1992). The thickness of layer ( ltr), responsible for contribution of translational eﬀectons in surface tension ( σtr) has the dimension of one molecule in all tem- perature interval for liquid water. The results of computer calculations of σ(eq.6.31) for water and experimen- tal data are presented at Fig.16. 70Fig. 16. Experimental ( ) and theoretical (-- -) temperature dependences of the surface tension for water, calculated fr om eq.(6.31). It is obvious, that the correspondence between theory and ex periment is very good, conﬁrming in such a way the correctness of our mode l and Hierarchic concept in general. 6.5. Mesoscopic theory of thermal conductivity Thermal conductivity may be related to phonons, photons, fr ee electrons, holes and [electron-hole] pairs movement. We will discuss here only the main type of thermal conductivi ty in condensed matter, related to phonons. The analogy with the known formula for thermal conductivity (κ) in the framework of the kinetic theory for gas is used: κ=1 3CvvsΛ (6.34) where C vis the heat capacity of condensed matter, vsis sound velocity, charac- terizing the speed of phonon propagation in matter, and Λ is t he average length of free run of phonons. The value of Λ depends on the scattering and dissipation of ph onons at other phonons and diﬀerent types of defects. Usually decrea sing temperature increases Λ. Diﬀerent factors inﬂuencing a thermal equilibrium in the sy stem of phonons are discussed. Among them are the so called U- and N- processe s describing the types of phonon-phonon interaction. However, the tradi tional theories are unable to calculate Λ directly. Mesoscopic theory introduce two contributions to thermal c onductivity: re- lated to phonons, irradiated by secondary eﬀectons and form ingsecondary 71translational and librational deformons ( κsd)tr,lband to phonons, irradiated by aandbconvertons [ tr/lb], forming the convertons-induced deformons ( κcd)ac.bc: κ= (κsd)tr,lb+ (κcd)ac.bc=1 3Cvvs[(Λsd)tr,lb+ (Λ cd)ac,bc] (6.35) where: free runs of secondary phonons (tr and lb) are represented as: 1/(Λsd)tr,lb= 1/(Λtr) + 1/(Λlb) = (νd)tr/vs+ (νd)lb/vs consequently: 1/(Λsd)tr,lb=vs (νd)tr+ (νd)lb(6.36) and free runs of convertons-induced phonons: 1/(Λcd)ac,bc= 1/(Λac) + 1/(Λbc) = (νac)/vs+ (νbc)/vs consequently: (Λ sd)tr,lb=vs (νd)tr+ (νd)lb(6.37) The heat capacity: CV=∂Utot/∂Tcan be calculated also from our theory (see Chapter 4 and 5). Fig. 17. Temperature dependences of total thermal conductiv- ity for water and contributions, related to acoustic deform ons and [lb/tr]convertons. The dependences were calculated, using eq. (3 7). Quantitative calculations show that formula (6.35), based on our mesoscopic model, works well for water (Fig. 17). It could be used for any other condensed matter also if positions of translational and librational b ands, sound velocity and molar volume for this matter at the same temperature inte rval are known. The small diﬀerence between experimental and theoretical d ata can reﬂect the contributions of non-phonon process in thermal conduct ivity, related to macrodeformons, superdeformons and macroconvertons, i.e . big ﬂuctuations. 726.6. Mesoscopic theory of viscosity for liquids and solids The viscosity is determined by the energy dissipation as a result of medium (liquid or solid) structure deformation. Viscosity corres ponding to the shift deformation is named shear viscosity . So- called bulk viscosity is related to deformation of volume parameters and corresponding dissip ation. These types of viscosity have not the same values and nature. The statistical theory of irreversible process leads to the following expression for shear viscosity (Prokhorov, 1988): η=nkTτ p+ (µ∞−nkT)τq (6.38) where [ n] is the concentration of particles, µ∞is the modulus of instant shift characterizing the instant elastic reaction of medium, τpandτqare the relaxation times of impulses and coordinates, respectively. However, eq.(38) is inconvenient for practical purposes du e to diﬃculties in determination of τp, τqandµ∞. Sometimes in a narrow temperature interval the empirical On drade equation is working: η=A(T)·exp(β/T) (6.39) A(T) is a function poorly dependent on temperature. A good results in study the microviscosity problem were obta ined by com- bining the model of molecular rotational relaxation and the Kramers equation (˚Akesson et al., 1991). However, the using of the ﬁt parameter s was necessarily in this case also. We present here our mesoscopic theory of viscosity. To this end the dissipation processes, related to ( A⇋B)tr.lbcycles of translational and li- brational macroeﬀectons and (a,b)- convertons excitations were used. The same approach was employed for elaboration of mesoscopic theory of diﬀusion in con- densed matter (see next section). In contrast to liquid state, the viscosity of solids is determined by the biggest ﬂuctuations: supereﬀectons andsuperdeformons , resulting from simultane- ous excitations of translational and librational macroeﬀe ctons and macrodefor- mons in the same volume. The dissipation phenomena and ability of particles or molec ules to diﬀusion are related to the local ﬂuctuations of the free volume (∆ vf)tr,lb. According to mesoscopic theory, the ﬂuctuations of free volume and that o f density occur in the almost macroscopic volumes of translational and librat ional macrodeformons and in mesoscopic volumes of macroconvertons , equal to volume of primary librational eﬀecton at the given conditions. Translationa l and librational types of macroeﬀectons determine two types of viscosity, i.e. tra nslational ( ηtr) and librational ( ηlb) ones. They can be attributed to the bulk viscosity. The con- tribution to viscosity, determined by (a and b)- convertons is much more local and may be responsible for microviscosity and mesoviscosit y. Let us start from calculation of the additional free volumes (∆vf) originat- ing from ﬂuctuations of density, accompanied the translati onal and librational macrodeformons (macrotransitons). 73For 1 mole of condensed matter the following ratio between fr ee volume and concentration ﬂuctuations is true: /parenleftbigg∆vf vf/parenrightbigg tr,lb=/parenleftbigg∆N0 N0/parenrightbigg tr,lb(6.40) where N0is the average number of molecules in 1 mole of matter and (∆ N0)tr,lb=N0/parenleftbiggPM D Z/parenrightbigg tr,lb(6.41) is the number of molecules changing their concentration as a result of transla- tional and librational macrodeformons excitation. The probability of translational and librational macroeﬀe ctons excitation (see eqs. 3.23; 3.24): /parenleftbiggPM D Z/parenrightbigg tr,lb=1 Zexp/parenleftbigg −ǫM D kT/parenrightbigg tr,lb(6.42) where Zis the total partition function of the system. Putting (6.41) to (6.40) and dividing to Avogadro number ( N0), we obtain the ﬂuctuating free volume, reduced to 1 molecule of matter: ∆v0 f=∆vf N0=/bracketleftbiggvf N0/parenleftbiggPM D Z/parenrightbigg/bracketrightbigg tr,lb(6.43) It has been shown above (eq.6.19) that the average value of fr ee volume in 1 mole of matter is: vf=V0/n2 Consequently, for reduced ﬂuctuating (additional) volume we have: (∆v0 f)tr,lb=V0 N0n21 Zexp/parenleftbigg −ǫM D kT/parenrightbigg tr,lb(6.44) Taking into account the dimensions of viscosity and its phys ical sense, it should be proportional to the work (activation energy) of ﬂu ctuation-dissipation, necessary for creating the unit of additional free volume: ( EM D/∆v0 f), and the period of ( A⇋B)tr.lbcycles of translational and librational macroeﬀectons τA⇋B,determined by the life-times of all intermediate states (eq .46). In turn, the energy of dissipation should be strongly depend ent on the structural factor (S): the ratio of kinetic energy of m atter to its total internal energy. We postulate here that this depen dence for viscosity is cubical: (Tk/Utot)3=S3. Consequently, the contributions of translational and libr ational macrodefor- mons to resulting viscosity we present in the following way: 74ηM tr,lb=/bracketleftBigg EM D ∆v0 fτM/parenleftbiggTk Utot/parenrightbigg3/bracketrightBigg tr,lb(6.45) where: reduced ﬂuctuating volume (∆ v0 f) corresponds to (44); the energy of macrodeformons: [ EM D=−kT(lnPM D)]tr,lb. The cycle-periods of the trandlibmacroeﬀectons has been introduced as: /bracketleftbig τM=τA+τB+τD/bracketrightbig tr,lb(6.46) where: characteristic life-times of macroeﬀectons in A, B- states and that of transition state in the volume of primary electromagnetic d eformons can be presented, correspondingly, as follows: /bracketleftBig τA= (τaτa)1/2/bracketrightBig tr,lband/bracketleftBig τA= (τaτa)1/2/bracketrightBig tr,lb(6.47) /bracketleftBig τD=\|(1/τA)−(1/τB)\|−1/bracketrightBig tr,lb Using (6.47, 6.46 and 6.44) it is possible to calculate the co ntributions of (A⇋B) cycles of translational and librational macroeﬀectons to viscosity sep- arately, using (6.45). The averaged contribution of Macroexcitations (tr and lb) i n viscosity is: ηM=/bracketleftbig(η)M tr(η)M lb/bracketrightbig1/2(6.48) The contribution of aandb convertons to viscosity of liquids could be pre- sented in a similar to (6.44-6.48) manner after substitutin g the parameters of tr and lb macroeﬀectons with parameters of a and b convertons: ηac,bc=/bracketleftBigg Ec ∆v0 fτc/parenleftbiggTk Utot/parenrightbigg3/bracketrightBigg ac,bc(6.49) where: reduced ﬂuctuating volume of ( aandb) convertons (∆ v0 f)ac,bccor- responds to: (∆v0 f)ac,bc=V0 N0n21 ZPac,bc (6.50) where: PacandPbcare the relative probabilities of tr/libinterconversions between aandbstates of translational and librational primary eﬀectons ( see Introduction); EacandEbcare the excitation energies of ( aandb) convertons correspondingly (see section 4 ); 75Characteristic life-times for ac-convertons and bc-convertons [ tr/lb] in the volume of primary librational eﬀectons (”ﬂickering cluste rs”) could be presented as: τac= (τa)tr+ (τa)lb= (1/νa)tr+ (1/νa)lb τbc= (τb)tr+ (τb)lb= (1/νb)tr+ (1/νb)lb(6.51) The averaged contribution of the both types of convertons in viscosity is: ηc= (ηacηbc)1/2(6.52) This contribution could be responsible for microviscosity or better term: meso- viscosity , related to volumes, equal to that of primary librational eﬀ ectons. The resulting viscosity (Fig.18) is a sum of the averaged con tributions of macrodeformons and convertons: η=ηM+ηc (6.53) Fig. 18. Theoretical and experimental temperature dependences of viscosities for water. Computer calculations were perfo rmed using eqs. (6.44 - 6.53) and (4.3; 4.36). The best correlation between theoretical and experimental data was achieved after assuming that only ( π/2 = 2 π/4) part of the period of above described ﬂuctuation cycles is important for dissipation and viscosi ty. Introducing this 76factor to equations for viscosity calculations gives up ver y good correspondence between theory and experiment in all temperature interval ( 0-1000C) for water (Fig.18). As will be shown below the same factor, introducing the eﬀect ive time of ﬂuctuations [τ π/2], leads to best results for self-diﬀusion coeﬃcient calcul ation. In the classical hydrodynamic theory the sound absorption c oeﬃcient ( α) obtained by Stokes includes share ( η) and bulk ( ηb) averaged microviscosity: α=Ω 2ρv3s/parenleftbigg4 3η+ηb/parenrightbigg , (6.54) where Ω is the angular frequency of sound waves; ρis the density of liquid. Bulk viscosity ( ηb) is usually calculated from the experimental ηandα. It is known that for water: (ηb/η)∼3. The viscosity of solids In accordance with our model, the biggest ﬂuctuations: supereﬀectons and superdeformons (see Introduction) are responsible for viscosity and diﬀus ion phenomena in solid state. Superdeformons are accompanied b y the emergency of cavitational ﬂuctuations in liquids and the defects in so lids. The presentation of viscosity formula in solids ( ηs) is similar to that for liquids: ηS=ES (∆v0 f)SτS/bracketleftbiggTk Utot/bracketrightbigg3 (6.55) where: reduced ﬂuctuating volume, related to superdeformo ns excitation (∆v0 f)sis: (∆v0 f)S=V0 N0n21 ZPS (6.56) where: Ps= (PM D)tr(PM D)lbis the relative probability of superdeformons, equal to product of probabilities of tr and lb macrodeformons excitation (see 42);Es=−kTlnPsis the energy of superdeformons (see Chapter 4); Characteristic cycle-period of ( A∗⇋B∗) transition of supereﬀectons is re- lated to its life-times in A∗,B∗and transition D∗states (see eq.6.46) as was shown τS=τA∗+τB∗+τD∗ (6.56a) The viscosity of ice, calculated from eq.(6.55) is bigger th an that of water (eq.6.53) to about 105times. This result is in accordance with available ex- perimental data. 776.7. Brownian diﬀusion The important formula obtained by Einstein in his theory of B rownian mo- tion is for translational motion of particle: r2= 6Dt=kT πηat (6.57) and that for rotational Brownian motion: ϕ2=kT 4πηa3t (6.58) where: a- radius of spherical particle, much larger than dimension o f molecules of liquid. The coeﬃcient of diﬀusion [D] for Brownian motion is equal to: D=kT 6πηa(6.59) If we take the angle ¯ ϕ2= 1/3 in (6.59), then the corresponding rotational correlation time comes to the form of the known Stokes- Einst ein equation: τ=4 3πa31 k/parenleftBigη T/parenrightBig (6.60) All these formulas (6.57 - 6.60) include macroscopic share v iscosity ( η) corre- sponding to our (6.53). In terms of our model, the Brownian mo vement is a consequence of macrodeformons and convertons. Putting our formula (6.53) for viscosity of liquid into (6.57 - 6.59), we get the possibi lity of quantitative analysis of corresponding parameters, using our computer p rogram. 6.8. Self-diﬀusion in liquids and solids Molecular theory of self-diﬀusion, as well as general conce pt oftransfer phe- nomena in condensed matter is extremely important, but still unres olved prob- lem. Simple semi-empirical approach developed by Frenkel leads to following ex- pression for diﬀusion coeﬃcient in liquid and solid: D=a2 τ0exp(−W/kT ) (6.61) where [a] is the distance of ﬂuctuation jump; τ0∼(10−12÷10−13)sis the average period of molecule oscillations between jumps; W - a ctivation energy of jump. The parameters: a,τ0andWshould be considered as a ﬁt parameters. In accordance with mesoscopic theory , the process of self-diﬀusion in liquids,like that of viscosity , described above, is determined by two contribu- tions: 78a) the collective, nonlocal contribution , related to translational and librational macrodeformons ( Dtr,lb); b) the local contribution, related to coherent clusters ﬂickering: [dissocia- tion/association] of primary librational eﬀectons ( aandb)- convertons ( Dac,bc). Each component of the resulting coeﬃcient of self-diﬀusion (D) in liquid could be presented as the ratio of ﬂuctuation volume cross-s ection surface: [∆v0 f]2/3to the period of macroﬂuctuation ( τ). The ﬁrst contribution to co- eﬃcient D,produced by translational and librational macrodeformons is: Dtr,lb=/bracketleftbigg/parenleftbig ∆v0 f/parenrightbig2/31 τM/bracketrightbigg tr,lb(6.62) where: the surface cross-sections of reduced ﬂuctuating fr ee volumes (see eq.43) ﬂuctuations in composition of macrodeformons ( tr and lb) are: (∆v0 f)2/3 tr,lb=/bracketleftBigg V0 N0n21 Zexp/parenleftbigg −ǫM D kT/parenrightbigg tr,lb/bracketrightBigg2/3 (6.63) (τM)tr,lbare the characteristic ( A⇔B) cycle-periods of translational and li- brational macroeﬀectons (see eqs. 6.46 and 6.47). The averaged component of self-diﬀusion coeﬃcient, which t akes into ac- count both types of nonlocal ﬂuctuations, related to transl ational and librational macroeﬀectons and macrodeformons, can be ﬁnd as: DM= [(D)M tr(D)M lb]1/2(6.64) The formulae for the second, local contribution to self-diﬀ usion in liquids, related to ( aandb) convertons ( Dac,bc) are symmetrical by form to that, presented above for nonlocal processes: Dac,bc=/bracketleftbigg (∆v0 f)2/31 τS/bracketrightbigg ac,bc(6.65) where: reduced ﬂuctuating free volume of ( aandb) convertons (∆ v0 f)ac,bc is the same as was used above in mesoscopic theory of viscosit y (eq.6.50): (∆v0 f)ac,bc=V0 N0n21 ZPac,bc (6.66) where: PacandPbcare the relative probabilities of tr/libinterconversions between aandbstates of translational and librational primary eﬀectons ( see Introduction and section 4) The averaged local component of self-diﬀusion coeﬃcient, w hich takes into account both types of convertons ( acandbc) is: DC= [(D)ac(D)bc]1/2(6.67) 79In similar way we should take into account the contribution o f macroconver- tons ( DMc): DMc=/parenleftbiggV0 N0n21 ZPMc/parenrightbigg2/31 τMc(6.67a) where: PMc=Pac·Pbcis a probability of macroconvertons excitation; the life-time of macroconvertons is: τMc= (τacτbc)1/2(6.67b) The cycle-period of ( ac) and ( bc) convertons are determined by the sum of life-times of intermediate states of primary translationa l and librational eﬀec- tons: τac= (τa)tr+ (τa)lb; and τbc= (τb)tr+ (τb)lb (6.67c) The life-times of primary and secondary eﬀectons (lb and tr) ina- and b- states are the reciprocal values of corresponding state fre quencies: [τa= 1/νa;τa= 1/νa; and τb= 1/νb;τb= 1/νb]tr,lb (6.67d) [νaandνb]tr,lbcorrespond to eqs. 4.8 and 4.9; [ νaandνb]tr,lbcould be calculated using eqs.2.54 and 2.55. The resulting coeﬃcient of self-diﬀusion in liquids (D) is a sum of nonlocal (DM) and local ( Dc, DMc) eﬀects contributions (see eqs.6.64 and 6.67): D=DM+Dc+DMc (6.68) The eﬀective ﬂuctuation-times were taken the same as in prev ious section for viscosity calculation, using the correction factor [( π/2)τ]. 80Fig. 19. Theoretical and experimental temperature dependences of self-diﬀusion coeﬃcients in water. Theoretical coeﬃcie nt was cal- culated using eq. 6.68. Like in the cases of thermal conductivity, viscosity and vap or pressure, the results of theoretical calculations of self-diﬀusion coeﬃ cient coincide well with experimental data for water (Fig. 19) in temperature interv al (0−1000C). The self-diﬀusion in solids In solid state only the biggest ﬂuctuations: superdeformons, representing simultaneous excitation of translational and librational macrodeformons in the same volumes of matter are responsible for diﬀusion and the v iscosity phenom- ena. They are related to origination and migration of the def ects in solids. The formal presentation of superdeformons contribution to sel f-diﬀusion in solids (Ds) is similar to that of macrodeformons for liquids: DS= (∆v0 f)2/3 S1 τS(6.69) where: reduced ﬂuctuating free volume in composition of sup erdeformons (∆v0 f)Sis the same as was used above in mesoscopic theory of viscosit y (eq.6.56): (∆v0 f)S=V0 N0n21 ZPS (6.70) where: PS= (PM D)tr(PM D)lbis the relative probability of superdeformons, equal to product of probabilities of tr and lb macrodeformons excitation (see 6.42). Characteristic cycle-period of supereﬀectons is related t o that of tr and lb macroeﬀectons like it was presented in eq.(6.56a): τs=τA∗+τB∗+τD∗ (6.71) The self-diﬀusion coeﬃcient for ice, calculated from eq.6. 69 is less than that of water (eq.6.53) to about 105times. This result is in accordance with available experimental data. Strong decreasing of D in a course of phase transition: [wate r→ice] pre- dicted by our mesoscopic theory also is in accordance with ex periment (Fig. 20). 81Fig. 20. Theoretical temperature dependences of self- diﬀusion coeﬃcients in ice. All these results allow to consider our hierarchic theory of transfer phe- nomena as a quantitatively conﬁrmed one. They point that the ”mesoscopic bridge” between Micro- and Macro Worlds is wide and reliable indeed. It gives a new possibilities for understanding and detailed descrip tion of very diﬀerent phenomena in solids and liquids. One of the advantages of our theory of viscosity and diﬀusion is the possibility of explaining numerous nonmonotonic tem perature changes, registered by a number of physicochemical methods in var- ious aqueous systems during the study of temperature depend ences (Drost-Hansen, 1976; 1992; Johri and Roberts, 1990; Aksnes , Asaad, 1989; Aksnes, Libnau, 1991; Kaivarainen, 1985; Kaivaraine n et al., 1993). Lot of them are related to diﬀusion or viscosity processes an d may be ex- plained by nonmonotonic changes of the refraction index, in cluded in our equa- tions: (6.44, 6.45, 6.50) for viscosity and eqs. (6.69, 6.70 ) for self-diﬀusion. For water these temperature anomalies of refraction index w ere revealed experi- mentally, using few wave lengths in the temperature interva l 3−950(Frontasev, Schreiber, 1966). They are close to Drost-Hansen temperatu res. The explana- tion of these eﬀects, related to periodic variation of prima ry librational eﬀectons stability with monotonic temperature change was presented as a comments to Fig.4a. Another consequence of our theory is the elucidation of a big diﬀerence be- tween librational ηlb(6.48), translational ηtr(6.45) viscosities and mesoviscosity, determined by [ lb/tr] convertons (6.49 and 6.52). The eﬀect of mesoviscosity can be checked as long as the volum e of a Brow- nian particle does not exceed much the volume of primary libr ational eﬀectons 82(eq. 6.15). If we take a Brownian particle, much bigger than t he librational pri- mary eﬀecton, then its motion will reﬂect only averaged shar e viscosity (eq.6.53). The third consequence of the mesoscopic theory of viscosity is the prediction of nonmonotonic temperature behavior of the sound absorpti on coeﬃcient α (6.51). Its temperature dependence must have anomalies in t he same regions, where the refraction index has. The experimentally revealed temperature anomalies of (n) a lso follow from our theory as a result of nonmonotonic ( a⇔b)lbequilibrium behavior, stabil- ity of primary lb eﬀectons and probability of [lb/tr] conver tons excitation (see Discussion to Fig.4a ). Our model predicts also that in the course of transition from the laminar type of ﬂow to the turbulent one the share viscosity ( η) will increases due to increasing of structural factor ( Tk/Utot) in eq. 6.45. The superﬂuidity ( η→0) in the liquid helium could be a result of inabil- ity of this liquid at the very low temperature for translatio nal and librational macroeﬀectons excitations, i.e. τM→0. In turn, it is a consequence of tending to zero the life-times of secondary eﬀectons and deformons in eq.(6.45), responsible for dissi pation processes, due to their Bose-condensation and transformation to primary o nes (Kaivarainen 1998). The polyeﬀectons, stabilized by Josephson’s juncti ons between primary eﬀectons form the superﬂuid component of liquid helium. 7. Osmose and solvent activity. Traditional and mesoscopic approach It was shown by Van’t Hoﬀ in 1887 that osmotic pressure (Π) in t he dilute concentration of solute (c) follows a simple expression: Π = RTc (7.1) This formula can be obtained from an equilibrium condition between a solvent and an ideal solution after saturation of diﬀusion process o f the solvent through a semipermeable membrane: µ0 1(P) =µ1(P+ Π, Xi) where µ0 1andµ1are the chemical potentials of a pure solvent and a solvent in solution; P- external pressure; Π - osmotic pressure; X1is the solvent fraction in solution. At equilibrium dµ0 1=dµ1= 0 and dµ1=/bracketleftbigg∂µ1 ∂P1/bracketrightbigg X1dP1+/bracketleftbigg∂µ1 ∂X1/bracketrightbigg P1dX1= 0 (7.1a) Because µ1=/parenleftbig ∂G/∂n 1/parenrightbig P,T=µ0 1+RTlnX1 (7.2) 83then /parenleftbigg∂µ1 ∂P1/parenrightbigg X1=/parenleftbigg∂2G ∂P∂n 1/parenrightbigg P,T,X=/parenleftbigg∂V ∂n1/parenrightbigg =V1 (7.3) where V1is the partial molar volume of the solvent. For dilute soluti on:¯V1≃ V0 1(molar volume of pure solvent). From (7.2) we have: ∂µ1 ∂X1=RT/parenleftbigg∂lnX1 ∂X1/parenrightbigg P,T(7.4) Putting (7.3) and (7.4) into (7.1a) we obtain: dP1=−RT V0 1X1dX1 Integration: p+π/integraldisplay PdP1=−RT V0 1x1/integraldisplay 1dlnX1 (7.5) gives: Π =−RT V0 1lnX1=−RT V0 1ln(1−X2) (7.6) and for the dilute solution ( X2≪1) we ﬁnally obtain Van’t Hoﬀ equation: Π =RT V0 1X2∼=RTn2/n1 V0 1= RTc (7.7) where X2=n2/(n1+n2)∼=n2/n1 (7.8) and n2/n1 V0 1=c (7.9) Considering a real solution, we only substitute solvent fra ction X1in (7.6) by solvent activity: X1→a1. Then taking into account (7.2), we can express osmotic pressure as follows: 84Π =−RT ¯V1lna1=∆µ1 ¯V1(7.10) where: ∆ µ1=µ0 1-µ1is the diﬀerence between the chemical potentials of a pure solvent and the one perturbed by solute at the starting m oment of osmotic process, i.e. the driving force of osmose; ¯V1∼=V1is the molar volume of solvent at dilute solutions. Although the osmotic eﬀects are widespread in Nature and are very impor- tant, especially in biology, the physical mechanism of osmo se remains unclear (Watterson, 1992). The explanation following from Van’t Hoﬀ equation (7.7) and pointing that osmotic pressure is equal to that induced by solute molecule s, if they are consid- ered as an ideal gas in the same volume at a given temperature i s not satisfactory. The osmoses phenomenon can be explained quantitatively on t he basis of our mesoscopic theory and state equation (see 6.6 an d 6.7). To this end, we have to introduce the rules of conservation of the main internal parameters of solvent in the presence of guest (solute) molecules or particles: 1. Internal pressure of solvent: Pin= const 2. The total energy of solvent: Utot= const/bracerightbigg (7.11) This conservation rules can be considered as the consequenc e of Le Chatelier principle. Using (6.6), we have for the pure solvent and the solvent pert urbed by a solute the following two equations, respectively: Pin=Utot V0 fr/parenleftbigg 1 +V Tk/parenrightbigg −Pext (7.12) P1 in=U1 tot V1 fr/parenleftbigg 1 +V1 T1 k/parenrightbigg −P1 ext, (7.13) where: V0 fr=V0 n2and V1 fr=V0 n2 1(7.14) are the free volumes of pure solvent and solvent in presence o f solute (guest) molecules as a ratio of molar volume of solvent to correspond ent value of refrac- tion index. The equilibrium conditions after osmotic process saturation , leading from our conservation rules (7.11) are Pin=Pinwhen Pext=Pext+ Π (7.15) Utot=V+Tk=V1+T1 k=U1 tot (7.16) 85From (7.16) we have: Dif = Tk−T1 k=V1−V (7.17) The index(1)denote perturbed solvent parameters. Comparing (7.12) and (7.13) and taking into account (7.14 - 7 .16), we obtain a new formula for osmotic pressure: Π =n2 V0Utot/bracketleftbiggn2 1Tk−n2T1 k TkT1 k/bracketrightbigg (7.18) where: n, V0, UtotandTkare the refraction index, molar volume, total energy and total kinetic energy of a pure solvent, respectiv ely;T∗ kandn1are the total kinetic energy and refraction index of the solvent in the presence of guest (solute) molecules; TkandT∗ kcan be calculated from our theory (eq.4.36). For the case of dilute solutions, when TkT1 k∼=T2 kandn∼=n1,the eq.(7.18) can be simpliﬁed: Π =n2 V0/parenleftbiggUtot Tk/parenrightbigg2/parenleftbig Tk−T1 k/parenrightbig (7.19) or using (7.17): Π =n2 V0/parenleftbiggUtot Tk/parenrightbigg2 (V1−V) (7.20) The ratio: S=Tk/Utot (7.21) is generally known as a structural factor. We can see from (7.19) and (7.20) that osmotic pressure is pro portional to the diﬀerence between total kinetic energy of a free solvent (Tk) and that of the solvent perturbed by guest molecules: ∆Tk=Tk−T1 k or related diﬀerence between the total potential energy of p erturbed and pure solvent: ∆V=V1−Vwhere: ∆ Tk= ∆V≡Dif (see Fig. 21). As far ∆T k>0 and ∆ V >0, it means that: Tk> T1 k or V1> V(7.22) 86Theoretical temperature dependence of the diﬀerence Dif= ∆Tk= ∆V calculated from (7.19) or (7.20) at constant osmotic pressu re: Π ≡Pos= 8 atm., pertinent to blood is presented on Fig. 21. The next Fig. 22 illustrate theoretical temperature depend ence of osmotic pressure (7.20) in blood at the constant value of Dif= 6.7·10−3(J/M), corre- sponding on Fig. 21 to physiological temperature (370). The ratios of this Difvalue to total potential (V) and total kinetic energy (Tk) of pure water at 370(see Fig. 21) are equal to: (Dif/V)≃6.7·10−3 1.3·104∼=5·10−7and (Dif/Tk)≃6.7·10−3 3.5·102∼=2·10−5 i.e. the relative changes of the solvent potential and kinet ic energies are very small. Fig. 21. Theoretical temperature dependence of the diﬀerence: Dif=V1−V=Tk−T1 kat constant osmotic pressure: Π ≡Pos= 8 atm., characteristic for blood. The computer calculation s were performed using eqs. (7.19) or (7.20). For each type of concentrated macromolecular solutions the optimum amount of water is needed to minimize the potential energy of the sys temdetermined mainly by clusterphilic interactions. The conservation ru les (7.11) and self- organization in solutions of macromolecules (clustron for mation) may be respon- sible for the driving force of osmose in the diﬀerent compartments of biological cells. Comparing (7.20) and (7.10) and assuming equality of the mol ar volumes V0=¯V1, we ﬁnd a relation between the diﬀerence in potential energi es and chemical potentials (∆ µ) of unperturbed solvent and that perturbed by the solute: 87∆µ=µ0 1−µ1=n2/parenleftbiggUtot Tk/parenrightbigg2 (V1−V) (7.23) Fig. 22. Theoretical temperature dependence of osmotic pres- sure (eq. 43) in blood at constant value of diﬀerence: Dif = ∆ T= ∆V= 6.7·10−3J/M. This value in accordance with Fig.3 corre- sponds to physiological temperature (370). The results obtained above mean that solvent activity (a1)and a lot of other thermodynamic parameters for solutions can be calc ulated on the basis of our hierarchic concept: a1= exp/parenleftbigg −∆µ RT/parenrightbigg = exp/bracketleftbigg −/parenleftBign S/parenrightBig2V1−V RT/bracketrightbigg (7.24) where: S=Tk/Utotis a structural factor for the solvent. The molar coeﬃcient of activity is: yi=ai/ci, (7.25) where ci=ni/V (7.26) is the molar quantity of i-component ( ni) in of solution (V - solution volume in liters). The molar activity of the solvent in solution is related to it s vapor pressure (Pi) as: ai=Pi/P0 i (7.27) where: P0 iis the vapor pressure of the pure solvent. Theoretical tempe rature dependence of water activity ( a1) in blood at constant diﬀerence: Dif = ∆ T= ∆V= 6.7·10−3J/M is presented on Fig. 23. 88Fig. 23. Theoretical temperature dependence of water activity (a1) (eq.7.24) in blood at constant diﬀerence: Dif= ∆T= ∆V= 6.7·10−3J/M. Another colligative parameter such as low temperature shift of freezing tem- perature of the solvent (∆ Tf) in the presence of guest molecules also can be calculated from (7.24) and the known relation (7.28a) betwe en water activity in solution and (∆ Tf): ∆Tf=−R(T0 f)2 ∆Hlna1(T0 f)2 ∆H T/parenleftBign S/parenrightBig2 (V1−V) (7.28) where: T0 fis the freezing temperature of the pure solvent; T is the temp er- ature at conditions of calculations of potential energies V1(T) and V(T) from eqs. 4.36 and 4.39; lna1=−[△H/R(T0 f)2]△Tf (7.28a) The partial molar enthalpy ( ¯H1) of solvent in solution are related to solvent activity like: H1=H0 1−RT2∂lna1 ∂T=H0 1+L0 1 (7.29) where H0 1is the partial enthalpy of the solvent at inﬁnitive dilution ; ¯L1=−RT2∂lna1 ∂T=T2∂ ∂T/bracketleftbigg/parenleftBign S/parenrightBig2V1−V T/bracketrightbigg (7.30) is the relative partial molar enthalpy of solvent in a given solution. From (7.29) we obtain partial molar heat capacity as: C1 p=∂ ∂T(H1) =C0 p−R/parenleftbigg T2∂2lna1 ∂T2+ 2T∂lna1 ∂T/parenrightbigg (7.31) 89An analogous equation exists for the solute of this solution as well as for partial molar volume and other important parameters of the solvent, including solvent activity (Godnev et al., 1982). It is obvious, the application of Hierarchic theory to solve nt activ- ity evaluation might be of practical importance for diﬀeren t processes in chemical and colloid technology. 8. New approach to theory of light refraction 8.1. Refraction in gas If the action of photons onto electrons of molecules is consi dered as a force, activating a harmonic oscillator with decay, it leads to the known classical equa- tions for a complex refraction index (Vuks, 1984). The Lorentz-Lorenz formula obtained in such a way is conveni ent for prac- tical needs. However, it does not describe the dependence of refraction index on the incident light frequency and did not take into account the intermolecular interactions. In the new theory proposed below we have tried to clear up the relationship between these parameters. Our basic idea is that the dielectric penetrability of matte rǫ, (equal in the optical interval of frequencies to the refraction index squ aredn2), is determined by the ratio of partial volume energies of photon in vacuum to similar volume energy of photon in matter: ǫ=n2=[E0 p] [Emp]=mpc2 mpc2m=c2 c2m(8.1) where mp=hνp/c2is the eﬀective photon mass, cis the light velocity in vacuum, cmis the eﬀective light velocity in matter. We introduce the notion of partial volume energy of a photon in vacuum [E0 p]and in matter [Em p]as a product of photon energy (Ep=hνp)and the volume (Vp)occupied by 3D standing wave of photon in vacuum and in matter, correspondingly: [E0 p] =EpV0 p [Em p] =EpVm p (8.2) The 3D standing photon volume as an interception volume of 3 d iﬀerent standing photons normal to each other was termed in our mesos copic model as a primary electromagnetic deformon (see Introduction). In vacuum, where the eﬀect of an excluded volume due to the spa tial incom- patibility of electron shells of molecules and photon is abs ent, the volume of 3 D photon standing wave (primary deformon) is: V0 p=1 np=3λ2 p 8π(8.3) 90We will consider the interaction of light with matter in this mesoscopic volume, containing a thousands of molecules of condensed matter. It is the reason why we titled this theory of light refraction as mesoscopic one. Putting (8.3) into (8.2), we obtain the formula for the parti al volume energy of a photon in vacuum: [E0 p] =EpV0 p=hνp9λ2 p 8π=9 4/planckover2pi1cλ2 p (8.4) Then we proceed from the assumption that waves B of photons ca n not exist with waves B of electrons, forming the shells of atoms a nd molecules in the same space elements. Hence, the eﬀect of excluded volume appears during the propagation of an external electromagnetic wave throug h the matter. It leads to the fact that in matter the volume occupied by a photo n, is equal to Vm p=V0 p−Vex p=V0 p−np M·VM e (8.5) where Vex p=np MVM eis the excluded volume which is equal to the product of the number of molecules in the volume of one photon standing w ave (np M) and the volume occupied by the electron shell of one molecule ( VM e). np Mis determined by the product of the volume of the photons 3D st anding wave in the vacuum (8.3) and the concentration of molecules ( nM=N0/V0): np M=9λ3 p 8π/parenleftbiggN0 V0/parenrightbigg (8.6) In the absence of the polarization by the external ﬁeld and in termolecular in- teraction, the volume occupied by electrons of the molecule : VM e=4 3πL3 e (8.7) where Leis the radius of the most probable wave B(Le=λe/2π) of the outer electron of a molecule. As it has been shown in (7.5) that the m ean molecule polarizability is: α=L3 e (8.8) Then taking (8.7) and (8.6) into account, the excluded volum e of primary elec- tromagnetic deformon in the matter is: Vex p=9λ3 p 8πnM4 3πα=3 2λ3 pnMα (8.9) Therefore, the partial volume energy of a photon in the vacuu m is determined by eq.(8.4), while that in matter, according to (8.5): [Em p] =EpVm p=Ep[V0 p−Vex p] (8.10) 91Putting (8.4) and (8.10) into (8.1) we obtain: ǫ=n2=EpV0 p Ep(V0p−Vexp)(8.11) or 1 n2= 1−Vex p V0(8.12) Then, putting eq.(8.9) and (8.3) into (8.12) we derive new equation for refraction index, leading from our mesoscopic theory: 1 n2= 1−4 3πnMα (8.13) or in another form: n2−1 n2=4 3πnMα=4 3πN0 V0α (8.14) where: nM=N0/V0is a concentration of molecules; In this equation α=L3 eis the average static polarizability of molecules for the case when the external electromagnetic ﬁelds as well as intermolecular interactions inducing the additional polarization are abs ent. This situation is realized at Ep=hνp→0 and λp→ ∞ in the gas phase. As will be shown below the value of resulting α∗in condensed matter is bigger. 8.2. Light refraction in liquids and solids According to the Lorentz classical theory, the electric com ponent of the outer electromagnetic ﬁeld is ampliﬁed by the additional inner ﬁe ld (Ead), related to the interaction of induced dipole moments in composition of condensed matter with each other: Ead=n2−1 3E (8.15) The mean Lorentz acting ﬁeld ¯Fcan be expressed as: F=E+Ead=n2+ 2 3E(atn→1,F→E) (8.16) ¯F- has a dimensions of electric ﬁeld tension and tends to E in th e gas phase when n→1. 92In accordance with our model, beside the Lorentz acting ﬁeld , the total internal acting ﬁeld, includes also two another contributi ons, increasing the molecules polarizability ( α) in condensed matter: 1. Potential intermolecular ﬁeld, including all the types o f Van- der-Waals interactions in composition of coherent collective excita tions, even without ex- ternal electromagnetic ﬁeld. Like total potential energy o f matter, this contri- bution must be dependent on temperature and pressure; 2. Primary internal ﬁeld, related with primary electromagn etic deformons (tr and lb). This component of the total acting ﬁeld also exis t without external ﬁelds. Its frequencies corresponds to IR range and its actio n is much weaker than the action of the external visible light. Let us try to estimate the energy of the total acting ﬁeld and i ts eﬀective frequency ( νf) and wavelength ( λf), that we introduce as: Af=hνf=hc λf=AL+AV+AD (8.17) where: AL, AVandADare contributions, related with Lorentz ﬁeld, po- tential ﬁeld and primary deformons ﬁeld correspondingly. When the interaction energy of the molecule with a photon ( Ep=hνp) is less than the energy of the resonance absorption, then it l eads to elastic polarization of the electron shell and origination of secon dary photons, i.e. light scattering. We assume in our consideration that the increme nt of polarization of a molecule ( α) under the action of the external photon ( hνp) and the total active ﬁeld ( Af=hνf) can be expressed through the increase of the most probable radius of the electron’s shell ( Le=α1/3), using our (eq. 7.6 from [Kaivarainen, 1995, 2000]): ∆Le=ωpme 2/planckover2pi1α (8.18) where the resulting increment: ∆L∗= ∆Le+ ∆Lf=(hνp+Af)me 2/planckover2pi12α (8.18a) where: α=L3 eis the average polarizability of molecule in gas phase at νf→ 0. For water molecule in the gas phase: Le=α1/3= 1.13·10−10m is a known constant, determined experimentally [4]. The total increment of polarizability radius (∆ L∗) and resulting polarizabil- ity of molecules ( α∗) in composition of condensed matter aﬀected by the acting ﬁeld α∗= (L∗)3(8.18b) can be ﬁnd from the experimental refraction index (n) using o ur formula (8.14): 93L∗= (α∗)1/3=/bracketleftbigg3 4πV0 N0n2−1 n2/bracketrightbigg1/3 (8.19) ∆L∗=L∗−Le (8.20) from (8.18) we get a formula for the increment of radius of pol arizability (∆ Lf), induced by the total internal acting ﬁeld: ∆Lf= ∆L∗−∆Le=Afme 2/planckover2pi12α (8.21) Like total internal acting ﬁeld energy (8.17), this total ac ting increment can be presented as a sum of contributions, related to Lorent z ﬁeld (∆ LF), potential ﬁeld (∆ LV) and primary deformons ﬁeld (∆ LD): ∆Lf= ∆LL+ ∆LV+ ∆LD (8.22) Increment ∆ Le, induced by external photon only, can be calculated from the known frequency ( νp) of the incident light (see 8.18a): ∆Le=hνpme 2/planckover2pi12α (8.23) It means that ∆ Lfcan be found from (8.21) and (8.17), using (8.23). Then from (8.21) we can calculate the energy ( Af), eﬀective frequency ( νf) and wave length ( λf) of the total acting ﬁeld like: Af=hνf=hc/λ f= 2∆Lf/planckover2pi12 meα(8.24) The computer calculations of α∗;L∗=Le+ ∆L∗= (α∗)1/3andAfin the temperature range (0 −950) are presented on Fig.24. One must keep in mind that in general case αandLare tensors. It means that all the increments, calculated on the base of eq.(8.18a ) must be considered as the eﬀective ones. Nevertheless, it is obvious that our ap proach to analysis of the acting ﬁeld parameters can give useful additional inf ormation about the properties of transparent condensed matter. 94Fig. 24. (a)- Temperature dependencies of the most probable outer electron shell radius of H2O(L∗) and the eﬀective polarizabil- ityα∗= (L∗)3in the total acting ﬁeld (eq. 8.19); (b)- Temperature dependence of the total acting ﬁeld ( Af) en- ergy (8.24) in water at the wavelength of the incident light λp= 5.461·10−5cm−1. The experimental data for refraction index n(t) were used in calculations. The initial electron shell radiu s is:Le= α1/3 H2O= 1.13·10−8cm. In graphical calculations in Fig.24a, the used experimental temperature dependence of the water refr action index were obtained by Frontas’ev and Schreiber (1966). The temperature dependencies of these parameters were comp uted using the known experimental data on refraction index n(t) for water and presented in Fig.24a. The radius L∗in the range 0 −950Cincreases less than by 1% at constant incident light wavelength ( λ= 546 .1nm). The change of ∆ Lfwith temperature is determined by its potential ﬁeld component c hange ∆ LV. The relative change of this component: ∆∆ LV/∆Lf(t= 00C) is about 9%. Corresponding to this change the increasing of the actin g ﬁeld energy Af(eq.8.23) increases approximately by 8 kJ/M (Fig 8.1 b) due to its potential ﬁeld contribution. It is important that the total potential energy of water in th e same tempera- ture range, according to our calculations, increase by the s ame magnitude (Fig. 3b). This fact points to the strong correlation between pote ntial intermolecular interaction in matter and the value of the acting ﬁeld energy . It was calculated that, at constant temperature (200) the energy of the acting ﬁeld ( Af),(eq.8.23) in water practically does not depend on the wavele ngth of incident light ( λp). At more than three time alterations of λp: from 12 .56· 10−5cmto 3.03·10−5cmwhen the water refraction index ( n) changes from 1.320999 to 1.358100 (Kikoin, 1976), the value of Afchanges less than by 1%. At the same conditions the electron shell radius L∗and the acting polar- izability α∗thereby increase from (1.45 to 1.5) ·10−10m and from (3.05 to 3.274)·10−30m3respectively (Fig.25). These changes are due to the inciden t photons action only. For water molecules in the gas phase and λp→ ∞ the initial polarizability ( α=L3 e) is equal to 1 .44·10−24cm3(Eisenberg, Kauz- mann,1969), i.e. signiﬁcantly less than in condensed matte r under the action of external and internal ﬁelds. Obviously, the temperature change of energy Af(Fig.24b) is determined by the internal pressure increasing (section ..), related to i ntermolecular interaction change, depending on mean distances between molecules and, hence, on the concentration ( N0/V0) of molecules in condensed matter. 95Fig. 25. Dependencies of the acting polarizability α∗= (L∗)3 and electron shell radius of water in the acting ﬁeld ( L∗) on incident light wavelength ( λp), calculated from eq. (8.14) and experimental datan(λp) (Kikoin, 1976). The initial polarizability of H2Oin the gas phase at λp→ ∞ is equal to α=L3 e= 1.44·10−24cm3. The corresponding initial radius of the H2Oelectron shell is Le= 1.13· 10−8cm. On the basis of our data, changes of Af,calculated from (8.24) are caused mainly by the heat expansion of the matter. The photon induce d increment of the polarizability ( α→α∗) practically do not change Af. The ability to obtain new valuable information about change s of molecule po- larizability under the action of incident light and about te mperature dependent molecular interaction in condensed medium markedly reinfo rce such a widely used method as refractometry. The above deﬁned relationship between the molecule polariz ability and the wave length of the incident light allows to make a new endeavo r to solve the light scattering problems. 9. Mesoscopic theory of Brillouin light scattering in conde nsed matter 9.1. Traditional approach According to traditional concept, light scattering in liqu ids and crystals as well as in gases takes place due to random heat ﬂuctuations. I n condensed media the ﬂuctuations of density, temperature and molecule orien tation are possible. Density ( ρ) ﬂuctuations leading to dielectric penetrability ( ǫ) ﬂuctuations are of major importance. This contribution is estimated by m eans of Einstein formula for scattering coeﬃcient of liquids: R=Ir2 I0V=π 2λ4kTβ T/parenleftbigg ρ∂ǫ ∂ρ/parenrightbigg T(9.1) 96where βTis isothermal compressibility. Many authors made attempts to ﬁnd a correct expression for th e variable (ρ∂ǫ ∂ρ). The formula derived by Vuks (1977, 1984) is most consistent w ith experi- mental data: ρ∂ǫ ∂ρ= (n2−1)3n2 2n2−1(9.2) 9.2. Fine structure of scattering The ﬁne structure - spectrum of the scattering in liquids is r epresented by two Brillouin components with frequencies shifted relativ ely from the incident light frequency: ν±=ν0±∆νand one unshifted band like in gases ( ν0). The shift of the Brillouin components is caused by the Dopple r eﬀect result- ing from a fraction of photons scattering on phonons moving a t sound speed in two opposite directions. This shift can be explained in diﬀerent way as well (Vuks, 197 7). If in the antinodes of the standing wave the density oscillation occu rs at frequency (Ω): ρ=ρ0cosΩt, (9.3) then the scattered wave amplitude will change at the same fre quency. Such a wave can be represented as a superposition of two monochroma tic waves having the frequencies:( ω+ Ω) and ( ω−Ω), where Ω = 2 πf (9.4) is the elastic wave frequency at which scattering occurs whe n the Wolf-Bragg condition is satisﬁed: 2Λ sin ϕ= 2Λ sinθ 2=λ′(9.5) or Λ =λ′/(2 sinθ 2) =c nν(2 sinθ 2) =vph/f (9.6) where Λ is the elastic wave length corresponding to the frequ encyf;λ′= λ/n=c/nν(λ′andλare the incident light wavelength in matter and vacuum, respectively); ϕis the angle of sliding; θis the angle of scattering; n is the refraction index of matter; cis the light speed. The value of Brillouin splitting is represented as: ±∆νM−B=f=Vph Λ= 2νVph cnsinθ 2(9.7) 97where: νn/c= 1/λ;nis the refraction index of matter; νis incident light frequency; vph=vS (9.8) is the phase velocity of a scattering wave equal to hypersoni c velocity. The formula (9.7) is identical to that obtained from the anal ysis of the Doppler eﬀect: ∆ν ν=±2VS cnsinθ 2(9.9) According to the classical theory, the central line, which i s analogous to that observed in gases, is caused by entropy ﬂuctuations in l iquids, without any changes of pressure (Vuks, 1977). On the basis of Frenkel the ory of liquid state, the central line can be explained by ﬂuctuations of ”hole” nu mber - cavitational ﬂuctuations (Theiner, 1969). The thermodynamic approach of Landau and Plachek leads to th e formula, which relates the intensities of the central (I) and two late ral (IM−B) lines of the scattering spectrum with compressibility and heat capa cities: I 2IM−B=Ip Iad=βT−βS βS=Cp−Cv Cv(9.10) where: βTandβSare isothermal and adiabatic compressibilities; CpandCv are isobaric and isohoric heat capacities. In crystals, quartz for example, the central line in the ﬁne s tructure of light scattering is usually absent or very small. However, instea d of one pair of shifted components, observed in liquids, there appear three Brillouin components in crystals. One of them used to be explained by scattering on th e longitudinal phonons, and two - by scattering on the transversal phonons. 9.3. New mesoscopic approach to problem In follows from our hierarchic theory that the thermal ”rand om” ﬂuctuations are ”organized” by diﬀerent types of superimposed quantum e xcitations. According to our Hierarchic model, including microscopic, mesoscopic and macroscopic scales of matter (see Introduction), the most p robable (primary) and mean (secondary) eﬀectons, translational and libratio nal are capable of quantum transitions between two discreet states: ( a⇔b)tr,lband (¯a⇔¯b)tr,lb respectively. These transitions lead to origination/anni hilation of photons and phonons, forming primary and secondary deformons. The mean heat energy of molecules is determined by the value o f 3kT, which as our calculations show, has the intermediate value betwee n the discreet en- ergies of a and b quantum states of primary eﬀectons (Fig...) , making, con- sequently, the non-equilibrium conditions in condensed ma tter. Such kind of instability is a result of ”competition” between classical and quantum distribu- tions of energy . 98The maximum deviations from thermal equilibrium and that of the dielec- tric properties of matter occur when the same states of prima ry and secondary quasiparticles, e.g. a,¯ aandb,¯boccur simultaneously. Such a situation corre- sponds to the A and B states of macroeﬀectons. The ( A⇔B)tr,lbtransitions represent thermal ﬂuctuations. The big density ﬂuctuation s are related to ”ﬂick- ering clusters” (macroconvertons between librational and translational primary eﬀectons) and the maximum ﬂuctuations correspond to Superd eformons. Only in the case of spatially independent ﬂuctuations the in ter- ference of secondary scattered photons does not lead to thei r total compensation. The probability of the event that two spatially uncorrelate d events coincide in time is equal to the product of their independent probabil ities. Thus, the probabilities of the coherent ( a,¯ a) and ( b,¯b) states of primary and secondary eﬀectons, corresponding to A and B states of the ma croeﬀectons (tr and lb), independent on each other, are equal to: /parenleftbigPA M/parenrightbigind tr,lb=/parenleftbigPa ef¯Pa ef/parenrightbigS tr,lb/parenleftbigg1 Z2/parenrightbigg =/parenleftbiggPA M Z2/parenrightbigg tr,lb(9.11) /parenleftbig PB M/parenrightbigind tr,lb=/parenleftbigPb ef¯Pb ef/parenrightbigS tr,lb/parenleftbigg1 Z2/parenrightbigg =/parenleftbiggPB M Z2/parenrightbigg tr,lb(9.12) where 1 Z/parenleftbig Pa ef/parenrightbig tr,lband1 Z/parenleftbig¯Pa ef/parenrightbig tr,lb(9.13) are the independent probabilities of aand¯ astates determined according to formulae (4.10 and 4.18), while probabilities/parenleftBig Pb ef/Z/parenrightBig tr,lband/parenleftBig ¯Pb ef/Z/parenrightBig tr,lbare determined according to formulae (4.11 and 4.19); Zis the sum of probabilities of all types of quasiparticles st ates - eq.(4.2). The probabilities of molecules being involved in the spatia lly independent translational and librational macrodeformons are express ed as the products (9.11) and (9.12): /parenleftbigPM D/parenrightbigind tr,lb=/bracketleftBig/parenleftbig PA M/parenrightbigind/parenleftbig PB M/parenrightbigind/bracketrightBig tr,lb=PM D Z4(9.14) Formulae (9.11) and (9.12) may be considered as the probabil ities of space- independent but coherent macroeﬀectons in A and B states, re spectively. For probabilities of space-independent supereﬀectons in A∗andB∗states we have: /parenleftbig PA∗ S/parenrightbigind=/parenleftbig PA M/parenrightbigind tr/parenleftbig PA M/parenrightbigind lb=PA∗ S Z4(9.15) /parenleftbig PB∗ S/parenrightbigind=/parenleftbigPB M/parenrightbigind tr/parenleftbigPB M/parenrightbigind tr=Pb∗ S Z4(9.15a) 99In a similar way we get from (9.14) the probabilities of spati ally independent superdeformons: /parenleftbig PD∗ S/parenrightbigind=/parenleftbigPD M/parenrightbig tr/parenleftbigPD M/parenrightbig lb=PD∗ S Z4(9.16) The concentrations of molecules, the states of which marked ly diﬀer from the equilibrium one and which cause light scattering in composi tion of spatially independent macroeﬀectons and macrodeformons, are equal t o: /bracketleftbigg NA M=N0 Z2V0/parenleftbigPA M/parenrightbig/bracketrightbigg tr,lb;/bracketleftbigg NB M=N0 Z2V0/parenleftbigPB M/parenrightbig/bracketrightbigg tr,lb(9.17) /bracketleftbigg ND M=N0 Z4V0/parenleftbigPD M/parenrightbig/bracketrightbigg tr,lb The concentrations of molecules, involved in a-convertons , b- convertons and Macroconvertons or c-Macrotransitons (see Introduction) are correspondingly: Nac M=N0 Z2V0Pac;Nbc M=N0 Z2V0Pbc;NC M=N0 Z4V0PcMt (9.18) The probabilities of convertons-related excitations are t he same as used in Sec- tion 4. The concentration of molecules, participating in the indep endent supereﬀec- tons and superdeformons: NA∗ M=N0 Z4V0PA∗ s;NB∗ M=N0 Z4V0PB∗ S (9.19) ND∗ M=N0 Z8V0PD∗ S (9.20) where N0andV0are the Avogadro number and the molar volume of the matter. Substituting (9.17 - 9.20) into well known Raleigh formula f or scat- tering coeﬃcient, measured at the right angle between incid ent and scattered beams: R=I I0r2 V=8π4 λ4α2nM(cm−1) (9.20a) we obtain the values of contributions from diﬀerent collect ive excitations: [tr] and [lb] macroeﬀectons in A and B states, macrodeformons and corresponding parameters of superdeformons - to the resulting scattering coeﬃcient: 100/parenleftbig RM A/parenrightbig tr,lb=8π4 λ4(α∗)2 Z2N0 V0/parenleftbigPA M/parenrightbig tr,lb;Rs A=8π4 λ4(α∗)2 Z4N0 V0PA∗ s (9.21) /parenleftbig RM B/parenrightbig tr,lb=8π4 λ4(α∗)2 Z2N0 V0/parenleftbig PB M/parenrightbig tr,lb;Rs B=8π4 λ4(α∗)2 Z4N0 V0PB∗ s (9.22) /parenleftbig RM D/parenrightbig tr,lb=8π4 λ4(α∗)2 Z2N0 V0/parenleftbig PB D/parenrightbig tr,lb;Rs D=8π4 λ4(α∗)2 Z4N0 V0PD∗ s (9.23) The contributions of excitations, related to [tr/lb]convertons are: Rac=8π4 λ4(α∗)2 Z2N0 V0Rbc=8π4 λ4(α∗)2 Z2N0 V0Pbc (9.23a) Rabc=8π4 λ4(α∗)2 Z4N0 V0PcMt (9.23b) where: α∗is the acting polarizability determined by eq.(8.24) and (8 .25). The resulting coeﬃcient of the isotropic scattering ( Riso) is deﬁned as the sum of contributions (9.21-9.23) and is subdivided into thr ee kinds of scattering: caused by translational quasiparticles, caused by librati onal quasiparticles and by the mixed type of quasiparticles: Riso= [RM A+RM B+RM D]tr+ [RM A+RM B+RM D]lb+ (9.24) + [Rac+Rbc+Rabc] + [Rs A+Rs B+Rs D] Total contribution, including all kind of convertons and Su perexcitations are correspondingly: RC=Rac+Rbc+RabcandRS=Rs A+Rs B+Rs D (9.24a) The polarizability of anisotropic molecules having no cubi c symmetry is a tensor. In this case, total scattering (R) consists of scatt ering at density ﬂuctua- tions ( Riso) and scattering at ﬂuctuations of the anisotropy/parenleftBig Ran=13∆ 6−7∆Riso/parenrightBig : R=Riso+13∆ 6−7∆Riso=Riso6 + 6∆ 6−7∆=RisoK (9.25) where R isocorresponds to eq.(9.24); ∆ is the depolarization coeﬃcient. The factor: /parenleftbigg6 + 6∆ 6−7∆/parenrightbigg =K 101was obtained by Cabanne and is called after him. In the case of isotropic molecules when ∆ = 0, the Cabanne factor is equal to 1. The depolarization coeﬃcient (∆) could be determined exper imentally as the ratio: ∆ =Ix/Iz, (9.26) where IxandIzare two polarized components of the beam scattered at right angle with respect to each other in which the electric vector is directed parallel and perpendicular to the incident beam, respectively. For e xample, in water ∆ = 0 .09 (Vuks, 1977). According to the proposed theory of light scattering in liqu ids the central un- shifted (like in gases) component of the Brillouin scatteri ng spectrum, is caused by ﬂuctuations of concentration and self-diﬀusion of molec ules, participating in the convertons, macrodeformons (tr and lib) and superdefor mons. The scatter- ing coeﬃcients of the central line ( Rcentr) and side lines (2 Rside) in transparent condensed matter, as follows from (9.24) and (9.25), are equ al correspondingly to: Rcent=K/bracketleftbig/parenleftbigRM D/parenrightbig tr+/parenleftbigRM D/parenrightbig lb/bracketrightbig +K(RC+RS) (9.27) and 2Rside=/parenleftbig RM A+RM B/parenrightbig tr+/parenleftbig RM A+RM A/parenrightbig lb(9.27a) where Kis the Cabanne factor. The total coeﬃcient of light scattering is: Rt=Rcent+ 2Rside (9.28) In accordance with our model the ﬂuctuations of anisotropy ( Cabanne factor) should be taken into account for calculations of the central component only. The orientations of molecules in composition of A and B states of Macroeﬀectons are correlated and their coherent oscillations are not accompa nied by ﬂuctuations of anisotropy of polarizability (see Fig.26). The probabilities of the convertons, macrodeformons and su perdeformons excitations in crystals are much lower than in liquids and he nce, the central line in the Brillouin spectra of crystals is not usually observed . The lateral lines in Brillouin spectra are caused by the scat tering on the molecules forming (A) and (B) states of spatially independe nt macroeﬀectons, as it was mentioned above. The polarizabilities of the molecules forming the independ ent macroeﬀec- tons, synchronized in (A) tr,lband (B)tr,lbstates and dielectric properties of these states, diﬀer from each other and from that of transiti on states (macrode- formons). Such short-living states should be considered as the non equilibrium ones. 102In fact we must keep in mind, that static polarizabilities in the more stable ground A state of the macroeﬀectons are higher than in B state , because the energy of long-term Van der Waals interaction between molec ules of the A state is bigger than that of B-state. If this diﬀerence may be attributed mainly to the diﬀerence i n the long-therm dispersion interaction, then from (8.33) we obtain: EB−EA=VB−VA=−3 2E0 r6/parenleftbigα2 B−α2 A/parenrightbig (9.29) where polarizability of molecules in A-state is higher, tha n that in B-state: α2 A>/bracketleftBig/parenleftbig α∗/parenrightbig2≈α2 D/bracketrightBig > α2 B The kinetic energy and dimensions of ”acoustic” and ”optic” states of macroef- fectons are the same: TA kin=TB kin. In our present calculations of light scattering we ignore th is diﬀerence (9.29) between polarizabilities of molecules in A and B states. But it can be taken into account if we assume, that polarizabi lities in (A) and (a), (B) and (b) states of primary eﬀectons are like: αA≃αa≃α∗;αB≃αb and the diﬀerence between the potential energy of (a) and (b) states is deter- mined mainly by dispersion interaction (eq.9.28). Experimental resulting polarizability ( α∗≃αa) can be expressed as: αa=faαa+fbαb+ftα (9.29a) where αt≃αis polarizability of molecules in the gas state (or transiti on state); fa=Pa Pa+Pb+Pt;fb=Pb Pa+Pb+Pt; and ft=fd=Pt Pa+Pb+Pt are the fractions of (a), (b) and transition (t) states (equa l to 9.66) as far Pt=Pd=Pa·Pb. On the other hand from (1.33) at r=const we have: ∆Vb→a dis=−3 4(2α∆α) r6·I0(ra=rb;Ia 0≃Ib 0) and ∆Vb→a dis Vb=hνp hνb=∆αa αor ∆αa=αaνp νb(9.29b) 103αb=αa−∆αa=αa(1−νp/νb) where:∆ αais a change of each molecule polarizability as a result of the primary eﬀecton energy changing: Eb→Ea+hνpwith photon radiation; νbis a frequency of primary eﬀecton in (b)- state (eq.9.28). Combining (9.29) and (9.29b) we derive for αaandαbof the molecules composing primary translational or librational eﬀectons: αa=ftα 1−/parenleftBig fa+fb+fbνp νb/parenrightBig (9.30) αb=αa/parenleftbigg 1−νp νb/parenrightbigg (9.30a) The calculations by means of (9.30) are approximate in the fr amework of our assumptions mentioned above. But they correctly reﬂect the tendencies of αa andαbchanges with temperature. The ratio of intensities or scattering coeﬃcients for the ce ntral component to the lateral ones previously was described by Landau- Plac hek formula (9.10). According to our mesoscopic theory this ratio can be calcula ted in another way leading from (9.27) and (9.28): Icentr 2IM−B=Rcent 2Rside(9.30b) Combining (9.30) and Landau- Plachek formula (9.10) it is po ssible to cal- culate the ratio ( βT/βS) and ( CP/CV) using our mesoscopic theory of light scattering. 9.4. Factors that determine the Brillouin line width The known equation for Brillouin shift is (see 9.7): ∆νM−B=ν0= 2vs λnsin(θ/2) (9.31) where: vsis the hypersonic velocity; λis the wavelength of incident light, nis the refraction index of matter, and θ- scattering angle. The deviation from ν0that determines the Brillouin side line half width may be expressed as the result of ﬂuctuations of sound velocity vsand n related to A and B states of tr and lib macroeﬀectons: ∆ν0 ν0=/parenleftbigg∆vs vs+∆n n/parenrightbigg (9.32) ∆ν0is the most probable side line width, i.e. the true half width of Brillouin line. It can be expressed as: 104∆ν0= ∆νexp−F∆νinc where ∆ νexpis the half width of the experimental line, ∆ νinc- the half width of the incident line, F- the coeﬃcient that takes into account apparatus eﬀects. Let us analyze the ﬁrst and the second terms in the right part o f (9.32) separately. Thevssquared is equal to the ratio of the compressibility modulus (K) and density ( ρ): v2 s=K2/ρ (9.33) Consequently, from (9.33) we have: ∆vs vs=1 2/parenleftbigg∆K K−∆ρ ρ/parenrightbigg (9.34) In the case of independent ﬂuctuations of K and ρ:: ∆vs vs=1 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆K K/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆ρ ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg (9.35) From our equation (8.14) we obtain for refraction index: n2=/parenleftbigg 1−4 3Nα∗/parenrightbigg−1 , (9.36) where N=N0/V0is the concentration of molecules. From (9.36) we can derive: ∆n n=1 2/parenleftbig n2−1/parenrightbig/parenleftbigg∆α∗ α∗+∆N N/parenrightbigg (9.37) where: (∆N/N) = (∆ ρ/ρ) (2.38) and /parenleftbigg∆α∗ α∗/parenrightbigg ≃/parenleftbigg∆K K/parenrightbigg (9.39) we can assume eq.(9.39) as far both parameters: polarizabil ity (α∗) and com- pressibility models (K) are related with the potential ener gy of intermolecular interaction. For the other hand one can suppose that the following relatio n is true: 105∆α∗ α∗≃\|¯Ea ef−3kT\| 3kT=∆K K(9.40) where: ¯Ea efis the energy of the secondary eﬀectons in (¯ a) state; E0= 3kT is the energy of an ”ideal” quasiparticle as a superposition of 3D standing waves. The density ﬂuctuations can be estimated as a result of the fr ee volume ( vf) ﬂuctuations (see 9.45): /parenleftbigg∆vf vf/parenrightbigg tr,lb=1 Z/parenleftbig PM D/parenrightbig tr,lb≃(∆N/N)tr,lb(9.41) Now, putting (9.40) and (9.41) into (9.37) and (9.34) and the n into (9.32), we obtain the semiempirical formulae for the Brillouin line ha lf width calculation: ∆νf νf≃n2 2/bracketleftBigg \|¯Ea ef−3kT\| 3kT+1 Z/parenleftbigPM D/parenrightbig/bracketrightBigg tr,lb(9.42) Brillouin line intensity depends on the half-width ∆ νof the line in following ways: for a Gaussian line shape: I(ν) =Imax 0exp/bracketleftBigg −0.693/parenleftbiggν−ν0 1 2∆ν0/parenrightbigg2/bracketrightBigg ; (9.43) for a Lorenzian line shape: I(ν) =Imax 0 1 +/bracketleftbig(ν−ν0)/1 2∆ν0/bracketrightbig2(9.44) The traditional theory of Brillouin line shape gives a possibility for calculation of ∆ν0taking into account the elastic (acoustic) wave dissipatio n. The fading out of acoustic wave amplitude may be expressed as : A=A0e−αxorA=A0e−αvs(9.45) where αis the extinction coeﬃcient; x=vst- the distance from the source of waves; vsandt- sound velocity and time, correspondingly. The hydrodynamic theory of sound propagation in liquids lea ds to the fol- lowing expression for the extinction coeﬃcient: α=αs+αb=Ω2 2ρv3s/parenleftbigg4 3ηs+ηb/parenrightbigg (9.46) 106where: αsandαbare contributions to α, related to share viscosity ( ηs) and bulk viscosity ( ηb), respectively; Ω = 2 πfis the angular frequency of acoustic waves. When the side lines in Brillouin spectra broaden slightly, t he following rela- tion between their intensity (I) and shift (∆ ω=\|ω−ω0\|) from frequency ω0, corresponding to maximum intensity ( I=I0) of side line is correct: I=I0 1 +/parenleftbigω−ω0 a/parenrightbig, (9.47) where: a=αvs. One can see from (9.46) that at I(ω) =I0/2, the half width: ∆ω1/2= 2π∆ν1/2=αvsand ∆ ν1/2=1 2παvs (9.48) It will be shown in Chapter 12 how one can calculate the values ofηsand consequently αson the basis of the mesoscopic theory of viscosity. 9.5. Quantitative veriﬁcation of hierarchic theory of Brillouin scattering The calculations made according to the formula (9.21 - 9.27) are presented in Fig.26-28. The proposed theory of scattering in liquids, ba sed on our hierarchic concept, is more adequate than the traditional Einstein, Br illouin, Landau- Plachek theories based on classical thermodynamics. It des cribes experimen- tal temperature dependencies and the Icentr/2IM−Bratio for water very well (Fig.28). The calculations are made for the wavelength of incident lig ht:λph= 546.1nm= 5.461·10−5cm. The experimental temperature dependence for the refraction index (n) at this wavelength was taken from th e Frontas’ev and Schreiber paper (1965). The rest of data for calculating of v arious light scat- tering parameters of water (density the location of transla tional and librational bands in the oscillatory spectra) are identical to those use d above in Chapter 6. 107Fig. 26. Theoretical temperature dependencies of the total scat- tering coeﬃcient for water without taking into account the a nisotropy of water molecules polarizability ﬂuctuations in the volum e of macroef- fectons, responsible for side lines: [ R(tot)] - eq.(9.27a; 9.28) and tak- ing them into account: [ KR(tot)], where Kis the Cabanne factor (eq.9.25). Fig. 27. Theoretical temperature dependencies of contributions to the total coeﬃcient of total light scattering (R) caused b y transla- tional and librational macroeﬀectons and macrodeformons ( without taking into account ﬂuctuations of anisotropy). 108Fig. 28. Theoretical temperature dependencies of central to side bands intensities ratio in Brillouin spectra (eq.9.30). Mesoscopic theory of light scattering can be used to verify t he correctness of our formula for refraction index of condensed matter we go t from our theory (eq. 1.14): n2−1 n2=4 3πN0 V0α∗(9.48a) and to compare the results of its using with that of the Lorent z-Lorenz formula: n2−1 n2+ 1=4 3πN0 V0α (9.49) From formula (9.48a) the resulting or eﬀective molecular po larizability squared ( α∗)2used in eq.(9.21-9.23) is: (α∗)2=/bracketleftbigg(n2−1)/n2 (4/3)π(N0/V0)/bracketrightbigg2 (9.50) On the other hand, from the Lorentz-Lorenz formula (9.49) we have another value of polarizability: α2=/bracketleftbigg(n2−1)/(n2+ 2) (4/3)π(N0/V0)/bracketrightbigg2 (9.51) It is evident that the light scattering coeﬃcients (eq.9.28 ), calculated using (9.50) and (9.51) taking refraction index: n= 1.33 should diﬀer more than four times as far: 109R(α∗) R(α)=(α∗)2 (α)2=(n2−1)/n2 (n2−1)/(n2+ 2)=/parenleftbiggn2+ 2 n2/parenrightbigg2 = 4.56 (9.52) At 250andλph= 546 nmthe theoretical magnitude of the scattering coeﬃcient for water, calculated from our formulae (9.28) is equal (see Fig.26) to: R= 11.2·10−5m−1(9.53) This result of our theory coincides well with the most reliab le experimental value (Vuks, 1977): Rexp= 10.8·10−5m−1 Multiplication of the side bands contribution (2 Rside) to Cabanne factor in- creases the calculated total scattering to about 25% and mak es the correspon- dence with experiment worse. This fact conﬁrms our assumpti on that ﬂuctua- tions of anisotropy of polarizability in composition of A an d B states of macroef- fectons should be ignored in light scattering evaluation du e to correlation of molecular dynamics in these states, in contrast to that of ma crodeformons. Fig. 29. Theoretical temperature dependencies of the contribu- tions of A and B states of translational Macroeﬀectons to the total scattering coeﬃcient of water (see also Fig.27); 110Fig. 30. Theoretical temperature dependencies of the contri- butions of the A and B states of librational Macroeﬀectons to the coeﬃcient of light scattering (R). It follows from the Fig.29 and Fig.30 that the light scatteri ng depends on (A⇔B) equilibrium of macroeﬀectons because ( RA)>(RB), i.e. scattering onAstates is bigger than that on Bstates. Fig. 31. Theoretical temperature dependencies of the contribu- tions to light scattering (central component), related to t ranslational (RD)trand librational ( RD)lbmacrodeformons. Comparing Figs. 26; 28, and 31 one can see that the main contri bution to central component of light scattering is determined by [ lb/tr] convertons Rc(see eq.9.27). 111Fig. 32. Theoretical temperature dependences for temperature derivative ( dR/dT ) of the total coeﬃcient of light scattering of water. Nonmonotonic deviations of the dependencies dR/dT (Fig.32) reﬂect the nonmonotonic changes of the refraction index for water nH2O(T), as indicated by available experimental data (Frontas’ev and Schreiber, 1965). The deviations of dependence nH2O(t) from the monotonic way in accordance with hierarchic theory, are a consequence of the nonmonotonic change in the s tability of water structure, i.e. nonlinear change of ( A⇔B)tr,lbequilibrium. Some possible reasons of such equilibrium change were discussed in Chapte r 6. It is clear from (9.52) that the calculations based on the Lor entz-Lorentz formula (9.51) give scattering coeﬃcient values of about 4. 5 times smaller than experimental ones. It means that the true α∗value can be calculated only on the basis of our mesoscopic theory of light refraction (eq.9 .50). The traditional Smolukhovsky-Einstein theory, valid for t he integral light scattering only (eq. 9.1), yield values in the range of R= 8.85·10−5m−1to R= 10.5·10−5m−1(Eisenberg, Kauzmann, 1969; Vuks, 1977). The results, discussed above, demonstrate that new theory o f light scattering works better and is more informative than the con ventional one. 9.6. Light scattering in solutions If the guest molecules are dissolved in a liquid and their siz es are much less than incident light wavelength, they do not radically alter the solvent properties. For this case the described above mechanism of light scatter ing of pure liquids does not changed qualitatively. For such dilute solutions the scattering on the ﬂuctuations of concentration of dissolved molecules ( Rc) is simply added to the scattering on the density ﬂuctuations of molecules of the host solvent (eq.9.28). Tak ing into account the ﬂuctuations of molecule polarizability anisotropy (see 9. 25) the total scattering coeﬃcient of the solution ( RS) is: 112RS=Rt+Rc (9.54) Eqs. (9.21 - 9.28) could be used for calculating Rtuntil critical concentra- tions ( Ccr) of dissolved substance when it start to destroy the solvent structure, so that the latter is no longer able to form primary libration al eﬀectons. Pertur- bations of solvent structure will induce low-frequency shi ft of librational bands in the oscillatory spectrum of the solution until these band s totally disappear. If the experiment is made with a two-component solution of li quids, soluble in each other, e.g. water-alcohol, benzol-methanol etc., a nd the positions of translational and librational bands of solution component s are diﬀerent, then at the concentration of the dissolved substance: C > C cr, the dissolved sub- stance and the solvent (the guest and host) can switch their r oles. Then the translational and librational bands pertinent to the guest subsystem start to dominate. In this case, Rtis to be calculated from the positions of the new bands corresponding to the ”new” host-solvent. The total ”m elting” of the pri- mary librational ”host eﬀectons” and the appearance of the d issolved substance ”guest eﬀectons” is like the second order phase transition and should be accom- panied by a heat capacity jump. The like experimental eﬀects take place indeed (Vuks, 1997). According to our concept, the coeﬃcient R cin eq.(9.54) is caused by the ﬂuc- tuations of concentration of dissolved molecules in the vol ume of translational and librational macro- and superdeformons of the solvent. I f the destabilization of the solvent is expressed in the low frequency shift of librational bands, then the coeﬃcients ( RAandRB)lbincrease (eq.9.21 and 9.22) with the prob- ability of macro-excitations.. The probabilities of conve rtons and macro- and superdeformons and the central component of Brillouin spec tra will increase also. Therefore, the intensity of the total light scatterin g increases correspond- ingly. The ﬂuctuations of concentration of the solute molecules, i n accordance with our model, occur in the volumes of macrodeformons and superd eformons. Con- sequently, the contribution of solute molecules in scatter ing (Rcvalue in eq.9.54) can be expressed by formula, similar to (9.23), but containi ng the molecule po- larizability of the dissolved substance (”guest”) ,equal to ( α∗ g)2instead of the molecule polarizability ( α∗) of the solvent (”host”), and the molecular con- centration of the ”guest” substance in the solution ( ng) instead of the solvent molecule concentration ( nM=N0/V0). For this case Rccould be presented as a sum of the following contributions: (Rc)tr,lb=8π4 λ4(α∗ g)2ng/bracketleftBig (PD M)tr,lb+PD∗ S/bracketrightBig (9.55) RD∗ c=8π4 λ4(α∗ g)2ng(PD∗ S) (9.55a) The resulting scattering coeﬃcient ( Re) on ﬂuctuations of concentration in (9.54) is equal to: 113Rc= (Rc)tr+ (Rc)lb+RD∗ c (9.56) Ifa few substances are dissolved with concentrations lower than ( Ccr), then theirRcare summed up additively. Formulae (9.55) and (9.56) are valid also for the dilute solu tions. Eqs.(9.21-9.28) and (9.54-9.56) should, therefore, be use d for calculating the resulting coeﬃcient of light scattering in solutions ( RS). The traditional theory represents the scattering coeﬃcient at ﬂuctuations of concentration as (Vuks, 1977): Rc=π2 2λ4/parenleftbigg∂ǫ ∂x/parenrightbigg2 ∆x2v (9.57) where ( ∂ǫ/∂x ) is the dielectric penetrability derivative with respect t o one of the components: ∆¯ x2is the ﬂuctuations of concentration of guest molecules squared in the volume element v. The transformation of (9.57) on the basis of classical therm odynamics (Vuks, 1977) leads to the formula: Rc=π2 2λ4N0/parenleftbigg 2n∂n ∂x/parenrightbigg/parenleftbigg9n2 (2n2+ 1)(n2+ 2)/parenrightbigg2 x1x2V12f, (9.58) where N0is the Avogadro number, x1andx2are the molar fractions of the ﬁrst and second components in the solution, V12is the molar volume of the solution, fis the function of ﬂuctuations of concentration determined exper- imentally from the partial vapor pressures of the ﬁrst ( P1) and second ( P2) solution components: 1 f=x1 P1∂P1 ∂x1=x2 P2∂P2 ∂x2(9.59) In the case of ideal solutions ∂P1 ∂x1=P1 x1;∂P2 ∂x2=P2 x2; and f= 1. For application the mesoscopic theory of light scattering t o study of crystals, liquids and solutions, the following information is needed : 1. Positions of translational and librational band maxima i n oscillatory spec- tra; 2. Concentration of all types of molecules in solutions; 3. Refraction index or polarizability in the acting ﬁeld of e ach component of solution at given temperature. Application of our theory to quantitative analysis of trans parent liquids and solids yields much more information about properties of mat ter, its mesoscopic and hierarchic dynamic structure than the traditional one. 11410. Hierarchic theory of M¨ ossbauer eﬀect 10.1. General background When the atomic nucleus with mass (M) in the gas phase irradia tesγ- quantum with energy of E0=hν0=mpc2(10.1) where: mpis the eﬀective photon mass, then according to the law of impu lse conservation, the nuclear acquires additional velocity in the opposite direction: v=−E0 Mc(10.2) The corresponding additional kinetic energy ER=Mv2 2=E2 0 2Mc2(10.3) is termed recoil energy. When an atom which irradiates γ-quantum is in composition of the solid body, then three situations are possible: 1. The recoil energy of the atom is higher than the energy of at om - lattice interaction. In this case, the atom irradiating γ-quantum would be knocked out from its position in the lattice. That leads to defects origi nation; 2. Recoil energy is insuﬃcient for the appreciable displace ment of an atom in the structure of the lattice, but is higher than the energy of phonon, equal to energy of secondary transitons and phonons excitation. I n this case, recoil energy is spent for heating the lattice; 3. Recoil energy is lower than the energy of primary transito ns, related to [emission/absorption] of IR translational and librationa l photons ( hνp)tr,lband phonons ( hνph)tr,lb. In that case, the probability (f) of γ-quantum irradiation without any the losses of energy appears, termed the probabi lity (fraction) of a recoilless processes. For example, when ER<< hν ph(νph- the mean frequency of phonons), then the mean energy of recoil: ER= (1−f)hνph (10.4) Hence, the probability of recoilless eﬀect is f= 1−ER hνph(10.5) According to eq.(10.3) the decrease of the recoil energy ERof an atom in the structure of the lattice is related to increase of its eﬀecti ve mass ( M). In our model Mcorresponds to the mass of the eﬀecton. 115The eﬀect of γ-quantum irradiation without recoil was discovered by M¨ os sbauer in 1957 and named after him. The value of M¨ ossbauer eﬀect is d etermined by the value of f≤1. The big recoil energy may be transferred to the lattice by por tions that areresonant to the frequency of IR photons (tr and lb) and phonons. The possibility of stimulation of superradiation of IR quanta a s a result of such recoil process is a consequence of our model. The scattering of γ-quanta without lattice excitation, when ER<< hν ph, is termed the elastic one.The general expression (Wertheim, 1964; Shpinel, 1969) for the probability of such phononless elastic γ-quantum radiation acts is equal to: f= exp/parenleftbigg −4π < x2> λ2 0/parenrightbigg (10.6) where λ0=c/ν0is the real wavelength of γ-quantum; <x2>- the nucleus oscillations mean amplitude squared in the direction of γ-quantum irradiation. Theγ-quanta wavelength parameter may be introduced like: L0=λ0/2π, (10.7) where: L0= 1.37·10−5cmforFe57, then eq.(10.6) could be written as follows: f= exp/parenleftbigg −< x2> L2 0/parenrightbigg (10.8) It may be shown (Shpinel, 1969), proceeding from the model of crystal as a system of 3N identical quantum oscillators, that when tempe rature (T) is much lower than the Debye one ( θD) then: < x2>=9/planckover2pi12 4Mkθ D/braceleftbigg 1 +2/planckover2pi12T2 3θ2 D/bracerightbigg , (10.9) where θD=hνD/kandνDis the Debye frequency. From (10.1), (10.3) and (10.7) we have: 1 L=E0 /planckover2pi1c(10.10) where: E0=hν=c(2ME R)1/2is the energy of γ-quantum Substituting eqs.(10.9 and 10.10) into eq.(10.8), we obtai n the Debye-Valler formula: f= exp/bracketleftbigg −ER kθD/braceleftbigg3 2+π2T2 θD/bracerightbigg/bracketrightbigg (10.11) when T→0, then 116f→exp/parenleftbigg −3ER 2kθD/parenrightbigg (10.12) 10.2. Probability of elastic eﬀects Mean square displacements <x2>of an atoms or molecules in condensed matter (eq. 10.8) is not related to excitation of thermal pho tons or phonons (i.e. primary or secondary transitons). According to our co ncept, < x2>is caused by the mobility of the atoms forming eﬀectons and diﬀe rs for primary and secondary translational and librational eﬀectons in ( a,¯a)tr,lband (b, b)tr,lb states. We will ignore below the contributions of macro- and supereﬀ ectons in M¨ ossbauer eﬀect as very small. Then the resulting probabil ity of elastic eﬀects atγ-quantum radiation is determined by the sum of the following contributions: f=1 Z/summationdisplay tr,lb/bracketleftbig/parenleftbigPa effa ef+Pb effb ef/parenrightbig +/parenleftbig¯Pa ef¯fa ef+¯Pb ef¯fb ef/parenrightbig/bracketrightbig tr,lb(10.13) where: Pa ef, Pb ef,¯Pa ef,¯Pb efare the relative probabilities of the acoustic and opticstates for primary and secondary eﬀectons; Z is the total par tition function (see 4.10-4.19 and 4.2). These parameters are calculated as described in Section 4 of this article. Each of contributions to resulting probability of the elast ic eﬀect can be calcu- lated separately as: /parenleftbigfa ef/parenrightbig tr,lb= exp −</parenleftbigxa/parenrightbig2 tr,lb> L2 0  (10.14) /parenleftbigfa ef/parenrightbig tr,lbis the probability of elastic eﬀect, related to dynamics of p rimary translational and librational eﬀectons in a-state; /parenleftbigfb ef/parenrightbig tr,lb= exp −</parenleftbig xb/parenrightbig2 tr,lb> L2 0  (10.15) /parenleftbigfb ef/parenrightbig tr,lbis the probability of elastic eﬀect in primary translationa l and libra- tional eﬀectons in b-state; /parenleftbig¯fa ef/parenrightbig tr,lb= exp/bracketleftbigg −</parenleftBig ¯xa/parenrightBig2 tr,lb> L2 0/bracketrightbigg (10.16) 117/parenleftbig¯fa ef/parenrightbig tr,lbis the probability for secondary eﬀectons in ¯ a-state; /parenleftbig¯fb ef/parenrightbig tr,lb= exp −</parenleftbig ¯xb/parenrightbig2 tr,lb> L2 0  (10.17) /parenleftbig¯fb ef/parenrightbig tr,lbis the probability of elastic eﬀect, related to secondary eﬀ ectons in ¯b-state. Mean square displacements within diﬀerent types of eﬀecton s in eqs.(10.14- 10.17) are related to their phase and group velocities. At ﬁr st we express the displacements using group velocities of the waves B(vgr) and periods of corre- sponding oscillations ( T) as: </parenleftbigxa/parenrightbig2 tr,lb>=<(va gr)2 tr,lb> < ν2a>tr,lb=</parenleftbigva grTa/parenrightbig2 tr,lb> (10.18) where ( Ta)tr,lb= (1/νa)tr,lbis a relation between the period and the frequency of primary translational and librational eﬀectons in a-state; (va gr=vb gr)tr,lbare the group velocities of atoms forming these eﬀectons equal in (a) and (b) states. In a similar way we can express the displacements of atoms for ming (b) state of primary eﬀectons (tr and lib): </parenleftbig xb/parenrightbig2 tr,lb>=<(vb gr)2 tr,lb> < ν2 b>tr,lb(10.19) where νbis the frequency of primary translational and librational eﬀectons in b-state. The mean square displacements of atoms forming secondary translational and librational eﬀectons in ¯ aand¯bstates: </parenleftbig¯xa/parenrightbig2 tr,lb>=<(va gr)2 tr,lb> <¯ν2a>tr,lb(10.20) </parenleftbig ¯xb/parenrightbig2 tr,lb>=<(vb gr)2 tr,lb> <¯ν2 b>tr,lb(10.21) where: (¯ va gr= ¯vb gr)tr,lb Group velocities of atoms in primary and secondary eﬀectons may be expressed using corresponding phase velocities ( vph) and formulae for waves B length as follows: /parenleftbigλa/parenrightbig tr,lb=h m < v gr>tr,lb=/parenleftbiggva ph νa/parenrightbigg tr,lb= (10.22) =/parenleftbigλb/parenrightbig tr,lb=/parenleftBigg vb ph νb/parenrightBigg tr,lb 118hence for the group velocities of the atoms or molecules form ing primary eﬀec- tons (tr and lb ) squared we have: /parenleftbigva,b gr/parenrightbig2 tr,lb=h2 m2/parenleftBigg νa,b va,b ph/parenrightBigg2 tr,lb(10.23) In accordance with mesoscopic theory, the wave B length, imp ulses and group velocities in aandbstates of the eﬀectons are equal. Similarly to (10.23), we obtain the group velocities of particles, composing second ary eﬀectons: /parenleftbig¯va,b gr/parenrightbig2 tr,lb=h2 m2/parenleftBigg ¯νa,b ¯va,b ph/parenrightBigg2 tr,lb(10.24) Substituting eqs.(10.23) and (10.24) into (10.18-10.21), we ﬁnd the important expressions for the average coherent displacements of part icles squared as a result of their oscillations in the volume of the eﬀectons ( tr, lib) in both discreet states (acoustic and optic): <(xa)2 tr,lb>= (h/mva ph)2 tr,lb (10.25) <(xb)2 tr,lb>= (h/mvb ph)2 tr,lb (10.26) <(xa)2 tr,lb>= (h/mva ph)2 tr,lb (10.27) <(xb)2 tr,lb>= (h/mvb ph)2 tr,lb (10.28) Then, substituting these values into eqs.(10.14-10.17) we obtain a set of diﬀerent contributions to the resulting probability of eﬀects witho ut recoil: /parenleftbigfa f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0va ph/parenrightBig2/bracketrightbigg tr,lb; /parenleftbigfb f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0vb ph/parenrightBig2/bracketrightbigg tr,lb;  (10.29) /parenleftbig¯fa f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0¯va ph/parenrightBig2/bracketrightbigg tr,lb; /parenleftbig¯fb f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0¯vb ph/parenrightBig2/bracketrightbigg tr,lb;  (10.30) where the phase velocities ( va ph, vb ph,¯va ph,¯vb ph)tr,lbare calculated from the resulting sound velocity and the positions of translationa l and librational bands in the oscillatory spectra of matter at given temperature us ing eqs.2.69-2.75. The wavelength parameter: 119L0=c 2πν0=hc 2πE0= 1.375·10−11m for gamma-quanta, radiated by nuclear of Fe57, with energy: E0= 14.4125 kev = 2 .30167·10−8erg Substituting eqs.(10.29) and (10.30) into (10.13), we ﬁnd t he total probabil- ity of recoilless eﬀects ( ftot) in the given substance. Corresponding computer calculations for ice and water are presented on Figs.33 and 3 4. As far the second order phase transitions in general case are accompanied by the alterations of the sound velocity and the positions of translational and librational bands, they should also be accompanied by alter ations of f totand its components. Fig. 33. Temperature dependences of total probability ( f) for elastic eﬀect without recoil and phonon excitation: (a) in ice; (b) in wate r; (c)-during phase transition. The calculations were performed using eq.(10. 13). 120Fig. 34 (a) - The contributions to probability of elastic eﬀect (f), presented at Fig.33, related to primary ( fa,b ef)trand secondary (¯f)trtranslational eﬀectons; (b)- contributions, related to primary (fa,b ef)lband secondary ( ¯f)lblibrational eﬀectons around the tem- perature of [ice ⇔water] phase transition. The total probability ( f) and its components, caused by primary and sec- ondary quasiparticles were calculated according to formul a (10.13). The value of (f) determines the magnitude of the M¨ ossbauer eﬀect register ed by γ-resonance spectroscopy. The band width caused by recoilless eﬀects is determined by t he uncertainty principle and expressed as follows: Γ =h τ≈10−27 1.4·10−7= 7.14·10−21erg = 4 .4·10−9eV (10.31) where τis the lifetime of nucleus in excited state (for Fe57τ= 1.4·10−7s). The position of the band depends on the mean square velocity o f atoms, i.e. on second order Doppler eﬀect. In the experiment, such an eﬀe ct is compen- sated by the velocity of γ-quanta source motion relative to absorbent. In the framework of our model this velocity is interrelated with th e mean velocity of the secondary eﬀectons diﬀusion in condensed matter. 10.3. Doppler broadening in spectra of nuclear gamma-reson ance (NGR) M¨ ossbauer eﬀect is characterized by the unbroadened compo nent of NGR spectra only, with probability of observation determined b y eq.(10.13). When the energy of absorbed γ-quanta exceeds the energy of thermal IR pho- tons (tr,lib) orphonons excitation, the absorbance band broadens as a result of Doppler eﬀect. Within the framework of our mesoscopic conce pt the Doppler broadening is caused by thermal displacements of the partic les during [ a⇔b and ¯a⇔¯b]tr,lbtransitions of primary and secondary eﬀectons, leading to o rig- ination/annihilation of the corresponding type of deformo ns (electromagnetic and acoustic). 121Theﬂickering clusters : [lb/tr] convertons ( aandb), can contribute in the NGR line broadening also. In that case, the value of Doppler broadening (∆Γ) of the band in the NGR spectrum could be estimated from corresponding kinetic ene rgies of these ex- citations, related to their group velocities (see eq. 4.31) . In our consideration we take into account the reduced to one molecule kinetic energies of pri- mary and secondary translational and librational transito ns,a-convertons and b-convertons. The contributions of macroconvertons, macro - and superdefor- mons are much smaller due to their small probability and conc entration: ∆Γ =V0 N0Z/summationdisplay tr,lb/parenleftbigntPtTt+ ¯nt¯Pt¯Tt/parenrightbig tr,lb+ (10.32) +V0 N0Z(nef)lb[PacTac+PbcTbc] where: N0andV0are the Avogadro number and molar volume; Zis the total partition function ( eq.4.2);ntand ¯ntare the concentrations of primary and secondary transitons (eqs.10.5 and 10.7); (nef)lb=nconis a concentration of primary librational eﬀectons, equal t o that of the convertons; Ptand¯Ptare the relative probabilities of primary and secondary transitons (eqs. 4.26 and 4 .27);PacandPbcare relative probabilities of (aandb) -convertons (see section 4 ); Ttand¯Ttare the kinetic energies of primary and secondary transiton s, re- lated to the corresponding total energies of these excitati ons (Etand¯Et), their masses ( MtandMt) and the resulting sound velocity ( vs, see eq.2.40) in the following form: (Tt)tr,lb=/summationtext3 1/parenleftBig E1,2,3 t/parenrightBig tr,lb 2Mt(vress)2(10.33) (Tt)tr,lb=/summationtext3 1/parenleftBig ¯E1,2,3 t/parenrightBig tr,lb 2¯Mt(vress)2(10.34) The kinetic energies of (a and b) convertons are expressed in a similar way: (Tac) =/summationtext3 1/parenleftbig E1,2,3 ac/parenrightbig tr,lb 2Mc(vress)2(10.34a) (Tbc) =/summationtext3 1/parenleftBig E1,2,3 bc/parenrightBig tr,lb 2Mc(vress)2(10.34b) where: E1,2,3 acandE1,2,3 bcare the energies of selected states of corresponding convertons; Mcis the mass of convertons, equal to that of primary libration al eﬀectons. 122The broadening of NGR spectral lines by Doppler eﬀect in liqu ids is generally expressed using the diﬀusion coeﬃcient (D) at the assumptio n that the motion of M¨ ossbauer atom has the character of unlimited diﬀusion ( Singvi, 1962): ∆Γ =2E2 0 /planckover2pi1c2D (10.35) where: E0=hν0is the energy of gamma quanta; c is light velocity and D=kT 6πηa(10.36) where: ηis viscosity, (a) is the eﬀective Stokes radius of the atom Fe57 The probability of recoilless γ-quantum absorption by the matter containing for example Fe57, decreases due to diﬀusion and corresponding Doppler broad - ening of band (∆Γ): fD=Γ Γ + ∆Γ(10.37) where ∆Γ corresponds to eq.(10.32).The formulae obtained h ere make it possible to experimentally verify a set of consequences of our mesosc opic theory using the gamma- resonance method. A more detailed interpretation of the data obtained by this method also becomes possible. The magnitude of (∆Γ) was calculated according to formula (1 0.32). It corresponds well to experimentally determined Doppler wid ening in the nuclear gamma resonance (NGR) spectra of ice. 123Fig. 35. The temperature dependences of the parameter ∆Γ, characterizing the nonelastic eﬀects and related to the exc itation of thermal phonons and IR photons: a) in ice; b) in water; c) near phase transition. 10.4. Acceleration and forces, related to thermal dynamics of molecules and ions. Vibro-Gravitational inter action. During the period of particles thermal oscillations (tr and lb), their in- stant velocity, acceleration and corresponding forces alt ernatively and strongly change. The change of wave B instant group velocity, averaged during the molecule oscillation period in composition of the (a) and (b) states o f the eﬀectons, de- termines the average acceleration: /bracketleftBigg aa,b gr=dva,b gr dt=va,b gr T=vgrνa,b/bracketrightBigg1,2,3 tr,lb(10.38) We keep in mind that group velocities, impulses and wave B len gth in (a) and (b) states of the eﬀectons are equal, in accordance with our m odel. Corresponding to (10.38) forces: /bracketleftbig Fa,b=maa,b gr/bracketrightbig1,2,3 tr,lb(10.39) The energies of molecules in (a) and (b) states of the eﬀecton s also can be expressed via accelerations: /bracketleftBig Ea,b=hνa,b=Fa,bλ=maa,b·λ=maa,b(va,b ph/νa,b)/bracketrightBig1,2,3 tr,lb(10.40) From (10.40) one can express the accelerations of particles in the primary eﬀec- tons of condensed matter, using their phase velocities as a w aves B: /bracketleftBigg aa,b gr=h(νa,b)2 mva,b ph/bracketrightBigg1,2,3 tr,lb(10.41) The accelerations of particles in composition of secondary eﬀectons have a sim- ilar form: /bracketleftBigg ¯aa,b gr=h(¯νa,b)2 m¯va,b ph/bracketrightBigg1,2,3 tr,lb(10.42) These parameters are important for understanding the dynam ic properties of condensed systems. The accelerations of the atoms, forming primary and sec- ondary eﬀectons can be calculated, using eqs.(3.35; 3.36) t o determine phase velocities and eqs. (3.5; 3.6 and 3.16; 3.17) to ﬁnd a frequen cies. 124Multiplying (10.41) and (10.42) by the atomic mass m, we derive the most probable and mean forces, acting upon the particles in both s tates of primary and secondary eﬀectons in condensed matter: /bracketleftBigg Fa,b gr=h(νa,b)2 va,b ph/bracketrightBigg1,2,3 tr,lb/bracketleftBigg ¯Fa,b gr=h(¯νa,b)2 ¯va,b ph/bracketrightBigg (10.43) The comparison of calculated accelerations with empirical data of the M¨ ossbauer eﬀect - supports the correctness of our approach. According to eq.(3.5) in the low temperature range, when hνa<< kT , the frequency of secondary tr and lb eﬀectons in the (a) state can be estimated as: νa=νa exp/parenleftbighνa kT/parenrightbig −1≈kT h(10.44) For example, at T= 200 Kwe have ¯ νa≈4·1012s−1. If the phase velocity in eq.(10.42) is taken equal to ¯ va ph= 2.1·105cm/s and the mass of water molecule: m= 18·1.66·10−24g= 2.98·10−23g, then from (10.42) we get the acceleration of molecules in com position of sec- ondary eﬀectons of ice in (a) state: ¯aa gr=h(¯νa)2 m¯va ph= 1.6·1016cm/s2(10.45) This value is about 1013times more than that of free fall acceleration ( g= 9.8·102cm/s2), which agrees well with experimental data, obtained for so lid bodies (Wertheim, 1964). Accelerations of H2Omolecules in composition of primary librational eﬀec- tons ( aa gr) in the ice at 200K and in water at 300K are equal to: 0 .6·1013cm/s2 and 2·1015cm/s2, correspondingly. They also exceed to many orders the free fall acceleration. It was shown experimentally (Sherwin, 1960), that heating o f solid body leads to decreasing of gamma-quanta frequency (red Doppler shift) i.e. increas- ing of corresponding quantum transitions period. This can b e explained as the relativist time-pace decreasing due to elevation of averag e thermal velocity of atoms. The thermal vibrations of particles (atoms, molecules) in c omposition of primary eﬀectons as a partial Bose-condensate are coherent . The increasing of such clusters dimensions, determined by most probable wa ve B length, as a result of cooling, pressure elevation or under magnetic ﬁe ld action leads to enhancement of coherent regions. Each coherently vibrating cluster of particles with big alt ernating accelerations, like librational and translational eﬀecto ns is a source of coherent gravitational waves. 125The frequency of these vibro-gravitational waves (VGW) is e qual to fre- quency of particles vibrations (i.e. frequency of the eﬀect ons in aorbstates). The amplitude of VGW ( AG) is proportional to the number of vibrating coher- ently particles ( NG) in composition of primary eﬀectons: AG∼NG∼Vef/(V0/N0) = (1 /nef)/(V0/N0) (10.46) The resonant long-distance gravitational interaction bet ween coherent clusters of the same body or between that of diﬀerent bodies is possibl e. The formal description of this vibro-gravitational interaction (VGI ) could be like that of distant macroscopic Van der Waals interaction. Diﬀerent patterns of virtual Bose-condensate of standing g ravitational waves in vacuum represent the vibro-gravitational replica (VGR) of condensed matter. Important role of proposed here distant resonant VIBRO - GRA V- ITATIONAL INTERACTION (VGI) in elementary acts of percepti on and memory can be provided by coherent primary librational w ater eﬀectons in microtubules of the nerve cells (see article: ”H ierarchic Model of Consciousness” by Kaivarainen, 2000). 11. ENTROPY-INFORMATIONAL CONTENT OF MATTER, SLOW RELAXATION, MACROSCOPIC OSCILLATIONS. AND EFFECTS OF MAGNETIC FIELD 11.1. Theoretical background One of the consequences of our concept is of special interest . It is the possibil- ity for oscillation processes in solids and liquids. The law of energy conservation is not violated thereupon because the energies of two quasip article subsystems related to eﬀectons and deformons, can change in opposite ph ases. The total internal energy of matter keeps almost constant. The equilibrium shift between subsystems of condensed matt er can be in- duced by any external factor, i.e. pressure or ﬁeld. The rela xation time, neces- sary for system to restore its equilibrium, corresponding t o minimum of potential or free energy after switching oﬀ external factor can be term ed ”memory” of system. The energy redistribution between primary and secondary eﬀ ecton and de- formon subsystems may have a periodical character, coupled with the oscillation of the ( a⇔b) equilibrium constant of primary eﬀectons ( Ka⇔b) and correlated oscillations of primary electromagnetic deformons concen tration if dissipation processes are weak or reversible. According to our model (Ta ble 1) the ( a→b) transition of primary eﬀecton is related to photon absorpti on, i.e. a decrease in primary electromagnetic deformon concentration, while the (b→a) transi- tion on the contrary, radiate photons. If, therefore, the [ a⇔b] and/bracketleftbig ¯a⇔¯b/bracketrightbig equilibriums are shifted right ward, and equilibrium const antsKa⇔band¯Ka⇔b decreases, then concentrations of primary and secondary de formons ( ndand ¯nd) also decreases. If Ka⇔bgrows up, i.e. the concentration of primary eﬀectons in a-states increases, then ndincreases. We remind that ( a) and ( b) states of the primary eﬀectons correspond to the more and less stable m olecular clusters 126(see Introduction). In accordance with our model, the stron g interrelation exists between dynamic equilibrium of primary and secondary eﬀect ons. Equilibrium of primary eﬀectons is more sensitive to any perturbations. However, the equi- librium shift of secondary eﬀectons aﬀect the total interna l energy, the entropy change and possible mass defect (see below) stronger than th at of primary ef- fectons. As we have shown (Fig. 29, 30), the scattering ability of A-st ates is more than two times as high as that of B-states. Their polarizabil ity, refraction index and dielectric permeability are also higher. It makes possi ble to register the oscillations in the condensed matter in diﬀerent ways. In accordance with our theory the oscillation of refraction index must induce the corresponding changes of viscosity and self-diﬀusion i n condensed matter. The diﬀusion variations are possible, for example, in solut ions of macromolecules or other Brownian particles. In such a way self-organizatio n in space and time gradually may originate in appropriate solvents, solution s, colloid systems and even in solid bodies. The period and amplitude of these oscillations depend on the times of relax- ation processes which are related to the activation energy o f equilibrium shifts in the eﬀectons, polyeﬀectons or coherent superclusters of primary eﬀectons subsystems. The reorganizations in the subsystems of translational and librational ef- fectons, macro- and supereﬀectons, as well as chain-like po lyeﬀectons, whose stabilities and sizes diﬀer from each other, must go on at diﬀ erent rates. It should, therefore, be expected that in the experiment the pr esence of several os- cillation processes would be revealed. These processes are interrelated but going with diﬀerent periods and amplitudes. Concomitant oscilla tions of self-diﬀusion rate also must be taken into account. In such a way Prigogin’s dissipative struc- tures could be developed (Prigogin, 1984). Instability in t he degree of ordering in time and space is accompanied by the slow oscillation of en tropy of the whole macroscopic system. The coherent extraterrestrial cosmic factors and gravitat ional instabilities can induce long relaxation and oscillation processes in wat er and other kind of condensed matter (Udaltsova, et. al., 1987). 11.2. The entropy - information content of matter as a hierar chic system The statistical weigh for macrosystem (P), equal to number o f microstates (W), corresponding to given macrostate, neces sary for entropy calculation could be presented as: W=N! N1!·N2!·. . .·Nq!(11.1) where: N=N1+N2+...Nq (11.2) is the total number of molecules in macrosystem; 127Niis the number of molecules in the i-th state; qis the number of independent states of all quasiparticles in macrosys- tem. We can subdivide macroscopic volume of 1 cm3into 24 types of quasiparticles in accordance with our hierarchic model (see Table 1). In turn, each type of the eﬀectons (primary, secondary, macr o- and super- eﬀectons) is subdivided on two states: ground (a,A) and exci ted (b,B) states. Taking into account two ways of the eﬀectons origination - du e to thermal trans- lations (tr) and librations (lb), excitations, related to [ lb/tr] convertons, macro- and super deformons, the total number of independent states is 24 also. It is equal to number of independent relative probabilities of ex citations, composing partition function Z (see eq.4.2 ). Consequently, we have: q= 24 Thenumber of molecules, in the unit of volume of condensed matter (1cm3), participating in each of 24 excitation states ( i) can be calculated as: Ni=(v)i V0/N0·ni·Pi Z=N0 V0Pi Z(11.3) where: ( v)i= 1/niis the volume of (i) quasiparticle, equal to reciprocal valu e of its concentration ( ni);N0andV0are Avogadro number and molar volume, correspondingly; Zis partition function and Piare relative probabilities of in- dependent excitations in composition of Z(eq.4.2). The total number of molecules of (i)-type of excitation in an y big volume of matter ( VMac) is equal to Ni Mac=NiVMac=VMacN0 V0Pi Z(11.3a) Now we can calculate the statistical weight and entropy from eqs.(11.1 and 11.4). For large values of Niit is convenient to use a Stirling formula: Ni= (2πN)1/2(N/e)N·exp(Θ /12N)∼(2πN)1/2(N/Θ)N(11.3b) Using this formula and (11.1), one can obtain the following e xpression for en- tropy: S=k·lnW=−k·q/summationdisplay i(Ni+1 2)lnNi+ const = S1+S2+...Si(11.4) From this eq. we can see that the temperature increasing or [s olid→liquid] phase transition will lead to the entropy elevation: ∆S=SL−SS=k·ln(WL/WS)>0 (11.5) 128It follows from (11.4) and (11.3) that under conditions when (Pi) and Niun- dergoes oscillations it can lead to oscillations of contrib utions of diﬀerent types of quasiparticles to the entropy of system and even to oscill ations of total en- tropy of system as an additive parameter. The coherent oscil lations of Piand Nican be induced by diﬀerent external ﬁelds: acoustic, electr omagnetic and gravitational. Macroscopic autooscillations may arise sp ontaneously also in the sensitive and highly cooperative systems. Experimental evidence for such phenomena will be discussed in the next section. The notions of probability of given microstate ( pi= 1/W), entropy ( Si) and information ( Ii) are strongly interrelated. The smaller the probability th e greater is information (Nicolis 1986): Ii= lg21 pi=−lg2pi= lg2Wi (11.6) where piis deﬁned from the Boltzmann distribution as: pi=exp(−Ei/kT)/summationtext∞ m=0exp(−nmhνi/kT)(11.7) where n mis quantum number; h is the Plank constant; Ei=hνiis the energy of (i)-state. There is strict relation between the entropy and informatio n, leading from comparison of (11.6), (11.1 and 11.4): Si= (kBln 2)Ii= 2.3·10−24Ii (11.8) The information entropy is given as expectation of the infor mation in the system (Nicolis,1986; Haken, 1988). < I > = ΣPilg2(1/pi) =−Σpilg2(pi) (11.9) From (26) and (22) we can see that variation of probability piand/or Niin (20) will lead to changes of entropy and information, characteri zing the matter as a hierarchical system. Thereduced information (entropy), characterizing its quality , related to selected collective excitation of any type of condensed m atter, we introduce here as a product of corresponding component of information [Ii] to the number of molecules (atoms) with similar dynamic properties in com position of this excitation: qi= (vi/vm) =N0/(V0ni) (11.9a) where: vi= 1/niis the volume of quasiparticle, reversible to its concentra tion (ni);vm=V0/N0is the volume, occupied by one molecule. 129The product of (11.9) and (11.9a), i.e. the reduced information gives the quantitative characteristic not only about quantity but al so about the quality of the information: (Iq)i=pilg2(1/pi)·N0/(V0ni) (11.9b) This new formula could be considered as a useful modiﬁcation of known Shennon equation. 11.3. Experimentally revealed macroscopic oscillations A series of experiments was conducted in our laboratory to st udy macro- scopically coherent oscillations in the buﬀer (pH 7.3) cont aining 0.15 M NaCl as a control system and immunoglobulin G solutions in this buﬀe r at the following concentrations: 3 ·10−3; 6·10−3; 1.2·10−2and 2.4·10−2mg/ml . The turbidity ( D∗) of water and the solutions were measured every 10 sec- onds with the spectrophotometer at λ= 350 nm. Data were obtained automat- ically with the time constant 5 s during 40 minutes. The numbe r ofD∗values in every series was usually equal to 256. The total number of t he fulﬁlled series was more than 30. The time series of D∗were processed by the software for time series analysis. The time trend was thus subtracted and the autocovariance fu nction and the spectral density were calculated. The empty quartz cuvette with the optical path about 1 cm were used as a base control. Only the optical density of water and water dissolved substa nces, which really exceeded background optical density in the control s eries were taken into account. It is shown that the noise of the photoelectronic mu ltiplier does not contribute markedly to dispersion of D∗. The measurements were made at temperatures of 17 ,28,32, and 370. The period of the trustworthily registered oscillation proces ses related to changes inD∗, had 2 to 4 discrete values over the range of (30 −600) sec under our conditions. It does not exclude the fact that the autooscill ations of longer or shorter periods exist. For example, in distilled water at 320Cthe oscillations of the scattering ability are characterized by periods of 30 , 120 and 600 s and the spectral density amplitudes 14, 38 and 78 (in relative un its), respectively. With an increase in the oscillation period their amplitude a lso increases. At 280Cthe periods of the values 30, 41 and 92 seconds see have the cor responding normalized amplitudes 14.7, 10.6 and 12.0. Autooscillations in the buﬀer solution at 280Cin a 1 cm wide cuvette with the optical way length 1 cm (i.e. square section) are charact erized with periods: 34,52,110 and 240 s and the amplitudes: 24 ,33,27 and 33 relative units. In the cuvette with a smaller (0.5 cm) or larger (5 cm) optical wa velength at the same width (1 cm) the periods of oscillations in the buﬀer cha nge insigniﬁcantly. However, amplitudes decreased by 50% in the 5 cm cuvette and b y 10-20% in the 0.5 cm-cuvette. This points to the role of geometry of space w here oscillations occur, and to the existence of the ﬁnite correlation radius o f the synchronous processes in the volume. But this radius is macroscopic and c omparable with the size of the cuvette. 130The dependence of the autooscillations amplitude on the con centration of the protein - immunoglobulin G has a sharp maximum at the conc entration of 1.2·10−2mg/ml . There is a background for considering it to be a manifesta- tion of the hydrodynamic Bjorkness forces between the pulsi ng macromolecules (K¨ aiv¨ ar¨ ainen, 1987). Oscillations in water and water solutions with nearly the sa me periods have been registered by the light-scattering method by Cherniko v (1985). Chernikov (1990d) has studied the dependence of light scatt ering ﬂuctua- tions on temperature , mechanic perturbation and magnetic ﬁ eld in water and water hemoglobin and DNA solution. It has been shown that an i ncrease in temperature results in the decline of long-term oscillatio n amplitude and in the increase of short-time ﬂuctuation amplitude. Mechanical m ixing removes long- term ﬂuctuations and over 10 hours are spent for their recove ry. Regular ﬂuc- tuations (oscillations) appear when the constant magnetic ﬁeld above 240 A/m is applied; the ﬂuctuations are retained for many hours afte r removing the ﬁeld. The period of long-term oscillations has the order of 10 minu tes. It has been assumed that the maintenance of long-range correlation of m olecular rotation- translation ﬂuctuation underlies the mechanism of long-te rm light scattering ﬂuctuations. It has been shown (Chernikov, 1990b) that a pulsed magnetic ﬁ eld (MF), like constant MF, gives rise to light scattering oscillations in water and other liquids containing H atoms: glycerin, xylol, ethanol, a mixture of u nsaturated lipids. All this liquids also have a distinct response to the constan t MF. ”Spontaneous” and MF-induced ﬂuctuations are shown to be associated with t he isotropic com- ponent of scattering. These phenomena do not occur in the non proton liquid (carbon tetrachloride) and are present to a certain extent i n chloroform (con- taining one hydrogen atom in its molecule). The facts obtain ed indicate an important role of hydrogen atoms and cooperative system of h ydrogen bonds in ”spontaneous” and induced by external perturbations macro scopic oscillations. The understanding of such phenomena can provide a physical b asis for of self-organization (Prigogin, 1980, 1984, Babloyantz, 198 6), the biological sys- tem evolution (Shnol, 1979, Udaltsova et al., 1987), and che mical processes oscillations (Field and Burger, 1988). It is quite probable that macroscopic oscillation processe s in biological liq- uids, e.g. blood and liquor, caused by the properties of wate r are involved in animal and human physiological processes. We have registered the oscillations of water activity in the protein-cell system by means of light microscopy using the apparatus ”Morphoqua nt”, through the change of the erythrocyte sizes, the erythrocytes being ATP -exhausted and fulﬁlling a role of the passive osmotic units. The revealed o scillations have a few minute-order periods. Preliminary data obtained from the analysis of oscillation processes in the human cerebrospinal liquor indicate their dependence on so me pathology. Per- haps, the autooscillations spectrum of the liquor can serve as a sensitive test for the physiological status of the organism. The liquor is a n electrolyte and its autooscillations can be modulated with the electromagn etic activity of the brain. The activity of the central nervous system and the biologica l rhythms of the organism may be dependent on the oscillation processe s in the liquor. If it is the case, then the directed inﬂuence on these autooscil- 131lation processes, for example, by means of external electro magnetic ﬁeld of resonant frequency makes it possible to regulate the state of the organism. Such way of correction of biorhythms could be s imple and eﬀective. During my stay in laboratory of G.Salvetty in the Institute o f Atomic and Molecular Physics in Pisa (Italy) in 1992, the oscillations of heat capacity [ Cp] in 0.1 M phosphate buﬀer (pH7) and in 1% solution of lysozyme in t he same buﬀer at 200Cwere revealed. The sensitive adiabatic diﬀerential microc alorimeter was used for this aim. The biggest relative amplitude changing: [∆Cp]/[Cp]∼(0.5± 0.02)% occurs with period of about 24 hours, i.e. correspondin g to circadian rhythm. Such oscillations can be stimulated by the variation of magn etic and gravitational conditions of the Earth in location of exp eriment during this 24h cycle. 11.4. Phenomena in water and aqueous systems, induced by magnetic ﬁeld In the works of (Semikhina and Kiselev, 1988, Kiselev et al., 1988, Berezin et al., 1988) the inﬂuence of the weak magnetic ﬁeld was revea led on the di- electric losses, the changes of dissociation constant, den sity, refraction index, light scattering and electroconductivity, the coeﬃcient o f heat transition, the depth of super-cooling for distilled water and for ice also. This ﬁeld used as a modulator a geomagnetic action. The absorption and the ﬂuorescence of the dye (rhodamine 6G) and protein in solutions also changed under the action of weak ﬁelds on wa ter. The latter circumstance reﬂects feedback links in the guest-host, or s olute -solvent system. The inﬂuence of constant and variable magnetic ﬁelds on wate r and ice in the frequency range 104−108Hzwas studied. The maximum sensitivity to ﬁeld action was observed at the frequency νmax= 105Hz. In accordance with our calculations, this frequency corresponds to frequency of superdeformons excitations in water (see Fig..12.d). A few of physical parameters changed after the long (nearly 6 hour) inﬂuence of the variable ﬁelds ( ˜H), modulating the geomagnetic ﬁeld of the tension [ H= Hgeo] with the frequency ( f) in the range of (1 −10)·102Hz(Semikhina and Kiselev, 1988, Kiselev et al., 1988): H=Hcos2πft (11.10) In the range of modulating magnetic ﬁeld (H) tension from 0 .08A/mto 212 A/m theeight maxima of dielectric losses tangent in the above mentioned ( f) range were observed. Dissociation constant decreases more than other param- eters (by 6 times) after the incubation of ice and water in mag netic ﬁeld. The relaxation time (”memory”) of the changes, induced in water by ﬁelds was in the interval from 0.5 to 8 hours. The authors interpret the experimental data obtained as the inﬂuence of magnetic ﬁeld on the probability of proton transfer along th e net of hydrogen bonds in water and ice, which lead to the deformation of this n et. 132Theequilibrium constant for the reaction of dissociation: H2O⇔OH−+H+(11.11) in ice is less by almost six orders ( ≃106) than that for water. On the other hand the values of the ﬁeld-induced eﬀects in ice are several times more than in water, and the time for reaching them in ice is less. So, the above in terpretation is doubtful. In the framework of our concept all the aforementioned pheno m- ena could be explained by the shift of the (a⇔b)equilibrium of primary translational and librational eﬀectons to the left .In turn, this shift stimulates polyeﬀectons or coherent superclust ers growth, under the inﬂuence of magnetic ﬁelds. Therefore, parameters such as t he refraction in- dex, dielectric permeability and light scattering have to e nhance in-phase, while theH2Odissociation constant depending on the probability of supe rdeformons must decrease. The latter correlate with declined electric conductance. As far, the magnetic moments of molecules within the coheren t su- perclusters or polyeﬀectons formed by primary librational eﬀectons are additive parameters, then the values of changes induced by mag- netic ﬁeld must be proportional to polyeﬀectons sizes. Thes e sizes are markedly higher in ice than in water and decrease with inc reas- ing temperature. Inasmuch the eﬀectons and polyeﬀectons interact with each o ther by means of phonons (i.e. the subsystem of secondary deformons), and the velocity of phonons is higher in ice than in water, then the saturation of all concomitant eﬀects and achievement of new equilibrium state in ice is fas ter than in water. The frequencies of geomagnetic ﬁeld modulation, at which ch anges in the properties of water and ice have maxima can correspond to the eigen-frequencies of the [ a⇔b] equilibrium constant of primary eﬀectons oscillations, d etermined by [assembly ⇔disassembly] equilibrium oscillations for coherent super clusters or polyeﬀectons. The presence of dissolved molecules (ions, proteins) in wat er or ice can inﬂu- ence on the initial [ a⇔b] equilibrium dimensions of polyeﬀectons and,consequentl y the interaction of solution with outer ﬁeld. Narrowing of1H-NMR lines in a salt-containing water and calcium bicar- bonate solution was observed after magnetic ﬁeld action. Th is indicates that the degree of ion hydration is decreased by magnetic treatment. On the other hand, the width of the resonance line in distilled water remains unchanged after 30 minute treatment in the ﬁeld (135 kA/m ) at water ﬂow rate of 60 cm/s (Klassen, 1982). The hydration of diamagnetic ions ( Li+, Mg2+, Ca2+) decreases, while the hydration of paramagnetic ions ( Fe3+, Ni2+, Cu2+) increases. It leads from corresponding changes in ultrasound velocity in ion soluti ons (Duhanin and Kluchnikov, 1975). There are numerous data which pointing to an increase the coa gulation of diﬀerent particles and their sedimentation velocity after magnetic ﬁeld treat- ment. These phenomena provide a reducing the scale formatio n in heating sys- tems, widely used in practice. Crystallization and polymer ization also increase in magnetic ﬁeld. It points to decrease of water activity. Increasing of refraction index (n) of water and its dielectr ic per- meability (ǫ≃n2)with in-phase enhancement of liquid viscosity (Mi- 133nenko, 1981) are in total accordance with our hierarchic vis cosity theory. It follows from our mesoscopic model that the increase of ( n) is related to the increase of molecular polarizability ( α) due to the shift of ( a⇔b)tr,lb equilibrium of primary eﬀectons leftward under the action o f magnetic ﬁeld. On the other hand, distant Van der Waals interactions and conse quently dimensions of primary eﬀectons depend on α. This explains the elevation of surface tension of liquids after magnetic treatment. The leftward shift of ( a⇔b)tr,lbequilibrium of primary eﬀectons must lead to decreasing of water activity due to (n2) increasing and structural fac- tor (T/U tot) decreasing its structure ordering. Corresponding change s in the vapor pressure, freezing, and boiling points, coagulation , polymerization and crystallization are the consequences of this shift and wate r activity decreasing. It follows from our theory that any changes in condensed matt er properties must be accompanied by change of such parameters as: 1) density; 2) sound velocity; 3) positions of translational and librational bands in osci llatory spectra; 4) refraction index. Using our equations and computer simulations by means of ela borated com- puter program: Comprehensive Analyzer of Matter Propertie s (CAMP), it is possible to obtain from these changes very detailed informa tion (more than 250 parameters) about even small perturbations of matter on mes o- and macroscopic levels. Available experimental data indicate that all of above ment ioned 4 exper- imental parameters of water have been changed indeed after m agnetic treat- ment. Minenko (1981) has shown that bidistilled water density increases by about 0 .02% after magnetic treatment (540 kA/m , ﬂow rate 80 cm/s).Sound velocity in distilled water increases to 0.1% after treatment under c onditions: 160kA/m and ﬂow rate 60 cm/s. Thepositions of the translational and librational bands of water were also changed after magnetic treatment in 415 kA/m (Klassen, 1982). 11.5. Coherent radio-frequency oscillations in water, rev ealed by C. Smith It was shown experimentally by Smith (1994) that the water di splay a co- herent properties in macroscopic scale and memory. He shows that water is capable of retaining the frequency of an alternating magnet ic ﬁeld. For a tube of water placed inside a solenoid coil, the threshold for the alternating magnetic ﬁeld, potentiating electromagnetic frequencies into wate r, is 7.6 µT(rms). He comes to conclusion that the frequency information is carri ed on the magnetic vector potential. He revealed also that in a course of yeast cells culture synch ronously di- viding, the radio-frequency emission around 1 MHz (1061/s), 7-9 MHz (7- 9×1061/s) and 50-80 MHz (5-9 ×1071/s) with very narrow bandwidth (˜50 Hz) might be observed for a few minutes. These frequencies could correspond to frequencies of diﬀer ent water collective excitations, introduced in our Hierarchic theory, like [lb /tr] macroconvertons, 134the [a⇋b]lbtransitons, etc. (see Fig. 12), taking into account the devi ation of water properties in the colloid and biological systems as re spect to pure one. Cyril Smith has proposed that the increasing of coherence ra dius in water could be a consequence of coherent water clusters associati on due to Joseph- son eﬀect (Josephson, 1965): tunneling of molecules betwee n clusters. As far primary librational eﬀectons are resulted from partial Bose-condensation of molecules, this idea looks quite acceptable in the framew ork of our Hierarchic theory. The coherent macroscopic oscillations in tube with water, r evealed by C.Smith could be induced by coherent electromagnetic radiation of m icrotubules of cells, produced by correlated intra-MTs water excitations in acco rdance with our Hi- erarchic model of consciousness (see Kaivarainen, 1998 and ”New Articles” in homepage: http://www.karelia.ru/˜alexk). The biological eﬀects of magnetically treated water can hav e very impor- tant applications. For example, hemolysis of erythrocytes is more vigorous in magnetically pretreated physiological solutions (Trinch er, 1967). Microwave radiation induces the same eﬀect (Il’ina et al., 1979). But a fter boiling such ef- fects in the treated solutions have been disappeared. It is s hown that magnetic treatment of water strongly stimulates the growth of corn an d plants (Klassen, 1982). Now it is obvious that a systematic research program is neede d to understand the physical background of multilateral eﬀects of magnetiz ed water. 11.6. Inﬂuence of weak magnetic ﬁeld on the properties of sol id bodies It has been established that as a result of magnetic ﬁeld acti on on solids with interaction energy ( µBH) much less than kT, many properties of matter such as hardness, parameters of crystal cells and others cha nge signiﬁcantly. The short-time action of magnetic ﬁeld on silicon semicondu ctors is followed by a very long (many days) relaxation process. The action of m agnetic ﬁeld was in the form of about 10 impulses with a length of 0.2 ms and an am plitude of about 105A/m. The most interesting fact was that this relaxation had an o scil- latory character with periods of about several days (Maslov sky and Postnikov, 1989). Such a type of long period oscillation eﬀects has been found i n magnetic and nonmagnetic materials. This points to the general nature of the macroscopic oscilla tion phenomena in solids and liquids. The period of oscillations in solids is much longer than in li quids. This may be due to stronger deviations of the energy of ( a) and ( b) states of primary eﬀectons and polyeﬀectons from thermal equilibrium and muc h lesser probabil- ities of transiton and deformon excitation. Consequently, the relaxation time of (a⇔b)tr,lbequilibrium shift in solids is much longer than in liquids. T he oscillations originate due to instability of dynamic equil ibrium between the sub- systems of eﬀectons and deformons. 13511.7. Possible mechanism of perturbations of nonmagnetic m aterials under magnetic treatment We shall try to discuss the interaction of magnetic ﬁeld with diamagnetic matter like water as an example. The magnetic susceptibilit y (χ) of water is a sum of two opposite contributions (Eisenberg and Kauzmann, 1969): 1) average negative diamagnetic part, induced by external m agnetic ﬁeld: ¯χd=1 2(χxx+χyy+χzz)∼=−14.6(±1.9)·10−6(11.12) 2) positive paramagnetism related to the polarization of wa ter molecule due to asymmetry of electron density distribution, existing wi thout external mag- netic ﬁeld. Paramagnetic susceptibility ( χp) ofH2Ois a tensor with the follow- ing components: χp xx= 2.46·10−6;χp yy= 0.77·10−6;χp zz= 1.42·10−6(11.13) The resulting susceptibility: χH2= ¯χd+ ¯χp∼=−13·10−6(11.14) The second contribution in the magnetic susceptibility of w ater is about 10 times lesser than the ﬁrst one. But the ﬁrst contribution to the mag netic moment of water depends on external magnetic ﬁeld and must disappear w hen it is switched out in contrast to second one. The coherent primary librational eﬀectons of water even in l iquid state con- tain about 100 molecules/bracketleftBig (nef M)lb≃100/bracketrightBig at room temperature (Fig.4a ). In ice (nef M)lb≥104. In (a)-state the vibrations of all these molecules are sync hro- nized in the same phase, and in (b)-state - in counterphase. C orrelation of H2O forming eﬀectons means that the energies of interaction of w ater molecules with external magnetic ﬁeld are additive: ǫef=nef M·µpH (11.15) In such a case this total energy of eﬀecton interaction with ﬁ eld may exceed thermal energy: ǫef> kT (11.16) In the case of polyeﬀectons formation this inequality becom es much stronger. It follows from our model that interaction of magnetic ﬁeld w ith (a)-state of the eﬀectons must be stronger than that with ( b)-state due to the additivity of the magnetic moments of coherent molecules: 136ǫef a> ǫef b(11.17) Consequently, magnetic ﬁeld shifts ( a⇔b)tr,lbequilibrium of the eﬀectons leftward. At the same time it minimizes the potential energy of matter, because potential energy of ( a)-state ( Va) is lesser than ( Vb): Va< Vband Ea< E b, (11.18) where Ea=Va+Ta kin;Eb=Vb+Tb kinare total energies of the eﬀectons. We keep in mind that the kinetic energies of ( a) and ( b)-states are equal: Ta kin=Tb kin=p2/2m. These energies decreases with increasing of the eﬀectons di mensions, deter- mined by the most probable impulses in selected directions: λ1,2,3=h/p1,2,3 The energy of interaction of magnetic ﬁeld with deformons as a transition state of eﬀectons must be even less than ǫef bdue to lesser order of molecules in this state and reciprocal compensation of their magnetic moment s: ǫd< ǫef b≤ǫef a (11.19) This important inequality means that as a result of external magnetic ﬁeld action the shift of ( a⇔b)tr,lbleftward is reinforced by leftward shift of equilib- rium [eﬀectons ⇋deformons] subsystems of matter. If water is ﬂowing in a tube it increases the relative orienta tions of all ef- fectons in volume and stimulate the coherent superclusters formation. All the above discussed eﬀects must increase. Similar ordering phe nomena happen in a rotating tube with liquid. After switching oﬀ the external magnetic ﬁeld the relaxatio n ofinduced ferro- magnetism in water begins. It may be accompanied by the oscillatory beh avior of (a⇔b)tr,lbequilibrium. All the experimental eﬀects discussed above c an be explained as a consequence of orchestrated in volume ( a⇔b) equilibrium oscillations. Remnant ferromagnetism in water was experimentally establ ished using a SQUID superconducting magnetometer by Kaivarainen et al. i n 1992 at Phys- ical department of University of Turku (unpublished data). In these experiments water was treated in constant magnetic ﬁeld 50 Gfor two hours. Then it was frozen and after switching oﬀ external magnetic ﬁeld the remnant ferromagnetism was registered at helium temperatu re. Even at this low temperature the slow relaxation of ferromagnetic signal am plitude was revealed. These results point to the correctness of the proposed mecha nism of magnetic ﬁeld - water interaction and perturbation. In the future thi s mechanism can be developed to quantitative level. The attempt to make a theory of magnetic ﬁeld inﬂuence on wate r based on other model were made already (Yashkichev, 1980). However, this theory does 137not take into account the quantum properties of water and can not be considered as a complete one. The comprehensive material obtained by group of S. Shnol (19 87, 1998) when studying macroscopic oscillations of very diﬀerent na ture reveals their fundamental character and their dependence on gravitation al coherent global perturbations. For more detailed discussion of like phenomena see my articl e: Dynamic model of wave-particle duality, Bi-vacuum and Superuniﬁca tion”, placed at http://arxiv.org/abs/physics/0003001 12. INNOVATION, BASED ON NEW THEORY: Comprehensive Analyzer of Matter Properties (CAMP) [see: www.karelia.ru/˜alexk (CAMP)] The set of formulae obtained in our theory allows to calculat e about 300 parameters of any condensed matter (liquid or solid). Most o f them are hidden, i.e. inaccessible for direct experimental measurements. Simulations evaluation of these parameters can be done usin g our computer program: CAMP (copyright 1997) and the following experimen tal methods: 1. Far-middle IR spectroscopy for determination the positi ons of transla- tional or librational bands: (30-2500) cm-1; 2. Sound velocimetry; 3. Dilatometry or densitometry, for molar volume or density registration; 4. Refractometry. Corresponding data should be obtained simultaneously at th e same temper- ature and pressure from the SAME SAMPLE in ideal case. Among t he param- eters of matter evaluated are so important as: internal ener gy, heat capacity, thermal conductivity, viscosity, coeﬃcient of self-diﬀus ion, surface tension, sol- vent activity, vapor pressure, internal pressure, paramet ers of all types of quasi- particles (concentration, volume, dimensions, energy, pr obability of excitation, life-time) and many others. This leads to idea of new optoacoustic device: Comprehensiv e Analyzer of Matter Properties (CAMP), which may provide a huge amount of data of any condensed system under study. The most complicated and expe nsive component of CAMP is FT-IR spectrometer for far and middle region. The m ost sensitive parameter is sound velocity. The less sensitive and stable p arameter is molar volume or density. One of possible CAMP conﬁguration should include special at tachment to FT-IR spectrometer, making it possible registration of reﬂ ection spectra in far/middle IR region. Such approach allows to study the prop erties of sam- ples with strong IR absorption (i.e. aqueous systems) and no n transparent mediums. The combination of modiﬁed FT-IR spectrometer wit h other equip- ment for simultaneous measurement of matter density and sou nd velocity and refraction index will provide 4 parameters, necessary for C AMP function. The sample cell for liquids and solids should have a shape, conve nient to make all these measurements simultaneously. The another conﬁguration of CAMP may include except FT-IR, t he Brillouin light scattering spectrometer. It makes possible simultan eous measurement of 138sound velocity (from the Doppler shift of side bands of Brill ouin spectra) and positions of intermolecular bands [tr and lb] in oscillator y spectra in the far IR. Our hierarchic theory of Brillouin light scattering gives m uch more information about condensed matter properties than conventional one. The interface of CAMP with personal computer will allow moni toring of very diﬀerent dynamic physical process in real time, using our co mputer program. Possible Applications for Comprehensive Analyzer of Matter Properties (CAMP) Applications to aqueous systems 1. Monitoring of drinking water and water based beverage phy sical proper- ties, related to taste and biological activity; 2. Monitoring of electromagnetic and acoustic pollution, u sing physical prop- erties of water as a test system (ecology problem); 3. In pharmaceutical technology - for monitoring of water pe rturbations, induced by vitamins and drugs at low physiologic concentrat ions. Correlation of water structure perturbations, induced by vitamins, dru gs, physical ﬁelds, with healing activity of solutions; 4. Study of colloid systems, related to paper technology. Mo nitoring of inﬂuence of electromagnetic and acoustic ﬁelds on physical parameters of the bulk and hydrated water for regulation of [coagulation - pep tization] equilibrium of colloids, aﬀecting the quality of paper; 5. In biotechnology and biochemistry: a wide range of proble ms, related to role of water in biosystems and water biopolymers interacti on (i.e. mechanism of cryoproteins action); 6. Mechanism of transition of ﬂow from the laminar to turbule nt one in pipe-lines and the ways of this process regulation by means o f electromagnetic and acoustic ﬁelds; 7. Evaluation of frequencies of cavitational ﬂuctuations o f water for the end of their eﬀective resonant stimulation. It may be useful for : a) de-infection of drinking water; b) stimulation of sonoluminiscense; c) d evelopment of pure energy technology; d) cold fusion stimulation. Application to nonaqueous systems 1. Fundamental research in all branches of condensed matter physics: ther- modynamics, dynamics, phase transitions, transport proce ss, surface tension, self-diﬀusion, viscosity, vapor pressure, etc. 2. Monitoring of new materials technology for searching the optimal condi- tions (T, P, physical ﬁelds) for providing the optimal param eters on mesoscopic and macroscopic scale for their best quality; 3. Study of mechanism of high-temperature superconductivi ty; 4. Study of mechanism of superﬂuidity. Comprehensive Analyzer of Matter Properties (CAMP) repres ents a basi- cally new type of scientiﬁc equipment, allowing to get incom parable big amount of information concerning physics of liquids or solids. It c an be very useful for investigation of dynamics and mesoscopic structure of pure matter as well as solid and liquid solutions, the colloid systems and host-gu est systems. 139The market for Comprehensive Analyzer of Matter Properties (CAMP) is free and due to its unique informational potential could be m uch bigger than that for IR, Raman or Brillouin spectrometers. CONCLUSION A quantum based new hierarchic quantitative theory, genera l for solids and liquids, has been developed. It is assumed, that anharmo nic oscillations of particles in any condensed matter lead to emergence of three -dimensional (3D) superposition of standing de Broglie waves of molecules, el ectromagnetic and acoustic waves. Consequently, any condensed matter could b e considered as a gas of 3D standing waves of corresponding nature. Our approa ch uniﬁes and develops the Einstein’s and Debye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, rep- resenting at certain conditions the molecular Bose condens ate, were analyzed, as a background of hierarchic model of condensed matter. The most probable de Broglie wave (wave B) length is determin ed by the ratio of Plank constant to the most probable impulse of molec ules, or by ratio of its most probable phase velocity to frequency. The waves B of molecules are related to their translations (tr) and librations (lb). As the quantum dynamics of condensed matter is anharmonic an d does not follow the classical Maxwell-Boltzmann distribution, the real most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular mesoscopic Bos e condensa- tion in solids and liquids at temperatures, below boiling po int. It is one of the most important results of new theory, which we have conﬁrmed by computer simulations on examples of water and ice and applying to Viri al theorem. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 1.Eﬀectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the eﬀectons (ﬂickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trandlb eﬀectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary eﬀectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of eﬀectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The ﬁrst condition m eans validity ofclassical approximation in description of the subsystems of the eﬀect ons. The second one points to quantum properties of coherent clusters due to mesoscopic Bose condensation (BC), in contrast to macrosco pic BC, pertinent for superﬂuidity and supercoductivity . The liquids are semiclassical systems because their primar y (tr) eﬀectons contain less than one molecule and primary (lb) eﬀectons - mo re than one 140molecule. The solids are quantum systems totally because both kind of t heir pri- mary eﬀectons (tr and lb) are mesoscopic molecular Bose cond ensates. These consequences of our theory are conﬁrmed by computer simulat ions. The 1st order [ gas→liquid ] transition is accompanied by strong decreas- ing of number of rotational (librational) degrees of freedo m due to emergence of primary (lb) eﬀectons and [ liquid →solid] transition - by decreasing of trans- lational degrees of freedom due to Bose-condensation of pri mary (tr) eﬀectons. In the general case the eﬀecton can be approximated by parall elepiped with edges determined by de Broglie waves length in three selecte d directions (1, 2, 3), related to symmetry of molecular dynamics. In the case of iso tropic molecular motion the eﬀectons’ shape is approximated by cube. The number of molecules in the volume of primary eﬀectons (tr and lb) is considered as the ”parameter of order” in our theory of 1st or der phase transi- tions. The in-phase oscillations of molecules in the eﬀectons corr espond to the eﬀecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the eﬀectons diﬀer in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the eﬀectons has a common feature with Fr¨ olich’s polar mode. The ( a→b) or (b→a) transition states of the primary eﬀectons (tr and lb), deﬁned as primary transitons, are accompanied by a chan ge in molecule polarizability and dipole moment without density ﬂuctuati ons. At this case they lead to absorption or radiation of IR photons, respecti vely. Superposition of three internal standing IR photons of diﬀe rent directions (1,2,3) - forms primary electromagnetic deformons (tr and l b). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density ﬂuctuations, leading to absorpt ion or radiation of phonons. Superposition of standing phonons in three directions (1,2 ,3), forms sec- ondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondar y eﬀectons and de- formons (tr and lb), localized in the volume of primary tr and lb electromag- netic deformons, lead to origination of macroeﬀectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeﬀec tons in the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereﬀectons. In turn, the simultaneous excitation of both: trandlb macrodeformons and macroconvertons in the same volume means origination of superdefor- mons. Superdeformons are the biggest (cavitational) ﬂuctuation s, leading to microbubbles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reﬂects all of possible combinations of the four basic ones [1-4], in troduced above. This set of collective excitations in the form of ”gas” of 3D stand ing waves of three types: de Broglie, acoustic and electromagnetic - is shown t o be able to explain virtually all the properties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which 141are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into accoun t their diﬀerent prop- erties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra ; [2]- Sound velocity; [3]- Density; [4]- Refraction index. The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coeﬃcient of self- diﬀusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. This is the ﬁrst theory able to predict all known experimenta l anomalies for water and ice. The conformity between theory and experiment is very good even without adjustable parameters. The hierarchic concept creates a bridge between micro- and m acro- phenom- ena, dynamics and thermodynamics, liquids and solids in ter ms of quantum physics. Computerized veriﬁcation of our Hierarchic theory of matte r on examples of water and ice has been performed, using special c omputer program: Comprehensive Analyzer of Matter Properties (CAM P, copyright, 1997, Kaivarainen). 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1 Relativistic integro-differential form of the Lorentz-Dirac equation in 3D without runaways Michael Ibison, Harold E. Puthoff Institute for Advanced Studies at Austin 4030 Braker Lane West, Suite 300 Austin, TX 78759, USA ibison@ntr.net , puthoff@aol.com Accepted for publication in Journal of Physics A 2 Abstract It is well known that the third-order Lorentz-Dirac equation admits ‘runaway’ solutions wherein the energy of the particle grows without limit, even when there is no external force. These solutions can be denied simply on physical grounds, and on the basis of careful analysis of the correspondence between classical and quantum theory. Nonetheless, one would prefer an equation that did not admit unphysical behavior at the outset. Such an equation - an integro-differential version of the Lorentz-Dirac equation – is currently available either in 1 dimension only, or in 3 dimensions only in the non-relativistic limit. It is shown herein how the Lorentz-Dirac equation may be integrated without approximation, and is thereby converted to a second-order integro-differential equation in 3D satisfying the above requirement. I.E., as a result, no additional constraints on the solutions are required because runaway solutions are intrinsically absent. The derivation is placed within the historical context established by standard works on classical electrodynamics by Rohrlich, and by Jackson. Introduction The Lorentz-Dirac equation (LDE) describes the motion of a classical charged particle subject to both an external force and self-interaction due to radiation. An undesirable characteristic is the prediction of an exponential (runaway) acceleration in the absence of an applied force. The source of the trouble may be traced to the third order derivative with respect to time. Since one would prefer a second order equation anyhow, a natural approach is to convert the original LDE into a second order equation by integrating over time. At the same time, one might take the opportunity to eliminate the runaway solution by a suitable choice for the constant of integration. This is the method cited by Jackson [1], as it applies to a non-relativistic (and thereby linearized) version of the LDE. It is successful in that runaway solutions are absent. The same approach was employed by Rohrlich [2] to the relativistic LDE, but without success; his resulting equation still permits runaway solutions. The attempt failed because he was unable to combine the acceleration and radiation parts (times an integrating factor) as a total differential in proper time. Jackson and Rohrlich are referred to herein because they are standard texts on classical theory. However, for an earlier review of the subject that is both lucid and thorough, the reader is referred to Erber [3]. The first appearance of the non-relativistic integro-differential form of the LDE is due to Haag [4], (and subsequently - in English - Plass [5]). It has been shown that the non-relativistic integro-differential form of the LDE is the finite-point limit of a finite-size (non-relativistic) model of the electron [6, 7, 8]. Since the latter is free of runaway solutions, this may be regarded as evidence in favor of the validity of the integro-differential form, over the original LDE. Also, (very importantly), Sharp [9] has shown that the non-relativistic integro-differential LDE corresponds to the quantum theory of a non-relativistic charge coupled to the quantized electromagnetic field (neither of which, therefore, display runaway solutions). Both these results point to the need for a relativistic generalization of the existing non-relativistic integro-differential version of the LDE. 3 Barut [10] has proposed a method to eliminate both the runaway and pre-acceleration behavior of the LDE by making the Abraham vector disappear when the external field disappears. However, as pointed out by Blanco [11], such an equation is essentially quite different from the original LDE. Jimenez and Hirsch [12] suggest that the non-relativistic LDE be supplemented by an external, stochastic, electromagnetic field, in the spirit of Stochastic Electrodynamics (see for instance [13])). This, they argue, has the effect of eliminating the undesirable runaway behavior without modification of the LDE (to an integro-differential form). Their program, though promising, potentially suffers from an externally-induced runaway problem unless the stochastic field is band-limited (which would be similar to supposing a finite-sized charge). Runaway solutions of the Lorentz-Dirac equation The Lorentz-Dirac equation in proper time is [1,2] ()2 00 0dama m au fdτττ−+ =, (1) where the force f can depend on τ explicitly, and implicitly via the position and its derivatives. {}{}0, aa aµ≡≡ ais the proper acceleration, {}{} { }0,, uu uµγ ≡≡ = uu is the proper velocity, and ()2 2 0 aa=− a.a, c = 1, and2 00 0 6emτπ ε= is (2/3) the time it takes for light to travel across the classical electron radius. The notorious runaway solution is most easily demonstrated in one dimension, wherein the LDE is easily linearized [2]. With the substitution ()() sinh dx d wττ= , one obtains from Eq. (1) 00 ww f mτ−=!! ! , (2) where f is the ordinary Newton force in the x direction. It is clear that even when there is no external force, (f = 0), w may increase without limit, since () 0 expw ττ " is a solution. This causes dx dτand γ to increase without limit, giving rise to the interpretation that the particle has accelerated to the speed of light and has acquired an infinite kinetic energy. The reason for the presence of such solutions may be traced to the intrinsically non-conservative nature of the equation of motion. It was conceived to account for losses due to radiation, but turns out to admit gains, presumably by the same mechanism. The non-relativistic integro-differential equation When the velocities are small compared to c, γ ≈ 1, dτ ≈ dt, and Eq. (1) becomes 00 0dmmdtτ−=aaf . (3) (This non-relativistic form of the LDE is also called the Abraham-Lorentz equation.) It suffers from the same runaway solution as Eq. (2) - the relativistic one-dimensional result written in hyperbolic co-ordinates. The traditional remedy [1] is to replace Eq. (3) with the integro-differential equation () 00 0smd s e t s τ∞ −=+∫af . (4) 4 It is readily verified upon substitution that the x that solve this equation are a subset of those that solve Eq. (3). It is also clear that, provided f vanishes in the remote future, the acceleration also vanishes in the remote future. Not only does this prescription eliminate the runaway solution, but it also restores the boundary condition requirements to those of a second order differential equation, e.g.: the position and velocity are given at some time. This time need not be when the force is zero (i.e. the remote past or the remote future); it may be any time. Though the runaway behavior is tamed, it is at the expense of an acausal connection between the applied force and the resulting acceleration. Specifically, it is seen from Eq. (4) that the acceleration depends on future forces (exhibits pre-acceleration). However, the temporal range, τ0, of that dependency, is such that pre-acceleration is too small to be observed on classical time scales. Rohrlich’s relativistic integro-differential equation It is carefully argued by Rohrlich [2] that runaway solutions must be denied by imposing a suitable constraint, i.e., a boundary condition on the acceleration. In this paper, we will be content with the condition 222 22 2lim 0 lim 0 lim 0 ttdx dx d dd t d tµµ ττ →+∞ →+∞ →+∞=⇔ =⇔ =x, (5) since we require an acceptable prediction of future behavior based on some ‘initial’ condition, given at some nominal but finite time. With the aim of integrating the constraint into the equation of motion, Rohrlich investigates a formal integration of Eq. (1), ()() () ()0 0 2 001aA e d e f a umττ τ ττ µµ µ µτττ τ ττ∞ ′−′′ ′ ′ =+ +  ∫, (6) where Aµ is a 4-vector constant of integration. He sets Aµ = 0, and considers the new equation as a possible replacement for Eq. (1). However, as he points out, setting Aµ to zero guarantees only that 0 lim 0eaττ µτ− →∞= which, clearly, is weaker than the requirement that the acceleration vanish, Eq. (5). Therefore we conclude that Eq. (6) with Aµ = 0 is unsatisfactory, since a supplemental constraint must still be imposed to filter out the unphysical behavior. An integrating factor for the Lorentz-Dirac equation A fully relativistic integro-differential form of the Lorentz-Dirac equation that does not admit runaway solutions (and therefore does not require supplemental constraints) is possible if a suitable integrating factor for the original LDE can be found. If it exists, an integrating factor (){}SSν µτ≡ satisfying () 001 dSa Sfdmττ=− , (7) 5 will permit - via the integration of Eq. (7) – the imposition of boundary conditions Eq. (5) on the acceleration. For this integrating factor to exist, by carrying out the differentiation in Eq. (7) and comparing with Eq. (1) left multiplied by S, it must be true that 2 0dS Saa S udττ+= , (8) where none of the elements of S can depend on the acceleration a. A substitution into Eq. (8) of 0 SR eττ−= (9) where (){}RRν µτ≡ , removes the exponential decay factor to give the requirement that R satisfy 2 dRaa R udτ= . (10) There are only three independent equations in Eq. (1) because the product of both sides with the four- velocity is identically zero. As a consequence, for any ()bµτ, () Rbuν ν µµτ= sets each side of Eq. (7) to zero, and so cannot be a candidate for the integrating factor. It follows that R cannot have a unique solution, since any candidate solution RCνν µµ= say will generate a family of solutions just by addition of this ‘null’ solution: () RC buνν ν µµ µ τ =+ . Of course, whatever form is chosen, that choice cannot impact the equation of motion for each component of xµ. With the sign convention {}{}0, qqµ≡− q, a particularly simple solution of Eq. (1) for the integrating factor is {} { }0123 10 0 20 3000 0000uuuu uuRR c u u cuuu uuνν νν µµ µ µ δ  == =− +  (11) where {}{}1,0,0,0 cµ≡ is a unit time-like vector. With this definition, one easily sees that Eq. (10) is satisfied, and in particular that the two terms are 22dRaa R ua cdν µ ν ν µν µτ== . (12) Recalling Eq.(9), it follows that the Lorentz-Dirac equation, Eq. (1), may be written ()0 0 00deeR a R fdmττ ττ ττ− −=− , (13) where R is given by Eq. (11), and the inverse of R, denoted here by ˆR, is {} ()2 00 1 0 2 0 3 2 01 1 12 13 1 200 02 12 2 23 2 03 13 23 31 11ˆ 2 1 1uu u u u u u uu u uu uuRR u u c cu cuu uu uu u uu uu uu uu uνν νν ν µµ µ µ µ δ− −−− −+ ≡= = − − +  −+   −+. (14) 6 R does not behave like a tensor under boosts, and is therefore not a Lorentz tensor. However, it does behave like a tensor under spatial rotations and space and time translations, and is therefore a Euclidean tensor. Nonetheless, the Lorentz invariance of the Lorentz-Dirac equation is preserved. This can be seen more readily if Eq. (13) is written as 00 0ˆda dRma m R a fddτττ−+ =, (15) whereupon it apparent that the requirement is not that R be a Lorentz tensor, but that ˆdRRadτ be a true 4- vector. The latter is guaranteed by design. Specifically it is equal to a2u, in conformity with Eq. (1), as may be confirmed using Eqs. (11) and (14). Integration and imposition of the boundary condition Formally, the first integral of Eq. (13) is ()() ()() ()() () () ( )( ) ()()()00 0 0 000 11 001 1c c c ccc cceR a e Ra d e Rfm ae RR a d e RR fmτ ττ τ τ τ τ τ ττ τττ ττ τττ τ τ τ τττ ττ τ τ τ τ τ ττ′ −− − ′ − − −−′′ ′ −= − ′′ ′ ⇒= −∫ ∫ (16) where τc is the time at which the proper acceleration is presumed known. We are now in a position to impose the requirement that the acceleration in the remote future, τc = +∞ - when the force has long since vanished - is zero. With a(τc) = 0, Eq. (16) becomes () ()()()0 1 001ad e R R fmττ τ τττ τ τ ττ′−∞ −′′ ′ =∫. (17) Upon the change of variable () 0 sττ τ′=− , this is ( ) ( ) () ()1 00 0 0sma d s e R R s f sττ τ τ τ τ∞ −−=+ +∫ (18) which may be recognized as a relativistic version of the non-relativistic form, Eq. (4). It is easily seen that, having isolated the second derivative on the left hand side, the acceleration is guaranteed to vanish in the remote future if the force also vanishes then. Therefore, the solution is evidently free of runaways. Further, it is evident that solutions of this equation are a subset of the solutions of the original Lorentz-Dirac equation, Eq. (1). Therefore, it can be concluded that the integro-differential equation Eq. (18) is the physically correct equation of motion for a classical charged particle; it retains the properties of the original Lorentz-Dirac equation without the unphysical behavior. Since it is not immediately evident from Eq. (18), we here confirm that, as required, the acceleration is orthogonal to the velocity. Taking the 4-vector product of Eq. (18) with the velocity gives () () () () ( ) ( ) 00 0ˆ sua d s e uR R s f sνλ µµ µµ ν λτ τ τ τ ττ ττ∞ −=+ +∫. (19) 7 Using Eq. (14) one finds that () 01ˆ 2 uR u uu cc uc cuννµµ νν ν ν µµ µ µ µ δ=− − + = . (20) Inserting this into Eq. (19) and then using Eq. (11) gives ( ) ( ) () () () () 00 0 0 0 000ssua d s e R s f s d s e u s f sλ µ λ µλ λτ τ ττ ττ ττ ττ∞∞ −−=+ + = + + =∫∫, (21) where the last step follows because the 4-force is required to be orthogonal to the velocity. Proper-time vector form The 3-vector form of Eq. (18) is obtained as follows. Given {} 0, fuλ=−u.f f , where f is the ordinary Newton force vector (i.e., borrowed from dp/dt = f), and 0 1 uγ== + u.u, then, using Eq. (11), one obtains () (){} (){}2 00 0 0 0, 0,TRf c u u c u f u f u f uλλ λλ νλ ν ν ν λ ν ν δ =− + = − + = −= × × − uu f u u f f . (22) Denoting the three-space part by ()≡× ×−wuu f f , Eq. (18) can be written () (){} ()1 003x30subsmR d s e sττ τ τ∞ −−=− + ∫α w (23) where αααα is the proper acceleration, and where the sub operation extracts the 3x3 (spatial) sub-matrix. Using Eq. (14) the latter is easily seen to be {}()1 3x301sub 1TRu−=+ uu (24) whereupon Eq. (23) gives the integro-differential version of the LDE in proper-time vector form: ()( )() ()()12 1 0 0011Ts T Tsmd s e d s eγγ γ∞∞ −− −−=+ − =+ − × × ∫∫α uu uu f uu f u u f , (25) where the functions in the integrand are to be evaluated at τ + sτ 0. In particular, if f is the Lorentz force, ()e γ =+fE u × B , then the proper acceleration is () ()()1 0 01Tsme d s eγγ∞ −−=+ − × × + ∫α uu E u u E u × B . (26) To write the proper acceleration in terms of vector cross-products, it is useful to define an intermediate quantity ()() 0sdse∞ −≡− × ×∫ff u u f , (27) where once again the functions in the integrand are to be evaluated at τ + sτ 0. With this substitution, an alternative form for Eq. (26) is therefore () 0mγγ=+ ××α fuu f . (28) 8 Proper-time series expansion in ττττ0 A series expansion of the integrand in ascending powers of τ 0 can be expected to converge rapidly if the projection of the force - ()2 Tγ−uu f - is slowly varying on the time scale of the classical time τ0. From Eq. (18), one has ()1 00 0n ndma R R fdττ∞ − == ∑ (29) where all functions are now evaluated at time τ. In vector form this is () ()12 00 01n TT ndmdγτ γτ∞ − ==+ − ∑ α uu uu f . (30) Ordinary-time vector form The integro-differential form of the LDE can be cast as a 3-vector equation in ordinary time as follows. From Eq. (17), one has () () ()0 001Ra d e R fmττ µµ τ νµ ν µ τττ τ ττ′−∞ ′′ ′ =∫, (31) the left hand side of which is () 0 23 0 00 0, 0, 0,du du d dRa u udd d d tµ νµ γττ τ  =− = − = −      u u βu , (32) where dd t=β x is the ordinary velocity. I.E., the left hand side of Eq. (31) is already in the direction of the ordinary acceleration. Further, noting that the product in the integrand is ()(){}20, Rfµ νµ γ=− ββ.f f , (33) then substitution of Eqs. (32) and (33) into Eq. (17) gives ()() ()0 0 2 33 00 0011e tde d t t mmττ ττ ττ ττγ τγ τγ′′−−∞∞ ′′ ′ =− =∫∫β fββ.f H!, (34) where the components of () ( )() tγ′=− Hf ββ.f are now redefined as functions of ordinary time. The transformation is complete once the exponential damping factor is explicitly cast as a function of ordinary time: ()()()()33 00 00 00 011exp exptt t tt tdt dtdt t dt t ttt mm τγ τγ τγ τγ′ ∞∞ + ′  ′′ ′′′′ ′ ′ == − +     ′′ ′′  ∫∫ ∫ ∫β HH!. (35) As for the proper-time form, the variable of integration can be rendered dimensionless, although here it does not result in a simplification. Letting 0 stτ′= : ()()()()0 00 33 00 00 00 011exp expts s tdd t d sd s ts d s tsdt t t s mmτ τττγ γ τ γγ+ ∞∞  ′′ ′ =− + =− +   ′′ ′ + ∫∫ ∫ ∫βHH . (36) 9 If f is the Lorentz force then ()()eγ=− + ×HE ββ.EβB. Ordinary-time series expansion in ττττ0 An ordinary-time series expansion of the integrand in ascending powers of τ0 can obtained from Eq. (36) by integrating by parts. The result is ()()()2 0 3 0 01n ndd dt dt mγτ γ γ==− ∑βfββ.f , (37) where the functions are of ordinary time, evaluated at time t. Summary A physically acceptable relativistic equation of motion for a classical charged particle in 3 spatial dimensions has been derived that has the properties desired of the original Lorentz-Dirac equation, but without the unphysical behavior. The exclusion of runaway solutions has been achieved by finding an integrating factor for the original Lorentz-Dirac equation so that the acceleration can be written as an integral operator on the force. 10 References [1] J. D. Jackson, Classical Electrodynamics , Chapter 17, (John Wiley, New York, NY, 1975). [2] F. Rohrlich, Classical Charged Particles , (Addison-Wesley, Reading, MA, 1965). [3] T. Erber, Fortschritte der Physik, 9, 343 (1961). [4] R. Haag, Zeitschrift für Naturforschung, 10A, 752 (1955). [5] G. N. Plass, Rev. Mod. Phys., 33, 37 (1961). [6] M. Sorg, Zeitschrift für Naturforschung, 31A, 683 (1976). [7] E. J. Moniz and D. H. Sharp, Phys. Rev. D 15, 2850 (1977). [8] H. Levine, E. J. Moniz, and D. H. Sharp, Am. J. Phys. 45, 75 (1977). [9] D. H. Sharp, Foundations of Radiation Theory and Quantum Electrodynamics , Ed. A. O. Barut, Chapter 10, (Plenum Press, New York, NY, 1980). [10] A.O. Barut, Phys. Lett. A, 145, 387, (1990). [11] R. Blanco, Phys. Lett. A, 169, 115, (1992). [12] J. L. Jimenez and J. Hirsch, Nuovo Cimento, 98 B , 87, (1986). [13] T. Boyer, Foundations of Radiation Theory and Quantum Electrodynamics , Ed. A. O. Barut, Chapter 5, (Plenum Press, New York, NY, 1980).
arXiv:physics/0102088v1 [physics.optics] 28 Feb 2001Coherent Control of Multiphoton Transitions with Femtosec ond pulse shaping S. Abbas Hosseini and Debabrata Goswami Tata Institute of Fundamental Research, Homi Bhabha Road, M umbai 400 005, India. (July 24, 2013) Abstract We explore the eﬀects of ultrafast shaped pulses for two-lev el systems that do not have a single photon resonance by developing a multiph oton density- matrix approach. We take advantage of the fact that the dynam ics of the intermediate virtual states are absent within our laser pul se timescales. Under these conditions, the multiphoton results are similar to th e single photon and that it is possible to extend the single photon coherent cont rol ideas to develop multiphoton coherent control. Typeset using REVT EX 1I. INTRODUCTION Use of optimally shaped pulses to guide the time evolution of a system and thereby control its future is an active ﬁeld of research in recent yea rs [1]- [16]. Such developments have been spurred by technological breakthroughs permitti ng arbitrarily amplitude mod- ulated laser pulses with 20-30 fs resolution and pulse energ ies ranging to almost hundred microjoules–either in the time domain or in the frequency do main. In most practical cases, computer optimizations are used to generate the useful shap es [1]- [7], since even approx- imate analytical solutions exist only for very specialized cases [7]- [12]. Such computer simulations have resulted in generating quite complicated theoretical waveforms that can break strong bonds [1]- [4], localize excitation [13]. Most of these interesting calculations involve intense pulses, which do not operate in the linear re sponse regime. Actual photo- chemical processes with such intense pulses that operate be yond the linear response region often involve multiphoton eﬀects. Unfortunately, multiph oton interactions typically induce additional complications and have not yet been explored muc h theoretically for coherent control purposes. In fact, most models for coherent control deal with light-matter interac- tion at the single-photon level. However, some recent exper iments show that they can even simplify quantum interference eﬀects; e.g., how Cs atoms ca n be made to absorb or not absorb light with non-resonant two-photon excitation with shaped optical pulses [14,15]. The experimental results have been treated with a perturbat ion model that works under the resonant condition. However, a more complete theoretic al treatment of multiphoton interactions for developing multiphoton coherent control is quite complex and is far from complete. In fact, the lack of such a theoretical basis is als o evident from the fact that in the classic demonstration of control of multiphoton ionizatio n process, an experimentally opti- mized feedback pulse shaping was found to provide the best-d esired yield [16]. In the present work, we develop a density matrix approach to multiphoton pr ocesses that do not have any lower-order process and demonstrate that can also explain t he oﬀ-resonance behavior. We present results, which show that this would be a promising approach. We ﬁrst apply 2the approach to the two-photon scenario in a simple two-leve l system (e.g., any narrow, single-photon transition line that is only multiphoton all owed). We then generalize the results to the case where only one N-photon (N ≥2, which is multiphoton) transition is possible and none of the (N-1) photon transition can exist. U nder these conditions, we show that most of the waveforms produce the same results as th e single photon case [18]. With care, therefore, we predict that it will be possible to e xtend some of the single photon coherent control ideas to develop multiphoton coherent con trol. We explore the various frequency-swept pulses into the multiphoton domain, which have been previously shown to be successful in inducing robust inversions under single-p hoton adiabatic conditions. We also investigate the case of phase modulated overlapped Gaussia n pulses for two-photon transition (in the spirit of a “dark” pulse of Meshulach and Silberberg, which they deﬁned as “a single burst of optical ﬁeld” that does not produce any net populati on transfer [15]). We show that the two-photon dark pulses, which are a result of smooth ly varying phase modulation, can be explained by invoking the well-established concept o f single photon adiabatic rapid passage (ARP) [17,18] to the multiphoton framework. In fact , the ARP explanation allows us to generalize the results to the N-photon case and show tha t such dark pulses are a result of the Stark shifting of the resonant Nthphoton transition. The extension of the concept of ARP into the multiphoton domain has very important conseque nces in generating inherent robust processes. II. FORMALISM The simplest model describing a molecular system is an isola ted two-level system or ensemble without relaxation or inhomogeneities. This simp le model often turns out to be a very practical model for most systems interacting with the f emtosecond laser pulses as the magnitude of the relaxation processes are immensely large a s compared to the femtosecond interaction time. Let us consider a linearly polarized puls e is being applied to the \|1>→\|2> transition, where \|1>and\|2>represent the ground and excited eigenlevels, respectivel y, 3of the ﬁeld-free Hamiltonian. In case of single photon inter actions (Fig. 1a), the total laboratory-frame Hamiltonian for such two-level system un der the eﬀect of an applied laser ﬁeld,E(t) =ε(t)ei[ω(t)t+φ(t)]=ε(t)ei[ω+˙φ(t)]tcan be written as [10,17]: H= E1V12 V21E2 = ¯hω1µ.E µ.E∗¯hω2 = ¯h −ωR 2µ.ε ¯hei(ω t+φ) µ.ε∗ ¯he−i(ω t+φ)ωR 2  (1) where ωR=ω2−ω1is resonance frequency, V12andV21are the negative interaction poten- tials and hω1, hω 2are the energies of ground ( E1) and excited state ( E2) respectively, and µ is the transition dipole moment of the \|1>→\|2>transition. In analogy to this single photon interaction as given in Eqn. (1), the interaction potential under the eﬀect of an applied laser ﬁeld, in two-photon absorption case (Fig. 1b) can be written as: V(t) =µ1mε(t)ei(ω t+φ(t))\|1/angbracketright /angbracketleftm\|µm2ε(t)ei(ω t+φ(t))\|m/angbracketright/angbracketleft2\|+c.c. (2) where mis the virtual state. Let us, for simplicity, take the transi tion dipole moment between the ground state to the virtual sate to be equal to the transition dipole moment between the virtual state and excited state ( µ1m=µm2=µ). In fact, we have veriﬁed in our simulations that the trend of the results is preserved even w hen we relax this simpliﬁcation. In any event, for developing the initial model, the above sai d simpliﬁcation allows us to take µas a common factor and we can rewrite Eqn. (2) as: V(t) = (µ ε(t))2e2i(ω t+φ(t))\|1/angbracketright/angbracketleft2\|+c.c. (3) since for normalized states, < m\|m >= 1. Using similar arguments for the N-photon case (Fig. 1c), the interaction potential can be written as: V(t) = (µ ε(t))NeiN(ω t+φ(t))\|1/angbracketright/angbracketleft2\|+c.c. (4) Thus, the total laboratory-frame N-photon Hamiltonian will be: H= ¯hω1 (µ.E)N (µ.E∗)N¯hω2 = ¯h −ωR 2(µ.ε)N ¯heiN(ω t+φ) (µ.ε∗)N ¯he−iN(ω t+φ)ωR 2  (5) 4The virtual levels for the two-photon (or N-photon) case can exist anywhere within the bandwidth ∆ ωof the applied laser pulse (Fig. 1) and the individual virtua l state dynamics is of no consequence. In analogy to the single photon case [11,12], there are two di ﬀerent ways to transform the elements of the above laboratory frame N-photon Hamiltonia n (Eqn. (5)) into a rotating frame of reference. Any time-dependent transformation fun ctionTcan be applied on both sides of the Schrodinger equation as follows: T/parenleftig i¯h∂ ∂tΨ =HΨ/parenrightig i¯h∂ ∂t(TΨ)−i¯h∂T ∂t(T−1T)Ψ = TH(T−1T)Ψ i¯h∂ ∂t(TΨ) =/bracketleftig THT−1+i¯h∂T ∂tT−1/bracketrightig (TΨ)(6) which results the following transformation equation: HTransformed=THT−1+i¯hT−1∂T ∂t(7) The usual frame of reference would be to rotate at Nω. This is the phase-modulated (PM) frame of reference, which can be derived from the Hamilt onian Hof Eqn. (5) by the transformation: TPM= e−iNω t 20 0 eiNω t 2  (8) Using of Eqn. (7), the transformed Hamiltonian in the PM fram e is: HPM= ¯h ∆µ(ε(t))N ¯heiNφ µ(ε∗(t))N ¯he−iNφ0  (9) under the assumption that the transient dipole moment of the individual intermediate virtual states in the multiphon ladder all add up constructively to t he ﬁnal state transition dipole moment and can be approximated to a constant ( µ) over the N-photon electric ﬁeld interac- tion. This approximation is particularly valid for the case of multiphoton interaction with femtosecond pulses where no intermediate virtual level dyn amics can be observed. Thus, we deﬁne multiphoton Rabi Frequency, as the complex conjuga te pairs: Ω(t)= µ.(ε(t))N/¯h 5and Ω∗(t)=µ.(ε∗(t))N/¯h, and the time-independent multiphoton detuning as: ∆ = ωR−Nω (Fig. 1c). However, in order to investigate the oﬀ-resonanc e behavior of continuously mod- ulated pulses, in the single photon case, it is useful to perf orm an alternate rotating-frame transformation to a frequency modulated (FM) frame with the transformation function: TFM= e−iNω t+φ 20 0 eiNω t+φ 2  (10) to transform the N-photon laboratory Hamiltonian in Eqn. (5 ) to the FM frame as: HFM= ¯h ∆ +N˙φ(t)µ.(ε(t))N ¯h µ.(ε∗(t))N ¯h0 = ¯h ∆ +N˙φ(t) Ω Ω∗0  (11) The time derivative of the phase function ˙φ(t),i.e., frequency modulation, appears as an additional resonance oﬀset over and above the time-ind ependent detuning ∆, while the direction of the ﬁeld in the orthogonal plane remains ﬁxe d. The time evolution of the unrelaxed two-level system can then be evaluated by inte grating the Liouville equation [10,17]: dρ(t) dt=i ¯h/bracketleftig ρ(t), HFM(t)/bracketrightig (12) where ρ(t) is a 2 ×2 density matrix whose diagonal elements represent populat ions in the ground and excited states and oﬀ-diagonal elements represe nt coherent superposition of states. This approach has been very successful in solving ma ny single-photon inversion processes for arbitrarily shaped amplitude and frequency m odulated pulses [12], [13]. We have just extended the same arguments to the multiphoton cas e. The simulations are performed with a laser pulse that either has (a) a Gaussian intensity proﬁle or (b) a hyperbolic secant intensity proﬁle which hav e the following respective forms: (a) I(t) =I0exp/bracketleftig −8ln2 (t/τ)2/bracketrightig implies ε (t) =ε0exp/bracketleftig −4ln2 (t/τ)2/bracketrightig (b) I(t) =I0sech2/bracketleftig/braceleftig 2ln/parenleftig 2 +√ 3/parenrightig/bracerightig (t/τ)/bracketrightig implies ε (t) =ε0sech/bracketleftig/braceleftig 2ln/parenleftig 2 +√ 3/parenrightig/bracerightig (t/τ)/bracketrightig(13) 6where τis the full width at half maximum, and I(t)is the pulse intensity. This is because most of the commercially available pulsed laser sources hav e these intrinsic laser parameters. We choose a range of frequency sweeps, such as (c) the linear f requency sweep for the Gaussian amplitude, (d) the hyperbolic tangent sweep for th e hyperbolic secant amplitude, and they have the following respective forms: (c)˙φ(t) =bt (d)˙φ(t) =b/braceleftig 2ln/parenleftig 2 +√ 3/parenrightig/bracerightig tanh/bracketleftig/braceleftig 2ln/parenleftig 2 +√ 3/parenrightig/bracerightig (t/τ)/bracketrightig (14) where bis a constant. Such pulses have been shown to invert populati on through ARP in single photon case and so we choose to use these particular sh apes for the multiphoton case. We also use the shaped overlapping Gaussian pulses for a two- photon transition similar to the ones used by Meshulach and Silberberg. In their case th e frequency sweep is given by: ˙φ(t) =  b t ≥t0 −b t < t 0(15) where t0is the midpoint of the pulse. This pulse does not satisfy the A RP condition and is quite susceptible to the changes in the pulse amplitude pr oﬁle and our results show this in the next section. However, if we instead use smoothly vary ing linear frequency sweeps, either changing monotonically as in Eqn. (14c), or linearly approaching and going away from resonance as given by: ˙φ(t) =bt,where bchanges sign at t0 (16) These pulses satisfy the ARP conditions as explained in the n ext section. Dark pulses given by Eqn (16) are thus quite insensitive to the changes in the pulse amplitude proﬁle. We also extend our calculations to the N-photon case in a simp le two-level type of system that supports only an Nthphoton transition and show how the phase switches eﬀect the population cycling. These generalizations would become ev ident when we examine the results based on the ARP extended to multiphoton case. 7III. RESULTS & DISCUSSION The population evaluation in a simple two level system witho ut relaxation for one photon absorption (N=1) is shown in Fig. 2 for the pulse shapes given by Eqns. (13) and (14). Excitation exactly on resonance creates a complete populat ion inversion when the pulse area (the time integral of the Rabi frequency) equals π. However, the population oscillates between the ground and excited state as sine function with re spect to the Rabi frequency. These oscillations are not desirable in most cases involvin g real atoms or molecules. They are washed out by inhomogeneous broadening, the transverse Gau ssian proﬁle of the laser, and (in the molecular case) diﬀerent values of µ.ε. For a single-photon case, as discussed in Ref. [18], frequency modulated pulses can instead produce adiab atic inversion, which avoids these complications. A linearly frequency swept (chirped) laser pulse can be generated by sweeping from far above resonance to far below resonance (blue to red s weeps), or alternatively from far below resonance to far above resonance (red to blue sweep s). When the frequency sweep is suﬃciently slow such that the irradiated system can evolve with the applied sweep, the transitions are “adiabatic”. If this adiabatic process is faster than the characteristic relaxation time of the system, a smooth population inversio n occurs with the evolution of the pulse, which is the well-known ARP. Let us now extend the eﬀect of such laser pulses (given by Eqns . (13) and (14)) to a two-photon (N=2) case as derived in our Hamiltonian of Eqn. ( 11). Fig. 3 shows the plots of the upper state population ( ρ22) as a function of applied Rabi frequency and detuning for two photon absorption case in the absence of one photon ab sorption. We ﬁnd that the results are qualitatively the same as the one-photon absorp tion. In fact, our simulations show that for such a simple case of a two-level system, where o nly an Nthphoton transition is possible, we can extend our single-photon results to the N -photon case. The diﬀerence lies in the Rabi frequency scaling. Thus, for this simple cas e as deﬁned here, we are able to invoke the concept of ARP for multiphoton interaction. We next use the overlapping Gaussian pulses (when the overla p is complete it collapses 8into a single Gaussian) with diﬀerent phase relationships. Our simulation shows that for shaped overlapping Gaussian pulses the excited sate popula tion depends on the form of the frequency sweep. In ﬁgure 4a, for the shaped pulse without sw eep the population of excited state oscillates symmetrically. For a simple monotonicall y increasing or decreasing sweep around resonance, it behaves like a Guassian pulse with line ar sweep (Fig. 4b). These results essentially conﬁrm another important implication of the adiabatic principle: that the exact amplitude of the pulse is not very important under t he adiabatic limit. Again, for this simple case, we are able to invoke the concept of ARP for m ultiphoton interaction to explain the inversion. The phase modulated overlapped Gaussian pulses are of inter est since Meshulach and Silberberg had experimentally switched the phase of the sec ond pulse with respect to the ﬁrst pulse and demonstrated two-photon excited state popul ation modulation. However, the phase switch in their pulse shapes was abrupt as given by Eqn. (15), and thus did not satisfy the ARP condition. As a result the population transfer with s uch pulses are very heavily dependent on the actual shape of the pulse. Figs. 5 shows that the upper state population for two photon absorption in the absence of one photon absorp tion is heavily dependent on the nature of the phase step, the intensity and the extent o f overlap of the pulses. At some particular phase switch, there is no excited-state pop ulation, and they called it the dark pulse. We show that it is indeed true for a speciﬁc overla pped amplitude proﬁle and intensity for a given phase switching position. These dark p ulses, however, are sensitive to the exact nature of the amplitude proﬁle and intensity. If instead we choose a smoothly varying linear frequency swe eping to the two-photon resonance and then away from resonance, as given by Eqn. (16) , the results are quite robust to the exact nature of the amplitude proﬁle and intensity (Fi g. 6). At detuning zero and for small values of Rabi frequency, we have some population in ex cited state. However, when the intensity of applied pulse increases, the excited state population returns to zero. In other words, we are sending shaped pulse into the two-level s ystem but ﬁnally there is no excited-state population. Curiously enough, for such puls es, the population is asymmetric 9about detuning from resonance. In fact, Fig. 6 clearly shows that the population transfer occurs at some non-zero detuning values at higher Rabi frequ encies when it does not have any excitation at resonance and behaves as a dark pulse. This result can be understood by examining the evolution of the dressed states [17]- [19] w ith time (Fig. 7). When the eﬀect of the pulse cannot be felt by the system at very early or and at very late times with respect to the presence of the pulse, each dressed state essentially corresponds to the single bare state ( \|α/angbracketright → \| 1/angbracketrightand\|β/angbracketright → \| 2/angbracketright). It is only during the pulse that the dressed states change in composition and evolve as a linear combinat ion of the two bare states. The proximity of these dressed states during the pulse essen tially determines the population exchange. The higher Rabi frequencies cause a stark shift in the dressed states so that at resonance there is no population exchange. Under such stark shifted condition, resonance occurs at some speciﬁc non-zero detuning value where Rabi os cillations are seen in Fig. 6. These results are completely general for a simple case of a tw o-level system, where only an Nthphoton transition is possible. The phase change of the overl apping Gaussian pulses essentially provide an additional parameter to control the population evolution of a simple two-level type of system that supports only an Nthphoton transition. IV. CONCLUSIONS In this paper, we have explored the eﬀects of ultrafast shape d pulses for two-level systems that do not have a single photon resonance by developing a mul tiphoton density-matrix approach. We took advantage of the fact that dynamics of the i ntermediate virtual states are absent in the femtosecond timescales, and demonstrated that many multiphoton results can be surprising similar to the well-known single photon re sults. When we extend the ARP to the multiphoton condition, robust population invers ion and dark pulses become possible that are insensitive to the exact proﬁle of the appl ied electric ﬁeld. We have shown, therefore, that it is possible to extend the single photon co herent control ideas to develop femtosecond multiphoton coherent control. 10REFERENCES [1] R. J. Gordon and S. A. Rice, Annu. Rev. Phys. Chem. 48, 601 (1997); S. Rice, Science 258, 412 (1992). [2] W.S. Warren, H. Rabitz, and M. Dahleh, Science 259,1581 (1993). [3] P. Brumer and M. Shapiro, Molecules in Laser Fields ed. A.D. Bandrauk, (Marcel Dekker, New York, 1994). [4] J. L Krause, R. M. Whitnell, K. R. Wilson, Y.J. Yan, and S. M ukamel, J. Chem. Phys. 99, 6562 (1993). [5] S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, Phys. Rev . Lett. 65, 2355 (1990); S. Chelkowski and A. D. Bandrauk, Chem. Phys. Lett. 186, 264 (1991). [6] R. Kosloﬀ, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Ta nnor, Chem. Phys. 139, 201 (1989). [7] W. S. Warren, Science 242, 878 (1988); W. S. Warren and M. Silver, Adv. Magn. Reson. 12, 247 (1988). [8] F. T. Hioe, Phys. Rev. A 30, 2100 (1984); F. T. Hioe, Chem. Phys. 73, 351 (1989). [9] J. F. McCann and A. D. Bandrauk, Phys. Lett. A 151, 509 (1990). [10] Allen and J. H. Eberly, Optical Resonance and Two Level Atoms (Dover, New York, 1975). [11] J. Baum, R. Tyco, A. Pines, Phys. Rev. A 32, 3435 (1985). [12] D. Goswami and W. S. Warren, Phys. Rev. A 50, 5190 (1994). [13] D. Goswami and W. S. Warren, J. Chem. Phys. 99, 4509 (1993). [14] D. Meshulach and Y. Silberberg, Nature 396, 239 (1998). [15] D. Meshulach and Y. Silberberg, Phys. Rev. A 60, 1287 (1999). 11[16] A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V . Seyfried, M. Strehle and G. Gerber, Science 282, 918 (1999). [17] See, for example, B. W. Shore, The Theory of Coherent Excitation (Wiley, New York, 1990). [18] J. S. Melinger, S. R. Gandhi, A. Hariharan, D. Goswami, a nd W. S. Warren, J. Chem. Phys.101, 6439 (1994). [19] Claude Cohen-Tannoudji, Bernard Dui, Frank Laloe, Quantum Mechanics (John Wiley & Sons, New York, 1978). 12FIGURES FIG. 1. Schematic of (a) single, (b) two and (c) multiphoton p rocesses, respectively. Symbols and notations are deﬁned in text.FIG. 2. Comparison of the excited state population for a sing le photon excitation as a function of Rabi frequency, for (a) a Gaussian pulse (solid curve: wit hout any frequency sweep; dashed curve: with linear frequency sweep), and (b) a hyperbolic secant pu lse (solid curve: without any frequency sweep; dashed curve: with hyperbolic tangent frequency swe ep). FIG. 3. Excited state population for 2-photon excitation as a function of Rabi frequency and detuning for: (a) transform-limited Guassian pulse; (b ) bandwidth equivalent linearly fre- quency-swept Gaussian pulse; (c) transform-limited hyper bolic secant pulse; and (d) hyperbolic secant pulse with hyperbolic tangent frequency sweep. FIG. 4. (a) Excited state population for 2-photon excitatio n as a function of Rabi frequency and detuning for Shaped overlapped Gaussian pulse without s weep. (b) Excited state population for 2-photon excitation as a function of Rabi frequency and detu ning for shaped overlapped Gaussian pulse with a monotonically increasing linear sweep. FIG. 5. Excited state population for 2-photon excitation as a function of phase step position (i.e., detuning) normalized to the pulse FWHM, τ, for two diﬀerent Rabi frequencies in the case of pulses with phase steps as given by Eqn. 15. The results are heavily subjective to the choice of parameters (as we show for the two Rabi frequencies used in th is Fig. that diﬀer by less than 5%), and are thus non-adiabatic, as discussed in the text. FIG. 6. Excited state population for 2-photon excitation as a function of Rabi frequency and detuning for shaped overlapped Gaussian pulse with a sweep l inearly approaching and going away from resonance as given by Eqn. 16. A contour plot (b) is shown for the 3-D surface plot (a) to better represent that the population exchange occurs at s ome detuned position for high Rabi frequencies. FIG. 7. Energies of the two dressed states evolving with time for the shaped Gaussian pulse whose population evolution is shown in Fig. 6 at a high Rabi fr equency for (a) no net population transfer at resonance, (b) the Stark-shifted frequency (de tuned from resonance on one direction) where the Rabi oscillations occur, (c) the Stark-shifted fr equency equally detuned from resonance to the other side where no Rabi oscillations occur. 13
WAVE UNIVERSE AND SPECTRUM OF QUASARS REDSHIFTS A.M. Chechelnitsky, Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna,Moscow Region, Russia E’mail: ach@thsun1.jinr.ru ABSTRACT In the framework of the Wave Universe concept it is shown, that the genesis of redshifts can be connected with the intra-system (endogenous) processes, which take place in astronomical systems. The existance of extremal redshift objects (quasars - QSO) with most probable z = 3.513 (3.847); 4.677; 6.947 (7.4); 10.524; 14.7; 27.79; is predicted. THE WAVE (MEGAWAVE) ASTRODYNAMICS CONCEPT A wide set of yet noninterpreted (enigmatic from the point of view of standard paradigma of celestial mechanics and astrophysics [1,2]) observed and experimental data, connected with the dynamical structure and geometry of the Solar system (in particular, with the arragement of planetary, satellite orbits, distribution its velocities, etc.) and other astronomical systems can be adequately interprated in the framework of Wave (Megawave) Astrodynamics (and Wave Universe concept) [2-6]. Accordingly to these representations, real objects observed in the Universe (in the megaworld, such as astronomical systems, for example, the Solar system) appear principally wave dynamic systems (WDS), in the some sence similar to the atom system (Micro - Mega analogy [2]), and can be described by Fundamental wave equations (in particular, Schrodinger-type equation) (Fig.1). The unique dimensional parameter /GFF/G0F/G03 /G5A/G4B/G4C/G46/G4B/G03 /G48/G51/G57/G48/G55/G56/G03 /G4C/G51/G57/G52/G03 /G56/G58/G46/G4B/G03 /G44/G03 /G5A/G44/G59/G48/G03 /G48/G54/G58/G44/G57/G4C/G52/G51/G0F/G03 /G4B/G44/G56/G03 /G57/G4B/G48/G03 dimension of sectorial velocity (circulation) [cm2/s] and corresponds to characteristic scale of system (Co-dimensional principle [2]). For the atom it has the order /GFF/G03/G20/G03/GFFe = /GAB /G12/G50e = 1.15767 cm2·s-1 ( /GAB /G03 /G20/G03 1.054572⋅10-27 g⋅ cm2·s-1 - Planck's constant, me = 9.109389⋅10-27 g – mass of electron), for the Solar system (SS) /GFF/G03/G20/G03/GFFSS ≈ 1019 cm2·s-1. EXTREMELY LOW MASS From the comparison of the circulation parameters, carried out in the end o f 70s in the monograph [2, p.245], naturally follows an evident, lying at surface, consequence. Representing the Solar system constant /GFF/G03/G20/G03/GFFSS = /GABSS/mSS /G4F/G4C/G4E/G48/G03/G44/G56/G03/G49/G52/G55/G03/G44/G57/G52/G50/G03/G0B/GFF/G20/GFF e= /GAB /G12/G50e), it is easy, for example, in the case /GABSS= /GAB /G0F/G03/G57/G52/G03/G52/G45/G57/G44/G4C/G51/G03/G57/G4B/G48/G03/G55/G48/G53/G55/G48/G56/G48/G51/G57 ation /GFF/G03/G20/G03/GFFSS = /GABSS/mSS and the order of mass mSS = /GAB /G12/GFFSS ≈ 1.054572⋅10-27/1019 ≈ 10-46 g. The physical sence of appearance of such an extremally low mass merits the special discussion. Meanwhile just note that this valuation is close to the upper limit of the experimental valuation of the photon mass [7]: mγ< 9⋅10-10eV/c2 = 1.604⋅10-42 g (Ryan, 1985), 4.73⋅10-12 ev/c2 = 0.8432·10-44 g (Chernikov, 1992), 1.0⋅10-14 ev/c2=1.782·10-47 g (Williams, 1971), etc. SPECTRUM OF ELITE VELOCITIES The Fundamental wave equation, described the Solar system (similarly to the atom system), separates the spectrum of physically distinguished, stationary - elite - orbits, corresponding to mean quantum numbers N, including the spectrum of permissible elite velocities vN. The following representation holds for the physically distinguished – elite – velocities vN in the G[s] Shells of wave dynamical (in particular, astronomical) systems [3-6]: vN = vN[s] = (2π)1/2⋅C∗[s]/N, C∗[s = C∗[1]⋅χ-(s-1), s = 0, ±1, ±2, … (χ - Fundamental parameter of Hierarchy (Chechelnitsky Number) χ=3.66(6), C∗[1] – Sound velocity of cosmic plasma in G[1] Shell [Chechelnitsky, [2-6], 1980-1992]), where elite values N (as it follows from observations) are close, in the general case, to the counting set of N - Integer (semi - integer). The most stable - dominant (strong elite) – orbits and the related dominant velocities correspond to the dominant values of quantum numbers, close to N = NDom = 8; 11; 13; (15.5) 16; 19.5; (21.5) 22.5. It can be shown, that NTR = χ(2π)1/2 ≅ 9.191 also is the physically distinguished (dominant) value N [6]. INVARIANCE (UNIVERSALITY) OF THE ELITE VELOCITIES SPECTRUM The spectrum of physically distinguished elite velocities vN and quantum numbers N of arbitrary wave dynamic systems (WDS) has some universal peculiarity. It is practically identical - invariant (universal) for all known observed systems of the Universe. In particular, the velocity spectra of experimentally well investigated Solar and satellite systems practically coincide for the observed planetary and satellite - dominant - orbits, corresponding to some (dominant) values of quantum numbers NDom. Thus it can be expected that the spectrum of elite (dominant - planetary) velocities of the Solar system (well identificated by observations) can be effectively used as quite representative - internal (endogenou s) - spectrum of elite (dominant) velocities, for example, of far astronomical systems of the Universe. PHYSICALLY DISTINGUISHED REDSHIFTS In the framework of the developed representations of Wave (Megawave) Astrodynamics and Wave Universe concept [2-6] the analytical representation can be ob tained for physically distinguished - preferably observed (elite) - redshifts of far astronomical objects (galaxies, quasars). The physically justified by experience (and correct consequences) relation z = f(v) between the velocity ν and the redshift z has the form z=β2 =(v/c) 2, β=v/c, where c=299792.458 km⋅s-1 - light velocity. This correlation between the redshift z and the (orbital) velocity v (as opposed to other relations) is carefully examined experimentally in laboratory conditions - on the Earth (Paund and Rebka’s experiment) and in Space - from the Sun (Brault’s experiment) [8]. It is also interesting to note that the used square dependen ce in the functional (mathematical) plane is in fact identical to the relation used in the calculation of the so-called gravity redshift [8] z=GM/c2r=(v/c) 2, where v2=GM/r, v - orbital velocity. THE PEAKS z IN OBSERVATIONS AND IN THEORY It may be shown that the most important peaks in histograms (of distribution) of the observed z unaccidentally and with sufficient reliability coincide with the physically distinguished-dominant - values zN[s] (at N=NDom). For example, the peaks widely known from observations peaks (Figs. 2, 3) from [9-10] z≈2, z≈1, z≈0.5, z≈0.35 (and other) coincide with the dominant (N=NDom) zN[-6] values of G[-6] Shell zN[-6] = z∗[-6]⋅⋅2π/N2, z∗[-6] = (C∗[-6]/c)2=[(C∗[1] /c)⋅χ7]2 , zN[-6] = 2.067; 1.093; 0.782; (0.55)0.516; 0.347; (0.286)0.261 and also (for NTR = 9.191) zTR[-6] =1.57. ENDOGENOUS NATURE OF z The set of large quantity of facts, agreement between of theory and observations, including the possibility of correct description of distinguishing peaks z over all the observed redshifts range (beginning from z=0) makes the next conclusion natural. Assertion. It seems very probable that the true genesis and physical nature of the observed redshifts is considerably closer connected with the own (inner) wave shell structure of astronomical systems (galaxies, quasars), than with the "kinematic" motion (translation) of their mass center - with the galaxies "expansion". ABOUT THE EXISTENCE OF OBJECTS WITH EXTREMAL z In the framework of the Wave Universe representations it must be expected that replenishing statistics of newly discovered astronomical objects will be characterized by the distribution peaks at z that correspond to the physically distinguished - dominant - values of redshifts, in particular, belonging to the G[-7] Shell: zN[-7] = z∗[-7]⋅⋅2π/N2, z∗[-7] = (C∗[-7]/c)2 = [(C∗[1]/c)⋅χ8]2 = 283.08668, zN[-7] = 27.79; 14.7; 10.524; (7.4)6.947; 4.677; (3.847)3.513. Already at the present it is interesting to note apparently, unaccidental compliance of the observed values z of (remotest for 1986) quasars z=3.53 (quasar OQ 172) and z=3.78 (quasar PKS 2000-330) with the pointed above z[-7] dominant values of the G[-7] Shell (z=3,513 and z=3.847). Thus, it is not excluded, that the quasars OQ 172 and PKS 2000 - 330 will become not as much the last from discovered quasars of preceding population QSO (G[-6]) with active G[-6] Shell as the first (and evidently having not the highest values of z) from discovered quasars of new population QSO (G[-7]) with active G[-7] Shell . THE PROBLEM OF SEARCH Basing on the above - discussed prognosis, we may also point to a set of supplementary physical orientating circumstances, that essentially shorten the search field for the objects with extermal z. One of them resides in the fact that the search must be carried out, in particular, among the astronomical objects having abundant radiation (peculiarities, peaks, radiation anomalies), besides gamma, in close infrared range, too. Really, for example, for hydrogen Lα - line λ(Lα) = 1215.67 /GD6 /G03/G20/G03/G14/G15/G14/G11/G18/G19/G1A/G03/G51/G50/G03/G20/G03/G13/G11/G14/G15/G14/G18/G19/G1A/G03 µm we have the system of shifted (by the redshift z=z[-7]) wave lenghts λN[-7] = λ(Lα)(1+zN[-7]) = 3.50; 1.90; 1.40; (1.02) 0.966; 0.69; (0.589) 0.548 µm that lay in IR-range. Purposeful search of objects (most probably having z that are close to the pointed above), in particular, among objects as Seyfert galaxies, Markarjan galaxies, may lead to discovery of new astronomical systems, which are characterized by extremal, so far unknown values of redshifts. FOLLOWING OBSERVATIONS Three years after the exposition of preceding results in 1986 [11-12] followed by a discussion between a confined circle of researchers - astrophysicists, in the end of 1989, american scientists from the Palomar Observatory M. Schmidt, J. Gunn, D. Schnaider discovered the extremely far object of the Universe - quasar in the Ursa Major constellation. It is interesting to note also that using of the experimental "solar" value N=19.43 (instead of N=19.5) indicates the more close (to discovered) value z=4.71 (instead of z=4 .677). REFERENCES 1. Roy A.E. Ovenden M.W. - On the Occurence of Commensurable Mean Motion in the Solar System, Mon.Not.Roy.Astron.Soc.,114, pp.232-241, (1954). 2. Chechelnitsky A.M., Extremum, Stability, Resonance in Astrodynamics and Cosmonautics, M., Mashinostroyenie, 312 pp. (1980) (Monograph in Russian). (Library of Congress Control Number: 97121007 ; Name: Chechelnitskii A.M.). 3. Chechelnitsky A.M., - The Shell Structure of Astronomical Systems, Astrononical Circular of the USSR Academy of Science, N1410, pp.3-7; N1411, pp.3-7, (1985). 4. Chechelnitsky A.M., - Wave Structure, Quantization, Megaspectroscopy of the Solar System; In the book:Spacecraft Dynamics and Space Research, M., Mashinostroyenie, 1986, pp.56-76 (in Russian). 5. Chechelnitsky A.M., - Uranus System, Solar System and Wave Astrodynamics; Prognosis of Theory and Voyager-2 Observations, Doklady AN SSSR, 1988, v.303, N5, pp.1082-1088. 6. Chechelnitsky A.M., - Wave Structure of the Solar System, Tandem-Press, 1992 (Monograph in Russian). 7. Review of Particle Physics, Phys.Rev. D, Part I, Vol. 54, N1, 1 July 1996. 8. Lang /G44. R. - Astrophysical Formulae, Mir, v.2, p.310, (in Russian) (1978). 9. Arp H., Bi H.G., Chu Y., Zhu X. - Periodicity of Quasar Redshifts, Astron. Astrophys. 239, p. 33-49 (1990). 10. Arp H. - Extragalactic Observations Requiring a Non-Standard Approach, Review given at IAU Symposium 124, Beijing, China, 29 Aug. 1986. 11. Chechelnitsky A.M. - Wave Universe and the Possibility of Existance of Extremal Redshift Quasars, Moscow, The original date of promulgation and discussion - November 30, 1986. 12. Chechelnitsky A.M. - Megawave and Shell Structure of Astronomical Systems and Redshift Quantization, Moscow, The original date of promulgation and discussion - December 4, 1986. Post Scriptum (2000) [From Chechelnitsky, 2000]: What Quasars with Record Redshifts Will be Discovered in Future? Megaquantization in the Universe. It is clear, Megaquantization (quantization “in the Large”), observed megaquantum effects are not monopolic privelege of only Solar system. Let us point the brief resume of research (prognosis), connected with problem of redshift quantization of far objects of Universe – quasars (QSO) [Chechelnitsky, (1986) 1977]: “Abstract: In the framework of the Wave Universe concept it is shown that the genesis of redshifts can be connected with the intra-system (endogenou s) processes which take place in astronomical systems. The existence of extremal redshift objects (quasars – QSO) with most probable z=3.513 (3.847); 4.677; 6.947 (7.4); 10.524; 14.7; 27.79; … is predicted.” Prognosis already had justified successively for extremal values of z redshifts ztheory = 3.513, zobs = 3.53 (quasar OQ172) ztheory = (3.847), zobs = 3.78 (quasar PKS2000-330) ztheory = 4.677, zobs = 4.71 (Schmidt, Gunn, Schnaider, 1989) zobs = 4.694 (4.672) (quasar BR1202-0725, Wampler et al., 1996) At the present time, apparently, also the object Q2203+29 G73 with record value z of redshift z=6.97 is discovered in special Astrophysical Observatory (SAO, Russia) ztheory = 6.947, zobs = 6.97 (Q2203+29 G73, Dodonov et al., 2000). The Quene – for objects with even more high redshifts z = 10.524; 14.7; … Consequences of such successfully realizable prognosis, imperatives of observations not only are unexpected for the Standard cosmology, but also, probably, its can stimulated the radical reconsideration of many habitual representations, having become as freezen dogmas. Chechelnitsky A.M.-Hot Points of the Wave Universe Concept: New World of Megaquantization, International Conference: Hot Points in Astrophysics, JINR, Dubna, Russia, August 22-26, 2000; http://arXiv.org/abs/physics/0102036. Dodonov S. N. et al., The Primeval Galaxy Candidate, Submitted to Astronomy and Astrophysics, 2000; Also JENAM 2000. Tifft W. G. – Global Redshift Periodicities and Periodicity Structure, Astrophysical Journal 468, pp. 491-518, Sept 10, 1996. Wampler E. J. et al., High Resolution Observations of the QSO BR 1202-0275: Deuterium and Ionic Abundances at Redshifts Above z = 4, Astronomy and Astrophysics, v.316, p.33-42, (1996). Figure 1 MICRO – MEGA ANALOGY MICROSYSTEM QUANTUM SYSTEM ATOM MEGASYSTEM ASTRONOMICAL SYSTEM SOLAR SYSTEM FUNDAMENTAL WAVE EQUATION ∇2/G0C /G03/G0E/G03/G0B/G15/G12/GFF2)[ε – /G03/G38/G40 /G0C /G03/G20/G03/G13/G03 /GFF/G20/GFFe = /GAB /G12/G50e =1.158 cm2s-1 U=V/me, V = - e2/a – - Electric Potential K = Ke = e2/me ε= E/me, E – Energy e – Electric Charge /GAB /G03– Planck’s Constant me – Electron Mass /GFF/G03– FUNDAMENTAL QUANTIZATION CONSTANT [cm2s-1/G40/G03/G0B/GFF/G03/G20/G03/G47/G12/G15 π) U = - K/a – Potential K – Dynamical Parameter [cm3s-2] a (=r) – Distance [cm] ε ∼ Normalized Energy (ε ∼ v2/2 [cm2s-2] /GFF/G03∼ 1019 cm2s-1 = 109 km2 s-1 U = - K/G7E/a K=K/G7E = 1.327·1011 km3 s-2 Gravitational Parameter of the Sun SCHRÖDINGER’S EQUATION ∇2/G0C /G03/G0E/G03/G0B/G15/G50 e/ /GAB2)[E-/G39/G40 /G0C /G03/G20 0 Relations of Quantum Mechanics DE BROGLIE: P = /GAB /G4E PLANCK-EINSTEIN: E= /GABω HEIZENBERG: ΔxΔp>(1/2) /GAB P = mv, k – Wave Number ω - Frequency BOHR’S STATE ORBITS CONSEQUENCES OF THE FUNDAMENTAL WAVE EQUATION Quantization of the Sectorial Velocity (Circulation) L [cm2s-1] L = LN=1N, N – Integer L = va = (Ka)1/2, LN=1 – Constant /GFF/G03∼ LN=1 /G03/G0B/GFF/G03/G20/G03ξLN=1, ξ - Constant) N = L/LN=1 – Normalized Sectorial Velocity – Quantum Number Relations of the Wave Astrodynamics /G59/G03/G20/G03/GFF/G2E~ ε /G03/G20/G03/GFFΩ ΔxΔ /G59/G03/G21/G03/G0B/G14/G12/G15/G0C/GFF K~ – Wave Number Ω - Frecuency ELITE ORBITS PLANETARY (DOMINANT) ORBITS This figure "Redfig2.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0102089v1This figure "Redfig3.gif" is available in "gif" format from: http://arxiv.org/ps/physics/0102089v1
arXiv:physics/0102090v1 [physics.plasm-ph] 28 Feb 2001EXPERIMENTAL INVESTIGATIONS OF SPATIAL DISTRIBUTION ANISOTROPY OF PULSED PLASMA GENERATOR RADIATION. Yu.A.Baurov∗, I.B.Timofeev, V.A.Chernikov, S.F.Chalkin* * Central Research Institute of Machine Building, 141070, P ionerskaya 4, Korolyov, Moscow region. ** Moscow State University named by M.V.Lomonosov, Departm ent of Physics, Chair of Physical Electronics, 119899, Vorob’evy Gory, Moscow. (January 13, 2014) Results of experimental investigation of plasma luminous emittance (integrated with respect to time and quartz trans - mission band spectrum) of a pulsed plasma generator depend- ing on its axis spatial position, are presented. It is shown that the spatial distribution of plasma radiant intensity i s of clearly anisotropic character, that is, there exists a co ne of the plasma generator axial directions in which the radiatio n of plasma reaches its peak. A possible explanation of the resul ts obtained is given based on a hypothesis of global anisotropy of space caused by the existence of a cosmological vectorial potential Ag. It is shown that the vector Aghas the follow- ing coordinates in the second equatorial coordinate system : right ascension α= 293◦±10◦, declination δ= 36◦±10◦. The experimental results are in accordance with those of the earlier experiments on determining the direction of Ag. 52.30, 12.60 I. INTRODUCTION. In Refs. [1–12], a new assumed interaction of objects in nature distinct from the four known ones (the strong, weak, electromagnetic, and gravitational interactions) i s predicted and investigated. The new force is caused by the existence of the cosmological vectorial potential Ag, a new fundamental vectorial constant entering into the deﬁnition of discrete objects, byuons. According to the hypothesis advanced in Refs. [1–4], in the process of minimization of potential energy of interaction between byuons in the one-dimensional space formed by them, the observable physical space as well as the world of ele- mentary particles together with their properties appear. The masses of particles in the model proposed are pro- portional to the modulus of the summary potential AΣ which contains Agand vectorial potentials of various magnetic sources as of natural origin (from the Earth, the Sun, etc.) so of artiﬁcial origin (for example, the vectorial potential Aof magnetic ﬁelds from solenoids, plasma generators, etc.). The value \|AΣ\|is always lesser than\|Ag\| ≈1,95×1011Gs·cm[1–7]. The vectors AΣ andAgare practically always collinear because of the ∗baurov@www.comgreat value of the latter. In the model of Refs. [1–4], the process of formation of the physical space and charge numbers of elementary particles is investigated. There- fore, in contrast with calibration theories (for example, the classical and quantum ﬁeld theories), the values of po- tentials acquire the physical sense, which is in tune with the known and experimentally tested Aharonoﬀ-Bohm eﬀect [13–16] as a particular case of quantum properties of space described in Refs. [1–4]. The on-earth experiments (with high-current magnets [1,2,5–7], with a gravimeter and an attached magnet [1,2,8]), investigations of changes in β-decay rate of ra- dioactive elements under the action of the new force [9], and astrophysical observations [10,11] have given the fol- lowing approximate coordinates (in the second equatorial system) for the direction of the vector Ag: right ascen- sionα≈270◦, declination β≈34◦. In the aggregate, the experiments carried out [1,2,4–12] have shown that if the vectorial potential of some current system is opposite in direction to the vector AΣthen the new force repels any substance out of the region of weak- ened\|AΣ\|mainly in the direction of Ag. The magnitude of the new force Fin the experiments with high-current magnets (magnetic ﬂux Bbeing up to 15 T) was equal to ∼0,01−0,08gfor the test body mass ∼30g. When in- vestigating the new interaction with the aid of a station- ary linear arc plasma generator (with ∼60kW, current ∼300A, voltage 220 V, mass ﬂow rate V≈120ms−1) po- sitioned on a special rotatable base, there were detected two special directions corresponding to energy release in the plasma jet up to 40% more than the average energy in the plasma ﬂow during rotation of the plasma gener- ator in the horizontal plane through nearly 360◦, with the summary experimental error of ±12%. These direc- tions laid left and right from the vector Agat an angle of ∼45◦−50◦with the latter, they corresponded with the most eﬃcient angle between the vector Aof the current system and the vector Ag(i.e. with the maximum mag- nitude of the new force) being equal to 135◦÷140◦[12]. The found directions of maximum action of the new force in the experiments with the plasma generator gave the following Ag- coordinates: α≈(280−297)◦,δ≈30◦. The aim of the present work is further experimental investigation of the global space anisotropy associated withAg. 1II. FORMULATION OF THE PROBLEM. The new force predicted in the Refs. [1–12] is of com- plex nonlinear and nonlocal character and can be rep- resented in the form of some series in ∆ A, a diﬀerence between changes in AΣat the location points of a sensor and test body. For the ﬁrst approximation of that series we have F∼N∆A∂∆A ∂x where xis the spatial coordinate, and Nis the number of stable elementary particles (electrons, protons, neutron s) in a space region with AΣvarying due to the vectorial potential of some current system. It was shown in the experiments [1,2] with rotating magnetic discs and an engine-generator as well as in the experiments with the plasma generator [12] that the force Fcan be substan- tially increased (tens and more times) when phasing the motion of the body with the process of physical space for- mation from byuons (i.e. the working body must change AΣby its own potential Aand move in the direction of Ag. Therewith the particles of the body must rotate in phase with the above-mentioned process of formation of the physical space). In such a case energy will be taken from the physical space through the elementary parti- cles of the working body. The law of energy conservation in the system ”working body - physical space” will be valid. As is known [17], the basic energy of the Universe (>90%) is determined by the ”dark” (virtual) matter. The model of formation of the physical space [1–4] de- scribes the phenomenon of the ”dark matter” reasonably well. Based upon the physics of the new assumed force and mechanisms of strengthening it to realize the aims of the present paper, the experimental installation should met the following requirements. First, it should realize a max- imum ∆ Acorresponding to maximum possible values of current. Second, to realize a maximum∂∆A ∂x, the current density should be as high as possible. Third, if a plasma generator is chosen for investigation of the new force, its discharge should be maintained in a medium with the most great value of N(for example, not in vacuum but in air at the atmospheric pressure or in water). Fourth, to realize the mechanism of strengthening the new force (i.e. phasing the motion of the working body with the process of space formation at the rate on the order of the light speed), the magnitude of the velocity Vin the discharge should be the maximum possible. Fifth, the experimen- tal installation on the base of a plasma generator chosen for investigating the global space anisotropy by way of scanning the celestial sphere should introduce into the ex- periment minimum systematic errors connected with the rotation of the plasma generator in space (for example, with an inﬂuence of curvature of hoses delivering water, air, and argon to a stationary plasma generator, on heat release in its jet as in Ref. [12], or with action of the Cori- olis force on the ﬂow of water in a measuring tube [12],etc.). All these requirements are most closely met by the pulsed plasma generator (magnetoplasma compressor). III. EXPERIMENTAL INSTALLATION AND TECHNIQUE. The experimental installation was comprised of a pulsed plasma generator and a system of measuring the plasma radiation. The plasma generator was fed from an energy-storage capacitor 100 mFin total capacity with operating voltage up to 5 kV. The total energy accu- mulated in the capacitor was equal to ∼1,25kJ. The battery was charged from a standard high-voltage power source GOR-100 and commutated to the load (pulsed plasma generator) with the aid of a trigatron type air spark gape activated by a short ( ∼1ns) high-voltage (∼30kV) pulse coming to the air-gap from a trigger cir- cuit. The design of the pulsed plasma generator (1) is shown in Fig.1. The case (2) of the generator (its outer electrode being anode) was made of thin-walled copper tube 11 mm in external diameter and 100 mmin length. The ax- ial electrode (3) (cathode) 4 mmin diameter made from copper bar was placed into an acrylic plastic tube (4) with inner diameter of 4 mmand outside diameter equal to that of the outer electrode. In the Fig.1 shown is also a statistic average pattern (5) of discharge currents of the plasma generator. The angle φwas equal to ∼30◦. The whole construction as a unit was an analogue of the coaxial plasma accelerator. The plasma generator was locked on a textolite plate (6) positioned on a special adjustment table (8) rotatable around its vertical axis (7). The table (8) was provided with a limb (9) allow- ing to control the angle of rotation of the whole system relative to some starting position. The plate (6) itself could rotate around the horizontal axis (10) through an arbitrary angle β. The system as an assembly made it possible to rotate the plasma generator during the ex- periment around the vertical axis (7) through any angle, and around the horizontal axis through any angle in the range −90◦< β < 60◦. It was assumed that at β <0 the discharge of the plasma generator turns to the sur- face of the Earth. The horizontal position of the plasma generator corresponded to the angle β= 0. The tra- jectory of motion of the face of plasma generator during its rotation in the horizontal plane represented a circle. All experiments were carried out in air at atmospheric pressure. The volt-ampere characteristics measured with the aid of Rogovsky belt and an induction-free voltage divider as- sembled from resistors of TVO-type, made it possible to judge the amount of energy put into the discharge chan- nel. Some typical volt-ampere characteristics are shown in Fig.2. One can see from the Figure that the discharge was of quasi-periodic character but with great damping. The quasi-period of the discharge current in conditions 2of the experiment was equal to ∼70ms. The amplitude of the current in the ﬁrst maximum reached 21 kA. The maximum voltage between the electrodes equaled 3 .5kV at 5kVcharging voltage on the energy-storage capacitor. The dynamics of plasma outﬂow (11 in Fig.1) as well as the characteristic dimensions of plasma jet were investi- gated with the aid of a super-high-speed photorecorder ofSFR-type operating in single-frame ﬁlming mode. A fragment of the record is given in Fig.3. The earlier stud- ies of the plasma generator in use have shown that about (30−40)% of its power were released in the optical fre- quency range [18]. The absolute value of radiation energy in quartz transmission band ( λ >220nm) was measured by a thermal detector of LETI -type (12 in Fig.1) rigidly ﬁxed on the plate (6 in Fig.1) so that the relative posi- tions of the plasma generator and thermal detector could not change as the plasma generator rotated. In so doing, the axis of the thermodetector was directed to the prior known range of maximum discharge glow lying on the axis of the plasma generator ∼2cmfrom its face. The thermodetector was calibrated with the use of a standard radiation source of IFP-1200 type giving Est= 35.64J/srof radiant energy per unit of solid angle. In the process of calibrating and measuring, the signal from the thermodetector came to the input of a mirror- galvanometer oscillograph K117 . The radiant energy of plasma was calculated from the formula E= 4πAl2n, where Ais the calibration coeﬃcient, lis the distance from the radiation source (in meters), nis the maximum magnitude of signal on the strip of oscillograph (in mil- limeters). At the characteristic dimensions of plasma cu- mulation zone of the order of 1 cmand the distance from the thermodetector to the plasma source l= 20cm,the radiation source could be taken as a point one. Because the radiative energy of plasma is proportional to T4, an insigniﬁcant change in temperature Tcould be sensed by the thermodetector. Despite a signiﬁcant increase in dis- charge current and voltage of the plasma generator con- sidered in comparison with those in the experiment of Ref. [12], the thermal eﬀect of the new force action was expected at a level of 10% owing to short duration of the discharge. The main parameter measured in the experi- ment was the deﬂection of the beam of the mirror- gal- vanometer oscillograph K117 . This deﬂection, propor- tional to the radiative intersity of plasma, was recorded on the photographic strip and gave information on the amount of energy released in the discharge of the plasma generator and, while scanning the celestial sphere, on the direction of maximum action of the new force upon the particles of the plasma discharge. To investigate the direction of the global anisotropy of physical space caused by the vector Ag, as well as the new interaction connected with this vector, the fol- lowing technique was used. In the ﬁrst of experiments (15.12.1999-3.05.2000), the plasma generator was rotated only around the vertical axis (7 in Fig.1).The start time of the experiment was determined by the position of the vector Agnear the horizontal plane. On the basis of previous experiments [1,2,4–12], the vec- torAgwas assumed to have the following approximate coordinates: 270◦< α < 300◦,20◦< δ < 40◦. The statistic average pattern of discharge currents shown in Fig.1 was assumed to correspond to direction of the max- imum current along the axis of the plasma generator. Therefore one could expect that the direction of axis of the plasma generator allowed to judge the eﬃcient an- gle of action of the new force, i.e. the eﬃcient angle between the main discharge current and the vector Ag. Recall that the new force acts when the vector of the dis- charge current has a component directed oppositely to Ag, and the vector of velocity Vof the particles points inAgdirection. In the experiments of 15.12.1999 and 20.01.2000, the luminous emittance of plasma jet was measured by the thermodetector LETI in 30 degree in- tervals through one complete revolution of the adjust- ment table. In all other experiments the measurements of the luminous emittance were made in 10 degree inter- vals. In the second sequence of experiments, to improve the direction of the new force in space, the celestial sphere was scanned in the vicinity of the extremum directions found in the previous experiments during rotation of the plasma generator in the horizontal plane. IV. RESULTS OF EXPERIMENTS AND DISCUSSION. From 15.12.1999 till 3.05.2000, 32 experiments with ro- tation of the plasma generator around the vertical axis through 360◦were carried out. The duration of one ex- periment (25 or 30 shots) was no more than 30-25 min. As an illustration, in Fig.4 shown are the values of de- ﬂection Lof the beam of the mirror-galvanometer oscillo- graph (in millimeters) in dependence on the angle of ro- tation θin the experiments of 15.12.1999 and 20.01.2000. In the experiment of 15.12.1999 conducted from 1640 till 1715, a burst in luminous emittance of plasma gen- erator discharge was observed at 1655. Therewith the angle θmeasured from an arbitrary direction H shown in Fig.5 was equal to 165◦, and the value of emittance was 25.7% above its average over the duration of the exper- iment with a root-mean-square error of ±3.7%. In the experiment of 20.01.2000 performed from 1535till 1610, the burst of plasma luminous emittance was detected at θ= 135◦(measured from that same direction H in Fig.5) with 24% excess over the average value of emittance at a root-mean-square error of ±3.3%. For qualitative understanding of the result obtained and the processing procedure, in Fig.5 are shown by ar- rows some projections of the plasma generator axis on the ecliptic plane corresponding to maximum luminous emittances of plasma discharge with indication of con- crete positions of the Earth in the process of its orbiting 3around the Sun, the data of the experiments, and the points in time at which the said maximum (with val- ues greater than the experimental error) were observed. The dotted line denotes secondary extremum directions for the emittance. In all experiments the plasma gen- erator rotated counter-clockwise if the plane of rotation seen from above. The direction of the axis of the plasma generator at θ= 0 is indicated by letter H for each ex- periment. At the center of Fig.5 (at the site of the Sun) a cir- cle diagram summarizing the results all experiments, is given. They were processed in the following manner. The circle was divided in ten-degree sectors so that the radius- vector passing from the center of the circle along the ini- tial boundary of the ﬁrst sector was aimed at the point of vernal equinox (21.03) from which the angular coordi- nateαis counted anticlockwise in the second equatorial system. In Fig.5 the heights of crosshatched triangles with a 20◦-angle at the center of the circle are proportional to the sums (in percentage) of extremum deﬂections of the oscillograph beam from its average coordinate which stand out above the standard error of measurements and fall within one or the other of the triangles, for all bursts in luminous emittance observed in all 32 experiments. As is seen from Fig.5, the maximum emittances were ob- served (more often and with maximum amplitudes) in the 25th,26thand 33th,34thsectors. Notice that when the vectorial potential of the plasma generator is directed exactly opposite to Ag, the change inAΣshould be maximum and, hence, the magnitude of the new force should be zero since∂∆A ∂x= 0. It fol- lows herefrom that the direction of the vector Agmust be related with the sectors 29,30 (Fig.5). This direction has the coordinates: α= 290◦±10◦, and an eﬃcient angle between the axial current of the plasma generator and the vector Agbeing equal to 140◦±10◦. The result obtained fully coincides (with an error above indicated) with that of Ref. [12] in which a stationary plasma gener- ator positioned on a special rotatable base and a copper measuring tube with water passed through as a sensing element located in the plasma jet, were used. The results of the present paper do not contradict those of earlier measurements of vector Ag[1,2,4–9] and are much more precise. It should be noted that a considerable number of bursts fall into the sectors 11 and 12, corresponding to the direction precisely opposite to that of Ag. This may be attributed to the action of side currents in the discharge of the plasma generator (see Fig.1) directed at an angle φ≈30◦to its axis. That is, in this case the side currents make an angle of ∼150◦withAgwhich is near to the most eﬃcient angle of ∼140◦found from the di- rection of the main axial current of the plasma generator. Therewith the vector of mass velocity Vof the discharge particles is in opposition to Ag. Hence, the mechanism of strengthening the new force is here ineﬀective as com- pared with the situations in which the axis of the plasma generator fall into the sectors 25,26 and 33,34 where thevectors VandAgare directed to the same side. From 10.05.2000 till 31.05.2000 and from 11.10.2000 till 3.11.2000, a run of experiments was carried out with scanning the celestial sphere in the vicinity of sectors from 25thtill 34thfor determining most eﬃcient angles of special position of the plasma generator axis relative to the vector Ag, i.e. the angles of maximum action of the new force. As an illustration, in Table 1 the results of the last experiment performed 3.11.2000 are presented. In all, 77 shots were made with the average duration of scan- ning the celestial sphere about 90min. The angle γin Table 1 corresponds to rotation of the plasma generator around the vertical axis, and the angle βdoes to its ro- tation around the horizontal axis. In each square of the Table, the magnitude of deﬂection of the beam of the mirror-galvanometer oscillograph proportional to the lu- minous emittance of the plasma discharge, is shown. In its turn, this emittance is proportional to the value of the new force acting on electrons and other particles of the discharge. As the duration of the experiment was more than 1.5 hours (the rotation of the Earth through approximately 23◦), the average deﬂections ( Lav) of the oscillograph beam were calculated for one passage of the plasma gen- erator through the angle γat the angle βﬁxed. The value Lavand root-mean-square deﬂections σ(in per- centage) are shown in Table 1 for each passage of the plasma generator. The start time of the experiment was chosen from an expected time of fall of Aginto the range of horizontal plane. In the experiment considered, the angles γ= 220◦andβ=−20◦correspond to a coinci- dence of the north-direction at the place of the installa- tion (Moscow Lomonosov University) with the direction of theAg- projection on the horizontal plane. A summary result of vernal and autumnal experiments with scanning the celestial sphere is shown in Fig.6. In this Figure given are the relative spatial coordinates and directions (arrows) of only those positions of the axis of the plasma generator at which the deﬂection of the oscillograph beam in the process of discharge was above the root-mean-square one. In Table 1 they are asterisked for the experiment of 3.11.2000. The results of the vernal ran of experiments are related to 11.25 of Moscow time of 31.05.2000 and noted by circles in Fig.6. The autumnal results (also asterisked) are related to 20.00 of Moscow time of 11.10.2000. When scanning the celestial sphere, the plasma gen- erator was positioned during the experiment (at various angles β) on circles of diﬀerent radii but with a deﬂec- tion from the level of some horizontal plane no more than ±12,5% in the range of angles −20◦< β < 40◦. There- fore, for clearness, all positions of the plasma generator are given in projection onto the horizontal plane as for vernal so for autumnal runs of experiments with indica- tion of angles of the slope of the generator axis to this plane (angle β). This is understandable since the sections of cone by the plane are well known. 4The ranges of an γ-angles 170◦−190◦and 250◦−270◦ in Fig.6 correspond in space to the sectors 34,33 and 26,25 in Fig.5, respectively. The horizontal planes for these data and the indicated related times of experiments (31.05.2000, 11.25 and 11.10.2000, 20.00) intersect at an angle of ∼35◦. As is seen from Fig.6, with the exception of one and only point ( γ= 25◦, β=−40◦), all other positions of the axis of the plasma generator in the vernal run of experiments form a section of a cone of directions. The dispersion of points in the autumnal run of experiments is more great but all experimental points except two ( γ= 220◦, β= +10◦;γ= 225◦, β= +10◦) ﬁt into the cone shown in Fig.6 with a dispersion of coordinates of axial directions equal to ±10◦. With that same error, the axial direction of the cone makes it possible to ﬁnd the direction of the vector Ag (since along this direction∂∆A ∂x= 0) from the vectorial potential of the axial current of the plasma generator and, hence, the new force should be zeroth, too, i.e. bursts should be absent along Ag. This direction is also shown in Fig.6. It has the coordinates α≈293◦±10◦and δ≈36◦±10◦in the second equatorial system. The di- rection indicated qualitatively coincides with the result s of earlier experiments [1,2,4–9] and speciﬁcally with the recent experiments carried out on the basis of stationary plasma generator [12] as well as with astrophysical ob- servation of anisotropy of distribution of solar ﬂares and galactic pulsars [1,2,10,11]. V. ANALYSIS OF EXPERIMENTAL ERRORS. The errors in determining the direction of Agcan be classiﬁed as systematic ( σsyst) and random, or statistic (σst) ones. In the experiment in consideration, the sys- tematic errors can be due to the following causes: ini- tial bursts at the beginning of experiment with rotation of the plasma generator associated with deﬁcient prior ”warming up” the system (i.e. without a previous ”rang- ing ﬁre” before the readings of the oscillograph ﬂatten out); a build up of the cathode of the plasma generator in the course of the experiment giving rise to a change in geometry of discharge currents (see Fig.1); burn-out of an isolator also changing the pattern of currents in the discharge; a glass turbidity of the thermodetector LETI; a withdrawal of its axis from the direction of the maxi- mum luminous emittance of the discharge; reﬂection of light from objects surrounding the experimental installa- tion; a limited resource of the capacitor. The statistic error σstis caused by the following rea- sons: the random character of geometric pattern of cur- rents (see Fig.1), the inaccuracy of setting the angles γ andβ; exactness of instrumentation (power unit GOR- 100, voltmeter etc.); inaccuracies of thermal detector and of constructions in Figs.5, 6, non-controllable over- heat of contacts; accidental changes in exterior condition s(convective ﬂows in the room, lighting ﬂuctuations, elec- tromagnetic background). Consider the systematic errors. In the early experi- ments, the systematic error caused by a prior warm-up of the plasma generator was included into the ﬁnal re- sult. As was further clariﬁed, there were necessary on the average 7-10 shots before start of the measurements (rotation of the plasma generator in space) to reach op- erating conditions with a minimum error. If this error taken into account, the heights of crosshatched triangles in sectors 17-20 (Fig.5) can be reduced by 25% making the result more pronounced (in Fig.5 these errors are not accounted). The inﬂuence of build up of the cathode in the course of the experiments at the rate of about 1mm per 30-40 shots on the deﬂection of the oscillograph beam is not yet clear. This factor led as to deterioration of sensitivity of measurements (decrease in average deﬂec- tionLavof the beam) so to an increase of sensitivity. In the vernal and autumnal runs of experiments, the said error was reduced to a minimum by way of returning the plasma generator to its initial condition (scraping bright the cathode and anode before each experiment). In the course of experiment, the plasma generator was not touched. The change of Lavdue to build-up of the cathode did not exceed 2 .5% (σk). In the long-term run of experiments performed from 15.12.1999 till 3.05.2000, the burn-out of the isolator led to a decrease of the total sensitivity of the experimental technique, i.e. to some stable drift at a level of 1 ÷2% per experiment ( ∼37 shots). This drift was taken into account and did not tell on the results of each separate experiment so as on the ﬁnal results, too, because the heights of cross-hatched triangles in Fig.5 are given in percentage. Before the new run of experiments with scanning the celestial sphere, the plasma generator was replaced by a new one with the same parameters. The grows of tur- bidity of the glass of the thermal detector was resulting in a drift of the beam deﬂection of the oscillograph and, hence, in a deterioration of total sensitivity of the mea- suring system, but cleaning the glass 1-2 hours before each experiment weakened this eﬀect down to a level of 2% which practically did not inﬂuence on the relative value of the amplitude of bursts analyzed. Withdrawal of the axis of the thermal detector from the direction of a maximum luminous emittance of the discharge could led only to a drift (or leaps) of Lavno more than 1% ( σm). The experiments with the plasma generator were car- ried out in a windowless underground room of the Phys- ical Department of the Moscow State University. The convective ﬂows were practically absent, the surround objects were always at the same places, there were no electromagnetic noise from exterior sources. In total ∼1800 shots were made in the course of experiments. The resource of the new capacitor was about 10000 shots, therefore its instability could not tell on the experimen- tal results. The summary systematic error ( σsyst) due to incontrollable processes during the experiment is repre- 5sentable in the form σsyst=/radicalBig σ2 k+σ2m=±2,7% since those processes were independent from each other. The statistic error caused by the random character of geometrical pattern of discharge currents (see Fig.1) en- tered into the summary error of experiment and was not determined separately. For the experiments carried out from 15.12.1999 till 3.05.2000 (with rotating the plasma generator in the horizontal plane), the summary error comprising the systematic and random ones is shown in Table 2. The accuracy of setting angles γandβin the course of experiments was no lesser than 0 ,5◦(σo<0,5%). The precision of the power GOR-100 and the voltmeter equaled 0 ,5% (σp), that of the mirror-galvanometer oscil- lograph ∼2% (σo). The error of calibration of the ther- mal detector was no more than 2% ( σc) (according to preliminary measurements). The accuracy of construc- tion was ∼1◦. An incontrollable overheat of contacts was impossible in the experiments considered. Random changes in luminous emittance of surround objects were absent. According to the data presented, the total statis- tic error ( σst) was no more than ∼2,8%. The total computation error in the experiments was ∼3,9%. As is seen, it laid near to the root-mean-square errors in- dicated in Tables 1,2 which enhanced the validity of the results obtained. As can be seen from the tables, in many experiments the amplitude of deﬂection of the beam ex- ceeded the root-mean-square error more than two times, this also strengthens the plausibility of results. VI. CONCLUSION. Thus it is shown in the present work that the spatial distribution of the intensity of plasma radiation of pulsed plasma generator is clearly anisotropic. A cone of direc- tions is observed in which plasma radiation reaches its maximum. The results of the experiments can be satis- factorily explained basing on a hypothesis about the ex- istence of cosmological vectorial potential, a new funda- mental vectorial constant determining a new anisotropic interaction of objects in nature. VII. ACKNOWLEDGMENTS. The authors are grateful to participants of sem- inars held in Moscow State University named by M.V.Lomonosov and in IOFRAN, as well as personally to prof. A.A.Rukhadze for fruitful discussion of results of investigation, to academicians of RAS S.T.Belyaev, V.A.Matveev, V.M.Lobashev for the discussion of the connection of the result obtained with the action of the new force manifesting itself in the β-decay [9].The authors give thanks also to the A.V.Chernikov for help in performing the experiments, E.P.Morozov, L.I.Kazinova, and A.Yu.Baurov for preparing the paper to print. [1] Yu. A. Baurov, Structure of physical space and new method of energy extraction (Theory, Experiment, Ap- plications), Moscow, ”Krechet”, 1998, (in Russian). [2] Yu. A. Baurov, On the structure of physical vacuum and a new interaction in Nature (Theory, Experiment and Applications), Nova Science, NY, 2000. [3] Yu. A. Baurov, Structure of physical space and nature of electromagnetic ﬁeld, in PHOTON: Old problems in light of new ideas , edited by V. Dvoeglazov, Nova Science, NY, 2000, p.259. [4] Yu. A. Baurov, Structure of physical space and new in- teraction in nature (theory and experiment), in Proceed- ings of conf. Lorentz group, CPT and Neutrinos , edited by A. Chubycalo, V. Dvoeglazov, V. Kadyshevsky et al., World Scientiﬁc, 2000, p.342. [5] Yu. A. Baurov, E. Yu. Klimenko, S. I. Novikov, Docl. Acad. Nauk SSSR, 315, 5, (1990), 1116. [6] Yu. A. Baurov, E. Yu. Klimenko, S. I. Novikov, Phys. Lett.,A162 , (1992), 32. [7] Yu. A. Baurov, Phys. Lett. A181 , (1993), 283. [8] Yu. A. Baurov, A. V. Kopaev, LANL E-print hep- ph/9601369. [9] Yu. A. Baurov, A. A. Konradov, V. F. Kushniruk and Yu. G. Sobolev LANL E-print hep-ex/9809014 Yu. A. Baurov, Yu. G. Sobolev, V. F. Kush- niruk, E. A. Kuznetzov and A. A. Konradov LANL E- print hep-ex/9907008. [10] Yu. A. Baurov, A. A. Eﬁmov, A. A. Shpitalnaya, Fizich- eskaya Mysl Rossii, 1, (1997), 1. [11] Yu. A. Baurov, A. A. Eﬁmov, A. A. Shpitalnaya, gr- qc/9606033. [12] Yu. A. Baurov, G. A. Beda, I. P. Danilenko and V. P. Ig- natko, Hadronic Jornal Supplement, 15, (2000), 195. [13] Y. Aharonov, D. Bohm, Phys. Rev., 115, 3, (1959), 485. [14] Y. Aharonov, D. Bohm, Phys. Rev., 123, 4, (1961), 1511. [15] A. Tonomura et. al. Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo, 1983, p.20. [16] N. Osakabe et. al. Phys. Rev. A34, (1986), 815. [17] Physical Encyclolopaedia, 4, Moscow, 1994, p.156 (in Russian). [18]Radiation plasmodinamics, v.1 , edited by Yu. Protasov, Energoatomizdat, 1991 (in Russian). 6β\γ270◦260◦250◦240◦230◦220◦210◦200◦190◦180◦170◦Lav[mm]σ% −20◦7070,5* 69,5 6767,5 65 6666676570 67,6 2,9 −10◦69,5* 6662,566,5 63 63 67666366,562,5 65,1 3,5 0◦65 64 6167,56966,5 6560,5 6365,5 66 64,8 3,8 +10◦63,5 6567 636562,5 66,5 61636363 63,9 2,7 +20◦65 65 6565,564,5 6766,563,563,56963,5 65,3 2,5 +30◦6464,5 666363,572,5 69,559,566,572* 68 66,3 5,6 +40◦71* 65 6367,567,5 65,5 6265,5 6569,5 66 66,1 3,8 Table 1. Deﬂection L of the oscillograph beam in the experime nt of 03.11.2000, 1722÷1900by Moscow time. N Date Time Amplitude Error 115.12.1999 1640−171525,7 % ±3,7 % 215.12.1999 1720−175011,8 % ±3,6% 320.01.2000 1545−161024 % ±3,3% 420.01.2000 1620−16403,5 % ±3% 521.01.2000 1445−15157 % ±4% 621.01.2000 1530−164022 % ±5,2% 72.02.2000 1400−144016,5 % ±5,2% 82.02.2000 1455−153017,9 % ±4,5% 99.02.2000 1325−140011,3 % ±6% 109.02.2000 1410−144010 % ±4,7% 1116.02.2000 1255−132011,8 % ±4,8% 1216.02.2000 1342−14108,7 % ±4% 1323.02.2000 1300−135519,8 % ±5,1% 1423.02.2000 1400−144511,6 % ±4,2% 151.03.2000 1150−124011 % ±4,5% 161.03.2000 1245−132012 % ±5% 179.03.2000 1120−12077,8 % ±2,6% 189.03.2000 1220−125813,9 % ±4,6% 1915.03.2000 1045−11305 % ±2,5% 2015.03.2000 1140−123011,3 % ±3% 2122.03.2000 1015−11079,3 % ±3% 2222.03.2000 1115−120010,2 % ±3,6% 2329.03.2000 1045−11558 % ±4% 2429.03.2000 1200−12357,6 % ±3,5% 255.04.2000 1025−11107 % ±3% 265.04.2000 1115−11558,6 % ±5% 2712.04.2000 1100−114012,2 % ±3% 2812.04.2000 1145−12229,5 % ±5% 2926.04.2000 1017−11004,5 % ±2,6% 3026.04.2000 1105−12454,6 % ±3% 313.05.2000 1107−11526,2 % ±2% 323.05.2000 1225−13074 % ±2% Table 2. Amplitudes of maximum deﬂection (in percent from the average) with indication of date, time (Moscow), and root-mean-square error of the experiments.ϕ − +1112 1016β 7 8 94 31 2 5 FIG. 1. The diagram of the measuring device. FIG. 2. The volt-ampere characteristics of the discharge of the plasma generator I - current, U - voltage. 7FIG. 3. A fragment of plasma jet rapid shooting ﬁlm. Fig.4020406080 0 30 60 90 120 150 180 210 240 270 300 330 360 θL [mm]15.12.1999 16.40-17.15 20.01.2000 15.45-16.20 FIG. 4. The magnitude L (mm) of beam deﬂection of the mirror-g alvanometer oscillograph in dependence of angle of rotatio n θfor the experiments of 15.12.1999 (1640−1715) (✷) and 20.01.2000 (1720−1750) (•). 8A30 31 32 33 34353621345678910111213141516171819202122 23 24 25 26 27 28 29A HH Ag 5515.12.1730 21.01.□1535 20.01. 1555 1.03.12 1.03.1230 5028.04.1030 3.05.1100H15.12.16H HH Hg g 21.03 FIG. 5. Directions of the axis of the plasma generator along w hich maximum deﬂections of the oscillograph beam (indicate d by arrows) when rotating the plasma generator in the horizon tal plane, were observed. By H start of rotation ( θ= 0) is denoted. Indicated are data and Moscow times of observation of maximums in beam deﬂection. At the center of the Figure, heights of cross-hatched triangles correspond to the sums o f beam deﬂection magnitudes (in percent) from their average value exceeding the error (for a given sector). Agis the cosmological vectorial potential. 9+40°+20° +20°-40° -30° -20°+10° -10°0°260° 250° 240° 230° 220° 210° 190°270° 180° 170°200°SN +30°+20° +40°+10° -20°+10°-10° +40°+30°+20°0° +10°-20° +10°-10° -10° -10°0°0° -20°-20°-20° -40° +40° +40° +40° +30° +20° -20°+10° +30°+30° +50° +50° +10°Ag γ ββγ FIG. 6. The relative spatial coordinates and directions of t he axis of the plasma generator corresponding to maximum deﬂections of the beam of the oscillograph in the experiment s carried out 10.05.2000 till 31.05.2000 (denoted by circle s◦) and from 11.10.2000 till 3.11.2000 (denoted by asterisks *) . The vernal experiments are related to 1125by Moscow time of 31.05.2000, and the autumnal experiments are to 2000of 11.10.2000 by the same time. Agis direction of the cosmological vectorial potential. 10
1The manipulation of massive ro-vibronic superpositions using time-frequency-resolved coherent anti-Stokes Raman scattering(TFRCARS): from quantum control to quantum computing R. Zadoyan, D. Kohen,† D. A. Lidar,§ V. A. Apkarian, Department of Chemistry, University of California, Irvine CA 92697-2025. ABSTRACT Molecular ro-vibronic coherences, joint energy-time distributions of quantum amplitudes, are selectively prepared, manipulated, and imaged in Time-Frequency-Resolved Coherent Anti-Stokes Raman Scattering (TFRCARS)measurements using femtosecond laser pulses. The studies are implemented iniodine vapor, with its thermally occupied statistical ro-vibrational densityserving as initial state. The evolution of the massive ro-vibronic superpositions,consisting of 10 3 eigenstates, is followed through two-dimensional images. The first- and second-order coherences are captured using time-integrated frequency-resolved CARS, while the third-order coherence is captured using time-gatedfrequency-resolved CARS. The Fourier filtering provided by time integrateddetection projects out single ro-vibronic transitions, while time-gated detectionallows the projection of arbitrary ro-vibronic superpositions from the coherentthird-order polarization. A detailed analysis of the data is provided to highlightthe salient features of this four-wave mixing process. The richly patternedimages of the ro-vibrational coherences can be understood in terms of phaseevolution in rotation-vibration-electronic Hilbert space, using time circuitdiagrams. Beside the control and imaging of chemistry, the controlledmanipulation of massive quantum coherences suggests the possibility ofquantum computing. We argue that the universal logic gates necessary forarbitrary quantum computing – all single qubit operations and the two-qubitcontrolled-NOT (CNOT) gate – are available in time resolved four-wave mixingin a molecule. The molecular rotational manifold is naturally “wired” forcarrying out all single qubit operations efficiently, and in parallel. We identifyvibronic coherences as one example of a naturally available two-qubit CNOTgate, wherein the vibrational qubit controls the switching of the targetedelectronic qubit. † Present address: Smith College, Chemistry Department, Northampton, MA 01063. § Permanent address: Chemistry Department, University of Toronto, 80 St. George St., Toronto, Ontario, Canada M5S 3H6.2I. INTRODUCTION Coherent anti-Stokes Raman scattering (CARS) is a well established and broadly applied spectroscopic tool,1,2 the subject of textbooks.3,4,5 When carried out with ultrafast lasers, within a single experiment, CARS combines the elements ofpreparation, manipulation, and interrogation of molecular coherences. These arethe key ingredients of quantum control, be it applied to quantum chemistry, 6,7,8 or quantum computing.9,10 It is from this perspective that we present our multi- dimensional time- and frequency-resolved CARS studies on the well-characterized system of diatomic iodine in the gas phase, and at roomtemperature. Quantum coherences can be defined by their two-dimensional mapping along relevant conjugate variables, such as time and energy. 1 This is familiar in practice in the characterization of optical pulses by such means as frequencyresolved optical gating (FROG). 11 An equivalent characterization of molecular coherences is enabled through joint time-frequency-resolved CARS (TFRCARS)spectroscopy. This, we recently demonstrated through time-gated, frequency-resolved detection of the anti-Stokes polarization (TGFCARS) in iodine vapor atroom temperature. 12 The experiment served to illustrate that the third-order coherence of the molecule could be imaged directly as an interferogram on thetime-frequency plane, and that ro-vibronic coherences consisting ofsuperpositions of 10 3 states could be rigorously decomposed in such maps. There, we emphasized that although fs pulses are used, the third-ordercoherence in resonant CARS can be detected with rotational resolution, since itevolves freely until the unset of decoherence. In room temperature vapor iodinero-vibrational decoherence would be determined by pure dephasing due to theDoppler inhomogeneous width of lines (t ~ 10 -9s). With regard to chemical dynamics, this implies that transients captured in fs time can be analyzed with high resolution ( ∆ω ~ 0.02 cm-1). Here, we extend the analysis to first- and second- order coherences, detected through time-integrated frequency-resolved CARS. As in our prior development, we maintain the intuitive languages ofdiscussing vibrational coherences in terms of wavepackets, 13 while ro-vibrational coherences14,15 are treated as phase evolution in vector space. As an example, in such a treatment, the counter-intuitive result that condensed phase CARS signalsshould be more deeply modulated than in the gas phase can be understood interms of the spatial extent of vibrational superpositions. 16 In contrast, the description of time evolution of rotational superpositions as purely phaseevolution of eigenstates in Hilbert space is more natural and suggestive incomputational applications. 1 In keeping with common practice energy, frequency, wavelength and angular frequency are interchangeably used in the text. The time-energy space is invariably refereed to as time-frequency while the experimental data are presented on a time-wavelength axis.3The concept that the controlled manipulation of quantum coherences can be used for quantum computation or information transfer, dates to anannunciation on general grounds by Feynman, 17 followed by the formal arguments presented by Deutsch.18 This very active field of research was more recently catapulted forward by the development of algorithms that takeadvantage of quantum parallelism to achieve exponential speed-up in particulartasks. The most notable among these being Shor ’s algorithm for factorization of a number into its primes, 19 and Grover ’s search algorithm,2 to search an arbitrarily large data base with a single query.20 The field has since advanced dramatically, with physical proofs of principle based on the demonstration of universalquantum logic gates, realized through a variety of experimental approaches.Among examples are: all optical interferometry, 21,22 cavity quantum electrodynamics,23 ion traps,24 superconducting Josephson junctions,25 and an explosion of studies in NMR following the initial propositions.26,27 More specific tasks, such as a search algorithm through the preparation and manipulation ofRydberg Superposition states in atomic beams, 28 have also been demonstrated. The latter belongs to the category that does not require entanglement,29 and therefore can be argued that does not allow exponential speed-up without theexpenditure of an exponential overhead in resources. 30 Physical realizations that involve entanglement,31 to-date, have been limited to several qubits (the quantum analog of a bit),32 with extensions to larger numbers being a nontrivial challenge. The multidimensional coherent spectroscopy of molecular ro-vibroniccoherences involves the controlled manipulation of massive Quantumsuperpositions; as such, it presents an opportunity for executing quantum logicon a very large manifold of states. After presenting the experimental data andtheir interpretation, we discuss the natural structure of the tensor space in whichthe ro-vibronic polarization is manipulated, and the naturally available “wiring ” of quantum logic gates for computational tasks. The massiveness of the ro-vibrational superposition that is manipulated, consisting of 10 3 states, and the precision with which coherences can be transferred between ro-vibronic states,makes the system particularly attractive for computation with massiveparallelism. Molecular iodine is chosen for these studies because of its spectroscopic convenience, well characterized spectroscopy, and the fact that it has alreadybeen scrutinized by fs CARS experiment and theory in the gas phase. 33,34,35,36 Although unintended, we discover several unexpected electronic scatteringchannels in iodine. We discuss these for completeness. The multiple electronicresonances that lead to complexity in the data can add significant flexibility inthe tailored manipulation of electronic coherences. Prior to presenting the experimental results and their analysis, we establish the frameworks to be used in the interpretation of vibronic and ro- 2 Although provably faster, Grover ’s algorithm gives square root speed-up over classical algorithms.4vibronic contributions to resonant CARS, along lines already introduced in two preceding papers.12,16 Coherent Raman scattering (CRS) is a four-wave mixing process that involves measuring the third-order material polarization in responseto the application of three input beams. 5 The fourth rank hyperpolarizability tensor that mediates the process allows selectivity by experimental control ontime-orderings and Cartesian components of the applied fields, hence theproliferation of acronyms devised to highlight specialization. In fs CARS, inputpulses of different color are chosen such that the detected coherent materialpolarization occurs to the blue of the input pulses. Single color fs CRS, oftenidentified by its frequency domain acronym of degenerate four-wave mixing(DFWM), has been implemented in small gas phase molecules to explore aspectsof molecular control. 37,38 In condensed media, the same measurements appear under the heading of three-pulse photon echo.39,40 Both time-gated41 and frequency-resolved42,43 stimulated photon echo measurements have been performed, which combined,44 would be equivalent to our TGFCARS experiments. The more complete four-wave mixing measurements involveheterodyne detection of the radiation, to yield amplitudes and phases of allfields. 45,46, Heterodyne detected three-pulse photon echo measurements have been extended to the infrared, and have been implemented to the study of peptides.47 Reviews of these inherently multi-dimensional fs spectroscopies of vibronicexcitations have recently appeared. 48,49 A) Vibronic Coherences In time resolved CARS, the third-order polarization, P(3), induced by the application of three short laser pulses is probed.3-5 As in the more common implementations, here too, two of the pulses are chosen to have the same color and are identified as pump, P and P ’; while the third, red-shifted pulse is identified as Stokes, S. The signal consists of the polarization propagating along the anti-Stokes wavevector, kAS=kp+kp’−kS. Choosing both pump and Stokes colors to overlap the dipole allowed X(1Σ0g+) ↔ B(3Π0u+) electronic transition, we may expect the rotating wave approximation to hold, and the consideration may be limited to scattering processes that are resonant in all orders. We will initiallyrestrict the consideration to the two-electronic state molecular Hamiltonian, H=XHXX+B(HB+Te)B (1) in which HB and HX are the vibrational Hamiltonians in the excited and ground electronic states, respectively. Within the interaction representation of time dependent perturbation theory, P(3) is the expectation value of the dipole operator:505P(3)(t)=ϕX(0)(t)ˆ µ ϕB(3)(t)+ϕX(2)(t)ˆ µ ϕB(1)(t)+c.c. (2) where ˆ µ =µ(ϕXϕB+ϕBϕX) (3) and /G1/G1ϕ(n)(t)=i /G61dt’e−iH(t−t’)ˆ µ ⋅E(t') −∞t ∫ϕ(n−1)(t') (4) with applied fields described by their envelopes El(t): El(t)=ˆ ε l[El(t)e−iωlt+El(t)e+iωlt] where l = P, P ’, S (5) The possible contributions to (2), which are generated by the permutations of the P, P’, and S pulses in (5), can be illustrated by the familiar double-sided Feynman diagrams, as in Figures 1 and 2. The equivalent time-circuit diagrams, whichprove useful in describing the material response in both state representation andin classical 51 or semiclassical propagations,52 are also shown in the figures. As it will become clearer, the time-circuit diagrams are quite useful in keeping track ofthe material response in state space. Choosing ωP to the red of the absorption maximum and ωS outside the absorption band ( /G61ωp−/G61ωs>kBT), only the first term in Eq. 2 (Fig. 1) can be electronically resonant in all three pulses. Further, by experimentally choosing the sequence P, followed by S, followed by P', we may transcribe the diagram inFig. 1 to the explicit third-order perturbation expression: 13 /G1/G1PkAS(3)(t)=PkAS(0,3)(t)+PkAS(3,0)(t) =k /G613dt3 −∞t ∫dt2 −∞t3 ∫dt1 −∞t2 ∫e−i(ωP’−ωS+ωP)t ×ϕXeiHXt/ /c61ˆ µ e−iHB(t−t3)/ /c61ˆ µ EP’(t3)e−iHX(t3−t2)/ /c61ˆ µ ES(t2)e−iHB(t2−t1)/ /c61ˆ µ EP(t1)e−iHXt1/ /c61ϕX + c.c. (6) The content of (6) can be readily visualized in the wavepacket picture of Fig. 3. For all times, the bra state vector ϕX(0)(t) evolves subject to the bare, ground state molecular Hamiltonian. The fields act on the ket state. At t1, the pump prepares a wavepacket in the B state, ϕB(1)(t), in the Franck-Condon window carved out by the pump laser. In this first order coherence, the packet on the B potential evolves until t = t2, when the Stokes pulse arrives. The portion of the packet that overlaps with the Stokes window can now be transferred to prepare the second- order (or the Raman) packet, ϕX(2)(t). The Raman packet evolves on the X state until t = t3, when the P ’ pulse acts. Now amplitude proportional to overlap of the6Raman packet with the pump window is transferred to the B-state, to prepare the third-order packet ϕB(3)(t). In this third-order coherence, the system evolves freely, radiating every time the vibrational packet reaches the anti-Stokes window, which is located at the inner turning point of the B-state potentialwhere the energy conservation condition δ[ωAS−(2ωP−ωS)] can be satisfied. This resonantly created third-order polarization will persist long after the termination of pulses, until destroyed by collisions. Recursions in the third-orderpolarization imply structure in the AS spectrum, as already argued anddemonstrated. 12 If we make an analogy with photon echo, the observable recursions in the third-order polarization would be more appropriately described as photon reverberations . Note, in first- and third-order the bra and ket are in separate electronic states, therefore the system is in a ro-vibronic (rotation-vibration-electronic) coherence. In second-order, the Raman packet is in a ro-vibrational coherence in an electronic population on the X-state. Finally, note thatthe complex conjugation indicated in (6) ensures that the third-order polarizationis real, and that a diagram conjugate to each of those illustrated in Figures 1 and2 is operative in the process. The concepts of wavepackets and coherences derivefrom somewhat different starting points; nevertheless, we marry the languagesto take advantage of the intuition contained in each. B) Ro-vibrational Coherences: The complete rotation-vibration contribution to P(3) can be given in terms of the density matrix in Hilbert space: ρ(t)= χ’,v’,j’∑ χ,v,j∑ m’=−j’j’ ∑ m=−jj ∑c(χ’,χ,v’,v,j’,j,m’,m)χ’,v’,j’,m’;tχ,v,j,m;t (7a) in which χ, v, j, m designate electronic, vibrational, rotational and magnetic quantum numbers of eigenstates ( χ,χ‘ = X, B). After three interactions with the laser field, the expectation value of the dipole over the third-order polarization Tr[ˆ µ ρ(3)(t)] is measured. Starting with a given initial eigenstate of the thermal density \|X,v,j><X,v,j\|, the dipole operator (3) ensures that after the first interaction with the radiation field the first-order coherence is prepared: ρ(1)(t)= v’,v,j∑c(v,v’,j+1,j)B,v’,j+1;tX,v,j+c(v,v’,j−1,j)B,v’,j−1;tX,v,j [] +c.c. =ϕB(1)(t)ϕX(0)(t)+ϕB(1)(t)ϕX(0)(t)+c.c. (7b)7in which the state coefficients are a function of the applied field (4), and contain Frank-Condon factors and rotational matrix elements. The second pulse prepares the electronic population, ρ(2)(t)=ϕX(2)(t)ϕX(0)(t)+c.c., which is a ro-vibrational coherence ρ(2)(t)=∑c(v",j")X,v",j";tX,v,j;t+c.c. with j” = j, j±2 and v” ≠ v since P and S pulses do not have any spectral overlap. Finally, according to Fig. 1, in the four-wave process a given eigenstate is forward propagated on excited B→X→B states over the intervals t21→t32→t43, then dipole projected back onto the original eigenstate and propagated in reverse-time over the t34 interval. This three-time correlation contributes a unity (reaches its maximum value) when the phase accumulated over the time-circuit is an integer multiple of 2 π:3 /G1/G1Ω(t;’,",’’’)=[Ev,jX(t1−t4)+Ev’’’,j’’’B(t4−t3)+Ev",j"X(t3−t2)+Ev’,j’B(t2−t1)]//G61 =ωv,jXt41+ωv’’’,j’’’Bt34+ωv",j"Xt23+ωv’,j’Bt12 =2πn(8) This defines the condition for phase coherence in CARS, for a given path in state space. Note, for a given path in vibrational space – v, v’, v”, v’’’ – the ∆j=±1 dipole selection rules lead to six rotational paths to close the time-circuit. This is illustrated in Fig. 4, in a standard diagram showing optical transitions withvertical arrows (Fig. 4a), and in a schematic “wiring ” diagram (Fig. 4b). The bra- state, <v,j;t\| , represented by the lower line in Fig. 4b, acts as a reference for defining instantaneous phases of the evolving ket-states. Thus, the instantaneous complex amplitudes in the coherent superposition a’\|v’,j+1><v,j\| + b ’\|v’,j- 1><v,j\| prepared by the P-pulse are given as: a’(t)=c’e−iΩ’(t) with Ω(t)=(ωv’,j+1B−ωv,jX)(t−t1), where c’ is determined by the laser field and transition matrix elements. For a laser pulse short in comparison to ro-vibrational periods, phase evolution under the pulses may be ignored. Thus, the amplitudes in the final superposition a’’’\|v’’’,j+1><v,j\| + b ’’’\|v’’’,j-1><v,j\| are controlled by the interference between the three paths that connect each of the final j±1 pair of eigenstates to the initial coherent j±1 pair, with path lengths controlled by the delay between laser pulses. The detectable polarization consists of contributionsfrom closed circuits from each of the thermally occupied initial eigenstates, as asquared, real quantity with a definite phase. Since it is the bulk polarization thatis detected experimentally, and the individual time-circuits starting from each statistical initial state is phase-locked by the sudden action of the P-pulse at t = t 1, all components of the radiation may interfere among themselves. Both phase andmagnitude information is contained in the polarization, and can be retrieved 46 3 Contrary to the practice of identifying the ground state quantum numbers with double primes, we will identify the ground state indices without primes while first, second and third order states are identified with one, two, and three primes, respectively.8(phase, to within a sign in the present measurements). We will discuss the information stored, manipulated, and retrieved in the various order quantumcoherences after presenting the experiment and its analysis. II. EXPERIMENTAL Experimentally, it is more natural to think about the four-wave mixing process interms of bulk polarization. 3 In the long-wave limit, the third-order bulk susceptibility probed in CARS, P(3), results from molecular contributions, P(3) = NP(3), where N is the molecular number density, and the third-order molecular polarization is the response to three applied fields mediated by the fourth rank hyperpolarizability tensor, γ: Pρ(3)(ω0,t)= (1,2,3)∑ στν∑γρστν(−ω0,ω1,ω2,ω3)Eσ(k3ω3,t3)Eτ(k2ω2,t2)Eν(k1ω1,t1) (9) We use two colors for the three input beams: ω1=ω3≡ωP=ωP’ and ω2≡ωS, in the forward BOXCARS arrangement illustrated in Figure 5.53 The three beams propagating along separate wavevectors, after passing through separate delaylines, are brought into focus using a single achromatic doublet (fl = 25 cm). Two pinholes serve to spatially filter, PkAS(3)(t)=drei(kP+kP’−kS)rP(3)∫, the forward-scattered coherent polarization along the anti-Stokes (AS) wavevector: kAS=kp+kp’−kS (10a) with associated energy conservation condition: ω0≡ωAS=2ωP−ωS (10b) In conventional implementations, the signal consists of the directional coherent AS polarization collected with a single element square-law detector, asa function of delay between pulses: S(t1,t2,t3)=dt4[∫PkAS(3)(t4;t3,t2,t1)]2(11) Instead, we record the spectrally resolved AS radiation in one of two modes: time-integrated or time-gated detection. In Time-integrated frequency-resolved CARS (TIFRCARS) , the spatially filtered AS polarization is dispersed in a 1/4-m monochromator and detected using a CCD array. The integration is over t≡t4, and since pulsed lasers are used, the limits of integration can be extended to ± ∞:9IAS(ω;t1,t2,t3)=[dte−iωt −∞∞ ∫PAS(3)(t)]2(12) The recorded AS spectrum is composed of the signal elements (pixels) of the array S(ω0)=[dω∫f(ω−ω0,δω)dte−iωt −∞∞ ∫PAS(3)(t)]2 =[dω∫f(ω−ω0,δω)PAS(3)(ω)]2(13) with f(ω-ω0,δω) defined as the amplitude bandpass of the spectrometer. By selecting the time delays between pulses, t1, t2, t3, the time integrated spectrum (12) yields images of the first and second coherences. Thus, following the time- circuit diagram in Fig. 1, the AS spectrum obtained as a function of t21 with t32 = 0 (i.e., with P pulse preceding the coincident S and P ’ pulses), yields the evolution in first-order coherence : S(ω,t21)=S(ω;δ[t3−t2],t2−t1)≡S(ω,t<) =[dte−iωt −∞∞ ∫PAS(3)(t,δ[t3−t2],t2−t1)]2(14a) Similarly, the time integrated spectrum obtained with coincident P and S pulses, δ[t2-t1], as a function of delay of the P ’ pulse, yields the evolution in second –order coherence : S(ω,t32)=S(ω;t3−t2,δ[t2−t1])≡S(ω,t>) =[dte−iωt −∞∞ ∫PAS(3)(t,t3−t2,δ[t2−t1])]2(14b) Since P and P ’ pulses are identical, in practice, one of the pump pulses is overlapped in time with the Stokes pulse and the second pump pulse is scanned in time. The experimental convention uses negative time, S(t<), to identify measurement in first-order coherence over t21, and positive time signal, S(t>), to identify measurement of evolution in second-order coherence, over t32. Experimentally, coincident pulses imply two pulses with a relative delay that issmall in comparison to their width. The relative delay is adjusted whileoptimizing a particular signal. Time-gated frequency-resolved CARS (TGFCARS) is implemented by passing the AS beam through a Kerr gate before entering the monochromator. The Kerrcell consists of a 5-mm long cuvette filled with CS 2, held between a cross- polarized pair of prisms (see Fig. 5). A few percent of the 800 nm fundamental of10the Ti:Sapphire laser, polarized at 45 ° relative to the CARS beam and passed through a separate delay line, is used to induce Kerr rotation in the cell. Spatialand temporal overlap of the gate pulse with the CARS pulse is obtained usingthe non-resonant signal from air. This also provides a calibration of the gate width, δt = 500 fs, in agreement with the known Kerr response of CS2.54 For a fixed time sequence of input pulses, the CARS spectrum as a function of gate delay, t4, yields the two-dimensional image of the evolving third-order coherence S(ω0,t4)=dtg(t4−t,δt)S(ω0,t) ∫ (15) Since, the width of the Kerr gate is comparable to vibrational periods in iodine; the signal will smooth over the vibrational modulation. The spectral resolution, δ, is determined by the broader of the spectrometer bandpass or the inverse gate width: δ2 = δω2+c-2δt-2. TGFCARS is similar to heterodyne detection, however, since the pulses used in the experiment are not phase locked, only an envelope function appears in (15), and phase information is lost. The fs laser source used in these experiments consists of a Ti:Sapphireoscillator, which is chirped pulse amplified at 1 kHz to an energy of 700 µJ/pulse, and compressed to a pulse width of 70 fs. The pulse is split with a 50% beam splitter to pump two three-stage optical parametric amplifiers (OPA). The OPA output is frequency up-converted by sum generation, to provide tunabilityin the 480-2000 nm spectral range. The time-frequency profiles of the pulses areadjusted with two-prism compressors following each OPA. The pulses are nottransform-limited. The OPA output is adjusted to yield a bandwidth of 450 cm -1. Critical alignment of beams in space and time is crucial for the successful execution of the experiments. This is achieved by first optimizing the non- resonant CARS signal obtained from a 200 µm thick glass plate, then removing the plate and optimizing the non-resonant CARS signal from air, and finally inserting the static quartz cell containing I2 heated to 50 °C in the beam overlap region. The cell is positioned to optimize the signal, then beam paths are adjustedto compensate for changes introduced by the cell windows. III. RESULTS AND ANALYSIS A) The ro-vibrational third-order coherence – TGFCARS: A time-gated, frequency-resolved CARS (TGFCARS) spectrum is shown in Figure 6. The image was obtained with coincident S+P ’ pulses, t23 =0, delayed from the P pulse by one vibrational period in the B state, t12 = 380 fs. A nearly identical image is obtained with all three pulses in coincidence, t12=t23=0. The AS polarization, which is patterned by interference among ~103 ro-vibrational11transitions is followed for 55 ps, with an effective resolution of 20 cm-1. This direct image of the third-order coherence was previously analyzed in somedetail. 12 The main features of the interferogram are easily understood: a) An intense signal occurs at t = 0 , which within 1.5 ps decays to a 12 ps period of silence indicating complete destructive interference. At t = 0 , since Ω = 0 for all eigenstates in the superposition, all states radiate in phase. Given the dense manifold of ro-vibrational states under the sampling gate, the initial decay is simply the dephasing time given by theinverse of the detection bandpass (20 cm -1). The period of silence corresponds to a flat distribution of phases of rotational transitionscollected under the time-frequency window of detection. b) As the highest observable rotational states execute an Ω = 2π phase rotation, Eq. 8, the polarization reappears, and develops into five chirped trails of rotational revivals in the five vibrational states v =31-36 of thethird-order wavepacket. The periodicity of recursions can be understoodas the beat between adjacent spectral components of the polarization. The hyperbolic trails result from the inverse dependence of recursion time, τr, on rotational quantum number: 1/τr = 2(B ”-B’)j/c (16) where B ” and B ’ are the rotational constants in the ground and excited states, respectively.12 c) The trails develop a complex pattern due to interference between different rotational transitions of a single vibrational transition, as thedifference between rotational recursions (winding numbers) of rotationalstates in a given vibration increases with time. Interference also occurs asrotational recurrences from different vibrations overlap in the time-frequency plane. All details of the two-dimensional interferogram can be reproduced using the accurately known spectroscopic constants of iodine, 55 as shown in the simulation in Fig. 6. The image of the third-order polarization is reconstructed byconsidering the AS radiation sampled under the time-frequency gate, consisting of P- and R-branch transitions ( j-1 →j and j+1→j ): /G1/G1S(ω,t4)=dωdt t+δtt−δt ∫g(ω,t−t4) ω−δωω−δω ∫ v,v’"∑ j∑e−βEv,jX [c(v,j,v’’’,j±1)e−i(Ev"’,j±1B−Ev,jX)t4/ /c61]      2 (17) with coefficients:12/G1/G1c(v,j,v’’’,j±1)=µBX4c(j,j±1)c(v,v’’’)δ[EL−Ev,j]3 =µBX4[D j,j±12D j±1,j±22+D j,j±14+D j,j±12D j,j /c6612]vv’’’v’’’v’’v’’v’v’v v’,v"∑ ×E(ωp)E(ωS)E(ωp)δ[/G61ωp−Ev’’’j’’’,v’’j’’]δ[/G61ωS−Ev"j",v’j’]δ[/G61ωp−Ev’j’,vj] (18a) where, for linearly polarized all-parallel fields56 Dj,j±1= m=−jj ∑j,m10j±1,m (18b) While their evaluation is straightforward, since within the sampling bandpass the transition coefficients are slow functions of wavelength, we set c(v,j,v ’’’,j±1) = 1 in the simulations. The good comparison between simulation and experimentin Fig. 6 validates this approximation, and suggests robustness of the detectablepolarization. The latter consideration is quite valuable in the quantumcomputational applications to be considered. The interference structure in the third-order polarization occurs on several time-frequency scales, therefore the detected image can vary substantively withthe choice of laser colors. The simulations in Fig. 7 and 8 are instructive in thisregard. They are used to illustrate the interference pattern on a spectral rangemuch larger than the experiment. In Fig. 7a the recursion trails of the P-branch oftransitions from v ’’’= 34 is shown. In Fig. 7b, both P- and R- branches of v ’’’ = 34 are included to show the interference between rotational branches of a singlevibrational band. Fig. 8 illustrates the third-order polarization for v ’’’ = 26-45 in the B state. The image in Fig. 8a is obtained by summing over the squaredtransition probabilities, as opposed the squared sum required by Eq. 17 andshown in Fig. 8b. This allows the illustration of the ro-vibrational revivals, withand without interference among the transitions. Remarkably, the intricate,predictable pattern that arises after a few rotational revivals (the slow pattern ofinterferences in 8b reflects the variation of vibration dependent rotational constants), is expected to last until decoherence, therefore until t ~10 -9s.57 B) First- and second-order coherences – TIFRCARS: Time-integrated frequency-resolved CARS (TIFRCARS) spectra, as a function of time delay between P and coincident S+P pulses are shown in Figure 9a; withrepresentative temporal and spectral slices shown in Figures 9b and 9c. The data were acquired with non-transform limited pulses, with pump centered at λ P = 553 nm (∆λP = 15 nm, ∆τP = 70 fs) and Stokes centered at λS = 577 nm ( ∆λS = 20 nm, ∆τS = 90 fs). The time-integrated spectra in Fig. 9c identify the ro-vibrational composition of the prepared third-order coherence. Vibrational recursions whichform the high frequency modulation of the time slices in Fig. 9b, appear as stripes in the time-frequency image in Fig. 9a. At t = 0 , when the three input13pulses overlap, a dense, nearly structureless spectrum which arises from multiple interaction diagrams is observed, see Fig. 9b. Upon introducing delays longerthan a vibrational period in the excited or ground electronic state, the scatteringsignal intensity drops, and significantly simplified spectra emerge. We will beconcerned with the signal at finite time delays, when the P pulse is delayed oradvanced relative to the coincident P+S pair, as defined by Eq. 14. In Figure 10, we show the same data as in Fig. 9, now stretched to time delays of t = ±25 ps . The time step in these scans is correspondingly coarser, ∆t = 100 fs , suitable to obtain images of rotational recursions. Evolution in first order coherence is followed over the t12 interval, or at negative time according to our convention (14a) . At ωP = 553 nm , the P pulse prepares a packet centered near v’ = 22 in the B state, and the observed period of 370 fs is consistent with the 87cm-1 spacing of vibrational levels at this energy: τ = /G61/[E(v=22)-E(v=21) ] = 380 fs. At 0 > t > -2ps , the waveform is deeply modulated; it consists of sharp vibrational recurrences which lose intensity with time, as the rotational density starts to evolve. Rotational revivals occur with a wide dispersion in time, for t < -2ps , leading to the slow rolling background. Once rotational phases spread, vibrational recurrences appear with greatly reduceddepth of modulation. As ro-vibrational states which have executed differentnumbers of periods (windings) overlap, the vibrational recursions blur, and the fast modulation is nearly eliminated at t< -6 ps, see Fig. 9b. The spectral slices provide complementary information and check the consistency of interpretation. The spectrum at t = - 2ps consists of the vibronic progression B( v’ = 27-34 ) → X(v = 0). At this early time in the evolution of rotational phases, the band profile of each vibration approximates the thermal distribution of rotational states, with population peaking near j = 50 . This spectrum also allows to establish that the contribution of transitions terminating on v = 1 is negligible (less than 5%). As time progresses, see the slice at t = - 4ps , the rotational band profiles sharpen and shift to lower j states, to states which have not yet evolved out of phase. At later time, at t = -6ps and t = -8ps , the spectra lose definition, retaining little resemblance to the trivially recognizable spectrum at t = -2 ps . The spectral features now consist of a congested set of ro-vibrational resonances, due tooverlapping ro-vibrational transitions that fall within the spectral bandpass ofthe monochromator. Evolution in the second-order coherence is followed over the t 32 time interval, or at positive time according to our convention (14b). This tracks the evolution of the Raman packet on the X state. The persistent shallow modulation observed in thewaveforms in Fig. 9b, occur with a period of 160 fs. This is consistent with apacket near X(v ” = 4), which could also be inferred from the Raman resonance condition ω = ωP-ωS,. In contrast, the set of prominent peaks at 0 < t < 1ps in the 541 nm waveform show a period of 350fs. Quite clearly, these vibrational resonances do not belong in the ground electronic state – they are not part of the Raman packet created by the P+S pulses. Evidently, an additional scattering14channel contributes to the signal. This is re-enforced by the congested spectra at positive time (see Fig. 9c), which do not show a simple vibrational progressioneven at early time. This additional scattering channel cannot be reconciled withthe two-electronic state Hamiltonian that we have assumed until now. Resonantscattering over yet another electronic state is implicated. To be certain, weconsider the explicit wavepacket simulation of the 2-D image of vibronic CARSwithin the two-electronic state Hamiltonian. C) Numerical simulation of vibronic TIFRCARS: Ignoring all rotational contributions, the vibronic CARS spectra are simulated by the wavepacket propagation method of Kosloff and Kosloff.58 The simulations were performed assuming that the laser fields only couple the X and B electronicpotentials, which are both described as Morse functions. 59 The various order wavepackets are obtained sequentially according to Eq. 4, by integrating thetime-dependent Hamiltonian, starting with: /G1/G1i/G61∂ ∂tϕX(0 )(t)=HXϕX(0)(t) (19) and evaluating all contributions to the third-order polarization: /G1/G1i/G61∂ ∂tϕα(n)(t)=Hαϕα(n)(t)+µαβEl(t)e±iωltϕβ(n−1)(t) (20) where l = P, S, P ’ and for n even, α = X and β = B while for n odd, α = B and β = X. The third-order polarization in this perturbation treatment is represented by the sequential absorption and emission of photons ( −ωl, and +ωl, respectively, when acting on the ket state). For the laser envelope, El(t), a Gaussian of fwhm = 70 fs is used, with a transform limited spectral width of 210 cm-1. Taking ωP = ωP’ = 550 nm, and ωS = 571.5 nm, all permutations of the fields in (20) generate ten different-order packets, which are simultaneously propagated for 20 ps. The time dependent spectra are then evaluated by Fourier transformation of the third-order time dependent polarization along the AS direction: S(ω,τ)=dte−iωtP2kP−kS(3)(t,τ) −∞∞ ∫2 (21) The calculated time-frequency image is shown in Figure 11. The simulated spectra have been convoluted with a Gaussian of fwhm = 70 fs to compensate forthe absence of rotations in the comparisons with experiment.15Since, all possible timing diagrams are explicitly included in the computation, their contribution to the final observables can be directly assessed.Thus, at negative time, the AS polarization formally consists of: P2kP−kS(3)(t)=ϕX(0)ˆ µ ϕB(3)(ωP’,−ωS,ωP,t)+ϕX(2)(ωS,−ωP,t)ˆ µ ϕB(1)(ωP’,t)+c.c. (22) while at positive time (see diagrams in Fig. 2): P2kP−kS(3)(t)=ϕX(0)ˆ µ ϕB(3)(ωP’,−ωS,ωP,t) +ϕX(2)(ωS,−ωP’,t)ˆ µ ϕB(1)(ωP,t)+ϕX(2)(ωS,−ωP,t)ˆ µ ϕB(1)(ωP’,t)+c.c. (23) For the chosen input pulses, the contribution from the first term in both (22) and(23) is three orders of magnitude larger than the rest of the terms. Hence theimportance of the time-circuit diagram in Fig. 1 which has been used in all of ourdiscussions. The closed nature of the time circuit diagrams implies that eachinitial statistical state can be separately evaluated, and co-added to obtain theoverall signal. As such, it is verified that although the thermal occupation ofB(v=1) is 35%, its contribution to the final spectrum is negligible (<1%). This isthe result of the strong selectivity of the nonlinear CARS process, as expressed bythe weights of states in Eq. 18. The agreement between simulation and data over the t 12 interval (negative time) is quite acceptable, given that rotations are not included in the simulation.The simulation predicts a rather simple vibrational progression. The firstrecursion occurs with a relatively compact packet, giving a pulse width limitedsignal. Due to the anharmonicity of the potential, the recurrences in the signal broaden with time and develop a chirp. By the tenth recurrence, near t = 4 ps , the packet splits and shows a doubling in the signal, in good agreement with theexperimental 2-D image of Fig 9a. The predicted image at positive time is rather similar to that at negative time. It consists of a simple vibrational progression, now with a recurrence givenby the X-state vibrational period. The simulation does not reproduce thecongested and strongly blue shifted spectrum observed in the experiment atpositive time (see Fig. 9c). Indeed, in the two-electronic state Hamiltonian, if timeevolution is ignored under the laser pulses, then the spectral composition of theCARS signal should be symmetric in time. This would be the expectation if wewere to note that states in the third-order superposition are determined by energy conservation, δ[E-(2ωP-ωS)]. Evolution of the molecular Hamiltonian under the laser pulses breaks this symmetry. The simulations show that at negative time, the spectral intensity peaks near 530 nm, near the wavelengthpredicted by the energy conservation condition (10a). At positive delay, thespectral maximum of the third order polarization shifts down by two vibrationalquanta, to 533 nm. The down-shift is the result of chirp generated by thesequential action of two pulses on a fast evolving packet, 60,61 which is a process16inherent to resonant Raman preparation. Despite the spectral shift of two vibrational quanta, inspection of the Wigner distribution of third-order packetcreated after one period of evolution in either first or second order coherencedoes not reveal any clear difference. The vibrational superposition is essentiallyunchanged. The simulated spectral asymmetry in time is in the opposite direction of what is observed in the experiment in Fig. 9c. We have verified experimentallythat the spectral envelopes and their time-asymmetry are sensitive to smalldelays in time overlap of the nominally coincident S+P pulses. Similarly,although not systematically studied, we have verified that spectral envelopes aresensitive to the chirp in the laser pulses. Either the chirp of the lasers must beincluded in the simulations, or chirp must be eliminated from the pulses to bemore quantitative in this comparison. At present, we do not fully understand the bi-modal spectral distribution observed at positive time (see spectrum at t = 2ps in Fig. 9). An intriguing possibility is the interference between preparation ofRaman packets via the B and B ” surfaces (see Fig. 12). Indeed, interference between these two channels has been identified previously in the analysis of theresonant Raman spectra of iodine in rare gas solids. 62 D) P(1,2)(t) contribution – Interference between vibronic packets: Laser chirp cannot explain the vibrational recurrences that appear at 0<t<1ps in the waveforms for λ = 532 nm – 541 nm in Fig. 9b. This set of recurrences appears most prominently in the 541 nm waveform, where as many as four resonances are observed with a period of 350 fs, over a background modulated at twicehigher frequency by the vibrational packet on the X state. The observed period istoo slow to be assigned to a packet on the X state, and it is distinctly faster than the 380 fs period of the packet prepared by the pump pulse near v = 22 of the B state. A packet near v= 19 of the B state would have the observed period. However, energetically, such a packet could not be prepared with the pulsesused. Moreover, if prepared, it would not be expected to decay as rapidly asobserved. Since the simulations succeed in capturing the negative time image,and since they contain all timing diagrams of the third-order polarization, wemay conclude that these anomalous vibronic resonances involve an additionalelectronic state in the molecular Hamiltonian. A consistent interpretation of this scattering channel is obtained by considering the interferometric signal between two vibrational packets in the Bstate, separately prepared by the pump and Stokes pulses near v = 22 and v = 14,and cross-correlated via a transition to a short-lived excited electronic state. Thesuggested time-circuit diagram for this process is given in Figure 12. It assumesoptical coupling between the B state and a higher lying electronic surface thatcan be reached with the P ’-photon. Energetically, a manifold of I+I and I + I* repulsive potentials are accessible. We assume in the figure that the transition is17B(0u)↔II(0g), which we have previously analyzed.63 To be clear, consider the transcription of the time-circuit diagram of Fig. 12 to the explicit perturbation expression for a given spectral component of the AS polarization: /G1/G1P(1,2)(ωAs)=dt −∞∞ ∫ϕB(1)(t;−ωs)ˆ µ CBeiωAStϕC(2)(t;ωp,ωp’) =dt −∞∞ ∫dt3 −∞t ∫ ×ϕB(1)(t2;−ωs)eiHB(t−t2)/ /c61ˆ µ BCeiωASte−iHC(t−t3)/ /c61ˆ µ CBEp’(t3)e−iωp’t3e−iHB(t3−t2)/ /c61ϕB(1)(t2;ωp) (24) in which the time origin, t2, is taken as that of the coincident arrival of P+S pulses. For a dissociative upper state, since the resonant scattering is in effect instantaneous, we may suppress evolution past t3. Since the duration of the laser is short in comparison to the period of motion, the snapshot limit is appropriate.Then in terms of the dipole dressed wave packets, φ, we have: P(1,2)(t)=φB(1)(t;−ωs)E(ωp’)δ[ωp−∆VCB]φB(1)(t;ωp)+c.c. =φB(1)(t;−ωs)W(q)φB(1)(t;ωp)+c.c. =dpdqdp’dq’∫φB(1)(t;−ωs)p’,q’p’,q’W(q)p,qp,qφB(1)(t;ωp)+c.c. (25) where t is the time delay between the arrival of S+P and P ’ pulses. Given an assumed form of the upper state potential, Eq. 24 is easily integratednumerically. Analytical evaluation of Eq. 25 is possible by taking a stationary Gaussian for the window function, W(q), and Gaussian packets for the P- and S- prepared superpositions. If we were to assume a window delta in space, then(25) reduces to the cross correlation between the S- and P-prepared packets. Thiscorrelation in energy representation, yields the strictly temporal evolution of thesignal: P (1,2)(t)= v’∑φB(1)(t;−ωs)v’v’v’ˆ P v" v"∑v"v"φB(1)(t;ωp)+c.c. =µCB2c(v’,v")av’ v’∑v’e−iωv’t+α      bv’ v’∑v’e+iωv’t+β      +c.c. (26) where18/G1/G1av’=µBX2 v∑e−βEvv’v2E(ωp)δ[/G61ωp−∆Evv’] and bv’=µBX2 v∑e−βEvv’v2E(ωS)δ[/G61ωS−∆Evv’](27) and the projector from v ” to v ’ in (26) is the stimulated Raman process spelled out in (24). Under the assumption that this process does not color the CARSspectrum we may set the coefficients c(v ’,v”) to a constant. Using the experimental spectral profiles of the P- and S-pulses, the vibrational amplitudesin each of the prepared superpositions are defined according to (27), and theexpected time-dependent CARS signal obtained according to (26). The simulated P (1,2)(t) contribution to the signal is shown in Fig. 13. Note, the scattering at t = 0 has contributions from several diagrams, and therefore will be modulated as afunction of AS wavelength. This is illustrated in Fig. 13 by showing thewaveforms obtained at two wavelengths. The spectral dispersion of thewaveform is given in (24). The observable signal can also be understood as thecross-correlation between the two packets. Thus, for an S-prepared packetcentered at v = 14 with a mean vibrational spacing of ω S = G(v=14) – G(v=13) = 102.6 cm-1 (period = 325 fs), and a P-prepared packet centered at v = 22, with ω p= G(v = 22) – G(v=21) = 87.6 cm-1 (period = 380 fs); the signal is obtained by squaring (26): I(1,2)(ωAs,t)=Icos(ω S+ω p 2t+ϑ)cos(ω S−ω p 2t+ϑ)e−t/τ(28) The signal contains three characteristic time constants: a) A fast modulation at the center frequency, τ1 ∼ 2/(ω S + ω p) = 350 fs, corresponding to the superposition moving out of the window as a whole; b) A slower modulation at the difference frequency, τ2 = 2/(ω S - ω p) = 2 ps, corresponding to decay of the signal as the S- and P-prepared packets split; c) An even slower decay envelope given by the dispersion of frequencies within each distribution, namely, the spreading of the individual packets dictated by the local anharmonicities on the B state, τ3 = 1/ωexe = 15 ps. The slower decay time, τ3, is now convoluted with the rotational dephasing which occurs on a similar time scale; and intensities in signal recurrences will now depend on the rotational phase distribution. Note, despite the decay of themodulation, this channel will contribute to the scattering process at all positivetimes, responsible in part for the broad time-integrated spectra at positive timesin Fig. 9c.19Additional electronic resonances can be inferred from the known dense manifold of electronic states of iodine, and from experiments in which we varythe time ordering in the sequence of non-overlapping pulses. As example, notethat the B ” state (Fig. 3) which is directly accessible from the ground, must contribute to CARS at positive delays. Under coincident P+S pulses, packetscreated by the P pulse on both the B and B ” surfaces may be transferred to the X state with the S pulse. Due to the difference in the curvature of potentials, the Band B ” channels of scattering would create a bimodal vibrational distribution in the X state, which would then be reflected in the AS spectrum. However, sincethe B ” state is dissociative, it cannot contribute to CARS at negative delays. A delay of <20 fs after the P pulse would be sufficient to ensure that the B ” packet escapes further interrogation. In short, there are several distinct electronic scattering channels that are observed to contribute to the CARS signal at positive delay, while a singlechannel dominates the signal at negative delay. E) The ro-vibrational contribution to first- and second-order images: The ro-vibrational contribution to the time-integrated CARS spectrum can be written down with the help of the diagram in Fig. 4. Since the detection involves integration over t4, the signal is determined by the time intervals t32 and t21. For a given path in vibrational state space, v’,v”,v’’’, three rotational circuits contribute to a given ro-vibronic transition in the CARS spectrum. Thus, for a P( j) transition we have: IP(j)(3)(v’,v",v’’’)={(2j+1)eβEv,j[c1e−i[ωv’,j+1t21+ωv",jt32−ωv,jt31] +c2e−i[ωv’,j−1t21+ωv",jt32−ωv,jt31] +c3e−i[ωv’,j−1t21+ωv",j−2t32−ωv,jt31]+c.c.]}2 (29) in which the weighting coefficients are the transition matrix elements encountered in (18). Ignoring the slowly varying coefficients, namely rotationalmatrix elements and F-C factors, but retaining the energy conservation condition derived from the spectral composition of the pulses c(v’,v”,v’’’) = δ[EL-Ev,j]3, it is possible to reproduce the experimental waveforms of Figure 8 and 9. At positive delay ( t32 = 0), the AS spectral line intensity will be modulated according to: IP(j)(3)(v’,v",v’’’,t21)=(2j+1){eβEv,jc(v’,v",v’’’) v’,v"∑ e−i(ωv’,j+1−ωv,j)t21[ +2e−i(ωv’,j−1−ωv,j)t21+c.c.]}2 (30a) while at negative time ( t21 = 0):20IP(j)(3)(v’,v",v’’’,t32)=(2j+1){eβEv,jc(v’,v",v’’’) v’,v"∑ 2[e−i(ωv",j−ωv,j)t32+e−i(ωv",j−2−ωv,j)t32+c.c.]}2 (30b) These impulse response functions, when convoluted with the laser cross- correlation function yield the finite pulse signals. Carrying out the square in (30)yields the rotational recursions at beat frequencies: /G1/G1[(Ev’,j+1−Ev,j)−(Ev’,j−1−Ev,j)]//G61=4B (v’)j (31a) /G1/G1[(Ev",j−Ev,j)−(Ev",j−2−Ev,j)]//G61=4B (v")j (31b) where B(v ’) and B(v ”) are the vibration dependent rotational constants in the excited and ground electronic states, respectively; and the over-bar impliesaveraging over the v ’,v” states that are accessible under the broadband laser pulses. In Fig 14 we provide a comparison between simulation (30) and an experimental waveform sliced from the two-dimensional image in Fig. 9. The simulation uses (30a) and (30b) separately, and joins them at t = 0 . This results in the mismatch of amplitude for the peak at t= 0. At t= 0, since t 21 = t32, the two diagrams shown as insets are degenerate, and therefore interfere (destructively).On this time-scale since the evolution is entirely vibrational, the effect isreproduced in the explicit wavepacket simulations of Fig. 11. To match the depthof modulation in experiment and simulation, we have used a convolution width of 70 fs in t 12, and a width of 140 fs in t23. This effective reduction of modulation depth for interrogating the Raman packet is expected, and discussed in somedepth previously. 16 It is a result of the separation between positive and negative momentum components of the Raman packet as it enters the resonance window. In Fig. 15 we provide the same comparison as in Fig. 14, but now for the coarse grain, long-time scan of Fig. 10a. Although, single rotational lines areresolvable, the experimental spectra are instrument limited by the spectrometerbandpass of 8 cm -1. Thus in the waveform of Fig. 14 which corresponds to the spectral slice at 538 nm, the main contributing transitions are: R54(29-0), P50(29-0), R89(30-0), P85(30-0), R111(31-0), P108(31-0) where in parenthesis we give thevibrational transition. Note, in contrast with the third-order coherence obtainedwith gated detection, in this case the lines that overlap in the observationwindow do not interfere - they are simply convoluted with the spectralbandpass. The use of short pulses forces the fan-out in vibrational state space. Inthe present v ’ = 23-28 and v ” = 2-5 are the main contributors to the signal. The vibration dependence of rotational constants (due to coriolis, centrifugal andhigher order distortions) generates dispersion in rotational recurrences evenwhen a single line is being monitored. Accordingly, the rotational recursions ofFig. 10 can be understood, as resulting from the overlap of several different ro-vibrational lines in the spectral bandpass of detection. The prominent recursion is that of the thermal maximum in the rotational population, j~50 , for which21theB (v’)≈B(v’=25)=0.02414 cm-1,55 which leads to a period of 6.9 ps (31a) as seen in Fig. 15 (we have not included higher order corrections to the rotational constant in the simulations). For the Raman packet of Fig. 10b, the dominantrecursion occurs at t ~ 4.5 ps, consistent with B (v")≈B(v"=3)=0.036966 cm-1.55 Due to approximations made in weighting coefficients and spectroscopic constants, the match between experiment and simulation in Figures 14 and 15 isnot exact. Nevertheless, the comparison is sufficiently detailed to confirm ouranalysis of the manipulated molecular coherences. IV. DISCUSSION The detailed exposition of the experimental time-frequency resolved CARSmeasurements and their analysis in vapor iodine serves mainly for the purposesof understanding the four-wave mixing process with the molecule acting asmixer. Despite the fact that we are interrogating a diatomic, the participation of alarge number of rotational states in the molecular coherence makes the analysisvaluable. The time-frequency images are understood and reproduced in terms ofphase evolution in multi-dimensional state space; formally, in the Hilbert space,/G43, consisting of the tensor product of electronic, vibrational, and rotational spaces: /G43 =/G3/G43 /c72/c79 ⊗/G3/G43 /c89/c76/c69 ⊗/G3/G43 /c85/c82/c87. The time-circuit diagrams and the phase coherence condition (8) provide the necessary bookkeeping to describe the various order coherences, as demonstrated by reconstructing the experimental two-dimensional images. The images contain information. The interferometric natureof the images, most clearly illustrated for the third-order coherence (figures 6-8),implies that the encoded information retains the wave nature inherent inquantum amplitudes. Such data can be used to reconstruct the molecularHamiltonian, by extracting accurate spectroscopic constants. Also, the significantcontrol exercised over the evolving molecular coherence through the three inputlaser fields can be taken as a paradigm for molecular control. The latter aspect,for the important case of single-color four-wave mixing has been recentlyexplored. 61,64 A useful application of quantum control is to be expected in quantum computation, or quantum information transfer, which we explore here. The observable coherent polarization is the outcome of operations on a quantum register consisting of the superposition of product states elvibrot. The nontrivial superposition of product states provides entanglement,65 an aspect unique to quantum information and key to scalable parallelism in quantumcomputing. 9,10 In the present case, the Born-Oppenheimer separated molecular Hamiltonian offers the ro-vibronic Hilbert space of 2 ×m×n (el⊗vib⊗rot) dimensions with the possibility of three natural entanglements. While thedimensionality is quite large, m~10 and n~10 2, the structure of this space does not conform to the theoretically optimal structure of 2N dimensional space, namely, the space of N qubits. Nevertheless, given efficient parallel logicaloperations on a register of 10 3 elements and a processor speed of 10-15s - 10-12s,22significant applications can be expected. Tasks that rely on few entanglements include Grover ’s search algorithm,20 and quantum cryptography.66 The mundane building blocks for such applications – reset, logical operation, and readout – can be readily inferred from the data presented. In what follows we first consider themeasurement and readout process more closely, then present the minimal butsufficient set of efficiently executable universal logic gates for all-purposequantum computing. It should be clear from the onset that the stored andretrieved information is in the complex amplitudes of eigenstates that define theevolving coherences. Process control is provided by the coherence of the laserfields, which can be thought to consist of a time sequence of discrete spectralcomponents. A) Measurement and Readout There is a fundamental difference between time-gated detection and time- integrated detection of the anti-Stokes polarization. The first allows readout byprojection on superposition states, while the latter projects out a singlecomponent of the evolving polarization for observation. This is directlyillustrated for the electronic-rotational entanglement as discerned from therotational recurrences in various order coherences. When using time-gated detection, the prominent rotational recurrence frequency in third-order coherence occurs at \| 2(B’-B”)j\|, at the beat between consecutive P-branch or R-branch transitions of the AS polarization (B ’ and B ” are the rotational constant of the molecule in the B and X electronic state, respectively). Noting that the classical rotation frequency of a diatomic is 2Bj, the observed recurrence is recognized as the difference between classical periods ofrotation on the ground and excited electronic surfaces. It would appear that over the t 43 interval the molecule rotates forward on one electronic surface and backward on the other. The observable period of rotational recurrence reports aproperty of an electronically maximally entangled state: ϕ=a∑Xvj+b∑Bv’’’j’’’, with a∑()2=b∑()2=0.5.67 In contrast, when using time-integrated detection, the rotational revivals occur (31) with a frequency of 4B’j in first-order coherence, during t21, and at 4B”j in second-order coherence, during t32, i.e., at twice the classical frequency of rotation in the excited and ground state, respectively. Despite the fact that infirst-order the evolution is in an electronic coherence while in second-order it is avibrational coherence that evolves as an electronic population on the groundelectronic surface, in both cases the property of a single electronic state ismeasured. It would appear that the molecule is rotating in the excited electronic state during t 21, just as it must rotate in the ground electronic state during t32. These results can be understood using the time-circuit diagram of Fig. 4b. With t4 integrated out, and either t12 = 0 or t23 = 0, only two circuits contribute to a given transition. In the observable beat between two such circuits, the common path23over <v,j\| is cancelled (31), yielding the beat between states separated by 2j’ or 2j”. While this explains the signal, more fundamental is the recognition that although over t21 the system evolves in an electronic superposition, Fourier filtering over t4 projects out the excited electronic state, as recognized by observing a rotational frequency that reflects the rotational constant of the Bstate. TGFCARS detection projects out all transitions that fall under the time- frequency bandwidth of detection. A dramatic manifestation of this is the period of silence at 2 < t 4 < 10 ps in the third-order image of Fig. 6. This is the time interval in which complete destructive interference occurs among the sampledtransitions. In terms of the Bloch sphere, by virtue of the equal electronic superposition (B+X)/ 2 the system evolves in the π/2 plane, and the silence occurs as the rotational phases span a flat distribution over all azimuthal angles. This occurs despite the fact that a thermal, statistical density serves as initial state, and therefore the radiators in different j states are not initially correlated. The sudden preparation of the initial state, and the fact that the CARS signal isdue to bulk polarization, establish a well defined phase among the independentmolecules. An even clearer demonstration of inter-molecular coherence in CARSis the observation of beating between vibrations of different molecules in liquidsolutions. 68 The above considerations of rotation-electronic entanglement apply equally well to vibronic coherences. For example, in time-integrated detection, the excited vibrational period is observed during t21 despite the fact that the system is in an electronic superposition. In time-gated detection, a vibrationalbeat between excited and ground state packets occurs, which is effectivelysmoothed out in the experiments due to the width of the Kerr-gate. A corollary to the observation that all transitions that fall under the time- frequency gate of detection interfere, is that any selected pair (or sets) oftransitions can be made to interfere as long as they can be brought under a giventime-frequency gate. Such “hard-wiring ” can be accomplished experimentally, e.g., by dispersing the AS polarization then recombining selected spectral ranges on a single detector element after a suitable delay line. Alternatively, instead ofusing an impulse to drive the time gate, it is possible to use a temporally modulated gate field to detect pre-selected superpositions, i.e., heterodyne detection of CARS using a specifically tailored local oscillator. Such hardwiringcreates selective quantum correlations, in effect, producing entanglementthrough measurement. B) Universal Logic Gates The universal logic gates sufficient to demonstrate a quantum computer consist of the one-qubit operations and the two-qubit controlled-not ( CNOT ) gate. 69 The full set of one-qubit operations is given by the Pauli spin matrices.70 Here, they can be readily demonstrated on the rotational coherences, and the24vibration-electronic entanglement is a readily available two-qubit CNOT gate. We first offer the more suggestive representation of the single qubit operations interms of classical logic gates. i) One qubit Operations Consider the time integrated signal, and in particular, of the spectrally resolved j-1→j transition illustrated in Fig. 4a. Let us assign the logical value “1” to the presence of this line, and “0” to its absence in the TIFRCARS spectrum. The upper level of this transition \|B,v ’’’,j-1> is connected to the two coherent inputs eiΩ(v",j)X,v",j and eiΩ(v",j−2)X,v",j−2, with phases Ω(v",j;t2) and Ω(v",j)=[ω(v",j)−ω(v,j)]t32+Ω(v",j;t2) determined relative to the freely evolving bra-state (see Fig. 1). As in the case of the simulations in Figures 14 and 15, here too, we neglect differences in amplitude due to transition matrix elements. Then for the experimental data where t21 = 0 , the relative phase between the two input lines is determined by the time interval t32, namely the time delay between S- and P ’-pulses. Assigning the logical values “1” for Ω = 2nπ and “0” to Ω = (2n-1) π for integer n, we may construct a truth table for the visibility of this particular spectral component: 1100 ⊕1010 (33) 1001 This bivalent representation of interference between two input channels can be recognized as the inverted exclusive-OR gate, XOR , (XOR , had we assigned “1” to the absence of the line from the spectrum). The XOR gate allows modulo 2 addition, which is described by the ⊕ operation. The same consideration applied to all nodes in Figure 4b produces the equivalent circuit of Figure 16. The figuredescribes the bivalent logic of the coherence transferred around the time-circuitdiagram of a single initial ro-vibrational eigenstate. If we were to consider the coherent superposition prepared by the P-pulse at t = t 1, as the input qubit: in≡a’B,v’,j+1;t1+b’B,v’,j−1;t1 (34a) and identify the output qubit as the coherent superposition prepared at t = t3, after action of the S- and P ’-pulses: out≡a’’’B,v’’’,j+1;t3+b’’’B,v’’’,j−1;t3 (34b) then using the diagram of Fig. 16 it is easily shown that if fully connected, the circuit flips the logical states between input and output ( a’→ b’’’ and b’→a’’’). Indeed, this is the bivalent prescription for the simulation of the TIFRCARS25signal for the second order coherence shown in Fig. 15 and given by (30) for the continuous representation. The circuit of Fig. 4b is more flexible than what is inferred by the bivalent logic. The relative phase in the superpositions evolve continuously . Moreover, in TGFCARS detection the relative phase in the output qubit is accessible information. To keep track of this information, we re-write Eq. 29 in the two-dimensional space of the qubit. out=ˆ µ EP’(t3)U(t32)ˆ µ ES(t2)U(t21)in =˜ U (t32)˜ U (t21)a' b'     =˜ U (t31)a' b'     (35) In the snapshot limit, since the ro-vibrational phases are frozen during the vertical transitions, the entire evolution reduces to one of phases over pathsprescribed by the timing of the spectrally broad, temporally sharp pulses: /G1/G1˜ U (t 21)=eiΩ(v’,j+1) eiΩ(v’,j−1)      =eiΩ1 ei2B’(2j+1)t21/ /c61      (36) with overall phase Ω=(Te+Ev’−Ev+Ej+1B,v’−EjX,v)t12//G61; ˜ U (t32)=eiχeiχ2+eiχ0eiχ0 eiχ0eiχ−2+eiχ0      =eiχeiχ01+σx[]eiχ21 ei(χ−2−χ2)      (37) with overall vibronic phase /G1/G1χ=(Te+Ev"−Ev)t23//G61; and /G1/G1χi=(Ej+iX,v"−EjX,v)t23//G61. All rotations under SU(2) are accessible under the timing conditions already realized in the presented data. Thus, for t32=0 and /G1/G1t21=(2n−1)π/G61/[2B’(2j+1)] ˜ U (t31)=˜ U (t21)=eiΩ1 −1      =2eiΩσz (38) This simply states that the free evolution of the qubit in the t21 interval corresponds to rotation around the z-axis on the Bloch sphere. This rotation occurs with 4B’j periodicity (38a), namely, the recursion period of the first order coherence (31a). If we do not take advantage of the overall vibronic phase, then to execute /G86z rotation on the manifold of j states, it would be necessary to use a chirped pulse, with a coherence similar to the third order polarization of Fig. 6. The fast vibronic phase, Ω, allows the execution of the transformation on all j states in parallel, by using a short pulse.26For t21=0, χ0=2nπ, χ2−χ−2=2nπ; therefore /G1/G1t32=nπ/G61/[2B’(2j−1)] and χ+χ2=(2n−1)π: ˜ U (t31)=˜ U (t32)=01 10      =2σx (39) Since /G1/G1χ0=(EjX,v"−EjX,v)t23//G61=(BX,v"−BX,v)t23//G61, and rotational constants of different vibrations within the same electronic state are similar, χ0~0 is easily fulfilled. As before, it is necessary to take advantage of the vibronic overall phase, χ, for parallel execution of /G86x rotation on the manifold of rotational qubits. The execution of a /G86y rotation requires finite t21 and t32. The required conditions are readily obtained. In principle this is superfluous, since arbitraryrotation around two independent axes is sufficient to span the full space of thetwo-dimensional qubit. ii) Two-qubit controlled gate A useful two-qubit CNOT gate would be one that operates on entangled states, using qubits in different spaces as target and control.71 Until now, the electronic and vibrational qubits have been used as simple state tags, with theirstates determined by the order of interaction with the laser field. Thisdeterminism in the electronic state is the result of choosing P. S. P ’ colors that yield only one diagram in the four-wave mixing process. To control electronicqubits arbitrarily, it is necessary to create a multiplicity of interfering paths. Thisis realized in single-color four-wave mixing, or for the condition where all threeinput fields overlap spectrally, since under such conditions CARS and CSRSpaths can interfere. As an illustrative example, consider the vibrational phase ofan input vibronic qubit to control the electronic superposition in the outputqubit. At the expense of greatly reducing the vibrational vector space, let us return to the wavepacket picture \| ϕv> and assign it a logical value \|1> when the packet is in the Frank-Condon window of the X ↔B transition, and \|0> otherwise. To further conform to quantum logic implementations, let us code the electronic qubit as \|0> ≡ \|X> and \|1> ≡ \|B>. Then, after preparation of the initial coherence with the first pulse, the action of the pulse at t= t2 can be defined as: ϕvϕelES(t2) →    ϕv(ϕv⊕ϕel) (40a) Which reads: if the vibrational packet is in the F-C window, then change the electronic state. This can be verified to be the two-qubit CNOT gate with the logical values2710 →  11 11 →  10 00 →  00 01 →  01(40b) Depending on t2, the operation in (40) controls the electronic superposition that will evolve in the t32 interval. Thus, taking ωX and ωB as the frequencies of vibrational packets in the X and B states, respectively; starting with the coherence ϕv’BXϕv prepared by the pump pulse; the possible outcomes of the action of the pulse at t2 are: ϕv’BXϕvE(t2) →    .ϕv’BBϕv if t21=2nπ/wX E(t2) →    .ϕv’XXϕv if t21=2nπ/wB E(t2) →    .aϕv’XXϕv+bϕv’BBϕv if t21=2nπ/wB=2mπ/wX (41) When t21 = 2nπ/ωX, the bra-packet on the X electronic surface is in the F-C window when the second pulse arrives, therefore an electronic population is created in the B state: 0BX1 → ϕv’BBϕv. When t21 = 2nπ/ωB, the ket- packet on the B electronic state is in the F-C window when the second pulse arrives, therefore an electronic population is created on the X surface: 1BX0 → ϕv’XXϕv. Finally, when t21 = 2nπ/ωB = 2mπ/ωX, when the packets on the B and X surfaces are simultaneously in the F-C window, an electronic superposition is created: 1BX1 →aϕv’XXϕv+bϕv’BBϕv. Thus, at a given delay, the electronic superposition to be prepared is controlled by the phase of the vibrational packet. In effect, in single-color four-wave mixing, the contributions from various diagrams can be controlled, P93)=aP(0,3)+bP(1,2), to drive the system to the desired state. Indeed, these timing conditions have beenexperimentally verified in degenerate four-wave mixing studies on I 2.37,37,61,64 Similar controlled gates are possible between rotation-vibration, and rotation-electronic qubits.72 Here, we are satisfied by indicating that the requisite two-qubit CNOT gate and the one-qubit operations are naturally wired in molecular four-wave mixing, sufficient to identify the potential of the system forquantum computing. A schematic of the general conceptual approach to usingthe present network for a single path in vibrational vector space is provided inFig. 17. In the example, a broad-band P-pulse resets the quantum register, astructured S-pulse writes, the P ’-pulse is used to process, and the output register is read either in TIFRCARS mode, or in TGFCARS with heterodyne detectionusing the gate pulse (G-pulse) as local oscillator.28V. CONCLUSIONS TFRCARS yields multi-dimensional images of molecular coherences, which we have presented here as a set of three two-dimensional plots. The images of thevarious order molecular ro-vibrational coherences contain sufficient detail toallow an accurate characterization of the molecular Hamiltonian. Although it hasbeen previously demonstrated that time resolved measurements can providespectroscopic constants of high accuracy, 73 and the present method of extracting two-dimensional images has the advantage of multiplexing, such an applicationto stable small molecules is of limited value. The unique advantage of themethod as a spectroscopic tool is in its ability to characterize transient spectra, ortransient species. This derives from the fact that the method permits transformlimited observation and interrogation of the conjugate time and frequencycoordinates. Consider the application of the method to image chemistry. Thesimplest example would be that of unimolecular dissociation. If the process isslower than the speed of the Kerr shutter in use, as in rotational predissociation, 74 then it should be possible to directly record the ro-vibrational tracks of thedissociating complex in third order-coherence using TGFCARS. An image,similar to that in Fig. 6 would now provide a map of the ro-vibrational channelsto bond-breaking. More generally, consider evolution along a reactive coordinateQ(t) that has been set in motion via a short optical pulse. Then the second-ordercoherence would image the evolution of frequencies orthogonal to Q, thestiffening or loosening of bonds due to chemistry (or any other change). If thepulses used are shorter than the time scale of evolution along Q, direct imagesare obtained. Otherwise, a deconvolution similar to that used in the frequencyresolved optical gating (FROG) to characterize laser pulses, 11 would be applied. Indeed, the time gated photon echo measurements in the liquid phaseaccomplish these very aims, with evolution along the solvation coordinate beingthe target of interest. 39-42 The same principles applied to protein unfolding and excitonic dynamics have been given recently.75,76 TFRCARS, or more generally four-wave mixing experiments, can be used as a method for molecular coherence control. This concept is usually associatedwith the possibilities of controlled chemistry. In practice, such applications arelikely to be somewhat limited. Time-frequency-resolved four-wave mixing using a molecule as mixer, allows the preparation, manipulation, and readout of massive quantumsuperpositions with sufficient control to consider applications to quantumcomputing. Parallels can be drawn between this approach and NMR, 77 which is the maturer field of coherence control.78 For example, the P(0,3) polarization can be regarded as stimulated photon reverberations, in analogy with the NMR photonecho. 79 The optical four-wave mixing approach has important advantages with regard to the manipulation and transfer of massive coherences, which is animportant step toward practice in quantum computation. Since the third-orderpolarization is the observable, the optical process does not require polarization of29the thermal initial ensemble, which is required in NMR and a major source of the limitation to few qubits.80 The optical four-wave mixing method circumvents this, since it allows the sequential manipulation of the initial coherence struck by thefirst pulse on a thermal background that does not interfere with the signal.Moreover, since the manipulations involve optical pulses, single operations canbe accomplished on fs-ps time scales. Given the Doppler control of decoherencein rarified media at room temperature, 10 3 – 104 operations can be completed prior to loss of signal. Clearly a major challenge in the proposed approach is incascading, to enable multiple sequential operations. Note, at least in oracle typeapplications single qubit logic gates can be implemented with efficiency and parallelism, and the required quantum CNOT gate can be readily implemented in one step. Recognizing that the observable output in the four-wave mixingexperiment is a coherent radiation field, it is not too difficult to imaginesequential processing, or networking, by having the AS beam from one stage actas one of the input fields for a second stage. The considerations we havepresented, we believe, are sufficient to encourage a search for useful algorithmsthat naturally map on molecular networks. VI. ACKNOWLEDGMENTS The support of this research through grants from the US AFOSR (F49620-98-1- 0163) and the NSF (9725462), is gratefully acknowledged. Discussions with Z.Bihary and J. Eloranta were most valuable in formulating some of the quantum-computational concepts presented here. Discussions with A. Ouderkirk onphysical designs of four-wave mixing cascades to construct logic networks, isfondly acknowledged.30REFERENCES 1. N. Bloembergen, Pure Appl. Chem. 10 (1987) 1229. 2. J. J. Valentini, in Spectrometric Techniques , vol 4., (Academic Press, New York, 1985). 3. D. L. 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Both time-circuit and Feynman diagrams are illustrated for a non-overlapping sequence of P, S, P ’ pulses, with central frequency of the S-pulse chosen to be outside the absorption spectrum of the B ←X transition, to ensure that only the P(0,3) component of the third-order polarization is interrogated. In this dominant contribution, all three pulses act on bra (ket) state while the ket (bra) state evolves field free. Note, forthe Feynman diagrams, we use the convention of Ref. 5, which is different thanthat of Ref. 4. Fig. 2: Diagrammatic representation of electronically all-resonant CARS with coincident Pump and Stokes pulses, P+S, followed by P ’. The first diagram is the same as that of Fig. 1. To be resonant in all fields, the ϕ(2)ˆ µ ϕ(1)≡P(2,1) contribution can only be initiated from vibrationally excited states of the ground electronic surface. The time-circuit diagrams make it clear that for this process to have significant cross section, ωp-ωS ~kBT must hold. Fig. 3: The wavepacket picture associated with the evolution of the ket-state in the diagram of Fig. 1, for resonant CARS in iodine. The required energy matching condition for the AS radiation, Eq. 10b of text, can only be met when the packet reaches the inner turning point of the B-surface. Once prepared, ϕ(3)(t) will oscillate, radiating periodically every time it reaches the inner turning point. Fig. 4 : The “wiring ” diagram of rotational eigenstates for a given path – v, v ’, v”, v’’’ – in vibrational state space. a) Conventional diagram: The rotational selection rule, ∆j=±1 , limits the possible paths. The dark solid lines connect the paths that lead to the P-branch ( j-1→j) transition in the CARS spectrum. These are the only35relevant paths when the measurement involves TIFRCARS of the j-1→j transition. The gray lines connect the path for the R-branch transition, j+1→j. Interference occurs when two transitions terminate on one eigenstate. Note also that the coherence transferred to <v’’’,j±3\| is lost from the detectable AS polarization. b) Schematic diagram, useful for the interferometric analysis of the four- wave mixing process. The diagram highlights the phases gained byevolution along different paths in ro-vibronic state space. Each innerloop is equivalent to a Mach-Zehnder interferometer, with nodescorresponding to a time at which the applied radiation field maystimulate interference between two paths. Fig. 5: Experimental arrangement for TGFCARS. The sample consists of iodine vapor contained in a quartz cell heated to 50 °C. The forward BOXCARS geometry is used, with the three input beams brought to focus using a 25 cmachromatic doublet, and a pinhole to spatially filter the anti-Stokes radiation. The AS beam is t-gated using a Kerr cell with CS 2 as active medium, then dispersed through a 1/4-m monochromator, and detected using a CCD array. An experimental spectral slice, at t = 2 ps, is shown in the inset. Fig. 6: Image of the third-order ro-vibronic coherence obtained by time gated frequency resolved CARS (TGFCARS): Experiment (right panel), simulation (left panel). The vibrational assignment is indicated at the population maxima, near j = 50, for each state. Note, the signal at t = 0 is strongly saturated. Fig. 7 : Simulation of the third-order coherence for v ’’’ = 34 showing the rotational revivals, or winding pattern of rotational recursions: a) P-branch transitions; b) P- and R-branch transitions. Note the interference betweenbranches, and within the same branch.36Fig. 8 : Simulation of the role of interference in the third-order polarization, for v’’’ = 26-45. a) The signal evaluated according to Eq. 17 of text, under the assumption of white spectra for the three pulses, and after setting all matrixelements to unity. b) The third order polarization with interferences suppressedby squaring contributions from each eigenstate before summing, as required byEq. 17. Fig. 9 : Time-integrated frequency-resolved CARS image of first- and second- order coherences and cross sections: a) Two-dimensional time-wavelength image; b) Waveforms (time-slices) at selected wavelengths; c) Spectra at selected times. Negative time corresponds to scanning t 21, with δ[t3-t2], during which the first-order electronic coherence is monitored. Positive time corresponds to scanning t32, with δ[t2-t1], during which the second-order vibrational coherence is monitored. Fig. 10 : Coarse grain images of time-integrated frequency resolved CARS: a) First-order coherence, where t = t21 ; b) Second-order coherence, where t = t32. Fig. 11 : Wavepacket simulations of first- and second-order coherence images of Fig. 9a, within the two-electronic state Hamiltonian. Fig. 12 : Time circuit diagram for the resonant P(2,1) contribution and the likely potential energy surfaces that are involved. Fig. 13: The P(1,2) contribution simulated, using Eq. 26 of text, based on the experimental spectral profile of the laser pulses, Eq. 27.37Fig. 14 : Spectral slice of the TIFRCARS signal in Fig. 9b at 538 nm. The peaks identified with stars are assigned to the P(1,2) contribution shown in Fig. 13. The simulation is according to Eq. 30 of text. Fig. 15 : Spectral slice from the coarse scan TIFRCARS of Fig. 10b. The simulation is according to Eq. 30. Fig. 16 : Two-level logic equivalent to the wiring diagram in Figure 4. Fig. 17 : The rotational network controlled with structured pulses for reset (=P), write (=S), process (=P ’), and read (G).S P P' AS \|v,j> <v,j\|t1t2t3t4 \|v',j'> \|v",j">\|v"',j"'> S PP'AS t1t2t3t4 \|v',j'>\|v",j">\|v"',j"'> \|v,j> ϕX(0)(t)ˆ µ ϕB(3)(t)<v,j\|a) b') b) c) a) b) c)δ[t2 – t1] δ[t3 – t2]Energy (cm -1) R (Å)B(0u) X(0g)B" P S P'ϕ(1)(t)ϕ(3)(t) ϕ(2)(t) 050001 1041.5 1042 1042.5 104 22.5 33.5 44.5 5\|v,j><v',j+1\| <v',j-1\| <v",j-2\|<v",j\|<v",j+2\|<v"',j-1\|<v"',j+1\| <v"',j-3\|<v"',j+3\| t 1t 2t 3t 4 e−iEjt/hj j−1eiEj−1t/h j+1eiEj+1t/h j−2eiEj−2t/h jeiEjt/h j+2eiEj+2t/h j−1eiEj−1t/h j+1eiEj+1t/hWavelengthTime (ps) 343332 31 3536a) b) Wavelength (nm)Time (ps)Wavelength (nm)Time (ps)20 060 40 Wavelength (nm)Time (ps)Wavelength (nm)Time (ps)t12 t32Time (ps) 024 -2-4-6-8Wavelength (nm) 520 530 540Time (ps) Wavelength (nm)Time (ps) Wavelength (nm)Time (ps) Wavelength (nm)01 1042 1043 1044 104 22.5 33.5 44.5 5Energy (cm -1) R (Å)B(0u) X(0g)B"II(0g) PSP'0 0.5 1 1.5 2 time (ps)λ AS=536 nm λ AS=538.8 nm P(1,2)Experiment Simulation-8 -6 -4 -2 0 2 4-8 -6 -4 -2 0 2 4 -10000100020003000AFJ Time (ps)SimulationExperimentCARS Intensityt 32t 21 * *0 510 15 20 25 30 t21 (ps)Experiment SimulationCARS Intensityt 21<v’,j+1\| <v’,j-1\|<v”,j\| <v”,j-21\|<v”,j+2\| <v’’’,j-1\|<v’’’,j+1\|PS P ’ G \|0>\|?>
arXiv:physics/0103001v1 [physics.class-ph] 1 Mar 2001About forces, acting on radiating charge Babken V. Khachatryan1 Department of Physics, Yerevan State University, 1 Alex Man oogian St, 375049 Yerevan, Ar- menia Abstract. It is shown, that the force acting on a radiating charge is sti pulated by two rea- sons - owing to exchange of a momentum between radiating char ge and electromagnetic ﬁeld of radiation, and also between a charge and ﬁeld accompanying t he charge. It is well known that the charged particle moving with accele ration radiates, and as a result an additional force (apart from the external one, /vectorF0) - force of radiation reaction acts on it. In present paper it is shown, that this force (we shall call it as a self-action force or simply by self-action) is a sum of two parts: the ﬁrst force is due to t he exchange of the momentum between a particle and radiation ﬁelds, i.e. the ﬁelds, whic h go away to inﬁnity. For the second force in the exchange of a momentum the ﬁelds, accompanying a charge participate as well. These ﬁelds do not go away to inﬁnity, i.e. at inﬁnity they hav e zero ﬂux of energy (details see below). We shall start with the momentum conservation law for a syste m of charge and electromag- netic ﬁeld [1], [2] d dt/parenleftbigg /vectorP+1 4πc/integraldisplay V/bracketleftBig/vectorE/vectorH/bracketrightBig dV/parenrightbigg =1 4π/contintegraldisplay S/braceleftBigg /vectorE/parenleftBig /vector n/vectorE/parenrightBig +/vectorH/parenleftBig /vector n/vectorH/parenrightBig −E2+H2 2/vector n/bracerightBigg dS, (1) where /vectorP- is the particle momentum, /vectorEand/vectorH- are the vectors for electromagnetic ﬁeld, /vector n- is the normal to the surface S, enclosing volume V. On the right of formula (1) the external force /vectorF0is omitted. From (1) we can see, that apart from external forc e, two forces act on the particle: force /vectorf1, expressed by a surface integral, and force /vectorf2expressed by a volume integral. As a surface Swe shall take sphere of a large radius R→ ∞, with the centre at the point of instantaneous place of the charge, then /vector n=/vectorR/R. For /vectorEand/vectorHwe shall use the known expressions for the ﬁelds created by a charged particle movi ng with arbitrary velocity /vector v(t) [2], [3] /vectorH= [/vector nE],/vectorE(/vector r,t) =e/parenleftBig /vector n−/vectorβ/parenrightBig γ2R2x3+e cRx3/bracketleftbigg /vector n/bracketleftbigg /vector n−/vectorβ,˙/vectorβ/bracketrightbigg/bracketrightbigg , (2) where c/vectorβ=/vector v,γ=/parenleftbig1−β2/parenrightbig−1 2,x= 1−/vector n/vectorβ,˙/vectorβ≡d/vectorβ/dt. Note, that all quantities in the right hand side of equation (2) are taken at the moment t′=t−R(t′)/c. Calculating the force /vectorf1we have to substitute in (1) the term with a lowest order of R−1 (the second term on the right in (2)), corresponding, to sphe rical electromagnetic ﬁelds going away to inﬁnity, i.e. radiation ﬁelds. Then, taking into acc ount the remark after formula (2), it is possible to write the force /vectorf1in the form /vectorf1=−/contintegraldisplay SE2 4π/vector ndS=−/contintegraldisplay /vector ndIn c, (3) where dIn- is the energy, radiated per unit of time in the element of the solid angle dΩ in an arbitrary direction /vector n[3] 1E-mail: saharyan@www.physdep.r.am 1dIn=e2 4πcx3  ˙β2+2 x/parenleftbigg /vector n˙/vectorβ/parenrightbigg/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg −/parenleftbigg /vector n˙/vectorβ/parenrightbigg2 γ2x2  dΩ. (4) The formula (3) allows the following clear interpretation o f the origin of the force /vectorf1: the radiation in a direction /vector nper unit time carries away with itself momentum /vector ndIn/c, and therefore, the charge acquires a momentum −/vector ndIn/c. As the change of a momentum per unit time is equal to the acting force, then as a result of radiation in a directi on/vector nthe force will act on the particle, equal to d/vectorf1=−/vector ndIn/c. Integrating over all directions (over total solid angle), we get the expression for the force /vectorf1(details for calculation see in [4]): /vectorf1=−I c/vectorβ;I=2e2 3cγ4/parenleftBigg ˙β2+γ2/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg2/parenrightBigg . (5) HereI- is the instantaneous power of radiation, being a relativis tic invariant and having the form [3], [5] I=−2 3ce2duk dsduk ds. (6) In this formula uk=dxk/dsis the four-velocity and ds=cdt/gamma is the Minkowskian interval (we follow the notations of the book [3]). Now we turn to the force /vectorf2. Here it is necessary to take into account the contribution o f both summands in formula (2). The calculations are too long a nd, as it is easy to see, lead to integrals, divergent at both small and long distances. The l atters are related to the divergences of the self-energy and momentum for the point charge ﬁeld. To avoid these diﬃculties, we shall act as follows. Let’s write a three-dimensional equation of motion d/vector p/dt =/vectorf=/vectorf1+/vectorf2in the four-dimensional (covariant) form dpi dt=gi=gi 1+gi 2, (7) by entering the four-dimensional momentum pi=mcui= (γmc, /vector p ) and force gi=/parenleftBigγ c/vectorf/vectorβ,γ/c /vectorf/parenrightBig . In formula (7) it is necessary to deﬁne gi 2. Taking into account (5) and 6, it is easy to see, that gi 1has the form gi 1=2e2 3cduk dsduk dsui. (8) As it follows from the deﬁnition of the force /vectorf2and formula (2), where the vectors /vectorβand ˙/vectorβenter only, four-dimensional vector gi 2can be expressed through the vectors ui,dui/dsand d2ui/ds2only. The ﬁrst possibility disappears as for /vector v=const, should be gi 2= 0. The summand containing dui/dsis united with a left-hand side of equation (7) and leads to th e renormalization of the charged particle mass, so that it remains the possibil itygi 2=αd2ui/ds2, where α= 2e2/3c is a number (four-dimensional scalar), which is determined from the requirement, that for an arbitrary four-dimensional force gishould be giui= 0 (to see this it is necessary to use identity uiui= 1 and its consequences as well). Hence gi 2=2e2 3cd2ui ds2. (9) From (9) the expression for three-dimensional force /vectorf2follows which we give for the reference purposes /vectorf2=2e2 3c2γ2/braceleftBigg·· /vectorβ+γ2˙β2/vectorβ+ 3γ2/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg˙/vectorβ+γ2/parenleftbigg /vectorβ¨/vectorβ/parenrightbigg /vectorβ+ 4γ4/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg2/vectorβ/bracerightBigg . 2The formulas (7), (8) and (9) lead to well-known expression ( see, for example, [3]) for the four-dimensional self-action force gi gi=2e2 3c2γ2/parenleftBigg d2ui ds2+duk dsduk dsui/parenrightBigg . Hence, for the three-dimensional self-action force /vectorfwe ﬁnd (compare to the corresponding formulas in [6], [7]) /vectorf=2e2 3c2/braceleftBig/vectorA+/bracketleftBig/vectorβ/bracketleftBig/vectorβ/vectorA/bracketrightBig/bracketrightBig/bracerightBig , (10) where /vectorA≡γ4/parenleftBigg·· /vectorβ+3γ2/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg˙/vectorβ/parenrightBigg . In the nonrelativistic case ( β≪1), at ﬁrst approximation over βfrom (10) we get the following expression for the self-action force (by the way w e shall indicate, that there was an error in the formula (6) in article [5]) /vectorf=2e2 3c2¨/vectorβ+2e2 c2/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg˙/vectorβ. (11) This force diﬀers from the conventional one /vectorf′=2e2 3c2·· /vectorβ, in which the essential defect is inherent: for uniformly accelerated motion/parenleftbigg¨/vectorβ= 0/parenrightbigg , the force of radiation reaction /vectorf′is zero, while the radiation is not equal to zero/parenleftbigg˙/vectorβ/negationslash= 0/parenrightbigg . The force (11) is deprived of this defect and always is nonzero, if the radiation is nonzero/parenleftbigg˙/vectorβ/negationslash= 0/parenrightbigg . If¨/vectorβ/negationslash= 0 and the ﬁrst summand in the right hand side of (11) dominates, then /vectorf=/vectorf′; depending on the law /vectorβ(t), the second summand can dominate. Generally, for β≪1, for self-action force it is necessary to use the formula (1 1). The above mentioned allows us to state that the total self-ac tion force acting on a radiating charge is determined by formula (10) and it is more appropria te to call a reaction force of radiation the force /vectorf1determined by formula (5). This force is always nonzero when the particle moves with acceleration and hence radiates. From this point of view let’s consider again uniformly accel erated motion (for arbitrary velocities). It is known that the condition for uniformly ac celerated motion has the form [7] d2ui ds2+duk dsduk dsui= 0, (12) (thence gi= 0) or in three-dimensional notations ¨/vectorβ+ 3γ2/parenleftbigg /vectorβ˙/vectorβ/parenrightbigg˙/vectorβ= 0. (13) As a result for this motion the vector /vectorAgoes to zero and this is the case for the self-action force. However the radiation and radiation reaction force a re nonzero, because the acceleration is nonzero. The latter can be easily obtained from the equati ond/vector p/dt =/vectorF0+/vectorfand is determined by the formula mcγ˙/vectorβ=/vectorF0+/vectorf−/vectorβ/parenleftBig/vectorβ/vectorF0/parenrightBig −/vectorβ/parenleftBig/vectorβ/vectorf/parenrightBig . (14) In our case for /vectorβ\|\|/vectorF0,/vectorF0=const, the acceleration is equal to c˙/vectorβ=/vectorF0 mγ3. (15) 3Hence, for the uniformly accelerated motion the only force a cting on charge is the external force /vectorF0(it can be easily checked that for the acceleration (15) the s elf-action force is zero). For /vectorβ→1 the acceleration tends to zero, and in the case /vectorβ→0 the acceleration, as it is expected, is equal to/vectorF0 m. I am grateful to the participants of the seminar of Chair of Th eoretical Physics of Yerevan State University. References [1] V. G. Levich, ”Course of Theoretical Physics ”. Vol.1, Mo scow, 1962 (in Russian). [2] J. D. Jackson, ”Classical electrodynamics”. John Wiley and Sons, inc, New York-London, 1962. [3] L. D. Landau and E. M. Lifshitz, ”Classical Theory of Fiel ds”. Pergamon, New York, 1972. [4] B. V. Khachatryan. Journal of Contemporary Physics (Arm enian Academy of Sciences) 32 (1997) 39. [5] B. V. Khachatryan. Journal of Contemporary Physics (Arm enian Academy of Sciences) 33 (1998) 20. [6] A. Sommerfeld, ”The Elektrodinamik”. Leipzid, 1949. [7] V. L. Ginzburg, ”Theoretical Physics and Astrophysics ” . Moscow, 1975 (in Russian). 4
arXiv:physics/0103002v1 [physics.plasm-ph] 1 Mar 2001Thermodynamics of hot dense H-plasmas: Path integral Monte Carlo simulations and analytical approximations V.S. Filinov1∗, M. Bonitz2, W. Ebeling3, and V.E. Fortov1 1Russian Academy of Sciences, High Energy Density Research C enter, Izhorskaya street 13-19, Moscow 127412, Russia 2Fachbereich Physik, Universit¨ at Rostock Universit¨ atsplatz 3, D-18051 Rostock, Germany 3Institut f¨ ur Physik, Humboldt-Universit¨ at Berlin Invalidenstrasse 110 D-10115 Berlin (February 2, 2008) Abstract This work is devoted to the thermodynamics of high-temperat ure dense hy- drogen plasmas in the pressure region between 10−1and 102Mbar. In par- ticular we present for this region results of extensive calc ulations based on a recently developed path integral Monte Carlo scheme (dire ct PIMC). This method allows for a correct treatment of the thermodynamic p roperties of hot dense Coulomb systems. Calculations were performed in a broad region of the nonideality parameter Γ /lessorsimilar3 and degeneracy parameter neΛ3/lessorsimilar10. We give a comparison with a few available results from other p ath integral calculations (restricted PIMC) and with analytical calcul ations based on Pad´ e approximations for strongly ionized plasmas. Good agreeme nt between the results obtained from the three independent methods is foun d. Typeset using REVT EX ∗Mercator guest professor at Rostock University 1I. INTRODUCTION The thermodynamics of strongly correlated Fermi systems at high pressures are of grow- ing importance in many ﬁelds, including shock and laser plas mas, astrophysics, solids and nuclear matter, see Refs. [1–4] for an overview. In particul ar the thermodynamical properties of hot dense plasmas under high pressure are of importace for the description of plasmas relevant for laser fusion [5]. Further among the phenomena o f current interest are Fermi liquids, metallic hydrogen [6], plasma phase transition, e .g. [7] and references therein, bound states etc., which occur in situations where both Coulomb andquantum eﬀects are relevant. There has been signiﬁcant progress in recent years to study t hese systems analytically and numerically, see e.g. [7–10,13–15]. Due to the enormeous di ﬃculties to develop analytical descriptions for hydrogen plasmas with strong coupling, e. g. [1–3], there is still an urgent need to test the analytical theory by an independent numeric al approach. An approach which is particularly well suited to describe th ermodynamic properties in the region of high pressure, characterized by strong coup ling and strong degeneracy, is the path integral quantum Monte Carlo (PIMC) method. Ther e has been remarkable recent progress in applying these techniques to Fermi syste ms, for an overview see e.g. Refs. [1,2,16–18]. However, these simulations are essenti ally hampered by the fermion sign problem. To overcome this diﬃculty, several strategies hav e been developed to simulate macroscopic Coulomb systems [8,19,20]: the ﬁrst is the rest ricted PIMC concept where additional assumptions on the density operator ˆ ρare introduced which reduce the sum over permutations to even (positive) contributions only. T his requires knowledge of the nodes of the density matrix which is available only in a few sp ecial cases, e.g. [19,20]. However, for interacting macroscopic systems, these nodes are known only approximately, e.g. [21], and the accuracy of the results is diﬃcult to asses s from within this scheme. An alternative are direct fermionic PIMC simulations which ha ve occasionally been attempted by various groups [22] but which were not suﬃciently precise and eﬃcient for practical purposes. Recently, three of us have proposed a new path inte gral representation for the N- particle density operator [23,24] which allows for direct fermionic path integral Monte Carlo simulations of dense plasmas in a broad range of densities an d temperatures. Using this concept we computed the pressure and energy of a degenerate s trongly coupled hydrogen plasma [24,26] and the pair distribution functions in the re gion of partial ionization and dissociation [26,27]. This scheme is rather eﬃcient when th e number of time slices (beads) in the path integral is less or equal 50 and was found to work we ll for temperatures kBT/greaterorsimilar 0.1Ry. In this paper we derive further improved formulas for the pr essure and energy and give, for the ﬁrst time, a detailed derivation of all main res ults and rigorously justify the use of the eﬀective quantum pair potential (Kelbg potential ) in direct PIMC simulations. Further, in the present work this method will be applied to hi gh-pressure plasmas ( p≃ 10−1−102Mbar) in such temperature regions were considerable deviat ions from the ideal behavior are observed. One diﬃculty of PIMC simulations is that reliable error esti mates are often not available, in particular for strongly coupled degenerate systems. Mor eover, in this region no reliable data from other theories such as density functional theory o r quantum statistics, e.g. [3,15], are available which would allow for an unambiguous test. Fur thermore, results from classi- cal molecular dynamics simulations exist, but they apply on ly to fully ionized and weakly 2degenerate plasmas, e.g. [28–30], which is outside the rang e of interest for this work. Also, new quantum molecular dynamics approaches are being develo ped, e.g. [10–12], but they are only beginning to produce accurate results. Therefore, it is of high interest to perform quantitative co mparisons of independent sim- ulations, such as restricted and direct fermionic PIMC, and to develop improved analytical approximations, which is the aim of this paper. We compare re cent results of Militzer et al. [32] for pressure and energy isochors ( n∼2.5·1023cm−3) of dense hydrogen to our own direct PIMC results. This is a non-trivial comparison since the two approaches employ indepen- dent sets of approximations. Nevertheless, we ﬁnd very good agreement for temperatures ranging from 106Kto as low as 50 ,000K. This is remarkable since there the coupling and degeneracy parameters reach rather large values, Γ ≈3 andneΛ3≈10, and the plasma contains a substantial fraction of bound states. Further, we use the new data to make a comparison with analyti cal estimates which are based on Pad´ e approximations for strongly ionized plas mas. These formulae were con- structed on the basis of the known analytical results for the limiting cases of low density [3,33] and high density [3]. These Pad´ e approximations are exact up to quadratic terms in the density and interpolate between the virial expansions a nd the high-density asymptotics [34–36]. We ﬁnd that the results for the internal energy and f or the pressure agree well with the PIMC results in the region of the density temperatur e plane, where Γ /lessorsimilar1.6 and nΛ3/lessorsimilar5. II. PATH INTEGRAL REPRESENTATION OF THERMODYNAMIC QUANTITIES We now our direct PIMC scheme. All thermodynamic properties of a two-component plasma are deﬁned by the partition function Zwhich, for the case of Neelectrons and Np protons, is given by Z(Ne,Np,V,β) =Q(Ne,Np,β) Ne!Np!, withQ(Ne,Np,β) =/summationdisplay σ/integraldisplay Vdqdrρ (q,r,σ;β), (1) whereβ= 1/kBT. The exact density matrix is, for a quantum system, in genera l, not known but can be constructed using a path integral representation [37], /integraldisplay VdR(0)/summationdisplay σρ(R(0),σ;β) =/integraldisplay VdR(0)...dR(n)ρ(1)·ρ(2)...ρ(n) ×/summationdisplay σ/summationdisplay P(±1)κPS(σ,ˆPσ′)ˆPρ(n+1), (2) whereρ(i)≡ρ/parenleftbig R(i−1),R(i); ∆β/parenrightbig ≡ /an}bracketle{tR(i−1)\|e−∆βˆH\|R(i)/an}bracketri}ht, whereas ∆ β≡β/(n+ 1) and ∆ λ2 a= 2π/planckover2pi12∆β/m a,a=p,e.ˆHis the Hamilton operator, ˆH=ˆK+ˆUc, containing kinetic and potential energy contributions, ˆKandˆUc, respectively, with ˆUc=ˆUp c+ˆUe c+ˆUep cbeing the 3sum of the Coulomb potentials between protons (p), electron s (e) and electrons and protons (ep)]. Further, R(i)= (q(i),r(i))≡(R(i) p,R(i) e), fori= 1,...n + 1,R(0)≡(q,r)≡(R(0) p,R(0) e), andR(n+1)≡R(0)andσ′=σ. This means, the particles are represented by fermionic loops with the coordinates (beads) [ R]≡[R(0);R(1);...;R(n);R(n+1)], whereqandrdenote the electron and proton coordinates, respectively. The spi n gives rise to the spin part of the density matrix S, whereas exchange eﬀects are accounted for by the permutati on operator ˆP, which acts on the electron coordinates and spin projection s, and the sum over the permutations with parity κP. In the fermionic case (minus sign), the sum contains Ne!/2 positive and negative terms leading to the notorious sign p roblem. Due to the large mass diﬀerence of electrons and ions, the exchange of the lat ter is not included. The matrix elementsρ(i)can be rewritten identically as /an}bracketle{tR(i−1)\|e−∆βˆH\|R(i)/an}bracketri}ht=/integraldisplay d˜p(i)d¯p(i)/an}bracketle{tR(i−1)\|e−∆βˆUc\|˜p(i)/an}bracketri}ht/an}bracketle{t˜p(i)\|e−∆βˆK\|¯p(i)/an}bracketri}ht/an}bracketle{t¯p(i)\|e−∆β2 2[ˆK,ˆUc]...\|R(i)/an}bracketri}ht. (3) To compute thermodynamic functions, the logarithm of the pa rtition function has to be diﬀerentiated with respect to thermodynamic variables. In particular, for the equation of statepand internal energy Efollows, βp=∂lnQ/∂V = [α/3V∂lnQ/∂α ]α=1, (4) βE=−β∂lnQ/∂β, (5) whereαis a length scaling parameter α=L/L 0. This means, in the path integral repre- sentation (2), each high-temperature density matrix has to be diﬀerentiated in turn. For example, the result for the energy will have the form βE=−1 Q/integraldisplay VdR(0)...dR(n) ×n+1/summationdisplay k=1ρ(1)...ρ(k−1)·/bracketleftbigg β∂ρ(k) ∂β/bracketrightbigg ·ρ(k+1)... ρ(n)/summationdisplay σ/summationdisplay P(±1)κPS(σ,ˆPσ′)ˆPρ(n+1),(6) and, analogously for other thermodynamic functions. There are two diﬀerent approaches to evaluate this expressi on. One is to ﬁrst choose an approximation for the high-temperature density matrice sρ(i)and then to perform the diﬀerentiation. The other way is to ﬁrst diﬀerentiate the op erator expression for ρ(k)and use an approximation for the matrix elements only in the ﬁnal result. As we checked, the second method is more accurate and will be used in the followi ng. To evaluate the derivatives in Eq. (6), it is convenient to in droduce dimensionless in- tegration variables η(k)= (η(k) p,η(k) e), whereη(k) a=κa(R(k) a−R(k−1) a) fork= 1,...,n and a=p,e, andκ2 a≡makBT/(2π/planckover2pi12) = 1/λ2 a, [24]. This has the advantage that now the dif- ferentiation of the density matrix aﬀects only the interact ion terms. Indeed, one can show that β∂ρ(k) ∂β=−β∂∆β·Uc(X(k−1)) ∂βρ(k)+β˜ρ(k) β, (7) 4whereX(0)≡(κpR(0) p,κeR(0) e),X(k)≡(X(k) p,X(k) e), withX(k) a=κaR(0) a+/summationtextk l=1η(l) a, andk runs from 1 to n. Further,X(n+1)≡(κpR(n+1) p,κeR(n+1) e) =X(0), and we denoted ˜ρ(k) β=/integraldisplay dp(k)/an}bracketle{tX(k−1)\|e−∆βˆUc\|p(k)/an}bracketri}hte−/angbracketleftp(k)\|p(k)/angbracketright 4π(n+1)/an}bracketle{tp(k)\|∂ ∂βe−(∆β)2 2[ˆK,ˆUc]...\|X(k)/an}bracketri}ht, (8) wherep(k) a= ˜p(k) a/(κa/planckover2pi1),p(k)≡(p(k) p,p(k) e) and use has been made of Eq. (3). For k=n+1, we have β∂ ∂β/summationdisplay σ/summationdisplay P(±1)κPS(σ,ˆPσ′)ˆPρ(n+1)=/summationdisplay σ/summationdisplay P(±1)κPS(σ,ˆPσ′)× ×/braceleftbigg −β∂∆β·Uc(X(n)) ∂βˆPρ(n+1)+ˆP/bracketleftBig β˜ρ(n+1) β/bracketrightBig/bracerightbigg . (9) Further,Uc(X(k−1))≡U(1) c(X(k−1)) +U(2) c(X(k−1)), withU(1) candU(2) cdenoting the interac- tion between identical and diﬀerent particle species, resp ectively,U(1) c(X) =Ue c(X)+Up c(X) andU(2) c(X) =Uep c(X). Using these results and Eq. (6), we obtain for the energy βE=3 2(Ne+Np)−1 Q1 λ3Np pλ3Nee/integraldisplay VdR(0)dη(1)...dη(n)/summationdisplay σ/summationdisplay P(±1)κPS(σ,ˆPσ′) ×/braceleftbiggn+1/summationdisplay k=1ρ(1)...ρ(k−1)/bracketleftBigg −β∂∆β·U(1) c(X(k−1)) ∂β−β∂∆β·U(2) c(X(k−1)) ∂β+β˜ρ(n+1) β/bracketrightBigg ×ρ(k)... ρ(n)ˆPρ(n+1)/bracerightbigg/vextendsingle/vextendsingle/vextendsingle X(n+1)=X(0), σ′=σ. (10) This way, the derivative of the density matrix has been calcu lated, and we turn to the next point - to ﬁnd approximations for the high-temperature dens ity matrix. III. HIGH-TEMPERATURE ASYMPTOTICS OF THE DENSITY MATRIX IN THE PATH INTEGRAL APPROACH. KELBG POTENTIAL In this section we derive an approximation for the high-temp ature density matrix which is suitable for direct PIMC simulations. Further, we demons trate that the proper choice of the eﬀective quantum pair potential is given by the Kelbg p otential. Following Refs. [16,38,39], we derive a modiﬁed representation for the dens ity matrix. The mains steps are: 1. The N-particle density matrix is expanded in terms of 2-pa rticle, 3-particle etc. con- tributions from which only the ﬁrst, ρab, is retained [16,38,39]; 2. In the high-temperature limit, ρabfactorizes into a kinetic ( ρ0) and an interaction term (ρab U),ρab≈ρ0ρab U, because it can be shown that [40,41] e−(∆β)2 2[ˆK,ˆUc]=ˆI+O/parenleftbigg1 (n+ 1)2/parenrightbigg , (11) 5where ˆIis the unity operator. In this way we get the following repres entation for the two-particle density matrix ρab=/parenleftbigg(mamb)3/2 (2π/planckover2pi1β)3/parenrightbigg exp[−ma 2/planckover2pi12β(ra−r′ a)2] exp[−mb 2/planckover2pi12β(rb−r′ b)2] exp[−βΦab] (12) where Φ ab(ra,r′ a,rb,r′ b) is the oﬀ-diagonal two-particle eﬀective potential. 3. In the following, the oﬀ-diagonal matrix elements of the e ﬀective binary potentials will be approximated by the diagonal ones by taking the Kelbg potential at the center coordinate, Φab(r,r′; ∆β)≈Φab(r+r′ 2; ∆β).; 4. For the plasma parameter region of interest, the protons c an be treated classically, and Φiimay be approximated by the Coulomb potential. We will now comment on these steps in some more detail. We calc ulated the eﬀective potential by solving a Bloch equation by ﬁrst order perturba tion theory. The method has been described in detail in [41]. This procedure deﬁnes an eﬀ ective oﬀ-diagonal quantum pair potential for Coulomb systems, which depends on the int er-particle distances rab,r′ ab. As a result of ﬁrst-order perturbation theory we get explici tely Φab(rab,r′ ab,∆β)≡eaeb/integraldisplay1 0dα dab(α)erf/parenleftBigg dab(α) 2λab/radicalbig α(1−α)/parenrightBigg , (13) wheredab(α) =\|αrab+ (1−α)r′ ab\|, erf(x) is the error function erf( x) =2√π/integraltextx 0dte−t2, and λ2 ab=/planckover2pi12∆β 2µabwithµ−1 ab=m−1 a+m−1 b. It is interesting to note, that a simple approximation of the complicated integral over αby the length of the interval multiplied with the integrand i n the center (Mittelwertsatz) leads us to the so-called KTR-p otential due to Klakow, Toepﬀer and Reinhard which (in the diagonal approximation) is often used in quasi-classical MD simulations [10,13] Φab(rab,r′ ab,∆β)≡eaeb dab(1/2)erf/parenleftbiggdab(1/2) λab/parenrightbigg , (14) In our direct PIMC calculations we used the full expression f or the interaction potential, keeping the α-integration but, in order to save computer time, we approxi mated the two- particle interaction potential by its diagonal elements. T he diagonal element ( r′ ab=rab) of Φabis just the familiar Kelbg potential, given by (we will use th e same notation for the potential) Φab(\|rab\|,∆β)≡Φab(rab,rab,∆β) =eaeb λabxab/bracketleftBig 1−e−x2 ab+/radicalbig {π}xab(1−erf(xab))/bracketrightBig ,(15) wherexab=\|rab\|/λab, and we underline that the Kelbg potential is ﬁnite at zero di stance. The error of the above approximations, for each of the high-t emperature factors on the r.h.s. of Eq. (2), is of the order 1 /(n+ 1)2. With these approximations, we obtain the result ρ(k)=ρ(k) 0ρ(k) U+O[(1/n+1)2], whereρ(k) 0 is the kinetic density matrix, while ρ(k) U=e−∆βU(X(k−1))δ(X(k−1)−X(k)), whereUdenotes 6the following sum of Coulomb and Kelbg potentials, U(X(k)) =Up c(X(k) p) +Ue(X(k) e) + Uep(X(k) p,X(k) e). Notice that special care has to be taken in performing the d erivatives with respect to βof the Coulomb potentials which appear in Eq. (10). Indeed, p roducts ρ(1)... ρ(n)ˆPρ(n+1)β∂∆β·Uc(X(k−1)) ∂βhave a singularity at zero interparticle distance which is integrable but leads to diﬃculties in the simulations. Due t o the Kelbg potential, for the e-e and p-p interaction, this singularity is weakenend, but it i s enhanced for the e-p interaction. In order to assure eﬃcient simulations we, therefore, furth er transform the e-p contribution in the following way: /integraldisplay1 0dα/integraldisplay dR(k−1)/an}bracketle{tR(k−2)\|e−∆βαˆK\|R(k−1)/an}bracketri}ht/bracketleftbigg −β∂ ∂β/parenleftbig ∆βU(2) c(R(k−1))/parenrightbig/bracketrightbigg ×/an}bracketle{tR(k−1)\|e−∆β(1−α)ˆK\|R(k)/an}bracketri}ht ≈ /an}bracketle{tR(k−1)\|e−∆βˆK\|R(k)/an}bracketri}ht/bracketleftbigg −β∂ ∂β/parenleftbig ∆βU(2)(R(k−1))/parenrightbig/bracketrightbigg +O/bracketleftbig (1/n+ 1)2/bracketrightbig . (16) This means, within the standard error of our approximation O[(∆β)2], we have replaced the e-p Coulomb potential U(2) cby the corresponding Kelbg potential U(2), which is much better suited for MC simulations. Thus, using λp≪λe, we ﬁnally obtain for the energy: βE=3 2(Ne+Np) +1 Q1 λ3Np p∆λ3NeeNe/summationdisplay s=0/integraldisplay dqdrdξρ s(q,[r],β)× /braceleftbiggNp/summationdisplay p<tβe2 \|qpt\|+n/summationdisplay l=0/bracketleftbiggNe/summationdisplay p<t∆βe2 \|rl pt\|+Np/summationdisplay p=1Ne/summationdisplay t=1Ψep l/bracketrightbigg +n/summationdisplay l=1/bracketleftbigg −Ne/summationdisplay p<tCl pt∆βe2 \|rl pt\|2+Np/summationdisplay p=1Ne/summationdisplay t=1Dl pt∂∆βΦep ∂\|xl pt\|/bracketrightbigg −1 det\|ψn,1 ab\|s∂det\|ψn,1 ab\|s ∂β/bracerightbigg , withCl pt=/an}bracketle{trl pt\|yl pt/an}bracketri}ht 2\|rl pt\|, Dl pt=/an}bracketle{txl pt\|yl p/an}bracketri}ht 2\|xl pt\|, (17) and Ψep l≡∆β∂[β′Φep(\|xl pt\|,β′)]/∂β′\|β′=∆βcontains the electron-proton Kelbg potential Φep. Here, /an}bracketle{t...\|.../an}bracketri}htdenotes the scalar product, and qpt,rptandxptare diﬀerences of two coor- dinate vectors: qpt≡qp−qt,rpt≡rp−rt,xpt≡rp−qt,rl pt=rpt+yl pt,xl pt≡xpt+yl pand yl pt≡yl p−yl t, withyn a= ∆λe/summationtextn k=1ξ(k) a. Here we introduced dimensionless distances between neighboring vertices on the loop, ξ(1),...ξ(n), thus, explicitly, [ r]≡[r;y(1) e;y(2) e;...].Further, the density matrix ρsin Eq. (17) is given by ρs(q,[r],β) =Cs Nee−βU(q,[r],β)n/productdisplay l=1Ne/productdisplay p=1φl ppdet\|ψn,1 ab\|s, (18) 7whereU(q,[r],β) =Up c(q)+{Ue([r],∆β)+Uep(q,[r],∆β)}/(n+1) andφl pp≡exp[−π\|ξ(l) p\|2]. We underline that the density matrix (18) does not contain an explicit sum over the per- mutations and thus no sum of terms with alternating sign. Ins tead, the whole exchange problem is contained in a single exchange matrix given by \|\|ψn,1 ab\|\|s≡ \|\|e−π ∆λ2e\|(ra−rb)+yn a\|2 \|\|s. (19) As a result of the spin summation, the matrix carries a subscr iptsdenoting the number of electrons having the same spin projection. For more details , we refer to Refs. [23,24]. In similar way, we obtain the result for the equation of state , βpV Ne+Np= 1 +1 Ne+Np(3Q)−1 λ3Np p∆λ3NeeNe/summationdisplay s=0/integraldisplay dqdrdξρ s(q,[r],β)× /braceleftbiggNp/summationdisplay p<tβe2 \|qpt\|+Ne/summationdisplay p<t∆βe2 \|rpt\|−Np/summationdisplay p=1Ne/summationdisplay t=1\|xpt\|∂∆βΦep ∂\|xpt\| +n/summationdisplay l=1/bracketleftBiggNe/summationdisplay p<tAl pt∆βe2 \|rl pt\|2−Np/summationdisplay p=1Ne/summationdisplay t=1Bl pt∂∆βΦep ∂\|xl pt\|/bracketrightBigg +α det\|ψn,1 ab\|s∂det\|ψn,1 ab\|s ∂α/bracerightbigg , withAl pt=/an}bracketle{trl pt\|rpt/an}bracketri}ht \|rl pt\|, Bl pt=/an}bracketle{txl pt\|xpt/an}bracketri}ht \|xl pt\|. (20) The structure of Eqs. (17, 20) is obvious: we have separated t he classical ideal gas part (ﬁrst term). The ideal quantum part in excess of the classica l one and the correlation contributions are contained in the integral term, where the second line results from the ionic correlations (ﬁrst term) and the e-e and e-i interaction at t he ﬁrst vertex (second and third terms respectively). The third and fourth lines are due to th e further electronic vertices and the explicit temperature dependence [in Eq. (17) and volume dependence in Eq. (20)] of the exchange matrix, respectively. The main advantage of Eqs. ( 17, 20) is that the explicit sum over permutations has been converted into the spin determin ant which can be computed very eﬃciently using standard linear algebra methods. Furt hermore, each of the sums in curly brackets in Eqs. (17, 20) is bounded as the number of ver tices increases, n→ ∞, and is thus well suited for eﬃcient Monte Carlo simulations. Notic e also that Eqs. (17, 20) contain the important limit of an ideal quantum plasma in a natural wa y [42]. IV. COMPARISON OF DIRECT AND RESTRICTED PIMC SIMULATIONS Expressions (17, 20) are well suited for numerical evaluati on using Monte Carlo tech- niques, e.g. [16,17]. In our Monte Carlo scheme we used three types of steps, where either electron or proton coordinates, riorqior inidividual electronic beads ξ(k) iwere moved un- til convergence of the calculated values was reached. Our pr ocedure has been extensively tested. In particular, we found from comparison with the kno wn analytical expressions for 8pressure and energy of an ideal Fermi gas that the Fermi stati stics is very well reproduced [26]. Further, we performed extensive tests for few–electr on systems in a harmonic trap where, again, the analytically known limiting behavior (e. g. energies) is well reproduced [43,44]. For the present simulations of dense hydrogen, we v aried both the particle number and the number of time slices (beads). As a result of these tes ts, we found that to obtain convergent results for the thermodynamic properties of den se hydrogen, particle numbers Ne=Np= 50 and beads numbers in the range of n= 6...20 are adequate [24,26]. 9FIGURES 103104105106107108 logT [K]102010211022102310241025102610271028log n [cm-3]T=50,000K rs=1.86=1.6 =0.2n3=5n3=0.2 FIG. 1. Density-temperature plane showing the parameter re gion for which calculations are performed. The data of Fig. 2 are along the dashed line (isoch orrs= 1.86). The data of Figs. 3 and 4 are inside the bold rhomb, along lines of constant Γ betw een the lines nΛ3= 2 and nΛ3= 5, respectively. Data for the vertical line (isotherm T= 50,000K) are given in Fig. 5. We will now compare our results with some available results o btained by the Monte Carlo technique developed by the Urbana group [19,32]. We ma y ﬁrst state that both Monte Carlo techniques diﬀer in several fundamental points , so that they are essentially independent approaches. Let us brieﬂy outline the main diﬀe rences between the technique developed in Urbana, known as the restricted PIMC scheme [32] and references therein, and the approach described here. These authors performed simul ations with 32 electrons and protons; their restricted PIMC scheme required to use a rath er small time step assuring 1/∆β∼2∗106K. Also, the treatment of the interactions diﬀers from our sch eme: the authors of Ref. [32] perform a numerical solution of the Bloc h equation for the two-particle density matrix whereas we use an analytical approximation f or the eﬀective pair interaction (based on the Kelbg potential, see above). Finally, Ref. [32 ] approximately computes the nodal surface of the density matrix using a variational ansa tz which is then used to restrict the integrations to the region of positive density matrix. F or more details regarding the restricted PIMC simulations, see Refs. [19,32]. 102512510251022Pressure[Mbar]n3 2 5 102 2 5 103 Temperature T, 103K-202468Energy[2NRy]PadeRestrictedPIMCDirect PIMC FIG. 2. Comparison of direct and restricted PIMC results and analytical results (PADE) for the pressure and total energy of dense hydrogen as a funct ion of temperature for rs= 1.86, corresponding to n= 2.5·1023cm−3. For illustration, also the coupling and degeneracy parame ters Γ and neΛ3are shown in the upper ﬁgure. Let us now turn to a comparison of the numerical results. The r estricted PIMC simulation data for dense hydrogen are taken from Ref. [32]. A compariso n of results for the pressure and the internal energy for a ﬁxed value of the density ( rs= 1.86) is shown in Fig. 1 and TABLE I. At high temperatures, above 50,000 K, where only a sm all fraction of atoms is expected, the agreement is rather good. This is remarkable s ince the nonideality and the degeneracy reach values of 3 and 10, respectively. This resu lt demonstrates that, at least for rs≃1 and forT≥50,000Kboth methods yield results which are more or less equivalent . At T <50,000K, where partial ionization is expected, we still observe a re asonable agreement of both approaches, however, we see also that the diﬀerences start to grow. The reasons for that are manyfold. From our results we conclude that the main problem is not the bound state formation - atoms and molecules are well described by t he two PIMC simulations which use a physical picture which does not involve any artiﬁ cial distinction between free and bound electrons, e.g. [26]. On the other hand, with growi ng degeneracy nΛ3, both PIMC methods become less reliable, and a detailed analysis, although being very desirable, will have to be based on more extensive calculations in the fu ture. 11Further we present in TAB. I also Pad´ e results for the weakly nonideal region. We ﬁnd good agreement with the PIMC results for T > 105K. Details on the method of these analytical calculations will be discussed in the next secti on. V. COMPARISON WITH ANALYTICAL APPROXIMATIONS FOR THE THERMODYNAMIC FUNCTIONS OF STRONGLY IONIZED DENSE PLASMAS In this section we give a comparison of the available data poi nts from direct PIMC calculations with analytical estimates based on Pad´ e appr oximations for strongly ionized plasmas [3,34–36]. The comparison concentrates on H-plasm as in a region in the density- temperature plane with the following borders 0.2≤Γ≤1.6, 0.2≤neΛ3 e≤5, (21) which will be called “rhomb of moderate nonideality and mode rate degeneracy“ (see bold rhomb in Fig. 1). With respect to analytical treatment, this rhombic region is of particular diﬃculty since none of the known analytic limiting expressi ons is valid. Further we calculated several points for rs= 1.86 and Γ /lessorsimilar2 which correspond to the PIMC data discussed in the previous section and also an isotherm at T= 50,000Kincluding some data at higher density, outside the rhomb, cf. Fig. 1 for an overviev. We demonstrate below that the Pad´ e approximations which in terpolate between the limits where theoretical results are available are a useful tool for the description of the available data points, at least for the case of moderate noni deality Γ /lessorsimilar1.6 and moderate degeneracy neΛ3/lessorsimilar5. The Pad´ e approximations which we use here were construct ed in earlier work, [34–36], from the known analytical results fo r limiting cases of low density [3,33] and high density [3]. The structure of the Pad´ e appro ximations was devised in such a way that they are analytically exact up to quadratic terms i n the density (up to the second virial coeﬃcient) and interpolate between the viria l expansions and the high-density asymptotic expressions [34–36]. The formation of bound sta tes was taken into account by using a chemical picture. This means the plasma is considere d as a mixture of free electrons, free ions, atoms and molecules which are in chemical equilib rium, being described by mass action laws or minimization of the free energy [36]. We follow in large here this cited work, only the contributio n of the ion-ion interaction which is, in most cases, the largest one, was substantially i mproved following recent work of Kahlbaum, who succeeded in describing the available classi cal Monte Carlo data for the ions by accurate Pad´ e approximations [46]. By using Kahlbaum’s formulas we achieve a rather accurate description of the thermodynamics in the region wh ere the plasma behaves like a classical one-component ion plasma imbedded into a sea of ne arly ideal electrons. This is the region where the electrons are strongly degenerate neΛ3 e≫1 andrs≪1, (22) and the ions are still classical but nonideal Γ≫1 andniΛ3 i≪1. (23) 12This region lies in the upper left corner of Fig. 1. With respect to the chemical picture we restrict ourselves t o the region of strong ioniza- tion where the number of atoms is still relatively low and whe re the fraction of molecules is small as well, see below. We will discuss here only the gene ral structure of the Pad´ e formulae. For example, the internal energy density of the pl asma is given by u=uid+uint. (24) Hereuidis the internal energy of an ideal plasma consisting of Fermi electrons, classical protons and classical atoms, and uintis the interaction energy uint=uii+uee+uie+uvdW. (25) The interaction contribution to the internal energy consis ts of four terms: •Ion-ion interaction contribution: this term which, in gene ral, yields the largest con- tribution is generated by the OCP subsystem of the protons. F or the OCP energy of protons many expressions are available, e.g. [45]. We have u sed here the most precise formula due to Kahlbaum [46] which interpolates between the Debye region, uii∼Γ3/2, and the high density ﬂuid, uii∼Γ. •Electron-electron interaction: This term corresponds to t he OCP energy of the electron subsystem. We used the rather simple expressions used in ear lier work [34,35]. •Electron-proton interaction: This term corresponds to the interaction between the two OCP subsystems which is mostly due to polarization eﬀects. A gain, we used the rather simple expressions proposed in earlier work [34,35]. •Van der Waals contribution: In the region of densities and te mperatures deﬁned above this contribution gives only a small correction. Therefore , this term was approximated here in the simplest way by a second virial contribution. The neutral particles were treated as hard spheres. In the region of densities which are studied here, molecules do not play a role, therefore, the formation of molecules was taken into account only in a ve ry rough approximation according to Ref. [34]. The number density of the neutrals wa s calculated on the basis of a nonideal Saha equation. We restricted this comparison to a r egion where the number density of neutrals is relatively small, the degree of ionization be ing larger than 75%. The contributions to the pressure were calculated, in part, from scaling relations e.g. we usedpii=uii/3, and, for the other (smaller) contributions, by numerical diﬀerentiation of the free energy given earlier [34,35]. In a similar way, the c hemical potential which appears in the nonideal Saha equation was obtained. For the partitio n function in the Saha equation we used the Brillouin-Planck-Larkin expression [3,36]. Th e solution of the nonideal Saha equation which determines the degree of ionization (the den sity of the atoms) was solved by up to 100 iterations starting from the ideal Saha equation. 130.60.91.21.51.8Energy[2Nk BT] = 0.8= 0.6= 0.4= 0.2= 0 0 1 2 3 4 5 Degeneracyn3-1.8-1.2-0.60.00.6Energy[2Nk BT] RPIMCPIMC= 1.6= 1.2= 1.0 FIG. 3. Comparison of Pad´ e calculations (lines without sym bols) for the internal energy with the direct PIMC results (lines with full circles). Since all the expression described so far are given in analyt ic form, the calculation of about 1000 data points for energy and pressure takes less tha n a minute on a PC. The result of our calculations for density-temperature points in the “ rhomb of moderate nonideality and moderate degeneracy” are given in Figs. 2,3. Further, we give in TAB. I several data points obtained from the Pad´ e formulas. Since the Pad´ e for mulas used here do not apply to low temperatures, we included in TAB. I only Pad´ e data for T >105K. 140.70.80.91.01.11.2Pressure [nk BT] = 0.8= 0.6= 0.4= 0.2= 0 0 1 2 3 4 5 Degeneracyn30.50.60.70.80.91.0Pressure[nk BT] RPIMCPIMC= 1.6= 1.2= 1.0 FIG. 4. Comparison of Pad´ e calculations (lines without sym bols) of the pressure (in units of the Boltzmann pressure) with direct PIMC simulation result s (lines with full circles). Summarizing the results for the internal energy and for the p ressure we ﬁnd that the Pad´ e results, with a few exceptions, agree well with the PIMC data in the region of the density temperature plane, where Γ ≤1.6 andnΛ3≤5. The agreement is particularly good for the energies. [The larger deviations for the pressure may be due to the numerical diﬀerentiation.] In fact, the Pad´ e formulas used here in combination with the chemical picture works only in the case that the plasma is strongly ionized, i.e. the degr ee of ionization is larger than 75%. The description of the region where a higher percentage of atoms and, due to this, also molecules is present needs a more reﬁned chemical pictu re [7,47,48]. 1510-510-410-310-210-1110102103104105106Pressure[Mbar] Padeideal plasmaDPIMC 1018101910201021102210231024102510261027 Densityn,cm-3-2024Energy [2N Ry] FIG. 5. Comparison of Pad´ e calculations (lines without sym bols) of the pressure (in units of the Boltzmann pressure) with direct PIMC simulation results (l ines with full circles) for an isotherm T= 50,000K. Finally, we compare the Pad´ e and PIMC data along the isother mT= 50,000Kwhich is given in Fig. 5. This ﬁgure shows the transition from a clas sical ideal gas (low density) to a nearly ideal quantum gas (limit of high density). In the c entral part, n/lessorsimilar1019cm−3/lessorsimilar 1025cm−3, Coulomb interaction leads to strong deviations from the be havior of an ideal plasma. The strong increase of the energy at high density is d ue to the Mott eﬀect and to the increase of the ideal quantum contribution to the electr on energy. Comparing the Pad´ e and PIMC results, we ﬁnd good agreement up to electron densit iesn= 1022cm3. For higher densities, the deviations are growing. For intermediate de nsities,n/lessorsimilar1022cm−3/lessorsimilar1024cm−3 the PIMC data are more reliable. On the other hand, in the limi t of very high density, rs≪1, 16the Pad´ e results are known to correctly approach the ideal q uantum plasma limit whereas the PIMC data should be regarded as preliminary due to the extrem ely high electron degeneracy. Interestingly, we ﬁnd that at high density the Pad´ e data app roaches the ideal curves earlier than the PIMC data which is important for further improvemen t of the presented Monte Carlo approach. VI. DISCUSSION This work is devoted to the investigation of the thermodynam ic properties of hot dense partially ionized plasmas in the pressure range between 0.1 and 100 Mbar. Most of the new results are based on a Quantum Monte Carlo study of a correlat ed proton-electron system with degenerate electrons and classical protons. In this pa per, we gave a detailed derivation of improved estimators for the internal energy and the equat ion of state for use in direct fermionic path integral simulations. Also, we gave a rigoro us justiﬁcation for the use of an eﬀective quantum pair potential (Kelbg potential) in PIMC s imulations. Further, we compared our direct PIMC results with independe nt restricted PIMC data of Militzer and Ceperley for one isochor corresponding to rs= 1.86, Fig. 2. We found very good quantitative agreement between the two PIMC methods fo r temperatures in the range of 50,000K≤T≤106K, where Γ /lessorsimilar3 andneΛ3 e/lessorsimilar10. This region is particularly com- plicated as here pressure and temperature ionization occur and, therefore, an accurate and consistent treatment of scattering and bound states is cruc ial. This agreement is remarkable because the two simulation methods are completely independ ent and use essentially diﬀerent approximations. We, therefore, expect that the results for the thermodynamic properties of high pressure hydrogen plasmas in this temperature-dens ity range are reliable within the limits of the simulation accuracy. This is the main result of the present paper. In future work, it will be important to extend the range of agr eement. To analyze the deviations between the simulation methods, we also incl uded some data for rs= 1.86 and lower temperatures, 10 ,000K≤T≤50,000K, Fig. 2. At this point, no conclusive answer about the reasons of the deviations can be given. For t hese parameters, the electron degeneracy is growing rapidly and, therefore, each of the si mulation methods is becoming less reliable. So these data should be regarded as prelimina ry results which will be useful for future improvements of the simulations. Furthermore, the Monte Carlo results allowed us to develop a nd test analytical approxi- mations of Pad´ e-type which are improvements of earlier app roximations [3,34–36] in a region in the density-temperature plane bounded by Γ ≤1.6 andneΛ3 e≤5. This is a region of moderate nonideality and degeneracy and high degree of ioni zation . We have shown that for these parameters, the Pad´ e approximations which interpol ate between the limits where the- oretical results are available agree well with the Monte Car lo data, cf. Figs. 2-4 and Table I. Thus, these approximations provide a useful tool for the des cription of these plasmas which include hydrogen at a pressur between 0.1 and 100 Mbar. At low er temperature, deviations from the Monte Carlo data are growing, cf. Fig. 2. This is most ly due to the growing role of bound states. Whether the Pad´ e approximations, in c ombination with an improved chemical picture (mass action law), continue to work at lowe r temperatures, has still to be explored, ﬁrst steps are under way [48]. 17Also, we showed some data for T= 50,000Kand higher pressure, up to p∼106Mbar, Fig. 5. Here the Monte Carlo simulations are particularly di ﬃcult due to the high electron degeneracy, and they can beneﬁt from the Pade simulations, a s the latter correctly reproduce the high-density limit, rs≪1. VII. ACKNOWLEDGEMENTS We acknowledge stimulating discussions with W.D. Kraeft, D . Kremp and M. Schlanges. We thank D.M. Ceperley and B. Militzer for discussions on PIM C concepts and for providing us with the data of Ref. [32] prior to publication. Further we thank J. Ortner for informing us about an alternative derivation for the oﬀ-diagonal elem ents of the interaction potential. This work was made possible by generous support from the Deut sche Forschungsgemein- schaft (Mercator-Programm) for VSF and by a grant for CPU tim e at the NIC J¨ ulich. 18REFERENCES [1]Strongly Coupled Coulomb Systems , G. Kalman (ed.), Pergamon Press 1998 [2] Proceedings of the International Conference on Strongl y Coupled Plasmas, W.D. Kraeft and M. Schlanges (eds.), World Scientiﬁc, Singapore 1996 [3] W.D. Kraeft, D. Kremp, W. Ebeling, and G. R¨ opke, Quantum Statistics of Charged Particle Systems , Akademie-Verlag Berlin 1986 [4]Progress in Nonequilibrium Green’s functions , M. Bonitz (ed.), World Scientiﬁc, Singa- pore 2000 [5] W. Ebeling, A.F¨ orster, H. Hess, M. Yu. Romanovsky, Plas ma Phys. Control. Fusion 38, A 31-A47 (1996) [6] I.B. Da Silva et al., Phys. Rev. Lett. 78, 783 (1997). [7] M. Schlanges, M. Bonitz, and A. Tschttschjan, Contrib. P lasma Phys. 35, 109 (1995) [8] For completeness, we mention interesting attempts to ov ercome the fermion sign prob- lem in path integral Monte Carlo simulations for few-electron systems by appropriately regrouping the sum over permutations, cf. R. Egger, W. H¨ aus ler, C.H. Mak, and H. Grabert, Phys. Rev. Lett. 82, 3320 (1999) and Refs. therein. [9] W. Ebeling, W. Stolzmann, A. F¨ orster, and M. Kasch, Cont r. Plasma Phys. 39, 287 (1999). [10] D. Klakow, C. Toepﬀer, and P.-G. Reinhard, Phys. Lett. A 192, 55 (1994); J. Chem. Phys.101, 10766 (1994). [11] M. Knaup et al., J. Phys. IV 10, 307 (2000) [12] V.S. Filinov, P.R. Levashov, V.E. Fortov, and M. Bonitz , J. Phys. IV France 10, 323 (2000) [13] W. Ebeling, and F. Schautz, Phys. Rev. E, 56, 3498 (1997). [14] M. Bonitz et al., J. Phys. B: Condensed Matter 8, 6057 (1996) [15] M. Bonitz, “Quantum Kinetic Theory”, B.G. Teubner, Stu ttgart/Leipzig 1998 [16] V.M. Zamalin, G.E. Norman, and V.S. Filinov, The Monte Carlo Method in Statistical Thermodynamics , Nauka, Moscow 1977 (in Russian). [17]The Monte Carlo and Molecular Dynamics of Condensed Matter S ystems , K. Binder and G. Cicotti (eds.), SIF, Bologna 1996 [18]Classical and Quantum Dynamics of Condensed Phase Simulati on, B.J. Berne, G. Cic- cotti and D.F. Coker eds., World Scientiﬁc, Singapore 1998. [19] D.M. Ceperley, in Ref. [17], pp. 447-482 [20] D.M. Ceperley, Rev. Mod. Phys. 65, 279 (1995) [21] B. Militzer, and R. Pollock, Phys. Rev. E 61, 3470 (2000) [22] As an example we mention the method of Imada, J. Phys. Soc . of Japan 53, 2861 (1984). However, we have found that his approach is not adequ ate for thermodynamic properties [31]. [23] V.S. Filinov, P.R. Levashov, V.E. Fortov, and M. Bonitz , in Ref. [4], (archive: cond- mat/9912055) [24] V.S. Filinov, and M. Bonitz, (Preprint, archive: cond- mat/9912049) [25] V.S. Filinov, J.Phys. A: Math. Gen. 34(2001) [26] V.S. Filinov, M. Bonitz, and V.E. Fortov, JETP Letters 72, 245 (2000) [27] V.S. Filinov, V.E. Fortov, M. Bonitz, and D. Kremp, Phys . Lett. A 274 , 228 (2000) [28] G. Zwicknagel, C. Toepﬀer, and P.-G. Reinhard, Phys. Re ports314, 671 (1999). 19[29] V. Golubnychiy, M. Bonitz, D. Kremp, and M. Schlanges, s ubm. to Phys. Rev. E. [30] V. Golubnychiy, M. Bonitz, D. Kremp, and M. Schlanges, C ontr. Plasma Phys. (2001). [31] The method of Imada [22] ignores the sign of the exchange determinant, cf. Eq. (19). We have found that Imada’s approximation leads to essential errors for strongly degen- erate systems. We underline that, in contrast, our method ri gorously takes into account the sign of the determinant of each Monte Carlo conﬁguration in the calculation of thermodynamic averages, such as pressure and energy, Eqs. ( 20, 17). [32] B. Militzer, and D.M. Ceperley, Phys. Rev. Lett. 85, 1890 (2000) [33] W. Ebeling, Ann. Physik (Leipzig) 21, 315 (1968); 22, 33, 383, 392 (1969); Physica 38, 378 (1968); 40, 290 (1968). [34] W. Ebeling, and W. Richert, Phys. Lett. A 7108, 80 (1985); phys. stat sol. (b) 128, 167 (1985). [35] W. Ebeling, Contr. Plasma Physics 29, 165 (1989); 30, 553 (1990). [36] W. Ebeling, A. F¨ orster, V. Fortov, V. Gryaznov, and A. P olishchuk, Thermophysical properties of hot dense plasmas , Teubner Stuttgart-Leipzig 1991 [37] R.P. Feynman, and A.R. Hibbs, Quantum mechanics and path integrals , McGraw-Hill, New York 1965 [38] V.S. Filinov, High Temperature 13, 1065 (1975) and 14, 225 (1976) [39] B.V. Zelener, G.E. Norman, V.S. Filinov, High Temperat ure13, 650, (1975) [40] G. Kelbg, Ann. Physik, 12, 219 (1963); 13, 354;14, 394 (1964). [41] W. Ebeling, H.J. Hoﬀmann, and G. Kelbg, Contr. Plasma Ph ys.7, 233 (1967) and references therein. [42] An energy estimator similar to Eq. (17) has been derived by Herman et al., J. Chem. Phys. (1982), but their result neglects the spin statistics and does not contain the correct noninteracting limit. [43] A.V. Filinov, Yu.E. Lozovik, and M. Bonitz, phys. stat. sol. (b) 221, 231 (2000) [44] A.V. Filinov, M. Bonitz, and Yu.E. Lozovik, Phys. Rev. L ett., accepted for publication [45] W.L. Slattery, G.D. Dolen, H.E. De Witt, Phys. Rev. A 21, 2087 (1980) [46] T. Kahlbaum, in Physics of Strongly Coupled Plasmas , W.D. Kraeft, M. Schlanges (eds.), World Scientiﬁc, Singapore 1996 [47] D. Beule et al. Phys. Rev. B 59, 14177 (1999 ); Contr. Plasma Phys. 39, 21 (1999) [48] B. Militzer, D. Beule et al., Preprint Inst. of Physics, Humboldt Univ. Berlin (2000) 20TABLES TABLE I. Direct versus restricted PIMC [23] simulation resu lts (upper and middle lines, respectively) and results of Pad´ e calculations (numbers i n the lowest lines) for the pressure p(Mbar) and energy E(2NRy) for dense hydrogen (deuterium [23]) for rs= 1.86 T,1000K nΛ3Γ p,Mbar E,2NRy 1000 0.10 0.169 67.74 ±0.02 9.050 ±0.005 66.86 ±0.08 9.018 ±0.015 67.38 9.063 500 0.29 0.339 32.85 ±0.03 4.169 ±0.003 32.13 ±0.05 4.114 ±0.007 31.91 4.162 250 0.83 0.679 15.37 ±0.01 1.654 ±0.005 14.91 ±0.03 1.629 ±0.007 14.40 1.679 125 2.33 1.350 6.98±0.01 0.412 ±0.005 6.66±0.02 0.404 ±0.004 6.47 0.471 62.5 6.58 2.701 3.07±0.02 -0.248 ±0.005 2.99±0.04 -0.140 ±0.007 31.25 18.48 5.376 2.20±0.01 -2.377 ±0.005 1.58±0.07 -0.360 ±0.010 21
arXiv:physics/0103003v1 [physics.atm-clus] 1 Mar 2001Preprint Quantum ﬂuid-dynamics from density functional theory S. K¨ ummel1,2and M. Brack1 1Institute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany 2Department of Physics and Quantum Theory Group, Tulane Univ ersity, New Orleans, Louisiana 70118, USA; e-mail: skuemmel@tulane.edu (December 18, 2012) Abstract A partial diﬀerential eigenvalue equation for the density d isplacement ﬁelds associated with electronic resonances is derived in the fra mework of density functional theory. Our quantum ﬂuid-dynamical approach is based on a varia- tional principle and the Kohn-Sham ground-state energy fun ctional. It allows for an intuitive interpretation of electronic excitations in terms of intrinsic lo- cal currents that obey a continuity equation. We demonstrat e the capabilities of our approach by calculating the photoabsorption spectra of small sodium clusters. The quantitative agreement between theoretical and experimental spectra shows that even for the smallest clusters, the reson ances observed experimentally at low temperatures can be interpreted in te rms of density vibrations. PACS: 31.15.Ew,36.40.Vz,71.15Mb Typeset using REVT EX 1I. INTRODUCTION Since its formal foundation as a theory of ground-state prop erties [1], density functional theory has developed into one of the most successful methods of modern many-body theory, today also with well-established extensions such as, e.g., time-dependent [2] and current [3,4] density-functional theory (DFT). In particular in the ﬁeld of metal cluster physics, DFT calculations have contributed substantially to a qualitat ive and quantitative description of both ground and excited state properties [5,6]. Understand ing the properties of small metal particles in turn oﬀers technological opportunities, e.g. , to better control catalysis [7], as well as fundamental insights into how matter grows [8,9]. Si nce the electronic and geometric structure of metal particles consisting of only a few atoms s till cannot be measured directly, photoabsorption spectra are their most accurate probes. Es pecially the spectra of charged sodium clusters have been measured with high accuracy for a b road range of cluster sizes and temperatures [10]. A distinct feature of these spectra i s that at elevated temperatures of several hundred K, in particular for the larger clusters, on ly a few broad peaks are observed, whereas at lower temperatures (100 K and less), a greater num ber of sharp lines can be resolved for clusters with only a few atoms. The peaks observ ed in the high-temperature experiments found an early and intuitive explanation as col lective excitations in analogy to the bulk plasmon and the giant resonances in nuclei: diﬀer ent peaks in the spectrum were understood as belonging to the diﬀerent spatial direct ions of the collective motion of the valence electrons with respect to the inert ionic backgr ound. On the other hand, the sharp lines observed in the low-temperature experiments we re interpreted as a hallmark of the molecule-like properties of the small clusters explica ble, in the language of quantum chemistry [11], only in terms of transitions between molecu lar states. In this work we present a density functional approach to the c alculation of excitations that leads us to a uniﬁed and transparent physical understan ding of the photoabsorption spectra of sodium clusters. We ﬁrst derive a general variati onal principle for the energy spectrum of an interacting many-body system. From this, we d erive an approximate solution 2in the form of quantum ﬂuid-dynamical diﬀerential equation s for the density displacement ﬁelds associated with the electronic vibrations around the ir ground state. By solving these equations, we obtain the eigenmodes within the DFT; hereby o nly the ground-state energy functional and the occupied Kohn-Sham orbitals are require d. We demonstrate the accuracy of our approach by calculating the photoabsorption spectra of small sodium clusters and comparing our results to low-temperature experiments and t o conﬁguration-interaction (CI) calculations. In this way we can show that also the spectra of the smallest clusters can be understood, without knowledge of the molecular many-body w avefunction, in an intuitive picture of oscillations of the valence-electron density ag ainst the ionic background. II. A VARIATIONAL PRINCIPLE Starting point for the derivation of the variational princi ple is the well-known fact that for a many-body system described by a Hamiltonian Hwith ground state \|0∝an}b∇acket∇i}htand energy E0, the creation and annihilation operators of all the eigensta tes obey the so-called equations of motion for excitation operators [12] ∝an}b∇acketle{t0\|Oν[H,O† ν]\|0∝an}b∇acket∇i}ht=/planckover2pi1ων∝an}b∇acketle{t0\|OνO† ν\|0∝an}b∇acket∇i}ht (1) ∝an}b∇acketle{t0\|Oν[H,Oν]\|0∝an}b∇acket∇i}ht=/planckover2pi1ων∝an}b∇acketle{t0\|OνOν\|0∝an}b∇acket∇i}ht= 0, (2) where OνandO† νare deﬁned by O† ν\|0∝an}b∇acket∇i}ht=\|ν∝an}b∇acket∇i}ht,Oν\|ν∝an}b∇acket∇i}ht=\|0∝an}b∇acket∇i}ht,and Oν\|0∝an}b∇acket∇i}ht= 0. (3) Of course, the exact solution of these equations are in gener al unknown. But a variety of approximations to the true excited states can be derived fro m them, e.g., the Tam-Dancoﬀ scheme and the small amplitude limit of time-dependent Hart ree-Fock theory (RPA). As discussed in [12], also higher-order approximations can be obtained. Related to these equations, we derive the following variati onal principle: solving the equations (1) and (2) for the lowest excited state is equival ent to solving the variational equation 3δE3[Q] δQ= 0, (4) whereE3is deﬁned by E3[Q] =/radicalBigg m3[Q] m1[Q], (5) andm1andm3are the multiple commutators m1[Q] =1 2∝an}b∇acketle{t0\|[Q,[H,Q]]\|0∝an}b∇acket∇i}ht (6) m3[Q] =1 2∝an}b∇acketle{t0\|[[H,Q],[[H,Q],H]]\|0∝an}b∇acket∇i}ht. (7) HerebyQis some general Hermitean operator that, as will be shown in t he course of the argument [see Eq. (15) below], can be interpreted as a genera lized coordinate. The minimum energyE3after the variation gives the ﬁrst excitation energy /planckover2pi1ω1. The second excitation with energy /planckover2pi1ω2can be obtained from variation in an operator space which has been or- thogonalized to the minimum Q, and in this way the whole spectrum /planckover2pi1ωνcan be calculated. The variation δQof an operator can be understood as a variation of the matrix e lements of the operator in the matrix mechanics picture. Therefore, 0 =δ δQ/parenleftbiggm3[Q] m1[Q]/parenrightbigg1 2 =1 2/parenleftbiggm3[Q] m1[Q]/parenrightbigg−1 2δ δQ/parenleftbiggm3[Q] m1[Q]/parenrightbigg , (8) and noting that the ﬁrst factors in the expression to the righ t are just 1/(2E3), 0 =δ δQ/parenleftbiggm3[Q] m1[Q]/parenrightbigg =1 m1[Q]δm3[Q] δQ−m3[Q] m1[Q]2δm1[Q] δQ(9) is obtained. With the deﬁnition E3=/planckover2pi1ω1, Eq. (9) turns into δm3[Q] δQ−(/planckover2pi1ω1)2δm1[Q] δQ= 0. (10) The variations δm3[Q] =m3[Q+δQ]−m3[Q] δm1[Q] =m1[Q+δQ]−m1[Q] (11) 4are evaluated by straightforward application of the commut ation rules (6) and (7), leading to ∝an}b∇acketle{t0\|[ [δQ,H ],/parenleftbig [H,[H,Q]]−(/planckover2pi1ω1)2Q/parenrightbig ]\|0∝an}b∇acket∇i}ht= 0. (12) WithδQHermitean, [ δQ,H ] is anti-Hermitean, and (12) therefore is an equation of the form c+c∗= 0 with c=∝an}b∇acketle{t0\|[δQ,H ]/parenleftbig [H,[H,Q]]−(/planckover2pi1ω1)2Q/parenrightbig \|0∝an}b∇acket∇i}ht ∈C. (13) Since \|0∝an}b∇acket∇i}htby deﬁnition is the exact ground state of H, and (13) must hold for any δQ, the equation /parenleftbig [H,[H,Q]]−(/planckover2pi1ω1)2Q/parenrightbig \|0∝an}b∇acket∇i}ht= 0 (14) is obtained. It resembles the equation of motion for a harmon ic oscillator. Therefore, Qis interpreted as a generalized coordinate, and in analogy to t he well-known algebraic way of solving the harmonic oscillator problem, Qis written as a linear combination Q∝ O† 1+O1 (15) of the creation and annihilation operator for the ﬁrst excit ed state. Inserting (15) into (14) leads to the two equations [H,[H,O† 1]]\|0∝an}b∇acket∇i}ht= (/planckover2pi1ω1)2O† 1\|0∝an}b∇acket∇i}ht (16) [H,[H,O1]]\|0∝an}b∇acket∇i}ht= (/planckover2pi1ω1)2O1\|0∝an}b∇acket∇i}ht= 0. (17) First consider (16). After closing with state ∝an}b∇acketle{t1\|, one exploits that, by deﬁnition, \|0∝an}b∇acket∇i}htand\|1∝an}b∇acket∇i}ht are eigenstates of Hand evaluates the outer commutator by letting Hact once to the left and once to the right. Recalling that ∝an}b∇acketle{t1\|=∝an}b∇acketle{t0\|O1, one ﬁnally obtains ∝an}b∇acketle{t0\|O1[H,O† 1]\|0∝an}b∇acket∇i}ht=/planckover2pi1ω1∝an}b∇acketle{t0\|O1O† 1\|0∝an}b∇acket∇i}ht. (18) This is exactly equation (1) for the ﬁrst excited state. In th e same way, (2) is obtained from (17), which completes the derivation of the variational pri nciple. 5We would like to point out that in earlier work [13], the RPA eq uations have been derived with a related technique that made use of both generalized co ordinate and momentum operators. The advantage of our present derivation is that – although within linear response theory – it goes beyond RPA and, due to the formulation in term s of a generalized coordinate only, is particularly suitable for the formulation of the va riational principle in the framework of density functional theory as shown below. III. QUANTUM FLUID DYNAMICS FROM THE GROUND-STATE ENERGY FUNCTIONAL: A LOCAL CURRENT APPROXIMATION In principle, the exact eigenenergies are deﬁned via Eqs. (1 ), (2) by the variational equation (4), provided that the operator Qis chosen in a suﬃciently general form. However, just as in the equations of motion technique, one is forced to make some explicit ansatz for the form of Q, which necessarily introduces approximations. In Ref. [13 ] it was shown that ifQis taken to be a one-particle-one-hole excitation operator , Eq. (4) leads to the RPA equations. Simpliﬁcations of the RPA, in which Qwas chosen from restricted sets of local operatorsQn(r), were proposed in connection with both semiclassical [14] and Kohn-Sham density functionals [13]. In the present paper, we derive a s et of quantum ﬂuid-dynamical equations from the variational principle (4) by choosing Qto a general local operator Q(r). These equations are then solved without any restriction oth er than Eq. (23) below. First we recall a relation that is well known in nuclear physi cs [15]: the commutator of Eq. (7) can be exactly obtained from m3[Q] =1 2∂2 ∂α2∝an}b∇acketle{tα\|H\|α∝an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle α=0, (19) whereSis the so called scaling operator deﬁned by S= [H,Q] (20) and\|α∝an}b∇acket∇i}htthe state that results from the unitary transformation 6\|α∝an}b∇acket∇i}ht=e−αS\|0∝an}b∇acket∇i}ht, (21) withαbeing a real and possibly time-dependent parameter. Assumi ng thatQis just a function of rand that the potentials in Hdo not contain derivatives with respect to r, as is the case for Coulombic systems, Eq. (20) is easily evaluated : S=Ne/summationdisplay i=1s(ri) =Ne/summationdisplay i=11 2(∇iu(ri)) +u(ri)· ∇i. (22) Here, the displacement ﬁeld u(r) =−/planckover2pi12 m∇Q(r) (23) has been introduced, and Neis the number of electrons. These equations can be related to DFT by noting that, ﬁrst, we can introduce a set of single particle orbitals {ψµ(ri)}, and from the scaled single particle orbitals, a scaled sing le particle density can be constructed via n(r,α) =Ne/summationdisplay µ=1/vextendsingle/vextendsinglee−αs(r)ψµ(r)/vextendsingle/vextendsingle2=e−αSnn(r), (24) with a density scaling operator Sn=/parenleftBig ∇u(r)/parenrightBig +u(r)· ∇. (25) Second, Eq. (6) can straightforwardly be evaluated for a loc alQ(r), m1[Q] =m 2/planckover2pi12/integraldisplay u(r)·u(r)n(r) d3r, (26) showing that m1depends only on nanduand is similar to a ﬂuid-dynamical inertial parameter. And third, we replace the expectation value in Eq . (19) by m3[Q] =1 2∂2 ∂α2∝an}b∇acketle{tα\|H\|α∝an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle α=0→1 2∂2 ∂α2E[n(r,α)]/vextendsingle/vextendsingle/vextendsingle α=0, (27) whereE[n] is the usual ground-state Kohn-Sham energy functional E[n;{R}] =Ts[n] +Exc[n] +e2 2/integraldisplay /integraldisplayn(r)n(r′) \|r−r′\|d3r′d3r +/integraldisplay n(r)Vion(r;{R}) d3r.(28) 7Eq. (26) is exact and also Eq. (24) can be veriﬁed order by orde r, but Eq. (27) goes beyond the safe grounds on which the energy functional is deﬁned. Ho wever, the replacement of an energy expectation value by the energy functional is intu itively very plausible, and its practical validity can be judged a posteriori by the results. A further strong argument for why really the density should be the basic variable can be mad e by calculating the derivative with respect to time of the scaled density, using Eqs. (24) an d (25), d dtn(r,α(t)) =−Sn˙α(t)n(r,α(t)) =−∇[ ˙α(t)u(r)n(r,α(t))], (29) where for the sake of clarity we now explicitly wrote the time dependence of α. Since j(r,t) = ˙α(t)u(r)n(r,α(t)), (30) is a current density, Eq. (29) is just the continuity equatio n dn(r,α(t))/dt +∇j(r,t) = 0. Thus, the variational principle Eq. (4) with a local functio nQ(r) describes excitations by intrinsic local currents. The time dependence of the parame terαis obviously harmonic, i.e., α(t)∝cos(ωνt), since the present derivation is based on linear response t heory. The physical signiﬁcance of the variational approach now be ing clear, it remains to derive the actual equations that determine the displacement ﬁelds u(r) and the energies /planckover2pi1ωthat are associated with particular excitations. Starting from Eq. (10) and using an explicit notation, δm3[u[Q(r)]] δQ(r′)−(/planckover2pi1ω1)2δm1[u[Q(r)]] δQ(r′)= 0 =/integraldisplay d3r′′/braceleftbiggδm3[u(r)] δu(r′′)−(/planckover2pi1ω1)2δm1[u(r)] δu(r′′)/bracerightbiggδu(r′′) δQ(r′) (31) follows by virtue of the chain rule for functional derivativ es. Thus, solutions of δm3[u(r)] δu(r′)= (/planckover2pi1ω1)2δm1[u(r)] δu(r′)(32) will also be solutions to Eq. (10) and thus Eq. (4). m1is already given as the functional m1[u] by Eq. (26), and m3[u] is readily obtained by inserting the scaled Kohn-Sham orbi tals and density from Eq. (24) into the energy functional Eq. (28) and calculating the second 8derivative with respect to the parameter α, Eq. (27). The ﬁnal equations are then derived in a lengthy but straightforward calculation from Eq. (32) by e xplicitly performing the variation onu. Using the usual deﬁnition δm3[u(r)] δu(r′)=δm3[u](r) δux(r′)ex+δm3[u(r)] δuy(r′)ey+δm3[u(r)] δuz(r′)ez, (33) whereeiare the unit vectors in the Cartesian directions, a set of thr ee coupled, partial diﬀer- ential eigenvalue equations of fourth order for the Cartesi an components uj(r) is obtained: δm3[u] δuj(r)= (/planckover2pi1ω)2δm1[u] δuj(r), j= 1,2,3, (34) where δm1[u] δuj(r)=m /planckover2pi12n(r)uj(r), (35) δm3[u] δuj(r)=δmkin 3[u] δuj(r)+δmKS 3[u] δuj(r)+δmh2 3[u] δuj(r)+δmxc2 3[u] δuj(r), (36) and δmkin 3[u] δuj(r)=−/planckover2pi12 2m1 2Ne/summationdisplay m=13/summationdisplay i=1ℜe/braceleftbigg/parenleftBig ∆ψm/parenrightBig/bracketleftBig (∂jui)(∂iψ∗ m) + (∂j∂iui)ψ∗ m+ui(∂j∂iψ∗ m)/bracketrightBig + /bracketleftBig (∂jui)(∂i∆ψm) +ui(∂j∂i∆ψm)/bracketrightBig ψ∗ m−ui/bracketleftBig (∂iψ∗ m)/parenleftBig ∂j∆ψm/parenrightBig +/parenleftBig ∂i∆ψm/parenrightBig (∂jψ∗ m)/bracketrightBig +2/bracketleftbigg (∂jψ∗ m)/bracketleftBig ∆/parenleftBig1 2(∂iui)ψm+ui(∂iψm)/parenrightBig/bracketrightBig −/bracketleftBig ∂j∆/parenleftBig1 2(∂iui)ψm+ui(∂iψm)/parenrightBig/bracketrightBig ψ∗ m/bracketrightbigg /bracerightbigg ,(37) δmKS 3[u] δuj(r)=1 23/summationdisplay i=1/bracketleftbigg n/parenleftBig (∂jui)(∂ivKS)−(∂iui)(∂jvKS)/parenrightBig +ui/parenleftBig n(∂i∂jvKS)−(∂in)(∂jvKS)/parenrightBig/bracketrightbigg , (38) δmh2 3[u] δuj(r)=n/integraldisplay/bracketleftBig3/summationdisplay i=1(∂′ iui(r′))n(r′) +ui(r′)(∂′ in(r′))/bracketrightBigrj−r′ j \|r−r′\|3d3r′, (39) δmxc2 3[u] δuj(r)=−n3/summationdisplay i=1/bracketleftbigg/parenleftBig ∂j((∂iui)n+ui(∂in))/parenrightBig∂vxc ∂n+/parenleftBig (∂iui)n+ui(∂in)/parenrightBig/parenleftBig ∂j∂vxc ∂n/parenrightBig/bracketrightbigg ,(40) 9where we used the shorthand notation ∂1=∂/∂x etc., and indicated the terms to which derivatives refer by including them in parenthesis. The usu al Kohn-Sham and exchange- correlation potential are denoted by vKSandvxc, respectively. Eqs. (34) – (40) are our quantum ﬂuid-dynamical equations. I n analogy to the local density approximation (LDA) used for vxc, we term our scheme the local current approxi- mation (LCA) to the dynamics, due to the use of a local function Q(r) in the variational principle (4). It should be noted that the above equations di ﬀer from the equations derived earlier in a semiclassical approximation [14] or by explici t particle-hole averaging [13]. Due to the fact that our approach is completely based on the Kohn- Sham density functional and therefore contains the full quantum-mechanical shell eﬀec ts in the ground-state density, it is also diﬀerent from some ﬂuid-dynamical approaches develop ed in nuclear physics [16] (and used in cluster physics [17]) which involved either schemat ic liquid-drop model densities or semiclassical densities derived from an extended Thomas-F ermi model. Although Eqs. (34) – (40) look rather formidable, they can be solved numerically with reasonable computational eﬀort, and we have done so for the s odium clusters Na 2and Na+ 5. The Kohn-Sham equations were solved basis-set free on a thre e-dimensional Cartesian real- space grid using the damped gradient iteration with multigr id relaxation [18]. The ionic coordinates were obtained by minimizing the total energy us ing a smooth-core pseudopo- tential [9]. For Exc, we employed the LDA functional of Ref. [19]. The uj(r) were expanded in harmonic oscillator wavefunctions and we explicitly enf orced Eq. (23). The convergence rate of the expansion can be improved by adding a few polynomi al functions to the basis. By multiplying Eqs. (32) and subsequently (34)–(40) from th e left with uand integrating over all space, a matrix equation for the expansion coeﬃcien ts is obtained which can be solved using library routines. The square roots of the eigen values then give the excitation energies and from the eigenvectors, the oscillator strengt hs can be computed. Fig. 1 shows the experimental photoabsorption spectrum [20 ] of Na 2in the upper left picture (adapted from Ref. [6]), and below the spectrum obta ined in the just described LCA. We introduced a phenomenological line broadening in the LCA results to guide the eye. The 10LCA correctly reproduces the electronic transitions, desp ite the fact that only two electrons are involved. Due to Eq. (29), one can very easily visualize h ow the electrons move in a particular excitation by plotting the corresponding ∇j(r), giving a “snapshot” picture of dn/dt. For the two main excitations, a crossection of this quant ity along the symmetry axis (zaxis) is shown in the lower left and upper right contourplots , and the ground-state valence electron density is shown in the lower right for reference. I n the plots of dn /dt, shadings darker than the background grey indicate a density increase , lighter shadings indicate a decrease. It becomes clear that the lower excitation corres ponds to a density oscillation along thezaxis whereas the higher excitation corresponds to two energ etically degenerate oscillations perpendicular to the symmetry axis. (For the s ake of clarity, we plotted the corresponding oscillator strengths on top of each other in t he photoabsorption spectrum.) This is exactly what one would have expected intuitively. Bu t the plots reveal that besides the expected general charge transfer from one end of the clus ter to the other, the presence of the ionic cores hinders the valence electrons to be shifte d freely, creating a density shift of reverse sign in between the ionic cores. Fig. 2 shows the ionic ground-state conﬁguration of Na+ 5with our labeling of axes in the upper left, the experimental low-temperature ( ≈100 K) photoabsorption spectrum [10] in the upper right, the LCA photoabsorption spectrum in the low er left, and the CI spectrum adapted from Ref. [11] in the lower right. Again, a phenomeno logical line broadening was introduced in the presentation of both the LCA and the CI resu lts. The LCA spectrum again is in close agreement with the experimentally observe d spectrum, showing three intense transitions. With our choice of the coordinate system, the l owest excitation corresponds to a density oscillation in zdirection, whereas the two higher excitations oscillate in bothx andydirections. In the interpretation of the LCA results, it mus t be kept in mind that due to our ﬁnite grid spacing the numerical accuracy for the exci tation energies is about 0.03 eV, which is absolutely suﬃcient in the light of the physical app roximations that we are making. But due to this ﬁnite numerical resolution and the fact that w e evaluate each direction of oscillation separately, the xandycomponents of the excitations at 2.7 eV and 3.4 eV, which 11really should be degenerate for symmetry reasons, appear as extremely close-lying double lines. However, since the symmetry of the cluster was in no wa y an input to our calculation, it is a reassuring test that the LCA, indeed, fulﬁlls the symm etry requirement within the numerical accuracy. Furthermore, it is reassuring to see th at with respect to the relative heights of the peaks the LCA is very close to the CI results, wi th diﬀerences observed only in the small subpeaks that are not seen experimentally anyway. And small diﬀerences to the CI calculation are already to be expected simply because of the use of diﬀerent pseudopotentials and the resulting small diﬀerences in the ionic ground-stat e structure. IV. CONCLUSION In summary, we have derived a set of quantum ﬂuid-dynamical e quations from a general variational principle for the excitations of a many-body sy stem. The equations describe here the eigenmodes of the system’s (valence) electrons and require only the knowledge of the occupied ground-state Kohn-Sham orbitals. From these e quations, we have computed the photoabsorption spectra for small sodium clusters and ﬁ nd quantitative agreement with the experimentally observed peak positions. Thus, even low -temperature photoabsorption spectra can be understood in an intuitive picture of density oscillations, without knowledge of the true (or any approximate) many-body wavefunction. ACKNOWLEDGMENTS We are grateful to P.-G. Reinhard for his vivid interest in th is work and for many stimulating discussions. This work was supported by the Deu tsche Forschungsgemeinschaft under grant No. Br 733/9 and by an Emmy-Noether scholarship. S.K. is grateful to J. Perdew for a warm welcome at Tulane University. 12REFERENCES [1] P. Hohenberg und W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn und L. J. Sham, Phys. Rev. 140, A1133 (1965). [2] For an overview see, e.g., E. K. U. Gross, J. F. Dobson, and M. Petersilka, in Density Functional Theory , edited by R. F. Nalewajski (Topics in Current Chemistry, Vo l. 181, Springer, Berlin, 1996). [3] B. M. Deb and S. K. Gosh, J. Chem. Phys. 77, 342 (1982). [4] G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4878 (1997). [5] For cluster excitations calculated in DFT, see, e.g., W. Ekardt, Phys. Rev. B 31, 6360 (1985); M. Madjet, C. Guet, and W. R. Johnson, Phys. Rev. A 51, 1327 (1995); A. Rubio, J. A. Alonso, X. Blase, L. C. Balb´ as, and S. G. Louie,, Phys. Rev. Lett. 77, 247 (1996); K. Yabana and G. F. Bertsch, Phys. Rev. B 54, 4484 (1996); A. Pohl, P.-G. Reinhard, E. Suraud, Phys. Rev. Lett. 84, 5090 (2000). [6] I. Vasiliev, S. ¨O˘ g¨ ut, and J. R. Chelikowsky, Phys. Rev. Lett. 82, 1919 (1999). [7] M. Moseler, H. Hakkinen, R.N. Barnett, and U. Landman, to appear in Phys. Rev. Lett. 2001; lanl preprint physics/0101069. [8] J. Akola, A. Rytk¨ onen, H. H¨ akkinen, and M. Manninen, Eu r. Phys. J. D 8, 93 (2000). [9] S. K¨ ummel, M. Brack, and P.-G. Reinhard, Phys. Rev. B 62, 7602 (2000). [10] C. Ellert, M. Schmidt, C. Schmitt, T. Reiners, and H. Hab erland, Phys. Rev. Lett. 75, 1731 (1995); M. Schmidt, C. Ellert, W. Kronm¨ uller, and H. Ha berland, Phys. Rev. B 59, 10970 (1999). [11] V. Bonaˇ cic-Kouteck´ y, J. Pittner, C. Fuchs, P. Fantuc ci, M. F. Guest, and J. Kouteck´ y, J. Chem. Phys. 104, 1427 (1996). 13[12] We particularly like the presentation of this techniqu e given in D. J. Rowe, Nuclear collective motion (Methuen and Co., London, 1970). [13] P.-G. Reinhard, M. Brack and O. Genzken, Phys. Rev. A 41, 5568 (1990). [14] M. Brack, Phys. Rev. B 39, 3533 (1989). [15] O. Bohigas, A. M. Lane, and J. Martorell, Phys. Rep. 51, 267 (1979); and references therein. [16] E. R. Marshalek and J. da Providˆ encia, Phys. Rev. C 7, 2281 (1973); J. da Providˆ encia and G. Holzwarth, Nucl. Phys. A 439, 477 (1985); E. Lipparini and S. Stringari, Phys. Rep.175, 103 (1989); P. Gleissl, M. Brack, J. Meyer, and P. Quentin, A nn. Phys. (N.Y.) 197, 205 (1990). [17] J. da Providˆ encia, Jr. and R. de Haro, Jr., Phys. Rev. B 49, 2086 (1994). [18] V. Blum, G. Lauritsch, J. A. Maruhn, and P.-G. Reinhard, J. of Comp. Phys. 100, 364 (1992); S. K¨ ummel, Structural and Optical Properties of Sodium Clusters studi ed in Density Functional Theory , (Logos Verlag, Berlin, 2000). [19] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). [20] W. R. Fredrickson and W. W. Watson, Phys. Rev. 30, 429 (1927). 14FIGURES Experiment 1234 eV 0LCA dn/dt5 10 15 20510152025dn/dt xz 5 10 15 20510152025 xzn z x510152025 5 10 15 20 FIG. 1. From top left to bottom right: Experimental photoabs orption spectrum [20] and LCA spectrum of Na 2in arbitrary units versus eV, density change associated wit h the ﬁrst excitation, density change associated with the second excitation, grou nd-state valence electron density. 15-2 0 2 x-202y -505 z -2 0 2 2 2.5 3 3.5eV 00.511.522.53arb . units 2 2.5 3 3.5eV 00.511.522.53arb . units 2 2.5 3 3.5eV 00.511.522.53arb . units FIG. 2. Upper left: ionic ground-state conﬁguration of Na+ 5, lower left: corresponding LCA photoabsorption spectrum, upper right: experimental low- temperature photoabsorption spectrum [10], lower right: Conﬁguration-Interaction photoabsorp tion spectrum from Ref. [11]. See text for discussion. 16
Properties of Pt Schottky Type Contacts On High-Resistivity CdZnTe Detectors Aleksey E. Bolotnikov, Steven E. Boggs, C. M. Hubert Chen, Walter R. Cook, Fiona A. Harrison, and Stephen M. Schindler California Institute of Technology, Pasadena, CA 91125 Abstract In this paper we present studies of the I-V characteristics of CdZnTe detectors with Pt contacts fabricated from high- resistivity single crystals grown by the high-pressure Brigman process. We have analyzed the experimental I-V curves using a model that approximates the CZT detector as a system consisting of a reversed Schottky contact in series with the bulkresistance. Least square fits to the experimental data yield 0.78-0.79 eV for the Pt-CZT Schottky barrier height, and <20 Vfor the voltage required to deplete a 2 mm thick CZT detector. We demonstrate that at high bias the thermionic current overthe Schottky barrier, the height of which is reduced due to an interfacial layer between the contact and CZT material, controlsthe leakage current of the detectors. In many cases the dark current is not determined by the resistivity of the bulk material, but rather the properties of the contacts; namely by the interfacial layer between the contact and CZT material. Keywords : X-ray astrophysics, CdZnTe pixel detectors, I-V curve measurements 1. Introduction The dark current is a critical parameter that for many configurations can be the primary factor limiting the energy resolution of CdZnTe (CZT) detectors. In the course of developing a focal plane detector for the balloon-borne High-EnergyFocusing Telescope (HEFT) [1], we carried out routine measurements of the dark current characteristics for a large number of CZT pixel detectors of a specific pixel contact design. Our detector anode pattern includes very thin strips (a grid) betwee n the pixel contacts, held at a small negative potential. The real purpose of this biased grid is to enhance the charge collectio n near the surface between pixel contacts. However, for the dark current measurements we can ground the grid, so that it servesas a guard ring to eliminate surface leakage currents, allowing accurate measurement of both surface and bulk leakage. We tested a large number of CZT detectors, measuring the surface and bulk I-V curves over a wide voltage range. We found large variations in the shapes and nominal surface dark currents for different detectors, as well as for different pixels of the same detector. This is the case even for detectors where the specific bulk resistivity, as defined by approximating I-V curves to Ohm’s law at very low bias, <0.5 V, varies only by 20-40%. In some detectors, the measured I-V characteristics also resemble a simple Ohm’s law at higher bias. The specific resistivity, evaluated by fitting the data from a high voltage region, significantly exceeds the upper limit established for the CZT material used in these measurements,~5x10 10 Ohm-cm at 26 C [2]. To understand these experimental leakage current measurements, we modeled the CZT detector as a metal- semiconductor-metal (MSM) system with two back-to-back Schottky barriers. Two simplified treatments have beenpreviously applied to such a system: Sze et al. [3] used the thermionic-limited approximation of the Schottky barrier, andCisneros et al. [4] treated the barrier in the diffusion-limited approximation. Neither of these approaches could explain ourdark current measurements. In our previous work [5] we briefly pointed out that the experimentally measured currents wereconsiderably smaller than the saturation thermionic current expected for the Pt-CZT, and the measured I-V curves differed in shape from the diffusion-limited current expected for two back-to-back Schottky barriers. Although the models described above are over-simplified, we also cannot explain our I-V curve measurements over the full voltage range even with a more general treatment of an MSM system. Crowell and Sze [6] demonstrated that thethermionic- and diffusion-limited models are not independent, but are in fact limiting cases of a more general thermionic-diffusion theory. Using this theory we can reproduce the measured I-V curves at a low voltages (in some cases up to 100 V), but at high voltages the measured current increases much faster than predicted by the theory. One might expect that the discrepancy could be explained by tunneling across the interface (normally the dominant current in highly doped Correspondence: Email: bolotnik@srl.caltech.edu; Telephone: 626-395-4488semiconductors at low temperatures). For this to be the case, our measurements show that tunneling would have to start tocontribute at ~50 V (for a 2 mm thick detector). At this low voltage, the total current across the CZT is much less than theexpected saturation thermionic current (see Eq. (21) and discussion below). The tunneling component should, however,become important at much higher biases, where the thermionic emission component is close to its saturation limit (~500 V). We find that to explain the shape of our I-V curves, we must assume the existence of a very thin (10-100 nm) insulating layer (residual oxide layer) between the contact and the semiconductor material, which could be formed before orafter metal contacts are deposited [7-12]. To include the effects of an interfacial layer in the Schottky barrier model, Wu [14 ] developed a combined interfacial layer-thermionic-diffusion (ITD) model. We show that adopting this ITD model allows usto accurately fit the experimental data without considering any other possible current components (such as tunneling, orgeneration recombination currents). We demonstrate that by taking into account the interfacial layer we can explain the fullvariety of measured I-V curves, and by fitting the data we can obtain for each detector a consistent set of parameters that characterize the Schottky barrier and CZT material.2. Theoretical background and fitting algorithm This section briefly describes the theoretical model of the Schottky barrier with a thin interfacial layer, as applied to the MSM system, which we employ in our analysis. For details we refer to the original work by Sze et al. [3], Sze [15],Cisneros et al. [4], Wu [14], and Cohen et al. [16]. From the mathematical point of view a Pt-CZT-Pt MSM system is rathercomplicated. Fortunately, because of the high bulk resistivity of semi-insulating material such as CZT, we can make somesimplifications. The series resistance of the undepleted bulk material is much higher than the resistance of the forward-biasedSchottky barrier at the anode, and the width of its depleted layer is much smaller than the total thickness of the CZT crystal.We can therefore neglect the effect of the anode contact. This simplification allows us to treat a CZT detector as a metal- semiconductor system consisting of a reversed-biased Schottky barrier at the cathode coupled to the series resistance of thebulk. The band diagram of this system is shown in Fig. 1. The detectors we have studied have rectangular pixel contacts surrounded by a grid on the anode side (see Fig. 2.) and a monolithic contact on the cathode side. We treat this as a one-dimensional system, where the electric field is uniform inboth X and Y directions. In the Schottky-depleted-layer approximation, if a small negative voltage, - V (V>0) is applied to the cathode, the electric field distribution, U(z), inside both the depleted and undepleted regions of the detector can be written as: U(z)=(eN D/2ε)(z-W)2-EA(z-W) + ∆V, 0<z<W (1) andU(z)=E A(W-z)+ ∆V, W<z<L . (2)Figure 1.Schottky contact with interfacial layer (a) unbiased and (b) reverse biased.Figure 2. Contact pattern with a focusing grid.WVInterfacial layer MetalReverse bias (b)V-DVFermi levels in metal contactsInterfacial layer SemiconductorMetal dVbi FoEqulibrium condition (a)Ec EvIn the above equations, W is the width of the depleted layer, L is the thickness of the CZT crystal, EA is the electric field strength inside the undepleted bulk (same as at the anode), ε is the permittivity of CZT, e is the electron charge, ND is the concentration of the ionized donor centers, and – ∆V (∆V>0) is the potential at the edge of the depleted layer ( ∆V=(L-W)E A). Using the boundary condition at the cathode and at the edge of the depleted layer, one can find the width of the depleted layer from: V+V bi=(eN D/2ε)W2+∆V, (3) where Vbi is the built-in voltage or diffusion potential at the cathode (see Fig. 1). From this equation, W can be calculated if EA or ∆V is known. If W/L<<1 and {(eN D/2ε)W2-Vbi}/V<<1 then EA≡∆V/(L-W)=V/L , and I~V, i.e. at small applied biases the current follows Ohm’s law. The voltage VRT required to deplete the whole volume of the crystal, defined as the reach-through voltage, is given by: VRT=(eN D/2ε)L2+E 0L-V bi (4) where E0 is the electric field strength at the anode when the cathode is at VRT, i.e. E0=E A(VRT). Notice, that when the bulk resistance is neglected, E0=0, and Eq. (4) becomes the standard expression for the flat-band voltage–a parameter usually defined to characterize the back-to-back barrier system [3,4]. For applied voltages higher than VRT: U(x)=V RT(z-L)2/2L+(z-L)V/L. (5) Correspondingly, the electric-field strength at the cathode EC–the parameter which we will need for further calculations–is given by: EC(V)=(eN D/ε)W+E A, V<V RT, (6) and EC(V)=(V RT+V)/L-E 0,V > V RT (7) In the combined ITD model, the reverse current, IR (A/cm2), over the barrier at the cathode is expressed as [13]: IR={ϑnAT2/(1+ϑnVR/VD)}exp(- ΦR/VTH)(1-exp{-(V-R SIR)/VTH}), (8) where A is the effective Richardson constant, T is the temperature, VR is the thermal velocity, ϑn is the transmission coefficient through the oxide layer, RS is the series resistance of the bulk, and VTH=kT/e. V D is an effective diffusion velocity [11,14] that can be calculated analytically if Eq. (1) is used to approximate the field distribution in the depleted layer. In this case, VD is simply the electron drift velocity at the cathode, namely: VD=µEC, (9) where µ is the electron mobility ( µ=1000 cm2/Vs). The effective Richardson constant is related to the thermal velocity VR by: AT2≡VRNC, (10) where NC is the effective density of the states in the conduction band given by: NC=2(2πm0kT/h2)3/2 (11) The Schottky barrier height, ΦR, is a function of the applied voltage and reflects the barrier lowering due to the voltage drop across the oxide layer. Again, following Wu [14], we assume that ΦR depends linearly on the applied voltage (the barrier lowering due to image-force is negligible in our case) given by: ΦR=Φ0-(1-1/n 0)V, (12)where Φ0 is the barrier height under thermal equilibrium conditions, with 1/n 0=εi/(εi+e2δDS). (13) Here εI and δ are the permitivity and thickness of the interfacial layer, and DS is the density of surface states per unit energy and area. The series resistance of the undepleted layer can be expressed as: RS=(L-W)/eN µ, (14) where N is the free electron concentration (we assume that CZT is an n-type semiconductor). Substituting Eq. (14) into Eq. (8) and using Eq. (5) for W and Eqs. (6,7) for EC we can numerically calculate the I-V dependence for the current across the whole system. The above equations contain too many free parameters, and the information contained in a single I-V curve is obviously insufficient to obtain the parameters from a fitting procedure. Our primary goal, however, is not to evaluate all these parameters explicitly, but to demonstrate that by assuming reasonable values for these parameters, the measured I-V curves can be explained with the ITD model. The effective Richardson constant can be calculated as A=120(m/m0) (in A-cm-2K-2), where m and m0 are the effective and free electron masses. Since the ratio m/m0 for ZnTe and CdTe are 0.11 [17] and 0.09 [10], respectively, we assume for CZT a similar ratio of 0.1. Thus, A=12 A-cm-2K-2. N can be evaluated from Eq. (14) after fitting the I-V curve at low voltages where the dependence follows Ohm’s law ( W<<L and ∆V=V ). For the typical intrinsic bulk resistivity of 3x1010 Ohm_cm, N=2.5x105 cm-3. The limits for the potential barrier height Φ0 can be found from results obtained for Pt-CdTe and Au-CdTe systems (see e.g. Refs. [10,18]) where 0.7< Φ0<0.9. As for VRT, ϑn and n0, these parameters depend on the contact fabrication process, and have to be found by fitting the I-V curves. In the high voltage region, where the crystal is fully depleted ( RS=0) Eq. (8) can be simplified: IR={C 0/(1+C 1/(VRT+V-E 0L))}exp(C 2V). (15) Here C0=ϑnAT2exp(-Φ0/VTH), (16) C1=ϑnLVR/µ, (17) and C2=1-1/n 0. (18) If the effect of interfacial layer is negligible, C2=0 and ϑn=1. From Eq. (10) one can find the following expression for the ratio C0/C1: C0/C1=(N Cµ/L)exp(-Φ0/VTH), (19) which allows us to estimate the potential barrier Φ0. To fit the experimental data, we first assume that the parameters VRT and C2 are known, and apply Eq. (15) to fit the I-V curve for the voltages above VRT (high enough that E0L/V RT<<1). We then evaluate the parameters C0 and C1 and use these to calculate the potential barrier height, Φ0, from Eq. (19), and the ϑnVR product from Eq. (17). ϑn and VR cannot be evaluated separately, however, since we assumed that A is known and equal to 12 A-cm-2K-2, then from Eq. (10): VR=8.5x106 cm/s. We then find E0 by solving Eqs. (7) and (8) with V=V RT, and RS=0. This allowed us to calculate ND from Eq. (4). Finally, we minimized the χ2(VRT,C2) function, given by: χ2(VRT,C2)=∑{(ICAL-IMEAS)/σ}2, (20) to obtain estimates for VRT and C2. Note that for V<VRT, we solve Eq. (3) and Eq. (8) numerically to calculate W, I and EA for each applied voltage V.3. Experimental setup We measured I-V dependencies using a probe stage with a GPIB-controlled HP 3458A multimeter and a EDC 521 DC calibrator. All measurements were taken at a steady state current condition. Because of the large number of deep traps inthe CZT material, it can take several minutes or even hours to reach equilibrium between free and trapped charge. Thesemeasurements are therefore very time consuming, and we use a computer-controlled setup. To reduce the waiting time before equilibrium is reached, we varied the bias on the cathode in small steps. After each step, we paused for several minutes before taking 10-20 sequential measurements of the current, separated in time by 1-min intervals. This sequence of data points allows us to verify that equilibrium has been actually achieved, and also toimprove the accuracy of the measurements. We took the majority of measurements at room temperature, (26 +/-1 C). For onedetector, we varied the temperature from 17 to 70 C. We place the detector on a hot-plate, covered by a super-insulatingscreen. During the measurement the temperature stability was +/-0.5 C, monitored with a thermocouple (accuracy +/-0.1 C)attached to the hot-plate in close proximity to the detector. We used four groups of CZT pixel detectors, fabricated by eV-Products over a two-year period. The first two groups, labeled D1 and D2, were fabricated (to the best of our knowledge) from different slices of the same ingot, and wetherefore expect them to have similar performance. Detectors from groups D1 and D4 are 12x12x2 mm CZT crystals, eachwith a single 8x8 mm contact enclosed inside a guard ring on one side, and a monolithic contact on the opposite side. Thegap between the contact and the guard ring is 0.2 mm. The detectors from group D2 are 8x8x2 mm single crystals, with fourpatterns of 8x8 pixel arrays (see Fig. 2). The physical size of a pixel is 650 by 680 µm. 50 µm wide orthogonal strips are placed between the pixel contacts. Each pixel from a pattern has the same gap between the contact and the grid, which variedbetween 100 and 250 µm from pattern to pattern. Finally, the detectors from the third group, D3, are 7.1x7.1x1.7 mm CZT crystals fabricated from a different ingot. The D3 detectors have pixel patterns similar to D2, except the pixel size is 400 by400 µm, and the gaps between contacts and grid are 50 and 75 µm, with a 25 µm grid width. We typically took the measurements from -100 V to +100 V between contacts and cathode, but for some detectors we increased the maximum applied voltage up to 1 kV. We eliminated the leakage current flowing over the side surfaces ofthe detector by using a guard ring.4. Results and discussion Figs. 3-5 show the I-V characteristics for the three groups of detectors, measured for bias voltage of -100 to +100 V. In these plots the currents are normalized to the effective area of the pixel contact (i.e. to the geometrical area with boundaries Figure 3. I-V characteristic measured for two D1 detectors. The contact size is 8x8 mm; the gap between the contact and the guardring is 200 µm; the effective contact area used to normalize the current is 0.672 cm 2.Figure 4. I-V characteristic measured for three D2 detectors; the pixel size is 650x680 µm; the gaps between the contact and the guard ring (grid) in µm are: (1) 100, (2) 200, (3) 250. The effective contact areas used to normalize the current in cm2are (1) 0.00264, (2) 0.00171 , and (3) 0.00132.0 50 -50 -100 100 Bias voltage, V0123 -1 -2 -3Current, nA/cm 2 1 2 1: 3.3x10 W-cm10 2: 3.0x10 W-cm10D1 detectors (8x8 mm contact size)1 23D2 detectors (650x680 mm pixel size) 1: 100 mm gap 2: 200 mm gap 3: 250 mm gap 1: 2.7x10 W-cm10 3: 1.0x10 W-cm11 0 50 -50 -100 100 Bias voltage, V0123 -1 -2 -3Current, nA/cm 2in the middle of the gap between contact and grid). This approximation works only for small gaps. Fig. 3 shows the curvesmeasured for the two pixels of one of the D1 (large contact) detectors. The shape of the curves clearly indicates the existenceof Schottky barriers on the anode and cathode sides of the detectors. At low applied biases (<1 V) the I-V curves follow Ohm’s law, with the slopes corresponding to a specific resistivity of 2.9x10 10 and 2.2x1010 Ohm-cm for detectors D1 and D2 respectively. These are typical values for high-resistivity CZT material grown by eV-Products. As the voltage increases, thelinear slope starts to change. When the absolute voltage is between 1 and 50 V, the I-V relations again becomes close to alinear law, but with a slope several times smaller. We observed similar behavior for the D2 detectors (small pixel contacts). Fig. 4 shows a set of curves measured for several different size pixels. Only the positive branches of the I-V curves (cathode is positive biased) exhibit the described behavior. The negative branches seem to be affected by the surface conductance in the gap between the guard ring and thecontact, and show a slightly different behavior. Here the current reaches a local maximum at around -25 V and then decreasesand starts rising again (negative dynamic resistance). This asymmetry of the positive and negative branches indicates that theCZT crystal is a n-type. Indeed, when a positive bias is applied to the cathode, a depleted layer starts to expand from a pixel contact (for an n-type CZT) toward the cathode and along the surface into the gap between the contact and the guard ring (afringe effect). Effectively, this increases the area of the contact until the whole area along the surface becomes depleted. Th is happens at relatively low biases, for which the measured current is still bulk resistance limited. At positive biases on thecathode, the fringe effect does not show up in the I-V curves. However, when a negative bias is applied to the cathode, the depleted layer starts to grow from the cathode, reaching the anode side (pixel contacts first) when the bulk resistancebecomes negligible. At high absolute bias (>100 V), the negative and positive branches of I-V curves behave similarly. Because of the surface effects, we cannot estimate the specific resistivity of the CZT for the pixels with large gaps betweencontacts and grid. For example, the bulk resistivity evaluated for a 250 µm gap pixel was greater that 10 11 Ohm-cm (curve 3 in Fig. 4) which is obviously an unrealistic value. For several I-V curves, measured for pixels of both the D1 and D2 detectors, we extended the maximum applied bias up to +/-400 V. These measurements revealed that above 100-150 V, thelinear portion of the I-V curves is followed by an exponential rise. Figure 5 shows typical I-V characteristics measured for the D3 detectors. At first glance, these curves look completely different from those measured for the D1 and D2 detectors. The curves have linear dependencies, with only slight diode-likebehavior at low biases. Nevertheless, as we describe below, we can in fact use the same physical model for all detectorgroups. For comparison, Fig. 6 shows two representative curves measured for the D1 and D3 detectors. Because of the smallpixel size of the D3 detectors, the measured currents were smaller than those measured for the D1 and D2 detectors at thesame bias. This is the reason for the fluctuation seen at low bias for the D3 detectors. The I-V curve measured for D4 detectors are very similar to those measured for D3 and we will discuss them later in Figure 5. I-V characteristic measured for the D3 detectors; the contact size is 400x400 µm; the gap between the contact and the guard ring is 50 µm; the effective contact area used to normalize the current is 0.00012 cm2.Figure 6. Comparison between representative I-V curves measured for the D1 and D3 detectors.0 50 -50 -100 100 Bias voltage, V0246 -2 -4 -6Current, nA/cm 2 121: 4.5x10 W-cm10 2: 4.1x10 W-cm10D3 detectors (0.275x0.275 mm contact size) D1 D3 0 50 -50 -100 100 Bias voltage, V0246 -2 -4 -6Current, nA/cm 2Figure 7. The measured (squares) and calculated (solid lines) I-V characteristics of the D1 detector. The curve labeled D is calculated for ϑn=1 and C2=0 (no interfacial layer), while the curve labeled T is calculated for ϑn≠1 and C2=0 (no potential barrier lowering). The curve ISAT represents the saturation current of the ideal Schottky barrier in the termionic approximation. Figure 8. Same as Fig. 7 but plotted for the D2 detector.0.1 1.0 10 100 1000 0.01 10000 Bias voltage, V110100100010000 Current, pA/cm2Isat TD Detector D2 V =20 VRT J=0.0082 F =0.782 eV O C =0.00015 2 r=2.5x10 Ohm-cm10 Ne=3.8x10 cm 5 -3 Nd=5.0x10 cm 9 -3110100100010000 Current, pA/cm2 0.1 1.0 10 100 1000 0.01 10000 Bias voltage, VIsat TDDetector D1 V =19 VRT J=0.021 F =0.789 eV O C =0.000094 2 r=3.0x10 Ohm-cm10 Ne=2.0x10 cm 5 -3 Nd=4.2x10 cm 9 -3conjunction with temperature dependence of dark currents. We applied the ITD model described in the previous section to fit the measured curves. We found that we can reproduce all the measured I-V characteristics accurately. To illustrate the fitting procedures, we selected three representative I-V characteristics; a positive branch of the I-V curve measured for the D1 detector (large contact); a negative branch measured for the D2 detector (small contacts), and a positive branch measured for the D3 detectors. The experimental curves(squares) and the evaluated theoretical curves (solid lines) are shown in Figs. 7-9 on a log-log scale. Table 1 summarizes themagnitude of the parameters obtained from the least square fit, used to calculate the theoretical curves. As seen, theagreement between the ITD theory and the experimental data is very good. For the D1 curve the χ2 function has a very broad minimum, and practically any value of VRT between 18 and 70 V provides a satisfactory fit to the data. For the D2 curve the acceptable values of VRT range between 12 and 25 V, with χ2 reaching the minimum at 19.9 V. Finally, the I-V curve measured for the D3 detector gives 9.7 V for VRT. For all groups of the detectors, the corresponding values of ND were between 0.2 and 2.5 x1010 cm-3. As seen, the effective concentration of the ionized donors in the depleted volume is much higher than the concentration of the free carriers(electrons) inside the undepleted bulk. This is typical for the highly compensated material. The correlation between theparameters ρ, N, and ND is also evident, This is probably related to the total impurity concentration. We found nearly the same barrier heights at zero field for all tested contacts, Φ0=0.78-0.79 eV, but very different magnitudes of ϑn and C2. Taking 0.8282 eV for the position of the Fermi level inside the CZT bandgap [19], one can find Vbi~0.03 eV. As seen from Table 1, there is correlation between the parameters ϑn and C2. This can be attributed to the fact that the larger the thickness of the interfacial layer, the smaller the transmission coefficient ϑn, and the higher the voltage drop across the interfacial layer (∆VI=C2V). The ITD theory allows us to understand the factors determining the bulk leakage currents in the high resistivity CZT detectors. At low voltages, current is always limited by the specific bulk resistivity of CZT, typically 1-5x1010 Ohm-cm. In the case of the ideal Schottky barrier, the maximum possible current, IMAX, would be equal to the saturation current ISAT across Figure 9. Same as Fig. 7 but plotted for the D3 detector.110100100010000 Current, pA/cm2 0.1 1.0 10 100 1000 0.01 10000 Bias voltage, VIsat TD Detector D3 V =10 VRT J=0.17 F =0.790 eV O C =0.00007 2 r=4.2x10 Ohm-cm 10 Ne=1.5x10 cm 5 -3 Nd=2.5x10 cm 9 -3the barrier, ISAT=A*T2exp(-Φ0/VTH). (21) For comparison, Figs. 7-9 show an ideal Schottky barrier characteristic with the saturation current ISAT. If the interfacial layer exists between the contact and semiconductor, the current will be significantly reduced due to the factor ϑn at low biases, and will rise exponentially at very high bias ( ϑnVR/VD<<1) because of the barrier height lowering: I=ϑnISATexp(C 2V/V TH). (22) As an example, for the D1 and D2 detectors, the measured current already exceeds ISAT. at biases above >500V. The current I, given by Eq. (22), is obtained in the thermionic limit ( ϑnVR/VD<<1), i.e. when all electrons entering the semiconductor are rapidly swept by the electric field. However, if the electron drift velocity VD is not fast enough to efficiently remove electrons from the near contact area, the resulting current will be smaller. In the diffusion limited current case, i.e. when ϑnVR/VD>>1, and V>V RT, then: I=eN CµECexp(-Φ0/VTH), (23) where EC is the electric field strength at the contact, and NC is given by Eq. (11). As for the actual current it is hard to say a priori if it is thermionic or diffusion-limited. In the general case the current is determined by Eq. (8) from which the diffusion and thermionic limits can be derived, depending on the ratio ϑnVR/VD. Table 1 D1, Fig. 7 D2, Fig. 8 D3, Fig. 9 D4, Fig. 10 ρ, x1010 Ohm_cm 2.9 2.2 4.2 4.5 N, x105 cm-32.1 3.0 1.5 1.3 ND, x1010 cm-30.4-2.7 0.5 0.25 0.25 VRT, V 18-70 20 9.7 12 Φ0, eV 0.78-0.79 0.782 0.790 0.788 ϑn 0.02-0.04 0.0082 0.17 0.12 C2, x10-59.2-9.4 15.0 6.8 6.0 To illustrate the effect of the interfacial layer on the dark current, we calculated the theoretical I-V curves for two cases: 1) ϑn=1 and C2=0, i.e. no interfacial layer, and 2) ϑn<1 and C2=0, i.e. no potential barrier lowering. The magnitudes of the remaining parameters were taken from the least square fit of the experimental data. If no interfacial layer exists (first case) the calculated current (curves D in Figs. 7-9) would be diffusion-limited up to very high biases, such that the condition VR/VD>>1 is satisfied. In other words, the dark current in high resistivity CZT detectors is diffusion-limited if no interfacial layer exists. Eq. (23) can be rewritten as: I=eN SµEC, (24) where NS is the free electron concentration near the contact. On the other hand, in the diffusion approximation the surface concentration NS can be expressed as: NS=N Bexp(-V bi/VTH), (25) where NB is the free electron concentration in the undepleted bulk. Eq. (24) resembles the Ohmic-like dependence but with a much smaller specific resistivity due to a reduction factor exp(-V bi/VTH), e.g. for Vbi=0.05 V exp(-V bi/VTH)=0.15. Thus, in theapplied bias range from 1 to 100V, the measured I-V curve could be misinterpreted as following Ohm’s law, and, as was first pointed out in Ref. [4], a significant overestimate of the bulk resistivity would be obtained. If no potential barrier lowering is assumed, i.e. C2=0, the calculated I-V curves, labeled T in Figs. 7-9, would correspond precisely to the termionic-limited current for the detectors D1 and D2, and still be diffusion-limited for D3. Asdiscussed previously, this is why, the I-V curves for the detectors from groups D1 and D2 are very different from those measured for D3. It appears that the D1 and D2 detectors have an interfacial layer which makes the condition ϑnVR/VD <<1 exist even at low bias. In contrast, we assume that the D3 detectors have a much thinner layer, with ϑn~1, and, as a result the current is diffusion-limited up to high bias. It is interesting to compare the I-V curves measured for D2 (thick interfacial layer) and D3 (thin interfacial layer). Below 1 V the current measured for D3 is approximately 2 times smaller than D2 because of the difference in bulkresistivities: 2.2x10 10 and 4.2x1010 Ohm-cm. On the contrary, around 200 V, the current measured for D2 becomes 3-4 times smaller, because of the transmission factor ϑn, than that measured for D3. At even higher biases, the exponential rise, due to the barrier lowering, dominates, and at some point the D2 current exceeds the D3 current again, as seen in Figs. 7-9. It isclear that for any operating voltage there should be an optimal thickness of the interfacial layer which provides the minimalleakage current. However, the most efficient way to reduce the leakage current is, of course, to use contacts with large barrie r heights. Figure 10 shows the I-V characteristics measured for a randomly selected D1 detector at different detector temperatures. We found that the least squares fit for each curve yields similar results within the fitting errors for allparameters of the Schottky barrier. The solid lines represents the theoretical curves calculated after substituting averagedvalues for the fitting parameters. The temperature dependence of the dark current in the range between 20 to 70 C is shown inFig. 11 for two cathode biases: 20 and 100 V. The solid line depicts the theoretical curves calculated by using the parametersfound from the previous fit shown in Fig. 10. Figure 10. The measured (squares) and calculated (solid lines) I-V characteristics of a D4 detector at six detector temperatures. The same set of free parameters was used to calculate the theoretical curves for each temperature.0.1 1.0 10 100 1000 0.01 Bias voltage, VDetector D4 V =12 VRT J=0.12 F =0.788 eV O C =0.00006 2 Nd=2.5x10 cm 9 -3Current, nA/cm2 0.0010.01 0.1110100 Temperature: 1: 314 K 2: 309 K 3: 305 K 4: 303 K 5: 297 K 6: 293 K12 34 565. Conclusions We have demonstrated that the bulk I-V characteristics measured for the CZT pixel detectors withPt contacts can be explained by applying a combinedinterfacial layer-thermionic-diffusion theory to a back-to-back Schottky barrier system. By fitting the measuredcurves over a 5 decade range we obtain consistentparameters for the Schottky barrier as well as for the CZTmaterial. For example, we found the potential barrier of thePt contact to be 0.78-0.79 eV. It appears that the interfacial layer, likely formed during the detector fabrication process, can significantlyaffect the I-V characteristics of CZT detectors with blocking contacts (Pt contacts in this case). The detectorleakage current is limited by the material bulk resistivity atlow bias (<1V). At high applied voltages, the current isdetermined by the potential barrier height, transmissioncoefficient through the interfacial layer, and by the barrierheight lowering effect due to the voltage drop across theinterfacial layer. If the effect of the interfacial layer issmall, the leakage current is diffusion-limited up to veryhigh bias, and can resemble ohmic behavior, with effectivebulk resistivity much higher than 5x10 10Ohm-cm. Acknowledgments This work was supported by NASA under grant No. NAG5-5289. The authors wish to thank K. Parhnam and C. Szeles from eV-Products, Inc. for fruitful discussions.6. References [1] F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, C. M. Hubert Chen, W. R. Cook, S. M. Schindler, Proc. of SPIE , vol. 4141 (2000) 137-143. [2] Cs.Szeles and M.C. Driver, Proc. of SPIE , vol. 3446 (1998) 1-8. [3] S. M. Sze, D.J. Coleman, and A. Loya, Solid-State Electronics, 14 (1971) 1209.[4] G. Cisneros and P. Mark, Solid-State Electronics, 18 (1975) 563-568.[5] A. E. Bolotnikov, S. E. Boggs, C. M. Hubert Chen, W. R. Cook, F. A. Harrison, and S. M. Schindler, Proc. of SPIE , vol. 4141 (2000) 243-251. [6] C.R. Crowell and S. M. Sze, Solid-State Electronics, 9 (1966) 1035.[7] M. Yousaf, D.Sands, C.G. Scott, Solid-State Electronics, 44 (2000) 923-927.[8] M. K. Hudait, S. B. Krupanidhi, Solid-State Electronics, 44 (2000) 1089-1097.[9] P. Cova, A. Singh, A. Media and R.A. Masut, Solid-State Electronics, 42 (1998) 477-485.[10] A. E. Rakhshani, Y. Makdisi, X. Mathew, and N. R. Mathews, Phys. Stat. Sol. A 168 (1998) 177-187.Figure 11. The temperature dependence of the dark current measured (squares) and calculated (solid lines) for 20 and 100 V biases on the cathode.20 40 60 80 0 Temperature, CV =12 VRT J=0.12 F =0.788 eV O C =0.00006 2 Nd=2.5x10 cm 9 -3Relative change of current, I(t)/I(t=26C) 0.1110100 Detector D4 Bias: - 20 V - 100 V[11] A. Turut, M. Saglam, H. Efeoglu, N. Yalcin, M. Yildirim, B. Abay, Physica B 205 (1995) 41-50. [12] O. Wada, A. Majerfeld, P.N. Robson, Solid-State Electronics, 25 (1982) 381-387. [13] G.W. Wright, R.B. James, D. Chinn, B.A. Brunett, R.W. Olsen, J.Van Scyoc III, M. Clift, A. Burger, K. Chattopadhyay, D. Shi, R. Wingfield, Proc. of SPIE , vol. 4141 (2000) 324-332. [14] Ching-Yuan Wu, J. Appl. Phys. 51 (1980) 3786-3789; J. Appl. Phys. 53 (1982) 5947-5950.[15] S. M. Sze, “Physics of Semiconductors Devices”, 1981.[16] S. S. Cohen, G. S. Gildenblat, Metal-Semiconductor Contacts and Devices, VLSI Electronics, Vol. 13, 1986.[17] H. Venghaus, P. J. Dean, P. E. Simmonda, and J. C. Pfister, Z. Phys. B30 (1978) 125. [18] I. M. Dharmadasa, C. J. Blomfield, C. G. Scott, R. Coratger, F. Ajustron and J. Beauvillain, Solid-State Electronics, 42, (1998) 595-604. [19] H. Y oon , M. S . Goo rs ky , B. A . Brun ett, J. M. Van S cy oc, J. C . Lu n d, an d R . B. Jam es , J. Electron ic Materials , 28, (1999), 838-842.
591 "... we want more than just a formula. First we have an observation, then we have numbers that we measure, thenwe have a law which summarizes all the numbers. Butthe real glory of science is that we can find a way of thinking such that the law is evident ." "The Feynman lectures on physics", Addison-Wesley, MA, 1966, p.26-3. THE GHOSTLY SOLUTION OF THE QUANTUM PARADOXES AND ITS EXPERIMENTAL VERIFICATION* Raoul Nakhmanson Frankfurt am Main, Germany† This conference is entitled "Frontiers of fundamental physics". What does this mean? Is it the frontiers of today's physical knowledge, or is it the frontiers of physics itself as a science? In my paper I shall try to show that today it is the same: the frontiers of contemporary physical knowledge coincide with the conceptual frontiers of physics as a science regardingthe behaviour of so-called inanimate matter and even cross over to invade into the kingdomof ghost. Such a point of view permits a very natural interpretation of quantum phenomena,and suggests essentially new experiments in which information plays the principal rôle. The microworld has surprised the "classical" physicists with the following paradoxes: 1,2 1) Before quantum mechanics (QM) was created: quantization of mass, charge, energy, angular momentum; the identity of particles of the same type; wave-particle duality. 2) In QM: statistical predictions, Heisenberg's uncertainty principle, Pauli's exclusion principle. 3) In standard (Copenhagen) interpretation of QM: rejection of the classical realism, a ban on speaking about non-measured parameters, trajectories, etc.; Bohr's complementarity prin-ciple, collapse of the wave function. The Copenhagen interpretation is only a translation of the mathematical formalism of QM to the ordinary language but not an interpretation in a common sense, because it does notexplain how, why, and in which frames this formalism works. Feynman told his students thatthe quantum world was not like anything that we know; and although everybody knows QM,many people use it, some of them develop it, but nobody understands it. In discussions about QM the "Gedankenexperimente" play an important rôle. We will discuss three of them which were really performed: 1) Delayed-choice experiment. 3 In one arm of an interferometer a Pockels cell is placed which closes the path of photons at the short moment when they can pass the cell. In accor-dance with old local-realistic concept each photon flies only in one arm of the interferometer.If it is the arm with the cell the photon will be absorbed and nothing will be registered. If it isanother arm, the short work of the cell placed far away does not act on the photon and thesame interference as without the cell must be registered. But no interference was found inaccordance with QM. * Shortened version of a report which was read on September 30, 1993 in Olympia, Greece. † Present address: Waldschmidtstrasse 131, 60314 Frankfurt, Germany. Frontiers of Fundamental Physics , Edited by M. Barone and F. Selleri, Plenum Press, New York, 19945922) Aharonov-Bohm effect. 4 In accordance with QM the frequency of wave-function oscillation depends on the energy. If the particle has different energies in different arms ofthe interferometer, it leads to an additional phase shift and changes the interference pattern.The experiments were performed with an electron interferometer and a magnetic vectorpotential and justified the predictions of QM. It is of interest that in the experiments theelectrons did not cross the magnetic field. From the old classical point of view it looks likenon-local action at a distance. 3) Einstein-Podolsky-Rosen (EPR) experiment. It was suggested in 5 and modernized by Bohm. 6 Here two particles emitted simultaneously have common non-factorisable wave function and are measured after parting by a large distance. There is some correlationbetween the results measured. Bell has shown 7 that any local realistic theory (i.e. theory with hidden parameters and restricted velocity of interaction) estimates the uppermost limit of suchcorrelation, and this limit is smaller than predicted by QM. The experiments beingperformed 2 are in accordance with QM, and today's dominant opinion is that local realism has been disproved and one must refuse either reality lying beyond the measurements (likeCopenhagen) or locality. Later I will show that this conclusion as well as Bell's theorem itselfdo not have the generality being ascribed to them. The EPR-scheme raises a question about separability. "Common sense" prompts that after some time and distance the "magic" correlation between particles must disappear, i.e. thefactorisation of the wave function must take place. But how? The analogous question isconnected with measuring procedure itself: If interaction between particle and apparatusallows several output results, the QM forecasted end state is a superposition of these results.But in practice the result of each measurement is a pure state, and the result of the series is astatistical mixture. It seems as if QM does not describe the whole measurement process. 8 There are some explanations of the EPR paradox. From the Copenhagen point of view it is so as it is. Speaking about some hidden parameters of particles, e.g. directions of spins,before the measurement, has no sense, and Bell's theorem and experiments justify this. Non-local theories with hidden parameters . 9 Here an instantaneous action at a distance is provided by instantaneous collapse of the wave function in all space. The critics emphasizethat these theories only rewrite the Schrödinger's equation in a more complex form, giving thesame results and nothing new. Action of future on the past . 10 If such action is possible, the future conditions of measu- rement can act on the hidden parameters of particles at the moment of their departure to tunethem for correct correlation. Up to now there is no complete theory ready to defy critique.But common sense prompts that such a world can not be stable. Fatalism . This possibility was noted particularly by Bell. 11 In the spirit of Laplace it is possible to think that everything is pre-determined, particularly our choice of position ofanalyser. Here we are confronted with the old problem of "free will". If free will exists man(and not only he) can control the choice of alternatives taking into account physical and socialconditions. The following chain of syllogisms supports the existence of free will: → Useful changes are selected and consolidated by evolution. + During evolution the volume of the human brain increases. = The volume of brain is a useful quantity. + Intelligence depends on the brain volume; as a rule, the greater the volume, the higher the intelligence. = Intelligence is useful. + Intelligence can develop itself only if it can choose among several alternatives; only in such situations can intelligence be useful. = Free choice, i.e. free will exists. One can reply that the increase of the brain volume as well as evolution itself are included in the fatalistic scenario. But if one considers the existence of free will ad hoc as an axiom, then, in accordance with these syllogisms, free will gives intelligence a chance to evolve.593The roots of free will do not lie in the macroworld which is ruled by deterministic laws. They lie in the microworld, and quantum uncertainty points to it. Human intelligence is notthe only product of free will. It is possible that earler, the free will created some intelligenceat the level of its roots, i.e. in microworld. Because the time (measured not in seconds but inevents) flowed there much faster, this intelligence had a longer evolution period. Perhaps thegolden age of it is over, and now we have to do it only with a "rudimentar" intelligence (socalled by Cochran 12). The additional pointers on intelligent matter are the Einstein's formula E = mc2 , the informational character of the wave function ψψψψ, the principle of the least action, and quantum-mechanical stochastics. 13 The development of quantum physics was a step across the boundary between matter and ghost drawn by Descartes. Physicists felt it and spoke about the free will of electrons andghost (spirit, consciousness, intelligence) in matter. Similar meanings were expressed byCharles Galton Darwin, Eddington, Heisenberg, Schrödinger, Pauli, Jordan, Margenau,Wigner, Charon, Cochran, and others. Feynman said that it looks as if a computer is in eachpoint of space. Cambrige University Press has published a book touching this theme 11 con- taining interviews with Bell, Bohm, Wheeler, Peierls, Aspect, and others. Some interesting analogies between microworld and people have been noticed. Niels Bohr saw the manifestation of his complementarity principle in human thinking. Margenau wroteabout Pauli's exclusion principle: 14 "Prior to that time, all theories had affected the individual nature of so-called 'parts'; the new principle regulated their social behaviour... The particles, though initially assumed to be free, are seen to avoideach other... In a crude manner of speaking, each particle wants to be alone; each runs away then it'smells' the other, and its sense of smell is keener the more nearly its velocity equals to the other's." This was said about Fermi-particles. Such behaviour is typical for scientists: each of them tries to find his own theme. Sometimes people's behaviour is like a Bose-particle. Pheno-mena such as fashion in dress or music, and applause or coughing in concert halls, areexamples of Bose-condensation. The same man can manifest himself as a Bose- or Fermi-person. For particles this was only possibe in "big bang" time. Are we now at the same stageof evolution? The next example concerns the EPR-experiment. Let us suppose there are twins, Ralf and Rolf, both of whom live in Frankfurt and work for Lufthansa as pilots. They fly all over theworld but mainly to England and Greece. For Lufthansa (not for their families!) they areindistinguishable "particles". The twins always try to dress alike, they believe that this bringsthem happiness. Because they are often in different countries, they agree on an order ofsartorial priority: cold before warmth and rain before dry spell. Figure 1. Einstein-Podolsky-Rosen (EPR) experiment and the apparent non-local interaction.594God, who is observing the twins, sees as a rule the striking correlation: the twins dress alike! For example, Ralf and Rolf arrive in England and Greece, respectively. If it is cold inEngland, not only Ralf but also Rolf wears the overcoat in spite of the warm weather; if it israining in Greece, not only Rolf but also Ralf hides beneath an umbrella, regardless ofwhether it is raining or not (Fig.1); etc.. "What is the matter?" - thinks God, - "I estimateexperimental conditions, namely, weather in England and Greece and the twins' financialstatus, telephoning is too expensive for them. It seems there is a non-local interactionbetween the twins. I am sure it is a new escapade of the devil!" God's conclusion was only half true. In his heavenly chariot he fell behind the technical progress of the 20th century. He was right suspecting the devil. But up to now the devildoes not realize non-local interaction. Instead, he has invented television, power computer formeteorology, and communication satellite. Because of it the twins watch TV at everyevening for a good tomorrow world weather forecast . Although our behaviour occurs in real space-time, the strategy of it is not there. It is in our consciousness, which controls our behaviour, taking into account physical and social lawsand circumstances. To develop a strategy we use our knowledge only about the past, andpropagate it on the future. The thoroughness of the forecast depends on the information takeninto account and the power of the intelligence. But let us come back to physics. Unfortunately the idea about intelligent matter is not developed up to now. They, who spoke about ghost in matter, did not go beyond such astatement and did not suggest any hypotheses and schemes which could be tested experimen-tally. From another side physicists using QM do not see the necessity of such an idea andfollow the principle thought as old as Aristoteles but named after William of Ockham"Ockham's Razor": "entities should not be multiplied beyond necessity". Niels Bohr said, that there are trivial and deep statements. To be asked "What is a deep statement?" he answered: "It is such a statement, that an opposite statement is also a deepone." If one accepts the Ockham's principle as a deep statement then, according to Bohr, "entities should not be canceled beyond necessity" must also be accepted as a deep statement. Besides, the practical necessity is not the only ormain criterion of theory. A consistent development of the idea of intelligent matter naturally interprets quantum paradoxes as well as QM itself within the limits of local realism, and suggests essentially newexperiments with microparticles and atoms in which information plays the principal rôle. In the new conception the wave function ψψψψ is a strategy-function. It reflects an optimal behaviour of particles. It is not in the real 3-dimensional space. It is in imaginary configura-tion space, which, in its turn, is in the imagination (consciousness) of the particle. When theparticle receives new information (it takes place by any interaction with micro- or macro-objects), it can change its strategy. Thus occurs the collapse of the wave function. It occursnot in the real (infinite) space, but in the consciousness of particles. The consequent time isdetermined by the rapidity of this consciousness. Therefore, compared with space-timeconditions of experiment, collapse is local and instantaneous. Von Neumann and Wigner suggested that human consciousness has influence on the collapse of ψψψψ- function. It is not so: in the human consciousness only the human knowledge about the ψψψψ- function collapses. The laws of both collapses lie beyond physics. The wave-particle duality is a mind-body one. In the space there exists only the particle; the wave exists in its consciousness, as well as the reflection of the whole world. If there are many particles, their distribution in accordance with the ψψψψ- function looks as a real wave in real space.595Particles are artifical things. Division into different sorts or species with internal identities is typical for mass products. It simplifies production, usage, repairs, and replacement of suchobjects. Technics, plants and animals illustrate it very well. In the last two cases theproduction is ruled at the genetic level. For example, people have a very narrow statisticaldistribution of sizes and masses; the world records in sport differ from the middle results notmore than twice. The identity of particles of one sort in QM is analogous to the identity ofvehicles of one sort with respect to traffic rules. The individual differences lies beyond QM. Because of free will the behaviour of particles is not strictly determined. In situations allowing alternative outputs the theory gives only a distribution of priorities. Taking this intoaccount the particle makes its choice. The optimal tactics of proportional proving of allpossibilities by an ensemble of disconnected particles is randomization of this choice. To doit the particles must have the generators of random signals. If some theory and random generator (RG) are used to choose the alternative, it looks like a complete algorithm. Well, but where is the free will now? Is it only to change the RG? The answer is, that purpose and means create a dependence. Really free is he who has no purpose and no desire including the desire of freedom. Therefore there is a danger that in the"Konsumgesellschaft" we transform ourselves into some kind of automata. Perhaps themicroworld did not avoid it. But the new turn of development can be connected with achange of purpose or new information. Besides, Gödel's theorem prompts, that the space ofcorrect statements can be manifold. In such a case to reach a new fold one must make a"quantum jump". "Do not sin against logic, one reaches nothing new", - said Einstein. Pauli's principle and Bose- and Fermi-particles were discussed above. These types of social behaviour are optimal for searching (fermions) and power action (bosons). In the lastcase some macroscopical effects can be observed (in superconductors, superfluids, lasers). With respect to Heisenberg's uncertainty principle: In the new conception it reflects not the reality but QM as a theory of measurement. In reality the particle has definite coordinates,impulse, trajectory, etc.. But during an interaction with the measurement apparatus it has apossibility to choose the next state. It solves this problem using its intelligence (reflected in the ψψψψ- function), random generator, and freedom (e.g. reflected in the choice of RG). Neither QM nor any other theory predicts a particular result: it would be a refusal of free will. In spite of this, the dream of Einstein and other realists, to know the values of all parame- ters included in a theory can become true. Particles remember what happened and tell it toothers. To do this, they must have synchronised clocks, measure rules, and reference pointsfor space and time. In this sense it is possible to speak about absolute coordinates and time,like Greenwich's ones. If we can communicate with particles 13,15 , they can say everything about their parameters and forecast their and our future. The new concept includes the previous realistic ones: empty waves 2 and parallel worlds 16 exist, but not in the real world: as virtual possibilities they exist in the consciousness of a particle. Not the real 10 but a forecasted future acts on the past . The above mentioned danger of total algorithmisation looks like a stochastic fatalism . The new explanation of delayed-choice and EPR experiments, and suggestions how to have "non-QM-results", were done in 13. The essence is, that particles are well informed about the world and its development. The Aharonov-Bohm effect has the same explanation.Besides, this effect emphasizes a priority of potential against field (in classical physics theyenjoy equal rights). From the new point of view it is natural, because potential just contribu-tes to the action function whose minimum as a function of trajectory is wanted. It should beobserved that the idea of forecasting the conditions on this trajectory is also included in the least action principle. The change from integral form to a differential one does not solve theproblem: the notion of derivative is connected with two points, and if we are in one of them,we know only the past conditions in the second point and must extend this into the present. The proof of Bell's theorem is based on the next assertion: if a particle 1 is measured in the point A having a condition (e.g. angle of analyser) α , and P a is a probability of result a ,596then a condition β existing in a distant point B , there is a measured particle 2 , has no influ- ence on the Pa , and vice versa. Here Bell and others saw the indispensable requirement of local realism. Mathematically it can be written as Pab(λ1i,λ2i,α,β) = Pa(λ1i,α)×Pb(λ2i,β) , (Bell) (1) where Pab is the probability of the join result ab , and λ1i and λ2i are hidden parameters of particles 1 and 2 in an arbitrary local-realistic theory. Under the influence of Bell's theorem and the following experiments some "realists" reject locality. In this case an instan-taneous action at a distance is possible, and one can write P ab(λ1i,λ2i,α,β) = Pa(λ1i,α,β)×Pb(λ2i,β,α) . (non-locality) (2) In principle such a relation permits a description of any correlation between a and b , particularly predicted by QM and observed in experiments. But in the frame of local realismthe condition (1) is not indispensable. Instead, one can write P ab(λ1i,λ2i,α,β) = Pa(λ1i,α,β´)×Pb(λ2i,β,α´) , (forecast) (3) where α´ and β´ are the conditions of measurements in points A and B , respectively, as they can be forecast by particles at the moment of their parting. If the forecast is good enough, i.e. α´ ≈ α and β´ ≈ β , then (3) practically coincides with (2) and has all its advan- tages plus locality. On the issue of separability: The EPR-particles have a common strategy. It can continue as long as they can forecast the future. But particles can also have so intensive interactions(e.g. with detectors) that initial strategy is not important anymore. In both cases the con-sciousness of the particle has an ability to cut off and forget the old partnership. QM is "microsociology". Like its humane sister, it makes only probabilistic forecasts. The transition to classical physics is the transition from sociology of persons to sociology ofcrowds: the level of freedom decreases and behaviour becomes deterministic. Feynman'sstatement "quantum world is not like anything that we know" is right only if we do not takeinto account living beings. If a baby, having more experience with his parents than with"inanimate" matter, could make experiments, the behavior of microparticles would appear toit to be very natural. REFERENCES 1. M.Jammer. The Conceptual Development of Quantum Mechanics , McGraw-Hill, New York (1966). 2. F.Selleri. Quantum Paradoxes and Physical Reality, Kluwer, Dordrecht (1990); Wave-Particle Duality , F.Selleri, ed., Plenum Press, New York (1992). 3. J.Baldzuhn, E.Mohler, and W.Martienssen. Z. Phys.-Cond. Matt . 77B, 347 (1989) and Refs. cited there. 4. Y.Aharonov, D.Bohm. Phys.Rev. 115, 485 (1959). 5. A.Einstein, B.Podolsky, and N.Rosen. Phys.Rev. 47, 777 (1935). 6. D.Bohm. Quantum Theory , Prentice Hall, New York (1951). 7. J.S.Bell. Physics 1, 195 (1965). 8. E.Wigner, in The Scientist Speculates , I.J.Good, ed., London (1962); G.Ludwig, in Werner Heisenberg und die Physik unserer Zeit , Vieweg, Braunschweig (1961). 9. L. de Broglie. J.Phys.Radium 8, 225 (1928); D.Bohm. Phys.Rev . 85, 166 (1952). 10. O.Costa de Beauregard. Nuovo Cimento B42, 41 (1977). 11. The Ghost in the Atom . P.C.W.Davies and J.R.Brown, eds., Cambridge University Press (1986). 12. A.A.Cochran. Found. Phys . 1, 235 (1971). 13. R.Nakhmanson, in Waves and Particles in Light and Matter , A.van der Merwe and A.Garuccio, eds., Plenum Press, New York (1994), p. 571. 14. H.Margenau. The Nature of Physical Reality. McGray-Hill, New York (1936). 15. R.Nakhmanson. Preprint 38 -79, Institute of Semiconductor Physics, Novosibirsk (1980); see also Ref.13 and A.Berezin and R.Nakhmanson. J. of Physics Essays 3, 331 (1990). 16. H.Everett III. Rev.Mod.Phys. 29 , 454 (1957); see also B.S. DeWitt. Phys.Today 9, 30 (1970).
arXiv:physics/0103007v1 [physics.atom-ph] 2 Mar 2001Classical calculation of high-order harmonic generation o f atomic and molecular gases in intense laser ﬁelds Chaohong Lee1∗,Yiwu Duan2,Wing-Ki Liu3, Jian-Min Yuan4, Lei Shi1,Xiwen Zhu1and Kelin Gao1 1Laboratory of Magnetic Resonance and Atomic and Molecular P hysics,Wuhan Institute of Physics and Mathematics,The Chinese Academy of Sciences,W uhan,430071, P.R.China. 2Department of Physics,Hunan Normal University,Changsha, 410081,P.R.China. 3Department of Physics,University of Waterloo,Waterloo,O ntario,N2L3G1,Canada. 4Department of Physics and Atmospheric Science,Drexel Univ ersity,Philadelphia, PA19104,USA. (February 2, 2008) Abstract Based upon our previous works ( Eur.Phys.J.D 6, 319(1999); C hin.Phys.Lett. 18, 236(2001)), we develop a classical approach to calculat e the high-order harmonic generation of the laser driven atoms and molecules . The Coulomb singularities in the system have been removed by a regulariz ation procedure. Action-angle variables have been used to generate the initi al microcanonical distribution which satisﬁes the inversion symmetry of the s ystem. The nu- merical simulation show, within a proper laser intensity, a harmonic plateau with only odd harmonics appears. At higher intensities, the spectra become noisier because of the existence of chaos. With further incr ease in laser in- tensity, ionization takes place, and the high-order harmon ics disappear. Thus chaos introduces noise in the spectra, and ionization suppr esses the harmonic ∗E-mail address: Chlee@wipm.whcnc.ac.cn. 1generation, with the onset of the ionization follows the ons et of chaos. PACS numbers: 42.65.Ky, 32.80.Wr, 32.80.Rm. Typeset using REVT EX 2I. INTRODUCTION The development of the high-power femtosecond laser has sti mulated the investigation of the multi-photon processes of atoms and molecules interact ing with intense laser ﬁelds [1-9]. Recently, there are many theoretical and experimental refe rences about these multiphoton process. Within a proper intensity region, lots of odd harmo nics of laser are generated by atomic and molecular gases [1-5,14]. The harmonic structur e distributes as a plateau which is cut oﬀ at a special high-order harmonic. When the intensity i ncreases, the ionization channel is opened and the high-order harmonics disappear. The above threshold ionization (ATI) occurs when the laser intensity is suﬃcient power (above 1013W/cm2). Various structures (plateau, angular distribution, etc.) in ATI spectra have b een detailed in Ref.[6-8]. The above threshold dissociation (ATD) and dissociation-ioni zation of molecular systems are also reported [9-10]. Studying the laser-matter interacti on deeply, not only can obtain new knowledge of the interacting mechanism, but also can provid e widely application in the generation of high-order coherent harmonics, X-rays laser and γ-rays laser. The classical dynamics of most laser-driven systems is gene rally chaotic, due to the ex- istence of nonlinearity. Chaos usually manifests itself as some control parameters (initial energy, laser intensity, laser frequency, etc.) are varied . The microscopic systems, in partic- ular those involving atoms and molecules, are governed by a H amiltonian. To study these systems are of great importance in the context of quantum-cl assical correspondence. At the investigated high power laser intensity (1013˜1015W/cm2), the electric ﬁeld of the laser is equal in strength to the Coulomb ﬁeld of the nuclei [8], and th e laser ﬁeld can not be looked as a perturbation to the ﬁeld-free system, for the states the mselves are no longer indepen- dent of the laser. Generally, the microscopic systems (atom s or molecules) are intrinsically quantum mechanical systems, thus they must be described by q uantum mechanics. How- ever, due to the presence of intense laser, the exact calcula tion even numerical simulation based upon quantum mechanics tend to be very diﬃcult to perfo rm. Fortunately, classical approach to these similar problems is useful for providing p hysical insight into dynamics 3processes [10-13]. Classical chaos associating with the mi crowave ionization of atomic hy- drogen reveals that the detailed mechanism of atomic ioniza tion in terms of transport in phase space [11-12]. Classical prediction for scaled frequ encies has been veriﬁed by quan- tum calculations [12-13], and, in turn, has conﬁrmed the dyn amical signiﬁcance of classical chaos. The harmonic generation (HG) of laser driven hydroge n atoms is simulated with classical Monte-Carlo method [14], the numerical results a re qualitatively consistent with the quantum mechanic results and the experimental observat ion. For the molecular systems, the classical calculation is a good ﬁrst step, since length a nd energy scales are often large enough for classical mechanics to be at least approximately valid. Using softened potential model, the classical dynamics of the one-dimensional hydro gen molecular ion H+ 2interacting with an intense laser pulse are detailed [15-16]. There are two important and realistic examples of laser driv en systems, where nonlinear dynamics has played a major role, which have been traditiona lly studied. One is the laser driven hydrogen atom that is the fundamental system in atomi c physics, the other is the laser driven hydrogen molecular ion H+ 2, which is the fundamental diatomic molecular system in molecular physics. Bellow, we shall show our classical calc ulation of the high-order harmonic generation of these two fundamental systems. The outline of this paper is as follows. The regularized model and the corresponding initial microcano nical distribution are presented in the next section. In section III, we show the numerical resul ts in details. A brieﬂy summary and discussion is given out in the last section. II. MODEL AND INITIAL MICROCANONICAL DISTRIBUTION A. Regularized model of laser driven hydrogen atom In this article, we consider a classical hydrogen atom, with an inﬁnite mass nucleus ﬁxed at the origin of coordinates, interacting with a high intens e laser ﬁeld which is linearly po- larized along the z−axis, and with the electric ﬁeld component ε(t). Thus the Hamiltonian 4in atomic units in Cartesian coordinates is H=H0+Hi, H0=1 2p2−1 r, Hi=−zε(t). (1) where, r=√x2+y2+z2andp2=p2 x+p2 y+p2 z. Apparently, the above Hamiltonian has a Coulomb singularity corresponding to electron-nucleus co llision. To remedy the singularity, we introduce the parabolic coordinates ( u, v, φ ) x=uvcosφ, y=uvsinφ, z= (v2−u2)/2. (2) and a new ﬁctive time scale τ dt/dτ =G(u, v) =v2+u2. (3) In order to implement regularization, following the notati on of Szebenhely [17], regarding the motion time tand the negative total energy −Eas generalized coordinate and generalized momentum respectively, then the Hamiltonian function in ex tended space becomes into H∗=H−E≡0 =H∗(x, y, z, t ;px, py, pz,−E) (4) For the sake of obtaining the regularized Hamiltonian, we in troduce the third category generating function F3(px, py, pz,−E;u, v, φ, t ), then obtain x(u, v, φ ) =−∂F3 ∂px, y(u, v, φ ) =−∂F3 ∂py, z(u, v, φ ) =−∂F3 ∂pz, t(u, v, φ ) =−∂F3 ∂E. (5) So the third category generating function F3(px, py, pz,−E;u, v, φ, t ) is in the form of F3=−xpx−ypy−zpz+Et, =−uvcosφpx−uvsinφpy−1 2(v2−u2)pz+Et. (6) The new momenta ( pu, pv, pφ,−E) are related with old momenta ( px, py, pz,−E) by pu=−∂F3 ∂u=vcosφpx+vsinφpy−upz, pv=−∂F3 ∂v=ucosφpx+usinφpy−vpz, pφ=−∂F3 ∂φ=−uvsinφpx+uvcosφpy, E=∂F3 ∂t=E. (7) 5Then the regularized Hamiltonian can be expressed as K=dt dτH∗=G(u, v)(H−E)≡0, =1 2[p2 u+p2 v+ (u−2+v−2)p2 φ]−2−E(u2+v2)−1 2(v4−u4)ε(t). (8) If we deﬁne Ku=1 2[p2 u+u−2p2 φ]−1−Eu2+1 2u4ε(t), Kv=1 2[p2 v+v−2p2 φ]−1−Ev2−1 2v4ε(t). (9) In the ﬁeld-free case, ε(t) = 0, Ku(=−Kv) is the component of the Laplace-Runge-Lenz vector along z−axis, i.e.,Ku=−Kv=Rz. The equations of motion can be derived from the above regularized Hamiltonian, as the following: dφ dτ=∂K ∂pφ=pφ(u−2+v−2),dpφ dτ=−∂K ∂φ= 0, (10) du dτ=∂K ∂pu=pu,dpu dτ=−∂K ∂u= 2Eu+p2 φu−3−2u3ε(t), (11) dv dτ=∂K ∂pv=pv,dpv dτ=−∂K ∂v= 2Ev+p2 φv−3+ 2v3ε(t), (12) and dE dτ=∂K ∂t=1 2(u4−v4)dε(t) dt,dt dτ=−∂K ∂E=G(u, v). (13) Apparently, pφis a constant which corresponding to the component of angula r momentum along z−axis. B. Regularized model of laser driven hydrogen molecular ion H+ 2 Within Born-Oppenheimer approximation, we can assume that two protons are ﬁxed at the positions AandB, with a distance Raway from each other, in the locations (0 ,0,−R/2) and (0 ,0, R/2) of the Cartesian coordinates system. A single electron at (x, y, z ) is subject 6to the Coulomb attraction of both protons and the interactio n of the laser. Let rAbe its distance from A, and rBbe its distance from B. Within the dipole approximation, the Hamiltonian in atomic units is H=H0+Hi, H0=1 2p2−1/rA−1/rB, Hi=−zε(t). (14) where, ε(t) is the electric ﬁeld of the laser pulse, Hiis the interacting Hamiltonian. Ap- parently, the above Hamiltonian has singularities at point srA= 0 and rB= 0, which corresponds to electron-proton collision. To overcome thi s barrier, regularization has to be performed. Deﬁne the new coordinates ( u, v, φ ) as x=−R 2sinusinhvcosφ, y=−R 2sinusinhvsinφ, z=R 2cosucoshv. (15) and introduce the new ﬁctive time scale τsatisfying dt/dτ =G(u, v) =rArB=R2(cosh2v−cos2u)/4. (16) Similar to the previous subsection, regarding the motion ti metand the negative total energy −Eas generalized coordinate and generalized momentum respec tively, then we can write the Hamiltonian function in extended space as H∗=H−E≡0 =H∗(x, y, z, t ;px, py, pz,−E) (17) Introducing the third category generating function F3(px, py, pz,−E;u, v, φ, t ), which satisﬁes F3=−R 2sinusinhvcosφpx−R 2sinusinhvsinφpy−R 2cosucoshvpz+Et. (18) Thus momenta ( pu, pv, pφ,−E) are related with old momenta ( px, py, pz,−E) by pu=−∂F3 ∂u=R 2cosusinhvcosφpx+R 2cosusinhvsinφpy−R 2sinusinhvpz, pv=−∂F3 ∂v=R 2sinucoshvcosφpx+R 2sinucoshvsinφpy−R 2sinucoshvpz, pφ=−∂F3 ∂φ=−R 2sinusinhvsinφpx+R 2sinusinhvcosφpy, E=∂F3 ∂t=E. (19) 7And the regularized Hamiltonian is K=dt dτH∗=G(u, v)(H−E)≡0, =1 2[p2 u+p2 v+ (csc2u+ csch2v)p2 φ]−Rcoshv−(E−Hi) 4R2(cosh2v−cos2u).(20) where, Hi=−zε(t) = (−R/2) cos ucoshvε(t). Obviously, pφis a constant, which corre- sponds to the component of the angular momentum which along t hez−axis. Ifpφis equal to zero, the electron is constrained in a plane which can be ch osen as y= 0, corresponding to a two-dimensional motion. In the ﬁeld-free case, ε(t) = 0, this two-dimensional model corresponds to the classical hydrogen molecular ion in grou nd-state. Equations of the mo- tion for the ground-state hydrogen molecular ion interacti ng with laser ﬁeld can be easily obtained from the regularized Hamiltonian, i.e., du dτ=∂K ∂pu=pu, dpu dτ=−∂K ∂u=ER2 4sin(2u) +ε(t)(z∂G ∂u+G∂z ∂u), (21) dv dτ=∂K ∂pv=pv, dpv dτ=−∂K ∂v=Rsinhv+ER2 4sinh(2 v) +ε(t)(z∂G ∂v+G∂z ∂v), (22) and dE dτ=∂K ∂t=zGdε(t) dt,dt dτ=−∂K ∂E=G(u, v). (23) In the ﬁeld-free case, deﬁning Ku=1 2(p2 u+p2 φcsc2u) +ER2 4cos2u, Kv=1 2(p2 v+p2 φcsch2v)−Rcoshv−ER2 4cosh2v. (24) thusKu(=−Kv) is a constant of the ﬁeld-free motion, they are related to γand Ω by Ku=−Kv=−γR2/4 =ER2/4 + Ω/2. (25) 8Constants Ω and γare ﬁrst introduced by Erickson [18] and Strand [19] respect ively, which have the following forms γ=−E−2Ω/(mR2), Ω =− →LA·− →LB+emR2(cosθA−cosθB). (26) Here,− →LAand− →LBare the angular momentum vector of the motion around nucleus Aand Brespectively, θAandθBare the angle from the vector− →rAand− →rBto positive z−axis respectively, mandeare the mass and the charge of the electron respectively. C. Action-angle variables and initial distributions The action-angle variables for separated systems are deﬁne d as Ii=1 2π/contintegraldisplay pidqi, θi=∂ ∂Ii/integraldisplay pidqi=/integraldisplay∂pi ∂H∂H ∂Iidqi. (27) The motion of the classical ﬁeld-free hydrogen atom is perio dic, then it need only a pair of conjugated action-angle variables [20] H0=E0=−1 2I2. (28) The corresponding angle is given by the Kepler’s equation θ=u−esinu. (29) This means θis the mean anomaly of the free orbits. The eccentric anomaly ucan be obtained from r=a(1−ecosu). Here, ris the distance from the electron to the origin, ais the instantaneous semimajor axis which satisﬁes a=−1/(2E0),eis the eccentricity of the free orbits satisfying e=√2E0L2+ 1,Lis the total angular momentum. The motion of the ground-state hydrogen molecular ion H+ 2is quasi-periodic, it need two pairs of action-angle variables. With elliptic integrals, action Iucan be expressed as follows. 9Iu=  √ 8Ku−2E0R2 πF1(π 2,/radicalBig 1 +4Ku E0R2−4Ku), for K u>0, /radicalBig −E0R2/π, for K u= 0,√ −2E0R2 π[F1(π 2,/radicalBig 1−4Ku E0R2)−4Ku E0R2F2(π 2,/radicalBig 1−4Ku E0R2)], for K u<0.(30) where, the ﬁrst category elliptic integral F1(ϕ, k) and the second category elliptic integral F2(ϕ, k) are in the form of F1(ϕ, k) =ϕ/integraldisplay 0/radicalBig 1−k2sin2xdx, F2(ϕ, k) =ϕ/integraldisplay 01//radicalBig 1−k2sin2xdx. Generally, a single trajectory lacks the inversion symmetr y of the real physical sys- tems. So the spectrum obtained from a single trajectory exhi bits unphysical even har- monics. A natural way to remedy the unphysical even harmonic s is to consider an en- semble of trajectories, evolving from an initial microcano nical distribution with inver- sion symmetry. For a chaotic system, the initial distributi on can be generated with Monte-Carlo method. However, for an integrable system, the distribution generated by Monte-Carlo method does not possess of ergodicity. We ﬁnd th at the points on the same equienergy surface generated by action-angle variabl es with regular steps possess of good ergodicity. To reconstruct the inversion symmetry, there must exist pairs of (x0, y0, z0;px0, py0, pz0) and ( −x0,−y0,−z0;−px0,−py0,−pz0) in the initial distribution. For the regularized model for hydrogen atom, it corresponds to p airs of ( u0, v0, φ0;pu0, pv0, pφ0) and (u0, v0, φ0+π;pu0, pv0, pφ0). And for the regularized hydrogen molecular ion H+ 2, it corresponds to pairs of ( u0, v0, φ0;pu0, pv0, pφ0) and ( u0+π, v0, φ0;pu0, pv0, pφ0). III. NUMERICAL SIMULATION The numerical computational procedure is based upon the cla ssical trajectory Monte Carlo (CTMC) method [20-21]. CTMC simulation procedure inv olves three stages, (i) choice of initial conditions, (ii) numerical integration of equat ion of motion, and (iii) categorization 10of each trajectory as excitation, charge transfer or ioniza tion. In the process of numerical integrating, numerical accuracy and computing time are two primary aspects that must be considered. We use the fourth-order Runge-Kutta method wit h variable steps to perform the numerical calculation. Note also that computer can not d eal with singularity, which corresponds to electron-nucleus collision, to overcome th is diﬃculty, we have implemented regularization. With complete regularization, numerical simulation can be established with required precision before, at, and, after collision succes sfully. To obtain the harmonic spectra, our procedure also calculat e the averaged dipole moment of the excited trajectories with pairs of inversion symmetr ic initial conditions in the same distribution. Having determined the actual trajectories o f the electron, one can easily obtain the component of the averaged dipole moment, which along the laser polarization direction. Then the harmonic spectra of the driven dipole is straightfo rwardly obtained from it’s power spectra D(ω) = lim t→+∞1 t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0/angbracketleftµ(τ)/angbracketrighteiωτdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (31) where, /angbracketleftµ(τ)/angbracketrightis the component of the averaged dipole moment along the lase r polarization direction. In our calculation, the electric ﬁelds of the ult rashort laser pulses are chosen as ε(t) =  EMsin2(ωLt/40) sin( ωLt), for 0≤t≤40π/ω L, 0, otherwise.(32) The maximum electric ﬁeld strength EMis related to the laser intensity I=/radicalBig ε0/µ0E2 M/2, andωLis the angular frequency, and period TLis equal to 2 π/ω L. The time-dependent dipole moment µz(t) of a single trajectory along the laser polariza- tion direction sensitively depends on the laser intensity. Fig.1 shows the time evolution of the dipole of the hydrogen molecular ion with initial energy E0=−1.1034hatree , inter- nuclear distance R= 2.00bohr, laser wavelength λ= 600 nmand diﬀerent laser intensity. And some dipoles of the hydrogen atom and their power spectra are presented in Fig.2, with initial energy E0=−0.5hatree , laser wavelength λ= 532 nmand diﬀerent laser 11intensity. With the increasing of the laser intensity, the o scillations of the dipole moments are modulated gradually, and it follows a regular pattern wi th both high and low frequency components. Such patterns are independent of initial condi tions. When the laser intensity is large enough, ionization takes place, which strongly modiﬁ es the subsequent time evolution of the dipole moment. The power spectra obtained from a single trajectory of the la ser driven hydrogen molecu- lar ion, which with initial energy E0=−1.1034hatree , internuclear distance R= 2.00bohr, laser wavelength λ= 600 nmand diﬀerent laser intensity, are presented in Fig.3. The ﬁr st one corresponds to the ﬁeld-free case. For the hydrogen atom , it possesses of only regularly decreasing peaks locating at multiples (harmonics) of Kepl er’s frequency, that is, ωn=nω0, forn= 1,2,3· ··, which manifests that the free motion is periodic. While for the hydrogen molecular ion, the free motion is quasi-periodic, it appear s regularly decreasing peaks locat- ing at two diﬀerent characteristic frequencies and their co mbinations, i.e., at nω01+mω02, forn= 0,1,2,3· ··,m= 0,1,2,3· ··andn+m >0. For the laser driven systems, the peaks of harmonic spectra depend strongly on the laser intensity. In addition to the origin peaks, a dominant line at the laser frequency ωLappears, which is the Rayleigh component in the light scattered by the atoms and molecular ions. High-order harmonics, which consist of both odd and even components, do appear in the spectra, and th eir orders and strengths increase with the increasing of the laser intensity. For a lo w laser intensity, i.e., in the perturbative regime, the characteristic peaks dominate th e power spectrum. For a proper laser intensity, the plateau structure containing both odd and even harmonics appears. For a larger laser intensity, because of the presence of chaos, t he spectra become noisier even the motion is bound. For a strong enough laser intensity, the motion becomes unbound and the corresponding spectrum is dominated by a very noise back ground, which is generated by the ionized electrons, the only line surviving being the o ne at the laser frequency, which corresponds to the light scattered by the asymptotically fr ee electron Thomson scattering. To eliminate the unphysical even harmonics, averaging an en semble of trajectories evolv- ing from an inversion symmetric distribution is necessary. The time evolution of the averaged 12dipole, which evolve from a microcanonical ensemble of 5000 trajectories, are presented in Fig.4, the ﬁrst one corresponds to the hydrogen atom with ini tial energy E0=−0.5hatree , laser wavelength λ= 532 nmand laser intensity I= 5.0×1014W/cm2, the second one corre- sponds to the hydrogen molecular ion with initial energy E0=−1.1034hatree , internuclear distance R= 2.00bohr, laser wavelength λ= 600 nmand laser intensity I= 1.0×1014 W/cm2. Comparing with the evolution of the dipole of a single traje ctory, one can easily ﬁnd that the oscillation of the averaged dipole is smooth and it globally follows the laser oscillation. The spectra obtained from an ensemble of trajectories are sh owed in Fig.5 and Fig.6. Fig.5 is the harmonic spectra of the ground-state hydrogen a toms interacting with the laser pulses with λ= 532 nmand diﬀerent laser intensity. Fig.6 is harmonic spectra of t he hydrogen molecular ion with initial energy E0=−1.1034hatree , internuclear distance R= 2.00bohr, laser wavelength λ= 600 nmand diﬀerent laser intensity. As a consequence of averaging process, the unphysical even harmonics are rem edied really. Within a proper laser intensity range, the plateau structure that only poss esses of odd harmonics appears. As pointed out previously, at a higher intensity, the spectra b ecome noisier even the ionization does not happen because of the eﬀects of chaos. This indicate s that the chaos cause the noise of the harmonic spectra. When the laser intensity is hi gh enough, the ionization takes place, thus the noise background conceals the high-order ha rmonics. This means that the onset of ionization follows the onset of chaos and the ioniza tion suppresses the harmonic generation. IV. SUMMARY AND DISCUSSION In summary, within the Born-Oppenheimer approximation and using the classical trajec- tory method,we have calculated the high-order harmonic gen eration spectra of the hydrogen atom and the hydrogen molecular ion interacting with ultras hort intense laser pulses. The other multi-photon phenomena, such as multi-photon ioniza tion and above threshold disso- 13ciation, can also be simulated. In our subsequent calculati ons, the dynamics of the electron is investigated by numerical integrating the equations of m otion using regularized coordi- nates. To eliminate the unphysical even harmonics of a singl e trajectory, averaging over an ensemble of trajectories evolving from an initial microcan onical distribution with inversion symmetry is necessary. Such distribution is constructed us ing action-angle variables. A plateau structure in the spectra with only odd harmonics is o bserved within a proper laser intensity range of about 1014W/cm2. From our numerical results, we observe that the high order harmonics are cut oﬀ at a special order harmonic. At hig her laser intensities, chaos introduces noise into the spectra even though the motion is s till bound. Finally as the intensity is further increased, ionization takes place, an d the harmonics disappear. These results are qualitatively consistent with recent qua ntum calculations [16] and ex- perimental observations [1-5], but the cutoﬀ order Nmof the plateau structure is not precisely consistent with formula Nm= (Ip+3.17Up)/¯hωL[27], here, Ipis the ionization potential and Up=e2E2 M/4meω2 Ldenotes the quiver energy or the ponderomotive energy of an e lectron. As an example, when the laser intensity I= 1014W/cm2and wavelength λ= 600 nm, the ionization potential Ipof the ground-state hydrogen molecular ion is 1 .1034hatree (29.77eV= 14.50 ¯hωL), the quiver energy Up= 3.36eV(= 1.63 ¯hωL), then Nm= 19, and when I= 7.5×1013W/cm2,Nm= 18; however, from our simulation, the harmonic plateau are both cut oﬀ at 17, and for very large intensity (above 1015W/cm2) the high-order har- monics are concealed by the noises. To obtain quantitative r esults, we have to integrate the time-dependent Schr..odinger equation, this can be realized with the split-operat or method [22]. For the hydrogen molecular ion, within BOA, a proper in ternuclear distance Rwill enhance the high-order harmonic generation[16], and it wil l be interesting to go beyond the Born-Oppenheimer approximation to investigate what furth er interesting insights can be obtained when the nuclear motion is taken into account [23-2 4]. In our model, we only consider the non-relativistic case wit h dipole approximation. When the laser is suﬃciently intense, photoelectrons of relativ istic energies can be produced, ne- cessitating a fully relativistic treatment [25]. The dipol e approximation is no longer valid, 14and the magnetic ﬁeld is not only present, but acquires an imp ortance similar to that of the electric ﬁeld. And even before the appearance of the relativ istic photoelectrons, the eﬀects of the magnetic ﬁeld may be very important too [26]. Due to the wiggly motion and the acceleration of the electron near the outermost turning poi nts induced by the magnetic ﬁeld, the cut-oﬀ order of the harmonic plateau maybe higher. Acknowledgment The work is supported by the National Natural Science Founda tions of China under Grant No. 19874019, 19904013 and 1990414. The author Chaoho ng Lee thanks very much for the help of Dr. Haoseng Zeng, Dr. Ming He and Dr. Zongxiu Ni e. 15FIGURES FIG. 1. Temporal variation of the dipole moment of a single el ectronic trajectory of the hy- drogen molecular ion for diﬀerent laser parameters with ini tial energy E0=−1.1034hatree and internuclear distance R= 2.00bohr. FIG. 2. The dipole and their power spectra of a single electro nic trajectory of the hydrogen atom for diﬀerent laser parameters with initial energy E0=−0.5hatree . FIG. 3. Power spectra of the dipole of a single electronic tra jectory of the hydrogen molecular ion for diﬀerent laser parameters with initial energy E0=−1.1034hatree and internuclear distance R= 2.00bohr. FIG. 4. Temporal variation of the averaged dipole moment for diﬀerent laser parameters. This ﬁrst one corresponds to the hydrogen atom with initial energ yE0=−0.5hatree , the other one corresponds to the hydrogen molecular ion with initial e nergy E0=−1.1034hatree and internuclear distance R= 2.00bohr. FIG. 5. Harmonic spectra of the ground-state hydrogen atoms interacting with diﬀerent laser pulses, the last two are magniﬁcations of the ﬁrst one. FIG. 6. Harmonic spectra of the ground-state hydrogen molec ular ion interacting with diﬀerent laser pulses, with initial energy E0=−1.1034hatree and nuclear internuclear distance R= 2.00 bohr. 16REFERENCES [1] Y. Liang, S. Augst, S. L. Chin, Y. Beaudion, and M. Chakert , J. Phys. B 27, 5119(1994). [2] T. E. Glover, et al., Phys. Rev. Lett. 74, 2468(1996). [3] C. Kan, et al., Phys. Rev. Lett. 79, 2971(1997). [4] M. Belini, et al., Phys. Rev. Lett. 81, 297(1998). [5] H. R. Lange, et al., Phys. Rev. Lett. 81, 1611(1998). [6] P. B. Corkum, Phys. Rev. Lett. 71, 1994(1993). [7] M. J. Nandor, M. A. Walker, and D. Van. Woerkom, J. Phys. B 3 1, 461(1998). [8] J. H. Eberly and Q. Su, J. Opt. Soc. Am. B 6, 1289(1989). [9] S. Chelkovski, et al., Phys. Rev. A 52, 2977(1995); Phys. Rev. A 54, 3235(1996). [10] J. Heagy and J. M. Yuan, Phys. Rev. A 41, 571(1990); J. Hea gy, Zi-Min Lu, J. M. Yuan, and M. Vallieres, Dynamics of Driven Molecular System s, in ”Quantum non- integrability: Direction in Chaos”, Vol. 4, edited by D. H. F eng and J. M. Yuan, World Scientiﬁc, Singapore, (1992). [11] M. Gajda, et al., Phys. Rev. A 46, 1638(1992). [12] J. Leopold and I. Pecival, J. Phys. B 12, 709(1979). [13] Yan Gu, Quantum Chaos, Shanghai Scientiﬁc and Technolo gical Education Publishing House, Shanghai, (1996). [14] G. Bandarage, A. Maquet, and J. Cooper, Phys. Rev. A 41, 1 744(1990); G. Bandarage and J. Cooper, Phys. Rev. A 46, 380(1992). [15] Weixing Qu, Suxing Hu, and Zhizhan Xu, Phys. Rev. A 57, 22 19(1998); Phys. Rev. A 57, 4528(1998). 17[16] Hengtai Yu, Tao Zuo, And Andre D. Bandrauk, Phys. Rev. A 5 4, 3290(1996). [17] V. Szebehely, Theory of Orbits, Academic Press, New Yor k, (1967). [18] H. A. Erickson and E. L. Hill, Phys. Rev. 75, 29(1949). [19] M. P. Strand and W. P. Reinhardt, J. Chem. Phys. 70, 3812( 1979). [20] J. E. Howard, Phys. Rev. A 46, 364(1992). [21] G. Bandarage and R. Parson, Phys. Rev. A 41, 5878(1990). [22] M. Nurhuda and F. H. M. Faisal, Phys. Rev. A 60, 3125(1999 ). [23] Chaohong Lee, Yiwu Duan, Wing-Ki Liu, and Jian-Min Yuan , Chin. Phys. Lett. 18, 236(2001). [24] Yiwu Duan, Wing-Ki Liu, and Jian-Min Yuan, Phys. Rev. A 6 1, 053403(2000); Eur. Phys. J. D 6, 319(1999). [25] H. R. Resis, Phys. Rev. A 63, 013409(2000). [26] J. R. Vazquez de Aldana and Luis Roso, Phys. Rev. A, 04340 3(2000). [27] J. L. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev . A 45, 4998(1992); Phys. Rev. Lett. 68, 3535(1992). 182 4 6 8-2.10-1.75-1.40-1.05-0.70-0.35 I=0 W/cm2,λ=600 nmDipole Amplitude Time (TL)2 4 6 8-2.10-1.75-1.40-1.05-0.70-0.35 I=1.01014 W/cm2,λ=600 nmDipole Amplitude Time (TL) 2 4 6 8-3.0-1.50.01.53.0 I=4.01014 W/cm2,λ=600 nmDipole Amplitude Time (TL)2 4 6 8-400-300-200-1000100 I=1.01015 W/cm2,λ=600 nmDipole Amplitude Time (TL) Fig.1 4 6 8 10 12-0.4-0.20.00.20.4µz Time(TL)0 10 20 30 40 501E-51E-41E-30.010.11 I=0.0 W/cm2, λ=532 nm Harmonic OrderPower 4 6 8 10 12-0.8-0.40.00.40.8µz Time(TL)0 10 20 30 40 501E-51E-41E-30.010.11 I=1013 W/cm2, λ=532 nm Harmonic OrderPower 4 6 8 10 12-1012 Powerµz Time(TL)0 10 20 30 40 501E-51E-41E-30.010.11 I=3.01014 W/cm2, λ=532 nm Harmonic Order Fig.20 4 8 12 16 20 24 281E-81E-61E-40.011 I=0 W/cm2, λ=600 nm Harmonic OrderPower 0 4 8 12 16 20 24 281E-81E-61E-40.011 I=1.01013 W/cm2, λ=600 nm Harmonic OrderPower 0 4 8 12 16 20 24 281E-81E-61E-40.011 I=5.01013 W/cm2, λ=600 nm Harmonic OrderPower 0 4 8 12 16 20 24 281E-81E-61E-40.011 I=1.01014 W/cm2, λ=600 nm Harmonic OrderPower 0 4 8 12 16 20 24 281E-81E-61E-40.011 I=4.01014 W/cm2, λ=600 nm Harmonic OrderPower 0 4 8 12 16 20 24 280.010.1110100 I=2.51015 W/cm2, λ=600 nm Harmonic OrderPower Fig.3 4 6 8 10 12-0.6-0.30.00.30.6 I=5.01014 W/cm2, λ=532 nmAveraged Dipole <µz> Time (TL) 2 4 6 8-0.10-0.050.000.050.10 I=1014 W/cm2, λ=600 nmAveraged Dipole <µz> Time (TL) Fig.4 0 10 20 30 40 50 6010-1310-1110-910-710-5I=1014 W/cm2, λ=532 nm Harmonic OrderHarmonic intensity 0 10 20 30 40 50 6010-1310-1110-910-710-5I=2.51014 W/cm2, λ=532 nm Harmonic OrderHarmonic intensity 0 10 20 30 40 50 6010-1310-1110-910-710-5I=5.01014 W/cm2, λ=532 nm Harmonic OrderHarmonic intensity 0 10 20 30 40 50 6010-1310-1110-910-710-510-3 I=7.51014 W/cm2, λ=532 nm Harmonic OrderHarmonic intensity 0 5 10 15 20 25 3010-1310-1110-910-710-5I=1014 W/cm2, λ=532 nm Harmonic OrderHarmonic intensity 30 35 40 45 50 55 6010-1310-1110-910-710-5I=1014 W/cm2, λ=532 nm Harmonic OrderHarmonic intensity Fig.5 0 4 8 12162024281E-121E-101E-81E-61E-4 I=7.51013 W/cm2, λ=600 nm Harmonic Order (ωL)Harmonic Intensity 0 4 8 12162024281E-101E-81E-61E-4I=1.0*1014 W/cm2, λ=600 nm Harmonic Order (ωL)Harmonic Intensity Fig.6
arXiv:physics/0103008v1 [physics.gen-ph] 2 Mar 2001Rest mass or inertial mass? R. I. Khrapko1 ABSTRACT Rest mass takes place of inertial mass in modern physics text books. It seems to be wrong. This topic has been considered by the author in the article [1, 2, 3 ]. Additional arguments to a conﬁrmation of such a thesis are presented here. “Einstein’s theory of the universe, based on the principle that all motion is relative, and showing that mass varies with its velocity, while space-time is a fourth dimension.” A. S. Hornby et al. [4] The end of 20-th century was marked by a great mish-mash of deﬁ nitions of mass. 1. REST MASS All was clear in the beginning of the century when the theory o f relativity was not yet created. Mass, m, denoted something like amount of substance or quantity of m atter. And at the same time mass was the quantitative measure of inertia of a body. Inertia of a body determines momentum Pof the body at given velocity vof the body, i. e. it is a proportionality factor in the formula P=mv. (1) The factor mis referred to as inertial mass. But mass as a measure of inertia of a body can be deﬁned also by t he formula F=ma: (2) By this formula, the more is mass, the less is the acceleratio n of a body at given force. Masses m deﬁned by the formulae (1) and (2) are equal because the formu la (2) is a consequence of the formula (1) if mass does not depend on time and speed. Thus, “mass is the quantitative or numerical measure of body’s ine rtia, that is of its resistance to being accelerated” [5]. The same value of mass can be measured by weighing a body, that is by measuring of the attraction to the Earth or to any other given body (which mass is designated M). Thus, the same massmappears in the Newton gravitational law F=γMm r2, (3) but here mis referred to as gravitational (passive) mass. This fact ex presses an equivalence of inertial and gravitational masses. Due to this equivalence, the acce leration due to gravity does not depend on the nature and the mass of a body: g=γM r2. (4) Thus, 1Moscow Aviation Institute, 4 Volokolamskoe Shosse, 125871 , Moscow, Russia. E-mail: tahir@k804.mainet.msk.su Subject: Khrapko 1“mass is the quantity of matter in a body. Mass may also be cons idered as the equivalent of inertia, or the resistance oﬀered by a body to change of mot ion (i. e. acceleration). Masses are compared by weighting them.” [6]. 2. INERTIAL MASS AT HIGH SPEEDS However, the special theory of relativity has shown that no b ody can be accelerated up to the speed of light because the acceleration of a body decreases t o zero when the speed of the body approaches the speed of light, however large the accelerati ng force is. This implies that inertia of a body increases to inﬁnity when the speed of the body tends to t he speed of light, though the “amount of substance” of the body obviously remains constant. More correctly, special relativity has shown that the momen tumPof a body at any speed is parallel to velocity v. Therefore the formula P=mvis valid at large speeds, if the coeﬃcient m, that is inertial mass, is accepted to be increased with speed in the fashion: m=m0/radicalBig 1−v2/c2, (5) where c is the speed of light. That is, the expression P=m0v/radicalBig 1−v2/c2, (6) is valid for the momentum of a body. In these formulae m0is the value of mass which was spoken about in the beginning. F or a determination of the value, the body should be slowed down an d, after it, the formula (1) or (2) must be applied at small speed. The value received by this met hod is called rest mass . This mass, by deﬁnition, does not vary on accelerating a body. Therefor e, the formulae (1), (2), (3) must be written as follow: P=m0v,F=m0a,F=γMm 0/r2. However, for small speeds, due to formula (5), inertial mass is equal to rest mass, m=m0, and consequently the record (1), (2), (3 ) is correct in the “before special relativity section”. To emphasize the fact that inertial mass mdepends on speed it is named relativistic mass : it appears to have diﬀerent values from points of view of variou s observers if the observers have relative velocities. Meanwhile, there is a preferred value of inerti al mass m0. This value is observed by an observer which has no velocity relative to the body. Such a pr operty of inertial mass is similar to the property of time: observers which are in motion relative to a clock measure longer time intervals then the time interval measured by an observer relative to wh om the clock is at rest. This time interval is called the proper time. Thus, “mass is the physical measure of the principal inertial prop erty of a body, i. e., its resistance to change of motion. At speeds small compared wit h the speed of light, the mass of a body is independent of its speed. At higher speeds, t he mass of a body depends on its speed relative to the observer according to the relati on: m=m0/radicalBig 1−v2/c2, where m0is the mass of the body by an observer at rest with respect to th e body, vis the speed of the body relative to the observer who ﬁnds its mas s to be m.” [7]. 2If you wish to check up the formula (6), you should measure vel ocityvand momentum Pof a body. The momentum of a body is measured by the following oper ation. A moving body is braked by a barrier, and during its braking the force F(t) acting on the barrier is measured. The initial momentum of the body, by deﬁnition, is equal to the integral P=/integraldisplay F(t)dt. (7) It is postulated that this integral does not depend on detail s of braking, that is on a form of function F(t). We should notice that the formulas (5) and (6) remain valid fo r object which has no rest mass, m0= 0, for example, for photon or neutrino (if one assumes that r est mass of neutrino is equal to zero). Such objects have inertial mass and momentum, but the y should move with the speed of light. It is impossible to stop them: they disappear if being stoppe d. Nevertheless, despite their speed is constant, their inertial mass appear to be diﬀerent for vari ous observers. However, in the case of such objects, no preferred value of inertial mass exists. Or , it is possible to say, the preferred value of inertial mass is equal to zero. We have detected the increase of inertia of a body at large spe ed by a reduction of its acceleration at large speed. Thus we have referred to formula (2). And it is allowable. However, just because of the increase of inertial mass with the body velocity, the for mula (2) can change its form. The point is that at ﬁxed acceleration, a force directed in parallel wi th the velocity should supply not only the increase of speed of available mass m=m0/radicalBig 1−v2/c2. (5) It should also supply an increase of mass: F=d dtP=d dt m0v/radicalBig 1−v2/c2 =m0a/radicalBig (1−v2/c2)3. (8) The coeﬃcientm0/radicalBig (1−v2/c2)3 is called “longitudinal mass” [8]. If the force is perpendicular to the velocity and so does not c hange speed and inertial mass of a body, the formula F=madoes not change its form: F=m0a/radicalBig 1−v2/c2. (9) Using this circumstance, R.Feynman put forward a simple ope rational deﬁnition of inertial mass m. “We may measure mass, for example, by swinging an object in a circle at a certain speed and measuring how much force we need to keep it in the circle.” [9] . When the force has an arbitrary direction, the proportional ity factor in formula (2) must be considered as a certain operator (tensor) which transforms vector ato vector F:F= ˆma. The operator ˆ mdepends on speed and a direction of the velocity of a body and, generally speaking, changes a direction of a vector. It is easy to accept. You see, velocity vis a property of a body, but 3a force Facting at the body is an external agent with respect to the bod y. It is clear that a result of the inﬂuence of the force, that is an acceleration of a body , can depend on a correlation between directions of the vectors Fandv. 3. GRAVITATIONAL MASS AT HIGHER SPEEDS At the same time the general theory of relativity has shown th at not only inertia of a body, but also its weight increases with speed by the law (5): P=γMm 0 r2·/radicalBig 1−v2/c2. For example, the formula (8) for a body falling downwards wit h speed vtake, roughly speaking, the form:γMm 0 r2·/radicalBig 1−v2/c2=m0g r2·/radicalBig (1−v2/c2)3. I. e. the acceleration due to gravity is g=γM(1−v2/c2) r2. So that inertial mass satisﬁes the principle of equivalence at any speed vof a body. The exact formula for acceleration can be received within th e framework of the general theory of relativity as is shown in Sec. 8: g=γM(1−v2/c2) r·/radicalBig r(r−rg), r g= 2γM/c2. (10) This formula is a relativistic generalization of the formul a (4). 4. ENERGY Furthermore, special relativity has shown that an incremen t of inertial mass, m−m0, multiplied on square of the speed of light is equal to kinetic energy of a b ody: (m−m0)c2=Ek. (11) “A result of the theory that mass can be ascribed to kinetic en ergy is that the eﬀective mass of the electron should vary with its velocities accordi ng to the expression m=m0/radicalBig 1−v2/c2. This has been conﬁrmed experimentally.” [6]. Therefore if we attach a rest energy E0=m0c2to a body at rest, the complete energy E=E0+Ek of a body appears to be proportional to inertial mass: E=mc2. (12) This famous Einstein formula proclaims an equivalence betw een inertial mass and energy. The two, up to now, diﬀerent concepts are incorporated in a singl e one. Thus, 4“the formula E=mc2equates a quantity of mass mto a quantity of energy E. The rela- tionship was developed from the relativity theory (special ), but has been experimentally conﬁrmed” [7]. We should notice that the formula (12), as well as formulae (5 ) and (6), are valid for an object which has no rest mass and rest energy, m0= 0. If you wish to check up the formula (11) and simultaneously to make sure that special relativity is valid, you must measure the inertial mass and the rest mass of a moving body as it was explained above and, besides this, you must measure kinetic energy of t he body: Ek=/integraldisplay F(l)dl. HereF(l) is the force acting on the barrier during the body braking an dF(l)dlis a scalar product of the force Fand an inﬁnitesimal vector dlof displacement of the barrier. (See [10]). The formula (11) connects inertial mass, rest mass and kinet ic energy. Using formula (6), it is easy to connect inertial mass, rest mass and momentum: m2 0=m2−P2/c2. (13) For zero rest mass particles we receive: mc=P,orE=Pc. 5. SYSTEM of BODIES If several bodies are considered to be a system of bodies, the n, as is known, their momenta and their inertial masses are summed up. For two bodies this take the form: P=P1+P2, m =m1+m2. (14) In other words, momentum and inertial mass are additive. The case of the rest mass is entirely diﬀerent. Equations (13 ), (14) imply that rest mass of a pair of bodies with rest masses m01,m02is equal not to the sum m01+m02but to a complex expression dependent on the momenta P1,P2: m0=/radicalBigg/parenleftbigg/radicalBig m2 01+P2 1/c2+/radicalBig m2 02+P2 2/c2/parenrightbigg2 −(P1+P2)2/c2. (15) Thus, rest mass is, generally speaking, not additive. For ex ample, a pair of photons each having no rest mass does have a rest mass if the photons move in diﬀere nt directions while the pair has no rest mass if the photons move in the same direction. Nevertheless, the three quantities, P,m,m0, satisfy the conservation law. That is, they remain constant with time for a closed system. However, it seems to be unsuitable to consider rest mass of a s ystem of bodies because of the nonadditivity of rest mass. It is meaningful to speak only ab out a sum of rest masses of separate bodies of system. So, when one speaks that “rest mass of ﬁnal s ystem increases in an inelastic encounter” [11], the rest mass after the encounter is compar ed with the sum of rest masses of 5bodies before the encounter, but not with the system rest mas s which is conserved thanks to the nonadditivity. Just so, when one speaks about the mass defec t at nuclear reactions, for example, at synthesis of deuterium, p+n=D+γ, the sum of the rest masses of proton and neutron is compared with the sum of the rest masses of deuterium and γ-quantum, but not with the system rest mass determined by the formula (15). 6. A COMPARISON of MASSES And here a problem arises. Which of the two masses, the rest ma ss or the inertial mass, must we name by a simple word mass, designate by the letter mwithout indexes, and recognize as a ”main” mass? It is not a terminological problem. A serious psycholo gic underlying reason is present here. To decide which of the masses is the main mass let us repeat onc e again properties of both masses. Rest mass is a constant quantity for a given body and denotes “ amount of substance of a body”. It corresponds to a rudimentary Newton belief that the masse s stayed constant. But, rest mass is not equivalent to the energy of a moving object, is not equi valent to gravitational mass, rest mass is nonadditive and is not used as a characteristic of a sy stem of bodies or particles. This last circumstance prevents the conservation law displaying. Pa rticles moving with the speed of light have no rest mass. The operational deﬁnition of rest mass of a part icle assumes its deceleration up to a small speed without use of an information about current cond ition of the particle. Inertial mass is relativistic mass. Its value depends on obs erver’s velocity. Inertial mass is equivalent to energy and to gravitational mass, Inertial ma ss is additive, it satisﬁes the conservation law. The operational deﬁnition of inertial mass is based on t he simple formula bf P = mv. From our point of view, inertial mass has to be called mass and to be designate m, as it is done in the present article. 7. UNDERLYING PSYCHOLOGIC REASON Unfortunately, plenty of physicists considers the rest mas s as a main mass, designates it by m, instead of m0, and discriminates the inertial mass. These physicists agr ee, for example, that the mass of gas which is at rest increases with temperature since its energy increases with temperature. But, probably, there is a psychologic barrier that prevents them from explaining this increase by an increase of masses of molecules owing to the increase of thei r thermal speed. These physics sacriﬁce the concept of mass as a measure of ine rtia, sacriﬁce the additivity of mass and the equivalence of mass and energy to a label attache d to a particle with information about “amount of substance” because the label corresponds t he customary Newton belief in invariable mass. And so they think that a radiation which, according to E instein [12], “transfers inertia between emitting and absorbing bodies” has no mass. Now inertial mass is excluded from textbooks and from popula r science literature [11, 13, 14], but this phenomenon is hidden by the fact that rest mass adher ents busily call rest mass mass, not rest mass , and the word massis associated with a measure of inertia. The main psychologic diﬃculty is to identify mass and energy (which varies), to accept these two essences as one. It is easy to accept the formula E0=m0c2for a body at rest. But it is more diﬃcult to accept a validity of the formula E=mc2for any speed. The famous Einstein relation between mass and energy, that is a symbol of 20-th century, seems “ugl y” to L. B. Okun’ [15]. Rest mass adherents are not, probably, capable to accept an i dea of relativistic mass the same as early opponents of special relativity could not accept th e relativity of time. The lifetime of an unstable particle varies with velocity as its inertial mass : τ=τ0//radicalBig 1−v2/c2. 6It is appropriate to quote here from Max Planck: “Great new scientiﬁc idea is seldom inculcated on opponents by means of a gradual persuasion. Saul seldom becomes Paul. As a matter of fact, op ponents gradually die out, and the new generation is accustomed to the new idea from the v ery beginning.” [16]. Unfortunately, the great idea of relativistic mass is caref ully isolated from youth. Now the article [1, 2, 3] is rejected by editors of the following journals: “Russ ian Physics Journal”, “Kvant” (Moscow), “American Journal of Physics”, “Physics Education” (Brist ol). “Physics Today”. 8. SCHWARZSCHILD SPACE Here we will arrive at the formula (10) considering Schwarzs child space-time [17] with the interval: ds2=r−rg rc2dt2−r r−rgdr2−r2(dθ2+ sin2θdϕ2) We get the equations of radial geodesic lines from the formul ae using the connection coeﬃcients Γi jk: d2t ds2+rg r(r−rg)·dr ds·dt ds= 0, (16) d2r ds2−rg 2r(r−rg)/parenleftBiggdr ds/parenrightBigg2 +(r−rg)c2rg 2r3/parenleftBiggdt ds/parenrightBigg2 = 0. (17) First integral of the equation (16) is: r−rg r·dt ds=ǫ=Const. (18) We will record now an expression for the acceleration a takin g into account (18) and the fact that relationships between distance land time t, on the one hand, and coordinates r,t, on the other, are given by the formulae dl=/radicalBiggr r−rgdr, dτ =/radicalBigg r−rg rdt: a=d dτdl dτ=/radicalBiggr r−rg·d dt/parenleftBiggr r−rg·dr dt/parenrightBigg =1 ǫ2·/radicalBigg r−rg r·d2r ds2. In this way, we have expressed the acceleration ain terms of d2r/ds2.Now we can use the equation (17) and then, having reverted to landt, we can arrive at a=−rg(c2−v2) 2r/radicalBig r(r−rg), v =dl dτ. (10) I thank G.S.Lapidus and V.P.Visgin. Their attention has hel ped me to improve this paper text. This paper has been published in Russian: http://www.mai.ru/projects/mai works/index.htm, No. 3. REFERENCES 71. R. I. Khrapko, ”What is Mass?”, Physics - Uspekhi, 43 (12), (2000). 2. R. I. Khrapko, ”What is Mass?”, Uspekhi Phizicheskikh Nau k, 170 (12), 1363 (2000) (in Russian). 3. R. I. Khrapko, ”What is mass?”, http://www.mai.ru/projects/mai works/index.htm, No. 2 (in Russian). 4. A. S. Hornby, E. V. Gatenby, H. Wakeﬁeld, The Advanced Lear ner’s Dictionary of Current English, Vol.3, p.48 (ISBN 5-900306-45-3(3)). 5. McGraw-Hill encyclopedia of science & technology, V. 10 ( McGraw-Hill Book Company, New York, 1987), p. 488. 6. Chambers’s Technical Dictionary, (W. & R. Chambers, Ltd. , London, 1956), p. 529. 7. Van Nostrand’s scientiﬁc encyclopedia (Van Nostrand Rei nhold, New York, 1989), p. 1796. 8. M. Jammer, Concept of mass (Harvard University Press, Cam bridge- Massachusetts, 1961). 9. R. P. Feynman et al., The Feynman Lectures on Physics, V. 1 ( Addison-Wesley, Massachusetts, 1963), p. 9-1. 10. R. I. Khrapko et al., Mechanics (MAI, Moscow, 1993) (in Ru ssian). 11. E. F. Taylor and J. A. Wheeler, Spacetime Physics (Freema n, San Francisco, 1966). 12. A. Einstein, ”Ist die Tragheit eines Korpers von seinem E nergiegehalt abhangig?” Ann. d. Phys. 18, 639, (1905). 13. R. Resnick et al., Physics, V.1 (Wiley, New York, 1992). 14. M. Alonso, E. J. Finn, Physics (Addison-Wesley, New York , 1995). 15. L. B. Okun’, ”The concept of mass (mass, energy, relativi ty)”, Physics - Uspekhi, 32(7), p. 637, (1989). 16. M. Planck, Vortrage und Erinnerungen. Stuttgart, 1949. 17. L. D. Landau, E. M. Lifshitz, The Classical Theory of Fiel ds (Pergamon, New York, 1975). 8
arXiv:physics/0103009 2 Mar 2001x/G0C 2 /G0A/G0A/G0B0(x1/G09v0t1),y/G0C 2 /G0A/G0Ay1,z/G0C 2 /G0A/G0Az1,t/G0C 2 /G0A/G0A/G0B0(t1/G09v0 c2x1), x1/G28/G0A/G0A/G0B0(x2/G0C/G08/G08v0t2/G0C),y1/G28/G0A/G0Ay2/G0C,z1/G28/G0A/G0Az2/G0C,t1/G28/G0A/G0A/G0B0(t2/G0C/G08/G08v0 c2x2/G0C), /G0B0 /G0A/G0A1 /G09v02 c2/G091 2. x/G0C 1 /G0A/G0A/G0B0(x2 /G08/G08v0t2),y/G0C 1 /G0A/G0Ay2,z/G0C 1 /G0A/G0Az2,t/G0C 1 /G0A/G0A/G0B0(t2 /G08/G08v0 c2x2), x/G28 2 /G0A/G0A/G0B0(x/G0C 1 /G09v0t/G0C 1),y/G28 2 /G0A/G0Ay/G0C 1,z° 2 /G0A/G0Az/G0C 1,t/G28 2 /G0A/G0A/G0B0(t/G0C 1 /G09v0 c2x/G0C 1).On some Implications of (Symmetric) Special Relativity to High Energy Physics Ernst Karl Kunst Im Spicher Garten 5 53639 Königswinter Germany Beside the rise of total cross sections or interaction radii of colliding high energetic particles and the shrinkage of mean-free-paths of ultra relativistic particles (nucleii) in material media (anomalons), which have been shown to be of special relativistic origin [1], still other phenomena in high energy physics may arise from relativistic kinematics. In particular this seems to be the case with the EMC-effect and the so called atmosperic neutrino anomaly. Key Words: Special Relativity - quantization of velocity, length and time - EMC- effect - relativistic aberration - atmosperic neutrino anomaly In the mentioned work on relativistic kinematics has been shown a preferred rest frame of nature (/G28) in any inertial motion to exist and any velocity (v) be0 0 symmetrically composite or quantized. From this a symmetric modification of the Lorentz transformation follows between a frame of reference S considered to be at1 rest according to the principle of relativity and a moving frame S2 where The dashed symbols designate the moving system S and the open circles the2 system Sat rest, now considered moving relative to /G28 and S' . Likewise the1 0 2 observer resting in Swill deduce the respective transformation:2 x2 /G0A/G0Ax1,y2 /G0A/G0Ay1,z2 /G0A/G0Az1,t2 /G0A/G0At1, x1/G0C/G0A/G0Ax2/G0C,y1/G0C/G0A/G0Ay2/G0C,z1/G0C/G0A/G0Az2/G0C,t1/G0C/G0A/G0At2/G0C, x2/G28/G0A/G0Ax1/G28,y2/G28/G0A/G0Ay1/G28,z2/G28/G0A/G0Az1/G28,t2/G28/G0A/G0At1/G28 x1/G28 /G12x1,t1/G28 /G12t1, x2/G28 /G12x2,t2/G28 /G12t2. V /G0C/G0C /G0A/G0A/G0Cx/G0C/G0Cy/G0C/G0Cz/G0C/G0A/G0A/G0Cx /G0B0/G0Cy /G0Cz/G0A/G0AV /G0B0, /G29/G0C 2 /G0A/G0A/G25( /G0Cr1/G0B1 9 0)2/G0A/G0A/G291/G0B2 9 0, /G29/G0C 1 /G0A/G0A/G25( /G0Cr2/G0B1 9 0)2/G0A/G0A/G292/G0B2 9 0 /G29geo /G0A/G0A/G29/G0C 2 /G08/G08/G29/G0C 1 /G0A/G0A2 /G291/G0B2 9 0,2 (1) (2)Furthermore, due to the absolute symmetry relative to /G28 must be valid:0 and always /G0Dv/G0D = /G0D-v/G0D. If the upper lines of the of the above tansformation0 0 equations are inserted into the second lines, the identity results: Further main results of the modified theory of relativistic kinematics among others are the Lorentz transformation not to predict the Fitzgerald-Lorentz contraction of the dimension (/G0Cx) parallel to the velocity vector, as invented by Fitzgerald and Lorentz to account for the null-result of the Michelson-Morley experiment on moving Earth, but rather an expansion /G0Cx’ = /G0Cx/G0B - analogously to the relativistic time0 dilation /G0Ct’ = /G0Ct/G0B. Accordingly the volume V’ of an inertially moving body will any0 observer resting in a frame considered at rest seem enhanced where V means volume. Among others it has been demonstrated, this expansion of /G0Cx (or V) be the cause of the experimentally observed increase of the interaction radius respectively cross section of elementary particles with rising energy (velocity), as determined in collision experiments and as is known from studies of cosmic radiation, according to the equations so that the mean total geometrical cross-section is given by/G0Cr/G0C 2 /G0A/G0A/G0Cr1/G0B1 9 0, /G0Cr/G0C 1 /G0A/G0A/G0Cr2/G0B1 9 0. 3nb R(x /G190) /G0A/G0A3nb n/G0C b2 9 /G0A/G0Anb n/G0C b2 27,3 (3)where /G29¯ = /G29¯/G29¯' = /G29¯'. 21 /G19 21 Hence the mean geometrical interaction- radius is given by Because v /G67 v (conventional velocity) it follows /G0B/G67/G0B (conventional Lorentz factor)0 0 so that predictions on the grounds of symmetric special relativity will deviate from the conventional view, the more the higher the velocity (see [1]). The Relativistic Origin of the “EMC-Effect” The enhan cement of the geometrical cross section or interaction radius according to the above equations also delivers an explanation of the so called EMC-effect in a direct way. Consider the simplest case if the dimensionless variable x /G19 0 so that the high energetic incident particle (electron, muon etc.) more or less traverses the nucleus, encountering on an average nucleons within the nucleus. In this case the loss of momentum or energy of the outbound pa rticle must be a relative one depend ing on the mean cross section the number (n) of nucleons constituting the nucleus presents to the moving particle.b1/3 Therefore, if ultra relativistic velocity or momentum per incident particle relative to the respective nucleus is assumed to be equal must the ratio of the relative dampening of momentum of the outwards moving particles within different nuclei independen tly of the respective ultra relativistc velocity or energy according to (2) be given by whereby n' > n. Our formula delivers at x = 0 the ratios D/He = 0.95, D/C = 0.88,bb D/Al = 0.82, D/Ca = 0.80, C/Su = 0.84, which results agree very well with experiment [2],[3]. On the other end of the scale, where x /G19 1, according to (3) the "shadowing effect" of the growing relative interaction radius of the respective nucleus relative to the incident particle must be considered. The scattering probability of the particle is dependen t on the growing of the interaction radius of the respective target nucleus and, therewith, its dampening of momentum in dependen ce of the number of (n)b1/3 nucleons constituting the relative diameter of the nucleus at a ratio of R(x /G191) /G0A/G0A3nb n/G0C b1 9 /G0A/G0Anb n/G0C b1 27. x/G0C 2 /G0A/G0Aux/G0C 2t/G0C 2,y/G0C 2 /G0A/G0Auy/G0C 2t/G0C 2,z/G0C 2 /G0A/G0A 0. u°x1 /G0A/G0Aux/G0C 2 /G08/G08v0 1/G08/G08v0ux/G0C 2 c2,u°y1 /G0A/G0Auy/G0C 2/G0B/G091 0 1/G08/G08v0ux/G0C 2 c2,u°z1 /G0A/G0A0, tan /G051 /G0A/G0Asin /G05/G0C 21 /G09v2 0 c2 cos /G05/G0C 2 /G08/G08v0 c.4 (4) (5)Comparison with experiment also shows excellent correspondence [2],[3]. Thus, the EMC-effect is of the same physical (relativistic) origin as the rise of the total cross section or interaction radius of hadrons in high energetic collisions (see [1]). Relativistic Aberration (Doppler Boosting) as a Possible Cause of the Atmospheric Neutrino Results from Super-Kamiokande and Kamiokande Consider a light signal or an ultra relativistic particle moving relative to S' according2 to the equations Transformation into the coordinates and the time of the moving system S°1 delivers wherefrom in connection with the above identity equations the aberration law of special relativity is deduced According to this theory (5) is valid as long as the systems S° and S' are1 2 considered freely moving relative to each other and no direct physical contact (collision) occurs. In the following will be shown that the relativistic aberration effect (5) predicts the short fall of muon neutrinos coming up through Earth, known as the atmosperic neutrino anomaly. Experimental results from the Super-Kamiokande atmospheric neutrino measurements show at large distances from the neutrino generation, especially from Earth’s far side, a significant suppression of the observed number of muon neutrinos with respect to the theoretical expectation [4]. For a relativistic analysis isarctan /G0B0sin /G05/G1Fµ 1 /G08/G08cos /G05/G1Fµ/G09sin /G05/G1Fe 1/G08/G08cos /G05/G1Fe×2 /G25l 360/G12d,5 of special interest that the muon neutrinos originate from two separate decay processes about 20 kilometers above: first a high energetic pion decays into a muon and a muon neutrino (/G1F) and in a further step the muon into an electron, an electronµ neutrino (/G1F) and a further muon neutrino. Thus, considering the decay modes only,e the ratio of muon to electron neutrinos generated in the atmosphere can be predicted with confidence to be R = 2. We underline especially the decay of the/G1F(µ/e) muon leading to the simultaneous generation of a /G1F and the /G1F. µ e The Super-Kamiokande team compared particularly neutrinos coming down (downgoing) from the sky (l /G11 20 km) with those coming upward (upgoing) through the Earth (l /G11 12800 km). Because the cosmic rays and the resulting neutrinos rain down from all directions, the ratio should be R = R /R = 1. For/G1F(µ/e)upgoing /G1F(µ/e)downgoing electron neutrinos Super-Kamiokande caught equal numbers going up and coming down: R/R /G11 1, however, for muon neutrinos in 535 ope ration days/G1F(e)upgoing /G1F(e)downgoing 256 downward and only 139 upward ones have been counted. Furthermore, the expected ratio R /G11 2 has been found, but only R /G11 1, with/G1F(µ/e)downgoing /G1F(µ/e)upgoing systematic variations depending on the the distance l to the point of neutrino generation or angle of the incoming neutrinos. The number of muon neutrinos decreases linearly from a maximum at l /G11 500 km to l /G11 6400 - 7000 km, leveling off and remaining roughly constant up to l /G11 12800 km at the far side of Earth. This observed short fall of muon neutrinos with increasing distance l from the detector is currently interpreted as evidence for /G1F oscillations [4].µ Consider a muon decaying at l /G11 20 km above the detector. At the time of /G07-decay its ultra relativistic velocity v may result in a Lorentz factor of /G0B = 10 (which seems0 04 quite reasonable) and, to consider a simple case, the electron shall be emitted at some small angle « /G25/2 relative to the muon’s direction. The neutrinos, counterbalancing the electron’s momentum, will be emitted at larger angles /G05 > /G05 if p > p (which we assume). If for instance in the rest frame of the/G1F(µ) /G1F(e) /G1F(µ) /G1F(e) decaying muon for reasons of simplicity /G05 /G11 0° and /G0C/G05 = /G05 - /G05 = 45° (90°,/G1F(e) /G1F(µ) /G1F(e) 135°), equation (5) predicts at l = 20 km a lateral displacement of 0.83 m (2 m, 4.83 m) and at l = 6400 km of 266 m (640 m, 1546 m) of the flight paths of both neutrino types. Even an improbably small angle of /G0C/G05 = 5° (10°) in the rest frame of the muon would at l = 6400 km result in a lateral displacement of 28 m (60 m) between both neutrinos. A Lorentz factor of 10 and an an gle /G0C/G05 = 45° (90°) would5 result in a lateral displacement of 0.08 m (0.2 m) at l = 20 km, 26.91 m (64 m) at l = 6400 km and 53 m (128 m) at l = 12800 km. If /G05 > 0° the respective differences/G1F(e) have to be considered. It is clear that in a broad band o f varying angles, muon energy or Lorentz factors we would for distances /G12 6400 km arrive at similiar results: where d means diameter of the detector, /G05 and /G05 are given in degrees and /G1F(µ) /G1F(e)ux1 /G0A/G0Aux/G0C 2 /G08/G08v0 1/G08/G08v0ux/G0C 2 c2,uy1 /G0A/G0Auy/G0C 2/G0B0 1/G08/G08v0ux/G0C 2 c2,uz1 /G0A/G0A0.6 v/c = 1. Even in the case of a very high energy muon would a respective large0 angle /G0C/G05 between the tracks of both neutrino types in its rest frame according to the above examples at l = 6400 km lead to a lateral displacement > d (Super- Kamiokande: d = 34 m and hight = 36 m). It is clear that in all these cases at l /G11 20 km the muon-type and electron-type neutrino would traverse jointly a sufficiently large detector so that for each of the two neutrino types there is an equal chance to become counted, whereas at l /G12 6400 km for each counted electron neutrino the accompanying muon neutrino will due to the large displacement mainly miss the detector and pass far away undetectably by.The exact distance dependen ce of this relativistic aberration effect also explains the linear decrease of the number of muon neutrinos with increasing distance from the maximum of counts to the point, wherefrom most upgoing muon neutrinos no more can reach the detector together with the electron neutrino in l /G12 6400 km, in a fully way. Thus, if, as already mentioned, the flux of upgoing and downgoing electron neutrinos is observed to be abou t equal so that R/R /G11 1, this/G1F(e)upgoing /G1F(e)downgoing conclusively implies that from the far side of Earth mainly those muon neutrinos reach the detector, which were generated by the pion decay. All or nearly all muon neutrinos generated together with the electron neutrinos in the course of the muon decay on the other hand can due to the growing distance from the electron neutrino’s flight path not be registered by the same detector, resulting in R /G11 1, exactly as observed./G1F(µ/e)upgoing Transversal Aberration Effects in Ultra Relativistic Collision Events The theory also explains independen tly of quantum mechanical models the steady increase of the mean transverse energy per particle and une xpected frequent appearance of events with very high transverse energy as well as their distribution normal to the beam direction in collision experiments. But owing to the strong relativistic elongation of the colliding particles in beam direction this transversal aberration effect will at lower energies be superimposed by a longitudinal alignment of the secondary particles - in agreement with experiment. Consider two identical particles in S' and S'' colliding elastically with equal but2 2 oppositely directed velocity at point S at rest, being the kinematical center and at1 the same time the center-of-mass. We restrict our analysis to the recoil particle in S' . In the time particle dt after collision, S is moving relative to S' according to the2 1 2 equations (4). Transformation into the coordinates and time of S delivers1 Thus, one expects an aberrational transversal deviation of ultra relativistic recoil particles and photons from the path pattern in the center-of-mass, which is given by tan /G051 /G0A/G0A sin /G05/G0C 2 cos /G05/G0C 2 /G08/G08vo c1 /G09v2 o c2. RT /G0A/G0A/G0B/G0C 0/G0B/G091 0, tan /G05°1 /G0A/G0Atan /G05°2. tan /G051 /G0A/G0A sin /G05/G0C 2 cos /G05/G0C 2 /G08/G08v0 c1 /G09v2 0 c2,7 (6) (7) (8)This transversal deviation in collisions is a purely relativistic effect in the kinematical center, which only depends on the velocity of the colliding particles (of equal mass). Division of the tangens function at two different velocities delivers where /G0B' < /G0B, implying v' > v. This ratio is of interest in extrapolating the increase00 0 0 of transversal momentum (energy) at different velocities and scattering angles in collision experiments in the center-of-mass frame (see below). If the collision of an ultra relativistic particle with a fixed target particle in the laboratory is analysed, evidently neither system, the particles rest within, can be considered at rest owing to the natural rest frame /G28 amidst them, implying both0 systems to move relative to each other. Obviously this requirement is fulfilled if the laboratory is considered as the moving system S° and the ultra relativistic particle as1 the oppositely moving system S ° according to the above transformation equations.2 According to the latter also is valid x = xetc. so that follows°° 21 Thus, no relativistic deviation is to expect in collision events with fixed targets. If we consider the above identity equations and transform the right hand member of (8) into the coordinates and time of S' , we receive again 2 the apparent increase of energetic particles transversal to the beam direction with growing velocity (energy), as compared with the expectations on the grounds of the validity of the aberration law (5) of special relativity for this kind of scattering experiment. In the following predictions on the grounds of (7) are compared with experiment.d /G29 d /G36 /G0A/G0Anumberofparticlesatscattering /G53 /G15, /G51/time×solid /G53 currentdensityofincidentparticles×numberofscatteringcenters. tan /G29T /G0A/G0ART/G0B/G0C2 9 0 /G0B/G092 9 0tanpT, 8 (9)The differential cross section is defined by Irrespective of quantum mechanical effects the "number of particles at scattering angle /G15" must directly depend on the rise of the geometrical cross section according to (1) and (2) and the transversal aberration effect in ultra relativistic collisions according to (6) in the case of colliders as well as accelerators. Therefore, at low transverse momentums the differential cross sections of scattering experiments at higher energies are extrapolatable with fair accuracy from low energy values. For this purpose simply the product of the ratio of the geometrical cross section at lower and higher velocity (energy), and of the the ratio (7) - the aberrational effects in collision events predicted by this theory as compared with special relativity - is to multiply by the tangens of the transverse momentum: where p means a given transverse momentum at lower velocity (p /G06 10 GeV) andT T /G0B' < /G0B. This simple geometrical derivation of aberrational effects at lower00 momentums from experimental values is possible because the differential cross section at a given velocity (energy) as a function of the four-momentum transfer squared is equivalent to plotting it as a function of scattering angle at fixed energy. The curve of the differential cross section at higher energies is geometrically approximated by the above formula by adding arctan /G29 units to the point p of the curve atT T lower scattering velocity (energy). The symmetrical transverse momentum p is computed from theT conventional momentum (see [1]). In the figure experimental results at the center-of-mass energy E* = 540 GeV are compared with extrapolations(crosses) from E* = 62 GeV to E* = 540 GeV according to the above formula. The approximation seems fairly good. Thus, it is predicted that the growing transversal deviation of the secondary particles out of collisions with ever growing energy of particles of whatever kind, as for instance found at the Fermilab’s Tevatron particle accelerator in protron-antiprotron collisions at 1800 GeV in the center-of-mass frame, is solely of relativistic origin. This also is true for high energetic collisions of all kinds of nucleii, where indeed this300 MeV ×E/G0C (E/G0B/G0A/G0A540 GeV)E/G0C (E/G0B/G0A/G0A540 GeV)2 9 E/G0C (E/G0B/G0A/G0A60 GeV)E/G0C (E/G0B/G0A/G0A60 GeV)2 92 9/G0A/G0A456 MeV.9 trend has been observed long since, as for instance at GSI in Darmstadt, Germany. In violent collisions (1 GeV/nucleon) between gold nuclei has been found that at polar emission angles of 90° in the center-of-mass frame, kaons as well as nucleons and pions emerge preferentially out of the plane of the collision, although kaons are expected to emerge isotropically [5]. However, according to the above formulas this effect clearly is to expect and must increase with ever growing energy (velocity). According to this theory the experimentally verified tendency of secondaries in high energetic collisions to deviate transversally to the beam direction with increasing energy (velocity) is merely of relativistic origin. This effect also comprises "jet" structures "seen" in protron-protron (antiprotron) and electron-positron collisions, which usually are interpreted as a manifestation of the interaction of the quarks constituting the hadrons. But according to this theory the observed jet structures are (mainly) fictitious and necessarily occur if v/G19c and the emitted particles -0 independen tly of their origin (elastic or inelastic scattering) - according to the above equations tend to fill out the transversal region. If high transverse momentums and jets are of the same kinematic origin, they also should exhibit similiar structures, regardless of the particles involved. Furthermore is clear that the bulk of high transverse energy events should have a non-jet like uniform azimuthal distribution. Indeed this is observed. Therefore, the probability of the production rate of jets should rise in accordance with this theory independen tly of the particles involved. And indeed: the UA1 experiment at CERN observed a rise of the jet cross section in protron-antiprotron collisions from /G11 5 mb at 350 GeV (center-of-mass frame) collision energy to /G11 10 mb at 900 GeV [6]. Extrapolation according to (9) also results in 10 mb. Comparable data were measured by the PLUTO experiment at DESY in Hamburg in electron-positron collisions. A rise of the mean square sums (p) of the transversalT2 momentum of jets as a function of the center-of-mass energy E from /G11 1.3 </G28(p)>(GeV) at E = 7.7 GeV to /G11 6.6 </G28(p)>(GeV) at E = 31.6 GeV has beenT T2 2 2 2 observed [7]. Extrapolation according to (9) results in 6.6 </G28(pT)>(GeV), too.2 2 It is clear that the mean total transverse energy (the sum of all Es in an event) orT mean transversal momentum must rise proportionally to the relativistic rise of the mean geometrical cross section in connection with the transversal aberration. Respective measurements were made at CERN, where a rise of the mean total transverse energy from /G11 300 MeV at ISR energy (60 GeV) to /G11 500 MeV at SPS energy (540 GeV) has been observed [8]. Extrapolation results in/G29T(150 GeV)/G1120 mb ×E/G0C (E/G0B/G0A/G0A150 GeV)E/G0C (E/G0B/G0A/G0A150 GeV)2 92 9 /G0A/G0A59.28 mb, /G29T(900 GeV)/G1120 mb ×E/G0C (E/G0B/G0A/G0A900 GeV)E/G0C (E/G0B/G0A/G0A900 GeV)2 92 9 /G0A/G0A84.83 mb,10 At CERN in classic protron-antiprotron scattering events a rise of the mean cross section of the individual particles transversal to the beam direction from 56.1 (±4.7) mb at 150 GeV total energy in the center-of-mass system to 85.5 (±6.4) mb at 900 GeV has been measured [9]. If the cross section of the protron (antiprotron) at rest = 10 mb our formulas deliver: where T in the left hand side means transversal, E’ energy based on (quantized) v0 and E* center-of-mass energy (see [1]). Respective measurements of electron-positron collisions were carried out by the PLUTO experiment at DESY in Hamburg. A rise of the mean transversal momentum of jets from /G11 0.3 GeV at 7.7 GeV total energy in the center-of-mass system to /G11 0.4 GeV at 31.6 GeV collision energy has been observed [10]. Extrapolation also results in 0.4 GeV and, thus, shows very good correspondence with experiment. Finally it is predicted that the excess of events (as compared with the expectations on the grounds of the standard model) in collisions between positrons of 27.5 GeV and protrons of 820 GeV center-of-mass energy with high momentum transfer or at a large angle, found at DESY’s HERA positron-protron collider, also owes its existence to the relativistic rise of the mean geometrical cross section in connection with relativistic transversal aberration in collisions. References [1] Kunst, E. K.: Is the Kinematics of Special Relativity incomplete?, physics/9909059 [2] Max-Planck-Inst. f. Kernphys., Jahresbericht, 116-117 (1990) [3] CERN Courier, 262 ( September 1983 ) [4] Fukuda, Y. et al., Phys. Rev. Lett. 81, 1562 (1998) [5] Shin, Y. et al., Phys. Rev. Lett. 81, 1576 (1998) [6] CERN Courier, 1 (March 1986) [7] DESY, Wissenschaftlicher Jahresbericht, 75 (1982) [8] Physics today, 19 (February 1982) [9] CERN Courier, 32 (Jan./February 1988) [10] DESY, Wissenschaftlicher Jahresbericht, 74 (1982)
K OENEMANN , F.H. Cauchy stress in mass distributions The thermodynamic definition of pressure P = ∂U/∂V is one form of the principle that in a given state, the mass in V and potential are proportional. Subject of this communication is the significance of this principle for the understanding of Cauchy stress. The stress theory as it is used today, was developed by Euler in 1776 and Cauchy in 1823. The following is a slightly edited quote of Truesdell [1]. p.164: Let f be pairwise equilibrated; let -S denote the contact having the same underlying set as S but opposite orientation; then t -S = - tS (1) p.170: Cauchy assumed that the tractions t on all like-oriented contacts with a common plane at x are the same at x, i.e. tS at x is assumed to depend on S only through the normal n of S at x: tS = t (x, n). This statement is called the Cauchy postulate. S is oriented so that its normal n points out of c( B) if S is a part of ∂c(B). Thus t (x, -n) is the traction at x on all surfaces S tangent to ∂c(B) and forming parts of the boundaries of bodies in the exterior c( Be) of c( B). In this sense t (x, n) is the traction exerted upon B at x by the contiguous bodies outside it. As a trivial corollary of (1) follows Cauchy´s fundamental lemma: t (x, -n) = -t (x, n). p.176: v1 and v2 are linearly independent. At a given place x0 the planes P 1 and P 2 normal to v1 and v2, respectively, are distinct. We set v3 = -(v1 + v2) and consider the wedge A that is bounded by these two planes and the plane P 3 normal to v3 at the place x0 + εv3. We suppose ε small enough that A be the shape of some part of B, and we denote by ∂iA the portion of the plane P i that makes a part of the boundary of A. We let ε approach 0. If we write Ai for the area of ∂iA, we see that .A )(V, )(O A,A A ,A A 2 0 as 3 333 32 2 3 31 1 v Avv vv ε =→ε ε== = (2) If /Gf2 ∂= Anxtv c 33 dA),(A, (3) from (2) and the assumption that t (·, n) is continuous we see that 0 as 3 1 →ε ε+/Gf7/Gf7 /Gf8/Gf6 /Ge7/Ge7 /Ge8/Ge6 =/Ge5 /Gf2 = ∂)(O dA ,A i ii ii iA vvxtv c . (4) Since t is a homogneous function of its second argument and a continuous function of its first argument, () 0 as 3 1 0 →ε →/Ge5 =kk,vxt c . (5) On the other hand, we see that c → 0 as ε → 0. Therefore, since the sum in (5) is independent of ε, it must vanish: () 0 vxt 3 1 0 = /Ge5 =kk, . [End of quote] (6) The key argument in the above text is: since the sum in (5) is independent of ε. It is an a priori condition; behind this is the assumption that Newtons 3rd law (1) is the proper equilibrium condition for the problem, and the Newtonian understanding of pressure, P = \|f\|/A which is believed to be universally scale-independent. Pressure is a state function, and a pressure increase requires that work is done on a system of mass distributed in V; it is a change of state in the sense of the First Law. Pressure is defined as energy density, P = ∂U/∂V. (7) The question is then: how are P = \|f\|/A and P = ∂U/∂V mathematically related? 2 The thermodynamic definition of P is scale-independent, and an explicit statement of the proportionality of mass (measured in V the radius of which is r = \|r\|) and potential U in a given state (Kellogg [2:80]). The thermodynamic equilibrium condition is P surr Psyst = 0 (8) in scalar form. If both terms are thought to be caused by forces f [Newton] acting from either side on the surface of the system V the equilibrium condition is fsurr fsyst = 0; (9) for isotropic conditions (subsequently implied), both f are radial force fields. Since the system contains mass, and since it interacts with the surrounding through exchange of work, it acts as a source of forces; i.e. its source density 0≠ϕ in some statically loaded state. ϕ is always proportional to the mass in the system (Kellogg [2:45]); an existence theorem requires that if there is some function f of a point Q such that ()/Gf2ϕ=dVQf , (10) both LHS and RHS must vanish with the maximum chord of V (Kellogg [2:147]). As with all of thermodynamics (Born [3]), the approach to stress must thus be based on a Poisson equation (Kellogg [2:156]). The equilibrium condition (8) thus can take the form 0syst surr =ϕ−ϕ . (11) It is therefore of interest how the volume functions relate to the surface functions if the domain of interest V is changed in scale. In /Gf2/Gf2ϕ=⋅∇=⋅ dV dA f nf , (12) f may be either one of the LHS terms in (9). If mass is continuously distributed, ϕ ∝ V, and ∇ ⋅ f is a constant that is characteristic of the energetic state in which the system is. Hence in (12), LHS ∝ V. Since V ∝ r3, but A ∝ r2, for LHS ∝ r3 to hold it follows that \| f\| ∝ r, or const = rf . (13) Thus as V → 0, \|f\|/A → ∞, yet ∆U/∆V → const . Both fsyst and fsurr vanish with r; the condition in (10) is observed, stating that a system V with zero magnitude cannot do work on its surrounding, and vice versa. ε (2-5) is an one-dimensional measure of the magnitude of the prism A (2), as is r for V in the subsequent discussion. It is to be taken into account that P = \|f\|/A is scale-independent if A is a free plane, yet both the surface of the prism A in (2) and the surface A in (12) are closed surfaces. The difference between (1) and (9) is that the latter distinguishes system and surrounding whereas the former does not. The thermodynamic system V represents a distributed source in the sense of potential theory (Kellogg [2:150 ff]). ε or r, respectively, is the zero potential distance (Kellogg [2:63]) which may have infinite length, or if it is finite it is set to have unit length by convention, but it cannot be zero or otherwise be let vani sh. References 1 T RUESDELL , C.A.: A first course in rational continuum mechanics. Academic Press, 1991. 2 K ELLOGG , O.D.: Foundations of potential theory. Springer Verlag, 1929. 3 B ORN, M.: Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik. Physik. Zeitschr., 22 (1921), 218-224, 249-254, 282-286. Address: FALK H. K OENEMANN , Im Johannistal 36, 52064 Aachen, Germany; peregrine@tonline.de
arXiv:physics/0103011v1 [physics.optics] 4 Mar 2001Analysis of Optical Pulse Propagation with ABCD Matrices Shayan Mookherjea∗ Department of Electrical Engineering, 136–93 California I nstitute of Technology, Pasadena, CA 91125 Amnon Yariv Department of Applied Physics, 128–95 California Institut e of Technology, Pasadena, CA 91125 (Dated: January 3, 2001) We review and extend the analogies between Gaussian pulse pr opagation and Gaussian beam diﬀraction. In addition to the well-known parallels betwee n pulse dispersion in optical ﬁber and CW beam diﬀraction in free space, we review temporal lenses a s a way to describe nonlinearities in the propagation equations, and then introduce further co ncepts that permit the description of pulse evolution in more complicated systems. These include the temporal equivalent of a spherical dielectric interface, which is used by way of example to deri ve design parameters used in a recent dispersion-mapped soliton transmission experiment. Our f ormalism oﬀers a quick, concise and powerful approach to analyzing a variety of linear and nonli near pulse propagation phenomena in optical ﬁbers. This paper introduces an ab-initio study of pulse prop- agation phenomena analogous to spatial CW diﬀraction behavior. We address both linear dispersive evolution as well the self-phase modulation eﬀects of the nonlin- ear index of refraction [1]. The latter is responsible for much of the current interest in nonlinear optical com- munications, since pulse shapes such as solitons and dispersion-managed solitons display much more attrac- tive transmission properties than linear transmission for - mats (e.g. NRZ) [2]. Such nonlinear pulses are usually self-consistent eigen- solutions of a wave equation, which is the primary reason for their robustness to uncompensated spectral broaden- ing and resultant dissipation into the continuum. The conventional hyperbolic secant soliton is an exact solu- tion of the nonlinear Schr¨ odinger equation [3], and prop- agates indeﬁnitely in a lossless medium without losing its shape. Lossless media can be realized in practice quite eﬀectively by using lumped ampliﬁcation stages, and erbium-doped ﬁber ampliﬁers oﬀer excellent charac- teristics in this regard. Breathers, sometimes called dispersion-managed soli- tons [4, 5], are also self-consistent ‘eigen solutions’ of the wave equation that propagate with periodic pulse width, chirp etc. While not strictly unchanging in shape, breathers evolve back to their initial conﬁguration, essen - tially traversing a closed, non-degenerate orbit in phase space [6]. Unlike pulse shapes designed for linear trans- mission channels, these pulses do not require periodic dispersion compensation along the transmission channel, and so oﬀer an attractive alternative to the strong control requirements of the nonlinear Schr¨ odinger soliton. Characterizing the solutions of the nonlinear wave equation is often simplest via direct numerical simula- tion, and this has been particularly true for dispersion ∗Electronic address: shayan@caltech.edu; URL: http://www.its. caltech.edu/~shayanmapped solitons [7]. In order to understand, capture and then predict and utilize the essential physics that guides this behavior, a more conceptually accessible framework is sometimes preferable, such as the variational approach with a pulse shape Ansatz [8]. The pulse shape is de- scribed as a dynamical system; we write the Hamilto- nian based on the action principle and seek solutions to the Euler-Lagrange equations of motion [9, 10]. This ap- proach is not always applicable, however, especially when the Ansatz is incapable of capturing some essential phys- ical behavior. Also, it is somewhat more of an analytical tool for probing the dynamics of systems that we already know something about, or can predict at least partially, and it may be convenient to have other approaches that can oﬀer quick insight into constructive aspects of non- linear propagation, so that diﬀerent geometries can be analyzed and compared quickly and easily. The parallels between dispersive pulse propagation in optical ﬁbers and paraxial CW Gaussian beam diﬀraction in free space have been identiﬁed for some time [11, 12, 13]. More recently, the analogies have been extended to include temporal lenses as a way to translate the imaging properties of spatial lenses into the temporal domain [14]. In this way, pulse correlation and convolution devices may also be constructed [15]. Still more recently, it was shown that temporal lenses can characterize nonlinear eﬀects in the wave equation, leading, for example to the formation of a class of steady-state repeating pulses [16]. We believe that this is perhaps the most potentially use- ful of the space-time analogies: in this paper, we further extend the use this formalism to describe still more pow- erful applications such as Gaussian pulse propagation in optical ﬁber systems, including dispersion mapped sys- tems, including the eﬀects of the nonlinear index of re- fraction. We ﬁrst outline the basic physics that motivates this discussion and sets the context for further development.2 I. SPACE-TIME ANALOGY OF BEAM DIFFRACTION AND PULSE PROPAGATION A. CW Gaussian beam diﬀraction The Fresnel-Kirchoﬀ diﬀraction integral is a well- founded approach to electromagnetic propagation prob- lems, and several textbooks cover the topic from a variety of approaches [17, 18, 19]. We will brieﬂy review only as much as necessary to establish our argument, limiting our argument to diﬀraction in 1+1 ( x,z) dimensions. An electromagnetic ﬁeld of radian frequency ωand scalar complex amplitude u(x,z) can be represented E(x,z,t) =u(x,z) exp(iωt) (1) whereu(x) obeys the wave equation, ∇2u+k2u= 0, k2=ω2µǫ=/parenleftbigg2πn λ/parenrightbigg2 .(2) This equation admits plane wave solutions of the form exp(±ikz) representing propagation along ∓zrespec- tively, and indeed, an arbitrary superposition of plane waves, each with the same wavelength, propagating along all possible directions, u(x,z) =/integraldisplay ˜u0(kx)exp[i(kxx)−i/radicalbig k2−k2xz]dkx(3) where ˜u0is the Fourier transform of the input ﬁeld u0(x,0). We consider optical beams whose plane wave compo- nents propagate at small angles to the zaxis (paraxial approximation), so that we can expand the square root in (3) in a Taylor series and keep the ﬁrst two terms, E(x,z) (4) =eiωt−ikz/integraltext/bracketleftBig ˜u0(kx)exp/parenleftBig ik2 x 2kz/parenrightBig/bracketrightBig exp(ikxx)dkx =eiωt−ikz/bracketleftbigg/radicalBig ik 2πz/integraltext u0(x′)exp/bracketleftBig −ik(x−x′)2 2z/bracketrightBig dx′/bracketrightbigg where the term in parentheses deﬁnes u(z,t), the ﬁeld envelope, u(z,t) =/radicalbigg ik 2πz/integraldisplay u0(x′)exp/bracketleftbigg −ik 2z(x−x′)2/bracketrightbigg dx′(5) The propagation of continuous-wave (CW) Gaussian beams in free space and rotationally-symmetric quadratic graded-index media is conveniently described by assum- ing that the envelope has the form [17] u= exp/braceleftbigg −i/bracketleftbigg P(z) +k 2q(z)r2/bracketrightbigg/bracerightbigg (6) where, we ﬁnd by substitution into the wave equation (2) thatdP/dz =−i/q(z) for such media. The q-parameter describes the Gaussian beam completely, 1 q(z)=1 R(z)−iλ πnw2(z). (7)In the above deﬁnition, R(z) describes the radius of cur- vature of the beam, and w(z) the beam spot size. The usefulness of the q-parameter lies in the bilinear transformation (ABCD law) that characterizes how this parameter evolves with propagation. For an optical sys- tem described by a real (or complex) ABCD matrix, the outputqparameter is given by qo=Aqi+B Cqi+D. (8) Separating the real and imaginary parts of qoenables us to calculate the radius of curvature and spot size of the Gaussian beam at the output of the optical sys- tem. Many practically important optical systems and their corresponding phenomena can be described by sim- ple ABCD matrices, such as propagation in a uniform medium, focusing via a thin lens, beam transformation at a dielectric interface, propagation through a curved di- electric interface and thick lens, propagation in a medium with a quadratic index variation etc. [17, Table 2-1] B. Gaussian pulse propagation Consider a single mode in an optical ﬁber, usually the lowest-order fundamental mode, excited at z= 0, and with an assumed temporal envelope of the Gaussian form, E(z= 0,t) =Re/bracketleftbig exp(−αt2+iω0t)/bracketrightbig (9) and write as a Fourier transform integral, E(0,t) =Re/bracketleftbigg exp(iω0t)/integraldisplay ˜u0(Ω)exp(iΩt)dΩ/bracketrightbigg (10) where ˜u0is the Fourier transform of the Gaussian enve- lopeu0= exp(−αt2). As in the spatial case, we can be think of this as a su- perposition of time-harmonic ﬁelds, each with frequency (ω0+Ω) and amplitude ˜ u0(Ω)dΩ. These waves will expe- rience a phase delay when propagating a distance z; we multiply each frequency component by its propagation delay factor exp[−iβ(ω0+ Ω)z] so that E(z,t) =/integraldisplay ˜u0(Ω)exp[i(ω0+ Ω)t−iβ(ω0+ Ω)z]dΩ. (11) Expanding β(ω0+ Ω) in a Taylor series about the op- tical frequency ω0, β(ω0+ Ω) =β(ω0) +dβ dω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=ω0Ω +1 2d2β dω2/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=ω0Ω2+... (12) we can write E(z,t) = exp[i(ω0t−β0z)]× (13)/integraldisplay ˜u0(Ω)exp/braceleftbigg i/bracketleftbigg Ωt−β′Ωz−1 2β′′Ω2z/bracketrightbigg/bracerightbigg dΩ3 where the integral deﬁnes the ﬁeld envelope u(z,t), so that E(z,t) =u(z,t) exp[i(ω0t−β0z)]. (14) The diﬀerential equation satisifed by uis, to second order of derivatives of β[3], ∂u ∂z+β′∂u ∂t+1 2β′′∂2u ∂t2= 0. (15) The solution (13) can be written using the inverse Fourier transform relationship, u(z,t) =/integraldisplay/bracketleftbigg ˜u0(Ω)exp/parenleftbigg −i 2β′′Ω2z−iβ′Ωz/parenrightbigg/bracketrightbigg ×exp(iΩt)dΩ =/radicalbigg1 i2πβ′′z ×/integraldisplay u0(t′)exp/bracketleftbiggi 2β′′z(T−t′)2/bracketrightbigg dt′(16) whereT=t−β′z=t−z/vgis the time coordinate in the frame of reference co-moving with the pulse envelope at the group velocity vg= 1/β′. Dispersion of the group- velocity (GVD) is represented by β′′. The formal similarity between (5) and (16) is the prin- cipal motivation for this analysis. We can write down a set of space-time translation rules (see Table I) to ap- ply results from spatial diﬀraction to temporal dispersion and vice versa. One family of results that can be derived from this space-time analogy correspond to spatial imag- ing e.g. the 2- fand 4-foptical systems. These can be applied to pulse compression or expansion experiments etc [14]. But we will see in later sections that many linear and nonlinear pulse propagation systems can be described by cascading simple ABCD matrices, and this can result in substantially simpler calculations and more direct phys- ical understanding of the physical processes involved in nonlinear pulse propagation. We will ﬁrst need to develop some additional facility in characterizing optical system s associated with the pulse propagation equations. The spatial q-parameter has a temporal equivalent qt in accordance with the space-time translation rules of Table I, deﬁned by 1 qt(z)=1 Rt(z)+i2β′′ τ2(z), (17) whereτ(z) represents the pulse width (scaled in the T frame by√ 2) andRt(z) its chirp. A Gaussian pulse in linearly dispersive ﬁbers is then represented by the envelope [16] u(z,t) =u0τ0 τ(z)(18) exp/bracketleftbigg itan−1z ζ0+it2 2β′′Rt(z)+β′′ \|β′′\|t2 τ2(z)/bracketrightbigg .where the pulse width and chirp satisfy evolution equa- tions in linear dispersive ﬁbers exactly analogous to their spatial counterparts, beam spot size and radius of curva- ture, in free space [16] τ2(z) =τ2 0/parenleftbigg 1 +z2 ζ2 0/parenrightbigg , Rt(z) =z/parenleftbigg 1 +ζ2 0 z2/parenrightbigg (19) withζ0=τ2 0/2\|β′′\|deﬁning the dispersion length [3]. II. COMPONENTS OF THE ABCD FORMALISM FOR GAUSSIAN PULSE PROPAGATION As a simple example of the application of the above translation rules, we consider the propagation of a Gaus- sian input pulse with envelope U(0,T) = exp/parenleftbigg −T2 2T2 0/parenrightbigg . (20) The transmission medium comprises of two concatenated sections of ﬁber with lengths z1andz2and with GVD coeﬃcients β′′ 1andβ′′ 2respectively. We ignore any non- linear eﬀects in this simple problem, and assume that the medium is lossless. What are the pulse characteristics at the output of the second medium i.e. what is the pulse width atz=z1+z2? One way of solving this problem is by recourse to the wave equation (16) solution by the Fourier transform technique. We have, ˜U(z2,ω) = ˜U(z1,ω)exp/parenleftbiggi 2β′′ 2z1ω2/parenrightbigg =˜U(z0,ω)exp/bracketleftbiggi 2(β′′ 1z1+β′′ 2z2)ω2/bracketrightbigg .(21) Taking the inverse Fourier transform, U(z1+z2,T) = 1 2π/integraldisplay∞ −∞˜U(0,ω)/bracketleftbiggi 2(β′′ 1z1+β′′ 2z2)ω2−iωT/bracketrightbigg dω =/bracketleftbiggT2 0 T2 0−i(β′′ 1z1+β′′ 2z2)/bracketrightbigg1 2 ×exp/bracketleftbigg −T2 2(T2 0−i(β′′ 1z1+β′′ 2z2)/bracketrightbigg , (22) from which we see that the ratio of the output to input pulse width, therefore, is T1 T0=/bracketleftBigg 1 +/parenleftbiggβ′′ 1z1+β′′ 2z2 T2 0/parenrightbigg2/bracketrightBigg1 2 . (23)4 TABLE I: Space-time translation rules spatial frequency (Fourier variable) kxΩ frequency (Fourier variable) transverse distance xt−z vgtime (in moving reference frame) propagation distance z z propagation distance wavevector (inverse) k−1−β′′GVD coeﬃcient (negative) We will now verify (23) using the ABCD matrix ap- proach. The system is described very simply by the prod- uct of three matrices, M=/parenleftBigg 1z2 0 1/parenrightBigg ./parenleftBigg 1 0 0β′′ 2 β′′ 1/parenrightBigg ./parenleftBigg 1z1 0 1/parenrightBigg =/parenleftBigg 1z1+β′′ 2 β′′ 1z2 0β′′ 2 β′′ 1/parenrightBigg (24) so that q2=Aq1+B Cq1+D=β′′ 1 β′′ 2q1+β′′ 1 β′′ 2z1+z2. (25) Using the shorthand notation R2≡R(z1+z2), R1≡R(0), τ2≡τ(z1+z2), τ1≡τ(0).(26) we have 1 R2−i2β′′ 2 τ2 2/parenleftBig 1 R2/parenrightBig2 +/parenleftBig 2β′′ 2 τ2 2/parenrightBig2=β′′ 1 β′′ 21 R1−i2β′′ 1 τ2 1/parenleftBig 1 R1/parenrightBig2 +/parenleftBig 2β′′ 1 τ2 1/parenrightBig2+β′′ 1 β′′ 2z1+z2 (27) The real and imaginary parts of both sides of the above equation have to be equal, leading to a pair of simulta- neous equations. For an unchirped input pulse, R1= 0 so that equality of the imaginary parts leads to /parenleftbigg1 R2/parenrightbigg2 +/parenleftbigg2β′′ 2 τ2 2/parenrightbigg2 =/parenleftbigg2β′′ 2 τ2τ1/parenrightbigg2 . Subsituting this expression into the equation of equality of the real parts of (27) and some algebraic manipulation leads to τ(z1+z2) τ(0)=τ2 τ1=/bracketleftBigg 1 +/parenleftbiggβ′′ 1z1+β′′ 2z2 τ2 1/2/parenrightbigg2/bracketrightBigg1 2 (28) which is the same as (23), since τ=√ 2 ∆T. In the above calculation, we have carried out some algebraic simpliﬁcations by hand in order to show that the result obtained by the ABCD matrix approach is the same as that obtained by the Fourier transform approach. Nevertheless, the former is computationally much sim- pler, and separating the real and imaginary parts of (27)as part of a numerical algorithm can be carried out with- out the notational complexity of, for example, rational- izing the denominator. While second-order dispersion is conveniently repre- sented by the ABCD matrix approach, there are prob- lems with extending the analysis to higher orders of dis- persion. The slowly-varying envelope equation analogous to (15) including the eﬀects of third-order dispersion i∂U ∂z=1 2β′′∂2U ∂T2+i 6β′′′∂3U ∂T3(29) or its solution in terms of the Fourier transformed vari- ables, ˜U(z,ω) =˜U(0,ω)exp/parenleftbiggi 2β′′zω2+i 6β′′′zω3/parenrightbigg (30) does not have an equivalent in the CW spatial diﬀraction context. To see this, consider the next term in the Taylor ex- pansion of/radicalbig k2−k2xin (3), which leads to an expression of the form u2(x) =e−ikz× (31)/integraldisplay u1(kx)exp/parenleftbigg ik2 x 2kz+ik4 x 8k3z/parenrightbigg exp(ikxx)dkx. Using the space-time translation rules, we ﬁnd that the above expression contains a description of second and fourth -order dispersion, not third-order dispersion. This is obviously a general characteristic of the above Taylor expansion; all odd-order dispersion terms have no spatial paraxial diﬀraction equivalent in the ABCD ma- trix content. Recall that the eﬀect of β′is accounted for by transforming to a moving reference frame T=t−β′z. For completeness, we derive the translation rule for any even-order dispersion in terms of the equivalent term in CW diﬀractive optics. A little algebra will show that the generalization of (30) yields ˜U(z,ω) = ˜U(0,ω)× (32) exp/parenleftbiggi 2β′′ω2z+i 6β′′′ω3z +...+i (2r)!β(2r)ω2rz/parenrightbigg ,(33) and, correspondingly, for the diﬀraction of a Gaussian5 beam, ˜U(x,kx) = ˜U(0,kx)exp/bracketleftbigg ik2 x 2kz+ik4 x 8k3z+... +i(−1)r r!r−1/productdisplay l=0/parenleftbigg1 2−l/parenrightbiggk2r x k2r−1z/bracketrightBigg .(34) Therefore, the translation rule for 2 r-order temporal dis- persion is given by β(2r)←/bracketleftBigg (−1)r(2r)! r!r−1/productdisplay l=0/parenleftbigg1 2−l/parenrightbigg/bracketrightBigg 1 k2r−1 =−/bracketleftBigg (2r)! r! 2rr−1/productdisplay l=1(2l−1)/bracketrightBigg 1 k2r−1. (35) Note that this corrects the statement in [14]: The slowly varying envelope equations corre- sponding to modulated plane waves in disper- sive media have the same form as the parax- ial equations describing the propagation of monochromatic waves of ﬁnite spatial extent (diﬀraction). We append that this correspondence holds for all even orders of dispersion, and of course, for β′as well, by transforming to a moving reference frame. Our ABCD formalism would be of limited interest if the only phenomena it could capture were that of disper- sive propgation. But, as mentioned in an earlier section, the development of the time-lens formalism lets us de- scribe nonlinear mechanisms as well. By analogy to spatial lenses which are characterized by a lens factor exp( ikr2/2f) which multiplies an incoming optical beam, we deﬁne a temporal lens as a device that multiplies the pulse envelope by a factor [14, 16] Lens Factor = exp/bracketleftbigg −it2 2β′′ft/bracketrightbigg ≡exp/bracketleftbig −ibt2/bracketrightbig (36) The ABCD matrix representing a temporal lens has the same form as that of a spatial lens, M=/parenleftBigg 1 0 −1 ft0/parenrightBigg (37) whereftrepresents the temporal “focal length”. A comparison of spatial and temporal lensing is shown in Figure 1. In the spatial case, the lens compensates for the spreading of the beam waist, and “ﬂips” the phase fronts to convert a diverging beam into a converging one. Similarly, a temporal lens reverses the sign of the chirp, so that further propagation in a β′′<0 dispersive ﬁber will compensate for the chirp (phase modulation) caused this far. This is also an interesting and physically illuminatin g approach to discussing the physics of the formation of solitons [17, Chapter 19].xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx phase frontsbeam waist Gaussian profilex z beam width increases Time Lens unchirped Gaussian pulse inputdispersive fiber chirped Gaussianunchirped Gaussian FIG. 1: (a) Spatial lens (b) Temporal lens One possible implementation, as proposed in [16], is to achieve temporal lensing by self-phase modulation during the passage of the pulse through a section of nonlinear ﬁber (β′′≈0,n2>0). For short distances, z≪πτ2 0/\|β′′\| and whenβ′′/τ2 0≪(2πn2/λ)Ipfor peak intensity Ip, a pulse with input electric ﬁeld envelope u(0,T) emerges from a length zof nonlinear ﬁber with phase modulation u(z,T) =u(0,T)exp/bracketleftbigg −iω0n2z 2cη\|u(0,T)\|2/bracketrightbigg (38) whereη=/radicalbig µ/ǫdeﬁnes the impedance of free space. If we write the pulse intensity as I=\|u\|2 2η=Ipexp/bracketleftBigg −2/parenleftbiggT τ0/parenrightbigg2/bracketrightBigg (39) and keep the ﬁrst two terms in the Taylor expansion of the exponential in (38), u(z,T) =u(0,T)exp/bracketleftbigg i2ω0n2Ipz cτ2 0T2/bracketrightbigg (40) modulo a phase term linear in zthat is independent of T. The eﬀect of propagation through length Lof nonlinear ﬁber is to impart a quadratic chirp to the pulse, which we represent by the multiplicative term exp( −ibt2) so that b=1 2β′′ft=−2ω0n2IpL cτ2 0. (41)6 Another method of obtaining time lensing is based on the principle of electro-optic modulation [14]. An electro - optic phase modulator driven by a sinusoidal bias voltage of angular frequency ωmresults in a phase modulation that is approximately quadratic under either extremum of the sinusoid. The phase shift can be written as exp[iφ(t)] = exp/bracketleftbigg −iK/parenleftbigg 1−ω2 mt2 2/parenrightbigg/bracketrightbigg (42) whereKis the modulation index [17, §9.4]. In this case, b=2 Kω2m(43) We have described our temporal lens by a section of nonlinear ﬁber of β′′≈0, analogous to a spatial thin lens, which is assumed to have no thickness. Just as practical lenses do have some thickness, practical ﬁbers have non- zeroβ′′. For those situations in which this cannot be ignored, or may even be utilized constructively, we derive the corresponding equivalent of a spatial “thick lens”. Our ﬁrst step is to characterize the temporal equivalent of a curved dielectric interface: a spatial lens comprises of two such interfaces separated by a length of material of enhanced refractive index. At a planar dielectric in- terface between two media of refractive indices n1and n2, a Gaussian beam undergoes a change in the radius of curvature, but is unchanged in beam width, R2=n2 n1R1, w 1=w2. (44) By analogy, a chirped Gaussian pulse at the interface be- tween two ﬁbers of GVD coeﬃcients β′′ 1andβ′′ 2trans- forms to a diﬀerent chirp, but with unchanged pulse width, 1 β′′ 2R2=1 β′′ 1R1, τ 1=τ2. (45) Of course, the pulse width evolves diﬀerently in the two sections of ﬁber, τ2 i(z) =τ2 0i/parenleftbigg 1 +z2 ζ2 0i/parenrightbigg i= 1,2 (46)whereζ0iis the dispersion length in ﬁber i. The ABCD matrix for a (spatial) spherical dielectric interface and its temporal translation are M:/parenleftBigg 1 0 n2−n1 n2Rn1 n2/parenrightBigg /ma√sto→/parenleftBigg 1 0 1−β′′ 2/β′′ 1 Rlβ′′ 2 β′′ 1/parenrightBigg (47) What does this represent? We use the ABCD bilinear transformation, q2=q1/slashbigg/bracketleftbigg1 Rl/parenleftbigg 1−β′′ 2 β′′ 1/parenrightbigg q1+β′′ 2 β′′ 1/bracketrightbigg (48) which implies that 1 q2=/parenleftbigg1 R2+i2β′′ 2 τ2 2/parenrightbigg =1 Rl/parenleftbigg 1−β′′ 2 β′′ 1/parenrightbigg +/parenleftbigg1 R1+i2β′′ 1 τ2 1/parenrightbiggβ′′ 2 β′′ 1(49) After some algebraic manipulation, we can write the above as β′′ 1/parenleftbigg1 Rl−1 R2/parenrightbigg =β′′ 2/parenleftbigg1 Rl−1 R1/parenrightbigg (50) showing explicitly how the chirp transforms at this inter- face. The ABCD matrix for a temporal lens of “thickness” dis written as the product of three ABCD matrices rep- resenting, when read from right to left, a transition from the input ﬁber to the ﬁber that deﬁnes the thin temporal lens, propagation in the second ﬁber, and a transition back to the input ﬁber, M=/parenleftBigg 1 0 1−β′′ 1/β′′ 2 R2β′′ 1 β′′ 2/parenrightBigg ./parenleftBigg 1d 0 1/parenrightBigg ./parenleftBigg 1 0 1−β′′ 2/β′′ 1 R1β′′ 2 β′′ 1/parenrightBigg and, multiplying the matrices together, we get a single ABCD matrix which deﬁnes the output qtparameter via the usual bilinear transformation ( Aqt+B)/(Cqt+D), M= 1 +d R1/parenleftBig 1−β′′ 2 β′′ 1/parenrightBig dβ′′ 2 β′′ 1 d R1(−R2)/parenleftBig β′′ 2 β′′ 1−1/parenrightBig/parenleftBig 1−β′′ 1 β′′ 2/parenrightBig −/parenleftBig 1 R1+1 −R2/parenrightBig/parenleftBig 1−β′′ 1 β′′ 2/parenrightBig 1 +d −R2/parenleftBig 1−β′′ 2 β′′ 1/parenrightBig  (51) The temporal focal length ˆftis analogous to the spatial focal length and is given by −A/C, ˆft=/bracketleftbigg 1 +d R1/parenleftbigg 1−β′′ 2 β′′ 1/parenrightbigg/bracketrightbigg/slashbigg/bracketleftbigg/parenleftbigg1 R1+1 −R2/parenrightbigg/parenleftbigg 1−β′′ 1 β′′ 2/parenrightbigg −d R1(−R2)/parenleftbiggβ′′ 2 β′′ 1−1/parenrightbigg/parenleftbigg 1−β′′ 1 β′′ 2/parenrightbigg/bracketrightbigg (52) The temporal focal length deﬁnes the time from the out- put pl ane at which an initially unchirped pulse becomes7 unchirped again. We can write the above in slightly simpler notation, for the speciﬁc case R1=−R2=R, and letκ=d/R, ∆β′′=β′′ 2−β′′ 1, ˆft= 1−κ∆β′′ β′′ 1 1−κ 2∆β′′ β′′ 1β′′ 2 ∆β′′ R 2(53) where the term in parentheses represents an enhancement factor over the “thin lens” formula. Forκ≪1, we can simplify the above expression keep- ing terms of O(κ), 1 ft≈/parenleftbigg 1−κ 2∆β′′ β′′ 1/parenrightbigg/parenleftbigg 1 +κ∆β′′ β′′ 1/parenrightbigg2∆β′′ Rβ′′ 2 ≈/parenleftbigg 1 +κ 2∆β′′ β′′ 1/parenrightbigg2∆β′′ Rβ′′ 2(54) The above relation conﬁrms our physical intuition that if β′′ 2−β′′ 1= ∆β′′<0, then we have reduced ft, the distance to the point of zero chirp from the output plane, for an initially unchirped input pulse. We now have the tools we need to analyze a reasonably complicated practical problem: designing the length of a dispersion map so as to get self-consistent eigen-pulses with periodic pulse width and chirp. III. DISPERSION-MANAGED SOLITON TRANSMISSION EXPERIMENT It has been recently found that a stable, self-consistent pulse solution exists in a dispersion-managed ﬁber trans- mission system [5]. While these are not solitons in the strict mathematical sense, they have been called dispersion-managed solitons, or perhaps more appropri- ately, breathers. They demonstrate periodic behaviour: the pulse width and chirp of Gaussian breathers, for instance, are periodic functions of the propagation dis- tance. Breathers share a property in common with soli- tons in that they can propagate indeﬁnitely without losing shape; even though the pulse shape undergoes changes within a disperion map period, the pulse does not disperse away to inﬁnity, or tend to self-focus to a point either of which invalidate the applicability of the nonlinear Schr¨ odinger equation after a certain distance. A dispersion-mapped (DM) soliton is closer to a Gaus- sian shape than the hyperbolic secant of the nonlinearSchr¨ odinger equation [20], and it is interesting to ask whether our analysis is capable of capturing the essen- tial aspects of its evolution along a dispersion-mapped transmission channel. We consider, as our example, the paper by Mu et al. [21] who have simulated DM soliton dynamics in a recirculating ﬁber loop. Their dispersion map consists of 100 km of dispersion shifted ﬁber (SMF-LS) with nor- mal dispersion D1equal to -1.10 ps/nm-km at 1551 nm, followed by an “approximately 7-km span” of standard single-mode ﬁber (SMF-28) with an anomalous disper- sionD2equal to 16.6 ps/nm-km at 1551 nm. The re- sults of the paper indicate that Gaussian shaped pulses of pulse duration 5.67 ps and peak power 9 dBm were used. We will derive the result that, for these parame- ters and given the length of SMF-LS ﬁber, the length of SMF-28 ﬁber that needs to be used is indeed “approx- imately 7-km”. In other words, we will show that this given dispersion map can support lowest-order chirped Gaussian self-consistent solutions, i.e. breathers. The dispersion map, shown schematically in Figure 2, consists of three ﬁber segments: a length z1/2 equal to 50 km of SMF-LS ﬁber, followed by a length z2of SMF- 28 ﬁber, whose numerical value is to be determined, and then the remainder z1/2 of SMF-LS ﬁber. Each segment of ﬁber has nonlinear characteristics, which we model via a time lens situated, for simplicity at the individual mid- points of the respective segments. Consequently, each segment is described by the cascaded product of three ABCD matrices, with two additional matrices represent- ing the transitions between ﬁbers of diﬀerent β′′. For simplicity, we will assume that the nonlinear properties of the ﬁbers are identical. The overall ABCD matrix for the system can be writ- ten down quite easily, M=/parenleftBigg 1z1/4 0 1/parenrightBigg ./parenleftBigg 1 0 −1/ft1/parenrightBigg ./parenleftBigg 1z1/4 0 1/parenrightBigg ./parenleftBigg 1 0 0β′′ 1/β′′ 2/parenrightBigg ./parenleftBigg 1z2/2 0 1/parenrightBigg ./parenleftBigg 1 0 −1/ft1/parenrightBigg ./parenleftBigg 1z2/2 0 1/parenrightBigg ./parenleftBigg 1 0 0β′′ 2/β′′ 1/parenrightBigg ./parenleftBigg 1z1/4 0 1/parenrightBigg ./parenleftBigg 1 0 −1/ft1/parenrightBigg ./parenleftBigg 1z1/4 0 1/parenrightBigg (55) which, after some algebra, can be written as an ABCD matrix with the following elements,8 A=/bracketleftbigg/parenleftbigg 1−z1 4f/parenrightbigg/parenleftbigg 1−z2 2f/parenrightbigg −z1 4fβ′′ 1 β′′ 2/parenleftbigg 2−z1 4f/parenrightbigg/bracketrightbigg/parenleftbigg 1−z1 4f/parenrightbigg −/bracketleftbiggβ′′ 2 β′′ 1/parenleftbigg 1−z1 4f/parenrightbigg/parenleftbigg 2−z2 2f/parenrightbiggz2 2f+/parenleftbigg 2−z1 4f/parenrightbigg/parenleftbigg 1−z2 2f/parenrightbiggz1 4f/bracketrightbigg (56) D=−z1 4f/bracketleftbigg/parenleftbigg 1−z2 2f/parenrightbigg +β′′ 1 β′′ 2/parenleftbigg 1−z1 4f/parenrightbigg/bracketrightbigg/parenleftbigg 2−z1 4f/parenrightbigg −/bracketleftbiggβ′′ 2 β′′ 1z2 2f/parenleftbigg 2−z2 2f/parenrightbigg −/parenleftbigg 1−z2 2f/parenrightbigg/parenleftbigg 1−z1 4f/parenrightbigg/bracketrightbigg/parenleftbigg 1−z1 4f/parenrightbigg (57) B=/bracketleftbigg/parenleftbigg 1−z1 4f/parenrightbigg/parenleftbigg 1−z2 2f/parenrightbigg −z1 4fβ′′ 1 β′′ 2/parenleftbigg 2−z1 4f/parenrightbigg/bracketrightbiggz1 4/parenleftbigg 2−z1 4f/parenrightbigg +/bracketleftbiggβ′′ 2 β′′ 1/parenleftbigg 1−z1 4f/parenrightbigg/parenleftbigg 2−z2 2f/parenrightbiggz2 2+/parenleftbigg 2−z1 4f/parenrightbigg/parenleftbigg 1−z2 2f/parenrightbiggz1 4/bracketrightbigg/parenleftbigg 1−z1 4f/parenrightbigg (58) C=−1 f/bracketleftbigg/parenleftbigg 1−z2 2f/parenrightbigg +β′′ 1 β′′ 2/parenleftbigg 1−z1 4f/parenrightbigg/bracketrightbigg/parenleftbigg 1−z1 4f/parenrightbigg −1 f/bracketleftbigg −β′′ 2 β′′ 1z2 2f/parenleftbigg 2−z2 2f/parenrightbigg +/parenleftbigg 1−z2 2f/parenrightbigg/parenleftbigg 1−z1 4f/parenrightbigg/bracketrightbigg (59) The algebraic complexity of writing out the expressions explicitly should not mask the simplicity of multiplying two-by-two matrices, usually numerically. Note that the expression (57) for Dis algebraically identical to that for A(56), and it may be veriﬁed that AD−BC= 1. Theq-parameter (we have dropped the tsubscript in this section for notational elegance) evolves according to the bilinear transformation law, and we require that the pulse repeat itself after propagation through one such ABCD matrix, 1 q=A+B/q C+D/q(60) which has the solution 1 q=D−A 2B±i/radicalBigg 1−/parenleftbiggD+A 2/parenrightbigg2 B(61) SinceD=Ain our above analysis, we already see that qis purely imaginary at z= 0 i.e. the pulse has zero chirp at the midplanes, as we would expect a breather to have. At this stage, we can substitute numerical values for the various parameters (except z2, which is what we seek) into the expressions for the A,B,CandDelements (56– 57) and solve (61) numerically for z2. While this is not diﬃcult, and already yields a quick solution to the prob- lem at hand, we can get further insight via a well-justiﬁed simpliﬁcation as follows. Theqparameter at the midplanes, where it is purely imaginary, is given by 1 q0=2\|β′′ 1\| τ2 0(62) whereβ′′ 1= 1.40×10−27s2/m and input pulse width τ0= 5.67×10−12s. Consequently, for such pulses, 1 /q≈ 0, and since A=D, this implies that A= 1 in (61).With the notational substitutions x=z2 2f, y=z1 4f, β′′ r=β′′ 1 β′′ 2(63) we get the necessary condition /bracketleftbig (1−y)2−(2−y)y/bracketrightbig (1−x)−β′′ ry(2−y)(1−y) −1 β′′r(1−y)(2−x)x= 1. (64) The solution of this equation is given by x=β′′ ry(2−y) 1−y(65) or, in terms of the initial variables, z2=/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ′′ 1 β′′ 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglez1 2/parenleftbigg 2−z1 4f/parenrightbigg /parenleftbigg 1−z1 4f/parenrightbigg (66) which is the necessary condition in order to have a stable self-consistent Gaussian eigen-pulse (breather) solutio n to the dispersion-map problem. All that remains is for us to interpret the variables in terms of the original problem and numerically evaluate this expression to get the desired length z2of SMF-28 ﬁber in this dispersion map. The various numerical val- ues are as follows: β′′ 1= 1.40×10−27s2/m, β′′ 2=−2.12×10−26s2/m, τ0= 5.67×10−12s, z1= 105m9 SMF-LS SMF-LS SMF-28 z12z12 z2 time lens GVD transition propagation(a) (b) FIG. 2: (a) Analytical schematic of dispersion map from [21] and (b) its representation to express in terms of ABCD matrix elements. Given the nature of the problem, we realize our time lens with the nonlinear ﬁber as described earlier (41), so that 1 f=−4β′′ NL/parenleftbigg2π λ/parenrightbiggn2IpLNL τ2(67) and takeβ′′ NL=β′′ 2,Ip= 3.62×106W/M2so that with ﬁber core area Aeﬀ= 47µm, we getP= 8 mW = 9 dBm. Also, we take LNL=z2consistent with our choice ofβ′′ NL. The numerical solution (of the quadratic equation) for z2is equal to 7.00 km which is indeed the value “approx- imately 7 km” stated in the paper [21]. In spite of appar- ent exact agreement, we should be careful to appreciate that this analysis is a characterization of only the most important processes in this experiment. Possible sources for approximation include the fact that a DM soliton is only approximately Gaussian, and that we have repre- sented the combined dispersive and nonlinear properties of the ﬁber segments by a single temporal lens. A better approximation may be to include several temporal lenses for each segment of ﬁber; this would make the algebraic expressions in this paper quite cumbersome to write down explicitly, but the numerical computation would not be much more diﬃcult, since the matrices are only two-by- two in size, and comprise of purely real elements. The experimental conﬁguration of [21] also includes several other elements which can aﬀect the pulse shape, such as ﬁlters, ﬁber ampliﬁers and polarization controllers.IV. HERMITE-GAUSSIAN BASIS Our ABCD matrix formalism for pulse propagation ap- plies to chirped Gaussian pulses. To analyze more com- plicated shapes, we can expand the given pulse shape in a basis of chirped Hermite-Gaussian functions, which form a complete orthonormal basis [18, 20]. The Hermite- Gaussian function (we consider only unchirped Gaussians here for simplicity) of order nis deﬁned as the product of the Hermite polynomial of order nwith a Gaussian function, ψn(t)≡Hn(t)exp(−t2/2), (68) where, for example, H0(t) = 1, H 1(t) = 2t, H 2(t) = 4t2−2.(69) We can expand an arbitrary input amplitude u0(t) in this basis, analogous to expanding a ﬁeld in terms of plane wave components, as in solution techniques of the standard parabolic diﬀraction equation by means of the Fourier transform, u0(t) =/summationdisplay n=0cnHn(t)exp(−t2/2) (70) where because of orthogonality of the Hermite-Gaussians, the expansion coeﬃcients are given by cn=1√π2nn!/integraldisplay∞ −∞u0(t)Hn(t)exp(−t2/2).(71) The propagation equation (16) deﬁnes the output pulse shape as the convolution of the input shape with a Gaus- sian kernel. Hermite-Gaussians, when convolved with a Gaussian, yield the product of a Hermite polynomial and a Gaussian [18], /integraldisplay∞ −∞dt0ψn(t0)exp/bracketleftBig −a 2(t−t0)2/bracketrightBig = /radicalbigg 2π a+ 1/parenleftbigga−1 a+ 1/parenrightbiggn 2 exp/bracketleftbiggat2 2(a2−1)/bracketrightbigg ψn/bracketleftbigga√ a2−1t/bracketrightbigg . (72) Taking as input the n-th Hermite-Gaussian mode u0= ψn(t) (which has width τ0=√ 2), we evaluate the am- plitude of this mode after propagation through distance z, un(z,T) =/radicalbigg1 1 +iβ′′z/bracketleftBigg −1 +i β′′z 1−i β′′z/bracketrightBiggn/2 ×exp/bracketleftBigg i 2β′′zT2 1 +1 (β′′z)2/bracketrightBigg ψn/bracketleftBigg T/radicalbig 1 + (β′′z)2/bracketrightBigg(73) which can be seen to agree with (19). A Hermite-Gaussian therefore maintains its shape dur- ing propagation, but adds a chirp (which is the same for10 all modes) and a scaling of the width according to (19). Power conservation implies that the amplitude corre- spondingly scales down. The only term that is dependent on the order of the Hermite-Gaussian is a phase term; higher-order modes have greater phase advances, since their spectral content is higher. The important observa- tion is that the orthogonality of the Hermite-Gaussian expansion is preserved, and so this expansion may be used to predict the pulse shape obtained by propagating an input pulse. Our formalism remains valid as long as the diﬀerential equation describing the propagation of a particular order Hermite-Gaussian is of the form (15), i.e. the slowly-varying envelope approximation is valid. Therefore, we can expect that the lower-order expansions are usually valid; the results of applying our analysis to higher-order expansion terms generate the residual ﬁeld corrections to the lower order results [22]. V. CONCLUSIONS We have developed a 2 ×2 ABCD matrix formalism for describing pulse propagation in media described by Maxwell’s equations, accounting for dispersion, nonlin- ear and gain/loss mechanisms. The method is analo- gous to techniques used in CW beam diﬀraction analy- sis, and correspondingly similar phenomena can be pre- dicted, such as chirp transformation, focusing, periodic pulse width expansion and narrowing etc. The spatial q parameter has a time equivalent qTin accordance with the given space-time translation rules. The real and imaginary parts of q−1 Trepresent the chirp and the width of the pulse as a function of propagation distance z. The propagation of various input pulse shapes can be described by expanding the given pulse in a basis of Hermite-Gaussian functions; the ABCD formalism ap- plies to each Gaussian wave function separately. Propa- gation through a complicated system of optical elementsis simple to calculate in terms of ABCD matrices: the resultant matrix is the cascaded product of the ABCD matrices of each of the individual elements with the ap- propriate ordering. The overall qTparameter is given by a bilinear transformation in terms of the ABCD elements of the overall product matrix, exactly analogous to the spatial case. We have formulated ABCD matrices for pulse propa- gation in dispersive ﬁbers, and for temporal lenses which can characterize self-phase modulation phenomena. A spatial dielectric interface translates to an interface be - tween ﬁber segments of dissimilar GVD coeﬃcient β′′. The temporal equivalent of a curved dielectric interface is useful for characterizing the transition between such dissimilarβ′′ﬁbers with the added presence of ﬁber non- linearities arising from the nonlinear index of refraction n2. We have used these tools to characterize a reason- ably complicated real-life system: calculation of the dis- persion map for self-consistent stable propagation of a dispersion-managed soliton. We believe this method of analysis forms a useful complement to conventional pulse propagation methods, such as the split-step Fourier transform numerical proce- dures [3] which are substantially more computationally intensive. The ABCD approach is useful for clarifying the important dispersive and nonlinear focusing eﬀects in dispersion-mapped nonlinear ﬁber segments. Together with the variational approach [8], based on modeling the pulse as a dynamical system characterized by a Hamilto- nian functional [9], the q-parameter oﬀers an insight into pulse evolution from a theoretical standpoint. Acknowledgments This work was supported by the Oﬃce of Naval Re- search and the Air Force Oﬃce of Scientiﬁc Research. [1] B. Crosignani, P. Di Porto, and E. Caglioti, in Nonlinear Waves in Solid State Physics , edited by A. Boardman, M. Bertolotti, and T. Twardowski (Plenum Press, 1990), NATO ASI Series B. Vol. 247. [2] G. Carter, R.-M. Mu, V. Grigoryan, C. Menyuk, P. Sinha, F. Carruthers, M. Dennis, and I. Duling III, Electronics Letters 35, 133 (1999). [3] G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989). [4] M. Suzuki, I. Morita, S. Yamamoto, N. Edagawa, H. Taga, and S. Akiba, Electronics Letters 31, 2027 (1995). [5] N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, Electronics Letters 32, 54 (1996). [6] J. Kutz, P. Holmes, S. Evangelides, and J. Gordon, J. Opt. Soc. Am. B 15(1), 87 (1998). [7] D. Marcure and C. Menyuk, Journal of Lightwave Tech- nology 17(4), 564 (1999).[8] D. Anderson, Phys. Rev. A 27, 3135 (1983). [9] D. Muraki and W. Kath, Physica D 48, 53 (1991). [10] H. Goldstein, Classical Mechanics (Addison-Wesley, 1950). [11] S. Akhmanov, A. Chirkin, K. Drabovich, A. Kovrigin, R. Khokhlov, and A. Sukhorukov, IEEE Journal of Quan- tum Electronics QE-4 (10) (1968). [12] A. Yariv and P. Yeh, Optics Communications 27(2) (1978). [13] P. B´ elanger and P. Mathieu, Optics Communications 67(6), 396 (1988). [14] B. Kolner and M. Nazarathy, Optics Letters 14(12) (1989). [15] A. Lohmann and D. Mendlovic, Applied Optics 31(9) (1992). [16] A. Yariv, Journal of Nonlinear Optical Physics and Ma- terials 8(1), 165 (1999). [17] A. Yariv, Optical Electronics in Modern Communications11 (Oxford University Press, 1997), 5th ed. [18] H. Haus, Waves and Fields in Optoelectronics (Prentice- Hall, 1984). [19] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), seventh ed. [20] T. Lakoba and D. Kaup, Physical Review E 58(5), 6728 (1998).[21] R.-M. Mu, V. Grigoryan, C. Menyuk, G. Carter, and J. Jacob, IEEE Journal of Selected Topics in Quantum Electronics 6(2) (2000). [22] P. Tchofo Dinda, K. Nakkeeran, and A. Moubissi, Optics Communications In press (2000).
arXiv:physics/0103012v1 [physics.atom-ph] 5 Mar 2001Absorption Line Shape of a One-Dimensional Bose Gas S. K. Yip Institute of Physics, Academia Sinica, Nankang, Taipei 115 29, Taiwan Abstract We discuss the line shape for an internal state transition fo r bosonic atoms conﬁned as a one-dimensional gas. Typical line shape is an ed ge singularity due to the absence of Bose-Einstein Condensation in such sys tems. PACS numbers: 03.75.Fi Recent progress in trapped bosonic gases has stimulated a lo t of activities in the study of many-body eﬀects in such systems. Most theoretical paper s concentrate on eﬀects due to the existence of the Bose-Einstein Condensate (BEC) and t he associated macroscopic wave-function. With current techniques, it is likely that o ne can as well study an eﬀectively one-dimensional (1D) trapped gas [1]. This situation occur s when the bosons are put in a trap which is tightly conﬁning in two directions ( yandz) but is essentially free in the third (x), and with particles occupying only the lowest quantized su bband for motions in they-zplane. This system is unique in that, for any ﬁnite interacti on among the bosons, there is no Bose-Einstein Condensation even at zero tempera ture. [2]. The long-wavelength quantum ﬂuctuations destroy the phase coherence of the syst em in the macroscopic limit. For a given density n, the occupation number N0of the lowest single particle state, though in general increases with the size and hence the total number of particles Nin the system, is nevertheless only a negligible fraction of the total (lim N→∞N0 N→0 ). Though many properties of 1D bose gas have been studied in the past [2], the subject re mains somewhat academic since there exists no physical realization. The possibilit y of realizing this 1D bose gas has already stimulated some more theoretical work on this syste m recently [3,4] The present 1D bosonic system is closely related to its fermi onic counterpart; and they are collectively known as the “Luttinger liquids” [5]. One-dim ensional fermionic systems have been studied much more extensively in the context of quantum wires, carbon nanotubes [6,7] etc. There the one-dimensional nature leads to the abs ence of fermi liquid behavior, so the properties of the system is qualitatively diﬀerent from its non-interacting counterpart. The atomically trapped 1D bose gas oﬀers us an excellent new o pportunity to study the peculiar properties of quantum systems in reduced dimensio ns, in particular the consequence due to the absence of BEC. Not only that this is the ﬁrst 1D boso nic system available, it provides advantages over the mentioned 1D fermionic cases i n that the present system is intrinsically clean, and moreover numerous atomic and opti cal tools can now be used. It may even be possible to further study novel systems such as mi xtures etc. In this paper we consider one such example, namely the absorp tion line shape of an optical transition in this 1D bose gas. We imagine initially a quasi-1d system of bosons in their internal ground state (referred to as the ‘a-atoms’ be low). An incident optical wave excites one of the atoms to a diﬀerent internal state. We are i nterested in the absorption 1line shape of this process; i.e., the probability that the absorption takes place at a given frequency Ω. We shall consider the problem at T= 0, paying particular attention to the special feature due to the absence of BEC. An analogous exper iment has already been done in athreedimensional trap for the 1s →2s transition in hydrogen [8]. For a uniform 3D bose gas atT= 0, the line is expected to be narrow, Lorentzian (see, e.g. [ 9], c.f. however [8]) with a width governed by the small ‘gaseous parameter(s)’ ( n3da3)1/2. Heren3dis the particle number density in 3d and a(’s) is (are) the s-wave scattering length(s) for scatterin g among the bosons. For this three dimensional case, the main weight of the transition comes from exciting a particle from momentum /vector p= 0. There is a macroscopic number of such particles. For a weakly interacting gas, the potential felt by the rest o f the bosons is only slightly modiﬁed after the transition. We shall see that the situatio n in 1D is very diﬀerent due to the absence of BEC. Exciting a single particle necessitat es a substantial rearrangement of the relative motion among the rest of the particles, i.e., emission of a large number of phonons. The basic line shape is typically an edge singulari ty. We shall then consider an atomic trap with tight conﬁnement i n they-zdirections. For deﬁniteness we shall assume that the conﬁnement potentials in these directions are isotropic and harmonic, with frequency ω⊥. We shall consider the case where the atoms are limited within 0<x<L but otherwise free. A weak harmonic trap potential along the x-axis has been considered in the past [10,11], and the low dimensional eﬀects have been shown to be signiﬁcantly reduced. We shall assume that such a potential is absent here. We shall further ignore the eﬀect due to ﬁniteness of L, a condition which we shall return below. The bosonic system is described by a Hamiltonian containing the kinetic energy and a short-range interaction among the bosons. Anticipating th at we shall eventually study the system in the quasi-1d limit, the ﬁeld operator for the origi nal bosons, referred to hereafter as the a-bosons, is expanded as ψa(/vector r) =ψa(x)χ0(/vector r⊥) +/summationtext jψaj(x)χj(/vector r⊥) whereψa(x) is the annihilation operator for the a-bosons in the lowest sub band (0) where the transverse wavefunction is χ0(/vector r⊥).ψaj(x)’s (j= 1,2,...) are the corresponding operators for the higher subbands (wavefunctions χj(/vector r⊥)). In the ground state \|G>,ψaj(x)\|G>= 0 since all particles reside in the lowest subband. We shall see that ψajforj= 1,2,...will not appear anywhere below, and can pretend that the expansion of ψa(/vector r) will consist thus only the term involving the lowest subband. The extent of the wavefunctio nχ0in the y-z direction will be denoted as a⊥. For harmonic trapping potentials in these directions a⊥=/radicalBig ¯h mω⊥. We shall not write down the interaction part of the Hamiltoni an involving the a-bosons explicitly since we won’t need it below. For delta-function type interactions it is possible to write down all the formal many-body wavefunctions [12]. Unf ortunately manipulations of such wavefunctions are typically very mathematically invo lved. Moreover we shall eventually be interested in ‘ﬁnal’ states where an a-atom has been excit ed internally. Such wavefunc- tions, where eﬀectively there is a ‘foreign’ atom present, a re not known (with an exception noted below). We thus proceed rather diﬀerently and follow H aldane [5], concentrating only on the low energy excitations which are density oscillation s in the system. The ﬁeld operator ψa(x) ( =/summationtext papeipx/√ L) describing the motion along xis re-expressed in terms of density n(x) and phase φ(x) byψa(x) = [n(x)]1/2eiφ(x). The eﬀective Hamiltonian for the 1d motion can be written as H0=¯h 2π/integraldisplay dx/bracketleftbigg vK(∇φ)2+v K(∇θ)2/bracketrightbigg (1) 2where ∇θis related to the number density ﬂuctuations δn(x) by∇θ=πδn.φ(x) and θ(x) must be considered as operators satisfying the commutatio n relation [φ(x),∇x′θ(x′)] = iπδ(x−x′).vis the density (sound) velocity wave of the system, and Kis the (dimen- sionless) Luttinger liquid interaction parameter. vK=πno¯h/mandKis related to the compressibility of the system via K=πv¯h(∂no/∂µ) wherenois the linear number density andµis the chemical potential of the system. Kdepends on the interaction among the bosons. For short range interactions, Kin principle have already been obtained [5,12] but there is no general analytic form known. We simply note here t hatK= 1 for the strongly repulsive (impenetrable) limit while K→ ∞ if the interaction among the bosons is weak. H0can be diagonalized easily [5] since it is quadratic. It is us eful to introduce bosonic operatorsbqandb† qwhich describe the sound modes of the system: θ(x) =−i/summationdisplay q/negationslash=0\|πK 2qL\|1/2sgn(q)eiqx(b† −q+bq) φ(x) =−i/summationdisplay q/negationslash=0\|π 2qLK\|1/2eiqx(b† −q−bq) (2) Then, apart from a constant, H0=/summationtext q/negationslash=0¯hωqb† qbqwhere the mode frequencies are ωq=v\|q\|. Now we consider the external optical ﬁeld responsible for th e transition in inter- nal state. Since the atom with a new internal state is disting uishable from the origi- nal atoms, we shall call the resulting atom the ‘c-atom’. For deﬁniteness we consider only the ‘Doppler-free’ part of the spectrum ( no momentum tr ansfer from the exciting laser beam(s)) and assume that the perturbation responsibl e for the excitation is uniform throughout the 1D bose gas. The relevant part of the Hamilton ian can be then written as Hex=w/integraltextd3/vector r/parenleftBig ψ† c(/vector r)ψa(/vector r) +ψ† a(/vector r)ψc(/vector r)/parenrightBig whereψc(/vector r) is the ﬁeld operator for the atom in the excited internal state. We are interested in the rate of t ransition as a function of the excitation frequency Ω. This rate I(Ω) is given simply by the golden rule: I(Ω) = 2π/summationdisplay F\|<F\|Hex\|G>\|2δ(Ω−(EF−EG)) (3) where\|F >are the set of ﬁnal states. To proceed further we need to know t he fate of the c-atom. There are many possibilities and we shall simply con sider two extreme limits: (1) An atom in the internal state c is not aﬀected at all by the t rapping potential for the atoms in state a. This is possible if, e.g., this potentia l is due to a laser with frequency near a dipole resonance for the a-atoms but far away from any o f those for c. In this case the c-atom escapes from the trap and no longer interacts with the remaining a-atoms. (2) The c-atom also feels the strong transverse conﬁnement p otential. It remains inside the trap and continues to interact with the a-atoms. We shall treat these two cases in turn. Case 1: In this case the ﬁnal state is simply a product state, \|F >=\|Φ>×\|/vector p>, with theN−1 a-atoms in state \|Φ>(with x momentum −px) and the c-atom in the plane wave state\|/vector p>with momentum /vector p. The rate for transition into states with given /vector pis I/vector p(Ω) = 2π\|f/vector p\|2/summationdisplay Φ\|<Φ\|apx\|G>\|2δ(Ω−(EΦ+/vector p2/2m+u−EG)) (4) 3wheref/vector p≡/integraltextd2/vector r⊥e−i/vector p⊥·/vector r⊥χ0(/vector r⊥) is a form factor, and uis the diﬀerence in internal energy between the a- and c- atoms. This rate is thus proportional to the spectral function for annihilation of a particle in the 1D bose gas (hereafter px→p) B(p,ω′)≡/summationdisplay Φ\|<Φ\|ap\|G>\|2δ(ω′−(EΦ−EG,N−1)) (5) withω′= Ω +EG−EG,N−1−/vector p2/2m−uand where EG,N−1is the ground state energy of N−1 atoms in the 1D trap. B(p,ω′) is given by −1 πImG1(p,ω′) where G1(x,t)≡i<ψ† a(0,0)ψa(x,t)>h(t) (6) The required Green’s function can be evaluated by ﬁnding the long distance and time behavior of G1(x,t). Substituting eq (2) into eq(6) one ﬁnds G1∼1 (x2+v2t2)1/4Kh(t). The Fourier transform G1(p,ω′) and hence B(p,ω′) for a given phas the scaling form ∼h(ω′−v\|p\|)/(ω′2−v2p2)1−αwithα=1 4K. Since 1< K < ∞the line shape is of the form of an edge singularity. For the three dimensional bose g as, the corresponding B(/vector p,ω′) is proportional to a delta function at the Bogoliubov mode fr equency (with an additional small incoherent part). The lack of a delta function in the pr esent case reﬂects the absence of a condensate. If /vector pcan be independently measured this experiment would be anal ogous to angle resolved photoemission spectroscopy employed ext ensively in recent studies of high T-c cuprates [13]. Case 2: In this case the ﬁnal result depends on the interaction betwe en the a and c atoms, which has the general form gac/integraltextd/vector rψ† c(/vector r)ψc(/vector r)ψ† a(/vector r)ψa(/vector r). Heregacis deﬁned in 3D and is related to the 3D scattering length aacbygac= 4π¯haac/m. We express ψc(/vector r) also in subbandsψc(/vector r) =/summationtext jψcj(x)χcj(/vector r⊥). The ﬁnal states in eq (3) in general can involve terms that are oﬀ-diagonal in the band index jfor the c-atom due to the interaction between a and c. Such processes correspond to the possibility that the transverse motion of the c- atom be modiﬁed due to the interaction with the a-atoms. For a tight trap however, these contributions are small in the dilute limit. They involve th e parameter nogac/πa2 ⊥ω⊥∼ aac/l. Herel= 1/nois the average interparticle spacing among the a-atoms. Ign oring thus these contributions, we have I(Ω) =/summationtext j\|fj\|2Ij(Ω),i.e., the transition line then becomes a superposition of ‘lines’ involving transition to ﬁnal stat es where the c-atom is within a given j-th subband, with a weight given by the form factor fj=/integraltextd/vector r⊥[χ0(/vector r⊥)χ∗ cj(/vector r⊥)]. The shape of each of these lines is determined by Im[ DR j(Ω)] where DR j(t) =−i/integraldisplay dx1/integraldisplay dx2<ψ† a(x1,t)ψcj(x1,t)ψ† cj(x2)ψa(x2)>h(t) (7) which we shall now compute. The eﬀective Hamiltonian is give n by the sum of H0involving only the a-atoms and Hcjwhich describes the motion of the c-atom within its j-th subband and its interaction with the a-atoms. H0has already been given in eq(1) and the eﬀective Hcjis given by H(1) cj+Hint cjwhere H(1) cj=/integraldisplay dx  ¯h2 2m∂ψ† cj(x) ∂x∂ψcj(x) ∂x+ (ǫcj+u)ψ† cj(x)ψcj(x)  (8) consists of the ‘one-body’ (kinetic, trap and internal ener gy) terms and 4Hint cj=gac,jj/integraldisplay dxψ† cj(x)ψcj(x)ψ† a(x)ψa(x) (9) is due to the interaction between the a and c atoms. Here ǫcjis the subband energy for the c-atom in its j-th subband and gac,jj=gac/integraltextd3/vector r\|χcjχ0(/vector r⊥)\|2depends on j. For small jit is of ordergac/πa2 ⊥if the trap potentials for a and c atoms are similar. It is convenient to express DR j(t) in the ‘interaction’ picture: DR j(t) =−i/integraldisplay dx1/integraldisplay dx2<ψ† a(x1,t)ψcj(x1,t)ˆTe−i/integraltextt 0Hint cj(t′)dt′ψ† cj(x2)ψa(x2)>h(t) (10) where ˆTis the time-ordering operator and the expectation value is t o be calculated with the Hamiltonion H0+H(1) cj. One can rewrite eq(10) in momentum space and expand the expo nent involving the interaction. Hint cjthen produces ‘scattering’ terms where the momentum of the c-atom is changed from one value to the other. The resulting c alculation does not seem to be tractable analytically in general. The problem however c an be simpliﬁed signiﬁcantly if we assume that gac,jjis small (condition given below) and concentrate again on th e low frequency limit, i.e., just above the threshold for transit ion into the j-th subband (given by ξcj=ǫcj−ǫ0−µ+u+gac,jjn1dto lowest order in gac). In this case transitions must be made to states where the momenta pof the c-atom are small. Provided that p<<mv , then the kinetic energy (of motion along x) transferred to the c-atom p2/2mis much less than that to the density oscillations of a-atoms vp. Thus so long as we are interested in frequency deviations ∆Ω from the threshold which satisfy ∆Ω << mv2, one can ignore the kinetic energy of the motion of the c-atoms along x. For an impenetrable bose gas this amounts to limiting ourselves to ∆Ω << π2¯h2/ml2. Ignoring thus the ﬁrst term in H(1) cjand rewriting the result back in real space, we obtain DR j(t) =−iLe−iξcjt<ψ† a(0,t)ˆTe−i/integraltextt 0˜Ha,int(t′)dt′ψa(0)>h(t) (11) where the eﬀective interaction Hamiltonian ˜Ha,intacts only on the a-atoms and is given by gac,jjψ† a(0)ψa(0). This result can be understood physically as follows: th e external optical ﬁeld produces an a →cj transition at t= 0 and at a general location say x0, annihilating an a-atom while producing a c-atom there in the j-th subband. Since thecatom moves with velocity p/mwhereas the sound waves move with the speed v, at smallpthe c-atom is essentially stationary. For t >0, due to the c-atom created, a delta-function interaction atx0acts on the a-atoms. We are interested in the overlap between the initial state with an a-atom destroyed at x0and the ﬁnal states with this extra interaction potential. T his overlap is independent of the location of the transition x0which then has been set to 0. The problem has thus become similar to that of X-ray absorption i n solid state physics. There initially (t <0) one has an electron gas in its ground state. At t= 0 an X-ray photon is absorbed which creates a charged nuclei with an extra electr on added to the electron gas, which att>0 feels an extra local potential due to the charged nuclei. No tice however there are slight diﬀerences. Here for t<0 we have an equilibrium 1D interacting bose gas and at t>0 there is one lessboson than initially. The correlation function in (11) and thus the line shape can b e found analogous to the X-ray problem [14]. The expectation value needed can be rewr itten as<ψ†(0)e−i˜Htψ(0)> where ˜H=H0+˜Hint.H0=/summationtext¯hωqb† qbqand˜Hint=gac,jjδn(0) can both be written in 5terms ofbqandb† q. The annihilation operator ψ∼eiφacts like a displacement operator for these bosons: eiφbqe−iφ=bq− \|π 2qLK\|1/2. Thus the expectation value is the same as <ˆTe−i/integraltextt 0(gac,jj−πvN)δn(0,t′)dt′>calculated with the Hamiltonian H0. Here we have deﬁned vN≡v/K. The calculation can be easily done since H0is quadratic in the boson operators bq andb† q. The long time dependence is given by, apart from a part oscil lating sinusoidally with twhich contributes to the line shift, DR j(t)∼1/tα′ jand thus the line shape ∼1/[∆Ω]1−α′ j withα′ j=1 2K[1−gac,jj πvN¯h]2. Notice that the line shape is in general diﬀerent for transi tion to diﬀerent subbands of the ‘c-atom’. It is instructive to compare this result with that of the X-ra y problem when the interac- tion between the nucleus after the X-ray and the electrons is modelled by a delta-function interaction of strength V. There the long time behavior of the corresponding correlat ion function is given by 1 /tαXand the line shape is ∼1/[∆Ω]1−αXwithαX= [1 +N(0)V]2, whereN(0) is the density of states at the fermi level for the electro n gas [14]. Here, the interaction strength is replaced by gac,jjand the density of states is replaced by∂n ∂µ=1 πvN¯h. In both cases the power law line shapes (for αX/negationslash= 0 andα′ j/negationslash= 0) are due to ‘orthogonal catastrophe’, that the overlap between the state just after the excitation and the new ground state of the system vanishes. There are however two diﬀerenc es: the + sign in αXis replaced by a−sign since now there is one less boson at t>0 rather than one more electron in the X-ray absorption problem. The decay of DR j(t) in time is slower ( α′ jis smaller) if gac>0: near the location of the excitation, the reduction in local d ensity of the a-atoms is also what an interaction with gac>0 prefers. The factor 1 /2Karises from the fact that we have a one-dimensional interacting bose gas rather than a three-d imensional fermi liquid. As the repulsive interaction gac,jjbetween the a and c atoms increases, α′ jdecreases and the line sharpens up. The above quantitive result for the exponentsα′ jhowever holds only for small gac,jj(<<πv N¯h). In the limit where interactions among allthe bosons are strongly repulsive, one reaches the impenetrable limit whe re the bosons cannot pass each other. In this limit the statistics, i.e., the indistinguis hability among the a-bosons and the distinguishability between the a- and c- bosons become irre levant. One can check that the wavefunctions written down in [15] for a system of identical bosons only can be generalized to the present case if we replace one of the bosons by the forei gn c-atom. It can be further veriﬁed that the absorption line shape becomes delta functi ons (one for each subband cj) in this impenetrable limit. The experimental observation of the line shape discussed ab ove should be feasible. The possibility of obtaining a quasi-1D bose gas has already bee n discussed in ref [1]. The above line-shape is applicable so long as ∆Ω< ∼π2¯h ml2≡ωl. For hydrogen atoms and with l∼1µm, ωl∼100kHz, much larger than the present available experimenta l resolutions. It is suﬃcient for the temperature Tto be small compared with the line-width. The condition T << ω l is satisﬁed for T < µK , a temperature readily achievable. A ﬁnite length Lof the system will break the line into a set of discrete sub-lines with sepa rations of order π2¯h/mL2, but the shape of the ‘line’ can still be observable as long as L>>l . I thank Ite Yu for a useful correspondence. 6REFERENCES [1] M. Olshanii, Phys. Rev. Lett. 82, 4208 (1998); and reference therein [2] see articles collected in D. C. Mattis, The Many Body Problem , World Scientiﬁc, 1993. [3] A. G. Rojo, J. L. Cohen and P. R. Berman, Phys. Rev. A 60, 1482 (1999) [4] M. D. Girardeau and E. M. Wright, Phys. Rev. Lett. 84, 5239 (2000) [5] F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981) [6] see, e.g., S. Tarucha, T. Honda and T. Saku, Sol State Comm .,94, 413 (1995) [7] Z. Yao, H. W. Ch. Postma, L. Balents and C. Dekker, Nature, 402, 273 (1999) [8] T. C. Killian et al, Phys. Rev. Lett. 81, 3807 (1998) [9] M. ¨O. Oktel and L. S. Levitov, Phys. Rev. Lett. 83, 6 (1999) [10] T. L. Ho and M. Ma, J. Low Temp. Phys. 115, 61 (1999) [11] D. S. Petrov, G. V. Shlyapnikov and J. T. M. Walraven, Phy s. Rev. Lett. 85, 3745 (2000) [12] E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963); E. H. Lieb, ibid, 1616. [13] see, e.g., M. Randeria and J. C. Campuzzano, in Proceedings of the International School of Physics “Erico Fermi”, Varenna, 1997 (North Holland) [14] S. Doniach and E. H. Sondheimer, Green’s Functions for S olid State Physicists, Frontiers in Physics , vol 44, Addison-Wesley, 1974. [15] M. Girardeau, J. Math. Phys. 1, 516 (1960) 7
arXiv:physics/0103013 6 Mar 2001Probability tree algorithm for general diffusion processes Lester Ingber1,2<ingber@ingber.com>, <ingber@alumni.caltech.edu> Colleen Chen1<cchen@drwin vestments.com> Radu Paul Mondescu1<rmondescu@drwtrading.com> David Muzzall1<dmuzzall@drwin vestments.com> Marco Renedo1<mrenedo@drwin vestments.com> 1DRWInv estments, LLC, 311 S Wacker Dr ,Ste 900, Chicago, IL 60606 2Lester Ingber Research, POB 06440 Sears T ower,Chicago, IL 60606 ABSTRACT Motivated by path-integral numerical solutions of diffusion processes, P ATHINT,w epresent anew tree algorithm, P ATHTREE, which permits e xtremely fast accurate computation of probability distributions of a large class of general nonlinear diffusion processes. Ke ywords: path integral; statistical mechanics PA CSNos.: 05.10.-a, 02.70.-c, 82.20.Wt, 89.90.+nProbability tree ... -2- I ngber,Chen, Mondescu, Muzzall, Renedo 1. INTRODUCTION 1.1. Path Integral Solution of Diffusion Processes There are three equi valent mathematical representations of a diffusion process, provided of course that boundary and initial conditions can be properly speciﬁed in each representation. In this paper we refer to all three representations. The Langevin rate equation for a stochastic process in dScan be written as in a prepoint discretization, dS=fdt+gdW, <dW>=0, <dW(t)dW(t′)>=dtδ(t−t′), ( 1) for general drift fand standard de viationgwhich may depend on Sandt,whereinfandgare understood to be e valuated at the prepoint t.Here, we just consider Sdependent, but our algorithm can easily be extended to time dependent cases and to multi variate systems. This corresponds to a F okker-Planck equation representing the short-time conditional probability P of evolving within time dt, ∂P ∂t=−∂(fP) ∂S+1 2∂2(g2P) ∂S2,( 2) where the diffusion is gi venbyg2. The path-integral representation for Pfor the short-time propagator is gi venby P(S′,t′\|S,t)=1 2πg2Δtexp(−Ldt) L=(dS dt−f)2 2g2 dS dt=S′−S dt,dt=t′−t.( 3) In the abo ve wehav eexplicitly used the prepoint discretization [1]. 1.2. PATHINT Moti vation From Pr evious Study In the abo ve wehav eexplicitly used the prepoint discretization, wherein fandgare understood to be evaluated at the prepoint t.Inthis paper,w ed on ot require multi variate generalizations, or issues dealing with other discretizations, or explication of long-time path-inte gral evaluations, or issues dealing with Riemannian in variance of our distrib utions. There exist other references dealing with these issues in the context of calculations presented here [2-5]. Our approach is moti vated by a multi variable generalization of a numerical path-inte gral algorithm [6-8], PATHINT,used to de velop the long-time e volution of the short-time probability distribution as used in se veral studies in chaotic systems[9,10], neuroscience [9,11,12], and ﬁnancial markets [4]. These studies suggested that we apply some aspects of this algorithm to the standard binomial tree. 1.3. PATHTREE Algorithms Tree algorithms are generally deri vedfrom binomial random w alks [13]. Formanyapplications, “tree” algorithms are often used, corresponding to the abo ve Langevin and F okker-Planck equations [14,15]. These algorithms ha ve typically been only well deﬁned for speciﬁc functional forms ofProbability tree ... -3- I ngber,Chen, Mondescu, Muzzall, Renedo fandg. We hav epreviously presented a powerful P ATHINT algorithm to deal with quite general fandg functions [4]. This general P ATHTREE algorithm can be used beyond previous speciﬁc systems, affording fast reasonable resolution calculations for probability distributions of a large class of nonlinear diffusion problems. 1.4. Organization of Paper Section 2 describes the standard tree algorithm. Section 3 de velops our probability P ATHTREE algorithm. Section 4presents our probability calculations. Section 5 is our conclusion. 2. STANDARD OPTION TREE ALGORITHM 2.1. Binomial Tr ee In a two-step binomial tree, the step up Suor step do wnSdfrom a gi vennode atSis chosen to match the standard deviation of the differential process. The constraints on uanddare chosen as ud=1, ( 4) If we assign probability pto the up step Su,andq=(1−p)t othe down step Sd,the matched mean and variance are pSu+(1−p)Sd=<S(t+Δt)>, S2((pu2+qd2−(pu+qd)2))=<((S(t+Δt)−<S(t+Δt)>))2>. ( 5) The right-hand-side can be deri vedfrom the stochastic model used. 2.2. Trinomial T ree The trinomial tree can be used as a rob ust alternate to the binomial tree. Assumepu,pmandpdare the probabilities of up jump Su,middle (no-)jump Sand down jump Sd,where the jumps are chosen to match the standard de viation. T omatch the variance of the process, the equations must satisfy pu+pm+pd=1, S(puu+pm+pdd)=<S(t+Δt)>, S2((puu2+pm+pdd2−(puu+pm+pdd)2))=<((S(t+Δt)−<S(t+Δt)>))2>. ( 6) 3. PROBABILITY TREE ALGORITHM 3.1. General Diffusion Process Consider the general Mark ov multiplicati ve diffusion process interpreted as an Ito ˆprepoint discretized process, Eq. (1) with drift fand diffusiong2.For ﬁnancial option studies the particular form of the drift bSand diffusion ( σS)2is chosen for lognormal Black-Scholes (BS) calculations [14]. For options, the coef ﬁcientbis the cost of carry ,e.g.,b=r,the risk-free rate, when Sis a stock price, and b=0whenSis a futures price [16]. The case of drift bSand constant dif fusion dif fusionσ2corresponds to the Ornstein-Uhlenbeck (OU) process [17]. Our formalism is general and can be applied to other functional forms of interest with quite general nonlinear drifts and diffusions, of course provided that the re gion of solution does not violate an y boundary or initial conditions. Statistical properties of the dSprocess and of an yderivative one based on nonlinear transformations applied to Sare determined once the transition probability distribution function P(S,t\|S0,t0)i sknown, where the 0 inde xdenotes initial values of time and of the stochastic v ariableS.Transformation are common and con venient for BS,Probability tree ... -4- I ngber,Chen, Mondescu, Muzzall, Renedo z=lnS,( 7) yielding a simple Gaussian distribution in z,greatly simplifying analytic and numerical calculations. The probability distribution can be obtained by solving the associated forw ard Fokker-Planck equation Eq. (2). Appropriate boundaries and initial condition must be speciﬁed, e.g., P(S\|S0)=δ(S−S0). In general cases, the F okker-Planck equation is rather difﬁcult to solve, although a v ast body of work is devoted to it [17]. The particular BS and OU cases possess exact results. Our goal is to obtain the solution of Eq. (1) for the more general process. Aquite general code, PATHINT [4], works ﬁne, but it is much slower than the P ATHTREE algorithm we present here. 3.2. Deﬁciency of Standard Algorithm to Order √ dt We brieﬂy describe the CRR construction of the binomial tree approximation [18]. Atree is constructed that represents the time e volution of the stochastic v ariableS.Sis assumed to takeonly 2 values,u,(up value), and d(down value) at moment t,giv enthe valueSat moment t−Δt. The probabilities for the up and do wn movements are pandq,respectively.The 4 unkno wns{u,d,p,q} are calculated by imposing the normalization of the probability and matching the ﬁrst tw omoments conditioned by the v alueSatt−Δt,using the variance of the exact probability distrib utionP(S,t\|S0,t0). One additional condition is arbitrary and is usually used to symmetrize the tree, e.g., ud=1. The main problem is that the abo ve procedure cannot be applied to a general nonlinear dif fusion process as considered in Eq. (1), as the algorithm in volves a previous knowledge of terms of O(Δt)i nthe formulas of quantities {u,p}obtained from a ﬁnite time expansion of the exact solution Psought. Otherwise, the discrete numerical approximation obtained does not con vergetothe proper solution. This observation can be check ed analytically in the BS CRR case by replacing the relation u=exp(σΔt)[15] with u=1+σ√ Δt,and deriving the continuous limit of the tree. This also can be checked numerically ,a swhen{u,p}are expanded to O(Δt), the proper solution is obtained. 3.3. Probability P ATHTREE As mentioned pre viously,ageneral path-inte gral solution of the F okker-Plank equation, including the Black-Scholes equation, can be numerically calculated using the P ATHINT algorithm. Although this approach leads to very accurate results, it is computationally intensi ve. In order to obtain tree variables valid up to O(Δt), we turn to the short-time path-inte gral representation of the solution of the F okker-Planck equation, which is just the multiplicati ve Gaussian- Markovian distribution [1,19]. In the prepoint discretization rele vant to the construction of a tree, P(S′,t′\|S,t)=1 √ 2πΔtg2exp  −(S′−S−fdt)2 2g2Δt   Δt=t′−t (8) valid for displacements S′fromS“reasonable” as measured by the standard de viationg√ Δt,which is the basis for the construction of meshes in the P ATHINT algorithm. The crucial aspects of this approach are: There is no a priori need of the ﬁrst moments of the e xact long-time probability distrib utionP,a sthe necessary statistical information to the correct order in time is contained in the short-time propag ator.The mesh in S at e very time step need not recombine in the sense that the prepoint-postpoint relationship be the same among neighboring Snodes, as the short-time probability density gi vesthe correct result up to order O(Δt)for anyﬁnal point S′.Instead, we use the natural metric of the space to ﬁrst lay down our mesh, then dynamically calculate the e volving local short- time distributions on this mesh. We construct an additi ve PATHTREE, starting with the initial value S0,with successi ve increments Si+1=Si+g√ Δt,Si>S0Probability tree ... -5- I ngber,Chen, Mondescu, Muzzall, Renedo Si−1=Si−g√ Δt,Si<S0,( 9) wheregis evaluated at Si.Wedeﬁne the up and down probabilities pandq,resp., in an abbre viated notation, as p=P(i+1\|i;Δt) P(i+1\|i;Δt)+P(i−1\|i;Δt) q=1−p.( 10) where the short-time transition probability densities P’s are calculated from Eq. (8). Note that in the limit of smallΔt, Δt→0limp=1 2.( 11) 3.3.1. Continuous Limit of the P ATHTREE Algorithm In either the upper or lower branches of the tree ( Si>S0orSi<S0,resp.), we al ways have the postpoint Si±1in terms of the prepoint Si,but we also need the in verses to ﬁnd the asymptotic Δ→0lim ofour PATHTREE building technique. Forexample, for the upper branch case, Si−1≈Si−g√ Δt+(g∂g ∂S)Δt+O(Δt3/2). ( 12) This expression must be used to extract the down change d(Si−1=Si+d), for comparison to the standard tree algorithm. The continuous limit of the previous tree building procedure is obtained by T aylor expanding all factors up to terms O(Δt)and as functions of the prepoint Si[15]. This leads to pu+qd Δt≈f+O(Δt1/2) pu2+qd2 Δt≈g2+O(Δt1/2), ( 13) from which the correct partial differential equation, Eq. (2), is reco vered up to O(Δt). In implementing the P ATHTREE algorithm, good numerical results are obtained in the parameter region deﬁned by the con vergence condition     g∂g ∂Si  dt+g√ dt  /Si<< 1.( 14) This insures the proper construction of the tree to order O(Δt). 3.3.2. Treating Binomial Oscillations Binomial trees exhibit by construction a systematic oscillatory beha vior as a function of the number of steps in the tree (equi valently,the number of time slices used to build the tree), and the ne wbuilding algorithm based on the short-time propag ator of the path-integral representation of the solution of the Fokker-Planck equation has this same problem. Acommon practice [20] is to perform a verages of runs with consecuti ve numbers of steps, e.g., C=CN+1+CN 2,( 15) whereCNsigniﬁes the value calculated with Nnumber of steps.Probability tree ... -6- I ngber,Chen, Mondescu, Muzzall, Renedo 3.3.3. Inappr opriate Trinomial T ree Another type of tree is the trinomial tree discussed abo ve,equivalent to the explicit ﬁnite dif ference method [14,15]. If we were to apply this approach to our ne wPATHTREE algorithm, we w ould allowthe stochastic v ariableSto remain unchanged after time step Δtwith a certain probability pm.Howev er, in our construction the trinomial tree approach is not correct, as the deterministic path is dominant in the construction of the probabilities {pu,pm,pd},and we would obtain Δt→0limpu=pd=0, Δt→0limpm=1. ( 16) 3.4. Linear and Quadratic Aspects of Numerical Implementation PATHTREE computes the expected value of a random v ariable at a later time gi venthe diffusion process and an initial point. The algorithm is similar to a binomial tree computation and it consists of three main steps: computation of a mesh of points, computation of transition probabilities at those points and computation of the expected value of the random variable. The ﬁrst step is the creation of a one dimensional mesh of points with g aps determined by the second moment of the short term distribution of the process. The mesh is created sequentially ,starting from the initial point, by progressi vely adding to the last point already determined (for the upward part of the mesh) the value of the standard de viation of the short term distribution with the same point as prepoint. In asimilar fashion we create the mesh do wnwards, this time by subtracting the standard deviations. The procedure tak es a linear amount of time on the number of time slices being considered and contributes very little to the o verall time of the algorithm. In the second step an array of up and down probabilities is created. These probabilities are the values of the short term transition probability density function obtained by using the current point as prepoint and the tw oneighboring points as post points. The probabilities are renormalized to sum to unity.This procedure takes a linear amount of time on the number of time slices. Notice that the probabilities only depend on the current point and not on time slice, hence only tw oprobabilities are computed per element of the array of points. The third step is the computation of the expected value of the random v ariable. F or example, the option price Cis developed by marching backwards along the tree and applying the risk-neutral evaluation C(Si,t−Δt)=e−rΔt[pC(Si+1,t)+qC(Si−1,t)] . (17) We emphasize ag ain that in Ito ˆterms the prepoint value is Si.This part works as a normal binomial tree algorithm. The algorithm uses the expected v alues at one time slice to compute the expected values at the previous one. The bulk of the time is spent in this part of the algorithm because the number of iterations is quadratic on the amount of time slices. We managed to optimize this part by reducing each iteration to about 10 double precision operations. In essence, this algorithm is not slo wer than standard binomial trees and it is very simple to implement. 4. CALCULA TION OF PROBABILITY 4.1. Direct Calculation of Probability We can calculate the probability density function by ﬁrst recursi vely computing the probabilities of reaching each node of the tree. This can be performed ef ﬁciently thanks to the Mark ov property.To compute the density function we need to rescale these probabilities by the distance to the neighboring nodes: the more spread the nodes are, the lower the density .Wecan estimate the probability density as follows: First we compute the probability of reaching each ﬁnal node of the tree. We dothis incrementally by ﬁrst computing the probabilities of reaching nodes in time slice 1, then time slice 2 andProbability tree ... -7- I ngber,Chen, Mondescu, Muzzall, Renedo so forth. At time slice 0, we kno wthat the middle node has probability 1 of being reached and all the others have probability 0. We compute the probability of reaching a node as a sum of tw ocontributions from the previous time slice. We reach the node with transition pufrom the node belo wa tthe previous slice, and with transition pdfrom the node abo ve.Each contribution is the product of the probability at the previous node times the transition to the current node. This formula is just a discretized version of the Chapman-Kolmogoro vequation p(xj,ti)=p(xj−1,ti−1)puj−1+p(xj+1,ti−1)pdj+1.( 18) Nowthat we ha ve computed the absolute probabilities at the ﬁnal nodes, we can gi ve a proper prepoint-discretized estimation of the density by scaling the probabilities by the spread of the S v alues. Forthe upper half of the tree we di vide the probability of each ﬁnal node by the size of the lower adjacent interval in the mesh: densityi=pi/(Si−Si−2). (Note: We use indexSi−2because the binomial tree is constructed o veratrinomial tree. In this way we can keep in memory all the nodes b ut only half of the nodes though are true ﬁnal nodes.) If there is a ﬁnal middle node we di vide its probability by the a verage of sizes of the tw oadjacent interv als, that is: densityi=pi/((Si+2−Si−2)/2). For the lower half of the mesh we divide the probability by the upper adjacent gap in the mesh: densityi=pi/(Si+2−Si). 4.2. Numerical Derivativeso fExpectation of Probability The probability Pcan be calculated as a numerical deri vative with respect to strik eXof a European Call option, taking the risk-free rate rto be zero, gi vena nunderlying S0evaluated at time t=0, with strikeX,and other variables such as v olatilityσ,cost of carry b,and time to e xpiration Tsuppressed here for clarity, C(S0,0;X), P[S(T)\|S(t0)] S(T)≡X=P[X\|S(t0)]=∂2C ∂X2(19) This calculation of the probability distrib ution is dependent on the same conditions necessary for anytree algorithm, i.e., that enough nodes are processed to ensure that the resultant e valuations are a good representation of the corresponding F okker-Planck equation, addressed abo ve,and that the number of iterations in P ATHTREE are sufﬁcient for con vergence. 4.2.1. Alter native First Derivative Calculation of Probability An alternati ve method of calculating the probability Paaﬁrst-order numerical deri vative,instead of as second-order deri vative,with respect to Xis to deﬁne a function CHusing the Hea viside step- function H(S,X)=1ifS≥Xand 0 otherwise, instead of the Max function at the time to e xpiration. This yields P[S(T)\|S(t0)] S(T)≡X=P[X\|S(t0)]=−∂CH ∂X(20) Sometimes this is numerically useful for sharply peaked distributions at the time of expiration, but we have found the second deri vative algorithm abo ve towork ﬁne with a sufﬁcient number of epochs. Our tests verify that the three methods abo ve giv ethe same density .Weconsider the numerical- derivative calculations a very necessary baseline to determine the number of epochs required to get reasonable accurac y. 4.2.2. Oscillatory Corrections Fig. 1 illustrates the importance of including oscillatory corrections in an ybinomial tree algorithm. When these are included, it is easy to see the good agreement of the BS P ATHTREE and OU P ATHTREE models. 4.3. Comparison to Exact Solutions Fig. 2 givesthe calculated probability distrib ution for the BS and OU models, compared to their exact analytic solutions.Probability tree ... -8- I ngber,Chen, Mondescu, Muzzall, Renedo 5. CONCLUSION We hav edeveloped a path-integral based binomial P ATHTREE algorithm that can be used in a variety of stochastic models. This algorithm is simple, fast and can be applied to dif fusion processes with quite arbitrarily nonlinear drifts and diffusions. As expected, this P ATHTREE algorithm is not as strong as P ATHINT [4], as PATHINT can include details of an extremely high dimensional tree with comple xboundary conditions. ForPATHINT,the time and space variables are determined independently .I.e., the ranges of the space variables are best determined by ﬁrst determining the reasonable spread of the distribution at the ﬁnal time epoch. ForPATHTREE just one parameter ,the number of epochs N,determines the mesh for both time and the space v ariables. This typically leads to a growth of the tree, proportional to √ N,much faster than the spread of the distribution, so that much of the calculation is not rele vant. However, this PATHTREE algorithm is surprisingly robust and accurate. Similar to P ATHINT,we expect its accurac yt ob ebest for moderate-noise systems. ACKNOWLEDGMENTS We thank Donald Wilson for his ﬁnancial support.Probability tree ... -9- I ngber,Chen, Mondescu, Muzzall, Renedo REFERENCES 1. F.Langouche, D. Roekaerts, and E. T irapegui, Discretization problems of functional integrals in phase space, Phys. Rev. D20,419-432 (1979). 2. L. Ingber,Statistical mechanical aids to calculating term structure models, Phys. Re v. A 42(12), 7057-7064 (1990). 3. L.Ingber and J.K. Wilson, Volatility of volatility of ﬁnancial mark ets,Mathl. Computer Modelling 29(5), 39-57 (1999). 4. L. 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Risken,The Fokker-Planc kEquation: Methods of Solution and Applications ,Springer-Verlag, Berlin, (1989). 18. J.C. Cox, S. A. Ross, and M. Rubenstein, Option pricing: A simpliﬁed approach, J. Fin. Econ. 7,229-263 (1979). 19. R. Graham, P ath-integral methods in nonequilibrium thermodynamics and statistics, in Stochastic Processes in Nonequilibrium Systems ,(Edited by L. Garrido, P .Seglar,and P.J. Shepherd), pp. 82-138, Springer ,New York, NY,(1978). 20. M. Broadie and J. Detemple, Recent advances in numerical methods for pricing deri vative securities, in Numerical Methods in F inance,(Edited by L.C.G Rogers and D. T alay), pp. 43-66, Cambridge Uni versity Press, Cambridge, UK, (1997).Probability tree ... -10- I ngber,Chen, Mondescu, Muzzall, Renedo FIGURE CAPTIONS FIG. 1. The oscillatory correction, an a verage ofNandN+1iteration solutions, provides a simple and effective ﬁxt othe well-known oscillations inherent to binomial trees. The uncorrected Black-Scholes binomial tree (a) can be compared to the Black-Scholes tree with oscillatory correction (b). In (c), the Ornstein-Uhlenbeck binomial tree also be robustly corrected as shown in (d). The BS P ATHTREE model shown in (e) can be compared to the Black-Scholes case shown in (b). The OU P ATHTREE model (f) is equivalent to the Ornstein-Uhlenbeck model in (d). Parameters used in these calculations are: S=50.0,X =55.0,T=1.0,r=0.0675,b=0,σ=0.20, andN=300. FIG. 2. Probability distributions for the P ATHTREE binomial model as described in the te xt. In (a), bar graphs indicate OU P ATHTREE agrees well with the exact Ornstein-Uhlenbeck distrib ution shown in the black line. In (b), the bar graphs indicate BS P ATHTREE agrees well with the exact Black- Scholes distribution shown in the black line. Parameters are the same as in Fig. 1.Probability tree ... -Figure 1 - Ingber ,Chen, Mondescu, Muzzall, Renedo 1.971.981.9922.012.022.032.042.05 50100150200250300 N(a) Black-Scholes 1.971.981.9922.012.022.032.042.05 50100150200250300 N(b) Black-Scholes with oscillatory correction 1.81.811.821.831.841.851.861.871.881.891.9 50100150200250300 N(c) Ornstein-Uhlenbeck 1.81.811.821.831.841.851.861.871.881.891.9 50100150200250300 N(d) Ornstein-Uhlenbeck with oscillatory correction 1.971.981.9922.012.022.032.042.05 50100150200250300 N(e) BS PATHTREE with oscillatory correction 1.81.811.821.831.841.851.861.871.881.891.9 50100150200250300 N(f) OU PATHTREE with oscillatory correctionProbability tree ... -Figure 2 - Ingber ,Chen, Mondescu, Muzzall, Renedo 00.0050.010.0150.020.0250.030.0350.040.0450.05 0 20 40 60 80 100 S0(b) Probability Distribution for X-Tree (x=1)00.0050.010.0150.020.0250.030.0350.040.0450.05 0 20 40 60 80 100 S0(a) Probability Distribution for X-Tree (x=0)
arXiv:physics/0103014v1 [physics.atom-ph] 6 Mar 2001Wave packet propagation study of the charge transfer interaction in the F−– Cu(111) and – Ag(111) systems A.G. Borisov, J.P. Gauyacq Laboratoire des Collisions Atomiques et Mol´ eculaires, Unit´ e mixte de recherche CNRS-Universit´ e Paris-Sud UMR 8 625, Bˆ atiment 351, Universit´ e Paris-Sud, 91405 Orsay CEDEX, F rance S.V. Shabanov Department of Mathematics, University of Florida, Little H all 358, Gainesville, FL 32611, USA Abstract The electron transfer between an F−ion andCu(111) andAg(111) surfaces is studied by the wave packet propagation method in order to determine spe ciﬁcs of the charge transfer interaction between the negative ion and the metal surface d ue to the projected band gap. A new modeling of the F−ion is developed that allows one to take into account the six q uasi- equivalent electrons of F−which are a priori active in the charge transfer process. The new model invokes methods of constrained quantum dynamics. The six-electron problem is transformed to two one-electron problems linked via a const raint. The projection method is used to develop a wave packet propagation subject to the mo deling constraint. The characteristics (energy and width) of the ion F−ion level interacting with the two surfaces are determined and discussed in connection with the surface projected band gap. 1I. INTRODUCTION When an atom or a molecule is close to a surface of a solid, its e lectrons interact with those of the solid, leading to the possibility of an electron trans fer between the atom (molecule) and the solid. This charge transfer process plays a very impo rtant role in a variety of diﬀerent situations. In particular, it often occurs as an in termediate step in reactions at sur- faces (desorption, fragmentation of adsorbates, chemical reactions, quenching of metastable species, etc.) [1–4]. The one-electron transfer between en ergetically degenerate electronic levels of the atom and the solid is called the Resonant Charge Transfer (RCT) process. It is usually considered as the most eﬃcient one among variou s possible charge transfer interactions. Since a few years the development of accurate theoretical approaches to the RCT in the case of free electron metal surfaces [5–11] has led to a successful description of a one-electron transfer in the interaction of ions (atoms ) with such surfaces [12–14]. All these approaches are based on the description of single elec tron being transferred between the atom and the surface. As an example, one can mention the ne utralisation of an alkali positive ion by RCT, even if the possibility of capturing the electron in two diﬀerent spin states somewhat alters the one-electron picture [15,16]. H owever, many atoms or molecules contain more than one electron which can possibly participa te in the charge transfer pro- cess, especially, when the active electrons occupy the same energy level. The latter has to be taken into account in any quantitative approach to the RCT [12–17]. For instance, a free ﬂuorine negative ion can be described as a closed 2p6outer electronic shell with six equivalent electrons, and all of them can part icipate in the RCT when the ion is close to a metal surface. The eﬀects of each electron canno t be simply added up to get a total eﬀect. Indeed, consider the loss of an electron by the ion. Any of the six electrons can be the one that is detached. However, the detachment of on e electron a priori precludes the detachment of another one. Clearly, the loss of a second e lectron would be a completely diﬀerent process because it corresponds to a formation of a p ositive ﬂuorine ion and, hence, has a diﬀerent energetics. Such correlations between the ou ter-shell electrons of the F− 2ion must be accounted for in any description of the RCT. In thi s work we show how such correlations can be modeled by two one-electron systems lin ked via a constraint . In classical dynamics, constraints appear as some algebrai c relations between generalized coordinates of a system and their time derivatives (velocit ies), which are to hold for any mo- ment of time. They are widely used to model, e.g., eﬀects of an environment on a system in question, or to develop theories with local symmetries (g auge theories) such as electro- dynamics, general relativity and Yang-Mills theories. Cla ssical constrained dynamics has been well studied in classical works by d’Alambert, Lagrang e, H¨ older and Gauss (see, e.g., Ref. [18] and references therein). In quantum mechanics con straints have been analyzed by Dirac [19], mainly because of the need of quantizing fundame ntal physical theories with local symmetries. The question addressed by Dirac was the followi ng:Given a classical system with constraints, construct a quantum theory which satisﬁe s the correspondence principle . A time evolution of a quantum system is described by the evolut ion operator. Its kernel in the coordinate representation is called a quantum mechanic al propagator. For constrained systems it is usually obtained by a reduction of the Feynman p hase space path integral onto the physical phase space, as proposed by Faddeev [20]. There are many subtleties associated with quantization of constrained systems (see for a review, e.g., [21]). Here we propose a quantum constrained system which can be use d to model the charge transfer interaction between an F−ion and a metal surface. The basic idea is to transform the original six-electron problem to two one-electron prob lems each of which describes a possible decay channel of the ion. As a consequence of the qua ntum equivalence of the outer-shell electrons, the eﬀective one-electron systems turn out to be linked onlyby a constraint. In other words, the entire interaction between the two systems occurs through a constraint rather than via a local potential. One can also sa y that such a kinematic coupling of the decay channels is induced by the symmerty of the origin al six-electron problem. The constraint has a remarkable feature: It does not have a cl assical limit, meaning that its eﬀects disappear in the formal classical limit ¯ h→0. That is, the dynamical eﬀect modeled by such a constraint is essentially quantum and cann ot be modeled by any classical 3potential force. In this regard, the canonical quantizatio n scheme for constrained systems due to Dirac is not applicable here. We develop below a novel c omputational scheme for a quantum evolution subject to such purely quantum constrai nts. The scheme is based on the projection formalism introduced earlier by one of us [ 22] within the framework of gauge theories. Our approach also comprises a novel method t o account for the intra-atomic correlations within a one-electron description of the char ge transfer interaction between an F−ion and the Cu(111) andAg(111) surfaces. We demonstrate that the approach turns out to be very eﬃcient in the wave packet propagation studies . The choice of the F−/Cu(111) andF−/Ag(111) systems is motivated by several reasons. The interaction of halogen negative ions with a free-electr on metal surface has already been studied theoretically within the Coupled Angular Mode appr oach which lead to a quite satisfactory description of the halogen negative ion forma tion in scattering halogen atoms by various metal surfaces [12,13]. Recently, on the example of theCu(111) surface it has been shown that the projected band gap of the (111) surfaces o f noble metals strongly aﬀects the RCT [23,24]. The point is that in a certain energy range el ectrons cannot propagate along the normal to the surface (L-band gap in the <111>direction [25]). On the other hand, the RCT process corresponds to an electron tunneling b etween the atom and the surface which is favored along the surface normal. In additi on, it was shown that the 2D electronic continuum of the surface state plays a signiﬁcan t role, often dominating the RCT process. The band gap has several consequences. First, ther e exist very long lived states in the alkali–Cu(111) systems [24,26,27]. Second, there is a parallel veloc ity dependence of the probability of an electron capture by the atom from Cu(111) surfaces in grazing scattering experiments that is characteristic of a 2D electronic conti nuum [28]. Third, there exits an avoided crossing between the energy level of the projectile and the bottom of the surface state continuum [23]. These eﬀects have been found to depend strongly on the interaction time [23,29,30]: They only appear for long interaction time s, i.e., for slow collisions. The freeF−ion energy level is slightly above the bottom of the surface s tate continuum of theCu(111) andAg(111) surfaces, and therefore, because of the image charge a ttraction, 4it could cross the bottom of the 2D continuum for a ﬁnite ion-s urface distance. In the present work, we investigate the eﬀects of the projected band gap and of the surface state continuum on theF−-metal charge transfer. In particular, we study the behavio r of the system when the ion energy level is very close to the bottom of the 2D surfa ce state continuum. A Wave Packet Propagation (WPP) approach to the RCT [8,23] provide s a quantitative description of dynamic and static aspects of the F−-surface charge transfer. The latter allows one to analyze the dependence of the band structure eﬀect upon an in teraction time in the RCT. II. METHOD A. A negative ion F− The free negative ion F−is usually described in the Hartree–Fock approximation as a closed shell ion with the electronic conﬁguration 1 s22s22p6. Its binding energy is 3.4 eV. In this approach the outer shell electrons are regarded as equi valent. Their wave function is given by the corresponding Slater determinant Φ =1√ 6!/vextendsingle/vextendsingle/vextendsingle2pα 02pβ 02pα 12pβ 12pα −12pβ −1/vextendsingle/vextendsingle/vextendsingle, (1) where 2psymbolizes a 2 porbital wave function, the superscripts αandβstand for the two possible spin directions, and the subscript indicates t he magnetic quantum number m corresponding to the projection of the 2 pangular momentum on the quantization axis. The Slater determinant (1) can be expanded into a sum of products of wave functions of an ionic core and an outer electron: Φ =1√ 66/summationdisplay j=1\|Fj\|Ajφj=6/summationdisplay j=1\|Gj\|φj, (2) where the ﬁve-electron determinant \|Fj\|describes a state of the ﬂuorine atom F(2P). The productAjφjcorresponds to the 2 pα,β morbital, which has been factorised into a spin factor Ajand a spatial wave function φj=φj(/vector r). The wave function (2) can also be used for an open shell description of the negative ion of the type 2 p52p′. In this case, the spatial 5wave function φjsingled out in (2) corresponds to the outer 2 p′orbital of the negative ion, whereas the core wave function \|Gj\|is formed by the inner 2 porbitals. When analyzing the electron detachment process in the open shell description, the 2p′orbital is regarded as an active one. The representation (2) is particularly well suited for an an alysis of the electron cap- ture/loss process between a ﬂuorine ion F−and a metal surface since it gives an expansion of the ion wave function over possible detachment channels. So, we retain the representation (2) to describe a loss or capture of an electron by the ﬂuorine core Ψ =/summationdisplay j\|Gj\|ψj. (3) Hereψj=ψj(/vector r) is a wave function of an electron (captured or lost) in the j-channel. Ne- glecting the spin-orbit interactions and assuming the ion- surface interaction to be invariant under translations of the ion parallel to the surface, the sy stem becomes invariant under the spin ﬂip and rotations about the z-axis which is set to be perpendicular to the metal surface and passing through the ion center. Next, the z-axis is chosen as the quantization axis. Therefore the states with the z-component of the electron angular momentum ±1 are degenerate. As a consequence, only two electron wave functi ons are distinct in the repre- sentation (3). They correspond to the states with m= 0 and \|m\|= 1. In what follows we assume that the charge transfer does not aﬀect the neutral co re wave function \|Gj\|. Only the outer electron is subject to the RCT dynamics. Any state of the system can be represented as a two dimensiona l isovector \|Ψ/angbracketrightwhose components are one-electron states corresponding to m= 0 and \|m\|= 1 (upper and lower elements of the isovector, respectively). In cylindrical c oordinates ( z,ρ,ϕ ), it can be written as /angbracketleft/vector r\|Ψ/angbracketright= ψ0(z,ρ) ψ1(z,ρ)eiϕ . (4) Note that, due to the symmetry of the problem, the ϕdependence can be explicitly given. Hence, one can limit oneself to studying the z- andρ-dependence of the electron wave packet. 6In our representation a free ion wave function is given by /angbracketleft/vector r\|ΨF−/angbracketright= 1√ 3p(r)Y10(θ,ϕ) /radicalig 2 3p(r)Y11(θ,ϕ)  (5) wherep(r) is the radial part of the free-ion orbital and Ylm(θ,ϕ) are the spherical harmonics. The diﬀerent normalization factors of the isovector compon ents are due to the fact that the free ion wave function (2) contains twice as many states with \|m\|= 1 as with m= 0. The functionsp(r)Ylm(θ,ϕ) have a unit norm so that /angbracketleftΨF−\|ΨF−/angbracketright= 1. Yet another remark is that the approach outlined above assum es that only an outer electron can be detached (the detachment occurs through the evolution of the wave functions ψ0andψ1), i.e., it assumes an open shell description (2 p52p′) of the ion F−. A closed shell description (2 p6) would correspond to an increase of the charge transfer inte raction by the factor√ 6, and, hence, to an increase of the width by the factor 6 (see a discussion in [16]). This appears to be better adapted for the halogen negative io n case. With all the above settings, a modeling of the charge transfe r in theF−-metal system implies ﬁnding a one-electron Hamiltonian that governs a ti me evolution in the Hilbert space spanned by vectors (4). We take it in the following form H=T+Vat+VS=Hat+VS. (6) The operators H,T,Vat,VSandHatare diagonal 2 ×2 matrices, in fact, we choose them to be proportional to a unit 2 ×2 matrix, with the diagonal elements denoted, respectively , H, T, V at, VSandHat. Here,Tis the electron kinetic energy operator, Vatthe potential of the interaction between an electron and the neutral core, VSthe potential of the electron- surface interaction, Hat=T+Vat, andH=Hat+VS. The wave functions p(r)Ylm(θ,ϕ) are eigenfunctions of the one-electron Hamiltonian Hat. The time evolution of the wave function (4) generated by the H amiltonian (6) is nothing but the evolution of two independent one-electron wave pack ets. On the other hand, the two components of the isovector (4) cannot evolve independentl y. Indeed, in the free ion case, the two components cannot be arbitrarily chosen in order for the state (4) to describe a physical 7free ion. Actually, the radial components of the column elem ents appear to be proportional to each other with a speciﬁc factor (cf. (5)). This relation b etween the two components comes from the expansion of the Slater determinant (1) which possesses a high symmetry being the symmetry of a quantum system of identical particle s occupying the same energy level. The very same symmetry must be preserved in the expans ion (2) and, hence, upon a reduction of the six-electron description to our one-elec tron formalism. In other words, there must be a correlation between evolving components of t he isovector corresponding to the electron states with m= 0 and \|m\|= 1 thanks to the symmetry of the six-electron problem. Therefore physically admissible states in the Hil bert space spanned by isovectors (4) must be subject to some constraints required by the symme try of the original six-electron problem. This will be a key point of our new approach to the RCT dynamics. To illustrate the necessity of constraints, consider the ca se when the detachment occurs in them= 0 channel. Then the bound part of the \|m\|= 1 channel must also disappear at the same time because there is only one ion F−which contains both m= 0 and \|m\|= 1 components. In the Eﬀective Range approach [31] used in the C oupled Angular Mode (CAM) method [6], this problem has been solved in the followi ng way. One only considers the wave function of the system given by expression (4) outsi de a spherical region of radius rc. TheFcore is contained in the region. The boundary condition on th e radial components ψ0andψ1atr=rccouples the two channels. This approach essentially relies on the use of spherical coordinates. Here we look for a coordinate indepe ndent description of the channel mixing that can be eﬃciently implemented in numerical calculations of the tim e evolution of the system (wave packet propagation) . The origin of the kinematic coupling of the channels is now sought in symmetries of the system. It is believed that such an approach is rather general and could be applied to other many-electron systems where a conventional mean ﬁeld approach does not provide a good approximation. The basic idea is that the Hilbert space spanned by isovector s (4) is too large and contains states which are physically not acceptable. It is clearly se en already from the fact that the radial components of the free ion state (5) are not independe nt. We shall then constrain the 8state (4) to allow only one bound ion F−of the form (5). Any other state with components ψ0∼p(r)Y10(θ,ϕ) andψ1∼p(r)Y11(θ,ϕ) should be forbidden. This corresponds to making the state (4) orthogonal to the vector /angbracketleft/vector r\|Q/angbracketright= /radicalig 2 3p(r)Y10(θ,ϕ) −/radicalig 1 3p(r)Y11(θ,ϕ) ≡ p0(z,ρ) −p1(z,ρ)eiϕ . (7) Since the state (7) cannot also occur as a virtual (or interme diate) state of the ion in the time evolution of the system, we demand that the time depende nt wave packet (4) must be orthogonal to the vector \|Q/angbracketright: /angbracketleftQ\|Ψ(t)/angbracketright=/angbracketleftp0\|ψ0(t)/angbracketright − /angbracketleftp1\|ψ1(t)/angbracketright= 0, (8) for anyt≥0, where /angbracketleftp0,1\|ψ0,1(t)/angbracketrightstands for a standard scalar product (written in cylindrica l coordinates because the functions p0,1andψ0,1depend onzandρonly). From the physical point of view the constraint (8) simply mea ns that there is an unwanted scattering mode in our eﬀective one-electron problem. Alth ough the Hamiltonian may allow for such a mode, we have given physical reasons to forbid it. T he constraint (8) implies that the time evolution of the two components of the isovecto r (4) is no longer independent even though the Hamiltonian (6) does not provide any direct c oupling of them. The link between the detachment channels with m= 0 and \|m\|= 1 steams directly from the free ion structure. It implicitly assumes that the correlation betw een the wave functions ψ0andψ1 in the ion perturbed by an interaction with a metal surface is the same as in the free ion. Thus, the electronic structure of F−has been modeled by two one-electron problems linked by the constraint (8). B. Physical Hamiltonian Now we face a problem of incorporating the constraint into qu antum dynamics gener- ated by the Hamiltonian (6). The diﬃculty is clear. Suppose a n initial state satisﬁes the constraint (8). Applying the evolution operator exp( −itH) to it, we immediately observe 9that the evolved state fails to satisfy the constraint. The p rocedure we propose is based on the projection operator formalism ﬁrst introduced for gaug e theories [22,32]. It has been generalized to general constrained systems [33,34] (see al so the review [21]). The key steps are as follows. Consider the projection operator P=I− \|Q/angbracketright/angbracketleftQ\|, (9) whereIis the unit operator, I\|Ψ/angbracketright=\|Ψ/angbracketrightfor any \|Ψ/angbracketright. It is easy to convince oneself that the operator (9) is self-adjoint, P†=P, and satisﬁes the characteristic property of a projection operator, P2=P. By construction, the state P\|Ψ/angbracketrightsatisﬁes the constraint (8) for any state \|Ψ/angbracketright. The operator (9) projects any state to the physical subspac e deﬁned by the condition (8). In particular, P\|Q/angbracketright= 0. To eliminate the state \|Q/angbracketrightas a possible intermediate state of the system in the time evolution, the Hamiltonian is project ed onto the physical subspace H→PHP =Hphys. (10) The physical Hamiltonian (10) is self-adjoint. Hence the ti me evolution generated by it is unitary. The state PHP \|Ψ/angbracketrightsatisﬁes the constraint (8). Clearly, the physical Hamilto nian is nonlocal, in general, even if the original Hamiltonian ha s a standard form of the sum of potential and kinetic energies. However, classical limits ofHandHphysare the same. The evolution operator has the form U(t1,t2) =PT exp/parenleftbigg −i/integraldisplayt2 t1dτPHP/parenrightbigg P=Pexp (−itPHP)P, (11) wheret=t2−t1and T exp stands for the time ordered exponential. The second equality holds when the Hamiltonian Hdoes not explicitly depend on time. The projection operator s before and after the exponential in (11) can be omitted if the initial state satisﬁes the constraint (8). The physical Hamiltonian provides the sought-for channel m ixing. To ﬁnd terms in the new Hamiltonian which give rise to the channel mixing, we com pute the action of PHP on a generic state \|Ψ/angbracketright. A straightforward computation leads to the following resu lt 10PHP \|Ψ/angbracketright= H\|ψ0/angbracketright −λ1(H+λ2)\|p0/angbracketright −λ2\|p0/angbracketright H\|ψ1/angbracketright −λ1(H+λ2)\|p1/angbracketright −λ2\|p1/angbracketright , (12) λ1=/angbracketleftQ\|Ψ/angbracketright=/angbracketleftp0\|ψ1/angbracketright+/angbracketleftp1\|ψ2/angbracketright, λ2=/angbracketleftQ\|VS\|Ψ/angbracketright=/angbracketleftp0\|VS\|ψ0/angbracketright+/angbracketleftp1\|VS\|ψ1/angbracketright. If the state \|Ψ/angbracketrightbelongs to the physical subspace then, according to (8), λ1= 0. However, the action of PHP even on the physical states is not reduced to that of Hbecause the amplitudeλ2is generally not zero. Thus, after the projection (10) the or iginal Hamiltonian (6) acquires an additional term PHP = (H+W)P (13) W=−\|Q/angbracketright/angbracketleftQ\|VS. (14) The operator Wprovides the channel mixing even if the initial state is in th e physical subspace. The correlation between the two components of \|Ψ/angbracketrightis a consequence of the unbalanced action of the operator VSon the components of \|Ψ/angbracketright. It vanishes in the limit of the free negative ion. C. Interaction potentials The potential Vatin the Hamiltonian (6) represents the interaction between t he active electron and the ﬂuorine neutral core. It is taken as a local m odel potential which depends on the distance rbetween the electron and the atom center. The potential also includes a long range polarization interaction. Its explicit form ha s been adjusted to reproduce the binding energy of the ion F−as well as the mean radius of the p-orbital. The potential reads Vat=−U0+gr2−ae−r2, r ≤1 ; (15) Vat=−α 2r4−ae−r2, r> 1, 11whereU0= 5.64,g= 3.76,a= 1.2558 and the atomic polarizability of a ﬂuorine atom α= 3.76 [35]. All constants are given in the atomic units. The potential VSin the Hamiltonian (6) describes the interaction of the acti ve electron with theCu(111) andAg(111) surfaces. It has been proposed by Chulkov et al on the ba sis of their ab initio studies [36]. This local potential depends only on the elect ron coordinate zalong the surface normal. Its explicit form can be found in Re f. [36]. Qualitatively, it is an image charge potential in vacuum which joins smoothl y an oscillating potential with the period being that of the (111) planes inside the meta l bulk. When describing an electron motion in the direction perpendicular to the Cu(111) orAg(111) surface, this potential represents rather well important features of the surface such as the projected band gap (between -5.83 and -0.69 eV with respect to vacuum for Cuand between -4.96 and -0.66 eV for Ag), the surface state (5.33 eV and 4.625 eV below vacuum for Cu(111) and Ag(111), respectively), and the image state energy positions (0.82 eV and 0.77 eV below vacuum, respectively, for Cu(111) andAg(111)). Since the RCT process mainly favors transitions around the surface normal, this potential is ex pected to account for the eﬀect of the pecularities of the Cu(111) andAg(111) surfaces. In order to illustrate the eﬀects of the projected band gap of theCu(111) andAg(111) surfaces, we also present results obtained for a free-elect ron description of the metal sur- face. In this case, the local electron-surface interaction potential corresponds to the Al(111) surface and is taken from the work of Jennings et al [37]. D. Wave packet propagation In the wave packet propagation (WPP) approach [8,23], one st udies the time evolution of an electron wave function /angbracketleft/vector r\|Ψ(t)/angbracketrightgenerated by the system Hamiltonian. Here we consider both the static and dynamic problems. In the former, the dist ance between the ion and the metal surface is ﬁxed, while in the latter the ion collides wi th the surface. In both cases, the initial state is the free ion state (5). 12The time evolution can be regarded as a sequence of inﬁnitesi mal time steps generated by the evolution operator (11) \|Ψ(t+ ∆t)/angbracketright=U(t,t+ ∆t)\|Ψ(t)/angbracketright. (16) Since the initial state \|Ψ(t= 0)/angbracketright=\|ΨF−/angbracketrightis orthogonal to the vector \|Q/angbracketright, i.e., it is in the physical subspace, we can omit the projection operators to the left and right of the exponential in (11). So in (11) we take U(t,t+ ∆t) =e−i∆PHP=e−i∆t(Hat−Ea\|Q/angbracketright/angbracketleftQ\|+PVSP), (17) where the property Hat\|Q/angbracketright=Ea\|Q/angbracketrightof the vector \|Q/angbracketrighthas been used; Eais the eigenvalue ofHatcorresponding to the eigenfunctions p(r)Yl,m(θ,ϕ) withl= 1 andm= 0,±1. For an inﬁnitesimal time step ∆ t, the action of the evolution operator (17) can be evaluated b y means of the split operator approximation [38,39] U(t,t+ ∆t) =e−i∆t 2PVSPe−i∆t(Hat−Ea\|Q/angbracketright/angbracketleftQ\|)e−i∆t 2PVSP+O(∆t3). (18) Making use of the commutation relation of Haand\|Q/angbracketright /angbracketleftQ\|, this representation can be further simpliﬁed U(t,t+ ∆t) =e−i∆t 2PVSP/braceleftig/parenleftig ei∆tEa−1/parenrightig \|Q/angbracketright /angbracketleftQ\|+I/bracerightig e−i∆tHate−i∆t 2PVSP+O(∆t3).(19) The action of the exponential involving VSis evaluated via a Taylor expansion in which four terms are typically kept for the time step ∆ t= 0.025 atomic units. The action of the kinetic energy operator is computed in the c ylindrical coordinates which are well suited to the symmetry of the problem T=−1 2∂2 ∂z2−1 2ρ∂ ∂ρρ∂ ∂ρ+m2 2ρ2≡Tz+Tρ, (20) whereTzcontains only the z-derivative. The exponential of Hatin (19) is then transformed as e−i∆tHat=e−i∆tHatI=e−i∆t 2(Tz+Vat)e−i∆tTρe−i∆t 2(Tz+Vat)I+O(∆t3). (21) 13Recall that the operator Hatis diagonal in the isotopic two-dimensional space. So, the operator (21) acts on both components of \|Ψ(t)/angbracketrightin the same way. Finally, all the exponentials in (21) are approximated by means of the Cayley representati on [40] e−i∆tA=1−i∆t 2A 1 +i∆t 2A+O(∆t3). (22) In order to accurately reproduce the wave packet variation c lose to the atom center, we use a mapping procedure [41,42] deﬁned by z=f(ξ) = 0.05ξ+0.95ξ3 400 +ξ2, (23) ρ=f(η) = 0.05η+0.95η3 400 +η2, /angbracketleftz,ρ\|Ψ(t)/angbracketright=1√ρ˜Ψ(t,z,ρ). The wave packet ˜Ψ(t,z,ρ) is evaluated on a 2D mesh of points ( ξk,ηj) of the size 1200×800 with the step size ∆ = 0 .2 atomic units for both the coordinates. At the grid boundary, an absorbing potential is introduced [43,44] in o rder to avoid the wave packet reﬂection. The kinetic energy operator (20) has to be written in the auxi liary variables ξandηand then discretized. After the change of variables (23) the ope ratorTρassumes the form Tρ=−1 21 J√f∂ ∂ηf J∂ ∂η1√f+1 2m2 f2, (24) whereJ(η) =f′(η) is the Jacobian. The grid in the η-coordinate is set as ηj= ∆/2+∆(j−1). For every value ξwe have ˜Ψj= Ψ(ξ,ηj), and the action of (24) is deﬁned by the following midpoint procedure /parenleftig Tρ˜Ψ/parenrightig j=−1 2∆21 Jj/radicalig fj fj+1/2 Jj+1/2 ˜Ψj+1/radicalig fj+1−˜Ψj/radicalig fj −fj−1/2 Jj−1/2 ˜Ψj/radicalig fj−˜Ψj−1/radicalig fj−1  +1 2m2 f2 j. (25) Here the subscript j±1/2 means that the corresponding function is taken at the midpo int ηj±∆/2. A similar expression can be obtained for the action of Tzon the grid ξk= ξ0+ ∆(k−1). 14In the ﬁrst series of calculations, we study the static probl em when the ion F−is at a ﬁxed distanceZfrom the metal surface. The survival amplitude of the ion (th e auto-correlation of the wave function) is given by A(t) =/angbracketleftΨ(t= 0)\|Ψ(t)/angbracketright. (26) From the Laplace transform of the function A(t), one can obtain the density of states (DOS) projected on the free ion wave function. The structure of the DOS yields the energy level and its width for the negative ion state interaction with the sur face. In what follows, this width is referred to as the “static width” to emphasize that it is ex tracted from static calculations. It gives the electron transfer rate between the negative ion and the metal surface in the static problem. In the second series of calculations, we study the evolution of the electron wave packet when a negative ﬂuorine ion collides with the surface. The io n is assumed to approach the surface along a straight line perpendicular to the surface a t a constant velocity v. Only the incoming part of the collision is studied. The time depen dence of the wave function is obtained in the projectile reference frame, i.e., the tim e dependence of the Hamiltonian occurs through the potential VS. For each collision velocity, the ion survival probability , P(t,v) =\|A(t,v)\|2, is computed. To analyze the dynamics of the charge transfer , we deﬁne an eﬀective width of the negative ion state by G(Z,v) =−∂log[P(t,v)] ∂t, (27) whereZ=Z0−vtwithZ0being an initial distance of the ion from the metal surface. It corresponds to an eﬀective decay rate of the ion when it app roaches the surface with a velocityv. Comparing G(Z,v) to the level width obtained in the static calculations allo ws us to see to what extent the dynamical evolution can be descri bed by the static width of the ion level. 15III. RESULTS AND DISCUSSION A. F−ions interacting with a surface Al(111) The interaction of an F−ion with an Al(111) surface, where the latter is regarded as a free-electron metal surface, has already been studied by th e CAM method associated with the eﬀective range treatment of the negative ion [12,13]. It lead to a successful description of the negative ion formation in a grazing angle scattering [ 12]. Similarly, for large angle scattering from Ag(110), and polycrystalline AgandAlsurfaces, a quantitative agreement with experiment results [14,45] has been obtained [13]. In Figures 1 and 2 we compare the results obtained in the stati c case by two diﬀerent methods: The CAM method with eﬀective range treatment of the negative ion and the present WPP results obtained with the projection formalism . In both cases, the Al(111) surface is described as a free-electron metal surface using the potential proposed in [37]. Figures 1 and 2 present, respectively, the energy position a nd the width of the F−ion level interacting with the surface as a function of the ion-surfac e distanceZmeasured from the image plane. The characteristics of the ion level as a functi on ofZdisplay the behavior common for atomic species in front of a free-electron surfac e: The energy of the negative ion state decreases as the ion is placed closer to the surface, wh ich can be anticipated because of the image charge attraction; The level width increases roug hly exponentially as Zdecreases. The results obtained by two diﬀerent methods are extremely c lose to each other. This gives conﬁdence in the equivalence of the two descriptions of the F−ion. The results for a free- electronAl(111) surface are used below as a “free electron” reference t o which compare the Cu(111) andAg(111) results. It appears that the free electron results are almost identical for the three metals, except at very small distances from the surface. 16B. F−ions interacting with a surface Cu(111). A static case Figures 3 and 4 present the F−ion level characteristics (energy (Fig.3) and width (Fig.4 )) as a function of the ion- Cu(111) surface distance. The negative ion level energy exhib its an avoided crossing around 4 a0from the surface, which is quite diﬀerent from the smooth behaviour seen in Figures 1 and 2 for the free-electron metal . This is a direct consequence of the peculiarities of the Cu(111) surface and a similar situation has already been obser ved in the case of H- interacting with the same surface [23]. A schematic picture of the electronic structure of the model Cu(111) surface is shown in Fig. 5. The energy of electronic levels is plotted as a functi on of the electron momentum, k/bardbl, parallel to the surface. For k/bardblequal to zero, the projected band gap lies within the energy range from -5.83 to -0.69 eV (with respect to vacuum). Inside the gap, there is a surface state at -5.33 eV. In the present model of a Cusurface, the dispersion curves of all the metal electronic states as functions of k/bardblare parabolic with a free-electron mass. The resonant charge transfer process corresponds to transi tions between the ion level and metal states of the same energy. At large distances, the F−ion level is degenerate with the band gap. Therefore it can only decay to metal states with a ﬁnite k/bardbli.e. either to the 2D surface state band, or into 3D propagating states. A sZdecreases, the energy of the negative ion state decreases and it comes close to the b ottom of the 2D surface state continuum. The ion can decay by ejecting an electron wi th the angular momentum m= 0,±1 (the quantization axis is normal to the surface). As explai ned in Ref. [23], a resonance cannot cross the bottom of a 2D continuum in the sym metrical case m= 0 and there always exists a bound state below the bottom of the cont inuum. This state has an avoided crossing with the state which becomes the free ion st ate asZtends to inﬁnity. The F−ion character is then found to be associated with two diﬀeren t states depending on the Z range. It is transferred from the upper to the lower state wh en going through the crossing region (decreasing Z). Far from the avoided crossing region, the ion energy level is rather close to that found in the free electron case. 17As for the width, at large Z, its absolute value for F−in front of the Cu(111) surface is larger than that in the case of a free-electron surface. This result might appear surprising since the projected band gap prohibits the electron transfe r from the projectile to the metal along the surface normal ( k/bardbl= 0) and blocks the RCT into the 3D bulk continuum. Indeed, as can be seen in Fig. 5, there are no electronic states of the m etal with small k/bardblwhich are in resonance with the negative ion state. The potential b arrier separating the ion and the surface attains its least value in the direction normal t o the surface. Therefore, the surface normal is the preferred direction of the resonant el ectron transfer. One would then expect that the eﬀect of the projected band gap would be to sta bilize the negative ion level as compared with the case of a free electron metal. This has in deed been found for H− interacting with Cu(111) where the width of the H−state was much reduced as compared to theH−/Al(111) - case [23]. In contrast, we have observed an increase of the ﬂuorine nega tive ion decay rate as compared to the free-electron metal case. The reason is twof old. First, the 2D surface state continuum contributes to the decay of F−. Second, a ﬂuorine has a much larger electron aﬃnity than a hydrogen. Thanks to a better overlap of the wave functions, the 2D surface state continuum, when energetically allowed, is a dominati ng decay channel for an ion state lying within the band gap [23,28,46]. Moreover, when the bin ding energy of a negative ion is close to that of the surface state, a coupling of the ion lev el with the 2D surface state continuum is more eﬃcient than its coupling with the 3D conti nuum of the free-electron metal states (see a discussion in [46]). The eﬃciency of the 2 D surface state continuum as a decay channel can also be deduced from the sharp decrease of t he level width when passing through the crossing region as Zdecreases, i.e., when this decay channel becomes closed. For small values of Z: (i) the 2D surface state continuum does not contribute to th e decay of the negative ion, and (ii) the energy of the negative ion state is very close to the bottom of the projected band gap for small k/bardblso that the band structure eﬀect for the decay into the 3D bulk continuum vanishes. Therefore, we ﬁnd the wi dth of the level with ionic character very close to the free-electron results. 18Finally, we can stress that the procedure of extracting the r esonance characteristics employed here is based on the autocorrelation function (26) using the free F−ion wave function as the initial state. It converges well for the stat es of an ionic type. Convergence of the resonance characteristics for other states is diﬃcul t to achieve. This is the reason for showing only one state far from the crossing region (small or largeZ). In the crossing region, the ionic character is shared between the lower and upper sta tes so that the characteristics (energy and width) of both of them can be extracted. Since the convergence is easier to achieve for the energy of the state, the interval of distance s Z where both states are presented is larger in Fig.3. C. F−ions interacting with an Ag(111) surface. Static and dynami c studies The electronic structures of Ag(111) andCu(111) look rather similar (cf. Figures 5 and 6). However, characteristic features of the electronic str uctures occur at diﬀerent energies. The surface state in Ag(111) is located higher in energy than in Cu(111). For this reason the avoided crossing appears at a larger Zwhere the ion-surface charge transfer interaction is smaller. As a consequence the avoided crossing could not b e resolved in the energy dependence because it is localised in a too small range of Z. Therefore, we have chosen to represent the results for the energy and the width by a single continuous line (Figures 7 and 8, respectively). In fact, the energy of the negative ion state is almost the same as for the free-electron metal surface. The characteristic chang e of the level width when passing the crossing region (decreasing Z) is however still perfectly visible. It fully conﬁrms the dominance of the 2D surface state channel in the F−ion decay at large Z. At smallZ, theF−ion level is embedded in the 3D propagating states of Agand its characteristics are very close to those of the free-electro n case. Very close to the surface, the decrease of the level width as compared to Al(111) free-electron results is caused by the closeness of the F−level to the bottom of the Ag(111) valence band. As can be seen in Fig. 3b the bottom of the Ag(111) valence band is located at −9.7 eV in our model description 19of theAg(111) surface. For the model free-electron Al(111) case it is located at −15.9 eV. By studying the time dependent problem, one can ﬁnd out wheth er the peculiarities of Ag(111) observed in the static case can still be visible in a col lision ofF−with theAg(111) surface. We have computed the eﬀective level width G, deﬁned by (27), as a function of the distance for an ion F−approaching the surface at diﬀerent velocities. The result s are displayed in Fig. 9 together with the results of the static ca se forAg(111) and the free- electron metal surface Al(111). As the collision velocity is increased, the eﬀective width becomes closer to the free-electron result. This feature is very similar to what we have found for ions H−interacting with a Cu(111) surface [23]. The system needs a ﬁnite time to react on the presence of the projected band gap. If the coll ision is too fast, the electron wave packet does not have enough time to “explore” the band st ructure of the metal, and the ion decay remains identical to that on a free-electron me tal surface. In contrast, as the collision velocity is decreased, the eﬀe ctive width comes nearer to the staticAg(111) width. For the smallest velocity used here, 0.0058 ato mic units which correspond to a collision energy about 16 eV, the eﬀective wi dth is very close to the static one at largeZ. WhenZis decreased, the eﬀective width fails to perfectly reprodu ce the change of the behavior associated with the crossing of the bottom of the surface state continuum for all collision velocities considered here. We can nevert heless see that this variation is better reproduced as the collision velocity is decreased. I n fact, the dynamical broadening introduced by the change of the negative ion state energy wit h time [47] makes it impossible to reproduce a sharp variation of the static width as the ion a pproaches the surface. This leads to rounded delayed structures displayed in Fig.9. Mor eover, the oscillations of the eﬀective width at small Zcan tentatively be attributed to the population transfer be tween the two adiabatic states at the crossing region. From these r esults one can conclude that in the studied collision energy range, 16 eV – 5 keV, the charge t ransfer rate is intermediate between the free-electron case and the static Ag(111) case. The formation of F−ions andH−ions by collision on a Ag(111) surface has been studied experimentally by Guillemot and Esaulov [29]. When compari ng with results obtained with 20free-electron like surfaces they found that the H−data presents a strong band gap eﬀect. In particular, the survival probability of the negative ion s leaving the surface was much larger forAg(111). This was attributed to the blocking of the resonant ch arge transfer in H−/Ag(111) system, in line with theoretical results obtained for H−/Cu(111) [23] where the RCT rates are reduced by orders of magnitude compared to t he free-electron case. At the same time, results obtained with F−ions were approximately consistent with a description based on the charge transfer rate obtained in th e framework of a free-electron description of the metal surface. In the energy range studie d in their work, as shown above, the dynamical behaviour of the charge transfer is intermedi ate between the free-electron and staticAg(111) limits. This prohibits the use of the simple classical treatment of the parallel velocity eﬀect [28,48] which was shown to be important for a f ormation of F−, even at rather low collision energy [13]. Consequently, we cannot quantit atively compare our results with theirs. Nevertheless, we can notice that in our study the ban d gap eﬀect is reversed and much smaller in the present system than in the H−/Cu(111) system; it is even partly suppressed by the ion motion. These ﬁndings qualitatively agree with th e experimental results [29]. IV. CONCLUSIONS We have reported on a study of the electron transfer in the ion -metal systems F−/Cu(111) andF−/Ag(111). We have developed a new method to describe the eﬀect of six quasi-equivalent outer-shell electrons of F−on the resonant charge transfer process. The original six-electron problem has been transformed int o two one-electron problems in which the dynamics are not independent but rather are linked via a constraint. The projec- tion formalism for quantum systems with constraints has bee n used to obtain the quantum mechanical propagator for such a system. This modeling of th e ionF−is simple, eﬃcient and can easily be implemented in the wave packet propagation approach. Both theCu(111) andAg(111) surfaces exhibit an electronic structure with a proje cted band gap in which the ion energy level lies at large ion-surfa ce distances. This peculiarity 21of the electronic structure inﬂuences the charge transfer i nteraction with the ion, leading to a few remarkable features: •The ion level presents an avoided crossing with the bottom of the surface state con- tinuum as predicted for a 2D continuum with m= 0. The avoided crossing is very clearly marked for the system F−/Cu(111). •Because of the correlation between the six electrons of the F−ion the avoided crossing, which is a characteristic feature of the symmetric case m= 0, appears in the present system where electrons in both states with m= 0 and \|m\|= 1 contribute to the charge transfer. •When the negative ion level is low in the projected band gap, t he band gap does not cause a drastic drop of the charge transfer rate as observed i n other systems [23,24]. 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Kokoouline, O. Dulieu, R. Kosloﬀ and F. Masnou-Seeuw s, J. Chem. Phys. 110 (1999) 9865 [43] R. Kosloﬀ and D. Kosloﬀ, J. Comp. Phys. 63 (1986) 363 [44] D. Neuhauser and M Baer, J. Chem. Phys. 91 (1989) 4651 [45] M. Maazouz, S. Ustaze, L. Guillemot and V.A. Esaulov, Su rf. Sci. 409 (1998) 189 [46] J.P. Gauyacq, A.G. Borisov, G. Raseev and A.K. Kazansky , Faraday Discussions 117 (2000) 15 [47] J.J.C. Geerlings, J. Los, J.P. Gauyacq and N.M. Temme, S urf. Sci. 172 (1986) 257 [48] J.N.M. van Wunnik, R. Brako, K. Makoshi and D.M. Newns, S urf. Sci. 126 (1983) 618 251 2 3 4 5 6 7 8 9-8.0-7.5-7.0-6.5-6.0-5.5-5.0-4.5-4.0 distance (a.u.)energy (eV) a) FIG. 1. Energy position of the F−ion level in front of the Al(111) free-electron-like surface, as functions of the ion-surface distance, measured from the image plane (atomic units). The solid line represents the results obtained with the present WPP ap proach. The black dots indicate the results obtained with the CAM method. 261 2 3 4 5 6 7 8 910-310-210-1100 distance (a.u.)width (eV)b) FIG. 2. Energy width for the same system as in Fig.1 271 2 3 4 5 6 7 8 9 10 11 12-8.5-8.0-7.5-7.0-6.5-6.0-5.5-5.0-4.5-4.0 distance (a.u.)energy (eV) a) FIG. 3. Energy position of the F−ion level in front of the model Cu(111) surface, as functions of the ion-surface distance, measured from the image plane. (atomic units) The energy reference is the vacuum level. Black dots: results for the free-electr on Al(111) surface. The horizontal dashed-dotted line indicates the energy position of the bot tom of the surface state continuum. Solid line: results for the highest lying resonance. Dashed line: results for the lowest lying resonance. 281 2 3 4 5 6 7 8 9 10 11 1210-410-310-210-1100 distance (a.u.)width (eV)b) FIG. 4. Energy width for the same system as in Fig.3. 290 0.2 0.4 0.6 0.8 1-12-10-8-6-4-202 k// (a.u.)energy (eV) Cu(111) FIG. 5. A schematic picture of the electronic structure of Cu(111) (work function 4.9 eV) as a function of the electron momentum parallel to the surface ( atomic units). The energy reference is the vacuum level. The shaded area represents the 3D valenc e band continuum. The dashed line represents the 2D surface state continuum. The energy o fF−level at some distance from the surface is displayed as the horizontal solid line. 300 0.2 0.4 0.6 0.8 1-12-10-8-6-4-202 k// (a.u.)energy (eV) Ag(111) FIG. 6. Same as Fig. 5 for the Ag(111) surface (work function 4.56 eV). 311 2 3 4 5 6 7 8 9 10 11 12-8.5-8.0-7.5-7.0-6.5-6.0-5.5-5.0-4.5-4.0 distance (a.u.)energy (eV) a) FIG. 7. Energy position of the F−ion level in front of the model Ag(111) surface (solid lines), as functions of the ion-surface distance, measured from the image plane (atomic units). Black dots represent the results obtained for the free-electron Al(111) surface. The horizontal dashed-dotted line indicates the energy position of the bottom of the surfa ce state continuum. 321 2 3 4 5 6 7 8 9 10 11 1210-310-210-1 distance (a.u.)width (eV)b) FIG. 8. Energy width for the same system as in Fig. 7. 333 4 5 6 7 8 9 10v = 0.1 a.u. v = 0.05 a.u. v = 0.02 a.u. v = 0.011 a.u. v = 0.0058 a.u.10-310-210-1 distance (a.u.)width (eV) FIG. 9. The eﬀective level width Gversus the ion-surface distance Zfor various collision velocities. The solid dots and solid squares stand for, resp ectively, the free electron and Ag(111) static widths. Continuous lines represent the results obta ined by the constrained wave packet propagation method for the model surface Ag(111) for various collision velocities (see insert). 34
arXiv:physics/0103015v1 [physics.ed-ph] 6 Mar 2001A Bubble Theorem Oscar Bolina University of California Davis, CA 95616-8633 bolina@math.ucdavis.edu J. Rodrigo Parreira Cluster Consulting Torre Mapfre pl 38 Barcelona, 080050 Spain Introduction It is always a good practice to provide the physical content o f an analytical result. The following algebraic inequality len ds itself well to this purpose: For any ﬁnite sequence of real numbers r1,r2,...,r N, we have (r3 1+r3 2+...+r3 N)2≤(r2 1+r2 2+...+r2 N)3. (1) A standard proof is given in [1]. An alternative proof follow s from the isoperimetric inequality A3≥36πV2, where Ais the surface area and Vthe volume of any three-dimensional body. Setting the area A=/summationtextN i=14πr2and the volume V=/summationtextN i=1(4/3)πr3yields (1). A Bubble Proof We give yet another proof, now using elements of surface tens ion theory and ideal gas laws to the formation and coalescence of bubbles. This proof, found in [2], runs as follows. According to a well-known result in surface tension theory, when a spherical bubble of radius Ris formed in the air, there is a diﬀerence of pressure between the inside and the outside of the surface ﬁlm given by p=p0+2T R, (2) where p0is the (external) atmospheric pressure on the surface ﬁlm of the bubble, p is the internal pressure, and Tis the surface tension that maintains the bubble [3].Author 2 Suppose initially that Nspherical bubbles of radii R1,R2,...,R Nﬂoat in the air under the same surface tension Tand internal pressures p1,p2,...pN. According to (2), pk=p0+2T Rk, k = 1,2,...N. (3) Now suppose that all Nbubbles come close enough to be drawn together by surface tension and combine to form a single spherical bubble of radi usRand internal pressure p, also obeying Eq. (2). When this happens, the product of the i nternal pressure pand the volume vof the resulting bubble formed by the coalescence of the initial bubbles is, according to the ideal gas law [3], gi ven by pv=p1v1+...+pNvN, (4) where vk(k=1,2,..., N) are the volumes of the individual bubbles bef ore the coales- cence took place. For spherical bubbles, (4) becomes pR3=p1R3 1+...+pNR3 N. (5) Substituting the values of pandpkgiven in (2) and (3) into (5), we obtain R3−R3 1−R3 2−...−R3 N=2T p0(R2 1+R2 2+...+R2 N−R2). (6) Now, if the total amount of air in the bubbles does not change, the surface area of the resulting bubble formed by the coalescence of the bubble s is always smaller than the sum of the surface area of the individual bubbles before c oalescence. Thus, R2 1+R2 2+...+R2 N≥R2. (7) Since the potential energy of a bubble is proportional to its surface area, (7) is a physical condition that the surface energy of the system is minimal after the coalescence. It follows from (7) and the fact that p0andTare positive constants that the left hand side of equation (6) satisﬁes R3 1+R3 2+...+R3 N≤R3. (8) The equality, which implies conservation of volumes, holds when the excess pressure in the bubble ﬁlm is much less the atmospheric pressure. Comb ining (7) and (8) yields the inequality (1), which is also valid for negative n umbers. Acknowledgment. O.B. would like to thank Dr. Joel Hass for pointing out the iso peri- metric proof of (1), and FAPESP for support under grant 97/14 430-2.Author 3 References [1] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities , Second Edition, Cambridge Mathematical Library, Cambridge, UK, 1988, p.4 [2] H. Bouasse, Capillarit´ e et Ph´ enom` enes Superﬁciels , Librairie Delagrave, Paris (1924) p.48 [3] A. Hudson and R. Nelson, University Physics , Harcourt Brace Jovanovich, Inc. NY, 1982, p. 371 and p. 418
Tight open knots /G33/G4C/G52/G57/G55/G03/G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C1, /G36/G5C/G4F/G5A/G48/G56/G57/G48/G55/G03/G33/G55/G5D/G5C/G45/G5C/GE11 and Andrzej Stasiak2 1/G33 /G52/G5D/G51/G44 /G14 /G38/G51 /G4C/G59/G48/G55/G56 /G4C/G57 /G5C /G52 /G49 /G37/G48/G46/G4B/G51/G52/G4F/G52/G4A /G5C/G0F /G31/G4C/G48/G56/G5D /G44/G5A/G56/G4E /G44 /G14/G16/G24/G0F /G19/G13 /G1C/G19/G18 /G33/G52/G5D/G51/G44 /G14 /G0F /G33/G52/G4F/G44/G51/G47 e-mail: Piotr.Pieranski@put.poznan.pl 2 Laboratory of Ultrastructural Analysis, University of Lausanne, Switzerland ABSTRACT The most tight conformations of prime knots are found with the use of the SONOalgorithm. Their curvature and torsion profiles are calculated. Symmetry of the knots isanalysed. Connections with the physics of polymers are discussed. PACS: 87.16 Ac 1. Introduction From the point of view of topology knots are closed, self-avoiding curves1,2. Tying a knot in practice we operate on a finite piece of a rope3. At the end of the knot tying procedure a compact knotted structure with the two loose ends is created. To fix the knot type we splice the ends of the rope (outside the knotted structure, of course); without cutting the rope, the knot type cannot be changed. The same knot type fixing effect is reached if instead of splicing the loose ends we pull them apart and attach to parallel walls. In what follows we shall refer to such structures as open knots. Open knots are more common in nature than the closed knots. As indicated by de Gennes4 such knots are spontaneously tied and untied on polymeric molecules. Their existence changes considerably macroscopic properties of the materials. In general, the topological aspects of the microscopic structure of polymeric materials prove to be an important issue of the physics of polymers5,6,7. As we know well from the everyday experience, knots tied on a piece of rope can be tightened by pulling apart its loose ends as much as possible. Obviously, there is always a limit to such a knot tightening process – a particular conformation of the knot is reached at which the loose ends cannot be pulled apart any more. We shallrefer to such conformations as the tight open knots. As laboratory experiments prove, the final conformation of a tightened knot depends on two major factors: the initial conformation from which the knot tightening procedure starts and the physical parameters of the rope on which the knot is tied. The rope-rope friction coefficient, its elasticity constants etc. are to be taken into account. To get rid of such material dependent parameters we consider below knots tied on the perfect rope. The perfect rope has from the physical point of view somewhat contradictory properties: i. it is perfectly hard - to squeeze its circular cross-section into an ellipse the infinite energy is needed, ii. it is perfectly flexible - as long as none of its circular cross-sections overlap no energy is needed to bend it, iii. it is perfectly slippery - no energy is needed to tighten a knot tied on it. Introduction of the notion of the perfect rope allows us to define better the subject of our study. In the same sense introduction of the notion of hard spheres clarified the formulation of the packing problems. Take a piece of the perfect rope of length L and tie a knot on it. Stretch the ends of the rope apart as much as possible and measure the distance L’ between them. Obviously, L’≤L. The difference L-L’ can be seen as the length of the rope engaged within the knot. Dividing this value by the diameter D of the used rope we obtain a dimensionless number Λ known as the open thickness energy8. In the case of closed knots, in which the ends of the rope are spliced, L’=0. Knots in conformations for which Λ reaches its global minimum are called ideal9. More detailed, rigorous analysis of the thickness energy functionals were performed by O ’Hara, Simon, Rawdon and Millett10. The most tight open conformation of a knot of a given type, i.e. the conformation at which Λ reaches its global minimum, is an interesting object of which very little is known. As suggested by the present authors11, peaks within its curvature profile indicate places at which the real rope would be most prone to rupture. This hypothesis was verified on the trefoil knots tied on starch gel (spaghetti) filaments. Below we describe in detail the curvature and torsion profiles of tight open knots.2. Open knots tied and tighened on the the perfect rope Perfect ropes do not exist in nature. However, they are easily simulated in numerical experiments. The algorithm we used in the numerical experiments described below is SONO (Shrink-On-No-Overlaps) used previously in the search for the most tight (ideal) conformations of closed knots12. To demonstrate how SONO algorithm performs the knot tightening task we present results of a numerical experiment in which the initial conformation of the rope is so entangled, that at the first sight one cannot decide if it is knotted or not. Fig. 1 Numerical simulation of axial shrinkage of the perfect rope entangled into a clumsy overhand knot (the open trefoil knot). Notice that nugatory crossings are easily removedwhile the entanglements due to knotting remain. As seen in the figure, SONO algorithm performs the disentangling task without any problems. That the perfect rope is perfectly flexible and that there is no friction at its self-contact points is here of primary importance. To compare the approximate values of Λ we obtained in numerical experiments with a Λ value known precisely, we considered also the case of the Hopf link for which the length of its open form would simply correspond to the minimal length of the unopened component threaded by the opened one. As easy to see Λ equals in this case exactly 2 π. See Figure 2.Figure 2. The ideal open Hopf link. As laboratory experiments performed by Diao et al. indicated8, Λ varies with the knot type – different knots reduce the rope length in a different manner. Figure 3 presents the most tight conformations of the open trefoil (3 1) and figure-eight (4 1) knots tied and tightened by SONO on pieces of rope of identical length L. Fig. 3 The most tight open conformations of the trefoil (3 1) and figure-eight (4 1) knots tied on pieces of the rope of the same length. Clearly, in accordance with laboratory experiments described in ref.8, the trefoil knot engages less rope than the figure-eight knot. The values of Λ we obtained for these knots are, respectively, 10.1 and 13.7 ( ± 0.05). Values provided by Diao et al.8 are significantly larger: 10.35 and 14.65. The difference between results obtained in laboratory and numerical experiments exceeds errors specified by the authors. Most probably problems with friction encountered in experiments performed on real ropes do not allow one to enter the most tight conformations of the knots accessible in numerical experiments performed on perfect ropes. Table I shows the Λ values we found for the most tight closed and open conformations of a few prime knots. Toprovide a natural Λ unit, we included into the table also the rigorous value of Λ known for the Hopf link Interestingly enough, the difference of length of the most tightclosed and open forms proves to be very close to 2 π also for several other chiral knots. On the other hand, for achiral knots, 4 1 and 6 3, the differences are significantly different. Knot type Open form Closed form Difference Hopf link 6.28 12.56 6.28 31 10.1 16.33 6.23 41 13.7 20.99 7.29 51 17.3 23.55 6.25 52 18.4 24.68 6.28 63 20.7 28.88 8.18 Table I The normalized minimal length Λ of the most tight conformations of a few prime knots in their open and closed forms. Λ values for the closed forms of the knots were taken from ref.12. Rolfsen notation was used to indicate the knot types13. 3. Symmetry of the curvature and torsion maps of the 3 1 and 4 1 open knots Calculating the curvature and torsion of a smooth (differentiable twice) curve defined by analytical formulae is a trivial task. On the other hand, the determination of the curvature and torsion maps of the knots simulated in numerical experiments is extremely susceptible to the inaccuracies with which positions of the consecutive points of the simulated knot are given. To find the curvature, the first and second derivatives must be known. In the case of torsion, the third derivative is also needed. The discrete differentiation procedures reveal a considerable noise present within the curvature and torsion maps. To get rid of the noise, we took averages over a large set of the maps calculated in long runs in which the tightened knot was moving slowly there and back along the simulated rope. The slow oscillatory motion of the knot was introduced on purpose to minimise effects stemming from the discrete structure of the simulated rope. Figure 4 presents consecutive curvature maps registered in a short interval cut out from one of such runs. Averaging the instantaneous curvature maps within a reference frame which moves together with the knot we obtain curvature maps which are much better defined. Figure 5 presents such maps found for the most tight 3 1 and 4 1 knots. The knots were tied on the numerically simulated perfect rope consisting of 200 segments of equal length.Fig.4 Curvature maps of the 3 1 knot registered in a numerical experiment in which the tightened knot was slowly moving there and back along the simulated rope. The plots display a few interesting features which we shall discuss below emphasising those of them, which according to us may survive in the limit of the most tight open conformations. Let us start with the discussion of the curvature profiles. Within some accuracy limits, both profiles can be seen as consisting of two mirror symmetrical parts. We assume here that the external zero curvature regions are extended to infinity. The curvature profile of e.g. the 3 1 knot seen from both ends of the knot looks almost identical. If by the curvature profile we understand the curvature κ as a function of the arc length l, this symmetry property can be expressed as follows: in the middle of the knot there exists such a point lm, that on the left and on the right of it curvature profiles are identical κ(lm+ε) = κ(lm-ε). In the case of the numerically simulated discretized knots the mirror symmetry is not exact. It seems to us, and we would like to express it as a conjecture, that in the case of the ideal open conformations of both 3 1 and 4 1 knots the mirror symmetry of the curvature profile should be exact. Mirror symmetry of the curvature profile reflects the twofold rotational symmetry of the open knot conformation. Let us note here that the point symmetry elements of the closed ideal conformations of the 3 1 and 41 knots are different. One of the most distinct local features of the both curvature landscapes are the double peaks visible at the entrance to the knots. As described previously, the inner peak develops only at the final stage of the tightening process11. Laboratoryexperiments prove that at the points of high curvature the filaments, on which knots are tied, are most susceptible to breaking. Fig. 5 The most tight open trefoil (3 1) and figure-eight (4 1) knots together with their maps of curvature and torsion. a) The trefoil knot shown in the figure (left) is of right-handed type and like other chiral knots cannot be converted into its mirror image. b) The map of curvature of the open trefoil knot shows a mirror symmetry (mirror plane is vertical and is in the centre of the map) thus, an identical map of curvature would beobtained for the left handed enantiomer of this knot. The curvature is normalised in respect to the diameter of tightly knotted tubes: value 1 is attained when the local radius of curvature corresponds to the diameter of the tube, value 2 would correspond to theradius of curvature being equal to the radius of the tube (points of sharp reversals). c) The map of torsion of the open trefoil knot shows also a mirror symmetry, thus, for left- handed enantiomer the map of torsion would be reversed along the vertical axis. d) Figure eight knot is achiral and is easily convertible into its mirror image (see Fig. 6). e) The map of curvature of the open figure eight knot shows perfect mirror symmetry and thus has no polarity. f) The map of torsion of the open figure eight knot shows a clear polarity - the passage from the left to right is different from this from the right to left. The total torsion ishowever zero as it is expected for an ideal form of achiral knot. The maps of torsion show important differences between chiral 31 knot and achiral 41 knot. The map of torsion of the 3l knot shows a mirror symmetry while the torsion map of the achiral knot 41 shows a symmetry of different kind: in the middle of the knot there exists such a point lm, that on the left and on the right of it torsion profiles are of identical magnitude but of an opposite sign κ(lm+ε) = -κ(lm-ε). Thus, observing the torsion we can distinguish between the two ends of this knot. Takinginto account curvature and torsion, which provide the complete descriptors of a given trajectory, we see that a time reversed travel through a chiral knot 31 is indistinguishable from the original one. In contrast to that, in the achiral 41 knot we can distinguish between time reversed travels – the signs of the torsion component appear during the travels in the reversed order. In the 41 knot the time reversed travel is identical with the not reversed travel along the mirror image of the original knot (the mirror reflection changes left-handed regions into right-handed and contrary). As could be expected for achiral knots the total torsion cancels to zero while this ofcourse is not the case for achiral knots. It may seem surprising that achiral knots 4 1 are in fact polar while this is not the case for a chiral knot 31. Interestingly, the ideal open configuration of 41 knot is congruent with its mirror image. Figure 6. Fig.6 Congruency of the open 4 1 knot with its mirror image. Rotation by π/2 is followed by a mirror reflection. Although, by definition, all achiral knots can be continuously converted into their mirror images there is only a small subset of their configurations which upon rigidtransformation are congruent with their mirror images 14. 4. The problem of the local minima Several independent starting configurations of such simple knots as 31 or 41 were converted during our simulations to configurations essentially identical to those shown in Fig. 3. This was, however, not the case for more complicated knots. Fig.7 shows what happens when the most tight closed configuration of 52 knot is opened in three different positions and the ends are pulled apart. Three different finalconfigurations are obtained and to pass from one to another the string has to be loosened and loops have to be moved along the knot. Apparently the three different configurations constitute different local minima in the configuration space of tight open knots. Interestingly the nice symmetrical configuration shown in Fig. 7a causes biggest effective shortening of the string and is thus furthest from the global minimum. The configuration shown on Fig. 7c may in fact represent a global minimum for this knot as its effective shortening of the tube is smallest. This configuration does not show any symmetry. Fig. 7 Three different local minima in the configuration space of open knots 52. (a) Knot 52 in its most tight closed configuration. Three distinct sites 1, 2 and 3 were used to open the knot. (b), (c), (d) The local minima configurations obtained upon opening the closed knot in sites 1, 2 and 3 respectively. Notice that the symmetrical conformation 1 engages more ofthe rope length than the other two configurations. The conformation 3 representspresumably a global minimum and thus would corresponds to an ideal open configuration of5 2 knot. Fig. 8 Local minima conformations of the 6 3 knot. Notice the symmetry of the conformation marked as (d). See text for its discussion.To check if symmetrical configurations may constitute local minima in configuration space of more complex tight open knots we took the most tight closed configuration of achiral knot 6 3 and opened it in different positions. See Fig. 8. Upon pulling apart the opened ends we noticed that one of the openings led to a nice regular form congruent with its mirror image. It seems to us that this form constitutes the global minimum. 5. Discussion Perfect ropes do not exist in nature, but polymeric molecules are not far from being perfect. Put into the perpetual motion by thermal fluctuations they never get blocked by friction which plays such an important role in the macroscopic world allowing, for instance, to stop the huge mass of a docking ship with a rope tied in an appropriate manner (round turn and two half hitches) to the bollard15. Time averaged conformations of the open knots tied on polymeric molecules are very similar to the ideal open knots we considered above. In particular, the length of the molecular rope engaged within such knots should be directly related to the Λ value we calculated. Properties of the knotted molecules are essentially different form the unknotted ones. For instance, their gel mobility coefficients are essentially different16,17. In general, polymeric materials in their phases in which the fraction of knotted molecules is considerable, should display interesting physical properties4. When the polymer technology will provide us with such materials is difficult to predict today. Certainly, the properties of a polymer, whose all molecules are tied into right-handed trefoil knots will be different from the polymer within which all the knots are left-handed. Acknowledgement We thank G. Dietler and J. Dubochet for helpful discussions. This work was carried out under projects: KBN 5PO3B01220 and SNF 31-61636.00. 1 L. H. Kauffman, Knots and Physics (World Scientific Publishing Co., 1993). 2 C. C. Adams, The Knot Book (W.H. Freeman and Company, New York, 1994). 3 C. W. Ashley, The Ashley Book of Knots (Doubleday, New York, 1993) 4 P.-G. de Gennes, Macromolecules, 17, 703 (1984). 5 M. D. Frank-Kamentskii and A. V. Vologodskii, Sov. Phys. Usp. 24 679 (1981); 6 A. Y. Grossberg, A. R. Khokhlov, Statistical physics of macromolecules , AIP Press, 1994. 7 A. Y. Grossberg, A. Feigel and Y. Rabin, Phys. Rev. A 54, 6618-6622 (1996). 8 Y. Diao, C. Ernst and E. J. Janse van Rensburg in Ideal Knots , eds. Stasiak, A., Katritch, V. and Kauffman, L. H. World Scientific, Singapore, 1998, p.52-69. 9 V. Katritch, J. Bednar, D. Michoud, R. G. Sharein, J. Dubochet and A. Stasiak, Nature 384, 142 -/G14/G17/G18/G03/G0B/G14/G1C/G1C/G19/G0C/G1E/G03/G39/G11/G03/G2E/G44/G57/G55/G4C/G57/G46/G4B/G0F/G03/G3A/G11/G03/G2E/G11/G03/G32/G4F/G56/G52/G51/G0F/G03/G33/G11/G03/G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C/G0F/G03/G2D/G11/G03/G27/G58/G45/G52/G46/G4B/G48/G57/G03/G44/G51/G47 A. Stasiak, Nature 388, 148 (1997). 10 See chapters by: E. Rawdon; J. A. Calvo and K. C. Millett; J. Simon; J. O ’Hara in Ideal Knots , eds. A. Stasiak , V. Katritch and L. H. Kauffman, (World Scientific, Singapore, 1998). 11 P. /G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C/G0F/G03/G36/G11/G03Kasas, G. Dietler, J. Dubochet and A. Stasiak, Localization of breakage points in knotted strings , submitted to New J. Phys. (2000). 12 P. /G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C/G03/G4C/G51/G03Ideal Knots , eds. A. Stasiak, V. Katritch and L. H. Kauffman, (World Scientific, Singapore, 1998) 13 D. Rolfsen D Knots and Links (Berkeley: Publish or Perish Press, 1976). 14 C. Liang and K. Mislow, J. Math. Chem. 15, 1 (1994). 15 L. H. Kauffman, Knots and Physics (World Scientific Publishing Co., 1993) p.325. 16 A. Stasiak, V. Katritch, J. Bednar, D. Michoud and J. Dubochet, Nature 384, 122 (1996) 17 V. Vologodskii, N. Crisona, B. Laurie, P. /G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C/G0F/G03/G39/G11/G03Katritch, J. Dubochet and A. Stasiak, J. Mol. Biol. 278, 1-3 (1998).
arXiv:physics/0103017v1 [physics.optics] 7 Mar 2001Resonant radiation pressure on neutral particles in a waveg uide R. G´ omez-Medina∗, P. San Jos´ e∗, A. Garc´ ıa-Mart´ ın∗‡, M. Lester∗a, M. Nieto-Vesperinas†& J.J. S´ aenz∗‡b ∗Departamento de F´ ısica de la Materia Condensada, Universi dad Aut´ onoma de Madrid, E-28049 Madrid, Spain. †Instituto de Ciencia de Materiales, CSIC, Campus de Cantobl anco, E-28049 Madrid, Spain. ‡Instituto de Ciencia de Materiales “Nicol´ as Cabrera”, Uni versidad Aut´ onoma de Madrid, E-28049 Madrid, Spain. (February 2, 2008) A theoretical analysis of electromagnetic forces on neutra l particles in an hollow waveguide is presented. We show that the eﬀective scattering cross secti on of a very small (Rayleigh) particle can be strongly modiﬁed inside a waveguide. The coupling of t he scattered dipolar ﬁeld with the waveguide modes induce a resonant enhanced backscattering state of the scatterer-guide system close to the onset of new modes. The particle eﬀective cross s ection can then be as large as the wavelength even far from any transition resonance. As we wil l show, a small particle can be strongly accelerated along the guide axis while being highly conﬁned in a narrow zone of the cross section of the guide. 42.50.Vk, 32.80.Lg, 42.25.Bs Demonstration of levitation and trapping of micron- sized particles by radiation pressure dates back to 1970 and the experiments reported by Ashkin and co-workers [1]. Since then, manipulation and trapping of neutral particles by optical forces has had a revolutionary im- pact on a variety of fundamental and applied studies in physics, chemistry and biology [2]. These ideas were ex- tended to atoms and molecules where radiation pressure can be very large due to the large eﬀective cross section (of the order of the optical wavelength) at speciﬁc res- onances [1,3]. When light is tuned close to a particular transition, optical forces involves (quantum) absorption and reradiation by spontaneous emission as well as co- herent (classical) scattering of the incoming ﬁeld with the induced dipole [4]. Selective control of the strong in- terplay between these two phenomena is the basis of laser cooling and trapping of neutral atoms [5]. However, far from resonance, light forces on atoms, molecules and nanometer sized particles are, in general, very small. Here we show that the scattering cross sec- tion of a very small (Rayleigh) particle can be strongly modiﬁed inside a waveguide. The coupling of the scat- tered dipolar ﬁeld with the waveguide modes induce a resonant enhanced backscattering state of the scatterer- guide system close to the onset of new modes. Just at the resonance, the eﬀective cross section becomes of the order of the wavelength leading to an enhanced resonant radiation pressure which does not involve any photon ab- sorption phenomena. As we will show, a small particle not only can be strongly accelerated along the guide axis but it can also be highly conﬁned in a narrow zone of the cross section of the guide. For the sake of simplicity we consider a two- dimensional xzwaveguide with perfectly conducting walls and cross section D. The particle is then repre- sented by a cylinder located at /vector r0= (x0, z0) with its axis along oyand radius much smaller than the wavelength(see top of Fig.1). However, apart from some depolar- ization eﬀects, the analysis contain the same phenomena as the full three-dimensional problem [6] and hence it permits an understanding of the basic physical processes involved in the optical forces without loss of generality. An s-polarized electromagnetic wave is assumed (the electric ﬁeld parallel to the cylinder axis), /vectorE(/vector r) = exp(−iωt)E0(/vector r)/vector ywith wavevector k=ω/c = 2π/λ. For a single-mode waveguide ( D/2< λ ≤ D), the incoming electric ﬁeld can be written as the sum of two interfering plane waves: E0(/vector r) = E0(exp(ikzz+ikxx)−exp(ikzz−ikxx)) where kz= kcos(θ),kx=ksin(θ) =π/D. The scatterer can be characterized by the scattering phase shift δ0or by its polarizability αand Rayleigh scattering cross section σ [7,8]. The time average force /vectorFcan be written as the sum of an optical gradient force and a scattering force: [9]: /vectorF=/braceleftBigg 1 4α/vector∇\|Einc\|2+/angb∇acketleft/vectorS/angb∇acket∇ight cσ/bracerightBigg /vector r=/vector r0(1) where Eincis the total incident ﬁeld on the particle and /angb∇acketleft/vectorS/angb∇acket∇ightis the time average Poynting vector. If we neglect the multiple scattering eﬀects between the scatterer and the waveguide walls, the interaction would be equivalent to that of a particle placed in the interfer- ence pattern of two crossed plane wave beams. In this caseEinc=E0and the theory of radiation pressure in a waveguide is straightforward. The longitudinal force F0 z (per unit length) can be written in terms of the average power density of the incident beams, /angb∇acketleftS/angb∇acket∇ight=ǫ0c\|E0\|2, as F0 z= 2σǫ0\|E0\|2cos(θ)sin2(πx0 D) (2) which is maximum ( F0 zmax= 2σǫ0\|E0\|2cos(θ)) just in middle of the waveguide. The transversal force induces an optical potential along xgiven by: 1U0 x=−α\|E0\|2sin2(πx0/D). (3) which, for α >0, conﬁnes the particle near the center of the waveguide. This is the two-dimensional analogue of previous approaches on laser-guiding of atoms and par- ticles in hollow-core optical ﬁbers [10–12] where the in- teraction of the dipole ﬁeld with the guide walls was ne- glected. The scattering with the waveguide walls may induce however a dramatic eﬀect on the optical forces on the particle. The scatterer radiates ﬁrst a dipole ﬁeld gen- erated by the ﬁeld of the incoming mode E0. Then, the scattered ﬁeld, perfectly reﬂected by the waveguide walls, goes back to the scatterer changing the ﬁeld inciding on the scatterer and so on. This multiple scattering process can be regarded as produced by a set of inﬁnite image dipoles [13,14]. From the exact solution for the total ﬁeld together with equation (1), we found that, for a sin- gle mode waveguide, the forward component of the force Fzcan be written in terms of the waveguide transmit- tance T(deﬁned as the ratio between the outgoing and incoming energy ﬂux; 0 ≤T≤1) as: Fz= 2Dǫ0\|E0\|2cos2(θ)(1−T(x0)) (4) where Tdepends on the transversal position of the par- ticlex0. The transmission coeﬃcient Thad been discussed be- fore in the context of electronic conductance of quasi-one- dimensional conductors with point-like attractive impuri - ties [13–16]. Tpresents two peculliar properties: i)when D/λis just at the onset of a new propagating mode, the scatterer becomes transparent; ii)interestingly, when D/λis close but still below a mode threshold, the trans- mittance of a single mode waveguide presents a dip down to exactly T= 0. This backscattering resonance, that was associated to the existence of a quasi-bound state in- duced by the attractive impurity [15,14], can be achieved for any attractive scattering potential of arbitrarily sma ll strength [14], i.e. for arbitrarily small polarizability α and cross section σ[8]. This resonance has a pronounced eﬀect on the radiation forces. In Fig. 2 we plot the transmission coeﬃcient as a function of both the waveguide width, D/λ, and the scatterer position x0forδ0= 10o. Near the thresh- old of the second mode ( D/λ<∼1),Tpresents sharp dips (down to T= 0) at some particular positions of the scatterer. Just at the enhanced backscattering res- onances, the longitudinal force present strong maxima Fzmaxwhich can be compared with F0 zmax(the maximum force obtained neglecting the interactions with the walls) Fzmax/F0 zmax= cos( θ)D/σ(see Fig. 3), i.e. at reso- nance the interaction cross section of the particle in the waveguide can be as large as the total cross section of the waveguide independently of its value σin the unbounded space. The force enhancement factor can be huge: for δ0= 10o(σ≈20nm) and at micron wavelengths, itis of the order of 50, while for a nanometer scale par- ticle ( δ0≈2o) this enhancement would be ≈103for two-dimensions, but ≈106in true three-dimensional sys- tems! This result is rigurously true for perfect walls. In actual waveguides however, the radiation losses through the walls and the scattering with surface defects [17] may modify the resonace behaviour for σvalues comparable to the surface roughness. The resonance is related to the strong coupling be- tween the incoming mode and the ﬁrst evanescent mode in the waveguide. This can be seen by plotting the ﬁeld intensity inside the waveguide for diﬀerent particle po- sitions (Fig. 1). At the resonance, the ﬁeld around the particle corresponds to that of the second mode in the waveguide (with a node in the waveguide axis) which de- cay far from the defect. When the particle is located at the middle of the waveguide, there is no coupling of the scattered ﬁeld with the ﬁrst evanescent mode and Fz presents a minimum. Transversal forces are also strongly aﬀected by the resonances. Although the main contribution to these forces come from polarization eﬀects (i.e. proportional to ∇\|Einc\|2), in contrast with the free space case, the lateral forces have also a contribution of pure scattering origin due to the reﬂections of the ﬂux from the walls. The in- duced transversal conﬁning potential far from the mode threshold presents a single well similar to that discussed for the unbound system. However, near the resonance condition it presents two strong minima reﬂecting the excitation of the evanescent mode. In Fig. 3 we plot the normalized longitudinal force Fzand transverse conﬁning potential Ux. The particle will be strongly conﬁned in a small region inside the waveguide where the forward lon- gitudinal force is maximum. For example, for δ0= 10o and at micron wavelengths, the potential well is more than one order of magnitude deeper than that obtained for the unbound system. In summary, we have discussed the electromagnetic forces on small neutral particles in a hollow waveguide. In contrast with standard resonance radiation forces, the waveguide-particle backscattering resonances discussed here do not involve photon absorption processes and, we believe, open intriguing posibilities of atom and molecule manipulation. Speciﬁcally, the depth of the potential wells for the particle in resonant conditions and its re- markably large cross section suggest stable guiding of the particle along the waveguide with extremely large accel- erations. We thank R. Arias, P. C. Chaumet, L. Froufe, F. J. Garc´ ıa-Vidal, R. Kaiser, T. L´ opez-Ciudad and L. Mart´ ın- Moreno for discussions. Work of M.L. was supported by a postdoctoral grant of the Comunidad Aut´ onoma de Madrid. This work has been supported by the Comu- nidad Aut´ onoma de Madrid and the DGICyT through Grants 07T/0024/1998 and No. PB98-0464. 2aPermanent address of M.L. is Instituto de F´ ısica Arroyo Seco, Facultad de Ciencias Exactas, UNCPBA, Pinto 399 (7000), Tandil, Argentina . bCorrespondence and requests for materials should be ad- dressed to J.J.S. (e-mail: juanjo.saenz@uam.es). [1] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970). [2] A. Ashkin, Science 210, 1081 (1980); Proc. Natl. Acad. Sci. USA 94, 4853 (1997). [3] V. Letokhov, Pis’ma Zh. Eksp. Teor. Fiz. 7, 348 (1968) [JETP Lett. 7, 272 (1968)]. [4] G.A. Askar’yan, Sov. Phys. JETP 15, 1088 (1962). [5] S. Chu, Rev. Mod. Phys. 70, 685 (1998); C. N. Cohen- Tannoudji, Rev. Mod. Phys. 70, 707 (1998); W. D. Phillips, Rev. Mod. Phys. 70, 721 (1998). [6] M. Lester, and M. Nieto-Vesperinas, Opt. Lett. 24, 936 (1999). [7] J. A. Stratton, Electromagnetic Theory , (MacGraw-Hill, New York, 1941). [8] Assuming a s-polarized electromagnetic plane wave and a cylindrical scatterer, the polarizability is related to t he scattering phase shift δ0byα=ǫ0(2/k2) sin(2 δ0) . A sim- ple application of the optical theorem gives a scattering cross section (cross length in this case) σ= (4/k) sin2(δ0). For example, for a dielectric rod of radius a, dielec- tric constant ǫandka≪1,δ0≈π(ǫ−1)(ka/2)2, α≈πǫ0(ǫ−1)a2andσ≈π2/(4k)(ǫ−1)2(ka)4. [9] J. P. Gordon, Phys. Rev. A 8, 14 (1973). [10] M. A. Ol’Shanii, Y. B. Ovchinnikov, and V. S. Letokhov, Opt. Commun. 98, 77 (1993). [11] M. J. Renn, et al., Phys. Rev. Lett. 75, 3253 (1995). [12] M. J. Renn, and R. Pastel, J. Vac. Sci. Technol. B 16, 3859 (1998). [13] C. S. Chu, and S. R. Sorbello, Phys. Rev. B 40, 5941 (1989). [14] Ch. Kunze, and R. Lenk, Sol. State Comm. 84, 457 (1992) [15] P. F. Bagwell, Phys. Rev. B 41, 10354 (1990). [16] E. Tekman, and S. Ciraci, Phys. Rev. B 43, 7145 (1991). [17] A. Garc´ ıa-Mart´ ın, J. A. Torres, J. J. S´ aenz, and M. Ni eto- Vesperinas , Appl. Phys. Lett. 71, 1912 (1997); Phys. Rev. Lett. 80, 4165 (1998) 3/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 a b c dx0D λ FIG. 1. Top: Sketch of the particle-waveguide system. Field intensity plots for diﬀerent particle positions x0across the waveguide ( a)x0/D= 0.15,b)x0/D= 0.22,c) x0/D= 0.25,d)x0/D= 0.5). The ﬁeld incides from the left side. /λDT x /D0FIG. 2. Transmittance of a single mode waveguide versus width D/λand scatterer position x0/Dfor a ﬁxed phase-shift value δ0= 10o. 01020304050 ab c d/BY /DE/BP/BY/BC /DE /D1/CP/DC 0 0.2 0.4 0.6 0.8 1 x0/D−15−10−50 a bcd/CD /BP/AC /AC /CD/BC /D1/CX/D2/AC /ACa b FIG. 3. Longitudinal force Fz(a)and lateral conﬁning potential U(b)versus particle position (normalized to the maximum force F0 zmaxand the minimum potential U0 minin the unbounded system). The values of x0/Dcorresponding to the dots are those of ( a, b, c, d ) in Fig. 1. 4
arXiv:physics/0103018v1 [physics.data-an] 8 Mar 2001Eﬀect of Trends on Detrended Fluctuation Analysis Kun Hu1, Plamen Ch. Ivanov12, Zhi Chen1, Pedro Carpena3, H. Eugene Stanley1 1Center for Polymer Studies and Department of Physics, Bosto n University, Boston, MA 02215 2Harvard Medical School, Beth Israel Deaconess Medical Cent er, Boston, MA 02215 3Departamento de F´ ısica Aplicada II, Universidad de M´ alag a E-29071, Spain Detrended ﬂuctuation analysis (DFA) is a scaling analysis m ethod used to estimate long-range power-law correlation exponents in noisy signals. Many noi sy signals in real systems display trends, so that the scaling results obtained from the DFA method beco me diﬃcult to analyze. We system- atically study the eﬀects of three types of trends — linear, p eriodic, and power-law trends, and oﬀer examples where these trends are likely to occur in real data. We compare the diﬀerence between the scaling results for artiﬁcially generated correlated nois e and correlated noise with a trend, and study how trends lead to the appearance of crossovers in the scalin g behavior. We ﬁnd that crossovers result from the competition between the scaling of the noise and the “apparent” scaling of the trend. We study how the characteristics of these crossovers depend on (i) the slope of the linear trend; (ii) the amplitude and period of the periodic trend; (iii) th e amplitude and power of the power-law trend and (iv) the length as well as the correlation properti es of the noise. Surprisingly, we ﬁnd that the crossovers in the scaling of noisy signals with trends al so follow scaling laws — i.e. long-range power-law dependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determ ined by the superposition of the separate results of the DFA on the noise and on the trend, assu ming that the noise and the trend are not correlated. If this superposition rule is not follow ed, this is an indication that the noise and the superimposed trend are not independent, so that remo ving the trend could lead to changes in the correlation properties of the noise. In addition, we s how how to use DFA appropriately to minimize the eﬀects of trends, and how to recognize if a cross over indicates indeed a transition from one type to a diﬀerent type of underlying correlation, or the crossover is due to a trend without any transition in the dynamical properties of the noise. I. INTRODUCTION Many physical and biological systems exhibit com- plex behavior characterized by long-range power-law cor- relations. Traditional approaches such as the power- spectrum and correlation analysis are not suited to accu- rately quantify long-range correlations in non-stationar y signals — e.g. signals exhibiting ﬂuctuations along poly- nomial trends. Detrended ﬂuctuation analysis (DFA) [1–4] is a scaling analysis method providing a simple quantitative parameter — the scaling exponent α— to represent the correlation properties of a signal. The ad- vantages of DFA over many methods are that it per- mits the detection of long-range correlations embedded in seemingly non-stationary time series, and also avoids the spurious detection of apparent long-range correla- tions that are artifact of non-stationarity. In the past few years, more than 100 publications have utilized the DFA as method of correlation analysis, and have uncov- ered long-range power-law correlations in many research ﬁelds such as cardiac dynamics [5–23], bioinformatics [1,2,24–34], economics [35–47], meteorology [48–50], ge- ology [51], ethology [52] etc. Furthermore, the DFA method may help identify diﬀerent states of the same system according to its diﬀerent scaling behaviors — e.g. the scaling exponent αfor heart inter-beat intervals is diﬀerent for healthy and sick individuals [14,16,17,53]. The correct interpretation of the scaling results ob- tained by the DFA method is crucial for understandingthe intrinsic dynamics of the systems under study. In fact, for all systems where the DFA method was applied, there are many issues that remain unexplained. One of the common challenges is that the correlation expo- nent is not always a constant (independent of scale) and crossovers often exist — i.e. change of the scaling expo- nentαfor diﬀerent range of scales [5,16,35]. A crossover usually can arise from a change in the correlation proper- ties of the signal at diﬀerent time or space scales, or can often arise from trends in the data. In this paper, we sys- tematically study how diﬀerent types of trends aﬀect the apparent scaling behavior of long-range correlated sig- nals. The existence of trends in times series generated by physical or biological systems is so common that it is al- most unavoidable. For example, the number of particles emitted by a radiation source in an unit time has a trend of decreasing because the source becomes weaker [54,55]; the density of air due to gravity has a trend at diﬀerent altitude [56]; the air temperature in diﬀerent geographic locations and the water ﬂow of rivers have a periodic trend due to seasonal changes [49,50,57–59]; the occur- rence rate of earthquakes in certain area has trend in diﬀerent time period [60]. An immediate problem fac- ing researchers applying scaling analysis to time series is whether trends in data arise from external conditions, having little to do with the intrinsic dynamics of the sys- tem generating noisy ﬂuctuating data. In this case, a possible approach is to ﬁrst recognize and ﬁlter out the trends before we attempt to quantify correlations in the 1noise. Alternatively, trends may arise from the intrinsic dynamics of the system, rather than being an epiphe- nomenon of external conditions, and thus may be corre- lated with the noisy ﬂuctuations generated by the sys- tem. In this case, careful considerations should be given if trends should be ﬁltered out when estimating correla- tions in the noise, since such ”intrinsic” trends may be related to the local properties of the noisy ﬂuctuations. Here we study the origin and the properties of crossovers in the scaling behavior of noisy signals, by ap- plying the DFA method ﬁrst on correlated noise and then on noise with trends, and comparing the diﬀerence in the scaling results. To this end, we generate artiﬁcial time series — anti-correlated, white and correlated noise with standard deviation equal to one — using the modiﬁed Fourier ﬁltering method introduced by Makse et al. [63]. We consider the case when the trend is independent of the local properties of the noise (external trend). We ﬁnd that the scaling behavior of noise with a trend is a su- perposition of the scaling of the noise and the apparent scaling of the trend, and we derive analytical relations based on the DFA, which we call “superposition rule”. We show how this “superposition rule” can be used to determine if the trends are independent of the noisy ﬂuc- tuation in real data, and if ﬁltering these trends out will no aﬀect the scaling properties of the data. The outline of this paper is as follows. In Sec.II, we re- view the algorithm of the DFA method, and in Appendix A we compare the performance of the DFA with the clas- sical scaling analysis —Hurst’s analysis (R/S analysis)— and show that the DFA is a superior method to quan- tify the scaling behavior of noisy signals. In Sec. III, we consider the eﬀect of a linear trend and we present an analytic derivation of the apparent scaling behavior of a linear trend in Appendix C. In Sec. IV, we study a periodic trend, and in Sec. V the eﬀect of power-law trend. We systematically study all resulting crossovers, their conditions of existence and their typical character- istics associated with the diﬀerent types of trends. In addition, we also show how to use DFA appropriately to minimize or even eliminate the eﬀects of those trends in cases that trends are not choices of the study, that is, trends do not reﬂect the dynamics of the system but are caused by some “irrelevant” background. Finally, Sec. VI contains a summary. II. DFA To illustrate the DFA method, we consider a noisy time series, u(i) (i= 1, .., N max). We integrate the time series u(i), y(j) =j/summationdisplay i=1(u(i)−< u > ), (1) where< u > =1 NmaxNmax/summationdisplay j=1u(i), (2) and is divided into boxes of equal size, n. In each box, we ﬁt the integrated time series by using a polynomial func- tion,yfit(i), which is called the local trend. For order- ℓ DFA (DFA-1 if ℓ= 1, DFA-2 if ℓ= 2 etc.), ℓorder poly- nomial function should be applied for the ﬁtting. We detrend The integrated time series, y(i) by subtracting the local trend yfit(i) in each box, and we calculate the detrended ﬂuctuation function Y(i) =y(i)−yfit(i). (3) For a given box size n, we calculate the root mean square (rms) ﬂuctuation F(n) =/radicaltp/radicalvertex/radicalvertex/radicalbt1 NmaxNmax/summationdisplay i=1[Y(i)]2(4) The above computation is repeated for box sizes n(dif- ferent scales) to provide a relationship between F(n) and n. A power-law relation between F(n) and the box size nindicates the presence of scaling: F(n)∼nα. The parameter α, called the scaling exponent or correlation exponent, represents the correlation properties of the sig - nal: if α= 0.5, there is no correlation and the signal is an uncorrelated signal (white noise); if α <0.5, the signal is anti-correlated; if α >0.5, there are positive correlations in the signal. III. NOISE WITH LINEAR TRENDS First we consider the simplest case: correlated noise with a linear trend. A linear trend u(i) =ALi (5) is characterized by only one variable — the slope of the trend, AL. For convenience, we denote the rms ﬂuctu- ation function for noise without trends by Fη(n), linear trends by FL(n), and noise with a linear trend by FηL(n). A. DFA-1 on noise with a linear trend Using the algorithm of Makse [63], we generate cor- related noise with standard deviation one, with a given correlation property characterized by a given scaling ex- ponent α. We apply DFA-1 to quantify the correlation properties of the noise and ﬁnd that only in certain good ﬁt region the rms ﬂuctuation function Fη(n) can be ap- proximated by a power-law function [see Appendix A] Fη(n) =b0nα(6) 2where b0is a parameter independent of the scale n. We ﬁnd that the good ﬁt region depends on the correlation exponent α[see Appendix A]. We also derive analyti- cally the rms ﬂuctuation function for linear trend only for DFA-1 and ﬁnd that [see Appendix C] FL(n) =k0ALnαL(7) where k0is a constant independent of the length of trend Nmax, of the box size nand of the slope of the trend AL. We obtain αL= 2. AL=2−16 AL=2−12 AL=2−8Correlated noise with linear trend: FηL(n) nxDFA−1 100101102103104105 n10−610−410−2100102104106F(n)Correlated noise : Fη(n) linear trends: FL(n) 22 FIG. 1. Crossover behavior of the root mean square ﬂuc- tuation function FηL(n) for noise (of length Nmax= 217and correlation exponent α= 0.1) with superposed linear trends of slope AL= 2−16,2−12,2−8. For comparison, we show Fη(n) for the noise (thick solid line) and FL(n) for the linear trends (dot-dashed line) (Eq.(7)). The results show that a crossov er at a scale n×forFηL(n). For n < n ×, the noise dominates andFηL(n)≈Fη(n). For n > n ×, the linear trend domi- nates and FηL(n)≈FL(n). Note that the crossover scale n× increases when the slope ALof the trend decreases. Next we apply the DFA-1 method to the superposi- tion of a linear trend with correlated noise and we com- pare the rms ﬂuctuation function FηL(n) with Fη(n) [see Fig.1]. We observe a crossover in FηL(n) at scale n=n×. Forn < n ×, the behavior of FηL(n) is very close to the behavior of Fη(n), while for n > n ×, the behavior of FηL(n) is very close to the behavior of FL(n). A sim- ilar crossover behavior is also observed in the scaling of the well-studied biased random walk [61, 62]. It is known that the crossover in the biased random walk is due to the competition of the unbiased random walk and the bias [see Fig.5.3 of [62]]. We illustrate this observa- tion in Fig. 2, where the detrended ﬂuctuation functions (Eq. (3)) of the correlated noise, Yη(i), and of the noise with a linear trend, YηL(i) are shown. For the box size n < n ×as shown in Fig. 2(a) and (b), YηL(i)≈Yη(i). Forn > n ×as shown in Fig. 2(c) and (d), YηL(i) has dis-tinguishable quadratic background signiﬁcantly diﬀerent fromYη(i). This quadratic background is due to the inte- gration of the linear trend within the DFA procedure and represents the detrended ﬂuctuation function YLof the linear trend. These relations between the detrended ﬂuc- tuation functions Y(i) at diﬀerent time scales nexplain the crossover in the scaling behavior of FηL(n): from very close to Fη(n) to very close to FL(n) (observed in Fig.1). 0 150 300−606YηLCorrelated noise + linear trend (b) n < nx 0 500 1000 i−20020YηL(d) n > nx0 150 300−606YηCorrelated noise (a) n < nx 0 500 1000 i−20020Yη(c) n > nx FIG. 2. Comparison of the detrended ﬂuctuation function for noise Yη(i) and for noise with linear trend YηL(i) at diﬀer- ent scales. (a) and (c) are Yηfor noise with α= 0.1; (b) and (d) are YηLfor the same noise with a linear trend with slope AL= 2−12(the crossover scale n×= 320 see Fig. 1). (a) (b) for scales n < n ×the eﬀect of the trend is not pronounced andYη≈YηL(i.e.Yη≫YL); (c)(d) for scales n > n ×, the linear trend is dominant and Yη≪YηL. The experimental results presented in Figs.1 and 2 sug- gest that the rms ﬂuctuation function for a signal which is a superposition of a correlated noise and a linear trend can be expressed as: [FηL(n)]2= [FL(n)]2+ [Fη(n)]2(8) We provide an analytic derivation of this relation in Ap- pendix B, where we show that Eq.(8) holds for the super- position of any two independent signals — in this particu- lar case noise and a linear trend. We call this relation the “superposition rule”. This rule helps us understand how the competition between the contribution of the noise and the trend to the rms ﬂuctuation function FηL(n) at diﬀerent scales nleads to appearance of crossovers [61]. Next, we ask how the crossover scale n×depends on: (i) the slope of the linear trend AL, (ii) the scaling ex- ponent αof the noise, and (iii) the length of the signal Nmax. Surprisingly, we ﬁnd that for noise with any given correlation exponent αthe crossover scale n×itself fol- lows a power-law scaling relation over several decades: n×∼(AL)θ(see Fig. 3). We ﬁnd that in this scaling re- lation, the crossover exponent θis negative and its value 3depends on the correlation exponent αof the noise — the magnitude of θdecreases when αincreases. We present the values of the “crossover exponent” θfor diﬀerent cor- relation exponents αin Table I. 10−610−510−410−310−210−1 AL101102103nxα=0.1 α=0.3 α=0.5 α=0.7 α=0.9θDFA−1 FIG. 3. The crossover n×ofFηL(n) for noise with a lin- ear trend. We determine the crossover scale n×based on the diﬀerence ∆ between log Fη(noise) and log FηL(noise with a linear trend). The scale for which ∆ = 0 .05 is the esti- mated crossover scale n×. For any given correlation exponent αof the noise, the crossover scale n×exhibits a long-range power-law behavior n×∼(AL)θ, where the crossover expo- nentθis a function of α[see Eq.(9) and Table I]. TABLE I. The crossover exponent θfrom the power-law relation between the crossover scale n×and the slope of the linear trend AL—n×∼(AL)θ—for dif- ferent values of the correlation exponents αof the noise [Fig. 3]. The values of θobtained from our simulations are in good agreement with the analytical prediction −1/(2−α) [Eq. (9)]. Note that −1/(2−α) are not always exactly equal to θbecause Fη(n) in simulations is not a perfect simple power-law function and the way we determine numerically n×is just approximated. α θ −1/(2−α) 0.1 -0.54 -0.53 0.3 -0.58 -0.59 0.5 -0.65 -0.67 0.7 -0.74 -0.77 0.9 -0.89 -0.91 To understand how the crossover scale depends on the correlation exponent αof the noise we employ the super- position rule [Eq.(8)] and estimate n×as the intercept between Fη(n) and FL(n). From the Eqs. (6) and (7), we obtain the following dependence of n×onα:n×=/parenleftbigg ALk0 b0/parenrightbigg1/(α−αL) =/parenleftbigg ALk0 b0/parenrightbigg1/(α−2) (9) This analytical calculation for the crossover exponent −1/(αL−α) is in a good agreement with the observed values of θobtained from our simulations [see Fig.3 and Table I]. Finally, since the FL(n) does not depend on Nmaxas we show in Eq.(7) and in Appendix C, we ﬁnd that n× does not depend on Nmax. This is a special case for linear trends and does not always hold for higher order polynomial trends [see Appendix D]. B. DFA-2 on noise with a linear trend Application of the DFA-2 method to noisy signals with- out any polynomial trends leads to scaling results identi- cal to the scaling obtained from the DFA-1 method, with the exception of some vertical shift to lower values for the rms ﬂuctuation function Fη(n) [see Appendix A]. How- ever, for signals which are a superposition of correlated noise and a linear trend, in contrast to the DFA-1 results presented in Fig. 1, FηL(n) obtained from DFA exhibits no crossovers, and is exactly equal to the rms ﬂuctuation function Fη(n) obtained from DFA-2 for correlated noise without trend (see Fig. 4). These results indicate that a linear trend has no eﬀect on the scaling obtained from DFA-2. The reason for this is that by design the DFA-2 method ﬁlters out linear trends, i.e. YL(i) = 0 (Eq.( 3)) and thus FηL(n) =Fη(n) due to the superposition rule (Eq. (8)). For the same reason, polynomial trends of or- der lower than ℓsuperimposed on correlated noise will have no eﬀect on the scaling properties of the noise when DFA-ℓis applied. Therefore, our results conﬁrm that the DFA method is a reliable tool to accurately quantify cor- relations in noisy signals embedded in polynomial trends. Moreover, the reported scaling and crossover features of F(n) can be used to determine the order of polynomial trends present in the data. 4100101102103104 n10−1100101102103F(n)α = 0.1 α = 0.3 α = 0.5 α = 0.7 α = 0.9 NoiseNoise with linear trend (AL=2−12): DFA−2α optimal fitting range FIG. 4. Comparison of the rms ﬂuctuation function Fη(n) for noise with diﬀerent types of correlations (lines) and FηL(n) for the same noise with a linear trend of slope AL= 2−12 (symbols) for DFA-2. FηL(n) =Fη(n) because the inte- grated linear trend can be perfectly ﬁltered out in DFA-2, thusYL(i) = 0 from Eq.(3). We note, that to estimate accu- rately the correlation exponents one has to choose an optima l range of scales n, where F(n) is ﬁtted. For details see Ap- pendix A . IV. NOISE WITH SINUSOIDAL TREND In this section, we study the eﬀect of sinusoidal trends on the scaling properties of noisy signals. For a signal which is a superposition of correlated noise and sinu- soidal trend, we ﬁnd that based on the superposition rule (Appendix B) the DFA rms ﬂuctuation function can be expressed as [FηS(n)]2= [Fη(n)]2+ [FS(n)]2, (10) where FηS(n) is the rms ﬂuctuation function of noise with a sinusoidal trend, and FS(n) is for the sinusoidal trend. First we consider the application of DFA-1 to a sinu- soidal trend. Next we study the scaling behavior and the features of crossovers in FηS(n) for the superposition of correlated noise and sinusoidal trend employing the su- perposition rule [Eq.(10)]. At the end of this section, we discuss the results obtained from higher order DFA.A. DFA-1 on sinusoidal trend 100101102103104105 n10−2100102104106FS(n)AS=64, T=211 AS=64, T=212 AS=32, T=211 AS=32, T=212 2 n2xDFA−1 FIG. 5. Root mean square ﬂuctuation function FS(n) for sinusoidal functions of length Nmax= 217with diﬀerent am- plitude ASand period T. All curves exhibit a crossover at n2×≈T/2, with a slope αS= 2 for n < n 2×, and a ﬂat region for n > n 2×. There are some spurious singularities at n=jT 2(jis a positive integer) shown by the spikes. Given a sinusoidal trend u(i) =ASsin (2πi/T) (i= 1, ..., N max), where ASis the amplitude of the signal and Tis the period, we ﬁnd that the rms ﬂuctuation func- tionFS(n) does not depend on the length of the signal Nmax, and has the same shape for diﬀerent amplitudes and diﬀerent periods [Fig. 5]. We ﬁnd a crossover at scale corresponding to the period of the sinusoidal trend n2×≈T, (11) and does not depend on the amplitude AS. We call this crossover n2×for convenience, as we will see later. For n < n 2×, the rms ﬂuctuation FS(n) exhibits an ap- parent scaling with the same exponent as FL(n) for the linear trend [see Eq. (7)]: FS(n) =k1AS TnαS(12) where k1is a constant independent of the length Nmax, of the period Tand the amplitude ASof the sinusoidal signal, and of the box size n. As for the linear trend [Eq.(7)], we obtain αS= 2 because at small scales (box sizen) the sinusoidal function is dominated by a linear term. For n > n 2×, due to the periodic property of the sinusoidal trend, FS(n) is a constant independent of the scalen: FS(n) =1 2√ 2πAS·T. (13) The period Tand the amplitude ASalso aﬀects the ver- tical shift of FS(n) in both regions. We note that in 5Eqs.(12) and (13), FS(n) is proportional to the ampli- tudeAS, a behavior which is also observed for the linear trend [Eq. (7)]. B. DFA-1 on noise with sinusoidal trend In this section, we study how the sinusoidal trend af- fects the scaling behavior of noise with diﬀerent type of correlations. We apply the DFA-1 method to a signal which is a superposition of correlated noise with a sinu- soidal trend. We observe that there are typically three crossovers in the rms ﬂuctuation FηS(n) at characteristic scales denoted by n1×,n2×andn3×[Fig. 6]. These three crossovers divide FηS(n) into four regions, as shown in Fig. 6(a) (the third crossover cannot be seen in Fig. 6(b) because its scale n3×is greater than the length of the sig- nal). We ﬁnd that the ﬁrst and third crossovers at scales n1×andn3×respectively [see Fig. 6] result from the com- petition between the eﬀects on FηS(n) of the sinusoidal signal and the correlated noise. For n < n 1×(region I) andn > n 3×(region IV), we ﬁnd that the noise has the dominating eﬀect ( Fη(n)> FS(n)), so the behavior of FηS(n) is very close to the behavior of Fη(n) [Eq. (10)]. Forn1×< n < n 2×(region II) and n2×< n < n 3×(re- gion III) the sinusoidal trend dominates ( FS(n)> Fη(n)), thus the behavior of FηS(n) is close to FS(n) [see Fig. 6 and Fig. 7]. (a) 100102104 n10−210−1100101102103F(n)Noise + sinusoidal trend Sinusoidal trend Correlatd Noise: α=0.9 n1xn2xn3xDFA−120.9(b) 100102104 n10−1100101102103F(n)Noise + sinusoidal trend Sinusoidal trend Anti−correlated Noise: α=0.1 n1xn2xDFA−1 20.1 FIG. 6. Crossover behavior of the root mean square ﬂuctu- ation function FηS(n) (circles) for correlated noise (of length Nmax= 217) with a superposed sinusoidal function charac- terized by period T= 128 and amplitude AS= 2. The rms ﬂuctuation function Fη(n) for noise (thick line) and FS(n) for the sinusoidal trend (thin line) are shown for compariso n. (a)FηS(n) for correlated noise with α= 0.9. (b) FηS(n) for anti-correlated noise with α= 0.9. There are three crossovers inFηS(n), at scales n1×,n2×andn3×(the third crossover can not be seen in (b) because it occurs at scale larger than the length of the signal). For n < n 1×andn > n 3×, the noise dominates and FηS(n)≈Fη(n) while for n1×< n < n 3×, the sinusoidal trend dominates and FηS(n)≈FS(n). The crossovers at n1×andn3×are due to the competition between the correlated noise and the sinusoidal trend [see Fig. 7], w hile the crossover at n2×relates only to the period Tof the sinu- soidal [Eq. (11)]. 0 200 400 600−5050Yη(e) n2x<n<n3x0 10 20 30 40 50−20020Yη(c) n1x<n<n2x1 2 3 4 5 6 7−202YηAnti−correlated noise (a)n<n1x 0 200 400 600−5050YηS(f) n2x<n<n3x0 10 20 30 40 50−20020YηS(d) n1x<n<n2x1 2 3 4 5 6 7−202YηSAnti. noise + sin. trend (b)n<n1x 60 2000 4000 i−5000500YηCorrelated noise (g) n>n3x 0 2000 4000 i−5000500YηSCorrelated noise + sin. trend (h) n>n3x FIG. 7. Comparison of the detrended ﬂuctuation function for noise, Yη(i) and noise with sinusoidal trend, YηS(i) in four regions as shown in Fig. 6. The same signals as in Fig. 6 are used. Panels (a)-(f) correspond to Fig. 6(b) for anti-corre lated noise with exponent α= 0.1, and panels (g)-(h) correspond to the Fig. 6(a) for correlated noise with exponent α= 0.9. (a)-(b) For all scales n < n 1×, the eﬀect of the trend is not pronounced and YηS(i)≈Yη(i) leading to FηS(n)≈Fη(n) (Fig. 6(a)). (c)(d) For n2×> n > n 1×, the trend is domi- nant, YηS(i)≫Yη(i) and FηS(n)≈FS(n). Since n2×≈T/2 (Eq. (11)), the scale n < T/ 2 and the sinusoidal behavior can be approximated as a linear trend. This explains the quadratic background in YηS(i) (d) [see Fig. 2(c)(d)]. (e)(f) Forn2×< n < n 3×(i.e. n≫T/2), the sinusoidal trend again dominates — YηS(i) is periodic function with period T. (g)(h) for n > n 3×, the eﬀect of the noise is dominant and the scaling of FηSfollows the scaling of Fη(Fig. 6(a)). To better understand why there are diﬀerent regions in the behavior of FηS(n), we consider the detrended ﬂuc- tuation function [Eq. (3) and Appendix B] of the corre- lated noise Yη(i), and of the noise with sinusoidal trend YηS. In Fig. 7 we compare Yη(i) and YηS(i) for anti- correlated and correlated noise in the four diﬀerent re- gions. For very small scales n < n 1×, the eﬀect of the sinusoidal trend is not pronounced, YηS(i)≈Yη(i), indi- cating that in this scale region the signal can be consid- ered as noise ﬂuctuating around a constant trend which is ﬁltered out by the DFA-1 procedure [Fig. 7(a)(b)]. Note, that the behavior of YηS[Fig. 7(b)] is identical to the be- havior of YηL[Fig. 2(b)], since both a sinusoidal with a large period Tand a linear trend with small slope ALcan be well approximated by a constant trend for n < n 1×. For small scales n1×< n < n 2×(region II), we ﬁnd that there is a dominant quadratic background for YηS(i) [Fig. 7(d)]. This quadratic background is due to the integration procedure in DFA-1, and is represented by the detrended ﬂuctuation function of the sinusoidal trend YS(i). It is similar to the quadratic background observed for linear trend YηL(i) [Fig. 2(d)] — i.e. for n1×< n < n 2×the sinusoidal trend behaves as a linear trend and YS(i)≈YL(i). Thus in region II the “lin- ear trend” eﬀect of the sinusoidal is dominant, YS> Yη, which leads to FηS(n)≈FS(n). This explains also why FηS(n) for n < n 2×(Fig. 6) exhibits crossover behav- ior similar to the one of FηL(n) observed for noise with a linear trend. For n2×< n < n 3×(region III) the sinusoidal behavior is strongly pronounced [Fig. 7(f)], YS(i)≫Yη(i), and YηS(i)≈YS(i) changes periodically with period equal to the period of the sinusoidal trend T. Since YηS(i) is bounded between a minimum and amaximum value, FηS(n) cannot increase and exhibits a ﬂat region (Fig. 6). At very large scales, n > n 3×, the noise eﬀect is again dominant ( YS(i) remains bounded, while Yηgrows when increasing the scale) which leads to FηS(n)≈Fη(n), and a scaling behavior corresponding to the scaling of the correlated noise. 102103104 T101102103n1xα=0.1 α=0.3 α=0.5 α=0.7 α=0.9(a) Noise + sin. trend (AS=5.0) θΤ1 DFA−1 10−1100101102 AS101102103n1xα=0.1 α=0.3 α=0.5 α=0.7 α=0.9(b) Noise + sin. trend (T=211) θA1 DFA−1 102103104 T102103104n2x(c) Noise + sin. trend 1.0 DFA−1 7101102 T102103104n3x α=0.4 α=0.5 α=0.6 α=0.7 α=0.8 α=0.9(d) Noise + sin. trend (AS=2) θT3 DFA−1 100101 AS102103104n3xα=0.4 α=0.5 α=0.6 α=0.7 α=0.8 α=0.9(e) Noise + sin. trend (T=16) θA3 DFA−1 FIG. 8. Dependence of the three crossovers in FηS(n) for noise with a sinusoidal trend (Fig. 6) on the period T, and amplitude ASof the sinusoidal trend. (a) Power-law rela- tion between the ﬁrst crossover scale n1×and the period T for ﬁxed amplitude ASand varying correlation exponent α: n1×∼TθT1, where θT1is a positive crossover exponent [see Table II and Eq. 14]. (b) Power-law relation between the ﬁrst crossover n1×and the amplitude of the sinusoidal trend ASfor ﬁxed period Tand varying correlation exponent α: n1×∼AθA1 Swhere θA1is a negative crossover exponent [Ta- ble II and Eq. (14)]. (c) The second crossover scale n2×de- pends only on the period T:n2×∼TθT2, where θT2≈1. (d) Power-law relation between the third crossover n3×and Tfor ﬁxed amplitude ASand varying αtrend: n3×∼TθT3. (e) Power-law relation between the third crossover n3×and ASfor ﬁxed Tand varying α:n3×∼(AS)θA3. We ﬁnd that θA3=θT3[Table III and Eq. (15)]. First, we consider n1×. Surprisingly, we ﬁnd that for noise with any given correlation exponent αthe crossover scale n1×exhibits long-range power-law dependence of the period T—n1×∼TθT1, and the amplitude AS— n1×∼(AS)θA1of the sinusoidal trend [see Fig. 8(a) and (b)]. We ﬁnd that the ”crossover exponents” θT1and θA1have the same magnitude but diﬀerent sign — θT1is positive while θA1is negative. We also ﬁnd that the mag- nitude of θT1andθA1increases for the larger values ofthe correlation exponents αof the noise. We present the values of θT1andθA1for diﬀerent correlation exponent αin Table II. To understand these power-law relations between n1×andT, and between n1×andAS, and also how the crossover scale n1×depends on the correlation exponent αwe employ the superposition rule [Eq. 10] and estimate n1×analytically as the ﬁrst intercept nth 1× ofFη(n) and FS(n). From Eqs. (12) and (6), we obtain the following dependence of n1×onT,ASandα: n1×=/parenleftbiggb0 k1T AS/parenrightbigg1/(2−α) (14) From this analytical calculation we obtain the fol- lowing relation between the two crossover expo- nents θT1andθA1and the correlation exponent α: θT1=−θA1= 1/(2−α), which is in a good agree- ment with the observed values of θT1,θA1obtained from simulations [see Fig. 8(a) (b) and Table II]. Next, we consider n2×. Our analysis of the rms ﬂuc- tuation function FS(n) for the sinusoidal signal in Fig. 5 suggests that the crossover scale FS(n) does not depend on the amplitude ASof the sinusoidal. The behavior of the rms ﬂuctuation function FηS(n) for noise with super- imposed sinusoidal trend in Fig. 6(a) and (b) indicates thatn2×does not depend on the correlation exponent αof the noise, since for both correlated ( α= 0.9) and anti-correlated ( α= 0) noise ( TandASare ﬁxed), the crossover scale n2×remains unchanged. We ﬁnd that n2× depends onlyon the period Tof the sinusoidal trend and exhibits a long-range power-law behavior n2×∼TθT2 with a crossover exponent θT2≈1 (Fig. 8(c)) which is in agreement with the prediction of Eq.(11). For the third crossover scale n3×, as for n1×we ﬁnd a power-law dependence on the period T,n3×∼TθT3, and amplitude AS,n3×∼(AS)θA3,of the sinusoidal trend [see Fig. 8(d) and (e)]. However, in contrast to the n1× case, we ﬁnd that the crossover exponents θTp3andθA3 are equal and positive with decreasing values for increas- ing correlation exponents α. In Table III, we present the values of these two exponents for diﬀerent correlation ex- ponent α. To understand how the scale n3×depends on T,ASand the correlation exponent αsimultaneously, we again employ the superposition rule [Eq. (10)] and estimate n3×as the second intercept nth 3×ofFη(n) and FS(n). From Eqs. (13) and (6), we obtain the following dependence: n3×=/parenleftbigg1 2√ 2πb0AST/parenrightbigg1/α . (15) From this analytical calculation we obtain θT3=θA3= 1/αwhich is in good agreement with the values of θT3 andθA3observed from simulations [Table III]. 8TABLE II. The crossover exponents θT1andθA1 characterizing the power-law dependence of n1×on the period Tand amplitude ASobtained from simulations: n1×∼TθT1andn1×∼(AS)θA1for diﬀerent value of the correlation exponent αof noise [Fig. 8(a)(b)]. The values of θT1andθA1are in good agreement with the analytical predictions θT1=−θA1= 1/(2−α) [Eq. (14)]. α θ T1 -θA1 1/(2−α) 0.1 0.55 0.54 0.53 0.3 0.58 0.59 0.59 0.5 0.66 0.66 0.67 0.7 0.74 0.75 0.77 0.9 0.87 0.90 0.91 TABLE III. The crossover exponents θT3andθA3 for the power-law relations: n3×∼TθT3and n3×∼(AS)θA3for diﬀerent value of the correlation exponent αof noise [Fig. 8(c)(d)]. The values of θp3 andθa3obtained from simulations are in good agree- ment with the analytical predictions θT3=θA3= 1/α [Eq. (15)]. α θ T3 θA3 1/α 0.4 2.29 2.38 2.50 0.5 1.92 1.95 2.00 0.6 1.69 1.71 1.67 0.7 1.39 1.43 1.43 0.8 1.26 1.27 1.25 0.9 1.06 1.10 1.11 Finally, our simulations show that all three crossover scales n1×,n2×andn3×do not depend on the length of the signal Nmax, since Fη(n) and FS(n) do not depend onNmaxas shown in Eqs. (6), (10), (12), and (13). C. Higher order DFA on pure sinusoidal trend In the previous Sec. IVB, we discussed how sinusoidal trends aﬀect the scaling behavior of correlated noise when the DFA-1 method is applied. Since DFA-1 removes only constant trends in data, it is natural to ask how the ob- served scaling results will change when we apply DFA of order ℓdesigned to remove polynomial trends of order lower than ℓ. In this section, we ﬁrst consider the rms ﬂuctuation FSfor a sinusoidal signal and then we study the scaling and crossover properties of FηSfor correlated noise with superimposed sinusoidal signal when higher order DFA is used. We ﬁnd that the rms ﬂuctuation function FSdoes not depend on the length of the signal Nmax, and preserves a similar shape when diﬀerent order- ℓDFA method is used [Fig. 9]. In particular, FSexhibits a crossover at ascalen2×proportional to the period Tof the sinusoidal: n2×∼TθT2withθT2≈1. The crossover scale shifts to larger values for higher order ℓ[Fig. 5 and Fig. 9]. For the scale n < n 2×,FSexhibits an apparent scaling: FS∼nαSwith an eﬀective exponent αS=ℓ+ 1 . For DFA-1, we have ℓ= 1 and recover αS= 2 as shown in Eq. (12). For n > n 2×,FS(n) is a constant independent of the scale n, and of the order ℓof the DFA method in agreement with Eq. (13). Next, we consider FηS(n) when DFA- ℓwith a higher order ℓis used. We ﬁnd that for all orders ℓ,FηS(n) does not depend on the length of the signal Nmaxand exhibits three crossovers — at small, intermediate and large scales — similar behavior is reported for DFA-1 in Fig. 6. Since the crossover at small scales, n1×, and the crossover at large scale, n3×, result from the “competi- tion” between the scaling of the correlated noise and the eﬀect of the sinusoidal trend (Figs. 6 and 7), using the superposition rule [Eq. (10)] we can estimate n1×and n3×as the intercepts of Fη(n) and FS(n) for the general case of DFA- ℓ. Forn1×we ﬁnd the following dependence on the pe- riodT, amplitude AS, the correlation exponent αof the noise, and the order ℓof the DFA- ℓmethod: n1×∼(T/AS)1/(ℓ+1−α)(16) For DFA-1, we have ℓ= 1 and we recover Eq. (14). In addition, n1×is shifted to larger scales when higher order DFA-ℓis applied, due to the fact that the value of FS(n) decreases when ℓincreases ( αS=ℓ+ 1, see Fig. 9). For the third crossover observed in FηS(n) at large scale n3×we ﬁnd for all orders ℓof the DFA- ℓthe following scaling relation: n3×∼(TAS)1/α. (17) Since the scaling function Fη(n) for correlated noise shifts vertically to lower values when higher order DFA- ℓis used [see the discussion in Appendix A and Sec. VB], n3×ex- hibits a slight shift to larger scales. For the crossover n2×inFηS(n) atFηS(n) at inter- mediate scales, we ﬁnd: n2×∼T. This relation is independent of the order ℓof the DFA and is identical to the relation found for FS(n) [Eq. (11)]. n2×also exhibits a shift to larger scales when higher order DFA is used [see Fig. 9]. The reported here features of the crossovers in FηS(n) can be used to identify low-frequency sinusoidal trends in noisy data, and to recognize their eﬀects on the scaling properties of the data. This information may be useful when quantifying correlation properties in data by means of scaling analysis. 9102103104 n10−210−1100101102103FS(n)DFA−1 DFA−2 DFA−32 3 4 FIG. 9. Comparison of the results of diﬀerent order DFA on a sinusoidal trend. The sinusoidal trend is given by the function 64 sin(2 πi/211) and the length of the signal isNmax= 217. The spurious singularities (spikes) arise from the discrete data we use for the sinusoidal function. V. NOISE WITH POWER-LAW TRENDS 101103 n10−2100102104F(n)Noise+ positive power−law trend Positive power−law trend: λ= 0.4 Correlated noise: α=0.9 αλ=1.9DFA−1(a) Positive λ α=0.9 nx101102103104 n10−210−1100101102103F(n)Noise+negative power−law trend Negative power−law trend: λ= −0.7 Correlated noise: α=1.5 DFA−1(b) Negative λ αλ=0.8 α=1.5 nx FIG. 10. Crossover behavior of the rms ﬂuctuation function FηP(n) (circles) for correlated noise (of length Nmax= 217) with a superimposed power-law trend u(i) =APiλ. The rms ﬂuctuation function Fη(n) for noise (solid line) and the rms ﬂuctuation function FP(n) (dash line) are also shown for com- parison. DFA-1 method is used. (a) FηP(n) for noise with correlation exponent αλ= 0.9, and power-law trend with am- plitude AP= 1000 /(Nmax)0.4and positive power λ= 0.4; (b) FηP(n) for Brownian noise (integrated white noise, αλ= 1.5), and power-law trend with amplitude AP= 0.01/(Nmax)−0.7 and negative power λ=−0.7. Note, that although in both cases there is a “similar” crossover behavior for FηP(n), the results in (a) and (b) represent completely opposite situa- tions: while in (a) the power-law trend with positive power λdominates the scaling of FηP(n) at large scales, in (b) the power-law trend with negative power λdominates the scaling at small scales, with arrow we indicate in (b) a weak crossove r inFP(n) (dashed lines) at small scales for negative power λ. In this section we study the eﬀect of power-law trends on the scaling properties of noisy signals. We consider the case of correlated noise with superposed power-law trend u(i) =APiλ, when APis a positive constant, i= 1, ..., N max, and Nmaxis the length of the signal. We ﬁnd that when the DFA-1 method is used, the rms ﬂuctuation function FηP(n) exhibits a crossover between two scaling regions [Fig. 10]. This behavior results from the fact that at diﬀerent scales n, either the correlated noise or the power-law trend is dominant, and can be predicted by employing the superposition rule: [FηP(n)]2= [Fη(n)]2+ [FP(n)]2, (18) where Fη(n) and FP(n) are the rms ﬂuctuation function of noise and the power-law trend respectively, and FηP(n) is the rms ﬂuctuation function for the superposition of the noise and the power-law trend. Since the behavior of Fη(n) is known (Eq. (6) and Appendix A), we can un- derstand the features of FηP(n), if we know how FP(n) depends on the characteristics of the power-law trend. We note that the scaling behavior of FηP(n) displayed in Fig. 10(a) is to some extent similar to the behavior of 10the rms ﬂuctuation function FηL(n) for correlated noise with a linear trend [Fig. 1] — e.g. the noise is dominant at small scales n, while the trend is dominant at large scales. However, the behavior FP(n) is more complex than that of FL(n) for the linear trend, since the eﬀec- tive exponent αλforFP(n) can depend on the power λ of the power-law trend. In particular, for negative val- ues of λ,FP(n) can become dominated at small scales (Fig. 10(b)) while Fη(n) dominates at large scales — a situation completely opposite of noise with linear trend (Fig. 1) or with power-law trend with positive values for the power λ. Moreover, FP(n) can exhibit crossover be- havior at small scales [Fig. 10(b)] for negative λwhich is not observed for positive λ. In addition FP(n) de- pends on the order ℓof the DFA method and the length Nmaxof the signal. We discuss the scaling features of the power-law trends in the following three subsections. A. Dependence of FP(n)on the power λ First we study how the rms ﬂuctuation function FP(n) for a power-law trend u(i) =APiλdepends on the power λ. We ﬁnd that FP(n)∼APnαλ, (19) where αλis the eﬀective exponent for the power-law trend. For positive λwe observe no crossovers in FP(n) (Fig. 10(a)). However, for negative λthere is a crossover inFP(n) at small scales n(Fig. 10(b)), and we ﬁnd that this crossover becomes even more pronounced with de- creasing λor increasing the order ℓof the DFA method, and is also shifted to larger scales [Fig. 11(a)]. 100101102103104105 n10−2100102104FP(n) λ = −0.6 λ = −1.6 λ = −2.6 λ = −3.6 DFA−3(a) αλ−4 −2 0 2 4 λ01234αλDFA−1 DFA−2 DFA−3(b) 100101102103104105 n10−1210−1010−810−610−410−2FP(n)λ =1.001 λ =1.0001 λ =1.00001 λ =1.000001 αλ∼2.5 DFA−2(c) FIG. 11. Scaling behavior of rms ﬂuctuation function FP(n) for power-law trends, u(i)∼iλ, where i= 1, ..., N max andNmax= 217is the length of the signal. (a) For λ <0, FP(n) exhibits crossover at small scales which is more pro- nounced with increasing the order ℓof DFA- ℓand decreasing the value of λ. Such crossover is not observed for λ >0 when FP(n)∼nαλfor all scales n[see Fig. 10(a)]. (b) Dependence of the eﬀective exponent αλon the power λfor diﬀerent order ℓ= 1,2,3 of the DFA method. Three regions are observed depending on the order ℓof the DFA: region I ( λ > ℓ−0.5), where αλ≈ℓ+ 1; region II ( −1.5< λ < ℓ −0.5), where αλ=λ+ 1.5; region III ( λ <−1.5), where αλ≈0. We note that for integer values of the power λ= 0,1, ..., ℓ−1, where ℓ is the order of DFA we used, there is no scaling for FP(n) and αλis not deﬁned, as indicated by the arrows. (c) Asymptotic behavior near integer values of λ.FP(n) is plotted for λ→1 when DFA-2 is used. Even for λ−1 = 10−6, we observe at large scales na region with an eﬀective exponent αλ≈2.5, This region is shifted to inﬁnitely large scales when λ= 1. Next, we study how the eﬀective exponent αλforFP(n) depends on the value of the power λfor the power-law trend. We examine the scaling of FP(n) and estimate 11αλfor−4< λ < 4. In the cases when FP(n) exhibits a crossover, in order to obtain αλwe ﬁt the range of larger scales to the right of the crossover. We ﬁnd that for any order ℓof the DFA- ℓmethod there are three regions with diﬀerent relations between αλandλ[Fig. 11(b)]: (i)αλ≈ℓ+ 1 for λ > ℓ−0.5 (region I); (ii)αλ≈λ+ 1.5 for−1.5≤λ≤ℓ−0.5 (region II); (iii)αλ≈0 forλ <−1.5 (region III). Note, that for integer values of the power λ(λ= 0,1, ..., m −1), i.e. polynomial trends of order m−1, the DFA- ℓmethod of order ℓ > m −1 (ℓis also an in- teger) leads to FP(n)≈0, since DFA- ℓis designed to remove polynomial trends. Thus for a integer values of the power λthere is no scaling and the eﬀective exponent αλis not deﬁned if a DFA- ℓmethod of order ℓ > λ is used [Fig. 11]. However, it is of interest to examine the asymp- totic behavior of the scaling of FP(n) when the value of the power λis close to an integer. In particular , we consider how the scaling of FP(n) obtained from DFA-2 method changes when λ→1 [Fig. 11(c)]. Surprisingly, we ﬁnd that even though the values of FP(n) are very small at large scales, there is a scaling for FP(n) with a smooth convergence of the eﬀective exponent αλ→2.5 when λ→1, according to the dependence αλ≈λ+ 1.5 established for region II [Fig. 11(b)]. At smaller scales there is a ﬂat region which is due to the fact that the ﬂuctuation function Y(i) (Eq. (3)) is smaller than the precision of the numerical simulation. B. Dependence of FP(n)on the order ℓof DFA Another factor that aﬀects the rms ﬂuctuation func- tion of the power-law trend FP(n), is the order ℓof the DFA method used. We ﬁrst take into account that: (1) for integer values of the power λ, the power-law trend u(i) =APiλis a polynomial trend which can be perfectly ﬁltered out by the DFA method of order ℓ > λ , and as discussed in Sec. III B and Sec. VA [see Fig. 11(b) and (c)], there is no scaling forFP(n). Therefore, in this section we consider only non-integer values of λ. (2) for a given value of the power λ, the eﬀective ex- ponent αλcan take diﬀerent values depending on the order ℓof the DFA method we use [see Fig. 11] — e.g. for ﬁxed λ > ℓ −0.5,αλ≈ℓ+ 1. There- fore, in this section, we consider only the case when λ < ℓ−0.5 (Region II and III).101102103104 n10−1100101102103FηP(n)DFA−1 DFA−2 DFA−3 nxα=0.1αλ=1.9(a) Noise with power−law trend 1 10 Order l of the DFA method10−1810−1410−1010−610−2102∆λ=−0.6 λ=−0.2 λ=0.2 λ=0.6 λ=1.2 λ=1.6τ(λ)(b) Dependence of vertical shift ∆ on l 2 3 4 5 6 7 8 9 −1 0 1 2 λ−7−5−3−1τ(λ)(c) 120 1 2 3 α−7−5−3−11τ Power law trend: τ vs. αλ Correlated noise: τ vs. α(d) FIG. 12. Eﬀect of higher order DFA- ℓon the rms ﬂuctua- tion function FηP(n) for correlated noise with superimposed power-law trend. (a) FηP(n) for anti-correlated noise with correlation exponent α= 0.1 and a power-law u(i) =APiλ, where AP= 25/(Nmax)0.4,Nmax= 217andλ= 0.4. Results for diﬀerent order ℓ= 1,2,3 of the DFA method show (i) a clear crossover from a region at small scales where the noise dominates FηP(n)≈Fη(n), to a region at larger scales where the power-law trend dominates FηP(n)≈FP(n), and (ii) a vertical shift ∆ in FηPwith increasing ℓ. (b) Dependence of the vertical shift ∆ in the rms ﬂuctuation function FP(n) for power-law trend on the order ℓof DFA- ℓfor diﬀerent val- ues of λ: ∆∼ℓτ(λ). We deﬁne the vertical shift ∆ as the y-intercept of FP(n): ∆≡FP(n= 1). Note, that we consider only non-integer values for λand that we consider the region λ < ℓ−0.5. Thus, for all values of λthe minimal order ℓ that can be used in the DFA method is ℓ > λ + 0.5. e.g. for λ= 1.6 the minimal order of the DFA that can be used is ℓ= 3 (for details see Fig. 11(b)). (c) Dependence of τon the power λ(error bars indicate the regression error for the ﬁts of ∆(l) in (b)). (d) Comparison of τ(αλ) forFP(n) and τ(α) for Fη(n). Faster decay of τ(αλ) indicates larger vertical shifts forFP(n) compared to Fη(n) with increasing order ℓof the DFA-ℓ. Since higher order DFA- ℓprovides a better ﬁt for the data, the ﬂuctuation function Y(i) (Eq. (3)) decreases with increasing order ℓ. This leads to a vertical shift to smaller values of the rms ﬂuctuation function F(n) (Eq. (4)). Such a vertical shift is observed for the rms ﬂuctuation function Fη(n) for correlated noise (see Ap- pendix A), as well as for the rms ﬂuctuation function of power-law trend FP(n). Here we ask how this vertical shift in Fη(n) and FP(n) depends on the order ℓof the DFA method, and if this shift has diﬀerent properties forFη(n) compared to FP(n). This information can help identify power-law trends in noisy data, and can be used to diﬀerentiate crossovers separating scaling regions wit h diﬀerent types of correlations, and crossovers which are due to eﬀects of power-law trends.We consider correlated noise with a superposed power- law trend, where the crossover in FηP(n) at large scales n results from the dominant eﬀect of the power-law trend —FηP(n)≈FP(n) (Eq. (18) and Fig. 10(a)). We choose the power λ <0.5, a range where for all orders ℓof the DFA method the eﬀective exponent αλofFP(n) remains the same — i.e. αλ=λ+1.5 (region II in Fig. 11(b)). For a superposition of an anti-correlated noise and power-law trend with λ= 0.4, we observe a crossover in the scaling behavior of FηP(n), from a scaling region characterized by the correlation exponent α= 0.1 of the noise, where FηP(n)≈Fη(n), to a region characterized by an eﬀective exponent αλ= 1.9, where FηP(n)≈FP(n), for all orders ℓ= 1,2,3 of the DFA- ℓmethod [Fig. 12(a)]. We also ﬁnd that the crossover of FηP(n) shifts to larger scales when the order ℓof DFA- ℓincreases, and that there is a vertical shift of FηP(n) to lower values. This vertical shift in FηP(n) at large scales, where FηP(n) =FP(n), appears to be diﬀerent in magnitude when diﬀerent or- derℓof the DFA- ℓmethod is used [Fig. 12(a)]. We also observe a less pronounced vertical shift at small scales where FηP(n)≈Fη(n). Next, we ask how these vertical shifts depend on the order ℓof DFA- ℓ. We deﬁne the vertical shift ∆ as the y-intercept of FP(n): ∆≡FP(n= 1). We ﬁnd that the vertical shift ∆ in FP(n) for power-law trend follows a power law: ∆ ∼ℓτ(λ). We tested this relation for orders up to ℓ= 10, and we ﬁnd that it holds for diﬀerent val- ues of the power λof the power-law trend [Fig. 12(b)]. Using Eq. (19) we can write: FP(n)/FP(n= 1) = nαλ, i.e.FP(n)∼FP(n= 1). Since FP(n= 1)≡∆∼ℓτ(λ) [Fig. 12(b)], we ﬁnd that: FP(n)∼ℓτ(λ). (20) We also ﬁnd that the exponent τis negative and is a decreasing function of the power λ[Fig. 12(c)]. Because the eﬀective exponent αλwhich characterizes FP(n) de- pends on the power λ[see Fig. 11(b)], we can express the exponent τas a function of αλas we show in Fig. 12(d). This representation can help us compare the behavior of the vertical shift ∆ in FP(n) with the shift in Fη(n). For correlated noise with diﬀerent correlation exponent α, we observe a similar power-law relation between the vertical shift in Fη(n) and the order ℓof DFA- ℓ: ∆∼ℓτ(α), where τis also a negative exponent which decreases with α. In Fig. 12(d) we compare τ(αλ) for FP(n) with τ(α) for Fη(n), and ﬁnd that for any αλ=α,τ(αλ)< τ(α). This diﬀerence between the vertical shift for correlated noise and for a power-law trend can be utilized to recognize eﬀects of power-law trends on the scaling properties of data. C. Dependence of FP(n)on the signal length Nmax Here, we study how the rms ﬂuctuation function FP(n) depends on the length Nmaxof the power-law signal 13u(i) =APiλ(i= 1, ..., N max). We ﬁnd that there is a vertical shift in FP(n) with increasing Nmax[Fig. 13(a)]. We observe that when doubling the length Nmaxof the signal the vertical shift in FP(n), which we deﬁne as F2Nmax P /FNmax P, remains the same, independent of the value of Nmax. This suggests a power-law dependence of FP(n) on the length of the signal: FP(n)∼(Nmax)γ, (21) where γis an eﬀective scaling exponent. Next, we ask if the vertical shift depends on the power λof the power-law trend. When doubling the length Nmaxof the signal, we ﬁnd that for λ < ℓ−0.5, where ℓ is the order of the DFA method, the vertical shift is a con- stant independent of λ[Fig. 13(b)]. Since the value of the vertical shift when doubling the length Nmaxis 2γ(from Eq. (21)), the results in Fig. 13(b) show that γis inde- pendent of λwhen λ < ℓ−0.5, and that −log2γ≈ −0.15, i.e. the eﬀective exponent γ≈ −0.5. Forλ > ℓ −0.5, when doubling the length Nmaxof the signal, we ﬁnd that the vertical shift 2γexhibits the following dependence on λ:−log102γ= log102λ−ℓ, and thus the eﬀective exponent γdepends on λ—γ=λ−ℓ. For positive integer values of λ(λ=ℓ), we ﬁnd that γ= 0, and there is no shift in FP(n), suggesting that FP(n) does not depend on the length Nmaxof the signal, when DFA of order ℓis used [Fig. 13]. Finally, we note that depending on the eﬀective exponent γ, i.e. on the order ℓof the DFA method and the value of the power λ, the vertical shift in the rms ﬂuctuation function FP(n) for power-law trend can be positive ( λ > ℓ ), negative (λ < ℓ), or zero ( λ=ℓ). 101102103104105 n10−610−410−2100102104FP(n)Nmax=217 Nmax=219 Nmax=221 DFA−1(a) Power−law trend: λ=0.4 αλ=1.9−2 0 2 4 λ−0.35−0.25−0.15−0.050.050.150.25−log10 [F2Nmax/FNmax] DFA−1 DFA−2 DFA−3 −log102(b) Vertical shift due to length doubling FIG. 13. Dependence of the rms ﬂuctuation function FP(n) for power-law trend u(i) =APiλ, where i= 1, ..., N max, on the length of the trend Nmax. (a) A vertical shift is ob- served in FP(n) for diﬀerent values of Nmax—N1maxand N2max. The ﬁgure shows that the vertical shift , deﬁned as FN1max P (n)/FN2max P (n), does not depend on Nmaxbut only on the ratio N1max/N2max, suggesting that FP(n)∼(Nmax)γ. (b) Dependence of the vertical shift on the power λ. For λ < ℓ−0.5 (ℓis the order of DFA), we ﬁnd a ﬂat (constant) region characterized with eﬀective exponent γ=−0.5 and negative vertical shift. For λ > ℓ−0.5, we ﬁnd an exponential dependence of the vertical shift on λ. In this region, γ=λ−ℓ, and the vertical shift can be negative (if λ < ℓ) or positive (if λ > ℓ). the slope of −log10/parenleftbig F2Nmax P (n)/FNmax P(n)/parenrightbig vs.λis −log102 due to doubling the length of the signal Nmax. This slope changes to −log10mwhen Nmaxis increased mtimes while γremains independent of Nmax. For λ=ℓthere is no vertical shift, as marked with ×. Arrows indicate integer values of λ < ℓ, for which values the DFA- ℓmethod ﬁlters out completely the power-law trend and FP= 0. D. Combined eﬀect on FP(n)ofλ,ℓandNmax We have seen that, taking into account the eﬀects of the power λ(Eq. (19)), the order ℓof DFA- ℓ(Eq. (20)) and the eﬀect of the length of the signal Nmax(Eq. (21)), we reach the following expression for the rms ﬂuctuation function FP(n) for a power-law trend u(i) =APiλ: FP(n)∼AP·nαλ·ℓτ(λ)·(Nmax)γ(λ), (22) For correlated noise, the rms ﬂuctuation function Fη(n) depends on the box size n(Eq. (6)) and on the order ℓ of DFA- ℓ(Sec. VB and Fig. 12(a), (d)), and does not depend on the length of the signal Nmax. Thus we have the following expression for Fη(n) Fη(n)∼nαℓτ(α), (23) To estimate the crossover scale n×observed in the apparent scaling of FηP(n) for a correlated noise su- perposed with a power-law trend [Fig. 10(a), (b) and 14Fig. 12(a)], we employ the superposition rule (Eq. (18)). From Eq. (22) and Eq. (23), we obtain n×as the inter- cept between FP(n) and Fη(n): n×∼/bracketleftBig Alτ(λ)−τ(α)(Nmax)γ/bracketrightBig1/(α−αλ) . (24) To test the validity of this result, we consider the case of correlated noise with a linear trend. For the case of a lin- ear trend ( λ= 1) when DFA-1 ( ℓ= 1) is applied, we have αλ= 2 (see Appendix C and Sec. VA, Fig. 11(b)). Since in this case λ=ℓ= 1> ℓ−0.5 we have γ=λ−ℓ= 0 (see Sec.VC Fig. 13(b)), and from Eq. (24) we recover Eq. (9). VI. CONCLUSION AND SUMMARY In this paper we show that the DFA method performs better than the standard R/S analysis to quantify the scaling behavior of noisy signals for a wide range of cor- relations, and we estimate the range of scales where the performance of the DFA method is optimal. We consider diﬀerent types of trends superposed on correlated noise, and study how these trends aﬀect the scaling behavior of the noise. We demonstrate that there is a competition be- tween a trend and a noise, and that this competition can lead to crossovers in the scaling. We investigate the fea- tures of these crossovers, their dependence on the proper- ties of the noise and the superposed trend. Surprisingly, we ﬁnd that crossovers which are a result of trends can exhibit power-law dependences on the parameters of the trends. We show that these crossover phenomena can be explained by the superposition of the separate results of the DFA method on the noise and on the trend, assum- ing that the noise and the trend are not correlated, and that the scaling properties of the noise and the appar- ent scaling behavior of the trend are known. Our work may provide some help to diﬀerentiate between diﬀer- ent types of crossovers — e.g. crossovers which separate scaling regions with diﬀerent correlation properties may diﬀer from crossovers which are an artifact of trends. The results we present here could be useful for identifying the presence of trends and to accurately interpret correlation properties of noisy data. ACKNOWLEDGMENTS We thank NIH/National Center for Research Re- sources (P41RR13622), NSF and the Spanish Govern- ment (BIO99-0651-CO2-01)for support, and C.-K. Peng, Y. Ashkenazy for helpful discussions. When concluding our work, we became aware of an independent study by J.W. Kantelhardt et. al [66], where similar issues are dis- cussed. We thank J.W. Kantelhardt and A. Bunde for sending us their preprint before publication.APPENDIX A: NOISE The standard signals we generate in our study are un- correlated, correlated, and anti-correlated noise. First we must have a clear idea of the scaling behaviors of these standard signals before we use them to study the eﬀects from other aspects. We generate noises by using a modiﬁed Fourier ﬁltering method [63]. This method can eﬃciently generate noise, u(i) (i= 1,2,3, ..., N max), with the desired power-law correlation function which asymp- totically behaves as: <\|i+t/summationtext j=iu(j)\|2>∼t2α. By default, a generated noise has standard deviation σ= 1. Then we can test DFA and R/S by applying it on generated noises since we know the expected scaling exponent α. 100101102103104105 n100101102103104R/Sα=0.1 α=0.3 α=0.5 α=0.7 α=0.9 1 2 3α2(a) R/S analysis 100101102103104 n10−1100101102103F(n)α=0.1 α=0.3 α=0.5 α=0.7 α=0.9 12 3(b) DFA−1 α2 15100101102103104 n10−1100101102103F(n)α = 0.1 α = 0.3 α = 0.5 α = 0.7 α = 0.9(c) DFA−2 FIG. 14. Scaling behavior of noise with the scaling ex- ponent α. The length of noise Nmax= 217. (a) Rescaled range analysis (R/S) (b) Order 1 detrended ﬂuctuation anal- ysis (DFA-1) (c) Order 2 detrended ﬂuctuation analysis. We do the linear ﬁtting for R/S analysis and DFA-1 in three re- gions as shown and get α1,α2andα3for estimated α, which are listed in the Table.IV and Table.V. We ﬁnd that the estimation of αis diﬀerent in the diﬀerent region. Before doing that, we want to brieﬂy review the algo- rithm of R/S analysis. For a signal u(i)(i= 1, ..., N max), it is divided into boxes of equal size n. In each box, thecumulative departure ,Xi(fork-th box, i=kn+ 1, ..., kn +n), is calculated Xi=i/summationdisplay j=kn+1(u(j)−< u > ) (A1) where < u > =n−1(k+1)n/summationtext i=kn+1u(i) , and the rescaled range R/Sis deﬁned by R/S=S−1/bracketleftbigg max kn+1≤i≤(k+1)nXi− min kn+1≤i≤(k+1)nXi/bracketrightbigg , (A2) where S=/radicalBigg n−1n/summationtext j=1(u(j)−< u > )2is the standard de- viation in each box. The average of rescaled range in all the boxes of equal size n, is obtained and denoted by < R/S > . Repeat the above computation over diﬀerent box size nto provide a relationship between < R/S > andn. According to Hurst’s experimental study [64], a power-law relation between < R/S > and the box size n indicates the presence of scaling: < R/S > ∼nα. Figure 14 shows the results of R/S, DFA-1 and DFA- 2 on the same generated noises. Loosely speaking, we can see that F(n) (for DFA) and R/S(for R/S analysis) show power-law relation with nas expected: F(n)∼nαandR/S∼nα. In addition, there is no signiﬁcant dif- ference between the results of diﬀerent order DFA except for some vertical shift of the curves and the little bend- down for small box size n. The bent-down for very small box of F(n) from higher order DFA is because there are more variables to ﬁt those few points. TABLE IV. Estimated αof correlation noise from R/S analysis in three regions as shown in Fig.14(a). α is the input value of the scaling exponent, α1is the estimated in the region 1 for 4 < n≤32,α2in the region 2 for 32 < n≤3162 and α3in the region 3 for 3126< n≤217. Noise are the same as used in Table.V. α α 1 α2 α3 0.1 0.44 0.23 0.12 0.3 0.52 0.37 0.23 0.5 0.62 0.52 0.47 0.7 0.72 0.70 0.45 0.9 0.81 0.87 0.63 TABLE V. Estimated αof correlation noise from DFA-1 in three regions as shown in Fig.14(b). αis the input value of the scaling exponent, α1 is the es- timated in the region 1 for 4 < n≤32,α2 in the region 2 for 32 < n≤3162 and α3 in the region 3 for 3126< n≤217. α α 1 α2 α3 0.1 0.28 0.15 0.08 0.3 0.40 0.31 0.22 0.5 0.55 0.50 0.35 0.7 0.72 0.69 0.55 0.9 0.91 0.91 0.69 Ideally, when analyzing a standard noise, F(n) (DFA) andR/S(R/Sanalysis) will be a power-law function with a given power: α, no matter which region of F(n) andR/Sis chosen for calculation. However, a careful study shows that the scaling exponent αdepends on scale n. The estimated αis diﬀerent for the diﬀerent regions ofF(n) and R/Sas illustrated by Figs. 14(a) and 14(b) and by Tables IV and V. It is very important to know the best ﬁtting region of DFA and R/S analysis in the study of real signals. Otherwise, the wrong αwill be obtained if an inappropriate region is selected. In order to ﬁnd the best region, we ﬁrst determine the dependence of the locally estimated α,αloc, on the scale n. First, generate a standard noise with given scaling exponent α; then calculate F(n) (or R/S), and obtain αloc(n) by local ﬁtting of F(n) (orR/S). Same random simulation is repeated 50 times for both DFA and R/S analysis. The resultant average αloc(n), respectively, are illustrated in Fig.15 for DFA-1 and R/S analysis. If a scaling analysis method is working properly, then the result αloc(n) from simulation with αwould be a 16horizontal line with slight ﬂuctuation centered about αloc(n) =α. Note from Fig.15 that such a horizontal behavior does not hold for all the scales nbut for a cer- tain range from nmintonmax. In addition, at small scale, R/S analysis gives αloc> αifα <0.7 and αloc< αif α >0.7, which has been pointed out by Mandelbrot [65] while DFA gives αloc> αifα <1.0 and αloc< αif α >1.0. It is clear that the smaller the nminand the larger the nmax, the better the method. We also perceive that the expected horizontal behavior stops because the ﬂuctua- tions become larger due to the under-sampling of F(n) orR/Swhen ngets closer to the length of the signal Nmax. Furthermore, it can be seen from Fig.15 that nmax≈1 10Nmaxindependent of α(if the best ﬁt region exists), which is why one tenth of the signal length is the maximum box size when using DFA or R/S analysis. 100101102103104 n0.10.30.50.70.91.1αlocα=0.1 α=0.3 α=0.5 α=0.7 α=0.9 integrated α=0.1 nmin(a) R/S Nmax=214 correlated anti uncorrelated 100102104 n0.10.30.50.70.91.1αlocα=0.1 α=0.3 α=0.5 α=0.7 α=0.9 integrated α=0.1 nmin(b) R/S Nmax=220 correlated anti uncorrelated100101102103104 n0.10.30.50.70.91.1αloc α=0.1 α=0.3 α=0.5 α=0.7 α=0.9 integrated α=0.1(c) DFA−1 Nmax=214 nmin correlated uncorrelatedanti 10−1101103105 n0.10.30.50.70.91.1αloc α=0.1 α=0.3 α=0.5 α=0.7 α=0.9 integrated α=0.1(d) DFA−1 Nmax=220 nmin correlated anti uncorrelated FIG. 15. The estimated αfrom local ﬁt (a) R/S anal- ysis, the length of signal Nmax= 214. (b)R/S analysis, Nmax= 220. (c) DFA-1, Nmax= 214(d) DFA-1, Nmax= 220. αloccome from the average of 50 simulations. If a technique is working, then the data for scaling exponent αshould be a weakly ﬂuctuating horizontal line centered about αloc=α. Note that such a horizontal behavior does not hold for all the scales. Generally, such a expected behavior begins from som e scalenmin, holds for a range and ends at a larger scale nmax. For DFA-1, nminis quite small α >0.5. For R/S analysis, nminis small only when α≈0.7. 170 0.5 1 1.5 α100102104106nminR/S DFA−1 minimum box size FIG. 16. The starting point of good ﬁt region, nmin, for DFA-1 and R/S analysis. The results are obtained from 50 simulations, in which the length of noise is Nmax= 220. The condition for a good ﬁt is ∆ α=\|αloc−α\|<0.01. The data forα >1.0 shown in the shading area are obtained by apply- ing analysis on the integrations of noises with α <1.0. It is clear that DFA-1 works better than R/S analysis because its nminis always smaller than that of R/S analysis. On the contrary, nmindoes not depend on the Nmax since αloc(n) at small nhardly changes as Nmaxvaries but it does depend on α. Thus, we obtain nminquan- titatively as shown in Fig.16. For R/S analysis, only forα≈0.7,nminis small; for αa little away from 0 .7 (for example, 0.5), nminbecomes very large and close to nmax, indicating that the best ﬁt region will vanish and R/S analysis does not work at all. Comparing to R/S, DFA works better since nminis quite small for α >0.5 correlated signals. One problem remains for DFA, nminfor small α(≤0.5) is still too large comparing to those for large α(>0.5). We can improve it by applying DFA on the integration of the noise with α <0.5. The resultant new expected α′for the integrated signal would be α′ 0=α+ 1, while thenminfor the integrated signal becomes much smaller as shown also in Fig.16(shading area α >1). Therefore, for a noise with α <0.5, it is best to estimate the scaling exponent α′of the integrated signal ﬁrst and then obtain αbyα=α′−1. This is what we did in the following sections to those anti-correlated signals. APPENDIX B: SUPERPOSITION LAW FOR DFA For two uncorrelated signals f(i) and g(i), their root mean square ﬂuctuation functions are Ff(n) and Fg(n)respectively. We want to prove that for the signal f(i) +g(i), its rms ﬂuctuation Ff+g(n) =/radicalBig Ff(n)2+Fg(n)2 (B1) Consider three signals in the same box ﬁrst. The in- tegrated signals for f,gandf+gareyf(i),yg(i) and yf+g(i) and their corresponding trends are yfit f,yfit g,yfit f+g (i= 1,2, ..., n,nis the box size). Since yf+g(i) = yf(i)+yg(i) and combine the deﬁnition of detrended ﬂuc- tuation function Eq.3, we have that for all boxes Yf+g(i) =Yf(i) +Yg(i), (B2) where Yf+gis the detrended ﬂuctuation function for the signal f+g,Yf(i) is for the signal fandYg(i) forg. Fur- thermore, according to the deﬁnition of rms ﬂuctuation, we can obtain Ff+g(n)=/radicaltp/radicalvertex/radicalvertex/radicalbt1 NmaxNmax/summationdisplay i=1[Yf+g(i)]2(B3) =/radicaltp/radicalvertex/radicalvertex/radicalbt1 NmaxNmax/summationdisplay i=1[Yf(i) +Yg(i)]2, where ℓis the number of boxes and kmeans the kthbox. Iffandgare not correlated, neither are Yf(i) and Yg(i) and, thus, Nmax/summationdisplay i=1Yf(i)Yg(i) = 0. (B4) From Eq.B4 and Eq.B4, we have Ff+g(n)=/radicaltp/radicalvertex/radicalvertex/radicalbt1 NmaxNmax/summationdisplay i=1[Yf(i)2+Yg(i)2] =/radicalBig [Ff(n)]2+ [Fg(n)]2. (B5) APPENDIX C: DFA-1 ON LINEAR TREND Let us suppose a linear time series u(i) =ALi. The integrated signal yL(i) is yL(i) =i/summationdisplay j=1ALj=ALi2+i 2(C1) Let as call Nmaxthe size of the series and nthe size of the box. The rms ﬂuctuation FL(n) as a function of n andNmaxis 18FL(n) =AL/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1 NmaxNmax/n/summationdisplay k=1kn/summationdisplay i=(k−1)n+1/parenleftbiggi2+i 2−(ak+bki)/parenrightbigg2 (C2) where akandbkare the parameters of a least-squares ﬁt of the k-th box of size n.akandbkcan be determined analytically, thus giving: ak= 1−1 12n2+1 2n2k+1 12n−1 2k2n2(C3) bk= 1−1 2n+kn+1 2(C4) With these values, FL(n) can be evaluated analytically: FL(n) =AL1 60/radicalbig (5n4+ 25n3+ 25n2−25n−30) (C5) The dominating term inside the square root is 5 n4and then one obtains FL(n)≈√ 5 60ALn2(C6) leading directly to an exponent of 2 in the DFA. An im- portant consequence is that, as F(n) does not depend onNmax, for linear trends with the same slope, the DFA must give exactly the same results for series of diﬀerent sizes. This is not true for other trends, where the expo- nent is 2, but the factor multiplying n2can depend on Nmax. APPENDIX D: DFA-1 ON QUADRATIC TREND Let us suppose now a series of the type u(i) =AQi2. The integrated time series y(i) is y(i) =AQi/summationdisplay j=1j2=AQ2i3+ 3i2+i 6(D1) As before, let us call Nmaxandnthe sizes of the se- ries and box, respectively. The rms ﬂuctuation function FQ(n) measuring the rms ﬂuctuation is now deﬁned as FQ(n) =AQ/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1 NmaxNmax/n/summationdisplay k=1kn/summationdisplay i=(k−1)n+1/parenleftbigg2i3+ 3i2+i 6−(ak+bki)/parenrightbigg2 (D2) where akandbkare the parameters of a least-squares ﬁt of the k-th box of size n. As before, akandbkcan be determined analytically, thus giving: ak=1 15n3+n3k2−7 15n3k+17 30n2k−7 60n2+1 20n−2 3k3n3−1 2n2k2+1 15kn (D3) bk=3 10n2+n2k2−n2k+kn−2 5n+1 10(D4) Once akandbkare known, F(n) can be evaluated, giving: FQ(n) =AQ1 1260/radicalbig −21 (n4+ 5n3+ 5n2−5n−6)(32n2−6n−81−210Nmax−140N2max) (D5) AsNmax> n, the dominant term inside the square root is given by 140 N2 max×21n4=AQ2940n4N2 max, and then one has approximately FQ(n)≈AQ1 1260/radicalbig 2940n4N2max=AQ1 90√ 15Nmaxn2 (D6) leading directly to an exponent 2 in the DFA analysis. An interesting consequence derived from Eq. (D6) is that, FQ(n) depends on the length of signal Nmax, and the DFA line (log FQ(n) versus log n) for quadratic seriesu(i) =AQi2of diﬀerent NmaxDO NOT overlap (as it happened for linear trends). [1] C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49, 1685 (1994). [2] S. V. Buldyrev, A. L. Goldberger, S. Havlin, C.-K. Peng, H.E. Stanley, and M. 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Bunde, Physica A (to be published) (2001). 21101102103104105 n0.00.10.20.30.40.50.60.70.80.91.0α(b) α0=0.9 α0=0.8 α0=0.7 α0=0.6 α0=0.5 α0=0.4 α0=0.3 α0=0.2 α0=0.1 correlated anti−corrrelated uncorrelated0 1 2 3 4 5 6 7 log10(n)00.10.20.30.40.50.60.70.80.91αout α=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9(c)100101102103104105106 n0.00.10.20.30.40.50.60.70.80.91.0αα0=0.9 α0=0.8 α0=0.7 α0=0.6 α0=0.5 α0=0.4 α0=0.3 α0=0.2 α0=0.1(d)101102103104105106107 n0.00.10.20.30.40.50.60.70.80.91.0α(d) α0=0.9 α0=0.8 α0=0.7 α0=0.6 α0=0.5 α0=0.4 α0=0.3 α0=0.2 α0=0.1 correlated anti−corrrelated uncorrelated
arXiv:physics/0103019v1 [physics.gen-ph] 8 Mar 2001Quantum Mechanical Description of Fluid Dynamics H. Y. Cui∗ Department of Applied Physics Beijing University of Aeronautics and Astronautics Beijing, 100083, China (December 31, 2012) In this paper, we deal with ﬂuid motion in terms of quan- tum mechanics. Mechanism accounting for the appearance of quantum behavior is discussed. Consider a ideal ﬂuid which is composed of discrete identical particles, its mass and charge are mandqre- spectively, it is convenient to consider the ﬂuid to be a ﬂow characterized by a 4-velocity ﬁeld u(x1,x2,x3,x4= ict) in a Cartesian coordinate system (in a laboratory frame of reference). The particle will be aﬀected by the 4-force due to in electromagnetic interaction. According to relativistic Newton’s second law, the motion of the particle satisﬁes the following governing equations mduµ dτ=qFµνuν (1) uµuµ=−c2(2) whereFµνis the 4-curl of electromagnetic vector poten- tialA. Since the reference frame is a Cartesian coordi- nate system whose axes are orthogonal to one another, there is no distinction between covariant and contravari- ant components, only subscripts need be used. Here and below, summation over twice repeated indices is implied in all case, Greek indices will take on the values 1,2,3,4, and regarding the mass mas a constant. Eq.(1) and (2) stand at every point for every particle. As is mentioned above, the 4-velocity ucan be regarded as a 4-velocity vector ﬁeld, then duµ dτ=∂uµ ∂xν∂xν ∂τ=uν∂νuµ (3) qFµνuν=quν(∂µAν−∂νAµ) (4) Substituting them back into Eq.(1), and re-arranging their terms, we obtain uν∂ν(muµ+qAµ) =uν∂µ(qAν) =uν∂µ(muν+qAν)−uν∂µ(muν) =uν∂µ(muν+qAν)−1 2∂µ(muνuν) =uν∂µ(muν+qAν)−1 2∂µ(−mc2) =uν∂µ(muν+qAν) (5) Using the notation Kµν=∂µ(muν+qAν)−∂ν(muµ+qAµ) (6)Eq.(5) is given by uνKµν= 0 (7) BecauseKµνcontains the variables ∂µuν,∂µAν,∂νuµ and∂νAµwhich are independent from uν, then a solution satisfying Eq.(7) is of Kµν= 0 (8) ∂µ(muν+qAν) =∂ν(muµ+qAµ) (9) The above equation allows us introduce a potential func- tion Φ in mathematics, further set Φ = −i¯hlnψ, we ob- tain a very important equation (muµ+qAµ)ψ=−i¯h∂µψ (10) We think it as an extended form of the relativistic New- ton’s second law in terms of 4-velocity ﬁeld. ψrepre- senting the wave nature may be a complex mathematical function, its physical meanings will be determined from experiments after the introduction of the Planck’s con- stant ¯h. Multiplying the two sides of the following familiar equation by ψ −m2c4=m2uµuµ (11) which stands at every points in the 4-velocity ﬁeld, and using Eq.(10), we obtain −m2c4ψ=muµ(−i¯h∂µ−qAµ)ψ = (−i¯h∂µ−qAµ)(muµψ)−[−i¯hψ∂ µ(muµ)] = (−i¯h∂µ−qAµ)2ψ−[−i¯hψ∂ µ(muµ)] (12) According to the continuity condition for the ﬂuid ∂µ(muµ) = 0 (13) we have −m2c4ψ= (−i¯h∂µ−qAµ)2ψ (14) Its form is known as the Klein-Gordon equation. On the condition of non-relativity, the Schrodinger equation form can be derived from the Klein-Gordon equation [2](P.469). However, we must admit that we are careless when we use the continuity condition Eq.(13), because, from Eq.(10) we obtain 1∂µ(muµ) =∂µ(−i¯h∂µlnψ−qAµ) =−i¯h∂µ∂µlnψ(15) where we have used the Lorentz gauge condition. Thus from Eq.(11) to Eq.(12) we obtain −m2c4ψ= (−i¯h∂µ−qAµ)2ψ+ ¯h2ψ∂µ∂µlnψ(16) This is of a perfect wave equation for describing accu- rately the motion of the ﬂow. In other wards, The Klein- Gordon equation form is ill for using the mistaken con- tinuity condition Eq.(13). Comparing with the Dirac equation result, we ﬁnd that the last term of Eq.(16) corresponds to the spin eﬀect of ﬂow (if it exists). In the following we shall show the Dirac equation form from Eq.(10) and Eq.(11). In general, there are many wave functions which sat- isfy Eq.(10) for the ﬂow, these functions and correspond- ing momentum components are denoted by ψ(j)and Pµ(j) =muµ(j), respectively, where j= 1,2,3,...,N, then Eq.(11) can be given by 0 =Pµ(j)Pµ(j)ψ2(j) +m2c4ψ2(j) =δµνPµ(j)ψ(j)Pν(j)ψ(j) +mc2ψ(j)mc2ψ(j) = (δµν+δνµ)Pµ(j)ψ(j)Pν(j)ψ(j)(µ≥ν) +mc2ψ(j)mc2ψ(j) = 2δµνPµ(j)ψ(j)Pν(j)ψ(j)(µ≥ν) +mc2ψ(j)mc2ψ(j) = 2δµνδjkδjlPµ(k)ψ(k)Pν(l)ψ(l)(µ≥ν) +δjkδjlmc2ψ(k)mc2ψ(l) (17) whereδis the Kronecker delta function, j,k,l = 1,2,3,...,N . Here, specially, we do not take jsum over; Prepresents momentum, not operator. Suppose there are two matrices aandbwhich satisfy aµjkaνjl+aνjkaµjl= 2δµνδjkδjl (18) aµjkbjl+bjkaµjl= 0 (19) bjkbjl=δjkδjl (20) then Eq.(17) can be rewritten as 0 = (aµjkaνjl+aνjkaµjl)Pµ(k)ψ(k)Pν(l)ψ(l)(µ≥ν) +(aµjkbjl+bjkaµjl)Pµ(k)ψ(k)mc2ψ(l) +bjkbjlmc2ψ(k)mc2ψ(l) = [aµjkPµ(k)ψ(k) +bjkmc2ψ(k)] ·[aνjlPν(l)ψ(l) +bjlmc2ψ(l)] = [aµjkPµ(k)ψ(k) +bjkmc2ψ(k)]2(21) Consequently, we obtain a wave equation: aµjkPµ(k)ψ(k) +bjkmc2ψ(k) = 0 (22) There are many solutions for aandbwhich satisfy Eq.(18-20), we select a familiar set of aandbas [2]:N= 4 (23) an= [anµν] =/bracketleftbigg 0σn σn0/bracketrightbigg =αn (24) a4= [a4µν] =I (25) b= [bjk] =/bracketleftbigg I0 0−I/bracketrightbigg =β (26) whereαnare the Pauli spin matrices, n= 1,2,3. Sub- stituting them into Eq.(22), we obtain [(−i¯h∂4−qA4) +αn(−i¯h∂n−qAn) +βmc2]ψ= 0 (27) whereψis an one-column matrix about ψ(k). The form of Eq.(27) is known as the Dirac equation. Of course, on the condition of non-relativity, the Schrodinger equation form can be derived from the Dirac equation [2](P.479). It is noted that Eq.(27), Eq.(22), Eq.(17) and Eq.(16) are equivalent despite they have the diﬀerent forms, be- cause they all originate from Eq.(10) and Eq.(11). It follows from Eq.(10) that the path of a particle is analogous to ”lines of electric force” in 4-dimensional space-time. In the case that the Klein-Gordon equation stands, i.e. Eq.(13) stands, at any point, the path can have but one direction (i.e. the local 4-velocity direction ), hence only one path can pass through each point of the space-time. In other words, the path never intersects it- self when it winds up itself into a cell about a nucleus. No path originates or terminates in the space-time. But, in general, the divergence of the 4-velocity ﬁeld does not equal to zero, as indicated in Eq.(15), so the Dirac equa- tion would be better than the Klein-Gordon equation in accuracy. Based on the above derivation, we conﬁrm that the dynamic condition of appearance of quantum behavior in ﬂuid is that the Planck’s constant ¯ his not relatively small in analogy with that for single particle, the condition of the appearance of spin structure in the ﬂuid is that Eq.(15) is un-negligeable. The mechanism profoundly accounts for the quantum wave natures such as spin eﬀect [4] [5]. The present work focus on the formalism and pursuing the correction and strictness in mathematics, its interpre - tation in physical terms remains to be discussed further in the future. ∗E-mail: hycui@public.fhnet.cn.net [1] E. G. Harris, Introduction to Modern Theoretical Physic s, Vol.1&2, (John Wiley & Sons, USA, 1975). [2] L. I. Schiﬀ, Quantum Mechanics, third edition, (McGraw- Hill, USA, 1968). [3] H. Y. Cui, College Physics (A monthly edited by Chi- nese Physical Society in Chinese), ”An Improvement in 2Variational Method for the Calculation of Energy Level of Helium Atom”, 4, 13(1989). [4] H. Y. Cui, eprint, physcis/0102073, (2001). [5] H. Y. Cui, eprint, quant-ph/0102114, (2001). 3
arXiv:physics/0103020v1 [physics.plasm-ph] 8 Mar 2001BGK Electron Solitary Waves Reexamined Li-Jen Chen and George K. Parks* Physics Department, University of Washington, Seattle, WA 98195 *Also at Space Science Laboratory, University of Californi a, Berkeley This paper reexamines the physical roles of trapped and pass ing electrons in electron Bernstein- Greene-Kruskal (BGK) solitary waves, also called the BGK ph ase space electron holes (EH). It is shown that the charge density variation in the vicinity of th e solitary potential is a net balance of the negative charge from trapped electrons and positive cha rge due to the decrease of the passing electron density. A BGK EH consists of electron density enha ncements as well as a density depletion, instead of only the density depletion as previously thought . The shielding of the positive core is not a thermal screening by the ambient plasma, but achieved b y trapped electrons oscillating inside the potential energy trough. The total charge of a BGK EH is th erefore zero. Two separated EHs do not interact and the concept of negative mass is not needed . These features are independent of the strength of the nonlinearity. BGK EHs do not require th ermal screening, and their size is thus not restricted to be greater than the Debye length λD. Our analysis predicts that BGK EHs smaller than λDcan exist. A width( δ)-amplitude( ψ) relation of an inequality form is obtained for BGK EHs in general. For empty-centered EHs with potential am plitude ≫1, we show that the width-amplitude relation of the form δ∝√ψis common to bell-shaped potentials. For ψ≪1, the width approaches zero faster than√ψ. PACS numbers: 52.35.Sb, 52.35.Mw, 52.35.Fp In 1957, Bernstein, Greene and Kruskal [1] solved the one-dimensional, time-independent Vlasov-Poisson equa- tions and obtained the general solutions for electrostatic nonlinear traveling waves, including solitary potential pulses. Their derivation emphasized the special role played by the particles trapped in the potential energy troughs. They demonstrated mathematically that one could construct waves of arbitrary shapes by assigning the distribution of trapped particles suitable for the de- sired wave form. In 1967, Roberts and Berk [2] provided a quasi-particle picture for the electron phase space holes (EH) based on the results of a numerical experiment on two-stream instability. They solved the time-dependent Vlasov- Poisson equations using the “water-bag” model in which the evolution of electron phase-space boundaries between f= 0 andf= 1 was followed, where fis the electron phase space density with values either 0 or 1. They in- terpreted the elliptical empty ( f= 0) region associated with a positive charge observed in the late stage of the nonlinear development as a BGK EH. In order to ex- plain the coalescence of neighboring EHs, a negative ef- fective mass was assigned to each EH to compensate for the Coulomb repulsion of two positive EHs. Thus, they suggested a quasi-particle picture that BGK EHs have positive charge and negative mass. This picture is in use even today to interpret results in computer simulations of EH disruptions due to ion motion [3], and to model the mutual interaction of electrostatic solitary waves in space plasma [4]. It was not until 1979 that BGK EHs were experimen- tally realized by the Risø laboratory experiments [5,6]. By applying large amplitude potential pulses in a plasma-loaded wave guide, solitary potential pulses were excited, including EHs and Korteweg-de Vries (KdV) solitons [7]. Investigations of the mutual interaction of EHs showed that two EHs close to each other would coalesce if they have almost equal velocity and they would pass through each other if their relative velocity was large [5]. The coalescence was interpreted in terms of the positive EH picture derived earlier [5,6]. The analytical work that followed the Risø experi- ments mainly focused on constructing solutions and ob- taining the corresponding width-amplitude relations to facilitate the comparison between BGK EHs and KdV solitons [8–10]. In addressing the quasi-particle picture of EHs, Schamel [9] concluded that EHs were positively charged, screened by the ambient electrons over many Debye lengths ( λD), and had negative mass [9]. This conclusion supports the positive EH picture previously obtained from the numerical experiments [2,11] and that the minimum size of EHs of several λDis a consequence of thermal screening by the ambient electrons. Turikov [8] followed the BGK approach and con- structed the trapped electron distribution for a Maxwellian ambient electron distribution and for sev- eral kinds of solitary potential proﬁles. He restricted his study to BGK EHs with phase space density zero in a hole center and the results showed that the potential width increases with increasing amplitude. This behav- ior is diﬀerent from that of the KdV solitons whose width decreases with increasing amplitude. He also numerically simulated the temporal evolution of the EHs for diﬀerent Mach numbers to study the EH stability and found that EHs are quasi-stable for Mach numbers less than 2. One of the main conclusions that Turikov made was that EHs 1are purely kinetic nonlinear objects in which trapped par- ticles play an important role, but exactly what physical role trapped particles played was not addressed. Space-borne experiments now show that electrostatic solitary waves (ESW) are ubiquitous in Earth’s magne- tospheric boundaries, shock, geomagnetic tail and auro- ral ionosphere [12–19]. While detailed properties of these solitary waves are continuously being studied, it has been shown that in a number of cases the ESWs have features that are consistent with BGK electron [20] and ion [21] mode solitary waves. A statistical study by FAST satel- lite observations [16,17] in the auroral ionosphere has re- vealed that solitary pulses with a positive potential typ- ically have a Gaussian half width ranging from less than oneλDto severalλDwith a mean of 1 .80λDand a stan- dard deviation of 1 .13λD. However, the statistical anal- ysis strongly favored large amplitude pulses [16,17] and smaller EHs if they existed were not sampled. This ques- tion of how small EHs can be is an important issue asso- ciated with how a collisionless plasma supports nonlinear waves and needs to be further investigated. To examine the physical roles played by trapped and passing electrons, we start ﬁrst with simple physical ar- guments and then perform analytical calculations using the same formulation used by Turikov [8], except we relax the restriction of empty-centered EHs and obtain a more general width-amplitude relation. The number density proﬁles of trapped and passing electrons are calculated to show that the negative charge density at the ﬂank of the EH comes exclusively from trapped electrons. It is argued that the BGK EH as a physical entity con- sists of a density enhanced region and a depletion region, and the total charge of the EH is zero. The screening of the positive core is achieved by trapped electrons oscil- lating between their turning points, and not the thermal screening by the ambient electrons as previously thought. There does not exist a minimum size for BGK EHs based on Debye shielding. In addition, two separated EHs do not interact and the concept of negative mass is not needed. These features are shown to be independent of the strength of the nonlinearity deﬁned by the amplitude of the potential. We ﬁrst discuss heuristically the behavior of electrons in the vicinity of a potential pulse. Figures 1(a)-1(c) show the general form of a positive solitary potential pulse (φ(x)), the corresponding bipolar electric ﬁeld ( E= −∂φ/∂x ) and the total charge density ( ρ=−∂2φ/∂x2). The charge density is positive at the core, negative at the boundary, and zero outside the solitary potential. Figure 1(d) shows the potential energy trough with an electron passing by (open circle) and a trapped electron (solid cir- cle) at its turning point. Consider the phase space tra- jectories of electrons passing by the potential and those that are trapped in the potential shown in Figure 1(e).−eφ XX V dxdφ2 2ρ= −d dxφE = −φ(a) (b) (c)(d) (e) FIG. 1. Please see the text for explanations. The dashed line marked by electrons with zero total en- ergy is the boundary of the trapping region inside which electrons are trapped and outside which electrons are un- trapped. The total energy, w=m 2v2−eφ, is a constant of motion. A passing electron ( w > 0) moves with a constant velocity outside the potential and the speed in- creases when it encounters the potential pulse and then decreases back to its original value as it moves away. A trapped electron ( w<0) bounces back and forth between its two turning points in the potential. Since there is no source or sink for the particles, the density is inversely proportional to the velocity. We can thus deduce that the density of passing electrons is constant outside and be- comes smaller as φincreases. No excess negative charge results from the passing electrons. On the other hand, trapped electrons have density maxima at their turning points, and so must be responsible for the excess nega- tive charge. The charge density variation (Figure 1(c)) needed to be self-consistent with the solitary potential pulse is thus a net balance of the negative charge from trapped electrons and the positive charge due to the den- sity decrease of passing electrons since the ion density is assumed uniform. From this simple picture, one can see that in BGK solitary waves it is the trapped electrons traveling with the solitary potential that screen out the positive core. Our picture is diﬀerent from the picture of a positive object in a plasma whose screening is achieved by the thermal motion of the plasma (Debye shielding). The entire trapping region that consists of the total electron density enhancement at the ﬂanks and deple- tion at the core is a physical entity produced by the self- consistent interaction between the plasma particles and the solitary potential pulse. This deﬁnes the physical identity of one BGK EH. The total charge of the entire trapping region is zero, and therefore it follows that two separated BGK EHs do not interact and the concept of negative mass is not needed. We now use the approach formulated by BGK [1] to quantify the above arguments and in addition demon- 2strate that the results are independent of the strength of the nonlinearity. The time-independent, coupled Vlasov and Poisson equations with the assumption of a uniform neutralizing ion background take the following form: v∂f(v,x) ∂x+1 2∂φ ∂x∂f(v,x) ∂v= 0, (1) ∂2φ ∂x2=/integraldisplay∞ −∞f(v,x)dv−1, (2) where f is the electron distribution function and the units have been normalized such that x is normalized by the Debye length, λD, energy by the ambient electron ther- mal energy, Te, velocity by vt=/radicalbig 2Te/m,φbyTe/e. The total energy w=v2−φunder this convention. f=f(w) is a solution to Eq. 1 as can be readily veriﬁed. Recognizing this, Eq. 2 can be re-written in the following form, ∂2φ ∂x2=/integraldisplay0 −φdwftr(w) 2√w+φ+/integraldisplay∞ 0dwfp(w) 2√w+φ−1,(3) whereftr(w) andfp(w) are the trapped and passing elec- tron phase space densities at energy w, respectively. The ﬁrst integral on the RHS of Eq. 3 is the trapped elec- tron density, and the second integral the passing electron density. Prescribe the solitary potential as a Gaussian, φ(ψ,δ,x ) =ψexp (−x2/2δ2), (4) and the passing electron distribution a Maxwellian where the density has been normalized to 1 outside the solitary potential, fp(w) =2√πexp (−w). (5) As in BGK approach, we write the trapped electron dis- tribution as ftr(w) =2 π/integraldisplay−w 0dg(φ) dφdφ√−w−φ, (6) where g(φ) =∂2φ ∂x2+ 1−/integraldisplay∞ 0dwfp(w) 2√w+φ, (7) is the trapped electron density. The trapped electron distribution obtained through Eq. 6 with the prescribed potential and passing electron distribution yields ftr(ψ,δ,w ) =4√−w πδ2/bracketleftbigg 1−2 ln(−4w ψ)/bracketrightbigg +2 exp( −w)√π/bracketleftbig 1−erf(√−w)/bracketrightbig . (8) The ﬁrst term on the RHS of Eq. 8 has been obtained numerically and the second term analytically by Turikov[8] except for the diﬀerence of an overall factor 2 because he deﬁned only half of the phase space density for energy wasftr(w). Turikov did not explore the details of ftr nor did he calculate the contributions from trapped and passing electrons to macroscopic quantities, such as the charge density, associated with EHs. As shown below, we have relaxed the restriction to empty-centered EHs and taken further steps to unfold the information contained inftrwith emphasis on understanding how a collisionless plasma kinetically supports solitary wave solutions. The ﬁrst term in ftrcomes from ∂2φ/∂x2term in Eq. 7 and has a single peak at w=−ψ 4e3/2. This term is 0 atw= 0−, goes negative at w=−ψ, and will al- ways be single peaked even for other bell-shaped soli- tary potentials (for example, sech2(x/δ) and sech4(x/δ), see Figure 4 of [8] for the special case of empty-centered EHs). Although the peak location may vary, it will not be at the end points, 0 and −ψ. The second term aris- ing from the integral of the passing electron distribu- tion monotonically decreases from w= 0−tow=−ψ. The end point behavior of the two terms implies that ftr(w= 0−)> ftr(w=−ψ). Combining the behav- ior of the two terms in ftr, it can be concluded that ftr(0> w≥ −ψ)≥ftr(w=−ψ). This feature of ftris essential in making a solitary pulse, and it manifests it- self at the peak of the potential as two counterstreaming beams. ftris subject to the constraint, ftr(ψ,δ,w =−ψ)≥0, (9) from which we obtain δ≥/bracketleftbigg2√ψ(2 ln 4 −1)√πeψ[1−erf(√ψ)]/bracketrightbigg1/2 . (10) Inequality 9 guarantees that ftr(ψ,δ,0> w≥ −ψ)≥0, sinceftr(0> w≥ −ψ)≥ftr(w=−ψ) as we have pointed out above. Turikov [8] only considered the spe- cial case of empty-centered EHs, which corresponds to the equal sign in inequalities 9 and 10. We plot the width-amplitude relation, Ineq. 10, in Figure 2. A point in the shaded region represents an allowed EH with a givenψandδ. The shaded region includes all of the allowedψandδfor the range of values shown. For a ﬁxedδ, allψ≤ψ0are allowed, where ψ0is such that ftr(ψ0,δ,w=−ψ0) = 0; while for a ﬁxed ψ, allδ≥δ0 are allowed, where δ0is such that ftr(ψ,δ0,w=−ψ) = 0. This width-amplitude relation is dramatically diﬀerent from that of KdV solitons whose width-amplitude rela- tion is a one-to-one mapping. There does not exist a minimum size for BGK EHs, and the size need not to be severalλD[22,9], since one can always adjust ψso that the RHS of 10 is smaller than the δthat’s picked up. We will come back to this issue later in the paper. Withfpandftr, we can now calculate the passing and trapped electron densities separately and obtain 3np(ψ,δ,x ) =/integraldisplay∞ √ φfp(v,x)dv+/integraldisplay−√ φ −∞fp(v,x)dv = exp (φ)/bracketleftBig 1−erf(/radicalbig φ)/bracketrightBig , (11) ntr(ψ,δ,x ) =/integraldisplay√ φ −√ φftr(ψ,δ,v,x )dv =−φ[1 + 2 ln(φ/ψ)] δ2+ exp (φ) erf(/radicalbig φ) +/integraldisplay√ φ −√ φdv−exp (φ−v2)√πerf(/radicalbig φ−v2).(12) The integration of the third term in Eq. 12 carried out by a change of variable y=/radicalbig φ−v2and integrating by parts sequentially yields 1 −exp(φ). Another way to obtain the expression for ntrcomes directly from solving Eq. 7 which is essentially the Poisson equation. This yields ntr=∂2φ ∂x2+ 1−np =−φ[1 + 2 ln(φ/ψ)] δ2+ 1−exp (φ)/bracketleftBig 1−erf(/radicalbig φ)/bracketrightBig , which is identical to Eq. 12. Solving Eq. 7 for ntris simpler, but without the knowledge of ftr, one is not guaranteed whether the particular set of ( ψ,δ) is physi- cally allowed ( ftr≥0). 1 2 3 4 512345 ψδ FIG. 2. the width-amplitude relation of BGK EHs that are not restricted to be empty-centered for a Gaussian potentia l and Maxwellian ambient electron distribution To study the contributions from trapped and passing electrons to the charge density ( −∂2φ/∂x2) and how such contributions are aﬀected by various parameters, we show in Figures 3-5 plots of ntr(ψ,δ,x ) andnp(ψ,δ,x ) and the charge density ρas a function of xfor several values ofψandδ. Figure 3 plots 100 ×ntr(2×10−5,0.1,x), 100×[np(2×10−5,0.1,x)−1], and 100 ×ρ. For an ambient plasma with Te= 700eVandλD= 100mas found at ionospheric heights by FAST satellite in the environmentof BGK EHs [17], this case corresponds to ψ= 1.4× 10−2Vandδ= 10m. As shown, in this weakly nonlinear case, the maximum perturbation in npis only 0.5% and inntr0.4%. The perturbation in the charge density ρis ≤0.2%, and occurs all within one λD. We plotnp(5,4.4,x) andntr(5,4.4,x), and the corre- spondingρin Figure 4. This choice corresponds to a point nearly located on the curve ftr(w=−ψ) = 0 in Figure 2 and is an extremely nonlinear case. One can see that the total charge density perturbation goes ∼10% negative and ∼25% positive, corresponding respectively to electron density enhancement and depletion. With similar format, Figure 5 plots a case with same δand ψ= 1 to illustrate the change in np,ntr, andρof an EH with equal width but smaller amplitude. By locating this case in Figure 2, one notices that farther away from the ftr(w=−ψ) = 0 curve, the dip in ntris ﬁlled up and the charge density perturbation only increases to 5% positive and 2% negative. These examples demonstrate how trapped electrons produce negative charge density perturbations and pass- ing electrons positive charge density perturbations owing to the decrease in their number density. It is always true thatntr≥0, since the number density cannot be nega- tive, and therefore trapped electrons always contribute to negative charge density regardless of the strength of the nonlinearity. This result disagrees with the picture that the positive core is due to a deﬁcit of deeply trapped elec- trons, and that this positive core is screened out by the ambient electrons [9]. It is also diﬀerent from the conclu- sion that the trapped electrons are screened out by the resonant or nonresonant passing electrons depending on the EH velocity in [22]. -0.6 -0.4 -0.2 0.2 0.4 0.6 -0.050.050.10.150.2 -0.6 -0.4 -0.2 0.2 0.4 0.6 -0.4-0.20.20.4 100n 100tr (np−1)100ρ FIG. 3. Trapped electron density ( ntr), passing electron density (np), and charge density ( ρ) forψ= 2×10−5and δ= 0.1. -20 -10 10 200.20.40.60.81 -20 -10 10 20 -0.1-0.050.050.10.150.20.25np ntrρ FIG. 4. Trapped electron density ( ntr), passing electron density (np), and charge density ( ρ) forψ= 5 andδ= 4.4. 4-20 -10 10 200.20.40.60.81 -20 -10 10 20 -0.02-0.010.010.020.030.040.05np ntrρ FIG. 5. Trapped electron density ( ntr), passing electron density (np), and charge density ( ρ) forψ= 1 andδ= 4.4. We now return to the issue of minimum size of EHs. From the illustrations of Figures 3-5, one sees that the trapped electrons have to distribute and oscillate in such a way to result in the desired negative charge at the ﬂanks to shield out the positive core. The entire solitary object is a self-consistent and self-sustained object with zero to - tal charge and does not require thermal screening. Thus, the size of EHs are not restricted to be greater than λD. The charge density variations ( ρ) in Figures 4 and 5 suggest that the maximum excursion of ρis proportional toψfor ﬁxedδ, asMax(ρ) decreases from 25% to 5% whenψvaries from 5 to 1. This relation can be quantiﬁed in the following way: Max(ρ)≡ρ(x= 0) =−ntr(x= 0)−np(x= 0) + 1 = ψ/δ2,(13) which shows that when δis ﬁxed,ρ(x= 0) varies lin- early with ψfrom 0 to ψ0/δ2, whereψ0is such that ftr(ψ0,δ,w =−ψ0) = 0 (the curve in Figure 2). Eq. 13 although obtained from a particular case of a Gaus- sian potential is actually a general relation that holds for bell-shaped potentials. We can examine the gener- ality of Eq. 13 by a dimensional analysis: since the potential amplitude and width are the only two char- acteristic scales involved in the second spatial derivativ e of the potential, the maximum excursion of the charge density (that is −∂2φ/∂x2\|x=0) is proportional to ψ/δ2. We can also examine how ρ(x= 0) varies along the curve ftr(w=−ψ) = 0 (empty-centered EHs) by writing δas a function of ψ. Eq. 13 then becomes ρ(x= 0)ec=√π√ψeψ[1−erf(√ψ)] 2(4 ln2 −1), (14) where the subscript ec stands for empty centered. The RHS of Eq. 14 tends to zero as√ψforψ≪1, and ap- proaches the constant 1 /2(4 ln2 −1) forψ≫1. In other words, for empty-centered EHs the maximum charge den- sity excursion does not increase indeﬁnitely with ψbut settles to a constant value. Since for a physical solu- tion, one would not expect Max(ρ) to increase indeﬁ- nitely, we can take Max(ρ) approaching a constant at largeψ(≫1) as a general behavior for bell-shaped soli- tary potentials. It then implies that for large ψ, the width-amplitude relation δ∝√ψholds in general. For ψ≪1, we can only deduce that δmust approach zeroin a manner faster than√ψin order to meet the physi- cal requirement that Max(ρ) goes to zero with ψ. This general width-amplitude relation that we obtained for empty-centered EHs is independent of the speciﬁc func- tional form of the solitary potential and is consistent with what Turikov [8] obtained for two special proﬁles of po- tentials, sech2(x/δ) and sech4(x/δ). In summary, we obtained the trapped electron dis- tribution function for a Gaussian potential and a Maxwellian passing electron distribution following the BGK approach, and showed that the charge density vari- ation is the net balance of the negative charge produced by trapped electrons and the positive charge density pro- duced by a depletion of passing electrons inside the soli- tary potential. It is not the thermal screening from the ambient electrons that shields out the positive core of the EH, but the oscillations of trapped electrons in be- tween their turning points that result in the excess neg- ative charge. We showed that a BGK EH consists of an electron density enhanced region and depletion region, and this means that the total charge for a BGK EH is zero. It thus follows that two separated EHs do not in- teract and the concept of negative mass is not needed. It also indicates that there does not exist a minimum size for BGK EHs. These features are independent of the particular choice of potential proﬁle and passing electron distribution, and also independent of the strength of non- linearity. The restriction to empty-centered EHs which was used previously is relaxed to obtain a more general width-amplitude relation that is not a one-to-one map- ping but of an inequality form. The maximum charge density excursion is shown to be proportional to ψand inversely proportional to δ, and approaches a constant forψ≫1. It is also argued that for BGK EHs that are empty-centered, the width-amplitude relation for ψ≫1 takes the common form δ∝√ψ, and forψ≪1,δap- proaches zero faster than√ψ. In the above calculation, the EHs do not have relative motion with respect to the ambient electrons. However, this does not restrict us to non-moving EHs, since the entire ambient electron population can have a ﬁnite mean velocity ∝angbracketleftv∝angbracketrighte−iwith respect to the ions (whose frame is taken to be the observer frame) as long as ∝angbracketleftv∝angbracketrighte−i/vtlies within the threshold of Buneman instability [23], where vtis the ambient electron thermal velocity. In fact, it has been shown that a ﬁnite ∝angbracketleftv∝angbracketrighte−iis needed for the long term stability of BGK EHs to ensure that ions do not participate in the dynamics of the solitary pulses [24]. The inclusion of a ﬁnite EH velocity with respect to the ambient electrons will only introduce an asymmetry in the partition between the two directions of the velocity of passing electrons, and will not alter the above conclusions as long as the BGK solutions remain valid, that is, before any instability sets in. Our result indicates that there is no long range inter- action between BGK EHs. Two EHs interact only when 5they are in contact. The only factor that can bring two separated EHs closer to each other is their relative veloc- ity. A system with multiple EHs will not evolve into a state with only one single EH, if the EHs are separated and move with equal velocity or if the faster EH moves in front of the slower ones. Finally, note that the arguments and results we ob- tained for BGK electron solitary waves can be analo- gously applied to BGK ion solitary waves. One of the authors (Chen) is grateful to Bill Peria for the discussions on the electric ﬁeld experiment onboard FAST satellite. The research at the University of Wash- ington is supported in part by NASA grants NAG5-3170 and NAG5-26580. [1] I. B. Bernstein, J. M. Greene, and M. D. Kruskal, Phys. Rev.,108, 546 (1957) [2] K. V. Roberts and H. L. Berk, Phys. Rev. Lett. ,19, 297 (1967) [3] K. Saeki and H. Genma, Phys. Rev. Lett. ,80, 1224 (1998) [4] V. L. Krasovsky, H. Matsumoto, and Y. Omura, Nonlin- ear Processes in Geophys. ,6, 205 (1999) [5] K. Saeki, P. Michelsen, H. L. P´ ecseli, and J. J. Ras- mussen, Phys. Rev. Lett. ,42, 501 (1979) [6] J. P. Lynov, P. Michelsen, H. L. P´ ecseli, and J. J. Ras- mussen, Phys. Scr. ,20, 328 (1979) [7] H. Washimi and T. Taniuti, Phys. Rev. Lett. ,17, 996 (1966) [8] V. A. Turikov, Phys. Scr. ,30, 73 (1984) [9] H. Schamel, Phys. Rep. ,140, pp172-173 (1986) [10] J. P. Lynov, P. Michelsen, H. L. P´ ecseli, J. J. Rasmusse n, and S. H. Sørensen, Phys. Scr. ,31, 596 (1985) [11] H. L. Berk, C. E. Nielsen, and K. V. Roberts, Phys. Flu- ids.,13, 980 (1970) [12] R. Bostr¨ om, et al., Phys. Rev. Lett. ,61, 82 (1988) [13] H. Matsumoto, et al., Geophys. Res. Lett. ,21, 2915 (1994) [14] H. Matsumoto, H. Kojima, Y. Omura, I. Nagano, Geo- phys. Monog. ,105, New Perspectives on the Earth’s Magnetotail, 259 (1998) [15] J. R. Franz and P. M. Kintner, J. S. Pickett, Geophys. Res. Lett. ,25, 1277, (1998) [16] R. E. Ergun, et al., Phys. Rev. Lett. ,81, 826 (1998) [17] R. E. Ergun, C. W. Carlson, L. Muschietti, I. Roth, and J. P. McFadden, Nonlinear Processes in Geophys. ,6, 187 (1999) [18] S. D. Bale, et al., Geophys. Res. Lett. ,25, 2929 (1998) [19] C. A. Cattell, et al., Geophys. Res. Lett. ,26, 425 (1999) [20] L. Muschietti, R. E. Ergun, I. Roth, and C. W. Carlson, Geophys. Res. Lett. ,26, 1093 (1999) [21] A. M¨ alkki, H. Koskinen, R. Bostr¨ om, and B. Holback, Phys. Scr. ,39, 787 (1989) [22] V. L. Krasovsky, H. Matsumoto, and Y. Omura, J. Geo- phys. Res. ,102, 22131 (1997)[23] O. Buneman, Phys. Rev. ,115, 503 (1959) [24] Y. Omura, H. Matsumoto, T. Miyake, and H. Kojima, J. Geophys. Res. ,101, 2685 (1996) 6
arXiv:physics/0103021v1 [physics.plasm-ph] 8 Mar 2001Kinetic theory of QED plasma in a strong electromagnetic ﬁeld I. The covariant hyperplane formalism A. H¨ olla,1V.G. Morozovb,2G. R¨ opkea,3 aPhysics Department, University of Rostock, Universit¨ ats platz 3, D-18051 Rostock, Germany bMoscow State Institute of Radioengineering, Electronics, and Automation, 117454 Vernadsky Prospect 78, Moscow, Russia Abstract We develop a covariant density matrix approach to kinetic th eory of QED plasmas subjected into a strong external electromagnetic ﬁeld. A ca nonical quantization of the system on space-like hyperplanes in Minkowski space and a covariant general- ization of the Coulomb gauge is used. The condensate mode ass ociated with the mean electromagnetic ﬁeld is separated from the photon degr ees of freedom by a time-dependent unitary transformation of both, the dynami cal variables and the nonequilibrium statistical operator. Therefore even in th e case of strong external ﬁelds a perturbative expansion in orders of the ﬁne structur e constant for the cor- relation functions as well as the statistical operator is ap plicable. A general scheme for deriving kinetic equations in the hyperplane formalism is presented. Key words: relativistic kinetic theory, QED plasma, hyperplane forma lism PACS: 05.20.Dd, 05.30.-d, 11.10.Ef, 52.27.Ny, 52.25.Dg 1 Introduction In recent years the theoretical study of dense relativistic plasmas is of in- creasing interest. Such plasmas are not only limited to astr ophysics, but can nowadays be produced by high-intense short-pulse lasers [1 ,2]. In view of the 1hoell@darss.mpg.uni-rostock.de 2vmorozov@orc.ru 3gerd@darss.mpg.uni-rostock.de Preprint submitted to Elsevier Preprint 25 September 2012inertial conﬁnement fusion, one has to consider a plasma und er extreme con- ditions which is created by a strong external ﬁeld. This new e xperimental progress needs a systematic approach based on quantum elect rodynamics and methods of nonequilibrium statistical mechanics. Considerable attention has been focussed on a mean-ﬁeld (Vl asov-type) kinetic equation for the fermionic Wigner function, which is an esse ntial step towards transport theory of laser-induced QED plasmas. Using the Wi gner operator deﬁned in four-dimensional momentum space [3–5], a manifes tly covariant mean-ﬁeld kinetic equation can be derived from the Heisenbe rg equations of motion for the ﬁeld operators. In this approach, however, it is diﬃcult to formulate an initial value problem for the kinetic equation since the four- dimensional Fourier transformation in the covariant Wigne r function includes integration of two-point correlation functions over time. This diﬃculty does not appear in the scheme based on the one-time fermionic Wigner function where the ﬁeld operators are taken at the same time and only th e spatial Fourier transformation is performed. In the context of QED, the one-time formulation was proposed by Bialynicki-Birula et al. [6] (r eferred to in the following as BGR) and used successfully in their study of the electron-positron vacuum. Within this approach one can explore a number of attr active features. The one-time Wigner function has a direct physical interpre tation and allows to calculate local observables, such as the charge density a nd the current density. The description in terms of one-time quantities is quite natural in kinetic theory based on the von Neumann equation for the stat istical operator and provides a consistent account of causality in collision integrals. It should be noted, however, that the one-time Wigner functi on does not con- tain a complete information about one-particle dynamics; t he spectral prop- erties of correlation functions can be described only in ter ms of two-point Green’s functions which are closely related to the covarian t Wigner function. Recently this aspect of relativistic kinetic theory was stu died within the mean- ﬁeld approximation [7,8]. The aforementioned incompleten ess of the one-time description is well known in non-relativistic kinetic theo ry, where two-time correlation functions can, in principle, be reconstructed from the one-time Wigner function by solving integral equations which follow from the Dyson equation for nonequilibrium Green’s functions [9]. The rec onstruction problem in relativistic kinetic theory remains to be explored. The s olution of this prob- lem requires a further development of the relativistic dens ity matrix method as well as the relativistic Green’s function technique. In this and subsequent papers we develop a density matrix app roach to kinetic theory of QED plasma subjected into a strong electromagneti c ﬁeld. From the conceptual point of view, our aim is to generalize the BGR sch eme [6] in two aspects. First, we wish to present the one-time formalism in covariant form. This removes a drawback of the BGR theory which is not manifes tly covariant. 2Second, we will develop a scheme which allows to go beyond the mean-ﬁeld approximation, including dissipative processes in QED pla sma and the inter- play between collisions and the mean-ﬁeld eﬀects. Whereas s ubsequent papers will concern with explicit kinetic equations, the present ﬁ rst part considers some general problems of the one-time covariant approach to relativistic ki- netic theory. In comparison to QED where the main object is th eS-matrix constructed from vacuum averages of the ﬁeld operators, kin etic theory of QED deals with averages over a nonequilibrium ensemble desc ribing a many- body system. Therefore we use the Hamiltonian formalism whi ch is typical for the density matrix method. In this case, however, one meets w ith some funda- mental problems which are considered in this paper. In order that the theory be manifestly covariant, canonical quantization of the sys tem will be carried out in a covariant fashion. Another point is that, in the pres ence of a strong electromagnetic ﬁeld, perturbation expansions in the ﬁne s tructure constant are not suitable. To overcome this diﬃculty, we will present a procedure which allows to separate the classical part of the electromagneti c (EM) ﬁeld and the photon degrees of freedom at any time. The paper is organized as follows. In Section 2 we brieﬂy sket ch a scheme of relativistic statistical mechanics in the form adapted t o kinetic theory. In our approach we use a manifestly covariant Schr¨ odinger pic ture on space- like hyperplanes in Minkowski space. Analogous formulatio ns of relativistic quantum mechanics and quantum ﬁeld theory can be found in lit erature for various applications (see, e.g., [10–14]). In this way, “eq ual-time” correlation functions are deﬁned with respect to the “invariant time” va riable on a hyper- plane. In Section 3 we perform canonical quantization of QED on space-like hyperplanes and derive the covariant quantum Hamiltonian. Section 4 deals with the condensate mode which corresponds to the electroma gnetic ﬁeld in- duced by the polarization in the system. The condensate mode is eliminated by a time-dependent unitary transformation of the statisti cal operator and dynamical variables. As a result, we obtain the eﬀective Ham iltonian, where the interaction of fermions with the mean electromagnetic ﬁ eld is incorpo- rated non-perturbatively at any time, while the interactio n between fermions and photons is described by a term which can be taken into acco unt within perturbation theory. It is shown how Maxwell equations for t he mean electro- magnetic ﬁeld are recovered in our scheme. In Section 5 the co variant one-time Wigner function and the photon density matrix are introduce d and a method for deriving kinetic equations in the hyperplane formalism is outlined. We conclude the paper with a few remarks concerning our results and further applications. We use the system of units with c= ¯h= 1. The signature of the metric tensor is (+,−,−,−). 32 Nonequilibrium statistical operator in the hyperplane fo rmalism 2.1 The relativistic von Neumann equation It is well known that in the special theory of relativity a qua ntum state of a system is deﬁned by a complete set of commuting observables w hich can be associated with a three-parameter space-like surface σin Minkowski space. Among these surfaces three-dimensional hyperplanes are es pecially easy to deal with [10–13]. Since the use of arbitrary space-like sur faces does not lead to new physics, in what follows we restrict our consideratio n to hyperplanes. A space-like hyperplane σ≡σn,τis characterized by a unit time-like normal vectornµand a scalar parameter τwhich may be interpreted as an “invariant time”. The equation of the hyperplane σn,τreads x·n=τ, n2=nµnµ= 1. (2.1) In the special Lorentz frame where nµ= (1,0,0,0), Eq. (2.1) reads x0=τ; in this frame the parameter τcoincides with the time variable t=x0. By treat- ing a state vector \|Ψ[σn,τ]/an}b∇acket∇i}htas a functional of σn,τ, the covariant Schr¨ odinger equation can be derived from the relation between the state v ector on the hy- perplaneσand the state vector on the hyperplane σ′=Lσwhich is obtained by an inhomogeneous Lorentz transformation L={a,Λ}: σ→σ′=Lσ:x→x′= Λx+a. (2.2) The relation between the state vectors is [15] U(L)\|Ψ[Lσ]/an}b∇acket∇i}ht=\|Ψ[σ]/an}b∇acket∇i}ht, (2.3) whereU(L) =U(a,Λ) is a unitary representation of the inhomogeneous Lorentz group. The generators of this representation, ˆPµandˆMµν, are the energy-momentum vector and the angular momentum tensor, re spectively. For our purposes, the only transformations of relevance are pur e time-like trans- lations which change the value of τ. Recalling the form of U(a,Λ) for pure translations U(a,1) = exp/braceleftBig iˆPµaµ/bracerightBig (2.4) and introducing the notation/vextendsingle/vextendsingle/vextendsingleΨ[σn,τ]/angbracketrightBig =\|Ψ(n,τ)/an}b∇acket∇i}ht, Eq. (2.3) can be written for an inﬁnitesimal time-like translation aµ=nµδτas \|Ψ(n,τ+δτ)/an}b∇acket∇i}ht+iδτ/parenleftBigˆPµnµ/parenrightBig \|Ψ(n,τ)/an}b∇acket∇i}ht=\|Ψ(n,τ)/an}b∇acket∇i}ht, (2.5) 4from which we obtain the relativistic Schr¨ odinger equatio n i∂ ∂τ\|Ψ(n,τ)/an}b∇acket∇i}ht=ˆH(n)\|Ψ(n,τ)/an}b∇acket∇i}ht (2.6) with the Hamiltonian on the hyperplane given by ˆH(n) =ˆPµnµ. (2.7) In the presence of a prescribed external ﬁeld, the energy-mo mentum vector and, consequently, the Hamiltonian ˆHτ(n) can depend explicitly on τ. Com- bining Eq. (2.6) with the adjoint equation for the bra-vecto r, one ﬁnds that the statistical operator ̺(n,τ) for a mixed quantum ensemble obeys the equation ∂̺(n,τ) ∂τ−i/bracketleftBig ̺(n,τ),ˆHτ(n)/bracketrightBig = 0, (2.8) which is analogous to the non-relativistic von Neumann equa tion. 2.2 Schr¨ odinger and Heisenberg pictures on hyperplanes The evolution of a mixed ensemble on space-like hyperplanes can be repre- sented in diﬀerent pictures. The statistical operator in th e Heisenberg picture does not depend on the parameter τand is associated with some ﬁxed hy- perplaneσn,τ0. Dynamical variables are represented by operators ˆOH([σn,τ]) which are functionals of the hyperplanes. Of particular int erest in quantum ﬁeld theory are local operators ˆOH(x) which depend on the space-time point x. In what follows it will be convenient to treat such operator s as functions of the parameter τ. To deﬁne this dependence, we introduce the transverse projector with respect to the normal vector nµ, ∆µ ν=δµ ν−nµnν, (2.9) and notice that a space-time four-vector xµcan be represented in the form xµ=nµτ+xµ ⊥, τ =n·x, (2.10) where xµ ⊥= ∆µ νxν(2.11) 5is the transverse (space-like) component of x. Geometrically, Eq. (2.10) means that the space-like vector xµ ⊥lies on the hyperplane σn,τpassing through the space-time point x. Using the decomposition (2.10), a local Heisenberg oper- atorˆOH(x) can be written as ˆOH(x) =ˆOH(nτ+x⊥)≡ˆOH(τ,x⊥). (2.12) Let us assume that ˆPµdoes not depend explicitly on τ. Then, recalling the well-known equation of motion for Heisenberg operators ∂µˆOH(x) =−i[ˆOH(x),ˆPµ], (2.13) one readily ﬁnds that the time-like evolution of such operat ors is described by the equation ˆOH(τ,x⊥) = ei(τ−τ0)ˆH(n)ˆOH(τ0,x⊥) e−i(τ−τ0)ˆH(n)(2.14) with the Hamiltonian (2.7). The generalization of Eq. (2.14 ) to situations in which the Hamiltonian ˆHτdepends explicitly on τis obvious. Deﬁning the evolution operator U(τ,τ′;n) as the ordered exponent U(τ,τ′;n) =Tτexp  −iτ/integraldisplay τ′ˆH¯τ(n)d¯τ  , (2.15) we have ˆOH(τ,x⊥) =U†(τ,τ0;n)ˆOH(τ0,x⊥)U(τ,τ0;n). (2.16) In the Schr¨ odinger picture, the statistical operator ̺(n,τ) isτ-dependent and its time-like evolution is governed by Eq. (2.8), whereas op erators ˆOSare deﬁned on a ﬁxed hyperplane. Assuming the Heisenberg and Sch r¨ odinger pic- tures to coincide on the hyperplane σn,τ0, Eq. (2.16) implies that the transition from the Schr¨ odinger picture to the Heisenberg picture is g iven by ˆOH(τ,x⊥) =U†(τ,τ0;n)ˆOS(x⊥)U(τ,τ0;n). (2.17) The mean values O(x) of local dynamical variables can be calculated in both pictures. Using a formal solution of Eq. (2.8) ̺(n,τ) =U(τ,τ0;n)̺(n,τ0)U†(τ,τ0;n), (2.18) 6we ﬁnd that O(x) =/an}b∇acketle{tˆOH(τ,x⊥)/an}b∇acket∇i}htτ0=/an}b∇acketle{tˆOS(x⊥)/an}b∇acket∇i}htτ. (2.19) Here and in what follows the symbol /an}b∇acketle{t· · ·/an}b∇acket∇i}htτstands for averages calculated with the statistical operator ̺(n,τ). In many problems one is dealing with partial derivatives ∂µO(x) which enter the equations of motion for local observables. In the hyperplane formalism, it is convenient to express the partial derivatives in terms of the derivatives with respect to τandx⊥. Recalling Eqs. (2.10) and (2.11), we write ∂µ=nµ∂ ∂τ+∇µ,∇µ= ∆ν µ∂ν= ∆ν µ∂ ∂xν ⊥. (2.20) Then, in the Heisenberg picture, Eq. (2.19) yields the equat ion of motion ∂µO(x) =/angbracketleftBig ∂µˆOH(τ,x⊥)/angbracketrightBigτ0, (2.21) where ∂µˆOH(x) =nµ∂ ∂τˆOH(τ,x⊥) +∇µˆOH(τ,x⊥) ≡ −inµ/bracketleftBigˆOH(τ,x⊥),ˆHH(n,τ)/bracketrightBig +∇µˆOH(τ,x⊥). (2.22) In the Schr¨ odinger picture, the τ-dependence of the mean values appears through the statistical operator which obeys the von Neuman n equation (2.8). Nevertheless, in this picture from Eq. (2.19) we obtain a sim ilar equation of motion ∂µO(x) =/angbracketleftBig ∂µˆOS(x⊥)/angbracketrightBigτ(2.23) with the analogous deﬁnition of the operator ∂µacting on local dynamical variables: ∂µˆOS(x⊥) =−inµ/bracketleftBigˆOS(x⊥),ˆH(n)/bracketrightBig +∇µˆOS(x⊥). (2.24) 2.3 “Equal-time” correlation functions Describing the evolution of the system in terms of hyperplan es, we can in- troduce “equal-time” correlation functions of local dynam ical variables with respect to the invariant time τ. Let ˆO1H(x),ˆO2H(x),...,ˆOkH(x) be some local 7Heisenberg operators. Then the “equal-time” correlation f unction for these operators can be deﬁned as F1···k(x1⊥,...,xk⊥;n,τ) =/an}b∇acketle{tˆO1H(x1)· · ·ˆOkH(xk)/an}b∇acket∇i}htτ0, (2.25) wheren·x1=n·x2=...=n·xk=τ. In the Schr¨ odinger picture this correlation function takes the form F1···k(x1⊥,...,xk⊥;n,τ) =/an}b∇acketle{tˆO1S(x1⊥)· · ·ˆOkS(xk⊥)/an}b∇acket∇i}htτ. (2.26) The covariant von Neumann equation (2.8) yields the equatio ns ∂ ∂τF1···k(x1⊥,...,xk⊥;n,τ) =−i/an}b∇acketle{t[ˆO1S(x1⊥)· · ·ˆOkS(xk⊥),ˆHτ(n)]/an}b∇acket∇i}htτ(2.27) which can serve as a starting point for constructing the quan tum hierarchy for the “equal-time” correlation functions. 3 Hamiltonian of QED on hyperplanes We will now apply the foregoing scheme to a relativistic syst em of charged fermions interacting through the EM ﬁeld. For deﬁniteness, we take these fermions to be electrons and positrons, so that protons will be treated as a positively charged background which ensures electric neut rality of the system. There is no diﬃculty in describing protons by an additional D irac ﬁeld. Having in mind applications to relativistic plasmas produced by hi gh-intense shot- pulse lasers, we assume the system to be subjected into a pres cribed external EM ﬁeld which is not necessarily weak. 3.1 The Lagrangian density The ﬁrst step in formulating the kinetic theory of QED plasma is to construct the Hamiltonian ˆH(n). We start with the classical Lagrange density L(x) =LD(x) +LEM(x) +Lint(x) +Lext(x), (3.1) where LD(x) and LEM(x) are the Lagrangian densities of free Dirac and EM ﬁelds respectively, Lint(x) is the interaction Lagrangian density, and the term Lext(x) describes the interaction of fermions with the external el ectromagnetic ﬁeld. In standard notation (see, e.g., [16]), we have 8LD(x) =¯ψ(x)/parenleftbiggi 2γµ↔ ∂µ−m/parenrightbigg ψ(x), (3.2) LEM(x) =−1 4Fµν(x)Fµν(x), (3.3) Lint(x) =−jµ(x)Aµ(x), (3.4) Lext(x) =−jµ(x)Aµ ext(x), (3.5) where↔ ∂µ=→ ∂µ−← ∂µ. In the following the electromagnetic ﬁeld tensor is taken in the form Fµν=∂µAν−∂νAµ. The current density four-vector will be expressed as jµ=e¯ψγµψwithe <0. We wish to remark that in our approach the four-potential of the EM ﬁeld is decomposed int o two terms. The variablesAµ(x) correspond to the EM ﬁeld caused by charges and currents in the system, while Aµ ext(x) is a prescribed external ﬁeld. In what follows, only the dynamical ﬁeld Aµ(x) will be quantized. 3.2 Canonical quantization on hyperplanes A canonical quantization implies that some gauge ﬁxing cond ition is im- posed onAµ. For many-particle systems studied in statistical mechani cs, the Coulomb gauge seems to be the most natural. However, the disa dvantage of this gauge is that it is not manifestly covariant. Therefore we will use a general- ization of the Coulomb gauge condition which is consistent w ith the covariant description of evolution in terms of space-like hyperplane s. To formulate this condition, we introduce for any four-vector Vµthe decomposition into the transverse and longitudinal parts by Vµ=nµV/bardbl+Vµ ⊥, V/bardbl=nνVν, Vµ ⊥= ∆µ νVν, (3.6) where ∆µ νis the projector (2.9). Then a natural generalization of the Coulomb gauge condition reads ∇µAµ ⊥= 0. (3.7) In the special frame where nµ= (1,0,0,0) andAµ= (A0,A), Eq. (3.7) reduces to∇·A= 0, which is the usual Coulomb gauge condition. To deﬁne canonical variables for the electromagnetic ﬁeld o n a hyperplane σn,τ, we ﬁrst perform the decomposition (3.6) of the ﬁeld variabl esAµand the decomposition (2.20) of the derivatives in the Euler-Lagra nge equations ∂L ∂Aµ−∂ν∂L ∂(∂νAµ)= 0. (3.8) 9A simple algebra shows that these equations are equivalent t o ∂L ∂A/bardbl−∂ ∂τ ∂L ∂˙A/bardbl − ∇ν∂L ∂(∇νA/bardbl)= 0, (3.9) ∆µν/bracketleftBigg∂L ∂Aν ⊥−∂ ∂τ/parenleftBigg∂L ∂˙Aν ⊥/parenrightBigg − ∇λ∂L ∂(∇λAν ⊥)/bracketrightBigg = 0, (3.10) where we use the notation ˙f≡∂f/∂τ for derivatives with respect to τ. Equa- tion (3.9) allows to eliminate the variable A/bardblin the Lagrangian. First we rewrite expressions (3.3) and (3.4) in terms of A/bardblandAµ ⊥using the decom- position procedure for the derivatives and the ﬁeld Aµ. As a result we obtain the Lagrangian density in the form L=−1 4F⊥µνFµν ⊥−1 2/parenleftBig ∇µA/bardbl−˙Aµ ⊥/parenrightBig/parenleftBig ∇µA/bardbl−˙A⊥µ/parenrightBig −j/bardblA/bardbl−j⊥µAµ ⊥+LD+Lext, (3.11) where we have introduced the notation Fµν ⊥=∇µAν ⊥− ∇νAµ ⊥. (3.12) Note that the last two terms in Eq. (3.11) do not contain A/bardblandAµ ⊥. Now using the expression (3.11) to calculate derivatives in Eq. (3.9) and taking into account that, according to the gauge condition (3.7), ∇µ˙Aµ ⊥= 0, we get ∇µ∇µA/bardbl=j/bardbl. (3.13) In the frame where nµ= (1,0,0,0), this reduces to the Poisson equation for A0. The solution of Eq. (3.13) is A/bardbl(τ,x⊥) =/integraldisplay σndσ′G(x⊥−x′ ⊥)j/bardbl(τ,x′ ⊥), (3.14) where the Green function G(x⊥) satisﬁes the equation ∇µ∇µG(x⊥) =δ3(x⊥) (3.15) with the three-dimensional delta function on a hyperplane σndeﬁned as δ3(x⊥) =/integraldisplayd4p (2π)3e−ip·xδ(p·n). (3.16) 10The solution of Eq. (3.15) for G(x⊥) is given by G(x⊥) =−/integraldisplayd4p (2π)3e−ip·xδ(p·n)1 p2 ⊥. (3.17) The variable A/bardblcan now be eliminated in the Lagrangian density (3.11) with the aid of Eq. (3.14). Terms like ∇ν(· · ·) can be dropped since they do not con- tribute to the Lagrangian L=/integraldisplay Ldσunder appropriate boundary conditions. Then a straightforward algebra leads to L=−1 4F⊥µνFµν ⊥−1 2˙A⊥µ˙Aµ ⊥−j⊥µAµ ⊥ +LD+Lext−1 2/integraldisplay σndσ′j/bardbl(τ,x⊥)G(x⊥−x′ ⊥)j/bardbl(τ,x′ ⊥). (3.18) We will treat the ﬁelds Aµ ⊥as dynamical variables for EM ﬁeld and follow the Dirac version of canonical quantization of theories with co nstraints [17,18]. The gauge condition (3.7) is one of the constraint equation i n this scheme. Another constraint equation follows directly from the deﬁn ition of transverse four-vectors, Eq. (3.6), and reads nµAµ ⊥(x) = 0. (3.19) We now deﬁne canonical conjugates for the ﬁeld variables Aµ ⊥by Π⊥µ=∂L ∂˙Aµ ⊥=−˙A⊥µ. (3.20) Obviously the Π’s are not independent variables since they s atisfy the con- straint equations ∇µΠµ ⊥= 0, andnµΠµ ⊥= 0. Thus, we have four constraints imposed on the canonical variables. Now, following the stan dard quantization procedure [18], the commutation relations for the ﬁeld oper ators ˆAµ ⊥andˆΠµ ⊥ can be derived. As shown in Appendix A, these commutation rel ations are /bracketleftBigˆAµ ⊥(τ,x⊥),ˆΠν ⊥(τ,x′ ⊥)/bracketrightBig =icµν(x⊥−x′ ⊥), (3.21) /bracketleftBigˆAµ ⊥(τ,x⊥),ˆAν ⊥(τ,x′ ⊥)/bracketrightBig =/bracketleftBigˆΠµ ⊥(τ,x⊥),ˆΠν ⊥(τ,x′ ⊥)/bracketrightBig = 0, (3.22) where cµν(x⊥−x′ ⊥) =/integraldisplayd4p (2π)3e−ip·(x−x′)δ(p·n)/bracketleftBigg ∆µν−pµ ⊥pν ⊥ p2 ⊥/bracketrightBigg .(3.23) 11In Appendix B the anticommutation relations for the Dirac ﬁe ld operators on hyperplanes are derived. The result can be written as /braceleftbigg ˆψa(τ,x⊥),ˆ¯ψa′(τ,x′ ⊥)/bracerightbigg =/bracketleftBig γ/bardbl(n)/bracketrightBig aa′δ3(x⊥−x′ ⊥), (3.24) /braceleftbigg ˆψa(τ,x⊥),ˆψa′(τ,x′ ⊥)/bracerightbigg =/braceleftbigg ˆ¯ψa(τ,x⊥),ˆ¯ψa′(τ,x′ ⊥)/bracerightbigg = 0, (3.25) wherea, a′are the spinor indices. The matrix γ/bardbl(n) is introduced through the following decomposition of the Dirac matrices γµ: γµ=nµγ/bardbl(n) +γµ ⊥(n), γ/bardbl(n) =nνγν, γµ ⊥(n) = (δµ ν−nµnν)γν.(3.26) In the special Lorentz frame where xµ= (t,r) andnµ= (1,0,0,0), we have γ/bardbl=γ0andδ3(x⊥−x′ ⊥) =δ(r−r′), so that Eq. (3.24) reduces to the well- known anticommutation relation for the quantized Dirac ﬁel d. 3.3 Derivation of the Hamiltonian The classical Hamiltonian on the hyperplane σn,τcan be derived in two ways. Following the canonical procedure, Hτ(n) is obtained by the Legendre trans- formation H(n) =/integraldisplay σn,τdσ/braceleftbigg Π⊥µ˙Aµ ⊥+ ¯π˙ψ+˙¯ψπ− L/bracerightbigg , (3.27) where Lis given by Eq. (3.18). To ﬁnd explicit expressions for the va riables πand ¯π, which are conjugates to the ﬁelds ¯ψandψ, we rewrite the Dirac Lagrangian density (3.2) using the decomposition (2.20) of derivatives: LD=¯ψ i 2 γ/bardbl↔ ∂ ∂τ+γµ ⊥↔ ∇µ −m ψ. (3.28) Then we have ¯π≡∂LD ∂˙ψ=i 2¯ψγ/bardbl, π ≡∂LD ∂˙¯ψ=−i 2γ/bardblψ. (3.29) Substituting expressions (3.28) and (3.29) into Eq. (3.27) , we arrive at the classical Hamiltonian. Another way is to start from the clas sical analog of 12Eq. (2.7) which reads H(n) =Pµnµ≡/integraldisplay σn,τdσnµTµνnν, (3.30) whereTµν(x) is the energy-momentum tensor. In order that the quantized Hamiltonian be hermitian, the energy-momentum tensor must be real. For instance, one can use the so-called Belinfante tensor [19]. When applied to the Lagrangian (3.1), the standard derivation of the Belinf ante tensor (see, e.g., [16]) gives Tµν(x) =−gµν/braceleftbigg ¯ψ/parenleftbiggi 2γλ↔ ∂λ−m/parenrightbigg ψ−jλ/parenleftBig Aλ+Aλ ext/parenrightBig −1 4FαβFαβ/bracerightbigg +FµλFλ ν+i 4¯ψ/parenleftbigg γν↔ ∂µ+γµ↔ ∂ν/parenrightbigg ψ−1 2/parenleftBig jνAµ+jµAν/parenrightBig .(3.31) Separating the longitudinal and transverse components wit h respect to the normal vector nµand then eliminating the τ-derivatives of the ﬁelds with the aid of Eqs. (3.20) and (3.29), the classical Hamiltonian on t he hyperplane is obtained from Eq. (3.30). It can be veriﬁed that in both cases we have the same expression for Hτ(n). The ﬁnal step is to replace the canonical variables Aµ ⊥,Πµ ⊥,ψ,¯ψby the corresponding quantum operators. As a result, we ﬁnd t he Hamiltonian in the form ˆHτ(n) =ˆHD(n) +ˆHEM(n) +ˆHint(n) +ˆHτ ext(n), (3.32) where ˆHD(n) and ˆHEM(n) are the Hamiltonians for free fermions and the polarization EM ﬁeld respectively, ˆHint(n) is the interaction term, and ˆHτ ext(n) describes the external EM ﬁeld eﬀects. In the Schr¨ odinger p icture the explicit expressions for these terms are ˆHD(n) =/integraldisplay σndσˆ¯ψ/parenleftbigg −i 2γµ ⊥(n)↔ ∇µ+m/parenrightbigg ˆψ, (3.33) ˆHEM(n) =/integraldisplay σndσ/parenleftbigg1 4ˆF⊥µνˆFµν ⊥−1 2ˆΠ⊥µˆΠµ ⊥/parenrightbigg , (3.34) ˆHint(n) =/integraldisplay σndσˆj⊥µˆAµ ⊥+1 2/integraldisplay σndσ/integraldisplay σndσ′ˆj/bardbl(x⊥)G(x⊥−x′ ⊥)ˆj/bardbl(x′ ⊥),(3.35) ˆHτ ext(n) =/integraldisplay σndσˆjµ(x⊥)Aµ ext(τ,x⊥). (3.36) 13In the EM ﬁeld Hamiltonian (3.34) the tensor operator ˆFµν ⊥=∇µˆAν ⊥− ∇νˆAµ ⊥ (3.37) contains only the transverse components of the ﬁeld operato rsˆAµwhich are decomposed as ˆAµ=nµˆA/bardbl+ˆAµ ⊥. (3.38) The longitudinal part ˆA/bardblhas been eliminated in the interaction Hamilto- nian (3.35) by the equation ∇µ∇µˆA/bardbl=ˆj/bardbl (3.39) which is analogous to Eq. (3.13). As usual, in Eqs. (3.33)–(3 .36) normal or- dering in operators is implied. The self-energy contributi on to the last term in Eq. (3.35) is omitted, so that the product : ˆj/bardbl(x⊥) : :ˆj/bardbl(x′ ⊥) : is under- stood. For simplicity, we have written the Hamiltonian for t he case that the fermionic subsystem is described by one Dirac ﬁeld. The gene ralization to a many-component case is obvious. 4 The condensate mode of EM ﬁeld An essential feature of the dynamical evolution of QED plasm a in a strong external ﬁeld is that the mean values of the canonical operat ors4,Aµ ⊥=/an}b∇acketle{tˆAµ ⊥/an}b∇acket∇i}ht and Πµ ⊥=/an}b∇acketle{tˆΠµ ⊥/an}b∇acket∇i}ht, just as the mean values of creation and annihilation bosoni c operators ˆaand ˆa†related to the canonical operators by plane-wave expan- sions, are not zero. Furthermore, they are macroscopic quan tities associated with the mean EM ﬁeld induced by the polarization in the syste m. In the language of statistical mechanics, the variables Aµ ⊥(x) and Πµ ⊥(x) describe a macroscopic condensate mode of the EM ﬁeld. This fact does not allow to apply perturbation theory directly to the Hamiltonian (3.3 2) because the in- teraction of fermions with the condensate mode or, what is th e same, with the mean EM ﬁeld is not weak. So, we have to separate the condensat e mode from the photon degrees of freedom in the Hamiltonian and the stat istical operator. 4From this point onwards symbols Aµ ⊥, Πµ ⊥,jµ, etc. denote mean values of the corresponding operators. 144.1 The time-dependent unitary transformation The condensate mode is most easily isolated by introducing t heτ-dependent unitary transformation ̺C(n,τ) = eiˆC(n,τ)̺(n,τ) e−iˆC(n,τ), (4.1) where the operator ˆC(n,τ) is given by ˆC(n,τ) =/integraldisplay σndσ/braceleftBig Aµ ⊥(x)ˆΠ⊥µ(x⊥)−Π⊥µ(x)ˆAµ ⊥(x⊥)/bracerightBig . (4.2) Note that the unitary transformation (4.1) does not aﬀect fe rmionic operators and has the properties eiˆC(n,τ)ˆAµ ⊥(x⊥) e−iˆC(n,τ)=ˆAµ ⊥(x⊥) +Aµ ⊥(x), eiˆC(n,τ)ˆΠµ ⊥(x⊥) e−iˆC(n,τ)=ˆΠµ ⊥(x⊥) + Πµ ⊥(x). (4.3) Taking now into account that, for any operator ˆO, /an}b∇acketle{tˆO/an}b∇acket∇i}htτ= Tr/braceleftBig eiˆC(n,τ)ˆOe−iˆC(n,τ)̺C(n,τ)/bracerightBig ≡/angbracketleftBig eiˆC(n,τ)ˆOe−iˆC(n,τ)/angbracketrightBigτ ̺C,(4.4) we ﬁnd that /angbracketleftBigˆAµ ⊥(x⊥)/angbracketrightBigτ ̺C=/angbracketleftBigˆΠµ ⊥(x⊥)/angbracketrightBigτ ̺C= 0. (4.5) Thus, in the state described by the transformed statistical operator (4.1), the canonical dynamical variables ˆAµ ⊥andˆΠµ ⊥have zero mean values and, hence, they are not related to the condensate mode. In other words, a fter the unitary transformation the EM ﬁeld operators correspond to the phot on degrees of freedom. Based on the above arguments, it is convenient to us e̺C(n,τ) as the statistical operator of the system. It should be noted, howe ver, that̺C(n,τ) does not satisfy the von Neumann equation (2.8) since the ope rator ˆCdepends onτ. In order to derive the equation of motion for ̺C(n,τ), we diﬀerentiate Eq. (4.1) with respect to τ. After some algebra which we omit, we ﬁnd that the transformed statistical operator satisﬁes the modiﬁed von Neumann equation ∂̺C(n,τ) ∂τ−i/bracketleftBig ̺C(n,τ),ˆHτ(n)/bracketrightBig = 0 (4.6) 15with the eﬀective Hamiltonian ˆHτ(n) = eiˆC(n,τ)ˆHτ(n) e−iˆC(n,τ) +/integraldisplay σndσ/braceleftBigg∂Π⊥µ(x) ∂τˆAµ ⊥(x⊥)−∂Aµ ⊥(x) ∂τˆΠ⊥µ(x⊥)/bracerightBigg .(4.7) The transformation of ˆH(n) in the ﬁrst term is trivial due to Eqs. (4.3) and the fact that the transformation does not aﬀect fermionic op erators. It is convenient to eliminate the derivatives in the last term of E q. (4.7) with the aid of the equations of motion for the condensate mode ∂Aµ ⊥(x) ∂τ=−Πµ ⊥(x),∂Πµ ⊥(x) ∂τ=∇λFλµ ⊥(x)−jµ ⊥(x), (4.8) which are easily derived using Eqs. (2.23), (2.24), and the c anonical commu- tation relations (3.21). The tensor Fµν ⊥in Eq. (4.8) is the mean value of the operator (3.37), and jµ ⊥(x) is the transverse part of the mean polarization current jµ(x) =/an}b∇acketle{tˆjµ(x⊥)/an}b∇acket∇i}htτ. (4.9) Inserting Eqs. (4.8) into Eq. (4.7), the eﬀective Hamiltoni an can be written as a sum ˆHτ(n) =ˆHτ 0(n) +ˆHτ int(n). (4.10) The main term ˆHτ 0(n) =ˆHD(n) +ˆHEM+/integraldisplay σndσˆjµ(x⊥)Aµ(x) (4.11) describes free photons and fermions interacting with the to tal electromagnetic ﬁeld Aµ(x) =Aµ ext(x) +Aµ(x), (4.12) where the mean polarization ﬁeld Aµ(x) is given by Aµ(x) =/angbracketleftBigˆAµ(x⊥)/angbracketrightBigτ. (4.13) The term ˆHτ int(n) in Eq. (4.10) describes a weak interaction between photons and fermions. The explicit expression for this term is 16ˆHτ int(n) =/integraldisplay σndσ∆ˆjµ ⊥(x⊥;τ)ˆAµ ⊥(x⊥) +1 2/integraldisplay σndσ/integraldisplay σndσ′∆ˆj/bardbl(x⊥;τ)G(x⊥−x′ ⊥) ∆ˆj/bardbl(x′ ⊥;τ),(4.14) where the operators ∆ˆjµ(x⊥;τ) =ˆjµ(x⊥)− /an}b∇acketle{tˆjµ(x⊥)/an}b∇acket∇i}htτ(4.15) represent quantum ﬂuctuations of the fermionic current. Th e essential point is that now the interaction term (4.14) does not contain a con tribution from the condensate mode and, consequently, one can use perturba tion expansions in the ﬁne structure constant. 4.2 Maxwell equations To complete our discussion of the condensate mode, we will sh ow how the Maxwell equations for the total mean EM are derived in our app roach. Ac- cording to Eq. (4.12), the total ﬁeld tensor can be written as Fµν(x) =∂µAν(x)−∂νAµ(x) =Fµν ext(x) +Fµν(x). (4.16) The external ﬁeld tensor Fµν extis assumed to satisfy the Maxwell equations ∂µFµν ext(x) =jν ext(x) (4.17) with some prescribed external current jµ ext. On the other hand, the polarization ﬁeld tensor Fµν(x) is the mean value of the operator ˆFµν(x⊥) =∂µˆAν−∂νˆAµ =ˆFµν ⊥+nν/parenleftBigˆΠµ ⊥+∇µˆA/bardbl/parenrightBig −nµ/parenleftBigˆΠν ⊥+∇νˆA/bardbl/parenrightBig . (4.18) Recalling Eqs. (2.23) and (2.24), straightforward algebra ic manipulations with the equations of motion for the ﬁeld operators lead to the Max well equations for the polarization ﬁeld tensor ∂µFµν(x) =jν(x). (4.19) 17Now Eqs. (4.17) and (4.19) can be combined into the Maxwell eq uations for the total ﬁeld tensor ∂µFµν(x) =jν(x) +jν ext(x). (4.20) A solution of these equations gives the total mean ﬁeld Aµin terms of the total mean current. 5 Kinetic description of QED plasma 5.1 The “one-time” Wigner function Within the hyperplane formalism a natural way of describing kinetic processes in the fermion subsystem is by the “one-time” Wigner functio n which depends on the variable τ. Since there is the gauge freedom for the mean ﬁeld Aµ, it is convenient to employ the gauge-invariant Wigner function o n the hyperplane σn,τdeﬁned as Waa′(x⊥,p⊥;τ) =/integraldisplay d4yeip·yδ(y·n) ×exp/braceleftBig ieΛ(x⊥+1 2y⊥,x⊥−1 2y⊥;τ)/bracerightBig faa′/parenleftBig x⊥+1 2y⊥,x⊥−1 2y⊥;τ/parenrightBig (5.1) with the gauge function Λ(x⊥,x′ ⊥;τ) =x⊥/integraldisplay x′ ⊥A⊥µ(τ,R⊥)dRµ ⊥ ≡1/integraldisplay 0ds(xµ ⊥−x′µ ⊥)A⊥µ(τ,x′ ⊥+s(x⊥−x′ ⊥)). (5.2) In Eq. (5.1) the one-particle density matrix faa′is the mean value faa′(x⊥,x′ ⊥;τ) =/angbracketleftBigˆfaa′(x⊥,x′ ⊥)/angbracketrightBigτ=/angbracketleftBigˆfaa′(x⊥,x′ ⊥)/angbracketrightBigτ ̺C(5.3) of some density operator ˆfaa′. In the literature one can ﬁnd diﬀerent deﬁnitions for the fermionic density operator. The most often used deﬁn itions are 18ˆfaa′(x⊥,x′ ⊥) =−1 2[ˆψa(x⊥),ˆ¯ψa′(x′ ⊥)], (5.4) ˆf′ aa′(x⊥,x′ ⊥) = :ˆ¯ψa′(x′ ⊥)ˆψa(x⊥):. (5.5) These two operators are related by ˆfaa′(x⊥,x′ ⊥) =ˆf′ aa′(x⊥,x′ ⊥) +Kaa′(x⊥,x′ ⊥), (5.6) where the last c-number term represents the vacuum expectation value of ˆf since the vacuum expectation value of ˆf′is zero. It can be shown, however, that the vacuum term in Eq. (5.6) does not contribute to local observables like the mean current jµ(x). The advantage of the deﬁnition (5.5) is that the mean values of one-particle dynamical variables (summatio n over repeated spinor indices) ˆO=/integraldisplay σndσdσ′Oa′a(x′ ⊥,x⊥) :ˆ¯ψa′(x′ ⊥)ˆψa(x⊥): (5.7) are conveniently expressed in terms of the density matrix f′=/an}b∇acketle{tˆf′/an}b∇acket∇i}htτ: /an}b∇acketle{tˆO/an}b∇acket∇i}htτ=/integraldisplay σndσdσ′Oa′a(x′ ⊥,x⊥)f′ aa′(x⊥,x′ ⊥;τ). (5.8) Unfortunately, the equation of motion for the density opera tor (5.5) with the Hamiltonian (4.10) contains vacuum (divergent) terms. On t he other hand, such terms do not appear in the equation of motion for the oper ator (5.4). For this reason, we shall take the operator (5.4) as the one-p article density operator in Eq. (5.3). An analogous deﬁnition was used previ ously for the phase-space description of the QED vacuum in a strong ﬁeld [6 ]. The Wigner function (5.1) is deﬁned on a given family of hyper planesσn,τand, hence, depends parametrically on the normal four-vector n. It should be noted, however, that local observables calculated from the Wigner function do not depend on the choice of n. As an important example, the mean polarization current (4.9) can be written in the form jµ(x) =e/an}b∇acketle{t:ˆ¯ψ(x⊥)γµˆψ(x⊥):/an}b∇acket∇i}htτ =e/integraldisplayd4p (2π)3δ(p·n) tr{γµW(x⊥,p⊥;τ=x·n)}, (5.9) where the symbol “tr” stands for the trace over spinor indice s. Geometrically, the above relation means that, in calculating the current, t he invariant time 19τhas a value such that the space-time point xlies on the hyperplane σn,τ. 5.2 The photon density matrix To deﬁne the photon density matrix, we start from the plane wa ve expansion of the vector potential operator ˆA⊥in terms of creation and annihilation op- erators. By analogy with the well-known representation for the free photon ﬁeld in the special Lorentz frame where nµ= (1,0,0,0), we write ˆAµ ⊥(τ,x⊥) =/integraldisplayd4q/radicalBig 2ωn(q⊥)(2π)3δ/parenleftBig q/bardbl−ωn(q⊥)/parenrightBig ×/summationdisplay l=1,2eµ(q⊥,l)/braceleftBig ˆal(q⊥) e−iq·x+ ˆa† l(q⊥) eiq·x/bracerightBig , (5.10) whereeµ(q⊥,l) are real-valued polarization four-vectors and ωn(q⊥) =ωn(−q⊥) =/parenleftBig −qµ ⊥q⊥µ/parenrightBig1/2(5.11) is the dispersion relation for free photons on the hyperplan e. The conditions ∇µˆAµ ⊥=nµˆAµ ⊥= 0 mean that the polarization vectors satisfy q⊥µeµ(q⊥,l) =nµeµ(q⊥,l) = 0. (5.12) The expansion of the operator ˆΠµ ⊥into plane waves is found from (5.10) by using ˆΠµ ⊥=−˙ˆAµ ⊥: ˆΠµ ⊥(τ,x⊥) =/integraldisplayd4q/radicalBig 2ωn(q⊥)(2π)3iωn(q⊥)δ/parenleftBig q/bardbl−ωn(q⊥)/parenrightBig ×/summationdisplay l=1,2eµ(q⊥,l)/braceleftBig ˆal(q⊥) e−iq·x−ˆa† l(q⊥) eiq·x/bracerightBig . (5.13) Assuming the commutation relations for the creation and ann ihilation opera- tors [ˆal(q⊥),ˆa† l′(q′ ⊥)] =δll′δ3(q⊥−q′ ⊥), (5.14) [ˆal(q⊥),ˆal′(q′ ⊥)] = [ˆa† l(q⊥),ˆa† l′(q′ ⊥)] = 0, 20and the completeness relation for the polarization vectors /summationdisplay l=1,2eµ(q⊥,l)eν(q⊥,l) =−/parenleftBigg ∆µν−qµ ⊥qν ⊥ q2 ⊥/parenrightBigg , (5.15) the commutation relation (3.21) for the ﬁeld operators is re covered. Note that Eqs. (5.10) and (5.13) give the ﬁeld operators in the interac tion picture. The corresponding expansions for the ﬁeld operators in the Schr ¨ odinger picture are obtained by setting τ= 0. In this case the delta-function δ(q/bardbl−ωn(q⊥)) may be replaced by δ(q/bardbl). The above considerations suggest that it is natural to deﬁne the photon density matrix in terms of the Schr¨ odinger operators ˆϕl(x⊥) =/integraldisplayd4q (2π)3/2δ(q/bardbl) e−iq·xˆal(q⊥), ˆϕ† l(x⊥) =/integraldisplayd4q (2π)3/2δ(q/bardbl) eiq·xˆa† l(q⊥),(5.16) which satisfy the commutation relations [ˆϕl(x⊥),ˆϕ† l′(x′ ⊥)] =δll′δ3(x⊥−x′ ⊥), [ˆϕl(x⊥),ˆϕl′(x′ ⊥)] = [ˆϕ† l(x⊥),ˆϕ† l′(x′ ⊥)] = 0.(5.17) The photon density matrix is deﬁned as Nll′(x⊥,x′ ⊥;τ) =/angbracketleftBigˆNll′(x⊥,x′ ⊥)/angbracketrightBigτ ̺C, (5.18) where ˆNll′(x⊥,x′ ⊥) = ˆϕ† l′(x′ ⊥) ˆϕl(x⊥) (5.19) is the photon density operator. It should be emphasized that in Eq. (5.18) the average is calculated with the transformed statistical operator̺C(n,τ) in which the condensate mode of EM ﬁeld has been eliminated. Whe n written in terms of the average with the statistical operator ̺(n,τ), the photon density matrix takes the form Nll′(x⊥,x′ ⊥;τ) =/angbracketleftBigˆNll′(x⊥,x′ ⊥)/angbracketrightBigτ− /an}b∇acketle{tˆϕl(x⊥)/an}b∇acket∇i}htτ/an}b∇acketle{tˆϕ† l′(x′ ⊥)/an}b∇acket∇i}htτ, (5.20) where the last term corresponds to the contribution from the condensate mode. 215.3 The covariant statistical operator in QED kinetics The evolution of the fermionic Wigner function (5.1) and the photon density matrix (5.18) is governed by kinetic equations which can be d erived from the equations of motions ∂ ∂τfaa′(x⊥,x′ ⊥;τ) =−iTr/braceleftBig [ˆfaa′(x⊥,x′ ⊥),ˆHτ 0(n) +ˆHτ int(n)]̺C(n,τ)/bracerightBig ,(5.21) ∂ ∂τNll′(x⊥,x′ ⊥;τ) =−iTr/braceleftBig [ˆNll′(x⊥,x′ ⊥),ˆHτ 0(n) +ˆHτ int(n)]̺C(n,τ)/bracerightBig .(5.22) There are two ways to express the right-hand sides of these eq uations in terms of the fermionic and photon density matrices using perturba tion expansions in the ﬁne structure constant. One method is by considering t he hierarchy for correlation functions which appear through the commuta tors with the interaction Hamiltonian ˆHτ int(n) and then employing some truncation proce- dure. Another method is to construct an approximate solutio n of Eq. (4.6) in terms of the density matrices fandN. In both cases one has to impose some boundary conditions of the retarded type on the correla tion functions or the statistical operator. The standard boundary conditi on in kinetic theory is Bogoliubov’s boundary condition of weakening of initial correlations which implies the uncoupling of all correlation functions to one- particle density ma- trices in the distant past, i.e., for τ→ −∞ . In the scheme developed by Zubarev (see, e.g., [20]), such boundary conditions can be i ncluded by using instead of Eq. (4.6) the equation with an inﬁnitesimally sma ll source term ∂̺C(n,τ) ∂τ−i/bracketleftBig ̺C(n,τ),ˆHτ(n)/bracketrightBig =−ε{̺C(n,τ)−̺rel(n,τ)},(5.23) whereε→+0 after the calculation of averages. Here ̺rel(n,τ) is the so- called relevant statistical operator which describes a Gibbs state for some given nonequilibrium state variables. In QED kinetics these vari ables are the Wigner function (5.1) and the photon density matrix (5.18). Theref ore, following the standard procedure [20], we obtain the relevant statistica l operator in the form (with summation over spinor and polarization indices) ̺rel(n,τ) =1 Zrel(n,τ)exp/braceleftBigg −/integraldisplay σndσdσ′/bracketleftbigg λ(f) aa′(x⊥,x′ ⊥;τ) :ˆ¯ψa(x⊥)ˆψa′(x′ ⊥): +λ(ph) ll′(x⊥,x′ ⊥;τ) ˆϕ† l(x⊥) ˆϕl′(x′ ⊥)/bracketrightbigg/bracerightBigg ,(5.24) 22whereZrel(n,τ) is the normalization constant (or the partition function i n the relevant ensemble) and λ(f) aa′(x⊥,x′ ⊥;τ),λ(ph) ll′(x⊥,x′ ⊥;τ) are Lagrange multipliers which are determined by the self-consistency conditions faa′(x⊥,x′ ⊥;τ) = Tr/braceleftBigˆfaa′(x⊥,x′ ⊥)̺rel(n,τ)/bracerightBig , Nll′(x⊥,x′ ⊥;τ) = Tr/braceleftBigˆNll′(x⊥,x′ ⊥)̺rel(n,τ)/bracerightBig .(5.25) Using Eq. (5.23) for the transformed statistical operator l eads to the hierarchy ∂ ∂τ/tildewideF1···k(x1⊥,...,xk⊥;n,τ) =−i/angbracketleftBig [ˆO1(x1⊥)· · ·ˆOk(xk⊥),ˆHτ(n)]/angbracketrightBigτ ̺C −ε/braceleftbigg /tildewideF1···k(x1⊥,...,xk⊥;n,τ)−/angbracketleftBigˆO1(x1⊥)· · ·ˆOk(xk⊥)/angbracketrightBigτ ̺rel/bracerightbigg ,(5.26) where ˆOi(xi⊥) are some Schr¨ odinger operators which may depend on the fermion operators as well as on the EM operators, and /tildewideF1···k(x1⊥,...,xk⊥;n,τ) =/angbracketleftBigˆO1(x1⊥)· · ·ˆOk(xk⊥)/angbracketrightBigτ ̺C(5.27) are the “equal-time” correlation functions in which the con densate mode of the EM ﬁeld is eliminated. Since the relevant statistical op erator (5.24) ad- mits Wick’s decomposition, the last term in Eq. (5.26) ensur es the boundary condition of complete weakening of initial correlations. N ote that the explicit knowledge of the statistical operator ̺C(n,τ) is not needed when considering the hierarchy for the correlation functions. Use of some tru ncation procedure is a standard practice in this case. The hierarchy for correl ation functions will be discussed in subsequent papers in the context of the deriv ation of collision integrals. Another method of handling Eq. (5.23) is by considering its f ormal solution ̺C(n,τ) =ετ/integraldisplay −∞dτ′e−ε(τ−τ′)U(τ,τ′)̺rel(n,τ′)U†(τ,τ′), (5.28) where the evolution operator can be written as the ordered ex ponent U(τ,τ′) =Tτexp  −iτ/integraldisplay τ′ˆH¯τ(n)d¯τ  . (5.29) 23After partial integration, the expression (5.28) becomes ̺C(n,τ) =̺rel(n,τ) + ∆̺(n,τ), (5.30) where ∆̺(n,τ) =−τ/integraldisplay −∞dτ′e−ε(τ−τ′) × U(τ,τ′)/braceleftBigg∂̺rel(n,τ′) ∂τ′−i/bracketleftBig ̺rel(n,τ′),ˆHτ′(n)/bracketrightBig/bracerightBigg U†(τ,τ′).(5.31) The representation (5.30) for the statistical operator all ows to separate the mean-ﬁeld terms and the collision terms in Eqs. (5.21) and (5 .22). Taking into account the self-consistency conditions (5.25) and the fac t that the Hamilto- nian (4.11) is bilinear in the fermion and photon operators, we arrive at the equations ∂ ∂τfaa′(x⊥,x′ ⊥;τ) =−i/angbracketleftBig [ˆfaa′(x⊥,x′ ⊥),ˆHτ 0(n)]/angbracketrightBigτ ̺rel+I(f) aa′(x⊥,x′ ⊥;τ),(5.32) ∂ ∂τNll′(x⊥,x′ ⊥;τ) =−i/angbracketleftBig [ˆNll′(x⊥,x′ ⊥),ˆHEM]/angbracketrightBigτ ̺rel+I(ph) ll′(x⊥,x′ ⊥;τ),(5.33) where the collision integrals for fermions and photons are g iven by I(f) aa′(x⊥,x′ ⊥;τ) =−i/angbracketleftBig [ˆfaa′(x⊥,x′ ⊥),ˆHτ int(n)]/angbracketrightBigτ ̺rel −iTr/braceleftBig [ˆfaa′(x⊥,x′ ⊥),ˆHτ int(n)]∆̺(n,τ)/bracerightBig , (5.34) I(ph) ll′(x⊥,x′ ⊥;τ) =−iTr/braceleftBig [ˆNll′(x⊥,x′ ⊥),ˆHτ int(n)]∆̺(n,τ)/bracerightBig . (5.35) In the presence of a strong EM ﬁeld, the evolution of the fermi on subsystem is governed predominantly by its interaction with the mean E M ﬁeld. Thus, the covariant mean-ﬁeld kinetic equation for the Wigner fun ction (5.1) can be derived from Eq. (5.32) neglecting the collision integra l. This kinetic equa- tion as well as the collision integrals (5.34) and (5.35) wil l be considered in subsequent papers. 246 Concluding remarks We have shown that the hyperplane formalism can serve as the b asis for ki- netic theory of QED plasma in the presence of a strong externa l ﬁeld. The formalism has the advantage that it is manifestly covariant and therefore al- lows to introduce diﬀerent approximations in covariant for m. The formalism makes only minor changes in the non-relativistic density ma trix method, so that many well-developed approaches can be directly applie d to QED plasma. For instance, the explicit construction of the statistical operator allows to incorporate many-particle correlations through the exten sion of the set of ba- sic state parameters (see, e.g.,[20,21]). Note also that, u sing the Heisenberg picture on hyperplanes, nonequilibrium Green’s functions can be introduced with respect to the invariant time parameter τ. In such a way, the spectral properties of microscopic dynamics can be incorporated. Finally, we would like to emphasize once again two key proble ms in a covariant density matrix approach to relativistic kinetic theory in t he presence of a strong mean ﬁeld. First, it is necessary to perform canonica l quantization of the system on a hyperplane in Minkowski space. Second, the co ndensate mode must be separated from the quantum degrees of freedom at any t ime. We have shown how these problems can be solved in the context of QED pl asmas. As a result, a general form of kinetic equations for fermions and photons was given. The scheme outlined in this paper is also applicable to some q uantum ﬁeld models used in QCD transport theory. In this case the non-Abe lian algebra must be worked out to describe the quark-gluon plasma. Appendix A Commutation relations for electromagnetic ﬁeld on hyperpl anes Let us write the constraint equations for the canonical vari ablesAµ ⊥and Πµ ⊥ in the form χN(x⊥) = 0, where χ1(x⊥) =∇µAµ ⊥(x⊥), χ2(x⊥) =∇µΠµ ⊥(x⊥), χ3(x⊥) =nµAµ ⊥(x⊥), χ4(x⊥) =nµΠµ ⊥(x⊥).(A.1) 25For any functionals Φ1and Φ2of the ﬁeld variables A⊥and Π⊥, we deﬁne the Poisson bracket [Φ1,Φ2]P≡/integraldisplay σn,τdσ/braceleftBiggδΦ1 δAµ ⊥(x⊥)δΦ2 δΠ⊥µ(x⊥)−δΦ2 δAµ ⊥(x⊥)δΦ1 δΠ⊥µ(x⊥)/bracerightBigg ,(A.2) where the constraints are ignored in calculating the functi onal derivatives. Applying this formula to the canonical variables we obtain [Aµ ⊥(x⊥),Π⊥ν(x′ ⊥)]P=δµ νδ3(x⊥−x′ ⊥) (A.3) with the three-dimensional delta function (3.16). All othe r Poisson brackets for the canonical variables are equal to zero. In the Dirac te rminology, func- tions (A.1) correspond to second class constraints since the matrix CNN′(x⊥,x′ ⊥) = [χN(x⊥),χN′(x′ ⊥)]P (A.4) is non-singular. A straightforward calculation of the Pois son brackets shows that the non-zero elements of Care C12(x⊥,x′ ⊥) =−C21(x⊥,x′ ⊥) =−∇µ∇µδ3(x⊥−x′ ⊥), C34(x⊥,x′ ⊥) =−C43(x⊥,x′ ⊥) =δ3(x⊥−x′ ⊥). (A.5) According to the general quantization scheme [17,18], comm utation relations for canonical operators are deﬁned by the Dirac brackets for classical canonical variables. In our case the Dirac brackets are written as [Φ1,Φ2]D= [Φ1,Φ2]P −/integraldisplay σn,τdσ/integraldisplay σn,τdσ′[Φ1,χN(x⊥)]PC−1 NN′(x⊥,x′ ⊥) [χN′(x′ ⊥),Φ2]P(A.6) (summation over repeated indices). The inverse matrix, C−1 NN′(x⊥,x′ ⊥), satisﬁes the equation /integraldisplay σn,τdσ′′CNN′′(x⊥,x′′ ⊥)C−1 N′′N′(x′′ ⊥,x′ ⊥) =δNN′δ3(x⊥−x′ ⊥). (A.7) Since the matrix elements (A.5) of Cdepend on the diﬀerence x⊥−x′ ⊥, Eq. (A.7) can be solved for C−1using a Fourier transform on σn,τ, which 26is deﬁned for any function f(x) as ˜f(τ,p⊥) =/integraldisplay d4xeip·xδ(x·n−τ)f(x). (A.8) The inverse transform is f(x)≡f(τ,x⊥) =/integraldisplayd4p (2π)3e−ip·xδ(p·n)˜f(τ,p⊥). (A.9) If we perform the Fourier transformation in Eq. (A.7), we ﬁnd by insert- ing (A.5) that the non-zero elements of C−1are C−1 12(x⊥,x′ ⊥) =−C−1 21(x⊥,x′ ⊥) =−/integraldisplayd4p (2π)3e−ip·(x−x′)δ(p·n)1 p2 ⊥, C−1 34(x⊥,x′ ⊥) =−C−1 43(x⊥,x′ ⊥) =−δ3(x⊥−x′ ⊥). (A.10) Now the Dirac brackets (A.6) for the canonical variables are easily calculated and we obtain [Aµ ⊥(x⊥),Πν ⊥(x′ ⊥)]D=cµν(x⊥−x′ ⊥), (A.11) [Aµ ⊥(x⊥),Aν ⊥(x′ ⊥)]D= [Πµ ⊥(x⊥),Πν ⊥(x′ ⊥)]D= 0, (A.12) where the functions cµν(x⊥−x′ ⊥) are given by Eq. (3.23). According to the general quantization rules, the commutation relations for canonical operators correspond to i[...]D. Thus, in the hyperplane formalism, the commutation relations for the operators of EM ﬁeld are given by (3.21) and (3.22). Obviously these relations are valid in the Schr¨ odinger and Heisenber g pictures. Appendix B Anticommutation relations for the Dirac ﬁeld on hyperplane s To ﬁnd the anticommutation relations for the fermion operat ors on the hyper- planeσn,τ, it is suﬃcient to consider a free Dirac ﬁeld. Our starting po int is the standard quantization scheme in the frame where xµ= (t,r) and nµ= (1,0,0,0) (see, e.g., [16]). In that case the ﬁeld operators ˆψaandˆ¯ψa 27can be written in terms of creation and annihilation operato rs according to ˆψa(x) =/integraldisplayd4p (2π)3/2δ(p0−ǫ(p))/radicalBig 2ǫ(p)/summationdisplay s=±1/bracketleftBigˆbs(p)uas(p)e−ip·x+ˆd† s(p)vas(p)eip·x/bracketrightBig , ˆ¯ψa(x) =/integraldisplayd4p (2π)3/2δ(p0−ǫ(p))/radicalBig 2ǫ(p)/summationdisplay s=±1/bracketleftBigˆds(p)¯vas(p)e−ip·x+ˆb† s(p)¯uas(p)eip·x/bracketrightBig , whereǫ(p) =/radicalBig p2+m2is the free fermion dispersion relation. Constructing the expression {ˆψa(x),ˆ¯ψa′(x′)}for two arbitrary space-time points and recall- ing the anticommutation relations /braceleftBigˆbs(p),ˆb† s′(p′)/bracerightBig =/braceleftBigˆds(p),ˆd† s′(p′)/bracerightBig =δss′δ3(p−p′), (B.1) as well as polarization sums /summationdisplay s=±1uas(p)¯ua′s(p) =/bracketleftBig γµpµ+m/bracketrightBig aa′,/summationdisplay s=±1vas(p)¯va′s(p) =/bracketleftBig γµpµ−m/bracketrightBig aa′, we arrive at /braceleftbigg ˆψa(x),ˆ¯ψa′(x′)/bracerightbigg =/integraldisplayd3p (2π)31 2ǫ(p)/braceleftBig/bracketleftBig γµpµ+m/bracketrightBig aa′e−ip·(x−x′) +/bracketleftBig γµpµ−m/bracketrightBig aa′eip·(x−x′)/bracerightBig , (B.2) wherep0=/radicalBig p2+m2. Using /integraldisplayd3p (2π)31 2ǫ(p)=/integraldisplayd4p (2π)3δ(p2−m2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle p0>0, (B.3) Eq. (B.2) can be rewritten in a Lorentz invariant form /braceleftbigg ˆψa(x),ˆ¯ψa′(x′)/bracerightbigg =/integraldisplayd4p (2π)3/braceleftbigg/bracketleftBig γµpµ+m/bracketrightBig aa′e−ip·(x−x′)δ(p2−m2)/vextendsingle/vextendsingle/vextendsingle p0>0 +/bracketleftBig γµpµ−m/bracketrightBig aa′eip·(x−x′)δ(p2−m2)/vextendsingle/vextendsingle/vextendsingle p0>0/bracerightbigg . (B.4) The anticommutation relation on the hyperplane σn,τis now obtained by set- tingx=nτ+x⊥andx′=nτ+x′ ⊥. In calculating the integrals, it is convenient to use a decomposition pµ=nµp/bardbl+pµ ⊥, (p/bardbl>0). Then we get 28/braceleftbigg ˆψa(τ,x⊥),ˆ¯ψa′(τ,x′ ⊥)/bracerightbigg =/integraldisplayd4p (2π)3δ(p/bardbl−ǫ(p⊥)) 2ǫ(p⊥) ×/braceleftBig/bracketleftBig γ/bardblp/bardbl+γµ ⊥p⊥µ+m/bracketrightBig aa′e−ip⊥µ(xµ ⊥−x′µ ⊥) +/bracketleftBig γ/bardblp/bardbl+γµ ⊥p⊥µ−m/bracketrightBig aa′eip⊥µ(xµ ⊥−x′µ ⊥)/bracerightBig (B.5) with the dispersion relation on the hyperplane ǫ(p⊥) =/radicalBig −p⊥µpµ ⊥+m2. 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arXiv:physics/0103022v1 [physics.bio-ph] 9 Mar 2001Selection for Fitness vs. Selection for Robustness in RNA Secondary Structure Folding Claus O. Wilke Digital Life Laboratory California Institute of Technology, Mail-Code 136-93 Pasadena, CA 91125 wilke@caltech.edu February 2, 2008 Abstract We investigate the competition between two quasis- pecies residing on two disparate neutral networks. Under the assumption that the two neutral networks have diﬀerent topologies and ﬁtness levels, it is the mutation rate that determines which quasispecies will eventually be driven to extinction. For small muta- tion rates, we ﬁnd that the quasispecies residing on the neutral network with the lower replication rate will disappear. For higher mutation rates, however, the faster replicating sequences may be outcompeted by the slower replicating ones in case the connec- tion density on the second neutral network is suﬃ- ciently high. Our analytical results are in excellent agreement with ﬂow-reactor simulations of replicat- ing RNA sequences. Keywords: quasispecies, mutant cloud, neutral networks, RNA secondary structure folding, selection of robustness At high mutation rates, the number of mutated oﬀspring generated in a population far exceeds the number of oﬀspring identical to their parents. As a result, a stable cloud of mutants, a so-called qua- sispecies (Eigen and Schuster 1979; Eigen et al. 1988; Eigen et al. 1989; Nowak 1992; Wilke et al. 2001), forms around the fastest replicating geno- types. Experimental evidence in favor of such a persistent cloud of mutants is available from RNA viruses (Steinhauer et al. 1989; Domingo and Hol- land 1997; Burch and Chao 2000) and in vitro RNAreplication (Biebricher 1987; Biebricher and Gardiner 1997); both are cases in which a high substitution rate per nucleotide is common (Drake 1993). The ex- istence of a quasispecies has important implications for the way in which selection acts, because the evo- lutionary success of individual sequences depends on the overall growth rate of the quasispecies they be- long to. As a consequence, organisms with a high replication rate that produce a large number of oﬀ- spring with poor ﬁtness can be outcompeted by or- ganisms of smaller ﬁtness that produce a larger num- ber of also-ﬁt oﬀspring (Schuster and Swetina 1988). Similarly, if a percentage of the possible mutations is neutral, and the majority of the non-neutral mu- tations is strongly deleterious, then the growth rate of a quasispecies depends signiﬁcantly on the con- nection density (the number of nearby neutral mu- tants of an average viable genotype) of the neutral genotypes (van Nimwegen et al. 1999). Therefore, a neutral network (a set of closely related mutants with identical ﬁtness) with high connectivity can be advantageous over one with higher ﬁtness, but lower connectivity. Here, we are interested in this latter possibility. In particular, we investigate the compe- tition of two quasispecies residing on separate neu- tral networks with diﬀerent connection densities and replication rates, and determine under what condi- tions selection favors the more ﬁt (i.e., of higher repli- cation rate) or the more robust (more densely con- nected) mutant cloud. Our approach is closely re- lated to the study of holey landscapes, in which all genotypes are classiﬁed into either viable or inviable ones (Gavrilets and Gravner 1997; Gavrilets 1997). 12 However, we extend this picture by further subdivid- ing the viable genotypes into two groups with diﬀer- ent replication rates. The paper is organized as follows. First, we de- scribe a simple model of a quasispecies on a single neutral network, and demonstrate that the model is consistent with simulations of RNA sequences. Then, based on this model, we present a model of two com- peting quasispecies, and compare the second model with simulation results as well. Following that, we study the probability of ﬁxation of a single advan- tageous mutant that arises in a fully formed quasis- pecies. Finally, we discuss the implications of our results and give conclusions. Population Dynamics on a Single Neutral Network Before we can address the competition of two qua- sispecies, we need a good description of a single qua- sispecies on a neutral network. A fundamental contri- bution to this problem has been made by van Nimwe- gen et al. (1999), who showed that the average ﬁtness of a population on a neutral network is determined only by the ﬁtness of the neutral genotypes, the mu- tation rate, and the largest eigenvalue of the neu- tral genotypes’ connection matrix. The connection matrix is a symmetric matrix with one row/column per neutral genotype. It holds a one in those posi- tions where the row- and the column-genotype are exactly one point-mutation apart, and a zero oth- erwise. In theory, the formalism of van Nimwegen et al. describes a population on a neutral network well. However, the exact connection matrix is nor- mally not known, which implies that we cannot cal- culate the population dynamics from ﬁrst principles. Nevertheless, we can base a very simple model on the fact – also established by van Nimwegen et al.– that the average neutrality in the population, which is exactly the largest eigenvalue of the connection ma- trix, is independent of the mutation rate. The main assumption of our simple model is that the popu- lation behaves as if all sequences in the population had the same neutrality ν, where νis given by the average neutrality in the population. Moreover, we consider genetic sequences of length l, and assume a per-symbol copy ﬁdelity of q. Then, the eﬀective copy ﬁdelity or neutral ﬁdelity (Ofria and Adami 2001) Q, i.e., the probability with which on average a viablesequence gives birth to oﬀspring that also resides on the neutral network, is given by Q= [1−(1−q)(1−ν)]l ≈e−l(1−q)(1−ν). (1) Now, we can devise a two-concentration model in which x1(t) is the total concentration of all sequences on the neutral network, and xd(t) is the concentra- tion of sequences oﬀ the network (these sequences are assumed to replicate so slowly that their oﬀspring can be neglected). The two quantities satisfy the equa- tions ˙x1(t) =w1Qx1(t)−e(t)x1(t), ˙xd(t) =w1(1−Q)x1(t)−e(t)xd(t),(2a) where w1is the ﬁtness of the sequences on the neutral network, and e(t) is the excess production (or mean ﬁtness in the population) e(t) =w1x1(t). Equa- tion (2a) can be integrated directly. We ﬁnd x1(t) =Qx1(0) x1(0) + [ Q−x1(0)]e−w1Qt. (3) In the steady state ( t→ ∞), this implies that the concentration of sequences on the network is equal to the eﬀective ﬁdelity Q, x1=Q=e−l(1−q)(1−ν). (4) Therefore, by measuring the decay of the concentra- tion of sequences on the neutral network as a function of the copy ﬁdelity q, we can estimate the population neutrality ν. Note that the above description of the evolving population is similar to the one presented by Reidys et al. (2001), with one important conceptual diﬀer- ence. The article by Reidys et al. (2001) was com- pleted before van Nimwegen et al.’s work was avail- able, and therefore it was not clear what their ef- fective ﬁdelity did actually relate to. Here, on the other hand, we know that Qdepends only on the copy ﬁdelity per nucleotide, q, and the average pop- ulation neutrality ν, which is independent of qand could be calculated exactly if the connection matrix of the neutral genotypes was known. We have measured the average equilibrium con- centration x1of sequences on the network for RNA3 secondary structure folding. RNA folding is a reli- able test case, and has been applied to a wide array of diﬀerent questions related to the dynamics of evo- lution (Fontana et al. 1993; Huynen et al. 1996; Fontana and Schuster 1998; Schuster and Fontana 1999; Ancel and Fontana 2000; Reidys et al. 2001). We simulated a ﬂow reactor using the Gillespie al- gorithm (Gillespie 1976), and performed the RNA folding with the Vienna package (Hofacker et al. 1994), version 1.3.1, which uses the parameters given by Walter et al. (1994). The carrying capacity was set to N= 1000 sequences, and the reactor was ini- tially ﬁlled with 1000 identical copies of a sequence that folded into a given target structure. Sequences folding into the target structure were replicating with rate one per unit time, and all other sequences with rate 10−6per unit time. We let the reactor equili- brate for 50 time steps, and then measured the aver- age concentration of correctly folding sequences over the next 150 time steps. Results for the two diﬀerent target structures de- picted in Fig. 1 are shown in Fig. 2. In both cases, we see a very clear exponential decay. Up to a muta- tion rate of 0 .05, which is quite high for the sequences of length l= 62 we are considering here, we cannot make out a signiﬁcant deviation from a straight line in the log-linear plot. This veriﬁes the applicability of our simple model to evolving RNA sequences. Note that our simulations also show a signiﬁcant diﬀerence in the eﬀective neutrality of the two structures, which will be of importance in the next section. Two Competing Quasispecies Analytical Model Above, we have established a simple description for a quasispecies residing on a single neutral net- work. In a similar fashion, we can treat the compe- tition of two quasispecies residing on separate net- works. We classify all sequences into three diﬀerent groups: sequences on network one, sequences on net- work two, and dead sequences (sequences that repli- cate much slower than sequences on either of the two networks, or do not replicate at all). We denote the respective relative concentrations by x1,x2, and xd. We make the further assumption that all sequences within a neutral network ihave the same probabil- ityQito mutate into another sequence on networki, and we neglect mutations from one network to the other. The probability to fall oﬀ of a network iis hence 1 −Qi. The diﬀerential equations for an inﬁ- nite population are then: ˙x1(t) =w1Q1x1(t)−e(t)x1(t), (5a) ˙x2(t) =w2Q2x2(t)−e(t)x2(t), (5b) ˙xd(t) =w1(1−Q1)x1(t) +w2(1−Q2)x2(t)−e(t)xd(t), (5c) where w1andw2are the ﬁtnesses of sequences on network one or two, respectively, and e(t) is the ex- cess production e(t) =w1x1(t)+w2x2(t). In order to solve Eq. (5), it is useful to introduce the matrix W= w1Q1 0 0 0 w2Q2 0 w1(1−Q1)w2(1−Q2) 0 .(6) We further need the exponential of W, which is given by exp(Wt) = ew1Q1t0 0 0 ew2Q2t0 1−Q1 Q1(ew1Q1t−1)1−Q2 Q2(ew2Q2t−1) 1  (7) Now, if we combine the concentrations x1,x2,xdinto a vector x= (x1,x2,xd)t, we ﬁnd x(t) = exp( Wt)·x(t)/[ˆe·exp(Wt)·x(0)] (8) withˆe:= (1,1,1). The denominator on the right- hand side of Eq. (8) corresponds to the cumulative excess production ecum(t) =/integraltextt 0e(t)dt, which is given by ecum(t) =ˆe·exp(Wt)·x(0) =x1(0) Q1(ew1Q1t+Q1−1) +x2(0) Q2(ew2Q2t+Q2−1) +xd(0).(9) The solution to Eq. (5) follows now as x1(t) =ew1Q1t ecum(t)x1(0), (10a) x2(t) =ew2Q2t ecum(t)x2(0), (10b) xd(t) =1 Q1Q2ecum(t)/bracketleftBig (ew1Q1t−1)(1−Q1)Q2x1(0) + (ew2Q2t−1)(1−Q2)Q1x2(0) +Q1Q2xd(0)/bracketrightBig . (10c)4 There exist two possible steady states. If w1Q1> w2Q2, then for t→ ∞ we have x1=Q1,x2= 0, xd= 1−Q1. Ifw1Q1< w2Q2, on the other hand, the steady state distribution if given by x1= 0,x2=Q2, xd= 1−Q2. The most interesting situation occurs when for a given w1andw2, the steady state depends on the mutation rate. This happens if w1> w2, but ν1< ν2, or vice versa. Namely, if we express Qias given in Eq. (1), we obtain from w1Q1=w2Q2the critical copy ﬁdelity qc= 1−ln(w2/w1) l(ν1−ν2). (11) Clearly, qccan only be smaller than one if either w1> w 2andν1< ν2or vice versa. Therefore, this is a necessary (though not suﬃcient) condition for the existence of two qualitatively diﬀerent steady states in diﬀerent mutational regimes. In the lan- guage of physics, the transition from one of the two steady state to the other is a ﬁrst order phase tran- sition (Stanley 1971). The transition is of ﬁrst order because the order parameter (which we can deﬁne to be either x1orx2) undergoes a discontinuous jump from a ﬁnite value to zero at the critical mutation rate. The two phases are not just a mathematical curios- ity, they have important biological interpretations. The phase in which the sequences with the larger wisurvive can be considered the “normal” selection regime, i.e., selection which favors faster replicating individuals. We will refer to this situation as the phase of “selection for replication speed”. In the other phase, however, the situation is exactly re- versed, and the sequences with the lower intrinsic replication rate wprevail. In this phase, the amount of neutrality (or the robustness against mutations) is more important, and we will consequently refer to this situation as the phase of “selection for ro- bustness”. In Fig. 3, we show two example phase diagrams. These diagrams demonstrate that the se- lection for robustness is not a pathological situation occurring only for extremely rare sets of parameters, but that in fact both phases have to be considered on equal grounds, none of them can be singled out as the more common one. In particular, as the ratio between w1andw2approaches unity, the selection for robustness becomes more and more important. Simulation ResultsAs in the case of a single quasispecies on a neutral network, we have tested our predictions with simu- lations of self-replicating RNA sequences in a ﬂow reactor. We assumed that sequences folding into ei- ther Fold 1 or 2 (Fig. 1) were replicating with rates w1= 1 and w2= 1.1, respectively, while all other folds had a vanishing replication rate. In all results presented below, we initialized the ﬂow reactor with 50% of the sequences folding into Fold 1, and the remaining sequences folding into Fold 2. Figure 4 shows a comparison between Eq. (10) and four example runs. Apart from ﬁnite size ﬂuctua- tions, which are to be expected in a simulation with N= 1000, the analytic expression predicts the actual population dynamics well. In Fig. 5, we present measurements of the concen- trations x1(t) and x2(t) as functions of the mutation rate 1−q, for a ﬁxed time t= 200. The points rep- resent results averaged over 25 independent simula- tions, and the lines stem from Eq. (10). In agreement with the predictions from our model, we observe two selection regimes, one in which the faster replicating sequences dominate, and one in which the sequences with the higher neutrality have a selective advantage. The transition between the two phases occurs in this particular case approximately at q= 0.98, and both the analytical model and the simulations agree well on this value. As is typical for a phase transition, the ﬂuctuations close to the transition point increase signiﬁcantly, and the time until either of the two qua- sispecies has gone extinct diverges (the latter point can be seen from the fact that close to the transition point, the disadvantageous fold is still present in a sizeable amount, while further away it has already vanished completely from the population). Figure 5 also shows that for very small popula- tions, the predictive value of the diﬀerential equation approach diminishes, presumably because the choice of a single eﬀective copy ﬁdelity Qis not justiﬁed anymore once a minimum population size has been reached. However, as long as we are dealing with pop- ulation sizes of several hundreds or more, our analyt- ical calculations predict the simulation results very well. Probability of Fixation In the previous subsection, we have established that selection acts on the product of replication rate5 wand ﬁdelity Q, rather than on the replication rate alone. In particular, for an appropriate choice of parameters, sequences with a lower replication rate can outcompete those with a higher replication rate. However, the competition experiments that we con- ducted in the previous section were unrealistic in so far that we assumed equal initial concentrations of the two competing types of sequences. A more real- istic assumption is that one type (the one with the lower product wiQi) dominates the population, while the second type is initially represented through only a single individual. The idea behind this scenario is of course that the second type (with higher product wiQi) has arisen through a rare mutation. The ques- tion in this context is whether the second type will be able to dominate the population, i.e., whether it will become ﬁxated. In a standard population genetics scenario, the an- swer to the above question is simple. If two sequences replicate with w1andw2, respectively, and mutations between the two sequences can be neglected, then a single sequence of type 2 ( w2> w1) will become ﬁx- ated in a background of sequences of type 1 with probability π= 1−e−2s≈2s, where s=w2/w1−1 is the selective advantage of the newly introduced sequence type (Haldane 1927; Kimura 1964; Ewens 1979). Note, however, that this celebrated result is only correct for a generational model with discrete time steps. In a continuous time model, the equiva- lent result reads π=s/(1 +s). This formula follows from the solution to the problem of the Gambler’s Ruin (Feller 1968; Lenski and Levin 1985) when tak- ing the limit of a large population size. Here, we are not dealing with individual sequences replicating with rate wi, but rather with quasispecies that grow with rate wiQi. A naive way to calculate the ﬁxation probability in this case is simply to re- place wiwithwiQiin the expression for the selective advantage, and hope that the result is correct. How- ever, it is not clear from the outset that this approach will work, because the factor Qidepends on the as- sumption that a fully developed quasispecies with the appropriate mean neutrality is already present. A single sequence struggling for ﬁxation does not satisfy this condition. Therefore, the actual ﬁxation prob- ability might deviate from the one thus calculated, in particular in circumstances in which a sequence with smaller replication rate is supposed to overtakean established quasispecies of sequences with higher replication rate. We performed ﬁxation experiments in both the “se- lection for replication speed” and the “selection for robustness” phase, in order to clarify whether the naive approach works. In both phases, we allowed a population of size N= 1000 to equilibrate, and then introduced a single sequence of the supposedly advantageous type. After 500 time steps, we deter- mined whether the advantageous type had vanished from the population or grown to a signiﬁcant propor- tion. By repeating this procedure 100 times, we ob- tained an estimate for the probability of ﬁxation. As in the previous section, we used w1= 1 and w2= 1.1. In Fig. 6, we compare our simulation results to the predicted ﬁxation probability π=s/(1 +s). Within the accuracy of our results, both agree well. This is particularly interesting for mutation rates above 0.02, where we introduce a sequence of lower replica- tion rate into a background of faster replicating se- quences. The increased neutrality of the introduced sequence is suﬃcient to let it rise to ﬁxation in a sig- niﬁcant proportion of cases. Moreover, the product wiQiis the sole determinant of the ﬁxation probabil- ity. Whether the value of the product wiQicomes mainly from the intrinsic growth rate wiof the se- quences or from the eﬀective ﬁdelity Qidoes not have an observable inﬂuence on the dynamics. Discussion The good agreement between our analytical model and our simulation results demonstrates that RNA sequences evolving on a neutral network of identi- cal secondary structure folds are well described by only two parameters, their intrinsic replication rate wand their eﬀective copy ﬁdelity Q. In the partic- ular context of two competing distinct folds, we ﬁnd furthermore that only the product of wandQis of importance. Indeed, it follows from Eq. (10) that the ratio between x1(t) and x2(t) depends only on the respective products of wandQ, but not on the individual values themselves. Unlike the intrinsic replication rate w, which is a property of the individual, the eﬀective ﬁdelity Qis a group property, as it is given by the average over all sequences in the population of the probability not to “fall oﬀ” the neutral network. Thus, in the regime in which Qdominates the evolutionary dynamics (the6 phase of selection for robustness in Fig. 3), the evo- lutionary success of an individual sequence depends strongly on the properties of the group it belongs to. In other words, we ﬁnd that selection acts on the whole group of mutants, rather than on individuals, despite the absence of standard factors supporting group selection such as spatial subdivision of the pop- ulation (Wilson 1979), altruistic behavior, parental care (Maynard Smith 1993), or mutual catalytic sup- port (Alves et al. 2001). Here, a sequence with a comparatively high neutrality embedded into a neu- tral network with a poor overall connection density will be at a disadvantage with respect to a sequence with a comparatively low neutrality that is, however, part of a neutral network with high connection den- sity. The overall higher ﬁdelity of a population on the second network results in a larger fraction of se- quences that actually reside on the network, which in turn increases the chance that a particular se- quence will be generated as mutant oﬀspring from some other sequence. Moya et al. (2000) noted that this type of group selection should follow from the quasispecies equations, and that populations under this type of selection would be best described by an eﬀective group replication rate r. In the present work, we have shown that this is indeed the case, and we can also derive r(which is simply r=wQ) from the quasispecies equations. Namely, the fact that the population neutrality ν(which determines Q) is given by the largest eigenvalue of the connection matrix of neutral genotypes is a direct consequence of the qua- sispecies equations (van Nimwegen et al. 1999). Schuster and Swetina (1988) were the ﬁrst to point out that at high mutation rates, the quasispecies around the highest peak in the landscape can disap- pear. They focused on situations in which the highest and the second-highest peak in a landscape were of almost equal height, while the immediate mutational neighborhood of the second peak was less deleteri- ous than the one of the ﬁrst peak. As a consequence, their results seemed to imply that the phase of ’selec- tion of robustness’ was only important in the case of very similar peaks. Our results, on the other hand, show that the diﬀerence in peak hight can be dra- matic, if balanced by an equally dramatic diﬀerence in robustness. While our analytical results apply strictly speaking only to inﬁnite populations, we have seen that in sim-ulations for population sizes as small as N= 500, the diﬀerential equation approach works well. Moreover, in our experiments on the probability of ﬁxation, we have seen that even very small numbers of the advan- tageous group (in the extreme only a single sequence) can rise to ﬁxation, despite their intrinsic replication rate being smaller than that of the currently dom- inating group. This result seems somewhat unin- tuitive at ﬁrst, but can be easily understood. The most important aspect of every ﬁxation event is the very ﬁrst replication of the new genotype, and the smaller its selective advantage, the more likely it is not to replicate even once. Now, if a new mutant with a poor replication rate wnewbut high eﬀective ﬁdelity Qnewarises in a population that is dominated by sequences with large intrinsic replication rate, we would intuitively assume that the mutant will hardly ever replicate even once, and therefore will never get a chance to employ its superior ﬁdelity. However, this is not correct if the eﬀective ﬁdelity of the dom- inating sequences, Qdom, is low. From Eq. (4), we ﬁnd that the concentration of sequences that actually replicate is given by Qdom. Therefore, even though the sequences that replicate do so at a high rate, the actual number of births that occur is small, compa- rable to the one in a population in which all individ- uals reproduce with rate wdomQdom. Therefore, the newly introduced genotype is relatively safe from be- ing washed out prematurely, and ﬁxation takes place at the predicted rate. Conclusions We have demonstrated that for a population in a landscape where neutral mutants abound, the prod- uct of intrinsic replication rate wand eﬀective copy- ﬁdelity Qis being maximized under selection, rather than the intrinsic replication rate alone. This ob- servation has led to the natural distinction between two modes of selection, one in which intrinsic repli- cation rate is favored, and one in which robustness (high Q) is more important. In the latter phase, the success of a single sequence depends strongly on the mutant cloud the sequence belongs to. Our results thus demonstrate that the unit of selection in molec- ular evolution is indeed the quasispecies, as proposed by Eigen and Schuster (1979), and not the individual replicating sequence. 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What is a quasispecies? TREE 7 , 118–121. Ofria, C. and C. Adami. 2001. Evolution of genetic organization in digital organisms. In L. Landweber and E. Winfree (Eds.), Evolu- tion as Computation , pp. 167. Springer, New York. Reidys, C., C. V. Forst, and P. Schuster. 2001. Replication and mutation on neutral networks. Bull. Math. Biol. 63 , 57–94. Schuster, P. and W. Fontana. 1999. Chance and ne- cessity in evolution: lessons from RNA. Physica D 133 , 427–452. Schuster, P. and J. Swetina. 1988. Stationary mu- tant distributions and evolutionary optimiza- tion.Bull. Math. Biol. 50 , 635–660. Stanley, E. H. 1971. Introduction to Phase Transi- tions and Critical Phenomena . Oxford Univer- sity Press, New York. Steinhauer, D. A., J. C. de la Torre, E. Meier, and J. J. Holland. 1989. Extreme heterogeneity in populations of vesicular stomatitis virus. J. of Virology 63 , 2072–2080. van Nimwegen, E., J. P. Crutchﬁeld, and M. Huy- nen. 1999. Neutral evolution of mutational ro- bustness. Proc. Natl. Acad. Sci. USA 96 , 9716– 9720. Walter, A. E., D. H. Turner, J. Kim, M. H. Lyttle, P. M¨ uller, D. H. Mathews, and M. Zuker. 1994. Coaxial stacking of helixes enhances binding of oligoribonucleotides and improves predictions of RNA folding. Proc. Natl. Acad. Sci. USA 91 , 9218–9222. Wilke, C. O., C. Ronnewinkel, and T. Martinetz. 2001. Dynamic ﬁtness landscapes in molecular evolution. Phys. Rep. . in press.Wilson, D. S. 1979. The Natural Selection of Popu- lations & Communities . Benjamin-Cummings, Menlo Park.9 Figure 1: The two diﬀerent folds used in this study. Both consist of the same number of base pairs ( l= 62), but Fold 1 has a higher neutrality ( ν= 0.442) than Fold 2 ( ν= 0.366). See also Fig. 2. Figure 2: Decay of the steady state concentration x1as a function of 1 −qfor two example secondary structures. The solid and the dashed line are given by exp[ −l(1−q)(1−νi)] with l= 62. The values forν1andν2have been obtained from a ﬁt of this expression to the measured data (shown as points with bars indicating the standard error). Figure 3: Typical phase diagram following from Eq. 11. We used l= 100, w2= 1, and ν1= 0.5, as well as ν2= 0.6 in graph a) and w1= 1.5 in graph b). Figure 4: Concentrations x1(t) and x2(t) as functions of the time tfor a copy ﬁdelity of q= 0.99. The thick lines represent the analytic predictions from Eqs. (10a) and (10b), and the thin lines stem from simulations with N= 1000. Figure 5: Concentrations x1(200) (dashed lines) and x2(200) (solid lines) as functions of the per-nucleotide mutation rate 1 −q. The lines represent the analytic predictions. The points represent the average over 25 independend simulation runs each, with bars indi- cating the standard error. We performed the simula- tions with four diﬀerent population sizes, N= 5000 (a),N= 1000 (b), N= 500 (c), and N= 100 (d). The initial concentrations in all simulations were x1(0) = x2(0) = 0 .5,xd(0) = 0. Figure 6: Probability of ﬁxation as a function of the mutation rate. Below 1 −q= 0.02, we are looking at the probability of ﬁxation of a single sequence of type 2 in a full population of sequences of type 1. Above 1 −q= 0.02, we are considering the reversed conﬁguration. The solid and dashed line represent the analytical prediction π=s/(1 +s), the points stem from simulations (bars indicate standard error).10 Figure 1: UCCGACGG GGUUGGA U C U A AAU U U G CA C GGUCAGCGAACAAAUAGCGGAGGGG UUGCUUAAUG G C C A GG A A GC G C G UG C G C CUA A UCG AAAGUCGGCGAAACGCACUGUUGGCAAAAUCUAAU G Fold 1 Fold 211 Figure 2: 1 0.5 0.1 00.010.020.030.040.05Rel. concentration x1 Mutation rate 1- qFold 1, ν1=0.442 Fold 2, ν2=0.36612 Figure 3: /CU/D3/D6 /D6/D3/CQ/D9/D7/D8/D2/CT/D7/D7/D7/CT/D0/CT /D8/CX/D3/D2 /D7/CT/D0/CT /D8/CX/D3/D2 /CU/D3/D6 /D6/CT/D4/D0/CX /CP/D8/CX/D3/D2 /D7/D4 /CT/CT/CS /D7/CT/D0/CT /D8/CX/D3/D2 /CU/D3/D6 /D6/CT/D4/BA /D7/D4 /CT/CT/CS /BC/BC/BA/BC/BE /BC/BA/BC/BG /BC/BA/BC/BI /BC/BA/BC/BK /BC/BA/BD/BC /BC/BA/BD/BE /BC/BA/BH /BD /BD/BA/BH /BE /BE/BA/BH /BF /BF/BA/BH /C5/D9/D8/CP/D8/CX/D3/D2 /D6/CP/D8/CT /BD /A0 /D5 /CA/CT/D4/D0/CX /CP/D8/CX/D3/D2 /D6/CP/D8/CT /DB /BD/D7/CT/D0/CT /D8/CX/D3/D2 /CU/D3/D6 /D6/D3/CQ/D9/D7/D8/D2/CT/D7/D7 /D7/CT/D0/CT /D8/CX/D3/D2 /CU/D3/D6 /D6/CT/D4/D0/CX /CP/D8/CX/D3/D2 /D7/D4 /CT/CT/CS /BC/BC/BA/BC/BE /BC/BA/BC/BG /BC/BA/BC/BI /BC/BA/BC/BK /BC/BA/BD/BC /BC/BA/BD/BE /BC/BA/BG /BC/BA/BH /BC/BA/BI /BC/BA/BJ /BC/BA/BK /BC/BA/BL /BD /C5/D9/D8/CP/D8/CX/D3/D2 /D6/CP/D8/CT /BD /A0 /D5 /C6/CT/D9/D8/D6/CP/D0/CX/D8 /DD /AN /BD b) a)13 Figure 4: /DC /BE/B4 /D8 /B5 /DC /BD/B4 /D8 /B5 /BC/BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /BC /BH/BC /BD/BC/BC /BD/BH/BC /BE/BC/BC /CA/CT/D0/BA /D3/D2 /CT/D2 /D8/D6/CP/D8/CX/D3/D2/D7 /CC/CX/D1/CT /D814 Figure 5: 00.20.40.60.81 00.010.020.030.040.05Rel. concentration Mutation rate 1- q00.20.40.60.81 00.010.020.030.040.05Rel. concentration Mutation rate 1- q00.20.40.60.81 00.010.020.030.040.05Rel. concentration Mutation rate 1- q00.20.40.60.81 00.010.020.030.040.05Rel. concentration Mutation rate 1- q N=500 N=100N=1000 N=500015 Figure 6: 00.10.20.3 00.010.020.030.040.05Fixation prob. π Mutation rate 1- qFixation of Fold 2 Fixation of Fold 1
arXiv:physics/0103023v1 [physics.gen-ph] 9 Mar 2001Quantization of Chaos for Particle Motion H. Y. Cui∗ Department of Applied Physics Beijing University of Aeronautics and Astronautics Beijing, 100083, China (May 27, 2013) We propose a formalism which makes the chaos to be quan- tized. Quantum mechanical equation is derived for describi ng the chaos for a particle moving in electromagnetic ﬁeld. Consider a particle of charge qand massmmoving a electromagnetic ﬁeld. Suppose the particle revolves about a nucleus and runs into a chaos state. It is con- venient to consider the bunch of its paths to be a ﬂow characterized by a 4-velocity ﬁeld u(x1,x2,x3,x4=ict) in a Cartesian coordinate system (in a laboratory frame of reference). The particle will be aﬀected by the 4-force due to in electromagnetic interaction. According to rel- ativistic Newton’s second law, the motion of the particle satisﬁes the following governing equations mduµ dτ=qFµνuν (1) uµuµ=−c2(2) whereFµνis the 4-curl of electromagnetic vector poten- tialA. Since the reference frame is a Cartesian coordinate system whose axes are orthogonal to one another, there is no distinction between covariant and contravariant com- ponents, only subscripts need be used. Here and below, summation over twice repeated indices is implied in all case, Greek indices will take on the values 1,2,3,4, and regarding the mass mas a constant. Eq.(1) and Eq.(2) stand at every point for the particle in the ﬁeld. As is mentioned above, the 4-velocity ucan be regarded as a 4-velocity vector ﬁeld, then duµ dτ=∂uµ ∂xν∂xν ∂τ=uν∂νuµ (3) qFµνuν=quν(∂µAν−∂νAµ) (4) Substituting them back into Eq.(1), and re-arranging their terms, we obtain uν∂ν(muµ+qAµ) =uν∂µ(qAν) =uν∂µ(muν+qAν)−uν∂µ(muν) =uν∂µ(muν+qAν)−1 2∂µ(muνuν) =uν∂µ(muν+qAν)−1 2∂µ(−mc2) =uν∂µ(muν+qAν) (5) Using the notation Kµν=∂µ(muν+qAν)−∂ν(muµ+qAµ) (6)Eq.(5) is given by uνKµν= 0 (7) BecauseKµνcontains the variables ∂µuν,∂µAν,∂νuµ and∂νAµwhich are independent from uν, then a solution satisfying Eq.(7) is of Kµν= 0 (8) ∂µ(muν+qAν) =∂ν(muµ+qAµ) (9) The above equation allows us introduce a potential func- tion Φ in mathematics, further set Φ = −i¯hlnψ, we ob- tain a very important equation (muµ+qAµ)ψ=−i¯h∂µψ (10) We think it as an extended form of the relativistic New- ton’s second law in terms of 4-velocity ﬁeld. ψrepre- senting the wave nature may be a complex mathematical function, its physical meanings will be determined from experiments after the introduction of the Planck’s con- stant ¯h. Multiplying the two sides of the following familiar equation by ψ −m2c4=m2uµuµ (11) which stands at every points in the 4-velocity ﬁeld, and using Eq.(10), we obtain −m2c4ψ=muµ(−i¯h∂µ−qAµ)ψ = (−i¯h∂µ−qAµ)(muµψ)−[−i¯hψ∂ µ(muµ)] = (−i¯h∂µ−qAµ)2ψ−[−i¯hψ∂ µ(muµ)] (12) According to the continuity condition for the particle mo- tion ∂µ(muµ) = 0 (13) we have −m2c4ψ= (−i¯h∂µ−qAµ)2ψ (14) Its is known as the Klein-Gordon equation. On the condition of non-relativity, the Schrodinger equation can be derived from the Klein-Gordon equation [2](P.469). However, we must admit that we are careless when we use the continuity condition Eq.(13), because, from Eq.(10) we obtain 1∂µ(muµ) =∂µ(−i¯h∂µlnψ−qAµ) =−i¯h∂µ∂µlnψ(15) where we have used the Lorentz gauge condition. Thus from Eq.(11) to Eq.(12) we obtain −m2c4ψ= (−i¯h∂µ−qAµ)2ψ+ ¯h2ψ∂µ∂µlnψ(16) This is of a perfect wave equation for describing accu- rately the motion of the particle. In other wards, The Klein-Gordon equation is ill for using the mistaken con- tinuity condition Eq.(13). Comparing with the Dirac equation result, we ﬁnd that the last term of Eq.(16) corresponds to the spin eﬀect of particle. In the follow- ing we shall show the Dirac equation from Eq.(10) and Eq.(11). In general, there are many wave functions which satisfy Eq.(10) for the particle, these functions and correspond- ing momentum components are denoted by ψ(j)and Pµ(j) =muµ(j), respectively, where j= 1,2,3,...,N, then Eq.(11) can be given by 0 =Pµ(j)Pµ(j)ψ2(j) +m2c4ψ2(j) =δµνPµ(j)ψ(j)Pν(j)ψ(j) +mc2ψ(j)mc2ψ(j) = (δµν+δνµ)Pµ(j)ψ(j)Pν(j)ψ(j)(µ≥ν) +mc2ψ(j)mc2ψ(j) = 2δµνPµ(j)ψ(j)Pν(j)ψ(j)(µ≥ν) +mc2ψ(j)mc2ψ(j) = 2δµνδjkδjlPµ(k)ψ(k)Pν(l)ψ(l)(µ≥ν) +δjkδjlmc2ψ(k)mc2ψ(l) (17) whereδis the Kronecker delta function, j,k,l = 1,2,3,...,N . Here, specially, we do not take jsum over; Prepresents momentum, not operator. Suppose there are two matrices aandbwhich satisfy aµjkaνjl+aνjkaµjl= 2δµνδjkδjl (18) aµjkbjl+bjkaµjl= 0 (19) bjkbjl=δjkδjl (20) then Eq.(17) can be rewritten as 0 = (aµjkaνjl+aνjkaµjl)Pµ(k)ψ(k)Pν(l)ψ(l)(µ≥ν) +(aµjkbjl+bjkaµjl)Pµ(k)ψ(k)mc2ψ(l) +bjkbjlmc2ψ(k)mc2ψ(l) = [aµjkPµ(k)ψ(k) +bjkmc2ψ(k)] ·[aνjlPν(l)ψ(l) +bjlmc2ψ(l)] = [aµjkPµ(k)ψ(k) +bjkmc2ψ(k)]2(21) Consequently, we obtain a wave equation: aµjkPµ(k)ψ(k) +bjkmc2ψ(k) = 0 (22) There are many solutions for aandbwhich satisfy Eq.(18-20), we select a familiar set of aandbas [2]:N= 4 (23) an= [anµν] =/bracketleftbigg 0σn σn0/bracketrightbigg =αn (24) a4= [a4µν] =I (25) b= [bjk] =/bracketleftbigg I0 0−I/bracketrightbigg =β (26) whereαnare the Pauli spin matrices, n= 1,2,3. Sub- stituting them into Eq.(22), we obtain [(−i¯h∂4−qA4) +αn(−i¯h∂n−qAn) +βmc2]ψ= 0 (27) whereψis an one-column matrix about ψ(k). Eq.(27) is known as the Dirac equation. Of course, on the condition of non-relativity, the Schrodinger equation can be derived from the Dirac equa- tion [2](P.479). It is noted that Eq.(27), Eq.(22), Eq.(17) and Eq.(16) are equivalent despite they have the diﬀerent forms, be- cause they all originate from Eq.(10) and Eq.(11). It follows from Eq.(10) that the path of a particle is analogous to ”lines of electric force” in 4-dimensional space-time. In the case that the Klein-Gordon equation stands, i.e. Eq.(13) stands, at any point, the path can have but one direction (i.e. the local 4-velocity direction ), hence only one path can pass through each point of the space-time. In other words, the path never intersects it- self when it winds up itself into a cell about a nucleus. No path originates or terminates in the space-time. But, in general, the divergence of the 4-velocity ﬁeld does not equal to zero, as indicated in Eq.(15), so the Dirac equa- tion would be better than the Klein-Gordon equation in accuracy. The condition of the appearance of spin structure for the chaos is that Eq.(15) is un-negligeable. The mecha- nism profoundly accounts for the quantum wave natures such as spin eﬀect [4] [5]. Its interpretation in physical terms remains to be discussed further in the future. In conclusion, in terms of the 4-velocity ﬁeld, the rela- tivistic Newton’s second law can be rewritten as a wave ﬁeld equation. By this discovery, the Klein-Gordon equa- tion, Schrodinger equation and Dirac equation can be derived from the Newtonian mechanics on diﬀerent con- ditions, respectively. Quantum mechanical equation is dominant in describing chaos for a particle moving in electromagnetic ﬁeld. ∗E-mail: hycui@public.fhnet.cn.net [1] E. G. Harris, Introduction to Modern Theoretical Physic s, Vol.1&2, (John Wiley & Sons, USA, 1975). [2] L. I. Schiﬀ, Quantum Mechanics, third edition, (McGraw- Hill, USA, 1968). 2[3] H. Y. Cui, College Physics (A monthly edited by Chi- nese Physical Society in Chinese), ”An Improvement in Variational Method for the Calculation of Energy Level of Helium Atom”, 4, 13(1989). [4] H. Y. Cui, eprint, physcis/0102073, (2001). [5] H. Y. Cui, eprint, quant-ph/0102114, (2001). 3
arXiv:physics/0103024v1 [physics.gen-ph] 9 Mar 2001Duality and Cosmology B.G. Sidharth B.M. Birla Science Centre, Adarshnagar, Hyderabad - 500 063 , India Abstract In some recent theories including Quantum SuperString theo ry we encounter duality - it arises due to a non commutative geomet ry which in eﬀect adds an extra term to the Heiserberg Uncertainity Pr inciple. The result is that the micro world and the macro universe seem to be linked. We show why this is so in the context of a recent cosmol ogical model and a physical picture emerges in the context of the Fey nman- Wheeler formulation of interactions. 1 Introduction Nearly a century ago several Physicists including Lorentz, Poincare and Abraham amongst others tackled unsuccessfully the problem of the extended electron[1, 2]. An extended electron appeared to contradic t Special Relativ- ity, while on the other hand, the limit of a point particle lea d to inexplicable inﬁnities. Dirac ﬁnally formulated an equation in which the physically rele- vant or ”renormalized” mass was ﬁnite and consisted of the ba re mass and the electromagnetic mass which become inﬁnite in the limit o f point parti- cles, no doubt, but the inﬁnities cancel one another. This ap proach lead to non-causal eﬀects, which were circumvented by a formalism o f Feynam and Wheeler, in which the interaction of a charge with the rest of the universe was considered, and also not just the point charge, but its ne ighbourhood had to be taken into account. These inﬁnities persisted for many decades. Infact the Heis enberg Uncertain- ity Principle straightaway leads to inﬁnities in the limit o f spacetime points. It was only through the artiﬁce of renormalization that ’t Ho oft could ﬁnally 1circumvent this vexing problem, in the 1970s. Nevertheless it has been realized that the concept of spacet ime points is only approximate[3, 4, 5, 6, 7]. We are beginning to realize that i t may be more meaningful to speak in terms of spacetime foam, strings, bra nes, non com- mutative geometry, fuzzy spacetime and so on[8]. This is wha t we will now discuss. 2 Duality We consider the well known theory of Quantum SuperStrings an d also an ap- proach in which an electron is considered to be a Kerr-Newman Black Hole, with the additional input of fuzzy spacetime. As is well known, String Theory originated from phenomenolo gical consider- ations in the late sixties through the pioneering work of Ven eziano, Nambu and others to explain features like the s-t channel dual reso nance scattering and Regge trajectories[9]. Originally strings were concei ved as one dimen- sional objects with an extension of the order of the Compton w avelength, which would fudge the point vertices of the s-t channel scatt ering graphs, so that both would eﬀectively correspond to one another (Cf.re f.[9]). The above strings are really Bosonic strings. Raimond[10], Scherk[11] and others laid the foundation for the theory of Fermionic strin gs. Essentially the relativistic Quantized String is given a rotation, when we get back the equation for Regge trajectories, J≤(2πT)−1M2+a0¯hwith a0= +1(+2)for the open (closed) string (1) When a0= 1 in (1) we have gauge Bosons while a0= 2 describes the gravi- tons. In the full theory of Quantum Super Strings, or QSS, we a re essentially dealing with extended objects rotating with the velocity of light, rather like spinning black holes. The spatial extention is at the Planck scale while fea- tures like extra space time dimensions which are curled up in the Kaluza Klein sense and, as we will see, non commutative geometry app ear[12, 13]. We next observe that it is well known that the Kerr Newman of ch arged spinning Black Hole itself mimics the electron remarkably w ell including the purely Quantum Mechanical anomalous g= 2 factor[14]. The problem is that there would be a naked singularity, that is the radius wo uld become 2complex, r+=GM c2+ıb, b≡/parenleftBiggG2Q2 c8+a2−G2M2 c4/parenrightBigg1/2 (2) where ais the angular momentum per unit mass. This problem has been studied in detail by the author in recen t years[15, 16]. Indeed it is quite remarkable that the position coordinate o f an electron in the Dirac theory is non Hermitian and mimics equation (2), be ing given by x= (c2p1H−1t+a1) +ı 2c¯h(α1−cp1H−1)H−1, (3) where the imaginary parts of (2) and (3) are both of the order o f the Comp- ton wavelength. The key to understanding the unacceptable imaginary part wa s given by Dirac himself[17], in terms of zitterbewegung. The point is that according to the Heisenberg Uncertainity Principle, space time point s themselves are not meaningful- only space time intervals have meaning, and we are really speaking of averages over such intervals, which are atleast of the order of the Compton scale. Once this is kept in mind, the imaginary term d isappears on averaging over the Compton scale. In this formulation, the mass and charge of the electron aris es due to zitter- bewegung eﬀects at the Compton scale[15, 16]. These masses a nd charges are renormalized in the sense of the Dirac mass in the classical t heory, alluded to in section 1. Indeed, from a classical point of view also, in the extreme re lativistic case, as is well known there is an extension of the order of the Compton wavelength, within which we encounter meaningless negative energies[1 8]. With this pro- viso, it has been shown that we could think of an electron as a s pinning Kerr Newman Black Hole. This has received independent support fr om the work of Nottale[19]. We are thus lead to the picture where there is a cut oﬀ in space t ime inter- vals. In the above two scenarios, the cut oﬀ is at the Compton scale ( l, τ) the Planck scale being a special case for the Planck mass. Such di screte space time models compatible with Special Relativity have been st udied for a long time by Snyder and several other scholars[20, 21, 22]. In thi s case it is well known that we have the following non commutative geometry [x, y] = (ıa2/¯h)Lz,[t, x] = (ıa2/¯hc)Mx, 3[y, z] = (ıa2/¯h)Lx,[t, y] = (ıa2/¯hc)My, (4) [z, x] = (ıa2/¯h)Ly,[t, z] = (ıa2/¯hc)Mz, where ais the minimum natural unit and Lx, Mxetc. have their usual sig- niﬁcance. Moreover in this case there is also a correction to the usual Q uantum Me- chanical commutation relations, which are now given by [x, px] =ı¯h[1 + (a/¯h)2p2 x]; [t, pt] =ı¯h[1−(a/¯hc)2p2 t]; [x, py] = [y, px] =ı¯h(a/¯h)2pxpy; (5) [x, pt] =c2[px,t] =ı¯h(a/¯h)2pxpt; etc. where pµdenotes the four momentum. In the Kerr Newman model for the electron alluded to above (or generally for a spinning sphere of spin ∼¯hand of radius l),Lxetc. reduce to the spin ¯h 2of a Fermion and the commutation relations (4) and (5) reduce to [x, y]≈0(l2),[x, px] =ı¯h[1 +βl2],[t, E] =ı¯h[1 +τ2] (6) where β= (px/¯h)2and similar equations. Interestingly the non commutative geometry given in (6) can be shown to lead to the representation of Dirac matrices and the Dirac eq uation[23]. From here we can get the Klein Gordon equation, as is well known[24 , 25], or al- ternatively we deduce the massless string equation. This is also the case with superstrings where Dirac spinors a re introduced, as indicated above. Infact in QSS also we have equations math ematically identical to the relations (6) containing momenta (Cf.ref. [13]). This, which implies (4), can now be seen to be the origin of non-commutati vity. The non commutative geometry and fuzzyness is contained in ( 6). Infact fuzzy spaces have been investigated in detail by Madore and o thers[26, 27], and we are lead back to the equation (6). The fuzzyness which i s closely tied up with the non commutative feature is symptomatic of th e breakdown of the concept of the spacetime points and point particles at small scales or high energies. As has been noted by Snyder, Witten, and sev eral other scholars, the divergences encountered in Quantum Field The ory are symp- tomatic of precisely such an extrapolation to spacetime poi nts and which 4necessitates devices like renormalization. As Witten poin ts out[28], ”in de- veloping relativity, Einstein assumed that the space time c oordinates were Bosonic; Fermions had not yet been discovered!... The struc ture of space time is enriched by Fermionic as well as Bosonic coordinates .” A related concept, which one encounters also in String Theor y is Duality. Infact the relation (6) leads to, ∆x∼¯h ∆p+α′∆p ¯h(7) where α′=l2, which in Quantum SuperStrings Theory ∼10−66. Witten has wondered about the basis of (7), but as we have seen, it is a con sequence of (6). This is an expression of the duality relation, R→α′/R This is symptomatic of the fact that we cannot go down to arbit rarily small spacetime intervals, below the Planck scale. There is an interesting meaning to the duality relation aris ing from (7) in the context of the Kerr-Newman Black Hole formulation. Whil e it appears that the ultra small is a gateway to the macro cosmos, we could look at it in the following manner. The ﬁrst term of the relation (7) which is the usual Heisenberg Uncertainity relation is supplemented by the se cond term which refers to the macro cosmos. Let us consider the second term in (7). We write ∆ p= ∆Nmc, where ∆ Nis the Uncertainity in the number of particles, N, in the universe. Also ∆ x=R, the radius of the universe where R∼√ Nl, (8) the famous Eddington relationship. It should be stressed th at the otherwise emperical Eddington formula, arises quite naturally in a Br ownian charac- terisation of the universe as has been pointed out earlier (C f. for example ref.[5]). Put simply (8) in the Random Walk equation We now get, using (2), ∆N=√ N 5Substituting this in the time analogue of the second term of ( 7), we immedi- ately get, Tbeing the age of the universe, T=√ Nτ (9) In the above analysis, including the Eddington formula (8), landτare the Compton wavelength and Compton time of a typical elementary particle, namely the pion. The equation for the age of the universe is al so correctly given above. Infact in the closely related model of ﬂuctuati onal cosmology (Cf. for example ref.[29]) all of the Dirac large number coin cidences as also the mysterious Weinberg formula relating the mass of the pio n to the Hubble constant, follow as a consequence, and are not emperical. In this formulation, in a nutshell,√ Nparticles are ﬂuctuationally created within the time τ, so that, dN dt=√ N τ(10) which leads to (9) (and (8)). Next use of the well known formula, ( M=Nm, M being the mass of the universe, and mthe pion mass) R≈GM/c2, gives on diﬀerentiation and use of (10) the Hubble law, with H=c l1√ N≈Gm3c ¯horm=/parenleftBigg¯hH Gc/parenrightBigg1/3 (11) (11) gives the supposedly mysterious and empirical Weinber g formula con- necting the pion mass to the Hubble constant. Using (11) we can deduce that there can be a cosmological cons tant Λ such that, Λ≤0(H2) Recent observations conﬁrm this ever expanding and possibl y accelerating feature of the universe[30]. All these relations relating l arge scale parameters to microphysical constants were shown to be symptomatic of w hat has been called, stochastic holism (Cf. also ref.[31]), that is a mic ro-macro connection with a Brownian or stochastic underpinning. Duality, or equ ivalently, rela- tion (7) is really an expression of this micro-macro link. 63 The Dirac and Feynman - Wheeler Formu- lations To appreciate this concept of holism in a more physical sense , we return to the classical description of the electron alluded to right a t the beginning. We will discuss very brieﬂy the contributions of Dirac, Feynma n and Wheeler. This was built upon the earlier work of Lorentz, Abraham, Fok ker and others. Our starting point is the so called Lorentz-Dirac equation[ 2]: maµ=Fµ in+Fµ ext+ Γµ(12) where Fµ in=e cFµv invv and Γµis the Abraham radiation reaction four vector related to the self force and, given by Γµ=2 3e2 c3(˙aµ−1 c2aλaλvµ) (13) Equation (12) is the relativistic generalisation for a poin t electron of an earlier equation proposed by Lorentz, while equation (13) i s the relativisitic generalisation of the original radiation reaction term due to energy loss by radiation. It must be mentioned that the mass min equation (12) consists of a neutral mass and the original electromagnetic mass of Lore ntz, which latter does tend to inﬁnity as the electron shrinks to a point, but, t his is absorbed into the neutral mass. Thus we have the forerunner of renorma lisation in quantum theory. There are three unsatisfactory features of the Lorentz-Dir ac equation (12). Firstly the third derivative of the position coordinate in ( 12) through Γµgives a whole family of solutions. Except one, the rest of the solut ions are run away - that is the velocity of the electron increases with time to t he velocity of light, even in the absence of any forces. This energy can be th ought to come from the inﬁnite self energy we get when the size of the electr on shrinks to zero. If we assume adhoc an asymptotically vanishing accele ration then we get a physically meaningful solution, though this leads to a second diﬃculty, that of violation of causality of even the physically meanin gful solutions. Let us see this brieﬂy. 7For this, we notice that equation (12) can be written in the fo rm[2], maµ(τ) =/integraldisplay∞ 0Kµ(τ+ατ0)e−αdα (14) where Kµ(τ) =Fµ in+Fµ ext−1 c2Rvµ, τ0≡2 3e2 mc3(15) and α=τ′−τ τ0, where τdenotes the time and Ris the total radiation rate. It can be seen that equation (14) diﬀers from the usual equati on of Newtonian Mechanics, in that it is non local in time. That is, the accele ration aµ(τ) depends on the force not only at time τ, but at subsequent times also. Let us now try to characterise this non locality in time. We obser ve that τ0given by equation (15) is the Compton time ∼10−23secs.So equation (14) can be approximated by maµ(τ) =Kµ(τ+ξτ0)≈Kµ(τ) (16) Thus as can be seen from (16), the Lorentz-Dirac equation diﬀ ers from the usual local theory by a term of the order of 2 3e2 c3˙aµ(17) the so called Schott term. So, the non locality in time is within intervals ∼τ0, the Compton time ex- actly what we encountered in section 2. It must also be reiterated that the Lorentz-Dirac equation m ust be supple- mented by the asymptotic condition of vanishing accelerati on in order to be meaningful. That is, we have to invoke not just the point elec tron, but also distant regions into the future as boundary conditions. Finally it must be borne in mind that the four vector Γµgiven in (13) can also be written as Γµ≡e 2c(Fµv ret−Fµv adv)vv (18) 8In (18) we can see the presence of the advanced or acausal ﬁeld which has been considered unsatisfactory. Infact this term, as is wel l known directly leads to the Schott term (17). Let us examine this non local fe ature. As is known, considering the time component of the Schott term ( 17) we get (cf.ref.[2]) −dE dt≈R≈2 3e2c r2(E mc2)4, where Eis the energy of the particle. whence intergrating over the period of non locality ∼τ0we can immediately deduce that r, the dimension of spatial non locality is given by r∼cτ0, that is of the order of the Compton wavelength. This follows i n any case in a relativistic theory, given the above Compton time. This ter m represents the eﬀects within the neighbourhood of the charge. What we have done is that we have quantiﬁed the space-time int erval of non locality - it is of the order of the Compton wavelength and time. This contains the renormalization eﬀect and gives the correct ph ysical mass. We now come to the Feynman-Wheeler action at a distance theor y[32, 33]. They showed that the apparent acausality of the theory would disappear if the interaction of a charge with all other charges in the univ erse, such that the remaining charges would absorb all local electromagnet ic inﬂuences was considered. The rationale behind this was that in an action a t a distance context, the motion of a charge would instantaneously aﬀect other charges, whose motion in turn would instantaneously aﬀect the origin al charge. Thus considering a small interval in the neighbourhood of the poi nt charge, they deduced, Fµ ret=1 2{Fµ ret+Fµ adv}+1 2{Fµ ret−Fµ adv} (19) The left side of (19) is the usual causal ﬁeld, while the right side has two terms. The ﬁrst of these is the time symmetric ﬁeld while the s econd can easily be identiﬁed with the Dirac ﬁeld above and represents the sum of the responses of the remaining charges calculated in the vic inity of the said charge. From this point of view, the self force or in the earlier Kerr- Newman for- mulation, eﬀects within the Compton scale, turns out to be th e combined 9reaction of the rest of the charges, or in the earlier duality and cosmological considerations, the holistic eﬀect. 4 Duality and Scale In a previous communication[34] it was shown that we could co nsider a scaled Planck constant h1≈N3/2¯h such that we would have R=h1 Mc It is interesting to note that these relations are essential ly the same as the second or extra term in (7) viz., l2∆p ¯h∼∆x with ∆ p=√ Nmand ∆ x=Ras before. This can be easily veriﬁed. In other words the two terms of the modiﬁed Heisenberg Uncert ainity relation (7) represents two scales. The ﬁrst term represents the micr o scale with the Planck constant, while the second term represents the macro scale with the scaled Planck constant h1, both being linked, as noted earlier. References [1] B.G. Sidharth, in Instantaneous Action at a Distance in M odern Physics: ”Pro and Contra” , Eds., A.E. Chubykalo et. al., Nova Science Publish- ing, New York, 1999. [2] F. Rohrlich, ”Classical Charged Particles”, Addison-W esley, Reading, Mass., 1965. [3] A.M. Polyakov, ”Gravitation and Quantizations”, Eds. B . Julia and J. Zinn-Justin, Les Houches, Session LVII, 1992, Elsevier Sci ence, 1995, p.786-804. [4] T.D. Lee, Physics Letters, Vol.122B, No.3,4, 10March 19 83, p.217-220. 10[5] B.G. Sidharth, Chaos, Solitons and Fractals, 12(2001), 173-178. [6] C. Wolf, Il Nuovo Cimento, Vol.100B, No.3, September 198 7, p.431-434. [7] J.A. Wheeler, ”Superspace and the Nature of Quantum Geom etrody- namics”, Battelles Rencontres, Lectures, Eds., B.S. De Wit t and J.A. Wheeler, Benjamin, New York, 1968. [8] A. Kempf, in ”From the Planck Length to the Hubble Radius” , Ed. A. Zichichi, World Scientiﬁc, Singapore, 2000, p.613-622. [9] G. Veneziano, Physics Reports, 9, No.4, 1974, p.199-242 . [10] P. Ramond, Phys.Rev.D., 3(10), 1971, pp.2415-2418. [11] J. Scherk, Rev.Mod.Phys., 47 (1), 1975, pp.1-3ﬀ. [12] W. Witten, Physics Today, April 1996, pp.24-30. [13] Y. Ne’eman, in Proceedings of the First Internatioinal Symposium, ”Frontiers of Fundmental Physics”, Eds. B.G. Sidharth and A . Burinskii, Universities Press, Hyderabad, 1999, pp.83ﬀ. [14] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitatio n”, W.H. Free- man, San Francisco, 1973, pp.819ﬀ. [15] B.G. Sidharth, Ind.J. Pure & Appld.Phys., Vol.35, 1997 , pp.456-471. [16] B.G. Sidharth, Int.J.Mod.Phys.A, 13 (15), 1998, p.259 9ﬀ. [17] P.A.M. Dirac, ”The Principles of Quantum Mechanics”, C larendon Press, Oxford, 1958, pp.4ﬀ, pp.253ﬀ. [18] C. Moller, ”The Theory of Relativity”, Clarendon Press , Oxford, 1952, pp.170 ﬀ. [19] L. Nottale, ”Scale relativity and gauge invariance”, t o appear in Chaos, Solitons and Fractals, special issue in honor of M. Conrad. [20] H.S. Snyder, Physical Review, Vol.72, No.1, July 1 1947 , p.68-71. 11[21] V.G. Kadyshevskii, Translated from Doklady Akademii N auk SSSR, Vol.147, No.6, December 1962, p.1336-1339. [22] L. Bombelli, J. Lee, D. Meyer and R.D. Sorkin, Physical R eview Letters, Vol.59, No.5, August 1987, p.521-524. [23] B.G. Sidharth, Chaos, Solitons and Fractals, 11 (2000) , 1269-1278. [24] V. Heine, ”Group Theory in Quantum Mechanics”, Pergamo n Press, Oxford, 1960, p.364. [25] J.R.Klauder, ”Bosons Without Bosons” in Quantum Theor y and The Structures of Time and Space, Vol.3 Eds by L. Castell, C.F. Va n Wei- izsecker, Carl Hanser Verlag, Munchen 1979. [26] J. Madore, Class.Quantum Grav. 9, 1992, p.69-87. [27] B.L. Cerchiai, J. Madore, S.Schraml, J. Wess, Eur.Phys .J. C 16, 2000, p.169-180. [28] J. Schwarz, M.B. Green and E. Witten, ”SuperString Theo ry”, Vol.I, Cambridge University Press, Cambridge, 1987. [29] B.G. Sidharth, Int.J.Th.Phys., 37 (4), 1998, p.1307ﬀ. [30] S. Perlmutter, et. al., Nature 3916662, 1998, 51-54. [31] B.G. Sidharth, ”The Chaotic Universe”, Nova Science Pu blishers, New York, In press. [32] F. Hoyle and J.V. Narlikar, ”Lectures on Cosmology and A ction at a Distance Electrodynamics”, World Scientiﬁc, Singapore, 1 996. [33] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys., 17, 157, 1945. [34] B.G. Sidharth, Chaos, Solitons and Fractals, 12, 2001, 613-616. 12
arXiv:physics/0103025v1 [physics.plasm-ph] 9 Mar 2001Trapping oscillations, discrete particle eﬀects and kinet ic theory of collisionless plasma F. Doveila†‡, M-C. Firpoa‡, Y. Elskensa‡, D. Guyomarc’ha, M. Poleniband P. Bertrandb‡ aEquipe turbulence plasma, Physique des interactions ioniq ues et mol´ eculaires, Unit´ e 6633 CNRS–Universit´ e de Provence, case 321, Centre de Saint-J´ erˆ ome, F-13397 Marseille cede x 20 bLaboratoire de physique des milieux ionis´ es et applicatio ns, Unit´ e 7040 CNRS–Universit´ e H. Poincar´ e, Nancy I, BP 239, F-54506 Vandœuvre cedex, France (preprint TP99.11) Eﬀects induced by the ﬁnite number Nof particles on the evolution of a monochromatic electrostatic perturbation i n a collisionless plasma are investigated. For growth as well a s damping of a single wave, discrete particle numerical simul a- tions show a N-dependent long time behavior which diﬀers from the numerical errors incurred by vlasovian approaches and follows from the pulsating separatrix crossing dynamic s of individual particles. Keywords : plasma kinetic theory wave-particle interaction self-consistent ﬁeld particle motions and dynamics PACS numbers : 05.20.Dd (Kinetic theory) 52.35.Fp (Plasma: electrostatic waves and oscillations) 52.65.-y (Plasma simulation) 52.25.Dg (Plasma kinetic equations) I. INTRODUCTION It is tempting to expect that kinetic equations and their numerical simulation provide a fair description of the time evolution of systems with long-range or ‘global’ interactions. A typical, fundamental example is oﬀered by wave-particle interactions, which play a central role in plasmas. In this Letter we test this opinion explicitly. Collisionless plasma dynamics is dominated by collec- tive processes. Langmuir waves and their familiar Lan- dau damping and growth [1] are a good example of these processes, with many applications, e.g. plasma heating in fusion devices and laser-plasma interactions. For sim- plicity we focus on the one-dimensional electrostatic case , traditionally described by the (kinetic) coupled set of Vlasov-Poisson equations [2,3]. The current debate on the long-time evolution of this system shows that further insight in this fundamental process is still needed [4]. The driving process (induced by the binary Coulomb interaction between particles) is the interaction of the electrostatic waves in the plasma with the particles at nearly resonant velocities, which one analyses canonicall y by partitioning the plasma in bulk and tail particles.Langmuir modes are the collective oscillations of bulk particles, with slowly varying complex amplitudes in an envelope representation ; their interaction with tail par- ticles is described by a self-consistent set of hamiltonian equations [5]. These equations already provided an eﬃ- cient basis [6] for investigating the cold beam plasma in- stability and exploring the nonlinear regime of the bump- on-tail instability [7]. Analytically, they were used to gi ve an intuitive and rigorous derivation of spontaneous emis- sion and Landau damping of Langmuir waves [8]. Be- sides, as it eliminates the rapid plasma oscillation scale ω−1 p, this self-consistent model oﬀers a genuine tool to investigate long-time dynamics. As we follow the motion of each particle, we can also address the inﬂuence of the ﬁnite number of particles in the long run. This question is discarded in the kinetic Vlasov-Poisson description, for which the ﬁnite- Ncor- rection is the Balescu-Lenard equation [9] formally de- rived from the accumulation of weak binary collisions, with small change of particle momenta. It implies a dif- fusion of momenta, driving the plasma towards equilib- rium. However, when wave-particle coupling is domi- nant, the Balescu-Lenard equation is not a straightfor- ward approach to ﬁnite- Neﬀects on the wave evolution. Here we investigate direct ﬁnite- Neﬀects on the self- consistent wave-particle dynamics. It is proved [10] that the kinetic limit N→ ∞ commutes with evolution over arbitrary times. As one might argue that ﬁnite Nbe analogous to numerical discretisation in solving kinetic equations, we also integrate the kinetic system with a ‘noise-free’ semi-lagrangian solver [11]. In this Letter w e compare ﬁnite grid eﬀects of the kinetic solver and gran- ular aspects of the N-particles system, whose evolution is computed with a symplectic scheme [7]. We discuss the case of one wave interacting with the particles. Though a broad spectrum of unstable waves is generally excited when tail particles form a warm beam, the single-wave situation can be realized experimentally [12] and allows to leave aside the diﬃcult problem of mode coupling mediated by resonant particles [13]. 1II. SELF-CONSISTENT WAVE-PARTICLE MODEL AND KINETIC MODEL Consider a one-dimensional electrostatic potential per- turbation Φ( z, τ) = [φk(τ)expi(kz−ωkτ) + c.c.] (where c.c. denotes complex conjugate), with complex enve- lopeφk, in a plasma of length Lwith periodic boundary conditions (and neutralizing background). Wavenum- berkand frequency ωksatisfy a dispersion relation ǫ(k, ωk) = 0. The density of N(quasi-)resonant elec- trons is σ(z, τ) = (nL/N )/summationtextN l=1δ(z−zl(τ)), where nis the electron number density and zlis the position at time τof electron labeled l(with charge eand mass m). Non- resonant electrons contribute only through the dielectric function ǫ, so that φkand the zl’s obey coupled equations [14] dφk/dτ=ine ǫ0k2N(∂ǫ/∂ω k)N/summationdisplay l=1exp[−ikzl+iωkτ] (1) d2zl/dτ2= (iek/m )φkexp[ikzl−iωkτ] + c.c. (2) where ǫ0is the vacuum dielectric constant. With α3= ne2/[mǫ0(∂ǫ/∂ω k)] [15], t=ατ, ˙=d/dt,xl=kzl−ωkτ andV= (ek2φk)/(α2m), this system deﬁnes the self- consistent dynamics (with N+ 1 degrees of freedom) ˙V=iN−1N/summationdisplay l=1exp(−ixl) (3) ¨xl=iVexp(ixl)−iV∗exp(−ixl) (4) for the coupled evolution of electrons and wave in di- mensionless form. This system derives from hamil- tonian H(x,p, ζ, ζ∗) =/summationtextN l=1(p2 l/2−N−1/2ζeixl− N−1/2ζ∗e−ixl), where a star means a complex conju- gate and ζ=N1/2V. An eﬃcient symplectic integration scheme is used to study this hamiltonian numerically [7]. The system (3)-(4) is invariant under two continuous groups of symmetries. Invariance under time translations implies the conservation of the energy H=H. The phase θofζ=\|ζ\|e−iθplays the role of a position for the wave, and system (3)-(4) is also invariant under translations θ′=θ+a,x′ l=xl+a. This translation invariance leads to the conservation of momentum P=/summationtext lpl+\|ζ\|2, where the contribution from the wave is analogous to the Poynt- ing vector of electromagnetic waves (which is quadratic in the electromagnetic ﬁelds) [16]. Conservation of these invariants constrains the evolution of our system, and we checked that the numerical integration preserves them. In the kinetic limit N→ ∞, electrons are distributed with a density f(x, p, t), and system (3)-(4) yields the Vlasov-wave system ˙V=i/integraldisplay e−ixf(x, p, t)dxdp (5) ∂tf+p∂xf+ (iV eix−iV∗e−ix)∂pf= 0 (6)For initial data approaching a smooth function fas N→ ∞, the solutions of (3)-(4) converge to those of the Vlasov-wave system over any ﬁnite time interval [10]. This kinetic model is integrated numerically by a semi- lagrangian solver, covering ( x, p) space with a rectangular mesh : the function f(interpolated by cubic splines) is transported along the characteristic lines of the kinetic equation, i.e. along trajectories of the original particle s [11]. Let us ﬁrst study linear instabilities. One solution of (3)-(4) corresponds to vanishing ﬁeld V0= 0, with par- ticles evenly distributed on a ﬁnite set of beams with given velocities. Small perturbations of this solution hav e δV=δV0eγt, with rate γsolving [8] γ=γr+iγi=iN−1N/summationdisplay l=1(γ+ipl)−2. (7) For a monokinetic beam with velocity U, (7) reads γ(γ+ iU)2=i; the most unstable solution occurs for U= 0 (with γ= (√ 3 +i)/2). For a warm beam with smooth initial distribution f(p) (normalized to/integraltext fdp= 1), the continuous limit of (7) yields γ=i/integraltext (γ+ip)−2f(p)dp. For a suﬃciently broad distribution ( \|f′(0)\| ≪1), we obtain \|γr\|γr=γrπf′(−γi), where f′=d f/dp, and γi≈πγrf′′(0) for \|f′′(0)\| ≪π−1. Except for the triv- ial solution γr= 0, other solutions can only exist for a positive slope f′(0). Then the perturbation is unsta- ble as the evolution of δVis controlled by the eigen- value γwith positive real part, i.e. with growth rate γr≈γL=πf′(0)>0. Negative slope leads to the lin- ear Landau damping paradox : the observed decay rate γL=πf′(0)<0 is not associated to genuine eigenvalues, but to phase mixing of eigenmodes [8,17,18], as a direct consequence of the hamiltonian nature of the dynamics. Now, this linear analysis generally fails to give the large time behavior. This is obvious for the unstable case as non-linear eﬀects are no longer negligible when the wave intensity grows so that the trapping frequency ωb(t) =/radicalbig 2\|V(t)\|becomes of the order of the linear rate γr. We used the monokinetic case as a testbed [18,19]. Finite- Nsimulations show that the unstable solution grows as predicted and saturates to a limit-cycle-like be- havior where the trapping frequency ωb(t) oscillates be- tween 1 .2γrand 2 γr. In this regime, some of the ini- tially monokinetic particles have been scattered rather uniformly over the chaotic domain, in and around the pulsating resonance, while others form a trapped bunch inside this resonance (away from the separatrix) [19]. This dynamics is quite well described by eﬀective hamil- tonians with few degrees of freedom [18,20]. In this Letter, we discuss the large time behavior of the warm beam case, with f′(p0)/negationslash= 0 at the wave nominal velocity p0= 0. Fig. 1 displays three distribution func- tions (in dimensionless form) with similar velocity width : (i)a function (CD) giving the same decay rate for all 2phase velocities, (ii)a function (CG) giving a constant growth rate for all phase velocities [7], (iii)a truncated Lorentzian (TL) with positive slope f′(0)>0. III. DAMPING CASE For the damping case, the linear description introduces time secularities which ultimately may break linear the- ory down : the ultimate evolution is intrinsically nonlin- ear, not only if the initial ﬁeld amplitude is large, as in O’Neil’s seminal picture [2], but also if one considers the evolution over time scales of the order of the trapping time (which is large for small initial wave amplitude). The question of the plasma wave long-time fate is thus far from trivial [4]. Though some simulations [21] infer that nonlinear waves eventually approach a Bernstein- Greene-Kruskal steady state [22] instead of Landau van- ishing ﬁeld, the answer should rather strongly depend on initial conditions. Our N-particle, 1-wave system is the simplest model to test these ideas. A thermodynamical analysis [17] predicts that, for a warm beam and small enough initial wave amplitude, ωb∼N−1/2at equilibrium in the limit N→ ∞. Fig. 2 shows the evolution of a small amplitude wave launched in the beam. The N-particle system (line N) and the kinetic system (line V) initially damp the wave exponen- tially as predicted by perturbation theory [8], for a time of the order of \|γL\|−1. After that phase-mixing time, trapping induces non- linear evolution and both systems evolve diﬀerently. For theN-particle system, the wave grows to a thermal level that scales as N−1/2, corresponding to a balance be- tween damping and spontaneous emission [8,17]. For the kinetic system, initial Landau damping is followed by slowly damped trapping oscillations around a mean value which also decays to zero, at a rate decreasing for reﬁned mesh size. Fig. 2 reveals that ﬁnite- Nand kinetic behaviors can considerably diverge as spontaneous emis- sion is taken into account. The time τNafter which the ﬁnite- Neﬀects force this divergence is found to diverge asN→ ∞. IV. UNSTABLE CASE Now consider an unstable warm beam ( f′(0)>0). Line N1 (resp. N2) of Fig. 3 displays ln( ωb(t)/γr) versus time for (3)-(4) with a CG distribution with N= 128000 (resp. 512000) and γr= 0.08. Line V1 (resp. V2) shows ln(ωb(t)/γr) versus γrtfor the kinetic system and the same initial distribution with a 32 ×128 (resp. 256 ×1024) grid in ( x, p) space. All four lines exhibit the same initial exponential growth of linear theory with less than 1% er- ror on the growth rate. Saturation occurs for ωb/γr≈3.1 [3]. Lines N1 and V1 do not superpose beyond the ﬁrsttrapping oscillation after saturation. Note that, in our system, oscillating saturation does not excite sideband Langmuir waves as our hamiltonian incorporates only a single wave, not a spectrum. After the ﬁrst trapping oscillation, kinetic simulations exhibit a second growth at a rate controlled by mesh size. Line V2 suggests that a kinetic approach would predict a level close to the trapping saturation level on a time scale awarded by reasonable integration time. This level is fairly below the equilibrium Vthpredicted by a gibbsian approach [17] ; such pathological relaxation properties in the N→ ∞ limit seem common to mean-ﬁeld long- range models [23]. Both kinetic simulations also exhibit a strong damping of trapping oscillations, which disap- pear after a few oscillations, whereas ﬁnite- Nsimulations show persistent trapping oscillations. One could expect that ﬁnite- Neﬀects would mainly damp these oscillations, so that the wave amplitude reaches a plateau. Actually, we observe persistent os- cillations for all N, and the wave amplitude slowly grows further, whereas the velocity distribution function ﬂat- tens over wider intervals of velocity. This spreading of particles is due to separatrix cross- ings, i.e. successive trapping and detrapping by the wave [19]. Indeed, when the wave amplitude grows (during its pulsation), it captures particles with nearby veloc- ity, i.e. with a relative velocity ∆ vin≈ ±/radicalbig 8\|V\|; the trapped particles start bouncing in the wave potential well. When the wave amplitude decreases, particles are released, but if they experienced only half a bouncing period, they are released with a relative velocity (with respect to the wave) opposite to their initial one, i.e. ∆vout≈ −∆vin. Now notice that a particle which has just been trapped would oscillate at a longer period than the nominal bouncing period (namely the one deep in the potential). Moreover, if the recently trapped particle had just adiabatic motion in the well, it would have to recross the separatrix when the resonance would enclose the same area as at its trapping [24]. Thus one expects the particle to be unable to complete a full bounce, and the fraction of particles for which ∆ vout≈ −∆vinis sig- niﬁcant. During this particle spreading process in ( x, p) space, the wave pulsation is maintained by the bunch of parti- cles which were initially trapped, and are deep enough in the potential well to remain trapped over a whole bounc- ing period. These particles form a macroparticle, as is best seen in the case of a cold beam [20]. Note that, over long times, the macroparticle must slowly spread in the wave resonance, following two processes. One acts if the trapped particle motion is regular : the trapped motions are anisochronous, i.e. have diﬀerent periods (only the harmonic oscillator has isochronous oscillations). The other one works if the motion is chaotic : nearby trajec- tories diverge due to chaos. Both processes contribute to the smoothing of the particle distribution for long times, 3but over much longer times than those over which we follow the system evolution and observe the wave modu- lation. This second growth after the ﬁrst trapping saturation depends on the shape of the initial distribution function. In Fig. 3(b), line N2 is the same as in Fig. 3(a), com- puted over a longer duration, and line N3 corresponds to N= 64000 with the TL distribution of Fig. 1. Although N3 corresponds to 8 times fewer particles than N2, the ﬁnal level reached at the end of the simulation is lower. In the second growth, particles are transported further in velocity, so that the plateau in f(p) broadens with time. As the wave grows, it can trap particles with ini- tial velocity further away from its phase velocity. Since the TL distribution reaches its maximum at v≈1.06 and decreases signiﬁcantly beyond this velocity (while CG is still growing for larger v), fewer particles (with TL than with CG) can give momentum to the wave when being trapped ( Pis conserved) ; hence the second growth is slower for the TL distribution. We followed the evolution of the wave amplitude for N3 up to γrt= 1750 : starting from the ﬁrst trapping sat- uration level (0 .4Vth), ﬂuctuations persist with a growth rate that slowly decreases as we reach 0 .78Vthat the end of the computation. Line N4 of Fig. 3 corresponds to the TL distribution with 2048000 particles and shows persis- tent oscillations with approximately the same amplitude as for N= 64000. V. CONCLUSION These observations clearly indicate that the kinetic models are an idealization and do not contain all the intricate behavior of a discrete particles system. Now, we must also admit that the kinetic simulation schemes do not exactly reproduce the analytic implications of the kinetic equation. It is then legitimate to ask whether the numerical implementation of the kinetic equations repro- duce the diﬀerence between the ﬁnite- Ndynamics and the kinetic theory. A basic property of the collisionless kinetic equation is that it transports the distribution function f(x, p) along the particle trajectories (or characteristic lines in ( x, p) space). As long as the kinetic calculation of fis accu- rate, one expects the kinetic scheme to follow closely the N-particle dynamics too. However, the kinetic scheme is bound to depart from the analytic predictions of the kinetic equation, because the (chaotic or anisochronous) separation of particle trajectories implies that constant - fcontours eventually evolve into complex, interleaved shapes. This ﬁlamentation is smoothed by numerical par- tial diﬀerential equation integrators, while N-body dy- namics follows the particles more realistically, sustaini ng the trapping oscillations. Hence both types of dynamicswill depart from each other when ﬁlamentation reaches scales below the semi-lagrangian kinetic code grid mesh. The onset of ﬁlamentation is easily evidenced in kinetic simulations. Indeed, whereas the kinetic equation analyt- ically preserves the 2-entropy/integraltext(1−f)fdxdp , numerical schemes increase entropy signiﬁcantly when constant- f contours form ﬁlaments in ( x, p)-space [25]. As this is also the time at which trapping oscillations are found to damp in our simulations, it appears that vlasovian sim- ulations must be considered with caution from that time on – and it turns out that it is also the time from which the second growth starts. In summary, discussing the basic propagation of a sin- gle electrostatic wave in a warm plasma, we presented ﬁnite- Neﬀects which do not merely result from nu- merical errors and elude a kinetic simulation approach. Their understanding depends crucially on the dynamics in phase space. The sensitive dependence of microscopic evolution to the ﬁne structure of the initial particle dis- tribution in phase space [18] implies that the interplay between limits t→ ∞ andN→ ∞ requires some cau- tion. Somewhat paradoxically, reﬁning the grid for the Vlasov simulations does not solve this problem. The driving process in the system evolution is sepa- ratrix crossing, which requires a geometric approach to the system dynamics. Further work in this direction [26] will also shed new light on the foundations of common approximations, such as replacing original dynamics (1)- (2) by coupled stochastic equations, in which particles undergo noisy transport. VI. ACKNOWLEDGMENTS The authors thank D.F. Escande for fruitful discus- sions, and J.R. Cary and I. Doxas for computational as- sistance. MCF and DG were supported by the French Minist` ere de la Recherche. Computer use at Institut M´ editerran´ een de Technologie and IDRIS was granted by R´ egion Provence-Alpes-Cˆ ote d’Azur and CNRS. This work is part of the european network Stability and uni- versality in classical mechanics and CNRS GdR Syst` emes de particules charg´ ees (SParCh). †Corresponding author : fax +33-491 28 82 25, phone +33-491 28 83 38. ‡Email : X@newsup.univ-mrs.fr (X = ﬁrpo, doveil, elskens), Pierre.Bertrand@lpmi.uhp-nancy.fr [1] L.D. Landau, J. Phys. USSR 10(1946) 25; J.H. Malm- berg and C.B. Wharton, Phys. Rev. Lett. 6(1964) 184; D.D. Ryutov, Plasma Phys. Control. Fusion 41(1999) A1. 4[2] T.M. O’Neil, Phys. Fluids 8(1965) 2255. [3] B.D. Fried, C.S. Liu, R.W. Means and R.Z. Sagdeev, Plasma Physics Group Report PPG-93, University of California, Los Angeles, 1971 (unpublished); A. Simon and M.N. Rosenbluth, Phys. Fluids 19(1976) 1567; P.A.E.M. Janssen and J.J. Rasmussen, Phys. Fluids 24 (1981) 268; J.D. Crawford, Phys. Rev. Lett. 73(1994) 656. [4] G. Brodin, Phys. Rev. Lett. 78(1997) 1263; G. Manfredi, ibid.79(1997) 2815; M.B. Isichenko, ibid. 78(1997) 2369,80(1998) 5237; C. Lancellotti and J.J. 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Holloway and J.J. Dorning, Phys. Rev. A 44(1991) 3856; M. Buchanan and J.J. Dorning, Phys. Rev. E 52(1995) 3015. [23] V. Latora, A. Rapisarda and S. Ruﬀo, Phys. Rev. Lett. 80(1998) 692. [24] Y. Elskens and D.F. Escande, Nonlinearity 4(1991) 615; Physica D 62(1992) 66. [25] M. Poleni, private report. [26] D. Benisti and D.F. Escande, Phys. Plasmas 4(1997) 1576; J. Stat. Phys. 92(1998) 909. 5−6 −4 −2 0 2 400.10.2CG CD TL p f(p) FIG. 1. Initial velocity distributions. FIG. 2. Time evolution of ln( ωb(t)/\|γL\|) for a CD velocity distribution and initial wave amplitude below thermal leve l : (N)N-particles system with N= 32000, (V) kinetic scheme with 32 ×512 (x, p) grid. Inset : short-time evolution. 0 10 20 30 40 5000.511.52 γr tln( ωb / γr )N1 N2V1 V2(a) 0 50 100 150 200 25011.11.21.31.41.5 γr t ln( ωb / γr )N2 N3 N4(b) FIG. 3. Time evolution of ln( ωb(t)/γr). (a) CG initial dis- tribution : kinetic scheme with (V1) 32 ×128, (V2) 256 ×1024 (x, p) grid ; N-particles system with (N1) N= 128000, (N2) N= 512000 ; (b) Comparison of CG (N2) with TL initial distribution for (N3) N= 64000, (N4) N= 2048000. 6
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/CF/CW/CT/D2 /DB /CT /D9/D7/CT /D8/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT/D2 Lab /CX/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CQ /DDLµν,e, /D8/CW/CT /D9/D7/D9/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D4/D9/D6/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /DB/CW/CX /CW /D3/D2/D2/CT /D8/D7/D8 /DB /D3 /D3 /D3/D6/CS/CX/D2/CP/D8/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7/B8 /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /B4/CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/B8 xµ e, xµ′ e /D3/CU /CP /CV/CX/DA /CT/D2/CT/DA /CT/D2 /D8/BA xµ e, xµ′ e /D6/CT/CU/CT/D6 /D8/D3 /D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7 /B4/DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX /D8/CT/D2/D7/D3/D6/D7/B5 S /CP/D2/CSS′, xµ′ e=Lµ′ ν,exν e, L0′ 0,e=γe, L0′ i,e=Li′ 0,e=−γevi e/c, Li′ j,e=δi j+ (γe−1)vi evje/v2 e, /B4/BE/B5/DB/CW/CT/D6/CT vµ e≡dxµ e/dτ= 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/D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /CX/D7 /CV/CX/DA /CT/D2 /CP/D7 r0=e0, ri=e0+ei, /B4/BF/B5/D7/CT/CT /CJ/BD/BD ℄/B8 /CJ/BH℄ /CP/D2/CS /CJ/BD/B8 /BE ℄/BA /CC/CW/CT /D1/CT/D8/D6/CX /D8/CT/D2/D7/D3/D6 gab /CQ /CT /D3/D1/CT/D7 gab=gµν,rdxµ r⊗dxν r /CX/D2 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B9/CQ/CP/D7/CT/CS/CV/CT/D3/D1/CT/D8/D6/CX /D0/CP/D2/CV/D9/CP/CV/CT /CP/D2/CS /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /D1/CT/D8/D6/CX /D8/CT/D2/D7/D3/D6/CP/D6/CT g00,r=g0i,r=gi0,r=gij,r(i/ne}ationslash=j) =−1, gii,r= 0. /B4/BG/B5 dxµ r, dxν r /CP/D6/CT /D8/CW/CT /CQ/CP/D7/CX/D7 /BD/B9/CU/D3/D6/D1/D7 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S, /CP/D2/CSdxµ r⊗dxν r /CX/D7 /CP/D2 /D3/D9/D8/CT/D6/D4/D6/D3 /CS/D9 /D8 /D3/CU /D8/CW/CT /CQ/CP/D7/CX/D7 /BD/B9/CU/D3/D6/D1/D7/B8 /CX/BA/CT/BA/B8 /CX/D8 /CX/D7 /D8/CW/CT /CQ/CP/D7/CX/D7 /CU/D3/D6 /B4/BC/B8/BE/B5 /D8/CT/D2/D7/D3/D6/D7/BA/CC/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC Tµν,r /DB/CW/CX /CW /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /D8/CW/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CU/D6/D3/D1 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/B9/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CP/D7 Tµµ,r=−T0i,r= 1, /B4/BH/B5/CP/D2/CS /CP/D0/D0 /D3/D8/CW/CT/D6 /CT/D0/CT/D1/CT/D2 /D8/D7 /D3/CUTµν,r /CP/D6/CT= 0 /BA /CD/D7/CX/D2/CV /D8/CW/CX/D7Tµν,r /DB /CT /AS/D2/CS xµ r=Tµ ν,rxν e, x0 r=x0 e−x1 e−x2 e−x3 e, xi r=xi e. /B4/BI/B5/BY /D3/D6 /D8/CW/CT /D7/CP/CZ /CT /D3/CU /D3/D1/D4/D0/CT/D8/CT/D2/CT/D7/D7 /DB /CT /CP/D0/D7/D3 /D5/D9/D3/D8/CT /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lµ′ ν,r /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/B9/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C1/D8 /CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /CU/D3/D9/D2/CS /CU/D6/D3/D1Lab /B4/BD/B5 /CP/D2/CS /D8/CW/CT /CZ/D2/D3 /DB/D2 gµν,r, /CP/D2/CS /D8/CW/CT /CT/D0/CT/D1/CT/D2 /D8/D7 /D8/CW/CP/D8 /CP/D6/CT/CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D6/D3/D1 /DE/CT/D6/D3 /CP/D6/CT x′µ r=Lµ′ ν,rxν r, L0′ 0,r=K, L0′ 2,r=L0′ 3,r=K−1, L1′ 0,r=L1′ 2,r=L1′ 3,r= (−βr/K), L1′ 1,r= 1/K, L2′ 2,r=L3′ 3,r= 1, /B4/BJ/B5/BG/DB/CW/CT/D6/CT K= (1 + 2 βr)1/2, /CP/D2/CSβr=dx1 r/dx0 r /CX/D7 /D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /D3/CU /D8/CW/CT /CU/D6/CP/D1/CT S′/CP/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /D8/CW/CT/CU/D6/CP/D1/CT S /B8βr=βe/(1−βe) /CP/D2/CS /CX/D8 /D6/CP/D2/CV/CT/D7 /CP/D7−1/2≺βr≺ ∞./BT/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU /CX/D7/D3/D1/CT/D8/D6/DD /CX/D7 /D8/CW/CT /D3 /DA /CP/D6/CX/CP/D2 /D8 /BG/BW /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lab /B4/BD/B5/BA /CF/CW/CT/D2 /D8/CW/CT /D3 /D3/D6/CS/CX/B9/D2/CP/D8/CT /CQ/CP/D7/CX/D7 /CX/D7 /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS /D8/CW/CT/D2/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /CX/D7/D3/D1/CT/D8/D6/DD Lab /B4/BD/B5 /DB/CX/D0/D0 /CQ /CT /CT/DC/D4/D6/CT/D7/D7/CT/CS /CP/D7 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lµ′ ν,e /B4/BE/B5 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D3/D6 /CP/D7Lµ′ ν,r /B4/BJ/B5 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/B9/D8/CX/DE/CP/D8/CX/D3/D2/BA/C6/D3 /DB /DB /CT /CP/D2 /CQ /CT/D8/D8/CT/D6 /CT/DC/D4/D0/CP/CX/D2 /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CS/CX/AR/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/D6/CT/CT /CP/D4/D4/D6/D3/CP /CW/CT/D7 /D8/D3 /CB/CA /CX/D2/D8/CW/CT /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D3/CU /D8/CW/CT /D3/D2 /CT/D4/D8 /D3/CU /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /D3/CQ/D7/CT/D6/DA /CT/D6/D7/BA /CF /CT /D7/CW/CP/D0/D0 /D3/D2/D7/CX/CS/CT/D6/D7/D3/D1/CT /D7/CX/D1/D4/D0/CT /CT/DC/CP/D1/D4/D0/CT/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BM /D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CU/D3/D6 /CP /D1/D3 /DA/CX/D2/CV /D6/D3 /CS /CP/D2/CS /D8/CW/CT/D2 /CU/D3/D6/CP /D1/D3 /DA/CX/D2/CV /D0/D3 /CZ/BA /CC/CW/CT /D7/CP/D1/CT /CT/DC/CP/D1/D4/D0/CT/D7 /DB/CX/D0/D0 /CQ /CT /CP/D0/D7/D3 /CT/DC/CP/D1/CX/D2/CT/CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/BE/BA/BD /CC/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CU/D3/D6 /CP /D1/D3 /DA/CX/D2/CV /D6/D3 /CS /CP/D2/CS /CP /D1/D3 /DA/CX/D2/CV /D0/D3 /CZ/C4/CT/D8 /D9/D7 /D8/CP/CZ /CT/B8 /CU/D3/D6 /D7/CX/D1/D4/D0/CX /CX/D8 /DD /B8 /D8/D3 /DB /D3/D6/CZ /CX/D2 /BE/BW /D7/D4/CP /CT/D8/CX/D1/CT/BA /CC/CW/CT/D2 /DB /CT /D3/D2/D7/CX/CS/CT/D6 /CP /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CP/CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 /B4/D8/CW/CT /B4/BD/B8/BC/B5 /D8/CT/D2/D7/D3/D6/B5 la AB=xa B−xa A /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /B4/DB/CX/D8/CW /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2/BG/B9/DA /CT /D8/D3/D6/D7 xa A /CP/D2/CSxa B /B5/BAla AB /CX/D7 /CW/D3/D7/CT/D2 /D8/D3 /CQ /CT /CP /D4/CP/D6/D8/CX /D9/D0/CP/D6 /BG/B9/DA /CT /D8/D3/D6 /DB/CW/CX /CW/B8 /CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BF/B7/BD/AH /D4/CX /D8/D9/D6/CT/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP/D2 /D3/CQ /CY/CT /D8/B8 /CP /D6/D3 /CS/B8 /D8/CW/CP/D8 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /CP/D2 /C1/BY/CAS /CP/D2/CS /D7/CX/D8/D9/CP/D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT /D3/D1/D1/D3/D2 x1 e, x1′ e−/CP/DC/CT/D7/BA /B4/CC/CW/CT /D7/CP/D1/CT /CT/DC/CP/D1/D4/D0/CT /CX/D7 /CP/D0/D6/CT/CP/CS/DD /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CJ/BD/B8 /BE ℄ /CP/D2/CS /CJ/BH℄/BA/B5 /CC/CW/CX/D7 /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 /CP/D2 /CQ /CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS/CX/D2 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B9/CQ/CP/D7/CT/CS /CV/CT/D3/D1/CT/D8/D6/CX /D0/CP/D2/CV/D9/CP/CV/CT /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /CQ/CP/D7/CT/D7/B8 {eµ} /CP/D2/CS{rµ} /CX/D2 /CP/D2 /C1/BY/CAS, /CP/D2/CS{eµ′}/CP/D2/CS{rµ′} /CX/D2 /CP /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CAS′, /CP/D7la AB=lµ eeµ=lµ rrµ=lµ′ eeµ′=lµ′ rrµ′, /DB/CW/CT/D6/CT/B8 /CT/BA/CV/BA/B8eµ/CP/D6/CT /D8/CW/CT /CQ/CP/D7/CX/D7 /BG/B9/DA /CT /D8/D3/D6/D7/B8 e0= (1,0,0,0) /CP/D2/CS /D7/D3 /D3/D2/B8 /CP/D2/CSlµ e /CP/D6/CT /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /DB/CW/CT/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /CW/D3/D7/CT/D2 /CX/D2 /D7/D3/D1/CT /C1/BY/CAS. /CC/CW/CT /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/D7 lµ eeµ /CP/D2/CSlµ rrµ /B4/CX/D2 /CP/D2 /C1/BY/CAS, /CP/D2/CS /CX/D2/D8/CW/CT /AH/CT/AH /CP/D2/CS /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD/B5 /CP/D2/CSlµ′ eeµ′/CP/D2/CSlµ′ rrµ′/B4/CX/D2 /CP /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CAS′/B8/CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /CP/D2/CS /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD/B5 /D3/CU /D8/CW/CT /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 la AB /CP/D6/CT /CP/D0/D0 /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0/D0/DD/CT /D5/D9/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CC/CW /D9/D7 /D8/CW/CT/DD /CP/D6/CT /D6/CT/CP/D0/D0/DD /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV/C1/BY/CA/D7 /CP/D2/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/BA /B4/CC/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6lµ r /CP/D2/CSlµ′ r /CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /CU/D3/D9/D2/CS /CU/D6/D3/D1/D8/CW/CT /CZ/D2/D3 /DB/D2 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC Tµ ν,r. /B5 /C8 /CP/D6/D8/CX /D9/D0/CP/D6/D0/DD /CU/D3/D6 /D8/CW/CX/D7 /CW/D3/CX /CT /D3/CU /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX /D5/D9/CP/D2 /D8/CX/D8 /DD la AB/CX/D8/D7 /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S /CX/D7la AB=l0 ee0+l1 ee1= 0e0+L0e1, /DB/CW/CX/D0/CT /CX/D2 S′, /DB/CW/CT/D6/CT /D8/CW/CT /D6/D3 /CS /CX/D7 /D1/D3 /DA/CX/D2/CV/B8 /CX/D8 /CQ /CT /D3/D1/CT/D7 la AB=−βeγeL0e0′+γeL0e1′, /CP/D2/CS/B8 /CP/D7 /CT/DC/D4/D0/CP/CX/D2/CT/CS /CP/CQ /D3 /DA /CT/B8 /CX/D8/CW/D3/D0/CS/D7 /D8/CW/CP/D8 la AB= 0e0+L0e1=−βeγeL0e0′+γeL0e1′. /B4/BK/B5/CF /CT /D7/CT/CT /CU/D6/D3/D1 /B4/BK/B5 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT/D6/CT /CX/D7 /CP /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8l1′ e=γeL0/DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3l1 e=L0. /C0/D3 /DA /CT/DB /CT/D6 /CX/D8 /CX/D7 /D0/CT/CP/D6 /CU/D6/D3/D1 /D8/CW/CT /CP/CQ /D3 /DA /CT /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /D8/CW/CP/D8 /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D3/D2/D0/DD/D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB /CX/D2S /CP/D2/CSS′/CX/D7 /D4/CW /DD/D7/CX /CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7/CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CF/CW/CT/D2 /D3/D2/D0/DD /D7/D3/D1/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /CP/D6/CT /D8/CP/CZ /CT/D2 /CP/D0/D3/D2/CT/D8/CW/CT/D2 /D8/CW/CT/DD /CS/D3 /D2/D3/D8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D7/D3/D1/CT /CS/CT/AS/D2/CX/D8/CT /D4/CW /DD/D7/CX /CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /D8/CW/CT /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/BA /CB/CX/D1/CX/D0/CP/D6/D0/DD /D8/CW/CT/CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/D7 /D3/CUla AB /CX/D2 /D8/CW/CT /AH/D6/AH /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D6/CT la AB=−L0r0+L0r1,=−KL0r0′+ (1 + βr)(1/K)L0r1′, /B4/BL/B5/DB/CW/CT/D6/CT K= (1+2 βr)1/2. /C1/D2 /D8/CW/CT /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D8/CW/CT /CV/CT /D3/D1/CT/D8/D6/CX /D5/D9/CP/D2/D8/CX/D8/DD la AB, /CX/BA/CT/BA/B8 /D8/CW/CT /D3 /D3/D6 /CS/CX/D2/CP/D8/CT/B9/CQ /CP/D7/CT /CS/CV/CT /D3/D1/CT/D8/D6/CX /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7 lµ eeµ=lµ′ eeµ′=lµ rrµ=lµ′ rrµ′, /D3/D1/D4/D6/CX/D7/CX/D2/CV /CQ /D3/D8/CW/B8 /D3/D1/D4 /D3/D2/CT/D2/D8/D7 /CP/D2/CS /D8/CW/CT /CQ /CP/D7/CX/D7/B8 /CX/D7/D8/CW/CT /D7/CP/D1/CT /BG/BW /D5/D9/CP/D2/D8/CX/D8/DD /CU/D3/D6 /CS/CX/AR/CT/D6 /CT/D2/D8 /D3/CQ/D7/CT/D6/DA/CT/D6/D7/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CX/CUl0 e= 0 /D8/CW/CT/D2lµ′ e /CX/D2 /CP/D2 /DD /D3/D8/CW/CT/D6 /C1/BY/CAS′/DB/CX/D0/D0 /D3/D2 /D8/CP/CX/D2 /D8/CW/CT /D8/CX/D1/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 l0′ e/ne}ationslash= 0. /CC/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D4 /D3/CX/D2 /D8/D7 /B4/CT/DA /CT/D2 /D8/D7/B5 /CX/D2 /BG/BW/D7/D4/CP /CT/D8/CX/D1/CT /CX/D7 /CS/CT/AS/D2/CT/CS /CP/D7 l= (gablalb)1/2. /B4/BD/BC/B5/CC/CW/CX/D7 /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /B4/BD/BC/B5 /CX/D7 /CU/D6/CP/D1/CT /CP/D2/CS /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CX/BA/CT/BA/B8 /CX/D8 /CW/D3/D0/CS/D7 /D8/CW/CP/D8 l= (lµ e,rlµe,r)1/2= (lµ′ e,rlµ′e,r)1/2=L0. /C1/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX /CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD l2 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /CX/D8/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 l2 e, /DB/CX/D8/CW /D8/CW/CT /D7/CT/D4/CP/D6/CP/D8/CT/CS /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7/B8 l2=l2 e= (li elie)−(l0 e)2/BA /CB/D9 /CW /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2 /D6/CT/D1/CP/CX/D2/D7 /DA /CP/D0/CX/CS /CX/D2 /D3/D8/CW/CT/D6 /CX/D2/CT/D6/D8/CX/CP/D0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/D7 /DB/CX/D8/CW /D8/CW/CT/C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX /D8/CT/D2/D7/D3/D6/B8 /CP/D2/CS /CX/D2S′/D3/D2/CT /AS/D2/CS/D7l2=l′2 e= (li′ eli′e)−(l0′ e)2, /DB/CW/CT/D6/CT lµ′ e /CX/D2S′/CX/D7 /D3/D2/D2/CT /D8/CT/CS/DB/CX/D8/CWlµ e /CX/D2S /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lµ′ ν,e /B4/BE/B5/BA /BY /D9/D6/D8/CW/CT/D6 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2 S, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D6/D3 /CS/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUlµ e /CX/D7l0 e= 0, /D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CX/D7 /CP/D1/CT/CP/D7/D9/D6/CT /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2 /CT/B8 /CX/BA/CT/BA/B8 /D3/CU /D8/CW/CT /D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D6/D3 /CS/B8 /CP/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /BH/D4/CW /DD/D7/CX /D7/BA /CB/CX/D2 /CT gµν,r, /CX/D2 /D3/D2 /D8/D6/CP/D7/D8 /D8/D3gµν,e, /CX/D7 /D2/D3/D8 /CP /CS/CX/CP/CV/D3/D2/CP/D0 /D1/CP/D8/D6/CX/DC/B8 /D8/CW/CT/D2 /CX/D2l2 r /B4/D8/CW/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU l2/CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7 /CP/D6/CT /D2/D3/D8 /D7/CT/D4/CP/D6/CP/D8/CT/CS/BA/C1/D2 /CP /D7/CX/D1/CX/D0/CP/D6 /D1/CP/D2/D2/CT/D6 /DB /CT /CP/D2 /CW/D3 /D3/D7/CT /CP/D2/D3/D8/CW/CT/D6 /D4/CP/D6/D8/CX /D9/D0/CP/D6 /CW/D3/CX /CT /CU/D3/D6 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB,/DB/CW/CX /CW /DB/CX/D0/D0 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /AH/D1 /D9/D3/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/B8/AH /CP/D2/CS /DB/CW/CX /CW /CX/D7 /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CX/D2 /D8/CW/CT /AH/BT /CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D8/CX/D1/CT /AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/AH/BA /B4/CC/CW/CX/D7 /CT/DC/CP/D1/D4/D0/CT /CX/D7 /CP/D0/D7/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS /CX/D2 /CJ/BD /B8 /BE ℄/BA/B5 /BY/CX/D6/D7/D8 /DB /CT /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CX/D7 /CT/DC/CP/D1/D4/D0/CT /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CC/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB /DB/CX/D0/D0 /CQ /CT /CT/DC/CP/D1/CX/D2/CT/CS /CX/D2 /D8 /DB /D3/D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7 S /CP/D2/CSS′/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT/braceleftbig eµ/bracerightbig/CP/D2/CS{eµ′} /CQ/CP/D7/CT/D7/BA /CC/CW/CTS /CU/D6/CP/D1/CT /CX/D7 /CW/D3/D7/CT/D2 /D8/D3 /CQ /CT/D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D1 /D9/D3/D2/BA /CC /DB /D3 /CT/DA /CT/D2 /D8/D7 /CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS/BN /D8/CW/CT /CT/DA /CT/D2 /D8 A /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /D8/CW/CT /D6/CT/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D1 /D9/D3/D2 /CP/D2/CS /D8/CW/CT /CT/DA /CT/D2 /D8 B /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /CX/D8/D7 /CS/CT /CP /DD /CP/CU/D8/CT/D6 /D8/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT τ0 /CX/D2S. /CC/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT /D8/D3/D6/D7 /D3/CU /D8/CW/CT/CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /CX/D2S /CP/D6/CT /D8/CP/CZ /CT/D2 /D8/D3 /CQ /CT /D3/D2 /D8/CW/CT /DB /D3/D6/D0/CS /D0/CX/D2/CT /D3/CU /CP /D7/D8/CP/D2/CS/CP/D6/CS /D0/D3 /CZ /D8/CW/CP/D8 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT/D3/D6/CX/CV/CX/D2 /D3/CUS. /CC/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB=xa B−xa A /D8/CW/CP/D8 /D3/D2/D2/CT /D8/D7 /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /CX/D7 /CS/CX/D6/CT /D8/CT/CS/CP/D0/D3/D2/CV /D8/CW/CTe0 /CQ/CP/D7/CX/D7 /DA /CT /D8/D3/D6 /CU/D6/D3/D1 /D8/CW/CT /CT/DA /CT/D2 /D8 A /D8/D3 /DB /CP/D6/CS /D8/CW/CT /CT/DA /CT/D2 /D8 B. /CC/CW/CX/D7 /CV/CT/D3/D1/CT/D8/D6/CX /D5/D9/CP/D2 /D8/CX/D8 /DD /CP/D2 /CQ /CT/DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B9/CQ/CP/D7/CT/CS /CV/CT/D3/D1/CT/D8/D6/CX /D0/CP/D2/CV/D9/CP/CV/CT/BA /CC/CW /D9/D7 /CX/D8 /CP/D2 /CQ /CT /CS/CT /D3/D1/D4 /D3/D7/CT/CS /CX/D2 /D8/CW/CT /CQ/CP/D7/CT/D7{eµ}/CP/D2/CS{eµ′} /CP/D7 la AB=cτ0e0+ 0e1=γcτ0e′ 0−βγcτ 0e′ 1. /B4/BD/BD/B5/CP/D2/CS /D7/CX/D1/CX/D0/CP/D6/D0/DD /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D7 la AB==cτ0r0+ 0r1=Kcτ0r′ 0−βrK−1cτ0r′ 1. /B4/BD/BE/B5/CF /CT /CP/CV/CP/CX/D2 /D7/CT/CT /D8/CW/CP/D8 /D8/CW/CT/D7/CT /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/D7/B8 /D3/D2 /D8/CP/CX/D2/CX/D2/CV /CQ /D3/D8/CW /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D2/CS /D8/CW/CT /CQ/CP/D7/CX/D7/DA /CT /D8/D3/D6/D7/B8 /CP/D6/CT /D8/CW/CT /D7/CP/D1/CT /CV/CT/D3/D1/CT/D8/D6/CX /D5/D9/CP/D2 /D8/CX/D8 /DD la AB. la AB /CS/D3 /CT/D7 /CW/CP /DA /CT /D3/D2/D0/DD /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7 /CX/D2S /B8 /DB/CW/CX/D0/CT /CX/D2 /D8/CW/CT {eµ′} /CQ/CP/D7/CX/D7la AB /D3/D2 /D8/CP/CX/D2/D7 /D2/D3/D8 /D3/D2/D0/DD /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /CQ/D9/D8 /CP/D0/D7/D3 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/BA /CC/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CX/D7 /CP/D0/DB /CP /DD/D7 /CP /DB /CT/D0/D0/B9/CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /CU/D3/D6 /D8/CW/CX/D7 /CT/DC/CP/D1/D4/D0/CT /CX/D8 /CX/D7l= (lµ elµe)1/2= (lµ′ elµ′e)1/2= (lµ rlµr)1/2= (lµ′ rlµ′r)1/2= (−c2τ2 0)1/2/BA /CB/CX/D2 /CT /CX/D2S /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 l1 e,r /D3/CUlµ e,r /CP/D6/CT /DE/CT/D6/D3/D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CX/D2S /CX/D7 /CP /D1/CT/CP/D7/D9/D6/CT /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CS/CX/D7/D8/CP/D2 /CT/B8 /CP/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D4/CW /DD/D7/CX /D7/BN/D3/D2/CT /CS/CT/AS/D2/CT/D7 /D8/CW/CP/D8c2τ2 0=−lµ elµe=−lµ rlµr./CC/CW/CT/D7/CT /CT/DC/CP/D1/D4/D0/CT/D7 /D4/D6/D3 /DA/CX/CS/CT /CP /D2/CX /CT /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /D8/D3 /CS/CX/D7 /D3 /DA /CT/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CX/D2 /D8/CW/CT /D3/D2 /CT/D4/D8 /D3/CU /D8/CW/CT /D7/CP/D1/CT/D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /D8/CW/CT /D9/D7/D9/CP/D0 /D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP /CW /D8/D3 /CB/CA/BA/CC/CW/CT /D9/D7/D9/CP/D0 /D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP /CW /CS/D3 /CT/D7 /D2/D3/D8 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CT/BA/CV/BA/B8 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB /B4/D3/D6 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD /D8/CW/CT /BV/BU/BZ/C9 lµ eeµ, /CT/D8 /BA/B5/B8 /CQ/D9/D8 /D3/D2/D0/DD /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/B8 lµ e /CP/D2/CSlν′ e, /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /CC/CW/CT /CQ /CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2/D8/D7 /B4/CT/BA/CV/BA/B8 lµ e /CP/D2/CSlν′ e /B5 /CP/D6 /CT /D3/D2/D7/CX/CS/CT/D6 /CT /CS /D8/D3 /CQ /CT /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2/D8/CX/D8/DD/CU/D3/D6 /CS/CX/AR/CT/D6 /CT/D2/D8 /D3/CQ/D7/CT/D6/DA/CT/D6/D7 /CU/D6 /D3/D1 /D8/CW/CT /D4 /D3/CX/D2/D8 /D3/CU /DA/CX/CT/DB /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /D3/DA/CP/D6/CX/CP/D2/D8 /CP/D4/D4/D6 /D3 /CP /CW /D8/D3 /CB/CA/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CX/D2 /D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/D5/D9/CP/D0/CX/D8/CX/CT/D7 /CU/D3/D6 /D8/CW/CT /BV/BU/BZ/C9/D7/B8 /D8/CW/CT /D7/CT/D8/D7 /D3/CU /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/B8 lµ e /CP/D2/CSlν′ e, /D8/CP/CZ /CT/D2 /CP/D0/D3/D2/CT/B8 /CP/D6/CT/D2/D3/D8 /CT/D5/D9/CP/D0/B8 lµ e/ne}ationslash=lν′ e, /CP/D2/CS /D8/CW /D9/D7 /D8/CW/CT/DD /CP/D6/CT /D2/D3/D8 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DA/CX/CT/DB/D4 /D3/CX/D2 /D8/BA/BY /D6/D3/D1 /D8/CW/CT /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU/B8 /CT/BA/CV/BA/B8 /CP(1,0) /D8/CT/D2/D7/D3/D6 /CP/D6/CT /CX/D8/D7 /DA /CP/D0/D9/CT/D7 /B4/D6/CT/CP/D0/D2 /D9/D1 /CQ /CT/D6/D7/B5 /DB/CW/CT/D2 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D2/CT/B9/CU/D3/D6/D1/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 eα, /CX/D7 /CX/D8/D7 /CP/D6/CV/D9/D1/CT/D2 /D8 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BL℄/B5/BA /CC/CW /D9/D7/B8 /CU/D3/D6/CT/DC/CP/D1/D4/D0/CT/B8 la AB(eα) =lµ eeµ(eα) =lα e /B4/DB/CW/CT/D6/CT eα/CX/D7 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D2/CT/B9/CU/D3/D6/D1 /CX/D2 /CP/D2 /C1/BY/CAS /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/B8 /DB/CW/CX/D0/CT la AB(eα′) =lµ′ eeµ′(eα′) =lα′ e /B4/DB/CW/CT/D6/CT eα′/CX/D7 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D2/CT/B9/CU/D3/D6/D1 /CX/D2S′/CP/D2/CS/CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD lα e /CP/D2/CSlα′ e /CP/D6/CT /D2/D3/D8 /D8/CW/CT /D7/CP/D1/CT /D6/CT/CP/D0 /D2 /D9/D1 /CQ /CT/D6/D7 /D7/CX/D2 /CT /D8/CW/CT /CQ/CP/D7/CX/D7/D3/D2/CT/B9/CU/D3/D6/D1/D7 eα/CP/D2/CSeα′/CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CQ/CP/D7/CT/D7/BA /C1/D8 /CX/D7 /D8/D6/D9/CT /D8/CW/CP/D8 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D7/D3/D1/CT /D8/CT/D2/D7/D3/D6 /D6 /CT/CU/CT/D6/D8/D3 /D8/CW/CT /D7/CP/D1/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7 S /CP/D2/CSS′/CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CQ/D9/D8 /D8/CW/CT/DD /CP/D6 /CT /D2/D3/D8 /CT /D5/D9/CP/D0 /D7/CX/D2 /CT /D8/CW/CT /CQ/CP/D7/CT/D7 /CP/D6/CT /D2/D3/D8 /CX/D2 /D0/D9/CS/CT/CS/BA/BE/BA/BE /CC/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /D8/CW/CT /BT /CC /D3/CU /D7/D4 /CT /CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /CS/CX/D7/D8/CP/D2 /CT/D7/BT/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS /D8/CW/CT /BT /CC /D6/CT/CU/CT/D6 /CT/DC /D0/D9/D7/CX/DA /CT/D0/DD /D8/D3 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /D3/CU /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D2/CS /CX/D2 /D8/CW/CP/D8 /CU/D3/D6/D1/D8/CW/CT/DD /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D3/D2/D0/DD /D7/D3/D1/CT /D3/D1/D4 /D3/D2/CT/D2/D8/D7 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /CB/D9 /CW /CP /D4/CP/D6/D8 /D3/CU /CP /BG/BW /D5/D9/CP/D2 /D8/CX/D8 /DD /B8/DB/CW/CT/D2 /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7 /B4/D3/D6 /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /D7/D3/D1/CT /C1/BY/CA/B5/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3/CS/CX/AR/CT/D6/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/BA /CC/CW/CT /D9/D7/D9/CP/D0/B8 /CX/BA/CT/BA/B8 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA /CX/D7 /CQ/CP/D7/CT/CS /D3/D2 /D8 /DB /D3/D4 /D3/D7/D8/D9/D0/CP/D8/CT/D7/BM /D8/CW/CT /D4/D6/CX/D2 /CX/D4/D0/CT /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CP/D2/CS /D8/CW/CT /D4 /D3/D7/D8/D9/D0/CP/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 /D3/D2/CT/B9/DB /CP /DD /B8 /D7/D4 /CT/CT/CS /D3/CU /D0/CX/CV/CW /D8/CX/D7 /CX/D7/D3/D8/D6/D3/D4/CX /CP/D2/CS /D3/D2/D7/D8/CP/D2 /D8/BA /C1/D2 /D8/CW/CP/D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D8/CW/CT /BT /CC /D3/CU /D8/CW/CT /D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7/D0/DD /CS/CT/AS/D2/CT/CS /D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/CJ/BF ℄ /CP/D2/CS /D8/CW/CT /BT /CC /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6 /CP/D0 /CS/CX/D7/D8/CP/D2 /CT /CJ/BF℄ /CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D8/CW/CT /D1/CP/CX/D2 /AH/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /AH /D3/D2/D7/CT/D5/D9/CT/D2 /CT/D7 /D3/CU/D8/CW/CT /D4 /D3/D7/D8/D9/D0/CP/D8/CT/D7/BA /C6/CP/D1/CT/D0/DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/CT/D6/CX/DA /CT/CS /CU/D6/D3/D1 /D8/CW/CT /D8 /DB /D3 /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D4 /D3/D7/D8/D9/D0/CP/D8/CT/D7/CP/D2/CS /D8/CW/CT/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D3/D2 /D8/D6/CP /D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT /CP/D6/CT /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CP/D7 /D8/CW/CP/D8 /D8/CW/CT/DD /CP/D6/CT /D8/CW/CT/C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /CS/CX/D7/D8/CP/D2 /CT/D7/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/CP/D6/CT /D8/CW/CT /CC/CC/B8 /CP/D7 /CP/D2 /CQ /CT /D7/CT/CT/D2 /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT /CT/CS/CX/D2/CV /D7/CT /D8/CX/D3/D2/D7/BN /D8/CW/CT/DD /CP/D0/DB /CP /DD/D7 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D8/CW/CT /DB/CW/D3/D0/CT /BG/BW /D8/CT/D2/D7/D3/D6/BI/D5/D9/CP/D2 /D8/CX/D8 /DD /CP/D2/CS /D8/CW /D9/D7 /D8/CW/CT/DD /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/B8 /D7/CT/CT/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BK/B5 /CP/D2/CS/B4/BD/BD/B5/B8 /D3/D6 /B4/BL/B5 /CP/D2/CS /B4/BD/BE/B5/BA /CB/CX/D2 /CT /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /CP/D6/CT 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/D6/CX/D4/D8/CX/D3/D2 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /CP/D1/D3 /DA/CX/D2/CV /CQ /D3 /CS/DD /CX/D7 /CS/CT/AS/D2/CT/CS /CP/D7 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D4 /D3/CX/D2 /D8/D7 /D3/D2 /CX/D8/B8 /CP/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD/CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CQ /D3 /CS/DD /BA /C6/CP/D1/CT/D0/DD /CX/D2 /D8/CW/CT /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB=xa B−xa A /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /B4/DB/CX/D8/CW /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT /D8/D3/D6/D7 xa A /CP/D2/CSxa B /B5 /CX/D7 /DB/D6/CX/D8/D8/CT/D2/D3/D2/D0/DD /CX/D2 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C1/D2S, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CQ /D3 /CS/DD /B8 /CX/D8/CX/D7 /B4/CX/D2 /BE/BW /D7/D4/CP /CT/D8/CX/D1/CT/B5 lµ AB= (0, L0) /B4L0 /CX/D7 /D8/CW/CT /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW /CP/D2/CS /CX/D8 /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7/D0/DD /CX/D2 S /B5/BA /C1/D2S′, /DB/CW/CT/D6/CT /D8/CW/CT /CQ /D3 /CS/DD /CX/D7 /D1/D3 /DA/CX/D2/CV/B8 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CUla AB /CX/D7 lµ′ AB= (−βeγeL0, γeL0). /C6/D3 /DB /D3/D1/CT/D7 /D8/CW/CT /D1/CP/CX/D2 /D4 /D3/CX/D2 /D8 /CX/D2 /D8/CW/CT /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/BA /C1/D8 /CX/D7 /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS/CX/D2 /D8/CW/CT /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA /D8/CW/CP/D8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8l1′ AB=γeL0=L′/D3/CUlµ′ AB /CX/D7 /D8/CW/CT/AH/CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7/AH /D0/CT/D2/CV/D8/CW L′/B8 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7/D0/DD /B4/D7/CX/D2 /CT /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /CX/D7/ne}ationslash= 0 /B5/B8 /CX/D2 /D8/CW/CT/CU/D6/CP/D1/CT S′/CX/D2 /DB/CW/CX /CW /D8/CW/CT /CQ /D3 /CS/DD /CX/D7 /D1/D3 /DA/CX/D2/CV/BA /C7/D2/CT /CP/D2 /D7/CP /DD /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP /C4/D3/D6/CT/D2 /D8/DE /D0/CT/D2/CV/D8/CW/CT/D2/CX/D2/CV /CX/D2 /D8/CW/CT/CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/B8 /CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /C4/D3/D6/CT/D2 /D8/DE /D3/D2 /D8/D6/CP /D8/CX/D3/D2 /D8/CW/CP/D8 /CT/DC/CX/D7/D8/D7 /CX/D2 /D8/CW/CT /AH/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7/B8/AH/CX/BA/CT/BA/B8 /D8/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA/BA /C1/D8 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8/CW/CT /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D8/CW/CP/D8L′/CX/D2 S′/CP/D2/CSL0 /CX/D2S /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /CC/CW/CT /D3/D1/D1/D3/D2 /CU/CT/CP/D8/D9/D6/CT /CU/D3/D6 /CQ /D3/D8/CW /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /CX/D7 /D8/CW/CP/D8 /D8/CW/CT/D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /CP /D1/D3/DA/CX/D2/CV /CQ /D3 /CS/DD /CX/D7 /CP/D7/D7/D9/D1/CT /CS /D8/D3 /CQ /CT /CP /DB/CT/D0 /D0 /CS/CT/AS/D2/CT /CS /D4/CW/DD/D7/CX /CP/D0 /D5/D9/CP/D2/D8/CX/D8/DD /CX/D2 /BG/BW /D7/D4 /CP /CT/D8/CX/D1/CT/BA/C7/D9/D6 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6/D7 /B4/D3/D6 /D8/CW/CT /BV/BU/BZ/C9/D7/B5 /D6/CT/DA /CT/CP/D0/D7 /D8/CW/CP/D8 /D8/CW/CX/D7 /CX/D7 /D2/D3/D8 /D8/D6/D9/CT/BN /CP /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS/D4/CW /DD/D7/CX /CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT /D8/CW/CP/D8 /CX/D7 /D3/D2/D2/CT /D8/CT/CS /DB/CX/D8/CW /CP /D1/D3 /DA/CX/D2/CV /CQ /D3 /CS/DD /CP/D2 /CQ /CT /D3/D2/D0/DD /CP /BG/BW /D8/CT/D2/D7/D3/D6/D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CT/BA/CV/BA/B8 /CT/CX/D8/CW/CT/D6 /D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /B4/BD/BC/B5/B8 /D3/D6 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB=xa B−xa A. /C1/CU/B8/CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D3/D2/CT /CS/D3 /CT/D7 /D2/D3/D8 /D9/D7/CT /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CQ/D9/D8 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D8/CW/CT/D2 /CQ /D3/D8/CW/CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /B4/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CP/D2/CS /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7/B5/B8 /DB/CW/CX /CW /CS/CT/CP/D0 /DB/CX/D8/CW /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /CP/D7 /CP /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS/BL/D4/CW /DD/D7/CX /CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CQ /CT /D3/D1/CT /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7/BA /C1/D8 /CX/D7 /D0/CT/CP/D6 /CU/D6/D3/D1 /D8/CW/CT /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /CX/D2 /CB/CT /D7/BA /BE /CP/D2/CS /BE/BA/BD /D8/CW/CP/D8 /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D3/D2/D0/DD /D7/D4/CP/D8/CX/CP/D0 /B4/D3/D6 /D8/CT/D1/D4 /D3/D6/CP/D0/B5 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6 la AB /CX/D2 S /CP/D2/CSS′/CX/D7 /D4/CW /DD/D7/CX /CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /D7/CX/D2 /CT /D7/D3/D1/CT /D3/D1/D4 /D3/D2/CT/D2/D8/D7 /D3/CU /CP /BG/BW /D8/CT/D2/D7/D3/D6/D5/D9/CP/D2/D8/CX/D8/DD/B8 /DB/CW/CT/D2 /D8/CW/CT/DD /CP/D6 /CT /D8/CP/CZ/CT/D2 /CP/D0/D3/D2/CT/B8 /CS/D3 /D2/D3/D8 /CP /D8/D9/CP/D0 /D0/DD /D6 /CT/D4/D6 /CT/D7/CT/D2/D8 /CP/D2/DD /BG/BW /D4/CW/DD/D7/CX /CP/D0 /D5/D9/CP/D2/D8/CX/D8/DD/BA /BT/D0/D7/D3 /DB /CT/D6/CT/D1/CP/D6/CZ /D8/CW/CP/D8 /D8/CW/CT /DB/CW/D3/D0/CT /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD la AB /D3/D1/D4/D6/CX/D7/CX/D2/CV /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D2/CS /D8/CW/CT /CQ/CP/D7/CX/D7 /CX/D7 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CQ /DD/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CU/D6/D3/D1S /D8/D3S′. /CC/CW/CX/D7 /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D3/CU /CB/CA /CP/D0/D7/D3 /CQ /CT/D0/D3/D2/CV/D7 /D8/D3 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/CC/CW/CT /D2/CT/DC/D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /DB/CW/CX /CW /DB/CX/D0/D0 /CQ /CT /CT/DC/CP/D1/CX/D2/CT/CS /CX/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /B4/D3/D6 /D6/CP/CS/CP/D6/B5 /D0/CT/D2/CV/D8/CW /CJ/BD/BL ℄/BA /B4/C7/D2/CT /CP/D2/D7/D4 /CT/CP/CZ /CP/CQ /D3/D9/D8 /D8/CW/CT /D6/CP/CS/CP/D6 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA/BA/B5 /C1/D8 /CX/D7 /CP/D7/D7/D9/D1/CT/CS /CX/D2 /CJ/BD/BL ℄ /D8/CW/CP/D8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D0/CT/D2/CV/D8/CW /B4/D8/CW/CT/D0/CT/D2/CV/D8/CW /D3/CU /CP /CU/CP/D7/D8/B9/D1/D3 /DA/CX/D2/CV /D6/D3 /CS/B5 /CX/D7 /CS/CT/AS/D2/CT/CS /CP/D7 /B4/D8/CW/CT /D8/CW/CX/D6/CS /CP/D6/D8/CX /D0/CT /CX/D2 /CJ/BD/BL ℄/B5/BM /AH/D8/CW/CT /CW/CP/D0/CU/B9/D7/D9/D1 /D3/CU /CS/CX/D7/D8/CP/D2 /CT/D7 /D3 /DA /CT/D6/CT/CS /CQ /DD /CP /D0/CX/CV/CW /D8 /D7/CX/CV/D2/CP/D0 /CX/D2 /CS/CX/D6/CT /D8 /CP/D2/CS /D3/D4/D4 /D3/D7/CX/D8/CT /CS/CX/D6/CT /D8/CX/D3/D2/D7 /CP/D0/D3/D2/CV /D8/CW/CT /D6/D3 /CS/BA/AH /C1/D2 /D8/CW/CT /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/CB/D8/D6/CT/D0/B3/D8/D7/D3 /DA /CS/CT/AS/D2/CT/D7 /D8/CW/CT /BG/B9/DA /CT /D8/D3/D6 /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D0/CT/D2/CV/D8/CW lµ rel /B4/CP /D8/D9/CP/D0/D0/DD /D8/CW/CX/D7 /D0/CT/D2/CV/D8/CW /CX/D7 /D2/D3/D8 /D8/CW/CT /BG/B9/DA /CT /D8/D3/D6 /CQ/D9/D8/CX/D8 /CX/D7 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CP /BG/B9/DA /CT /D8/D3/D6/B5 /CP/D7/BM /AH/D8/CW/CT /CW/CP/D0/CU/B9/CS/CX/AR/CT/D6/CT/D2 /CT /D3/CU /D8 /DB /D3/D0/CX/CV/CW /D8 /BG/B9/DA /CT /D8/D3/D6/D7 /B4/CX/BA/CT/BA/B8 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1/B5 lµ d /CP/D2/CSlµ b /DB/CW/CX /CW /CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D4/D6/D3 /CT/D7/D7/CT/D7 /D3/CU/D0/CX/CV/CW /D8 /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2 /B4/CX/D2 /D8/CW/CT /CS/CX/D6/CT /D8 /CP/D2/CS /D3/D4/D4 /D3/D7/CX/D8/CT /CS/CX/D6/CT /D8/CX/D3/D2/D7/B5/BA/AH /CC/CW/CT/D2 , /CX/D2S, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D6/D3 /CS/B8 lµ d= (cL0/c, L 0,0,0) /CP/D2/CSlµ b= (cL0/c,−L0,0,0), /DB/CW/CX/D0/CT /CX/D2S′, /DB/CW/CT/D6/CT /D8/CW/CT /D6/D3 /CS /CX/D7 /D1/D3 /DA/CX/D2/CV/B8 /D8/CW/CT/DD /CP/D6/CTlµ′ d= (cγL0(1+β)/c), γL0(1+β),0,0), /CP/D2/CSlµ′ b= (cγL0(1−β)/c),−γL0(1−β),0,0). /CC/CW/CT/D2 /CT /CX/D2S /D3/D2/CT /AS/D2/CS/D7 lµ rel= (lµ d−lµ b)/2 = (0 , L0,0,0) /CP/D2/CS /CX/D2S′/D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /D3/CU /D8/CW/CX/D7 /BG/B9/DA /CT /D8/D3/D6 /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D0/CT/D2/CV/D8/CW/CX/D7lµ′ rel= (γβL0, γL0,0,0). /C6/D3 /DB /CB/D8/D6/CT/D0/B3/D8/D7/D3 /DA/B8 /CX/D2 /D8/CW/CT /D7/CX/D1/CX/D0/CP/D6 /DB /CP /DD /CP/D7 /CX/D2 /D8/CW/CT /CP/D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/B8 /D3/D1/D4/CP/D6/CT/D7 /D3/D2/D0/DD /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 /D3/CUlµ′ rel /CP/D2/CSlµ rel /CP/D2/CS /CS/CT/AS/D2/CT/D7 /D8/CW/CP/D8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D0/CT/D2/CV/D8/CW /CX/D2S′/CX/D7 l′ rel≡l1′ rel, /DB/CW/CX /CW /CX/D7 /D6/CT/D0/CP/D8/CT/CS /DB/CX/D8/CWlrel≡l1 rel /CX/D2S /CQ /DD /D8/CW/CT /AH/CT/D0/D3/D2/CV/CP/D8/CX/D3/D2 /CU/D3/D6/D1 /D9/D0/CP/AH l′ rel=γlrel. /CC/CW/CT/D7/CT/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 l1′ rel /CP/D2/CSl1 rel /CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/D3 /CQ /CT /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D3/D6 /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2S′/CP/D2/CS /CX/D2S. /C1/D8/CX/D7 /CP/D6/CV/D9/CT/CS /CX/D2 /CJ/BD/BL ℄ /D8/CW/CP/D8 /D7/D9 /CW /AH/CP/D4/D4/D6/D3/CP /CW /CW/CP/D7 /CP /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D3 /DA /CP/D6/CX/CP/D2 /D8 /CW/CP/D6/CP 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/CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D6/CP/CS/CX/D3 /D8/CX/DA /CT/B9/CS/CT /CP /DD /D4/D6/D3 /CT/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT/D3/D6/DD /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7 dN/dt =−λN, N s=Nmexp(−t/τ). /B4/BD/BL/B5/CC/CW/CT /D8/D6/CP /DA /CT/D0 /D8/CX/D1/CTtE /CX/D7 /D2/D3/D8 /CS/CX/D6/CT /D8/D0/DD /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /D0/D3 /CZ/D7/B8 /CQ/D9/D8/B8 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B8 /CX/D8 /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/CP/D7 /D8/CW/CT /D6/CP/D8/CX/D3 /D3/CU /D8/CW/CT /CW/CT/CX/CV/CW /D8 /D3/CU /D8/CW/CT /D1/D3/D9/D2 /D8/CP/CX/D2 HE /CP/D2/CS /D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /D3/CU /D8/CW/CT /D1 /D9/D3/D2/D7 v /B8tE=HE/v./CC/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/BL/B5 /CW/D3/D0/CS/D7 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /D8/D3 /D3/B8 /D7/CX/D2 /CT /D8/CW/CT /D8 /DB /D3 /CU/D6/CP/D1/CT/D7 /CP/D6/CT /D3/D2/D2/CT /D8/CT/CS /CQ /DD /D8/CW/CT /BZ/CP/D0/CX/D0/CT/CP/D2 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/B8 /CP/D2/CS/B8 /CP/D7 /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CP/CQ /D3 /DA /CT/B8 /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/CX/D1/CT/D7 /CP/D6/CT/CT/D5/D9/CP/D0/B8 tE=tµ /CP/D2/CSτE=τµ. /C0/CT/D2 /CT /DB /CT /D3/D2 /D0/D9/CS/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT/D3/D6/DD /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0/CU/CP /D8/D3/D6/D7 /CP/D6/CT /D8/CW/CT /D7/CP/D1/CT /CX/D2 /CQ /D3/D8/CW /CU/D6/CP/D1/CT/D7 /CP/D2/CS /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /AT/D9/DC/CT/D7 /CX/D2 /D8/CW/CT /D8 /DB /D3 /CU/D6/CP/D1/CT/D7/CP/D6/CT /CT/D5/D9/CP/D0/B8 Nsµ /BPNsE /CP/D2/CSNmµ=NmE /B8 /CP/D7 /CX/D8 /D1 /D9/D7/D8 /CQ /CT/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D7/CW/D3 /DB /D8/CW/CP/D8 /D8/CW/CT/BD/BD/CP /D8/D9/CP/D0 /AT/D9/DC 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/BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B8 /CQ/D9/D8 /CX/D2 /D8/CW/CT /D1 /D9/D3/D2/CU/D6/CP/D1/CT /D3/D2/CT /CT/DC/D4/D0/CP/CX/D2/D7 /D8/CW/CT /CS/CP/D8/CP /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /AH /D3/D2 /D8/D6/CP /D8/CX/D3/D2/BA/AH /C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /CT/DC/D4/D0/D3/CX/D8 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7/D3/CU /CB/CT /D7/BA /BE/BA/BD /CP/D2/CS /BE/BA/BE /DB /CT /CP/D2/CP/D0/DD/D7/CT /D8/CW/CT /AH/D1 /D9/D3/D2/AH /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /D2/D3/D8 /D3/D2/D0/DD /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CQ/D9/D8/CP/D0/D7/D3 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /BT/D7 /D7/CW/D3 /DB/D2 /CX/D2 /CB/CT /BA /BE/BA/BE /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CP/D8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0/CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CP/D6/CT /DB /CT/D0/D0/B9/CS/CT/AS/D2/CT/CS /D4/CW /DD/D7/CX /CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/BA/CC/CW/CT/D2/B8 /CP/D7 /CX/D2 /D8/CW/CT /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT/D3/D6/DD /B8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/B9/D8/CX/DA/CX/D8 /DD/AH /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7 dN/dx0=−λN, N s=Nmexp(−λx0). /B4/BE/BC/B5/CC/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BC/B5 /D3/D2 /D8/CP/CX/D2/D7 /CP /D7/D4 /CT /CX/AS /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 /D8/CW/CTx0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 /DB/CW/CX /CW /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/B4/BE/BC/B5 /DB/CX/D0/D0 /D2/D3/D8 /D6/CT/D1/CP/CX/D2 /D9/D2 /CW/CP/D2/CV/CT/CS /D9/D4 /D3/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/B8 /CX/BA/CT/BA/B8 /CX/D8 /DB/CX/D0/D0 /D2/D3/D8 /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT/CU/D3/D6/D1 /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7 /B4/CP/D2/CS /CP/D0/D7/D3 /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/B5/BA /BU/D9/D8 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D8 /CX/D7/D2/D3/D8 /D6/CT/D5/D9/CX/D6/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D4/CW /DD/D7/CX /CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D1 /D9/D7/D8 /CQ /CT /D8/CW/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D8/CW/CP/D8 /D3/D6/D6/CT /D8/D0/DD /D8/D6/CP/D2/D7/CU/D3/D6/D1/D9/D4 /D3/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/BA /CC/CW /D9/D7 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /B4/BE/BC/B5 /CP/D6/CT /D2/D3/D8 /D8/CW/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CX/BA/CT/BA/B8/D8/CW/CT/DD /CP/D6/CT /D2/D3/D8 /D8/CW/CT /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6/D7 /D3/D6 /D8/CW/CT /BV/BU/BZ/C9/D7/BA /CC/CW/CX/D7 /DB/CX/D0/D0 /CP/D9/D7/CT /D8/CW/CP/D8 /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/CW/CT/D2/D3/D1/CT/D2/CP /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8/C1/BY/CA/D7 /DB/CX/D0/D0 /D2/CT/CT/CS /D8/D3 /CQ /CT /CX/D2 /DA /D3/CZ /CT/CS /D8/D3 /CT/DC/D4/D0/CP/CX/D2 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX /CP/D0 /CT/AR/CT /D8/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /D7/CP/D1/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/CS/CP/D8/CP/BA /C1/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /DB /CT /CP/D2 /DB/D6/CX/D8/CT /CX/D2 /B4/BE/BC/B5 /D8/CW/CP/D8x0 E=ctE, λE= 1/cτE, /DB/CW/CX /CW /CV/CX/DA /CT/D7 /D8/CW/CP/D8 /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D0/CP /DB /CQ /CT /D3/D1/CT/D7 NsE=NmEexp(−tE/τE). /C1/D2 /D8/CW/CT/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BF ℄NsE, NmE, /CP/D2/CStE=HE/v /CP/D6/CT /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /B4/D8/CP /CX/D8/D0/DD /CP/D7/D7/D9/D1/CX/D2/CV/D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT /D3/CU /D1 /D9/D3/D2/D7 /CX/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8/CW/CT/CX/D6 /D6/CT/D7/D8 /CU/D6/CP/D1/CT/BA /C6/D3 /DB/B8/CX/D2 /D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT/D3/D6/DD /DB/CW/CT/D6/CT τE=τµ /CP/D2/CStE=tµ, /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D7/D7/D9/D1/CT/D7/D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT/D6/CT /CX/D7 /D8/CW/CT /D8/CX/D1/CT /AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/AH /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /B4/BD/BJ/B5/B8 /DB/CW/CX /CW /CV/CX/DA /CT/D7 /D8/CW/CT /D3/D2/D2/CT /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT/D7 /D3/CU /D1 /D9/D3/D2/D7 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT τE /CP/D2/CS /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D0/CX/CU/CT/D8/CX/D1/CT /CX/D2 /D8/CW/CT/D1 /D9/D3/D2 /CU/D6/CP/D1/CT τµ /CP/D7 τE=γτµ. /B4/BE/BD/B5/CD/D7/CX/D2/CV /D8/CW/CP/D8 /D6/CT/D0/CP/D8/CX/D3/D2 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CP/D8 /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D0/CP /DB/B8 /DB/CW/CT/D2 /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D1/CT/CP/B9/D7/D9/D6/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CQ /CT /D3/D1/CT/D7 NsE=NmEexp(−tE/τE) =NmEexp(−tE/γτµ). /B4/BE/BE/B5/CC/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D9/D7/CT/CS /CX/D2 /CJ/BE/BF ℄ /D8/D3 /D1/CP/CZ /CT /D8/CW/CT /AH/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CP/D2/CS /D3/D1/D4/CP/D6/CT /CX/D8 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/B9/D1/CT/D2 /D8/CP/D0 /CS/CP/D8/CP/BA /C1/D2 /CU/CP /D8/B8 /CX/D2 /CJ/BE/BF ℄/B8 /D8/CW/CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /CX/D7 /D1/CP/CS/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D4/D6/CT/CS/CX /D8/CT/CS /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /CU/CP /D8/D3/D6 γ /D3/CU /D8/CW/CT /D1 /D9/D3/D2/D7 /CP/D2/CS /CP/D2 /D3/CQ/D7/CT/D6/DA /CT/CS γ. /CC/CW/CT /D4/D6/CT/CS/CX /D8/CT/CS γ /CX/D78.4±2, /DB/CW/CX/D0/CT /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS γ /CX/D7 /CU/D3/D9/D2/CS /D8/D3 /CQ /CT 8.8±0.8 /B8 /DB/CW/CX /CW /CX/D7 /CP /D3/D2 /DA/CX/D2 /CX/D2/CV /CP/CV/D6/CT/CT/D1/CT/D2 /D8/BA /CC/CW/CT /D4/D6/CT/CS/CX /D8/CX/D3/D2 /D3/CUγ /CX/D7 /D1/CP/CS/CT /CU/D6/D3/D1 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /CT/D2/CT/D6/B9/CV/CX/CT/D7 /D3/CU /D1 /D9/D3/D2/D7 /D3/D2 /D8/CW/CT /D1/D3/D9/D2 /D8/CP/CX/D2 /CP/D2/CS /CP/D8 /D7/CT/CP /D0/CT/DA /CT/D0/BN /D8/CW/CT/D7/CT /CT/D2/CT/D6/CV/CX/CT/D7 /CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS/CP/D1/D3/D9/D2 /D8 /D3/CU /D1/CP/D8/CT/D6/CX/CP/D0 /DB/CW/CX /CW /D1 /D9/D3/D2/D7 /D4 /CT/D2/CT/D8/D6/CP/D8/CT/CS /DB/CW/CT/D2 /D7/D8/D3/D4/D4 /CT/CS/B8 /CP/D2/CS /D8/CW/CT/D2 /D8/CW/CT /CT/D2/CT/D6/CV/CX/CT/D7 /CP/D6/CT /D3/D2 /DA /CT/D6/D8/CT/CS /D8/D3/D8/CW/CT /D7/D4 /CT/CT/CS/D7 /D3/CU /D8/CW/CT /D1 /D9/D3/D2/D7 /D9/D7/CX/D2/CV /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8/D3/D8/CP/D0 /CT/D2/CT/D6/CV/DD /CP/D2/CS /D8/CW/CT /D7/D4 /CT/CT/CS/BA /CC/CW/CT/D3/CQ/D7/CT/D6/DA /CT/CS γ /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BE/BE/B5/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D6/CP/D8/CT/D7 /DB /CT/D6/CTNsE= 397 ±9 /CP/D2/CS NmE= 550 ±10, /CP/D2/CS /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /CW/CT/CX/CV/CW /D8 /D3/CU /D8/CW/CT /D1/D3/D9/D2 /D8/CP/CX/D2 /CX/D7HE= 1907 m. /CC/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT /D3/CU /D1 /D9/D3/D2/D7 τµ /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /CX/D7 /D8/CP/CZ /CT/D2 /CP/D7 /D8/CW/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CU/D6/D3/D1 /D3/D8/CW/CT/D6 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /B4/CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D3/CQ/D8/CP/CX/D2 /D1/D3/D6/CT/CP /D9/D6/CP/D8/CT /D6/CT/D7/D9/D0/D8/B5 /CP/D2/CS /CX/D8 /CX/D7τµ= 2.211·10−6s./C4/CT/D8 /D9/D7 /D2/D3 /DB /D7/CT/CT /CW/D3 /DB /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CP/D6/CT /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT/BA /B4/CF /CT /D2/D3/D8/CT /D8/CW/CP/D8 /CJ/BE/BF ℄ /D3/D1/D4/CP/D6/CT/CS /D8/CW/CT /D8/CW/CT/D3/D6/DD /B4/D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B5 /CP/D2/CS /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D3/D2/D0/DD /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B8 /CQ/D9/D8 /D9/D7/CX/D2/CV τµ/CU/D6/D3/D1 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT/BA/B5 /BY/CX/D6/D7/D8 /DB /CT /CW/CP /DA /CT /D8/D3 /AS/D2/CS /D8/CW/CT /CU/D3/D6/D1 /D3/CU /D8/CW/CT /D0/CP /DB /CU/D3/D6 /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D4/D6/D3 /CT/D7/D7/CT/D7/B4/BE/BC/B5 /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT/BA /BT/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/CQ /D3 /DA /CT /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D0/CP /DBNsE=NmEexp(−tE/τE)/CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BC/B5 /D9/D7/CX/D2/CV /D8/CW/CT/D6/CT/D0/CP/D8/CX/D3/D2/D7 x0 E=ctE /CP/D2/CSλE= 1/cτE. /BU/D9/D8/B8 /CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BC/B5 /CS/D3 /CT/D7 /D2/D3/D8 /D6/CT/D1/CP/CX/D2/D9/D2 /CW/CP/D2/CV/CT/CS /D9/D4 /D3/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA /BT /D3/D6/CS/CX/D2/CV/D0/DD /CX/D8 /CP/D2/D2/D3/D8 /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT /CU/D3/D6/D1 /CX/D2 /D8/CW/CT/BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT/BA /CB/D3/B8 /CP /D8/D9/CP/D0/D0/DD /B8 /CX/D2 /D8/CW/CT /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/B8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT/BD/BE/D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D4/D6/D3 /CT/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /D3/D9/D0/CS /CW/CP /DA /CT/B8 /CX/D2 /D4/D6/CX/D2 /CX/D4/D0/CT/B8 /CP /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D9/D2 /D8/CX/D3/D2/CP/D0 /CU/D3/D6/D1/D8/CW/CP/D2 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BE/B5/B8 /DB/CW/CX /CW /CS/CT/D7 /D6/CX/CQ /CT/D7 /D8/CW/CT /D7/CP/D1/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9 /CS/CT /CP /DD /D4/D6/D3 /CT/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/BA/C0/D3 /DB /CT/DA /CT/D6/B8 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CS/CT/D7/D4/CX/D8/CT /D3/CU /D8/CW/CT /CU/CP /D8 /D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT/D1 /D9/D3/D2 /CU/D6/CP/D1/CT /CP/D6/CT /D2/D3/D8 /D3/D2/D2/CT /D8/CT/CS /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/B8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D4/D6/D3 /CT/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BC/B5 /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7 /CX/D2 /D8/CW/CT/BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B8 /CX/BA/CT/BA/B8 /DB/D6/CX/D8/D8/CX/D2/CV /D8/CW/CP/D8x0 µ=ctµ, /CP/D2/CSλµ= 1/cτµ, /DB/CW/CT/D2 /CT Nsµ=Nmµexp(−tµ/τµ). /B4/BE/BF/B5/CC/CW/CT /CY/D9/D7/D8/CX/AS /CP/D8/CX/D3/D2 /CU/D3/D6 /D7/D9 /CW /CP /D4/D6/D3 /CT/CS/D9/D6/CT /CP/D2 /CQ /CT /CS/D3/D2/CT /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /DB /CP /DD /BA /C1/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CT/D4/D6/CX/D2 /CX/D4/D0/CT /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CP /D8/D7 /CP/D7 /D7/D3/D1/CT /D7/D3/D6/D8 /D3/CU /AH/BW/CT/D9/D7 /CT/DC /D1/CP /CW/CX/D2/CP/B8/AH /DB/CW/CX /CW /D6/CT/D7/D3/D0/DA /CT/D7 /D4/D6/D3/CQ/D0/CT/D1/D7/BN /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/B4/BE/BC/B5 /CX/D7 /D4/D6 /D3 /D0/CP/CX/D1/CT /CS /D8/D3 /CQ /CT /D8/CW/CT /D4/CW /DD/D7/CX /CP/D0 /D0/CP /DB /CP/D2/CS /D8/CW/CT /D4/D6/CX/D2 /CX/D4/D0/CT /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D6/CT/D5/D9/CX/D6/CT/D7 /D8/CW/CP/D8 /CP /D4/CW /DD/D7/CX /CP/D0/D0/CP /DB /D1 /D9/D7/D8 /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT /CU/D3/D6/D1 /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/BA /B4/CC/CW/CX/D7 /CX/D7 /D8/CW/CT /D9/D7/D9/CP/D0 /DB /CP /DD /CX/D2 /DB/CW/CX /CW /D8/CW/CT /D4/D6/CX/D2 /CX/D4/D0/CT/D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D7 /D9/D2/CS/CT/D6/D7/D8/D3 /D3 /CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/B5 /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D3/D2/CT /CP/D2 /DB/D6/CX/D8/CT /CX/D2 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BC/B5/D8/CW/CP/D8x0 E=ctE /CP/D2/CSλE= 1/cτE /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D2/CSx0 µ=ctµ, /CP/D2/CSλµ= 1/cτµ /CX/D2 /D8/CW/CT /D1 /D9/D3/D2/CU/D6/CP/D1/CT/BA /CF/CX/D8/CW /D7/D9 /CW /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2/D7 /D8/CW/CT /CU/D3/D6/D1 /D3/CU /D8/CW/CT /D0/CP /DB /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CX/D2 /CQ /D3/D8/CW /CU/D6/CP/D1/CT/D7/B8 /CP/D7 /CX/D8 /CX/D7 /D6/CT/D5/D9/CX/D6/CT/CS /CQ /DD/D8/CW/CT /D4/D6/CX/D2 /CX/D4/D0/CT /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA /CC/CW/CT/D2/B8 /CP/D7 /DB /CT /CW/CP /DA /CT /CP/D0/D6/CT/CP/CS/DD /D7/CT/CT/D2/B8 /DB/CW/CT/D2 /D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /CX/D7 /CS/D3/D2/CT /CX/D2 /D8/CW/CT/BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BE/BD/B5 /CU/D3/D6 /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /CX/D7 /D9/D7/CT/CS /D8/D3 /D3/D2/D2/CT /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8 /DB /D3 /CU/D6/CP/D1/CT/D7/B8/CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/D3 /D3/D2/D2/CT /D8 /D8/CW/CT/D1 /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/BA /CF/CW/CT/D2 /D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /CX/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS/CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /CP/D2/D3/D8/CW/CT/D6 /D6/CT/D0/CP/D8/CX/D3/D2 /CX/D7 /CX/D2 /DA /D3/CZ /CT/CS /D8/D3 /D3/D2/D2/CT /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8 /DB /D3 /CU/D6/CP/D1/CT/D7/BA /C6/CP/D1/CT/D0/DD /CX/D8 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /D8/CW/CT /D1/D3/D9/D2 /D8/CP/CX/D2 /CX/D7 /D1/D3 /DA/CX/D2/CV /CP/D2/CS /D8/CW/CT /D1 /D9/D3/D2/AH/D7/CT/CT/D7/AH /D8/CW/CT /CW/CT/CX/CV/CW /D8 /D3/CU /D8/CW/CT /D1/D3/D9/D2 /D8/CP/CX/D2 /C4/D3/D6/CT/D2 /D8/DE /D3/D2 /D8/D6/CP /D8/CT/CS/B8 Hµ=HE/γ, /B4/BE/BG/B5/DB/CW/CX /CW /CX/D7 /BX/D5/BA /B4/BD/BG/B5 /CU/D3/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D3/D2 /D8/D6/CP /D8/CX/D3/D2/B8 /CV/CX/DA/CX/D2/CV /D8/CW/CP/D8 tµ=Hµ/v=HE/γv=tE/γ. /B4/BE/BH/B5/CC/CW/CX/D7 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /CU/CP /D8/D3/D6 /CX/D2 /B4/BE/BF/B5 /CP/D7 /D8/CW/CP/D8 /D3/D2/CT /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CX/D2 /B4/BE/BE/B5/B8exp(−tµ/τµ) = exp(−tE/(γτµ)). /BY /D6/D3/D1 /D8/CW/CP/D8 /D6/CT/D7/D9/D0/D8 /CX/D8 /CX/D7 /D3/D2 /D0/D9/CS/CT/CS /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /AT/D9/DC/CT/D7 /CP/D6/CT /CT/D5/D9/CP/D0 /CX/D2 /D8/CW/CT /D8 /DB /D3 /CU/D6/CP/D1/CT/D7/B8 Nsµ /BPNsE=Ns /CP/D2/CS Nmµ=NmE=Nm. /CB/D8/D6/CX /D8/D0/DD /D7/D4 /CT/CP/CZ/CX/D2/CV/B8 /CX/D8 /CX/D7 /D2/D3/D8 /D8/CW/CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CT/D5/D9/CP/D0/CX/D8 /DD /D3/CU /AT/D9/DC/CT/D7/B8 /CQ/D9/D8 /D8/CW/CT /CT/D5/D9/CP/D0/CX/D8 /DD /D3/CU/D6/CP/D8/CX/D3/D7 /D3/CU /AT/D9/DC/CT/D7/B8 NsE/NmE=Nsµ/Nmµ /B8 /DB/CW/CX /CW /CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1 /D8/CW/CT /CT/D5/D9/CP/D0/CX/D8 /DD /D3/CU /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /CU/CP /D8/D3/D6/D7/CX/D2 /B4/BE/BE/B5 /CP/D2/CS /B4/BE/BF/B5/BA /C1/D2 /CJ/BE/BF ℄ /D8/CW/CT /D8/CX/D1/CTtµ /D8/CW/CP/D8 /D8/CW/CT /D1 /D9/D3/D2/D7 /D7/D4 /CT/D2 /D8 /CX/D2 /AT/CX/CV/CW /D8 /CP /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT/CX/D6 /D3 /DB/D2 /D0/D3 /CZ/D7/DB /CP/D7 /CX/D2/CU/CT/D6/D6/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /CS/CT /CP /DD /D8/CX/D1/CT/D7 /D3/CU /D1 /D9/D3/D2/D7 /CP/D8 /D6/CT/D7/D8/BA /CB/CX/D2 /CT /D8/CW/CT /D4/D6/CT/CS/CX /D8/CT/CS/AT/D9/DC/CT/D7 NsE /CP/D2/CSNmE /CP/D6/CT /CX/D2 /CP /D7/CP/D8/CX/D7/CU/CP /D8/D3/D6/DD /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D3/D2/CT/D7/B8 /CP/D2/CS /D7/CX/D2 /CT /D8/CW/CT /D8/CW/CT/D3/D6/DD/B4/DB/CW/CX /CW /CS/CT/CP/D0/D7 /DB/CX/D8/CW /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D3/D2 /D8/D6/CP /D8/CX/D3/D2/B5 /D4/D6/CT/CS/CX /D8/D7 /D8/CW/CT/CX/D6 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/D2/D8/CW/CT /CW/D3/D7/CT/D2 /CU/D6/CP/D1/CT/B8 /CX/D8 /CX/D7 /CV/CT/D2/CT/D6/CP/D0/D0/DD /CP /CT/D4/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D6/D6/CT /D8/D0/DD /CT/DC/D4/D0/CP/CX/D2/D7 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS/CS/CP/D8/CP/BA/CC/CW/CT /CP/CQ /D3 /DA /CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /CX/D7 /DB /D3/D6/CZ /CT/CS /D3/D9/D8 /D3/D2/D0/DD /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CQ/D9/D8 /D8/CW/CT /D4/CW /DD/D7/CX /D7 /CS/CT/D1/CP/D2/CS/D7/D8/CW/CP/D8 /D8/CW/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /D8/CW/CT /AT/D9/DC/CT/D7 /D3/D2 /D8/CW/CT /CW/D3/D7/CT/D2 /CU/D6/CP/D1/CT /D1 /D9/D7/D8 /CW/D3/D0/CS /CX/D2 /CP/D0/D0 /D4 /CT/D6/D1/CX/D7/D7/CX/CQ/D0/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/B9/D8/CX/D3/D2/D7/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT /DB /CT /D2/D3 /DB /CS/CX/D7 /D9/D7/D7 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BF ℄ /CU/D6/D3/D1 /D8/CW/CT /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /D3/CU /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/CQ/D9/D8 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /CC/CW/CT/D2/B8 /D9/D7/CX/D2/CV /B4/BE/BC/B5/B8 /DB /CT /CP/D2 /DB/D6/CX/D8/CT /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /AT/D9/DC/CT/D7 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CP/D7 Nr,sE=Nr,mEexp(−λr,Ex0 r,E) =Nr,mEexp(−x0 r,E/x0 r,E(τE)),/DB/CW/CT/D6/CT x0 r,E(τE) = 1 /λr,E. /BT/CV/CP/CX/D2/B8 /CP/D7 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /DB /CT /CW/CP /DA /CT /D8/D3 /CT/DC/D4/D6/CT/D7/D7 x0 r,E(τE) /CX/D2/D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD x0 r,µ(τµ) /D9/D7/CX/D2/CV /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BD/BK/B5 /CU/D3/D6 /D8/CW/CT /D8/CX/D1/CT/AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/AH /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 x0 r,E(τE) = (1 + 2 βr)1/2cτµ./C0/CT/D2 /CT/B8 /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D0/CP /DB /B4/BE/BC/B5/B8 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CP/D2/CS /DB/CW/CT/D2 /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CU/D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CQ /CT /D3/D1/CT/D7 Nr,sE=Nr,mEexp(−x0 r,E/(1 + 2 βr)1/2cτµ), /B4/BE/BI/B5/BD/BF/CP/D2/CS /CX/D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BE/BE/B5 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C1/CU /DB /CT /CT/DC/D4/D6/CT/D7/D7 βr /CX/D2 /D8/CT/D6/D1/D7 /D3/CUβ= v/c /CP/D7βr=β/(1−β) /B4/D7/CT/CT /B4/BJ/B5/B5 /CP/D2/CS /D9/D7/CT /B4/BI/B5 /D8/D3 /D3/D2/D2/CT /D8 /D8/CW/CT /AH/D6/AH /CP/D2/CS /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/B8 x0 r,E=x0 E− x1 E=ctE−HE, /D8/CW/CT/D2 /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /CU/CP /D8/D3/D6 /CX/D2 /B4/BE/BI/B5 /CQ /CT /D3/D1/CT/D7 = exp/braceleftBig −(ctE−HE)/[(1 +β)/(1−β)]1/2cτµ/bracerightBig ./CD/D7/CX/D2/CV HE=vtE /D8/CW/CX/D7 /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /CU/CP /D8/D3/D6 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT /CU/D3/D6/D1 /D8/CW/CP/D8 /D6/CT/D7/CT/D1 /CQ/D0/CT/D7 /D8/D3 /D8/CW/CP/D8 /D3/D2/CT /CX/D2/B4/BE/BE/B5/B8 /CX/BA/CT/BA/B8 /CX/D8 /CX/D7= exp( −tE/ΓrEτµ), /CP/D2/CS /B4/BE/BI/B5 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7 Nr,sE=Nr,mEexp(−tE/ΓrEτµ). /B4/BE/BJ/B5/CF /CT /D7/CT/CT /D8/CW/CP/D8γ= (1−β)−1/2/CX/D2 /B4/BE/BE/B5 /B4/D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5 /CX/D7 /D6/CT/D4/D0/CP /CT/CS /CQ /DD /CP /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/CP /D8/D3/D6 ΓrE= (1 + β)1/2(1−β)−3/2= (1 + β)(1−β)−1γ /B4/BE/BK/B5/CX/D2 /B4/BE/BJ/B5 /B4/D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/BA /CC/CW/CT /D3/CQ/D7/CT/D6/DA/CT /CS ΓrE /CX/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BF ℄ /D1 /D9/D7/D8 /D6/CT/D1/CP/CX/D2 /D8/CW/CT/D7/CP/D1/CT/B8 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS ΓrE= 8.8±0.8, /B4/CX/D8 /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/D6/D3/D1 /B4/BE/BJ/B5 /DB/CX/D8/CW /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /DA /CP/D0/D9/CT/D7 /D3/CU Nr,sE, Nr,mE, tE /CP/D2/CSτµ /B5/B8 /CQ/D9/D8 /D8/CW/CT /D4/D6 /CT /CS/CX /D8/CT /CS ΓrE, /D9/D7/CX/D2/CV /D8/CW/CT /CP/CQ /D3 /DA /CT /D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6Γr /CP/D2/CS /D8/CW/CT /CZ/D2/D3 /DB/D2/B8/D4/D6/CT/CS/CX /D8/CT/CS/B8 γ= 8.4±2, /CQ /CT /D3/D1/CT/D7 ≃250γ, ΓrE≃250γ. /B4/BE/BL/B5/CF /CT /D7/CT/CT /D8/CW/CP/D8 /CU/D6/D3/D1 /D8/CW/CT /D3/D1/D1/D3/D2 /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /CP /D5/D9/CX/D8/CT /D9/D2/CT/DC/D4 /CT /D8/CT/CS /D6/CT/D7/D9/D0/D8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BN /D8/CW/CT /D3/CQ/D7/CT/D6/DA/CT /CS ΓrE /CX/D7 /CP/D7 /CQ /CT/CU/D3/D6/CT = 8.8, /DB/CW/CX/D0/CT /D8/CW/CT /D4/D6 /CT /CS/CX /D8/CT /CS ΓrE /CX/D7≃250·8.4 = 2100 ./CB/CX/D1/CX/D0/CP/D6/D0/DD /B8 /D3/D2/CT /CP/D2 /D7/CW/D3 /DB /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP /CV/D6/CT/CP/D8 /CS/CX/D7 /D6/CT/D4/CP/D2 /DD /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /AT/D9/DC/CT/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /CJ/BE/BF ℄ /CP/D2/CS /D8/CW/CT/AT/D9/DC/CT/D7 /D4/D6/CT/CS/CX /D8/CT/CS /DB/CW/CT/D2 /D8/CW/CT /AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/AH /D3/CU /D8/CX/D1/CT /CX/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /CQ/D9/D8 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/CP/D2/CS /CP/D0/D0 /CX/D7 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 /CX/D8 /CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /D4/D6/D3 /DA /CT/CS /D8/CW/CP/D8 /D4/D6/CT/CS/CX /D8/CT/CS /DA /CP/D0/D9/CT/D7 /CX/D2 /D8/CW/CT/AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2 /D8/CW/CT /D1 /D9/D3/D2 /CU/D6/CP/D1/CT /DB/CX/D0/D0 /CP/CV/CP/CX/D2 /CV/D6/CT/CP/D8/D0/DD /CS/CX/AR/CT/D6 /CU/D6/D3/D1 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D3/D2/CT/D7/BA /CB/D9 /CW/D6 /CT/D7/D9/D0/D8/D7 /CT/DC/D4/D0/CX /CX/D8/D0/DD /D7/CW/D3/DB /D8/CW/CP/D8 /D8/CW/CT /AH/BT /CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /CX/D7 /D2/D3/D8 /CP /D7/CP/D8/CX/D7/CU/CP /D8/D3/D6/DD /D6 /CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT /D3/D6/DD/BN /CX/D8 /D4/D6 /CT /CS/CX /D8/D7/B8/CT/BA/CV/BA/B8 /CS/CX/AR/CT/D6 /CT/D2/D8 /DA/CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /AT/D9/DCNs /B4/CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /D1/CT /CP/D7/D9/D6 /CT /CS Nm /B5 /CX/D2 /CS/CX/AR/CT/D6 /CT/D2/D8 /D7/DD/D2 /CW/D6 /D3/D2/CX/DE/CP/D8/CX/D3/D2/D7/CP/D2/CS /CU/D3/D6 /D7/D3/D1/CT /D7/DD/D2 /CW/D6 /D3/D2/CX/DE/CP/D8/CX/D3/D2/D7 /D8/CW/CT/D7/CT /D4/D6 /CT /CS/CX /D8/CT /CS /DA/CP/D0/D9/CT/D7 /CP/D6 /CT /D5/D9/CX/D8/CT /CS/CX/AR/CT/D6 /CT/D2/D8 /CQ/D9/D8 /D8/CW/CT /D1/CT /CP/D7/D9/D6 /CT /CS /D3/D2/CT/D7/BA/CC/CW/CT/D7/CT /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /CS/CX/D6/CT /D8/D0/DD /D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/D0/DD /CP /CT/D4/D8/CT/CS /D3/D4/CX/D2/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /DA /CP/D0/CX/CS/CX/D8 /DD /D3/CU /D8/CW/CT /AH/BT /CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/BG/BA/BF /CC/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D4/D4/D6/D3/CP /CW/C4/CT/D8 /D9/D7 /D2/D3 /DB /CT/DC/CP/D1/CX/D2/CT /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BF ℄ /CU/D6/D3/D1 /D8/CW/CT /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /D3/CU /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /C1/D2 /D8/CW/CT /AH/CC/CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CT/D2 /D8/CT/D6/CX/D2/CV /CX/D2 /D8/D3 /D4/CW /DD/D7/CX /CP/D0 /D0/CP /DB/D7 /D1 /D9/D7/D8 /CQ /CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CP/D2/CS /D8/CW /D9/D7 /DB/CX/D8/CW /D3/D6/D6/CT /D8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7/BN /D8/CW/CT /D7/CP/D1/CT /BG/BW /D5/D9/CP/D2/D8/CX/D8/DD /CW/CP/D7 /D8/D3 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7 /CP/D2/CS/CS/CX/AR/CT/D6/CT/D2 /D8 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/BA /C1/D2 /D8/CW/CT /D9/D7/D9/CP/D0/B8 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /AH/D1 /D9/D3/D2/AH /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/B8 /CU/D3/D6/CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT/D7 τE /CP/D2/CSτµ /CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /BT/D0/D8/CW/D3/D9/CV/CW /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/D2/D2/CT /D8/CX/D2/CV τE /CP/D2/CSτµ /B4/D8/CW/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT /B4/BE/BD/B5/B5 /CX/D7 /D3/D2/D0/DD /CP /D4 /CP/D6/D8 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CX/D8 /CX/D7 /CQ /CT/D0/CX/CT/DA /CT/CS /CQ /DD /CP/D0/D0 /D4/D6/D3/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CP/D8τE/CP/D2/CSτµ /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CS/CX/D7/D8/CP/D2 /CT /B4/D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD/B5 /CQ/D9/D8 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2/D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /D4/D6/CT /CT/CS/CX/D2/CV /D7/CT /D8/CX/D3/D2/D7 /CP/D2/CS /CX/D2 /CJ/BD ℄ /B4/D7/CT/CT /BY/CX/CV/BA/BG/B5/B8/CX/D2 /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT τE /CP/D2/CSτµ /D6/CT/CU/CT/D6 /D8/D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /DB/CW/CX /CW /CP/D6/CT /D2/D3/D8 /D3/D2/D2/CT /D8/CT/CS /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA /CC /D3 /D4/CP/D6/CP/D4/CW/D6/CP/D7/CT /BZ/CP/D1 /CQ/CP /CJ/BJ ℄/BM /AH/BT/D7 /CU/CP/D6 /CP/D7 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D7 /D3/D2 /CT/D6/D2/CT/CS/B8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D0/CX/CZ /CTτE/CP/D2/CSτµ /CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /D2/D3/D8 /D2/CT /CT/D7/D7/CP/D6/CX/D0/DD /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D3/D2/CT /CP/D2/D3/D8/CW/CT/D6/BA /CC /D3 /CP/D7/CZ /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 τE /CP/D2/CSτµ /CU/D6/D3/D1 /D8/CW/CT /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8 /CX/D7 /D0/CX/CZ /CT /CP/D7/CZ/CX/D2/CV /DB/CW/CP/D8 /CX/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D6/CP/CS/CX/D9/D7 /D3/CU /D8/CW/CT /BX/CP/D6/D8/CW /D1/CP/CS/CT /CQ /DD /CP/D2 /D3/CQ/D7/CT/D6/DA /CT/D6 S /CP/D2/CS /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D6/CP/CS/CX/D9/D7/D3/CU /CE /CT/D2 /D9/D7 /D1/CP/CS/CT /CQ /DD /CP/D2 /D3/CQ/D7/CT/D6/DA /CT/D6 S′. /CF /CT /CP/D2 /CT/D6/D8/CP/CX/D2/D0/DD /D8/CP/CZ /CT /D8/CW/CT /D6/CP/D8/CX/D3 /D3/CU /D8/CW/CT /D8 /DB /D3 /D1/CT/CP/D7/D9/D6/CT/D7/BN /DB/CW/CP/D8 /CX/D7/DB/D6/D3/D2/CV /CX/D7 /D8/CW/CT /D8/CP /CX/D8 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D8/CW/CP/D8 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CW/CP/D7 /D7/D3/D1/CT/D8/CW/CX/D2/CV /D8/D3 /CS/D3 /DB/CX/D8/CW /D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /CY/D9/D7/D8 /CQ /CT /CP/D9/D7/CT/D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /DB /CT/D6/CT /D1/CP/CS/CT /CQ /DD /D8/DB/D3 /D3/CQ/D7/CT/D6/DA /CT/D6/D7/BA/AH/C0/CT/D2 /CT/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BC/B5/B8 /DB/CW/CX /CW /CT/DC/D4/D0/CX /CX/D8/D0/DD /D3/D2 /D8/CP/CX/D2/D7 /D3/D2/D0/DD /D8/CW/CT/D7/D4 /CT /CX/AS /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 x0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 /DB /CT /CU/D3/D6/D1 /D9/D0/CP/D8/CT /D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D0/CP /DB /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /BV/BU/BZ/C9/D7/B8 /CP/D7 dN/dl =−λN, N =N0exp(−λl). /B4/BF/BC/B5 l /CX/D7 /D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW 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/D1/CT/CP/D7/D9/D6/CT/CS /DA /CP/D0/D9/CT/D7 /D3/CUτµ, tµ, Ns /CP/D2/CSNm /CP/D2/CS /D3/D1/D4/CP/D6/CT /D8/CW/CT/D1 /DB/CX/D8/CW /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D4/D6/CT/CS/CX /D8/CT/CS/CQ /DD /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BF/BE/B5/BA /C1/D2 /CJ/BE/BF ℄τµ /CX/D7 /D8/CP/CZ /CT/D2 /D8/D3 /CQ /CTτµ= 2.211µs, N s= 397 ±9, Nm= 550 ±10, /CQ/D9/D8tµ/CX/D7 /D2/D3/D8 /D1/CT/CP/D7/D9/D6/CT/CS /D8/CW/CP/D2 /CX/D8 /CX/D7 /CT/D7/D8/CX/D1/CP/D8/CT/CS /CU/D6/D3/D1 /BY/CX/CV/BA /BI/B4/CP/B5 /CX/D2 /CJ/BE/BF ℄ /D8/D3 /CQ /CTtµ= 0.7µs. /C1/D2/D7/CT/D6/D8/CX/D2/CV /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU τµ, tµ /CP/D2/CSNm /CU/D6/D3/D1 /CJ/BE/BF ℄ /B4/CU/D3/D6 /D8/CW/CX/D7 /D7/CX/D1/D4/D0/CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /DB /CT /D8/CP/CZ /CT /D3/D2/D0/DD /D8/CW/CT /D1/CT/CP/D2 /DA /CP/D0/D9/CT/D7 /DB/CX/D8/CW/D3/D9/D8 /CT/D6/D6/D3/D6/D7/B5/CX/D2 /D8/D3 /B4/BF/BE/B5 /DB /CT /D4/D6/CT/CS/CX /D8 /D8/CW/CP/D8Ns /CX/D7Ns= 401 , /DB/CW/CX /CW /CX/D7 /CX/D2 /CP/D2 /CT/DC /CT/D0/D0/CT/D2 /D8 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS Ns= 397 . /BT/D7 /CX/D8 /CX/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /D8/CW/CT /D7/D4/CP /CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /D8/CP/CZ /CT/D7 /D8/CW/CT /D7/CP/D1/CT /DA /CP/D0/D9/CT /CX/D2 /CQ /D3/D8/CW /CU/D6/CP/D1/CT/D7 /CP/D2/CS/CQ /D3/D8/CW /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/B8 le,µ=le,E=lr,µ=lr,E. /C0/CT/D2 /CT/B8 /CU/D3/D6 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS Nm= 550 /CP/D2/CS /CX/CU /D8/CW/CT/CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6/D7 la OA /CP/D2/CSla OT /DB /D3/D9/D0/CS /CQ /CT /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B8 /CP/D2/CS /CX/D2 /CQ /D3/D8/CW /CU/D6/CP/D1/CT/D7 /CX/D2/D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /DB /CT /DB /D3/D9/D0/CS /AS/D2/CS /D8/CW/CT /D7/CP/D1/CT Ns= 401 . /CC/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /D9/D2/CS/D3/D9/CQ/D8/CT/CS/D0/DD /D3/D2/AS/D6/D1/D7 /D8/CW/CT /D3/D2/D7/CX/D7/D8/CT/D2 /DD /CP/D2/CS /D8/CW/CT /DA /CP/D0/CX/CS/CX/D8 /DD /D3/CU /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/CC/CW/CT /D2/D3/D2/D6 /CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT /D3/D6/DD /D4/D6 /CT /CS/CX /D8/D7 /D8/CW/CT /D7/CP/D1/CT /DA/CP/D0/D9/CT /D3/CU /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2/D8/CX/CP/D0 /CU/CP /D8/D3/D6 /CX/D2 /CQ /D3/D8/CW /CU/D6 /CP/D1/CT/D7/B8 exp(−tE/τE) = exp( −tµ/τµ), /D7/CX/D2 /CT /CX/D8 /CS/CT /CP/D0/D7 /DB/CX/D8/CW /D8/CW/CT /CP/CQ/D7/D3/D0/D9/D8/CT /D8/CX/D1/CT/B8 /CX/BA/CT/BA/B8 /DB/CX/D8/CW /D8/CW/CT /BZ/CP/D0/CX/D0/CT /CP/D2 /D8/D6 /CP/D2/D7/B9/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/BA /BU/D9/D8/B8 /CU/D3/D6 /D8/CW/CT /D1/CT /CP/D7/D9/D6 /CT /CS Nm /D8/CW/CT /D2/D3/D2/D6 /CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CW/CT /D3/D6/DD /D4/D6 /CT /CS/CX /D8/D7 /D8/D3 /D3 /D7/D1/CP/D0 /D0 Ns. /CC/CW/CT /AH/BT /CC/D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D3/D6/D6 /CT /D8/D0/DD /D4/D6 /CT /CS/CX /D8/D7 /D8/CW/CT /DA/CP/D0/D9/CT /D3/CUNs /CX/D2 /CQ /D3/D8/CW /CU/D6 /CP/D1/CT/D7 /CQ/D9/D8 /D3/D2/D0/DD /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8/DB/CW/CX/D0/CT /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2/D8/CP/D0 Ns /CP/D2/CS /D8/CW/CT /D8/CW/CT /D3/D6 /CT/D8/CX /CP/D0 /D0/DD /D4/D6 /CT /CS/CX /D8/CT /CS Ns /CS/D6 /CP/D7/D8/CX /CP/D0 /D0/DD/CS/CX/AR/CT/D6/BA /CC/CW/CT /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D3/D1/D4/D0/CT/D8/CT/D0/DD /CP/CV/D6 /CT /CT/D7 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2/D8/D7 /CX/D2 /CP/D0 /D0 /C1/BY/CA/D7 /CP/D2/CS /CP/D0 /D0 /D4 /CT/D6/D1/CX/D7/D7/CX/CQ/D0/CT /D3/B9/D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/BA /CC/CW/D9/D7/B8 /D8/CW/CT /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/B8/AH /CP/D7 /D8/CW/CT /D8/CW/CT /D3/D6/DD /D3/CU /BG/BW /D7/D4 /CP /CT/D8/CX/D1/CT /DB/CX/D8/CW /D8/CW/CT /D4/D7/CT/D9/CS/D3/B9/BX/D9 /D0/CX/CS/CT /CP/D2/CV/CT /D3/D1/CT/D8/D6/DD/B8 /CX/D7 /CX/D2 /CP /D3/D1/D4/D0/CT/D8/CT /CP/CV/D6 /CT /CT/D1/CT/D2/D8 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2/D8/D7/BA/BD/BH/BG/BA/BG /BT/D2/D3/D8/CW/CT/D6 /D8/CX/D1/CT /AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/AH /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/CC/CW/CT /D7/CP/D1/CT /D3/D2 /D0/D9/D7/CX/D3/D2 /CP/D2 /CQ /CT /CP /CW/CX/CT/DA /CT/CS /D3/D1/D4/CP/D6/CX/D2/CV /D8/CW/CT /D3/D8/CW/CT/D6 /D4/CP/D6/D8/CX /D0/CT /D0/CX/CU/CT/D8/CX/D1/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/B8 /CT/BA/CV/BA/B8 /CJ/BE/BH ℄/B8/D3/D6 /CU/D3/D6 /D8/CW/CT /D4/CX/D3/D2 /D0/CX/CU/CT/D8/CX/D1/CT /CJ/BE/BI ℄/B8 /DB/CX/D8/CW /CP/D0/D0 /D8/CW/D6/CT/CT /D8/CW/CT/D3/D6/CX/CT/D7/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CP/D7 /CX/D8 /CX/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /CP/D0/D0 /D8/CW/CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/B8 /CP/D2/CS /D2/D3/D8 /D3/D2/D0/DD /D8/CW/CT/D1 /CQ/D9/D8 /CP/D0/D0 /D3/D8/CW/CT/D6 /D8/D3 /D3/B8 /DB /CT/D6/CT /CS/CT/D7/CX/CV/D2/CT/CS /D8/D3 /D8/CT/D7/D8 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CC/CW /D9/D7/CX/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BH ℄/B8 /DB/CW/CX /CW /D4/D6/CT /CT/CS/CT/CS /D8/D3 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BF ℄ /CP/D2/CS /CJ/BE/BG ℄/B8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /D7/CX/D1/CX/D0/CP/D6 /D8/D3/B4/BE/BE/B5 /CX/D7 /D9/D7/CT/CS /CQ/D9/D8 /DB/CX/D8/CWtE /D6/CT/D4/D0/CP /CT/CS /CQ /DDHE /B4/BPvtE /B5 /CP/D2/CSτE /B4/D8/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT /D3/CU /D1 /D9/D3/D2/D7 /CX/D2 /D8/CW/CT /BX/CP/D6/D8/CW /CU/D6/CP/D1/CT/B5/D6/CT/D4/D0/CP /CT/CS /CQ /DDL=vτE /B4L /CX/D7 /D8/CW/CT /AH/CP /DA /CT/D6/CP/CV/CT /D6/CP/D2/CV/CT /CQ /CT/CU/D3/D6/CT /CS/CT /CP /DD/AH/B5/B8 /CP/D2/CS /CP/D0/D7/D3 /D8/CW/CT /D3/D2/D2/CT /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D0/CX/CU/CT/D8/CX/D1/CT/D7 /B4/BE/BD/B5 /B4τE=γτµ /B5 /CX/D7 /CT/D1/D4/D0/D3 /DD /CT/CS/BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD /D8/CW/CT /D4/D6 /CT /CS/CX /D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/CJ/BE/BH ℄ /DB/CX/D0/D0 /CS/CT/D4 /CT/D2/CS /D3/D2 /D8/CW/CT /CW/D3/D7/CT/D2 /D7/DD/D2 /CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/B8 /D7/CX/D2 /CT /D8/CW/CT/DD /CS/CT/CP/D0 /DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /D9/D7/CT/D8/CW/CT /D6/CP/CS/CX/D3/CP /D8/CX/DA /CT/B9/CS/CT /CP /DD /D0/CP /DB /CX/D2 /D8/CW/CT /CU/D3/D6/D1 /D8/CW/CP/D8 /D3/D2 /D8/CP/CX/D2/D7 /D3/D2/D0/DD /CP /D4/CP/D6/D8 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /BG/B9/DA /CT /D8/D3/D6/BA /CC/CW/CT/D4/D6 /CT /CS/CX /D8/CX/D3/D2/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DB/CX/D0/D0 /CQ /CT /CP/CV/CP/CX/D2 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/D2 /D8/CW/CT /CW/D3/D7/CT/D2 /C1/BY/CA/CP/D2/CS /D8/CW/CT /CW/D3/D7/CT/D2 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT/D7/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BH ℄ /DB/CX/D8/CW /D8/CW/CT /AH/CC/CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D7 /CS/CXꜶ /D9/D0/D8 /D7/CX/D2 /CT/B8 /CT/BA/CV/BA/B8 /D8/CW/CT/DD /CW/CP /DA /CT /D2/D3 /CS/CP/D8/CP /CU/D3/D6tµ. /CB/CX/D1/CX/D0/CP/D6/D0/DD /CW/CP/D4/D4 /CT/D2/D7 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/D6/CT/D4 /D3/D6/D8/CT/CS /CX/D2 /CJ/BE/BI ℄/BA/CC/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D3/CU /D1 /D9/D3/D2/D7 /CX/D2 /D8/CW/CT /CV/B9/BE /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BJ ℄ /CP/D6/CT /D3/CU/D8/CT/D2 /D5/D9/D3/D8/CT/CS /CP/D7 /D8/CW/CT /D1/D3/D7/D8 /D3/D2 /DA/CX/D2 /CX/D2/CV /CT/DA/CX/CS/CT/D2 /CT /CU/D3/D6 /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/B8 /CX/BA/CT/BA/B8 /D8/CW/CT/DD /CP/D6/CT /D0/CP/CX/D1/CT/CS /CP/D7 /CW/CX/CV/CW/B9/D4/D6/CT /CX/D7/CX/D3/D2 /CT/DA/CX/CS/CT/D2 /CT /CU/D3/D6/CB/CA/BA /C6/CP/D1/CT/D0/DD /CX/D2 /D8/CW/CT /D0/CX/D8/CT/D6/CP/D8/D9/D6/CT /D8/CW/CT /CT/DA/CX/CS/CT/D2 /CT /CU/D3/D6 /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /CX/D7 /D3/D1/D1/D3/D2/D0/DD /D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D8/CW/CT/CT/DA/CX/CS/CT/D2 /CT /CU/D3/D6 /CB/CA/BA /CC/CW/CT /D1 /D9/D3/D2 /D0/CX/CU/CT/D8/CX/D1/CT /CX/D2 /AT/CX/CV/CW /D8 τ /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /AS/D8/D8/CX/D2/CV /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /CS/CT /CP /DD/CT/D0/CT /D8/D6/D3/D2 /D8/CX/D1/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /D7/CX/DC/B9/D4/CP/D6/CP/D1/CT/D8/CT/D6 /D4/CW/CT/D2/D3/D1/CT/D2/D3/D0/D3/CV/CX /CP/D0 /CU/D9/D2 /D8/CX/D3/D2 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT /D2/D3/D6/D1/CP/D0/D1/D3 /CS/D9/D0/CP/D8/CT/CS /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /CS/CT /CP /DD /D7/D4 /CT /D8/D6/D9/D1 /B4/D8/CW/CT/CX/D6 /BX/D5/BA/B4/BD/B5/B5/BA /CC/CW/CT/D2 /CQ /DD /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 τ=γτ0/CP/D2/CS /D3/CUτ0 /B4/D3/D9/D6τµ /B5/B8 /D8/CW/CT /D0/CX/CU/CT/D8/CX/D1/CT /CP/D8 /D6/CT/D7/D8 /B4/CP/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D3/D8/CW/CT/D6 /DB /D3/D6/CZ /CT/D6/D7/B5/B8 /D8/CW/CT/DD /D3/CQ/D8/CP/CX/D2/CT/CS /D8/CW/CT /D8/CX/D1/CT/B9/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /CU/CP /D8/D3/D6 γ, /D3/D6 /D8/CW/CT /CZ/CX/D2/CT/D1/CP/D8/CX /CP/D0 γ. /CC/CW/CX/D7γ /CX/D7 /D3/D1/D4/CP/D6/CT/CS /DB/CX/D8/CW /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CS/DD/D2/CP/D1/CX /CP/D0 γ/CU/CP /D8/D3/D6 /B4γ= (p/m)dp/dE /B5/B8 /DB/CW/CX /CW /D8/CW/CT/DD /CP/D0/D0/CT/CS γ /B4/D8/CW/CT /CP /DA /CT/D6/CP/CV/CT γ /DA /CP/D0/D9/CT/B5/BA γ /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT/D1/CT/CP/D2 /D6/D3/D8/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD frot /CQ /DD /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /CU/D3/D6 /CT /D0/CP /DB /B4/D8/CW/CT /AH/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /AH /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/B5/BN/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX /AS/CT/D0/CS /DB /CP/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D4/D6/D3/D8/D3/D2 /C6/C5/CA /CU/D6/CT/D5/D9/CT/D2 /DD fp /B4/CU/D3/D6 /D8/CW/CT /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /D3/CU g−2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /DB/CX/D8/CW/CX/D2 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D7/CT/CT /CP/D0/D7/D3 /CJ/BE/BK ℄/B5/BA /C4/CX/D1/CX/D8/D7 /D3/CU /D3/D6/CS/CT/D6 10−3/CX/D2 (γ−γ)/γ /CP/D8 /D8/CW/CT /CZ/CX/D2/CT/D1/CP/D8/CX /CP/D0 γ= 29.3 /DB /CT/D6/CT /D7/CT/D8/BA /C1/D2 /D8/CW/CP/D8 /DB /CP /DD /D8/CW/CT/DD /CP/D0/D7/D3 /D3/D1/D4/CP/D6/CT/CS /D8/CW/CT /DA /CP/D0/D9/CT /D3/CU/D8/CW/CTµ+/D0/CX/CU/CT/D8/CX/D1/CT /CP/D8 /D6/CT/D7/D8τ+ 0 /B4/CU/D6/D3/D1 /D8/CW/CT /D3/D8/CW/CT/D6 /D4/D6/CT /CX/D7/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/B5 /DB/CX/D8/CW /D8/CW/CT /DA /CP/D0/D9/CT /CU/D3/D9/D2/CS /CX/D2 /D8/CW/CT/CX/D6/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 τ+/γ, /CP/D2/CS /D3/CQ/D8/CP/CX/D2/CT/CS (τ+ 0−τ+/γ)/τ+ 0= (2±9)×10−4, /B4/D8/CW/CX/D7 /CX/D7 /D8/CW/CT /D7/CP/D1/CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /CP/D7/D8/CW/CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CUγ /DB/CX/D8/CWγ /B5/BA /CC/CW/CT/DD /D0/CP/CX/D1/CT/CS/BM /AH/BT /D895% /D3/D2/AS/CS/CT/D2 /CT /D8/CW/CT /CU/D6/CP /D8/CX/D3/D2/CP/D0 /CS/CX/AR/CT/D6/CT/D2 /CT/CQ /CT/D8 /DB /CT/CT/D2 τ+ 0 /CP/D2/CSτ+/γ /CX/D7 /CX/D2 /D8/CW/CT /D6/CP/D2/CV/CT (−1.6−2.0)×10−3/BA/AH /CP/D2/CS /AH/CC /D3 /CS/CP/D8/CT/B8 /D8/CW/CX/D7 /CX/D7 /D8/CW/CT /D1/D3/D7/D8 /CP /D9/D6/CP/D8/CT/D8/CT/D7/D8 /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/CX/D1/CT /CS/CX/D0/CP/D8/CX/D3/D2 /D9/D7/CX/D2/CV /CT/D0/CT/D1/CT/D2 /D8/CP/D6/DD /D4/CP/D6/D8/CX /D0/CT/D7/BA/AH /CC/CW/CT /D3/CQ /CY/CT /D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT /D4/D6/CT /CX/D7/CX/D3/D2 /D3/CU /D8/CW/CT/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BE/BJ℄/B8 /CP/D2/CS /D8/CW/CT /D6/CT/D1/CP/D6/CZ /D8/CW/CP/D8 /CP /D3/D2 /DA/CX/D2 /CX/D2/CV /CS/CX/D6/CT /D8 /D8/CT/D7/D8 /D3/CU /CB/CA /D1 /D9/D7/D8 /D2/D3/D8 /CP/D7/D7/D9/D1/CT /D8/CW/CT /DA /CP/D0/CX/CS/CX/D8 /DD/D3/CU /CB/CA /CX/D2 /CP/CS/DA /CP/D2 /CT /B4/CX/D2 /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /AH/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /AH 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/CP /DD/D7 /D8/CW/CT /D7/CP/D1/CT /BG/BW /D5/D9/CP/D2/D8/CX/D8/DD /B8 /CT/BA/CV/BA/B8ka, /D3/D6lb, /D3/D6φ, /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS/BA /B4/BX/D5/BA /B4/BF/BI/B5 /D7/CW/D3 /DB/D7/CX/D8 /CU/D3/D6φ /BA/B5 /CC/CW/CX/D7 /CX/D7 /D2/D3/D8 /D8/CW/CT /CP/D7/CT /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CC/CW/CT/D6/CT/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D8/CX/D1/CT/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 t′ 1=γt1, /DB/CW/CX /CW /CX/D7 /D9/D7/CT/CS /CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BE/BD ℄/B8 /CJ/BE/BE ℄ /CP/D2/CS /CJ/BF/BH ℄/B5 /D3/CU /D8/CW/CT/C5/CX /CW/CT/D0/D7/D3/D2/B9/C5/D3/D6/D0/CT/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/B8 /CX/D7 /D2/D3/D8 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D7/D3/D1/CT /BG/BW /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CP/D2/CSt′ 1 /CP/D2/CS t1 /CS/D3 /D2/D3/D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /D7/CP/D1/CT /BG/BW /D5/D9/CP/D2 /D8/CX/D8 /DD /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2S′/CP/D2/CSS /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /CQ/D9/D8 /D8/D3 /CS/CX/AR/CT/D6/CT/D2 /D8/BG/BW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CP/D7 /CP/D2 /CQ /CT /D0/CT/CP/D6/D0/DD /D7/CT/CT/D2 /CU/D6/D3/D1 /CB/CT /BA /BE/BA/BE /B4/D7/CT/CT /BD/BJ/B5/BA /C7/D2/D0/DD /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CTωt/CP/D2/CSkl /CU/CP /D8/D3/D6/D7 /CP/D2 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /CP/D2/CS /CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D6/CT/D8/CP/CX/D2 /D8/CW/CT /D7/CX/D1/CX/D0/CP/D6/CX/D8 /DD /DB/CX/D8/CW/BD/BK/D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /CP/D2/CS /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2/D7/B8 /DB /CT /AS/D6/D7/D8 /CS/CT/D8/CT/D6/D1/CX/D2/CT φ /B4/BF/BG/B5/B8 /B4/BF/BI/B5/B8 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2 /D8/CW/CTS /CU/D6/CP/D1/CT /B4/D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/D3/D1/CT/D8/CT/D6/B5/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8φ /DB/CX/D0/D0/CQ /CT /CP/D0 /D9/D0/CP/D8/CT/CS /CU/D6/D3/D1 /B4/BF/BI/B5 /CP/D7 /D8/CW/CT /BV/BU/BZ/C9 φ=kµ egµν,elν e./C4/CT/D8 /D2/D3 /DBA, B /CP/D2/CSA1 /CS/CT/D2/D3/D8/CT /D8/CW/CT /CT/DA /CT/D2 /D8/D7/BN /D8/CW/CT /CS/CT/D4/CP/D6/D8/D9/D6/CT /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /D6/CP /DD /CU/D6/D3/D1 /D8/CW/CT /CW/CP/D0/CU/B9/D7/CX/D0/DA /CT/D6/CT/CS /D1/CX/D6/D6/D3/D6 O, /D8/CW/CT /D6/CT/AT/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CX/D7 /D6/CP /DD /D3/D2 /D8/CW/CT /D1/CX/D6/D6/D3/D6 M1 /CP/D2/CS /D8/CW/CT /CP/D6/D6/CX/DA /CP/D0 /D3/CU /D8/CW/CX/D7 /CQ /CT/CP/D1 /D3/CU /D0/CX/CV/CW /D8/CP/CU/D8/CT/D6 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4 /D3/D2 /D8/CW/CT /CW/CP/D0/CU/B9/D7/CX/D0/DA /CT/D6/CT/CS /D1/CX/D6/D6/D3/D6 O, /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /C1/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /DB /CT /CW/CP /DA /CT/B8 /CU/D3/D6/D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CP/D6/D1 /D3/CU /D8/CW/CT /CX/D2 /D8/CT/CU/CT/D6/D3/D1/CT/D8/CT/D6/B8 /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CT/DA /CT/D2 /D8/D7 A, C /CP/D2/CSA2. /CC /D3 /D7/CX/D1/D4/D0/CX/CU/DD /D8/CW/CT/D2/D3/D8/CP/D8/CX/D3/D2 /DB /CT /D3/D1/CX/D8 /D8/CW/CT /D7/D9/CQ/D7 /D6/CX/D4/D8 /B3/CT/B3 /CX/D2 /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CC/CW/CT/D2 kµ AB /CP/D2/CSlµ AB /B4/D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/D3/CUka AB /CP/D2/CSla AB /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S /B5 /CU/D3/D6 /D8/CW/CT /DB /CP /DA /CT /D3/D2 /D8/CW/CT /D8/D6/CX/D4OM1 /B4/D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /B5 /CP/D6/CTkµ AB= (ω/c,0,2π/λ,0), lµ AB= (ctM1,0,L,0) /BA /BY /D3/D6 /D8/CW/CT /DB /CP /DA /CT /D3/D2 /D8/CW/CT /D6/CT/D8/D9/D6/D2 /D8/D6/CX/D4 M1O, /B4/D8/CW/CT /CT/DA /CT/D2 /D8/D7 B /CP/D2/CSA1 /B5kµ BA1= (ω/c,0,−2π/λ,0) /CP/D2/CSlµ BA1= (ctM1,0,−L,0) /B4/D8/CW/CT /CT/D0/CP/D4/D7/CT/CS/D8/CX/D1/CT/D7 tOM1 /CP/D2/CStM1O /CU/D3/D6 /D8/CW/CT /D8/D6/CX/D4/D7OM1 /CP/D2/CSM1O /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /CP/D6/CT /CT/D5/D9/CP/D0 /CP/D2/CS /CS/CT/D2/D3/D8/CT/CS /CP/D7tM1 /B8 tOM1=tM1O=tM1 /B5/BA /C0/CT/D2 /CT /D8/CW/CT /CX/D2 /D6/CT/D1/CT/D2 /D8 /D3/CU /D4/CW/CP/D7/CT φ1 /CU/D3/D6 /D8/CW/CT /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4OM1O, /CX/D7 φ1=kµ ABlµAB+kµ BA1lµBA1= 2(−ωtM1+ (2π/λ)L), /B4/BF/BJ/B5/DB/CW/CT/D6/CT ω /CX/D7 /D8/CW/CT /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/D5/D9/CT/D2 /DD /BA L /CX/D7 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D7/CT/CV/D1/CT/D2 /D8 OM2 /CP/D2/CSL=L(1+ε) /B4ε≪1 /B5 /CX/D7/D8/CP/CZ /CT/D2 /D8/D3 /CQ /CT/B8 /CP/D7 /CX/D2 /CJ/BF/BI ℄/B8 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CP/D6/D1OM1. /BT/D7 /CT/DC/D4/D0/CP/CX/D2/CT/CS /CX/D2 /CJ/BF/BI ℄/BM /AH/CC/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT L−L=εL/CX/D7 /D9/D7/D9/CP/D0/D0/DD /CP /CU/CT/DB /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/D7 /B4≺25 /B5 /CP/D2/CS /CX/D7 /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CU/D3/D6 /D3/CQ/D8/CP/CX/D2/CX/D2/CV /D9/D7/CT/CU/D9/D0 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2 /CT /CU/D6/CX/D2/CV/CT/D7/BA/AH L,L/CP/D2/CSν /CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CX/D2S /B8 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/D3/D1/CT/D8/CT/D6/BA /CD/D7/CX/D2/CV /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lµ′ ν,e /B4/BE/B5 /D3/D2/CT /CP/D2 /AS/D2/CSkµ′/CP/D2/CSlµ′/CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S′/CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /D8/D6/CX/D4/D7 /CP/D7 /CX/D2S /BA/CC/CW/CT/D2 /CX/D8 /CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /D7/CW/D3 /DB/D2 /D8/CW/CP/D8φ′ 1 /CX/D2S′/CX/D7 /D8/CW/CT /D7/CP/D1/CT /CP/D7 /CX/D2S, φ′ 1=φ1. /BT/D0/D7/D3 /D9/D7/CX/D2/CV /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D1/CP/D8/D6/CX/DC Tµν,r /B4/BH/B5/B8 /DB/CW/CX /CW /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D3/D2/CT /CP/D2 /CV/CT/D8/CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S /B8 /CP/D2/CS /D8/CW/CT/D2 /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lµ′ ν,r /B4/BJ/B5/D8/CW/CT/D7/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D2 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S′/BAφ1 /DB/CX/D0/D0 /CQ /CT /CP/D0/DB /CP /DD/D7 /D8/CW/CT /D7/CP/D1/CT/CX/D2 /CP /D3/D6/CS/CP/D2 /CT /DB/CX/D8/CW /B4/BF/BI/B5/BA /C6/D3/D8/CT /D8/CW/CP/D8gµν,r /B4/BG/B5 /CU/D6/D3/D1 /CB/CT /BA /BE /CW/CP/D7 /D8/D3 /CQ /CT /D9/D7/CT/CS /CX/D2 /D8/CW/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CUφ /CX/D2 /D8/CW/CT/AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /BT/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT /DB /CT /D5/D9/D3/D8/CT kµ AB,r /CP/D2/CSlµ AB,r /BMkµ AB,r= ((ω/c)−2π/λ,0,2π/λ,0)/CP/D2/CSlµ AB,r= (ctM1−L,0,L,0). /C0/CT/D2 /CT/B8 /D9/D7/CX/D2/CV gµν,r /D3/D2/CT /CT/CP/D7/CX/D0/DD /AS/D2/CS/D7 /D8/CW/CP/D8 φAB,r=kµ rgµν,rlν r= (−ωtM1+ (2π/λ)L) =φAB,e./BY /D3/D6 /CU/D9/D6/D8/CW/CT/D6 /D4/D9/D6/D4 /D3/D7/CT/D7 /DB /CT /D7/CW/CP/D0/D0 /CP/D0/D7/D3 /D2/CT/CT/CSkµ′ AB,r /CP/D2/CSlµ′ AB,r. /CC/CW/CT/DD /CP/D6/CTkµ′ AB,r= ((γω/c)(1 + β)− 2π/λ,−βγω/c, 2π/λ,0) /CP/D2/CSlµ′ AB,r= (γctM1(1 +β)−L,−βγct M1,L,0) /DB/CW/CX /CW /DD/CX/CT/D0/CS/D7 φ′ AB,r=φAB,r=φ′ AB,e=φAB,e./C1/D2 /CP /D0/CX/CZ /CT /D1/CP/D2/D2/CT/D6 /DB /CT /AS/D2/CSkµ AC /CP/D2/CSlµ AC /CU/D3/D6 /D8/CW/CT /DB /CP /DA /CT /D3/D2 /D8/CW/CT /D8/D6/CX/D4OM2, /B4/D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CT/DA /CT/D2 /D8/D7 /CP/D6/CT A /CP/D2/CSC /B5 /CP/D7kµ AC= (ω/c,2π/λ,0,0) /CP/D2/CSlµ AC= (ctM2, L,0,0). /BY /D3/D6 /D8/CW/CT /DB /CP /DA /CT /D3/D2 /D8/CW/CT /D6/CT/D8/D9/D6/D2 /D8/D6/CX/D4M2O/B4/D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CT/DA /CT/D2 /D8/D7 /CP/D6/CTC /CP/D2/CSA2 /B5kµ CA2= (ω/c,−2π/λ,0,0) /CP/D2/CSlµ CA2= (ctM2,−L,0,0) /B5/B4tOM2=tM2O=tM2 /B5/B8 /DB/CW/CT/D2 /CT φ2=kµ AClµAC+kµ CA2lµCA2= 2(−ωtM2+ (2π/λ)L). /B4/BF/BK/B5/C7/CU /D3/D9/D6/D7/CT /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CT /D7/CP/D1/CTφ2 /CX/D2S /CP/D2/CSS′/CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /CP/D2/CS /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/BA /C0/CT/D2 /CT φ1−φ2=−2ω(tM1−tM2) + 2(2 π/λ)(L−L). /B4/BF/BL/B5/C8 /CP/D6/D8/CX /D9/D0/CP/D6/D0/DD /CU/D3/D6L=L, /CP/D2/CS /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD tM1=tM2, /D3/D2/CT /AS/D2/CS/D7 φ1−φ2= 0. /C1/D8 /CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD/D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /D8/CW/CT /D7/CP/D1/CT /CS/CX/AR/CT/D6/CT/D2 /CT /D3/CU /D4/CW/CP/D7/CT /B4/BF/BL/B5 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /CP/D7/CT /DB/CW/CT/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/D3/D1/CT/D8/CT/D6/CX/D7 /D6/D3/D8/CP/D8/CT/CS /D8/CW/D6/D3/D9/CV/CW 900, /DB/CW/CT/D2 /CT /DB /CT /AS/D2/CS /D8/CW/CP/D8△(φ1−φ2) = 0 , /CP/D2/CS△N= 0. /BT /D3/D6 /CS/CX/D2/CV /D8/D3 /D8/CW/CT /D3/D2/D7/D8/D6/D9 /D8/CX/D3/D2 φ /B4/BF/BG/B5/B8 /D3/D6 /B4/BF/BI/B5/B8 /CX/D7 /CP /CU/D6 /CP/D1/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2/D8 /D5/D9/CP/D2/D8/CX/D8/DD /CP/D2/CS /CX/D8 /CP/D0/D7/D3 /CS/D3 /CT/D7 /D2/D3/D8 /CS/CT/D4 /CT/D2/CS /D3/D2 /D8/CW/CT /CW/D3/D7/CT/D2 /D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /CP /D3/D2/D7/CX/CS/CT/D6 /CT /CS /C1/BY/CA/BA /CC/CW /D9/D7 /DB /CT /D3/D2 /D0/D9/CS/CT /D8/CW/CP/D8 △Ne=△N′ e=△Nr=△N′ r= 0. /B4/BG/BC/B5/CC/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /CX/D7 /CX/D2 /CP /D3/D1/D4/D0/CT/D8/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /C5/CX /CW/CT/D0/D7/D3/D2/B9/C5/D3/D6/D0/CT/DD /CJ/BF/BG ℄ /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/BA/BD/BL/BW/D6/CX/D7 /D3/D0/D0 /CJ/BF/BI ℄ /CX/D1/D4/D6/D3 /DA /CT/CS /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3/CP /D3/D9/D2 /D8 /D8/CW/CT /CW/CP/D2/CV/CT/D7 /CX/D2 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8/BA /CC/CW/CX/D7 /CX/D1/D4/D6/D3 /DA /CT/D1/CT/D2 /D8 /D6/CT/D7/D9/D0/D8/CT/CS /CX/D2 /CP/AH/D7/D9/D6/D4/D6/CX/D7/CX/D2/CV/AH /D2/D3/D2/B9/D2 /D9/D0/D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 △N′=△(φ′ 2−φ′ 1)/2π= 4(Lν/c)β2, /B4/BG/BD/B5/CP/D2/CS /DB /CT /D7/CT/CT /D8/CW/CP/D8 /D8/CW/CT /CT/D2 /D8/CX/D6/CT /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /B4/D7/CT/CT /CJ/BF/BI ℄ /CP/D2/CS /CJ/BE ℄/B5/BA /C1/D8 /CX/D7 /CT/DC/D4/D0/CX /CX/D8/D0/DD/D7/CW/D3 /DB/D2 /CX/D2 /CJ/BE℄ /D8/CW/CP/D8 /BW/D6/CX/D7 /D3/D0/D0/B3/D7 /D6/CT/D7/D9/D0/D8 /CP/D2 /CQ /CT /CT /CP/D7/CX/D0/DD /D3/CQ/D8/CP/CX/D2/CT /CS /CU/D6 /D3/D1 /D3/D9/D6 /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /CP/D4/D4/D6 /D3 /CP /CW /D8/CP/CZ/CX/D2/CV/D3/D2/D0/DD /D8/CW/CT /D4/D6 /D3 /CS/D9 /D8 k0′ el0′e /CX/D2 /D8/CW/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CX/D2 /D6 /CT/D1/CT/D2/D8 /D3/CU /D4/CW/CP/D7/CT φ′ e /CX/D2S′/CX/D2 /DB/CW/CX /CW /D8/CW/CT /CP/D4/D4/CP/D6/CP/D8/D9/D7/CX/D7 /D1/D3 /DA/CX/D2/CV/BA/CF /CT /D6/CT/D1/CP/D6/CZ /D8/CW/CP/D8 /D8/CW/CT /D2/D3/D2/B9/D2 /D9/D0/D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /B4/BG/BD/B5 /DB /D3/D9/D0/CS /CQ /CT /D5/D9/CX/D8/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CX/D2 /CP/D2/D3/D8/CW/CT/D6 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/B9/D8/CX/D3/D2/B8 /CT/BA/CV/BA/B8 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D7/CX/D2 /CT /D3/D2/D0/DD /CP /D4/CP/D6/D8k0′ el0′e /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD φ/B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS/BA /CC/CW /D9/D7 /DB/CW/CT/D2 /D3/D2/D0/DD /CP /D4/CP/D6/D8 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3/CP /D3/D9/D2 /D8 /D8/CW/CT/D2 /CX/D8 /D0/CT/CP/CS/D7 /D8/D3 /CP/D2 /D9/D2/D4/CW /DD/D7/CX /CP/D0 /D6/CT/D7/D9/D0/D8/BA/BT/D7 /D7/CW/D3 /DB/D2 /CX/D2 /CJ/BE℄ /D8/CW/CT /D7/CP/D1/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CUki′li′, /D8/CW/CT /D3/D2/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /D4 /CP/D6/D8/D7 /D3/CUkµ′/CP/D2/CSlµ′/D8/D3△N′ e, /D7/CW/D3/DB/D7 /D8/CW/CP/D8 /D8/CW/CX/D7 /D8/CT/D6/D1 /CT/DC/CP /D8/D0/DD /CP/D2 /CT/D0 /D8/CW/CTk0′l0′ /D3/D2/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /B4/BW/D6/CX/D7 /D3/D0 /D0/B3/D7 /D2/D3/D2/B9/D2/D9/D0 /D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8/B4/BG/BD/B5/B5/B8 /DD/CX/CT/D0/CS/CX/D2/CV /D8/CW/CP/D8△N′ e=△Ne= 0. /CC/CW /D9/D7 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/CP/D4/D4/D6/D3/CP /CW /D8/D3 /CB/CA /D2/CP/D8/D9/D6/CP/D0/D0/DD /CT/DC/D4/D0/CP/CX/D2/D7/D8/CW/CT /D6/CT/CP/D7/D3/D2 /CU/D3/D6 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /BW/D6/CX/D7 /D3/D0/D0/B3/D7 /D2/D3/D2/B9/D2 /D9/D0/D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /B4/BG/BD/B5/BA/CC/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CP/D2 /CQ /CT /CT /CP/D7/CX/D0/DD /CT/DC/D4/D0/CP/CX/D2/CT /CS /CU/D6 /D3/D1 /D3/D9/D6 /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6/CU/D3/D6/D1/D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA /D8/CP/CZ/CX/D2/CV /D3/D2/D0/DD /D8/CW/CT /D4 /CP/D6/D8k0 el0′e /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D2 /D8/CW/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D3/CU /D8/CW/CT /CX/D2 /D6 /CT/D1/CT/D2/D8 /D3/CU /D4/CW/CP/D7/CT φ′ e /CX/D2S′. /C1/D2 /D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /BW/D6/CX/D7 /D3/D0/D0/B3/D7 /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CT /D4/CP/D6/D8k0 el0e /B4/D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5/B8 /B4/BF/BI/B5/B5 /CX/D2S, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/D3/D1/CT/D8/CT/D6/B8/CP/D2/CSk0 el0′e /CX/D2S′/B8 /CX/D2 /DB/CW/CX /CW /D8/CW/CT /CP/D4/D4/CP/D6/CP/D8/D9/D7 /CX/D7 /D1/D3 /DA/CX/D2/CV/BA k0 e /CX/D7 /D2/D3/D8 /CW/CP/D2/CV/CT /CS /CX/D2 /D8/D6 /CP/D2/D7/CX/D8/CX/D3/D2 /CU/D6 /D3/D1S /D8/D3S′/BA/CC/CW /D9/D7 /D8/CW/CT /CX/D2 /D6/CT/D1/CT/D2 /D8 /D3/CU /D4/CW/CP/D7/CT φ1 /CU/D3/D6 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4OM1O /CX/D2S /B8 /CX/D7 φ1=k0 ABg00,el0 AB+k0 BA1g00,el0 BA1=−2(ω/c)(ctM1) =−2ωtM1. /B4/BG/BE/B5/C1/D2 /D8/CW/CTS′/CU/D6/CP/D1/CT /DB /CT /AS/D2/CS /CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /D8/D6/CX/D4 /D8/CW/CP/D8 φ′ 1=k0 ABl0′AB+k0 BA1l0′BA1=−2(ω/c)(γctM1) =−2ω(γtM1). /B4/BG/BF/B5/CC/CW/CX/D7 /CX/D7 /CT/DC/CP /D8/D0/DD /D8/CW/CT /D6/CT/D7/D9/D0/D8 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /B4/D7/CT/CT /CJ/BE/BD ℄ /D3/D6 /CJ/BE/BE ℄/B5 /DB/CW/CX /CW /CX/D7 /CX/D2/CT/D6/D4/D6/CT/D8/CT/CS/CP/D7 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP /D8/CX/D1/CT /AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/AH t′ 1=γt1 /BA /C1/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /DB /CT /AS/D2/CS /D8/CW/CP/D8 /D8/CW/CT /CX/D2 /D6/CT/D1/CT/D2 /D8 /D3/CU /D4/CW/CP/D7/CT φ2 /CU/D3/D6 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4OM2O /CX/D2S /B8 /CX/D7 φ2=k0 ACl0AC+k0 CA2l0CA2=−2ωtM2, /B4/BG/BG/B5/CP/D2/CSφ′ 2 /CX/D2S′/CX/D7 φ′ 2=k0 ACl0′AC+k0 CA2l0′CA2=−2(ω/c)(γctM2) =−2ω(γtM2). /B4/BG/BH/B5/CC/CW/CX/D7 /CX/D7 /CP/CV/CP/CX/D2 /D8/CW/CT /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7/B8 /D8/CW/CT /D8/CX/D1/CT /AH/CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/B8/AH t′ 2=γt2 /BA /BY /D3/D6t1=t2 /B8/CX/BA/CT/BA/B8 /CU/D3/D6L=L, /D3/D2/CT /AS/D2/CP/D0/D0/DD /AS/D2/CS/D7 /D8/CW/CT /D2 /D9/D0/D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /D8/CW/CP/D8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 △N′ e=△Ne= 0. /CF /CT /D7/CT/CT /D8/CW/CP/D8 /D7/D9 /CW /CP /D2/D9/D0 /D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT /CS /D8/CP/CZ/CX/D2/CV /CX/D2/D8/D3 /CP /D3/D9/D2/D8 /D3/D2/D0/DD /CP /D4 /CP/D6/D8 /D3/CU/D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5/B8 /CP/D2/CS /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D0/DD/B8 /CX/D2 /D8/CW/CP/D8 /D4 /CP/D6/D8/B8k0 e /CX/D7 /D2/D3/D8 /CW/CP/D2/CV/CT /CS /CX/D2 /D8/D6 /CP/D2/D7/CX/D8/CX/D3/D2 /CU/D6 /D3/D1 S /D8/D3S′/BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD /D8/CW/CX/D7 /D3/D6/D6/CT /D8 /D6/CT/D7/D9/D0/D8 /CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1 /CP /D4/CW /DD/D7/CX /CP/D0/D0/DD /CX/D2 /D3/D6/D6/CT /D8 /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D4/CW/CP/D7/CT φ/B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5/BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT /CX/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D2/D3/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D9/D7/D9/CP/D0 /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CX/D7 /CP/D0/DB /CP /DD/D7 /CS/D3/D2/CT /D3/D2/D0/DD /CX/D2 /D8/CW/CT/AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/CB/CX/D2 /CT /D3/D2/D0/DD /D8/CW/CT /D4/CP/D6/D8k0 el0e /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /B4/CP/D2/CS /CP/D0/D7/D3 k0′ e=k0 e /B5 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CP/D6/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/CT/D2 /D8/BA /CF /CT/CT/DC/D4/D0/CX /CX/D8/D0/DD /D7/CW/D3 /DB /CX/D8 /D9/D7/CX/D2/CV /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/C1/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /CX/D2 /D6/CT/D1/CT/D2 /D8 /D3/CU /D4/CW/CP/D7/CT φr /CX/D7 /CP/D0 /D9/D0/CP/D8/CT/CS /CU/D6/D3/D1φr=k0 rg00,rl0 r /CX/D2S/CP/D2/CS /CU/D6/D3/D1φ′ r=k0 rg00,rl0′ r /CX/D2S′. /C0/CT/D2 /CT /DB /CT /AS/D2/CS /D8/CW/CP/D8φ1r /CU/D3/D6 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4OM1O /CX/D2S /CX/D7 φ1r=−2(ωtM1+ (2π/λ)L), /B4/BG/BI/B5/CP/D2/CSφ2r /CU/D3/D6 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4OM2O /CX/D2S /CX/D7 φ2r=−2(ωtM2+ (2π/λ)L). /B4/BG/BJ/B5/BE/BC/BY /D3/D6L=L, /CP/D2/CS /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD tM1=tM2, /DB /CT /AS/D2/CS /D8/CW/CP/D8φ1r−φ2r= 0 /B8 /DB/CW/CT/D2 /CT △Nr= 0. /CA/CT/D1/CP/D6/CZ /D8/CW/CP/D8/D8/CW/CT /D4/CW/CP/D7/CT/D7 φ1r /CP/D2/CSφ2r /CS/CX/AR/CT/D6 /CU/D6/D3/D1 /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D4/CW/CP/D7/CT/D7 φ1e /CP/D2/CSφ2e /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/BT/D7 /D7/CW/D3 /DB/D2 /CP/CQ /D3 /DA /CT /D8/CW/CX/D7 /CX/D7 /D2/D3/D8 /D8/CW/CT /CP/D7/CT /DB/CW/CT/D2 /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8/BA/C0/D3 /DB /CT/DA /CT/D6/B8 /CX/D2S′, /DB /CT /AS/D2/CS /CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /D8/D6/CX/D4/D7 /D8/CW/CP/D8 φ′ 1r=−2(γωtM1(1 +β) + (2 π/λ)L), /B4/BG/BK/B5 φ′ 2r=−2γ2(1 +β2)(ωtM2+ (2π/λ)L). /B4/BG/BL/B5/C7/CQ /DA/CX/D3/D9/D7/D0/DD φ′ 1r−φ′ 2r/ne}ationslash= 0 /CP/D2/CS /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /CX/D8 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /D2/D3/D2/B9/D2/D9/D0 /D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 △N′ r/ne}ationslash= 0, /B4/BH/BC/B5/DB/CW/CX /CW /CW/D3/D0/CS/D7 /CT/DA /CT/D2 /CX/D2 /D8/CW/CT /CP/D7/CT /DB/CW/CT/D2 tM1=tM2. /CC/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /D0/CT/CP/D6/D0/DD /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2/D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /C5/CX /CW/CT/D0/D7/D3/D2/B9/C5/D3/D6/D0/CT/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /CX/D7 /D3/D2/D0/DD /CP/D2 /AH/CP/D4/D4/CP/D6/CT/D2 /D8/AH/CP/CV/D6/CT/CT/D1/CT/D2 /D8/BA /C1/D8 /CX/D7 /CP /CW/CX/CT/DA /CT/CS /CQ /DD /CP/D2 /CX/D2 /D3/D6/D6/CT /D8 /D4/D6/D3 /CT/CS/D9/D6/CT /CP/D2/CS /CX/D8 /CW/D3/D0/CS/D7 /D3/D2/D0/DD /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/CF /CT /CP/D0/D7/D3 /D6/CT/D1/CP/D6/CZ /D8/CW/CP/D8 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CV/CX/DA /CT/D7 /CS/CX/AR/CT/D6/CT/D2 /D8 /DA /CP/D0/D9/CT/D7 /CU/D3/D6 /D8/CW/CT/D4/CW/CP/D7/CT/D7/B8 /CT/BA/CV/BA/B8φ1e, φ′ 1e, φ1r /CP/D2/CSφ′ 1r, /D7/CX/D2 /CT /D3/D2/D0/DD /CP /D4/CP/D6/D8 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS/BA/CC/CW/CT/D7/CT /D4/CW/CP/D7/CT/D7 /CP/D6 /CT /CU/D6 /CP/D1/CT /CP/D2/CS /D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/CT/D2/D8 /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7/BA /CF/CW/CT/D2 /D8/CW/CT /DB/CW/D3/D0/CT /D4/CW/CP/D7/CT φ /B4/BF/BG/B5/D3/D6 /B4/BF/BI/B5 /CX/D7 /D8/CP/CZ/CT/D2 /CX/D2/D8/D3 /CP /D3/D9/D2/D8/B8 /CX/BA/CT/BA/B8 /CX/D2 /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/B8/AH /CP/D0 /D0 /D8/CW/CT /D1/CT/D2/D8/CX/D3/D2/CT /CS /D4/CW/CP/D7/CT/D7 /CP/D6 /CT /CT/DC/CP /D8/D0/DD /CT /D5/D9/CP/D0/D5/D9/CP/D2/D8/CX/D8/CX/CT/D7/BN /D8/CW/CT/DD /CP/D6 /CT /D8/CW/CT /D7/CP/D1/CT/B8 /CU/D6 /CP/D1/CT /CP/D2/CS /D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2/D8/B8 /D5/D9/CP/D2/D8/CX/D8/DD/BA/BH/BA/BE /CC/CW/CT /D1/D3 /CS/CT/D6/D2 /D0/CP/D7/CT/D6 /DA /CT/D6/D7/CX/D3/D2/D7/CC/CW/CT /D1/D3 /CS/CT/D6/D2 /D0/CP/D7/CT/D6 /DA /CT/D6/D7/CX/D3/D2/D7 /D3/CU /D8/CW/CT /C5/CX /CW/CT/D0/D7/D3/D2/B9/C5/D3/D6/D0/CT/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/B8 /CT/BA/CV/BA/B8 /CJ/BF/BK ℄ /CP/D2/CS /CJ/BF/BL ℄/B8 /CP/D6/CT /CP/D0/DB /CP /DD/D7/CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CP /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CC/CW/CT/DD /D6/CT/D0/DD /D3/D2 /CW/CX/CV/CW/D0/DD /D1/D3/D2/D3 /CW/D6/D3/D1/CP/D8/CX /B4/D1/CP/D7/CT/D6/B5 /D0/CP/D7/CT/D6/CU/D6/CT/D5/D9/CT/D2 /DD /D1/CT/D8/D6/D3/D0/D3/CV/DD /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D2 /D3/D4/D8/CX /CP/D0 /CX/D2 /D8/CT/D6/CU/CT/D6/D3/D1/CT/D8/D6/DD/BN /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D7 /D2/D3/D8 /D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1/D7/CW/CX/CU/D8 /CX/D2 /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CU/D6/CX/D2/CV/CT/D7 /D8/CW/CP/D2 /CP /CQ /CT/CP/D8 /CU/D6/CT/D5/D9/CT/D2 /DD /DA /CP/D6/CX/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /CP/D7/D7/D3 /CX/CP/D8/CT/CS /B4/D1/CP/D7/CT/D6/B5 /D0/CP/D7/CT/D6/B9/CU/D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8/BA /C1/D2 /CJ/BF/BK ℄ /D8/CW/CT /CP/D9/D8/CW/D3/D6/D7 /D6/CT /D3/D6/CS/CT/CS /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2/D7 /CX/D2 /CQ /CT/CP/D8 /CU/D6/CT/D5/D9/CT/D2 /DD /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D3/D4/D8/CX /CP/D0/D1/CP/D7/CT/D6 /D3/D7 /CX/D0/D0/CP/D8/D3/D6/D7 /DB/CW/CT/D2 /D6/D3/D8/CP/D8/CT/CS /D8/CW/D6/D3/D9/CV/CW 900/CX/D2 /D7/D4/CP /CT/BN /D8/CW/CT /D8 /DB /D3 /D1/CP/D7/CT/D6 /CP /DA/CX/D8/CX/CT/D7 /CP/D6/CT /D4/D0/CP /CT/CS /D3/D6/D8/CW/D3/CV/D3/D2/CP/D0/D0/DD/D3/D2 /CP /D6/D3/D8/CP/D8/CX/D2/CV /D8/CP/CQ/D0/CT /CP/D2/CS /D8/CW/CT/DD /CP/D2 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D8 /DB /D3 /D0/CX/CV/CW /D8 /D0/D3 /CZ/D7/BA /C1/D8 /CX/D7 /D7/D8/CP/D8/CT/CS /CX/D2 /CJ/BF/BK ℄ /D8/CW/CP/D8 /D8/CW/CT /CW/CX/CV/CW/D0/DD/D1/D3/D2/D3 /CW/D6/D3/D1/CP/D8/CX /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/CU /D1/CP/D7/CT/D6/D7/BN /AH/BA/BA/BA/CP/D0/D0/D3 /DB /DA /CT/D6/DD /D7/CT/D2/D7/CX/D8/CX/DA /CT /CS/CT/D8/CT /D8/CX/D3/D2 /D3/CU /CP/D2 /DD /CW/CP/D2/CV/CT /CX/D2 /D8/CW/CT /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4 /D3/D4/D8/CX /CP/D0 /CS/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D6/CT/AT/CT /D8/CX/D2/CV /D7/D9/D6/CU/CP /CT/D7/BA/AH /CP/D2/CS /D8/CW/CP/D8 /D8/CW/CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7/D3/CU /D8 /DB /D3 /D1/CP/D7/CT/D6/D7 /CP/D0/D0/D3 /DB/D7/BM /AH/BA/BA/BA/CP /DA /CT/D6/DD /D4/D6/CT /CX/D7/CT /CT/DC/CP/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CX/D7/D3/D8/D6/D3/D4 /DD /D3/CU /D7/D4/CP /CT /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D0/CX/CV/CW /D8/D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2/BA/AH /CC/CW/CT /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CX/D7 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /DB /CP/D7/BM /AH/BA/BA/BA /D8/CW/CT/D6/CT /DB /CP/D7 /D2/D3 /D6/CT/D0/CP/D8/CX/DA /CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D1/CP/D7/CT/D6/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CP/D7/D7/D3 /CX/CP/D8/CT/CS /DB/CX/D8/CW /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/CP/D6/D8/CW /CX/D2 /D7/D4/CP /CT /CV/D6/CT/CP/D8/CT/D6 /D8/CW/CP/D2 /CP/CQ /D3/D9/D8 /BF /CZ /BB/D7/CT /BA/AH /CB/CX/D1/CX/D0/CP/D6/D0/DD/CJ/BF/BL ℄ /D3/D1/D4/CP/D6/CT/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/CU /CP /C0/CT/B9/C6/CT /D0/CP/D7/CT/D6 /D0/D3 /CZ /CT/CS /D8/D3 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2 /D8 /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /CP /CW/CX/CV/D0/DD /D7/D8/CP/CQ/D0/CT/BY /CP/CQ/D6/DD/B9/C8 /CT/D6/D3/D8 /CP /DA/CX/D8 /DD /B4/D8/CW/CT /D1/CT/D8/CT/D6/B9/D7/D8/CX /CZ/B8 /CX/BA/CT/BA/B8 /AH/CT/D8/CP/D0/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW/AH/B5 /CP/D2/CS /D3/CU /CPCH4 /D7/D8/CP/CQ/CX/D0/CX/DE/CT/CS /AH/D8/CT/D0/CT/D7 /D3/D4 /CT/B9/D0/CP/D7/CT/D6/AH 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/CJ/BG/BE ℄ /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CP /D4/D6/CT /CX/D7/CX/D3/D2 /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /CX/D2 /DB/CW/CX /CW /D8/CW/CT/DD /D9/D7/CT/CS /CP /CQ /CT/CP/D1 /D3/CU/CT/DC /CX/D8/CT/CS /CW /DD/CS/D6/D3/CV/CT/D2 /D1/D3/D0/CT /D9/D0/CT/D7 /CP/D7 /CP /D1/D3 /DA/CX/D2/CV /D0/CX/CV/CW /D8 /D7/D3/D9/D6 /CT/BA /CC/CW/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/CU /D8/CW/CT /D0/CX/CV/CW /D8 /CT/D1/CX/D8/D8/CT/CS /D4/CP/D6/CP/D0/D0/CT/D0/CP/D2/CS /CP/D2 /D8/CX/D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CT /CQ /CT/CP/D1 /CS/CX/D6/CT /D8/CX/D3/D2 /DB /CT/D6/CT /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /CP /D7/D4 /CT /D8/D3/CV/D6/CP/D4/CW /B4/CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD/B5/BA/BE/BE/CC/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /D8/CW/CX/D7 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /CX/D7 △f/f0= (△fb− △fr)/f0, /B4/BH/BD/B5/DB/CW/CT/D6/CT f0 /CX/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /D8/CW/CT /D0/CX/CV/CW /D8 /CT/D1/CX/D8/D8/CT/CS /CU/D6/D3/D1 /D6/CT/D7/D8/CX/D2/CV /CP/D8/D3/D1/D7/BA △fb=\|fb−f0\| /CP/D2/CS△fr= \|fr−f0\|, /DB/CW/CT/D6/CT fb /CX/D7 /D8/CW/CT /CQ/D0/D9/CT/B9/BW/D3/D4/D4/D0/CT/D6/B9/D7/CW/CX/CU/D8/CT/CS /CU/D6/CT/D5/D9/CT/D2 /DD /D8/CW/CP/D8 /CX/D7 /CT/D1/CX/D8/D8/CT/CS /CX/D2 /CP /CS/CX/D6/CT /D8/CX/D3/D2 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 v /B4v /CX/D7 /D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /D3/CU /D8/CW/CT /CP/D8/D3/D1/D7 /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD/B5/B8 /CP/D2/CSfr /CX/D7 /D8/CW/CT /D6/CT/CS/B9/BW/D3/D4/D4/D0/CT/D6/B9/D7/CW/CX/CU/D8/CT/CS/CU/D6/CT/D5/D9/CT/D2 /DD /D8/CW/CP/D8 /CX/D7 /CT/D1/CX/D8/D8/CT/CS /CX/D2 /CP /CS/CX/D6/CT /D8/CX/D3/D2 /D3/D4/D4 /D3/D7/CX/D8/CT /D8/D3v. /CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DD △f/ f0 /D1/CT/CP/D7/D9/D6/CT/D7 /D8/CW/CT /CT/DC/D8/CT/D2 /D8/D8/D3 /DB/CW/CX /CW /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /D8/CW/CT /D0/CX/CV/CW /D8 /CU/D6/D3/D1 /D6/CT/D7/D8/CX/D2/CV /CP/D8/D3/D1/D7 /CU/CP/CX/D0/D7 /D8/D3 /D0/CX/CT /CW/CP/D0/CU/DB /CP /DD /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 fr /CP/D2/CSfb. /C1/D2 /D8/CT/D6/D1/D7 /D3/CU /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BH/BD/B5 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7 △λ/λ0= (△λr− △λb)/λ0, /B4/BH/BE/B5/DB/CW/CT/D6/CT △λr=\|λr−λ0\| /CP/D2/CS△λb=\|λb−λ0\|, /CP/D2/CS/B8 /CP/D7 /DB /CT /D7/CP/CX/CS/B8λr /CP/D2/CSλb /CP/D6/CT /D8/CW/CT /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/D7 /D7/CW/CX/CU/D8/CT/CS/CS/D9/CT /D8/D3 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /D8/D3 /D8/CW/CT /AH/D6/CT/CS/AH /CP/D2/CS /AH/CQ/D0/D9/CT/AH /D6/CT/CV/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D7/D4 /CT /D8/D6/D9/D1/BA /C1/D2 /D8/CW/CP/D8 /DB /CP /DD /C1/DA /CT/D7 /CP/D2/CS/CB/D8/CX/D0/DB /CT/D0/D0 /D6/CT/D4/D0/CP /CT/CS /D8/CW/CT /CS/CXꜶ /D9/D0/D8 /D4/D6/D3/CQ/D0/CT/D1 /D3/CU /D8/CW/CT /D4/D6/CT /CX/D7/CT /CS/CT/D8/CT/D6/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /DB/CX/D8/CW /D1 /D9 /CW/D7/CX/D1/D4/D0/CT/D6 /D4/D6/D3/CQ/D0/CT/D1 /D3/CU /D8/CW/CT /CS/CT/D8/CT/D6/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP/D7/DD/D1/D1/CT/D8/D6/DD /D3/CU /D7/CW/CX/CU/D8/D7 /D3/CU /D8/CW/CT /AH/D6/CT/CS/AH /CP/D2/CS /AH/CQ/D0/D9/CT/AH /D7/CW/CX/CU/D8/CT/CS/D0/CX/D2/CT/D7 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /D9/D2/D7/CW/CX/CU/D8/CT/CS /D0/CX/D2/CT/BA /CC/CW/CT/DD /CJ/BG/BE ℄ /D7/CW/D3 /DB /CT/CS /D8/CW/CP/D8 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS /D6/CT/D7/D9/D0/D8/D7 /CP/CV/D6/CT/CT /DB/CX/D8/CW /D8/CW/CT/CU/D3/D6/D1 /D9/D0/CP /D4/D6/CT/CS/CX /D8/CT/CS /CQ /DD /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CP/D2/CS /D2/D3/D8 /DB/CX/D8/CW/D8/CW/CT /D0/CP/D7/D7/CX /CP/D0 /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8/BA /C4/CT/D8 /D9/D7 /CT/DC/D4/D0/CP/CX/D2 /CX/D8 /CX/D2 /D1/D3/D6/CT /CS/CT/D8/CP/CX/D0/BA/BJ/BA/BD /CC/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2/C1/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D2/CT /D9/D7/D9/CP/D0/D0/DD /D7/D8/CP/D6/D8/D7 /DB/CX/D8/CW /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 kµ(ω/c,k=nω/c) /D3/CU /D8/CW/CT /BG/B9/DA /CT /D8/D3/D6 ka/D3/CU /D8/CW/CT /D0/CX/CV/CW /D8 /DB /CP /DA /CT /CU/D6/D3/D1 /CP/D2 /C1/BY/CAS /D8/D3 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV/B4/CP/D0/D3/D2/CV /D8/CW/CT /D3/D1/D1/D3/D2 x, x′− /CP/DC/CT/D7/B5 /C1/BY/CAS′/BA /C6/D3/D8/CT /D8/CW/CP/D8 /D3/D2/D0/DD /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D9/D7/CT/CS /CX/D2 /D7/D9 /CW/D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /D8/D6/CT/CP/D8/D1/CT/D2 /D8/BA /CC/CW/CT/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CUkµ /CP/D2 /CQ /CT/DB/D6/CX/D8/D8/CT/D2 /CP/D7 k0′=ω′/c=γ(ω/c−βk1), k1′=γ(k1−βω/c), k2′=k2, k3′=k3, /B4/BH/BF/B5/D3/D6 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D9/D2/CX/D8 /DB /CP /DA /CT /DA /CT /D8/D3/D6 n /B4/DB/CW/CX /CW /CX/D7 /CX/D2 /D8/CW/CT /CS/CX/D6/CT /D8/CX/D3/D2 /D3/CU /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB /CP /DA /CT/B5 ω′=γω(1−βn1), n1′=N(n1−β), n2′= (N/γ)n2, n3′= (N/γ)n3, /B4/BH/BG/B5/DB/CW/CT/D6/CT N= (1−βn1)−1. /C6/D3 /DB /D3/D1/CT/D7 /D8/CW/CT /D1/CP/CX/D2 /D4 /D3/CX/D2 /D8 /CX/D2 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2/BA /BT/D0/D8/CW/D3/D9/CV/CW /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/B9/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 kµ/D3/CU /D8/CW/CT /BG/B9/DA /CT /D8/D3/D6 ka/CU/D6/D3/D1S /D8/D3S′, /BX/D5/D7/BA/B4/BH/BF/B5 /CP/D2/CS /B4/BH/BG/B5/B8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7/CP/D0/D0 /CU/D3/D9/D6 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUkµ/D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUkµ, /CX/BA/CT/BA/B8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /B8 /CP/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D3/CUkµ, /CX/BA/CT/BA/B8 /D8/CW/CT /D9/D2/CX/D8 /DB /CP /DA /CT /DA /CT /D8/D3/D6 n. /CC/CW /D9/D7 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CS/CT/CP/D0/D7 /DB/CX/D8/CW /D8 /DB /D3 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2/D8 /D4/CW /DD/D7/CX /CP/D0/D4/CW/CT/D2/D3/D1/CT/D2/CP /B9 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /CP/D2/CS /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW /D8/BA /B4/CA/CT /CP/D0/D0 /D8/CW/CP/D8 /DB /CT /CW/CP /DA /CT /CP/D0/D6/CT/CP/CS/DD /D1/CT/D8 /D7/D9 /CW/D3/D1/CX/D7/D7/CX/D3/D2 /D3/CU /D3/D2/CT /D4/CP/D6/D8 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /CP /BG/B9/DA /CT /D8/D3/D6 /B4/DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/CX/D2 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D3/D2 /D8/D6/CP /D8/CX/D3/D2 /B4/BD/BG/B5 /CP/D2/CS /D8/CW/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT /B4/BD/BJ/B5/CX/D2 /CB/CT /BA /BE/BA/BE/BA/B5 /CF /CT /D2/D3/D8/CT /D3/D2 /CT /CP/CV/CP/CX/D2 /D8/CW/CP/D8 /D7/D9 /CW /CS/CX/D7/D8/CX/D2 /D8/CX/D3/D2 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D3/D2/D0/DD /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BN/CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D1/CT/D8/D6/CX /D8/CT/D2/D7/D3/D6 gµν,r /CX/D7 /D2/D3/D8 /CS/CX/CP/CV/D3/D2/CP/D0 /CP/D2/CS /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /D8/CW/CT /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2/D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CP/D2/CS /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 /CS/D3 /CT/D7 /D2/D3/D8 /CT/DC/CX/D7/D8/BA /CC/CW /D9/D7 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CX/D7 /D6/CT/D7/D8/D6/CX /D8/CT/CS/D8/D3 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C1/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D7/D9 /CW /D8/CW/CT/D3/D6/CT/D8/CX /CP/D0 /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D8/CW/CT /CT/DC/CX/D7/D8/CX/D2/CV /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/B4/CX/D2 /D0/D9/CS/CX/D2/CV /D8/CW/CT /D1/D3 /CS/CT/D6/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CQ/CP/D7/CT/CS /D3/D2 /D3/D0/D0/CX/D2/CT/CP/D6 /D0/CP/D7/CT/D6 /D7/D4 /CT /D8/D6/D3/D7 /D3/D4 /DD/BN /D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BG/BF /B8 /BG/BG /B8 /BG/BH ℄/B8 /D3/D6/D8/CW/CT /D6/CT/DA/CX/CT/DB /CJ/BG/BI ℄/B5 /CP/D6/CT /CS/CT/D7/CX/CV/D2/CT/CS /CX/D2 /D7/D9 /CW /CP /DB /CP /DD /D8/D3 /D1/CT/CP/D7/D9/D6/CT /CT/CX/D8/CW/CT/D6 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /D3/D6 /D8/CW/CT /CP/CQ /CT/D6/D6/CP/B9/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW /D8/BA /C4/CT/D8 /D9/D7 /DB/D6/CX/D8/CT /D8/CW/CT /CP/CQ /D3 /DA /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CU/D3/D6/D1 /CU/D6/D3/D1 /DB/CW/CX /CW /D3/D2/CT /CP/D2 /CS/CT/D8/CT/D6/D1/CX/D2/CT/D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /B4/BH/BE/B5 /CP/D2/CS /D8/CW/CT/D2 /D3/D1/D4/CP/D6/CT /D8/CW/CT/D1 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA /CC/CW/CT /D7/D4 /CT /D8/D3/CV/D6/CP/D4/CW /CX/D7 /CP/D8 /D6/CT/D7/D8/CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /B4/D8/CW/CTS /CU/D6/CP/D1/CT/B5 /CP/D2/CS /D8/CW/CT /D0/CX/CV/CW /D8 /D7/D3/D9/D6 /CT /B4/CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CTS′/CU/D6/CP/D1/CT/B5 /CX/D7 /D1/D3 /DA/CX/D2/CV /DB/CX/D8/CWv/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3S. /CC/CW/CT/D2 /CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D4/D4/D6/D3/CP /CW /D3/D2/D0/DD /D8/CW/CT /AS/D6/D7/D8 /D6 /CT/D0/CP/D8/CX/D3/D2 /CU/D6 /D3/D1 /B4/BH/BF/B5/B8 /D3/D6 /B4/BH/BG/B5/B8/CX/D7 /D9/D7/CT /CS/B8 /DB/CW/CX /CW /D1/CT/CP/D2/D7 /D8/CW/CP/D8/B8 /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D4/D6/CT/DA/CX/D3/D9/D7 /CP/D7/CT/D7/B8 /D8/CW/CT /AH/BT /CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /CS/CT /CP/D0/D7/DB/CX/D8/CW /D8/DB/D3 /CS/CX/AR/CT/D6 /CT/D2/D8 /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7 /CX/D2 /BG/BW /D7/D4 /CP /CT/D8/CX/D1/CT/B8 /CW/CT/D6 /CTω /CP/D2/CSω′/BA /CC/CW/CT/D2 /DB/D6/CX/D8/D8/CX/D2/CV /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU/D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUkµ, /CX/BA/CT/BA/B8 /D3/CUω, /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW λ /DB /CT /AS/D2/CS λ=γλ0(1−βcosθ), /B4/BH/BH/B5/BE/BF/DB/CW/CT/D6/CT λ /CX/D7 /D8/CW/CT /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /D6/CT /CT/CX/DA /CT/CS /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /CU/D6/D3/D1 /D8/CW/CT /D1/D3 /DA/CX/D2/CV /D7/D3/D9/D6 /CT /B4/D8/CW/CT /D7/CW/CX/CU/D8/CT/CS /D0/CX/D2/CT/B5/B8 λ0 /B4=λ′/B5 /CX/D7 /D8/CW/CT /D2/CP/D8/D9/D6/CP/D0 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /B4/D8/CW/CT /D9/D2/D7/CW/CX/CU/D8/CT/CS /D0/CX/D2/CT/B5 /CP/D2/CSθ /CX/D7 /D8/CW/CT /CP/D2/CV/D0/CT /D3/CUk /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT/CS/CX/D6/CT /D8/CX/D3/D2 /D3/CUv /CP/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /BA /CC/CW/CT /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8/D4/D6/CT/CS/CX /D8/D7 λ=λ0(1−βcosθ), /CP/D2/CS /CX/D2 /D8/CW/CT /D0/CP/D7/D7/CX /CP/D0 /CP/D7/CT /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /CS/D3 /CT/D7 /D2/D3/D8 /CT/DC/CX/D7/D8 /CU/D3/D6θ=π/2 /BA/CC/CW/CX/D7 /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /B4θ=π/2, λ=γλ0, /D3/D6ν=ν0/γ /B5 /CX/D7 /CP/D0/DB /CP /DD/D7/B8 /CX/D2 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0/B8 /AH/BT /CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CP/D4/D4/D6/D3/CP /CW /D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/D3 /CQ /CT /CP /CS/CX/D6/CT /D8 /D3/D2/D7/CT/D5/D9/CT/D2 /CT /D3/CU /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2/BN /CX/D8 /CX/D7 /CP/D7/D7/CT/D6/D8/CT/CS/B4/CT/BA/CV/BA /CJ/BE/BE ℄/B5 /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D1 /D9/D7/D8 /CQ /CT /D6/CT/D0/CP/D8/CT/CS /CP/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D3/CU /D8/CW/CT /D8/CX/D1/CT/D7 /CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2/CU/D3/D6 /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 △t=△t0γ /BA /C1/D8 /CX/D7 /D9/D7/D9/CP/D0/D0/DD /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CJ/BG/BI ℄/BM /AH/CC/CW/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA/BA/BA /D3/D1/D4/CP/D6/CT /D8/CW/CT /D6/CP/D8/CT/D7 /D3/CU /D8 /DB /D3 /AH /D0/D3 /CZ/D7/AH /D8/CW/CP/D8 /CP/D6/CT /CX/D2 /D1/D3/D8/CX/D3/D2 /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /CT/CP /CW /D3/D8/CW/CT/D6/BA /CC/CW/CT/DD /D1/CT /CP/D7/D9/D6 /CT/D8/CX/D1/CT /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /B4/D1 /DD /CT/D1/D4/CW/CP/D7/CX/D7/B5 /CP/D2/CS /CP/D2 /D8/CT/D7/D8 /D8/CW/CT /DA /CP/D0/CX/CS/CX/D8 /DD /D3/CU /D8/CW/CT /D7/D4 /CT /CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D2 /D8/CW/CX/D7 /D6/CT/D7/D4 /CT /D8/BA/AH/CB/CX/D1/CX/D0/CP/D6/D0/DD /CX/D8 /CX/D7 /CS/CT /D0/CP/D6/CT/CS /CX/D2 /CJ/BG/BF ℄/BM /AH/CC/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /CP /D1/D3/D6/CT /D8/CW/CP/D2 /D8/CT/D2/CU/D3/D0/CS /CX/D1/D4/D6/D3 /DA /CT/D1/CT/D2 /D8 /D3 /DA /CT/D6/D3/D8/CW/CT/D6 /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D2/CS /DA/CT/D6/CX/AS/CT/D7 /D8/CW/CT /D8/CX/D1/CT /CS/CX/D0/CP/D8/CX/D3/D2 /CT/AR/CT /D8 /B4/D1 /DD /CT/D1/D4/CW/CP/D7/CX/D7/B5 /CP/D8 /CP/D2 /CP /D9/D6/CP /DD/D0/CT/DA /CT/D0 /D3/CU /BE/BA/BF /D4/D4/D1/BA/AH /C7/CQ /DA/CX/D3/D9/D7/D0/DD /B8 /CP/D7 /DB /CT /D7/CP/CX/CS/B8 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CP/D6/CT /D8/CW/CT/D3/D6/CT/D8/CX /CP/D0/D0/DD /CP/D2/CP/D0/DD/D7/CT/CS/D3/D2/D0/DD /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /DB/CW/CX /CW /D8/D6/CT/CP/D8/D7 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUkµ/CP/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8 /D3/CUkµ, /CP/D2/CS /D1/D3/D6/CT/D3 /DA /CT/D6 /D3/D1/D4/D0/CT/D8/CT/D0/DD /D2/CT/CV/D0/CT /D8/D7/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8 /D3/CUkµ./C1/D2 /D8/CW/CT /C1/DA /CT/D7 /CP/D2/CS /CB/D8/CX/D0/DB /CT/D0/D0 /D8 /DD/D4 /CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D6/CT /D3/D2/CS/D9 /D8/CT/CS /CP/D8 /D7/DD/D1/D1/CT/D8/D6/CX /D3/CQ/D7/CT/D6/DA /CP/B9/D8/CX/D3/D2 /CP/D2/CV/D0/CT/D7 θ /CP/D2/CSθ+1800; /D4/CP/D6/D8/CX /D9/D0/CP/D6/D0/DD /CX/D2 /CJ/BG/BE ℄θ /CX/D7 /CW/D3/D7/CT/D2 /D8/D3 /CQ /CT≃00/BA /CC/CW/CT /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /CX/D2 /D8/CW/CT /CS/CX/D6/CT /D8/CX/D3/D2/D3/CU /D1/D3/D8/CX/D3/D2 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /B4/BH/BH/B5 /CP/D7λb=γλ0(1−βcosθ), /DB/CW/CX/D0/CT /D8/CW/CP/D8 /D3/D2/CT /CX/D2 /D8/CW/CT /D3/D4/D4 /D3/D7/CX/D8/CT /CS/CX/D6/CT /D8/CX/D3/D2/B4/D8/CW/CT /CP/D2/CV/D0/CT θ+ 1800/B5 /CX/D7λr=γλ0(1 +βcosθ), /CP/D2/CS /D8/CW/CT/D2△λb=\|λb−λ0\|=\|λ0(1−γ+βγcosθ)\|, △λr=\|λr−λ0\|=\|λ0(γ−1 +βγcosθ)\|, /CP/D2/CS /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CX/D2 /D7/CW/CX/CU/D8/D7 /CX/D7 △λ=△λr− △λb= 2λ0(γ−1)≃λ0β2, /B4/BH/BI/B5/DB/CW/CT/D6/CT /D8/CW/CT /D0/CP/D7/D8 /D6/CT/D0/CP/D8/CX/D3/D2 /CW/D3/D0/CS/D7 /CU/D3/D6β≪1. /C6/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D6/CT/CS/D7/CW/CX/CU/D8 /CS/D9/CT /D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /BW/D3/D4/D4/D0/CT/D6/CT/AR/CT /D8 /B4λ0β2/B5 /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/D2 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT θ /BA /C1/D2 /D8/CW/CT /D2/D3/D2/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX /CP/D7/CT△λ= 0 /B8 /D8/CW/CT/D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /CX/D7 /DE/CT/D6/D3/BA /C1/DA /CT/D7 /CP/D2/CS /CB/D8/CX/D0/DB /CT/D0/D0 /CU/D3/D9/D2/CS /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7/DB/CX/D8/CW /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BH/BI/B5 /CP/D2/CS /D2/D3/D8 /DB/CX/D8/CW /D8/CW/CT /D0/CP/D7/D7/CX /CP/D0 /D6/CT/D7/D9/D0/D8 △λ= 0./C0/D3 /DB /CT/DA /CT/D6/B8 /CP /D1/D3/D6/CT /CP/D6/CT/CU/D9/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D4/D6/CT/B9/CS/CX /D8/CX/D3/D2 /BX/D5/BA/B4/BH/BI/B5 /CP/D2/CS /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BE ℄ /CX/D7/B8 /D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 /CQ /CT/D0/CX/CT/CU/B8 /D3/D2/D0/DD /CP/D2 /AH/CP/D4/D4/CP/D6/CT/D2 /D8/AH/CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CP/D2/CS /D2/D3/D8 /D8/CW/CT /AH/D8/D6/D9/CT/AH /D3/D2/CT/BA /CC/CW/CX/D7 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CP /D8/D9/CP/D0/D0/DD /CW/CP/D4/D4 /CT/D2/D7 /CU/D3/D6 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D6/CT/CP/D7/D3/D2/D7/BA/BY/CX/D6/D7/D8/B8 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX /CP/D0 /D6/CT/D7/D9/D0/D8 /B4/BH/BI/B5 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /DB/CW/CX /CW /D3/D2/CT /CP/D2 /D7/D4 /CT/CP/CZ/CP/CQ /D3/D9/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD ω /CP/D2/CS /D8/CW/CT /DB /CP /DA /CT /DA /CT /D8/D3/D6 k /CP/D7 /DB /CT/D0/D0/B9/CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CD/D7/CX/D2/CV /D8/CW/CT /D1/CP/D8/D6/CX/DC Tµν,r/B4/BH/B5 /DB/CW/CX /CW /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 kµ r=Tµν,rkν e /B4/D3/D2/D0/DD /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS/B5/B8 /D3/D2/CT /AS/D2/CS/D7k0 r=k0 e−k1 e−k2 e−k3 e, ki r=ki e, /DB/CW/CT/D2 /CT /DB /CT /D3/D2 /D0/D9/CS/CT/D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX /CP/D0 /D4/D6/CT/CS/CX /D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2/D8/D7 /D3/CU /CP /BG/B9/DA /CT /D8/D3/D6/B8 /CX/BA/CT/BA/B8/CU/D3/D6λ, /DB/CX/D0/D0 /CQ /CT /D5/D9/CX/D8/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CQ/D9/D8 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CX/BA/CT/BA/B8 /CQ/D9/D8 /D8/CW/CT /D6/CT/D7/D9/D0/D8 /B4/BH/BI/B5/B8 /CP/D2/CS /D8/CW /D9/D7 /D2/D3/D8/CX/D2 /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /CJ/BG/BE ℄/BA /BY /D9/D6/D8/CW/CT/D6/B8 /D8/CW/CT /D7/D4 /CT /CX/AS /CW/D3/CX /CT /D3/CUθ /B4θ≃00) /CX/D2 /D8/CW/CT/CT/DC/D4 /CT/D6/CX/D1/CT/D2/D8/D7 /CJ/BG/BE ℄ /CX/D7 /D8/CW/CT /D2/CT/DC/D8 /D6 /CT /CP/D7/D3/D2 /CU/D3/D6 /D8/CW/CT /CP/CV/D6 /CT /CT/D1/CT/D2/D8 /DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D6 /CT/D7/D9/D0/D8 /B4/BH/BI/B5/BA /C6/CP/D1/CT/D0/DD /B8/CX/CUθ= 00/D8/CW/CT/D2n1= 1, n2=n3= 0 /B8 /CP/D2/CSkµ/CX/D7(ω/c, ω/c, 0,0). /BY /D6/D3/D1 /B4/BH/BF/B5 /D3/D6 /B4/BH/BG/B5 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CP/D8 /CX/D2 S′/D8/D3 /D3θ′= 00, n1′= 1 /CP/D2/CSn2′=n3′= 0 /B4/D8/CW/CT /D7/CP/D1/CT /CW/D3/D0/CS/D7 /CU/D3/D6θ= 1800, n1=−1, n2=n3= 0 /B8/D8/CW/CT/D2θ′= 1800/CP/D2/CSn1′=−1, n2′=n3′= 0 /B5/BA /C1/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BE ℄ /D8/CW/CT /CT/D1/CX/D8/D8/CT/D6 /CX/D7 /D8/CW/CT /D1/D3 /DA/CX/D2/CV/CX/D3/D2 /B4/CX/D8/D7 /D6/CT/D7/D8 /CU/D6/CP/D1/CT /CX/D7S′/B5/B8 /DB/CW/CX/D0/CT /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6 /CX/D7 /D8/CW/CT /D7/D4 /CT /D8/D6/D3/D1/CT/D8/CT/D6 /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /B4/D8/CW/CT S /CU/D6/CP/D1/CT/B5/BA /CB/CX/D2 /CT /CX/D2 /CJ/BG/BE ℄ /D8/CW/CT /CP/D2/CV/D0/CT /D3/CU /D8/CW/CT /D6/CP /DD /CT/D1/CX/D8/D8/CT/CS /CQ /DD /D8/CW/CT /CX/D3/D2 /CP/D8 /D6/CT/D7/D8 /CX/D7 /CW/D3/D7/CT/D2 /D8/D3 /CQ /CTθ′= 00/B41800/B5/B8 /D8/CW/CT/D2 /D8/CW/CT /CP/D2/CV/D0/CT /D3/CU /D8/CW/CX/D7 /D6/CP /DD /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /B8 /DB/CW/CT/D6/CT /D8/CW/CT /CX/D3/D2 /CX/D7 /D1/D3 /DA/CX/D2/CV/B8 /DB/CX/D0/D0 /CQ /CT /D8/CW/CT/D7/CP/D1/CT θ= 00/B41800/B5/BA /B4/CB/CX/D1/CX/D0/CP/D6/D0/DD /CW/CP/D4/D4 /CT/D2/D7 /CX/D2 /D8/CW/CT /D1/D3 /CS/CT/D6/D2 /DA /CT/D6/D7/CX/D3/D2/D7 /CJ/BG/BF /B8 /BG/BH ℄ /D3/CU /D8/CW/CT /C1/DA /CT/D7/B9/CB/D8/CX/D0/DB /CT/D0/D0 /CT/DC/D4 /CT/D6/B9/CX/D1/CT/D2 /D8/BN /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BF /B8 /BG/BH ℄ /D1/CP/CZ /CT /D9/D7/CT /D3/CU /CP/D2 /CP/D8/D3/D1/CX /D3/D6 /CX/D3/D2/CX /CQ /CT/CP/D1 /CP/D7 /CP /D1/D3 /DA/CX/D2/CV /D0/CX/CV/CW /D8 /CP/D2/CP/D0/DD/DE/CT/D6/B4/D8/CW/CT /CP /CT/D0/CT/D6/CP/D8/CT/CS /CX/D3/D2 /CX/D7 /D8/CW/CT /AH/D3/CQ/D7/CT/D6/DA /CT/D6/AH/B5 /CP/D2/CS /D8 /DB /D3 /D3/D0/D0/CX/D2/CT/CP/D6 /D0/CP/D7/CT/D6 /CQ /CT/CP/D1/D7 /B4/D4/CP/D6/CP/D0/D0/CT/D0 /CP/D2/CS /CP/D2 /D8/CX/D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3/D8/CW/CT /D4/CP/D6/D8/CX /D0/CT /CQ /CT/CP/D1/B5 /CP/D7 /D0/CX/CV/CW /D8 /D7/D3/D9/D6 /CT/D7 /B4/D8/CW/CT /CT/D1/CX/D8/D8/CT/D6/B5/B8 /DB/CW/CX /CW /CP/D6/CT /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /BA/B5 /BY /D6/D3/D1 /D8/CW/CX/D7 /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /DB /CT /D3/D2 /D0/D9/CS/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT/D7/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D3/D2/CT /CP/D2 /D3/D2/D7/CX/CS/CT/D6 /D3/D2/D0/DD /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8/B8 /D8/CW/CP/D8/CX/D7/B8 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CUω /B4/D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUkµ; /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /D3/CU /D8/CW/CT /D8/D6/D9/CT /BG/B9/DA /CT /D8/D3/D6 ka/CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5/B8 /CP/D2/CS /D2/D3/D8 /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW /D8/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CUn, /CX/BA/CT/BA/B8k,/B4/D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8 /D3/CUkµ/B5/BA /BU/CT /CP/D9/D7/CT /D3/CU /D8/CW/CP/D8 /D8/CW/CT/DD /CU/D3/D9/D2/CS /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BH/BH/B5 /B4/D3/D6/B4/BH/BI/B5/B5 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BH/BF/B5 /CP/D2/CS /B4/BH/BG/B5 /D6/CT/DA /CT/CP/D0 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU /CP/D2/CP/D6/CQ/CX/D8/D6/CP/D6/DD θ /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUkµ /CP/D2/D2/D3/D8 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU/D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /CX/D2 /D7/D9 /CW /CP/D7/CT /D3/D2/CT /CP/D2/D2/D3/D8 /CT/DC/D4 /CT /D8 /D8/CW/CP/D8 /D8/CW/CT/BE/BG/D6/CT/D0/CP/D8/CX/D3/D2 /B4/BH/BI/B5/B8 /D8/CP/CZ /CT/D2 /CP/D0/D3/D2/CT/B8 /DB/CX/D0/D0 /CQ /CT /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CP/D8 /D7/D3/D1/CT /CP/D6/CQ/CX/D8/D6/CP/D6/DD θ. /CB/D9 /CW /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /DB /CT/D6/CT/B8 /CX/D2 /CU/CP /D8/B8 /D6/CT /CT/D2 /D8/D0/DD /D3/D2/CS/D9 /D8/CT/CS /CP/D2/CS /DB /CT /CS/CX/D7 /D9/D7/D7 /D8/CW/CT/D1 /CW/CT/D6/CT/BA/C8 /D3/CQ /CT/CS/D3/D2/D3/D7/D8/D7/CT/DA /CP/D2/CS /D3/D0/D0/CP/CQ 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/D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D3/CU /D7/D9 /CW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/DB /CT/D6/CT /D1/CP/CS/CT /CQ /DD /D8/DB/D3 /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CS/D3 /CT/D7 /D2/D3/D8 /D1/CT/CP/D2 /D8/CW/CP/D8 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CW/CP/D7 /D7/D3/D1/CT/D8/CW/CX/D2/CV /D8/D3 /CS/D3 /DB/CX/D8/CW /D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1/BA/C1/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CT /CT/D2 /D8/CX/D6/CT /BG/BW /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /D8/CW/CT /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 /D3/D6 /D8/CW/CT /BV/BU/BZ/C9/B8 /CW/CP/D7 /D8/D3 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS/CQ /D3/D8/CW /CX/D2 /D8/CW/CT /D8/CW/CT/D3/D6/DD /CP/D2/CS /CX/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2/D8/D7 /BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D8/CW/CT/D3/D6/CT/D8/CX /CP/D0/D0/DD /CS/CX/D7 /D9/D7/D7 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/D3/CU /D8/CW/CT /C1/DA /CT/D7/B9/CB/D8/CX/D0/DB /CT/D0/D0 /D8 /DD/D4 /CT /DB /CT /CW/D3 /D3/D7/CT /CP/D7 /D8/CW/CT /D6/CT/D0/CT/DA /CP/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8 /DD /D8/CW/CT /DB /CP /DA /CT /DA /CT /D8/D3/D6 ka, /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /DB/CW/CX /CW /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B9/CQ/CP/D7/CT/CS /CV/CT/D3/D1/CT/D8/D6/CX /D0/CP/D2/CV/D9/CP/CV/CT /CP/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BF/BH/B5/B8 ka=kµ′eµ′=kµeµ=kµ′ rrµ′=kµ rrµ. /BX/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD /D3/D2/CT /CP/D2 /D3/D2/D7/CX/CS/CT/D6 /CX/D8/D7 /D7/D5/D9/CP/D6/CT /CU/D3/D6 /DB/CW/CX /CW /CX/D8 /CW/D3/D0/CS/D7/D8/CW/CP/D8 kagabkb= 0; /B4/BH/BJ/B5/D8/CW/CX/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CX/D7 /CP /C4/D3/D6/CT/D2 /D8/DE /D7 /CP/D0/CP/D6 /CP/D2/CS /CX/D8 /CX/D7 /CP/D0/D7/D3 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT /CW/D3/CX /CT /D3/CU /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/CC/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BF/BH/B5 /CP/D2/CS /B4/BH/BJ/B5 /D7/CW/D3 /DB /D8/CW/CP/D8 /DB /CT /CP/D2 /CP/D0 /D9/D0/CP/D8/CT ka/B4/D3/D6kagabkb/B5 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/CP/D2/CS /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CT/D1/CX/D8/D8/CT/D6 /B4/D8/CW/CTS′/CU/D6/CP/D1/CT/B5/BN /D8/CW/CT /CT/D1/CX/D8/D8/CT/D6 /CX/D7 /D8/CW/CT /CX/D3/D2 /D1/D3 /DA/CX/D2/CV /CX/D2S, /D8/CW/CT /D6/CT/D7/D8/CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D7/D4 /CT /D8/D6/D3/D1/CT/D8/CT/D6/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /CU/D6/CP/D1/CT/BA /C1/D2 /D3/D8/CW/CT/D6 /D4 /CT/D6/D1/CX/D7/D7/CX/CQ/D0/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/CP/D2/CS /CX/D2 /D3/D8/CW/CT/D6 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7 /D8/CW/CT/D7/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /DB/CX/D0/D0 /CQ /CT /CT/DC/CP /D8/D0/DD /D8/CW/CT /D7/CP/D1/CT /CP/D7 /CX/D2S′/CP/D2/CS /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /CC/CW/CP/D8 /CX/D7 /CP /CV/D6/CT/CP/D8 /D4/D6/CP /D8/CX /CP/D0 /CP/CS/DA /CP/D2 /D8/CP/CV/CT /D3/CU /D8/CW/CT /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CB/CA/BN /DB/CW/CT/D2/D8/CW/CT /DB/CW/D3/D0/CT /B4/CX/D2 /D0/D9/CS/CX/D2/CV /D8/CW/CT /CQ /CP/D7/CX/D7/B5 /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2/D8/CX/D8/DD /CX/D7 /D3/D2/D7/CX/CS/CT/D6 /CT /CS /D8/CW/CT/D2 /CX/D8 /CX/D7 /CP/D2 /CX/D2/DA/CP/D6/CX/CP/D2/D8 /D5/D9/CP/D2/D8/CX/D8/DD/BA/BY/CX/D6/D7/D8 /DB /CT /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BJ ℄ /CP/D2/CS /CJ/BG/BK ℄ /D7/CX/D2 /CT /D8/CW/CT/DD /D7/CW/D3 /DB /CT/CS /D8/CW/CT /CS/CX/D7/CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT/D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /D8/CW/CT/D3/D6/DD /B8 /CX/BA/CT/BA/B8 /DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CC/CW/CT/D2 ka/CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2S′/CX/D7/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CQ /DD /D8/CW/CT /BV/BU/BZ/C9 kµ′eµ′/DB/CW/CT/D2 /CT /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 kµ′/CP/D6/CTkµ′= (ω′/c)(1,cosθ′,sinθ′,0)/CP/D2/CSkµ′kµ′= 0. /CC/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6 /B4/D8/CW/CT /D7/D4 /CT /D8/D6/D3/D1/CT/D8/CT/D6/B5 /CX/D2 /D8/CW/CT /D0/CP/CQ /D3/D6/CP/D8/D3/D6/DD /CU/D6/CP/D1/CT /DB/CX/D0/D0 /D0/D3 /D3/CZ /CP/D8 /D8/CW/CT /D7/CP/D1/CT/BG/BW /D5/D9/CP/D2/D8/CX/D8/DD ka, /D3/D6 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD /D8/CW/CT /BV/BU/BZ/C9 kµeµ /B8 /CP/D2/CS /AS/D2/CSkµ, /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /D3/D1/D4 /D3/D2/CT/D2 /D8/CU/D3/D6/D1 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB /CP /DA /CT /DA /CT /D8/D3/D6 kµeµ, /CP/D7 kµ= [γ(ω′/c)(1 +βcosθ′), γ(ω′/c)(cosθ′+β),(ω′/c)sinθ′,0],/DB/CW/CT/D2 /CT kµkµ /CX/D7 /CP/D0/D7/D3= 0. /BY /D6/D3/D1 /D8/CW/CP/D8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/D2/CT /CP/D2 /AS/D2/CS /D8/CW/CP/D8 n1= (n1′+β)/(1 +βn1′), n2=n2′/γ(1 +βn1′), n3=n3′/γ(1 +βn1′),/BE/BH/D3/D6 /D8/CW/CP/D8 sinθ= sinθ′/γ(1 +βcosθ′),cosθ= (cos θ′+β)/(1 +βcosθ′), tanθ= sinθ′/γ(β+ cos θ′). /B4/BH/BK/B5/CC/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BH/BK/B5 /D6/CT/DA /CT/CP/D0 /D8/CW/CP/D8 /D2/D3/D8 /D3/D2/D0/DDω /CX/D7 /CW/CP/D2/CV/CT/CS /B4/D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8/B5 /DB/CW/CT/D2 /CV/D3/CX/D2/CV /CU/D6/D3/D1S′/D8/D3 S /CQ/D9/D8 /CP/D0/D7/D3 /D8/CW/CT /CP/D2/CV/D0/CT /D3/CUk /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT /CS/CX/D6/CT /D8/CX/D3/D2 /D3/CUv /CX/D7 /CW/CP/D2/CV/CT/CS /B4/D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW /D8/B5/BA /CC/CW/CX/D7/D1/CT/CP/D2/D7 /D8/CW/CP/D8 /CX/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D9/D2/D7/CW/CX/CU/D8/CT/CS /D0/CX/D2/CT /B4/CX/BA/CT/BA/B8 /D3/CU /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD ω′=ω0 /CU/D6/D3/D1 /D8/CW/CT /CP/D8/D3/D1/CP/D8 /D6/CT/D7/D8/B5 /CX/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CP/D8 /CP/D2 /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT θ′/CX/D2S′, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CT/D1/CX/D8/D8/CT/D6/B8 /D8/CW/CT/D2 /D8/CW/CT/D7/CP/D1/CT /D0/CX/CV/CW/D8 /DB/CP/DA/CT /B4/CU/D6/D3/D1 /D8/CW/CT /D7/CP/D1/CT /CQ/D9/D8 /D2/D3 /DB /D1/D3 /DA/CX/D2/CV /CP/D8/D3/D1/B5 /DB/CX/D0/D0 /CW/CP /DA /CT /D8/CW/CT /D7/CW/CX/CU/D8/CT/CS /CU/D6/CT/D5/D9/CT/D2 /DD ω /CP/D2/CS /DB/CX/D0 /D0/CQ /CT /D7/CT /CT/D2 /CP/D8 /CP/D2 /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT θ /B4/CV/CT/D2/CT/D6/CP/D0/D0/DD /B8 /ne}ationslash=θ′/B5 /CX/D2S, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D7/D4 /CT /D8/D6/D3/D1/CT/D8/CT/D6/BA /C1/D2 S′/D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 ω′/CP/D2/CSθ′/CS/CT/AS/D2/CT /D8/CW/CT /BV/BU/BZ/C9 kµ′eµ′, /CP/D2/CS /D8/CW/CX/D7 /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2 /BG/B9/DA /CT /D8/D3/D6 /D7/CP/D8/CX/D7/AS/CT/D7 /D8/CW/CT/D6/CT/D0/CP/D8/CX/D3/D2 kµ′kµ′= 0, /DB/CW/CX /CW /CX/D7 /D8/CW/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BH/BJ/B5 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/CP/D2/CS /CX/D2 /D8/CW/CTS′/CU/D6/CP/D1/CT/BA /CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 ω′/CP/D2/CSθ′/B4/D8/CW/CP/D8 /CS/CT/AS/D2/CT /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV kµ′eµ′/CX/D2S′/B5 /CP/D6/CT /D3/D2/D2/CT /D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV ω /CP/D2/CSθ /B4/D8/CW/CP/D8 /CS/CT/AS/D2/CT /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV kµeµ /CX/D2S /B5 /CQ /DD /D1/CT/CP/D2/D7 /D3/CU/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 Lµ′ ν,e /B4/BE/B5 /B4/CP/D2/CS /CX/D8/D7 /CX/D2 /DA /CT/D6/D7/CT/B5 /D3/CUkµ′eµ′. /CC/CW/CT/D2 kµeµ /CX/D7 /D7/D9 /CW /D8/CW/CP/D8 /CX/D8 /CP/D0/D7/D3/D7/CP/D8/CX/D7/AS/CT/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 kµkµ= 0, /D8/CW/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /B4/BH/BJ/B5 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /D2/D3 /DB /CX/D2/D8/CW/CTS /CU/D6/CP/D1/CT/BA /CC/CW/CT /CP/D9/D8/CW/D3/D6/D7 /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BJ ℄ /B4/CP/D2/CS /CJ/BG/BK ℄/B5 /D1/CP/CS/CT /D8/CW/CT /D3/CQ/D7/CT/D6/DA/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D6 /CP/CS/CX/CP/D8/CX/D3/D2/CU/D6 /D3/D1 /D8/CW/CT /CP/D8/D3/D1 /CP/D8 /D6 /CT/D7/D8 /B4/D8/CW/CT /D9/D2/D7/CW/CX/CU/D8/CT /CS /D0/CX/D2/CT/B5 /CP/D2/CS /CU/D6 /D3/D1 /CP /D1/D3/DA/CX/D2/CV /CP/D8/D3/D1 /CP/D8 /D8/CW/CT /D7/CP/D1/CT /D3/CQ/D7/CT/D6/DA/CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT/BA/CC/CW/CT /D4/D6/CT /CT/CS/CX/D2/CV /CS/CX/D7 /D9/D7/D7/CX/D3/D2 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /CX/CU /D8/CW/CT/DD /D7/D9 /CT/CT/CS/CT/CS /D8/D3 /D7/CT/CTω′=ω0 /B4/CX/BA/CT/BA/B8λ0 /B5 /CU/D6/D3/D1 /D8/CW/CT /CP/D8/D3/D1 /CP/D8/D6/CT/D7/D8 /CP/D8 /D7/D3/D1/CT /D7/DD/D1/D1/CT/D8/D6/CX /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT/D7 θ′/B4/ne}ationslash= 0 /B5 /CP/D2/CSθ′+ 1800/B4/CX/BA/CT/BA/B8 /D7/D3/D1/CT kµ′eµ′/B5 /D8/CW/CT/D2 /D8/CW/CT/DD /D3/D9/D0/CS /D2/D3/D8 /D7/CT/CT /D8/CW/CT /CP/D7/D7/DD/D1/CT/D8/D6/CX /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /B4/CU/D6/D3/D1 /D1/D3 /DA/CX/D2/CV /CP/D8/D3/D1/D7/B5 /CP/D8 /D8/CW/CT /D7/CP/D1/CT /CP/D2/CV/D0/CT/D7 θ=θ′/B4/CP/D2/CS θ+ 1800=θ′+ 1800/B5/BA /CC/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CS/D3 /CT/D7 /D2/D3/D8 /D3/D2/D2/CT /D8 /D7/D9 /CW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CC/CW/CX/D7 /DB /CP/D7 /D8/CW/CT/D6/CT/CP/D7/D3/D2 /D8/CW/CP/D8 /D8/CW/CT/DD /CS/CT/D8/CT /D8/CT/CS △λ≃0 /CP/D2/CS /D2/D3/D8△λ≃λ0β2. /BU/D9/D8 /DB /CT /CT/DC/D4 /CT /D8 /D8/CW/CP/D8 /D8/CW/CT /D6/CT/D7/D9/D0/D8 △λ≃λ0β2 /CP/D2 /CQ /CT /D7/CT/CT/D2 /CX/CU /D8/CW/CT /D7/CX/D1/CX/D0/CP/D6 /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8/D7 /D3/CU /D8/CW/CT /CU/D6 /CT /D5/D9/CT/D2 /CX/CT/D7/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /DB/CP/DA/CT/D0/CT/D2/CV/D8/CW/D7/B8 /D3/CU /D8/CW/CT /D6 /CP/CS/CX/CP/D8/CX/D3/D2/CU/D6 /D3/D1 /D1/D3/DA/CX/D2/CV /CP/D8/D3/D1/D7 /DB/D3/D9/D0/CS /CQ /CT /D4 /CT/D6/CU/D3/D6/D1/CT /CS /D2/D3/D8 /CP/D8θ=θ′/CQ/D9/D8 /CP/D8θ /CS/CT/D8/CT/D6/D1/CX/D2/CT /CS /CQ/DD /D8/CW/CT /D6 /CT/D0/CP/D8/CX/D3/D2 /B4/BH/BK/B5/BA/C7/D2/D0/DD /CX/D2 /D8/CW/CP/D8 /CP/D7/CT /D3/D2/CT /DB/CX/D0 /D0 /D1/CP/CZ/CT /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8 /D3/CU /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2/D8/CX/D8/DD ka=kµ′eµ′=kµeµ /CU/D6 /D3/D1 /D8/DB/D3/CS/CX/AR/CT/D6 /CT/D2/D8 /D6 /CT/D0/CP/D8/CX/DA/CT/D0/DD /D1/D3/DA/CX/D2/CV /C1/BY/CA/D7/BA/CA/CT /CT/D2 /D8/D0/DD /B8 /BU/CT/CZ/D0/CY/CP/D1/CX/D7/CW/CT/DA /CJ/BG/BL ℄ /CP/D1/CT /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D3/D2 /D0/D9/D7/CX/D3/D2/D7 /B4/CQ/D9/D8 /CS/CT/CP/D0/CX/D2/CV /D3/D2/D0/DD /DB/CX/D8/CW /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/CU/D3/D6/D1 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5 /CP/D2/CS /CT/DC/D4/D0/CP/CX/D2/CT/CS /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BJ ℄ /CP/D2/CS /CJ/BG/BK ℄ /D8/CP/CZ/CX/D2/CV/CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW /D8 /D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8/BA /C1/D8 /CX/D7 /CP/D6/CV/D9/CT/CS /CX/D2 /CJ/BG/BL℄ /D8/CW/CP/D8/BX/D5/BA/B4/BH/BH/B5 /CU/D3/D6 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /CP/D2 /CQ /CT /D6/CT/CP/D0/CX/DE/CT/CS /D3/D2/D0/DD /DB/CW/CT/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT /CX/D7/CU/D9/D0/AS/D0/D0/CT/CS/B8 △θ=βsinθ′, /B4/BH/BL/B5/DB/CW/CT/D6/CT △θ=θ′−θ, /CP/D2/CSβ /CX/D7 /D8/CP/CZ /CT/D2 /D8/D3 /CQ /CTβ≪1. /CC/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /B4/BH/BL/B5 /CS/CX/D6/CT /D8/D0/DD /CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1 /D8/CW/CT/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6sinθ /CX/D2 /B4/BH/BK/B5 /D8/CP/CZ/CX/D2/CV /D8/CW/CP/D8β≪1. /CC/CW/CT /CP/D7/D7/DD/D1/CT/D8/D6/CX /D7/CW/CX/CU/D8 /DB/CX/D0 /D0 /CQ /CT /D7/CT /CT/D2 /DB/CW/CT/D2 /D8/CW/CT /D3/D0 /D0/CX/D1/CP/D8/D3/D6/CP/D7/D7/CT/D1/CQ/D0/DD /CX/D7 /D8/CX/D0/D8/CT /CS /CP/D8 /CP /DA/CT/D0/D3 /CX/D8/DD /CS/CT/D4 /CT/D2/CS/CT/D2/D8 /CP/D2/CV/D0/CT △θ. /C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/D3 /DB /D3/D6/CZ/B8 /CP/D7 /D9/D7/D9/CP/D0/B8 /DB/CX/D8/CW /D8/CW/CT /CP/D6/D1/D7/D3/CU /D8/CW/CT /D3/D0/D0/CX/D1/CP/D8/D3/D6 /CP/D8 /AS/DC/CT/CS /CP/D2/CV/D0/CT/D7 θ /CP/D2/CSθ+ 1800, /BU/CT/CZ/D0/CY/CP/D1/CX/D7/CW/CT/DA /CJ/BG/BL ℄ /D4/D6/D3/D4 /D3/D7/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D3/D0/D0/CX/D1/CP/D8/D3/D6/CP/D7/D7/CT/D1 /CQ/D0/DD /D1 /D9/D7/D8 /CQ /CT /D3/D2/D7/D8/D6/D9 /D8/CT/CS /CX/D2 /D7/D9 /CW /CP /DB /CP /DD /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /D8/CW/CT /D3/D6/D6/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT/D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8/D0/DD /CU/D3/D6 /CQ /D3/D8/CW /CP/D6/D1/D7/BN /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /CP/D6/D1 /CP/D8 /CP/D2/CV/D0/CT θ /B4θ+ 1800/B5 /CW/CP/D7 /D8/D3/CQ /CT /D8/CX/D0/D8/CT/CS /D0/D3 /CZ/DB/CX/D7/CT /B4 /D3/D9/D2 /D8/CT/D6/B9 /D0/D3 /CZ/DB/CX/D7/CT/B5 /CQ /DD /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT△θ. /C7/D8/CW/CT/D6/DB/CX/D7/CT /D8/CW/CT /CP/D7/D7/DD/D1/CT/D8/D6/DD /CX/D2 /D8/CW/CT/BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8/D7 /DB/CX/D0/D0 /D2/D3/D8 /CQ /CT /D3/CQ/D7/CT/D6/DA /CT/CS/BA /CC/CW /D9/D7 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BJ ℄ /CP/D2/CS /CJ/BG/BK ℄ /DB /D3/D9/D0/CS /D2/CT/CT/CS /D8/D3 /CQ /CT /D6/CT/D4 /CT/CP/D8/CT/CS/D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /BU/CT/CZ/D0/CY/CP/D1/CX/D7/CW/CT/DA/B3/D7 /D4/D6/D3/D4 /D3/D7/CX/D8/CX/D3/D2/BA /CC/CW/CT /D4 /D3/D7/CX/D8/CX/DA/CT /D6 /CT/D7/D9/D0/D8 /CU/D3/D6 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8△λ /B4/BH/BI/B5/B8/DB/CW/CT/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CP/CQ /CT/D6/D6 /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT △θ /B4/BH/BL/B5 /CX/D7 /CU/D9/D0/AS/D0 /D0/CT /CS/B8 /DB/CX/D0 /D0 /CS/CT/AS/D2/CX/D8/CT/D0/DD /D7/CW/D3/DB /D8/CW/CP/D8 /CX/D8 /CX/D7 /D2/D3/D8/D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D8/D6 /CT /CP/D8 /D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /CP/D2/CS /D8/CW/CT /CP/CQ /CT/D6/D6 /CP/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW/D8 /CP/D7 /D7/CT/D4 /CP/D6 /CP/D8/CT/B8 /DB/CT/D0 /D0/B9/CS/CT/AS/D2/CT /CS/B8 /CT/AR/CT /D8/D7/B8 /CX/BA/CT/BA/B8/D8/CW/CP/D8 /CX/D8 /CX/D7 /D8/CW/CT /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/B8/AH /CP/D2/CS /D2/D3/D8 /D8/CW/CT /AH/BT /CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/B8/AH /DB/CW/CX /CW /D3/D6/D6 /CT /D8/D0/DD /CT/DC/D4/D0/CP/CX/D2/D7 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2/D8/D7/D8/CW/CP/D8 /D8/CT/D7/D8 /CB/CA/BA/BK /BV/C7/C6/BV/C4/CD/CB/C1/C7/C6/CB /BT/C6/BW /BW/C1/CB/BV/CD/CB/CB/C1/C7/C6/C1/D2 /D8/CW/CT /AS/D6/D7/D8 /D4/CP/D6/D8 /D3/CU /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /DB /CT /CW/CP /DA /CT /CS/CX/D7 /D9/D7/D7/CT/CS /CP/D2/CS /CT/DC/D4 /D3/D7/CT/CS /D8/CW/CT /D1/CP/CX/D2 /CS/CX/AR/CT/D6/CT/D2 /CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/D6/CT/CT/D8/CW/CT/D3/D6/CT/D8/CX /CP/D0 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /CB/CA/B8 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /D8/CW/CT /D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP /CW /D8/D3 /CB/CA /CP/D2/CS /D8/CW/CT /AH/BT /CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /C1/D2 /D8/CW/CT /D7/CT /D3/D2/CS /D4/CP/D6/D8 /DB /CT /CW/CP /DA /CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /D8/CW/CT /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT/D7/CT /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /D8/CW/CT/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA /CC/CW/CT /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /DB/CW/CX /CW /D8/CT/D7/D8 /CB/CA /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT/DD /CP/CV/D6/CT/CT /DB/CX/D8/CW /D8/CW/CT/D4/D6/CT/CS/CX /D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /D2/D3/D8/B8 /CP/D7 /D9/D7/D9/CP/D0/D0/DD /D7/D9/D4/D4 /D3/D7/CT/CS/B8 /DB/CX/D8/CW /D8/CW/D3/D7/CT /D3/CU /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/BE/BI/C1/D2 /D8/CW/CT /AH/D1 /D9/D3/D2/AH /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /D8/CW/CT /AT/D9/DC/CT/D7 /D3/CU /D1 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/CU/D3/D6 /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2S /CP/D2/CSS′/CU/D6/CP/D1/CT/D7/B8 /CP/D2/CS /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /D8/CW/CP/D8 /D8/CW/CT/D7/CT /D8 /DB /D3 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /D3/D2/D2/CT /D8/CT/CS /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA /CB/CX/D2 /CT /D3/D2/D0/DD /CP /D4/CP/D6/D8k0′ el0′e /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD φ /B4/BF/BG/B5 /D3/D6 /B4/BF/BI/B5 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/CW/CT /D2/D3/D2/B9/D2 /D9/D0/D0 /CU/D6/CX/D2/CV/CT /D7/CW/CX/CU/D8 /B4/BG/BD/B5 /CP/D2 /CQ /CT /D7/CW/D3 /DB/D2 /D8/D3 /CQ /CT /D5/D9/CX/D8/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CX/D2/CP/D2/D3/D8/CW/CT/D6 /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CT/BA/CV/BA/B8 /CX/D2 /D8/CW/CT /AH/D6/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /B4/D7/CT/CT /CJ/BE℄/B5/BA/BE/BJ/CC/CW/CT /D7/CP/D1/CT /D3/D2 /D0/D9/D7/CX/D3/D2/D7 /CP/D2 /CQ /CT /CS/D6/CP /DB/D2 /CU/D3/D6 /D8/CW/CT /C3/CT/D2/D2/CT/CS/DD/B9/CC/CW/D3/D6/D2/CS/CX/CZ /CT /D8 /DD/D4 /CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA/C1/D2 /D8/CW/CT /C1/DA /CT/D7/B9/CB/D8/CX/D0/DB /CT/D0/D0 /D8 /DD/D4 /CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CU/D3/D6/D8/CW/CT /BW/D3/D4/D4/D0/CT/D6 /CT/AR/CT /D8 /CP/D2/CS /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CX/D7 /CP/CV/CP/CX/D2 /D3/D2/D0/DD /CP/D2 /AH/CP/D4/D4/CP/D6/CT/D2 /D8/AH /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CP/D2/CS /D2/D3/D8 /D8/CW/CT /AH/D8/D6/D9/CT/AH/D3/D2/CT/BA /C6/CP/D1/CT/D0/DD /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /BW/D3/D4/D4/D0/CT/D6 /D7/CW/CX/CU/D8 /B4λ0β2/B8 /B4/BH/BI/B5/B5 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2/DB/CW/CX /CW /D3/D2/CT /CP/D2 /D7/D4 /CT/CP/CZ /CP/CQ /D3/D9/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD ω /CP/D2/CS /D8/CW/CT /DB /CP /DA /CT /DA /CT /D8/D3/D6 k /CP/D7 /DB /CT/D0/D0/B9/CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA/BY /D9/D6/D8/CW/CT/D6 /CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D4/D4/D6/D3/CP /CW /D3/D2/D0/DD /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CUω /B4/D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CU kµ/B5 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS/B8 /DB/CW/CX/D0/CT /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CX/CV/CW /D8/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CUn, /CX/BA/CT/BA/B8k, /B4/D8/CW/CT /D7/D4/CP/D8/CX/CP/D0/D4/CP/D6/D8 /D3/CUkµ/B5 /CX/D7 /D2/CT/CV/D0/CT /D8/CT/CS/BA /B4kµ/CX/D7 /D8/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 /CU/D3/D6/D1 /CX/D2 /D8/CW/CT /AH/CT/AH /D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D6/D9/CT /D8/CT/D2/D7/D3/D6 ka/B4/BF/BH/B5/BA/B5 /CC/CW /D9/D7 /CX/D2 /D8/CW/CX/D7 /CP/D7/CT /D8/D3 /D3 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CS/CT/CP/D0/D7 /DB/CX/D8/CW /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /BG/BW /D7/D4/CP /CT/D8/CX/D1/CT/B8 ω /CP/D2/CSω′/B8 /DB/CW/CX /CW /CP/D6/CT /D2/D3/D8 /D3/D2/D2/CT /D8/CT/CS /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CU/D3/D6 /D8/CW/CT /D7/D4 /CT /CX/AS /CW/D3/CX /CT/D3/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CP/D2/CV/D0/CT/D7 θ′= 00/B41800/B5 /CX/D2S′/B4/D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CT/D1/CX/D8/D8/CT/D6/B5/B8 /D3/D2/CT /AS/D2/CS/D7 /CU/D6/D3/D1 /D8/CW/CT/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CUkµ/D8/CW/CP/D8θ /CX/D2S /CX/D7 /CP/CV/CP/CX/D2 = 00/B41800/B5/BA /CB/CX/D2 /CT /CX/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CJ/BG/BE ℄/B8 /CP/D2/CS /CX/D8/D7 /D1/D3 /CS/CT/D6/D2/DA /CT/D6/D7/CX/D3/D2/D7 /CJ/BG/BF /B8 /BG/BH ℄/B8 /CY/D9/D7/D8 /D7/D9 /CW /CP/D2/CV/D0/CT/D7 /DB /CT/D6/CT 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arXiv:physics/0103027v1 [physics.ins-det] 9 Mar 2001An insulating grid spacer for large-area MICROMEGAS chambe rs D. Bernard, H. Delagrange, D. G. d’Enterria, M. Le Guay, G. Ma rt´ ınez∗, M.J. Mora, P. Pichot, D. Roy and Y. Schutz SUBATECH (Ecole des Mines de Nantes, IN2P3/CNRS, Universit ´ e de Nantes), BP20722, 44307 Nantes Cedex 3, France A. Gandi and R. de Oliveira CERN, CH - 1211 Geneva 23, Switzerland Abstract We present an original design for large area gaseous detecto rs based on the MICROMEGAS technology. This tech- nology incorporates an insulating grid, sandwiched betwee n the micro-mesh and the anode-pad plane, which provides an uniform 200 µm ampliﬁcation gap. The uniformity of the ampliﬁcation gap t hickness has been veriﬁed. The gain performances of the detector are presented and compared to t he values obtained with detectors using cylindrical micro spacers. The new design presents several technical and ﬁnan cial advantages. Particle detectors based on the MICROMEGAS technology [1–8 ] consist of a parallel-plate gas-chamber split into two asymmetric stages by an intermediate electrode made out of a micro-mesh. By applying appropriate voltages between the three electrodes, a very high electric ﬁeld ( ∼30 kV/cm) in the thinnest stage (the “ampliﬁcation gap”) and a low electric ﬁeld (3 kV/cm) in the thicker stage (the “drift ” or “conversion gap”) can be simultaneously established. When a charged particle traverses the conversion gap, it gen erates primary electrons which are subsequently multiplie d in the ampliﬁcation gap. The ampliﬁed electron cloud is coll ected on the anode generating a fast electric signal. The associated ion cloud is quickly collected on the micro-mesh , whereas only a small fraction of the ion cloud enters into the conversion region. Based on this technology, we had earl ier developed [8] large area particle detectors. In the current detector, the main modiﬁcations with respect our pr evious developments [8] are: •The total area of the anode plane has been enlarged to 400 ×400 mm2, providing an active area of 387 ×387 mm2. The anode electrode consists of a 1.0 mm thick printed circu it board. Its inner surface is segmented in rectangular 12 ×12 mm2gilded copper pads. The inter-pad width is 100 µm and the total number of pads is 1024. •The electro-formed micro-mesh1consists of a 7 µm thick grid of 508 ×508 mm2made of pure Ni. The 106 µm squared holes grid are outlined by a 20 µm thick border of Ni in steps of 126 µm, i.e. 200 LPI (lines per inch). The optical transparency reaches 70%. The micro-mesh is str etched on the Plexiglas frame whose height deﬁnes the 6 mm thick (compared to the 3 mm thick in the earlier design ) conversion gap between the micro-mesh and the cathode plane. The ampliﬁcation gap between the micro-m esh and the anode plane has been doubled to 200µm. In this letter, two diﬀerent designs to keep an uniform ampli ﬁcation gap of 200 µm have been compared: •Old design. The ampliﬁcation gap is deﬁned by cylindrical micro-spacer s of 200 µm high and 250 µm in diameter, glued on to the anode-pads with a pitch of 2 mm in bot h directions. •New design. The micro-spacers are replaced by an insulating grid sandwi ched between the micro-mesh and the anode plane (see Fig.1). The new design presents several decisive technical but also ﬁnancial advantages when compared to the cylindrical micro-spacer design. Indeed, since the grid is independent of the printed-circuit board, boards and grids can be interchanged in case of malfunction of one of the two element s. Moreover, the printed-circuit board without cylindrica l micro-spacers is more robust. This particular feature faci litates the production process, in particular the polishin g of ∗Corresponding author: martinez@in2p3.fr 1BMC Industries, 278 East 7th Street, St. Paul, MN 55101, USA. 1the anode-pad surface. In addition, the connectors can be so ldered more easily to the back of the board by standard industrial techniques, thus reducing the cost of the detect ors. The choice of the grid material is dictated by the insulation properties of the material and by its mechanical properties: It must be resistant to heat and rigid enough to be manipulated. A 200 µm thick FR4 grid of the same size as the printed board (394 ×394 mm2) was chosen. The grid consists of 1024 squared cells of 11.8 mm width and 300 µm pitch (see Fig.2). On the edge of the grid, 28 2.4 mm diameter holes provides wide opening for the gas to ﬂow ins ide the detector. The critical parameter in the new design is the uniformity of the ampliﬁcation-gap across the grid cells. We have therefore measured the ampliﬁcation-gap thickness at diﬀe rent positions across the grid cell and for various voltages applied to the micro-mesh. For that purpose, we used a magnif ying camera whose position in the three spatial directions can be accurately measured. With this technique the thickness can be determined with a 5 µm accuracy. For non-zero voltages the mesh ﬂattens against the grid near its edge of the grid. For usual operating voltages from -440 V to -600 V (establishing electric ﬁelds from 22 kV/cm to 30 kV/cm), the average ampliﬁcation gap thickness is (203±5)µm and the diﬀerence in the gap thickness between the edge and t he center of a grid hole is less than 10 µm at -440 V and less than 7 µm at -600V (see Fig.3). Such diﬀerences do not strongly aﬀect the gain uniformity along the grid cell. In the worse case a gain variation of no more tha n 40% might occur. This would induce a minor impact on the detection of minimum ionizing particles. At voltages above -700 V (electric ﬁeld larger than 35 kV/cm) the mesh is deformed by the electric force and the ampliﬁcation g ap thickness reaches a minimum value of (190 ±5)µm at the center of the cell. The gain of this newly designed detector has been measured fo r diﬀerent gas mixtures and micro-mesh voltages. A relative determination of the gain has been obtained by mea suring the electric current induced in the micro-mesh when the detector is irradiated by a90Sr radioactive source. The source was placed on a 50 µm mylar foil replacing the anode of the detector. We have compared the two designs, c ylindrical micro-spacers2and insulating grid. To eliminate most of systematic diﬀerences, the measurements have been performed simultaneously, the two detectors being supplied with gas from the same source. To estimate the absolute gain in the ampliﬁcation gap, we have assumed that the number of electrons collected by the cathode is equa l to the number of ions collected by the micro-mesh. The resulting electric current gives a relative measure of the d etector gain and the following expression gives an estimati on of the gain in the ampliﬁcation gap: G=I e < N > A(1) where Iis the measured electrical current, ethe electron charge, Athe activity of the source expressed in Bq (5.6 ×105 Bq in the present measurement) and < N > the average number of primary electrons created in the 6 mm co nversion gap. This later number obviously depends on the gas mixture, being < N > = 27 for Ne+CO 2(5%), < N > = 29 for Ne+CO 2(10%) and < N > = 56 for Ar+CO 2(10%). The sensibility of the apparatus to the micro-mesh cu rrent was 1nA. The results (see Fig.4) indicate that the two designs be have similarly, although for a given voltage systematicall y higher gains are reached. with the micro-spacer design: abo ut 10 V more are needed for the grid design to reach the same gain as with the micro-spacers. This could be relate d with a systematic diﬀerence of the ampliﬁcation gap thickness between both detector designs. As a matter of f act, a variation on the micro-mesh voltage of 10 V represents a gain variation of around 40%. This gain diﬀeren ce between both designs would correspond to a gap- thickness systematic diﬀerence of only 10 µm. In addition, the electron source was placed on a mylar foil 50µm thick for the new design and on a Ni micro-mesh 8 µm thick for the old design. In summary, we have presented a new design for a gas chamber ba sed on Micromegas technology, where the 200 µm uniform ampliﬁcation gap is provided by an insulating grid s andwiched by the micro-mesh and the cathode-pad board. The micro-mesh is not strongly deformed for usual operating electric ﬁelds. The measured gain in the ampliﬁcation gap for diﬀerent gas mixtures and micro-mesh voltages shows results similar to those obtained with cylindrical micro- spacers. The grid design presents several decisive technic al as well as ﬁnancial advantages when compared to the cylindrical micro-spacers design, thus allowing for the in dustrial production of inexpensive particle detectors. This work has been supported by the IN2P3/CNRS, France and th e “Conseil R´ egional des Pays de la Loire”, France. 2In this design, the90Sr radioactive source was placed on a 8 µm thick micro-mesh Ni foil. 2[1] Y. Giomataris, Ph. Rebourgeard, J.P. Robert and G. Charp ak, Nucl. Inst. and Meth. A376 (1996) 29. [2] Y. Giomataris, Nucl. Inst. and Meth. A419 (1998) 239. [3] G. Charpak, Nucl. Inst. and Meth. A412 (1998) 47. [4] G. Barouch et al., Nucl. Inst. and Meth. A423 (1999) 32. [5] J.P. Cussonneau et al., Nucl. Inst. and Meth. A419 (1998) 452. [6] Proceedings 2nd MICROMEGAS Workshop, 24 Feb. - 5 March 19 99, Saclay, France. [7] Proceedings 3rd MICROMEGAS Workshop (http://www-iphe .unil.ch/micromegas), 8 - 9 March 2000, Lausanne, Suisse. [8] L. Aphecetche et al., Nucl. Instr. and Meth. A459 (2001) 502 3FIG. 1. Sketch of the new design for a large area particle dete ctor with an insulating grid providing the ampliﬁcation gap . 4Thickness : 200 microns 5,005,005,0010,005,005,005,0010,005,005,005,005,005,003,55 2,00 1,20 R 2,40 MACHINING MARK A0,30 11,80 3,55 3,55 11,80 0,3032 x 32 holes (11,80 x 11,80 mm)3,5592,00 75,00394,00 3,55 MACHINING MACHINING MACHINING MACHININGMARK A MARK A MARK A MARK A394,00 FIG. 2. Technical drawing illustrating the machining appli ed on a 200 µm insulating plate to obtain the grid. The circular zoom details the rectangular holes which delimit active are as. The rectangular zoom shows the holes which permit the gas to ﬂow freely inside the detector. 5100120140160180200220240 Vmesh = -400 V 100120140160180200220240Vmesh = -600 V 100120140160180200220240Vmesh = -700 V100120140160180200220240 Vmesh = -200 V Cell position (mm)Amplification gap (µµm) 0.0 2.95 5.9 8.85 11.8 0.0 2.95 5.9 8.85 11.8 FIG. 3. Ampliﬁcation gap thickness on several points of the m esh across one hole of the grid as a function of voltage applie d in the micro-mesh: -200 V, -400 V, -600 V and -700 V 61,E+021,E+031,E+041,E+05 -720-670-620-570-520-470-420-370-320 MICRO-MESH VOLTAGES (V)GAINNe5%CO2-Micro-spacers Ne5%CO2-Grid Ne10%CO2-Micro-spacers Ne10%CO2-Grid Ar10%CO2-Micro-spacers Ar10%CO2-Grid FIG. 4. Measured gain as a function of the voltage applied to t he micro-mesh for two detector designs: one based on the micro-spacers design and the insulating-grid design. Vari ous gas mixtures have been studied. 7
arXiv:physics/0103028v1 [physics.bio-ph] 10 Mar 2001NSF-ITP-01-18 Compositional Representation of Protein Sequences and the Number of Eulerian Loops Bailin Hao∗ † Institute for Theoretical Physics, UCSB, Santa Barbara, CA 93106-4030, USA Huimin Xie Department of Mathematics, Suzhou University, Suzhou 2150 06, China Shuyu Zhang Institute of Physics, Academia Sinica, P. O. Box 603, Beijin g 100080, China (January 12, 2014) An amino acid sequence of a protein may be decomposed into con secutive overlapping strings of length K. How unique is the converse, i.e., reconstruction of amino a cid sequences using the set ofK-strings obtained in the decomposition? This problem may be transformed into the problem of counting the number of Eulerian loops in an Euler graph, th ough the well-known formula must be modiﬁed. By exhaustive enumeration and by using the modiﬁ ed formula, we show that the reconstruction is unique at K≥5 for an overwhelming majority of the proteins in pdb.seq database. The corresponding Euler graphs provide a means to study the s tructure of repeated segments in protein sequences. PACS number: 87.10+e 87.14Ee I. INTRODUCTION The composition of nucleotides in DNA sequences and the amin o acids composition in protein sequences have been widely studied. For example, the g+ccontents or CpGislands in DNAs have played an important role in gene-ﬁnding programs. However, this kind of study usually has been restr icted to the frequency of single letters or short strings, e.g., dinucleotide correlations in DNA sequences [1], amin o acids frequency in various complete genomes [2]. However, in contrast to DNA sequences amino acid correlations in prot eins have been much less studied. A simple reason might be that there are 20 amino acids and it is diﬃcult to comp rehend the 400 correlation functions even at the two-letter level. A more serious obstacle consists in that p rotein sequences are too short for taking averages in the usual deﬁnition of correlation functions. For short sequences like proteins one should naturally appr oach the problem from the other extreme by applying more deterministic, non-probabilistic methods. In fact, t he presence of repeated segments in a protein is a strong manifestation of amino acid correlation. This problem has a nice connection to the number of Eulerian loops in Euler graphs. Therefore, we start with a brief detour to graph theo ry. II. NUMBER OF EULERIAN LOOPS IN AN EULER GRAPH Eulerian paths and Euler graphs comprise a well-developed c hapter of graph theory, see, e.g., [3]. We collect a few deﬁnitions in order to ﬁx our notation. Consider a connected , directed graph made of a certain number of labeled nodes. A node imay be connected to a node jby a directed arc. If from a starting node v0one may go through ∗Corresponding author. E-mail: hao@itp.ac.cn †On leave from the Institute of Theoretical Physics, Academi a Sinica, P. O. Box 2735, Beijing 100080, China 1a collection of arcs to reach an ending node vfin such a way that each arc is passed once and only once, then it is called an Eulerian path . Ifv0andvfcoincide the path becomes an Eulerian loop . A graph in which there exists an Eulerian loop is called an Eulerian graph . An Eulerian path may be made an Eulerian loop by drawing an au xiliary arc from vfback to v0. We only consider Euler graphs deﬁned by an Eulerian loop. From a node there may be doutarcs going out to other nodes, doutis called the outdegree (fan-out) of the node. There may be dinarcs coming into a node, dinbeing the indegree (fan-in) of the node. The condition for a g raph to be Eulerian was indicated by Euler in 1736 and consists in din(i) =dout(i)≡di= an even number for all nodes i. Numbering the nodes in a certain way, we may put their indegre es as a diagonal matrix: M= diag( d1, d2,· · ·, dm). (1) The connectivity of the nodes may be described by an adjacent matrix A={aij}, where aijis the number of arcs leading from node ito node j. From the MandAmatrices one forms the Kirchhoﬀ matrix: C=M−A. (2) The Kirchhoﬀ matrix has the peculiar property that its eleme nts along any row or column sum to zero:/summationtext icij= 0,/summationtext jcij= 0. Further more, for an m×mKirchhoﬀ matrix all ( m−1)×(m−1) minors are equal and we denote it by ∆. A graph is called simple if between any pairs of nodes there ar e no parallel (repeated) arcs and at all nodes there are no rings, i.e., aij= 0 or 1 ∀i, jandaii= 0∀i. The number Rof Eulerian loops in a simple Euler graph is given by The BEST Theorem [3] (BEST stands for N. G. de Bruijn, T. van Aardenne- Ehrenfest, C. A. B. Smith, and W. T.Tutte): R= ∆/productdisplay i(di−1)! (3) For general Euler graphs, however, there may be arcs going ou t and coming into one and the same node (some aii/ne}ationslash= 0) as well as parallel arcs leading from node itoj(aij>1). It is enough to put auxiliary nodes on each parallel arc and ring to make the graph simple. The derivation goes jus t as for simple graphs and the ﬁnal result is one has the original graph without auxiliary nodes but with aii/ne}ationslash= 0 and aij>1 incorporated into the adjacent matrix A. However, in accordance with the unlabeled nature of the para llel arcs and rings one must eliminate the redundancy in the counting result by dividing it by aij!. Thus the BEST formula is modiﬁed to R=∆/producttext i(di−1)!/producttext ijaij!(4) As 0! = 1! = 1 Eq. (4) reduces to (3) for simple graphs. III. EULERIAN GRAPH FROM A PROTEIN SEQUENCE We ﬁrst decompose a given protein sequence of length Linto a set of L−K+ 1 consecutive overlapping K-strings by using a window of width K, sliding one letter at a time. Combining repeated strings in to one and recording their copy number, we get a collection {WK j, nj}M j=1, where M≤L−K+ 1 is the number of diﬀerent K-strings. Now we formulate the inverse problem. Given the collection {WK j, nj}M j=1obtained from the decomposition of a given protein, reconstruct all possible amino acid sequenc es subject to the following requirements: 1. Keep the starting K-string unchanged. This is because most protein sequences s tart with methionine ( M); even the tRNA for this initiation Mis diﬀerent from that for elongation. This condition can eas ily be relaxed. 2. Use each WK jstring njtimes and only njtimes until the given collection is used up. 3. The reconstructed sequence must reach the original lengt hL. 2Clearly, the inverse problem has at least one solution — the o riginal protein sequence. It may have multiple solutions. However, for Kbig enough the solution must be unique as evidenced by the ext reme case K=L−1. We are concerned with how unique is the solution for real proteins. Our guess is for most proteins the solution is unique at K≥5. In order to tell the number of reconstructed sequences we tra nsform the original protein sequence into an Euler graph in the following way. Consider the two ( K−1)substrings of a K-string as two nodes and draw a directed arc to connect them. The same repeated ( K−1)-strings are treated as a single node with more than one inc oming and outgoing arcs. Take the SWISS-PROT entry ANPA PSEAM as an example [4]. This antifreeze protein A/B precurs or of winter ﬂounder has a short sequence of 82 amino acids and some repeat ed segments related to alanine-rich helices. Its sequence reads: MALSLFTVGQ LIFLFWTMRI TEASPDPAAK AAPAAAAAPA AAAPDTASDA A AAAALTAAN AKAAAELTAA NAAAAAAATA RG Consider the case K= 5. The ﬁrst 5-string MALSL gives rise to a transition from node MALS toALSL. Shifting by one letter, from the next 5-string ALSLF we get an arc from node ALSL to node LSLF, and so on, and so forth. Clearly, we get an Eulerian path whose all nodes have even ind egree (outdegree) except for the ﬁrst and the last nodes. Then we draw an auxiliary arc from the last node TARG back to the ﬁrst MALS to get a closed Eulerian loop. In order to get the number of Eulerian loops there is no need to generate a fully-ﬂedged graph with all the M distinct ( K−1)-strings treated as nodes. The number of nodes may be reduc ed by replacing a series of consecutive nodes with din=dout= 1 by a single arc, keeping the topology of the graph unchange d. In other words, only those strings in {WK−1 j, nj}withnj≥2 are used in drawing the graph. In our example it reduces to a s mall Euler graph consisting of 9 nodes: {AKAA,2;AAPA,2;APAA,2;PAAA,2;AAAA,10;AAAP,2;LTAA,2;TAAN,2;AANA,2}. The Kirchhoﬀ matrix is: C= 2−1 0 0 0 0 −1 0 0 0 2 −2 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 −1 0 0 0 4 −2−1 0 0 0−1 0 0 −1 2 0 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 2 −2 −1 0 0 0 −1 0 0 0 2 , (5) The minor ∆ = 192 and R(5) =∆9! 6!26= 1512 . We write R(K) to denote the number of reconstructed sequences from a deco mposition using K-strings. We note, however, precautions must be taken with spurious re peated arcs caused by the reduction of number of nodes. In calculating the/producttext ijaijin the denominator of Eq. (4) one must subtract the number of s purious repeated arcs from the corresponding matrix element of the adjacent m atrix. This remark applies also to the auxiliary arc obtained by connecting the last node to the ﬁrst. Fortunatel y, there are no such spurious arcs in the example above. We have written a program to exhaustively enumerate the numb er of reconstructed amino acid sequences from a given protein sequence and another program to implement th e Eq. (4). The two programs yield identical results whenever comparable — the enumeration program skips the seq uence when the number of reconstructed sequences exceeds 10000. IV. RESULT OF DATABASE INSPECTION We used the two programs to inspect the 2820 proteins in the sp ecial selection pdb.seq [4]. The summary is given in Table I. As expected most of the proteins lead to unique rec onstruction even at K= 5. At K= 10 such proteins make 99% of the total. 3TABLE I. Distribution of the 2820 proteins in pdb.seq by the number of reconstructed sequences at diﬀerent K. Percentages in parentheses are given in respect to the total number 2820. K Unique 2-10 11-100 101-1000 1001-10000 >10000 5 2164 (76.7%) 404 90 45 21 93 6 2651 (94.0%) 77 29 10 4 49 7 2732 (96.9%) 32 16 3 2 44 8 2740 (97.1%) 23 10 3 0 44 9 2763 (97.9%) 13 7 1 0 36 10 2793 (99.0%) 11 7 2 1 6 11 2798 (99.2%) 12 2 1 1 6 The fact that most of the protein sequences have unique recon struction is not surprising if we note that for a random amino acid sequence of the length of a typical protein one would expect R= 1 at K= 5, as it is very unlikely that its decomposition may yield repeated pairs of K-strings among the 205= 3200000 possible strings. A more positive implication of this uniqueness is one may take the c ollection of {WK j}L j=1as an equivalent representation of the original protein sequence. This may be used in inferri ng phylogenetic relations based on complete genomes when it is impossible to start with sequence alignment. We wi ll report our on-going work along this line in a separate publication [5]. A more interesting result of the database screening consist s in there exists a small group of proteins which have an extremely large number of reconstructed sequences. The n umber Ris not necessarily related to the length of the protein. As a rule, long protein sequences, say, with 2000 or more amino acids, tend to have larger RatK= 5 or so, but the number drops down quickly. In fact, all 29 protein s inpdb.seq with more than 2115 amino acids have unique or a small number of reconstructed sequences. Some no t very long proteins have much more reconstructions than the long ones. We show a few ”mild” examples in Table II. TABLE II. A few examples of protein decomposition with compa ratively large RatK= 5. AA is the number of amino acids in the protein. Protein MCMI YEAST PLMN HUMAN CENB HUMAN CERU HUMAN AA 286 810 599 1065 R(5) 7441920 3024000 491166720 3507840 R(6) 39312 384 17421 512 R(7) 1620 192 90 21 R(8) 252 96 12 6 R(9) 16 5 4 1 R(10) 2 1 1 R(11) 1 4The inspection is being extended to all available protein se quences in public databases. V. DISCUSSION In this paper we have given some precise construction and num bers associated with real protein sequences. Their biological implications have to be yet explored. As mentioned in Section IV, we have been using the uniqueness of the reconstruction for most protein sequences to justify the compositional distance approach to infer phylo genetic relations among procaryotes based on their complet e genomes [5]. Most of the phylogenetic studies so far conside r mutations at the sequence level. Sequences of more or less the same length are aligned and distances among species are derived from the alignments. However, mutations from a common ancestral sequence reﬂect only one way of evolu tion. There might be another way of protein evolution — short polypeptides may fuse to form longer proteins. Perha ps our approach may better capture the latter situation. The decomposition and reconstruction described in this pap er provide a way to study polypeptide repeats and amino acid correlations. The reconstruction problem natur ally singles out a small group of proteins that have a complicated structure of repeated segments. One may introd uce further coarse-graining by reducing the cardinality of the amino acid alphabet according to their biological pro perties. This makes the approach closer to real proteins. Investigation along these lines are under way. We note that the Eulerian path problem has been invoked in the study of sequencing by hybridization, i.e., in the context of RNA or DNA sequences, see [6] and references there in. To the best of our knowledge the modiﬁcation of the BEST formula to take into account parallel arcs and rings has not been discussed so far. ACKNOWLEDGMENTS This work was accomplished during the Program on Statistica l Physics and Biological Information at ITP, UCSB, supported in part by the National Science Foundation under G rant No. PHY99-07949. It was also supported partly by the Special Funds for Major State Basic Research Project o f China and the Major Innovation Research Project ”248” of Beijing Municipality. BLH thanks Prof. Ming Li for c alling his attention to [6]. [1] W. Li, The study of correlation structures in DNA sequenc es — a critical review, Computer & Chemistry 21(1997) 257-172. [2] See, for example, the Proteome page of EBI: http://www.ebi.ac.uk/proteome/ [3] H. Fleischner, Eulerian Graphs and Related Topics , Part 1, vol. 2, p. IX80, Elsevier, 1991. [4]pdb.seq is a collection of SWISS-PROT entries that have one or more po inters to the PDB structural database. In the ﬁle associated with SWISS-PROT Rel. 39 (May 2000) there are 2821 entries. In our calculation we excluded a protein with too manyXs (undetermined amino acids). We fetched the ﬁle from: ftp://ftp.cbi.pku.edu.cn/pub/database/swissprot/spe cialselections/pdb.seq [5] Bin Wang, and Bailin Hao, Procaryote phylogeny based on c omplete genomes (in preparation). [6] P. Pevzner, Computational Molecular Biology. An Algorithmic Approach , SS5.4, MIT Press, 2000. 5
arXiv:physics/0103029v1 [physics.class-ph] 11 Mar 2001The Inertial Polarization Principle: The Mechanism Underlying Sonoluminescence ? Marcelo Schiﬀer∗ The College of Judea and Samaria Ariel, 44837, Israel February 9, 2008 Abstract In this paper we put forward a mechanism in which imploding sh ock waves emit electromagnetic radiation in the spectral region λ0∼=2πR0., where R 0is the radius of the shock by the time it is ﬁrst formed. The mechanism relies on three diﬀerent pieces of Phy sics: Maxwell’s equations, the existence of corrugation instabilities of imploding shock waves and, last but not least, the Inertial Polarization Principle . The principle is extensively discussed: how it emerges fro m very elementary physics and ﬁnds experimental support in shock waves propagating in wat er. The spectrum of the emitted light is obtained and depends upon two free parameters, the amplit ude of the instabilities and the cut-oﬀ Rmax, the shocks’ spatial extension. The spectral intensity is d etermined by the former , but its shape turns out to have only a mild dependence on the latter, i n the region of physical interest. The matching with the observed spectrum requires a ﬁne tunin g of the perturbation amplitude ε∼10−14,indicating a quantum mechanical origin. Indeed, we support this conjecture with an order of magnitude estimative. The Inertial Polarization Princi ple clues the resolution of the noble gas puzzle in SL. PACS:78.60. Mq,42.50Fx,34.80Dp,03.65.Bz ∗On leave of absence from Campinas State University 1The Inertial Polarization Principle In this paper we put forward a mechanism responsible for tran sducing the kinetic energy stored in an imploding spherical shock wave into electromagnetic radia tion, which is based solely upon Maxwell’s equations, the existence of very small instabilities away f rom the spherically symmetric ﬂow and the inertial polarization paradigm. Based on these premisses w e obtained the spectral intensity of the outgoing radiation. The mechanism turns out to be so eﬃcient that the observed energy emission rate of P(λ)∼10−10Watt/nm calls for perturbation amplitudes no larger than ε= 10−14! Maxwell’s equations are a pillar of theoretical physics while inertial polariza tion is a consequence of very elementary physics : an atom that undergoes an acceleration, say a, develops in its interior polarized electromagnetic ﬁelds. The issue is made clear for an observer sitting in the f rame of the molecule, where he sees inertial forces acting both upon the nucleus FN=MNaand on the electronic cloud Fe=Mea. The gradient between these forces tends to sag the cloud away from the nucl eus, and the atom develops internal polarization ﬁelds, say E0,to compensate this gradient e E0∼(MN−Me)a. The role of inertial polarization remained hitherto unnoticed only because det ectable polarization ﬁelds call for tremendous accelerations, say, E0∼1V/mwould require a∼(e/M p)E0∼10−2(eV/(Mpc2))(c2/cm)∼1010cm/sec2 which are absent in every day life experiments. Nevertheles s, there are two instances where such large accelerations manifest: i.) in the realm of very strong grav itational ﬁelds where inertial polarization was shown to be the working mechanism that rescues the second law of thermodynamics from bankruptcy (otherwise super-luminal motion of black-holes inside die lectric media would entail a violation of the generalized second law [1],[2]); ii) in the realm of shock wa ves, because shocks are powerful accelerators of ﬂuid molecules: a ﬂuid molecule that crosses the shock und ergoes a macroscopic velocity change (of the order of the ﬂuid velocity itself ) within a microscopic d istance – the shock width (of the order of the mean free path for the atomic collisions [3]). The inertial polarization principle is the single non-very -well-established piece of physics in our recipe and we proceed by making our case for it. Consider a planar str ong shock wave propagating within a perfect gas. Let v2andv1represent the ﬂuid velocity in the back and in front the shock , respectively ( likewise, the index 2 (1) refer to physical quantities behi nd (in front) the shock ). As the ﬂuid molecules cross the shock they experience a mean acceleration ¯ a= (v2−v1)(¯∆t), where ¯∆tis the mean time it takes the gas to cross the shock-width δ. Clearly ¯∆t=δ/¯v, where ¯ v∼=(v1+v2)/2, is the mean velocity. Putting these pieces together ¯a=v2 2−v2 1 2δ(1) For a strong shock propagating in a perfect gas [3]: v2 2−v2 1=−2γ γ+ 1p2V1 (2) where V1is the gas’ speciﬁc volume. The compression rate satisﬁes V1/V2= (γ+ 1)/(γ−1)[3] ¯a=−γ γ−1p2V2 δ(3) The shock width δis known to be of order of the mean free path for collisions of a toms in the ﬂuid, δ≈(nσ)−1, where nstands for the number density of atoms and σfor the collision’s cross section. Bearing in mind that nV=A/µwhere Ais Avogadro’s number and µis the molecular weight of the gas, we obtain the colossal ﬁgure for the mean acceleration atoms experience as they cross the shock: ¯a≈ −γ γ−16×1013/parenleftigp2 atm/parenrightig /parenleftigσ 10−16cm2/parenrightig/parenleftbigggram µ/parenrightbigg cm/sec2. (4) The mean electric polarization developed across the shock ¯E0≈(Mp/e)¯ais also sizeable ¯E0≈6×103V meterγ γ−1/parenleftigp2 atm/parenrightig/parenleftigσ 10−16cm2/parenrightig/parenleftbigggram µ/parenrightbigg . (5) 2Unfortunately, the shock is so thin that the voltage develop ed across its ends is very small V∼¯aMpδ e∼1.2×10−6V oltγ γ−1/parenleftigp atm/parenrightig/parenleftbiggcm3 g ̺/parenrightbigg (6) Shock Polarization was ﬁrst observed in the early sixties [4 ] for shock waves propagating inside water. Since then, both quality and range of the measurements impro ved considerably [6]. Harris [5, 6] credits the eﬀect to the fact that large pressure gradients inside th e shock results in a torque ﬁeld acting upon the water molecule causing the molecule’s dipole to align. W e reproduce his results via the table: p(kbar) 98 75 74.558 54 45 36 20 V(mV)/p(kbar)1.971.330.970.770.430.890.680 0.8 The underlying Physics for a shock propagating in water is th e very same as for a gas and we infer the averaged electric potential across the shock from eq.(6 ) bearing in mind that: i.)˜the compression rate for water is of order one, therefore we go one step back in this equation by replacing γ/(γ−1)→ γ/(γ+ 1)≃1/2); ii.)˜the equation was obtained for a gas and for liquids i t should be regarded as the linear expansion of the function V(p). Then it follows that V(mV)/p(kbar)∼0.6 , in agreement with the lower pressure region of the experimental data. Detecti on of shock polarization for non-polar ﬂuids would vindicate the Inertial Polarization Principle. The acceleration ﬁeld inside planar shocks is space and time independent (and so the corresponding polarized electromagnetic ﬁelds) . Nevertheless, planar s hocks are known to develop corrugation instabili- ties [3], small deformations of the planar geometry that det ach from the shock and propagate throughout the ﬂuid. They correspond to the spontaneous emission of sou nd from the shock. These instabilities will cause a space time dependent acceleration ﬁeld inside the sh ock, and by the Inertial Polarization Principle a wiggling 6 ×103V/melectric ﬁeld vector that is radiated away: sound and light a re emitted simulta- neously , provided the Inertial Polarization relaxation ti me is small enough. This brings to one’s mind the famous and intriguing sonoluminescence eﬀect [7] in whi ch under heavy bombarding of ultra-sound waves, a little ( 5 µm) bubble of air cavitating within a ﬂask of water undergoes a s pectacular collapse, attains the supersonic regime and glows (mainly) violet lig ht. The eﬀect has been around for sixty years or so ( [8],[ ?]) and proper understanding of the problem remains elusive. The most popular mechanism is the Bremsstrahlung from free electrons in the gas where th e ionization is caused by two successive heating processes: ﬁrst the adiabatic collapse of the bubbl e which is then followed by the motion of a shock wall inside the bubble (the shock’s Mach number contro ls the temperature rate T2/T1∼M2) [10]. The formation of a shock wall, a collapsing spherical front o f radius R(t) =A(−t)α(α <1),happens by the time supersonic regime is attained inside the bubble [ 7]. The acceleration of the shock front surface a(t)∼A(−t)α−2, becomes very large at focusing ( t→0) engendering very large space and time depen- dent Inertial-Polarization ﬁelds. Nevertheless, the sphe rical symmetric geometry of the problem prevents these ﬁelds to be radiated away: pursuing the present avenue seems to require some supplementary mech- anism to account for the radiation ﬂash (a sparking mechanis m was proposed [11, 12]). Fortunately, no supplementary mechanism is needed: numerical calculation s ([13]) have shown the existence of unstable perturbations of the collapsing shock which provide the mul tipole time-dependent inertial-polarization ﬁelds that are radiated away. The purpose of this paper is to c alculate the spectral distribution of the emitted light . The paper is organized as follows. The following section rev iews the dynamics of imploding shocks, and the existence of unstable multipole perturbation modes is rigorously proved. As a bonus, we obtain the energy and the power carried away by the sound waves that d etach from the shock (corrugation instabilities). A novel semi-analytical procedure for sol ving the diﬀerential equations for the perturbations is developed, which nevertheless, is displayed in the appen dix in order prevent the disruption of the main argument line with technicalities. In section II , we ob tain the polarization ﬁelds engendered by the corrugation instabilities and show that they act as a sou rce term in Maxwell’s equations. Then we calculate the spectrum of the outgoing radiation. The spe ctrum depends on the dynamics of the corrugation instabilities, but fortunately it is possible to obtain the main structure of the spectrum without having to delve too deeply into the dynamics. The int ensity of the outgoing radiation turned 3out to be proportional to pε2where εis the corrugation instability amplitude and p=E2 p/(2α/planckover2pi1),is Inertial-Polarization power-constant ( Epstands for the proton’s rest energy and αfor the ﬁne structure constant). This constant is of the order ⋍1.47×1016Watt (!): collapsing shock waves are the most eﬃcient power-stations in nature, with the sole possible ex ception of astrophysical objects! Agreement with the experimental data calls for amplitudes of the order ε∼10−13orδr∼10−19m! These tiny perturbations must have a quantum mechanical origin, and we support this conjecture by an order of magnitude estimative. Finally we suggest the resolution of the noble gas puzzle in SL. 1 Dynamics of Imploding Shocks The non-viscous implosion of a spherical shock cannot be cha racterized by any dimensional parameter . Consequently the ﬂow admits a self-similar symmetry. Let R(t) =Ai(−t)αrepresent the radius of the shock front, where Aiandαare two constants and vshock=αR(t)/t, its implosion velocity. The self similar parameter here is ξ=r/R(t); the surface of the shock is given by ξ= 1. Self-similarity constrains the form of the speed of sound, radial ﬂow velocity and densit y [14] : c2 2=/parenleftigαr t/parenrightig2 Z(ξ) (7) v2=/parenleftigαr t/parenrightig V(ξ) (8) ρ2=ρ0G(ξ) (9) When expressed in terms of the self similar quantities Z, VandG, the boundary conditions for a strong shock /vector n·/vector vshock>> c read, G(1) =γ−1 γ+ 1, V(1) =2 γ+ 1, Z(1) =2γ(γ−1) (γ+ 1)2(10) The equations that govern the ﬂow are the entropy and mass con servation laws and Euler’s equation . They provide a set of non-linear coupled equations for G(ξ), V(ξ) and Z(ξ), which when solved for Z(V) andξ(V),yield the pair of equations [3] dZ dV=Z 1−V/bracketleftigg/parenleftbig Z−(1−V)2/parenrightbig (2/α−(3γ−1)V) (3V−κ)Z−V(1−V)(1/α−V)+γ−1/bracketrightigg (11) and dlnξ dV=−Z−(1−V)2 (3V−κ)Z−V(1−V)(1/α−V)(12) where κ= 2(1 −α)/(αγ). Inspection of these equations reveals the existence of a s ingular point at Z= (1−V)2(dV/dξ → ∞?). Clearly, all physical quantities, and their derivative s must be ﬁnite across the singular point, meaning that the conditions (3 V−κ)Z−V(1−V)(1/α−V) = 0 and Z= (1−V)2are simultaneous to each other at this point, such as to keep thei r ratio ﬁnite. Call Vc(α), Zc(α) the solution of this pair of algebraic equations. The parameter αis obtained by numerically integrating Z(V) from V=V(1) to Vcfor diﬀerent values of αuntil the matching Z(Vc(α)) =Zc(α) is obtained. The good values for αare 0.688376 /0.71717 for a monatomic/diatomic gas. The limit t→0−corresponds to the shock’s focusing time, after which the shock reﬂects and ree xpands .For latter reference, we mention the asymptotic behavior V∼ξ−1/αasξ→ ∞ [3]. We are seeking now perturbations away from this ﬂow. Let δ=δρ/ρbe the contrast function and δ/vector v the velocity ﬂuctuation. The latter can be decomposed into i ts normal and perpendicular components δvn=/vector n·/vector v,δ/vector v⊥=δ/vector v−δvn/vector n. The linearized mass and entropy conservation equations rea d /parenleftbigg∂ ∂t+v∂ ∂r/parenrightbigg δ+δvn∂lnρ ∂r+/vector∇ ·δ/vector v= 0 (13) 4/parenleftbigg∂ ∂t+v∂ ∂r/parenrightbigg δs+δvn∂s ∂r= 0 (14) while perturbing Euler’s equation yields /parenleftbigg∂ ∂t+v∂ ∂r/parenrightbigg δ/vector v+δvn∂v ∂r/vector n+v rδ/vector v⊥=δ/vector∇p−/vector∇δp ρ(15) Next, we introduce the self-similar ansatz δvn=εαr t0/parenleftbiggt t0/parenrightbiggαβ−1 (1−V)Φ(ξ)Ylm(θ, φ) (16) δ/vector v⊥=εαr t0/parenleftbiggt t0/parenrightbiggαβ−1 τ(ξ)/parenleftig r/vector∇/parenrightig Ylm(θ, φ) δ=ε/parenleftbiggt t0/parenrightbiggαβ ∆(ξ)Ylm(θ, φ) δs=εcp/parenleftbiggt t0/parenrightbiggαβ σ(ξ)Ylm(θ, φ) where cpis the speciﬁc heat of the gas , t0is shock formation time and εthe amplitude of the perturbation at this moment. After some tedious algebra we translate the p revious equations in terms of the self-similar quantities. The mass and entropy conservation yield (13,[ ?]) (1−V)ξ(∆′−Φ′) =β∆ + 3Φ −l(l+ 1)τ (17) (1−V)ξσ′=βσ−κΦ (18) where κ= 2(1 −α)/(αγ). The projection of Euler’s equation (15) into the perpendi cular direction yields a compact form (1−V)ξτ′= (2V+β−1 α)τ+Z(∆ +σ). (19) but the normal projection gives a more cumbersome expressio n (1−V)ξ/parenleftbig (1−V)2Φ′−Z(∆′+σ′)/parenrightbig = (20) =/bracketleftbigg (1−V)2(2V+ 2ξV′+β−1 α)/bracketrightbigg Φ +Z((γ−1)∆ + γσ)(3V+ξV′−κ) Equation (17) suggests the deﬁnition of a new dynamical vari able Π = ∆ −Φ. We display these equations in matrix formd dV\|Y(V)/angb∇acket∇ight=M(V)\|Y(V)/angb∇acket∇ight; 0≤V≤V(1)≡V1 (21) where \|X(V)/angb∇acket∇ight= (φ(V), τ(V), π(V), σ(V)), ;\|X(V)/angb∇acket∇ight= exp[ β/integraltextV V1m(V)dV]\|Y(V)/angb∇acket∇ightand, furthermore M(V) =m(V) P(V)φ1(V)P(V)φ2(V)P(V)φ3(V)P(V)φ4(V) Z 2V−1 αZ Z 3 +β −l(l+ 1) 0 0 −κ 0 0 0 (22) with m(V) =1 1−Vdlnξ dV;P(V) =1 (1−V)2−Z(23) φ1(V) = Z[5−2/α+ 2β+ (γ−1)(3V+dV)] + (1 −V)2[−1/α+ 2V+ 2dV] (24) φ2(V) = −Z l(l+ 1) φ3(V) = Z[(γ−1)(3V+dV−κ) +β) (25) φ4(V) = Z[γ(3V+dV−κ) +β] 5where dV(V)≡ξV′. Clearly this set of diﬀerential equations possess a regula r singular point when Z−(1−V)2= 0, that is to say, at Vc.The limit V→0 (ξ→ ∞ ,m(V)→ −α/V, P (V)→1;φ1→ −1/α;φ2,3,4→0), reveals an additional singularity d dV\|Y(V)/angb∇acket∇ight ≈1 V 1 0 0 0 0 1 0 0 −α(3 +β)αl(l+ 1) 0 0 ακ 0 0 0 \|Y(V)/angb∇acket∇ight (26) The matrix on the right-hand-side of this equation deﬁnes an eigenvalue problem whose solution λ1,2= 0 →/braceleftbigg\|θ1>= (0,0,1,0) \|θ2>= (0,0,0,1) λ3,4= 1→/braceleftbigg \|θ3>= (l(l+ 1),3 +β,0, ακl(l+ 1)) \|θ4>= (0,1, αl(l+ 1),0)(27) yields the asymptotic form \|X(V)/angb∇acket∇ight ≈V−αβ[(a1\|θ1>+a2\|θ2>) +V(a3\|θ3>+a4\|θ4>)] ; V→0, (28) where anare integration constants. Asymptotically regular ﬁelds r equire ℜ(β)≤0 (except for the the particular mode a1=a2= 0 which calls for a less stringent condition ℜ(β)≤1/α). A further constraint onβarises from energetic considerations. The energy of a polyt ropic gas is E=/integraldisplay ρ[v2+c2 γ(γ−1)]dV. (29) The lowest order contribution ( in the perturbation paramet erǫ) to the energy stored in the perturbed- shock is the second order expression δEl(t) =/integraldisplay δρ[vδvn+δc2 γ(γ−1)]4πr2dr, (30) or after some algebra δEl(t) = 4πα2ε2ρ0R5 0t2αβ+5α−2 tα(2β+5) 0Cl, (31) where Cl=/integraldisplayξc 1G(ξ)[Φ(ξ) + Π( ξ)][V(1−V)Φ +Z γ(γ−1)(γσ(ξ) + (γ−1)(Φ(ξ) + Π( ξ))]ξ4dξ (32) andR0stands for the radius of the shock by the time it is ﬁrst formed t0. Note that for ξ >> ˜1,Φ(ξ)+ Π(ξ)∼V−αβ∼ξβ, G(ξ)∼const: the integral diverges as ξ5+2β, vindicating the introduction of the cut oﬀ ξc,which represents the boundary of the self-similarity solut ion. Clearly, this energy has to remain ﬁnite at any time and at focusing it requires that 1 /α−2.5≤Re(β)≤0. In the appendix we develop a semi-analytical method for solving eq.(21) and ob taining the correspondent spectrum for βl,n. In consonance with previous numerical calculations ([13]) we conﬁrm that βlies in this interval. By the way, the most unstable modes are shown to lie in the interval .5 + 1/α < Re(β)<−2.5 + 3/(2α) , even for very large values of l. For these modes, the energy emission rate Pl(t) = 4π(2αβ+ 5α−2)α2ε2ρ0R5 0t2αβ+5α−3 tα(2β+5) 0Cl (33) diverges. This means that, in analogy with the corrugation i nstabilities in planar shocks, a burst of sound is emitted at the focusing. The total energy carried away dur ing the shock-collapse is Esound=/summationdisplay l=1δEl(t0) =4πα2ε2ρ0R5 0 t2 0C (34) where we deﬁned C= Re[/summationtext l=1,βCl(β)]. 62 Inertial Polarization At Work As discussed already, electromagnetic bounded systems who se constituents have sizeable mass diﬀerences, say ∆ M, and which are subjected to a strong acceleration ﬁeld d/vector v/dt engender polarization ﬁelds /vectorE0,/vectorB0 that tend to restore the balance between electromagnetic an d inertial forces. Clearly, these polarization ﬁelds satisfy ∆Md/vector v dt=Ze/parenleftbigg /vectorE0+/vector v c×/vectorB0/parenrightbigg (35) where AandZcorrespond to the atomic and proton numbers and eis the electronic charge. Clearly, ∆M≈AMp, where Mpis the proton mass. Deﬁning a polarized potential-vector (Φ 0,/vectorA0) in the usual way, allows us to write the balance equation in the form /bracketleftbigg∂/vector v ∂t−/vector v×(/vector∇ ×/vector v) +/vector∇v2 2/bracketrightbigg =−Ze AMpc/bracketleftigg ∂/vectorA0 ∂t−/vector v×(/vector∇ ×/vectorA0) +/vector∇(cΦ0)/bracketrightigg (36) that suggests the identiﬁcation /vectorA0→ −AM pc Ze/vector vand Φ 0→ −AM p Zev2/2 . Other possible identiﬁcations exist, but they are gauge equivalent. The corresponding pol arization ﬁelds are /vectorE0=AMp Ze/bracketleftbigg∂/vector v ∂t+/vector∇v2 2/bracketrightbigg ;/vectorB0=−AMpc Ze/vector∇ ×/vector v (37) The time varying inertial-polarization ﬁelds engender the radiation ﬁelds /vectorE,/vectorBand their superposition must satisfy the sourceless Maxwell’s equations: /vector∇ ·(/vectorE+/vectorE0) = 0 →/vector∇ ·/vectorE= 4π̺eff /vector∇ ·(/vectorB+/vectorB0) = 0 →/vector∇ ·/vectorB= 0 /vector∇ ×(/vectorE+/vectorE0) +1 c∂ ∂t(/vectorB+/vectorB0) = 0 →/vector∇ ×/vectorE+1 c∂/vectorB ∂t= 0 (38) /vector∇ ×(/vectorB+/vectorB0)−1 c∂ ∂t(/vectorE+/vectorE0) = 0 →/vector∇ ×/vectorB−1 c∂/vectorE ∂t=4π c(− →Jeff+− →jeff) with ̺eff=−AMp 4πZe/bracketleftigg ∂/vector∇ ·/vector v ∂t+∇2v2 2/bracketrightigg − →jeff=AMp 4πZe/bracketleftbigg∂2/vector v ∂t2+1 2∂ ∂t(/vector∇v2)/bracketrightbigg clearly satisfying the conservation equation ∂̺eff/∂t+− →▽·− →jeff= 0 and − →J=AMpc2 4πeZ/vector∇ ×/vector∇ ×/vector v For non-relativistic ﬂows \|jµ\|/\|Jµ\| ∼(L/T)2/c2∼v2/c2, and the ﬁeld equations reduce to /vector∇ ·/vectorE= 0 (39) /vector∇ ·/vectorB= 0 /vector∇ ×/vectorE+1 c∂/vectorB ∂t= 0 /vector∇ ×/vectorB−1 c∂/vectorE ∂t=η/vector∇ ×/vector∇ ×/vector v 7where η=AMpc/Ze. Next we expand/parenleftbigg/vectorE /vectorB/parenrightbigg =/summationtext3 n=1/parenleftbiggEn Bn/parenrightbigg /vector enwhere /vector enis the familiar vector basis [16]: E= (/vector e1,/vector e2,/vector e3) =/parenleftig /vector nYlm(θ, φ),(r/vector∇)Ylm(θ, φ),(/vector r×/vector∇)Ylm(θ, φ)/parenrightig (40) For latter reference we mention the following identities: /vector∇ · E=Ylm r(2,−l(l+ 1),0);/vector∇ × E =1 r(−/vector e3,/vector e3,−/vector e2−l(l+ 1)/vector e1) (41) The unperturbed ﬂow is rotation free and the leading contrib ution to Maxwell’s equations [eq. (39)] comes from the perturbed ﬂow δ/vector v=δvn/vector e1+δv⊥/vector e2, ∂(r2B1) ∂r−l(l+ 1)B2r= 0 (42) ∂(r2E1) ∂r−l(l+ 1)E2r= 0 (43) r c˙B1−l(l+ 1)E3= 0 (44) r c˙B2−∂(rE3) ∂r= 0 (45) r c˙B3−E1+∂(rE2) ∂r= 0 (46) r c˙E1+l(l+ 1)B3=ηl(l+ 1)f(r, t) (47) r c˙E2+∂(rB3) ∂r=η∂(rf) ∂r(48) −r c˙E3−B1+∂(rB2) ∂r= 0 (49) where f(r, t) =∂δv⊥ ∂r+δv⊥−δvn r. Notice that B2, B1andE3are independent of the source term, and are taken to vanish identically. The other mode is /vectorE=E1/vector e1+E2/vector e2;/vectorB=B3/vector e3. (50) Averaging the Poynting vector /vectorS=c 8π(/vectorE×/vectorB∗) =cB∗ 3 8π/bracketleftig −rE1(Y/vector∇Y∗) +r2E2(/vector∇Y·/vector∇Y∗)/vector n/bracketrightig , (51) over all directions gives the radial energy ﬂux Sr=cl(l+ 1) 8πℜ(E2B∗ 3). (52) The corresponding spectral intensity is Il(ω) =cr2l(l+ 1) 2\|E2(ω)B∗ 3(ω)\| (53) We obtain the wave equation for Λ ≡E1(ω)rby combining eqs.(42)-(49) (∇2 r+k2)Λ(ω) =ikηl(l+ 1)f(ω, r) (54) 8and in terms of Λ ,the spectral intensity reads Il(ω) =ω 2l(l+ 1)/vextendsingle/vextendsingle/vextendsingle/vextendsinglerΛ(ω)∂(rΛ∗(ω)) ∂r/vextendsingle/vextendsingle/vextendsingle/vextendsingle(55) The wave equation is solved through the Green’s function met hod in the region away from the near zone: Λ(ω) =ikh(1) l(kr)/integraldisplay [−ikηl(l+ 1)f(ω, r′)]jl(kr′)r′2dr′. (56) In the radiation zone, Λ( ω) reduces to : Λ(ω)r≈ −eikr(−i)l+1kηl(l+ 1)/integraldisplay f(ω, r′)jl(kr′)r′2dr′ (57) Putting these pieces together, Il(ω) =1 2cη2k4l(l+ 1)\|Al(k)\|2(58) with Al(k) =/integraldisplay /integraldisplay f(r, t)e−iωtjl(kr)r2drdt (59) The function f(r, t) can be expressed in terms of the ﬂuctuation functions [eqs. (16)], f(r, t) =αε t0/parenleftbiggt t0/parenrightbiggαβ−1 [ξτ′(ξ) + 2τ(ξ)−(1−V(ξ))Φ(ξ). (60) Calling x=krand performing a change of integration variables we obtain r adiation emission rate per wave-length λ: Pl(λ) =p λε2α2l(l+ 1)\|Wl(k)\|2(61) with Wl(k) =/integraldisplay∞ 0jl(x)x2dx/integraldisplay1 0[ξτ′+ 2τ−(1−V)Φ]yαβ−1exp[−iQy]dy, (62) where Q≡kR0(t0/R0c) and p≡c3η2/2 . According to Barber ([15]) the ratio αR0/t0=c0, the speed of sound, and Q=α kR 0(c0/c)∼10−5(kR0).The asymptotic behavior given by eq.(28) and the fact that V∝ξ−1/αsuggests the expansion: [ξτ′+ 2τ−(1−V)Φ] =/summationdisplay n=1bnξβ−n/α=/summationdisplay n=1bn/parenleftbiggx kR0/parenrightbiggβ−n/α yn−αβ(63) where the coeﬃcients bnare determined by the dynamics of perturbations. Note that t he sum does not contain the n= 0 term because the leading term of the series [see again eq.( 28 )] for the velocity components Φ , τisV1−αβ.Therefore, Wl(k) =/summationdisplay n=1bn(kR0)n/α−β/integraldisplay1 0yn−1exp[−iQy]dy/integraldisplaykRmax 0jl(x)x2+β−n/.αdx. (64) The cutoﬀ kRmaxin the x-integral was introduced because the shock does not e xtend beyond Rmax, the ambient radius of the bubble. For Q << 1 we might transform this expression into Wl(k) = (kR0)−β/summationdisplay n=1bn n/bracketleftigg (kR0)n/α/integraldisplaykRmax kR0jl(x)x2+β−n/.αdx+/integraldisplaykRmax 0jl(x)x2+βdx/bracketrightigg (65) The detailed form of the spectrum requires a full knowledge o fbn, that is to say, dynamics of the ﬂuctu- ations must be speciﬁed (this can be done analytically by usi ng the method developed in the appendix). 9Fortunately, the major features of the spectrum can be obtai ned without delving into the diﬀerential equations. For instance, in the region where kRmax<1 we can approximate jl(x)≃(2x)ll!/(2l+ 1)! and then Wl(k)≃2ll! (2l+ 1)!(kR0)l+3/summationdisplay n=1bn n/braceleftigg 1 l+ 3 + β−n/α/bracketleftigg/parenleftbiggRmax R0/parenrightbiggl+3+β−n/α −1/bracketrightigg −1 (l+ 3 + β)/parenleftbiggRmax R0/parenrightbiggl+β+3/bracerightigg (66) In the other end of the spectrum kR0>1, taking the asymptotic expression jl(x)≈1/xsin(x−lπ/2) is justiﬁed, either because in the ﬁrst integral the integra tion variable x >1 or because in the second integral the measure x2+β(with 2 + β >1) ensures that important contributions to the integral com es from the large arguments. Thus, Wl(k)≃(kR0)/parenleftbiggRmax R0/parenrightbiggβ+1/summationdisplay n=1bn n/bracketleftigg f(β;kRmax) +/parenleftbiggR0 Rmax/parenrightbiggn/α f(β−n/α;kRmax)−f(β−n/α;kR0)/bracketrightigg ; (67) where f(β;x) = Im/bracketleftigg e−ilπ/2∞/summationdisplay m=0(ix)m+1 (m+β+ 2−n/α)m!/bracketrightigg . (68) The dominant power low contribution to Wl(k) in the region kR0>1 comes from the linear term (kR0) because the series f(β;x) behaves nearly like sin( x), for x >1.Taking the following ﬁgures Rmax∼5µm, the ambient radius of the bubble and R0∼0.15µm,(we shall explain in a moment) and deﬁning λ0= 2πR0,we display our asymptotic expressions in the form Pl(λ)∼pε2/braceleftbigg Alλ−1(λ0/λ)2l+6;λ >> λ 0 λ2 0/λ3gl(λ);λ < λ 0(69) where Al=l(l+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleα2ll! (2l+ 1)!/summationdisplay n=1bn n/braceleftigg 1 l+ 3 + β−n/α/bracketleftigg/parenleftbiggRmax R0/parenrightbiggl+3+β−n/α −1/bracketrightigg −1 (l+ 3 + β)/parenleftbiggRmax R0/parenrightbiggl+β+3/bracerightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (70) and gl(λ) =l(l+ 1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/parenleftbiggRmax R0/parenrightbiggβ+1 hl(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (71) with hl(k) =/summationdisplay n=1bn n/bracketleftigg f(β;kRmax) +/parenleftbiggR0 Rmax/parenrightbiggn/α f(β−n/α;kRmax)−f(β−n/α;kR0)/bracketrightigg (72) The apparent divergence of gl(λ) at large angular momenta [see eq. (71)] seems to endanger th e present results. This worry is removed studying the asympto tic behavior gl(λ), bearing in mind that in this limit β≃ ±il/radicalbig (γ−1)/(γ+ 1),[?]. This yields that gl(λ)→0 asl→ ∞, regardless of the speciﬁc form of the dynamical coeﬃcients bnmay take. 3 Assessment of the Results The present SL mechanism relies on very basic pieces of physi cs, the existence of corrugation instabilities in spherical shocks, whose existence is well known, Maxwell ’s equations and the inertial polarization 10paradigm. As we had the opportunity to explain, this paradig m stems from very elementary physics and it has remained hitherto unnoticed only because huge accele rations are required for sizeable polarizations. The detection of shock polarization in non-polar liquids wo uld lend an undisputable status to the inertial polarization principle . In the transduction of sound into r adiation , the ﬂash of light must be coincident with a burst of sound since the emission of radiation is cause d by corrugation instabilities . According to eq.(50), only one ﬁeld-mode is related to the sonoluminesce nt light. This mode has a longitudinal electric ﬁeld component E1, and some experiment must be devised to detect it .The transv ersal component E2points into the direction of the vector − →e2=/radicaligg 2l+ 1 4π(l−m)! (l+m)!eimϕsin (θ)(imPm l(cos(θ))− →eϕ−P′m l(cos(θ))− →eθ) and this (weird) polarization should be observed in sonolum inescent light. Physics is seldom controlled by cut-oﬀ parameters, and we ex pect the cut-oﬀ parameter Rmax( the bubble’s ambient radius) to play a marginal role in delimiti ng the frequency band where light is emitted. The main features of the spectrum should be controlled by the remaining parameters: R0,the radius of the shock-wave when it is ﬁrst formed and the perturbation am plitude ε. Thus, R0should characterize the typical wave-length of the emitted light λ≈λ0= 2πR0. Our asymptotic results [eq.(69)] conﬁrms this feeling. Numerical and theoretical studies of the dynamics of imploding shocks support the picture that the bubble collapses at the speed of sound by the time it passe s through its ambient radius as the right criterion both for shock formation and the existence of SL ([ 15]- [17]). According to these investigations, at 100 psbefore the bubble reaches its minimum size, a shock wave of in itial radius R0= 0.15µmdevelops: by this time the interface is imploding with 4 to 5 times the am bient speed of sound. With these ﬁgures, we predict the emitted light to lie in λ≈λ0= 900 nmspectral region, regardless the kind of gas present in the bubble; in SL experiments light is observed in the 200 nm/lessorsimilarλ/lessorsimilar800nminterval. According to this result, it is legitimate to infer the spectrum in this wave-length interval through the asymptotic formula for λ≼λ0[see eq.(69)]. How does the particular kind of gas present in the bubble impact on the the emission power? The dependence of the emitted light upon the particular type of gas present in the bubble stems from two diﬀerent factors: i.) diﬀerent values of the adiabatic index γleads to a diﬀerent shock-wave and corrugation instability dynamics; ii.) diﬀerent gases have diﬀerent dielectric per meability ǫ. The dielectric nature of the gas is implemented through the r eplacement E→Din the Poynting vector, which corresponds to the replacement of \|Wl(k)\|2by/tildewideǫ(k)\|Wl(k)\|2, orgl(λ)→/tildewideǫ(k)gl(λ) . Diﬀerent adiabatic indexes would cause hl(k) to change because both the spectrum of βand the dynamical coeﬃcients bn, depend upon γ. These two conditions will cause a change on the shape of the f unction gl(λ) . Assuming that after taking these corrections into account , the function gl(λ) still remains marginally dependent upon the wave-length (non power law), the overall change produced by diﬀerent gases in the shape on the logartithmic representation of the spectrum ln P∼ −3 lnλ+ lngl(λ) +const forλ/lessorsimilarλ0 , is a displacement of the nearly parallel lines of inclinati onm∼=−3. This behaviour is changed as we approach the λ << λ 0region because then the dielectric constant being governed by the plasma frequency of the gas, causes the function ln gl(λ) to strongly depend upon λ. Infering the uncorrected spectra for transmission by the su rrounding medium observed by Hiller in SL experiments for bubbles trapping pure noble gases bubbles a t 00C([18]) we infered m∼=−2.7. For pure He,m∼=−2.5.Inspection of the spectra shows the nearly linear dependenc e forAr, He, He3,andNe. The agreement is less accurate for XeandKr,for reasons which are presently unclear: it might well be the that heavier noble gases cannot be handled with the naive classical Inertial Polarization picture, they have too much internal structure and must be handled with a fu ll quantum mechanical approach. The spectrum for a mixture of 1% of HeandN2closely resemble the behavior of pure He 2([18]). Diﬀerences might be credited to the superimposition of the Bremsstrahl ung spectrum of free electrons of the ionized 11N2gas in the mixture to the original spectrum, or even the eﬀect of the Inertial-Polarization ﬁelds upon these electrons. Regarding now the intensity of the outgoing radiation, it is governed by the product pε2. A small p would require large corrugation instabilities, invalidat ing the linear regime approximations. Surprisingly, p=E2 p/(2/planckover2pi1α)⋍1.47×1016Watt , imploding shocks are fantastic power stations ! Actua lly, we have to worry to have suﬃciently small perturbations to ﬁt the exp erimental data! Typical power emissions are of of the order of 10−11Watt/nm in the λ0region [18], calling for an amplitude ε∼10−12or δr=εR0∼10−19m, which being much smaller then the nuclear dimensions can ha ve only a quantum mechanical origin. Now, the radius of the shock at the moment it is formed R0is governed by the radius of the bubble wall Rb, by the time it is collapsing at 4-5 times the ambient speed of sound. The dependence of the former on the latter is linear. In a semi-classical app roach, it is to be expected that the ﬂuctuations on the shape of the imploding shock are also governed by bubbl e wall ﬂuctuations, ǫ=δR0/R0=δRb/Rb. The ﬂuctuations of the bubble interface should be of the orde r of the bubble’s Compton wave-length λb andδr= (R0/Rb)λb∼λp/N, where λp∼10−15mis the Compton wave-length of the proton and N is the number of gas atoms trapped inside the bubble , N∼107.Thus, in this scenario δr∼10−22m, which is close to the amplitude needed to ﬁt the observed inte nsity of the radiation. One of the most intriguing issues in SL is beyond any doubt the noble gas puzzle: only bubbles containing noble gas,even at very small concentrations, gl ow. What can we say in this respect? Does our paradigm shed some light in this direction? Here is a clue . As the bubble collapses and the attains supersonic regime the adiabatic heating raises the gas temp erature to ∼0.4eV([19]) . The gas is further heated when it crosses the shock front, the temperature is in creased by a factor M4. This is more than enough to bring diatomic gases to their excited states, but n ot for noble gases. The dipole contribution /angb∇acketleftΨlmn\|/hatwidep\|Ψlmn/angb∇acket∇ight ·r−3of the excited states being much larger than the Inertial Pol arization Fields will wash away information regarding the latter. A full quantum m echanical calculation should resolve this issue. There are immense challenges ahead. From the theoretical po int of view, one needs to calculate the detailed spectrum taking full account of the shock dynamics , study the back reaction of the polarized ﬁelds upon the dynamics, clarify whether the polarization c aused by quantum mechanical transitions of a diatomic molecule are the culprits for washing out the Iner tial Polarization ﬁelds, etc. The immediate experimental challenge is to detect Shock Polarization in n on-polar ﬂuids. If the eﬀect is conﬁrmed in non-polar ﬂuids then it will be very hard to defuse the presen t transduction mechanism. Appendix – A semi-analytical solution of the diﬀerential eq uations for the perturbed ﬂow In order to solve the set of diﬀerential equations we split th e matrix into its regular and divergent parts M(V) =A (V−Vc)+B(V) (73) with A=m(Vc) (dP/dV )Vc φ1φ2φ3φ4 0 0 0 0 0 0 0 0 0 0 0 0 ;B=m(V) ˜φ1(V)˜φ2(V)˜φ3(V)˜φ4(V) Z 2V−1 αZ Z 3 +β−l(l+ 1) 0 0 −κ 0 0 0 (74) where φαis a short notation for φα(Vc) and φα(V) =φα(V)/P(V)−m(Vc)/(m(V)(dP/dV )Vc(V−V c)). Assuming B(V) and \|Y(V)/angb∇acket∇ightregular functions at the critical point Vcpermits the expansions B(V) =/summationtext nBn(V−Vc)n;\|Y(V)/angb∇acket∇ight=/summationtext kYk(V−Vc)k. Substitution into the diﬀerential equation yields the recurrence formulae: 12AY0= 0 (75) Yn+1= [(n+ 1)I − A ]−1/summationdisplay m≤nBn−mYm (76) The matrix Apossess three distinct null-eigenvectors: Y(2) 0= (−φ2, φ1,0,0) Y(3) 0= (−φ3,0, φ1,0) (77) Y(4) 0= (−φ4,0,0, φ1). Associated to each one of these eigenvectors we can construc t through the recurrence relations/vextendsingle/vextendsingleY(i)(V)/angbracketrightbig .The solution of the diﬀerential equation is the linear combi nation \|Y(V)/angb∇acket∇ight=/summationtext i=2,4ci/vextendsingle/vextendsingleY(i)(V)/angbracketrightbig . The fulﬁllment of the boundary requires that \|X(V1)/angb∇acket∇ight=\|Y(V1)/angb∇acket∇ight=/summationdisplay k[c2Y(2) k+c3Y(3) k+c4Y(4) k](V1−Vc)k. (78) This equation constitutes a set of four equations for the unk nown ( ci, β), which can be solved in a perturbational approach in powers ( V1−Vc), once the state \|X(V1)/angb∇acket∇ightis known. The only missing piece of information is the set of boundary conditions for the pertur bed ﬁelds. The boundary conditions for the perturbed ﬂow Supersonic motion produces a discontinuity in the ﬂuid ﬂow k nown as a shock wave or simply shock. Let us call /vector v2andρ2the ﬂuid velocity and density, and c2the speed of sound behind the shock, as measured in the laboratory frame(likewise, the subscript 1 refers to the same quantities in the front of the shock). The normal to the shock is /vector nand its velocity in the lab frame is /vector vshock. The discontinuities have to fulﬁll the following conditions at the shock surface [3] /vector n×[/vector v1−/vector vshock] =/vector n×[/vector v2−/vector vshock] (79) /vector n·[/vector v2−/vector vshock] /vector n·[/vector v1−/vector vshock]=ρ1 ρ2=γ−1 γ+ 1(80) c2 2=γ−1 γ+ 1/bracketleftbigg c2 1+2γ γ+ 1(/vector n·/vector vshock)2/bracketrightbigg (81) In the perturbed ﬂow the shock front is displaced from Σ 0:r−R(t) = 0 to (Σ 0+δΣ) :r−R(t)− δr(t, θ, φ) = 0. The corresponding perturbed normal is δ/vector n=−/vector∇δr, the perturbed shock velocity is ˙δr while the location of the shock itself in self-similar coord inate is 1 + δξ,δξ=δr/R(t). Accordingly, δξ=ε/parenleftbiggt t0/parenrightbiggαβ Ylm(θ, φ) δr=εR(t)/parenleftbiggt t0/parenrightbiggαβ Ylm(θ, φ) δ/vector n=−ε/parenleftbiggt t0/parenrightbiggαβ (R(t)/vector∇)Ylm(θ, φ) δ/vector vs=εα(1 +β)R(t)/parenleftbiggt t0/parenrightbiggαβ−1 Ylm(θ, φ)/vector n (82) 13The ﬁrst order corrections to the boundary conditions [eqs. (79)-(81) ] are: /vector n×/bracketleftbigg/parenleftbigg∂v ∂ξδξ+δvn−δvs/parenrightbigg /vector n+δ/vector v⊥/bracketrightbigg +δ/vector n×/vector n(v−vs) = −/vector n×δ/vector vs−δ/vector n×/vector n vs δρ2(1) +∂ρ2 ∂ξδξ= 0 (83) /vector n·/bracketleftbigg/parenleftbigg∂v ∂ξδξ+δvn−δvs/parenrightbigg /vector n+δ/vector v⊥/bracketrightbigg +δ/vector n·/vector n(v−vs) = −γ−1 γ+ 1(δvs+/vector n·δ/vector n vs) δc2 2+∂c2 2 ∂ξδξ= 2Z(1)vs(δ/vector n·/vector vs+/vector n·δ/vector vs) Inserting eqs.(82)-(16) into these boundary conditions, y ields Φ1=βV1−V′ 1 1−V1; τ1=−V1; ∆1=−G′ 1 G1= δZ1 Z1= (2 β−Z′ 1 Z1) (84) or, equivalently \|X(V1)/angb∇acket∇ight=ε (βV1−V′ 1)/(1−V1) −V1 −(β+ 3)V1/(1−V1) 2 (β+ (1/α−V1)/(1−V1))/γ (85) Numerical Procedure Our procedure for resolving the spectrum of βconsists of ﬁrst ﬁtting the unperturbed ﬂow ( Z(V), dV(V)) by a polynomial in V, from which we extract the matrices AandB(V) as power series in V. Then through the recurrence formulae ( ??) we obtain the expansion coeﬃcients Yn(β) up to a given order and insert then into eq.(78), in conjunction with the above b oundary condition \|X(V1)/angb∇acket∇ight[eq.(85) ] . This procedure yields a polynomial equation for β, which is solved numerically. We display the results for γ= 7/5, l= 1,2,3,4. Acknowledgments: I am thankful to N. Shnerb, J.Bekenstein, M.Chacham, S. Oliv eira and J. Portnoy for the enlightening conversations. References [1] Jacob D. Bekenstein and Marcelo Schiﬀer, 1998, Phys. Rev . D58 ,6414. [2] Alberto Saa and Marcelo Schiﬀer, 1998, Mod. Phys. Lett. A 13, 1557. [3] L. D. Landau and E. M Lifshitz, 1987, Fluid Mechanics,(Pe rgamon, Oxford) [4] R. J. Eichelberg and G. E. Hauber, 1962, ” Solid State Tran sducers for Recording of Intense Pressure Pulses” , in Les Ondes de Detonation (Centre National de la Recherche Scientiﬁque, Paris) 14Figure 1: Real and Imaginary parts of βfor l=1,2,3,4. 15[5] Paul Harris, 1964, J. App. Phys. 36,739. [6] Paul Harris and Henri-No˘ el Presles,1982, J. Chem. Phys . 77, 5157. [7] B.P. Barber et al.,(1997), Phys. Rep. 281, 65. [8] N. Marinesco and J.J. Trillat,1933, Proc. R. Acad. Sci, 1 96,858. [9] H. Frenzel and H. Schuyltes, 1934, Z. Phys. Chem.27B, 421 . [10] C. C. Wu and P. H. Roberts, 1993, Phys. Rev. Lett.70, 3424 . [11] N. Garcia and A. P. Levanyuk, 1996, JETP Lett. 64, 909. [12] N. Garcia and A. Hasmy, 1998, JETP Lett. 66, 472. [13] C. C. Wu and P. H. Roberts, 1996, Q. Jl. Mech. appl. Math, 4 9, 501. [14] G. Guderley,1942, Luftfahrforschung 19, 302. [15] B. P. Barber et al. ,1994, Phys. Rev. Lett.72, 1380. [16] Jacskon, J. D. (1975) Classical Electrodynamics,(NY: John Willey and Sons). [17] C. C. Wu and P. H. Roberts, 1994, Proc. R. Soc. A445,323. [18] R. Hiller et al. ,1994, Science 266,248. [19] R. L¨ ofstedt et al.,1992, J. Acoust. Soc. Am. 90, 2027. 16
arXiv:physics/0103030v1 [physics.data-an] 12 Mar 2001A Good Measure for Bayesian Inference Hanns L. Harney Max-Planck-Institut f¨ ur Kernphysik∗ Heidelberg January 16, 2014 Abstract The Gaussian theory of errors has been generalized to situat ions, where the Gaussian distribution and, hence, the Gaussian ru les of error propagation are inadequate. The generalizations are based on Bayes’ theorem and a suitable measure. The following text sketches some chapters of a monograph1that is presently prepared. We concentrate on the material that is — to the best of our knowledge — not yet i n the statistical literature. See especially the extension of fo rm invariance to discrete data in section 4, the criterion on the compatibili ty between a proposed distribution and sparse data in section 7 and the “d iscovery” of probability amplitudes in section 9. 1 The Prior Distribution Bayes’ theorem [1] allows one to deduce the distribution P(ξ\|x) of the pa- rameter ξconditioned by the data x. The distribution p(x\|ξ) of the data conditioned by the parameter ξmust be given. The theorem reads P(ξ\|x)m(x) = p(x\|ξ)µ(ξ) (1) m(x) =/integraldisplay dξ p(x\|ξ)µ(ξ). (2) See e.g. [2]. Here, µ(ξ) is called the prior and Pthe posterior distribution ofξ. The posterior can be used to deduce an interval Iof error: We deﬁne it as the smallest interval in which ξis with probability K. This is called ∗Postfach 103980, D-69029 Heidelberg, Germany; harney@mpi -hd.mpg.de; http://www.mpi-hd.mpg.de/harney 1submitted to Springer Verlag, Heidelberg 1the Bayesian interval I=I(K). In order to make it independent of any reparametrisation η=T(ξ), one has to judge the size Aof an interval Iby help of a measure µ(ξ), i.e. A=/integraldisplay Idξ µ(ξ). (3) We identify this measure with the prior distribution of µ. 2 Form Invariance Ideally the conditional distribution p(x\|ξ) possesses a symmetry called form invariance. This family of distributions then emerges by a m athematical group of transformations Gξxfrom one and the same basic distribution w, i.e. p(x\|ξ)dx=w(Gξx)dGξx. (4) It is not required that every acceptable phas this symmetry. But the sym- metry guarantees an unbiased inference in the sense of secti on 3. If there is no form invariance, unbiased inference can be achieved only approximately. The prior distribution is deﬁned as the invariant measure of the group of transformations. Symmetry arguments were ﬁrst discusse d in [3, 4, 5, 6]. They were not generally accepted because not all reasonable distributions possess the symmetry (4). It cannot exist at all if xis discrete. Since ξis assumed to be continuous, it can be changed inﬁnitesimally. However, no inﬁnitesimal transformation of a discrete variable is poss ible. In section 4, we generalize form invariance to this case. Form invariance is a property of ideal, well behaved distrib utions. How- ever, its existence is not a prerequisite of statistical inf erence, see section 6. The invariant measure can be found from p— without analysis of the group — by evaluating the expression µ(ξ)∝det/parenleftbigg/integraldisplay dxp(x\|ξ)∂ξL∂T ξL/parenrightbigg1/2 . (5) Here, the function Lis L(ξ) = ln p(x\|ξ) (6) and∂ξL∂T ξLmeans the dyadic product of the vector ∂ξLof partial deriva- tives with itself. Eq.(5) is known as Jeﬀreys’ rule [7]. One shall see in section 6 that this expression deﬁnes µin any case that is to say in the absence of form invariance, too. 23 Invariance of the Entropy of the Posterior Dis- tribution The posterior distribution P(ξ\|x) has the same symmetry as the conditional distribution p(x\|ξ) if form invariance exists. The entropy H(x) =−/integraldisplay dxP(ξ\|x)lnP(ξ\|x) µ(ξ)(7) is then independent of the true value ˆξof the parameter ξbecause one has H(x) =H(Gρx) (8) for every transformation Gρof the symmetry group. This entails that H(x) does not depend on ˆξbut only on the number Nof the data x1... x N. One can say that all values of the parameter ξare equally diﬃcult to measure. In this sense, form invariance guarantees unbiased estimat ion of ξand by the same token the invariant measure µis the parametrization of ignorance about ξ. 4 Form Invariance for Discrete x If the variable xis discrete — e.g. a number of counts — then form invariance cannot exist in the sense of eq.(4) since an inﬁnitesimal shi ft ofξcannot be compensated by an inﬁnitesimal transformation of x. One then has to deﬁne a vector a(ξ) the components of which are labelled by x. The probability p(x\|ξ) must be a unique function of ax(ξ). Form invariance then means that a(ξ) =Gξa(ξ= 0). (9) Again µis the invariant measure of the group. The transformation Gξshall be linear so that it is the linear representation of the symme try group of form invariance. It is necessarily unitary. The choice ax(ξ) =p(x\|ξ) is precluded because a group of transforma- tions cannot — for all of its elements — map a vector with posit ive elements onto one with the same property. With the choice ax(ξ) =/radicalBig p(x\|ξ) (10) one succeeds. That means: Important discrete distribution s — such as the Poisson and the binomial distributions — possess form invar iance. Further- more the property (4) can be recast into a relation correspon ding to eq.(9), 3i.e. it can be written as a linear transformation of the space of functions (p(x\|ξ))1/2. Hence, (9) is not diﬀerent from (4); it is a generalization. Note that (10) is a probability amplitude as it is used in quan tum me- chanics. However, it is real up to this point. The generaliza tion to complex probability amplitudes is sketched in section 8. 5 The Poisson Distribution Form invariance in the sense of section 4 does not seem to have been treated in the literature on statistics. As an example let us conside r the Poisson distribution p(x\|ξ) =λx x!exp(−λ) x= 0,1,2... (11) With ξ=λ1/2(12) one obtains the amplitudes ax(ξ) =ξx √ x!exp(−ξ2/2). (13) The derivative of ais found to be ∂ ∂ξa(ξ) = (A+−A)a(ξ), (14) where A,A+are linear operators independent of ξ. They have the commu- tator [A,A+] = 1. (15) Hence, A,A+are destruction and creation operators of numbers of counts or events. Integrating the diﬀerential equation (14) one ﬁn ds a(ξ) = exp/parenleftbigξ/parenleftbigA+−A/parenrightbig/parenrightbig\|0∝angbracketright. (16) Here, the vacuum \|0∝angbracketrightis the vector that provides zero counts with probability 1. Equation (16) means that the linear transformation Gξis Gξ= exp/parenleftbigξ/parenleftbigA+−A/parenrightbig/parenrightbig. (17) The measure µof this group of transformations is µ(ξ)≡const. (18) 4It can also be obtained by straightforward application of Je ﬀreys’ rule (5) without analysis of the symmetry group. This can be generalized to the joint Poisson distribution p(x1... x M\|ξ1... ξM) =M/productdisplay k=1ξ2xk k xk!exp(−ξ2 k) (19) of the numbers xkof counts in a histogram with Mbins. One ﬁnds the amplitude vector a(ξ1... ξM) = exp/parenleftBiggM/summationdisplay kξk(A+ k−Ak)/parenrightBigg \|0∝angbracketright (20) and again the uniform measure µ(ξ)≡const. As a further generalization, one can introduce destruction and creation operators Bν,B+ νof quasi-events ν= 1... nvia Bν=M/summationdisplay k=1ckνAk. (21) If the vectors \|cν∝angbracketrightforν= 1... nare orthonormal then [Bν,B+ ν′] =δνν′, (22) whence Bν,B+ νare destruction and creation operators. One ﬁnds the am- plitude vector a(ξ) = exp/parenleftBiggn/summationdisplay ν=1ξν/parenleftbigB+ ν−Bν/parenrightbig/parenrightBigg \|0∝angbracketright (23) The amplitude axto ﬁnd the event xis given by ax(ξ) =M/productdisplay k=11√xk!(Ξk)xkexp/parenleftBigg −1 2/summationdisplay νξ2 ν/parenrightBigg . (24) Here, the amplitude Ξk=n/summationdisplay ν=1ξνckν (25) to ﬁnd events in the k-th bin is given by an expansion into the orthogonal system of amplitude vectors \|cν∝angbracketright. More precisely: By working with the creation operators B+ ν, one infers an expansion of the vector \|Ξ∝angbracketrightin terms of 5the orthogonal system \|cν∝angbracketright. The prior distribution of the amplitudes ξνis again uniform, µ(ξ1... ξν)≡const. (26) On Summary: The problem of ﬁnding the expansion coeﬃcients ξνfrom the counting rates xkis form invariant and thus guarantees unbiased inference. One should therefore expand probability amplitudes and not probabilities in terms of an orthogonal system if one performs e.g. a Fourie r analysis. 6 The Prior Probability in the Absence of Form Invariance Jeﬀreys’ rule (5) can be rewritten in the form µ(ξ)∝det/parenleftbigg/integraldisplay dx∂ξa∂T ξa/parenrightbigg1/2 . (27) The integral means a summation if xis discrete. In diﬀerential geometry [8, 9], it is shown that (27) is the me asure on the surface deﬁned by the parametrisation a(ξ). A prerequisite for this measure is the assumption that one has the same uniform measure on eac h coordinate axis in the space; more precisely, the metric tensor of the sp ace must be proportional to the unit matrix. Since the coordinates axare probability amplitudes, this is justiﬁed by the last result of section 5. Hence, Jeﬀreys’ rule provides the prior distribution in any case. In the absence of form invariance, however, one cannot guarantee t hat all values of the parameter ξare equally diﬃcult to measure, i.e. one cannot guarantee unbiased inference. 7 Does a Proposed Distribution Fit an Observed Histogram? The Poisson distribution (19) yields the posterior P(ξ1... ξM\|x1... x M)∝M/productdisplay k=1ξ2xk kexp(−ξ2 k) (28) We want to decide whether — in the light of the data — the propos alτkis a reasonable estimate of ξk,k= 1... M. This is equivalent to the question whether τis in the Bayesian Interval I=I(K). The Bayesian interval is 6bordered by the “contour line” Γ( K) which is — in the case at hand — deﬁned as the set of points with the property P(ξ\|x) =C(K). This means thatτ∈Iexactly if P(τ\|x)> C(K) (29) or that τis accepted if and only if (29) holds. The number C(K) can be calculated. If the count rates xkare large in every bin k, the procedure essentially yields the well-known χ2-criterion. If, however, M≥N=/summationtext kxk, i.e. if the data are sparse, then this leads to the condition 1 NM/summationdisplay k=1xk/parenleftBigg N xkτ2 k−1−lnNτ2 k xk/parenrightBigg < ln/parenleftbigg 1 +M 2N/parenrightbigg +N−1/2Φ−1(K). (30) Here, Φ−1is the inverse of the probability function. Note that the exp ression in brackets ( ...) on the l.h.s. is ≥0 if /summationdisplay kτ2 k= 1. (31) Hence, the inequality (30) sets an upper limit to a positive e xpression. This criterion is new. It is needed because the situation M≥Nis surely met if kis a multidimensional variable i.e. if the observable is mul tidimensional. See [10]. Any attempt to apply Gaussian arguments is hopeles s in this case. 8 Does a Proposed Probability Density Fit Ob- served Data? Suppose that the data x1... x Nhave been observed. Each xkis supposed to follow, say, an exponential distribution p(x\|ξ) =ξ−1exp(−x/ξ). (32) They shall all be conditioned by one and the same hypothesis p arameter ξ. If this is true, the posterior P(ξ\|x1... x N) yields the distribution of ξand, hence, the Bayesian interval for ξ. It is intuitively clear that — at least for largeN— one can learn from the data not only the best ﬁtting values of ξ but one can even decide whether the exponential (32) is justi ﬁed at all. I.e. 7one can ﬁnd out whether the model is satisfactory. How does th is work? We do not want to produce a histogram by binning the data. This would reduce the problem to the one solved in section 7 but it would i ntroduce an arbitrary element into the decision: The deﬁnition of the bi ns. The basic idea is to determine ξfrom every data point, i.e. Ntimes, and to decide whether this result is compatible with ξhaving the same value everywhere. One deﬁnes the distribution qof the N-dimensional event ( x1... x N) conditioned by the N-dimensional hypothesis ( ξ1... ξN) as the product q(x1... x N\|ξ1... ξN) =N/productdisplay k=1p(xk\|ξk). (33) One writes down the posterior distribution Q(ξ1... ξN\|x1... x N) of the N- dimensional hypothesis ( ξ1... ξN). One studies its Bayesian interval I(K). A proposed hypothesis ( τ1... τN) is acceptable exactly if it is an element of I. In the case at hand, one determines the best value αof the hypothesis ξfrom the model that assigns one and the same hypothesis to all the data. One then asks whether the N-dimensional τwithτk=αfor all kis inI. The criterion (30) has been derived by help of this argument. Note, however, that the argument fails, when one wants to kno w whether the data ( x1... x N) follow the proposed distribution t(x). There is no hy- pothesis ξ. The family of distributions is not deﬁned from which t(x) is taken. Indeed the above argument does not judge the distribu tionp(x\|α) all by itself. It actually judges whether the family of distr ibutions, i.e. the whole model p(x\|ξ), is compatible with the data. The question whether t(x) ﬁts the data, is too general to be answered. One must specify w hich features of the distribution are important — its form in a region, wher e one ﬁnds most events or in a region where there are very few events? The relevant features are expressed by the parametric dependence on ξand the measure derived from it. 9 The Logic of Quantum Mechanics The results of section 5 show that probability amplitudes ra ther than prob- abilities can be inferred in an unbiased way from counting ev ents. Alterna- tivesν,ν′are deﬁned by two vectors \|cν∝angbracketrightand\|cν′∝angbracketright. Each vector characterizes a distribution over the bins k= 1... M of a histogram. A decision between νandν′amounts to assess the amplitudes ξνandξν′. They determine the strength with which the distributions νandν′are present in the data. 8However, the amplitudes can interfere — the probabilities c annot. The real amplitudes introduced so far can be generalized to complex o nes: We arrive at the quantum mechanical way to treat alternatives. The parameters ξdeduced from counting events are then completely analogous with quantum mechanical probability amplitudes . It may be bet- ter to turn this statement around and to say: The logic of quan tum me- chanics is the logic of unbiased inference from random event s; it is not a collection of the rules according to which the microworld “e xists”. The generalization of real amplitudes to complex ones is ach ieved by generalizing the amplitude vector (23) to a(ξ,ζ,φ) = exp/parenleftBigg in/summationdisplay ν=1Dν/parenrightBigg \|0∝angbracketright, (34) where the operator Dνis Dν=ζν(Bν+B+ ν) +iξν(Bν−B+ ν) +φν. (35) Here, the three generators do generate a group since one has t he commutator /bracketleftbigBν−B+ ν, Bν+B+ ν/bracketrightbig= 2. (36) The invariant measure is µ(ξ,ζ,φ)≡const. (37) By explicit evaluation of eq.(34) one ﬁnds ax=M/productdisplay k=11√xk!/parenleftBiggn/summationdisplay ν=1Ξk/parenrightBiggxk exp/parenleftBigg −1 2/summationdisplay ν(ξ2 ν+ζ2 ν−2iφν)/parenrightBigg (38) This is a generalization of expression (24). It is again a Poi sson distribution, but now the amplitude Ξ kto ﬁnd events in the k-th bin is Ξk=n/summationdisplay ν=1(ξν+iζν)c∗ kν. (39) This is an expansion of the probability amplitude in terms of the system of mutually orthogonal vectors \|c∗ ν∝angbracketrightwhich may be complex. The expansion coeﬃcients ξν+iζνmay be complex, too. 9The phase/summationtext νφνthat appears in (38) cannot be measured since only the modulus of (38) is accessible. The Poisson distribution possesses form invariance with re spect to the probability amplitudes even if these are complex. Put diﬀer ently, one should expand the square root of a distribution into a system of orth onormal vec- tors. They may be complex. The expansion coeﬃcients deduced from the data may also be complex. Inference on the real and imaginary parts of the expansion coeﬃcients is unbiased. The Fourier expansio n is an example; however, it must be the square root of the probability distri bution that is expanded. 10 Alternatives that cannot Interfere In quantum physics alternatives can interfere. Suppose tha t a cross section σ=σ(E) is observed as a function of energy E— e.g. in neutron scattering by heavy nuclei. Suppose that this excitation function show s a resonance line plus a smooth background. The book [11] is full of exampl es. Look e.g at the middle part of page 691. There is a ﬂat background with sup erimposed resonances. The resonance lines destructively and constru ctively interfere with the background. Speaking in the language of section 5, the ﬁgure oﬀers a simpl e alter- native ν= 1,2. The ﬁrst possibility ( ν= 1) is that the incoming neutron together with the target forms a compound system which decay s after some time. The second possibility ( ν= 2) is the reaction to occur without de- lay. The probability amplitudes ξν+iζνfor these two possibilities interfere. The interference pattern is visible if the resolution of the detection system is better than the width of the resonance. If the resolution i s much worse, the interference pattern disappears and the cross section d ue to the reso- nance is added to the cross section due to the background, i.e . one adds the probabilities πν=ξ2 ν+ζ2 νinstead of the amplitudes. The situation of insuﬃcient resolution is the situation of c lassical physics and classical statistics: Alternatives do not interfere. T heir probabilities are added. The typical situation of classical physics is that the detec tion system lumps many events together that have distinguishable prope rties. In our example: It does not well enough discriminate the energies o f the scattered particles. The events recorded in classical physics can in p rinciple be diﬀer- entiated according to more properties than are actually use d to distinguish them. The tacit assumption of classical physics was that thi s were always 10so. If objects are observed that allow for a small number of disti nctions only, one is lead to the logic of interfering probability amp litudes by the way sketched in sections 5 and 9. Consider the two slit experiment as a further example. If it i s performed with polarized electrons, an impressive interference patt ern appears. Use of unpolarized electrons reduces the contrast of the pattern. Had the scattered particles more than two “ways to be”, the contrast of the inte rference would be reduced up to the point, where the probability of a particl e going through the ﬁrst slit would be added to the probability of the particl e going through the second slit. See chapter 1 of [12]. Suppose that we know that there is interference between the t wo possi- bilities in the above neutron scattering experiment. The am plitudes ξν+iζν for the possibilities ν= 1,2 would be inferred from the data x1... x kas follows. The distribution of the data is p(x1... x N\|ξ1ζ1ξ2ζ2) =M/productdisplay k=1λxk k xk!exp(−λk), (40) where the expectation value λkin the k-th bin is a function of ξν,ζν, namely λk=\|(ξ1+iζ1)Line(k) + (ξ2+iζ2)Bg(k)\|2. (41) Here, Line(k) is the line shape and Bg(k) is the shape of the background. By section 9, this is a form invariant model allowing for unbi ased inference. Suppose on the contrary that there cannot be any interferenc e between the two possibilities in the neutron experiment. The probab ilities π1andπ2 are inferred via the model p(x1...N\|π1π2) which is again given by eq. (40). But now λkis the incoherent sum λk=π1\|Line(k)\|2+π2\|Bg(k)\|2. (42) The prior distribution for this model must be calculated by h elp of (5). The model is not form invariant, whence unbiased inference cann ot be guaran- teed. A closer inspection shows that the model “has a prejudi ce against” very small values of π1orπ2. This means: Small values are harder to establish than large ones. 11 Summary The basis of the foregoing work is twofold: (i) All statement s and relations in statistical inference must be invariant under reparamet rizations and (ii) to state ignorance about ξmeans to claim a symmetry. 11It is the symmetry of form invariance that guarantees unbias ed infer- ence of the hypothesis ξ, if the invariant measure of the symmetry group is identiﬁed with the prior distribution in Bayesian inferenc e. The invariant measure is obtained in a straightforward way — i.e. without a nalysis of the group — by Jeﬀreys’ rule. We have shown that even distribu tions of counted numbers possess form invariance. A study of the Poisson distribution shows that the basic quan tities in statistical inference are probability amplitudes not prob abilities. The am- plitudes may even be complex. This is not only an analogy to th e logic of quantum mechanics. This says that the logic of quantum mecha nics is the logic of unbiased inference from counted events. These considerations do not mean that form invariance is a co ndition for the possibility of inference. Lack of form invariance pr ecludes unbiased inference; it does not preclude inference. In the absence of form invariance, the prior distribution is deﬁned as the diﬀerential geometr ical measure on a suitably deﬁned surface: The surface must lie in a space of p robability amplitudes. The measure on the surface is again given by Jeﬀr eys’ rule. As a practically useful result, we have formulated the decis ion whether a proposed distribution ﬁts an observed histogram. The deci sion covers the case of sparse data. This case does not allow a Gaussian appro ximation and, hence, no χ2-test. References [1] Thomas Bayes, Phil. Trans. Roy. Soc. 53(1763)330–418. R eprinted in Biometrika 45(1958)293–315 and in Studies in the History of Statis- tics and Probability , E.S. Pearson and M.G. Kendall eds., C. Griﬃn & Co., London 1970, and in Two Papers by Bayes with Commentaries , W.E. Deming ed., Hafner Publishing, N.Y. 1963 [2] P.M. Lee. Bayesian Statistics: An Introduction Arnold, London 1997 [3] J. Hartigan, Ann. Math. Statist. 35(1964)836–845 [4] C.M. Stein, Approximation of Improper Prior Measures by Proper Prob- ability Measures in Neyman et al. [13] p. 217–240 [5] , E.T. Jaynes, IEEE Transactions on Systems Science and C ybernetics, SSC-4(3)227–241, September 1968 [6] C. Villegas, in Godambe and Sprott eds. [14] p. 409–414 12[7] H. Jeﬀreys, Theory of Probability Oxford University Press, Oxford 1939; 2nd edition 1948; 3rd edition 1961, here Jeﬀreys’ rule is fou nd in iii$ 3.10 [8] Shun-ichi Amari, Diﬀerential Geometrical Methods in Statistics , Vol- ume 28 of Lecture Notes in Statistics Springer, Heidelberg 1985 [9] C.C. Rodriguez, Objective Bayesianism and Geometry in Foug` ere ed. [15] p. 31–39 [10] J. Levin, D. Kella, and Z. Vager, Phys. Rev. A53(1996)14 69–1475 [11] V. McLane, C.L. Dunford, and Ph.F. Rose Neutron Cross Sections , Volume 2, Academic Press, Boston 1988 [12] M. Sands, R.P. Feynman, and R.B. Leighton The Feynman Lectures on Physics. Quantum Mechanics Volume III, Addison-Wesley, Reading 1965. Reprinted 1989 [13] J. Neyman et al. eds. Bernoulli, Bayes, Laplace. Proceedings of an In- ternational Research Seminar. Statistical Laboratory. Springer, N.Y. 1965 [14] V.P. Godambe and D.A. Sprott eds., Foundations of Statistical Infer- ence. Waterloo, Ontario 1970. Holt, Rinehart & Winston, Toronto 1971 [15] P.F. Foug` ere ed., Maximum Entropy and Bayesian Methods, Dartmouth 1989. Kluwer, Dordrecht 1990 13
arXiv:physics/0103031v1 [physics.gen-ph] 12 Mar 2001Virtual Replica of Matter in Bivacuum & Possible Mechanism of Distant Mind - Matter and Mind - Mind Interaction Alex Kaivarainen H2o@karelia.ru http://www.karelia.ru/˜alexk The original mechanism of bivacuum mediated Mind-Matter and Mind- Mind interaction, proposed here is based on the following st ages of long term eﬀorts (http://arXiv.org/ﬁnd/physics/1/au:+Kaivarain enA/0/1/0/all/0/1). - New dynamic models of bivacuum, sub-elementary particles and corpuscle- wave [C-W] duality, as a background of Superuniﬁcation; - New Hierarchic theory of liquids and solids, veriﬁed on exa mples of water and ice; - New Hierarchic model of elementary act of consciousness, b ased on microtubules of distant neurons exchange interaction; - Virtual Replica (VR) of matter, including living organism s in bivacuum; - The distant resonant [Mind-Bivacuum-Matter] and [Mind-B ivacuum- Mind] interaction, mediated by Bivacuum oscillation (BvO) with Golden mean frequency, accompanied by virtual particles/antipar ticles pressure os- cillation. The latter factor is related to oscillation of va cuum permittivity (ε0) and permeability ( µ0).These kinds of interaction may be realized also by modulation of energy of neutrino or antineutrino and [neu trino ⇋an- tineutrino] equilibrium. Our theory of Superuniﬁcation is based on new models of bivac uum, neu- trino/antineutrino, sub-elementary particles, their sel f-assembly to particles and [corpuscle (C)⇋wave(W)] duality. It elucidates the quantum back- ground of non-locality, principle of Least Action and Golde n mean, uniﬁes the quantum and relativist theories. Bivacuum is considere d as two nonmix- ing superﬂuid oceans of subquantum particles of positive (r eal) and negative (mirror) energy. The primordial bivacuum in the absence of m atter and ﬁelds is symmetric in contrast to secondary one. It is shown, that self-organization and evolution of system s at huge range of scales: from microscopic to cosmic ones - drives them to Go lden Mean conditions under the inﬂuence of Bivacuum oscillations (Bv O) with Golden Mean (GM) frequencies. This occur, as a result of tending of [ C⇋W] pulsation of matter elementary particles to resonance with fundamental (GM) frequencies ( ωi 0=mi 0c2//planckover2pi1), of BvO. The principle of Least action can be a consequence of corresponding ”Harmonization” driving for ce of bivacuum. 1The virtual replica (VR) of condensed matter (living organi sms in private case), may inﬂuence the properties of uncompensa ted (eﬀective) virtual pressure of asymmetric bivacuum in foll owing manner: 1) changing the amplitude of virtual pressure waves (VPW) in -phase with Bivacuum oscillations (BvO). This factor is dependent on fr action of coherent particles in system with in-phase [ C⇋W] transitions. The important role of Mind-Matter and Mind-Mind interaction is related to cohe rent fraction of water in microtubules in state of mesoscopic molecular Bo se condensate. This fraction is a variable parameter, dependent on kind of e lementary act of consciousness and number of simultaneous acts; 2) changing bivacuum symmetry shift, related to [ BV F↑⇋BV F↓]≡ [neutrino ⇋antineutrino ] equilibrium shift, induced by magnetic ﬁeld of matter variation. Decreasing/increasing of the vacuum s ymmetry shift will be accompanied by decreasing/increasing of the eﬀecti ve uncompensated VPW energy; 3) shifting the Golden mean resonance conditions of [matter - bivacuum] interaction by exchange of BvO, as a result of spatial pertur bation of matter, changing the frequency of [ C⇋W] pulsations of its elementary particles. This factor may increase or decrease the amplitude of BvO and , consequently, the amplitude of virtual pressure waves (VPW). Key words: vacuum, duality, Superuniﬁcation, act of consci ous- ness, virtual replica, Golden mean, bivacuum oscillations , Mind- Matter and Mind-Mind interaction. I. New Model of Bivacuum Our Dynamic model of Corpuscle -Wave [ C⇋W] duality is based on the new notion of bivacuum. We postulate the existence of POS ITIVE (real) and NEGATIVE (mirror) vacuum as two non mixing ’oceans’ of su perﬂuid liquid, formed by virtual sub-quantum particles of the opposite energies. The uniﬁed system of positive (real) and negative (mirror) v acuum is termed: BIVACUUM. It is assumed to be an inﬁnitive source of bivacuum fermions with positive (BVF↑ S=1/2) and negative (BVF↓ S=−1/2) half-integer spins and bivacuum bosons (BVB± S=0) of two possible polarization ( ±) and zero spin. The (BVF↑) and (BVF↓) are introduced in our model as a correlated pairs ofin-phase circulations of quantum liquid, but in two opposite directi ons: i.e. clockwise and anticlockwise ( ⇈and/dblarrowdwn), like: BV F↑ (S=1/2)= [real rotor (V+ ↑) +mirror rotor (V− ↑)]≡[V+⇈V−] (1) 2and: BV F↓ (S=−1/2)= [real antirotor (V+ ↓) +mirror antirotor (V+ ↓)]≡[V+/dblarrowdwnV−] (2) correspondingly. For the other hand, the (BVB±) of two possible polarization: (+) and ( −) is formed by the pair of counter phase real and mirror circulations: BV B± S=0= [real rotor (V+ /arrowbothv) +mirror antirotor (V− /arrowbothv)]≡[V+/arrowbothvV−] (3) The BVB±with properties of Falaco soliton (Kiehn, 1998) is the inter me- diate transition state between (BVF↑) and (BVF↓): BV F↑ (S=1/2)⇋BV B± S=0⇋BV F↓ (S=−1/2)(4) 1.1. Quantization of bivacuum The energies of real and mirror rotors, forming bivacuum fer mions ( BV F/arrowbothv) and bivacuum bosons ( BV B±) are quantized as quantum harmonic oscilla- tors of positive and negative energy: /parenleftbig E+ V/parenrightbigi n= +/planckover2pi1ωi 0(1 2+n) = +mi 0c2(1 2+n) =m+ Vc2=/planckover2pi1c L+ V(4a) /parenleftbig E− V/parenrightbigi n=−/planckover2pi1ωi 0(1 2+n) =−mi 0c2(1 2+n) =−m− Vc2=−/planckover2pi1c L− V(4b) where: ωi 0andmi 0correspond to grand values of angle frequency and eﬀective mass of real and mirror rotors with quantum number n= 0.The radiuses of corresponding rotors are: L+ V=/planckover2pi1/ m+ VcandL− V=/planckover2pi1/m− Vc It leads from our theory, that the values of mi 0are equal to the rest mass of (i) basic elementary particles, like electron, positron, qua rks, etc. The symmetric bivacuum (in absence of vacuum symmetry shift ) is char- acterized by the equality of resulting energy of bivacuum fe rmions and bi- vacuum bosons to zero: Ei V=/parenleftbig E+ V/parenrightbigi n+/parenleftbig E− V/parenrightbigi n= 0 (5) This condition (5) of energetic symmetry means the absence o f matter andprimordial bivacuum existing. It follows from our theory, that in secondary vacuum, existing in pres- ence of matter or antimatter and ﬁelds, the dynamic equilibr ium (4) is shifted 3to the left or to the right. This results in corresponding biv acuum symmetry shift: (∆mV) =/parenleftbig/vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingle/parenrightbig /negationslash= 0 (6) For such a case, real in presence of matter and ﬁelds, the eqs. (4a and 4b) transform to: /parenleftbig E+ V+ ∆E+ V/parenrightbigi n= +/planckover2pi1(ωi 0+ ∆ωi 0)(1 2+n) = +/parenleftbig mi 0+ ∆mi 0/parenrightbig c2(1 2+n) = (m+ V+1 2∆m+ V)c2 (6a) /parenleftbig E− V+ ∆E− V/parenrightbigi n=−/planckover2pi1(ωi 0−∆ωi 0)(1 2+n) =−(mi 0−∆mi 0)c2(1 2+n) =−(m− V−1 2∆m− V)c2 (6b) The total energy of asymmetric secondary bivacuum in contrast to primordial (5) is nonzero and dependent on the sign of vacuum shift (+ or−) /vextendsingle/vextendsingleEi V/vextendsingle/vextendsingleas= 2∆/parenleftbig E− V/parenrightbigi n= 2/planckover2pi1∆ωi 0(1 2+n) = 2∆ mi 0c2(1 2+n) = ∆ m± Vc2(6c) However, the diﬀerence between sublevels of positive and ne gative vac- uum sublevels (energetic gaps) is independent on vacuum sym metry shift. This fact is responsible for keeping permanent the quantize d frequencies of Bivacuum oscillations (BvO), radiating/absorbing as a res ult transitions be- tween corresponding sublevels (section 1.4). 1.2. Neutrino and antineutrino: what is it ? Neutrino and antineutrino are the neutral fermions with opp osite spins (spirality) and very small (or even zero) mass, propagating in vacuum with light velocity. Due to neutrality and small probability of s cattering any kind of screens are transparent for neutrino/antineutrino. We suppose that the uncompensated BV F↑andBV F↓, originated from the equilibrium (4) shift to the left or to the right, cor respondingly, may represent three generations of neutrino and antineutrino: the electron’s ( νe,/tildewideνe), the muon’s ( νµ,/tildewideνµ) and tau-electron’s ( ντ,/tildewideντ).Their energy and eﬀective mass are directly related to zero-point mass of the electrons of corresponding generation: ( m0)e,µ,τand may be quantized in similar way like (4a and 4b): 4Eν,/tildewideν e,µ,τ=±/planckover2pi1ων,/tildewideν e,µ,τ(1 2+n) =±/parenleftbig m± ν/parenrightbign=0 e,µ,τc2(1 2+n) =±βe,µ,τ(m0)e,µ,τc2(1 2+n) (7) where ( m0)e,µ,τare the rest mass of [ e, µ, τ ] electrons; βe,µ,τ= (m0/MPl)2 e,µ,τ is a gravitational ﬁne structure constants, introduced in o ur theory of gravi- tation. Consequently, neutrinos ( νe,µ,τ) and antineutrinos ( /tildewideνe,µ,τ) represent cer- tain perturbations of bivacuum symmetry (nonlocal or almos t nonlocal) of opposite sign, induced by uncompensated BV F↑andBV F↓,correspond- ingly: (∆mV) =/parenleftbig/vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingle/parenrightbig >0for ν e,µ,τ, when K ν⇋/tildewideν=BV F↑ BV F↓>1 (7a) (∆mV) =/parenleftbig/vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingle/parenrightbig <0for/tildewideνe,µ,τ, when K ν⇋/tildewideν=BV F↑ BV F↓<1 (7b) The curvature of vorticity for neutrinos at their eﬀective m ass [(m± ν)n=0 e,µ,τ= βe,µ,τ(m0)e,µ,τ] tending to zero may be of cosmic scale: Le,µ,τ=/planckover2pi1//bracketleftBig βe,µ,τ(m0)e,µ,τc/bracketrightBig →∞ (8) In the interaction of neutrino with real target - only vortic es (V+) of uncompensated BVF↑(eq.1),corresponding to real positive energy of bivac- uum, may be eﬀective. It is shown, that the internal velocity of circulation of sub-quantum par- ticles, forming vorticities [ V+and V−]e,µ,τ, of superﬂuid bivacuum is luminal (Kaivarainen, 2000). 1.3. Virtual Bose condensation in bivacuum as a background o f nonlocality The condition of primordial bivacuum in the absence of matter is: /vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle=/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingleand ∆mV=/vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingle= 0 (8a) Theexternal resulting impulses (momentum) of all of three kind of sym- metric bivacuum excitations: PBVB±,PBVF↑andPBVF↓are equal to zero, 5as far their external group velocity is zero. This means that external vir- tual wave B length of these excitations, as a ratio of Plank co nstant to their impulse is tending to inﬁnity: λext BC=h/Pext BV B±, BV F/arrowbothv→∞ (9) It is shown in our work, using Virial theorem , that corresponding to (9) inﬁnive virtual Bose condensation (BC) of bivacuum, whe reλextis a parameter of order, coincide with condition of nonlocality . We deﬁne non- locality, as independence of any potential in BC (real or vir tual) on distance: V(r) =const. Deviation of secondary vacuum from ideal symmetry (8a) lead s toPext BV B±, BV F/arrowbothv> 0 and disassembly of inﬁnitive virtual BC to huge, but ﬁnite v irtual BC: λext BC=h/Pext BV B±, BV F/arrowbothv/negationslash=∞The values of λext BCare dependent on potentials of gravitational, electromagnetic and torsion ﬁelds. 1.4. Quantization of energetic gap of bivacuum. Bivacuum gap Oscillation (BvO) The rotors ( V+ /arrowbothv) and antirotors ( V+ /arrowbothv) of two possible polarization ( ↑and ↓) in realms of positive (real) and negative (mirror) vacuum, are separated from each other by quantized energetic gap, the same (or very close) in primordial and secondary vacuum (see eqs. 4a, 4b and 6a, 6b ): (An V)i=/bracketleftbig/parenleftbig m+ V/parenrightbign−/parenleftbig m− V/parenrightbign/bracketrightbigic2=/planckover2pi1ωi 0(2n+ 1) = mi 0c2(2n+ 1) = /planckover2pi1c/(Ln V)i (10) with radius of corresponding circulations of BVF/arrowbothvand BVB±: (Ln V)i=/planckover2pi1//bracketleftbig mi 0(2n+ 1)c/bracketrightbig =/planckover2pi1c/(An V)i(11) The important notion of Bivacuum oscillations (BvO) is intro- duced as a symmetric oscillations of bivacuum energetic gap (An V)i, resulting from transitions between the ground gap with n= 0 and n= 1,2,3... The energy of ground BvO for selected basic level ( i) from (10) is: Ai BvO= (An V−A0 V)i= 2n/planckover2pi1ωi 0= 2n mi 0c2(12) The resonant frequencies of BvO ( ωi 0) are dependent on the kind of basic level ( i) of bivacuum, related directly to the rest mass of basic elem entary particles (three generation of the electrons and positrons (e, µ, τ ), quarks, etc.): ωi 0=mi 0c2//planckover2pi1 (13) 6The symmetric bivacuum excitations: BVB±,BVF↑and BVF↓may have a broad spectra of energetic gaps and related radiuses (from microscopic to cosmic ones), determined by the eﬀective mass of excitation s (m+ Vandm− V) in 4a and 4b and corresponding resonant frequencies (2 n ωi 0). The vacuum symmetry shift (6) in the presence of matter or ﬁel ds is not accompanied by BvO symmetry perturbation as far the diﬀeren ce between sublevels remains unchanged. However, the frequency and am plitude of BvO may be modulated by vibro-gravitational waves (VGW), excit ed by collective particle oscillations (see section 4.4). 1.5. Virtual particles and antiparticles of bivacuum. Excitation of Virtual pressure waves (VPW) by Bivacuum oscillations The virtual particles and antiparticles [ origination ⇋annihilation ] is a result of correlated transitions between rotors in realms o f positive and neg- ative energy:/bracketleftbig V+ j−V+ k≡VT+ j,k/bracketrightbig and/bracketleftbig V− j−V− k≡VT− j,k/bracketrightbig ,correspond- ingly, accompanied by ﬂuctuations of bivacuum virtual pres sure. These quan- tum transitions occur between BV B± jandBV B± kor between BV F/arrowbothv jand BV F/arrowbothv kwith diﬀerent energetic gaps and radiuses. Such transition states, re- sponsible for virtual particles and antiparticles (virtua l bosons and fermions) origination/annihilation, are termed Virtual Transitons (VT±). It is obvious, that corresponding virtual density/pressur e oscillations are related directly to radiation/absorption of the Bivacuum o scillations (BvO) with energy: ∆Aj,k BvO= (Aj V−Ak V)i= 2n/planckover2pi1ωi 0(j−k) = 2mi 0c2(j−k) (14) In private case, when k= 0 the (eq.14) turns to formula for basic BvO (12). Virtual pressure, related to virtual particles and those, r elated to antiparticles totally compensate each other in conditio n of pri- mordial bivacuum, when vacuum symmetry shift is zero. Howev er, in secondary bivacuum, in presence of matter and ﬁelds when i n- equality (6) take a place, such compensation is broken and pr essure of virtual particles or antiparticles becomes nonzero. This displays, for example, in Casimir eﬀect. Propagation of BvO of resonant amplitude, accompanied by virtual par- ticles and antiparticles origination and annihilation, oc cur in bivacuum with velocity, limited by the life-times of corresponding virtu al particles/antiparticles. 7The relation between the life time of virtual pair (∆ t) and energy of pair (∆Aj,k BvO) is determined by uncertainty principle: ∆t∆Aj,k BvO≥/planckover2pi1 (15) The nonresonant sub-quantum BvO in bivacuum virtual Bose condensate, unable to excite BVF/arrowbothvof BVB±to next quantum state, are nondissipative and nonlocal. At certain quantum boundary conditions, satisfying the con servation laws, thetransaction between emitter and absorber of nonlocal BvO via bivac- uum may lead to system of standing BvO origination (Cramer, 1 988). 2. Origination of matter as a result of bivacuum symmetry shi ft. 2.1. Sub-elementary particles and antiparticles Our model postulates, that the sub-elementary particles and sub- elementary antiparticles in corpuscular [C] phase represent the asymmet- rically excited bivacuum fermions: /parenleftbig BVF↑/parenrightbig∗≡F− ↑or/parenleftbig BVF↓/parenrightbig∗≡F+ ↓ (16) in form of [real ( m+ C) + mirror ( m− C)] mass-dipole with opposite spins ( S= ±1 2) and charge ( e±). The spatial image of this mass-dipole is a correlated dynamic pair: [real vortex + mirror rotor] (Kaivarainen, 20 00). The real inertial ( m+ C) and inertialess ( m− C) mass are the result of bi- vacuum symmetry shift, accompanied by sub-elementary part icle or sub- elementary antiparticle origination. The latter is depend ent on the sign of shift. The asymmetrically excited bivacuum bosons: (BVB±)∗≡B±(17) represent the intermediate or transition state between sub -elementary fermions of opposite spins: F− ↑⇋B±⇋F+ ↓ (17a) The electron, in accordance to our model, is a triplet e−=/angbracketleft[F− ↑⊲ ⊳F+ ↓] +F− ↓/angbracketright (18) formed by two negatively charged sub-elementary fermions o f opposite spins (F− ↑andF− ↓) and one sub-elementary antifermion ( F+ ↓) with positive charge .The 8symmetric pair of standing sub-elementary fermion and sub- elementary an- tifermion: [wave+antiwaveB ]≡/bracketleftbig F− ↑⊲ ⊳F+ ↓/bracketrightbig (18a) are pulsing between Corpuscular [C] and Wave [W] states in-p hase, compen- sating the inﬂuence of energy, spin and charge of each other. Notation ( ⊲ ⊳) means such compensation. However it is not total due to vacuu m symmetry shift, produced by the uncompensated sub-elementary parti cle. Deviation of total compensation increases with particle velocity and ﬁelds tension, in- creasing symmetry shift. The positron may be presented as asymmetric form of the elect ron (18), however, with uncompensated sub-elementary antifermion ( F+ ↑): e+=/angbracketleft[F− ↑⊲ ⊳F+ ↓] +F+ ↑/angbracketright (18b) Theu−quark is considered as superposition of two µor/and τpositron like structures u= [e++e+]µ,τZ= +2/3 (18c) andd−quarks may be composed from two electron and one positron ( µ or/and τ) like structures: d= [(e−+e−) +e+]µ,τ Z=−1/3 (18d) It is supposed, that each uncompensated sub-elementary fer mions/antifermion of quarks has elementary charge: ±Z=±1/3. It leads from our model, that elementary particles with ferm ion properties are composed from non equal number of sub-elementary partic les and with boson properties - from their equal number. For example photon, resulting from annihilation of the elec tron with positron may be considered as a system of three pairs of sub-e lementary fermions and antifermions: photon = 3[F− ↑⊲ ⊳F+ ↓]e= 0; S= 1 (18e) The value of spin of photon S= 1 may be explained by the in-phase rotation of both sub-elementary particles, forming one pai r [F− ↑⊲ ⊳F+ ↓] in contrast to other two pairs with antiphase rotation of F− ↑andF+ ↓. The symmetry of our bivacuum as respect to probability of ele - mentary particles and antiparticles origination, makes it principally diﬀerent from asymmetric Dirac’s vacuum, with its realm of n ega- tive energy saturated with electrons. Positrons in his model represent the ’holes’, originated as a result of the electrons jumps in realm of positive energy. 92.2. Dynamic model of Corpuscle ⇋Wave: [C ⇋W] duality I supposed, that duality do not display itself, depending si mply on the experimental way of particle detection, when both properti ed are embedded in particle permanently, as it was generally accepted. In my model it is assumed, that the corpuscular [C] and wave [W] phases of sub- elementary particles/antiparticles represent two alternative phase of de Broglie wave (wave B), which are in dynamic equilibrium (Kaivarainen, 19 93, 1995, 2000). The frequency of [ C⇋W] pulsation (ωB) is equal to frequency of quantum beats between asymmetric (BVF/arrowbothv)∗≡F± /arrowbothvand symmetric (BVF/arrowbothv) states of bivacuum sub-elementary fermions (F− ↑) and antifermions (F+ ↓). The energy of [C] phase, deﬁned by this frequency, is a sum of energies of real ( EC+) and mirror ( EC−) corpuscular states of asymmetrically excited bivacuum fermion (F/arrowbothv). For example: EC=E(V+ /arrowbothv)∗+EV− /arrowbothv=3 2/planckover2pi1ω+ (−1 2)/planckover2pi1ω=/planckover2pi1ωB (19) where: EC+corresponds to energy of excited rotor ( V+ /arrowbothv)∗;EC−corre- sponds to energy of antirotor in ground state ( V− /arrowbothv), forming with ( V+ /arrowbothv)∗the asymmetric sub-elementary fermion: (F+ ↑)≡[(V+ /arrowbothv)∗+ (V− /arrowbothv)] (20) or the asymmetric sub-elementary antifermion: (F− ↓)≡[(V− /arrowbothv)∗+ (V+ /arrowbothv)] (20a) The energy of [W] phase of (F+ ↑),existing in form of cumulative virtual cloud (CVC) , is deﬁned as the energy of transition between rotors in excited real (V+ /arrowbothv)∗and ground (V+ /arrowbothv) states: EW=E(V+ /arrowbothv)∗−EV+ /arrowbothv=3 2/planckover2pi1ω−1 2/planckover2pi1ω=/planckover2pi1ωB (21) It is equal to energy of [C] phase ( EC). Consequently, the energy of both phase: [C] and [W] are equal to energy of wave B: EB=EC=EW≡ECV C . 2.3. Extension of special theory of relativity. Corpuscular and Wave phase of sub-elementary particles 10Postulated in our work mass of rest ( m0)conservation law for sub- elementary fermion (antifermion), presented by (20 and 20a ), interrelates the real inertial mass ( m+ C), corresponding to asymmetrically excited rotor (antirotor) and mirror mass ( m− C),corresponding to rotor (antirotor) in realm of negative (positive) energy m+ Cm− C=m2 0 (22) Thereal (inertial) and mirror (inertialess) masses - change with external group velocity ( v≡vgr) of sub-elementary particles, composing particles in the counterphase manner, compensating each ot her: real mass: m+ C=±m0/[1−(v/c)2]1/2(23) mirror mass: m− C=±m0[1−(v/c)2]1/2(23a) The real mass ( m+ C) corresponds to the energy of excited positive vacuum and the mirror mass ( m− C) to the ground state of the negative vacuum. Dividing eq.(23a) to (23), we get important relation betwee n real and mirror mass: m− C m+ C= 1−(v/c)2(23b) The eqs. 23 and 23a can be transformed to following shapes: /parenleftbig E+ C/parenrightbig2= (m+ C)2c4=m2 0c4+ (m+ Cv)2c2(23c) /parenleftbig E− C/parenrightbig2= (m− C)2c4=m2 0c4−(m0v)2c2(23d) where: E+ CandE− Care the real and mirror energy of wave B. The ﬁrst of these eqs. coincides with those, obtained by Dira c, the second is a new one. The another of (19-21) way to express the energy of sub-eleme ntary wave B, following from 23b, is to consider it as a result of energy o f beats between the real and mirror states with frequency ( ωB): EB=EC=EW=/planckover2pi1ωB=m+ Cc2−m− Cc2=/parenleftbig m+ C−m− C/parenrightbig c2=m+ Cv2= 2Tk (24) Corresponding transition state between real [C+] and mirror [C−] states we deﬁne as a wave [W]-phase of wave B. This phase, in contrast to [C]-phase, represents cumulative virtual cloud (CVC) of subquantum pa rticles, forming superﬂuid vacuum. 11It easy to see from (23 and 23a) that/parenleftbig m+ C−m− C/parenrightbig c2is equal to the dou- bled kinetic energy (2 Tk) of sub-elementary particle, related to corresponding hidden impulse (momentum) P±as: (2Tk) =/parenleftbig m+ C−m− C/parenrightbig c2=m+ Cv2=P±c (25) where :P± W=/parenleftbig m+ C−m− C/parenrightbig c=P± C=m+ Cv(v/c) (25a) The hidden ( L±) and real external ( L+ C) spatial dimensions of sub-elementary particle as mass dipole are L±≡L± W=/planckover2pi1/parenleftbig m+ C−m− C/parenrightbig c=L± C=/planckover2pi1 m+ Cv(v/c)(26) and L+ C=/planckover2pi1 m+ Cv(26a) We can see from (26), that the characteristic hidden dimensi on of [C] phase and hidden dimension of [W] phase in form of CVC are equa l:L±≡ L± C=L± W. For nonrelativist elementary particles ( v << c ),the external L+ Cis much shorter, than hidden L± CV C, as it leads from 25a and 26: L+ C/L±=v/c (27) It may be shown from canonical representation of (23c) and (2 3d), that spatial image, corresponding to real [C+] state is equilateral hyperbola and spatial image of mirror [C−] state of [C] phase is a circle . Spatial image of CVC, corresponding to [W] phase, considere d as a dif- ference between images of [C+] and [C−] states is a parted (two-cavity) hyperboloid (Kaivarainen, 2000). The restoration of [C] - PHASE in form of [ real+mirror ] mass-dipole is a result of binding of CVC to BVF in ground state, accompanied b y asymmetric excitation of bivacuum fermion: [BV F/arrowbothv+CV C][W→C]→F± /arrowbothv . This [ W→C] transition is totally reversible with the opposite one [ C→ W] : F± /arrowbothv[C→W]→[BV F/arrowbothv+CV C] Oscillations between [C] and [W] phase of wave B are accompan ied by oscil- lations of kinetic energy and time, in accordance to our theo ry of time (see eq. 46). 12In general case mass of rest ( m0) has the intermediate value between real and mirror masses: /vextendsingle/vextendsinglem+ C/vextendsingle/vextendsingle≥m0≥/vextendsingle/vextendsinglem− C/vextendsingle/vextendsingle (27a) The energy of CVC may be presented as a sum of energies of Vacuu m Density Waves ( EV DW) and Vacuum Symmetry Waves ( EV SW): ECV C=/parenleftbig m+ C−m− C/parenrightbig c2=EV DW+EV SW (27b) where: EV DW=/parenleftbig m+ C−m0/parenrightbig c2(27c) and E V SW=/parenleftbig m0−m− C/parenrightbig c2(27d) Propagation of fermion in bivacuum in a course of [ C⇋W] cycling is a jump-way process, termed ’kangaroo eﬀect’, because the [W] phase is luminal in contrast to sub-luminal [C] phase. Our model uniﬁ es electromag- netic and gravitational potentials of elementary charge (e lectron) with its real kinetic energy (equal to energy of CVC) and bivacuum sym metry shift in very clear way. Explanation of two-slit experiment The bunched character of the electron’s trajectory can be a r esult of impulses, produced by uncompensated sub-elementary particle ( F− ↓) in a course of its [ C⇋W] pulsation. In accordance to our model, such pulsation is accompanied by outgoing and incoming Cumulative Virtual Cloud (CVC). Another possible explanation of bunched trajectory of the e lectron is the interaction of pair/bracketleftbig F− ↑⊲ ⊳F+ ↓/bracketrightbig with Bivacuum oscillations (BvO), excited in bivacuum spontaneously or by torsion and curling magnetic ﬁ elds. The BvO may be generated also by [ C⇋W] pulsations of other particles, including those of two-slit screen. The energy of resonant bivacuum Bv O (ABvO,see eqs. 12 and 14) may be absorbed by symmetric pair of sub-eleme ntary par- ticles/bracketleftbig F− ↑⊲ ⊳F+ ↓/bracketrightbig in their wave [W] phase in triplets, turning them back to [C] phase. Interaction of BvO with [C] phase of/bracketleftbig F− ↑⊲ ⊳F+ ↓/bracketrightbig may perturb their properties, i.e. increasing momentum and kinetic ene rgy. The BvO may change, consequently, the frequency of their [ C⇋W] pulsation near resonance conditions (see eq.24). It is a consequence of our model, that the energy and momentum of the electron and positron (18 and 18b), are determined mostl y by uncom- pensated sub-elementary particle ( F− ↓). These parameters are related with 13change of similar parameters of pair/bracketleftbig F− ↑⊲ ⊳F+ ↓/bracketrightbig due to conservation of sym- metry of properties of each sub-elementary particle/antip article in triplets .It means that properties of uncompensated sub-elementary fer mion ( F− ↓) and, consequently, the whole particle may be modulated by the out coming and incoming Bivacuum oscillations (BvO) in a course of [ C⇋W] pulsation of pair/bracketleftbig F− ↑⊲ ⊳F+ ↓/bracketrightbig . 2.4. The electric and magnetic components of electromagnet ic charge The CVC, representing [W] phase is composed from Virtual Density Waves (VDW) , responsible for electric component ( i) of elementary electro- magnetic charge and from Virtual Symmetry Waves (VSW) , related to magnetic component ( η) of resulting elementary charge. In contrast to VDW, accompanied by real energy reversible change (eq.27c), the VSW (eq.27d) are excited by oscillations of negative mirror energy in a cours e of [C⇋W] pul- sation. We relate ( i) to real mass ( m+ C) and ( η) to mirror mass ( m− C). The product of two components is equal to resulting charge squar ed i×η=e2=α/planckover2pi1c (28) The electromagnetic ﬁne structure constant is: α=e2//planckover2pi1c=e2/Q2, where the total charge we deﬁne like: Q= (/planckover2pi1c)1/2. 2.5. Uniﬁcation of electromagnetism and gravitation (Kaivarainen, 2000; 2001) We deﬁne the maximum of electromagnetic potential as the inter- nal interaction energy between electric and magnetic fract ions of elementary charge on the distance, determined by the mass-dipole radiu s (26): Emax el=i×η L±=e2 L±= (29) =α/parenleftbig m+ C−m− C/parenrightbig c2=αm+ Cv2=α2Tk (29a) It leads from equations obtained, that small part of W-phase energy in form of CVC, determined by electromagnetic ﬁne structure co nstant ( α) as a factor, is responsible for [ Emax el] at the region of particle localization, deter- mined by its Compton radius. 14The doubled real kinetic energy of elementary particle in general case of its translational and rotational movement with angle frequency ( ωrot) on the orbit with radius ( Lrot) may be presented as: 2Tk=/parenleftbig m+ Cv2/parenrightbig tr+/parenleftbig m+ Cω2 rotL2/parenrightbig rot=m+ C(v2+ω2 rotL2 rot) (30) The maximum of gravitational potential of uncompensated sub- elementary particle, close to that of elementary particle, we deﬁne as the energy of gravitational attraction between real and corpus cular mass of [C] phase, separated by wave B hidden mass-dipole dimension (26 ): Emax G=Gm+ Cm− C L±=Gm2 0 L±= (31) =βG/parenleftbig m+ C−m− C/parenrightbig c2=/parenleftbig m+ V−m− V/parenrightbig c2=βG2Tk= (31a) =βGm+ C(v2+ω2 rotL2 rot) =Etr G+Erot G (31b) where the contribution to gravitation, related to translat ional component of mass is Etr G=βGm+ Cv2and contribution of particle rotation (torsion) is: Erot G=βGm+ Cω2 rotL2 rot. In contrast to Emax el, deﬁned by electromagnetic ﬁne structure constant (α=e2//planckover2pi1c), maximum of gravitational potential ( Emax G) is determined by introduced in our work (Kaivarainen, 1995, 2000) gravitational ﬁne struc- ture factor (βG=m2 0/M2 Pl). We assume that the electromagnetic and gravitational poten tials de- creases with distance ( R) like: [Eel(r)and E G(r)] ˜− →r /r (31c) where− →ris the unitary radius-vector. For the case of macroscopic body as a system of interacting at oms and molecules, the translational and rotational contribution s should be subdi- vided to internal (microscopic) and external (macroscopic ) subcontributions. The resulting gravitational potential of body, containing (i) par- ticles, with total mass ( M), rotating on orbit with radius ( Rext rot), will have a shape: − →EG=/bracketleftBig/parenleftbig Etr G/parenrightbigin+/parenleftbig Etr G/parenrightbigext/bracketrightBig +/bracketleftBig/parenleftbig Erot G/parenrightbigin+/parenleftbig Erot G/parenrightbigext/bracketrightBig = (31d) =− →r rβG/parenleftBiggi/summationdisplay/parenleftBig m+ Cv2/parenrightBigin i+Mv2 ext/parenrightBiggtr +− →r rβG/parenleftBiggi/summationdisplay/parenleftBig m+ Cω2 rotL2 rot/parenrightBigin i+M/parenleftbig ωext rotRext rot/parenrightbig2/parenrightBiggrot 15The contribution of internal translational and rotational (librational) dy- namics may be comparable or bigger than the external ones. Ou r theory predicts that the increasing of temperature of solid body ma y increase its gravitational potential due to activation of thermal dynam ics of atoms and molecules. For the other hand, interaction of molecules of body with ele ctromagnetic ﬁeld (photons), increasing their polarizability and, cons equently, Van der Waals interactions, will reduce molecular thermal dynamic s and the internal kinetic energy of body. It should reduce also its gravitatio nal potential. There are some experimental evidence, pointing, that the ab ove predictions of my theory are right. The gravitational factor ( βG=m2 0/M2 Pl) relates the mass symmetry shift/parenleftbig m+ C−m− C/parenrightbig with vacuum symmetry shift ∆ mV= (m+ V−m− V), which, in turn, is dependent on equilibrium constant between bivacuu m fermions of opposite spins ( KBV F↑⇋BV F↓=N+[BV F↑] N−[BV F↓]): ∆mV=/vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingle≡βG/parenleftbig m+ C−m− C/parenrightbig =N+mBV F↑−N−mBV F↓=mBV F/arrowbothv[N+−N−] (32) This shift is accompanied by the local [ BV F↑⇋BV F↓] equilibrium shift, leading to similar shift of [ neutrino ⇋antineutrino ]. It means that local gravitational potential may be regulate d by torsion (spin) ﬁeld, inﬂuencing on vacuum symmetry shift, as a resul t of rotational kinetic energy of particle or system of particles change in a ccordance to (31d). In our approach, the spin (torsion) ﬁeld (− →ES) may be deﬁned simply as a part of gravitational ﬁeld, dependent on rotational kin etic energy of body (internal and external). Using (31d), we get the formula of torsion ﬁeld: − →ET≡Erot G=/bracketleftBig/parenleftbig Erot G/parenrightbigin+/parenleftbig Erot G/parenrightbigext/bracketrightBig (32a) =− →r rβG/parenleftBiggi/summationdisplay/parenleftBig m+ Cω2 rotL2 rot/parenrightBigin i+M/parenleftbig ωext rotRext rot/parenrightbig2/parenrightBiggrot Like the gravitational ﬁeld, the torsion ﬁeld is not nonloca l. Only nonres- onant Bivacuum oscillations (BvO) are nonlocal in the scale of virtual Bose condensate, where they are excited. The inﬂuence of magnetic ﬁeld, generated by gravitating bod y, on bivacuum symmetry 16The [BV F↑⇋BV F↓] equilibrium may be changed also due to diﬀer- ence of interaction energy of curled magnetic ﬁeld (− →H),generated by rotat- ing body, with magnetic moments of virtual fermions (− →µV F) and virtual antifermions (− →µV aF): ∆− →EH G=− →H(N−− →µF− ↑−N+− →µF+ ↓) (33) This contribution to resulting gravitational potential is dependent on rel- ative orientation of vectors of gravitational ﬁeld polariz ation between two interacting mass [− →E(M←→m)] and− →Hand of course the value of− →H tension. Consequently, in the presence of magnetic ﬁeld, generated b y rotating body the resulting gravitational potential may be expresse d as a sum of three contributions: EG= [Etr G+Erot G]in,ext±∆EH G (33a) Contribution of magnetic ﬁeld to resulting gravitational p otential, deﬁned by (33), may have the opposite sign, than [ Etr G+Erot G]in,ext.It means, that magnetic ﬁeld, generated by body, may inﬂuence its gravitat ional potential and eﬀective mass, changing EG. This important result of our theory is in total accordance with Searl eﬀect, conﬁrmed in experimen ts of de Palma, Baurov (1998), Roshin and Godin (2000). These experiments a nd our theory point to possibility of extraction of ’free’ energy from secondary bivacuum due to its symmetry shift. The symmetry shift of primordial bivacuum in the absence of matter and ﬁelds is zero in accordance to our theory (see section 1.1). The Einstein’s theory of general relativity did not take int o account such factors as body rotation, its internal dynamics and generat ed by body mag- netic ﬁeld. 2.6. Interrelation between hidden and external parameters of elementary particles. Hidden Harmony as a Golden mean condition It is shown in our work, that the internal ( vin gr) and external ( v) group velocities of sub-elementary particles, uniﬁed with light velocity via corre- sponding phase velocities: vin grvin ph=vgrvph=c2(34) are interrelated with each other in a following manner: 17c vingr=1 [1−(v/c)2]1/4(35) at the Hidden Harmony conditions , when the internal (hidden) and external group and phase velocities are equal: vin gr=vext grand vin ph=vext ph (36) equation transforms (35) to simple quadratic equation: S2+S−1 = 0 (37) or:S (1−S)1/2= 1 (37a) with solution, corresponding to Golden Mean S= (v/c)2=v/vph= 0.618 (38) At the Golden Mean realization, the mass symmetry shift of un compensated sub-elementary particle is equal to the rest mass of element ary particle: /bracketleftbig/parenleftbig/vextendsingle/vextendsinglem+ C/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− C/vextendsingle/vextendsingle/parenrightbig =m0/bracketrightbigS(39) and:S= (v/c)2= (vgr/vph)in,ext=/bracketleftbigg2Tkin Etot/bracketrightbiggs = 0.618 (39a) where the total energy of wave B is a sum of kinetic and potenti al ones: Etot=Tkin+V. Using (39) and (24) we get, that the frequency of [ C⇋W] pulsation at Golden mean condition is determined by the rest mass of elementary particle: ωS 0=m0c2//planckover2pi1 (40) We deﬁne this fundamental frequency of elementary particle s pulsation asGolden mean frequency. Form0,equal to mass of rest of the electron, the Golden mean frequen cy is:ω0= 9.03·1020s−1.For quarks it is about three order higher. The latter is close to Golden mean frequency of τ−electrons, as a possible components of quarks in form of corresponding standing waves. 18The expressions for electromagnetic (29) and gravitationa l (31) potentials also change in corresponding way at Hidden Harmony conditio ns: ES el=αm0c2ES G=βm0c2(41) Similarity between ES Gand the energy of neutrino, as that of uncompen- sated bivacuum fermion (eq. 7), points to participation of neutri no/antineutrino in mechanism of gravitation. ¿From eq.(19), taking into account (40) we get the following expressions for real ( m+ C) and mirror ( m− C) mass at Golden mean conditions: /bracketleftbig m+ Cv2=m0c2/bracketrightbig2→(m+ C)2v4=m+ Cm− Cc4(42) or:/bracketleftbiggm− C m+ C/bracketrightbiggS =/parenleftbiggvS c/parenrightbigg4 =S2(42a) where :/bracketleftbig m+ C/bracketrightbigS=m0 S2and/bracketleftbig m− C/bracketrightbigS=m0S2(42b) 3. Bivacuum - Matter Interaction 3.1. Inﬂuence of bivacuum quantum oscillations on matter properties We put forward a hypothesis, that any kind of selected system, able to self-assembly, self-organization and evolution: from a toms to living organisms and from galactics to Universe - are tendin g to condition of Hidden Harmony (28), displaying in Golden Mean re- alization. Corresponding driving force may be named ”Harmonization Force (HF)” . This force is a consequence of minimization of diﬀerence be - tween external and internal action, i.e. minimization of re sulting action: ∆S=/vextendsingle/vextendsingleSin−Sext/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsinglem+ C/parenleftbig vin gr/parenrightbig2−m+ C(vext gr)2/vextendsingle/vextendsingle/vextendsinglet→0 (43) This is accompanied by: /vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle→m0 (43a) In accordance to our model (Kaivarainen, 1995; 2000), the in ternal (hid- den) kinetic energy is a constant, speciﬁc for each kind of su b-elementary particle: 2Tin k=m+ C/parenleftbig vin gr/parenrightbig2=const (44) 19Consequently, condition (43) may be achieved only by change of external kinetic energy: m+ C(vext gr)2.The mechanism of corresponding driving Harmo- nization force, in accordance to our theory, is related to in teraction of sub- systems: bivacuum andmatter due to inﬂuence of Bivacuum oscillations (BvO) with basic frequencies ( ωi 0), deﬁned by the rest mass of elementary particles (eq. 13), on frequency of [ C⇋W] pulsation of elementary particles, forming matter. Under the permanent action of BvO synchroni zation of total system [ bivacuum +matter ] occur. It is shown in theory of autooscillations of nonlinear systems with many degrees of freedom (matter in our case), that the resonance may take a place not only at the equality of exte rnal frequency (ωi 0of BvO in our case) and internal Golden mean frequency (ωi B=ωi 0) of [C⇋W] pulsation, but at following combinations of external and i nternal frequencies: pωi 0=qω(1) B (45) pωi 0=qω(1) B+rω(2) B p, q, r = 1,2,3...(integer numbers) The resonant energy exchange between interacting elements of nonlinear systems with many degrees of freedom, i.e. between nuclears and electrons, atoms and molecules may occur. The energy exchange between d iﬀerent degrees of freedom at resonance conditions may be displayed in coherent change of external kinetic energy of clusters of atoms and mo lecules and ﬁnally in acceleration of even macroscopic bodies, if the am plitude of BvO is big enough. The latter may be achieved as a result of bivacuum transitons excitation by the curled magnetic ﬁeld, like in Searl eﬀect ( Roshin and Godin, 2000). Except the deﬁnite relations between the frequency of exter nal and in- ternal frequencies (31), the resonant excitation of system (matter) needs certain geometrical conditions , which exclude possible compensation of the external ﬁeld action by diﬀerent elements of this system . The latter con- ditions means that resonant interaction of BvO with macromo lecules, like DNA and proteins, as well as with macroscopic systems may dir ect the evo- lution of such systems geometry to Golden mean, inducing the changes of not only in their dynamics, but also of spatial parameters. Cert ain demands to three-dimensional parameters of systems with dissipation , interacting with BvO, may be determined by the autooscillations regime conditions. The described directed inﬂuence of Bivacuum oscillations w ith funda- mental frequencies ( ωi 0) on elementary particles, their assembles in form of atoms and molecules, aﬀecting the dynamics and geometry of m icroscopic, mesoscopic and macroscopic systems - could be a physical background 20for realization of Principle of Least Action. 3.2. Uniﬁcation a of Electromagnetism and Gravitation with Time, Space and Mass Analysis of principle of Least action in Lagrange form and pr inciple of uncertainty in coherent form for free particle ( V= 0) leads us to formulation of pace of time ( dt/t=dlnt) as a measure of the system’s real kinetic energy pace of change: dlnt=−dlnTkin=dlnm+ C+ 2dlnL+(46) where the real kinetic energy T+ kin=/planckover2pi12/2m+ CL2is related with space parameter - the radius of wave B length as: L+=/planckover2pi1/m+ Cv It is easy to show, that at permanent velocity v=const, the real mass m+ Cand real space L+are also constant and, consequently, the time: t= const. Increasing of wave B length of elementary systems (particle s) means in- creasing the probability of their Bose condensation and uni ﬁcation. At crit- ical values of wave B length, the process, like 1st order phas e transition of matter, takes a place (Kaivarainen, 1995, 2000). Taking into account (29) and (31), we get from (46) simple, symmetric and very important formula of uniﬁcation of temporal ﬁeld change of any closed system with changes of its electromagne tic and gravitational potential, mass and space: dlnt=−dlnTkin=−dlnEel=−dlnEG=dlnm+ C+ 2dlnL+(47) Resistance of Bivacuum Symmetry to Perturbation, as Reason of Inertia The inertial property of real mass (m+ C) of [C] phase in our model is a consequence of bivacuum symmetry reaction to real kinetic e nergy increasing (m+ Cv2) and corresponding increasing of CVC energy of [W] phase, ne cessary to keep [ C⇋W] equilibrium. Such tendency to keep dynamic [ C⇋W] equilibrium of sub-elementary particles, corresponding t o certain bivacuum symmetry shift, in spite of external perturbation of this eq uilibrium, we termed generalized principle of Le Chatelier (Kaivarainen, 2000) . The bivacuum symmetry resistance to perturbation, respons ible for in- ertia, is a distant, but not nonlocal eﬀect. In contrast to no nlocal Mach’s 21principle, our theory explains the existence of inertial ma ss even for only one particle in the empty Universe. We do not need to apply in o ur the- ory to mass producing Higgs-like ﬁelds also. The equality of inertial and gravitational mass leads naturally from our theory of gravi tation and inertial mass. 4. Inﬂuence of Matter on Bivacuum Properties. 4.1. Positive and negative Casimir eﬀects, Virtual Jet Gene rator The decreasing of bivacuum virtual pressure in space betwee n close and parallel conducting plates, as respect to virtual pressure outside the plates, explains the Casimir attractive (posit ive) ef- fect. Increasing the Bivacuum oscillations (BvO) symmetry and ma king them more coherent in space between plates may be induced by the coherent [ C⇋W] pulsation of the electrons of conducting plates. It leads to decreasing of probability of uncompensated (asy mmetric) vir- tual transitons (VT) excitation. This determines the eﬀect ive value of den- sity/pressure of virtual particles or antiparticles betwe en conducting plates. In contrast to this situation, it is known that cavities with some special geometry, like two close hemispheres - increase the probabi lity of virtual par- ticles and antiparticles origination in bivacuum, leading to repulsion between hemispheres (Lamoreaux, 1997). We termed like phenomena as negative Casimir eﬀect (Kaivarainen, 2000). It was supposed, that each of two hemispheres or other asym- metric structures, like open cones, pyramids, etc. may serv e as Virtual Jet Generators (VJG), increasing the uncompensate d frac- tion of virtual pressure (VP). If so, they may be used for extr ac- tion of free energy from bivacuum and propulsion in bivacuum (Kaivarainen, 2000; 2001). Consequently, between subsyste ms: bivacuum and matter the feedback reaction is existing. 4.2. Virtual replica (VR) of condensed matter Three contributions to Virtual Replica (VR), generated by m at- ter, are introduced: 1)local - electromagnetic contribution in form of IR photons radiat ion of any bodies with absolute temperature T >0). This contribution is dissipating quickly with distance; 2)distant - vibro-gravitational in form of modulated resonant Bivacu um oscillations (BvO), generated the virtual pressure waves ( VPW). Another distant factors of VR are modulated energy of neutrino, as un compensated 22bivacuum fermions (BVF↑) and oscillation of [ neutrino ⇋antineutrino ] equilibrium constant. The/bracketleftbig BV F↑⇋BV F↓/bracketrightbig equilibrium constant, equal to that of [ neutrino ⇋ antineutrino ] (see eqs. 7a and 7b) is a function of bivacuum symmetry shift : KBV F↑⇋BV F↓=[BV F↑] [BV F↓]=Kν⇋/tildewideν=f(∆mV) (48) Corresponding equilibrium may be modulated by vibro-gravi tational and magnetic ﬁeld of matter; 3)nonlocal contribution to VR may be related to nonresonant pertur- bation of Bivacuum oscillations (BvO) without changing the scale of Virtual Bose Condensate, formed by BVF and BVB. The second and third type of VR represent superposition of N- dimensional standing vibro-gravitational waves (VGW) and modulated VP W, correspond- ingly. The N is a number of virtual degrees of freedom in bivac uum, excited by matter in bivacuum. Consequently, VR has the N-dimension alhologram properties. In this point our theory is close to ideas, devel oped by Bohm. At resonant conditions BvO are responsible for energy, dens ity of virtual particles (virtual transitons) and virtual pressure, prov iding the feedback inﬂuence of BVR on matter, including living organisms. All k inds of BVR are the result of coherent [ C⇋W] transitions of quasisymmetric pairs [ F− ↑⊲ ⊳ F+ ↓]∗of triplets/angbracketleftbig [F− ↑⊲ ⊳F+ ↓]∗+F− ↑/angbracketrightbig , modulated by molecular dynamics of condensed matter, related with properties of uncompensate d sub-elementary fermion ( F− ↑). The resulting Pointing vector (− →Pres e−m) of quasisymmetric pair [ F− ↑⊲ ⊳ F+ ↓]∗, in contrast to ideally symmetric one [ F− ↑⊲ ⊳F+ ↓],is nonzero, because the electromagnetic components of CVC in former case do not c ompensate each other totally: − →Pres e−m=− →PF− ↑ e−m+− →PF+ ↑ e−m= [E×H]F− ↑+ [E×H]F+ ↑/negationslash= 0 (48a) This inequality determines the diﬀerence between energy an d pressure of virtual particles and antiparticles, due to asymmetry of Bv O, generated by coherent [ C⇋W] pulsation of [ F− ↑andF+ ↓]∗. The bigger is gravitational potential of body , the bigger is induced by it resulting bivacuum symmetry shift: EG˜ ∆mVc2and absolute value of excessive/vextendsingle/vextendsingle/vextendsingle− →Pres e−m/vextendsingle/vextendsingle/vextendsinglein (48a). The 3D spatial structure of body and its composition may be re- sponsible for value and sign of Casimir eﬀect, decreasing or increasing the 23resulting virtual pressure of bivacuum, generated by gravi tating body. Ro- tation of this body will increase the above eﬀect as a result its particles kinetic energy increasing. The electromagnetic ﬁeld , generated by rotating body may increase the probability of transitions between bivacuum energy sub levels, density of virtual particles/antiparticles and resulting (noncompe nsated) virtual pres- sure around the body. Our notion of bivacuum symmetry shift and its consequences h ave some similarity with notion of Polarizable Vacuum (PV), introdu ced by Puthoﬀ (1999). The PV approach means, in principle, the possibilit y of space-time metric ”engineering” by changing vacuum permittivity: ε0→Kε0and per- meability: µ0→Kµ0, where Kis a variable vacuum dielectric constant (Puthoﬀ, Little and Ibison, 2000). If K >1 (K≈1 + 2GM/rc2in solar system), this decreases the values of light velocity, frequ ency of photons, en- ergy, length, pace of time and increases the mass of body, as r espect to the values of same parameters of body in the absence of gravitati onal ﬁeld, when K= 1. In terms of our theory K >1,corresponds to positive vacuum symmetry shift and K= 1 corresponds to vacuum symmetry shift equal to zero. At condition K < 1 (negative vacuum symmetry shift) all the above listed parameters, related to space-time metric, change in opposi te direction as respect to K >1. It leads from our Hierarchic theory of condensed matter (see http://arXiv.org/abs/physics/0003044 that each of 24 collective quantum excitations, introduced in new theory, is characterized by speciﬁc coherent oscillations and corres ponding averaged kinetic energy ( Ti kin).Each of these contributions to resulting gravitational potential may be evaluated separately. Our Hierarchic theo ry of condensed matter in combination with described model of duality allow s the quantita- tive evaluation of distant component of vacuum replica: Vib ro - Gravitational Replica (VGR). 4.3. Main features of Hierarchic theory of condensed matter A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed (Kaivarainen, 2000a). It was as sumed, that anharmonic oscillations of particles in any condensed matt er lead to emer- gence of three-dimensional (3D) superposition of standing de Broglie waves of molecules, electromagnetic and acoustic waves. Consequen tly, any condensed matter could be considered as a gas of 3D standing waves of cor responding nature. Our approach uniﬁes and develops strongly the Einst ein’s and De- bye’s models. 24Collective excitations in form of coherent clusters, repre senting at certain conditions the mesoscopic molecular Bose condensate, were analyzed, as a background of hierarchic model of condensed matter. The most probable de Broglie wave (wave B) length is determin ed by the ratio of Plank constant to the most probable impulse of molec ules or by ratio of its most probable phase velocity to frequency. The waves B are related to molecular translations (tr ) and librations (lb). As the qua ntum dynamics of condensed matter does not follow in general case the class ical Maxwell- Boltzmann distribution, the real most probable de Broglie w ave length can exceed the classical thermal de Broglie wave length and the d istance between centers of molecules many times. This makes possible the atomic and molecular mesoscopic Bose condensation in solids and liqui ds at temperatures, below boiling point. It is one of the most important results of new theory, which we have conﬁrmed by computer sim ulations on examples of water and ice and the Virial theorem. Four strongly in- terrelated new types of quasiparticles (collective excita tions) were introduced in our hierarchic model: 1.Eﬀectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2Convertons , corresponding to interconversions between trandlbtypes of the eﬀectons (ﬂickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trand lbeﬀectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons and [lb ⇋tr]convertons. Primary eﬀectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating i ons, atoms or molecules. The volume of eﬀectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The ﬁrst condit ion means validity of classical approximation in description of the s ubsystems of the eﬀectons. The second one points to quantum properties of coh erent clusters due to molecular Bose condensation. It leads from our computer simulations, that liquids are sem iclassical systems because their primary (tr) eﬀectons contain less th an one molecule and primary (lb) eﬀectons - more than one molecule. The solid s are quan- tum systems totally because both kind of their primary eﬀect ons (tr and lb) are mesoscopic molecular Bose condensates. It is shown, tha t the 1st order [gas→liquid ] transition is accompanied by strong decrease of libration al (rotational) degrees of freedom due to emergence of primary (lb) eﬀectons. In turn, the [ liquid→solid] transition is followed by decreasing of transla- 25tional degrees of freedom due to molecular mesoscopic Bose- condensation in form of primary (tr) eﬀectons. In the general case the eﬀecton can be approximated by paral- lelepiped with edges corresponding to de Broglie waves leng th in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. The in-phase oscillations of molecules in the eﬀectons corr espond to the eﬀecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the eﬀectons diﬀer in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The ( a→b) or (b→a) transition states of the primary eﬀectons (tr and lb), deﬁned as primary transitons, are acco mpanied by a change in molecule polarizability and dipole moment withou t density ﬂuc- tuation. In this case the transitions lead to absorption or r adiation of IR photons, respectively. Superposition of three internal standing IR photons, penet rating in dif- ferent directions (1,2,3) - forms primary electromagnetic deformons (tr and lb). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density ﬂuctuations, leading to absorption or radiation of phonons . Superposition of standing phonons in three directions (1,2,3), forms sec- ondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary eﬀectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromagnetic deformons ,lead to origina- tion of macroeﬀectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeﬀec tons in the volume of superimposed trandlbelectromagnetic deformons lead to orig- ination of supereﬀectons. In turn, the coherent excitation of both: tr andlbmacrodeformons and macroconvertons in the same volume mean s cre- ation of superdeformons. Superdeformons are the biggest (cavitational) ﬂuctuations, leading to microbubbles in liquids and to loca l defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, re- ﬂects all of possible combinations of the four basic ones [1- 4], introduced above. This set of collective excitations - is proved to be ab le to explain virtually all the properties of condensed matter. It is quan titatively veriﬁed on examples of water and ice in wide T-interval: 5-373 K, usin g new the- ory based computer program (copyright, 1997, Kaivarainen) . Our hierarchic concept creates a bridge between micro- and macro- phenomen a, dynamics and thermodynamics, liquids and solids in terms of quantum p hysics. 264.4. Modulation of matter-induced Bivacuum oscillations by Vibro-Gravitational Waves (VGW). The vibro-gravitational waves Ai V GWwith frequency of order: νV GW˜ 1012s−1, related with thermal vibrations of atoms and molecules of co ndensed matter, modulate the high-frequency [ C⇋W] pulsation ( νC⇋W˜ 1021s−1˜ω0) of elementary particles. In turn, the matter-generated and thermally mod ulated Bivac- uum oscillation (BvO) pattern superimpose with basic BvO of Golden mean quantized frequency ( ω0): (n+1 2)ω0= (n+1 2)m0c2//planckover2pi1 (49) Corresponding resulting superposition contains informat ion about matter properties and may be termed ”Virtual replica (VR)” of matter . For each of 24 selected collective excitation of condensed m atter, con- sidered in our Hierarchic theory of condensed matter (Kaiva rainen, 2000b), the averaged thermal vibrations contribution to gravitati onal potential of particles, can be evaluated: Ai V GW=β2Ti kin (50) The equation for total internal kinetic energy of condensed matter is a sum of contributions of each of 24 excitation. It may be calcu lated, using our computer program (Kaivarainen, 1995; 2000a). The most eﬀective source of vibro-gravitational waves (VGW ) are coher- ent clusters, existing in liquids ( librational primary eﬀectons ) and solids (librational and translational primary eﬀectons ) as a result of high - temperature mesoscopic Bose condensation. Primary transitons , represent- ing transition state between optic (b) and acoustic (a) mode s of the primary eﬀectons and convertons - transition states between primary librational and translational eﬀectons also may generate VGW and vibro- gravitational replica (VGR) in bivacuum. Due to coherency of VGW, excited i n bivacuum by listed kind of excitations, they may form a hologram-like system of stand- ing waves - VGR. Other excitations of condensed matter are no t so coherent. Their VGW can not form standing waves and their VGR are not sta ble. This means that corresponding ’memory’ of bivacuum is very short . Taking this into account, the energy of vibro-gravitationa l waves (as a part of CVC energy), generated by one mole of condensed matte r may be calculated (Kaivarainen, 2000a): 27AV GW= 2β(Ttot kin) = 2β[Teff kin+Tt kin+Tcon kin] = (51) =βV02 Z/summationdisplay tr,lb/bracketleftBigg nef/summationtext(Ea)2 1,2,3 2Mef(va ph)2/parenleftbig Pa ef+Pb ef/parenrightbig/bracketrightBigg +/bracketleftBigg nt/summationtext(Et)2 1,2,3 2Mt(vress)2Pd/bracketrightBigg (52) +V0ncon Z/parenleftbig Eac/parenrightbig2 6Mc(vress)2Pac+/parenleftbig Ebc/parenrightbig2 6Mc(vress)2Pbc+/parenleftbig EcMd/parenrightbig2 6Mc(vress)2 In accordance to our model, between the mass of two sub-eleme ntary particles, forming coherent pair : [V+⊲ ⊳V−] and the third sub-elementary particle ( V±) oftriplet: /angbracketleftbig [V+⊲ ⊳V−] +V±/angbracketrightbig e−, e+ (53) the direct correlation is existing. Such correlation is a re sult of highly correlated dynamics of sub-elementary particles, composi ng particles. The mass, charge and other properties of elementary particle ar e determined by uncompensated sub-elementary particle ( V±) of triplet. The sum of contributions, generated by [C⇋W] pulsations of number (i) of coherent pairs of sub-elementary particles: [ V+⊲ ⊳V−], to amplitude of basic Golden mean bivacuum oscillations ( ABvO) - determines the amplitude of matter-induced bivacuum gap oscillations (BvO): [ABvO(t)]mat=/summationdisplay i/parenleftbig ∆m+ V+ ∆m− V/parenrightbigmatc2(54) Corresponding instant resulting amplitude of energetic la ndscape of bi- vacuum gap is a sum of instant values of basic BvO amplitude [ ABvO(t)] and matter-induced BvO [ Amat BvO(t)], modulated by thermal vibrations of atoms and molecules with gravitational contribution [ βACV C V GW(t)]: Ares BvO(t) =ABvO(t) +/bracketleftBigg/summationdisplay iAmat BvO(t) +βACV C V GW(t)/bracketrightBiggmat (54a) The energetic landscape, determined by Ares BvO(t) may be very unsmooth, depending on kinetic energy distribution of particles and p airs [V+⊲ ⊳V−] in composition of oscillating atoms and molecules of condense d matter . The N-dimensional superposition of thermally modulated ma tter-generated Bivacuum oscillations/bracketleftbigg/summationtext iAmat BvO(t)/bracketrightbigg and virtual pressure waves (VPW), ex- cited by basic Bivacuum oscillations (BvO), may have the N-d imensional 28hologram properties. We termed this hologram as Virtual rep lica (VR) of condensed matter. It reﬂects the matter dynamic properties . Any changes of these properties are accompanied by change of VR, i.e. holomovement after Bohm. 4 .5. Diﬀerent components of biological cells as a possible v irtual jet generators Our Hierarchic Model of Consciousness - HMC (Kaivarainen, 2 000c) is based on Hierarchic theory of condensed matter (Kaivaraine n, 1995; 2000b). In accordance to this theory, coherent properties of water c lusters in micro- tubules (MT) and distant exchange by IR photons, radiated by these clusters (mesoscopic molecular Bose condensate - MBC) may be respons ible for dis- tant interaction between MT of diﬀerent neurons and neuron e nsembles with similar orientation of MT. In accordance to our HMC, each speciﬁc kind of neuron ensembl es exci- tation - corresponds to hierarchical system of three-dimen sional (3D) stand- ing waves of following interrelated kinds: thermal de Brogl ie waves (waves B), produced by anharmonic vibrations of molecules; electr omagnetic (IR) waves; acoustic waves and vibro-gravitational waves (Kaiv arainen, 2000b). Corresponding complex hologram may be responsible for dist ant quantum neurodynamics regulation and for morphogenetic ﬁeld. In our model we consider quantum collective excitations, re sulted from coherent anharmonic translational and librational oscill ations of water in the hollow core of the microtubules. It was shown, that water fra ction, related to librations, represent mesoscopic molecular Bose condensa te (MBC) in form of coherent clusters. The dimensions of water clusters (nan ometers) and frequency of their IR radiation may be enhanced by interacti on with walls of MT. It is most organized and orchestrated fraction of conden sed matter in biological cells. The Brownian eﬀects, which inﬂuence reor ientation of MT system and probability of cavitational ﬂuctuations, stimu lating [gel - sol] transition in nerve cells - may be responsible for non-compu tational element of consciousness. Other models (Wigner, 1955 and Penrose, 1 994) relate this element to wave function collapse. Change of the ordered fraction of water in microtubules in fo rm of MBC, leads to [gel-sol] transition, related to reversible assem bly - disassembly of actin microﬁlaments, change of osmotic pressure, pulsatio n of cells volume and membranes deformation. Corresponding ”holomovement” of Virtual replica (VR) of living organism may be responsible for mind- matter in- teraction, telepathy and other phenomena, related to parap sychology. The 29bigger is number of MTs with similar orientation of coherent ly interacting cells, the bigger is corresponding fraction of ordered wate r, very sensitive to nerve excitation. There are evidence, pointing that spat ial properties of DNA and MTs follow the Golden mean rule (see web site of Dan Win ter http://www.danwinter.com/). In accordance to our results , it is a condition, optimal for exchange interaction of matter with bivacuum by Bivacuum os- cillations (BvO). Consequently, DNA, chromosomes, microtubules and bunches of MTs may serve as eﬀective virtual jet generators (VJG), increas ing virtual pres- sure in selected direction. It may be due to existing of libra tional eﬀectons and high frequency conversions between librational and tra nslational water eﬀectons in MT. The collective contribution of MT, related t o librational kinetic energy of coherent water, in Virtual replica of living organisms may be signiﬁcant. This contribution for one mole of water ma y be calculated like (Kaivarainen, 2000): (2Tlb k)in=V02 Z/summationdisplay lb/bracketleftBigg nef/summationtext(Ea)2 1,2,3 2Mef(va ph)2/parenleftbig Pa ef+Pb ef/parenrightbig/bracketrightBigg (55) The doubled kinetic energy of [lb/tr] convertons also may be evaluated: (2Tcon k)in=V0ncon Z/bracketleftBigg/parenleftbigEac/parenrightbig2 6Mc(vress)2Pac+/parenleftbigEbc/parenrightbig2 6Mc(vress)2Pbc+/parenleftbigEcMd/parenrightbig2 6Mc(vress)2PcMd/bracketrightBigg (56) The charged bilayer membranes of biological cells, includi ng neurons and axons have a properties of system of Casimir cham bers of variable geometry. At certain conditions (i.e. depolari zation) they may provide also the cumulative virtual jet eﬀect. At the ”rest” condition of cells the resulting concentratio n of internal anions of neurons is bigger than that of external ones, provi ding the diﬀerence of potentials equal to 50-100mV. As far the thickness of memb rane is only about 5nm or 50 ˚A it means that the gradient of electric tension is about: 100.000V/sm i.e. it is extremely high. Depolarization of membrane usual ly is related to penetration of Na+ions into the cell. The processes of depolarization, ac- companied by pulsation of nerve cell body, -change the properties of mem- branes as Casimir chambers and, consequently, the virtual r eplica of cell. The virtual replica of all cells, involved in nerve excitati on, in- cluding acupuncture points, change in-phase with correspo nding elementary acts of consciousness. 305. Possible mechanism of Bivacuum mediated Matter-Matter and Mind-Matter interaction The virtual replica (VR) of condensed matter (living organi sms in private case), may inﬂuence properties of uncompensated (eﬀec- tive) virtual pressure in following ways: 1) changing the amplitude of virtual pressure waves (VPW) in -phase with Bivacuum oscillations (BvO). This factor is dependent on fr action of coherent particles in system with in-phase [ C⇋W] transitions. The important role of Mind-Matter and Mind-Mind interaction is related to cohe rent fraction of water in microtubules in state of mesoscopic molecular Bo se condensate. This fraction is a variable parameter, speciﬁc for kind of el ementary act of consciousness; 2) changing bivacuum symmetry shift, related to [ BV F↑⇋BV F↓]≡ [neutrino ⇋antineutrino ] equilibrium shift, induced by magnetic ﬁeld of matter variation. Decreasing/increasing of the vacuum s ymmetry shift will be accompanied by decreasing/increasing of the eﬀecti ve uncompensated VPW energy; 3) shifting the Golden mean resonance conditions of [matter - bivacuum] interaction by exchange of BvO, as a result of spatial pertur bation of matter, changing the frequency of [ C⇋W] pulsations of its elementary particles. This factor may increase or decrease the amplitude of BvO and , consequently, the amplitude of virtual pressure waves (VPW). If geometry and other properties of matter provide the nega- tive (repulsion) Casimir eﬀect, corresponding virtual jet generation (VJG) increases the above described manners of inﬂuence of m at- ter on virtual pressure waves (VPW). The deviation of virtual replica (VR) in form of standing VPW from ”Virtual Noise” of bivacuum and a life-time of VR are de- pendent on the scale of coherent molecular/atomic excitati ons in biosystems, i.e. amplitude of VR and proximity of character is- tic frequencies of VR to fundamental Golden mean frequencie s of BvO, i.e. the eﬀectiveness of resonant matter-bivacuum ene rgy exchange. Combination of the described above three factors of virtual replica (VR), generated, for example by ’sender’: crystal, water or human Mind, may change a strong, weak and electromagnetic interaction betw een quarks, el- ementary particles, atoms and molecules of ’receiver’, its inertial mass and gravitational potential. 31Distant Mind-Matter interaction, including telekinesis, may be related to dependent on human will changes of cumulative virtual pressure waves (VPW) parameters. It is obvious that parameters of VR of Mind are much more varia ble than those of Matter. They are dependent of human will and are more adjustable for maximum of Mind-Matter interaction. In more conventional terminology (Puthoﬀ, Little, Ibison, 2000) the Mind activity may change vacuum dielectric constant to value ( ±∆K), vacuum permittivity to value ( ±∆ε0) and vacuum permeability to value ( ±∆µ0).In turn, these vacuum changes aﬀect the matter properties, exi sting in vacuum medium. Opposite by sign deviation of bivacuum properties from thos e, corre- sponding to bivacuum symmetry shift equal to zero (∆ mV= 0), as a result of complicated hierarchical processes (from microscopic t o macroscopic scale) may be resulted in opposite change of velocity of radioactiv e decay, process of phase transitions, kinetics of self-organization, rate of microorganisms divi- sion, deviations of random number generator data from norma l distribution and perturbation of many other cooperative collective proc esses. It is predictable, that the value of Lamb shift, dependent on virtual par- ticles screening of electromagnetic interaction in hydrog en like atoms, may be used as indicator of Mind-Matter interaction, if the mech anism proposed is right. However, the strong magnetic ﬁeld, used in radiosp ectroscopy ex- periments may ’screen’ the eﬀect, induced by Mind. Another p ossible exper- imental approach to detect bivacuum perturbation, related to Mind activity, is the precise measurement of Casimir eﬀect and its Mind indu ced variations. 6. Possible Mechanism of Mind-Mind Interaction (telepathy ) We assume, that the uncompensated due to bivacuum symmetry shift virtual pressure waves (VPW) , excited by Bivacuum oscillations (BvO) and dependent on their amplitude, may be responsible for dis tant subtle interaction between living organisms. The amplitude of BvO, modulated by matter is dependent on the : 1) number of coherent particles, participating in this spec iﬁc kind of in- teraction; 2) sharpness of resonance between [ C⇋W] pulsation of elementary particles of matter and fundamental frequencies of BvO, equ al to Golden mean frequencies ( ω0=m0c2//planckover2pi1)e,µ,τ.The latter factor is dependent, in turn, on spatial-dynamic properties of matter or neuron’s organe lles in private case. 32The VR of orchestrated system of nerve cells -’senders’ , excited by coherent water of their microtubules, membranes and synapt ic contacts and perturbations of this VR as a result of series of elementary a cts of con- sciousness induce corresponding changes in similar organe lles ofnerve cells -’receivers’. There are experimental evidence that cavita tional ﬂuc- tuations, accompanied by sonoluminescence, playing in our Hier- archic model of consciousness the important role in [gel-so l] transi- tions, are related to quantum vacuum radiation (virtual pre ssure) (Eberlein, 1996). As a result, the probability of the same co nse- quence of consciousness acts as in Mind - Sender increases in Mind of Receptor. Modulated by vibro-gravitational ﬁeld of microtubules and membranes the all-penetrating neutrino is another probable mediator of subtle Mind- Mind interaction. This modulation is most eﬀective in a cour se of nerve cells, including axons, excitation. Modulated by ’sender’ neutri no (a consequence of uncompensated bivacuum fermion), leads to correspondin g modulation of vacuum symmetry shift. Vacuum symmetry shift oscillation induce in-phase oscilla tion of eﬀec- tive,uncompensated virtual pressure in water structure of microtubules, membranes as Casimir chambers, etc. of ’receiver’. The charged virtual particles density oscillation change c orrespondingly the electromagnetic Van der Waals interaction between mole cules and the rate of physicochemical processes in nerve cells of ’receiv er’. Just this sec- ondary eﬀect of modulated neutrino is important for Mind-Mi nd interaction. In such a way the transmission of emotions and images from Mind sender to Mind receiver may occur. The closer are VR frequency- phase parameters of two or more interacting biosystems, the higher is prob- ability of resonant VPW exchange interaction between such s ystems. The Mind-Mind interaction, mediated by modulated BvO and ne utrino, may be much more speciﬁc than in Mind-Matter interaction. It may be most eﬀective, when interacting persons are geneti cally close, i.e. their nerve systems are tuned to each other on molecular , cell/subcell levels, generating very close Virtual Replicas and with sim ilar reaction on them. 7. Audio/Video Signals Skin Transmitter, based on Hierarch ic model of consciousness I proposed the idea of new device, where the laser beam with fr equency of cavitational ﬂuctuations and/or convertons and ultrawe ak intensity will be modulated by acoustic and/or video signals. The modulate d output optic 33signals will be transmitted from laser to the nerve nodes of s kin, using wave- guides. It is supposed that the nerve impulses, stimulated b y modulated laser beam, can propagate via complex axon-synapse system t o brain centers, responsible for perception and processing of audio and vide o information. The long-term memorizing process also can be stimulated eﬀe ctively by Skin Transmitter. The direct and feedback reaction between brain centers, res ponsible for audio and video information processing and certain nerve no des on skin is predictable. The coherent electromagnetic radiation of th ese nodes, including the acupuncture one can be responsible for so-called aura. One of the important consequence of our Hierarchic model of c onscious- ness is related to radiation of ultraviolet and visible phot ons (”biophotons”) as a result of water molecules recombination after their dis sociation. Disso- ciation can be stimulated by cavitational ﬂuctuation of wat er in the volume of superdeformons, inducing reversible disassembly of mic roﬁlaments and [gel-sol] transition. The frequency and intensity of this e lectromagnetic com- ponent of bioﬁeld, in turn, can aﬀect the kinetic energy of th e electrons, emitted by skin in the process of Kirlian eﬀect measurement. Our model predicts, that the above mentioned stimulation of psi-acti vity by resonant external radiation, should inﬂuence on colors and characte r of Kirlian pic- ture, taken even from distant untreated by skin-transmitte r points of human body. There are another resonant frequencies also, calcula ted from my Hier- archic theory of matter, enable to stimulate big ﬂuctuation s of water in MTs and their disassembly. Veriﬁcation of these important consequences of our model an d making a prototype of Audio/Video Signals Skin Transmitter is the in triguing task of future. The practical realization of Audio/Video Signals S kin Transmitter will be a good additional evidence in proof of HMC and useful f or lot of people with corresponding disabilities. The existence of distant Mind-Mind interaction may be prove d by encephalogram registration and Kirlian eﬀect. It is pred ictable, that application of audio/video signals skin transmitter ( Kaivarari- nen, 1999, 2000b), to acupuncture points, should be eﬀectiv e reg- ulator and stimulator of psi-abilities. REFERENCES Aspect A. and Grangier P. Experiments on Einstein-Podolsky - Rosen type correlations with pairs of visible photons. In: Q uan- tum Theory and Measurment. Eds. 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arXiv:physics/0103032v1 [physics.atm-clus] 12 Mar 2001LETTER TO THE EDITOR Spurious oscillations from local self-interaction correction in high energy photoionization calculations for metal clusters M.E. Madjet, Himadri S. Chakraborty §, and Jan-M. Rost Max-Planck-Institut f¨ ur Physik Komplexer Systeme, N¨ oth nitzer Strasse 38, D-01187 Dresden, Germany Abstract. We ﬁnd that for simple metal clusters a single-electron desc ription of the ground state employing self-interaction correction (SIC) in the framework of local- density approximation strongly contaminates the high ener gy photoionization cross sections with spurious oscillations for a subshell contain ing node(s). This eﬀect is shown connected to the unphysical structure that SIC genera tes in ensuing state- dependent radial potentials around a position where the res pective orbital density attains nodal zero. Non-local Hartree-Fock that exactly eliminates the electron self- interaction is found entirely free from this eﬀect. It is inf erred that while SIC is largely unimportant in high photon-energies, any implementation o f it within the local frame can induce unphysical oscillations in the high energy photo spectra of metal clusters pointing to a general need for caution in choosing appropria te theoretical tools. PACS numbers: 31.15.Ew, 36.40.Cg, 36.40.Vz The local-density approximation (LDA), along with its time -dependent version, is a standard theoretical technique to describe the structure a nd dynamics of large systems. From a practical standpoint, LDA is typically preferred to o ther conventional many- body methods (such as, Hartree-Fock (HF) or techniques base d on conﬁguration- interactions) because of its relatively low computational costs. In the context of the studies involving static and dynamical properties of si mple metal clusters LDA has proved to be particularly successful [1-3]. However, a w ell known drawback of LDA is that it only partially accounts for unphysical electr on self-interactions. As a consequence, the resulting potential for a ﬁnite system dec ays exponentially at large distance instead of producing the physical 1 /rbehavior. To render the long distance behavior of the LDA potential realistic, therefore, approx imation schemes have been suggested [4]. The most general and widely applied to remedy the error is the one proposed by Perdew and Zunger [5], which concerns an orbit-b y-orbit elimination of self-interaction, although the scheme immediately makes t he potential state-dependent. The self-interaction corrected LDA (LDA-SIC) improves rem arkably the vast variety of results related to many structural properties of physica l systems: for instance, §To whom correspondence should be addressed (himadri@mpipk s-dresden.mpg.de)Letter to the Editor 2 improvements in total energies of atoms, allowance for self -consistent bound solutions for negative ions, prediction of orbital energies that are c lose to electron removal energies thus restoring Koopmans’ theorem, ensuring dissociation o f heteronuclear molecules to neutral fragments, improvement of the band gap in solids etc . (a good account in this regard may be found through Ref. 4). In the dynamical regime t oo, especially in the context of low-energy photoionization of simple metal clus ters, the description of the electronic ground state via LDA-SIC results in important ma ny-body eﬀects including single electron Rydberg resonances [6,7]. At photon-energies well beyond the ionization threshold th e photospectrum shows special qualitative behavior. For spherical jellium clust ers over this energy range theory predicts a characteristic oscillation in the cross section with a frequency connected to the cluster diameter. The mechanism behind this oscillator y pattern is the interference of electron waves emanated from equivalent sites of the clus ter edge [8]. While there has been no experimental study on metal clusters, oscillati ons in the photoelectron intensity are indeed observed for fullerene molecules [9]. Generically, the high energy photoionization process should be rather sensitive to the d egree of accuracy in the description of the ground state. This can be understood from the fact that in an independent particle model the high-energy transition mat rix element has a leading contribution from the Fourier transform of the ground state wavefunction to the photoelectron momentum space (or retarded-momentum space if non-dipole interactions are included) [10]. From such an elementary viewpoint, LDA- SIC may also appear to be a suitable tool for the high energy photoionization stu dies of various cluster systems. However, this paper shows that while the correctio n for self-interaction is virtually unimportant in the study of energetic photoioniz ation of metal clusters, any approximation to it in a local frame can generate spurious os cillations in the cross section for photoelectrons emerging from subshells having orbital node(s). The point is illustrated by presenting calculations on Na 20, which can be well described by a spherical jellium model, and which is the smallest system (1 s21p61d102s2) having one subshell (2 s) with a node. The usual single electron potential in the Kohn-Sham LDA for malism is VKS(/vector r) =Vjel(/vector r) +Vd[ρ(/vector r)] +Vxc[ρ(/vector r)] (1) where the terms on the right-hand-side are respectively jel lium, direct (Hartree) Vd[ρ(/vector r)] =/integraltextd/vector r′ρ(/vector r′)/\|/vector r−/vector r′\|, and exchange-correlation potentials. The ground state electronic density ρ(/vector r) is deﬁned in terms of single-electron densities ρiand orbitals φi: ρ(/vector r) =N/summationdisplay i=1ρi(/vector r) =/summationdisplay i\|φi(/vector r)\|2 As mentioned earlier, an approximate prescription for SIC t o this LDA potential (1) is to eliminate orbitalwise from the outset those terms whic h represent an electron i interacting to itself. The resulting orbital-speciﬁc pote ntials, therefore, are Vi SIC(/vector r) =Vjel(/vector r) +/integraldisplay d/vector r′ρ(/vector r′)−ρi(/vector r′) \|/vector r−/vector r′\|+Vxc[ρ(/vector r)]−Vxc[ρi(/vector r)] (2)Letter to the Editor 3 As the exact form of Vxcis unknown a widely used scheme is to employ the formula[11]: Vxc[ρ(/vector r)] =−/parenleftBigg3ρ(/vector r) π/parenrightBigg1/3 −0.0333 log 1 + 11.4/parenleftBigg4πρ(/vector r) 3/parenrightBigg1/3  (3) The ﬁrst term on the right-hand-side in the above expression is exactly derivable by a variational approach from the HF exchange energy of a unifor m electron system with a uniform positively charged background; the second term is the so called correlation potential, a quantity not borne in HF formalism. We use LDA po tentials both with and without SIC approximation to calculate the dipole photo ionization cross sections upto approximately 1 KeV photon-energy for each subshell of the Na 20cluster in the independent particle frame †. Calculations are also performed in the self-consistent HF scheme to better identify the origin of the resulting discre pancy between the two LDA predictions. Quantities are in atomic units throughout, ex cept where speciﬁed otherwise. LDA and LDA-SIC cross sections for each of the 1 s, 1pand 1dsubshells are found to be almost identical at high enough energies showing a sing le monotonic oscillation. Results using HF for these subshells yield similar qualitat ive behavior. The situation, however, is quite diﬀerent for the 2 sphotoionization. Figure 1 presents 2 scross sections as obtained through LDA, LDA-SIC, and HF, as a funct ion of 2sphotoelectron momentum k2s=/radicalBig 2(E−I2s), withI2s(∼3.5 eV) being the 2 sionization threshold. Generally, in the low-energy range for all subshells of Na 20HF predictions are diﬀerent from LDA owing to the partly non-identical ground state corr elation they account for and this causes a constant phase diﬀerence between them at hi gher energies, where such correlation eﬀects are insigniﬁcant. Bearing this in m ind we ﬁnd in ﬁgure 1 that while LDA and HF again maintain the same trend oscillationwi se, LDA-SIC points to a progressively strong qualitative diﬀerence starting r oughly from 40 eV photon- energy. To identify closely the discrepancy between σ2swith and without SIC we have evaluated the Fourier transforms of σ2s(k2s) (see ﬁgure 2). Both LDA and HF are seen to have approximately the same Fourier spectrum with just on e peak. But LDA-SIC contains three additional peaks beside the one that is commo n to all three spectra. This common frequency is connected to the diameter of the clu ster. In fact, a simple theoretical analysis shows that high energy photo cross sec tions of a spherical jellium cluster oscillate in the respective photoelectron momentu m space at a frequency 2 Rc, whereRcis the cluster radius [8]. However, where do the other freque ncies in the LDA-SIC 2 scross section come from? In order to answer this we need to take a close look at the singl e-electron ground state LDA and LDA-SIC radial potentials. As pointed out earl ier, in LDA formalism “all” electrons of the system in the ground state experience the same potential deﬁned by equation (1) which for Na 20is denoted by the dotted curve in ﬁgure 3. The potential, as is typical for a cluster, is ﬂat in the interior region (reg ion of de-localized quasi-free electrons) while showing a strong screening at the edge arou ndRc; the unphysical †Of course at such high energy the Na+core will ionize. However, the inclusion of this eﬀect, goin g beyond the jellium frame, will not change our result qualita tively.Letter to the Editor 4 exponential decay at the long range may be noted. Switching t o the LDA-SIC scheme, electrons in every orbital now feel a distinctly diﬀerent po tential (see equation (2)) with an approximately correct long range behavior as represente d by four solid curves in ﬁgure 3. In this group of four SIC potentials the ones for 1 s, 1p, and 1dlook qualitatively similar to the LDA potential but are slightly deeper. The 2 spotential, on the other hand, exhibits a unique feature: a strong local variation ar ound the position r=Rn. To pin down how this structure in the 2 sLDA-SIC potential comes about we need to focus on the SIC exchange correction Vxc[ρ2s(/vector r)]. This quantity, with reference to expression (3), can be explicitly written as: Vxc[ρ2s(/vector r)] =−/parenleftBigg3ρ2s(/vector r π/parenrightBigg1/3 −0.0333 log 1 + 11.4/parenleftBigg4πρ2s(/vector r) 3/parenrightBigg1/3 (4) The 2sorbital density, ρ2s(/vector r) =\|φ2s(/vector r)\|2, in the above equation, vanishes at r=Rn as the 2sradial wavefunction passes through its node at Rn. Consequently, Vxc[ρ2s(/vector r)] generates a cusp-like structure in the neighborhood of Rnthat shows up in the LDA- SIC potential proﬁle for the 2 sorbital. Since the behavior here is directly connected to the zero in the 2 selectron density we stress that it must also occur in any alte rnate prescription for Vxc[ρ2s(/vector r)] diﬀerent from formula (2). We further emphasize that this structure is entirely an artifact of an externally imposed S IC in a purely local frame, which certainly is an approximation since a complete cancel lation of self-interactions requires an appropriate non-local treatment of the electro n-exchange phenomenon as in the HF formalism. In fact, a forced localization of the exc hange (Fock) term in the HF scheme does indeed produce an inﬁnite singularity in t he potential at the zero of the corresponding one-electron state function [12]. Nev ertheless, the structure from LDA-SIC has a direct bearing on the subsequent 2 sphotoionization matrix element by producing an unphysical oscillation. To behold the underlying mechanism let us consider the photo ionization dipole matrix element. We use for convenience the acceleration gau ge representation of the dipole interaction that involves the gradient of the potent ial seen by the outgoing electron. After carrying out angular integration with the a ssumptions of spherical symmetry and unpolarized light, one is left with a reduced ra dial matrix element for a dipole transition nl→ǫl′that in the acceleration formalism is < ψ ǫl′\|dV/dr \|ψnl>. Figure 4 shows that the derivatives of both the LDA potential and the LDA-SIC 2 s potential peak close to r=Rc. In fact, the ﬁrst derivative of any general cluster potential always peaks at the edge Rc, and therefore, the overlap integral in the radial matrix element has dominating contribution coming from the edge [8]. Further, for high enough energy ψǫl′can be described in the ﬁrst Born picture as a spherical wave w ith asymptotic form cos( knlr+δl′). This immediately suggests that the matrix element will oscillate in the knlspace with roughly a frequency that is equal to the distance o f the peak derivative point from the origin. As a consequence, resulting cross sections should exhibit an oscillation as a function of knlwith a frequence 2 Rc(since the cross section is the squared modulus of the matrix element). As men tioned before, thisLetter to the Editor 5 eﬀect is already known and can be related to the common freque ncy peak in ﬁgure 2. However, something additional happens for the LDA-SIC case . The structure induced by the wavefunction node in the LDA-SIC potential for the 2 sorbital produces a sharp discontinuity at RnindV2s/dr, as also seen in ﬁgure 4. Such a derivative-discontinuity induces a second oscillation in the respective overlap inte gral with a frequency about Rn [13]. Subsequently, the 2 scross section with SIC acquires four oscillation frequenci es: Rc−Rn, 2Rn,Rc+Rn, and 2Rc(see ﬁgure 2) as a result of the interference. Evidently, the ﬁrst three frequencies are artiﬁcial being connected to the unphysical structure in the potential. Non-local HF, which exactly eliminates elec tron self-interaction terms, is free from this eﬀect (as is seen from ﬁgures 1 and 2). Moreov er, the qualitative agreement of HF with LDA suggests that SIC is practically uni mportant at high enough photon-energy. This is simply because with predominant con tribution coming from the potential edge for higher energies any improvement in the as ymptotic behavior of the wavefunction does not signiﬁcantly inﬂuence the overlap in tegral. Therefore, for large systems where HF becomes computationally impracticable th e usual LDA may be a safe choice in the high-energy regime. On the other hand, the fact that slower photoelectrons with their longer wavelength can hardly “resolve” this noda l structure explains why low- energy cross sections in LDA-SIC are practically uncontami nated. It is also simple to understand that there is nothing special about 2 sphotoelectrons, for the eﬀect must also be present in the case of subshells having more than one n ode. The characteristic potential for any de-localized electro n system, as in a metal cluster, has a nearly ﬂat interior region. Any rapid variati on in this potential occurring in a small range can, therefore, have considerable eﬀect on t he photoionization overlap integral by signiﬁcantly altering the amplitude of the cont inuum wave across this range. For atomic systems, however, electrons are far more localiz ed owing to the strong nuclear attraction, and therefore, wavefunctions are far more comp act around the nucleus. The near-Coulombic shape of a typical atomic potential with ste ep slope close to the origin can practically overwhelm any local variation as the one dis cussed in this paper. In order to verify this, we applied LDA-SIC for some typical cases of a tomic photoionization without any problem. It is true that SIC in the LDA frame induces certain extensive ness in the calculations by making the potential state-dependent. One possible simp liﬁcation is to average over all such state-dependent potentials and use the averaged on e for all electrons. We applied such an average-SIC potential to examine whether or not the e ﬀect reduces. We ﬁnd that not only the eﬀect survives but that it also now substant ially aﬀects photoelectrons from subshells without a node because the wavefunction over lap across the nodal zone is rather strong for them since their ground state wavefunct ions are large in this region. However, it remains to be seen what happens if the potential i s further approximated by a simpliﬁed-implementation of SIC, namely, the optimize d eﬀective potential method [14]. Finally, it has recently been found in the context of at oms that the independent particle model breaks down for the high energy photoionizat ion due to the interchannel coupling eﬀect [15]. There is no apriori reason to assume that this will not be the caseLetter to the Editor 6 for cluster systems, although no study has yet been made. Nev ertheless, in the future even if a multi-channel frame (namely, the time-dependent L DA which is akin to the random-phase approximation) is needed to characterize the energetic photoionization of clusters, this spurious eﬀect will remain, at least qualita tively, and may also aﬀect those channels whose single channel description is otherwise err or-free. To summarize, we have shown that the theoretical analysis in the framework of LDA with SIC incorporated may invoke unphysical strong qual itative variations in high energy photospectra of metal clusters for electrons em itted from a subshell with node(s); although there is no denying that LDA-SIC is one of t he strong methodologies available to address low-energy processes. Through a compa rison with the results via non-local HF, that is intrinsically free from the self-inte raction error, we conclude that the diﬃculty is connected to an inexact footing of SIC in the L DA formalism. Hence, it is important to choose appropriate theoretical techniqu es suitable for a given energy range to avoid mis-interpretation of various eﬀects in clus ter photo-dynamical studies. We thank Professor Steven T. Manson of GSU-Atlanta, USA, for making useful comments on the manuscript. References [1] Calvayrac F, Reinhard P -G, Suraud E, and Ullrich C A 2000 Phys. Rep. 337493 [2] 1999 Metal Clusters , edited by Ekardt W (New York: Wiley) [3] Brack M 1993 Rev. Mod. Phys. 65677 [4] Perdew J P and Ernzerhof 1998 Electronic Density Functional Theory Recent Progress and N ew Directions , edited by Dobson John F, Vignale Giovanni and Das Mukunda P ( New York: Plenum Press) p 31 [5] Perdew J P and Zunger A 1981 Phys. Rev. B235048 [6] Madjet M E and Hervieux P A 1999 European Phys. J. D9217 [7] Pacheco J M and Ekardt W 1992 Z. Phys. D2465 [8] Frank Olaf and Rost Jan M 1996 Z. Phys. D3859; 1997 Chem. Phys. Letts. 271367 [9] Xu Y B, Tan M Q, and Becker U 1996 Phys. Rev. Letts. 763538; Liebsch T, Hentges R, R¨ udel A, Viefhaus J, Becker U, and Schl¨ ogl R 1997 Chem. Phys. Letts. 279197; Becker Uwe, Gessner Oliver and R¨ udel Andy 2000 J. Elec. Spect. Rel. Phen 108189 [10] Bethe Hans A and Salpeter Edwin E, in Quantum Mechanics of One- and Two-Electron Atoms (Plenum) 1977, p 299. [11] Gunnerson O and Lundqvist B I 1976 Phys. Rev. B134274 [12] Hansen M and Nishioka H 1993 Z. Phys. D2873 [13] Oscillation in the cross section from derivative disco ntinuity in the single electron potential is known in the context of atomic photoionization. Ref: Amusia M Ya, Band I M, Ivandov V K, Kupchenko V A, and Trzhashovskaya M B 1986 Iz. Akad. Nauk SSSR 501267; Kuang Y, Pratt R H, Wu Y J, Stein J, Goldberg I B, and Ron A 1987 J. Phys. (Paris) Colloq. 48C9-527; Zhou Bin and Pratt R H 1992 Phys. Rev. A456318 [14] Ullrich C A, Reinhard P -G, and Suraud E 2000 Phys. Rev. A62053202-1 [15] Chakraborty H S, Hansen D L, Hemmers O, Deshmukh P C, Fock e P, Sellin I A, Heske C, Lindle D W, and Manson S T 2001 Phys. Rev. A (slated for April issue), and references therein.Letter to the Editor 7 0 2 4 6 8k2s (a.u.)10−2010−1510−1010−5100σ2s (a.u.)LDA LDA−SIC HFNa20 Figure 1. Photoionization cross sections for 2 ssubshell as a function of 2 s photoelectron momentum calculated in LDA, LDA-SIC and HF ap proximations. 0 5 10 15 20 25 30 35 40r (a.u.)0.00e+00Fourier magnitude of σ2s (arb.u.)LDA LDA−SIC HFNa20Rc−Rn 2RnRc+Rn 2Rc Figure 2. Fourier spectra of the same cross sections presented in ﬁgur e 1.Letter to the Editor 8 0 5 10 15 20 r (a.u.)−0.4−0.3−0.2−0.10V(r) (a.u.)Na20 2s1p1s 1dLDA LDA−SIC Rn Rc Figure 3. Comparison among LDA and four state-dependent LDA-SIC radi al potentials. 0 5 10 15 20r (a.u.)−0.4−0.3−0.2−0.100.1V2s(a.u.) dV2s/drNa20 Rn RcLDA LDA−SIC Figure 4. LDA and LDA-SIC-for-2 spotentials and their derivatives.
arXiv:physics/0103033v1 [physics.bio-ph] 13 Mar 2001Eﬀects of thermal ﬂuctuation and receptor-receptor intera ction in bacterial chemotactic signalling and adaptation Yu Shi∗ Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Abstract Bacterial chemotaxis is controlled by receptor conformati onal changes in response to the change of ambient chemical concentration. I n a statistical mechanical approach, the signalling is a thermodynamic ave rage quantity, de- termined by the temperature and the total energy of the syste m, including both ligand-receptor interaction and receptor-receptor i nteraction. The con- formation of a receptor dimer is not only inﬂuenced by whethe r it is bound to a ligand, but also inﬂuenced by the conformation-dependent interaction with its neighbors. This physical theory suggests to biology a ne w understand- ing of cooperation in ligand binding and receptor signallin g problems. How much experimental support of this approach can be obtained f rom the cur- rent available data? What are the parameter values? What is t he practical information for experiments? Here we make comparisons betw een the theory and recent experimental results. Although currently compa risons can only be semi-quantitative or qualitative , consistency can clearl y be seen. The theory also helps to sort a variety of data. PACS number: 87.10.+e,87.16.-b.05.20.-y ∗Email: ys219@phy.cam.ac.uk 1I. INTRODUCTION Bacterial chemotaxis refers to the phenomenon that a bacter ium such as Escherichia coli swims towards higher concentration of attractant and lower concentration of repellent [1–4]. This is because with the rate determined by the change of the a mbient chemical concen- tration, the motors switch between counterclockwise and cl ockwise rotations, consequently the cell switches between tumbling and running. The ratio be tween the frequencies of the two rotation modes is determined by the rate at which kinase C heA phosphorylates CheY, which binds the base of a motor. CheA phosphorylation rate is regulated by the receptor conformational state, which is inﬂuenced by ligand binding . The receptors are dimeric and is joined to a CheA dimer by a CheW dimer, furnishing a signallin g complex. Hence a receptor dimer can be regarded as a basic unit, as supported by the ﬁndi ng that a receptor dimer with a damaged subunit can still work [5]. Because of thermal ﬂuct uation, even in the absence of ligand binding, or in a fully adapted situation, there is sti ll a certain probability distribution of the receptor conformational states; microscopically a r eceptor dimer stochastically ﬂips between the two states. Attractant binding changes the prob ability distribution, causing the receptor dimer to be more likely in the state corresponding t o lower CheA phosphorylation rate. On a longer time scale, after an initial response to lig and concentration change, the activity of the system returns to the pre-stimulus level. A c areful consideration of such a basic picture already ﬁnds the ideas of statistical mechani cs necessary: with the presence of thermal ﬂuctuation, it is the probability distribution of t he the receptor states, rather than a deﬁnite state, that is monitored by ligand concentration c hange and monitors the motor rotation bias. However, this point is not universally appre ciated in biological literature. The chemotactic response is very sensitive [6], and it had be en conjectured that there might be cooperation between receptors or the signalling co mplex so that the signal could be ampliﬁed [7,3]. The fact that most of the receptors cluste r together at a pole of the cell provides further clues for cooperation between receptors [ 8,9]. It was found experimentally that the clustering of receptors was not to be favorable for c ounting statistics and that the 2receptor cluster does not favor a special end of the cell [10] . This is an indication that there is a special reason, which may well be to have the receptor-re ceptor interaction. With a detailed analysis on the possibility of cooperation b etween receptor dimers, we constructed a statistical mechanical theory to provide a pi cture of how the receptors cooper- ate through physical interaction and how the thermal ﬂuctua tion makes statistical mechanics important in the signalling process [11,12]. As will be stre ssed here, the ﬁrst message from this approach is an emphasis on thermal ﬂuctuation. Moreove r, thermal ﬂuctuation helps to distinguish diﬀerent stimuli. Because of large separati on of time scales, the thermal ﬂuc- tuation can be treated as quasi-equilibrium, so equilibriu m statistical mechanical can give a reasonable response-stimuli relation. Hence the basic of our theory is useful no matter whether there is interaction between receptor dimers. The s econd message of this theory is that the anticipated cooperation is just physical recept or-receptor interaction between nearest-neighboring receptor dimers. Therefore the confo rmational state of a receptor dimer is not only inﬂuenced by ligand binding of itself, but also by the receptor-receptor inter- action which is dependent on conformations of the two neighb oring receptor dimers. The third message is that the large separation of time scales lea ds to a complementary usage of equilibrium statistical mechanics for the calculation of r esponse in a shorter time scale and a non-equilibrium description of the adaptation in a longer time scale. Dynamics on the longer time scale determines whether randomness of ligand b inding is quenched or annealed on the shorter time scale of quasi-equilibrium state, as wil l be elaborated later on. In the high temperature limit, this does not make a diﬀerence on the average signalling. Based on some aspects of the theory [11], a numerical simulation was m ade [13]. Recently there appeared some experimental data which are mo re directly relevant for the many-body nature of the receptor cluster and the possible co operation [14–16]. Therefore it is interesting and important to make comparisons between the theory and the experi- mental results, testing the theory on one hand, and providin g some information on what experimental data are wanted on the other hand. However, we d o not expect the model in the current form can ﬁt perfectly all data on this complex sys tem, rather, what we provide 3is a theoretical framework amenable for reﬁnements. For exa mple, for simplicity, we have only considered the cooperation between the receptor dimer , while extensions to possible cooperations among other components at later stages of the s ignalling process, for exam- ple, CheA, CheY, CheZ and the switch complex, is straightfor ward if concrete information is available. The idea of receptor-receptor interaction br oadens the view on cooperation, which previously largely refers to the existence of more tha n one binding sites, and thus the occupancy is larger than that with one binding site, as descr ibed by the model presented by Hill a century ago [18]. For simplicity, we try to preserve th e scenario of one binding site, while the extension to the situation of more binding sites is straightforward if needed. Our strategy is to start with the minimum model. With improvement and simpliﬁcation, we ﬁrst synthesis vari ous aspects of the theory. Then we make comparisons with the experimental results, fol lowed by summary and discus- sions. II. THEORY Consider a lattice of receptor dimers, as shown in Fig. 1. Let the coordinate number beν, which is 6 for a honeycomb lattice and is 4 for a square lattic e. The exact coordinate number in reality is subject to experimental investigation s. The behavior of the system is determined by its energy function, or Hamiltonian, which ca n be written as H(t) =−/summationdisplay <ij>TijViVj−/summationdisplay iHiVi+/summationdisplay iWiVi. (1) Viis a variable characterizing the conformation of receptor d imeri, so it is likely the position of the receptor molecule with respective to a certain equili brium position. In the popular two-state approach, Viassumes one of two values V0orV1.Hiis the inﬂuence, or force, due to ligand binding and the modulation of methylation leve l,Hi= 0 if there is no ligand binding, while Hi=Hif there is a ligand binding. −HiViis the energy due to ligand binding, hence ligand binding causes the energy diﬀerence b etween the two conformations 4to make a shift of H(V1−V0).Wi(V0−V1) is the original energy diﬀerence between the two conformations. /angbracketleftij/angbracketrightdenotes nearest neighbouring pairs, −TijViVjis the interaction energy between the neighboring receptor dimers. For convenience, deﬁning Si= 2(Vi−V0)/∆V−1, where ∆ V=V1−V0, one transforms the Hamiltonian to H(t) =−/summationdisplay /angbracketleftij/angbracketrightJijSiSj−/summationdisplay iBi(t)Si+/summationdisplay iUiSi, (2) where Si= 1,−1 represents the two conformational states of the receptor d imer at site i, Jij=Tij∆V2/4,Bi=Hi∆V/2,Ui= ∆V W i/2−∆V2/summationtext jTij. We refer to Bias ﬁeld. For simplicity, it is assumed that Jij=JandUi=Uare independent of iandj.Bi= 0 if there is no ligand binding, while Bi=B=H∆V/2 if there is a ligand binding. Hence energy diﬀerence due to ligand binding between the two conformatio ns are 2 Bi.USirepresents the original energy in the absence of ligand binding. Eq. (1) and (2) can be justiﬁed as follows. It is reasonable to assume an interaction energy proportion al to ( Vi−Vj)2, which can be reduced to −TijViVj, with constant terms neglected and the terms proportional t oSiorSj included in/summationtext iUiSi. On the other hand, this assumption is simple enough to allow a feasible treatment which captures the essential features. From now on, we focus on Eq. (2). Suppose that before time t= 0, there is no ligand bound to the system, or there are bound ligands, but the syste m is fully adapted. Hence Bi(t <0) = 0. Afterward, at time t= 0, the occupancy, i.e. the fraction of receptor dimers with ligands bound, changes to c. Hence the occupancy change is δc=c. This means Bi(t= 0) = B0 i, with B0 i=  B,with probability c 0,with probability 1 −c(3) The occupancy cis determined by the ligand concentration L,c=L/(L+Kd), where the dissociation constant Kdis on a time scale during which the receptor has undergone man y ﬂips between diﬀerent conformations, hence it is an average and phenomenological quantity. 5On the other hand, through the modulation of methylation lev el by CheB and CheR, there is a negative feedback from the receptor state Sito the ﬁeld Bi, with a time delay tr. A simple quantitative representation of this feedback is dBi(t) dt=−σ[Si(t−tr)−m0], (4) where σ >0,m0is the pre-stimulus average of Si. If she likes, one might call this self-tuning. A remarkable feature of this system is the large separation o f time scales. Ligand bind- ing and conformation change occur within only millisecond, while overall time needed to complete the adaptation, through the slow modulation of met hylation level, is on the scale of many seconds to minutes [19,2]. We note that in most cases, ligand debinding is on a much longer time scale than ligand binding, seen as follows. Consider the kinetics of the following reaction L+R⇀↽RL, (5) where Rrepresents the receptor without ligand binding, while RLrepresents liganded re- ceptor. k+andk−are reaction rates for the binding and debinding, respectiv ely. The ratio between the time scales of debinding and binding is k+L/k −≡L/K d, where Kdis the dissociation constant. A typical value is Kd∼1.2µM[2]. Usually, Lis much larger, so the debinding time scale is much longer than the time scale of ligand binding and receptor conformational change. In extreme cases when Lis comparable to Kd, debinding time scale is comparable to binding time scale. With the large separation of time scales, the treatment unde r the above formulation becomes easier. One may discretize the time on the scale of ad aptation, according to the feedback delay time. tis thus replaced by an integer τ, which is the integer part of t/tr. On the other hand, each instant τis still very long compared with the time scale of conformati onal change . Hence the activity at each τis an average quantity m(τ), which can be calculated from the Hamiltonian in (2) by standard methods of statistic al mechanics. Note that the average activity mjust corresponds to the time scale of the measured quantitie s such as 6motor bias, longer than the very short period in which the rec eptor is in either of the two conformations, but shorter than the adaptation time. In mak ing the average, an important thing is that the randomness of the ﬁeld is usually quenched s inceL >> K d, and is annealed otherwise. In fact we obtain a generalized version of the so- called random-ﬁeld Ising model; in a conventional random-ﬁeld Ising model, the average ﬁeld vanishes, but it is generically non-zero in our model. In the long time scale, the ﬁeld change s because of feedback. It can be expressed as Bi(τ) =B0 i+M(τ), where M(τ) is an induced ﬁeld due to methylation modulation, M(τ) =−στ−1/summationdisplay k=0[m(k)−m0]. (6) Before being stimulated, m(τ <0) =m0is determined by U.m0= 0 if and only if U= 0.m= 0 means that each receptor is in either of the two conformati ons with equal probability, and thus the rates of counterclockwise and clo ckwise rotations of the motors are equal. In most cases, the randomness of B0 iis quenched, the general relation between m(τ) and δcis then m(τ) =2δc 1+exp[ −2β(νJm(τ)−θ(τ−1)σ/summationtextτ−1 k=τ0(m(k)−m0)+U+B)] +2(1−δc) 1+exp[ −2β(νJm(τ)−θ(τ−1)σ/summationtextτ−1 k=τ0(m(k)−m0)+U)]−1, (7) where β= 1/kBT,θ(x) is 1 if x≥0, and is 0 otherwise. On the other hand, when the ligand concentration is lower than Kd, the randomness of B0 iis annealed, it can be found that m=δc[eβ(f(m)+B)−e−β(f(m)+B)] + (1 −δc)[eβf(m)−e−βf(m)] δc[eβ(f(m)+B)+e−β(f(m)+B)] + (1 −δc)[eβf(m)+e−βf(m)], (8) where f(m) =νJm−θ(τ−1)σ/summationtextτ−1 k=0m(k) +U. m(τ= 0) corresponds to the response-stimulus relation, as usua lly referred to. After the step increase at τ= 0,m(τ) always decrease back towards the pre-stimulus value m0. This is the robustness of exact adaptation [20]. Practically the adaptation time is obtained when m−m0reaches the detection threshold m∗. 7The results can be simpliﬁed under the condition that the the rmal noise is so strong that βνJandβBare not large. Then both Eq. (7) and Eq. (8) can be simpliﬁed to m(τ≥0)−m0=βBδc 1−βνJ/parenleftBigg 1−βσ 1−βνJ/parenrightBiggτ , (9) with m0=βBU 1−βνJ. (10) 1−βνJrepresents the enhancement of response compared with non-i nteracting scenario. One may obtain the adaptation time t∗, after which m−m0is less than the detection threshold m∗: τ∗=logδc+ log(βB 1−βνJ)−logm∗ −ln(1−βσ 1−βνJ). (11) m∗can be related to the lower bound of detectable occupancy cha nge,δc∗by m∗=βBδc∗ 1−βνJ, (12) hence τ∗=logδc−logδc∗ −ln(1−βσ 1−βνJ). (13) At exact adaptation, setting m(τ) =m0, one may obtain the total induced ﬁeld due to methylation modulation M∗=Bc. Then for the next stimulus, suppose the occupancy changes from δctoδc+ ∆cat a later time τ1, it can be found that the result with the occupancy δc+ ∆cand the induced ﬁeld M∗is the same as that with the occupancy ∆ c and without M∗, that is, the previous occupancy change has been canceled by M∗, therefore the fully adaptation with ligand binding is equivalent to no ligand binding. So m(τ≥τ1) is given by the above relevant equations with τchanged to τ−τ1, and δcsubstituted by ∆c. One can thus simply forget the pre-adaptation history, and re-start the application of the above formulation with τ1shifted to 0. The cancellation holds exactly only under the assumption of small βνJandβB, which is likely the reality. The ﬁniteness of detection threshold further widens the practical range of its validit y. 8III. COMPARISONS BETWEEN EXPERIMENTS AND THE THEORY A. Clustering. The clustering was recently studied in greater details [16] . The observed clustering of receptors and the co-localization of the CheA, CheY, and Che Z with the receptors is a favor for the eﬀects of interactions. An in vitro receptor lattice formation was also observed (Ying and Lai, 2000). B. Response-stimulus relation. A basic prediction of our theory is the response-stimulus re lation. Note that the time scale of the response, corresponding to min our theory , is longer than the very short lifetime of the individual conformations, but is only transient on th e time scale of the adaptation process. A remarkable thing is that min our theory is measurable. Motor rotation bias was measured [14]. From this result we can obtain m, as follows. The population motor bias is b=fccw/(fccw+fcw), where fccwandfcware rates of counterclockwise and clockwise rotations, respectively. Suppose the value of bisr1for conformational state 1, and is r−1 for conformational state −1. Hence the the average bias should be: b=r1x+r−1(1−x), (14) where xis the average fraction of receptors with state 1. xis related to mbym=x−(1−x) = 2x−1. So if we know r1andr−1, we can obtain mfrom the average b. In literature, there is no investigation on r1andr−1. A simple assumption which is often implicitly assumed in literature is that r1= 1,r−1= 0, that is, state 1 corresponds to CCW, state −1 corresponds to CW. We follow this assumption here. But it should be kept in mind that an experimental investigation on r1andr−1would be very valuable. Therefore, for the time being, we use b=m+ 1 2, (15) 9Thus from the pre-stimulus value of b, one may determine m0, and thus βU. An empirical formula is b= 1−0.0012(rcd−360), where rcdis the absolute angular rate of change of direction of the cell centroid in degree ·s−1[14,24]. From [24], the pre-stimulus value of rcd is known as ∼600, so the pre-stimulus value of mbis∼0.712. Hence m0≈βU 1−βνJ≈0.424. (16) The occupancy change used in [14] was calculated from the con centrations by assuming that the ligand randomly binds one of two possible binding si tes: in addition to the site with Kd∼1.2µM, as widely acknowledged [19], there is another site with Kd∼70µM. This was based on an earlier attempt to have a better ﬁtting for the ada ptation time [21]. However, as told above, we try to make things as simple as possible in th e ﬁrst instance, so prefer to preserve the scenario of one binding site with Kd∼1.2µM. Actually with one binding site, as discussed later on, it seems that our theory can ﬁt th e adaptation time by choosing appropriate parameter values, thus improve the coherence b etween various data. So we ﬁrst transform the occupancy given in [14]. One has cJ=1 2(c1+c2), (17) where cJrepresents the occupancy used by Jasuja et al.,c1corresponds to dissociation constant K1= 1.2µM,c2corresponds to dissociation constant K2= 70µM. From cl= L/(L+Kl) forl= 1,2, one obtains the change of the occupancy δcl=KlδL (L+δL+Kl)(L+Kl), (18) where δLis the change of ligand concentration. Since δL << L , one may obtain δc1= 2δcJ/(1 +α), where α≈K1(L+K1)2/K2(L+K2)2. With L≈10µM,α≈1, one has δc1≈δcJ. Therefore under this condition, we may simply use the occup ancy used in [14]. Eq. (15) leads to the relation between the initial change of mand that of the motor bias, δb, δm= 2δb, (19) 10where δm=m(δc, τ= 0)−m0. So the data in Fig. 3 of [14] can be transformed to δm−δcrelation as shown in our Fig. 2. Unfortunately, it is notable that the data is limited to very low values of occ upancy change ! Nevertheless, a qualitative ﬁtting can be made. According Eq. (9), where τis set to 0, we ﬁt the data with a straight line δm=aδc. From the slop of the ﬁtting line, we obtain a=βB 1−βνJ≈10.49. (20) C. Adaptation time. Eq. (18) tells us that with a same concentration change, the o ccupancy change and thus the response decreases with the increase of pre-stimulus ligand concentration. Th is is veriﬁed by Fig. 7 of [21]. Eq. (11) predicts that the adaptati on time increases linearly with, but not proportional to, the logarithm of occupancy change. This is consistent with the available experimental results. It had been thought that th e adaptation time is proportional to the occupancy change [22,23,21]. We found that a logarith mic relation is also consistent with the current available data. As an example, using Kd= 3×10−7, we transform the better set of the data, the left plot, in Fig. 4 of [23] to the oc cupancy change. For accuracy, the data points at the highest and lowest concentration chan ges are dropped since they are close to the detection limit. and it is hard to recognize t he diﬀerence in adaptation time with the the other data points closest to them, though th e concentration changes are quite diﬀerent. The transformed data is shown in our Fig 3(a) . While there could be a proportional (not only linear) ﬁtting, as usually done, the y may be ﬁtted by a logarithmic relation, t∗=τ∗·tr=glog10δc+h, with g= 95.151 and h= 124 .0574. From Eq. (11), we have tr −log10(1−βσ 1−βνJ)=g. (21) and 11tr[log10δc∗] log10(1−βσ 1−βνJ)=h. (22) We use δc∗≈0.004 [21]. and suppose tr≈0.1s. Then one may ﬁnd βσ 1−βνJ≈0.0024 to 0 .0045. (23) where the ﬁrst value estimated from (21), and the second from (22). They are quite close, as an indication of the consistency of the theory. Furthermore our predicted logarithmic relation may explai n the discrepancy in analysis of data in Fig. 4 of [21] about a relation between the adaptati on time and the concentra- tion. The logarithm can simply decrease the predicted value of adaptation time, without resorting to the assumption of the existence of two binding s ites. We have tried to make a quantitative ﬁtting for the data in Fig. 4 of [21]. Using Kd= 1.2µM, we transform the ligand concentration to the occupancy change, as shown in ou r Fig. 4. To make better use of the data, we ignore data point for δc >0.95, because the ﬁniteness of detection threshold may cause uncertainty in deciding the adaptation time; the d ata for δc >0.95 show too large variation for so close values of δc. The ﬁtting straight line is t∗=τ∗·tr=glog10δc+h, withg= 156 .3513 and h= 114 .9912. From (21) and (22), one may ﬁnd βσ 1−βνJ≈0.0015 to 0 .0047. (24) Again, they are quite close. It is very impressive that (23) a nd (24) are very close, though they are obtained for diﬀerent sets of data. D. CheA activity. Bornhorst and Falke studied relative CheA activity and made analyses using Hill model with non-integer coeﬃcient [15]. Here we analyze the data fr om the viewpoint of our theory. Suppose S= 1,−1 correspond respectively to CheA activity A1andA−1. Then the average CheA activity is1 2(A1+A−1) +m 2(A1−A−1). Consequently the relative CheA activity, as measured in [15]. is 12R=(A1+A−1) + (A1−A−1)m(δc) (A1+A−1) + (A1−A−1)m(δc= 0)= 1−FL L+Kd, (25) where F=a E+aU B, with E= (A−1+A1/(A−1−A1)>0. Note that A−1> A1. It is constrained that for attractant binding, F≤1, since R≥0. Setting F= 0.95 and Kd= 20µM, we obtains a reasonable ﬁtting to Fig. 1 of [15]. as shown in o ur Fig. 4. Therefore E≈a(1 0.95−U B). (26) Combined with Eqs. (16) and (20), it tells that the ratio betw een the two levels of CheA activity is A−1/A1≈164.77.Very interestingly, this result of deduction is in good cons istence with the available experimental information that this rati o is more than 100 [2]. Again, this is an indication of the consistency of the theory. However, there is discrepancy in the ﬁtting. This may be beca use of high temperature approximation, and may be because of some other minor factor s not considered here for simplicity. IV. SUMMARY AND DISCUSSIONS We suggest that statistical mechanics is helpful and import ant in understanding receptor signalling and adaptation. We have made semi-quantitative comparisons between the theory and recent experiments to obtain estimations of parameter v alues. However, for such a complex system, we do not expect the ﬁtting is perfect. The th ermal ﬂuctuation in a cell is very strong, kBT≈4pN˙nm≈0.025eV, comparable to the energy scales, so we simplify the formulation by using high temperature approxi mation. Then Eqs. (9) and (10) essentially contain all the information we need. 1 −βνJcharacterizes the enhancement of signalling by receptor-receptor interaction. With this simpliﬁed formulation, we look at recent experimental results. From the data on pre-stimulus motor rotation bias [24], we obtain the pre-stimulus activity, as in Eq. (16), implying t hat there are approximately 70% receptor dimers are at the state corresponding the lower rat e of CheA autophosphorylation. 13Although the data of the response-stimulus relation are not very limited, from this we estimate that βB/(1−βνJ)≈10.5. Eq. (20), which characterizes the eﬀect of ligand binding. We study adaptation time for two diﬀerent sets of da ta [23,21], and ﬁnd the feedback strength compared with coupling, βσ/(1−βνJ), is approximately 0 .0024 to 0 .0045, or 0.0015 to 0 .0047, respectively. These numbers obtained from diﬀerent d ata or by using diﬀerent methods are impressively close, as a good sign of th e consistency of theory. From the data on the relative CheA activity [15], we obtain Eq. (26 ), which gives the relation between the two levels of CheA activity corresponding to the two conformations of the receptor dimer. Combined with other results, it tells that t he ratio between the two levels of CheA activity is A−1/A1≈164.77, in good consistence with the available experimental information on this ratio. We need an improvement of other already available data, espe cially we need a signiﬁ- cant increase of the range of occupancy change in response-s timulus relation. We also need a clearer relation between adaptation time and occupancy ch ange. More accurate mea- surement of A−1/A1can provide more accurate test and reﬁnement of the theory. M ore information is also needed on the relation between the confo rmational state and the relative rate of the two rotation modes of the motor. Independent determination of the dissociate constant is al so important. Most exciting experiments might be direct measurements of the conformati onal states V0,V1, and the coupling coeﬃcient Tij. First, a clariﬁcation on whether the conformational chang e is rota- tion or a vertical displacement is needed. For the former, V0andV1are angles, while H, the eﬀect of ligand binding, is a torque. For the latter, V0andV1are positions, while His a force. The receptor-receptor interaction can be determin ed by measuring the relation of force or torque on one receptor dimer and the conformations o f its neighbors. This would be a direct test of the conformation-dependent interaction. A determination of the geometry of the lattice is also interesting, from which we can obtain the value of βνJ, and consequently other parameter values. Our theory is entirely diﬀerent from Hill model. An integer H ill coeﬃcient is understood 14as the number of ligands bound to a receptor. A non-integer Hi ll coeﬃcient, as often used, is not clear conceptually though could be tuned to ﬁt the data . Nonetheless, from mean ﬁeld point of view, the eﬀect of receptor-receptor interact ion could be viewed as eﬀective additional ligand binding. Therefore from this perspectiv e, the conclusion of Bornhorst and Falke on limited cooperativity is consistent with strong th ermal ﬂuctuation in our theory. Here we specialize in chemotactic receptors, however, the t heory also applies to many other receptor systems. For example, state-dependent co-i nhibition between transmitter- gated cation channels was observed [25]. Clustering of GABA Areceptors and the decrease of aﬃnity was also studied [26], in a way similar to the analys es of Bohrnhost and Falke for chemotactic receptors, and can also be explained by our theo ry as an indication of receptor- receptor interaction and thermal noise. In many receptor sy stems, clustering, or called oligomerization, together with signalling, occurs as a res ponse to stimulus. This situation is dealt with elsewhere. In ﬁnishing this paper, let me list some new experiments anti cipated from the point of view of this theory. (1) Direct determination of conformati onal change due to interaction with another receptor dimer. (2) Independent determinatio n of dissociate constant using other methods. (3) Investigations on the responses corresp onding to ﬁxed conformational states, thus r1andr−1discussed above is determined. (4) Direct measurements on C heA and CheY activities. (5) More clariﬁcation on the relation betw een the receptor state and CheA activity. (6) Increasing the range of occupancy change in re sponse-stimulus relations, and more accurate determination of pre-stimulus occupancy and occupancy change. (7) More accurate determination on adaptation time as a function of b oth pre-stimulus occupancy and the occupancy change. (8) Quantitative determination o f the details of feedback due to change of the methylation level. 15Figure captions: Fig. 1. An illustrative snapshot of the conﬁguration of rece ptor dimers on a 50 ×50 square lattice. Up triangles represent the conformation state Si= 1, down triangles represent Si=−1, ﬁlled triangles represent binding a ligand, empty triang les represent no ligand binding. Fig. 2. Response-stimulus relation δm−δc. The data points are transformed from [14]. The range of receptor occupancy change is too small, so only q ualitative comparison is possible. The straight line is the least square ﬁtting δm= 10.49δc. Fig. 3. (a) Normal-normal plot of the relation between adapt ation time t∗and occupancy change δc. The data points are adopted from [21], with the concentrati on transformed to occupancy. (b) Normal-log plot of the same data, showing tha t they can be ﬁtted to a logarithmic relation. Fig. 4. Relation between adaptation time t∗and occupancy change δc. The data points are adopted from [21], with the concentration transformed t o occupancy. The straight line is the least square ﬁtting t∗= 156 .3513 log10δc+ 114.9912. Fig. 5. The relation between the relative CheA autophosphor ylation rate Rand ligand concentration L. The data points are adopted from [15]. The theoretical curv e isR= 1−FL L+Kd, with F= 0.95 and Kd= 20µM. 161 50150 17/BC /BC/BA/BD /BC/BA/BE /BC/BA/BF /BC/BA/BG /BC/BA/BH /BC/BA/BI /BC/BA/BJ /BC/BA/BK /BC/BA/BL /BD/BC /BC/BA/BC/BD /BC/BA/BC/BE /BC/BA/BC/BF /BC/BA/BC/BG /BC/BA/BC/BH /BC/BA/BC/BI /BC/BA/BC/BJ /BC/BA/BC/BK /BC/BA/BC/BL Æ /D1Æ /BR /BR /BR/BR/BR /BR/BR /BR/BR/BR /BR/BR/BR/BR /BR/BR/BR/BR/BR/BR /BR/BR/BR /BR/BR/BR /BR/BR /BR/BR/BR/BR/BR /BR/BR/BR/BR/BD18/BC /BD/BC /BE/BC /BF/BC /BG/BC /BH/BC /BI/BC /BJ/BC /BK/BC /BL/BC /BD/BC/BC/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD /D8 /A3Æ /BF /BF /BF/BF/BF /BF/BF /BF /BF/BF /BF /BF/BD19/BC /BD/BC /BE/BC /BF/BC /BG/BC /BH/BC /BI/BC /BJ/BC /BK/BC /BL/BC /BD/BC/BC/BC/BA/BD /BD /D8 /A3Æ /BF /BF /BF/BF/BF /BF/BF /BF /BF/BF /BF /BF/BD20/BC /BE/BC /BG/BC /BI/BC /BK/BC /BD/BC/BC /BD/BE/BC/BC/BA/BE /BC/BA/BF /BC/BA/BG /BC/BA/BH /BC/BA/BI /BC/BA/BJ /BC/BA/BK /BC/BA/BL /BD /D8 /A3Æ /BR /BR /BR /BR /BR /BR/BR/BD21/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD /BD/BA/BE/BD/CT/B9/BC/BL /BD/CT/B9/BC/BK /BD/CT/B9/BC/BJ /BD/CT/B9/BC/BI /BD/CT/B9/BC/BH /BC/BA/BC/BC/BC/BD /BC/BA/BC/BC/BD /BC/BA/BC/BD /CA/C4 /BF /BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF/BF /BF/BF /BF/BF/BD22REFERENCES [1] H. 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In the so-called raindrops calculation therein, overlap between diﬀerent d rops was actually not taken into account, and this calculation, even if done in a right way, wi ll not give the maximum concentration, as expected by the authors, but only an impro vement on the minimum one; what was thought to be the minimum one is that without con sideration of the over- lap eﬀect. Nevertheless, these details are not important, s ince most of the chemotactic receptors form a single cluster, while the assumption in thi s paper that ligand binding of one receptor equally change the activity of each receptor in a cluster would lead to a constant response to diﬀerent stimulus. [10] H. Berg and L. Turner, Proc. Natl. Acad. Sci. USA 92, 447 (1975). 23[11] Y. Shi and T. Duke, Phys. Rev. E 58, 6399 (1998); also obtainable at http://xxx.lanl.gov/abs/physics/9901052. We apologize that we have not drawn a pic- ture of a lattice, though it is apparent that a lattice model i s discussed therein. [12] Y. 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arXiv:physics/0103034v1 [physics.bio-ph] 13 Mar 2001Designer Gene Networks: Towards Fundamental Cellular Control Jeﬀ Hasty1, Farren Isaacs1, Milos Dolnik2, David McMillen1, and J. J. Collins1 August 20, 2000 1Center for BioDynamics and Dept. of Biomedical Engineering , Boston University, 44 Cummington St., Boston, MA 02215 2Dept. of Chemistry and Center for Complex Systems, Brandeis University, Waltham, MA 02454 Submitted to ChaosABSTRACT The engineered control of cellular function throu gh the design of synthetic genetic networks is becoming plausible. Here we show how a na turally occurring network can be used as a parts list for artiﬁcial network design, and h ow model formulation leads to computational and analytical approaches relevant to non linear dynamics and statistical physics. We ﬁrst review the relevant work on synthetic gene n etworks, highlighting the important experimental ﬁndings with regard to genetic swit ches and oscillators. We then present the derivation of a deterministic model describing the temporal evolution of the concentration of protein in a single-gene network. Bistabi lity in the steady-state protein concentration arises naturally as a consequence of autoreg ulatory feedback, and we focus on the hysteretic properties of the protein concentration as a function of the degradation rate. We then formulate the eﬀect of an external noise source which interacts with the protein degradation rate. We demonstrate the utility of such a formu lation by constructing a protein switch, whereby external noise pulses are used to switch the protein concentration between two values. Following the lead of earlier work, we show how th e addition of a second network component can be used to construct a relaxation oscillator, whereby the system is driven around the hysteresis loop. We highlight the frequency depe ndence on the tunable parameter values, and discuss design plausibility. We emphasize how t he model equations can be used to develop design criteria for robust oscillations, and ill ustrate this point with parameter plots illuminating the oscillatory regions for given param eter values. We then turn to the utilization of an intrinsic cellular process as a means of co ntrolling the oscillations. We consider a network design which exhibits self-sustained os cillations, and discuss the driving of the oscillator in the context of synchronization. Then, a s a second design, we consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In this case, we show how resonance can lead to the induction of oscil lations and ampliﬁcation of a cellular signal. Finally, we construct a toggle switch from positive regulatory elements, and compare the switching properties for this network with thos e of a network constructed using negative regulation. Our results demonstrate the utility o f model analysis in the construction of synthetic gene regulatory networks. 1Lead Paragraph Many fundamental cellular processes are governed by geneti c programs which employ protein-DNA interactions in regulating function. O wing to recent tech- nological advances, it is now possible to design synthetic g ene regulatory net- works. While the idea of utilizing synthetic networks in a th erapeutic setting is still in its infancy, the stage is set for the notion of engine ered cellular control at the DNA level. Theoretically, the biochemistry of the feedb ack loops associated with protein-DNA interactions often leads to nonlinear equ ations, and the tools of nonlinear analysis become invaluable. Here we utilize a n aturally occurring genetic network to elucidate the construction and design po ssibilities for syn- thetic gene regulation. Speciﬁcally, we show how the geneti c circuitry of the bacteriophage λcan be used to design switching and oscillating networks, an d how these networks can be coupled to cellular processes. Thi s work suggests that a genetic toolbox can be developed using modular design concepts. Such advancements could be utilized in engineered approaches to the modiﬁcation or evaluation of cellular processes. 1 Introduction Remarkable progress in genomic research is leading to a comp lete map of the building blocks of biology. Knowledge of this map is, in turn, fueling the stu dy of gene regulation, where proteins often regulate their own production or that of othe r proteins in a complex web of interactions. Post-genomic research will likely center on the dissection and analysis of these complex dynamical interactions. While the notions of prote in-DNA feedback loops and network complexity are not new [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], experimental advances are inducing a resurgence of interest in the quantitative de scription of gene regulation [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. These advances a re beginning to set the stage for 2amodular description of the regulatory processes underlying basic c ellular function [13, 26, 27, 28, 29, 34]. In light of nearly three decades of parallel p rogress in the study of complex nonlinear and stochastic processes, the project of quantit atively describing gene regulatory networks is timely. The concept of engineering genetic networks has roots that d ate back nearly half a cen- tury [30, 31]. It is relatively recent, however, that experi mental progress has made the design and implementation of genetic networks amenable to quantit ative analysis. There are two dominant reasons for constructing synthetic networks. Fir st, simple networks represent a ﬁrst step towards logical cellular control, whereby biolog ical processes can be manipulated or monitored at the DNA level [32]. Such control could have a s igniﬁcant impact on post- genomic biotechnology. From the construction of simple swi tches or oscillators, one can imagine the design of genetic code, or software, capable of p erforming increasingly elabo- rate functions [33, 34]. A second complementary motivation for network construction is the scientiﬁc notion of reduced complexity; the inherently red uctionist approach of decoupling a simple network from its native and often complex biologica l setting can lead to valuable information regarding evolutionary design principles [35 ]. Ultimately, we envision the implementation of synthetic ne tworks in therapeutic appli- cations. However, such a utilization depends on concurrent progress in eﬀorts to uncover basic genomic and interspecies information. For example, b road applicability will only arise with detailed information regarding tissue-speciﬁc promo ters, proteins, and genes. Likewise, quantitative network design is contingent on a ﬁrm understa nding of cellular diﬀerentiation and fundamental processes such as transcription, translat ion, and protein metabolism. More crucially, delivery is a major hurdle; without identiﬁable cell-speciﬁc recognition molecules, there is no method for introducing a network to a speciﬁc type of cell. Since, in many re- gards, therapeutic applications are somewhat premature, w e focus on the implementation of synthetic networks in less complicated organisms. The desi gn of synthetic circuits and opti- mization of their function in bacteria, yeast, or other plan t organisms should reveal nonlinear 3properties that can be employed as possible mechanisms of ce llular control. In this paper, we develop several models describing the dyna mics of the protein concen- tration in small self-contained synthetic networks, and de monstrate techniques for externally controlling the dynamics. Although our results are general , as they originate from networks designed with common gene regulatory elements, we ground th e discussion by considering the genetic circuitry of bacteriophage λ. Since the range of potentially interesting behavior is wide, we focus primarily on the concentration of the λrepressor protein. We ﬁrst show how bistability in the steady-state value of the repressor p rotein can arise from a single-gene network. We then show how an external noise source aﬀecting p rotein degradation can be introduced to our model, and how the subsequent Langevin equ ation is analyzed by way of transforming to an equation describing the evolution of a probability function. We then obtain the steady-state mean repressor concentration by so lving this equation in the long- time limit, and discuss its relationship to the magnitude of the external perturbation. This leads to a potentially useful application, whereby one util izes the noise to construct a genetic switch. We next show how the addition of a second network comp onent can lead to a genetic relaxation oscillator. We study the oscillator model in det ail, highlighting the essential de- sign criteria. We introduce a mechanism for coupling the osc illator to a time-varying genetic process. In the model equations, such coupling leads to a dri ven oscillator, and we study the resulting system in the framework of synchronization. W e illustrate the utility of such driving through the construction of an ampliﬁer for small pe riodic signals. Finally, we turn to the construction of a genetic toggle switch, and compare s witching times for our network with those of a network constructed using negative regulati on. 2 Background Many processes involving cellular regulation take place at the level of gene transcription [36]. The very nature of cellular diﬀerentiation and role-speciﬁ c interaction across cell types im- plicates a not yet understood order to cellular processes. V arious modeling approaches 4have successfully described certain aspects of gene regula tion in speciﬁc biological sys- tems [9, 12, 13, 14, 18, 24, 25, 38, 39]. It is only recent, howe ver, that designed network experiments have arisen in direct support of regulatory mod els [21, 22, 23]. In this section, we highlight the results of these experimental studies, and set the stage for the discussion of the network designs described in this work. For completeness, we ﬁrst discuss the basic concepts of prom oters and regulatory feedback loops [40, 41]. A promoter region (or, simply, a promoter) denotes a segment of DNA where an RNA polymerase molecule will bind and subsequently trans cribe a gene into an mRNA molecule. Thus, one speaks of a promoter as driving the trans cription of a speciﬁc gene. Transcription begins downstream from the promoter at a part icular sequence of DNA that is recognized by the polymerase as the start site of transcri ption. A chemical sequence of DNA known as the start codon codes for the region of the gene that is converted into amino acids, the protein building blocks. Feedback arises w hen the translated protein is capable of interacting with the promoter that drives its own production or promoters of other genes. Such transcriptional regulation is the typical method utilized by cells in controlling expression [42, 43], and it can occur in a positive or negativ e sense. Positive regulation, or activation, occurs when a protein increases transcripti on through biochemical reactions that enhance polymerase binding at the promoter region. Neg ative regulation, or repression, involves the blocking of polymerase binding at the promoter region. Proteins commonly exist as multi-subunits or multimers which perform regulat ory functions throughout the cell or serve as DNA-binding proteins. Typically, protein homod imers (or heterodimers) regulate transcription, and this fact is responsible for much of the n onlinearity that arises in genetic networks [19]. Recently, there have been three important experimental stu dies involving the design of synthetic genetic networks. All three employ the use of repr essive promoters. In order of increasing complexity, they consist of (i) a single autorep ressive promoter utilized to demon- strate the interplay between negative feedback and interna l noise [23], (ii) two repressive 5promoters used to construct a genetic toggle switch [22], an d (iii) three repressive promoters employed to exhibit sustained oscillations [21]. We now bri eﬂy review the key ﬁndings in these three studies. In the single gene study, both a negatively controlled and an unregulated promoter were utilized to study the eﬀect of regulation on variations in ce llular protein concentration [23]. The central result is that negative feedback decreases the c ell-to-cell ﬂuctuations in protein concentration measurements. Although the theoretical not ion of network-induced decreased variability is not new [44], this study empirically demonst rates the phenomenon through the measurement of protein ﬂuorescence distributions over a po pulation of cells. The ﬁndings show that, for a repressive network, the ﬂuorescence distri bution is signiﬁcantly tightened, and that such tightening is proportional to the degree to whi ch the promoter is negatively controlled. These results suggest that negative feedback i s utilized in cellular design as a means for mitigating variations in cellular protein conce ntrations. Since the number of proteins per cell is typically small, internal noise is thou ght to be an important issue, and this study speaks to issues regarding the reliability of cel lular processes in the presence of internal noise. The toggle switch involves a network where each of two protei ns negatively regulates the synthesis of the other; protein “ A” turns oﬀ the promoter for gene “ B”, and protein B turns oﬀ the promoter for gene A[22]. In this work, it is shown how certain biochemical parameters lead to two stable steady states, with either a hi gh concentration of A(low B), or a high concentration of B(lowA). Reliable switching between states is induced through the transient introduction of either a chemical or t hermal stimulus, and shown to be signiﬁcantly sharper than for that of a network designed wit hout co-repression. Additionally, the change in ﬂuorescence distributions during the switchi ng process suggests interesting statistical properties regarding internal noise. These re sults demonstrate that synthetic toggle switches can be designed and utilized in a cellular en vironment. Co-repressive switches have long been proposed as a common regulatory theme [45], an d the synthetic toggle serves 6as a model system in which to study such networks. In the oscillator study, three repressible promoters were u sed to construct a network ca- pable of producing temporal oscillations in the concentrat ions of cellular proteins [21]. The regulatory network was designed with cyclic repressibilit y; protein Aturns oﬀ the promoter for gene B, protein Bturns oﬀ the promoter for gene C, and protein Cturns oﬀ the promoter for gene A. For certain biochemical parameters, the “repressilator” was shown to exhibit self- sustained oscillations over the entire growth phase of the h ostE. coli cells. Interestingly, the period of the oscillations was shown to be longer than the bac terial septation period, suggest- ing that cellular conditions important to the oscillator ne twork were reliably transmitted to the progeny cells. However, signiﬁcant variations in oscil latory phases and amplitudes were observed between daughter cells, and internal noise was pro posed as a plausible decorrela- tion mechanism. These variations suggest that, in order to c ircumvent the eﬀects of noise, naturally-occurring oscillators might need some addition al form of control. Indeed, an im- portant aspect of this study was its focus on the utilization of synthetic networks as tools for biological inference. In this regard, the repressilato r work provides potentially valuable information pertaining to the design principles of other os cillatory systems, such as circadian clocks. These studies represent important advances in the engineer ing-based methodology of synthetic network design. In all three, the experimental be havior is consistent with pre- dictions which arise from continuum dynamical modeling. Fu rther, theoretical models were utilized to determine design criteria, lending support to t he notion of an engineering-based approach to genetic network design. These criteria include d the use of strong constitutive promoters, eﬀective transcriptional repression, coopera tive protein interactions, and similar protein degradation rates. In the immediate future, the con struction and analysis of a circuit containing an activating control element (i.e., a positive feedback system) appears to be a next logical step. In this work, we present several models describing the desig n of synthetic networks in 7prokaryotic organisms. Speciﬁcally, we will utilize genet ic components from the virus bacte- riophage λ. While other quantitative studies have concentrated on the switching properties of the λphage circuitry [9, 12, 18, 38], we focus on its value as a part s list for designing synthetic networks. Importantly, the biochemical reactio ns that constitute the control of λ phage are very well characterized; the fundamental biochem ical reactions are understood, and the equilibrium association constants are known [9, 46, 47, 48, 49, 50]. In its naturally occurring state, λphage infects the bacteria Escherichia coli (E. coli ). Upon infection, the evolution of λphage proceeds down one of two pathways. The lysispathway entails the viral destruction of the host, creating hundreds of phage progeny in the process. These progeny can then infect other bacteria. The lysogenous pathway involves the incorporation of the phage DNA into the host genome. In this state, the virus is abl e to dormantly pass on its DNA through the bacterial progeny. The extensive interest i nλphage lies in its ability to perform a remarkable trick; if an E. coli cell infected with a lysogen is endangered (i.e. exposure to UV radiation), the lysogen will quickly switch t o the lysis pathway and abandon the challenged host cell. The biochemistry of the viral “abandon-ship” response is a t extbook example [36] of cellular regulation via a naturally-occurring genetic swi tch. The lytic and lysogenic states are controlled by the croandcIgenes, respectively. These genes are regulated by what are known as the PRM(cIgene) and PR(crogene) promoters. They overlap in an operator region consisting of the three binding sites OR1, OR2, and OR 3, and the Cro and λrepres- sor (“repressor”, the cIproduct) protein actively compete for these binding sites. When the Cro protein (product of crogene) binds to these sites, it induces lysis. When repressor binds, lysogeny is maintained and lysis suppressed. When po tentially fatal DNA damage is sensed by an E. coli host, part of the cellular response is to attempt DNA repair t hrough the activation of a protein called RecA. λphage has evolved to utilize RecA as a signal; RecA degrades the viral repressor protein and Cro subsequently a ssumes control of the promoter region. Once Cro is in control, lysis ensues and the switch is thrown. 83 Bistability in a Single-Gene Network In this section, we develop a quantitative model describing the regulation of the PRMoperator region of λphage. We envision that our system is a DNA plasmid consistin g of the promoter region and cIgene. As noted above, the promoter region contains the three opera tor sites known as OR1, OR2, and OR3. The basic dynamical properties of this network , along with a categorization of the biochemical reactions, are as follows. The gene cIexpresses repressor (CI), which in turn dimerizes and binds to the DNA as a transcription factor . This binding can take place at one of the three binding sites OR1, OR2, or OR3. The binding aﬃnities are such that, typically, binding proceeds sequentially; the dimer ﬁrst b inds to the OR1 site, then OR2, and lastly OR3 [37]. Positive feedback arises due to the fact that downstream transcription is enhanced by binding at OR2, while binding at OR3 represses transcription, eﬀectively turning oﬀ production and thereby constituting a negative f eedback loop. The chemical reactions describing the network are naturall y divided into two categories – fast and slow. The fast reactions have rate constants of ord er seconds, and are therefore assumed to be in equilibrium with respect to the slow reactio ns, which are described by rates of order minutes. If we let X,X2, and Ddenote the repressor, repressor dimer, and DNA promoter site, respectively, then we may write the equilibr ium reactions X+XK1⇀↽X2 (1) D+X2K2⇀↽D1 D1+X2K3⇀↽D2D1 D2D1+X2K4⇀↽D3D2D1 where Didenotes dimer binding to the OR isite, and the Ki=ki/k−iare equilibrium constants. We let K3=σ1K2andK4=σ2K2, so that σ1andσ2represent binding strengths relative to the dimer-OR1 strength. The slow irreversible reactions are transcription and degr adation. If no repressor is bound 9to the operator region, or if a single repressor dimer is boun d to OR1, transcription proceeds at a normal unenhanced rate. If, however, a repressor dimer i s bound to OR2, the binding aﬃnity of RNA polymerase to the promoter region is enhanced, leading to an ampliﬁcation of transcription. Degradation is essentially due to cell grow th. We write the reactions governing these processes as D+Pkt→D+P+nX (2) D1+Pkt→D1+P+nX D2D1+Pαkt→D2D1+P+nX Xkx→ where Pdenotes the concentration of RNA polymerase, nis the number of repressor proteins per mRNA transcript, and α >1 is the degree to which transcription is enhanced by dimer occupation of OR2. Deﬁning concentrations as our dynamical variables, x= [X],x2= [X2],d0= [D], d1= [D1],d2= [D2D1], and d3= [D3D2D1], we can write a rate equation describing the evolution of the concentration of repressor, ˙x=−2k1x2+ 2k−1x2+nktp0(d0+d1+αd2)−kxx (3) where we assume that the concentration of RNA polymerase p0remains constant during time. We next eliminate x2and the difrom Eq. (3) as follows. We utilize the fact that the reactions in Eq. (1) are fast compared to expression and degr adation, and write algebraic expressions x2=K1x2(4) d1=K2d0x2= (K1K2)d0x2 d2=K3d1x2=σ1(K1K2)2d0x4 d3=K4d2x2=σ1σ2(K1K2)3d0x6 10Further, the total concentration of DNA promoter sites dTis constant, so that mdT=d0(1 +K1K2x2+σ1(K1K2)2x4+σ1σ2(K1K2)3x6) (5) where mis the copy number for the plasmid, i.e., the number of plasmi ds per cell. We next eliminate two of the parameters by rescaling the repr essor concentration x and time. To this end, we deﬁne the dimensionless variables /tildewidex=x√K1K2and/tildewidet= t(ktp0dTn√K1K2). Upon substitution into Eq. (3), we obtain ˙x=m(1 +x2+ασ1x4) 1 +x2+σ1x4+σ1σ2x6−γxx (6) where γx=kx/(dTnktp0√K1K2), the time derivative is with respect to /tildewidet, and we have suppressed the overbar on x. The equilibrium constants are K1= 5.0×107M−1and K2= 3.3×108M−1[9, 46, 48, 49], so that the transformation from the dimensio nless vari- able x to the total concentration of repressor (monomeric an d dimeric forms) is given by [CI]= (7 .7x+ 3.0x2) nM. The scaling of time involves the parameter kt, and since transcrip- tion and translation are actually a complex sequence of reac tions, it is diﬃcult to give this lump parameter a numerical value. However, in Ref. [58], it i s shown that, by utilizing a model for the lysogenous state of the λphage, a consistency argument yields a value for the product of parameters ( dtnktp0) = 87 .6 nM min−1. This leads to a transformation from the dimensionless time /tildewidetto time measured in minutes of t(minutes ) = 0.089/tildewidet. Since equations similar to Eq. (3) often arise in the modelin g of genetic circuits (see Refs. [53]) of this Focus Issue), it is worth noting the speci ﬁcs of its functional form. The ﬁrst term on the right hand side of Eq. (6) represents production o f repressor due to transcription. The even polynomials in xoccur due to dimerization and subsequent binding to the prom oter region. As noted above, the σiprefactors denote the relative aﬃnities for dimer binding t o OR1 versus that of binding to OR2 ( σ1) and OR3 ( σ2). The prefactor α >1 on the x4term is present because transcription is enhanced when the two op erator sites OR1 and OR2 are occupied ( x2x2). The x6term represents the occupation of all three operator sites, and arises 11in the denominator because dimer occupation of OR3 inhibits polymerase binding and shuts oﬀ transcription. For the operator region of λphage, we have σ1∼2,σ2∼0.08, and α∼11 [9, 46, 48, 49], so that the parameters γxandmin Eq. (6) determine the steady-state concentration of repressor. The parameter γxis directly proportional to the protein degradation rate, a nd in the construction of artiﬁcial networks, it can be utilize d as a tunable parameter. The integer parameter mrepresents the number of plasmids per cell. While this param eter is not accessible during an experiment, it is possible to design a p lasmid with a given copy number, with typical values in the range of 1-100. The nonlinearity of Eq. (6) leads to a bistable regime in the s teady state concentration of repressor, and in Figure 1A we plot the steady-state concent ration of repressor as a function of the parameter γx. The bistability arises as a consequence of the competition between the production of xalong with dimerization and its degradation. For certain pa rameter values, the initial concentration is irrelevant, but for those that more closely balance production and loss, the ﬁnal concentration is determined by the initial va lue. Before turning to the next section, we make one additional ob servation regarding the synonymous issues of the general applicability of a synthet ic network and experimental mea- surement. In experimental situations, a Green Fluorescent Protein (GFP) is often employed as a measurement tag known as a reporter gene. This is done by i nserting the gene encoding GFP adjacent to the gene of interest, so that the reporter pro tein is produced in tandem with the protein of interest. In the context of the formulati on given above, we can generalize Eq. (6) to include the dynamics of the reporter protein, ˙x=f(x)−γxx (7) ˙g=f(x)−γgg where f(x) is the nonlinear term in Eq. (6), γg=kg/(dTnktp0√K1K2), and the GFP con- centration is scaled by the same factor as repressor ( /tildewideg=g√K1K2). In analogy with the equation for x,kgis the degradation rate for GFP, and we have assumed that the n umber 12of proteins per transcript nis the same for both processes. This ability to co-transcrib e two genes from the same promoter and transcribe in tandem has two important consequences. First, since proteins are typically very stable, it is often desirable to substantially increase their degradation rate in order to access some nonlinear reg ime [21, 22]. Such a high degra- dation rate typically will lead to a low protein concentrati on, and this, in turn, can induce detection problems. The utilization of a GFP-type reporter protein can help to mitigate this problem, since its degradation rate can be left at a relative ly low value. Second, and perhaps more importantly, are the signiﬁcant implications for the g enerality of designer networks; in prokaryotic organisms, anyprotein can be substituted for GFP and co-transcribed, so th at one network design can be utilized in a myriad of situations. 4 A Noise-Based Protein Switch We now focus on parameter values leading to bistability, and consider how an external noise source can be utilized to alter the production of protein. Ph ysically, we take the dynamical variables xandgdescribed above to represent the protein concentrations wi thin a colony of cells, and consider the noise to act on many copies of this col ony. In the absence of noise, each colony will evolve identically to one of the two ﬁxed poi nts, as discussed above. The presence of a noise source will at times modify this simple be havior, whereby colony-to-colony ﬂuctuations can induce novel behavior. Noise in the form of random ﬂuctuations arises in biochemica l networks in one of two ways. As discussed elsewhere in this Focus Issue [52], internal noise is inherent in biochem- ical reactions, often arising due to the relatively small nu mbers of reactant molecules. On the other hand, external noise originates in the random variation of one or more of the externally-set control parameters, such as the rate consta nts associated with a given set of reactions. If the noise source is small, its eﬀect can often b e incorporated post hoc into the rate equations. In the case of internal noise, this is done in an attempt to recapture the lost information embodied in the rate-equation approximation. But in the case of external noise, 13one often wishes to introduce some new phenomenon where the d etails of the eﬀect are not precisely known. In either case, the governing rate equatio ns are augmented with additive or multiplicative stochastic terms. These terms, viewed as a random perturbation to the deterministic picture, can induce various eﬀects, most not ably, switching between potential attractors (i.e., ﬁxed points, limit cycles, chaotic attra ctors) [54]. In previous work, the eﬀects of coupling between an external noise source and both the basal production rate and the transcriptional enhancem ent process were examined [55]. Here, we analyze the eﬀect of a noise source which alters prot ein degradation. Since the mathematical formulation is similar to that of Ref. [55], ou r goal here is to reproduce the phenomenology of that work under diﬀerent assumptions. As i n Ref.[55], we posit that the external noise eﬀect will be small and can be treated as a rand om perturbation to our existing treatment; we envision that events induced will be interact ions between the external noise source and the protein degradation rate, and that this will t ranslate to a rapidly varying protein degradation embodied in the external parameters γxandγg. In order to introduce this eﬀect, we generalize the model of the previous section s uch that random ﬂuctuations enter Eq. (7) multiplicatively, ˙x=f(x)−(γx−ξx(t))x (8) ˙g=f(x)−(γg−ξg(t))g (9) where the ξi(t) are rapidly ﬂuctuating random terms with zero mean ( < ξi(t)>= 0). In order to encapsulate the independent random ﬂuctuations, we make the standard requirement that the autocorrelation be “ δ-correlated”, i.e., the statistics of the ξi(t) are such that < ξi(t)ξj(t′)>=Dδi,j(t−t′), with Dproportional to the strength of the perturbation, and we have assumed that the size of the induced ﬂuctuations is the s ame for both proteins. Since, in Eqs. (8) and (9), the reporter protein concentrati ongdoes not couple to the equation for the repressor concentration, the qualitative behavior of the set of equations may be obtained by analyzing x. We ﬁrst deﬁne a change of variables which transforms the 14multiplicative Langevin equation to an additive one. Letti ngx=ez, Eq. (8) becomes, ˙z=1 +e2z+ 22e4z ez+e3z+ 2e5z+.16e7z−γx+ξx(t) (10) ≡g(z) +ξx(t) Eq. (10) can be rewritten as: ˙z=−∂φ(z) ∂z+ξx(t) (11) where the potential φ(z) is introduced: φ(z) =−/integraldisplay g(z)dz (12) φ(z) can be viewed as an “energy landscape”, whereby zis considered the position of a particle moving in the landscape. One such landscape is plot ted in Fig. 1B. Note that the stable ﬁxed points correspond to the minima of the potential φin Fig. 1b, and the eﬀect of the additive noise term is to cause random kicks to the partic le (system state point) lying in one of these minima. On occasion, a sequence of kicks may en able the particle to escape a local minimum and reside in a new valley. In order to analyze Eq. (11), one typically introduces the pr obability distribution P(z, t), which is eﬀectively the probability of ﬁnding the system in a statezat time t. Then, given Eq. (11), a Fokker-Planck (FP) equation for P(z, t) can be constructed [56]. The steady-state solution for this equation is given by Ps(z) =Ae−2 Dφ(z)(13) where Ais a normalization constant determined by requiring the int egral of Ps(z) over all z be unity. Using the steady-state distribution, the steady-state mea n (ssm), < z > ss, is given by < z > ss=/integraldisplay∞ 0zAe−2 Dφ(z)dz (14) 15In Fig. 1C, we plot the ssm value of zas a function of D, obtained by numerically integrating Eq. (14). It can be seen that the ssm of z increases with D, corresponding to the increasing likelihood of populating the upper state in Fig. 1B. Figure 1C indicates that the external noise can be used to con trol the ssm concentration. As a candidate application, consider the following protein switch. Given parameter values leading to the landscape of Fig. 1B, we begin the switch in the “oﬀ” position by tuning the noise strength to a very low value. This will cause a high popu lation in the lower state, and a correspondingly low value of the concentration. Then at so me time later, consider pulsing the system by increasing the noise to some large value for a sh ort period of time, followed by a decrease back to the original low value. The pulse will ca use the upper state to become populated, corresponding to a concentration increase and a ﬂipping of the switch to the “on” position. As the pulse quickly subsides, the upper state rem ains populated as the noise is not of suﬃcient strength to drive the system across either ba rrier (on relevant time scales). To return the switch to the oﬀ position, the upper-state popu lation needs to be decreased to a low value. This can be achieved by applying a second noise pu lse of intermediate strength. This intermediate value is chosen large enough so as to enhan ce transitions to the lower state, but small enough as to remain prohibitive to upper-st ate transitions. Figure 1D depicts the time evolution of the switching proces s for noise pulses of strengths D= 1.0 and D= 0.1. Initially, the concentration begins at a level of ∼0.4µM, correspond- ing to a low noise value of D= 0.01. At 40 minutes, a noise burst of strength D= 1.0 is used to drive the concentration to a value of ∼2.2µM. Following this burst, the noise is returned to its original value. At 80 minutes, a second noise burst of strength D= 0.1 is used to return the concentration to its original value. 5 A Genetic Relaxation Oscillator The repressillator represents an impressive step towards t he generation of controllable in vivo genetic oscillations. However, there were signiﬁcant cell -to-cell variations, apparently arising 16from small molecule number ﬂuctuations [21, 35]. In order to circumvent such variability, the utilization of hysteresis-based oscillations has rece ntly been proposed [35]. In this work, it was shown how a model circadian network can oscillate reli ably in the presence of internal noise. In this section, we describe an implementation of suc h an oscillator, based on the repressor network of Section 3. The hysteretic eﬀect in Fig. 1A can be employed to induce osci llations, provided we can couple the network to a slow subsystem that eﬀectively drive s the parameter γx. This can be done by inserting a repressor protease under the control o f a separate PRMpromoter region. The network is depicted in Fig. 2A. On one plasmid, we have the network of Section 3; the repressor protein CI, which is under the control of the promoter PRM, stimulates its own production at low concentrations and shuts oﬀ the promot er at high concentrations. On a second plasmid, we again utilize the PRMpromoter region, but here we insert the gene encoding the protein RcsA. The crucial interaction is b etween RcsA and CI; RcsA is a protease for repressor, eﬀectively inactivating its ab ility to control the PRMpromoter region [57]. The equations governing this network can be deduced from Eq. (6) by noting the following. First, both RcsA and repressor are under the control of the sa me promoter, so that the functional form of the production term f(x) in Eq.(6) will be the same for both proteins. Second, we envision our network as being constructed from tw o plasmids – one for repressor and one for RcsA, and that we have control over the number of pl asmids per cell (copy number) of each type. Lastly, the interaction of the RcsA and repressor proteins leads to the degradation of repressor. Putting these facts together, an d letting ydenote the concentration of RcsA, we have ˙x=mxf(x)−γxx−γxyxy (15) =mxf(x)−γ(y)x ˙y=myf(x)−γyy 17where γ(y)≡γx+γxyy, and mxandmydenote the plasmid copy numbers for the two species. In Fig. 2B, we present simulation results for the concentrat ion of repressor as a function time. The nature of the oscillations can be understood using Fig. 1A. Suppose we begin with a parameter value of γ(y) = 4 and on the upper branch of the ﬁgure. The large value of repressor will then serve to activate the promoter f or the RcsA, and thus lead to its increased production. An increase in the RcsA acts as an a dditive degradation term for repressor (see Eq. 15), and thus eﬀectively induces slow motion to the right on the upper branch of Fig. 1A. This motion will continue until the r epressor concentration falls oﬀ the upper branch at γ(y)∼5.8. At this point, with the repressor concentration at a very low value, the promoters are essentially turned oﬀ. The n, as RcsA begins to degrade, the repressor concentration slowly moves to left along the l ower branch of Fig. 1A, until it encounters the bifurcation point at γ(y)∼3.6. It then jumps to its original high value, with the entire process repeating and producing the oscillation s in Fig. 2B. The oscillations in Fig. 2B are for speciﬁc parameter values ; of course, not all choices of parameters will lead to oscillations. The clariﬁcation o f the speciﬁc parameter values leading to oscillations is therefore important in the desig n of synthetic networks [21]. For proteins in their native state, the degradation rates γxandγyare very small, corresponding to the high degree of stability for most proteins. For exampl e, a consistency argument applied to a similar model for λphage switching [58] leads to γx∼0.004. However, using a temperature-sensitive variety of the repressor protein, γxcan be made tunable over many orders of magnitude. Other techniques, such as SSRA tagging or titration, can be employed to increase the degradation rate for RcsA. The copy numbers mxandmycan be chosen for a particular design, and the parameter γxy, which measures the rate of repressor degradation by RcsA, is unknown. In Fig. 3A, we present oscillatory regimes for Eq. (15) as a fu nction of γxandγy, and for two ﬁxed values of the parameter γxy. We see that the oscillatory regime is larger for 18smaller values of the parameter γxy. However, the larger regime corresponds to larger values of the degradation rate for RcsA. Interestingly, if we take t he native (i.e., without tuning) degradation rates to be γx∼γy∼0.005, we note that the system is naturally poised very near the oscillatory regime. In Fig. 3B, we present the oscil latory regime as a function of γxandγxy, and for two ﬁxed values of γy. The regime is increased for smaller values of γy, and, in both cases, small values of γxare preferable. Moreover, both Figs. 3A and 3B indicate that the system will oscillate for arbitrarily small values of the repressor degradation parameter γx. In Fig. 3C we depict the oscillatory regime as a function of t he copy numbers mxandmy, and for ﬁxed degradation rates. Importantly, one can adjus t the periodic regime to account for the unknown parameter γxy. Figure 3C indicates that, for oscillations, one should choose as large a copy number as possible for the plasm id containing the repressor protein ( mx). Correspondingly, one should design the RcsA plasmid with a signiﬁcantly smaller copy number my. We now turn brieﬂy to the period of the oscillations. If desig ned genetic oscillations are to be utilized, an important issue is the dependence of the osci llation period on the parameter values. In Fig. 4A we plot the oscillation period for our CI-R csA network as a function of the degradation parameter γy, and for other parameter values corresponding to the lower wedge of Fig. 3A. We observe that an increase in γywill decrease the period of oscillations. Further, since the cell-division period for E. coli is∼35−40 minutes, we note that the lower limit roughly corresponds to this period, and that, at the up per limit, we can expect four oscillations per cell division. The utilization of tuning t he period of the oscillations to the cell-division time will be discussed in the next section. In Fig. 4B, we plot the period as a function of the copy number mx. We observe that the period depends very weakly on the copy number. 196 Driving the Oscillator We next turn to the utilization of an intrinsic cellular proc ess as a means of controlling the oscillations described in the previous section. We will ﬁrs t consider a network design which exhibits self-sustained oscillations (i.e., with paramet ers that are in one of the oscillatory regions of Fig. 3), and discuss the driving of the oscillator in the context of synchronization. As a second design, we will consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In that case, we will sho w how resonance can lead to the induction of oscillations and ampliﬁcation of a cellular si gnal. We suppose that an intrinsic cellular process involves osci llations in the production of protein U, and that the concentration of Uis given by u=u0sinωt. In order to couple the oscillations of Uto our network, we imagine inserting the gene encoding repre ssor adjacent to the gene encoding U. Then, since Uis being transcribed periodically, the co-transcription of repressor will lead to an oscillating source term in Eq. (1 5), ˙x=mxf(x)−γxx−γxyxy+ Γ sin( ωt) (16) ˙y=myf(x)−γyy We ﬁrst consider parameter values as in Fig. 2, so that the con centrations xandyoscillate in the absence of driving. Here, we are interested in how the d rive aﬀects the “internal” oscillations. Although there are many interesting propert ies associated with driven nonlinear equations such as Eq. (16), we focus on the conditions whereb y the periodic drive can cause the dynamics to shift the internal frequency and entrain to t he external drive frequency ω. We utilize the numerical bifurcation and continuation pack age CONT [59] to determine the boundaries of the major resonance regions. These boundarie s are depicted in the parameter- space plot of Fig. 5A, where the period of the drive is plotted versus the drive amplitude. The resonance regions form the so-called Arnold tongues, wh ich display an increasing range of the locking period as the amplitude of drive is increased. Without the periodic drive, the period of the autonomous oscillations is equal to 14.6 minut es. As one might expect, the 20dominant Arnold tongue is found around this autonomous peri od. Within this resonance region, the period of the oscillations is entrained, and is e qual to the external periodic force. The second largest region of frequency locking occurs for pe riods of forcing which are close to half of the period of the autonomous oscillations. As a res ult of the periodic driving, we observe 1:2 locking, whereby the system responds with one oscillatory cycle, while the drive has undergone two cycles. Other depicted resonant reg ions (3:2, 2:1, 5:2, 3:1) display signiﬁcantly narrower ranges for locking periods. This sug gests that higher order frequency locking will be less common and probably unstable in the pres ence of noise. Outside the resonance regions shown in Fig. 5A one can ﬁnd a rich structur e of very narrow M:N locking regions with M and N quite large, together with quasiperiodi c oscillations. The order of resonances along the drive period axis is given by the Farey s equence [60], i.e. in between two resonance regions characterized by rational numbers, M 1:N1and M 2:N2, there is a region with ratio (M 1+M2):(N1+N2). The preceding notions correspond to the driving of genetic n etworks which are intrinsi- cally oscillating. We now turn to a network designed with par ameter values just outside the oscillatory region, and consider the use of resonance in the following application. Suppose there is a cellular process that depends critically on oscil lations of a given amplitude. We seek a strategy for modifying the amplitude of this process i f, for some reason, it is too small. For concreteness, consider a cellular process linked to the cell-division period of the host for our synthetic network. For E. coli cells at a temperature of ∼37 degrees C, this period is of order 35 −40 minutes. Using Figs. 3A and 4A, we can deduce parameter val ues that will cause a CI-RscA network to oscillate, when driven, with this period. The lower wedge of Fig. 3A implies that, for γxy= 0.1, we should design the network with values of γxandγy just below the lower boundary of the wedge. Fig. 4A implies th at, for γx= 0.1, a choice ofγy= 0.004 will yield oscillations with a period close to the cell-d ivision period. In order to stay outside the oscillatory region, we therefore choose γyjust below this value. Taken together, these choices will yield a network whereby oscill ations can be induced by cellular 21processes related to cell division. In Fig. 6A, we plot the dr ive versus response amplitudes (Γ vs Γ x) obtained from numerical integration of Eq. (16). We see tha t, depending on the prox- imity to the oscillatory region, oscillations are triggere d when the drive reaches some critical amplitude. In Fig. 6B, we plot the gain g≡(Γ + Γ x)/Γ as a function of the drive amplitude Γ, and observe that, for certain values of the amplitude of th e drive, the network can induce a signiﬁcant gain. 7 Harnessing the Lambda Switch The ability to switch between multiple stable states is a cri tical ﬁrst step towards sophis- ticated cellular control schemes. Nonlinearities giving r ise to two stable states suggest the possibility of using these states as digital signals to be pr ocessed in cellular-level computa- tions (see, for example, [33, 34]). One may eventually be abl e to produce systems in which sequences of such switching events are combined to control g ene expression in complex ways. In any such application, the speed with which systems make tr ansitions between their sta- ble states will act as a limiting factor on the time scales at w hich cellular events may be controlled. In this section, we describe a bistable switch b ased on the mechanism used by λ phage, and show that such a system oﬀers rapid switching time s. The genetic network of λphage switches its host bacterium from the dormant lysogeno us state to the lytic growth state in roughly twenty minutes [51 ]). As discussed in Section 2, the regulatory network implementing this exceptionally fast s witch has two main features: two proteins (CI and Cro) compete directly for access to promote r sites; and one of the proteins (CI) positively regulates its own level of transcription. H ere, we compare a synthetic switch based on the λphage’s switching mechanism to another two-protein switch (the toggle switch described in Ref. [22]), and numerically show that the λ-like system oﬀers a faster switching time under comparable conditions. To implement the synthetic λswitch, we use the plasmid described in Section 3, on which the PRMpromoter controls the expression of the λrepressor protein, CI. To this, 22we add a second plasmid on which the PRpromoter is used to control the expression of Cro. The operator regions OR1, OR2, and OR3 exist on each plas mid, and both proteins are capable of binding to these regions on either of the plasm ids. On the PRM-promoter plasmid, transcription of CI takes place whenever there is n o protein (of either type) bound to OR3; when CI is bound to OR2, the rate of CI transcription is enhanced. On the PR- promoter plasmid, Cro is transcribed only when operator sit e OR3 is either clear, or has a Cro dimer bound to it; either protein being bound to either OR 1 or OR2 has the eﬀect of halting the transcription of Cro. Letting yrepresent the concentration of Cro, the competition for ope rator sites leads to equations of the form ˙ x=f(x, y)−γxx, ˙y=g(x, y)−γyy. We derive the form of these equations by following the process described in Section 3. A s with the CI plasmid of that section, we have Eq. 1 describing the equilibrium reactions for the binding of CI to the various operator sites. To these, we add the reactions entai ling the binding of Cro, and the reactions in which both proteins are bound simultaneously t o diﬀerent operator sites: Y+YK3⇀↽Y2 (17) D+Y2K4⇀↽DY 3 DY 3+Y2β1K4⇀↽DY 3DY 2 DY 3+Y2β2K4⇀↽DY 3DY 1 DY 3DY 2+Y2β3K4⇀↽DY 3DY 2DY 1 DX 2DX 1+Y2β4K4⇀↽DY 3DX 2DX 1 DX 1+Y2β5K4⇀↽DY 3DX 1, where Yrepresents the Cro monomer, and Dp irepresents binding of protein pto the OR i site. For the operator region of λphage, we have β1≃β2≃β3∼0.08, and β4≃β5∼1 [46, 48, 49] The transcriptional processes are as follows. Transcripti on of repressor takes place when 23there is no protein (of either type) bound to OR3. When repres sor is bound to OR2, the rate of repressor transcription is enhanced, and Cro is tran scribed only when OR3 is either vacant, or has a Cro dimer bound to it. If either repressor or C ro is bound to either OR1 or OR2, the production of Cro is halted. These processes, alo ng with degradation, yield the following irreversible reactions, D+Pktx→D+P+nxX (18) DX 1+Pktx→DX 1+P+nxX DX 2DX 1+Pαktx→DX 2DX 1+P+nxX D+Pkty→D+P+nyY DY 3+Pkty→DY 3+P+nyY Xkdx→ Ykdy→ Following the rate equation formulation of section 3, we obt ain ˙x=mx(1 +x2+ασ1x4) Q(x, y)−γxx (19) ˙y=myρy(1 +y2) Q(x, y)−γyy where Q(x, y) = 1 + x2+σ1x4+σ1σ2x6+y2+ (β1+β2)y4+β1β3y6+σ1β4x4y2+β5x2y2. The derivatives are with respect to dimensionless time, wit h scaling as in Section 3; ˜t= t(ktxp0dTnx√K1K2), where ktxis the transcription rate constant for CI, and nxis the number of CI monomers per mRNA transcript. The integers mxandmyrepresent the plasmid copy numbers for the two species; ρyis a constant related to the scaling of yrelative to x. The parameters γxandγyare directly proportional to the decay rates of CI and Cro, re spectively; 24we will tune these values to cause transitions between stabl e states. The system exhibits bistability over a wide range of parameter values, and we plo t the null-clines in Fig. 7A. For comparison, we now consider the co-repressive toggle sw itch brieﬂy reviewed in sec- tion 2 [22]. This switch uses the CI and Lac proteins, where ea ch protein shuts oﬀ transcrip- tion from the other protein’s promoter region. The experime ntal design was guided by the model equations, ˙u=α1 1 +vδ−u (20) ˙v=α2 1 +/bracketleftbigg u (1+[IPTG ] K)η/bracketrightbiggµ−kv where uandvare dimensionless concentrations of the Lac and CI proteins , respectively, and the time derivatives are with respect to a dimensionless time: τ=kdt, with kd= 2.52h−1[12, 18] being the protein decay rate. The dimensionless par ameters α1,α2,δ, and µdeﬁne the basic model. The CI protein used in the experiments is temperature-sensitive, changing its rate of degradation with temperature[66, 67]; we modify the original model slightly to include the factor k, which represents a varying decay rate for the CI protein. Switching is induced either by changing k, or by adjusting the concentration of isopropyl- β- D-thiogalactopyranoside (IPTG); the parameters K= 2.9618×10−5M and η= 2.0015 deﬁne the eﬀect of the inducer molecule IPTG on the Lac protein. Ove r a wide range of parameter values, the system has two stable ﬁxed points; the null-clin es are shown in Figure 7B. The time courses of switching between stable states in the tw o models are shown in Figure 8; transitions are induced by eliminating the bistab ility, then restoring it. The precise time course of switching from one stable state to another is d etermined by the way in which the model parameters are adjusted to eliminate the bistabil ity. In each case, some parameter is increased until the system passes through a saddle node bi furcation: two stable ﬁxed points and one unstable ﬁxed point collapse into a single stable poi nt. In an eﬀort to examine the behaviour of the two systems under analogous conditions, we eliminate the bistability in every case by setting the system to a parameter value 10% past the bifurcation point. 25The transitions shown in Figure 8 are generated as follows. T he system begins (0-1 hour) in its default bistable state, sitting at one of the two stable ﬁxed points. Then (1–4 hours) the bistability is eliminated (as described above), with the only remaining ﬁxed point being such that the other protein has a high concentration. O nce the concentrations have switched, the default parameters are restored and the syste m moves to the nearby stable ﬁxed point (4–7 hours). Finally, the system is rendered mono stable again (7–10 hours), causing another transition, followed by a period (10–11 hou rs) during which the bistable parameters are restored. Under the conditions shown, the λswitch model displays signiﬁcantly more rapid tran- sitions between its stable states than those seen in the mode l of the toggle switch. The numerical results indicate that the properties of the λswitch do oﬀer an advantage in terms of the speed of transitions, indicating that it may be fruitf ul to study synthetic models based on this natural system. Future analytical work on models suc h as the one presented in this section may allow us to make more precise statements regardi ng the source of this advantage. 8 Conclusion From an engineering perspective, the control of cellular fu nction through the design and manipulation of gene regulatory networks is an intriguing p ossibility. Current examples of potential applicability range from the use of genetically e ngineered microorganisms for envi- ronmental cleanup purposes [68], to the ﬂipping of genetic s witches in mammalian neuronal cells [69]. While the experimental techniques employed in s tudies of this nature are certainly impressive, it is clear that reliable theoretical tools wou ld be of enormous value. On a strictly practical level, such techniques could potentially reduce the degree of “trial-and-error” ex- perimentation. More importantly, computational and theor etical approaches will lead to testable predictions regarding the current understanding of complex biological networks. While other studies have centered on certain aspects of natu rally-occurring genetic regu- latory networks [9, 12, 13, 14, 18, 24, 25, 38, 39], an alterna tive approach is to focus on the 26design of synthetic networks. Such an engineering-based ap proach has signiﬁcant technolog- ical implications, and will lead, in a complementary fashio n, to an enhanced understanding of biological design principles. In this work, we have shown how several synthetic networks can be designed from the genetic machinery of the virus λphage. We have highlighted some of the possible behavior of these networks through the discu ssion of the design of two types of switches and a relaxation oscillator. Additionally, in t he case of the oscillator, we have coupled the network to an existing cellular process. Such co upling could lead to possible strategies for entraining or inducing network oscillation s in cellular protein levels, and prove useful in the design of networks that interact with cellular processes that require precise timing. With regard to model formulation, there are several intrigu ing areas for further work. For one, the number of molecules governing the biochemistry of genetic networks is of- ten relatively small, leading to interesting issues involv ing internal noise. Recent pivotal work [17, 18, 27] has led to a systematic modeling approach wh ich utilizes a Monte Carlo- type simulation of the biochemical reactions [61]. While th is approach is impressively com- plete, its complexity makes analysis nearly impossible. An alternative approach could entail the use of Langevin equations, whereby the eﬀects of interna l noise are incorporated into stochastic terms whose magnitudes are concentration-depe ndent. Indeed, in the context of genetic switches, this approach has recently been suggeste d [62]. The advantage of this formulation is that stochastic eﬀects can be viewed as a pert urbation to the deterministic picture, so that analytic tools can be utilized. A potentially important technical issue involves the impli cit assumption that the reactions take place in three-dimensional space. While this assumpti on is perhaps the most natural, proteins have been observed sliding along a DNA molecule in s earch of a promoter region [64], so that protein-DNA reactions might eﬀectively take place o n a surface. While this would not alter the qualitative form of Eq. (6), the exponents on th e variable xcould take on other values [65], and this, in turn, could lead to signiﬁcant quan titative diﬀerences. 27It has been nearly 30 years since the pioneering theoretical work on interacting genetic networks [1]-[8]. Due, in part, to the inherent complexity o f regulatory networks, the true signiﬁcance of these studies had to await technological adv ances. 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The corresponding concentration will increas as the noise causes the upper sta te of (B) to become increasingly populated. (D) Simulation of Eqs. (8) and (9) demonstrating the utilization of external noise for protein switching. Initially, the concentration begins at a level of [GFP] ∼0.4µM corresponding to a low noise value of D= 0.01. After 40 minutes, a large 2-minute noise pulse of strength D= 1.0 is used to drive the concentration to ∼2.2µM. Following this pulse, the noise is returned to its original value. At 80 minu tes, a smaller 10-minute noise pulse of strength D= 0.1 is used to return the concentration to near its original val ue. The simulation technique is that of Ref. [63]. FIG. 2. The relaxation oscillator. (A) Schematic of the circ uit. The PRMpromoter is used on two plasmids to control the production of repressor ( X) and RcsA (Y). After dimer- ization, repressor acts to turn on both plasmids through its interaction at PRM. As its promoter is activated, RcsA concentrations rise, leading t o an induced reduction of repres- sor. (B) Simulation of Eqs. (15). Oscillations arise as the R csA-induced degradation of repressor causes a transversal of the hysteresis diagram in Fig. 1A. The parameter values are mx= 10, my= 1,γx= 0.1,γy= 0.01, and γxy= 0.1. FIG. 3. Oscillatory regimes for the relaxation oscillator. (A) The bifurcation wedge is larger for smaller values of the parameter γxy. This larger regime corresponds to larger values of the RcsA degradation parameter γy. Note that the native (i.e., without tuning) 36degradation rates of γx∼γy∼0.005 are very near the oscillatory regime. (B) Bifurcation diagrams as a function of γxandγxy, and for two ﬁxed values of γy. The oscillatory regime is increased for smaller values of γy, and, in both cases, small values of γxare preferable for oscillations. (C) The bifurcation diagram as a function of the copy numbers mxand my, and for ﬁxed degradation rates. Importantly, one can adjus t the periodic regime to account for the unknown parameter γxy. The ﬁgure also indicates that, for oscillations, one should choose as large a copy number as possible for the plasm id containing the repressor protein ( mx). In (A) and (B), constant parameter values are mx= 10 and my= 1, and in (C)γx= 1.0 and γy= 0.01. FIG. 4. Parameter dependence of the oscillatory period. (A) An increase in γydecreases the period of oscillations. (B) The period depends very weak ly on the copy number. In (A) mx= 10 and in (B) γy= 0.01, and for both plots, other parameter values are γx= 0.1, γxy= 0.1, and my= 1. FIG. 5. Dynamics of a periodically driven relaxation oscill ator - Eq.(15); γx= 0.1,γy= 0.01, γxy= 0.1. (A) Resonant regions in period-amplitude parametric pla ne. Solid lines (limit lines of periodic solutions) together with dashed (period d oubling) lines deﬁne boundaries of stable periodic solutions for a given phase locking regio n M:N, where M is the number of relaxation oscillation and N is the number of driving sinuso idal oscillations. (B,C,D) Oscilla- tions in periodically driven repressor (top curve) concent ration together with the oscillations of the sinusoidal driving (bottom curve). (B) 1:1 synchroni zation; the 14.6 minute period of cI oscillations is equal to the driving period. (C) 1:2 phase locking; the 29.2 minute period of the cI oscillations is twice as long as the driving period. (D) 2:1 phase locking; the 7.6 minute period of the cI oscillations is equal to one half of th e driving period. FIG. 6. (A) As a function of the driving amplitude Γ, the ampli tude Γ xof the induced 37network oscillations shows a sharp increase for a critical v alue of the drive. The critical value corresponds to a drive large enough to induce the hyste retic oscillations, and it in- creases as one decreases γyand moves away from the oscillatory region in parameter spac e. The three curves denoted 1,2, and 3 are for γyvalues of 0.0038, 0.0036, and 0.0034. (B) The gain as a function of the drive amplitude for γy= 0.0038. Close to the oscillatory region, a signiﬁcant gain in the drive amplitude can be induced. Param eter values for both plots are γx= 0.1,γxy= 0.1,mx= 10, and my= 1. Note that, corresponding to these values, the network does not oscillate (without driving) for γy<0.004 (see the bottom wedge of Fig. 3A). FIG. 7 Null-clines for the two-protein bistable switch syst ems. Stable ﬁxed points are marked with circles, and unstable ﬁxed points are marked with squar es. (A) Null-clines for the syn- thetic λswitch, Eqs. (19). Solid line: ˙ x= 0 cline. Dashed line: ˙ y= 0 cline. Parameter values: γx= 0.004;γy= 0.008;ρy= 62.92;α= 11; mx=my= 1;σ1= 2;σ2= 0.08; β1=β2=β3= 0.08; and β4=β5= 1. (B) Null-clines for the toggle switch, Eqs. (20). Solid line: ˙u= 0 cline. Dashed line: ˙ v= 0 cline. Parameter values (from Ref. [22]): α1= 156 .25; α2= 15.6;δ= 2.5;µ= 1;η= 2.0015; [ IPTG ] = 0; k= 1. FIG. 8. Transitions between stable states for the two-prote in bistable switch systems. The protein concentrations have been normalized (the trace for each protein is normalized rela- tive to its own maximum value). The system parameters are var ied over time, altering the stability of the system and causing transitions, as describ ed in the text. Upper Plots: The switching of Lac (solid) and CI (dashed) in the synthetic tog gle model [22]. The parameter values are as given in the caption to Fig. 7, except as follows . (1–4 hours): [ IPTG ] = 2 mM,k= 1.0. (7–10 hours): [ IPTG ] = 0.0,k= 50.81. Lower Plots: The switching of CRO (dashed line) and CI (solid) in the synthetic λmodel. The parameter values are as given in the caption to Fig. 7, except as follows. (1–4 hours) :γx= 0.004,γy= 21.6. (7–10 hours): γx= 18.0,γy= 0.008. 383 4 5 6 7010203040 0.1 0.2 0.3 0.4-1.75-1.50-1.25-1.00-0.75 0100 200 300 40001234[GFP] (mM)[CI] (nM)< z > Time (Minutes)g DA B C D-2.5 -1.5 -0.5 0.5 1.54.55.56.57.5 zf Figure 1 - Hasty et al.PX + P+Y - Time (Minutes)[CI] (nM)A B 0 20 40 600102030RM RM Figure 2 - Hasty et al.gyOscillatory Regimes gxg xyg xy= 0.2 g xy= 0.1 gy=0.02 gy=0.03gx0 10 20 300.000.010.020.030.040.05 0 10 20 300.00.20.40.60.8 0 50 100 150 20001020304050 m mxyg xy= 0.1 g xy= 0.2A B C Figure 3 - Hasty et al.5.06.07.08.09.010.010203040 40 60 80 100101214164.0A BT0 mx(Minutes) T0(Minutes)gy (x10 )-3 Figure 4 - Hasty et al.0 20 40 60 800102030 Time (Minutes)CI [nM] 0 20 40 600102030 Time (Minutes) Time (Minutes)0 20 40 60 80 100010203000.20.40.60.811.21.4 0 9.0 18.0 27.0 36.0 45.01:21:1 2:13:2 5:23:1 Period (Minutes)Driving AmplitudeA B C D Figure 5 - Hasty et al.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0A x 0.0 0.2 0.4 0.6 0.8 1.00102030 gBG G G1 2 3 Figure 6 - Hasty et al.0 40 80 120 1600510150 10 20 3005101520 uvxyA B Figure 7 - Hasty et al.0 2 4 6 8 1000.20.40.60.81 Time [h]Normalized concentration 0123456789101100.20.40.60.81 Time [h]Normalized concentration Figure 8 - Hasty et al.
arXiv:physics/0103035v1 [physics.atom-ph] 13 Mar 2001S-, P- and D-wave resonances in positronium-sodium and positronium-potassium scattering Sadhan K Adhikari†and Puspajit Mandal†$ †Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista, 01.405-900 S˜ ao Paulo, S˜ ao Paulo, Brazil $Department of Mathematics, Visva Bharati, Santiniketan 73 1 235, India (February 21, 2014) Abstract Scattering of positronium (Ps) by sodium and potassium atom s has been investigated employing a three-Ps-state coupled-cha nnel model with Ps(1s,2s,2p) states using a time-reversal-symmetric regu larized electron- exchange model potential ﬁtted to reproduce accurate theor etical results for PsNa and PsK binding energies. We ﬁnd a narrow S-wave singlet resonance at 4.58 eV of width 0.002 eV in the Ps-Na system and at 4.77 eV of width 0.003 eV in the Ps-K system. Singlet P-wave resonances in bot h systems are found at 5.07 eV of width 0.3 eV. Singlet D-wave structures ar e found at 5.3 eV in both systems. We also report results for elastic and Ps- excitation cross sections for Ps scattering by Na and K. PACS Number(s): 34.10.+x, 36.10.Dr 1Recent successful high precision measurements of positron ium (Ps) scattering by H 2, N2, He, Ne, Ar, C 4H10, and C 5H12[1,2] have enhanced theoretical activities [3–6] in this su bject. We suggested [7] a regularized, symmetric, nonlocal electr on-exchange model potential and used it in the successful study of Ps scattering by H [8], He [7 ,9–11], Ne [11], Ar [11], H 2 [12] and Li [13]. Our results were in agreement with experime ntal total cross sections [1,2], specially at low energies for He, Ne, Ar and H 2. Moreover, these studies yielded correct results for resonance and binding energies for the S wave ele ctronic singlet state of Ps-H [4,8] and Ps-Li [13] systems in addition to experimental pic k-oﬀ quenching rate in Ps-He [10] scattering. In the present work we use the above exchange potential to stu dy Ps-Na and Ps-K scattering using the three-Ps-state coupled channel metho d. We ﬁnd resonances in the singlet channel at low energies in S, P and D waves of both syst ems near the Ps(2) excitation threshold. We also report angle-integrated elastic and Ps- excitation cross sections for both systems. The appearance of resonances in electron-atom [14] and posi tron-atom [15] scattering, and in other atomic processes in general, is of great interes t. Several resonances in the electron-hydrogen system have been found in the close-coup ling calculation and later recon- ﬁrmed in the variational calculation [16]. Resonances have also been found in the close- coupling calculation of electron scattering by Li, Na and K [ 17]. These resonances provide the necessary testing ground for a theoretical formulation , which can eventually be detected experimentally. Detailed dynamical description of the imp ortant degrees of freedom in a theoretical formulation is necessary for the appearance of these resonances. The ability of the present exchange potential to reproduce the resonances in diverse Ps-atom systems [8,13] assures its realistic nature. The theory for the coupled-channel study of Ps scattering wi th the regularized model potential has already appeared in the literature [7,8,11] a nd we quote the relevant working equations here. For target-elastic scattering we solve the following Lippmann-Schwinger scattering integral equation in momentum space f± ν′,ν(k′,k) =B± ν′,ν(k′,k) −/summationdisplay ν′′/integraldisplaydk′′ 2π2B± ν′,ν′′(k′,k′′)f± ν′′,ν(k′′,k) k2 ν′′/4−k′′2/4 + i0(1) where the singlet (+) and triplet ( −) “Born” amplitudes, B±, are given by B± ν′,ν(k′,k) = gD ν′,ν(k′,k)±gE ν′,ν(k′,k),where gDandgErepresent the direct and exchange Born amplitudes and the f±are the singlet and triplet scattering amplitudes, respect ively. The quantum 2states are labeled by the indices ν, referring to the Ps atom. The variables k,k′,k′′etc denote the appropriate momentum states of Ps; kν′′is the on-shell relative momentum of Ps in the channel ν′′. We use atomic unit (a.u.) where ¯ h=m= 1 with mis the electron mass. To avoid complication of calculating exchange potential wi th a many-electron wave func- tion, we consider a frozen-core one-electron approximatio n for the targets Na and K. Such wave functions have been successfully used for scattering o f alkali metal atoms in other contexts and also for positronium scattering by Li [5]. The N a(3s) and K(4s) frozen-core hydrogen-atom-like wave functions are taken as φNa(r) =1 9√ 3/radicalBig 4π¯a3 0(6−6ρ+ρ2)e−ρ/2(2) φK(r) =1 96/radicalBig 4π¯a3 0(24−36ρ+ 12ρ2−ρ3)e−ρ/2(3) where ρ= 2rαwithα= 1/(n¯a0). Here n= 3 for Na and = 4 for K and ¯ a0= (2n2Ei)−1a0 withEithe ionization energy of the target in a.u. and a0the Bohr radius of H. Here we use the following experimental values for ionization energies for Na and K, respectively: 5.138 eV and 4.341 eV [18]. The direct Born amplitude of Ps scattering is given by [7,9] gD ν′,ν(kf,ki) =4 Q2/integraldisplay φ∗(r) [1−exp(iQ.r)]φ(r)dr ×/integraldisplay χ∗ ν′(t)2isin( Q.t/2)χν(t)dt, (4) where φ(r) is the target wave function and χ(t) is the Ps wave function. The exchange amplitude corresponding to the model potential is given by [ 8] gE ν′,ν(kf,ki) =4(−1)l+l′ D/integraldisplay φ∗(r) exp(iQ.r)φ(r)dr ×/integraldisplay χ∗ ν′(t) exp(iQ.t/2)χν(t)dt (5) with D= (k2 i+k2 f)/8 +C2[α2+ (β2 ν+β2 ν′)/2] (6) where landl′are the angular momenta of the initial and ﬁnal Ps states and Cis the only parameter of the exchange potential. The initial and ﬁn al Ps momenta are kiandkf, Q=ki−kf, and β2 νandβ2 ν′are the binding energies of the initial and ﬁnal states of Ps i n a.u., respectively. It has been demonstrated for the Ps-H sy stem that at high energies the 3model-exchange amplitude (5) reduces to [19] the Born-Oppe nheimer exchange amplitude [20]. This exchange potential for Ps scattering is consider ed [7] to be a generalization of the Ochkur-Rudge exchange potential for electron scattering [ 21]. After a partial-wave projection, the system of coupled equa tions (1) is solved by the method of matrix inversion. Forty Gauss-Legendre quadratu re points are used in the dis- cretization of each momentum-space integral. The calculat ion is performed with the exact Ps wave functions and frozen-core orbitals (2) and (3) for Na and K ground state. We consider Ps-Na and Ps-K scattering using the three-Ps-stat e model that includes the fol- lowing states: Ps(1s)Na(3s), Ps(2s)Na(3s), Ps(2p)Na(3s) , and Ps(1s)K(4s), Ps(2s)K(4s), Ps(2p)K(4s), for Na and K, respectively. The parameter Cof the potential given by (5) and (6) was adjusted to ﬁt the accurate theoretical results [ 22] for PsNa and PsK binding energies which are 0.005892 a.u. and 0.003275 a.u., respect ively. We ﬁnd that C= 0.785 ﬁts both binding energies well in the three-Ps-state model a nd this value of Cis used in all calculations reported here. The proper strength of the mode l potential is obtained by ﬁtting the binding energies of the Ps-Na and Ps-K systems, and it is e xpected that this choice of Cwould lead to a good overall description of scattering in the se systems. We recall that this value of Calso reproduced the accurate variational result of PsH reso nance energy in a recent ﬁve-state model for Ps-H scattering [8]. A similar va riation of the parameter Cfrom unity also led to a good overall description of the total scat tering cross section in agreement with experiment [2] in the Ps-He system in the energy range 0 e V to 70 eV [9]. The Ps-Na and Ps-K systems have an eﬀective attractive inter action in the electronic singlet channel as in Ps-H [8] and Ps-Li [13] systems. The tar gets of these systems have one active electron outside a closed shell. In the present th ree-Ps-state calculation we ﬁnd resonances in the singlet channel in both systems. No resona nces appear in the triplet channel possibly because of a predominantly repulsive inte raction in this channel. For the resonances to appear, the inclusion of the excited states of Ps is fundamental in a coupled- channel calculation. The static-exchange model with both t he target and Ps in the ground state does not lead to these resonances. A detailed study of t hese resonances in coupled- channel study of Ps-H [4,8] and Ps-Li [13] systems in the sing let channel has appeared in the literature. Here to study the resonances, ﬁrst we calculate the S-, P- and D-wave elastic phase shifts and cross sections in the singlet channel of the Ps-Na and Ps-K systems using the 3-Ps-state model. In ﬁgure 1 we show the low-energy singlet S -wave cross sections. The singlet S-wave phase shifts near the resonance energies are shown in the oﬀ-set of ﬁgure 1. The Ps-Na system has a resonance at 4.58 eV of width 0.002 eV. T he resonance in the Ps-K 4system appears at 4.77 eV and has a width 0.003 eV. The phase sh ift curves clearly show the resonances where the phase shifts jump by π. In ﬁgure 2 we show the singlet P-wave Ps-Na and Ps-K elastic ph ase shifts; the cor- responding singlet P-wave cross sections are shown in the oﬀ -set. Both systems possess resonances at 5.07 eV of width of 0.3 eV. The cross sections cl early exhibit these resonances. In ﬁgure 3 we plot the D-wave singlet elastic cross sections f or Ps-Na and Ps-K systems at low energies. There is a structure in both systems at 5.3 eV wh ich is more diﬀuse than in S and P waves. Next we calculate the diﬀerent partial cross sections of Ps- Na and Ps-K scattering. The convergence of the cross sections with respect to partial wa ves is slower in this case than in the case of Ps-H scattering. At a incident Ps energy of 50 eV, 4 0 partial waves were used to achieve convergence of the partial-wave scheme. In ﬁgures 4 and 5 we plot diﬀerent angle- integrated partial cross sections of Ps-Na and Ps-K scatter ing, respectively. Speciﬁcally, we plot the elastic, Ps(2s) and Ps(2p) excitation cross sectio ns using the three-Ps-state model. For comparison we also plot the elastic cross section obtain ed with the static-exchange model. The elastic cross section is large at low energies in b oth systems. The eﬀect of the inclusion of highly polarizable Ps(2) states in the couplin g scheme could be considerable, specially at low energies. The local minima in the three-Ps- state elastic cross section for both systems at about 4 ∼5 eV are manifestations of the P- and D-wave resonances in thi s energy region. Similar minima found in electron scattering by alkali-metal atoms are also consequences of resonances [17]. To summarize, we have performed a three-Ps-state coupled-c hannel calculation of Ps-Na and Ps-K scattering at low energies using a regularized symm etric nonlocal electron-exchange model potential [7] successfully used [7–13] previously in diﬀerent Ps scattering problems. The only parameter of the model potential was adjusted to ﬁt a ccurate theoretical result for PsNa and PsK binding [22]. We present the results of angle-in tegrated partial cross sections at diﬀerent Ps energies. We ﬁnd resonances in S, P and D waves n ear the Ps(2) excitation threshold. In this study we have used a three-Ps-state model . Similar resonances have been observed in the coupled-channel study of electron-H [1 6], electron-Na, electron-K [17], positron-hydrogen [15], Ps-H [4,8] and Ps-Li [13] systems. In most cases, a more complete calculation and (in some cases) experiments have reconﬁrme d these resonances. Hence we do not believe that the appearance of resonances in the prese nt three-state calculation to be so peculiar as to have no general validity. On the contrary, i n view of the correlation found between resonance and binding energies in the singlet Ps-H s ystem [8], it is expected that the reproduction of correct binding energies of the Ps-Na an d Ps-K systems in the present 5model should lead to correct resonance energies in these sys tems. However, the resonance energies might change slightly after a more complete calcul ation (with accurate many-body wave functions of the target and including the excited state s of the target) and it would be intersting to study the present resonances using more compl ete theoretical models in the future in addition to compare the present results with futur e experiments. The work is supported in part by the Conselho Nacional de Dese nvolvimento - Cient´ ıﬁco e Tecnol´ ogico, Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado d e S˜ ao Paulo, and Financiadora de Estudos e Projetos of Brazil. 6REFERENCES [1] Charlton M and Laricchia G 1991 Comments At. Mol. Phys. 26253 Tang S and Surko C M 1993 Phys. Rev. A 47R743 [2] Garner A J, Laricchia G and ¨Ozen A 1996 J. Phys. B: At. Mol. Opt. Phys. 295961 Garner A J, ¨Ozen A and Laricchia G 2000 J. Phys. B: At. Mol. Opt. Phys. 331149 Garner A J and Laricchia G 1996 Can. J. Phys. 74518 Nagashima Y, Hyodo T, Fujiwara K and Ichimura A 1998 J. Phys. B: At. Mol. Opt. Phys.31329 Zafar N, Laricchia G, Charlton M and Garner A 1996 Phys. Rev. Lett. 761595 Skalsey M, Engbrecht J J, Bithell R K, Vallery R S and Gidley D W 1998Phys. Rev. Lett.803727 [3] Andersen H H, Armour E A G, Humberston J W and Laricchia G 19 98Nucl. Instrum. & Methods Phys. Res. B 143U10 Laricchia G 1995 Nucl. Instrum. Methods Phys. Res. B 99363 Hara S, Hyodo T, Nagashima Y and Rehn L 2000 Nucl. Instrum. & Methods Phys. Res. B1711 Biswas P K 2000 Nucl. Instrum. & Methods Phys. Res. B 171135 [4] Frolov A M and Smith Jr. V H 1997 Phys. Rev. A 552662 Yan Z C and Ho Y K 1999 Phys. Rev. A 592697 Yan Z C and Ho Y K 1999 Phys. Rev. A 605098 Jiang N and Schrader D M 1997 Mat. Sc. Forum 255-2 312 Jiang N and Schrader D M 1998 J. Chem. Phys. 1099430 [5] Ray H 1999 J. Phys. B: At. Mol. Opt. Phys. 325681 Ray H 2000 J. Phys. B: At. Mol. Opt. Phys. 334285 [6] Barker M I and Bransden B H 1968 J. Phys. B: At. Mol. Opt. Phys. 11109 Fraser P A 1968 J. Phys. B: At. Mol. Opt. Phys. 11006 [7] Biswas P K and Adhikari S K 1999 Phys. Rev. A 59363 [8] Adhikari S K and Biswas P K 1999 Phys. Rev. A 592058 7[9] Adhikari S K 2000 Phys. Rev. A 62062708 [10] Adhikari S K, Biswas P K and Sultanov R A 1999 Phys. Rev. A 594829 [11] Biswas P K and Adhikari S K 2000 Chem. Phys. Lett. 317129 [12] Biswas P K and Adhikari S K 1998 J. Phys. B: At. Mol. Opt. Phys. 31L315 Biswas P K and Adhikari S K 1998 J. Phys. B: At. Mol. Opt. Phys. 31L737 Biswas P K and Adhikari S K 2000 J. Phys. B: At. Mol. Opt. Phys. 331575 [13] Biswas P K 2000 Phys. Rev. A 61012502 [14] Callaway J 1978 Phys. Rep. 4589 Pathak A, Burke P G and Berrington K A 1989 J. Phys. B: At. Mol. Opt. Phys. 22 2759 Fon W C, Berrington K A, Burke P G and Kingston A E 1989 J. Phys. B: At. Mol. Opt. Phys. 223939 [15] Seiler G J, Oberoi R S and Callaway J 1971 Phys. Rev. A 32006 [16] Burke P G and Schey H M 1962 Phys. Rev. 126147 Shimamura I 1971 J. Phys. Soc. (Japan) 31852 Das J N and Rudge M R H 1976 J. Phys. B: At. Mol. Opt. Phys. 91131 [17] Moores D L and Norcross D W 1972 J. Phys. B: At. Mol. Opt. Phys. 51482 Moores D L 1976 J. Phys. B: At. Mol. Opt. Phys. 91329 Burke P G and Joanna Taylor A 1969 J. Phys. B: At. Mol. Opt. Phys. 2869 [18] Hart G A and Goodfriend P L 1970 Chem. Phys. Lett. 53448 [19] Adhikari S K and Mandal P 2000 J. Phys. B: At. Mol. Opt. Phys. 33L761 [20] Oppenheimer J R 1928 Phys. Rev. 32361 [21] Rudge M R H 1965 Proc. Phys. Soc. London 86763 Ochkur V I 1963 Zh. Eksp. Teor. Fiz. 45734 [English Transl. 1964 Sov. Phys. JETP 18503 ] [22] Mitroy J and Ryzhikh G 1999 J. Phys. B: At. Mol. Opt. Phys. 323839 Ryzhikh G and Mitroy J 1998 J. Phys. B: At. Mol. Opt. Phys. 31L401 8Figure Caption: 1. Singlet S-wave elastic cross sections at diﬀerent Ps ener gies for Ps-Na (dashed line) and Ps-K (full line) scattering. The corresponding phase sh ifts near resonance are shown in the oﬀ-set. 2. Singlet P-wave elastic phase shifts at diﬀerent Ps energi es for Ps-Na (dashed line) and Ps-K (full line) scattering. The corresponding cross secti ons are shown in the oﬀ-set. 3. Singlet D-wave elastic cross sections at diﬀerent Ps ener gies for Ps-Na (dashed line) and Ps-K (full line) scattering. 4. Partial cross sections for Ps-Na scattering at diﬀerent P s energies: three-Ps-state elastic (full line), three-Ps-state Ps(2s) (dashed-dotte d line), three-Ps-state Ps(2p) (short- dashed line), static-exchange elastic (long-dashed line) . 5. Same as in ﬁgure 4 for Ps-K scattering. 90 2 4 6 Energy (eV)020406080100Singlet S-wave Cross Section ( a02)πFigure 1 4.5 4.6 4.7 4.8 Energy (eV)024Phase shift (rad)0 1 2 3 4 5 Energy (eV)01234P-wave Singlet Phase Shift (rad)Figure 2 0 2 4 6 Energy (eV)0816Cross Section ( a02)π0 4 8 Energy (eV)0123Singlet D-wave Cross Section ( a02) πFigure 30 10 20 30 40 50 Energy (eV)0.010.101.0010.00100.00 Cross Section (units of a02)πFigure 40 10 20 30 40 50 Energy (eV)0.010.101.0010.00100.00 Cross section (units of a02)πFigure 5
arXiv:physics/0103036v1 [physics.atom-ph] 13 Mar 2001Convergent variational calculation of positronium-hydro gen-atom scattering lengths Sadhan K Adhikari†and Puspajit Mandal†,$ †Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista 01.405-900 S˜ ao Paulo, S˜ ao Paulo, Brazil $Department of Mathematics, Visva-Bharati, Santiniketan 7 31 235, India (February 20, 2014) Abstract We present a convergent variational basis-set calculation al scheme for elas- tic scattering of positronium atom by hydrogen atom in S wave . Highly cor- related trial functions with appropriate symmetry are need ed for achieving convergence. We report convergent results for scattering l engths in atomic units for both singlet (= 3 .49±0.20) and triplet (= 2 .46±0.10) states. PACS Number(s): 34.90.+q, 36.10.Dr 1Lately, there has been interest in the experimental [1] and t heoretical [2–8] studies of ortho positronium (Ps) atom scattering by diﬀerent neutral atomic and molecular targets. The Ps-H system is theoretically the most simple and fundame ntal and a complete under- standing of this system is necessary before a venture to more complex targets [7–9]. There have been R-matrix [3], close-coupling (CC) [4,10] and mode l-potential [5,11] calculations for Ps-H scattering. Here we present a convergent variation al basis-set calculational scheme for low-energy Ps-H scattering in S wave below the lowest Ps- excitation threshold at 5.1 eV. Using this method we report numerical results for scatterin g length of electronic singlet and triplet states. A recent study based on a regularized nonlocal electron-exc hange model potential [6] yielded low-energy (total) cross sections in agreement wit h experiment for Ps scattering by He [6], Ne [7], Ar [7] and H 2[8]. For the Ps-H system the model-potential results for S-wave singlet binding and resonance energies are in agreem ent with accurate variational estimates [12]. It would be interesting to see if the model-p otential result for the Ps-H singlet scattering length agrees with the present convergent calcu lation. Because of the existence of two identical fermions (electro ns) in the Ps-H system, one needs to antisymmetrize the full wave function before attem pting a solution of the scattering problem. The position vectors of the electrons −r1(Ps) and r2(H)−and positron ( x) with respect to (w.r.t.) the massive proton at the origin are as sh own in ﬁgure 1. We also use the position vectors sj= (x+rj)/2,ρj=x−rj,j= 1,2,r12=r1−r2. The fully antisymmetric stateψA kof Ps-H scattering is given by \|ψA k/an}bracketri}ht=A1\|ψ1 k/an}bracketri}ht= (1±P12)\|ψ1 k/an}bracketri}ht=\|ψ1 k/an}bracketri}ht ± \|ψ2 k/an}bracketri}htwhere kis the incident momentum, the antisymmetrizer A1is (1 +P12) for the singlet state and (1−P12) for the triplet state with P12the permutation operator of electrons 1 and 2. The functionψ1 krefers to the Ps-H wave function with electron 1 forming the P s as in ﬁgure 1 andψ2 krefers to the same with the two electrons interchanged. The full Ps-H Hamiltonian Hcan be broken in two convenient forms as follows H= H1+V1=H2+V2whereH1includes the full kinetic energy and intracluster interact ion of H and Ps for the arrangement shown in ﬁgure 1 and V1is the sum of the intercluster interaction between H and Ps in the same conﬁguration, H2andV2refer to the same quantities with the two electrons interchanged: 2V1=/bracketleftbigg1 x−1 r1+1 r12−1 ρ2/bracketrightbigg , V 2=P12V1=/bracketleftbigg1 x−1 r2+1 r12−1 ρ1/bracketrightbigg . (1) The fully antisymmetric state satisﬁes the Lippmann-Schwi nger equation [13] \|ψA k/an}bracketri}ht=\|φ1 k/an}bracketri}ht+G1V1\|ψA k/an}bracketri}ht. (2) where the channel Green’s function G1≡(E+i0−H1)−1and the incident wave \|φ1 k/an}bracketri}htsatisﬁes (E−H1)\|φ1 k/an}bracketri}ht= 0.The incident Ps energy E= 6.8k2eV. We are using atomic units (au) in whicha0=e=m= ¯h= 1, where e(m) is the electronic charge (mass) and a0the Bohr radius. Using the deﬁnition of the antisymmetrized state we rewrite (2) as [13] \|ψ1 k/an}bracketri}ht=\|φ1 k/an}bracketri}ht+G1M1\|ψ1 k/an}bracketri}ht (3) M1=V1A1+ (E−H1)(1− A 1)≡ A 1V1+ (1− A 1)(E−H1). (4) The properly symmetrized transition matrix for elastic sca ttering is deﬁned by /an}bracketle{tφ1 k\|TA\|φ1 k/an}bracketri}ht= /an}bracketle{tφ1 k\|V1\|ψA k/an}bracketri}ht=/an}bracketle{tφ1 k\|V1A1\|ψ1 k/an}bracketri}ht=/an}bracketle{tψ1 k\|A1V1\|φ1 k/an}bracketri}ht[13]. A basis-set calculational scheme for the transition matrix can be obtained from the following expres sion [14] /an}bracketle{tφ1 k\|TA\|φ1 k/an}bracketri}ht=/an}bracketle{tψ1 k\|A1V1\|φ1 k/an}bracketri}ht+/an}bracketle{tφ1 k\|A1V1\|ψ1 k/an}bracketri}ht − /an}bracketle{tψ1 k\|A1V1−M1G1A1V1\|ψ1 k/an}bracketri}ht. (5) Using (3), it can be veriﬁed that (5) is an identity if exact sc attering wave fumctions ψ1 k are used. If approximate wave functions are used, (5) is stat ionary w.r.t. small variations of\|ψ1 k/an}bracketri}htbut not with /an}bracketle{tψ1 k\|. This one-sided variational property emerges because of th e lack of symmetry of the formulation in the presence of explicit an tisymmetrization operator A1. However, this variational property can be used to formulate a basis-set calculational scheme with the following trial functions [14] \|ψ1 k/an}bracketri}ht=N/summationdisplay n=1an\|fn/an}bracketri}ht,/an}bracketle{tψ1 k\|=N/summationdisplay m=1bm/an}bracketle{tfm\|. (6) Substituting (6) into (5) and using this variational proper ty w.r.t. \|ψ1 k/an}bracketri}htwe obtain [14] /an}bracketle{tψ1 k\|=N/summationdisplay m=1/an}bracketle{tφ1 k\|A1V1\|fn/an}bracketri}htDnm/an}bracketle{tfm\| (7) (D−1)mn=/an}bracketle{tfm\|A1V1−[A1V1+ (1− A 1)(E−H1)]G1A1V1\|fn/an}bracketri}ht. (8) 3Using the variational form (7) and deﬁnition /an}bracketle{tφ1 k\|TA\|φ1 k/an}bracketri}ht=/an}bracketle{tψ1 k\|A1V1\|φ1 k/an}bracketri}htwe obtain the following basis-set calculational scheme for the transiti on matrix /an}bracketle{tφ1 k\|TA\|φ1 k/an}bracketri}ht=N/summationdisplay m,n=1/an}bracketle{tφ1 k\|A1V1\|fn/an}bracketri}htDnm/an}bracketle{tfm\|A1V1\|φ1 k/an}bracketri}ht. (9) (8) and (9) are also valid for the K matrix and in partial-wave form where the momentum- space integration over the Green’s function G1should be performed with the principal-value prescription. In the calculation, the basis functions are taken in the foll owing form fm(r2,ρ1,s1) =ϕ(r2)η(ρ1)e−δmr2−αmρ1−βms1−γm(ρ2+r12)−µm(x+r1)sin(ks1) ks1(10) whereϕ(r) = exp( −r)/√πandη(ρ) = exp( −0.5ρ)/√ 8πrepresent the H(1s) and Ps(1s) wave functions, respectively. For elastic scattering the d irect Born amplitude is zero and the exchange correlation dominates scattering. To be consiste nt with this, the direct terms in the form factors /an}bracketle{tfm\|A1V1\|φ1 k/an}bracketri}htand/an}bracketle{tφ1 k\|A1V1\|fn/an}bracketri}htare zero with the above choice of correlations in the basis functions via γmandµm. This property follows as the above function is invariant w.r.t. the interchange of xandr1whereas the remaining part of the integrand in the direct terms changes sign under this transformation. A proper choi ce of the correlation parameters γmandµmis crucial for obtaining good convergence. In the following we specialize to the K-matrix formulation i n S wave at zero energy, when sin(ks1)/(ks1) = 1 in (10). The useful matrix elements of the present approa ch are explicitly written as [14] /an}bracketle{tφ1 p\|A1V1\|fn/an}bracketri}ht=±1 2π/integraldisplay ϕ(r1)η(ρ2)sinps2 ps2[V1]fn(r2,ρ1,s1)dr2dρ1ds1 (11) /an}bracketle{tfm\|A1V1\|φ1 p/an}bracketri}ht=±1 2π/integraldisplay fm(r1,ρ2,s2)[V1]ϕ(r2)η(ρ1)sinps1 ps1dr2dρ1ds1 (12) /an}bracketle{tfm\|A1V1\|fn/an}bracketri}ht=±1 4π/integraldisplay fm(r1,ρ2,s2)[V1]fn(r2,ρ1,s1)dr2dρ1ds1 (13) /an}bracketle{tfm\|M1G1A1V1\|fn/an}bracketri}ht ≈ −2 π/integraldisplay∞ 0dp/an}bracketle{tfm\|A1V1\|φ1 p/an}bracketri}ht/an}bracketle{tφ1 p\|A1V1\|fn/an}bracketri}ht (14) 4where the so called oﬀ-shell term (1 − A 1)(E−H1) has been neglected for numerical sim- pliﬁcation in this calculation. This term is expected to con tribute to reﬁnement over the present calculation. In this convention the on-shell K-mat rix element at zero energy is the scattering length: a=/an}bracketle{tφ1 0\|KA\|φ1 0/an}bracketri}ht. All the matrix elements above can be evaluated by a method pre sented in [15]. We describe it in the following for /an}bracketle{tφ1 p\|A1V1\|fn/an}bracketri}htof (11). By a transformation of variables from (r2,ρ1,s1) to (s1,s2,x) with Jacobian 26and separating the radial and angular integrations, the form factor (11) is given by /an}bracketle{tφ1 p\|A1V1\|fn/an}bracketri}ht=±26 16π3/integraldisplay∞ 0s2 2ds2sin(ps2) ps2/integraldisplay∞ 0s2 1ds1e−βns1/integraldisplay∞ 0x2dxe−µnx ×/integraldisplay e−(ar1+bρ1/2)e−(cr2+dρ2/2)e−γnr12[V1]dˆs1dˆs2dˆx (15) wherea= 1 +µn,b= 2αn+ 1,c= 1 +δnandd= 2γn+ 1. Recalling that rj= 2sj−x, r12= 2(s1−s2),ρj= 2(x−sj),j= 1,2, we employ the following expansions of the exponentials in (15) e−a\|2s−x\|−b\|x−s\|=4π sx/summationdisplay lmG(a,b) l(s,x)Y∗ lm(ˆs)Ylm(ˆx) (16) e−a\|2s−x\|−b\|x−s\| \|2s−x\|=4π sx/summationdisplay lmJ(a,b) l(s,x)Y∗ lm(ˆs)Ylm(ˆx) (17) e−a\|2s−x\|−b\|x−s\| \|s−x\|=4π sx/summationdisplay lmK(a,b) l(s,x)Y∗ lm(ˆs)Ylm(ˆx) (18) e−a\|s1−s2\| \|s1−s2\|=4π s1s2/summationdisplay lmA(a) l(s1,s2)Y∗ lm(ˆs1)Ylm(ˆs2) (19) e−a\|s1−s2\|=4π s1s2/summationdisplay lmB(a) l(s1,s2)Y∗ lm(ˆs1)Ylm(ˆs2) (20) where theYlm’s are the usual spherical harmonics. Using (16) −(20) in (15) we get /an}bracketle{tφ1 p\|A1V1\|fn/an}bracketri}ht=±28/integraldisplay∞ 0e−βns1ds1/integraldisplay∞ 0ds2sin(ps2) ps2/integraldisplay∞ 0dxe−µnxL/summationdisplay l=0(2l+ 1) ×/bracketleftbigg1 xG(a,b) l(s1,x)G(c,d) l(s2,x)B(2γn) l(s1,s2)−J(a,b) l(s1,x) ×G(c,d) l(s2,x)B(2γn) l(s1,s2) +1 2G(a,b) l(s1,x)G(c,d) l(s2,x) ×A(2γn) l(s1,s2)−1 2G(a,b) l(s1,x)K(c,d) l(s2,x)B(2γn) l(s1,s2)/bracketrightbigg . (21) 5where thel-sum is truncated to l=L. This evaluation avoids complicated angular integra- tions involving s1,s2andx. These integrals take a simple form requiring straightforw ard numerical computation of certain radial integrals only, wh ich must, however, be carried out carefully. The functions Gl,Jl,Kletc. are easily calculated using (16) −(20): G(a,b) l(s,x) =sx 2/integraldisplay1 −1dtPl(t)e−a\|2s−x\|−b\|x−s\|(22) wherePl(t) is the usual Legendre polynomial and tis the cosine of the angle between sand x. The integrals (12) and (13) can be evaluated similarly. For example /an}bracketle{tfm\|A1V1\|φ1 p/an}bracketri}ht=±28/integraldisplay∞ 0e−βms2ds2/integraldisplay∞ 0ds1sin(ps1) ps1/integraldisplay∞ 0dxe−µmxL/summationdisplay l=0(2l+ 1) ×/bracketleftbigg1 xG(c,d) l(s1,x)G(a,b) l(s2,x)B(2γm) l(s1,s2)−J(c,d) l(s1,x) ×G(a,b) l(s2,x)B(2γm) l(s1,s2) +1 2G(c,d) l(s1,x)G(a,b) l(s2,x) ×A(2γm) l(s1,s2)−1 2G(c,d) l(s1,x)K(a,b) l(s2,x)B(2γm) l(s1,s2)/bracketrightbigg (23) /an}bracketle{tfm\|A1V1\|fn/an}bracketri}ht=±27/integraldisplay∞ 0e−βns1ds1/integraldisplay∞ 0ds2e−βms2/integraldisplay∞ 0dxe−(µn+µm)xL/summationdisplay l=0(2l+ 1) ×/bracketleftbigg1 xG(e,f) l(s1,x)G(g,h) l(s2,x)B(2γmn) l(s1,s2)−J(e,f) l(s1,x) ×G(g,h) l(s2,x)B(2γmn) l(s1,s2) +1 2G(e,f) l(s1,x)G(g,h) l(s2,x) ×A(2γmn) l(s1,s2)−1 2G(e,f) l(s1,x)K(g,h) l(s2,x)B(2γmn) l(s1,s2)/bracketrightbigg (24) wheree= 1 +δm+µn,f= 2αn+ 2γm+ 1,g= 1 +δn+µm,h= 2αm+ 2γn+ 1 and γmn=γm+γn. We tested the convergence of the integrals by varying the num ber of integration points in thex,s1ands2integrals in (21), (23) and (24) and the tintegral in (22). The evaluation of (21), (23) and (24) essentially involves four-dimensional integration, which is performed with caution. The xintegration was relatively easy and 20 Gauss-Legendre quad rature points appropriately distributed between 0 and 16 were enough for c onvergence. In the evaluation of integrals of type (22) 40 Gauss-Legendre quadrature poin ts were suﬃcient for adequate convergence. The convergence in the numerical integration overs1ands2was achieved with 300 Gauss-Legendre quadrature points between 0 and 12. The maximum value of l 6in the sum in (21), (23) and (24), L, is taken to be 6 which is suﬃcient for obtaining the convergence with the partial-wave expansions (16) −(20). Table 1: Singlet ( as) and triplet ( at) Ps-H scattering lengths for diﬀerent LandN. L= 0L= 2L= 4L= 6 Natasatasatasatas 64.00 3.61 3.66 4.00 3.07 3.75 2.92 3.74 73.98 4.11 3.25 3.88 2.99 4.06 2.85 4.00 83.93 4.12 3.35 3.96 2.66 3.72 2.54 3.73 93.97 4.15 3.42 3.83 2.65 3.92 2.55 4.06 103.97 4.22 3.42 3.92 2.57 4.42 2.50 3.45 114.01 3.82 3.44 3.91 3.55 3.75 2.67 3.80 12 2.54 3.72 2.46 3.73 13 2.48 3.47 2.46 3.49 In a numerical calculation a judicial choice of the paramete rs in (10) is needed for rapid convergence. As the variational method does not pr ovide a bound on the result, the method could converge to a wrong scattering leng th if an inappropriate (incomplete) basis set is choosen. After some experimentat ion we ﬁnd that to ob- tain proper convergence the parameters δnandαnshould be taken to have both positive and negative values, γnandµnshould include values close to unity and βnshould have progressively increasing values till about 1.5 . The results reported in this work are obtained with the following parameters for t he functions fn,n= 1,...,13:{δn,αn,βn,γn,µn} ≡ {− 0.5,−0.25,0.3,0.01,0.02},{−0.5,−0.25,0.5,0.04,0.02}, {−0.5,−0.25,0.7,0.03,0.06}, {−0.2,−0.1,0.6,0.2,0.2}, {−0.1,0.1,0.8,0.25,0.25}, {0.2,−0.2,0.6,0.35,0.35},{−0.1,−0.1,0.7,0.4,0.4},{0.15,0.2,0.8,0.5,0.5}, {0.12,−0.12,1,0.7,0.7},{0.2,0.2,1.2,0.9,0.9},{0.1,0.2,1.3,1,0.7},{0.2,0.1,1.4,0.7,1}, {0.3,0.15,1.5,1,1} In table 1 we show the convergence pattern of the present calc ulation w.r.t. the number of partial waves Land basis functions Nused in the calculation. The convergence is smooth with increasing L. However, as the present calculation does not produce a boun d on the 7result, the convergence is not monotonic with increasing N. The lack of an upper bound in this calculation is clearly revealed in table 1 where the res ults do not decrease monotonically as the number of terms in the trial wave function is increased . The unbounded nature of the results is consistent with the noted oscillation of the s cattering lengths as Nincreases. The oscillation is larger in the singlet state where it is mor e diﬃcult to obtain convergence. Although, the ﬁnal result for the largest NandLis supposed to be the most accurate, it is not known whether this result is larger or smaller than the exact one. Also, an estimate of error of this result is not known. It is diﬃcult to provide a quantitative measure of convergence. However, from the noted ﬂuctuation of the resu lts for large NforL= 6 we believe the error in the triplet scattering length to be le ss than 0.10 and in the singlet scattering length to be less than 0.20. The ﬁnal results of th e present calculation are those for N= 13 andL= 6 with the above estimate of error: as= 3.49±0.20 au andat= 2.46±0.10 au. The maximum number of functions ( N= 13) used in this calculation is also pretty small, compared to those used in diﬀerent Kohn-type variati onal calculations for electron- hydrogen (N= 56) [16] and positron-hydrogen ( N≤286) [17] scattering. Because of the explicit appearance of the Green’s function, the present ba sis-set approach is similar to the Schwinger variational method. Using the Schwinger method, convergent results for electron- hydrogen [18] and positron-hydrogen [19] scattering have b een obtained with a relatively small basis set ( N∼10). These suggest a more rapid convergence in these problem s with a Schwinger-type method. Now we compare the present results with those of other calcul ations. While a static- exchange calculation by Hara and Fraser [10] yielded as= 7.28 au and at= 2.48 au a model potential calculation by Drachman and Houston [11] pr oducedas= 5.33 au and at= 2.36 au. A 22-state R-matrix calculation by Campbell et al. [3] and a six-state CC calculation by Sinha et al. [4] yielded as= 5.20 au,at= 2.45 au andas= 5.90 au,at= 2.32 au, respectively. These triplet scattering lengths are in q ualitative agreement with the present result: at= 2.46±0.10. However, the singlet scattering lengths of these calcul ations have not yet converged. The disagreement in the singlet chan nel shows that it is more diﬃcult to get the converged result in the attractive single t channel than in the repulsive 8triplet channel. This is consistent with the common wisdom t hat a scattering model at low energies is more sensitive to the detail of an eﬀective attra ctive interaction than to that of a repulsive interaction. The nonconvergence of these result s for the singlet scattering length was conjectured before [5]. Using a correlation between the S-wave scattering length and binding energy for the Ps-H system, the value as= 3.5 au was predicted in that study [5] in excellent agreement with the present result: as= 3.49±0.20. To summarize, we have formulated a convergent basis-set cal culational scheme for S-wave Ps-H elastic scattering below the lowest inelastic thresho ld using a variational expression for the transition matrix. We illustrate the method numericall y by calculating the singlet and triplet scattering lengths: as= 3.49±0.20 au andat= 2.46±0.10 au. The work is supported in part by the Conselho Nacional de Dese nvolvimento - Cient´ ıﬁco e Tecnol´ ogico, Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado d e S˜ ao Paulo, and Financiadora de Estudos e Projetos of Brazil. 9REFERENCES [1] Garner A J, Laricchia G and ¨Ozen A 1996 J. Phys. B: At. Mol. Opt. Phys. 295961 Garner A J, ¨Ozen A and Laricchia G 2000 J. Phys. B: At. Mol. Opt. Phys. 331149 Skalsey M, Engbrecht J J, Bithell R K, Vallery R S and Gidley D W 1998Phys. Rev. Lett.803727 Nagashima Y, Hyodo T, Fujiwara K and Ichimura A 1998 J. Phys. B: At. Mol. Opt. Phys.31329 [2] Andersen H H, Armour E A G, Humberston J W and Laricchia G 19 98Nucl. Instrum. & Methods Phys. Res. B 143U10 Biswas P K 2000 Nucl. Instrum. & Methods Phys. Res. B 171135 [3] Campbell C P, McAlinden M T, MacDonald F G R S and Walters H R J 1998 Phys. Rev. Lett. 805097 [4] Sinha P K, Basu A and Ghosh A S 2000 J. Phys. B: At. Mol. Opt. Phys. 332579 [5] Adhikari S K and Biswas P K 1999 Phys. Rev. A 592058 Adhikari S K 2001 Phys. Rev. A 63in press [6] Biswas P K and Adhikari S K 1999 Phys. Rev. A 59363 Adhikari S K 2000 Phys. Rev. A 62062708 [7] Biswas P K and Adhikari S K 2000 Chem. Phys. Lett. 317129 [8] Biswas P K and Adhikari S K 2000 J. Phys. B: At. Mol. Opt. Phys. 331575 [9] Barker M I and Bransden B H 1968 J. Phys. B: At. Mol. Opt. Phys. 11109 [10] Hara S and Fraser P A 1975 J. Phys. B: At. Mol. Opt. Phys. 8L472 [11] Drachman R J and Houston S K 1975 Phys. Rev. A 12885 Drachman R J and Houston S K 1976 Phys. Rev. A 14894 [12] Frolov A M and Smith Jr. V H 1997 Phys. Rev. A 552662 Yan Z C and Ho Y K 1999 Phys. Rev. A 592697 10Jiang N and Schrader D M 1998 J. Chem. Phys. 1099430 [13] Adhikari S K 1982 Phys. Rev. C 25118 [14] Adhikari S K and Sloan I H 1975 Phys. Rev. C 111133 Adhikari S K and Sloan I H 1975 Nucl. Phys. A241 429 Adhikari S K 1974 Phys. Rev. C 101623 Adhikari S K 1998 Variational Principles and the Numerical Solution of Scatt ering Problems (New York: John-Wiley) [15] Adhikari S K and Mandal P 2000 J. Phys. B: At. Mol. Opt. Phys. 33L761 [16] Shimamura I 1971 J. Phys. Soc. Japan 301702 [17] Humberston J W, VanReeth P, Watts M S T and Meyerhof W E 199 7J. Phys. B: At. Mol. Opt. Phys. 302477 Armour E A G and Humberston J W 1991 Phys. Rep. 2041734 [18] Takatsuka K and McKoy V 1984 Phys. Rev. A 301734 [19] Kar S and Mandal P 2000 J. Phys. B: At. Mol. Opt. Phys. 332379 Kar S and Mandal P 1999 J. Phys. B: At. Mol. Opt. Phys. 322297 Kar S and Mandal P 1999 Phys. Rev. A 591913 Figure Caption: 1. Diﬀerent position vectors for the Ps-H system w.r.t the pr oton (p) at the origin in arrangement 1 with electron 1 forming the Ps. 11/G86/G20 /G91/G85/G21/G85/G20 e+e_ (1)e_ (2) pρ1/G85/G20/G21 HPs Figure 1
- 1 -On the Dependence of Electromagnetic Phenomena on the Relativity of Simultaneity DOUGLAS M. S NYDER LOS ANGELES , CALIFORNIA ABSTRACT Maxwell's equations hold in inertial reference frames in uniform translational motion relative to one another. In conjunction with the Lorentz coordinate transformation equations, the transformation equations for the electric and magnetic field components in these reference frames can be derived. As the derivation of the Lorentz coordinate transformation equations dependson the relativity of simultaneity, and indeed on the argument on the relativity ofsimultaneity, electromagnetic phenomena indicate that human cognition is involved in the structure and functioning of the physical world. T EXT It is known that the relativity of simultaneity underlies the Lorentz coordinate transformation equations for two inertial reference frames in uniformtranslational motion relative to one another. It has been shown that an arbitrary decision on the part of the individual considering the two inertial reference frames as to which is the "stationary" and which the "moving" reference frame is involved in arguing the relativity of simultan eity. This arbitrary decision leads to the result that cognition is involved in the relativity of simultaneity andtherefore in the structure and functioning of the physical world described with the use of the Lorentz coordinate transformation equations (Snyder, 1994). The consideration of electromagnetic phenomena in terms of the special theory is particularly important because the special theory allows for a clear explanation of these common phenomena and, in contrast, explanations based in Newtonian mechanics do not. The arbitrary decision regarding the direction in which the relativity of simultaneity is argued is at the core of electromagnetic phenomena. This last result follows from the fact that the Lorentz coordinate transformation equations allow for the correct determination of electric and magnetic field components for inertial reference frames in uniform translational motion relative to one another. As Einstein wrote (1910/1993) in a quote that will be given later, one can hold that Maxwell's laws of electromagnetism are valid in two such inertial frames and use the Lorentz coordinate transformationOn the Dependence - 2 -to deduce the electric and magnetic field components in one of the reference frames once these field components are specified in the other reference frame. Thus, the relative character of forces due to electric and magnetic fields that intrigued Einstein are dependent on the argument on the relativity ofsimultaneity because this argument underlies the Lorentz coordinate transformation equations. This paper will attempt to demonstrate this point in some detail. T HE RELATIVE CHARACTER OF FORCES DUE TO ELECTRIC AND MAGNETIC FIELDS Allow that the electrically charged test particle is at rest in one of the inertial reference frames and that both electric and magnetic fields are present. In the inertial reference frame where the test particle is at rest, only a force associated with an electric field is exerted on the test particle. In the inertial reference frame where the particle is moving in a uniform translational manner, both the magnetic and the electric fields in general affect the test particle. In the words with which Einstein began his first paper proposing the special theory of relativity: It is known that Maxwell's electrodynamics--as usually understood at the present time--when applied to moving bodies, leads to asymmetries which do not appear to be inherent in thephenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor [corresponding to the electrically charged test particle]. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor is at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor is in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force [due to the motion of the conductor in the magnetic field associated with the magnet], to which in itself there is no corresponding energy, but which gives rise--On the Dependence - 3 -assuming equality of relative motion in the two cases discussed- -to electric currents of the same path and intensity as those produced by the electric forces in the former case. (Einstein, 1905/1952, p. 37) Considerations of the type just noted in the quote from Einstein were central to his development of the special theory of relativity. In a statement prepared for a meeting of the Cleveland Physics Society in 1952 honoring the centenary of Michelson's birth, Einstein wrote: What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field. But I was also guided by the result of the Fizeau- experiment and the phenomenon of aberration. (Shankland, 1964, p. 35) The Lorentz Force Law To provide a bit of context for the quote immediately above, consider the following. The basic outline of the program in Newtonian mechanics is that a general law relates an external force applied to an object to the motion of thisobject, specifically that the object accelerates in direct proportion to the force applied to the object in the direction in which the force is applied. There are, in addition, various laws that specify the different forces, Newton's law of gravitational force being a prominent example of such a law. In Newtonian mechanics, the general force law is F = ma, where F is the external force applied to an object, m is the object's mass, and a is the acceleration associated with the application of the force. For electromagnetic phenomena, the basic specification of the force on an electrically charged particle is given by the Lorentz Force Law: F = (qE) + (B x v) where F is the force experienced by the particle, q is the charge of the particle, E is the electric field, B is the magnetic field, and v is the uniform translational velocity, if any, of the electrically charged particle (Halliday & Resnick, 1960/1978). The term qE indicates that the particle experiences an electric force, irrespective of the particle's motion. The term B x v indicates that if the particle is moving and there is a magnetic field, the particle experiences a force orthogonal to the direction of the field and its motion. (The exception is where the motion of the particle and the direction of the field are in the same, orOn the Dependence - 4 -opposite, directions.) Thus in an inertial reference frame where the particle is moving in uniform translational motion through an electric field and a magnetic field, the particle experiences a force with two components, one due to the electric field and another associated with the magnetic field. If one considers the force exerted on the particle from the perspective of an inertial frame movingin the same manner as the particle, the particle experiences a force which is dueonly to an electric field. Irrespective of the inertial reference frame from which the particle is considered, as Einstein noted, "The observable phenomenon here depends only on the relative motion of the conductor [in our case, the electrically charged particle] and the magnet" (p. 37), not on the inertial reference frame from which it is considered. A M ORE DETAILED ANALYSIS OF THE RELATIVE CHARACTER OF ELECTRIC AND MAGNETIC FIELDS THEMSELVES Now here is Einstein's quote, alluded to earlier, in which he noted the dependence of the transformation equations for the field components in Maxwell's laws on the Lorentz coordinate transformation equations. He noted explicitly the consequences of this dependence. Let us apply the [Lorentz] transformation equations...to the Maxwell-Lorentz equations representing the magnetic field [and electromagnetic phenomena in general]. Let E x, Ey, Ez be the vector components of the electric field, and M x, My, Mz the components of the magnetic field, with respect to the system S [where the system SÕ is in uniform translational motion relative to S along the x and xÕ axes]. Calculation shows that the transformed [Maxwell-Lorentz] equations will be of the same form as the original ones if one sets Ex' = ExEy' = §(Ey - v/c Mz) Ez' = §(Ez + v/c My) Mx' = MxMy' = §(My + v/c Ez) Mz' = §(Mz - v/c Ey) [where § = (1 /(1 - v 2/c2)1/2 ].On the Dependence - 5 -The vectors (E x', Ey', Ez') and (M x', My', Mz') play the same role in the [Maxwell-Lorentz] equations referred to S' as the vectors (E x, Ey, Ez) and (M x, My, Mz) play in the equations referred to S. Hence the important result: The existence of the electric field, as well as that of the magnetic field, depends on the state of motion of the coordinate system. The transformed [Maxwell-Lorentz] equations [for inertial reference frames in uniform translational motion relative to one another] permit us to know an electromagnetic field with respect to any arbitrary system in nonaccelerated motion S ' if the field is known relative to another system S of the same type. These transformations would be impossible if the state of motion of the coordinate system played no role in the definition of the [electric and magnetic field] vectors. This we will recognize at once if we consider the definition of the electric field strength: the magnitude, directions, and orientation of the field strength at a given point are determined by the ponderomotive force exerted by the field on the unit quantity ofelectricity [the charged particle], which is assumed to be concentrated in the point considered and at rest with respect to the system of axes . The [Lorentz] transformation equations [used to transform the Maxwell-Lorentz equations for electromagnetism] demonstrate that the difficulties we have encountered...regarding the phenomena caused by the relative motions of a closed [electric] circuit and a magnetic pole [associated with a magnetic field] have been completely averted in the new theory. For let us consider an electric charge moving uniformly with respect to a magnetic pole. We may observe this phenomenon either from a system of axes S linked with the magnet [where B x v results in a force but no electric field], or from a system of axes S' linked with the electric charge [where a changing magnetic field generates an electric field]. With respect to S there exists only a magnetic field (M x, My, Mz), but not any electric field. In contrast, with respect to S' there exists--as can be seen from the expression for E' y and E' z--an electric fieldOn the Dependence - 6 -that acts on the electric charge at rest relative to S'. Thus, the manner of considering the phenomena varies with the state of motion of the reference system: all depends on the point of view, but in this case these changes in the point of view play noessential role and do not correspond to anything that one could objectify, which was not the case when these changes were being attributed to changes of state of a medium filling all of space. (Einstein, 1910/1993, pp. 140-141) It is useful to provide the set of transformation equations for the electric and magnetic field components for E instein's example where there is only a magnetic field in S where an electric charge is at rest. According to the electric and magnetic field transformation equations given by Einstein, the result for thescenario just described results in the following equations: Ey' = §(- v/c M z) Ez' = §(v/c M y) Mx' = M x My' = §(My) Mz' = §(Mz) . Thus, where no electric field exists in S and no electric force is thus exerted on the electric charge, in S' there are forces associated with both electric and magnetic fields on the electric charge. It should be noted that where S' is the "stationary" frame and S the "moving" frame, for an electric charge at rest in S', a similar situation exists except for certain changes in certain electric field components in S due to the change in direction of the velocity of the reference frames relative to one another. Ey = §(v/c M z') Ez = §(- v/c M y') Mx = M x' My = §(My') Mz = §(Mz') .On the Dependence - 7 -The similarity between this set of equations and the set of equations when S is the "stationary" inertial reference frame is due to the ability to consider either S or S' the "stationary" reference frame and the other frame the "moving" reference frame in arguing the relativity of simultaneity and employing particular set of the Lorentz coordinate transformation equationsdependent on a particular direction in which the relativity of simultaneity is argued. In sum, that Maxwell's equations hold in inertial reference frames supports the special theory, in particular the Lorentz coordinate transformation equations that allow for the derivation of the transformation equations for the electric and magnetic field components in these reference frames. As the derivation of the Lorentz coordinate transformation equations depends on the relativity of simultaneity, and indeed on the argument on the relativity ofsimultaneity, electromagnetic phenomena indicate that human cognition is involved in the structure and functioning of the physical world. One need go no further than to note that the integrity of the special theory depends on the ability to argue the relativity of simultaneity with either one of two inertial reference frames in uniform translational motion relative to one another the reference frame in which simultaneity is first defined in the argument. If this were not the case, then the fundamental tenet of the special theory that inertial reference frames are equivalent for the description of physical phenomena would not hold. R EFERENCES Einstein, A. (1952). On the e lectrodynamics of moving bodies. In H. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (Eds.), The principle of relativity, a collection of o riginal memoirs on the special and general theories of relativity (pp. 35-65) [W. Perrett and G. B. Jeffrey, Trans.]. New York: Dover. (Original work published 1905) Einstein, A. (1993). The principle of relativity and its consequences in modern physics. In The Collected Papers of Albert Einstein: Vol.3 , (pp. 117-142) (A. Beck, Trans.). Princeton, New Jersey: Princeton University Press. (Original work published 1910) Halliday, D. & and Resnick, R. (1978). Physics: Part 2 (3rd ed.). New York: John Wiley & Sons. (Original work published 1960) Shankland, R. S. (1964). Michelson-Morley experiment. American Journal of Physics , 32, 16-35. Snyder, D. M. (1994). On the arbitrary choice regarding which inertial reference frame is "stationary" and which is "moving" in the special theory of relativity. Physics Essays , 7 , 297-334.
arXiv:physics/0103038v1 [physics.ed-ph] 14 Mar 2001Mecˆ anica Relacional: A prop´ osito de uma resenha C. O. Escobarae V. Pleitezb aInstituto de F´ ısica Gleb Wataghin Universidade Estadual de Campinas, UNICAMP 13084-971 – Campinas, SP, Brazil bInstituto de F´ ısica Te´ orica– Universidade Estadual Paul ista Rua Pamplona, 145 011405-900–S˜ ao Paulo, SP, Brazil RESUMO Neste artigo fazemos uma an´ alise cr´ ıtica ` a proposta da Me cˆ anica Rela- cional tal como apresentada no livro de mesmo nome, objeto de uma resenha recente nesta revista. ABSTRACT We present a critical analysis of what is called Relational M echanics, as it has been presented in a book thus entitled, which has bee n recently reviewed in this journal.1 Introdu¸ c˜ ao Ainda que n˜ ao seja parte do dia-a-dia de um pesquisador, de v ez em quando a quest˜ ao do m´ etodo cient´ ıﬁco aparece para ser considerada, mesmo que seja de maneira breve, instantˆ anea. Aﬁnal, tantas coisas para f azer e a vida ´ e t˜ ao curta! No entanto, seja motivado pela leitura de um trabalho ex´ otico colo- cado na rede eletrˆ onica de preprints , seja pelo artigo confuso de uma revista especializada, vez ou outra somos levados a nos perguntar: O que distingue a ciˆ encia de outras atividades? Como fazem os cientistas pa ra eliminar ou conﬁrmar as teorias? ´E poss´ ıvel distinguir ciˆ encia dapseudociˆ encia ? Existe ciˆ encia patol´ ogica ? Tem alguma importˆ ancia estas quest˜ oes? Por exem- plo, este tipo de preocupa¸ c˜ ao teria alguma implica¸ c˜ ao n a nossa vida de pesquisador? ´E (ou deve ser) a ciˆ encia conservadora? ´E freq¨ uente lembrar dos casos de persegui¸ c˜ ao cient´ ıﬁca: Giordano Bruno, Gal ileu ou, de pelo menos cegueira coletiva da comunidade cient´ ıﬁca: Boltzma nn por exemplo ou, mais recentemente, Alfred Wegener [AL88]. Deveria isso imobilizar a comunidade cient´ ıﬁca? aﬁnal quem ´ e essa comunidade?1 A maioria das atividades que podemos classiﬁcar com os adjet ivos adi- cionais ao substantivo ciˆ encia , mencionados no par´ agrafo anterior, s˜ ao re- alizadas fora das universidades. Assim, com exce¸ c˜ ao de al guns cientistas como C. Sagan [SA96], os pesquisadores n˜ ao se d˜ ao o trabalh o de discutir e criticar essas atividades da mesma maneira como criticam os pr´ oprios tra- balhos cient´ ıﬁcos. Aﬁnal, uma das caracter´ ısticas do dia -a-dia da ciˆ encia ´ e essa tens˜ ao entre propostas alternativas como explica¸ c˜ ao dos fenˆ omenos naturais. Mas, e quando isso acontecer numa universidade? S ˜ ao as cr´ ıticas necess´ arias? Violariam a liberdade acadˆ emica? A liberda de acadˆ emica deve ser ampla e irrestrita? Se sim, ´ e isso compat´ ıvel com um bom crit´ erio de utiliza¸ c˜ ao dos fundos p´ ublicos? Recentemente foi publicado o livro Mecˆ anica Relacional (MR) [AK99]. Nesse livro pretende-se colocar uma nova vis˜ ao da mecˆ anic a. Seria mais um livro de ensino dessa disciplina ou um livro de divulga¸ c˜ ao cient´ ıﬁca? Nenhum desses casos, sen˜ ao vejamos. Um livro que aﬁrme no pr ef´ acio: Este livro tem como objetivo apresentar as propriedades e carater´ ısticas desta nova vis˜ ao da mecˆ anica [···] ﬁca f´ acil fazer uma compara¸ c˜ ao com as vis˜ oes anteriores2do mundo 1Estes casos n˜ ao s˜ ao exatamente como os manuais descrevem m as n˜ ao ´ e nosso objetivo aqui dar detalhes deles. 2Os negritos s˜ ao nossos. 2(newtoniana e einsteiniana ),3 e que, al´ em disso, ´ e editado pelo Centro de L´ ogica, Episte mologia e Hist´ oria da Ciˆ encia da UNICAMP n˜ ao pode passar desapercebido pela c omunidade cient´ ıﬁca do pa´ ıs. Ele deve ser analisado, comentado, cri ticado pelos cien- tistas da mesma forma que o s˜ ao as teorias e resultados exper imentais da ciˆ encia normal . N˜ ao ´ e poss´ ıvel que algu´ em chegue dizendo que as vis˜ oes de Newton e Einstein est˜ ao erradas e ningu´ em da comunidade dos f´ ısicos diga nada. Conﬁrme-se e aceite-se seu impacto na f´ ısica e ci ˆ encias aﬁns ou coloque-se esta obra em merecido ostracismo. Esperamos d eixar claro neste artigo que, corretamente analisado, o assunto levant ado pelo livro em quest˜ ao sequer polˆ emico ´ e. Por´ em, depois das caracter´ ısticas acima men- cionadas, o livro tem de ser analisado criticamente. Tamb´ e m, acrescente-se, de maneira deﬁnitiva. ´E necess´ ario saber se de fato representa uma vis˜ ao nova da f ´ ısica porque, se for verdade, j´ a imaginaram, leitores? Ter´ ıamos que rev er tudo que foi feito nas ´ ultimas d´ ecadas, v´ arios prˆ emios Nobel deviam ser de volvidos, o Brasil estaria na vanguarda da ciˆ encia. Mas, e se n˜ ao fosse? Seria um exemplo deciˆ encia patol´ ogica ? Enﬁm ... deﬁnitivamente n˜ ao pode passar sem ser percebido, ainda que n˜ ao seja polˆ emico. J´ a foi feita uma resenha sobre o referido livro, publicada n esta revis- ta [SO99], da´ ı a “resenha” do t´ ıtulo. Nessa resenha n˜ ao se poupam elogios ` a nova vis˜ ao da f´ ısica pretendida no livro MR. No entanto, ´ e interessante notar que na vers˜ ao publicada dessa resenha foi acrescenta da uma nota de rodap´ e onde se agradece a um ´ arbitro anˆ onimo, o qual pedia para o autor da resenha ler o livro de A. Pais [PA95]. Nesse livro encontra-s e uma hist´ oria mais detalhada sobre a inﬂuˆ encia do princ´ ıpio de Mach no pe nsamento de Einstein. No entanto, a leitura do livro de Pais deveria ter i nduzido o autor da resenha a revˆ e-la toda e mesmo mudar sua opini˜ ao sobre o l ivro. Mas no pr´ oprio livro MR, apenas s˜ ao citadas as palavras de Eins tein sobre a inﬂuˆ encia que Mach teve sobre ele num certo per´ ıodo de sua v ida. Omite-se outras, e que n´ os incluimos aqui, nas quais Einstein revˆ e a sua posi¸ c˜ ao com rela¸ c˜ ao ao princ´ ıpio de Mach .4 Assim, ´ e nosso objetivo fazer uma cr´ ıtica ` a proposta da MR baseada nas teorias cient´ ıﬁcas desenvolvidas nos ´ ultimos 100 ano s, mais ou menos. Tentamos deixar claro para o leitor que: 1) n˜ ao ´ e verdade qu e as teorias da relatividade especial e geral (TRE e TRG, respectivamente) estejam erradas, 3Observe-se o tempo passado com rela¸ c˜ ao a f´ ısica newtonia na e einsteiniana. 4Dever´ ıamos dizer, em geral, da ﬁlosoﬁa de Mach. 3elas s˜ ao de fato muito bem veriﬁcadas experimentalmente; a l´ em do que, conceitualmente elas tˆ em permitido avan¸ cos t´ ecnicos e t e´ oricos em diversas ´ areas como a astronomia, a astrof´ ısica e, principalmente , na ´ area da f´ ısica das intera¸ c˜ oes fundamentais. 2) ´E sim, a MR que n˜ ao descreve os fenˆ omenos naturais observados. Nosso objetivo n˜ ao ´ e convencer o autor do livro que a sua pro posta n˜ ao concorda com os dados experimentais, mas procurar convence r o leitor, que por pouca familiaridade com as teorias da f´ ısica do Sec. XX p ode pensar estar diante de uma proposta que na sua opini˜ ao ´ e, na pior da s hipˆ oteses, pelo menos “cient´ ıﬁca”, perceber por si s´ o que o que n´ os co locamos aqui ´ e correto: n˜ ao s´ o as cr´ ıticas ` as TRE e TRG est˜ ao erradas, m as a pr´ opria MR est´ a h´ a muito tempo descartada pela experiˆ encia. Claro, n˜ ao esperamos que apenas a leitura desta resenha sej a suﬁciente para tal efeito. Ser´ a necess´ ario que o leitor que ainda tiv er d´ uvidas pro- cure consultar algumas das referˆ encias aqui citadas que po der˜ ao ser-lhe de utilidade, ainda que n˜ ao pretendamos ser exaustivos nesse aspecto. Na Sec. 2 revisamos o princ´ ıpio de Mach visando esclarecer q ual foi a sua inﬂuˆ encia sobre Einstein. Ficar´ a claro que a partir de certo momento Einstein afastou-se dele. Veriﬁca-se tamb´ em que esse prin c´ ıpio n˜ ao seria necess´ ario para a elabora¸ c˜ ao daquelas teorias (TRE e TRG ). Na Sec. 3 enfatizamos que as teorias da relatividade, especial e gera l, s˜ ao teorias bem estabelecidas experimentalmente e que n˜ ao procedem as cr´ ıticas a ambas feitas no livro MR [AK99]. Pelo contr´ ario, mostramos na Sec . 4 que a MR ´ e a teoria que est´ a errada. Alguns coment´ arios ﬁnais est˜ ao na Sec. 5. 2 O princ´ ıpio de Mach Ernst Mach (1838-1916) foi um cientista polivalente, mas a s ua maior in- ﬂuˆ encia foi na mecˆ anica de ﬂuidos e na ﬁlosoﬁa. Foi um cr´ ıt ico do conceito de espa¸ co absoluto da mecˆ anica newtoniana. No pref´ acio d a primeira edi¸ c˜ ao (alem˜ a) do seu livro disse [MA83] The present volume is not a treatise upon the application of the principles of mechanics. Its aim is to clear up ideas, exp ose the real signiﬁcance of the matter, and get rid of metaphysic al obscurities. Essas palavras devem ser entendidas no contexto do empirism o radical que muitos cientistas defendiam nas ´ ultimas d´ ecadas do Se c. XIX. A ter- modinˆ amica era ent˜ ao rainha absoluta como paradigma de ci ˆ encia. Estava 4baseada apenas em quantidades que podiam ser medidas no labo rat´ orio, outro tipo de abordagem era considerado metaf´ ısico . Isso inﬂuenciaria muito o pensamento de Einstein, mas depois ele aceitou que “´ e a teo ria que diz o que ´ e observ´ avel e o que n˜ ao ´ e” [HE78]. A. Pais, referindo-se ` a critica que Mach ﬁzera em seu livro d e 1883 [MA83] ` a mecˆ anica de Newton, disse [PA95a] As mencionadas referˆ encias mostram que Mach reconhecia claramente os aspectos cl´ assicos da mecˆ anica cl´ assica e que n˜ ao esteve longe de exigir uma teoria da relatividade geral, ist o h´ a cerca de meio s´ eculo antes! Por´ em Mach disse em 19135 Devo [ ···] com igual intensidade recusar ser precursor dos relativistas, como me retirei da cren¸ ca atomista da atuali dade. A vis˜ ao de Mach da mecˆ anica est´ a bem resumida na aﬁrma¸ c˜ a o de que quando [ ···] aﬁrmamos que um corpo conserva sem altera¸ c˜ ao sua dire¸ c˜ ao e velocidade no espa¸ co , nossa aﬁrma¸ c˜ ao n˜ ao ´ e nem mais nem menos do que uma referˆ encia abreviada ao universo inteiro (os it´ alicos s˜ ao de Mach) [ ···] Comentando as palavras de Mach acima Pais disse [PA95b]: N˜ ao encontramos no livro de Mach como se manifesta esta importˆ ancia de todos os corpos, pois ele nunca propˆ os um es - quema dinˆ amico expl´ ıcito para esta nova interpreta¸ c˜ ao da lei de in´ ercia. Isso ainda ´ e verdade: o princ´ ıpio de Mach n˜ ao foi implemen tado de maneira consistente por nenhuma teoria. O conhecido astron ˆ omo H. Bondi ´ e bem claro a respeito [BO68] ... the postulate of relativity of inertia (Mach’s principl e) is intelectually agreable in many ways, and seems to some autho rs to be inescapably true. Others regard it with suspicion, sin ce it has not been possible so far to express it in mathematical 5Esta frase est´ a traduzida de maneira diferente por diferen tes autores. Aqui queremos somente lembrar a intransigˆ encia de Mach, sendo que ele mes mo acreditava ser “n˜ ao dogm´ atico”. 5form (not even in the general relativity), and since it has no t so far been veriﬁed experimentally [ ···] Even if Mach’s princi- ple is correct, other theories are therefore required to dea l with experimental and observational invariance. Como dissemos antes, a constru¸ c˜ ao da TRE foi muito inﬂuenc iada pela ﬁlosoﬁa pragm´ atica de Mach: foram usadas apenas quantidad es pass´ ıveis de serem medidas. Einstein posteriormente tamb´ em se afastou dessa ﬁlosoﬁa, mas n˜ ao consideraremos isso aqui. Por outro lado, o mesmo ac onteceu com a TRG. Em 1912, usando uma vers˜ ao rudimentar da teoria da gra vita¸ c˜ ao, Einstein mostrou que se uma esfera oca massiva ´ e acelerada e m torno de um eixo que passa pelo centro no qual se encontra uma massa inerc ial pontual, ent˜ ao a massa inercial desta ´ ultima ´ e aumentada. Nas pr´ o prias palavras de Einstein [PA95c] Esta [conclus˜ ao] fornece plausabilidade ` a conjectura de que a in´ ercia totalde um ponto com massa ´ e um efeito que decorre da presen¸ ca de todas as outras massas, gra¸ cas a um tipo de intera¸ c˜ ao com estas ´ ultimas [ ···].´E este justamente o ponto de vista sustentado por Mach nas suas investiga¸ c˜ oes profu ndas sobre este tema. Vemos que Einstein tinha de fato o princ´ ıpio de Mach como gui a para a constru¸ c˜ ao das teorias da relatividade.6 Em 1917 Einstein, no que seria o primeiro trabalho da hist´ or ia sobre cosmologia relativista [EI17], ainda pensava de acordo com a cita¸ c˜ ao acima a respeito das id´ eias de Mach [PA95c]. Ele ainda tentava imp lementar uma origem inteiramente material da in´ ercia, isto ´ e, que a m´ e tricagµνdo espa¸ co- tempo seria determinada apenas pela mat´ eria [PA95c]. De fa to, nesse tra- balho Einstein introduz o “termo cosmol´ ogico” para estar e m acordo com o princ´ ıpio de Mach, isto ´ e, para ter um universo fechado, e tamb´ em para conseguir um universo homogˆ eneo, isotr´ opico e est´ atico e tal que gµν= 0 na ausˆ encia de mat´ eria. Provavelmente a demostra¸ c˜ ao de de Sitter em 1917 sobre a ex istˆ encia de solu¸ c˜ oes das equa¸ c˜ oes da TRG no v´ acuo: gµν∝negationslash= 0 e T= 0, isto ´ e, solu¸ c˜ oes para as equa¸ c˜ oes da TRG sem mat´ eria (que Einst ein acreditava n˜ ao existirem) ´ e que come¸ cou a minar sua credibilidade ne sse princ´ ıpio. A outra motiva¸ c˜ ao, de um universo homogˆ eneo, isotr´ opic oeest´ atico, foi 6De fato foi Einstein quem chamou a conjetura da origem da in´ e rcia de Mach como o “Princ´ ıpio de Mach”. 6eliminada quando em 1922 A. Friedmann demonstra que era poss ´ ıvel um universo homgˆ eneo e istr´ opico se ele estivesse se expandi ndo (e n˜ ao est´ atico como assumia Einstein). Mas o acontecimento crucial foi ent ˜ ao a descoberta de de Sitter que as equa¸ c˜ oes de Einstein com o termo cosmol´ ogico tinham solu¸ c˜ ao mesmo no vazio: a in´ ercia ´ e diferente de zero mes mo sem a presen¸ ca da mat´ eria. Inicialmente Einstein, que antes tinha dito qu e “um corpo num universo vazio n˜ ao poderia ter in´ ercia”, objetou a solu¸ c ˜ ao de de Sitter mas logo ele se convenceu que aquele tinha raz˜ ao. N˜ ao era mais p oss´ ıvel que gµνpudesse ser determinado completamente pela mat´ eria. Vemo s ent˜ ao que n˜ ao se pode fazer uma cita¸ c˜ ao de Einstein de 1917 sem levar em conta que alguns anos depois ele se convenceria de seu pr´ oprio erro! Segundo Pais [PA95f] Anos mais tarde, o entusiasmo de Einstein pelo princ´ ıpio de Mach esmoreceu e, ﬁnalmente desapareceu. Por exemplo, em 1954 em uma carta a Felix Pirani ele disse [HO8 2, PA95d] Na minha opini˜ ao nunca mais dever´ ıamos falar do princ´ ıpi o de Mach. Houve uma ´ epoca na qual pensava-se que os ‘corpos ponder´ aveis’ eram a ´ unica realidade f´ ısica e que, numa te oria todos os elementos que n˜ ao estiverem totalmente determina dos por eles, deveriam ser escrupulosamente evitados. Sou cons ciente que durante um longo tempo tamb´ em fui inﬂuenciado por essa id´ eia ﬁxa. Pouco tempo depois ele disse [SC49] ...So, if one regards as possible, gravitational ﬁelds of ar bi- trary extension which are not initially restricted by spati al lim- itations, the concept ‘acceleration relative to space’, th en loses every meaning and with it the principle of inertia together w ith the entire problem of Mach. Em geral os cosm´ ologos aceitam o ponto de vista posterior de Einstein, por exemplo, segundo Bondi [BO68] for this reason he introduced the so-called cosmological co n- stant in the hope of reconciling general relativity with Mac h’s principle [ ···] This hope was, however, not fulﬁlled. 7A origem da in´ ercia (das massas) continua a ser um ponto em ab erto em qualquer teoria fundamental das part´ ıculas elementare s. Assim segundo Pais [PA95d]: Do meu ponto de vista, at´ e agora o princ´ ıpio de Mach n˜ ao fez avan¸ car decisivamente a f´ ısica, e a origem da in´ ercia ´ e, e continua a ser, o assunto maisobscuro na teoria de part´ ıculas e campos. O princ´ ıpio de Mach pode, conseq¨ uentemente ter futuro, ma s n˜ ao sem a teoria quˆ antica. Podemos concluir que o princ´ ıpio de Mach n˜ ao foi at´ e agora conﬁr- mado nem te´ orica nem experimentalmente. Que todas as teori as atuais n˜ ao tenham sido capazes de implement´ a-lo pesa mais contra ele que con- tra as pr´ oprias teorias. O estudo da inﬂuˆ encia de Mach sobr e Einstein pertence mais ao que Holton chama de “el peregrinaje ﬁlos´ oﬁ co de Albert Einstein” [HO82b] un peregrinaje desde una ﬁlosof´ ıa de la ciencia en la que el sensacionalismo y el empirismo ocupaban una posici´ on cent ral, hasta otra que est´ a fundada en un realismo racional. Deﬁnitivamente ent˜ ao, a partir de um certo momento, Einste in e outros f´ ısicos bem conhecidos n˜ ao levaram mais em conta o princ´ ı pio de Mach como guia na constru¸ c˜ ao das suas teorias. 3 As Teorias da Relatividade est˜ ao erradas? N˜ ao. Muito pelo contr´ ario. Vide, por exemplo, o amplo arti go de Will [WI79], onde se resume os testes experimentais de ambas teorias da re latividade, a especial e a geral. No caso da relatividade especial, que ´ e s em d´ uvida a melhor testada, Will diz [WI79b]: A lot of experiments in the high-energy laboratory have ver- iﬁed and reveriﬁed the validity of special relativity in the limit when gravitational eﬀects can be ignored. Those experiment s range from direct test of time-dilation to tests of esoteric predic- tions of Lorentz-invariant quantum ﬁeld theory. No entanto o autor de MR insiste [AK99b]: 8Defendemos aqui que as teorias de Einstein n˜ ao implemen- taram as id´ eias de Mach e que a Mecˆ ancia Relacional ´ e uma teoria melhor do que as de Einstein para descrever os fenˆ ome nos observados na natureza [ ···] Einstein e seus seguidores criaram muitos problemas com esta teoria. A TRE tornou a hip´ otese do ´ eter sup´ erﬂua, n˜ ao mostrou que este n˜ ao existia. Isso ´ e t´ ıpico do conhecimento cient´ ıﬁco. O auto r do texto MR n˜ ao entendeu como funciona a ciˆ encia. A ciˆ encia n˜ ao mostra qu e os deuses da chuva e anjos carregando os planetas n˜ ao existem. Ela apena s n˜ ao usa essas hip´ oteses. Claro, se acreditamos que existe uma realidade independente de n´ os mesmos e que ´ e, pelo menos parcialmente, desvendada pela ciˆ encia, ent˜ ao o fato de o ´ eter n˜ ao ser necess´ ario para as teorias f ´ ısicas pode ser interpretado como indicativo de sua inexistˆ encia. As cr´ ıticas do autor ` a TRE n˜ ao s˜ ao corretas e mostram a pou ca familia- ridade dele com o tema. Por exemplo [AK99d] ...h´ a muitos problemas com as teorias da relatividade espe cial e geral. Enfatizamos alguns aqui. 1) elas s˜ ao baseadas na formula¸ c˜ ao de Lorentz da eletrodi nˆ amica de Maxwell, formula¸ c˜ ao que apresenta diversas assimetri as como as apontadas por Einstein e muitos outros [ ···] H´ a uma teoria do eletromagnetismo que evita todos estas assimetrias de fo rma natural [ ···] a eletrodinˆ amica de Weber... ´E sabido, faz mais de 100 anos, que a eletrodinˆ amica de Weber n˜ ao ´ e uma descri¸ c˜ ao dos fenˆ omenos eletromagn´ eticos: na sua vers ˜ ao original n˜ ao prevˆ e a existˆ encia de ondas eletromagn´ eticas! Da maneira como ´ e comparada com “a formula¸ c˜ ao de Lorentz da eletrodinˆ amica de Maxwell” p arece que a de Weber ´ e uma outra formula¸ c˜ ao da mesma. A formula¸ c˜ ao de L orentz a que se refere o autor ´ e a das equa¸ c˜ oes de Maxwell microsc´ opic as. Nela todos os fenˆ omenos eletromagn´ eticos podem ser vistos como send o produzidos por portadores de cargas elementares como os el´ etrons e os n´ uc leos atˆ omicos. As equa¸ c˜ oes de Maxwell macrosc´ opicas podem, em casos simpl es, ser deduzidas a partir das equa¸ c˜ oes de Maxwell-Lorentz. Na verdade ´ e a f ormula¸ c˜ ao de Lorentz que ´ e generaliz´ avel para a mecˆ anica quˆ antica re lativista. A assimetria a que se refere o autor ´ e aquela mencionada no pr imeiro artigo de Einstein de 1905 sobre a TRE [EI05, EI05b]: Como ´ e sabido, a Eletrodinˆ amica de Maxwell–tal como atual - mente se concebe– conduz, na sua aplica¸ c˜ ao a corpos em movi - 9mento, a assimetrias que n˜ ao parecem ser inerentes aos fe- nˆ omenos .7Consideremos, por exemplo, as a¸ c˜ oes eletrodinˆ amicas entre um ´ ım˜ a e um condutor. O fenˆ omeno observ´ avel de- pende unicamente do movimento relativo do condutor e do ´ ım˜ a , ao passo que, segundo a concep¸ c˜ ao habitual, s˜ ao ni- tidamente distintos os casos em que o m´ ovel ´ e um, ou outro, desses corpos. Assim, se for m´ ovel o ´ ım˜ a e o condutor estiv er em repouso, estabelecer-se-´ a em volta do ´ ım˜ a campo el´ etri co com determinado conte´ udo energ´ etico, que dar´ a origem a uma c or- rente el´ etrica nas regi˜ oes onde estiverem colocadas por¸ c˜ oes do condutor. Mas, se ´ e o ´ ım˜ a que est´ a em repouso e o condutor q ue est´ a em movimento, ent˜ ao, embora n˜ ao se estabele¸ ca em vo lta do ´ ım˜ a nenhum campo el´ etrico, h´ a no entanto uma for¸ ca el etro- motriz que n˜ ao corresponde a nenhuma energia, mas que d´ a lu - gar a correntes el´ etricas de grandeza e comportamento igua is ` as do primeiro caso, produzidas por for¸ cas el´ etricas–desde que, nos dois casos considerados, haja identidade no movimento rela tivo. Mais adiante, depois de apresentar a sua teoria, Einstein di z [EI05c] Como se vˆ e, na teoria que se desenvolveu, a for¸ ca eletromot riz apenas desempenha o papel de conceito auxiliar, que deve a su a introdu¸ c˜ ao ao fato de as for¸ cas el´ etricas e magn´ eticas n˜ ao terem existˆ encia independente do estado de movimento do sistema de coordenadas. ´E tamb´ em claro que a assimetria mencionada na introdu¸ c˜ ao , que surge quando se consideram as correntes el´ etricas prov o- cadas pelo movimento relativo de um ´ ıman e de um condutor, desaparece agora. Vemos ent˜ ao que o autor da MR n˜ ao entendeu o argumento de Ein - stein no seu artigo de 1905. Hoje dir´ ıamos que as equa¸ c˜ oes de Maxwell, usando a nota¸ c˜ ao de 3-vetores, introduzida por Heaviside , n˜ ao s˜ ao manifes- tamente invariantes sob as transforma¸ c˜ oes de Lorentz. Ma s essa assimetria n˜ ao ocorre, como observado pelo pr´ oprio Einstein, nos fen ˆ omenos obser- vados, como sabemos desde Faraday. A assimetria desaparece porque no sistema de referˆ encia que acompanha o condutor, do ponto de vista da TRE, temos tamb´ em um campo el´ etrico: /vectorE′∝v×/vectorB′. A inter-rela¸ c˜ ao entre campos el´ etricos e magn´ eticos na eletrodinˆ amica d e Maxwell s´ o foi 7Os negritos s˜ ao nossos. 10descoberta por Einstein. Ainda que a teoria microsc´ opica ( que ´ e de Lorentz mas continua sendo a eletrodinˆ amica de Maxwell) seja relat ivisticamente invariante, o fenˆ omeno mencionado foi percebido por Einst ein mesmo. As- sim, ´ e apenas quando se descobre a invariˆ ancia das equa¸ c˜ oes de Maxwell sob transforma¸ c˜ oes de Lorentz que a assimetria desaparec e. Isso est´ a bem explicado em livros b´ asicos como o de Purcell [PU78] apenas para dar um exemplo. Outro ponto que deve ser enfatizado ´ e que o autor da MR n˜ ao co m- preendeu a covariˆ ancia geral, confundindo-a com a covariˆ ancia introduzida por Minkowski que se refere apenas ` as transforma¸ c˜ oes de L orentz [AK99j]. Na TGR sim, temos uma covariˆ ancia geral, no sentido que as eq ua¸ c˜ oes s˜ ao as mesmas em qualquer sistema de referˆ encia, inercial ou n˜ ao. Como j´ a foi dito acima, a TRE n˜ ao ´ e veriﬁcada somente pelas ex- periˆ encias diretas. Todo o edif´ ıcio conceitual da f´ ısic a de part´ ıculas ele- mentares e as suas t´ ecnicas te´ oricas e experimentais est˜ ao baseados nela. Mesmo que algu´ em mostrasse que as experiˆ encias cl´ assica s n˜ ao s˜ ao suﬁ- cientes para testar com a precis˜ ao necess´ aria a teoria, es ta n˜ ao seria facil- mente abandonada porque j´ a foi conﬁrmada na pr´ atica em out ras ´ areas. O mesmo ocorre com a TRG: nos anos de 1939-40 Einstein, com Leo pold Infeld e Banesh Hoﬀmann, tratou o problema do movimento de N c orpos com a relatividade geral. Segundo Misner et al. [MI73] Equations [ ···] are called the Einstein-Infeld-Hoﬀman (EIH) equations for the geometry and evolution of a many-body syst em. They are used widely in analyses of planetary orbits in the so lar system. For example, the Caltech Jet Propulsion Laboratory uses them, in modiﬁed form, to calculate ephemerides for hig h- precision tracking of planets and spacecraft. Vemos ent˜ ao que j´ a existem aplica¸ c˜ oes da TRG (ver mais so bre isso maisa- diante). Al´ em disso novos testes mais acadˆ emicos s˜ ao obtidos. Por exemplo, em 1993 R. A. Hulse e J. H. Taylor ganharam o prˆ emio Nobel de F´ ıs ica: ”for the discovery of a new type of pulsar, a discovery that has ope ned up new possibilities for the study of gravitation” [NO93]. Mas o que isso signiﬁca? Bem, Hulse e Taylor observaram duran te quase 20 anos, de 1975 a 1993, um pulsar bin´ ario com o ex´ otico nome de PSR 1913+168–e que consiste de um par de estrelas de nˆ eutrons, com um raio 8PSR signiﬁca “pulsar” e 1913+16 especiﬁca a posi¸ c˜ ao do pul sar no c´ eu. 11de algumas dezenas de quilˆ ometros e com massa da ordem da mas sa do Sol e com uma distˆ ancia relativa da ordem da algumas vezes a distˆ ancia Terra-Lua girando ao redor de seu centro de massa. Eles determinaram qu e a perda de energia do sistema era consistente com os c´ alculos basea dos na teoria da relatividade geral [PE98]. Este foi um teste de TRG mais deﬁn itivo que os trˆ es testes cl´ assicos: o perih´ elio de Merc´ urio, o desvi o da luz pelo Sol e o atraso de rel´ ogios em campos gravitacionais. Estes testes estavam restritos ao nosso sistema solar onde o campo gravitacional ´ e fraco. O s resultados de Hulse e Taylor foram os primeiros testes de grande precis˜ ao da TGR. Segundo a TRG, objetos em ´ orbita, como no caso do pulsar acim a men- cionado, irradiam energia sob a forma de ondas gravitaciona is (ondula¸ c˜ oes no espa¸ co-tempo). Isto implica numa perda de energia do sis tema que pode ser calculada usando a TRG. Os resultados de Hulse e Taylor co ncordaram muito bem (´ e um n´ umero da ordem de 10−14e medido com uma precis˜ ao de 0.5% !) com as previs˜ oes te´ oricas da TGR. Em 100 anos de prˆ emios Nobel apenas em 6 ocasi˜ oes n˜ ao foi en tregue. Deste total, 27 est˜ ao relacionados de alguma maneira com a r elatividade especial e pelo menos 1 com a TGR. Estes s˜ ao: P. A. M. Dirac (19 33, teoria relativista do el´ etron), J. Chadwick (1935, descob erta do nˆ eutron), C. D. Anderson (1936, descoberta da anti-mat´ eria); E. O. La wrence (1939, inven¸ c˜ ao do ciclotron); W. Pauli (1945, princ´ ıpio de exc lus˜ ao); H. Yukawa (1949, pelo m´ esons π); J. D. Cockcroft e E. T. S. Walton (1951, acelerado- res de part´ ıculas); W. E. Lamb e P. Kush (1955, efeitos relat iv´ ısticos nos ´ atomos); C. N. Yang e T. D. Lee (1957, viola¸ c˜ ao da paridade ); E. G. Segr´ e e O. Chamberlein (1959, descoberta de anti-mat´ eria hadrˆ on ica: anti-pr´ oton); E. P. Wigner (1963, princ´ ıpios de simetria); S-I. Tomonaga , J. Schwinger e R. Feyman (1965, pela eletrodinˆ amica relativista); H. A. B ethe (1967, pelos mecanismo relativistas da cria¸ c˜ ao da energia nas estrela s); L. W. Alvarez (1968, descobertas experimentais em f´ ısica de part´ ıcula s elementares); M. Gell-Mann (1969, contribui¸ c˜ oes te´ oricas ` a f´ ısica de p art´ ıculas elementares); B. Richter e S. C. C. Ting (1976, descoberta do quark b); S. Glashow, A. Salam e S. Weinberg (1979, modelo de intera¸ c˜ oes eletrof racas); J. W. Cronin e V. L. Fitch (1980, descoberta da viola¸ c˜ ao da simet ria discreta CP); S. Chandrasekhar (1983, evolu¸ c˜ ao e estrutura das est relas) C. Rubbia e S. van der Meer (1984, descoberta dos b´ osons intermedi´ ar iosW±,Z0); L. M. Lederman, M. Schwartz e J. Steinberg (1988, descoberta do segundo neutrino, νµ); G. Charpac (1992, detetores de part´ ıculas relativistas ); M. Perl e F. Reines (1995, descobertas do l´ epton τe dete¸ c˜ ao do neutrino do eletron, νe, respectivamente); G. ’t Hooft e M. J. G. Veltman (1999, corr e¸ c˜ oes 12quˆ anticas ao modelo eletrofraco de Glashow-Salam-Weinbe rg). Todas estas descobertas te´ oricas ou experimentais somente tˆ em senti do no contexto de teorias quˆ antico-relativistas. Com rela¸ c˜ ao ` a TGR pode mos colocar os j´ a acima mencionados R. A. Hulse e J. H. Taylor (1993, dete¸ c˜ ao indireta de ondas gravitacionais). N˜ ao mencionamos aqui alguns resul tados tamb´ em premiados que de maneira indireta usam a eletrodinˆ amica de Maxwell, que sendo relativista poderia ser considerado como teste indir eto da TRE. Se as teorias da relatividade estivessem erradas todos esses prˆ emios Nobel teriam que ser devolvidos. O leitor interessado pode visitar a p´ ag ina WWW da Funda¸ c˜ ao Nobel [NO00]. Com rela¸ c˜ ao ` a dilata¸ c˜ ao do tempo nas TRE e TRG, o astrˆ on omo real Martin Rees diz que mesmo n˜ ao sendo percept´ ıvel nos movime ntos e tempos do dia-a-dia [RE00] Esse pequeno efeito [dilata¸ c˜ ao do tempo da TRE] foi agora, contudo, medido por experimentos com rel´ ogios atˆ omicos c om precis˜ ao de um bilion´ esimo de segundo, e est´ a de acordo co m as previs˜ oes de Einstein... uma “dilata¸ c˜ ao do tempo” semel hante ´ e causada pela gravidade: nas proximidades de uma grande mass a, os rel´ ogios tendem a andar mais devagar.... essa dilata¸ c˜ ao deve ser levada em conta, juntamente com os efeitos do movimento orbital, na programa¸ c˜ ao do notavelmente preciso sistema GPS (Global Positioning Satellite)... De fato, atualmente o sistema GPS tem uma precis˜ ao de milime tros, uma discordˆ ancia de uma milhon´ esima de segundo implica num er ro da ordem de 300 metros! [HE96]. Al´ em disso as cada vez mais precisas medidas do fator ( g−2)µs˜ ao compat´ ıveis com dilata¸ c˜ oes da vida m´ edia do m´ uon de at´ eγ= 29.3 [BR01] Isso mostra que o efeito nos m´ uons n˜ ao apenas os observados na atmosfera e o argumento do autor da MR n˜ ao se sustenta (ver a proxima se¸ c˜ ao). Poder´ ıamos mencionar outras situa¸ c˜ oes onde ﬁca claro o p ouco conheci- mento que o autor de MR tem das teorias da relatividade. Basta m mais um exemplos: o autor da MR n˜ ao sabe que n˜ ao existe “paradoxo do s gˆ emeos”, n˜ ao entendeu o atraso do rel´ ogio [AK99l]. N˜ ao comentamos mais sobre este ponto porque ´ e bastante bem considerado em livros elementa res de relativi- dade [GE78]. 134 Est´ a a Mecˆ anica Relacional errada? Sim. Ap´ os criticar a TRG de Einstein por n˜ ao ter implementa do a rota de construir a teoria apenas em termos de distˆ ancias relativa s, diz [AK99c] ... como veremos neste livro, ´ e poss´ ıvel seguir esta rota c om sucesso utilizando uma lei de Weber para a gravita¸ c˜ ao. Por que usar uma lei da gravita¸ c˜ ao baseada numa lei da eletr ost´ atica que n˜ ao deu certo? Mesmo que algu´ em acredite na MR, dever-se-ia per guntar: por que essa for¸ ca e n˜ ao outra? Assim, existiriam tantas MR qua nto poss´ ıveis autores. O papel desempenhado pelas simetrias nas leis da F´ ısica n˜ ao foi comprendido pelo autor da MR: ele ignora os trabalhos de cien tistas como E. Wigner, H. Weyl, C. N. Yang etc! As simetrias tˆ em desempen hado um papel importante na descoberta de novas leis da natureza, ma s na MR lemos que tudo isso n˜ ao ´ e necess´ ario na nova f´ ısica ali proposta! No momento que se abre m˜ ao dos princ´ ıpios de simetria tudo ´ e v´ alido! A MR est´ a baseada em trˆ es postulados. Os dois primeiros s˜ a o com- pat´ ıveis com as leis de Newton. J´ a o terceiro postulado, di z [AK99e] A soma de todas as for¸ cas de qualquer natureza (gravita- cional, el´ etrica, magn´ etica, el´ astica, nuclear,...) a gindo sobre qualquer corpo ´ e sempre nula em todos os sistemas de referˆ e ncia. Bom, sabemos que cada uma das for¸ cas mencionadas no postula do III tem uma intensidade carater´ ıstica bem diferente. Por exemplo , a for¸ ca gravita- cional ´ e 10−40vezes mais fraca que a for¸ ca eletromagn´ etica. Assim, se a s ua soma se anula, ent˜ ao devem existir outras for¸ cas tais que f a¸ cam a soma ser zero. Onde est˜ ao essas for¸ cas? Compare com o postulado de Einstein “a luz, no espa¸ co vazio, se propaga sempre com uma velocidade determinada, independente do est ado de movi- mento da fonte luminosa”. Na melhor das hip´ oteses, em 1905 estes dois postulados pode riam ter sido considerados como alternativas poss´ ıveis. Hoje, dep ois de tantos testes experimentais e te´ oricos, n˜ ao mais. Mas, na MR se insiste na eletrodinˆ amica de Weber, por exempl o [AK99f] As propriedades e vantagens da teoria eletromagn´ etica de Weber foram consideradas em outro livro. 14Essa teoria n˜ ao tem nenhuma vantagem, ela j´ a foi descartad a como pro- posta cient´ ıﬁca. Enfatizamos, um s´ eculo de experimentos e aplica¸ c˜ oes tec- nol´ ogicas e, n˜ ao menos importante, os esquemas conceitua is constru´ ıdos a partir da eletrodinˆ amica de Maxwell n˜ ao deixam espa¸ co pa ra ela. Lembre-se disso, caro leitor, quando assistir televis˜ ao ou ouvir a su a m´ usica favorita no seuCD player . Da eletrodinˆ amica se passa ` a gravita¸ c˜ ao, o autor contin ua ... em analogia a eletrodinˆ amica de Weber, propomos como a base para a mecˆ anica relacional que a lei de Newton da gravit a¸ c˜ ao universal seja modiﬁcada para ﬁcar nos moldes da lei de Weber . O leitor deve se convencer por ele mesmo que teorias de campo n ˜ ao re- lativ´ ısticas n˜ ao est˜ ao de acordo com a experiˆ encia. O de slocamento Lamb e o momento magn´ etico do el´ etron e do m´ uon s˜ ao exemplos, en tre outros, da validade dessas teorias.9 Mais ainda, os testes mais fortes, repetimos, de uma teoria s ˜ ao os indi- retos. Por exemplo, toda a f´ ısica de aceleradores n˜ ao seri a poss´ ıvel sem a TRE. Como foi mencionado na se¸ c˜ ao anterior, mesmo o sistem a GPS est´ a usando ambas TRE e TRG. Como explicar esse sucesso no context o da MR? O fato que os testes indiretos passam a ser mais importantes q ue os dire- tos (que s˜ ao importantes quando se est´ a propondo uma teori a) faz com que caso algu´ em hoje repetisse as experiˆ encias de Michelson- Morley ou Fizeau e aﬁrmasse ter achado resultados opostos aos das experiˆ enc ias originais, o experimento ser´ a encarado como errado! [De fato isso acont eceu com a experiˆ encia de Michelson-Morley: em 1926 um f´ ısico chego u a conclus˜ oes opostas. Nunca se conﬁrmou onde estava o erro mas j´ a n˜ ao era mais necess´ ario ach´ a-lo!]. ´E isso que quer dizer o conhecido f´ ısico, premio No- bel de 1977, P. W. Anderson quando aﬁrma que [AN90] It is the nature of physics that its generalizations are cont in- ually tested for correctness and consistency not only by car eful experiments aimed directly at them but usually much more severely10, by the total consistency of the entire structure of physics [ ···] My moral ﬁnally, is that physics–in fact all of science– is a pretty seamless web. 9Estes s˜ ao os c´ alculos esot´ ericos de teoria quˆ antica de c ampos mencionada por Will acima. 10Os negritos s˜ ao nossos. 15A MR ´ e uma teoria de tempo absoluto e n˜ ao passa por testes que evi- denciam a “dilata¸ c˜ ao do tempo”. E mais, os argumentos (fra cos) contra a TRE e TGR parecem ser motivados pelo fato do autor perceber qu e quem as aceita n˜ ao pode aceitar a MR. A dilata¸ c˜ ao do tempo em campos gravit´ atorios ´ e particul armente im- portante para demonstrar que os resultados da MR s˜ ao incons istentes com as observa¸ c˜ oes. Seja por exemplo o caso de um corpo preso a u ma mola oscilando horizontalmente. Este caso ´ e considerado no MR [ AK99m] e o resultado ´ e que a freq¨ encia de oscila¸ c˜ ao ´ e dada por ω=/radicalBigg k mg(1) ondek´ e a constante el´ astica da mola. Se observa no MR que a difere n¸ ca com o resultado na mecˆ anica newtoniana ´ e que na MR aparece mge n˜ ao mi. O problema ´ e quando se usa o resultado da Eq. (1) para aﬁrmar [ AK99m]: Dobrando a quantidade de gal´ axias do universo, mantendo inalteradas a mola, a Terra e o corpo de prova, diminuiria a freqˆ’encia de oscila¸ c˜ ao em√ 2. Isto ´ e equivalente a dobrar a massa inercial newtoniana do corpo de prova. Isto ´ e, se extrapola um resultado que na pr´ atica coincide c om o da mecˆ anica de Newton (´ e por isso n˜ ao ´ e importante) para o Un iverso todo! Qualquer sistema peri´ odico ´ e um relo´ ogio. Acontece que s e usamos isso para calcular a diferen¸ ca de tempos de 2 sistemas de molas, um na b ase e outro no alto de uma torre, a diferen¸ ca de tempos segundo a MR ´ e: ze ro! Segundo a MR teriamos τ2−τ1 τ1=N/ω 2−N/ω 1 N/ω 1≡0 (2) ondeN´ e o n´ umero de oscila¸ c˜ oes e ω1,2s˜ ao a freq¨ uencias na base e no alto da torre. A identidade decorre da igualdade entre ω1eω2uma vez que pela Eq. (1) as freq¨ uˆ encias so dependem da massa gravitaci onal do corpo a qual ´ e inalter´ avel.11No entanto como mencionado acima essa diferen¸ ca dos rel´ ogios em campos gravitat´ orios j´ a foi bem testada e est ´ a em acordo com as teorias da relatividade. De fato, experimentos que medem o desvio para 11Os autores agradecem G. E. A. Matsas discus˜ oes sobre este pu nto. De fato, a Eq. (2) foi colocada pela primeira vez no debate realizado no IFGW-U NICAMP entre o autor da MR e Matsas. 16o vermelho gravitacional usando rel´ ogios em torres e o sist ema GPS, como comentado acima, conﬁrmam a TRG [MI73b]. ´E interessante a aﬁrma¸ c˜ ao com rela¸ c˜ ao ` a dilata¸ c˜ ao do tempo necess´ aria para explicar a chegada de p´ ıons e m´ uons produzidos na atmo sfera at´ e a Terra [AK99g]: o mesmo pode ser aplicado na experiˆ encia dos m´ esons. Ao inv´ es de aﬁrmar que o tempo anda mais lentamente para o corpo em movimento, nos parece mais simples e de acordo com a ex- periˆ encia aﬁrmar que a meia-vida do m´ eson depende ou dos ca m- pos eletromagn´ eticos a que foi exposto nesta situa¸ c˜ ao ou ao seu movimento (velocidade ou acelera¸ c˜ ao) em rela¸ c˜ ao ao lab orat´ orio e aos corpos distantes. Acontece que a dilata¸ c˜ ao do tempo foi medida em circunstˆ a ncias diversas: em aceleradores, em experimentos em avi˜ oes e sat´ elites, e m experiˆ encias que medem o fator g−2 do m´ uon, etc. Quais campos eletromagn´ aticos se aplicariam nestes casos? Mesmo na atmosfera, se existissem campos eletro- magn´ eticos teriam outros efeitos por exemplo nas comunica ¸ c˜ oes via sat´ elite. Vemos ent˜ ao que ´ e a MR que n˜ ao d´ a conta dos fatos observado s como demostrado acima, um rel´ ogio na base de uma torre atrasa com rela¸ c˜ ao a um rel´ ogio no topo da mesma devido ` a i nﬂuˆ encia do campo gra vitacional da Terra. A Eq. (2) mostra ent˜ ao que segundo a mecˆ anica rela cional, mo- las oscilando horizontalmente a Terra n˜ ao tˆ em sua freq¨ uˆ encia, dadas pela Eq. (1), alterada estejam elas na base ou no topo da torre. Tai s osciladores seriam apenas um exemplo de rel´ ogios que n˜ ao atrasariam de vido ao campo gravitacional! A express˜ ao dada na MR para a oscila¸ c˜ ao de pende apenas da massa gravitacional e da constante da mola, que s˜ ao conce itos primitivos em sua teoria e portanto n˜ ao sofrem altera¸ c˜ ao das estrela s ﬁxas. Finalmente, a MR n˜ ao ´ e uma teoria de campos e portanto n˜ ao p revˆ e a emiss˜ ao de ondas gravitacionais de maneira natural. Apes ar dos grandes detectores terrestres de ondas gravitacionais ainda n˜ ao e starem em funciona- mento, ondas gravitacionais j´ a foram indiretamente obser vadas em sistemas astrof´ ısicos bin´ arios, como mencionado antes. 5 Coment´ arios ﬁnais O fato de uma teoria satisfazer ou n˜ ao o Princ´ ıpio de Mach (e m qualquer uma de suas formula¸ c˜ oes) n˜ ao pesa a favor ou contra a teori a, visto que n˜ ao 17h´ a qualquer experimento comprovando a validade dele, mesm o porque seria bastante dif´ ıcil mover todas as estrelas do ﬁrmamento! A ex periˆ encia do balde n˜ ao pode ser considerada uma veriﬁca¸ c˜ ao experimen tal do princ´ ıpio como ´ e aﬁrmado na MR. Sabemos que a lei de Coulomb tem corre¸ c˜ oes de origem quˆ ant ica, mas nem por isso dizemos que a lei deve ser mudada, apenas reconhe ce-se que num determinado contexto (o ˆ atomo de hidrogˆ enio, por exem plo) outros fa- tores s˜ ao importantes. No caso da lei da gravita¸ c˜ ao de New ton, que tinha sido testada para distˆ ancias maiores que 1 cm, pensava-se q ue poderiam ocorrer desvios para distˆ ancias da ordem de µm. Especula-se por exemplo, que, se existissem das dimens˜ oes espaciais extras, o poten cial gravitacional de Newton seria substitu´ ıdo por uma express˜ ao mais geral. Trata-se de uma proposta te´ orica, por´ em medidas recentes na escala de 200µm n˜ ao mostram desvios da lei de gravita¸ c˜ ao de Newton [HO01]. Mes mo que esse tipo de teorias venha a ser conﬁrmada no futuro ainda assim co ntinuaremos a usar o potencial de Newton em muitas das aplica¸ c˜ oes em dis tˆ ancias de µmetros at´ e milhares de quilˆ ometros ou, dependendo da prec is˜ ao, a TRG. Mesmo que desvios da lei da gravita¸ c˜ ao fossem um dia observ ados, cabe ressaltar que seriam oriundos em teorias consistentes na ma ioria dos as- pectos com a TRG. Estas teorias tˆ em de acrescentar outros in gredientes te´ oricos, como simetrias extras ou mais dimens˜ oes espaci ais. Por outro lado, na eletrodinˆ amica quˆ antica temos o deslocamento La mb, efeito bem medido no atˆ omo de hidrogˆ enio, que implica numa corre¸ c˜ a o ao potencial de Coulomb [HA84]. O que aprendemos com estes exemplos? A respo sta ´ e que temos de ter sempre em mente em que contexto uma modiﬁca¸ c˜ ao ´ e feita numa lei b´ asica. Um aspecto que deve ser notado ´ e que no pref´ acio da MR [AK99h ] aparece o seguinte Este livro ´ e direcionado a f´ ısicos, matem´ aticos, engenh eiros, ﬁl´ osofos e historiadores da ciˆ encia [ ···] Acima de tudo, ´ e escrito para as pessoas jovens e sem preconceitos que tˆ em interesse nas quest˜ oes da f´ ısica. Deveria ser acrescentado e com pouco senso cr´ ıtico , porque para aceitar a MR, depois das observa¸ c˜ oes acima discutidas, ´ e preciso, isto sim, ter precon- ceitoa favor da mecˆ anica relacional. Ainda no pref´ acio podemos ler [AK 99i] Ap´ os compreender a mecˆ anica relacional entraremos num novo mundo, enxergando os mesmos fenˆ omenos com olhos difer - entes e sob uma nova perspectiva. ´E uma mudan¸ ca de paradigma. 18O autor se refere ao conceito de paradigma cient´ ıﬁco introd uzido po T. Kuhn. Sinceramente, leitor, se vocˆ e tem interesse nas ques t˜ oes da f´ ısica, por acaso leu em algum lugar que Einstein, Heisenberg, Bohr, Dirac, Fermi, Pauli, e tantos outros conhecidos cientistas ﬁzeram logo de in´ ıcio esse tipo de aﬁrma¸ c˜ ao? Vamos al´ em: esses autores escreveram livro s sobre as suas teorias somente depois de alguns anos e quando a comunidade d e f´ ısicos era majoritariamente a favor delas. Enﬁm, a verdadeira f´ ısica nova s´ o se percebe depois de certo tempo, mesmo para aqueles que a pro puseram. Para Pais [PA95e] A nova dinˆ amica contida nas equa¸ c˜ oes relativistas gener ali- zadas n˜ ao foi completamente dominada, nem durante a vida de Einstein, nem no quarto de s´ eculo que se seguiu ` a sua morte [ ···] nem mesmo num n´ ıvel puramente cl´ assico, ningu´ em pode hoj e em dia gabar-se de ter um dom´ ınio completo do rico conte´ udo dinˆ amico da dinˆ amica n˜ ao linear designada por relativid ade geral. Apenas na ciˆ encia patol´ ogica as coisas s˜ ao enganosamente claras de uma vez por todas. Ali´ as essa ´ e, de fato, uma maneira de identiﬁ c´ a-la. Os cien- tistas tˆ em preconceitos, mas mesmo estes est˜ ao, na maiori a dos casos, bem fundamentados. Agora sabemos, por exemplo, que o esquema de Copernico- Kepler-Galileu precisava de uma f´ ısica nova, aﬁnal formul ada por Newton; que o que Boltzmann queria n˜ ao era poss´ ıvel sem a mecˆ anica quˆ antica, que ainda n˜ ao tinha sido descoberta.12 Assim, n˜ ao ´ e apenas pelos testes diretos, desde Michelson e Morley, que a relatividade ´ e aceita como correta em certo dom´ ınio de fe nˆ omenos. Mais importante ainda ´ e a consistˆ encia que ela trouxe para dive rsos dom´ ınios: astronomia, o sistema GPS, aceleradores de part´ ıculas, f´ ısica sub-atˆ omica, etc. Como exemplo de ciˆ encia patol´ ogica podemos lembrar do cas o Velikovsky [LA99, EB00]. Immanuel Velikovsky (1895-1879) propˆ os uma teoria as- tronˆ omica em seu livro “Worlds in Collision”. Ali ele dava a rgumentos so- bre uma s´ erie de cat´ astrofes ocorridas na Terra, uma delas teria provocado a abertura do Mar Vermelho para que os judeus vindos de Egito p udessem atravessar o mar. Ao ser questionado sobre a inexistˆ encia d e outros registros al´ em da B´ ıblia sobre esse tipo de cat´ astrofe ele argument ava: “amn´ esia co- letiva provocada pelas mesmas cat´ astrofes”! S˜ ao esses ar gumentos de na- tureza ad hoc que caraterizam a ciˆ encia patol´ ogica . Em particular, segundo 12Mas seus m´ etodos e princ´ ıpios estavam corretos. Apenas a n atureza n˜ ao os realizava da maneira que ele acreditava. 19o qu´ ımico e prˆ emio Nobel I. Langmuir [ST00], a ciˆ encia pat ol´ ogica tem a seguintes carater´ ısticas (existem outras mas estas s˜ ao m ais relacionadas com a ciˆ encia experimental): Fantastic theories contrary to experiences and criticisms are met by ad hoc excuses thought up on the spur moment. As cr´ ıticas no livro MR ` as TRE e TRG s˜ ao desse tipo, o argume nto de “campos magn´ eticos” para expilcar a dilata¸ c˜ ao do tempo n o caso dos raios c´ osmicos ´ e caracter´ ıstico desse tipo de argumenta¸ c˜ ao ad hoc . J´ a foi discu- tido na se¸ c˜ ao anterior que isso n˜ ao procede, porque a dila ta¸ c˜ ao do tempo j´ a foi medida em diversas situa¸ c˜ oes e concorda bem com a TR E e a TRG. Ao obstinadamente negar estas teorias e tudo que elas implic am, o autor da MR ´ e obrigado a recorrer a processos misteriosos e invocar f ontes ainda n˜ ao investigadas, mas que tenham, para o incauto leitor,uma aur a de plausibili- dade (efeitos novos, campos magn´ eticos desconhecidos, et c.). Por exemplo, no problema da precess˜ ao das ´ orbitas dos planetas para ﬁxa r o valor ob- servado ´ e introduzido de forma ad hoc um parˆ ametro extra, ξ[AK99k] que deve valer ξ= 6 para concordar com as medi¸ c˜ oes. Por ´ ultimo e n˜ ao menos importante, o que dizer com rela¸ c˜ a o ao ensino de f´ ısica no terceiro grau? Como apresentar para os estudan tes, sob o ponto de vista da MR, aﬁrma¸ c˜ oes como as que seguem (tomadas do res peitado e muito usado livro texto de Purcell [PU78b]) Today we see in the postulate of relativity and their implica - tions a wide framework, one that embraces all physical laws a nd not solely those of electromagnetism. We expect any complet e physical theory to be relativistically invariant. Ensinar-se-ia a um grupo de estudantes “sem preconceitos” q ue estas frases est˜ ao erradas?, que toda a f´ ısica do Sec. XX tamb´ em esta? N ˜ ao seria um crime deixar os estudantes nessa ignorˆ ancia? Mas n˜ ao apenas no ensino a n´ ıvel de segundo e terceiro grau. Por exem- plo, n˜ ao ´ e conceb´ ıvel que um bi´ ologo, qu´ ımico ou f´ ısic o de outra especial- idade, digamos de estado s´ olido ou ciˆ encia dos materiais, use ferramentas como luz s´ ıncroton e acredite que a eletrodinˆ amica de Webe r ainda pode- ria ser considerada uma teoria rival ` aquela que permitiu a c onstru¸ c˜ ao do aparelho que usa nas suas pesquisas. Finalmente, gostar´ ıamos de observar o seguinte. Mesmo se n os restringir- 20mos ` a ciˆ encia normal13podemos distinguir, numa mesma ´ area, diferentes comunidades. A primeira divis˜ ao ´ e pela especializa¸ c˜ ao . Em geral uma co- munidade tem uma ou v´ arias revistas nas quais publica assun tos de um interesse que serve para deﬁnir essa comunidade. A maioria d as referˆ encias usadas no livro MR est˜ ao em revistas onde n˜ ao s˜ ao usualmen te encontrados trabalhos da ciˆ encia normal . Se algu´ em tem argumentos v´ alidos de que as TRE e TRG est˜ ao erradas (esse ali´ as j´ a seria um resultado i mpressionante) deveria publicar em revistas como Physical Review Letters. De nada adi- anta argumentar que essas revistas n˜ ao publicariam, que tˆ em preconceito etc. Isso mostra que as pessoas que apoiam os pontos de vista d a MR per- tencem a uma comunidade marginal, isolada das principais te ndˆ encias da f´ ısica. Tudo bem em outro lugar, mas ...na UNICAMP? Para terminar, esperamos ter deixado claro duas coisas: 1)n˜ ao ´ e apenas pelos testes diretos, desde Michelson e Morley, que as teori as da relatividade s˜ ao aceitas como corretas em certo dom´ ınio de fenˆ omenos. Mais importante ainda ´ e a consistˆ encia que ela trouxe para diversos dom´ ın ios: astronomia, aceleradores de part´ ıculas, f´ ısica sub-atˆ omica, o sist ema GPS etc. 2)Que a proposta da mecˆ anica relacional [AK99] ´ e errada e a resenh a anterior [SO99] ´ e, por isso, inconseq¨ ente. Agradecimentos Agradecemos ao CNPq pelo auxilio ﬁnanceiro parcial; a L. F. d os Santos pela leitura do manuscrito e a G. E. A. Matsas por ´ uteis discu s˜ oes sobre as teorias da relatividade. 13Aqui usamos esse termo para rotular uma atividade de pesquis a que se publica em revistas com razo´ avel parˆ ametro de impacto. 21References [AK99] A. K. Assis, Mecˆ anica Relacional , Centro de L´ ogica, Epistemologia e Hist´ oria da Ciˆ encia-UNICAMP, Campinas, 1998. [AK99b] Ref. [AK99], p. 145. [AK99c] Ref. [AK99], p. 178. [AK99d] Ref. [AK99], p. 190. [AK99e] Ref. [AK99], p. 200 [AK99f] Ref. [AK99], p. 205. [AK99g] Ref. [AK99], p. 157. [AK99h] Ref. [AK99], p. xviii. [AK99i] Ref. [AK99], p. xvix. [AK99j] Ref. [AK99], p. 179. [AK99k] Ref. [AK99], p. 280. [AK99l] Ref. [AK99], p. 157. [AK99m] Ref. [AK99], p. 254. [AL88] C. All` egre, A Espuma da Terra , Gradiva, Lisboa, 1988. [AN90] P. W. Anderson, On the nature of the physical laws , Physics Today, 43(12), 9 (1990). [BO68] H. Bondi, Cosmology , Cambridge University Press, Cambridge, 1968; p. 31-33. [BR01] H. N. Brown et al. (Muon g−2 Collaboration), hep-ex/0102017 [DI00] M. Dine, The Utility of Quantum Field Theory , Palestra plen´ aria no ICHEP 2000, Osaka, Japon; hep-ph/0010035. [EI05] A. Einstein, Ann. d. Phys. 17, 805 (1905). Tradu¸ c˜ ao portuguesa em “Sobre a eletrodinˆ amica dos corpos em movimento”, A. Ein stein, H. Lorentz, H. Weyl e H. Minkowski, O Princ´ ıpio da Relatividade , Funda¸ c˜ ao Caloust Gulbenkien, Lisboa, 1978; p. 47. 22[EI05b] Ref. [EI05] p. 47-48. [EI05c] Ref. [EI05] p. 72. [EI17] A. Einstein, Kosmologische Betrachtungen zur allge meinen Rel- ativit¨ atstheorie, Preuss. Akad. Wiss. Berlin, Sitzer. , 142 (1917). Tradu¸ c˜ ao portuguesa: Considera¸ c˜ oes cosmol´ ogicas so bre a teoria da relatividade geral, em Ref. [EI05], p. 141. [EB00] Immanuel Velikovsky, Encyclopædia Britannica Online . [GE78] R. Geroch, General Relativity from A to B , University of Chicago Press, Chicago, 1978. [HA84] F. Halzen e A. D. Martin, Quarks and Leptons , John-Willey, New York, 1984; p. 158. [HE78] W. Heisenberg, A Parte e o Todo , Contraponto, Rio de Janeiro, 1996; p. 78. [HE96] T. A. Herring, Sci. Am. 274, 32 (1996). [HO82] G. Holton, Ensayos sobre el pensamiento cient´ ıﬁco en la ´ epoca de Einstein , Alianza Editorial, Madrid, 1982; p. 177. [HO82b] Ref. [HO82], p. 165. [HO01] C. D. Hoyle, et al., Phys. Rev. Lett. 86, 1518 (20010. [LA99] I. Lakatos e P. Feyerabend, For and Against Methos , University of Chicago Press, Chicago, 1999; p. 21-22. [MA83] A primeira edi¸ c˜ ao alem˜ a ´ e de 1883. Aqui usaremos a vers˜ ao inglesa: The Science of Mechanics , The Open Court Publishing Co., Chicago, 1942. [MI73] C. W. Misner, K. S. Thorne e J. A. Wheeler, Gravitation , Freeman, New York, 1963; p.1095. [MI73b] Ref. [MI73] p. 1055 e as referˆ encias ali citadas. [NO93]http://www.nobel.se/physics/laureates/1993/press.ht ml. [NO00]http://www.nobel.se/index.html 23[PA95] A. Pais, “S´ util ´ e o Senhor”... A Ciˆ encia e a Vida de Albert Einstein , Editora Nova Fronteira, Rio de Janeiro, 1995. [PE98] R. Penrose, O Grande, o Pequeno e a Mente Humana , Editora UN- ESP, S˜ ao Paulo, 1998; p. 37-38. [PA95a] Ref. [PA95], p. 333. [PA95b] Ref. [PA95], p. 335. [PA95c] Ref. [PA95], p. 336. [PA95d] Ref. [PA95], p. 340. [PA95e] Ref. [PA95], p. 314. [PA95f] Ref. [PA95], p. 339. [PU78] E. M. Purcell, Electricity and Magnetism . McGraw-Hill, berkeley, 1978. [PU78b] Ref. [PU78], p. 148. [RE00] M. Rees, Apenas seis n´ umeros: as for¸ cas profundas que controlam o universo , Rocco, Rio de Janeiro, (2000); p. 44. [SA96] C. Sagan, O mundo assombrado pelos demonios , Companhia das Letras, S˜ ao Paulo, 1996. [SC49] P. A. Schipp, Albert Einstein, Philosopher-Scientist , Library of Liv- ing Philosophers, Evanston, 1949; p.67. [SO99] D. S. L. Soares, Revista Brasileira de Ensino de F´ ısi ca,21(4), 556 (1999). [ST00] S. Stone, Pathological Science , hep-ph/0010295. [WI79] C. M. Will, The confrontation between gravitation th eory and ex- periment, em General Relativity, An Einstein Centenary Survey , S. W. Hawking e W. Israel (Eds.), Cambridge University Press, Cam bridge, 1979. [WI79b] Ref. [WI79], p. 29. 24
arXiv:physics/0103039v1 [physics.comp-ph] 14 Mar 2001The ideal trefoil knot P. Pieranski and S. Przybyl Poznan University of Technology Nieszawska 13A, 60 965 Poznan Poland e-mail: Piotr.Pieranski@put.poznan.pl February 2, 2008 Abstract The most tight conformation of the trefoil knot found by the S ONO algorithm is presented. Structure of the set of its self-con tact points is analyzed. 1 Introduction Finding the best way of packing a tube within a box seems to be r ather a gar- dening than a scientiﬁc problem. However, the optimal singl e helix, discovered in a computer simulation study of this problem, [1] and [2], p roves to be ubiqui- tous in many proteins as their α-helical parts. It seems, as suggested in [3], that also the closely packed double helix appearing in the proces s of twisting two ropes together [4] have been already discovered and applied by nature. Labora- tory experiments allow one to observe in the real time how the optimal helices are formed in various systems e.g. the bacterial ﬂagellas [5 ] or phospholipid membranes [6]. Both processes, of packing the ropes and twisting them toget her, occur si- multaneously when a knot tied on a rope becomes tightened. Th e problem of ﬁnding the most tight, least rope consuming conformations o f knots was inde- pendently posed and indicated as essential by diﬀerent auth ors; for references see [7]. Knots in such optimal, most tight conformations are often called ideal, a term proposed by Simon [8], and introduced into the literat ure by Stasiak [9]. Ideal conformations minimize the value of the size-invaria nt variable Λ = L/D, where LandDare, respectively, the length and the diameter of the perfect rope (deﬁned below) on which the knot is tied. The only knot whose i deal confor- mation is known at present is the trivial knot (unknot). See F ig.1. Its length in the ideal, circular conformation equals πD, thus Λ = π. Finding the ideal conformation of a nontrivial knot is a nontrivial task. Init iated a few years ago search for the ideal conformations of nontrivial knots cont inues. 1Figure 1: Ideal unknot. One of the algorithms used in the search is SONO (Shrink-On-N o-Overlaps)[10]. SONO simulates a process in which the rope, on which a knot is t ied, slowly shrinks. The rope is allowed to shrink only when no overlaps o f the rope with itself are detected within the knot. When such overlaps occu r, SONO modiﬁes the knot conformation to remove them. If this is no more possi ble, the pro- cess ends. Unfortunately, ending of the tightening process does not mean that the ideal conformation of a given knot was found. The tighten ing process could have stopped also because a local minimum of the thickness en ergy was entered. The possibility that there exists a diﬀerent, less rope cons uming conformation, cannot be excluded. SONO has been used in the search of ideal conformations of bot h prime and composite knots. Parameters of the least rope consuming con formations found by the algorithm were listed in [11] and [12]. In a few cases, S ONO managed to ﬁnd better conformations than the simulated annealing proc edure [9]. However, for the most simple knots, in particular, the trefoil knot, t he simulated annealing and SONO provided identical results; the Λ values are identi cal within experi- mental errors. It seems obvious, that no better conformatio ns of the knot exist. We feel obliged to emphasize, however, that it is only an intu itively obvious conclusion - no formal proofs have been provided so far. As in dicated in [3], we are in a situation similar to that, which lasted in the proble m of the best pack- ing of spheres for 400 years. That the face centered cubic and hexagonal close packed lattices were among the structures which minimize th e volume occupied by closely packed hard spheres seemed to be obvious since the times of Kepler, however the formal proof of the conjecture was provided but a few years ago [13]. Waiting for the formal proofs that what we have observed in th e knot tighten- ing numerical experiments is the ideal conformation of the t refoil, seems to be a too cautious attitude. Thus, after a few years of experimen ting, we decided to present the best, least rope consuming conformation of th e trefoil knot we managed to ﬁnd. We compare it with the most tight conformatio n of the knot which can be found within the analytically deﬁned family of t orus knots. In particular, we describe the qualitative change in the set of self-contacts which 2Figure 2: The perfect rope. Perpendicular sections of the ro pe are of the disk shape. None of the disks are allowed to overlap. This puts a li mit not only on the spacial distance of diﬀerent fragments of the curve into which the rope is shaped, but also on its local curvature. takes place within the trefoil knot during the tightening pr ocess. We believe that some of the features of the self-contact set we have foun d may be present also in ideal conformations of other knot types. An alternative method of searching for the most tight confor mations of knots consists in inﬂating the rope on which the knot has been tied I n such a process the length of the rope is kept ﬁxed. The maximum radius to whic h the rope in a given conformation of a knot can be inﬂated is closely rel ated with the injectivity radius considered in detail by Rawdon [14]. 2 The perfect rope It is the aim of the computer simulations we perform to simula te the tightening process of knots tied on the perfect rope : perfectly ﬂexible, but at the same time perfectly hard in its circular cross-section. The surface o f the perfect rope can be seen as the union of all circles centered on and perpendicu lar to the knot axisC. See Fig.2. We assume that Cis smooth and simple, i.e. self-avoiding, what guaranties that at each of its points rthe tangent vectors τ(r),and thus the circular cross-section, are well deﬁned. The surface remains smooth as long as: A. the local curvature radius rκof the knot axis is nowhere smaller than D/2, B. the minimum distance of closest approach d∗is nowhere smaller then D/2 . Theminimum distance of closest approach d∗, known also as the doubly critical self-distance , see [8], is deﬁned in [16], as the smallest distance between all pairs of points ( r1,r2) on the knot axis, having the property, that the vector (r2−r1) joining them is orthogonal to the tangent vectors τ(r1),τ(r2) located 3Figure 3: The trefoil knot is a torus knot - it can be tied on the surface of a torus. at the points: d∗(C) = min r1,r2∈C{\|r2−r1\|:τ(r1)⊥(r2−r1),τ(r2)⊥(r2−r1)} (1) As shown by Gonzalez and Maddocks [16], the two conditions ca n be gath- ered into a single one providing that the notion of the global curvature radius ρGis introduced: ρG(r1) = min r2,r3∈C r1/negationslash=r2/negationslash=r3/negationslash=r1ρ(r1,r2,r3) (2) where, ρ(r1,r2,r3) is the radius of the unique circle (the circumcircle) which passes through all of the three points: r1,r2andr3. Using the notion of the global curvature, the condition which guaranties smoothne ss of the knot surface can be reformulated as follows: C. the global curvature radius ρGof the knot axis is nowhere smaller than D/2. Analysis of the conformations produced by the SONO algorith m proves that conditions A and B, (and C) are fulﬁlled. 3 Parametrically tied trefoil knot The trefoil knot can be tied on the surface of a torus. See Fig. 3Consider the set 4of 3 periodic functions: x= [R+rcos(2ν1π t)]sin(2ν2π t) (3) y= [R+rcos(2ν1π t)]cos(2ν2π t) (4) z=rsin(2ν1π t) (5) The trajectory determined by equations 3, 4 and 5 becomes clo sed as tspans a unit interval. For the sake of simplicity we shall consider the [0 ,1) interval. For all relatively prime integer values of ν1,ν2equations 3, 4 and 5 deﬁne self- avoiding closed curves located on the surface of a torus. Rdenotes here the radius of the circle determining the central axis of the toru s while rdenotes the radius of its circular cross-sections. For the trefoil k not, frequencies ν1,ν2 equal 2 and 3, respectively. In what follows we consider knot s tied on a rope; trajectories deﬁned by equations 3, 4 and 5 determine positi on of its axis. The (ν1, ν2) and the ( ν2, ν1) torus knots are ambient isotopic, i.e. they can be transformed one into another without cutting the rope on w hich they are tied [17]. As shown previously, the (2 ,3) version of the trefoil is less rope consuming [12]. Thus, the (3 ,2) version will not be discussed below. Assume that the trefoil knot whose axis is deﬁned by equation s 3, 4 and 5 is tied on a rope of diameter D= 1. In what follows we shall refer to it as the parametrically tied trefoil (PTT) knot. In such a case, radius rof the torus on which the axis of knot is located, cannot be smaller than 1 /2 ; below this value overlaps of the rope with itself will certainly appear; at r= 1/2 the rope remains in a continuous self-contact along the torus axis. To keep th e self-contacts we assume in what follows that r= 1/2. To check, if the knot is free of overlaps in other regions, one can analyze the map of its internal distan ces. Let t1andt2be two values of the parameter t, both located in the [0 ,1) interval. Let ( x1, y1, z1) and (x2, y2, z2) be the coordinates of two points indicated within the knot a xis byt1andt2, respectively. Let d(t1, t2) be the Euclidean distance between the points: d(t1, t2) =/radicalbig (x2−x1)2+ (y2−y1)2+ (z2−z1)2 (6) The map of the function, see Fig.4 displays a mirror symmetry induced by the equality d(t1, t2) =d(t2, t1). Looking for possible overlaps within the knot one looks for r egions within the internal distances landscape, where d(t1, t2)<1. The most visible depression within the landscape of the interknot distances is located a round the diagonal where t1=t2. As easy to see, d(t1, t2) = 0 along the line, but for obvious reasons this does not implies any overlaps within the knot. Another valley within which d(t1, t2) may go down to the critical 1 value is localized in the vicinity of lines deﬁned by equality \|t2−t1\|= 1/2. To see, if in the vicinity of the lines the height really drops to or even be low 1, we plotted the map of the d(t1, t2) function in such a manner, that regions lying below the arbitrarily chosen 1 .005 level were cut oﬀ. 5Figure 4: The map of the intraknot distances of the most tight PPT knot. As seen in Fig.5 there are four such regions within the PTT kno t: one in the shape of a sinusoidal band and three in shapes of almost circu lar patches. The band contains in its middle the mentioned above continuous l ine of self-contacts points; it is the axis of the torus on which the knot is tied. Th e circular patches contain 3 additional contact points; when Rbecomes too small, overlaps appear around the points. Numerical analysis we performed reveals that (with the 5 decimal digits accuracy we applied) the overlaps occurring within these regions vanish above R= 1.1158. For R= 1.1159 the distance between the closest points located within these regions of the knot equals 0 .9999. For R= 1.1158 the distance is equal 1 .0000. Where, within the PTT knot the self-contact points are located is shown in Fig.6 4 SONO tied trefoil knot Considerations presented above indicated the value of R, at which the PTT knot reaches its most tight conformation. The length Ltof the rope engaged in this conformation of the trefoil knot equals 17 .0883. Can one tie the trefoil knot using a shorter piece of the rope? Theoretical consider ations indicate that this possibility cannot be excluded. As proven in [18] t he piece of rope used to tie the trefoil knot cannot be shorter than Lm= (2 +√ 2)π≈10.72. Such a location of this lower limit leaves a lot of place for a p ossible further tightening of the knot. Application of SONO reveals that the tightening is possible providing the conformation of the knot is allowed t o leave the subspace of the parametrically tied torus conformations. This happe ns spontaneously in 6Figure 5: The map of the intraknot distances. Left - the most t ight PTT knot. Right - the most tight STT knot. The map was cut from below at th e height 10.005. numerical simulations in which the most tight PTT knot is sup plied to SONO as the initial conformation. SONO algorithm manages to make it shorter. In the simulations we performed, SONO reduced the length of the kno t by about 4% toLexp= 16.38. The discrete representation of the knot used in the simul ations contained N= 327 nodes. Below we describe the ﬁnal conformation. For the sake of simplicity we shall refer to trefoil knots processed by the SONO algorithm as the SONO tied trefoil (STT) knots. The diﬀerences in the conformation of the most tight conform ations of the PTT and STT knots is a subtle one. The essential diﬀerence lie s in the structure of the sets of their self-contact points. As mentioned above , the circular line of self-contact points present in the family of the PTT knots st ays intact as Ris changed within the family. Tightening of a PTT knot achieved by decreasing the radius Rof the torus stops when additional discrete points of contac ts appear at three locations within the knot. This happens as Rbecomes equal 1 .1158. Further tightening of the knot within the family of PTT knots is not possible, 7Figure 6: Localization of the set of self-contact points wit hin the most tight PPT knot. it becomes possible within the family of the STT knots. During the tightening process carried out by SONO, the set of the self- contact points undergoes both qualitative and quantitativ e changes. First of all, the line of contacts present in the PTT knot changes its s hape becoming distinctly non-circular. Secondly, the three contact poin ts give birth to pieces of new line of self-contacts. Unexpectedly, the new pieces d o not connect into a new line, wiggling around and crossing the old line, but they are mounted into the old line in such a manner, that a single, self-avoiding an d knotted line of self-contacts is created. That this is the case was revealed by a precise analysis of the interknot distances function. A map covering the inte rknot distances only within the very thin [1 .00000 ,1.00002] interval shows two separated lines, see Fig.7, corresponding to a single, self-avoiding and knotte d line of contact. In addition to the line, a set of three points of self-contact s is formed. The points are located at places where the line of self-contacts becomes almost tan- gent to itself. The self-contact line runs twice around the k not. As a result, each of the circular cross-sections of the rope stays here in touch with another two such sections. The close packed structure formed in such a manner is much more stable than the structure of the most tight PTT knot, whe re single con- tacts were predominant. Let us note, that ﬁgure 1e presented in ref. [16] a similar self-contact line structure can be seen. Unfortuna tely, inspecting the ﬁgure one cannot see, if the ”self-contact spikes” shown the re form a single, self-avoiding, knotted or a double, crossing itself line. T he problem was not discussed in the text. Let us emphasize, however, that the di ﬀerence between 8Figure 7: The set of the self-contact points in the most tight STT knot as seen within the map of the intraknot distances. the two possibilities is conﬁned to a zero-measure set. 5 Discussion Ideal knots are objects of which very little is known still. T he only knot whose ideal conformation is known rigorously is the unknot. Its id eal conformation, a circle of a radius identical with the radius of the rope on whi ch it is tied, can be conveniently described parametrically. The set of the se lf-contact points is here limited to a single point: the center of the circle. All c ircular sections of the rope meet at this point. The maximum local curvature and t he minimum double critical self-distance limiting conditions are sim ultaneously met. The situation in the case of the trefoil knot, the simplest no n-trivial prime knot, is radically diﬀerent. Here the most tight parametric ally deﬁned con- formation proves to be not ideal. As demonstrated by the pres ent authors, it can be tightened more with the use of the SONO algorithm. The s et of the self-contact points becomes rebuilt during the tightening process. Its topology becomes diﬀerent. In the case the PPT knot the set of the self- contact points consists of acircle and 3 separated points. As the numerical experiments we per- formed suggest, in the case of the STT knot, the set of the self -contact points turns unexpectedly into a single line. Which the structure o f the set of self 9Figure 8: Position of the line of the self-contact points wit hin the ideal trefoil knot. To make the line more visible, a part of the knot was cut o ut. contact points in other prime knots is, remains an open quest ion. Acknowledgment PP thanks Andrzej Stasiak, John Maddocs, Robert Kus- ner, Kenneth Millet, Jason Cantarella and Eric Rawdon for he lpful discussions. This work was carried out under Project KBN 5 PO3B 01220. References [1] A. Maritan, C. Micheletti, A. Trovato and J. R. Bonavar, N ature406, 287 (2000) [2] S. Przybyl and P. Pieranski E. Phys. J. E, (2000, in print) [3] A. Stasiak and J. H. Maddocks, Nature 406, 251 (2000). [4] S. Przybyl and P. Pieranski, Pro Dialog 6, 87 (1998). [5] R. E. Goldstein, A. Goriely, G. Huber and C.Wolgemuth, Ph ys. Rev. Let- ters84, 1631 (2000). [6] I. Tsafrir, M.-A. Guedeau-Boudeville, D. Kandel and J. S tavans, Phys. Rev. E, submitted for publication. [7]Ideal Knots , eds. A. Stasiak, V. Katritch and L. H. Kauﬀman, World Sin- gapore 1998. [8] J. K. Simon, a talk at KNOTS’96, Waseda University, Tokyo (1996). 10[9] V. Katritch, J. Bednar, J. Michoud, R. G. Scherein, J. Dub ochet and A. Stasiak, Nature 384, 142 (1996). [10] P. Pieranski, Pro Dialog 5, 111 (1996). [11] V. Katritch, W. K. Olson, P. Pieranski, J. Dubochet and A . Stasiak, Nature 388, 148 (1997). [12] P. Pieranski in [7] [13] N. J. Sloane, Nature 395, 435-436 (1998). [14] E. Rawdon in [7] [15] J. Simon in [7] [16] O. Gonzalez and J. H. Maddocks, Proc. Nat. Acad. Sci. 96, 4769 (1999). [17] C. C. Adams, The Knot Book , W. H. Freeman and Co., New York 1994, p.111. [18] Private communication. 11This figure "31torus.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1This figure "DoubleLineInKnotGrey.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1This figure "HELICE2.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0103039v1This figure "Fig5.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1This figure "Fig3_11.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1This figure "Fig4_1005.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1This figure "IdealUnknot.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1This figure "map100002.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103039v1
arXiv:physics/0103040v1 [physics.optics] 14 Mar 2001Absolute frequency measurement of the 435.5 nm171Yb+clock transition with a Kerr-lens mode-locked femtosecond laser J¨ orn Stenger, Christian Tamm, Nils Haverkamp, Stefan Weye rs, and Harald. R. Telle Physikalisch-Technische Bundesanstalt, Bundesallee 100 , 38116 Braunschweig, Germany We have measured the frequency of the 6 s2S1/2−5d2D3/2electric- quadrupole transition of171Yb+with a relative uncertainty of 1 ×10−14, νY b= 688 358 979 309 312 Hz ±6 Hz. A femtosecond frequency comb gen- erator was used to phase-coherently link the optical freque ncy derived from a single trapped ion to a cesium fountain controlled hydrogen maser. This mea- surement is one of the most accurate measurements of optical frequencies ever reported, and it represents a contribution to the developme nt of optical clocks based on an171Yb+ion standard. Frequency comb generators based on Kerr-lens mode- locked femtosecond lasers [1, 2] have dramatically sim- pliﬁed absolute optical frequency measurements. Such measurements of optical clock transitions in samples of cold atoms or single trapped ions mark an important step towards the realization of future optical clocks. Several optical transition frequencies have already been directly compared with primary cesium standards by the femtosecond comb technique, such as the hydro- gen Lyman- αtransition [3], the 657 nm intercombi- nation transition in Ca [4, 5], or a mercury ion tran- sition at 282 nm [5]. Recently, the indium ion clock transition at 237 nm was measured with respect to a cesium-fountain calibrated He-Ne standard [6]. Here we report the ﬁrst absolute frequency measurement of the 6 s2S1/2(F= 0)−5d2D3/2(F= 2) transition of 171Yb+at 435.5 nm (688 THz). This transition is attractive for optical clocks due to its small natural linewidth of 3.1 Hz and a ∆ mF= 0 component with vanishing low-ﬁeld linear Zeeman fre- quency shift. A single171Yb+ion is laser cooled in a spherical radiofrequency Paul trap so that the Lamb- Dicke condition is satisﬁed at 435.5 nm. The clock transition is alternately probed on both sides of the resonance line with the frequency-doubled output of an extended-cavity diode laser emitting at 871.0 nm. One laser cooling and probe excitation cycle lasts 80 ms. The frequency of the probe laser is stabilized to the line center with an eﬀective time constant of 30 s through a second-order integrating servo algorithm. Short-timeﬂuctuations are reduced by stabilization to an environ- mentally isolated high-ﬁnesse cavity. More details are given in Ref. [7]. In the measurements reported here the171Yb+clock transition was resolved with an es- sentially Fourier-limited linewidth of 30 Hz. A frequency comb was generated by a Kerr-lens mode-locked femtosecond laser. The emitted periodic pulse train corresponds in the frequency domain to a comb-like spectrum, which can be completely charac- terized by only three numbers: the line spacing equal to the repetition rate frep, the longitudinal mode order m of a line and an oﬀset frequency νceo, which reﬂects the frequency oﬀset of the whole comb with respect to the frequency origin. Thus, an external optical frequency νextcan be written νext=νceo+mfrep+ ∆x , (1) where ∆ xis the beat frequency of the external optical signal with the mth comb mode. In the experiment the radiofrequencies νceo,frep, and ∆ xare referenced to a hydrogen maser, which in turn is compared with a cesium fountain. The experimental setup is schematically shown in Fig. 1. The frequency comb generator comprises a 10 fs Kerr-lens mode-locked Ti:Sapphire laser and a microstructure ﬁber [8] for external broadening of the spectrum via self-phase modulation. More details are given in Ref. [4]. By coupling approximately 30 mW of the laser output into a 10 cm long piece of ﬁber we achieved a spectrum ranging from 500 nm to 1200 nm. 1Figure 1: Schematic of the setup. LBO denotes the LiB 3O5–frequency doubling crystal, PD 1-3 photo detectors, IF interference ﬁlter, and PLL phase-locked loop, respecti vely. Details are described in the text. Two additional servo loops (not shown in the Figure) were used for a slow stab ilization of frepand ∆ xin order to keep the beat signals within the hold-in range of the PLL tracking oscilla tors. The 871 nm light from the171Yb+standard was guided to the frequency comb generator via a 250 m long single-mode polarization preserving ﬁber. About 1 mW of that light was combined with light from the femtosecond laser. After spectral ﬁltering by a 10 nm (FWHM) interference ﬁlter and spatial ﬁltering in a short piece of single mode ﬁber the beat note ∆ xwas detected with a fast Si PIN photodiode (PD 1), ﬁl- tered by a phase-locked loop (PLL) and counted by a totalizing counter. Special care had to be taken for a phase-resolved determination of the repetition rate frep, which, ac- cording to eqn. (1), enters the optical frequency mea- surement with a large multiplication factor m. Thus we detected the 103rd harmonic of frepat 10 GHz with a fast InGaAs PIN photodiode (PD 2) after spectral ﬁl- tering with a fused-silica etalon. This microwave signal was downconverted, ﬁltered and frequency-multipliedby 288. Owing to the resulting large overall multi- plication factor of 29664 the digitization error was re- duced below the instability of the hydrogen maser. The wavelength of the 871 nm signal was pre-measured by a lambdameter with absolute accuracy corresponding to 1.5 MHz, thus determining the longitudinal mode order m. The frequency νceowas measured by detecting the beat note between frequency-doubled comb modes around 1070 nm and modes around 535 nm. According to eqn. (1) the frequencies of the comb modes are shifted by νceowhereas the harmonics are shifted by 2 νceo. The resulting beat note νceowas detected by a photo mul- tiplier (PD 3) after spatial and spectral ﬁltering both ﬁelds with a single mode ﬁber and a 600 l/mm grating, respectively. The signal was tracked with a third PLL and ﬁnally counted. Data were taken on three diﬀerent days. By averag- 2ing we derive the following value for the 6 s2S1/2(F= 0)−5d2D3/2(F= 2) electric-quadrupole clock tran- sition of the171Yb+ion: νY b= 688 358 979 309 312 Hz ±6 Hz.(2) This frequency includes the frequency shift of the 171Yb+transition due to isotropic blackbody radiation at an ambient temperature of 298 K. This shift is cal- culated to −0.4 Hz using tabulated atomic data [9]. Fig. 2 shows the Allan standard deviation of one day’s data. The typical instability of the hydrogen maser (open circles in Fig 2) is approached. Figure 2: Allan standard deviation of the171Yb+clock transition measurement and that of the typical perfo- mance of the hydrogen maser. The inset shows the dis- tribution of the measured frequency values (averaging time 1 s, bin-width = 40 Hz, σ= 111 Hz). The Allan standard deviation of the clock transition measurement approaches the typical performance of the hydrogen maser. The maser was operated in a self-tuning mode which reduced the available frequency stability for av- eraging times in the range of 200 s. The combined 1 σuncertainty of 6 Hz is given by the random and systematic contributions listed in Ta- ble 1. Sources of systematic uncertainties are the ce- sium fountain frequency standard [10] and the171Yb+ frequency standard. The contributions due to ther- mally and acoustically induced length ﬂuctuations both of the coaxial cable carrying the 100 MHz hydrogen maser signal and of the optical ﬁber guiding the 871 nm light was measured to be below 1 ×10−15and thus are negligible. For the171Yb+frequency standard we take into account a servo uncertainty of 1 Hz due to probe laser frequency drifts which are in the rangeof - 0.05 ±0.03 Hz/s and a 3 Hz uncertainty due to conﬁnement-related shifts of the atomic transition fre- quency. The dominant source of these is the electric- quadrupole interaction of the upper state of the clock transition with stationary electric ﬁeld gradients. In order to avoid quadrupole shifts by the trap ﬁeld, the applied trap voltage contained no dc component. The residual shift due to uncompensated stray ﬁeld gra- dients is estimated to be not larger than 0.1 Hz for atomic D3/2andD5/2states [11]. An arrangement of compensation coils was used to adjust a magnetic ﬁeld of 1±0.2 mT in the trap region during excitation of the 171Yb+clock transition. The corresponding quadratic Zeeman shift of the ∆ mF= 0 reference transition is in the range of only 0.05 Hz. The trap region was pro- tected from ambient heat sources by the light shield of the trap setup. We assume that the thermal radia- tion ﬁeld in the trap region represented an equilibrium blackbody ﬁeld at room temperature and neglect the corresponding contribution to the uncertainty budget. In conclusion, we measured the frequency of the electric-quadrupole clock transition of the171Yb+ion with a relative uncertainty of 1 ×10−14. This demon- strates the potential of the171Yb+standard as an ultraprecise optical frequency reference. Simultane- ously, we demonstrated the capability of a femtosecond comb generator of measuring optical frequencies with Cs clock accuracy. A future application can be the di- rect frequency comparison of the171Yb+standard with another optical standard such as the cold-atom based calcium standard, aiming e.g. to measure a possible variation of fundamental constants [12]. A drift of the relative frequencies of these optical standards due to a drift of the ﬁnestructure constant αby more than 10−15per year appears excluded [12]. However, the unprecedented measurement accuracy achievable with femtosecond comb generators encourages one to pursue such fundamental, ultra-precise measurements. We gratefully acknowledge ﬁnancial support from the Deutsche Forschungsgemeinschaft through SFB 407 and contributions of Burghard Lipphardt, Uwe Sterr, Andreas Bauch, G¨ unter Steinmeyer and Ursula Keller in several stages of the experiment. We are also in- debted to Robert Windeler of Lucent Technologies for providing the microstructure ﬁber. References 3corrected standard uncertainties arising from: combined results random eﬀects systematic eﬀects uncertainty ν−νY b measurement reference Yb+reference day1: + 0.5 12.4 5.4 3.2 1.0 13.9 day2: + 2.6 7.8 6.3 3.2 1.8 10.7 day3:−2.5 5.6 4.2 3.2 1.8 7.9 Table 1: Deviations of the frequency measurement results fr om the weighted mean νY b= 688 358 979 309 312 ±6 Hz, and uncertainties. The results are corrected for recog nized systematic eﬀects of the H-maser and of the171Yb+standard. All numbers are given in Hz. [1] T. Udem, J. Reichert, R. Holzwarth, and T. W. H¨ ansch; Opt. Lett. 24881 (1999). [2] S. A. Diddams, D. J. Jones, J. Ye, S. T. Cun- diﬀ, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holtzwarth, T. Udem, and T. W. H¨ ansch; Phys. Rev. Lett. 845102 (2000). [3] M. Niering et al.; Phys. Rev. Lett. 845496 (2000). [4] J. Stenger, T. Binnewies, G. Wilpers, F. Riehle, H. R. Telle, J. K. Ranka, R. S. Windeler, A. J. Stentz; Phys. Rev. A. 63021802(R) (2001). [5] T. Udem, S. A. Diddams, K. R. Vogel, C. W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg; submitted (eprint arXiv:physics/0101029) (2001). [6] J. von Zanthier et al.; Opt. Lett. 251729 (2000). [7] Chr. Tamm, D. Engelke, V. B¨ uhner; Phys. Rev. A61053405 (2000). [8] J. K. Ranka, R. S. Windeler, and A. J. Stentz; Opt. Lett. 2525 (2000). [9] B. C. Fawcett, M. Wilson; At. Data Nucl. Data Tabl. 47 (1991) 241, J. W. Farley, W. H. Wing; Phys. Rev. A232397 (1981). [10] S. Weyers, U. H¨ ubner, R. Schr¨ oder, Chr. Tamm, A. Bauch; to be published Metrologia 38(4) (2001), S. Weyers, U. H¨ ubner, B. Fischer, R. Schr¨ oder, Chr. Tamm, A. Bauch; Proc. 14th EFTF 2000, Torino-Italy, 53 (2000) [11] D. J. Wineland, J. C. Bergquist, W. M. Itano, F. Diedrich, C. S. Weimer, in ”The Hydrogen Atom”, ed. by G. F. Bassani, M. Inguscio, T. W. H¨ ansch (Springer Verlag, Berlin, 1989) 123, W. M. Itano, eprint arXiv:physics/0010056) (2000).[12] J. D. Prestage, R. L. Tjoelker, and L. Maleki; Phys. Rev. Lett. 743511 (1995), V. A. Dzuba, V. V. Flambaum, and J. K. Webb; Phys. Rev. A59 230 (1999), V. A. Dzuba and V. V. Flambaum; Phys. Rev. A61034502 (2000). 4
arXiv:physics/0103041v1 [physics.flu-dyn] 15 Mar 2001Chemical eﬃciency of reactive microﬂows with heterogeneus catalysis: a lattice Boltzmann study S. Succi1,3∗, A. Gabrielli2, G. Smith3, E. Kaxiras3 1Istituto di Applicazioni Calcolo, viale Policlinico 137, 0 0161 - Roma, Italy 2INFM, Dipartimento di Fisica, Universit` a di Roma ”La Sapie nza”, P.le A. Moro 2, 00185 - Roma, Italy 3Lyman Laboratory of Physics, Harvard University, Cambridg e, USA 3∗Visiting Scholar, Lyman Lab. of Physics, Harvard Universit y (November 19, 2013) We investigate the eﬀects of geometrical micro-irregulari ties on the conversion eﬃciency of reactive ﬂows in narrow channels of millimetric size. Three-dimensi onal simulations, based upon a Lattice- Boltzmann-Lax-Wendroﬀ code, indicate that periodic micro -barriers may have an appreciable eﬀect on the eﬀective reaction eﬃciency of the device. Once extrap olated to macroscopic scales, these eﬀects can result in a sizeable increase of the overall react ion eﬃciency. I. INTRODUCTION The formulation of mathematical models and atten- dant simulational tools for the description of complex phenomena involving multiple scales in space and time represents one of the outstanding frontiers of modern applied physics/mathematics [1]. One such example of complex multiscale phenomena is the dynamics of reac- tive ﬂows, a subject of wide interdisciplinary concern in theoretical and applied science, with several application s in molecular engineering, material science, environmen- tal and life sciences alike. The complexity of reactive ﬂow dynamics is parametrized by three dimensionless quanti- ties: the Reynolds number Re=UL/ν , theDamkohler number Da=τh/τc, and the Peclet number Pe=UH D. HereU,LandHdenote the macroscopic ﬂow speed and longitudinal/transversal lengths of the ﬂow, respectivel y, νthe ﬂuid kinematic viscosity and Dthe pollutant molec- ular diﬀusivity. The quantities τcandτhrepresent typical timescales of chemical and hydrodynamic phenomena. High Reynolds numbers are associated with turbu- lence, namely loss of coherence of the ﬂow ﬁeld in both space and time. High Damkohler numbers imply that chemistry is much faster than hydrodynamics, so that re- actions are always in chemical equilibrium and take place in tiny regions (thin ﬂames, reaction pockets) of evolving ﬂow conﬁgurations. The opposite regime (“well-stirred” reactor) characterizes situations where the chemistry is slow and always takes place at local mechanical equilib- rium. Finally, high Peclet numbers imply that the trans- ported species stick tightly to the ﬂuid carrier (in the limitPe→ ∞ the tracer ﬁeld is “frozen-in” within ﬂow streamlines). Navigation across the three dimensional Re−Da−Peparameter space meets with an enormous variety of chemico-physical behaviours, ranging from tur- bulent combustion to hydrodynamic dispersion and oth- ers [2]. The picture gets further complicated when ge- ometry is taken into account, since boundary conditions select the spatio-temporal structures sustaining the non-linear interaction between the various ﬁelds. In this work we shall deal with low-Reynolds, fast-reacting ﬂows with heterogeneus catalysis . In particular we wish to gain in- sights into the role of geometric micro-irregularities on the eﬀective rate of absorption of tracer species (pollu- tant hereafter) at catalytic boundaries. This is a theme of broad interest, with applications in biology, physics, chemistry, environmental sciences and more. It is there- fore hoped that such kind of theoretical-computational studies may promote a better understanding of the com- plex phenomena behind these important applications [3]. II. MATHEMATICAL MODEL OF REACTIVE MICROFLOW DYNAMICS We shall deal with an incompressible, isothermal ﬂow with soluted species which are transported (advect and diﬀuse) by the ﬂow and, upon reaching solid walls, they undergo catalytic chemical reactions . The basic equa- tions of ﬂuid motion are: ∂tρ+div(ρ/vector u) = 0 (1) ∂t(ρ/vector u) +div(ρ/vector u/vector u) =−∇P+div(µ∇/vector u) (2) where ρis the ﬂow density, /vector uthe ﬂow speed, P=ρT the ﬂuid pressure, Tthe temperature and µ=ρνthe dynamic viscosity and /vector u/vector udenotes the dyadic tensor uaub, a, b=x, y, z . Multispecies transport with chemical reactions is de- scribed by a set of generalized continuity-diﬀusion equa- tions: ∂tCs+div(Cs/vector us) =div[Ds∇(Cs/ρ)] +˙Ωs (3) where Csdenotes the mass density of the generic s- th species, Dsits mass diﬀusivity and ˙Ωsis a surface- chemical reaction term to be detailed shortly. In the fol- lowing we indicate with the subscripts wandgthe “wall” (solid) and “gas” in contact with the wall respectively. 1According to Fick’s law, the outgoing (bulk-to-wall) dif- fusive mass ﬂux is given by: /vectorJg→w=−D∇Cg\|w. (4) Upon contact with solid walls, the transported species react according to the following empirical rate equation (the species index being removed for simplicity): ˙Ω≡dCw dt= Γw−KcCw(5) where the wall-ﬂux is taken in the simple linear form: Γw=Kw(Cg−Cw) (6) where Kwis the wall to/from ﬂuid mass transfer rate andKcis the chemical reaction rate dictating species consumption once a molecule is absorbed by the wall. The subscripts wandgmean “wall” (solid) and “gas” in a contact with the wall respectively. The above rate equation serves as a dynamic boundary condition for the species transport equations, so that each boundary cell can be regarded as a microscopic chemical reactor sus- tained by the mass inﬂow from the ﬂuid. In the absence of surface chemical reactions the species concentration in the solid wall would pile up in time, up to the point where no outﬂow would occurr, a condition met when Cg=Cw. Chemistry sets a time scale for this pile-up and ﬁxes the steady-state mass exchange rate. At steady state we obtain: Cw=Kw Kw+KcCg(7) hence Γw=Cg τw+τc(8) where τw= 1/Kwandτc= 1/Kc. These expres- sions show that ﬁnite-rate chemistry ( Kc>0) ensures a non-zero steady wall outﬂux of pollutant. At steady state, this mass ﬂow to the catalytic wall comes into bal- ance with chemical reactions, thus ﬁxing a relation be- tween the value of the wall-gradient concentration and its normal-to-wall gradient: /ba∇dblD∂⊥Cg\|w/ba∇dbl=p Cg/(τc+τw), where ∂⊥means the normal to the perimeter compo- nent of the gradient and pis the perimeter (volume/area) of the reactive cell. This is a mixed Neumann-Dirichlet boundary condition and identiﬁes the free-slip length of the tracer as ls=D(τw+τc)/p. III. THE COMPUTATIONAL METHOD The ﬂow ﬁeld is solved by a lattice Boltzmann method [4–7] while the multispecies transport and chemical re- actions are handled with a variant of the Lax-Wendroﬀ method [8]. A few details are given in the following.A. Lattice Boltzmann equation The simplest, and most popular form of lattice Boltz- mann equation (Lattice BGK, for Bahtnagar, Gross, Krook) [7], reads as follows: fi(/vector x+/vector ci, t+ 1)−fi(/vector x, t) =−ω[fi−fe i](/vector x, t) (9) where fi(/vector x, t)≡f(/vector x,/vector v=/vector ci, t) is a discrete population moving along the discrete speed /vector ci. The set of discrete speeds must be chosen in such a way as to guarantee mass, momentum and energy conservation, as well as ro- tational invariance. Only a limited subclass of lattices qualiﬁes. In the sequel, we shall refer to the nineteen- speed lattice consisting of zero-speed, speed one c= 1 (nearest neighbor connection), and speed c=√ 2, (next- nearest-neighbor connection). This makes a total of 19 discrete speeds, 6 neighbors, 12 nearest-neighbors and 1 rest particle ( c= 0). The right hand side of (9) represents the relaxation to a local equilibrium fe iin a time lapse of the order of ω−1. This local equilibrium is usually taken in the form of a quadratic expansion of a Maxwellian: fe i=ρ/bracketleftbigg 1 +/vector u·/vector ci c2s+/vector u/vector u·(/vector ci/vector ci−c2 sI) 2c4s/bracketrightbigg (10) where csis the sound speed and Idenotes the identity. Once the discrete populations are known, ﬂuid density and speed are obtained by (weighted) sums over the set of discrete speeds: ρ=m/summationdisplay ifi, ρ/vector u =m/summationdisplay ifi/vector ci (11) LBE was historically derived as the one-body kinetic equation resulting from many-body Lattice Gas Au- tomata, but it can mathematically obtained by standard projection upon Hermite polynomials of the continuum BGK equation and subsequent evaluation of the kinetic moment by Gaussian quadrature [9]. It so happens that the discrete speeds /vector ciare nothing but the Gaussian knots, showing that Gaussian integration achieves a sort of au- tomatic “importance sampling” of velocity space which allows to capture the complexities of hydrodynamic ﬂows by means of only a handful of discrete speeds. The LBE proves a very competitive tool for the numerical studies of hydrodynamic ﬂows, ranging from complex ﬂows in porous media to fully developed turbulence. B. Modiﬁed Lax-Wendroﬀ scheme for species transport Since species transport equation is linear in the species concentration, we can solve it on a simple 6-neighbors cubic lattice. Within this approach, each species is as- sociated with a species density Cs, which splits into six separate contributions along the lattice links. 2With these preparations, the transport operator in 3 dimensions reads as follows (in units of ∆ t= 1)): Cs(/vector x, t) =6/summationdisplay j=0pj(/vector x−/vector cj, t−1)Cs(/vector x−/vector cj, t−1) (12) The index jruns over /vector xand its nearest-neighbors (hence simpler than the LBE stencil) spanned by the vectors /vector x+/vector cj,j= 1,6,j= 0 being associated with the node /vector x itself. The break-up coeﬃcient pjrepresents the proba- bility that a particle at /vector xj≡/vector x−/vector cjat time t−1 moves along link jto contribute to Cs(/vector x) at time t. For instance in a one dimensional lattice the exact expression of these coeﬃcients (in lattice units /vector cj=±1, j= 1,2, ∆t= 1) is: pi(x±1, t−1) =1∓u′ 2+D′ s, i= 1,2 (13) p0(x, t−1) =−2D′ s (14) where u′= (u+ρ−1∂xρ) is the eﬀective speed, in- clusive of the density gradient component, and D′ s= Ds(1−u′2)/2 is the eﬀective diﬀusion, the square u′de- pendence being dictated by arguments of numerical sta- bility. C. Multiscale considerations The simulation of a reactive ﬂow system is to all eﬀects amulti-physics problem involving four distinct physical processes: 1. Fluid Motion (F) 2. Species Transport (T) 3. Fluid-Wall interaction (W) 4. Wall Chemical Reactions (C) Each of these processes is characterized by its own timescale which may diﬀer considerably from process to process depending on the local thermodynamic condi- tions. Loosely speaking, we think of FandTas to macroscopic phenomena, and WandCas of microscopic ones. The relevant ﬂuid scales are the advective and momentum-diﬀusive time, and the mass-diﬀusion time of the species respectively: τA=L/U, τν=H2/ν,(15) where L, Hare the length and height of the ﬂuid domain. The relevant time scales for species dynamics are: τD=H2/D, τw=K−1 w, τc=K−1 c(16)As discussed in the introduction, they deﬁne the major dimensionless parameters Re=UH/ν ≡τA/τν, (17) Pe=UH/D ≡τA/τD, (18) Dac=τc/τA, Da w=τw/τA (19) To acknowledge the multiscale nature in time of the problem, a subcycled time-stepper is adopted. This is organized as follows. The code ticks with the hopping time of the ﬂuid populations from a lattice site to its neighbors dt=dx/c= 1. Under all circumstances dt is much smaller than both diﬀusive and advective ﬂuid scales in order to provide a faithful description of ﬂuid ﬂow. Whenever dtexceeds the chemical time-scales (high Damkohler regime), fractional time-stepping , i.e. subcy- cling of the microscopic mechanisms, namely chemical- wall transfer is performed. This means that the chemi- cal and wall transfer operators are performed dt/τ c,dt/τ w times respectively at each ﬂuid cycle. As it will be ap- preciated shortly, since the ﬂow solver ticks at the sound speed, the present microﬂow simulations proceed in very short time steps, of the order of tens of nanoseconds. This means that they can be in principle coupled to meso- scopic methods, such as kinetic Monte Carlo, aﬀording a more realistic description of the ﬂuid-wall interactions . In particular, a Kinetic Monte Carlo update of a sin- gle boundary cell could proceed in parallel with a cor- responding hydrodynamic treatment of the entire pile of ﬂuid cells on top of the wall. The ﬂip side of the medal is that in order to draw quantitative conclusions at the scale of the macroscopic devices a two-three decade ex- trapolation is required. This commands a robust scaling argument. IV. CATALYTIC EFFICIENCY: QUALITATIVE ANALYSIS Ideally, we would like to synthetize a universal func- tional dependence of the catalytic eﬃciency as a function of the relevant dimensionless numbers and geometrical design parameters: η=f(Re, Da, Pe ; ¯g). (20) where ¯ grepresents a vector of geometric parameters char- acterizing the boundary shape. The question is to as- sess the sensitivity of ηto ¯gand possibly ﬁnd an op- timal solution (maximum η) within the given parame- ter space. Mathematically, this is a complex non-linear functional optimization problem for the geometrical pa- rameters. We ﬁnd it convenient to start from a simple- and yet representative-baseline geometry as an “unper- turbed” zero order approximation, which is easily acces- sible either analytically or numerically. Perturbations t o 3this baseline situation can then be parametrized as “topo- logical excitations” on top of the geometrical “ground state”. In the present study, the unperturbed geometry is a straight channel of size Lalong the ﬂow direction and H×Hacross it. Perturbations are then deﬁned as micro- corrugations in the bottom wall of the form z=h(x, y), h≡0 being the smooth-wall unperturbed case. In this work, the perturbation is taken in the form of delta-like protrusions (barriers) h(x, y, z ) =/summationtext ihiδ(x−xi). From a macroscopic point of view the device eﬃciency is deﬁned as amount of pollutant consumpted per unit mass injected: η=Φin−Φout Φin(21) where Φ(x) =/integraldisplay [uC](x, y, z )dydz (22) is the longitudinal mass ﬂow of the pollutant at section x. The in-out longitudinal ﬂow deﬁcit is of course equal to the amount of pollutant absorbed at the catalytic wall, namely the normal-to-wall mass ﬂow rate: Γ =/integraldisplay S/vector γ(x, y, z )·d/vectorS (23) where the ﬂux consists of both advective and diﬀusive components: /vector γ=/vector uC−D∇C (24) and the integral runs over the entire absorbing surface S The goal of the optimization problem is to maximize Γ at a given Φ in. As it is apparent from the above ex- pressions, this means maximizing complex conﬁguration- dependent quantities, such as the wall distribution of the pollutant and its normal-to-wall gradient. For future pur- poses, we ﬁnd it convenient to recast the catalytic eﬃ- ciency as η= 1−T, where Tis the channell transmittance T≡Φout/Φin (25) From a microscopic viewpoint, Tcan be regarded as the probability for a tracer molecule injected at the inlet to exit the channel without being absorbed by the wall and consequently it ﬁxes the escape rate from the chemi- cal trap. Roughly speaking, in the limit of fast-chemistry, this is controlled by the ratio of advection to diﬀusion timescales. More precisely, the escape rate is high if the cross-channel distance walked by a tracer molecule in a transit time τAis much smaller than the channel cross- length H/2. Mathematically: DτA≪H2/4, which is: Pe≫4L/H (26) The above inequality (in reverse) shows that in order to achieve high conversion eﬃciencies, the longitudinal aspect ratio L/Hof the device has to scale linearly with the Peclet number.A. The role of micro-irregularities We now discuss the main qualitative eﬀect of geomet- rical roughness on the above picture from a microscopic point of view, i.e. trying to resolve ﬂow features at the same scale of the micro-irregularity. In the ﬁrst place, geometric irregularities provide a po- tential enhancement of reactivity via the sheer increase of the surface/volume ratio. Of course, how much of this potential is actually realized depends on the resulting ﬂow conﬁguration. Here, the ﬂuid plays a two-faced role. First, geomet- rical restrictions lead to local ﬂuid acceleration, hence less time for the pollutant molecules to migrate from the bulk to the wall before being convected away by the mainstream ﬂow. This eﬀect, usually negligible for macroscopic ﬂows, may become appreciable for micro- ﬂows with h/H≃0.1 (like in actual catalytic convert- ers),hbeing the typical geometrical micro-scale of the wall corrugations. Moreover, obstacles shield away part of the active surface (wake of the obstacle) where the ﬂuid circulates at much reduced rates (stagnation) so that less pollutant is fed into the active surface. The size of the shielded region is proportional to the Reynolds number of the ﬂow. On the other hand, if by some mechanism the ﬂow proves capable of feeding the shielded region, then eﬃcient absorption is restored simply because the pollu- tant is conﬁned by recirculating patterns and has almost inﬁnite time to react without being convected away. The ordinary mechanism to feed the wall is molecular diﬀu- sion/dispersion, which is usually rather slow as compared to advection. More eﬃcient is the case where the ﬂow de- velops local micro-turbulence which may increase bulk- to-wall diﬀusive transport via enhanced density gradients and attendant density jumps Cg−Cw: Γtur w=−[w′C]w (27) where w′is the normal-to-wall microturbulent velocity ﬂuctuation. This latter can even dominate the picture whenever turbulent ﬂuctuations are suﬃciently energetic, a condition met when the micro-Peclet number exceeds unity: Peh=w′h D≫1 (28) where his the typical geometrical micro-scale. Given this complex competition of eﬃciency-promoting and eﬃciency-degrading interweaved eﬀects it is clear that assessing which type of micro-irregularities can promote better eﬃciency is a non-trivial task. B. Eﬃciency: analytic and scaling considerations For a smooth channel, the steady state solution of the longitudinal concentration ﬁeld away from the inlet 4boundary factors into the product of three independent one-dimensional functions: C(x, y, z ) =X(x)Y(y)Z(z). Replacing this ansatz into the steady-state version of the equation (3) we obtain: X(x) =X0e−x/l Y(y) =Y0 Z(z) =Z0cos(z/l⊥)(29) with the longitudinal and cross-ﬂow absorption lengths related via: l=l2 ⊥¯U D(30) where ¯Uis the average ﬂow speed ¯U(x) =/summationdisplay y,zu(x, y, z )C(x, y, z )//summationdisplay y,zC(x, y, z ) (31) Note that the proﬁle along the spanwise coordinate y remains almost ﬂat because we stipulate that only the top and bottom walls host catalytic reactions. To determine the cross-ﬂow absorption length l⊥we impose that along all ﬂuid cells in a contact with the wall, the diﬀusive ﬂux is exactly equal to ﬂuid-to-wall outﬂow, namely: C l2 ⊥=Cg τ2 Nz(32) where τthe eﬀective absorption/reaction time scale, 1 τ≃1 τD+1 (τc+τw), (33) andNz=H2is the number of cells ( dx= 1 in the code) in a cross-section x=const. of the channel. Therefore the factor 2 /Nzis the fraction of reactive cells along any given cross-section x=const. of the channel. The form factor Cg/Cis readily obtained by the third of Eq. (29) which yields Cg C≃cos(H/2l⊥) (34) Combining this equation with Eq. (32) we obtain a non- linear algebraic equation for l⊥: λ−2cos(λ/2) =Dτ H2Nz 2(35) where we have set λ≡H/l⊥. For each set of parameters this equation can be easily solved numerically to deliver l⊥, hence lvia the Eq. (30). Given the exponential dependence along the stream- wise coordinate x, the eﬃciency can then be estimated as: η0≃1−e−L/l(36)Note that in the low absorption limit L≪l, the above relation reduces to η0≃L/l, meaning that doubling, say, the absorption length implies same eﬃciency with a twice shorter catalyzer. In the opposite high-absorption limit, L≫l, the relative pay-oﬀ becomes increasingly less signiﬁcant. C. Corrugated channel: Analytical estimates Having discussed tha baseline geometry, we now turn to the case of a “perturbed” geometry. Let us begin by considering a single barrier of height h. The reference situation is a smooth channel at high Damkohler with η0= 1−e−L/l. We seek perturbative corrections in the smallness parameter g≡h/H, the coupling-strength to geometrical perturbations. The unperturbed wall-ﬂux is Γ0≃2DCh hLH (37) where Chis the concentration at the tip of the barrier calculated in the smooth channel. Therefore Ch/his an estimate of the normal-to-wall diﬀusive gradient. The geometrical gain due to extra-active wall surface is Γ1≃ChuhhH (38) where uh≃4U0(g−g2) (39) is the average longitudinal ﬂow speed in front of the bar- rier along a section x=const. . The shadowed region of sizewin the wake of the obstacle yields a contribution Γ2≃a DCh hwH (40) where ais a measure of the absorption activity in the shielded region. Three distinctive cases can be identiﬁed: •a= 0: The wake region is totally deactivated, ab- sorption zero. •a= 1: The wake absorption is exactly the same as for unperturbed ﬂow •a >1: The wake absorption is higher than with unperturbed ﬂow (back-ﬂowing micro-vortices can hit the rear side of the barrier) Combining these expressions we obtain the following compact expression: δη η0=Γ1+ Γ2−Γ2(h= 0) Γ0≃A 2h HReh[Sc+K(a−1)] (41) 5where A=H/Lis the aspect ratio of the channel and Sc=ν/Dis the Schmidt number (ﬂuid viscosity/tracer mass diﬀusivity) and the wake length can be estimated asw/h=KRe hwithK≃0.1. The above expression shows a perturbative (quadratic) correction in hover the unperturbed (smooth chan- nel situation). However, since the eﬀective absorption in the shielded region is aﬀected by higher order com- plex phenomena, the factor amay itself exhibit a non- perturbative dependence on h, so that departures from this quadratic scaling should not come as a surprise. Apart from its actual accuracy, we believe expressions like (41) may provide a qualitative guideline to esti- mate the eﬃciency of generic/random obstacle distribu- tions [ xi, hi]: In particular, they should oﬀer a semi- quantitative insights into non-perturbative eﬀects due to non-linear ﬂuid interactions triggered by geometrical micro-irregularities. V. APPLICATION: REACTIVE FLOW OVER A MICROBARRIER The previous computational scheme has been applied to a ﬂuid ﬂowing in a millimeter-sized box of of size 2 × 1×1 millimeters along the x, y, z directions with a pair of perpendicular barriers of height ha distance sapart on the bottom wall (see Fig. 1 for a rapid sketch). The single-barrier set up corresponds to the limit s= 0. The ﬂuid ﬂow carries a passive pollutant, say an exhaust gas ﬂow, which is absorbed at the channel walls where it disappears due to heterogeneus catalysis. The ﬂow is forced with a constant volumetric force which mimics the eﬀects of a pressure gradient. The exhaust gas is continuously injected at the inlet, x= 0, with a ﬂat proﬁle across the channel and, upon diﬀusing across the ﬂow, it reaches solid walls where it gets trapped and subsequently reacts according to a ﬁrst order catalytic reaction: C+A→P (42) where Adenote an active catalyzer and Pthe reaction products. The initial conditions are: C(x, y, z ) = 1, x= 1 (43) C(x, y, z ) = 0,elsewher e (44) ρ(x, y, z ) = 1 (45) u(x, y, z ) =U0, v(x, y, z ) =w(x, y, z ) = 0 (46) The pollutant is continuously injected at the inlet and released at the open outlet, while ﬂow periodicity is im- posed at the inlet/outlet boundaries. On the upper and lower walls, the ﬂow speed is forced to vanish, whereasthe ﬂuid-wall mass exchange is modelled via a mass trans- fer rate equation of the form previously discussed. We explore the eﬀects of a sub-millimeter pair of barri- ers of height ha distance sapart on the bottom wall. The idea is to assess the eﬀects of the interbarrier height ,h, and interbarrier separation son the chemical eﬃciency. Upon using a 80 ×40×40 computational grid, we obtain a lattice with dx=dy=dz= 0.0025 (25 microns), and dt=csdx/V s≃50 10−9(50 nanoseconds). Here we have assumed a sound speed Vs= 300 m/s and used the fact that the sound speed is cs= 1/√ 3 in lattice units. Our simulations refer to the following values (in lattice units ): U0≃0.1−0.2,D= 0.1,ν= 0.01,Kc=Kw= 0.1. This corresponds to a diﬀusion-limited scenario: τc=τw= 10< τA≃800< τD= 16000 < τν= 160000 (47) or, in terms of dimensionless numbers: Pe≃40, Re≃400, Da > 80 (48) As per the interbarrier separation, we consider the fol- lowing values: h/H= 0.2 and s/L= 0,1/8,1/4,1/2, and h/H= 0.05,0.1,0.2 fors/L= 0. For the sake of com- parison, the case of a smooth wall ( s= 0, h= 0) is also included. The typical simulation time-span is t= 32000 time- steps, namely about 1 .6 milliseconds in physical time, corresponding to two mass diﬀusion times across the channel. The physico-chemical parameters given above are not intended to match any speciﬁc experimental con- dition, but rather to develop a generic intuition for the interplay of the various processes in action under the fast chemistry assumption. A. Single barrier: eﬀects of barrier heigth We consider a single barrier of height hplaced in the middle of the bottom wall at x=L/2, z= 0. With the above parameters we may estimate the reference eﬃ- ciency for the case of smooth channel ﬂow. With ¯U≃0.1, andτ= 20, we obtain l≃200, hence η0≃0.5. A typical two-dimensional cut of the ﬂow pattern and pollutant spatial distribution in the section y=H/2 is shown in Figs. 2 and 3, which refer to the case h= 8, s= 0 (h/H= 0.1, s/L= 0.0). An extended (if feeble) recir- culation pattern is well visible past the barrier. Also, en- hanced concentration gradients in correspondence of the tip of the barrier is easily recognized from Fig. 3. A more quantitative information is conveyed by Fig. 4, where the integrated longitudinal concentration of the pollutant: C(x) =/summationdisplay y,zC(x, y, z ) (49) 6is presented for the cases h= 0,2,4,8 (always with s= 0). The main highlight is a substantial reduction of the pollutant concentration with increasing barrier height. This is qualitatively very plausible since the bulk ﬂow is richer in pollutant and consequently the tip of the barrier “eats up” more pollutant than the lower region. In order to gain a semi-quantitative estimate of the chem- ical eﬃciency, we measure the the pollutant longitudinal mass ﬂow: Φ(x) =/summationdisplay y,z[Cu](x, y, z ) (50) The values at x= 1 and x=Ldeﬁne the eﬃciency ac- cording to Eq. (21) (to minimize ﬁnite-size eﬀects actual measurements are taken at x= 2 and x= 70). The corresponding results are shown in Table I, where subscript Arefers to the analytical expression (41) with a= 1. These results are in a reasonable agreement with the analytical estimate Eq. (41) taken at a= 1 (same absorption as the smooth channel). However, for h= 8 the assumption a= 1 overestimates the actual eﬃciency, indicating that the shielded region absorbs signiﬁcantly less pollutant than in the smooth-channel scenario. In- deed, inspection of the transversal concentration proﬁles (Fig. 5) along the chord x= 3L/4, y=H/2 reveals a neat depletion of the pollutant in the wake region. This is the shielding eﬀect of the barrier. Besides this eﬃciency-degrading eﬀect, the barrier also promotes a potentially beneﬁcial ﬂow recirculation, which is well visible in Figs. 6 and 7. Figure 6 shows the time evolution of the streamwise velocity u(z) in the mid-line x= 3L/4, y=H/2. It clearly reveals that recir- culating backﬂow only sets in for h= 8, and also shows that the velocity proﬁle gets very close to steady state. A blow-up of the recirculating pattern in the near-wall back-barrier region is shown in Fig. 7. However these recirculation eﬀects are feeble (the intensity of the recir - culating ﬂow is less than ten percent of the bulk ﬂow) and depletion remains the dominant mechanism. In fact forh= 8 the measured local Peclet number is of the or- der 0.01·8/0.1 = 0 .8, seemingly too small to promote appreciable micro-turbulent eﬀects. In passing, it should be noticed that raising the barrier height has an appre- ciable impact on the bulk ﬂow as well, which displays some twenty percent reduction due to mechanical losses on the barrier. Finally, we observe that the measured eﬃciency is smaller than the theoretical ηcfor smoth channel. This is due to the fact that the ﬂow Φ( x= 2) is signiﬁcantly enhanced by the imposed inlet ﬂat proﬁle C(z) = 1 at x= 1 (as well visible in Fig. 4). Leaving aside the initial portion of the channel, our numerical data are pretty well ﬁtted by an exponential with absorption length l= 160, in a reasonable agreement with the theoretical estimate l≃200 obtained by solving Eqs. (30) and (32).B. Eﬀects of barrier separation Next we examine the eﬀect of interbarrier separation. To this purpose, three separations s= 10,20,40 sym- metric around x0=L/2 are been considered. A typical two-barrier ﬂow pattern with s= 40 is shown in Fig. 8. From this picture we see that even with the largest sepa- ration s= 40, the second barrier is still marginally in the wake of the ﬁrst one. As a result, we expect it to suﬀer seriously from the aforementioned depletion eﬀected pro- duced by the ﬁrst barrier. This expectation is indeed con- ﬁrmed by the results reported in Table II. These results show that, at least on the microscopic scale, the presence of a second barrier does not seem to make any signiﬁcant diﬀerence, regardless of its separation from the ﬁrst one. As anticipated, the most intuitive explanation is again shadowing: the ﬁrst barrier gets much more “food” than the second one, which is left with much less pollutant due to the depletion eﬀect induced by the ﬁrst one. Inspec- tion of the longitudinal pollutant concentration (Fig. 9) clearly shows that the ﬁrst barrier, regardless of its lo- cation, “eats up” most of the pollutant (deﬁcit with re- spect to the upper-lying smooth-channel curve is almost unchanged on top of the second barrier). Of course, this destructive interference is expected to go away for “well- separated” barriers with s≫w. Indeed, the ultimate goal of such investigations should be to devise geomet- rical set-ups leading to constructive interference . This would require much larger and longer simulations which are beyond the scope of the present work. C. Eﬀects of barrier height on a longer timescale Since the previous simulations only cover a fraction of the global momentum diﬀusion time, one may wonder how would the picture change by going to longer time scales of the order of H2/ν. Longer single-barrier simula- tions, with t= 160 ,000, up to 10 diﬀusion times, namely about 15 milliseconds, provide the results exposed in Ta- ble III. We observe that the quantitative change is very minor, just a small eﬃciency reduction due to a slightly higher ﬂow speed. Indeed, the spatial distribution of the pollu- tant does not show any signiﬁcant changes as compared to the shorter simulations. and a similar conclusion ap- plies to the ﬂow pattern (see Figs. 10 and 11). This is because in a Poiseuille ﬂow, the ﬂuid gets quickly to, say, 90 percent of its total bulk speed (and even quicker to its near-wall steady conﬁguration), while it takes much longer to attain the remaining ten percent. Since it is the near-wall ﬂow conﬁguration which matters mostly in terms of a semi-quantitative estimate of the chemical eﬃ- ciency, we may conclude that the simulation span can be contained to within a fraction of the global momentum equilibration time. 7VI. UPSCALING TO MACROSCOPIC DEVICES It is important to realize that even tiny improvements on the microscopic scale can result in pretty sizeable cu- mulative eﬀects on the macroscopic scale of the real de- vices, say 10 centimeters. Assuming for a while the ef- ﬁciency of an array of Nserial micro-channels can be estimated simply as ηN= 1−TN, (51) it is readily recognized that even low single-channel ef- ﬁciencies can result in signiﬁcant eﬃciencies of macro- scopic devices with N= 10−100 (see Fig. 12). In particular, single-channel transmittances as high as 90 percent can lead to appreciable macroscopic eﬃciencies, around 60 percent, when just ten such micro-channels are linked-up together. Such a sensitive dependence implies that extrapolation to the macroscopic scales, even when successfull in matching experimental data [11,12], must be taken cautiously. In fact, the above expression (51) represents of course a rather bold upscaling assumption. As a partial supporting argument, we note that unless the geometry itself is made self-aﬃne (fractal walls [10]), or the ﬂow develops its own intrinsic scaling structure (fully developed turbulence), the basic phenomena should re- main controlled by a single scale l, independent of the device size L. Since both instances can be excluded for the present work, extrapolation to macroscopic scales is indeed conceivable. Nonetheless, it is clear a tight sin- ergy between computer simulation and adequate analyt- ical scaling theories is in great demand to make sensible predictions at the macroscopic scale. VII. CONCLUSIONS This work presents a very preliminary exploratory study of the complex hydro-chemical phenomena which control the eﬀective reactivity of catalytic devices of mil - limetric size. Although the simulations generally conﬁrm qualitative expectations on the overall dependence on the major physical parameters, they also highlight the exis- tence of non-perturbative eﬀects, such as the onset of micro-vorticity in the wake of geometrical obstrusions, which are hardly amenable to analytical treatment. It is hoped that the ﬂexibility of the present computer tool, as combined with semi-analytical theories, can be of signiﬁ- cant help in developing semi-quantitative intuition about the subtle and fascinating interplay between geometry, chemistry, diﬀusion and hydrodynamics in the design of chemical traps, catalytic converters and other related de- vices.VIII. ACKNOWLEDGEMENTS Work performed under NATO Grant PST.CLG.976357. SS acknowledges a scholarship from the Physics Department at Harvard University. [1] F. Abraham, J. Broughton, N. Bernstein, E. Kaxiras, Comp. in Phys., 12, 538 (1998). [2] E. Oran, J. Boris, Numerical simulation of reactive ﬂows , Elsevier Science, New York, 1987. [3] G. Ertl, H.J. Freund, Catalysis and surface science, Phy s. Today, 52, n.1, 32 (1999). [4] G. Mc Namara, G. Zanetti, Phys. Rev. Lett., 61, 2332 (1988). [5] F. Higuera, S. Succi, R. Benzi, Europhys. Lett., 9, 345 (1989). [6] R. Benzi, S. Succi and M. Vergassola, Phys. Rep., 222, 145 (1992). [7] Y. Qian, D.d’Humieres, P. Lallemand, Europhys. Lett., 17, 149 (1989). [8] S. Succi, G. Bella, H. Chen, K. Molvig, C. Teixeira, J. Comp. Phys., 152, 493 (1999). [9] X. He, L. Luo, Phys. Rev. E, 55, 6333 (1997). [10] B. Sapoval, Europhys. Lett., in press, 2001. [11] S. Succi, G. Smith, E. Kaxiras, J. Stat. Phys., 2001, sub - mitted. [12] A. Bergmann, R. Bruck, C. Kruse, Society of Automotive Engineers (SAE) technical paper SAE 971027, Proceed- ings of the 1997 International SAE Congress, Detroit, USA, February 1997. 8Run h/H ηδη η,δηA ηA R00 0 0.295 0.00 R02 1/20 0.301 0.02,0.025 R04 1/10 0.312 0.06,0.10 R08 2/10 0.360 0.22,0.40 TABLE I. Single barrier at x= 40: the eﬀect of barrier height. Run s/L η R00 0 0.30 R08 1/8 0.36 R28 2/8 0.37 R48 4/8 0.375 TABLE II. Two barriers of height h= 8: Eﬀect of inter- separation s. Run h/H ηδη η,δηA ηA L00 0 0.290 0,0 L02 0/20 0.296 0.02,0.025 L04 1/10 0.307 0.06,0.10 L08 2/10 0.360 0.24,0.40 TABLE III. s= 0,h= 0,4,8: 10 mass diﬀusion times FIG. 1. Sketch of the of a section at y=const. of a typical channel with two microbarriers. Two barriers of height h= 3 a distance s= 10 apart: F=ﬂuid, B=buﬀer. u(x,z) at y=L/2: t=32000 U=-0.01U=0.15 01020304050607080 X0510152025303540 Z FIG. 2. Typical two-dimensional cut of the ﬂow pattern with a single barrier of heigth h= 8. Streamwise ﬂow speed in the plane y=H/2. 9C(x,z) at y=L/2: t=32000 0.250.651.0 01020304050607080 X0510152025303540 Z FIG. 3. Concentration isocontours with a single barrier of heigth h= 8. 60080010001200140016001800 10 20 30 40 50 60 70C(X) Streamwise coordinate XLongitudinal pollutant concentration: single barrier h=8h=4h=0 FIG. 4. Integrated longitudinal concentration C(x) of the pollutant with a single barrier of height h= 8 after 32000 steps.0510152025303540 0.10.20.30.40.50.60.70.80.9 11.11.2Z Concentration C(Z) Transverse pollutant concentration at x=3L/4,y=H/2: 32000 steps h=8 h=4 h=0 FIG. 5. Transverse pollutant concentration C(z) at x= 3L/4 and y=H/2. Single barrier of varying height. The four curves for each of the three diﬀerent heigths are taken at t= 3200 ,6400,29800 ,32000. 0510152025303540 -0.01 0.04 0.09 0.14 0.19 0.24Z U(Z)Streamwise speed U(Z) at x=3L/4,y=L/2 h=8 h=4 h=0 FIG. 6. Time evolution of the transversal streamwise speed u(z) atx= 3L/4 and y=L/2. Single barrier of varying height. 10Stream function at y=L/2: h=8, 160000 timesteps \|U\|=0.0 \|U\|=0.01\|U\|=0.10 30354045505560657075 X024681012 Z FIG. 7. Blow-up of the streamlines of the ﬂow ﬁeld past a barrier of height h= 8 located at x= 40. The velocity direction in the closed streamlines of the vortex is clockwi se. Streamwise u(x,z) at y=L/2 U=0.15 U=-0.0025 01020304050607080 X0510152025303540 Z FIG. 8. Isocontours of the streamwise ﬂow speed with two barriers with h= 8, s= 20 at t= 32000. 60080010001200140016001800 10 20 30 40 50 60 70C(X) Streamwise coordinate XLongitudinal concentration: two barriers at various separations s=20s=40s=0s=0,h=0FIG. 9. Longitudinal concentration C(x) for h= 8, s= 0,20,40 all at t= 32000. 0510152025303540 0 0.2 0.4 0.6 0.8 1 1.2Z Concentration C(Z) Transversal pollutant concentration at x=3L/2,y=H/4: 160000 time-steps h=8 h=4 h=0 FIG. 10. Integrated longitudinal concentration C(x) of the pollutant with a single barrier of height h= 8 after 160000 steps. 0510152025303540 -0.02 0.03 0.08 0.13 0.18 0.23z u(z)Streamwise speed U(Z) at x=3L/4,y=L/2 h=8 h=4 h=0 FIG. 11. Time evolution of the transversal streamwise speed u(z) atx= 3L/4 and y=L/2 after 160000 steps. Single barrier of varying height. 110.50.550.60.650.70.750.80.850.90.951 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Efficiency Single-channel transmittance TEfficiency of N serial channels N=1N=2N=10N=50 N=5 FIG. 12. Eﬃciency of a series of Nmicro-channels as a function of the single-channel transmittance. 12
arXiv:physics/0103042v1 [physics.bio-ph] 15 Mar 2001Information transport by sine-Gordon solitons in microtub ules Elcio Abdalla1, Bouchra Marouﬁ1,2, Bertha Cuadros Melgar1and Moulay Brahim Sedra2,3 1Instituto de F´ ısica, Universidade de S˜ ao Paulo C.P.66.318, CEP 05315-970, S˜ ao Paulo, Brazil 2Laboratoire-UFR, de Physique des Hautes Energies, Facult´ e des Sciences, Av Ibn Batouta, BP 1014, Rabat, Morocco. 3Laboratoire de Physique Th´ eorique et Appliqu´ ee (L.P.T.A ), D´ epartment de Physique Facult´ e des Sciences, BP 133, K´ enitra, Morocco (01/12/01) We study the problem of information propagation in brain mic rotubules. After considering the prop- agation of electromagnetic waves in a ﬂuid of permanent elec tric dipoles, the problem reduces to the sine-Gordon wave equation in one space and one time dimen sions. The problem of propagation of information is thus set. PACS number(s): 87.17.Aa 87.17.-d 89.70.+c 89.75.Fb 89.75 .-k Information is a central question in the understand- ing of the mechanisms regulating the brain. Questions as bounds on information, capacity of communication channels are of extreme relevance and a theory of in- formation and communication is of utter value [1]. Thus, how much information can be stored by a cube is an im- portant question. Foreseeable technology making use of atomic manipulation would suggest an upper bound of around 1020bits. But as technology takes advantage of unforeseen paradigms, this number could grow up. Could the bound grow without limit? With black hole ther- modynamics some deﬁnite answers are forthcoming [2]. In Quantum Field Theory the question developed unex- pectedly in the framework of black hole thermodynamics and Quantum Gravity [3]. Some knowledge was already known from Shannon’s Theory of Information as well [4]. Shannon imagined a system capable of storing infor- mation by virtue of it possessing many distinguishable states. A state ais not known a priori but its probabil- itypais known. The measure of uncertainty corresponds to an entropy Swhich formally coincides with the corre- sponding statistical interpretation of entropy: S=−K/summationdisplay apaln pa (1) On the other hand, it has been conjectured [5] that the brain microtubules might be an active component of the brain functioning. It is thus natural to consider elec- tromagnetic waves moving in that cavity as transporting and carrying information. With these matters in mind, we consider here the ef- fective electromagnetic wave obtained when the second quantized electromagnetic ﬁeld interacts with the per- manent dipole moment of the vicinal water in brain mi- crotubules. The second quantized electromagnetic ﬁeld shall be given by the usual development in frequency components [6]. The electric dipoles can be seen as two-component spinors described as eﬀective spin-ﬁelds. Therefore, we have, for the eﬀective Hamiltonian,H=HQED−µN/summationdisplay j=1[/vectorEtr(/vector xj, t)/vector s] +ǫN/summationdisplay j=1sz,(2) where /vectorEtris the electric ﬁeld transversal to the wave propagation direction and /vector sis a spin ﬁeld describing the electric dipole moment degree of freedom. The last term represents an eﬀective interaction of the z-component of the electric dipole with an average electric ﬁeld, that is, it represents a two-energy eigenstates system. The value ofǫ≈50cm−1= 6.3×10−3eV= 1 .0×10−14erg has been claimed in [7]. Such a problem has been considered by [8]. We derive some results which have not been explicitely obtained in [8]. We suppose that the electromagnetic ﬁeld has a fast dependence on z−ctand a slow dependence on zandt, allowing us to write the expansion /vectorE(/vector x, t) =/summationdisplay/vectorEn tr(z, t)eikn(z−ct)(3) Using the equations of motion derived from the Hamil- tonian (2), that is, Maxwell equations with sources, we arrive at ∂E± ∂z+1 c∂E± ∂t=±i2πǫµ ¯hVs∓(4) This is a quantum equation of motion. However we do not have any practical means to either measure the vari- ables, or take care of its detailed dynamics, therefore we take its quantum average. Such an average is easily obtained due to the simple description of sin terms of Pauli matrices, leading to a result written in terms of the exponential of the ﬁeld θ, deﬁned by θ±(z, t) =µ ¯h/integraldisplayt 0/angbracketleftE±(z, t)/angbracketrightqudu , (5) where we take the quantum average /angbracketleft/angbracketrightqu. Following uch a procedure in equation (4) leads to the semiclassical equa- tion of motion 1∂2θ± ∂t∂σ=−4πǫNµ2 ¯h2Vsinθ±(6) where N/Vis the number of dipoles (molecules) per unit volume, and σ=t+z c. Above, the indices ±correspond to the usual combinations of the transversal direction, and we supposed also that the longitudinal direction does not propagate. This is a variant of the well known sine- Gordon equation. The one-soliton solution is given by the expression E=¯h µAsechA (t−z ν0) (7) where the angular frequency characteristic of the model is A=/radicalBigg 2πǫµ2Nν0 ¯h2V(c−ν0)(8) andν0is the velocity of the soliton. In order to understand the propagation of information in such a device, we follow [1] and consider small pertur- bations around the soliton, which amounts to solving the equation ω2η+iω∂η ∂t−A2 0cosθ0η= 0 (9) where A0=/radicalBig 2πǫN Vµ ¯h≈3.1×1014s−1. Plugging in back the solution θ0= 4 arctanexp[ −Az/ν 0], that is, cos θ0= 1−2sech2(Az/ν 0), we obtain the equation iω∂η ∂t=−2A2 0sech2(Az/ν 0)η (10) where we chose the boundary conditions such that ω= A0. The only solution is η=exp[2i/radicalbiggν0 ctanhAz ν0] (11) Let us discuss the physics behind the problem and the consequences for the constants appearing in the solution. First, the constant ǫis a free parameter and represents the energy of a dipole in the vicinal water. It is of the order of magnitude of diﬀerence of two molecular energy levels. The study of vibration in water indicates the value ǫ≈50cm−1= 6.3×10−3eV= 1 .0×10−14erg [7]. The constant µrepresents the permanent electric dipole mo- ment of the water, which is the electron charge times 0.2×10−8cm, that is, in CGS units, µ≈6.8×10−18. Fi- nally, the number of molecules per unit volume is easily obtained for the water, it is of the order of 0 .3×1023cm−3. In order to ﬁx the velocity of the wave, we integrate the solitonic electric ﬁeld imposing that it is the unit synapti c potential coming from the quantum of transmitter pro- ducing a postsynaptic potential, typically 0 .5 to 1 .0mV,as discussed in [9], who has proposed it to be a quantum unit of potential in such a context. We have V≈/integraldisplay Edz=π 2¯hν0 µ, (12) thus obtaining, for the velocity parameter, the value ν0≈ 1.4×104cm/s, orν0 c≈0.5×10−6. With this result for ν0we obtain for the constant Athe result A≈2.2× 1011s−1. Estimating the time to send information as that necessary to pass the bulk of the soliton (7), we get a rough estimate for the frequency of the waves as ν≈ A 6≈3.7×1010s−1. On the other hand, taking the average electric ﬁeld in the brain as corresponding to the quantum unit of elec- tric potential value as discussed above, namely ∼1mV, divided by the lenght of the typical microtubule, that is, ∼10nm we are led, for the average electric ﬁeld, to the value Eave≈10−1V2×102×10−9m≈3statV/cm (13) Now, the constants AandEaveare related by A=Emaxµ ¯h=√ 2Eaveµ ¯h≈4×1010s−1(14) This is compatible with the previous value for A, giving us some conﬁdence on the result. The corresponding wavelenght is λ=c 2πA≈3mm. This corresponds to the order of magnitude of the pineal gland. Whether this is just a coincidence or whether it has a deeper meaning is a question that deserves further study. Moreover, it corresponds to a typical frequency already obtained for phonon transitions in the brain, al- lowing for new theoretical models of the interaction of electromagnetism with the biological cells [10]! Furthermore, there are bounds on information storage. Theoretically, in a problem of completely diﬀerent char- acter, one arrives at the maximum entropy a cache can hold, with the result Imax<2πRE ¯hcln 2(15) where Ris the overall radius of the object under study andEits energy. In the theory of solitons, the number of possible information-holding conﬁgurations based on the soliton equals the number of quanta that might populate the ﬁrst excited level. To this number we must add unity to account for the soliton conﬁguration itself. So, the pos- sible conﬁgurations within an energy budget Eabove the soliton energy is N(E) = 1+[[ E/¯hω]] ([[x]] stands for the integral part of x), then Imax=ln(1 + [[ E/ω1]])log2e bits (16) consistent with the bound (15). In our case, for a micro- tubule, we have R≈10−4cm. Taking the energy Eas cor- responding to a quantum of energy ¯ hA0, from the source 2of the electromagnetic ﬁeld, we ﬁnd E= ¯hA0≈0.2eV. In such a case, Imax≈1 (17) while the bound (15) corresponds to Imax<6. It would not be too original to call such an information aquantum information unit sent via the microtubuiles, in view of similar considerations, in a diferent context by Gabor [11]. In the present case, the soliton, formed by the interaction of Quantum Electrodynamics with the elec- tric dipole moment of the background water in the one- dimensional device oﬀered by microtubules is the natural way chosen by nature to send information bits. Another interesting point is the fact that the frequency parameters which showed up naturally in the course of the computations have natural interpretations in terms of brain structures. The frequency A0≈3.1×1014s−1 is compatible with the size of the microtubules. But the real frequency ν≈A0 6/radicalbigν0 c≈3.7×1010s−1is compatible with the transition period observed for the socalled con- formational changes connected with tubulin dimer pro- tein (namely ≈109to 1011s−1) [8]. This is a further example of the application of Quan- tum Field Theory to general aspects of matter interac- tions in complex systems. It is clear that this is not a way of achieving comprehension of the complexity as- pects of such a sophisticated system, but it certainly pro- vides valuable tools for working in this ﬁeld as well. Acknowledgements: this work has been partially supported by CNPq (Conselho Nacional de Desenvolvi- mento Cient´ ıﬁco e Tecnol´ ogico) and FAPESP (Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo). B.Marouﬁ would like to thank the Instituto de F´ ısica, Universidade de S˜ ao Paulo, for the hospitality. E.A. thanks Drs. I. Prates de Oliveira and S. F. de Oliveira for discussions concerning aspects of the brain functions. [1] J. Bekenstein and M. Schiﬀer Int. J. Mod. Phys. C 1 (1990) 355-422. [2] J.D.Bekenstein, Lett. Nuov. Cim. 4(1972) 737; Phys. Rev.D7(1973) 2333; D9(1974) 3292; S. Hawking, Com- mun. Math. Phys. 43(1975) 199. [3] G. ’t Hooft, gr-qc/9310026; L. Susskind, J. Math. Phys. 36(1995) 6377. [4] C. Shannon and W. Weaver, The Mathematical Theory of Communication , Univ. of Illinois Press, 1949. [5] R. Penrose The Emperor’s New Mind , Oxford Univer- sity Press, 1989, Shadows of the Mind , Oxford University Press, 1994. [6] C. Itsykson and J.B. Zuber Quantum Field Theory , Mcgraw-hill, New York, (1980).[7] F. Franks Water: A comprehensive Treatise , Plenum Press, New York, 1972. [8] M. Jibu S. Hagan, S. R. Hameroﬀ, K. H. Pribram and K. Yasue, Bio Systems 32(1994) 195-209. [9] E. R. Kandel in Essentials of Neural Science and Behav- ior, ed. E. R. Kandel, J. H. Schwartz and T. M. Jessel, Prentice Hall International, INC, 1995. [10] S. R. Hameroﬀ and R. C. Watt J. Theor. Biol. 98(1982) 549; E. Del Giudice, G. Preparata and G. Vitiello Phys. Rev. Lett. 61(1988) 1085. [11] D.Gabor Nature 161(1948) 777-778. 3
1Analytic tomography of the mantle in a spherically Earth. The technique MZY Yoël Lana-Renault Geophysics. Department of Theoretical Physics. University of Zaragoza. 50009 Zaragoza. Spain e-mail: yoel@kepler.unizar.es Abstract. An explicit expression for P-wave velocity is proposed to develop a novel tomographic technique in a spherically symmetric model of the Earth (MZY). The distribution of the P velocity structure in the mantle is determined using only 34 P- and 2 PcP- observed traveltimes. By applying a non-linear inversion, the P-residuals in the range between 0º and 100º are minimised up to a maximum value of 0.015 s. Furthermore, from the high quality computation of PcP traveltimes, with residuals much better than 0.13 s., it is possible to infer the existence of a brief low velocity layer in the D” region. This is then followed by a gradual increasing in the velocity profile towards the core, which begins at a depth of 2893.9 km. Key words: D” shell, Earth’s mantle, P-wave velocity, tomography, traveltimes. Introduction To date, numerous studies use the arrival times of seismic waves to explore the Earth structure. Seismic arrival times have provided a fundamental constraint on the radial and lateral velocity structure of our planet. Sengupta and Toksoz (1976), Clayton and Comer (1983), Dziewonski (1984) among others, studied the variation of the P-wave velocity in the lower mantle. These works have been extended rapidly to the whole mantle (Pulliam et al., 1993) from many different viewpoints and perspectives, but concluding in almost all cases in interesting correlations with the structure predicted by the plate tectonics. On the other hand, reference models constitute the common basis for all the different studies concerning the Earth. Some of them are fairly relevant and well known in the seismological literature, as PREM (Dziewonski and Anderson, 1981), IASP91 (Kennet and Engdahl, 1991) and SP6 (Morelli and Dziewonski, 1993). They constitute the starting point for a number of applications, including seismic tomography and synthetic seismogram calculations. The strategy of finding an agreement between physical meaningful and2achieving observations is of crucial importance. A decreasing of the relative error between the reproduced and measured data becomes in an increasing knowledge of the main features concerning the Earth structure. In this sense, any effort made to improve the available reference models, will benefit on the current seismological knowledge, especially those concerning local deviations in boundary interfaces in the Earth’s interior. From this viewpoint, in this preliminary work we pretend to improve the fitting of reference traveltime tables (JB: Jeffreys and Bullen, 1958; BSSA: Herrin et al., 1968) to observed traveltimes and, as consequence of that, to infer the slight deviations of the whole structure with respect to the average models. We have focused our attention on the tomography of the mantle, using and developing a non-linear inversion technique based on the analytical solution of the elliptical integrals involved in the theory of wave propagation. In this sense the approach described in this paper cannot be viewed like an empirical model. We demonstrate that the range of the achievement is large enough and, therefore, the real interpretation is to be an improvement for the reference model derived from the Herrin et al. traveltime tables, used in this work. Eventually, this sort of agreement to respect the observed data (errors not larger than a particular threshold) has been imposed as a first objective of this study, but it is not unique. The use of an analytical function avoids the common strategy of deriving spherical averages from seismological observations via an inversion procedure (i.e., the least-square approach). An interesting comment of this performing can be found in Morelli and Dziewonski (1993). In our scheme, the inherent biased data distribution is largely overcome since only traveltime tables are taking into account. This absence of real data is a major lack in the model we present in this paper, and we agree. However, we keep the opinion that the results should be interpreted in a different way as those derived from a reference model, because they maintain internal consistency and do not pretend to be an alternative to PREM, ISAP91 or SP6 models. The use of analytical functions to derive a model that globally reproduces the observed traveltimes by acting locally on a multilayer and spherical mantle does not prescribe the meaning, from a physical viewpoint, of the new model. Indeed, the analytical tomography results in an improved understanding of some particular areas, for example the D’’ layer at the base of the mantle. These features are the most relevant conclusions of our work as they provide some slight differences to the current knowledge of the mantle. Methodology The trial P-wave velocity function used in this work to analyse the structure of the mantle can be summarised by the expression v()r.r( ) B.Aln()r, (1)3where r is the radius, and (A,B) two independent parameters to be determined. This formula can be simplified by defining the function w()r( ) B .Aln()r , (2) and then: v()r.rw()r. (3) The P-wave velocity function expressed in Eq. (1) has been used (Lana- Renault and Cid, 1991; Lana-Renault, 1998) to obtain different Earth models by varying the different parameters. The smoothness of this function makes it adequate to tomographic studies of the Earth’s mantle, once a proper parameterisation is applied, i.e. a division in many spherical layers, which is the one followed in this work. Another useful property of the function described in Eq. (1) is that converts the elliptical integrals arisen during the hamiltonian formulation of ray propagation, into analytical functions. For example, for a ray crossing the first layer ( i = 1) of the mantle, who radius of the top surface is R1 (Earth’s radius), the epicentral distance D can be expressed as D..2w1 A1senh.A1T 2, (4) where: w1w1R1B1.A1lnR1. The general analytical expressions for D and T can be obtained using the classical integral expressions (Bullen and Bolt, 1985) D .p d rpro r .r1h2p2, (5) T d rpro r..h2 r1 h2 p2 . Denoting the angle of incidence at the top surface of the ith layer by Ii and its radius by Ri , and similarly for the variables at the bottom ( I´i and R´i ), see figure 1, it is always possible to write4wiwiRiBi.AilnRisenIi p, (6) and w´iw´iR´iBi.AilnR´isenI´i p. (7) Fig. 1. P-trajectory traveling through a layer i In general, for a point P(r) we have wi()rBi.Ailn()rsen()I p. (8) Hence, through its derivative, dr .rcos()IdI .pAi(9) we can calculate the expressions (5) for a P-trajectory which travels from Ri to R´i DicosIicosI´i .pAi, (10) Ti.1 AilntgI´i 2 tgIi 2. (11) 5Therefore, the observables at the Earth surface can be computed as a result of several additions of these computed values at each layer. That is, if one ray travels along k layers, the final epicentral distance and traveltime are calculated through of the following 2k+1 equations: D.2 p icosIicosI´i AicosIk Ak, (12) T.2 i.1 AilntgI´i 2 tgIi 2.1 AklntgIk 2(13) and these 2k-1 auxiliary equations psenIi wiRisenI´i wiR´isenIk wkRk, (14) where : i = 1, 2, ..., k-1 . On the other hand, the observables for a PcP-trajectory are calculated by the following 2(k+1) equations: D.2 p icosIicosI´i Ai, (15) T.2 i.1 AilntgI´i 2 tgIi 2(16) and these 2k auxiliary equations psenIi wiRisenI´i wiR´i. (17) where : i = 1, 2, ..., k . Finally, by integrating Eq. (9) between P(Ri) and P(r), it is easy to calculate the radius of any single point P(r) along the trajectory: r.RiexpsenIisen()I .pAi.Riexpwiwi()r Ai. (18) 6Results With a single collection of observed traveltimes, it is possible to reproduce the observations on the Earth’s surface for any event. For the sake of simplicity, as an example of the versatility and functionality of the proposed methodology, we have selected the datasets reproduced in Herrin et al. (1968) . The sequence of calculations consists of determining the specific constants Ai , wi and w´i (i = 1,...,N), for each layer , N being the number of layers. (Note that w´i is a measure of the thickness of the ith layer and that we don’t use Bi . The parameter Bi is calculated after using the Eq. (6)) Let suppose these quantities are already known for the first k-1 layers, except w´k-1, the starting point for the kth layer. The inverse problem can be posed as a system of non-linear equations (12-14) that will provide the parameters Ak and w k of the layer k. We must use three P-observed trajectories reproducing three fixed points ( Dl , Tol ; l = 1,2,3) as boundary conditions for the system of 3(2k+1) non linear equations with 3(2k+1) unknowns ( pl , Iil , I´il , Ikl , w´k-1 , wk , Ak). The solution is then iterated till assure a convergence criterion, in our case, a threshold for the computed residuals less than a certain value (10-15). Known the values w´k-1 , wk , y Ak , we prove that the residual times To-Tc (observed minus computed time) of the all the others P-trajectories which also return to the surface-focus from the kth layer are smaller than a determined e . If it is not so, we begin again taking others three observables ( Dl , Tol) nearer among them. It is to be noted that the algebra applied in our methodology permits a discontinuity of the 1st kind ( w´k-1 ≠ wk) in the velocity function. This last property can be analysed through the study of the derivative ( dT/dD) (Herrin et al. , 1968), in order to detect jumps in the selected velocity pattern. If we have the security that only a discontinuity of the 2nd kind ( w´k-1 = wk) is present, then it is possible to work with only 2 observables ( D , To) or fixed boundary conditions, thus eliminating 2k+1 redundant equations from the global system. In this case, the experience tells us that is much better to work with one observable ( D , To) and, thus, fixing the final of the k-1th layer by a value for w´k-1. and insuring that the residual times of all the P-trajectories which return from the k-1th layer to the surface-focus are less than our e . Thereby, we resolve a non linear system with only 2k+1 equations. We have performed a complete description of the Mantle using a maximum residual time ε = 0.015 s. and only 34 P-observed traveltimes. The total number of layers used in this description is 28. The last one finishes at a depth of 2810.1 km., maximum for the last P-observed trajectory at D = 100º according to Herrin et al. (1968), with To = 826.7303 s. Once known the problematic of the lack of information for D > 100º and the special case of the D” shell, we have worked with7data available from D > 88º and maximum residuals of 0.002 s. See Table 1 and Residuals of P-travel times (Figure 2) for a graphic representation and further details. Fig. 2. P-residual times To-Tc . D: 0 – 100º every 0.5º. Maximum residual 0.015 sec. at D = 18.5º.P times -0,02-0,010,000,010,02 0 10 20 30 40 50 60 70 80 90 100 Delta (degrees)Residual (s) The 29th layer . Since the derivative associated with the surface-focus travel time ( dT/dD) is effectively constant ( Herrin et al. , 1968) beyond 99.0º, and that our residuals are practically zero for those points, we consider the boundary condition w´28 = 1/p(100º) = w29 produces the best results. Thus, our problem is reduced to calculate the parameters w´29 (final layer) and A29. For this purpose, we pose a non linear system with two PcP observables. One of these fixed points should always be the axial trajectory ( D = 0º ; To = 511.3 sec. ) that allows us to use only one equation: T.2 i.1 Ailnw´i wi(19) We have considered that the other observable should be very separated from the first, and thus, selected D = 93º ; To = 795.2 s as second observable . Once obtained the values w´29 and A29, all the PcP residuals were balanced with an error less than 0.13 s. (Figure 3, Table 2 of PcP-travel times). By applying Eq. (18) we found the outer core at 2893.9 km. Further details can be seen in Figures 4, 5, 6 and Table 3 of P-wave velocity.8Fig. 3. PcP-residual times To-Tc . D: 0 – 96º every 1º. Maximum residual 0.13 sec. at D = 75º. Depth Outer Core: 2893.9 km. Fig. 4. P-wave velocity in the mantle. Herrin et al. versus MZY.PcP times -0,2-0,10,00,10,2 0 10 20 30 40 50 60 70 80 90 100 Delta (degrees)Residual (s) P - wave velocity 567891011121314 0 400 800 1200 1600 2000 2400 2800 Depth (km)km / s Herrin et al. MZY Fig. 4 shows a comparison between our velocity distribution and the one provided by Herrin et al . (1968). The most conspicuous difference is observed at 2749.8 km ., final of the 25th layer, where the last trajectory returns to the surface at D = 92º and the corresponding residual is null. Figure 5 exhibits residuals of computed velocity Herrin et al. minus MZY. Let us note how the maximum residual is 0.0044 km/s at a depth of 755 km.9P - wave velocity -0,006-0,0030,0000,0030,006 0 300 600 900 1200 1500 1800 2100 2400 2700 Depth (km)Residual (km/s) Fig. 5. Residual velocity Herrin et al. minus MZY in the mantle until a depth of 2745 km. Maximum residual: 0.0044 km/sec at a depth of 755 km. From 2749.8 km. until the core-mantle boundary ( region D” ), our velocity distribution begins to be completely different to Herrin et al . (1968), as can be seen in Fig. 6. This is due that Herrin et al. adopted a special smooth velocity distribution to the region D” to explain the last results from Taggart and Engdahl, (1968) , which indicated a slow increase of velocity towards the core. Morelli and Dziewonski (1993), in their SP6 model, obtained a continuous decrease from 2741 km. In our MZY model, we propose that D” region begins at 2780.7 km. with a brief (29.4 km.) negative gradient (layers number 27 and 28) followed by a slow increase until the core. With this profile we insure the residuals of all the observables P and PcP from Herrin et al . (1968) are minima, and reproduce accurately the observed times. Fig. 6. P-wave velocity in D” region. Herrin et al. versus MZYP - wave velocity 13,5613,5813,6013,6213,6413,6613,6813,7013,72 2720 2740 2760 2780 2800 2820 2840 2860 2880 Depth (km)km / s Herrin et al. MZY10Concluding remarks We have presented a new technique to tomography the interior of the Earth for which one is able to obtain residual times less than a determined value e for all observed trajectories P. The more minor is the value of e, more genuine and real is the tomography. Also, we have seen that the technique MZY developed is very easy to apply. Its potentiality is based in the function velocity found that it provides us analytical solutions for Δ and T. Acknowledgements The author is grateful to Dr. Javier Sabadell by the comments and suggestions made during the writing of the manuscript.11 Table 1 P travel times (sec.) p. 1/2 La- Δ initial Δ Observed Computed Residual La- Δ initial Δ Observed Computed Residual yers Δ final (Herrin & al. ) MZY To - Tc yers Δ final (Herrin & al. ) MZY To - Tc 00.00 0.0000 0.0000 0.0000 27.14 27.50 347.2025 347.1930 0.0095 1 0.50 9.2663 9.2662 0.0001 28.00 351.6796 351.6743 0.0053 7.85 1.00 18.5323 18.5321 0.0002 15 28.50 356.1456 356.1490 -0.0034 20.52 1.50 26.9525 26.9525 0.0000 29.00 360.6048 360.6144 -0.0096 10.66 29.50 365.0596 365.0682 -0.0086 0.989 2.00 34.8630 34.8630 0.0000 3030.00 369.5086 369.5086 0.0000 2.50 41.7231 41.7213 0.0018 3030.50 373.9477 373.9401 0.0076 3.00 48.5813 48.5782 0.0031 31.00 378.3751 378.3650 0.0101 3.50 55.4373 55.4334 0.0039 31.50 382.7900 382.7810 0.0090 4.00 62.2906 62.2864 0.0042 32.00 387.1923 387.1867 0.0056 4.50 69.1410 69.1367 0.0043 32.50 391.5831 391.5808 0.0023 5.00 75.9880 75.9840 0.0040 33.00 395.9621 395.9627 -0.0006 5.50 82.8312 82.8278 0.0034 33.50 400.3281 400.3315 -0.0034 6.00 89.6703 89.6676 0.0027 34.00 404.6807 404.6864 -0.0057 6.50 96.5049 96.5030 0.0019 16 34.50 409.0193 409.0270 -0.0077 7.00 103.3346 103.3337 0.0009 35.00 413.3435 413.3527 -0.0092 3 7.50 110.1591 110.1591 0.0000 35.50 417.6532 417.6629 -0.0097 8.00 116.9779 116.9789 -0.0010 36.00 421.9479 421.9573 -0.0094 8.50 123.7908 123.7926 -0.0018 36.50 426.2269 426.2353 -0.0084 9.00 130.5973 130.5998 -0.0025 37.00 430.4894 430.4967 -0.0073 9.50 137.3970 137.4000 -0.0030 37.50 434.7347 434.7411 -0.0064 10.00 144.1896 144.1930 -0.0034 38.00 438.9626 438.9681 -0.0055 10.50 150.9747 150.9783 -0.0036 38.50 443.1730 443.1775 -0.0045 11.00 157.7519 157.7554 -0.0035 39.00 447.3662 447.3690 -0.0028 11.50 164.5209 164.5240 -0.0031 39.5 39.50 451.5425 451.5425 0.0000 12.00 171.2813 171.2836 -0.0023 39.5 40.00 455.7020 455.7008 0.0012 12.50 178.0326 178.0340 -0.0014 40.50 459.8449 459.8440 0.0009 13.15 13.00 184.7746 184.7746 0.0000 41.00 463.9710 463.9709 0.0001 413.15 13.50 191.4964 191.4929 0.0035 41.50 468.0802 468.0808 -0.0006 14.14 14.14 200.0582 200.0655 -0.0073 42.00 472.1723 472.1736 -0.0013 514.14 14.50 204.8555 204.8480 0.0075 42.50 476.2473 476.2488 -0.0015 15.13 15.13 213.1831 213.1946 -0.0115 43.00 480.3051 480.3064 -0.0013 615.13 15.50 218.0429 218.0320 0.0109 43.50 484.3454 484.3462 -0.0008 16.11 16.11 225.9644 225.9771 -0.0127 44.00 488.3680 488.3680 0.0000 716.11 16.50 230.9845 230.9726 0.0119 44.50 492.3728 492.3716 0.0012 17.09 17.09 238.4697 238.4824 -0.0127 45.00 496.3596 496.3571 0.0025 817.09 17.50 243.6096 243.5956 0.0140 45.50 500.3285 500.3244 0.0041 18.08 18.08 250.7491 250.7627 -0.0136 46.00 504.2791 504.2733 0.0058 918.08 18.50 255.8408 255.8261 0.0147 46.50 508.2111 508.2038 0.0073 19.06 19.06 262.4864 262.4973 -0.0109 47.00 512.1242 512.1159 0.0083 1019.06 19.50 267.6136 267.6016 0.0120 17 47.50 516.0178 516.0096 0.0082 20.04 20.04 273.7653 273.7721 -0.0068 48.00 519.8920 519.8847 0.0073 1120.04 20.50 278.9036 278.8956 0.0080 48.50 523.7469 523.7415 0.0054 21.02 21.02 284.5832 284.5863 -0.0031 49.00 527.5828 527.5797 0.0031 1221.02 21.50 289.7160 289.7114 0.0046 49.50 531.4001 531.3995 0.0006 2222.00 294.9501 294.9501 0.0000 50.00 535.1992 535.2008 -0.0016 2222.50 300.0806 300.0759 0.0047 50.50 538.9802 538.9837 -0.0035 23.00 305.1134 305.1050 0.0084 51.00 542.7433 542.7482 -0.0049 13 23.50 310.0533 310.0470 0.0063 51.50 546.4887 546.4944 -0.0057 24.00 314.9070 314.9070 0.0000 52.00 550.2164 550.2222 -0.0058 24.50 319.6818 319.6890 -0.0072 52.50 553.9266 553.9317 -0.0051 25.46 25.00 324.3869 324.3961 -0.0092 53.00 557.6192 557.6230 -0.0038 25.46 25.46 328.6614 328.6630 -0.0016 53.50 561.2941 561.2962 -0.0021 14 26.00 333.6295 333.6262 0.0033 54.00 564.9510 564.9512 -0.0002 26.50 338.1848 338.1873 -0.0025 54.50 568.5899 568.5882 0.0017 27.14 27.14 343.9656 343.9643 0.0013 55.00 572.2107 572.2072 0.003512 Table 1 P travel times (sec.) p. 2/2 La- Δ initial Δ Observed Computed Residual La- Δ initial Δ Observed Computed Residual yers Δ final (Herrin & al. ) MZY To - Tc yers Δ final (Herrin & al. ) MZY To - Tc 55.50 575.8137 575.8082 0.0055 81.8 81.80 740.2336 740.2409 -0.0073 56.00 579.3986 579.3915 0.0071 82.50 743.9007 743.8974 0.0033 56.50 582.9653 582.9569 0.0084 22 83.00 746.4926 746.4852 0.0074 17 57.00 586.5135 586.5047 0.0088 83.50 749.0611 749.0544 0.0067 57.50 590.0430 590.0348 0.0082 84.04 84.04 751.8075 751.8092 -0.0017 58.00 593.5538 593.5475 0.0063 84.04 84.50 754.1271 754.1271 0.0000 58.50 597.0462 597.0427 0.0035 85.00 756.6260 756.6263 -0.0003 5959.00 600.5205 600.5205 0.0000 85.50 759.1042 759.1061 -0.0019 5959.50 603.9770 603.9803 -0.0033 23 86.00 761.5636 761.5676 -0.0040 60.00 607.4162 607.4225 -0.0063 86.50 764.0064 764.0117 -0.0053 60.50 610.8385 610.8471 -0.0086 87.00 766.4338 766.4390 -0.0052 61.00 614.2444 614.2545 -0.0101 87.50 768.8465 768.8502 -0.0037 61.50 617.6343 617.6447 -0.0104 8888.00 771.2455 771.2455 0.0000 62.00 621.0084 621.0179 -0.0095 8888.50 773.6315 773.6301 0.0014 62.50 624.3668 624.3743 -0.0075 24 89.00 776.0056 776.0053 0.0003 63.00 627.7094 627.7138 -0.0044 89.50 778.3687 778.3695 -0.0008 63.50 631.0356 631.0367 -0.0011 9090.00 780.7222 780.7222 0.0000 18 64.00 634.3452 634.3430 0.0022 9090.50 783.0673 783.0673 0.0000 64.50 637.6379 637.6329 0.0050 25 91.00 785.4049 785.4061 -0.0012 65.00 640.9137 640.9064 0.0073 91.50 787.7356 787.7371 -0.0015 65.50 644.1724 644.1637 0.0087 9292.00 790.0597 790.0597 0.0000 66.00 647.4142 647.4049 0.0093 9292.50 792.3774 792.3767 0.0007 66.50 650.6392 650.6301 0.0091 26 93.00 794.6891 794.6892 -0.0001 67.00 653.8477 653.8395 0.0082 93.50 796.9953 796.9961 -0.0008 67.50 657.0398 657.0330 0.0068 94.03 94.03 799.4344 799.4344 0.0000 68.00 660.2151 660.2108 0.0043 94.03 94.50 801.5937 801.5930 0.0007 68.50 663.3731 663.3731 0.0000 27 95.00 803.8872 803.8873 -0.0001 69.08 69.08 667.0129 667.0220 -0.0091 95.50 806.1777 806.1784 -0.0007 69.08 69.50 669.6355 669.6387 -0.0032 96.123 96.123 809.0282 809.0279 0.0003 70.00 672.7383 672.7354 0.0029 96.123 96.50 810.7518 810.7503 0.0015 19 70.50 675.8202 675.8127 0.0075 97.00 813.0361 813.0345 0.0016 71.00 678.8805 678.8716 0.0089 97.50 815.3192 815.3185 0.0007 71.50 681.9193 681.9128 0.0065 28 98.00 817.6016 817.6021 -0.0005 72.35 72.00 684.9366 684.9366 0.0000 98.50 819.8838 819.8851 -0.0013 72.35 72.35 687.0340 687.0432 -0.0092 99.00 822.1660 822.1676 -0.0016 73.00 690.9092 690.9170 -0.0078 99.50 824.4481 824.4494 -0.0013 73.50 693.8665 693.8729 -0.0064 100100.00 826.7303 826.7303 0.0000 74.00 696.8054 696.8096 -0.0042 100 99.5 824.4481 824.4498 -0.0017 74.50 699.7264 699.7279 -0.0015 99.0 822.1660 822.1694 -0.0034 20 75.00 702.6299 702.6283 0.0016 98.5 819.8838 819.8892 -0.0054 75.50 705.5159 705.5115 0.0044 98.0 817.6016 817.6094 -0.0078 76.00 708.3843 708.3776 0.0067 29 97.5 815.3192 815.3302 -0.0110 76.50 711.2346 711.2272 0.0074 97.0 813.0361 813.0520 -0.0159 77.00 714.0661 714.0604 0.0057 96.5 810.7518 810.7757 -0.0239 77.9 77.50 716.8776 716.8776 0.0000 96.0 808.4658 808.5028 -0.0370 77.9 77.90 719.1107 719.1200 -0.0093 95.618 806.7177 806.7715 -0.0538 78.50 722.4405 722.4426 -0.0021 96.0 808.4658 808.4805 -0.0147 79.00 725.1920 725.1893 0.0027 96.289 96.289 809.7871 809.7642 0.0229 21 79.50 727.9234 727.9171 0.0063 interpolated 80.00 730.6349 730.6267 0.0082 80.50 733.3270 733.3190 0.0080 81.00 735.9998 735.9944 0.0054 81.8 81.50 738.6533 738.6533 0.000013 Table 2 PcP travel times (sec.) Δ Observed Computed Residual Δ Observed Computed Residual (Herrin & al.) MZY To - Tc (Herrin & al.) MZY To - Tc 0 511.3 511.300 0.00 49 611.9 611.870 0.03 1 511.4 511.348 0.05 50 615.5 615.511 -0.01 2 511.5 511.492 0.01 51 619.2 619.191 0.01 3 511.7 511.731 -0.03 52 622.9 622.910 -0.01 4 512.1 512.066 0.03 53 626.7 626.666 0.03 5 512.5 512.497 0.00 54 630.4 630.458 -0.06 6 513.0 513.022 -0.02 55 634.3 634.284 0.02 7 513.6 513.641 -0.04 56 638.1 638.144 -0.04 8 514.4 514.355 0.04 57 642.0 642.037 -0.04 9 515.2 515.163 0.04 58 646.0 645.961 0.04 10 516.1 516.063 0.04 59 649.9 649.914 -0.01 11 517.1 517.055 0.04 60 653.9 653.896 0.00 12 518.1 518.139 -0.04 61 657.9 657.906 -0.01 13 519.3 519.314 -0.01 62 661.9 661.943 -0.04 14 520.6 520.578 0.02 63 666.0 666.005 0.00 15 521.9 521.932 -0.03 64 670.1 670.091 0.01 16 523.4 523.373 0.03 65 674.2 674.201 0.00 17 524.9 524.901 0.00 66 678.3 678.333 -0.03 18 526.5 526.515 -0.01 67 682.5 682.486 0.01 19 528.2 528.214 -0.01 68 686.6 686.660 -0.06 20 530.0 529.996 0.00 69 690.8 690.853 -0.05 21 531.9 531.861 0.04 70 695.1 695.065 0.04 22 533.8 533.806 -0.01 71 699.3 699.294 0.01 23 535.8 535.832 -0.03 72 703.5 703.540 -0.04 24 537.9 537.936 -0.04 73 707.9 707.802 0.10 25 540.1 540.117 -0.02 74 712.1 712.079 0.02 26 542.4 542.374 0.03 75 716.5 716.370 0.13 27 544.7 544.705 -0.01 76 720.7 720.675 0.03 28 547.1 547.110 -0.01 77 725.0 724.992 0.01 29 549.6 549.586 0.01 78 729.3 729.320 -0.02 30 552.1 552.132 -0.03 79 733.6 733.660 -0.06 31 554.7 554.747 -0.05 80 738.0 738.010 -0.01 32 557.4 557.428 -0.03 81 742.4 742.370 0.03 33 560.2 560.176 0.02 82 746.7 746.739 -0.04 34 563.0 562.988 0.01 83 751.1 751.116 -0.02 35 565.9 565.863 0.04 84 755.5 755.500 0.00 36 568.8 568.798 0.00 85 759.9 759.892 0.01 37 571.8 571.794 0.01 86 764.3 764.290 0.01 38 574.8 574.848 -0.05 87 768.7 768.693 0.01 39 578.0 577.959 0.04 88 773.1 773.102 0.00 40 581.1 581.125 -0.02 89 777.5 777.515 -0.01 41 584.3 584.345 -0.04 90 781.9 781.932 -0.03 42 587.6 587.617 -0.02 91 786.3 786.352 -0.05 43 590.9 590.940 -0.04 92 790.8 790.775 0.03 44 594.3 594.313 -0.01 93 795.2 795.200 0.00 45 597.7 597.734 -0.03 94 799.6 799.627 -0.03 46 601.2 601.202 0.00 95 804.0 804.055 -0.05 47 604.7 604.714 -0.01 96 808.5 808.484 0.02 48 608.3 608.271 0.03 96.289 809.76414 Table 3 Data MZY P-wave velocity (km/s) Data MZY p. 1/4 P-wave velocity (km/s) Radius (km) vi (km/s) Radius of surface-focus = 6371.028 Radius (km) vi (km/s) Layers Bi ( x 10-2 )Depth Layers Bi ( x 10-2 )Depth Ai ( x 10-3 )(km) Herrin & al. MZY Residual Ai ( x 10-3 )(km) Herrin & al. MZY Residual Depth (km) v´i (km/s) H. & al. - MZY Depth (km) v´i (km/s) H. & al. - MZY 6371.028 6.0000 0 6.0000 6.0001 -0.0001 6021.290062 8.8862 350 8.8905 8.8875 0.0030 1 0.92341045 5 6.0000 6.0001 -0.0001 355 8.9131 8.9114 0.0017 0.94666548 10 6.0000 6.0001 -0.0001 360 8.9360 8.9352 0.0008 15.001533 6.0001 15 6.0000 6.0001 -0.0001 365 8.9590 8.9591 -0.0001 6356.026467 6.7500 20 6.7500 6.7501 -0.0001 370 8.9823 8.9829 -0.0006 2 1.04491768 25 6.7500 6.7501 -0.0001 10 5.59329351 375 9.0058 9.0067 -0.0009 1.07194400 30 6.7500 6.7502 -0.0002 6.25724227 380 9.0294 9.0304 -0.0010 35 6.7500 6.7502 -0.0002 385 9.0532 9.0542 -0.0010 40.053935 6.7502 40 6.7500 6.7502 -0.0002 390 9.0773 9.0779 -0.0006 6330.974065 8.0540 45 8.0582 8.0593 -0.0011 395 9.1015 9.1016 -0.0001 50 8.0642 8.0645 -0.0003 400 9.1258 9.1252 0.0006 55 8.0698 8.0698 0.0000 405 9.1503 9.1489 0.0014 60 8.0753 8.0751 0.0002 411.322422 9.1787 410 9.1750 9.1725 0.0025 65 8.0806 8.0803 0.0003 5959.705578 9.1787 415 9.1999 9.1977 0.0022 3 2.16670136 70 8.0859 8.0856 0.0003 420 9.2248 9.2236 0.0012 2.32998530 75 8.0911 8.0908 0.0003 425 9.2499 9.2494 0.0005 80 8.0962 8.0960 0.0002 430 9.2752 9.2752 0.0000 85 8.1013 8.1012 0.0001 435 9.3007 9.3010 -0.0003 90 8.1064 8.1064 0.0000 11 5.99325871 440 9.3262 9.3267 -0.0005 95 8.1115 8.1116 -0.0001 6.71735442 445 9.3519 9.3524 -0.0005 104.957687 8.1219 100 8.1165 8.1168 -0.0003 450 9.3778 9.3781 -0.0003 6266.070313 8.1219 105 8.1219 8.1220 -0.0001 455 9.4038 9.4038 0.0000 4 2.51841026 110 8.1285 8.1292 -0.0007 460 9.4299 9.4294 0.0005 2.73226453 115 8.1356 8.1363 -0.0007 465 9.4562 9.4550 0.0012 120 8.1432 8.1435 -0.0003 472.071820 9.4911 470 9.4826 9.4805 0.0021 125.320 8.1511 125 8.1513 8.1506 0.0007 5898.956180 9.4911 475 9.5091 9.5072 0.0019 6245.707939 8.1511 130 8.1599 8.1602 -0.0003 480 9.5358 9.5345 0.0013 135 8.1690 8.1699 -0.0009 485 9.5626 9.5618 0.0008 5 2.97405843 140 8.1786 8.1796 -0.0010 490 9.5895 9.5891 0.0004 3.25362200 145 8.1886 8.1893 -0.0007 495 9.6165 9.6164 0.0001 150 8.1991 8.1990 0.0001 12 6.31396167 500 9.6437 9.6436 0.0001 155.027311 8.2087 155 8.2101 8.2087 0.0014 7.08672019 505 9.6709 9.6709 0.0000 6216.000689 8.2087 160 8.2214 8.2213 0.0001 510 9.6981 9.6980 0.0001 165 8.2332 8.2340 -0.0008 515 9.7255 9.7252 0.0003 6 3.50244082 170 8.2454 8.2467 -0.0013 520 9.7530 9.7523 0.0007 3.85853274 175 8.2580 8.2593 -0.0013 525 9.7805 9.7794 0.0011 180 8.2710 8.2719 -0.0009 531.559687 9.8149 530 9.8080 9.8064 0.0016 185 8.2843 8.2845 -0.0002 5839.468313 9.8149 535 9.8356 9.8342 0.0014 193.795314 8.3067 190 8.2980 8.2971 0.0009 540 9.8632 9.8623 0.0009 6177.232686 8.3067 195 8.3120 8.3104 0.0016 545 9.8908 9.8903 0.0005 200 8.3264 8.3260 0.0004 550 9.9185 9.9184 0.0001 205 8.3410 8.3415 -0.0005 555 9.9462 9.9464 -0.0002 7 4.03019188 210 8.3560 8.3571 -0.0011 560 9.9740 9.9743 -0.0003 4.46315376 215 8.3713 8.3726 -0.0013 565 10.0018 10.0023 -0.0005 220 8.3870 8.3881 -0.0011 570 10.0296 10.0302 -0.0006 225 8.4029 8.4036 -0.0007 575 10.0574 10.0580 -0.0006 230 8.4191 8.4191 0.0000 580 10.0853 10.0859 -0.0006 239.227039 8.4476 235 8.4357 8.4345 0.0012 585 10.1132 10.1137 -0.0005 6131.800961 8.4476 240 8.4525 8.4504 0.0021 590 10.1411 10.1414 -0.0003 245 8.4696 8.4689 0.0007 595 10.1690 10.1692 -0.0002 250 8.4870 8.4873 -0.0003 600 10.1970 10.1969 0.0001 255 8.5047 8.5057 -0.0010 605 10.2249 10.2246 0.0003 8 4.55770255 260 8.5227 8.5241 -0.0014 13 6.50401417 610 10.2528 10.2522 0.0006 5.06801092 265 8.5410 8.5424 -0.0014 7.30586668 615 10.2807 10.2798 0.0009 270 8.5595 8.5607 -0.0012 620 10.3086 10.3074 0.0012 275 8.5783 8.5791 -0.0008 625 10.3364 10.3350 0.0014 280 8.5973 8.5974 -0.0001 630 10.3642 10.3625 0.0017 285 8.6167 8.6156 0.0011 635 10.3920 10.3900 0.0020 291.392218 8.6390 290 8.6362 8.6339 0.0023 640 10.4197 10.4174 0.0023 6079.635782 8.6390 295 8.6561 8.6543 0.0018 645 10.4474 10.4449 0.0025 300 8.6762 8.6756 0.0006 650 10.4750 10.4723 0.0027 305 8.6966 8.6969 -0.0003 655 10.5024 10.4996 0.0028 310 8.7172 8.7182 -0.0010 660 10.5297 10.5270 0.0027 9 5.09632938 315 8.7380 8.7394 -0.0014 665 10.5570 10.5542 0.0028 5.68621977 320 8.7591 8.7606 -0.0015 670 10.5840 10.5815 0.0025 325 8.7804 8.7818 -0.0014 675 10.6109 10.6087 0.0022 330 8.8020 8.8029 -0.0009 680 10.6375 10.6359 0.0016 335 8.8238 8.8241 -0.0003 685 10.6638 10.6631 0.0007 340 8.8458 8.8452 0.0006 690 10.6899 10.6903 -0.0004 349.737938 8.8862 345 8.8680 8.8663 0.0017 695 10.7157 10.7174 -0.0017 700.177664 10.7454 700 10.7412 10.7444 -0.003215 Table 3 Data MZY P-wave velocity (km/s) Data MZY p. 2/4 P-wave velocity (km/s) Radius (km) vi (km/s) Radius of surface-focus = 6371.028 Radius (km) vi (km/s) Layers Bi ( x 10-2 )Depth Layers Bi ( x 10-2 )Depth Ai ( x 10-3 )(km) Herrin & al. MZY Residual Ai ( x 10-3 )(km) Herrin & al. MZY Residual Depth (km) v´i (km/s) H. & al. - MZY Depth (km) v´i (km/s) H. & al. - MZY 5670.850336 10.7454 705 10.7664 10.7679 -0.0015 1115 11.6288 11.6296 -0.0008 710 10.7911 10.7912 -0.0001 1120 11.6367 11.6375 -0.0008 715 10.8154 10.8144 0.0010 1125 11.6446 11.6455 -0.0009 14 5.85891401 720 10.8392 10.8376 0.0016 1130 11.6525 11.6534 -0.0009 6.55949029 725 10.8624 10.8608 0.0016 1135 11.6604 11.6613 -0.0009 730 10.8850 10.8840 0.0010 1140 11.6684 11.6692 -0.0008 735 10.9068 10.9071 -0.0003 1145 11.6763 11.6771 -0.0008 744.109784 10.9492 740 10.9279 10.9302 -0.0023 1150 11.6842 11.6849 -0.0007 5626.918216 10.9492 745 10.9479 10.9515 -0.0036 1155 11.6921 11.6928 -0.0007 750 10.9663 10.9645 0.0018 1160 11.7000 11.7006 -0.0006 15 4.12047826 755 10.9819 10.9775 0.0044 1165 11.7080 11.7084 -0.0004 4.54632080 760 10.9933 10.9904 0.0029 1170 11.7159 11.7162 -0.0003 765 11.0029 11.0033 -0.0004 1175 11.7238 11.7239 -0.0001 772.052042 11.0215 770 11.0134 11.0163 -0.0029 1180 11.7316 11.7317 -0.0001 5598.975958 11.0215 775 11.0240 11.0273 -0.0033 1185 11.7395 11.7394 0.0001 780 11.0348 11.0370 -0.0022 1190 11.7473 11.7471 0.0002 785 11.0455 11.0467 -0.0012 1195 11.7551 11.7548 0.0003 790 11.0561 11.0564 -0.0003 1200 11.7629 11.7624 0.0005 795 11.0666 11.0660 0.0006 1205 11.7707 11.7701 0.0006 800 11.0766 11.0757 0.0009 1210 11.7785 11.7777 0.0008 805 11.0865 11.0853 0.0012 1215 11.7862 11.7853 0.0009 810 11.0962 11.0949 0.0013 1220 11.7938 11.7929 0.0009 815 11.1060 11.1045 0.0015 1225 11.8015 11.8005 0.0010 820 11.1156 11.1141 0.0015 1230 11.8091 11.8081 0.0010 825 11.1252 11.1236 0.0016 1235 11.8167 11.8156 0.0011 830 11.1348 11.1332 0.0016 1240 11.8242 11.8231 0.0011 835 11.1443 11.1427 0.0016 1245 11.8318 11.8306 0.0012 840 11.1538 11.1522 0.0016 1250 11.8393 11.8381 0.0012 845 11.1632 11.1617 0.0015 1255 11.8467 11.8456 0.0011 850 11.1726 11.1711 0.0015 1260 11.8542 11.8530 0.0012 16 3.57557893 855 11.1819 11.1806 0.0013 1265 11.8616 11.8605 0.0011 3.91494425 860 11.1912 11.1900 0.0012 1270 11.8689 11.8679 0.0010 865 11.2004 11.1994 0.0010 1275 11.8763 11.8753 0.0010 870 11.2096 11.2088 0.0008 1280 11.8836 11.8826 0.0010 875 11.2189 11.2182 0.0007 1285 11.8909 11.8900 0.0009 880 11.2281 11.2276 0.0005 1290 11.8982 11.8973 0.0009 885 11.2373 11.2369 0.0004 1295 11.9054 11.9046 0.0008 890 11.2466 11.2462 0.0004 17 3.48245516 1300 11.9126 11.9119 0.0007 895 11.2558 11.2555 0.0003 3.80663410 1305 11.9198 11.9192 0.0006 900 11.2651 11.2648 0.0003 1310 11.9269 11.9265 0.0004 905 11.2743 11.2741 0.0002 1315 11.9341 11.9337 0.0004 910 11.2835 11.2833 0.0002 1320 11.9412 11.9409 0.0003 915 11.2927 11.2926 0.0001 1325 11.9483 11.9481 0.0002 920 11.3019 11.3018 0.0001 1330 11.9554 11.9553 0.0001 925 11.3110 11.3110 0.0000 1335 11.9624 11.9625 -0.0001 930 11.3201 11.3202 -0.0001 1340 11.9695 11.9696 -0.0001 935 11.3291 11.3293 -0.0002 1345 11.9765 11.9768 -0.0003 940 11.3381 11.3385 -0.0004 1350 11.9836 11.9839 -0.0003 945 11.3470 11.3476 -0.0006 1355 11.9906 11.9910 -0.0004 950.864911 11.3583 950 11.3558 11.3567 -0.0009 1360 11.9976 11.9980 -0.0004 5420.163089 11.3583 955 11.3646 11.3654 -0.0008 1365 12.0046 12.0051 -0.0005 960 11.3734 11.3739 -0.0005 1370 12.0116 12.0121 -0.0005 965 11.3820 11.3824 -0.0004 1375 12.0185 12.0191 -0.0006 970 11.3907 11.3909 -0.0002 1380 12.0255 12.0261 -0.0006 975 11.3993 11.3994 -0.0001 1385 12.0324 12.0331 -0.0007 980 11.4079 11.4079 0.0000 1390 12.0393 12.0401 -0.0008 985 11.4164 11.4163 0.0001 1395 12.0462 12.0470 -0.0008 990 11.4249 11.4247 0.0002 1400 12.0531 12.0539 -0.0008 995 11.4333 11.4331 0.0002 1405 12.0599 12.0608 -0.0009 1000 11.4418 11.4415 0.0003 1410 12.0668 12.0677 -0.0009 1005 11.4502 11.4499 0.0003 1415 12.0736 12.0746 -0.0010 1010 11.4585 11.4582 0.0003 1420 12.0805 12.0814 -0.0009 1015 11.4668 11.4666 0.0002 1425 12.0873 12.0882 -0.0009 1020 11.4751 11.4749 0.0002 1430 12.0941 12.0950 -0.0009 1025 11.4834 11.4832 0.0002 1435 12.1009 12.1018 -0.0009 1030 11.4917 11.4915 0.0002 1440 12.1077 12.1086 -0.0009 1035 11.4999 11.4998 0.0001 1445 12.1145 12.1153 -0.0008 17 3.48245516 1040 11.5081 11.5080 0.0001 1450 12.1213 12.1221 -0.0008 3.80663410 1045 11.5163 11.5162 0.0001 1455 12.1281 12.1288 -0.0007 1050 11.5244 11.5245 -0.0001 1460 12.1349 12.1354 -0.0005 1055 11.5326 11.5326 0.0000 1465 12.1417 12.1421 -0.0004 1060 11.5407 11.5408 -0.0001 1470 12.1485 12.1488 -0.0003 1065 11.5488 11.5490 -0.0002 1475 12.1553 12.1554 -0.0001 1070 11.5569 11.5571 -0.0002 1480 12.1620 12.1620 0.0000 1075 11.5649 11.5652 -0.0003 1485 12.1688 12.1686 0.0002 1080 11.5730 11.5734 -0.0004 1490 12.1755 12.1752 0.0003 1085 11.5810 11.5814 -0.0004 1495 12.1822 12.1817 0.0005 1090 11.5890 11.5895 -0.0005 1500 12.1889 12.1882 0.0007 1095 11.5970 11.5976 -0.0006 1505 12.1956 12.1948 0.0008 1100 11.6049 11.6056 -0.0007 1510 12.2023 12.2013 0.0010 1105 11.6129 11.6136 -0.0007 1516.957451 12.2103 1515 12.2089 12.2077 0.0012 1110 11.6208 11.6216 -0.000816 Table 3 Data MZY P-wave velocity (km/s) Data MZY p. 3/4 P-wave velocity (km/s) Radius (km) vi (km/s) Radius of surface-focus = 6371.028 Radius (km) vi (km/s) Layers Bi ( x 10-2 )Depth Layers Bi ( x 10-2 )Depth Ai ( x 10-3 )(km) Herrin & al. MZY Residual Ai ( x 10-3 )(km) Herrin & al. MZY Residual Depth (km) v´i (km/s) H. & al. - MZY Depth (km) v´i (km/s) H. & al. - MZY 4854.070549 12.2103 1520 12.2155 12.2142 0.0013 4526.927109 12.5946 1845 12.5982 12.5957 0.0025 1525 12.2221 12.2207 0.0014 1850 12.6037 12.6018 0.0019 1530 12.2287 12.2272 0.0015 1855 12.6093 12.6079 0.0014 1535 12.2352 12.2337 0.0015 1860 12.6149 12.6141 0.0008 1540 12.2417 12.2402 0.0015 1865 12.6205 12.6201 0.0004 1545 12.2482 12.2466 0.0016 1870 12.6262 12.6262 0.0000 1550 12.2547 12.2530 0.0017 1875 12.6319 12.6322 -0.0003 1555 12.2611 12.2594 0.0017 1880 12.6376 12.6383 -0.0007 1560 12.2675 12.2658 0.0017 1885 12.6433 12.6443 -0.0010 1565 12.2738 12.2721 0.0017 1890 12.6491 12.6502 -0.0011 1570 12.2801 12.2784 0.0017 1895 12.6549 12.6562 -0.0013 1575 12.2864 12.2848 0.0016 1900 12.6607 12.6621 -0.0014 1580 12.2926 12.2910 0.0016 19 3.65834366 1905 12.6666 12.6680 -0.0014 1585 12.2988 12.2973 0.0015 4.01545527 1910 12.6725 12.6739 -0.0014 1590 12.3050 12.3036 0.0014 1915 12.6784 12.6798 -0.0014 1595 12.3111 12.3098 0.0013 1920 12.6843 12.6856 -0.0013 1600 12.3172 12.3160 0.0012 1925 12.6902 12.6914 -0.0012 1605 12.3232 12.3222 0.0010 1930 12.6962 12.6972 -0.0010 1610 12.3292 12.3284 0.0008 1935 12.7022 12.7030 -0.0008 1615 12.3352 12.3345 0.0007 1940 12.7082 12.7087 -0.0005 1620 12.3411 12.3407 0.0004 1945 12.7142 12.7144 -0.0002 1625 12.3471 12.3468 0.0003 1950 12.7202 12.7202 0.0000 1630 12.3530 12.3529 0.0001 1955 12.7262 12.7258 0.0004 1635 12.3589 12.3589 0.0000 1960 12.7323 12.7315 0.0008 1640 12.3648 12.3650 -0.0002 1965 12.7383 12.7371 0.0012 1645 12.3706 12.3710 -0.0004 1970 12.7444 12.7427 0.0017 1650 12.3765 12.3770 -0.0005 1975 12.7505 12.7483 0.0022 1655 12.3823 12.3830 -0.0007 1981.009473 12.7550 1980 12.7565 12.7539 0.0026 1660 12.3882 12.3890 -0.0008 4390.018527 12.7550 1985 12.7626 12.7601 0.0025 1665 12.3940 12.3950 -0.0010 1990 12.7687 12.7664 0.0023 18 3.49547729 1670 12.3999 12.4009 -0.0010 1995 12.7747 12.7726 0.0021 3.82197670 1675 12.4057 12.4068 -0.0011 2000 12.7808 12.7789 0.0019 1680 12.4115 12.4127 -0.0012 2005 12.7868 12.7851 0.0017 1685 12.4173 12.4186 -0.0013 2010 12.7928 12.7913 0.0015 1690 12.4231 12.4244 -0.0013 2015 12.7988 12.7975 0.0013 1695 12.4289 12.4302 -0.0013 2020 12.8048 12.8037 0.0011 1700 12.4347 12.4360 -0.0013 2025 12.8108 12.8098 0.0010 1705 12.4405 12.4418 -0.0013 2030 12.8167 12.8159 0.0008 1710 12.4463 12.4476 -0.0013 2035 12.8227 12.8220 0.0007 1715 12.4521 12.4533 -0.0012 2040 12.8286 12.8280 0.0006 1720 12.4578 12.4591 -0.0013 2045 12.8345 12.8341 0.0004 1725 12.4636 12.4648 -0.0012 2050 12.8403 12.8401 0.0002 1730 12.4693 12.4705 -0.0012 2055 12.8462 12.8461 0.0001 1735 12.4751 12.4761 -0.0010 2060 12.8520 12.8520 0.0000 1740 12.4808 12.4818 -0.0010 2065 12.8578 12.8580 -0.0002 1745 12.4865 12.4874 -0.0009 2070 12.8636 12.8639 -0.0003 1750 12.4922 12.4930 -0.0008 2075 12.8693 12.8698 -0.0005 1755 12.4978 12.4986 -0.0008 20 3.78936074 2080 12.8751 12.8757 -0.0006 1760 12.5035 12.5041 -0.0006 4.17166810 2085 12.8808 12.8815 -0.0007 1765 12.5091 12.5097 -0.0006 2090 12.8865 12.8873 -0.0008 1770 12.5148 12.5152 -0.0004 2095 12.8922 12.8931 -0.0009 1775 12.5204 12.5207 -0.0003 2100 12.8979 12.8989 -0.0010 1780 12.5259 12.5262 -0.0003 2105 12.9036 12.9046 -0.0010 1785 12.5315 12.5316 -0.0001 2110 12.9092 12.9104 -0.0012 1790 12.5371 12.5371 0.0000 2115 12.9149 12.9161 -0.0012 1795 12.5426 12.5425 0.0001 2120 12.9205 12.9217 -0.0012 1800 12.5481 12.5479 0.0002 2125 12.9262 12.9274 -0.0012 1805 12.5537 12.5533 0.0004 2130 12.9318 12.9330 -0.0012 1810 12.5592 12.5586 0.0006 2135 12.9375 12.9386 -0.0011 1815 12.5648 12.5639 0.0009 2140 12.9431 12.9442 -0.0011 1820 12.5703 12.5693 0.0010 2145 12.9487 12.9497 -0.0010 1825 12.5759 12.5745 0.0014 2150 12.9544 12.9552 -0.0008 1830 12.5814 12.5798 0.0016 2155 12.9601 12.9607 -0.0006 1835 12.5870 12.5851 0.0019 2160 12.9657 12.9662 -0.0005 1844.100891 12.5946 1840 12.5926 12.5903 0.0023 2165 12.9714 12.9717 -0.0003 2170 12.9771 12.9771 0.0000 2175 12.9829 12.9825 0.0004 2180 12.9886 12.9879 0.0007 2185 12.9944 12.9932 0.0012 2190 13.0002 12.9985 0.0017 2195 13.0059 13.0038 0.0021 2201.417284 13.0106 2200 13.0117 13.0091 0.002617 Table 3 Data MZY P-wave velocity (km/s) Data MZY p. 4/4 P-wave velocity (km/s) Radius (km) vi (km/s) Radius of surface-focus = 6371.028 Radius (km) vi (km/s) Layers Bi ( x 10-2 )Depth Layers Bi ( x 10-2 )Depth Ai ( x 10-3 )(km) Herrin & al. MZY Residual Ai ( x 10-3 )(km) Herrin & al. MZY Residual Depth (km) v´i (km/s) H. & al. - MZY Depth (km) v´i (km/s) H. & al. - MZY 4169.610716 13.0106 2205 13.0175 13.0151 0.0024 3887.018610 13.3469 2485 13.3499 13.3482 0.0017 2210 13.0234 13.0214 0.0020 2490 13.3562 13.3549 0.0013 2215 13.0292 13.0277 0.0015 2495 13.3626 13.3615 0.0011 2220 13.0350 13.0339 0.0011 2500 13.3690 13.3680 0.0010 2225 13.0408 13.0401 0.0007 2505 13.3753 13.3746 0.0007 2230 13.0466 13.0462 0.0004 2510 13.3817 13.3811 0.0006 2235 13.0525 13.0524 0.0001 2515 13.3881 13.3876 0.0005 2240 13.0583 13.0585 -0.0002 2520 13.3945 13.3940 0.0005 2245 13.0641 13.0646 -0.0005 2525 13.4009 13.4004 0.0005 2250 13.0700 13.0707 -0.0007 2530 13.4073 13.4068 0.0005 2255 13.0758 13.0767 -0.0009 2535 13.4137 13.4132 0.0005 2260 13.0817 13.0827 -0.0010 2540 13.4200 13.4195 0.0005 2265 13.0876 13.0887 -0.0011 2545 13.4264 13.4258 0.0006 2270 13.0934 13.0947 -0.0013 2550 13.4327 13.4321 0.0006 2275 13.0993 13.1006 -0.0013 2555 13.4391 13.4383 0.0008 21 3.96471462 2280 13.1052 13.1065 -0.0013 2560 13.4454 13.4445 0.0009 4.38203609 2285 13.1110 13.1124 -0.0014 23 4.28150596 2565 13.4516 13.4507 0.0009 2290 13.1169 13.1182 -0.0013 4.76460375 2570 13.4578 13.4568 0.0010 2295 13.1228 13.1241 -0.0013 2575 13.4640 13.4629 0.0011 2300 13.1287 13.1298 -0.0011 2580 13.4702 13.4690 0.0012 2305 13.1345 13.1356 -0.0011 2585 13.4763 13.4750 0.0013 2310 13.1404 13.1414 -0.0010 2590 13.4823 13.4810 0.0013 2315 13.1462 13.1471 -0.0009 2595 13.4883 13.4870 0.0013 2320 13.1521 13.1528 -0.0007 2600 13.4942 13.4930 0.0012 2325 13.1579 13.1584 -0.0005 2605 13.5000 13.4989 0.0011 2330 13.1637 13.1641 -0.0004 2610 13.5058 13.5048 0.0010 2335 13.1696 13.1697 -0.0001 2615 13.5116 13.5106 0.0010 2340 13.1754 13.1753 0.0001 2620 13.5172 13.5164 0.0008 2345 13.1812 13.1808 0.0004 2625 13.5229 13.5222 0.0007 2350 13.1871 13.1863 0.0008 2630 13.5285 13.5280 0.0005 2355 13.1929 13.1918 0.0011 2635 13.5340 13.5337 0.0003 2360 13.1987 13.1973 0.0014 2640 13.5394 13.5394 0.0000 2365 13.2046 13.2028 0.0018 2645 13.5448 13.5451 -0.0003 2372.501241 13.2109 2370 13.2104 13.2082 0.0022 2650 13.5501 13.5507 -0.0006 3998.526759 13.2109 2375 13.2163 13.2141 0.0022 2655 13.5554 13.5563 -0.0009 2380 13.2221 13.2205 0.0016 2660.414706 13.5623 2660 13.5606 13.5619 -0.0013 2385 13.2280 13.2268 0.0012 3710.613294 13.5623 2665 13.5657 13.5666 -0.0009 2390 13.2339 13.2332 0.0007 2670 13.5707 13.5712 -0.0005 2395 13.2397 13.2395 0.0002 2675 13.5756 13.5758 -0.0002 2400 13.2456 13.2458 -0.0002 2680 13.5804 13.5804 0.0000 2405 13.2516 13.2520 -0.0004 24 4.13634936 2685 13.5851 13.5849 0.0002 2410 13.2575 13.2582 -0.0007 4.58799171 2690 13.5898 13.5894 0.0004 2415 13.2635 13.2644 -0.0009 2695 13.5942 13.5938 0.0004 22 4.13594750 2420 13.2694 13.2706 -0.0012 2700 13.5986 13.5983 0.0003 4.58849794 2425 13.2755 13.2767 -0.0012 2705 13.6027 13.6027 0.0000 2430 13.2815 13.2828 -0.0013 2711.399520 13.6083 2710 13.6067 13.6071 -0.0004 2435 13.2876 13.2889 -0.0013 3659.628480 13.6083 2715 13.6105 13.6106 -0.0001 2440 13.2937 13.2950 -0.0013 2720 13.6140 13.6137 0.0003 2445 13.2998 13.3010 -0.0012 25 3.94351119 2725 13.6173 13.6168 0.0005 2450 13.3060 13.3070 -0.0010 4.35296986 2730 13.6205 13.6199 0.0006 2455 13.3122 13.3129 -0.0007 2735 13.6234 13.6229 0.0005 2460 13.3184 13.3188 -0.0004 2740 13.6261 13.6259 0.0002 2465 13.3247 13.3247 0.0000 2749.829382 13.6318 2745 13.6287 13.6289 -0.0002 2470 13.3310 13.3306 0.0004 3621.198618 13.6318 2750 13.6312 13.6318 -0.0006 2475 13.3372 13.3365 0.0007 2755 13.6336 13.6335 0.0001 2484.009390 13.3469 2480 13.3436 13.3423 0.0013 26 3.72963470 2760 13.6359 13.6350 0.0009 4.09197173 2765 13.6381 13.6366 0.0015 2770 13.6402 13.6382 0.0020 2775 13.6422 13.6397 0.0025 2780.705225 13.6413 2780 13.6441 13.6411 0.0030 3590.322775 13.6413 2785 13.6460 13.6409 0.0051 27 3.40172802 2790 13.6478 13.6403 0.0075 3.69140151 2795 13.6495 13.6397 0.0098 2800.782857 13.6390 2800 13.6512 13.6391 0.0121 3570.245143 13.6390 2805 13.6528 13.6332 0.0196 28 2.39586620 2.46180014 2810.096720 13.6263 2810 13.6544 13.6264 0.0280 3560.931280 13.6263 2815 13.6559 13.6310 0.0249 2820 13.6574 13.6358 0.0216 2825 13.6588 13.6405 0.0183 2830 13.6601 13.6452 0.0149 2835 13.6613 13.6499 0.0114 2840 13.6625 13.6545 0.0080 2845 13.6636 13.6592 0.0044 2850 13.6646 13.6637 0.0009 29 4.30127189 2855 13.6655 13.6683 -0.0028 4.79178003 2860 13.6663 13.6728 -0.0065 2865 13.6670 13.6772 -0.0102 2870 13.6677 13.6817 -0.0140 2875 13.6683 13.6861 -0.0178 2880 13.6689 13.6904 -0.0215 2885 13.6694 13.6948 -0.0254 2890 13.6698 13.6991 -0.0293 Radius outer core = 3477,114454 2893.913546 13.7024 2894 13.670018REFERENCES Bullen K. E. & Bolt Bruce A., 1985, An introduction to the theory of seismology , Cambridge University Press, Cambridge, 499 pp. Clayton, R. W. & Comer, R. P., 1983, A tomographic analysis of mantle heterogeneities from body wave travel times , Eos Trans. AGU , 776. Dziewonski, A. M. & Anderson, D. L., 1981, Preliminary reference Earth model, Phys. Earth. planet. Interiors , 25, 297-356. Dziewonski, A. M., 1984, Mapping the lower mantle: Determination of lateral heterogeneity in P velocity up to degree and order 6, J. Geophys. Res. , 89, 5929-5952. Herrin et al., 1968, 1968 seismological tables for P phases, Bull. Seism. Soc. Am., Vol. 58, No 4, pp. 1193-1235. Jeffreys, H. & Bullen, K. E., 1958, Seismological Tables , British Association for the Advancement of Science, London. Kennett, B. L. N. & Engdahl, E. R., 1991, Traveltimes for global earthquake location and phase identification, Geophys. J. Int. , 105, 429-465. Lana-Renault, Yoël & Cid Palacios, Rafael, 1991, On the problem of the internal constitution of the Earth , Academia de Ciencias de Zaragoza, Univ. Zaragoza, Vol. 4, 158 pp. Lana-Renault, Yoël, 1998, Modelo de constitución interna de la Tierra , Doctoral Dissertation, Departamento de Física Teórica, Univ. Zaragoza, 146 pp. Morelli, A. & Dziewonski, A. M., 1993, Body wave traveltimes and a spherically symmetric P- and S-wave velocity model, Geophys. J. Int. 112, 178-194. Pulliam, R. J., Vasco, D. W. & Johnson L. R., 1993, Tomographic inversions for mantle P wave velocity structure based on the minimisation of l^2 and l^1 norms of International Seismological Centre travel time residuals, J. Geophys. Res., 98, 699-734. Sengupta, M.K. & Toksoz, M. N., 1976, Three-dimensional model of seismic velocity variation in the Earth’s mantle, Geophys. Res. Lett. , 3, 84-86. Taggart, J. N. & Engdahl, E. R., 1968, Estimation of PcP travel times and the depth to the core, Bull. Seism. Soc. Am. , 58, 1243-1260.
1On the Relativistic Origin of Inertia and Zero-Point Forces Charles T. Ridgely charles@ridgely.ws Abstract Current approaches to the problem of inertia attempt to explain the inertial properties of matter by expressing the inertial mass appearing in Newton's second law of motion in terms of some other morefundamental interaction. One increasingly popular approach explains inertial and gravitational forces as drag forces arising due to quantum vacuum zero-point phenomena. General relativity, however, suggests that gravitational and inertial forces are manifestations of space-time geometry. Based on this, the presentanalysis demonstrates that inertia and zero-point forces are ultimately relativistic in origin. Additionally, it isargued that body forces induced on matter through zero-point interactions are resistive forces acting in addition to inertia. 1. Introduction One of the longest standing questions in physics certainly has been by what means do material bodies resist changes to their states of motion (inertia). One approach to this problem has been to view inertia asmerely a fundamental property of all matter with no further explanation attainable. Another approach has been to view inertia as arising due to a gravitational coupling among all matter in the universe. This approach, first proposed by Mach (ca. 1883), stems from the notion that relative motion is meaningless inthe absence of surrounding matter. With this in mind, one is then led to the idea that the inertial propertiesof matter must somehow be related to the cosmic distribution of all matter in the universe [1]. Unfortunately, Mach's principle leads to action at a distance phenomena which even today appear irreconcilable with accepted theory. More recent studies have sought to explain inertia by expressing the inertial mass m appearing in Newton's second law of motion, ()dmdt=fv ,( 1 ) in terms of some other more fundamental entity or interaction. At present, there appears to be several approaches to this end. One approach put forth by J. F. Woodward and T. Mahood seeks to preserve, andapparently expand upon, the Machian view mentioned above [1]-[2]. Another approach, put forth withinthe Standard Model of particle physics, proposes a scalar Higgs field which assigns specific quantities of energy to elementary particles. With this approach, elementary particles possess inertial properties simply because they possess energy, which is equivalent to mass. Unfortunately, the Higgs mechanism stops shortof actually explaining why mass, or energy, resists acceleration. Consequently, the Higgs mechanismleaves us with the above-mentioned notion, mentioned above, that inertia is a fundamental property of all matter with no further explanation attainable. Still another approach to the problem of inertia has been put forth by B. Haisch, A. Rueda, H. E. Puthoff, et al [3]-[4]. With their approach, it is proposed that theinertial properties of ordinary matter are due to local interactions between a vacuum electromagnetic zero- point field (ZPF) and subatomic particles, such as quarks and electrons, constituting ordinary matter. In2essence, the ZPF approach asserts that when a material object accelerates through the zero-point radiation pervading all of space, some of the radiation is scattered by the quarks and electrons constituting the object. This scattering of radiation exerts a reactive body force on the object, which, according to the ZPFproposal, can be associated with the inertial mass of the object. Although we challenged the ZPF proposal in previous papers [5]-[6], the ZPF proposal seems increasingly appealing. Not only does the ZPF proposal s uggest a local basis for gravitational and inertial forces, thus avoiding action at a distance phenomena often associated with Mach's principle, but also seemsto suggest an intimate relationship between electromagnetism, inertia, and gravitation [3]-[4]. However,while current ZPF theory does show that zero-point phenomena give rise to body forces on ordinary matter, this is only part of the story. According to the ZPF proposal, the intrinsic rest mass-energy content of matter is induced in large part through interaction with zero-point radiation. Incident zero-point radiation imparts an ultrarelativisticjittering motion, or zitterbewegung [4], to the quarks and electrons comprising ordinary matter, thereby endowing matter with large quantities of internal kinetic energy. With this paradigm, the rest mass-energy of matter predicted by the familiar expression 2Em c= is interpreted as entirely ZPF-induced kinetic energy. But this seems to suggest that subatomic particles have no intrinsic rest mass-energy aside from their ZPF-induced motion. Of course subatomic particles are, indeed, well known to possess definite quantities of intrinsic rest mass-energy. A more traditional interpretation is that 2Em c= i s s i m p l y a statement that all forms of energy exhibit inertial properties [7]-[8], which can be associated with inertial mass, and that mass is merely one particular embodiment of energy. Herein, both interpretations are utilized; the energy content of matter comprises intrinsic rest mass-energy as well as internal kinetic energy due to ZPF-induced zitterbewegung [4] of subatomic particles. Another important point to recognize is that general relativity suggests that gravitational and inertial forces alike arise due to the behavior of space-time [9]-[14]. This implies that greater insight into the origin of inertia cannot be obtained simply by eliminating the inertial mass m appearing in Eq. (1) in favor of some other entity or interaction. Rather, space-time is a fundamental participant in the generation ofinertial and gravitational forces, and as such must be taken into account, as well [5]. Thus, while theenergy content of matter may indeed be largely ZPF-induced, one must still explain why energy resists acceleration. On this basis, a primary objective of the present analysis is to provide an unequivocal demonstration that the inertia of energy is ultimately a purely relativistic manifestation [5]. This isachieved by treating the mass, and more generally the energy, of matter as entirely generic, thereby singling out the role played by relativity while avoiding an introduction of zero-point phenomena. A secondary objective is to demonstrate that forces induced on matter by zero-point radiation are additional body forcesalso arising due to the relativistic properties of space-time [5]. In the next Section, general relativity is used to derive the proper force experienced by an observer accelerating due to the action of an external force. This is carried out by expressing the geodesic equation in accelerating coordinates and taking into account that the accelerating observer remains positioned at the3origin of the local accelerating coordinate system at all times. It is also pointed out that so long as the observer’s acceleration is uniform, the components of the metric tensor in the accelerating system are independent of time. Using these observations leads to an expression for the proper force suggesting thatthe behavior of space-time in the accelerating system is an active participant in the generation of inertia [5].Next, the expression for the proper force is used to derive an expression for the inertial resistance of the accelerating observer as experienced by a force-producing agent who exerts a force on the accelerating observer. This is done by considering the case of weak, Newtonian acceleration and then using Newton’sthird law of motion to obtain an expression for the reaction force experienced by the force-producing agent.The resulting expression for the inertial resistance supports a conventional notion that all forms of energy actively resist acceleration [7]-[8], and suggests that such resistance is purely relativistic in origin [5]. In Section 3, special relativity is used to derive the inertial resistance experienced by a stationary observer, residing in flat space-time, who exerts a constant force on a moving observer. Unlike theprevious Section, however, the moving observer considered in this Section does not undergo linear acceleration; rather, the moving observer accelerates tangentially along a circular path of constant radius. Upon each revolution the moving observer passes by the stationary observer with greater velocity. Fromthe perspective of the stationary system, time in the moving system appears increasingly dilated. Thechange in time dilation arising between the two observers is used to derive an expression for the inertial resistance of the moving observer. The resulting expression for the inertial resistance is identical to that derived in Section 2. It is concluded that all forms of energy possess inertial properties, arising due torelativity [5],[7]-[8]. Section 4 demonstrates that ZPF-induced forces acting on accelerating matter are ultimately relativistic in origin, and that such forces act in addition to inertial forces. This is accomplished by considering auniformly accelerating block of matter. Observers residing on the accelerating block detect a flux of zero-point radiation passing through the block [4]. The force density due to this flux of radiation is derived for the case of a small, particle-sized block. Upon expressing the force in terms of the relativistic Doppler-shift of zero-point radiation occurring within the accelerating system, it becomes straightforward to see thatZPF-induced forces on matter are purely relativistic in origin. Next, the expression for the force is used toderive an expression for the total resistance acting on a particle accelerating uniformly through zero-point radiation. The form of the resulting expression makes it clear that ZPF-induced forces act in addition to the inertia of matter. 2. Proper Force Experienced by a Uniformly Accelerating Observer As pointed out in the Introduction, general relativity suggests that inertial forces arise due to the behavior of space-time in accelerating systems [9]-[14]. This is demonstrated herein by using general relativity to derive the proper 3-force experienced by an accelerating observer. Let an observer be accelerating uniformly under the action of a constant, external force. From the vantage of the accelerating reference system, the observer’s local coordinate system appears as though4characterized by the presence of a uniform gravitational field [15]. The line element in the local accelerating coordinate system can be expressed generally in the form ()22 2 2 2 2 00,, ds c g x y z dt dx dy dz=− − − ,( 2 ) in which ()00,, gx y z contains information about the force on the accelerating observer. So long as the observer’s acceleration is uniform, ()00,, gx y z remains independent of time [16]-[17]. Upon noticing that the accelerating observer remains stationary at the origin of the accelerating coordinate system at all times, Eq. (2) leads to ()2 00,,dgx y zdtτ=.( 3 ) In this expression, dτ is an interval of proper time experienced by the accelerating observer and dt is an interval of time experienced by a momentarily comoving observer whose coordinate origin is momentarily coincident with that of the accelerating observer at a time 0 t=. The proper force experienced by the accelerating observer can be derived by using the geodesic equation: 2 20d x dx dx dd dαµ ν α µνττ τ+Γ = ,( 4 ) in which the Christoffel symbol α µνΓ is given by ()1,,,2gg g gαα β µν µβνβνµµνβ Γ= + − .( 5 ) For the case of the accelerating observer residing within the accelerating coordinate system suggested by Eq. (2), Eq. (4) is easily reduced to 2 2 2 00 20dx d tcd dα α τ τ+Γ =.( 6 ) Expressing the Christoffel symbol in terms of the accelerating coordinate system leads directly to 2 22 00 2,02j jdx c d tggd dα α τ τ−=,( 7 ) wherein Latin indices are carried over the set of values {1, 2, 3}. This expression simplifies further upon noticing that partial differentiation of Eq. (3) yields500,2jjddgdt dtττ =∂  ,( 8 ) in which j∂ denotes partial differentiation with respect to jx. As a further simplification, the inverse metric tensor jgα can be expressed in the form 0, 0 ,j ijgiα α δα= =−=,( 9 ) wherein ijδ equals unity when ij=, and equals zero when ij≠. Substituting Eqs. (8) and (9) into Eq. (7) leads to 2 2 20i ij jdx d t d tcdd dδττ τ+∂ = . (10) Next, expressing the second term in terms of a natural logarithm, Eq. (10) can be recast in the form 2 2 2ln 0i ij jdx d tcd dδτ τ−∂ =. (11) This is the proper acceleration experienced by the accelerating observer. Equation (11) can be used to derive an expression for the force experienced by the accelerating observer upon noticing that the external force acting on the observer is expressible as 2 2i i dxfmdτ= , (12) where m is the observer’s proper mass. Using Eq. (12), and simplifying a bit, Eq. (11) then becomes 2lnii j jdtfm cdδτ=∂ . (13) This is the proper force experienced by the accelerating observer [5]. With Eq. (13) in hand, the inertial resistance of the observer can be derived. One approach is to transform Eq. (13) from the accelerating frame to the reference frame from which the force on the observer arises. Another approach is to consider the case of weak acceleration, thereby placing the force within theNewtonian realm, and then appeal to Newton’s third law of motion. Choosing the latter approach, one will notice that when the observer accelerates weakly, the scalar function dt dτ assumes values very close to unity. For the case of weak acceleration, therefore, the natural logarithm of the scalar function dt dτ may be approximated by use of the expression [18]6ln 1dt dt ddττ≈−. (14) Using this approximation in Eq. (13) leads directly to 2 ii j jdtfm cdδτ=∂ . (15) This is the force experienced by the accelerating observer in the limit of weak, Newtonian acceleration. Expressing Eq. (15) in vector notation, the force experienced by the accelerating observer becomes dtEdτ=f∇∇∇∇ , (16) where 2Em c= has been used, and ∇∇∇∇ represents the three dimensional gradient operator. Equation (16) expresses the force imparted to the accelerating observer by an external force-producing agent. Assuming that the force-producing agent resides in an inertial frame, Newton’s third law of motion suggests that the reaction force experienced by the force-producing agent is given by in=−ff . The subscript on inf indicates that the reaction force is due to the inertia of the accelerating observer. Using this force with Eq. (16) leads directly to indtEdτ=−f∇∇∇∇ . (17) This is the inertial resistance of the accelerating observer experienced by the force-producing agent [5]. It is straightforward to see that the inertial resistance is solely dependent upon the total proper energy content of the accelerating observer and the gradient of the scalar function dt dτ arising due to relativity. The form of Eq. (17) suggests that all forms of energy resist acceleration and possess inertial properties which are entirely relativistic in origin [5],[7]-[8]. On this basis, it may be concluded that inertia is purely relativistic. 3. Inertial Resistance of an Observer Accelerating in Flat Space-Time In the previous Section, general relativity was used to demonstrate that all forms of energy possess inertial properties arising as purely relativistic manifestations [5],[7]-[8]. In this Section, special relativity is used to derive the inertial resistance of an observer accelerating uniformly through flat space-time. Consider a stationary observer residing in an inertial system S who applies a force to a second, moving observer such that the moving observer accelerates tangentially along a circular path of radius r. It is envisioned that the stationary observer exerts the force on the moving observer through the use of some mechanical means, which exerts a constant torque on the moving observer. The stationary observer compares the lengths of time intervals in the moving and stationary systems each time the moving observer7reaches the point of closest approach; that is, when the relative velocity is entirely transverse. As the moving observer’s velocity increases, the stationary observer finds that time in the moving system becomes increasingly dilated according to the familiar expression [19]: 221 1dt d vc τ= −, (18) where dt is a time interval in system S, and dτ is an interval of proper time experienced by the moving observer. The stationary observer determines an initial time dilation ()idt dτ; and then at some later time, the stationary observer determines a final time dilation ()fdt dτ: 221 1 iidt d vc τ= −, (19a) 221 1 ffdt d vc τ= −, (19b) where iv and fv are initial and final velocities of the moving observer at the instants when the initial and final time dilation are respectively measured. The inertial resistance of the moving observer can be derived by first using Eqs. (19) to express the change in time dilation as 22 2 211 11 fifidt dt dd vc vc ττ−= − −−, (20) in which the magnitude of the moving observer’s tangential acceleration remains constant. In order to derive the inertial resistance by use of Eq. (20), one will notice that the change in the scalar function dt dτ on the left hand side of Eq. (20) may be expressed as a line integral of the form f i fidt dt dtddd dττ τ −= ⋅  ∫l ∇∇∇∇ , (21) where dl is an infinitesimal length vector aligned along the circular path on which the moving observer travels. Also, the right-hand side (RHS) of Eq. (20) can be expressed as ()23 222 2 2 2211 1 11 1f i fid c vc vc vc⋅−= −− −∫vv, (22) where v is an instantaneous velocity of the moving observer at a point between the velocities iv and fv. Using Eqs. (21) and (22) in Eq. (20) gives8()23 2221 1ff iidt ddd c vc τ⋅ ⋅= −∫∫vvl ∇∇∇∇ . (23) Upon expanding the RHS of this expression by use of the relation ()() ()32 3222 22 2 2211 1 11 vc vc c vc⋅=+ − −−vv, (24) and performing some algebraic manipulation, Eq. (23) then becomes () ()23 222 22 21 1 1ff iid dt dddc vc cv c τ⋅  ⋅= ⋅ +  − −∫∫vv v vlv ∇∇∇∇ . (25) This expression can be simplified upon noticing that () ()3222 22 22 211 1d dd vc vc cv c ⋅ +=−− − vv v vv. (26) Using this, and expressing the velocity of the moving observer as dd t=vl , Eq. (25) can be rewritten in the form 2221 1ff iidt d mdddd t mc vc τ ⋅= ⋅  −∫∫lvl ∇∇∇∇ , (27) in which the moving observer’s proper mass, m, has been introduced on the RHS. Rearranging Eq. (27) a bit leads to 2221 1ff iidt d mdddd t mc vc τ ⋅= ⋅  −∫∫vll ∇∇∇∇ . (28) It is now straightforward to see that the integrand on the RHS of this expression contains the familiar relativistic generalization of Newton’s second law of motion [20]-[21]. Taking this into account, Eq. (28)can be recast in the form 1 ff iidtdddEτ⋅= ⋅∫∫lf l ∇∇∇∇ , (29) wherein 2Em c= has been used, and f is the external force imparted to the moving observer by the stationary observer.9While the moving observer accelerates, there are at least two forces detectable by the stationary observer. One force is the external force f applied to the moving observer by the stationary observer. A second force is the moving observer’s inertial resistance inf. Assuming that no other external forces are present, the inertial resistance is related to the external force according to in=−ff . Substituting this into Eq. (29), and eliminating the integration on both sides of the expression leads directly to indtEdτ=−f∇∇∇∇ . (30) This is the inertial resistance of the moving observer, experienced by the stationary observer residing in flat space-time [5]. The form of Eq. (30) is identical to that of Eq. (17), derived on the basis of generalrelativity, and suggests that all forms of energy, regardless of embodiment, exhibit inertial properties arising as purely relativistic phenomena [5],[7]-[8]. 4. Resistance Force on Accelerating Matter due to Zero-Point Radiation According to the preceding Sections, special and general relativity suggest that the inertial properties of matter, as well as of all other forms of energy, are purely relativistic in origin. In this Section, it isdemonstrated that forces induced on matter through interaction with zero-point radiation are ultimately relativistic in origin, and that such forces arise in addition to inertial forces. Consider a block of matter, of proper volume V, undergoing uniform acceleration through flat space- time due to an external force. Observers residing on the block detect a flux of zero-point radiation entering the block through the front side, which may be called wall A, and passing out of the block through the back side, called wall B. According to these observers, the radiation within the volume of the block possesses a momentum density of the form ()21 ABc∆= − SS pppp , (31) where AS and BS are Poynting vectors corresponding to the flux of zero-point radiation detected at walls A and B, respectively. While the block accelerates, the Poynting vectors are not equal in magnitude; zero- point radiation becomes Doppler-shifted as it passes through the accelerating frame of the block. Taking this into consideration, the Poynting vectors may be expressed in terms of the energy density of the ZPFobserved at each wall: AAcu=Sn , (32a) BBcu=Sn , (32b)10where the subscripts A and B indicate the walls at which the energy density of the ZPF is detected, and n is a unit vector pointing in the direction of the block's acceleration. Since the block is accelerating, zero-point radiation gains energy as it passes from wall A to wall B. As a result, radiation detected at wall B appears blue-shifted relative to radiation detected at wall A. Taking this into account, the Poynting vector given by Eq. (32b) can be expressed as 2 A BA Bdcudτ τ= Sn , (33) where Adτ and Bdτ are intervals of proper time experienced by the observers situated at walls A and B, respectively. Using Eqs. (32a) and (33) in Eq. (31), the momentum density of zero-point radiation within the block becomes 2 1AA Bud cdτ τ∆= − n pppp . (34) Expressing this in terms of time experienced by observers residing in a momentarily comoving reference frame (MCRF), and simplifying a bit, leads to 22 2 AA ABud dt dt cd t d dτ ττ  ∆= −   n pppp , (35) where dt is an interval of time experienced by the observers in the MCRF. With Eq. (35) in hand, the force density due to zero-point radiation passing through the block can be derived by using τ=∆ ∆fpfpfpfp , where τ∆ is an interval of proper time in the accelerating frame of the block. Supposing that the x′−axis of the accelerating coordinate system is oriented in the direction of the block’s acceleration, observers positioned at wall A can express τ∆ in terms of the block's length, x′∆, along the x′−axis of the accelerating frame. This is carried out by observing that the time taken for a light signal to complete a round trip across the block is 2xcτ′ ∆=∆ . Using this time interval and Eq. (35), the force density can then be expressed as 22 2 ˆ 2AA ABud dt dt xd t d dτ ττ ′  =−  ′∆ x ffff , (36) where ˆ′x is a unit vector aligned along the x′−axis of the local accelerating coordinate system. Equation (36) holds for volumes in which ABddττ≠ and x′∆ assumes measurable values. When the volume of the block is on the order of a particle-sized volume, however, then Adτ is approximately equal to Bdτ and x′∆11is infinitesimally small. The force density within a particle-sized volume can be obtained by taking the limit of Eq. (36) as Bdτ tends to Adτ and x′∆ tends to zero: 0BAdd xLim ττ→ ′∆→′=ffffffff . (37) Carrying out the limit for the case of a particle and noting that BA dt d dt dττ> , the force density assumes the form [5]-[6] ˆ lndtuxdτ∂ ′′=− ′∂x ffff , (38) where u is the proper energy density of the ZPF according to observers moving with the particle. Equation (38) expresses the force density due to zero-point radiation passing through and interacting with a particle of matter undergoing substantial acceleration. It will be noticed that Eq. (38) is substantiallysimilar to Eq. (13) of Section 2. In Section 2, it was pointed out that when the acceleration is weak, the scalar function dt dτ can be approximated by use of the expression given by Eq. (14). Upon using Eq. (14) with the force density, embodied by Eq. (38), one immediately obtains [5]-[6] 0 ˆdtuxdτ∂′′=− ′∂x ffff . (39) Upon transforming this expression from the accelerating system to the inertial frame from which the external force on the particle arises, the force density becomes [5]-[6] dtudτ=−∇∇∇∇ ffff , (40) where the prime has been dropped from ′ffff for simplicity. Equation (40) is a resistance force on the particle due to interaction with zero-point radiation. According to a force-producing observer residing in flat space-time, when an external force is applied to the particle, a resistance force ZPFf arises due to interaction between the particle and zero-point radiation. Using Eq. (40), such as observer can express this resistance force as ZPFdtuVdτ=−f ∇∇∇∇ , (41) where V is the proper volume of the accelerating particle. Equation (41) gives the resistance force acting on the accelerating particle due to the scattering of zero-point radiation. According to Eq. (41), the force on the particle arises not only due to the presence of zero-point radiation, but also due to the gradient of the scalar function dt dτ. It is interesting to note that when the scalar function dt dτ assumes a constant12value (i.e., the particle is no longer accelerating), the force given by Eq. (41) is then zero. This suggests that while zero-point phenomena may give rise to a large portion of the mass-energy content of matter, the inertia of such energy is purely relativistic [5]. At this point, it is worthwhile to show that Eq. (41) can be used to derive an expression for the total resistance force acting on the accelerating particle discussed above. According to the ZPF proposal, when a material body undergoes acceleration due to an external force, the quarks and electrons constituting the body scatter a portion of the zero-point radiation passing through the body. The scattered portion ofradiation imparts an energy density to the accelerating body, which is expressed in the form [4] ()3 232ZPFudcωηω ωπ=∫!, (42) In this expression, ()ηω is a spectral function that governs the fraction of zero-point radiation that actually interacts with the accelerating body. Those who support the ZPF proposal interpret the energy density, given by Eq. (42), as the sole origin of the inertial mass of matter [3]-[4]. However, as discussed in the Introduction, another form of energy that must be taken into account is the intrinsic proper mass-energy ofthe accelerating body [5]-[8]. On this basis, the total energy content of the body must be expressed as Total ZPFEE E U=+ + , where E is the intrinsic mass-energy of the body, ZPFE is the internal kinetic energy due to ZPF-induced zitterbewegung [4] of the subatomic particles comprising the body, and U includes additional forms of energy that may be present. Using this expression for the accelerating particle discussed above leads to ()3 2 232TotalEm c V d Ucωηω ωπ=+ +∫!, (43) where 2Em c= has been used in the first term, Eq. (42) has been used to obtain the second term, and V is the proper volume of the particle. In order to derive an expression for the total resistance force acting of the accelerating particle, Eq. (41) must be amended to include all forms of energy possessed by the particle [7]-[8]. This calls for replacing uV in Eq. (41) with the total energy given by Eq. (43). Carrying this out gives the force in the form ()3 2 232dtmc V d Ud cωηω ωτ π =− + +  ∫f!∇∇∇∇ . (44) This expresses the total resistance force acting on the uniformly accelerating particle. Equation (44) can be simplified by considering the case of weak, uniform acceleration along the x−coordinate axis of an inertial system. For this case, the scalar function dt dτ may be expressed in the form1321dt ax dcτ≈+ , (45) where a is the acceleration and x is a distance traveled along the x−axis due to the acceleration. Carrying out the gradient of this expression leads to 2dt dcτ=a∇∇∇∇ , (46) where the acceleration is expressed in vector notation as ˆa=ax . Substituting Eq. (46) into Eq. (44), and simplifying a bit, then leads to ()3 22 3 22VUmdcc cωηω ωπ=− − − ∫fa a a!. (47) Equation (47) is the total resistance force acting on a particle of proper mass m that accelerates uniformly through zero-point radiation. With the exclusion of the first and last terms, Eq. (47) is identicalto an expression for the force derived on the basis of ZPF theory [4]. Supporters of the ZPF proposal claimthat the inertial mass of accelerating matter ought to be expressed entirely by the expression within parentheses, in the second term of Eq. (47). Clearly, this can be the case only if the energy content of matter is entirely due to ZPF-induced zitterbewegung [4] of the subatomic particles comprising matter,thereby allowing one to drop the first and last terms in Eq. (47). According to the Introduction, however, one cannot disregard the intrinsic mass-energy content of matter. As shown herein, the intrinsic mass- energy content of matter gives rise inertial phenomena arising solely due the relativistic properties of space-time [5]. Based on this, it appears that the only way to reconcile the intrinsic inertial properties of ordinarymatter with the ZPF proposal is to accept that ZPF-induced forces are resistive forces acting in addition to inertia. 5. Discussion and Conclusions One approach to the problem of inertia has been an attempt to express the inertial mass appearing in Newton's second law of motion in terms of some other more fundamental entity or interaction. Anincreasingly popular, and very appealing, approach to this end is the zero-point field (ZPF) proposal, which asserts that inertia and gravitation arise due to scattering of zero-point radiation by quarks and electrons constituting ordinary matter [3]-[4]. An objective of the present analysis, however, has been to show thatthe inertial properties of ordinary matter are ultimately relativistic in origin [5]. As pointed out in the Introduction, general relativity asserts that the forces of gravitation and inertia arise directly out the structure of space-time [9]-[14]. Based on this, both special and general relativity were used to derive an expression for the inertial resistance of an observer undergoing acceleration due toan external force. Both approaches led to an expression for the inertial resistance suggesting that all forms14of energy possess inertial properties manifesting chiefly due to the relativistic properties of space-time [5],[7]-[8]. Although electromagnetic zero-point phenomena may well be the origin of the energy content of matter [4], neither of the above-mentioned approaches required a specific appeal to zero-pointphenomena. On this basis, it was concluded that inertia is a purely relativistic manifestation [5]. Another objective of the present analysis was to demonstrate that forces induced on matter due to zero- point radiation are additional body forces arising due to the relativistic properties of space-time. This was carried out by deriving the resistance force acting on an accelerating block of matter that scatters a portionof the zero-point radiation passing through the block. Expressing the force in terms of the Doppler-shift ofradiation in the accelerating system led directly to an expression for the force identical to that derived herein on the basis of special and general relativity. In addition, the expression for the force was used in conjunction with an expression for the energy density of zero-point radiation that interacts with anaccelerating particle [4]. This led to an expression suggesting that ZPF-induced forces are resistive forcesacting in addition to inertia. Notes and References [1] See, for example, James F. Woodward, “Maximal acceleration, Mach's principle and the mass of the electron,” Found. Phys. Lett. 6, 233 (1993); and “A stationary apparent weight shift from a transient Machian mass fluctuation,” Found. Phys. Lett. 5, 425 (1992). [2] J. F. Woodward and T. Mahood, “What is the cause of inertia?” Found. Phys. 29, 899 (1999). [3] Y. Dobyns, A. Rueda and B. Haisch, “The case for inertia as a vacuum effect: a reply to Woodward and Mahood,” Found. Phys. 30, 59 (2000). [4] B. Haisch, A. Rueda, and Y. Dobyns, “Inertial mass and the quantum vacuum fields,” Annalen der Physik, in press (2000). [5] C. T. Ridgely, “On the nature of inertia,” Gal. Elect. 11, 11 (2000). [6] C. T. Ridgely, “Can zero-point phenomena truly be the origin of inertia?” Gal. Elect. , in review (2001). [7] For an interesting discussion on the inertia of energy, see A. Einstein, “Does the inertia of a body depend upon its energy-content?,” in Einstein, The Principle of Relativity (Dover, New York, 1952), pp. 69-71; and H. Weyl, Space-Time-Matter (Dover, New York, 1952), 4th ed., p.202. [8] A discussion of the inertia of energy can be found in Max Born, Einstein’s Theory of Relativity (Dover, New York, 1965), pp. 283, 286. [9] A. Einstein, The Meaning of Relativity, Including the Relativistic Theory of the Non-Symmetric Field (Princeton, New Jersey, 1988), 5th ed., pp. 57-58. [10] Max Born, Ref. [8], pp. 313-317. [11] P. G. Bergmann, Introduction to the Theory of Relativity (Dover, New York, 1976), p. 156. [12] B. F. Schutz, A first course in general relativity (Cambridge, New York, 1990), p.122. [13] H. Ohanian and R. Ruffini, Gravitation and Spacetime (Norton, New York, 1994), 2nd ed., pp. 53-54.15 [14] See, for example, I. R. Kenyon, General Relativity (Oxford, New York, 1990), p. 10. [15] A discussion of accelerating observers can be found in H. Ohanian and R. Ruffini, Ref. [13], pp. 356- 357, 431. [16] See, for example, Max Born, Ref. [8], p. 276. [17] See, for example, R. Resnick, Introduction to Special Relativity (Wiley, New York, 1968), p. 125. [18] See, for example, H. B. Dwight, “ Tables of Integrals and Other Mathematical Data, ” (Macmillan, New York, 1961), p. 138. [19] From an experimental standpoint, one may suppose that while accelerating the moving observer carries a light source of proper frequency 0ν. The stationary observer can then measure the frequency of light, ν, each time the moving observer reaches the point when the relative velocity is entirely transverse. Following this, the stationary observer can use 0 dt dτνν= to determine the dilation of time in the moving system relative to the inertial system S. See, for example, C. Lanczos, The Variational Principles of Mechanics (Dover, New York, 1968), 4th ed., p. 339; and R. Resnick, Ref. [17], pp. 90-91. [20] See, for example, R. Resnick, Ref. [17], p. 119. [21] See, for example, P. G. Bergmann, Introduction to the Theory of Relativity (Dover, New York, 1976), pp. 103-104.
arXiv:physics/0103045v1 [physics.atom-ph] 16 Mar 2001Large Faraday rotation of resonant light in a cold atomic cloud G. Labeyrie, C. Miniatura and R.Kaiser Laboratoire Ondes et D´ esordre, FRE 2302 CNRS 1361 route des Lucioles, F-06560 Valbonne January 15, 2014 Abstract We experimentally studied the Faraday rotation of resonant light in an optically-thick cloud of laser-cooled rubidium atoms. Mea surements yield a large Verdet constant in the range of 200000◦/T/mm and a maximal polarization rotation of 150◦. A complete analysis of the polarization state of the transmitted light was necessary to account for the rol e of the probe laser’s spectrum. PACS numbers : 33.55.Ad, 32.80.Pj 1 Introduction During the past two years, we have been theoretically and exp erimentally in- vestigating coherent backscattering (CBS) of near-resona nt light in a sample of cold rubidium atoms [1, 2, 3]. CBS is an interference eﬀect in the multiple scattering regime of propagation inside random media, yiel ding an enhance- ment of the backscattered light intensity [4]. This interfe rence is very robust and can be destroyed only by a few mechanisms, including Fara day rotation [5] and dynamical eﬀects [6]. In the particular case of atomic sc atterers, we have shown that the existence of an internal Zeeman structure sig niﬁcantly degrades the CBS interference [1, 3]. The breakdown of CBS due to the Fa raday eﬀect in classical samples has been recently observed and studied in details [7], in a situation where the scatterers are embedded in a Faraday-ac tive matrix. We are currently exploring the behavior of CBS when a magnetic ﬁ eld is applied to the cold atomic cloud. Since the Faraday eﬀect is expected to be large even at weak applied ﬁelds (of the order of 1 G= 10−4T), it seems relevant to evaluate its magnitude in the particular regime of near-resonant exc itation. The Faraday eﬀect, i.e. the rotation of polarization experi enced by light propagating inside a medium along an applied magnetic ﬁeld, is a well-known phenomenon [8]. Faraday glass-based optical insulators ar e widely used in laser experiments to avoid unwanted optical feedback. Due to the p resence of well- deﬁned lines (strong resonances), the Faraday eﬀect is pote ntially strong in 1atomic systems, and has been extensively studied in hot vapo rs [9]. In addition, light can modify the atomic gas as it propagates and induce al ignment or ori- entation via optical pumping, yielding various non-linear eﬀects [10]. However, our experiment is quite insensitive to these eﬀects and the s cope of this paper will only be the linear, ”standard” Faraday rotation. Even t hough laser-cooled atomic vapors appear interesting due to suppression of Dopp ler broadening and collisions, few experiments on cold atoms are, to our knowle dge, reported in the litterature [11]. In Section 2 we expose a simple formalism to understand the ma in character- istics of optical activity in an atomic system. This model, a dapted to the atomic structure of Rb, will be used in the quantitative analysis of the experimental results. We also brieﬂy recall in this Section the principle s of the Stokes analysis of a polarization state. The experimental setup and procedu re are described in Section 3. The results are presented in Section 4, and compar ed to the model. 2 Faraday eﬀect and dichroism Let us consider a gas of two-level atoms excited by a near-res onant monochro- matic light ﬁeld of frequency ν. The induced electric dipole has a component in phase with the excitation, which corresponds to the real p art of the succep- tibility of the atomic medium (with a dispersive behavior), and a component in quadrature, which corresponds to the imaginary part of th e succeptibility (absorptive behavior). The former thus relates to the refra ctive index of the gas, while the later corresponds to the absorption or scatte ring. At low light intensity I≪Isat(where Isat=1.6 mW/cm2is the saturation intensity for Rubidium), the refractive index nof a dilute gas is given by : n(δ)≃1−ρ6π k3δ/Γ 1 + 4 ( δ/Γ)2(1) where ρis the atomic gas density, k = 2 π/λthe light wave number in vacuum, δ=ω−ωatthe light detuning, and Γ the natural width (Γ /2π= 5.9 MHz for Rb). On the other hand, the imaginary part of the succepti bility yields the atomic scattering cross-section σ: σ(δ) =3λ2 2π1 1 + 4 ( δ/Γ)2(2) with the usual Lorentzian line shape. This term will result i n an attenuation exp(−ρσL) of the incident light as it propagates over a distance Linside the medium ; the quantity b=ρσLis the optical thickness of the atomic sample. We thus see that the wave will experience both a phase shift an d an attenuation as it propagates. Let us now consider a J = 0 →J’= 1 transition excited by a linearly po- larized light ﬁeld. A magnetic ﬁeld Bis applied along the wavector k, whose 2direction is taken as the quantization axis. The magnetic ﬁe ld displaces the resonance frequencies of the excited state Zeeman sublevel s of magnetic number me=±1 by an amount meµB, where µ= 1.4 Mhz/G. The incident linearly- polarized light decomposes as the sum of σ+andσ−waves of equal amplitudes, which couple the unshifted ground state to excited state sub levels of magnetic number ±1 respectively. These waves thus propagate in media with diﬀ erent refractive indices n+andn−and scattering cross-sections σ+andσ−, and expe- rience diﬀerent phase shifts and attenuations. If, in a ﬁrst step, we neglect the absorption term, the two transmitted waves recombine in a li nearly-polarized wave, rotated by an angle θ=1 2(ϕ+−ϕ−) =π λ(n+−n−)L. This is the Fara- day rotation angle. At small magnetic ﬁeld µB/Γ≪1, the rotation angle is simply : θ≈b×µB/(Γ/2π). Thus, at small B, the Faraday angle is simply the optical thickness btimes the Zeeman shift expressed in units of the nat- ural width Γ (however, the proportionality between θandbremains valid for arbitrary magnetic ﬁeld). It should be emphasized that, in a tomic vapors, the Faraday eﬀect is very strong compared to that of standard Far aday materials (like Faraday glasses), due to the high sensitivity of atomi c energy levels to magnetic ﬁeld : for a density ρ=1010cm−3, the Verdet constant is 40◦/G/mm (4×105◦/T/mm ), more than four orders of magnitude above that of typical Faraday glasses. However, for cold atomic gases, the linear increase of rotation angle with magnetic ﬁeld is restricted to a small ﬁeld range o f a few Gauss (the Zeeman splitting must remain smaller than the natural width ), above which the Faraday eﬀect decreases. Of course, in the regime of near-resonant excitation we are d ealing with, one usually can not neglect absorption. The diﬀerent attenu ations for the σ+ andσ−components cause a deformation of the transmitted polariza tion as well as rotation. The polarization thus becomes elliptic with an ellipticity (ratio of small to large axis) determined by the diﬀerential absorp tion between σ+ andσ−light. This eﬀect is known as circular dichroism. The angle θis then the angle between the initial polarization and the large axi s of the transmitted ellipse. We will see how the Stokes analysis allows to extrac tθand the ellipticity from the measurements. Although the simple picture developed above gives access to the main mech- anisms of optical activity in a an atomic gas, it does not accu rately describe the F = 3 →F’= 4 transition of the D2 line of Rb85used in this experiment. To expand the description to the case of a ground state Zeeman structure, we will make the simplifying assumption that all the transitio ns between diﬀer- ent ground state Zeeman sublevels are independent. We thus n eglect optical pumping and coherences. The refractive index for, say, σ+light, then writes as n+(δ, B) =3/summationtext m=−3pmc+2 mn+ m(δ, B), where the p mare the ground state sublevels populations, the c+ mthe Clebsch-Gordan coeﬃcients for the various σ+transi- tions, and the n+ m(δ, B) the refractive indices for the Zeeman-shifted transition s (a transition from a ground state of magnetic number mgto an excited state meis frequency-shifted by ( mege−mggg)µB, where ge= 1/2 and gg= 1/3 3are the Land´ e factors for the F = 3 →F’= 4 transition of the D2 line of Rb85). We can express in the same way the total scattering cross-sec tion for each cir- cular polarization. In the absence of magnetic ﬁeld and assu ming a uniform population distribution among the ground state sublevels, the total scattering cross-section on resonance is σ(δ= 0)≃0.43×3λ2/2π, the 0.43 prefactor being the average of the squared Clebsch-Gordan coeﬃcient (or the degeneracy factor of the transition1 3(2F’+1) (2F+1)). As discussed above, the polarization of the transmitted lig ht can diﬀer from the incident linear polarization. It is thus necessary to fu lly characterize the polarization state of the transmitted light. This can be don e using the Stokes formalism [12]. Four quantities need to be measured : the (li near) component of the transmitted light parallel to the incident polarizat ion (I//), the (linear) orthogonal component ( I⊥), the (linear) component at 45◦(I45◦), and one of the two circular components ( Icirc). The sum of the ﬁrst two is the total intensity s0; the three other Stokes parameters are : s1=I//−I⊥= 2I//−s0,s2= 2I45◦−s0ands3= 2Icirc−s0. One can then compute the three quantities which characterize any polarization state : P=/radicalbig s2 1+s2 2+s2 3 s0(3) sin 2χ=s3 s0P(4) tan 2θ=s2 s1(5) Here, Pis the degree of polarization of the light, i.e. the ratio of t he intensities of the polarized component to the unpolarized one (a pure pol arization state yields P= 1 while P= 0 corresponds to totally unpolarized light). Even though we would not a priori expect any unpolarized component, we will see that this analysis is indeed necessary in our case. The quant itye= tan ( χ) is the ellipticity of the polarized component ( e=±a/b, where aandbare the small and large axis of the ellipse respectively and the + or - sign denotes the sense of rotation of the electric ﬁeld). The Faraday angle θis the angle between the large axis of the ellipse and the direction of the inciden t polarization. 3 Description of experiment 3.1 Preparation of cold atoms The experimental setup, essentially the same as in our coher ent backscattering experiment, is described in detail elsewhere [2]. A magneto -optical trap (MOT) is loaded from a dilute Rb85vapor (P ∼10−8mbar) using six laser beams (di- ameter 2.8 cm, power 30 mW), two-by-two counter-propagatin g and tuned to the red of the F= 3 →F’= 4 of the Rb85D2 line (wavelength λ= 780 nm). The 4applied magnetic ﬁeld gradient is typically 10 G/cm. During the experiment, the MOT (trapping beams, repumper, and magnetic ﬁeld gradie nt) is continu- ously turned on and oﬀ . The ”dark” period is short enough (8 ms ) so that the cold atoms do not leave the capture volume and are recaptured during the next ”bright” period (duration 20 ms). To characterize the cold atomic cloud, we measure its optica l thickness as described in the next subsection. The shape and size of the cl oud is recorded in 3D using ﬂuorescence imaging, by illuminating the sample wi th a laser beam de- tuned by several Γ. We use a time-of-ﬂight technique to measu re the atom’s rms velocity, typically 10 cm/s. The atomic cloud contains typi cally 3 ×109atoms with a quasi-gaussian spatial distribution ∼5 mm FWHM (on average, the cloud being usually slightly cigar-shaped), yielding a pea k density of ∼1010 cm−3. 3.2 Optical thickness and transmitted polarization mea- surements The laser probe used for transmission measurements lies in t he horizontal plane containing 4 of the trapping beams, at an angle of 25◦. It is produced by a 50 mW SDL diode laser injected by a Yokogawa DBR diode laser, who se linewidth is 2-3 MHz FWHM as estimated from the beatnote between 2 ident ical diodes. This laser is passed through a Fabry-Perot cavity (transmis sion peak FWHM 10 MHz) before being sent through the atomic cloud. Although th is ﬁltering does not signiﬁcantly reduce the linewidth of the laser, it stron gly suppresses the spec- tral components in the wings of the initial lineshape which l imit the accuracy of the transmission measurement. The frequency of the probe can be scanned in a controlled way by ±50 MHz around the 3 →4 transition of the D2 line. The probe beam diameter is 1-2 mm, and its polarization is lin ear (vertical). The power in the probe is typically 0.1 µW, yielding a saturation parameter s = 2×10−3. It is turned on for 2 ms (yielding a maximum of about 80 photon s exchanged per atoms), typically 5 ms after the MOT is switche d oﬀ. The trans- mitted beam is detected by a photodiode after a rough spatial mode selection by two diaphragms of diameter 3 mm, distant of 1 m. The optical thickness measurement is performed without applied magnetic ﬁeld. As we emphasized in [2], simply measuring the on-resonance transmission yie lds a strongly biased estimate for the optical thickness b, due to the oﬀ-resonant components in the probe laser’s spectrum. To overcome this problem, one solut ion is to scan the laser detuning δand record the transmission line shape. We describe this cur ve as the convolution product of the transmission line for a pur ely monochromatic laser T( δ) = exp ( −b(δ)) with the laser line shape. If the later is known, one can extract the optical thickness from the transmission dat a (for instance from the FWHM of the transmission curve). This method is quite acc urate for large values of b, where the width of the transmission curve is only weakly dep endent on the laser’s linewidth. For small values of b, measuring the transmission at δ= 0 is more accurate but still requires some knowledge of the p robe laser’s spectrum. When working with dense atomic clouds at non-zero detuning, one 5should also keep in mind the possible inﬂuence of ”lensing” e ﬀect (focussing or deﬂection of the transmitted beam), due to the spatially-in homogeneous refrac- tive index of the sample. In our case, the rather large cloud s ize (∼5 mm) and moderate density ( ∼1010cm−3) yield a large focal length of about 25 m for the cloud, and a small (but still measurable) lensing eﬀect. Using the measured size of the cloud and assuming a uniform po pulation distribution in the ground state, we can then obtain the peak atomic density and the number of atoms in the sample. The maximal optical thickn ess measured in our trap is b= 24 (yielding a FWHM for the transmission curve ∆ δ∼6Γ). As mentioned in Section 2, the Stokes analysis relies on four transmission measurements. To perform the polarization measurement, we insert a polarime- ter in the path of the transmitted beam, as shown in ﬁg.1. It co nsists of a quarter-wave plate (only used for the circular component), a half-wave plate and a glan prism polarizer (ﬁxed). The rejection factor of th e polarizer is ∼10−3. The four transmission signals ( I//,I⊥,I45◦andIcirc) are recorded as a function of the laser detuning. The degree of polarizati on, ellipticity and rotation angle can then be computed using expressions (3), ( 4), and (5). 4 Results and discussion 4.1 Role of detuning Fig.2 shows a typical example of the raw signals obtained in t he four polarization channels necessary for the Stokes analysis detailed in Sec. 2. The transmitted intensity is recorded as a function of the detuning (express ed in units of Γ) in the parallel ( A), orthogonal ( B), 45◦(C) and circular ( D) polarization channels. All curves have been scaled by the incident intensity. These data were obtained for a sample of optical thickness b= 4.6 and an applied magnetic ﬁeld B= 3G. We see on curve ( B) that more than 10% of the incident light is transferred to the orthogonal channel. On curves ( C) and ( D), the oﬀ-resonant detected intensities are close to 0.5, since the incident linear pola rization projection on each of these channels is 1/2. The transmission curve ( D), which corresponds to the σ+component, presents a minimum shifted towards positive det unings by the Zeeman eﬀect ; the position of the minimum corresponds roughly to the splitting of the mg= +3→me= +4 transition (1.4 MHz/G), which has a maximum Clebsch-Gordan coeﬃcient of 1. Curves ( B) and ( C) exhibit noticeable asymmetries, which we will discuss later. From the data of ﬁg.2, we computed the three curves P(δ) (A),e(δ) (B) and θ(δ) (C) characterizing the polarization state of the transmitted light (ﬁg.3). We note on the curve ( A)of ﬁg.3 that the degree of polarization Pis not always equal to unity, and can be substantially smaller depending o n the parameters. This unexpected observation is due to the ﬁnite linewidth of our probe beam : light components at diﬀerent frequencies, initially all l inearly polarized, ex- perience diﬀerent rotations and deformations while passin g through the cloud. Because these components have diﬀerent frequencies, the re sult of their recom- 6bination, when integrated over a time long compared to their beatnote time, is a loss of polarization (for example, two orthogonal linearl y-polarized waves of diﬀerent frequencies and same intensity yield a totally unp olarized light P= 0). The result of the recombination of all the spectral componen ts is not straight- forward to predict, since each frequency is transmitted wit h a diﬀerent intensity, ellipticity and rotation angle. However, if we assume that a ll the spectral com- ponents are mutually incoherent, the total intensity detec ted in each channel is the sum of the intensities corresponding to all the spectral components. Thus, in order to compare the experimental data with the model, we c onvolute the transmission curves I//(δ), I⊥(δ),I45◦(δ) and Icirc(δ), as calculated with the model of Sec.2, with the power spectrum of the probe laser. We can then com- pute the curves P(δ),e(δ) and θ(δ) using expressions (3), (4), and (5). We stress that the inﬂuence of the laser’s linewidth is quite st rong : even for a (lorentzian) linewidth of a tenth of the natural width, for a n optical thickness b= 5 and a magnetic ﬁeld B= 1G, the loss of polarization is already 32% (P= 0.68). This phenomenon also aﬀects the estimates for the ellip ticity and the rotation angle, since these quantities reﬂect mainly th e polarization state of the dominant transmitted spectral component. Indeed, if one again consid- ers the example of the superposition of two waves of diﬀerent frequencies and polarizations (pure states), the results of the Stokes anal ysis will vary continu- ously from one polarization state to the other depending on t he intensity ratio of the two waves. In the intermediate regime of comparable in tensities, the Stokes analysis will not describe adequately any of the two p olarizations. It should be noted that this situation diﬀers from the case wher e the studied light consist of a polarized component plus a depolarized one [12] ; in this case, the Stokes analysis provides the ”correct” result (that is, the ellipticity and angle of the polarized component), even for arbitrarily small pro portion of polarized light. We experimentally tested the inﬂuence of a polychrom atic excitation by superimposing to the normal probe beam a weaker one, obtaine d from the same laser but detuned by 80 MHz with an acousto-optic modulator. The calcu- lated degradation of P(δ),e(δ) and θ(δ) account well for the experimentally observed behavior. Fig.3 ( B)shows how the ellipticity eof the polarized component of the transmitted light varies with laser detuning. For δ >0, it is mainly the σ+ component of the incident light which is absorbed, yielding a mostly σ−trans- mitted polarization (negative ellipticity). On resonance (δ= 0) both compo- nents are absorbed in the same proportion, an the transmitte d polarization is linear ( e= 0). Since the ellipticity depends on the diﬀerential absor ption be- tween circular components, it is a direct measurement of the dichroism in the sample. The curve on ﬁg.3 ( C)is the Faraday rotation angle computed from ex- pression (5). The on-resonance rotation in this case is abou t 40◦. The solid lines in ﬁg.3 are obtained with our model using the convolution pro cedure discussed above ; the probe light spectrum is described by the product o f a lorentzian laser lineshape (FWHM = 3 MHz) by a lorentzian Fabry-Perot tr ansmission (FWHM =10 MHz). We account for the observed asymmetry in the e xperi- mental curves by introducing a linear variation in the popul ations of the ground 7state sublevels (i.e. a partial orientation of the sample), with maximum vari- ation±20% between extreme magnetic numbers mg=±3.We have checked some other possible mechanisms for this asymmetry, such as t he proximity of the F= 3 →F’= 3 transition (121 MHz to the red) or optical pumping, but both eﬀects seem to play a small role. The fact that we were als o able to in- vert the asymmetry by varying the orientation of the magneti c ﬁeld produced by the compensation coils also favors the hypothesis of a par tial orientation of the medium. We actually do see some optical pumping eﬀects , manifested by variations of the measured transmission signals with tim e during the probe pulse (duration 2 ms). The overall eﬀect of optical pumping i s to increase the Faraday rotation as the number of exchanged photons (i.e. ti me) increases, i.e. the measured rotation immediatly after turning on the probe is smaller than after 2 ms of presence of the light (by about 6%). The comparison of ﬁg.3 between experiment and theory shows t hat, despite some discrepancies due to our rather vague knowledge of the p robe lineshape and to the simplicity of our model, the overall agreement is q uite satisfying. 4.2 Role of magnetic ﬁeld To determine the Verdet constant, it is necessary to measure the Faraday an- gle as a function of the magnetic ﬁeld. Fig.4 shows such curve s obtained for two diﬀerent optical thicknesses : b= 0.75 (solid circles) and b= 9 (open cir- cles). For each curve, the on-resonance Faraday angle θis scaled by the optical thickness of the cloud, since one expects the rotation to be p roportional to the optical thickness (thus, all experimental data should lie o n the same ”universal” curve). The solid line is the prediction of the model with a in ﬁnitely narrow probe laser. Its slope around B= 0 is about 10◦/G, yielding a Verdet constant V= 20◦/G/mm for an optical thickness b= 10 and a sample diameter of 5 mm. The dashed curve represents the small optical thickness limit when the laser linewidth is taken into account, which lowers the slop e at around 6◦/G. Both curves exhibit the expected dispersive shape, with a li near increase of the Faraday angle at small magnetic ﬁeld values (where the Ve rdet constant is deﬁned), and then a decreasing rotation when the splittin g between the σ+ andσ−transitions becomes larger than the natural width. The expe rimental curve for small optical thickness is quite close to the model prediction (dashed line). For larger values of the optical thickness, the measu rements depart from this ideal situation due to the ﬁnite linewidth of the laser : the curve for b= 9 presents a smaller slope around B= 0 and the scaled rotation is globally re- duced. As the optical thickness is further increased, the tr ansmitted light be- comes increasingly dominated by oﬀ-resonant components of the laser spectrum and the information about the central (resonant) frequency is lost. This process is further illustrated in the following subsection. 84.3 Role of optical thickness In the ideal case of a monochromatic laser, one expects the Fa raday rotation to increase linearly with optical thickness. We thus recorded the rotation at δ= 0 and a ﬁxed value of B, as a function of the optical thickness which was varied by detuning the trapping laser. The result of such an experimen t is shown in ﬁg.5. The open circles correspond to an applied magnetic ﬁeld B= 2 G, while the solid circles are for B= 8 G. The solid line is the expected evolution for B= 8 G and a monochromatic laser; the dashed line is the prediction of t he same model for B= 2 G. We see that, for the higher value of B, the expected linear behavior is indeed obtained, yielding a slope of about 8◦/G. The (absolute) rotation angle increases up to ∼150◦. However, the evolution observed at smaller magnetic ﬁeld (B= 2 G, circles) is quite diﬀerent, where the data quickly depa rt from the linear evolution even at low optical thickness and suddenly drop towards positive values of the angle at large optical thickness. Qualitative ly, this reﬂects the fact that, at small applied ﬁeld and large optical thickness, the central (resonant) frequency of the probe laser is strongly attenuated and can b ecome smaller than other (oﬀ-resonant) spectral components. The measure d rotation angle then passes continuously from the angle of the central frequ ency component to that of the dominant detuned component (in the wings of the ab sorption line), which can be negative (see ﬁg.3 ( C)). The large dispersion of the data for B= 2 G above b∼12 is due to the important relative error in this low transmis sion regime. At larger Bﬁeld, the on-resonance transmission increases due to the Zeeman splitting, and the central frequency component rema ins dominant for larger values of the optical thickness (for instance, the to tal transmission at b= 20 is around 0.1 for B= 8 G, while it is only 3 ×10−3forB= 2 G). Thus, the simple model of Section 2 provides us with a good des cription for the various behaviors observed experimentally. A fair quan titative agreement is obtained for moderate optical thickness or high magnetic ﬁe ld. The model is also helpful to understand the important role played by the lines hape of the probe laser in this experimental situation of optically-thick sa mple and resonant light. The experimental data conﬁrm the occurrence of large Farada y eﬀect inside our atomic cloud, with a Verdet constant in the range of 20◦/G/mm for a typical optical thickness of 10. 5 Conclusion We have reported in this paper the measurement of large Farad ay eﬀect in an optically-thick sample of cold rubidium atoms. Due to near- resonant excitation, we need to take into account both Faraday rotation (diﬀerent ial refractive in- dex) and dichroism (diﬀerential absorption) to analyze the experimental data. Using a very simple model for our F = 3 →F’= 4 transition, we obtain a good agreement with the experimental data. We measure large Verd et constant of the order of 20◦/G/mm . We have shown that the ﬁnite width of the laser spec- trum plays a crucial role in the signals obtained for an optic ally-thick sample. 9A complete analysis of the transmitted light polarization s tate is then necessary to correctly interpret the data. The determination of the Verdet constant Vin the cloud is an important step in our current study of the eﬀect of an applied magnetic ﬁeld o n the coherent backscattering of light by the cold atoms. For Faraday eﬀect to seriously aﬀect the CBS cone, one needs the phase diﬀerence between time-rev ersed waves, accumulated on a distance of the order of the light mean-free pathl, to be of the order of π(i.e. a rotation of π/2 for a linear polarization). This corresponds to a situation where V Bl∼1 [7]. However, the main diﬀerence between the situation of ref.[7] (scatterers in a Faraday-active matri x) and our atomic cloud situation is that, in our case, the Verdet constant is determ ined by the density of scatterers ρ(Vproportional to ρ), which in turn ﬁxes the mean-free path (lproportional to1 /ρ). Thus, there is a maximum rotation per mean-free path length scale , which is about 13◦(forB= 3 G) according to the curve of ﬁg.4 (solid line). It seems to us interesting to study this unusua l situation. Our aim is also to understand how the Faraday eﬀect combines to the ot her eﬀects due to the atom’s internal structure to determine the CBS enhanc ement factor in the presence of an external magnetic ﬁeld. Acknowledgement 1 This research program is supported by the CNRS and the PACA Region. We also thank the GDR PRIMA. The contributio n of J.-C. Bernard was determinant in the development of the experimen t. We are grateful to D. Delande for some very fruitful discussions. 6 Bibliography References [1] G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. M¨ uller, C . Miniatura & R. Kaiser, Phys. Rev. Lett. 835266 (1999). [2] G. Labeyrie, C.A. M¨ uller, D.S. Wiersma, Ch. Miniatura a nd R. Kaiser, J. Opt. B: Quantum Semiclass. Opt 2672-685(2000). [3] T. Jonckheere, C.A. M¨ uller, R. Kaiser, Ch. Miniatura an d D. Delande, Phys. Rev. Lett. 85, 4269 (2000). [4] M.P. van Albada, A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985) ; E. Akkermans, P.E. Wolf and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986). [5] F.C. MacKintosh and Sajeev John, Phys. Rev. B 37, 1884 (1988); A.S. Martinez and R. Maynard, Phys. Rev. B 50, 3714 (1994); D. Lacoste and B.A. van Tiggelen, Phys. Rev. B 61, 4556 (2000). [6] A.A. Golubentsev, Sov. Phys. JETP 59, 26 (1984). [7] R. Lenke and G. Maret, Eur. Phys. J. B 17, 171 (2000). 10[8] L.D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, Cambridge, 1982). [9] F. Schuller, M.J.D. Macpherson, D. N. Stacey, R.B. Warri ngtonand K.P. Zetie, Optics Comm. 86, 123 (1991); M. Kristensen, F.J. Blok, M.A. van Eijkelenborg, G. Nienhuis, and J.P. Woerdman, Phys. Rev. A 51, 1085 (1995). [10] G.G. Adonts, D.G. Akopyan, and K. V. Arutunyan, J. Phys. B : At. Mol. Phys.19, 4113 (1986); W. V. Davis, A.L. Gaeata, and R.W. Boyd, Opt. Lett.17(18), 1304 (1992); S.I. Kanorsky, A. Weis, J. Wurster, and T. W. H¨ ansch, Phys. Rev. A 47, 1220 (1993); D. Budker, D.F. Kimball, S.M. Rochester, and V.V. Yashchuk, Phys. Rev. Lett. 85, 2088 (2000). [11] T. Isayama, Y. Takahashi, N. Tanaka, K. Ishikawa, and T. Yabuzaki, Phys. Rev. A 59, 4836 (1999). [12] Max Born and Emil Wolf in Principles of Optics , sixth edition, Pergamon Press, p. 554. Figures captions: Figure 1 : Simpliﬁed experimental setup. A probe laser beam of linear polarization Eiand wave vector kis sent through the cold atomic cloud, where a magnetic ﬁeld Bis applied along k. The polarization Etof the transmitted probe light is deformed and rotated. A polarimeter measures the transmitted intensities in four polarization chan- nels : I//(parallel to the incident polarization), I⊥(orthogonal to the incident polarization), I45◦(at 45◦from the incident polarization), Icirc(circular polar- ization). These quantities allow to determine the degree of polarization P, the ellipticity e, and the rotation angle θof the transmitted light. Figure 2 : Typical transmission curves in the four polarization chann els. The transmission is measured as a function of the laser detun ing (in units of the natural width Γ), for a sample of optical thickness (at zero ﬁeld) b= 4.6 and an applied magnetic ﬁeld B= 3 G. All data are scaled by the total incident light intensity. A: intensity I//in the linear parallel channel. B: intensity I⊥ in the linear orthogonal channel. C: intensity I45◦in the linear 45◦channel. D : intensity Icircin the circular channel. Figure 3 : Typical results from the Stokes polarization analysis and c om- parison with model. These curves are obtained from the data of ﬁg.2. A: degree of polarization P(see expressions (3)). B: ellipticity e.C: Faraday rotation angle θ. The symbols correspond to experimental data and the solid lines to the predictions of the model described in Sec. II. To reproduce the experimen tal asymmetry, the model assumes a linear variation of the ground state popu lations p mwith magnetic number m, with a total variation amplitude of 40% between extreme ground state sublevels mg=±3. 11Figure 4 : Scaled Faraday angle as a function of the applied magnetic ﬁe ld B. The rotation angle θis scaled by the optical thickness bof the sample. The symbols correspond to samples with two diﬀerent optical thi cknesses : b= 0.75 (solid circles) and b= 9 (open circles). The solid line is the model prediction assuming a monochromatic probe laser (and a uniform populat ion distribution in the ground state). The dotted line is the small optical thi ckness limit of the model when taking into account the lineshape of the probe las er. Figure 5 : On-resonance Faraday rotation angle θas a function of the optical thickness. The symbols correspond to experiments with two diﬀerent val ues of the mag- netic ﬁeld : B= 2 G (open circles) and B= 8 G (solid circles). The optical thickness of the atomic cloud is varied by scanning the detun ing of the MOT laser. The lines correspond to the predictions of the model w ith a monochro- matic laser, for B= 2 G (dashed line) and B= 8 G (solid line). For the largest Bvalue, we observe the expected linear increase of Faraday an gle with optical thickness. The measured rotation is close to the pre diction of the ideal, monochromatic model (solid line). On the other hand, the beh avior for B= 2 G is quite diﬀerent : the rotation angle quickly departs from the linear increase, saturates and then decreases. Indeed, as optical thickness increases, the mea- sured rotation becomes increasingly aﬀected by other spect ral components of the laser, until these oﬀ-resonant frequencies become domi nant causing a sharp drop of the angle. At large Bthe central, resonant frequency component of the laser is always dominant in the optical thickness range inve stigated, and the linear behavior is recovered. 12figure 1x yz EikB// ⊥45° polarimeter Etθ ab cold atomic cloud0,00,51,0 AI // 0,000,050,10 I ⊥B -6 -4 -2 0 2 4 60,00,51,0 figure 2C I 45° δ / Γ-6 -4 -2 0 2 4 60,00,51,0 I circD δ / Γ-8 -6 -4 -2 0 2 4 6 8-2002040CFaraday angle θ (°) δ / Γ-0,50,00,5B ellipticity e0,00,51,0 figure 3Adegree of polarization P-15 -10 -5 0 5 10 15-15-10-5051015 figure 4θ / b (°) magnetic field (G)0 5 10 15 20-50050100150 figure 5Faraday rotation θ (°) optical thickness
arXiv:physics/0103046v1 [physics.atom-ph] 16 Mar 2001Anharmonic parametric excitation in optical lattices R. J´ auregui Instituto de F´ ısica, Universidad Nacional Aut´ onoma de M´ exico, Apdo. Postal 20-364, M´ exico, 01000, D.F., M´ exico N. Poli, G. Roati, and G. Modugno INFM-European Laboratory for Nonlinear Spectroscopy (LEN S), Universit` a di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy (February 2, 2008) We study both experimentally and theoretically the losses induced by parametric excitation in far–oﬀ–resonance opti cal lattices. The atoms conﬁned in a 1D sinusoidal lattice prese nt an excitation spectrum and dynamics substantially diﬀeren t from those expected for a harmonic potential. We develop a model based on the actual atomic Hamiltonian in the lattice and we introduce semiempirically a broadening of the width of lattice energy bands which can physically arise from in- homogeneities and ﬂuctuations of the lattice, and also from atomic collisions. The position and strength of the paramet ric resonances and the evolution of the number of trapped atoms are satisfactorily described by our model. 32.80.Pj, 32.80.Lg I. INTRODUCTION The phenomenon of parametric excitation of the mo- tion of cold trapped atoms has recently been the sub- ject of several theoretical and experimental investigatio ns [1–3]. The excitation caused by resonant amplitude noise has been proposed as one of the major sources of heating in far–oﬀ–resonance optical traps (FORTs), where the heating due to spontaneous scattering forces is strongly reduced [4]. In particular, the eﬀect of resonant excita- tion is expected to be particularly important in optical lattices, which usually provide a very strong conﬁnement to the atoms, resulting in a large vibrational frequency and in a correspondingly large transfer of energy from the noise ﬁeld to the atoms [1]. Nevertheless, parametric excitation is not only a source of heating, but it also represents a very useful tool to characterize the spring constant of a FORT or in general of a trap for cold particles, and to study the dynamics of the trapped gas. Indeed, the trap frequencies can be mea- sured by intentionally exciting the trap vibrational modes with a small modulation of the amplitude of the trap- ping potential, which results in heating [5] or losses [2,6] for the trapped atoms when the modulation frequency is tuned to twice the oscillation frequency. This proce- dure usually yields frequencies that satisfactorily agree with calculated values, and are indeed expected to be accurate for the atoms at the bottom of the trapping po- tential. From the measured trap frequencies is then pos- sible to estimate quantities such as the trap depth and the number and phase space densities of trapped atoms.We note that this kind of measurement is particularly important in optical lattices, since the spatial resolutio n of standard imaging techniques is usually not enough to estimate the atomic density from a measurement of the volume of a single lattice site. Recently, 1D lattices have proved to be the proper envi- ronment to study collisional processes in large and dense samples of cold atoms, using a trapping potential inde- pendent for the magnetic state of the atoms. In this systems, the parametric excitation of the energetic vi- brational mode along the lattice provides an eﬃcient way to investigate the cross-dimensional rethermalization dy - namics mediated by elastic collisions [7,6]. Most theoretical studies of parametric excitation rely on a classical [8] or quantum [1] harmonic approximation of the conﬁning potential. Under certain circumstances these expressions show quite good agreement with exper- imental results [3]. However, general features of the opti- cal lattice could be lost in these approaches. For example, a sinusoidal potential exhibits an energy band structure and a spread of transition energies, while harmonic oscil- lators have just a discrete equidistant spectrum. Thus, we might expect that the excitation process may happen at several frequencies, and with a non-negligible band- width. Such anharmonic eﬀects can be important when- ever the atoms are occupying a relatively large fraction of the lattice energy levels. The purpose of this paper is to give a simple description of parametric excitation in a sinusoidal 1D lattice. In the next Section, we brieﬂy discuss general features of the stationary states on such a lattice. Then, we summarize the harmonic description given in Ref. [1] and extend it to the anharmonic case. By a numerical evaluation of transition rates, we make a temporal description of parametric excitation which is compared with experimental results. We discuss about the relevance of broadening of the spectral lines to un- derstand the excitation process in this kind of systems. Some conclusions are given in the last Section. II. STATIONARY STATES OF A SINUSOIDAL OPTICAL LATTICE. The Hamiltonian for an atom in a red detuned FORT is H=P2 2M+Veﬀ(/vector x), (2.1) 1with Veﬀ(/vector x) =−1 4α\|E(/vector x)\|2, (2.2) where αis the eﬀective atomic polarizations and E(x) is the radiation ﬁeld amplitude. For the axial motion in a sinusoidal 1D lattice we can take Hax=P2 z 2M+V0cos2(kz) (2.3) =P2 z 2M+V0 2/parenleftbig 1 + cos(2 kz))/parenrightbig . (2.4) The corresponding stationary Schr¨ odinger equation −¯h2 2Md2Φ dz2+V0 2/parenleftbig 1 + cos(2 kz)/parenrightbig Φ =EΦ (2.5) can be written in canonical Mathieu’s form d2Φ du2+ (a−2qcos2u)Φ = 0 (2.6) with a=/parenleftbig E−V0 2/parenrightbig/parenleftbig2M ¯h2k2/parenrightbig 2q=V0 2/parenleftbig2M ¯h2k2/parenrightbig . (2.7) It is well known that there exists countably inﬁnite sets of characteristic values {ar}and{br}which respectively yield even and odd periodic solutions of Mathieu equa- tion. These values also separate regions of stability. In particular, for q≥0 the band structure of the sinusoidal lattice corresponds to energy eigenvalues between arand br+1[11]. The unstable regions are between brandar. Forq >> 1, there is an analytical expression for the band width [11]: br+1−ar∼24r+5/radicalbig 2/πq1 2r+3 4e−4√q/r!. (2.8) The quantities deﬁned above can be expressed in terms of a frequency ω0deﬁned in the harmonic approximation of the potential 1 2Mω2 0=V0 2(2k)2 2!, (2.9) thus obtaining a=/parenleftbig E−V0 2/parenrightbig/parenleftbig4V0 ¯h2ω2 0/parenrightbig ;q=/parenleftbigV0 ¯hω0/parenrightbig2. (2.10) Thus, the width of the r-band can be estimated using Eq. (2.8) whenever the condition ( V0/¯hω0)2>>1 is sat- isﬁed. In the experiment we shall be working with a 1D optical lattice having V0∼10.5¯hω0. While the lowest bandr= 0 has a negligible width ∼10−18¯hω0, the band widths for highest lying levels r= 10,11,12,and 13 would respectively be 0 .0065 ,0 .1036, 1 .52, and 20 .56 in units of ¯hω0.In order to determine the energy spectrum, a varia- tional calculation can be performed. We considered a harmonic oscillator basis set centered in a given site of the lattice, and with frequency ω0. The diagonalization of the Hamiltonian matrix associated to (2.4) using 40 basis functions gives the eigenvalues En< V0shown in Table I for V0= 10.5¯hω0. According to the results of last paragraph, the eigenvalues 12 and 13 belong to the same band while the band width for lower levels is smaller than 0.11¯hω0 III. PARAMETRIC EXCITATION As already mentioned, parametric excitation of the trapped atoms consists in applying a small modulation to the intensity of the trapping light, H=P2 2M+Veﬀ[1 +ǫ(t)]. (3.1) Within ﬁrst order perturbation theory, this additional ﬁeld induces transitions between the stationary states n andmwith an averaged rate Rm←n=1 T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−i ¯h/integraldisplayT 0dtT(m, n)ǫ(t)eiωmnt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =π 2¯h2\|T(m, n)\|2S(ωmn);ωmn=Em−En ¯h(3.2) where T(m, n) =∝an}bracketle{tm\|Veﬀ\|n∝an}bracketri}ht =Enδnm−1 2M∝an}bracketle{tm\|ˆP2\|n∝an}bracketri}ht (3.3) is the matrix element of the space part of the perturba- tion and S(ω) =2 π/integraldisplayT 0dτcosωτ∝an}bracketle{tǫ(t)ǫ(t+τ)∝an}bracketri}ht (3.4) is the one-sided power spectrum of the two-time correla- tion function associated to the excitation ﬁeld amplitude. If the conﬁning potential is approximated by a har- monic well, the transition rates diﬀerent from zero are Rn←n=πω2 0 16S(0)(2n+ 1) (3.5) Rn±2←n=πω2 0 16S(2ω0)(n+ 1±1)(n±1) (3.6) The latter equation was used in [1] to obtain a simple expression for the heating rate, ∝an}bracketle{t˙E∝an}bracketri}ht=π 2ω2 0S(2ω0)∝an}bracketle{tE∝an}bracketri}ht, (3.7) showing its exponential character. The dependence on 2ω0is characteristic of the parametric nature of the exci- tation process. The fact that ¯ his not present is consistent with the applicability of Eq. (3.7) in the classical regime. 2Classically, parametric harmonic oscillators exhibit resonances not just at 2 ω0but also at 2 ω0/nwithnany natural number [8]. In fact, the resonances corresponding ton=2,i.e.at an excitation frequency ω0, have been ob- served in optical lattices [2,6]. A quantum description of parametric harmonic excitation also predicts resonances at the same frequencies via n-th order perturbation the- ory [10]. In particular, the presence of the resonance at ω0can be justiﬁed with the following argument. Accord- ing to the standard procedure, the second order correc- tion to the transition amplitude between states \|n∝an}bracketri}htand \|m∝an}bracketri}htis given by R(2) m←n=∝an}bracketle{tn\|U(2)(t0, t)\|m∝an}bracketri}ht=/summationdisplay k/parenleftbig−i ¯h/parenrightbig2T(n, k)T(k, m) /integraldisplayt t0dt′eiωnkt′ǫ(t′)/integraldisplayt t0dt′′eiωkmt′′ǫ(t′′) (3.8) withU(2)(t0, t) the second order correction to the evo- lution operatorU. Therefore, the transition may be de- scribed as a two step procedure \|m∝an}bracketri}ht←\|k∝an}bracketri}ht←\|n∝an}bracketri}ht. For harmonic parametric excitation the matrix element of the space part of the perturbation diﬀers from zero just for transitions\|n∝an}bracketri}ht←\|n∝an}bracketri}htand\|n±2∝an}bracketri}ht←\|n∝an}bracketri}ht. Consider a tran- sition in Eq. (3.8) involving a ”ﬁrst” step in which the state does not change \|n∝an}bracketri}ht←\|n∝an}bracketri}htand a ”second” step for which\|n±2∝an}bracketri}ht←\|n∝an}bracketri}ht. Then resonance phenomena occur when the total energy of the two excitations, 2¯ hΩ, coin- cides with that of the second step transition, i. e.for an excitation frequency Ω = ωo. These ideas can be directly extended to anharmonic potentials: the corresponding transition probability rat es R(n, m) would be determined by the transition matrix T(n, m), by the transition frequencies ωnmand by the time dependence of the excitation ǫ(t). In general, an- harmonic transition matrix elements T(n, m) will be dif- ferent from zero for a wider set of pairs ( n, m). Besides, the transition energies will not be unique so that the exci- tation process is notdetermined by the excitation power spectrum at a single given frequency 2 ω0and its sub- harmonics 2 ωo/n. As an example the transition energies for the speciﬁc potential considered in this work are re- ported in Table I. Therefore, within the model Hamilto- nian of Eq. (3.1), resonance eﬀects can occur for several frequencies that may alter the shape of the population distribution within the trap. However, in general these resonant excitations will not be associated with the es- cape of trapped atoms. Here we are interested in a 1D lattice; the direct ex- tension of the formalism mentioned above requires the evaluation of the matrix elements T(n, m) among the diﬀerent Mathieu states that conform a band. This in- volves integrals which, to our knowledge, lack an ana- lytical expression and require numerical evaluation. As an alternative, we consider functions which variationally approximate the Mathieu functions. They are the eigen- states of the Hamiltonian (2.4) in a harmonic basis set of frequency ω0:\|n∝an}bracketri}ht=imax/summationdisplay i=1cni\|i∝an}bracketri}htω0, (3.9) These states are ordered according to their energy: En≤ En+1as exempliﬁed in Table I. Within this scheme one obtains a very simple expression for T(n, m) T(n, m) =Enδnm−imax/summationdisplay i,j=1cnicnj1 2M∝an}bracketle{ti\|ˆP2\|j∝an}bracketri}ht.(3.10) It is recognized that any discrete basis set approxima- tion to a system with a band spectrum will lack fea- tures of the original problem which have to be care- fully analyzed. Anyway, alternatives to a discrete ba- sis approach may be cumbersome and not necessarily yield a better approach to understand general proper- ties of experimental data. While the discrete basis ap- proach is exact for transitions between the lowest levels, which have a negligible width, eigenstates belonging to a band of measurably width should be treated with spe- cial care. Thus, we shall assume that matrix elements T(ν, µ) involving states with energies EνandEµ, so that En−(En−En−1)/2≤Eν≤En+ (En+1−En)/2 with an analogous expressions for Eµ, are well approximated byT(n, m). Within this scheme the equations which describe the probability P(n) of ﬁnding an atom in level n, given the transition rates Rm←nare ˙P(n) =/summationdisplay mR(1) m←n(P(m)−P(n)) (3.11) in the ﬁrst order perturbation theory scheme, and the ﬁnite diﬀerence equations Pn(t) =Pn(t0) +/summationdisplay mR(1) m←n(Pm(t0)−Pn(t0))(t−t0) + /summationdisplay mR(2) m←n(Pm(t0)−Pn(t0))(t−t0)2,(3.12) valid up to second order time-dependent perturbation theory whenever t∼t0. Both sets of equations are sub- jected to the condition /summationdisplay nP(n) = 1. (3.13) Now, according to Eqs. (3.2) and Eq (3.8), the evalua- tion of R(r) n←malso requires the speciﬁcation of the spec- tral density S(ω). In the problem under consideration, the discrete labels m, nare used to calculate interband transitions which are actually spectrally broad. This broadening might arise not only from the band structure of the energy spectra associated with the Hamiltonian Eq. (2.4), but also from other sources, which we will dis- cuss below. Broad spectral lines can be introduced in our 3formalism by deﬁning an eﬀective spectral density Seﬀ(ω) which should incorporate essential features of this broad- ening without simulating speciﬁc features. Having this in mind, an eﬀective Gaussian density of states Sn(ω) is as- sociated to each level \|n∝an}bracketri}htof energy En Sn(ω) =1√ 2πσne−(¯hω−En)2 2(¯hσn)2. (3.14) The spectral eﬀective density Seﬀ(ωnm) associated to the transition m←nis obtained by the convolution of Sn(ω) withSm(ω) and with the excitation source spectral den- sityS(ω). For a monochromatic source the latter is also taken as a Gaussian centered at the modulation fre- quency that once integrated over all frequencies yields the square of the intensity of the modulation source. The net result is that Seﬀ(ωnm) has the form Seﬀ(ω) =S0e−(ω−ωeﬀ)2 2σ2 eﬀ (3.15) withωeﬀdetermined by the modulation frequency Ω and the energies EnandEm. The eﬀective width σeﬀcontains information about the frequency widths of the excitation source and those of each level. IV. COMPARISON WITH EXPERIMENTAL RESULTS. We have tested the procedure described in last section to model parametric excitation in a speciﬁc experiment conducted at LENS. In this experiment40K fermionic atoms are trapped in a 1D lattice, realized retroreﬂect- ing linearly polarized light obtained from a single–mode Ti:Sa laser at λ=787nm, detuned on the red of both the D 1and D 2transition of potassium, respectively at 769.9nm, and 766.7nm. The laser radiation propagates along the vertical direction, to provide a strong conﬁne- ment against gravity. The laser beam is weakly focused within a two-lens telescope to a waist size w0≃90µm, with a Rayleigh length zR=3cm; the eﬀective running power at the waist position is P=350mW. The trap is loaded from a magneto-optical trap (MOT), thanks to a compression procedure already de- scribed in [6], with about 5 ×105atoms at a density around 1011cm−3. The typical vertical extension of the trapped atomic cloud, as detected with a CCD camera (see Fig. 1), is 500 µm, corresponding to about 1200 oc- cupied lattice sites with an average of 400 atoms in each site. Since the axial extension of the atomic cloud is much smaller than zR, we can approximate the trap potential to V(r, z) =V0e−2r2 w2 0cos2(kz);k= 2π/λ, (4.1)thus neglecting a 5% variation of V0along the lattice. The atomic temperature in the lattice direction is mea- sured with a time–of–ﬂight technique and it is about 50µK. In order to parametrically excite the atoms we modu- late the intensity of the conﬁning laser with a fast AOM for a time interval T ≃100ms, with a sine of amplitude ǫ=3% and frequency Ω. The variation of the number of trapped atoms is measured by illuminating the atoms with the MOT beams and collecting the resulting ﬂuo- rescence on a photomultiplier. In Fig.2 the fraction of atoms left in the trap after the parametric excitation is reported vsthe modulation frequency Ω /2π. Three res- onances in the trap losses are clearly seen at modulation frequencies 340kHz, 670kHz and 1280kHz. By identify- ing the ﬁrst two resonances with the lattice vibrational frequency and its ﬁrst harmonic, respectively, we get as ﬁrst estimate ω0≃2π×340 kHz. As we will show in the following, these resonance are actually on the redofω0 and 2ω0, respectively, and therefore a better estimate is ω0≃2π×360 kHz. Therefore the eﬀective trap depth is, from Eq. (9), V0≃185µK≃10.5 ¯hω0. Since the atomic temperature is about V0/3.5, we expect that most of the energy levels of the lattice have a nonnegligible popu- lation and therefore the anharmonicity of the potential could play an important role in the dynamics of paramet- ric excitation. Note that the third resonance at high fre- quency, close to 4 ω0, is not predicted from the harmonic theory. It is possible to observe also a much weaker reso- nance in the trap losses around 1.5kHz, which we inter- pret to be twice the oscillation frequency in the loosely conﬁned radial direction. Anyway, in the following we will focus our attention just on the axial resonances. As discussed in the previous Section, the overall width of the excitation assumed for our model system could play an important role in reproducing essential features of experimental data. Since the source used in the ex- periment has a negligible line width, it is necessary to model just the broadening of the atomic resonances. The spread of the transition energies due to the axial anhar- monicity is reported in Table I, while the broadening of each energy level, due to the periodic character of the sine potential, is estimated using Eq. (2.8). We now note that the 1D motion assumed in Section II is not com- pletely valid in our case, since the atoms move radially along a Gaussian potential. Since the period of the radial motion is about 500 times longer than the axial period, the atoms see an eﬀective axial frequency which varies with their radial position, resulting in a broadening of the transition frequency. Other sources of broadening are ﬂuctuations of the laser intensity and pointing, and inhomogeneities along the lattice. We note that also elas- tic collisions within the trapped sample, which tend to keep a thermal distribution of the trap levels population, can contribute to an overall broadening of the loss res- onances. Since it is not easy to build a model which involve all these sources, we introduce semiempirically an eﬀective broadening for the r-th level (see Eq. (3.14)). 4Recognizing that the width could be energy dependent we considered the simple expression σ2 r=λ1/parenleftbigEr V0/parenrightbigp+λ0 (4.2) for several values of the constants λ1, λ0andp. When p= 0, i.e. for a constant value of the band width we were not able to reproduce the general experimental be- havior reported in Fig.2. The best agreement between the simulation and the experimental observations is ob- tained for λ0= 0.0002, λ1= 0.0135, in units of ω2 0, andp= 3. Similar results are obtained also for slightly higher (lower) values of λ0,1together with slightly higher (lower) values of the power p. In Table I the resulting widths are shown for the lower twelve levels. Note that we have intentionally excluded levels 11, 12 and 13 from the calculation, since their intrinsic width is so large tha t the atoms can tunnel out of the trap along the lattice in much less than 100ms [9]. Anyway, the inclusion of these levels proved not to change substantially the result of the simulation. The comparison of experimental and theoretical results is made in Fig. 3; the abscissa for the experimental data has been normalized by identifying ω0with 2 π×360 kHz. As already anticipated, the principal resonance in trap losses appears at Ω ≃1.85ω0. This result follows from the fact that the excitation of the lowest trap levels is not resulting in a loss of atoms, as it would happen for a harmonic potential. On the contrary, the most energetic atoms, which have a vibrational frequency smaller than the harmonic one, are easily excited out of the trap. The asymmetry of the resonances, which has been observed also in [2], is well reproduced in the calculations and it is a further evidence of the spread of the vibrational frequen- cies. The ﬁrst interesting result obtained by our study of parametric excitation is therefore the correction nec- essary to extract the actual harmonic frequency from the loss spectrum. For the speciﬁc conditions of the present experiment, we ﬁnd indeed that the principal resonance in the trap losses appears at Ω ≃1.85ω0. Anyway, the calculation shows that the resonance is nearby this po- sition for all the explored values of λ0,1andpalso for deeper traps, up to V0=25¯hω0, and therefore it appears to be an invariant characteristic of the sinusoidal poten- tial. The result of the numerical integration of Eqs. (3.12) reported in Fig.3 reproduces relatively well the subhar- monic resonance, which in the harmonic case would be expected at ω0. On the contrary, both experiment and calculation show that the actual position of the resonance is Ω≃0.9ω0. It must be mentioned that the accuracy of these results is restricted by the ﬁnite diﬀerence charac- ter of Eqs. (3.12) and by the fact that some noise sources which have not been included could be resonant at a nearby frequency. In particular, a possible modulation of the laser pointing associated to the intensity modula- tion is expected to be resonant at Ω = ω0in the harmonicproblem [1], and it could play an analogous role in our sinusoidal lattice. The higher order resonance around 3.5 ω0observed in the experiment is also well reproduced by the calcula- tions based on ﬁrst order perturbation theory. Note that a simpler approximation to the conﬁning potential by a quartic potential VQ(z) =k2z2+k4z4would yield a reso- nance around 4 ω0and not 3 .5ω0. Anyway, it is possible to understand qualitatively one of the features of this resonance considering a quartic perturbation of the form ǫ(t)VQto a harmonic potential. In this case the ratio of the transition rates at the 2 ω0and 4ω0resonances is set by Eqs.3.2 to \|T(n±2, n)\|2/\|T(n±4, n)\|2∝V2 0 ω2 0. (4.3) This result can qualitatively explain the absence of the corresponding high order resonance in the radial excita- tion spectrum (see Fig.2): since the radial trap frequency is a factor 500 smaller than the axial one, the relative strength of such radial anharmonic resonance is expected to be suppressed by a factor (500)2. In conclusion, high harmonics resonances, which certainly depend on the ac- tual shape of the anharmonic potential, are expected to appear only if the spring constant of the trap is large. In Fig.4 theoretical and experimental results for the evolution of the total population of trapped atoms at the resonant exciting frequency Ω = 2 ω0are shown. Al- though there is a satisfactory agreement between the model and the experiment, we notice that experimental data exhibit a diﬀerent rate for the loss of atoms be- fore and after 100ms. This change is probably due to a variation of the collision rate as the number of trapped atoms is modiﬁed, which cannot be easily included in the model. The comparison of the experimental evolution of the trap population with and without modulation shows the eﬀectiveness of the excitation process in emptying the trap on a short time-scale. We have also simulated the energy growth of the trapped atoms due to the parametric excitation, which is reported in Fig. 5. Our calculations show nonexponen- tial energy increase in contrast with what expected in the harmonic approximation, Eq. (3.7). The fast energy growth at short times is related to the depopulation of the lowest levels, which are resonant with the 2 ω0para- metric source. The saturation eﬀect observed for longer times is due to the fact that the resonance condition is not satisﬁed for the upper levels so that they do not de- populate easily. V. CONCLUSIONS. We have studied both theoretically and experimentally the time evolution of the population of atoms trapped in a 1D sinusoidal optical lattice, following a parametric excitation of the lattice vibrational mode. In detail, we 5have presented a theoretical model for the excitation in an anharmonic potential, which represents an extension of the previous harmonic models, and we have applied it to the actual sinusoidal potential used to trap cold potassium atoms. The simulation seems to reproduce relatively well the main features of both the spectrum of trap losses, including the appearance of resonances beyond 2 ω0, and the time evolution of the total number of trapped atoms. By comparing the theoretical predictions and the ex- perimental observations the usefulness of a parametric excitation procedure to characterize the spring constant of the trap has been veriﬁed. Although the loss reso- nances are red-shifted and wider than what expected in the harmonic case, the lattice harmonic frequency can be easily extracted from the experimental spectra, to esti- mate useful quantities such as the trap depth and spring constant. We have also made emphasis on the need of modeling the broadening of bands with a negligible natural width in order to reproduce the observed loss spectrum. In a harmonic model this broadening is not necessary since the equidistant energy spectrum guarantees that a sin- gle transition energy characterizes the excitation proces s. We think that most of the broadening in our speciﬁc ex- periment is due to the fact that the actual trapping po- tential is not one-dimensional, and also to possible ﬂuc- tuations and inhomogeneities of the lattice. To conclude, we note that the dynamical analysis we have made can be easily extended to lattices with larger dimensionality, and also to other potentials, such as Gaussian potentials, which are also commonly used for optical trapping. We acknowledge illuminating discussions with R. Brecha. This work was supported by the European Com- munity Council (ECC) under the Contracts HPRI-CT- 1999-00111 and HPRN-CT-2000-00125, and by MURST under the PRIN 1999 and PRIN 2000 Programs. [1] T. A. Savard, K. M. O’Hara, and J. E. Thomas, Phys. Rev.A56, R1095 (1997). [2] S. Friebel, C. D’Andrea, J. Walz, M. Weitz, and T. W. H¨ ansch, Phys. Rev. A 57, R20 (1998). [3] C. W. Gardiner, J. Ye, H. C. Nagerl, and H. J. Kimble, Phys. Rev. A61, 045801 (2000). [4] J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A47, R4567 (1993). [5] V. Vuletic, C. Chin, A. J. Kerman, and S. Chu, Phys. Rev. Lett. 81, 5768 (1998). [6] G. Roati, W. Jastrzebski, A. Simoni, G. Modugno, and M. Inguscio, to appear in Phys. Rev. A; e-print: arXiv physics/0010065. [7] V. Vuletic, A. J. Kerman, C. Chin, and S. Chu, Phys.Rev. Lett. 82, 1406 (1999). [8] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, Oxford, 1976). [9] We estimate the mean velocity of the atoms along the lattice from the energy width of the levels as ¯ v= ∆ω/2k. [10] R. J´ auregui, submitted for publication. [11]Handbook of Mathematical Functions , edited by M. Abramowitz and I. A. Stegun (Dover Publications, New York, 1965). r Er Er+1−Er σr 0 0.494 0.976 0.014 1 1.470 0.95 0.015 2 2.420 0.923 0.019 3 3.343 0.897 0.025 4 4.240 0.867 0.032 5 5.107 0.837 0.042 6 5.944 0.802 0.051 7 6.746 0.767 0.062 8 7.513 0.727 0.072 9 8.240 0.680 0.082 10 8.920 0.624 0.092 11 9.544 0.551 – 12 10.095 0.402 – 13 10.497 – – TABLE I. Energy spectrum in units of ¯ hω0obtained from the diagonalization of the Hamiltonian Eq. (2.4) for V0=10.5 ¯ hω0in a harmonic basis set with the lowest 40 functions. The third column shows the band widths σr, Eqs. (3.14)and (4.2), used in the numerical simulations re- ported in Section IV. FIG. 1. Absorption image of the atoms in the optical lat- tice, and shape of the optical potential in the two relevant directions. 601234400 800 1200 1600 20000.00.20.40.60.81.0 Frequency (kHz)Fraction of trapped atoms FIG. 2. Experimental spectrum of the losses associated to parametric excitation of the trap vibrational modes. For th e low and high frequency regions two diﬀerent modulation am- plitudes of 20% and 3% respectively, were used. 0.00.20.40.60.81.0 4 2 1 Frequency (ω0)Fraction of trapped atoms FIG. 3. Experimental (circles) and theoretical (lines) fra c- tion of atoms left in the trap after parametric excitation vs the modulation frequency. The continuous line corresponds to the numerical integration of the ﬁrst order perturbation theory equations (3.11) and the dashed line to the numerical integration of the ﬁnite diﬀerence second order perturbati on theory equations (3.8).0 50 100 150 2000.00.20.40.60.81.0 Fraction of trapped atoms Time (ms) FIG. 4. Theoretical (continuous line) and experimen- tal (triangles) results for the evolution of the population of trapped atoms at the resonant exciting frequency Ω = 2 ω0. The circles show the evolution of the population in absence of modulation. 0 50 100 150 200012345Average energy per atom (hω0) Time(ms) FIG. 5. Calculated evolution of the average energy of the trapped atoms during parametric excitation at Ω = 2 ω0. 7
arXiv:physics/0103047v1 [physics.med-ph] 16 Mar 2001Military use of depleted uranium: assessment of prolonged population exposure C. Giannardiaand D. Dominicib,c aFisica Ambientale, Dipartimento di Firenze, ARPAT c.giannardi@arpat.toscana.it bDipartimento di Fisica, Universit` a di Firenze cI.N.F.N., Sezione di Firenze dominici@ﬁ.infn.it Abstract This work is an exposure assessment for a population living i n an area contaminated by use of depleted uranium (DU) weapons . RESRAD 5.91 code is used to evaluate the average eﬀective dos e delivered from 1, 10, 20 cmdepths of contaminated soil, in a resi- dential farmer scenario. Critical pathway and group are ide ntiﬁed in soil inhalation or ingestion and children playing with the s oil, respec- tively. From available information on DU released on target ed sites, both critical and average exposure can leave to toxicologic al hazards; annual dose limit for population can be exceeded on short-te rm pe- riod (years) for soil inhalation. As a consequence, in targe ted sites cleaning up must be planned on the basis of measured concentr ation, when available, while special cautions have to be adopted al together to reduce unaware exposures, taking into account the amount of the avertable dose. 11 Introduction Munitions containing depleted uranium (DU) have been used b y NATO and US forces during the war operations in Iraq (1991), Bosnia (1 994), Kosovo and Serbia (1999). Recently some information on 112 sites ta rgeted by DU weapons in Kosovo has been supplied by NATO to the United Nati on En- vironmental Program Balkans Task Force (UNEP BTF); on Novem ber 2000 measurements to detect contamination have been undertaken by a UNEP team in 11 among the 112 sites. Aim of this paper is outlining some aspects of the exposure of people living in an area contaminated by DU, on the basis of oﬃcial av ailable infor- mation and of simulations, looking for main pathways of aver age and critical exposure. Individuation of pathways of high exposure could allow to ad vice to popu- lation; average dose assessment, together with measures of DU concentration in soil, will make delimitation of areas to be cleaned up poss ible. 2 Military use of depleted uranium The Gulf war against Iraq in 1991 was the ﬁrst one known where D U rounds have been used in large quantity (approximately 300 tonnes) [1, 2]. The consequences on the health of the Iraqi population and of the US veterans are still under study. DU exposure at the moment is not consid ered the most probable cause of the Gulf War Syndrome experienced by hundr eds thousand veterans [3]; on the other hand, the eﬀects of the DU left over the Iraqi territory are diﬃcult to show, due to the large number of toxi c substances dispersed in the environment during the war and the deterior ation of the sanitary situation caused by the embargo to which the country is submitted from 1991 (cited work in [4], app.3). Reports on potential eﬀects on human health and environment from the use of DU have appeared during the last years: studies on risk assessment for the Jeﬀerson Proving Ground, a US facility for testing DU munitions, have been performed [5]; the risk for population for the Koso vo conﬂict and for the Gulf war has been also considered [4, 6]. DU can be obtained as by-product in the enrichment process of natural uranium for the production of nuclear fuel and for military a pplications; as 2the ore extracted natural uranium, DU is associated to a redu ced chain of radioactive isotopes, formed by238Uand235Udecay products having shorter decay times:234Th(24 days),234mPa(1.17 min) and234Pa(6.7 hours),231 Th(25.5 hours). DU can also be obtained by the reprocessing o f nuclear power plant spent fuel, and so traces of transuranic element s and236Ucan be present. According to oﬃcial information, DU used by the U .S. Depart- ment of Defence contains approximately 0.2% of235Uand traces of234U, 236U. Following the indications in [7, 8], we will assume the uran ium iso- topic composition of DU given in Table 1. DU speciﬁc activity is in part due Table 1: Assumed depleted uranium composition. Aiis the speciﬁc isotopic activity, ADUis the activity concentration per mgof DU. % T1/2 Ai ADU (y)(Bq/mg )(Bq/mg ) 238U99.796 4.5 10912.4 12.375 235U 0.2 0.7 10980 0.160 234U0.001 2.5 1052.3 1052.300 236U0.003 2.3 1072.4 1030.072/summationtext U100 14.907 to uranium isotopes (14.9 Bq/mg , 36%), and for the residual part to beta emitting short-life decay products (64%); among the transu ranic elements oﬃcial information is available only for239Pu(2.4 104years), whose content is estimated in 11 ppb [9]. DU speciﬁc activity is not substan tially aﬀected by the declared amount of traces elements. Metallic uranium has a high density (19 g/cm3), is pyrophoric and cheaper than tungsten, and so has been attractive for U.S. Army for th e production of armor piercing ammunition since 1960s. Tungsten alloys hav e been preferred up 1973, when a DU alloy with 0.75% of titanium (U-3/4Ti) was a dopted for ammunition made by a thin cylinder in DU alloy encased with li ghter ma- terial. Systems of DU weapons are owned or under development in diﬀerent countries (Saudi Arabia, France, United Kingdom, Israel, P akistan, Russia, Thailand and Turkey) [8]. Use of DU ammunition causes exposure of people soon and after , because DU is dispersed as aerosol when the projectile strikes a hard target and 3then falls out on a limited area [7]. Contamination of all env ironmental matrices takes place and health eﬀects on people living near by must be taken in account, both for toxicological damage and for radiologi cal risk. Among diﬀerent isotopes present in DU as declared,238U,234Uand235Uare of concern in risk assessment. For chemical hazard, kidney is identiﬁe d as the target organ, whatever the path of assumption [10]. Due to prevalen t short-range emitted radiation, the risk associated with exposure to ion izing radiation mainly derives from ingestion and inhalation of radioactiv e material; external irradiation from soil is less relevant. 3 Dispersion of DU in the environment and exposure of the population DU contained in projectiles, spread out as aerosol in air aft er striking the tar- get, falls out producing environmental and food chain conta mination. Possi- ble occurring of chemical hazard and entity of radiation dos e must be assessed for people living in the area, taking into account both avera ge and critical group exposure. DU concentration in the soil is the starting point; while wai ting for mea- surements of contamination in Iraq, Bosnia, Kosovo and Serb ia, we present computed radiation doses and associated concentrations fo r diﬀerent con- taminated soil thickness, as soil mixing will extend the ini tial superﬁcial deposition to underlying layers in not undisturbed areas. A vailable soil mea- sured DU concentrations in contaminated sites that we are aw are of, are the following: •at Jeﬀerson Proving Ground area an average/summationtextUconcentration of 318 Bq/kg was reported [11]; more recently a lower and an upper bound of the concentration ranging from 592 Bq/kg to 13690 Bq/kg was also measured [5]; •among the areas where the US personnel lived in the Gulf regio n (out- side Iraq) the highest DU concentration (433 Bq/kg ) was measured in the Iraqi Tank Yard (the area where captured Iraqi equipment is stored in Kuwait) [12]; 4•in some sample analyzed by the RFY scientists a speciﬁc activ ity of 238Uup to 2 .35 105Bq/kg was detected [13]. Following the hypothesis assumed in the BTF report, we have a ssumed as a reference value a DU contamination of 1000 Bq/kg of soil over an area of A= 10000 m2, in the hypothesis of 10 kgof DU entirely dispersed in the impact as aerosol of uranium oxides, contaminating 1 cmof soil. With the composition given in Table 1 initial activities per kgof soil for238U,235U, 234Uand236Uare respectively 830 Bq, 11Bq, 154Bqand 5 Bq. Average eﬀective dose is conservatively assessed using the residential farmer scenario. The following pathways are considered: ex ternal irradia- tion from soil, inhalation from resuspended dust, ingestio n of contaminated soil and water, ingestion of plants and animal products grow n in site and ingestion of ﬁsh grown in a pond contaminated by groundwater . Diﬀerent pathways are considered for plant contamination due to ﬁrst root uptake (wa- ter independent) and due to secondary root uptake from use of contaminated water (water dependent). Radon inhalation is excluded. RES RAD 5.91 [14] code is used, all parameters default except for the ones give n in Table 2. Es- timates of dose to individuals and population for risk in con taminated sites have been performed by EPA employing primarily the code RESR AD (for related work see [15, 16]). RESRAD default libraries values have been corrected to give eﬀective dose [17] rather than equivalent eﬀective dose [18]: due to the al gorithm used by RESRAD, anyway, values for external irradiation EGin Tables 4 and 5 have been impossible to modify, and are approximate by 10% maximu m defect. In Tables 3, 4, 5 and 6 we show average annual eﬀective doses an d corre- sponding DU concentrations in water and vegetables for thre e diﬀerent soil thickness, respectively 1 cm, 10cmand 20 cm. The following quantities are given at diﬀerent times, from the ﬁrst year to about two hundr ed years after maximum dose, for main pathways: the total dose ( Etot), the dose from exter- nal irradiation from the soil ( EG), from inhalation of contaminated dust ( EI), from consumption of edible plants (water independent EP, water dependent Ew P) and of water ( EH2O). The dependence of tmaxandEmaxon some hydrogeological parameters, mainly aﬀecting the water dependent pathways, is shown in Ta ble 7 and 8. The maximum value of the dose is not much aﬀected by most of t he parameters considered in Table 8 except Kd. This parameter is deﬁned as 5Table 2: RESRAD parameters diﬀerent from the default value. this paper RESRAD def indoor time fraction 0.6 0.5 outdoor time fraction 0.2 0.25 exposure duration 50years 30years well pump intake depth 3m 10m drinking water intake 730l/y 510l/y the ratio of the mass of solute species observed in the solids per unit of dry mass of the soil to the solute concentration in the liquids. A wide range has been observed for uranium Kdvalues [19]. For largest value of Kdthe DU is retained in surface and does not reach at least within the ﬁrs t 1000 years the watertable. A measurement of the local value of this paramet er is therefore necessary to reduce the uncertainty on the dose assessment. Strong dependence of maximum inhalation dose has been found , as ex- pected, on the dust loading parameter, as shown in Table 9. As already outlined, presented doses and concentrations ha ve been ob- tained from an average value of soil contamination, in order to assess the average exposure of population. Whatever the average value considered, anyway, highly inhomogeneous soil concentrations must be e xpected in the contaminated area, both for sparse aerosol deposition and f or oxidation of DU fragments: concentrations up to 12% in weight have been re ported [20]. In order to assess the dose to critical population group, thi s must be taken in account, especially if inhalation of soil was the critical p athway: inhalation of 0.1 gof soil with maximum reported DU contamination, equal to 12 mg DU, corresponds to 1.44 mSv; ingestion of 1 gof soil, equal to 120 mgDU, corresponds to 0.08 mSv. A scenario, in which permanence in dusting air and ingestion of soil are possible, is the one for children playing with soil. From the presented dose assessment and considerations children playing with soil m ay be identiﬁed as the critical population group, with inhalation and/or inge stion of contami- nated soil as critical pathway. Evidently, average and crit ical doses are some- how competitive, because the higher fraction of DU is disper sed as aerosol, the lower part of it can rest in soil as fragment, being presen ce of fragments 6Table 3: Eﬀective doses ( µSv) for contaminated soil thickness 1cm.Etot, EG,EI,EP,EH2O,Ew Pare the total dose, the ground, inhalation, plant (water independent), water, plant (water independent) dos es. The initial contamination is assumed of 1000Bq/kg over an area of A= 10000 m2. The symbol - means doses less than 1 µSv. All not speciﬁed parameters as in Table 2. t(y)EmaxEGEIEPEH2OEw P 0 4 4-- - - 1 3 3-- - - 3 - --- - - 300 - --- - - 485 4 --- 4 - 500 4 --- 4 - 700 - --- - - the main cause of hot spots in soil contamination. It must be outlined that the amount of DU considered in the sim ulation corresponds to 37 A-10 /GAU-8 ammunitions. According to the available information, a much larger number of projectiles has been ﬁr ed on each site (between 50 and 2320, average 300) and up to now unknown is the extension of targeted sites. Both for average and critical exposure, a nyway, more realis- tic dose assessment will be possible only when measured cont amination data will be known, scaling the values in the tables for the approp riate factor. Increment of inhalation dose attributable to239Pupresence in DU is oﬃcially estimated in 14% [9]: with 11 ppb of239Puin DU RESRAD gives a maximum dose increment of 0.6%. 4 Normative and recommendations framework Before discussing compliance of average assessed doses and exposure with international standards set to prevent from toxicological damage and limit ionizing radiation risk, we shortly line out an aspect relat ive to radioprotec- tion system, maybe useful even in wider considerations on ri sk. 7Table 4: Eﬀective doses ( µSv) for contaminated soil thickness 10cm. All not speciﬁed parameters as in Table 2. t(y)EmaxEGEIEPEH2OEw P 0 18 15 -1 - - 1 17 14 -1 - - 3 15 12 -1 - - 10 9 8-1 - - 30 2 2-- - - 100 - --- - - 300 - --- - - 486 44 --- 41 2 500 44 --- 41 2 700 - --- - - Due to accepted linear-no-threshold model for eﬀects produ ced by ioniz- ing radiation, justiﬁcation of a practise has to be the ﬁrst o ne posed, that is if the population exposure from military use of DU is justiﬁe d or not. Com- parison between dose estimates in such a scenario and dose li mits and dose constraints stated by regulations is anyway useful, for a qu antitative per- ception of risk. In order to assess the need for remediation i n contaminated areas, once again the question of justiﬁcation has to be cons idered; speciﬁc reference levels , linked to the avertable annual dose, have to be deﬁned by national authorities. ” Generic reference levels ... should be used with great caution” and their use ”should not prevent protective actio ns from being taken to reduce ... dominant components [of existing annual dose]” [21]. We next report a comment to assessed doses, comparing them with radiological and toxicological reference values, in order not to hold the question narrowed to exceeding of dose limits. Values of annual dose in Tables 3, 4 and 5 show the same tempora l shape, with an initial prevalent dose from irradiation by soil and a maximum from ingestion of contaminated drinking water occurring after a bout ﬁve hundreds years, when contamination reaches the acquifer serving the population. Max- imum dose, progressively increasing as inventary of DU incr eases, is always lower than annual population limit (1 mSv/y ), starts to be comparable with 8Table 5: Eﬀective doses ( µSv) for contaminated soil thickness 20cm. All not speciﬁed parameters as in Table 2. t(y)EmaxEGEIEPEH2OEw P 0 24 19 12 - - 1 23 18 12 - - 3 21 17 12 - - 10 17 13 -2 - - 30 8 7-1 - - 100 1 1-- - - 300 - --- - - 499 87 --- 82 4 700 1 --- 1 - Table 6: DU concentrations in the water CH2Oand in the edible plants (water dependent) Cw Pat the maximum dose time. CH2O(Bq/l)Cw P(Bq/kg ) 1cm 1.11 1.48 10cm 1.15 1.85 20cm 2.25 3.74 EPA cleanup limit criterion (150 µSv/y , [22]) for 20 cmdepth. Exceeding of dose constraint of 0 .1mSv/y indicated in [21] for longlived isotopes may not be excluded. This in general happen only after long times, du e to the low mobility of the uranium oxides (the mean transit times for in soluble uranium in the top 10 cm of soil range from 7.4 to 15.4 years with an aver age of 13.4 years [23]; soluble forms have a mean transit times of one mon th). At the maximum dose time concentration of DU in the water reaches th e provisional value of WHO guideline for drinkable water (0.05 Bq/l [24]) already for 1 cmdepth. The concentration of DU in leafy vegetables at time of maximum dose ranges from 2 to 4 Bq/kg ; no derived limit is deﬁned for consumption of dietary parts. 9Table 7: Eﬀective doses for contaminated soil thickness 10cm, unsaturated zone thickness 3.90 m, for diﬀerent values of the well pump intake depth (WPID). ( CH2OandCw Pare the concentrations of DU in the water and in the plants (water dep)). All not speciﬁed parameters as in Ta ble 2. WPID( m)tmax(y)Emax(µSv)CH2O(Bq/l)Cw P(Bq/kg ) 1 398 103 2.7 4.5 2 417 65 1.7 2.8 4 564 33 0.7 1.4 Table 8: Contaminated soil thickness 10cm. All not speciﬁed parameters as in Table 2. tmax(y)Emax(µSv) prec.rate (0 .9−1.1)m 537−435 43.0−44.1 watershed area (106±105)m2486 43.6 well pumping rate (200 −300)m3/y 486 43.6 distrib.coeﬀ. Kd(20−100)cm3/g 215−0 118−19 Inhalation of highly contaminated soil may leave to exceedi ng of annual dose limit, with possible occurring of toxicological damag e: maximum al- lowed concentration in air for workplaces stated by NRC, 45 µg/m3for sol- uble and 200 µg/m3for insoluble uranium forms, would be exceeded if dust loading was more than 1700 µg/m3, a high but not extreme value. Less im- portant seems ingestion of contaminated soil, due to the low er value of dose conversion factor with respect to the inhalation one. Anywa y, ingestion of 1 gmaximum contaminated soil would result in 120 mgDU ingestion, when maximum daily ingestion of uranium, due to toxicological eﬀ ects, was stated in 150 mgby italian legislation till year 2000. 10Table 9: Average dose from inhalation at t= 0for diﬀerent values of the dust loading parameter. Contaminated soil thickness 10cm. All not speciﬁed parameters as in Table 2. 100µg/m31mg/m35mg/m3 inhal. dose ( µSv) - 3 16 5 Conclusions DU contained in projectiles, spread out in air after strikin g the target, falls out producing environmental and food chain contamination. Possible occur- ring of chemical hazard and entity of radiation dose must be a ssessed for diﬀerent kind of exposure of people living in the area, takin g into account both average and critical group exposure. While waiting for measurements of contamination in Iraq, Bosnia, Kosovo and Serbia, we have computed ra- diation doses and concentrations for diﬀerent contaminate d soil thickness, as soil mixing will extend the initial superﬁcial deposition t o underlying layers in not undisturbed areas. In order to assess the average exposure of population, doses and con- centrations have been obtained from an average value of soil contamination. For the individuation of the critical group inhomogeneous s oil concentration has been considered. The presented dose assessment suggest s a short term exposure due to inhalation and/or ingestion of contaminate d soil and a long term exposure due to ingestion of contaminated water and foo d; the propa- gation of the superﬁcial contamination to the watertable cr itically depends on various hydrogeological parameters to be evaluated on th e site. In sites targeted by DU munitions special cautions have to be adopted to reduce unaware exposures and cleanup must be planned on the b asis of the measured concentrations. References [1] Department of Defense, Exposure Report - August 4, 1998, En- vironmental Exposure Report Depleted Uranium in the Gulf, 11http://www.gulﬂink.osd.mil/du [2] Tabella F in [1] http://www.gulﬂink.osd.mil/du/du tabf.htm [3] M.J. Hodgson and H. M. Hansen, JOEM, 41(1999) 443 [4] UNEP/UNCHS, Balkan Task Force, The Potential Eﬀects on H uman Health and Environment Arising from Possible Use of Deplete d Ura- nium DUring the 1999 Kosovo Conﬂict. A preliminary Assessme nt. October 1999 http://balkans.unep.ch/ ﬁles/du ﬁnalreport.pdf [5] M. H. Ebinger, LA-UR-98-5053 http://lib-www.lanl.gov/la-pubs/00418777.pdf [6] S. Fetter and F. von Hippel, Science and Global Security 8(1999) 125 [7] Health and Environmental Consequences of Depleted Uran ium Use in the U.S.Army: Technical Report . Army Environmental Policy Insti- tute, Atlanta, Georgia 1995 [8] N. H. Harley, E. C. Foulkes, L. H. Hilborne, A. Hudson and C . R. Anthony, Rand Report, A Review of the Scientiﬁc Literature A s it Pertains to Gulf War Illness, vol. 7, Depleted Uranium, RAND [9] Letter of DOE to Ms. Thornton, 20/1/2000, http://www.miltoxproj.org/DU/DOE.pdf [10] BeirV, Health Eﬀects of Exposure to Low Levels of Ionizi ng Radiation, National Research Council, National Academy Press, Washin gton, DC, 1990 [11] M. H. Ebinger and W. R. Hansen, LA-UR-94-1809, april 199 4 [12] Environmental Exposure Report, Depleted Uranium in th e Gulf (II), http://www.gulﬂink.osd.mil/du ii/ [13] Federal Ministry for Development Science and Environm ent, FR Yu- goslavia Report, Belgrade, February 2000 12[14] RESRAD, Yu et al [15] A.B. Wolbarst et al, Health Phys. 71(1996) 644 [16] J.L. Wood et al, Health Phys. 76(1999) 413 [17] ICRP Publication 60, 1990 [18] ICRP Publication 26, 1977 [19] M.I. Sheppard and D.H. Thibauld, Health Phys. 59(1990) 471 [20] M. H. Ebinger et al., LA-11790-MS, june 1990 [21] ICRP Publication 82, 2000 [22] U.S. Environmental Protection Agency. Memorandum: Es tablishment of cleanup levels for CERCLA sites with radioactive contami nation. Washington,D.C.: U.S.EPA; OSWER 9200.4-18;1997- [23] G. G. Killogh, et al., J. of Environmental Radioactivity 45(1999) 95. [24] World Health Organization: Guidelines for Drinking-w ater Quality, Second edition, Addendum to Volume 2: Health Criteria and Ot her Supporting Information, WHO/EOS/98.1, Geneva 1998. http://www.who.int/water sanitation health/GDWQ/Chemicals/uraniumfull.htm 13
arXiv:physics/0103048v1 [physics.data-an] 16 Mar 2001Strange Attractors in Multipath propagation: Detection an d characterisation. C. Tannous∗and R. Davies Alberta Government Telephones Calgary, Alberta, Canada T2 G 4Y5 A. Angus NovAtel Communications Calgary, Alberta, Canada T2E 7V8 (Dated: March 16, 2001) Abstract Multipath propagation of radio waves in indoor/outdoor env ironments shows a highly irregular behavior as a function of time. Typical modeling of this phen omenon assumes the received signal is a stochastic process composed of the superposition of var ious altered replicas of the transmitted one, their amplitudes and phases being drawn from speciﬁc pr obability densities. We set out to explore the hypothesis of the presence of deterministic cha os in signals propagating inside various buildings at the University of Calgary. The correlation dim ension versus embedding dimension saturates to a value between 3 and 4 for various antenna polar izations. The full Liapunov spectrum calculated contains two positive exponents and yields thro ugh the Kaplan-Yorke conjecture the same dimension obtained from the correlation sum. The prese nce of strange attractors in multipath propagation hints to better ways to predict the behaviour of the signal and better methods to counter the eﬀects of interference. The use of Neural Networ ks in non linear prediction will be illustrated in an example and potential applications of sam e will be highlighted. ∗Electronic address: tannous@univ-brest.fr; Present addr ess: Laboratoire de Magntisme de Bretagne, UP- RES A CNRS 6135, Universit de Bretagne Occidentale, BP: 809 B rest CEDEX, 29285 FRANCE 1I. INTRODUCTION Multipath propagation of radio waves in indoor or outdoor en vironments shows a highly irregular behavior as a function of time [1]. The characteri zation of radio channels in mobile or in building propagation is important for addressing issu es of design, coding, modulation and equalization techniques tailored speciﬁcally to comba t time and frequency dispersion eﬀects. Irregular behavior of the received signal has prompted rese archers in the past to model the channel with stochastic processes. One of the earliest line ar models in this vein is the Turin et al. model [2] in which the impulse response of the channel i s written as a superposition of replicas of the transmitted signal delayed and having alt ered amplitudes and phases. A number of models exist diﬀering in the numbers of replicas o f the signal or in the type of probability distributions from which the amplitudes and phases are drawn. Also, diﬀer- ent stochastic processes are used in the generation of delay times of received replicas. The popular choice for the amplitude probability density funct ions (PDF) are Rayleigh or Rice PDFs depending on whether a weak or strong line of sight propa gation exists; nevertheless other PDFs have been used such the lognormal, Nakagami-m or u niform. The phases ought to be drawn from PDFs compatible with the ones selected for th e corresponding ampli- tudes; nevertheless the most popular choice found in the lit erature is the uniform [0 −2π] distribution. Delay times are usually extracted from eithe r stationary of stationary Poisson processes although in some cases the Weibull PDF is used. II. A HYPOTHESIS OF DETERMINISTIC CHAOS Although an assumption of stochastic behavior in mobile or i ndoor propagation is ubiqui- tous, in the present work we set out to explore the hypothesis that the indoor communication channel displays deterministically chaotic behavior. Thi s question is important in many re- spects and the tools to answer it readily exist. These tools a re based on the determination of the correlation dimension of the strange attractor assoc iated with the multipath proﬁle considered as a real valued time-series x(t). Using delay co ordinates [3] one forms the m- dimensional delay vector X(t) = [x(t), x(t−T)...x(t−(m−1)T] with delay Tand computes the correlation sum C(r) which is the ratio of the number of pa irs of delay vectors (the 2distance between which is less than r) to the total number of p airs. From this, the corre- lation dimension νis deﬁned as the logarithmic slope of C(r) versus r for small r . For a true stochastic process, νincreases with m without showing any saturation. In contras t, for deterministic chaos, νsaturates at a value, the next integer greater than which, re p- resent the minimum number of non-linear recursion or diﬀere ntial equations from which it originates. If the proﬁle turns out to be deterministic, t he modulation, coding and de- tection/demodulation techniques ought to be adapted accor dingly in order to account for this fact; otherwise one has to rely upon techniques capable of handling stochastic signals. Let us illustrate this by an analysis of multipath measureme nts we have made in an indoor environment. III. EXPERIMENT AND CORRELATION DIMENSION ANALYSIS The propagation environment from which the data are collect ed are hallways in the En- gineering Building at The University of Calgary [4]. The tra nsmitted power was 10 dBm fed into a half wave dipole antenna with a matching balun. The receiving antenna was a cross-polarized dipole array. The co-polarized antennas ( CPA) and crosspolarized antennas (XPA) proﬁles referred to in Figure 1 are from a single measur ement run and points from both proﬁles were obtained in coincident pairs. The terms CP A and XPA simply refer to the relative state of polarisation between the transmit and receive antennas. The receiving hardware was specially developed to measure diversity char acteristics and gives an accu- rate reference between distance from transmitting to recei ving antennas and received signal strength. The same measurement procedure was employed as fo r the arbitrarily polarized data set. We have estimated the correlation dimension for th e sets of data: Arbitrarily Polarized Antennas (APA, 6000 data points), co-polarized ( CPA, 3800 points) and Cross- polarized antennas (XPA, 3800 points) by the box-counting m ethod of Grassberger and Procaccia [3]. There are a number of limitations and potenti al pitfalls with correlation dimension estimation that have been discussed by various au thors [5]. In addition, the num- ber of operations it takes to estimate C(r) is O(N2) where Nis the number of collected experimental points. Recently Theiler [5] devised a powerf ul box-assisted correlation sum algorithm based on a lexicographical ordering of the boxes c overing the attractor, reducing the number of operations to O(Nlog(N)) and incorporating several test procedures aimed 3at avoiding the previous pitfalls. Before we used Theiler’s algorithm, we made some pre- liminary tests against well known cases. We generated unifo rm random numbers, Gaussian random numbers, and numbers z(n) according to the logistic o ne-dimensional map at the onset of Chaos: z(n+1) = a z(n)(1-z(n)) with a = 3.5699456 and in the fully developed Chaotic regime at a=4. In the ﬁrst two cases we found νapproximately equal to m as expected in purely stochastic series whereas νsaturated respectively at 0.48 and 0.98 (we used 2000 points only) for the logistic map indicating the pr esence of a low dimensional attractor and deterministic Chaos (the exact correlation d imensions for the logistic map is 0.5 at a=3.5699456 and 1 for a=4.). We tested as well the H´ eno n two dimensional map, the Lorenz three dimensional system of non-linear diﬀerent ial equations, the R¨ ossler three and four dimensional systems as well as an inﬁnite dimension al system, the delay-diﬀerential Mackey-Glass equation whose attractor dimension is tunabl e with the delay time. The re- sults we found for the various correlation dimensions agree d with all the results known in the literature to within a few percents. Then we went ahead and ex amined the νvs. m curves for the three sets of experimental data along with a set of 600 0 Rayleigh and band-limited Rayleigh distributed numbers which constitute prototypic received envelopes. Our results, in Figure 1 show that, while ν∼mfor the pure Rayleigh case (with a slope equal to one), andν∝mfor the band limited Rayleigh case (with a slope smaller than one) the νvs. m curves for the three examined experimental sets of data sta rt linearly with m then show saturation indicating the presence of a low dimensional att ractor (whose dimension is about 4 for CPA and XPA data whereas it is slightly above 4 for the APA situation). This ﬁnding is in line with the Ruelle criterion [6] that sets an upper bou nd on the possible correlation dimension one can get from any algorithm of the Grassberger- Procaccia type. This upper bound is set by the available number of data points Nin the time series. The dimension that can be detected should be much smaller than 2 log10(N). Since we used 3800 and 6000 points respectively, the upper bound for the detectable cor relation dimension in our case is about 7.16 to 7.56. We respect this bound since our correla tion dimensions are around 4. Nevertheless the presence of Chaos is going to be conﬁrmed through another route, the spectrum of the Liapunov exponents that will be discusssed n ext. 4IV. SPECTRUM OF THE LIAPUNOV EXPONENTS The spectrum of Liapunov exponents is very important in the s tudy of dynamical systems. If the largest exponent is positive, this is a very strong ind ication for the presence of Chaos in the time series originating from the dynamical system. The r eciprocal of this exponent is the average prediction time of the series and the sum of all the po sitive exponents (if more than one is detected like in hyper-chaotic systems such as the R¨ o ssler four dimensional system of non-linear diﬀerential equations or the large delay Mackey -Glass equation) is the Kolmogorov entropy rate of the system. The latter gives a quantitative i dea about the information processes going on in the dynamical system. In addition, wit h the Kaplan-Yorke conjecture, the full spectrum gives the Hausdorﬀ dimension of the strang e attractor governing the long time evolution of the dynamical system. We have calculated t he Liapunov exponents of the data with four diﬀerent methods. Firstly, we determined the largest exponent λmax from the exponential separation of initially close points o n the attractor and averaging over several thousand iterations. Second, we determined the lar gest Liapunov exponent from the correlation sum with the help of the relation C(r)∼rνexp(−mTλ max) valid for large values of the embedding dimension mand small values of r. Finally, we determined all exponents with two diﬀerent methods: the Eckmann et al. method [7] and t he Brown et al.’s [8]. Our results for the spectrum of exponents is shown in ﬁgures 2 and 3. We tried several embedding delay times T and several approximation degrees for the tang ent mapping polynomial (as allowed in the Brown et al. [8] algorithm ) without observing major changes in the spectrum. Several time series (Logistic map, H´ enon, Lorenz, R¨ ossle r and Mackey-Glass) were tested for the sake of comparison to results obtained with the exper imental data. In addition, the Liapunov exponents saturate smoothly as they should for lar ge embedding dimension. Then we applied the Kaplan-Yorke conjecture to get the dimension of the attractor: Using the following typical numbers we obtained for the exponents λmax=λ1= 18.06, λ2= 1.88, λ3= −8.85, λ4=−24.94, λ5=−68.80 and using the formula: D=j+/summationtext iλi \|λj+1\|(1) where the summation is over i=1,2...j. The λi’s are ordered in a way such that they decrease as i increases. We determine j from the conditions/summationtext iλi>0 and: λj+1+/summationtext iλi<0. We get j=3 and D=3.44 for the strange attractor dimension (ca lled its Liapunov dimension). 5The total sum of the Liapunov exponents is negative ( equal to -82.65) as it should be for dissipative systems with a strange attractor. The value of t he attractor dimension will be conﬁrmed from the spectrum of singularities or the mutifrac tal spectrum in the next section. V. MULTIFRACTAL SPECTRUM The generalized dimension may be used to characterize non-u niform fractals for which there are diﬀerent scaling exponents for diﬀerent aspects o f the fractal, so-called multifrac- tals. For these, there are two scaling exponents, one genera lly called τ, for the support of the fractal, and one called q, for the measure of bulk of the fract al. In general, τ(q) = (q−1)Dq, where Dqis the generalized dimension. Multifractals have been empl oyed to characterize multiplicative random processes, turbulence, electrical discharge, diﬀusion-limited aggrega- tion, and viscous ﬁngering [9]. Multifractals have this in c ommon: there is a non-uniform measure (growth rate, probability, mass) on a fractal suppo rt. Besides the exponents, τand q, and the generalized dimension Dq, there is another method for characterizing multifrac- tals. This depends upon the use of the mass exponent α, and the multifractal spectrum, f(α) [9]. A graph of the multifractal spectrum explicitly shows the fractal dimension, f, of the points in the fractal with the mass exponent (or scaling i ndex), α. We have estimated Dqby use of the generalised moments of the correlation sum with a window chosen carefully enough to avoid temporal correlation eﬀects. We have develo ped a program that computes the generalized correlation sum using a box-assisted metho d. Our program is based on one written by Theiler [5]. Several modiﬁcations had to be made t o the straightforward box- assisted correlation sum method. In addition, our program a llows for logarithmic scaling with the distance parameter r. From a log-log graph of the gen eralized correlation sums, appropriate scaling regions can be identiﬁed, for each orde r, q. Least-squares ﬁts to these scaling regions yields a sequence of generalized correlati on dimensions, Dq, for values of q between ±∞. We have found computation of Dqfor integers in the interval [-10,10] and D−∞andD+∞to be suﬃcient. From the Dq, we calculate the τ(q) = (q−1)Dq. We then perform a Legendre transform to obtain the f(α) curve. We do this by ﬁrst ﬁtting a smooth curve (a hyperbola was considered to be adequate) to the τ(q) curve. With an analytic expression for the τ(q) curve, we can compute the Legendre transform in closed form . The domain of f(α) may be found from D−∞andD+∞; we assume that αis conﬁned to this 6region, and that f(α) is 0 at these points. The values of f(α) forD−∞≤α≤D+∞are cal- culated as min[ qα−τ(q)], the minimum being taken over q. We found that this procedu re, although complex, corrects for the known numerical sensiti vities of the Legendre transform. We checked that our method for obtaining f(α) gave the same results as those found in the literature for the Logistic map and the strongly dissipa tive circle map [9]. The f(α) curve for the multipath data is shown in Figure 4. It may be see n that the peak value of f(α), corresponding to the box-counting dimension at q=0, is ab out 3.7. This is consistent with our above ﬁndings from the correlation dimension and th e full spectrum of Liapunov exponents. Our further research in this area concerns the pr ediction of the received signal intensity, from our above hypothesis of the presence of dete rministic chaos. VI. NON LINEAR PREDICTION We applied the above ﬁndings to the non linear prediction of m ultipath proﬁles consider- ing that each point on the envelope of the measured signal is a function of some number of past points in the series, deviating from the traditional wa ve superposition approach. More precisely, we write: y(n+ 1) = F[y(n), y(n−1), y(n−2), y(n−3)...y(n−m+ 1)] (2) with F an mdimensional map and y(n) the value of the signal x(t) sampled at timestep n. Expressing F as a sum of sigmoidal and linear functions [10 ] we determine the unknown weights through the Marquardt least squares minimisation m ethod [11] in order to achieve the best ﬁt to the data. Our results comparing the onestep pre diction to the multipath data analysed above are displayed in ﬁgure 5. The goodness of ﬁt between the measured and predicted envelopes for a map dimension m=5 is another in dication of the soundness of the approach. This is conﬁrmed in Figure 6 where we display the normalised prediction error versus the embedding dimension. A minimum is observed in the prediction error for an embedding equal to 5 or 6. In Figure 6, we started always fro m the same initial weights and let the system run through 150 iterations searching for t he least squares minimum for 200 data points and later on making 300 one step ahead pred ictions. For embedding dimension larger or equal to 7 the minimisation procedure st opped because of the presence of zero-pivot in the least-squares matrices. The presence o f a minimum in the prediction 7error around an embedding equal to 5 or 6 complies again with t he value of embedding dimension used previously in the correlation dimension ana lysis, the Liapunov spectrum and the Kaplan-Yorke conjecture. VII. DISCUSSION We stress that although the proﬁles we examined were found to be chaotic in all three experimental conﬁgurations with conﬁrmations from the Lia punov spectrum and non linear prediction studies, indicating that we would be able to desc ribe our data with a set of at most 5 non-linear diﬀerential or algebraic equations, inve stigation in other propagation situations is needed. Nevertheless, in our investigations we observed a signiﬁcant amount of consistency between the various methods of detecting Cha os and characterising it using embedding dimensions mbeyond the minimum mminrequired by the Takens theorem (mmin>2d+ 1, where d is the dimension of the strange attractor). Assum ing the hypothesis of the presence of Chaos in a given multipath proﬁ le is ﬁrmly established, many avenues become possible. For instance, one might consider d evising ways for controlling the signal propagation by altering slightly some accessibl e system parameter and improving the performance characteristics of the channel [12]. Shaw [ 13] has introduced the concept that a deterministically chaotic system can generate entro py. The consequences of this observation is important for the design of communication eq uipment when the channel is a chaotic system. For one, it implies that information at the receiver about the state of the channel is lost at a mean rate given by the Kolmogorov entr opy. For another it implies that a channel estimator should be adapted to the mathematic al nature of the set of non linear equations describing the channel as shown in the prev ious paragraph. Our studies up to this date have shown that this approach is valid in an ind oor situation but not in an outdoor one. This might be due to the conﬁned geometry one enc ounters inside buildings and the boundary conditions for the electromagnetic ﬁelds l eading to a low dimensional system of non linear equations giving birth to the observed c haotic behaviour. Our studies in this direction are in progress. Acknowledgements 8We thank James Theiler, Jean-Pierre Eckmann and Reggie Brow n for sending us their computer programs and correspondance, as well as Halbert Wh ite for some unpublished material. C.T. thanks Sunit Lohtia and Bin Tan for their frie ndly help with the manuscript. [1] A.A.M. Saleh and R. A.Valenzuela: ”A Statistical Model F or Indoor Multipath Propagation”, IEEESAC-5 , 138 (1987). [2] G. L.Turin, F. D. Clapp, T. L. Johnston, S. B. Fine and D. La vry: ”A Statistical Model of Urban Multipath Propagation”, IEEE VT-21 , 1 (1972). [3] P. Grassberger and I. Procaccia: ”Characterisation of S trange Attractors ”, Phys. Rev. Lett. 50, 346 (1983). [4] R. J. Davies: ”In-Building UHF Propagation Studies”, MS c Thesis, University of Calgary (1989) Unpublished. [5] J. Theiler: ”Eﬃcient algorithm for estimating the corre lation dimension from a set of discrete points”, Phys. Rev. A36, 4456 (1987). [6] D. Ruelle: ”Deterministic Chaos: the science and the ﬁct ion”, Proc. R. Soc. London A427 , 241 (1990). [7] J-P Eckmann, S. Oliﬀson Kamphorst, D. Ruelle and S. Cilib erto: ”Liapunov exponents from time series”, Phys. Rev. A34, 4971 (1986). [8] R. Brown, P. Bryant and H.D.I. Abarbanel:” Computing the Liapunov spectrum of a dynam- ical system from observed time series’ Phys. Rev. A43, 2787 (1991). [9] G. Paladin and A. Vulpiani: ”Anomalous scaling laws in mu ltifractal objects”, Phys. Rep. 156, 141 (1987). [10] H. White: ”Some asymptotic results for learning in sing le hidden-layer feedforward network models” J. Am. Stat Association 84, 1003 (1989). [11] D. W. Marquardt: ” An algorithm for least squares estima tion of non-linear parameters”, J. Soc. Ind. App. Math. 11, 431 (1963). [12] E. Ott, C. Grebogi and J. A. Yorke: ”Controlling Chaos”, Phys. Rev. Lett. 64, 1196 (1990). [13] R. S. Shaw: ”Strange attracctors, Chaotic behaviour an d information ﬂow”, Z. Naturforsch 36a, 80 (1981). 9Figure Captions Fig. 1: Correlation dimensions vs embedding dimension: Ful l squares are for Rayleigh dis- tributed points; full triangles are for handlimited Raylei gh distributed points. Full diamonds are for experimental results in the XPA case wherea s open diamonds corre- spond to the CPA case and open squares to the APA case. Fig. 2: Liapunov exponent spectrum from Eckmann et al. [7] me thod versus embedding di- mension for the APA data (since APA data consist of the larges t number of points, 6000). Fig. 3: Liapunov exponent spectrum from Brown et al. method [ 8] for the same data as those of Fig.2. A linear tangent mapping is used to ﬁt the dyna mics. A very similar spectrum is obtained for a second order polynomial. Fig. 4: Spectrum of generalised dimensions f(α) versus a for the APA data used in ﬁg.2. The value at the maximum of f(α) corresponding to the Hausdorﬀ dimension of the strange attractor agrees with the minimum bound obtained fr om ﬁg.1 and with the Kaplan-Yorke conjecture (see text). The spectrum is obtain ed through embedding in 10 dimensions. This happens to be enough, given the values ob tained for the various generalised dimensions. Fig. 5: Measured envelope (APA data used in Fig.2 continuous curve) and its one step predic- tion (dashed curve) from the Neural Network ﬁt to the ﬁve dime nsional map F (eq.2). The training is over the ﬁrst 200 ﬁrst points. Fig. 6: Normalised prediction error versus embedding. Star ting from the same initial weights, we trained the neural network, for a given embedding, over th e ﬁrst 200 points with a Marquardt minimisation standard deviation parameter equa l to 0.01 and total number of 150 iterations. Once, the parameters at the minimum error are found we made a one-step ahead prediction over the next 300 points and calcu lated the resulting squared error divided by the total of points. One sees a minimum for an embedding dimension around 5 or 6. For an embedding equal to 7 or larger, a large err or or no convergence (null pivot encountered in the least square error matrices) were observed. 10corelation.dim.XLC Page 1Correlation Dimension versus Embedding 024681012 0 1 2 3 4 5 6 7 8 9 10 11 12 mννbob.seq.liapeck.XLs Page 1-80-60-40-20020406080 1 2 3 4 5 6 7 8 m-1λbobnorm.liapl.1.XLC Page 1Liapunov Exponents of bobnorm.seq ( Brown method) -40-20020406080100 1 2 3 4 5 6 7 8 9 10 mλ2 3 4 5 6 701234Multifractal spectrum for APA data Bf α00.511.522.5 050100150200250300350400450500 Sample numberAPA envelope Predicted 0.010.020.030.04 1 2 3 4 5 6 7Prediction Error m
arXiv:physics/0103049v1 [physics.class-ph] 18 Mar 2001OBLIQUE SURFACE WAVES ON A PAIR OF PLANAR PERIODIC SLOTTED WAVEGUIDES. C. Tannous∗ TRLabs, Suite 108, 15 Innovation Boulevard Saskatoon SK, S7 N 2X8, Canada R. Lahlou and M. Amram Dpartement de Gnie physique, Ecole Polytechnique de Montra l C.P. 6079, Succursale A, Montral, PQ, H3C 3A7, Canada (Dated: March 16, 2001) Abstract The dispersion relation and mode amplitudes of oblique surf ace waves propagating on an acoustic double comb ﬁlter are obtained with a method based on the calc ulus of residues. We obtain a better agreement (below 480 Hz) between theoretical predictions a nd measurements reported previously when the ﬁlter was being supposed to be made of a single comb st ructure. ∗Electronic address: tannous@univ-brest.fr; Present addr ess: Laboratoire de Magntisme de Bretagne, UP- RES A CNRS 6135, Universit de Bretagne Occidentale, BP: 809 B rest CEDEX, 29285 FRANCE 1I. INTRODUCTION The behavior of a slow wave ﬁlter made of a pair of planar perio dic waveguides subjected to low frequency acoustic waves incident upon the aperture s eparating the waveguides has been investigated theoretically and experimentally for it s potential use in acoustic ﬁltering devices [1]. Each waveguide has a comb structure consisting of a periodic array of blades perpendicular to a base plane (Figure 1). Using a mathematical model borrowed from the study of electr ical ﬁlters, a ﬁlter having the same geometric structure of a single comb waveguide has b een analyzed previously [1]. The dispersion relation, amplitude and phase as functions o f frequency and wave number were derived and compared to experiment. In this work, we ext end our previous theoretical results and consider the actual nature of the ﬁlter consisti ng of the two waveguides facing each other. We derive the dispersion relation and reﬂection (transmission) coeﬃcients of surface waves propagating along any oblique wave number in t he plane parallel to the comb structure base planes. Our calculations are based on a weak-coupling approximatio n and in the limit of large distance separating the two structures. This means th e separation is much larger than the inter-blade distance. The blades are supposed to ha ve a vanishingly small thickness and we neglect possible reﬂections from the plana r base aﬀecting the propa- gating modes, by direct analogy with the electromagnetic ca se [2]. This is equivalent to assuming a slot depth large with respect to the inverse lowes t attenuation of the structure [2]. Our work is organized as follows: In section II, we discuss th e geometry, propagating modes dispersion and amplitude relation for the surface wav es. Section III covers the com- parison with the experimental results and the conclusion is in Section IV. II. DISPERSION RELATION, MODES AND AMPLITUDES Periodic arrays of slotted waveguides stacked to form a rect angular [3] or prismatic [4] structure are good candidates for reducing environment al noise (0.1 to 2 kHz). Their 2properties have been analyzed theoretically and experimen tally [1, 3, 4] such as their reﬂection scattering of sound waves harmful to the general p opulation living near highways or other sources of damaging sources of low frequency noise. It is important to understand how these structures absorb, reﬂect, transmit or phase dela y the incoming sound waves reaching them with arbitrary time dependent angles. For the rectangular structure, we have already undertaken such study from the experimental po int of view as well as from the theoretical one. In this work, we set out to investigate a new type of structure introduced in detail in Ref. 1 theoretically and experimentally (Fig. 1 ). We have studied the dispersion relation of acoustic waves im pinging on the structure at an arbitrary ﬁxed angle in the base plane, and measured the soun d reﬂection and transmission with respect to the incident angle. Our prior theoretical in vestigation took account of a single comb structure only. Here we extend it and deal with a s ymmetrical weakly- coupled double comb structure [5] in the limitb d≫1 where b is half the distance between the tip of the blades belonging to each of the waveguides and d is the i nter-blade distance in any waveguide (Fig. 1). 2b AB x zy dh θ FIG. 1: Geometry of the double comb structure waveguide. Following our notation [1], we write for the acoustic ﬁelds i n region A (free space) keeping the symmetric modes only: ΦA(x, y, z ) =∞/summationdisplay n=−∞Ane−jβnx−jτzcosh(αny) (1) where βnandτare the propagation constants along x and z and αnis the attenuation constant along y. The propagation constant β0deﬁning the fundamental mode is determined 3from the propagation geometry (Fig. 1 of [6]). It is equal toτ tg(θ)where θis the angle, the surface wave vector makes with the x-axis [Fig. 1]. The surfa ce wave has a smaller velocity than in true free space by the ratio β2+τ2 k. In region B, the acoustic ﬁeld in the n-th slot deﬁned by the inequalities: νd−d 2≤x≤νd+d 2is given by: Φν B(x, y, z ) =∞/summationdisplay n=−∞Bν me−jτzcos(mπx ν d)cos[γm(y+b+h)] (2) The coeﬃcients Bν mare determined with the help of Floquet’s [7] theorem Bν m= Bme−jνβ0dand the abscissae xνare equal to x-( ν-1/2)d. In order to ﬁnd the dispersion equation of the surface waves and the coeﬃcients Am, Bm, we will proceed as we did in our previous work following the approach pioneered by Whitehea d [7] and Hurd [2]. It consists of writing the equations of continuity for the ﬁelds Φ Aand Φ Band their derivatives along the vertical y axis on the boundaries y=±b. These equations are considered as originating from Cauchy’s theorem of residues for a meromorphic function f(w) taken along some contour and the contribution of each pole is identiﬁed with the contr ibution of some corresponding mode. The contour and f(w) should be such that the presumed theorem of residues is satisﬁed. Moreover, the asymptotic behavior of f(w) is tailored by the underlying physical problem and is basically dictated by the scattering of the wa ves by the edges of the blades [7]. We obtain the following meromorphic function f(w) of the complex variable w: f(w) =dB0γ0e−jγ0h [e−jβ0d−1](jγ0−α0 w−α0)/producttext 1(w) /producttext 1(jγ0)/producttext 2(jγ0) /producttext 2(w)exp[(jγ0−w)d ln(2) π] (3) where/producttext 1(w)and/producttext 2(w) are the following inﬁnite products: /producttext 1(w) =∞/productdisplay p=1(w−jγp)(−d pπ)edw pπ (4) and: /producttext 2(w) =∞/productdisplay p=1(w−αp)(w+α−p)(d 2pπ)2edw pπ (5) The propagation constants γmalong y, are given by: γ2 m=k2−τ2−(mπ d)2withm=0,1... (6) In order to derive the dispersion relation, we form the ratio : 4f(−jγ0) f(jγ0)=−e2jγ0h(7) Taking the logarithm and using trigonometric identities [R ef. 2], we obtain: γ0h−γ0d ln(2) π=π 2−sin−1(γ0 β0)+∞/productdisplay p=1[tg−1(γ0 α−p)+d γ0 2πp]+∞/productdisplay p=1[tg−1(γ0 \|γp\|)−d γ0 pπ]−∞/productdisplay p=1[sin−1(γ0 βp)−d γ0 2πp] (8) This equation is the same as that obtained by Hougardy and Han sen [6] who treated a single comb structure from the electromagnetic point of vi ew. Here, we are dealing with the weak coupling symmetric case limit and with the addi tional simplifying as- sumptions:b d≫1, α0b≫1 and α−p∼β−p. We ﬁnd that the dispersion relation is essentially the same as in the case of a single comb structure . The double comb structure simply behaves as a single one from the dispersion relation p oint of view. This justiﬁes our assumptions in Ref. 1 where we found very good agreement b etween theory and experiment up to frequencies on the order of 400 Hz. Neverthe less this is not true for reﬂection (transmission) coeﬃcients of the single/double comb structures as discussed below. In order to calculate the mode amplitudes and obtain from the m the reﬂection (trans- mission) coeﬃcients of the structure, we use the residue of f(w) atw=αn: Res[f(w)]w=αn=Anβnejβnd/2cosh(αnb) (9) to obtain (n=0, 1, 2...): \|An\| \|B0\|=dγ0eαnb 16πcosh (αnb)\|αn+α0\| \|αnβn\|\|αn+α1\| \|αn−α−1\| \|αn+jγ1\|Γ[2 +dαn π]exp(−αnd ln(2) π) Γ[2 +d 2π(αn+β0)] Γ[2 +d 2π(αn−β0)] (10) where Γ stands for the Euler Gamma function. For negative val ues of n, it suﬃces to change αninto−αnin the above expression. Let us note that when the separation 2b between the two parts of the structure, becomes very large we recover exactly the expression found by Hougardy and Hansen [6] corresponding to a single co mb structure. In order to calculate the Bncoeﬃcients, we use: 5f(−jγn) =d 2Bnγnejγnh [(−)nexp(−jβ0d)−1](11) and the deﬁnition (3) of f(w) to obtain: \|Bn\| \|B0\|=2γ0ǫ \|γnejγnh\|\|jγ0−α0\| \|jγn+α0\|\|/producttext 1(−jγn)\| \|/producttext 2(−jγn)\|sin(β0d 2) (β0d 2)exp(jγnd ln(2) π) (12) where ǫ= 1 for n even, and ǫ=1 \|tg(β0d) 2)\|for n odd. Let us note that the Bncoeﬃcients are the same as those obtained by Hougardy and Hansen [6] reﬂecting the fact, the weak- coupling approxima tion aﬀects in a diﬀerent way theAnand the Bncoeﬃcients. This has important implications on our measure ments of the amplitude proﬁle. III. COMPARISON WITH EXPERIMENT In our previous work, we derived the dispersion relation, tr ansmission and reﬂection coeﬃcients and found excellent agreement between the singl e comb structure theory and experiment up to 400 Hz [1]. This work shows that a weak coupli ng between two comb structures does not aﬀect the surface wave dispersion relat ions and the Bnamplitude coeﬃcients but it does aﬀect the Anamplitude coeﬃcients. We are going to evaluate how our theory modiﬁes the amplitude ratio\|A0\| \|B0\|associated with the fundamental mode (n=0) in relation (10) compared wi th same given in Lahlou et al. [1]. The double comb over single comb structure ratio of t he two expressions is given by: F(θ) =eα0b 2cosh(α0b)(13) For a given frequency and a given incident angle θwe solve the dispersion relation given by equation (8), obtain the propagation factor α0and use it in (13). The corrections F( θ) in dB are plotted versus θin the interval [1, 80] degrees for various frequencies [400 -600 Hz] in Fig.2. The correction comprised between 0 and -3 dB is small for high er frequencies and small incident angles. It decreases rapidly for angles larger tha n 10 to 20 degrees and by a larger 6-3.5-3-2.5-2-1.5-1-0.50 01020304050607080Corrections in dB Angle in degrees400 Hz 450 Hz 500 Hz 550 Hz 600 Hz FIG. 2: Corrections F(θ) to the fundamental mode amplitude ratio\|A0\| \|B0\|with the following values (from the experimental setup) b=0.0125 m, d=0.05 m, h=0.112 m. The corrections calculated from 10log10(F(θ=0) F(θ)) are evaluated as a function of the incident angle θat a ﬁxed frequency varying from 400 to 600 Hz by steps of 50 Hz. amount for higher frequencies. A comparison to the experime ntal data reveals that the correction is pronounced mostly at higher frequency (336 Hz ) and for the largest angle of incidence (47 degrees). For the highest experimental frequ encies (480 and 496 Hz), the correction introduces more disagreement between the exper imental and theoretical single comb structure theory. This behavior may be explained by the fact that there are several sources of errors associated with the measurements at these higher frequencies. IV. CONCLUSION We have developed a weak coupling theory based on the calculu s of residues in order to model the oblique propagation of acoustic waves propagatin g through a slow wave ﬁlter made of a pair of comb structured waveguides separated by a distan ce that is large with respect to the inter-blade distance. The correction arising from the s ymmetrical coupling between the two waveguides has been evaluated and shown to improve sligh tly the agreement between the theoretical and the experimental values of Lahlou et al. [1] being at the most 3 dB for the largest frequency and angle evaluated. Those results sh ow that the approximation taken in our previous investigation is quite acceptable and that t he new theory does not bring 7substantial additional accuracy to our previous single com b structure model. Our studies of the strong coupling case ( b < d) being mathematically much more complicated, and intended for improving the agreement between the theoretical result s and the experimental ones at the higher frequencies are in progress and will be reported i n the near future. [1] R. Lahlou, M. Amram and G. Ostiguy, 1989, J. Acoust. Soc.A m.85, 1449-1455, ”Oblique acoustic wave propagation through a slotted waveguide”. [2] R.A. Hurd, 1954, Can. J. Phys. 32, 727-734, ”Propagation of an electromagnetic wave along an inﬁnite corrugated surface”. [3] L. Mongeau, M. Amram and J. Rousselet, 1985, J. Acoust. So c.Am. 80, 665-671, ”Scattering of sound waves by a periodic array of slotted waveguides”. [4] M. Amram and R. Stern, 1981, J. Acoust. Soc. Am. 70, 1463-1472. ”Refractive and other acoustic eﬀects produced by a prism-shaped network of rigid strips”. [5] L. Brillouin, 1948, J. Appl. Physics 19, 1023-1041. ”Waveguides for slow waves”. [6] R.W. Hougardy and R.C. Hansen, 1958, IRE Trans. Antennas and Propag. AP-2 , 370-376, ”Scanning surface wave antenna - oblique surface waves over a corrugated conductor”. [7] E.A.N. Whitehead, 1951, Proc. IEEE 98, (III) , 133-140, ”Theory of parallel plate media for microwave lenses”. 8
arXiv:physics/0103050v1 [physics.flu-dyn] 19 Mar 2001The Inverse Energy Cascade of Two-Dimensional Turbulence by Michael K. Rivera B.S., University of Miami, 1995 M.S., University of Pittsburgh, 1997 Submitted to the Graduate Faculty of Arts and Sciences in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2000University of Pittsburgh Faculty of Arts and Sciences This dissertation was presented by Michael K. Rivera. It was defended on and approved by Dr. W. I. Goldburg Dr. D. Jasnow Dr. J. Mueller Dr. A. Robertson Dr. X.L. Wu Committee Chairperson iic∝ci∇cleco√†∇tCopyright by Michael K. Rivera 2000 iiiThe Inverse Energy Cascade of Two-Dimensional Turbulence Michael K. Rivera, Ph.D. University of Pittsburgh, 2000 This thesis presents an experimental study of the inverse en ergy cascade as it occurs in an electromagnetically forced soap ﬁlm. It focuses on cha racterizing important features of the inverse cascade such as it’s range, how energ y is distributed over the range and how energy ﬂows through the range. The thesis also p robes the assumption of scale invariance that is associated with the existence of an inverse cascade. These investigations demonstrate that the extent of the inverse c ascade range and the be- havior of the energy distribution are in agreement with dime nsional predictions. The energy ﬂow in the inverse cascade range is shown to be well des cribed by exact math- ematical predictions obtained from the Navier-Stokes equa tion. At no time does the energy ﬂow in the inverse cascade range produced by the e-m ce ll behave inertially or in a scale invariant manner. Evidence that the cascade could become scale invariant should an inertial range develop is presented, as are the req uirements that a system must satisfy to create such an inertial range. ivContents 1 Introduction 1 1.1 The Inverse Energy Cascade . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 History of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 The E-M Cell 10 2.1 The E-M Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 The Magnet Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 External Dissipation: The Air Friction . . . . . . . . . . . . . . . . . 18 2.4 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Particle Tracking Velocimetry . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Cell Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Modeling Flows in the E-M Cell 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 The Karman-Howarth Relationship . . . . . . . . . . . . . . . . . . . 28 3.3 Testing Karman-Howarth . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Experimental Considerations . . . . . . . . . . . . . . . . . . . 3 0 3.3.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.3 Consistency Check: Energy Balance . . . . . . . . . . . . . . . 4 1 4 Energy Distribution and Energy Flow 43 4.1 Distribution of Energy and the Outer Scale . . . . . . . . . . . . . . . 43 4.2 Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.1 The Anisotropic Third Moment . . . . . . . . . . . . . . . . . 52 4.2.2 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 v4.2.3 The Inertial Range and The Integral Scale . . . . . . . . . . . 60 5 High Order Moments 65 5.1 Scale Invariance and Moments . . . . . . . . . . . . . . . . . . . . . . 6 5 5.2 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 The PDF of Longitudinal Velocity Diﬀerence . . . . . . . . . . . . . . 68 6 Conclusion 78 A Particle Tracking Velocimetry: Program Listing 80 Bibliography 107 viList of Tables 4.1 Global constants for several runs of the e-m cell using Ko lmogorov forcing 48 viiList of Figures 1.1 Two pictures of the “eddy” concept for a 2D ﬂuid: (a) a sing le large eddy and (b) a large eddy made from many interacting smaller e ddies. 2 1.2 A 3D wind tunnel creating turbulence and it’s 2D equivale nt: the soap ﬁlm tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 A 2D soap ﬁlm tunnel creating forced 2D turbulence. . . . . . . . . . 7 2.1 Basic operation of the e-m cell. . . . . . . . . . . . . . . . . . . . . . 11 2.2 Replenishing ﬂuid lost to evaporation. . . . . . . . . . . . . . . . . . . 13 2.3 Film curvature near an edge and a plate. . . . . . . . . . . . . . . . . 14 2.4 Array of magnets creating the spatially varying externa l magnetic ﬁeld. 15 2.5 Top view of the magnet arrays which create the spatially v arying ex- ternal magnetic ﬁeld in the e-m cell: (a) the Kolmogorov arra y, (b) the square array, (c) stretched hexagonal array, (d) pseudo-ra ndom array. The direction of the current Jis shown as is the coordinate axis. . . . 16 2.6 Fluid between a top plate moving with velocity Uand ﬁxed bottom plate produces a linear velocity proﬁle. . . . . . . . . . . . . . . . . . 18 2.7 A thick ﬁlm droops under the action of gravity, as shown by the dotted line. A box enclosing the top of the e-m cell frame is brought t o a lower pressure than the surrounding environment to balance gravi ty. . . . . 20 2.8 Timing of the CCD camera frames and laser pulses used in PT V. Frame 1 and Frame 2 denote a single PTV image pair from which velocit y ﬁelds are extracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.1 Thin ﬁlm interference fringes demonstrate that the thic kness of the soap ﬁlm in the e-m cell is not constant but varies from point t o point in the ﬂow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 viii3.2 Typical velocity (a) and pressure (b) ﬁelds obtained fro m the e-m cell. In the pressure ﬁeld green denotes positive and blue negativ e values. . 33 3.3 (a) Time dependence of urmsandωrmsfor a single run in the e-m cell. This demonstrates that the e-m cell is in an approximately st eady state. (b) Time dependence of the enstrophy normalized mean square divergence, D2/ω2 rms, for a single run in the e-m cell. The fact that D2/ω2 rmsis small indicates negligible compressibility. . . . . . . . . . 35 3.4 (a) The mean ﬂow in the e-m cell averaged over 1000 vector ﬁ elds. The length of the reference vector in the upper right corresp onds to 2 cm/s. (b) The decay of the ﬂuctuations in the mean ﬂow as the number of ﬁelds, N, in the average increases. The line corresponds to the expected N−1/2decay of a centered Gaussian variable. . . . . . . 36 3.5 The RMS ﬂuctuations of (a) uxand (b) uyas a function of position in the e-m cell. Green denotes large values of the ﬂuctuations w hile blue denotes small values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6 Measured values of (a) Ay,y, (b)By,y, (c)Ay,y+By,y, and (d) −2αb(2) y,y from Eq. 3.13. Green denotes positive values and blue denote s negative values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.7 Cross sections of Ay,y+By,y(−·) and −2αb(2) y,y(−) along the lines (a) r=rx(ry= 0), (b) r=ry(rx= 0) and (c) r=rx=ry. . . . . . . . . 40 3.8 Cross section of Ax,x+Bx,x(−·) and−2αb(2) x,x(−) along the line r=rx (ry= 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1 The energy spectrum, U(⊳vectork), for (a) Kolmogorov forcing, (b) square forcing using 6 mm round magnets, (c) square forcing using 3 m m round magnets and (d) stretched hexagonal forcing. Green de notes large values of U(⊳vectork) while blue denotes small values. . . . . . . . . . . 45 4.2 The circularly integrated energy spectrum, E(k), for (a) Kolmogorov forcing, (b) square forcing using 6 mm round magnets, (c) squ are forcing using 3 mm round magnets and (d) stretched hexagonal forc- ing. The dashed lines correspond to the Kraichnan predictio n that E(k)∝k−5/3[1]. The arrows indicate the injection wavenumber kinj. 46 4.3 The circularly integrated energy spectrum, E(k), for the four cases of Kolmogorov ﬂow labeled in Table 4.1. . . . . . . . . . . . . . . . . . . 49 ix4.4 Comparison of the outer scale obtained from the energy sp ectra by rout= 2π/koutwith that obtained using the dimensional prediction rout= (ǫinj/α3)1/2for all of the data sets in Table 4.1. . . . . . . . . . 49 4.5 The measured linear drag coeﬃcient, α, versus the magnet-ﬁlm dis- tance, d, for the data sets reported in Table 4.1. The dotted line represents the ﬁt α=ηair/ρhd+CwithC= 0.25 Hz. . . . . . . . . . 50 4.6 Comparison of S(3) a(r) (⋄),J(r) ( ⊳), and K(r) (◦) for the data sets la- beled (a)-(d) in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . 57 4.7 Spatial variation of velocity ﬂuctuation for the four da ta sets labeled in (a)-(d) Table 4.1. Green denotes large values of the ﬂuctuat ions while blue denotes small values. . . . . . . . . . . . . . . . . . . . . . . . . 58 4.8 Typical streamlines for the four cases labeled (a)-(d) i n Table 4.1. . . 59 4.9 Comparison of K(r) (◦) with the independently measured right hand side of Eq. 4.10 , R(r) (−),for the data sets labeled (a)-(d) in Table 4.1. 61 4.10 The right hand side of Eq. 4.10 , R(r),for the data sets labeled (a)-(d) in Table 4.1: (a) ⋄, (b)⊲, (c)⊳, (d)◦.R(r) has been normalized so that the peak value just after rinjis unity. . . . . . . . . . . . . . . . 63 4.11 The dimensionally predicted outer scale, ( ǫinj/α3)1/2vs. integral scale rint(inset is the same plot on log-log scales). The line correspo nds to the power law ﬁt of r2 int. . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.1P(δul(r), r) calculated from data set (c) in Table 4.1. Divisions in the coloration increase on an exponential scale. The inject ion and outer scale are marked by lines, in between which is the inver se energy cascade range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Cross sections at various rforP(δul(r), r) shown in Fig 5.1. . . . . . 69 5.3 (a) S2(r) (log-log) and (b) S3(r) (lin-lin) calculated from data set (c) in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.4 The normalized high order moments, Tn(r), evaluated from data set (c) of Table 4.1 for 4 ≤n≤11. . . . . . . . . . . . . . . . . . . . . . 72 5.5Tn(r)/bnfor odd n≥3 evaluated using the data set (c) in Table 4.1. . 73 5.6 The multiplicative constants anandbn/b3for the data set (c) in Ta- ble 4.1. The dotted line corresponds to the exact values of a p urely Gaussian distribution given by Eq. 5.3. . . . . . . . . . . . . . . . . . 74 x5.7Tn(r)/bnfor odd nfor (a) case (d) in Table 4.1 and (b) a run of the e-m cell with a square array. . . . . . . . . . . . . . . . . . . . . . . . 76 5.8 The measured ananddn(=bn/b3) for three data sets using diﬀerent α and diﬀerent types of forcing. . . . . . . . . . . . . . . . . . . . . . . 77 xiChapter 1 Introduction One of the curious predictions in turbulence theory is that t here might possibly exist a range of length scales in a two-dimensional (2D) turbulent ﬂuid over which kinetic energy is transferred from small to large length scales. Tha t this range could exist was ﬁrst predicted in the late 1960’s by Kraichnan[1]. Numerica l simulations that followed yielded varying degrees of agreement with this prediction[ 2, 3, 4]. Experimental veriﬁcation of the existence of such a range did not come abou t until much later due to the diﬃculty inherent in building and maintaining a syste m which approximates a 2D ﬂuid[5, 6, 7]. This thesis presents an investigation of th e 2D inverse energy cascade in a new apparatus, the electromagnetically forced soap ﬁlm . What follows in this chapter is a description of the phenomenology surrounding t he inverse energy cascade and a discussion of the experiments that have attempted to pr obe it’s properties. 1.1 The Inverse Energy Cascade The phenomenology of turbulence, in three-dimensions (3D) or 2D, is usually phrased in terms of “eddies”. An eddy itself is not a well deﬁned objec t, though there have been many recent attempts using wavelets to better deﬁne the concept[8]. Loosely speaking it is a region in a ﬂuid that is behaving coherently. The extent of an eddy is dictated by boundaries within which an arbitrary determina tion is made that some sort of structure exists. Thus an eddy can be a single large re gion of rotation, such as the whirlpool which forms above a bathroom drain. Or an edd y can be a large region containing many smaller eddies which are interactin g with one another while behaving distinctly (again by an arbitrary determination) from other neighboring 1Chapter 1. Introduction 2 Figure 1.1: Two pictures of the “eddy” concept for a 2D ﬂuid: ( a) a single large eddy and (b) a large eddy made from many interacting smaller eddie s. clusters of eddies. These two ideas are drawn in Fig. 1.1 for t he case of a 2D ﬂuid. Eddies in 3D are much more diﬃcult to picture. Two important p roperties that are associated with an eddy are size and energy. These two proper ties allow predictions about energy motion in ﬂuids to be made if some knowledge of ho w eddies interact in the system is known. In 2D ﬂuids, one way in which eddies (which assume the more fam iliar label “vortices” in 2D) interact with each other is through a proce ss known as “vortex cannibalization”. A cannibalization event is when two neig hboring eddies of like rotational sense merge to form a single larger eddy. When can nibalization occurs energy ﬂows out of the length scales of the initial eddies and into the length scale of the ﬁnal eddy. Since the ﬁnal eddy is larger than the initia l ones, cannibalization results in the ﬂow of energy from small to large length scales . In a 2D turbulent ﬂuid, many eddies are generally created at a small length scale called the energy injection scale, rinj. The expectation is that through interaction by cannibalization these small eddies cluster and merge int o larger eddies. These larger eddies are also expected to cluster and merge to form e ven larger eddies and so on. This means that energy, initially injected into the turb ulence at the length scale rinjshould gradually be moved by consecutive cannibalization e vents to larger length scales. This type of energy motion constitutes an inverse en ergy cascade[1]. Using the eddy concept has the advantage of highlighting two important features associated with the existence of an inverse energy cascade: scale invariance and local- ity of interaction. The ﬁrst of these can be understood by loo king again at Fig. 1.1(b) which shows many smaller like signed eddies clustering to fo rm a single larger eddy.Chapter 1. Introduction 3 Presumably, the small eddies in the ﬁgure are themselves for med by the clustering of even smaller eddies, which in turn are formed by even small er eddies. Likewise, the large eddy cluster in the ﬁgure is most likely interactin g with other eddy clusters in the system. As long as the eddies at the very smallest scale , the injection scale, are being continuously created to replenish those which are cannibalized, the inverse cascade range is scale-invariant. That is to say that no leng th scale in the inverse cascade range can be distinguished from any other length sca le that is also in the range. Scale invariance is exceedingly important from a the oretical stand point. The assumption of scale invariance of ﬁelds, such as the probabi lity of velocity diﬀerence on a length scale r, allow important predictions about turbulence to be made (s ee chapter 5)[9]. Before discussing locality, a delicate point must be made. I f the eddies at the injection length scale are not being continuously replenis hed then the number of eddies at the smallest scales gradually begins to decrease a s more and more eddies are lost to cannibalization events. To maintain an inverse c ascade range, then, the turbulence has to be continuously forced. That is, eddies mu st be continuously created at the energy injection scale. If the turbulence is not force d then the cascade range will eventually consume itself from small scales up, ultima tely leading to a state which can be described as a diﬀuse gas of large individual edd ies (eddies not made of clusters of smaller eddies)[10, 11]. The term “coarsening” is used to describe decaying 2D turbulence’s behavior in order to distinguish it from the inverse energy cascade. The second property assumed to hold in the inverse cascade is locality of interac- tion. This property refers to constraints on the manner in wh ich eddies interact. If an eddy of very small size is close to, or embedded in, an eddy o f exceedingly large size, the small eddy will merely be swept along by the large ed dy and not strongly deformed. Likewise the large eddy will not be signiﬁcantly e ﬀected by it’s small com- panion. Since neither of the eddies is strongly deformed, th e cannibalization process is expected to happen over a long period of time, if at all[9]. On the other hand, two neighboring eddies of similar size interact and deform one a nother strongly, and thus the cannibalization happens swiftly. Energy transfer by ca nnibalization is therefore most eﬃcient when occurring between similarly sized length scales; this is what is meant by locality. Due to locality, the kinetic energy at sma ll scales in the inverse cascade is expected to be moved to large scales in a continuou s manner, stepping through the intervening length scales by local interaction s rather than making largeChapter 1. Introduction 4 length scale jumps by the merger of a small and large eddy. Hen ce the term cascade. The picture of 2D turbulence and it’s inverse cascade is now a lmost complete. Energy is continually injected into a ﬂuid in the form of smal l eddies. These small eddies cluster to form large eddies moving energy to larger s cales. In turn the eddy clusters themselves cluster to form larger clusters of eddy clusters, etcetera. It is the etcetera that is of concern. At what point does the vortex merger process and growth of larger and larger eddies stop? That is, how is the en ergy injected into a 2D turbulent ﬂuid thermostated? Consider ﬁrst the thermostating mechanism in 3D turbulence . In 3D turbulence there exists a direct energy cascade, where instead of energ y being moved from small to large scales by eddy merger the opposite happens; energy i s moved from large to small scales by eddy stretching (commonly called vortex s tretching). Eventually, through continuous vortex stretching, a smallest eddy scal e is reached, at which point the kinetic energy contained in these small eddies is dissip ated into heat by the ﬂuids internal viscosity. All of the energy that is injected into t he large length scales of a 3D turbulent ﬂuid is eventually exhausted by viscosity at sm all length scales[9]. Internal viscosity is a short range force, only becoming a go od thermostat when the kinetic energy reaches small length scales[9]. Thermos tating is not an issue in 3D where the direct cascade takes energy down to such small sc ales. In 2D, however, the inverse cascade moves energy away from small scales. The refore viscosity has no chance to exhaust the injected energy. An ideal 2D turbule nt ﬂuid driven to a state of turbulence with a continuous forcing would never be in a steady state since the total energy in the ﬂow would continue to build up as large r and larger eddies form[12]. What is needed to maintain 2D forced turbulence in a steady state and stop the inverse cascade process is some sort of external dissipa tion mechanism which is an eﬀective thermostat at large length scales. In other word s, some sort of dissipation mechanism that is not internal to the ﬂuid itself must exist t o take energy out of large length scales and dictate the largest size eddies that can be formed by the cascade. Fortunately, 2D experiments are almost always coupled to th e surrounding 3D environment by frictional forces[13, 14, 7]. In these exper iments this external fric- tional force provide the turbulence with an eﬀective large s cale thermostat and sets the largest length scales which can be reached by the inverse cascade process. The inclusion of an external thermostat completes this phenome nological description of the inverse energy cascade.Chapter 1. Introduction 5 1.2 History of Experiments Laboratory experiments which have attempted to probe the in verse energy cascade of 2D turbulence fall into two major categories: soap ﬁlms an d stratiﬁed shallow layers of ﬂuid. The soap ﬁlm experiments in 2D ﬂuid mechanics were initiated by Couder in the early 1980’s[15]. This early work investigate d coarse features of both 2D turbulence and 2D hydrodynamics. Further attempts at usi ng the soap ﬁlm to measure 2D turbulence in the search for an inverse cascade we re done by Gharib and Derango a few years later[16]. The experimental system that was used by Gharib and Derango, called a soap ﬁlm tunnel, was later perfected by Kellay et al.[17] and Rutgers et al.[14]. The soap ﬁlm tunnel is a 2D equivalent of the wind tunnel, whic h is the mainstay of 3D turbulence research. The manner in which turbulence is created in each is identical. A grid, or some other obstacle, is placed in the pa th of a swiftly moving mean ﬂow. If the ﬂow speed is fast enough, the ﬂuid becomes tur bulent downstream from the grid. Such 2D and 3D tunnels are shown in Fig. 1.2. Soa p ﬁlm tunnels would seem to be ideal 2D ﬂuids for performing turbulence research in because their aspect ratios are exceedingly large (many cm across to a few microme ters thick) and thus the ﬂuid ﬂow is almost entirely two-dimensional. There are, how ever, diﬃculties inherent in the use of soap ﬁlms. For example thin ﬁlms couple strongly to the air and the magnitude of their internal viscosity is large compared to t hat of water. For the most part these diﬃculties are thought to be mitigated by clever e xperimental techniques, such as the use of vacuum chambers[14] or by using thick ( ≈10µm) ﬁlms[11]. Every attempt to study the inverse energy cascade in a 2D soap ﬁlm tunnel with the conﬁguration shown in Fig. 1.2 has met with failure. This is not a disparaging comment about the researchers involved in the eﬀort. Indeed their considerable skill eventually tamed the delicate and whimsical soap ﬁlms into a useful experimental system. The lack of inverse cascade in these systems reﬂects the fact that the con- ﬁguration shown in Fig. 1.2 creates decaying 2D turbulence. The eddies that are injected at the grid are not replenished as the ﬂuid moves dow nstream. By the dis- cussion in the last section this means that the system does no t have the ability to form the eddy clusters that is expected of an inverse cascade . Once the understanding that 2D turbulence needs to be forced for an inverse cascade to be present was reached, the ﬁlm tunnel design was m odiﬁed to createChapter 1. Introduction 6 Figure 1.2: A 3D wind tunnel creating turbulence and it’s 2D e quivalent: the soap ﬁlm tunnel forced turbulence[6]. The design of the ﬁlm tunnel is identi cal to that shown before except for the orientation of the turbulence producing grid . Instead of having a single grid oriented perpendicular to the ﬂow direction, two grids were oriented at angles to the ﬂow direction so that they formed two sides of a triangl e with the tip of the triangle oriented upstream. This modiﬁed form is shown in Fi g. 1.3. An area of forced turbulence exists in the interior between the two gri ds since it is here that vortices created at the grids are able to diﬀuse into the inte rior and replenish those lost to cannibalization. Though certain properties of the inverse cascade can be inve stigated with such a modiﬁed ﬁlm tunnel, the setup is not ideal because the turbu lence it creates is inhomogeneous. One can imagine that the ﬂuctuation in the dr iven turbulence area near the grids are quite large compared to those in the interi or. Homogeneity is a critical simplifying assumption in almost all areas of turb ulence theory. The inhomo- geneity of the modiﬁed ﬁlm tunnel, then, has devastating con sequences with regard to comparing results with theory. A more ideal setup would in volve the injection of vortices directly into the body of the ﬂuid by some sort of ext ernal force, rather than injecting from the boundaries. This is a diﬃcult task to do in soap ﬁlm tunnels. A system which does achieve such an injection of vortices fal ls into the second class of 2D turbulence experiments: stratiﬁed shallow layers of ﬂ uid. For the most part, theChapter 1. Introduction 7 Figure 1.3: A 2D soap ﬁlm tunnel creating forced 2D turbulenc e. use of such layers in turbulence research has been pioneered by Tabeling and others in the early 1990’s. A successful observation of an inverse c ascade regime was reached a few years later by the same group[18]. The stratiﬁed shallo w layer apparatus, as it’s name suggest, suspends a layer of pure water, above a hig her density layer of salt water. The layers in question are only a few millimeters thic k, and the area tends to be on the order of ten centimeters so that the aspect ratio, wh ile not nearing that of soap ﬁlms, is still large. The salt water layer is subject to a current ﬂowing in it’s plane, and placed in a spatially varying magnetic ﬁeld. The r esultant Lorentz force acts directly on the ﬂuid layer driving it to a state of turbul ence. Stratiﬁcation helps to impose a measure of two-dimensionality to the ﬂuctuation s, thus one has 2D forced turbulence. Note that in this system the force is acting directly on the ﬂu id, unlike the modi- ﬁed ﬁlm tunnel where forcing happened near the grid. This res tores homogeneity to the system, allowing accurate comparison with theory. The p lace that shallow layers suﬀer is in their approximation of a two-dimensional ﬂuid. T he bottom of the ﬂuid container in shallow layer systems enforces a no slip bounda ry on the lower surface of the ﬂuid. If the velocity in the ﬂuid becomes large, a stron g shearing can develop between the upper and lower layers of ﬂuid. This inevitably c auses mixing which destroys the stratiﬁcation and sacriﬁces two-dimensional ity. Thus, shallow layer sys- tems are severely limited in the strength of the turbulent ﬂu ctuations which they can successfully explore.Chapter 1. Introduction 8 This brief experimental history, then, shows that there hav e been two systems used to investigate the inverse cascade, neither of which are ide al. However, each system complements the others diﬃculties almost perfectly. Film t unnels are great 2D ﬂuids, but not easily forced homogeneously. Stratiﬁed layers are m arginal 2D ﬂuids, but easily forced homogeneously. What is needed then, as an idea l test apparatus for the inverse cascade, is a combination of these two systems that t akes advantage of their beneﬁts without inheriting their diﬃculties. What is neede d is the electromagnetically forced soap ﬁlm, simply called the e-m cell. 1.3 Thesis Overview This thesis presents an experimental study of the inverse en ergy cascade as it occurs in an electromagnetically forced soap ﬁlm. In particular it focuses on characterizing important features of the inverse cascade such as it’s range , how energy is distributed over the range and how energy ﬂows through the range. The thes is also probes the assumption of scale invariance that is associated with the e xistence of an inverse cascade. Chapter 2 describes the workings of the e-m cell and measurem ent apparatus. The basic design and implementation of the e-m cell is reviewed i n the ﬁrst two sections. How the frictional coupling of the soap ﬁlm to the air is contr olled is described in the following section. The fourth section explains certain lim itations on apparatus size that are imposed by the existence of gravity. This size const raint is one of the chief diﬃculties that limit results throughout the rest of the the sis. The ﬁfth section in the chapter is an overview of the measurement system that was developed to extract velocity information from the e-m cell. The chapter is concl uded with a brief overview of the operation of the e-m cell. Chapter 3 is a systematic check that the e-m cell does behave a s a 2D ﬂuid. The ﬁrst section motivates the need for this test and the seco nd derives the relevant mathematical relationships necessary for such a test. The t hird section compares the data from the cell to these derived relationships to verify t hat the e-m cell behaves as a 2D ﬂuid. This section also establishes a model for the extern al dissipation mechanism in the e-m cell. A self consistency check is also performed to help strengthen this veriﬁcation. Chapter 4 begins the systematic investigation of the proper ties of the inverseChapter 1. Introduction 9 energy cascade. The extent of the range and how energy is dist ributed over this range is measured and presented in the ﬁrst section. The mann er of energy transfer is described in the following section. In particular, this s econd section attempts to determine if the energy ﬂow is “inertial”, a property which i s necessary if one wants results which are universal to all 2D turbulence systems. Finally, chapter 5 attempts to determine if the inverse casc ade is scale invariant. Predictions for structure functions in a scale invariant ﬂu id are presented in the ﬁrst section. The next two sections attempt to extend results fro m the e-m cell to test these predictions, as well as explain why one should be critical of such extensions. While conclusions presented in earlier chapters are quite strong , the conclusions drawn in this chapter are weak. This weakness stems from the size limi tations presented in chapter 2 and can not be overcome in the current apparatus.Chapter 2 The E-M Cell A ﬂuid which carries a current density, ⊳vectorJ, in the presence of a magnetic ﬁeld, ⊳vectorB, is subject to a force per unit mass ⊳vectorF=⊳vectorJ×⊳vectorBwhich drives ﬂuid motion. This principle has been used in earlier experiments to excite motion in shal low layers of electrolytic ﬂuid[5, 19], and the techniques used in these experiments ma y be readily adapted for use in soap ﬁlms. The result of such adaptation is the elec tromagnetically forced soap ﬁlm, called brieﬂy an e-m cell. The e-m cell is a useful to ol in the study of 2D hydrodynamics, and in particular 2D turbulence[7]. This ch apter reviews the design and operation of the e-m cell, as well as the measurement tech nique used to obtain velocity data from it. 2.1 The E-M Cell The main component of the e-m cell is a free standing soap ﬁlm d rawn from an electrolytic soap solution across a square frame. The frame has two opposing sides made from stainless steel, while the remaining sides are con structed from plastic or glass. A voltage diﬀerence applied to the stainless steel si des results in a current which lies in the plane of the ﬁlm. A spatially varying extern al magnetic ﬁeld is then created and oriented so that it penetrates the ﬁlm plane perpendicularly. The resulting Lorentz force lies in the plane of the ﬁlm and drive s the ﬂuid motion. A diagram of this is shown in Fig. 2.1. The electrolytic soap solution which the ﬁlm is drawn from is made from 400 ml distilled water, 80 g ammonium chloride salt, 40 ml glycerol and 5 ml commercial liquid detergent (regular Dawn or Joy). To this solution par ticles are added up to a 10Chapter 2. The E-M Cell 11 Figure 2.1: Basic operation of the e-m cell. volume fraction of about 10−3. These particles, either 10 µm hollow glass spheres or lycopodium mushroom spores which appear as ∼40µm particles, are of a density comparable to the soap solution so that they closely follow t he surrounding ﬂow. These particles will be used in the measurement technique to be described later. The salt used to make the solution electrolytic, ammonium chlor ide, was chosen after numerous trials which included sodium chloride and potassi um chloride. Ammonium chloride was chosen because high concentrations could be re ached with little eﬀect on the stability of the ﬁlm. This was found not to be the case wi th potassium or sodium chloride. High concentrations of salt are necessary to keep the ﬁlms electrical resistance as small as possible. This limits ﬁlm evaporatio n due to Joule heating caused by the driving current, a necessity if a ﬁlm is to be stu died for long periods of time. The soap ﬁlm, once drawn, is maintained at a thickness of arou nd 50 µm. Earlier soap ﬁlm work tended to use thin ﬁlms with thicknesses of 5 µm or less. There are a number of advantages that come from using thick ﬁlms. First, for the e-m cell, it is desirable to keep the electrical resistance low, again to li mit evaporation due to Joule heating. The larger the cross sectional area through which t he current is passed, then, the better. Another reason to use thick ﬁlms is that they lose a smaller percentage of their energy than thin ﬁlms to frictional rubbing against th e surrounding air. Recall from chapter 1 that 2D driven turbulence requires an externa l dissipation mechanism to maintain energy balance. Air friction plays this role in t he e-m cell. If the air friction is strong then it can easily dissipate large amount s of energy. Therefore the energy injected from the electromagnetic force must be e xceptionally high to maintain a state of strong turbulence. This would require la rge currents which wouldChapter 2. The E-M Cell 12 enhance the Joule heating and evaporation of the ﬁlm. Using t hick ﬁlms reduces this coupling to the air and allows strong turbulence to be ma intained for reasonable values of the driving current. A ﬁnal reason for using thick ﬁ lms arises from the observation that a soap ﬁlms kinematic viscosity, ν, is dependent on it’s thickness. Trapeznikov predicted that the eﬀective kinematic viscosi ty should depend on the thickness, h, asν=νbulk+2νsurf/h, where νbulkis the kinematic viscosity of the soap solution and νsurfis the 2D viscosity due to the soap ﬁlm surfaces[20]. The surf ace viscosity was recently measured1to be around νsurf≈1.5×10−5cm3/s for soap ﬁlms similar to that in the e-m cell[21]. Since the soap solution’ s viscosity is .01 cm2/s the eﬀective viscosity of a 50 µm thick soap ﬁlm is approximately 0 .016 cm2/s. Using thick ﬁlms, then, reduces the amount of energy lost to viscou s dissipation, and by the same argument given above, allows strong turbulence to be ma intained for reasonable values of driving current. There are, of course, disadvantages to the use of thick soap ﬁ lms. One is that the speed of a 2D density wave in the soap ﬁlm is dependent on th e thickness of the ﬁlm2. Thicker ﬁlms contain more interstitial ﬂuid than thin ﬁlms , and therefore have a lower wave velocity due to their increased mass. Therefore , as the ﬁlm becomes thicker it begins to be more easily compressed. That is to say that it’s 2D density couples more strongly to the velocity ﬁeld in the ﬁlm. A measu re of the importance of such compressibility eﬀects is the Mach number M=urms/c, where cis the density wave speed in the medium. In the case of a 50 µm thick ﬁlm the wave velocity is approximately 2 m/s. Velocity ﬂuctuations in the e-m cell ar e therefore kept less than 20 cm/s so that M <0.1. With such a small Mach number the system does not develop shock waves and behaves approximately as an incompr essible ﬂuid. As noted earlier, Joule heating evaporates ﬂuid from the ﬁlm . This causes the average thickness of the soap ﬁlm to change over time. Since t hick ﬁlms were used to minimize this eﬀect, thickness changes due to evaporation h appen at a slow rate and can be balanced by injecting small amounts of ﬂuid into the ﬁl m. In the e-m cell, soap solution is injected by a syringe pump through a small needle inserted through the plastic side of the frame as in Fig. 2.2. It is important that t he ﬂuid be burst in, as 1This measurement utilized a ﬂowing soap ﬁlm tunnel to analyz e the shedding of vortices from a cylinder placed in the ﬂow. The vortex shedding frequency i s sensitive to changes in the ﬂuids viscosity and through proper normalization allows νsurfto be approximated. 2A 2D density wave in a soap ﬁlm is a thickness wave, where the ﬁl m bulges or shrinks in the third dimension.Chapter 2. The E-M Cell 13 Figure 2.2: Replenishing ﬂuid lost to evaporation. a brief jet, and not slowly injected. Without initial moment um, ﬂuid builds up near the injection point forming a droplet which eventually pops the ﬁlm. Fluid injected with a burst shoots to the center of the e-m cell where it is qui ckly mixed into the ﬁlm replenishing the lost ﬂuid. Though not used in the experi ments reported here, this process can be automated by monitoring the ﬁlm resistan ce. When the resistance becomes to high a small burst of ﬂuid is shot into the ﬁlm raisi ng the thickness and lowering the resistance to an acceptable level. The square frame that the ﬁlm is drawn across is limited in siz e to 7 ×7 cm2in all of the experiments reported here for reasons which will b e made clear later. Both the stainless steel and plastic sides are milled to a sharp ed ge of about 45◦, though the stainless steel edge generally dulls over time by corros ion. Recent eﬀorts have attempted to replace these edges with sheets of stainless st eel and glass that are less than 100 µm thick. The intent of this is to limit the size of the wetting r egion near the edge of the frame, as shown in Fig. 2.3. Since the ﬁlm has a ﬁ nite surface tension, there must be a pressure jump across the ﬁlm surface if it is cu rved. A wetting region induces a negative curvature near the edge of the frame, ther efore the pressure inside the ﬁlm near the frame edge is less than at the ﬁlm center where there is no curvature. This causes ﬂuid to be forced through the ﬁlm to the edges, cau sing the center of the ﬁlm to drain. Elimination of these wetting regions elimi nates such drainage. This eﬀect is relatively weak compared to other eﬀects which cause drainage so using sheets instead of edges is a low priority, sheets being diﬃcu lt to work with due to their delicacy. All experiments reported in this thesis use edges instead of thin sheets. However, the preliminary work with sheets which is not repor ted here has provided encouraging early results. Using a 50 µm ﬁlm made from the above ammonium chloride solution on the 7 ×7Chapter 2. The E-M Cell 14 Figure 2.3: Film curvature near an edge and a plate. cm2frame results in resistance values of around 1 kΩ. Driving cu rrents of between 10 and 45 mA are used depending on the strength of turbulence des ired, strength of the air dissipation, and strength of externally applied magnet ic ﬁeld. In all turbulence experiments, the applied voltage oscillates at 3 Hz with a sq uare waveform. There are several reasons for this, one is that the current in the ﬁlm ca uses chlorine gas bubbles to accumulate on the positive electrode. By oscillating the current, the bubbles form at both electrodes, slowing the inevitable bubble build up w hich eventually invades the ﬁlm and renders it useless. Another reason for current os cillation is to eliminate the formation of large vortices which form and dominate regi ons of the ﬂuid containing a force ﬁeld which is sympathetic to it’s motion. Once formed , such structures induce a spatially varying mean ﬂow, and thus a mean shear, renderin g the turbulence in the e-m cell inhomogeneous.3 A ﬁnal note: due to the conﬁguration of the e-m cell, the curre nt which is driven in the plane of the ﬁlm is unidirectional, namely parallel to th e plastic edges of the frame running from one electrode to another. For the rest of the pap er the current direction will be assumed to lie along the yaxis. By the Lorentz law, this unidirectional 3Oscillating the current may also eliminate any polarizatio n of charge in the e-m cell caused by the net motion of ions in the ﬂuid. At this time it is unclear wh ether this polarization is important in the e-m cell. Experiments done in both laminar and turbule nt ﬂow with D.C. forcing do not exhibit signs of charge polarization, such as the gradual de crease in applied force due to shielding of the electrodes. However, these experiments where done over short (minutes) time scale. What eﬀect polarization may have over a typical experiment time scale ( around ﬁfteen minutes) is unknown since such eﬀects are masked by the eﬀects of evaporation.Chapter 2. The E-M Cell 15 Figure 2.4: Array of magnets creating the spatially varying external magnetic ﬁeld. current creates a unidirectional electromagnetic force wh ich lies on the xaxis. Due to this unidirectionality, the e-m cell can not behave isotrop ically, that is the system is not rotationally invariant, regardless of the symmetry of t he spatial variation of the magnetic ﬁeld. This lack of isotropy will be an exceedingly i mportant consideration later in the thesis. 2.2 The Magnet Array The spatially varying magnetic ﬁeld used in the creation of t he Lorentz force is gen- erated by an array of Neodymium rare earth magnets (NdFeB) pl aced just below the ﬁlm as shown in Fig. 2.4. These magnets produce quite powerfu l ﬁelds, typically on the order of 0 .1 T at their surface. This ﬁeld decays away from the surface, h owever, with a typical length scale of the order of the magnet size. Si nce the magnets are small, generally less than 0 .5 mm, they must be brought very close to the ﬁlm surface to generate ﬁelds strong enough to drive ﬂow. One could use la rger magnets to induce motion. This is undesirable, though, since the magnet size d ictates the energy injec- tion scale, rinj. Recall from the discussion in chapter 1 that rinjmust be signiﬁcantly smaller than the system size for an inverse cascade region to exist in an experimental 2D turbulence system. For the e-m cell described above, the s ystem size is the frame size, which is limited to 7 cm. Therefore magnet arrays with i njection scales less than 0.7 cm must be used to allow for a decade of inverse cascade range . Several types of magnet arrays are used in the e-m cell, each d istinguished by the type of ﬂow it induces at small Reynolds number (it’s laminar behavior) or equiva- lently by the characteristics of the force ﬁeld it produces i n the e-m cell. The ﬁrst type of array, and by far the most important, is the Kolmogorov arr ay. This array is madeChapter 2. The E-M Cell 16 Figure 2.5: Top view of the magnet arrays which create the spa tially varying external magnetic ﬁeld in the e-m cell: (a) the Kolmogorov array, (b) t he square array, (c) stretched hexagonal array, (d) pseudo-random array. The di rection of the current J is shown as is the coordinate axis.Chapter 2. The E-M Cell 17 of alternating north-south layers of long rectangular magn ets of approximately 0 .3 cm in width, as shown in Fig. 2.5 a. The magnetic ﬁeld it produc es in the ﬁlm varies approximately sinusoidally in space. The magnets are orien ted so that this variation lies in the direction of the current, causing the Lorentz for ce in the ﬁlm to have the formFx=f0sin(k0y) (Recall Fy= 0 because the force is unidirectional). Two mag- nets make a single north-south oscillation, therefore the w avelength of the sinusoidal variations in the force ﬁeld (which is the energy injection s cale) is rinj= 0.6 cm. The associated injection wavenumber is k0= 2π/rinj= 10 cm−1. The laminar ﬂow this forcing produces is one of alternating shear layers, a ﬂow wh ich Kolmogorov proposed investigating as an interesting model to study a ﬂuids trans ition from laminar to tur- bulent motion (hence the name). The Kolmogorov array has sev eral properties which will be of importance in later analysis. The ﬁrst is that the f orcing is divergence free, i.e.⊳vector∇ ·⊳vectorF= 0. This property will allow pressure ﬁelds to be approximat ed from velocity ﬁelds using Fourier techniques. Another property of importance is that the forcing is invariant to translation along the ˆ xdirection. Chapter 3 will demonstrate how this property may be used to obtain the energy injection r ate from the forcing without having to explicitly measure the magnitude of the fo rcingf0. The next type of magnet array, shown in Fig. 2.5 b, is called th e square array. It is made of round magnets oriented in a tick-tack-toe arran gement with like poled magnets along diagonals. There are two such arrays in the e-m cell arsenal made from 0 .3 cm and 0 .6 cm magnets. The force ﬁeld created by these arrays has the fo rm Fx=f0(sin(k0(x+y)) +sin(k0(x−y))). Here the wavelength along the diagonals isrinj=√ 2wwhere wis the magnet diameter, and again k0= 2π/rinj. The 0 .3 cm square array has the smallest injection scale of all the arra ys used with rinj= 0.42 cm. Unfortunately these magnets are exceptionally weak, li miting the magnitude of the forcing one can create with reasonable currents. Also, t he forcing created by these arrays are not divergence free. Without the ability to accou nt for pressure much the usefulness of this array is limited to testing ideas of unive rsality (Chapter 5). The ﬁnal two arrays are seldom used and are mentioned here onl y for completeness. The ﬁrst, shown in Fig. 2.5 c, is a stretched hexagonal array, called the hex array, and the second, shown in Fig. 2.5 d, is a pseudo-random array. The former is constructed from a mixture of both 0 .3 cm round and 0 .6 cm round magnets. The result could be though of as a Kolmogorov ﬂow with a periodic v ariation along every other magnet. The injection wavenumber associated with thi s array is quite large,Chapter 2. The E-M Cell 18 Figure 2.6: Fluid between a top plate moving with velocity Uand ﬁxed bottom plate produces a linear velocity proﬁle. limiting it’s usefulness. The laminar behavior a hex array p roduces is a triangular vortex crystal. The pseudo-random array is constructed fro m 0.3 cm round magnets placed at random on an iron sheet. The positions of the magnet s where generated via a random number generator which attempted to maximize th e magnet density. It was constructed in a naive attempt to obtain more homogeno us turbulence in the e-m cell. The pseudo-random ﬂow has no well-deﬁned laminar ﬂ ow behavior. 2.3 External Dissipation: The Air Friction The air friction has already been described as the external d issipation mechanism in the e-m cell. Chapter 3 will demonstrate that the air frict ion can be adequately modeled as a linear friction, that is if an element of the ﬁlm i s moving with velocity ⊳vector uit experiences a drag force from the air of the form ⊳vectorFdrag=−α⊳vector u. The following discussion sets the basis for this linear drag model. A ﬂuid between two parallel plates separated in the x2direction by a small distance d, one ﬁxed and the other moving with velocity Uin the x1direction, assumes a linear proﬁle of the form, u1=Ux2/d(This assumes there are no other forces acting on the ﬂuid, the plates are inﬁnite and boundary conditions are no slip at either plate). This is shown in Fig. 2.6. The drag force per unit area exerted on the top plate by the ﬂuid is fdrag=η∂u1 ∂x2\|d=ηU dwhere ηis the ordinary 3D shear viscosity of the ﬂuid between the plates. In the e-m cell the role of the top plate is played by the soap ﬁlm, while the role of the lower plate is played by the magnet a rray. The ﬂuid is the surrounding air and the length dis just the distance of the magnets to the ﬁlm. Thus raising the magnets closer to the ﬁlm increases the strength of the air drag in the e-m cell.Chapter 2. The E-M Cell 19 The drag force must be normalized by the ﬁlms 2D density so tha t the force per unit mass can be considered. The 2D density of the ﬁlm is given byρhwhere ρis the density of the soap solution (about that of water) and his the ﬁlms thickness. The force per unit mass caused by the air drag on the ﬁlm is then ⊳vectorFdrag=−η ρhd⊳vector u. This demonstrates a point that was made earlier in this chapt er, that the frictional coupling to the air depends not only on the magnet-ﬁlm distan ce but on the thickness of the soap ﬁlm as well. Keeping the ﬁlm as thick as possible, t hen, minimizes this coupling. There are a couple of reasons to worry about the applicabilit y of this linear friction model. First is that the soap ﬁlm in the e-m cell is not an inﬁni te ﬂat plate moving with a constant velocity, but has a velocity which ﬂuctuates from point to point in the ﬂow. As long as the size of these velocity ﬂuctuations, th at is the size of the vortices, is larger than both the magnet-ﬁlm distance, d, and the ﬁlm thickness, h, the approximation as an inﬁnite plate should apply. Recall f rom the discussion above that the magnetic ﬁeld created by the array quickly decays wi thin the width of a single magnet. Therefore the magnets are always kept within one magnet width of the ﬁlm, otherwise the magnetic ﬁeld would be too weak to driv e turbulence. Since the magnet width also dictates the smallest vortex size, rinj, the requirement that d < r injis always met in the e-m cell. The second complication is the fact that there is more than on e side to the ﬁlm. The linear drag model accounts for the drag force exerted on t he lower surface of the soap ﬁlm. The upper surface of the ﬁlm is also dragging along a layer of air. The velocity proﬁle of the air above the ﬁlm is not a simple linear shear as it is below the ﬁlm, but a more complex Prandtl-like boundary layer due to th e absence of a second ﬁxed plate. Chapter 4 will show that this causes a non-neglig ible correction to the above linear drag model. 2.4 Gravity To this point there has been no discussion about the orientat ion of the soap ﬁlm with respect to the earth’s gravitational ﬁeld. Vertical or ientation, that is the ﬁlm plane lying along the gravitational ﬁeld direction, is not d esirable because it would strongly stratify the soap ﬁlm, making it thinner on top than on the bottom. This is tantamount to both a severe change in the 2D density and a ch ange in the ﬁlmsChapter 2. The E-M Cell 20 Figure 2.7: A thick ﬁlm droops under the action of gravity, as shown by the dotted line. A box enclosing the top of the e-m cell frame is brought t o a lower pressure than the surrounding environment to balance gravity. kinematic viscosity from the top of the ﬁlm to the bottom. In o ther words the ﬁlm becomes an inhomogeneous ﬂuid. Vertical orientation must t herefore be discarded and a horizontal orientation used to allow the ﬁlm to approxi mate a homogenous ﬂuid. A horizontally oriented soap ﬁlm droops under the force of th e earth’s gravitational ﬁeld. This eﬀect is exacerbated by the fact that the ﬁlm is 50 µm thick. To balance gravity, a box enclosing the region on the top surface of the s oap ﬁlm is evacuated of a small amount of air, as in Fig. 2.7. This lowers the pressu re in the box causing a pressure gradient across the ﬁlm plane and thus a force oppo sing gravity. Enough pressure is drawn to almost exactly balance the gravitation al ﬁeld. Unfortunately the larger the soap ﬁlm, the more delicate this balance becomes. This is the reason that soap ﬁlms used in the e-m cell are limited to sizes under 7 ×7 cm. This pressure balance is delicate and can be disrupted by the evaporation of ﬂuid from the soap ﬁlm into the container above the ﬁlm. It must be c onstantly monitored to make sure that the ﬁlm stays at the same level. This is done i n the e-m cell by reﬂecting a laser light oﬀ a portion of the ﬁlm near the middle of one of the edges. A drooping ﬁlm deﬂects the beam, and this deﬂection can be mon itored by various techniques, for example using a position sensitive detecto r or a linear CCD array. A feedback loop based on the deﬂection measurement can be easi ly constructed and ﬁlm level kept steady, even for high current and large amounts of evaporation.Chapter 2. The E-M Cell 21 2.5 Particle Tracking Velocimetry Velocity information is obtained from the e-m cell using a pa rticle tracking method (PTV) which is similar to the standard technique known as par ticle imaging velocime- try (PIV). The PIV technique uses a camera to capture images o f small particles that seed the ﬂuid ﬂow. Two consecutive images are separated in ti me by a small amount ∆t. These images are sectioned into small regions by a discrete grid. Corresponding regions from the two consecutive images are compared to obta in the average motion ⊳vector∆xof the particles in that region over the time ∆ t. Each region is then assigned a velocity vector ⊳vector u=⊳vector∆x/∆t, yielding an entire velocity ﬁeld on a grid. There are many papers and review articles which describe the PIV techn ique [11, 22, 23, 24], interested readers should refer to these for a complete desc ription. Where PIV attempts to track the average displacement of a num ber of particles (usually around 10) in a square region formed by a grid, PTV at tempts to determine the displacement of individual particles. This allows PTV t o obtain ﬁner spatial resolution than PIV. Equivalently one could say that PTV has higher vector density. Since the algorithms used in PTV and PIV to determine transla tion are identical it might be expected that this increased resolution comes on ly with increased noise. This is not the case, however, because PTV has a built-in self correction that allows noise to be suppressed. There is one sacriﬁce though. Since P TV attempts to track individual particles the technique is much more sensitive t o particles leaving the measurement volume than PIV. This is not a issue in 2D ﬂows suc h as are studied here (except near the boundaries of the images), though in 3D it could be a signiﬁcant problem. A copy of the PTV program is listed in Appendix A. There are three main steps to PTV by which one goes from CCD ima ges of particles to velocity information: particle identiﬁcatio n, neighborhood comparison, and matching. Before going into these, the manner in which im ages are acquired should be described. As stated earlier the soap ﬁlm in the e-m cell is seeded with small particles. These particles are illuminated by two pul sed Nd:Yag lasers that yield 12 mJ of energy per nano-second pulse. Images of the par ticles illuminated by the lasers are obtained using a 30 Hz, 8 bit CCD camera of resol ution 768 ×480 rectangular pixels with aspect ratio 1 : 1 .17. The lasers are slaved to the camera so that the ﬁrst Nd:Yag laser pulses at the end of the ﬁrst image a nd the second laser pulses a time ∆ tlater in the second image. In this manner two images of partic lesChapter 2. The E-M Cell 22 Figure 2.8: Timing of the CCD camera frames and laser pulses u sed in PTV. Frame 1 and Frame 2 denote a single PTV image pair from which velocity ﬁelds are extracted. separated by time ∆ tare obtained on a camera with a ﬁxed time resolution of .03 ms. This timing is shown in Fig. 2.8. To determine the positions of the particles in the ﬂow the bac kground of each image must be subtracted oﬀ. There are many routines by which one can do this; in this thesis a high pass ﬁlter is used since the particles are s patially small compared to the image size. Once the background is subtracted the part icles in each image are found by an exhaustive nearest neighbor searching algorith ms. If the ( i, j) pixel is found to be non-zero then this serves as the base of a particle group. The four pixels at (i+ 1, j), (i−1, j),(i, j+ 1), and ( i, j−1) are said to neighbor the base pixel, and any of these which are non-zero are added to the group. Any nei ghbor of a pixel in the group is then considered, and if it is non-zero and not alr eady part of the group, it is added to the group. This process continues until no pixe ls are being added to the group. The ﬁnal group of non-zero pixels is called a parti cle. This process is performed until all non-zero pixels in each image are accoun ted for in a particle. In what follows the particles in the ﬁrst image will be indexed b yaand those in the second image by b. This manner of ﬁnding a particle does not distinguish betwee n an individual par- ticle and particles that are so close together that they form a continuous image on the CCD camera. Since the turbulence in the e-m cell is only mildl y compressible, par- ticles which start initially very close should stay close ov er the short period of time, ∆t, between laser ﬂashes. The indistinguishability, then, sh ould not be an issue. There are three quantities which need to be determined for ea ch particle: it’sChapter 2. The E-M Cell 23 centroid, pixel centroid and size. The centroid is the cente r of the group of pixels which make the particle weighted by the intensity, I(z) (i,j), of the pixels in the group (z denotes the image). If there are N pixels, indexed by n, in the pixel group of particle ain image one, then the centroid, ( x(a) c, y(a) c), is given by: (x(a) c, y(a) c)≡/summationtextN n=1(i(n), j(n))I(1) (i(n),j(n)) /summationtextN n=1I(1) (i(n),j(n)). Note that where the pixels themselves are discretized on a ﬁn ite grid, the center of mass of a particle need not be if there is more than one pixel co ntained in it’s group. This phenomenon is called sub-pixel resolution since it all ows particle position to be tracked with a resolution ﬁner than the pixel size of the came ra. Sub-pixel resolution can be used to enhance the dynamic range of PTV measurement, t hough it is not relied upon for results in this thesis. The pixel centroid of particle ais deﬁned as the nearest pixel to the particles centroid, and will be denoted as (x(a), y(a)). Finally the size of particle ais simply the 2ndmoment of the intensity distribution given by R(a)≡ /summationtextN n=1((i(n)−x(a) c)2+ (j(n)−y(a) c)2)I(1) (i(n),j(n)) /summationtextN n=1I(1) (i(n),j(n)) 1/2 . The particle size is used as a ﬁlter to discard particles whic h are too big or too small. Once particles have been identiﬁed the challenge is to track them from one image to the next. This is done by comparing the regions surroundin g particles in the ﬁrst image with regions around particles in the second to establi sh how well they correlate. For particle ain the ﬁrst image and bin the second image the correlation number ca,b is determined by ca,b=/summationtextl m=−l/summationtextl n=−l˜I(1) ((x(a)+m),(y(a)+n))˜I(2) ((x(b)+m),(y(b)+n)) (/summationtextl m=−l/summationtextl n=−l(˜I(1) ((x(a)+m),(y(a)+n)))2)1/2(/summationtextl m=−l/summationtextl n=−l(˜I(2) ((x(b)+m),(y(b)+n)))2)1/2. In the above 2 lis called the correlation box size. The intensities, ˜I(z) i,j, used in deter- mining the correlation number are the intensities of the ima gesI(z) i,jless their average over their respective correlation boxes so that ca,bassumes a value between −1 and 1. If the correlation number is close to 1 then the region arou nd particle ais similar to the region around particle b. The closer to 1 the more similar the regions. One need not compare all particles in the ﬁrst image to all par ticles in the second. Only a subset of particles within a certain distance of one an other need be considered.Chapter 2. The E-M Cell 24 That is if a particle ais found at ( x(a) c, y(a) c) in the ﬁrst frame and bis at ( x(b) c, y(b) c) in the second, their correlation number need only be calculate d if ((x(a)−x(b))2+(y(a)− y(b))2)1/2< s, where sis some reasonable threshold displacement based on the RMS velocity ﬂuctuations and ﬂash spacing. Once all the particle comparisons have been performed, one n eed only match particles in the ﬁrst frame to those in the second. This is don e by an iterative mutual maxima technique. Look at correlation number ca,band determine if it is above some initial threshold climit. If it is, then look to see if ca,bis the maximum correlation value for both particle aand particle b. That is make sure ca,b> ca,xfor any x∝negationslash=b andca,b> cy,bfor any y∝negationslash=a. If so then then ca,bis the mutual maximum for particles aandband they are considered a match. Thus particle ahas moved from position (x(a) c, y(a) c) in the ﬁrst frame to ( x(b) c, y(b) c) in the second. The fact that we have checked that not only is particle athe best ﬁt for b, but that bis best ﬁt by ais the self- correction that PTV has that PIV does not (PIV can only check t hatais best for b), and the reason that PTV can achieve higher vector density wit hout much sacriﬁce in velocity resolution. Once matched these particles and their correlation numbers are removed from consideration. This is done for all correlation numbers abo veclimit, and all mutual maxima are obtained in this way. The climitis then lowered slightly and the procedure performed over all particles which have not already been mat ched. This is done until climithits some speciﬁed lowest bound and the particles which have not been matched at this point are discarded. This leaves a ﬁnal list of particles which have been tracked f rom a point in the ﬁrst image to a point in the second. The average velocity of th e particle is determined by the motion of it’s center of mass. This velocity is assigne d to the average particle position. This yields a ﬁeld of average velocities which is t he ﬁnal ﬁeld from the PTV technique. These ﬁelds are generally interpolated to a ﬁnit e grid (binned) so that derivatives may be taken. This interpolation can be perform ed by any number of weighted averaging techniques. 2.6 Cell Operation This section describes the procedure that was used to employ the features of the e-m cell described above. First a marginally thick ﬁlm is pulled across the square frame.Chapter 2. The E-M Cell 25 The plastic edges of the frame are dried to remove any ﬂuid bri dges that might short the electrical current. A small amount of air is removed from the box until the ﬁlm is just level to the eye. This ﬁlm is then placed above the magn et array and a mixing current, generally around 15 mA, is turned on. The feedback l oops to inject ﬂuid and pressure balance the ﬁlm are initiated. After a balance is re ached the magnets are raised (or lowered) to the desired level, and the current is a djusted until the target urmsis reached. Particle images are then acquired at a rate on the order of a few Hz until a large number (between 500-1000) image pairs are obta ined. These pairs are then interrogated using the PTV algorithm to obtain velocit y information.Chapter 3 Modeling Flows in the E-M Cell A frustrating problem that arises when soap ﬁlms are used as a n experimental system for studying 2D ﬂuid dynamics is the lack of direct evidence t hat these ﬁlms obey the 2D incompressible Navier-Stokes equation, ∂ui ∂t+us∂ui ∂xs=−∂p ∂xi+ν∂2ui ∂x2s+Fext i, (3.1) ∂us ∂xs= 0. (3.2) In the above equations uiis the ithcomponent of the ﬂuid’s velocity ﬁeld, pis the internal pressure ﬁeld normalized by ﬂuid density, and Fext irepresents the ithcompo- nent of any external force ﬁeld (per unit mass) acting on the ﬂ uid. The constant ν is the ﬂuid’s kinematic viscosity. Einstein summation conv ention is used, and will be used throughout the thesis unless otherwise noted. These eq uations are the governing equations for the time evolution of the velocity ﬁeld of an in compressible ﬂuid. If the soap ﬁlm does not obey this equation then it is not behaving as an incompressible Navier-Stokes ﬂuid and is therefore useless as a system for s tandard turbulence in- vestigations. The purpose of this chapter is to demonstrate that the e-m cell does indeed approximate a 2D Navier-Stokes ﬂuid. 3.1 Introduction There are a number of reasons to be skeptical about soap ﬁlms b ehaving as an in- compressible Navier-Stokes ﬂuids. Most of the problems ari se from the presence of 26Chapter 3. Modeling Flows in the E-M Cell 27 Figure 3.1: Thin ﬁlm interference fringes demonstrate that the thickness of the soap ﬁlm in the e-m cell is not constant but varies from point to poi nt in the ﬂow. thickness ﬂuctuations which are caused by the motion of the ﬁ lm. Such thickness ﬂuctuations can be imaged using thin ﬁlm interference as in F ig. 3.1 and are indica- tive of compressibility in the soap ﬁlm. Compressibility co nstitutes a failure of Eq. 3.2 in soap ﬁlms. This has already been discussed somewhat in chapter 2 where it was implied that keeping the Mach number small eliminates th is problem. This is a little misleading; a small Mach number does not mean that th ere are no thickness ﬂuctuations. Rather it means that the thickness ﬂuctuation s do not develop shock fronts and vary in a smooth manner from point to point in the ﬂo w. In the range of Mach number present in the e-m cell the thickness ﬂuctuati ons tend to be around 20% the mean thickness of the ﬁlm [11]. One would like to deter mine if this is small enough to allow Eq. 3.2 to hold approximately. Aside from the incompressibility issue, thickness ﬂuctuat ions could cause soap ﬁlms to deviate from a Navier-Stokes ﬂuid in a more sinister w ay. Recall from the discussion in chapter 2 that the kinematic viscosity of the s oap ﬁlm, ν, is dependent on thickness. Since the ﬁlm thickness varies from point to po int in the ﬂow, so shouldChapter 3. Modeling Flows in the E-M Cell 28 the eﬀective viscosity of the ﬂuid. These thickness ﬂuctuat ions respond to velocity gradients in the ﬁlm, therefore the viscosity is dependent o n the local shear rate. A ﬂuid with such a shear dependent viscosity is said to be non-N ewtonian and does not obey Eq. 3.1. Here again one would like to determine if the vis cosity ﬂuctuations are small enough to be considered negligible and Eq. 3.1 to hold a pproximately. Another problem when dealing with soap ﬁlms is the external f rictional coupling to the air. It’s presence is important to attain an energy balan ce in 2D forced turbulence, as discussed in chapter 1. Indeed its strength and form shoul d dictate the outer scale of the turbulence and aﬀect energy transfer at large scales. Because of its importance the eﬀects of this coupling must be modeled and tested. The si mplest model of the frictional eﬀects of the air is to assume it acts as a linear dr ag on the ﬁlm. Therefore the external force ﬁeld Fext iacting on the e-m cell can be broken into two parts, the Lorentz force caused by current and magnetic ﬁeld, Fi, and the air drag Fair i=−αui. The constant αrepresents the strength of the frictional coupling of the ai r to the ﬁlm. This model must be tested if it is to be used in later investiga tions. Though a direct test of the Navier-Stokes equations by inver se methods is not easily performed, it is possible to test an equation known as the Karman-Howarth relationship. This relationship can be derived from the Nav ier-Stokes equation with a single assumption, and easily tested with data from the e-m cell. It’s failure or success in describing data from the e-m cell would then const itute an indirect test of the Navier-Stokes equation as well as the linear drag mode l proposed for the air friction. 3.2 The Karman-Howarth Relationship Although the derivation of the Karman-Howarth relationshi p can be found in a num- ber of texts on turbulence, it is performed here for two purpo ses: the relationship is used extensively in later chapters and to present notation w hich will be used through- out the thesis. It is also performed with the inclusion of a li near damping term in the Navier-Stokes equation to represent the air friction as discussed earlier. The derivation here, with the exception of the air drag, closely follows that found in Hinze [25]. The Karman-Howarth relationship governs the time evolutio n of the two-point velocity correlation, ∝angb∇acketleftui(⊳vector x)uj(⊳vectorx′)∝angb∇acket∇ight, for a ﬂuid in a state of homogenous turbulence. TheChapter 3. Modeling Flows in the E-M Cell 29 brackets ∝angb∇acketleft...∝angb∇acket∇ightrepresent an ensemble average. This relationship can be der ived from the incompressible Navier-Stokes equation using only the a ssumption of homogeneity in the following manner. Multiply Eq. 3.1 which is evaluated for the ithcomponent of the velocity ﬁeld at the point ⊳vector xwith the jthcomponent of the ﬁeld at point ⊳vectorx′: u′ j∂ui ∂t+∂ ∂xs(uiusu′ j) =−∂ ∂xi(pu′ j) +ν∂2 ∂x2s(uiu′ j) +u′ jFi−αuiu′ j. (3.3) In the above, ﬁeld quantities at the point ⊳vectorx′are denoted by a ′and the linear drag model has been explicitly inserted into the equation. The fa ct that the derivative at point⊳vector xcommutes with multiplication by ﬁelds evaluated at ⊳vectorx′has been used to move u′ jinside spatial derivatives. Incompressibility has also be en used in the second term on the left-hand-side to move usinto the derivative. Add Eq. 3.3 with the corresponding equation evaluated by mul tiplying Eq. 3.1 evaluated for the jthcomponent of the velocity ﬁeld at the point ⊳vectorx′with the ith component of the ﬁeld at point ⊳vector x. This allows both velocity terms to be brought into the time derivative, ∂ ∂t(uiu′ j) = −∂ ∂xs(uiusu′ j)−∂ ∂x′s(uiu′ su′ j)−∂ ∂xi(pu′ j)−∂ ∂x′ j(p′ui) +ν∂2 ∂x2 s(uiu′ j) +ν∂2 ∂x′2 s(uiu′ j) +u′ jFi+uiF′ j−2αuiu′ j.(3.4) A coordinate transformation to relative, ri≡x′ i−xi, and absolute, ξi≡1/2(x′ i+xi), coordinates can now be performed. Grouping the appropriate terms yields: ∂ ∂t(uiu′ j) = −1 2∂ ∂ξs(uiusu′ j+uiu′ su′ j)−∂ ∂rs(uiu′ su′ j−uiusu′ j) −1 2∂ ∂ξi(pu′ j)−1 2∂ ∂ξj(p′ui) +∂ ∂ri(pu′ j)−∂ ∂rj(p′ui) +1 2ν∂2 ∂ξ2s(uiu′ j) + 2ν∂2 ∂r2s(uiu′ j) +u′ jFi+uiF′ j−2αuiu′ j.(3.5) Now an ensemble average is performed. Using the assumption o f homogeneity eliminates the derivative of averages with respect to absol ute position, ξi, leaving only derivatives with respect to relative position, ri. What remains is called the Karman-Howarth relationship, ∂ ∂t∝angb∇acketleftuiu′ j∝angb∇acket∇ight=−∂ ∂rs∝angb∇acketleftuiu′ su′ j−uiusu′ j∝angb∇acket∇ight+∂ ∂ri∝angb∇acketleftpu′ j∝angb∇acket∇ight −∂ ∂rj∝angb∇acketleftp′ui∝angb∇acket∇ight +2ν∂2 ∂r2 s∝angb∇acketleftuiu′ j∝angb∇acket∇ight+∝angb∇acketleftu′ jFi∝angb∇acket∇ight+∝angb∇acketleftuiF′ j∝angb∇acket∇ight −2α∝angb∇acketleftuiu′ j∝angb∇acket∇ight. (3.6)Chapter 3. Modeling Flows in the E-M Cell 30 In the future the n-term two-point velocity correlation fun ctions will be denoted by b(n) ij...,k... ≡ ∝angb∇acketleftuiuj...u′ k...∝angb∇acket∇ight. Using this notation, the correlation on the left hand side o f Eq. 3.6 is given by b(2) i,j, while the ﬁrst term in the ﬁrst derivative on the right is giv en byb(3) i,sj. This relationship can also be used to derive the energy balan ce equation for ho- mogenous turbulence. Energy balance will be used in what fol lows as a self consis- tency check for data that attempts to ﬁt the Karman-Howarth r elationship. Taking the limit of Eq. 3.6 as ⊳vector r→0 (or equivalently as ⊳vector x→⊳vectorx′)yields ∂ ∂t∝angb∇acketleftuiuj∝angb∇acket∇ight= Π ij−ǫij+∝angb∇acketleftuiFj∝angb∇acket∇ight −2α∝angb∇acketleftuiuj∝angb∇acket∇ight. (3.7) In the above the tensors Π ijandǫijare deﬁned as Πij≡lim r→0(∂ ∂ri∝angb∇acketleftpu′ j∝angb∇acket∇ight −∂ ∂rj∝angb∇acketleftp′ui∝angb∇acket∇ight), (3.8) ǫij≡lim r→02ν∂2 ∂r2 s∝angb∇acketleftuiu′ j∝angb∇acket∇ight. (3.9) Taking half the trace of Eq. 3.7 eliminates the pressure term Πijand yields the energy balance relationship 1 2∂ ∂tu2 rms=−νω2 rms+∝angb∇acketleftusFs∝angb∇acket∇ight −αu2 rms, (3.10) where the vorticity, ω, is deﬁned as the curl of the velocity ﬁeld (i.e. ω=⊳vector∇ ×⊳vector u). The ﬁrst term on the right, νω2 rms≡ǫν, is the amount of energy changed to heat by internal viscous dissipation. The second, ∝angb∇acketleftusFs∝angb∇acket∇ight ≡ǫinj, is the work done by the external force. The ﬁnal term, αu2 rms≡ǫair, is the energy lost to the linear drag. Equation 3.10 is the statement that the change in energy in th e system is simply the amount gained from the external forcing less the amount lost to dissipative eﬀects. 3.3 Testing Karman-Howarth 3.3.1 Experimental Considerations Recall that the objective of the experiments presented in th is section is to demon- strate that the dynamics of the e-m cell are governed by the Na vier-Stokes equation,Chapter 3. Modeling Flows in the E-M Cell 31 Eq. 3.1, with the eﬀects of air friction modeled as a linear dr ag. This will be demon- strated indirectly by showing that the Karman-Howarth rela tionship, Eq. 3.6, holds for homogenous turbulence in the e-m cell. The number of term s which must be measured to check Eq. 3.6 can be simpliﬁed by using speciﬁc ch aracteristics of the e-m cell. The ﬁrst is the elimination of the time derivative. This term can be ignored if the turbulence is in a statistically steady state. Since t he e-m cell was designed speciﬁcally to study steady state turbulence it is easy to ma intain energy and enstro- phy approximately constant. Elimination of the time deriva tive in this manner is the main reason that testing of the Karman-Howarth relationshi p is signiﬁcantly easier than directly testing the Navier-Stokes equation. Another term which can be dropped is the viscous term, if one c onsiders length scales, r, greater than the viscous scale rν≈(ν3/ǫinj)1/4. For typical values of energy injection in the e-m cell rν= 200 µm. Since the particle tracking measurements focus on the inverse cascade regime, which occurs at length s cales greater than a millimeter in the e-m cell, most of the measurement resoluti on lies well above this criteria. Since the viscous term is being ignored these expe riments can draw no conclusions about how thickness changes may be aﬀecting cha nges in viscosity. Small scale investigations, outside the scope of this thesis, wou ld have to be performed to draw conclusions about this eﬀect. What remains of the Karman-Howarth relationship after usin g these assumptions is ∂ ∂rs(b(3) i,sj−b(3) is,j)−∂ ∂ri∝angb∇acketleftpu′ j∝angb∇acket∇ight+∂ ∂rj∝angb∇acketleftp′ui∝angb∇acket∇ight=∝angb∇acketleftu′ jFi∝angb∇acket∇ight+∝angb∇acketleftuiF′ j∝angb∇acket∇ight −2αb(2) i,j. (3.11) Normally the assumption of isotropy would also be made to eli minate the pressure- velocity correlations on the left-hand-side. Recall from t he discussion in chapter 2 that this assumption cannot be made in the e-m cell due to unid irectional forcing. Therefore to check Eq. 3.11 a pressure-velocity correlatio n must be measured, indi- cating that not only is a velocity ﬁeld needed for the check bu t a pressure ﬁeld as well. To obtain the pressure ﬁeld, the divergence operator is appl ied to Eq. 3.1 and Eq. 3.2 is used. What is left has the form ∇2p= 2Λ + ∇ ·⊳vectorF (3.12) where Λ ≡∂ux ∂x∂uy ∂y−∂ux ∂y∂uy ∂x. If the Kolmogorov magnet array is used, the divergenceChapter 3. Modeling Flows in the E-M Cell 32 of the electromagnetic force on the right may be dropped (see chapter 2). This allows the pressure to be approximated from the velocity ﬁeld using Fourier techniques1. For this reason the Kolmogorov array will be used in these exp eriments. With Kolmogorov forcing and the assumptions above, all the t erms in Eq. 3.11 may be measured and tested as an indirect test of the Navier-S tokes equation and linear drag model. One ﬁnal simpliﬁcation can be made. An exa ct measure of the external electromagnetic forcing is diﬃcult at best. Howev er, since the forcing is uni- directional, along the ˆ xdirection, the force-velocity correlation terms can be dro pped if the ( i, j) = (y, y) component of the tensor equation is considered. This is eas ily done leaving the ﬁnal equation to be tested: ∂ ∂rs(b(3) y,sy−b(3) ys,y)−∂ ∂ry∝angb∇acketleftpu′ y∝angb∇acket∇ight+∂ ∂ry∝angb∇acketleftp′uy∝angb∇acket∇ight=−2αb(2) y,y. (3.13) All of the terms in the above can be measured, and the constant αcan be used as a single free ﬁtting parameter. The quality of the ﬁt will det ermine if the Karman- Howarth relationship holds in the e-m cell, and therefore by implication the Navier- Stokes equation and linear drag model. Such a detailed compa rison between theory and experiment has not been performed before for 2D soap ﬁlm s ystems. 3.3.2 The Data A single run of the e-m cell using Kolmogorov forcing was perf ormed over a time span of ∼300s during which one thousand vector ﬁelds were obtained by PTV. The cell was driven at a voltage 40 V with a current of 40 mA oscilla ting with a square wave form at 5 Hz. The magnet array was placed approximately 1 mm below the ﬁlm. This resulted in velocity ﬂuctuations of around 11 cm/s over the time of the experiment. A typical velocity ﬁeld that is obtained by binn ing the particle tracks is shown with the associated pressure ﬁeld derived using the me thod described above in Fig. 3.2. The ﬁrst order of business is to check that the assumptions of incompressibility and that the system is in a steady state are accurate. Shown in Fig. 3.3a is the time dependence of the velocity and vorticity ﬂuctuations for th e run. The ﬂuctuations are 1This approximation of the pressure ﬁeld assumes periodic bo undary conditions. Though the velocity ﬁelds extracted from the e-m cell do not satisfy thi s boundary condition it is hoped that the solution for the pressure ﬁeld will be insensitive to this ap proximation away from the boundariesChapter 3. Modeling Flows in the E-M Cell 33 Figure 3.2: Typical velocity (a) and pressure (b) ﬁelds obta ined from the e-m cell. In the pressure ﬁeld green denotes positive and blue negative v alues.Chapter 3. Modeling Flows in the E-M Cell 34 not exactly constant, but the change is negligible due to the fact that it happens over a long time, i.e. the average time derivative is approximate ly zero. The steady state assumption is therefore approximately correct. The incomp ressibility assumption can be tested by measuring the divergence of the ﬂow, D≡⊳vector∇ ·⊳vector u, and normalizing its square by the enstrophy Ω ≡ω2 rms. This forms a dimensionless quantity which must be small if the divergence eﬀect is to be ignorable. In the e-m cellD2/Ω≈0.1 over the time of the run as shown in Fig. 3.3b. This indicates that t he divergence is not overly large and incompressibility can be assumed. Coincid entally this number is also close to the Mach number of the system, which the reader will r ecall was kept small for the purpose of minimizing compressibility. The ﬁnal assumption necessary to check before Karman-Howar th is applicable to the system is homogeneity. That is the average quantities in the turbulence should be invariant with respect to translation. A crude test of hom ogeneity is to measure the mean, ∝angb∇acketleft⊳vector u(⊳vector x)∝angb∇acket∇ightN, and RMS ﬂuctuation, ( ∝angb∇acketleft\|⊳vector u(⊳vector x)\|2∝angb∇acket∇ightN)1 2, of the velocity as a function of position, where Ndenotes the number of ﬁelds the quantity is averaged over. Bo th should be independent of position for homogeneity to hold. M oreover, since the ﬁlm in the e-m cell does not have a net translation, the mean ﬂow ev erywhere should be identically zero. Figure 3.4a shows the the mean ﬂow for th e run after having averaged over the thousand images ( N= 1000). One can see that there still exists a small mean. Though at ﬁrst this is discouraging, it is misle ading since a ﬁnite amount of data will almost never converge exactly to zero. Ra ther the magnitude of the ﬂuctuations in the mean shear should decrease as N−1/2if one assumes the ﬂuctuations away from the mean are behaving as a centered Gau ssian variable. To this end, the RMS ﬂuctuations of the mean ﬂow, ∝angb∇acketleft⊳vector u(⊳vector x)∝angb∇acket∇ightN rms, is plotted as a function Nin Fig. 3.4b. It is clear that the magnitude of the ﬂuctuation s in the mean is decreasing almost perfectly as N−1/2, indicating that the mean ﬂow as N→ ∞ should go to zero as required by homogeneity. Although the mean ﬂow is constant (since it’s zero) and satis ﬁes the requirement for homogeneity, the RMS ﬂuctuations, ( ∝angb∇acketleft\|⊳vector u(⊳vector x)\|2∝angb∇acket∇ightN)1 2, does not. This can be seen by looking at Fig. 3.5 which shows the RMS ﬂuctuations of the two velocity components averaged over the thousand images. Although the ˆ ycomponent of the velocity ﬂuctu- ations is approximately constant, the ˆ xﬂuctuations display strong striations. These striations reﬂect the Kolmogorov forcing, as they must if th e electromagnetic force is to inject energy into the system. That is, some part of uxmust be non-randomChapter 3. Modeling Flows in the E-M Cell 35 Figure 3.3: (a) Time dependence of urmsandωrmsfor a single run in the e-m cell. This demonstrates that the e-m cell is in an approximately steady state. (b) Time depen- dence of the enstrophy normalized mean square divergence, D2/ω2 rms, for a single run in the e-m cell. The fact that D2/ω2 rmsis small indicates negligible compressibility.Chapter 3. Modeling Flows in the E-M Cell 36 Figure 3.4: (a) The mean ﬂow in the e-m cell averaged over 1000 vector ﬁelds. The length of the reference vector in the upper right correspond s to 2 cm/s. (b) The decay of the ﬂuctuations in the mean ﬂow as the number of ﬁelds ,N, in the average increases. The line corresponds to the expected N−1/2decay of a centered Gaussian variable.Chapter 3. Modeling Flows in the E-M Cell 37 Figure 3.5: The RMS ﬂuctuations of (a) uxand (b) uyas a function of position in the e-m cell. Green denotes large values of the ﬂuctuations w hile blue denotes small values.Chapter 3. Modeling Flows in the E-M Cell 38 and in phase with the forcing otherwise ∝angb∇acketleft⊳vectorF·⊳vector u∝angb∇acket∇ight= 0 and the e-m cell could not be maintained in an energetically steady state. Since the forc ing is oscillating in time so must this in-phase component; thus it shows up in the RMS ﬂuct uations as a function of position and not in the mean ﬂow. Fortunately the magnitud e of the in phase part of the ﬂuctuations is small, around 1 .5 cm/s, compared to the total RMS ﬂuctuations of 12 cm/s. Therefore they will be assumed to be ignorable. Ot her than these os- cillations, the cell appears approximately homogenous in t he RMS ﬂuctuations as a function of position. The assumption of homogeneity can be s aid to weakly hold for the turbulence in the e-m cell. This approximation will be re ﬁned in later chapters. The Karman-Howarth relationship is now in a position to be te sted. For simplicity, deﬁne Ai,j≡∂ ∂rs(b(3) i,sj−b(3) is,j), (3.14) Bi,j≡ −∂ ∂ri∝angb∇acketleftpu′ j∝angb∇acket∇ight+∂ ∂rj∝angb∇acketleftp′ui∝angb∇acket∇ight, (3.15) so that the ( y, y) component of the Karman-Howarth relationship, Eq. 3.13, m ay be written Ay,y+By,y=−2αb(2) y,y. The three separate terms Ay,y,By,yandb(2) y,ywere measured and a least squares algorithm used to obtain the αvalue which best ﬁt the measured data to the Karman-Howarth equation. In this ca seα≈0.7 Hz. The results of these measurements are shown in Fig. 3.6 a,b,d. In Fig. 3.6c the sum Ay,y+By,yis shown. First note that By,y, the term involving pressure velocity correlations is non- zero, as it would be if the turbulence were anisotropic. This conﬁr ms what was earlier assumed to be the case, that the unidirectional forcing does not allow isotropy to be assumed. Next note that the images in Fig. 3.6c and d have ve ry similar forms, namely a central negative trough with two positive peaks on t herxaxis. This is evidence that the Karman- Howarth relationship is indeed ho lding in the e-m cell. To get a better feel for the degree to which there is agreement , several cross-sections of plots c and d are shown in Fig. 3.7. The noise in the terms Ay,yandBy,yarises from the fact that these terms are derivatives, which are alw ays noisy and converge slowly in experiment. In spite of the noise there is clearly a greement, and it is therefore concluded that Karman-Howarth, and hence the Navier-Stoke s equation with a linear drag, holds for the e-m cell.Chapter 3. Modeling Flows in the E-M Cell 39 Figure 3.6: Measured values of (a) Ay,y, (b)By,y, (c)Ay,y+By,y, and (d) −2αb(2) y,y from Eq. 3.13. Green denotes positive values and blue denote s negative values.Chapter 3. Modeling Flows in the E-M Cell 40 Figure 3.7: Cross sections of Ay,y+By,y(−·) and −2αb(2) y,y(−) along the lines (a) r=rx(ry= 0), (b) r=ry(rx= 0) and (c) r=rx=ry.Chapter 3. Modeling Flows in the E-M Cell 41 A quick check to see if this conclusion is correct is to see if t he measured coeﬃcient for the linear drag, α, is viable. Using the discussion in chapter 2 the linear drag coeﬃcient can be approximated as α=η/ρhd . Given a 50 µm thick ﬁlm, a magnet- ﬁlm distance of 1 mm the coeﬃcient assumes the value 0 .36 Hz. This value is at least the right order of magnitude, and the diﬀerence between this predicted value and the measured one may be accounted for by recalling that the drag o n the top surface of the ﬁlm has been ignored(see chapter 4). 3.3.3 Consistency Check: Energy Balance The previous section has checked that the ( y, y) component of the Karman-Howarth equation is consistent with measurements made in the e-m cel l. However, the ﬁt to the data was somewhat noisy in spite of a thousand ﬁelds being used in the average. What is needed to bolster conﬁdence in the equations is some s ort of consistency check. This is provided by the energy balance relationship, Eq. 3.10. The energy balance statement for the time independent ﬂow si mply states that the energy injected into the system by the electromagnetic forc e,ǫinj, must be balanced by the energy lost to the air friction, ǫair, and viscosity, ǫν. The later two of these can now be measured using the deﬁnitions of the various ǫ’s given earlier, the extracted value of α≈0.7 Hz and the kinematic viscosity of ν≈0.016 cm2/s. The energy dissipated by air is found to be ǫair≈85 cm2/s3, while the energy dissipated by viscous forces is ǫν≈55 cm2/s3. Using energy balance this suggests that the energy injected by the electromagnetic force should be ǫinj≈140 cm2/s3. An independent measure of ǫinjwould then yield a consistency check of the measured value of αand the quality of agreement of data to the ( y, y) component of the Karman-Howarth relationship. This check is provided by the ( x, x) component of the Karman-Howarth relation- ship which allows a measure of ǫinj. This component of the Karman-Howarth equation has the form Ax,x+Bx,x=∝angb∇acketleftu′ xFx∝angb∇acket∇ight+∝angb∇acketleftuxF′ x∝angb∇acket∇ight −2αb(2) x,x. (3.16) Recall that the Kolmogorov forcing is invariant to translat ion in the ˆ xdirection, that is, along the forcing. Let us then restrict the displacement vector ⊳vector rto lie along this direction. Then ∝angb∇acketleftu′ xFx∝angb∇acket∇ight=∝angb∇acketleftu′ xF′ x∝angb∇acket∇ightwhich by homogeneity equals ∝angb∇acketleftuxFx∝angb∇acket∇ight=∝angb∇acketleftuxF′ x∝angb∇acket∇ight. ButChapter 3. Modeling Flows in the E-M Cell 42 Figure 3.8: Cross section of Ax,x+Bx,x(−·) and −2αb(2) x,x(−) along the line r=rx (ry= 0). ǫinj=∝angb∇acketleftuxFx∝angb∇acket∇ight. Using these relationships in Eq. 3.16 yields Ax,x+Bx,x= 2ǫinj−2αb(2) x,x along the ⊳vector r=rxcross-section. Thus the plots of Ax,x+Bx,xand 2αb(2) x,xshould look similar to the ones shown earlier except oﬀset by a constant w hich is equal to 2 ǫinj. Figure 3.8 shows the plot of the rxcross-section of the ( x, x) components of Ax,x+ Bx,xand−2αb(2) x,x. Clearly the plots are oﬀset by positive constant of 2 ǫinj≈240 cm2/s3. Thus ǫinj≈120 cm2/s3which is close to the expected value of 140 cm2/s3. Systematic data discussed in the next chapter will show that the extracted values of α andǫinjﬂuctuate around ±20%, thus the measured value of ǫinjis within experimental error of the expected value. This is further supporting evid ence that the Karman- Howarth relationship does indeed work for data extracted fr om the e-m cell.Chapter 4 Energy Distribution and Energy Flow In the last chapter the energy balance relationship, ǫinj−ǫν−ǫair= 0, was used as a consistency check to determine if the Karman-Howarth rela tionship was applicable to data from the e-m cell. Though it was not discussed there, t hese measurements are the ﬁrst indication that an inverse cascade is present. R ecall from the discussion in chapter 1 that if an inverse cascade exists then energy is m oved from small length scales to large ones, away from length scales at which viscou s dissipation is eﬀective. Thus the bulk of the energy is expected to be dissipated by the external dissipation mechanism acting at large scales, which in the case of the e-m cell is the air dissipation. This is exactly the result that the measured energy rates in t he e-m cell demonstrate, i.e.ǫair> ǫν. This chapter begins the investigation of the inverse energ y cascade in the e-m cell by quantifying the length scales over which it ex ists (it’s range), measuring how the energy is distributed over these length scales and de termining the rate at which energy ﬂows through these length scales. 4.1 Distribution of Energy and the Outer Scale The energy spectrum, U(⊳vectork), provides a means for describing the manner in which turbulent kinetic energy is distributed over diﬀerent wave numbers (inverse length scales) in the e-m cell. It is deﬁned as the average square mod ulus of the Fourier 43Chapter 4. Energy Distribution and Energy Flow 44 transform of the velocity ﬂuctuations, U(⊳vectork)≡ ∝angb∇acketleft˜u(⊳vectork)˜u†(⊳vectork)∝angb∇acket∇ight1, and it’s circular inte- gral,E(k)≡/integraltext2π 0\|k\|U(⊳vectork)dθ, denotes the average amount of kinetic energy stored in wavenumbers of modulus k. If an inverse cascade is present in the e-m cell, the expectation is that energy would build up in wavenumbers sma ller than the energy injection wavenumber kinj. This expectation proves to be the case in the e-m cell. Fig. 4. 1 are plots of U(⊳vectork) calculated from transforms of the PTV velocity ﬁelds for var ious types of driving force in the e-m cell. Note that the peaks corresponding to th e injection wavenumber diﬀer as expected for the diﬀerent types of forcing. For exam ple the two square arrays produce peaks along the line kx=±ky, with the distance from 0 being dictated by the size of the magnets used in the array. Also note that energ y contained in small wavenumbers is greater than that contained in large wavenum bers, as expected for an inverse cascade. This can be better seen in Fig. 4.2 where t he circular integrals have been taken and a build up of energy at wavenumbers smalle r than kinjcan be seen. The Kraichnan prediction for the inverse energy cascade ran ge is that E(k)∼ ǫ2/3k−5/3fork < k injandǫa typical energy rate (in the case of the e-m cell this is ǫair) [1]. This result is consistent with dimensional predictio ns for the scaling behavior ofE(k). Lines corresponding to this prediction have been drawn on the plots in Fig. 4.2. These lines should not be interpreted as a ﬁt to the data a nd are drawn only as a guide. Only the Kolmogorov data set, (a), appears to be in qu alitative agreement with the dimensional prediction over slightly less than hal f a decade of wavenumbers below the injection wavenumber. All the remaining data sets have a small range directly below the injection wavenumber which could be inte rpreted as k−5/3, but this is quickly lost to a broad peak in the spectrum at small k. This type of behavior in the spectrum is in agreement with results reported in [13] where the build up of energy at small kis associated with the saturation of energy in the largest le ngth scales in the system. Such saturation is not included in the K raichnan prediction and is therefore not indicative of failure of the theory, but rat her a failure of the system to satisfy the assumptions of the theory2. Since case (a) satisﬁes the assumptions of 1†denotes a complex conjugate. 2The assumption which is violated in a saturated system is tha t the velocity ﬂuctuations are homogenous. A saturated system occurs when the outer scale w hich is determined by external dissipation exceeds the system size, as it did for the data in Fig. 4.2(b)-(d). When this happens large structures attempt to pack into a small area near the ce nter of the system away from systemChapter 4. Energy Distribution and Energy Flow 45 Figure 4.1: The energy spectrum, U(⊳vectork), for (a) Kolmogorov forcing, (b) square forcing using 6 mm round magnets, (c) square forcing using 3 mm round m agnets and (d) stretched hexagonal forcing. Green denotes large values of U(⊳vectork) while blue denotes small values.Chapter 4. Energy Distribution and Energy Flow 46 Figure 4.2: The circularly integrated energy spectrum, E(k), for (a) Kolmogorov forcing, (b) square forcing using 6 mm round magnets, (c) squ are forcing using 3 mm round magnets and (d) stretched hexagonal forcing. The dash ed lines correspond to the Kraichnan prediction that E(k)∝k−5/3[1]. The arrows indicate the injection wavenumber kinj.Chapter 4. Energy Distribution and Energy Flow 47 Kraichnan’s theory one might conclude that the k−5/3prediction is correct. However, it should also be noted that the behavior in the measured E(k) depends on the type of window function used when Fourier transforming the veloc ity ﬁelds. In the data shown in the Fig. 4.2, a three term Blackman-Harris window ha s been used. By changing this window one could get up to a 20% change in the slo pe of the energy spectra. Due to the limitations imposed by windowing no conc lusion may be drawn from this data about possible corrections to the Kraichnan s caling prediction. The low klimit of the inverse cascade range, denoted kout, is determined by the size of the largest vortices which result from the inverse ca scade, and corresponds to the low kpeak in E(k). The position of this peak should depend on the strength of the air friction since this is the large scale external dis sipation mechanism in the e-m cell. In the chapter 3 the linear damping model for the air friction possessed a coeﬃcient αwhich determines it’s strength. Using dimensional analysi s,α, andǫinj, a length scale called the outer scale can be calculated by rout≡(ǫinj/α3)1/2. The outer scale represents the size of the vortices at which more energ y is lost to air friction than is transferred to the next size larger vortices. The out er scale should be related to the low kpeak in E(k) bykout= 2π/rout. To check this dimensional prediction, a systematic set of da ta using the Kol- mogorov forcing with various magnet-ﬁlm distances was take n holding urmsapproxi- mately constant. Kolmogorov forcing was used so that αandǫinjcould be extracted using the techniques of chapter 3. Between 400 and 500 vector ﬁelds where obtained for each magnet-ﬁlm distance. As in chapter 3, the energy in t he e-m cell remained approximately constant during the data acquisition time fo r each run. Table 4.1 lists the various constants associated with each of the diﬀerent d ata sets. The ﬁrst four data sets listed in Table 4.1 may be compared for the purpose of error analysis. The ﬁrst and second of these were obtained us ing identical values of the external control parameters and thus the extracted valu es ofαandǫinjshould be identical. This is found to be the case up to two signiﬁcant ﬁgures for the ﬁrst two data sets. The third and fourth data sets are also taken un der identical control conditions diﬀerent from those used for the ﬁrst and second d ata sets (the magnet-ﬁlm distance was slightly smaller). In these data sets the value s ofαandǫinjvary around the mean by about 10%. Thus, to be conservative, the error in t he two quantities α boundaries. This will be investigated in some detail later i n this chapter.Chapter 4. Energy Distribution and Energy Flow 48 ǫinj(cm2/s3)α(s−1)urms(cm2/s2)ω2 rms(s−2)2π/kout(cm)rint(cm)case 63 0.45 7.81 2049 3.14 0.63 a 63 0.45 7.80 2014 3.14 0.64 101 0.6 8.96 2901 2.72 0.59 b 110 0.65 9.18 3103 2.63 0.58 150 0.9 9.18 3579 1.56 0.51 c 154 1.25 7.81 3211 1.31 0.41 197 1.55 7.95 3567 1.31 0.38 d Table 4.1: Global constants for several runs of the e-m cell u sing Kolmogorov forcing andǫinjwill be assumed to be as much as 20% of the measured value. TheE(k) measured from the data sets labeled a,b,c and d are displaye d in Fig 4.3. The low wavenumber peak, kout, moves to smaller and smaller wavenumber as αdecreases. This is in qualitative agreement with the dimens ional prediction. Using the position of this peak the outer scale is calculated by rout= 2π/koutand compared to that obtained from the dimensional prediction using the m easured values of α andǫinj. The results of this comparison are shown for all the data set s in Fig. 4.4. The vertical error bars reﬂect the propagated error of ǫinjandαwhile the horizontal represent the discretization inherent in ﬁnite Fourier tra nsforms. From Fig. 4.4 one can see that the dimensional prediction of the outer scale is not inconsistent with the measured outer scale using the low kpeak in E(k). The measurements presented here clearly indicate that an in verse cascade is present in the e-m cell for all types of forcing. The inverse c ascade range is shown to exist over wavenumbers ksuch that kout< k < k inj, where kinjis the energy in- jection wavenumber determined by the electromagnetic forc ing and koutis the outer wavenumber determined by the external dissipation. koutis found to be not inconsis- tent with the dimensional prediction, kout∼(ǫinj/α3)1/2. No strong conclusion can be draw about the manner in which energy is distributed over t his range due to win- dowing diﬃculties, however the Kraichnan prediction of E(k)∼k−5/3superﬁcially holds for data sets in agreement with the assumptions of the p rediction. Before leaving this section, the systematic data set used in obtaining the outer scale allow the external dissipation to be compared to the pr edicted value α= ηair/ρhd. The measured values of αversus the magnet-ﬁlm distance are shown inChapter 4. Energy Distribution and Energy Flow 49 Figure 4.3: The circularly integrated energy spectrum, E(k), for the four cases of Kolmogorov ﬂow labeled in Table 4.1. Figure 4.4: Comparison of the outer scale obtained from the e nergy spectra by rout= 2π/koutwith that obtained using the dimensional prediction rout= (ǫinj/α3)1/2for all of the data sets in Table 4.1.Chapter 4. Energy Distribution and Energy Flow 50 Figure 4.5: The measured linear drag coeﬃcient, α, versus the magnet-ﬁlm distance, d, for the data sets reported in Table 4.1. The dotted line repr esents the ﬁt α= ηair/ρhd+CwithC= 0.25 Hz. Fig. 4.5. The vertical error bars again reﬂect the 20% error i n measurement while the horizontal error bars denote the limit of control over ma gnet-ﬁlm distance. A line corresponding to ηair/ρhd+C, where C= 0.25 Hz is also plotted with the data and for the most part is within error of the measured values. Acco rding to this data the prediction for the magnitude of the linear dissipation must be oﬀset by a small posi- tive constant to be accurate. This constant can be accounted for by recalling that the eﬀect of air friction on the top surface of the ﬁlm has been ign ored. Approximations of the frictional force on the top surface of the ﬁlm indicate that the measured value of the oﬀset is appropriate, though more experimentation ne eds to be done to more accurately account for this oﬀset. 4.2 Energy Flow A simpliﬁed viewpoint of how kinetic energy might be transfe red through the inverse cascade range in the e-m cell is to imagine that the energy is ﬂ owing like a liquid through a pipe. Energy produced by the forcing is poured in at one end of the pipe which characterizes the injection scale. It is then moved al ong the pipe to larger length scales by the mixing of ﬂuid (i.e. by vortex cannibali zation). Finally energyChapter 4. Energy Distribution and Energy Flow 51 is exhausted from the pipe at the opposite end which characte rizes the outer length scale. Since the total energy in the system is constant, the a mount of energy poured into the pipe must be equal to the amount exhausted from the pi pe. It is also expected that the rate of energy being poured into the pipe is equivale nt to the rate of energy transferred across any length scale in the middle of the pipe . That is to say that the energy ﬂux through the pipe is not dependent on the position i n the pipe and remains constant. This viewpoint is a severe simpliﬁcation of what actually ha ppens in 2D turbu- lence. First it ignores losses of energy to viscous dissipat ion and assumes all energy is lost to external dissipation. Since viscous losses are pr esumably smaller than losses to external dissipation, let us accept for now that this simp liﬁcation is valid. A more important simpliﬁcation, the one which is of concern in this section, is that the pipe doesn’t leak. That is there are no holes drilled along the len gth of the pipe. That is to say energy cannot be exhausted from the pipe by external dissipation until the outer length scale is reached. To hope that this is actually t he case in the e-m cell, or for that matter any other laboratory 2D turbulence system is quite a stretch. It is more likely that there exists a range in the pipe, probab ly close to the injec- tion scale since external dissipation increases at large le ngth scales, over which the amount of energy lost to leaks is negligible compared to the e nergy ﬂux through the pipe. This range will be called an “inertial” range since the energy ﬂow is almost entirely dictated by the ﬂuid’s inertia and not the energy di ssipation. The inertial range is of interest due to it’s universality. Presumably tw o 2D turbulent systems with completely diﬀerent external dissipation mechanisms will behave identically in their inertial ranges. To use the pipe analogy, the inertial range of both systems is completely closed and thus ﬂuid mixing and energy ﬂux shou ld be the same in this region. Outside of this range the pipe leaks, and the ene rgy ﬂux depends on how many holes of what size are drilled at what position in the pipe. Therefore the characteristics of the turbulence may not be universal outs ide of the inertial range. To determine if a range is inertial or not, a measurement of en ergy ﬂow must be made. One way in which energy ﬂow may be characterized is by th e third moment of velocity diﬀerence. The third moment, labeled S(3)(r) for now, can be thought of as the average energy per unit mass advected over a circle of r adius rcentered at ⊳vector xChapter 4. Energy Distribution and Energy Flow 52 per unit circumference of the circle: S(3)(r) =1 2πr/integraldisplay2π 0∝angb∇acketleftE(⊳vector x+⊳vector r)⊳vector u(⊳vector x+⊳vector r)·⊳vector r∝angb∇acket∇ightdθ. Assume also that the reference frame is moving so that the vel ocity⊳vector u(⊳vector x) = 0. As- suming homogeneity the circle can be placed anywhere in the t urbulence, so that S(3)does not depend on ⊳vector x.Ein the above is the energy per unit mass. If the third moment is positive, then on average energy is being advected from inside the circle to outside the circle, and vice versa if the third moment is nega tive. It is interesting to note what happens if ris assumed to be in an inertial range. If this is the case then no energy is lost to external dissipation at that length scal e. Since the energy held in the turbulence at a scale ris in a steady state then all of the energy injected by the forcing into the circle must be advected over the surface of t he circle. Call ǫthe rate of energy injection per unit mass into the system. The rate at which energy is injected into the circle is then πr2ǫ. Replacing the integral with this yields S(3)(r)≈ǫr. Thus a linear range in the third moment indicates inertial behavi or of the energy transfer. The above derivation of a linear behavior in the third moment for an inertial range can be put on much more solid foundation. Indeed the third mom ent is one of the few quantities for which an exact prediction can be derived f rom the Navier-Stokes equation. This derivation was ﬁrst done by Kolmogorov for 3D homogenous and isotropic turbulence. To apply to results from the e-m cell K olmogorov’s derivation must be relaxed to the case of 2D homogenous but anisotropic t urbulence with an external linear drag. This relaxation is given below. Follo wing this are measurement and analysis of the third moment in the e-m cell for the data se ts in Table 4.1. 4.2.1 The Anisotropic Third Moment The starting point for the derivation of the third moment rel ationship for homogenous anisotropic 2D turbulence is the Karman-Howarth relations hip, Eq. 3.6, which was derived in section 3.2. All of the notation and conventions u sed in that section will be carried over without alteration. The notation and the ﬁrs t step of the derivation follows that given in a recent paper by Lindborg [26]. Add Eq. 3.6 evaluated for ( i, j) to that evaluated for ( j, i), and use Eq. 3.7 to obtain 2Πij−2ǫ(ν) ij=∂ ∂rsB(3) isj+∂ ∂tB(2) ij−∂ ∂rjPi−∂ ∂riPjChapter 4. Energy Distribution and Energy Flow 53 −2ν∂2 ∂rs∂rsB(2) ij+ 2αB(2) ij−Wij−Wji. (4.1) In the above, the nterm moments B(n) ij...k(⊳vector r)≡ ∝angb∇acketleftδuiδuj...δu k∝angb∇acket∇ightof velocity diﬀerence δui≡u′ i−uihave been deﬁned. Also deﬁned are Pi≡ ∝angb∇acketleftuip′∝angb∇acket∇ight − ∝angb∇acketleftu′ ip∝angb∇acket∇ightandWij≡ ∝angb∇acketleftδuiδFj∝angb∇acket∇ightwithδFj≡F′ j−Fj. Note that the homogeneity assumption has been used in a number of places in the above step. Most notably, it sets ∂ ∂rs(b(3) i,sj−b(3) is,j) =∂ ∂rsB(3) isj. (4.2) Contracting Eq. 4.1 with the unit vectors ni(≡ri/r=ri/\|⊳vector r\|) and njand using the identities ninjB(2) ij=B(2) rr, (4.3) ninj∂ ∂rsB(3) isj=∂ ∂rs(ninjB(3) isj)−2 rB(3) rtt, (4.4) ninj∂ ∂rjPi=∂ ∂rj(ninjPi)−1 rniPi, (4.5) ninj∂2 ∂rs∂rsB(2) ij=2 r2(B(2) rr−B(2) tt) +4 r∂ ∂rB(2) rr +∂2 ∂rs∂rsB(2) rr, (4.6) where the subscripts randtdenote longitudinal, i.e. along ⊳vector r, and transverse direc- tional coordinates, we obtain ninj(Πij−ǫ(ν) ij)−1 2∂ ∂tB(2) rr=1 2∂ ∂rs(ninjB(3) isj)−1 rB(3) rtt−∂ ∂rj(ninjPi)−1 rniPi −ν/parenleftBigg2 r2(B(2) rr−B(2) tt) +4 r∂ ∂rB(2) rr+∂2 ∂rs∂rsB(2) rr/parenrightBigg +αB(2) rr−1 2ninj(Wij+Wji). (4.7) Eq. 4.7 is now in a form which may be easily integrated over a ci rcle of radius r. This procedure, along with incompressibility and the assu mption of homogeneity eliminates the pressure terms. Using the divergence theore m and rearranging the terms yields S(3) rrr(r)−2 r/integraldisplayr 0dr′S(3) rtt(r′) = −ǫνr−1 r/integraldisplayr 0dr′r′∂ ∂tS(2) rr(r′)−2α r/integraldisplayr 0dr′r′S(2) rr(r′) +2ν/parenleftBigg4 rS(2) rr+∂ ∂rS(2) rr+2 r/integraldisplayr 0dr′(S(2) rr(r′)−S(2) tt(r′))/r′/parenrightBigg +1 2πr/integraldisplayr 0dr′/integraldisplay2π 0r′dθninj(Wij(⊳vectorr′) +Wji(⊳vectorr′)). (4.8)Chapter 4. Energy Distribution and Energy Flow 54 Here the circular averages of velocity moments have been den oted as S(n) ij..k(r)≡ 1 2πr/integraltext2π 0rdθB(n) ij...k(⊳vector r). Though this equation seems to explicitly contain an ǫrterm, it is somewhat superﬁcial as this term exactly cancels with t erms contained in the ﬁrst and ﬁnal expressions on the right hand side. Removing th ese terms yields S(3) rrr(r)−2 r/integraldisplayr 0dr′S(3) rtt(r′) = −1 2πr/integraldisplayr 0dr′/integraldisplay2π 0r′dθninj∝angb∇acketleftu′ iFj+uiF′ j+u′ jFi+ujF′ i∝angb∇acket∇ight +2ν/parenleftBigg4 rS(2) rr+∂ ∂rS(2) rr+2 r/integraldisplayr 0dr′(S(2) rr(r′)−S(2) tt(r′))/r′/parenrightBigg +/parenleftBigg1 πr∂ ∂t+2α πr/parenrightBigg/integraldisplayr 0dr′/integraldisplay2π 0r′dθb(2) r,r(⊳vectorr′). (4.9) Note that using the notation introduced here b(2) r,r(⊳vector r) is simply the two-point longitudi- nal velocity correlation. The terms on the left hand side are deﬁned as the anisotropic third moment of velocity diﬀerence, S(3) a, and up to a constant play the role of the third moment of the longitudinal velocity diﬀerence in the f ully developed isotropic- homogeneous turbulence derived by Kolmogorov [9]. The term s on the right account for the energy ﬂux at some length scale due to external forces . Eq. 4.9 is essentially the scale-by-scale energy balance relationship for the sys tem. For large rthe viscous term in Eq. 4.9 can be ignored, as can the time derivative if th e system is in an energetically steady state, leaving the ﬁnal form which wil l be used in this thesis S(3) rrr(r)−2 r/integraldisplayr 0dr′S(3) rtt(r′) = −1 2πr/integraldisplayr 0dr′/integraldisplay2π 0r′dθninj∝angb∇acketleftu′ iFj+uiF′ j+u′ jFi+ujF′ i∝angb∇acket∇ight +2α πr/integraldisplayr 0dr′/integraldisplay2π 0r′dθb(2) r,r(⊳vectorr′). (4.10) Limits of this equation are now ready to be taken to establish that a linear range can exist. First note that the force-velocity correlation t erm in Eq. 4.10 (ﬁrst term on the right) should decay in magnitude as 1 /rforr≫rinjforFperiodic in space. One limit which can be considered is the case where this force-ve locity term is neglected. In this case the only remaining term arises from the linear di ssipation. Assuming S(2) rr(r)∝r2/3over some range of length scales for r > r inj, then b(2) r,r(r) =u2 rms− S(2) rr(r)/2∝u2 rms−Ar2/3in that range, where A is a constant. Since the longitudinal velocity correlation, b(2) r,r(r), remains positive the ﬁrst term is dominant. Thus b(2) r,r(r)≈ u2 rmsin this range. Inserting this approximation into Eq. 4.10 an d integrating yields S(3) a= 2αu2 rmsr= 2ǫairr, which is the extension of the earlier mentioned Kolmogorov 4/5 result for 3D turbulence to 2D anisotropic turbulence. Ano ther limit of interest is when both the force-velocity and velocity-velocity correl ations have disappeared. InChapter 4. Energy Distribution and Energy Flow 55 this case the integrals on the right hand side become constan t and the third moment decays as S(3) a≈r−1. Notice that the ǫfound in the third moment relationship is ǫair, the energy dissi- pated by the e-m cell’s external dissipation mechanism of ai r friction. One might have expected that this should be ǫinj, the total energy injection rate. Recall that in the previous discussion of the third moment, energy lost to visc ous forces were ignored so thatǫinj=ǫair. When viscosity is reintroduced, then the energy ﬂowing ove r a circle of radius ris the energy injected less the amount dissipated by viscosi ty, i.e. ǫinj−ǫν. By energy conservation this is just ǫair, which dictates the remaining energy which must ﬂow over the circle. This is why ǫairdetermines the third moment and not ǫinj. 4.2.2 Homogeneity Before sliding headlong into S(3) ameasurements and blindly searching for positive linear ranges, a word of caution is warranted. The assumptio n of homogeneity has been used a number of times in the preceding derivation of S(3) a, not to mention the fact that it was used in deriving the Karman-Howarth equatio n. This makes S(3) a measurements extremely sensitive to inhomogeneity in the e -m cell.3 Fortunately the derivation of S(3) ahas provided a simple test of homogeneity. If the turbulence in the e-m cell is homogenous then the followi ng should be equivalent representations for the anisotropic third moment, S(3) a: S(3) a(r)≡S(3) rrr(r)−2 r/integraldisplayr 0drS(3) rtt(r), (4.11) =1 2πr/integraldisplayr 0dr/integraldisplay2π 0dθninj∂ ∂xsB(3) isj(⊳vector r), (4.12) =1 2πr/integraldisplayr 0dr/integraldisplay2π 0dθninj∂ ∂xs(b(3) i,sj(⊳vector r)−b(3) is,j(⊳vector r)). (4.13) The latter two forms will be denoted J(r) and K(r), respectively. Plots of all three of these quantities for the data sets labeled (a)-(d) in Tabl e 4.1 are shown in Fig. 4.6. It is clear that only case (d) of extremely heavy damping (large α) produces approximate agreement for all three forms at length scales l arger than the injection 3There might be some concern that results presented in chapte r 3 might be inaccurate due to inhomogeneity. Recall that in that chapter homogeneity was assumed and not exactly checked. Using the analysis presented in this section on the data of ch apter 3 demonstrates that these data sets are approximately homogenous.Chapter 4. Energy Distribution and Energy Flow 56 scale. It is therefore the only approximately homogenous da ta set. It is also evident that case (a) and (b) are strongly inhomogeneous, and case (c ) is marginal. To understand the nature of the discrepancies, ensemble ave rages of individual velocity ﬁelds were performed over Nimages. Again the ensemble averaged velocity at any given point in the ﬂow, ∝angb∇acketleft⊳vector u(⊳vector x)∝angb∇acket∇ightN, though not identically zero was found to decrease in magnitude as N−1/2for all of the data sets indicating negligible mean ﬂow in the system. The inhomogeneity, then, stems from the sp atial variation of the velocity ﬂuctuations, ( ∝angb∇acketleft\|⊳vector u(⊳vector x)\|2∝angb∇acket∇ightN)1 2, which is shown in Fig. 4.7 for the four data sets (a)-(d). The oscillating light and dark bands, correspondi ng to high and low urms and inhomogeneity at the injection scale are again present i n all four sets to a greater or lesser extent. As before these oscillations will be ignor ed. A closer inspection of theurmsﬁelds for the four data sets also reveals a large-scale inhom ogeneity which increases in magnitude as αdecreases. Note that for the most weakly damped case (a), the ﬂuctuations near the corners are weak compared to th ose near the box center. This large-scale inhomogeneity is the main source of the dis crepancies for the three diﬀerent forms of S(3) a. The reason that the velocity ﬂuctuations begin to form this l arge scale inhomo- geneity for weak damping is connected to the growth of the out er scale of the turbu- lence. As discussed before the outer scale indicates the lar gest sized vortices present in the turbulence. This can be seen in Fig. 4.8 which shows typ ical streamlines for the four cases (a)-(d). Clearly the heavy damping, case (d), has many small vortices but few large ones compared to the case of weak damping, (a). Thes e large vortices prefer to exist in regions removed from solid boundaries, otherwis e a large shear builds up between the vortex and the boundary and quickly dissipates t he vortex. Since the largest vortices are of diameter rout, this preference causes an absence of ﬂuctuations for distances smaller than routfrom the wall. Should this boundary region invade the PTV measurement area homogeneity is sacriﬁced. For all of the data sets in Table 4.1 the distance between the P TV measurement area and the boundary was approximately 1 .5 cm, this sets the largest outer scale possible before sacriﬁcing homogeneity. Table 4.1 reveals that the spectrally measured outer scale, 2 π/kout, of the case (a) and (b) are well above this value explaining t heir strong inhomogeneity. Case (d) is below this value therefor e the measurement volume is homogenous. Case (c) has an outer scale just exceeding thi s limit, which explains it’s marginal homogeneity.Chapter 4. Energy Distribution and Energy Flow 57 Figure 4.6: Comparison of S(3) a(r) (⋄),J(r) ( ⊳), and K(r) (◦) for the data sets labeled (a)-(d) in Table 4.1.Chapter 4. Energy Distribution and Energy Flow 58 Figure 4.7: Spatial variation of velocity ﬂuctuation for th e four data sets labeled in (a)-(d) Table 4.1. Green denotes large values of the ﬂuctuat ions while blue denotes small values.Chapter 4. Energy Distribution and Energy Flow 59 Figure 4.8: Typical streamlines for the four cases labeled ( a)-(d) in Table 4.1.Chapter 4. Energy Distribution and Energy Flow 60 4.2.3 The Inertial Range and The Integral Scale In the last section only one of the data sets which were analyz ed strictly satisﬁed the homogeneity condition, case (d). Case (c) was marginal in it ’s homogeneity, so it can be considered as well. Considering only these two cases it is apparent that there is no inertial range in the e-m cell. Recall that from Eq. 4.10 an in ertial range is indicated by a linear range in S(3) a(r). Neither case (c) or (d) shows such a range in S(3) a. Since both (c) and (d) have an inverse cascade range, this is the ﬁrs t indication that the inertial range is behaving distinctly from the inverse casc ade range. However, since neither case (c) or (d) has an extensive inverse cascade rang e, it would be helpful to determine if either case (a) or (b), both of which exhibit lar ge inverse cascade ranges, has an inertial range in spite of their lack of homogeneity. To that end consider Eq. 4.10. The left-hand side of this equa tion can be rep- resented by any of the three forms S(3)(r),J(r), orK(r) from the last section if the turbulence is homogenous. For the inhomogeneous case it would be useful to ﬁnd if any of these three forms satisfy Eq. 4.10, eﬀectively r elaxing the condition of homogeneity on the left of the equation. To test this idea t he right-hand side of Eq. 4.10, denoted R(r), was independently measured and is displayed in Fig. 4.9 along with K(r) the third representation of the anisotropic third moment f or the four cases discussed earlier. The strongly and moderately dampe d cases produce mod- erate agreement between K(r) and R(r) , while for the weakly damped cases there is some disagreement which increases for large scales. The a greement for all cases, however, is better for K(r) than if S(3)(r) had been used to represent the left hand side. One can weakly conclude then that the homogeneity assu mption used to obtain Eq. 4.10 can be relaxed on the left for cases of moderate inhom ogeneity if the K(r) representation of the anisotropic third moment is used. For this reason we call K(r) the quasi-homogenous part of the third moment. Now the determination of the inertial range for weakly inhom ogeneous turbulence boils down to whether or not a linear range exists in the quasi -homogenous part of the anisotropic third moment for r > r inj. Clearly, even for the cases of weak damping, this does not happen. Indeed over the majority of th e range displaying inverse cascade, the quasi-homogenous part of the anisotro pic third moment decays asr−1, which is indicative of a linear dissipation dominated regi me. Therefore, none of the inverse cascade ranges produced in the e-m cell are inert ial, they are all dissipationChapter 4. Energy Distribution and Energy Flow 61 Figure 4.9: Comparison of K(r) (◦) with the independently measured right hand side of Eq. 4.10 , R(r) (−),for the data sets labeled (a)-(d) in Table 4.1.Chapter 4. Energy Distribution and Energy Flow 62 dominated. In the previous section, the outer scale of turbulence was de termined to behave as rout∝(ǫinj α3)1/2. This length scale, obviously, does not determine the upwar d extent of the range over which energy transfer is inertial, otherwi se, a few of the data sets should display a linear range. It is reasonable to ask what co ndition is required in order for any linearly damped 2D system to have an inertial range. Equation 4.10 indicates that if a linear range is to exist it arises fro m the dominance of the second integral on the right-hand side over the ﬁrst integra l at large scales. The ﬁrst integral, the force-velocity correlation, decays (as 1/r) at length scales larger than the injection scale. Thus, if a typical length scale of t he second integral greatly exceeds the injection scale, the third moment should have so me linear range. The second integral contains the longitudinal two-point veloc ity correlation as one of its functional arguments. This suggests that a measure of the ty pical length scale of this integral is the so-called integral scale, rint≡1 b(2) r,r(0)/integraldisplay∞ 0drb(2) r,r(r). The size of the integral scale for the data sets is displayed i n Table 4.1. Note that all but the most weakly damped data sets have integral scales sma ller than the injection scale ( rinj≈0.6 cm). For the case of very weak damping the integral scale has just exceeded the injection scale. This is reﬂected in the quasi- homogenous part of the third moment for the weakly damped cases (a) and (b), which sh ow an extended plateau right after the injection peak. To better visualize this plateau growth, the right hand side of Eq. 4.10 for the four cases is displayed in t he Fig. 4.10 with the plots normalized by the value of the peak after the injection scale. Such a feature is absent in the more heavily damped cases. Thus, the integral s cale seems to govern the upward extent of the inertial range, in contrast to the ou ter scale that governs the upward extent of the inverse cascade in 2D. One might believe that the integral scale and the outer scale should be linearly proportional to each other. This may not be the case. Figure 4 .11 is a comparison of the the outer scale calculated using the dimensional argume nt with the integral scale. Though the error in the plot does allow for the possibility of a linear ﬁt, it is more likely a power law growth as shown by the r2 intcurve drawn in the ﬁgure. Within the error indicated on the plots, the integral scale seems to behave as the geometricChapter 4. Energy Distribution and Energy Flow 63 Figure 4.10: The right hand side of Eq. 4.10 , R(r),for the data sets labeled (a)-(d) in Table 4.1: (a) ⋄, (b)⊲, (c)⊳, (d)◦.R(r) has been normalized so that the peak value just after rinjis unity.Chapter 4. Energy Distribution and Energy Flow 64 Figure 4.11: The dimensionally predicted outer scale, ( ǫinj/α3)1/2vs. integral scale rint(inset is the same plot on log-log scales). The line correspo nds to the power law ﬁt ofr2 int. average of the outer scale and injection scale (see the inset ), rint=√rinjrout=/parenleftBiggr2 injǫinj α3/parenrightBigg1/4 . (4.14) It seems that this relationship should be predictable by ins erting ﬁnite ranges in the energy spectrum and inverse transforming to get the two-poi nt correlation. At this time such a calculation has not been performed.Chapter 5 High Order Moments Recall from the introduction that the inverse energy cascad e range was expected to have two important properties: locality and scale invarian ce. The experiments in the e-m cell do not have enough inverse cascade range for any conc lusion to be directly drawn about either of these properties. However, some indir ect conclusions pertaining to scale invariance might be drawn if one assumes that certai n measured properties of the moments of velocity diﬀerence can be extended to the ca se of a large inverse cascade range. Displaying these properties and demonstrat ing how such conclusions may be drawn is the purpose of this chapter. First, to facilit ate the extension of mea- sured results, a brief discussion which describes the expec ted behavior for moments of velocity diﬀerence assuming scale invariance is presented . This discussion is similar to one in [9]. 5.1 Scale Invariance and Moments In the introduction the velocity diﬀerence over a length sca ler,δ⊳vector u(⊳vector r)≡(⊳vector u(⊳vector x+⊳vector r)− ⊳vector u(⊳vector x)), was said to be scale invariant in the inverse cascade rang e (Note that though δ⊳vector udepends on the absolute position, ⊳vector x, any statistical quantities formed from δ⊳vector u does not if homogeneity is assumed). Here, scale invariance means that there exists a unique scaling exponent hsuch that P(δ⊳vector u(λ⊳vector r)) =P(λhδ⊳vector u(⊳vector r)), where Pdenotes a probability distribution function (PDF). For the moment, consider homogenous isotropic 2D turbulence. Instead of using both components o fδ⊳vector u(⊳vector r) inP, isotropy allows for the simpliﬁcation to only a single component. For various reasons this component is usually the longitudinal component, δu\|\|(⊳vector r) =δ⊳vector u(⊳vector r)·ˆr. Also note that 65Chapter 5. High Order Moments 66 isotropy allows dependence on ⊳vector rto be reduced to only dependence on r=\|⊳vector r\|. From this the scaling relationship becomes P(δu\|\|(λr)) =P(λhu\|\|(r)). Thenthorder moment of longitudinal velocity diﬀerence is deﬁned a s S(n) rrr...(r)≡ ∝angb∇acketleft(δu\|\|(r))n∝angb∇acket∇ight=/integraldisplay+∞ −∞dδu\|\|(r) (δu\|\|(r))nP(δu\|\|(r)). (5.1) Note that these functions were already deﬁned in chapter 4. S ince longitudinal ﬂuctuations will only be considered here the notation can be simpliﬁed by deﬁn- ingSn(r) =S(n) rrr...(r). The scale invariance of the PDF’s directly results in the s cale invariance of the moments of velocity diﬀerence, with the sc aling exponent nhfor the nthorder moment. That is Sn(λr) =/integraldisplay+∞ −∞dδu\|\|(λr) (δu\|\|(λr))nP(δu\|\|(λr)) =/integraldisplay+∞ −∞dλhδu\|\|(r) (λhδu\|\|(r))nP(λhδu\|\|(r)) =λnh/integraldisplay+∞ −∞λhdδu\|\|(r) (δu\|\|(r))nλ−hP(δu\|\|(r)) =λnhSn(r). (5.2) Thus scale invariance implies that the moments of longitudi nal velocity diﬀerence behave as Sn(r)∝rnh. Recall from the discussion in chapter 4 that there exists an e xact result for the third moment of velocity diﬀerence in an inertial range, S3(r) =3 2ǫr. If an inertial range exists, and if it is scale invariant, then this exact re sult ﬁxes the scaling exponent h= 1/3 and the ﬁnal expectation for the scaling behavior of the nthorder moment is that Sn(r)∝rn/3. This result was ﬁrst reached by Kolmogorov in his 1941 theor y of homogenous isotropic turbulence and will be called the K4 1 theory. The result can be ﬂeshed out a little more if one assumes that the only var iables of importance in the inertial range are the constant energy ﬂow rate, ǫ, and the length scale, r. Under these assumptions dimensional analysis predicts Sn(r)∝Cn(ǫr)n/3, where the dimensionless Cnare assumed to be universal constants. Incompressibility s etsC1= 0 and the previously mentioned exact result sets C3= 3/2. This analysis yields a simple way in which the scale invarian ce of the turbulent ﬁelds can be veriﬁed. Simply make sure that the PDF’s of longi tudinal velocity diﬀerence for various rin the inertial range, when properly normalized, collapse t o a single curve. Equivalently, make sure that the nthorder moments of longitudinal velocity diﬀerence scale as rn/3for all n. It should be pointed out that this analysisChapter 5. High Order Moments 67 holds for homogenous isotropic 3D turbulence, except for a c hange in coeﬃcients Cn. Although one might expect the velocity diﬀerences in 3D tur bulence to be scale invariant as expressed above, it is an experimental fact tha t the moments steadily deviate from the expected scaling behavior of the K41 theory as the order of the moment is increased. The possibility of such deviations hap pening in 2D turbulence is still an open experimental question, though what follows in this chapter might be considered clues as to what the answer may be. 5.2 Disclaimer The analysis in this chapter must begin by pointing out a coup le of reasons notto extend certain conclusion presented in it to cases of 2D turb ulence beyond the e-m cell. The ﬁrst of these reasons stems from the type of statist ical quantities which will be used in the determination of scale invariance. The an alysis in the previous section was phrased solely in terms of longitudinal velocit y diﬀerences. In order to go from full velocity diﬀerences to longitudinal diﬀerences w ithout loss of information it is necessary that the turbulence be fully isotropic. The f act that the forcing is unidirectional, as discussed in chapter 2 does not allow iso tropy to be assumed in the e-m cell. This means that the analysis given above must be don e for fully anisotropic turbulence to be absolutely correct. Unfortunately, this analysis becomes prohibitive for anis otropic turbulence as the order of the moments increase since the number of moments con taining transverse velocity diﬀerences that need to be measured grows. Earlier experiments show that the deviations from scaling prediction in 3D become apparen t only for moments of large order. Assuming that 2D might be similar means that a la rge number of quan- tities would have to be accurately measured and compared, ma king fully anisotropic analysis exceedingly diﬃcult. The fact that anisotropy doe s not allow the dependence on the diﬀerence vector ⊳vector rto be simpliﬁed to dependence on r=⊳vector rexacerbates this diﬃculty. From an experimental point of view this is devasta ting since dependence only on rallows statistics to be averaged over circles, increasing t he number of points used in evaluating the PDF’s by 2 πrfor each scale r. Without this buﬀering of the statistics one cannot hope to obtain enough data to measure h igh order moments of the PDF. For these reasons a fully anisotropic analysis must be abandoned. All results in this chapter use only longitudinal diﬀerences, and it is h oped that the anisotropyChapter 5. High Order Moments 68 will not seriously aﬀect the ﬁnal results. The second reason was hinted at in chapter 4. The results that will be discussed will be coming from an inverse cascade range that is not inert ial. This means that the form of the external damping may be aﬀecting the results. Since the external damping in the e-m cell is known to have a linear form, the resu lts can strictly be said to only apply to other linearly damped 2D turbulent ﬂu ids. This fact, along with the e-m cells anisotropy, raises serious questions abo ut the universality of certain results which are obtained. In spite of this, the author feel s that many of the results presented are robust due to similarities with data obtained from other experiments [5] and simulation [27]. 5.3 The PDF of Longitudinal Velocity Diﬀerence Of the data sets presented in Table 4.1 the only two to display homogeneity or marginal homogeneity by the analysis of chapter 4 are cases ( c) and (d). These will be the main data sets from which conclusions in this chap ter will be drawn. In particular, focus will be given to (c) since it has the larges t inverse cascade range of the two as shown in Fig. 4.3. The PDF of longitudinal velocity diﬀ erence, P=P(δu\|\|(r)) was calculated in a straightforward manner from data set (c) and is presented in the color plot of Fig. 5.1. Several cross-sections of the plot fo r various rare shown in Fig. 5.2. It is clear from the cross-sections that Pis approximately Gaussian at all r. Of course it cannot be perfectly Gaussian since there must be so me third moment as was measured in chapter 4. The magnitude of the odd moments, such as the third, must be small compared to the even order moments for Pto have such a strongly Gaussian character. The tails of Pdo not seem to strongly deviate from Gaussian decay into either exponential or algebraic decay at any rin the inverse cascade range. In 3D, deviations of the high order moments from the K41 theory is as sociated with slower than Gaussian decay in the PDF tails. This behavior is common ly termed “intermit- tency” since it indicates intermittent bursty behavior in t he velocity ﬁeld. That no such deviation in the PDF tails is seen in these experiments i s an indication that the scale invariant result may hold. The approximately Gaussia n PDF’s measured here are in agreement with both recent experiments [5] and simula tions [27]. The moments of the longitudinal velocity diﬀerence are calc ulated from PusingChapter 5. High Order Moments 69 Figure 5.1: P(δul(r), r) calculated from data set (c) in Table 4.1. Divisions in the coloration increase on an exponential scale. The injection and outer scale are marked by lines, in between which is the inverse energy cascade rang e. Figure 5.2: Cross sections at various rforP(δul(r), r) shown in Fig 5.1.Chapter 5. High Order Moments 70 Eq. 5.1. Since the two lowest order moments in 3D do not measur ably deviate from the K41 theory, and it is expected that this will be the case in 2D, the low order moments are analyzed in the e-m cell ﬁrst. Displayed in Fig: 5 .3 are S2(r) and S3(r) for the data set (c). Clearly there is no range in the second mo ment which scales asr2/3in between the injection and outer scale as K41 predicts. The re is no linear behavior of S3(r) in this range either. This later result is hardly surprisin g considering that a linear range is indicative of an inertial range which h as already been shown not to exist in the e-m cell (see chapter 4). These graphs clearly violate the scaling prediction in the K 41 theory, therefore the e-m cell’s inverse cascade range is not scale invariant. It might become scale invariant if an inertial range was allowed to build in the cel l. To better understand what might happen, the higher order moments (i.e. n >3) of the longitudinal velocity diﬀerence are evaluated. These moments are more easily disp layed if they are made dimensionless by dividing out the appropriate power of the s econd moments. Deﬁne Tn(r) =Sn(r)/(S2(r))n/2, so that T3is the skewness, T4is the ﬂatness etc. These normalized moments are shown in Fig. 5.4 for 4 ≤n≤11. The error bars are calculated by truncating the PDF wherever noise dominates t he PDF measurement. First consider the moments of even order. With the exception of a blow up at small r, the even Tn(r) are essentially constant for r > r injwith the exception of logarithmically small ﬂuctuations between rinjandrout. The constant values that the even Tnassume at large rare only slightly less than those of a pure Gaussian distribution, namely an=n! 2(n/2)(n/2)!, (5.3) forn≥2. These are shown on the even plots of Fig. 5.4 as dotted lines . This conﬁrms the earlier visual observation that Pwas approximately Gaussian. The large blow up at small scales r < r injis due to poor statistics. The ﬂuctuations of the even Tn at scales in between rinjandroutseems to force the value slightly higher than the Gaussian value. It should be pointed out that a similar rise i s seen in [5] though without the ﬂuctuating character. Fluctuations are also se en in [27] near the outer scale, though these settle back down to the Gaussian values o nce the inertial range is reached. Since the ﬂuctuations are small the Gaussian value s will be assumed to hold for all r > r inj. From the measured constant values of the even Tnthe higher order even moments in the e-m cell can be approximated as Sn(r)≈an(S2(r))n/2with theChapter 5. High Order Moments 71 Figure 5.3: (a) S2(r) (log-log) and (b) S3(r) (lin-lin) calculated from data set (c) in Table 4.1.Chapter 5. High Order Moments 72 Figure 5.4: The normalized high order moments, Tn(r), evaluated from data set (c) of Table 4.1 for 4 ≤n≤11.Chapter 5. High Order Moments 73 Figure 5.5: Tn(r)/bnfor odd n≥3 evaluated using the data set (c) in Table 4.1. anassuming the values of a Gaussian distribution. Now consider the odd moments. Unlike the even moments, the od dTndisplay a complicated behavior for r > r inj. However, up to a multiplicative constant, which will be denoted bn, the odd Tnhave similar behaviors. To see this the multiplicative constant has been removed and all odd Tnforn≥3 have been plotted in Fig. 5.5. The bnwhere chosen to be the value of Tnatrinj. This was completely arbitrary and more complicated procedures could be performed to get better ﬁts . Though the plots are not identical, they clearly have the same trends, and agreem ent is within measurement error. Using the T3as a base function this experimental data indicates that Tn(r)≈ bn b3T3(r). Or, in terms of the unnormalized moments Sn(r)≈bn b3S3(S2)(n−3)/2. The coeﬃcientsbn b3behave in a diﬀerent manner from their even counterparts, th ean. The two sets of coeﬃcients are plotted in Fig. 5.6 along with a dot ted line representing Gaussian values. Where the anare following the Gaussian prediction quite nicely, the bn/b3behave as an almost perfect exponential. The experimental results lead to the conclusion that the hig her order moments can be expressed approximately in terms of the two lowest ord er moments as Sn(r) = an(S2(r))n 2:neven, (5.4) Sm(r) = dmS3(r)(S2(r))m−3 2:modd. (5.5)Chapter 5. High Order Moments 74 Figure 5.6: The multiplicative constants anandbn/b3for the data set (c) in Table 4.1. The dotted line corresponds to the exact values of a pure ly Gaussian distribution given by Eq. 5.3. where dm≡bm/b3. We are now in a position to draw some conclusions about scale invariance in the inertial range. In the inertial range it is clear that S3(r)∝ras the exact result from chapter 4 shows. Further, the range S2(r) is not expected to deviate in an inertial range from it’s r2/3behavior. Assuming these two scaling laws hold in an inertial range and assuming also that the above res ults can be extended into the inertial range yields Sn(r)∝rn 3:∀n. (5.6) This is precisely the statement of the K41 theory for rin an inertial range. Assuming, then, that the extensions and assumptions made are correct, the inertial range of 2D turbulence should behave in a scale invariant manner. One might ask about the universality of the coeﬃcients ananddn. Recall that one of the predictions of K41 is that these dimensionless numbers should be independen t of any external parameters, such as αor⊳vectorFwhich govern the turbulence. To test this the procedure presented above is performed on data set (d) of Table 4.1 as well as a set of data taken using a square magnet array instead of a Kolm ogorov array. SimilarChapter 5. High Order Moments 75 to the earlier data sets, the even moments display strong Gau ssian characteristics. The underlying structure of the odd normalized moments has c hanged, but they are still marginally collapsible by extracting an arbitrary co nstant. The odd Tn/bnare shown in Fig. 5.7(a) and (b) for the strongly damped Kolmogor ov ﬂow (case (d) in Table 4.1) and square array respectively. In Fig. 5.8 the coe ﬃcients for all of the data sets are displayed simultaneously. Note that the measured v alues do not signiﬁcantly diﬀer from one data set to the next, the anremain close to the Gaussian values given by Eq. 5.3, and the dnare exponential. This indicates that the coeﬃcients in the K41 theory are indeed universal as predicted.Chapter 5. High Order Moments 76 Figure 5.7: Tn(r)/bnfor odd nfor (a) case (d) in Table 4.1 and (b) a run of the e-m cell with a square array.Chapter 5. High Order Moments 77 Figure 5.8: The measured ananddn(=bn/b3) for three data sets using diﬀerent α and diﬀerent types of forcing.Chapter 6 Conclusion The inverse energy cascade of 2D turbulence as it occurs in th e e-m cell has been measured and quantiﬁed in the preceding chapters. Clearly a n inverse cascade exists and it’s energy distribution and extent behave as predicted by dimensional analysis. The energy ﬂow, when homogenous enough to be accurately comp ared with theory agrees almost perfectly with exact predictions made using t he 2D incompressible Navier-Stokes equation. However, at no time is this energy ﬂ ow inertial. That the inertial range and inverse cascade range are not coi ncident for a lin- early damped 2D ﬂuid is perhaps the most important result in t he thesis, from an experimental and numerical stand point. This is because pre vious 2D turbulence ex- periments and simulations did not attempt to diﬀerentiate b etween the two ranges, and merely assumed that a −5/3 range in E(k) must be accompanied by the nec- essary linear range in S3(r) (orS(3) adepending on if the turbulence was isotropic or not). This is now known not to be the case and casts doubt on som e of the earlier ex- perimental results. Finally, the inverse cascade range mea sured in the e-m cell is not scale invariant, though there is some evidence that it might become so if an inertial range ever developed. This is perhaps as far as experimental results on the inverse cascade can be taken in the current incarnation of the e-m cell. The lack of an iner tial range being the most signiﬁcant limiting feature. To get an idea of what would be n eeded to eliminate this feature we can use the knowledge that has been gained that the inertial range grows as the geometric average of the injection and outer scale (Eq . 4.14). If a decade of inertial range is desired for accurate measurements of scal ing behavior then by Eq. 4.14 the system size would have to be two decades larger than t he injection scale, 78Chapter 6. Conclusion 79 that is rout= 100 ×rinj. To get an idea what this would mean in the e-m cell, recall that the Kolmogorov magnets produced an injection scale of 0 .6 cm. Thus the e-m cell frame would need to be 60 cm across to support a large enou gh system to have a decade of inertial range while maintaining homogeneity. M oreover, the outer scale would have to be forced higher by either increasing ǫinj, which is risky because velocity ﬂuctuations would grow and sacriﬁce incompressibility, or by reducing α. Since αhas almost reached it’s asymptotic limit for the weakly damped c ases considered in this thesis, a partial vacuum must be employed to achieve lower α. The author need not emphasize the diﬃculty in creating a half meter sized soap bu bble, balancing it with respect to gravity, and ﬁnally putting the whole system in a p artial vacuum. Some sort of happy medium might be reached by marginally incr easing the system size, say by a factor of two, and reducing the magnet size a bit . Some inertial range would exist, though hardly enough to draw conclusions about scaling behavior. However, if use of the e-m cell in inverse cascade investigat ions is to continue such compromises must be made to obtain an inertial range.Appendix A Particle Tracking Velocimetry: Program Listing Included in this appendix is the program code, piv chk.cpp, for the particle tracking routine presented in chapter 2. It has been written in the C pr ogram language for no other reason than personal preference. The program has be en successfully com- piled with both the GNU C compiler and the Microsoft compiler . No performance enhancement was found through use of diﬀerent compilers. Us e of the code after compilations is done from the command line by pivchk foo1.tif foo2.tif outfoo.vec where “foo1.tif” and “foo2.tif” are the ﬁrst and second tif i mages to be compared respectively and “outfoo.vec” is the output ﬁle holding the particle positions and dis- placements. In “outfoo.vec” the origin of the coordinate sy stem is assumed to be at the upper left corner of the tif image, with the ˆ xdirection denoting the vertical coor- dinate in the image and the ˆ ydirection denoting the horizontal coordinate. Values of xincrease as one goes down the tif image and values of yincrease as one moves right. If a matched particle set is found at position ( x1, y1) in the ﬁrst image and ( x2, y2) in the second image it is saved in “outfoo.vec” as: ξy,ξx,uy,ux where ξx= 1/2(x1+x2),ξy= 1/2(y1+y2),ux=x2−x1anduy=y2−y1. All units in the output ﬁle are in pixels. 80Appendix A. Particle Tracking Velocimetry: Program Listing 81 The various parameters in the routine that determine the cha racteristics of the particles, the search radius, the correlation box size and o ther parameters are set by #deﬁne statements at the beginning of the routine. These are commented in the source code for clarity. pivchk.cpp: #include <stdio.h> #include <stdlib.h> #include <conio.h> #include <math.h> /***Control Parameters**/ /The three #define statements below determine the size and properties of the pictures. For example, if your picture is (640x480) wi th 30 header bits prefacing the contiguous data block in the .tif f ile, then header is 30, row is 480, and col is 640. (use 8 bit .tif)/ #define col 768 #define row 480 #define header 234 /The two variables below set your background subtraction p roperties. abox is the size of the box to take a local average over and must be odd. thresh is the multiple of the background to be subtrac ted from the picture for the purposes of particle identificatio n. Once the background has been subtracted, any contiguous group of points in the picture with pixel values greater than zero is a candid ate to be a particle./ #define abox 21 #define thresh 1.1 /minsize (maxsze) is the minimum (maximum) rms of a candida te group of points intensity distribution for that group to be labele d a particle. One can think of this as the particle size in pixels ./ #define minsize .25 #define maxsze 600Appendix A. Particle Tracking Velocimetry: Program Listing 82 /The three variables below are the meat of the routine. srad is the distance in pixels to search for the particle from frame one to frame two. Set this as low as is reasonable. If you know the particles travel no farther than 5 pixels from frame one to frame two set srad to 5. cbox is the correlation box size . This box must be big enough to contain at least 4 neighbors of a particle and must be odd. cthresh is the lower bound of the correlation. Any correlation above cthresh will be conside red for a match, and saved./ #define srad 15 #define cbox 17 #define cthresh 0.5 /Most bad interogations happen near the boundaries. Set brdr to approx cbox/2 to eliminate these./ #define brdr 12 /The below just allocates memory. maxnum is the maximum # of particles in a picture and maxsize is the maximum number of contiguous pixels in the candidate block. These are set unreasonably high since computer memory is cheap./ #define maxnum 30000 #define maxsize 10000 /Do not go below here unless you have strongly correlated pa rticle motion. If you do not have strongly correlated motion cvstat should be zero, which turns the below variables off./ #define cvstat 0 #define nbrstat 2 #define nrad 15 #define cvthresh 1.25 #define lbnd .6 /**Declare Global Variables**/Appendix A. Particle Tracking Velocimetry: Program Listing 83 struct connection{ float val; int plc; }; struct object{ float size; float x,y; float theta; int numnbr,numcan; struct connection nbr; struct connection can; int status; }; /**Declare Functions**/ void sort(struct connection ,int); int mutmax(struct object ,struct object ,int,int); void background(unsigned char ,float ,int); struct object findparts(unsigned char ,unsigned char ,int,int); void connect(struct object ,struct object ,int,int,int ); float correl(unsigned char , unsigned char , int); void getbox(unsigned char , unsigned char , int,int,in t); void clean(struct object ,struct object ,int,int); int check_vec(struct object ,struct object ,int,int); /***Main Function**/ void main(int argv, char argc[]) { FILE fin1, fin2, fout;Appendix A. Particle Tracking Velocimetry: Program Listing 84 int i,j,k,l,m,n; int x0,y0,r,s,t; int n1,n2,flag; float val; float x,y,u,v; float theta0,theta1,thetadiff; float vorticity; float dist1,dist2,xdiff,ydiff; double rem,pos; unsigned char pic1,pic2,bfr1,bfr2,hdr; float *mean; struct object list1, list2; if(argv<4){ printf("\nsyntax: piv <tif file #1> <tif file #2> <vec file> "); exit(0); } /**Open tif and output files*/ if((fin1=fopen(argc[1],"rb"))==NULL){ printf("Could not open %s",argc[1]); exit(0); } if((fin2=fopen(argc[2],"rb"))==NULL){ printf("Could not open %s",argc[2]); exit(0); } if((fout=fopen(argc[3],"w"))==NULL){ printf("Could not open %s",argc[3]); exit(0); }Appendix A. Particle Tracking Velocimetry: Program Listing 85 /*Allocate some memory**/ hdr = new(unsigned char[header]); pic1 = new(unsigned char [row]); pic2 = new(unsigned char [row]); bfr1 = new(unsigned char [row]); bfr2 = new(unsigned char [row]); mean = new(float [row]); for(i=0;i<row;i++){ pic1[i] = new(unsigned char[col]); pic2[i] = new(unsigned char[col]); bfr1[i] = new(unsigned char[col]); bfr2[i] = new(unsigned char[col]); mean[i] = new(float[col]); } /***Read picture files*/ printf("Reading files\n"); if(fread(hdr,sizeof(unsigned char),header,fin1)!=hea der){ printf("Error stripping header from %s",argc[1]); exit(0); } if(fread(hdr,sizeof(unsigned char),header,fin2)!=hea der){ printf("Error stripping header from %s",argc[2]); exit(0); } for(i=0;i<row;i++){ if(fread(pic1[i],sizeof(unsigned char),col,fin1)!=co l){ printf("Error reading %s",argc[1]); exit(0); } if(fread(pic2[i],sizeof(unsigned char),col,fin2)!=co l){ printf("Error reading %s",argc[2]); exit(0); } }Appendix A. Particle Tracking Velocimetry: Program Listing 86 /*Begin background subtraction**/ printf("\nSubtracting background"); background(pic1,mean,abox); for(i=0;i<row;i++){ for(j=0;j<col;j++){ if(threshmean[i][j] >= pic1[i][j]) bfr1[i][j] = 0; else{ val = mean[i][j]; rem = modf(val,&pos); if(rem > .5) pos++; val =(float)pos; bfr1[i][j] = pic1[i][j] - (unsigned char)val; } } } background(pic2,mean,abox); for(i=0;i<row;i++){ for(j=0;j<col;j++){ if(threshmean[i][j] >= pic2[i][j]) bfr2[i][j] = 0; else{ val = mean[i][j]; rem = modf(val,&pos); if(rem > .5) pos++; val =(float)pos; bfr2[i][j] = pic2[i][j] - (unsigned char)val; } } } /**A little housekeeping*/ delete[] mean; list1 = new(struct object[maxnum]); list2 = new(struct object[maxnum]); n1 = 0; n2 = 0;Appendix A. Particle Tracking Velocimetry: Program Listing 87 /*Begin Finding particles*/ printf("\nFinding particles"); for(i=0;i<row;i++){ for(j=0;j<col;j++){ if(bfr1[i][j] > 0){ list1[n1]=findparts(bfr1,pic1,i,j); if(list1[n1].size < maxsze && list1[n1].size > minsize/2 ) n1++; } if(bfr2[i][j] > 0){ list2[n2]=findparts(bfr2,pic2,i,j); if(list2[n2].size < maxsze && list2[n2].size > minsize/2 ) n2++; } } } /*A little housekeeping*/ delete[] bfr1; delete[] bfr2; /*Start finding neighbors and candidates**/ printf("\n%d\t%d\nEstablishing connections",n1,n2); connect(list1,list2,n1,n2,srad); bfr1 = new(unsigned char [cbox]); bfr2 = new(unsigned char [cbox]); for(i=0;i<cbox;i++){ bfr1[i] = new(unsigned char[cbox]); bfr2[i] = new(unsigned char[cbox]); for(j=0;j<cbox;j++){ bfr1[i][j] = 0; bfr2[i][j] = 0; } }Appendix A. Particle Tracking Velocimetry: Program Listing 88 /**Calculate correlations for the candidates*/ printf("\nBeginning correlations"); for(i=0;i<n1;i++){ if(i%100 == 0) printf("."); m = list1[i].numcan; if(m==0) continue; if(kbhit()) break; val = list1[i].x; rem = modf(val,&pos); if(rem > .5) pos++; x0 =(int)pos; val = list1[i].y; rem = modf(val,&pos); if(rem > .5) pos++; y0 =(int)pos; getbox(pic1,bfr1,x0,y0,cbox); for(j=0;j<m;j++){ n=list1[i].can[j].plc; val = list2[n].x; rem = modf(val,&pos); if(rem > .5) pos++; x0 = (int)pos; val = list2[n].y; rem = modf(val,&pos); if(rem > .5) pos++; y0 = (int)pos; getbox(pic2,bfr2,x0,y0,cbox); val = correl(bfr1,bfr2,cbox); list1[i].can[j].val = val; x0 = 0; while(list2[n].can[x0].plc != i) x0++; list2[n].can[x0].val = val; } }Appendix A. Particle Tracking Velocimetry: Program Listing 89 /*Sort by size of correlations*/ printf("\nsorting"); for(i=0;i<n1;i++){ m = list1[i].numcan; sort(list1[i].can,m); m = list1[i].numnbr; sort(list1[i].nbr,m); } for(i=0;i<n2;i++){ m = list2[i].numcan; sort(list2[i].can,m); m = list2[i].numnbr; sort(list2[i].nbr,m); } /*Start matching*/ printf("\nmatching"); delete[] pic1; delete[] pic2; for(val=.9;val>cthresh;val-=0.01){ for(i=0;i<n1;i++){ if(list1[i].numcan==0) continue; if(list1[i].status==1) continue; m = list1[i].numcan - 1; n = list1[i].can[m].plc; if(mutmax(list1,list2,i,n)==1 && list1[i].can[m].val> =val){ if(cvstat == 1 && val < lbnd){ if(check_vec(list1,list2,i,n)==1){ list1[i].status = 1; list2[n].status = 1; clean(list1,list2,i,n); }Appendix A. Particle Tracking Velocimetry: Program Listing 90 else if(check_vec(list1,list2,i,n)==0){ list1[i].numcan--; list2[n].numcan--; } } else{ list1[i].status = 1; list2[n].status = 1; clean(list1,list2,i,n); } } } } n=0; /*Print results*/ for(i=0;i<n1;i++){ if(list1[i].status == 1){ j = list1[i].numcan-1; m = list1[i].can[j].plc; if(list1[i].size < minsize \|\| list2[m].size < minsize) con tinue; x = (list2[m].x + list1[i].x)/2; y = (list2[m].y + list1[i].y)/2; u = list2[m].x - list1[i].x; v = list2[m].y - list1[i].y; if(x > brdr && x <= row-brdr && y > brdr && y<= col-brdr){ fprintf(fout,"%f\t%f\t%f\t%f\n",y,x,v,u); n++; } } } printf("\n%d",n); /*Some final housekeeping**/Appendix A. Particle Tracking Velocimetry: Program Listing 91 delete[] list1; delete[] list2; } /This function attempts to loosen the correlation limit by comparing candidate motions with previously matched neighbors./ int check_vec(struct object list1,struct object list2, int n1, int n2) { int num1,num2; int nbrs,matched; int i,j,m,n; float x,y,u,v,val; float xm,ym,um,vm; float xp,yp,up,vp; float jtr; matched=0; nbrs = list1[n1].numnbr; xp = (list2[n2].x+list1[n1].x)/2; yp = (list2[n2].y+list1[n1].y)/2; up = (list2[n2].x-list1[n1].x); vp = (list2[n2].y-list1[n1].y); for(i=0;i<nbrs;i++){ m=list1[n1].nbr[i].plc; if(list1[m].status == 1){ j=list1[m].numcan-1; n=list1[m].can[j].plc; x = (list2[n].x+list1[m].x)/2 - xp; y = (list2[n].y+list1[m].y)/2 - yp; val = sqrt(xx + yy); if(val < nrad && x+xp > brdr && x+xp<=row-brdr && y+yp > brdr && y+yp <= col-brdr) matched++; } } if(matched < nbrstat) return(2);Appendix A. Particle Tracking Velocimetry: Program Listing 92 xm=0;ym=0;um=0;vm=0; for(i=0;i<nbrs;i++){ m=list1[n1].nbr[i].plc; if(list1[m].status == 1){ j=list1[m].numcan-1; n=list1[m].can[j].plc; x = (list2[n].x+list1[m].x)/2 - xp; y = (list2[n].y+list1[m].y)/2 - yp; val = sqrt(xx + yy); if(val < nrad && x+xp > brdr && x+xp<=row-brdr && y+yp > brdr && y+yp <= col-brdr){ xm += (list2[n].x+list1[m].x)/(2(float)matched); ym += (list2[n].y+list1[m].y)/(2(float)matched); um += (list2[n].x-list1[m].x)/(float)matched; vm += (list2[n].y-list1[m].y)/(float)matched; } } } jtr=0; for(i=0;i<nbrs;i++){ m=list1[n1].nbr[i].plc; if(list1[m].status == 1){ j=list1[m].numcan-1; n=list1[m].can[j].plc; x = (list2[n].x+list1[m].x)/2 - xp; y = (list2[n].y+list1[m].y)/2 - yp; val = sqrt(xx + yy); if(val < nrad && x+xp > brdr && x+xp<=row-brdr && y+yp > brdr && y+yp <= col-brdr){ u = (list2[n].x-list1[m].x) - um; v = (list2[n].y-list1[m].y) - vm; jtr += (uu+vv)/(float)matched; }Appendix A. Particle Tracking Velocimetry: Program Listing 93 } } jtr=sqrt(jtr); u = up-um; v = vp-vm; val = sqrt(uu + vv); if(val < cvthreshjtr) return(1); else return(0); } /Cleans the objects i and j from any candidate list since the y have presumably been matched./ void clean(struct object list1,struct object list2,int n1, int n2) { int num1,num2; int i,j,k,l,m; struct connection temp; num1 = list1[n1].numcan - 1; if(num1 > 0){ for(i=0;i<num1;i++){ j = list1[n1].can[i].plc; m = list2[j].numcan; temp = list2[j].can[m-1]; k=0; while(list2[j].can[k].plc != n1) k++; list2[j].can[k] = temp; list2[j].numcan--; m--; sort(list2[j].can,m); } } num2 = list2[n2].numcan - 1;Appendix A. Particle Tracking Velocimetry: Program Listing 94 if(num2 > 0){ for(i=0;i<num2;i++){ j = list2[n2].can[i].plc; m = list1[j].numcan; temp = list1[j].can[m-1]; k=0; while(list1[j].can[k].plc != n2) k++; list1[j].can[k] = temp; list1[j].numcan--; m--; sort(list1[j].can,m); } } } /Returns a 1 if the maximum correlation of list1[n1] and lis t2[n2] point to one another. 0 otherwise. This routine assumes we have already sorted the connections using "sort"./ int mutmax(struct object list1,struct object list2,int n1,int n2) { int num1,num2; int i,j; num1 = list1[n1].numcan - 1; num2 = list2[n2].numcan - 1; i = list1[n1].can[num1].plc; j = list2[n2].can[num2].plc; if(i == n2 && j == n1) return(1); else return(0); } /Sorting routine used in object lists. This is used to sort t he nbr connections in order from closest to farthest away and th e can connection from lowest correlation to highest./Appendix A. Particle Tracking Velocimetry: Program Listing 95 void sort(struct connection ptr,int num) { int i,j; struct connection temp; for(j=1;j<num;j++){ temp = ptr[j]; i=j-1; while(i>=0 && ptr[i].val > temp.val){ ptr[i+1]=ptr[i]; i--; } ptr[i+1] = temp; } } /Finds all the parts of an particle given that there is a brig ht spot at x0,y0. Return the value of the centroid and the rms./ struct object findparts(unsigned char ptr1,unsigned ch ar ptr2, int x0,int y0) { int i,m=1,n=1; int x,y; int j,k,val; unsigned char intensity; int brght=0; float tx,ty,std; struct object out; x = new(int[maxsize]); y = new(int[maxsize]); x[0]=x0; y[0]=y0; ptr1[x0][y0]=0; while(n<maxsize){Appendix A. Particle Tracking Velocimetry: Program Listing 96 for(i=0;i<m;i++){ if(x[i] - 1 >= 0){ if(ptr1[x[i]-1][y[i]] > 0){ x[n] = x[i] - 1; y[n] = y[i]; ptr1[x[n]][y[n]]=0; n++; } } if(x[i] + 1 < row){ if(ptr1[x[i]+1][y[i]] > 0){ x[n] = x[i] + 1; y[n] = y[i]; ptr1[x[n]][y[n]]=0; n++; } } if(y[i] + 1 < col){ if(ptr1[x[i]][y[i]+1] > 0){ x[n] = x[i]; y[n] = y[i]+1; ptr1[x[n]][y[n]]=0; n++; } } if(y[i] - 1 >= 0){ if(ptr1[x[i]][y[i]-1] > 0){ x[n] = x[i]; y[n] = y[i]-1; ptr1[x[n]][y[n]]=0; n++; } } }Appendix A. Particle Tracking Velocimetry: Program Listing 97 if(n==m) break; else m=n; } intensity = new(unsigned char[n]); for(i=0;i<n;i++){ intensity[i] = ptr2[x[i]][y[i]]; brght += (int)intensity[i]; } out.x=0; out.y=0; for(i=0;i<n;i++){ out.x += (float)intensity[i](float)x[i]/(float)brght ; out.y += (float)intensity[i](float)y[i]/(float)brght ; } std = 0; for(i=0;i<n;i++){ tx = (float)x[i]-out.x; ty = (float)y[i]-out.y; std += (float)intensity[i](txtx+tyty)/(float)brght ; } std = sqrt(std); out.size = std; out.numnbr = 0; out.numcan = 0; out.status = 0; delete[] intensity; delete[] x; delete[] y; return(out); }Appendix A. Particle Tracking Velocimetry: Program Listing 98 /Creates float ** mean which contains the value of the mean of a box of sizexsize around each point in pic./ void background(unsigned char pic,float back,int siz e) { int i,j; int x,y,val=0; int half = (size-1)/2; float mean; mean=0; for(i=0;i<=half;i++){ for(j=0;j<=half;j++){ mean = (valmean +(float)pic[i][j])/((float)val+1); val++; } } back[0][0]=mean; for(j=0;j<row;j+=2){ if(j%64==0) printf("."); for(i=1;i<col;i++){ y=i-half-1; for(x=j-half;x<=j+half;x++){ if(y < 0 \|\| x < 0 \|\| x >= row) continue; mean = (meanval - (float)pic[x][y])/((float)val-1); val--; } y=i+half; for(x=j-half;x<=j+half;x++){ if(y >= col \|\| x < 0 \|\| x >= row) continue; mean = (meanval + (float)pic[x][y])/((float)val+1); val++; }Appendix A. Particle Tracking Velocimetry: Program Listing 99 back[j][i]=mean; } x = j-half; for(y=col-1-half;y<col;y++){ if(x < 0) continue; mean =(meanval - (float)pic[x][y])/((float)val-1); val--; } x = j+1+half; for(y=col-1-half;y<col;y++){ if(x >= row) continue; mean =(meanval + (float)pic[x][y])/((float)val+1); val++; } back[j+1][col-1] = mean; for(i=col-2;i>=0;i--){ y=i+half+1; for(x=j+1-half;x<=j+1+half;x++){ if(y >= col \|\| x < 0 \|\| x >= row) continue; mean = (meanval - (float)pic[x][y])/((float)val-1); val--; } y=i-half; for(x=j+1-half;x<=j+1+half;x++){ if(y < 0 \|\| x < 0 \|\| x >= row) continue; mean = (meanval + (float)pic[x][y])/((float)val+1); val++; } back[j+1][i]=mean; }Appendix A. Particle Tracking Velocimetry: Program Listing 100 x = j+1-half; for(y=0;y<=half;y++){ if(x < 0) continue; mean =(meanval - (float)pic[x][y])/((float)val-1); val--; } x = j+2+half; for(y=0;y<=half;y++){ if(x >= row) continue; mean =(meanval + (float)pic[x][y])/((float)val+1); val++; } if((j+2) >= row) continue; back[j+2][0] = mean; } } /Finds all the neighbors and candidates for a particle and t hen stores this info in the appropriate spot in the lists./ void connect(struct object list1,struct object list2, int n1,int n2,int maxdist) { int x0,y0; int i,j; int x,y; int m1,m2; int pic1,pic2; int temp1,temp2; float val,dist,xdiff,ydiff; double rem,pos; pic1 = new(int [row]); pic2 = new(int [row]);Appendix A. Particle Tracking Velocimetry: Program Listing 101 for(i=0;i<row;i++){ pic1[i] = new(int[col]); pic2[i] = new(int[col]); for(j=0;j<col;j++){ pic1[i][j] = -1; pic2[i][j] = -1; } } for(i=0;i<n1;i++){ val = list1[i].x; rem = modf(val,&pos); if(rem >.5) pos++; x = (int)pos; val = list1[i].y; rem = modf(val,&pos); if(rem >.5) pos++; y = (int)pos; pic1[x][y] = i; } for(i=0;i<n2;i++){ val = list2[i].x; rem = modf(val,&pos); if(rem >.5) pos++; x = (int)pos; val = list2[i].y; rem = modf(val,&pos); if(rem >.5) pos++; y = (int)pos; pic2[x][y] = i; } temp1 = new(int[maxdistmaxdist]); temp2 = new(int[maxdistmaxdist]);Appendix A. Particle Tracking Velocimetry: Program Listing 102 for(x0=0;x0<row;x0++){ if(x0 % 64 == 0) printf("."); for(y0=0;y0<col;y0++){ if(pic1[x0][y0] == -1 && pic2[x0][y0]==-1) continue; if(pic1[x0][y0] != -1){ m1=0; m2=0; for(i=-maxdist;i<=maxdist;i++){ x = x0 + i; if(x < 0 \|\| x >= row) continue; for(j=-maxdist;j<=maxdist;j++){ y = y0 + j; if(y < 0 \|\| y >= col) continue; if((ii + jj)> maxdistmaxdist) continue; if(pic2[x][y] != -1){ temp2[m2] = pic2[x][y]; m2++; } if(pic1[x][y] != -1){ if(x==x0 && y==y0) continue; temp1[m1] = pic1[x][y]; m1++; } } } list1[pic1[x0][y0]].nbr = new(struct connection[m1]); for(i=0;i<m1;i++){ xdiff = list1[temp1[i]].x - list1[pic1[x0][y0]].x; ydiff = list1[temp1[i]].y - list1[pic1[x0][y0]].y; dist = sqrt(xdiffxdiff+ydiffydiff); list1[pic1[x0][y0]].nbr[i].plc = temp1[i]; list1[pic1[x0][y0]].nbr[i].val = dist; }Appendix A. Particle Tracking Velocimetry: Program Listing 103 list1[pic1[x0][y0]].numnbr = m1; list1[pic1[x0][y0]].can = new(struct connection[m2]); for(i=0;i<m2;i++){ list1[pic1[x0][y0]].can[i].plc = temp2[i]; } list1[pic1[x0][y0]].numcan = m2; } if(pic2[x0][y0] != -1){ m1=0; m2=0; for(i=-maxdist;i<=maxdist;i++){ x = x0 + i; if(x < 0 \|\| x >= row) continue; for(j=-maxdist;j<=maxdist;j++){ y = y0 + j; if(y < 0 \|\| y >= col) continue; if((ii + jj)> maxdistmaxdist) continue; if(pic1[x][y] != -1){ temp2[m2] = pic1[x][y]; m2++; } if(pic2[x][y] != -1){ if(x==x0 && y==y0) continue; temp1[m1] = pic2[x][y]; m1++; } } } list2[pic2[x0][y0]].nbr = new(struct connection[m1]);Appendix A. Particle Tracking Velocimetry: Program Listing 104 for(i=0;i<m1;i++){ xdiff = list2[temp1[i]].x - list2[pic2[x0][y0]].x; ydiff = list2[temp1[i]].y - list2[pic2[x0][y0]].y; dist = sqrt(xdiffxdiff + ydiffydiff); list2[pic2[x0][y0]].nbr[i].plc = temp1[i]; list2[pic2[x0][y0]].nbr[i].val = dist; } list2[pic2[x0][y0]].numnbr = m1; list2[pic2[x0][y0]].can = new(struct connection[m2]); for(i=0;i<m2;i++){ list2[pic2[x0][y0]].can[i].plc = temp2[i]; } list2[pic2[x0][y0]].numcan = m2; } } } delete[] pic1; delete[] pic2; delete[] temp1; delete[] temp2; } /Returns the correlation number between two arrays of size x size./ float correl(unsigned char ptr1,unsigned char ptr2,i nt size) { int i,j; float mean1,mean2; float std1,std2,cor; mean1=0; mean2=0;Appendix A. Particle Tracking Velocimetry: Program Listing 105 for(i=0;i<size;i++){ for(j=0;j<size;j++){ mean1 += (float)ptr1[i][j]/(float)(sizesize); mean2 += (float)ptr2[i][j]/(float)(sizesize); } } for(i=0;i<size;i++){ for(j=0;j<size;j++){ std1 += (((float)ptr1[i][j] - mean1)((float)ptr1[i][j] - mean1)) /(float)(sizesize); std2 += (((float)ptr2[i][j] - mean2)((float)ptr2[i][j] - mean2)) /(float)(sizesize); cor += (((float)ptr1[i][j] - mean1)((float)ptr2[i][j] - mean2)) /(float)(sizesize); } } cor /=(sqrt(std1)sqrt(std2)); return(cor); } /Gets a box from pic1 centered at x0,y0 and stores it in ptr1. ptr1 should be at least size x size and size must be odd!!!./ void getbox(unsigned char pic1, unsigned char *ptr1, int x0,int y0,int size) { int i,j,x,y; int half = (size-1)/2; for(i=0;i<size;i++){ x = x0 + i - half; if(x<0 \|\| x>=row){ for(j=0;j<size;j++){ ptr1[i][j] = 0; } continue; }Appendix A. Particle Tracking Velocimetry: Program Listing 106 for(j=0;j<size;j++){ y = y0 + j - half; if(y<0 \|\| y>=col){ ptr1[i][j] = 0; continue; } ptr1[i][j] = pic1[x][y]; } } }BibliographyBibliography [1] Robert H. Kraichnan. Inertial ranges in two-dimensiona l turbulence. Physics of Fluids , 10(7):1417–1423, July 1967. [2] E.D. Siggia and H. Aref. Point-vortex simulation of the i nverse energy cascade in two-dimensional turbulence. Physics of Fluids , 24:1981, 1981. [3] U. Frisch and P.L. Sulem. Numerical simulation of the inv erse cascade in two- dimensional turbulence. Physics of Fluids , 27:1921, 1984. [4] M.E. Maltrud and G.K. Vallis. Energy spectra and coheren t structures in forced two-dimensional and beta-plane turbulence. Journal of Fluid Mechanics , 228:321, 1991. [5] J. Paret and P. Tabeling. Intermittency in the two-dimen sional inverse cascade of energy: Experimental observations. Physics of Fluids , 10(12):3126–3136, De- cember 1998. [6] Maarten A. Rutgers. 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McGraw-Hill, New York, 1968.This figure "airdrag.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "alphavsd.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "arrays.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "consts.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "eddies.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "experimental1.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "experimental2.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "experimental4.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "filmcurve.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "highmoms.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "homog1.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "karmanhowarth_xx_cross.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "karmanhowarth_yy.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "karmanhowarth_yy_cross.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "longitudinalpdf1.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "longitudinalpdf2.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "lowmoms.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "lscales.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "magnetarray.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "meanflow.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "meanrmsflow.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "normoddmom.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "outerscale.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "outervelocity.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "rmsfluc.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "spectra.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "steadystate.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "thinfilm.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "thirdmom.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "thirdmom_norm.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "timing.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "tunnel1.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "tunnel2.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "typicalfields.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "universality1.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "universality2.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "var2dspectra.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1This figure "varspectra.png" is available in "png" format from: http://arXiv.org/ps/physics/0103050v1
arXiv:physics/0103051v1 [physics.gen-ph] 19 Mar 2001What is mass? R. I. Khrapko1 Abstract Does the mass of bodies depend on their velocity? Is the mass a dditive if separate bodies are joined together to form a composite system? Is the mass of an isolated system conserved? Diﬀerent teachers of physics and specialists gi ve diﬀerent answers to these questions because there is no general agreement on the deﬁni tion of mass..We shall show that the notion of the velocity-dependent relativistic mas s should be given preference over that of the rest mass. 1. Introduction One of the achievements of the special theory of relativity i s the statement about the equiv- alence of mass and energy in a sense that the mass of a body incr eases with its energy including kinetic energy; therefore, the mass depends on the velocity of the body. This relationship is unambiguously interpreted in the works of renowned physici sts. Max Born (1962): “The mass of one and the same body is a relativ e quantity. It is to have diﬀerent values according to the system of reference fr om which it is measured, or, if measured from a deﬁnite system of reference, according to th e velocity of the moving body. It is impossible that mass is a constant quantity peculiar to ea ch body.” [1]. Richard Feynman (1965): “Because of the relation of mass and energy the energy associated with the motion appears as an extra mass, so things get heavie r when they move. Newton believed that this was not the case, and that the masses staye d constant.” [2]. Statements to the same eﬀect can be also found in textbooks. S. P. Strelkov (1975): “The dependence of mass on velocity is a principal proposition of Einstein’s mechanics.” [3]. However, recently there had been a return to the Newton’s bel ief. According to this belief the mass of a body does not change with increasing velocity an d remains equal to the rest mass. L B Okun’ is a dedicated mouthpiece of this tendency [4, 5]. Ea rlier, a similar viewpoint was advocated in the book [6). L. B. Okun’ (1989): “The mass that increases with speed – that was truly incomprehensible. The mass of a body mdoes not change when it is in motion and, apart from the factor c, is equal to the energy contained in the body at rest. The mass mdoes not depend on the reference frame. At the end of the twentieth century one should bid fare well to the concept of mass dependent on velocity. This is an absolutely simple matter! ” [4]. J. Wheeler at al. (1966): “The concept of relativistic mass i s subject to misunderstanding” ([6], p. 137). This opinion is shared by the authors of certain textbooks fo r university students published abroad. R. Resnick et al. (1992): “‘The Concept of Mass’ by Lev B. Okun (see Ref. [5] of this letter) summarizes the views held by many physicists and ado pted for use in this book. But there is not universal agreement on the interpretation of Eq . E0=mc2. (35) 1Moscow Aviation Institute, 4 Volokolamskoe Shosse, 125871 , Moscow, Russia. E-mail: tahir@k804.mainet.msk.su Subject: Khrapko 1This equation tells us that a particle of mass mhas associated with it a rest energy E0. Nevertheless Eq. 35 asserts that energy has mass” [7] A serious confusion that arose from the reversion to the Newt onian concept of mass is reﬂected in the following dialogue: “Schoolboy: ‘Does mass really depend on velocity, dad?’ Fat her physicist: ‘No! Well, yes... Actually, no, but don’t tell your teacher.’ The next day the s on dropped physics.” [8]. We hope that we shall succeed in this letter to formulate a rat ional approach to the deﬁnition of mass. 2. A splitting of a deﬁnition of mass There are two diﬀerent deﬁnitions of the inertial mass, coin cident in the non-relativistic context. Deﬁnition 1. “In ordinary language the word mass denotes something like amount of sub- stance. The concept of substance is considered self-eviden t.” (See [1] p. 33.) More precisely: mass is deﬁned “as a number attached to each particle or body o btained by comparison with a standard body whose mass is deﬁne as unity” [9]. Deﬁnition 2. Mass is a measure of the inertia of a body, i.e. th e coeﬃcient of proportionality in the formula F=ma. (1) or in the formula p=mv. (2) Because F,a,pandvhave indisputable operational deﬁnitions, formulas (1) an d (2) give the operational deﬁnition of mass. These formulas will be us ed to make the aforementioned comparison (see Def. 1) in order to obtain the number mattached to a body. For the operational deﬁnition of momentum, see [10]. Here is an extract from this work: ”The meaning of the operational deﬁnition consists in the id entiﬁcation of two terms: ‘deﬁni- tion’ and ‘determination’. The operation used to deﬁne a mom entum is essentially as follows. When a certain obstacle causes a moving particle to stop, a fo rceF(t) is measured with which the particle acts on the obstacle during retardation. The pa rticle’s initial momentum equals the integral p=/integraltextF(t)dt,by deﬁnition. It is postulated that this integral is indepen dent of retardation characteristics, i.e. the form of the function F(t).” Unfortunately, the attached number determined by formulas (1) and (2) using the oper- ational deﬁnitions of F,a,p,vfor one and the same body, i.e. for the same ‘amount of substance’, turns out to be dependent on the speed of the body ; when the body has a speed, it also depends on the choice of the formula, (1) or (2). There fore, the deﬁnition of mass for a body in motion splits in three. ‘The amount of substance’ sp eciﬁed by the attached number from Def. 1 is no longer a measure of a inertia of the moving bod y. (a) In order to determine the ‘amount of substance’, i.e. the attached number from Def. 1, the body must be stopped and formula (1) or (2) used for a low sp eed. The number received by this method is called the rest mass. By deﬁnition, this mas s does not change when the body undergoes acceleration. (b) If the body is not stopped to measure its mass, formula (1) is known to give no unam- biguous result. Because the force and acceleration are not p roperties of the body, the coeﬃcient in formula (1) depends on the direction of the force relative to the body’s velocity. As a matter of fact, this coeﬃcient, in general, becomes a tensor. There fore, the deﬁnition of the mass by 2formula (1) is completely inadequate. It is even not worth co nsidering if the body’s speed is not suﬃciently low. (c) In contrast, formula (2) is valid at any speed including t hat of light. For this reason, it and only it gives the operational deﬁnition of the mass of a moving body. Such a mass is a measure of the inertia of a moving body. It is called the relativistic mass . It appears appropriate to cite M. Born once again: “In physic s, as we must very strongly emphasize, the word masshas no meaning other than that given by formula p=mv. It is the measure of resistance of a body to changes of velocity.” (See [1], p. 33) At this point, a problem arises. Which of the two masses, the r est mass of (a) or the relativistic mass of (c), is to be called simply mass and denoted by the letter mwithout a subscript and thus regarded as the ‘chief’ mass. This is not a matter of terminology. The problem has serious psychological and methodological impl ications. It can be resolved through the comparison of the properties o f diﬀerent masses. The rest mass will be denoted by the symbol m0and the relativistic mass by the symbol m(otherwise, the latter will have no simple designation at all). 3. System of particles If two particles having momenta p1=m1v1andp2=m2v2join together into a single whole system, the momenta are known to add up so that p=p1+p2.Moreover, the four-dimensional momenta are also summed giving P=P1+P2.The 4-momentum Pis by deﬁnition tangential to the world line of a particle in Minkowski space and its spat ial component equals an ordinary momentum p. Hence, the time component is equal to the relativistic mass m: P={m,p}. This immediately leads to the conclusion that the relativis tic masses are simply summed up:m=m1+m2,when particles join together into a system. Things diﬀer when rest masses come into question. In the 4-di mensional sense, the rest mass of a particle is the modulus of its 4-momentum (to an accu racy of c): m0=/radicalBig m2−p2/c2. Therefore, the rest mass of a pair of bodies with rest masses m01, m 02is not equal to the sum m01+m02but is determined by a complicated expression dependent on m omenta p1,p2[4]: m0=/radicalBigg/parenleftbigg/radicalBig m2 01+p2 1/c2+/radicalBig m2 02+p2 2/c2/parenrightbigg2 −(p1+p2)2/c2. (3) A similar formula for the rest mass is presented in [6] (c = 1): M2= (Esystem)2−(px system)2−(py system)2−(pz system)2. (4) It follows from formulas (3) and (4) that the rest mass is lack ing the property of additivity. We think that physicists do not mean the rest mass when they sp eak about beauty as a criterion for truth. 4. A violence to mind 3The thing is that both the relativistic mass (a time componen t of 4-momentum) and the rest mass (its modulus) obey the conservation law. This is as certained in [4]. However, it is not so simple to accept that a non-additive qua ntity is conserved. Indeed, according to (3) and (4), the rest mass of a system does not cha nge as a result of particle collisions or nuclear reactions. But as soon as a system of tw o moving bodies is mentally divided into two separate bodies, the rest mass will change b ecause the rest mass of the pair is not equal to the sum of the rest mass of the bodies of the pair . In our opinion, the use of non-additive notions entails a serious intellectual burde n: a pair of photons, each having no rest mass, does have a rest mass. Another very diﬃcult question is: “Does energy have a rest ma ss?” The correct answer may be as follows: the energy of two photons will have a rest ma ss when they move in opposite directions. A system of two photons will have zero rest mass i f they move in the same direction [4]. Thus, it appears that even the authors of the textbook [7 ] failed to solve the problem. Furthermore, photons moving in the same direction have no re st mass while the rest mass of the body which emitted them decreases. Therefore, it may b e suggested that some of the body’s rest mass has been converted into the massless energy of photons. However, according to (3), (4) the rest mass of the system body-photons has been c onserved during radiation! Unable to bear such an intellectual burden, the advocates of the rest mass concept refuse to adopt the law of conservation of the rest mass of a system, i n deﬁance of the formulas (3), (4). Now, they state that “rest mass of ﬁnal system increases in an inelastic encounter” ([6], p. 121). In contrast, nuclear reactions lead to ‘the mass defec t’. For example, in the synthesis of deuteron, p + n = D + 0.2 MeV, its rest mass is less than that of th e neutron and proton. At the same time, it follows from formulas (3), (4) that there must be no rest mass ‘defect’ during nuclear reactions. In our example, the allegedly lac king rest mass of the system at stage D + 0.2 MeV is actually provided by a massless γ-quantum with the energy of 0.2 MeV. This disturbs the additivity of the system’s rest mass. It is easy to understand why the schoolboy dropped physics in the face of such a confusion concerning the rest mass. 5. Underlying psychologic reason For all that, many physicists consider the rest mass to be the ‘chief’ one and denote it by the symbol minstead of m0.Simultaneously, they discriminate against the relativist ic mass and leave it without notation. This causes an additional confus ion making it sometimes diﬃcult to understand which mass is really meant. This situation is exe mpliﬁed by the statement from [7] cited above. These physicists agree that the mass of a gas in a state of rest increases upon heating because the energy contained in it grows. However, there seems to exi st a psychological barrier which prevents relating this rise to a larger mass of individual mo lecules due to their high thermal velocity. The said physicists sacriﬁce the concept of a mass as a measur e of inertia, sacriﬁce the additivity of mass and the equivalence of mass and energy to a label attached to each particle and bearing information about a constant ‘amount of substan ce’, just because such a label is in line with the deeply ingrained Newtonian concept of mass. Fo r them, radiation that ”transmits inertia between emitting and absorbing bodies” (according to A Einstein [11]) has no mass. The main psychological problem is how to establish the ident ity between mass and energy (which varies) and regard these two entities as one. It is eas y to accept that E0=m0c2for a 4body at rest. The authors of Ref. [6] entitled Chapter 13 as “T he equivalence of energy and rest mass”2It is more diﬃcult to admit that the formula E=mc2is valid for any speed. The remarkable formula E=mc2is described by L. B. Okun’ as ‘ugly’ [4]. Transition from the rest mass to the relativistic one in the r elativistic theory appears to encounter the same psychological problems as transition fr om proper to relative time. It is appropriate to quote from Max Planck here: “An important scientiﬁc innovation rarely makes its way by g radually winning over and converting its opponents: it rarely happens that Saul be comes Paul. What does happen is that its opponents gradually die out and that the gr owing generation is familiarized with the idea from the beginning: another inst ance of the fact that the future lies with youth. For this reason a suitable planning o f school teaching is one of the most important conditions of progress in science.” [1 2]. Unfortunately, the important concept of relativistic mass is carefully isolated from youth: the present paper has been rejected by editors of the followi ng journals: “Russian Physics Journal”, “Kvant” (Moscow), “American Journal of Physics” , “Physics Education” (Bristol). “Physics Today”. 6. Conclusions Thus, the relativistic mass has a natural operational deﬁni tion based on the formula p=mv. It is additive and obeys the law of conservation. Also, it is e quivalent to both energy and gravitational mass. It should be referred to as mass and deno ted by the letter m. The rest mass is not conserved or lacks the property of additi vity. Here, the advocates of the rest mass concept contradict themselves; at ﬁrst, they j ustly maintain that the rest mass is conserved but not additive, then they say that it is additi ve but not conserved. It is not equivalent to energy. It should be denoted as m0and used with caution especially if the notion is applied to a system of bodies. The relativistic mass together with momentum are transform ed as coordinates of an event during transition to a new inertial laboratory: m=m′+p′v/c2 /radicalBig 1−v2/c2, p=p′+m′v/radicalBig 1−v2/c2. Speciﬁcally, if p′= 0 then m′=m0,and m=m0/radicalBig 1−v2/c2, p=m0v/radicalBig 1−v2/c2. It is worthwhile to note in conclusion that if instead of the c oordinates t, x,... we use the coordinates t′, x′,... the relativistic mass mand the rest mass m0, which are both scalars, will be expressed by the formulas mc=ui′pj′gi′j′, m 0c=/radicalBig pi′pj′gi′j′, 2The title is characteristically ambiguous implying the equ ivalence between the restenergy and the rest mass. 5which are valid for the curved space of GTR. Here, ui′, pj′andgi′j′are the unit vector of the ex- perimentalist, 4-momentum of the body, and metric tensor of the new coordinates, respectively. It is assumed that for the initial coordinates t, x,...,ui=δi 0, g00= 1, gii=−1,... A photon has no rest mass-energy, hence no proper frequency. But its mass-energy and frequency can be measured in experiment as E=hν=cuipjgijand prove to be of any value depending on the experimenter’s speed. I thank G. S. Lapidus whose comments helped to improve the text of this paper. This paper has been published in Physics - Uspekhi 43(12) 1267 (2000), http://www.ufn.ru Uspekhi Fizicheskikh Nauk 170(12) 1363 (2000), http://www.ufn.ru http://www.mai.ru/projects/mai works/index.htm This topic is elaborated in physics/0103008 . References 1. Born M., Einstein’s Theory of Relativity (New York: Dover Publ., 1962) p. 269. 2. Feynman R., Character of Physical Law (London: Cox and Wym an, 1965) p. 76. 3. Strelkov S. P., Mechanics (Moscow: Nauka, 1975) p. 533 (in Russian). 4. Okun’ L. B., “The concept of mass”, Soviet Physics Uspekhi 32(7), 629–638 (1989). 5. Okun’ L. B., “The concept of mass”, Physics Today 42(6), 31 (1989) 6. Taylor E. F., Wheeler J. A., Spacetime Physics (San Franci sco: W.H. Freeman, 1966). 7. Resnick R., Halliday D., Krane K. S., Physics, Vol. 1 (New Y ork Wiley, 1992), p. 166, 167. 8. Adler C. G. “Does mass really depend on velocity, dad!” Am. J. Phys. 55, 739 (1987). 9. Alonso M., Finn E. J., Physics (Wokingham, England: Addis on- Wesley, 1992) p. 96. 10. Khrapko R. I., Spirin G. G., Ramrenov V. M., Mechanics (Mo scow: Izd. MAI, 1993). 11. Einstein A. “Ist die Tragheit eines Korpers von seinem En ergiegehalt abhangig?’ Ann. d. Phys. 18, 639 (1905). 12. Planck M., The Philosophy of Physics (George Allen & Unwi n Ltd, London, 1936), p. 90. 6
arXiv:physics/0103052v1 [physics.optics] 19 Mar 2001NEW IMPROVEMENTS FOR MIE SCATTERING CALCULATIONS V. E. Cachorro Departamento de F´ ısica Aplicada I Valladolid University, 47071 Valladolid, SPAIN L. L. Salcedo Departamento de F´ ısica Moderna Granada University, 18071 Granada, SPAIN ABSTRACT New improvements to compute Mie scattering quantities are p resented. They are based on a detailed analysis of the various sources of error i n Mie computations and on mathematical justiﬁcations. The algorithm developed on th ese improvements proves to be reliable and eﬃcient, without size ( x= 2πR/λ) nor refractive index ( m=mR−imI) limitations, and the user has a choice to ﬁx in advance the des ired precision in the results. It also includes a new and eﬃcient method to initiate the down ward recurrences of Bessel functions. 11. INTRODUCTION The Mie theory of light scattering by a homogeneous sphere is used for many prob- lems of atmospheric optics and also in other ﬁelds in Physics . The application of Mie theory still needs modern computers for numerical calculat ions of the many functions and coeﬃcients involved. The primary diﬃculty is in the precise evaluation of expansion coef- ﬁcientsanandbn. This is further aggravated as xgets large, and when the calculation of size distribution is needed. An optimization of computer ti me for reliable computation is clearly of necessity. The formulas for Mie scattering are well known1,2. Here we follow the notation of Bohren and Huﬀman3. The scattering and extinction eﬃciency factors are given b y Qs=2 x2N/summationdisplay n=1(2n+ 1)/parenleftbig \|an\|2+\|bn\|2/parenrightbig Qe=2 x2N/summationdisplay n=1(2n+ 1)Re(an+bn)(1) wherex= 2πR/λ is the size parameter of the problem, Rbeing the radius of the sphere, λ the wavelength of the light and Na large enough number. The Mie scattering coeﬃcients anandbnare functions of xand the relative refractive index m=mR−imI, with mR≥1,mI≥0. an=xψn(x)ψ′ n(y)−yψ′ n(x)ψn(y) xζn(x)ψ′n(y)−yζ′n(x)ψn(y) bn=yψn(x)ψ′ n(y)−xψ′ n(x)ψn(y) yζn(x)ψ′n(y)−xζ′n(x)ψn(y)(2) wherey=mxandψn(z),ζn(z) are the Riccati-Bessel functions related to the spherical Bessel functions jn(z) andyn(z): ψn(z) =zjn(z) ζn(z) =zjn(z)−izyn(z)(3) These functions are known in closed form (Ref. 4, p. 437) but i t is more convenient to use the recurrence relation Xn+1(z) =Fn(z)Xn(z)−Xn−1(z), Fn(z) = (2n+ 1)/z .(4) whereXis any of the functions in eqn. (3). Presently, there are many versions of Mie scattering comput er codes (Dave5,6, Blattner7, Grehan and Gouesbet8,9, Wiscombe10,11, Goedecke et al.12, Miller13) and au- thors who had been doing Mie calculations (Kattawar and Plas s14, Deirmendjian15, Quen- zel and M¨ uller16, Bohren and Huﬀman3). These are reﬂected in performing our work. 2Some essential points should be addressed by any Mie scatter ing algorithm: 1) How to determine the number Nfor truncating a Mie series. 2) Whether the Riccati-Bessel functions will be computed by upward recursion or by downward recursion. 3) If downward recursion is used, how to initialize it. 4) How to structure the algorithm in an eﬃcient way. Answers to all the above questions constitute the objective of this paper. We focus particularly on analyzing the numerical error sources and s how that our Mie algorithm permits users to prescribe a precision ǫbeforehand, to eﬀect an eﬃcient, reliable Mie coeﬃcients calculation. Needless to say, the precisely eva luated Mie coeﬃcients an, bnare required for calculating the angular scattering amplitude s1,2,3,5,6,10. 2. CONVERGENCE PROPERTIES OF THE MIE SERIES In this section we shall estimate the error introduced in som e typical quantity such as the eﬃciency factors, by keeping a ﬁnite number Nof partial waves in the Mie series. We shall also ﬁnd a criterion for choosing the value of N. In this section the quantities an,bnthemselves are assumed to be computed exactly. In order to investigate the convergence properties of the sc attering coeﬃcients an,bn we shall make use of very well known properties of the spheric al Bessel functions (e.g. ref. 4, p. 438 and ﬀ.). Let us recall some properties which are rele vant for us: i) lim n→∞ψn(z) = 0,lim n→∞ζn(z) =∞. (5) ii) Forz=xreal,ψn(x) andζn(x) have two distinct regimes as functions of n: a) oscillating regime for n < x .ψn(x) andζn(x) keep changing their sign regularly, and\|ψn(x)\|and\|ζn(x)\|are bounded by slowly changing functions of n. b) exponential regime for n>x .ψn(x) becomes exponentially decreasing and \|ζn(x)\| becomes exponentially increasing. In view of these considerations one concludes from eqn. (2), that all the partial wavesn < x (xbeing the size parameter from now on) will contribute to the M ie series and convergence will appear only after nenters in the exponential regime. This is so becauseψn(x),ψ′ n(x) go very quickly to zero in the numerator and ζn(x),ζ′ n(x) go to inﬁnity in the denominator. On the other hand ψn(y),ψ′ n(y) appear both in numerator and denominator and therefore seem to play no role in the conv ergence. We can emphasize 3this fact by writing an=ψn(x) ζn(x)[a]n=ψn(x) ζn(x)n(y/x−x/y) +xAn(y)−yAn(x) n(y/x−x/y) +xAn(y)−yBn(x) bn=ψn(x) ζn(x)[b]n=ψn(x) ζn(x)yAn(y)−xAn(x) yAn(y)−xBn(x)(6) where we have extracted the factor ψn(x)/ζn(x) responsible for the convergence of anand bnand also we have reexpressed the ratios ψ′ n(z)/ψn(z) andζ′ n(x)/ζn(x) in terms of (ref. 4, p. 439) An(z) =ψn−1(z) ψn(z), B n(x) =ζn−1(x) ζn(x)(7) Let us state more clearly our assumption: we shall assume tha t the quantities [a]n,[b]nare bounded by slowly varying functions of nin the exponential regime n > x . The validity of this assumption will be analyzed in a later se ction. If [a]nand [b]nare well behaved for large n, we can approximate them by their asymptotic values in order to discuss the convergence of anandbn. In order to take ad- vantage of this approximation we can use the asymptotic expa nsion of the Bessel functions for large orders (ref. 4, p. 365), An(z)∼Fn(z), B n(x)∼F−1 n(x) asn→ ∞ (8) where the next term in the expansion has a higher power of 1 /n. We obtain [a]n∼1−m2 1 +m2+O/parenleftbig1 n/parenrightbig ,[b]n∼O/parenleftbig1 n/parenrightbig (9) In practice, for x≤n≤N, [a]nand [b]nare both of the order of unity, (unless mis nearly 1, in which case [ a]n,[b]n≈0). On the other hand, recalling that mR≥1, it can be proved that\|1−m2 1+m2\|<2, therefore a good enough estimate is [a]n,[b]n≈1 (10) Using this and the asymptotic values (8), it can be shown that the truncation error in Qe is bounded by δQe≤ \|aN\| (11) The proof is presented in Appendix I where it is shown that the series/summationtext∞ n=N+1\|an\| converges faster than some geometric series. Let us note tha t what actually appears in Qe is Rean, not\|an\|, therefore the bound (11) will usually be conservative. Thi s is especially true for small mIbecause in this case Re an∼ \|an\|2(i.e.Qe∼Qs) and\|an\|2≪ \|an\|for n>N . 4Let us now ﬁnd a criterion for choosing the number Nof partial waves that should be taken into account. For this purpose let ǫbe the error allowed in the calculation, and let us take δQeas a typical quantity in the problem. Then Nshould be taken so that δQe≤ǫ (12) Taking the quantity Qehas the advantage of being simple and also that δQs≤δQe, because \|an\|2<Rean(i.e.Qs≤Qefor each partial wave). Other interesting quantities, such as the scattering amplitudes, have similar convergenc e properties as QeandQs. Putting together the bound (11), the criterion (12) and the e stimate (10) we ﬁnd the following prescription/vextendsingle/vextendsingle/vextendsingle/vextendsingleψN(x) ζN(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ǫ (13) In order to ﬁnd something more convenient let us make use of th e Wronskian identity (ref. 4, p.439) ψn(x)ζn−1(x)−ψn−1(x)ζn(x) = i, (14) and the asymptotic values of An(x) andBn(x). In this way we obtain (within approxima- tions keeping the order of magnitude) ψn(x)ζn(x)≈ −iF−1 n(x), (15) This allows us to remove ψn(x) from (13) and ﬁnally we obtain the prescription for N \|ImζN(x)\| ≥/radicalbigg 1 ǫ, (16) which has been written in a form convenient for being checked whileζn(x) is being com- puted by upward recurrence. In getting (16) we have neglecte d a factorFN(x) from (15) be- cause by doing so Nmay increase at most by one unit (recall that ζn(x)/ζn−1(x)≈Fn(x)). Also we have used that Re ζn(x) =ψn(x) is negligible as compared to Im ζn(x) in the ex- ponential region. It is remarkable that the value of Nobtained from (16) for ǫ= 10−8is virtually identical to the standard prescription N=x+cx1/3+ 1, withc= 4.3. It is shown in Appendix II that it must be so using asymptotic expansions fo rζn(x), and also how to modifycif some other precision ǫis desired. To know N(x) in advance is necessary if the computer code is to be vectorized10,11. 3. NUMERICAL ERROR AND UPWARD RECURRENCE In this section we shall discuss the propagation of numerica l error through the calculation. 5It is known that the determination of ψn(z) by upward recursion is intrinsically unstable (see e.g. ref. 5). Let us clarify this point.* For th e sake of simplicity let us assume that the numerical error is coming from the initial va lues ˜ψ0(z) =ψ0(z) +ǫ0,˜ψ1(z) =ψ1(z) +ǫ1 (17) but the recursion itself is free of roundoﬀ error, i.e. ˜ψn+1(z) =Fn(z)˜ψn(z)−˜ψn−1(z) (18) ǫ0,ǫ1being small numbers depending on the precision of the comput er, and ˜ψn(z) being the numerical sequence that is actually obtained instead of the exact one,ψn(z). Subtracting the exact recursion for ψn(z) from (18) we ﬁnd δψn+1(z) =Fn(z)δψn(z)−δψn−1(z) (19) whereδψn(z) =˜ψn(z)−ψn(z) is the error in our numerical sequence. Any sequence satisfying the recurrence relation (4) is a linear combinat ion ofψn(z) andζn(z), therefore δψn(z) =ηψn(z) +η′ζn(z) (20) The small numbers η,η′are directly related to ǫ0,ǫ1through eqn. (17), namely η= i(ǫ0ζ1−ǫ1ζ0) η′=−i(ǫ0ψ1−ǫ1ψ0)(21) Recalling now that ζn(z) diverges for large nwe conclude that the absolute error in ˜ψn(z) will eventually blow up. More generally, if the recursion it self is not exact due to computer roundoﬀ error, ˜ψn(z) is rather given by δψn(z) =ηnψn(z) +η′ nζn(z) (22) whereηn,η′ nare of the order of the roundoﬀ error or the initial values err or, whichever the largest. In any case the conclusion is still that δψn(z) is small for small n(or whilenis in the oscillating regime for znearly real), but blows up when nenters in the exponentially increasing regime of ζn(z). Sinceψn(z) itself goes to zero in the exponential regime, ˜ψn(z) has less and less correct ﬁgures at each step. We can extract some corollaries from the previous discussio n: 1) The upward recursion is always unstable for computing ψn(z) for large n, de- pending on z. The error δψn(z) grows as \|ζn(z)\|. On the other hand the upward recursion is perfectly stable for computing ζn(x) for any value of n. This is because δζn(x) still * We thank one of the referees for providing us with a simpler p roof of this statement. 6grows as \|ζn(x)\|, therefore the relative error in ζn(x) is kept small. Note however that the relative error in the quantity Re ζn(x) =ψn(x) is not at all small. 2) A downward recursion is stable for computing ψn(z), because \|ζn(z)\|is either slowly changing (in the oscillating regime) or quickly decr easing with decreasing n(in the exponential regime). This allows for taking even very rough estimates for the initial values ofψn(z) in the downward recursion and the ratio ˜ψn−1(z)/˜ψn(z) will still quickly approach the exact value An(z). On the other hand, a downward recursion is not appropriate for computing ζn(x) or the ratio Bn(x) if it starts in the exponential regime. Now let us study the inﬂuence of the numerical error on the an,bncoeﬃcients, and hence onQeif an upward recursion is used to compute ψn(x). In this analysis ζn(x) andBn(x) are assumed to be exact due to previous considerations. On t he other hand An(y) is also assumed to be exact. The eﬀect of using approximate v alues ofψn(y) will be considered later. We can make the discussion for an. Similar conclusions will hold for bn. Eqn. (6) can be rewritten as an=ψn(x) ζn(x)f(An(x)), (23) where only the An(x) dependence is shown explicitly as it is the only relevant on e for error analysis. The relative error in anwill be given by δan an≈δψn ψn+f′ fδAn An. (24) Recalling the deﬁnition (7), the relative error in Ancan be estimated to be of the same order of magnitude as that of ψn, and taking into account that fis a smooth function of the order of unity (cf. eqn. (10)), one gets the estimate δan≈anδψn ψn≈anη′ζn ψn=η′f≈η′. (25) where use has been made of eqn. (22) and η′is some typical value of η′ n. This means that the absolute error in anorbn, remains roughly constant throughout the computation. Of course eqn. (24) holds only for small δψn, but this is guaranteed asNis of the order of xand so the recurrence does not go deep inside the exponential region. The important consequence of eqn. (25) is that the up ward recursion can be used to obtainψn(x) because the error introduced is of the order of the roundoﬀ e rror (see however the comment at the end of Section 6). Let us note that t his fact is consistent with available algorithms for doing Mie calculations, wher eψn(x) andζn(x) are always computed by upward recursion (e.g. refs. 5,11). Let us consider now the eﬀect of the numerical error coming fo rmψn(y). We have argued before that an upward recursion would not be appropri ate for computing ψn(z) in general, however we have just shown that it can be used in the c ase ofψn(x). The reason 7for this was that the relative error in ψn(x) grew asζn(x)/ψn(x) but the quantities anand bnthemselves converged to zero as ψn(x)/ζn(x). Both factors cancel rendering δanand δbnbounded. We cannot apply a similar argument to δψn(y) and therefore an upward recursion is not reliable to compute ψn(y) for arbitrary y. We can consider two limiting cases a)mI= 0. In this case yis real and greater than x, thus the instability in ψn(y) starts only after that in ψn(x), therefore the upward recursion can be used. b) LargemI. From the initial values4 ψ0(z) = sin(z), ψ 1(z) =1 zsin(z)−cos(z) ζ0(z) = i exp( −iz), ζ 1(z) =/parenleftbiggi z−1/parenrightbigg exp(−iz)(26) one can see that ψn∼exp(mIx), ζn∼exp(−mIx), for small n, thusψnis much larger than ζn. On the other hand ǫ0,1are related to the computer precision, typicallyǫ0,1∼rψ0,1withr≈10−16in double precision. Upon substitution in (21) we ﬁnd that ηis small but η′∼rexp(2mIx) which is not necessarily small. The relative error in ˜ψn(z) goes as δψn(z) ψn(z)≈r/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ0(z) ζ0(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleζn(z) ψn(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(27) For smallnthe relative error is small, of the order of r, however for n∼ \|z\|, where ψnandζnare of the order of unity, the relative error is r\|ψ0/ζ0\| ∼rexp(2mIx) which is large for large mI. Therefore the upward recursion is not stable in this case. To summarize, the upward recursion to compute ψn(y) can be used if mIis small enough but becomes unstable for large mI. We have not analyzed in any detail in which cases the upward recursion for ψn(y) is reliable, therefore we shall only consider downward recurrences for this quantity. See however refs. 10,11 for a n extensive analysis of this problem through computer experiments. Noting that all we ne ed is the ratio An(y), for 1≤n≤N, we can use the downward recursion An(y) =Fn(y)−1 An+1(y). (28) Computing the initial value AN(y) requires some algorithm such as that of Lentz17or the one we present in the next section. Let us estimate now the pre cision required in AN(y) in order not to introduce an error in Qelarger than the prescribed precision ǫ. By arguments similar to those used for ψn(x), we have δan an≈δAn(y) An(y)(29) 8whereδanis the error introduced by δAn(y). Given that the downward recursion is stable we can assume that /vextendsingle/vextendsingle/vextendsingle/vextendsingleδan an/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleδAN(y) AN(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingleforn≤N (30) Using this relationship one gets for the numerical error in Qe δQe≈1 x2N/summationdisplay n=1(2n+ 1)δan≤Qe/vextendsingle/vextendsingle/vextendsingle/vextendsingleδAN(y) AN(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (31) Therefore the numerical error from An(y) will be under control by imposing /vextendsingle/vextendsingle/vextendsingle/vextendsingleδAN(y) AN(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ǫ Qe. (32) Let us note that this criterion will be conservative in gener al. An exception would be the case ofybeing real and bigger than N. In this case the recurrence (28) has no healing properties (for it already starts in the oscillatory regime ) and hence the equal sign is reached in (30). 4. INITIALIZATION OF THE DOWNWARD RECURRENCE In this section we present a new method to compute AN(z), of similar eﬃciency to that due to Lentz17(actually ours needs one multiplication less at each step). This method has the advantage of being able to implement a precision cond ition as that in eqn. (32), hence controlling the required precision in An(y). LetXn(z) andYn(z) be two sequences satisfying the recurrence (4) for some val ue ofz(the dependence on zis irrelevant here). Then they will satisfy the Wronskian id entity C=XnYn+1−Xn+1Yn (33) whereCis independent of n. We can rewrite it as a diﬀerence equation C=YnYn+1/braceleftbigg/parenleftbiggX Y/parenrightbigg n−/parenleftbiggX Y/parenrightbigg n+1/bracerightbigg , (34) and solve it in Xn Xn=DYn+CYn∞/summationdisplay k=n(YkYk+1)−1, (35) Dbeing a constant. To write (35) we have assumed that Ynis a sequence going to inﬁnity for largen, which is true for almost any solution of the recurrence (4). If we takeYnas a ﬁxed sequence and regard C,Das free parameters, then Xnis the most general solution of 9the recurrence relation (4). In particular for D= 0,Xngoes to zero as ngoes to inﬁnity, as a consequence it must be proportional to ψn, ψn(z) =C(z)Yn(z)∞/summationdisplay k=n(Yk(z)Yk+1(z))−1(36) The constant Ccancels after computing the ratio An(z) An(z) =Y−1 n/braceleftbig Yn−1+Y−1 n/bracketleftbig∞/summationdisplay k=n(YkYk+1)−1/bracketrightbig−1/bracerightbig . (37) Finally, a simpler formula can be obtained for AN(z) by choosing as starting values for the sequence Yn YN−1= 0, Y N= 1 (38) AN(z) =/bracketleftbig∞/summationdisplay k=N(Yk(z)Yk+1(z))−1/bracketrightbig−1. (39) About the convergence of the series in (39), we note that it is very fast when Ykenters in its exponential regime. Note that for real ythe convergence begins only after k≥y. A similar conclusion was reached by other authors11in Lentz’s method which basically follows the same principle as ours and so has similar converg ence properties. The sequence in eqn. (39) must be truncated at some value k=Min such a way as to fulﬁll the requirement (32). This can be easily done by noting that the error introduced in A−1 N(y) is of the order of the last term taken into account (this foll ows from \|Yk/Yk−1\| ≈ \|Fk\|>2 for largek), δA−1 N≈/parenleftbig YMYM+1/parenrightbig−1. (40) On the other hand we should require \|δA−1 N(y)\| ≈/vextendsingle/vextendsingle/vextendsingle/vextendsingleA−1 NδAN AN/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 FN(y)ǫ Qe/vextendsingle/vextendsingle/vextendsingle/vextendsingle(41) where we have made use of eqns. (8) and (32). Recall now that fo rx≥1,FNandQeare of the order of unity whereas for x≪1 the product of FNQeis still of the order of unity, therefore the ﬁnal criterion to truncate (39) is /vextendsingle/vextendsingle/parenleftbig YM(y)YM+1(y)/parenrightbig−1/vextendsingle/vextendsingle≤ǫ. (42) To ﬁnish this section we shall show how to avoid ill-conditio ning in (39), which will appear ifYkgets too near to zero for some value of k. To do this we can use the recurrence relation (4) to write 1 Yk−1Yk+1 YkYk+1=Yk−1+Yk+1 Yk−1YkYk+1=Fk Yk−1Yk+1, (43) 10which is well behaved even for Yk= 0. 5. COMPUTATIONAL ALGORITHM Using the previous ideas, we have developed a computational algorithm which we shall brieﬂy describe now. The input is x,mandǫand the main output are the coeﬃcients anandbn, andN. To start with, analytic expressions for ζ0(x) andζ1(x) are taken to initiate an upward recurrence for ζn(x). This quantity is kept in a (complex) array variable. The recurrence stops when the condition (16) is fu lﬁlled, providing the value ofN. The quantities ψn(x) are automatically obtained as the real part of ζn(x). As a second step, AN(y) is computed using eqns. (38), (39) and (42). Here we note tha t from a computational point of view an equivalent form of (42) is mor e convenient, which consist in doing the check for the absolute values of the real and imag inary parts. This is much faster than computing the modulus of a complex number. Then a downward recurrence is performed for An(y), eqn. (28), until n= 1. Si- multaneously, anandbnare computed using ζn(x) andAn(y). The quantities QsandQe can then be computed. We have not developed any especial algo rithm for computing the scattering amplitudes S1andS2. To do this eﬃciently see for instance ref. 11. The criteria developed above are intended to be robust, henc e they are rather con- servative. As a consequence the error in Qeis smaller than the prescribed precision ǫ. This is especially true for small values of x, whereas for x≫1, about two more ﬁgures than expected are obtained. We point out also that Qsis always obtained as accurately as Qe or more. This fact was expected because the criteria were sta ted for \|an\|whileQsgoes as \|an\|2which converges faster. 6. RESONANT TERMS IN THE MIE SERIES Let us recall that after eqn. (7) we stated a smoothness assum ption for the quantities [a]n,[b]n, namely that they are nearly constant in the xexponential regime and do not play any role in the convergence of the Mie series, which is on ly controlled by the ratio ψn(x)/ζn(x). In particular this assumption implied that the highest pa rtial wave with a relevant contribution is independent of m(cf. eqn. (16)). In other words, Nis a function of xonly. This result is also supported numerically, (see for in stance refs. 10,11). Therefore it was a surprise for us to discover that strictly speaking su ch a statement must be false. Moreover, for any choice of Nas a function of xonly, and for any prescribed value of n, n>N , one can always pick a value of m(in fact inﬁnitely many of them) in such a way that then-th term in the Mie series is not negligible, for instance one can makean= 1. The consequence of this that in order to guarantee that the nu merical value of Qeis correct within some prescribed precision, Nshould depend on mas well as on x. In order to clarify the point let us consider the worst case, w hich is also the simplest, namelymI= 0, i.e.yreal. This is the only case in which \|an\|or\|bn\|can reach the value 1. The point can be made for an: recalling that for zreal Reζn(z) =ψn(z), eqn. (2) can 11be rewritten as an=ReDn Dn Dn=xζn(x)ψ′ n(y)−yζ′ n(x)ψn(y)(44) whereDnis a complex quantity. Obviously an= 1 if and only if ImDn= 0. (45) Let us regard xandnas given and look for solutions of (45) in the variable y. The equation can be rewritten as 1 yψ′ n(y) ψn(y)=1 xImζ′ n(x) Imζn(x)(46) In the interval y>n,ψn(y) is a real oscillating function of ywith inﬁnitely many zeroes. Between two zeroes of ψn(y), the l.h.s. of eqn. (46) takes every real value, therefore t here are inﬁnitely many solutions to our equation for any values o fxandn, no matter how large isnas compared to x. For these values of x,m, andn,anwill not at all be negligible. Let us now show that these resonances do not occur for unreali stic values of m. Typically (and asymptotically for large y) the distance between two consecutive zeroes of ψn(y) is of the order of π, therefore for given xandnthe lowest resonant value of mwill occur near the interval (n x,n+π x) approximately. For large xthis happens for mnear to unity, and all the other resonant values will follow at a dist ance of about π/xfrom each other. From a rigorous point of view these ﬁndings would invalidate the estimates (10) and their consequences. They would also invalidate any algorit hm in which Ndepends on x only, namely every existent algorithm known to us. In fact th e only practical way to make sure that the resonant partial waves have been accounted for would be to take Ngreater thanyin order to guarantee that ψn(y) has no zeroes for n>N . Nevertheless it is clear that in practice the existent algor ithms to do Mie scattering calculations work. To account for this fact we should consid er not only the existence of resonant partial waves but also their width. Let us show that for sensible choices of N(as a function of x) and forn>N the resonances are so narrow that they will not normally show up. Let y0be one the values of ysuch thatan= 1. A look to eqn. (44) shows that for generic y, ReDngoes asψn(x) whereasDngoes asζn(x), therefore anis very small. However for the especial value y0there is a cancellation between two huge numbers in ImDn, leavinganof the order of unity. The range of values of yfor which a partial cancellation takes place is related to the slope of Dniny=y0, namely Γ≈/vextendsingle/vextendsingle/vextendsingle/vextendsingleDn D′n/vextendsingle/vextendsingle/vextendsingle/vextendsingle y=y0=/vextendsingle/vextendsingle/vextendsingle/vextendsingleReDn D′n/vextendsingle/vextendsingle/vextendsingle/vextendsingle y=y0≈/vextendsingle/vextendsingle/vextendsingle/vextendsingleψn(x) ζn(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (47) WhereD′ n= dDn/dy. In other words, if Nis large enough only by a very careful choice of morxcan one ﬁnd one these resonant contributions. More precisel y, recalling eqn. (13), 12we can see that morxshould be ﬁne tuned at least with a precision ǫin order to pick a resonant term for some n > N . On the other hand, except for these rare cases, an,bn are indeed small and of the order of ψn(x)/ζn(x), therefore our analysis applies. If mis allowed to be complex, a more involved analysis would be need ed, but we expect that the conclusion would not diﬀer. Let us ﬁnally note another consequence of the resonant terms on the calculation, even when they are taken into account. For one of these terms t he quantity fin eqn. (23) is no longer of the order of unity, on the contrary it is rather large, and the last step in eqn. (25) cannot be taken. This means that a resonant term amp liﬁes the error due to ψn(x). The cure is simply to compute ψn(x) by downward recursion for x<n<N . This has in fact been observed in selected quantities such as the b ackscattering eﬃciency for suitable values of xandm(Ref. 5). 7. CONCLUSIONS In this paper we have addressed several points relevant to Mi e scattering calcula- tions. To be speciﬁc: a) We have estimated the error introduced in the calculation by truncating the Mie series, thereby ﬁnding a prescription for choosing N. We have found that in the generic caseNdepends on xonly. b) The possible instabilities in the recursions used to comp uteψnandζnhave been analyzed. We have found that upward recursion is always unst able for computing ψn(z) ifnis large enough. However it can be used to compute ψn(x) in Mie calculations. As a matter of fact ψn(x) is computed in this way in nowadays available algorithms. W e have also found that upward recursion can be used for ψn(y) ifmIis small enough, but no criterion is given for how small mIshould be. c) A criterion has been established for the allowed error in ψn−1(y)/ψn(y). d) A new method to compute ψn−1(y)/ψn(y) is presented which is eﬃcient and allows for controlling the error and removing ill-conditio ning. e) It has been shown the existence of resonant terms in the Mie series which can also appear forn>N . Strictly speaking the existence of these terms invalidate s any algorithm in whichNis a function of xonly. However we have also shown that those resonant terms are extremely rare, namely they appear with a probability of the order of ǫ. A speciﬁc algorithm is also described. It is meant to be robus t and eﬃcient for a wide range of size parameters and refractive indices. With t his algorithm we have written the computer program LVEC-MIE18, which is available both in single and double precision contacting V.E. Cachorro. 13APPENDIX I Let us justify the bound (11). To do so we shall study the conve rgence rate of the terms left out in the series, n>N . In this region we can make use of the estimate (10), δQe=2 x2∞/summationdisplay n=N+1(2n+ 1)Re (an+bn) ≤2 x2∞/summationdisplay n=N+1(2n+ 1)/parenleftbig \|an\|+\|bn\|/parenrightbig ≈8 x2∞/summationdisplay n=N+1n/vextendsingle/vextendsingle/vextendsingle/vextendsingleψn(x) ζn(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle =8 x2∞/summationdisplay n=N+1n/vextendsingle/vextendsingle/vextendsingle/vextendsingleBn(x) An(x)Bn−1(x) An−1(x)...BN+1(x) AN+1(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleψN(x) ζN(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(I.1) Now making use of (8) and recalling that Fn(x) is a monotonically increasing function of n, we obtain δQe≤8 x2∞/summationdisplay n=N+1n1 F2n(x)1 F2 n−1(x)...1 F2 N+1(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleψN(x) ζN(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤8 x2∞/summationdisplay n=N+1n/parenleftbig FN(x)/parenrightbig2(N−n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleψN(x) ζN(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle =8 x2/parenleftbiggN F2 N(x)−1+F2 N(x) /parenleftbig F2 N(x)−1/parenrightbig2/parenrightbigg \|an\|(I.2) For smallx,N= 1 andFN(x) is large, hence δQe≤2\|an\| (I.3) on the other hand, for large x,N≈xandFN(x)∼2, δQe≤8 31 x\|an\|. (I.4) In both cases eqn. (11) is valid (up to factors of the order of u nity). 14APPENDIX II In order to know in advance the value of Nthat will be obtained from the prescrip- tion (16) for given xandǫ, let us recall that Im ζn(x) =/radicalbig πx/2YN+1 2(x),Yν(z) being the Bessel function of the second kind. Let νandcbe deﬁned by N=ν−1 2 x=ν−cν1/3.(II.1) Note that for large ν, eqn. (II.1) can be inverted to give N≈x+cx1/3. Now we can make use of the leading order term in the asymptotic expansion of Yνfor largeνand ﬁxedc, ref. 4, p. 367: ImζN(x)∼ −√π/parenleftbiggν 2/parenrightbigg1/6 Bi/parenleftbig 21/3c/parenrightbig , (II.2) where Bi(z) is the Airy function of the second kind, ref. 4, p. 446. This f unction is given by Bi(z) =z−1/4f(z) exp(2 3z3/2), (II.3) wheref(z) is nearly constant for z >1 withf(z)≈1/√π, ref. 4, p. 449. Thus /vextendsingle/vextendsingle/vextendsingle/vextendsingleImζN(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≈/parenleftbiggν 2√ 2/parenrightbigg1/6 c−1/4exp/parenleftbig1 3(2c)3/2/parenrightbig . (II.4) The right hand side of (II.4) has a very strong dependence on cwhereas it depends very smoothly on ν. Actually ( ν/2√ 2)1/6is of the order of unity for ν= 1 up to 105. Therefore using eqn. (16), cwill be determined by ǫ. We ﬁnd that c= 4.3 corresponds to ǫ= 10−8. Other values are c= 4.0, ǫ= 10−7, andc= 5.0, ǫ= 10−10, computed for ν= 100 in (II.4). 15REFERENCES 1. H. C. van de Hulst, Light Scattering by Small Particles , John Wiley, N. Y. 1957. 2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiatio n, Academic Press. N. Y., 1969. 3. C. F. Bohren and D. R. Huﬀman, Absorption and Scattering of Light by Small Particles , Wiley Interscience, N. Y. 1983. 4. M. Abramowitz and I. A. Stegun ed., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables , Dover Pub. Inc., N. Y., 1965. 5. J. V. Dave, Subroutines for Computing the Parameters of Electromagnet ic Radiation Scattered by a Sphere , Report No. 320-3237, IBM Scientiﬁc Center, Palo Alto, California, USA, 1968. 6. J. V. Dave, Scattering of Electromagnetic Radiation by Large Absorbin g Spheres , IBM J. Res. Develop., Vol.13, 1302-1313, 1969. 7. W. Blattner, Utilization Instruction for Operation of the Mie Programs o n the CDC- 6600 Computer at AFCRL , Radiation Center Associates, Ft. Worth, Texas, Res. Note RRA-N7240, 1972. 8. G. Grehan and G. Gouesbet, The Computer Program SUPERMIDI for Mie Theory Calculation, without Practical Size nor Refractive Index L imitations , Internal Re- port TTI/GG/79/03/20, Laboratoire de G´ enie Chemique Anal ytique, U. de Rouen, 76130 Mt-St-Aignan (France), 1979. Also Private communica tion. 9. G. Grehan and G. Gouesbet, Mie theory calculations: new progress, with emphasis on particle sizing , Appl. Opt., Vol. 18, 3489-3493, 1979. 10. W. J. Wiscombe, Mie scattering calculations: Advances in technique and fas t vec- tor speed computer codes . NCAR Technical Note NCAR/TN-140+STR (National Center for Atmospheric Research) Boulder, Colorado, 80307 , 1979, and private communication. 11. W. J. Wiscombe, Improved Mie Scattering Algorithms , Appl. Opt., Vol. 19, 1505- 1509, 1980. 12. G. H. Goedecke, A. Miller and R. C. Shirkey, Simple Scattering Code Agausx , in Atmospheric Aerosols: Their Formation, Optical Propertie s and Eﬀects. Ed. A. Deepak, Spectrum Press, Hampton, Virginia, 1982. 1613. A. Miller, Comments on Mie Calculations , Am. J. Phys., Vol. 54, 297-297, 1986. Also private communication. 14. G. W. Kattawar and G. N. Plass, Electromagnetic Scattering from Absorbing Spheres , Appl. opt., vol. 6, 1377, 1967. 15. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersion , Elsevier, N. Y. 1969. 16. H. Quenzel and H. M¨ uller, Optical properties of single particles diagrams of inten- sity, extinction scattering and absorption eﬃciencies , Wissenschaftliche Mitteilung, n. 34. Metereologisches Institut, Universit¨ at M¨ unchen, 1978. 17. W. J. Lentz, Generating Bessel Functions in Mie Scattering Calculation s using Continued Fractions , Appl. Opt., vol. 15, 668-671, 1976. 18. V. E. Cachorro, L. L. Salcedo and J. L. Casanova, Programa LVEC-MIE para el c´ alculo de las magnitudes de la teor´ ıa de esparcimiento de Mie, Anales de F´ ısica, vol. 85, Serie B, 198-211, 1989. 17
arXiv:physics/0103053v1 [physics.chem-ph] 19 Mar 2001Energy dissipation and scattering angle distribution anal ysis of the classical trajectory calculations of methane scattering from a Ni(111) surface Robin Milot Schuit Institute of Catalysis, ST/SKA, Eindhoven Universi ty of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands. A.W. Kleyn Leiden Institute of Chemistry, Department of Surfaces and C atalysis, Leiden University, P.O. Box 9502, NL-2300 RA Leiden, The Netherlands. A.P.J. Jansen Schuit Institute of Catalysis, ST/SKA, Eindhoven Universi ty of Technology. (February 2, 2008) We present classical trajectory calculations of the rota- tional vibrational scattering of a non-rigid methane molec ule from a Ni(111) surface. Energy dissipation and scattering angles have been studied as a function of the translational k i- netic energy, the incidence angle, the (rotational) nozzle tem- perature, and the surface temperature. Scattering angles a re somewhat towards the surface for the incidence angles of 30◦, 45◦, and 60◦at a translational energy of 96 kJ/mol. Energy loss is primarily from the normal component of the transla- tional energy. It is transfered for somewhat more than half to the surface and the rest is transfered mostly to rotationa l motion. The spread in the change of translational energy has a basis in the spread of the transfer to rotational energy , and can be enhanced by raising of the surface temperature through the transfer process to the surface motion. 34.50.Dy,31.15.Qg,34.50.Ez,34.20.Mq,79.20.Rf I. INTRODUCTION The dissociative adsorption of methane on transition metals is an important reaction in catalysis; it is the rate limiting step in steam reforming to produce syngas, and it is prototypical for catalytic C–H activation. There- fore the dissociation is of high interest for many surface scientists. (See for a recent review Ref. 1.) Molecular beam experiments in which the dissociation probability was measured as a function of translational energy have observed that the dissociation probability is enhanced by the normal incidence component of the incidence trans- lational energy.2–12This suggests that the reaction oc- curs primarily through a direct dissociation mechanism at least for high translational kinetic energies. Some ex- periments have also observed that vibrationally hot CH 4 dissociates more readily than cold CH 4, with the energy in the internal vibrations being about as eﬀective as the translational energy in inducing dissociation.2–4,8,7,13,9,10 A molecular beam experiment with laser excitation of the ν3mode did succeed in measuring a strong enhancement of the dissociation on a Ni(100) surface. However, thisenhancement was still much too low to account for the vi- brational activation observed in previous studies and in- dicated that other vibrationally excited modes contribute signiﬁcantly to the reactivity of thermal samples.14 It is very interesting to simulate the dynamics of the dissociation, because of the direct dissociation mecha- nism, and the role of the internal vibrations. Wave packet simulations of the methane dissociation reaction on tran- sition metals have treated the methane molecule always as a diatomic up to now.15–20Apart from one C–H bond (a pseudo ν3stretch mode) and the molecule surface dis- tance, either (multiple) rotations or some lattice motion were included. None of these studies have looked at the role of the other internal vibrations, so there is no model that describes which vibrationally excited mode might be responsible for the experimental observed vibrational activation. In previous papers we have reported on wave packet simulations to determine which and to what extent in- ternal vibrations are important for the dissociation in the vibrational ground state of CH 4,21and CD 4.22We were not able yet to simulate the dissociation including all internal vibrations. Instead we simulated the scatter- ing of methane in ﬁxed orientations, for which all inter- nal vibrations can be included, and used the results to deduce consequences for the dissociation. These simula- tions indicate that to dissociate methane the interaction of the molecule with the surface should lead to an elon- gated equilibrium C–H bond length close to the surface, and that the scattering was almost elastic. Later on we reported on wave packet simulations of the role of vibra- tional excitations for the scattering of CH 4and CD 4.23 We predicted that initial vibrational excitations of the asymmetrical stretch ( ν3) but especially the symmetri- cal stretch ( ν1) modes will give the highest enhancement of the dissociation probability of methane. Although we have performed these wave packet simulations in ten di- mensions, we still had to neglect two translational and three rotational coordinates of the methane molecule and we did not account for surface motion and corrugation. It is nowadays still hard to include all these features into 1a wave packet simulation, therefore we decided to study these with classical trajectory simulations. In this article we will present full classical trajectory simulations of methane from a Ni(111) surface. We have especially interest in the eﬀect of the molecular rota- tions and surface motion, which we study as a function of the nozzle and surface temperature. The methane molecule is ﬂexible and able to vibrate. We do not include vibrational kinetic energy at the beginning of the simulation, because a study of vibrational excita- tion due to the nozzle temperature needs a special semi- classical treatment. Besides its relevance for the disso- ciation reaction of methane on transition metals, our scattering simulation can also be of interest as a refer- ence model for the interpretation of methane scattering itself, which have been studied with molecular beams on Ag(111),24,25Pt(111),26–29and Cu(111) surfaces.30 It was observed that the scattering angles are in some cases in disagreement with the outcome of the classi- cal Hard Cube Model (HCM) described in Ref. 31.26,27 We will show in this article that the assumptions of this HCM model are too crude for describing the processes obtained from our simulation. The time-of-ﬂight experi- ments show that there is almost no vibrational excitation during the scattering,28,29which is in agreement with our current classical simulations and our previous wave packet simulations.21,22 The rest of this article is organized as follows. We start with a description of our model potential, and an expla- nation of the simulation conditions. The results and dis- cussion are presented next. We start with the scattering angles, and relate them to the energy dissipation pro- cesses. Next we will compare our simulation with other experiments and theoretical models. We end with a sum- mary and some general conclusions. II. COMPUTATIONAL DETAILS We have used classical molecular dynamics for sim- ulating the scattering of methane from a Ni(111) sur- face. The methane molecule was modelled as a ﬂexible molecule. The forces on the carbon, hydrogen, and Ni atoms are given by the gradient of the model potential energy surface described below. The ﬁrst-order ordinary diﬀerential equations for the Newtonian equations of mo- tion of the Cartesian coordinates were solved with use of a variable-order, variable-step Adams method.32We have simulated at translational energies of 24, 48, 72, and 96 kJ/mol at normal incidence, and at 96 kJ/mol for inci- dence angles of 30◦, 45◦, and 60◦with the surface normal. The surface temperature and (rotational) nozzle temper- ature for a certain simulation were taken independently between 200 and 800 K.A. Potential energy surface The model potential energy surface used for the clas- sical dynamics is derived from one of our model poten- tials with elongated C–H bond lengths towards the sur- face, previously used for wave packet simulation of the vibrational scattering of ﬁxed oriented methane on a ﬂat surface.21,22In this original potential there is one part responsible for the repulsive interaction between the sur- face and the hydrogens, and another part for the in- tramolecular interaction between carbon and hydrogens. We have rewritten the repulsive part in pair potential terms between top layer surface Ni atoms and hydrogens in such a way that the surface integral over all these Ni atoms give the same overall exponential fall-oﬀ as the original repulsive PES term for a methane molecule far away from the surface in an orientation with three bonds pointing towards the surface. The repulsive pair interac- tion term Vrepbetween hydrogen iand Ni atom jat the surface is then given by Vrep=A e−αZij Zij, (1) where Zijis the distance between hydrogen atom iand Ni atom j. The intramolecular potential part is split up in bond, bond angle, and cross potential energy terms. The single C–H bond energy is given by a Morse function with bond lengthening towards the surface Vbond=De/bracketleftBig 1−e−γ(Ri−Req)/bracketrightBig2 , (2) where Deis the dissociation energy of methane in the gas phase, and Riis the length of the C–H bond i. Disso- ciation is not possible at the surface with this potential term, but the entrance channel for dissociation is mim- icked by an elongation of the equilibrium bond length Req when the distance between the hydrogen atom and the Ni atoms in the top layer of the surface become shorter. This is achieved by Req=R0+S/summationdisplay je−αZij Zij, (3) where R0is the equilibrium C–H bond length in the gas phase. The bond elongation factor Swas chosen in such a way that the elongation is 0.054 nm at the classical turning point of 93.2 kJ/mol incidence translational en- ergy for a rigid methane molecule, when the molecule approach a surface Ni atom atop with one bond pointing towards the surface. The single angle energy is given by the harmonic expression Vangle=kθ(θij−θ0)2, (4) where θijis the angle between C–H bond iandj, and θ0the equilibrium bond angle. Furthermore, there are 2some cross-term potentials between bonds and angles. The interaction between two bonds are given by Vbb=kRR(Ri−R0)(Rj−R0). (5) The interaction between a bond angle and the bond angle on the other side is given by Vaa=kθθ(θij−θ0)(θkl−θ0). (6) The interaction between a bond angle and one of its bonds is given by Vab=kθR(θij−θ0)(Ri−R0). (7) The parameters of the intramolecular potential energy terms were calculated by ﬁtting the second derivatives of these terms on the experimental vibrational frequencies of CH 4and CD 4in the gas phase.33,34 The Ni-Ni interaction between nearest-neighbours is given by the harmonic form VNi−Ni=1 2λij[(ui−uj)·ˆ rij] +1 2µij/braceleftBig (ui−uj)2−[(ui−uj)·ˆ rij]2/bracerightBig .(8) Theu’s are the displacements from the equilibrium po- sitions, and ˆ ris a unit vector connecting the equilibrium positions. The Ni atoms were placed at bulk positions with a nearest-neighbour distance of 0.2489 nm. The pa- rameters λijandµijwere ﬁtted on the elastic constants35 and cell parameters36of the bulk. The values of all pa- rameters are given in Table I. B. Simulation model The surface is modelled by a slab consisting of four layers of eight times eight Ni atoms. Periodic boundary conditions have been used in the lateral direction for the Ni-Ni interactions. The methane molecule has interac- tions with the sixty-four Ni atoms in the top layer of the slab. The surface temperature is set according to the following procedure. The Ni atoms are placed in equilib- rium positions and are given random velocities out of a Maxwell-Boltzmann distribution with twice the surface temperature. The velocities are corrected such that the total momentum of all surface atoms is zero in all direc- tions, which ﬁxes the surface in space. Next the surface is allowed to relax for 350 fs. We do the following ten times iteratively. If at the end of previous relaxation the total kinetic energy is above or below the given surface tem- perature, then all velocities are scaled down or up with a factor of√ 1.1 respectively. Afterwards a new relaxation simulation is performed. The end of each relaxation run is used as the begin condition of the surface for the actual scattering simulation. The initial perpendicular carbon position was chosen 180 nm above the equilibrium z-position of the top layeratoms and was given randomly parallel ( x,y) positions within the central surface unit cell of the simulation slab for the normal incidence simulations. The methane was placed in a random orientation with the bonds and an- gles of the methane in the minimum of the gas phase potential. The initial rotational angular momentum was generated randomly from a Maxwell-Boltzmann distribu- tion for the given nozzle temperature for all three rota- tion axis separately. No vibrational kinetic energy was given initially. Initial translational velocity was given to all methane atoms according to the translational energy. The simulations under an angle were given parallel mo- mentum in the [110] direction. The parallel positions have been translated according to the parallel velocities in such a way that the ﬁrst collision occurs one unit cell before the central unit cell of the simulation box. We tested other directions, but did not see any diﬀerences for the scattering. Each scattering simulation consisted of 2500 trajecto- ries with a simulation time of 1500 fs each. We calculated the (change of) translational, total kinetic, rotational a nd vibrational kinetic, intramolecular potential, and total energy of the methane molecule; and the scattering an- gles at the end of each trajectory. We calculated for them the averages and standard deviations, which gives the spread for the set of trajectories, and correlations co- eﬃcients from which we can abstract information about the energy transfer processes. III. RESULTS AND DISCUSSION We will now present and discuss the results of our sim- ulations. We begin with the scattering angle distribu- tion. Next we will explain this in terms of the energy dissipation processes. Finally we will compare our sim- ulation with previous theoretical and experimental scat- tering studies, and discuss the possible eﬀects on the dis- sociation of methane on transition metal surfaces. TABLE I. Parameters of the potential energy surface. Ni–H A 971.3 kJ nm mol−1 α 20.27 nm−1 S 0.563 nm2 CH4 γ 17.41 nm−1 De 480.0 kJ mol−1 R0 0.115 nm kθ 178.6 kJ mol−1rad−2 θ0 1.911 rad kRR 4380 kJ mol−1nm−2 kθθ 11.45 kJ mol−1rad−2 kθR -472.7 kJ mol−1rad−1nm−1 Ni–Ni λnn 28328 kJ mol−1nm−2 µnn -820 kJ mol−1nm−2 3A. Scattering angles Figure 1 shows the scattering angle distribution for dif- ferent incidence angles with a initial total translational energy of 96 kJ/mol at nozzle and surface temperatures of both 200 and 800 K. The scatter angle is calculated from the ratio between the normal and the total paral- lel momentum of the whole methane molecule. We ob- serve that most of the trajectories scatter some degrees towards the surface from the specular. This means that there is relatively more parallel momentum than normal momentum at the end of the simulation compared with the initial ratio. This ratio change is almost completely caused by a decrease of normal momentum. The higher nozzle and surface temperatures have al- most no inﬂuence on the peak position of the distribu- tion, but give a broader distribution. The standard de- viation in the scattering angle distribution goes up from 2.7◦, 2.4◦, and 2 .2◦at 200K to 4 .4◦, 3.8◦, and 3 .4◦at 800K for incidence angles of 30◦, 45◦, and 60◦respec- tively. This means that the angular width is very nar- row, because the full width at half maximum (FWHM) are usually larger than 20◦.37(The FWHM is approxi- mately somewhat more than twice the standard devia- tion.) The broadening is caused almost completely by raising the surface temperature, and has again primarily an eﬀect on the spread of the normal momentum of the molecule. This indicates that the scattering of methane from Ni(111) is dominated by a thermal roughening pro- cess. We do not observe an average out-of-plane diﬀraction for the non normal incidence simulations, but we do see some small out-of-plane broadening. The standard de- viations in the out-of-plane angle were 0.9◦, 1.8◦, 3.4◦ at a surface temperature of 200K, and 1.7◦, 3.3◦, and 6.0◦at 800K for incidence angles of 30◦, 45◦, and 60◦ with the surface normal. Raising the (rotational) noz- zle temperature has hardly any eﬀect on the out-of-plane broadening. B. Energy dissipation processes 1. Translational energy Figure 2 shows the average energy change of some en- ergy components of the methane molecule between the end and the begin of the trajectories as a function of the initial total translational energy. The incoming angle for all is 0◦(normal incidence), and both the nozzle and sur- face are initially 400K. If we plot the normal incidence translational energy component of the simulation at 96 kJ/mol for the diﬀerent incidence angles, then we see a similar relation. This means that there is normal transla- tional energy scaling for the scattering process in general , except for some small diﬀerences discussed later on.0150300450600 0 15 30 45 60 75 90Intensity 0150300450600 0 15 30 45 60 75 90Intensitya) T = 200 K b) T = 800 Kangle [degrees] angle [degrees]0 3045600 3045 60 FIG. 1. The distribution of the scattering angle for a total initial translational energy of 96 kJ/mol with incidence an gles of 0◦, 30◦, 45◦, and 60◦with the surface normal. Both the nozzle and surface temperature are: a) 200K, and b) 800K. Most of the initial energy of methane is available as translational energy, so it cannot be surprising that we see here the highest energy loss. The translational energy loss takes a higher percentage of the initial translational energy at higher initial translational energies. Since al- most all of the momentum loss is in the normal direction, we also see that the loss of translational energy can be found back in the normal component of the translational energy for the non-normal incidence simulations. The average change of the total energy of the methane molecule is less negative than the average change in trans- lational energy, which means that there is a net transfer of the initial methane energy towards the surface dur- ing the scattering. This is somewhat more than half of the loss of translational energy. The percentage of trans- fered energy to the surface related to the normal inci- dence translational energy is also enhanced at higher in- cidence energies. There is somewhat more translational energy loss, and energy transfer towards the surface for the larger scattering angles, than occurs at the compa- rable normal translational energy at normal incidence. This is caused probably by interactions with more sur- face atoms, when the molecule scatters under an larger angle with the surface normal. In Fig. 2 we also plotted the average change of methane potential energy and the change of rotational and vibra- tional kinetic energy of methane. We observe that there is extremely little energy transfer towards the potential energy, and a lot of energy transfer towards rotational 4and vibrational kinetic energy. Vibrational motion gives an increase of both potential and kinetic energy. Rota- tional motion gives only an increase in kinetic energy. So this means that there is almost no vibrational inelastic scattering, and very much rotational inelastic scattering . -20-15-10-50510 24 48 72 96Energy change [kJ/mol] Initial translational energy [kJ/mol]-25TranslationalTotalPotentialRotational and vibrational kinetic FIG. 2. The average energy change (kJ/mol) of the methane translational energy, the methane total energy, th e methane potential energy, and the methane rotational and vibrational kinetic energy as a function translational kin etic energy (kJ/mol) at normal incidence. The nozzle and surface temperature were 400K. Figure 3 shows the standard deviations in the energy change of some energy components of methane versus the initial translational energy at normal incidence for a noz- zle and surface temperature of 200K. (The temperature eﬀects will be discussed below.) The standard deviations in the energy changes are quite large compared to the average values. The standard deviations in the change of the methane translational energy and in the change of methane rotational and vibrational kinetic energy in- crease more than the standard deviation in the change of methane total energy, when the initial translational en- ergy is increased. We ﬁnd again an identical relation if we plot the standard deviations versus the initial normal energy component of the scattering at diﬀerent incidence angles. The standard deviations are much smaller in the parallel than in the normal component of the transla- tional energy, so again only the normal component of the translational energy is important. Although the stan- dard deviations in the translational energy is smaller at larger incidence angles than at smaller incidence angles, we see in Fig. 1 that the spread in the angle distributionis almost the same. This is caused by the fact that at large angles deviations in the normal direction has more eﬀect on the deviation in the angle than at smaller angles with the normal. 10 24 48 72 96Standard deviation [kJ/mol] Initial translational energy [kJ/mol]5 TotalTranslational Rotational and vibrational kinetic FIG. 3. The standard deviation in the energy change (kJ/mol) of the methane translational energy, the methane total energy, and the methane rotational and vibrational ki - netic energy as a function of the initial translational ener gy (kJ/mol) at normal incidence. The surface and nozzle tem- perature are both 200K. 2. Surface temperature An increase of surface temperature gives a small re- duction of average translational energy loss (around 5 % from 200K to 800K at 96 kJ/mol normal incidence). This is the reason why we do not observe a large shift of the peak position of the scattering angle distribution. However, an increase of surface temperature does have a larger eﬀect on the average energy transfer to the sur- face, but this is in part compensated through a decrease of energy transfer to rotational energy. Figure 4 shows the standard deviations in the energy change of the translational energy, the methane total en- ergy, and the methane rotational and vibrational kinetic energy as a function of the surface temperature. We ob- serve that the standard deviation in the change of rota- tional and vibrational kinetic energy hardly changes at increasing surface temperature. At a low surface tem- perature it is much higher than the standard deviation in the change of the methane total energy. So the base- line broadening of translational energy is caused by the transfer of translational to rotational motion. The stan- dard deviation in the change of the methane total energy increases much at higher surface temperature. This re- sults also in an increase of the standard deviation in the change of translational energy, which means that the sur- face temperature inﬂuences the energy transfer process between translational and surface motion. The spread in the change of translational energy is related to the spread in the scattering angle distributions. It is now clear that the observed broadening of the scattering angle distribu- tion with increasing surface temperature is really caused by a thermal roughening process. 551015 200 400 600 800Standard deviation [kJ/mol] Surface temperature [K]TotalRotational and vibrational kineticTranslational FIG. 4. The standard deviation in the energy change (kJ/mol) of the methane translational energy, the methane total energy, and the methane rotational and vibrational ki - netic energy as a function of the surface temperature (K). Th e nozzle temperature is 400K, and the translational energy is 96 kJ/mol at normal incidence. 3. Nozzle temperature Figure 5 shows the dependency of the standard de- viations for the diﬀerent energy changes on the nozzle temperature. From this ﬁgure it is clear that the noz- zle temperature has relative little inﬂuence on the stan- dard deviations in the diﬀerent energy changes. There- fore we observe almost no peak broadening in the scat- tering angle distribution due to the nozzle temperature. The nozzle temperature has also no inﬂuence on the av- erage change of rotational and vibrational kinetic energy, which means that this part of the energy transfer process is driven primarily by normal incidence translational en- ergy. 51015 200 400 600 800Standard deviation [kJ/mol] Nozzle temperature [K]TotalRotational and vibrational kineticTranslational FIG. 5. The standard deviation in the energy change (kJ/mol) of the methane translational energy, the methane total energy, and the methane rotational and vibrational ki - netic energy as a function of the nozzle temperature (K). The surface temperature is 400K, and the translational energy i s 96 kJ/mol at normal incidence. We have to keep in mind that we only studied the rotational heating by the nozzle temperature, and that we did not take vibrational excitation by nozzle heating into account. From our wave packet simulations we know that vibrational excitations can contribute to a strongenhancement of vibrational inelastic scattering.23So the actual eﬀect of raising the nozzle temperature can be diﬀerent than sketched here. C. Comparison with other studies 1. Scattering angles and the Hard Cube Model The angular dependnece of scattered intensity for a ﬁxed total scattering angle has only been measured at Pt(111).26,27The measurement has been compared with the predictions of the Hard Cube Model (HCM) as de- scribed in Ref. 31. There seems to be more or less agreement for low translational energies under an angle around 45◦with the surface, but is anomalous at a trans- lational energy of 55 kJ/mol. The anomalous behaviour has been explained by altering the inelastic collision dy- namics through intermediate methyl fragments. Although our simulations are for Ni(111) instead of Pt(111) and we calculate real angular distributions, we will show now that the HCM is insuﬃcient for describing the processes involved with the scattering of methane in our simulation. The HCM neglects the energy transfer to rotational excitations, and overestimates the energy transfer to the surface. This is not surprising, because the HCM is constructed as a simple classical model for the scattering of gas atoms from a solid surface. The basic assumptions are that (1) the interaction of the gas atom with a surface atom is represented by an impulsive force of repulsion, (2) the gas-surface intermolecular po- tential is uniform in the plane of the surface, (3) the sur- face is represented by a set of independent particles con- ﬁned by square well potentials, (4) the surface particles have a Maxwellian velocity distribution.31Assumption 1 excludes inelastic rotational scattering, because the gas particle is an atom without moment of inertia. So the HCM misses a large part of inelastic scattering. How- ever, it still predicts scattering angles much more below the incidence angles than we found from our simulation. For example: The HCM predicts an average scattering angle with the surface normal of 64◦from Ni(111), at an incidence angle of 45◦at a surface temperature four times lower than the gas temperature. This is much more than for Pt(111), because the mass ratio between the gas par- ticle and the surface atom is higher for Ni(111). There are several explanations for this error. First, the assump- tion 3 is unreasonable for atomic surfaces with low atom weight, because the surface atoms are strongly bound to each other. This means that eﬀectively the surface has a higher mass than assumed.38Second, there is no one- on-one interaction between surface atom and methane molecule, but multiple hydrogen atoms interacting with diﬀerent Ni atoms. Third, the methane molecule is not rigid in contrast to assumption 1. We have followed the energy distribution during the simulation for some tra- jectories and ﬁnd that the methane molecule adsorbs ini- 6tial rotational and translational energy as vibrational en - ergy in its bonds and bond angles when close the surface, which is returned after the methane moves away from it. It would be nice to test our model with molecular beam experiment of the scattering angles on surfaces with rel- atively low atom weight, which also try to look at rota- tional inelastic scattering. 2. Wave packet simulations Let us now compare the full classical dynamics with our ﬁxed oriented wave packet simulations,21–23because this was initial the reason to perform the classical dynam- ics simulations. Again we observe very little vibrational inelastic scattering. This is in agreement with the obser- vations in the time-of-ﬂight experiments on Pt(111).28,29 Since we used our wave packet simulations to deduce consequences for the dissociation of methane, we have to wonder whether the observed inelastic scattering in our classical simulations changes the picture of the dissoci- ation in our previous publications. Therefore we have to look at what happens at the surface. We did so by following some trajectories in time. We ﬁnd approximately the same energy rearrange- ments for the classical simulations as discussed for the wave packet simulations for the vibrational groundstate in Refs. 22 and 23. Again most of the normal transla- tional energy is transfered to the potential energy terms of the surface repulsion [see Eq. 1]. This repulsive poten- tial energy was only given back to translational energy in the wave packet simulations, because the orientations and surface were ﬁxed. For the classical trajectory sim- ulations presented in this article, the repulsive potentia l energy is transfered to translational, rotational, and sur - face energy through the inherent force of the repulsive energy terms. We observe almost no energy transfers to translational energy parallel to the surface, so exclusion of these translational coordinates in the wave packet sim- ulations do not eﬀect our deduction on the dissociation. The energy transfers to the rotational and surface en- ergy during the collision make it harder for the molecule to approach the surface. This will have a quantitative eﬀect on the eﬀective bond lengthening near the surface, but not a qualitative. The remaining problem deals with the eﬀect of rota- tional motion on the dissociation probability and steer- ing. Our ﬁrst intension was to look for the favourable orientation at the surface, but from following some tra- jectories it is clear that steering does not seem to occur. There is always some rotational motion, and the molecule leaves the surface often with another hydrogen pointing towards to surface than when it approaches the surface. This indicates that multiple bonds have a chance to dis- sociate during one collision. However, it will be very speculative to draw more conclusion on the dissociation of methane based on the scattering in these classical tra-jectory simulations. Classical trajectory simulation wit h an extension of our potentials with an exit channel for dissociation can possibly learn us more. IV. CONCLUSIONS We have performed classical dynamics simulations of the rotational vibrational scattering of non-rigid methan e from a corrugated Ni(111) surface. Energy dissipation and scattering angles have been studied as a function of the translational kinetic energy, the incidence angle, the (rotational) nozzle temperature, and the surface temper- ature. We ﬁnd the peak of the scattering angle distribution somewhat below the incidence angle of 30◦, 45◦, and 60◦ at a translational energy of 96 kJ/mol. This is caused by an average energy loss in the normal component of the translational energy. An increase of initial normal trans- lational energy gives an enhancement of inelastic scat- tering. The energy loss is transfered for somewhat more than half to the surface and the rest mostly to rotational motion. The vibrational scattering is almost completely elastic. The broadening of the scattering angle distribution is mainly caused by the energy transfer process of transla- tional energy to rotational energy. Heating of the noz- zle temperature gives no peak broadening. Heating of the surface temperature gives an extra peak broadening through thermal roughening of the surface. The Hard Cube Model seems to be insuﬃcient for de- scribing the scattering angles of methane from Ni(111), if we compare its assumptions with the processes found in our simulations. ACKNOWLEDGMENTS This research has been ﬁnancially supported by the Council for Chemical Sciences of the Netherlands Or- ganization for Scientiﬁc Research (CW-NWO), and has been performed under the auspices of the Netherlands Institute for Catalysis Research (NIOK). 1J. H. Larsen and I. Chorkendorﬀ, Surf. Sci. Rep. 35, 163 (2000). 2C. T. Rettner, H. E. Pfn¨ ur, and D. J. Auerbach, Phys. Rev. Lett.54, 2716 (1985). 3C. T. Rettner, H. E. 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arXiv:physics/0103054v1 [physics.optics] 19 Mar 2001Superluminal Localized Waves of Electromagnetic Field in Vacuo Peeter Saari Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia July 23, 2013 Abstract Presented is an overview of electromagnetic versions of the so-called X-type waves intensively studied since their invention in e arly 1990.-ies in ultrasonics. These waves may be extremely localized both laterally and longitudinally and – what has been considered as most sta rtling – propagate superluminally without apparent spread. Spotli ghted are the issues of the relativistic causality, variety of mathemati cal description and possibilities of practical applications of the waves. PACS numbers: 42.25.Bs, 03.40.Kf, 42.65.Re, 41.20.Jb. TO BE PUBLISHED IN Proceedings of the conference ”Time’s Arrows, Quantum Measurements and Superluminal Behaviour” (Naples , October 2-6, 2000) by the Italian NCR. 1 Introduction More often than not some physical truths, as they gain genera l acceptance, en- ter textbooks and become stock rules, loose their exact cont ent for the majority of the physics community. Moreover, in this way superﬁciall y understood rules may turn to superﬂuous taboos inhibiting to study new phenom ena. For ex- ample, conviction that ”uniformly moving charge does not ra diate” caused a considerable delay in discovering and understanding the Ch erenkov eﬀect. By the way, even the reﬁned statement ”uniformly moving charge does not radiate in vacuum” is not exact as it excludes the so-called transiti on radiation known an half of century only, despite it is a purely classical eﬀec t of macroscopic electrodynamics. In this paper we give an overview of electromagnetic version s of the so- called X-type waves intensively studied since 1990.-ies [1 ]-[13]. The results obtained have encountered such taboo-fashioned attitudes sometimes. Indeed, these waves, or more exactly – wavepackets, may be extremely localized both lat- erally and longitudinally and, what is most startling, prop agate superluminally 1without apparent diﬀraction or spread as yet. Furthermore, they are solutions - although exotic - of linear wave equations and, hence, have n othing to do with solitons or other localization phenomena known in contempo rary nonlinear sci- ence. Instead, study of these solutions has in a sense reinca rnated some almost forgotten ideas and ﬁndings of mathematical physics of the p revious turn of the century. X-type waves belong to phenomena where a naive s uperluminality taboo ”group velocity cannot exceed the speed of light in vac uum” is broken. In this respect they fall into the same category as plane waves i n dispersive reso- nant media and the evanescent waves, propagation of which (p hoton tunneling) has provoked much interest since publication of papers [14] ,[15],[16] .Therefore it is not surprising that tunneling of X waves in frustrated i nternal reﬂection has been treated in a recent theoretical paper [17] . Indeed, studies conducted in diﬀerent subﬁelds of physics, which are dealing with superluminal movements, are interfering and merging f ruitfully. A convinc- ing proof of this trend is the given Conference and the collec tion of its papers in hand. This is why in this paper we spotlight just superluminality o f the X waves, which is now an experimentally veriﬁed fact [8],[9],[13], b ut which should not be considered as their most interesting attribute in genera l. Their name was coined within theoretical ultrasonics by the authors of the paper [1] which initi- ated an intensive study of the X waves, particularly due to ou tlooks of applica- tion in medical ultrasonic imaging. Possible superluminal ity of electromagnetic localized waves was touched by the authors of Ref. [2] – who ha d derived the waves under name ”slingshot pulses” independently from the paper [1] – and be- came the focus of growing interest thanks to E .Recami (see [1 0] and references therein), who pointed out physically deeply meaningful res emblance between the shape of the X waves and that of the tachyon [18]. The paper [18 ] was published in times of great activity in theoretical study of these hypo thetical superlumi- nal particles. To these years belongs paper [19] where a doub le-cone-shaped ”electromagnetic tachyon” as a result of light reﬂection by a conical mirror was considered. This a quarter-of-century-old paper seems to b e the very pioneering work on X-waves, though this and the subsequent papers of the same author have been practically unknown and only very recently were re discovered for the X wave community (see references in the review [11]). Last bu t not least, if one asked what was the very ﬁrst sort of superluminal waves imple mented in physics, the answer would be – realistic plane waves. Indeed, as it is w ell known, the most simple physically feasible realization of a plane wave beam is the Gaussian beam with its bounded cross-section and, correspondingly, a ﬁnite energy ﬂux. However, much less is known that due to the Gouy phase shift th e group veloc- ity in the waist region of the Gaussian beam is slightly super luminal, what one can readily check on the analytical expressions for the beam (see also Ref. [20]). For all the reasons mentioned, in this paper we present – afte r an introduction of the physical nature of the X-type waves (Section 2) - quite in detail a new representation of the localized waves (Section 3). This rep resentation – what we believe is a new and useful addition into the theory of X-ty pe waves – in a sense generalizes the Huygens principle into superluminal domain and directly 2relies on superluminality of focal behavior of any type of fr ee-space waves, which manifests itself in the Gouy phase shift. The startling supe rluminality issues are brieﬂy discussed in the last Section. Figures showing 3-dimensional plots have been included for a vivid compre- hension of the spatio-temporal shape of the waves, however, only few of the animations showed in the oral presentation had sense to be re produced here in the static black-and-white form. The bibliography is far fr om being complete, but hopefully a number of related references can be found in o ther papers of the issue in hand. 2 Physical nature of X-type waves In order to make the physical nature of the X-type superlumin al localized waves better comprehensible, we ﬁrst discuss a simple representa tion of them as a result of interference between plane wave pulses. Fig.1. X-type scalar wave formed by scalar plane wave pulses containing three cosinusoidal cycles. The propagation direction (along the axisz) is indi- cated by arrow. As linear gray-scale plots in a plane of the pr opagation axis and at a ﬁxed instant, shown are (a) the ﬁeld of the wave (r eal part if the plane waves are given as analytic signals) and (b) its amp litude (mod- ulus). Note that the central bullet-like part of the wave wou ld stand out even more sharply against the sidelobes if one plotted the th e distribution of the intensity (modulus squared) of the wave. With reference to Fig.1(a) let us consider a pair of plane wav e bursts pos- sessing identical temporal dependences and the wave vector s in the plane y= 0. Their propagation directions given by unit vectors n/=[sin θ,0,cosθ] and n\=[−sinθ,0,cosθ] are tilted under angle θwith respect to the axis z.In spatio-temporal regions where the pulses do not overlap the ir ﬁeld is given sim- ply by the burst proﬁle as Ψ P(η−ct), where ηis the spatial coordinate along 3the direction n/orn\, respectively. In the overlap region , if we introduce the radius vector of a ﬁeld point r= [x, y, z ] the ﬁeld is given by superposition ΨP(rn/−ct) + Ψ P(rn\−ct) = Ψ P(xsinθ+zcosθ−ct) + + Ψ P(−xsinθ+zcosθ−ct), (1) which is nothing but the well-known two-wave-interference pattern with dou- bled amplitudes. Altogether, the superposition of the puls e pair – as two branches \and / form the letter X – makes up an X-shaped propagation-inv ariant interference pattern moving along the axis zwith speed v=c/cosθwhich is both the phase and the group velocity of the wave ﬁeld in the di rection of the propagation axis z. This speed is superluminal in a similar way as one gets a faster-than-light movement of a bright stripe on a screen wh en a plane wave light pulse is falling at the angle θonto the screen plane. Let us stress that here we need not to deal with the vagueness of the physical mea ning inherent to the group velocity in general – simply the whole spatial di stribution of the ﬁeld moves rigidly with vbecause the time enters into the Eq.(1) only together with the coordinate zthrough the propagation variable zt=z−vt. Further let us superimpose axisymmetrically all such pairs of waves whose propagation directions form a cone around the axis zwith the top angle 2 θ, in other words, let the pair of the unit vectors be n/=[sin θcosφ,sinθsinφ,cosθ] andn\=[sin θcos(φ+π),sinθsin (φ+π),cosθ], where the angle φruns from 0 to 180 degrees. As a result, we get an X-type supeluminal loc alized wave in the following simple representation ΨX(ρ, zt) =/integraldisplayπ 0dφ/bracketleftbig ΨP/parenleftbig rtn//parenrightbig + Ψ P/parenleftbig rtn\/parenrightbig/bracketrightbig =/integraldisplay2π 0dφΨP/parenleftbig rtn//parenrightbig ,(2) where rt= [ρcosϕ, ρsinϕ, z t] is the radius vector of a ﬁeld point in the co- propagating frame and cylindrical coordinates ( ρ, ϕ, z ) have been introduced for we restrict ourselves in this paper to axisymmetric or so-ca lled zeroth-order X- type waves only. Hence, according to Eq.(2) the ﬁeld is built up from interfering pairs of identical bursts of plane waves. Fig.1 gives an exam ple in which the plane wave proﬁle Ψ Pcontains three cycles. The less extended the proﬁle Ψ P, the better the separation and resolution of the branches of the X-shaped ﬁeld. In the superposition the p oints of completely constructive interference lie on the zaxis, where the highly localized energy ”bullet” arises in the center, while the intensity falls oﬀ a sρ−1along the branches and much faster in all other directions (note that in contras t with Fig.1(a) in the case of interference of only two waves the on-axis and o ﬀ-axis maxima must be of equal strength). The optical carrier manifests it self as one or more (depending on the number of cycles in the pulse) halo toroids which are nothing but residues of the concentric cylinders of intensity chara cteristic of the Bessel beam. That is why we use the term ’Bessel-X pulse’ (or wave) to draw a distinction from carrierless X waves. By making use of an int egral representation of the zeroth-order Bessel function J0(v) =π−1/integraltextπ 0cos[vcos(φ−ϕ)]dφ, where 4ϕis an arbitrary angle, and by reversing the mathematical pro cedure described in Ref. [9], we get the common representation of X-type waves as superposition of monochromatic cylindrical modes (Bessel beams) of diﬀer ent wavenumber k=ω/c ΨX(ρ, z, t) =/integraldisplay∞ 0dk S(k)J0(ρksinθ) exp [ i(zkcosθ−ωt)], (3) where S(k) denotes the Fourier spectrum of the proﬁle Ψ P. Again, the Eq.(3) gives for both the phase and the group velocities (along the a xisz– in the direc- tion of the propagation of the packet of the cylindrical wave s) the superluminal value v=c/cosθ. 3 X-waves as wakewaves Although the representation Eq.(2) of the X-type waves as bu ilt up from two- plane-wave-pulse interference patterns constitutes an ea sily comprehensible ap- proach to the superluminality issues, it may turn out to be co unter-intuitive for symmetry considerations as will be shown below. In this s ection we deve- lope another representation introduced in Ref. [21], which is, in a sense, a generalization of the Huygens principle into superluminal domain and allows ﬁguratively to describe formation of superluminal localiz ed waves. As known in electrodynamics, the D’Alambert (source-free) wave equation possesses a particular solution D0, which is spherically symmetric and can be expressed through retarded and advanced Green functions of the equation, G(+) andG(−), respectively, as D0=c−2/bracketleftBig G(+)−G(−)/bracketrightBig =1 4πRc[δ(R−ct)−δ(R+ct)], (4) where Ris the distance from the origin and δis the Dirac delta function. Thus, the function D0represents a spherical delta-pulse-shaped wave, ﬁrst (at n egative times t) converging to the origin (the right term) and then (at posit ive times t) diverging from it. The minus sign between the two terms, whic h results from the requirement that a source-free ﬁeld cannot have a singular p ointR= +0, is of crucial importance as it assures vanishing of the function a tt= 0. This change of the sign when the wave goes through the collapsed stage at t he focus is also responsible for the 90 degrees phase factor associated with the Huygens-Fresnel- Kirchhoﬀ principle and for the Gouy phase shift peculiar to a ll focused waves. From Eq.(4), using the common procedure one can calculate Li ´ enard-Wiechert potentials for a moving point charge qﬂying, e.g. with a constant velocity v along axis z. However, as D0includes not only the retarded Green function but also the advanced one and therefore what is moving has to b e considered as a source coupled with a sink at the same point. For such a Huy gens-type source there is no restriction v≤cand for superluminal velocity v/c > 1 5we obtain axisymmetric scalar and vector potentials (in CGS units, Lorentz gauge) Φ( ρ, z, t) andA(ρ, z, t) = Φ( ρ, z, t)v/c, where ρis the radial distance of the ﬁeld point from the axis z, the velocity vector is v=[0,0, v], and Φ(ρ, z, t) =2q/radicalbig (z−vt)2+ρ2(1−v2/c2)  Θ/bracketleftBig −(z−vt)−ρ/radicalbig v2/c2−1/bracketrightBig − −Θ/bracketleftBig (z−vt)−ρ/radicalbig v2/c2−1/bracketrightBig  , (5) where Θ( x) denotes the Heaviside step function. Here for the sake of si mplicity we do not calculate the electromagnetic ﬁeld vectors EandH(orB), neither will we consider dipole sources and sinks required for obtaining non-axisymmetric ﬁelds. We will restrict ourselves to scalar ﬁelds obtained a s superpositions of the potential given by the Eq.(5). The ﬁrst of the two terms in the Eq.(5) gives an electromagnetic Mach cone of the superluminally ﬂy ing charge q. In other words – it represents nothing but a shock wave emitted b y a superluminal electron in vacuum, mathematical expression for which was f ound by Sommer- feld three decades earlier than Tamm and Frank worked out the theory of the Cherenkov eﬀect, but which was forgotten as an unphysical re sult after the spe- cial theory of relativity appeared [22]. The second term in t he Eq.(5) describes a leading and reversed Mach cone collapsing into the superlu minal sink coupled with the source and thus feeding the latter. Hence, the parti cular solution to the wave equation which is given by the Eq.(5) represents a do uble-cone-shaped pulse propagating rigidly and superluminally along the axi sz. In other words – it represents an X-type wave as put together from (i) the con e of incoming waves collapsing into the sink, thereby generating a superl uminal Huygens-type point source and from (ii) wakewave-type radiation cone of t he source. Let it be recalled that the ﬁeld given by the Eq.(5) had been found for δ-like spatial distri- bution of the charge. That is why the ﬁeld diverges on the surf ace of the double cone or on any of its X-shaped generatrices given by ( z−vt) =±ρ/radicalbig v2/c2−1 and the ﬁeld can be considered as an elementary one constitut ing a base for constructing various X-type waves through appropriate lin ear superpositions. Hence, any axisymmetric X-type wave could be correlated to i ts speciﬁc (con- tinuous and time-dependent) distribution ρ=δ(x)δ(y)λ(z, t) of the ”charge” (or the sink-and-source) with linear density λ(z, t) on the propagation axis, while the superluminal speed of the wave corresponds to the veloci tyvof propagation of that distribution along the axis, i.e. λ(z, t) =λ(z−vt). Let us introduce a superluminal version of the Lorentz trans formation co- eﬃcient γ= 1//radicalbig v2/c2−1 = cot θ, where θis the Axicon angle considered earlier and let us ﬁrst choose the ”charge” distribution λ(z−vt) =λ(zt) as a Lorentzian. In this case the ﬁeld potential is given as convo lution of Eq.(5) with the normalized distribution, which can be evaluated using F ourier and Laplace transform tables: 6Φ(ρ, zt)⊗1 π∆ z2 t+ ∆2=−q/radicalbigg 2 πIm/parenleftBigg 1/radicalbig (∆−izt)2+ρ2/γ2/parenrightBigg , (6) where ∆ is the HWHM of the distribution and zt=z−vtis, as in the preceding section, the axial variable in the co-propagating frame. Th e resulting potential shown in Fig.2 (a) moves rigidly along the axis z(from left to right in Fig.2) with the same superluminal speed v > c. The plot (a) depicts qualitatively also the elementary potential as far as the divergences of the Eq. (5) are smoothed out in the Eq.(6). Im U ( )□□□(a)□□ Re U ( )□□□(b)□□ Fig.2. Dependence on the longitudinal coordinate zt=z−vt(increasing from the left to the right) and a lateral one x=±ρof the imaginary (a) and real (b) parts of the ﬁeld of the simplest X-wave. The velocity v= 1.005cand, correspondingly, the superluminality parameter γ= 10. Distance between 7grid lines on the basal plane is 4∆ along the axis ztand 20∆ along the lateral axis, the unit being the half-width ∆. We see that an unipolar and even ”charge” distribution gives an odd and bipolar potential, as expected, while the symmetry of the pl ot diﬀers from what might be expected from superimposing two plane wave pulses u nder the tilt angle 2 θ. Indeed, in the latter case the plane waves are depicted by ea ch of the two diagonal branches ( \and /) of the X-shaped plot and therefore the proﬁle of the potential on a given branch has to retain its sig n and shape if one moves from one side of the central interference region to ano ther side along the same branch. Disappearance of the latter kind of symmetry, w hich can be most distinctly followed in the case of bipolar single-cycle pul ses – just the case of Fig.2 a – is due to mutual interference of all the plane wave pa irs forming the cone as φruns from 0 to π. Secondly, let us take the ”charge” distribution as a dispers ion curve with the same width parameter ∆ , i.e. as the Hilbert transform of t he Lorentzian. Again, using Fourier and Laplace transform tables, we readi ly obtain: Φ(ρ, zt)⊗1 πzt z2 t+ ∆2=q/radicalbigg 2 πRe/parenleftBigg 1/radicalbig (∆−izt)2+ρ2/γ2/parenrightBigg . (7) The potential of the Eq.(7) depicted in Fig.2 (b) is – with acc uracy of a real constant multiplier – nothing but the well-known zeroth-or der unipolar X wave, ﬁrst introduced in Ref. [1] and studied in a number of papers a fterwards. Hence, we have demonstrated here how the real and imaginary part of t he simplest X- wave solution ΦX0(ρ, zt)∝1/radicalbig (∆−izt)2+ρ2/γ2(8) of the free-space wave equation can be represented as ﬁelds g enerated by cor- responding ”sink-and-source charge” distributions movin g superluminally along the propagation axis. The procedure how to ﬁnd for a given axi symmetric X- type wave its ”generator charge distribution” is readily de rived from a closer inspection of the Eq.(5). Namely, on the axis z, i. e. for ρ= 0, the Eq.(5) consti- tutes the Hilbert transform kernel for the convolution. The refore, the ”charge” distribution can be readily found as the Hilbert image of the on-axis proﬁle of the potential and vice versa. Hence, we have obtained a ﬁgurative representation in which the superlu- minal waves can be classiﬁed viathe distribution and other properties of the Huygens-type sources propagating superluminally along th e axis and thus gen- erating the wave ﬁeld [24]. Such representation – which may b e named as Sommerfeld representation to acknowledge his unfortunate result of 1904 – has been generalized to nonaxisymmetric and vector ﬁelds and ap plied by us to various known localized waves [23]. 8Re U ( )□□□(a)□□ U→□□□(b)□□ Fig.3. The longitudinal-lateral dependences of the real pa rt (a) and the mod- ulus (b) of the ﬁeld of the Bessel-X wave. The parameters vandγare the same as in Fig.2. The new parameter of the wave – the wavele ngth λ= 2π/kzof the optical carrier being the unit, the distance between g rid lines on the basal plane is 1 λalong the axis ztand 5λalong the lateral axis, while the half-width ∆ = λ/2. For visible light pulses λis in sub- micrometer range, which means that the period of the cycle as well as full duration of the pulse on the propagation axis are as short as a couple of femtoseconds. For example, in optical domain one has to deal with the so-cal led Bessel-X wave [3]-[9], which is a band-limited and oscillatory versi on of the X wave. It is obvious that for the Bessel-X wave the ”charge” distributio n contains oscillations corresponding to the optical carrier of the pulse. Bandwidt h (FWHM) equal to ( or narrower than) the carrier frequency roughly corresp onds to 2-3 ( or more) distinguishable oscillation cycles of the ﬁeld as wel l as of the ”charge” along the propagation axis. Fortunately, few-cycle light p ulses are aﬀordable in 9contemporary femtosecond laser optics. On the other hand, i f the number of the oscillations in the Bessel-X wave pulse (on the axis z) is of the order n≃10, the X-branching occurs too far from the propagation axis, i.e. i n the outer region where the ﬁeld practically vanishes and, with further incre ase of n, the ﬁeld becomes just a truncated Bessel beam. The analytic expressi on for a Bessel-X wave depends on speciﬁc choice of the oscillatory function o r, equivalently, of the Fourier spectrum of the pulse on the axis z. One way to obtain a Bessel- X wave possessing approximately noscillations is to take a derivative of the order m=n2from the Eq.(8) with respect to zt(orzort), which according to Eqs.(6),(7) is equivalent to taking the same derivative f rom the distribution function. The mth temporal derivative of the common X wave can be expressed in closed form through the associated Legendre polynoms [5] . Another way is to use the following expression, which for n/greaterorsimilar3 approximates well the ﬁeld of the Bessel-X wave with a near-Gaussian spectrum [3],[6] ΦBX0(ρ, zt)∝/radicalbig Z(zt)exp/bracketleftbigg −1 ∆2/parenleftbig z2 t+ρ2/γ2/parenrightbig/bracketrightbigg ·J0[Z(zt)kzρ/γ]·exp(ikzzt), (9) where complex-valued function Z(zt) = 1 + i·zt/kz∆2makes the argument of the Bessel function also complex .The longitudinal wavenum berkz=kcosθ= (ω/c)cosθtogether with the half-width ∆ (at 1 /e-amplitude level on the axis z) are the parameters of the pulse. Again, dependence on z, tthrough the single propagation variable zt=z−vtindicates the propagation-invariance of the wave ﬁeld shown in Fig.3. 4 Application prospects of the X-type waves Limited aperture of practically realizable X-type waves ca uses an abrupt decay of the interference structure of the wave after ﬂying rigidl y over a certain dis- tance. However, the depth of invariant propagation of the ce ntral spot of the wave can be made substantial – by the factor cot θ=γ= 1//radicalbig v2/c2−1 larger than the aperture diameter. Such type of electromagnetic pu lses, enabling di- rected, laterally and temporally concentrated and nonspre ading propagation of wavepacket energy through space-time have a number of poten tial applications in various areas of science and technology. Let us brieﬂy con sider some results obtained along this line. Any ultrashort laser pulse propagating in a dispersive medi um – even in air – suﬀers from a temporal spread, which is a well-known obstac le in femtosecond optics. For the Bessel-X wave with its composite nature, how ever, there exists a possibility to suppress the broadening caused by the group -velocity disper- sion [3],[7]. Namely, the dispersion of the angle θ, which is to a certain extent inherent in any Bessel-X wave generator, can be played again st the dispersion 10of the medium with the aim of their mutual compensation. The i dea has been veriﬁed in an experimental setup with the lateral dimension and the width of the temporal autocorrelation function of the Bessel-X wave pulses, respectively, of the order of 20 microns and 200 fs [8]. Thus, an application of optical X-type waves has been worked out – a method of designing femtosecond pulsed light ﬁelds that maintain their strong (sub-millimeter range) lo ngitudinal and lateral localization in the course of superluminal propagation int o a considerable depth of a given dispersive medium. Optical Bessel-X waves allow to accomplish a sort of diﬀract ion-free trans- mission of arbitrary 2-dimensional images [3],[6]. Despit e its highly localized ”diﬀraction-free” bright central spot, the zeroth-order m onochromatic Bessel beam behaves poorly in a role of point-spread function in 2-D imaging. The reason is that its intensity decays too slowly with lateral d istance, i.e. as ∼ρ−1. On the contrary, the Bessel-X wave is oﬀering a loop hole to ov ercome the problem. Despite the time-averaged intensity of the Bessel -X wave possesses the same slow radial decay ∼ρ−1due to the asymptotic behavior along the X-branches, an instantaneous intensity has the strong Gaus sian localization in lateral cross-section at the maximum of the pulse and theref ore it might serve as a point-spread function with well-constrained support but also with an extraor- dinary capability to maintain the image focused without any spread over large propagation depths. By developing further this approach it is possible to build a speciﬁc communication system [25]. Ideas of using the wave s in high-energy physics for particle acceleration – one of such was proposed already two decades ago [26] – are not much developed as yet. It is obvious that for a majority of possible applications th e spread-free central spot is the most attractive peculiarity of the X-typ e waves. The better the faster the intensity decay along lateral directions and X-branches is. In this respect a new type of X waves – recently discovered Focused X w ave [12] – seems to be rather promising. As one can see in Fig.4. and by inspecting an analytical expre ssion for the wave ΦFX0(ρ, z, t) =∆exp( k0γ∆) R(ρ, zt)exp [ik0γ[iR(ρ, zt) + (v/c)z−ct]], (10) where R(ρ, zt) =/radicalBig [∆ +izt]2+ (ρ /γ)2andk0is a parameter of carrier wavenumber type, the wave is very well localized indeed. How ever, like luminal localized waves called Focus Wave Modes [2],[12], for this w ave propagation- invariant is the intensity only, while the wave ﬁeld itself c hanges during propaga- tion in an oscillatory manner due to the last phase factor exp [ik0γ[(v/c)z−ct]] in the Eq.(10), which has another z, t-dependence than through the propaga- tion variable zt=z−vt. An animated version of Fig.4(a), made for the oral presentation of the paper, shows that the oscillatory modul ation moves in the direction −z, i.e. opposite to the pulse propagation. 11Re U ( )□□□(a)□□ U→□□□(b)□□ Fig.4. The longitudinal-lateral dependences of the real pa rt (a) and the modu- lus (b) of the ﬁeld of the Focused X wave. The parameters v= 1.01cand γ= 7. Distance between grid lines on the basal plane is 4∆ along both axes. λ= 2π/k0= 0.4∆. To our best knowledge, the superluminal Focused X wave has no t realized experimentally yet, but probably the approach worked out fo r luminal Focus Wave Modes recently [27] may help to accomplish that. 5 Discussion and conclusions Let us make ﬁnally some remarks on the intriguing superlumin ality issues of the X-type wave pulses. Indeed, while phase velocities greater thancare well known in various ﬁelds of physics, a superluminal group velocity m ore often than not is considered as a taboo, because at ﬁrst glance it seems to be at variance with the special theory of relativity, particularly, with the relat ivistic causality. However, 12since the beginning of the previous century – starting from S ommerfeld’s works on plane-wave pulse propagation in dispersive media and pre cursors appearing in this process – it is known that group velocity need not to be a physically profound quantity and by no means should be confused with sig nal propagation velocity. But in case of X-type waves not only the group veloc ity exceeds cbut the pulse as whole propagates rigidly faster than c. A diversity of interpretations concerning this startling b ut experimentally veriﬁed fact [9],[13] can be encountered. On the one end of th e scale are claims, based on a sophisticated mathematical considerati on, that the relativis- tic causality is violated in case of these pulses [11]. A rece nt paper [28] devoted to this issue proves, however, that the causality is not viol ated globally in the case of the X-type waves, but still the author admits a possib ility of noncausal signalling locally. On the opposite end of the scale are statements insisting tha t the pulse is not a real one but simply an interference pattern rebuilt at e very point of its propagation axis from truly real plane wave constituents tr avelling at a slight tilt with respect to the axis. Such argumentation is not wron g but brings us nowhere. Of course, there is a similarity between superlumi nality of the X wave and a faster-than-light movement of the cutting point in the scissors eﬀect or of a bright stripe on a screen when a plane wave light pulse is f alling at the angle θonto the screen plane. But in the central highest-energy par t of the X wave there is nothing moving at the tilt angle. The phase plan es are perpendic- ular to the axis and the whole ﬁeld moves rigidly along the axi s. The Pointing vector lays also along the axis, however, the energy ﬂux is no t superluminal. Hence, to consider the X waves as something inferior compare d to ”real” pulses is not sound. Similar logic would bring one to a conclusion th at femtosecond pulses emitted by a mode-locked laser are not real but ”simpl y an interference” between the continuous-wave laser modes. In other words, on e would ignore the superposition principle of linear ﬁelds, which implies reversible relation be- tween ”resultant” and ”constituent” ﬁelds and does not make any of possible orthogonal basis inferior than others. Moreover, even plan e waves, as far as they are truly real ones, suﬀer from a certain superluminality. I ndeed, as it is well known, the most simple physically feasible realization of a plane wave beam is the Gaussian beam with its constrained cross-section and, c orrespondingly, a ﬁnite energy ﬂux. However, one can readily check on the analy tical expressions for the beam (see also Ref. [20]) that due to the Gouy phase shi ft the group velocity in the waist region of the Gaussian beam is slightly superluminal. We are convinced that the X-type waves are not – and cannot be – at vari- ance with the special theory of relativity since they are der ived as solutions to the D’Alambert wave equation and corresponding electromag netic vector ﬁelds are solutions to the Maxwell equations. The relativistic ca usality has been in- herently built into them as it was demonstrated also in the pr esent paper, when we developed the Sommerfeld representation basing upon the relativistically invariant retarded and advanced Green functions. An analys is of local evolu- tion and propagation of a ”signal mark” made, e. g. by a shutte r onto the X wave is not a simple task due to diﬀractive changes in the ﬁel d behind the 13”mark”. Therefore conclusions concerning the local causal ity may remain ob- scured. However, a rather straightforward geometrical ana lysis in the case of inﬁnitely wideband X wave (with the width parameter ∆ →0 ) shows that the wave cannot carry any causal signal between two points along its propagation axis. So, we arrive at conclusion that the X-type waves const itute one exam- ple of ”allowed” but nontrivial superluminal movements. As a matter of fact – although perhaps it is not widely known – superluminal move ments allowed by the relativistic causality have been studied since the mi ddle of the previous century (see references in [22]). For example, the reﬂectio n of a light pulse on a metallic planar surface could be treated as 2-dimension al Cherenkov-Mach radiation of a supeluminal current induced on the surface. I n the same vain, the representation of the X waves as generated by the Huygens-ty pe sources might be developed further, vis. we could place a real wire along th e propagation axis and treat the outgoing cone of the wave as a result of cylindri cal reﬂection of (or of radiation by the superluminal current in the wire indu ced by) the leading collapsing cone of the wave. In conclusion, superluminal movement of individual materi al particles is not allowed but excitations in an ensemble may propagate with an y speed, however, if the speed exceeds cthey cannot transmit any physical signal. Last two decades have made it profoundly clear how promising and fruitful is s tudying of the superluminal phenomena instead of considering them as a sor t of trivialities or taboos. We have in mind here not only the localized waves or ph oton tunneling or propagation in inverted resonant media, etc., but also – o r even ﬁrst of all – the implementation and application of entangled states of Einstein-Podolsky- Rozen pairs of particles in quantum telecommunication and c omputing. This research was supported by the Estonian Science Foundat ion Grant No.3386. The author is very grateful to the organizers of thi s exceptionally interesting Conference in warm atmosphere of Naples. References [1] J. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroele ctr. Freq. Control 39, 19 (1992). [2] R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, J. Op t. Soc. Am. A 10, 75 (1993). [3] P. Saari, in Ultrafast Processes in Spectroscopy (Edited by O. Svelto, S. De Silvestri, and G. Denardo), Plenum, p.151 (1996). [4] J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, an d M. M. Salo- maa, Phys. Rev. E, 54, 4347 (1996). [5] A T. Friberg, J. Fagerholm, and M M. Salomaa, Opt. Commun. 136, 207, (1997). [6] P. Saari, H. S˜ onajalg, Laser Physics, 7, 32 (1997). 14[7] H. S˜ onajalg, P. Saari, Optics Lett., 21, 1162 (1996). [8] H. S˜ onajalg, M. R¨ atsep, and P. Saari, Opt. Lett. 22, 310 (1997). [9] P. Saari, K. Reivelt, Phys. Rev. Lett., 79, 4135 (1997). [10] E. Recami, Physica A, 252, 586 (1998) [11] W. A. Rodriguez and J. Y. Lu, Found. Phys, 27, 435 (1997). [12] Almost complete list of all publications on the localiz ed ﬁelds is given in a review article by I. Besieris, M. Abdel-Rahman, A. Shaar awi, and A. Chatzipetros, Progr. in Electromagn. Research, 19, 1 (1998). [13] D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. ,84, 4830 (2000). [14] A. Enders and G. Nimtz, Phys. Rev. B, 47,9605 (1993), Phys. Rev. E, 48, 632 (1993). [15] A. Ranfagni, P.Fabeni, G. P. Pazzi, and D. Mugnai, Phys. Rev. E, 48, 1453 (1993). [16] A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, Phys. Rev. Lett ., 71, 708 (1993). [17] A. M. Shaarawi and I. M. Besieris, Phys. Rev. E, 62,7415 (2000). [18] E .Recami, Rivista Nuovo Cimento, 9, 1 (1986). [19] M. I. Faingold, in Einsteinovski Sbornik (in Russian), Nauka, Moscow, p.276 (1976). [20] Z. L. Horv´ ath and Zs. Bor, Phys. Rev. E, 60, 2337 (1999). [21] P.Saari, in: Ultrafast Phenomena XI (Edited by T. Elsaesser, J. G. Fuji- moto, D. A. Wiersma, and W. Zinth), Springer, p. 121 (1998). [22] V. L. Ginsburg, in Progress in Optics (Edited by E. Wolf), 32, 267 (1993) and references therein, where A. Sommerfeld’s pioneering b ut forgotten publication in G¨ ottinger Nachrichten (1904) is considered. [23] P. Saari (to be published). [24] Mathematical procedures of derivation of wave ﬁelds as if they are gen- erated by sources from complex locations (i.e. a coordinate is formally made a complex number) are well known in the theory of the Gaus sian beams already as well as in treating the localized waves. For example, in the Ref.[2] an expression for the zeroth-order X wave had bee n obtained as Li´ enard-Wiechert potentials of an electron moving alon g the complex zaxis displaced from real space, but this comparatively form al approach remained undeveloped. [25] J. Lu and S. He, Opt. Comm., 161, 187 (1999). 15[26] M. I. Faingold, J. Technical Phys. (in Russian), 50, 915 (1980). [27] K. Reivelt and P. Saari, JOSA, 17, 1785 (2000). [28] A. M. Shaarawi and I. M. Besieris, J. Phys. A: Math.Gen., 33, 7255 (2000). 16
PREPRINT Extreme Ultraviolet (EUV) Sources for Lithography based on Synchrotron Radiation Guiseppe Dattoli1, Andrea Doria1, Gian Piero Gallerano1, Luca Giannessi1, Klaus Hesch2, Herbert O. Moser6, Pier Luigi Ottaviani1, Eric Pellegrin3, Robert Rossmanith2, Ralph Steininger2, Volker Saile4, Jürgen Wüst5 1ENEA INN-FIS-LAC, Frascati, Italy 2Synchrotron Radiation Research Group, 3 Institute of Solid State Physics, 4 Institute of Microstructure Technology, 5 Technology Transfer and Marketing, Forschungszentrum Karlsruhe 6 Singapore Synchrotron Light Source SSLS, National University of Singapore Corresponding author, Forschungszentrum Karlsruhe, Synchrotron Radiation Research Group FGS, P. O. Box 3640, D-76021 Karlsruhe, Germany, Tel. ++49 7247 82 6179, Fax ++49 7247 82 6172, e-mail rossmanith@anka.fzk.de Submitted to Nuclear Instruments and Methods A ______ Work supported by the German Ministry for Research and Education BMB+F under contract No. 01 M 3103 A Page 2 of 32 Abstract: The study presented here was initiated by a discussion to investigate the possibility of using synchrotron radiation as a source for the Next Generation Lithography (NGL) based on the EUV-concept (Extreme Ultra-Violet; here 13.5 nm or 11.3 nm radiation, respectively). The requirements are: 50 W, 2% bandwidth and minimal power outside this bandwidth. Three options were investigated. The first two deal with radiation from bending magnets and undulators. The results confirm the earlier work by Oxfords Instrument and others that these light-sources lack in-band power while emitting excessive out-of-band radiation. The third approach is a FEL (Free Electron Laser) driven by a 500 MeV linear accelerator with a superconducting mini-undulator as radiation emitting device. Such a device would produce in-band EUV-power in excess of 50 W with negligible out-of-band power. Author Keywords: Synchrotron Radiation; Lithography; Radiation by moving charges PACS classification codes: 07.85.Qe; 81.16.Nd; 41.60.-m Page 3 of 32 1. Introduction Lithography, the technique for manufacturing microelectronic semiconductor devices such as processors or memory chips, presently uses deep UV (DUV) radiation. The main radiation source is the 193 nm line of an ArF excimer laser [1]. Future sources will be F 2 lasers at a wavelength of 157 nm and eventually H 2 lasers at a wavelength of 127 nm. In addition, advanced lithography technologies (Next Generation of Lithography: NGL) based on EUV, X-ray photons, electrons, and ions are being investigated by chip makers and equipment manufacturers. The competing technologies are: SCALPEL electron lithography (Scattering with Angular Limitation in Projection Electron-Beam Lithography) [2], Ion Projection Lithography [3], X-ray Proximity Lithography [4] and Extreme UV Lithography [5]. The latter is being considered as one of the most promising. In the US a program to develop this technology was set up as early as in 1994 by the EUV LLC (Limited Liability Corporation) in cooperation with the VNL (Virtual National Lab). Members of VNL are LLNL (Lawrence Livermore National Lab), LBNL (Lawrence Berkeley National Lab) and Sandia National Lab. In Japan the ASET consortium was funded (Association of Super-Advanced Electronics Technologies) [6]. During the research phase the needs for a EUV source suitable for future production lines were identified. The main requirements are: Wavelength: 13.5 nm (=92 eV) or 11.3 nm Page 4 of 32 Bandwidth: 2% Output power: 25 W (first step) and later 50 W In addition, the power radiated outside this band has to be less than 500 W to avoid thermal problems on the optics. The development of a suitable source is one of the big challenges in EUV lithography. Basically, powerful sources of EUV photons may be based on either plasmas [7] (produced by laser irradiation of matter or by gas discharges) or on relativistic electrons (synchrotron radiation). In Europe, the development of synchrotron radiation-based EUV sources [8] was partly supported by the European Union within the framework of the EUCLIDES program [9]. Similar investigations were performed in Japan [10] and the USA [11]. The studies showed that conventional storage rings with and without additional magnets (normal conducting or superconductive wigglers or undulators) do not fulfil all the above-mentioned specifications for the EUV source. The German Federal Ministry of Education and Research initiated at the beginning of 2000 a program on plasmas generated by lasers or gas discharges as sources of EUV light for the next generation lithography (NGL). In the initial phase of this project it was felt that sources based on synchrotron radiation should be reconsidered. In the first quarter of 2000 the authors presented their report. The present paper is a shortened version of this report. The result was that among all possible sources based on synchrotron radiation only a Free Electron Laser can meet the above mentioned stringent requirements at 13.5 nm. In summer 2000 a group at DESY published independently a paper in which the design of a SASE Free Electron Laser source for lithography at 70 nm is described [33 ] confirming at least in principle the viability of the FEL concept. Page 5 of 32 2. Incoherent radiation from the storage rings In the following the results already obtained in the EUCLIDES study are summarized for reference. A model storage ring is shown in fig.1. The parameters which are needed to calculate the emitted photon intensity at 13.5 nm within the required bandwidth of 2% are - electron energy – magnetic field strength – electron current The maximum storable current depends on two limitations: beam instabilities and intrabeam scattering (Touschek-Effect) [12]. Beam instabilities can be defeated by feedback systems. The Touschek lifetime for an unpolarized beam is approximately VcNCr acc xe 232 ')( [sec]1 εγσζ π τ= (1) where r e = 2,8.10-15 m (classical electron radius). N is the number of particles per bunch. Assuming a 500 MHz RF frequency N is equal to 1,25.1010 for a stored beam of 1 A. γ is the ratio between energy and rest energy. The rest of the parameters describes the particle density in relation to the region in which the particles are stable, the so-called energy acceptance. ε acc is the energy acceptance of the storage ring. A particle gets lost when the scattering is so violent that a particle changes its energy by more than that. The scattering probability depends on the density of the particles in the bunch. The parameter ζ = (εacc/γσ´x)2. σ´x is the divergence in the beam. For ζ ≤ 10-2 the following approximation is valid C( ζ) ≅ -ln(1,732 ζ)-1.5. The bunch volume V is 8 π3/2σxσyσL. Typical Touschek life times for a 1 A beam are summarized in Table I. Page 6 of 32 The strong dependence of the Touschek effect on the energy indicates that the preferred storage rings are operating at higher energies: 0.3 GeV and higher. The spectral power ∆P of the emitted synchrotron radiation in Watts per eV, per mrad horizontal angle ϑ and integrated over the vertical angle is given by formula (2) [12], [13]. [] )(][][] [73.8 / /24 yGmrAI GeVEeV mrad WattP=∆∆ϑϑ (2) with /Gb3∞ = yd Ky yG ηη)( )(3/52 2 and c Phot E Ey / = I is the stored beam current, E is the energy of the stored beam and K is the modified Bessel function. E Phot is the photon energy and E c is the so-called critical photon energy E c [eV] = 2218.3 ][] [3 mrGeVE r is the bending radius. r and the bending field B in Tesla are related by the equation: r[m] = 3.34][] [ TBGeVE (3) The power emitted at 13.5 nm per mrad horizontal angle within a 2% bandwidth is according to (2) [] )(][][] [1746.0 /24 yGmrAI GeVEmrad WattP=∆∆ϑϑ (4) Figs. 2 and 3 show the results of equation (4). The assumed current is 1 A in all cases. In conclusion it can be said that for ca. 100 eV photons the spectral density has a maximum at fields near 1.5 to 2 T. It follows from fig. 3 that the spectral power increases the higher the energy is. The optimum values can be reached with room temperature magnets (1.5 T) and high energies (in other words with fairly large machines). Page 7 of 32 The maximum angle over which photons can be collected is 6.28 rad (the full circumference of the storage ring). The maximum in-band power as a function of energy and field strength is shown in fig. 4 for a stored beam of 1A. Fig. 5 shows 2 D cuts of figure 4. From these curves it is obvious that the 50 W requirement with a stored beam of 1 A can only be met at energies significantly above 1 GeV. The maximum collectible power at an energy of 0.6 GeV is 27 W. This in-band power has to be compared with the total radiated power: P T[kW] = 88 .5 [] [ ] []mrAI GeVE4 (5) which is for 0.6 GeV and 1.5 T (r = 1.336 m) ca. 8.6 kW. The power ratio (defined as in-band power P/total power P T) is )(. 01239.02yGPP T= (6) The power ratio is shown in fig. 6. The maximum values are obtained at low fields. This argument confirms that high beam energy and low magnetic fields are the optimum parameters. A storage ring of 0.6 GeV and a field of 1.5 T might be a fair compromise to obtain a total in-band power of more than 25 W. It is obvious all the formulas mentioned in the previous chapter valid for bending magnets are also valid for wigglers. The wiggler has two advantages over a bending magnet. Firstly, the photons are emitted into a cone centred around the direction of motion of the beam. The collection of photons is easier with a wiggler than with a bending magnet. Secondly, since there is no net deflection, it is easier to choose the optimum field. Wigglers with a small maximum beam deflection angle α are called undulators. The K-value is defined in the following way: Page 8 of 32 K= α.γ = 0.94. B[T]. λu [cm] (7) Constructive interference in the vicinity of the beam axis happens when: λPhot = ()22 2 22/ 1 2θγ γλ+ +K nu (8) θ is the angle between electron beam axis and the photon detector. A measured spectrum of the first harmonics of an undulator depending on the angle is shown in fig. 7. In fig. 7 angle and photon energy are clearly related (depending on the emittance of the beam). This is described by (8) [28]. The undulator condition (8) has to be fulfilled for 13.5 nm. This condition limits the number of possible solutions for the period length. In addition, a general rule states that the period length should not be shorter than 4 times the gap width of the undulator. If this rule is not observed, than the field acting on the beam becomes too small [14]. The formula used in Table I for the total power radiated from an undulator is P[W] = ][][] [ 26.72 2 cmKNAI GeVE uu λ (9) Nu is the number of periods. The calculated in-band power for an undulator is shown (as an example) in fig. 8. The parameters of different undulators are summarized in Table II. The first harmonics of all undulators is close to 13.5 nm. Despite the fact that the maximum obtainable power does not fulfil the stringent requirement the undulator has clear advantages over wigglers. The K-values in Table I are in the order of 1 to 2. According to formula (7) the magnetic fields of the undulator are larger than 1 T. These values are larger than those achieved with conventional permanent magnet undulators. In Brookhaven [15 ] and Karlsruhe [16] independent concepts for using superconductors rather than arrays of permanent magnets have Page 9 of 32 been under discussion. Recently Karlsruhe together with a group at Mainz [20] have been able to demonstrate the viability of such a concept under normal beam conditions. Fig. 7 shows the measured spectrum from these experiments. Fig. 9 shows the principle of a superconductive undulator. The field is generated by a superconductive wire in an iron matrix (darker parts in fig. 9). The superconductive wires are close to the beam. The undulator is indirectly cooled by liquid helium not shown in figure 9. The parameters for this specific undulator are: period length 14 mm and K=2 (1.5 T) [30]. The calculated undulator field is shown in fig. 10. 3. The Free Electron Laser approach It has been shown in the previous chapters that the specifications for the source defined in the introduction, 50 W within a 2 % bandwidth at 13.5 nm, is barely achievable to obtain with a conventional synchrotron radiation source. In 1951 Motz was the first to point out that the intensity of a photon beam emitted by electrons can be increased by coherent superposition [18]. The logic is as follows. If each electron emits a photon the resulting electric field E total is: /Ga6= nn total E E (10) En is the electric field of the individual photons. The intensity is proportional to E2 total. . When the phases of the photons have a random distribution (incoherent light) the cross terms cancel and the averaged sum is I = 2. . EN E E jj ii =/Ga6/Ga6 (11) where N is the number of electrons. Page 10 of 32 When the phases of the electrons are identically and they are not randomly distributed the cross-terms do not disappear and the intensity is N2 times the intensity of a single electron. This is obviously the case when the electrons are concentrated in bunches. The length of these bunches (so-called micro-bunches) must be smaller than the wavelength. The micro-bunches are separated by a multiple of a wavelength. If this argument is turned around, then most of the intensity of a conventional synchrotron radiation source is destroyed by incoherence or, if expressed in other terms, by the random distances of the emitting electrons. When the electrons have distances which are smaller than the emitted light wave the intensity can be increased by an enormous factor (N is a very big number). Since 1951 this principle has been experimentally investigated with great success by various groups and this has changed dramatically the design of synchrotron light sources [19], [20] and beam diagnostics tools [21]. In order to operate a FEL effectively the following conditions on the emittance have to be fulfilled [29]. a.) Particles with an angle to the beam axis do not fulfil the resonance condition (8). The electron velocity in the direction of the axis is changed by –x´ 2/2 ( x´ is the angle relative to the axis). In order to keep the electron within a half-wave over the whole undulator length, the condition 2/1 2/Gb8 /Gb9/Gb7/Ga8 /Ga9/Ga7≤′ LxPhotλ (12) has to be fulfilled, where L is the length of the undulator [31] . b.) The spot-size of the optical mode is given by 2/1 /Gb8 /Gb9/Gb7/Ga8 /Ga9/Ga7≤ πλRxPhot (13) Page 11 of 32 where R is the Raleigh length (the distance in which the area of a diffracted wave doubles). Typically the Raleigh length is one-half of the interaction length L. The restrictions c.) and d.) are usually combined to one requirement [22] πλε4Phot≤ (14) Equation (14) requires that the horizontal and vertical emittance of the beam has to be smaller than 1.07 nm. In a linac the emittance shrinks with energy (adiabatic damping) ε = εn / γ (15) where εn is the so-called normalized emittance. The magnitude of the normalized emittance depends on the gun. For a photo-cathode gun εn is close to 10-6 m.rad (depending on current, bunch length etc.) [23]. Following equation (15) γ has to be 1000 or higher (linac energy equal or above 500 MeV). Assuming a gradient of 20 MeV/m, the linac is 27.5 m (or close to 30 m) long. For 0.52 GeV and K = 1.4 the period length of the undulator is 1.43 cm according to equation (8). The peak field is circa 1.05 T according to equation (7). Linacs with more than 40 MeV/m are available, so that the minimum length of the linac is about 15 m. The SASE FEL [32] is generally described by analytical methods. The fundamental parameter in this description is the gain length L G, the length in which the FEL power increases by a factor of e. The gain length depends on the power density of the emitted light. The power density is a function of the undulator properties (K-value, period length), the beam properties (peak current, energy, β-functions, emittance, energy spread etc.) and the properties of the optical beam (diffraction). In order to separate the different influences the following parameters are introduced: Page 12 of 32 L G = Su ρχ πλ 34 = SLG χ0 (16) where ρ is the so called Pierce parameter 3/1 2 32 21).].,1[(2 41 /Gbb /Gbc/Gba /Gab /Gac/Gaa=πβελ γπ πρ Apeak uIIK K JJ (17) The Pierce parameter describes the emission of synchrotron radiation. I A is 17 kA (Alven current) and the rest of the parameters are explained in previous equations. χ and S are correction parameters describing the influence of the diffraction and the energy spread. The diffraction effects are described by the parameter S in (16): /Gbb/Gbb /Gbc/Gba /Gab/Gab /Gac/Gaa /Gbb/Gbc/Gba /Gab/Gac/Gaa+=2 22 0 41 /1πελ βPhot GLS (18) and produce the curve shown in fig. 11 [24]. The curve has a minimum close to a beta of 0.3 m with a slight slope towards higher beta-functions. A beta function of 2 m throughout the whole undulator is an acceptable compromise. The undulator shown in fig. 9 has to be modified in such a way that it focuses in both directions. The focusing in both directions in an undulator with permanent magnets was demonstrated at DESY for the first time [35]. A similar effect can be achieved for the superconductive undulator either by shaping the iron poles in an appropriate way. A study on SASE and superconductive undulators can be found in [26]. Up to now the influence of the energy spread was not taken into account. The energy spread leads to a broadening of the laser line and finally to a loss of gain. The influence of the energy spread is described by the function χ in a complex way. χ is determined by three parameters: Page 13 of 32 ()()βρσβµε ε2= ()() 222 2 003 un G K L λγβε π λββµ = ()() γβε λββµn GL 00 1 3= (19) All 3 parameters depend on β and on the energy spread σε. The parameters have to fulfil an integral equation and a solution is only possible by numerical techniques. The function )(βχ is a solution of the integral equation: χπµ πµεπµχ=− − /Gb3∞ −+− 0 12) ( 23 ) 1(1 ) 1(12 dssi sie essi (20) χ depends somewhat on β but strongly on the energy spread. The FEL process starts from the radiation emitted in the first gain length of the undulator. The number N of undulator periods in the first gain length (equation (16)) is ρ π341=GN (21) and the spontaneous peak power P emitted during the first gain length (peak current PeakI): () PhotGPeakn G electronLI K JJN GeV E x W Pλβεβ22 2 19 ],1[] [ 1048.1][= (22) For a relative energy spread of 10-4 this value is circa 19 W for a beta-function of 2.5 m. The development of the power along the undulator axis z is described by ()()GLz ePzP9β= (23) Page 14 of 32 The amplification is stopped by an undulator with constant period length at the saturation length z sat Gpeak G sat LPPL z +/Gb8/Gb8 /Gb9/Gb7 /Ga8/Ga8 /Ga9/Ga7=)(9lnβ (24) with ] [][ 109GeVEA I PPeak Peak ρ = In the following it is assumed that the beta-function is 0.5 m. As shown before, all results depend strongly on the beta function. The development of the peak power, including saturation, is /Gb8/Gb8 /Gb9/Gb7 /Ga8/Ga8 /Ga9/Ga7 − += 19)(19)( GG Lz PeakLz z ePPeP P ββ (25) The dependence of the peak power on the undulator length is shown for two cases (energy spread of 10-4 and an energy spread of 5.10-4) in figs. 12 and 13. The peak current I peak is 200 A. Obviously, the final peak power is the same in both cases. The energy spread only defines the length of the undulator [27]. The peak power of one pulse is ca. 1.33. 10 8 W. One pulse with an assumed bunch length of 3 psec produces an energy of π2.3.10-12 1.33 108 J or about 1 mJ. In order to produce a cw power of 50 W, a pulse repetition rate of 50 kHz is required. The required average current is fairly modest. Working on the basis of a 1.5 GHz linac RF system (bucket repetition of time of 0.67 nsec) the average current is 200(3/667)(5.10 4/1,5.109) A = 30 µA. Page 15 of 32 4. Possible Layout of the FEL source A possible layout of the EUV laser system for a wafer fab is shown in fig. 14. It is assumed that the EUV source (linac, undulator etc.) will be located in the basement of the factory. The EUV radiation enters the clean room via evacuated pipes which come up through the floor. The normalized emittance of the beam (assumed to be10 -6) determines the energy of the linac: 500 MeV. The accelerating structures can be either superconductive (Nb-cavities) or normal conducting (Cu-cavities). Normal conductive cavities allow simple and short structures: energy gains of up 40 MeV/m and higher are possible. The length of the linac would be less than 15 m. The accelerating gradient for superconductive linacs is at the moment ≥20 MeV/m and, as a result, a superconducting linac will be almost twice as long as a room temperature linac. The accelerated beam is directly sent to a 11m long SASE undulator. All bends along the trajectory have to be isochronous in order to prevent bunch lengthening. The installation of most of the equipment in auxiliary and/or distant rooms, such as the basement is an integral part of the following layout considerations. Fig. 14 shows one possible way of distributing the EUV light. A central linac provides a distributed undulator system with an electron beam. The beam is switched by magnets to the undulators. The fact that several SASE superconducting undulators are fed from one linac reduces the capital cost per stepper. Conclusion The German Federal Ministry of Education and Research initiated a program on plasmas generated by lasers or gas discharges as sources of EUV light (50 W at 13.5 nm, bandwidth 2%) for the next generation lithography (NGL). In the initial phase of this project it was felt Page 16 of 32 that sources based on synchrotron radiation should be reconsidered. The aim of this report is to investigate such sources. The report starts with investigations into the emitted power of small storage rings with energies of less than 0.6 GeV. The total emitted power collected over the entire circumference is less than 50 W (stored current of 1 A). In a next step storage rings were equipped with wigglers. It is easier to collect the wiggler radiation but the conclusions are similar: the total emitted power is insufficient. In the following step storage rings with undulators have been studied. Under certain circumstances these devices have clear advantages over the wiggler system. In undulators the emitted photons can interfere coherently. This fact makes it possible to amplify the intensity within the required bandwidth and minimize it outside. In order to optimise the output power, superconducting mini-undulators are required. Free Electron Lasers (FELs) consisting of linacs and undulators produce light with a high degree of coherence and of high power. Unwanted out-of-band-radiation is almost completely eliminated. The study shows that the so-called SASE technique (Self Amplified Stimulated Emission) can easily produce the required EUV power. The SASE effect was already observed experimentally at wavelengths as low as 80 nm. As a result, synchrotron radiation (mainly FELs) can easily fulfil the stringent requirements for the Next Generation of Lithography based on EUV when suficet space is forseen in a wafer fab. Page 17 of 32 Acknowledgements This study is based on numerous discussions with and contributions from many colleagues. The authors would like to thank them. It is impossible to mention all names of the individuals who contributed to this study. Our special thanks go to DESY (Prof. Schneider, Prof. Materlik, Dr. Rossbach, Dr. Pflueger and the SASE FEL team), to ESRF (Dr. Elleaume and his team), BNL (Dr. Ben-Zvi and colleagues), Swiss Light Source (Prof. Wrulich, Dr. Ingold), ENEA (Prof. Renieri), Elettra (Dr. Walker and colleagues), JLab (Dr. Neil), University of Virginia (Prof. Norum, Prof. Gallagher), LBNL (Dr. Jackson, Dr. Robin and colleagues), Duke University (Profs. Edwards and Litivenko), UCLA (Prof. C. Pellegrini), ACCEL (Drs. Klein, Krischel, Schillo and Geisler), and many others. Page 18 of 32 9. Literature [1] B. Nikolaus, O. Semprez, G.Blumenstock, P. Das, 193 nm Microlithography and DUV Light Source Design, Lithography Resource, Edition 9, March 1999, ICG Publishing Ltd., London UK [2] G. R. Bogart, et al., 200 mm SCALPEL mask development, Proc.SPIE, Vol 3676, Emerging Lithography Techniques III, p. 171, Y.Vladimirski, Editor, Santa Clara1999 [3] R. Mohondro, Ion Projection Lithography, Semiconductor Fabtech, Edition 3, Oct. 1995, p. 177 [4] R. A. Selzer and Y. Vladimirski, X-ray lithography, a system integration effort , Proc.SPIE, Vol 3676, Emerging Lithography Techniques III, p. 10, Y.Vladimirski, Editor, Santa Clara1999 [5] R. H. Stulen, Progress in the development of extreme ultraviolet lithography, Proc. SPIE, Vol 3676, Emerging Lithography Techniques III, Y.Vlad imirski, Editor, Santa Clara 1999 [6] S. Okazaki, EUV Program in Japan, Proc. SPIE, Vol 3676, Emerging Lithography Techniques III, Y.Vlad imirski, Editor, Santa Clara1999 [7] R. L. Kauffmann, D.W. Phillion and R. C.Spitzer, X-ray production, 13 nm from laser- produced plasmas for soft-x-ray projection lithography, Applied Optics, Vol. 32, No 34, p. 6897 [8] J. P. Benschop, EUV overview from Europe, Proc SPIE, Vol 3676, Emerging Lithography Techniques III, Y.Vlad imirski, Editor, Santa Clara1999 [9] J. P. Benschop et al., EUCLIDES: European EUVL Program, J. Vac. Sci. Technol. B17(6), Nov/Dec 1999 [10] S. Masui et al., Applications of the superconducting compact ring AURORA, Rev.Sci.Instrum. 66:2352-2354,1995 [11] J. B. Murphy, D. L. White, A. A. MacDowell and O. R. Wood II, Synchrotron Radiation Sources and condensers for projection X-ray lithography, Appl. Optics, Vol 32, No 34, Dec. 1993, 6920 J. B. Murphy, X-ray lithography sources, a review, Proc. 1989 IEEE Particle Accelerator Conference, NY 1987, 757 [12] J. Murphy, Synchrotron Light Source Data Book, Internal report BNL 42333J. [13] H. Wiedemann, Particle Accelerator Physics, Berlin, Germany: Springer (1993 and 1995) Page 19 of 32 [14] S. H. Kim and Y. Cho, IEEE Trans. Nucl. Sc, Vol. NS-32, No. 5 , p 3386 (1985) K. Wille, Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen, Teubner- Verlag, 1992, page 251 [15] Ben-Zvi, I., et al. , The performance of a superconductive micro-undulator prototype, Nucl. Instr. Meth. A 297 (1990) 301 G. Ingold, et al. Fabrication of a high field short-period superconductive undulator, Nucl. Instr. and Meth. A375 (1996) 451 [16] T. Hezel et al., Proc. of the 1999 Particle Accelerator Conference, New York 1999 T. Hezel et al. J. Synchrotron Radiation (1998), 5, p448 H. O. Moser, R. Rossmanith et al., Design Study of a superconductive in-vacuo undulator for storage rings with an electrical tunability of k between 0 and 2, Proc. of EPAC 2000, in preparation R. P. Walker and B. Diviacco, Insertion Devices: recent developments and future trends, Synchr. Rad. News., Vol. 13, 1 (33) [17] R.P. Walker, B. Diviacco, URGENT, A computer program for calculating undulator radiation spectral, angular, polarization and power density properties, ST-M-91-12B, July 1991, Presented at 4th Int. Conf. on Synchrotron Radiation Instrumentation, Chester, England, Jul 15-19, 1991 [18] H. Motz, Applications of the Radiation from Fast Electron Beams, J. Appl. Physics, Vol 22, No. 5 (1951) 527 H. Motz, W. Thon, R. N. Whitehurst, Experiments on Radiation by fast Electron Beams, J. Appl. Physics, Vol. 24 (1953)826 [19] For an overview of worldwide FEL activities see http://sbfel3.ucsb.edu/www/ [20] J. Rossbach et al., A VUV free electron laser at the TESLA test acility at DESY, Nucl. Instr. Meth. 375 (1996) 269 R.Tatchyn et al., Research and development toward a 4.5-1.5 Angstroem linac coherent Light source (LLS) at SLAC, A375 (1996) 274 S. V.Milton et al., Status of the Advanced Photon Source low-energy undulator test line, Nucl. Instr.Meth. A 407 (1998)8 V. N.Litivenko et al., First UV/visible lasing with the OK-4/Duke storage ring FEL ; Nucl. Instr.Meth. A 407 (1998)8 Page 20 of 32 [21] W. Barry, Measurement of subpicosecond bunch profile using coherent transition radiation, Talk given at 7th Beam Instrumentation Workshop (BIW 96), Argonne, IL, 6-9 May 1996. In Argonne 1996, Beam instrumentation 173-185. Hung-chi Lihn, P. Kung, Chitrlada Settakorn, H. Wiedemann, David Bocek, Measurement of sub- picosecond electron pulses. Phys.Rev.E53:6413-6418,1996 [22] R. Bonifacio, C. Pellegrini, L. Narducci, Opt. Comm. 50 (1984)373 C. Pellegrini, Laser Handbook, Vol. 6, Free Electron Lasers, North Holland (1990) [23] A. Tremaine et al., Status and Initial commisioning of a high gain 800 nm SASE FEL, Nucl. Instr.Meth. A 445 (2000) 160 S. Reiche, Compensation of FEL gain reduction by emittance effects in a strong focusing lattice, Nucl.Instr.Meth. A 445 (2000) 90 [24] G. Dattoli, A. Doria, G. P. Gallerano, L. Giannessi , P. L. Ottaviani, A note on a FEL operating at 13.5 nm – 50W (CW) output power, Internal technical note ENEA-Frascati, to be published [25] J. Pflüger, Undulators for SASE FEL, Nucl. Instr. Meth. A 445 (2000) 366 [26] P. Elleaume, J. Chavanne, Design Considerations for a 1 A SASE Undulator, ESRF Note/MACH ID 00/59 January 2000 [27] G. Dattoli, L. Giannessi, P. L. Ottaviani and M. Carpanese, A simple model of gain saturation in high gain single pass free electron lasers, Nucl. Instrum. Meth. A393 (1997) 133-136 [28] H. Winick, G. Brown, K. Halbach, J. Harris, Wiggler and Undulator Magnets- a review, Nucl. Instr. Meth. 208,65,1983 S. Krinsky, Undulators as Sources for Synchrotron Radiataion, IEEE Trans. Nucl. Sc., Vol. NS-30, No4, 3078 K. Wille, Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen, Teubner- Verlag, 1996 P. Elleaume, Insertion Devices for the new generation of synchrotron sources, a review. Rev. Sc. Instr. 63 (1), January 1992, 321 H. Winick and S. Doniach (editors), Synchrotron Radiation Research, Plenum Press, New York [29] W. B. Colson, C. Pellegrini, A. Renieri, The Laser Handbook, Vol VI, North Holland, Amsterdam 1990 Page 21 of 32 M. Poole, Storage Ring based FELS, Synchr. Rad. News, Vol 13 No.1, 4 H.-D. Nuhn, J. Rossbach, Short Wavelength FELs, Synchr. Rad. News, Vol 13 No.1, 18 [30] K. J. Kim, in X-ray data booklet, ed. By D. Vaughan, LBL PUB-490 (1985) [31] R.H. Pantell , Free Electron Lasers, In Batavia 1987/Ithaca 1988, Proceedings, Physics of particle accelerators 1707-1728 [32] G. Dattoli, A. Renieri, A. Torre. Lectures on the Free Electron Laser Theory and related topics. World Scientific Singapore 1993 [33] C. Pagani, E.L.Saldin, E.A. Schneidmiller, M. V. Yurkow, Design Considerations of 10 kW-Scale Extreme Ultraviolet SASE FEL for Lithography, DESY report 00-115 Page 22 of 32 Energy [GeV] γ ζ C(ζ) τ [sec] 0.6 1174 1,85.10-4 6.54 24064 0.4 783 4,13.10-4 5.74 8117 0.3 587 7,35.10-4 5.17 3805 0.2 391 1,65.10-3 4.36 1334 0.1 196 6,00.10-3 3.01 246 Table I: Touschek lifetime for various beam energies. Beam current 1 A. εacc= 5.10-3, εhor=500nm, εvert = 10 nm.rad, βhor = 5 m, βvert = 10 m, σL =3 cm . Energy [GeV] K λu [cm] Total emitted power [W] In-band power [W] Power ratio 0.3 1 0.62 105 2.5 2.4.10-2 0.4 1.5 0.78 335 3.9 1.2.10-2 0.4 1 1.1. 106 2.5 2.4.10-2 0.6 2 1.24 843 4.3 5.1.10-3 0.6 1.5 1.49 544 3.9 7.2 10-3 0.6 1 2.48 105 2.5 2.4.10-2 Table II Characteristics of various 100 period long undulators. The current is 1 A in all cases. Page 23 of 32 Fig. 1: Model storage ring source Fig. 2: Spectral power at constant field with the energy as parameter Fig. 3: Spectral power at constant energy with the magnetic field as a parameter Fig. 4: In-band power versus energy and magnetic field as in fig. 5 ( energy range 0.2 to 0.6 GeV) Fig. 5: In-band power versus magnetic field with the energy as a paramete (2D cuts of fig .4) Fig. 6: Power ratio as a function of energy and bending field strength Fig. 7 Angle dependance of the measured X-ray spectrum of an undulator [16] Fig. 8 Undulator: beam energy 0.6 GeV, K=2, λu =1.24 cm. The maximum in-band output power isabout 4.8 Watt. Calculated with the program URGENT [17] Fig. 9: Layout of the superconductive miniundulator (shown from two perspectives). The dark red material is iron, the lighter coloured material depicts superconductive wires. The beam travels in the gap between the two undulator poles. The current direction through the wires alters from wire to wire generating the undulator field. Fig. 10: Undulator field (calculated) Fig. 11: Influence of diffraction effects on the gain length Fig. 12: Peak power versus undulator length for an energy spread of 1 .10-4 Fig. 13: Peak power versus undulator length for an energy spread of 5 .10-4 Fig 14: Chain-type layout of an FEL source Page 24 of 32 Wiggler or undulator Fig.1 Model storage ring source. Fig.1 Wiggler or Undulator Wiggler or Undulator Page 25 of 32 Fig. 2 Fig. 3 E E [GeV] B [T] r [m] E c [eV] A 0.6 4 0.5 957.6 B 0.4 4 0.33 425.6 C 0.3 4 0.25 239.4 D 0.2 4 0.16 106.4 E 0.1 4 0.08 26.6 1 10 100 1000 1000 photon energy in eV E D C B A Watt/eV/mrad/1A 10-3 10-4 10-5 10-6 E [GeV] B [T] r [m] E c [eV] A 0.6 8 0.25 1915 B 0.6 6 0.334 1436 C 0.6 4 0.5 957 D 0.6 2 1.0 478 E 0.6 1.5 1.336 359 1 10 100 1000 10000 Photon energy in eV Watt/eV/mrad/1 A 10 -3 10-4 10-5 E D C B A Page 26 of 32 0.2 0.3 0.4 0.5energy in GeV246810 field in T01020 Watt /G732e V /G731A at 13.5 nm 0.2 0.3 0.4 0.5energy in GeV Fig. 4 Fig. 5 Field in T Energy in GeV Watt/ 2 eV/ 1A at 13.5 nm 300 MeV 400 MeV 500 MeV 600 MeV 0 1 2 3 4 5 6 Field in Tesla Watt/2 eV/1 A at 13.5 nm 30 20 10 5 Page 27 of 32 Fig. 6 Field in T Energy in GeV Ratio In-band/ Out-band power Beam energy 885 MeV Period length λu 3.8 mm Number of periods N u 100 Field 0.3 T Gap 2 mm Fig. 7 Page 28 of 32 0 500 1000 1500 2000012345W/ 2 eV Photon energy [eV] Fig. 8 Page 29 of 32 Fig. 9 Page 30 of 32 Fig. 10 Fig. 11 0 20 40 60 80 mm 0.5 1 1.5 2 2.5 3 Beta function in m L G in m 1 0.8 0.6 0.4 Page 31 of 32 Fig. 12 Fig. 13 Peak power in Wat t 109 107 105 10 3 10 5 10 15 m Undulator length Peak power inWatt 109 107 10 5 103 10 10 30 50 m Undulator length Page 32 of 32 Fig 14 Undulator
arXiv:physics/0103056v1 [physics.gen-ph] 19 Mar 2001 /C9/BX/BW /B9 /D0/CP/D7/D7/CX /CP/D0/D0/DD /BA /BT/D2 /CX/D2 /D8/D9/CX/D8/CX/DA /CT /BT/D4/D4/D6/D3/CP /CW/CC/CX/CQ /D3/D6 /BW/D9/CS/CP/D7/BU/D6/CX/CT/CU /CX/D2 /D8/D6/D3 /CS/D9 /D8/CX/D3/D2/CC/CW/CX/D7 /D4/CP/D4 /CT/D6 /CS/CT/CP/D0/D7 /DB/CX/D8/CW /C9/BX/BW/B9/D4/CP/D6/D8/CX /D0/CT/D7 /CP/D2/CS /D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D1 /D3/D2 /CP /D0/CP/D7/D7/CX /CP/D0 /D0/CT/DA /CT/D0/BA /CC/CW/CT /C5/CP/DC/DB /CT/D0/D0/B9/CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D9/D7/CT/CS /D1/CP/CX/D2/D0/DD /BA /B4/C8/D6/D3 /D3/CU/D7 /CP/D6/CT /D2/D3/D8 /D9/D7/CT/CS/CX/D2 /CP /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /CQ/D9/D8 /CX/D2 /D8/D9/CX/D8/CX/DA /CT /D7/CT/D2/D7/CT/BA/B5 /C1/D2 /D8/CW/CT /AS/D6/D7/D8 /D7/D8/CT/D4 /D8/CW/CT /D1/CP/CX/D2 /D7/D8/CP/D8/CT/D1/CT/D2 /D8/D7/CP/D6/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS/BA /CC/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D4/D6/D3 /D3/CU/D7 /CP/D6/CT /CV/CX/DA /CT/D2 /CX/D2 /D8/CW/CT /D7/CT /D3/D2/CS /CP/D2/CS /AS/D2/CP/D0/D7/D8/CT/D4/BA/CB/D8/CP/D8/CT/D1/CT/D2 /D8/D7/C1/BA /BT/D2 /CT/D0/CT /D8/D6/D3/D2/BB/D4 /D3/D7/CX/D8/D6/D3/D2 /CX/D7 /CP /D7/CX/D2/CZ/BB/D7/D3/D9/D6 /CT /D3/CU /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /D5/D9/CP/D2 /D8/CP/BA/C1 /C1/BA /CC/CW/CT/D7/CT /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CS /D5/D9/CP/D2 /D8/CP /D7/CW/D3/D9/D0/CS /D0/CT/CP/D6/D0/DD /CQ /CT /CX/CS/CT/D2 /D8/CX/AS/CT/CS/DB/CX/D8/CW /D4/CW/D3/D8/D3/D2/D7/BA/C1 /C1 /C1/BA /BV/CW/CP/D6/CV/CT/B9 /CW/CP/D6/CV/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D7 /CT/CP/D7/CX/D0/DD /D3/D2 /CT/CX/DA /CP/CQ/D0/CT/BA/C1/CE/BA /CC/CW/CT /D1/CP/CV/D2/CT/D8/CX /DA /CT /D8/D3/D6 /AS/CT/D0/CS /D3/D2/D7/D8/CX/D8/D9/D8/CT/D7 /CP /AT/D3 /DB /D3/CU /D4/CW/D3/D8/D3/D2/D7/BA/CE/BA /C1/D8 /CQ /CT /D3/D1/CT/D7 /D0/CT/CP/D6/B8 /DB/CW /DD /CP /CW/CP/D6/CV/CT/CS /D4/CP/D6/D8/CX /D0/CT /CX/D7 /CS/CT/AT/CT /D8/CT/CS /D4 /CT/D6/D4 /CT/D2/CS/CX /D9/D0/CP/D6 /D8/D3/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX /AS/CT/D0/CS /D0/CX/D2/CT/D7/BA/C8/D6/D3 /D3/CU/D7/C1/BA /C4/CT/D1/D1/CP/BA /C1/D8 /CX/D7 /D7/D9Ꜷ /CX/CT/D2 /D8 /D8/D3 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CS /D3/D2/D0/DD/DB/CX/D8/CW/D3/D9/D8 /D8/CW/CT /DA /CT /D8/D3/D6 /AS/CT/D0/CS/BA/C8/D6 /D3/D3/CU/BA /BT /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE/B9 /D3/D2/CS/CX/D8/CX/D3/D2 ∂µAµ= 0⇔∂0A0+/vector∇/vectorA= 0/C1/D8 /CX/D7 Aµ=/parenleftbiggΦ /vectorA/parenrightbigg/BD/CP/D2/CS xµ=/parenleftbiggt /vector x/parenrightbigg/DB/CX/D8/CW /vectorB=/vector∇ ×/vectorA /CP/D2/CS /B4/BD/B5 /vectorE=−/vector∇φ−1 c∂t/vectorA. /B4/BE/B5/C1/D2 /D8/CT/CV/D6/CP/D8/CT /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE/B9 /D3/D2/CS/CX/D8/CX/D3/D2 /D3 /DA /CT/D6 /CP /AS/D2/CX/D8/CT /DA /D3/D0/D9/D1/CT /CP/D2/CS /D9/D7/CT /BZ/CP/D9/D7/D7/B3 /D0/CP /DB/CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D3/CQ/D8/CP/CX/D2 O=/integraldisplay V∂tφd3x+/integraldisplay V/vector∇/vectorAd3x=/integraldisplay V∂tφd3x+/integraldisplay ∂V/vectorAd/vector n ⇔/integraldisplay V∂tφd3x=−/integraldisplay ∂V/vectorAd/vector n /BY/CX/CV/D9/D6/CT /BD/BM /C1/CUΦ /CW/CP/D2/CV/CT/D7 /DB/CX/D8/CW /D8/CX/D1/CT/B8 /CX/D8 /CX/D7 /CS/D9/CT/D8/D3 /CP /AT/D3 /DB /D3/CUΦ /CX/D2 /D8/D3 /D3/D6 /D3/D9/D8 /D3/CU /D8/CW/CT /DA /D3/D0/D9/D1/CT/BA/C6/D3 /DB/B8 /CX/CUφ /CX/D7 /CX/D2 /D6/CT/CP/D7/CX/D2/CV /DB/CX/D8/CW /D8/CX/D1/CT/B8 /D8/CW/CT/D6/CT /CX/D7 /CP /D2/CT/D8 /CX/D2 /DB /CP/D6/CS /AT/D3 /DB /D3/D6 /CP /D2/CT/D8/D2/D3/D2 /DA /CP/D2/CX/D7/CW/CX/D2/CV /D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU/vectorA /D4 /D3/CX/D2 /D8/CX/D2/CV /CX/D2 /DB /CP/D6/CS /D8/CW/CT /DA /D3/D0/D9/D1/CT/BA ✷/C6/D3/D8/CX /CT/BM ∂tρ+/vector∇/vector = 0 /CU/D3/D6 /CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ /CP/D2/CS/vector =ρ/vector v /CQ /CT/CX/D2/CV /D8/CW/CT /CW/CP/D6/CV/CT/AT/D9/DC/BA/C0/CT/D2 /CT/B8 /D0/CT/D8 /D9/D7 /D1/CP/CZ /CT /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CP/D7/D9/D1/D4/D8/CX/D3/D2/D7/BM /B4/BD/B5/vectorA /CX/D7 /CP /AT/D3 /DB /D3/CU /D8/CW/CT /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CSφ /BA /B4/BE/B5φ /CX/D7 /D5/D9/CP/D2 /D8/CX/DE/CT/CS/BA/C8/D6 /D3/D3/CU /B4/C1/BA /CB/D8 /CP /D8/CT/D1/CT/D2/D8/B5/BA /BY /D3/D6 /CP /D4 /D3/CX/D2 /D8 /CW/CP/D6/CV/CT φ∼1 r /BA/BE/CC/CW/CT /D7 /CP/D0/CP/D6 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D7/CW/D3/D9/D0/CS /DB /CT/CP/CZ /CT/D2/B8 /CX/CU /D8/CW/CT /D7/CT/D4 /CT/D6/CP/D8/CX/D3/D2 ds /CQ /CT/D8 /DB /CT/CT/D2 /CX/D8/D7 /D5/D9/CP/D2 /D8/CP/CX/D2 /D6/CT/CP/D7/CT/D7/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /DB /CT /CT/DC/D4 /CT /D8 φ∼1 ds=1 r,/DB/CW/CX /CW /CX/D7 /CP /D3/D2/AS/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D3/D9/D6 /CX/D1/CP/CV/CT/BA /BY/CX/CV/D9/D6/CT /BE/BM /CC/CW/CT /D4/CW/D3/D8/D3/D2/D7 /AT/D3 /DB/BB/CT/D1/CT/D6/CV/CT/CX/D2 /D8/D3/BB/CU/D6/D3/D1 /D8/CW/CT /CT/D0/CT /D8/D6/D3/D2/BB/D4 /D3/D7/CX/D8/D6/D3/D2/BA Φ∼1 ds= 1 r /BA /CC/CW/CT /D7 /CP/D0/CP/D6 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D7/CW/D3/D9/D0/CS /DB /CT/CP/CZ /CT/D2/B8 /CX/CU /D8/CW/CT/D7/CT/D4 /CT/D6/CP/D8/CX/D3/D2 ds /CQ /CT/D8 /DB /CT/CT/D2 /CX/D8/D7 /D5/D9/CP/D2 /D8/CP /CX/D2 /D6/CT/CP/D7/CT/D7/BA/BT /D3/D6/CS/CX/D2/CV /D8/D3/vectorE=−/vector∇φ /CU/D3/D6 /CP/D2 /CT/D0/CT /D8/D6/D3/D2 /D8/CW/CT /D3/D2 /CT/D2 /D8/D6/CP/D8/CX/D3/D2 /CX/D2 /D6/CT/CP/D7/CT/D7 /CU/D3/D6/CQ/CX/CV/CV/CT/D6 /CS/CX/D7/D8/CP/D2 /CT/D7 /CU/D6/D3/D1 /D8/CW/CT /D3/D6/CT/BA /C0/CT/D2 /CT/B8 /CP/D2 /CT/D0/CT /D8/D6/D3/D2 /CX/D7 /CP /D7/CX/D2/CZ/BA /BU/DD /D6/CT/DA /CT/D6/D7/CT/CP/D6/CV/D9/D1/CT/D2 /D8/CP/D8/CX/D3/D2/B8 /CP /D4 /D3/D7/CX/D8/D6/D3/D2 /CX/D7 /CP /D7/D3/D9/D6 /CT /D3/CU /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CS /D5/D9/CP/D2 /D8/CP/BA ✷/CC/CW/CT /D7 /CP/D0/CP/D6 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D7/CW/D3/D9/D0/CS /DB /CT/CP/CZ /CT/D2/B8 /CX/CU /D8/CW/CT /D7/CT/D4 /CT/D6/CP/D8/CX/D3/D2 ds /CQ /CT/D8 /DB /CT/CT/D2 /CX/D8/D7 /D5/D9/CP/D2 /D8/CP/CX/D2 /D6/CT/CP/D7/CT/D7/BA/C1 /C1/BA /CC /D6/CX/DA/CX/CP/D0/B8 /CP/D7 /D2/CT/D7 /CT/D7/D7/CP/D6/CX/D0/DD /B8 /D8/CW/CT/D7/CT /D5/D9/CP/D2 /D8/CP /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT /D4/CW/D3/D8/D3/D2/D7/B8 /CP/D7 /D8/CW/CT/DD /D6/CT/CP/D8/CT /D8/CW/CT /CT/D0/CT /D8/D6/CX /AS/CT/D0/CS /DA/CX/CP /B4/BE/B5/BA ✷/C1 /C1 /C1/BA /BV/CW/CP/D6/CV/CT/B9/BV/CW/CP/D6/CV/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2/BA/B4/BD/B5 /BX/D0/CT /D8/D6/D3/D2/B9/C8/CW/D3/D8/D3/D2/BA /CC/CW/CT /D1/D3 /DA/CX/D2/CV /CT/D0/CT /D8/D6/D3/D2 /CX/D7 /CS/D6/CP/CV/CV/CT/CS /D8/D3 /DB /CP/D6/CS/D7 /D8/D3 /D7/D3/D9/D6 /CT /D3/CU/CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CS /CP/D2/CS /DA/CX /CT /DA /CT/D6/D7/CP/BA/C8/D6 /D3/D3/CU/BA /CC/CW/CX/D7 /CX/D7 /CP /CU/CP /D8/B8 /CY/D9/D7/D8 /D8/CW/CT /CX/D1/CP/CV/CT /CX/D7 /CS/CX/AR/CT/D6/CT/D2 /D8/BA/B4/BE/B5 /BX/D0/CT /D8/D6/D3/D2/B9/BX/D0/CT /D8/D6/D3/D2/BA /CC/CW/CT /CP/CV/CV/D0/D3/D1/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D4/CW/D3/D8/D3/D2/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /D7/D3/D9/D6 /CT/D7/D4/D9/D7/CW/CT/D7 /D8/CW/CT/D1 /CP/D4/CP/D6/D8 /CU/D6/D3/D1 /CT/CP /CW /D3/D8/CW/CT/D6/B8 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /D0/CX/CZ /CT /D8 /DB /D3 /AS/D6/CT/D1/CT/D2 /D7/CX/D8/D8/CX/D2/CV /D3/D2/D1/D3 /DA /CP/CQ/D0/CT /CW/CP/CX/D6/D7 /DB /D3/D9/D0/CS /D7/CT/D4 /CT/D6/CP/D8/CT /CQ /DD /CS/CX/D6/CT /D8/CX/D2/CV /D8/CW/CT/CX/D6 /D6/D9/D2/D2/CX/D2/CV /DB /CP/D8/CT/D6/CV/D9/D2/D7 /D8/D3 /DB /CP/D6/CS/D7/CT/CP /CW /D3/D8/CW/CT/D6/BA/BF/B4/BF/B5 /BT/D7 /D8/CW/CT /D3/D2 /CT/D2 /D8/D6/CP/D8/CX/D3/D2 /D3/CU /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CS /D5/D9/CP/D2 /D8/CP /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D8 /DB /D3 /D7/CX/D2/CZ/D7 /CX/D7 /D0/D3 /DB /CT/D6 /D8/CW/CP/D2 /CP/D7/CX/CS/CT /CU/D6/D3/D1 /D8/CW/CT/D1/B8 /D8/CW/CT/DD /CP/D6/CT /CS/D6/CP/CV/CV/CT/CS /CP/D4/CP/D6/D8 /D8/D3 /DB /CP/D6/CS/D7 /D8/CW/CT/CW/CX/CV/CW/CT/D6 /D3/D2 /CT/D2 /D8/D6/CP/D8/CX/D3/D2/B8 /D8/CW/CT/D6/CT/CU/D3/D6/CT /D7/CT/D4 /CT/D6/CP/D8/CT/BA ✷/C1/CE/BA /C5/CP/CV/D2/CT/D8/CX/D7/D1/BB/C4/D3/D6/CT/D2 /D8/DE/B9/CU/D3/D6 /CT/BA /vectorB=/vector∇ ×/vectorA /CP/D2/CS/vectorA /CX/D7 /AT/D3 /DB /D3/CU /CT/D0/CT/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6 /AS/CT/D0/CS /D5/D9/CP/D2 /D8/CP /B4/D7/CT/CT/C4/CT/D1/D1/CP /CX/D2 /C1/B5/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /DB /CT /CW/CP /DA /CT /CP /D2/D3/D2 /DA /CP/D2/CX/D7/CW/CX/D2/CV /DB/CW/CX/D6/D0 /D3/CU /D1/D3 /DA/CX/D2/CV /D4/CW/D3/D8/D3/D2/D7/B8 /D3/D2/D7/D8/CX/D8/D9/D8/CX/D2/CV /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX /AS/CT/D0/CS/BA ✷/CE/BA /C4/D3/D6/CT/D2 /D8/DE/B9/CU/D3/D6 /CT /BY/CX/CV/D9/D6/CT /BF/BM /BT /DB/CW/CX/D6/D0 /D3/CU /CT/D0/CT /D8/D6/D3/D1/CP/CV/D2/CT/D8/CX /D7 /CP/D0/CP/D6/AS/CT/D0/CSΦ /D5/D9/CP/D2 /D8/CP /D3/D2/D7/D8/CX/D8/D9/D8/CT /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX /AS/CT/D0/CS/BA −/integraltext ∂V/vectorAd/vector n=/integraltext V∂tφd3x /BA /vectorB=/vector∇ ×/vectorA, /vectorF=q/vector v×/vectorB./D6/D3/D8/vectorA /CX/D7 /CP /DB/CW/CX/D6/D0 /D3/CU /D1/D3 /DA/CX/D2/CV /D4/CW/D3/D8/D3/D2/D7/BA /BT/D2 /CT/D0/CT /D8/D6/D3/D2 /CX/D7 /CS/CT/AT/CT /D8/CT/CS /CS/D3 /DB/D2 /DB /CP/D6/CS/D7 /D3/D1/CX/D2/CV/CU/D6/D3/D1 /D8/CW/CT /CU/D6/D3/D2 /D8/B8 /CP/D7 /CX/D8 /D6/D9/D2/D7 /D8/D3 /DB /CP/D6/CS/D7/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D1/D3/D7/D8 /D4/CW/D3/D8/D3/D2/D7 /CP/D6/CT /D3/D1/CX/D2/CV /CU/D6/D3/D1/BA/BT /D4 /D3/D7/CX/D8/D6/D3/D2 /CX/D7 /CS/CT/AT/CT /D8/CT/CS /D9/D4 /DB /CP/D6/CS/D7/B8 /CP/D7 /D4/CW/D3/D8/D3/D2/D7 /CP/D6/CT /D3/D1/CX/D2/CV /D7/D0/CX/CV/CW /D8/D0/DD /CU/D6/D3/D1 /CQ /CT/D0/D3 /DB/BA/CC/CW/CX/D7 /CX/D7 /DB/CW/CP/D8 /D8/CW/CT /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D7/CP /DD /D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D3/D2/CS/CX/D8/CX/D3/D2/D8/CP/CZ /CT/D2 /D4/CW /DD/D7/CX /CP/D0/D0/DD /D6/CT/D0/CT/DA /CP/D2 /D8/BA/C5/CP/CV/D2/CT/D8/CX/D7/D1 /CP/D2 /D8/CW/CT/D6/CT/CU/D3/D6/CT /CQ /CT /D6/CT/CS/D9 /CT/CS /D8/D3 /D4/CW/D3/D8/D3/D2/B9 /CW/CP/D6/CV/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /D8/CW/CT/D7/CP/D1/CT /DB /CP /DD /D0/CX/CZ /CT /CW/CP/D6/CV/CT/B9 /CW/CP/D6/CV/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D8/CP/CZ /CT/D7 /D4/D0/CP /CT/B8 /D0/CP/D7/D7/CX /CP/D0/D0/DD /BA ✷/BG
1 LABORATÓRIO DE INSTRUMENTAÇÃO E FÍSICA EXPERIMENTAL DE PARTÍCULAS Preprint LIP/00-04 16 October 2000 High-Resolution TOF with RPCs P. Fonte 1,2,, V. Peskov3 1 – CERN-EP, Geneva, Switzerland 2 – LIP, Coimbra, Portugal 3 – Royal Institute of Technology, Stockholm, Sweden. Abstract In this work we describe some recent results concerning the application of Resistive Plate Chambers operated in avalanche mode at atmospheric pressure for high-resolution time-of- flight measurements. A combination of multiple, mechanically accurate, thin gas gaps and state-of-the-art electronics yielded an overall (detector plus electronics) timing accuracy better than 50 ps σ with a detection efficiency up to 99% for MIPs. Single gap chambers were also tested in order to clarify experimentally several aspects of the mode of operation of these detectors. These results open perspectives of affordable and reliable high granularity large area TOF detectors, with an efficiency and time resolution comparable to the existing scintillator-based TOF technology but with a significantly, up to an order of magnitude, lower price per channel. Presented at the PSD99-5th International Conference on Position-Sensitive Detectors 13-17 th September 1999, University College, London, Corresponding author: Paulo Fonte, LIP - Coimbra, Departamento de Física da Universidade de Coimbra, 3004-516 Coimbra, PORTUGAL. tel: (+351) 239 833 465, fax: (+351) 239 822 358, email: fonte@lipc.fis.uc.pt21. - Introduction Heavy-ion collision physics at very high energies is emerging in many accelerator centres around the world (RHIC at BNL, SIS at GSI, and LHC-HI at CERN), emphasising the need for large area particle identification systems able to cope with high particle multiplicities. Time of Flight (TOF) sub-detectors, in particular, are foreseen for many such experiments (STAR, FOPI, ALICE), stimulating R&D on new, more cost effective, approaches to the timing of MIPs. In this paper we describe some recent work done in the framework of the ALICE experiment on the development of Resistive Plate Chambers ( RPCs) for TOF measurements (timing RPCs). The work included beam tests of single chambers and of a multichannel TOF prototype equipped with such chambers. Some studies aimed to clarify experimentally the origin of the very good detection efficiency (99%) observed for MIPs in our timing RPCs will be also described. 2. - Experimental setup Details on the mechanical construction, electronics, experimental setup and data analysis for the timing RPCs were already given elsewhere [1] and we will describe here only the few modifications introduced for the single cell tests. Timing RPCs were made with glass [2] and aluminium electrodes forming a pair of double-gap chambers. The four gas gaps of 0.3 mm were accurately defined by glass optical-fiber spacers. The single gap RPCs were built essentially along the same lines, being a schematic drawing shown in Figure 1. The signals were sensed by a custom made pre -amplifier [1], whose output was split in 3 identical channels via analogue buffers. One of the outputs was directly fed to a LeCroy 2249W ADC that measured a charge proportional to the total signal charge. Another output had the ion signal component (1 to 3 µs long) cancelled by forming the difference between the signal and it´s image delayed by 16 ns. A LeCroy 2249A ADC integrated the resulting short pulse, measuring a charge proportional only to the electron (fast) component of the signal. The third output was further amplified by a factor 10 and fed to a custom-made fixed threshold discriminator (typically set at a level equivalent to a total signal charge of 0.2 pC), followed by a LeCroy 2229 TDC with a 50 ps bin width. The total charge was calibrated by injecting to the test input of the pre-amplifier a current pulse of an intensity and width similar to the ion current pulse from the chambers. 3. – Results 3.1 – Timing RPCs Timing RPCs in a single channel configuration have shown timing resolutions below 50 ps σ with an efficiency of 99% for MIPs [1]. A typical signal charge distribution is shown in Figure 2. A 32-channel prototype equipped with similar chambers and suitable multichannel electronics has shown an average time resolution of 88 ps σ with a spread of 9 ps and an average efficiency of 98 % with a spread of 0.5 % [3]. The crosstalk between neighbouring channels generally did not exceed 1%.33.2 - Single gap chambers In Figure 3 we show the signal charge distributions measured at different applied voltages in single gas gaps of 0.1 and 0.3 mm filled with methane, isobutane or a “standard RPC mixture” containing C 2H2F4+10%SF 6+5%isobutane, also used for the timing RPCs. In pure isobutane and in the “standard mixture”, for both gap sizes, the distributions show an extended flat region for the larger applied voltages. This is quite surprising because it can be shown theoretically1 that the event by event variations in the position of the leading cluster (closer to the cathode) causes strong fluctuations in the final avalanche size, resulting in a charge distribution almost proportional to 1/Q (being Q the signal charge). The observed distribution is much more favourable for an efficient particle detection than the expected “1/Q” distribution, allowing for the excellent detection efficiency measured in chambers with four gaps (see also Figure 2). In methane only a modest gas gain could be reached due to the onset of discharges, presumably caused by photon feedback. Indeed, while for the other gases all results were essentially independent of the cathode material (aluminium or glass), in methane larger gains could be reached with the glass cathode, presumably due to the smaller quantum efficiency of glass. Even for a single 0.1 mm gas g ap the detection efficiency was still around 45 % (isobutane). From the corresponding inefficiency figure and the gap length it can be calculated that the mean number of primary clusters per unit length must be at least 6 mm-1 (at least one cluster must be produced for a particle to be detected). However, due to the exponential dependence of the final avalanche size on the cluster position, only a small region of the gas gap (closer to the cathode) will be sensitive to the ionising particles. Therefore, to explain the observed efficiency, the ionisation density must be a few times larger than the figure calculated above, being quite doubtful whether this is physically possible [4]. In fact there is some indication that a process other than gas ionisation may be also contributing to the observed detection efficiency (Figure 4). Some further information can be obtained by plotting the average signal charge as a function of the applied field (Figure 5). There is a strongly sub -exponential growth of the average charge with the applied field, indicating the presence of a gas gain saturation effect. Although similar effects have been observed in parallel geometry counters at low pressures [5] and wire counters, being generally attributed to space -charge effects, su ch strong saturation has never been, to the author’s knowledge, observed at atmospheric pressure in proportional -mode parallel geometry counters. This may be related to the fact that our detectors work in an E/p range (around 100 V/cm Torr) similar to those typically found in low-pressure parallel -plate counters or at the surface of wires in cylindrical counters and MWPCs. Further evidence of a strong space charge effect is presented in Figure 6, were correlation plots between the fast (electron) signal charge and the total signal charge were drawn for the same experimental conditions studied above. Standard detector theory [5] shows that the ratio between these quantities should be independent of the avalanche size and equal to (av)-1, where a is the First Townsend Coefficient and v is the electron drift velocity. This situation is observed for the case of a 0 .1 mm gap filled with methane. For all other cases the upward 1 See for instance the Appendix in ref. [6].4curving correlation is compatible with a space charge effect that would reduce the effective value of a for the larger avalanches [5]. 4. – Conclusions Timing RPCs made with glass and metal electrodes, forming four of accurately spaced gas gaps of 0.3 mm, have reached time resolutions below 50 ps σ with a detection efficiency of 99% for MIPs [1]. A 32-channel prototype equipped with such chambers has shown an average resolution of 88 ps σ with a spread of 9 ps and an average efficiency of 98 % with a spread of 0.5 %. The crosstalk between neighbouring channels generally did not exceed 1% [3]. It was found that in single gas gaps the signal charge distribution departs strongly from the theoretically expected shape and that the gas amplification process seems to be strongly influenced by a space charge effect. This effect may be related to the unexpected charge distribution observed. The relatively large detection efficiencies observed in single gas gaps (up to 45% for 0.1 mm gaps and up to 75% for 0.3 mm gaps) seem to be incompatible with a primary detection process based uniquely in the ionisation of the gas by the incoming particles. These conclusions seem to be quite independent on the nature of the filling gas, applying both to the operation in pure isobutane and in a strongly electronegative mixture containing Freon and SF 6. 5. – Acknowledge ments The use of the instrumented beam line installed by the ALICE experiment for the tests in the T10 PS beam under the supervision of W. Klempt is acknowledged, as well as the kind support of the CERN EP/AIT group. The data acquisition infrastructure and the tracking system were implemented and managed by Paolo Martinengo. We are also very grateful to P.G. Innocenti, W. Klempt, C. Lourenço, G. Paic, F. Piuz, R. Ribeiro and J. Schukraft for their comments and suggestions and for their interest on our work. We benefited also from many discussions and from the accumulated experience of the members of the ALICE-TOF project and from the technical expertise of Dave Williams. This work was partially supported by the FCT research contract CERN/FAE/1197/98. One of us (V. Peskov) acknowledges the financial support of LIP- Coimbra. 6. - References [1]P. Fonte, R. Ferreira Marques, J. Pinhão, N. Carolino, A. Policarpo, “High Resolution RPCs for Large TOF Systems”, Nucl. Instr. and Meth. in Phys. Res. A , 449 (2000) 295 .. [2]SCHOTT ATHERMAL . [3]A. Akindinov et al.,”A Four-Gap Glass-RPC Time of Flight Array with 90 ps Time Resolution”, ALICE note ALICE-PUB-99-34, preprint CERN-EP-99-166. [4]F.Sauli, CERN Yellow Report 77-09 (1977). [5]H. Raether, “Electron Avalanches and Breakdown in Gases” (London, Butterworths, 1964). [6]M. Abbrescia et al, Nucl. Instr. and Meth. in Phys. Res. A431 (1999) 413.5Figure captions Figure 1 - Structure of a single-gap detector cell. Figure 2 – Comparison between the signal charge distri bution observed in a timing RPC (histogram) with four gas gaps and the 4 -fold self -convolution of the charge distribution from a single-gap chamber (solid line) measured in similar operating conditions. The overall good agreement between both distributions suggests that the charge distribution observed in the four-gaps chamber can be interpreted as the analog sum of four independent single gaps. Figure 3 – Distribution of the signal charge in single gap chambers for several applied voltages, filling gases and gap widths. The innermost distributions correspond to the ADC pedestal and the peak close to 0 pC corresponds to the detector inefficiency. The efficiency figures were measured by the method described in [2]. Figure 4 – The extrapolation of the observe d detection efficiency to a vanishing gas gap width suggests that some additional process, other than gas ionization, may be contributing to the detection efficiency. Figure 5 – Average signal charge as a function of the applied electrical field, calculat ed from the data presented in Figure 3. The solid lines correspond to exponential functions fitted to the lower 3 points of each experimental series and extrapolated to the larger fields, evidencing the sub-exponential character of experimental data (gain saturation). The onset of gain saturation for the 0 .1 mm gaps occurs at a charge level that is an order of magnitude smaller than for the 0.3 mm gaps. Figure 6 – Correlation plots between the fast (electron) signal charge and the total signal charge. Standard detector theory shows that the ratio between these quantities should be a constant, being the observed upward curving correlation compatible with a space charge effect.6Figures7Aluminium electrode Glass fiber spacers Glass electrode with back- metalization Figure 1 - Structure of a single-gap detector cell.80 0.5 1 1.5 2 2.5 3 3.5 40100200300400500600700Counts/50 fC Total signal charge ( pC)Events/ 20 fC Figure 2 – Comparison between the signal charge distri bution observed in a timing RPC (histogram) with four gas gaps and the 4 -fold self -convolution of the charge distribution from a single-gap chamber (solid line) measured in similar operating conditions. The overall good agreement between both distributions suggests that the charge distribution observed in the four-gaps chamber can be interpreted as the analog sum of four independent single gaps.90 0.5 1 1.5 2100101102103104 pCIsobutane - 0.1 mm V=1700 V - Eff = 45.82% V=1500 V - Eff = 30.92% V=1300 V - Eff = 2.52% 0 0.5 1 1.5 2100101102103104 pCMethane - 0.1 mm V=1300 V - Eff = 15.88% V=1200 V - Eff = 12.4% V=1100 V - Eff = 6.8% 0 0.5 1 1.5 2100101102103104 pCStandard mixture 0.1 mm V=1600 V - Eff = 38.44% V=1400 V - Eff = 23.42% V=1200 V - Eff = 0.68% 0 2 4 6100101102103 pCMethane - 0.3 mm V=2100 V - Eff = 28.98% V=1900 V - Eff = 21.88% V=1800 V - Eff = 13.4%V=2100 V - Eff = 28.98% V=1900 V - Eff = 21.88% V=1800 V - Eff = 13.4% 0 2 4 6100101102103 pCIsobutane - 0.3 mm V=2800 V - Eff = 74% V=2600 V - Eff = 65.36% V=2400 V - Eff = 37.18% 0 2 4 6100101102103 pCStandard mixture 0.3 mm V=2800 V - Eff = 73.18% V=2500 V - Eff = 55.1% V=2300 V - Eff = 18.86% Total signal chargeCounts/bin Figure 3 – Distribution of the signal charge in single gap chambers for several applied voltages, filling gases and gap widths. The innermost distributions correspond to the ADC pedestal and the peak close to 0 pC corresponds to the detector inefficiency. The efficiency figures were measured by the method described in [2].100%10%20%30%40%50%60%70%80% 0 0,1 0,2 0,3 0,4 Single gas gap (mm)Detection efficiency for MIPsIsobutane Standard mixture Linear (Standard Figure 4 – The extrapolation of the observe d detection efficiency to a vanishing gas gap width suggests that some additional process, other than gas ionization, may be contributing to the detection efficiency.110.00010.0010.010.1110100 0 20 40 60 80 100 120 140 160 180 Electric field (kV/cm)Average charge ( pC)Standard mix - 0.3 Isobutane - 0.3 Methane - 0.3 Standard mix - 0.1 mm Isobutane - 0.1 Methane - 0.1 0.1 mm0.3 mm Figure 5 – Average signal charge as a function of the applied electrical field, calculat ed from the data presented in Figure 3. The solid lines correspond to exponential functions fitted to the lower 3 points of each experimental series and extrapolated to the larger fields, evidencing the sub-exponential character of experimental data (gain saturation). The onset of gain saturation for the 0.1 mm gaps occurs at a charge level that is an order of magnitude smaller than for the 0.3 mm gaps.120 0.5 100.20.40.60.81Methane - 0.1 mm 0 0.5 100.20.40.60.81Isobutane - 0.1 mm 0 0.5 100.20.40.60.81 Standard mixture 0.3 mm 0 2 4 60123456Isobutane - 0.3 mm 0 2 4 600.511.52 0 2 4 600.511.522.53Methane - 0.3 mm Total charge ( pC)Fast charge ( unid .arb.)Standard mixture 0.1 mm Figure 6 – Correlation plots between the fast (electron) signal charge and the total signal charge. Standard detector theory shows that the ratio between these quantities should be a constant, being the observed upward curving correlation compatible with a space charge effect.
ON NON-MEASURABLE SETS AND INVARIANT TORI by Piotr Pierański +and Krzysztof W. Wojciechowski ++ +Poznań University of Technology Piotrowo 3, 60 965 Poznań, Poland ++Institute of Molecular Physics, Polish Academy of Sciences M. Smoluchowskiego 17, 60 159 Poznań, Poland ABSTRACT The question: " How many different trajectories are there on a single invariant torus within the phase space of an integrable Hamiltonian system ?" is posed. A rigorous answer to the question is found both for the rational and the irrational tori. The relevant notion of non-measurable sets is discussed. I. INTRODUCTION Irrational invariant tori play a crucial role in physics of Hamiltonian systems. In contrast to the rational tori, they prove to be to some extent resistant to the destructive action of the non-integrable perturbations and, as the KAM theorem establishes it, the measure of the set of those tori which remain intact, though obviously distorted, is non-zero at low levels of the perturbation.1 It is the aim of this paper to indicate a peculiar property of irrational tori: non-measurability of sets of points which initiate on them all possible (and different) trajectories. To make the considerations which follow as clear as possible, we fix our attention on the simplest nontrivialcase - a Hamiltonian system with but two degrees of freedom q 1 and q2 whose trajectories are located within a four-dimensional phase space Γ. (Generalisation for more degrees of freedom istrivial.) We assume that the system is integrable, i.e. there exist two integrals of motion I1 and I2 which allow one to describe any motion of the system as two independent rotations on a two- dimensional torus; see Figure1. t t1 1 1 +(0)=)( ω ϕ ϕ (1) t t2 2 2 +(0) =)( ω ϕ ϕ (2) In the most general case, the two frequencies are different on each of the tori into which the whole phase space of the system is partitioned. There are two basic types of the tori: those for which the ratio ω1/ω2 is rational and those for which the ratio is irrational. Let us ask a question: How many different trajectories are there on a single: (i) rational and (ii) irrational torus ? To observe trajectories which move on the torus T in a more convenient manner, we cut it with a Poincaré section S. In this plane a single trajectory is seen as a sequence of points which mark all its consecutive (past and future) passages through S. All the points are located, of course, on the circle C = T ∩ S. A single point from such a sequence determines the whole (and single) trajectory. There are many different (disjoint) trajectories moving on the torus. Each of them defines on C a sequence of points. Choosing single points from all such sequences allows one to construct the required set of points which initiate on the torus different trajectories. Let us denote the set by M0. Thus, the question we posed above can be reduced to the following one: How big is the set M0?. Fig. 1 Invariant torus and a Poincare section of it. A trajectory starting form point P0 pierces plane S in consecutive points P1, P2, P3 … . See text.By "how big" we mean here two things : 1. Is the set M0 countable ? 2. Which is its measure µ(M0) ? Below we shall answer the questions, first, for the case of rational tori, then, for the case of the irrational ones. II. µµµµ(M0) ON RATIONAL TORI. Let Tr be a rational torus, i.e. a torus for which ω1/ω2 = r = m/n, where m and n are integers. Any trajectory on Tr marks in S as a cycle of n points { Pk}, k=0, 1, …, n-1, whose angular coordinates ϕk are given by 1mod2 20krk+πϕ=πϕ(4) As easy to note, all trajectories which start from those points on C whose ϕ coordinates are located within any interval [ ϕ0, ϕ0+2π/n), where ϕ0 is arbitrary, are different and, as such a choice is made, there are no other different trajectories. Consequently, the set )}2[0,)(: {=0nP CTP Mπ∈ϕ∩∈ (5) can be seen as the simplest realisation of the set of points which initiate on T all possible different trajectories. Obviously, in this case, the set is uncountable and its measure : nCM)(=)(0µµ (6) Any other choice of M0 provides the same answer.III. µµµµ(M0) FOR IRRATIONAL TORI. Let ω1/ω2 = ρ be an irrational number. Now, any trajectory on Tρ is seen within the Poincar é section S as an infinite, never repeating itself sequence of points { Pk}, k= -∞ …, -1, 0, 1, 2, … +∞, whose angular coordinates are given by 1) (mod+2=20 kρπϕ πϕk (7) The countable set { Pk} covers C in a dense manner but is different from it: C\{Pk}≠∅. Thus, there are on C some points which initiate other trajectories. How to find all of them i.e. define set M0 ? To reach the aim we shall proceed in three steps. Step 1 . We define within the [0,1) interval a countable, everywhere dense set } :1modρ{ Nk k E ∈ = (8) Step 2 . We define in C a relation ℜ: PℜQ if and only if there exists in E an x such that: xQP=π− 2(9) In plain words the physical meaning of ℜ can be expressed as follows : P and Q stay in relation ℜ when they belong to the same trajectory. One can prove that ℜ is an equivalence relation, thus, ℜ divides C into a family of equivalence classes C/ℜ. In view of the physical meaning of ℜ the classes are simply Poincar é section images of trajectories which move on the torus T. Since the classes are disjoint, the trajectories they represent are all different. Since the classes cover all C - there are no other trajectories. Step 3. From each class from the family C/ℜ we take one point and put it into a set M0. Obviously, the set M0 can be seen as the set of points which initiate on T different and all possible trajectories. Let us have a closer look at it.First of all, we may check what happens when trajectories initiated by all points from M0 pass through the Poincar é section S. Let sets Mk , k=-∞, …, -1, 0, 1, …, +∞, be the images of M0 which appear in C as the trajectories make consecutive turns on T. All the sets are disjoint: ji M Mj k ≠∅=, (10) and their union covers whole C: C M kk= . (11) Since C is uncountable (continuum) and the family of sets { Mk} is only countable, the equipollent sets Mk cannot be countable. In particular, M0 is uncountable. This answers the first part of the question. Now, let us consider its second part. Since each Mk can be obtained from M0 by a rotation of the latter along C by an angle 2 π[kρ (mod 1)], all of the sets are mutually congruent. Thus, if the measure of the set M0 is µ(M0), the measure µ(Mk) of each Mk must be the same: )( )(j i M M µ= µ for all i, j (12) On the other hand, in view of Eq.11 , we have )( ) ( C Mk kµ= µ∑ =1. (13) How much is µ(M0) ? It cannot be zero, since in that case Eq.13 would give µ(C)=0, which is not true. On the other hand, it cannot be finite either, since in that case we would obtain from Eq.13 µ(C)=∞ . Thus: How much is it ? We cannot say. A number which would describe the value of µ(M0) does not exist : the set M0 is non-measurable . IV. DISCUSSION The reasoning we described in Section III can be seen as a direct translation of the Vitali construction of a non-measurable set 2,3 onto the language of the Hamiltonian mechanics. As a careful reader may have noted, step 3 of the construction we presented makes use of the Axiom ofChoice (AC) , the most controversial and at the same time, the most thoroughly studied pillar from the few ones on which the theory of sets can be built4. We write "can", since, being both relatively consisten t5 with other axioms of the set theory and independent6 of them, the axiom can be used or not. Taking the former attitude, i.e. deciding to use the Axiom, one is able not only to prove a few most useful theorems (impossible to prove without the axiom), such as that the union of countably many countable sets is countable or that every infinite set has a denumerable subset , but, and this is most disturbing, a number of theorems which seem to stay in contradiction with what our common sense tells. The most famous from such theorems, proven by Banach and Tarski7, says that it is possible to dismount a sphere into a few (at least five) such subsets, from which, using but translations and rotations i.e. transformations which certainly preserve measure, one can mount two spheres identical with the initial one . Obviously, the Banach-Tarski theorem stays in conflict with our common sense. But was it not like that already once with the theorems of the non-Euclidean geometry? The problem is not, we emphasise it, if the Banach-Tarski theorem is true or not; its proof is clean and completely legitimate within a certain mathematical environment. One should rather ask if we need this branch of mathematics to describe things which happen in the world and which we study in physics laboratories . (Non-Euclidean geometry has proven its usefulness in the description of our world at scales somewhat larger from that at which our common sense is formed.) Do we know phenomena whose description would necessarily require the use of the axiom of choice? There have been so far but a few attempts of applying the axiom of choice based mathematics to describe physical reality. The first one was the Pitowsky ’s work on a possible resolution of the Einstein-Podolsky-Rosen paradox via the Banach-Tarski one 8,9. The work by Pitowsky indicated the possibility that physical paradoxes encountered in quantum mechanics can be reduced to mathematical pathologies8. The aim of the present note is somewhat different. It shows that such a mathematical pathology (non-measurability) appears in the formal analysis of a very simple problem of classical mechanics and cannot be avoided there; one cannot answer the question concerning the number of trajectories on irrational tori without using the axiom of choice. Consequently, if one decides to answer the question one must necessarily get in touch with the paradoxical concept of sets without measure. A similar problem and a similar way of solving it is described in [10], where Svozil and Neufeld analyse the concept of linear chaos. A general, very vivid exposition of the problem of applicability of the set theory in the description of the physical world can be found in [11]. As Svozil argues there, the prohibition on the use of paradoxical results of the set theory cannot be accepted. Such a “No-Go ” attitude, as he calls it, has no justification. According to Svozil, the No-Go attitude should be rejected in favour of the “Go-Go ” attitude, according to which results of any consistent mathematical theory may be used in the description of the physical world. From the Svozil ’s point of view, the present authors took a full advantage of the Go-Go attitude: using the based on the Axiom of Choice notion of non-measurable set they answered a concrete, sensible question formulated within the frames of classical mechanics. An extensive study of the links between physics and set theory was presented also by Augenstein [12], who among other examples draws our attention to the use by El Naschie of the paradoxical decomposition technique in the analysis of the Cantorian micro space-time [13]. Concluding the present work, we admit that defining the set of points which on an irrational invariant torus initiate different trajectories we did use the Axiom of Choice. From the formal point of view the definition must be seen as a nonconstructive one. In Svozil ’s wording: throwing out the nonconstructive bath water we would throw with it the nonmeasurable baby [10]. We thinkit would be not right. Acknowledgements One of the Authors (K.W.W.) is grateful to Professor William G. Hoover and to Professor Janusz Tarski for encouragement. He is also grateful to Professor M. P. Tosi and Professor Yu Lu for hospitality at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. REFERENCES 1. See e.g.: A. J. Lichtenberg and M.A.Lieberman, Regular and Stochastic Motion , Springer- Verlag, New York, 1983; M. Berry, in Topics in Nonlinear Dynamics , S.Jorna (ed.), Vol.46., American institute of Physics, New York (1978); M. Henon, in Chaotic Behaviour of Deterministic Systems , G.Iooss et all. (eds.) , North-Holland (1983); G. M. Zaslavsky, Stochasticity in Dynamical Systems , Nauka (1984), in Russian. 2. G. Vitali, " Sul problema della misura dei gruppi di punti di una retta ", Gamberini and Parmeggiani, Bologna (1905).3. For a concise description of the Vitali construction see e.g. T. J. Jech, The Axiom of Choice , North Holland (1973). 4. For a general presentation of the role the axiom of choice plays in the set theory see: A. A. Fraenkel, Set Theory and Logic , Addison-Wesley, Reading (1965). A detailed analysis of the axiom and its significance in various branches of mathematics can be found in: T. J. Jech, The Axiom of Choice , North-Holland, Amsterdam (1965). The history of the axiom in all its various versions, the fate of people involved in bitter controversies which arose around it are vividly described in: G. H. Moors, Zermelo's Axiom of Choice; Its Origin, Development and Influence , Springer-Verlag, New York (1982). 5. Axiom A is said to be relatively consisten t with the axiomatic system Σ if , under the assumption that Σ is consistent itself, the negation of A is not provable from the axioms of Σ. 6. Axiom A is said to be independent from a set Σ of axioms, if it cannot be proven from the set. 7. S. Banach and A. Tarski, Found. Math., 6, 244 (1924), in French. For generalized versions of the Banach-Tarski paradox analyzed within the framework of the group theory see: S. Wagon, The Banach-Tarski Paradox , Cambridge Univ. Press (1985). An elementary and detailed proof can be found in: K. Stromberg, Amer. Math. Monthly 86, 151 (1979). 8. I. Pitowsky, Phys. Rev. Letters, 48, 1299 (1982); I. Pitowsky, Phys. Rev. D27, 2316 (1983). For critical comments of the Pitowsky ideas see: N. D. Mermin, Phys. Rev. Lett. 49, 1214 (1982); A. L. MacDonald, Phys. Rev. Lett. 49, 1215 (1982); and Pitovsky's response: I. Pitovsky, Phys. Rev. Lett. 49, 1216 (1982). 9. S. P. Gudder, Int. J. Theor. Phys., 24, 343 (1985) and references therein. 10. K. Svozil, Chaos, Solitons & Fractals 7, 785-793 (1996). 11. K. Svozil, Found. Phys. 25, 1541 (1995). 12. B. W. Augenstein, Chaos, Solitons and Fractals 7, 1761 (1996). 13. M. S. El Naschie, Chaos, Solitons and Fractals 5, 1503 (1995).
arXiv:physics/0103059v1 [physics.atom-ph] 20 Mar 2001Calculations of collisions between cold alkaline earth ato ms in a weak laser ﬁeld Mette Machholm Department of Computational Science, The National Univers ity of Singapore, Singapore 119260 Paul S. Julienne National Institute for Standards and Technology, 100 Burea u Drive, Stop 8423, Gaithersburg, MD 20899-8423 Kalle-Antti Suominen Department of Applied Physics, University of Turku, FIN-20 014 Turun yliopisto, Finland Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin y liopisto, Finland Ørsted Laboratory, NBIfAFG, University of Copenhagen, Uni versitetsparken 5, DK-2100 Copenhagen Ø, Denmark (July 29, 2013) We calculate the light-induced collisional loss of laser-c ooled and trapped magnesium atoms for detunings up to 50 atomic linewidths to the red of the1S0-1P1cooling transition. We evaluate loss rate coeﬃcients due to both radiative and nonradiative stat e-changing mechanisms for temperatures at and below the Doppler cooling temperature. We solve the Sc hr¨ odinger equation with a complex potential to represent spontaneous decay, but also give ana lytic models for various limits. Vibrational structure due to molecular photoassociation is present in t he trap loss spectrum. Relatively broad structure due to absorption to the Mg 21Σustate occurs for detunings larger than about 10 atomic linewidths. Much sharper structure, especially evident at low temperature, occurs even at smaller detunings due to of Mg 21Πgabsorption, which is weakly allowed due to relativistic ret ardation corrections to the forbidden dipole transition strength. W e also perform model studies for the other alkaline earth species Ca, Sr, and Ba and for Yb, and ﬁnd simil ar qualitative behavior as for Mg. 34.50.Rk, 34.10.+x, 32.80.Pj I. INTRODUCTION A. Background Laser cooling and trapping of neutral atoms has re- cently opened many new research areas in atomic physics. One can cool a gas of neutral atoms in magneto-optical traps (MOT) down to temperatures of 1 mK and be- low, and obtain densities up to 1012atoms/cm3. Evap- orative cooling methods have allowed the cooling of al- kali species to much lower temperatures below 1 µK so that Bose-Einstein condensation (BEC) occurs. Binary atomic collisions play an important role in the physics of cold trapped atomic gases, and have been widely in- vestigated [1]. One of the ﬁrst cold collisional process to be studied is the heating and loss of trapped atoms which result from tuning laser light to near-resonance with the atomic cooling transition [2]. Here we take near- resonance, or small-detuning, to mean detuning ∆ up to 50 natural linewidths to the red of atomic resonance. Studies of small-detuning trap loss, extensively re- viewed by Weiner et al. [1], have mainly concentrated on alkali atoms [3], for which it has been very diﬃcult to de- velop quantitative theoretical models to compare with ex- periment. This is because alkali atoms have extensive hy- perﬁne structure, and thus the number of collision chan- nels is simply too large for accurate theoretical modeling. It has even been diﬃcult to estimate the relative weight of the diﬀerent possible loss processes. Although one can develop simpliﬁed models, these are diﬃcult to testadequately with complex alkali systems. On the other hand, trap loss photoassociation spectra in alkali systems for large detuning can be modeled quite accurately [4–6]. This is because the underlying molecular physics of alkali dimer molecules is well-known, and the spectra are deter- mined by isolated molecular vibrational-rotational level s, for which the photoassociation line shapes can be well- characterized even in the presence of hyperﬁne structure. Quantitative analysis of such spectra have permitted the determination of scattering lengths for ground state colli - sions [1]. These scattering lengths are critical parameter s for BEC studies. Alkaline earth cooling and trapping have recently been of considerable experimental interest. Trap loss collisio ns have been studied in a Sr MOT [7], and intercombination line cooling of Sr has resulted in temperatures below 1 µK and raised the prospects of BEC of Sr [8–11]. Ca is of interest for possible applications as an optical frequency standard [12–16], and photoassociation spectroscopy in a Ca MOT has been reported [17]. Intercombination line cooling has also been reported for Yb [18,19], which we have included in our discussion because of its similarity in structure to alkaline earth atoms. Alkaline earth species provide an excellent testing ground for cold collision theories, especially given the rapidly developing experimental interest in the subject. Since the main isotopes of alkaline earth atoms have no hyperﬁne structure, the number of collision channels be- comes low enough to allow theoretical calculations even in the small detuning trap loss regime. Consequently, 1this paper presents theoretical predictions for small de- tuning trap loss spectra in cold and trapped Mg gas in the presence of near-resonant weak laser light tuned near the 1S0→1P1atomic transition, and discusses the nature of similar processes for Ca, Sr, Ba, and Yb. This work extends our previous note on Mg trap loss [20] to other species and lower temperatures, and shows the relation between small detuning trap loss and photoassociation theory. It is necessary to include spontaneous emission in modeling trap loss collision dynamics because of the long time scale of cold collisions. In addition, relativis- tic retardation eﬀects play a prominent role at small de- tunings by allowing transitions to the dipole forbidden 1Πgstate, which exhibits resolved vibrational structure, especially at very low temperature. Although we treat the long-range molecular interactions accurately, the po- tential energy curves and coupling matrix elements for the dimer molecules in the short-range region of chemical bonding are not suﬃciently well-known to determine all aspects of trap loss. Therefore, we examine the uncer- tainties associated with the unknown phases developed in the short-range region of chemical bonding, and show which features are robust with respect to such uncertain- ties and which must be measured or later determined from improved theory. B. Trap loss collisions Light-induced trap loss takes place as a molecular pro- cess. Two colliding cold atoms form a quasimolecule, and their motion can be described in terms of the elec- tronic (Born-Oppenheimer) potentials of the molecular dimer with light-induced transitions between the molec- ular states. We consider only red detuning, which excites attractive potentials at long range. Such potentials sup- port a number of bound vibrational states. Figure 1 schematically indicates the nature of the trap loss process in a weak radiation ﬁeld. An excitation laser with energy hνis tuned a few atomic linewidths below the atomic transition energy hν0. The ground \|g/an}bracketri}htand excited \|e/an}bracketri}htquasimolecular electronic states are thus cou- pled resonantly by the laser at some long-range Condon radius RC, where the photon energy matches the diﬀer- ence between the excited and ground potential energy curves. The excited state decays to a loss channel \|p/an}bracketri}ht due to interactions at short range. The fully quantum mechanical description in Ref. [21] shows that the overall probability Ppgof a trap loss collision can be factored as follows: Ppg=PpeJeg. (1) Here Jegrepresents an excitation-transfer probability which is proportional to the scattering ﬂux that reaches the short range region near Rpdue to long-range exci- tation near the Condon point followed by propagationon the excited state to the short-range region. Pperep- resents the probability of a transition from the excited state to the loss channel at short range during a single cycle of oscillation through the short-range region. Figure 1 indicates the qualitative behavior of the trans- fer function Jegversus detuning ∆. At very small detun- ing of a few atomic linewidths Jegbecomes very small if there is a high probability of spontaneous emission dur- ing transit from the outer to inner regions. At suﬃciently large detuning Jegexhibits resonance structure due to the bound vibrational levels in the excited state poten- tial, where the vibrational period is much shorter than the decay time [22]. This is the domain of photoassocia- tion to individual vibrational levels. Section III C below will show simple analytic formulas for Jegthat apply in these two limiting cases of small detuning with fast decay or isolated photoassociation lines. These formulas show howJegcan in turn be factored as Jeg=JeePeg, (2) where Pegrepresents the probability of excitation from the ground to excited state in the outer zone near the Condon point, and Jeerepresents an excited state trans- fer function between the long-range outer excitation zone and the short-range zone. The factorization in Eq. (2) is also schematically indicated in Fig. 1. PpePegJeeLoss Transfer Excitation 0Rhν Jeg\|e〉 \|g〉\|p〉 RCRp FIG. 1. Schematic representation of trap loss collisions. Each of the factors in Eqs. (1) and (2) can be af- fected by unknown phases associated with the short- range molecular physics of the dimer molecules. (1) Peg is sensitive to the asymptotic phase of the ground state wave function. (2) Vibrational resonance structure in Jegis sensitive to the position and widths of vibrational features. (3) Ppeis sensitive to St¨ uckelberg oscillations in short-range curve crossing probabilities. These eﬀects are discussed in detail in Secs. II D and II E. The overall eﬀect of such sensitivities will depend on the temperature and the alkaline earth species. There are two possible inner zone loss processes char- acterized by Ppe: the state change (SC) and radiative es- cape (RE) mechanisms, both represented schematically by the loss channel pin Fig. 1. In the SC process the 2excited state couples to another molecular state near a short-range crossing point Rp, and population trans- fer between them is possible. The products of the col- lision emerge on a state that correlates asymptotically with other atomic states of lower energy, such as3P+1S, thereby releasing a large amount of kinetic energy to the separating atoms. In RE the excited state can decay by spontaneous emission after the atoms have been acceler- ated towards each other on the excited state potential. The ground state atoms then separate with this gain in kinetic energy. If enough kinetic energy has been gained to exceed the trap depth, this is a loss process (early de- cay after insuﬃcient acceleration only leads to radiative heating). Here Rprepresents the distance at which the atoms have received suﬃcient acceleration to be lost from the trap. Any emission with R < R pleads to trap loss. In this paper we use the fully quantum complex po- tential method of Ref. [21] and do not resort to semiclas- sical methods (with one exception), although semiclas- sical concepts are often useful for interpretation. Our quantum methods are fully capable of describing the s-wave limit and the quantum threshold properties of the extremely low collision energies near or below the critical temperature for BEC. We describe the sponta- neous emission processes with a complex potential, and solve the corresponding time-independent multichannel Schr¨ odinger equation in the molecular electronic state basis. This takes any vibrational state structure into ac- count automatically without the need to calculate wave functions or Franck-Condon factors, but limits our study to weak laser ﬁelds only, where only a single photon is ex- changed with the ﬁeld. Typical cooling lasers are strong and detuned only a few linewidths. By alternating with a weaker probe laser one can access the particular range of detuning and intensity that we study. Future studies are needed to address the eﬀects of strong laser ﬁelds and the consequent revision of our results due to saturation and power broadening [23,24]. We also include rotational states into our model, which for the molecular ground state correspond to the partial waves of a standard scattering problem (angular momen- tum quantum number l). The symmetry of spinless al- kaline earth dimers permits only even partial waves. It should be pointed out that for near-resonant light the Condon point is at relatively large distances. This means that although the collision energy is low, one needs to go to relatively large lbefore the ground state centrifugal barrier stops the quasimolecule from reaching RC. We correct here some mistakes that we discovered in the sum over partial waves in Ref. [20]. The probe laser can be tuned over a wide range from a few to many atomic linewidths. For suﬃciently large detuning the rotational structure becomes sharp, and it should be possible to resolve the vibrational and rota- tional states. However, even at 27 GHz detuning, rota- tional features in the Ca 21Σg→1Σuphotoassociation spectrum are only partially resolved [17]. Photoassocia- tion studies can yield precise information on the molecu-lar potentials. Also, the photoassociation line shapes are sensitive to the near-threshold ground state wave func- tion, especially if it has nodes in the region swept by the detuning-dependent Condon point. Analysis of such data would hopefully give a value for the s-wave scat- tering length for Mg or other alkaline earth species, and consequently determine whether a stable Bose-Einstein condensate is possible. In this paper we present the calculated estimates for trap loss rate coeﬃcients in Mg at temperatures around and below the Doppler temperature. The contributions from diﬀerent mechanisms and states are identiﬁed and compared. We also calculate predictions for other al- kaline earth atoms and Yb by combining the appropri- ate atomic properties with model molecular potentials. Sec. II presents in detail the atomic data, molecular po- tentials, and laser couplings used for Mg and other alka- line earth atoms. Sec. III describes our theoretical ap- proach. The results for Mg are given in Sec. IV and for the other atoms (Sr, Ca, Ba, Yb) in Sec. V. Finally we present some conclusions in Sec. VI. II. THE MOLECULAR PHYSICS OF ALKALINE EARTH DIMER MOLECULES A. Alkaline earth atomic structure Table I gives the basic atomic data for the alkaline earth atoms and Yb, which has an electronic structure similar to the Group II elements. The major isotopes have no hyperﬁne structure, except for Be. Alkaline- earth atoms have a1S0ground state, and excited1P1 and3P1states that are optically connected to the ground state, the latter weakly. Figure 2 sketches the energy lev- els of the Group II atoms. The1,3P ﬁrst excited states are the most important for laser cooling. The1,3D states shift downwards as the atomic number increases. For Mg the1,3D states are above the1,3P states. For Ca and Sr they are between the3P and the1P states, and for Ba the1,3D states are below the1,3P states. In laser cooling one uses typically the1S0-1P1transition, which is the situation studied in this paper. For Mg this re- quires a UV laser source, and for the heavier elements requires repumping to recycle atoms that decay to lower levels. The weak1S0-3P1intercombination transition has a very narrow linewidth, and is within the optical range. Thus alkaline earth atoms are good candidates for opti- cal atomic clocks when cooled to low temperatures. For clock applications we need to understand their laser cool- ing properties, including the magnitude and nature of laser-induced collisional trap loss. Beryllium is not likely to be a serious candidate for laser cooling. It is toxic, has a very short wavelength cool- ing transition, and the intercombination “clock” transi- tion is so weak as to be eﬀectively forbidden. Therefore, we will not consider Be in the rest of this paper. 3TABLE I. Atomic data for major isotopes of Group II elements a nd Yb without hyperﬁne structure (natural abundance shown). Most data is derived from [25]. The lifetime τfor1P1is taken to be the inverse of the1P1-1S0spontaneous emission rate Γ at/¯h, thus neglecting weak transitions to other states for Ba and Yb. The3P lifetimes are from several sources [26]. The linewidth in frequency units is Γ at/h= (2πτ)−1. The wavelengths λand ﬁne structure splittings for Sr and Ba are taken from [27]. The Doppler temperature is deﬁned as TD= Γ at/(2kB). We take the recoil temperature to be TR= (¯h2)/(mλ2kB), where λ=λ/2πandmis the atomic mass. The dipole moment is d0=/radicalbig 3Γatλ3/4 (in a.u.). The atomic units for dipole moment, length, and energy are ea0= 8.4783×10−30Cm,a0= 0.0529177 nm, and e2/(4πǫ0a0) = 4.3597482 ×10−18J respectively. Be Mg Ca Sr Ba Yb Major isotopes without hyperﬁne structure9Be(100%)24Mg(78.99%)40Ca(96.94%)88Sr(82.58%)138Ba(71.70%)174Yb(31.8%) (abundance)26Mg(11.01%)86Sr(9.86%)172Yb(21.9%) τ 1P1(ns) 1.80 2.02 4.59 4.98 8.40 5.68 3P1(ms) 5.1 0.48 0.021 0.0014 0.00088 Γat/h 1P1(MHz) 88.5 78.8 34.7 32.0 18.9 28.0 3P1(kHz) 0.031 0.33 7.5 120 181 Doppler-cooling limit 1P1(mK) 2.1 1.9 0.83 0.77 0.45 0.67 3P1(nK) 0.75 8.0 179 2.8 1034.4 103 Recoil limit 1P1(µK) 39 9.8 2.7 1.0 0.45 0.69 3P1(µK) 3.8 1.1 0.46 0.22 0.36 d0(a.u.) 1S0−1P1 1.89 2.38 2.85 3.11 3.16 2.35 λ(nm) 1S0−1P1 234.861 285.21261 422.6728 460.733 553.548 398.799 λ=λ/(2π) (a0) 1S0−1P1 706.4 857.8 1271.2 1385.7 1664.9 1199.4 FS splitting 3P2−3P0(a.u.) 1.36 10−52.77 10−47.20 10−42.65 10−35.69 10−31.10 10−2 Because of the lack of hyperﬁne structure the ba- sic laser cooling mechanism for alkaline earth atoms is Doppler cooling, for which the temperature limit TD, de- ﬁned in the caption of Table I, is set by the linewidth Γatof the cooling transition (widths in this paper are ex- pressed in energy units, so that the decay rate is Γ at/¯h). The lifetime of the alkaline earth1P1state is between 1.8 and 8.4 ns, giving a Doppler-cooling limit between 2.1 and 0.45 mK for the elements in Table I. On the other hand the3P1state has a long lifetime with Doppler- cooling limits in the nK range. This can be compared to the photon recoil limit TR, deﬁned in the Table caption; Table I shows that TRis between 0.2 and 4 µK. Thus the recoil limit is above the intercombination line Doppler cooling limit for Mg and Ca, nearly coincident with it for Sr, and below it for Ba and Yb. Although Sisyphus cooling and magnetic trapping is not available for1S0atoms as it is for alkali atoms, in- tercombination line cooling is possible for some Group II species. This has been used to cool Sr to ∼400 nK with relatively high phase space density >0.1 [8,9]. If com- bined with far oﬀ-resonant optical traps and evaporative cooling it may become possible to obtain Bose-Einstein condensation with optical methods alone.Be Mg Ca Sr Ba Yb1S01S01S01S01S01S03P1P 3P 3P 3P3P3P1P 1P1P 1P1P1S 3S 1S 3S1S 3S 1D 3D1D 3D 1D 3D0.10.2 0Energy [a.u.] FIG. 2. Energy levels of Group II atoms and Yb. 4B. Alkaline earth dimer molecular structure Figure 3 shows the lowest electronic potentials for Mg2[28]. There are only two states with attractive long- range potentials correlating with1P1+1S0that can be resonantly coupled to the ground state by laser light, namely1Σ+ uand1Πg. Both states oﬀer the possibil- ity for the SC and RE mechanisms. When comparing Figs. 2 and 3 one can see that Mg is special. For other alkaline earth atoms the molecular state picture is fur- ther complicated by the atomic D states below the1P1 state. This increases the number of molecular states and thus the number of energetically available exit channels. The small number of molecular states is one reason we have chosen Mg as the basis for our studies. Theoretical calculations require precise information about the molec- ular potentials and couplings over a wide range of inter- atomic distance. Ab initio calculations of ground and excited molecular potentials are also available for Sr [29] and Ba [30]. While working on the manuscript we received new ab initio data on Mg 2from E. Czuchaj, including both im- proved potential curves and spin-orbit couplings [31]. Al- though the new data diﬀer to some extent from the re- sults of Stevens and Krauss [28] because of improved cal- culations of electron correlation eﬀects, we do not expect that use of the new data would lead to any strong mod- iﬁcation of our basic results, which should be viewed as qualitative model calculations for the reasons to be dis- cussed in the following sections. 0.080.10.120.140.160.18 4 6 8 10 12 14V(R) [a.u.] R / a01S+1P 1S+3P1Πu 21Σg 1Σu 1Πg 3Πu 3Σg 3Πg3Σu FIG. 3. The molecular states of Mg 2in atomic units cor- responding to the asymptotic atomic states1S0+1P1and 1S0+3P0,1,2[28]; the zero of energy is at the ground state 1S0+1S0asymptote. There are four states correlating with each asymptote, of which two are attractive and two are re- pulsive at large R, where the system is expected to be resonant with the laser ﬁeld. The linewidth of the excited1Σustate depends on theinteratomic distance Rwith a magnitude on the order of twice the atomic linewidth. Thus the vibrational levels of the1Σustate near the1S+1P dissociation limit overlap strongly, and one does not expect to resolve them. One interesting point is that the dipole-forbidden1Πg−1Σg transition becomes allowed at large Rdue to retardation corrections. This means that the1Πgstate can be ex- cited at RC, but the spontaneous emission probability goes down quickly as Rdecreases. Consequently,1Πg vibrational states near the dissociation limit have nar- row emission linewidths. Thus the assumption that the vibrational states overlap strongly and can not be re- solved at detunings of a few linewidths is not necessarily valid. One must determine if resolvable features persist when one sums over all involved rotational states/partial waves, and takes the energy average over a thermal dis- tribution. We show that one can indeed see vibrational structure, especially if the temperature is well below the 1S0-1P1Doppler limit. As mentioned above, this is by no means impossible, if one does the cooling using the 1S0-3P1transition. C. Long-range properties of states correlating with 1P1+1S0atoms There are four molecular states correlating with 1P1+1S0atoms: two long-range attractive states,1Σ+ u and1Πg, and two long-range repulsive states,1Σ+ gand 1Πu. Here long-range means that Ris large compared to the short-range region of chemical bonding and van der Waals interactions shown in Fig. 3 so that the potential is determined by the ﬁrst-order dipole-dipole-interactio n withC3=∓2d2 0and∓d2 0for the1Σ and1Π states, re- spectively. Here d0is the z-component of the atomic transition dipole matrix element, which is related to the atomic linewidth Γ atand the reduced wavelength of the 1S0−1P1transition ( λ=λ/2π) byd2 0= 3λ3Γat/4. The exact long-range (lr) potentials including relativistic r e- tardation corrections are [32] Vlr(u;1Σ+ u) =−3Γat 2u3[cos(u) +usin(u)], (3) Vlr(u;1Πg) =−3Γat 4u3[cos(u) +usin(u)−u2cos(u)], where u=R/λis the scaled distance. The molec- ular linewidths with relativistic retardation correction s are [32] Γ(u;1Σ+ u) = Γ at/braceleftbigg 1−3 u3[ucos(u)−sin(u)]/bracerightbigg , (4) Γ(u;1Πg) = Γ at/braceleftbigg 1−3 2u3/bracketleftbig ucos(u)−(1−u2)sin(u)/bracketrightbig/bracerightbigg . In the region with R <λ, the potentials vary as 1 /u3, and Γ(1Σ+ u) and Γ(1Πg) respectively vary as 2Γ atand Γatu2/5. 5The very long-range excited state potentials result in large Condon points for excitation of the attractive states. If the laser detuning relative to the atomic tran- sition is expressed in units of Γ atand the distance in units ofλ, the scaled Condon point ( uC) for the ground-excited state transition becomes independent of atomic species. Table II shows uCfor several detunings ∆, where we de- ﬁne red detuning to be positive. For Mg at ∆=Γ atwe haveRC(1Σ+ u)=1132 a 0andRC(1Πg)=728 a 0. TABLE II. Condon points in scaled distance for selected detunings Detuning ∆ uC(1Σ+ u) uC(1Πg) Γat 1.32 0.849 5Γat 0.716 0.511 10Γat 0.555 0.411 30Γat 0.376 0.288 The attractive potentials support a series of vibrational levels leading up to the dissociation limit. Assuming a potential with a long-range form −C3/R3gives the bind- ing energy of vibrational level v[33]: ε1/3 v=/parenleftbiggπ 2a3/parenrightbigg2¯h2 2µC2/3 3(v−vD)2, (5) where a3= (√π/2)Γ(5/6)/Γ(4/3) = 1 .120251, vDis the vibrational quantum number at the dissociation limit ( vD is generally nonintegral) and µis the reduced mass. The vibrational spacing function, which we need later, is ∂εv ∂v=hνv=3π a3/parenleftBigg ¯h2 2µC2/3 3/parenrightBigg1/2 ε5/6 v, (6) where hνvis the vibrational frequency. D. Ground state The ground-state van der Waals potential varies at long-range as R−6and is essentially ﬂat for the range of Condon points we consider. The energy-normalized ground state scattering wave function for collisional mo- mentum ¯ hk∞and partial wave lhas the long-range form Ψ(R, l, k ∞) =/parenleftbigg2µ π¯h2k∞/parenrightbigg1 2 sin/bracketleftBig k∞R−π 2l+ηl(k∞)/bracketrightBig . (7) We deﬁne the collisional energy as ε= ¯h2k2 ∞/(2µ). The short-range potentials are not suﬃciently well-known for alkaline earth dimers to determine accurately the scatter- ing phase shifts ηl(k∞). Therefore, our calculations will have to be model calculations. However, we test the sen- sitivity of our trap loss spectra to the unknown phases,and show that this is not a serious limitation. There are two reasons for this. One is that the ground state poten- tial is ﬂat in the long-range region, and the amplitude of Ψ(R, l, k ∞) has its asymptotic value in Eq. (7) indepen- dent of Ras long as R > x 0=1 2(2µC6/¯h2)1/4[34,35]; the use of x0(or a closely related length) as an appropri- ate length scale for van der Waals potentials is described in the Appendix of Ref. [6]. The condition RC≫x0is easily satisﬁed in our case. Second, at the Doppler limit TDfor1S0→1P1cooling, a number of partial waves l contribute to trap loss in our detuning range (1-50 Γ at). We demonstrate in Sec. IVA that a sum over lremoves the dependence on the short-range potential. Our trap loss spectra for s-wave scattering at the low temperatures available via intercombination line cooling will be sensitive to the actual scattering length A0of the ground1Σ+ gpotential. However, we demonstrate a sim- ple scaling relationship that will allow our low T s-wave results to be scaled to any value of the scattering length. We may expect that the1Σ+ gscattering length can be determined from one or two-color photoassociation spec- tra, as has been done for alkali species [1]. However, such analysis will require an accurate C6coeﬃcient and for optimum results needs a reasonably accurate short-range potential as well. E. Excited state short-range potentials Trap loss spectra depend strongly on the excited state short-range potential structure in three ways: (1) The curve crossings leading to SC trap loss occur at short range, and determine Ppe. In a Landau-Zener interpretation, Ppe(J′) = 4e−2πΛ/parenleftbig 1−e−2πΛ/parenrightbig sin2(βJ′), (8) where Λ = \|Vpe(Rp)\|2/(¯hvpDp) and J′designates excited state total angular momentum (See Section II F). Here Vpe(Rp),vp,Dpare the respective coupling matrix ele- ment, speed, and slope diﬀerence at the Rpcrossing, and βJ′is a semiclassical phase angle [36]: βJ′=/integraldisplayRp R0eke(R, J′)dR−/integraldisplayRp R0pkp(R, J′)dR+π 4,(9) where R0iand ¯hki(R, J′) are the respective inner clas- sical turning point at zero energy and local momentum for state i=eorp. The attractive singlet potentials may have one or more crossings with repulsive potentials from lower-lying states, e.g., those correlating to3P+1S. Thus the number, positions of crossings and the coupling between states with crossing potentials are important for the magnitude of Ppe. Although we can make reasonable estimates for Λ, the phase angle βJ′is sensitive to details of the potentials and can only be calculated accurately if very accurate potentials are available [37]. 6(2) The vibrational structure in the trap loss spectra depends on both short and long-range potentials. The spacing between the vibrational levels given by Eq. (6) depends only on the long-range potentials, but the exact positions of the levels in Eq. (5) are determined by the short-range potentials in the region of chemical bonding (through the vDparameter). Thus, the magnitude of the vibrational spacings in our model calculations will be cor- rect, whereas the actual positions can only be determined by measurement. (3) The short-range SC process also contributes to the width of vibrational features in the trap loss spectra [38]. Depending on species, temperature, and detuning, the widths may be primarily determined by natural or ther- mal broadening or by the predissociation decay rate re- lated to Ppe. However, we are able to place approximate bounds on the magnitude of Ppe. Sections III C, IV, and V discuss the contributions to feature widths and show that natural and thermal broadening tends to be domi- nant at small detuning, whereas SC broadening may be- come dominant at large detuning. The available data on short-range potentials varies among the Group II elements. Ab initio potentials are available for Mg [28,31]. The structure of the molecular potentials is fairly simple because only states correlat- ing to1P+1S (4 states) and3P+1S atoms (4 states) are present (see Fig. 3) and the potentials provide quali- tative data for possible SC mechanisms. On the other hand, Ca, Sr and Ba have a very complex short-range structure, because of large ﬁne structure splittings in the triplet states, and the states correlating to1D+1S and 3D+1S coming into play. For example, Boutassetta et al.[29] and Allouche et al.[30] calculated the short-range potentials for Sr and Ba respectively, but the large num- ber of states (e.g., 38 states for Ba) makes it excessively complicated to treat the short-range SC mechanism even qualitatively. The coupling between the singlet and triplet states at short-range is due to spin-orbit couplings. The exact magnitude of the couplings is unknown for all Group II elements, but can be estimated using Table A1 of Ref. [39], which relates the coupling matrix elements to the3P2−3P0ﬁne structure splitting. This approxima- tion ignores any R-dependence of the spin-orbit matrix elements due to chemical bonding. The ﬁne structure splitting of the3P state increases with atomic number (see Table I). We have opted for Mg as a model system, because short-range potentials are available. Each singlet state with attractive potential couples to only one triplet state in the inner zone. Thus the SC trap loss problem for Mg decouples into two three-state calculations, one for the1Σ+ uand one for the1Πgexcited state. For Mg we provide qualitatively correct SC trap loss spectra and even give some quantitative estimates. In the case of the other Group II elements and Yb, we treat their complicated inner zone physics as one ef- fective crossing, that is, we use three-state calculationsbased on the Mg-model, and explore the eﬀects of mass, radiative properties, and coupling strength (size of spin- orbit splitting) in these model calculations. Thus we do not use the ab initio potentials of Refs. [29,30] discussed in the paragraph above, because even if we were to in- clude all the curves, there would still be uncertainties associated with unknown phases and unknown coupling matrix elements. Rather, our goal is to indicate qualita- tive diﬀerences in magnitude and spectral shapes among the various species. We trust that these will be helpful in providing guidance for future experimental and the- oretical studies of these systems. As we discuss in Sec- tion III B, our calculation of the excitation-transfer coef - ﬁcient κwill be to a large extent independent of details of the complicated short-range molecular physics and curve crossings. F. Molecular rotational structure and coupling to the laser ﬁeld The full three-dimensional treatment of the collision of a1S atom coupled to a1P atom by a light ﬁeld is worked out in Ref. [40]. We adapt this treatment to our simpliﬁed model with molecular Hund’s case (a) transi- tions between two molecular states with the usual rota- tional branch structure. The angular momentum J′′in the ground state can only be that of molecular axis rota- tion,J′′=l′′with projection m′′on a space-ﬁxed axis. The excited rotational levels in a Hund’s case (a) molec- ular basis for a1Σ+ uor1Πgstate do not have mechanical rotation l′as a good quantum number but instead the total angular momentum J′with space projection M′. The three possible transition branches have J′=l′′+B, where the branch labels P, Q, and R respectively desig- nate the cases B= -1, 0, and 1. The quasi-molecule ground state can couple to the1Σ+ ustate only by P and R branches, but to the1Πgstate by P, Q, and R branches. All potentials have a centrifugal term added, n(n+ 1)/(2µR2), depending on ground state n=l′′or excited states n=J′=l′′+B. The radiative coupling terms are VA,B,l′′,m′′,q(u) =/parenleftbigg2πI c/parenrightbigg1/2 /an}bracketle{tA, J′M′\|ˆeq·/vectord\|l′′m′′/an}bracketri}ht, (10) where A=1Σ+ uor1Πglabels the excited state, Iis light intensity, /vectordis the lab frame dipole operator, ˆ eqis the polarization vector of light with polarization q= 0,±1, andM′,m′′are lab frame angular momentum projection quantum numbers. In our weak-ﬁeld case, where transi- tion probabilities are linear in \|VA,B,l′′,m′′(u),q\|2, we can deﬁne an m′′-averaged radiative coupling matrix element (the sum over m′′removes the dependence on q): Veg,A(u, l′′, B, I) 7= 1 2l′′+ 1l′′/summationdisplay m′′=−l′′\|VA,B,l′′,m′′,q(u)\|2 1/2 =/parenleftbigg2πI c/parenrightbigg1/2 αA,B,l′′dA(u) = 2.669×10−9αA,B,l′′/radicalBig I(W/cm2)dA(u)(a.u.). (11) The molecular electronic transition dipole moment dAis dA(u)(a.u.) =/radicalbigg 3λ(a.u.)3 4ΓA(u)(a.u.). (12) where Γ A(u) is the molecular linewidth of the excited state as in Eq. (4). The factors√ 2l′′+ 1αA,B,l′′are given in Table III. TABLE III. The rotational line strength factors√ 2l′′+ 1αA,B,l′′. State A Branch B l′′= 0 (s-wave) l′′/negationslash= 0 Σ P 0/radicalbig l′′/3 Σ R/radicalbig 2/3/radicalbig (l′′+ 1)/3 Π P 0/radicalbig (l′′−1)/3 Π Q 0/radicalbig (2l′′+ 1)/3 Π R 2 /√ 3/radicalbig (l′′+ 2)/3 G. Model potentials for Mg Our results are not sensitive to the detailed form of the ground state potential, for the reasons given in Sec. II D. Thus, we model the ground state potential by a Lennard- Jones 6-12 form Vg(u) = 4ǫ/bracketleftbigg/parenleftBigσ λu/parenrightBig12 −/parenleftBigσ λu/parenrightBig6/bracketrightbigg . (13) For Mg we model the ground state potential from [28] with a well depth of ǫ= 0.002825 a.u. and an inner turn- ing point of σ= 6.23 a0. This potential has a scat- tering length of −95 a0for the24Mg reduced mass of 23.985042/2 atomic mass units. The scattering length A0is determined from the k→0 behavior of the s-wave phase shift: η0=−k∞A0. Since the excited ab initio potentials of Ref. [28] do not permit a quantitative calculation of spectroscopic accu- racy, for the reasons given in Sec. II E, the speciﬁc forms of the short-range (sr) excited-state potentials are not important for our purposes of modeling the qualitative structure and magnitude of the collisional loss. How- ever, it is important to retain the correct long-range form. Therefore for simplicity in the calculations and because we want to model the other alkaline earth systems to ex- plore the eﬀect of diﬀerent C3, Γat,λand mass, we havemodeled the ab initio potentials with Lennard-Jones 3-6 potentials keeping the long-range form ﬁxed to its known C3value given in Eq. (3) Vsr(u) = 4ǫ/bracketleftbigg/parenleftBigσ λu/parenrightBig6 −/parenleftBigσ λu/parenrightBig3/bracketrightbigg . (14) We have two ﬁtting parameters ǫandσ, and three given values: well depth of the ab initio potential (the depth of the model potential is ǫ), position of minimum uminandC3(1Σ+ u) =−2d2 0orC3(1Πg) =−d2 0. Because we want to ﬁx the long-range potential C3we can not ﬁtuminand the well depth at the same time and have chosen the latter: Ve(u;1Σ+ u) = 4ǫ(Σ)/parenleftbiggσ(Σ) λu/parenrightbigg6 (15) −2d2 0 λ3u3[cos(u) +usin(u)], Ve(u;1Πg) = 4ǫ(Π)/parenleftbiggσ(Π) λu/parenrightbigg6 (16) −d2 0 λ3u3[cos(u) +usin(u)−u2cos(u)], where ǫ(Σ) = 0 .0347 a.u., σ(Σ) = 4 .339 a 0, ǫ(Π) = 0 .0681 a.u., σ(Π) = 2 .751 a 0. The well minima are λumin= 5.5 a0and 3.5 a 0for1Σ+ u and1Πg, respectively, compared to the ab initio values λumin= 6.1 a0and 5.4 a 0. Both excited states have a SC mechanism in the short- range region with coupling to a triplet state. The triplet states are purely repulsive, and modeled with Vp,A(u) =C6 λ6u6+V∞, (17) where A=1Σuor1Πglabels the molecular state, and V∞=−0.0601 a.u. The SC from1Σ+ uto3Πutakes place around the inner turning point of the1Σ+ upotential well. We have chosen C6= 392 a.u. for the model of the3Πustate. The SC from1Πgto3Σgtakes place about 1.5 a 0out- side and 0.019 a.u. above the minimum of the1Πgstate potential well. With C6= 81 a.u. we have a model of the crossing where the corresponding values are: 1.0 a 0and 0.019 a.u. The diﬀerence in slope of the crossing poten- tials is 0.030 a.u. for the ab initio potentials and 0.037 a.u. for the model. The coupling between the crossing states are approx- imated using Table A1 of Ref. [39]. The1Σ+ u-3Πuand 1Πg-3Σ+ gmatrix elements are ζ/√ 2 and ζ/2, respectively, where ζ= 1.84×10−4a.u. is 2/3 of the atomic3P2- 3P0splitting. For example, we estimate an upper bound (sin2βJ′= 1) to the Landau-Zener version of Ppein Eq. (8) for the1Πg-3Σgcrossing to be 2.6 ×10−3for the model potentials and 2.8 ×10−3estimated from the ab initio potentials. This upper bound is consistent with 8the calculated Ppeas a function of J′from our complex potential calculation described below. Since we will use a semiclassical method to determine thePpefactor for the RE process via the1Σ+ ustate (see Sec. III D below), we do not need an explicit probe chan- nel for RE. However, we introduce a probe channel to simulate RE in order to show that the same Jegfactor in Eq. (1) applies for both SC and RE processes, irre- spective of the choice of the short-range Rp. Since we may take any form we like for a RE probe state, we use a probe state potential which crosses the excited state at a distance up, where the kinetic energy gained by the collision pair is 1 K, corresponding to a trap depth of 0.5 K. The RE probe potential has a repulsive inner wall VRE,probe(u) =C12 (λu)12−Vkin,RE,∞, (18) where Vkin,RE,∞= 3.17×10−6a.u. and C12= 5×106a.u. No rotational term is included in this probe channel. The same probe state potential is used for all collision ener- gies, which are small (mK range and below) compared to the 1 K kinetic energy at up. The coupling between the excited state and the probe state is chosen to be weak: 10−9a.u. H. Model potentials for Ca, Sr, Ba and Yb Since the diﬀerent ground state values of C6and inner potential shape make no diﬀerence for these model stud- ies, for the reasons given in Sec. II G, we take the same ground state Lennard-Jones 6-12 potential, Eq. (13), as in the Mg case to model the other alkaline earth ground states, and only change the reduced mass. This proce- dure yields respective model scattering lengths of 67 a 0, -65 a 0, -41 a 0, and 97 a 0for40Ca,88Sr,138Ba, and174Yb. Thus, \|A0\| ≪RCin all model cases. The long-range of the excited state potentials for Ca, Sr, Ba and Yb is still exact, using the data from Table I with the form in Eq. (3). Due to lack of accurate ex- cited state short-range molecular potentials and because of their more complicated structure, we model the trap loss for Ca, Sr, Ba and Yb by scaling the potentials from the Mg model Eqs. (15) and (16). The well depth ǫof the1Σ+ uand1Πgpotentials are scaled by the size of the singlet-triplet states splitting compared to that splitti ng in Mg, e.g.: ǫCa=ǫMgE(1P,Ca)−E(3P,Ca) E(1P,Mg)−E(3P,Mg). (19) The short-range structure is treated as one eﬀective crossing. The probe states are qualitatively like those in the Mg model. The position (in energy) of the crossing between the1Πgand probe state potentials is scaled as above. The1Σ+ uand probe state potentials come very close at the inner wall of the1Σ+ upotential around the classical turning point Ve(u) =ε.The spin-orbit coupling constant ζscales with the spin- orbit splitting of the3P atomic states. We use the same deﬁnition of the matrix elements as for Mg. The Landau- Zener adiabaticity parameter 2 πΛ in Eq. (8) for Ba and Yb is larger than unity, leading to a modiﬁed shape of the short-range adiabatic potentials and very small Ppe,SC. Thus for Ba the1Σ+ u-probe state coupling and for Yb the1Πgand the1Σ+ u-probe state couplings have been reduced by about a factor 5 to obtain values of Ppe,SC close to unity, in order to test the limit of very strong broadening of the vibrational structure. We believe this limit is physically more realistic. The variety of crossing s in these systems might lead to a strong SC process. III. COMPLEX POTENTIAL CLOSE COUPLED CALCULATIONS A. Description of method The weak ﬁeld approximation assumed in this study allows us to apply a complex potential method [21,41], since re-excitation of any decayed quasimolecular popu- lation can be ignored. Furthermore, the weak ﬁeld only couples each ground-state partial wave to at most 3 rota- tional states of the1Σuor1Πgstate through the P, Q, or R branches. However, in the weak ﬁeld the excited-state rotational states do not couple further to other ground- state partial waves. Therefore we can ignore any partial wave ladder climbing. Thus, for each trap loss mecha- nism we have three dressed states: a ground state g, an excited state e, and a probe state, p. We solve the three- channel, time-independent radial Schr¨ odinger equation for ground state collision energy ε, partial wave l=l′′, for each transition branch Band for a given intensity I d2 dR2φ(ε, R) +2µ ¯h2[ε1−V(R, l, B, ∆, I)]φ(ε, R) =0, (20) Vis the 3 ×3 potential matrix V(u, l, B, ∆, I) = (21) ∆ +Ve(u, l, B )−iΓ(u) 2Veg(u, l, B, I )Vpe(u) Veg(u, l, B, I ) Vg(u, l) 0 Vpe(u) 0 Vp(u, l, B ) . The elements of Vare described in Sec. II above. A complex term −iΓ(u)/2 is added to the excited-state po- tential to simulate the eﬀect of excited-state decay. The full retarded form of the molecular linewidth, Eq. (4), is used. Application of standard asymptotic scattering bound- ary conditions to the three-component state vector φ gives the S-matrix elements Sij(ε, l, B, ∆, I). Ifε >∆, all three channels are open: i, j=g,p, ore. When ε <∆, as is normally the case in our model, channel e 9is closed, and Sijis only deﬁned for i, j=gorp. We choose the light intensity Ilow enough that the results are in the weak ﬁeld limit where the Ppg=\|Spg\|2matrix element scales linearly in I. Our results are normalized to a standard intensity of I= 1 mW /cm2. We ﬁnd that we can make a change in the asymptotic shape of the artiﬁcial probe potential to make the model much more manageable computationally. The deep po- tential well of the excited state and the large kinetic en- ergy in the probe channel require a small stepsize in u (λ∆u≈0.005 a 0). However, with the large range of u (λumax≈1500-3000 a 0) a small ∆ uincreases the com- putation time and may compromise the numerical sta- bility. Therefore, we modify the probe state potential to bring Vp(u) to a small negative value at intermediate and asymptotic u. This results in a small asymptotic momen- tum ¯hkin the probe channel, and allows us to gradually increase the stepsize to λ∆u≈0.5 a0asuincreases. The coupling between the excited and probe states is turned oﬀ exponentially before the change in Vp(u). The prob- abilities \|Spg\|2andPpeare completely independent of the asymptotic properties of the probe potential if there are no asymptotic barriers. Since we have no centrifu- gal potential in the asymptotic probe channel, there are no asymptotic centrifugal barriers. The presence of such barriers in our previously published model [20] resulted in some errors at larger l′′which we have now corrected in the present model. The thermally averaged loss rate coeﬃcient via state e is: K(∆, T) =kBT hQT/integraldisplay∞ 0dε kBTe−ε/kBT(22) ×/summationdisplay l′′even,B(2l′′+ 1)\|Spg(ε, l′′, B,∆, I)\|2, where QT= (2πµkBT/h2)3/2is the translational parti- tion function. Identical particle exchange symmetry en- sures that only even partial waves exist for the ground state. We also deﬁne a non-averaged rate coeﬃcient for a ﬁxed collision energy ε, where we deﬁne Tε≡ε/kB. Only the sum over partial waves and branches is performed: K(∆, ε) =kBTε hQTε/summationdisplay l′′even,B(2l′′+ 1)\|Spg(ε, l′′, B,∆, I)\|2. (23) There are two possible cutoﬀs l′′ maxto the partial wave sum provided by the ground and excited-state centrifu- gal potentials, respectively. For the ground state we can takel′′ maxto be the largest integer for which ¯ h2l′′ max(l′′ max+ 1)/2µR2 C< εat the Condon point RC. Thus, the Con- don point is classically accessible for l′′≤l′′ maxand clas- sically forbidden for l′′> l′′ max. For the excited state, the centrifugal potential may create a barrier inside the Condon point for the g→eexcitation. The position and the height of the barrier depend on J′. For collision en- ergies around ε=kBTDthis barrier may prevent allowedground-state partial waves from contributing to the loss, because the excited-state population never reaches the in- ner zone where RE decay and SC take place. In this case, l′′ maxmay decrease from the value deﬁned by the above inequality. We ﬁnd that the ground state cutoﬀ applies except for the case of high energy and small detuning. In either case \|Spg\|2decreases many orders of magnitude asl′′varies from l′′ maxover the next few l′′-values. The upper limit for the sum in Eqs. (22) and (23) is set to the l′′-value where (2 l′′+1)\|Spg(ε, l′′, B,∆, I)\|2is 10−6of the maximum previous (2 l′′+ 1)\|Spg(ε, l′′, B,∆, I)\|2value. B. Factorization of trap loss probability The factorization in Eq. (1) allows us to separate the physics of the long-range excitation and the short-range decay to the trap loss channel. Reference [21] shows how to determine the short-range probability Ppefrom a dif- ferent coupled channels calculation where the complex decay term −iΓ/2 in Eq. (21) is omitted. Since Ppeis determined in a region near Rpwhere the local kinetic energy is very high in relation to ε, this probability is nearly independent of εover a wide range. Therefore, we calculate [21] Ppe(J′) =\|Spe(ε >∆, l′′, B= 0,∆, I= 0)\|2.(24) Hereεis taken above the threshold energy ∆ where the echannel becomes open and Speis deﬁned. Our numerical calculations show, as expected, that Ppe is independent of εover a wide range, typically of ε/kB from 0.3 mK to 300 mK at low J′and 3 mK to 300 mK at high J′, and also independent of ∆ in our small range of detuning. The Landau-Zener interpretation of Ppein Eq. (8) leads us to expect that Ppewill vary with J′. This variation should be stronger for the outer1Πg- 3Σgcrossing than for the inner1Σu-3Πucrossing. For the latter crossing, our calculations do give Ppevalues which vary slowly with J′. We calculate Ppe(J′= 1) to be 0.024, 0.44, 0.31, 0.34 and 0.64 for Mg, Ca, Sr, Ba, and Yb respectively. These probabilities are all large (order unity) except for the case of Mg. This qualitative conclusion is likely to be robust, even though our model calculations are only quite approximate. In contrast to our results for the1Σu-3Πucross- ing, Fig. 4 shows that the calculated Ppe(J′) values for the outer1Πg-3Σgcrossing indeed depend much more strongly on J′. A test of the Landau-Zener formula for Mg shows that the result of Eq. (8) is indistinguishable from the calculated line on the ﬁgure. The qualitative feature of a dip in Ppeas it goes near zero for some J′ is associated with the phase factor in the LZ formula, Eq. (9). Since the speciﬁc J′-range where this dip occurs is sensitive to the potentials used [37], our model calcu- lations can only be a qualitative guide even for Mg. The relative values for the other species are also only qualita- tive guides, since other curve crossings are also involved. 10In any case, Sr is likely to have a large (order unity), perhaps the largest, Ppefor the1ΠgSC process. 10-510-410-310-210-1100 0 20 40 60 80 100 120MgCaSr BaYbPpe J' FIG. 4. Calculated probabilities Ppe(J′) versus J′for the 1Πg-3ΣgSC crossing. We can use PpgandPpefrom the close coupling calcu- lations to divide out the inner zone probability so as to deﬁne a numerical excitation-transfer function from the long-range region [21] Jeg(ε, l′′, B,∆, I) =Ppg(ε, l′′, B,∆, I) Ppe(J′), (25) where J′=l′′+B.Jegmay be interpreted as the prob- ability of reaching the short-range region due to opti- cal excitation at long-range and propagation to short- range, including return after multiple vibrations across the short-range well. This interpretation follows from the fact that one gets the total trap loss probability Ppg(ε, l′′, B,∆, I) by multiplying Jeg(ε, l′′, B,∆, I) by the probability Ppe(J′) in Eq. (24) of a trap loss event in a single complete cycle across the well. Using Eq. (25), we can deﬁne an excitation-transfer rate coeﬃcient κ(∆, ε) κ(∆, ε) =kBTε hQTε/summationdisplay l′′even,B(2l′′+ 1)Jge(ε, l′′, B,∆, I).(26) This rate coeﬃcient κ(∆, ε) is related to the ordinary rate coeﬃcient K(∆, ε) in Eq. (23) through a mean inner zone probability /an}bracketle{tPpe(ε)/an}bracketri}ht, which we can deﬁne by the relation K(∆, ε) =/an}bracketle{tPpe(ε)/an}bracketri}htκ(∆, ε). (27) Clearly, we can also deﬁne a thermal average κ(∆, T) analogous to that in Eq. (22), and deﬁne a thermal aver- age/an}bracketle{tPpe(T)/an}bracketri}ht=K(∆, T)/κ(∆, T). The usefulness of the factorization in Eq. (1) is that it allows us to deﬁne an excitation-transfer rate coeﬃcient κfrom which the inner zone SC probability has been re- moved (however, see the discussion in Section III C3 be- low about how a large Ppe(J′) may aﬀect the width of res- onance features). We can predict much more conﬁdently the properties of the long-range excitation and vibrationthan we can the short-range SC probabilities. Thus, once we have a better knowledge of these short-range proba- bilities, either through measurements or through better theoretical knowledge of potential curves and couplings, we can multiply our κcoeﬃcients by /an}bracketle{tPpe(T)/an}bracketri}htto get the SC rate coeﬃcients. C. Limiting cases of the excitation-transfer probability The attractive molecular potentials support molecu- lar vibrational levels with vibrational quantum numbers v, as described in Section II C. We can ﬁnd simple analytic expressions for the excitation-transfer functio n Jeg=JeePegfactored according to Eq. (2) for two limit- ing cases: (1) strongly overlapping resonances where the probability is large for spontaneous decay during a sin- gle vibrational cycle, i.e., the level width is larger than the level spacing, and (2) non-overlapping, or isolated, resonances, where many vibrations occur during a vibra- tional decay lifetime, i.e., the level width is much smaller than the level spacing. For Group II species,1Σutran- sitions at small detuning tend to be of the former type, but never become fully isolated in the detuning range we consider. On the other hand,1Πgtransitions tend to be of the latter type unless the detuning is very small or the SC probability is very large. 1. Small detuning and fast spontaneous decay The quantum mechanical theory of the ﬁrst limiting case for trap loss for small detuning and fast radiative decay has been worked out in detail in Refs. [21,41,42,24], where: Jeg(ε, l′′, B,∆, I) =Jee(ε, l′′, B,∆)Peg(ε, l′′, B,∆, I). (28) The factor Peg(ε, l′′, B,∆, I) = 1−e−2πΛ, Λ =\|Veg(RC)\|2/(¯hvCDC), (29) where vCandDCare the speed and slope diﬀerence at the Condon point, represents the Landau-Zener proba- bility of excitation from the ground state to the excited state in a one-way passage through the Condon point atRC. In this limit, radiative decay is faster than the vibrational period (Γ v≫hνv), there are no multiple vi- brations, and the excited state transfer factor Jee(ε, l′′, B,∆) = e−aout≪1, aout=λ/integraldisplayup uCduΓ(u) ¯hv(u)(30) 11represents the probability of survival along the clas- sical trajectory from the Condon point of excitation to the point Rpof inner zone curve crossing; v(u) = {2[ε−Ve(u)−∆]/µ}1/2is the local classical speed. Note that the Landau-Zener expression for Pegin Eq. (29) does not have the proper Wigner law threshold behavior, since Jegshould be proportional to kat low col- lision energy. However, our numerical Jegwill have the proper Wigner law form. Note also that the quantum mechanical calculations in Ref. [21,24,41,42] support thi s semiclassical picture of localized excitation at the Con- don point, not the delocalized excitation picture of the Gallagher-Pritchard (GP) model [43], which for small de- tuning predicts a dominant contribution to trap loss from oﬀ-resonant excitation at distances much less than RC. The GP model also does not satisfy the Wigner law at lowT. We defer detailed comparisons with semiclassical theories to a future publication. 2. Non-overlapping resonances The second limiting case is that of non-overlapping vi- brational resonances, that is, the spacing hνv[see Eq. (6)] between vibrational levels vis much larger than their to- tal width Γ v. This is typical of large detuning. Then \|Spg\|2is given by an isolated Breit-Wigner resonance scattering formula for photoassociation lines [38,44]: Ppg=ΓvpΓvg [ε−(Ev+sv)]2+ (Γ v/2)2. (31) HereEv= ∆−εvis the detuning-dependent position of the vibrational level in the molecule-ﬁeld picture relativ e to the ground state separated atom energy (when ∆ = εv, then Ev= 0 and the vibrational level is in exact resonance with colliding atoms with zero kinetic energy), svis a level shift due to the laser-induced coupling, and the total width Γ v= Γ vp+ Γvg+ Γv,radis the sum of the decay widths into the probe (Γ vp) and ground state (Γvg) channels and the radiative decay rate (Γ v,rad). In the weak decay limit (Γ v≪hνv), we can write the Fermi golden rule decay widths as [45,46] Γvi= 2π\|/an}bracketle{tv\|Vvi\|ε, l/an}bracketri}ht\|2= ¯hνvPvi, (32) where i=gorp,lis the partial wave for channel i, and Pvirepresents the probability of decay during a single cycleof vibration from level vto channel i. For the SC process, Pvpis a very weak function of energy as long as the detuning is not too large, and we can take Pvp= Ppe, where Ppeis the energy-independent SC probability discussed in Sec. III B. In the weak-ﬁeld limit, Γ vgis very small in relation to Γ v,rad, and we can ignore it (that is, there is no power broadening). Using Eq. (32) in Eq. (31), we get the factorization in Eq. (28) with the resonant-enhanced transfer functionJee=(¯hνv)2 [ε−(Ev+sv)]2+ (Γ v/2)2. (33) If we use the reﬂection approximation for the Franck- Condon factor in Γ vg[34,35,44], then Peg= 4π2\|Veg(RC)\|21 DC\|φg(ε, l′′, RC)\|2. (34) Equation (34) may be used throughout the whole cold collision domain (mK to nK). It satisﬁes the the correct Wigner threshold law behavior at low energies because of the\|φg\|2factor. In the s-wave limit for low temperature, we may take the asymptotic form of the ground state wavefunction and obtain: Peg= 16π2\|Veg(RC)\|2 hv∞DCsin2k(RC−A0). (35) This looks just like the Landau-Zener result in Eq. (8), except that the asymptotic speed v∞appears in the de- nominator instead of vC[34], and the correct quantum phase appears in the sine factor instead of a semiclassical phase. 3. Contributions to the linewidths The expression, Eq. (28), for Jegin the limit of small detuning and fast decay does not depend in any way on the short-range SC probability. However, in the expres- sion for Jegin the isolated resonance limit, the width Γ v in the Jeefactor, Eq. (33), does depend on Ppethrough the contribution of Γ vp= ¯hνvPpe. As long as Γ vpis small compared to Γ v,rad, the total width Γ vis determined pri- marily by Γ v,rad, and the shape of trap loss spectral lines will still be nearly independent of Ppe. However, if Γ vp makes a signiﬁcant contribution to the total width, the long-range excitation-transfer function Jegwill show ad- ditional broadening dependent on the magnitude of Ppe. The total radiative decay width Γ v,radcan be calcu- lated from the long-range form of the decay rates in Eqs. (4), using the excellent semiclassical approxima- tion [47], /an}bracketle{tv\|Γ(u)\|v/an}bracketri}ht=νv/contintegraltext v(Γ(u)/v(u))λdu, where the semiclassical integral is over a complete vibrational cy- cle. When RC<λ, we can use the lead term in the expansion of Γ( u) inuin Eqs. (4), so that Γv,rad(1Σu) = 2Γ at= constant , (36) Γv,rad(1Πg) = Γ atπ 20a3u2 C= 0.701Γ C(1Πg), (37) where a3is deﬁned after Eq. (5) and Γ C(1Πg) is evalu- ated at the outer turning point of the vibration, which is almost the same as the Condon point. For the detuning range we consider, the radiative width of1Σulevels, 2Γ at, is much larger than Γ vp, which can be calculated from Eq. (32) using the probabilities listed in Sec. III B. Thus, Γ v≈Γv,radso that the shape of1Σu 12features (that is, their spacings and widths) should be well-determined in our calculations. Figure 5 shows Γ v,radand Γ vpfor1Πgfeatures for Mg, Ca, and Sr. In our detuning range, Γ v,rad≫Γvpfor Mg. Thus, the shape of Mg1Πgfeatures should also be well- determined in our calculations. On the other hand, for Ca and Sr, the Γ vpis larger due to the larger Ppe. Γvp increases as ∆5/6due to the νvfactor in Eq. (32), and be- comes larger than Γ v,radnear ∆ = 5Γ atin our model for Sr and near 20Γ atfor Ca. Thus, we can expect predisso- ciation broadening of1Πgfeatures to become observable for Ca or Sr at relatively small detunings. Measurements of such widths could lead to experimental information about Ppefor the1Πgstate. On the other hand, our calculated model line shapes should only be viewed as a qualitative guide in a region where Γ vp≫Γv,rad. 00.020.040.060.080.1 0 10 20 30 40 50Γv,rad(1Πg) Γvp: Mg Γvp: Ca Γvp: SrΓ(Δ) / Γat Δ / Γat FIG. 5. Radiative width Γ v,radfor Mg and widths Γ vpfor Mg, Ca and Sr versus ∆ for the1Πg-3ΣgSC crossing. D. Radiative escape calculations The calculation of the RE trap loss rate coeﬃcient fol- lows the factorization procedure in Eq. (1) as for the SC process. The RE loss is not due to a single curve crossing, but rather to excited state emission from the distance range R < R p=λup, where upis the point for which a kinetic-energy increase of 1 K for the atom pair has been gained after excitation (the 1 K is arbitrary– we only choose it to represent a “standard” loss energy). Clearly, RE can only be signiﬁcant for the1Σustate, be- cause of the negligible short-range emission from the1Πg state. We calculate the total probability of radiative es- cape, Ppe=Pdecayduring a complete cycle of vibration across the region u < u pby integrating along the classical trajectory: Pdecay(ε, J′,∆) = 1 −exp(−a), a= 2λ/integraldisplayuin upduΓ(u) ¯hv(u). (38) Pdecaydepends only weakly on ∆, J′. Variations with ε at the highest collision energies also play a role when cal-culating the thermally averaged rate. The main contribu- tion to Pdecaycomes from the long-range region where the potential is determined by its analytic long-range form. ThePdecayprobability is insensitive to collision energy and detuning. In the detuning range ∆ /Γatfrom 1 to 50 and for a collision energy of kBTD, we ﬁnd that Pdecay ranges from 0.157 to 0.144 for Mg, 0.103 to 0.100 for Ca, 0.147 to 0.142 for Sr, 0.113 to 0.110 for Ba, and 0.149 to 0.145 for Yb. These hardly change at all at a collision energy of kBTD/1000, for example, changing to 0.105 to 0.100 for Ca. We have also used a calculation with an artiﬁcial probe state crossing the excited state potential at Rp(Rp≈ 150 a 0for our Mg model for J′= 1), as described in Sec. II G, to calculate the excitation-transfer function Jeg appropriate to the RE process. We ﬁnd, as expected, that the numerical Jegfunction calculated this way is very nearly the same as the one calculated using the SC Rpat much shorter range. For our detuning range the radiative contribution to the total width Γ vof1Σulevels is much larger than contribution due to predissociation to the SC channel. We expect that our radiative escape trap loss calcu- lations are reliable in magnitude, since only long-range properties are relevant in determining both Jegand Pdecay. Therefore, in the next Section we can conﬁdently give absolute magnitudes for the RE contribution to the total trap loss rate coeﬃcient K(∆, T) for all alkaline earth atoms we study here. IV. RESULTS A. Trap loss for Mg at T=TD Our calculated results for TD= 1.9 mK for Mg colli- sions are shown in Figs. 6(a), 7(a), and 8(a). These re- sults are diﬀerent from the results presented in Ref. [20], since we have corrected some errors we made in the sum over partial waves in that reference [48]. Figure 6(a) shows on a logarithmic scale the separate contributions of each SC or RE process to the thermally averaged rate constant K(∆, T) from Eq. (22), whereas Fig. 7(a) shows the corresponding results for K(∆, ε) at a single collision energy ε. Figure 8(a) shows on a linear scale the sum of contributions from all loss processes, and shows what one might expect to see in a laboratory spectrum. The dominant loss process for Mg at 1.9 mK is due to RE from the1Σustate. The spectra for1ΣuRE and SC processes have the same shape, since they have the same excitation-transfer function κ. The RE and SC processes diﬀer only by a multiplicative factor that is nearly in- dependent of ∆, due to the diﬀerent Ppefactors for RE and SC. The1ΣuRE probability only varies by 0.157 to 0.144 from detunings of 1 to 50 Γ at, whereas the1ΣuSC probability is constant over this range. The1Σuspec- tra in Figs. 6(a) and 7(a) are nearly the same, since the 13κSC,Σ 10-1710-1610-1510-140 1000 2000 3000SC 1ΠgSC 1ΣuRE 1ΣuK(Δ,T) [cm3/s]10-11 10-12 10-13 10-1410-12 10-13 10-14 10-15κSC,Π[cm3/s] Δ/h [MHz] T = 1.9 mK 3029282726 31vt=25 24 23222120vt=19 10-1710-1610-1510-14 0 10 20 30 40 50 60K(Δ,T) [cm3/s] T=190µK10-11 10-12 10-13 10-1410-12 10-13 10-14 10-15 Δ / Γat(a) (b) FIG. 6. Contributions from the1ΣuRE and1Σuand1Πg SC processes to the thermally averaged loss rate coeﬃcient K(∆, T) at (a) 1.9 mK and (b) 190 µK as a function of laser detuning ∆ for Mg at a standard laser intensity I= 1 mW/cm2. The scales for the excitation-transfer coeﬃcients, κ(∆, T), for the SC processes are indicated by the vertical axes to the right. The vibrational quantum numbers from the top of the potential, vt, are indicated for1Σuand1Πg features. broad features do not change much upon thermal averag- ing. The rate coeﬃcient becomes very small as detuning decreases below 2 or 3 Γ at. This is because Jee≪1 for very small detuning due to spontaneous emission during the long-range approach of the two atoms. Spontaneous emission losses become small for detunings larger than around 10 Γ at, and vibrationally resolved, but rotation- ally unresolved, photoassociation structure begins to de- velop as detuning increases. This occurs as the spacing between adjacent1Σuvibrational levels from Eq. (6) be- comes larger than the radiative decay width. Several ro- tational features with diﬀerent J′may contribute to each of the broad photoassociation resonances, with the range ofJ′depending on detuning. Each individual1Σurota- tional line has a width on the order of (2Γ at+kBT)/h≈ 200 MHz. There is negligible predissociation broadening due to SC processes in this region of the spectrum. Using Eq. (5) in Section II C, the vibrational quantum number vt, as counted down from the top of the potential at the dissociation limit, can be given for the resolved, or partially resolved, features in the trap loss spectrum. We deﬁne vtto be vD−vrounded up to the next inte-κSC,Σ 10-1710-1610-1510-140 1000 2000 3000SC 1ΠgSC 1ΣuRE 1ΣuK(Δ,ε) [cm3/s]10-11 10-12 10-13 10-1410-12 10-13 10-14 10-15T = 1.9 mKκSC,Π[cm3/s] Δ/h [MHz] J'=7531 10-1710-1610-1510-14 0 10 20 30 40 50 60K(Δ,ε) [cm3/s]10-11 10-12 10-13 10-1410-12 10-13 10-14 10-15 Δ / ΓAtT=190µK(a) (b) FIG. 7. Contributions from the1ΣuRE and1Σuand1Πg SC processes to the loss rate coeﬃcients K(∆, ε) at a ﬁxed collision energy (a) ε=kB(1.9 mK) (b) ε=kB(190µK) as a function of laser detuning ∆ for Mg at laser intensity I= 1 mW/cm2.K(∆, ε) is a sum over partial waves and branches forε=kBT. The corresponding excitation- transfer coeﬃ- cients, κ(∆, ε), are indicated by the vertical axis to the right. Excited state rotational quantum numbers are indicated for thevt= 241Πgfeature in (a). ger. Each integer vtvalue deﬁnes an energy range which contains only one vibrational level for a given J. The vtquantum numbers for1Σuand1Πgfeatures are indi- cated on Fig. 7. Note that there are many levels (not calculated) within the range ∆ /Γat<1, a range where Eq. (5) is not meaningful due to retardation eﬀects on the potential. The broad1Σufeatures provide an exam- ple of overlapping resonances, analogous to those treated by Bell and Seaton [49] for the case of dielectronic recom- bination where the spacing between collisional resonance levels becomes less than their radiative decay width. The contribution to K(∆, T) from the1ΠgSC pro- cess shows much sharper vibrational structure than the corresponding1Σuspectrum. This is because of the small radiative widths of the1Πglevels, which become even smaller as ∆ increases. The individual contribution from a number of narrow rotational levels is evident in Fig. 7(a). Figure 6(a) shows that this1Πgstructure even survives thermal averaging. Figure 8(a) shows that sharp 1Πgfeatures can even survive thermal averaging at 1.9 mK, although such features are quite weak for Mg and would be hard to see (However, see below for Ca and 1400.10.20.30 500 1000 1500K(Δ,T) [10-13 cm3/s]Δ/h [MHz] T = 1.9 mK272625 vt=24 012345 0 5 10 15 20 25K(Δ,T) [10-13 cm3/s] Δ / ΓatT = 1.9 µK14 26 2524vt=23vt=2120 19 18 17 16 15 27(a) (b) FIG. 8. Total thermally averaged Mg spectrum, K(∆, T) summed over all RE and SC contributions, on a linear scale. (a) At TD=1.9 mK, (b) In the s-wave limit at T= 1.9µK. The vibrational quantum numbers vtare indicated for the 1Σuand1Πgfeatures. Only excited J′= 1 levels contribute R-branch transition from s-waves in panel (b). Ba, where such features might be observable). We ﬁnd that there are sharp J′= 11Πgfeatures due to s-wave collisions that can be much narrower than kBT(which is about 40 MHz at 1.9 mK), whereas features due to l′′>0 collisions have widths on the order of kBT. This s-wave behavior is evident in our numerical calculations, but can be easily explained in terms of the analytic behavior of the isolated line shapes using Eqs. (31), (32), and (34). We will discuss this s-wave resonance narrowing feature in more detail elsewhere. Figures 6 and 8 both show that at very small detuning, on the order of 1 or 2 Γ at, the trap loss is dominated by SC due to the1Πgstate. The increasing radiative transition probability as detuning decreases, and the near absence of spontaneous emission losses for the weakly emitting state, ensures that the1Πgcontribution to trap loss must be dominant at very small ∆. We will show in the next section that this is even more important for the heavier species. Our conclusion concerning the role of the1Πgstate at small ∆ agrees with the ﬁndings of Refs. [7,20]. Figures 6(b) and 7(b) show the contributions to SC and RE processes for Mg at 190 µK. The broad1Σufea- tures are not very sensitive to changing the temperature. They narrow slightly at the lower temperature. However, the1Πgfeatures simplify and clearly have contributions from fewer partial waves. The eﬀect of thermal averag- ing on1Πgfeatures is to cause some broadening, with consequent decrease in peak height.0123 10.35 10.4 10.45 10.5 10.55 10.6κ(Δ,T) [10-10 cm3/s] Δ / ΓatMg 1Πg T = 1.9 µK Γv,rad 0123456 11.25 11.3 11.35 11.4 11.45 11.5κ(Δ,T) [10-10 cm3/s] Δ / ΓatCa 1Πg T = 830 nK Γv,rad(a) (b) 00.20.40.60.81 10.35 10.4 10.45 10.5 10.55 10.6κ(Δ,T) [10-10 cm3/s] Δ / ΓatSr 1Πg T = 770 nK Γv,rad Γvp(c) FIG. 9. Single1Πgvibrational feature in the vicinity of ∆≈10Γatfor (a) Mg, (b) Ca, and (c) Sr. The ﬁgure shows the quality of the isolated resonance approximation for the excitation-transfer line shape κ(∆, T). The solid line is the complex potential numerical calculation, and the dashed li ne is the analytic line shape based on Eqs. (28), (33), and (35). B. Trap loss for Mg near 1 µK Figure 8(b) shows K(∆, T) summed over all compo- nents at the extremely cold temperature of 1.9 µK. This is deeply in the Wigner law domain, where only s-wave collisions contribute to the spectrum, and the rate con- stant K(∆, T) becomes independent of T[1]. The broad 1Σufeatures are similar to the ones at higher tempera- ture, but are due only to absorption by a single R branch line from l′′= 0 to a J′= 11Σulevel. The only signiﬁ- cant broadening is due to radiative decay. On the other hand, the1Πgfeatures, also due to a single R branch line from l′′= 0 to a J′= 11Πglevel, become promi- nent sharp features in the spectrum, having widths on the order of a few MHz due to radiative decay. Even the 15level near 1 Γ atdetuning is quite sharp and isolated. Sec- tion II E discusses why we expect to get the vibrational spacings right, although we do not expect to predict cor- rectly the actual position of levels, which depend on an unknown phase due to the short-range1Πgpotential. Figure 9(a) shows the excellent quality of the isolated resonance approximation for a Mg1Πgs-wave absorp- tion feature due to a single vibrational level. This ap- proximation should be good in this case, since the mean vibrational spacing near this level is 280 MHz, which is much larger than the width. The Figure compares the numerical line shape with that calculated using the iso- lated resonance formulas discussed in Section III C. The analytic formula calculates the factors in Eqs. (28), which are used in Eq. (26), by making the isolated resonance ap- proximation, Eq. (33), and the reﬂection approximation, Eq. (35). The linewidth in the denominator of Eq. (33), calculated to be 1.6 MHz from Eq. (37), is almost entirely due to weak spontaneous decay of this1Πglevel, as dis- cussed Section III C in relation to Fig. 5. Any broaden- ing due to thermal averaging is negligible, since kBT/h= 0.04 MHz. We have veriﬁed that our thermal spectrum at rela- tively high temperature, 1.9 mK, is to a good approxima- tion independent of the choice of ground state potential, as discussed in Section II D. This is because of the need to sum over several partial waves, for which the phase of the ground state wavefunction varies by more than π. In addition, the need to average over a range of collision energies also contributes a range of phase variation to the ground state wavefunction. The spectrum at very low temperature, on the other hand, is sensitive to the phase of the ground state wave- function, which is generally unknown for Group II species and strongly dependent of the details of the ground state potential. This sensitivity is explained by the reﬂection approximation in Eq. (35), which shows Pegis propor- tional to sin2k(RC−A0). We have just seen that the reﬂection approximation is excellent for isolated reso- nance line shapes. Therefore, if we know K(∆, T) in thes-wave domain for one scattering length A0, and if we have a diﬀerent potential with a diﬀerent scat- tering length A′ 0, the K(∆, T) for the new case can be scaled from the original one by multiplying by the ratio sin2k(RC−A′ 0)/sin2k(RC−A0). Figure 10 compares this scaling (dashed lines) to numerical calculations (sol id lines) for several diﬀerent model ground state potentials with diﬀerent A′ 0. The former are scaled from our original calculation, for which A0=−95 a0. Figure 10 demon- strates that this scaling is a good approximation, even when the scattering length is unusually large and even for overlapping1Σufeatures. The scaling relation is excel- lent at small ∆ for scattering lengths having magnitudes up to a few times x0(deﬁned in Sec. II D and having a value of 36 a 0for Mg). The scaling is even a reasonable approximation for the case where A′ 0= 400 a 0and the ground state wavefunction has a node at RC=A′ 0near∆/Γat= 15. The node for the A′ 0= 930 a 0case occurs for ∆/Γat<1 and is oﬀ scale in Fig. 10 for the A′ 0= 99 a0case. 10-1610-1510-14 0 10 20 30 40 500 1000 2000 3000 Δ / ΓatΔ/h [MHz] E/kB=1.9µK A0=-95 A0=930 A0=99 A0=400RC=400K(Δ,ε) [cm3/s] FIG. 10. Scaling with diﬀerent scattering lengths of K(∆, ε) for the1Σutransition in Mg. The bold solid line shows the numerically calculated K(∆, ε) atε=kB(1.9 µK) for the “standard” ground state model potential with A0=−95 a0. The other solid lines show the calculated K(∆, ε) for three other model potentials with diﬀerent scat- tering lengths of 99 a 0, 400 a 0, and 930 a 0. The dashed line shows the scaled K(∆, ε) calculated from the “standard” one using the scaling relation discussed in the text. The detuni ng for which the Condon point is 400 a 0is indicated by the ar- row. The eﬀect of the node in the ground state wavefunction is evident for the A0= 400 a 0case. V. OTHER ALKALINE EARTH ATOMS Our calculations for the other Group II species and Yb are shown in Figs. 9(b), 9(c), 11, 12, and 13. We trust that these model calculations, which can only provide order of magnitude estimates for SC probabilities and predissociation contributions to linewidths, will provid e a useful qualitative guide to diﬀerences and similarities among the various cases to guide future experiments on these systems. Our calculations should be fairly robust with respect to qualitative expectations as to the diﬀerent kinds of features to expect in trap loss spectra. Figure 9(b) shows that a very low temperature Ca 1Πgfeature is very similar to the Mg one previously discussed. The total width is slightly larger than the radiative width due to weak predissociation of this fea- ture (see Fig. 5). The eﬀect of the large predissociation width, where Γ vp>Γrad, is evident for the Sr feature in Fig. 9(c). The isolated resonance approximation is also beginning to fail for Sr lines because of the strong predis- sociation broadening in our model with Ppe= 0.22 (see Fig. 4). Although our model calculation for Sr predissoci- ation widths should not be considered to be reliable, the model does show that if the widths of1Πgfeatures like 1610-1410-1310-12 0 10 20 30 40 50Mg Ca Sr Ba Ybκ(Δ,T) [cm3/s] Δ / Γat 10-1310-1210-1110-10 0 5 10 15 20 25κ(Δ,T) [cm3/s] Δ / Γat(a) (b) FIG. 11. Excitation transfer coeﬃcients κ(∆, ε) on a loga- rithmic scale for the (a)1Σuand (b)1Πgstates as a function of laser detuning ∆ for Mg, Ca, Sr, Ba, and Yb at a laser intensity of I= 1 mW/cm2. the one in Fig. 9(c) could be measured, the data should allow a value to be determined for Ppe. Since temper- atures in the nK regime have already been reported for intercombination line cooling of Sr, it may be quite fea- sible to measure such widths. Figure 11 shows the thermally averaged excitation- transfer coeﬃcients κ(∆, T) (see Eq. (26) and following) for the1Σuand1Πgstates in these systems at TDfor the 1S→1P cooling transition. In spite of the fact that the inner zone SC probability Ppeis divided out of the expres- sion for κ(∆, T), there are still a number of diﬀerences among the diﬀerent species. The diﬀerences in spacing and contrast of the individual vibrational features that appear at larger detuning is clearly related to the vibra- tional spacing, Eq. (6), which decreases with increasing mass. The diﬀerences in magnitude can be qualitatively related to the scaling of the diﬀerent factors that make up κ(∆, ε) in Eq. (26). There are four factors that contribute to the scaling: (1) 1 /QT→µ−3/2, (2) the sum over l′′→l2 max→µd4/3 0/∆2/3, (3)Jee(peak) →(νv/Γv)2→ λ6∆5/3/(µd16/3 0), and (4) Peg→ \|Veg(RC)\|2/(vDC)→ µ1/2d8/3 0/∆4/3. The net scaling of the peak magnitude ofκ(∆, T) thus scales approximately as λ6/(µd4/3 0∆1/3). This gives scaling factors at the same ∆ of 1, 5.0, 3.4, 6.4, 1.1 for Mg, Ca, Sr, Ba, and Yb respectively (thesefactors should be scaled by an additional factor of λ/d2/3 0 if evaluated at the same scaled detuning, ∆ /Γat). These scaling factors account for the relative magnitudes of the peakκ(∆, T) for the1Σustate in Fig. 11(a) in the rel- atively ﬂat region from 20 to 50 ∆ /Γat. The scaling for the1Πgspectra in Fig. 11(b) also needs to take into ac- count the predissociation contribution to the width Γ v, which was taken to be purely radiative for the scaling of the1Σuspectrum in Fig. 11(a). For example, this extra predissociation broadening lowers the peak of Sr features below those for Mg in Fig. 11(b). 00.20.40.60.81Mg 1.9mK [10-13 cm3/s] Ca 830µK [10-12 cm3/s]Sr 770µK [10-11 cm3/s] Ba 450µK [2 10-12 cm3/s] 0 5 10 15 20 25K(Δ,T) FIG. 12. Spectrum K(∆, T) summed over all RE and SC contributions at TDfor Mg, Ca, Sr, and Ba at a laser intensity ofI= 1 mW/cm2. Figure 12 shows our model thermally averaged K(∆, T) summed over all contributions. With the caveat that the relative contributions of SC processes are not likely to be reliable in our model calculations, these model spectra show qualitative features that one might observe in laboratory spectra. In particular,1ΠgSC processes make a dominant contribution to the small detuning trap loss for ∆ <a few Γ at. This has already been discussed in Refs. [7,20]. We can compare our results to the mea- sured 2 K(∆, T) = 4 .5(0.3)(1.1)×10−10cm3/s [50] for Sr at ∆ /Γat= 1.75,I= 60 mW/cm2, and T≈4TD [7]. Although the eﬀect of a strong laser ﬁeld needs to be investigated for this case, Ref. [24] suggests that near-linear scaling may apply to small detuning trap loss even in the strong ﬁeld domain (see Fig. 6 of that refer- ence). If we assume linear scaling with I, our calculated value for T=TDatI= 1 mW/cm2scales to a value of 2K= 6×10−10cm3/s atI= 60 mW/cm2. The agree- ment of our very approximate model to within a factor of two with the measured result for Sr is gratifying and lends conﬁdence to the usefulness of our estimates. At the present, there are no other data on Sr or other Group II species to which we can compare our calcula- tions directly. The Ca 2photoassociation spectra in a 3 mK Ca MOT reported by Zinner et al. in Ref. [17] ex- tend over a detuning range from about 50 to 2700 Γ at, which is larger than we calculate. They observed1Σu features and gave a detailed analysis of partially resolved 17rotational substructure for a feature near ∆ = 27 GHz = 780 Γ at. The fact that the width of this feature could be explained by a combination of natural and thermal broadening of several rotational lines implies that pre- dissociation broadening makes a small contribution to the linewidth of this feature. If we assume that 20 MHz or less of the observed 150 MHz feature width is due to predissociation, we would then estimate the1ΣuSC Ppe<0.05, which is much less than the value 0.4 esti- mated by our model for Ca. Although our model should not be extrapolated to such large detuning without care- ful testing, this apparent inconsistency points out that much more detailed knowledge of potentials and matrix elements is needed for accurate calculations. It is an in- teresting fact to be explained why the apparent predisso- ciation rate of1Σulevels in Ca 2is relatively small, given the likelihood of several curve crossings with moderately large matrix elements (see Fig. 2). Our calculations in Fig. 12 suggest that resolved struc- ture due to1Πgfeatures may be seen at small detunings below around 25Γ atfor Ca and Ba. Structure for Sr is predicted to be suppressed by strong predissociation broadening. No1Πgstructure was reported for detun- ings larger than around 50Γ atin Ref. [17]. Such1Πg structure in Ca 2at these larger detunings may be hard to see due to masking by the strong1Σufeatures. Figure 13 shows our predictions for Ca and Sr features at extremely low T=TD/1000. This is in the s-wave limit where the1Πgstructure becomes quite sharp, as discussed in relation to Fig. 9 above. In this domain sharp1Πgfeatures should be the dominant features in the trap loss spectrum. It is noteworthy that this structure is predicted to persist even to very small detunings on the order of Γ at. Thus, if the Sr trap loss experiments of Ref. [7] could be repeated at these low temperatures, such features might be measurable. Figure 5 predicts that predissociation widths may be large enough for Sr 2 1Πgfeatures at even a few Γ atdetuning that observed broadening in the spectra might be able to determine Ppefor the Sr1ΠgSC process. Thus, low temperature measurements provide for tests of consistency with high temperature measurements. VI. CONCLUSIONS We have carried out model calculations of the small- detuning collisional trap loss spectrum of laser cooled Group II species Mg, Ca, Sr, and Ba and also Yb. We consider detunings ∆ up to 50 atomic linewidths Γ atto the red of the1S0→1P1laser cooling transition for these species and treat both inelastic state-changing col- lisions and radiative loss. Although our calculations are only model calculations because the short-range molec- ular potentials are not known to suﬃcient accuracy, we do incorporate the correct long-range aspects of the po- tentials and spectra. These calculations are intended as246810121416K(Δ,T) [10-12 cm3/s]Ca T = 830 nK 0123456 0 5 10 15 20 25K(Δ,T) [10-11 cm3/s] Δ / ΓatSr T = 770 nK(a) (b) FIG. 13. Spectrum K(∆, T) summed over all RE and SC contributions at s-wave domain at TD/1000 for (a) Ca and (b) Sr at a laser intensity of I= 1 mW/cm2. a guide for developing experimental studies on these sys- tems, which have the advantage that the collisions are not complicated by molecular hyperﬁne structure. We consider both the mK range for Doppler cooling on the allowed1S0→1P1transition, and µK range for Doppler cooling on the1S0→3P1intercombination tran- sition. Collisions in the mK range involve many partial waves, whereas µK collisions only involve s-wave colli- sions. Our quantum mechanical calculations avoid semi- classical approximations and properly account for the threshold properties of the collisions. Our interpretatio n of trap loss collision dynamics is based on a factorization of the overall probability into parts that represent long- range excitation, propagation to the short-range region, and short-range radiative or curve crossing processes that lead to loss. Thus, we can deﬁne an excitation-transfer coeﬃcient κ(∆, T), which, unlike the conventional rate coeﬃcient K(∆, T), oﬀers a signiﬁcant degree of indepen- dence from the details of unknown short-range processes. Our analysis shows how analytic formulas in the limits of small or large detuning can be used to interpret the trap loss spectrum. The trap loss spectra in all the Group II systems are inﬂuenced by two molecular transitions, the dipole- allowed1Σg→1Σutranstion and the dipole-forbidden 1Σg→1Πgtransition. The latter becomes allowed at long-range because of retardation corrections to the tran- sition matrix element. The1Σufeatures are structure- less at small detuning and reduced in magnitude due to spontaneous decay of the excited state as the atoms ap- proach one another on the excited state molecular poten- tial. They show broad vibrationally resolved but rota- 18tionally unresolved photoassociation structure as detun- ing increases away from atomic resonance. On the other hand, the1Πgabsorption always dominates at small de- tuning. Resolved1Πgvibrational and rotational pho- toassociation structure can persist even to small detun- ing, and should be especially prominent at very low tem- perature. Measurement of the widths of such features could lead to information about the short-range prob- ability of the state-changing collisions, at least for the heavier Group II elements. There are only very limited data with which we can compare our calculations. Our model calculations agree within a factor of two with the measured Sr trap loss rate coeﬃcient at a single detuning. Photoassociation spectra for Ca only exist for much larger detuning than we consider here, but suggest that the probability of the1Σustate changing process may be much smaller than our model calculations indicate. The time is right for more detailed and complete experimental studies on these Group II systems. Recent experimental advances in Group II cooling and trapping suggest that such stud- ies will be forthcoming. A number of other directions are also open for continuing experimental and theoret- ical studies, for example, trap loss collisions near the 1S0→3P1intercombination line, or collisions associated with two1P1atoms or two3P atoms. ACKNOWLEDGMENTS This work has been supported by the Academy of Fin- land (projects 43336 and 50314), Nordita, NorFA, the Carlsberg Foundation, and the U. S. Oﬃce of Naval Re- search. 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arXiv:physics/0103060v1 [physics.gen-ph] 20 Mar 2001Vacuum charges within a teleparallel weyl tensor: a new apporach to quantum gravity Edward F. Halerewicz, Jr.∗ Lincoln Land Community College† 5250 Shepherd Road Springﬁeld, IL 62794-9256 USA February 5, 2001 Abstract A comparison is given between the Newtonian and Einsteinian frames of gravitation. From this it is shown that there exist a weak c onnection to gravitation and electromagnetism. This connection is then studied more thoroughly with the Weyl tensor and with the electromagneti c vacuum Λ. Which dictates General Relativity should be reformulated t o confer to a ‘Einstein-Cartan-Weyl’ geometry. Where it is seen that the Gravitational Constant is the inverse of the Compton wavelength shown thro ugh a Weyl gauge potential of form [ Fαβ+Aα aβ];β. The gauge potential along with Einstein-Cartan geometry is argued to explain superlumina l velocities ob- served within General Relativity. PACS numbers: 04.20.-q, 95.30.Sf, 98.80.Es, 04.60.-m, 06. 20.Jr “The acceleration of motion is ever proportional to the motiv e force impressed; and is in the direction of the right line in which that force is impressed. ” –Newton (1687) 1 Introduction There are currently two viable models for the gravitational ﬁeld, being the classical and the relativistic gravitational ﬁeld. Of cour se this refers to the law of universal gravitation proposed by Newton and General Rel ativity (GR) by Einstein. The two theories are certainly correlated and att empt to describe the same phenomena, however mathematically they are treated qu ite diﬀerent from one another. It thus becomes natural for the sake of coherenc e to give a uniﬁed structure of the two theories, even if only ad hock. What is pe culiar, or even ∗email: ehj@warpnet.net †This address is given for mailing purposes only, since I’m a s tudent and do not hold a professional position at the above address. This work was ma de possible through my own personal research and studies, and does not reﬂect the above listed institution. 1far more obvious is that both theories of gravitation are sub tle variations of the second law of motion . The development of this work was made possible through study ing the rela- tionship between classical and relativistic gravitationa l ﬁelds where a weak con- nection can be shown to electromagnetism. In previous works it has been shown that the cosmological constant Λ can be represented by a cova riant electromag- netic ﬁeld [1], [2]. It has also been suggested that the cosmo logical constant can be derived from the quantum vacuum [3], [4]. Using an anal og of quantum theory and electromagnetism an empirical uniﬁcation with g ravitation is quickly realized with the classical Kaluza-Klien (KK) theory [6]. F rom an empirical KK geometry a connection with gravitation and Newton’s second law of motion can be explained by means of the Weyl tensor within the GR formali sm. The general conclusion that can be arrived from this analysis is that GR i s not a complete theory of gravitation when only considering Ricci curvatur e of the Riemannian manifold. The organization of this work is given rather straight forwa rd in §1.1 the Newtonian theory for gravitation is explained through the second law of mo- tion. In§1.2 Newton’s view of the world is brieﬂy discussed, where as i n§1.3 Einstein’s ‘world view’ is given, where the two are related w ithin§1.4. In §1.5 G is reformulated through the second law, where a relation is shown to elec- trostatics in §1.6, which is shown to be a superposition ﬁeld within §2. In§2.1 empirical equations of this ﬁeld are given, which is resembl es a KK space which is discussed in §3. A gauge ﬁeld is considered for gravitation in §4, which sug- gest a correlation to the vacuum energy explored in §5. In§6 the GR analog is discussed with the Weyl tensor. In §6.1 ﬁrst order Lagrangians are presented, which leads to teleparallel Weyl tensor in §6.2. In §7 a relationship between vac- uum energy and the gravitational constant are given. A stand ing wave is shown in§7.1 which gives an allusion of the classical KK space. The exp lanation for the Vacuum charges in the title is seen in the Appendices. App endix A describes the relation between the gravitational constant and the Com pton wavelength, as well as explaining superluminal velocities observed in ast rophysics. Appendix B gives an alternative origin for mass increase, Appendix C br ieﬂy discusses other theoretical values for G. Finally in Appendix D there is show n a need to modify the deﬁnition of the planck length under this work. 1.1 Newton-Einstein Gravity It is obvious to start such a modeling with the widely known Ne wton-Einstein action: d2xµ dt2=κc2 8π∂ ∂xµ/integraldisplayσdV0 t(1) Where the left term is deﬁned by the second law of motion: m/braceleftbiggd2xµ dt2+ Γα βydxβ dtdxy dt/bracerightbigg =Fα(2) Here another generalization of the second law can be given wi th the equation /vectorF=md2/vector r dt2. When taking the convection /vector r= (x,y,z ) a gravitational acceleration 2is derived by: md2/vector r dt2=−GMm/vextendsingle/vextendsingle/vextendsingle/vector r−/vectorR′/vextendsingle/vextendsingle/vextendsingle2/vector e(/vector r,/vectorR) (3) The gravitational ﬁeld is then deﬁned through ϕ=GM/R , thus a gravitational ﬁeld is produced through poisson’s equation through ∇ϕ= 4πGρ. Where for simplicity sake we receive the standard deviation for Newto n’s gravitational ﬁeld: /vectorFg=Gm1m2 d/vector r2(4) It is quite clear that Newton’s formulation of gravitation i s formed through the his second law of motion. Which is explained as an external fo rce mechanism which causes masses to accelerate one another. 1.2 the absolute spacetime Newton considered space and time as separate and ﬁnite invar iant dimensions. We can see this deﬁnition early on in Book I of Princpa by means of Scholium I: “absolute, true, and mathematical time, of itself, and from i ts own nature ﬂows equably without regard to anything external, an d by an- other name is called duration: relative, apparent, and comm on time is some sensible and external (whether accurate or unequala ble) mea- sure of duration by the means of motion, which is commonly use d instead of true time; such as an hour, a day, a month, a year .” A conclusion that is drawn from the roots of Euclidean geomet ry which can be expressed in the form ds=/radicalbig dx2+dy2+dz2 (5) where the following spatial identities arise by means of an i nﬁnitesimal rotation: Fx=Fxcosθ+Fysinθ, Fy=Fxsinθ+Fycosθ, Fz′=Fz(6) again we see how the spatial deﬁnition directly relates to th e second law of motion. 1.3 Special Relativity With a suggestion from Minkowski Einstein transformed the c herished absolute description of space and time to a relative space-time of the form: ds2=dct2−dx2−dy2−dz2(7) with the invention of a ‘spacetime’ continuum, one can notic e subtle changes with an inﬁnitesimal rotation: ct′=coshθct −sinhθx, x′=sinhθct +coshx, y′=y, z′=z(8) With Einstein’s fundamental postulate acceleration in suc h a frame would be limited to the speed of light, lending the beta function: γ=γ(v) =1 /parenleftbig 1−v2 c2/parenrightbig1/2(9) 3working backwards now we can see that kinetic energy of a body in this frame is given in the form KE=u/integraldisplay 0mγ3udu=mc2(γ−1) (10) Thus at this point we see that these two diﬀerent formulation s of space will produce very diﬀerent forms of acceleration. In classical t erms mass is deﬁned as a focused point of force, while in relativistic terms it is deﬁned as ‘stress- energy’ within an arbitrary manifold. 1.4 forces to ﬁelds Taking a new look at Newton-Einstein gravity one may make a se cond order generalization of the ﬁeld by: Ti j/braceleftbiggd2xµ dt2−Γα βydxβ dtdxy dt/bracerightbigg =kc2 8πgi j∂ ∂xµ(11) Notice how this equation describes gravitation not as a forc e but as a manifold. Furthermore mass is no longer an intrinsic property but a loc al ﬁeld in the geometry, i.e. the change results in going from a point parti cle theory (mass) to a ﬁeld theory (tensors). This force is taken equivalent to a manifold of the form∂f ∂xµ=∂ ∂xµ, assuming a Riemannian manifold and lines of calculations o ne results in the Einstein-Field-Equation (EFE) Ri j−1 2gi jR=−kc2 8πTi J. (12) Thus it can naturally be seen that the roots of GR can be origin ated through the second law of motion. In a larger since, the “gravitation al force” is in reality a consequence of the second law of motion, expressed diﬀerently only in the terms of mathematical dimensions. Therefore one may w ish to ex- plain the gravitational force as a two dimensional accelera tion of the form F′ g=a(E/c2)(E/c2)/dr2= 4πaρ=∇ϕ. Of course for a gravitational ﬁeld a is replaced by Newton’s Gravitational constant G. 1.5 dimensional analysis of the gravitational constant One can not deny the similarity between a classical gravitat ional ﬁeld and the Coloumb Law Fcol=kQ1Q2/r2. Suggesting classically what Kaluza, Klein, Weyl, and others have proposed, a uniﬁcation with the electr omagnetic force. In GR the ﬂat or massless gravitational ﬁeld is given with Ri j−1 2gi jR= 0.This is not entirely correct because the kappa term does not entirel y vanish leading to Ri j−1 2gi jR= 8πke.... It is seen that the interpretation of a geometrical manifol d neglects gravitational acceleration. If it is a property of the electromagnetic ﬁeld however, the second law of motion and the vacuum ﬁeld equatio ns still hold true. To elaborate more on the gravitational constant1one must be familiar with 1Modern values given the gravitational constant as [5]: G= (6.74215 ±0.000092) × 10−11m3kg−1s−2. 4its roots, where one begins with Kepler’s third law of motion : T= 2π/radicalbigg a3 M. (13) In keeping with the relationship between the gravitation an d the second law of motion this must be rewritten in the form T2=kr3, which is analogous to pendulum motion T2= 4π2l/gwhere k≡√ G=2π T/radicalbigg a3 M. (14) Through dimensional analysis we can reduce this in a form whi ch relates to the Coloumb law: F=m×(2πr/T)2 r=m×4π2mr T2=m×(2πr)4/kr3 r(15) =m×16πr/k r=m×16πmr kr3=m×16πm kr2=G. Once again we see the relevance of the second law of motion, pe rhaps more relevently with Kepler’s Laws. Furthermore the gravitatio nal constant G can be represented by k≡√ kr2such that Newton’s Law of universal gravitation becomes: F=/radicalbig km1m2r (16) 1.6 electrostatics In more simpler language the force of gravitation can be deri ved through the gaussian gravitational constant kof a line charge by means of a Coloumb ﬁeld. δ/contintegraldisplay Ldscol= (kQ1Q2/vector r)1/2(17) Where an electric ﬁeld is propagated perpendicularly by: E⊥=λs+L/2/integraldisplay −L/2(z2+s2)−3/2dz=λs1 s2L /parenleftbigL2 4+s2/parenrightbig1/2(18) or simply E⊥=2λL s(L2= 4s2)−1/2(19) with poisson’s equation a general electrostatic potential is given by ∇2φ= −4πρ(/vector r) whence by the fundamental theorem of vector ﬁelds we have an inverse square relationship φ=/contintegraldisplay dVρ R=ρ Rdxdydz =/integraldisplayλdz R(20) for simplicity we will look at a charge conﬁguration of the fo rmEr=∂φ ∂r/vector r. We now notice a direct relationship between an electrostatic ﬁ eld line and gravita- tional acceleration by /vector g=∂ϕ ∂rˆr. Empirically the combination of the two ﬁelds would represent a force of the ﬁrst order /vectorFg=∂φ ∂r/vector r+∂ϕ ∂rˆr=/summationdisplay∂2φ2 ∂2r2ˆr2=/integraldisplayr ∞Gm1+m2r r2=/radicalbig k∂Q1∂Q2(/vector r).(21) 5From here it can bee seen that a (neutral) static charge conﬁg uration can yield gravitational acceleration. gij=−∂2φ2 ∂2xij=−φij (22) Such that it is now seen that relative acceleration of two par ticles can be given in pseudo Levi-Civta coordinates d4xij dt4= ∆gij=−φijklηkl. (23) Where a generalized pseudo Riemannian ﬁeld is produced R∗ abcd−1 2gcdpqRpgra=−8πGTabcd (24) which reduces to directly to the Einstein Field Equation (12 ). 2 superposition Equation (22) can be represented by an operator of the form i¯hsuch that −φij=h i∂2φ2 ∂2xij+Hψ (25) with Schr¨ odinger’s equation one has i¯h=∂ψ ∂t=−h2 2m/parenleftbigg∂2φ ∂xij+∂2ϕ ∂xij/parenrightbigg (26) from the Laplacian ∇2awe note this represents the original ﬁeld, and which yields two gradients in spherical coordinates of the form /vector∇a=∂a rˆr+1 r∂a ∂θˆθ+1 rsinθ∂a ∂φˆφ Which gives rise to electrostatic conﬁgurations and gravit ational acceleration. Which naturally lends itself to the Schwartzschild solutio n when the ﬁelds are given in the ﬁrst order approximation in the classical ﬁeld ds2= (1−2ϕ)dt2−dr2 (1−2ϕ)−r2dθ2−r2sin2θdφ2. (27) The gravitational and electrical ﬁelds in equation (26) can be related more clearly through superposition. This also means that the ﬁel d equations (22-26) are really superposition ﬁelds. A superposition of electric and gravitational ﬁelds can be g iven through ψ(x) =ψφ(xs) +ψϕ(xs), with Huygens principle yields: ψ(x)∼/integraldisplay φexp[2πi(x−xs)/λ] \|x−xs\|ψφ(xs)dxs+/integraldisplay ϕexp[2πi(x−xs)/λ] \|x−xs\|ψϕ(xs)dxs. (28) 6Where through quantum mechanics an interference between th e two ﬁelds arises from the probability P(x) =\|ψφ(x)ψϕ(x)\|2=Pφ(x) +Pϕ(x) +ψ∗ φ(x)ψϕ(x) +ψ∗ ϕ(x)ψφ(x) (29) thus equation (22) may be reevaluated in the form: gi=−∂ψ(x) ∂xi=−ψ(x)i (30) lending d2xi dt2=−∆gi=−ψ(x)ijηj(31) 2.1 empirical equations From the above the empirical gravitational ﬁeld that transl ates is Rµν−1 2δµνR=8πG c4Tµν M+8πke c4a mTµν CC (32) or Rµν−1 2δµνR=8πG c4Tµν ψ(x)(33) Since this ﬁeld describes a quantum superposition, imagina ry coordinates are required lending: ∗R∗ abcd−1 4ǫabpgǫcdraRpgra=i/braceleftbigg8πG c4[Tabcd M+Tabcd EM(Q1) +Tabcd EM(Q2)]/bracerightbigg ¯h(34) Of course this would correspond to a complex spacetime φ(x,y,z,t ) =/integraldisplayπ −πF(xcosθ +ysinθ +iz,y+izsinθ +cosθ,θ )dθ (35) Maintaining the Minkowski metric, the background manifold Mone has (ω,z2) =ωct2−(ω)z2 1−(ω)z2 2−(ω)z2 3 (36) Without the superposition of the mass-energy tensor, the va cuum ﬁeld equation becomes: Rν µ−1 2gν µR=8πke c2Tµν EM(Q1) +Tµν EM(Q2). (37) From equation (25) it is seen that a quantum interpretation m ust be given to G. With electrodynamics in mind one might consider a form whi ch pertains to the ﬁne structure constant αe=2πe2 ¯hc→αg=−1 24πGm2 ¯hc. (38) This interpretation can be made when one takes the Weyl tenso r, and compares it to the mechanical properties of an electromagnetic ﬁeld: ∂Tα i ∂k−1 2∂gαβ ∂xiTαβ= 0. (39) As suggested in the beginning of this work the above ﬁeld is im plicitly implied by the second law of motion. 73 expanding KK-space On taking Klien’s method of compactiﬁcation one begins with a tensor of order [6]: g(5) IJ=/parenleftbiggg(4) µν+∨AµAν∨Aν ∨Aµ ∨/parenrightbigg (40) From equation (36) and with an earlier work [7], I choose to wr ite a Minkowski metric of form: \|(ω,z)\|2= (φ)c∧z1−ωz2 2−ωz2 3−ωz2 4−(φ)c∧z5≡ (41) I(ct)2−i(x)2−j(x)2−k(x)2(42) which is representative of a fractal spacetime of the form 4 ∧φ2. In tensorial terms leads to ˆM= i0 0 −1 0−i1 0 0−i i 0 −i0 0 −i ⇒ M = 1−1 1 −1 1−1−1 1 −1−1−1−1 −1−1 1 1 (43) such that the interpretation then transverses to: Mdiag= 2 0 0 0 0−2 0 0 0 0 2i0 0 0 0 −2i ∧˜M(4)≡ 2 0 0 0 0 2 0 0 0 0 −2i0 0 0 0 2 i (44) Here the time dimension is given statute through quaternion rotations in C* space. The superposition of electromagnetism and gravitat ion can be seen within a relativistic frame in a accordance with ˆ ηIK=diag(−2,−2,2i,−2i), in the ﬁfth coordinate this corresponds to η(5) IK=diag(−1,−1,−1,−1,−1). In essence (44) is a combination of two metrics, a similar metri c was inferred in Ref. [11] in relation to quantum gravity: dτ2=a rdt2+a rdr2−dx2 1−dx2 2 (45) Through some work made by Weyl [8] one can write a solution to E FE which corresponds to Rk i−1 2δk iR=−1 2∇ψk i... (46) which can be reduced for convenience as1 2∇ψk i=−Tk i. Furthermore this action can be represented with advanced and retarded potent ials. When one conveniently exchanges the ψterm from equation (28), one is left with the potentials ψk i(x)−=−/integraldisplayTk i(t−r) 2πrdVand, ψk i(x)+=−/integraldisplayTk i(t+r) 2πrdV (47) Therefore meaning that the superposition of the ﬁeld is made possible through an advanced wave. Thus one has the compactiﬁcation of a Fouie r series of form gIK=/summationdisplay ng(n) IK(xµ)einx5/λ5. (48) 8Which under compactiﬁcation yields ψ(x,x5) =1/radicalbig lp/summationdisplay n∈2ψn(◦,x)einx5/R5(49) where ◦represents quarternions. The advanced Fouier sine wave is: ψ(x,x5) =/radicalbigg 2 π/integraldisplay∞ 0f(◦,x)sindxtdt (50) which undergoes the quantum transform Ψ(◦,k,t) =1 ¯hΦ(◦,k,t)eikωdk and, Φ(◦,k,t) =1 ¯hΨ(◦,k,t)eikωdk. (51) This action creates a cascade motion within the ﬁfth coordin ate and resulting in torsion within four-dimensional spacetime. Torsion wou ld appear to be in form of gravitational waves through the action /parenleftbigg DµDµ−n2 R2 5/parenrightbigg ψn= 0. (52) Thus it is seen that an observation will only occur in a quantu m system if two anti-symmetric η(5) IJtensors come in contact (which one might expect from the Weyl tensor). This wave equation suggest KK-space expan ds into four- dimensions, resulting in self interaction. Furthermore wh en one compares the chargeqn=n(k/R5) with the planck length, one sees the relation with the ﬁne structure constant. R5=2√αlp (53) From equation (38), from this it may be seen that the second la w produces ﬁne structure which in turn yields the planck length. 4 gauge backgrounds The gravitational force is a collection of interacting forc es connected in some form by the second law (e.g. the ﬁne structure constant). When one separates the properties of a given force from the Einstein equations, its fundamental principle break resulting in only a weak equivalence princi ples (which can be interpreted as a gravitational pressure). Thus lending a ma nifold whos proper- ties depend on the pressures applied to it by external factor s. By the methods implied thus far it makes sense to make use of the semi-classi cal approach to gravitation Gµν(γ) =<ψ\|Tµν(g,ˆφ)\|ψ >. To begin let us apply a gauge ﬁeld of form −k(Fµ;ψ ν−1 2δµν ;ψ(x)+Aαν µF;ψ)/negationslash= 0 (54) which resembles a convection made seventy year ago by Einste in [9]: Gµα;α−Fµν;ν+ ΛµστFστ≡0. (55) 9Thus it may be viewed that the above equation is the solution f or ﬂat spacetime which implies that the canonical approach γαβ(x) =ηαβ+khαβ(x) should be utilized. Such that the gauge ﬁeld equation becomes: −k(Fµ;ψ ν−1 2δ;ψ(x) µν+Aµ ανF;ψ)≥i¯h∂ψ ∂t/braceleftbigg8π√−˜gTµ;ψ(x) ν (x)/bracerightbigg (56) where i¯h∂ψ ∂t= [1 2m(ˆp−eA)2+eV]ψ. (57) From this it is seen that the right of the equation is governed by the laws of quantum mechanics giving a pseudo uniﬁcation through mea ns of a complex gauge ﬁeld. Meaning that the ﬁfth coordinate is false, howev er through complex ﬁelds, torsion becomes an integral part of both sides of the g auge inequality. The stress-energy tensor can have torsion along with electroma gnetic ﬁeld through the classical connection Tµν= (Qc2+p)uµuν+pgµν+1 c2(FµαFα ν+1 4gµνFµνFµν). (58) Where torsion is given through Sµνσ=ψ[µνσ], implying the inequality has torsion in ﬂat spacetime; where one may utilize the action pr inciple [10]: δ/integraldisplay√−gd4x/parenleftbiggR k+L/parenrightbigg = 0. (59) Therefore a pseudo superposition can take place within ﬂat s pacetime, explain- ing the relationship between Newtonian gravitation and ele ctrostatic potentials in previous sections. 5 vacuum energy and geodesics From the Dirac ﬁeld i¯h∂ψ ∂t, matter would act as a void within the QED vacuum. This would thus cause the virtual energy1 2¯hωof the quantum vacuum, to adapt a negative energy term. This process would then act to collap se the space around it, in the presence of n≥1 ‘false vacuum’ mass acts on the ﬁelds to adopt anegative energy requirement , which violates the weak energy condition (WEC)TµνVµVν≥0. Here we take this to mean a cosmological constant, such that the gauge inequality (56) becomes: −k(Fµ;ψ ν−1 2δ;ψ(x) µν+Aµ ανF;ψ) +λ≥i¯h∂ψ ∂t/braceleftbigg8π√−˜gTµ;ψ(x) ν (x)/bracerightbigg (60) The cosmological constant can be given through Λ = −1 16πFµνFµν, so that the inequality suggest that ∆ xµ∆xν≥1 2Λgµν. From this we may conclude that there exist an uncertainty within the ﬁeld. This is impart be cause the vacuum can be described through: Rµν−1 2gµνR=−Λgµν (61) 10we can also see that this formalism closely resembles (46), i .e. Weyl’s deﬁnition. Which suggest electrostatic energy is lost through the unce rtainty which exist through the pseudo geometry and vacuum. With Fµν=∂Aν ∂xµ−∂Aµ ∂xν, (62) the geodesic for the vacuum becomes ∂2Aν ∂S2+ Γν µ/parenleftbigg∂xµ ∂S/parenrightbigg /parenleftbigg∂xν ∂S/parenrightbigg =−e mc2Aµxµ(63) we note that under this pseudo connection the Gamma term appe ars to be under torsion, through an action of Γb a=dΛb a+ Λc a∧Λb c. Thus (62) would appear to take the form: Fν µ=∂Aν ∂xµ−∂Aµ ∂xν(64) such a geodesic path is remarkably similar to a sphere geodes ic of an electron traveling through gravitational and magnetic ﬁelds d2xµ ds2+ Γµ αdxα dsdxβ ds=q mc2Fµ αdxα ds. (65) However, the accepted geodesic for an electromagnetic ﬁeld is that of mc/parenleftbigg∂ui ∂S+ Γi klukul/parenrightbigg =e cFikuk (66) From the above equation, it can be seen that at least empirica lly the vacuum electrostatic potential (the true vacuum) is responsible f or curvature of space- time. If one were to block the vacuum energy as in the case of th e Casimir eﬀect, it will create an inequality within the pseudo geometry resu lting in a gravita- tional pressure. Therefore so to speak, a matter Lagrangian (false vacuum) shields (true) vacuum (zero-point-ﬁeld) energy, thus resu lting in negative en- ergy, which may be interpreted through the Weyl tensor as tor sion. Speciﬁcally the interaction in time produced by the pseudo Kaluza-Klien space produces the relationship between the vacuum states by [21]: φout(k,η) =αφ+ k+βφ− k(67) thus acting as the advanced and retarded Fouier series seen i n section (2.1). Therefore the Riemannian tensor Rµνρσcontracts via the Levi-Civita connection to conserve the vacuum term, resulting in symmetric Ricci sp acetime curvature, along with an antisymmetric Weyl torsion. 6 EM vacuum and the Weyl tensor If one takes the coeﬃcients of the Cosmological Constant and the Weyl tensor one has the antisymetric ﬁeld C[ρσ][µν]FµνFµν(note: for simplicity the con- trivariant term Fµν, will be removed, it will be reinstated in (79))2. Which has 2It is exactly from this action that we see the gauge condition envisioned by Einstein [9] appear in a more coherent form. 11the following form: C[ρσ]Fµν=Cµ [ρσ]µFµν=C[ρσ][µν]FµνFµν (68) =CFµν=Cµ µFµν=CµνFµνFµν (69) =CρσFµνFµν (70) Cρ[σµν]= 0 (71) from this it may be seen that the Weyl tensor is an electromagn etic version of GR. One may now apply a jacobi identity in order to from a pseudo Bi anchi Identity of the form Cαβ[µν;λ]= 0. Which can be reduced to ∇λCαβµν+∇νCαβλµ+∇µCαβνλ= 0 (72) where we can contract with Fαµ ∇λCβν− ∇νCβλ+∇µCµ βνλ= 0 (73) which can be contracted further with Fβλ ∇λCλ ν− ∇νC+∇µCµ ν= 0 (74) or ∇µ(Cµν−1 2FµνC) = 0 (75) From this we may conclude that a similar transformation will be made with the anitsymmetric covariant term Cρσ. When we restore the ﬁeld with equation (70), we have the following ﬁeld equation: C[ρσ]−1 4Fµν+Aµ ανC=−Λgµν (76) where the gauge term is assumed to be an Ansatz Aµ α=ηµν α∂νlnφ(x), thus equation (46) is only an approximation of the above ﬁeld. To o btain ﬁeld density one is left with Fµν α=−/parenleftbig x2+ρ2/parenrightbig2 4ρ2ηµν α(77) which is similar to an anitsymmetric gauge for a Yang-Mills ﬁ eld see Ref. [13]: SYM=−1 4/integraldisplay d4xFα µν⋆Fαµν(78) We recall from section (3) that an observation of a gravitati onal ﬁeld will only occur when two antisymmetric tensor come in contact. Th us it is precisely the below ﬁeld equation which bridges the gap between quantu m theory and GR. /bracketleftbigg Cαβ−1 4˜Fαβ+Aα aβC/bracketrightbigg ;β= 0 (79) The gauge potential Aα aβrepresents the second component of the cosmological termFµνunder torsion. So that with (77), we have /bracketleftigg Cαβ−1 4−4ρ2ηa αβ (x2+ρ2)2−/parenleftbig x2+ρ2/parenrightbig2 4ρ2ηa αβC/bracketrightigg ;β= 0 (80) 12which reduces to /bracketleftbigg Cαβ−1 4gαβ+ 2C/bracketrightbigg ;β= 0 (81) this solution is parallel to the Einstein ﬁeld equation, whe n considering anti- symmetric scalar curvatures. Thus it is seen that the cosmological constant is the source o f torsion in the antisymetric Weyl tensor. In essence the false electrom agnetic vacuum is responsible for gravitation within the Weyl-tensor. Our ne w variable action is a Weyl-Hilbert action: SWH=δ/integraldisplay√−gd4x/parenleftbigg(R+ 2Λ) 16πG+LM+LYM/parenrightbigg = 0 (82) resulting in an uncertainty of the form ∆ xµ∆xν=1 2\|θµν\|(note: such an empir- ical uncertainty was give in section (5)). Which suggest tha t the Weyl tensor should be given within a complex gauge ﬁeld. Such that the Wey l-Hilbert action transforms to a Complex-Hilbert action of form: SCH=√gd4x/parenleftbigg(C+ 2Λ) 8πG+LM/parenrightbigg +i/bracketleftbigg√gd4x/parenleftbigg(C[ρσ]+ 2Λ) 8πG+LM)/parenrightbigg/bracketrightbigg /negationslash= 0 (83) This was suggested in section (2.1), meaning within a quantu m frame the electromagnetic and gravitational properties of the Weyl t ensor may interact through a weak superposition. However, we also note from sec tion (4) a com- plex solution is only empirical, thus (83) is only a pseudo ac tion. 6.1 gravitational Lagrangians Let us form a Lagrangian for the vacuum solution −Λgµν. In order to describe such an action principle we will start with the GR Lagrangian for matter in the form: SM=/integraldisplay√gd4x(gµν∂µφ∂νφ+...) (84) from our approach thus far we would like to consider perturba tions from the vacuum such that: δmetricSEM=−/integraldisplay√gd4xΛgαβδ˜Fαβ(85) which can simply be given by Λgαβ:=1√gδ δ˜FαβSEM. (86) From section (6) we now make the modiﬁcation Λgαβ:=1√−gδLEH δ˜FαβSWH= ([Cρσ−1 4gαβ+ 2C];β). (87) It is seen from this Lagrangian that the cosmological consta nt in the EFE, is in fact an electromagnetic Weyl tensor. 13On taking the conditions Tµν(x) = 0 ⇒Rµνρσ=Cµνρσ(x), in the absence of a matter Lagrangian torsion is carried by the symmetric We yl tensor or gen- erated from the electromagnetic vacuum −Λgµν. Resulting in an uncertainty of the form ∆ xµ∆xν=1 2\|θµν/iαha i∂µ αhν a\|. With this one has torsion resulting in a teleparallel description of gravitation in ﬂat spacetime . Thus the cosmolog- ical constant is a perturbation within the curvature connec tion made possible through virtual particles (i.e. the false vacuum). 6.2 teleparallel geometry The Cartan torsion connection is given by: Tσµν= Γσνµ−Γσµν, (88) with tetrad form Γρ µν=haρ∂νha µ (89) Thus the vacuum-energy tensor has torsion of the order −ΛGβ α=haβ/parenleftbigg −1√−gδLEH δhaα/parenrightbigg (90) Hence the gravitational ﬁeld equation from Weyl torsion is s een through /bracketleftbigg Cαβ−1 4gαβ+ 2C/bracketrightbigg ;β=−ΛGβ α (91) Thus the Weyl torsion tensor within GR can be given through th e metric gα=ηαβ+λGβ α (92) such that within the EFE the Cosmological Constant takes the form Rαβ−1 2gαβR=−8πGTαβ+λgα (93) It is known that the electromagnetic ﬁeld and the stress-ene rgy tensor can feel torsion through the action Tµν=−2√−gδLEM δgµν=1 4/bracketleftbigg Fρ µFµρ−1 4gµνFρσFρσ/bracketrightbigg (94) Therefore the torsion of Gαcan be carried not only through the stress energy tensor but onto G itself. Covariant Maxwell’s potentials can be given through torsio n by F∗:=−ων∧θν=Aν µ∧ων∧ωµ(95) with current potentials J∗:=dF∗=dθµ∧ωµ−θµ∧dωµ (96) The Weyl tensor can represent such a current through ∇µCµνρσ=Jνρσ (97) 14with Jνρσ=kn−3 n−2/bracketleftbigg ∇ρTνσ− ∇σTνρ−1 n−1/bracketleftbig ∇ρTλ λgνσ− ∇σTλ λgνρ/bracketrightbig/bracketrightbigg (98) Thus it is seen that the uncertainty by the relation ∆ xµ∆xν≥1 2\|θµν\|causes the Weyl tensor to carry a cosmological charge current. Ther eby the ﬁnal form of the Weyl gravitational ﬁeld is /bracketleftbig Cαβ−1 4gαβ+ 2C/bracketrightbig ;β=−JβΛGα(99) The right side of the above equation can be written in the form −JΛGβ α. From our pseudo Weyl Curvature (75), the relationship between ch arges is given by: ∇µCµν+ [∇ρTνσ− ∇σTνρ...]≡Jν[ρσ]−→ (100) ∇µFµν[ρσ]=Jν−[∇ρTνσ− ∇σTνρ...] (101) This approximation can be given through: Fµν=∂Φν ∂xµ−∂Φµ xν+Cαβ µνΦαΦβ (102) which is made possible through a Ricci symmetric tensor of fo rm ∂Λσ µ ∂xν=ηµνσΦνΛσ µ (103) which in a constant ﬁeld is given by Rµν= Λσ µFσν, such a ﬁeld can transpose to the relation Fµν=∂µΦν−∂νΦµ (104) such a ﬁeld describes two opposed electromagnetic ﬁelds thr ough the connection Γσ µν= Λσ µΦν. Which has the geodesic relation d2xµ ds2+/parenleftbigg Φσdxσ ds/parenrightbigg Λµ νdxµ ds= 0 (105) the above geodesic also has the form Φσ∂Λσ µ ∂xν=Cαβ νσΦαΦβΛσ ν (106) this term is thus antisymmetric and yields Rµν=/parenleftbigg∂Φν ∂xσ−∂Φσ xν+Cαβ νσΦαΦβ/parenrightbigg Λσ µ=FσνΛσ µ (107) One should also note that a four vector line element under thi s prescription is given by uq=Φ;q/radicalbig −Φ;Φq(108) Finally a like wise connection is given through Cρσµν=Rρσµν−/parenleftbigg2 (n−2)gρ[µRν]σ−gσ[µRν]ρ +2 (n−1)(n−2)Rgρ[µgν]σ/parenrightbigg (109) 15Thus allowing a metric to remain conformaly invariant throu gh the rescaling operationgµν(x)→ef(x)gµν(x). This would make the torsion term appear to disappear within classical GR by means of Ricci curvature , or through the Einstein-Hilbert action S=/integraltext√gdx4R. However, the cosmological charge cur- rent term would still remain connected to the stress-energy tensor. This result may be obtained through the Tucker-Wang action S=/integraldisplay λ2R⋆1 (110) whereR⋆1 =Ra b∧(ea∧eb), is a scalar torsion corresponding to Ta=deaΛa b∧eb. With this one can have an action principle which resembles a B rans-Dicke space by: S=δ/integraldisplay λ2/parenleftbiggR⋆1 16πG+LYM/parenrightbigg = 0 (111) Thus we have a action corresponding to a Cosmological Consta nt, without Λ. This is made possible because we have been considering a Weyl action, empiri- cally given by: SW=−α/integraldisplay CλµνκCλµνκ√−gdx4. (112) Since the Cosmological coeﬃcient is included in the Weyl ten sor we have been considering, it vanishes under a Cosmological model. There fore the action for Weyl gravitation is not governed by the pseudo action (83), b ut by (111). 7 The vacuum and the meaning of G An alternative interpretation of mass was assumed by de Brog lie by the Einstein- de Broglie equation: ¯hωC=m0c2. (113) This formalism has be restated recently by Haisch and Rueda [ 14] as a possible explanation for the origin of inertial mass. Where C is given by the Compton wavelength λC=h/mc, thereby asserting the origin of inertia through the Compton wavelength. If we take the equivalence principle by heart then, one must assume that gravitational inertial would arise throug h a similar action. From Einstein-Cartan geometry we can assume that this ﬁeld w ould be given through torsion. Speciﬁcally we will assume an equivalence of order Λ C= 2πG−1(see A for details), this is validated by the quantization λC/2π. From this we see that G is the inverse charge of an electron’s Compt on wavelength. In terms of EFE we have Rαβ−1 2gαβR=−8π/angb∇acketleftGQ/angb∇acket∇ight c2Tαβ+/angb∇acketleftλQ/angb∇acket∇ight Gα (114) in essence it appears that this is simply a post Einsteinian s emi-classical cor- rection to the ﬁeld equations (with teleparallel Weyl torsi on acting as a gauge background) . From our supposed relation, one would have an identity of for mλG=I. Therefore the above relation transverses to Rµν−1 2gµνR+λ−G=−8πITµν (115) 16Let us now suppose that the identity is equivalent to the de Br oglie wavelength I=h/p. Thus: /angb∇acketleftGQ/angb∇acket∇ight ≡h/p h/mc=λd λC(116) whereλdis the de Broglie wavelength given by λd=h/p. From Compton scattering we may assume that gravitational waves can pass t hrough particles as though they were a wave. When we view the geodesic (66) one m ay rewrite the identity in eq. (115), such that I= 1/√gkk. This results from a geodesic of order [17]: 1√gkk/bracketleftbigg∂2ui ∂S2+ Γi kl/parenleftbigg∂xk ∂S/parenrightbigg /parenleftbigg∂xl ∂S/parenrightbigg/bracketrightbigg =e mc2Fik(117) thus it is viewed from this that the term e/mc2is nearly a classical approxima- tion of the Compton wavelength λC=h/mc. We see this when we compare the energy sources to that of the classical ﬁne structure consta nt σrad∼Z2 137/parenleftbigge2 m0c2/parenrightbigg2 cm2/nucules (118) which is given by an electrons radius r=e2/m0c2= 2.818×10−13cm. Where through the quantum correction one has 2 πe2/hc=1 137. Further the inversion by the Cosmological Constant3yields the Gravitational Constant through the identity 1/√gkk, or speciﬁcally through the geodesic coordinate uk=∂S/∂xk. Thus it is seen that the gravitational constant, ﬁne structu re, and the second law of motion appear to arrive from quantum charges! 7.1 standing waves and the ﬁfth coordinate We note that a superposition of a sinusoidial wave yields a st anding wave of the formp= 2asin(2πx/λ)cos(2πvt/λ ). With this the standing wave of the gravitational constant would be given through: pG−λ0= 2acos(2πvt/λd) sin(2πx/λC)= cot 2avtλd xλC(119) Thus G is the inverse ratio between the superposition of de Br oglie and Compton wavelengths4. This standing wave can be seen as potential barrier, which r esults in the interaction of advanced and retarded potentials5. This action results in the violation of the WEC, i.e. results in a false vacuum and We yl gravitation. 3This prescription of the Cosmological Constant as the Compt on wavelength λC, may have bearing on modern cosmological theories. For example it has been proposed by recent obser- vations from type IA supernova, there may be something causi ng the universe to accelerate its expansion. Under this scenario, the Universe would be co asting from the initial ’big bang,’ however through a cosmological Compton scattering this eﬀe ct would appear to increase, thus giving the allusion of an ‘accelerating’ universe. 4The idea of a quantum connection to the Gravitational Consta nt and the Cosmological constant is not new idea, and neither is a superposition rela tion see Ref.[21]. 5An alternative to this interpretaion arises through Quantu m Mechanics, from SR an electromagnetic ﬁeld at k reading locally as electric may re ad as a magnetic ﬁeld in the frame k’. Through the action of the Weyl tensor the electric and mag netic terms may become superimposed, thus initially one has a superposition of for mψ(x) =E(x) +H(x), which doesn’t take on its SR form until the wave function has been ca nceled. 17From this we understand that a gravitational constant is an i nverse charge of a particles Compton wavelength. Meaning that each partic le has its own local isotropic gravitational ﬁeld, which is induced by mas s and acceleration. This also leads to a startling corollary under relativistic velocities the charge of spacetime Q would be altered. In such a situation the Compton charge would be altered by L=mc2/radicalbigg 1−v2 c2−mΦc−qΦE+q/vector v/vectorA, (120) causing an inverse relation in the G (the consequences of suc h an eﬀect is brieﬂy mentioned in B). Since the electromagnetic force has an inve rse relationship squared to inﬁnity, thus is the gravitational ﬁeld. Forces s uch as the Yang-Mills ﬁeld are conﬁned within the Coloumb barrier, thus allowing t he gravitational ﬁeld at a ﬁrst approximation to adopt the Newton’s Gravitati onal constant GN. When compared to classical gravitation one has the ﬁeld ∇ϕ= 4π[2avtλd xλC]ρ. From this a four-vector is required, thus gravitation is a ch arge in spacetime! Through Ref. [15] we see our prior assumed equivalence with t he Compton wavelength pops up again through the ﬁve-dimensional actio n: Ψ(xµ, x5) = exp/bracketleftbigg ik2π λCx5/bracketrightbigg ψ(xµ) (121) where it is interpreted that the gravitational force arises through torsion. While electromagnetism is derived through the ﬁfth gauge-compon ent of the torsion tensor (which has been shown to be false in previous sections ). This is a pseudo complex interpretation produced by (119) and (67), thereby giving the allu- sion to a ‘ﬁfth-coordinate,’ through a superposition mecha nism. 8 discussion This analysis of a gravitational background space reveals t he following sub- tle quantum aspects of the gravitational ﬁeld. The dual inte rpretation of a causal trajectory in the Feynman school, is responsible for the appearance of a pseudo ‘ﬁfth coordinate.’ Thus causing true vacuum energy to translate into false vacuum energy converting the potential virtual energ y into kinetic energy. This results in torsion within the background space, which a cts to conserve the negative energy created by the false vacuum. Torsion then ac ts to produce a gravitational metric by means of a quantum charge, where by t heequivalence principle thesecond law becomes valid for a classical body. Secondarly torsion alters the de Broglie wavelength which causes electrostati c potentials to lower, acting as a relativistic gravitational ﬁeld. This analysis showed the importance of the often neglected W eyl component of Riemannian geometry. It is the antisymmetric Weyl tensor acting along with an Einstein-Cartan geometry that is responsible for the gra vitational constant. Speciﬁcally pertaining to an electrons Compton wavelength for long range grav- itation. However, for ﬁeld of varying charge one would expec t the gravitational constant and the cosmological constants to accept diﬀerent values, thus gravi- tation in its true form would carry more than the background e lectromagnetic vacuum, and gravitation would be expected to have ranges lim ited to there local ﬁelds6. 6For example, nuclear ﬁelds do not correspond to the inverse s quare relationship 1 /r2, thus 18Acknowledgement I would like to thank Fernando Loup for his correspondence on the Cosmological Constant, and for the support of this work. A The Compton wavelength and ‘superluminal’ shifts The relationship between the gravitational constant G, and the Compton wave- length can be seen, when we except the value of G to be of order [ 5]: G= (6.74215 ±0.000092) ×10−11m3kg−1s−2. (122) We can now compare that to the Compton wavelength7of an electron given by λC=h/mec= 2.426×10−12m. However from ﬁeld density relationship shown in (80), we are left with the crude relation λC= 2πG−1, we must consider the quantized wave form: h·λC 2π= 386.1592642(28) ×10−15m (123) Thus the inverse of G is that of 2πG−1= 0.02361 =λC×10−1(124) Such that it is seen that 2 πGis the inverse of the Compton wavelength. Accel- eration will altered the ‘charge’ or Compton wavelength suc h that gravitational constant(s) would be altered upon relativistic velocities . This could very well explain the propagation of observed superluminal jets eman ating from Active Galactic Nuclei (AGN). Since classically electromagnetic waves propagate via the relationC=λν, we expect a gravitational shift from ∆v≈vi(GaM)/parenleftbigg1 ri+1 rf/parenrightbigg . (125) thus when viewed parallel to the direction of travel one is le ft with a blue shift by C0/parenleftbigg 1 +Φg c2/parenrightbigg =λ0/parenleftbigg ν0/parenleftbigg 1 +Φg c2/parenrightbigg/parenrightbigg . (126) Where Φg=−GQm0/radicalbig x2+ (y2+z2)(1−v2/c2)(127) the wavelength would appear to be altered by Lλ=λ′/radicalbigg 1−v2 c2(128) gravitation would be expected to behave fundamentally diﬀe rent here. Since the gravitational force is one of a collective nature, all vacuum ﬁeld sources s hould be included within such a potential formalism. 7Data on the Compton wavelength comes from [18]. 19this is because (126), makes it appear that the wavelength λis increasing while it is really of function of C0andν0. This would appear to yield superluminal travel, such a result is in accordance with [19]. This eﬀect d oes not seem to be limited to AGN either, a superluminal source was also detect ed near SN1987A, see [20]. B origin for inertia and mass increase? From the previous section we have seen that the gravitationa l constant could be considered as the inverse of the Compton wavelength. From (1 26), we now may consider an inverse of the quantum energy E=hvand the classical wavelength C=λν: λd=E C0/parenleftig 1 +Φg c2/parenrightig=hν0 λ/parenleftig ν0/parenleftig 1 +Φg c2/parenrightig/parenrightig→h λ/parenleftig νg/parenleftig 1 +Φg c2/parenrightig/parenrightig≡I (129) such that acceleration yields a mass increase through the de Borglie relation p0/parenleftbigg 1 +Φg c2/parenrightbigg =mv0=λC/parenleftbigg ν0/parenleftbigg 1 +Φg c2/parenrightbigg/parenrightbigg (130) thus mass increase is governed by the action seen in (128). Th is is thus a veriﬁcation of the equivalence principle, i.e. inertial an d gravitational masses are equivalent! From (129) we can now consider a Lagrangian o f form: λ′ d=m0c2−qΦ′E C0/parenleftig 1 +Φg c2/parenrightig =m0c2−q/parenleftbigg ΦE C0/parenleftbig 1+Φg c2/parenrightbig−v·A/c/parenrightbigg /radicalig 1−v2 c2(131) this interpretation runs parallel with [12]. However, this work diverges with equation (126), thus it is λwhich creates observable gravitational eﬀects and notν. With this in mind one can have an action of ∆S=m0c2 h/radicalbigg 1−v2 c2+λC h/integraldisplay Φgdt, (132) therefore the appearance of inertia only appears for partic les with a correspond- ing Compton wavelength through the action ∆S=λC h/integraldisplay (Φg−v·A/c). (133) C gravitation within the QED vacuum It is known that particles such as the proton have a value of λc=h/mpc= 1.321...×10−15m, for the Compton wavelength. Meaning that Gravitation is not a force directed by one term, but all terms of vacuum. Thus it maybe seen 20that gravitation within an nucleus behaves quite diﬀerentl y than the Newtonian prescription. We assume from elementary data that a ‘nuclea r’ gravitational ﬁeld would be conﬁned to the nucleus, not manifesting its eﬀe cts in the global sense. However, for the early universe, one may have a spacet ime with quite diﬀerent cosmological constant(s) then the ones observed t oday, possibly giving new justiﬁcation for inﬂation theory. Lastly in comparison to Appendix A, a beam of protons being accelerated from an AGN source would re sult in another prediction. The proton/electron ‘acceleration’ rates, an d for any particle in general is directly proportional to their Compton waveleng ths. D implications for the planck length It is believed that the planck length lp= (Gh/c3)1/2, is the fundamental cut oﬀ point for the gravitational ﬁeld. Two problems arise with th is work 1) the planck length is determined by the Compton wavelength of the mass in question. 2) the gravitational constant and thus the planck length are alter ed upon acceleration. The ﬁrst problem is not a problem it is simply a modiﬁcation re quired by the theory, and for the large scale universe this result is negli gible under a ﬁrst approximation. The second problem is still a problem, howev er in an earlier work [7], I modiﬁed the planck length with disconcern. Howev er, with that work in mind problem two is easily solved and is given by lp= (GCh/mp0c3)1/2·ψ. (134) WhereGCis the gravitational constant given by the Compton waveleng th, for an electron this becomes Newton’s gravitational constant GN. Andmp0is the rest momentum of the mass in question, which is given by mp0=∓(pc/c−2). (135) This deﬁnition is given by the relativistic wave equation E=±(pc+m0c2). From (134) the gravitational constant can also be considere d in the form GC=2πc3l2 p h= 2πc3(G2 Ch/m2 p0c6)1/2= 2πGC1 2hc3 mp0c3= 2πGC1 2h mp0(136) With this we see a gravitational uncertainty through ∆ x∆p≥1 2Gh. Finally after quantization of (136) we have a pure quantum charge, i. eGCh, thus grav- itation carries the uncertainty of the Compton wavelength. References [1] Pitts J. and Schieve W. Slightly Bimetric Gravitation Preprint gr- qc/0101058 [2] Moﬀat J. Lagrangian Formulation of a Solution to the Cosm ological Con- stant Problem Preprint astro-ph/9608202 [3] Carrol S. The Cosmological Constant Preprint astro-ph/0004075 [4] Roberts M. Vacuum Energy. Preprint hep-th/0012062 21[5] Gundlach J. and Merkowitz M. Measurement of Newton’s Con stant Using a Torsion Balance with Angular Accleration Feedback Phys. Rev. Lett. 85 (2000) 2869–72 Preprint gr-qc/0006043) [6] O’Raifeartaigh L. Early History of Gauge Theories and Ka luza-Klien The- ories, with a Glance at Recent Developments Preprint hep-ph/9810524 [7] Halerewicz E. The quantum vacuum, fractal geometry, and the quest for a new theory of gravity Preprint physics/0008094 [8] Weyl H. Space-Time-Matter . 1952 (New York: Dover) [9] Einstein A. Uniﬁed Field Theory based on Riemannian Metr ics and distant Parallelism. Math. Annal. 102(1930) [10] Hammond R. Class. Quantum Grav. 13(1996) L73–79 [11] Bell S, Cullerne J, and Diaz B. A new approach to Quantum G ravity Preprint gr-qc/0010106 [12] Krough K. Gravitation Without Curved Space-time Preprint astro- ph/9910325 [13] Moﬀat J. Noncommutative Quantum Gravity. Phys. Lett. B491 (2000) 345–52 Preprint hep-th/0007181 [14] Haisch B. and Rueda A. On the relation between a zero-poi nt-ﬁeld-induced inertial eﬀect and the Einstein-de Broglie formula. Phys. Lett. A268 (2000) 224–27 Preprint gr-qc/9906084 [15] Andrade V, Guillen L, and Pereira J. Teleparallel Equiv alent of the Kaluza- Klein Theory. Phys. Rev. D61(2000) 084031 Preprint gr-qc/9909004 [16] Unzicker A. Teleparallel Space-Time with Defects yiel ds Geometrization of Electrodynamics with quantized Charges Preprint gr-qc/9612061 [17] Loup F. The Alcubierre Warp Drive: Hypefast travel with in an electro- magnetic version of general relativity (to appear in gr-qc) [18] Mohr P and Taylor N. CODATA Recommended Values of the Fun damental Physical Constant: 1998. J. Phys. Chem. No. 628(1999), and Rev. Mod. Phys. No. 272(2000). avaible online http://physics.nist.gov/constants [19] Yuan M, et al. The Physics of Blazar Optical Emission Reg ions II: Magnetic Field Oreintation, Viewing Angle and Beaming. Preprint astro-ph/0010215 [20] Nisenson P. A Second Bright Source Detected Near SN1987 A.Preprint astro-ph/9904109 [21] Sahni V. The Case for a Positivie Cosmological Λ-term. Int. J. Mod. Phys. D9(2000) 373–444 Preprint astro-ph/9904398 22
arXiv:physics/0103061v1 [physics.gen-ph] 20 Mar 2001The Nonlinear Maxwell Theory—an Outline Artur Sowa 841 Orange Street New Haven, CT 06511∗ March 2nd, 2001 Abstract The goal of this paper is to sketch a broader outline of the mat hematical struc- tures present in the Nonlinear Maxwell Theory in continuati on of work previously presented in [11], [12] and [13]. In particular, I display ne w types of both dynamic and static solutions of the Nonlinear Maxwell Equations (NM ). I point out how the resulting theory ties to the Quantum Mechanics of Correl ated Electrons inas- much as it provides a mesoscopic description of phenomena li ke nonresistive charge transport, static magnetic ﬂux tubes, and charge stripes in a way consistent with both the phenomenology and the microscopic principles. In a ddition, I point at a bunch of geometric structures intrinsic for the theory. On o ne hand, the presence of these structures indicates that the equations at hand can be used as ‘probing tools’ for purely geometric exploration of low-dimensiona l manifolds. On the other hand, global aspects of these structures are in my view prere quisite to incorporat- ing (quantum) informational features of Correlated Electr on Systems within the framework of the Nonlinear Maxwell Theory. 1 Introduction The general goal of this paper is to examine broader ramiﬁcat ions of the Nonlinear Maxwell Equations (NM) as introduced by me in 1992/93 and fur ther developed in [11], [12], [13]. To this end, I ﬁrst point out that the theory is con siderably richer than that of the classical linear Electromagnetism. In particular, I describe here several distinct types of both static and dynamic solutions on a spacetime of t he formM3×R. On the technical side, I have essentially avoided heavier anal ysis as the solutions are either obtained by means of elementary calculation, or are otherwi se based on deeper analytic work described in [13]. One should be aware that the possibil ities opening in consequence ∗The author is currently with the Pegasus Imaging Corporatio n. This work is beyond the scope of his obligations there and has been performed in his free time. No other institution has been helpful to the author in conducting this research. 1of the introduction of these new structures have not been ful ly exploited in this paper, thus postponing many potential developments into the futur e. More precisely, in the ‘dynamic’ part of the paper I display a solution in the form of a charge-carrying electromagnetic wave. It is a soliton typ e wave that transports charge with constant speed and without resistance. In addition, on e notes existence of a speciﬁc to dimension four nonlinear Fourier type transform—an inte resting structure whose role within the theory is twofold. On one hand, it can be used to ﬁnd and analyze new solutions of the Nonlinear Maxwell Equations. On the other hand, the tr ansform deﬁnes an exotic duality—a (quadratic) generalization of the (linear) Hodg e duality. Consequences of this new duality for the four-geometry will be exploited in the fu ture. The second set of results in this paper is focused around the q uestion of existence and properties of static solutions. To this end, I ﬁrst examine t he situation on the Euclidean three-space. In particular, one takes note of the occurrenc e of global structures in the form of magnetic ﬂux tubes as well as the so-called charge str ipes. It is interesting from the point of view of geometry that these objects exist in gene ral only on three-manifolds whose fundamental group is not ﬁnite. This is tied to the geom etric fact that the nonlinear gauge theory at hand induces an additional structure on M3—namely a taut codimension one foliation. These global aspects of static solutions pro mpt an assumption of topological point of view. Accordingly, I sketch the possibility of cons tructing ‘nonlinear cohomology’ that would account for a sort of ‘ﬂux tube’ invariant of a thre e manifold. The discussion here is based on two particular examples that I feel provide a n optimal illustration of the underlying concept. The Nonlinear Maxwell Equations, cf. (1-3) below, involve a vector potential that encodes the electric and magnetic ﬁelds in the usual way as we ll as an additional scalar f. The function fcontains information, extractable in a certain simple cano nical way, about the local value of the ﬁlling factor (also known as the ﬁlling fraction). (The ﬁlling factor is deﬁned as the number of quanta of the magnetic ﬁeld p er electron charge in the ﬁrst Landau level. It is then natural and eﬀective to think of the electrons as forming in conjunction with the corresponding magnetic ﬂux quanta c omposite particles—either bosons or fermions, or Laughlin particles depending on the a ctual value of the ﬁlling factor.) It is thus postulated that the ﬁlling fraction—typ ically an input of a microscopic theory that is always assumed constant microlocally—is all owed to slowly vary in the coarser scale. In fact, it was shown in [13] that NM predict oc currence of phase changes that lead to formation of vortices in f, anda fortiori in the magnetic ﬁeld. This picture conforms with the well known analogy between the Quantum Hal l Eﬀects and the High- TcSuperconductivity. An inquisitive reader might now point a t the following seeming conundrum. The physical interpretation of fas a ﬁlling factor requires the presence of two-dimensional geometric structures that endow us with a p ossibility of including the lowest Landau level in the basic dictionary. Thus, it may app ear a priori puzzling, how we are going to retain this interpretation of fin three or four spatial dimensions? The answer is provided by the intrinsic structure of the NM themselves. On one hand, it is shown below that the ﬁlling factor variable may be completely fact ored out of the equations when viewed in the complete four dimensions of spacetime. Needle ss to say, if one attempted to analyze such f-free form in two dimensions the fvariable would reemerge without 2change as it is there encoded in the magnetic ﬁeld B=b/f,b= const. On the other hand, one notes that a remnant or a generalization of the ﬁlli ng factor interpretation carries over to three dimensions. Namely, the NM in three dim ensions imply existence of a codimension one foliation of the three-space associate d with the static solutions. Moreover, one notes here that the NM do not a priori introduce any restrictions as to the type of the resulting foliation—in fact any regular foli ation and even foliations with singularities introduced by degenerating leaves are admit ted by the equations. However, as already mentioned above the existence of solutions of a sp ecial type, namely the ﬂux- tube type, implies geometrical restrictions on the foliati on and topological restrictions on the three-manifold. Indeed, in this case foliation must be t aut. It also seems reasonable to expect that the composite particle interpretation remai ns valid in this setting and the number of participating electrons in each leaf is again dete rmined byf, virtually leading to the notion of an eﬀective Landau level. In the last words of this section I would like to admit that, th e subject matter at hand being both new and inherently interdisciplinary as well as b y way of my own background and limitations, it is not always easy to pick the optimal ter minology. Realizing I will unavoidably fail to satisfy in this respect one group of read ers or another, I can only ask the readership to be as tolerant as they can aﬀord and hope tha t in the end substance will triumph over form. 2 Nonlinear Maxwell Equations in Spacetime In what follows, in order to get around rather tedious algebr a while not compromising our understanding of what is essentially involved, I presen t a shortcut style exposition of the necessary calculations. I believe that readers who are w ell familiar with diﬀerential geometry will ﬁnd it easy to reinterpret this calculation in its natural invariant setting, while those who are less familiar with the abstract setting m ay in fact appreciate its absence here. Consider the following system of equations—the Nonlinear M axwell Equations (NM) in the form in which they have appeared in my previous papers. dFA= 0 (1) δ(fFA) = 0 (2) ✷f+a\|FA\|2f=νf. (3) wherefis a real valued function and Ais the electromagnetic vector potential, so that the corresponding electromagnetic ﬁeld is FA=dA. Here,a >0 is a physical constant with unit/bracketleftig Tesla−2m−2/bracketrightig ; I will not discuss the precise physical interpretation of a2in this paper. Further, dis the exterior derivative and δ=⋆d⋆its adjoint. Here, it is assumed that the Hodge star ⋆and the D’Alembertian ✷are induced by the Lorentzian metric tensor on a spacetime of 3 + 1 dimensions. Let me point out that assumingf= const and dragging it to zero one recovers the classical Maxwell eq uations. In this sense, all phenomena of the classical electromagnetism are included i n the present model. 3Now the goal is to better understand the essential ingredien ts of the NM in terms of the classical ﬁeld variables. To this end, let us say the spaceti me is in fact the ﬂat Minkowski space (with the speed of light 1) so that in particular one can identify coeﬃcients of the electric ﬁeld /vectorEand the magnetic ﬁeld /vectorBwith the coeﬃcients of the curvature tensor FA by the formula FA=B1dy∧dz+B2dz∧dx+B3dx∧dy+E1dx∧dt+E2dy∧dt+E3dz∧dt.(4) For the sake of our discussion below it is good to keep in mind t he well known fact that the components of FAare not Lorentz invariant. This property leads one to the der ivation of the Lorentz force, so that the latter one is logically inde pendent of the particular form of a gauge theory formulated in terms of FA. In other words, the Lorentz force remains unchanged and valid as one attempts to modify the ﬁeld equati ons. With this understood, let us continue the discussion of equations (1-3). It would be rather straightforward to rewrite equations (1- 3) in the anticipated Maxwellian form by following the usual procedures for translating (2) i nto the ´Ampere and Gauss laws after having replaced /vectorEbyf/vectorEand/vectorBbyf/vectorB. In fact, this would lead to an ad hoc inter- pretation of fas a material constant—a route taken in our older, one might s aynaive, paper [11]. However, this form of the system oﬀers little ins ight as to the more essential implications of the NM, and one needs to ﬁnd a less obvious ref ormulation. One notes that equation (2) may be equivalently written in th e form fδF A=FA(∇3+1f,.) (5) where ∇3+1stands for the gradient in spacetime. For a reason that will b ecome clear later, one identiﬁes FAwith a skew-symmetric matrix in a standard way F= 0−B3B2−E1 B3 0−B1−E2 −B2B1 0−E3 E1E2E3 0 . It is important to note that by the miraculous property of ske w-symmetric matrices in four dimensions, det F=/vectorE·/vectorBand F−1=1 /vectorE·/vectorB 0E3−E2B1 −E3 0E1B2 E2−E1 0B3 −B1−B2−B30 =1 /vectorE·/vectorBˆF. (I emphasize that ˆFis not the matrix corresponding to the Hodge-dual of FAin the given metric with signature (+ + + −).) On the other hand, representing both the 1-forms and vectors as columns so that in particular ∇3+1f= fx fy fz −ft andδFA= E1,t+B2,z−B3,y E2,t+B3,x−B1,z E3,t+B1,y−B2,x E1,x+E2,y+E3,z , 4one checks directly that FA(∇3+1f,.) =F∇3+1f. This enables us to rewrite equation (5) in the form F−1δFA=∇3+1ln(f). (6) It is perhaps worthwhile to realize that in this context Fis a ﬁberwise-linear mapping from the tangent bundle to the cotangent bundle. Here one assumes f >0 a.e. This conforms with the principle that one will be consistently looking for strong solutions so that in particularfmay always be replaced with \|f\|in (1-3). Next one recalls that on one hand the ﬁrst part of the NM (1) is identical with the analogous par t of the classical Maxwell equations and it encodes the Faraday’s law of magnetic induc tion and the fact that there are no spatially extensive magnetic charges. This gives the ﬁrst four (scalar) equations below, namely (7) and (8). On the other hand, by direct multip lication and regrouping in (6), one obtains four further scalar equations that happen t o radically modify the ´Ampere law. Written in the familiar three-space vector notation, t he NM assume the form ∂/vectorB ∂t+∇ ×/vectorE= 0 (7) ∇ ·/vectorB= 0 (8) (∂/vectorE ∂t− ∇ ×/vectorB)×/vectorE+ (∇ ·/vectorE)/vectorB=−(/vectorE·/vectorB)∇lnf (9) (∂/vectorE ∂t− ∇ ×/vectorB)·/vectorB=−(/vectorE·/vectorB)∂ ∂tlnf (10) (∂2 ∂t2− △)f+a(\|/vectorB\|2− \|/vectorE\|2)f=νf, (11) provided/vectorE·/vectorB/negationslash= 0. In fact, also the case when /vectorE·/vectorB= 0 is worth our attention and will be discussed below. As much as one should avoid indulgin g in formal manipulations of formulas, here the advantage of having the equations rewr itten in several equivalent forms is that they all lead to the discovery of new types of sol utions, the existence of which would be otherwise obscured by notation. This will bec ome more evident in the following sections. As already mentioned in the introduction, I postulate the fo llowing physical interpre- tation:fis the spatially varying ﬁlling factor—a notion central to t he modern composite- particle theories. In fact, the canonical microscopic-the ory interpretation of the ﬁlling factor is valid in two spatial dimensions only, in which case it signiﬁes the ratio of the number of quanta of the ambient magnetic ﬁeld to the number of electrons in the ﬁrst Landau level, cf. [7], [17]. Moreover, the microscopic theo ry oﬀers no hints as to the existence and relevance of an analogous notion in the three- space. A description of the 5interaction of the electromagnetic ﬁeld with fermions in th e ﬁrst Landau level provided by the equations above is valid in the mesoscopic scale. Here , as one ‘zooms out’ from the microscopic scale, the ﬁlling factor is neither a rational n umber nor is it a constant any- more. In fact, as it has been communicated in previous papers the spatially varying ﬁlling factor may assume the form of a vortex lattice, cf. [13]. For t he time being, this point of view is validated by the well known analogy between the Quant um Hall Eﬀect and the High Temperature Superconductivity and it awaits experime ntal conﬁrmation. Moreover, the NM extend the notion of the ﬁlling factor to three spatial dimensions. However, as we will see below, the presence of the ﬁlling factor introduc es an especially interesting modiﬁcation of the laws of Electromagnetism only if the thre e-space comes equipped with a codimension one foliation. This latter fact makes it possi ble to talk about Landau levels in a certain sense, anyhow. Finally, fcan be completely eliminated from the NM in 3+1 dimensions. (In general, this requires that the ﬁrst cohomo logy group of the spacetime vanishes.) In that case the NM can be written in the f-free form dFA= 0 d/parenleftig F−1δFA/parenrightig#= 0 (12) δ/parenleftig F−1δFA/parenrightig#−/vextendsingle/vextendsingle/vextendsingleF−1δFA/vextendsingle/vextendsingle/vextendsingle2+a\|FA\|2−ν= 0, (13) where # is the isomorphism of the tangent and the cotangent bu ndles given by the metric. Indeed, under the assumption of vanishing ﬁrst de-Rham coho mology, equation (6) is equivalent to its integrability condition (12). Moreover, since the last scalar equation of the system can be written in terms of dlnfin the form δdlnf− \|dlnf\|2+a\|FA\|2−ν= 0, equation (6) also implies (13). Computation of the symbol sh ows that the system obtained in this way is non-hyperbolic—in fact its degeneracy is of hi gher order. Thus, this form of the NM appears impractical for any mathematical work, and an introduction of the dimensionless scalar fis necessary also from the point of view of analysis. Neverth eless, as indicated in the Introduction and the discussion above, p hysical implications of the existence of an f-free form of the NM are important. 3 Geometry Behind the Equations The geometrical arena of the Maxwell equations consists of a spacetime, say N, and a principalU(1)-bundle, say P, stack up above N. In addition, it seems any description of the interaction of the electromagnetic ﬁeld with fermion s requires, at least within this framework, a principal connection, i.e. a smooth (at least a .e.) distribution of horizontal planes that is invariant with respect to the circle action. T his distribution can be written as kerA= kerfAforf/negationslash= 0. In addition, if U(1) is to remain the elemental symmetry group of Electromagnetism, then fmust be constant along the ﬁbers so that it eﬀectively 6descends to a function on N. In particular, within this dictionary one can construct a Kaluza-Klein metric on P, which is given by µA(X,Y) =g(π∗X,π ∗Y) +aA(X)A(Y), where the unit of a >0 must be/bracketleftig Tesla−2meter−2/bracketrightig if the unit of length on Pis to be [meter] and the unit of FA=dAis to remain, say, [Tesla]. Let us say the corresponding Laplace-Beltrami operator on forms is then △µA=△A. Calculation shows that the condition △A(fA) =νfA is equivalent to the system of equations (1–3), cf. [11]. 4 Exotic Duality For the sake of discussion in this section, consider the NM on either a Lorentzian or a Riemannian four-manifold as the metric signature plays a se condary role. In particular, it is preferable to replace the ✷-notation with the △-notation. Assume for the sake of simplicity that the second cohomology group of the manifold is trivial. Omitting the constanta, write the system one more time in the form δ(fdA) = 0 (14) − △f+\|dA\|2f=νf. (15) Since (14) implies d(f ⋆dA ) = 0, one has f ⋆dA =d˜Aso that dA=±1/f ⋆d ˜A (16) and the new form ˜Asatisﬁes a dualsystem of equations δ(1 fd˜A) = 0 (17) − △f+\|d˜A\|21 f=νf. (18) This is a functional transform reminiscent of the Fourier or the Backlund transforms, notwithstanding the fact that all transforms are somewhat r eminiscent of one another. In particular, the resulting dualistic perspective has the expected property that trivial solutions of one of the systems lead to more complex solution s of the dual system. To illustrate the idea, let me now present a few examples of dual solutions on R4with either the Euclidean or the Minkowski metric as speciﬁed in the disc ussion. Example 1. Let the metric be Euclidean and take dA=Edz∧dt,E= const, and f=f(x,y). Equation (14) is automatically satisﬁed and (15) assumes the form −fxx− fyy= (ν−E2)fso thatf= cos (k1x+k2y+α) fork2 1+k2 2=ν−E2solves the problem. Nowd˜A=±f(x,y)dx∧dyand it staisﬁes equations (17-18). 7Example 2. Departing for a while from the assumption of vanishing secon d cohomology, let us reinterpret the previous example on a four-torus assu ming periodicity of coordinates (x,y,z,t ) with period 2 π. Note that the ﬁrst bundle is necessarily nontrivial as the cohomology class [ dA]/negationslash= 0. Let us allow the function fdrop its dependence on yso that, say,f= cosx, provided the ‘right choice’ of νhas been made. Now, d˜A=d(sinxdy) is an exact form so the second bundle is topologically trivial. Example 3. ConsiderdA=Bdx∧dyandf=f(z,t) so that (14) is satisﬁed. Let us now look at the metric with signature (+++ −) so that (15) means ftt−fzz= (ν−B2)f. The general solution of this equation is a standing wave with variable amplitude. This pattern is inherited by d˜A=f(z,t)dz∧dt(up to the sign again) which satisﬁes (17-18). Example 4. Let us for a change begin on the other side and take, say, d˜A=edz∧dtand f=f(x,y). Again, the ﬁrst equation (14) is automatically satisﬁed w hile (15) becomes −fxx−fyy+e2/f=νf. As explained in [13] (see also remarks at the end of section 6 below) apart from the trivial constant solution, this probl em also has a solution in the form of a vortex lattice. In the latter case dA=f(x,y)dx∧dysatisﬁes (14-15) and represents static magnetic ﬂux tubes. I emphasize that only the vector potential Aand the ﬁlling fraction variable fthat appear in the ﬁrst set of equations have physical interpreta tion. Reassuringly, the presence of a nontrivial fin examples 1and3did not contribute anything unexpectedly strange to the constant electric and magnetic ﬁelds in these example s, while it ‘introduced’ ﬂux tubes in example 4. Although one could consider similar interpretation of the transformed vector potential ˜A, just as one can for any U(1)-connection, I feel this is uncalled for and would probably be unjustiﬁable at this point. Nevertheless , the existence of the transform is a remarkable fact whose possible applications to four-ma nifolds will be explored more thoroughly in the future. In a way, this new duality is a gener alization of the regular Hodge-star duality that may be compared to the projective ge neralization of the Euclidean reﬂection. This analogy may be justiﬁed in the following way . Projective duality is induced by a ﬁxed quadratic form. What is the NM analog of that object? Introduce notation ϕ= lnf. A direct calculation shows that (14-15) may be written in th e form of a system of quadratic equations δdA+⋆(dϕ∧⋆dA) = 0 (19) − △ϕ− \|dϕ\|2+\|dA\|2−ν= 0. (20) This form of the equation has one other advantage. Suppose on e has found a solution (A,ϕ) of (19-20). One can now use gauge invariance of the equation s in the following way. Letχbe a solution of the equation δdχ=−δA. The existence of a solution χfollows from the Fredholm alternative when the metric is positive deﬁnite, and it amounts to solving a linear wave equ ation in a Lorentzian metric. One can now replace AwithA+dχ(and denote the resulting form by Aagain). In the new gauge δA= 0, so that Ain fact satisﬁes △A+⋆(dϕ∧⋆dA) = 0. (21) 8The system that consists of (21) and (20) is either quasiline ar elliptic or hyperbolic, depending on the metric. Solving the latter system may not be helpful at all in ﬁnding solutions of the original (19-20), since one cannot guarant ee that a solution satisﬁes the Lorentz gauge condition δA= 0. However, solutions of (19-20) a fortiori satisfy (21) and (20) so that in particular they will obey all a priori estimat es on the solutions of, say, quasilinear hyperbolic systems. In particular, this point of view may justify the claim that the phenomena described in this paper shed some light on the complex nature of quasilinear systems of PDE of certain types in general. 5 Charge Transport and Charge Stripes I will now take full advantage of the (7-11) form of the NM. In a nalogy to the electromag- netic wave in vacuum, that one recalls is counted among the so lutions of this system, one wants to look for a solution with /vectorE·/vectorB= 0. In the end I will check that the new solution of (7-11) in fact satisﬁes (1-3) which is not a priori guarant ied. Make an Ansatz /vectorB=B1∂ ∂x+B2∂ ∂y,/vectorE=e/parenleftigg −B2∂ ∂x+B1∂ ∂y/parenrightigg , (22) wheree,B1andB2are a priori functions of ( x,y,z,t ) that are smooth a.e. and neither one of them vanishes identically. As an immediate consequen ce, one obtains that (7) and (8) are equivalent to B1,t= (eB1),z (23) B2,t= (eB2),z (24) (eB1),x+ (eB2),y= 0 (25) B1,x+B2,y= 0 (26) which implies e,xB1+e,yB2= 0. (27) On the other hand, (9) and (10) are equivalent to (B2,x−B1,y)eB1= (eB2),xB1−(eB1),yB1 (28) (B2,x−B1,y)eB2= (eB2),xB2−(eB1),yB2 (29) (−(eB2),t+B2,z)B1+ ((eB1),t−B1,z)B2= 0. (30) Equations (23), (24) and (30) imply that eis in fact constant e=±1. (31) Using (23) and (24) again, one obtains B1=B1(x,y,t +ez), B 2=B2(x,y,t +ez). 9In particular, /vectorBand/vectorEare not compactly supported. At this point, the only conditi on left a priori unfulﬁlled is the vanishing divergence condit ion. Thus, all equations (23-30) above are satisﬁed iﬀ there is a function ψ=ψ(x,y,t +ez) such that B1=−ψy(x,y,t +ez), B 2=ψx(x,y,t +ez). (32) Deﬁning the electric and magnetic ﬁelds by (22) with e=±1 , so that in particular \|/vectorE\|=\|/vectorB\|, and choosing fthat satisﬁes the linear wave equation (11), one obtains a solution of (7-11) . However, physical solutions must in addition satisfy the a priori more restrictive sys- tem (1-3). Consider FAas given in (4). Equation (1) is satisﬁed automatically sinc e it is equivalent to (7-8). On the other hand, (2) becomes (fB2),x−(fB1),y= 0 (33) (feB 2),t−(fB2),z= 0 (34) −(feB 1),t+ (fB1),z= 0 (35) −(fB1),y+ (fB2),x= 0 (36) Now, (34) and (35) imply via (32) that f=f(x,y,t +ez). In particular, f,tt−f,zz= 0. Thus, (1-3) has been reduced to the following system of tw o equations: −f,xx−f,yy=νf (37) (fψ,x),x+ (fψ,y),y= 0. (38) The ﬁrst equation above admits three types of classical solu tions. Namely, f=  A(t+ez) ln (x2+y2) ν= 0 A(t+ez) cos (k1x+k2y+α(t+ez))ν=k2 1+k2 2 A(t+ez) exp (k1x+k2y) ν=−k2 1−k2 2.(39) Observe that each solution eﬀectively depends on one harmon ic variable in the ( x,y)- domain—either, u=k1x+k2yoru= lnr2= ln (x2+y2). Thus, equation (38) is satisﬁed if ψu=C(t+ez)/f(u,t+ez), for an arbitrary function Cof one variable. Therefore, in view of (32) one obtains three types of solutions (redeﬁning C) [B1,B2] =  C(t+ez)/(r2lnr2)[−y,x] C(t+ez) sec (k1x+k2y+α)[−k2,k1] C(t+ez) exp (−k1x−k2y)[−k2,k1](40) in correspondence with (39). Since one is looking for strong solutions, one has the freedom to cut oﬀ pieces of the classical solutions (by restricting t he domain) and to put them 10back together. In this way, one obtains solutions that are ei ther continuous or have jump discontinuities but may be guarantied to remain bounde d. Last but not least, it is physically correct to interpret the divergence of the e lectric ﬁeld as charge ρand −∂ ∂t/vectorE+∇ ×/vectorBas the electric current. One checks that for solutions as abo ve the (x,y)- component of current vanishes while the z-component jis equal to −eρ. More precisely, one obtains that piecewise eρ=−j=  4C(t+ez)/(r2ln2r2) νC(t+ez) sec (k1x+k2y+α)tan(k1x+k2y+α) −νC(t+ez) exp (−k1x−k2y)(41) in correspondence with (39) and (40). In addition to the piec ewise smooth distribution of charge, one should include charge concentrated on singular surfaces where the electric ﬁeld has jump discontinuities as indicated by the distributiona l derivative ∇ ·/vectorE. Therefore, charge is transported along the z-axis with the speed e=±1 and without resistance as the vector of current is perpendicular to the electric ﬁeld. Charge is mostly concentrated along charge stripes where the electric and magnetic ﬁelds have singularities. T he net current depends on the particular choice of a (strong) solut ion. Of course, the theory does not tell us how to solve the practical problem of electro nics—namely, how to create conditions for a particular function C=C(t+ez), constant νand a desired mosaic of singularities to actually occur in a physical system. 6 Static Solutions and Magnetic Flux Tubes The classical Maxwell equations admit static solutions of t wo types only: the uniform ﬁeld solutions, and the unit charge or monolpole-type solutions , as well as superpositions of these fundamental types of solutions. As we will see below, t he nonlinear theory encom- passes a larger realm including the magnetic-ﬂux-tube type and the charge-stripe type solutions. These additional conﬁgurations require nonlin earity and cannot be superposed, which gives them more rigidity. In the next section we will se e what can be said about the variety of such solutions, while in this section I will only d isplay a single example of this type. Apart from the applicable goal, the idea is to present a n example that possesses all the essential features of the general class of solutions yet the required calculation is free of more subtle geometric technicalities. Time-independent solutions of the NM posses physical inter pretation only if they satisfy the equations in the classical sense almost everywh ere. Assuming that all ﬁelds are independent of time (7-11) takes on the form ∇ ×/vectorE= 0 (42) ∇ ·/vectorB= 0 (43) −(∇ ×/vectorB)×/vectorE+ (∇ ·/vectorE)/vectorB=−(/vectorE·/vectorB)∇lnf (44) (∇ ×/vectorB)·/vectorB= 0 (45) 11− △f+a(\|/vectorB\|2− \|/vectorE\|2)f=νf, (46) under the assuption that /vectorE·/vectorB/negationslash= 0 a.e. Adopt an Ansatz that the integral surfaces of the planes perpendicular to the ﬁeld /vectorBare ﬂat, say, /vectorB=b(x,y)∂ ∂z. One easily checks that equations (43) and (45) are satisﬁed. Assume in addition that the electric ﬁeld is potential, i.e. /vectorE=∇ψ(x,y,z ),whereψz/negationslash= 0 a.e. so that (42) is satisﬁed. Remembering notation ϕ= lnf, one calculates directly that (∇ ×/vectorB)×/vectorE=−ψzbx∂ ∂x−ψzby∂ ∂y+ (ψxbx+ψyby)∂ ∂z, while (∇ ·/vectorE)/vectorB=△ψb∂ ∂z, and (/vectorE·/vectorB)∇ϕ. Thus, equation (44) is equivalent to the following system of three equations ψz(bϕx+bx) = 0 ψz(bϕy+by) = 0 b△ψ−ψxbx−ψyby−bψzϕz= 0 and sinceψz/negationslash= 0 one obtains from the ﬁrst two equations ϕ(x,y,z ) =ϕ1(x,y) +ϕ2(z) andb=βexp (−ϕ1), while the third equation assumes the form △ψ+∇ψ· ∇ϕ= 0 (47) At this point the NM have been reduced to the system of just two scalar equations (46) and (47). Denote f1= expϕ1andf2= expϕ2and assume in addition ψ=ψ(z) so that ψ′(z) =ǫexp (−ϕ2) =ǫ f2 It now follows from (46) and (47) that the triplet /vectorB=β f1(x,y)∂ ∂z,/vectorE=ε f2(z)∂ ∂z, (48) 12and f(x,y,z ) =f1(x,y)f2(z) (49) is a solution of the NM if only f1andf2satisfy a decoupled system of semi-linear elliptic equations −f′′ 2(z)−ε2 f2(z)=ν2f2(z) (50) − △f1(x,y) +β2 f1(x,y)=ν1f1(x,y). (51) At this point, I would like to emphasize one more time that in a ﬁeld theory one looks for strong solutions, i.e. solutions that satisfy equations in the cla ssical sense almost everywhere. Typically, such solutions are smooth except fo r singularities supported on a union of closed submanifolds. Furthermore, geometricall y invariant derivatives of the resulting ﬁelds in the distributional sense signify charge s. With this understood, let us brieﬂy turn attention to equation (50). One wants to avoid ho lding the reader hostage to the formal analysis of this elementary equation which mig ht be somewhat distracting. Thus, I have chosen to brieﬂy describe the solutions qualita tively leaving aside technical details that can be easily reconstructed aside by the reader . First, one notes that if ν2>0 then a solution is concave, while for ν2<0 it will be convex for large values where f2 2>−ε2/ν2. Assuming formally that f2is a function of f′ 2(piecewise), one reduces (50) to the ﬁrst order equation df2 dz=±/radicalig c−ν2f2 2−ε2lnf2 2. Thus, there are essentially two types of positive solutions , depending on the actual values of constants c,ε,ν 2. The ﬁrst type includes solutions that assume value 0 at a cer tain pointz0and increase monotonously to inﬁnity as z→ ∞ as well as the symmetric solutions deﬁned between −∞and some point, say z0again, where they reach 0. These solutions require ν2<0 and they asymptotically look like exp ( ±(−ν2)1/2z) One can use both branches in order to put together a strong solution that forms a cusp or a jump discontinuity at z0. The second type consists of solutions that are concave, ris e to the highest peak at f2=m, whenc−ν2f2 2−ε2lnf2 2= 0, and fall oﬀ to 0 on both sides in ﬁnite time while being diﬀerentiable in-between. Selectin g the constants and combining both types of solutions piecewise segment-by-segment one o btains strong solutions f2that in turn provide electric ﬁelds according to formula (48). Since, with the exception of the trivial constant solution, there are no global smooth solutions, one concludes that either /vectorEis constant or there exist charge stripes located at planesz= const where f2(z) has singularities. The distributional derivative is in ea ch case equal to the Dirac measure concentrated at z= const as above and scaled by the size of the jump, and classical derivatives on both sides of the si ngularity. Even in absence of a jump, the charge will switch from negative to positive th us forming what can be amenably called a charge-stripe. An example of this is shown inFig.1. It is much more diﬃcult to ﬁgure out solutions of the second eq uation. I refer the reader to [13] for a more thorough analysis, while here I will just brieﬂy summarize my 13previous ﬁndings. Solutions of equation (51) correspond to critical points of the functional L(f1) =1 2/integraltext\|∇f1\|2+β2/integraltextln(f1) /integraltextf2 1 which is neither bounded below nor above, so that one is looki ng at the problem of ex- istence of localextrema. The equation always admits a trivial constant solu tion. But, as it is shown in [13], it also possesses nontrivial vortex latt ice solutions. More precisely, if βis larger than a certain critical value then there is a noncon stant doubly periodic func- tionfwhich satisﬁes the ﬁnite diﬀerence version of (51) everywhe re except at a periodic lattice of isolated points, one point per each cell. In this w ay, a lattice of ﬂux tubes, cf.Fig.2, emerges as a solution of the NM. For the time being, the proof of this fact relies on ﬁnite-dimensionality essentially, and does not a dmit a direct generalization to the continuous-domain case. However, physical parameters , like/integraltextf2andβ, are asymp- totically independent of the density of discretization. Th us, I conjecture existence of the continuous domain solutions that satisfy the equation a.e. in the classical sense and re- tain the particular vortex morphology. Presently, the esse ntial obstacle to proving this conjecture is lack of a regularity theory for the discrete vo rtex solutions. The proof in [13] is carried out in the (discretized) torus setting. One belie ves that vortex type solutions exist on any closed (orientable) surface. 7 Topological Quantum Numbers Every gauge theory comes equipped with an associated set of t opological invariants— usually characteristic classes of the bundles used to intro duce the gauge ﬁeld. Articles [4], [5], [6] teach us how such topological invariants may be mani fested in an electronic system as observable quantum numbers. The Nonlinear Maxwell Theor y is naturally equipped with two kinds of topological invariants. On one hand, one ha s the ﬁrst Chern class of the originalU(1)-bundle. Additionally, we will see below that in the case of static solutions the NM give us an additional set of invariants deﬁned directl y by the foliated structure of the underlying three-manifold. (In the discussion below , I generally assume for the sake of simplicity that Mis a closed orientable manifold unless stated otherwise.) I n this section I will make an eﬀort only to identify rather than exploit to the fullest the geometric and topological ramiﬁcations of this nonlinear t heory of Electromagnetism. To gain some initial impetus, let us be guided by the following q uestion What are the necessary and suﬃcient conditions on a Riemanni an three- manifoldMfor the NM to admit a separation of variables of the type seen in the previous section, i.e. for the equation (51) to de couple so that its solutions will generate magnetic-ﬂux-tube type soluti ons onM? A question of this type is typical in algebraic topology wher e one is asking about global obstructions to the presence of certain algebraic factoriz ation properties of analytic ob- jects, like linear diﬀerential equations as it is the case fo r, say, the de-Rham cohomology groups. In our case, the equations are nonlinear, but the pri nciple remains the same. The 14importance of these questions for practical issues of Elect romagnetism is twofold. First, one wants to know how big is the set of possible conﬁgurations —especially in the absence of the superposition principle. Secondly, I believe the top ological invariants displayed below are directly on target in an eﬀort to explain and descri be the nature of certain rigid structures, like the Quantum Hall Eﬀects, that physically o ccur in electronic systems. First, it needs to be emphasized that the static ﬁeld equatio ns I want to consider, i.e. the equations that descend from the four-dimensional s pacetime via time-freezing coeﬃcients, are distinct from the equations (1-3) consider ed directly on a three manifold. Secondly, the equations (42-46) are only valid on a Euclidea n space. The geometry behind these equations is easier to identify when they are rewritte n in an invariant form that can be considered on any three-manifold in a coordinate indepen dent setting. Fix a Riemannian metric on Mwith scalar product < .,. > extended to include measuring diﬀerential forms. Denote by BandEthe forms dual to the magnetic and electric ﬁeld vectors; recall notation ϕ= lnfand puta= 1. The static NM assume the form dE= 0 (52) δB= 0 (53) ⋆(⋆dB∧E) + (δE)B=−<E,B >dϕ (54) dB∧B= 0 (55) △ϕ+\|dϕ\|2+\|E\|2− \|B\|2+ν= 0. (56) Equation (55) is the familiar Frobenious condition on integ rability of the distribution of planes given by ker B. One always assumes Bis nonsingular a.e. so that the distribution isa priori also deﬁned a.e. For convenience, it is assumed throughout t his section that the foliation determined by ker Bis smooth. (It is quite clear that for the ﬂux-tube type solutions the distribution extends through the singular po ints and is deﬁned everywhere. At this stage, however, it is hard to make a formal argument to this eﬀect, hence the a priori assumption.) The condition of smoothness implies that the t hree-manifold Mmust have vanishing Euler characteristic. In particular, singu lar foliations, some of which may be associated with other types of solutions of the NM, are exc luded from the discussion below. It follows that there is a 1-form α, known as the Godbillon-Vey form, such that dB=α∧B. This form is not deﬁned uniquely. However, as is well known, d(α∧dα) = 0 and the Godbillon-Vey (GV) cohomology class [α∧dα]H3(M) is uniquely deﬁned. On a three manifold this class can be eval uated by integration result- ing in a Godbillon-Vey number Q=/integraldisplay Mα∧dα. 15This invariant poses many interesting questions that have n ot been fully resolved by geometers yet. Below, I will justify two observations. Firs t, the condition of existence of the magnetic ﬂux-tube solutions imposes both local and gl obal restrictions on the foliation. Second, magnetic ﬂux-tube solutions exist in to pologically nontrivial situations with nonzero GV-number Q. This is formally summarized in the two propositions that follow. They are far from the most general statements that ca n be anticipated in this direction, but are also nontrivial enough to suggest a conje cture regarding quantization of the GV-number that I will formulate following Propositio n 1. Consider a priori a foliation given by ker Blocally. First, one introduces a local coordinate patch ( x,y,z ) such that the foliation is given by the ( x,y)-planes and \|dz\|= 1. In particular B=β(x,y,z )dz. Letγ=g(x,y,z )dx∧dydenote the volume element on a leaf. One has ⋆B=β(x,y,z )γ. Equation (53) becomes d(β(x,y,z )g(x,y,z )dx∧dy) = 0. Thus, there is a function χ= χ(x,y) such that β(x,y,z ) =χ(x,y) g(x,y,z ). A calculation analogous to that in the previous section show s that the whole system (53-56) is reduced to /parenleftigg lnχ(x,y) g(x,y,z )/parenrightigg x=−ϕx,/parenleftigg lnχ(x,y) g(x,y,z )/parenrightigg y=−ϕy (57) δE+<E,dϕ> = 0 (58) △ϕ+\|dϕ\|2+\|E\|2−/parenleftiggχ(x,y) g(x,y,z )/parenrightigg2 +ν= 0. (59) Observe that in order to obtain factorization ϕ(x,y,z ) =ϕ1(x,y) +ϕ2(z) (60) it is necessary and suﬃcient that g=g(x,y), (61) i.e. a priori dependence of gonzis dropped. If that holds, the equations (58) and (59) can be decoupled with an additional Ansatz E=e(z)dz. One also has that χ/g=bexp (−ϕ1) for a constant band △x,yϕ1+\|dϕ1\|2−b2exp (−2ϕ1) +ν= 0. (62) Conversely, if (62) and (60) hold, then so must (61) and the me an curvature hof a leaf vanishes. Indeed, by deﬁnition h=δ/parenleftigg1 \|B\|B/parenrightigg =−⋆d(g(x,y)dx∧dy) = 0. This implies 16Proposition 1 For the existence of ﬂux-tube type solutions—in the sense of existence of factorization (60) and decoupling of equation (62)—it is ne cessary that the foliation given bykerBbe taut, i.e. the mean curvature of leaves must vanish. In par ticularπ1(M)must be inﬁnite. Proof. The ﬁrst part has been shown above. The second part follows fr om a result of D. Sullivan [14] that he deduced from the result of Novikov on th e existence of a closed leaf that is a torus (cf. [8], and [16] for additional general mate rial and references). ✷ In particular, there are no ﬂux-tube type solutions of the NM that would conform with the Reeb foliation [9]. This is a practical issue since t he Reeb foliation exists on a solid torus, so that in principle it might be observed exper imentally which would be inconsistent with the theory at hand. This fact is also inter esting for another reason. Namely, according to the celebrated theorem by Thurston in [ 15] each real number may be realized as the Godbillon-Vey number for a certain codime nsion one foliation on the three-sphere S3. The known proof of this result uses the Reeb foliation in an e ssential way. I do not know if this fact is canonical, i.e. if the presen ce of the Reeb foliation is necessary for the result to hold, but if it turns out to be so then excluding the Reeb foliation from the game should result in a reduction of the ra nge of the G-V number, possibly to a discrete subset of the real line. In such a case, the resulting set of the G-V numbers accompanying ﬂux-tube type solutions of the NM w ould also be discrete. This is consistent with my expectation that these invariant s must be related with (both the integer and the fractional) Quantum Hall Eﬀects. Future research should bring a resolution of this problem. Another observation is that the factorization given by (60) and (62) does exist in topologically nontrivial situations. More precisely, I wa nt to consider solutions of the NM onPSL(2,R) and its compact factors. These three-manifolds are equipp ed with interesting codimension one foliations known as the Roussa rie foliation [10]. Let the Lie algebra sl(2,R) be given by [X,Y+] =Y+,[X,Y−] =−Y−,[Y+,Y−] = 2X. Pick a metric on PSL(2,R) in which the corresponding left-invariant vector ﬁelds X,Y+, andY−are orthonormal and let µ,ν+, andν−be the corresponding dual 1-forms. One checks directly that dν−=µ∧ν−. so that the distribution ker ν−is integrable and µis the GV-form of the resulting fo- liation. In particular, one can introduce local coordinate s (x,y,z ) such that ∂x=X, ∂y=Y+,∂z=Y−. This foliation descends to compact factors of PSL(2,R) that can each be identiﬁed with T1Mg—the total space of the unit tangent bundle of the hyper- bolic Riemann surface of genus gthat depends on our choice of the co-compact subgroup acting onPSL(2,R) by isometries. Moreover, the GV-integrand µ∧dµis proportional to the natural volume form on the three-manifold. As a result of this, the corresponding GV-numbers Q=/integraldisplay T1Mgµ∧dµ=−2Vol(Mg) 17assume values in a discrete set. I want to look for solutions o f the NM that satisfy the Ansatz B=βν−. (63) In particular, (the Frobenious) equation (55) is satisﬁed a utomatically. Moreover, since ⋆dB= (Y−β)µ∧ν+∧ν−, equation (53) implies β=β(x,y). (64) As before, one checks that (54) implies (58) as well as Xϕ=−XlnβandY+ϕ= −Y+lnβ. In consequence, one again has (60) and assuming E=e(z)ν−as before one obtains (62). In consequence, the following holds true. Proposition 2 The Roussarie folitions on PSL(2,R)and its compact factors satisfy the factorization condition for the existence of magnetic ﬂux- tube type solutions in the sense that the tangent distribution can be expressed as kerBa.e. and one can reduce the NM to the form (60-62). In a similar way one can obtain factorization (60) and (62) fo r other foliations, like the natural foliation on say S2×S1. 8 More on the Physical Framework of the NM It is natural to ask if the NM descend from a Lagrangian functi onal depending on the two variables Aandf, say Φ(A,f), via the Euler-Lagrange calculus of variations. The answer is negative as one can easily see considering that in g eneral a gradient must pass the second derivative test: δ2 δAδfΦ =δ2 δfδAΦ —a condition that cannot be satisﬁed by the expressions in (1 -3) viewed as the gradient, say (δ δAΦ,δ δfΦ), of an unknown functional Φ. This suggests that the NM may c onstitute just a part of a broader theory that would encompass addition al physical ﬁelds. In other words, the equations (1-3) would have to be coupled to some ot her equations via additional ﬁelds. In addition, such coupling would have to induce only a very small perturbation of the present picture that one believes is essentially accura te. Such possibilities may become more accessible in the future. Among other, perhaps related goals is that of deriving the NM equation directly from the microscopic principles. The well-known analogy between the Quantum Hall Eﬀects and H igh-TcSupercon- ductivity suggests that there should exist vortex lattices involving the so-called ﬁlling factor (microlocally a constant scalar) that plays a major role in t he description of Com- posite Particles . The NM describe exactly this type of a vortex-lattice. Simu lation and theory show that this system conforms with the experimental ly observed physical facts. It stretches the domain of applicability of the Maxwell theo ry to encompass phenom- ena such as the Magnetic Oscillations ,Magnetic Vortices ,Charge Stripes that occur in low-temperature electronic systems exposed to high magnet ic ﬁelds. 18There are other systems of PDE that admit vortex-lattice sol utions and are conceptu- ally connected with Electromagnetism, like the well known G inzburg-Landau equations valid within the framework of low Tctype-II superconductivity, or the Chern-Simons ex- tension of these equations which, some researchers have sug gested, may be more relevant to the Fractional Quantum Hall Eﬀect and/or High- TcSuperconductivity, cf. [18]. The free variables of these equations are the so-called order parameter (a section of a com- plex line-bundle) and a U(1)-principal connection, both of them containing topolog ical information. In the case of NM, all the topological informat ion is contained in one of the variables, i.e. the principal connection, while the other i s a scalar function. An additional advantage of the NM is in that it remains meaningful in three- plus-one dimensions just as well as in the two-dimensional setting. I would also like t o mention that recently other researchers have introduced nonlinear Maxwell equations o f another type in the context of the Quantum Hall Eﬀects, cf. [3]. The NM theory presented i n this and the preceding articles of mine is of a diﬀerent nature. Finally, although t his is far from my areas of expertise and the remark should be received as completely ad hoc , I would also like to mention that yet another context in which foliations come in touch with the Quantum Hall Eﬀect is that of noncommutative geometry, cf. [1]. Let me conclude with a question that may suggest yet another p oint of view. Namely, is there a coalescence between the nonlinear PDEs (in the for m of the NM) and the (Quantum) Information Theory? As it was pointed out, constr uction of error correcting codes may unavoidably require manipulating quantum inform ation at the topological level. Anyhow, this is how I have understood the essential th ought in [2]. Adopting this paradigm would strongly suggest that the eﬀective language of quantum computation should be costructed at many levels, including that of the me soscopic ﬁeld theory in parallel with the language derrived from the basic principl es as it is done now. Future research will likely better clarify these issues. References [1] A. Connes, Noncommutative Geometry, Academic Press, 19 94 [2] M. H. Freedman, plenary talk at the Mathematical Challenges of the 21st Century Conference, Los Angeles, August, 2000 [3] J. Fr¨ ohlich and B. Pedrini, in: A. Fokas, A. Grigorian, T . Kibble, B. Zegarlinski, eds.,Mathematical Physics 2000 , Imperial College Press [4] R. B. Laughlin, Phys. Rev. B 23 (1981), 5632-5633 [5] R. B. Laughlin, Phys. Rev. B 27 (1983), 3383-3389 [6] R. B. Laughlin, Phys. Rev. Lett. 50 (1983), 1395-1398 [7] R. B. Laughlin, Science 242 (1988), 525-533 19[8] S. Novikov, Trudy Moskov. Mat. Obsc. 14 (1965), 248-278, AMS translation, Trans. Moscow Math. Soc. 14 (1967), 268-304 [9] G. Reeb, Actualit ´e Sci. Indust. 1183, Hermann, Paris (1952) [10] R. Roussarie, Ann. Inst. Fourier 21 (1971), 13-82 [11] A. Sowa, J. reine angew. Math. , 514 (1999), 1-8 [12] A. Sowa, Physics Letters A 228 (1997), 347-350 [13] A. Sowa, cond-mat/9904204 [14] D. Sullivan, Comment. Math. Helv. 54 (1979), 218-223 [15] W. Thurston, Bull. Amer. Math. Soc. 78 (1972), 511-514 [16] Ph. Tondeur, Geometry of Foliations, Birkh¨ auser Verl ag, 1997 [17] R. E. Prange, S. M. Girvin, Eds., The Quantum Hall Eﬀect, Springer-Verlag, 1990 [18] S. C. Zhang, Int. J. Mod. Phys. B 6, No. 1 (1992), 25-58 20Fig.1 An example of a strong solution of (50). f=f2(z) is a positive function, the electric ﬁeld is given by formula (48). The resulting charge distribution is obtained by evaluating ∇ ·/vectorE. (In general, ∇ ·/vectorEis understood in the distributional sense). Charge is concentrated along certain plains z= const. This is the basic appearance of charge stripes—intertwining concentrations of positive and nega tive charges. (One should com- pare this static picture with the description of moving char ge stripes in section 5.) Strong solution f=f(z) The corresponding electric field along the z−axis The resulting charge distribution along the z−axis 21Fig.2 The luminance graph of f=f1(x,y) that solves (51). The corresponding magnetic ﬁeld on the right is obtained via (48). Magnetic flux−tubes Vortex−lattice type f 22
arXiv:physics/0103062v1 [physics.gen-ph] 20 Mar 2001A Proposed New Test of General Relativity and a Possible Solu tion to the Cosmological Constant Problem Murat ¨Ozer∗ CIENA Corporation, 991-A Corporate Boulevard Linthicum MD 21090-2227 (February 2, 2008) Following a conjecture of Feynman, we explore the possibili ty that only those energy forms that are associated with (massive or massless) particles couple to t he gravitational ﬁeld, but not others. We propose an experiment to deﬂect electrons by a small charged sphere to determine if the standard general relativity or this modiﬁed one corresponds to reali ty. The outcome of this experiment may also solve the cosmological constant problem. 04.80.Cc, 98.80.Es, 04.20.Cv, 04.50.+h The equivalence of mass and energy, expressed in his celebra ted formula E=mc2, led Einstein to postulate that the energy-momentum tensor Tµνin the ﬁeld equation of general relativity [1] Rµν−1 2gµνR=8πG c4Tµν, (1) contains all kinds of energies, such as matter, radiation, e lectromagnetic, vacuum, etc. Thus assuming that the vacuum energy is negligible, the ﬁeld equation (1) outside a n object of total mass Mand static electric charge Q containg no neutral and charged masses or other ﬁelds around it reduces to Rµν=8πG c4Tµν EM, (2) where Tµν EMis the traceless energy-momentum tensor of the electric ﬁel d due to the charge Qof the object. For the purpose of this letter we shall classify diﬀerent energy types into two. The ﬁrst class is the set of energy types with which massive or massless particles are associated. Th us the energy of an already existing mass distribution is obviously of this class1. Since the energy in an electromagnetic wave (electromagne tic radiation) is carried in packages that behave like massless particles (photons) the electromagnetic radiation energy is also of this class2. Energies associated with other massless particles like neu trinos and gravitons are further examples. Each energy type in this class may rightly be called ‘mass energy’ or ‘particl e energy’. The second class is the set of energy types with which no particles are associated3. The energies in the electric ﬁelds of a static charge distri bution and between the plates of a capacitor as well as the vacuum energy are of the se cond class. There is plenty of emprical proof, such as the successes of th e big-bang cosmology and the deﬂection of light by the sun, that the ﬁrst class energies couple to the gravitationa l ﬁeld. But there does not exist any emprical proof at present for the coupling of the second class energies to the gravitat ional ﬁeld. Therefore, we do not know with certainty if Eq. (2) corresponds to a fact of nature. It lacks experimental su pport. There is the intriguing possibility that we shall consider in this letter, as ﬁrst hinted by Feynman [2] when he said’...Now gravity is supposed to interact with every form of energy and should interact then with this vacuum ener gy. And therefore, so to speak, a vacuum would have a weight-an equivalent mass energy-and would produce a grav itational ﬁeld. Well, it doesn’t! The gravitational ﬁeld produced by the energy in the electromagnetic ﬁeld in a vacuu m-where there’s no light, just quiet, nothing-should be ∗E-mail: mhozer@hotmail.com 1The mass of the distribution may be constantly changing due t o its mechanical energy, its absorbtion or loss of heat energ y, etc. Furthermore, while we can estimate how much electromag netic binding energy of an atom contributes to its rest mass we do not know, for example, how much weak and gravitational e nergies contribute to it. We also know from the E¨ otv¨ os experiment that electromagnetic binding energy contribut es equally to inertial and gravitational massess. We assume this is the case for the other energy types. 2Recall that though massless, an ‘eﬀective mass’ can be assig ned to photons. 3Of course, there is an ‘equivalent mass’ through E=mc2for such energies too. But the crucial point is that there is n o already existing massive or massless particles associated with them. 1enermous, so enermous, it would be obvious. The fact is, it’s zero! Or so small that it’s completely in disagreement with what we’d expect from the ﬁeld theory. This problem is so metimes called the cosmological constant problem. It suggests that we’re missing something in our formulation of the theory of gravity. It’s even possible that the cause of the trouble-the inﬁnities-arises from the gravity inter acting with its own energy in a vacuum. And we started oﬀ wrong because we already know there’s something wrong with t he idea that gravity should interact with the energy of a vacuum. So I think the ﬁrst thing we should understand is how to formulate gravity so that it doesn’t interact with the energy in a vacuum...’ According to this conjecture of Feynman there is the possibi lity that the right side of Eq. (2) may be zero: Rµν= 0. (3) The importance of confronting with experiment the predicti ons of equations (2) and (3) is not merely academic. If it turns out that Eq. (3) is the one favored by nature, we then h ave a very simple solution [2] to the cosmological constant problem [3]. It would mean that being of the second c lass, the vacuum energy does not couple to the gravitational ﬁeld. The present value of the vacuum energy d ensity is as large as it had been in the early universe. The cosmological constant, however, is simply zero, as it ha s always been. The purpose of this letter is to propose a deﬂection of electr ons by a positively charged sphere experiment to distinguish between equations (2) and (3). To this end, we sh all need the solutions of these equations. The solution of Eq. (2) for a static and spherical distribution of mass Mand electric charge Qlocated at r= 0 is known as the Reissner-Nordstrøm solution [4,5]. It is given by ds2=/parenleftbigg 1−2GM c2r+GkeQ2 c4r2/parenrightbigg c2dt2−/parenleftbigg 1−2GM c2r+GkeQ2 c4r2/parenrightbigg−1 dr2− r2dθ2−r2sin2θdφ2, (4) where keis the electric(Coulomb) constant. It should be noted that a ccording to Eq. (3), the electric ﬁeld of the sphere does not contribute to its gravitational ﬁeld and hence must assert itself separately and independently. Therefore, fo r weak ﬁelds Eq. (3) must reduce to Laplace’s equation ∇2(ΦG+ ΦE) = 0, (5) where Φ Gand Φ Eare the gravitational and electric potentials of the sphere . Finding the solution of Eq. (3) proceeds along the lines of the Schwarzschild solution [6]. We ﬁnd ds2=/parenleftbigg 1−2GM c2r−2e mkeQ c2r/parenrightbigg c2dt2−/parenleftbigg 1−2GM c2r−2e mkeQ c2r/parenrightbigg−1 dr2− r2dθ2−r2sin2θdφ2, (6) where−eandmare the charge and the mass of an electron-a test particle-in the viscinity of the spherical object4. To ﬁnd the trajectory of an electron deﬂected by a positively charged sphere we also need the equations describing the trajectory according to equations (2) and (3). They are d2xµ ds2+ Γµ αβdxα dsdxβ ds=−e mc2Fµ αdxα ds, (7) according to Eq. (2), and d2xµ ds2+ Γµ αβdxα dsdxβ ds= 0, (8) 4Eq. (6) can also be obtained intuitively by classical energy considerations. Consider an electron moving radially away from a sphere of mass Mand charge Q. For the electron to escape from this object at a distance rfrom its center and reach inﬁnity with zero speed, the escape velocity vescsatisﬁes mv2 esc/2−GmM/r −ekeQ/r= 0.Replacing vescwithc, the speed of light, (so that the electron cannot escape from the surface of radiu s r) and dividing it by mc2/2 the left side of this equation becomes (1−2GM/c2r−2ekeQ/mc2r), which is the g00in Eq. (6) . 2according to Eq. (3)5. Here Γµ αβandFµα=∂Aα/∂xµ−∂Aµ/∂xαare the connection coeﬃcients and the electro- magnetic ﬁeld strength tensor, with Aµ= (keQ/r,0) being the electromagnetic four-potential of the sphere. Before we indulge in obtaining the orbit equations in experimental ly relevant form, we propose the following experiment. Consider a rectangular vacuum chamber. Let a small metallic sphere of radius R≈2−5cmpositively charged to a voltage V(R) =keQ/Rbe hanged freely from an insulating thread. Let an electron g un be located at a distance d away from the equator (the θ=π/2 plane) of the sphere with an impact parameter bwhich is the horizontal distance between the initial path of the ejected electron beam and the center of the sphere. Thus the initial position of the beam is ( xi, yi) = (−b, d) at an angle φi=π/2 +arctan (b/d). Put a calibrated ﬂuorescent screen on the negative y axis at φ= 3π/2. The initial conditions for solving the diﬀerential equat ions that we shall obtain (see equations (15) and (16) ) are u(φi) =r−1 ianddu/dφ (φi) =/radicalbig r2 i/b2−1/riwithri=√ b2+d2. Make a large enough glass window on the side of the box facing the screen (or monitor the positi on of the electron beam on the screen electronically). Compare the reading of the position of the beam with the predi ctions of the equations that we obtain now. Using spherical coordinates, we write the line element in th e form ds2=eηc2dt2−e−ηdr2−r2dθ2−r2sin2θdφ2. (9) Inserting dθ/ds = 0 in Eq. (7) and integrating the equations obtained for the c oordinates x0=ctandx3=φwe get dt ds=e−η c/parenleftbigg −qkeQ mc21 r+a/parenrightbigg , (10) r2dφ ds=h, (11) where aandhare integration constants. Using equations (10) and (11) in the equation obtained from the condition of timelike geodesics gµν(dxµ/ds)(dxν/ds) = 1, putting eη≈16, and then diﬀerentiating with respect to du/dφ we get d2u dφ2+u=mE h2+m2 E h2u, (12) where u= 1/rand the constant ahas been set to 1 so that when h=l/mc, with l=mr2˙φbeing the ordinary angular momentum, the ﬁrst term on the right side of Eq. (12) agrees wi th the corresponding Newtonian expression. Here mE=ekeQ/mc2=eRV(R)/mc2has the dimension of length and corresponds to mG=GM/c2in the Schwarzschild solution. On the other hand, we obtain from Eq. (8) dt ds=e−η c(13) and Eq. (11) remains intact. By putting eη= (1−2mG/r−2mE/r)≈(1−2mE/r) and proceeding as above we get d2u dφ2+u=mE h2+ 3mEu2. (14) mchis the conserved angular momentum of the electron in its rest frame. hcan be expressed in terms of l, the angular momentum in the laboratory frame, using equations (10) and ( 11) in the Reissner-Nordstrøm case and equations (13) and (11) in our case. They are, respectively, h=l(1 +mEu)/mcandh=l(1−2mEu)−1/mc. Inserting these in equations (12) and (14) we ﬁnally obtain d2u dφ2+u=m2c2 l2mE (1 +mEu), (15) 5Note that charged particles follow the geodesics, Eq. (8), o f the metric gµν. This is a consequence of Eq. (3). Note also that there is a diﬀerent metric for each particle with a diﬀer ent charge-to-mass ratio. The resulting theory, therefore , is a multi-metric theory. 6For a sphere of M= 1kg,R= 5cm,V(R) = 103V, we have for an electron just grazing the sphere gRN 00= (1−2mG/R+ GV(R)2/kec4) = (1−1.48×10−26+ 9.19×10−49)≈1. 3which is the orbit equation for the Reissner-Nordstrøm solu tion, and d2u dφ2+u=m2c2 l2mE(1−2mEu)2+ 3mEu2, (16) which is the orbit equation in our case which we call modiﬁed g eneral relativity. By using the initial conditions stated above these equations can be solved numerically for r= 1/u, the predicted position of the electron beam on the screen from the center of the sphere. By comparing the experimental value with these predictions, the correct theory can be determined. In Figures 1 and 2 we depict the trajectory of t he electrons according to the two theories. It is seen that the diﬀerence between the two predictions is large enou gh. Hence the experimentally favored one can be picked up rather easily. Before we conclude, we wish to clarify the implications of th e E¨ otv¨ os experiment in regard to the theory presented here. The electromagnetic energy of the atoms, or any other f orm of energy, in an object has already been converted to mass ( and thus belongs to the ﬁrst type in our classiﬁcation o f the energy types above). What the E¨ otv¨ os experiment tells us is that the electromagnetic energy contributes in e qual amounts to gravitational and inertial masses. It does not tell us that this energy couples to the gravitational ﬁel d independently as energy. Had the electromagnetic energy coupled to the gravitational ﬁeld independently, a deﬂecti on in the balance of the E¨ otv¨ os apparatus would have been seen when two equal massess having considerably diﬀerent el ectromagnetic binding energies were used. Of course this does not happen. In conclusion, we have explored the conjecture of Feynman on a reformulation of general relativity. In this new scheme only the ‘mass energy’ couples to the gravitational ﬁ eld, but not other energy forms. We have proposed a deﬂection of electrons by a charged sphere experiment. The s igniﬁcance of this experiment is that it not only provides a new test of general relativity but also may point out to the s olution of the cosmological constant problem. Not being a wave, the energy of the vacuum is not associated with quanti zed packages and a formula like E∝f, with fbeing the frequency, cannot be written and an eﬀective mass meff∝f/c2cannot be deﬁned. Thus according to the scheme presented here the ﬁeld equation for vacuum is Rµν= 0, implying that the cosmological constant λ= (8πG/c4)ρVof standard general relativity, with ρVbeing the vacuum energy density, is λ= 0×ρV= 0 here. λhas always been equal to zero! Keeping on mind that (i) standard general relativit y remains one of the least tested of scientiﬁc theories, and (ii) the theory presented here oﬀers a very simple soluti on to the cosmological constant problem, the immediate performance of the experiment suggested here cannot be over emphasized. We wish to thank Dr. Bahram Mashhoon for an e-mail correspond ence on the implications of the E¨ otv¨ os experiment. [1] A. Einstein, Ann. d. Phys. 49, 769 (1916). [2] P. C. W. Davies and J. Brown (ed.), Superstrings, A Theory of Everything, (1988) Cambridge University Press, p.201. [3] See the reviews L. Abbott, Sci. Am. May 1988 , 82 (1988); S. Weinberg, Rev. Mod. Phys. 61,1 (1989); S. M. Carroll, W. H. Press, E. L. Turner, Annu. Rev. Astron. Astrophys. 499, (1 992); V. Sahni and A. Starobinsky, astro-ph/9904398. [4] H. Reissner, Ann. d. Phys. 50, 106 (1916). [5] G. Nordstrøm, Proc. Kon. Ned. Akad. Wet. 20, 1238 (1918). [6] K. Schwarzschild, Berl. Ber. 189 (1916). 4-7.5 -5 -2.5 0 2.5 5 7.5 10 x(cm)-100-80-60-40-20020y(cm) FIG. 1. The trajectories of the electron beam according to th e Reissner-Nordstrøm (the bottom curve) and the Modiﬁed General Relativity (the top curve) theories for an anode-ca thode voltage of 30 kVfor the electron gun located at a vertical distance of 20 cmwith an impact parameter of 7 cm, for a sphere of R= 2.5cmandV(R) = 5kV. 5-7.5 -5 -2.5 0 2.5 5 7.5 10 x(cm)-50-40-30-20-1001020y(cm) FIG. 2. Same as Fig.1, but R= 5cm. 6
arXiv:physics/0103063v1 [physics.atom-ph] 20 Mar 2001Penning collisions of laser-cooled metastable helium atom s F. Pereira Dos Santos, F. Peralesa, J. L´ eonard, A. Sinatra, Junmin Wangb, F. S. Pavonec, E. Raseld, C.S. Unnikrishnane, M. Leduc Laboratoire Kastler Brossel∗, D´ epartement de Physique, Ecole Normale Sup´ erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France () We present experimental results on the two-body loss rates in a magneto-optical trap of metastable helium atoms. Abso- lute rates are measured in a systematic way for several laser detunings ranging from -5 to -30 MHz and at diﬀerent inten- sities, by monitoring the decay of the trap ﬂuorescence. The dependence of the two-body loss rate coeﬃcient βon the ex- cited state (23P2) and metastable state (23S1) populations is also investigated. From these results we infer a rather unif orm rate constant Ksp= (1±0.4)×10−7cm3/s. PACS 32.80.PjOptical cooling of atoms;trapping, 34.50.Rk - Laser modiﬁed scattering and reactions I. INTRODUCTION Helium atoms in the metastable triplet state 23S1(He) appear to be a good candidate for Bose-Einstein Condensa- tion (BEC) according to theoretical predictions [1]. The cr oss section for elastic collisions between spin-polarised met astable helium atoms is expected to be large, allowing eﬃcient ther- malization and evaporation in a magnetostatic trap, which is the standard technique to reach BEC [2–5]. On the other hand, very high autoionization rates (Penning collisions) pre- vent reaching high densities of metastable helium atoms, bo th in the presence and in the absence of light, unless the sample is spin polarized. If a metastable helium atom collide either with an other metastable atom, or with an helium atom excited in the 23P2 state, the quasi molecule formed can autoionize according t o the following reactions: He (23S1) + He (23S1)→/braceleftbigg He(11S0) + He++e− He+ 2+e− (1) He (23P2) + He (23S1)→/braceleftbigg He(11S0) + He++e− He+ 2+e− (2) A ﬁrst experiment at subthermal energy ( E= 1.6 meV) with the metastable helium system was performed by M¨ uller et al.[6], allowing the determination of the interaction poten- tials. Using those potentials the rate βSSfor the reactions (1) has been calculated [7–9] to be a few 10−10cm3/s, which agrees with measurements performed in Magneto-Optical Traps (MOT) [10,7,11]. According to theoretical predictio ns [1], the ionization rate corresponding to the reactions (1) should be suppressed by four orders of magnitude in a magne- tostatic trap. Spin polarization of the atoms and spin conse r- vation in the collisional process are the causes of this supp res- sion, which makes the quest of BEC reasonable. Actually, a reduction of more than a factor of 20 in the two-body loss ratein an optically polarized sample was observed experimental ly [12]. In presence of light exciting the transition 23S1→23P2, the reaction (2) is dominant. “Optical collisions” with metast able helium atoms were measured to have surprisingly large cross sections when compared with alkali systems [13]. The study of optical collisions is of fundamental importance in order to optimize the ﬁrst step towards BEC, consisting in pre-cooli ng and trapping the atoms in a MOT. The goal is to transfer a cloud as dense as possible in a magnetic trap, in order to increase the elastic collision rate and start evaporation. The experimental study of optical collisions is the subject of t his paper. Several groups reported measurements of optical collision s rates, by studying losses in the MOT at small detunings [10,8,11] around−5 MHz and at large detunings [11] at −35 MHz and−45 MHz. Measurements over a broad range of detunings, from−5 MHz to−20 MHz, were reported in [14] and the dependence of the loss rate on the intensity of the MOT laser beams was investigated. In reference [7] a theo- retical model for optical collisions is also proposed predi cting rates in good agreement with the measurements, but diﬀer- ing by more than one order of magnitude with all the other measurements previously quoted. Our measurements are performed in a MOT loaded with 109atoms, at a peak density of 1010atoms/cm3. With respect to previous works, we extend the measurements of the two- body loss rate to a wider range of detunings and intensities with a good precision, by measuring the number of atoms and the size of the trap using absorption techniques. Also, by measuring accurately the excited state population in eac h trapping condition, we are able to interpret our data with a simple model, expressing the two-body loss rate in terms of the excited state population and of a rate constant Ksp, found to be independent of the laser detuning and intensity. Our experimental setup is described in section II, while in section III we explain our detection system and we give the working conditions and performance of our magneto-optical trap. In section IV we describe in detail the experimental procedure used to measure the two-body loss rate and the ex- cited state population for diﬀerent trapping conditions. T he results are given in section V, and the conclusions in sectio n VI. II. EXPERIMENTAL SET-UP A beam of metastable helium atoms is generated by a con- tinuous high voltage discharge in helium gas, cooled to liqu id nitrogen temperature. Radiation pressure on the metastabl e beam allows one to increase its brightness, and to deﬂect it 1from the ground state helium beam [15]. The metastable atoms are then decelerated by the Zeeman slowing technique and loaded in a magneto-optical trap (MOT) in a quartz cell at a background pressure of 5 ×10−10torr. More details on the experimental setup will be given in a forthcoming paper [16] . MOT parameters for optimal loading of the trap are listed in table I. For the laser manipulation of the atoms, we use the line at 1083 nm, connecting the metastable triplet state 23S1to the radiative state 23P2. The saturation intensity Isat for this transition is 0,16 mW/cm2and the linewidth Γ /2π is 1.6 MHz. Our laser system consists of a DBR laser diode (SDL-6702-H1) in an extended cavity conﬁguration, injecti ng a commercial Ytterbium doped ﬁber ampliﬁer (IRE-POLUS Group). The diode is stabilized by saturation spectroscopy at -240 MHz from resonance. At the ﬁber output we obtain 600 mW of power, in a TEM00 mode at the same frequency. The estimated linewidth is around 300 kHz. All the frequen- cies required for collimation, deﬂection, trapping and pro bing are generated by acousto-optical modulators in a double pas s conﬁguration, while we use directly part of the ﬁber output beam for slowing the atoms. TABLE I. Optimal loading parameters of the He∗mag- neto-optical trap. Laser detuning -45MHz Laser beam diameter 2 cm Vertical laser intensity (Ox) 2×9 mW/cm2 Longitudinal laser intensity (Oy) 2×9 mW/cm2 Transverse laser intensity (Oz) 2×7 mW/cm2 Total intensity 50 mW/cm2 Weak axis magnetic ﬁeld gradient bx=by= 20 G/cm Strong axis magnetic ﬁeld gradient bz= 40 G/cm III. DETECTION SYSTEM AND CHARACTERIZATION OF THE MOT zy x λ/2Absorption PD1Fluorescence PD2 CCD camera for absorptionHe λ/4λ/4 FIG. 1. Detection set-up. By rotating the λ/2 plate, one can create either a progressive plane wave for measuring the abs orption on the photodiode PD1, or a standing wave, with both beams cir - cularly polarized in the cell region, for imaging the cloud o nto the CCD camera. PD2 monitors the ﬂuorescence of the MOT.In order to fully characterize the cloud, we use a probe laser beam on resonance, whose diameter is about 1 cm, which is turned on 100 µs after the MOT ﬁeld and light beams have been turned oﬀ. Our detection setup (see ﬁg. 1) allows dif- ferent measurements. With the combination of λ/2 plates and polarization beam splitter cubes, we can create either ( i) a progressive wave, circularly polarized, passing through the atomic cloud towards a photodiode (PD1 in ﬁgure 1), giving the total absorption by the atoms, or (ii) a stationary wave, also circularly polarized, one arm of which is sent to a CCD camera, allowing spatially resolved absorption pictures o f the cloud. A second photodiode (PD2 in ﬁgure 1) is used to col- lect the cloud ﬂuorescence. We use the absorption photodiod e PD1 to measure N, the number of atoms in the steady state of the MOT. The probe beam saturates the transition when the incident power exceeds 10 mW (see ﬁg. 2). The maximum absorbed power is then P=NhνΓ 2. Our Watt-meter (Co- herent lab-master) is calibrated to 3% accuracy and allows a rectilinear calibration ﬁt of the photodiode voltage. We me a- sure a maximum total absorption of 1 mW, corresponding to (1±0.1)×109atoms. We estimate the accuracy for the mea- surement of N to be about 10%. 0 1 2 3 4 5 6 7 8 9 10 110,00,20,40,60,81,0109 atoms Incident power (mW) FIG. 2. Absorbed power by the MOT versus incident power of the laser probe beam. The absorbed power saturates at 1 mW for an incident power of 10 mW. The corresponding number of atoms is (1±0.1)×109atoms The typical parameters of our magneto-optical trap with the operating conditions of table I are listed in table II. TABLE II. Characterization of the MOT with parameters of table I. Number of atoms N= (1±0.1)×109 RMS size (weak axis) σx=σy= (2±0.1) mm RMS size (strong axis) σz= (1.6±0.1) mm Density at the center (1±0.25)×1010atoms/cm3 Temperature 1 mK 2We stress the fact that the case of He* diﬀers of that of alkalis, for which the imaging method gives a direct measure - ment of both the two-dimensional column density and the rms sizes of the MOT, by absorption of a brief and low intensity probe pulse ( I≪Isat). In the case of He, the quantum eﬃciency of the CCD camera (10−3at 1.083 µm) is too low to provide images with a suﬃcient signal to noise ratio. We need instead to illuminate the atoms with a 200 µspulse whose intensity is about 0.1 mW/cm2(I∼Isat), and use a moderate magniﬁcation of 1/5. Another diﬃculty with He occurs from the large recoil momentum ¯ hk/m (9.2 cm/s) due to the light mass of the atoms : the atoms are pushed out of resonance during the 200 µspulse if a traveling wave pulse is used. The solution we adopted is to illuminate the atoms in a standing wave with the set-up shown in ﬁgure 1. Though this scheme allows us to obtain pictures with a good contrast, the drawback is that the images obtained in the standing wave conﬁguration for I∼Isatare more diﬃcult to analyze than in the low intensity case. In order to interpret the absorpti on pictures in the standing wave conﬁguration, and for any sat- uration parameter, we developed a handy theoretical model (see appendix A) giving the column density of the atoms for each pixel of the CCD camera. The resulting density is then ﬁtted by a Gaussian curve to extract the size of the cloud. IV. MEASURING THE TRAP DECAY BY FLUORESCENCE Once the loading of the MOT is interrupted, the evolution of the number of trapped atoms Nis given by the following equation: dN dt=−αN−β/integraldisplay n2(r, t)d3r (3) where n(r, t) is the atomic density at position rand time t, αis the decay rate due to collisions between trapped atoms and the residual gas, and βis the two body intra-MOT loss factor. Assuming that the spatial distribution is independ ent of the time evolution of the number of atoms, which is valid at low enough densities, one can write the density as n(r, t) =N(t) (2π)3 2σxσyσze−x2 2σ2x−y2 2σ2y−z2 2σ2z (4) At low enough pressure and high enough density, losses due to background gas are negligible, so that the equation reduc es to dN dt=−βN2(t) (4π)3 2σxσyσz(5) whose solution is N(t) =N(t0) 1 +β 2√ 2n(0, t0)(t−t0)(6) where t0is the initial time. In order to follow the evolution of the number of trapped atoms, we monitor the ﬂuorescence de- cay of the MOT with a photodiode (PD2 in ﬁgure 1). As theﬂuorescence signal is proportional to the number of atoms, w e obtain a ﬂuorescence decay curve reproducing equation (6), which we ﬁt to get the parameter βn(0, t0). In order to de- termine β, one still has to measure n(0, t0), which means that one has to measure the rms size of the cloud along the three directions and the initial number of atoms N(t0). Our goal is to measure the loss rate for a wide range of de- tunings and intensities. The experimental procedure, divi ded in three successive steps, is the following. (1) First, we load the trap for 1 s at δ=−45 MHz and at the highest intensity in the trapping beams (I/Isat=50 per lase r arm). Then, we stop the loading by blocking the slowing beam with a mechanical shutter. 20 ms later, we “compress” the MOT by suddenly changing its detuning and intensity using acousto-optical modulators. We record the ﬂuorescen ce signal during this procedure. A typical ﬂuorescence curve i s shown in ﬁgure 3. -40 -20 020 40 60 80 100 120 140 160012345Trapping δ = - 45 MHz End of the loadingδ = -20 MHz Starting point for the fit Time (ms) FIG. 3. Evolution of the ﬂuorescence signal. Once the loading is stopped, scattered light from the slowing beam is blocked , which explains the drop of the signal at t=-10 ms. The detuning is th en set to δ=−20 MHz at t=0 ms and the ﬂuorescence decays. The loading is stopped at t=-20 ms and the photodiode sig- nal drops by a factor of 2 at t=-10 ms because the background light from the slowing beam is blocked. The ﬂuorescence is greatly enhanced in the beginning of the compression phase at t=0 ms, as expected when the detuning is set closer to res- onance (the detuning is set here to -20 MHz), but decays to almost zero in about 100 ms because of the two-body losses. Figure 4 shows the time evolution of the size of the cloud dur- ing this phase of compression, showing that 10 ms are enough to reach the new equilibrium size. Thus, we extract the pa- rameter βn(0, t0) from a ﬁt of the ﬂuorescence decay starting fromt=t0= 10 ms. At this very time we measure the sizes of the MOT along x and y and the number of atoms in order to calculate n(0, t0). 30 2 4 6 8 10 120,00,51,01,52,02,5 σx σy Time (ms) FIG. 4. Size of the MOT during the compression phase. The new equilibrium is reached after 10 ms (2) Then, the sizes along the weak axis of the magnetic ﬁeld gradient are measured by absorption on the CCD camera as explained in section III. Figure 5 shows the rms size along x for various laser detunings and intensities. The size along z (strong axis of the quadrupole ﬁeld) is inferred from measur e- ments of the sizes along x and y with a magnetic ﬁeld gradient btwice as large. We ﬁnd a typical size along z 20% smaller than along the weak axes of the quadrupole. We did not cor- rect the sizes for the expansion of the cloud during the pulse lasting 200 µs, as this would have required the measurement of the temperature for all the detunings and intensities. Ne v- ertheless, we performed some time of ﬂight measurements, giving temperatures ranging from 0.3 mK at -10 MHz to 1 mK at -40 MHz, from which we estimate that the sizes are overestimated at most by 5 % at -25 MHz and by 15 % at -5 MHz. In addition, we measured the statistical error on the sizes to be relatively small at large detunings, 2 to 3%, but larger at small detunings (about 10% at -5 MHz). This is due (i) to the poor spatial resolution of our imaging system (pix el dimension 80 µm×130µm), and (ii) to a low signal to noise ratio for small detunings where the loss rate is larger, as mo st of the atoms are lost during the compression phase. 0 50 100 150 200 2500,00,20,40,60,81,01,21,41,6 - 25 MHz - 20 MHz - 15 MHz - 10 MHz - 5 MHz I/Isat FIG. 5. Rms size of the MOT cloud as a function of the intensity of the MOT laser beams for various detunings.(3) Finally, to determine the number of atoms that were still trapped at t0= 10 ms, we simultaneously switch oﬀ the magnetic ﬁeld and set the trapping beams on resonance at t0, instead of letting the trap decay as in ﬁgure 3. The laser intensity is set to a high enough value to strongly saturate t he transition. We get a peak of ﬂuorescence, whose amplitude is proportional to the number of atoms. We compare it with the peak obtained with the same procedure but for the MOT in the best loading conditions of ﬁgure 2, for which we measured the number of atoms precisely. From this comparison, we infe r the number of atoms at t=t0in the compressed MOT, and thus determine n(0, t0). This measurement also gives access to the value of the averag e population of the excited state πp. Indeed, πpis given by F Fmax=πp 1/2= 2×πp (7) where Fis the ﬂuorescence signal we measure in the com- pressed MOT at t0, and Fmaxthe ﬂuorescence signal at res- onance, when the transition is saturated, and πpexpected to be 1/2. Figure 6 shows the results of the ﬂuorescence measurements, giving Fmax/Fas a function of the inverse of the laser inten- sity I for various detunings. It is interesting to note that t he inverse of Fis found to vary linearly with the inverse of I. 0 1 2 3 405101520253035 -5 MHz -10 MHz -15 MHz -20 MHz -25 MHz Isat/I (×10 -2) FIG. 6. Fluorescence signal F from the MOT as a function of intensity I of the laser beams. The inverse of the ﬂuorescenc eF is found to vary linearly with the inverse of the intensity I. The results are used for the calibration of the number of atoms. Following [17], the ﬂuorescence of N atoms in the com- pressed MOT can be modeled by the following equation: F=η N hνΓ 2C1I Isat 1 +C2I Isat+ 4δ2 Γ2(8) where ηis the detection eﬃciency, Iis the total intensity of the six MOT beams, and C1andC2phenomenological factors. C1andC2would be 1 for a two-level atom, but they are expected to be smaller for an atom placed at the intersection of 6 diﬀerently polarized laser beams, as happens in a MOT. In reference [17], C1andC2are found to be equal, and slighly larger than the average of the squares of the Clebsch-Gordan 4coeﬃcients over all possible transitions. For a J= 1←→J= 2 transition, this average is 0.56. We can rewrite equation ( 8) as Fmax F=C2 C1+1 + 4δ2 Γ2 C1Isat I(9) where Fmax=ηNhνΓ 2. The results of ﬁgure 6 show a good agreement with (9). But, C2andC1are not found equal, and both depend on the de- tuning. For example, C1is found to be 0.58, 0.48, 0.46, 0.44, 0.22 for δ=-25, -20, -15, -10, -5 MHz respectively. We stress the fact that, for the ﬂuorescence at resonance, and for full saturation, C1 and C2 are expected to be equal. V. RESULTS The results of the Penning collisions rate βare shown in ﬁgures 7 and 8. 0 50 100 150 200 2500,01,0x102,0x10-83,0x104,0x10 -30 MHz -20 MHz -10 MHz -5 MHz I/Isat-8-8-8 FIG. 7. Two-body loss rate factor as a function of laser power for several detunings. Figure 7 presents the loss parameter βas a function of the laser intensity for diﬀerent detunings δ, from -30 to -5 MHz. The uncertainty of the measurements varies from 25 % for large detunings to 60 % for small detunings. For all detun- ings,βincreases with power, which shows that S-P collisions are dominant.-45 -40 -35 -30 -25 -20 -15 -10 -501010-8I/Isat=80 δ (MHz)-9 FIG. 8. Two-body loss rate factor as a function of detuning for a ﬁxed intensity I= 80Isatof the laser. Fig. 8 shows the loss parameter as a function of detuning for a ﬁxed intensity (I Isat= 80). For the same reason, the rate increases when the detuning goes to zero, as the population in the P state increases. Our results for βagree with previous results [10,8,11,14] within the given error bars, extendin g the measurements to a wider range of parameters. For example, at -5 MHz and in an intensity range for which βis not ex- pected to vary strongly ( I= 140 to 200 Isat), Kumakura et al. [8] ﬁnd β= (4.2±1.2)×10−8cm3/s, Browaeys et al. [14] β= 2×10−8cm3/s with an uncertainty of a factor 2 and Tol et al. [11] β= (1.3±0.3)×10−8cm3/s. Our measurement β= (3.5±1.4)×10−8cm3/s agrees best with [8]. One should also note that we ﬁnd neither a decrease of βfor high intensi- ties at small detunings, nor a decrease of βat small detunings for a given intensity : this diﬀers from the results of [14]. I n fact, we ﬁnd that βincreases with intensity at small detun- ings, and also increases with decreasing detunings at a give n intensity. We also disagree with the results of [7] where muc h smaller rates are found. Finally, we also measured the loss rate in the trapping con- ditions ( δ=−45 MHz, I= 310 Isat) : the decay rate of the number of atoms was found to be βn(0) = 30s−1at a density of 1010at/cm3, which gives β= 3×10−9cm3/s. One can further analyze these data following the simple mode l of [10] which relates the decay constant βto the constant rate coeﬃcients Kss,KspandKppand to the populations of the excited and ground state levels, πpandπsrespectively: β=Kssπsπs+ 2Kspπsπp+Kppπpπp (10) Experiments [7,11] or theory [7–9] have shown that the contr i- butions Kssπ2 sandKppπ2 pto the total rate βare smaller than theKspterm by approximately two orders of magnitude. From the measurements of the ﬂuorescence signal in ﬁgure 6, we derive πpfor each experimental point, as F/F maxin eq. (7) is equal to 2×πp. 50 50 100 150 200 2500,02,0x104,0x106,0x108,0x101,0x10-71,2x10-71,4x10-71,6x10-71,8x10-7 -5 MHz -10 MHz -15 MHz -20 MHz -25 MHz -30 MHz I/Isat-8-8-8-8 FIG. 9. Rate coeﬃcient Kspfor all our measurements, as a function of the laser intensity I for several detunings. In ﬁgure 9, we then plot Kspfor the ensemble of our data. We do not see clear evidence for a dependence of Ksp with the detuning or the intensity within the dispersion of our data. To a good approximation, we estimate then that Kspis actually constant in the explored range of parameters: Ksp= (1.0±0.4)×10−7cm3/s, with a dispersion that roughly agrees with the error bars we claim. This result agrees with the ﬁrst measurement ever performed [10], but the precision is now much improved. It also agrees well with the measure- ments of [8] where the authors found Ksp= (8.3±2.5)×10−8 cm3/s, assuming that for their parameters ( δ=−5 MHz and I= 30 mW/cm2),πs=πp= 0.5. An important point is that, in contrast with the measure- ment of the ﬂuorescence at resonance where the transition is assumed to be saturated, πpin the compressed MOT never reaches 0.5 in our measurements : even for the smallest de- tuning and the highest intensity, πpis only 0.2. This explains why the results for βin ﬁgure 7 at δ=−5 MHz strongly in- crease for increasing intensity over the whole explored ran ge. VI. CONCLUSION We measured the absolute two-body loss rate between metastable atoms in a magneto-optical trap as a function of detuning and intensity. We extended the range of these pa- rameters and compared the results to those of previous mea- surements, mostly performed at small detunings. Using a new experimental approach, we obtained reliable values for the two-body loss rates with an improved accuracy as compared to most earlier results. In the region of overlap of parame- ters, we ﬁnd a good agreement with previous measurements, within the quoted uncertainties. We ﬁnd a loss rate monotoni - cally increasing as a function of intensity and decreasing w ith detuning. Our measurements are interpreted with a simple model, giving a rather constant loss rate Ksp, with an aver- age value of (1±0.4)×10−7cm3/s, as already found in the very ﬁrst measurement of [10]. We believe that the quality and the extended range of our measurements should moti- vate more theoretical work, in order to understand better th e peculiar dynamics of Penning collisions between metastabl ehelium atoms in the presence of light. Acknowledgments: The authors wish to thank C. Cohen- Tannoudji for helpful discussions and careful reading of th e manuscript. aPermanent address: Laboratoire de Physique des Lasers, UMR 7538 du CNRS, Universit´ e Paris Nord, Avenue J.B. Cl´ ement, 93430 Villetaneuse, France. bPermanent address: Institute of Opto-Electronics, Shanxi University, 36 Wucheng Road, Taiyuan, Shanxi 030006, China. cPermanent address : Dept. of Physics, Univ. of Perugia, Via Pascoli, Perugia, Italy; Lens and INFM, L.go E. Fermi 2, Firenze, Italy dPresent address : Universit¨ at Hannover, Welfengarten 1, D-30167 Hannover, Germany. ePermanent address : TIFR, Homi Bhabha Road, Mumbai 400005, India. ∗Unit´ e de Recherche de l’Ecole Normale Sup´ erieure et de l’Universit´ e Pierre et Marie Curie, associ´ ee au CNRS (UMR 8552). APPENDIX A: MODEL OF THE ABSORPTION In this appendix we describe the method we used to quan- titatively interpret the absorption images of the atomic cl oud when a standing wave conﬁguration of the probe beam is used, and for an arbitrary saturation parameter. We describe the atoms as two-level atoms characterized by a non linear sus- ceptibility: χ=n(x,y, z)/bracketleftbigg −d2 ¯hǫ0δ−i(Γ/2) (Γ/2)2+δ2+\|Ω\|2/2/bracketrightbigg (A1) where n(x, y,z) is the atomic density, dthe atomic dipole, δ the detuning, Γ the inverse lifetime of the excited state and Ω is the Rabi frequency given by ¯hΩ 2=−dE(+)E=E(+)e−iωt+c.c. (A2) where E is the electric ﬁeld. The direction of propagation of the beam is zand the ﬁeld is supposed to be uniform in the plane (x,y). The propagation of the ﬁeld is then described by the Maxwell equations: /bracketleftbig ∆ +k2 0(1 +χ)/bracketrightbig Ω(z) = 0 (A3) where k0is the wavevector of the light. The principle of the model is to use the slowly varying en- velope approximation generalized to the case of a standing wave. We then decompose the probe beam ﬁeld as: Ω(z) =A+(z)eik0z+A−(z)e−ik0z(A4) where A+,A−are the slowly varying envelopes of the wave going towards positive zand negative zrespectively. A simi- lar decomposition holds for the nonlinar susceptibility of the atoms: χ(z) =χ0(z) +χ+(z)e2ik0z+χ−(z)e−2ik0z+. . . (A5) 6where χ0,χ+andχ−are slowly varying envelopes, and where we neglect terms in the expansion describing generation of frequencies others than the probe frequency via the non line ar interaction. If we insert the expansions (A4) and (A5) into the propagatio n equation (A3) and use the rotating wave appoximation, we obtain a set of two coupled diﬀerential equations for the slo wly varying ﬁeld amplitudes A+,A−. By splitting the complex amplitudes into modulus and phase: A+=\|A+\|eiφ+A−=\|A−\|eiφ−(A6) and by introducing the phase diﬀerence ( φ+−φ−) in the deﬁnition of the slowly varying susceptibilities χ+andχ−: χ+= ˜χ+ei(φ+−φ−)χ−= ˜χ−e−i(φ+−φ−),(A7) one can write : d\|A+\| dz=k0 2(Im˜χ+\|A−\|+ Im˜χ0\|A+\|) (A8) d\|A−\| dz=−k0 2(Im˜χ−\|A+\|+ Im˜χ0\|A−\|). (A9) By using expressions (A1) and (A4), the quantities k0Im˜χ+, k0Im˜χ−andk0Im˜χ0are readily calculated: k0Im˜χ0=3λ2 2πn(x, y,z)α f0 (A10) k0Im˜χ+=k0Im˜χ−=3λ2 2πn(x, y, z )α f1 (A11) where α=(Γ/2)2 (Γ/2)2+δ2+1 2(\|A+\|2+\|A−\|2)(A12) f0=1√ 1−ǫ2; f1=1−f0 ǫ(A13) ǫ=\|A+\|\|A−\| (Γ/2)2+δ2+1 2(\|A+\|2+\|A−\|2). (A14) As a last step we eliminate the atomic density n(x, y, z ) from the equations by changing variable: Z(z) =/integraldisplayz −∞n(x, y,z′)dz′(A15) and we obtain the ﬁnal coupled equations: d\|˜A+\| dZ=3λ2 4πα/parenleftbig f1\|˜A−\|+f0\|˜A+\|/parenrightbig (A16) d\|˜A−\| dZ=3λ2 4πα/parenleftbig f1\|˜A+\|+f0\|˜A−\|/parenrightbig , (A17) where: ˜A−=A−/(Γ/2) ˜A+=A+/(Γ/2). (A18) Forδ= 0 and in the limit of small saturation parameters, one has α= 1, f0≃1,f1≃0 and one recovers the usual decoupled equation for low saturation absorption. We have now to solve equations (A16) and (A17). More precisely we wish to calculate the column density Z∞=/integraldisplay+∞ −∞n(x, y,z′)dz′(A19)for each eﬀective pixel (x,y) of our image of the cloud. For each eﬀective pixel, we can measure the initial conditions: \|˜A+\|2(Z(−∞) = 0) = Ii (A20) \|˜A−\|2(Z(−∞) = 0) = If (A21) corresponding respectively to the intensity of the probe be am before passing through the cloud, or equivalently without t he atoms, and to the intensity of the probe beam that passed through the atomic cloud. For symmetry reasons, the column density (A19) is given by 2 Z0=Z(0), where Z0veriﬁes \|˜A+(Z0)\|2=\|˜A−(Z0)\|2. (A22) For each pixel (x,y), we then integrate equations (A16)-(A1 7) numerically using the initial conditions (A20)-(A21) unti l \|˜A+(Z)\|2=\|˜A−(Z)\|2. The corresponding Zmultiplied by 2 gives the column density. 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arXiv:physics/0103064v1 [physics.atom-ph] 20 Mar 2001Eﬃcient magneto-optical trapping of a metastable helium ga s F. Pereira Dos Santos, F. Peralesa, J. L´ eonard, A. Sinatra, Junmin Wangb, F. S. Pavonec, E. Raseld, C.S. Unnikrishnane, M. Leduc Laboratoire Kastler Brossel∗, D´ epartement de Physique, Ecole Normale Sup´ erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France Abstract This article presents a new experiment aiming at BEC of metas table helium atoms. It describes the design of a high ﬂux discharge source of atoms a nd a robust laser system using a DBR diode coupled with a high power Yb doped ﬁber ampliﬁer for manipulating the beam of metastable atoms. The atoms are trapped in a small quartz cel l in an extreme high vacuum. The trapping design uses an additional laser (repumper) and allows the capture of a large number of metastable helium atoms (approximately 109) in a geometry favorable for loading a tight magnetostatic trap. PACS 32.80.PjOptical cooling of atoms;trapping 1 Introduction The main goals of this experiment is ﬁrst to produce a gas of4He atoms in the metastable 23S1state with a density as large as possible, and then to bring do wn the temperature of the gas to ultralow temperatures. This experiment replaces an e arlier experiment based on the VSCPT cooling method [1] which, despite the achievement of u ltra low temperatures, could not reach high phase space densities, due to the small number of atoms being cooled. Let us mention that several other groups are involved in experim ents with similar goals dealing with trapped ultracold metastable helium atoms [2, 3, 4]. Th e present experiment, as well as those of references [3, 4], aim at achieving Bose-Einstein c ondensation (BEC) by combination of laser cooling, magnetic trapping and subsequent evapora tive cooling, following the route successfully taken for BEC of alkali atoms (Rb,Na,Li) since 1995 [5, 6, 7] and atomic hydrogen in 1998 [8]. A metastable helium condensate would be the ﬁrst one with atoms in an excited state of high internal energy (19.8 eV) and long lifetime (ap proximately 8000 s). It should be interesting to compare the properties of such a gaseous dilu te helium condensate to those of superﬂuid liquid helium, dominated by interactions betwee n particles. Helium BEC should be capable of forming a helium atom laser as it was the case for alkali BEC. There are many applications of such coherent matter waves. For instance a m etastable atom laser [9] could be a valuable source for lithography [10]. Another applicatio n for helium atoms is metrological [11], since its energy levels can be calculated with a high de gree of accuracy. Let us ﬁnally mention an interesting property of these metastable helium atoms. They can transfer their high internal energy when they collide with surfaces or mole cules. This property can be used for highly eﬃcient detection, almost ”one by one”, with good spatial and temporal resolution [12], using microchannel plates for example. It thus appears that the metastable helium atom displays app ealing features as a candi- date for BEC. According to theoretical predictions [13], th e cross section for elastic collisions between cold metastable atoms should be large, ensuring rap id thermalisation for eﬃcient evaporative cooling in the magnetic trap. Furthermore, Pen ning collisions which are the main source of inelastic losses in a magneto-optical trap, a re expected to have a rate slow enough to allow the formation of BEC in a magnetostatic trap. However, reaching BEC with 1metastable helium remains uncertain. The predicted value o f approximately 10 nm for the scattering length could be inaccurate as it is very sensitiv e to the details of long range elastic potential between atoms, which is not known to a high accurac y. It should also be mentioned that the rates of the inelastic collisions, which are likely to heat up and empty the trap at low temperatures [13, 14, 15], have not yet been measured. Fr om an experimental point of view, one ﬁrst needs to trap a dense cloud of these atoms in an u ltra high vacuum. A second step is to construct a strongly conﬁning magnetostatic trap . Thus, we choose to trap the atomic cloud in a quartz cell of small dimension while having the conﬁning magnets external to the cell and close to the vacuum chamber. Section 2 of this a rticle gives details of the discharge source and of the optimal parameters chosen to ach ieve a high ﬂux of metastable helium atoms. Section 3 shows the laser system consisting in a DBR diode laser coupled with a high power ﬁber ampliﬁer. Section 4 describes the lase r techniques used to increase the brightness of the beam of atoms, to deﬂect it from the beam of ground state atoms and to slow it down in a spatially varying magnetic ﬁeld. Section 5 demonstrates the advantages of our particular trapping scheme. 2 The source of metastable atoms 2.1 Principle The development of an intense and slow beam of metastable hel ium atoms requires to solve several problems. First, helium atoms have to be eﬃciently e xcited to the metastable state. Secondly, the beam has to be cooled to a low enough temperatur e to avoid diﬃculties with the subsequent deceleration, knowing that the small mass of the atom results in a high ve- locity at room temperature. In a previous experiment [1], me tastable atoms were produced at a moderate rate by electron bombardment and they were cool ed down to liquid helium temperature. In the present setup, a diﬀerent strategy is us ed: metastable atoms are pro- duced in a gas discharge and cooled to liquid nitrogen temper ature, reaching mean velocities of approximately 1000 m/s. It is known that the most eﬃcient w ay to produce high rates of metastable helium atoms is to start with a pulsed or contin uous gas discharge, where atoms are excited to upper states by electronic collisions a nd then decay to the long-lived metastable state. A fraction between 10−6and 10−4of the ground-state atomic ﬂux can be produced in the metastable state with an intense discharge. However, the heat generation in the discharge makes it diﬃcult to obtain both intense and col d atomic beam of metastable helium. Several attempts [16, 17, 18] have been made to solve this problem. The source developed at ENS results from a design which combines severa l advantages of the sources developed by the other groups. The setup is compact and robus t and gives a reliable large ﬂux of atoms, a signiﬁcant number of which have velocities be low 1000 m/s. All discharge sources for metastable helium consist basica lly of a gas reservoir ﬁlled with helium gas to a pressure of a few tens of mbar. The discharge oc curs between the cathode inside the reservoir through the outlet channel to the anode , placed on the high vacuum side, or directly to the skimmer. The design of Fahey et al. [16] ach ieves low gas temperatures by cooling only the nozzle with liquid nitrogen. The source des cribed by Kawanaka [17] makes use of an elaborate scheme to cool down all of the source and to remove the hot gas by a roughing pump. Similar results have been obtained regardi ng ﬂuxes reaching up to 1014 atoms per second and steradian [16] and [17]. Velocities are found to be slightly above 1000 m/s in [16] and slightly below 1000 m/s in [17]. For the design presented here and shown in ﬁg. 1, the complete source, including the discharge elect rodes, is cooled in a simple and eﬃcient way. No additional eﬀort such as removing the hot gas is required. As a result the entire source is very compact. A careful design of the shape o f the electrodes and of the gas outlet ensures that the discharge is partially burning into the high vacuum (see ﬁg. 1) rather than inside the reservoir. This provides a high ﬂux of metast able atoms, as they do not hit walls at the place where they are produced. The design is such that a reliable and stable operation mode has been achieved for several weeks of contin uous operation. 22.2 The source design liquid nitrogen Helium copperaluminum anodeIsolator (Araldite) cathode dischargeHeliumboron nitridehigh voltage 4 cmHelium indium o-ring liquid nitrogenflangea) skimmerb) 0.4 mmanode boron nitride outlet channel 4 cm Source chamberDetection chamber P=10-4 mbar P=10-6 mbar Figure 1: a) Compact discharge source of metastable helium atoms. The discharge burns in the outlet channel and outside the anode. The metallic anode is cooled to liquid nitrogen temperature. b) Zoom of the outlet channel region. The source and detection chambers are separated by t he skimmer. The source shown in ﬁg. 1 consists of a cylindrical gas reserv oir entirely made of boron nitride, a material which has a suﬃciently high heat conduct ivity as well as a high electrical and chemical resistance. This material was already used by F ahey et al. for the nozzle of their discharge source [16]. The external part of the gas res ervoir is covered with vacuum grease for better thermal contact with the copper cylinder c ooled by liquid nitrogen, into which it is pressed. A copper ﬂange, held by an electrically i solating support of araldite and tightened by an indium o-ring, closes the reservoir on the ba ckside. The ﬂange serves as the inlet of the helium gas, as well as for the mount of the cathode . The cathode is a stainless steel tip adjustable in position during mounting. The dista nce between cathode and anode is typically between 2 and 3 mm but it is not found critical. The particular shape of both electrodes allows them to be cen tered with respect to each other by mounting them on the reservoir. Particular care was taken to design the gas outlet, a 2 mm long channel of 0.4 mm in diameter drawn in the boron nitr ide reservoir and located in front of the cathode. The outlet channel diameter is chose n in such a way that a stable discharge and a high ﬂux of metastable atoms are reached, eve n at low operating discharge currents and for gas loads adapted to the speed of the pumps. I t is directly followed by the anode, a 1 mm thick aluminum disc with a hole of 0.4 mm in dia meter. A particular feature of the present design is that the anode is cooled to li quid nitrogen temperature. For eﬃcient cooling it is tightly ﬁxed on the copper container. C rucial for reliable operation of the discharge is the cleanliness of the cathode and anode. Du ring operation both parts suﬀer from impurities of the gas or from the oil vapor of the diﬀusio n pumps. Our design allows easy dismantling of the source to clean the electrodes. The material, the size, the depth and the alignment are cruci al for an eﬃcient excitation of helium atoms. Best performances were obtained with an alu minum anode with a hole of diameter at least as large as that of the outlet channel in the boron nitride reservoir. This ensures that the electrical ﬁeld lines reach into the source chamber (see ﬁg. 1 b), so that the discharge extends also into the vacuum at the source exit . The cylindrical design and the length (30 mm) of the reservoir have been chosen such that no parasitic discharge can occur competing with the discharge at the source exit when op erated in a pressure range of 10 torrs. The source reservoir is ﬁlled with gas via a plastic tube, which isolates the source from the vacuum chamber. A throttle valve in front of the gas i nlet keeps the pressure high in the gas tube and suppresses parasitic discharge in the tub e. To avoid contamination of the source, the helium gas is ﬁltered with charcoal. 32.3 Measurement of the atomic ﬂux and velocity The source has been tested in a vacuum apparatus consisting o f two vacuum chambers (see ﬁg. 1), one for the source and one for the beam diagnostic. Bot h chambers are separated by a skimmer (1 mm diameter) and evacuated by diﬀusion pumps ( pumping speed of ap- proximately 800 l/s) equipped with liquid nitrogen traps. D uring operation the pressure in the source chamber rises to approximately 10−4mbar, in the detection chamber to approx- imately 10−6mbar. To characterize the atomic beam an in-situ detector wa s constructed, consisting of a gold mirror and a channeltron (ﬁg. 2). Upon co llision with the mirror surface, metastable atoms decay down to the ground state and release o ne electron out of the surface, with a high eﬃciency [19]. Assuming that each metastable ato m hitting the mirror releases an electron, one gets a lower limit for the atomic ﬂux by measu ring the current with a pi- coammeter. The metastable beam can also be pulsed with a mech anical chopper, in order e-metastable helium beam ChanneltronElectronPico- ammeter Oscilloscope Figure 2: In-situ detection of the atomic beam to perform time of ﬂight measurements. In this case, the curr ent on the gold mirror is too weak to be detected. The released electrons need to be accele rated towards a channeltron which detects the ampliﬁed current. The pulsed signal of the channeltron is then sent to an oscilloscope to record the time of ﬂight distribution. To se parate the metastable triplets 23S1 from other species produced by the discharge (UV light, ions , metastable singlet 21S0), the beam is collimated by diaphragms, and deﬂected by a laser bea m tuned to the 23S1→23P2 transition (see section 3). From the time of ﬂight measureme nts with and without deﬂection, it has been observed that the source essentially produces me tastable atoms which are in the triplet state. We assume that the singlet state atoms are que nched by the radiation emitted by the discharge. 2.4 Choice of discharge current and pressure The eﬃciency of the source depends on a large variety of param eters: the gas pressure and temperature, the discharge current, the purity of helium an d the geometry of the discharge. All these parameters have to be carefully optimized to obtai n the highest maximum ﬂux with moderate heating. The discharge current and the pressure in side of the gas reservoir were separately varied as shown in ﬁg. 3. For a given pressure, the production rate of metastable atoms increases linearly with the current up to 4 mA (ﬁg. 3 a). For higher currents the rate starts to saturate. With increasing current, the gas te mperature rises locally in the discharge region due to resistive heating resulting in a hig her atomic velocity (see ﬁg. 3 c). A current of approximately 6 mA was chosen as a compromise between high ﬂux and low velocities. In ﬁgure 3 b), there are two regimes for the pr essure. At lower pressures, an increase of pressure inside the reservoir results in an in crease of the metastable helium ﬂux. At higher pressures, an increase of pressure inside the reservoir leads to a decrease of ﬂux due to quenching of metastable helium atoms by collision s between metastable atoms in the nozzle region and by collisions with the background ga s. Eventually, if the pressure is increased passed the background pressure of 3 ×10−4mbar, the metastable helium beam can be completely quenched. For the vacuum setup described abov e, the optimum pressure was achieved at approximately 10−4mbar. After optimization of all parameters of the atomic 4P=1.25 10 -4 mbar 9009501000105011000 2 4 6 8 10 12 14 16 0,5 1,0 1,5 2,0 2,5 3,0 Pressure (10 -4 mbar)I=6mA 949698000204 0001 01 01 0I=6mA Pressure (10 -4 mbar)Discharge current ( mA) 0,5 1,0 1,5 2,0 2,5 3,0 0 2 4 6 8 10 12 14 16 Discharge current ( mA)P=1.25 10 -4 mbara) b) c) d) Figure 3: Curves a) and b) show the atomic ﬂux, in arbitrary units, infe rred from the maximum of the time of ﬂight distribution as a function of the discharge current an d the pressure in the vacuum chamber. Curves c) and d) display the corresponding mean velocity as a function of t he discharge current and the pressure in the vacuum chamber. Curves a) and c) are taken at a pressure of 1 .25×10−4mbar, curves b) and d) for a current of 6 mA. Note : the pressure in the reservoir is proportional to the pr essure in the vacuum chamber (see ﬁgure 1). source, ﬂuxes of triplet metastable atoms of the order of 2 ×1014atoms/sec/steradian were found, with a mean velocity of 1000 m/s. Using this highly com pact source, the measured values compare well with the ones obtained in other experime nts [16, 17, 18]. 3 The laser system Earlier experiments on laser cooling and trapping of helium at 1083 nm (transition 23S1→ 23P2) were performed using a LNA ring laser, pumped by an argon ion laser [20]. In this experiment, we use an optical ampliﬁer based on Ytterbium do ped ﬁber (IRE-POLUS) and seeded by a diode laser at 1083 nm. This laser source is especi ally eﬃcient to manipulate metastable helium atoms. Historically, the ﬁrst Ytterbium ﬁber ampliﬁer was developed and characterized in a single stage low ampliﬁcation conﬁgu ration [21]. Later, a double core prototype of this MOPFA system (Master Oscillation Power Fi ber Ampliﬁer) was built by S.V. Chernikov [22]. The laser system used in the present exp eriment is shown in ﬁgure 4. DBR LDSRM OIPZTC1 Power Amplifier600 mW 1 mW P1 P2OI OI APCDCF V-SPC2 OI Master OscillatorB1 B4B3B275mW, δ = 0 MHz 15mW, δ = -240 MHz 150 mW, δ = -45 MHz 10 mW, δ = 0 MHz Figure 4: Laser setup. A master oscillator at 1.083 µm (DBR Laser Diode) injects an optical Yb doped ﬁber power ampliﬁer. C1 : collimator, PZT : piezo-transducer, SR M : semi-reﬂecting mirror, OI : optical isolator, P1 and P2 : λ/2 and λ/4 plates, V-SP : V-groove side-pumping by diode arrays, DCF : double clad ﬁber, APC : angle polished connector, C2 : collimator. The box represen ting the power ampliﬁer, commercially available from IRE-POLUS, has two ﬁber connections for input and output. Th e output beam is split up in four independent beams : B1 represents the collimation-deﬂection beam, B2 th e slowing beam, B3 the MOT beams and B4 the probe beam. The seed laser is a single mode DBR laser diode (SDL-6702-H1) emitting at 1083nm and delivering a maximum power of 50mW. The line width of the lase r diode is reduced from 53 MHz to 250 kHz by an external cavity using a semireﬂecting mi rror of transmission 80 %, as has already been observed [23]. The laser diode is coupl ed with the ﬁber ampliﬁer using bulk optics. Two optical isolators providing a total i solation of 60 dB prevent optical feedback in the DBR diode. In addition, a set of two birefring ent plates ( λ/2 and λ/4) of adjustable orientation compensates for the birefringen ce of the ampliﬁer, which slightly varies with temperature changes and mechanical stress. Thi s adjustment provides the proper linear polarization at the output of the ampliﬁer. An additi onal 30 dB optical isolator is placed at its output because the ampliﬁer is sensitive to fee dback from the experiment. The Yb doped ﬁber ampliﬁer consists of two ampliﬁcation stages, both pumped by diode arrays operating at 970 nm (V-groove side pumping). The second ampl ifying stage (called booster) is designed for 600 mW saturated output power. It consists of a double clad ﬁber (DCF) with an optimized bidirectional side pumping. The angle pol ished connector (APC) at the output end prevents the ampliﬁer from oscillating. An input power of 1mW is suﬃcient to saturate the ampliﬁer and achieve performances independen t of the input level. The ampliﬁer provides a collimated beam in a TEM00 mode of 0.4 mm waist. A preliminary study of the frequency noise of the laser sourc e was performed using an autocorrelation setup and heterodyne detection. It was fou nd that the ﬁber ampliﬁer did not cause additional noise to the frequency spectrum of the i njection diode when the diode is frequency narrowed in an external cavity. The laser diode is locked -240 MHz away from resonance by satu rated absorption in a low pressure discharge cell. The ﬁber-laser light is split up in to four independent beams (see ﬁg. 4): the ﬁrst one is used to collimate and deﬂect the atomic bea m, the second one to slow the beam down, the third one to trap the atoms, and the last one to p robe the trap (see sections 4 and 5). Required frequency for each arm is set by acousto-op tical modulators used in a double-pass conﬁguration. 4 Laser manipulation: collimation, deﬂection and decel- eration The need for extremely high vacuum in the present experiment requires that the intense ground state helium beam is prevented from reaching the cell . The metastable beam, initially merged with the ground-state beam, has then to be spatially s eparated and directed towards a diﬀerent axis than the nozzle-skimmer axis. Radiation pre ssure forces are used for this purpose [24]. For the collimation and the deﬂection, a power of 75 mW of laser light is used (beam B1 in the ﬁgure 4). This power is evenly split in three : t he ﬁrst one for vertical collimation, the second one for horizontal collimation and the last one for deﬂection. The eﬀusive beam coming out from the source is highly divergent ( 0.1 rd) and has a uniform spatial intensity proﬁle. Collimation is thus performed sl ightly oﬀ axis, 1◦upwards with respect to the horizontal axis. Two apertures (a diaphragm a nd a tube) placed oﬀ axis selectively blocks the ground state beam while allowing the metastable state beam to be deﬂected by the laser in order to pass through (see ﬁg. 5). The circular aperture (Ø = 5 mm) and the separating tube (Ø = 1 cm, length 10 cm) are 1.2 m apa rt from each other. They deﬁne the new axis of the experiment starting 5 mm above t he nozzle-skimmer one all the way down towards the cell. The separating tube provid es diﬀerential pumping in the chamber connected to the main slowing magnet and pumped by a t urbo molecular pump (1000 l/s). We use a vertically movable Faraday cup (Ø = 7 mm) l ocated 1.15 m downstream from the nozzle to monitor the intensity of the metastable he lium beam (detector D1 in ﬁgure 5). For the collimation of the atomic beam in the two transver se directions, we use the so- called ”zig-zag” conﬁguration of the laser beam [24]. It use s a resonant beam (Ø = 8 mm) reﬂecting between two mirrors (3 ×15 cm) sligthly tilted from being parallel to cross the atomic beam about 10 times. The capture range of the transver se velocity is approximately 20 m/s. The increase in the metastable ﬂux is measured on a pic oammeter (Keithley) connected to the Faraday cup 1. Although ”white light” can be used to achieve collimation [24], we did not use it to prevent the broadening of the laser- source linewidth which has 6multiple uses in the experiment. To deﬂect the collimated be am, we used a curved-wavefront laser beam in a ribbon shape at resonance. The optimised radi us of curvature is of 5 m. The deﬂection keeps the beam collimated with nearly a 100% eﬃcie ncy. The second Faraday cup D2 (Ø = 8 mm) (see ﬁg. 5), located 2.4 m aw ay form the tube entry, is used to optimize the ﬂux of the collimated-deﬂ ected beam. Typical currents measured on D2 are of 20 nA in comparison with 1 nA measured on D 1 when the metastable beam is neither collimated nor deﬂected. This corresponds t o a collimated ﬂux of approxi- mately 2 ×1011atoms/s. To characterize the velocity of the pure triplet me tastable beam, To turbo pump 1To diffusion pump 2He flux Discharge sourceCollimationDetector D1 To diffusion pump 1Deflection Circular apertureTo turbo pump 2First Zeeman SlowerDetector D2Second Zeeman slower Cell zy xChamber 1 Mirror MSeparating tube ViewportCompensation Coil Figure 5: Experimental setup. The metastable helium beam is collimat ed, deﬂected, decelerated and trapped at the end of the setup in a small dimension quartz cell. we performed a time of ﬂight measurement. We used a resonant l aser beam (Ø = 1 cm) with a light chopper, crossing at right angle with the collim ated-deﬂected metastable beam, and a channeltron mounted besides the Faraday cup (D2). The l ight beam acting as an atom pusher is hidden for a short time period (50 µs) every 10 ms. We recorded the time of ﬂight spectrum and found a peak velocity of 930 m/s and a rel ative spread (FWHM) of approximately 30%, which is signiﬁcantly lower than the unc ollimated beam : this shows that the collimation-deﬂection process acts more eﬃcientl y on slow atoms because their in- teraction time with the laser beams is longer. The metastabl e helium beam is decelerated by the Zeeman tuning technique [25]. For this purpose, a lase r beam with 15 mW of total power is increased to a diameter of approximately 2 cm, with a right circular polarisation and a detuning from resonance of δslo=−240 MHz. The laser beam enters the cell and propagates anti-parallel to the atom beam. It is resonant wi th atoms having longitudinal velocity of 1000 m/s at the entrance of the ﬁrst Zeeman slower . This ﬁrst Zeeman slower has a length of 2 m, an inner diameter of 2.2 cm and a ﬁeld of 540 G . At the end of this ﬁrst Zeeman slower, the atomic velocity is approximately 240 m/s . The second Zeeman slower is approximately 15 cm long with an enclosed tube of 40 mm inner d iameter and creates a ﬁeld going from 0 to -140 G. The atoms are slowed down to a ﬁnal veloc ity of 40 m/s as they exit the second Zeeman slower. A compensating coil minimizes the magnetic ﬁeld leakage from the second Zeeman slower into the cell region. Control of the successive decelerations was done by a Dopple r sensitive absorption spec- troscopy method, using a laser-probe beam crossing the cell with an angle of approximately 20◦. Time of ﬂight measurements of the unslowed beam was used to c alibrate the Doppler detuning with respect to the atom velocity. Velocity measur ements are in good agreement with simulations of the slowing process. 5 The trapping scheme 7MOTMOT CoilsM O TMOT σ − zy xM O Tσ + σ −σ +R σ + ZS σ + Second Zeeman SlowerCompensation Coil Figure 6: MOT setup. The MOT beams are perpendicular to the surfaces of the quartz cell. The trapping scheme requires an extra laser, the repumper (R), which is su perimposed to the Zeeman slowing (ZS) and MOT beams along the axis of the atomic beam. 5.1 The laser beams geometry In order to optimize the number of trapped atoms in the MOT, we use a far detuned (δmot/2π=−45 MHz) and high intensity laser (total intensity I= 50 mW/cm2) (beam B3 in ﬁg. 4). This laser detuning minimizes inelastic Pennin g collisions between atoms in the 23S1metastable state and atoms in the 23P2excited state [3, 26, 27, 28]. Our scheme aims at trapping the gas at the center of a quartz cell of high q uality commercially available from Hellma (5cm ×5cm×4cm). We use large diameter laser beams (Ø = 2 cm) in order to capture a large number of atoms. The MOT is as close as possibl e to the slowing magnet end to allow a higher loading rate. The MOT beams are 6 independen t laser beams crossing the cell perpendicular to its faces. The two MOT beams along the z direction (see ﬁgure 6) are nearly superimposed with the slowing-laser beam and merged with the atomic beam. The σ+MOT beam is directed by the mirror M at 45◦incidence and the glass viewport placed on the vacuum chamber 1 (see ﬁgure 5). It propagates along the z-axis through the vacuum chamber and the Zeeman slowers. The two contrapropagating b eams along the z-axis are focused onto the mirror M with an edge separated by 1 cm from th e center of the atomic beam. In this geometry, both the σ+and the σ−of the MOT beams along the atomic beam axis aﬀect the slowing process. Consequently, several precautions such as the use of a repumper beam are required for optimization of the MOT as ex plained in the following section. 5.2 Optimization of the MOT On one hand, the σ+MOT beam along the z-axis is resonant with the atoms at a given position in the slowing magnet. It can thus be absorbed by the atoms and accelerate them. On the other hand, the σ−MOT beam along the z-axis is likely to depolarize the traveli ng atoms at another position. These two eﬀects have to be correc ted for. The slowing beam detuning from resonance is δslo/2π=−240 MHz from resonance (beam B2 in ﬁg. 4). During the slowing process, the velocity of the atom decreases acco rding to the following equation (1) δslo+kv=µbB/¯h (1) where Bandvare the projection along the z-axis of the magnetic ﬁeld and a tom velocity respectively. The atoms are spin polarized in the mJ= +1 level during the slowing process, cycling between the states 23S1, gs= 2, mJ= +1 and 23P2, gp= 3/2, mJ= +2 (see ﬁg. 7). Theσ+MOT beam, parallel to the atomic beam (see ﬁg. 6), can also ind uce transitions between these magnetic sublevels, if the following resonan ce condition is fulﬁlled: δmot−kv=µbB/¯h (2) 8Eq. (1) and (2) are both satisﬁed for 2kv+=δmot−δslo (3) which gives v+= 105 m/s. So, when the velocity becomes v+, which occurs before the end of the slowing process, in the second part of the Zeeman sl ower, the σ+MOT beam accelerates the atoms. The net eﬀect results from the intens ity unbalance between the σ+ MOT beam and the slowing beam. If the intensity of the slowing beam is less than the MOT beam intensity, the σ+MOT beam accelerates the atoms so much that the slowing proce ss is stopped. One needs to adjust the intensity of the slowing b eam to be higher than the intensity of the MOT beam to prevent this undesirable phenom enon. We typically use 15 mW/cm2for the slowing beam, which corresponds to 1.5 times the inte nsity of each of the MOT beams. In addition, the σ−MOT beam along the z-axis can induce transitions between 23S1, mJ= +1 and 23P2, mJ= 0 sublevels, which depolarize the atoms when they de- cay to the 23S1, mJ= 0 and mJ=−1 sublevels (see ﬁg. 7). Once the atoms have decayed, they are no longer resonant with the slowing beam and the slow ing process is stopped. This actually happens when the following resonance condition is fulﬁlled: δmot+kv=−2µbB/¯h (4) Eq. (1) and (4) are both satisﬁed for 3kv−=−(δmot+ 2δslo) (5) which gives v−= 190 m/s. This velocity is also reached in the second part of t he Zeeman slower. To avoid this problem, we repump the atoms from the mJ= 0 and mJ=−1 sublevels mJ = -2 mJ = -1mJ = 0 mJ = +2mJ = +1 mJ = -1 mJ = 0 mJ = +1ZS+ M-R1+R2+ Figure 7: Repumping scheme between 23S1and 23P2states of helium. ZS+ corresponds to the Zeeman slower beam, σ+polarized, M- to the σ−MOT beam along the atomic beam, and R1+ and R2+ are the two repu mping transitions required to bring the atoms back into the mJ= +1 sublevel and restore the slowing process. back to the mJ= +1 sublevel. Two repumping beams are both σ+polarized and resonant with the transitions 23S1, mJ=−1→23P2, mJ= 0 and 23S1, mJ= 0→23P2, mJ= +1, at exactly the same magnetic ﬁeld and the same atomic velocit y at which the depumping happens (see ﬁg. 7). One calculates that the required freque ncies are detuned by -272.5 MHz and -305 MHz from the atomic resonance at zero magnetic ﬁeld. To generate the required frequencies, we lock a unique additional DBR laser tuned -28 9 MHz from resonance. We RF- modulate the diode current at 16 MHz. This generates sideban ds into its spectrum. The level of modulation is optimized to get the maximum intensity into the ﬁrst two lateral sidebands, whose frequencies are the ones required for repumping. The p ower in the repumping beam is approximately 20 mW. It is checked using the absorption me asurement explained earlier that the repumping process brings back nearly the same ﬂux of slow atoms as in the abscence of the MOT beams. 95.3 Characterization of the MOT The atoms are ﬁnally conﬁned in a magneto-optical trap. Two c ylindrical coils, separated by 5.2 cm along the y-axis (see ﬁgure 6), create a magnetic gra dient of 40 G/cm along the symmetry axis for a given current of 5A. The repumper beam typ ically increases the number of trapped atoms by a factor of 3. Losses are dominated by intr a MOT Penning collisions [3, 26, 27, 28]. In this regime, the number of trapped atoms go es as the square root of the loading rate. This increase in the number of atoms implies th at the loading rate is increased by a factor of 32= 9. Using the repumper beam, we routinely trap approximatel y 109 atoms, in a volume of 0.1 cm3, at a temperature of approximately 1 mK. The temperature is measured by a time of ﬂight technique [28]. The number of trap ped atoms is inferred from the measurement of the absorption in a 1 cm diameter probe las er beam, intense enough to saturate the transition. The size of the MOT is measured by ab sorption imaging on a CCD Camera. 6 Conclusion In this article, we report on a new experiment aiming at reach ing BEC with metastable he- lium atoms. We demonstrate the eﬃciency and the robustness o f a new discharge source of metastable atoms and of a bright laser setup using a high powe r ﬁber ampliﬁer at 1.083 nm for the manipulation of atoms. For the MOT the original trapp ing scheme requires an addi- tional laser beam used to repump the atoms during the slowing process. The atomic cloud is trapped at the center of a small quartz cell. The present se tup has several advantages. First, it gives a good optical access to the atomic cloud. Thi s allows to further trap atoms in a strongly conﬁning magnetic trap placed as close possible t o the cell as. Secondly, it allows to reach extremely low pressures inside the small volume of t he cell. However, the present setup makes it diﬃcult to use ion detectors or channel plates to detect the metastable atoms. Further developements of the experiment could include such detectors in an appropriate cell. Detection is performed in the present setup by purely optica l means. The infrared line or other visible lines for which CCD cameras have a better eﬃcie ncy can be use for detection. The present experiment allows to routinely trap approximat ely 109helium atoms in the 23S1 metastable state inside a MOT of 2 mm rms radius. The setup is currently being modiﬁed to add magnetic coils fo r a magnetostatic trap that will be used in the search for BEC. This Ioﬀe type trap consist s of three asymmetric coils plus two large compensation Helmholtz coils, giving a ﬁeld c onﬁguration similar to the QUIC trap [29]. The gradients are approximately 280 G/cm, the cur vature is 200 G/cm2and the depth of 33 mK for a current of 50 A. Before loading in the QUIC t rap, the atoms will be ﬁrst cooled into a molasse phase, where the ﬁeld gradient of the MO T is turned oﬀ, which should allow to reach lower temperatures (50 to 100 µK). When a large density of ultracold atoms is loaded into the Ioﬀe trap, it will be possible to check the the oretical predictions of [13, ?] on elastic and inelastic collision rates between metastable a toms at very low temperature. The measured collision rates will then indicate whether one can achieve BEC using evaporative cooling, as successfully used with alkali atoms. Acknowledgments: The authors thank C. Cohen-Tannoudji for very helpful discu ssions, and for his input in the experiment. aPermanent address: Laboratoire de Physique des Lasers, UMR 7538 du CNRS, Uni- versit´ e Paris Nord, Avenue J.B. Cl´ ement, 93430 Villetane use, France. bPermanent address: Institute of Opto-Electronics, Shanxi University, 36 Wucheng Road, Taiyuan, Shanxi 030006, China. cPermanent address : Dept. of Physics, Univ. of Perugia, Via P ascoli, Perugia, Italy; Lens and INFM, L.go E. Fermi 2, Firenze, Italy dPresent address : Universit¨ at Hannover, Welfengarten 1, D -30167 Hannover, Germany. ePermanent address : TIFR, Homi Bhabha Road, Mumbai 400005, I ndia. 10∗Unit´ e de Recherche de l’Ecole Normale Sup´ erieure et de l’U niversit´ e Pierre et Marie Curie, associ´ ee au CNRS (UMR 8552). References [1] B. Saubam´ ea, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M . Leduc, and C. Cohen- Tannoudji, Phys. Rev. Lett. 79, 3146 (1999) [2] W. Rooijakkers, W. Hogervorst, W. Vassen, Opt. Comm., 135, 149 (1997) [3] P.J.J. Tol, N. Herschbach, E.A. Hessels, W. Hogervorst, W. Vassen, Phys. Rev. A 60, R761 (1999) [4] S. Nowak, A. Browaeys, J. Poupard, A. Robert, D. Boiron, C . Westbrook, A. Aspect, Appl. Phys. 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arXiv:physics/0103065v1 [physics.class-ph] 21 Mar 2001DELAY EQUATION FOR CHARGED BROWN PARTICLE Alexander A. Vlasov High Energy and Quantum Theory Department of Physics Moscow State University Moscow, 119899 Russia In previous work ( physics/0004026) was shown, with the help of numer- ical calculations, that the eﬀective Brown temperature for charged particle is lower than that for particle without charge. Here we derive t his result with- out numerical calculations, integrating the delay equatio n analytically, as for zero, so for nonzero viscosity. 03.50.De 1. To describe motion of charged Brown particle in so called ”ex tended quasi-stationary approximation”[1] in [2] was used the Som merfeld model [3] of charged rigid sphere. The equation of straightline motio n of such Brown particle in dimensionless form reads [2]: ˙y(x) =f(x) +γ·[y(x−δ)−y(x)] (1) here y(x) - is dimensionless velocity of the particle; x- is dimensionless ”time”; f(x) - is some external (stochastic) force; δ- is ”time” delay; γ- is coeﬃcient: γ·δis proportional to the ratio of particle’s electromag- netic mass to the mechanical mass: γ·δ= (2/3)(Q2/a)/(mc2) ( 2a- is the size of Sommerfeld particle of charge Qand mass m); the viscosity Γ of the surrounding medium is zero. In [2] was shown, with the help of numerical calculations, th at the eﬀective Brown temperature for charged particle is lower than that fo r particle without charge. Here we derive this result without numerical calcul ations, integrating the delay equation (1) analytically. 1With zero initial conditions: y= ˙y= 0 for x <0 dividing the x-axis into δ- intervals ( i−1)δ≤x≤iδ,i= 1, ..., and integrating eq. (1) step by step with boundary condition syi(x= iδ) =yi+1(x=iδ), we ﬁnally get the recurrence formula: for (N−1)δ≤x≤Nδ y(x) =yN(x) = /integraldisplayx 0dzf(z) expγ(z−x) +γ/integraldisplayx−δ (N−2)δdz y N−1(z) expγ(z+δ−x) +γN−2/summationdisplay i=1/integraldisplayiδ (i−1)δdz y i(z) expγ(z+δ−x) (2) with y1(x) =/integraldisplayx 0dzf(z) expγ(z−x),0< x≤δ Let’s consider one interesting case: f(x) for intervals ( i−1)δ≤x≤iδis constant and is equal to fi. Then the eq.(2) for x=Nδ≡xNyields y∗ N≡yN(x=xN) =1 γN/summationdisplay k=1fk[1−C(N−k;p)]≡N/summationdisplay k=1fkDk (3) where the function C(n;p) is deﬁned as C(n;p) = exp ( −p(n+ 1))n/summationdisplay m=0(pexpp)m(n+ 1−m)m/(m!); (4) herep≡γδ. Function C(n;p) is positive and for suﬃciently large n(for ex., n >20 forp= 1.0 ) is equival to1 1+p: C(n;p)n≫1=1 1 +p→Dn≈p (1 +p)γ(5) Thus if fi=f0=const ∀i, then from (3,5) we get for N≫1 y∗ N≈f0N·p (1 +p)γ=f0 1 +pxN 2in accordance with the exact solution of (1) for f=f0=const: y(x) =f0 1 +px Also for N≫1 one can rewrite (3) in the form y∗ N≈p (1 +p)γN/summationdisplay k=1fk=δ (1 +p)N/summationdisplay k=1fk (6) This result resembles the classical Brown result: from eq.( 4) with γ= 0 one immediately gets y(x) =/integraldisplayx 0f(z)dz, (7) dividing x-interval of integration in (7) into δ- intervals with f(x) =fkfor (k−1)δ≤x≤kδ, one can take the integral in (7) in the following manner: y(x=xN) =δN/summationdisplay k=1fk (8) This result diﬀers from (6) only in the multiplier1 (1+p). Thus one can say that the eﬀect of delay (eﬀect of retardation ) for eq.(1) reduces to the eﬀect of mass renormalization: m→m/(1+p), or consequently to the eﬀect of reduction of the external force: f→f/(1 +p) (9) This result also says that the reduction of the external forc e is model-independent one, and instead of γδone can write the classical ratio of self-electromagnetic mass to the mechanical mass min its general form: γδ→1 mc2/integraldisplay d/vector rd/vectorr′ρ(/vector r)ρ(/vectorr′) \|/vector r−/vectorr′\|(10) hereρ- is distribution of charge of a particle. Iffk, k= 1, ...- is the range of stochastic numbers with average value fa: < fk>=fa(here brackets <>denote time average with the same deﬁnition as in the classical theory of Brownian motion), then eq.(3) y ields < y∗ N>=faN/summationdisplay k=1Dk≈faxN/(1 +p) (11) 3Consequently the dispersion Dis D= (y∗ N−< y∗ N>)2=N/summationdisplay k=1N/summationdisplay m=1DkDm<(fk−fa)(fm−fa)> =N/summationdisplay k=1N/summationdisplay m=1DkDmR(k−m) (12) hereR(k−m) - is correlation function of stochastic force f. IfRis compact: R(k−m) =R0δmk/δ (13) then the dispersion (12) is D=R0/δN/summationdisplay k=1(Dk)2≈R0xn/(1 +p)2(14) This result should be compared with classical one. The theory of Brownian motion without viscosity tells ( eq. ( 1) with γ= 0 ) that the dispersion DBis DB=/integraldisplayx 0dz1/integraldisplayx 0dz2·R(z1−z2) (15) hereR(z1−z2) =<(f(z1)−fa)(f(z2)−fa)>- is the correlation function. If R(z1−z2) =R0δ(z1−z2) then DB=R0x (16) Consequently we see that (eqs. (16) and (14) ) the dispersion of the Som- merfeld charged particle is lower than that of the classical Brown particle without electric charge: D=DB(1 +p)−2. Thus one can say that the ef- fective temperature of Sommerfeld particle is lower than th at of the Brown one. This result is model independent one (see the remark mad e above - eq. (10) ). So we conﬁrm the result of the work [2]. 42. If the viscosity Γ is not zero, the main equation reads: ˙y(x) + Γ·y(x) =f(x) +γ·[y(x−δ)−y(x)] (17) Forf=f0=const eq.(17) has the exact solution y(x) =f0 Γ(1−exp (−ax)) (18) andais determined by the eq. Γ +γ−a=γexp (aδ) (19) Iterative solution y(xN) =y∗ Nof eq.(17), if f(x) =fi=const for intervals (i−1)δ≤x≤iδ, can be put in the form: y∗ N=N/summationdisplay k=1fkDk (20) hereDk- some discrete function which can be found from recurrence f or- mula, analogous to (2). But it is convenient to ﬁnd Dkfrom the following considerations, using exact results (18,19). Solution (20 ) must tend to the exact solution (18) (in the case fi=f0=const∀i) if the x-axis is divided into inﬁnitesimally small δ-intervals: δ→0 and N→ ∞ in such a way that xN=δ·N=const. Thus one can rewrite y∗ N=N/summationdisplay k=1fkDk=f0N/summationdisplay k=1Dk=f0 Γ(1−exp (−axN)) so N/summationdisplay k=1Dk=1 Γ(1−exp (−aδN)) (21) Ifδ→0 we can replace the sum in lhs of (21) by the integral: N/summationdisplay k=1Dk≈/integraldisplayN Dkdk=1 Γ(1−exp (−aδN)) (22) Diﬀerentiation of (22) with respect to Nprovides us with this expression for DN: 5DN≈aδ Γexp (−aδN) (23) Substitution of (23) back into (21) gives N/summationdisplay k=1aδ Γexp (−aδk) =aδ Γ·1−exp (−aδN) exp (aδ)−1(24) Consequently the required result (rhs of (21) ) is reproduce d if we expand the denominator in (24) in the following way: exp (aδ)−1≈aδ (24) Using this representation of Dk, one can ﬁnd the dispersion D. For cor- relation function (13) we have D=R0 δN/summationdisplay k=1(Dk)2≈ R0 δ·(aδ Γ)2·1−exp (−2aδN) exp (2 aδ)−1≈R0a 2(Γ)2(1−exp (−2aδN)) (25) here we expanded the expression exp (2 aδ)−1 in the same manner as in (24): exp (2 aδ)−1≈2aδ. Solving the eq.(19) in approximation (24), we ﬁnd a≈Γ (1 +γδ)(26) So with (26) and (25) the dispersion D takes the form D=R0 2(Γ)(1 + γδ)(1−exp (−2aδN)) (27) Dispersion (27) for γ≡0 is exactly the same as Brownian dispersion DB: DB=R0 2Γ(1−exp (−2ΓxN)) IfaxN≪1, solution (27) yields D≈R0xN (1 +γδ)2 6i.e. the solution we have got earlier (14). IfaxN≫1, (27) yields D≈R0 2Γ(1 + γδ)=DB (1 +γδ) Thus dispersion Ddiﬀers from the Brownian one. Consequently the eﬀec- tive temperature of charged particle, undergoing Brownian motion, is lower then that of particle without charge. Now we have proved this result in general case of nonzero viscosity. Of course, our general co nclusion is model- independent one - see the above remark (10). REFERENCES 1. T.Erber, Fortschr. Phys., 9, 343 (1961). 2. Alexander A.Vlasov, physics/0004026. 3. A.Sommerfeld, Gottingen Nachrichten, 29 (1904), 363 (19 04), 201 (1905). 7
arXiv:physics/0103066 21 Mar 2001 ELEMENTAR OBJECTS OF MATTER: UNIVERSALITY, HIERARCHY, ISOMORPHYSM, DYNAMICAL SPECTRUM A.M. Chechelnitsky Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia E’mail: ach@thsun1.jinr.ru ABSTRACT In the frame of Wave Universe Concept (WU Concept) it is presented the alternative approach to the effective description of Elementar Objects of Matter (EOM) of micro and megaworld hierarchy, in particular, of particles in subatomic physics. According to the Wave Universe Concept (WU Concept), discrete spectrum of EOM is close connected, generate by universal spectrum of physically preferable, elite velocities in the Universe. The special attantion to analysis and precise description of central set of EOM - stationary states (of EOM) is payed. In particular, sufficiently precise representations for mass values, cross relations between masses of main important objects of particle physics (proton, pion, main mesons, etc.) are obtained. Obtained representations for the hierarchy of characteristic dimensional parameters, for instance, for the mass spectrum - mass formula are not contained any divergencies - its are simple, compact, possess clear physical sence and ha ve not any kind of fitting parameters. With this, the competence field of these representations is practically indefinitely - apparently, "all" Wave Universe for wide set EOM of micro and megaworld. ELEMENTAR OBJECTS OF MATTER Astohishing diversity of real objects of Universe, observed on different Levels of matter, may be considered as manifestation of nonexhausting creative capacities of Nature. The most characteristic, wide representable, as possible, the simplest from its with most probability are attracting in some known concepts as candidates to fundamental, elementary objects of matter (EOM), representing (and organizing) the observed appearance of Universe. It is considered evidently, that compositions, combinations such fundamental constituents create and demonstrate all observed variety of complex systems - at all Levels of the Universe hierarchy. Elementary Objects of Matter (EOM) – As Wave Dynamic System (WDS) With any way - speculative, dinamical, physical - of attempt to describe, qualify these or another characteristic objects or all its taxanomy, we suppose, that the most frequantly asking question: "From what are consist...?" - don't has the special sense and real perspective. It is appear as more constructive, fundamental the following conclusion, having far-reaching conseqences [Chechelnitsky, 1980]. Proposition. Observed in Universe real objects and most fundamental from its - elementary objects of matter (EOM)- represent itself, in conceptual plane - the principal Wave dynamic systems (WDS). Wave (Megawave) aspect of structure of any observed systems of Universe at all Levels of its hierarchy is not external formal supplement, but is deep internal fundamental basis of its dynamical and physical structure. VELOCITIES HIERARCHY AND UNIVERSALITY Hierarchy and Spectrum of Elite Velocities. The Fundamental wave equation [Chechelnitsky, 1980], described of Solar system (similarly to the atom system), separates the spectrum of physically distinguished, stationary - elite - orbits, corresponding to mean quantum numbers N, including the spectrum of permissible elite velocities vN. It is the follow representation for the physically distinguished - elite velocities vN[s] in G[s] Shells of wave dynamical (in particular, astronomical) systems (WDS) [Chechelnitsky, 1986] vN[s] = C∗[s](2π)1/2/N, s=...,-2,-1,0,1,2,... C∗ [s] = (1/χs-1)⋅C∗ [1]. Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 2 Here C∗[1] = 154.3864 km⋅s-1 is the calculated value of sound velocity of wave dynamic system (WDS) in the G[1] Shell, that was made valid by observations, χ = 3.66(6) - the Fundamental parameter of hierarchy (Chechelnitsky Number χ = 3.66(6)) [Chechelnitsky, 1980 - 1986], s - the countable parameter of Shells, N - (Mega)Quantum numbers of elite states, a) Close to NDom = 8; 11; 13; (15.5)16; (19,5); (21,5) 22,5 - for the strong elite (dominant) states (orbits); b) Close to N - Integer, Semi-Integer - for the week elite (recessive) states (orbits). In the wave structure of the Solar System for planetary orbits of Mercury (ME), Venus (V), Earth (E), Mars (MA), we have, in particular, N = (2πa/a∗)1/2 (a - semi-major axes of planetary orbits, a∗[1] = 8R/G7E - semi-major axis of TR∗[1] - Transsphere, R - radius of Sun) [Chechelnitsky, 1986] N = 8.083; 11.050; 12.993; 16.038, close to integer N = 8; 11; 13; 16. Taking into account Ceres (CE) orbit and transponating in G[1] (from G[2]) planetary orbits of Uranus - (U), Neptune - (NE), Pluto - (P), it may be received the general representation for observational dominant N TR∗ ME TR V E (U) MA (NE) CE (P) N= (2π)1/2=2.5066 8.083 (2π)1/2χ=9.191 11.050 12.993 15.512 16.038 19.431 21.614 22.235 It may be show, that N = N∗ = (2π)1/2=2.5066 (critical - transspheric value) and NTR=χ(2π)1/2 ≅ 9.191 also are physically distinguished (dominant) N values [Chechelnitsky, 1986]. Extended Representation It is possible, in principle, examine the following substitution 1/N → ς / N# or N → N#/ς and extended formula for elite velocities vN[s] = C∗[s](2π)1/2(ς / N#), s=...,-2,-1,0,1,2,... ς , N# - integer. In that case, for instance, the previouns condition N - semi - integer will be indicate (for the set of integer numbers) the condition ς =2, N# - integer, and thus, - the substitution N → N#/2. General Dichotomy Very close (to discussed above) variant of description of physically distinguished states may be possible with using of effective approximation, proposing by the General Dichotomy Law [Chechelnitsky, 1992]. Connected with it compact representation for the N quantum number has the explicit form Nν = Nν=0·2ν/2, Nν=0 = 6.5037 that depends from countable parameter ν = k/2, k=0,1,2,3,... It follows, in perticular, to exponential, (power) dependen ce for a semi-major axes aν[s] = aν=0[s] 2ν, aν=0[s] = a∗[s] (Νν=0)2/2π, In the some sense - this is expansion and gene ralization to all WDS of Universe of the well-known Titius-Bode Law for the planetary orbits. Such idealazing model representation - the General Dychotomy Law (GDL) - gives approximate, but easy observed description of the set of distinguished (dominant) orbits. Universal Spectrum of Elite Velocities in the Universe. Megaworld and Microworld (Quasars and Particles). Proposition. The spectrum of physically distinguished elite (dominant) velocities vN[s] and quan tum numbers N of arbitrary Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 3 wave dynamic systems (WDS) have the some universal peculiarity. It practically is identical - universal (invariant) for all known observed systems of Universe (of megaworld and microworld). In particular, velocities spectrum of experimentally well investigated Solar and satellite systems practically coincide for observed planetary and satellite - dominant orbits, corresponding to some (dominant) values of quantum numbers NDom. Thus it may be expected, that spectrum of elite (dominant - planetary) velosities of the Solar system (well identificated by observations) may be effectivelly used as quite representive - internal (endogenic) - spectrum of physically distinguished, well observed - elite (dominant) velocities, for example, of far astronomical systems of Universe [Chechelnitsky, 1986, 1997] and of wave dynamic systems (WDS) - elementary objects of subatomic physics. Quantization of Circulation and Velocity. We once more repit in the compact form the important conclusion which was obtained in the monograph (Chechelnitsky, 1980) and repeatedly underlined afterwards. Proposition (Quantization of Velocities). In the frames of Wave Universe Concept and Universal wave dynamics # The fundamental properties of discreteness, quantization of wave dynamic systems (WDS) - objects both mega and microworld - are connected not only with discreteness, quantization of i ) Kinetic momentum (angular momentum) Km= mva, ii ) And momentum (impuls) P = mv (as that is discrabed in well known formalism of quantum mechanics), # But - on the fundamental level - are connected with discreteness, quantization of v) Sectorial velocity (circulation) L = Km/m = va, (L∗ = ξ /GFF/G03/G20/G03ξ /GAB /G12/G50/G0C/G0F ξ - nondimensial coefficient, vv) And (Keplerian) velocity v = P/m. vvv) Together with the relating to its sizes (lengths) - a - semi-major axes of orbits and T - periods (frequencies). Universality of Observed, Physically Distinguished Velocities From the point of view of experimental investigations of real systems of Universe the Law of Universality of Elite (Dominant) velocities may be briefly formulated as follows Proposition (Universality of Elite – (Dominant) Velocities in Universe). # Detectable in experiments and observations velocities of real systems of Universe - from objects of microworld (subatomic physics) to objects of megaworld - astronomical systems - with the most probability belong to the Universal Spectrum of elite (dominant) velocities of Universe. # This Universal Spectrum of Velocities in the sufficient approximation may be represented in the form: vN[s] = C∗[s](2π)1/2/N, s=...,-2,-1,0,1,2,... C∗[s] = (1/χs-1)⋅C∗[1]. General Homological Series of Sound Velocities Once more let pay our attention to the hierarchy of sound velocities, that is definded by the recurrence relation C∗[s] = (1/χs-1)⋅C∗[1] s=...,-2,-1,0,1,2,... In view of its special important significance and possibility of following generalizations we will to name it "The General Homological Series (GHS) of sound velocities". By the quality of generative member in that series essentially it is used, for instance, the C∗[1]=154.3864 km⋅s-1 - value of sound velocity in G[1] Shell of WDS. As a matter of fact, that is primary source (eponim) of that series. Of course, as the capacity of primary source may be used any member of that series. Testimony (Evidence) for that is only most knowlege reliability of that value - its experimental definiteness (determination). Fundamental Parameter of Hierarchy. At 70-th in investigation of wave structure of Solar system [Chechelnitsky, 1980] it have been d iscovered significent arguments for existance of Shell structure, hierarchy and similarity - dynamical isomorphysm - of Solar system Shells. First of all, that concerned to dynamical isomorphysm of clearly observed G[1] and G[2] Shells, connecting Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 4 respectively with I (Earth's) and II (Jovian) groups of planets. It was determined that arrangement of physically distinguished - elite (particularly powerful, strong -dominant) orbits of Mercury in G[1] (and Jupiter in G[2]), Venus in G[1] (and Saturn in G[2]) Shells brightly underline the similarity of geometry and dynamics of processes, flowing in these Shells, with accuracy up to the some scale factor. As the quantitative characteristics of that isomorphysm, the recalculation coefficient χ - Fundamental parameter of hierarchy (FPH) - may be used the ratio, for instance, of # (Keplerian) orbital velocities v vME/vJ=47.8721 km⋅s-1/13.0581 km⋅s-1=3.66608 ⇒ χ , vV /vSA=35.0206 km⋅s-1/9.6519 km⋅s-1=3.62836 ⇒ χ , # Sectorial velocities L LJ/LME=1.01632⋅1010km2⋅s-1/0.27722⋅1010 km2⋅s-1=3.66608 ⇒ χ , LSA /LV=1.37498 ⋅1010 km2⋅s-1/0.37895 km2⋅s-1=3.628357⇒ χ , # Semi-major axes a aJ /aME=5.202655 AU /0.387097 AU = = 13.440164=(3.666082)2 ⇒ χ2 , aSA/aV= 9.522688 AU /0.723335 AU = = 13.164975 = (3.628357)2 ⇒ χ2 , # Orbital periods T (d - days) TJ/TME=4334.47015d/87.96892d=49.272744=(3.666082)3 ⇒ χ3 , TSA/TV=10733.41227d/224.70246d = 47.76722=(3.6283568)3 ⇒ χ3. In the published at 1980 monograph [Chechelnitsky,1980] (date of manuscript acception - 11 May 1978) this dynamical isomorphysm, similarity of geometry and dynamics of physically distinguished orbits of I (Earth's) and II (Jovian) groups were analized. According to the content of "Heuristic Analysis" division [Chechelnitsky, 1980, pp.258-263, Fig.17,18] similarity coefficient - recalculation scale coefficient of megaquants ΔΙ =LME/3=0.924⋅109 km2⋅s-1 ΔΙΙ =L J /3=3.388⋅109 km2⋅s-1 of L - sectorial velocities (actions, circulations) of I and II groups of planets is equal Δ ΙΙ / ΔΙ = L J /LME =3.66(6) ⇒χ It was not surprise, that transition to another Shells of Solar (planetary) system (to Trans-Pluto and Intra- Mercurian Shells) would be characterized with the same χ − Fundamental parameter of hierarchy (FPH) χ=3,66(6). Universality of FPH Analysis of (mega)wave structure of physically autonomous satellite systems of Jupiter, Saturn, etc., indicated, that discovered χ Fundamental parameter of hierarchy (FPH) plays in its the similar essential role, as in the Solar (planetary) system, characterizing the hierarchy, recursion and isomorphysm of Shells. Thus, it takes shape the essentially universal character of (FPH) - its validity for the analysis of (mega)wave structure of any WDS. That corresponds to representations, connected with co-dimension principle [Chechelnitsky, 1980, p.245]: "...fundanental fact is that when we pass on to another WDS, the value of /GFF [character value of sectorial velocity (action, circulation)] doesn't remain constant, but varies according scales of these systems. This fact is the consequence of co-dimension principle ..." "Magic Number"("Chechelnitsky Number", FPH) χ χ=3,66(6). Role and Status of Fundamental Parameter of Hierarchy in Universe. Previous after primary publications [Chechelnitsky, 1980-1985] time and new investigations to the full extent convince the theory expectations, in particular, connected with the G[s] Shells hierarchy in each of such WDS, with the hierarchy of Levels of matter (and WDS) in Universe, with the exceptional role of the introduced in the theory χ FPH [Chechelnitsky, (1978) 1980-1986]. The very brief resume of some aspects of these investigations may be formulated in frame of following short suggestion. Proposition (Role and Status of χ FPH in Universe) [Chechelnitsky, (1978) 1980-1986] # Τhe central parameter, which organizes and orders the dynamical and physycal structure, geometry, hierarchy of Universe ∗ "Wave Universe (WU) Staircase" of matter Levels, ∗ Internal structure each of real systems - wave dynamic systems (WDS) at any Levels of matter, is (manifested oneself) χ - the Fundamental Parameter Hierarchy (FPH) - nondimensional number χ =3,66(6). Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 5 # It may be expected, that investigations, can show in the full scale, that χ - FPH, generally speeking, presents and appea res everywhere - in any case, - in an extremely wide circle of dynamical relations, which reflect the geometry, dynamical structure, hierarchy of real systems of Universe. We aren't be able now and at once to appear all well-known to us relations and multiple links, in which oneself the [Chechelnitsky] χ=3.66(6) "Magic Number" manifests. We hope that all this stands (becomes) possible in due time and with new opening opportunities for the publications and communications. STATIONARY STATES. Spectrum of Masses (of EOM). Mass Formulae. In the frame of Wave Universe Concept even for extent time (for the own perspective investigations of WDS of various hierarchy Levels of Universe) we use the following representation for the spectrum of mass of stationary states. We cite it with hope in potentially wide employment in various, occasionally, far (distant) extending domain of knowledge (for instance, - in particle physics, astrophysics, cosmology). MN[s]=M∗[s]N2/2π, s=...,-2,-1,0,1,2,... M∗[s]=(χ2)sM∗[0]=(χ2)s-1M∗[1] More detail representation is possible. Spectrum of Masses. Mass Formulae. The description of mass of stationary states is possible in the frame of following assertion. Proposition Characteristic spectrum of masses of certain U(k) matter Level may be represented in form MN(k)=M∗(k)N2/2π, k=...,-2,-1,0,1,2,... Here k - contable parameter, that determines U(k) Level of matter, N - main quantum number. Preferable values of N belong to set physically distinguished ∗ Elite states; and among them - to the more restricted subset of elite states - ∗ Dominant states - strong elite states. Generative ("Transspheric", critical, General) M∗(k) mass, which formes the mass spectrum of examined U(k) local Level of matter, itself belong to the General Homological series (GHS) of masses M∗(k)=χkM∗(0), k = ...,-2,-1,0,1,2,..., M∗(0) - physically distinguished, certain existing, real observed generative mass, will be say, primary-image (eponim) mass (it may be any the well-known from M∗(k)), χ - Fundamental Parameter of Hierarchy (FPH) (Chechelnitsky Number χ = 3.66(6)). Matter Levels and Shells. By special, preferable - more rare - set of matter Leveles U(k) it may be considered the sequence - hierarchy of matter Levels, when generative mass M∗[s] belongs to the General Homological Series (GHS) of mass M∗[s]=(χ2)sM∗[0], s = ...,-2,-1,0,1,2,... In other words - this is Even subset of matter Levels U(k) at k = 2s, M∗(0) = M∗[0] Such hierarchy of U(2s) = U[s] matter Levels corresponds (in some sense equivalent) to G[s] Shells hierarchy, that is wide analysed in structure of WDS. So, the mass spectrum, close connected with G[s] Shell structure, may be represented at following compact form MΝ[s]=M∗[s]N2/2π, s=...,-2,-1,0,1,2,... M∗[s]=(χ2)sM∗[0] General Homological Series of Masses What value must be choosed (selected) as M∗[0] - generative ("transspheric", critical) - General value of mass? It must be comprehend also, that sampling of only one value of the primary - image (eponim) M∗(0) = M∗[0] (or, for instance, M∗[2]), essentially, signifies also sampling of the whole of General Homological Series (GHS) of masses (or General Homology of masses) M∗(k)=χkM∗(0), k = ...,-2,-1,0,1,2,.... Such choosing is not only formal, only mathematical operation. It must be dictated by physics, objective reality, Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 6 observations. It may be show by "appearance order" (i.e. by justified, convincing consequences), that such mass is mπ (the mass of π±-meson) or M2p=2mp mass of Di-proton (or Deuteron). In terms of unique Homological series - that is, essentially, the same. Di-proton (Deuteron) and pion (π± - meson) belong to the same (General) Homological series (GHS), in other words, its generate the same General Homology of masses. Each of its may be considered as primary-image (eponim) of Homological series. This affirmation may be convinced to true immediately. We have, according to [RPP], for Di - proton mass M2p = 2mp = 2·938.27231 = 1876.5446 Mev/c2 and for π± - meson (pion) mass mπ = 139.56995 Mev/c2 the ratio M2p/mπ = 13.44519 = χ2. The following from that quantity value χ = 3.666(7684) coinsides with standard, accepted value of χ = 3.66(6) - Fundamental parameter of hierarchy (FPH). This result may be also considered as one of possible experimental determination of χ - FPH (in microworld). Pion, Di-proton and χ χ constant. Once more we point importance of the following observation. Proposition # Masses of mπ pion, m2p Di-proton and value of χ FPH are connected (group together) by the following relation M2p/mπ =2mp/mπ=13.44519 = χ2. # That relation in some specific sense may be considering as experimental definition of χ FPH . It will be good for consequent calculations the following relation between mp and me. Proposition # By using equation mπ/me=β2α-1 # The relation between fundamental masses mp and me may be express by the formula mp/me=βχ2/α=1836.1527, where β=0.996623 is approximating coefficient (β∼1). Thus, the following assertions open possibilities of the wide using of mass spectrum representation in various ranges (spans) of masses. Proposition The General Homological series (GHS) of masses M∗(k)=χkM∗(0), k = ...,-2, -1, 0, 1, 2,..., and M∗[s]=(χ2)sM∗[0] , M∗(0) = M∗[0] s = ...,-2, -1, 0, 1, 2,..., are is completely represented by the π± - meson - Di - proton GHS (or π - D Homology) of masses. For the definiteness (and there have specific physical sense) it may be considered, that for GHS of masses M∗[s]=(χ2)sM∗[0] s = ...,-2, -1, 0, 1, 2,... valid M∗[1] = mπ, M∗[2] = M2p = 2mp = 1876.5446 Mev/c2 and then M∗[2] = χ2M∗[1] = M2p = 2mp In this circumstances GHS of masses M∗[s] = (χ2)s-1M∗[1]] = (χ2)s-2M∗[2], s = ...,-2, -1, 0, 1, 2,... become fully definite (by Chechelnitsky Number χ=3,66(6)) and containing following set - hierarchy of masses (fragment) M∗[s] ⇒ ..., M∗[0]=10.3816, M∗[1]=139.575, M∗[2]=1876.5446, M∗[3]=25228.6 Mev/c2,... Even Homology. It is interesting to point, that discussed above U(k)=U(2s) subset of matter Levels may qualified as Even (k=2s) subset, and GHS - as Even Homology of mass related to M∗[0] prime image (eponim). Odd Homology. It is clear, that (residual) remaining (in U(k) set) the U(k)=U(2q+1) subset of matter Levels, may consider as Odd subset, and in M∗(k) Homological series (M∗(k) HS) of mass the remaining set - special series Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 7 M∗[q] = χ(χ2)qM∗[0] = χ 2q+1M∗[0] is Odd (k=2q+1) Homology of mass. It may be shown, that this Odd Homological series (Odd HS) have the nontrivial physical sense. MICROWORLD (SUBATOMIC WORLD). STATIONARY STATES  MASS SPECTRUM. THEORY AND EXPERIMENTS. Calculations that use the discussed above representations for the mass hierarchy discover, essentially, new world of dynamical accordances, propose possibilities for constructive interpretations of dynamical structure of known from experiments stationary states and resonances of subatomic world. Stationary States of G[1] Shell: Population of Pion (π π± ± meson). Presentation of stationary states dynamical spectrum we shall from, it will be say, population of π± meson. This is matter Level, that corresponds to the G[1] Shell (s=1), matter Level G[1] (or k=2s=2, U(k)=U(2)). Mass spectrum is generated by one of components (π± meson mass) M∗[1] = mπ ≅ 139.575 Mev/c2 that belongs to π–D General Homological Series (π–D GHS). Mass spectrum of stationary states seems stonishingly saturated [Table 1]. The ϒ ϒ Family. It is interesting to point, that spectrum of ϒ states, that is detected in experiments (see RPP), also belongs to periphery of G[1] Shell - with large (periphery) values of N quantum number. Stationary States of G[2] Shell: Population of Di-proton. The corresponding to G[2] Shell (s=2, k=2s=4, U(k)=U(4)) matter Level is generated by physically distinguished state of Di-proton (Deuteron). In the Table 2 it is use mass value M∗[2]=2mp=1876.51 Mev/c2 that belongs to π–D GHS (π–D Homology). It is presented the comparision of theoretically calculated masses of stationary states (also is presented the theoretical calculation by General Dichotomy) with collected data of experiments (estimations also, etc.) from [RPP] (Table 2). At initial stage of search investigations it is hardly advisible to develop too rigid selection, based on customary preferences of the past. So, to the comparison with the theory it is attracted, as it possible, most wide material. Stationary States of G[3] Shell. Mass spectrum of stationary states, connected with G[3] Shell, is represented in Table 3. It is generated by the mass M∗[3] = χ22mp = 25.2286 Gev/c2 that belongs to π–D GHS (π–D Homology) [Table 3]. Stationary States of G[4] Shell. Mass spectrum of stationary states, connected with G[4] Shell, is represented in Table 4. It is generated by the mass M∗[4] = χ42mp=0.339 Tev/c2 that belongs to π–D GHS (π– D Homology). It is possible that modern and future HEP in high degree will be connected with manifestation of stationary states of G[3], G[4] and later Shells. Another Levels of Matter. Stationary States of G[-2] Shell. Mass spectrum of stationary states of G[-2] Shell is generated by mass M∗[-2] = χ -82mp =0.05743 Mev/c2 that belongs to π–D GHS. The state ME (N=8.083) of G[-2] Shell M = 0.5973 Mev/c2 , that is close to electron mass me = 0,51099906 Mev/c2 draws the most attention.. In frame of this G[-2] Shell to the experimental value of electron quantum number N the value N = (2πM∧)1/2 = 7.477054, M∧ = me/M∗[-2] = 8.89777, P∧ = 2πM∧ = 55.9053 corresponds. It lies at the interval permissible, often observing N values, for instance, of elite states in Solar (planetary) system and satellite systems of planets. Observed in the system of Saturn S1 (Mimas) satellite has N=7.380, in the Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 8 system of Jupiter J2 (Europe) satellite - N=7.680. It is interesting to point that M∧, P∧ close to integer and (N) - to semi-integer (that is patterns of wave stability). Stationary States of G[0] Shell. The mass spectrum of G[0] Shell also demonstrates the physical significantness of description of mass spectrum. Physically preferable mass, that belongs to π−D GHS (π−D Homology) M∗[0] = 10.3816 Mev/c2 generates the mass spectrum of stationary states, that corresponds to (elite) dominant NDom values (see Table 5). First of all, it may be pointed, that in mass spectrum the stationary states are discovered, that correspond to ones detected in experiments π-meson and ρ−meson. It may be waited that many of other mesons correspond also to periphery elite (may be, not so strong, as dominant) values of N quantum number. For instance, for the η′ (958) meson with m = 957.77 Mev/c2 we have quantum number N = (2πm/M∗[0])1/2 = 24.0762 that is close to integer. Subset of Neutrino. It is possible, that with increasing of mass precision it may be stated, that part of neutrino ντ (for which now only up limits of masses is indicated) indeed belongs to stationary states of G[0] Shell (see Table 5, TR state). Stationary state - muon. Finally, the last but not least, in the population of dominant - stationary states of G[0] Shell (Table 5) it is discovered the state with M = 107.97 Mev/c2 mass, evidently close to indeed corresponding (as indicate analysis) muon state. Even for concidered only for first main approximation, the achieved precision must be concidered in sufficient degree acceptable, especially at background of low accuracies of few known in particle physics mass formulae (as Gell-Mann-Okubo, etc.) Nevertheless, problem of more precise corresponding of theoretical and experimental masses of muon must be specially considered. Mistery of muon. The physical nature of muon, latent sense of it existence, it's true status in theoretical physics lies in the center of attention at even many decades. That is how this that problem is sounded b y Nobel Prizer M.Perl [Perl, 1995(1996)]: "There are two puzzles, connected with electron and muon. The first puzzle: ... properties of these particles relate to interactions the same, but the muon at 206,8 times more heavy. Why? The second puzzle was connected with that muon is not stable and desintegrate (decay) by the time 2,2⋅10-6 sec... To the end of 1950 electron-muon problem (e−µ problem) consisted from two parts: 1. Why the muon at 206,8 times more heavy then electron? 2.Why the muon is not desintegrated by the way µ− → e− + γ ?" In reality that expression is continued the tradition which exist (before) him. In the frame of discussed approach it may be discovered, that "experimental" value of N quantum number in the G[0] Shell for the observed mass of muon mµ = 105.658389 Mev/c2 is equal to N = (2πmµ /M∗[0])1/2 = 7,9966806, that is close to integer. In that case the P∧ = 2πmµ /M∗[0] = 63.9469 is the value of azimutal quantization, that is also close to integer. These kind of properties in the Wave Universe Concept are patterns, the indicators of increased stability of wave configurations. The e - µ µ Similarity Our answer to questions, connected with pointed by more investigatores similarity of electron and muon properties, in the limited brief form may be stated as follows. Proposition. # The electron and muon similarity is close connected with that both belong (close to) ME dominant level (in the N = 7.6 - 8 region). # The difference of electron and muon properties is connected with that its belong to ME dominant levels of different G[-2] and G[0] Shells. # With that distinction, evidently, the known difficult in µ− → e− + γ - decay is connected, becouse that is decay Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 9 in the e stay, that belongs to comparatively far distant G[-2] Shell. # In general it may be stated, that the muon is neighboring (in the sense of ME dominant level) recurrention of electron in the another distant G[0] Shell. Stationary States of G[-7] Shell. Set of Neutrino. As the information for the following analysis it may be shown the mass spectrum of stationary states, that belongs to some (decade) ev/c2 range. This is the region (of G[-7] Shell, Table 6), that with most probability corresponds to νe electron neutrino states [see RPP]. It may be waited, that with increasing of experiments precision it will be discovered the more correspondence of experimental values not of only one, but all spectrum of electron neutrino, to the theoretically predicted mass spectrum of G[-7] Shell (and lieing down by masses G[s] Shells). Hierarchy, Recurrences, Isomorphysm. Transponation - As Effective Tool of Analysis and Extrapolation. Wide potential possibilities of EOM investigations in the frame of Wave Universe Concept are close connected ∗ With - constructing by theory and observing in reality - WDS Hierarchy at each discrete Level of matter, ∗ With dynamical isomorphism of real objects - as similar (in structure) WDS, ∗ With recurrent appearence of analogous properties at different Levels of matter. That open possibility of wide use of the effective tool of analysis and extrapolation - it will be say, (Tool of) Transponation. Shortly saying, all this is signify the possibility of constructive carry - resonable (controlled by experiments and observations) extrapolation - of clearly observed properties of WDS, its stationary states at some U(k) Level of matter (at some G[s] Shell) - to another U(k+p) Level of matter (to another G[s+r] Shell). For instance, values of NNE, NP quantum numbers, corresponding to dominant (planetary) orbits of Neptune and Pluto, definited in G[2] Shell of the Solar system may be transponate in it G[1] Shell as N(NE), N(P). In general case, some properties of components of Homological (by χ - FPH) series (HS) may in some sense be considered as similar. The carry-over - Transponation of knowledge abou t this - at large "distances" by "Wave Universe (WU) Staircase" (by different scales) can give "board" for special examinations for initiative, euristical searches. Alternative Aspect of Z0 Gauge Boson According to RPP, Z0 gauge boson has the mass m(Z0)= 91.187± 0.007 Gev/c2 Its charge is equal to zero. # From the point of view of discussed here approach WU Concept it is not difficult to prove in validity of following relations M = χ5m(π0) = 3.66665⋅134.9764 = 89457.136 Mev/c2 M = χ5m(π±) = 3.66665⋅139.56995 = 92501.56 Mev/c2 As it is easily seen, Z0 boson mass lies in the interval (range) between these calculated values. So, it may be concluded follow Proposition. # In the principal aproximation the mass of Z0 heavy boson is represented in form M = χ5mπ = χ3 2mp = 92506.67Mev/c2=92.506Gev/c2 # It belongs to Odd (k=2q+1) subset of M(k) mass General Homological series of Di-proton M(k)= χk M∗(0), k=2q+1, q = -2,-1,0,1,2,3,... at q=3 (k=7), if M∗(4) =M∗[2]=2mp, # That is coinside also with fact, that heavy Z0 boson is the elite state of G[2] Shell (M∗[2]=2mp=1.8765 Gev/c2) with N quantum number close to N≅17.5. # Heavy W± boson is also elite state the same G[2] Shell with N close to and N≅16.5 (see Table 2). Universal Invariant of Energy – Temperature. It is interesting to point the following nontrivial fact. Proposition # Universal Hierarchies of - physically distinguished, elite (dominant) vN[s] velocities - and MN[s] masses in Universe are not independen t, but are generated by some, with its connected, Universal Invariant (UI) E = 2TKin = 2(1/2)MN[s](vN[s])2 = MN[s](vN[s])2 =const ⇒ Invar Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 10 # It is essentially, that Universal Invariant (UI) of E Energy (H Hamiltonian) H=E=2TKin also may be considered - and as Universal Invariant of T Temperature T=(1/3k)E=const, where k =1.3807⋅10-16 erg⋅K-1 - Boltzmann constant. # Numerical value (of right part - const) of that Energy UI is equal to E=const=E∗[2]=M∗[2](C∗[2])2=2mp(C∗[2])2=0.593066⋅10-10 erg=37.016 ev, where mp =1.672623⋅10-24g, C∗[2] = 42.10538 km⋅s-1 = 0.4210538 ⋅107cm⋅s-1, 1 erg = 6.2415⋅1011 ev, and of that Temperature UI is equal to T = const = T∗[2] = (1/3k)E∗[2] = 143180.12 K°. Genesis, sense, significance of that astonishing, mysterious Invariant in Universe may be represented as the object of special examination. There are immence amount of evident and less evident consequances, effects, associations, continuations, that immediately imply or connected with approaches, ideas of WU Concept. By virtue of clear causes, we are not able to present its in all totality, at once, simultaneously. We hope, its will make up the object of consequent publications. DISCUSSION Main ideas and results of discussed aproach are obtained by author long ago. Its for a long time kept lie, subjected to the critical analysis, comprehend, overgrowning by details and by more convinced argumentation - and waited till own hour for a publication. Previous several decades of intensive investigations, connected with development of basic ideas of the Wave Universe Concept, created the fundamental base for break in new, early unexperienced range of knowledge. Value of receiving results is extremely extensive. Majority from its still remain nonpublished. Suggesting continuations, consequances of WU Concept ideas, often, such natural and evident, that it may be waited in not far future appearences of works and pape rs of another advanced researchers, where these results will be rediscovered, developed in details. The alternative character of the aproach too evident, it opens unexpected perspectives and those possible circumstanies and new problems, which, frequently, arise with proposals and appea rences of principally new ideas. Why, as Niels Bohr said, - "Problems are more important, then decisions - solutions can be ob solete, but problems - never". Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 11 REFERENCES Chechelnitsky A.M., Extremum, Stability, Resonance in Astrodynamics and Cosmonautics, M., Mashinostroyenie, 1980, 312 pp., (Monograph in Russian). (Library of Congress Control Number: 97121007 ; Name: Chechelnitskii A.M.). Chechelnitsky A.M., On the Quantization of the Solar System, Astronomical Circular of the USSR Academy of Science, N1257, pp.5-7, (1983); N1260, pp.1-2, (1983); N1336, pp.1-4, (1984). Chechelnitsky A.M., The Shell Structure of Astronomical Systems, Astrononical Circular of the USSR Academy of Science, N1410, pp.3-7; N1411, pp.3-7, (1985). Chechelnitsky A.M., Wave Structure, Quantization, Megaspectroscopy of the Solar System; In the book: Spacecraft Dynamics and Space Research, M., Mashinostroyenie, pp. 56-76, (in Russian) (1986); Chechelnitsky A.M., Uranus System, Solar System and Wave Astrodynamics; Prognosis of Theory and Voyager-2 Observations, Doklady AN SSSR, v.303, N5 pp.1082-1088, (1988). Chechelnitsky A.M., Wave Structure of the Solar System, Report to the World Space Congress, Washington, DC, (1992) (Aug.22-Sept.5). Chechelnitsky A.M., Neptune - Unexpected and Predicted: Prognosis of Theory and Voyager-2 Observations, Report (IAF-92-0009) to the World Space Congress, Washington, DC, (Aug.22-Sept.5), Preprint AIAA, (1992). Chechelnitsky A.M., Wave Structure of the Solar System, Report to the World Space Congress, Washington, DC, (Aug.22-Sept.5), (1992). Chechelnitsky A. M., Wave Structure of the Solar System, (Monograph), Tandem-Press, 1992 (in Russian). Chechelnitsky A.M., Wave World of Universe and Life: Space - Time and Wave Dynamics of Rhythms, Fields, Structures, Report to the XV Int. Congress of Biomathematics, Paris, September 7-9, 1995; Bio-Math (Bio- Mathematique & Bio- Theorique), Tome XXXIV, N134, pp.12-48, (1996). Chechelnitsky A.M., On the Way to Great Synthesis of XXI Century: Wave Universe Concept, Solar System, Rhythms Genesis, Quantization "In the Large", pp. 10-27: In the book: Proceedings of International Conference "Systems Analysis on the Threshold of XXI Century: Theory and Practice", Intellect Publishing House, Moscow, (1996-1997). Chechelnitsky A.M., Mystery of the Fine Structure Constant: Universal Constant of Micro and Megaworld, Wave Genesis, Theoretical Representation, pp. 46-47: In the book: Proceedings of International Conference "Systems Analysis on the Threshold of XXI Century: Theory and Practice", Intellect Publishing House, Moscow, (1996-1997); http:// arXiv.org/abs/physics/0011035. Chechelnitsky A.M., Wave Universe and Spectrum of Quasars Redshifts, Preprint E2-97-259, Lab. Theor. Physics, Joint Institute for Nuclear Research, (1997); http://arXiv.org/abs/physics/0102089. Perl M.N. Nobel Lecture, Stokholm, 1995, (in Uspekhy Fis. Nauk, v. 166, N12, pp. 1340-1351, (Dec. 1996). RPP - Review of Particle Properties, Physical Review D Particles and Fields, Part I, v. 50, N3, 1 Aug. (1994). ADDITIONAL REFERENCES (2000 – 2001) Acciarri M. Et al. Higgs Candidates in e+ e- Interactions at √s=206.6 Gev, arXiv: hep-ex/0011043, v. 2, (16 Nov 2000). Chechelnitsky A. M., Large - Scale Homogeneity or Principle Hierarchy of the Universe? Report to 32 COSPAR Assembly, Warsaw, 14-21 July 2000; http://arXiv.org/abs/physics/0102008. Chechelnitsky A.M., Hot Points of the Wave Universe Concept: New World of Megaquantization, Proceedings of International Conference “Hot Points in Astrophysics”, JINR, Dubna, Russia, August 22-26, (2000); http://arXiv.org/abs/physics/0102036. Felcini M. Status of the Higgs Search with L3, LEPC Meeting, CERN, (November 3, 2000). Tully C. L3 Higgs Candidates, CERN Meeting, 14 November (2000). Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 12 TABLE 1 MASS SPECTRUM: STATIONARY STATES - G[1] Shell States Quantum Number N Mass M = M∗ ∗(N2/2π π) [Mev/c2] Mass M (Experiment) [Mev/c2] TR∗ ∗ 2.5066 M∗ ∗=139.575 π, m=139.56995 ME 8.083 1451.64 f (1420), m=1426.8±2.3; ω(1420)[m], 1419±31; ρ(1450)[0], m=1465±25; η(1440), m=1420±20; ΝΝ, 1468±6; TR 9.191 1876.51 D, m=1869.4±0.4; D[o], m=1864.6±0.5; η2(1870), m=1881 ± 32 ± 40; NN, m=1873 ± 2.5; X, m=1870.0±40 V 11.050 2712.56 NN, m=2710.0±20; X, m=2747±32 ηc(1S), m=2978.8±1.9; J/ψ(1S), m=3096.88±0.04 E 12.993 3750.10 ψ(3770), m=3769.9±2.5 (U) 15.512 5345.36 B+, m=5278.7 ± 2.0; Bo, m=5279.0±2.0; Bso, m=5375±6; MA 16.038 5714.18 (NE) 19.431 8387.17 (1S), m=9460.37±0.21; CE 21.614 10378.34 (3S) = (10355), m=(10355.3±0.0005 (P) 22.235 10982.55 (10860), m=10865±0.008; (11020), m=11019±0.008; TABLE 2 MASS SPECTRUM: STATIONARY STATES - G[2] SHELL T H E O R Y EXPERIMENT Micro – Mega (MM) Analogy General Dichotomy Experiment [RPP,1994,p.1367;RPP,2000] States Quantum Number N Mass M= M∗ ∗(N2/2π π) ) M∗ ∗=1.8675 States ν ν Quantum Number N=Nν ν= Nν ν=02ν ν/2, Nν ν=0=6.5037 Mass M= M∗ ∗(N2/2π π) ) M∗ ∗=1.8675 Mass M [Gev/c2] [Gev/c2] [Gev/c2] TR∗ ∗ 2.5066 1.8675 2.5066 1.8675 ν=0.0 6.5037 12.6329 Exclude m=0.04 ÷12 Gev/c2 10-8+25 Ellis,93B; 10-8+60 Novikov, 93B; ME 8.083 19.516 0.5 7.734 17.865 TR 9.191 25.228 1.0 9.197 25.265 25-19+275 Ellis, 92E V 11.050 36.468 1.5 10.938 35.731 35.4 ± 5 Abreu, 92J; 35-26+205 Ellis, 94 E 12.992 50.418 2.0 13.007 50.531 50-0+353 Renton, 92 (U) 15.512 71.865 2.5 15.468 71.462 73-13+178 Blondel, 93 MA 16.038 76.823 16.5 80.918 W±: M = 80.84 ± 0.22 ± 0.83 Alitti N = (2πM/M∗)1/2 = 16.452 W±: M=79.91 ± 0.39 Abe N=16.357 17.5 91.024 Z0 : M = 91.187 ± 0.007 N=(2πM/M∗)1/2 = 17.473 18.5 19.0 101.588 107.297 3.0 18.395 101.063 103.7 L3 Collaboration [Felcini, 2000] 108.9 L3 Collaboration (NE) 19.431 19.5 112.760 113.0186 114.5 L3 Collaboration [Felcini, 2000; Tully,2000;Acciarry et al., 2000] CE 21.614 3.5 21.876 142.925 (P) 22.235 147.654 4.0 26.015 202.126 Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 13 TABLE 3 MASS SPECTRUM: STATIONARY STATES - G[3] Shell T H E O R Y General Dichotomy States Mass M=M∗ ∗(N2/2π π) ) States ν ν Mass M=M∗ ∗ × × (Nν ν2/2π π) ), , Nν ν= Nν ν=0 2ν ν/2, Nν ν=0 =6.5 E X P E R I M E N T MASS M [Tev/c2] [Tev/c2] [Tev/c2] TR∗ ∗ M∗=0.025228 M∗=0.0252 0.0 0.16984 ME 0.26238 0.5 0.24019 TR 0.33918 1.0 0.33968 V 0.49030 1.5 0.48038 E 0.67784 2.0 0.67936 (U) 0.96619 2.5 0.96077 MA 1.03285 3.0 1.35873 (NE) 1.516 CE 1.87591 3.5 1.92154 (P) 1.98512 4.0 2.71747 L H C TABLE 4 MASS SPECTRUM: STATIONARY STATES - G[4] Shell T H E O R Y General Dichotomy States Mass M=M∗ ∗(N2/2π π) ) States ν ν Mass M=M∗ ∗ × × (Nν ν2/2π π) ), , Nν ν= Nν ν=0 2ν ν/2, Nν ν=0 = 6.5 E X P E R I M E N T S M MASS [Tev/c2] [Tev/c2] [Tev/c2] TR∗ ∗ M∗=0.339 M∗=0.339 0.0 2.283 ME 3.527 0.5 3.229 TR 4.560 1.0 4.566 V 6.591 1.5 6.458 E 9.113 2.0 9.133 (U) 12989 2.5 12.917 MA 13.886 3.0 18.267 (NE) 20.381 CE 25.220 3.5 25.834 (P) 26.688 4.0 36.534 L H C Chechelnitsky A. M. Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum 14 TABLE 5 MASS SPECTRUM: STATIONARY STATES - G[0] Shell States Quantum Number N Mass M= M∗ ∗(N2/2π π) ) [Mev/c2] Mass M (Experiment) [Mev/c2] TR* 2.5066 M∗ ∗=10.3816 ME 8.083 107.97 µ , m = 105.658398 TR 9.191 139.57 π, m = 139.56995 ντ , m <125, m < 143, m < 157 V 11.050 201.76 E 12.993 278.93 (U) 15.512 397.58 MA 16.038 425.02 (NE) 19.431 623.83 CE 21.614 771.90 ρ (770), m = 769.9±0.8 (P) 22.235 816.88 ω (782), m = 781.94±0.12 η′ (958), m = 957.77±0.14 TABLE 6 MASS SPECTRUM: STATIONARY STATES - G[-7] Shell States Quantum Number N Mass M= M∗ ∗(N2/2π π) ) [ev/c2] Mass M Experiment [RPP] ν νe Neutrino mass [ev/c2] TR∗ 2.5066 M∗ ∗=0.130757 ME 8.083 1.3599 TR 9.191 1.7579 V 11.050 2.5411 E 12.993 3.5131 (U) 15.512 5.0076 MA 16.038 5.3531 (NE) 19.431 7.8573 <7.2; <8.0 CE 21.614 9.7226 <9.3 (P) 22.235 10.2887 <11.7 <13.1; <14.0
arXiv:physics/0103067 21 Mar 2001 Chechelnitsky A.M. PHANTOM OF HIGGS BOSON VERSUS HIERARCHY OF STATIONARY STATES OF SUPERHIGH ENERGIES Dubna 2001 Chechelnitsky A.M. Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia E’mail: ach@thsun1.jinr.ru  Chechelnitsky A.M. March 2001 Chechelnitsky Albert Michailovich – is an astrophysicist, cosmologist, expert in space research, theoretical physics, theory of dynamic systems, automatic control, optimization of large systems, econometrics, constructive sociology, anthropology; COSPAR Associate: Member of International organization – Committee on Space Research (COSPAR) – Member of B, D, E Scientific Commissions. (COSPAR – most competent international organization, connected with fundamental interdisciplinary investigations of Space). Author of the (Mega) Wave Universe Concept. Chechelnitsky A.M. Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies. ABSTRACT As is known, the Standard Model mainly ideologically and qualitatively focuss the experimenters in their search of new mass states (of EP- elementary particles). The exact quantitative prognosis of their properties, especially of masses, lays outside opportunities of the usual theory. Model of Stationary states of EP within the framework of the Wave Universe Concept [Chechelnitsky, 1980-2001] points on existence of Hierarchy of physically distinguished - stationary (elite, dominant) states described by the mass formulas, in particular, in a range 10÷210 Gev/c2: The states close to…, 101.5; 107.3; 112.76÷113; 139.5÷143; 147.6; 202 Gev/c2 should be observed. Apparently, the experiment already confirms this prognosis in a range up to 100 Gev/c2. You see preferable states, observable already now in experiment, it - not rejected by the usual theory as the candidates in constituents of Standard model (for example, not holding Higgs bosons), but quite real displays of stationary (first of all, - dominant) mass states. Last data of L3 (CERN) Collaboration really specify displays of new mass states and close to 103.7; 108.9; 114.5 Gev/c2. March 2001 Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 5 RANGE 12-210 Gev/c2: "GREAT DESERT"? On a bounda ry of centuries formed in physics of high energies situation is characterized by some remarkable circumstances: # According to prevailing representations of Standard model the extreme efforts of physical community are still concentrated on search and judgement of postulated basic constituents of the theory – of bosons, realizing weak interaction (Z0 - neutral currents, W± - charged currents) and of Higgs bosons /G470, responsible for observable in experiments manifestation of real masses. # It is considered, that observable mass states in area of masses /G46=80.3 Gev/c2, /G46=91.2 Gev/c2 is just those carriers (quantums) of interaction, which are responsible for weak interaction. # Nowadays at the centre of attention there is a search of Higgs boson /G470 [Felcini, 2000; Tully, 2000; Acciarri et al., 2000]. # As the theory essentially is not capable precisely to point, in particular, major parameter – mass of /G470 boson (as however – and mass of top-quark), the search is conducted by the tested way - "At random". # This "method" already is compelled was applied at total "combing" of a range /G46=80.3 Gev/c2, resulted to detection of mass configurations in area /G46=80.3 Gev/c2, /G46=91.2 Gev/c2. You see here again there were no precise and exact indications of the theory on localization of required states on a scale of masses. # The rather specific characteristics of postulated Higgs boson /G470 result to hard selection of the potential candidates. Many really observable mass states are rejected as unsufficiently valid the candidates in /G470 bosons. By virtue of a similar sort of the factors, traditions and preferences in representation of physical community there is a following picture of HEP on a bounda ry of centuries. # From 10 up to 210 Gev/c2 and, probably, further “Great Desert " reaches, where, as it is considered, there are no mass states, deserving attention. # Above it the peaks W±, Z0 of bosons tower as area of Everest only. 6 Chechelnitsky A. M. # Till now fruitless search of Higgs boson is conducted (for the present). # But still adepts of Standard model are complete of optimism. As is known, - the theorists frequently are mistaken, but never doubt. Crisis of Belief. We suppose, that such picture, dictating by preferences of the prevailing theory, in many respects, will disorient not only theorists, but, main, - the experimenters conducting intense, extremely difficult search in a rich fog of unverified opinions, were guided only by assurances of authorities. The situation too obviously reminds fantastic "Go there - I do not know where". But main, - the dominated dogmas of habitual representations extremely narrow prospects of experimental search. There is a hunt only at widely known Phantoms of the prevailing theory. Other Horizons. Other prospects are offered by system of representations connected with Wave Universe Concept (WU Concept) [Chechelnitsky (1978) 1980-2000]. As against Standard model not capable to point exactly, for example, localization of new states on a scale of mass, the offers of WU Concept are rather critical. The offered mass spectrum of new stationary states [Chechelnitsky, 2000] is quite certain and is unequivocal. It can be veryfied- confirmed or denied by an obvious way by experiment. Panorama Represented by an G[2] Shell. The picture offered by WU Concept for forward edge of HEP is rather certain [see also Chechelnitsky, 2000]. # There is a whole set of physically distinguished - elite (among them - strongest - dominant) states in area of masses M = 10 - 210 Gev/c2. # This cluster of stationary states we shall present by an G[2] Shell. The mass spectrum of dominant (elite) states is Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 7 described by the Mass formulas for stationary states [Chechelnitsky, 2000] (see. the Tables 1,2,3). # In a range up to /G46 ∼100 Gev/c2 should be observed the dominant states in area 12.6, 17.8÷19.5, 25.2, 35.7÷36.5, 50.4, 71.4÷71.8, 76.8 Gev/c2. # Separate theme - true physical sense and na ture of detected in experiment states with masses close to /G46 = 80.84 Gev/c2 and to /G46 = 91.2 Gev/c2. Its - states laying close to elite values of the Main quantum number N = 16.5 and 17.5. # On periphery of an G[2] Shell in area of masses 100 - 200 Gev/c2 extend the Transitive zone (it - dynamic analogue of Transitive zones of asteroids and comets in Shells G[1] and G[2] of Solar system [Chechelnitsky,1986,1992,1999]). It is necessary to expect, that detected in this range the mass states will be, generally speaking, less steady, than states in another (with smaller masses) half of G[2] Shell. Perspective Search. # New Renessance – manifestation of new mass states (following behind a Transitive zone of fading, less steady states in an G[2] Shell) it is necessary to expect at detecting of physically distinguished - (elite) dominant states in area of the following G[3] Shell. It is a range of masses /G46 = 170 - 2700 Gev/c2. # Mass spectrum of dominant states laying in the subsequent range /G46 = 2.28 ÷ 36.5 /G4Cev/c2 is represented by an G[4] Shell. All this - perspective field of researches of HEP of new century. The spectrum of potentially arising mass configurations, which will be met by the experimenters in forthcoming search, is described by the mass formulas of WU Concept for stationary states (see. the Tables 1,2,3). Reference Points for Experiment. 8 Chechelnitsky A. M. As is known, the basic lesson of a History (science, including) is, that nobody takes from it of the special lessons. Nevertheless, we shall try to comprehend the future. # The experimenters substantially will facilitate to themselves life, and, main, will achieve decisive results, if, whenever possible, get rid from tyrannical influence of habitual dogmas of the settled theory (Standard model). You see, - on the one hand , it does not give the exact instructions, where (in what place on a scale of masses) to search, for example, Higgs boson (top-quark, etc). On the other hand, it approves, that it is not enough of the required candidates (on a role of Higgs boson) ab definito. # Tactics, used by the experimenters, of severe extreme selection from here follows. And consequently, it is ((probable) quite steady mass states are exposed to rather rigid selection, are denied already during experiments and its are wrongful eliminated from a field of consideration of the experimenters and independen t theorists. # As against the usual representations, WU Concept approves, that in a range /G46=10÷210 Gev/c2 (and higher) there is a rather advanced Hierarchy of the physically distinguished states, on a variety, probably, not yielding to observable in experiment Hierarchy of states in a range up to /G46=10 Gev/c2 (nowadays – to basic contents of Particle Data Group). # At presence of such polar, alternative representations it is best to the experimenters to not trust finally anybody, but to give steadfast attention to each of mass states, opening in experiment, - without preliminary theoretical selection and imposed assumptions. # Received the advanced experimental spectrum of the physically distinguished states (all - bar none) will ensure, except for other, also objective verification of competing theoretical models. The Future History of HEP. Analyzing the latent tendencies, social aspects and human, psychological motives accompanying to development of exact sciences - cosmology, physics, including, - physics of high energy (HEP), it is possible to Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 9 imagine and picture of the future development of HEP. The enormous, extreme intellectual and material efforts persistently require the subsequent justification. The social order is those. The physical community can not permit itself to admit that the caravan of highly experimentally equipped science long time moved not in right direction or has lost the way because the theory in next time gave the incorrect instructions. The expectations and searches of the justification are so great, that, is possible, the Higgs boson will be, by and large, "open" in foreseeable time. Suitable, the experimentally found out mass state can be soon successor of "empty throne " and will be coronated (and interpreted) by the prevailing theory in the main generator (the Higgs mechanism, "moderator" or "Emperor") of masses. So, actually, has taken place and with experimentally observable mass states in area /G46=80.8 Gev/c2 and /G46=91.2 Gev/c2. Its were announced (are interpreted) as so long and iintense expected by the theory (quantums) of weak interaction - W± and Z0 bosons - with all accompanying attributes, activity of scientific and social mass-media and "with distribution of prizes and elephants ". In contrast with it there are serious bases to believe, that # Do not exist the postulated by the Standard theory the special Higgs mechanism and appropriate (set) of Higgs bosons. Such thought up the ad ho c concept and connected with it constituents (H0, etc) are not by the necessary functional basis of the effective, serviceable theory. Especially as such concepts do not follow from dynamic and physical base principles. # (As) there are no carriers (quantums) of weak interaction laying in area of high masses (and energies) /G46=80.8 Gev/c2 and /G46=91.2 Gev/c2. Such the conceptual inversion is hardly viable and is hardly realized in Hierarchy of masses and interactions in the Universe. # The states with masses /G46=80.8 Gev/c2 and /G46=91.2 Gev/c2 are quite independen t and, generally speaking, 10 Chechelnitsky A. M. ordinaries states - same as many other of compendium of the data of Particle Data Group [RPP] [see the Tables 1,2,3]. What is Farther? New Horizons. Peering in the Future, it is necessary to hope, that the boundary of centuries will appear also time of deep, critical doubts and choice of new ways, with which the physics of new time will follow. Is possible, as a result of the severe analysis that the physics of high energies long time went in a fog, following for phantoms externally attractive, for fantastically beautiful constructions of the usual theory. But when the fog of biases eventually will dissipate, we shall see completely other bright picture of HEP, sated by set of mass states (resonances) demonstrating an advanced spectrum of Hierarchy. Brightest of them will correspond to physically distinguished – dominant states of the Fundamental spectrum (of masses) of stationary states. This spectrum arises not in result each time again thought out ad hoc mechanisms and theories. Were based on fundamental principles, Nature, the Wave Universe with use only of simple and universal receptions in recurrent regime builds all observable Hierarchy of the physically distinguished states of micro - and megaworld. REFERENCES Acciarri M. Et al. Higgs Candidates in e+ e- Interactions at √s=206.6 Gev, arXiv: hep-ex/0011043, v. 2, (16 Nov 2000). Chechelnitsky A.M., Extremum, Stability, Resonance in Astrodynamics and Cosmonautics, M., Mashinostroyenie, 1980, 312 pp., (Monograph in Russian). (Library of Congress Control Number: 97121007 ; Name: Chechelnitskii A. M.). Chechelnitsky A.M., On the Quantization of the Solar System, Astronomical Circular of the USSR Academy of Science, N1257, pp.5-7, (1983); N1260, pp.1-2, (1983); N1336, pp.1-4, (1984). Chechelnitsky A.M., The Shell Structure of Astronomical Systems, Astrononical Circular of the USSR Academy of Science, N1410, pp.3-7; N1411, pp.3-7, (1985). Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 11 Chechelnitsky A.M., Wave Structure, Quantization, Megaspectroscopy of the Solar System; In the book: Spacecraft Dynamics and Space Research, M., Mashinostroyenie, pp. 56-76, (in Russian) (1986). Chechelnitsky A.M., Uranus System, Solar System and Wave Astrodynamics; Prognosis of Theory and Voyager-2 Observations, Doklady AN SSSR, v.303, N5 pp.1082-1088, (1988). Chechelnitsky A.M., Wave Structure of the Solar System, Report to the World Space Congress, Washington, DC, (1992) (Aug.22-Sept.5). Chechelnitsky A.M., Neptune - Unexpected and Predicted: Prognosis of Theory and Voyager-2 Observations, Report (IAF-92-0009) to the World Space Congress, Washington, DC, (Aug.22-Sept.5), Preprint AIAA, (1992). Chechelnitsky A.M., Wave Structure of the Solar System, Report to the World Space Congress, Washington, DC, (Aug.22-Sept.5), (1992). Chechelnitsky A.M., Wave Structure of the Solar System, (Monograph), Tandem-Press, 1992 (in Russian). Chechelnitsky A.M., Wave World of Universe and Life: Space - Time and Wave Dynamics of Rhythms, Fields, Structures, Report to the XV Int. Congress of Biomathematics, Paris, September 7-9, 1995; Bio-Math (Bio- Mathematique and Bio- Theorique), Tome XXXIV, N134, pp.12-48, (1996). Chechelnitsky A.M., On the Way to Great Synthesis of XXI Century: Wave Universe Concept, Solar System, Rhythms Genesis, Quantization " In the Large ", pp. 10-27: In the book: Proceedings of International Conference " Systems Analysis on the Threshold of XXI Century: Theory and Practice ", Intellect Publishing House, Moscow, (1996- 1997). Chechelnitsky A.M., Mystery of the Fine Structure Constant: Universal Constant of Micro and Megaworld, Wave Genesis, Theoretical Representation, pp. 46-47: In the book: Proceedings of International Conference " Systems Analysis on the Threshold of XXI Century: Theory and Practice ", Intellect Publishing House, Moscow, (1996-1997); http:// arXiv.org/abs/physics/0011035. 12 Chechelnitsky A. M. Chechelnitsky A.M., Wave Universe and Spectrum of Quasars Redshifts, Preprint E2-97-259, Lab. Theor. Physics, Joint Institute for Nuclear Research, (1997); http:// arXiv.org/abs/physics/0102089. Chechelnitsky A.M., Wave Astrodynamics Concept and It Consequences, In book: Search of Mathematical Laws of Universe: Physical Ideas, Approaches, Concepts, Selected Proceedings of II Siberian Conference on Mathematical Problem of Complex Systems Space - Time (PST - 98), Novosibirsk, 19-21 June 1998 , Publishing House of Novosibirsk Mathematical Institute, p.74-91, (1999) (In Russian) Chechelnitsky A.M., Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) - Mass Spectrum, Dubna, Publishing House "Geo", Preprint N126, 2 August (2000). Chechelnitsky A.M., Elementar Objects of Matter: Universality, Hierarchy, Isomorphysm, Dynamical Spectrum, Dubna, Publishing House "Geo", Preprint N127, 2 August (2000). Felcini M. Status of the Higgs Search with L3, LEPC Meeting, CERN, (November 3, 2000). RPP - Review of Particle Properties, Physical Review D Particles and Fields, Part I, v. 50, N3, 1 Aug. (1994), RPP, 2000. Tully C. L3 Higgs Candidates, CERN Meeting, 14 November (2000). Chechelnitsky A.M. Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 13 Table 1 MASS SPECTRUM: STATIONARY STATES - G[2] SHELL T H E O R Y EXPERIMENT Micro – Mega (MM) Analogy General Dichotomy Experiment [RPP,1994,p.1367;RPP,2000] States Quantum Number N Mass M= M∗∗(N2/2ππ)) M∗∗=1.8675 States νν Quantum Number N=Nνν= Nνν=02νν/2, Nνν=0=6.5037 Mass M= M∗∗(N2/2ππ)) M∗∗=1.8675 Mass M [Gev/c2] [Gev/c2] [Gev/c2] TR∗∗ 2.5066 1.8675 2.5066 1.8675 ν=0.0 6.5037 12.6329 Exclude m=0.04÷12 Gev/c2 10-8+25 Ellis,93B; 10-8+60 Novikov, 93B; ME 8.083 19.516 0.5 7.734 17.865 TR 9.191 25.228 1.0 9.197 25.265 25-19+275 Ellis, 92E V 11.050 36.468 1.5 10.938 35.731 35.4 ± 5 Abreu, 92J; 35-26+205 Ellis, 94 E 12.992 50.418 2.0 13.007 50.531 50-0+353 Renton, 92 (U) 15.512 71.865 2.5 15.468 71.462 73-13+178 Blondel, 93 MA 16.038 76.823 16.5 80.918 W±: M=80.84 ± 0.22 ± 0.83 Alitti N=(2πM/M∗)1/2 = 16.452 W±: M=79.91 ± 0.39 Abe N=16.357 17.5 91.024 Z0 : M=91.187 ± 0.007 N=(2πM/M∗)1/2 = 17.473 18.5 19.0 101.588 107.297 3.0 18.395 101.063 103.7 L3 Collaboration [Felcini, 2000] 108.9 L3 Collaboration (NE) 19.431 19.5 112.760 113.0186 114.5 L3 Collaboration [Felcini, 2000; Tully,2000;Acciarry et al., 2000] Chechelnitsky A.M. Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 14 Table 2 MASS SPECTRUM: STATIONARY STATES - G[3] Shell T H E O R Y General Dichotomy States Mass M=M∗∗(N2/2ππ)) States νν Mass M=M∗∗××(Nνν2/2ππ)),, Nνν= Nνν=0 2νν/2, Nνν=0 = 6.5 E X P E R I M E N T MASS M [Tev/c2] [Tev/c2] [Tev/c2] TR∗∗ M∗=0.339 M∗=0.339 0.0 0.16984 ME 0.26238 0.5 0.24019 TR 0.33918 1.0 0.33968 V 0.49030 1.5 0.48038 E 0.67784 2.0 0.67936 (U) 0.96619 2.5 0.96077 MA 1.03285 3.0 1.35873 (NE) 1.516 CE 1.87591 3.5 1.92154 (P) 1.98512 4.0 2.71747 L H C Chechelnitsky A.M. Phantom of Higgs Boson Versus Hierarchy of Stationary States of Superhigh Energies 15 Table 3 MASS SPECTRUM: STATIONARY STATES - G[4] Shell T H E O R Y General Dichotomy States Mass M=M∗∗(N2/2ππ)) States νν Mass M=M∗∗××(Nνν2/2ππ)),, Nνν= Nνν=0 2νν/2, Nνν=0 = 6.5 E X P E R I M E N T S MASS M [Tev/c2] [Tev/c2] [Tev/c2] TR∗∗ M∗=0.339 M∗=0.339 0.0 2.283 ME 3.527 0.5 3.229 TR 4.560 1.0 4.566 V 6.591 1.5 6.458 E 9.113 2.0 9.133 (U) 12989 2.5 12.917 MA 13.886 3.0 18.267 (NE) 20.381 CE 25.220 3.5 25.834 (P) 26.688 4.0 36.534 L H C
arXiv:physics/0103068v1 [physics.atom-ph] 22 Mar 2001Testing Lorentz and CPT symmetry with hydrogen masers M.A. Humphrey, D.F. Phillips, E.M. Mattison, R.F.C. Vessot , R.E. Stoner and R.L. Walsworth Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 (February 2, 2008) We present details from a recent test of Lorentz and CPT symme try using hydrogen masers [1]. We have placed a new limit on Lorentz and CPT violation of the proton in terms of a recent standard model extension by placing a bound on sidereal vari ation of the F= 1, ∆ mF=±1 Zeeman frequency in hydrogen. Here, the theoretical standa rd model extension is reviewed. The operating principles of the maser and the double resonance t echnique used to measure the Zeeman frequency are discussed. The characterization of systemat ic eﬀects is described, and the method of data analysis is presented. We compare our result to other re cent experiments, and discuss potential steps to improve our measurement. I. INTRODUCTION A theoretical framework has recently been developed that in corporates Lorentz and CPT symmetry violation into the standard model and quantiﬁes their eﬀects. [2–14]. One branch of this framework emphasizes low energy, experimental searches for symmetry violating eﬀects in ato mic energy levels [13,14]. In particular, Lorentz and CPT violation in hydrogen has been examined and sidereal variat ions in the F= 1, ∆mF=±1 Zeeman frequency have been quantiﬁed [15]. Motivated by this work, we have conduct ed a search for sidereal variation in the hydrogen Zeeman frequency, and have placed a new clean bound of 10−27GeV on Lorentz and CPT violation of the proton [1]. Here we provide additional details of the theoretical frame work, experiment and analysis. In Sec. II we discuss the standard model extension. In Sec. III we describe the basic c oncepts of hydrogen maser operation and our Zeeman frequency measurement technique. In Sec. IV we describe the procedure used to collect data and extract a sidereal bound on the Zeeman frequency. In Sec. V we describe eﬀorts to reduce and characterize systematic eﬀects. Finally, in Sec. VI we compare our result to other clock-comparison te sts of Lorentz and CPT symmetry, and discuss potential means of improving our measurement. II. LORENTZ AND CPT SYMMETRY VIOLATION IN THE STANDARD MODEL Experimental investigations of Lorentz symmetry provide i mportant tests of the standard model of particle physics and general relativity. While the standard model successfu lly describes particle phenomenology, it is believed to be t he low energy limit of a fundamental theory that incorporates g ravity. This underlying theory may be Lorentz invariant, yet contain spontaneous symmetry-breaking that could resu lt in small violations of Lorentz invariance and CPT at the level of the standard model. A theoretical framework has been developed to describe Lore ntz and CPT violation at the level of the standard model by Kosteleck´ y and coworkers [2–14]. This standard-m odel extension is quite general: it emerges as the low- energy limit of any underlying theory that generates the sta ndard model and contains spontaneous Lorentz symmetry violation [2–4]. For example, such characteristics might e merge from string theory [5–8]. A key feature of the standard model extension is that it is formulated at the level of the kn own elementary particles, and thus enables quantitative comparison of a wide array of searches for Lorentz and CPT vio lation [9–12]. “Clock comparison experiments” are searches for temporal v ariations in atomic energy levels. According to the standard model extension considered here, Lorentz and CPT v iolation may produce shifts in certain atomic levels, whose magnitude depends on the orientation of the atom’s qua ntization axis relative to a ﬁxed inertial frame [13,14]. Certain atomic transition frequencies, therefore, may exh ibit sinusoidal variation as the earth rotates on its axis. New limits can be placed on Lorentz and CPT violation by bound ing sidereal variation of these atomic transition frequencies. Speciﬁcally, the description of Lorentz and CPT violation i s included in the relativistic Lagrange density of the constituent particles of the atom. For example, the modiﬁed electron Lagrangian becomes [13] 1L=1 2i¯ψΓν∂νψ−¯ψMψ +LQED int (1) where Γν=γν+/parenleftbigg cµνγµ+dµνγ5γµ+eν+ifνγ5+1 2gλµνσλµ/parenrightbigg (2) and M=m+/parenleftbigg aµγµ+bµγ5γµ+1 2Hµνσµν/parenrightbigg . (3) The parameters aµ,bµ,cµν,dµν,eν,fν,gλµνandHµνrepresent possible vacuum expectation values of Lorentz tensors generated through spontaneous Lorentz symmetry br eaking in an underlying theory. These are absent in the standard model. The parameters aµ,bµ,eν,fνandgλµνrepresent coupling strengths for terms that violate both CPT and Lorentz symmetry, while cµν,dµν, andHµνviolate Lorentz symmetry only. An analogous expression exi sts for the modiﬁed proton and neutron Lagrangians (a superscri pt will be appended to diﬀerentiate between the sets of parameters). The standard model extension treats only th e free particle properties of the constituent particles, estimating that all interaction eﬀects will be of higher ord er [13]. As a result, the interaction term LQED intis unchanged from the conventional, Lorentz invariant, QED interaction term. Within this phenomenological framework, the values of thes e parameters are not calculable; instead, values must be determined experimentally. The general nature of this th eory ensures that diﬀerent experimental searches may place bounds on diﬀerent combinations of Lorentz and CPT vio lating terms, while direct comparisons between these experiments are possible (see Table III and Ref. [13]). The leading-order Lorentz and CPT violating energy level sh ifts for a given atom are obtained by summing over the individual free particle shifts of the atomic constituents . From the symmetry violating correction to the relativisti c Lagrangian, a non-relativistic correction Hamiltonian δhis found using standard ﬁeld theory techniques [13]. Assumi ng Lorentz and CPT violating eﬀects to be small, the energy leve l shifts are calculated perturbatively by taking the expectation value of the correction Hamiltonian with respe ct to the unperturbed atomic states, leading to a shift in an atomic ( F,m F) sublevel given by [13] ∆EF,m F=/angbracketleftF,m F\|neδhe+npδhp+nnδhn\|F,m F/angbracketright. (4) Herenwis the number of each type of particle and δhwis the corresponding correction Hamiltonian. Note that for most atoms, the interpretation of energy level shifts in terms of this standard model extension is reliant on the particular model used to describe the atomic nucleus (e.g., the Schmidt model). One key advantage of a study in hydrogen is the simplicity of the nuclear structure (a singl e proton), with its results uncompromised by any nuclear model uncertainty. Among the most recent clock comparison experiments are Penn ing trap tests by Dehmelt and co-workers with the electron and positron [16,17] which place a limit on electro n Lorentz and CPT violation at 10−25GeV. A recent re-analysis by Adelberger, Gundlach, Heckel, and co-worke rs of existing data from the “E¨ ot-Wash II” spin-polarized torsion pendulum [18,19] has improved this to a level of 10−29GeV [20], the most stringent bound to date on Lorentz and CPT violation of the electron. A new limit on neutron Lore ntz and CPT violation has been placed at 10−31GeV by Bear et al. [21] using a dual species noble gas maser and com paring Zeeman frequencies of129Xe and3He. The current limit on Lorentz and CPT violation of the proton is 10−27GeV, as derived from an experiment by Lamoreaux and Hunter [22] which compared Zeeman frequencies of199Hg and133Cs. Figure 1 shows the Lorentz and CPT violating corrections to t he energy levels of the ground state of hydrogen [15]. The shift in the F= 1, ∆mF=±1 Zeeman frequency is [23]: \|∆νZ\|=1 h\|(be 3−de 30me−He 12) + (bp 3−dp 30mp−Hp 12)\|. (5) The subscripts denote the projection of the tensor coupling s onto the laboratory frame. Therefore, as the earth rotates relative to a ﬁxed inertial frame, the Zeeman freque ncyνZwill exhibit a sidereal variation. We have recently published the result of a search for this variation of the F= 1, ∆mF=±1 Zeeman frequency in hydrogen using hydrogen masers [1]. This search has placed a new, clean boun d on Lorentz and CPT violation of the proton at a level of 10−27GeV. 21000 500 0 magnetic field [Gauss]hνHFS F = 1 F = 0 mF = +1 mF = 0 mF = -1 mF = 0 1 2 3 4 FIG. 1. Hydrogen hyperﬁne structure. The full curves are the unperturbed hyperﬁne levels, while the dashed curves illus - trate the shifts due to Lorentz and CPT violating eﬀects with the exaggerated values of \|be 3−de 30me−He 12\|= 90 MHz and \|bp 3−dp 30mp−Hp 12\|= 10 MHz. This work reports a bound of less than 1 mHz for these t erms. A hydrogen maser oscillates on the ﬁrst-order magnetic ﬁeld-independent \|2/angbracketright ↔ \|4/angbracketrighthyperﬁne transition near 1420 MHz. The maser typically oper ates with a static ﬁeld less than 1 mG. For these low ﬁeld strengths, the t woF= 1, ∆ mF=±1 Zeeman frequencies are nearly degenerate, andν12≈ν23≈1 kHz. 3III. HYDROGEN MASER CONCEPTS The electronic ground state in hydrogen is split into four le vels by the hyperﬁne interaction, labeled (following the notation of Andresen [24]) \|1/angbracketrightto\|4/angbracketrightin order of decreasing energy (Fig. 1). The energies of atoms in\|1/angbracketrightand \|2/angbracketrightdecrease as the magnetic ﬁeld decreases; these are therefor e low-ﬁeld seeking states. Conversely, \|3/angbracketrightand\|4/angbracketrightare high-ﬁeld seeking states. In low ﬁelds, \|2/angbracketrightand\|4/angbracketrightare only dependent on magnetic ﬁeld in second order. The mase r oscillates on the \|2/angbracketright ↔ \|4/angbracketrighttransition (ﬁeld-independent to ﬁrst-order). This transi tion frequency, as a function of static ﬁeld, is given by ν24=νhfs+ 2750B2(νin Hz with Bin Gauss, with νhfs≈1420.405751 MHz the zero- ﬁeld hyperﬁne frequency). Hydrogen masers typically opera te with low static ﬁelds (less than 1 mG), where ν24is shifted from νhfsby about 3 mHz, or 2 parts in 1012. The two F= 1, ∆mF=±1 Zeeman frequencies are given byν12= 1.4×106B−1375B2andν23= 1.4×106B+ 1375B2. AtB= 1 mG these are nearly degenerate, with ν12−ν23≈3 mHz, much less than the Zeeman linewidth of approximately 1 Hz. A. Maser operation In a hydrogen maser [25–27], molecular hydrogen is dissocia ted in an rf discharge and a beam of hydrogen atoms is formed, as shown in Fig. 2. A hexapole state selecting magn et focuses the low-ﬁeld-seeking hyperﬁne states \|1/angbracketright and\|2/angbracketrightinto a quartz maser bulb at about 1012atoms/sec. Inside the bulb (volume ∼103cm3), the atoms travel ballistically for about 1 second before escaping, making ∼104collisions with the bulb wall. A Teﬂon coating reduces the atom-wall interaction and thus inhibits decoherence of the masing atomic ensemble by wall collisions. The maser bulb is centered inside a cylindrical TE 011microwave cavity resonant with the 1420 MHz hyperﬁne transi tion. The microwave ﬁeld stimulates a small, coherent magnetization in the atomic ensemble, and this magnetization acts as a source to stimulate the microwave ﬁeld. With suﬃciently hi gh atomic ﬂux and low cavity losses, this feedback induces active maser oscillation. The maser signal is induc tively coupled out of the microwave cavity and ampliﬁed with an external receiver. Surrounding the cavity, a soleno id produces the weak static magnetic ﬁeld ( ≈1 mG) that establishes the quantization axis inside the maser bulb and sets the Zeeman frequency ( ≈1 kHz). A pair of Helmholtz coils produces the oscillating transverse magnetic ﬁeld th at drives the F= 1, ∆mF=±1 Zeeman transitions. The cavity, solenoid, and Zeeman coils are all enclosed within s everal layers of high permeability magnetic shielding. A well engineered hydrogen maser can have fractional stabil ities approaching 10−15over intervals of hours. This stability is enabled by a long atom-ﬁeld interaction time (1 s), a low atom-wall interaction (due to the low atomic polarizability of H and the wall’s Teﬂon coating), reduced D oppler eﬀects (the atoms are conﬁned to a region of uniform microwave ﬁeld phase, eﬀectively averaging their v elocity to zero over the interaction time with the ﬁeld), and multiple layers of thermal control of the cavity (stabil izing cavity pulling shifts). B. Maser characterization Among the quantities used to characterize a hydrogen maser, those most relevant to this experiment are the atomic line-QQl, the population decay rate γ1, the hyperﬁne decoherence rate γ2, the atomic ﬂow rate into and out of the bulbγb, and the maser Rabi frequency \|X24\|. We describe here a comprehensive set of measurements to cha racterize hydrogen maser P-8. The results discussed here are summariz ed in Table I. Our Lorentz and CPT symmetry test data were taken with a similar but newer hydrogen maser, P-28 [28]. A few of the maser characterization parameters for P-28, while not directly measured, have been inferred us ing ﬁtting parameters from the double resonance method used to measure the F= 1, ∆mF=±1 Zeeman frequency, described in Sec. IVA. These values are i ncluded in Table I in italics. To determine these parameters of an operating H maser, the ca vity volume VC, bulb volume Vb, cavity quality factorQC, ﬁlling factor η, and output coupling coeﬃcient βmust be known. For both masers, VC= 1.4 ×10−2m3, Vb= 2.9 ×10−3m3,QC≈40,000, and β= 0.23 [29]. The ﬁlling factor, deﬁned as [26] η=/angbracketleftHz/angbracketright2 bulb /angbracketleftH2/angbracketrightcavity, (6) quantiﬁes the ratio of average magnetic ﬁeld energy inside t he bulb to the average magnetic ﬁeld energy in the cavity. This has a value of η= 2.14 for masers P-8 and P-28 [29]. 4H2 dissociator hexapole magnet solenoidmagnetic shields microwave cavityquartz bulbto receiver B0 M H C Zeeman coils FIG. 2. Hydrogen maser schematic. The solenoid generates a w eak static magnetic ﬁeld B0which deﬁnes a quantization axis inside the maser bulb. The microwave cavity ﬁeld HC(dashed ﬁeld lines) and the coherent magnetization Mof the atomic ensemble form the coupled actively oscillating system. 5parameter symbol P-8 P-28 cavity volume VC 1.4×10−2m31.4×10−2m3 bulb volume Vb 2.9×10−3m32.9×10−3m3 cavity-Q QC 39,346 ﬁlling factor η 2.14 2.14 line-Q Ql 1.6×1091.6×109 maser quality parameter q 0.100 maser relaxation rate γt 1.83 rad/s bulb escape rate γb 0.86 rad/s 0.86 rad/s population decay rate γ1 4.04 rad/s 2.88 rad/s maser decoherence rate γ2 2.77 rad/s 2.8 rad/s spin-exchange decay rate γse 1.06 rad/s radiated power P 600 fW threshold power Pc 250 fW output coupling β 0.23 0.23 output power Po 112 fW ≈100 fW total ﬂux Itot 15.0×1012atoms/s ﬂux of \|2/angbracketrightatoms I 3.13×1012atoms/s threshold ﬂux Ith 0.54×1012atoms/s atomic density n 2.8×1015atoms/m3 maser Rabi frequency \|X24\| 2.77 rad/s 2.14 rad/s TABLE I. Maser characterization parameters. The italicize d values for P-28 were inferred from double resonance ﬁt para m- eters as described in Sec. IVA. All other values were either c alculated or extracted from direct measurements as describ ed in this section. For canonical hydrogen maser operation, there are two impor tant relaxation rates [27,29]. For a room temperature H maser, the decay of the population inversion is described b y the longitudinal relaxation rate γ1=γb+γr+ 2γse+γ′ 1, (7) and the decay of the atomic coherence is described by the tran sverse relaxation rate γ2=γb+γr+γse+γ′ 2. (8) Here,γbis the atomic ﬂow rate into the bulb, γris the rate of recombination into molecular hydrogen at the b ulb wall, γseis the hydrogen-hydrogen spin-exchange decay rate, and γ′ iincludes all other sources of decay, such as decoherence during wall collisions and eﬀects of magnetic ﬁeld gradient s. In the steady state, the atom ﬂow rate into the bulb is equal to the geometric escape rate from the bulb, given by γb= ¯vA/4KVb, where ¯v= 2.5 ×105cm/s is the mean thermal velocity of atoms in the bulb, A= 0.254 cm3is the area of the bulb entrance aperture, and K≈6 is the Klausing factor [30]. Thus, γb= 0.86 rad/s for both P-8 and P-28. The spin exchange decay rate is given approximately by [27,29] γse=1 2n¯vrσ (9) where ¯vr= 3.6 ×105cm/s is the mean relative velocity of atoms in the bulb and σ= 21 ×10−16cm2is the hydrogen-hydrogen spin-exchange cross section. The hydro gen density is given by [27,29] n=Itot (γb+γr)Vb(10) whereItotis the total ﬂux of hydrogen atoms into the storage bulb. The atomic line-Q is related to the transverse relaxation ra te and the maser oscillation frequency ωby [26,29] Ql=ω 2γ2. (11) 6It is measured using the cavity pulling of the maser frequenc y: neglecting spin-exchange shifts, the maser frequency is given by [26] ω=ω24+QC Ql(ωC−ω24). (12) By measuring the maser frequency as a function of cavity freq uency setting, the line-Q can be determined. For both P-8 and P-28, we ﬁnd Ql= 1.6 ×109, and therefore γ2= 2.8 rad/s. A convenient single measure of spin-exchange-independent relaxation in a hydrogen maser is given by “gamma-t” [27,29] γt= [(γb+γr+γ′ 1)(γb+γr+γ′ 2)]1 2. (13) Using this, a more useful form for the longitudinal relaxati on rate,γ1, can be found. By combining Eqn. 13 with Eqns. 7 and 8, we ﬁnd γ1=γ2 t γ2−γse+ 2γse. (14) Using Eqns. 8-11, we can relate the line-Q to I, the input ﬂux of atoms in state \|2/angbracketrightas [29] 1 Ql=2 ω/bracketleftbigg γb+γr+γ′ 2+qI Ithγt/bracketrightbigg (15) using the threshold ﬂux required for maser oscillation (neg lecting spin-exchange) Ith=¯hVCγ2 t 4πµ2 BQCη, (16) and the maser quality parameter q=/bracketleftbiggσ¯vr¯h 8πµ2 b/bracketrightbiggγt γb+γr/bracketleftbiggVC ηVb/bracketrightbigg/parenleftbigg1 QC/parenrightbiggItot I. (17) The ratioI/Itotis a measure of the eﬀectiveness of the state selection of ato ms entering the bulb. While Iis not directly measurable, it can be related to the power Pradiated by the atoms by [27,29] P Pc=−2q2/parenleftbiggI Ith/parenrightbigg2 + (1−3q)/parenleftbiggI Ith/parenrightbigg −1 (18) wherePc= ¯hωIth/2. The maser power is also related to the maser Rabi frequency by [27] P=I¯hω 2\|X24\|2 γ1γ2/parenleftbigg 1 +\|X24\|2 γ1γ2/parenrightbigg−1 . (19) The power coupled out of the maser is given by [29] Po/P=β/(1 +β). Generally, the parameter qis less than 0.1, while I/Ithis approximately 2 or 3. Hence, the ﬁrst term of Eqn. 18 can be neglected relative to the others. If we make the reason able approximation that γ′ 1=γ′ 2, then we can rewrite Eqn. 15 using Eqns. 13, 16-18 as [29] 1 Ql=mP+b (20) where b=2 ωγt/bracketleftbigg 1 +q 1−3q/bracketrightbigg (21) and 7m=16πµ2 bQCη ω2¯h2VC/bracketleftbiggq 1−3q/bracketrightbigg1 γt. (22) Therefore, by measuring the line-Q as a function of maser pow er and extracting the slope mand the y-intercept b, we can determine qandγt. For maser P-8, q= 0.100 and γt= 1.83 rad/s. With these values of qandγt, we found Ith= 0.54 ×1012atoms/s (using Eqn. 16), and Pc= 250 fW. With a measured output power of Po= 112 fW, the atoms were radiating P= 599 fW, and the ﬂux of state \|2/angbracketrightatoms was I= 3.13 ×1012atoms/s (Eqn. 18). Under the assumption that γt≈γb+γrwe found that the total ﬂux was Itot= 15.0×1012atoms/s (Eqn. 17) and the density was n= 2.8 ×1015atoms/m3(Eqn. 10). The spin-exchange decay rate was then found to be γse= 1.06 rad/s (Eqn. 9). Finally, the population decay rate was γ1= 4.04 rad/s (Eqn. 14) and the maser Rabi frequency was \|X24\|= 2.77 rad/s (Eqn. 19). C. Zeeman frequency determination TheF= 1, ∆mF=±1 Zeeman frequency is measured using a double resonance tech nique [24,31,32]. As the frequency of an audio frequency magnetic ﬁeld ωZ, applied perpendicular to the quantization axis, is swept t hrough theZeeman frequency, a shift in the maser frequency is observed (Fig. 3). When the applied ﬁeld is near the Zeeman frequency, two-photon transitions (one audio photon plus o ne microwave photon) link states \|1/angbracketrightand\|3/angbracketrightto state \|4/angbracketright, in addition to the single microwave photon transition betwe en states \|2/angbracketrightand\|4/angbracketright. This two photon coupling shifts the maser frequency antisymmetrically with respect to the detu ning of the applied ﬁeld about the Zeeman resonance [32]. To second order in the Rabi frequency of the applied Zeeman ﬁe ld,\|X12\|the small static-ﬁeld limit of the maser frequency shift from the unperturbed frequency is given by [ 24] ∆ω=−\|X12\|2(ρ0 11−ρ0 33)δ(γ1γ2+\|X0 24\|2)(γZ/γb) (γ2 Z−δ2+1 4\|X0 24\|2)2+ (2δγZ)2(23) +\|X12\|2/parenleftbiggωC−ω24 ω24/parenrightbiggQCγZ(1 +K) γ2 Z(1 +K)2+δ2(1−K)2 whereγZis the Zeeman decoherence rate, δ=ωZ−ω23is the detuning of the applied ﬁeld from the atomic Zeeman frequency,K=1 4\|X0 24\|2/(γ2 Z+δ2), andρ0 11−ρ0 33=γb/(2γ1) is the steady state population diﬀerence between states \|1/angbracketrightand\|3/angbracketrightin the absence of the applied Zeeman ﬁeld. The ﬁrst term in Eqn . 23 results from the coherent two-photon mixing of the F= 1 levels as described above [32], while the second term is a m odiﬁed cavity pulling term that results from the reduced line-Q in the presence of the applied Zeeman ﬁeld. We compared Eqn. 23 to experimental data from P-8, inserting the independently measured values of \|X0 24\|,γb,γ1, andγ2. By matching the ﬁt to the data we extracted the Zeeman ﬁeld parameters \|X12\|andγZshown in Fig. 3. In addition to the shift given by Eqn. 23, there is a small symm etric frequency shift due to the slight non-degeneracy of the twoF= 1, ∆mF=±1 Zeeman frequencies. This term oﬀsets the zero crossing of t he maser shift resonance away from the average Zeeman frequency1 2(ν12+ν23), however the contribution is negligible at small static ﬁe lds. Also, a reanalysis of the double resonance maser shift [31], which included the eﬀects of spin-exchange collisions [33] , showed that there is an additional hydrogen density-depend ent oﬀset of the zero crossing of the maser shift resonance from the average Zeeman frequency. Using the full spin-exch ange corrected formula for the maser frequency shift [31], we calculated this oﬀset and found that for typical hydrogen maser densities ( n≈3×1015m−3), the oﬀset varied with average maser power as approximately -50 µHz/fW (assuming a linear relation between maser power and at omic density of∆P ∆n≈100fW 3×1015m−3). As described below, our masers typically have sidereal po wer ﬂuctuations less than 1 fW, making this eﬀect negligible. The applied Zeeman ﬁeld also acts to diminish the maser power , as shown in Fig. 4, and to decrease the maser’s line-Q. By driving the F= 1, ∆mF=±1 Zeeman transitions, the applied ﬁeld depletes the populat ion of the upper masing state \|2/angbracketright, thereby diminishing the number of atoms undergoing the mas er transition and reducing the maser power. Also, by decreasing the lifetime of atoms in state \|2/angbracketright, the line-Q is reduced. A very weak Zeeman ﬁeld of about 35 nG (as was used in our Lorentz symmetry test) decreas es the maser power by less than 2% on resonance and reduces the line-Q by 2% (as calculated using Eqn. 6 of [24]). The standard method of determining the average static magnetic ﬁeld strength is to scan the Zeeman resonance with a large applied ﬁeld and record the power diminishment (such as that shown in Fig. 4, open circles). From the applied ﬁeld frequency at the center of the power resonance, which typically has a width of about 1 Hz, the magnetic ﬁeld ca n be found with a resolution of about 1 µG. 810 0 -10maser shift [mHz] -3 -2 -1 0 1 2 3 Zeeman detuning [Hz]10 5 0 -5 -10 1012 ∆ω / ω24 \|X240\| = 2.82 rad/s \|X12\| = 1.02 rad/s γ1 = 4.13 rad/s γ2 = 2.80 rad/s γΖ = 2.30 rad/s γb = 0.86 rad/s FIG. 3. Double resonance maser frequency shifts. The large o pen circles (maser P-8) are compared with Eqn. 23 (full curve ) using the parameter values shown. The values of \|X12\|andγZwere chosen to ﬁt the data, while the remaining parameters were independently measured as described in subsection III B. The experimental error of each measurement (about 40 µHz) is smaller than the circle marking it. The solid square data poi nts are data from the Lorentz symmetry test (maser P-28). The large variation of maser frequency with Zeeman detuning nea r resonance, along with the excellent maser frequency stabi lity, allows the Zeeman frequency ( ≈800 Hz) to be determined to 3 mHz in a single resonance (requir ing 18 minutes of data acquisition). The inversion of the shift between the two is d ue to the fact that for the P-8 data (open circles), the maser operated with an input ﬂux of \|2/angbracketrightand\|3/angbracketrightatoms, while for the P-28 data (solid points), the typical in put of \|1/angbracketrightand\|2/angbracketrightatoms was used. Changing between these two input ﬂux modes is done b y inverting the direction of the static solenoid ﬁeld, while maintaining a ﬁxed quantization axis for the state selectin g hexapole magnet (see Sec. IV D). 9130 120 110 100 90maser power [fW] 869.0 868.0 867.0 866.0 865.0 864.0 Zeeman field frequency [Hz] Zeeman field strength 80 nG 560 nG FIG. 4. Double resonance maser power diminishment. The open circles, taken with an applied Zeeman ﬁeld strength of about 560 nG, represent typical data used to determine the va lue of the static magnetic ﬁeld in the maser bulb. The ﬁlled circles are maser power curves with an applied ﬁeld strength of about 80 nG. Our Lorentz symmetry test data were taken with a ﬁeld strength of about 35 nG, where the power diminishment i s less than 2%. IV. EXPERIMENTAL PROCEDURE A. Zeeman frequency measurement To measure the F= 1, ∆mF=±1 Zeeman frequency, we applied an oscillating ﬁeld of about 3 5 nG near the Zeeman frequency. This ﬁeld shifted the maser frequency by a few mHz (at the extrema), a fractional shift of about 2 parts per trillion. Because of the excellent fractional mas er stability (2 parts in 1014over our averaging times of 10 s), the shift was easily resolved (see the solid data in Fig. 3). A s the frequency of the applied ﬁeld was stepped through the Zeeman resonance, the maser frequency (of perturbed maser P -28) was compared to a second, unperturbed hydrogen maser frequency (P-13). The two maser signals at ≈1420 MHz were phase locked to independent voltage controlle d crystal oscillator receivers. The exact value of the receiv ers’ outputs were set by tunable synthesizers, which were se t such that there was a 1.2 Hz oﬀset between them. The two receiv er outputs were combined in a heterodyne mixer and the resulting 0.8 s period beat note was averaged for 10 s ( about 12 periods) with a Hewlett-Packard Model HP 5334B frequency counter. The full double resonance spectru m consisted of 100 such points. For each spectrum, 80% of the points were taken over the middle 40% of the scan range, where the frequency shift varies the most. Once an entire spectrum of beat period vs applied Zeeman freq uency was obtained, it was ﬁt to the function Tb=A0+A3δ(1−κ) A1(1 +κ)2+δ2(1−κ)2−A3(δ+τ)(1−κ) A1(1 +κ)2+ (δ+τ)2(1−κ)2(24) +A5δ (A1−δ2+A4)2+ 4δ2A1+A6(1 +κ) A1(1 +κ)2+δ2(1−κ)2 to determine the Zeeman frequency. Here δ=ν−νZis the Zeeman detuning of the applied ﬁeld νaway from the Zeeman frequency νZ,κ=A4/(A1+δ2) is the analog of the parameter Kfrom Eqn. 23, and τ= (1.403×10−9)×ν2 Z is the small diﬀerence between the two Zeeman frequencies ν12andν23. The ﬁrst term A0is the constant oﬀset representing the unperturbed beat period between the two ma sers. The second and third terms comprise the ﬁrst- order symmetric maser shift (not included in Eqn. 23 but desc ribed in the text above); these two terms nearly cancel at low static ﬁeld where τvanishes. The ﬁnal two terms account for the two shifts given in Eqn. 23. 1040 30 20 10 0number -10 -5 0 510 Zeeman frequency shift [mHz]σν12 = 2.7 mHz FIG. 5. Results from a Monte Carlo analysis. The horizontal a xis represents the shift of the Zeeman frequency as determin ed by our ﬁts of over 100 synthetic data sets, the vertical axis i s the number within each shift bin. The width of the Gaussian ﬁ t to the data is 2.7 mHz, representing the resolution of a singl e Zeeman frequency measurement. For our spectra in maser P-28 with small applied ﬁeld amplitu de (solid square data points of Fig. 3), typical ﬁt parameters were: A0= 0.84550 ±0.00001,A1= 0.141 ±0.005,νZ= 857.063 ±0.003,A3= 0.006 ±0.010,A4= 0.029 ±0.003,A5= (3.2 ±0.1)×10−4, andA6= (-1 ±5)×10−6. The uncertainty in the Zeeman frequency was 3 mHz. Also, A3andA6, the amplitude coeﬃcients of the residual ﬁrst order eﬀect a nd the cavity pulling term, were consistent with zero. With our known value of γb= 0.86 rad/s, and our measured value of γ2= 2.77 rad/s (from line-Q), the above set of ﬁt parameters were consistent with the reasonable values X0 24= 2.14 rad/s, γZ= 2.36 rad/s, γ1= 2.88 rad/s, and X12= 0.40 rad/s (since A3had such a large error bar, the value of X12was chosen such the ratio of the maser shift amplitude in P-28 to P-8, shown in Fig. 3, is equal to the ratio of the squares of X12for P-28 to P-8). To determine the number of points and length of averaging tha t optimized the Zeeman frequency resolution, we recorded several spectra with 50, 100, and 150 points at 5 s and 10 s averaging. We also varied the “density distribution” of points, including spectra where the middl e 40% of the scan contained 80% of the points and those where the middle 30% contained 80% of the points (thus increa sing the number of points in the region where the antisymmetric shift varies the most). With each of these spe ctra, we ran the following Monte Carlo analysis [34]: after ﬁtting each scan to Eqn. 24, we constructed 100 synthet ic data sets by adding Gaussian noise to the ﬁt, with noise amplitude determined by the unperturbed maser freque ncy resolution of about 40 µHz. Each of these synthetic data sets was ﬁt and a histogram of the ﬁtted Zeeman frequenci es was constructed. The resolution of each spectrum was taken as the width of the Gaussian curve that ﬁt the histog ram (see Fig. 5). As the total length of the scans increased, the resolution improved and converged to a limit of around 2.5 mHz. While the resolution improved slowly with increased acquisition time, it would have eventually b egun to degrade due to long term drifting of the Zeeman frequency. (As will be described below, we found that the Zee man frequency exhibited slow drifts of about 10-100 mHz/day). We therefore chose a scan of 100 points at 10 s avera ging, for a total length of about 18 minutes for our Lorentz symmetry test spectra. The results from the Mont e Carlo analysis for one of these spectra indicated a Zeeman frequency resolution is 2.7 mHz (see Fig. 5). B. Data analysis Our net result combines data from three runs. During each dat a run, the 18 minute Zeeman frequency scans were automated and run consecutively. After every 10 scans, 20 mi nutes of “unperturbed” maser frequency stability data was taken to track the maser’s stability. Each run contained about 10 continuous days worth of data, and each set contained more than 500 Zeeman frequency measurements, tak en at ≈18 minute intervals. For each run, the long term Zeeman frequency data was ﬁt to a fu nction of the form fit= (piecewise continuous linear function ) +δνZ,αcos(ωsidt) +δνZ,βsin(ωsidt) (25) 110.15 0.10 0.05 0.00 -0.05 νΖ - 857.061 Hz 12 108 6 4 2 0 sidereal days(a) -40040residuals [mHz] 12 108 6 4 2 0 sidereal days(b) FIG. 6. (a) Run 1 data (November 1999), with solenoid current ﬂuctuations subtracted. From the measured Zeeman frequencies, we subtracted 857.061 Hz. (b) Residuals after ﬁtting the data to Eqn. 25. whereδνZ,αandδνZ,βrepresent the cosine and sine components of the sidereal sin usoid. The time origin of the sinusoids for all three runs was taken as midnight (00:00) of November 19, 1999. The subscripts αandβrefer to two non-rotating orthogonal axes perpendicular to the rota tion axis of the earth. The total sidereal amplitude was determined by adding δνZ,αandδνZ,βin quadrature. During each run, the Zeeman frequency drifte d hundreds of mHz over tens of days. The piecewise continuous linear funct ion, consisting of segments one sidereal day in length, was included to account for these long term Zeeman frequency drifts. This function was continuous at each break, while the derivative was discontinuous. The result of this analysis, where the ﬁtting function (Eqn. 25) was applied to the full data set, was found to be in good agreement with a second analysis, where each individua l day of data was ﬁt to a line plus the sidereal sinusoid and the cosine and sine amplitudes of each day were averaged s eparately and then combined in quadrature to ﬁnd the total sidereal amplitude. C. Run 1 The cumulative data from the ﬁrst run (November 1999) are sho wn in Fig. 6(a) and the residuals from the complete ﬁt (Eqn. 25) are shown in Fig. 6(b). The data set consisted of 1 1 full days of data and had an overall drift of about 250 mHz. To avoid a biased choice of ﬁtting, we allowed the location of the slope discontinuities in the piecewise continuous linear function to shift throughout a sidereal day. We made e ight separate ﬁts, each with the location of the slope discontinuities shifted by three sidereal hours. The total sidereal amplitude and reduced chi square for each is shown in Fig. 7. We chose our result from the ﬁt with minimum reduced chi square. 122.0 1.5 1.0 0.5 0.0 -0.5sidereal amplitude [mHz] -1.0 -0.8 -0.6 -0.4 -0.2 0.0 shift of slope discontinuities [sidereal days](a) 1.4 1.3 1.2 1.1 1.0 0.9reduced chi square -1.0 -0.8 -0.6 -0.4 -0.2 0.0 shift of slope discontinuities [sidereal days](b) FIG. 7. (a) Total sidereal amplitudes for the ﬁrst run. The di ﬀerent points are from diﬀerent choices of slope discontinu ity locations. (b) Corresponding reduced chi square parameter s. The minimum value occurs with a slope break origin of midni ght (00:00) of November 19, 1999. 13Run δνZ,α[mHz] σα[mHz] δνZ,β[mHz] σβ[mHz] 1 0.43 0.36 -0.21 0.36 2 -2.02 1.27 -2.75 1.41 3 4.30 1.86 1.70 1.94 TABLE II. Sidereal amplitudes from all runs. As noted above, the error bar on a single Zeeman frequency det ermination was about 3 mHz. However, when analyzing a smooth region of long term Zeeman data (about 1 da y) we calculate a standard deviation of about 5 mHz. We believe this error bar is due mainly to residual thermal ﬂu ctuations (see Fig. 14). For our choice of slope discontinuity with minimum reduced c hi square [35], the cosine amplitude was 0.43 mHz ± 0.36 mHz, and the sine amplitude was -0.21 mHz ±0.36 mHz. The total sidereal amplitude was therefore 0.48 mH z ±0.36 mHz. D. Field-inverted runs 2 and 3 In runs 2 and 3, the static solenoid ﬁeld orientation was oppo site that of the initial run to further study the double resonance technique and any potential systematics associa ted with the solenoid ﬁeld. With the static ﬁeld inverted, and therefore directed opposite the quantization axis in th e state selecting hexapole magnet, the input ﬂux consists of atoms in states \|2/angbracketrightand\|3/angbracketright(rather than the states \|1/angbracketrightand\|2/angbracketright). Thus, reversing the ﬁeld inverts the steady state population diﬀerence ( ρ0 11−ρ0 33) of Eqn. 23 and acts to invert the antisymmetric double reson ance maser frequency shift [32]. Operating the maser in the ﬁeld reversed mode degrades the ma ser performance and subsequently the Zeeman frequency data. With opposed quantization ﬁelds inside the maser bulb and at the exit of the state selecting hexapole magnet, a narrow region of ﬁeld inversion is created. Where t he ﬁeld passes through zero, Majorana transitions between the diﬀerent mFsublevels of the F= 1 manifold can occur. This can alter the number of atoms in th e upper maser state ( F= 1,mF= 0, state \|2/angbracketright), which diminishes the overall maser amplitude and stabili ty. In the ﬁeld-inverted conﬁguration, the maser amplitude was reduc ed by 30%, and both the maser frequency and Zeeman frequency were less stable. In addition, the ﬁeld-inverted runs were each conducted soon after a number of rather invasive repairs were made to the maser [28]. Thus, the quali ty of the latter two data sets was somewhat degraded from the ﬁrst run (see Figs. 8(a) and 9(a)). The overall drift was larger (nearly 800 mHz over about 10 days), and the scatter in the data was increased, as can be seen from the r esidual plots from these runs (Figs. 8(b) and 9(b)) which have been plotted on the same scale as the residuals fro m the ﬁrst run (Fig. 6(b)). The latter two runs were also less suitable for the piecewise continuous linear drift model used in the ﬁrst run. In that case, the large slope changes were coincidentally sepa rated by an integer number of sidereal days; in the last two runs, the larger and more frequent changes in slope were not. Therefore, only certain selected sections could be ﬁt to the same model (Eqn. 25), signiﬁcantly truncating the data s ets. Due to all of these factors, the sidereal amplitudes and the associated error bars were up to an order of magnitude larger for the ﬁeld-inverted runs than the ﬁrst run. All values are shown together in Table II. E. Combined result The ﬁnal sidereal bound, combining all three runs, was calcu lated using the data in Table II. First, the weighted averages of the cosine and sine amplitudes, ¯δνZ,αand¯δνZ,β, were found using the standard formula for weighted mean [36] µ′=/parenleftbigg Σxi σ2 i/parenrightbigg //parenleftbigg Σ1 σ2 i/parenrightbigg , (26) and their uncertainties were given by 14-0.8-0.6-0.4-0.20.0 νΖ + 894.942 Hz -6 -4 -2 0 2 4 sidereal days(a) -40040residual [mHz] -6 -4 -2 0 2 4 sidereal days(b) FIG. 8. (a) Run 2 data (December 1999), with solenoid current ﬂuctuations subtracted. To the measured Zeeman frequencie s, we added 894.942 Hz. (Note the sign reversal from run 1 to acco unt for the inverted ﬁeld). (b) residuals after ﬁtting the da ta to Eqn. 25. 150.8 0.6 0.4 0.2 0.0 νΖ + 849.674Hz 121086420 sidereal days(a) -40040residual [mHz] 121086420 sidereal days(b) FIG. 9. (a) Run 3 data (March 2000), with solenoid current ﬂuc tuations subtracted. To the measured Zeeman frequencies, we added 849.674 Hz. (Note the sign reversal from run 1 to acco unt for the inverted ﬁeld). (b) residuals after ﬁtting the da ta to Eqn. 25. 16σ2 µ′=/radicaligg 1//parenleftbigg Σ1 σ2 i/parenrightbigg . (27) The sign reversal due to the ﬁeld inversion was accounted for in the raw data, before the data were ﬁt. Thus, the runs are combined using conventional (i.e., additive) averagin g. The ﬁnal sidereal amplitude Awas calculated by adding the mean cosine and sine amplitudes in quadrature, A=/radicalig ¯δν2 Z,α+¯δν2 Z,β. We measure a sidereal variation of the F= 1, ∆mF=±1 Zeeman frequency of hydrogen of A= 0.49±0.34 mHz. We note that since we are measuring an amplitude, and therefo re a strictly positive quantity, this result is consistent with no sidereal variation at the 1-sigma level: in the case w here¯δνZ,αand¯δνZ,βhave zero mean value and the same varianceσ, the probability distribution for Atakes the form P(A) =Aσ−2exp(−A2/2σ2), which has the most probable value occurring at A=σ. V. ERROR ANALYSIS In addition to our automated acquisition of Zeeman frequenc y data, we continuosly monitored the maser’s external environment. At every ten second step, in addition to applie d frequency and maser beat period, we recorded room temperature, maser cabinet temperature, solenoid current , maser power, ambient magnetic ﬁeld, and active Helmholtz coil current (see Sec. VA). A. Magnetic systematics TheF= 1,mF=±1 Zeeman frequency depends to ﬁrst-order on the z-component of the magnetic ﬁeld in the storage bulb. Thus, all external ﬁeld ﬂuctuations must be su ﬃciently screened to enable a sensitivity to shifts from Lorentz and CPT symmetry violation. The maser cavity and bul b are therefore surrounded by a set of four nested magnetic shields that reduce the ambient ﬁeld by a factor of a bout 32,000. We measure unshielded ﬂuctuations in the ambient ﬁeld of about 3 mG (peak-peak) during the day, and even when shielded, these add signiﬁcant noise to a single Zeeman scan, as illustrated in Fig. 11(a). Furtherm ore, the amplitude of the ﬁeld ﬂuctuations is signiﬁcantly reduced late at night, which could generate a diurnal system atic eﬀect in our data. To reduce the eﬀect of ﬂuctuations in the ambient magnetic ﬁe ld, we installed an active feedback system (see Fig. 10) consisting of two pairs of large Helmholtz coils (2. 4 m diameter). The ﬁrst pair of coils (50 turns) produced a uniform ﬁeld that cancelled most of the z-component of the a mbient ﬁeld, leaving a residual ﬁeld of around 5 mG. A magnetometer probe that sensed the residual ambient ﬁeld w as placed partially inside the maser’s magnetic shields near the maser cavity. This probe had a sensitivity of s= 1.7 mG/V. Due to its location partially inside the magnetic shields, the probe was screened by a factor of about six from e xternal ﬁelds, reducing the sensitivity to s′= 0.3 mG/V, and producing a diﬀerential screening of 5300 between the ma gnetometer probe and the atoms. The magnetometer output was passed into a PID servo (Linear Research model LR- 130), which contained a proportional stage (gain G = 33), an integral stage (time constant T i= 0.1 s) and a derivative stage (time constant = 0.01 s). The co rrection voltage was applied to the second pair of Helmholtz coils (3 t urns) which produced a uniform ﬁeld ( p= 14 mG/V) along the z-axis to nullify the residual ﬁeld and actively co unter any ﬁeld ﬂuctuations. Neglecting the small eﬀect of the derivative stage, the overall time constant of this syst em was given by τ=Ti(1 +s′/pG)≈0.1 s, about 100 times shorter than the averaging time of our maser frequency shift measurements (10 s). With this system we were able to further reduce ambient ﬁeld ﬂ uctuations at the magnetometer by a factor of 3,000. The resulting unshielded ﬂuctuations were less than 1 µG peak-peak. The ﬁeld recorded by the partially screened magnetometer probe is shown in Fig. 12. The noise on a single Z eeman scan was reduced below our Zeeman frequency resolution, as shown in Fig. 11(b). During our Lorentz symme try test, we monitored the ﬁeld at the magnetometer probe and placed a bound of ∼5 nG on the sidereal component of the variation. This corresp onds to a shift of less than 0.2 µHz on the hydrogen Zeeman frequency, three orders of magnitu de smaller than the sidereal Zeeman frequency bound measured. The magnetometer [37] used in the feedback loop was a ﬂuxgate magnetometer probe (RFL industries Model 101) which consisted of two parallel high-permeability magneti c cores each surrounded by an excitation coil (the excitatio n coils were wound in the opposite sense of each other). A separ ate pickup coil was wound around the pair of cores. An AC current (about 2.5 kHz) in the excitation coils drove th e cores into saturation, and, in the presence of any slowly varying external magnetic ﬁeld oriented along the ma gnetic cores’ axes, an EMF was generated in the pickup 1710 k10 k10 k 330 k 10 µFmagnetometer probemagnetic shieldsmaser bulb2 pairs Helmholtz coils FIG. 10. Schematic of the active Helmholtz control loop. A la rge set of Helmholtz coils (50 turns) cancelled all but a resi dual ∼5 mG of the z-component of the ambient ﬁeld. This residual ﬁel d, detected with a ﬂuxgate magnetometer probe, was actively cancelled by a servoloop and a second pair of Helmholtz coils (3 turns). The servoloop consisted of a proportional stage ( gain = 33), and integral stage (time constant = 0.1 s) and a derivat ive stage (time constant = 0.01, not shown). The overall time constant of the loop was about τ= 0.1 s. 180.842 0.840 0.838 0.836 0.834beat period [s] -2 -1 0 1 2 Zeeman detuning [Hz](a) 0.850 0.848 0.846 0.844 0.842beat period [s] -2 -1 0 1 2 Zeeman detuning [Hz](b) FIG. 11. (a) Zeeman scan without the active Helmholtz feedba ck loop. The noise on the data is due to the left and right shifting of the antisymmetric resonance as the Zeeman frequ ency shifts due to 3 mG ambient ﬁeld ﬂuctuations. (b) Zeeman scan with active Helmholtz control. Ambient ﬁeld ﬂuctuatio ns were reduced to less than 1 µG. 190.6 0.4 0.2 0.0 -0.2 -0.4magnetometer [ µG] 7260483624120 time [hours] FIG. 12. Residual ambient magnetic ﬁeld, after cancellatio n by the active Helmholtz control loop, sensed at the magneto meter probe. Each point is a 10 s average. These three days worth of d ata depict a Sunday, Monday and Tuesday, with the time origin corresponding to 00:00 Sunday. From these data it can be seen that for three hours every night the magnetic noise dies out dramatically, and that the noise level is signiﬁcantly lowe r on weekends than weekdays. Nevertheless, with the active f eedback system even the largest ﬂuctuations (1 µG peak-peak) causes changes in the Zeeman frequency well bel ow our sensitivity (∆B= 1µG⇒∆νZ= 40µHz). coil at the second and higher harmonics of the excitation fre quency. The magnitude of the time-averaged EMF was proportional to the external ﬁeld. The probe had a sensitivi ty of approximately 1 nG. Any Lorentz violating spin-orientation dependence of the e nergy of the electrons in the magnetic cores would induce a sidereal variation in the cores’ magnetization and could g enerate, or mask, a sidereal variation in the hydrogen Zeeman frequency through the feedback circuit. However, ba sed on the latest bound on electron Lorentz violation [18] (10−29GeV), the Lorentz violating shift would be less than 10−11G, far below the level of residual ambient ﬁeld ﬂuctuations. Also, the additional shielding factor of 5300 between the probe and the atoms further reduced the eﬀect of any Lorentz violating shift in the probe electrons’ energ ies. With the ambient ﬁeld kept nearly constant near zero, the Zee man frequency was set by the magnetic ﬁeld generated by the solenoid, and hence by the solenoid current. We monito red solenoid current ﬂuctuations by measuring the voltage across the current-setting 5 kΩ resistor with a 5 1/2 digit multimeter (Fluke model 8840A/AF). By measuring the Zeeman frequency shift caused by large current changes, we found a dependence of around 10 mHz/nA. When acquiring Lorentz symmetry test data, we measured long term drifts in the current of about 5 nA (see Fig. 13), signiﬁcant enough to produce detectable shifts in the Zeema n frequency. Thus, we subtracted these directly from the Zeeman data. We measured a sidereal variation of 25 ±10 pA on the solenoid current, corresponding to a sidereal variation of 0.16 ±0.08 mHz on the Zeeman frequency correction. This systemati c uncertainty in the Zeeman frequency was included in the net error analysis, as describ ed in Sec. V C. B. Other systematics The maser resided in a closed, temperature stabilized room w here the temperature oscillated with a peak-peak amplitude of slightly less than 0.5˚ C with a period of around 15 minutes. The maser was contained in an insulated and thermally controlled cabinet, which provided a factor o f ﬁve to ten shielding from the room, and reduced the ﬂuctuations to less than 0.1˚ C peak-peak, as shown in Fig. 14 . By making large changes in the maser cabinet temperature and measuring the eﬀect on the Zeeman frequency , we found a temperature coeﬃcient of about 200 2096.2690 96.2680 96.2670 96.2660 96.2650solenoid current [ µA] 24020016012080400 time [hours] FIG. 13. Solenoid current during the ﬁrst data run. Each poin t is an average over one full Zeeman frequency measurement (18 mins). Since the Zeeman frequency is directly proportio nal to the solenoid current, we subtracted these solenoid cu rrent drifts directly from the raw Zeeman data, using a measured ca libration. We ﬁnd a sidereal component of 25 ±10 pA to that correction, corresponding to a signal of 0.16 ±0.08 mHz on the Zeeman frequency. This systematic uncertain ty has been included in our overall error analysis. mHz/˚ C. We believe this frequency shift was due mainly to the resistors which set the solenoid current, which had 100 ppm/˚ C temperature coeﬃcients. We monitored the cabine t temperature and placed a bound on the sidereal component of the temperature ﬂuctuations at 0.5 mK, which wo uld produce a systematic sidereal variation of 100 µHz on the Zeeman frequency, about a factor of 3 smaller than th e measured limit on sidereal variation in Zeeman frequency. As mentioned in Sec. III C, spin-exchange eﬀects induce a sma ll oﬀset of the Zeeman frequency given by Eqn. 23 from the actual Zeeman frequency [31]. This would imply that ﬂuctuations in the input atomic ﬂux (and therefore the maser power) could cause ﬂuctuations in the Zeeman frequ ency measurement. We measured a limit on the shift of the Zeeman frequency due to large changes in average maser power at less than 0.8 mHz/fW. (Expected shifts from spin-exchange are ten times smaller than this level (Sec. II I C). We believe the measured limit is related to heating of the maser as the ﬂux is increased). During long-term opera tion, the average maser power drifted approximately 1 fW/day (see Fig. 15). The sidereal component of the variatio ns of the maser power were less than 0.05 fW, implying a variation in the Zeeman frequency of less than 40 µHz, an order of magnitude smaller than our experimental boun d for sidereal Zeeman frequency variation. C. Final result We measured systematic errors in sidereal Zeeman frequency variation (as described in Secs. VA and VB) due to ambient magnetic ﬁeld (0.2 µHz), solenoid ﬁeld (80 µHz), maser cabinet temperature (100 µHz), and hydrogen density induced spin-exchange shifts (40 µHz). Combining these errors in quadrature with the 0.34 mHz s tatistical uncertainty in Zeeman frequency variation, we ﬁnd a siderea l variation of the F= 1, ∆mF=±1 Zeeman frequency in hydrogen of 0.44 ±0.37 mHz at the 1- σlevel. This 0.37 mHz bound corresponds to 1.5 ×10−27GeV in energy units. 2125.4 25.2 25.0 24.8 24.6 24.4 24.2temperature [oC] 5.04.54.03.53.02.52.01.51.00.50.0 time [hours]room temperature cabinet temperature FIG. 14. Temperature data during the ﬁrst run. Each point is a 10 second average. The top trace shows the characteristic 0.5˚ C peak-peak, 15 minute period oscillation of the room te mperature. The bottom trace shows the screened oscillation s inside the maser cabinet. The cabinet is insulated and tempe rature controlled with a blown air system. In addition, the innermost regions of the maser, including the microwave cav ity, are further insulated from the maser cabinet air temper ature, and independently temperature controlled. The residual te mperature variation of the maser cabinet air had a sidereal v ariation of 0.5 mK, resulting in an additional systematic uncertaint y of 0.1 mHz on the Zeeman frequency. This value is included in the net error analysis. 78 76 74 72 70average maser power [fW] 109876543210 time [days] FIG. 15. Average maser power during the ﬁrst data run. Each po int is an average over one full Zeeman frequency mea- surement (18 mins). We measure a sidereal variation in this p ower at less than 0.05 fW, leading to an additional systemati c uncertainty in the Zeeman frequency of 0.04 mHz, which is inc luded in the net error analysis. 22XYZ xyz χ χΩt FIG. 16. Coordinate systems used. The (X,Y,Z) set refers to a ﬁxed reference frame, and the (x,y,z) set refers to the laboratory frame. The lab frame is tilted from the ﬁxed Z-axi s by our co-latitude, and it rotates about Z as the earth rotat es. Theαandβaxes, described in Sec. IV, span a plane parallel to the X-Y pl ane. VI. DISCUSSION A. Transformation to ﬁxed frame Our experimental bound of 0.37 mHz on sidereal variation of t he hydrogen Zeeman frequency may be interpreted in terms of Eqn. 5 as a bound on vector and tensor components of th e standard model extension. To make meaningful comparisons to other experiments, we transform our result i nto a ﬁxed reference frame. Following the construction in reference [13], we label the ﬁxed frame with coordinates (X, Y,Z) and the laboratory frame with coordinates (x,y,z), as shown in Fig. 16. We select the earth’s rotation axis as the ﬁxed Z axis, (declination = 90 degrees). We then deﬁne ﬁxed X as declination = right ascension = 0 degrees, and ﬁxed Y as declination = 0 degrees, right ascension = 90 degrees. With this convention, the X and Y axes lie in the plan e of the earth’s equator. Note that the α,βaxes of Sec. IVE, also in the earth’s equatorial plane, are rotated a bout the earth’s rotation axis from the X,Y axes by an angle equivalent to the right ascension of 71◦7’ longitude at 00:00 of November 19, 1999. For our experiment, the quantization axis (which we denote z ) was vertical in the lab frame, making an angle χ≈ 48 degrees relative to Z, accounted for by rotating the entir e (x,y,z) system by χabout Y. The lab frame (x,y,z) rotates about Z by an angle Ω t, where Ω is the frequency of the earth’s (sidereal) rotation . These two coordinate systems are related through the transf ormation  t x y z = 1 0 0 0 0 cosχcosΩtcosχsinΩt−sinχ 0−sinΩt cosΩt 0 0 sinχcosΩtsinχsin Ωtcosχ  0 X Y Z =T 0 X Y Z . (28) 23Experiment ˜be X,Y[GHz] ˜bp X,Y[GHz] ˜bn X,Y[GHz] anomaly frequency of e−in Penning trap [16] 10−25- - 199Hg and133Cs precession frequencies [22] 10−2710−2710−30 this work [1] 10−2710−27- spin polarized torsion pendulum [20] 10−29- - dual species129Xe/3He maser [21] - - 10−31 TABLE III. Electron, proton and neutron experimental bound s. Then, vectors transform as /vectorblab=T/vectorbfixed, while tensors transform as dlab=T d fixedT−1. As shown in equation (5), our signal depends on the following combination of terms (for both electron and proton): ˜b3=b3−md30−H12. (29) Transforming these to the ﬁxed frame, we see b3=bZcosχ+bXsinχcosΩt+bYsinχsin Ωt, d30=dZ0cosχ+dX0sinχcosΩt+dY0sinχsin Ωt, (30) H12=HXYcosχ+HY ZsinχcosΩt+HZXsinχsin Ωt, so our observable is given by ˜b3= (bZ−mdZ0−HXY)cosχ + (bY−mdY0−HZX)sinχsinΩt (31) + (bX−mdX0−HY Z)sinχcosΩt. The ﬁrst term on the right is a constant oﬀset, not bounded by o ur experiment. The second and third terms each vary at the sidereal frequency. Combining Eqn. 31 (for both e−and p) with Eqn. 5, we see \|∆νZ\|2= [(be Y−mede Y0−He ZX) + (bp Y−mpdp Y0−Hp ZX)]2sin2χ h2(32) + [(be X−mede X0−He Y Z) + (bp X−mpdp X0−Hp Y Z)]2sin2χ h2. Insertingχ= 48 degrees, we obtain the ﬁnal result /radicalbigg/parenleftig ˜be X+˜bp X/parenrightig2 +/parenleftig ˜be Y+˜bp Y/parenrightig2 = (3±2)×10−27GeV. (33) Our 1-sigma bound on Lorentz and CPT violation of the proton a nd electron is therefore 2 ×10−27GeV. B. Comparison to previous experiments We compare our result with other recent tests of Lorentz and C PT symmetry in Table III. Although our bounds are numerically similar to the those from the199Hg/133Cs experiment, the simplicity of the hydrogen atom allows us to place bounds directly on the electron and proton; uncerta inties in nuclear structure models do not complicate the interpretation of our result. The recent limit set by the tor sion pendulum experiment of Adelberger et. al. [20] on electron Lorentz and CPT violation casts our result as a clea n bound on Lorentz and CPT violation of the proton. C. Future work To make a more sensitive measure of the sidereal variation of the Zeeman frequency in a hydrogen maser, it will be important to clearly identify and reduce the magnitude of the long term drifts of the Zeeman frequency. Possible 24sources of these drifts are magnetic ﬁelds near the maser bul b caused by stray currents in heaters or power supplies in the inner regions of the maser. Also, the scatter of the Zee man data points, believed to be due mainly to residual thermal ﬂuctuations, should be reduced. Both of these objec tives could be accomplished by carefully rebuilding a hydrogen maser, with better engineered power and temperatu re control systems. VII. ACKNOWLEDGMENTS We gratefully acknowledge the encouragement of Alan Kostel eck´ y. 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[28] The maser used for our Lorentz symmetry test, maser P-28 was only temporarily available for our use, as it was in our lab only to undergo several repairs. As a result, the amount o f data we could acquire was limited. Maser P-8 is housed permanently in our laboratory, so much of our characterizat ion was done with this maser. However, this maser was not suitable for a Lorentz symmetry test because it suﬀered from large, long term Zeeman frequency drifts, attributed to les s magnetic shielding and larger extraneous ﬁelds (e.g., from heating elements). [29] E.M. Mattison, W. Shen, and R.F.C. Vessot, in Proceedings of the 39th Annual Frequency Contol Symposium (IEEE, New York, 1985), p. 72. [30] N.F. Ramsey, Molecular Beams (Clarendon Press, Oxford, 1956), Chap. 2. [31] J.-Y. Savard, G. Busca, S. Rovea, M. Desaintfuscien, an d P. Petit, Can. J. Phys. 57, 904 (1979). [32] M.A. Humphrey, D.F. Phillips, and R.L. Walsworth, Phys . Rev. A 62, 063405 (2000). 25[33] P.L. Bender, Phys. 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arXiv:physics/0103069v1 [physics.ed-ph] 22 Mar 2001ADDING RESISTANCES AND CAPACITANCES IN INTRODUCTORY ELECTRICITY C.J. Efthimiou1and R.A. Llewellyn2 Department of Physics University of Central Florida Orlando, FL 32826 Abstract We propose a uniﬁed approach to addition of resistors and cap acitors such that the formulæ are always simply additive. This approach has the ad vantage of being consistent with the intuition of the students. To demonstrate our point of view, we re-work some well- known end-of-the-chapter textbook problems and propose so me additional new problems. 1 Introduction All introductory physics textbooks, with or without calcul us, cover the addition of both resis- tances and capacitances in series and in parallel. The formu læ for adding resistances R=R1+R2+. . . , (1) 1 R=1 R1+1 R2+. . . , (2) and capacitances 1 C=1 C1+1 C2+. . . , (3) C=C1+C2+. . . , (4) are well-known and well-studied in all the books. In books with calculus there are often end-of-chapter probl ems in which students must ﬁnd RandCusing continuous versions of equations (1) and (4) [2, 5, 6, 7 , 8]. However, we have found nonewhich includes problems that make use of continuous version s of equations (2) and (3) [2, 3, 4, 5, 6, 7, 8]. Students who can understand and solve the ﬁrst class of problems should be able to handle the second class of the problems, as w ell. We feel that continuous problems that make use of all four equations should be shown t o the students in order to give them a global picture of how calculus is applied to physical p roblems. Physics contains much more than mathematics. When integrating quantities in phys ics, the way we integrate them 1costas@physics.ucf.edu 2ral@physics.ucf.edu 1is motivated by the underlying physics. Students often forg et the physical reasoning and they tend to add (integrate) quantities only in one way. In this paper, we introduce an approach to solving continuou s versions of equations (2) and (3) that is as straightforward and logical for the studen ts as solving continuous versions of equations (1) and (4). We then present some problems in which the student must decide which formula is the right one to use for integration. We hope that t his article will motivate teachers to explain to students the subtle points between ‘straight i ntegration’ as taught in calculus and ‘physical integration’ to ﬁnd a physical quantity. 2 Adding Resistances Problem [Cylindrical Resistor] The cylindrical resistor shown in ﬁgure 1 is made such that th e resistivity ρis a function of the distance rfrom the axis. What is the total resistance Rof the resistor? rdr a l Figure 1: The ﬁgure shows a cylindrical wire of radius a. A potential diﬀerence is applied between the bases of the cylinder and therefore electric cur rent is running parallel to the axis of the cylinder. Towards a Solution : We divide the cylindrical resistor into inﬁnitesimal resis tors in the form of cylindrical shells of thickness dr. One of these shells is seen in red in ﬁgure 1. If we apply equat ion (1) naively, we must write R=/integraldisplay cylinderdR . The inﬁnitesimal resistance of the red shell is given by dR=ρ(r)ℓ dA, where dA= 2πrdris the area of the base of the inﬁnitesimal shell. Since dris small, dRis huge, which is absurd. Where is the error? Discussion : When the current is ﬂowing along the axis of the cylinder, the inﬁnitesimal resistors are not connected in series. Therefore, the naive approach R=/integraldisplay dR 2does not work since this formula assumes that the shells are c onnected in series. Instead, all of the inﬁnitesimal cylindrical shells of width drare connected at the same end points and, therefore, have the same applied potential. In other words, the shells are connected in parallel and it is the inverse resistance that is importnat, not Ritself. Speciﬁcally 1 R=/integraldisplay cylinderd/parenleftbigg1 R/parenrightbigg . Students may feel unconfortable with this equation as at the begining since it may seem ‘con- tradictory’ to their calculus knowledge; therefore, some d iscussion may be helpful. Equation (1) states that when resistors are connected in ser ies, they make it harder for the current to go through. Their resistances add to give the t otal restistance. However, when resistors are connected in parallel, many ‘paths’ are avail able simultaneously; the current is ﬂowing easily and ‘resistance’ —which is a measure of ﬂow diﬃ culty— is not a good quantity to use. Maybe an analogy from everyday life is useful here. Pa ying tolls at toll booths is in direct analogy. When only a single booth is available, then a ll traﬃc has to go through that lane and no matter how dense the traﬃc is, there will be a relat ive delay. The traﬃc encounters some ‘resistance’ in the ﬂow. However, when multiple booths are open, the drivers choose to go through the lanes that are free at the time of their approac h to the booths and thus the delays encountered are minimal. In this case, the ‘availabi lity’ of booths is a better quantity to be used to descibe what is happening instead of the ‘resistan ce’ at the booths. Ultimately, the two quantities are related, but intuitely it is more satisfy ing to use one over the over depending on the situation. In direct analogy, for resistors connecte d in parallel, the relevant quantity is notRany longer, but S, where S=1 R=σA ℓ, andσ= 1/ρis the conductivity. We may call Stheconduction of the resistor. When resistors are connected in parallel, they make it easier for the curren t to go through. Their conductions add to give the total conduction: S=S1+S1+···. Thus, the conduction follows the usual addition S=/integraldisplay dS when inﬁtesimal resistors are connected in parallel. We are now in position to compute the answer to the posed probl em in a way that is consistent with the intuition of the students. Solution : Since the cylindrical shells are connected in parallel, con duction is the additive quantity. For the inﬁnitesimal shell dS=σ(r)2πrdr ℓ. Therefore S=/integraldisplay cylinderdS=2π ℓ/integraldisplaya 0σ(r)rdr . 3For example, if σ(r) =σ0a r, then S= 2σ0πa2 ℓ, where σ0= 1/ρ0. The resistance is therefore R=ρ0 2ℓ πa2. Problem [Truncated-Cone Resistor] A resistor is made from a truncated cone of material with unif orm resistivity ρ. What is the total resistance Rof the resistor when the potential diﬀerence is applied betw een the two bases of the cone? Solution : This is a well-known problem found in many of the introductor y physics textbooks [2, 5, 6, 7, 8]. We can partition the cone into inﬁtesimal cylindric al resistors of length dz. One representative resistor at distance zfrom the top base is seen in ﬁgure 2. The area of the resistor is A=πr2and therefore its inﬁnitesimal resistance is given by dR=ρdz πr2. From the ﬁgure we can see that z h=r−b c−b⇒dz=h c−bdr . hb cz dzr Figure 2: A truncated cone which has been sliced in inﬁtesima l cylinders of height dz. The inﬁnitesimal resistors are connected in series and ther efore R=/integraldisplay conedR=ρh π(c−b)/integraldisplayc bdr r2=ρh πbc. (5) 43 Adding Capacitances A similar discussion may be given for capacitors. When capac itors are connected in parrallel, capacitance is the the relative additive quantity: C=C1+C2+···. For a parallel-plate capacitor of area Aand distance dbetween the plates C=ε0A d. When the capacitor is ﬁlled with a uniform dielectric of diel ectric constant κthen C=ε0κA d. However, when capacitors are connected in series, the inver se capacitance D=1 C. is the additive quantity. We may call it the incapacitance . For a parallel-plate capacitor D=1 ε0κd A. In other words, when capacitors are connected in series D=D1+D2+···. Problems like this are encountered when we ﬁll a capacitor wi th a dielectric for which the dielectric constact is a function of the distance from the pl ates of the capacitor. Students are familiar with such problems for a parallel-plate capacitor in the discrete case. For example, problems asking students to compute the total capacitance i n cases as those shown in ﬁgure 3 are found in several textbooks [5, 6, 8]. However, continuou s problems are not found in any textbook [2, 3, 4, 5, 6, 7, 8]. We can easily construct new problems or re-work old problem u sing this idea. For example, the well-known formula for the capacitance of a cylindrical capacitor can be found this way. As shown in the left side of ﬁgure 4, the capacitor is partitione d into small cylindrical capacitors for which the distance between the plates is dr. For such small capacitors, the formula of a parallel-plate capacitor is valid. We notice though that al l inﬁntesimal capacitors are connected in series. Therefore dD=1 ε0dr 2πrh. and D=/integraldisplay cylinderdD=1 2πε0h/integraldisplayb adr r=1 2πε0hlna b. The total capacitance is then C=1 D=2πε0h lnb a. 5/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 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In the left picture, we have sliced it in inﬁnitesimal cylindrical shells, while in the right picture we have sliced it in inﬁnitesimal annuli. Comment :One might be tempted to partition the cylindrical capacitor into inﬁnitesimal capacitors as seen in the ﬁgure to the left (blue section). Such capacitors look simpler than the inﬁtesimal cylindrical shell we used above. Furthermore, they are connected in parallel (no tice that each capacitor is carrying an inﬁnitesimal charge dQand/integraltext cylinderdQ=Q) and therefore it is enough to deal with capacitance, C=/integraltext cylinderdC, and not incapacitance D. However, with a minute’s reﬂection the reader will see that i n order to use the parallel-plate capacitor formula in the inﬁnitesimal case, the distance between the p lates must be inﬁnitesimal which indicates that the inﬁnitesimal capacitors must be connected in series. In the proposed (blue) slicing, the distance between 6the plates of the inﬁnitesimal capacitor is ﬁnite, namely b−a. The inﬁnitesimal capacitor is still a cylindrical capacitor of inﬁtesimal height and therefore its capacitan ce should be expressed in a form that is not known before the problem is solved. In other words, dC=2πε0 ln(b/a)dz (6) from which C=2πε0 ln(b/a)/integraldisplayh 0dz=2πε0 ln(b/a)h . (7) But the expression (6) is unkown until the result (7) is found . Problem [Truncated-cone Capacitor] A capacitor is made of two circular disks of radii bandcrespectively placed at a distance hsuch that the line that joins their centers is perpendicular to the disks. Find the capacitance of this arrangement (Seen in ﬁgure 2). Solution : We partition the capacitor into inﬁnitesimal parallel-pl ate capacitors of distance dzand plate area A=πr2exactly as seen in ﬁgure 2. These inﬁtesimal capacitors are c onnected in series and therefore the incapacitance is the relevant ad ditive quantity: dD=1 ε0dz πr2. Notice that the computation is identical to that of Rwith ﬁnal result: D=1 ε0h πbc⇒C=ε0πbc h. (8) When b=c, we recover the result of the parallel-plate capacitor. 4 Conclusions In this paper, we have tried to argue that when the right varia bles are used then the law of addition for capacitances and resistances is always additi ve. Table 1 summarizes our main formulæ. This is in agreement with the intuition of students when they solve continuous problems on the subject who like to add quantities in a simple way. We must point out that our discusssion by no means is restrict ed to capacitances and resistances only. Similar addition laws are encountered in many other areas of physics. For example, when connecting springs in parallel the total stiﬀ ness constant is given by the sum of the individual stiﬀness constants: k=k1+k2+···. When the springs are connected in series, then 1 k=1 k1+1 k2+···, pointing out that in this case, not the stiﬀness constant but theelasticity constant ℓ=1 k is the relevant additive constant. Our discussion can thus b e repeated verbatim in all similar cases. Table 2 lists the most common cases found in introduct ory physics. 7resistors capacitors series R=/summationtext iRiD=/summationtext iDi parallel S=/summationtext iSiC=/summationtext iCi Table 1: When viewed in the right physical quantities, addit ion of resistors and capacitors is always simple. In the above table, Ris the resistance, Sis the conduction, Cis the capacitance, andDthe incapacitance of a circuit element. resistors capacitors inductors springs thermal conductors seriesresistance Rincapacitance Dinductance Lelasticity ℓthermal resistance R parallelconduction Scapacitance Cdeductance Kstiﬀness kthermal conduction S Table 2: This table summarizes the additive physical quanti ties in the most common cases encountered in introductory physics. The quantities that a re not usually deﬁned in the in- troductory books are the conduction S= 1/R, the incapacitance D= 1/C, the deductance K= 1/L, the elasticity constant ℓ= 1/k, and the the thermal conduction S= 1/R. 8A Suggested Problems We end our article with some suggested problems which the rea der may wish to solve. 1. Re-derive the well-known expression for the capacitance of a spherical capacitor C= 4πε0ab b−a, (where a, bare the radii of the spheres with b > a) by partitioning it into inﬁnitesimal capacitors. 2. Show that the capacitance of a cylindrical capacitor whic h is ﬁlled with a dielectric having dielectric constant κ(r) =crn, where ris the distance from the axis and c,n/negationslash= 0 are constants, is given by C= 2πε0hcnanbn bn−an. 3. Show that the capacitance of a cylindrical capacitor whic h is ﬁlled with a dielectric having dielectric constant κ(z) =czn, where zis the distance from the basis and c,n≥0 are constants is given by C= 2πε0chn+1 (n+ 1) ln( b/a). 4. Show that the capacitance of a spherical capacitor which i s ﬁlled with a dielectric having dielectric constant κ(r) =crn, where ris the distance from the center and c, nare constants is given by C= 4πε0c ln(b/a) forn=−1 and C= 4πε0c(n+ 1)an+1bn+1 bn+1−an+1 forn/negationslash=−1. b cha Figure 5: A hollow truncated cone . 95. (a) Two metallic ﬂat annuli are placed such that they form a capacitor with the shape of a hollow truncated cone as seen in ﬁgure 5. Partition the capac itor in inﬁtesimal capacitors and show that the capacitance is given by C= 2πε0h a(c−b)/bracketleftBigg lnc−a c+a−lnb−a b+a/bracketrightBigg . Show that this result reduces to that of a cylindrical capaci tor for c=b. Also, show that it agrees with (8) when a= 0. (b) Now, ﬁll the two bases with disks of radius aand argue that the capacitance of the hollow truncated cone equals that of the truncated cone minu s the capacitance of the parallel-plate capacitor that we have removed (superposit ion principle). This means that the capacitance of the hollow truncated cone should equal to C=πε0bc−a2 h. How is it possible that this result does not agree with that of part (a)? 6. A capacitor with the shape of a hollow truncated cone is now formed from two ‘cylindrical’ shells. Show that the capacitance in this case is C= 2πε0ah c−b/bracketleftBigg li/parenleftbiggc a/parenrightbigg −li/parenleftBiggb a/parenrightBigg/bracketrightBigg , where li is the logarithmic function[1] li(X)≡/integraldisplayX 0dx lnx, X > 1. 7. (a) A conductor has the shape seen in ﬁgure 5. Show that the r esistance when the voltage is applied between the upper and lower bases is given by R=ρh 2πa(c−b)/bracketleftBigg lnc−a c+a−lnb−a b+a/bracketrightBigg . Show that this result reduces to equation (5) for a= 0. (b) Argue now that the resistance of the hollow truncated-co nical wire is the diﬀerence between the the resistance of the truncated-conical wire an d a cylindrical wire of radius a(superposition principle). This implies that R=ρh bc−a2. Explain why this does not agree with part (a). 8. A conductor has the shape of a hollow cylinder as seen in ﬁgu re 4. Show that the resistance when the voltage is applied between the inner and outer surfa ces is given by R=ρ 2πhlnb a. 109. A conductor has the shape seen in ﬁgure 5. Show that the resi stance when the voltage is applied between the inner and outer surfaces is given by R=ρ 2πhac−b li(c/a)−li(b/a). Show that, for c=b, this result agrees with that of the previous problem. 10. A conductor has the shape of a truncated wedge as seen in ﬁg ure 6. Show that the resistance of the conductor when the voltage is applied betw een the left and right faces is R=ρ aℓln(c/b) c−b, while the resistance when the voltage is applied between the top and bottom faces is R=ρ ac−b ℓln(c/b). xyz abc ℓ Figure 6: A conductor with the shape of a truncated wedge. 11References [1]M. Abramowitz, I.A. Stegun ,Handbook of Mathematical Functions with Formulas , Dover. [2]D. Haliday, R. Resnick, J. Walker ,Fundamentals of Physics , 6th ed., John-Wiley & Sons. [3]E. Hecht ,Physics , Brooks/Cole 1996. [4]P. Nolan ,Fundamentals of College Physics , Wm. C. Brown Communications 1993. [5]R.A. Serway ,Physics for Scientists and Engineers , 4th ed., Saunders College Publishing. [6]P.A. Tipler ,Physics for Scientists and Engineers , 3rd ed., Worth Publishers. [7]R. Wolfson, J.M. Pasachoff ,Physics for Scientists and Engineers , 3rd ed., Addison-Wesley. [8]H.D. Young, R.A. Freedman ,University Physics , 9th ed., Addison-Wesley 1996. 12
arXiv:physics/0103070v1 [physics.ao-ph] 22 Mar 2001Acoustic scattering by a cylinder near a pressure release su rface Zhen Ye and You-Yu Chen Department of Physics, National Central University, Chung li, Taiwan 32054, Republic of China (February 2, 2008) Abstract This paper presents a study of acoustic scattering by a cylin der of either inﬁnite or ﬁnite length near a ﬂat pressure-release surface. A novel self-consistent method is developed to describe the multiple scattering interactions between the cylinder and the surface. The comp lete scattering amplitude for the cylinder is derived from a set of equations, and is numerically evalua ted. The results show that the presence of the surface can either enhance or reduce the scattering of th e cylinder, depending on the frequency, the composition of the cylinder, and the distance between the cy linder and the surface. Both air-ﬁlled and rigid cylinders are considered. PACS number: 43.30.Gv., 43.30.Bp., 43.20.Fn. INTRODUCTION Acoustic scattering by underwater objects near a pressure r elease boundary is a very important issue in a number of current research and applications, including the modeling of scattering from surface dwelling ﬁsh, the understanding of oceanic ﬂuxes and ambient noises gener ated at ocean surface layers. It may also be of great help in models of acoustic scattering by submarines ne ar the ocean surface. In the literature, the research on sound scattering by under water objects near a pressure release surface has been mainly focused on the scattering by a spherical obje ct such as an air bubble (Refs. e. g. [1, 2, 3, 4, 5, 6, 7]). In many important applications, however, underwa ter objects may not take the spherical geometry. Rather they often take elongated shapes. This includes, for example, the surface dwelling ﬁsh, the ﬂoating logs in rivers, military objects, and so on. For these situat ions, it is desirable to study acoustic scattering by an elongated object near a boundary. By searching the lite rature, we ﬁnd that the research along this line is surprisingly scarce. The purpose of the the present p aper is to present an investigation of acoustic scattering by a cylinder of either inﬁnite or ﬁnite length ne ar a ﬂat pressure-release boundary. We consider acoustic scattering by an elongated object near a ﬂat pressure release surface; the sea or river surface can be regarded as one of such surfaces when the acoustic wavelength is long compared to the surface wave. As a ﬁrst step, for simplicity yet not to compro mising the generality, we assume the object as a straight cylinder. Due to the presence of the surface, the w ave will be scattered back and forth between the surface and the object before it reaches a receiver. The resc attering from the scatterer and rereﬂection from the surface are studied using a self-consistent method by ex pressing all the waves in terms of modal series. The scattering by the cylinder is thus exactly evaluated, an d analyzed. The theory is ﬁrst developed for an inﬁnite cylinder, then extended to ﬁnite cylinders using th e genuine approach given by Ref. [8]. Although the theory allows us to consider a variety of cylinders, in or der to show the essence of the theory in its most transparent way we focus on two important types of cylin ders, that is, the air-ﬁlled and the rigid cylinders. The former can be used to model the ﬁsh while the la tter may resemble some acoustic scattering characteristics of military objects. I. FORMULATION OF THE PROBLEM The problem considered in this paper is depicted in Fig. 1. A s traight cylinder is located in the water at a depth dbeneath a pressure release plane which can be the sea surface . For simplicity, we assume that the axis of the cylinder is parallel to the plane. The radius of th e cylinder is a. The acoustic parameters of the cylinder are taken as: the mass density ρ1and sound speed c1, while those of the surround water are ρand c; therefore the acoustic contrasts are g=ρ1/ρandh=c1/c. A parallel line acoustic source transmitting a wave of frequency ωis at/vector rssome distance away from the surface. The transmitted wave is scattered by 1the cylinder and reﬂected from the surface, as shown in Fig. 1 . The reﬂected wave is also scattered by the cylinder. The wave scattered by the cylinder is again reﬂect ed by the surface. Such a process is repeated, establishing an inﬁnite series of rescattering and rereﬂec tion between the cylinder and the surface. This multiple scattering process can be conveniently treated by a self-consistent manner. The rectangular frame is set up in such a way that the z-axis is parallel to the axis of the cylinder. the x-axis and y-axis are shown in Fig. 1. To solve the scattering problem, however, we use th e cylindrical coordinates in the rectangular system. We note that in the present paper, for brevity we do no t consider the case that the incident direction is oblique to the axis of the cylinder; the extension to obliq ue cases is straightforward. The setting in the problem is by analogy with that described in Ref. [7], where a spherical air bubble is placed beneath the ﬂat boundary. A.Scattering by a cylinder of inﬁnite length In this section, we present a formulation for sound scatteri ng by an inﬁnite cylinder near a pressure-release boundary. For succinctness, we only show the most essential steps in the derivation. First the direct wave from the line source can be written as pinc=iπH(1) 0(k\|/vector r−/vector rs\|), (1) withkbeing the wave number of the transmitted wave ( k=ω/c), and H(1) 0being the zero-th order Hankel function of the ﬁrst kind. The reason why we choose to use the l ine source is that it can easily used to include the usual plane wave situation; for this we just need to put th e source at a place so that k\|/vector r−/vector rs\|>>1. Due to the presence of the pressure release surface, the reﬂe ction from the surface of the direct wave can be regarded as coming from an image source located symmetrical ly about the surface, and is written as pr=−iπH(1) 0(k\|/vector r−/vector rsi\|), (2) where /vector rsiis the vector coordinate for the image, which is at the parity position about the plane. The scattered wave from the cylinder can be generally writte n as ps1=∞/summationdisplay n=−∞AnH(1) n(k\|/vector r−/vector r1\|)einφ/vector r−/vector r1, (3) where Anare the coeﬃcients to be determined later, H(1) nare the n-th order Hankel functions of the ﬁrst kind, and φis the azimuthal angle that sweeps through the plane perpend icular to the longitudinal axis of the cylinder. According to Brekhovskikh[9], the eﬀect of th e boundary on the cylinder can be represented by introducing an image cylinder located at the mirror symmetr y site about the plane surface. The rereﬂection and rescattering between the surface and the cylinder can be represented by the multiple scattering between the cylinder and its image. The scattered wave from this imag e can be similarly written as ps2=∞/summationdisplay n=−∞BnH(1) n(k\|/vector r−/vector r2\|)einφ/vector r−/vector r2, (4) where /vector r2is the location of the image of the cylinder, which is symmetr ic about the pressure-release plane. At the pressure release surface, the boundary condition req uiresps1+ps2= 0, leading to Bn=−A−n, (5) where we have used the relations φ/vector r−/vector r1=π−φ/vector r−/vector r2,andH(1) n(x) = (−1)nH(1) −n(x). Similarly the wave inside the cylinder can be written as pin=∞/summationdisplay n=−∞CnJn(k\|/vector r−/vector r1\|)einφ/vector r−/vector r1. (6) Again, Cnare the unknown coeﬃcients, and Jnare the n-th order Bessel functions of the ﬁrst kind. 2To solve for the unknown coeﬃcients An(thus Bn) and Cn, we employ the boundary conditions at the surface of the cylinder. For the purpose, we express all wave ﬁelds in the coordinates with respect to the position of the cylinder. This can be achieved by using the ad dition theorem for the Hankel functions H(1) n(k\|/vector r−/vector r′\|)einφ/vector r−/vector r′=einφ/vector r1−/vector r′∞/summationdisplay l=−∞H(1) n−l(k\|/vector r1−/vector r′\|)e−ilφ/vector r1−/vector r′Jl(k\|/vector r−/vector r1\|)eilφ/vector r−/vector r1, (7) where /vector r′can either be the location of the source by setting /vector r′=/vector rs, the location of the image of the source with/vector r′=/vector rsi, or the location of the image of the cylinder with /vector r′=/vector r2. The boundary conditions on the surface of the cylinder state that both the acoustic ﬁeld and the radial displacement be continuous across the interface. Applying the addition theorem to the express ions for the concerned waves in Eqs. (1), (2), (4), and (6), then plugging them into the boundary condition s, and after a careful calculation, we are led to the following equation Dl−∞/summationdisplay n=−∞A−nei(n−l)φ/vector r1−/vector r2H(1) n−l(k\|/vector r1−/vector r2\|) = Γ lAl, (8) where we have used Bn=−A−n. In Eq. (8), we derived Γl=−H(1) l(ka)J′ l(ka/h)−ghH(1) l′(ka)Jl(ka/h) Jl(ka)J′ l(ka/h)−ghJ′ l(ka)Jl(ka/h), (9) and Dl=iπ/bracketleftBig H(1) −l(k\|/vector r1−/vector rs\|)e−ilφ/vector r1−/vector rs−H(1) −l(k\|/vector r1−/vector rsi\|)e−ilφ/vector r1−/vector rsi/bracketrightBig . (10) The coeﬃcients Anare thus determined by a set of self-consistent equations in (8). Once Anare found, the total scattered wave can be evaluated from ps=ps1+ps2 =∞/summationdisplay n=−∞/bracketleftBig AnH(1) n(k\|/vector r−/vector r1\|)einφ/vector r−/vector r1+BnH(1) n(k\|/vector r−/vector r2\|)einφ/vector r−/vector r2/bracketrightBig . (11) In the far ﬁeld limit, r→ ∞, by expanding the Hankel functions, we have ps≈/radicalbigg 2 πreikr∞/summationdisplay n=−∞e−i(nπ/2+π/4)/bracketleftbig Ane−ik/vector r1·ˆr+Bne−ik/vector r2·ˆr/bracketrightbig einφ/vector r =/radicalbigg 2 πrQeikr, (12) where we deﬁne Q≡∞/summationdisplay n=−∞e−i(nπ/2+π/4)/bracketleftbig Ane−ik/vector r1·ˆr+Bne−ik/vector r2·ˆr/bracketrightbig einφ/vector r, (13) withBn=−A−n, as a measure of the scattering strength. B.Scattering by a cylinder of ﬁnite length In practice, we are often concerned with acoustic scatterin g by objects of ﬁnite length. Here we consider the scattering by a ﬁnite cylinder beneath a ﬂat pressure releas e surface such as the sea plane. The problem of acoustic scattering by a ﬁnite object has been diﬃcult enoug h, let alone the presence of a boundary. Exact solutions only exist for simply shaped objects. Approximat e methods have been developed. A review on various methods for computing sound scattering by an isolat ed elongated object is presented in Ref. [8]. In this section, we extend the cylinder-method proposed in R ef. [8], devised for an isolated cylinder, to the present case of a cylinder near a boundary. The reason for choosing this method is that it has been veriﬁed both theoretically and experimentally that this me thod is reasonably accurate for a wide range of situations[10, 11]. This is particularly true for the scena rios discussed in the present paper. From the Kirchhoﬀ integral theorem, the scattering functio n from any scatter can be evaluated from f(/vector r,/vector ri) =−e−ik/vector r1·ˆr 4π/integraldisplay Sds′e−ik/vector r′·ˆr/vector n·[∇r′ps(/vector r′) +ikˆrps(/vector r′)], (14) 3where /vector nis an outwardly directed unit vector normal to the surface, a nd ˆris the unit vector in the scattering direction deﬁned as ˆ r=/vector r/r. Function f(/vector r,/vector ri) refers to the scattering function for incident direction a t/vector ri implicit in the scattering ﬁeld ps(/vector r) and the scattering direction ˆ r. First we consider the scattering from the cylinder. Then in E q. (14), the ﬁeld psis the scattering ﬁeld taking values at the surface of scatterer. According to [8], this can be mimicked by that of an inﬁnite cylinder of the same radius. On the surface of the cylinder (not the ima ge), from Eq. (3) the scattered ﬁeld can be expressed as ps1=∞/summationdisplay n=−∞AnH(1) n(ka)einφ, (15) and /vector n· ∇r′ps1=∞/summationdisplay n=−∞AnkH(1) n′(ka)einφ. (16) Then the integral for the scattering function of the cylinde r, using Eq. (14), becomes fc(/vector r,/vector ri) =∞/summationdisplay n=−∞fn(/vector r,/vector ri), (17) with fn(/vector r,/vector ri) =−aLA ne−ik/vector r1·ˆr 4π/integraldisplay2π 0dφe−ikacos(φscat−φ) ×/bracketleftBig ikcos(φscat−φ)H(1) n(ka)einφ+kH(1) n′(ka)einφ/bracketrightBig , (18) where φscatis the scattering angle with respect to x−axis (i. e. φscat=φ/vector r). Using integral identities /integraldisplay2π 0dφe−ikacos(φ−φscat)einφ= 2π(−i)nJn(ka)einφ scat, (19) and/integraldisplay2π 0dφe−ikacos(φ−φscat)cos(φ−φscat)einφ= 2π(−i)niJ′ n(ka)einφ scat, (20) we can reduce Eq. (18) to fn(/vector r,/vector ri) =−kaL(−i)nAne−ik/vector r1·ˆr 2einφ scat/bracketleftBig H(1) n(ka)′Jn(ka)−H(1) n(ka)J′ n(ka)/bracketrightBig . (21) By the Wronskian identity [Jn(x)H(1) n′(x)−J′ n(x)H(1) n(x)] =2i πx, (22) Eq. (21) becomes fn(/vector r,/vector ri) =−i(−i)nLAne−ik/vector r1·ˆr πeinφ scat. (23) The scattering from the image of the cylinder can be consider ed in the same spirit. We thus obtain fi(/vector r,/vector ri) =∞/summationdisplay n=−∞−i(−i)nLBne−ik/vector r2·ˆr πeinφ scat. (24) The total scattering function is f(/vector r,/vector ri) =∞/summationdisplay n=−∞/bracketleftbigg(−i)n+1LAne−ik/vector r1·ˆr π+(−i)n+1LBne−ik/vector r2·ˆr π/bracketrightbigg einφ scat =∞/summationdisplay n=−∞/parenleftbig Ane−ik/vector r1·ˆr+Bne−ik/vector r2·ˆr/parenrightbig(−i)n+1Leinφ scat π. (25) Thereduced diﬀerential scattering cross section is σ(/vector r,/vector ri) =\|f(/vector r,/vector ri)/L\|2. (26) 4The reduced target strength is evaluated from TS = 10 log10(σ). (27) This equation bears much similarity with the scattering str ength for the inﬁnite cylinder given in Eq. (13). In the following section, we should compute the target stren gth for ﬁnite cylinders near a pressure release boundary. In particularly, we are interested in the situati on of backscattering, in which the scattering direction is opposite to the incident direction, i. e. /vector r=−/vector ri. II. NUMERICAL RESULTS Some interesting properties are found for acoustic scatter ing by a cylindrical object beneath a ﬂat pressure release plane. Two kinds of cylinders are considered: air-ﬁ lled and rigid cylinders. Let us ﬁrst consider the sound scattering by an air-ﬁlled cyl inder of length L. Although the theory developed in the last section allows the study of scattering for arbitrary incident and scattering angles, we will ﬁrst concentrate on backscattering. In addition, with out notiﬁcation we will consider the incident at an angle of π/4 with respect to the normal to the ﬂat surface. Fig. 2 shows th e reduced backscattering target strength in an arbitrary unit as a function of frequen cy in terms of the non-dimensional parameter ka. The cylinder is placed at the depths of d/a= 1,2,4,8, and 16 respectively. For comparison, the situation that the boundary is absent is also plotted. Witho ut boundary, the scattering by a single cylinder has a resonant peak at about ka= 0.005. When a ﬂat pressure-plane is added, the scattering from the cylinder will be greatly suppressed for most frequencies un der consideration, except for the resonance. At the resonance, the scattering is in fact enhanced by the pres ence of the surface. This is a unique feature for the cylinder situation. Another eﬀect of the boundary is to s hift the resonance peak of the cylinder towards higher frequencies. As the distance between the cylinder an d the surface is decreased, the position of the peak moves further towards higher frequencies, and the reso nance peak is becoming narrower and narrower. Before the resonance peak, there is a prominent dip in the sca ttering strength. For the extreme case that the cylinder touches the boundary, the signiﬁcant dip appea rs immediately before the resonance. This dip is not observed in the case of a spherical bubble beneath a bou ndary[7]. When the distance between the cylinder and the surface is inc reased, the resonance peak moves to lower frequencies until reaching that of the cylinder without a bo undary. In Fig. 3, the reduced target strength is plotted against kaford/a= 25,50,and 100. Here we see that, as the cylinder is moved further fro m the surface, regular oscillatory features appear in the sca ttering strength around the values without the boundary. The observed peaks and nulls are mainly due to inte rference eﬀects between the cylinder and the boundary, as these oscillatory features persist even wh en the multiple scattering is turned oﬀ. The nulls, appearing at some frequency intervals, are more nume rous and are spaced more closely together as the cylinder is moved away from the boundary. The peak and null st ructures are somewhat in accordance with the Lloyd’s mirror eﬀect. These features are in analogy with the results shown for the case of a spherical bubble beneath the boundary [7]. However, there is a distinc t diﬀerence. Namely, the separation between the peaks or between the nulls decreases as the frequency inc reases. We have also studied the contributions from diﬀerent oscill ation modes of a cylinder to the scattering. From Eq. (27), it is clear that the scattering is contributed from various vibration modes and the contributions are represented by the summation in which the index ndenotes the modes. We ﬁnd that when the cylinder is located far enough from the surface, the scattering is dom inated by n= 0 mode for low frequencies (e. g. ka <1); mode n= 0 is the omni-directional pulsating mode of the cylinder, i . e. its scattering is uniform in every direction. When the cylinder is moved close to the surf ace, higher vibration modes become important. These properties are illustrated in Fig. 4. For the extreme c ase that the cylinder touches the boundary as shown in Fig. 4(a), the result from including only n= 0 mode is compared with that including all modes. It is interesting to see that the eﬀect of coupling the pulsat ing mode with other modes is only to shift the resonance and dip peaks. For low frequencies away from the re sonance and the dip, the eﬀect from higher models is not evident. As the cylinder is move away from the su rface, the eﬀect of higher modes gradually decreases. For the case d/a= 4, the eﬀect of higher modes (i. e. \|n\| ≥2) virtually diminished. The eﬀects of the incident angle on the back scattering is sho wn by Fig. 5. The results show that the scattering is highly anisotropic except at the scattering d ip and peak positions; note the scale used in plotting Fig. 5. The fact that the scattering dip does not rely on the in cident angle implies that it is not caused by the Lloyd mirror eﬀect. This is because if it were due to the Ll oyd mirror eﬀect, diﬀerent incident angles would lead to diﬀerent acoustic paths in reﬂection and incid ence and thus result in diﬀerent phases, causing the scattering pattern to vary. Next we consider scattering from a rigid cylinder beneath a p ressure release boundary. For the rigid cylinder, in contrast to the air cylinder case, the scatteri ng is not so signiﬁcantly reduced by the presence 5of the surface. Instead, it is interesting that the presence of the surface in fact can enhance the scattering strength for most frequencies, except for the frequencies a t which the Lloyd eﬀect comes into function. This enhancement is particularly obvious in the low frequency re gime. Similar to the air cylinder case, when the distance is large enough, the Lloyd mirror eﬀect causes the s cattering strength to oscillate around the values without the boundary for low frequencies. Fig. 6 shows that f or low frequencies, the frequency dependence of the scattering is similar for diﬀerent distances between the cylinder and the surface. For high frequencies, e. g.ka >0.4, the multiple scattering is evident and is shown to increas e the scattering strength. The backscattering by the rigid cylinder under the boundary is anisotropic. This is illustrated in Fig. 7, which shows the backscattering target strength as a functio n ofkafor diﬀerent incidence angles. The separation between he cylinder and the surface is d/a= 4, and the incidence angle is measured with respect to the x-axis, referring to Fig. 1. For low frequencies, i. e. ka < 0.1, the scattering is strongest when the incidence is normal to the surface (i. e. for the zero degree i ncidence). Diﬀerent from the above air cylinder case, the dips in the scattering strength depend on the incid ent angles. Finally we consider the bistatic scattering. The scatterin g is in the x−yplane (See Fig. 1). We ﬁx the incident angle at 45 degree with respect to the normal to the b oundary. The scattering azimuthal angle is measured from the negative direction of the x-axis (Referring to Fig. 1). Fig. 8 shows the scattering angl e dependence of the bistatic scattering target strength for t he air ﬁlled and rigid cylinders respectively. It is interesting to see that when the frequency is low, the scatte ring tends to be symmetric around the normal to the boundary, i. e. the zero degree scattering angle, for bot h the air-ﬁlled and rigid cylinders. The scattering is strongest at the zero scattering angles. This result indi cates that when the frequency is low, the scattering from a cylinder near a boundary bears similar properties of t he acoustic radiation from a dipole source, independent of the incident angle. This feature seems again st the intuition at the ﬁrst sight, but can be understood as follows. The scattering from a target can be re garded as a second source radiating waves into the space. From, for instance, Eq. (3), we know that the radia ted wave consists of the contributions from all vibration modes of the cylinder. The mode of n= 0 is the monopole which radiates an omni-directional wave. At low frequencies, this monopole radiation dominates. In t he low frequency regime, both the cylinder and its image radiate waves but in the opposite phase. If the mono pole mode dominates, the resulting radiation should appear as that from a dipole source: the strongest rad iation is along the dipole axis. This is in fact exactly what is shown by Fig. 8. Comparing Figs. 5 with 7, howe ver, the fact that the bacskscattering relies on the incident angle indicates that the overall bistatic sc attering does depend on the incident angle. When the frequency is increased to a certain extent, the bistatic scattering pattern is no longer symmetric around the normal to the boundary. III. SUMMARY In this paper, we considered acoustic scattering by cylinde rs near a pressure-release boundary. A novel method has been developed to describe the multiple scatteri ng between the boundary and the cylinder in terms of an inﬁnite modal series. The complete solution has b een derived. Although the theory developed allows for study of various cylinders, for brevity only the c ases of air-ﬁlled and rigid cylinders are considered. The numerical results show that the presence of the boundary modiﬁes the scattering strength in various ways. One of the most signiﬁcant discoveries is that the pres ent of the surface can greatly suppress the scattering from ‘soft’ targets while may enhance rigid bodi es. In addition, comparison has been made with the previously investigated case of a spherical air-bubble beneath a pressure-release boundary. The study presented here may link to various applications such as acou stic scattering from ocean-surface dwelling ﬁsh or from any underwater elongated objects including submari ne. ACKNOWLEDGEMENT The work received support from the National Science Council . References [1] M. Strasburg, “The pulsating frequency of non-spherica l gas bubbles in liquids”, J. Acoust. Soc. Am. 25, 536-537 (1953). [2] H. N. Oguz and A. Prosperetti, “Bubble oscillation in the vicility of a nearly plane surface”, J. Acoust. Soc. Am. 87, 2085-2092 (1990). [3] I. Tolstoy, “Superresonant systems of scatterers I.”, J . Acoust. Soc. Am. 80, 282-294 (1986). 6[4] G. C. Gaunaurd and H. Huang, “Acoustic scattering by an ai r-bubble near the sea surface”, IEEE J. Ocean. Eng. 20, 285-292 (1995). [5] M. Strasburg, “Comments on ‘Acoustic scattering by an ai r-bubble near the sea surface’,”, IEEE J. Ocean. Eng. 21, 233 (1996). [6] G. C. Gaunaurd and H. Huang, “Reply to “Comments on ‘Acous tic scattering by an air-bubble near the sea surface’,”,”, IEEE J. Ocean. Eng. 21, 233 (1996). [7] Z. Ye and C. Feuillade, “Sound scattering by an air bubble near a plane sea surface”, J. Acoust. Soc. Am.102, 789-805 (1997). [8] Z. Ye, “A novel approach to sound scattering by cylinders of ﬁnite length”, J. Acoust. Soc. Am. 102, 877-884 (1997). [9] L. M. Brekhovskikh, Waves in Layered Media , (Academic, New York, 1980). [10] Z. Ye, E. Hoskinson, R. Dewey, L. Ding, and D. M. Farmer, “ A method for acoustic scattering by slender bodies. I. Theory and veriﬁcation”, J. Acoust. Soc. Am.102, 1964-1976 (1997). [11] L. Ding and Z. Ye, “A method for acoustic scattering by sl ender bodies. II. Comparison with laboratory measurements”, J. Acoust. Soc. Am. 102, 1977-1981 (1997). 7dPressure release plane xr 1r 2 yImage cylinder CylinderAcoustic source Transmitted wave Water Air Figure 1: Schematic diagram for an cylinder near a ﬂat pressu re release surface −4 −3.5 −3 −2.5 −2 −1.5 −1−400−350−300−250−200−150−100−50050 Log10(ka)TSNo Boundary d/a=1 2 4 8 16 Figure 2: Air Cylinder: Backscattering target strength ver sus frequency for various d/avalues. The incident angle is π/4. 800.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−200−150−100−50050 kaTS d/a=25 d/a=50 d/a=100 Figure 3: Air Cylinder: Backscattering target strength ver sus frequency for larger d/avalues. The incident angle is π/4. 9−4−3.5 −3−2.5 −2−1.5 −1−350−300−250−200−150−100−50050TS −4−3.5 −3−2.5 −2−1.5 −1−350−300−250−200−150−100−50050 Log10(ka)TS convergence mode=0 (a) d/a = 1 (b) d/a = 4 Figure 4: Air Cylinder: Backscattering target strength ver sus frequency for diﬀerent modes. The incident angle is π/4. 10−4 −3.5 −3 −2.5 −2 −1.5 −1−350−300−250−200−150−100−50050 Log10(ka)TS 0o 30o 45o 60o Figure 5: Air Cylinder: Backscattering target strength ver sus frequency for various incident angles. The incidence angle is measured with respect to the x-axis, referring to Fig. 1. Here d/a= 4. −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−220−200−180−160−140−120−100−80−60−40−20 Log10(ka)TS No Boundary d/a=1 d/a=16 Figure 6: Rigid cylinder: Backscattering target strength v ersus frequency for various d/a. The incident angle is π/4. 11−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−250−200−150−100−500 Log10(ka)TS 0o 30o 45o 60o Figure 7: Rigid cylinder: Backscattering target strength v ersus frequency for various incident angles with d/a= 4. 12−100 −50 0 50 100−120−100−80−60−40−20TS ka=0.01 ka=1.0 −100 −50 0 50 100−180−160−140−120−100−80−60−40−20 Angle (degree)TS ka=0.01 ka=1.0 (a) Air (b) Rigid Figure 8: Bistatic scattering target strength versus scatt ering angle for two frequencies ka= 0.01,0.1: (a) Air-ﬁlled cylinder, (b) Rigid cylinder. Here d/a= 4 and the incidence angle is 45 degree. The scattering angle is measured with respect to the negative x-axis referring to Fig. 1 13
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- 1 -On Whether People Have the Capacity to Make Observations of Mutually Exclusive Physical Phenomena Douglas M. Snyder Los Angeles, California Abstract It has been shown by Einstein, Podolsky, and Rosen that in quantum mechanics either one of two different wave functions can characterize the same physical existent, without a physical i nteraction responsible for which wave function occurs. This result means that one can make predictions regarding mutually exclusive features of a physical existent. It is important to ask whether people have the capacity to make observations of mutually exclusive phenomena? Our everyday experience informs us that a human observer is capable of observing only one set of physical circumstances at a time. Evidencefrom psychology, though, indicates that people may have the capacity to make observations of mutually exclusive physical phenomena, even though this capacity is not generally recognized. Working independently, Sigmund Freud and William James provided some of this evidence. How the nature of the quantum mechanical wave function is associated with the problem posed by Einstein, Podolsky, and Rosen is addressed at the end of the paper. Text In this paper, information has been assembled from a number of sources in physics and psychology in order to explore an issue in quantum mechanical measurement, namely the possibility of measuring mutually exclusive physical phenomena. The resolution of this issue has important implications for psychology as well as physics and, indeed, what their future relationship with one another will be. The problem that occasioned the topic ofthis paper was initially addressed by Einstein, Podolsky, and Rosen (1935). Their problem concerned the ability in quantum mechanics to make predictionsregarding quantities of a physical existent that from a physical standpoint are mutually exclusive. The question is whether human observers have the capacity to make observations that would confirm the mutually exclusive features characterizing the physical existent? The roots of Einstein, Podolsky, and RosenÕs problem, as well as the primary concern of this paper, lie in theOn Whether People - 2 -broad principles of quantum mechanics, in particular as they concern the nature of the wave function associated with a physical entity. Writings by Sigmund Freud and William James indicate that people may have the capability to make observations on mutually exclusive physical phenomena. These writings are explored. Also, current descriptions of the mental disorders discussed by James are provided as additional evidence to support FreudÕs and JamesÕs conclusions. How the nature of the quantum mechanical wave function is associated with the problem posed by Einstein, Podolsky, and Rosen is addressed at the end of the paper. QUANTUM MECHANICS Einstein, Podolsky, and Rosen showed that in quantum mechanics an individual may know quantities of a physical existent that from a physical standpoint are mutually exclusive.1 An example would be the spin angular momentum components of an electron along orthogonal spatial axes. In one situation, if the spin component of the electron along one of the axes is known precisely (e.g., the z axis), knowledge of the spin component along one of the other axes (e.g., the y axis) is completely uncertain. The spin component along the y axis has a 50-50 chance of having either of two possible values. In another situation, the spin component along the y axis can be known precisely, and the spin component along the z axis is then completely uncertain. The spin component along the z axis in this situation has a 50-50 chance of having either of two possible values. According to Einstein, Podolsky, and Rosen, either of these situations can characterize the electron. Which one does depends on an event that cannot physically affect the electron. It cannot physically affect the electron because the change in the quantum mechanical wave function occasioned by the event occurs instantaneously throughout space and is therefore not subject to thevelocity limitation of the special theory of relativity. The question arises: Do human observers have the capacity to make observations of mutually exclusive physical phenomena, such as those that characterize the electron? There are a number of points supporting the importance of human observation in quantum mechanical measurement. The first point is that a quantum mechanical measurement does not take a final form until a human 1 Bohm (1951) and then Bell (1964) detailed out Einstein, Podolsky, and RosenÕs proposal. Experimental evidence (e.g., Aspect, Dalibard, & Roger, 1982) supports this proposal.On Whether People - 3 -observer records an observation, due to the ability to consider all of the physical ÒmeasurementÓ interactions as quantum mechanical interactions (Wigner, 1961/1983; Snyder, 1992). A second point is that the occurrence of what are called ÒnegativeÓ observations indicates that a person, in his or her observational capacity, is centrally involved in making measurements in quantum mechanics (Bergquist, Hulet, Itano, & Wineland, 1986; Epstein, 1945; Nagourney, Sandberg, & Dehmelt, 1986; Renninger, 1960; Sauter, Neuhauser, Blatt & Toschek, 1986). In negative observations, there is no physical interaction in a measurement of some physical quantity, the physical existent generally changes, and yet a human observer is involved in the measurement process. Thus, the human observer is central to measurement in quantum mechanics. How can we understand that there then exists the possibility of a person making mutually exclusive observations on a physical existent in quantum mechanics? Our everyday experience informs us that a human observer is capable of observing only one set of physical circumstances at a time. EVIDENCE FROM PSYCHOLOGY Evidence from the discipline of psychology indicates that individuals indeed have the capacity to simultaneously observe mutually exclusive features of a physical existent. Research on the experiential and behavioral adaptation toinversion of incoming light can be combined with experimental scenarios from quantum mechanics involving mutually exclusive physical circumstances toshow the aforementioned capacity of human observers. It can be shown for example in analogous circumstances to that usually discussed in the Schrdinger cat gedankenexperimient that essentially SchrdingerÕs cat can both be alive and dead for different observers and very likely for the same observer (Snyder, 1992, 1993, 1995a, 1995b, 1997). There is other evidence as well that is relevant to the proposed enhanced observational capacities of humans that has not previously been brought to bear on the issues before us. The evidence in each case is not new, and it stems from the observations of keen observers of the human mind, James and Freud. The quotes that follow are long ones. Both James and Freud stated their positions very well, and the matter before them was subtle as evidenced by FreudÕs own question on the subject regarding whether the matter of his concern was real or illusory and by James noting how much easier conceptuallyOn Whether People - 4 -things would be if mutually exclusive selves and/or consciousnesses were not supported by empirical data. The psychological tendency in people to maintain a sense of wholeness and integrity is a strong one. It has been considered, for example, a hallmark of mental health. Yet the data indicate that the mind has a larger capability for maintaining diverse, or mutually exclusive, viewpoints at the same time. It is this feature of the mind that both James and Freud were concerned with in the work to be discussed. They saw evidence of it in those diagnosed as having amental illness, and they extended the results of their investigations with these individuals to those who are normal. It should be emphasized that these mutually exclusive viewpoints may exist simultaneously. Yet, they maintain some connection to each other. That is they affect one another, and it may be said that each would not exist without the other. But this is very different than stating that these viewpoints are really simply modes of expression of a unified co nsciousness. Instead they are entities existing in a common world and thus maintaining certain relations with one another, but nonetheless existing as distinct and separate entities. Near the end of his life, Freud (1940a/1964) wrote a description of key elements of psychoanalysis entitled An Outline of Psychoanalysis . In a section of this work entitled ÒThe External World,Ó he took up a topic he had begun in a paper that he had written about a few years before but which he never completed. In this earlier paper, ÒSplitting of the Ego in the Process of Defence,Ó Freud (1940b/1964) began by wondering whether the topic he was about to discuss was really of significance or whether he had explored it in depth before. This statement is quite curious. In this paper and in An Outline of Psychoanalysis , Freud attempted to grapple in a new way with the issue of there being two distinct psychological elements working within an individual, oftentimes working at odds with one another. This, of course, seems to be a basic premise of psychoanalysis and generally falls under the rubric of intrapsychic dynamics. Indeed, the direction of an individualÕs personality afterchildhood is in FreudÕs psychoanalysis determined by the relative balance of energy at the dis posal of the ego on the one hand and other psychological structures such as the id and super-ego as well as the demands put on the ego by the external world. FreudÕs curiosity though was drawn anew to this incongruity that a single individual could simultaneously have these conflicting psychologicalOn Whether People - 5 -elements. He approached the topic of mutually exclusive situations involving the mind initially from areas where this mutual exclusivity is readily apparent and then proceeded to work toward the less extreme types of mental disorder and finally to normal psychological functioning. FreudÕs discussion is compelling and points toward the need to systematically explore the potential inindividuals to simultaneously manifest mutually exclusive modes of psychological functioning. What generally passes as ÒnormalityÓ in our own experience masks these distinct modes in some sort of integration or unification. But this ÒintegrationÓ is not actually a fusing of the mutually exclusive modes. It is more of an enveloping of the modes as the ego exerts as much effort as it can to enwrap them in a skin that makes them appear integrated, consistent, and understandable. Thus the act of eating for example can express both destructive and constructive features of the mind and be seenas a single act, though these mutually exclusive features cannot reduced to a single feature of the mind. Psychopathological conditions, though, present situations where a single act does not allow for fusing, or ÒintegrationÓ of, these different psychological features and pointedly s hows that the veneer of unification in everyday experience masks the fundamental and simultanaeous presence of mutually exclusive psychological features. In The Principles of Psychology , James (1899) suggested the same point concerning ÒintegrationÓ in discussing certain behaviors of hysterics that today would be found in individuals diagnosed with Conversion Disorder and/or Dissociative Identity Disorder. The simultaneously existing, mutually exclusive modes of psychological functioning have their own perceptual systems. This being the case, FreudÕs and JamesÕs work is significant to the problem explored in this paper. Though their work was concerned largely with mental disorder, there is a current running through it indicating that the mutually exclusive modes of psychological functioning are general factors, not limited to mental disorder. The denial that may characterize one perceptual mode in some forms of mental illness turns into different kinds and levels of attention, or different forms of adaptation, concerning features of the physical world for the mutually exclusivemodes of functioning in the ÒnormalÓ mind. The mutually exclusive perceptualphenomena that the mind can support can be tied to the mutually exclusive features of a physical existent that can occur in quantum mechanics. First, a portion of ÒThe External WorldÓ from FreudÕs An Outline of Psychoanalysis is presented. His unfinished paper, ÒSplitting of the Ego inOn Whether People - 6 -DefenceÓ is then presented. From The Principles of Psychology , part of the chapter entitled ÒThe Relation of Minds to Other ThingsÓ is presented. AN OUTLINE OF PSYCHOANALYSIS : THE EXTERNAL WORLD According to Freud, the ego is the psychological structure that has the executive duties for mental functioning. It manages the conflicting demands exerted by other psychological structures and those imposed by the external world. In its executive role, the ego is responsible for rational thought as well as consciousness. We have repeatedly had to insist on the fact that the ego owes its origin as well as the most important of its acquired characteristics to its relation to the real external world. We are thus prepared to assume that the egoÕs pathological states, in which it most approximates once again to the id, are founded on cessation or slackening of that relation to the external world. This tallies very well with what we learn from clinical experience -namely, that the precipitating cause of the outbreak of psychosis is either that reality has become intolerably painful or that the instincts have become extraordinarily intensified -both of which, in view of the rival claims made on the ego by the id and the external world, must lead to the same result. (p. 201) This is a straightforward view of psychoses in which the ego is simply too weak to stand up to and manage the demands of the id or the external world and the individual simply withdraws from rational interaction with the physical world. The ego, which is the psychological structure that interfaces with the external world, is overwhelmed. The pleasure principle dominates psychological functioning according to which the individual seeks immediate gratification without distinguishing whether the source of the gratification is realor illusory. The problem of psychoses would be simple and perspicuous if the egoÕs detachment from reality could be carried through completely. (p. 201) But Freud says the situation is not so simple. Rather we get a situation like two executive structures, two egos each functioning independently of the other. Freud discussed this situation in the case of psychoses.On Whether People - 7 -But that seems to happen only rarely or perhaps never. Even in a state so far removed from the reality of the external world as one of hallucinatory confusion, one learns from patients after their recovery that at the time in some corner of their mind (as they put it) there was a normal person hidden, who, like a detached spectator, watched the hubbub of illness go past him [italics added]. I do not know if we may assume that this is so in general, but I can report the same of other p sychoses with a less tempestuous course. I call to mind a case of chronic paranoia in which after each attack of jealousy a dream conveyedto the analyst a correct picture of the precipitating cause, free from any delusion. An interesting contrast was thus brought to light: while we are accustomed to discover from the dreams of neurotics jealousies which are alien to their waking lives, in this psychotic case the delusion which dominated the patient in the day-time was corrected by his dream. We may probably take it as being generally true that what occurs in all these cases is a psychical split. Two psychical attitudes have been formed instead of a single one -one, the normal one which takes account of reality, and another which under the influence of the instincts detaches the ego from reality. The two exist alongside of each other [italics added except for ÒsplitÓ]. The issue depends on their relative strength. If the second is or becomes the stronger, the necessary precondition for a psychosis is present. If the relation is reversed, then there is an apparent cure of the delusional disorder. Actually it has only retreated into the unconscious just as numerous observations lead us to believe that the delusion existed ready-made for a long time before its manifest irruption. (. p. 201-202) Freud then extended his discussion of the splitting of the ego to other psychopathological conditions. He began by discussing fetishes. The view which postulates that in all psychoses there is a splitting of the ego could not call for so much notice if it did notturn out to apply to other states more like the neuroses and, finally, to the neuroses themselves [italics added except for Òsplitting of the egoÓ]. I first became convinced of this in casesof fetishism . This abnormality, which may be counted as one ofOn Whether People - 8 -the perversions, is, as is well known, based on the patient (who is almost always male) not recognizing the fact that females have no penis -a fact which is extremely undesirable to him since it is a proof of the possibility of his being castrated himself. He therefore disavows his own sense-perception which showed him that the female genitals lack a penis and holds fast to the contrary conviction. The disavowed perception does not, however, remain entirely without influence for, in spite of everything, he has not the courage to assert that he actually saw a penis. He takes hold of something else instead -a part of the body or some other object -and assigns it the role of the penis which he cannot do wi thout. It is usually something that he in fact saw at the moment at which he saw the female genitals, or itis something that can suitably serve as a symbolic substitute forthe penis. Now it would be incorrect to describe this process when a fetish is constructed as a splitting of the ego; it is a compromise formed with the help of displacement, such as we have been familiar with in dreams. (pp. 202-203) It is a compromise where the underlying concern to the individual is unconscious and yet there is some allowance for his concern and the perception accompanying it, namely the existence of a penis in a woman. But our observations show us still more. The creation of the fetish was due to an intention to destroy the evidence for the possibility of castration, so that fear of castration could be avoided. If females, like other living creatures, possess a penis, there is no need to tremble for the continued possession of oneÕsown penis. (p. 203) Here we have the origin of the development of two executive functions, the splitting of the ego. Also, note that there is a consistent and thorough basisfor the development of an alternative executive function. The individualÕs intention is to get rid of the possibility of castration, an attempt that cannot wholly succeed because according to Freud, this possibility is a basic ele ment of psychosexual development. Thus we arrive at the beginning of two mutuallyexclusive attitudes toward the same phenomenon. Note though that this split ego nonetheless relies on an some acknowledgement by at least one part of theego (in this case, the part of the ego that disavows the individualÕs perception)On Whether People - 9 -of the other executive function that together with the former constitute the split ego. Then Freud showed that we indeed have two executive functions that appear to be functioning independently of each other. The ÒnormalÓ executive function develops independently of the ÒabnormalÓ one. Now we come across fetishists who have developed the same fear of castration as non-fetishists and react in the same way to it. Their behaviour is therefore simultaneously expressing two contrary premisses [italics added]. On the one hand they are disavowing the fact of their perception -the fact that they saw no penis in the female genitals; and on the other hand they are recognizing the fact that females have no penis and are drawingthe correct conclusions from it. (p. 203) Note that both attitudes involve perception and thus the involvement of the ego in both is essential. The ego is split, maintaining contrary attitudes in response to the fear generated by sexual impulses seeking uninhibited expression. The two attitudes persist side by side throughout their lives without influencing each other. Here is what may rightly be called a splitting of the ego [italics added]. This circumstance also enables us to understand how it is that fetishism is so often only partially developed. It does not govern the choice of object exclusively but leaves room for a greater or lesser amount of normal sexual behaviour; sometimes, indeed, it retires into playing a modest part or is limited to a mere hint. In festishists, therefore, the detachment of the ego from the reality of the external world has never succeeded completely. (p. 203) Freud then proceeded one step further by showing how the splitting of the ego is not limited to psychoses and fetishes. He showed how in the general process of psychological development, an individual may disavow aspects of their perceptions that ameliorate some demand being made on a child. Note that these demands from the external world assume importance in large measure because of instinctual demands that are not acceptable in the external world. Freud wrote that this disavowal is at the heart of the development of the two independently functioning executive functions.On Whether People - 10 -It must not be thought that fetishism presents an exceptional case as regards a splitting of the ego; it is merely a particularly favourable subject for studying the question. Let us return to our thesis that the childish ego, under the domination of the real world, gets rid of undesirable instinctual demands by what are called repressions. We will now supplement this by further asserting that, during the same period of life, the ego often enough finds itself in the position of fending off some demand from the external world which it feels distressing and that this is effected by means of a disavowal of the perceptions which bringto knowledge this demand from reality. Disavowals of this kindoccur very often and not only with fetishists; and whenever weare in a position to study them they turn out to be half measures, incomplete attempts at detachment from reality. The disavowal is always supplemented by an acknowledgement; two contrary and independent attitudes always arise and result in the situationof there being a splitting of the ego. Once more the issue depends on which of the two can seize hold of the greater intensity. (pp. 203-204) Freud then noted again that the process of the splitting of the ego, is not uncommon to psychological development. Freud then noted the existence of distinct and opposing attitudes that are represented in behaviors of the neurotic. These are found in neurotic symptoms. The facts of this splitting of the ego, which we have just described, are neither so new nor so strange as they may at first appear. It is indeed a universal characteristic of neuroses that there are present in the subjectÕs mental life, as regards some particular beh aviour, two different attitudes, contrary to each other and independ ent of each other. In the case of neuroses, however, one of these attitudes belongs to the ego and the contrary one, which is repressed, belongs to the id [italics added]. (p. 204) Freud then noted that neurosis and, I believe, fetishism are different topographically or structurally regarding the splitting of the ego, not in terms ofprocess. He noted that they both involve compromise between two distinct andopposing attitudes without fully distinguishing what the essential difference is. Freud implied that as far as our general awareness is concerned, individualsOn Whether People - 11 -function with a unified sense of our experience and that the simultaneous existence of the distinct and opposing attitudes is not what we generally feel. Ifwe become aware of such attitudes, they generally are in a sequence, one at a time, not all at the same time. The difference between this case and the other [discussed in the previous paragraph] is essentially a topographical or structuralone, and it is not always easy to decide in an individual instancewith which of the two possibilities one is dealing. They have, however, the following important characteristic in common. Whatever the ego does in its efforts of defence, whether it seeksto disavow a portion of the real external world or whether it seeks to reject an instinctual demand from the internal world, its success is never complete and unqualified. The outcome always lies in two contrary attitudes, of which the defeated, weaker one, no less than the other, leads to psychical complications. In conclusion, it is only necessary to point out how little of all these processes becomes known to us through our conscious perception [where we act with a unified sense of experience]. (p. 204) 2 SPLITTING OF THE EGO IN THE PROCESS OF DEFENSE According to Strachey (1964), this paper was written shortly before An Outline of Psychoanalysis .3 Freud (1940b/1964) began this work by noting that he was unsure whether the conflicting attitudes in neurosis, and in normal behavior as well, are fundamentally different than that found for a splitting of the ego in psychoses and fetishes. I find myself for a moment in the interesting position of not knowing whether what I have to say should be regarded as 2 [discussed in the previous paragraph] is from the original text. 3 ÒSplitting of the Ego in the Process of DefenceÓ is presented after the text from An Outline of Psychoanalysis because in the latter work Freud extends his notion of the splitting of the ego to neurosis and and by implication to general psychological functioning. Strachey (1964) wrote that in An Outline of Psychoanalysis , Freud Òextends the application of the idea of a splitting of the ego beyond the cases of fetishism and of the psychoses to neuroses in general. Thus the topic links up with the wider question of the Ôalteration of the egoÕ which is invariably brought about by the processes of defenceÓ (p. 274).On Whether People - 12 -something long familiar and obvious or as something entirely new and puzzling. But I am inclined to think the latter. I have at last been struck by the fact that the ego of a person whom we know as a patient in analysis must, dozens of years earlier, when it was young, have behaved in a remarkable manner in certain particular situations of pressure. We can assign in general and somewhat vague terms the conditions under which this comes about, by saying that it occurs under theinfluence of a psychical trauma. I prefer to select a single sharply defined special case, though it certainly does not cover all the possible modes of causation. (p. 275) Freud began by talking about the general process of development that may lead to psychopathology. The distinguishing characteristic of psychopathology for Freud is that the instinctual demand is stronger than the capability of the ego to manage it in the face of reality. Let us suppose, then, that a childÕs ego is under the sway of a powerful instinctual demand which it is accustomed to satisfyand that it is suddenly frightened by an experience which teachesit that the continuance of this satisfaction will result in an almostintolerable real danger. It must now decide either to recognize the real danger, give way to it and renounce the instinctual satisfaction, or to disavow reality and make itself believe that there is no reason for fear, so that it may be able to retain the satisfaction. Thus there is a conflict between the demand by theinstinct and the prohibition by reality. (p. 275) Here is the prototypical developmental situation c onfronting a child where the desire for instinctual satisfaction must be managed because of perceived negative consequences from the environment that will result from thecontinuation of behavior directed toward this satisfaction. The path to psychosis lies in a strong disavowal of reality. The path to psychological maturity and to neurosis lies in reducing instinctual satisfaction. 4 4 These alternatives toward satisfying the demands of the environment or of instinct actually blend with one another in the individual. There may well be a simple denial of some event while intact reality testing is maintained. There is very often instinctual satisfaction where the primary reaction of the individual is to control it and even to minimize it, particularly its primitive expression.On Whether People - 13 -But Freud was headed somewhere else, somewhere that can only be found by a more subtle consideration of psychodynamics. Freud presents the situation where two executive agencies take different approaches toward handling the drive to instinctual expression in the face of limitations imposed bythe environment. One of these agencies appears in some ways as the ego that istoo weak to stand up to the instinctual demands. This agency develops a symptom, the fetish, that allows for some disguised primitive sexual expression. The other agency appears like the normal ego engaging in normal mature sexual expression. But this agency is anything but normal, relying on the other one to provide the Òcover,Ó a way of dealing with the fear generated by the desired instinctual expression so that this agency can go on its merry way without being aware of this fear. This situation Freud refers to as a splitting of the ego. The executive agencies are intert wined, but yet for all intents and purposes are also independent as each embodies what generally is considered either a normal, healthy ego or a neurotic one. But in fact the child takes neither course, or rather he takes both simultaneously, which comes to the same thing. He replies to the conflict with two contrary reactions, both of which are valid and effective. On the one hand, with the help of certain mechanisms he rejects reality and refuses to accept any prohibition; on the other hand, in the same breath he recognizes the danger of reality, takes over the fear of that danger as a pathological symptom and tries subsequently to divest himself of the fear. It must be confessed that this is a very ingenious solution of the difficulty. Both of the parties to the dispute obtain their share: the instinct is allowed to retain its satisfactionand proper respect is shown to reality. But everything has to be paid for in one way or another, and this success is achieved at the price of a rift in the ego which never heals but which increases as time goes on. The two contrary reactions to the conflict persist as the centre-point of a splitting of the ego. The whole process seems so strange to us because we take for granted the synthetic nature of the processes of the ego. But we are clearly at fault in this. The synthetic function of the ego, though it is of such extraordinary importance, is subject to particular conditions and is liable to a whole number of disturbances. (pp. 275-276)On Whether People - 14 -Freud then introduced the specific features of the case history he presented that illustrates how the general principles discussed in the previous quoted paragraph may be manifested. It will assist if I introduce an individual case history into this schematic disquisition. A little boy, while he was between three and four years of age, had become acquainted with the female genitals through being seduced by an older girl. After these relations had been broken off, he carried on the sexual stimulation set going in this way by zealously practicing manual masturbation; but he was soon caught at it by his energetic nurseand was threatened with castration, the carrying out of which was, as usual, ascribed to his father. There were thus present inthis case conditions calculated to produce a tremendous effect offright. A threat of castration by itself need not produce a great impression. A child will refuse to believe in it, for he cannot easily imagine the possibility of losing such a highly prized part of his body. His [earlier] sight of the female genitals might have convinced our child of that possibility. But he drew no such conclusion from it, since his disinclination to doing so was too great and there was no motive present which could compel him to. On the contrary, whatever uneasiness he may have felt was calmed by the reflection that what was missing would yet make its appearance: she would grow one (a penis) later. Anyone whohas observed enough small boys will be able to recollect having come across some such remark at the sight of a baby sisterÕs genitals. But it is different if both factors are present together. Inthat case the threat revives the memory of the perception which had hitherto been regarded as harmless and finds in that memorya dreaded confirmation. The little boy now thinks he understands why the girlÕs genitals showed no sign of a penis and no longer ventures to doubt that his own genitals may meet with the same fate. Thenceforward he cannot help believing in the reality of the danger of castration. (pp. 276-277) The usual result of the fright of castration, the result that passes as the normal one, is that, either immediately or after some considerable struggle, the boy gives way to the threat and obeys the prohibition either wholly or at least in part (that is, byOn Whether People - 15 -no longer touching his genitals with his hand). In other words, he gives up, in whole or in part, the satisfaction of the instinct. We are prepared to hear, however, that our present patient foundanother way out. He c reated a substitute for the penis which he missed in females -that is to say, a fetish. In so doing, it is true that he had disavowed reality, but he had saved his own penis. So long as he was not obliged to acknowledge that females have lost their penis, there was no need for him to believe the threat that had been made against him: he need have no fears for his own penis, so he could proceed with his mastu rbation undisturbed. (p. 277) Freud noted that the fetish involved a loosening of the tie to reality, but unlike the psychoses, involved a displacement of value only. This behaviour on the part of our patient strikes us forcibly as being a turning away from reality -a procedure which we should prefer to reserve for psychoses. And it is in fact not very different. Yet we will suspend our judgement, for upon closer inspection we shall discover a not unimportant distinction. Theboy did not simply contradict his perceptions and hallucinate a penis where there was none to be seen; he effected no more than a displacement of value -he transferred the importance of the penis to another part of the body, a procedure in which he was assisted by the mechanism of regression (in a manner which need not here be explained). This displacement, it is true, related only to the female body; as regards his own penis nothing was changed. (p. 277) Freud wrote the final part of this unfinished piece. He discussed the widely differing behaviors reflective of the functioning of two independent psychological agencies. This way of dealing with reality, which almost deserves to be described as artful, was decisive as regards the boyÕs practical behaviour. He continued with his masturbation as though it implied no danger to his penis; but at the same time, incomplete contradiction to his apparent boldness or indifference, he developed a symptom which showed that he nevertheless didrecognize the danger. He had been threatened with beingOn Whether People - 16 -castrated by his father, and immediately afterwards, simultaneously with the creation of his fetish, he developed an intense fear of his father punishing him, which it required the whole force of his masculinity to master and overcompensate. This fear of his father, too, was silent on the subject of castration: by the help of regression to an oral phase, it assumedthe form of a fear of being eaten by his father. At this point it is impossible to forget a primitive fragment of Greek mythology which tells how Kronos, the old Father God, swallowed his children and sought to swallow his youngest son Zeus like the rest, and how Zeus was saved by the craft of his mother and later on castrated his father. But we must return to our case history and add that the boy produced yet another symptom, though it was a slight one, which he has retained to this day. This was an anxious susceptibility against either of his little toes being touched, as though, in all the to and fro between disavowal and acknowledgement, it was nevertheless castration that found the clearer expression.... (pp. 277-278) Now JamesÕs work will be explored concerning the possibility of the mind simultaneously manifesting what James referred to as mutually exclusive consciousnesses. T HE PRINCIPLES OF PSYCHOLOGY Immediately before beginning his section of ÒUnconsciousness in Hysterics,Ó James (1899) wrote: a lot of curious observations made on hysterical and hypnotic subjects, which prove the existence of a highly developed consciousness in places where it has hitherto not been suspectedat all. These observations throw such a novel light upon human nature that I must give them is some detail. That at least four different and in a certain sense rival observers should agree in the same conclusion justifies us in the accepting the conclusion as true [italics added]. (p. 202) 5 5 James listed more than four individuals in the material presented here who could meet this criterion. Among them are Pierre Janet, Jules Janet, Binet, Gurney, Beaunis, Bernheim, andPitres.On Whether People - 17 -ÔUnconsciousnessÕ in Hysterics James began by describing a common symptom of severe cases of what was known as hysteria, namely some alteration of the bodyÕs sensibility that did not correspond to a meaningful organic disease pattern. One of the most constant symptoms in persons suffering from hysteric disease in its extreme forms consists in alte rations of the natural sensibility of various parts and organs of the body. Usually the alteration is in the direction of defect, or an¾sthesia. One or both eyes are blind, or color-blind, or there is hemianopsia (blindness to one half the field of view), or the field is contracted. Hearing, taste, smell may similarly disappear, in part or in totality. Still more striking are the cutaneous an¾sthesias. The old witch-finders looking for the ÔdevilÕs sealsÕ learned well the existence of those insensible patches on the skin of their victims, to which the minute physical examinations of recent medicine have but recently attracted attention again. They may be scattered anywhere, but are very apt to affect one side of the body. Not infrequently they affect an entire lateral half, from head to foot; and the insensibleskin of, say, the left side will then be found separated from the naturally sensitive skin of the right by a perfectly sharp line of demarcation down the middle of the front and back. Sometimes, most remarkable of all, the entire skin, hands, feet, face, everything, and the mucous membranes, muscles and joints so far as they can be explored, become completely insensible without the other vital functions becoming gravely disturbed. (pp. 202-203) James explained the ways known at the time that could make hysterical anesthesia disappear, in particular by noting the use of a hypnotic trance. These hysterical an¾sthesias can be made to disappear more or less completely by various odd processes. It has been recently found that magnets, plates of metal, or the electrodes of a battery, placed against the skin, have this peculiar power. And when one side is relieved in this way, the an¾sthesia is often found to have transferred itself to the opposite side, which until then was well. Whether these strange effects of magnets andOn Whether People - 18 -metals be due to their direct physiological action, or to a prior effect on the patientÕs mind (Ôexpectant attentionÕ orÔsuggestionÕ) is still a mooted question. A still better awakener of sensibility is the hypnotic trance, into which many of these patients can be very easily placed, and in which their lost sensibility not infrequently becomes entirely restored. Such returns of sensibility succeed the times of insensibility and alternate with them. (p. 203) Then James went one step further and noted that hysterical anesthesia and its absence, alt ernating in some kind of cyclical pattern, need not be the only scheme of presentation of these different states of cons ciousness. Also, these different states may be simultaneous, giving rise to JamesÕs contention that there may exist different forms of consciousnesses simultaneously which are independent of one another. But Messrs. Pierre Janet and A. Binet have shown that during the times of an¾sthesia, and coexisting with it, sensibility to the an¾esthetic parts is also there, in the form of a secondary consciousness entirely cut off from the primary or normal one, but susceptible of being tapped and made to testify to its existence in various odd ways. (p. 203) James noted that hysterics may have a very limited scope of attention and that these individuals are open to a Òmethod of distractionÓ that allows both consciousnesses to express themselves simultaneously. Chief amongst these is what M. Janet calls Ôthe method of distraction .Õ These hysterics are apt to possess a very narrow field of attention, and to be unable to think of more than onething at a time. When talking with any person they forget everything else. ÒWhen Lucie talked directly with anyone,Ó says M. Janet, Òshe ceased to be able to hear any other person. You may stand behind her, call her by name, shout abuse into her ears, without making her turn round; or place yourself before her, show her objects, touch her, etc., without attracting her notice. When finally she becomes aware of you, she thinks you have just come into the room again, and greets you accordingly. This singular forgetfulness makes her liable to tellOn Whether People - 19 -all her secrets aloud, unrestrained by the presence of unsuitable auditors.Ó (p. 203) James noted how Janet used this Òmethod of distractionÓ to show the simultaneous existence of different, independent consciousnesses. Note that here the same sense, hearing, is being used by both consciousnesses simultaneously. Now M. Janet found in several subjects like this that if he came up behind them whilst they were plunged in conversation with a third party, and addressed them in a whisper, telling them to raise their hand or perform other simple acts, they would obey the order given, although their talking intelligence was quite unconscious of receiving it. Leading them from one thing to another, he made them reply by signs to his whispered questions, and finally made them answer in writing, if a pencil were placed in their hand. The primary consciousness meanwhile went on with the conversation, entirely unaware of these performances on the handÕs part. The consciousness which presided over these latter appeared in its turn to be quite as little disturbed by the upper consciousnessÕs concerns. This proof by ÔautomaticÕ writing , of a secondary consciousnessÕs existence, is the most cogent and striking one; but a crowd of other facts prove the same thing. If I run through them rapidly, the reader will probably be convinced. (p. 204) James proceeded to discuss ways in which there could be a psychological split in perception that underlaid the simultaneous existence of mutually exclusive situations. Note the similarity to FreudÕs example of a splitting of the ego where one executive agency appears normal but is in fact notbecause of its co-existence, and relationship, with a clearly disordered executiveagency. Also, note that the sense of touch is being used by both consciousnesses simultaneously. One claimed that it cannot feel with one hand.The other relied on the sense of touch in that same hand to adjust to the object init. The apparently an¾sthetic hand of these subjects, for one thing, will often adapt itself discriminatingly to whatever object may be put into it. With a pencil it will make writing movements; into a pair of scissors it will put its fingers and willOn Whether People - 20 -open and shut them, etc., etc. The primary consciousness, so to call it, is meanwhile unable to say whether or no [sic] anything is in the hand, if the latter be hidden from sight. ÒI put a pair of eyeglasses into LonieÕs an¾sthetic hand, this hand opens it andraises it towards the nose, but half way thither it enters the field of vision of Lonie, who sees it and stops stupefied: ÔWhy,Õ says she, ÔI have an eyeglass in my left hand!ÕÓ M. Binet found a very curious sort of connection between the apparently an¾sthetic skin and the mind in some Salptrire-subjects. Things placed in the hand were not felt, but thought of (apparently in visual terms) and in no wise referred by the subject to their starting point in the handÕs sensation. A key, a knife, placed in the hand occasioned ideas of a key or a knife, but the hand felt nothing. Similarly the subject thought of the number 3, 6, etc., if the hand or finger was bent three or six times by the operator, or if he stroked it three, six, etc., times. (p. 204) James also pointed out that one consciousness may have certain ideas due to the other consciousness having had a certain experience, with the former not knowing about the latter. James provided more examples of the phenomenon of one consciousness being affected by an experience of the other without the former consciousness knowing of the latterÕs experience. In certain individuals there was found a still odder phenomenon, which reminds one of that curious idiosyncrasy ofÔcolored hearingÕ of which a few cases have been lately described with great care by foreign writers. These individuals,namely, saw the impression received by the hand, but could not feel it; and the thing seen appeared by no means associated withthe hand, but more like an independent vision, which usually interested and surprised the patient. Her hand being hidden by ascreen, she was ordered to look at another screen and to tell of any visual image which might project itself thereon. Numbers would then come, corresponding to the number of times the insensible member was raised, touched, etc. Colored lines and figures would come, corresponding to similar ones traced on thepalm; the hand itself or its fingers would come when manipulated; and finally objects placed in it would come; but onOn Whether People - 21 -the hand itself nothing would ever be felt. Of course simulation would not be hard here; but M. Binet disbelieves this (usually very shallow) explanation to be a probable one in cases in question. (p. 205) In a footnote on this last quote, James shows clearly that he maintained that more than one self may exist for a person. This whole phenomenon shows how an idea which remains itself below the threshold of a certain conscious self [italics added] may occasion associative effects therein. The skin- sensations unfelt by the patients primary consciousness awaken nevertheless their usual visual associates therein. (p. 205) James continued by explicitly pointing out differences in perception of the same stimuli among the various consciousnesses that exist at the same time for the same sensory modality as evidenced by explicit responses from the subject. The usual way in which doctors measure the delicacy of our touch is by the compass-points. Two points are normally felt as one whenever they are too close together for discrimination; but what is Ôtoo closeÕ on one part of the skin may seem very far apart on another. In the middle of the back or on the thigh, less than 3 inches may be too close; on the finger-tip a tenth of aninch is far enough apart. Now, as tested in this way, with the appeal made to the primary con sciousness, which talks through the mouth and seems to hold the field alone, a certain personÕs skin may be entirely an¾sthetic and not feel the compass-points at all; and yet this same skin will prove to have a perfectly normal sensibility if the appeal be made to that other secondary or sub-consciousness, which expresses itself automatically by writing or by movements of the hand. M. Binet, M. Pierre Janet, and M. Jules Janet have all found this. The subject, whenever touched, would signify Ôone pointÕ or Ôtwo points,Õ asaccurately as if she were a normal person. She would signify itonly by these movements; and of the movements themselves herprimary self would be as unconscious as of the facts they signified, for what the submerged consciousness makes the hand do automatically is unknown to the consciousness which uses the mouth. (pp. 205-206)On Whether People - 22 -James discussed similar phenomena to that presented for discriminative touch in the case of visual perception. Messrs. Bernheim and Pitres have also proved, by observations, too complicated to be given in this spot, that the hysterical blindness is no real blindness at all. The eye of an hysteric which is totally blind when the other or seeing eye is shut, will do its share of vision perfectly well when both eyes are open together. But even where both eyes are semi-blind from hysterical disease, the method of automatic writing proves that their perceptions exist, only cut off from communication with the upper consciousness. M. Binet has found the hand of his patients unconsciously writing down words which their eyeswere vainly endeavoring to Ôsee,Õ i.e., to bring to the upper consciousness. Their submerged consciousness was of course seeing them, or the hand could not have written as it did. Colors are similarly perceived by the sub-conscious self, which the hysterically color-blind eyes cannot bring to the normal consciousness. Pricks, burns, and pinches on the an¾sthetic skin, all unnoticed by the upper self, are recollected to have beensuffered, and complained of, as soon as the under self gets a chance to express itself by the passage of the subject into hypnotic trance. (p. 206) James then summarized the results and stated his conclusion that an individual may show mutually exclusive consciousnesses that exist simultaneously. Importantly, he noted that these consciousnesses were not aware of the same object at the same time. He called this characteristic Òcomplementary.Ó He exemplified this with the case of the post-hypnotic trancebehavior of Lucie. It must be admitted, therefore, that in certain persons, at least, the total possible consciousness may be split into parts which coexist but mutually ignore each other, and share the objects of knowledge between them. More remarkable still, theyare complementary. Give an object to one of the consciousnesses, and by that fact you remove it from the other or others [italics added]. Barring a certain common fund of information, like the command of language, etc., what the upperOn Whether People - 23 -self knows the under self is ignorant of, and vice versa . (p. 206)6 James then provided an example of the conclusion that he just drew that led to his discussion of a second self, or executive agency, besides the one that is shown publicly. M. Janet has proved this beautifully in his subject Lucie. The following experiment will serve as the type of the rest: In her trance he covered her lap with cards, each bearing a number. He then told her that on waking she should not see any card whose number was a multiple of three. This is the ordinary so- called Ôpost-hypnotic suggestion,Õ now well known, and for which Lucie was a well-adapted subject. Accordingly, when shewas awakened and asked about the papers on her lap, she counted and said she saw those only whose number was not a multiple of 3. To the 12, 18, 9, etc., she was blind. But the hand, when the sub-conscious self was interrogated by the usualmethod of engrossing the upper self in another conversation, wrote that the only cards in LucieÕs lap were those numbered 12, 18, 9, etc., and on being asked to pick up all the cards which were there, picked up these and let the others lie. Similarly when the sight of certain things was suggested to the sub-conscious Lucie, the normal Lucie suddenly became partially or totally blind. ÒWhat is the matter? I canÕt see!Ó the normal personage suddenly cried out in the midst of her conversation, when M. Janet whispered to the secondary personage to make use of her eyes. The an¾sthesiaÕs, paralyses, contractions and other irregularities from which hysterics suffer seem then to be due to the fact that their secondary personage has enriched itself by robbing the primary one of a function which the latter ought to have retained. (pp. 206-207) James then indicated how Jules Janet attempted to resolve the symptoms of a patient that were under the control of the less accessible executive structure. 6 ÒIn certain persons, at least, the total possible consciousness may be split into parts which coexist but mutually ignore each otherÓ and ÒcomplementaryÓ are italicized in the original text.On Whether People - 24 -The curative indication is evident: get at the secondary personage, by hypnotization or in whatever other way, and make her give up the eye, the sk in, the arm, or whatever the affected part may be. The normal self thereupon regains possession, sees, feels, or is able to move again. In this way M. Jules Janet easily cured the well known subject of the Salptrire, Witt., of all sorts of afflictions which, until he discovered the secret of her deeper trance, it had been difficult tosubdue. ÒCessez cette mauvaise plaisanterie,Ó he said to the secondary self -and the latter obeyed. The way in which the various personages share the stock of pos sible sensations between them seems to be amusingly illustrated in this young woman. When awake, her skin is insensible everywhere except on a zone about the arm where she habitually wears a gold bracelet. This zone has feeling; but in the deepest trance, when all the rest of her body feels, this particular zone becomes absolutely an¾sthetic. (p. 207) James provided an example of the incongruent sets of coordinated behaviors that an individual may display, providing support for the existence ofmutually exclusive consciousnesses that simultaneously exist. Sometimes the mutual ignorance of the selves leads to incidents which are strange enough. The acts and movements performed by the sub-conscious self are withdrawn from the conscious one, and the subject will do all sorts of incongruous things of which he remains quite unaware. ÒI order Lucie [by the method of distraction ] to make a pied de nez , and her hands go forthwith to the end of her nose. Asked what she is doing, she replies that she is doing nothing, and continues for a long time talking, with no apparent suspicion that her fingers are moving in front of her nose. I make her walk about the room; she continues to speak and believes herself sitting down.Ó (p. 208) James provided other examples, examples that show the degree to which an individual may go to maintain the incongruent sets of behaviors and his own witnessing such behaviors.On Whether People - 25 -M. Janet observed similar acts in a man in alcoholic delirium. Whilst the doctor was questioning him, M. J. made him by whispered suggestion walk, sit, kneel, and even lie down on his face on the floor, he all the while believing himself to be standing beside his bed. Such bizarreries sound incredible, until one has seen their like. Long ago, without understanding it, I myself saw a small example of the way in which a personÕs knowledge may be shared by the two selves. A young woman who had been writing automatically was sitting with a pencil in her hand, trying to recall at my request the name of a gentleman whom she had once seen. She could only recollect the first syllable. Her hand meanwhile, without her knowledge, wrotedown the last two syllables. In a perfectly healthy young man who can write with the planchette, I lately found the hand to be entirely an¾sthetic during the writing act; I could prick it severely without the Subject knowing the fact. The writing on the planchette , however, accused me in strong terms of hurting the hand. Pricks on the other (non-writing) hand, meanwhile, which, awakened strong protest from the young manÕs vocal organs, were denied to exist by the self which made the planchette go. (p. 208) James discussed hypnosis specifically, and the evidence that individuals who are given directions when in a hypnotic trance to engage in certain actions indeed perform these actions when they are no longer in a trance and have no recollection of the actions having been suggested to them in a trance. We get exactly similar results in the so-called post-hypnotic suggestion . It is a familiar fact that certain subjects, when told during a [hypnotic] trance to perform an act or to experience an hallucination after waking, will when the time comes, obey the command. How is the command registered? How is its performance so accurately timed? These problems were long a mystery, for the primary personality remembers nothing of the trance or the suggestion, and will often trump up an improvisedpretext for yielding to the unaccountable impulse which possesses the man so suddenly and which he cannot resist. Edmund Gurney was the first to discover, by means of automatic writing, that the secondary self is awake, keeping itsOn Whether People - 26 -attention constantly fixed on the command and watching for the signal of its execution. (pp. 208-209) James then combined post-hypnotic trance with Òautomatic writers,Ó those apparently suffering from hysteria. Certain trance-subjects who were also automatic writers, when roused from trance and put to the planchette, -not knowing then what they wro te, and having their upper attention fully engrossed by reading aloud, talking, or solving problems in mental arithmetic, -would inscribe the orders which they had received, together with notes relative to the time elapsed and the time yet to run before the execution. It is therefore to no ÔautomatismÕ in the mechanical sense that such acts are due: a self presides over them, a split-off, limited and buried, but yet afully conscious, self [italics added]. More than this, the buried self often comes to the surface and drives out the other self whilst the acts are performing. In other words, the subject lapses into trance again when the moment arrives for execution, and has no subsequent recollection of the act which he has done.Gurney and Beaunis established this fact, which has since been verified on a large scale; and Gurney also showed that the patient became suggestible again during the brief time of the performance. M. JanetÕs observations, in their turn well illustrate the phenomenon. (p. 209) We see then that James noted that there were two executive agencies, often unaware of each other and each corresponding to FreudÕs concept of ego, governing their respective psychological agencies. James used JanetÕs subject Lucie to support his point that post-hypnotic trance behavior demonstrates thesetwo executive agencies. He quoted Janet: ÒI tell Lucie to keep her arms raised after she shall have awakened. Hardly is she in the normal state, when up go her arms above her head, but she pays no attention to them. She goes, comes, converses, holding her arms high in the air. If asked what her arms are do ing, she is surprised at such a question, and says very sincerely: ÔMy hands are doing nothing;they are just like yours.Õ... I command her to weep, and when awake she really sobs, but continues in the midst of her tears toOn Whether People - 27 -talk of very gay matters. The sobbing over, there remained no trace of this grief, which seemed to have been quite sub- conscious.Ó (pp. 209-210) In the following, James expressed his own sense of the unusual character of the behavior of Lonie and Lucie. The primary self often has to invent an hallucination by which to mask and hide from its own view the deeds which the other self is enacting. Lonie 3 (M. Janet designates by numbersthe different personalities which the subject may display.) writesreal letters, whilst Lonie 1 believes that she is knitting; or Lucie 3 really comes to the doctorÕs office, whilst Lucie 1 believes herself to be at home. This is a sort of delirium. The alphabet, orthe series of numbers, when handed over to the attention of the secondary personage may for the time be lost to the normal self. Whilst the hand writes the alphabet, obediently to command, theÔsubject,Õ to her great stupefaction, finds herself unable to recall it, etc. Few things are more curious than these relations of mutual exclusion, of which all gradations exist between the several partial consciousnesses [italics added]. 7 (p. 210) James then began to discuss opinions regarding whether these mutually exclusive consciousnesses characterize those of use who are ÒnormalÓ as well as those who suffer from hysteria. The ÒnormalÓ mind to this day is generally considered to be unitary in nature integrating disparate experiences within itselfand with a cohesive sense that is called oneÕs identity. How far this splitting up of the mind into separate consciousnesses may exist in each one of us is a problem. M. Janet holds that it is only possible where there is abnormal weakness, and consequently a defect of unifying or co- ordinating power. An hysterical woman abandons part of her consciousness because she is too weak nervously to hold it together. The abandoned part meanwhile may solidify into a secondary or sub-conscious self. In a perfectly sound subject, on the other hand, what is dropped out of mind at one moment keeps coming back at the next. The whole fund of experiences 7 The text in parentheses appears as a footnote in The Principles of Psychology .On Whether People - 28 -and knowledges remains integrated, and no split-off portions of it can get organized stably enough to form subordinate selves. (p. 210) Attempting to provide further evidence for the existence of a second consciousness, or executive agency, James provided evidence for certain characteristics of the executive agency that dwelled mostly in the background. The stability, monotony, and stupidity of these latter is often very striking. The post-hypnotic sub-consciousness seems to think of nothing but the order which it last received; the cataleptic sub-consciousness, of nothing but the last position imprinted on the limb. M. Janet could cause definitely circumscribed reddening and tumefaction of the skin on two of his subjects, by suggesting to them in hypnotism the hallucination of a mustard-poultice of any special shape. ÒJÕaitout le temps pens votre sinapisme,Ó says the subject, when put back into trance after the suggestion has taken effect. A manN.,...whom M. Janet operated on at long intervals, was betweenwhiles tampered with by another operator, and when put to sleep again by M. Janet, said he was Ôtoo far away to receive orders, being in Algiers.Õ The other operator, having suggested that hallucination, had forgotten to remove it before waking the subject from his trance, and the poor passive trance- personality had stuck for weeks in the stagnant dream. LonieÕssub-conscious performances having been illustrated to a caller, by a Ô pied de nez Õ executed with her left hand in the course of conversation, when, a year later, she meets him again, up goes the same hand to her nose again, without LonieÕs normal self suspecting the fact. (pp. 210-211) James, though, appeared to differ from Janet and maintain that these mutually exclusive consciousnesses may characterize anyone. All these facts, taken together, form unquestionably the beginning of an inquiry which is destined to throw a new light into the very abysses of our nature. It is for that reason that I have cited at such length in this early chapter of the book. They prove one thing conclusively, namely, that we must never take a personÕs testimony, however sincere, that he has felt nothing, asOn Whether People - 29 -proof positive that no feeling has been there. It may have been there as part of the consciousness of a Ôsecondary personage,Õ of whose experiences the primary one whom we are consulting can naturally give no account.8 (p. 211) Next James focused on the relationship between these distinct consciousnesses. In particular, he pointed out that there is a recognition of at least one consciousness of the other whereby the former consciousness actively excludes some feature of the world from its own experience. In hypnotic subjects (as we shall see in a later chapter) just as it is the easiest thing in the world to paralyze a movement or member by simple suggestion, so it is easy to produce what is called a systematized an¾sthesia by word of command. A systematized an¾sthesia means an insensibility, not to any oneelement of things, but to some one concrete thing or class of things. The subject is made blind or deaf to a certain person in the room and to no one else, and thereupon denies that that person is present, or has spoken, etc. M. P. JanetÕs Lucie,blind to some of the numbered cards in her lap (p. 207 above), is a case in point. Now when the object is simple, like a red wafer or a black cross, the subject, although he denies that he sees it when he looks straight at it, nevertheless gets a Ônegative after-imageÕ of it when he looks away again, showing that the optical impression of it has been received. Moreover reflection shows that such a subject must distinguish the object from others like it in order to be blind to it . Make him blind to one person in the room, set all the persons in a row, and tell him to count them. He will count all but that one. But how can he tell which one not to count without recognizing who he is? In like manner, make a stroke on paper or blackboard, and tell him it is not there, and he will see nothing but the clean paper or board. Next (he not looking) surround the original stroke with other strokes exactly like it, and ask him what he sees. He will point out one by one all the new strokes, and omit the original one every time, no matter how numerous the new strokes may be, or 8 ÒWe must never take a personÕs testimony, however sincere, that he has felt nothing, as proof positive that no feeling has been there .Ó is from original text.On Whether People - 30 -in what order they are arranged. Similarly, if the original single stroke to which he is blind be doubled by a prism of some sixteen degrees placed before one of his eyes (both being kept open), he will say that he now sees one stroke, and point in the direction in which the image seen through the prism lies, ignoring still the original stroke. (pp. 211-212) Having discussed the point implied in FreudÕs writings on splitting of the ego, namely that one consciousness or ego must distinguish the other/s in order to beÒblindÓ to it, James then discussed this very important point in more detail. Obviously, then, he is not blind to the kind of stroke in the least. He is blind only to one individual stroke of that kind in a particular position on the board or paper -that is to a particular complex object; and, paradoxical as it may seem to say so, he must distinguish it with great accuracy from others like it, in order to remain blind to it when the others are brought near. He discriminates it, as a preliminary to not seeing it at all. Again, when by a prism before one eye [and only that eye] a previously invisible line [presumably through hypnosis] has been made visible to that eye, and the other eye is thereupon closed or screened, its closure makes no difference; the line still remains visible. But if then the prism be removed, the line will disappear even to the eye which a moment ago saw it, and both eyes will revert to their original blind state. We have, then, to deal in these cases neither with a blindness of the eye itself, nor with a mere failure to notice, but with something much more complex; namely, an active countingout and positive exclusion of certain objects. It is as when one ÔcutsÕ an acquaintance, ÔignoresÕ a claim, or Ôrefuses to beinfluencedÕ by a consideration. But the perceptive activity which works to this result is disconnected [italics added] from the consciousness which is personal, so to speak, to the subject, and makes of the object concerning which the suggestion is made, its own private possession and prey. (pp. 212-213) Notice that the consciousness that employs this ÒblindnessÓ to some particular feature of the world is not aware of its own blindness. These consciousnesses are mutually exclusive since one is concerned with theOn Whether People - 31 -occurrence of some experience and the other is concerned with its denial. It is really to say that two executive agencies are functioning in one mind. As if to reinforce the radical nature of the these he discussed, James wrote in a footnote: How to conceive of this state of mind is not easy. It would be much simpler to understand the process, if adding new strokes made the first one visible. There would then be two different objects apperceived as totals,-paper with one stroke, paper with many strokes; and, blind to the former, he would see all that was in the latter, because he would have apperceived it asa different total in the first instance. A process of this sort occurs sometimes (not always) when the new strokes, instead of being mere repetitions of the original one, are lines which combine with it into a total object, say a human face. The subject of the trance then may regain his sight of the line to which he had previously been blind, by seeing it aspart of the face. (p. 213) James was struggling in the above quote with the mutual exclusivity of the two different consciousnesses, implying two different executive functions. He tried to show that if some perceptual totality underlaid the phenomena he discussed, one could argue that there is but one executive mental agency and that the apparently mutually exclusive phenomena reflect are perceptual wholes of which they are part. He discussed the human face in this regard. It should be remembered that James maintained, difficult as it was for him, that distinct and different consciousnesess could simultaneously exist for the same person. James concludes by giving an example from everyday life for a normal person of the phenomenon discussed. The mother who is asleep to every sound but the stirrings of her babe, evidently has the babe-portion of her auditory sensibility systematically awake. Relatively to that, the rest of her mind is in a state of systematized an¾sthesia. That department, split off and disconnected from the sleeping part, can none the less wake the latter up in case of need. So that on the whole the quarrel between Descartes and Locke as to whether the mind ever sleeps is less near to solution than ever. On a priori speculative grounds LockeÕs view that thought and feeling may at times wholly disappear seems the more plausible.On Whether People - 32 -As glands cease to secrete and muscles to contract, so the brain should sometimes cease to carry currents, and with this minimum of its activity might well coexist a minimum of consciousness. On the other hand, we see how deceptive are appearances, and are forced to admit that a part of consciousnessmay sever its connections with other parts and yet continue to be[italics added]. On the whole it is best to abstain from a conclusion. The science of the near future will doubtless answerthis question more wisely than we can now. (p. 213) Whether JamesÕs questions concerning whether the mind sleeps has or has not been answered conclusive is secondary to the point that evidence comesfrom physics that mutually exclusive consciousnesses exist, supporting the findings of James and Freud. T HE CONTEMPORARY CONSIDERATION OF HYSTERIA In order to demonstrate that the mental disorder that James referred to as hysteria is present today, some quotes are presented from the Diagnostic and Statistical Manual of Mental Dis orders (1994), known as DSM-IV . Today, hysterical symptoms are considered within two disorders: conversion disorder and dissociative identity disorder. Conversion Disorder Following are quotes from the DSM-IV on conversion disorder. Conversion Disorder involves unexplained symptoms or deficits affecting voluntary motor or sensory function that suggest a neurological or other general medical condition. Psychological factors are judged to be associated with the symptoms or deficits.... Conversion symptoms typically do not conform to known anatomical pathways and physiological mechanisms, but insteadfollow the individualÕs conceptualization of a condition. A ÒparalysisÓ may involve inability to perform a particularmovement or to move an entire body part, rather than a deficit corresponding to patterns of motor innervation. Conversionsymptoms are often inconsistent. A ÒparalyzedÓ extremity will be moved inadvertently while dressing or when attention is directed elsewhere. If placed above the head and released, aOn Whether People - 33 -ÒparalyzedÓ arm will briefly retain its position, then fall to the side, rather than striking the head. Unacknowledged strength inantagonistic muscles, normal muscle tone, and intact reflexes may be demonstrated. An electromyogram will be normal. Difficulty swallowing will he equal with liquids and solids. Conversion ÒanesthesiaÓ of a foot or a hand may follow a so- called stocking-glove distribution with uniform (no proximal to distal gradient) loss of all sensory modalities (i.e., touch, temperature, and pain) sharply demarcated at an anatomical landmark rather than according to dermatomes. A conversionÒseizureÓ will vary from convulsion to convulsion, and paroxysmal activity will not be evident on an EEG.... Conversion symptoms are related to voluntary motor or sensory functioning and are thus referred to as Òpseudoneurological.Ó Motor symptoms or deficits include impaired coordination or balance, paralysis or localized weakness, aphonia, difficulty swallowing or a sensation of a lump in the throat, and urinary retention. Sensory symptoms ordeficits include loss of touch or pain sensation, double vision, blindness, deafness, and hallucinations. Symptoms may also include seizures or convuls ions. The more medically naive the person, the more implausible are the presenting symptoms. More sophisticated persons tend to have more subtle symptomsand deficits that may closely simulate neurological or other general medical conditions.... Reported rates of Conversion Disorder have varied widely, ranging from 11/100,000 to 300/100,000 in general population samples. It has been reported as a focus of treatment in 1%-3% of outpatient referrals to mental health clinics. ( Diagnostic and Statistical Manual of Mental Disorders , 1994, pp. 445, 452- 455) Dissociative Identity Disorder Following are quotes from the DSM-IV on dissociative identity disorder. Dissociative Identity Disorder (formerly Multiple Personality Disorder) is characterized by the presence of two or moreOn Whether People - 34 -distinct identities or personality states that recurrently take control of the individualÕs behavior accompanied by an inability to recall important personal information that is too extensive to be explained by ordinary forgetfulness.... Dissociative Identity Disorder reflects a failure to integrate various aspects of identity, memory, and consciousness. Each personality state may he experienced as if it has a distinct personal history, self-image, and identity, including a separate name. Usually there is a primary identity that carries the individualÕs given name and is passive, dependent, guilty, and depressed. The alternate identities frequently have different names and characteristics that contrast with the primary identity (e.g., are hostile, controlling, and self-destructive). Particular identities may emerge in specific circumstances and may differin reported age and gender, vocabulary, general knowledge, or predominant affect. Alternate identities are experienced as takingcontrol in sequence, one at the expense of the other, and may deny knowledge of one another, be critical of one another, or appear to be in open conflict. Occasionally, one or more powerful identities allocate time to the others. Aggressive or hostile identities may at times interrupt activities or place theothers in uncomfortable situations. Individuals with this disorder experience frequent gaps in memory for personal history, both remote and recent. The amnesia is frequently asymmetrical. The more passive identities tend to have more constricted memories, whereas the more hostile, controlling, or ÔprotectorÓ identities have more complete memories. An identity that is not in control may nonetheless gain access to consciousness by producing auditory or visual hallucinations (e.g., a voice giving instructions). Evidence of amnesia may be uncovered by reports from others who have witnessed behavior that is disavowed by the individual or by theindividualÕs own discoveries (e.g., finding items of clothing at home that the individual cannot remember having bought). There may be loss of memory not only for recurrent periods of time, but also an overall loss of biographical memory for some extended period of childhood. Transitions among identities areOn Whether People - 35 -often triggered by psychosocial st ress. The time required to switch from one identity to another is usually a matter of seconds, but, less frequently, may be gradual. The number of identities reported ranges from 2 to more than 100. Half of reported cases include individuals with 10 or fewer identities.... The sharp rise in reported cases of Dissociative Identity Disorder in the United States in recent years has been subject to very different interpretations. Some believe that the greater awareness of the diagnosis among mental health professionals has resulted in the identification of cases that were previously undiagnosed. In contrast, others believe that the syndrome has been overdiagnosed in individuals who are highly suggestible. (Diagnostic and Statistical Manual of Mental Disorders , 1994, pp. 477, 484-486) The problem discussed by Einstein, Podolsky, and Rosen on the possibility of there existing simultaneously mutually exclusive situations in thephysical world has been noted. Now the root of this problem in terms of the nature of the wave function in quantum mechanics will be discussed. H OW THE NATURE OF THE WAVE FUNCTION IN QUANTUM MECHANICS UNDERLIES THE PROBLEM POSED BY EINSTEIN , PODOLSKY , AND ROSEN How the nature of the wave function is central to the problem posed by Einstein, Podolsky, and Rosen in 1935 will be discussed through a presentation of EinsteinÕs view of the wave function in quantum mechanics. Einstein ( 1949/1969) addressed the relevant broad principles of quantum mechanics as well as the argument he developed with Podolsky and Rosen in his ÒAutobiographical Notes.Ó BohrÕs response to this problem is discussed asit is central to understanding the underlying issues. Einstein first noted that Newtonian mechanics is readily understood in terms of the realistic basis of physics. Physics is an attempt conceptually to grasp reality as it is thought independently of its being observed. In this sense one speaks of Òphysical reality.Ó In pre-quantum physics there was no doubt as to how this was to be understood. In Newton's theory reality was determined by a material point in space andOn Whether People - 36 -time [functioning in a deterministic manner independent of cognition]; in Maxwell's theory, by the field in space and time. (pp. 81, 83) Einstein (1949/1969) continued: In quantum mechanics it is not so easily seen [i.e., the realistic basis of physics]. If one asks: Does a Y-function of the quantum theory represent a real factual situation in the same sense in which this is the case of a material system of points or of an electromagnetic field, one hesitates to reply with a simple ÒyesÓ or ÒnoÓ; why? What the Y-function (at a definite time) asserts, is this: What is the probability for finding a definite physical magnitude q (or p) [of a physical system] in a definitely given interval, if I measure it at time t? The probability is here to be viewed as an empirically determinable, and therefore certainlyas a ÒrealÓ quantity which I may determine if I create the same Y-function very often and perform a q-measurement each time. But what about the single measured value of q? Did the respective individual system have this q-value even before the measurement? To this question there is no definite answer within the framework of the [existing] theory, since the measurement is a process which implies a finite disturbance of the system from the outside [generally resulting in the change inwave function that occurs immediately thro ughout space]; it would therefore be thinkable that the system obtains a definite numerical value for q (or p), i.e., the measured numerical value, only through the measurement itself. (p. 83) 9 Then Einstein presented the essence of a gedankenexperiment that he had proposed earlier with Podolsky and Rosen (Einstein, Podolsky, and Rosen, 1935). We now present...the following instance: There is to be a system which at the time t of our observation consists of two partial systems S1 and S2, which at this time are spatially separated [without limit on the separation] and (in the sense of 9 The term existing , along with the brackets that enclose it, that are found in the quote are actually part of the quoted material and not added by myself.On Whether People - 37 -the classical physics) are without significant reciprocity. The total system is to be completely described through a known Y- function Y12 in the sense of quantum mechanics. All quantum theoreticians now agree upon the following: If I make a complete measurement of S1, I get from the results of the measurement and from Y12 an entirely definite Y-function Y2 of the system S2 [immediately]. The character of Y2 then depends upon what kind of measurement I undertake on S1. Now it appears to me that one may speak of the real factual situation of the partial system S2. Of this real factual situation, we know to begin with, before the measurement of S1, even less than we know of a system described by the Y-function. But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S2 is independent of what is done with the system S1, which is spatially separated from the former. According to the type of measurement which I make of S1, I get, however, a very different Y2 for the second partial system ( Y2, Y21,...). Now, however, the real situation of S2 must be independent of what happens to S1. For the same real situation of S2 it is possible therefore to find, according to oneÕs choice, different types of Y-function. (One can escape from this conclusion only by either assuming that the measurement of S1 ((telepathically)) changes the real situation of S2 or by denying independent real situations as such to things which are spatially separated from each other. Both alternatives appear to me entirely unacceptable.) If now...physicists...accept this consideration as valid, then B [a particular physicist] will have to give up his position that the Y-function constitutes a complete description of a real factual situation. For in this case [i.e., the case of a complete description] it would be impossible that two different types of Y-functions [representing mutually exclusive situations] could be co-ordinated [simultaneously] with the identical factual situation of S2 [the same concrete physical circumstances]. (Einstein, 1949/1969, pp. 85; 87)On Whether People - 38 -BohrÕs (1935) response to Einstein, Podolsky, and RosenÕs gedankenexperiment was that there is an unavoidable interaction between the physical existent measured and the measuring instrument in their gedankenexperiment that cannot be ignored. Essentially, BohrÕs response was that the situation Einstein, Podolsky, and Rosen were referring to is quantum mechanical in its structure. That is, the structure of the gedankenexperiment presented by Einstein, Podolsky, and Rosen was based on: (1) probabilistic prediction rooted in the quantum mechanical wave function that describes the physical system, and (2) the in general immediate change throughout space of the wave function upon measurement of the physical system. Furthermore, Bohr was saying that because the situation described by Einstein, Podolsky, and Rosen is framed within the theory of quantum mechanics, their result that two very different wave functions (really two mutually exclusive views of the world) can characterize the same concrete physical circumstances in quantum mechanics is incorrect. According to Bohr, the particular interaction of the measuring apparatus and S1 is associated with a specific state of S2 upon the measurement of S1. But Einstein, Podolsky, and RosenÕs result is basically correct. Two very different wave functions can indeed characterize the same concrete physical circumstances, even if the state of S2 depends on the measurement result at S1. The velocity limitation of the special theory of relativity, the velocity of light in vacuum, is not a limiting factor in the change of the wave function throughout space when a measurement is made. Where Bohr was correct was in noting that the conception of physical reality that they indeed adopted in their gedankenexperiment was that Òphysics is an attempt conceptually to grasp reality as it is thought independently of its being observedÓ (Einstein, 1949/1969, p. 81), and this conception of physical reality is not part of quantum mechanics. Yet it is quantum mechanics that Einstein, Podolsky, and Rosen used to structure their gedankenexperiment. According to Bohr, in not allowing for the interaction between the physical existent measured and the measuring process, of which the observer is the chief component, in defining an element of physical reality, they were able to frame their argument so as to obtain the result that quantum mechanics is not a complete theory of the physical world. Thus, Bohr was correct in his criticism up to a point, and Einstein, Podolsky, and Rosen were correct without the artificial constraint of their realistic definition of the physical world and its essential independence of the physical theory describing it.On Whether People - 39 -JAMES ÕS INFLUENCE ON BOHR In the mid-1920Õs, Bohr came to believe that in quantum mechanics, certain quantities (such as the position and momentum) of certain physical existents (such as an electron) cannot both be known with arbitrary precision. He argued that in principle descriptions of these quantities are mutually exclusive. In that the mutually exclusive descriptions of these quantities could both describe the existent and as these descriptions together could simultaneously apply to the existent in pre-quantum physics, Bohr called these descriptions complementary. Bohr anchored complementarity to the physical world because, for Bohr, the mutually exclusive descriptions were determined by the concrete experimental arrangements that the physicist had selected (e.g., one experimental arrangement to measure position and another experimental arrangement to measure momentum of an electron) (Bohr, 1935). Bohr was significantly influenced by William James in his development of the concept of complementarity, a central concept in the theory of quantum mechanics. Jammer (1966/1989) argued that JamesÕs work had a significant impact on BohrÕs work in physics, specifically in his development of complementarity. Jammer noted, ÒBohr repeatedly admitted how impressed he was particularly by the psychological writings of this American philos opherÓ (p. 182). It appears that Bohr was well-acquainted with certain ideas discussed in JamesÕs The Principles of Psychology . Jammer argued that it was probably JamesÕs use of the term complementarity in the context of his discussion of work on hysteria that we have reviewed that had the major impact on Bohr. In contrast to BohrÕs above stated view of complementarity, Bohr (1934/1961) seems to have suggested that complementarity itself might fundamentally involve the fundamental structuring of perception, namely the essential separation between that which is perceived and the perceiving person. In this separation only a part of the world is accessible to the perceiving person because this person of necessity maintains a particular stance in the world. For describing our mental activity [which includes perceptions of the physical world], we require, on one hand, an objectively given content to be placed in opposition to a perceiving subject, while, on the other hand, as is already implied in such an assertion, no sharp sensation between object and subject can be maintained, since the perceiving subject also belongs to our mental content. From these circumstances follows...that aOn Whether People - 40 -complete elucidation of one and the same object may require diverse points of view which defy a unique description. (Bohr, 1934/1961, p. 96) Consider the following quote. Here Bohr appeared to attribute the extent of the realm of the measurable space upon which physics depends, and rooted in basic psychological experience, to whether a physical entity is considered part of the physical world that can be measured by an observer or instead is an extension of the human observer who is attempting to measure the physical world. It is very instructive that already in simple psychological experiences we come upon fundamental features not only of therelativistic but also of the reciprocal [complementary] view. Therelativity of our perception of motion, with which we become conversant as children when travelling by ship or by train, corresponds to common-place experiences on the reciprocal character of the perception of touch. One need only remember here the sensation, often cited by psychologists, which every one has experienced when attempting to orient himself in a darkroom by feeling with a stick. When the stick is held loosely, it appears to the sense of touch to be an object. When, however, it is held firmly, we lose the sensation that it is a foreign body, and the imp ression of touch becomes immediately localized at the point where the stick is touching the body under investigation. It would scarcely be an exaggeration to maintain,purely from psychological experiences, that the concepts of space and time by their very nature acquire a meaning only because of the possibility of neglecting the interaction with the means of measurement. (Bohr, 1934/1961, pp. 98-99) It appears that the inescapable interaction between the measuring apparatus and the physical entity measured that was at the heart of BohrÕs concept of complementarity might indeed be subsumed in the more fundamental structure of perception. In addition, with a bit more attention to JamesÕs description of hysteria, Bohr may well have recognized that Einstein, Podolsky, and RosenÕs experiment really afforded the possibility of mutually exclusive situations characterizing the same concrete physical circumstances. As discussed, JamesOn Whether People - 41 -(1899) acknowledged that the hysteric manifests Òpossible consciousnesses...[that n onetheless may] coexist....[even though you may] give an object to one of the consciousnesses, and by that fact you remove it from the other or othersÓ (p. 204). CONCLUSION Einstein, Podolsky, and Rosen showed that in quantum mechanics two different wave functions can characterize a physical existent. This raises the question whether observations of mutually exclusive physical phenomena are possible? Evidence from psychology has been presented that indicates that people may indeed have the capacity to make such observations. There is additional evidence supporting this conclusion from research on adaptation to inversion of incoming light. The evidence from psychology presented here in large part stemmed originally from the study of mental illness. Both Freud and James saw the relevance of their insights to normal mental functioning as well. One might question the usefulness of insights originally gained from a study of mental phenomena characterizing a small percentage of people. But this circumstance is not different than situations concerning physical phenomena where broadphysical principles are developed initially on the basis of evidence from physical phenomena that at least at first are not frequently encountered and seemto have little to do with understanding the vast majority of physical phenomena that we encounter in our daily lives. Thus, stellar aberration, the propagation oflight in moving media, the invariant velocity of light in vacuum, and even what appeared as the curious properties of electric and magnetic fields in EinsteinÕs day were significant factors in the development and verification of the special theory of relativity (Resnick, 1968). The odd findings that subatomic particles had wave-like properties, and that light had particle-like properties led to quantum theory, a theory much more powerful than classical physics, includingNewtonian mechanics, in understanding physical phenomena. R EFERENCES Aspect, A., Dalibard, J., and R oger, G. (1982). Experimental test of BellÕs inequalities using time-varying an alyzers. Physical Review Letters , 49, 1804-1807.On Whether People - 42 -Bergquist, J. C., Hulet, R. G., Itano, W. M., and Wineland, D. J. (1986). Observation of quantum jumps in a single atom. Physical Review Letters , 57, 1699-1702. Diagnostic and Statistical Manual of Mental Di sorders (4th ed.). (1994). (pp. 445, 452-457, 477, 484-487). Washington, DC: American Psychiatric Association. Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics , 1, 195- 200. Bohm, D. (1951). Quantum Theory . New York: Prentice-Hall. Bohr, N. (1935). Can quantum-mechanical description of nature be considered complete? Physical Review , 49, 1804-1807. Bohr, N. (1961). Atomic Theory and the Description of Nature . Cambridge, England: Cambridge University Press. (Original work published 1934) Einstein, A. (1969). Autobi ographical notes. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-scientist (Vol. 1) (pp. 1-94). La Salle, Illinois: Open Court. (Original work published 1949) Einstein, A., Podolsky, B., and Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review , 47, 777-780. Epstein, P. (1945). The reality problem in quantum mechanics. American Journal of Physics , 13, 127-136. Freud, S. (1964a). An outline of psychoanalysis. In J. Strachey (Ed. and Trans), The Standard Edition of the Complete Psychological Works of Sigmund Freud (Vol. 23, pp. 141-207). London: The Hogarth Press and the Institute of Psycho-Analysis. (Original work published 1940) Freud, S. (1964b). Splitting of the ego in the process of defence. In J. Strachey (Ed. and Trans), The Standard Edition of the Complete Psychological Works of Sigmund Freud (Vol. 23, pp. 271-278). London: The Hogarth Press and the Institute of Psycho-Analysis. (Original work published 1938) James, W. (1899). The Principles of Psychology (Vol. 1). New York: Henry Holt and Company.On Whether People - 43 -Jammer, M. (1989). The Conceptual Development of Quantum Mechanics . New York: Tomash Publishers and American Institute of Physics. (Original work published 1966) Nagourney, W., Sandberg, J., and Dehmelt, H. (1986). Shelved optical electron amplifier: ob servation of quantum jumps . Physical Review Letters , 56, 2797-2799. Renninger, M. (1960). Messungen ohne strung des me§objekts [Observations without disturbing the object]. Zeitschrift fr Physik , 158, 417-421. Resnick, R. (1967). Introduction to Special Relativity . New York: Wiley. Sauter, T., Neuhauser, W., Blatt, R. and Toschek, P. E. (1986). Observation of quantum jumps. 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arXiv:physics/0103073v1 [physics.optics] 22 Mar 2001Nonlocal reﬂection by photonic barriers G¨ unter Nimtz and Astrid Haibel Universit¨ at zu K¨ oln, II. Physikalisches Institut, Z¨ ulpicher Str.77, D-50937 K¨ oln, Germany The time behaviour of partial reﬂection by opaque photonic b arriers was measured with microwaves. It was observed that unlike the duration of partial reﬂection by dielectric sheets, the measured reﬂection duration of barr iers is independent of their length. The experimental results point to a nonlocal b ehaviour of evanescent modes at least over a distance of some ten wavelengths. I. INTRODUCTION We are used to measuring a reﬂection time determined by sheet thickness from partial reﬂection of light by a sheet of glass. The reﬂection is observed only af ter a time span corresponding to twice the layer thickness multiplied by the group velocity o f light in glass. Three hundred years ago Newton conjectured that light was composed of corpuscle s and argued in the case of partial reﬂection by two or more surfaces: Light striking the ﬁrst su rface sets oﬀ a kind of wave or ﬁeld that travels along with the light and predisposes it to reﬂec t or not reﬂect oﬀ the second surface. He called this process ’ﬁts of easy reﬂection or easy transmi ssion’ [1]. As theory and experiments have shown this is not true in the case of dielectric media wit h a real part of the refractive index. Amazingly in the case of reﬂection by the surface of an opaque photonic barrier, where the refractive index is purely imaginary, Newton’s conjecture seems to be c lose to reality: The partial reﬂection by barriers suﬀers a short and constant time delay independe nt of length. For the photonic barriers investigated here we found that the reﬂection duration equa ls the transmission time observed in photonic tunnelling experiments [2] (see Fig. 1). We are going to explain the experimental set-up and the exper iments and discuss the unexpected observation.2 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 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(a) Two one–dimensional periodic quarter w avelength hetero–structures of perspex and air which are separated by a distance of 189 mm forming a re sonant cavity (cross–section is 400×400 mm2) and (b) the evanescent air gap between a double–prism in the case of frustrated total internal reﬂection. The latter represents the analog y of one-dimensional quantum mechanical tunnelling [3]. II. EXPERIMENTAL SET-UP The experimental set-up is displayed in Fig. 2. Pulse-like m icrowave signals with a half width of 8.5 ns are transmitted from a parabolic antenna. The carrier frequency of the pulse is 9.15 GHz (λ= 32.8 mm) the frequency-band width is 80 MHz. The reﬂected signal is received with a second antenna and detected by an HP 54825 oscilloscope . The measur ement is performed asymptotically, so that any coupling between generator, detector, and devic es under test (a perspex sheet and photonic barriers) is avoided by the long optical distances (≈3m) and by uniline devices in the microwave circuit. The time resolution of the set-up is ±10 ps. To check the experimental arrangement we measured the time r esponse of partial reﬂection by the two surfaces of a perspex sheet of 0.8 m thickness. The second peak of the signal corresponding to the partial reﬂection by the back surface arrived ≈8.5 ns later than the ﬁrst one related to the front surface reﬂection. For a clear demonstration of th e partial reﬂection by the two surfaces, the intensity reﬂected by the back surface was adjusted to th at by the front side as shown in Fig. 3. This was obtained by a partial metallic coating of the perspex layer’s back surface. The measured time delay of the two peaks is in agreement with the c alculated propagation time 8.53 ns considering the refractive index n=1.6 of perspex.3 Tunnel (Barrier) ∆t = 0Detector(Oscilloscope)Detector x1 2xModulator (Signal)Generator (Carrier) FIG. 2: Experimental set-up for the periodic dielectric qua rter wavelength heterostructure to measure the group velocity. The diagram shows two resonant p hotonic barriers with 2 ×4 and 2 ×2 perspex sheets separated by the same air gap. III. PARTIAL REFLECTION BY PHOTONIC BARRIERS The investigated photonic barrier device is sketched in Fig . 1(a). It consists of two photonic lattices each 45.5 mm long which are separated by an air gap of 189 mm resulting in a total length of 280 mm. Each lattice consists of four perspex slabs each 5 m m thick in an equidistant distance of 8.5 mm air. For such a structure in the ’normal dielectric case’ we would expect a broadened pulse composed of the partial reﬂections of all the slabs similar to the signal shown in Fig. 3. The last surface reﬂected signal should be seen ≈1.9 ns after the reﬂected one by the front surface. However, t he partial reﬂection by photonic barriers revealed a strange behaviou r. If the barrier is shortened to 4 or 2 sheets (see Figs. 2 and 4) the reﬂection duration keeps const ant whereas the amplitude decreases as a result of the increasing transmission. The measured reﬂ ection duration is ≈100 ps. A back surface reﬂected signal from opaque barriers, however, has never been detected (see Fig. 4). Transmission and phase-time velocity dispersion relation s of the long barrier are displayed in Fig. 5. There are ﬁve pronounced forbidden bands separated by reson ance transmission peaks in the frequency range displayed. The phase-time velocity vϕis deﬁned by [5]:4 0.0050.010.0150.020.0250.030.0350.04 0 510 15 20 25 30Intensity [a.u.] Time [ns] FIG. 3: Partial reﬂection of a microwave pulse by a perspex sh eet. The dashed pulse is the result of reﬂection by a metallic front surface only. The layer is 40 0 mm thick, its refractive index is 1.6. The double peak is due to the superposition of reﬂection of the signal pulse by front and back surfaces. The delay of the second peak is 8.5 ns in agreem ent with the propagation time in perspex. In order to enhance reﬂection the back surface was p artially coated with a metal ﬁlm. 00.010.020.030.040.050.060.070.08 2468101214161820Intensity [a.u.] Time [ns]reflection by a mirror at barrier's end Reflection by the barrier (8 sheets) (4 sheets) (2 sheets) FIG. 4: Signals reﬂected by barriers of diﬀerent length. The largest one had a total barrier length of 280 mm, the two smaller one were recorded after the barrier was shortened to a total length of 226 mm and 199 mm, respectively . The procedure of shorteni ng is illustrated in Fig. 2. For comparison the reﬂection time of a mirror at the back surface position of the longest photonic barrier is displayed. The expected travel time between fron t and end position of 1.87 ns has been in fact measured.5 vϕ=xdϕ dω≡vgr (1) where xis the travelled distance, ϕis the phase shift, ωis the angular frequency, and vgris the group velocity. 8 9 10 11 12 Frequency [GHz]0.00.20.40.60.81.0Transmission (a)8 9 10 11 12 Frequency [GHz]0246810vϕ/c (b) FIG. 5: The graphs show the dispersion relations for the reso nant heterostructure vs frequency of Fig. 1(a). The transmission dispersion of the periodic he terostructure displays ﬁve forbidden gaps, which correspond to the photonic tunnelling regime, f or details see Ref. [4]. (b) shows the calculated phase-time velocity vϕ,cis the vacuum velocity of light. The calculated phase-time velocity equals the experimental value of the group velocit y as has been measured at 9.15 GHz. The same strange behaviour as in the case of reﬂection by phot onic lattices has been observed in the case of frustrated total internal reﬂection by a doubl e prism (Fig. 1(b)). The measured reﬂection time was 117 ±10 ps and is equal to the transmission time [2]. IV. CONCLUSIONS In measuring the reﬂection duration by photonic barriers we observed that the partial reﬂection by the back surface is an instantaneous eﬀect on the amplitude, whereas the reﬂection duration is not changed. This strange behaviour is opposite to the measured partial reﬂection by a perspex layer (see Fig. 3). The behaviour may be explained by a nonlocality of evanescent modes. As a result6 of our experiments evanescent modes constituted by ensembl es of photons behave like a quantum mechanical particle. Nonlocality and causality were inves tigated in Ref. [6] and quite recently with respect to superluminal photonic tunnelling nonlocality w as discussed by Perel’man (Ref. [7]). In our experiments the applied signal pulse had a carrier fre quency of 9.15 GHz in the center of a forbidden band gap (see Fig. 5) and a narrow 1 % frequency- band width. Consequently all frequency components of the signal were evanescent. In this case there is no ﬁnite phase-time expected nor observed for a signal inside a barrier [8, 9]. Su ch a behaviour seem to explain the experimental data of reﬂection by opaque barriers. Obviously the information on photonic barrier length is ava ilable at the front surface already. This is a property which Newton suggested erroneously to explain partial reﬂection of corpuscles by dielectric layers [1]. Evanescent modes appear to be nonloc al at least within a range of some ten wavelengths as experiments have shown in this study [7]. The distance of observing nonlocality eﬀects is limited by the exponential decay of the ﬁeld intens ity of evanescent modes, i.e. of the probability in the wave mechanical tunnelling analogy. We gratefully acknowledge discussions with H. Aichmann, P. Mittelstaedt, A. Stahlhofen, and R.-M. Vetter. We thank M. E. Perel’man for giving us the paper on his investigation prior to publication. REFERENCES [1] R. P. Feynman, QED, The strange Theory of Light and Matter , p.22, Princeton University Press, Princeton NJ (1988) [2] A. Haibel and G. Nimtz, Ann. Phys. (Leipzig) 10, number 8 (2001) [3] R. P. Feynman, R. B. Leighton, and M. Sands, The Feyman Lec tures on Physics, Addison– Wesley Publishing Company, II33–12 (1964) [4] G. Nimtz, A. Enders, and H. Spieker, J. Phys. I., France 4, 565 (1994) [5] E. Merzbacher, Quantum Mechanics, 2nd ed., John Wiley & S ons, New York (1970) 6 [6] G. C. Hegerfeldt and S. N. M. Ruijsenaars, Phys. Rev. D 22, 377 (1980) [7] M. E. Perel’man, preprint (2001) [8] Th. Hartman, J. Appl. Phys. 33, 3427 (1962)7 [9] G. Nimtz and W. Heitmann, Prog. Quantum Electronics 21, 81 (1997)
arXiv:physics/0103074v1 [physics.acc-ph] 23 Mar 2001Multi-bunch generationby thermionic gun M.Kuriki,H.Hayano, T.Naito, KEK,Tsukuba,Ibaraki, Japan K. Hasegawa,ScientiﬁcuniversityofTokyo,Noda, Chiba,Ja pan Abstract KEK-ATFisstudyingthelow-emittancemulti-bunchelec- tronbeamforthefuturelinearcollider. InATF, thermionic gun is used to generate20 buncheselectronbeam with the bunch spacing of 2.8 ns. Due to a distortion of the gun emission and the beam loading effect in the bunching sys- tem, the intensity for each bunch is not uniform by up to 40 % at the end of the injector. We have developed a sys- temtocorrectthegunemissionbypreciselycontrollingthe cathode voltage with a function generator. For the beam loading effect, we have introduced RF amplitude modula- tion on Sub Harmonic Buncher, SHB. By these technique, bunch intensity uniformity was improved and beam trans- missionforlaterbuncheswasrecoveredfrom67%to91%, but intensityforﬁrst ﬁvebunchesisstill lowerthanothers . 1 INTRODUCTION KEK-ATF is a test facility to develop the low emittance multi-bunchbeamandbeaminstrumentationtechniquefor the future linear collider. That consists from 1.5 GeV S- band linac, a beam transport line, a damping ring, and a diagnosticextractionline. In the linac, the electron beam is generated by a thermionic electron gun. Typical intensity is 1.0×1010 electron/bunch. The bunch length is compressed from 1 ns to 10 psby passing a coupleof sub-harmonicbunchers, a TW buncher, and the ﬁrst S-band accelerating structure. This area is called as injector part. After the injector part , electronenergybecomes80MeV. Theelectronbeamisthenacceleratedupto1.3GeVby8 oftheS-bandregularacceleratingsection. Onesectionhas twoacceleratingstructuresdrivenbyaklystron-modulato r. KlystronisToshibaE3712generating80MWwithapulse duration of 4.5µsRF. A peak power of 400 MW with a pulse duration of 1.0µsis obtained by SLED cavity and makesa highgradientacceleratingﬁeld,30MeV/m. 20ofbunchesseparatedby2.8nsareacceleratedbyone RF pulse. Thismulti-bunchmethodis one ofthe keytech- niqueinthelinearcollider. In April 2000, we achieved horizontal emittance 1.3× 10−9rad.m,verticalemittance 1.7×10−11rad.m(bothfor 2.0×109electron/bunch , single bunch mode )[1] which are almostourtarget. In November2000,we have started the multi-bunchop- eration. The commissioning was successfully done. Due to lack of the instrumentation device for the multi-bunch diagnostic,emittanceforeachbunchisnotmeasuredyet.2 MULTI-BUNCH BEAM GENERATION The gun assembly consists from a thermionic gun, Grid pulser,anda highvoltagegunpulser. Thethermionicgun,isatriodetype,EIMACY796. The electroncurrentis controlledbyGridbias. Tomakeamulti-bunchelectronbeamwithabunchspac- ing of 2.8 ns, 357 MHz RF signal is applied to the GUN cathode. 357MHz ECL level RF signal is ampliﬁed by a power ampliﬁer. This output has a pulse height of 400 V peak-to-peak, but the amplitude is gradually changing at the riseandfalledgeasshownin FIG.1. Rectangular signalGrid bias357MHz RF Combined pulse Figure 1: To omit the rise and fall edge of 357 MHz RF signal,arectangularsignaliscombined. Gridbiasisdeter - minedtoclipuniformmulti-bunchbeam. IftheRFsignalisdirectlyappliedtoGuncathode,bunch intensity becomesnot ﬂat. To get uniformbunches,an ad- ditional rectangular pulse is combined as shown in FIG 1. The gridbiasis determinedthat the rectangularpulseclips out the ﬂat part of the RF signal. Finally, only the ﬂat part oftheRF pulseisobtainedasrealbeamcurrent. 3 EMISSION CORRECTION FIG. 2 shows the multi-bunch beam generated by thermionicgun. Theverticalandhorizontalaxesshowtime in nsandthe beamcurrentinA respectively. Thegridbias was set to 240 V. The left side is early bunch. The beam current is measured by a current transformer which is set right after the gun exit. The current transformer measures thebeamcurrentastheinductionvoltage,sotheoutputde- cayswitha timeconstant. Intensity for the ﬁrst three bunches is still increasing. Thisbehaviorisduetothe roundedrisingedgeoftheclip- pingrectangularpulse. In addition, several bunches around 13th and 14th have lower intensity than others. A study for the gun emission [2] demonstrated that the gun response to the rectangular(ns)(A) -0.5-0.4-0.3-0.2-0.100.1 -80 -60 -40 -20 Figure 2: Multi-bunch beam measured by a current trans- former. Vertical axis shows the beam current in A. Grid biaswas240V. pulse reproducedthis dip, but the reason was not fully un- derstood. This is not any problem on the electrical circuit such as reﬂectionsignal because anydip was not observed in direct measurement of the rectangular pulse applied to the cathode. (ns)(A) -0.5-0.4-0.3-0.2-0.100.1 -80 -60 -40 -20 Figure3: Multi-bunchbeamapplyingthecorrectionsignal. Gridbiaswas240V. To correct this dip, an additional signal source was in- troduced. The correction signal is produced by a function generator, Tektronix AWG 510 which can make an arbi- trary waveform with 1 GHz clock speed. The signal is transfered to the gun high voltage station through an op- tical cable , ampliﬁed 20 W RF ampliﬁer, and combined with the main signal through a resistive power combiner. Typicalamplitudeofthecorrectionsignal is30Vwhichis roughly10%ofthemainsignalappliedtotheguncathode. FIG. 3 shows the gun output by applying the correc- tion signal. The grid bias was set to 240 V. The ﬁrst three bunches have still current lower than others, but the large dipon12-15thbunchesin FIG.2waswell compensated.4 SHBAMPLITUDE MODULATION Electronbeamgeneratedbythethermionicgunhasapprox- imately 1 ns bunch length which is larger than acceptance of S-band acceleration. A couple of 357 MHz standing waveSub-harmonicbunchers,andatravelingwaveS-band buncher are placed to gather electrons into the S-band ac- ceptance, 10−20ps. Inmulti-bunchoperation,thebunchingﬁeldisdecreased by beam induced ﬁeld, i.e. wake ﬁeld. This is the beam loading effect. Beam loading effect is larger for later bunch, so the condition becomes worse for the later bunches. Beam loadingAmplitude modulationCavity amplitudeActual cavity amplitude time Figure 4: In amplitude modulation method, amplitude of input RF is changed synchronouslywith the beam timing. Cavity RF amplitude is then gradually increased with the ﬁllingtimeasshownbythedashedline. Ontheotherhand, RF amplitude is decreased by the beam loading effect as shown by the dotted line. Totally, cavity RF amplitude is keptﬂat. To compensate the beam loading effect, we have in- troduced amplitude modulation on pulsed RF for SHBs. In amplitude modulation, the amplitude of pulsed RF is changed synchronously with the beam timing. FIG. 4 shows the beam loading compensation by amplitude mod- ulation schematically. Cavity RF amplitude is then gradu- allyincreasedwithﬁllingtimeasshownbythedashedline. On the otherhand,RF amplitudeis decreasedby thebeam loading effect as shown by the dotted line. Totally, cavity RF amplitude is kept ﬂat. The bunching quality becomes equalforallbunches. Thebeamloadingeffectalwaysdeceleratesthefollowed bunches. In ATF, the ﬁrst SHB, SHB1 is operated in de- celeration mode and the second, SHB2 is in acceleration mode. The beam loading effectively increases RF ampli- tude in SHB1 and decreasesin SHB2 . Modulationsign is thennegativeforSHB1andpositiveforSHB2. Optimization for the amplitude modulation has been done by looking beam transmission at the end of injector part. A wall current monitor is placed at the exit of the injector part to observe the beam current. FIG. 5 shows theresponseofthewallcurrentmonitortothemulti-bunch beam. The dotted and solid curvesindicate those obtained withtheconventionalpulsedRF andtheamplitudelymod- ulated pulsed RF on SHBs respectively. Transmission fornsWC signal -5051015 20 40 60 80 Figure 5: Multi-bunch beam proﬁle by wall current mon- itor. Horizontalaxis showstime in ns. Vertical axis shows wall current monitor response in V. The dotted and solid curveswere obtainedwithout and with amplitude modula- tiononSHBRF. the later buncheswas recoveredby the amplitude modula- tion. The beam loading effect affects the transmission for the later bunches, then we should investigate the bunch trans- mission to examine the beam loading effect. Since the ab- solute transmission for each bunch is hard to measure ex- actly, the intensity ratio of the early bunch and later bunch canbe usedinsteadofthe absolutetransmission. Intensityofthelastbunchismuchlowerthanothersdue to the less sharpnessof the clippingrectangularpulse. Be- cause of that, effect of the amplitude modulation should be examined by the last second bunch rather than the last bunch. Thetransmissionratioofthesecondlastbunchwas0.67 for the conventionalSHB RF and 0.91 for the amplitudely modulated SHB RF respectively. The most intense bunch was used as the reference. Improvement of the transmis- sion bythe amplitudemodulationwas24%. FIG.6showsdistributionsofbunchintensityfor6thand later bunches. The peak voltage of wall current monitor is here used instead of the real beam current. The solid and hatchedhistogramsare those with the amplitudemod- ulation and the conventional pulsed RF on SHB respec- tively. With the amplitude modulation, most bunches are distributed more than 18 V, but with the conventional RF, bunches are spread widely from 12 V to 20 V. The ampli- tudemodulationimprovedtheﬂatnessofintensityforthese later bunches. FIG.6doesnotincludetheﬁrst ﬁvebunches. Thelower intensityofthesebunchesisduetotheroundedrisingedge oftheclippingpulse. Thatwillbeoneofthemainissueon the multi-bunchoperation. 5 SUMMARY In KEK-ATF, multi-bunch beam was successfully gener- ated by a thermionic electron gun with bunch spacing ofWC output (V)0510 0 5 10 15 20 Figure 6: The horizontal axis shows bunch intensity mea- sured by wall currentmonitor. The vertical axis is number of bunches per 1.0 V. The solid and hatched histograms showthosewiththeamplitudemodulationandtheconven- tionalRF onSHB respectively. 2.8 ns. The beam already reached to the extraction line, but the emittance was not measured yet due to lack of the instrumentationformulti-bunchbeam. Intensity for each bunch is not uniform because of ; 1) gunemissionun-uniformity;2)beamloadingeffect. For gun emission problem, we have applied a correc- tion signal generated by an arbitrary function generator to Guncathode. Bunchintensityﬂatnesswassigniﬁcantlyim- proved by this emission correction. However, Gun emis- sion for ﬁrst ﬁve bunches is still lower than others. That will beoneofthemainissue infuture. For beam loading effect, we have introduced amplitude modulation on SHB RF. The amplitude modulation com- pensated the beam loading effect and recovered the beam transmissionfrom67%to91%. 6 REFERENCES [1] http://lcdev.kek.jp/ATF/ [2] M. Kuriki et al., ’Multi-bunch beam generation by Thermionic Electron Gun’,24th Linac meeting at Sapporo, 1999
arXiv:physics/0103075 23 Mar 2001HIERARCHY OF FUNDAMENTAL INTERACTIONS IN WAVE UNIVERSE A.M. Chechelnitsky, Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna,Moscow Region, Russia E’mail: ach@thsun1.jinr.ru ABSTRACT Fundamental interactions (FI) represent the core of physical World Picture, compose the basis of observed in Universe phenomena and flowing in it processes. In the frame of Wave Universe Concept (WU Concept) it is pointed to essential features of observed FI set - its hierarchy, isomorphysm, recurrence character. Hierarchy of α(k) fundamental interactions (FI) is shown by the (infinite) Homological series α(k) = χkα(0) k=...-2, -1, 0, 1, 2, ..., where χ - Fundamental parameter of hierarchy - Chechelnitsky Number χ = 3.66(6), α(0) = α =e2/ /GAB /G46/G03/G03/G03- Fine Structure constant. Available experimental data sufficiently confirm the analitical representions of the theory (WU Concept). IN SEARCH OF WORLD PICTURE. Earth and Heaven, Universe, phenomena and its constituted objects - that are invariable subjects of observation by man - Homo Sapiens and Homo Instrumentalis - in the course of the whole of history. Modern science, essentially, is only continuation of this nonexpire, all swallowing tendency. The actual science differs from the past maybe only by possibility to set up of more comlex, often, grand experiments, by possibility to use of more wide data base and array of accumulated knowledge. Setting up of more new and new experiments, of course, - is not a self - aim. Beyond all of this invariable tendency remains to create the ordered, well-proportioned World Picture (or, as say ancients - Imago Mundi), among them, and picture of early inaccessible to experiments phenomena of subatomic world. That is aim, undoubtly, worthy of man, great and, possible, completely impracticable. COMPONENTS OF MODERN PHYSICAL WORLD PICTURE. FUNDAMENTAL INTERACTIONS. Set of observed objects and pheno mena of Universe strikes an imaginations. The modern science, will be say, the Standard Model suggests, that beyond all this brightly and infinite phenomenology the resrict set only four fundamental interactions (FI) - strong, weak, electromagnetic and gravitational (interactions) stands. Nondimensional Constants of Interactions. According to experimental data, characteristic nondimensional constants of αi fundamental interactions estimate as follow αs=es2/ /GAB /G46 ∼ ∼ 1/10=0.1 αw=ew2/ /GAB /G46 ∼ ∼ 1/27 αem=eem2/ /GAB /G46 ∼ ∼ 1/137÷ ÷1/128 αg=eg2/ /GAB /G46 ∼ ∼ 4.6⋅10-40 (for the protons interaction [Perkins, 1991, p. 25), where /GAB /G03/G20/G03/G4B/G12/G15π - Planck const, e - electron charge, c - light speed. Fundamental Questions. In connection with the physical World Picture, which is described to us by the modern science - physics and cosmology, some natural questions of unbiased reseachers may be arised: # Why, according to modern physics representation, only four (not less, not more) fundamental interactions (FI) exist? # Why charges of known FI have namely these, not another values? # By what its are motivated? # If exists any causal (and analytical) connection between FI of different ranges? # Why so extremal difference between absolute values of electromagnetic and gravitational FI exists? Questions may be multipliced... Chechelnitsky A. M. Hierarchy of Fundamental Interactions in Wave Universe 2 WAVE UNIVERSE CONCEPT AND FUNDAMENTAL INTERACTIONS. Hierarchycal Structure of Universe. One of most bright, evident phenomenon of real Universe is its hierarchycal structure. In Universe obviously elementar object of matter (EOM) of different scale - atoms, stars, galaxies, etc. are observing. Matter Levels. ″ ″Wave Universe Staircase″ ″ of Matter. That incontrovertible fact takes inself natural reflection in the Wave Universe concept [Checheknitsky, 1980-1997]. Hierarchy of U(k), k=...-2.-1,0,1,2,... matter Levels form the ″Wave Universe Staircase″ of matter. In the analitical, mathematical plane this hierarchy may be represented by Homological series (GS) of characteristic parameters EOM of each Level of matter. Layers of Matter. It is evident fact too, that not all Levels of matter equally brightly represent in observations. In that sense most brightly in real Universe its are manifested only some clasters of close situated matter Levels, it will be say, Layers of matter. Most characteristic from its are # Atomic (subatomic) Layer of matter. It contain some Levels of matter, connected with populations of atoms, particles, etc. # Stellar (Star) Layer of matter. It contain some matter Levels, connected with populations of stars, planets, etc. # Galactical Layer of matter. Objects of matter - different galaxies, etc. Fundamental Parameter of Hierarchy. At 70-th in investigation of wave structure of Solar system [Chechelnitsky, 1980] it have been d iscovered significent arguments for existance of Shell structure, hierarchy and similarity - dynamical isomorphysm - of Solar system Shells. First of all, that concerned to dynamical isomorphysm of clearly observed G[1] and G[2] Shells, connecting respectively with I (Earth's) and II (Jovian) groups of planets. It was determined that arrangement of physically distinguished - elite (particularly powerful, strong - dominant) orbits of Mercury in G[1] (and Jupiter in G[2]), Venus in G[1] (and Saturn in G[2]) Shells brightly underline the similarity of geometry and dynamics of processes, flowing in these Shells, with accuracy up to the some scale factor. As the quantitative characteristics of that isomorphysm, the recalculation coefficient χ - Fundamental parameter of hierarchy (FPH) - may be used the ratio, for instance, of # (Keplerian) orbital velocities v vME/vJ=47.8721 km⋅s-1/13.0581 km⋅s-1=3.66608 ⇒ χ , vV /vSA=35.0206 km⋅s-1/9.6519 km⋅s-1=3.62836 ⇒ χ , # Sectorial velocities L LJ /LME=1.01632⋅1010 km2⋅s-1/0.27722⋅1010 km2⋅s-1=3.66608 ⇒ χ, LSA /LV=1.37498 ⋅1010 km2⋅s-1/0.37895 km2⋅s-1=3.628357⇒ χ, # Semi-major axes a aJ /aME=5.202655 AU /0.387097 AU = 13.440164 = (3.666082)2⇒ χ2, aSA/aV =9.522688 AU /0.723335 AU = 13.164975 = (3.628357)2⇒ χ2, # Orbital periods T (d - days) TJ /TME=4334.47015 d/87.96892 d = 49.272744=(3.666082)3 ⇒ χ3, TSA /TV=10733.41227d / 224.70246 d = 47.76722=(3.6283568)3 ⇒ χ3. In the published at 1980 monograph [Chechelnitsky,1980] (date of manuscript acception - 11 May 1978) this dynamical isomorphysm, similarity of geometry and dynamics of physically distinguished orbits of I (Earth's) and II (Jovian) groups were analized. According to the content of "Heuristic Analysis" division [Chechelnitsky, 1980, pp.258-263, Fig.17,18] similarity coefficient - recalculation scale coefficient of megaquants DI =LME/3=0.924⋅109 km2⋅s-1 DII =L J /3=3.388⋅109 km2⋅s-1 of L - sectorial velocities (actions, circulations) of I and II groups of planets is equal DII / DI = L J /LME = 3.66(6) ⇒ χ. It was not surprise, that transition to another Shells of Solar (planetary) system (to Trans-Pluto and Intra- Mercurian Shells) would be characterized with the same χ - Fundamental parameter of hierarchy (FPH) χ=3.66(6). Chechelnitsky A. M. Hierarchy of Fundamental Interactions in Wave Universe 3 Universality of FPH. Analysis of (mega) wave structure of physically autonomous satellite systems of Jupiter, Saturn, etc., indicated, that discovered χ Fundamental parameter of hierarchy (FPH) plays in its the similar essential role, as in the Solar (planetary) system, characterizing the hierarchy, recursion and isomorphysm of Shells. Thus, it takes shape the essentially universal character of (FPH) - its validity for the analysis of (mega) wave structure of any WDS. That corresponds to representations, connected with co-dimension principle [Chechelnitsky, 1980, p.245]: "...fundanental fact is that when we pass on to another WDS, the value of d− − [character value of sectorial velocity (action, circulation)] doesn't remain constant, but varies according scales of these systems. This fact is the consequence of co-dimension principle..." "Magic Number" ("Chechelnitsky Number", FPH) χ = 3.66(6). Role and Status of Fundamental Parameter of Hierarchy in Universe. Previous after primary publications [Chechelnitsky, 1980-1985] time and new investigations to the full extent convince the theory expectations, in particular, connected with the G[s] Shells hierarchy in each of such WDS, with the hierarchy of Levels of matter (and WDS) in Universe, with the exceptional role of the introduced in the theory c FPH [Chechelnitsky, (1978) 1980-1986]. The very brief resume of some aspects of these investigations may be formulated in frame of following short suggestion. Proposition (Role and Status of c FPH in Universe) [Chechelnitsky, (1978) 1980-1986] # The central parameter, which organizes and orders the dynamical and physycal structure, geometry, hierarchy of Universe ∗ "Wave Universe Staircase" of matter Levels, ∗ Internal structure each of real systems - wave dynamic systems (WDS) at any Levels of matter, is (manifested oneself) χ - the Fundamental Parameter Hierarchy (FPH) - nondimensional number χ = 3.66(6). # It may be expected, that investigations, can show in the full scale, that χ - FPH, generally speeking, presents and appea res everywhere - in any case, - in an extremely wide circle of dynamical relations, which reflect the geometry, dynamical structure, hierarchy of real systems of Universe. We aren't be able now and at once to appear all well-known to us relations and multiple links, in which one self the [Chechelnitsky] χ = 3.66(6) "Magic Number" manifests. We hope that all this stands (becomes) possible in due time and with new opening opportunities for the publications and communications. HIERARCHY OF MATTER LEVELS AND FUNDAMENTAL INTERACTIONS. In the frame of Wave Universe Concept (WU Concept) it may be suggested that hierarchy of α(k) fundamental interactions (FI) corresponds to hierarchy of U(k), k=...-2,-1,0,1,2,... matter Levels. In particular, claster of neighbouring, fundamental interactions corresponds to Atomic (Subatomic) Layer of matter - its some matter Levels. Among its the well-known in modern physics strong, weak, electromagnetic interactions are most brightly manifested. Hierarchy of Fundamental Interactions. Proposition (α α Hierarchy - α α Homology). Set of observed interactions, phenomena, objects dynamical structures of Universe # Connects with infinite hierarchy U(k), k=...-2.-1,0,1,2,... matter Levels, # Connects and is defined by the infinite hierarchy of Fundamental Interactions (FI) α(k), k=...-2.-1,0,1,2,..., it will be say, by α Hierarchy of FI # α Hierarchy, genarally say, is infinite, # This α Hierarchy is represented by Homological series of FI (by the α Homology) in form α(k) = χkα(0), k=...-2.-1,0,1,2,... As prime image (eponim) it is reason to choose the Fine Structure Constant (FSC) - nondimensional constant of electromagnetic interaction α(0) = αem= α =e2/ /GAB /G46 where e - electron charge, /GAB /G03- Planck constant, c - light speed. # Well-known from Standard Model - strong, weak, electromagnetic, gravitational FI belong to α Hierarchy and its are represented by α Homology in form Strong FI α(2) =αs=es2/ /GAB /G46/G03/G20/G03χ2α(0) = χ2α = χ2e2/ /GAB /G46/G03/G20/G03/G13/G11/G13/G1C/G1B/G14/G0F Weak FI α(1) =αw=ew2/ /GAB /G46/G03/G20/G03χα(0) = χα = χe2/ /GAB /G46/G03/G20/G03/G13/G11/G13/G15/G19/G1A/G18/G0F/G03 Electromagnetic FI α(0) =αem=α=e2/ /GAB /G46/G03/G20/G03/G14/G12/G14/G16/G1A/G11/G13/G16/G19/G03/G20/G03/G13/G11/G13/G13/G1A/G15/G1C/G1A Chechelnitsky A. M. Hierarchy of Fundamental Interactions in Wave Universe 4 and - for the one of near arrangeing FI in the Gravitational Layer of matter (for the electrons interaction) - Gravitational FI α(-75) =αg(-75) = (eg(-75))2/ /GAB /G46/G03/G20/G03χ-75α = χ-75e2/ /GAB /G46/G03/G20/G03/G13/G11/G16/G17/G1B/G1C/G1B/G19 ⋅10-44 THEORY AND EXPERIMENT. Stationary Values of FI Constants. In the world physical literature variety of experimental estimations of α(k) FI values is circulated. We take (spare) the special attention to those of its, which describe some asimptotical, limitational (apparently, convergent, fixed, stable) its states. For the definitness it would be later named these experimental estimations - as stationary (values). STRONG INTERACTION. Constant of Strong FI. The developed now experimental situation most brightly and evidently represents the well-known Figure of [RPP, 1997, Fig.9.2 in Division ″Lattice QCD″]. It is not difficult to point, that observed in many experiments asimptotical, limitational, apperently, stationary value of αs FI constant lie at region αs ≈ 0.098 ÷ 0.1 Let cite some αs values - result of concrete experiments, - quite corresponding to predictions of theory (WU Concept). References take from RPP # RPP, p.79: "...The result can be combined to give αs(Mz)=0.112 ± 0.002 ± 0.04... # RPP, p.80: "...A fit to ϒ, ϒ′ and ϒ " gives αs(Mz)=0.108 ± 0.001 (expt.) # RPP, p. 81: "... the Standard Model is used αs(Mz)=0.104 # RPP, p. 90: αs(Mz)=0.101 ± 0.008 # RPP, p.90: "...Nonsuper symmetric unified theories predict the low value αs(Mz)=0.073 ± 0.001±0.001 # RPP, p. 91: αs=0.101(8) # RPP, p.105 : αs=0.103 ± 0.008 It is possible another, not less effective way to the determination of the αs FI constant. Electron - Positron Annigilation to Adrons and Lepton - Lepton Decay. Dynamical Isomorphysm. Examination of process of e+e− annigilation in hadrons e+e− → hadrons is possible with idea of it "close analogy" [see, for instance, Perkins, 1991, p.271] with process of lepton- lepton decay e+e− → µ+µ− The characteristical, definitable parameter - relation of total cross sections R=σ(e+e− → hadrons)/σ(e+e− → µ+µ−), as result of many experiments on high energy e+e− collaiders , is practically stationary at E>10 Gev. This "...confirms the point character of e+e− adrons process, happened ana logous to e+e− →µ+µ− process..." [Perkins, 1991, p.255]. The experimental value of R is equal R=σ(e+e− → hadrons)/σ(e+e− → µ+µ−)=11/3. Therefore, it may be assumed the some dynamical isomorphism (similarity, "close analogy") of these two processes with accuracy to similarity parameter (recalculation parameter) R=11/3. Strong Interaction and (FPH) Constant It is not difficult to comprehend, that experimentally obtaining value of R=11/3=3.66(6) similarity parameter, in reality, completly coincides with value of χ=3.66(6) - Fundamental parameter of hierarchy R = 11/3 = 3/66(6) ⇒ χ = 3.66(6). Therefore, it may be justified the relation R=σ(e+e− → adrons)/σ(e+e− → µ+µ−) = χ = αs/αw = es2/ew2. Experiment spontaneous, directly and confidently fixed and confirm following from the theory (WU Concept) value of αs strong and αw weak FI relation. Chechelnitsky A. M. Hierarchy of Fundamental Interactions in Wave Universe 5 WEAK INTERACTION. Constant of Weak FI. It is observed comparatively wide field of αw weak FI estimations. Apparently, this connect with situation when point some "intermediate", not asimptotical evaluations, in more degree depend ing from dynamical circumstan cies of experiment (transfer momentum, etc). Influence of Model Representations. It may be suppose that incertainity in αw estimations is connected also with nonjustified propositions, limitations and links of developed theoretical models of weak FI (for instance, Weinberg-Salam model), in frame of which the experimental data are calculated and comprehended. This interesting aspect deserves the special deep discussion. Weak Angle of Mixing. Comparision with Experiment. Let take as definition the following representation for the θw - Weak angle of mixing (Weinberg angle) e/ew = sinθw It is wide used in Standard Model (Weinberg - Salam model), and θw - for the comparision of theory and experimental data. In frame of discussed WU Concept representations it is not difficult to receive numerical, theoretical representation for this angle. From the comparision ew2/e2 = sin-2θw ⇒ χ = 3.66(6) it is followed sin2θw = 1/χ = 0.272(272), sinθw = 0.522(232), θw = 31°.48. As the case of preliminary reason for reflection let take only one reference [Perkins, 1991]. In division "Assimetries of polarized electrons decay on deuterons", it is constated [p.320]: "...The final result has the form sin2θw = 0.22± 0.02 and is coordinated with previous estimations. However, if save the ρ relation of neutral and charged currents as a free parameter, then result will be another: ρ = 1.74± 0.033, sin2θw = 0.293 ± 0.033 ± 0.100, because ρ and sin2θw values are strong correlated...". From one side, it is not difficult to point the comporativeness of experimental data sin2θw = 0.22÷0.293 and WU Concept result sin2θw = 0.272. From another side, these data of experiments reflect still surviving inconsistency, sqeezed in the interpetation of experimental data. It seems, that results of experiments highly nonwillingly invide in imperatives of Standard Models - in restricting relations of developed theory. Problem is so interesting and entertaining that we intend to return to it with detailed critical analysis of developed sutiation. Weak Interaction and χ χ FPH. The Fundamental parameter hierarchy (χ FPH) plays decisive role in hierarchy of weak and electromagnetic FI. To be convinced in it, let cite data of experiment, which point another way for experimental definition of αw constant of weak FI. Dynamical Isomorphysm of Electron and Neutrino Decay (at Nucleons). Experiments with leptons (electrons and neutrino) decay on nucleons are characterized by standard functions of nucleons F1eN, F2eN - for electromagnetic, F1νN, F2νN, F3νN - for weak interactions. Accumulated information leads, in particular, to relation F2νN ≤ (18/5)F2eN Commentary [Perkins,1991, Fig.8.11] sounds as follow: "...This is the first comparision of F2νN function, mesured by neutrino - nucleon decay in CERN neutrino bearn..., with the SLAC data of F2eN function in electron - nucleon decay with the same q2... Both data sets coincide each to other, if points of electron decay are multipliced by 18/5..." Thus, it is observed the dynamical isomorphysm (similarity) of weak and electromagnetic decay processes with accuracy to similarity parameter F2νN/F2eN = r ≅18/5 It is not difficult to comprehand, that these orienting data of experiment, in reality, correspond to the χ FPH r ≅ 18/5 = 3.6 ⇒ 3.66(6). Chechelnitsky A. M. Hierarchy of Fundamental Interactions in Wave Universe 6 Another words, experimental data directly and confidently fixed the weak and electromagnetic constant FI relation F2νN/F2eN ≅ χ = 3.66(6) = αw/αem = ew2/e2, characterized by FPH χ=3.66(6). ELECTROMAGNETIC INTERACTION. Constant of Electromagnetic FI. Experimental value of αem FI in the HEP (high energy physis) is estimated as lieing in region αem ∼ ∼ 1/137 ÷ ÷ 1/128 in dependen ce of transfering momentum (with growth of transfering momentum - the αem grows - in difference of αem and αw). It is natural to consider that the well-known from QED and macroworld value of electromagnetic FI constant is fixed, stationary and is equal to αem =α =e2/ /GAB /G46/G20/G14/G12/G14/G16/G1A/G11/G13/G16/G19/G0F where α =e2/ /GAB /G46/G03- Fine Structure constant, /GAB /G03/G20/G03/G4B/G12/G15π - Planck const, e - electron charge, c - light speed. GRAVITATIONAL INTERACTION. Constant of Gravitational FI. Existing estimations of αg gravitational FI constant scarcely may be considered as based on specially created experiments. So, accepted now (inderect) experimental estimation point the value (for the protons interaction - Perkins, 1991, p.25) αg ∼ 4.6⋅10-40. In frame of WU Concept we point (more definitely) the following value of αg gravitational FI constant (for the electrons interaction) α(-75) =αg(-75) = (eg(-75))2/ /GAB /G46/G03/G20/G03χ-75α = χ-75e2/ /GAB /G46/G03/G20/G03/G16/G11/G17/G1B/G1C/G1B ⋅10-45. It corresponds to one of matter Levels, it will be say, of Gravitational Layer of matter. This is claster of near lieing Levels of matter. Of course, the ″force″ of gravitational FI, corresponding to one of matter Levels of Gravitational Layer of matter, is very small in comparision to ″forces″ of strong, weak and electromagnetic FI. Gravitational FI - Some Additional Aspects. The base for that representation for αg gravitational FI constant is the following assumption of WU Concept Proposition (Gravitation and Electromagnetism). # Gravitation and Electromagnetism (as and another FI) in Wave Universe passess by fundamental stable wave link, characterized, in particular, by properties of commensurability, stable resonance. # Fundamental constants of gravitation and electromagnetism submit to the relation e2/2Gme2 = χ75, where e, me - charge and mass of electron, G - gravitational constant. In fact, that astonishing relation points to the new theoretical (not experimental) representation of G gravitational constant over characteristic constants of microworld e, me - charge and mass of electron. Simultaneously, it opens the possibility to receive the explicit representation for the αg gravitational FI constant αg = 2Gme2/ /GAB /G46/G03⇒ eg2/ /GAB /G46/G03 and corresponding gravitational charge eg = (2G)1/2me UNKNOWN POTENTIAL POSSIBLE FI. Comparison approaches, connected with Standard Model and WU Concept, are not assist to well-being in connection with developed, prevailind representations. Evidently, that value - ″wealth″ (″power″) of FI, Described by α Homology, extremaly more, then ″wealth″ of FI, proposed by Standard Model. Its are related as ∞ :4. What must we do with this ″wealth″? It is so far remain unknown in the modern physical World Picture and, therefore, still non investigated by all available in a modern science intellectual and experimental potential. It will be reasonable treat with attention to the indications of theory, a’priori not reject a possibility of search of new laws in new areas and, at first, begin purposeful experimental investigations, connected with early unknown fundamental interactions. Nearest FI.
arXiv:physics/0103076v1 [physics.data-an] 23 Mar 2001Complexity Through Nonextensivity William Bialek1, Ilya Nemenman1, and Naftali Tishby1,2 1NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540 2School of Computer Science and Engineering, and Center for Neural Computation, Hebrew University, Jeru salem 91904, Israel (February 2, 2008) The problem of deﬁning and studying complexity of a time seri es has interested people for years. In the context of dynamical systems, Grassberger has suggeste d that a slow approach of the entropy to its extensive asymptotic limit is a sign of complexity. We in vestigate this idea further by information theoretic and statistical mechanics techniques and show th at these arguments can be made precise, and that they generalize many previous approaches to comple xity, in particular unifying ideas from the physics literature with ideas from learning and coding t heory; there are even connections of this statistical approach to algorithmic or Kolmogorov complex ity. Moreover, a set of simple axioms similar to those used by Shannon in his development of inform ation theory allows us to prove that the divergent part of the subextensive component of the entr opy is a unique complexity measure. We classify time series by their complexities and demonstrate that beyond the ‘logarithmic’ complexity classes widely anticipated in the literature there are qual itatively more complex, ‘power–law’ classes which deserve more attention. PACS The problem of quantifying complexity is very old. In- terest in the ﬁeld has been fueled by three sorts of ques- tions. First, one would like to make precise an impression that some systems, such as life on earth or a turbulent ﬂuid ﬂow, evolve toward a state of higher complexity, and one would like to classify these states; this is the realm of dynamical systems theory. Second, in choosing among diﬀerent models that describe an experiment, one wants to quantify a preference for simpler explanations or, equivalently, provide a penalty for complex models that can be weighed against the more conventional good- ness of ﬁt criteria; this type of question usually is inves- tigated in statistics. Finally, there are questions about how hard it is to compute or to describe the state of a complex system; this is the area of formal mathematics and computer science. Research in each of these three directions has given birth to numerous deﬁnitions of complexity. The usual objective is to make these deﬁnitions focused enough to be operational in particular contexts but general enough to connect with our intuitive notions. For many years the dominant candidate for a universal measure has been the mathematically rigorous notion of Kolmogorov oral- gorithmic complexity that measures (roughly) the min- imum length of a computer program that can recreate the observed time series [1]. Unfortunately there is no algorithm that can calculate the Kolmogorov complexity of all data sets. Therefore, for applications to statistics , Rissanen [2] and others have developed a new concept: stochastic complexity of the data with respect to a par- ticular class of models, which measures the shortest total description of the data and the model within the class, but cannot rule out the possibility that a diﬀerent model class could generate a shorter code. The main diﬃculty of all these approaches is that theKolmogorov complexity is closely related to the Shannon entropy, which means that it measures something closer to our intuitive concept of randomness than to the intu- itive concept of complexity [3]. A true random string can- not be compressed and hence requires a long description, yet the physical process that generates this string may be very simple. As physicists, our intuitive notions of com- plexity correspond to statements about the underlying process, and not directly to the description length or Kol- mogorov complexity: a dynamics with a predictable con- stant output (small algorithmic complexity) is as trivial as one for which the output is completely unpredictable and random (large algorithmic complexity), while really complex processes lie somewhere in between. The two extreme cases, however, have one feature in common: the entropy of the output strings (or, equiva- lently, the Kolmogorov complexity of a typical one) ei- ther is a ﬁxed constant or grows exactly linearly with the length of the strings. In both cases, corrections to the asymptotic behavior do not grow with the size of the data set . This allowed Grassberger [4] to identify the slow approach of the entropy to its extensive limit as a sign of complexity. He has proposed several functions to analyze this slow approach and studied systems that exhibited a broad range of complexity properties. To deal with the same problem, Rissanen has empha- sized strongly that ﬁtting a model to data represents an encoding of those data, or predicting future data. Shorter encodings generally mean better prediction or generaliza- tion. However, much of the code usually describes the meaningless, nongeneralizable “noise”—statistical ﬂuc- tuations within the model. Only model description is relevant to prediction, and this part of the code has been termed the model complexity [2]. While systems with model complexity of very diﬀerent types are known, the 1two extreme examples above are similar: it only takes a ﬁxed number of bits to code either a call to a random number generator or to a constant function. The present work may be viewed as expanding on the notions of subextensivity and eﬀective prediction. We construct a coherent theory that brings these ideas to- gether in an intuitive way, but nonetheless is suﬃciently general to be applied in many diﬀerent contexts. We will show that with only a little bit of work Grassberger’s def- initions may be made as mathematically precise as they are aesthetically pleasing. Finally, we will argue that the deﬁnitions are unique if one accepts a set of simple ax- ioms in the spirit of Shannon’s original work, and that these deﬁnitions relate to the usual Kolmogorov complex- ity in a straightforward way. Much of this paper follows closely a more detailed analysis in Ref. [5], to which we refer for calculation details and a thorough discussion of the relevant literature. Our path to connecting the various complexity mea- sures begins by noticing that the subextensive compo- nents of entropy identiﬁed by Grassberger in fact deter- mine the information available for making predictions. This also suggests a connection to the importance or value of information, especially in a biological or eco- nomic context: information is valuable if it can be used to guide our actions, but actions take time and hence ob- served data can be useful only to the extent that those data inform us about the state of the world at later times. It would be attractive if what we identify as “complex” in a time series were also the “useful” or “meaningful” components. While prediction may come in various forms, depend- ing on context, information theory allows us to treat all of them on the same footing. For this we only need to recognize that all predictions are probabilistic, and that , even before we look at the data, we know that certain fu- tures are more likely than others. This knowledge can be summarized by a prior probability distribution for the futures. Our observations on the past lead us to a new, more tightly concentrated distribution, the distri- bution of futures conditional on the past data. Diﬀerent kinds of predictions are diﬀerent slices through or aver- ages over this conditional distribution, but information theory quantiﬁes the “concentration” of the distribution without making any commitment as to which averages will be most interesting. Imagine that we observe a stream of data x(t) over a time interval −T < t < 0; let all of these past data be denoted by the shorthand xpast. We are interested in saying something about the future, so we want to know about the data x(t) that will be observed in the time interval 0 < t < T′; let these future data be called xfuture. In the absence of any other knowledge, futures are drawn from the probability distribution P(xfuture), while observations of particular past data xpasttell us that futures will be drawn from the conditional distri-bution P(xfuture\|xpast). The greater concentration of the conditional distribution can be quantiﬁed by the fact that it has smaller entropy than the prior distribution, and this reduction in entropy is Shannon’s deﬁnition of the information that the past provides about the future. We can write the average of this predictive information as Ipred(T, T′) =/angbracketleftBigg log2/bracketleftbiggP(xfuture\|xpast) P(xfuture)/bracketrightbigg/angbracketrightBigg (1) =−∝an}bracketle{tlog2P(xfuture)∝an}bracketri}ht − ∝an}bracketle{tlog2P(xpast)∝an}bracketri}ht −[−∝an}bracketle{tlog2P(xfuture, xpast)∝an}bracketri}ht],(2) where ∝an}bracketle{t· · ·∝an}bracketri}htdenotes an average over the joint distribution of the past and the future, P(xfuture, xpast). Each of the terms in Eq. (2) is an entropy. Since we are interested in predictability or generalization, which are associated with some features of the signal persist- ing forever, we may assume stationarity or invariance under time translations. Then the entropy of the past data depends only on the duration of our observations, so we can write −∝an}bracketle{tlog2P(xpast)∝an}bracketri}ht=S(T), and by the same argument −∝an}bracketle{tlog2P(xfuture)∝an}bracketri}ht=S(T′). Finally, the entropy of the past and the future taken together is the entropy of observations on a window of duration T+T′, so that −∝an}bracketle{tlog2P(xfuture, xpast)∝an}bracketri}ht=S(T+T′). Putting these equations together, we obtain Ipred(T, T′) =S(T) +S(T′)−S(T+T′). (3) In the same way that the entropy of a gas at ﬁxed den- sity is proportional to the volume, the entropy of a time series (asymptotically) is proportional to its duration, s o that lim T→∞S(T)/T=S0; entropy is an extensive quan- tity. But from Eq. (3) any extensive component of the entropy cancels in the computation of the predictive in- formation: predictability is a deviation from extensivity . If we write S(T) =S0T+S1(T), (4) then Eq. (3) tells us that the predictive information is related onlyto the nonextensive term S1(T). We know two general facts about the behavior of S1(T). First, the corrections to extensive behavior are positive, S1(T)≥0. Second, the statement that entropy is extensive is the statement that the limit limT→∞S(T)/T=S0exists, and for this to be true we must also have lim T→∞S1(T)/T= 0.Thus the nonex- tensive terms in the entropy must be subextensive, that is they must grow with Tless rapidly than a linear func- tion. Taken together, these facts guarantee that the pre- dictive information is positive and subextensive. Further , if we let the future extend forward for a very long time, T′→ ∞, then we can measure the information that our sample provides about the entire future, Ipred(T) = lim T′→∞Ipred(T, T′) =S1(T), (5) 2and this is precisely equal to the subextensive entropy. If we have been observing a time series for a (long) timeT, then the total amount of data we have collected in is measured by the entropy S(T), and at large Tthis is given approximately by S0T. But the predictive infor- mation that we have gathered cannot grow linearly with time, even if we are making predictions about a future which stretches out to inﬁnity. As a result, of the total information we have taken in by observing xpast, only a vanishing fraction is of relevance to the prediction: lim T→∞Predictive Information Total Information=Ipred(T) S(T)→0.(6) In this precise sense, most of what we observe is irrele- vant to the problem of predicting the future. Since the average Kolmogorov complexity of a time series is related to its (total) Shannon entropy, this result means also that most of the algorithm that is required to encode the data encodes aspects of the data that are useless for prediction or for guiding our actions based on the data. This is a strong indication that the usual notions of Kolmogorov complexity in fact do not capture anything at all like the (intuitive) utility of the data stream. Consider the case where time is measured in discrete steps, so that we have seen Ntime points x1, x2,· · ·, xN. How much is there to learn about the underlying pattern in these data? In the limit of large number of observa- tions, N→ ∞ orT→ ∞ the answer to this question is surprisingly universal: predictive information may eithe r stay ﬁnite, or grow to inﬁnity together with T; in the latter case the rate of growth may be slow (logarithmic) or fast (sublinear power). The ﬁrst possibility, lim T→∞Ipred(T) = constant, means that no matter how long we observe we gain only a ﬁnite amount of information about the future. This sit- uation prevails, in both extreme cases mentioned above. For example, when the dynamics are too regular, such as it is for a purely periodic system, complete prediction is possible once we know the phase, and if we sample the data at discrete times this is a ﬁnite amount of informa- tion; longer period orbits intuitively are more complex and also have larger Ipred, but this doesn’t change the limiting behavior lim T→∞Ipred(T) = constant. Similarly, the predictive information can be small when the dynamics are irregular but the best predictions are controlled only by the immediate past, so that the corre- lation times of the observable data are ﬁnite. This hap- pens, for example, in many physical systems far away from phase transitions. Imagine, for example, that we observe x(t) at a series of discrete times {tn}, and that at each time point we ﬁnd the value xn. Then we always can write the joint distribution of the Ndata points as a product, P(x1, x2,· · ·, xN) =P(x1)P(x2\|x1)P(x3\|x2, x1)· · ·.(7)For Markov processes, what we observe at tndepends only on events at the previous time step tn−1, so that P(xn\|{x1≤i≤n−1}) =P(xn\|xn−1), (8) and hence the predictive information reduces to Ipred=/angbracketleftBigg log2/bracketleftbiggP(xn\|xn−1) P(xn)/bracketrightbigg/angbracketrightBigg . (9) The maximum possible predictive information in this case is the entropy of the distribution of states at one time step, which in turn is bounded by the logarithm of the number of accessible states. To approach this bound the system must maintain memory for a long time, since the predictive information is reduced by the entropy of the transition probabilities. Thus systems with more states and longer memories have larger values of Ipred. More interesting are those cases in which Ipred(T) di- verges at large T. In physical systems we know that there are critical points where correlation times become inﬁnite, so that optimal predictions will be inﬂuenced by events in the arbitrarily distant past. Under these condi- tions the predictive information can grow without bound asTbecomes large; for many systems the divergence is logarithmic, Ipred(T→ ∞)∝logT. Long range correlation also are important in a time se- ries where we can learn some underlying rules. Suppose a series of random vector variables {/vector xi}are drawn inde- pendently from the same probability distribution Q(/vector x\|α), and this distribution depends on a (potentially inﬁnite dimensional) vector of parameters α. The parameters are unknown, and before the series starts they are cho- sen randomly from a distribution P(α). In this set- ting, at least implicitly, our observations of {/vector xi}pro- vide data from which we can learn the parameters α. Here we put aside (for the moment) the usual problem of learning—which might involve constructing some es- timation or regression scheme that determines a “best ﬁt”αfrom the data {/vector xi}—and treat the ensemble of data streams P[{/vector xi}] as we would any other set of con- ﬁgurations in statistical mechanics or dynamical systems theory. In particular, we can compute the entropy of the distribution P[{/vector xi}] even if we can’t provide explicit algorithms for solving the learning problem. As is shown in [5], the crucial quantity in such anal- ysis is the density of models in the vicinity of the target ¯α—the parameters that actually generated the sequence. For two distributions, a natural distance measure is the Kullback–Leibler divergence D(¯α\|\|α) =/integraltext d/vector xQ(/vector x\|¯α)log [Q(/vector x\|¯α)/Q(/vector x\|α)], and the density is ρ(D;¯α) =/integraldisplay dKαP(α)δ[D−DKL(¯α\|\|α)]. (10) Ifρis large as D→0, then one easily can get close to the target for many diﬀerent data; thus they are not very 3informative. On the other hand, small density means that only very particular data lead to ¯α, so they carry a lot of predictive information. Therefore, it is clear that the density, but not the number of parameters or any other simplistic measure, characterizes predictability a nd the complexity of prediction. If, as often is the case for dimα<∞, the density behaves in the way common to ﬁnite dimensional systems of the usual statistical me- chanics, ρ(D→0,¯α)≈AD(K−2)/2, (11) then the predictive information to the leading order is Ipred(N)≈K/2 logN . (12) The modern theory of learning is concerned in large part with quantifying the complexity of a model class, and in particular with replacing a simple count of pa- rameters with a more rigorous notion of dimensionality for the space of models; for a general review of these ideas see Ref. [6], and for discussion close in spirit to ours see Ref. [7]. The important point here is that the dimension- ality of the model class, and hence the complexity of the class in the sense of learning theory, emerges as the coeﬃ- cient of the logarithmic divergence in Ipred. Thus a mea- sure of complexity in learning problems can be derived from a more general dynamical systems or statistical me- chanics point of view, treating the data in the learning problem as a time series or one dimensional lattice. The logarithmic complexity class that we identify as being associated with ﬁnite dimensional models also arises, for example, at the Feigenbaum accumulation point in the period doubling route to chaos [4]. As noted by Grassberger in his original discussion, there are time series for which the divergence of Ipred is stronger than a logarithm. We can construct an exam- ple by looking at the density function ρin our learning problem above: ﬁnite dimensional models are associated with algebraic decay of the density as D→0, and we can imagine that there are model classes in which this decay is more rapid, for example ρ(D→0)≈Aexp [−B/Dµ], µ > 0. (13) In this case it can be shown that the predictive informa- tion diverges very rapidly, as a sublinear power law, Ipred(N)∼Nµ/(µ+1). (14) One way that this scenario can arise is if the distribution Q(/vector x) that we are trying to learn does not belong to any ﬁnite parameter family, but is itself drawn from a distri- bution that enforces a degree of smoothness [8]. Under- standably, stronger smoothness constraints have smaller powers (less to predict) than the weaker ones (more to predict). For example, a rather simple case of predictinga one dimensional variable that comes from a continuous distribution produces Ipred(N)∼√ N. As with the logarithmic class, we expect that power– law divergences in Ipredare not restricted to the learn- ing problems that we have studied in detail. The gen- eral point is that such behavior will be seen in prob- lems where predictability over long scales, rather then being controlled by a ﬁxed set of ever more precisely known parameters, is governed by a progressively more detailed description—eﬀectively increasing the number of parameters—as we collect more data. This seems a plausible description of what happens in language, where rules of spelling allow us to predict forthcoming letters of long words, grammar binds the words together, and compositional unity of the entire text allows to make pre- dictions about the subject of the last page of the book after reading only the ﬁrst few. Indeed, Shannon’s clas- sic experiment on the predictability of English text (by human readers!) shows this behavior [9], and more re- cently several groups have extracted power–law subex- tensive components from the numerical analysis of large corpora of text (see, for example, [10], [11]). Interestingly, even without an explicit example, a sim- ple argument ensures existence of exponential densities and, therefore, power law predictive information models. If the number of parameters in a learning problem is not ﬁnite then in principle it is impossible to predict anything unless there is some appropriate regularization. If we let the number of parameters stay ﬁnite but become large, then there is moreto be learned and correspondingly the predictive information grows in proportion to this num- ber. On the other hand, if the number of parameters becomes inﬁnite without regularization, then the predic- tive information should go to zero since nothing can be learned. We should be able to see this happen in a regu- larized problem as the regularization weakens: eventually the regularization would be insuﬃcient and the predictive information would vanish. The only way this can hap- pen is if the predictive information grows more and more rapidly with Nas we weaken the regularization, until ﬁ- nally it becomes extensive (equivalently, drops to zero) at the point where prediction becomes impossible. To realize this scenario we have to go beyond Ipred∝logT withIpred∝Nµ/(µ+1); the transition from increasing predictive information to zero occurs as µ→1. This discussion makes it clear that the predictive infor- mation (the subextensive entropy) distinguishes between problems of intuitively diﬀerent complexity and thus, in accord to Grassberger’s deﬁnitions [4], is probably a good choice for a universal complexity measure. Can this in- tuition be made more precise? First we need to decide whether we want to attach mea- sures of complexity to a particular signal x(t) or whether we are interested in measures that are deﬁned by an av- erage over the ensemble P[x(t)]. One problem in assign- ing complexity to single realizations is that there can be 4atypical data streams. Second, Grassberger [4] in par- ticular has argued that our visual intuition about the complexity of spatial patterns is an ensemble concept, even if the ensemble is only implicit. The fact that we admit probabilistic models is crucial: even at a colloquial level, if we allow for probabilistic models then there is a simple description for a sequence of truly random bits, but if we insist on a deterministic model then it may be very complicated to generate precisely the observed string of bits. Furthermore, in the context of probabilis- tic models it hardly makes sense to ask for a dynamics that generates a particular data stream; we must ask for dynamics that generate the data with reasonable prob- ability, which is more or less equivalent to asking that the given string be a typical member of the ensemble generated by the model. All of these paths lead us to thinking not about single strings but about ensembles in the tradition of statistical mechanics, and so we shall search for measures of complexity that are averages over the distribution P[x(t)]. Once we focus on average quantities, we can provide an axiomatic proof (much in the spirit of Shannon’s [12] arguments establishing entropy as a unique information measure) that links Ipredto complexity. We can start by adopting Shannon’s postulates as constraints on a mea- sure of complexity: if there are Nequally likely signals, then the measure should be monotonic in N; if the sig- nal is decomposable into statistically independent parts then the measure should be additive with respect to this decomposition; and if the signal can be described as a leaf on a tree of statistically independent decisions then the measure should be a weighted sum of the measures at each branching point. We believe that these constraints are as plausible for complexity measures as for informa- tion measures, and it is well known from Shannon’s orig- inal work that this set of constraints leaves the entropy as the only possibility. Since we are discussing a time de- pendent signal, this entropy depends on the duration of our sample, S(T). We know of course that this cannot be the end of the discussion, because we need to distinguish between randomness (entropy) and complexity. The path to this distinction is to introduce other constraints on our measure. First we notice that if the signal xis continuous, then the entropy is not invariant under transformations of x that do not mix point at diﬀerent times (reparameteri- zations). It seems reasonable to ask that complexity be a function of the process we are observing and not of the coordinate system in which we choose to record our observations. However, that it is not the whole function S(T) which depends on the coordinate system for x; it is only the extensive component of the entropy that has this noninvariance. This can be seen more generally by not- ing that subextensive terms in the entropy contribute to the mutual information among diﬀerent segments of the data stream (including the predictive information deﬁnedhere), while the extensive entropy cannot; mutual infor- mation is coordinate invariant, so all of the noninvariance must reside in the extensive term. Thus, any measure complexity that is coordinate invariant must discard the extensive component of the entropy. If we continue along these lines, we can think about the asymptotic expansion of the entropy at large T. The extensive term is the ﬁrst term in this series, and we have seen that it must be discarded. What about the other terms? In the context of predicting in a parameterized model, most of the terms in this series depend in detail on our prior distribution in parameter space, which might seem odd for a measure of complexity. More generally, if we consider transformations of the data stream x(t) that mix points within a temporal window of size τ, then for T >> τ the entropy S(T) may have subextensive terms which are constant, and these are not invariant under this class of transformations. On the other hand, if there are divergent subextensive terms, these areinvariant un- der such temporally local transformations [13]. So if we insist that measures of complexity be invariant not only under instantaneous coordinate transformations, but also under temporally local transformations, then we can dis- card both the extensive and the ﬁnite subextensive terms in the entropy, leaving only the divergent subextensive terms as a possible measure of complexity. To illustrate the purpose of these two extra conditions, we may think of the following example: measuring veloc- ity of a turbulent ﬂuid ﬂow at a given point. The condi- tion of invariance under reparameterizations means that the complexity is independent of the scale used by the speedometer. On the other hand, the second condition ensures that the temporal mixing due to the ﬁniteness of the inertia of the speedometer’s needle does not change the estimated complexity of the ﬂow. In our view, these arguments (or their slight variation also presented in [5]) settle the question of the unique deﬁnition of complexity. Not only is the divergent subex- tensive component of the entropy the unique complexity measure, but it is also a universal one since it is con- nected in a straightforward way to many other measures that have arisen in statistics and in dynamical systems theory. A bit less straightforward is the connection to the Kolmogorov’s deﬁnition that started the whole dis- cussion, but even this can also be made. To make this connection we follow the suggestion of Standish [14] that one should focus not on the complex- ity of particular strings but of equivalence classes. In the present case it is natural to deﬁne an equivalence class of data x(−T < t ≤0) as those data that generate indis- tinguishable conditional probability distributions for t he future, P[x(t >0)\|x(−T < t ≤0)]. If this conditional distribution has suﬃcient statistics, then there exists a compression of the past data x(−T < t ≤0) into exactly Ipred(T) bits while preserving all of the mutual informa- tion with the future. But this means that the ensemble of 5data in an equivalence class can be described, on average, using exactly this many bits. Thus, for dynamics such that the prediction problem has suﬃcient statistics, the average Kolmogorov complexity of equivalence classes de- ﬁned by the indistinguishability of predictions is equal to the predictive information. By the arguments above, pre- diction is theuseful thing which we can do with a data stream, and so in this case it makes sense to say that the Kolmogorov complexity of representing the useful bits of data is equal to the predictive information. Note also that Kolmogorov complexity is deﬁned only up to a con- stant depending on the computer used [1]. A computer independent deﬁnition requires ignoring constant terms and focusing only on asymptotic behavior. This agrees very well with our arguments above that identiﬁed only the divergent part of the predictive information with the complexity of a data stream. In the terminology suggested by Grassberger, the statement that the prediction problem has suﬃcient statistics means that the True Measure Complexity is equal to the Eﬀective Measure Complexity [4]; simi- larly, the statistical complexity deﬁned by Crutchﬁeld and coworkers [15] then also is equal to predictive informa- tion deﬁned here. These are strong statements, and it is likely that they are not true precisely for most natural data streams. More generally one can ask for compres- sions that preserve the maximum fraction of the relevant (in this case, predictive) information, and our intuitive notion of data being “understandable” or “summariz- able” is that these selective compressions can be very eﬃcient [16]—here eﬃciency means that we can com- press the past into a description with length not much larger than Ipred(T) while preserving a ﬁnite fraction of the (diverging) information about the future; an exam- ple is when we summarize data by the parameters of the model that describes the underlying stochastic process. The opposite situation is illustrated by certain crypto- graphic codes, where the relevant information is accessi- ble (at best) only from the entire data set. Thus we can classify data streams by their predictive information, but additionally by whether this predictive information can be represented eﬃciently. For those data where eﬃcient representation is possible, the predictive information an d the mean Kolmogorov complexity of future–equivalent classes will be similar; with more care we can guarantee that these quantities are proportional as T→ ∞. Per- haps Wigner’s famous remarks about the unreasonable eﬀectiveness of mathematics in the natural sciences could be rephrased as the conjecture that the data streams oc- curring in nature—although often complex as measured by their predictive information—nonetheless belong to this eﬃciently representable class.[1] M. Li and P. Vit´ anyi. An Introduction to Kolmogorov Complexity and its Applications , Springer–Verlag, New York (1993). [2] J. Rissanen. Stochastic Complexity and Statistical In- quiry, World Scientiﬁc, Singapore (1989); J. Rissanen, IEEE Trans. Inf. Thy. 42, 40–47 (1996). [3] C. Bennett, in Complexity, Entropy and the Physics of Information , W. H. Zurek, ed., Addison–Wesley, Red- wood City, pp. 137–148 (1990). [4] P. Grassberger, Int. J. Theor. Phys. 25, 907–938 (1986). [5] W. Bialek, I. Nemenman, and N. Tishby, to appear in Neural Computation (2001). E-print: physics/0007070 . [6] V. Vapnik. Statistical Learning Theory , John Wiley & Sons, New York (1998). [7] V. Balasubramanian, Neural Comp. 9, 349–368 (1997). [8] W. Bialek, C. Callan, and S. Strong, Phys. Rev. Lett. 77, 4693–4697 (1996). [9] C. E. Shannon, Bell Sys. Tech. J. 30, 50–64 (1951). W. Hilberg, Frequenz 44, 243–248(1990). [10] W. Ebeling, T. P¨ oschel, Europhys. Lett. 26, 241–246 (1994). [11] T. Schurmann and P. Grassberger, Chaos ,6, 414–427 (1996). [12] C. E. Shannon, Bell Sys. Tech. J. 27, 379–423, 623–656 (1948). [13] Throughout this discussion we assume that the signal xat one point in time is ﬁnite dimensional. There are subtleties if we allow xto represent the conﬁguration of a spatially inﬁnite system. [14] R. K. Standish, submitted to Complexity International . E-print: nlin.AO/0101006 . [15] C. R. Shalizi and J. P. Crutchﬁeld, to appear inJournal of Statistical Physics (2001). E-print: cond-mat/9907176 . [16] N. Tishby, F. Pereira, and W. Bialek, in Proceedings of the 37th Annual Allerton Conference on Communication, Control and Computing , B. Hajek and R. S. Sreenivas, eds., University of Illinois, pp. 368–377 (1999). E-print: physics/0004057. 6
arXiv:physics/0103077v1 [physics.atom-ph] 23 Mar 2001Solution of the two identical ion Penning trap ﬁnal state W. Blackburn 3633 Iron Lace Drive, Lexington, KY 40509 T. L. Brown Department of Electrical Engineering, Washington Univers ity, P.O. Box 1127, St. Louis, MO 63130 E. Cozzo 201 Ellipse Street, #12, Berea, Kentucky 40403 B. Moyers Wine.com, Inc., 665 3rd Street Suite 117 San Francisco, CA 94 107 M. Crescimanno Center for Photon Induced Processes, Department of Physics and Astronomy, Youngstown State University, Youngstown, O H, 44555-2001 (July 2, 2011) We have derived a closed form analytic expression for the asy mptotic motion of a pair of identical ions in a high precision Penning trap. The analytic solution includes the eﬀects of special relativity and the Coulomb interaction between the ions. The existence and physical relevance of such a ﬁnal state is supported by a conﬂuence of theoretical, experimen tal and numerical evidence. PACS numbers: 32.80.Pj, 02.20+b, 33.80.Ps High precision Penning traps are ideal for studying phys- ical characteristics of individual ions. These traps, as described for example in Ref.[1], have magnetic ﬁelds that over the trajectories of the ions vary by less than a part per billion. In consequence, the motional frequency linewidths can be made so narrow that eﬀects of special relativity are readily apparent even at these relatively lo w velocities2. To remove systematic eﬀects it is often desirable to ﬁll the trap with two ions and much is known about the re- sulting frequency perturbations caused by the Coulomb interaction between dissimilar ions3. The situation with two identical ions has also been extensively studied much (see Ref.[4,5] and references therein). The solution and approach that we describe here are rather diﬀerent than those references however, since they include the electric trap ﬁeld but ignore relativistic mass increase. Includ- ing this eﬀect of special relativity may be crucial for un- derstanding the observation6of cyclotron mode-locking between identical ions (see also Ref.[7]). We present details of an analytical model of two iden- tical ions in a high precision Penning trap. The model is asymptotically solvable in terms of elliptic functions. This solution is, in practical terms for protons and heav- ier ions, a generic ﬁnal state of two identical ions in a precision Penning trap. We begin with a symmetry argument detailing what is special about the two identical ion system and then we introduce and solve the model. For two dissimilar ions the center of charge is diﬀerent than the center of mass. The motion of the center of charge causes currents to run in the detection circuit and in the walls of the trap itself causing a force to act back on the ions. This retarding force acts on the center of charge and so if the center of charge is diﬀerent than the center of mass thesedamping forces act always on a mixture of the center of mass motion and the relative motions of the ion pair. This is not the case for identical ions in the trap. In that case the center of mass and the center of charge are the same and so the retarding force acts only on the center of mass motion. Thus, the relative motion of the ions is relatively undamped, being subject only to the weaker quadrupolar damping (which is associated with timescales generally longer than typical experiments). In this sense we speak of this ﬁnal state of the two identical ion system as a decoupled, or, dark state. One way to understand the existence of this cyclotron dark state is with a symmetry argument. Neglect dissi- pation, relativity and interaction and consider the Pois- son algebra of two ions moving in a horizontal plane (we shall describe why this is relevant to experiment later) in a uniform perpendicular magnetic ﬁeld. The Hamilto- nian is proportional to H=p2 1+p2 2+α(p2 3+p2 4), where α=ma/mbis the mass ratio and p1,2(resp. p3,4) are the canonical momenta of particle a(resp. particle b). For α/negationslash= 1 the subalgebra commuting with Hisso(2) xso(2) whereas if α= 1 the algebra is so(2) x so(3). The fact that there are additional commuting generators in the equal mass case indicates that there is a ﬂat direction in the dynamics of that case, corresponding to degeneracy between cyclotron dark states of diﬀerent total angular momentum. There is a straightforward geometrical way of under- standing the special qualities of the two identical ion Penning trap. Again consider the ions conﬁned to a plane perpendicular to the magnetic ﬁeld and ignore tem- porarily the eﬀects of relativity and interaction. The to- tal angular momentum of the two ion system is L= p2 1+p2 2+p2 3+p2 4(note independent of the mass ratio α). Now, turning on relativity and interactions pertur-batively, we learn that the motion is essentially restricte d to the intersection of iso- Hand iso- Lsurfaces. A generic intersection of these surfaces in R4for the α/negationslash= 1 case is a two-dimensional torus (and so has an isometry group so(2) x so(2)) whereas when α= 1 the intersection is not generic, but is the whole S3. Although the isome- try group of S3, being so(4), is isomorphic to so(3) x so(3) the physically relevant isometry group is that which preserves not only the geometry but also the underly- ing Poisson structure, which is sp(4) in this case. The canonical intersection8in the group of matrices GL(4) of so(4) and sp(4) is the algebra u(2), which is isomorphic toso(2) xso(3), which again is the enhanced symmetry discussed above. We note that both the geometrical and algebraic picture can be easily generalized to the case of Nidentical ions9. Having described the symmetry properties unique to two identical ions in a Penning trap, we now introduce the interacting model by starting with the following three assumptions. 1) The ions are very near the center of the trap, and ignore eﬀects due to the spatial gradient of the electro- static ﬁelds of the trap (that is, we completely ignore the trap magnetron motion). The cyclotron frequency shifts in an isolated ion’s cyclotron motion is entirely due to relativistic eﬀects. 2) the ions are mode locked already in the trap’s axial drive and so their motions may be thought of as being conﬁned to a plane6,7. 3) The energy loss mechanism is entirely due to the dissipation of image charge currents induced in the trap/detection system, and thus couple only to the center of mass of the ion pair). Under these assumptions, the equations of motion for the ion pair are the formidable looking non-linear coupled diﬀerential equations; ¨/vector r1+ω0(1−f1)ˆz×˙/vector r1+γ˙/vectorXcm−e2ˆR m0R2= 0 (1) ¨/vector r2+ω0(1−f2)ˆz×˙/vector r2+γ˙/vectorXcm+e2ˆR m0R2= 0 (2) where /vectorXcm= (/vector r1+/vector r2)/2, and /vectorR=/vector r1−/vector r2and where f1=\|˙/vector r1\|2 2c2is just the ratio of the kinetic energy to the rest mass-energy of ion 1 (similar expression for f2is in terms of the kinetic energy of the second particle). This term, due entirely to special relativistic mass increase, causes the cyclotron frequency to depend on the kinetic energy of the ion(s). We add and subtract Eq. (1) and Eq. (2) to rewrite them in terms of the center of mass co-ordinate /vectorXcmand the relative coordinate /vectorR, ¨/vectorXcm+ 2γ˙/vectorXcm+ω0(1−f1+f2 2)ˆz×˙/vectorXcm =ω 4(f1−f2)ˆz×˙/vectorR (3)¨/vectorR+ω0(1−f1+f2 2)ˆz×˙/vectorR−2e2/vectorR m0R3=ω0(f1−f2)˙/vectorXcm (4) Let/vectorV=˙/vectorXcmbe a symbol for the center of mass velocity. As expected, only the center of mass velocity enters into the equations. Conﬁned as they are to the same vertical plane, this becomes a six-dimensional (phase-space) sys- tem. Let /vectorU=˙/vectorR. In these variables, the combinations f1−f2=/vectorU·/vectorV c2andf1+f2=/vectorU2+4/vectorV2 4c2. As per earlier discussion, from Eq. (3) and Eq. (4), it is clear that the center of mass motion is damped but the relative motion is not. Thus, after suﬃcient time, it is consistent to assume that the center of mass motion damps out completely, that is, /vectorV→0. The coupling term between the /vectorRmotion and the /vectorV(center of mass) motion is through the term proportional to f1−f2(itself proportional to V), and so Eq. (3) and Eq. (4) quickly decouple as /vectorV→0. The resulting motion can be treated perturbatively in small /vectorV. To ﬁnd the zeroth order term we ignore the cou- pling term completely, resulting in exponential decay for /vectorVand the total center-of-mass kinetic energy. Asymp- totically for the relative co-ordinate Eq. (4) becomes ¨/vectorR+ω0(1−f1+f2 2)ˆz×˙/vectorR−2e2/vectorR m0R3= 0 . (5) This is a system of two coupled non-linear second or- der diﬀerential equations. Generally such systems do not admit closed-form, analytical solution. Somewhat sur- prisingly, we now point out that Eq. (5) admits a general solution in terms of elliptic functions. The approach is standard. First we ﬁnd two integrals of the motion, reducing the four (phase space) dimen- sional system in Eq. (5) to a two dimensional (phase space) system. The integrals are the energy and a gen- eralization of angular momentum. The inter-ion energy results from taking the dot product of Eq. (5) with˙/vectorR, forming the total diﬀerential, and integrating to ﬁnd the integration constant, u0=1 2\|˙/vectorR\|2+2e2 m0R(6) Since the equations have manifest rotational symmetry, there is a conserved angular momentum. As always with a magnetic ﬁeld, the total angular momentum receives a contribution from the magnetic ﬁeld. Proceed by taking the vector cross product of /vectorRand Eq. (5) to ﬁnd dL dt−ω0 2(1−f)dR2 dt= 0 (7) where, as always, R=\|/vectorR\|, andf= (\|˙/vectorR\| 2c)2is the term due to special relativity. The angular momentum per unit massL= ˆz·(/vectorR×˙/vectorR) =R2dφ dtis the standard deﬁnition. Now, using the inter-ion energy integral Eq. (6), fcan be written entirely as a function of R. Doing so for finEq. (7) and integrating leads to the integration constant L0, L0=L−w0 2/parenleftbig 1−u0 2c2/parenrightbig R2−ω0e2 2m0c2R (8) L0represents the generalized angular momentum. Since they are independent, the constants of motion in equations Eq. (6) and Eq. (8) constrain the motion to lie in a two-dimensional surface in the original four- dimensional phase space. Of course, that fact by itself is insuﬃcient to guarantee integrability of the equations of motion in closed form. However additional peculiarities of this system Eq. (5) result in closed form solution. In polar co-ordinates the kinetic energy in the potential energy equation can be written /parenleftbiggd/vectorR dt/parenrightbigg2 =/parenleftbiggdR dt/parenrightbigg2 +L2 R2(9) and solving Eq. (8) for Land substituting we ﬁnd that Eq. (6) becomes, /parenleftbiggdR dt/parenrightbigg2 = 2u0−4e2 m0R−(L0+αR+βR2)2 R2(10) where α=ω0e2 2m0c2andβ=ω0 2/parenleftbig 1−u0 2c2/parenrightbig . Since the RHS involves only ﬁve consecutive powers of R(namely, R2, R, R0, ...R−2). the equation is that of an elliptic func- tion. More explicitly, we now compute the orbital period of the dark state and ﬁnd the orbit trajectory parametri- caly. To compute the period we rewrite Eq. (10) as dt=dR/radicalbig ˜u−L2 0/R2−n/R−2αβR−β2R2(11) with ˜u= 2u0−2L0β−α2, and n= 2L0α+4e2 m0. The integral is a combination of standard elliptic func- tions. In lab co-ordinates R, φthe orbits will in general be open (with some precession rate which can be written in terms of complete elliptic integrals) just as viewing the orbits in the R, tco-ordinates, where now “precession” in tin simply the period of the orbit. The period Tof these orbits is thus given by a contour integral of the RHS of Eq. (11) around the cut running between the classical turning points (we label) a0anda1, namely, T=/integraldisplay dt=/contintegraldisplaydR/radicalbig ˜u−L2 0/R2−n/R−2αβR−β2R2 =1 iβ/contintegraldisplayRdR/radicalbig (R−a0)(R−a1)(R−a2)(R−a3)(12) where the aiare the roots of the fourth degree polyno- mial written in Eq. (17). By looking at the signs of terms in the polynomial we can see that there can be at most two real positive roots. Physically we expect there to be exactly two real positive roots which we have called a0 anda1. These are the classical turning points of the mo- tion, and represent the furthest and nearest approaches of the particles.Furthermore, in the system we are working with, for typical values of parameters, we ﬁnd that all roots are real, with two positive and two negative. We may then order the roots a0≥a1≥a2≥a3. Note also that the canonical choice of phase for the square root on the cut between a0anda1isiand so the period in Eq. (12) is real and positive. Finally, computing the integral in Eq. (12) yields (no- tation is that in Ref. [10]), T=2 ρβ/bracketleftbigg (a0−a3)Π(a1−a0 a1−a2, k) +a3K(k)/bracketrightbigg (13) where Kand Π are respectively the complete ellip- tic integrals of the ﬁrst and third kind, and ρ=/radicalbig (a0−a2)(a1−a3) and k=√ (a0−a1)(a2−a3) ρis the square root of the cross-ratio of the roots. Note that the ﬁrst argument in the Π is negative, as it should be on physical grounds, since Π is convergent for any negative argument. One of the most striking experimental surprises of the two identical ion system is the discovery of cyclotron mode-locking6. In these events the two frequency traces corresponding (approximately) to the individual ions mo- tions meld into one trace. This visible trace is the cen- ter of mass motion of the dark state. Our analysis in- dicates that there is another invisible (as a dipole) fre- quency branch associated with the inter-ion motion and that it has frequency2π TwithTof Eq. (13). For the case of two protons in a typical precision Penning trap (at ω0∼5x108) we ﬁnd that Eq. (13) yields frequencies are some tens of Hertz diﬀerent than ω0. It would be an inter- esting test to apply a sequence of dipole and quadrupolar ﬁelds to make transitions between dark states and (visi- ble) center of mass states. By standard means we now derive explicit formulae for the shape of the dark state orbits. Recall that, by deﬁnition of the angular momentum, L, and Eq. (8) dφ dt=L0 R2+α R+β (14) Thus, eliminating time between this and Eq. (11) we ﬁnd φ=1 iβ/integraldisplayR(L0 R2+α R+β) dt/radicalbig (R−a0)(R−a1)(R−a2)(R−a3)(15) which may be evaluated in terms of incomplete elliptic functions. We ﬁnd φ−φ0=2 ρ/bracketleftbigg/parenleftbigL0 a2+α+βa2/parenrightbig F(θ(R), k) +/parenleftbiga2 a1−1/parenrightbig/braceleftbigL0 a2Π/parenleftbig θ(R),a2(a0−a1) a1(a0−a2), k/parenrightbig −βa2Π/parenleftbig θ(R),a0−a1 a0−a2, k/parenrightbig/bracerightbig/bracketrightbigg (16) where, again, the aiare the (ordered) roots of the poly- nomial P(R) =−β2R4−2αβR3+ (2u0−α2−2L0β)R2−(2L0α+4e2 m0)R−L2 0 (17) withαandβas deﬁned previously and where sinθ(R) =/radicalBigg (a0−a2)(R−a1) (a0−a1)(R−a2)(18) Note directly from Eq. (16) and Eq. (18) that the pre- cession of these orbits is given by twice the RHS Eq. (16) with each incomplete elliptic functions replaced by its complete elliptic counterpart. We have completed a numerical simulation of the sys- tem Eq. (1) and Eq. (2) for a range of initial conditions. To abet numerical stability those equations were rewrit- ten in the co-rotating frame and integrated using com- mercial (IDLtm) routines on a DEC Alpha workstation. Some of these IDLtmprograms link compiled versions of CERN’s Mathlib elliptic function routines. The results from a typical run are shown in Figures 1 (resp. 2) where both the u0of Eq. (6) (resp. L0of Eq. (8)) are plotted as functions of time. The ﬁgures show that initially the motions of the ions are essentially independent as the energy dissipates. Dur- ing this regime the total energy of the system is split between the center of mass motion and the inter-ion mo- tion. Note that due to the large dynamic range of these simulations we have plotted the logarithm of the energy. Thus, the linear decay of the envelope of the inter-ion en- ergyu0in this initial regime is the exponential damping of the energy of the system as a whole. Eventually the center of charge motion damps away appreciably and the remaining inter-ion motion persists. As described earlier, in real experiments of this type the dark state we are describing is likely to be eﬀectively the ﬁnal state since we expect the inter-ion motion to decay via quadrupole radiation on a timescale long compared with typical two-ion experiments. For our simulation this ﬁnal state is reached at simulated time 150, after which bothu0andL0are essentially constant (up to numerical accuracy of the simulations). In conclusion, we have derived closed form analytic for- mulae for the dark state of two identical ions in a Pen- ning trap. To ﬁnd this solution, we assumed that the pair is near the center of the trap (we have completely neglected the eﬀect of the trap’s electrostatic ﬁelds) and that the motion of the ions is conﬁned to the same az- imuthal plane. It is straightforward to include in this analysis the eﬀects of the trap’s electric ﬁeld and also a ﬁxed average vertical oﬀset between the cyclotron planes of the ions. This results in formulae for the two integrals of motion that have additional terms compared with the Eq. (6) and Eq. (8). However, the resulting equations of motion for the dark state are no longer solvable in terms of known functions. This research was supported in part by Research Corporation Cottrell Science Award #CC3943 and #CC5285 in part by the National Science Foundation un- der grants PHY 94-07194 and EPS-9874764 and in partby Appalachian Colleges Association Mellon Foundation Student-Faculty Grants. We would like to thank CERN Mathlib for the use of the elliptic function libraries. We are delighted to thankfully acknowledge G. Gabrielse, C. H. Tseng, D. Phillips, L. J. Lapidus, A. Khabbaz and A. Shapere for many interesting and stimulating discus- sions and the theory group at the University of Kentucky where much of this work was done. [1] L. S. Brown and G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986). [2] G. Gabrielse, Am. J. Phys. 63, 568 (1995). [3] E. A. Cornell, K. R. Boyce, D. L. K. Fygenson and D. E. Pritchard, Phys. Rev A 45, 3049, (1992). [4] G. Baumann and T.F. Nonnenmacher, Phys. Rev. A , 46, 2682 (1992). [5] D. Farrelly and J. E. Howard, Phys. Rev. A ,49. 1494 (1994). [6] G. Gabrielse, Private Communication (1994). [7] L.J. Lapidus, C. H. Tseng and G. Gabrielse, “The Dy- namics of Two Particles in a Penning Trap,” (1997), un- published. [8] M. Gourdin, “Basics of Lie Groups,” Editions Frontieres , (1982), pg. 62. [9] M. Crescimanno and A. S. Landsberg, Phys. Rev A 63, 035601-1, (2001). [10] I. S. Gradshteyn and I.M. Ryzhik, “Tables of Integrals, Series and Products,” Academic Press, NY, (1980), pg. 243.FIG. 1. The internal energy u0as a function of time. FIG. 2. The internal angular momentum L0as a function of time.
1Gravitation and Forces Induced by Zero-Point Phenomena Charles T. Ridgely charles@ridgely.ws Galilean Electrodynamics, submitted (2001) Abstract A recent proposal asserts that gravitational forces arise due to an interaction between matter and vacuum electromagnetic zero-point radiation. The present analysis demonstrates that forces induced on matter byzero-point radiation arise in addition to gravitational forces. It is argued that zero-point radiation should bered-shifted near large gravitational sources while remaining essentially undetectable within freely falling reference frames. On this basis, an effective weight of an observer stationed near the surface of the Earth is derived for the case when zero-point radiation is present. It is then argued that the weight of matter may beaffected by altering the gravitational anisotropy of zero-point radiation. 1. Introduction In recent times, there have been several attempts to shed greater insight into the origin of inertial and gravitational forces. According to one recent proposal, inertial and gravitational forces alike arise entirely due to an interaction between vacuum electromagnetic zero-point radiation and subatomic particles comprising ordinary matter [1]. For the case of inertia, this proposal suggests that when an objectaccelerates through the zero-point radiation field (ZPF), pervading all of space, the quarks and electronscomprising the object scatter a portion of the radiation passing through the object. This in turn exerts an electromagnetic drag force on the object, which, according to this ZPF proposal, can be associated with the object’s inertia. Additionally, those who support the ZPF proposal also seek to ascribe gravitation entirelyto interaction with zero-point radiation. However, such an approach is certainly not without conceptual difficulties. One source of difficulty surrounds the question of what percentage of the force on an object is actually ZPF induced. This question seems to be hinged on the origin of the rest mass-energy of matter. Accordingto the ZPF proposal, the energy content of matter is entirely internal kinetic energy due to ZPF-induced jittering motion, or zitterbewegung [1], of quarks and electrons comprising ordinary matter. As pointed out in [2], however, ascribing the energy content of subatomic particles entirely to internal kinetic energyeffectively neglects the rest mass-energy content of these particles. It is straightforward to see that quarksand electrons each possess charge and spin, and hence are each surrounded by an electromagnetic field. These electromagnetic fields possess energy. Therefore, while ZPF-induced zitterbewegung must certainly give rise to kinetic energy of quarks and electrons, such particles also possess intrinsic quantities of restmass-energy due at least in part to their electromagnetic fields. Another seemingly important issue is the space-time curvature, or distortion, existing near gravitational sources. The ZPF proposal appears to do away with the notion of space-time distortion, opting in favor of electrodynamic properties of space-time. One example of this can be seen in an attempt to explain thegravitational bending of light near gravitational sources by ascribing variable dielectric properties to space[1]. In essence, the ZPF proposal treats space as a polarizable vacuum. As is very well known, however, general relativity predicts that space and time intervals are affected by the presence of gravitating matter. It2is difficult to imagine how the ZPF proposal can account for the well-documented dilation of time arising near gravitational sources [5]. In our opinion it is precisely the behavior of space-time predicted by general relativity that gives rise to inertial and gravitational forces. In previous analyses, we have used special and general relativity to demonstrate that inertia is purely relativistic in origin [2]. Specifically, for the case of inertia, we have shown that a force-producing agent who exerts a constant force on an object experiences a reactive force of the form dtEdτ=−f∇∇∇∇ ,( 1 ) wherein E is the total energy content of the object, dτ is an interval of proper time experienced by the accelerating object, and dt is a corresponding interval of time experienced by the force-producing agent. Using this expression, it has been further shown that ZPF-induced forces acting on accelerating matter arise in addition to the intrinsic inertial properties of matter [2]. The present analysis uses the same approach for the case of ZPF-forces induced due to gravitation. Herein it is demonstrated that gravitationally induced ZPF-forces acting on an observer residing near a gravitational source arise in addition to the gravitationalforce due to the intrinsic energy content of the observer and the source. One noteworthy point to notice is that electromagnetic zero-point radiation possesses energy. Thus, it makes sense that zero-point radiation must be subject to gravitation just as are other forms of energy, and so one would expect the ZPF to be red-shifted near large gravitational sources. Such a red-shift ought toarise simply because gravitational space-time curvature gives rise to anisotropy in the electromagneticmode structure of the ZPF. Additionally, electromagnetic zero-point radiation is Lorentz invariant [1], which suggests that zero-point radiation should be essentially uniformly distributed within freely falling reference frames. Based on this line of reasoning, we suspect that the ZPF should exert some level of forceon an observer stationed near a gravitational source, while remaining substantially undetectable to freelyfalling observers. This observation is used herein to demonstrate that ZPF-induced forces act in addition to gravitational forces. In the next Section, an effective weight is derived for the case of an observer stationed near the Earth in the presence of zero-point radiation. The resulting expression for the observer’s weight contains threeterms. The first term is the usual Newtonian expression for the gravitational force between two massive bodies. The second term is a body force acting on the observer due to interaction with zero-point radiation. The third term is a gravitational force arising due to other forms of energy that may be present, such as dueto the small and weak forces within the observer. Based on this expression, it is concluded that gravitationally induced ZPF-forces on matter are additional body forces that contribute to the observable weight of ordinary matter. In Section 3, it is argued that zero-point radiation may be manipulated to create observable forces on material objects. It is pointed out that since zero-point radiation is electromagnetic in origin, and contributes to the weight of matter, it may in fact be possible to alter the effective weight of matter through3electromagnetic manipulation of the ZPF. The key to doing this appears to be to alter the gravitational anisotropy of the electromagnetic mode structure of the ZPF [6]. To illustrate this, the expression for the effective weight derived in Section 2 is considered. It is argued that a local manipulation of thegravitational anisotropy of zero-point radiation correspondingly alters the quantity of passive gravitationalenergy imparted to the observer, which in turn affects the effective weight of the observer. 2. Weight due to Gravitational Anisotropy of Zero-Point Radiation As pointed out in the Introduction, electromagnetic zero-point radiation does not exert forces on bodies undergoing free-fall motion simply because such radiation is uniformly distributed within freely fallingreference frames. This is not the case, however, when an object is held stationary near a large gravitationalsource. The reference frame of such an object is non-inertial, and so all modes comprising the zero-point field (ZPF) are gravitationally red-shifted. This anisotropy in the mode structure of the ZPF must give rise to some level of force on the stationary object [1]. However, as shown herein, such a ZPF-induced forcenecessarily arises in addition to the gravitational force on the object. Let an observer, of proper mass m, be stationed a distance r from the center of a weak gravitational source such as the Earth. Within the stationary reference frame of the observer, the ZPF is observably red- shifted due to space-time distortion near the Earth. According to the reasoning presented in theIntroduction, such a red-shift of the ZPF gives rise to a body force on the observer, which acts in additionto the gravitational force on the observer [2]. According to [2], the total force acting on the observer may be expressed in the form dtEdτ′=−f∇∇∇∇ ,( 2 ) where E′ is the total energy content of the observer, and dt dτ is a scalar function that characterizes the gravitational distortion of space-time near the Earth. One important point to notice is that E′ represents all forms of energy possessed by the observer, and is not limited to merely the ZPF-induced energy. Thus, the total energy content of the observer must be expressed as ZPF EE E U′=+ + , where E is the intrinsic mass- energy of the observer, ZPFE is the internal ZPF-induced kinetic energy of the subatomic particles comprising the observer, and U includes additional forms of energy that may be possessed by the observer, such as due to the strong and weak forces. Using this expression for the energy content of the observer, Eq.(2) can be recast in the form ()ZPFdtEE Udτ=− + + f ∇∇∇∇ .( 3 )4The sum enclosed within parentheses makes it clear that the interaction between zero-point radiation and the subatomic particles comprising the observer contributes positively to the passive gravitational mass- energy of the observer. In order to evaluate Eq. (3), an expression for the scalar function dt dτ must first be derived. As is very well known, the geometry of space-time exterior to a large spherical source is described by the Schwarzschild coordinate system, having a space-time interval of the form 2 22 2 2 2 2 2 2 2 22121GM drds c dt r d r Sin dGM cr crθθ φ=− − − − −,( 4 ) in which G is the gravitational constant, M is the total mass of the source, and { , , } rθφ are spherical coordinates exterior to the source. Noting that the observer remains stationary and using Eq. (4), the scalar function dt dτ is easily found to be 21 21dt d GM crτ= −,( 5 ) wherein dτ is an interval of proper time experienced by the stationary observer and dt is an interval of time experienced by a free fall observer whose coordinate origin is momentarily coincident with that of the stationary observer at a time 0 t=. For the case of small, weakly gravitating sources, Eq. (5) simplifies to an approximate form given by 21dt GM dc rτ≈+ .( 6 ) This expression for the scalar function dt dτ holds for the case of small, weakly gravitating sources such as the Earth. Using Eq. (6) and expressing the gradient operator in terms of the Schwarzschild coordinate system, Eq. (3) is easily simplified to ()2ˆ 1ZPFGMEE U rrc r∂=− + + + ∂f ,( 7 ) where ˆr is a unit vector in the radial direction relative to the Earth, and only first order terms have been retained. Carrying out the partial differentiation, and simplifying somewhat, Eq. (7) reduces to 22 2 2 2ˆˆ ˆZPFGMm GM GMrE rU rrc r c r=− − −f ,( 8 )5wherein 2Em c= has been used to simplify the first term. Equation (8) can be further simplified upon noticing that, according to the ZPF proposal, the energy contributed to the observer due to interaction with zero-point radiation is expressed as [1] ()3 232ZPFEV dcωηω ωπ=∫!,( 9 ) where V is the proper volume of the observer, and ()ηω is a spectral function that governs the extent to which zero-point radiation actually interacts with the observer. Using Eq. (9) in Eq. (8), and rearranging a bit, leads to ()3 22 2 2 3 2 2ˆˆ ˆ 2GMm GMV GMrr dU rrc r c c rωηω ωπ=− − − ∫f!. (10) This is the effective weight of an observer stationed near the Earth in the presence of zero-point radiation. The first term is easily identified as the usual Newtonian expression for the gravitational force between twomassive bodies. The second term is an additional force on the observer arising due to gravitationally induced scattering of zero-point radiation. The third term arises due to additional forms of energy with which the observer may be endowed, such as due to the strong and weak forces. The Newtonian forcearises simply because both the observer and the source possess intrinsic gravitational mass-energy. The force due to zero-point radiation is clearly an additional downward-acting body force that contributes to the measurable weight of the stationary observer. It is straightforward to see that if the gravitational mass-energy of the observer and source were strictly ZPF-induced, then the first and last terms in Eq. (10) wouldbe zero, and the force would then be described solely by the second term. This is precisely the objective of those who support the ZPF proposal. However, without experimental evidence it must be presently concluded that gravitationally induced ZPF-forces are additional forces that contribute to the observableweight of ordinary matter. 3. Affecting Weight Through Manipulation of Zero-Point Radiation In the preceding Section, it was shown that zero-point radiation exerts additional forces on observers stationed near gravitational sources. Such gravitationally induced ZPF-forces increase the observable weights of such observers. With that in mind, it seems that the next natural question to ask is can the ZPFbe manipulated in such a manner that ZPF-induced forces act in opposition to gravitational forces?According to [6], both gravitational and inertial forces alike may be affected by altering the electromagnetic mode structure of the ZPF. This suggests that ZPF-induced forces may be affected through electromagnetic means. As pointed out in the preceding Section, the ZPF contributes to thepassive gravitational energy content of stationary observers simply because the ZPF is anisotropic in thenon-inertial frames of those observers. It is not difficult to imagine that some sort of electromagnetic process might be performed that counteracts the gravitationally induced anisotropy of the ZPF. More6specifically, if the anisotropy of the ZPF were annulled or at least substantially reduced within a region in which a stationary observer resides, near a gravitational source, the energy contributed to the observer due to zero-point radiation would be substantially reduced, as well. Assuming that such an alteration canindeed be carried out, it is straightforward to see that in the limit as the ZPF-induced force tends towardzero, Eq. (10) reduces to 22 2ˆˆGMm GMrU rrc r=− −f . (11) This implies that when the ZPF is affected in such a manner that it imparts zero energy to the stationary observer, the force on the observer should reduce to a purely gravitational force. Continuing along theselines, it can be further imagined that through some process the ZPF might be affected to such an extent thatthe gravitational anisotropy of zero-point radiation becomes inverted, thus occurring in a direction opposite to that normally caused by gravitation. The passive gravitational energy imparted to the observer then becomes negative, and Eq. (10) assumes the form ()3 22 2 2 3 2 2ˆˆ ˆ 2GMm GMV GMrr dU rrc r c c rωηω ωπ=− + − ∫f!. (12) The force given by the second term in this expression clearly acts in opposition to the purely gravitational forces given by the first and third terms. This implies that when the ZPF is altered so that the ZPF-inducedforce acts upward, the weight of the stationary observer should appear somewhat smaller than when the ZPF is unaltered. Clearly then, as the alteration in the anisotropy of the ZPF is further inverted, the observable weight of the stationary observer should be further decreased, as well. Based on this line ofreasoning, when the ZPF is manipulated to such an extent that the resulting ZPF-induced force is equal, butoppositely directed, to the downward gravitation force, the observable weight of the observer should drop to zero. The next natural question to ask is how can such an alteration of the ZPF be carried out? One possible answer may be found in [7] wherein it was shown that when the ZPF performs positive electromagneticwork within a localized region, the energy density of that region appears negative relative to the rest of the universe. More simply stated, when the ZPF performs positive work, the energy density of the ZPF decreases. Based on this, it is not difficult to speculate that any process through which the ZPF iscompelled to perform positive work may alter the local field geometry of the ZPF. And with certain field configurations [7], it just may be possible to manipulate the weights of material objects. One problem, however, is that the ZPF comprises an infinite number of electromagnetic frequency modes [1]. At first thought, this seems to imply that an unmanageably large number of modes must bealtered in order to affect the ZPF. But this may not necessarily be the case. A very interesting discussion given by [8] proposes that drawing energy from lower frequency modes of the ZPF may cause energy to flow down the spectrum from the higher frequency modes. This seems to suggest that if, through some7process, the low frequency portion of the ZPF can be set into performing work, then the higher frequency portion may be affected as well, thereby altering a substantial portion of the ZPF spectrum. Such an approach may be the key to successfully manipulating the zero-point field mode structure. As a final thought, it is interesting to ponder the work of Podkletnov and Nieminen [9] wherein a levitating superconducting ring was alleged to have reduced the weights of objects suspended above the ring. It may be that the superconducting ring somehow came into interaction with a portion of the ZPF, causing the ZPF to perform work, and thereby reducing the gravitational anisotropy of the ZPF above thering. If the superconducting ring did indeed bring about a small reduction of the anisotropy of the ZPF,then the weights of objects placed above the ring should appear slightly smaller than when weighed elsewhere. According to Podkletnov and Nieminen [9], such a minute weight-shift was indeed observed; a 5.47834-g sample was observed to loose between 0.05% and 2.0% of its weight when placed above thelevitating superconducting ring. It would be interesting to see if this small weight-shift can be derivedsolely on the basis of ZPF theory. 4. Discussion A recent proposal asserts that the origin of gravitational forces is an interaction between the vacuum electromagnetic zero-point field (ZPF) and the subatomic particles comprising ordinary matter [1]. On thecontrary, the present analysis has argued that zero-point radiation is not the source of gravitation, but rather that ZPF-induced forces comprise additional gravitationally induced forces on matter. To demonstrate this, an expression for the effective weight of an observer stationed near the Earth in the presence of zero-point radiation was derived. The derivation was performed on the basis that since zero-point radiation iselectromagnetic, the ZPF is red-shifted near gravitational sources while remaining undetectable in the inertial reference frames of free fall observers. The resulting expression for the effective weight suggests that gravitationally induced ZPF-forces act in addition to gravitational forces, thereby increasing theweights of stationary observers. Of particular interest was the question as to whether the weights of material objects may be affected through manipulation of zero-point radiation. It was argued that such a task might be carried out by somehow altering the gravitationally induced anisotropy of zero-point radiation [6]. Altering theanisotropy of zero-point radiation within a localized region necessarily alters the interaction energyimparted to observers residing within that region. This in turn affects the magnitude of ZPF-induced forces on those observers. It was argued that reducing the gravitational anisotropy of the ZPF in the presence of a stationary observer, residing near a gravitational source, may give rise to a corresponding reduction in theobservable weight of the observer. In closing, the observations of Podkletnov and Nieminen [9] were citedas one example in which a reduction of the gravitational anisotropy of zero-point radiation may have been successfully achieved.8Notes and References [1] B. Haisch, A. Rueda, and Y. Dobyns, “Inertial mass and the quantum vacuum fields,” Annalen der Physik , in press (2000). [2] C. T. Ridgely, “On the relativistic origin of inertia and zero-point forces,” submitted to Annales de Physique (2001), physics/0103044. [3] C. T. Ridgely, “Can zero-point phenomena truly be the origin of inertia?” Gal. Elect. , in review (2000), physics/0010018. [4] See, for example, I. R. Kenyon, General Relativity (Oxford, New York, 1990), p. 17. [5] C. T. Ridgely, “On the nature of inertia,” Gal. Elect. 11, 11 (2000). [6] B. Haisch, A. Rueda, and H. E. Puthoff, “Advances in the proposed electromagnetic zero-point-field theory of inertia,” 34th AIAA/ASME/SEA/ASEE Joint Propulsion Conference, AIAA paper 98-3134, (1998). [7] C. T. Ridgely, “A macroscopic approach to the origin of exotic matter,” Gal. Elect. , accepted for publication (2001), physics/0010027. [8] Moray B. King, “Stepping down high frequency energy,” in Tapping The Zero-Point Energy (Paraclete, Provo, 1992). [9] E. Podkletnov and R. Nieminen, “A possibility of gravitational force shielding by bulk YBa 2Cu3O7 − v superconductor,” Physica C 203, 441 (1992).
arXiv:physics/0103079v1 [physics.atom-ph] 24 Mar 2001version 1.5 Hyperﬁne splitting of 23S1state in He3 Krzysztof Pachucki∗ Institute of Theoretical Physics, Warsaw University, Ho˙ z a 69, 00-681 Warsaw, Poland Abstract Relativistic corrections to the hyperﬁne splitting are cal culated for the triplet 23S1state of the helium isotope He3. The electron-electron correlations are fully incorporated. Due to the unknown nuclear structure co ntribution a comparison with the experimental result is performed via He3+hyperﬁne splitting. A signiﬁcant discrepancy with the experiment ar ises and a possible explanation is proposed. PACS numbers 31.30 Jv, 12.20 Ds, 06.20 Jr, 32.10 Fn Typeset using REVT EX ∗E-mail address: krp@fuw.edu.pl 1The calculation of higher order relativistic and QED eﬀects in few electron systems is a long standing problem. Various measurements of transitio n frequencies have reached the precision of few ppb, while no theoretical predictions are y et so accurate [1]. The usual approaches, which incorporate most of relativistic eﬀects from the beginning, are not capa- ble to include electron-electron correlations in a complet e way. The alternative approach starts from the Schr¨ odinger equation and incorporates rel ativistic eﬀects perturbatively in the eﬀective hamiltonian. The principal advantage is the si mplicity and high accuracy of nonrelativistic wave functions, which allows for precise c alculations of higher order relativis- tic eﬀects. The cost one pays in the perturbative approach is the complexity and singularity of the eﬀective hamiltonian. The He3hyperﬁne splitting [2] is a nice example of diﬃculties in the bound state QED and perturbative approaches. The Ferm i interaction, as given by Dirac δ3(r) function is already quite singular. Incorporation of furt her relativistic eﬀects leads to even more singular operators, which have to be prope rly handled. In a recent work onm α6contribution [3] to the helium 23S1ionization energy, we have shown the way to handle singular eﬀective operators. In this approach the Co ulomb interaction have to be smoothed at small distances with the use of some parameter λ, to avoid small rnoninte- grable singularities. The key point of this approach lies in the fact, that dependence on λ cancel out between all matrix elements, and correct energy l evels are restored in the limit λ→ ∞. However, our result obtained here is in the disagreement wi th experiments on the helium hyperﬁne splitting. The possible explanation is pos tponed to the discussion at the end of this work. Hyperﬁne splitting in He3of 23S1state is due to the interaction of electron and helion (nucleus) magnetic moments. In the 23S1state both electron spins are parallel and sum to S= 1 in contrast to the ground state were S= 0. Magnetic moment of helion comes mainly from the neutron particle. It means that helion g-factor is n egative and hyperﬁne sublevels are inverted with respect to hydrogen, the upper one has S+I= 1/2 and the lower one S+I= 3/2. Therefore, by a hyperﬁne splitting one means here Ehfs=E(1/2)−E(3/2). According to the perturbative approach the general express ion for the hyperﬁne splitting up to the order m α6is: Ehfs=/an}b∇acketle{tH(4) hfs/an}b∇acket∇i}ht+/an}b∇acketle{tH(5) hfs/an}b∇acket∇i}ht+/an}b∇acketle{tH(6) hfs/an}b∇acket∇i}ht+ 2/an}b∇acketle{tH(4) 1 (E−H)′H(4) hfs/an}b∇acket∇i}ht. (1) H(4) hfsis: H(4) hfs=HA hfs+HB hfs+HD hfs, (2) HA hfs=8Z α 3m M/bracketleftbiggσn·σ1 4π δ3(r1) +{1→2}/bracketrightbigg (1 +k) (1 + a), (3) HB hfs=Z α 2m M/bracketleftbiggr1×p1 r3 1+r2×p2 r3 2/bracketrightbigg ·σn(1 +k), (4) HD hfs=−Z α 4m M/bracketleftbiggσi nσj 1 r3 1/parenleftbigg δij−3ri 1rj 1 r2 1/parenrightbigg +{1→2}/bracketrightbigg (1 +k), (5) where a, kare anomalous magnetic moments of the electron and the helio n respectively. However, it is not commonly accepted such a notion for a nucle us. The relation of kwith the magnetic moment of the nucleus with charge Z eisµ= 2 (1 + k)Z e/(2M)I. Masses 2mandMare of the electron and helion respectively. The expectatio n values of HB hfsand HD hfsvanish in 23S1state, however they contribute in the second order, last ter m in Eq. (1). H(4)is a Breit hamiltonian in the nonrecoil limit: H(4)=HA+HB+HD, (6) HA=−1 8m3(p4 1+p4 2) +Z α π 2m2[δ3(r1) +δ3(r2)]−α 2m2pi 1/parenleftbiggδij r+rirj r3/parenrightbigg pj 2, (7) HB=/bracketleftbiggZ α 4m2/parenleftbiggr1×p1 r3 1+r2×p2 r3 2/parenrightbigg −3α 4m2r r3×(p1−p2)/bracketrightbiggσ1+σ2 2, (8) HD=α 4m2σi 1σj 2 r3/parenleftbigg δij−3rirj r2/parenrightbigg , (9) where r=r1−r2andr=\|r\|. Since H(4)contributes at order m α6of hfs, see last term in Eq. (1), we neglect here small recoil corrections. However, the possible signiﬁcance of these corrections is discussed at the end of this work. /an}b∇acketle{tH(5) hfs/an}b∇acket∇i}htis delta-like term with the coeﬃcient given by the two-photon forward scattering amplitude. It is the same like in hydrogen and strongly depends on the nuclear structure. It also automati cally includes nuclear recoil ef- fects and inelastic contribution (nuclear polarizability )./an}b∇acketle{tH(5) hfs/an}b∇acket∇i}hthas been considered in detail for the case of hydrogen and muonic hydrogen. However, there are no suﬃcient experimental data available for helion, He3nucleus, therefore we were unable to estimate these contrib u- tions. Moreover, it would be incorrect to apply this correct ion for point-like nucleus, because high energy photon momenta are involved, where nucleus, cou ld not be approximated as a point like particle. Therefore we will leave this contribut ion unevaluated, and at the end for the comparison with an experiment, will subtract the app ropriately scaled hydrogenic value for hyperﬁne splitting The last term H(6) hfsincludes spin dependent operators, which contribute at order m α6. It is only this term, which derivation was not performed so f ar in the literature and is presented here. The detailed descri ption on derivation of eﬀective hamiltonian, reader may ﬁnd in former works, for example in [ 4]. There are four time or- dered diagrams, which contributes to H(6) hfsand are presented in Fig 1. The ﬁrst two are the same as in hydrogen, other two are essentially three body ter ms. One derives the following expressions, which corresponds to these diagrams. H(6) hfs=V1+V2+V3+V4, (10) V1=−(1 +k)σn·σ1 48M m3/braceleftbigg 2p2 14π Z α δ3(r1) + 2·4π Z α δ3(r1)p2 1+/bracketleftbigg p2 1,/bracketleftbigg p2 1,Z α r1/bracketrightbigg/bracketrightbigg/bracerightbigg +{1→2}, (11) V2= (1 + k)(Z α)2 r4 1σn·σ1 6M m2+{1→2}, (12) V3=−(1 +k)σn·σ1 6M m2Z αr1 r3 1·αr r3+{1↔2}, (13) V4= (1 + k)σn·σ2 6M m2Z αr1 r3 1·αr r3+{1↔2}. (14) It is worth noting the cancellation of electron-electron te rms,/an}b∇acketle{tV3+V4/an}b∇acket∇i}ht= 0. Only the electron-nucleus interaction terms remains, which are the same as in hydrogen. Matrix 3element of H(6) hfsand last term in Eq. (1) are separately divergent for 23S1state. Therefore we introduce the following regulator λto the electron-nucleus Coulomb interaction Z α ri→Z α ri(1−e−λ m Z α r i), (15) in all hamiltonians in Eq.(1), as well as in the nonrelativis tic one. This leads to the following further replacements in H(6) hfs 4π Z α δ3(ri)≡ −∇2Z α ri→ −∇2Z α ri(1−e−λ m Z α r i), (16) (Z α)2 r4 i≡/parenleftbigg ∇Z α ri/parenrightbigg2 →/parenleftbigg ∇Z α ri(1−e−λ m Z α r i)/parenrightbigg2 . (17) Once the interaction is regularized, one can calculate all m atrix elements and take the limitλ→ ∞. As a ﬁrst step, using formulas from [4], we rederived the kno wn relativistic correction to hfs in hydrogen δEhfs= (1 + k)µ3 m M(Z α)6 n3σnσe 4/parenleftbigg44 9+4 n−44 9n2/parenrightbigg , (18) where nis a principal quantum number. It agrees with that, obtained directly from the Dirac equation. Since for helium, all matrix elements could be calculated only numerically, we will transform eﬀective operators to the regular form, wh ereλcould be taken to inﬁnity before the numerical calculations. The initial expression for a complete set of relativistic corrections in atomic units is (with implicit λregularization): δEhfs=\|1 +k\|µ3 m Mα6E, (19) E=EA+EB+ED+EN, (20) EA= 2/angbracketleftbigg/braceleftbigg −1 8(p4 1+p4 2) +Z π 2[δ3(r1) +δ3(r2)]−1 2pi 1/parenleftbiggδij r+rirj r3/parenrightbigg/bracerightbigg 1 (E−H)′2Z π[δ3(r1) +δ3(r2)]/angbracketrightbigg , (21) EB= 2/angbracketleftbigg/braceleftbiggZ 4/bracketleftbiggr1×p1 r3 1+r2×p2 r3 2/bracketrightbigg −3 4r×(p1−p2) r3/bracerightbigg1 (E−H)′ Z 2/bracketleftbiggr1×p1 r3 1+r2×p2 r3 2/bracketrightbigg/angbracketrightbigg , (22) ED= 2/angbracketleftbigg1 4/parenleftbiggδij r3−3rirj r5/parenrightbigg1 (E−H)′/parenleftbigg −Z 4/parenrightbigg/bracketleftbigg/parenleftbiggδij r3 1−3ri 1rj 1 r5 1/parenrightbigg +/parenleftbiggδij r3 2−3ri 2rj 2 r5 2/parenrightbigg/bracketrightbigg/angbracketrightbigg ,(23) EN=/angbracketleftbigg −1 4/parenleftbigg p2 1δ3(r1) +p2 2δ3(r2)/parenrightbigg −1 16/parenleftbigg/bracketleftbigg p2 1,/bracketleftbigg p2 1,Z r1/bracketrightbigg/bracketrightbigg +/bracketleftbigg p2 2,/bracketleftbigg p2 2,Z r2/bracketrightbigg/bracketrightbigg/parenrightbigg +1 2/parenleftbiggZ2 r4 1+Z2 r4 2/parenrightbigg/angbracketrightbigg , (24) where EN=/an}b∇acketle{tH(6) hfs/an}b∇acket∇i}ht, and we used the following formulas for hfs of3S1states: 4/an}b∇acketle{tσn·σ1/an}b∇acket∇i}ht=/an}b∇acketle{tσn·(σ1+σ2)/2/an}b∇acket∇i}ht=−3, (25) /an}b∇acketle{tσi 1σj 2Qij 1σa n(σ1+σ2)bQab 2/an}b∇acket∇i}ht=−2Qij 1Qij 2, (26) for symmetric and traceless Qij. There is also a one loop radiative correction ER= 2Z2/parenleftbigg ln 2−5 2/parenrightbigg /an}b∇acketle{tπ δ3(r1) +π δ3(r2)/an}b∇acket∇i}ht, (27) which is similar to that in hydrogen. It will not contribute t o the special diﬀerence between the helium and hydrogen-like helium hfs, therefore we will n ot consider it any further. The initial expression is rewritten to the regular form, where λregularization is not necessary. The operators in second order terms EAare transformed with the use of H′A≡HA−1 4/parenleftbiggZ r1+Z r2/parenrightbigg (E−H)−1 4(E−H)/parenleftbiggZ r1+Z r2/parenrightbigg , (28) 4π Z[δ3(r1) +δ3(r2)]′≡4π Z[δ3(r1) +δ3(r2)] +2/parenleftbiggZ r1+Z r2/parenrightbigg (E−H) + 2 ( E−H)/parenleftbiggZ r1+Z r2/parenrightbigg . (29) This transformation leads to new form for E′ AandE′ N, such that EA+EN=E′ A+E′ N, (30) E′ A= 2/angbracketleftbigg H′A1 (E−H)′2Z π[δ3(r1) +δ3(r2)]′/angbracketrightbigg , (31) E′ N=/angbracketleftbigg/parenleftbigg E−1 r/parenrightbigg2/parenleftbiggZ r1+Z r2/parenrightbigg +/parenleftbigg E−1 r/parenrightbigg /parenleftbiggZ2 r2 1+Z2 r2 2+ 4Z r1Z r2/parenrightbigg + 2Z r1Z r2/parenleftbiggZ r1+Z r2/parenrightbigg −/parenleftbigg E−1 r+Z r2−p2 2 2/parenrightbigg 4π Z δ3(r1)−Z 4ri r3/parenleftbiggri 1 r3 1−ri 2 r3 2/parenrightbigg +pi 1Z2 r2 1pi 1−p2 2Z r1p2 1+ 2pi 2Z r1/parenleftbiggδij r+rirj r3/parenrightbigg pj 1/angbracketrightbigg −1 4/angbracketleftbiggZ r1+Z r2/angbracketrightbigg /an}b∇acketle{t4π Z(δ3(r1) +δ3(r2))/an}b∇acket∇i}ht+ 2/angbracketleftbiggZ r1+Z r2/angbracketrightbigg /an}b∇acketle{tHA/an}b∇acket∇i}ht. (32) In the numerical calculations of these matrix elements we fo llow the approach developed by Korobov [5]. The Swave function is expanded in the sum of pure exponentials φ=N/summationdisplay i=1vi(e−αir1−βir2−γir−(r1↔r2)), (33) with randomly chosen αi, βi, γiin some speciﬁed limits. This basis set has been proven to give excellent results for the nonrelativistic energy and t he wave function. Moreover, its simplicity allows for the calculations of relativistic cor rections. With basis set N= 1200 we obtained the nonrelativistic energy (without the mass pola rization term p1p2µ/M) E=−2.1752293782367913057(1) , (34) slightly below the previous result in [6]. Expectation valu es of Dirac delta function without and with the mass polarization term are correspondingly: 5/an}b∇acketle{t4π(δ3(r1) +δ3(r2))/an}b∇acket∇i}ht= 33.184142630(1) , (35) /an}b∇acketle{t4π(δ3(r1) +δ3(r2))/an}b∇acket∇i}htMP= 33.184152589(1) . (36) The last one gives the leading hfs in helium, which is Ehfs= 2Z α4µ3 m M\|1 +k\|(1 +a)/an}b∇acketle{tπ(δ3(r1) +δ3(r2))/an}b∇acket∇i}htMP≈6 740 451 kHz , (37) where use values of physical constants from Ref. [7] with one exception [8]. Numerical results for EXwithX=A, B, D, N are presented in Table I. The inversion of H−Ein EAis performed in the similar basis set as for 23S1wave function, however the nonlinear parameters had have to be properly chosen, to obtain a suﬃcie ntly accurate result. Namely, if 0< X, Y, Z < 1 are independent pseudo random numbers with homogeneous di stribution, then: α=A2X−n+A1, (38) β= (B2−B1)Y+B1, (39) γ= (C2−C1)Z+C1. (40) Parameters A, B, C andnare found, by minimization of the second order term with regu - larized Dirac delta on both sides. The inversion of H−EinEBis performed in the basis set of the form φ=r1×r2N/summationdisplay i=1vi(e−αir1−βir2−γir+ (r1↔r2)). (41) Unfortunately, we have not been able to get a reliable number forEDwith this numerical approach. The reason is that operators in EDare so singular, that this basis set gives a very pure convergence. The result presented in Table I, is ob tained analytically within 1 /Z approximation, namely we neglected completely electron-e lectron interaction and corrected this value by factor /an}b∇acketle{tπ(δ3(r1)+δ3(r2))/an}b∇acket∇i}ht/9. The estimated uncertainty is of the order of 10%. The total contribution of m α6term to hfs is E(6)=\|1 +k\|µ3 m Mα6201.0297(5) = 2171 .930(5) kHz , (42) what could compared to the leading Fermi contact interactio n in Eq. (37), E(6)/Ehfs≈ 0.000322. Between all the contributions to He3(23S1) hyperﬁne splitting in Eq.(1), H(5), essentially the nuclear structure contribution, requires input from the nuclear physics to be reliable evaluated. This is the reason we do not present ﬁnal theoretical predictions for hfs, to compare with the precise measurement in [2] Ehfs(He) = 6 739701 .177(16) kHz . (43) Instead, we can compare our result indirectly by subtractin g the ground state hfs of helium ion as measured in [9] Ehfs(He+) = 8 665 649 .867(10) kHz , (44) 6by composing the following diﬀerence ∆Eexp=Ehfs(He)−3 4/an}b∇acketle{tπ(δ3(r1) +δ3(r2))/an}b∇acket∇i}htMP 8Ehfs(He+) =−38.998(19) kHz . (45) In this way, nuclear structure contribution, of order m α5cancels out, as well as the leading Fermi contact interaction. What remain are electron–elect ron correlation eﬀects. Theoreti- cal predictions for this diﬀerence are ∆Eth=\|1 +k\|µ3 m Mα61.9248(5) = 20 .796(5) kHz . (46) We do not associate here the uncertainty due to higher order t erms. A strong disagreement of the theoretical result in Eq. (46) with the experimental o ne in Eq. (45) indicates that the calculations presented here are incomplete or incorrec t. The set of operators in H(6) hfsis the same as in hydrogen, and in fact we rederived for checking the hydrogenic result. We have shown that there are no extra three-body terms due to int ernal cancellations. The whole numerics was performed using multiple precision libr ary by Bailey [10] with 48 digits to avoid possible round-oﬀ error. The higher order QED corre ctions could not explain this discrepancy, because it would require a very large coeﬃcien t∼ −4α/π(Z α)6ln(Z α)−2. However the old calculations in [11] using an approximate Di rac-Hartree wave function, led to the result which is in a much better agreement with experim ent ∆Eth,old=−32(22) kHz. Nevertheless, we think, this result might not reliable at th e precision level of 3 %, which is the discrepancy in question. The most probable explanati on is the correction discovered by Sternheim [12]. It is the second order contribution due to Fermi interaction HA hfsin Eq. (3) with singlet Sintermediate states. It could be understood as a recoil corr ection since it includes additional small factor m/M. The nonvanishing oﬀ–diagonal matrix element between the singlet and triplet S-state is given by HA hfs→δH=Z α 3m Mσn·(σ1−σ2) [δ3(r1)−δ3(r2)] (1 + k) (1 + a), (47) and the correction is δE=/an}b∇acketle{tδH1 (E−H)′δH/an}b∇acket∇i}ht. (48) One would expect the largest contribution coming from 21S0state, since it has the closest energy. Sternheim result is δE=−66.7(3) kHz, what nicely would explain the discrepancy of 69 kHz. However, the inclusion of higher excited states, w ill lead to the inﬁnite result. Moreover, this correction is partially included in /an}b∇acketle{tH(5) hfs/an}b∇acket∇i}ht, and only after proper subtraction it becomes ﬁnite. The complete calculation of the recoil cor rection to helium hyperﬁne splitting is beyond the scope of this work. Nevertheless, it would be a surprising result that relativistic recoil correction which has additional facto rm/M gives larger contribution than the leading relativistic one, to the hyperﬁne structure diﬀ erence between He and He+. ACKNOWLEDGMENTS I gratefully acknowledge helpful information about experi mental results from Peter Mohr. This work was supported by Polish Committee for Scientiﬁc Re search under Contract No. 2P03B 057 18. 7REFERENCES [1] G.W.F. Drake and W.C. Martin, Can. J. Phys. 76, 679 (1998). [2] S.D. Rosner and F.M. Pipkin, Phys. Rev. A, 1, 571 (1970). [3] K. Pachucki, Phys. Rev. Lett. 84, 4561 (2000). [4] K. Pachucki, Phys. Rev. A 56, 297-304 (1997). [5] V.I. Korobov, Phys. Rev. A 61, 064503 (2000). [6] G.W.F. Drake and Z. C. Yan, Chem. Phys. Lett. 229, 486 (199 4). [7] P.J. Mohr and B.N. Taylor, Rev. Mod. Phys. 72, 351 (2000). [8] The experimental value for the helion magnetic moment is somehow problematic. The reference [7] presents values for the screened helion in He3, which involves unknown binding corrections. Therefore we have decided to follow th e reference [2] and adopt the value µh/µp=−0.761 812 0(7) for the helion–proton magnetic moment ratio. [9] H.A. Schuessler, E.N. Fortson, and H.G. Dehmelt, Phys. R ev.187, 5 (1969). [10] D.H. Bailey, ACM Trans. Math. Softw. 19, 288 (1991); 21, 379 (1995); see also www.netlib.org/mpfun. [11] A.M. Sessler and H.M. Foley, Phys. Rev. 98, 6 (1955). [12] M.M. Sternheim, Phys. Rev. Lett. 15, 545, (1965). 8FIGURES III IVI II FIG. 1. Time ordered diagrams contributing to helium hyperﬁ ne structure at order m α6. Dashed line is a Coulomb photon, the wavy line is the transver se photon, the thicker vertical line denotes nucleus, two other electrons. 9TABLES contribution \|1 +k\|m2/M α6 EA 202.6761 EB 0.0059 ED 0.0054(5) EN -1.6577 E 201.0297(5) 24/an}b∇acketle{tπ(δ3(r1) +δ3(r2))/an}b∇acket∇i}ht 199.1049 ∆E 1.9248(5) ∆E(exp) -3.6095(18) TABLE I. Numerical results for contributions at order m α6to helium hyperﬁne structure. The factor 24 in the above comes from Breit correction Eq.(18) wi thn= 1 times 3/4 from spin algebra times Z6/8. 10
arXiv:physics/0103080v1 [physics.comp-ph] 26 Mar 2001Gordian unknots P. Pieranski1, S. Przybyl1and A. Stasiak2 1Poznan University of Technology e-mail: Piotr.Pieranski@put.poznan.pl Nieszawska 13A, 60 965 Poznan, Poland 2University of Lausanne, Switzerland February 9, 2008 Abstract Numerical simulations indicate that there exist conformat ions of the unknot, tied on a ﬁnite piece of rope, entangled in such a mann er, that they cannot be disentangled to the torus conformation witho ut cutting the rope. The simplest example of such a gordian unknot is pre sented. Knots are closed, self-avoiding curves in the 3-dimensiona l space. The shape and size of a knot, i.e. its conformation, can be changed in a v ery broad range without changing the knot type. The necessary condition to k eep the knot type intact is that during all transformations applied to the kno t the curve must remain self-avoiding. From the topological point of view, a ll conformations of a knot are equivalent but if the knot is considered as a physic al object, it may be not so. Let us give a simple example. Take a concrete, knott ed space curve K. Imagine, that Kis inﬂated into a tube of diameter D. IfKis scaled down without scaling down D, then there is obviously a minimum size below which one cannot go without changing the shape of K. Diminishing, in a thought or computer experiment, the size of a knot one arrives to the lim it below which in some places of the knot the impenetrability of the tube on whi ch it has been tied would be violated. Consider a knot tied on a piece of a rope. If the knot is tied in a loose manner, one can easily change its shape. However, the range o f transformations available in such a process is much more narrow than in the cas e of knots tied on an inﬁnitely thin rope. Limitations imposed on the transf ormations used to change the knot shape by the ﬁxed thickness and length of the r ope may make some conformations of the knot inaccessible from each other . The limitations can be in an elegant manner represented by the single conditi on that the global curvature of the knot cannot be larger than 2 /D[1]. That it is the case we shall try to demonstrate in the most simple case of the unknot . The knot is a particular one since we know for it the shape of the ideal, lea st rope consuming conformation [2]. The simplest shape of the unknot is obviou sly circular. If the knot is tied on the rope of diameter Dthe shortest piece of rope one must use to 1Figure 1: SONO disentagles an unknot entagled in a simple man ner. How the length of the rope changes in this process is shown in Fig.2 (l ower curve). form it has the length Lmin=πD. If one starts from the circular conformation of the unknot tied on a longer piece of rope, the length of the r ope can be subsequently reduced without changing the circular shape u ntil the Lminvalue is reached. Consider now a diﬀerent, entangled conformation of the unkn ot tied on a piece of rope having the length L > L min. Can it be disentangled to the canonical circular shape? Are there such conformations of t he unknot, which cannot be disentangled to a circle without elongating the ro pe? For obvious reasons we propose to call such conformations gordian. In wh at follows we shall report results of numerical experiments suggesting e xistence of the gordian conformations of the unknot. Imagine that the entangled conformation of the unknot is tie d on piece the ideal rope of diameter Dand length L > Lmin . The ideal rope is perfectly ﬂexible but at the same time perfectly hard. Its perpendicul ar cross-sections remain always circular. The diameters of all the cross-sect ions are equal D. None of the circular cross-sections overlap. The surface of the rope is per- fectly slippery. In such conditions one may try to force the k not to disentangle itself just by shortening the rope length. Such a process, in which the knot is tightened, can be easily simulated with a computer. The de tails of SONO (Shrink-On-No-Overlaps), the simulation algorithm we dev eloped, are described elsewhere[3]. As shown in [3], SONO disentangles some simpl e conformations of the unknot. See Fig.1. It manages to cope also with the more complex con- formation proposed by Freedman [4] disentangled previousl y by the Kusner and Sullivan algorithm minimizing the M¨ obius energy [5]. The steps of the construction of the Freedman conformation, are as follows 2Figure 2: Evolution of the lenght of the rope in a process in wh ich SONO disen- tagles the Freedman’s F(31,31) conformation of the unknot. Initially, the loose F(31,31) conformation is rapidly tightened. Then, the evolution sl ows down. At the end of the slow stage one of the end knots becomes untied . Subsequently, the other of the end knots becomes untied. Eventually the con formation be- comes disentagled and the unknot reaches its ideal, circula r shape. The lower curve shows the evolution of the rope lenght in the much faste r process in which the unknot shown in Fig.1 becomes disentangled. [6]: 1. Take a circular unknot and splash it into a ﬂat double rope b and. 2. Tie overhand knots on both ends of the band and tighten them . (From the point of view of the knot theory, the overhand knots are op en trefoil knots.) 3. Open and slip the end loops over the bodies of the overhand k nots, so that they meet in the central part of the band. 4. Move the rope through both overhand knots so that the loops become smaller. In what follows we shall refer to the conformation as F(31,31). To disentan- gleF(31,31), one must slip the loops back all around the bodies of the ove rhand knots, which is diﬃcult, since the move needs ﬁrst making the loops bigger. How the SONO algorithm copes with this task is shown in Fig.2, where con- 3Figure 3: SONO tightens the F(51,51) conformation of the unknot, but does not manage to disentangle it. secutive stages of the disentangling process are shown. Tig htening the F(31,31) conformation SONO algorithm brings it to the very compact st ate, which seems at the ﬁrst sight to be impossible to disentangle. The end loo ps are very tight and they seem to be too small to slip back over the bodies of the overhand knots. However, as the computer simulations prove, there exists a p ath in the conﬁg- urational space of the knot along which the loops slowly beco me bigger and one of them slips over the body of the overhand knot. Then, the disentangling process proceeds without any problems. Results of the compu ter experiments we performed suggest strongly, that the F(31,31) conformation is not gordian. The construction of original Freedman entanglement may be m odiﬁed mak- ing it more diﬃcult to disentangle. The simplest way of doing this is to change the end trefoil knots to some more complex knots. For the sake of brevity we will use F(K(1), K(2)) symbols to indicate with what kind of the Freedman con- formation of the unknot we are dealing with. Results of compu ter simulations we performed prove that the F(41,41) conformation is also disentangled in the knot tightening process. However, the F(51,51) conformation proves to be re- sistant to SONO algorithm. Fig.3 shows consecutive stages o f the tightening process. The initial conformation, is loose, it becomes tig ht soon. Then the evolution process slows down and eventually stops. The ﬁnal conformation is proves to be stable. The gordian conformation has been reach ed. Eperimenting with knots tied on real, macroscopic ropes or t ubes is by no means easy [7]. First of all, the surface of any real rope is ne ver smooth and strong friction often stops the walk within the conﬁguratio nal space of a knot tied on such a rope. The role of friction was exposed by Kauﬀma n [8]. Fric- tion can be signiﬁcantly reduced, however, when a knot is tie d on a smooth nanoscopic ﬁlament, e.g. a nanotube, or on a thermally ﬂuctu ating polymer 4molecule [9]. There exists another, less obvious, factor wh ich makes laboratory experiments on knots diﬃcult: the Berry’s phase [10], to be m ore precise, its classical counterpart - the Hannay’s angle [11]. Modern rop es are often con- structed in the following manner: a parallel bundle of smoot h ﬁlaments is kept together by a tube-like, plaited cover. As easy to check, suc h ropes are much easier to bend than to twist. Forming a knot on a rope, one has t o deform it. In view of what was said above, the deformation applied is rat her bending than twisting. Avoiding the twist deformations one follows the p rocedure known as the parallel transport. As a result, when at the ﬁnal stage of the knot tying procedure the ends of the rope meet, they are in general rotat ed in relation to each other: the misﬁt angle Ais the Hannay’s angle. As shown in [12] and [13], the Hannay’s angle Astays in a simple relation, 1 +Wr= (A/2π)mod 2 with the writhe Wrof the knot into which the rope has been formed. Splicing the ends of the rope one ﬁxes the misﬁt angle A. Consequently, the writhe value Wrbecomes ﬁxed as well. As a result, any further changes of the c onformation of the knot become very diﬃcult and are basically restricted to the manifold of constant writhe. (The speciﬁc construction of the Freedm an conformations makes them achiral [14]. Their writhe is equal zero.) The natural question arises, if the impossibility of disent angling the gordian conformation does not stem from the described above frictio n and writhe factors. We feel emphasize, that it is not the case. The rope simulated by the SONO algorithm is perfect: it is frictionless and utterly ﬂexibl e. It has no internal, parallel bundle structure and it accepts any twist. Problem s with disentangling the gordian conformations are purely steric. Tightening th eF(51,51) Freedman conformation SONO brings it into a cul-de-sac of what mathem aticians call thickness energy [15]. To get out of it, one needs elongate th e rope. By how much? We do not know yet the answer to this question. We thank Jacques Dubochet, Giovanni Dietler, Kenneth Mille tt, Robert Kusner, Alain Goriely, Eric Rawdon, Jonathan Simon, Gregor y Buck and Joel Hass for helpful discussions and correspondence. PP thanks the Herbette Foun- dation for ﬁnancial support during his visit in LAU. This wor k was carried out under Project KBN 5 PO3B 01220. References [1] O. Gonzalez and J. H. Maddocks, Proc. Natl. Acad. Sci. 96, 4769 (1999). [2] V. Katritch et al. Nature 384, 142 (1996). [3] P. Pieranski in Ideal Knots , edited by A. Stasiak, V. Katritch and L. H. Kauﬀman (World Scientiﬁc, Singapore, 1998). [4] M. Freedman, Z.-X. He, Z. Wang, Annals of Math. 139, 1 (1994). 5[5] R. B. Kusner and J. M. Sullivan in Ideal Knots , edited by A. Stasiak, V. Katritch and L. H. Kauﬀman (World Scientiﬁc, Singapore, 199 8). [6] An equivalent prescription for creation of the Freedman conformations of the unknot was formulated by Joel Hass (private communicati on). [7] G. Buck in Ideal Knots , edited by A. Stasiak, V. Katritch and L. H. Kauﬀ- man (World Scientiﬁc, Singapore, 1998). [8] L. H. Kauﬀman, Knots and physics , (World Scientiﬁc, 1993. [9] P.-G. de Gennes, Macromolecules 17, 703 (1985) [10] M. V. Berry, Nature 326, 277 (1987). [11] J. H. Hannay, J. Phys. A 31,L321 (1998). [12] Phys. Rev. Lett. 85, 472 (2000). [13] J. Aldinger, I. Klapper and M. Tabor, (unpublished). [14] C. Liang and K. Mislow, J. Math. Chem. 15, 1 (1994). [15] See chapters by O. Hara, Simon and Rawdon in Ideal Knots , edited by A. Stasiak, V. Katritch and L. H. Kauﬀman (World Scientiﬁc, S ingapore, 1998). 6This figure "Fig1.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103080v1This figure "Fig2.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103080v1This figure "Fig3.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103080v1
arXiv:physics/0103081v1 [physics.plasm-ph] 26 Mar 2001Instability of Shear Waves in an Inhomogeneous Strongly Cou pled Dusty Plasma Amruta Mishra∗, P. K. Kaw and A. Sen Institute for Plasma Research, Bhat – 382 428, India It is demonstrated that low frequency shear modes in a strong ly coupled, inhomogeneous, dusty plasma can grow on account of an instability involving the dy namical charge ﬂuctuations of the dust grains. The instability is driven by the gradient of the equilibrium dust charge density and is associated with the ﬁnite charging time of the dust grains. T he present calculations, carried out in the generalized hydrodynamic viscoelastic formalism, als o bring out important modiﬁcations in the threshold and growth rate of the instability due to collecti ve eﬀects associated with coupling to the compressional mode. PACS numbers: 52.25G, 52.25Z, 52.35F I. INTRODUCTION Dusty plasmas are of great interest because of their possibl e applications to a number of ﬁelds of contemporary research such as plasma astrophysics of interplanetary and interstellar matter, fusion research, plasmas used for semiconductor etching, arc plasmas used to manufacture ﬁne metal and ceramic powders, plasmas in simple ﬂames etc. [1]. It is now widely recognized that the dust component in these plasmas is often in the strongly coupled coulomb regime with the parameter, Γ ≃(Zde)2/Tdd, typically taking values much greater than unity (( −Zde) is the charge on the dust particle, d≃n−1/3 dis the interparticle distance and Tdis the temperature of the dust component). This leads to many novel physical eﬀects such as the formation of d ust plasma crystals [2], modiﬁed dispersion of the compressional waves [3,4], the existence of the transverse shear waves [4] etc. Many of these novel features have now been veriﬁed by experiments and computer simulations [5]. Recently, an experiment on the self–excitation of the verti cal motion of the dust particles trapped in a plasma sheath boundary, has been reported [6]. The physics of this e xcitation is related to charging of the dust particles by the inﬂow of ambient plasma currents in the inhomogeneous pl asma sheath and the delay resulting because of the ﬁnite time required by the charging process to bring the dust charge to its ambient steady state value. In this paper, we demonstrate that the same physical mechanism can be used f or the excitation of the transverse shear modes in an inhomogeneous strongly coupled dusty plasma. Using a gen eralized hydrodynamic viscoelastic formalism [7] to describe the strongly coupled dusty plasma and incorporati ng the novel feature of time variation of the dust charge through a charge dynamics equation [8], we have derived a gen eral dispersion relation for low frequency shear and compressional modes in the plasma. We ﬁnd that in a plasma wit h ﬁnite gradients of the equilibrium dust charge density, the two modes are coupled and we show that the shear m ode is driven unstable if certain threshold values are exceeded. Our paper is organized as follows. In the next section we brie ﬂy discuss the equilibrium of an inhomogeneous dusty plasma that is conﬁned against gravity by the electric ﬁeld o f a plasma sheath. In such a conﬁguration dust particles of varying sizes and charges arrange themselves in horizont al layers at diﬀerent heights to form a nonuniform cloud [9,10]. In section 3 we carry out a linear stability analysis of such an equilibrium in the framework of the generalized hydrodynamic equations. The dispersion relation of the cou pled shear wave and compression wave is solved analyt- ically (in simple limits) as well as numerically in section 4 . The physical mechanism of the shear wave instability is also discussed and the modiﬁcations in the threshold and gro wth rate brought about by the coupling to compressive waves are elucidated. Section 5 is devoted to a summary and di scussion of the principal results. ∗Electronic mail: am@plasma.ernet.in 1II. DUST CLOUD EQUILIBRIUM We consider an inhomogeneous sheath equilibrium in which th e dust particles are suspended with electric ﬁeld forces balancing the gravitational force on the particle an d in which the dust charge ( −Zde) and dust size rdare both functions of the vertical distance z. Then the force balance equation gives, Zd(z)eE0(z) =4 3πrd(z)3ρg, (1) where, ρ,g,E0refer respectively to the dust mass density, gravitational acceleration and the sheath electric ﬁeld. For particle sizes of the order of a few microns, other f orces acting on the particle (such as the drag and viscous forces) are about an order of magnitude smaller than the elec tric and gravitational forces and can be neglected for the equilibrium calculation [10]. Note that for dust partic les of a uniform size (monodispersive size distribution) th e above equilibrium can only be attained at one vertical point leading to a monolayer of dust. A dispersion in sizes leads to a large number of layers resulting in a nonuniform dust clo ud with a gradient in the equilibrium charge ( −Zde) and the dust size rd. The electric ﬁeld E0is determined by the sheath condition, dE0 dz=−4πe(ne−ni+Zdnd) (2) where ne,i,d are the local electron, ion and dust densities respectively . The charge ( −Zde) on a dust particle in the sheath region is given by ( −Zde) =Cd(φf−φ) where Cdis the capacitance, φfis the ﬂoating potential at the surface of the dust particle and φis the bulk plasma potential. For a spherical dust particle Cd=rd, and the ﬂoating potential can be determined by the steady state condition fr om the dust charging equation, namely, [8] Ie+Ii= 0 (3) where the electron and ion currents impinging on the dust par ticle are given by [1] Ie=−πr2 de/parenleftbigg8kTe πme/parenrightbigg1/2 neexp/bracketleftbigge kTe(φf−φ)/bracketrightbigg , (4a) Ii=πr2 de/parenleftbigg8kTi πmi/parenrightbigg1/2 ni/bracketleftbigg 1−2e kTi+miv2 id(φf−φ)/bracketrightbigg . (4b) HereTeandTiare the electron and ion temperatures, miis the ion mass and vidis the mean drifting velocity of the ions in the electric ﬁeld of the sheath (it is assumed to be the ion sound velocity at the sheath edge). We also assume that the dust particles have much smaller thermal velocitie s than the electrons and ions. Equations (1 - 3) selfconsistently determine the equilibri um of the dust cloud. Such clouds have been experimentally observed in a number of experiments [9,10]. In [10] theoreti cal modeling along the lines discussed above, agree very well with the experimental observations of clouds formed wi th polydispersive particle size distribution of dust parti cles trapped in the plasma sheath region. A typical equilibrium v ariation of the dust particle size with the vertical distanc e, when the Child Langmuir law holds for the plasma sheath poten tial, is given as [10], rd=/parenleftbigg3(φf−φ) 4πρg/parenrightbigg1/2/parenleftbigg6πensCs µi/parenrightbigg1/3 (δ−z)1/3(5) where ns,Csare the plasma density and the ion sound velocity, δis the thickness of the sheath and µi= (eλi−n/mi)1/2withλi−nrepresenting the mean free path of ions colliding with the ba ckground neutrals. Using (5) we can obtain the corresponding zvariation for Zd. As discussed in detail in [10], this dust cloud equilibrium i s conﬁned to the plasma sheath boundary region in the potential well created from the upward electrostatic and do wnward gravitational forces. Note that the force balance equation (1) does not prevent the particles from oscillatin g about their mean positions especially if they have signiﬁ- cant kinetic energy or temperature. However their mean posi tions are at various vertical distances and the mean Zd is a function of z. This is reminiscent of particle gyrations in a magnetic ﬁel d. If we consider wave motions in which dust oscillation excursions are much smaller than waveleng ths, we can use a ﬂuid theory to analyze such behaviour. In the next section, we adopt this view point and carry out a li near stability analysis of the equilibrium discussed above to low frequency wave perturbations. 2III. LINEAR STABILITY ANALYSIS For low frequency perturbations in the regime kvthd<< ω << kv the, kvthi, where vthd,vtheandvthiare the thermal velocities of the dust, electron and ion components respectively, the electron and ion responses obey the Boltzmann law which can be simply obtained from an ordinary h ydrodynamic representation. The dust component on the other hand can be in the strongly coupled regime for whi ch a proper description is provided by the generalized viscoelastic formalism. Using such a description a general dispersion relation for low frequency waves (with typical wavelengths longer than any lattice spacings) was obtained in [4] for longitudinal sound waves and transverse shear waves. The shear modes exist in a strongly coupled dusty plas ma because of elasticity eﬀects introduced by strong correlations [4]. Our objective in this work is to look for th e eﬀect of dust charge dynamics on these shear modes in the strongly coupled regime. As demonstrated in our earlier work [4], the coupling of the low frequency shear modes to transverse electromagnetic perturbations is ﬁnite but n egligibly small; we ignore this coupling here. However, introduction of the dust charge dynamics in the inhomogeneo us plasma leads to a coupling of the low frequency shear and compressional modes; thus the space charge dynamics and quasineutrality condition play an important role in describing the perturbations. The basic equations for the d ust ﬂuid [7] we work with, are the continuity equation, ∂ ∂tδnd+nd0/vector∇ ·δ/vector ud+nd0 M(δ/vector ud·/vector∇)M= 0, (6) the equation of motion, /parenleftBig 1 +τm∂ ∂t/parenrightBig/bracketleftBigg/parenleftBig∂ ∂t+ν/parenrightBig /vectorδud+/vector▽δP Mnd0+Zde M/vectorδE +δZd Me/vectorE0/bracketrightBigg =1 Mnd0/bracketleftBig η/vector▽2/vectorδud+/parenleftbig ζ+η 3/parenrightbig/vector▽(/vector▽ ·/vectorδud)/bracketrightBig , (7) and the equation of state, ( ∂P/∂n )T≡MC2 d, given in terms of the compressibility, µd, as [4] µd≡1 Td/parenleftBig∂P ∂n/parenrightBig T= 1 +u(Γ) 3+Γ 9∂u(Γ) ∂Γ, (8) with the excess internal energy of the system given by the ﬁtt ing formula [11] u(Γ) = −0.89Γ + 0 .95Γ1/4+ 0.19Γ−1/4−0.81. (9) In the above, Mis the dust mass, νis the dust–neutral collision frequency, δud,δndandδZdare the perturbations in the dust velocity, number density and dust charge, δP,δEare the pressure and electric ﬁeld perturbations, nd0 andZdare the equilibrium number density and charge for the dust an dE0is the unperturbed electric ﬁeld. ηandζ refer to the coeﬃcients of the shear and bulk viscosities and τmis the viscoelastic relaxation time. Note that in the continuity equation we have a contribution from the equilib rium inhomogeneity in the dust mass distribution (arising from the size dispersion of the particles). This term as we sh all see later modiﬁes the real frequency of the shear waves. These equations are supplemented with the dynamical equati on for the dust charge perturbations which, for per- turbations with phase velocity much smaller than the electr on and ion thermal velocities, is given as [8] ∂ ∂t(δZd) +/vectorδud·/vector▽Zd+ηcδZd=−\|Ie0\| e/parenleftBigg δni ni0−δne ne0/parenrightBigg , (10) where, ηc=/parenleftBig e\|Ie0\|/C/parenrightBig/parenleftBig 1/Te+ 1/w0/parenrightBig is the inverse of charging time of dust grains and w0=Ti−e(φf−φ)0. Note that the second term on the left hand side of eq.(10) arises be cause of the inhomogeneity of the mean charge on the dust particles; as shall be shown later, this is the criti cal term responsible for the instability. It is also obvious that the dust charge variation in space will lead to shieldin g by electrons and ions with the associated coupling of the perturbation to dust compressional modes. We must thu s extend the above set of equations to include the quasi-neutrality condition, δne+Zdδnd+nd0δZd−δni≃0, (11) 3and the equation describing the electron and ion density per turbations in terms of the potential, as δne ne0=eδφ Te;δni ni0=−eδφ Ti. (12) These are the Boltzmann relations which arise whenever the p erturbations satisfy ω << kv the, kvthi. We shall next derive the dispersion relation for the low freq uency mode. We may note that the typical time scale for the decay of the charge ﬂuctuations for the dust can be ver y small [6], with ηc>> ω and we shall work in that limit. We use the local approximation (wave lengths smaller than characteristic equilibrium scale lengths) and choose the propagation vector for the wave perturbation as /vectork= (k,0,0), the perturbed dust velocity, /vectorδud= (δu1,0, δu3) and the perturbation in the electric ﬁeld as /vectorδE=−ikδφ(1,0,0). Using the continuity equation (6) and the equations (10) – (12), and after some simple algebra, one obtains the ﬂuctua tion in the dust charge and the potential as δZd=a1 D/parenleftbiggk ω/parenrightbigg δu1+/parenleftBiga2 D+a3 (iω)D/parenrightBig δu3, (13a) δφ=−Zdnd0ηc eD/parenleftbiggk ω/parenrightbigg δu1+nd0 eD/parenleftBig Z′ d−ZdM′ηc M(iω)/parenrightBig δu3, (13b) where, a1=−\|Ie0\| e/parenleftBigg 1 Te+1 Ti/parenrightBigg Zdnd0;a2=−Z′ d/parenleftBigg ne0 Te+ni0 Ti/parenrightBigg , a3=−\|Ie0\| e/parenleftBigg 1 Te+1 Ti/parenrightBigg M′ Mnd0Zd D=ηc/parenleftBigg ne0 Te+ni0 Ti/parenrightBigg +nd0\|Ie0\| e/parenleftBigg 1 Te+1 Ti/parenrightBigg , (14) and the primes denote derivatives with respect to zthe vertical direction. We then write down the longitudinal and transverse components of the dust momentum equation (i.e. o f equation (7)), as (1−iωτm)/bracketleftBig (−iω+ν)δu1+ikδP Mnd0−Zde M(ikδφ)/bracketrightBig =−1 Mnd0ηlk2δu1 (15a) (1−iωτm)/bracketleftBig (−iω+ν)δu3+δZd MeE0/bracketrightBig =−1 Mnd0ηk2δu3, (15b) where, ηl=4 3η+ζ. In the limit ωτm>>1, using equations (13)– (15), we obtain the dispersion rela tion for the coupled shear–compressional mode, as /bracketleftBig ω2+iων+iωeE0 MDa2+eE0 MDa3−C2 shk2/bracketrightBig/bracketleftBig ω2+iων−C2 DAk2/bracketrightBig −iωk2eE0 MDa1ZdZ′ dnd0 MD+k2eE0 MDa1M′ M/parenleftbig C2 d+C2 da/parenrightbig = 0, (16) where C2 sh= (η/Mn d0τm),C2 da= (Z2 dnd0ηc/MD) and C2 DA=C2 d+C2 da+ (ηl/Mn d0τm). In the above equation the expression in the ﬁrst set of brackets represents the disper sion relation for the transverse shear wave, the second set of brackets contains the compressive mode dispersion relat ion and the ﬁnal two terms denote the coupling between the two branches. We will now study the behaviour of the shear mode in the presence of the charge inhomogeneity and the coupling to the compressive mode. 4IV. SHEAR WAVE INSTABILITY In the limit when the coupling to the compressive wave is weak , so that the last two terms in the dispersion relation (16) can be neglected, we can obtain the roots for the shear br anch as, ω=−i 2/parenleftBig ν+eE0 MDa2/parenrightBig ±/bracketleftBig k2C2 sh−eE0 MDa3−1 4/parenleftBig ν+eE0 MDa2/parenrightBig2/bracketrightBig1/2 . (17) In the absence of the inhomogeneities and the collision term , this is the basic shear wave described in [4]. The collisional term introduces wave damping. The inhomogeneo us terms introduce two important modiﬁcations. The term proportional to the mass (size) inhomogeneity contrib utes to the real part of the frequency whereas the charge inhomogeneity term can drive the wave unstable if E0a2<0 (i.e., E0Q′ 0<0) and the threshold condition ν <\|eE0 MDa2\| is satisﬁed. Physically, this instability arises because o f delayed charging eﬀect, the same physical mechanism which was used by Nunomura et al[6] to explain the observed instability of single particle v ertical displacement in their sheath experiments. Speciﬁcally, the charge on the vertica lly oscillating dust particle in the shear wave propagating in the inhomogeneous plasma, is always diﬀerent from the equ ilibrium value Zdbecause of the ﬁnite charging time η−1 c. This perturbation is of order δZd≃Z′ dδu3/ηcand leads to an energy exchange between the shear wave and the ambient electric ﬁeld at a rate δZdE0δu∗ 3≈ \|E0Z′ d\|\|δu3\|2/ηc. When this energy gain by the shear wave exceeds the loss rate due to collisions ≈νM 2\|δu3\|2, we have an instability. This gives us the approximate thres hold condition described above. If we express the dust neutral collision fr equency, νin terms of the ambient neutral pressure as ν=p/parenleftbig2mn Tn/parenrightbig1/2πa2 M, our threshold condition is functionally identical to that derived by Nunomura et al[6] on the basis of physical arguments. The only substantial diﬀerence is th eir use of exponential charging time which follows from our equation (10) viz. δZd≈(δu3Z′ d/ηc)[1−exp(−ηct)]; since we have assumed the frequency of the shear mode ω << η c, we use the asymptotic condition described above. We now demonstrate that for the collective shear mode being d escribed here, the coupling to the compressional dust acoustic wave due to the last two terms in equation (16) i s very crucial; thus the above single particle results are strongly modiﬁed by the hydrodynamic treatment. A simple an alytic result clearly demonstrating the modiﬁcation is obtained by neglecting ω2+iωνcompared to k2C2 DAin the second bracket of equation (16); this is reasonable wh en the wave–vector kis not too small. In this limit, the shear modes are described by the root ω=−i 2/parenleftBig ν+eE0 MD/parenleftBig a2+a1ZdZ′ d MDnd0 C2 DA/parenrightBig/parenrightBig ±/bracketleftBig k2C2 sh−eE0 MD/parenleftBig a3−a1M′ M(C2 d+C2 da) C2 DA/parenrightBig −1 4/parenleftBig ν+eE0 MD/parenleftBig a2+a1ZdZ′ d MDnd0 C2 DA/parenrightBig/parenrightBig2/bracketrightBig1/2 . (18) We thus note that the threshold condition and the growth rate s are signiﬁcantly modiﬁed by the inclusion of coupling to compressional waves. In order to quantitatively illustr ate the eﬀect of coupling terms, we now present a detailed numerical investigation of the dispersion relation equati on (16). It is generally the case that the bulk viscosity coeﬃcient ζis negligible compared to the shear viscosity coeﬃcient, η, particularly in the one component plasma (OCP) limit [7] and so we shall drop it in our calculations. Fu rther, the viscoelastic relaxation time, τm, is given as [4], τm=4η 3nd0Td(1−γdµd+4 15u)(19) withγdas the adiabatic index and the compressibility, µddeﬁned through (8). We assume the gradient of the equilibriated dust charge to be of the form, Z′ d=Zd/LZ, the mass gradient to be of the form M′=M/L Mwhere LZ∼LM=Lis a few Debye lengths. In our computations, we choose L≈5 times the Debye length, which is the typical order of magnitude as observed experimentally [10] . For further computations, we introduce the dimensionless quantities, ˆω=ω/ω pd; ˆν=ν/ωpd;ˆk=kd; ˆτm=τmωpd; ˆη=η Mnd0ωpdd2;ˆC2 α=C2 α/(ω2 pdd2);α≡sh, d, da, DA, e0=eE0 MDa2 ωpd;e1=eE0 MDa3 ω2 pd; e01=a1ZdZ′ dnd0 a2MD1 ω2 pdd2;e11=eE0 MDa1M′ M(ˆC2 d+ˆC2 da)1 ω2 pd, (20) 5where ωpdanddare the dust plasma frequency and the inter–grain distance r espectively. The dispersion relation for the shear mode (16) can then be written as /bracketleftBig ˆω2+iˆωˆν+iˆωe0+e1−ˆC2 shˆk2/bracketrightBig/bracketleftBig ˆω2+iˆωˆν−ˆC2 DAˆk2/bracketrightBig −iˆωˆk2e0e01+ˆk2e11= 0, (21) Equation (21) has been solved numerically for the shear mode roots and some typical results are presented in ﬁgures (1) and (2). Figures (1a) and (1b) display a comparison of the dispersion curve for the shear mode (ˆ ωRvsˆkand ˆγvsˆk for ﬁxed values of e0=−0.0008 and e1=−0.05), with and without the inclusion of the coupling to the com pressional mode. The various ﬁxed parameter values corresponding to th ese curves are ˆC2 sh= 0.02,ˆC2 DA= 0.4, ˆν= 0.0004 and e01= 0.3,e11=−0.01 when the coupling is on. The choice of these numerical valu es for the dimensionless parameters ˆν,ˆk,e0,e01,ˆCshandˆCDAhas been guided by the magnitude of these quantities observe d in some of the laboratory plasmas [9,10]. It is seen from these plots that there is a sub stantial inﬂuence of the compressional mode coupling, described through the parameter, e01, e11, on the growth rate and the real frequency of the shear wave em phasizing the importance of the collective physics of coupling to the c ompressional mode. We next plot in ﬁgure (2) the gas pressure, p, versus ne0proﬁles for various values of γ, the imaginary part of ω. Plotting the γ= 0 curve, we get a threshold relation between pandne0, where we ﬁx the other parameters as follows – dust radius, rd=2.5 microns, the inter–grain distance, d=430 microns, Te=Ti≃1eV,kd= 1, and dust mass density, ρd=2.5 gms/cm3. We see that the qualitative trend of the curve is similar to that observe d in the single particle instability studies of [6] illustra ting the commonality of the underlying physical mechanism. Howe ver it should be emphasized that the experiment in [6] did not observe any collective excitations and their equ ilibrium consisted of a monolayer of equal sized particles. The equilibria of [9,10] are more appropriate for observing collective excitations of shear waves and our theoretical results can be usefully employed in such a situation. In Fig. (2) we have once again highlighted the signiﬁcance of the coupling to the compressive wave, in this case for its eﬀe ct on the threshold values, by displaying the uncoupled threshold and growth rate curves (dashed curves). Note that the inﬂuence of the coupling is to raise the threshold value at low values of ne0(i.e. a higher value of pis needed to excite the instability) whereas it reduces the t hreshold at the higher end of the ne0scale. The rest of the curves displayed in the ﬁgure (2) corre spond to the various positive values of γ, which correspond to the situation where the shear mode is ex cited and saturates at some values. These ﬁgures are again qualitatively similar to the curves obtain ed in [6] for various saturation amplitudes. However a direc t comparison is again not appropriate for the reason discusse d above and also because our calculations are linear and cannot provide any quantitative results about nonlinearly saturated amplitudes. V. CONCLUSION AND DISCUSSION To summarize, in this paper we have investigated the stabili ty of a low frequency shear mode in an inhomogeneous dusty plasma in the strongly correlated regime. The equilib rium dust cloud has both an inhomogeneity in the dust charge distribution and in the dust mass distribution (aris ing from a distribution in the sizes of the dust particles). The shear mode in such a plasma undergoes two signiﬁcant modi ﬁcations. Its real frequency is shifted by a contri- bution from the mass inhomogeneity and the dust charge inhom ogeneity can drive it unstable through the dynamics of dust charge ﬂuctuations in a manner very similar to the ins tability of the vertical motion of single particles in a plasma sheath as observed in the recent experiment of Nunom uraet al[6]. The ﬁnite charging time, η−1 cof the dust particles plays a critical role in the instability. We a lso show how collective eﬀects due to coupling with the compressional modes strongly modify the threshold conditi ons for the instability as well as its growth rate and real frequency. Our calculations have been carried out in the hyd rodynamic formalism including viscoelastic eﬀects and we have neglected any kinetic eﬀects. Our results are therefor e strictly valid in the low frequency limit. Finite correcti ons arising from kinetic eﬀects can occur at higher frequencies and wave numbers. This has recently been demonstrated for the compressive dust acoustic mode in a dusty plasma from a kinetic calculation based on the dynamic local ﬁeld correction (DLFC) method [12]. Such corrections, if any, fo r the transverse shear mode has not yet been done and needs to be examined. Finally we would like to remark that the transverse dust shea r mode which is a collective mode of the strongly coupled plasma regime has only been observed in computer sim ulations till now; its detailed experimental investigatio n is therefore of great current interest. Such waves can be exc ited in inhomogeneous dust clouds that have been obtained in the experiments carried out with varying grain sizes [9,1 0]. It would be of interest therefore to look for the wave features discussed in our model calculations in controlled propagation experiments on such equilibria. It is also apparent that free energy sources, such as ion beams, which m ay readily couple with the compressional waves may also 6be useful for exciting the more interesting shear waves in th e strongly coupled inhomogeneous plasma. Investigation of these and related eﬀects are in progress. [1] E. C. Whipple, T. G. Northrop and D. A. Mendis, J. Geophys. Res. 90, 7405 (1985); E. C. Whipple, Rep. Prog. Phys. 44, 1197 (1981); D. A. Mendis and M. Rosenberg, Ann. Rev. Astron. Astrophys. 32,419 (1994); U. de Angelis, Phys. Scr. 45, 465 (1992); C. K. G¨ oertz, Rev. Geophys. 27, 271 (1989); M. Ho ranyi, H. L. F. Houpis and D. A. Mendis, Astrophys. Space Sci. 144, 215 (1988); V. N. Tsytovich, G. E. Morﬁll, R. Bingha m and U. de Angelis, Comments Plasma Phys. Controlled Fusion 13, 153 (1990); G. S. Selwyn, J. Singh and R. S. Bennet, J. Vac. Sci. Technol A7, 2758 (1989). [2] H. Ikezi, Phys. Fluids 29, 1764 (1986); H. Thomas, G. E. Mo rﬁll, V. Demmel, J. Goree, B. Feuerbacher and D. Mohlmann, Phys. Rev. Lett. 73, 652 (1994). [3] M. Rosenberg and G. Kalman, Phys. Rev. E56, 7166 (1997). [4] P. K. Kaw and A. Sen, Phys. Plasmas 5, 3552 (1998). [5] J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996) ; P. Schmidt, G. Zwickmagel, P. G. Reinhard and C. Toepﬀer, Phys. Rev. E56, 7310 (1997). [6] S. Nunomura, T. Misawa, N. Ohno and S. Takamura, Phys. Rev . Lett. 83, 1970 (1999). [7] S. Ichimaru, H. Iyetomi and S. Tanaka, Phys. Rep. 149, 91 ( 1987); M. A. Berkovsky, Phys. Lett. A166, 365 (1992); S. Tanaka and S. Ichimaru, Phys. Rev. A 35, 4743 (1987). [8] M. R. Jana, A. Sen and P. K. Kaw, Phys. Rev. E48, 3930 (1993) ; J. R. Bhatt and B. P. Pandey, Phys. Rev. E 50, 3980 (1994). [9] J. H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). [10] S. Nunomura, N. Ohno and S. Takamura, Phys. Plasmas 5, 35 17 (1998). [11] W. L. Slattery, G. D. Doolen and H. E. DeWitt, Phys. Rev. A 21, 2087 (1980);ibid, 26, 2255 (1982). [12] M.S. Murillo, Phys. Plasmas 5, 3116 (1998); M.S. Murill o, Phys. Plasmas 7, 33 (2000). 70 0.5 10.20.30.40.50.6 k dωR / ωpd (a) 0 0.5 1−4−2024x 10−4 k dγ / ωpd (b) FIG. 1. (a) The normalized real frequency and (b) the normali zed imaginary frequency, vs.the normalized wave number for the shear mode with e0=−0.0008, e01= 0.3,e1=−0.05,e11=−0.01 (solid curves). The dashed curves are for e01=e11= 0 and correspond to the uncoupled shear mode. 80 0.5 1 1.5 201234567 p (in mtorr)ne0 (in 108 /cm3)γ = 0 Hz γ = 0.02 Hz γ = 0.04 Hz FIG. 2. The electron number density ne0(in units of 108/cm3) is plotted as a function of the gas pressure, p(in mtorr) for various values of γ. For comparison, the accompanying dashed curves display th e situation when the coupling to the compressional mode is neglected. 9
1High pressure operation of the triple-GEM detector in pure Ne, Ar and Xe A. Bondar, A. Buzulutskov ∗, L. Shekhtman Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia Abstract We study the performance of the triple-GEM (Gas Electron Multiplier) detector in pure noble gases Ne, Ar and Xe, at different pressures varying from 1 to 10atm. In Ar and Xe, the maximum attainable gain of the detector abruptly dropsdown for pressures exceeding 3 atm. In contrast, the maximum gain in Ne wasfound to increase with pressure, reaching a value of 10 5 at 7 atm. The results obtained are of particular interest for developing noble gas-based cryogenicparticle detectors for solar neutrino and dark matter search. ∗ Corresponding author. Tel: +7-3832-302024; fax: +7-3832-342163. Email: buzulu@inp.nsk.su2This study is motivated by the growing interest in developing cryogenic double-phase particle detectors for solar neutrino and dark matter search [1,2]. Insuch detectors, the ionization produced in a noble liquid by a neutrino or weakly ionizing particle interaction, in Ne or Xe correspondingly, is extracted from the liquid to a gas phase, where it is detected with the help of the gas multiplier. In traditional gaseous detectors, namely in the multi-wire proportional and parallel-plate avalanche chambers, the maximum gain obtained in pure noblegases is by far too low due to photon- and ion-mediated secondary processes. Themulti-GEM (Gas Electron Multiplier [3]) multiplier could provide a solution: ithas been recently shown that the triple- and quadruple-GEM structures caneffectively operate in pure Ar and its mixtures with Ne and Xe, reaching ratherhigh gains, up to 10 5 , at atmospheric pressure [4]. Another problem is that the density of noble gases near the boiling point, at normal pressure, is higher compared to that at room temperature. For example, inNe, Ar and Xe the density difference is as large as a factor of 10.5, 2.8 and 1.6,correspondingly [5]. This means, that the operation of gas detectors at lowtemperature and atmospheric pressure can be equivalent to that at high pressureand room temperature. On the other hand, it was shown that the maximum GEMgain rapidly decreases with pressure in Ar/CO 2 and Xe/CO 2 [6]. In this paper we report on the performance of a triple-GEM detector in pure noble gases at high pressures, varying from 1 to 10 atm. The noble gases investigated are Ne, Ar and Xe. We show that the gain dependence on pressure is strongly affected by the gas nature. The experimental setup is shown in Fig.1. 3 GEM foils (50 µm thick kapton, 80 µm diameter and 140 µm pitch holes, 28×28 mm2 active area) and a printed curcuit board (PCB), mounted in cascade with 1.6 mm gaps, were installed within a stainless steel vessel. The vessel was filled with Ne, Ar or Xe at a certainpressure. The noble gases purity was 99.99%. The detector was irradiated with anX-ray tube through an Al window. The GEM and PCB electrodes were connected to a resistive high-voltage divider, as shown in Fig.1. The divider was optimized in such a way as to3maximize the gain in Ar at 1 atm and at the same time to prevent the parallel- plate amplification mode in inter-GEM (transfer) and GEM-PCB (induction) gaps.In particular, the voltage drops across GEMs were not equal and increased fromthe first to last GEM, similar to that used in [4]. Typical electric fields in the transfer and induction gaps, at 1 atm, were below 1.2, 3.0 and 2.8 kV/cm in Ne, Ar and Xe correspondingly. The same divider was used in the measurements withother pressures and gases. It should be remarked, however, that the optimizeddivider for them might be different. The anode signal was readout from the PCB either in a current or pulse- counting mode. The anode current value was always kept below 100 nA, using X-ray attenuation filters, to prevent charging-up effects. The ratio of the anodecurrent to the current recorded in the drift gap provides the gain value. Themaximum attainable gain was defined as that at which neither dark currents nor anode current instabilities (discharges) were observed for at least about 1 min. Fig.2 shows the gain-voltage characteristics of the triple-GEM detector in Ar, at different pressures. One can see that there are two types of the gaindependence on pressure. Below 3 atm, the maximum gain weakly depends onpressure, varying from 4×10 4 to 105 . In this pressure range it was limited by the onset of the dark current, of the order of few hundreds nA, most probably arisendue to the ion feedback between GEM elements [4]. At higher pressures, themaximum gain rapidly dropped down to below 10 at 7 atm. Here the limitation onthe maximum gain was imposed by GEM discharges. In Xe, the pressure dependence of the maximum gain also consisted of two parts: a slow decrease below 2 atm and very fast drop at higher pressures (Fig.3).On the other hand, the maximum operation gain was lower: it did not exceed 10 4. The maximum gain in Xe was limited by discharges in the whole pressure range.In addition, among other gases studied Xe was found to be the worst in terms ofthe discharge detrimental effect: at least in two cases all 3 GEMs werecompletely destroyed after few discharges when operated in Xe, while in Ne andAr even hundreds discharges did not result in noticeable degradation of thetriple-GEM structure.4It is interesting, that the maximum gain (discharge) boundary in Ar and Xe at higher pressures looks like a barrier in the voltage drop across a GEM, of about700 V, which cannot be overcome. This is probably related to the properties ofthe discharge mechanism in given gases. Neon showed quite different behavior compared to Ar and Xe (see Fig.4). Unlike Ar and Xe, the maximum gain in Ne turned out to be a growing functionof the pressure: it increased from 10 3 at 1 atm to above 105 a t 7 a t m . T h e limitation on the maximum gain in Ne was imposed by discharges. Note that theoperation voltages in Ne are considerably lower compared to those in Ar and Xe.Another interesting observation is that the gain-voltage characteristics in Nealmost do not change with pressure, for above 5 atm, in contrast to Ar and Xe.This is unusual for traditional gaseous devices, for which one would ratherexpect the E/p behavior of the detector characteristics. The detector performance in Ne was studied in a pulse-counting mode as well, using a charge-sensitive amplifier: the data were in coherence with those obtained in the current mode. We do not aware at the moment of any consistent explanation of GEM behavior at high pressures. We can only speculate that the violation of E/p scaling in Necould indicate on the existence of some geometrical factors governing the gasamplification mechanism at high pressures, similar to that of the avalancheconfinement in GEM holes considered in [8]. We also believe that the rather lowcross-section of electron-atomic collisions in Ne, as compared to other gases [7],may play an important role. In conclusion, we have studied for the first time the high-pressure operation of a triple-GEM detector in pure Ne, Ar and Xe. Neon showed quite differentpressure dependence of the maximum gain as compared to Ar and Xe: in Ar andXe the maximum gain drastically drops down for pressures exceeding 3 atm,while in Ne it increases with pressure up to 7 atm. In all the gases studied thereexist an optimal pressure at which the triple-GEM detector has the maximumgain: 10 4 at 1 atm, 105 at 3 atm and 105 at 7 atm in Xe, Ar and Ne correspondingly. One can see that the optimal pressures are close to thosecorresponding to appropriate gas densities near the boiling points. This means that the triple-GEM detector, in terms of gain characteristics, is a good candidate5for the proposed cryogenic double-phase particle detectors. At the same time, the gas amplification mechanism in GEM at high pressures is still unclear. Furtherinvestigations are required. We thank Drs. M. Leltchouk and D. Tovey for useful discussions. References [1] V. Radeka, P. Rehak, V. Tcherniatine, J. Dodd, M. Leltchouk, W. J. Willis, Private communication and the report on “The Nevis Laboratories Summer 2000 Education Workshop”, BNL and Columbia University (Nevis Laboratories), 2000, unpublished. [2] D. Tovey, Private communication and UK Dark Matter Collaboration Proposal on Galactic Dark Matter Search, Sheffield University, 2000, unpublished. [3] F. Sauli, Nucl. Instr. and Meth. A 386 (1997) 531. [4] A. Buzulutskov, A. Breskin, R. Chechik, G. Garty, F. Sauli, L. Shekhtman, Nucl. Instr. and Meth. A 443 (2000) 164. [5] The Infrared Handbook, Eds. W. Wolfe, G.Zissis, ERIM, Ann Arbor, Mich., 1993. [6] A. Bondar, A. Buzulutskov, F. Sauli, L. Shekhtman, Nucl. Instr. and Meth. A 419 (1998) 418. [7] Y. P. Raizer, Gas Discharge Physics, Nauka, Moscow, 1987 (in Russian). [8] A. Buzulutskov, L. Shekhtman, A. Bressan, A. Di Mauro, L. Ropelewski, F. Sauli, S. Biagi, Nucl. Instr. and Meth. A 433 (1999) 471.6X-ray sourceAl window GEM3GEM2GEM1 1.6 mm4 mmDrift gap 1.6 mm 1.6 mm PCB St. steel vessel R 0.9R RR R 0.95R R+H.V. Fig.1 A schematic view of the triple-GEM detector operated at high pressures. Fig.2 Detector gain as a function of the voltage drop across the last GEM, in Ar at different pressures.7Fig.3 Detector gain as function of the voltage drop across the last GEM, in Xe at different pressures. Fig.4 Detector gain as a function of the voltage drop across the last GEM, in Ne at different pressures.
arXiv:physics/0103083v1 [physics.flu-dyn] 26 Mar 2001Time resolved tracking of a sound scatterer in a complex ﬂow: non-stationary signal analysis and applications Nicolas Mordant and Jean-Fran¸ cois Pinton∗ ´Ecole Normale Sup´ erieure de Lyon & CNRS umr 5672, Laboratoire de Physique 46, all´ ee d’Italie, F-69364 Lyon, France Olivier Michel Universit´ e de Nice Laboratoire d’astrophysique & CNRS umr 5525, Parc Valrose, F-06108 Nice, France (Dated: December 22, 2013) It is known that ultrasound techniques yield non-intrusive measurements of hydrodynamic ﬂows. For example, the study of the echoes produced by a large numbe r of particles insoniﬁed by pulsed wavetrains has led to a now standard velocimetry technique. In this paper, we propose to extend the method to the continuous tracking of one single particle embedded in a complex ﬂow. This gives a Lagrangian measurement of the ﬂuid motion, which is of impo rtance in mixing and turbulence studies. The method relies on the ability to resolve in time t he Doppler shift of the sound scattered by the continuously insoniﬁed particle. For this signal processing problem two classes of approache s are used: time-frequency analysis and parametric high resolution methods. In the ﬁrst class we con sider the spectrogram and reassigned spectrogram, and we apply it to detect the motion of a small be ad settling in a ﬂuid at rest. In more non-stationary turbulent ﬂows where methods in the second c lass are more robust, we have adapted an Approximated Maximum Likelihood technique coupled with a generalized Kalman ﬁlter. PACS numbers: 43.30.Es, 43.60.-c, 47.80.+v, 43.60.Qv I. INTRODUCTION In several areas of ﬂuid dynamics research, it is desirable t o study the motion individual ﬂuid particles in a ﬂow, i.e. the Lagrangian dynamics of the ﬂow. The properties of th is motion governs the physics of mixing, the behavior of binary ﬂows and the Eulerian complexity of chaotic and turbu lent ﬂows. Lagrangian studies are possible in numerical experiments where chaotic1and turbulent2,3,4,5ﬂows have been studied. For turbulence, the numerical studi es are limited to small Reynolds number ﬂows whose evolution is onl y followed during a few large-eddy turnover times. In addition only the small scales properties of homogeneous tu rbulence are captured; the inﬂuence of inhomogeneities (such as large scale coherent structures) are not taken into account. It does not seem possible at the moment to extend high resolution turbulent DNS computations to long periods of time or to high Reynolds number ﬂows. Experimental studies are thus needed. They diﬀer from the numerical studi es because one cannot tag and follow individual ﬂuid particles; most techniques aim at recording the motion of solidparticles carried by the ﬂow motion. The degree of ﬁdelity with which solid particles can act as Lagrangian t racers is an open problem; it depends on the size and density of the particle. While the interaction between the p article and its wake can be important for large particles or particles with a large density diﬀerence with the surroundi ng ﬂuid6,7,8, it is generally admitted that density matched particles with a size smaller that the Kolmogorov length fol low the ﬂuid. Measurements of small particle motion have been made, using optical techniques that follow individual particle motion over short times/distances9,10. We propose here an acoustic technique that can resolve an individual pa rticle motion over long periods of time (compared with the characteristic time of ﬂow forcing). The principle of the technique is to monitor the Doppler shif t of the sound scattered by a particle which is con- tinuously insoniﬁed. This is an extension of the pulsed Doppler method that has been developed to measure velocity proﬁles and that has many applications in ﬂuid mechanics and medicine11. The main advantage of the continuous insoniﬁcation is to improve the time resolution of the measu rement, although it is limited to the tracking of a very small number of particles (the tests reported here are made w ith only one particle in the ﬂow). The measurement relies on the ability to track a Doppler frequency and its var iation in time. For this signal processing problem two classes of approaches have been developed: (i) time-freque ncy analysis and (ii) high resolution parametric spectral analysis. Time frequency methods mainly rely upon the quadr atic Wigner-Ville transform, or smoothed versions of it. Numerous studies and papers have recently been published, i n which the theoretical issues are presented (see e.g. the textbooks by Flandrin12or Cohen13). These non parametric techniques are convenient and well- suited for weakly non- stationary signals with a good signal-to-noise ratio (SNR) . However, time frequency representations present numerou s2 FIG. 1: Principle of measurement. A large 3D measurement zon e is achieved by using a transducer size of a few wavelengths. drawbacks when it comes to extract trajectory information. Their quadratic nature give rise to numerous spurious interference terms that require post processing. For signa ls with a faster frequency modulation and a low SNR, we show here that an optimized parametric approach is a better c hoice. Parametric high resolution spectral analysis methods take advantage of an a priori knowledge of the spectr al content of the recorded signal, namely the emitted signal frequency plus one or many doppler-shifted echoes. F urthermore, a time-recursive frame for the estimation of the Doppler shift is proposed here, where the evolution of th e frequency is taken into account in the algorithm. The two methods are tested in two experiments, in which the ac oustic signals have diﬀerent time scales and noise levels. The ﬁrst experiment is a study of the transient accel eration of a heavy sphere settling under gravity in a ﬂuid at rest. In this case the characteristic time scale of veloci ty variations is slow ( τ∼50 ms) and the signal to noise ratio is fair (about 20 dB); we show that a technique of reassignmen t of the spectrogram gives good results. The second experiments deals with the motion of a neutrally buoyant sph ere embedded in a turbulent ﬂow. In this case, velocity variations occur over times of about 1 ms and the signal to noi se ratio is low (less than 6 dB). We show that the AML parametric method yields very good results in that situatio n. The paper is organized as follows: in section II we present th e acoustic technique and measurement procedure. In section III we describe the signal processing techniques, w ith a particular emphasis on the AML method which has been developed and optimized to this particle tracking prob lem. Examples of applications to measurements in real ﬂows are given in section IV. II. ACOUSTICAL SET-UP A. Principle of the measurement In the experimental technique proposed here, a particle is c ontinuously insoniﬁed. It scatters a sound wave whose frequency is shifted from the incoming sound frequency due t o the Doppler eﬀect. This Doppler shift is directly related to the particle velocity vp: ∆ω=q·vp, (1) where qis the scattering wavevector (the diﬀerence between the inc ident and scattered wavevectors q=kscat−kinc) andωis the wave pulsation. We choose a backscattering geometry (see ﬁgure 1) so that q=−2kincand the frequency shift becomes ∆ω(t) =−2v(t) cω0, (2) where cis the speed of sound, ω0is the incident pulsation, and v(t) is the component of the velocity on the incident direction at time t. We continuously insonify the moving particle and record th e scattered sound. If need be, the particle position can be obtained by numerical integration of the velocity signal.3 00.050.10.150.20.250.3−40−2002040 time ( s )amplitude ( µV ) 0.10.11 0.12−20020 −2000 −1000 01000 2000−170−150−130−110 frequency ( Hz )amplitude ( dB ) FIG. 2: Data from a steel bead (diameter 1 mm) settling in wate r at rest. (a) Typical time series; (b)power spectral densit y of the inset ﬁgure. On the x-axis, zero corresponds to the emiss ion frequency. B. Transducers characteristics and acquisition We use a Vermon array of ultrasonic transducers made of indiv idual elements of size 2 ×2 mm each, separated by 100µm. Their resonant frequency is about 3.2 MHz and their bandwi dth at -3 dB is 1.5 MHz. Sound emission is set at 3 MHz or 3.5 MHz; experiments are performed in water so t hat the wavelength is λ=0.50 mm or 0.43 mm. The corresponding emission cone for each d= 2 mm square element is 29◦at 3 MHZ and 24◦at 3.5 MHz. In our measurements, the particle to transducer distance lies bet ween 5 cm and 40 cm, so that measurements are made in the far ﬁeld ( d2/λ > 10 mm). Given maximum ﬂow and particle velocities of the orde r of 1.5 m.s−1, we expect a maximum sound frequency shift of the order of 5 kHz or 6 kHz, de pending on whether the emission is at 3 MHz or 3.5 MHz. This yields a frequency modulation rate of at most 0. 25%. One element of the transducer array is used for continuous sound emission and another for scattered sound d etection. As the operation is continuous (as opposed to pulsed) and the elements are located close to one-another, w e observe a coupling between the emitter and the receiver of the order of 60 dB (this is due both to electromagnetic and a coustic surface waves cross-talk). The sound scattered by the moving particle is detected by a pi ezoelectric transducer. Upon connection to a 50 Ω impedance, it yields an electrical signal of about 2 to 30 µV. In comparison, the noise is 1 µV and the electromagnetic coupling with the emitter is 8 mV. Hence the signal to noise ra tio is between 0 dB and 30 dB. The transducer output is sampled at 10 MHz over a 21 bit dynamical range (input range 31.25 mV) and numerically heterodyned at the emitting frequency. Then it is decimated at the ﬁnal samplin g frequency of 19531 Hz. The acquisition device is a HP-e1430A VXI digitizer. C. Scattering by an elastic sphere The study of sound scattered by a ﬁxed solid sphere is a classi c but continuing area of study and diﬃculties arise in the interpretation of observed phenomena especial ly when trying to deal with elasticity and absorption14,15,16.4 Complex behaviour is observed linked with resonances of Ray leigh waves at the surface of the sphere. As a consequence the scattered pressure distribution varies both in directi vity and amplitude. A generic expression for the far ﬁeld pressure is the following : pscat(r, θ) =pincaf(ka, θ) 2reikr, (3) where ris the distance from the center of the sphere, aits radius, pincthe incident pressure on the sphere, kthe incident wavenumber in the ﬂuid, θthe scattering angle and fis a form function which depends on the physical properties of the solid medium. Under very general assumpti ons,fcan be developed as a series of partial waves: f(ka, θ) =2 ika∞/summationdisplay n=0(2n+ 1)Bn(ka) Dn(ka)Pn(cosθ), (4) where Pnis the a Legendre polynomial, BnandDnare determinants of matrices composed of spherical Bessel a nd Hankel function and their derivatives14. Physically, frepresents the sum of the specular echo and of interferences due to the radiation by Rayleigh waves15,16. As a result, fis a strongly varying function, particularly for high value s of ka. In our experiments we used spheres of diﬀerent material (po lypropylene PP, steel, tungsten carbide, glass) with corresponding kabetween 7 and 15. The ﬂow acts on the sphere motion, thus causi ng its acceleration and, eventually, its rotation. These eﬀects may change the radiation diagram : ﬁrst there is Doppler shift for the sound received by the sphere, and, perhaps more importantly, the sphere rotat ion may change the Rayleigh emission. For these reasons, the evolution of the amplitude of the scattered sound during the particle motion is quite complex. However, the observed amplitude modulation (see ﬁgures 2 and 8) varies sl owly enough to allow a correct estimate of the frequency modulation of the scattered sound. III. SIGNAL PROCESSING Numerous spectral estimation techniques are based on the id eas behind Fourier analysis of linear time invariant (LTI) diﬀerential equations. These techniques may be divided int o (i) non-parametric techniques where the basis functions are implicitly the harmonically related complex exponenti als of Fourier analysis and (ii) parametric techniques whos e task is the estimation of the parameters of a (sub)set of comp lex exponentials. The spectrogram and the reassigned spectrogram belong the former category, whereas the maximu m likelihood and its approximate form belong to the latter. A. Time-Frequency analysis The most common time frequency distribution (TFD), the spec trogram, involves a moving time window. This window attempts to capture a portion of the signal which is su ﬃciently restricted in time so that stationarity and LTI assumptions are approximately met. To overcome the inhe rently poor localization in the time-frequency plane, a method has been proposed by Gendrin et al.17, and extended more recently by Auger and Flandrin12,18. The idea is to locally reassign the energy distribution to the local cen ter of gravity of the Fourier transform. Despite its ability to exhibit clear and well localized trajectories in the time-f requency plane, this technique requires an additional imag e processing step to extract the TF trajectory. For rapidly ﬂu ctuating frequency modulations and/or low SNR spurious clusters appear which makes this extraction diﬃcult. The pa rametric method presented below is more robust. B. AML spectral estimation This approach is largely based upon maximum likelihood spec tral estimation (see e.g. Kay19). The fundamentals are brieﬂy recalled, as they serve as a basis for the approxim ate likelihood scheme, originally developed by Clergeot and Tressens20. This work is extended here within a recursive estimation fr ame, thus allowing to track the variations of the Doppler frequency shift induced by fast velocity chan ges of a scattering sphere imbedded in a turbulent ﬂow. Michel and Clergeot have developed a similar approach for no n stationary spectral analysis in an array processing frame21,22.5 1. Introduction In this section, we address the problem of estimating the fre quencies f1, . . ., f MofMharmonic signals embedded in noise, from a small number of samples x(t) =M/summationdisplay m=1am(t)exp(j(2πfmt+φm) +n(t). (5) As the number of sampling points that are supposed to be avail able is low, classical Fourier based approaches fail to provide good results. We focus our attention on parametri c approach, where an a priori knowledge about the structure of the signal is taken into account to improve the a nalysis. The following assumptions are made: •The time series is regularly sampled with time period Ts, so as to insure1 Ts>fmax 2where fmaxstands for the bandwidth of the anti-aliasing ﬁlter used in the recording p rocess. For convenience, Tswill be set to Ts= 1 and the term frequency will refer to normalized frequency (i.e. the actual frequency, divided by Fs=1 Ts. •The amplitudes ai(t) are deterministic but unknown. •The noise is a complex white gaussian circular and iid (indep endent increment identically distributed) with (unknown) variance σ2; the distribution function of a k-dimensional vector /vectorNdeﬁned by /vectorN(t) = [n(t), n(t+ 1), n(t+ 2), . . ., n (t+ (K−1))]T(6) reads p(/vectorN) =1 (√ 2πσ)Kexp/parenleftigg \|/vectorN\|2 2σ2/parenrightigg . (7) Furthermore, the noise and the signal are independent. •The term observation refers to a set of Q K-dimensional vectors constructed from the sampled time ser ies, according to /vectorX(tj) = [x(tj), x(tj+ 1), x(tj+ 2), . . . , x (tj+ (K−1))]Tj= 1, . . ., Q (8) •The frequency fis supposed to remain constant during an observation. Under the assumption that the noise process is iid, and that t he observed vectors are corrupted by independent reali- sations of the noise process, the likelihood of the observat ion is given by the product of the likelihood of each vector. L et Pbe the set of searched parameters ( Pcontains σ2,F={f1, . . . , f M}and the /vectorA= [a1exp(jφ1), . . . , a Mexp(jφM]T), the loglikelihood of an observation is simply given by L(P) =−KQlog(2πσ2)−1 2σ2Q/summationdisplay q=1\|/vectorX(q)−S(/vectorF)/vectorA(q)\|2, (9) where S(/vectorF) = [/vectorS1, . . . ,/vectorSM] = 1 exp(2 πf1). . .exp(2π(K−1)f1 ...... 1 exp(2 πfM. . .exp(2π(K−1)fM T . (10) According to the maximum likelihood principle, the set Pof parameters must be chosen in order to maximize expression (9).6 2. Reduced expression Minimizing (9) jointly for all the parameters is usually unt ractable. Most authors propose a separate maximisation for each of the parameters. For our application, the spectra l components (i.e. /vectorAand/vectorF) are the relevant variables. We ﬁrst maximize with respect to /vectorAand derive an expression for the optimal /vectorF;σ2is estimated independently. The value of vector /vectorAwhich minimizes the norm \|/vectorX(q)−S(/vectorF)/vectorA(q)\|2is easily obtained : /vectorA(q) = (S+S)−1S+(/vectorF)/vectorX(q). (11) Note that the “signal only” vector /vectorY=/vectorX−/vectorNappears to be the orthogonal projection of /vectorXon the signal subspace spanned by the row vectors of S: /vectorY=S(/vectorF)./vectorA(q) =S((S+S)−1S+(/vectorF)/vectorX= Π s(/vectorF)/vectorX , (12) where Π s(/vectorF) stands for the parametric projector on the signal subspace27. Let Π n(/vectorF) =I−Πs(/vectorF) be the noise subspace, Iis the identity matrix. By substituting (12) and using the de ﬁnition of Π n(/vectorF) in the expression of the log-likelihood (9), one gets the following simpliﬁed expre ssion to minimise L(/vectorF) =1 σ2Q/summationdisplay q=1\|Πn(/vectorF)/vectorX\|2. (13) Using the properties of the trace operator (hereafter denot ed Tr) and those of the projection matrix Π n(/vectorF), the maximum likelihood estimation of /vectorFtakes the common form : minimize L(/vectorF) =Q σ2Tr/bracketleftig Πn(/vectorF)ˆRx/bracketrightig , (14) where ˆRxis an estimate of the correlation matrix Rxof the vector process /vectorX(q). Minimizing L(/vectorF) in (14) leads to the exact value /vectorFMLwhich has the maximum likelihood. It is important here to emp hasize the following : if the vectors /vectorXare obtained by time-shift over the recorded time series, an observation runs over K+Q−1 samples, i.e. the actual duration of one observation is Tobs= (Q+K−2)Ts. In this case, the observed vectors may not be considered as b eing corrupted by independent realisations of the noise process , as some ’time integration’ is performed in the estimation o f Rx. The consequences and interest of such smoothing have been s tudied by Clergeot and Tressens20, and Ouamri23, in the frame of array processing (in this context, ’time inte gration’ becomes ’spatial smoothing’). In the remainder of this paper, the development are based on equation (14), no matter how Rxis estimated; see appendix for the practical implementation. 3. Approximate Max-likelihood Equation (14) is still too complicated to be solved analytic ally in a simple way. A minimization can be easily performed if L(/vectorF) has a quadratic dependance in S20. LetRybe the correlation matrix of the signal vectors /vectorY(q), the assumption that signal and noise are independent allow t o establish the following equalities ˆRx=Ry+ ˆσ2I, (15) Ry=SPS+, (16) P=E[/vectorA/vectorA+], (17) where Estands for the mathematical expectation. Substituting in e quation (14) leads to: L(/vectorF) =Q ˆσ2Tr/bracketleftig Πn(/vectorF)SˆPS+/bracketrightig . (18) Clergeot and Tressens20propose a second order approximation of L(/vectorF): LAML(/vectorF) =Q ˆσ2Tr/bracketleftig ˆΠnS(/vectorF)ˆPS+(/vectorF)/bracketrightig , (19)7 in which ˆΠnis estimated by computing the projector spanned by the ( K−N) smallest eigenvalues of the estimated covariance matrix ˆRx. They prove that this approach leads to more reliable estima tes of /vectorFat low signal to noise ratio (SNR), and that the minimization of LAMLis asymptotically eﬃcient. In practice, the following set o f equations is used ˆσ2=1 K−MTr(ˆΠnˆRx), (20) Πs(/vectorF) =S(/vectorF)(S+(/vectorF).S(/vectorF))−1.S+(/vectorF), (21) S(/vectorF).ˆP.S+(/vectorF) = Π s(/vectorF)(ˆRx−σ2I).Πs(/vectorF). (22) The approximately quadratic dependence of LAMLinS(/vectorF), allows a fast convergence of the minimization algorithm by using a simple Newton-Gauss algorithm: /vectorF(k+ 1) = /vectorF(k)−H−1./vectorgrad(LAML)\|/vectorF=/vectorF(k), (23) where kstands for the iteration step in the minimization process, /vectorgrad and Hare the gradient and hessian respectively (see expressions in the appendix). 4. Combining new measurements and estimates In this section, it is assumed that new measurements do not al low by itself the derivation of a good estimate. The variance of such an estimate varies as1 Tobs≃(K+Q−1)−1, whereas integrating new measurements to this estimate allow to derive a better estimation. Letˆ/vectorF(t) be an estimate of /vectorFat time t, and N(ˆ/vectorF(t),Γ(t)) its density, assumed to be normal with variance Γ( t)28. If a linear evolution model is known for /vectorF(t), one has /vectorF(t+ 1) = M/vectorF(t) +ε(t), (24) pt+1\|t(/vectorF) =N(Mˆ/vectorF(t),MΓ(t)M++Rε), (25) where Mis the evolution matrix; εis a perturbation term, which is statistically independent from /vectorF, and , Rεis its covariance matrix. pt+1\|tis the probability density function that can be derived for t imet+ 1, if the observations are made until time tonly. As such an evolution equation is usually unknown, Mwill be set to the identity matrix in the rest of the paper (see Michel22) for a detailed discussion). Applying the Bayes rule over co nditional probabilities gives : pt+1\|t+1(/vectorF) =pt+1\|t(/vectorF). pt+1(/vectorX\|/vectorF) pt+1(/vectorX). (26) Noting that log( pt+1(/vectorX\|/vectorF)) is the loglikelihood function for which a reduced express ion has been derived in the previous section, one gets after all reductions and identiﬁ cations the simple following expressions ˆ/vectorF(t+ 1\|t) =ˆ/vectorF(t), (27) Γ(t+ 1\|t) = Γ( t) +Rε, (28) Γ(t+ 1)−1=H+ Γ(t+ 1\|t)−1, (29) ˆ/vectorF(t+ 1\|t+ 1) =ˆ/vectorF(t+ 1) =ˆ/vectorF(t+ 1\|t)−Γ(t+ 1)−1./vectorgrad, (30) where it can be shown that the gradient function has the same e xpression as in the previous section. Rεis an unknown matrix which will be practically set to v2I, where v2will be tuned in order to allow the algorithm to take slight ch anges in/vectorFinto account. Furthermore, it is interesting that the set of expression above expresses a generalized Kalman ﬁlter for estimating /vectorF(in the sense that it relies upon second order expansion of th e loglikelihood functions). The statistical convergence properties and numerical eﬃcienc y of these approaches are described in the work of Michel & Clergeot21and Michel22.8 FIG. 3: Experimental setup in the case of the settling sphere . IV. EXPERIMENTAL RESULTS We ﬁrst describe the simple case of a particle settling in a ﬂu id at rest. It is well adapted to the reassigned spectrogram method because the acoustic signal has a good SN R and a slow frequency modulation. We show that it allows to extract the subtle interaction between the fallin g particle and its wake. We then study the more complicated case of the motion of a particle embedded in a turbulent ﬂow, w here the dynamics of motion is much faster and the SNR is poor. We show that the AML method is well suited. A. The settling sphere 1. Motivation and experimental setup When a particle is released in a ﬂuid at rest, its developing m otion creates a wake. The particle velocity is then set by the balance between buoyancy forces and drag, and addi tional subtle eﬀects: ﬁrst, ‘added mass’ corrections because the particles ‘pushes’ the ﬂuid, and second, a ‘hist ory’ force because the wake reacts back on the particle. Formally, one can write the equation of motion as6,8,24: (mp+1 2mf)dvp dt= (mp−mf)g−1 2πa2ρf/bardblvp/bardblvpcD(Re) +Fhistory . (31) where mpis the particle mass, mfis the mass of a ﬂuid particle of the same size, vpis the particle velocity, gis the acceleration of gravity, ais the sphere radius, ρfis the ﬂuid density, cDis the static empiric drag coeﬃcient, Reis the Reynolds number Re=2avp νandFhistory is the so-called history force. In this expression, the drag coeﬃcient is usually obtained from measurement of the forces acting on a b ody at rest in an hydrodynamic tunnel. The history term, however, is largely unknown. Analytic expression can only be derived in the limit of small Reynolds numbers (less than 10) and cannot be applied for real ﬂow conﬁguratio ns (e.g. multiphase ﬂows) where Re≫1. We perform measurements of the motion of a settling sphere, w ith the aim of evaluating the inﬂuence of the history forces. We use a water tank of size 1.1 m ×0.75 m and depth 0.65 m, ﬁlled with water at rest (ﬁgure 3). The bead is held by a pair of tweezers, ﬁve centimeters below the transdu cers. It is released a time t=0 without initial velocity and its trajectory is about 50 cm long. The data acquisition i s started before the bead is released in order to capture the onset of motion. 2. Results Let us use as a ﬁrst example, the fall of steel bead, 0.8 mm in di ameter. The Doppler shift during the bead motion is detected using the spectrogram representation and a subs equent reassignment scheme. The simple spectrogram and reassigned version are shown in ﬁgure 4. The reassignmen t technique drastically improves the localization of the energy in the time-frequency plane. In this case, the image p rocessing step computes vp(t) as the line of maxima. The precision of the overall measurement depends on two fact ors : ﬁrst on the intrinsic precision of the reassignment method and second on the dispersion of the measurements (the reproducibility of the bead motion over several experiments). The intrinsic precision of the reassignment method has been empirically studied using synthetic signal s modelling the particle dynamics plus a noise that mimics the experimental data. We observed that for our choice of9 FIG. 4: ( a) Spectrogram of the backscattered sound, after heterodyne detection. ( b) Reassigned spectrogram. In each ﬁgure the inset shows a normalized cross-section of the spectrogr am. The algorithm is that of the tfrrsp function of the MATLAB time-frequency toolbox25. To get rid of the spectral components at zero frequency due t o the coupling between transducers and at small frequencies around zero due to slow motion of the water surface, we use a high pass ﬁfth order Butterworth ﬁlte r of cut-oﬀ frequency 25 Hz (corresponding to a velocity of 5 mm /s). Data of 0.8 mm steel bead settling in water at rest. parameters (a time-frequency picture with 256 ×256 pixels) the rmsprecision is about one half pixel both in time and frequency directions. The method thus allows a precise anal ysis of the dynamics of the fall; we describe below two sets of experiment that illustrate the potential of the reas signment technique. First, we show in ﬁgure 5 the velocity of a 1 mm steel bead (aver age over ten falls) together with two numerical simulations based on equation 31, ﬁrst without the memory fo rce and second with the expression of the memory force derived at low Reynolds numbers (called the Stokes mem ory term, as in Maxey & Riley6). The precision of the detection technique is suﬃcient for the measured proﬁle to be compared to the simulated curves and to draw physical conclusions about the hydrodynamical forces. At e arly times, the trajectory is close to the simulation with memory force. This is due to the diﬀusion away from the bead su rface of the vorticity generated at the boundary6,8,24. However, as the instantaneous Reynolds number increases, t he curve deviates from this simple regime: vorticity is advected into the wake. Memory is progressively lost and the sphere reaches a terminal velocity in a ﬁnite time as does the simulation without memory. The measurement and signal processing techniques are then t ested on a more non-stationary motion, as in the case of a bead whose density is close to that of the ﬂuid. In this sit uation a stronger interaction is expected between the particle motion and the development of its wake. Formally, t his traces back to diﬀerences in the eﬀective inertial mass and buoyancy mass of the particle – see equation (31). In ﬁgur e 6, we show the velocity variation for a light glass sphere (density 2.48) compared to a tungsten bead (density 1 4.8). We observe that the velocity of the glass oscillates before reaching a constant terminal value whereas the other particle has a regular acceleration. In the case of light10 010020030040050060000.10.20.30.4Vp ( ms−1 ) time ( ms )0 20 40 6000.10.20.3 FIG. 5: Velocity measurement of a steel bead of diameter 1 mm ( solid line), compared to numerical simulations without mem ory force (dashed) and with Stokes memory (dash-dotted). The in set shows an enlargement near the onset of motion. The Reynol ds number, based on the limit velocity is 430. The sphere veloci ty proﬁle results from averaging n= 10 successive experiments. 010020030040050060000.20.40.60.811.2 time ( ms ) V* FIG. 6: Fall of a tungsten carbide sphere D=1 mm (dashed) comp ared to a glass bead D=2 mm (solid), at Re∼400. The velocity is non-dimensionalized by the limit velocity. Cur ves are not averaged over several experiments. beads the hydrodynamic forces may be large enough to overcom e the gravity and change the sign of the acceleration . This is linked with the non-stationarity of the wake, as vor tex shedding is known to occur for Reynolds number above critical ( Rec∼250). B. Turbulent ﬂow : Lagrangian velocity measurement 1. Experimental set-up The turbulent ﬂow is generated in a von K´ arm´ an geometry : th e water is set into motion by two coaxial counter rotating disk in a cylindrical tank (ﬁgure 7). The Reynolds n umber Re=2πR2f ν(where ν= 0.8910−6m2s−1is the water kinematic viscosity) is equal to 106. To prevent cavitation in the ﬂow, we boil the water before ﬁl ling the tank by lowering the pressure with a vacuum pump and during the exp eriment the pressure is increased to two bars. For the acoustic measurement, we use the same array of transduce rs as in the previous experiments, at emitting frequency 3 MHz. The cylinder and the surface of the disks are covered by 3 cm of Ciba Ureol 5073A and 6414B. Its density is 1.1 and the sound velocity is 1460 m.s−1so that its acoustic impedance is close to that of the water, r educing drastically the reﬂections at the interface water/ureol co mpared to water/steel. The attenuation at 2.5 MHz is about 6 dB per cm. With a 3 cm layer and after the reﬂection on steel th e total absorption is about 36 dB. The total11 FIG. 7: Experimental setup. The inner radius of the cylinder is 10 cm (disks radius R=9.5 cm) and the distance between the disks is 18 cm. The disks are driven by two 1 kW motors at a const ant rotation frequency of f=18 Hz. The transducers are placed 18 cm oﬀ axis, in order to increase the volume of the mea surement region. reﬂection at the interfaces is reduced by a factor 60. The par ticle is a polypropylene (PP) sphere of radius 1 mm and density 0 .9. 2. Results We show in ﬁgure 8 the time series when one particle is in the ul trasonic beam and the corresponding spectrogram and reassigned spectrogram. The signal to noise ratio is ver y poor, typically less than 6dB (to give an idea, in ﬁgure 8(a) the bead enters the ultrasonic beam at t∼20 ms). One can also see some events localized in the time frequency plane that may be considered as noise and that may h ave several origin (noise of the motors, external electromagnetic noise, cavitation in the ﬂow.. . ). Altoget her, the time-frequency pictures show the trajectory of the particle but the low SNR prevents it from being easily extrac ted. In particular, the trajectory in the reassigned pictur e becomes quite lacunar and extracting it would require sophi sticated (and CPU greedy) image processing techniques. The result of the AML algorithm is plotted in ﬁgure 9. The extr acted frequency modulation is of course within the estimation in the spectrogram as in ﬁgure 8(b), but one obser ves that ﬁne variations in the velocity of the bead are now detected. The algorithm provides also an estimate of the amplitude of the source (ﬁgure 9(c)). It can be seen that there is a strong amplitude modulation and that the SNR i s at most 6 dB and may become less than 0 dB. As the hessian is related to the Fisher information matrix26, its inverse square root is linked with the variance of the estimation: a large value of the hessian indicates an a ccurate estimation of the modulation frequency and, hence, of the bead velocity. The inverse square root of the he ssian is plotted in ﬁgure 9(b): very large values are calculated in the absence of a bead in the measurement volume at the beginning and end of the time series (as a signature of the mismatch between the model which is compose d of one source at least and the reality: no source). Local lower values (typically less than 0.1) are observed wh en the variance on the estimation is small. Spurious eﬀects are generated when the frequency modulation approaches zer o as the hessian also becomes very small because of the ﬁltering operation made in order to get rid of the coupling pa rt of the signal. Finally, one observes that the hessian decreases as the signal to noise ratio increases (see at time 0.55 s). V. CONCLUDING REMARKS As can be seen in the previous section, both methods, time-fr equency analysis and parametric spectral analysis are suited for extracting the time-varying frequency modulati on due to a Doppler eﬀect. The domain of application of each method depends on the degree of non-stationarity and on the SNR. For high SNR and weakly non-stationary signals, the time-fr equency approach yields very good results. One drawback is the need of a second processing stage to extract t he trajectory from the time-frequency picture. This stage may become increasingly diﬃcult if there is more than o ne spectral component or if the SNR degrades. In both cases the quadratic nature of the algorithm produces in terference patterns in the image: spurious clusters and12 0 0.2 0.4 0.6 0.8 1−6−4−20246 time ( s )amplitude ( µV ) time ( s )frequency ( Hz ) 0 0.2 0.4 0.6 0.8 1−4000−2000020004000 time ( s )frequency ( Hz ) 0 0.2 0.4 0.6 0.8 1−4000−2000020004000 FIG. 8: Sound scattered by a 2 mm diameter PP bead in a turbulen t ﬂow at Re= 106. (a) Typical time series; (b) and (c) corresponding spectrogram and reassigned spectrogram.13 0 0.2 0.4 0.6 0.8 1−0.500.51 time ( s )velocity ( m.s−1 ) 0 0.2 0.4 0.6 0.8 100.10.20.30.40.5 time ( s )hessien−1/2 0 0.2 0.4 0.6 0.8 1012345 time ( s )amplitude ( µV ) FIG. 9: Velocity measurement for the motion of a 2 mm diameter PP bead in a turbulent ﬂow at Re= 106. Output of the AML algorithm: (a) velocity (b) corresponding inverse squa re root of the hessian, (c) amplitude of the source (the rms va lue of the noise is 0.9 µV). AML algorithm parameters: M=1, K=7, Q=13, v2=10−5.14 a lacunar trajectory result. Another, more fundamental, li mitation is that the length of the time window must be long enough to preserve an acceptable frequency resolution , even with the reassigned spectrogram. This limits the methods to weakly non-stationary signals. For signals with a rapid frequency modulation, the AML spect ral estimation is well suited, as long as the noise is near iid. The size of the time window can be decreased because of the parametric nature of the method, since a priori knowledge has been taken into account. The performance is fu rther increased by the use of a Kalman-like ﬁlter. The drawback is the necessity to ﬁnd a good dynamical model for th e evolution of the spectral components. We have chosen here the simplest model which works well for our exper iments but the approach can be reﬁned by increasing the number of parameters in order to consider more precisely the variation of the frequency. The AML algorithm also provides a quantitative estimation of the quality of the dem odulation and the instantaneous power of the spectral component. Finally, the AML method has the advantage to prov ide directly frequency modulation as a function of time, in one stage. Acknowledgments We are indebted to Pascal Metz for the development of the sign al conditioning electronics. We thank Marc Moulin for his help in the design of the von K´ arm´ an setup, VERMON fo r continuous assistance in the development of the transducer array. This work is partially supported by ACI gr ant No. 2226. AML ALGORITHM •First step : calculate ˆRxusing the following expression22: ˆRx=1 2Qt+Q/summationdisplay i=t+1/parenleftig /vectorX(i)/vectorX(i)T+˜/vectorX(i)˜/vectorX(i)T/parenrightig , (32) with ˜/vectorX(i) = [x(i+K−1), x(i+K−2), . . ., x (i)]∗T, (33) where∗stands for complex conjugate.˜/vectorXis the complex conjugate of the time reversed version of /vectorX. •Second step : diagonalize ˆRx; one obtains the eigenvectors ( /vectorVi)i=1..Kand eigenvalues ( λi)i=1..Ksorted in decreasing order. •Third step : Compute ˆΠnand ˆσ2, using the set of equations ˆΠn=K/summationdisplay i=M+1/vectorVi/vectorVT i, (34) ˆσ2=1 K−MTr(ˆΠnˆRx) =1 K−MK/summationdisplay i=M+1λi. (35) •Forth step : choose /vectorF=ˆ/vectorF(t) as candidate value. Compute /vectorgrad and Husing21 /vectorgrad =2Q σ2Re/braceleftig Diag(S′+(/vectorF).Πn(/vectorF).ˆΠn.S(/vectorF).ˆP)/bracerightig , (36) H=2Q σ2Re/braceleftig Diag/parenleftig (S′+(/vectorF).Πn(/vectorF).ˆΠn.Πn(/vectorF).S′)/parenrightig ⋆ˆP∗/bracerightig , (37) where the operator ⋆stand for the term to term matrix multiplication, and P∗is the conjugate of P, and S′= [d/vectorS1 d f1, . . .,d/vectorSM d fM]T. (38)15 •Fifth step : using equations (27) to (30), computeˆ/vectorF(t+ 1) and Γ( t+ 1). The initialization of the algorithm is done by either (i) set ting an initial value of /vectorF(1) or (ii) estimating this value using the maxima of the amplitude of the FFT of a small window o f signal (of length 64 or 128 samples) and using the iterative algorithm described in section III B 3 to conve rge towards /vectorF(1). For example, the extracted velocity of ﬁgure 9 is obtained by starting at the maximum of energy of the signal and applying the algorithm forward and backward in time. The alg orithm is stopped as the mean of the inverse square root of the hessian over a window of size 400 samples exceeds 0 .5 for more than 400 samples. ∗e-mail : pinton@ens-lyon.fr 1G. O. Fountain, D. V. Khakhar, I. Mezic, and J. M. Ottino, “Cha otic mixing in a bounded 3D ﬂow”, J. Fluid Mech. 417 (2000). 2P. K. Yeung and S. B. Pope, “Lagrangian statistics from direc t numerical simulations of isotropic turbulence”, J. Fluid Mech. 207, 531 (1989). 3K. D. Squires and J. K. Eaton, “Lagrangian and eulerian stati stics obtained from direct numerical simulation of homogen eous turbulence”, Phys. Fluids A3, 130 (1991). 4P. K. Yeung, “Direct numerical simulation of two-particle r elative diﬀusion in isotropic turbulence”, Phys. Fluids 6(10), 3416 (1994). 5P. K. Yeung, “One- and two-particle lagrangian acceleratio n correlations in numerically simulated homogeneous turbu lence”, Phys. Fluids 9(10), 2981 (1997). 6M. R. Maxey and J. J. Riley, “Equation of motion for a small rig id sphere in a nonuniform ﬂow”, Phys. Fluids 26, 883 (1983). 7G. Sridhar and J. Katz, “Drag and lift forces on microscopic b ubbles entrained by a vortex”, Phys. ﬂuids 7, 389 (1995). 8N. Mordant and J.-F. Pinton, “Velocity measurement of a sett ling sphere”, Eur. Phys. J. B, in press (2000). 9M. Virant and T. Dracos, “3D PTV and its application on lagran gian motion”, Meas. Sci. Technol. 8, 1539 (1997). 10G. A. Voth, K. Satyanarayan, and E. Bodenschatz, “Lagrangia n acceleration measurements at large Reynolds numbers”, Phys. Fluids 10(9), 2268 (1998). 11O. F. Bay and I. G¨ uler, “Tissue blood ﬂow assessment of pulse d Doppler ultrasound using autoregressive modeling”, J. Me d. Syst.23(1), 77 (1999). 12P. Flandrin, Time-Frequency / Time Scale analysis , vol. 10 of Wavelet analysis and its application (Academic Press, 1998). 13L. Cohen, Time-Frequency analysis (A. V. Oppenheim Ed., Prentice Hall, 1995). 14G. Gaunard and H. Uberall, “RST analysis of monostatic and bi static acoustic echoes from an elastic sphere”, J. Acous. Soc. Am. 73(1), 1 (1983). 15P. D. Thorne, T. J. Brudner, and K. R. Waters, “Time-domain an d frequency-domain analysis of acoustic scattering by spheres”, J. Acoust. Soc. Am. 95(5), 2478 (1994). 16B. T. Hefner and P. L. Marston, “Backscattering enhancement s associated with subsonic Rayleigh waves on polymer sphere s in water : observation and modeling for acrylic spheres”, J. Acoust. Soc. Am. 107(4), 1930 (2000). 17R. Gendrin and C. de Villedary, “Anambiguous determination of ﬁne structures in multicomponents time-varying signals ”, Ann. Telecom. 35(3–4), 122 (1979). 18F. Auger and P. Flandrin, “Improving the readability of time -frequency and time-scale representations by the reassign ment method”, IEEE Trans. on Signal Processing 43, 1068 (1995). 19S. Kay, Modern Spectral Estimation, Theory and Application , Prentice Hall Signal Processing Series (A. V. Oppenheim Ed ., Prentice Hall, 1988). 20H. Clergeot and S. Tressens, “Comparison of two eﬃcient algo rithms for HR source tracking: time recursive implementati on.”, inICASSP’90, Albuquerque, USA (1990). 21O. Michel and H. Clergeot, “Multiple source tracking using a high resolution method”, in ICASSP’91, Toronto, Canada (1991), pp. 1277–1280. 22O. Michel, Application des m´ ethodes haute r´ esolution ` a la localisa tion et la poursuite de sources sonores , Ph.D. thesis, Universit´ e Paris XI, Orsay (1991). 23A. Ouamri, Etude des performances des m´ ethodes d’identiﬁcation ` a ha ute r´ esolution et application ` a l’identiﬁcation des ´ ech os par une antenne lin´ eaire multicapteurs , Ph.D. thesis, Universit´ e Paris-Sud, Orsay (1986). 24C. J. Lawrence and R. Mei, “Long-time behaviour of the drag on a body in impulsive motion”, J. Fluid Mech. 283, 307 (1995). 25Trademark, The Mathworks Company. The time-frequency tool box can be downloaded at http://crttsn.univ- nantes.fr/˜auger/tftbftp.html. 26L. Scharf, Statistical signal processing; detection, estimation and time series analysis (Addison-Wesley, 1991). 27It is straightforward to establish that Π s/vectorY=/vectorYfor any vector /vectorYlying in the signal subspace. Here ‘parametric projector’ must be understood as the projector calculated for the vecto r of frequencies /vectorF16 28This is generally not the case, but this assumption remains v alid as long as the loglikelihood is well approximated by its second order expansion aroundˆ/vectorF.
arXiv:physics/0103084v1 [physics.flu-dyn] 26 Mar 2001Scaling and intermittency of Lagrangian velocity in fully developed turbulence N. Mordant(1), P. Metz(1), O. Michel(2), J.-F. Pinton(1) (1)CNRS & Laboratoire de Physique, ´Ecole Normale Sup´ erieure, 46 all´ ee d’Italie, F-69007 Lyon, France (2)Laboratoire d’Astrophysique, Universit´ e de Nice Parc Valrose, F-06108, Nice, France Abstract We have developed a new experimental technique to measure th e Lagrangian velocity of tracer particles in a turbulent ﬂow. We observe that the Lagrangian velocity spectrum has an inertial scaling range EL(ω)∼ω−2, in agreement with a Kolmogorov picture. Single particle ve locity increments display an intermittency that is as pronounced a s that in the Eulerian framework. We note that in the Lagrangian case, this intermittency can be d escribed as a stochastic additive process. PACS numbers: 47.27.Gs, 43.58.+z, 02.50.Fz 1Lagrangian characteristics of ﬂuid motion are of fundament al importance in the under- standing of transport and mixing. It is a natural approach fo r reacting ﬂows or pollutant contamination problems to analyze the motion of individual ﬂuid particles. Another char- acteristic of mixing ﬂows is their high degree of turbulence . For practical reasons, most of the experimental work concerning high Reynolds number ﬂows has been obtained in the Eulerian framework. Lagrangian measurements are challeng ing because they involve the tracking of particle trajectories: enough time resolution , both at small and large scales, is required to describe the turbulent ﬂuctuations. With this i n mind, we have developed a new experimental method, based on sonar techniques to obtain a m easurement of single particle velocities for times up to the ﬂow large scale turnover time [ 1]. Our aim in this Letter is to compare the statistical properties of the Lagrangian vel ocity ﬂuctuations to well known characteristics in the Eulerian domain, which we ﬁrst brieﬂ y recall. Eulerian velocity measurements are usually obtained as the evolution in time of the ve- locity ﬁeld sampled at a ﬁxed point. In this framework one is i nterested in velocity proﬁles in space which are derived using the Taylor hypothesis. Whil e some issues regarding the inﬂuence of isotropy and homogeneity are still debated [3, 4 , 5], the following statistical prop- erties of the Eulerian velocity ﬁeld are generally accepted : (i) the spectrum has an inertial range EE(k)∼k−5/3, as predicted by Kolmogorov’s original K41 mean ﬁeld approa ch, (ii) the probability density function (PDF) of the velocity incr ements ∆ ur(x) =u(x+r)−u(x) have functional forms that evolve from Gaussian at integral scales to strongly non-Gaussian with wide tails near the dissipative scale (a phenomenon ref erred to as ‘intermittency’) , (iii) this evolution can be described as being the result of a multi plicative cascade as originally proposed by Kolmogorov and Obukhov (K62 model) and much deve loped since [2]. We show here that the Lagrangian velocity ﬂuctuations have s imilar properties, and that the intermittency in that frame can be interpreted in terms o f an additive process. Experimentally, Lagrangian measurement have been quite sc arce. Recent data have been obtained by optical detection of particles tracks, using ei ther Particle Tracking Velocime- try [6] or high speed detectors [7, 8]. In the ﬁrst case the res ults concentrate on the particle trajectories while in the second case a high time resolution has been used to analyze the statistics of particle acceleration. We propose a compleme ntary technique that gives a di- rect access to the Lagrangian velocity across the inertial r ange of time scales. It is based on the principle of a continuous sonar. A small (2mm ×2mm) emitter continuously insoniﬁes 2the ﬂow with a pure sine wave, at frequency f0= 2.5 MHz (in water). The moving parti- cle backscatters the ultrasound towards an array of receivi ng transducers, with a Doppler frequency shift related to the velocity of the particle: 2π∆f=q.v. (1) The scattering wavevector qis equal to the diﬀerence between the incident and scattered directions. A numerical demodulation of the time evolution of the Doppler shift gives the component of the particle velocity along the scattering wav evector q. It is performed using a high resolution parametric method which relies on an Appro ximated Maximum Likelihood scheme coupled with a generalized Kalman ﬁlter [1]. The stud y reported here is made with a single array of transducers so that only one Lagrangian vel ocity component is measured. The turbulent ﬂow is produced in the gap between counter-rot ating discs [9, 10]. This setup has the advantage to generate a strong turbulence in a c ompact region of space, with no mean advection. In this way, particles can be tracked duri ng times comparable to the large eddy turnover time. Smooth discs of radius R= 9.5 cm are used to set water into motion inside a cylindrical vessel of height H= 18 cm. In the measurement reported here, the power input is ǫ∼13 W/kg. The integral Reynolds number is Re=R2Ω/ν= 1.75 105, where Ω = 1 /Tis the rotation frequency of the discs (17.5 Hz), and ν= 10−6m2/s is the kinematic viscosity of water. The turbulent Reynolds nu mber is computed using the measured rmsamplitude of velocity ﬂuctuations ( urms= 0.32 m/s) and an estimate of the Taylor microscale ( λ=/radicalBig 15νu2rms/ǫ= 350 µm); we obtain Rλ= 110. This value is consistent with earlier studies in the same geometry [11]. The ﬂow is seeded with a small number of neutrally buoyant (de nsity 1.06) polystyrene spheres with diameter d= 500 µm . It is expected that the particles follow the ﬂuid motion up to characteristic times of the order of the turbulence edd y turnover time, at a scale corresponding to their diameter, i.e. τmin∼d/ud∼ǫ−1/3d2/3, using standard Kolmogorov phenomenology. For beads of diameter 500 µm, one estimates τmin∼3 ms. This value is within the resolution of the demodulation algorithm, so tha t both the time and space scales of the measurement cover the inertial range of the turbulent motion. One of the ﬁrst quantity of interest is the Lagrangian veloci ty auto-correlation function: ρL v(τ) =∝angbracketleftv(t)v(t+τ)∝angbracketrightt ∝angbracketleftv2∝angbracketright. (2) 3We observe – Fig.1a – that it has a slow decrease which can be mo deled by an exponential function ρv(τ)∝e−τ/TL, in the range t/T∈[0.1,2]. The characteristic decorrelation time TL is of the order of the integral time scale (the ﬁt in Fig.1a yie ldsTL= 54 ms while T= 57 ms). This observation is in agreement with numerical simulation s [12, 13]. 0.5 1 1.5 2 2.5 3−80−70−60−50−40−30−20 log 10 ( Frequency Hz)E , PSD [dB]01234567−0.200.20.40.60.81 00.10.20.30.40.50.60.50.60.70.80.91Lτ / Tρ ( τ ) L v FIG. 1: (a) Autocorrelation function, the exponential ﬁt is ρL v(τ) = 1.045e1.05τ/Tand (b) Spectrum; the solid line is a power law ω−2behavior. The corresponding Lagrangian velocity power spectrum EL(ω) is plotted in Fig.1b. One observes a range of frequencies consistent with a power law s caling EL(ω)∝ω−2. This is in agreement with a Kolmogorov K41 picture in which the spectra l density at a frequency ωis a 4dimensional function of ωandǫ:EL(ω)∝ǫ ω−2. To our knowledge, this is the ﬁrst time that it is directly observed experimentally although it has been reported in DNS by Yeung [14]. We note that the ‘inertial range’ of scales extends for the en tire range of frequencies where the velocity of the particle is expected to correctly reprod uce the Lagrangian velocity of ﬂuid elements: at high frequency, the observed cut-oﬀ may be due t o particle inertia, whereas at low frequency the measurement is limited to the longest time that particles remain in the detection zone (about 4 T). We now consider the Lagrangian velocity increments ∆ vτ=v(t+τ)−v(t). We emphasize that these are time increments, and not space increments as i n the Eulerian studies. The spectrum gives the variance of ∝angbracketleft∆v2 τ∝angbracketright; the scaling region EL(ω)∝ω−2is equivalent to ∝angbracketleft∆v2 τ∝angbracketright ∝τ. As usual for ﬂows with Rλless than about 500, the range of scale motion is not suﬃciently wide for a true scaling region to develop. The plot of ∝angbracketleft∆v2 τ∝angbracketrightas a function ofτis rather rounded (Fig.2), with a trivial scaling ∝angbracketleft∆v2 τ∝angbracketright ∝τ2in the dissipative range and∝angbracketleft∆v2 τ∝angbracketright ∼2u2 rmsat integral time and over (at such time lags, v(t) and v(t+τ) are uncorrelated). In between, the variance of the velocity inc rements increases monotonously with the increment’s width; the inset in Fig.2 shows ∝angbracketleft∆v2 τ∝angbracketright/τand no plateau can be detected. 00.511.522.533.5400.020.040.060.080.10.120.140.160.180.2 τ / T DL 2 0 1 2 3 400.050.10.150.20.25 FIG. 2: Second order structure function, ∝angbracketleft∆v2 τ∝angbracketright. The inset shows ∝angbracketleft∆v2 τ∝angbracketright/τ. Note that quantities are plotted in linear coordinates. 5Turning to the question of intermittency, we show in Fig.3 th e PDFs of the Lagrangian increments. Their functional form, normalized to the varia nce, changes clearly with the −10 −5 0 5 10−8−7−6−5−4−3−2−10 ∆ v / Πτστ σττ FIG. 3: PDF στΠτof the normalized increment <∆vτ> /σ τ. The curves are shifted for clarity. From top to bottom: τ= 1,2,4,8,16,32,64,128,256 ms. increment’s width: they are almost Gaussian at integral tim e scales and exhibit wide tails at small scales. One measure of that evolution is given by the ﬂatness factor F(τ) = ∝angbracketleft∆v4 τ∝angbracketright/∝angbracketleft∆v2 τ∝angbracketright2; in our case Fvaries from 16 at smallest time lag to 3 .5 forτ∼T. In this respect, the intermittency is as developed in the Lagrangia n frame as it is in the Eulerian one [15]. More generally, one can choose to describe the PDFs evolutio n by the behavior of their moments (or ‘structure functions’). A consequence of the ch ange of shape of the PDFs with scale is that their moments, as the ﬂatness factor above , vary with scale. One way to compensate for the lack of a true inertial range is to use one s tructure function as a reference and to study the evolution of the others relative to that refe rence (ESS ansatz [16]). In the spirit of numerous studies in the Eulerian frame, we use the s econd order structure function as a reference; indeed, the dimensional Kolmogorov-like ar gument yields: ∝angbracketleft∆v2 τ∝angbracketright=CLǫ τ . (3) 6This expression shows that the second order structure funct ion is not aﬀected by spatial or temporal inhomogeneities of the dissipation ǫ. It is the analogue of the third order structure function in the Eulerian domain. In that respect, CLis expected to be a universal constant, although there is no known equivalent of the K´ arm´ an-Howar th relationship in Lagrangian coordinates. −3.5 −3 −2.5 −2 −1.5 −1 −0.5−6−5−4−3−2−10 log10 ( )2∆ v τ< > log 10 ( )p∆ v τ<\| \| > FIG. 4: ESS plots of the structure function variation (in dou ble log coordinates). The solid curves are best linear ﬁts with slopes equal to ξL p= 0.56,1.35,1.64 for p= 1,3,4. The relative scaling of the structure functions is evidence d in Fig.4, where they are plotted up to order 4 (higher orders would require more stati stics to converge than currently available). We observe that they follow a relative scaling l aw, i.e. ∝angbracketleft\|∆vτ\|p∝angbracketright ∝ ∝angbracketleft ∆v2 τ∝angbracketrightξL p. (4) The scaling domain extends from τ/T∼0.02 to τ/T∼1.3, a wider range than the scaling domain detected in the spectrum (hence the name ‘Extended Se lf Similar’ range) . The relative exponents are ξL 1= 0.56, ξL 3= 1.35, ξL 4= 1.64. These Lagrangian exponents follow a sequence close to, but slightly more intermittent than the c orresponding Eulerian quantity. Indeed, we obtain: ξL 1/ξL 3= 0.42, ξL 2/ξL 3= 0.74, ξL 4/ξL 3= 1.21, to be compared to the commonly accepted Eulerian values [17] ξE 1/ξE 3= 0.36, ξE 2/ξE 3= 0.70, ξE 4/ξE 3= 1.28. 7In the Eulerian context, the ESS property has been regarded a s the sign of inﬁnite divisibility for the multiplicative cascade underlying th e ﬂuctuations of dissipation [18]. We propose that in the Lagrangian framework, the intermittenc y of the increments result from an additive process. Indeed, the increment at time lag τcan be written as: ∆vτ(t) =/integraldisplayt+τ ta(t′)dt′, (5) where a(t) is the Lagrangian acceleration. It is known that its auto-c orrelation time is quite small, of the order of τη[7, 8, 12, 13]. Thus one could view the velocity increment at t ime lagτ≫τηas resulting from a sum of uncorrelated contributions. In th is case the PDF at time lag τwould results from successive convolutions of the PDF at a ti me interval equal to a few units of τη. The validity of this assumption can be checked by computing the 00.020.040.060.080.10.120.140.1600.0020.0040.0060.0080.010.0120.0140.0160.0180.02 00.020.040.0600.0020.0040.0060.0080.01C [(m/s) ]4L C [(m/s) ]2 L 24 FIG. 5: Relative variation of the fourth cumulant with the se cond ones. Quantities are displayed in linear, dimensional, units and calculated with absolute values in the inset. cumulants CL p(τ) of the probability distributions of the Lagrangian veloci ty increments. A simple convolution law by a ﬁxed kernel means that the cumula nts of any two orders are an aﬃne function of one another. Indeed, in such an additive pro cess, one has for the PDF Π τ at time lag τ: Πτ= Π τ0⊗Π⊗[n(τ)−n(τ0)] 0 , (6) 8where τ0is the starting scale, Π 0is the propagator and n(τ) the number of steps of the process. This implies for the cumulants: CL p(τ) =CL p(τ0) + [n(τ)−n(τ0)]C0 p, (7) so that for any two orders ( p, q) on has: CL p(τ) =CL q(τ)C0 p C0q+/bracketleftBigg CL p(τ0)−CL q(τ0)C0 p C0q/bracketrightBigg . (8) This aﬃne behavior is indeed observed in Fig.5 for the second and fourth cumulants. We note that when the cumulants of the absolute values are compu ted, the range of linearity is extended to the entire range where ESS is veriﬁed. The additi ve process is not a completely uncorrelated one because the number of convolution steps n(τ) is not a simple linear function ofτ; its shape is given by CL 2– equal to the second order structure function, shown in Fig. 2. In conclusion, we have observed experimentally that there i s much resemblance between the Eulerian velocity and the Lagrangian velocity. In both c ase the power spectra obey scaling laws that are given by Kolmogorov similarity argume nts. Also, the velocity incre- ments are intermittent, but one consequence of our observat ions is that the intermittency of the Lagrangian (1-point) velocity increments can be desc ribed by an additive process. This raises several questions which may deserve further inv estigations: (i) the additive process results from the statistical properties of the Lagr angian acceleration. Could some of its statistical characteristics be directly derived fro m the Navier-Stokes equation? What are the relative contributions of the pressure and viscous t erms? (ii) what is the behavior of this additive process in the limit of inﬁnite Reynolds num bers? acknowledgements: We thank Bernard Castaing for interesting discussions and V ermon Corporation for the design of the ultrasonic transducers. T his work is supported by grant ACI No.2226 from the French Minist` ere de la Recherche. [1] Mordant N., Michel O., Pinton J.-F., submitted to JASA , (2000) and ArXiv:physics/0103083. [2] See for instance Frisch U., Turbulence , Cambridge U. Press, (1995) and references therein. [3] Toschi F., L´ ev` eque E., Ruiz-Chavarria G., Phys. Rev. Lett. ,85, 1436, (2000). 9[4] Shen X., Warhaft Z., Phys. Fluids ,12, 2976, (2000). Arad I., Biferale L., Mazzitelli I., Pro- caccia I., Phys. Rev. Lett. ,82, 5040, (1999). [5] Simand C., Chill` a F., Pinton J.-F. , Europhys. Lett. ,49, 336, (2000). [6] Virant M., Dracos T., Meas. Sci. Technol. ,8, 1539, (1997). [7] Voth G.A., Satyanarayan K., Bodenschatz E., Phys. Fluids ,10, 2268, (1998). [8] La Porta A., Voth G.A., Crawford A., Alexender J., Bodens chatz E., Nature , (2001). [9] Zandbergen P. J. and Dijkstra D., Ann. Rev. Fluid Mech. ,19, 465-491, (1987). [10] Douady S., Couder Y. and Brachet M.-E., Phys. Rev. Lett. 67, 983-986 (1991). [11] Mordant N., Pinton J.-F., Chill` a F., J. Phys. II France ,7, 1729-1742, (1997). [12] Yeung P.K., Pope S.B., J. Fluid Mech. ,207, 531, (1989). [13] Yeung P.K., Phys. Fluids ,9, 2981, (1997). [14] Yeung P.K., J. Fluid Mech. ,427, 241, (2001). [15] Anselmet F., Gagne Y., Hopﬁnger E.J., Antonia R.A. J. Fluid Mech. ,140, 63, (1984). [16] Benzi R., Ciliberto S., Baudet C., Ruiz-Chavarria G., T ripiccione C., Europhys. Lett ,24, 275, (1993). [17] Arneodo A. et al., Europhys. Lett ,34, 411, (1996). [18] Gagne Y., Castaing B., Marchand M., J. Phys. II France ,4, 1-8, (1994). 10
arXiv:physics/0103085v1 [physics.atom-ph] 26 Mar 2001EPJ manuscript No. (will be inserted by the editor) An Atom Faucet W. Wohllebena, F. Chevy, K. Madison, J. Dalibard, Laboratoire Kastler Brosselb, D´ epartement de Physique de l’Ecole Normale Sup´ erieure, 24 rue Lhomond, 75005 Paris, France the date of receipt and acceptance should be inserted later Abstract. We have constructed and modeled a simple and eﬃcient source o f slow atoms. From a background vapour loaded magneto-optical trap, a thin laser beam extra cts a continuous jet of cold rubidium atoms. In this setup, the extraction column that is typical to leaki ng MOT systems is created without any optical parts placed inside the vacuum chamber. For detailed analys is, we present a simple 3D numerical simulation of the atomic motion in the presence of multiple saturating l aser ﬁelds combined with an inhomogeneous magnetic ﬁeld. At a pressure of PRb87= 1×10−8mbar, the moderate laser power of 10mW per beam generates a jet of ﬂux Φ= 1.3×108atoms/s with a mean velocity of 14m/s and a divergence of <20 mrad. PACS. 32.80.Lg Mechanical eﬀects of light on atoms, molecules, an d ions – 32.80.Pj Optical cooling of atoms; trapping 1 Introduction Experiments on trapped cold atom clouds require in most cases high particle numbers and long trapping lifetimes. In order to restrict the lifetime limiting collisions with bac k- ground gas, an ultra-high vacuum (UHV) environment is necessary. In turn, at these pressures a purely background vapour charged magneto-optical trap (VCMOT) is limited to very small atom numbers and long loading times and thus needs to be loaded by an additional jet of cold atoms. As to the simplest possible cold atom sources, a laser- free velocity ﬁlter [1] is elegant, but its maximum ﬂux can be greatly improved upon by adding a laser cooling stage. The Zeeman slower is a widely used technique espe- cially for light and thus thermally non capturable fast species. For heavier elements, one can accumulate atoms into a MOT in a vapour cell, with various strategies for subsequent transfer to a recapture MOT in the UHV cell. These strategies can be categorized into either a pulsed [2,3] or continuous transfer scheme. The latter category involves either a moving molasses [4] or a ’leaking MOT’ scheme [5,6]. This paper presents the construction and numerical modeling of a cold atom jet whose ﬂux is continuous, ad- justable in a given direction, and velocity tunable. The device we present is based on an ordinary VCMOT. It captures and cools atoms from the low velocity part of the room temperature Maxwell-Boltzmann distribution in a high pressure cell of P∼10−8mbar. From the center of aPresent address: Max-Planck-Institut f¨ ur Quantenoptik, 85748 Garching, Germany. bUnit´ e de Recherche de l’Ecole normale sup´ erieure et de l’Universit´ e Pierre et Marie Curie, associ´ ee au CNRS.this source MOT, an additional pushing beam of ∼1mm spot size extracts a continuous jet that is slow enough to be recaptured in a MOT in the UHV region. The jet passes through a tube that maintains the pressure diﬀer- ential between the two cells, and the atom number transfer between the two MOTs is found to be typically 50 % and as high as 60 % eﬃcient. The Atom Faucet is closely related to the LVIS [5] and the 2D+MOT [6]. The common concept which relates them in the ’leaking MOT’ family is the creation of a thin extraction column in the center of the MOT where the radiation pressure is imbalanced and through which leaks a continuous jet of cold atoms. Operation in a continu- ous mode maximizes the mean ﬂux up to a value ideally equal to the source trap capture rate. Since a leaking trap operates at a low trap density, once captured, an atom has much higher probability to leave the trap via the jet rather than undergoing a collision that expels it. The LVIS and 2D+MOT place a mirror inside the vac- uum for retroreﬂection of one of the MOT beams. By piercing a hole in this mirror, one creates a hollow retrore- ﬂection beam, and the jet exits through the hole. By con- trast, the Atom Faucet requires no optical parts inside the vacuum system. Here, we superimpose an additional colli- mated ’pushing beam’ that pierces the extraction column through the MOT. In these complex magneto-optical arrangements the behavior of the system is no longer intuitively obvious. On its way into the jet, a thermal atom undergoes sub- sequent phases of strong radiation pressure (capture from vapour), overdamped guidance to the magnetic ﬁeld min- imum (MOT molasses) and 1D strong radiation pressure with transverse 2D molasses cooling (extraction process).2 W. Wohlleben, F. Chevy, K. Madison, J. Dalibard,: An Atom Fa ucet Theoretical estimates for near-resonant atom traps con- centrate either on the capture [7] or on the cooling [8]. We develop a simple and heuristic generalization of the semiclassical radiation pressure expression for the case o f multiple saturating laser ﬁelds and inhomogeneous mag- netic ﬁeld. The new approach of integrating the atomic trajectory through both capture and cooling mechanisms (neglecting optical pumping and particle interaction) re- produces the parameter dependences of the Atom Faucet. The trajectories indicate the physical mechanisms of the 7- beam-interplay. However, the simpliﬁcations made to the Rubidium level scheme lead to an overestimation of the absolute value of the radiation pressure force and hence an overestimate for the capture velocity of the MOT. This paper is organized as follows: In section 2 we give details on the experimental realization of the Atom Faucet. In section 3 we present the numerical model. Sec- tion 4 discusses the parameter dependences of the device in the experiment and in the simulations and ﬁnally in sec- tion 5 we compare this scheme to other vapour cell cold atom sources. 2 Experimental Realisation The vacuum system consists of two glass cells separated vertically by 67 cm with a MOT aligned at the center of each cell. Using an appropriate pumping scheme and a diﬀerential pumping tube of diameter 5 mm and length 15 cm the pressure in the lower recapture cell is less than 10−11mbar while in the source cell it is ∼10−8mbar. We deduce the87Rb pressure in the source cell from the reso- nant absorption of a multi-passed probe beam. A heated reservoir connected to the upper source cell supplies the Rubidium vapour. A grating stabilized diode laser locked to the \|5S1/2, F= 2/angbracketright → \| 5P3/2, F= 3/angbracketrighttransition injects into three slave lasers, two for the source MOT and one for the recapture MOT. The Atom Faucet (see ﬁg. 1) is based on a standard MOT conﬁguration: two Anti-Helmholtz-coils maintain a magnetic ﬁeld gradient of 15G/cm along their axis, which is horizontal in this setup. A pair of axial beams with pos- itive helicity counterpropagate along the axis of the coils and two mutually orthogonal pairs of radial beams with negative helicity counterpropagate in the symmetry plane of the coils. The radial beams are inclined by 45◦. The ra- dial trap beams have an 8 mm spot size and the axial beam 11mm respectively, all clipped to a diameter of 1 inch by our quarterwaveplates. The axial beam carries 20mW be- fore retroreﬂection, and the radial beams each have 5 mW each before retroreﬂection. The repumping light on the \|5S1/2, F= 1/angbracketright → \|5P3/2, F= 2/angbracketrighttransition from an inde- pendent grating stabilized laser is mixed only in the axial beam and has a power of ∼5mW. In addition to these trapping lasers, a permanent push- ing beam on the \|5S1/2, F= 2/angbracketright → \|5P3/2, F= 3/angbracketrighttran- sition with linear polarization [9] and optimal power of 200µW is aligned vertically onto the trap. It is focused to a waist of 90 µm 30cm before entering the source cell such that it diverges to a size of 1 .1mm at the source trap and Fig. 1. The Atom Faucet setup (with the recapture MOT below). A permanent pushing beam with ∼1mm spot size pierces an extraction column into an ordinary vapour charge d MOT. The high pressure region is separated from the ultra- high-vacuum region by a diﬀerential pumping tube. The pres- sure in the source cell is monitored by the absorption of an additional multi-passed probe beam (not shown). 3.3mm at the recapture trap. Its intensity at the center of the source MOT and detuning are comparable to those of the MOT beams and hence its radiation pressure is also comparable with the trapping forces in the MOT. Because of the divergence of the pushing beam, the intensity in the lower MOT is lower by a factor of 10. It decenters the re- capture MOT by ≃1 mm but does not destabilize it. Note that the pushing beam carries no repumping light, so that it acts on the atoms only where it intersects the MOT beams. By studying the loading characteristics of the recap- ture MOT, we deduce the main features of the atom jet: –When the recapture MOT is empty the initial recap- ture rate gives directly the recaptured ﬂux since the density dependent intrinsic losses in the MOT are not yet important. The absolute number of atoms is deter- mined using an absorption imaging technique. –The time dependence of the recapture loading rate pro- vides a measurement of the longitudinal velocity dis- tribution of the jet. More precisely, by suddenly disin- jecting the source MOT slave lasers and then recording the recapture ﬁlling rate via the ﬂuorescence, the char- acteristics of the tail of the moving extraction columnW. Wohlleben, F. Chevy, K. Madison, J. Dalibard,: An Atom Fau cet 3 Time since source deinjection (ms)020 40 60 80 100recapture MOT fluorescence ( mV) 1.52.02.53.03.5 Fig. 2. Development of the ﬂuorescence of the recapture trap (circles are photodiode signal) after sudden disinjection of the source MOT beams. The pushing beam is not changed in order to keep constant its inﬂuence on the lower trap ﬂuorescence. The ﬁt (solid line) Φ(v) =Φ0×exp/parenleftbig −(v−¯v)2/2δv2/parenrightbig with a Gaussian envelope for the jet velocity distribution yields v= 14±9m/s. are measured. The jet transfer distance D= 67cm and the time delay Tof the ﬁlling rate response gives the mean longitudinal velocity ¯ v=D/T in the jet, and the time width ∆tof this response gives access to the longitudinal velocity dispersion δv(see Fig. 2). For the determination of the transfer eﬃciency, the loading rate of the source MOT is determined by its ﬂuo- rescence and compared with the measured recapture ﬂux. The ﬂuorescence measurement is done at resonance and we assume full saturation of the transition under the in- ﬂuence of all six laser beams and thus a photon scattering rate of Γ/2photons/atom/second. We observe a typical transfer eﬃciency of 50 % (see below). Since the radius of the recapture MOT beams is r= 5 mm and the transfer distance is D= 67cm, less than 50% of the atoms are emitted with a divergence larger thanr/D∼10 mrad. 3 Theoretical Description for Numerics In order to model both the capture of the atoms from the vapor into the source MOT and the subsequent cool- ing and pushing processes, we have developed a numeri- cal simulation which integrates the equation of motion for atoms chosen with random initial positions and velocities. We describe the atomic motion using classical dynamics. The action of the seven laser beams (6 MOT beams + 1 pushing beam) on an atom located at rwith velocity v is taken into account through an average radiation force F(r,v). We neglect any heating or diﬀusion caused by spontaneous emission. The calculation of the semi-classical force acting on an atom in this multiple beam conﬁguration is a priori verycomplex. For simplicity, we model the atomic transition as a\|g, Jg= 0/angbracketright ↔ \|e, Je= 1/angbracketrighttransition with frequency ¯ hωA, where \|g/angbracketrightand\|e/angbracketrightstand for the ground and excited state respectively. We denote Γ−1the lifetime of e. Consider a single plane-wave beam with wave vector k, detuning δ= ωL−ωA, intensity I, and polarisation σ±along the local magnetic ﬁeld Binr. The radiation pressure force [10] reads F= ¯hkΓ 2s(r,v) 1 +s(r,v)(1) where the saturation parameter is given by s(r,v) =I IsatΓ2 Γ2+ 4(δ−k·v±µB/¯h)2. µis the magnetic moment associated with level \|e/angbracketrightand Isatis the saturation intensity for the transition ( Isat= 1.62 mW/cm2for the D2resonance line in Rb). Still re- stricting our attention to a single traveling wave, we con- sider now the case where the light couples \|g/angbracketrightto two or three Zeeman sublevels \|em/angbracketright. The calculation is in this case more involved since the solution of the optical Bloch equa- tions requires the study of 16 coupled diﬀerential equa- tions. A simple approximation is obtained in the low sat- uration limit ( s≪1): F= ¯hkΓ 2/summationdisplay m=−1,0,1sm(r,v) (2) with sm=Im IsatΓ2 Γ2+ 4(δ−k·v+mµB/ ¯h)2 and where Imis the intensity of the laser wave driving the\|g/angbracketright ↔ \|em/angbracketrighttransition. We can sum up the three forces associated with the three possible transitions, each calcu - lated with the proper detuning taking into account the Zeeman eﬀect. Still working in the low intensity limit, we can gener- alize eq. (2) to the case where Nlaser beams with wave vectors kjand detunings δj,(j= 1, ..., N ) are present. The force then reads F=/summationdisplay j¯hkjΓ 2/summationdisplay m=−1,0,1sj,m(r,v) (3) with sj,m=Ij,m IsatΓ2 Γ2+ 4(δj−kj·v+mµB/ ¯h)2 Note that in establishing eq. (3) we have taken the spatial average of the radiative force over a cell of size λ= 2π/k, neglecting thus all interference terms varying asi(kj−kj′)·r. We therefore neglect any eﬀect of the dipole force associated with the light intensity gradients on the wavelength scale. This is justiﬁed in the case of a leaking MOT since the associated dipole potential wells are much less deep than the expected residual energy of the atoms before extraction.4 W. Wohlleben, F. Chevy, K. Madison, J. Dalibard,: An Atom Fa ucet At the center of the capture MOT, we can no longer neglect inter-beam saturation eﬀects since the saturation parameter for each of the 7 beams is equally ∼1/7. In principle, accounting for this saturation eﬀect requires a step-by-step numerical integration of the 16 coupled Bloch optical equations (for a \|g, Jg= 0/angbracketright ↔ \|e, Je= 1/angbracketrighttransi- tion), as the atom moves in the total electric ﬁeld result- ing from the interference of all the laser beams present in the experiment. Such a calculation is unfortunately much too computationally intensive to lead to interesting pre- dictions for our Atom Faucet in a reasonable time. We therefore decided to turn to a heuristic and approximate expression for the force, demanding: –In the case of a single traveling wave, σ±polarized along the magnetic ﬁeld, we should recover expression (1). –In the low intensity limit, the force should simplify to expression (3). –The magnitude of the force should never exceed ¯ hk Γ/ 2, which is the maximal radiation pressure force in a sin- gle plane wave. There are of course an inﬁnite number of expressions which fulﬁll these three conditions. We have taken the simplest one: F=/summationdisplay i¯hkiΓ 2/summationtext msi,m 1 +/summationtext j,msj,m(4) with partial saturation parameters sj,mas deﬁned in eq. (3). This equation is the generalization of the heuristic expression used by Phillips and co-workers [8] to account for saturation eﬀects in an optical molasses. In the simulation, the MOT beams are chosen to have Gaussian proﬁles truncated to the diameter of the quar- terwaveplates. Also they are chosen to be equally strong with a central intensity of 5 Isatand to have the proper polarizations and directions. The pushing beam’s intensit y is of the same order. We assume that because of optical pumping into the lower hyperﬁne ground state, an atom sees no forces when it is out of the repumper light mixed in the axial beams. Finally, the magnetic quadrupole ﬁeld isB(x) =b′(−2x, y, z ). In the simulation the initial position of each atom is chosen on one of the cell windows following a uniform spa- tial distribution. The initial velocity is given by a Maxwel l Boltzmann distribution for T= 300K. The trajectory is then integrated using a Runge-Kutta method. From these trajectories (see ﬁg. 3), one obtains a probability for an atom to be captured and transferred into the jet, as well as the jet’s characteristics: velocity distribution, diverg ence, and total ﬂux. The absolute ﬂux of the simulated jet is cal- ibrated using the real number of atoms emitted per unit time and per unit surface of the cell at a pressure Pwhich isP/√2πmkBT[11,12]. The simulation neglects interaction eﬀects like colli- sions and multiple light scattering. The validity of the lin - ear scaling with pressure is limited to the low pressure regime ( P <10−7mbar) where the characteristic extrac- tion time of ≃20ms is shorter than the collision time, which is in turn of the order of the trap lifetime.-10-50510 -10010 -40-2002040-10010 Fig. 3. Some simulated trajectories of atoms in the VCMOT + pushing beam light ﬁeld that are captured and transfered to the jet (distances in mm). 4 Results Inspecting qualitatively the trajectories, we ﬁnd that an atom that enters the beam intersection is ﬁrst decelerated by radiation pressure on a distance much smaller than the trapping beam radius. It then slowly moves to the center of the trap where it enters the extraction column. The ﬁnal transverse cooling of the jet takes place during extraction , so that the divergence of the jet grows if the extraction happens too fast. We believe that this is the principal loss mechanism of any leaking MOT system, which have in common an extraction column and a transverse molasses provided by the trapping beams. 4.1 Total Flux For a typical choice of parameters, the simulation ﬁnds 90 % transfer from the source MOT through the diﬀeren- tial pumping tube to the recapture MOT. The remaining 10 % of the atoms leave the source at a divergence too large to be recaptured and are lost. Experimentally, we have achieved a transfer eﬃciency of at most 60 ±10 %. This value is most probably limited by the diﬀerential pumping tube diameter.W. Wohlleben, F. Chevy, K. Madison, J. Dalibard,: An Atom Fau cet 5 0 1 2 3 4 87Rb pressure (10-8mbar)Recapture flux (108atoms/s) 0246 Fig. 4. Recaptured ﬂux versus source cell pressure. The linear ﬁt yields Φjet exp= 1.3±0.2×108atoms /s×PRb87(10−8mbar) . Concerning the total ﬂux, we explored the pressure regime of 10−9< P < 4×10−8mbar and found no devia- tion from a linear dependence (see ﬁg. 4) Φjet exp= 1.3±0.4×108atoms /s×PRb87(10−8mbar) . The uncertainty primarily comes from the atom number determination in the recapture MOT by absorption imag- ing. Deviation from linear scaling with pressure is to be expected when the collision time with background gas be- comes of the order of the typical extraction time from the MOT center into the diﬀerential pumping tube. This will be the case for PRb87≥10−7mbar. In comparison we found that the simulation overesti- mates the capture velocity of the MOT, so that we need to calibrate its predictions. Therefore we simulate a pure MOT without pushing beam and compare the predicted capture rate of τMOT sim= 13×108atoms /s×PRb87(10−8mbar) with the value we measured in the initial regime of linear growth of the vapour charged source MOT, τMOT exp= 2.5±0.6×108atoms /s×PRb87(10−8mbar) We believe that the disagreement between these two results corresponds to an overestimation of the source MOT capture velocity vc. Since the number of atoms cap- tured in a VCMOT varies as v4 c, our simple model over- estimates vcby (13 /2.5)1/4∼1.5. In the graphs 5,6,7, we normalize the absolute value of the ﬂux and concentrate on its variation with system parameters. Simulated VCMOT Optimisation. Using the simulation of a pure MOT without pushing beam, we can readily ﬁnd the parameters which optimise the capture rate from the background vapour. The total laser power is taken to be 20 mW, equally distributed among three beams which arethen retroreﬂected. We calculate an optimal detuning of −3Γ. The capture rate is divided by more than 2 when the detuning is beyond −4.5Γor smaller than −1.5Γ. This is the typical MOT operation range. The magnetic gradient seems to have little inﬂuence as long as it is between 8 and 20 G/cm. It is particularly helpful to calculate the optimal beam waist for a given laser power since in the optical setup this parameter is tiresome to change and demands subsequent trap realignment. In our case, a 9 mm spot size gives the best simulated capture rate, with half maximum values at 4 mm and 16mm. For a ﬁxed laser power, having a large intersection volume is preferable to increasing the satura - tion beyond ∼4Isat. The experiment uses an 8 mm spot size, and the optimum parameters do not change signiﬁ- cantly if the retroreﬂection loss of 20 % is included. Finall y, the simulation reproduces the smoothly decreasing slope of the capture rate versus the MOT beam power of ref [7]. 4.2 Pushing Beam Parameters We now add the thin pushing beam to the MOT light ﬁeld. Doing so does not modify the optimal parameters of the capture MOT, neither in experiment nor in the simulation. Remember that the volume aﬀected by the thin beam is very small compared to the total capture volume of the source MOT. We investigate the inﬂuence of pushing beam power, detuning, and size on the atomic jet emerging from the MOT. The following discussion shall directly combine experimental ﬁndings and the results from the theoretical model. Power. For very low pushing beam power the trap is de- centered but not yet leaking. At Ppush= 80 µW (cor- responding to a pushing beam intensity 1 /4 of a MOT beam intensity), the ﬂux increases sharply and then falls oﬀ with increasing power (see ﬁg. 5). The simulation pre- dicts exactly the same critical power, without adjustable parameters (see ﬁg. 5). The decrease at higher power can be understood if one examines the simulated divergence of the atomic jet, which grows with increasing pushing beam power. This eﬀect is attributed to an insuﬃcient short transverse cooling time due to the strong acceler- ation (see discussion below). Experimentally the jet ve- locity is deduced from measurements like ﬁg. 2. With in- creasing pushing beam power it grows from 12 to 15m/s with an average width of 10m/s. In the simulation, we ﬁnd a smaller width of 1 m/s. This discrepancy is proba- bly due to the fact that we have completely neglected the heating due to spontaneous emission. The longitudinal ve- locity width is larger than that of the LVIS or 2D MOT; however, for the purpose of loading a recapture MOT the velocity width does not matter. Detuning. The complex behaviour of the ﬂux on the push- ing beam detuning ( δpush) is qualitatively very well repro- duced by the simulation (see ﬁgs. 6 and 7). If the pushing6 W. Wohlleben, F. Chevy, K. Madison, J. Dalibard,: An Atom Fa ucet Pushing power (mW)00.2 0.4 0.6 0.8Flux (normalized ) 00.20.40.60.81.0 Fig. 5. Dependence of the atomic ﬂux on the pushing beam power. Flux is normalized, see text. The dots are experiment al, the solid line is simulation. Pushing beam detuning (Γ)-4 -2 0 2 4Experimental flux (normalized ) 00.20.40.60.81.0 Fig. 6. Dependence of the atomic ﬂux on the pushing beam detuning. The ﬂux is normalized as indicated in the text. beam detuning is negative and exceeds the MOT beam detuning \|δpush\|>\|δMOT\|, the trap is decentered, but not yet leaking. Remember that the intensity of the pushing beam is about the same as for the MOT beams, so that as the detuning is increased the pushing radiation pres- sures becomes weaker than the trapping pressure. With zero or small blue detuning, atoms are resonantly acceler- ated, and their extraction is too fast to allow for eﬃcient transverse cooling. These atoms leave at high divergence and are lost. Generally the simulation ﬁnds a 1 : 1 cor- relation of extraction time (ﬂight time ∼10ms from the center of the trap to the depumping region) with diver- gence. Clearly, transverse cooling takes a certain time, an d if the extraction acceleration is too strong, losses due to a high beam divergence are inevitable. For a blue detuning of the pushing beam such that δpush≃ \|δMOT\|, a prominent peak in the ﬂux appears in both the experiment and the simulation. To interpret this result we use the model of a \|g, Jg= 0/angbracketright ↔ \| e, Je= 1/angbracketright transition in a one dimensional magneto-optical trap (thePushing beam detuning (Γ)-4 -2 0 2 4Predicted flux (normalized ) 00.20.40.60.81.0 Fig. 7. Simulation of the dependence of the atomic ﬂux on pushing beam detuning. The ﬂux is normalized as indicated in the text. actual beam inclination and polarisation make the sit- uation a bit more complicated). For an atom traveling downwards in the extraction column, the \|e, m=−1/angbracketright level approaches the MOT beam resonance at negative detuning. At the same time, the \|e, m= +1/angbracketrightlevel ap- proaches the pushing beam resonance at positive detun- ing. When δpush≃ \|δMOT\|, the accelerating pushing beam and the decelerating MOT beams stay equally close to resonance throughout the extraction, and the atoms leave slowly. The extraction time is ∼8 ms and the atoms are cooled transversely leading to a large recapture ﬂux in the lower MOT. Finally if δpush>\|δMOT\|, the detuning of the \|e, m=−1/angbracketrightlevel from the recentering MOT light is al- ways less than the detuning of the \|e, m= +1/angbracketrightlevel from the pushing beam light, and so the trap is decentered but not destabilized (analogous to the behaviour at a large red detuning). Complementary Numerical Study: Waist. With a very small pushing beam size <0.4mm atoms drift out of the extraction column and are decelerated. They are recycled forever or leave the trap with high divergence. For a large spot size of >1.5mm atoms are not all extracted from the center and so many are not cooled suﬃciently transversely. Both cases induce losses. 5 Comparison and Conclusion Certainly, there are other techniques for the directed tran s- fer of cold atoms from a VCMOT into a jet. A moving molasses launch [2] provides a rather cold beam but low ﬂux. A pulsed MOT launched by a resonant beam push is heated in the absence of transverse cooling beams [3]. Dur- ing the launch ∼√ 1000photons are spontaneously emit- ted into the transverse plane, while in continuous schemes there is transverse cooling during extraction. As a resultW. Wohlleben, F. Chevy, K. Madison, J. Dalibard,: An Atom Fau cet 7 there is then no need for magnetic guiding [13], to achieve an elevated transfer eﬃciency. Continuous schemes suﬀer less from interparticle in- teractions, since the steady state source cloud stays small . Leaking MOTs therefore accumulate atoms with the ini- tial capture rate of the MOT. The Atom Faucet provides a 50 % transfer eﬃciency from ﬁrst capture, through the diﬀerential pumping tube, and to a recapture MOT in an UHV cell. It creates an extraction column that is typical of leaking MOT systems with a ﬂexible design and without optical parts inside the vacuum chamber. The ﬂux of Φ= 1×108atoms/s at a background vapour pressure of PRb87= 7.6×10−9mbar is equal to that of the low power version of the LVIS in [6] and su- perior to the 2D+MOT in this pressure region. The later design in turn provides very high ﬂux at high pressure, since it minimizes the source trap density. We did not explore pressures that were incompatible with the UHV requirements in our recapture cell and found no devia- tion from the linear scaling of the ﬂux with pressure up to pressures of 4 ×10−8mbar. Essentially, the Atom Faucet transplants to a MOT at 10−11mbar the loading rate of a MOT at few 10−8mbar. We have also presented a 3D simulation of the atomic motion in multiple laser ﬁelds with an inhomogeneous magnetic ﬁeld, neglecting interactions and ﬂuctuations. We ﬁnd that the transverse cooling inside the extraction column turns out to be a crucial element for the satis- factory performance of leaking MOT atom sources. Our simulation overestimates the capture rate, but predicts well the measured parameter dependences. Moreover, it is readily adapted to an arbitrary laser and B-ﬁeld conﬁgu- ration. We are indebted to F. Pereira dos Santos for coming up with the child’s name and to the ENS Laser Cool- ing Group for helpful discussions. This work was partially supported by CNRS, Coll` ege de France, DRET, DRED, and EC (TMR network ERB FMRX-CT96-0002). This material is based upon work supported by the North At- lantic Treaty Organisation under an NSF-NATO grant awarded to K.M. in 1999. W.W. gratefully acknowledges support by the Studienstiftung des deutschen Volkes and the DAAD. Note added: After this work was completed, we became aware that a similar setup has been successfully achieved in Napoli, in the group of Prof. Tino. References 1. B. Ghaﬀari, J. M. Gerton, W. L. McAlexander, K. E. Strecker, D. M. Homan and R. G. Hulet, Phys. Rev. A 60, 3878 (1999) 2. S. Weyers, E. Aucouturier, C. Valentin and N. Dimarcq, Optics Comm. 143, 30 (1997) 3. J.J. Arlt, O. Marag´ o, S. Webster, S. Hopkins and C. J. Foot, Optics Comm. 157, 303 (1998) 4. H. Chen and E. Riis, Appl. Phys. B 70, 665 (2000) 5. Z.T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996)6. K. Dieckmann, R. J. C. Spreeuw, M. Weidem¨ uller and J. T. M. Walraven, Phys. Rev. A 58, 3891 (1998) 7. K. Lindquist, M. Stephens and C. Wieman, Phys. Rev. A 46, 4082 (1992) 8. P.D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts and C. I. Westbrook, J. Opt. Soc. Am B 6, 2084 (1989) 9. We checked that neither in experiment nor in simulation does the direction of the linear polarization have any eﬀect . 10. C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-Photon Interactions, Basic Processes (Wiley 1992). 11. F. Reif, Fundamentals of statistical and thermal physics (McGraw-Hill, New York, 1965). 12. In order to increase the eﬃciency of the simulation we only evolve atoms with an initial velocity lower than vmax= 45 m/s. We checked that atoms with a larger veloc- ity cannot be captured in the MOT, whatever the direction of their initial velocity. 13. C.J. Myatt, N. R. Newbury, R. W. Ghrist, S. Loutzenhiser and C. E. Wieman, Opt. Lett. 21, 290 (1995)
LABORATÓRIO DE INSTRUMENTAÇÃO E FÍSICA EXPERIMENTAL DE PARTÍCULAS Preprint LIP/01 -04 19 March 2001 A Large Area Timing RPC A. Blanco1,2, R. Ferreira -Marques1,3, Ch. Finck4, P. Fonte1,5,, A. Gobbi4, A. Policarpo1,3, M. Rozas2. 1-LIP, Coimbra, Portugal. 2-GENP, Dept. Fisica de Particulas, Univ. de Santiago de Compostela, Espanha 3-Departamento de Física da Universidade de Coimbra, Portugal. 4-GSI, Darmstadt, Germany. 5-ISEC, Coimbra, Portugal. Abstract A large area Resisti ve Plate Chamber (RPC) with a total active surface of 160×10 cm2 was built and tested. The surface was segmented in two 5 cm wide strips readout on both ends with custom, very high frequency, front end electronics. A timing resolution between 50 and 75 ps σ with an efficiency for Minimum Ionizing Particles (MIPs) larger than 95% was attained over the whole active area, in addition with a position resolution along the strips of 1.2 cm . Despite the large active area per electronic channel, the observed timing resolution is remarkably close to the one previously obtained (50 ps σ) with much smaller chambers of about 10 cm2 area. These results open perspectives of extending the application of timing RPCs to large area arrays exposed to moderate particle multip licities, where the low cost, good time resolution, insensitivity to the magnetic field and compact mechanics may be attractive when compared with the standard scintillator -based Time -of-Flight (TOF) technology. Submitted to Nucl. Instrum. and Meth. in P hys. Res. A Corresponding author: Paulo Fonte, LIP - Coimbra, Departamento de Física da Universidade de Coimbra, 3004 -516 Coimbra, PORTUGAL. tel: (+351) 239 833 465, fax: (+351) 239 822 358, email: fonte@lipc.fis.uc.pt 21 Introduction The development of timing Resistive Plate Chambers (RPCs) [1] opened the possibility to build large -granularity high -resolution TOF systems at a quite reduced cost per channel when compar ed with the standard scintillator -based technology. Previous work has shown a timing resolution better than 50 ps σ at 99% efficiency in single four -gap chambers [2] and an average timing resolution of 88 ps σ at and average efficiency of 97% in a 32 channel system [3]. It has also been shown that each amplifying gap of 0.3 mm thickness has a detection efficiency close to 75% and that the avalanche develops under the infl uence of a strong space charge effect [4]. A Monte - Carlo model of the avalanche development reproduced well the observed data, confirming the dominant role of space charge effects in these detectors [5]. Although timing RPCs have so far been built with relatively small active areas per electronic readout channel (on the order of 10 cm2), compatible with the high-multiplicity requirements of High Energy Heavy Ion Physics, ther e is a number of possible applications in lower multiplicity environments ( [6], [7]) for more coarsely segmented counters. Having in view such applications, we describe in this paper the structure a nd performance of a large counter, with an active area of 160 ×10 cm2, readout by only 2 or 4 electronic channels. 2 Detector description The detector was built from 3 mm thick float -glass plates with an area of 160 ×12 cm2 and a measured bulk resistivity of 2×1012 Ω cm. The stack of plates, with attached copper foil1 electrode strips (see Figure 1), was mounted on a supporting 1 cm thick acrylic plate and a controlled pressure was applied to the stack by means of regularly spaced spr ing-loaded plastic bars. Four gas gaps were defined by glass fibres of 0.3 mm diameter placed between the glass plates, beneath the pressing bars. There were six individual electrode strips, with dimensions of 160 ×5 cm2, connected in two independent multil ayer groups with a 1 mm wide interstrip distance. The arrangement defined an active area of 160 ×10 cm2, leaving uncovered a 1 cm wide region on the outer rim of the glass plates. The ends of each group were connected via a short 50 Ω coaxial cable ( Figure 1) to preamplifiers placed inside the gas volume, whose signal was fed through gas tight connectors to external amplifier -discriminator boards. The front -end chain was custom-made from commercially available analogue and digital i ntegrated circuits, yielding measured timing and charge resolutions of respectively 10 ps σ and 3.2 fC σ [8]. For stability reasons the discriminators had a built -in dead time of 1 µs after each detected pulse that will contrib ute to the overall counter inefficiency. High voltage, around 6 kV, was applied to the outer strips via 10 M Ω resistors and the signal travelling in these strips was fed to the shielding of the signal cables via 2.2 nF high voltage capacitors (see Figure 1). The central wire of the signal cables was connected directly to the central strips and to the preamplifiers inputs. To avoid the use 1 3M cond uctive-adhesive copper tape. 3of floating electrodes, the glass plates placed in the middle of each (upper and lower) half of the detector had thin copper electrodes glued along their lateral edges, kept by a resistive voltage divider at half of the voltage applied to the outer strips. The detector, high voltage distribution network and preamplifiers were placed inside a gas-tight aluminium enclosure that was kept under a continuous flow of a non-flammable gas mixture consisting of 85% C2H2F4† + 10% SF6 + 5% iso-C4H10 [9], at a flow rate close to 100 cm3/min. 3 Test Setup and Data Acquisition The tests were made at the CERN PS using a secondary beam (T11) of 3.5 GeV particles, mainly negative pions, in August 2000. The spills were 0.3 s long and spaced by a few seconds. Most tests were done with a defocused beam that illuminated the detector over a regi on of a few hundred cm2. A pair of plastic scintillation counters (Bicron BC420) measuring 8 ×3×2 cm3 and viewed on each 3 ×2 cm2 face by a fast photomultiplier (Hamamatsu H6533) provided the reference time information. Both counters had a timing resolution close to 35 ps σ and defined a coincidence (trigger) area of 2 ×2 cm2, being placed upstream from the RPC. The data acquisition system, based on the CAMAC standard, was triggered by the coincidence of both timing scintillation counters, being additionally required that no signal was present (veto) in a fourth wide scintillation counter that surrounded the coincidence -selected beam. After a valid trigger the four timing signals from the reference scintillation counters and the four timing signals from the RP C were digitised by a LeCroy 2229 TDC. A LeCroy 2249w ADC, operated with a gate of 300 ns, digitised the corresponding charge information. The TDC was calibrated using an Ortec 240 time calibrator and the ADC was calibrated by injecting in the preamplifier s a known amount of charge via the test inputs. 4 Data Analysis Prior to any analysis the TDC digitisation error was taken into account by adding to each data value a random number distributed in the interval [ -0.5, 0.5] and the events were selected by exter nal and internal cuts. After cuts the events were attributed to the strip showing the largest charge signal and the data from each strip was analysed separately. 4.1 External Cuts To clear the data from beam -related artefacts, like multi -particle events or sc attered particles, several cuts were made based on the information collected from the timing scintillators. Events selected for further analysis had to comply with the following requirements: † Commercially known as R134a. 4 the difference between the TOF information from both timing scin tillators (time average between both ends of a counter) should agree within ±2σ of the mean value; the position information from each timing scintillator (time difference between both ends of a counter) should be within ±2σ of the respective mean value2; the charge measured on each timing scintillator should lie between the 5% and the 95% percentiles of the respective (strongly non -gaussian) distribution. All these distributions showed wide tails beyond the specified cuts, generating corresponding tails in the time response of the RPC (see section 4.5). Typically about 50% of the events were accepted after external cuts. 4.2 Internal Cuts The time difference between both strip ends (that should be independent of the details of the av alanche development process and depend only on its position) shows large bilateral tails that were cut at ±2σ of the mean value3. This cut had a negligible effect on the time resolution of the counter but it strongly reduced the amount of timing tails (see section 4.5) close to the strip ends. Tighter cuts have little further influence and presumably the remaining timing tails are mostly due to the physical process related to the avalanche onset and development (depending, for i nstance, on the applied voltage - see section 5.4). It should be noted that when the quality of the trigger was enhanced by requiring an additional coincidence with a small (0.3 ×0.6 cm2) scintillator placed downstream from the RPC the amount of tails became negligible (see Figure 8) and the cut mentioned above had little effect on the results, suggesting that the need for such cut arises mainly due to beam quality limitations. For accurate correction of the measured time as a function of charge and to decrease the amount of timing tails seen by the RPC without significant efficiency loss, a 1% cut was made in the lower part of the RPC charge distribution ( Figure 2). (The RPC charge was defined as the sum of the charges sensed on both ends of each strip.) 4.3 Detection efficiency The detection efficiency was determined for each strip after the external cuts and before the internal cuts. The impact of the (optional) int ernal cuts should be subtracted from the efficiency values given in all figures. Two different definitions of detection efficiency were formed by the ratio of the following quantities to the total number of events: the number of events yielding an amount of charge in the RPC larger than the upper limit of the ADC pedestal distribution (charge efficiency); the number of events for which a valid time was measured in the RPC (time efficiency). 2 The position spread was mainly related with the width of the 2 ×2 cm2 coincidence -selected trigger region. 3 Note that the same type of cut was applied to the timing scintillator data. 5In principle both definitions should yield similar results except if a considerable amount of inter -strip crosstalk will be present. In this case the crosstalk signal, not being galvanically coupled, will not contribute to any net charge but may induce a voltage level above the discriminating threshold, generating a vali d-time event. 4.4 Position accuracy A position -dependent timing information can be formed for each event by the time difference between both strip ends. This information was calibrated with respect to a known displacement of the RPC and it was charge -corrected in a manner similar to the TOF information (section 4.5). 4.5 Timing accuracy In principle a position -independent timing (TOF) information can be formed for each event by averaging the time measured on both strip ends. However, it was found that close to the extremes of the counter the TOF information was correlated with the position information and a linear correction could be applied to the former as a function of the later. Data will be presented with and without this position c orrection. The TOF information also strongly correlates (see Figure 3) with the measured signal charge and a correction was made along the lines described in [2]. The method automatically determines an d corrects the contribution of the reference counters time jitter. The resulting time distribution was not purely gaussian, showing a bilateral excess of events (timing tails) exemplified in Figure 4. We opted to characterize sepa rately the main timing resolution figure ( σ) and the amount of timing tails, since the later can be important or not, depending on the application. The main resolution figure was determined by a gaussian fit to the corrected time distribution (with 5 ps bins) within ±1.5 σ of the mean value and the timing tails were characterized as the fraction of events whose distance to the mean value exceeds 300 ps. For a purely gaussian distribution with σ ≤ 100 ps this fraction should be smaller than 0.3%. For each ru n the time vs. log(charge) correlation curve, which is almost linear, was characterised by the average time, charge and slope (see Figure 3). This information was used to assess the stability of the time -charge correlation and the need for separate correction curves at different positions along the counter. 5 Results and Discussion In the following discussion we will refer to the position of the centre of the trigger region along and across the strips as, respectively, the X and Y co ordinates, defining the origin of the coordinate system in the geometrical centre of the counter. 5.1 Detection Efficiency and Crosstalk Charge and time efficiency curves are shown in Figure 5 a) as a function of Y (for X=0). The frac tion of events inducing a measurable amount of charge in both strips is also shown, probably corresponding to avalanches occurring close to the inter -strip region. 6When the trigger region was fully contained within a single strip the charge and time efficiencies closely match for that strip, while the opposing strip shows a very reduced charge efficiency and a considerable (from 80% to 90%) time efficiency. This large crosstalk level (see section 4.3), actually to be expected on such long strips, did not significatively affect the timing characteristics of the device, but would affect its multi - hit capability. In Figure 6 a) the time efficiency is shown as a function of X (for Y= ±3 cm, corresponding esse ntially to the centre of each strip). The measurements were generally taken in steps of 7.5 cm except for a region of strip A, between 0 and 20 cm, that was scanned in steps of 1.5 cm to assure that at least one measurement contained a spacer. The values r ange between 95% and 98%, being slightly larger for the strip A. This small difference can be attributed to slight differences in the gain of the front -end electronics chain. However it should be noted that a smaller chamber of similar construction has sho wn a time efficiency above 99% [2]. The slightly reduced efficiency found in the present counter can be attributed to a poorer trigger quality, evidenced by the tails visible in the scintillators time and position information ( see section 4.1) and to a much larger sensitive area that collects a larger event rate from the wide beam (see the discussion about the discriminators dead -time in section 3). The combination of bot h strip signals into a single amplifier for each end of the counter, doubling the active area per amplifier, caused absolutely no degradation in the detection efficiency. Also no influence from the spacers could be found in the fine -step scan. 5.2 Timing Accur acy 5.2.1 Timing resolution The timing resolution is shown as a function of Y (for X=0) in Figure 5 b). It ranges from 58 to 76 ps σ across the counter, including the outer edges and the inter -strip region. Since in a real application t here would be no possibility to determine the avalanche position along Y, in the same figure we present also (horizontal lines) the resolution figure obtained by analysing for each strip a data set containing an equal number of events from each position, y ielding 67 and 76 ps σ for the strips A and B, respectively. In Figure 6 b) we show the timing resolution as a function of X (for Y= ±3 cm). Most data points range from 50 to 70 ps σ, except for two regions, around -20 cm and -70 cm, where the resolution was degraded to, respectively, 80 and 90 ps σ. This degradation most probably has a local mechanical origin, since it is not symmetrical with the counter geometry and not identical for both strips. The combination of both strip sig nals into a single amplifier for each end of the detector, doubling the active area per amplifier, caused absolutely no degradation in the time resolution of the device. Also no clear influence from the spacers could be found in the fine-step scan. 5.2.2 Timing tails The magnitude of the timing tails is shown as a function of Y (for X=0) in Figure 5 c). The tails do not exceed 2% of the total number of events, being smaller than 71% in the centre of the strips. The effect is larger in the outer edges than in the inner strip edges, possibly because no attempt was made to sharply reduce the electric field at the strip edges (the glass plates extend up to 1 cm beyond the copper strips). Avalanches occurring in this space will induce currents both in the strips and in the enclosing gas box, creating a position -dependent induced charge fluctuation that may cause timing errors. A similar phenomenon can be perceived in the space between the strips, where the induced charge was shared among the str ips in a position -dependent manner (see also Figure 5 a)). In Figure 5 c) we present also (horizontal lines) the values obtained by analysing for each strip a data set containing an equal number of events from each position, yielding tails of 1% for both strips. In Figure 6 c) we show the amount of timing tails as a function of X (for Y= ±3 cm). Strip B shows tails generally below 1.0 %, raising up to 1.5 % close to the counter extremities. Strip A shows larger tails, up to 2% over the whole counter. A possible reason for this difference, that doesn’t appear in the Y -scan (Figure 5 c)), could be a momentary beam quality fluctuation. The combination of both strip signals into a single amplifier for each side of the detector, doubling the active area per amplifier, caused absolutely no degradation in the amount of tails and no clear influence from the spacers could be found in the fine -step scan. 5.2.3 Variations of the measured time and charge along the counter Due to mechanical or electrical inhomogeneities there will be position dependent charge and time variations along the counter. This effect should be corrected by calibration using the counter’s position resol ution (discussed in section 4.4), being however interesting to determine how finely segmented this calibration should be. In Figure 7 we show the variations of time (a), charge (b) and of the slope of the time vs. log(charge) correlation curve (c), represented in Figure 3, as a function of X (for Y= ±3 cm). Variations of the average time by about 400 ps are apparent along with large changes of the time vs. log(charge) correlatio n slope, while the average charge remains relatively stable. It should be noted that the large slope variations visible in the left-hand side of Figure 7 c) correlate well with the degradation of the time resolution visible in the same region of Figure 6 b). To evaluate directly the influence of these position dependencies on the time resolution of the counter as a function of the segmentation of the calibration procedure along X, we have combined an equal number of events from adjacent positions along the strip B and analysed jointly the resulting data set. The results are shown in Figure 7 d): events from the right -hand side of the counter could be jointly analysed without much d egradation arising from position dependent effects, while the left hand side was severely affected by such effects, calling for a finer segmentation of the calibration procedure. Such features have probably a local, mechanical, origin, since all other vari ables are equal along the counter. 5.3 Position Resolution In Figure 8 a) the time difference between both strip ends is plotted as a function of X (for Y= ±3 cm) showing an accurately linear dependency with a slope of 70.9 ps/cm, 8which corresponds to a signal propagation velocity of 14.1 cm/ns. In Figure 8 b) the time difference distribution is plotted for two trigger positions 5.0 cm apart, yielding a position accuracy of 12 mm σ. For this measurement the wid th of the trigger region in the X direction was reduced to 3 mm via an extra coincidence scintillator. 5.4 Behaviour as a function of the applied voltage It is interesting to study how some of the quantities mentioned above change as a function of the applied voltage, as illustrated in Figure 9. Figure 9 a) shows the evolution of the time resolution and efficiency, the later showing a plateau of 98% above 6.1 kV while the former shows a broad minimum at 52 ps σ close to 6.1 kV. This voltage has been chosen as the optimum operating point and most of the data presented in the previous sections has been taken at this setting. Figure 9 b) shows the evolution of the average fast charge and of the amount of timing tails. The behaviour of charge as a function of voltage was already discussed at length in [4] and we will not further elaborate on this subject here. The amount of timing tails decreases with increasing voltage and reaches a plateau of 1% above 5.9 kV. Figure 9 c) shows the variation of the absolute measured time and of the time vs. log(charge) correlation slope. As expected, the measured time shows a negative variation, compati ble with a larger value of the avalanches exponential growth parameter; further details on this subject can be found in ( [4], [5], [8]). The time vs. log(charge) correla tion slope becomes less steep with increasing voltage, reaching a plateau above 6.3 kV. 5.5 Behaviour as a function of the counting rate Being the counting rate capability an important characteristic of RPCs, several quantities of interest were studied as a f unction of the counting rate per unit area, as show in Figure 10. The vertical arrow indicates the standard operating point (140 Hz/cm2) at which most of the measurements presented above were taken. Figure 10 a) shows the variation of the time resolution and efficiency. Both quantities are constant below 140 Hz/cm2 and degraded at larger counting rates. Operation up to 500 Hz/cm2 may be possible if a slight degradation of the counter performance is accepte d (comparable with the performance variations observed as a function of position). Figure 10 b) shows the evolution of the average fast charge and of the amount of timing tails. The average fast charge shows a continuous decrease with increasing counting rate, suggesting a counting -rate induced reduction of the average electric field in the amplifying gap, while the timing tails remain quite small (less than 4 %) up to 1 kHz/cm2. Figure 10 c) shows the var iation of the absolute measured time and of the time vs. log(charge) correlation slope. The average measured time shows a positive variation, compatible the observed reduction of the gas gain, which is nevertheless quite small (±10 ps) around the standard operating point, while the slope remains essentially unchanged around this point. It should be stressed that any variations of these quantities can be taken into account by appropriate calibration. 9The general behaviour of the detector characteristics as a function of the counting rate indicates that there is no degradation of the performance up 140 Hz/cm2 and that a counting rate of 500 Hz/cm2 could be handled if a slight performance degradation of will be accepted. 6 Conclusions We built and tested a large area timing RPC, with an active surface of 160 ×10 cm2, to be applied in medium (e.g. [6]) or low multiplicity (e.g. [7]) TOF counters. The active area was segmented in two readout strips, each measu ring 160 ×5 cm2, sensed in both ends by identical custom -made, very high -frequency, front -end electronic channels [8]. A timing resolution between 50 and 90 ps σ with an efficiency between 95% and 98% for MIPs was attained over the whole active area. The performance could be improved close to the strip ends by correcting the measured time as a function of the measured avalanche position, thus improving the time resolution range to lie between 50 and 75 ps σ. The combination of both strip signals into a single amplifier for each end of the detector, doubling the active area per amplifier, caused absolutely no degradation in the efficiency or in the time resolution of the device. Also no clear influence from the spacers could be found in a fine -step scan. The avalanche position along the counter could be determined from the time difference between both strip ends, yielding a position resolution of 1.2 cm σ along the strips with very good linearity. Timing tails, defined as the fraction of events whose absolute time deviation from the average was larger than 300 ps, were smaller than 2%, occurring the larger values in the outer edges of the detector and close to the strip ends. The general behaviour of the detector characteristics as a fu nction of the counting rate indicates that there is no degradation of the performance up 140 Hz/cm2 and that a counting rate of 500 Hz/cm2 could be handled if a slight degradation is accepted. Probably due to structural inhomogeneities there were considera ble variations of the average measured time along the counter, requiring a calibration procedure segmented every few tens of centimetres. In an experimental array this segmentation would be achieved with the help of the counter’s position resolution. The large inter -strip crosstalk level observed (80%) does not seem to influence the time resolution of the counter, affecting only its multi -hit capability. It should be pondered whether for a given application it is not preferable to base the detector on multiple layers of single -strip chambers, reaching full geometrical coverage and avoiding any crosstalk. A multilayer configuration, providing multiple measurements for each particle, would also have the advantages of being self -calibrating and allowing an improved rejection of timing tails 7 Acknowledgements We are grateful to Paolo Martinengo and Piotr Szimanski of the ALICE test beam support team for their efficient and friendly cooperation; to Juan Garzon from 10University of Santiago de Compostela for his inte rest and support; to José Pinhão, Américo Pereira and Fernando Ribeiro from our technical staff for their competent collaboration. This work was done in the framework of the FCT project CERN/P/FIS/15198/1999. 8 References [1] P.Fonte, A. Smirnitski and M.C.S. Wi lliams, “A New High -Resolution Time -of- Flight Technology”, Nucl. Instr. and Meth. in Phys. Res. A , 443 (2000) 201. [2] P. Fonte, R. Ferreira Marques, J. Pinhão, N. Carolino and A. Policarpo “High - Resolution RPCs for Large TOF Systems“, Nucl. Instr. and Meth. i n Phys. Res. A, 449 (2000) 295. [3] A. Akindinov et al.,”A Four -Gap Glass -RPC Time of Flight Array with 90 ps Time Resolution”, ALICE note ALICE -PUB-99-34, preprint CERN -EP-99-166. [4] P.Fonte and V.Peskov. “High -Resolution TOF with RPCs”, presented at the “PSD99 - 5th International Conference on Position -Sensitive Detectors”, 13 - 17 th September 1999, University College, London, preprint LIP/00 -04 http://xxx.lanl.gov/abs/physics/0103057. [5] P.Fonte, “High -Resolution Timing of MIPs with RPCs – a Model”, presented at the “RPC99 - 5th International Workshop on Resistive Plate Chambers”, 28 - 29th October 1999, Bari, Italy, Nucl. Instr. and Meth. in Phys. Res. A , 456 (2000) 6. [6] FOPI-Collaboration, "Upgrading the FOPI Detector System", GSI-Scientific Report, (1998), pp. 177. [7] The HARP collaboration ( PS214 ), "The Hadron Production Experiment at the PS", CERN -SPSC/99 -35, SPSC/P315, 15 November, 1999. [8] A. Blanco, N. Carolino, P. Fonte, A. Gobbi, “A Simplified and Accurate Front - End Electronics Chain for Timing RPCs”, presented a t the “LEB 2000 -6th Workshop on Electronics for LHC Experiments”, 11 -15 September 2000, Cracow, Poland, published in the conference proceedings CERN 2000 -010 CERN/LHCC/2000 -041. A. Blanco, N. Carolino, P. Fonte, A. Gobbi, “A New Front -End Electronics Chain for Timing RPCs”, presented at the “2000 IEEE Nuclear Science Symposium and Medical Imaging Conference”, 15 -20 October 2000, Lyon, France, accepted for publication in IEEE Trans. Nucl. Sci. [9] P. Camarri et al, “Streamer suppression with SF 6 in RPCs operat ed in avalanche mode”, Nucl. Instr. and Meth. in Phys. Res. A 414 (1998) 317. 11 9 Figure Captions Figure 1: Pictures and schematic drawings of the detector. Figure 2: Typical fast charge4 distribution in log arithmic and linear (inset) scales. Figure 3: Typical time vs. log(charge) correlation plot, showing the calculated time-charge correction curve (thin line) and the average slope (thick line). Figure 4: Typical time distribution (from strip B at X= -30 cm) after charge correction in logarithmic and linear (inset) scales. The thick line corresponds to a gaussian curve fitted within ±1.5 σ to determine the main resolution figure (after correction for the cont ribution of the start counters – see section 4.5 – the timing resolution for this example is st=53 ps). The dashed line corresponds to the extension of the fitted gaussian to ±3.5 σ. Timing tails were defined as the fraction of events whose absolute time deviation from the average was larger than 300 ps and amount in this example to 0.4 %. Figure 5: Several quantities of interest as a function of the position of the centre of the trigger region across t he strips. a) Charge and time efficiency. The lower curve corresponds to the fraction of events that show a measurable amount of charge in both strips. The superimposed dashed lines indicate the position of the copper strips and the outer dotted line the e dge of the glass plates. b) Timing resolution with separate analysis in each position and when all events for each strip are analysed simultaneously. The solid triangles correspond to data taken with both strips connected together. c) Same as b), for the t iming tails. Figure 6: Several quantities of interest as a function of the position of the centre of the trigger region along the strips. a) Time efficiency, better than 95%; b) Time resolution with and without position correction , ranging from 50 to 90 ps σ (50 to 75 ps σ with position -dependent time correction). c) Timing tails with and without position correction, smaller than 2%. In all figures the solid triangles correspond to data taken with both strips connected together. 4 Electron ic component of the signal. 12Figure 7: Several quantities related with the time -charge correlation curve (see Figure 3) are plotted as a function of the position of the trigger region along the strips5. a) Variations of the average mea sured time, covering a range of 400 ps. b) Average measured charge, which shows little variation along the counter. c) The average slope shows considerable variations along the counter, particularly in the left hand side, where a poorer time resolution is also visible ( Figure 6). d) Joint analysis of a data set containing an equal number events from each of the positions indicated by the extent of the horizontal lines: the right hand side is only weakly affected by position depende nt effects, while the left hand side would require a more finely segmented calibration procedure. Figure 8: a) Time difference between both strip ends as a function of the position of the trigger region along the strips. There is an accurately linear dependency (evidenced by the small residues shown in b)), with a slope of 70.9 ps/cm, corresponding to a signal propagation velocity of 14.1 cm/ns. c) The width of the trigger region was reduced to 3 mm in the X direction and the sprea d of the time difference was compared with a measured displacement of 5.0 cm, yielding a position accuracy of 12 mm σ. Figure 9: Several quantities of interest plotted as a function of the applied voltage. a) Time resolution and e fficiency. b) Average amount of fast charge and the amount of timing tails. c) The variation of the absolute value of the measured time and the slope of the time vs. log(charge) correlation curve. Figure 10: Several quantities of interest plotted as a function of the counting rate density. a) Time resolution and efficiency. b) Average fast charge and amount of timing tails. c) The variation of the absolute value of the measured time and the slope of the time vs. log(charge) correla tion curve. The vertical arrow indicates the counting rate at which most of the measurements presented were taken. 5 It should be stressed that these variations can be corrected by calibration using the position information given by the time difference between both strip ends ( Figure 8). 133 mm float glass Copper foil3 mm float glass Copper foilTop view 5 cm1.6 m 5 cm1.6 m Side viewa) b)...1 GΩ 10 MΩ1 GΩHV 1.6 m2.2 nF......1 GΩ1 GΩ 10 MΩ10 MΩ1 GΩ1 GΩHV 1.6 m2.2 nF Figure 1: Pictures and schematic drawings of the detector. 14 0 2 4 6 8100101102103 0 2 4 6 8100101102103 02460100200300 Fast charge (pC)Events / 20 fC 0 2 4 6 8100101102103 0 2 4 6 8100101102103 02460100200300 0 2 4 6 8100101102103 0 2 4 6 8100101102103 02460100200300 Fast charge (pC)Events / 20 fC Figure 2: Typical fast charge distribution in log arithmic and linear (inset) scales. 1500.5 11.5 22.5 310001200140016001800200022002400Measured time (ps) log10(Charge /bin) Figure 3: Typical time vs. log(charge) correlation plot, showing the calculated time -charge correction curve (thin line) and the average slope (thick line). 16 -1000 -500-300 0300500 1000100101102103 Time diference ( ps)Events /20 ps s= 63.4 ps 0200400600 -1000-50005001000 -1000 -500-300 0300500 1000100101102103 Time diference ( ps)Events /20 ps s= 63.4 ps 0200400600 -1000-500050010000200400600 0200400600 -1000-50005001000 -1000-50005001000 Figure 4: Typical time distribution (from strip B at X= -30 cm) after charge correction in logarithmic and linear (inset) scales. The thick line corresponds to a gaussian curve fitted within ±1.5 σ to determine the main resolution figure (after correction for the contribution of the start counters – see section 4.5 – the timing resolution for this example is σt=53 ps). The dashed line corresponds to the extension of the fitted gaussian to ±3.5 σ. 17Strip A 0,0%0,5%1,0%1,5%2,0%2,5% -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Center of the trigger region across the strips (cm)Timing tailsStrip A 405060708090100 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Center of the trigger region along Y (cm)Resolution (ps σσ)Strip A Strip B Strip A - all events Strip B - all events Strips A+Ba) b) c)0%10%20%30%40%50%60%70%80%90%100% -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7EfficiencyStrip A-Charge Strip B-Charge Strip A-Time Strip B-Time Strips A&B-ChargeSize of the trigger region Strip A Strip B Figure 5: Several quantities of interest as a function of the position of the centre of the trigger region across t he strips. a) Charge and time efficiency. The lower curve corresponds to the fraction of events that show a measurable amount of charge in both strips. The superimposed dashed lines indicate the position of the copper strips and the outer dotted line the e dge of the glass plates. b) Timing resolution with separate analysis in each position and when all events for each strip are analysed simultaneously. The solid triangles correspond to data taken with both strips connected together. c) Same as b), for the timing tails. 1894%95%96%97%98%99%100% -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strips (cm)Timing efficiencyStrip A Strip B Strips A+B c)405060708090100 -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strips (cm)Timing resolution (ps σσ) b)a) 0.0%0.5%1.0%1.5%2.0%2.5%3.0% -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strips (cm)Timing tailsStrip A Strip B Strip A - position corrected Strip B - position corrected Strips A+B Figure 6: Several quantities of interest as a function of the position of the centre of the trigger region along the strips. a) Time efficiency, better than 95%; b) Time resolution with and without position correction , ranging from 50 to 90 ps σ (50 to 75 ps σ with position - dependent time correction). c) Timing tails with and without position correction, smaller than 2%. In all figures the solid triangles correspond to data taken with both strips connected together. 19-800-700-600-500-400-300-200 -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strips (cm)Time-charge correlation slope (a.u.)00.10.20.30.40.50.60.70.80.91 -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strips (cm)Average charge (pC) c)b)a)-300-200-1000100200300 -80-70-60-50-40-30-20-10 01020304050607080Time walk (ps)Strip A Strip B 406080100120140 -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strip B (cm)Timing resolution (ps σσ) d) Figure 7: Several quantities related with the time -charge correlation curve (see Figure 3) are plotted as a function of the position of the trigger region along the strips . a) Variations of the average mea sured time, covering a range of 400 ps. b) Average measured charge, which shows little variation along the counter. c) The average slope shows considerable variations along the counter, particularly in the left hand side, where a poorer time resolution is also visible ( Figure 6). d) Joint analysis of a data set containing an equal number events from each of the positions indicated by the extent of the horizontal lines: the right hand side is only weakly affected by position depende nt effects, while the left hand side would require a more finely segmented calibration procedure. 20a) c)-8-6-4-202468 -80-70-60-50-40-30-20-10 01020304050607080ΔΔt/2 (ns) Strip A Strip By=0.0709 + 0.0001 -0,2-0,100,10,20,3 -80-70-60-50-40-30-20-10 01020304050607080 Center of the trigger region along the strips (cm)Fit residuals (ns) b) 0050100150200250300350400 ΔΔt/2 (ps)Events/25 psσX= 12 mm 5 cm -1000 -500 500 1000 0050100150200250300350400 ΔΔt/2 (ps)Events/25 psσX= 12 mm 5 cm -1000 -500 500 1000 Figure 8: a) Time difference between both strip ends as a function of the position of the trigger region along the strips. There is an accurately linear dependency (evidenced by the small residues shown in b)), with a slope of 70.9 ps/cm, corresponding to a signal propagation velocity of 14.1 cm/ns. c) The width of the trigger region was reduced to 3 mm in the X direction and the sprea d of the time difference was compared with a measured displacement of 5.0 cm, yielding a position accuracy of 12 mm σ. 2100,511,5 5,1 5,3 5,5 5,7 5,9 6,1 6,3 6,5 6,7 6,9 Tensão (KV)Average fast charge (pC)0,0%1,0%2,0%3,0% Timing tails TailsCharge -300-100100300500 5,1 5,3 5,5 5,7 5,9 6,1 6,3 6,5 6,7 6,9 Applied voltage (KV)Time walk (ps) -600-500-400-300-200 Time-charge correlation slope (a.u.)Time Walk Slope405060708090100 5,1 5,3 5,5 5,7 5,9 6,1 6,3 6,5 6,7 6,9Time resolution (ps σσ) 88%90%92%94%96%98%100% Time efficiencyResolutionEfficiency a) b) c) Figure 9: Several quantities of interest plotted as a function of the applied voltage. a) Time resolution and e fficiency. b) Average amount of fast charge and the amount of timing tails. c) The variation of the absolute value of the measured time and the slope of the time vs. log(charge) correlation curve. 22 140140a) b) c)255075100125 10 100 1000 10000Time resolution ( ps σ) 80%85%90%95%100% Time efficiencyEfficiency Resolution 255075100125 10 100 1000 10000Time resolution ( ps σ) 80%85%90%95%100% Time efficiencyEfficiency Resolution 0.00.51.01.52.0 10 100 1000 10000Average fast charge ( pC) 0%2%4%6%8% TailsCharge Tails 0.00.51.01.52.0 10 100 1000 10000Average fast charge ( pC) 0%2%4%6%8% TailsCharge Tails 020406080100120140 10 100 1000 10000 Counting rate density (Hz/cm2)Time variation ( ps) -400-350-300-250-200-150-100-50 Time charge correlation slope (a.u.) TimeSlope 020406080100120140 10 100 1000 10000 Counting rate density (Hz/cm2)Time variation ( ps) -400-350-300-250-200-150-100-50 Time charge correlation slope (a.u.) TimeSlope Figure 10: Several quantities of interest plotted as a function of the counting rate density. a) Time resolution and efficiency. b) Average fast charge and amount of timing tails. c) The variation of the absolute value of the measured time and the slope of the time vs. log(charge) correla tion curve. The vertical arrow indicates the counting rate at which most of the measurements presented were taken.
arXiv:physics/0103087v1 [physics.class-ph] 27 Mar 2001Thoughtful comments on ‘Bessel beams and signal propagation’ E. Capelas de Oliveira∗, W. A. Rodrigues, Jr.∗,⋆, D. S. Thober⋆ and A. L. Xavier⋆ ∗Institute of Mathematics, Statistics and Scientiﬁc Comput ation, IMECC-UNICAMP CP 6065, 13083-970, Campinas, SP, Brazil ⋆Center for Research and Technology CPTec-UNISAL Av. A. Garret, 267, 13087-290, Campinas, SP, Brazil February 20, 2014 Abstract In this paper we present thoughtful comments on the paper ‘Be ssel beams and signal propagation’ showing that the main claims o f that paper are wrong. Moreover, we take the opportunity to show th e non trivial and indeed surprising result that a scalar pulse (i. e., a wave train of compact support in the time domain) that is solution of the homogeneous wave equation ( vector ( /vectorE,/vectorB) pulse that is solution of Maxwell equations) is such that its wave front in some cases does travel with speed greater thanc, the speed of light . In order for a pulse to posses a front that travels with speed c, an additional condition must be satisﬁed, namely the pulse must have ﬁnite energy. Wh en this condition is fulﬁlled the pulse still can show peaks pro pagating with superluminal (or subluminal) velocities, but now its w ave front travels at speed c. These results are important because they explain several experimental results obtained in recent experimen ts, where superluminal velocities have been observed, without imply ing in any breakdown of the Principle of Relativity. 1In this paper we present some thoughtful comments ( C1−C4) concerning statements presented in the paper ‘Bessel beams and signal p ropagation’ [1] and also some non trivial results concerning superlumin al propagation of peaks in particular electromagnetic pulses in nondispersi ve media. In [1] the author recalls that the experimental results pres ented in [2] showed that Bessel beams generated at microwave frequencie s have a group velocity greater than the velocity of light c(in what follows we use units such thatc= 1)1. His intention was then to show that the signal velocity, deﬁ ned according to Brillouin and Sommerfeld ( B&S) was also superluminal. We explicitly shows that the particular example used by the aut hor of [1], given by the Bessel beam of his eq.(3) does not endorse his claim. Co ntrary to the author’s conclusion this beam has no fronts in both space and time domains, hence cannot satisfy B&Sdeﬁntion of a signal. Moreover, the beam given by eq.(3) of [1] travels rigidly with a superluminal speed. W e prove then that there are two classes of general Bessel pulses satisfyi ngB&Sdeﬁnition of signal. A solution of the HWE corresponding to class I is such that the group speed is always less than cwhereas its front moves with speed c.2A solution of the HWE of the class II travels rigidly at superluminal speed if care is not taken of the energy content of the pulse. We presen t also some necessary comments concerning solutions of Maxwell equati ons associated with Bessel beams of classes I and II. We start by recalling the general solution of the HWE /squareΦ = 0 in Minkowski spacetime ( M, η, D ) [10-12]. In a given Lorentz reference frame [10-12] I=∂/∂t∈secTM, we choose cylindrical coordinates ( ρ, ϕ, z ) natu- rally adapted to the Ireference frame, where ρ= (x2+y2)1 2andx=ρcosϕ 1In [3] we scrutinized the experimental results of [2]. We pre sented there a simple model showing that all particulars of the data (including th e slowing of the superluminal velocity of the peak along the propagation direction) can be qualitatively and quantita- tively understood as a scissor’s like eﬀect. Moreover in [3] we called the readers attention that in [4] peaks of ﬁnite aperture approximations (FAA) to particular acoustical Bessel pulses called X-waves (ﬁrst discoverd by Lu and Greenleaf ([5,6]) have been see to travel at supersonic speed i.e., with velocity greater than cs, the sound speed parameter ap- pearing on the homogenous wave equation ( HWE). In [4] and [7] it is also predicted the possibilty of launching FAAto superluminal electromagnetic X-waves, a fact that has been conﬁrmed experimentally in the microwave region in [2] and in the optical region in [8]. A review concerning the diﬀerent facets of ‘superlumin al’ wave motion under diﬀerent physical conditions can be found in [9]. 2Of course, this is a kind of generalized reshaping phenomena which cannot endures for ever. It lasts until the peak of the wave catches the front . 2andy=ρsinϕ, with ( x, y, z ) being the usual cartesian coordinates naturally adapted to I. Writting Φ(t, ρ, ϕ, z ) =f1(ρ)f2(ϕ)f3(t, z), (1) and substituting eq.(1) in the HWE we get the following equations (where ν and Ω are separation parameters ),   /bracketleftBig ρ2d2 dρ2+ρd dρ+ (ρ2Ω2−ν2)/bracketrightBig f1= 0,/parenleftBig d2 dϕ2+ν2/parenrightBig f2= 0,/parenleftBig ∂2 ∂t2−∂2 ∂z2+ Ω2/parenrightBig f3= 0.. (2) The ﬁrst of eqs.(2) is Bessel’s equation, the second one impl ies that νmust be an integer and the third is a Klein-Gordon equation in two d imensional Minkowski spacetime.3In what follows ( without loss of generality for the objectives of the present paper) we choose ν= 0 (and also Ω >0). Then, we obtain as a solution of eqs.(2) a wave propagating in the z-direction, i.e., ΦJ0(t, ρ, z) =J0(ρΩ) exp[ −i(ωt−¯kz)], (3) where the following dispersion relation must necessarily b e satisﬁed, ω2−¯k2= Ω2. (4) The dispersion relation given by eq.(4) may look strange at ﬁ rst sight, but there are evidences that it can be realized in nature (see below) in some special circunstances. C1. It is quite clear that the wave described by eq.(3), called i n [1] a Bessel beam4, has phase velocity vph=ω/¯k >1. However, we point out that the statement done in [1] is false, namely: ‘As known, in the absence of dispersion the group velocity vgrof a Bessel pulse is equal to the phase one [4,5]5since all the components at diﬀerent frequencies propagate with the same velocity’. To prove its falsity recall that there exist s a Lorentz reference frame [10-12] I′= (1−v2 gr)1 2(∂/∂t+vgr∂/∂z)∈secTM, (5) 3In 4-dimensional spacetime the Klein-Gordon equation poss ess families of luminal and superluminal solutions, besides subluminal solutions. Se e [4] and references therein. 4Note that in [1] the author writes Ω = ωsinθand¯k=ωcosθ. 5The references [4,5] in [1] are the references [8,13] in the p resent paper. 3which is moving with velocity vgr=dω/d¯k <1 in relation to the frame Iin the z-direction. In the coordinates naturally adapted to the fra meI′ the frequency of the wave is ω′= Ω, which means that in the frame I′the Bessel beam is stationary. This proves our statement that fo r Bessel beam the group velocity is always less than the velocity of light c. C2. Now, we show how to build two diﬀerent classes (I and II) of so lutions of the HWE by appropriate linear superpositions of waves of the form gi ven by our eq.(3). Class I . Suppose, following B&S[13,14 ] that a signal is deﬁned as a pulse with a ﬁnite time duration at the origin z= 0 where a physical de- vice generated it. We model our problem as a Sommerfeld problem [15] for theHWE (with cylindrical symmetry), i.e., we want to ﬁnd the soluti on of theHWE with the following conditions (called in what follows Somme rfeld conditions), Φ(t, ρ, ϕ, 0) = AJ0(ρΩ)[Θ( t)−Θ(t−T)] sinω0t =AJ0(ρΩ)1 2πℜ/integraldisplay Γdωe−iωt/braceleftbig eiωT−1/bracerightbig ω−ω0, ∂Φ(t, ρ, z) ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=0=AJ0(ρΩ)1 2πℜ/integraldisplay Γdω¯k(ω) e−iωt/braceleftbig eiωT−1/bracerightbig ω−ω0.(6) In eq.(6) Θ( t) is the Heaviside function, Aandω0= Ω are constants, ℜmeans real part and ¯k(ω) is given below and for simplicity we take T= Nτ0= 2πN/ω 0, with Nan integer. Now, to solve our problem it is enough to get a solution of the third of eqs.(2). We have, f3(t, z) =1 2πℜ/integraldisplay Γdω ω−ω0/braceleftbig e−iω(t−T−vgrz)−e−iω(t−vgrz)/bracerightbig (7) where vgr=¯k(ω)/ωand Γ is an appropriate path in the complex ω-plane. We note lim ω→∞vgr= 1.Putting eq.(7) into the third of eqs.(2) we see that the dispersion relation given by eq.(4) must be satisﬁed. To continue we write, ¯k(ω) =/radicalbig (ω+ Ω)(ω−Ω). (8) There are two branch points at ω=±Ω. The corresponding branch cuts can be taken as the segment ( −Ω,Ω) in the real ω-axis. Following Γ from positive values of ℜωabove and close to the real axis, the root in eq.(8) 4acquires a phase factor eiπ=−1 when passing from ℜω >Ω toℜω <−Ω. Then, on the real ω-axis we have, ¯k(ω) =/braceleftbigg \|√ ω2−Ω2\|, ω > Ω −\|√ ω2−Ω2\|, ω < −Ω(9) a result that is necessary in order to calculate the value of f3for (t−vgrz)>0. We are not going to investigate this case here, since we are in terested in the behavior of f3for the case where ( t−z <0). In this case, we must close the contour Γ in the upper half plane. Since there are no poles inside the contour we get that f3(t, z) = 0 for t−z <0. (10) Now, it is easy to verify the intensity of the wave which is sol ution of the HWE and satisﬁes the Sommerfeld conditions given by eq.(6) has a maximum forω=ω0, i.e., the waves with frequency near ω0have always a much greater amplitude than all others. Under these conditions let us wri te, ωt−¯kz= (ω0t−¯k0z) + (t−z vgr0)(ω−ω0), (11) where vgr0= (dω/d¯k)\|ω=ω0<1 and vph0=ω0/¯k0>1. We can write an approximation for the function f3(t, z) denoted by ˜f3(t, z) as, ˜f3(t, z) =1 2πℜ  e−iω0(t−z/vph0)ω0+△ω/integraldisplay ω0−△ωdω ω−ω0/braceleftbig e−iω(t−T−z/vgr0)−e−iω(t−z/vgr0)/bracerightbig  . (12) We see that ˜f3(t,0) is equal to f3(t,0) if we suppress in the expression for this function the frequencies very diﬀerent from ω0. Now, ˜f3(t,0) has support on the whole temporal axis, i.e., in the interval −∞< t <∞, but it is taken by some authors (like, e.g., [16]) as representing a wave that begin gradually at t= 0 and ends gradually at t=T. Of course, no wave of the kind of ˜f3can be build by any physical device. The importance of the function ˜f3(t, z) is that, as emphasized by B&S[13,14] it shows that we can associate a group velocity to pulse peaks in general (and of Bessel beams in particular) satisfying the Sommerfeld conditons (eq.(6)) and that the group 5velocity in this case is lessthan the velocity of light. This means that after a while the backend of the wave that is travelling at speed c(= 1) will catch the peak. The wave reshapes even when propagating in vacuum. A general subluminal J0-Bessel beam can be written as, ΦB(t, ρ, z) =J0(ρω)F−1[T(ω)]ei¯kz(13) where T(ω) is an appropriate transfer function and F−1is the inverse Fourier transform. Now, the peaks ofFAAto acoustical pulses of the form given by eq.(13) (i.e., the waves at z= 0 are not zero only in the time interval 0< t < T ) have been seen travelling at subluminal speed6in an experiment described in [4], thus endorsing the above analysis. Class II . We now return to the dispersion relation given by eq.(4) and write, ¯k=kcosθ,Ω =ksinθ, (14) where θis a constant called axicon angle [5,6,17]. It results that ω=±k. (15) We immediately verify that J0(ωρsinθ)e−i(ωt−kzcosθ), (16) is a solution of the HWE whose beam width is proportional to 1 /ωsinθ, thus being frequency dependent. The dependency of the beam width on frequency will cause the beam to have a pulse response that is independe nt of position. Indeed, suppose that the source is driven by a frequency dist ribution B(ω), i.e., we have a pulse ΦX(t, ρ, z) =∞/integraldisplay −∞dωB(ω)J0(ωρsinθ)e−i(ωt−kzcosθ), ω=k. (17) IfJ0were not dependent on frequency the integral in eq.(17) woul d be simply the inverse Fourier transform of the source spectrum and we return 6Of course, in this case the speed paramenter appearing in the HWE must be cs, the sound speed in the medium, and the word subluminal speed used must be understood as a speed less than cs. 6to class I solutions. However, here J0is dependent on frequency and also on position and consequently modiﬁes the pulse spectrum in s uch a way to make the time response of the pulse dependent on radial posit ion. We put an index Xin the wave given by eq.(17) because pulses of this kind have b een named X-waves by Lu and Greenleaf since 1992 [5,6]. Even more, takin g B(ω) =Ae−a0\|ω\|(Aanda0>0 being constants), we can easily verify (c.r., pages 707 and 763 of [18]) that we can write for sin θ >0, ΦX(t, ρ, z) =A/integraldisplay∞ −∞dωe−a0\|ω\|J0(ωρsinθ)e−iω(t−zcosθ)(18a) =A/integraldisplay∞ 0dωe−a0ωJ0(ωρsinθ) cos(ωµ) =A /bracketleftbig ρ2sin2θ+ [a0+iµ]2/bracketrightbig1 2+A /bracketleftbig ρ2sin2θ+ [a0−iµ]2/bracketrightbig1 2(18b) =A√ 2/braceleftbigg/bracketleftBig/bracketleftbig ρ2sin2θ+a2 0−µ2/bracketrightbig2+ 4a2 0µ2/bracketrightBig1 2+ρ2sin2θ+a2 0−µ2/bracerightbigg1 2 /braceleftBig/bracketleftbig ρ2sin2θ+a2 0−µ2/bracketrightbig2+ 4a2 0µ2/bracerightBig1 2, (18c) where µ= (t−zcosθ). Eq.(18c) shows that this wave is a real solution of the HWE. We recall that if in eq.(18a) we use as integration interval 0 < ω < ∞, we get only the ﬁrst term in eq.(18b). In this case we have a complex wave that has b een called thebroad band X-wave in [4-6]. These waves and the more general ones given by eq.(18b) propagate without distortion with superlumina l velocity given by 1/cosθ, but of course they cannot be produced in the physical world b ecause (like the plane wave solutions of the HWE) they have inﬁnity energy , as it is easy to verify. Waves that are solutions of the linear rela tivistic wave equations and that propagate in a distortion free mode, have been called UPWs (undistorted progressive waves) in [4]. 7Now, we show that a X-pulse even if it has compact support in th e time domain (thus being of the form of a B&Ssignal) is such that its front propa- gates with superluminal speed. To prove our statement we loo k for a solution of the HWE satisfying the following Sommerfeld conditions7, ΦX(t, ρ,0) = [Θ( t)−Θ(t−T)]∞/integraldisplay −∞dωB(ω)J0(ωρsinθ)e−iωt, ∂Φ(t, ρ, z) ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=0= [Θ( t)−Θ(t−T)]∞/integraldisplay −∞dωB(ω)J0(ωρsinθ)e−iωt,(19) andk(ω) =ω. Proceeding in the same way as in the Sommerfeld problem of class I solution presented above we obtain as a solution of theHWE (for z >0), ΦX(t, ρ, z) =1 2π∞/integraldisplay −∞d¯ωB(¯ω)J0(¯ωρsinθ)∞/integraldisplay −∞dωe−iω(t−zcosθ)/bracketleftbiggei(ω−¯ω)T−1 i(ω−¯ω)/bracketrightbigg =  ∞/integraltext −∞dωB(ω)J0(ωρsinθ)e−iω(t−zcosθ)for\|t−zcosθ\|> T 0 for \|t−zcosθ\|< T. (20) of Γ We see that for \|t−zcosθ\|> Tthe integral in eq.(20) is notzero. Since the axicon angle θ >0, then 1 >cosθ >0 and it follows that the pulse is not zero for z > t andt > T , what means that the wave front of our pulse propagates with superluminal speed! Of course, the pulse is zero for z >(t−T)/cosθorz <(t+T)/cosθ. We observe that the above result is true even a single Bessel pulse, i.e., when B(ω) =δ(ω−ω0), a result that we mentioned in [3]. How to compare this ﬁnding with the famous B&Sresult [13,14] stating that a wave pulse which propagates in a dispersive medium wit h loss has a front propagating at maximum speed c? Some things are to be recalled 7B(ω) is taken in this example as a function such that∞/integraltext −∞dωB(ω)J0(ωρsinθ)e−iωthas support in the interval −∞< t <∞. 8in order to get a meaningful answer. The ﬁrst is that B&Sexample refers to a propagation of a ‘plane’ wave truncated in time (which, o f course, has inﬁnite energy) satisfying the Sommerfeld conditions (analogous t o eq.(6)) and propagating in a dispersive medium with loss. A careful a nalysis [19] shows that the same problem in a dispersive medium with gain r eveals that in this case we can ﬁnd two kinds of solutions ( both of of inﬁni te energy). In one of these kinds, by appropriately choosing the integra tion path in the complex ω-plane we obtain as result that the front of the wave may trave l with superluminal speed. This situation is somewhat analogous t o what happen with some possible mathematical solutions of the tachyonic Klein-Gordon equation in two dimensional Minkowski spacetime [20,21]. T his equation is important because it can be associated with the so called t elegraphist equation. The reason for our ﬁnding that the X-pulse propagating in a nondisper- sive medium, although of compact support in the time domain, is such that its front travel at superluminal speed is the following; the solution given by eq.(20) is not of compact support in the space domain and as such has inﬁnite energy as can be easily veriﬁed. Only for a pulse of ﬁnite ener gy we can war- ranty that its front always travel with a speed that cannot be greater than maximum speed. Indeed, suppose we produce on the plane z= 0 a pulse like the one given by eq.(20), except that it has a ﬁnite lateral ci rcular width of radius a, i.e.,it is taken as zero for ρ > a . Such a pulse is called a FAAto the pulse given by eq.(20) and as can be easily veriﬁed has ﬁnite energy . If such a pulse does not spread with inﬁnite velocity during its build up, then after it is ready, i.e., at t=Tit occupies a region of compact support in space given by \|/vector x\|< R, where Ris the maximum linear dimension involved. Such a ﬁeld conﬁguration can then be taken as part of the initi al conditions for astrictly hyperbolic Cauchy problem at t=T. For such a problem it is well known the mathematical theorem that stablishes that [2 2,23] the time evolution of the pulse must be such that it is nullfor\|/vector x\|> R+c(t−T). In conclusion, it is not suﬃcient for a wave to be of compact supp ort in the time domain (i.e., to be a pulse) to assure that the wave front of th e pulse moves in a nondispersive medium at maximum speed c. In order for the wave front to move with velocity cit is necessary that the pulse possess ﬁnite energy , and in order for this condition to be satisﬁed the pulse must h ave compact support in the space domain after its build up. We recall here that in [4] the peaks of FAAto acoustical pulses given by eq.(18) (with appropriated B(ω)) have been seen traveling with velocities cs/cosθ, thus conﬁrming the theory 9developed above. C3. We now examine the claim of [1] that a wave given by our eq.(17 ), withB(ω) = 1, i.e., U(t, ρ, z) =∞/integraldisplay −∞dωJ0(ωρsinθ)e−i(ωt−kzcosθ), ω=k. (21) is a pulse with support only in the z-axis at points z=t/cosθand with value at that points δ(0). The calculations presented in [1] are wrong. Before we prove our statement let us recall that [1] quotes Brillouin: ‘a signal can be deﬁned as a pulse of ﬁnite temporal extension, that is, of inﬁ nite extension in the frequecy domain’.8The wave given by eq.(21) has an inﬁnite extension in the frequency domain but it is not a pulse of ﬁnite time doma in (for a ﬁxedz). Indeed, as theorem 11 on page 22 in Sneddon’s book [24] stab lishes: a function which is bounded in the time domain has an inﬁnite e xtension in the frequency domain, but it is not true that a function wit h an inﬁnite frequency spectrum is necessarily bounded in the time domai n. A trivial example of the last statement is the case of a Gaussian pulse, whose Fourier transform is itself a Gaussian. In the particular case of the wave given by eq.(21) it is immediate to realize that the integral is nothi ng more than the Fourier transform of aJ0function, and the value of the integral is given in many books, in particular on page 523 of Sneddon’s book [24]. We have, ∞/integraldisplay −∞dωJ0(ωρsinθ)e−i(ωt−kzcosθ)(22a) =/braceleftBigg2 √ ρ2sin2θ−(t−zcosθ)2for\|t−zcosθ\|< ρsinθ 0 for\|t−zcosθ\|> ρsinθ(22b) Eq.(22b) shows that U(t, ρ, z) has support in the entire time axis provided that\|t−zcosθ\|< ρsinθ. When ρ= 0, since Uis real (as can be seen directly from eq.(22a) we must have that \|t−zcosθ\|= 0 and the function U is singular. We see that the result expressed by eq.(22b) is c ompatible with the one given by eq.(18b) if we take the limit for a0→0. 8This deﬁnition is due to Sommerfeld. See [13,14]. 10C4. Finally, we investigate the claim (done in [1] and attribut ed to [8]) that the wave function given by eq.(3) represents an electri c ﬁeld. This claim is a nonsequitur. Indeed, the scalar solutions of the HWE can be used to generated solutions of the Maxwell system using the Hertz potential method (see, e.g.[25,26]). In particular, superluminal solution s of the HWE can be used to produce superluminal solutions of Maxwell equation s [4,7,9]. If we choose a magnetic Hertz potential /vectorΠm= Φ J0ˆzit is a simple exercise to show that the transverse electric and magnetic ﬁelds do not show a ny dependence onJ0. Only the Bzcomponent of the electromagnetic ﬁeld conﬁguration has aJ0dependence, but has also two other terms showing a J1and a J2depen- dence. Explicitly we have from the well known formulas /vectorE=−∂/∂t(∇×/vectorΠm) and/vectorB=∇ × ∇ × /vectorΠmthat, Eρ= 0, Eϕ=−iωΩJ1(Ωω) ρe−i(ωt−¯kz), Ez= 0, Bρ=−kΩJ1(Ωρ)e−i(ωt−¯kz), Bz=/bracketleftbigg −Ω ρJ1(Ωω)−Ω2 2J0(Ωω) +Ω2 2J2(Ωω)/bracketrightbigg e−i(ωt−¯kz), ω2−¯k2= 0. (23) With an electric Hertz potential we obtain a solution where o nly the Ez component has a J0dependence. As such, we conclude that the electromag- netic beams observed in [2] and also in [8,17] are not J0beams. A careful analysis of the solutions of Maxwell equations in cylindric al symmetry shows that there are not J0solutions representing transverse electric ﬁelds. The existence of only one peak observerd in the experiments done in [2] must be due to the J1/ρterm in Eϕ. A more detailed analysis will be reported elsewhere. Our conclusions are as follows: (i) our results show that the main claims of [1] are wrong and/or misleading and leads to equivocated c onclusions con- cerning recent experimental results showing superluminal motion of peaks of particular electromagnetic ﬁeld conﬁgurations in nondispersive media; (ii) we also prove a non trivial result, namely that the condition that a wave is ofﬁnite time duration is not a suﬃcient condition for its front to propagate with the speed c. It is necessary in order for the front to travel with speed c that the pulse possess ﬁnite energy, and thus as explained above it must (af- ter being prepared by the launching device) have support onl y in a compact 11space region when ready;9(iii) only FAAto superluminal solutions of the HWE (acoustical case) and to superluminal solutions of Maxwell equations can be produced in nature, because only waves of this kind hav e ﬁnite en- ergy. These FAAexhibit peaks propagating with superluminal speeds even in the vacuum, but since their fronts propagate with speed cthis kind of phenomenom does not implies in any danger for the Theory of Re lativity. 9We mention here that any electromagnetic pulse fulﬁlling th is condition spreads, a result that may be called the non focusing theorem [27]. 12Acknowledgments: W.A.R., D.S.T. and A.L.X.Jr. are gratefu l to Mo- torola Industrial Ltda. for a research grant. A. L. X. Jr. wou ld like also to thank FAPESP (Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado de S ˜ ao Paulo) for ﬁnancial support under contract 00/03168-0. The author s are also grate- ful to Dr. J. E. Maiorino and Professor J. Vaz Jr. for useful di scussions. References [1] D. Mugnai, Bessel beams and signal propagation , in publication in Phys. Lett A (2001). [2] D. Mugnai, A. Ranfagni and R. Ruggeri, Phys. Rev. Lett. 80(2000) 4830. [3] W. A. Rodrigues Jr, D. S. Thober and A. L. Xavier, Causal explanation of observed superluminal behavior of microwave propagatio n in free space , http://arXiv.org/abs/physics/0012032, subm. for public ation (2001). [4] W. A. Rodrigues, Jr. and J. Y. Lu, Found. Phys. 27(1997) 435. [5] J.Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelec. Freq. Cont.39(1992) 19. [6] J.Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelec. Freq. Cont.39(1992) 441. [7] E. Capelas Oliveira and W. A. Rodrigues, Jr, Ann. der Physik 7 (1998) 654. [8] P. Saari and K. Reivelt, Phys. Rev. Lett .21(1997) 4135. [9] J. E. Maiorino and W. A. Rodrigues, Jr., What is Superluminal Wave Motion ?, (electronic book at http://www.cptec.br/stm, Sci. and Tech. Mag. 4(2) 1999 ). [10] R. K. Sachs and H. Wu, General Relativity for Mathematicians , Spring Verlarg, New York, 1977. [11] W.A . Rodrigues, Jr. and M. A. F. Rosa, Found. Phys. 19(1989) 705. [12] W. A. Rodrigues, Jr. and E. Capelas de Oliveira, Phys. Lett. A 140 (1989) 479. [13] A. Sommerfeld, Optics , Academic Press, New York, 1952. [14] L. Brillouin, Wave Propagation and Group Velocity , Academic press, New York, 1960. [15] F. A. Mehmeti, Transient Tunnel Eﬀect and Sommerfeld Problem , Akademie Verlag, Berlin, 1996. [16] G. Nimtz, Ann. der Physik 7(1998), 618. 13[17] J. Durnin, J. J. Miceli, Jr. and J. H. Eberly, Phys. Rev. Lett .58 (1987)1499. [18] I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products , 4th edition, prepared by Yu.V. Geronimus and M. Yu. Tseytin , translated by A. Jeﬀrey, Academic Press, New York, 1965. [19] X. Zhou, Possibility of a light pulse with speed greater than c, in publ. inPhys. Lett. A (2001). [20] R. Fox, C. G. Kuper and S. G. Lipson, Proc. Roy. Soc. London A 316(1970) 515. [21] P. Moretti and A. Agresti, N. Cimento B 110(1995) 905. [22] M. E. Taylor, Pseudo Diﬀerential Operators , Princeton Univ. Press, Princeton, 1981. [23] R. Courant and D. Hilbert, Methods of Mathematical Physics , vol. 2, John Wiley and Sons, New York, 1966. [24] I. N. Sneddon, Fourier Transforms , Dover Publ. Inc., New York, 1995 [25] J. A. Stratton, Electromagnetic Theory , McGraw-Hill, New York, 1941. [26] W. K. H. Panofski and M. Phillips, Classical Electricity and Mag- netism , 2nd edition, Addison-Wesley, Reading, MA, 1962. [27] T. T. Wu and H. Lehmann, J. Appl. Phys .58(1985) 2064. 14Figure 1: Contour for integration of eq.(7) for t−z <0 15Figure 2: Contours for integration of eq.(20). Γ 1for\|t−zcosθ\|< Tand Γ 216
arXiv:physics/0103088v1 [physics.bio-ph] 27 Mar 2001Neural coding of naturalistic motion stimuli G. D. Lewen, W. Bialek and R. R. de Ruyter van Steveninck NEC Research Institute 4 Independence Way Princeton, New Jersey 08540Neural coding of naturalistic motion stimuli 2 Abstract. We study a wide ﬁeld motion sensitive neuron in the visual sys tem of the blowﬂy Calliphora vicina . By rotating the ﬂy on a stepper motor outside in a wooded area , and along an angular motion trajectory representative of na tural ﬂight, we stimulate the ﬂy’s visual system with input that approaches the natura l situation. The neural response is analyzed in the framework of information theory , using methods that are free from assumptions. We demonstrate that information abo ut the motion trajectory increases as the light level increases over a natural range. This indicates that the ﬂy’s brain utilizes the increase in photon ﬂux to extract mor e information from the photoreceptor array, suggesting that imprecision in neura l signals is dominated by photon shot noise in the physical input, rather than by noise generated within the nervous system itself. 1. Introduction One tried and tested way to study sensory information proces sing by the brain is to stimulate the sense organ of interest with physically app ropriate stimuli and to observe the responses of a selected part of the system that le nds itself to measurement. Within that framework there are strong incentives, both pra ctical and analytical, to simplify stimuli. After all, short lightﬂashes or consta nt tones are easier to generate and to capture mathematically than the everchangi ng complex world outside the laboratory. Fortunately, sense organs and brains are ex tremely adaptive, and they apparently function in sensible ways, even in the artiﬁ cial conditions of typical laboratory experiments. A further reason for using simpliﬁ ed stimuli is that they are presumed to elicit simple responses, facilitating interpr etation of the system’s input- output behaviour in terms of underlying mechanism. Typical ly, these simple stimuli are repeated a large number of times, the measured outputs are av eraged, and this average is deﬁned to be the ‘meaningful’ component of the response. T his is especially helpful in the case of spiking neurons, where we face the embarrasmen t of the action potential: Because they are an extremely nonlinear feature of the neura l response we often do not really know how to interpret sequences of action potentials (Rieke et al, 1997). One way to evade the question and save tractability is to work wit h derived observables, in particular with smooth functions of time, such as the averag e ﬁring rate. Although it certainly is useful to perform experiments with simpliﬁed inputs, one also would like to know how those stimuli are processed and en coded that an animal is likely to encounter in nature. We expect animals to be ‘desig ned’ for those conditions, and it will be interesting to see to what extent the brain can k eep up with the range and strength of these stimuli. In the present work we are conc erned primarily with the question of how noisy neural information processing really is. This question cannot be answered satisfactorily if we do not study the problem that t he brain is designed to solve, because for us it is hard to distinguish willful negle ct on the part of the brain in solving artiﬁcial tasks, from noisiness of its components.Neural coding of naturalistic motion stimuli 3 As soon as we try to characterize the behaviour of a sensory sy stem in response to the complex, dynamic, nonrepeated signals presented by the natural world, we lose many of the simpliﬁcations mentioned earlier. To meet the challe nge we must modify both our experimental designs and our methods for analyzing the resp onses to these much more complicated inputs. Recent examples of laboratory based ap proaches to the problem of natural stimulation are studies of bullfrog auditory neuro ns responding to synthesized frog calls (Rieke et al , 1995), insect olfactory neurons responding to odour plume s (Vickers et al, 2001), cat LGN cells responding to movies (Dan et al, 1996, Stanley et al, 1999), primate visual cortical cells during free viewing o f natural images (Gallant et al , 1998, Vinje and Gallant, 2000), auditory neurons in song bi rds stimulated by song and song–like signals (Theunissen and Doupe, 1998, The unissen et al, 2000, Sen et al, 2000), the responses in cat auditory cortex to signals with naturalistic statistical properties (Rotman et al, 1999), and motion sensitive cells in the ﬂy (Warzecha and Egelhaaf, 2001, de Ruyter van Steveninck et al, 2001). In each case compromises are struck between well controlled stimuli with understandabl e statistical properties and the fully natural case. A more radical approach to natural stimulation was taken by R oeder in the early sixties (see Roeder, 1998). He and his coworkers made record ings from moth auditory neurons in response to the cries of bats ﬂying overhead in the open ﬁeld. More recently the visual system of Limulus was studied with the animal moving almost free on the sea ﬂoor (Passaglia et al, 1997). Here we study motion sensitive visual neurons in the ﬂy, and— in the spirit of Roeder’s work—rather than trying to construct approximati ons to natural stimuli in the laboratory, we take the experiment into nature. We recor d the responses of H1, a wide ﬁeld direction selective neuron that responds to horiz ontal motion, while the ﬂy is being rotated along angular velocity trajectories repre sentative for free ﬂying ﬂies. These trajectories indeed are quite wild, with velocities o f several thousand degrees per second and direction changes which are complete within t en milliseconds. In analyzing the responses to these stimuli we would like to use methods that do not depend on detailed assumptions about what features of the st imulus are encoded or about what features of the spike sequences carry this coded s ignal. Recently, information theoretical methods were developed for analysing neural re sponses to repeated sequences of otherwise arbitrarily complex stimuli (de Ruyter van Ste veninck et al, 1997, Strong et al, 1998). In our experiments we repeat the same motion trace, l asting several seconds, and this provides us with the raw data for computing the relev ant information measures, as explained in section 2.3. We emphasize that although we re peat the stimulus many times to estimate the relevant probability distributions o f responses, the measures we derive from these distributions characterize the informat ion coded by a single example of the neural response.Neural coding of naturalistic motion stimuli 4 2. Methods 2.1. Stimulus design considerations The giant motion sensitive interneurons in the ﬂy’s lobula p late are sensitive primarily to such rigid rotational motions of the ﬂy as occur during ﬂig ht (Hausen, 1982, Krapp et al, 1997), and these cells typically have very large visual ﬁel ds. It is this wide ﬁeld rigid rotation that we want to reproduce as we construct a naturali stic stimulus. But what pattern of rotational velocities should we use? As a benchma rk we will present data from an experiment where the ﬂy was rotated at velocities tha t remained constant for one second each. We would, however, also like to present the ﬂ y with stimuli that are more representative for natural ﬂight. Free ﬂight trajecto ries were recorded in the classic work of Land and Collett (1974), who studied chasing behavio ur inFannia canicularis and found body turning speeds of several thousand degrees pe r second. A recent study (van Hateren and Schilstra, 1999) reports ﬂight measuremen ts from Calliphora at high temporal and spatial resolution. In these experiments ﬂies made of order 10 turns per second, and turning speeds of the head reached values of over 3000◦/s. In general, high angular velocities pose problems for visual stimulus d isplays, because even at relatively high frame rates they may give rise to ghost image s. In our laboratory we use a Tektronix 608 display monitor with a 500 Hz frame rate,. The n 3000◦/s corresponds to jumps from frame to frame of 6◦, four times larger than the spacing between photoreceptors. Although the frame rate used here is well ab ove the photoreceptor ﬂicker fusion frequency (de Ruyter van Steveninck and Laugh lin, 1996) the presence of ghost images may have consequences for the encoding of mot ion signals. Further, the light intensity of the typical displays used in the labor atory is much lower than outside. As an example, the Tektronix 608 induces of order 5 ·104photoconversions/s in ﬂy photoreceptors at its maximum brightness. The brightn ess outside can easily be a factor of a hundred higher (Land, 1981), although the photor eceptor pupil mechanism will limit the maximum photon ﬂux to about 106photoconversions/s (Howard et al, 1987). Finally, the ﬁeld of view of H1 is very large, covering essentially the ﬁeld of one eye (Krapp and Hengstenberg, 1997), which is about 6.85 sr or 55% of the full 4 πsr in female Calliphora vicina (estimates based on Beersma et al, 1977. See also Fig. 1). In practice with a display monitor it is hard to stimulate the ﬂy with coherent motion over such a large area and in most of our laboratory experimen ts we stimulate less than about 20% of the full visual ﬁeld of H1. 2.2. Stimulus apparatus All the factors mentioned above suggest an experimental des ign in which the visual world can be made to move more or less continuously relative to the ﬂ y, and this is easiest to accomplish by moving the ﬂy relative to the world as occurs during free ﬂight. We therefore constructed a light and compact assembly consist ing of a ﬂy holder, electrode manipulator, and preampliﬁer that can be mounted on a steppe r motor, as shown inNeural coding of naturalistic motion stimuli 5 ﬁgure 1. This setup is rigid enough to allow high speed rotati ons around the vertical axis while extracellular recordings are made from the H1 cel l. Because it is powered by batteries the setup can be taken outside, so that the ﬂy’s vis ual system is stimulated with natural visual scenes. The mounting and recording stag e inevitably covers some area in the ﬂy’s visual ﬁeld. During the experiment this rota tes along with the ﬂy, and so does not contribute to motion in the ﬂy’s visual ﬁeld. B y tracing the contours of the setup as seen from the ﬂy, we estimate the shape and size of this overlap, as depicted in ﬁgure 1. The setup was designed to minimize the ov erlap in the visual ﬁeld of the left eye. In the experiments presented here, recordin gs were therefore made from contralateral H1, on the right side of the head. The setup occ ludes only 1.52 sr, or 22%, of the visual ﬁeld of the left eye, most of it ventral-caudal, as indicated by the heavy mesh in the right panel of ﬁgure 1. The stepper motor (Berger-Lahr RDM 564/50) was driven by a Di vistep 331.1 controller in microstep mode, that is, at 104steps per revolution, corresponding to a smallest step size of 0 .036◦or roughly 1/30th of an interommatidial angle. The Divistep controller in turn was driven by pulses from a custo m designed interface that produced pulse trains by reading pulse frequency values fro m the parallel port of a laptop computer. Pulse frequency values were refreshed eve ry 2 ms. To generate naturalistic motion stimuli we used published t rajectories of chasing Fannia from Land and Collett (1974), interpolated smoothly betwee n their 20 ms sample points. For technical reasons we had to limit the accelerati ons of the setup, and we chose therefore to rotate the ﬂy at half the rotational velocities derived from the Land and Collett data. This may be reasonable as Calliphora is a larger ﬂy than Fannia , and is likely to make slower turns. The constant velocity data pr esented in ﬁgure 2 were taken with rotation speeds ranging from about 0 .28◦/s to 4500◦/s. To avoid extreme accelerations during high velocity presentations the puls e program for the stepper motor delivered smooth 100 ms pulse frequency ramps to switch betw een velocities. For velocities below 18◦/s pulses were sent to the controller at intervals longer tha n 2 ms. At the lowest constant velocity used in our experiments, 0 .28◦/s, pulses were delivered at 128 ms intervals. The step size was small enough so that a mo dulation of the PSTH was undetectable in the experiment. The experiment of ﬁgure 2 compares data from the outdoor setu p to data taken inside with the ﬂy observing a Tektronix 608 CRT. The stimulu s displayed on this monitor consisted of 190 vertical lines, with intensities d erived from a one-dimensional scan of the scene viewed by the ﬂy in the outdoor experiment. T he moving scene was generated by a digital signal processor, and written at a 500 Hz frame rate. As mentioned above, this gives rise to ghosting at high image speeds when t he pattern makes large jumps from frame to frame. The DSP produced the coarse part of motion essentially by stepping through lines in a buﬀer memory. On top of this, ﬁn e displacements were produced by moving the entire image by fractions of a linewid th at each frame. The resulting motion was smooth and not limited to integer steps . The ﬂy was positioned so that the screen subtended a rectangular area of 67◦horizontal by 55◦vertical, withNeural coding of naturalistic motion stimuli 6 the left eye facing the CRT and rightmost vertical edge of the CRT approximately in the sagittal plane of the ﬂy’s head. 2.3. Information theoretic analysis of neural ﬁring patter ns We describe brieﬂy a technique for quantifying information transmission by spike trains (de Ruyter van Steveninck et al, 1997, Strong et al, 1998, de Ruyter van Steveninck et al, 2001). We consider segments of the spike train with length Tdivided in a number of bins of width ∆ t, where ∆ tranges from one millisecond up to ∆ t=T. Each such bin may hold a number of spikes, but within a bin no distinctio n is made on where the spikes appear. However, two windows of length Tthat have diﬀerent combinations of ﬁlled bins are counted as diﬀerent ﬁring patterns. Also, t wo windows in which the same bins are ﬁlled but with diﬀerent count values, are disti nguished. We refer to such ﬁring patterns as words, WT,∆t. From an experiment in which we repeat a reasonably long naturalistic stimulus a number of times, Nr(hereNr= 200 repetitions of a Tr= 5 seconds long sequence) we get a large number of these words, WT,∆t(t), with tthe time since the start of the experiment. Here we discretize tin 1 ms bins, giving us 5000 words per repetition period, and 106words in the entire experiment. From this set of words we set up word probability distributions, from which w e calculate total and noise entropies, and their diﬀerence, according to Shannon’s deﬁ nitions: (i) The total entropy, Stot(T,∆t). From the list of words WT,∆t(t), for all t(0≤t≤ Nr·Tr), we directly get a distribution, P(WT,∆t) describing the probability of ﬁnding a word anywhere in the entire experiment. The total entropy i s now: Stot(T,∆t) =−/summationdisplay WP(WT,∆t)·log2[P(WT,∆t)] (1) This entropy measures the richness of the ‘vocabulary’ used by H1 under these experimental conditions, hence the time of occurrence of th e pattern within the experiment is irrelevant. (ii) The average noise entropy, ¯Snoise(T,∆t). If the neuron responded perfectly reproducibly to repeated stimuli, then the information con veyed by the spike train would equal the total entropy deﬁned above. There is noise, h owever, and this leads to variations in the responses, as can be seen directly from t he rasters in Fig. 3. ¯Snoise(T,∆t) gives us an estimate of how variable the response to identic al stimuli is. We ﬁrst accumulate, for each instant trin the stimulus sequence, the distribution of all those ﬁring patterns P(WT,∆t\|tr), taken across all trials, that begin at tr(note that 0 ≤tr≤Tr). The entropy of this distribution measures the (ir)reprod ucibility of the response at each instant tr: Snoise(T,∆t, tr) =−/summationdisplay WP(WT,∆t\|tr)·log2[P(WT,∆t\|tr)]. (2) Calculating this for each point in time and averaging all the se values we obtain the average noise entropy: ¯Snoise(T,∆t) =1 Tr/integraldisplayTr 0Snoise(T,∆t, tr)dtr. (3)Neural coding of naturalistic motion stimuli 7 (iii) The information conveyed by words at the given length Tand resolution ∆ tis the diﬀerence of these two entropies: I(T,∆t) =Stot(T,∆t)−¯Snoise(T,∆t). (4) The coding eﬃciency of the spike train is the fraction of the t otal entropy that is utilized to convey information: η(T,∆t) =I(T,∆t) Stot(T,∆t). (5) Small values of η(T,∆t) indicate a loose coupling between stimulus and spike train , whereas values close to 1 imply that there is little noise ent ropy, so that most of the structure of the spike train is meaningful, and carries a mes sage. Here we will not be interested in the decoding question, that is in whatthat message is, but only in how much information is conveyed about the stimulus. We will then com pare these values in diﬀerent conditions. It should be stressed that the information values we derive b y these methods are not strictly about velocity. They are potentially about anything in the stimul us that is repetitive with period Tr. It is our job as experimenters to construct inputs that we th ink will stimulate the neuron well, and for H1, naturalistic wid e ﬁeld motion seems to be a good choice. But that does not necessarily mean that that is t he best choice. Further, the motion pattern is dynamic, and any noiseless time invari ant operation on this signal will produce a result that has the same repeat period as the or iginal. Our information measures do not distinguish these cases; speciﬁcally, our d iscussion is unaﬀected by the question of whether H1 encodes velocity, acceleration, or some nonlinear function of these variables. Questions of decoding are highly intere sting, but at the same time diﬃcult to tackle for stimuli of the type studied here, and we will leave them aside in this paper. It is interesting to try and estimate I(T,∆t) as we let Tbecome very long, and ∆tvery short, as this limit is the average rate of information t ransmission. Because calculating this limit requires very large data sets, we foc us here on the information transmitted in constant time windows, T= 30 ms, as a function of ∆ t. We choose T= 30ms because that amounts to the delay time with which a chas ing ﬂy follows turns of a leading ﬂy during a chase (Land and Collett, 1974); the end result, that is the dependence of information transmission on ∆ t, was found not to depend critically on the choice of T. To quantify noise entropy, the method described above requi res that a stimulus waveform be repeated. Although it is possible in principle t o quantify information transmission based only on one repetition, using many repet itions is easier in practice. In a sense this mode of stimulation is still removed from the r ealistic situation in which stimuli are not repeated at all. Indeed, in our experiments t here are hints that the ﬂy adapts to the stimulus somewhat over the ﬁrst few presentati ons of the 5 second long stimulus. The eﬀects of adaptation to dynamic stimuli are ce rtainly interesting (Brenner et al , 2000, Fairhall et al , 2001), but in the data we present here we skip the ﬁrstNeural coding of naturalistic motion stimuli 8 few presentations, and only analyze that part of the experim ent in which the ﬂy seems fully adapted to the ongoing dynamic stimulus. Inspection o f the rasters in that phase shows no obvious trends, so that the ﬂy seems to be close to sta tionary conditions. In this regard our information measures are lower bounds, as de viations from stationarity will increase our estimate of the noise entropy, lowering in formation estimates. 3. Results 3.1. Operating range for naturalistic motion stimuli In order to be sure that H1 receives no dominant motion relate d signals from other modalities than vision we rotated the ﬂy either in darkness o r under a cover that turned along with the ﬂy. This did not produce discernible motion re sponses in H1. Strictly speaking that does not exclude possible modulatory mechano sensory input, which could be investigated in principle by presenting conﬂicting visu al and mechanosensory stimuli. The possibility seems remote, however, and even if true it wo uld not invalidate our conclusions about the dependence of H1’s information trans mission on parameters of the visual stimulus. As a ﬁrst comparison between laboratory and natural conditi ons we present data from an experiment in which H1 was excited by one second long e pisodes of motion at constant velocity. These were presented at a range of velo cities from about 0.28◦/s to 4500◦/s. Outdoors the ﬂy was placed in a wooded environment and rot ated on the stepper motor. In the laboratory the same ﬂy watched a vertic al bar pattern derived from a one dimensional scan of the natural environment in whi ch the outdoor experiment was done. This pattern was displayed on a standard Tektronix 608 monitor, with a rectangular stimulated visual area of 67◦horizontal by 55◦vertical. The pattern moved at the same settings of angular velocity as were used outdoor s, but the indoor and outdoor stimuli diﬀered both in average light level and in st imulated area. Figure 2 shows the average ﬁring rates obtained from the last half second of each velocity presentation. At low velocities, up to about 20◦/s, the spike rates for both conditions are not very diﬀerent, despite the large change i n total motion signal present in the photoreceptor array. Apparently the ﬂy adapts these d iﬀerences away (see Brenner et al, 2000). In both experiments the rate depends roughly logari thmically on velocity over an appreciable range and this is partly a result of adapt ation as well (de Ruyter van Steveninck et al, 1986). In the laboratory experiment the motion response pe aks at about 100◦/s, whereas in natural conditions the ﬂy encodes velocities monotonically for an extra order of magnitude, its response peaking in the neig hbourhood of 1000◦/s. This brings H1’s encoding of motion under natural conditions in t he range of behaviourally relevant velocities. A lack of sensitivity to high speeds ha s been claimed both to be an essential result of the computational strategy used by the ﬂ y, and to be advantageous in optomotor course control (Warzecha and Egelhaaf, 1998). These conclusions do not pertain to the conditions in the outdoor experiment, where H 1 responds robustly andNeural coding of naturalistic motion stimuli 9 reliably to angular velocities of well over 1000◦/s. 3.2. Motion detection throughout the day Figure 3 shows spike train rasters generated by H1 in three ou tdoor experiments, focusing on a short segment that illustrates some qualitati ve points. Trace (a) shows the velocity waveform, which was the same in all three cases. The experiments were performed at noon (b), half an hour before sunset (c), and abo ut half an hour after sunset (d). Rough estimates of the photon ﬂux in a blowﬂy phot oreceptor looking at zenith are shown beside the panels. In all experiments the ﬂy saw the same scene, with a spatial distribution of intensities ranging from about 5% to 100% of the zenith value. The ﬁgure reveals that some aspects of the response are quite reproducible, and further that particular events in the stimulus can be associ ated reliably with small numbers of spikes. More dramatically, the timing precision of the spike trains gradually decreases going from the noon experiment to the one after sun set. Higher photon rates imply a more reliable physical input to the visual system. Th e ﬁgure therefore strongly suggests that the ﬂy’s visual system utilizes this increase d input reliability to compute and encode motion more accurately when the light intensity i ncreases. This statement is ecologically relevant, as the conditions of the experime nt correspond to naturally occurring light levels and approximately to the naturally s timulated visual area. To get a feeling for the spike timing precision in the three conditi ons we can simply look at the distribution of timing of the ﬁrst spike generated after a ﬁx ed criterion time (for which we choose t=0.28 s in (b) and (c) and t=0.30 s in (d)). The jitte r in the spike timing across diﬀerent trials has a standard deviation of 0.95 ms in (b), 1.4 ms in (c), and 5.8 ms in (d). The relative timing of spikes can be even more accur ate: the interval from the ﬁrst to the second spike ﬁred after the criterion time is 2 .3±0.23 ms in (b), 5.0 ±0.6 ms in (c), and 16 ±2.4 ms in (d). Compared to the rapid onsets and oﬀsets of the sp ike activity at the higher photon ﬂuxes, the stimulus varies rat her smoothly, which means that the time deﬁnition of spikes with respect to the stimulu s can be much better than might be suggested by the stimulus bandwidth. An example can be seen in the rather smooth hump in the velocity waveform at about t=0.43 s, which induces on most trials a well deﬁned response consisting of a sharply deﬁned pair of spikes. We quantify these impressions using the information theore tic approach described brieﬂy in section 2.3. The result of this analysis is shown in Fig. 4a-d, for the three diﬀerent experiments discussed above. Figure 4c clearly sh ows that the information in a 30 ms window increases both when the light intensity goes up, and when the spikes are timed with higher accuracy. The increase in information wit h increasing spike timing precision is most dramatic for the highest light levels, ind icating that coding by ﬁne spike timing becomes more prominent the better the input sig nal to noise ratio. A comparison of ﬁgures 4a (total entropy) and ﬁgure 4b (noise e ntropy) reveals that the increase in information content with increasing light leve ls is primarily due to an increase in total entropy: The neuron’s vocabulary increases in size as its input becomes betterNeural coding of naturalistic motion stimuli 10 deﬁned. Figure 4d shows that at the two highest light intensi ties the coding eﬃciency is of order 0.5 at time resolution ∆ t=1 ms, increasing slightly for larger values of ∆ t. In the darkest condition the eﬃciency decreases markedly for a ll values of time resolution. The right column of ﬁgure 4 compares experiments in which we t ook data both outdoors and in the laboratory. These data are from another ﬂ y, but the conditions of the outdoor experiment were similar to those for the ﬁrst ﬂy at th e highest light level. After the outdoor experiment the ﬂy was taken inside the laborator y, and the same velocity stimulus as the one used outside was repeated inside. In the l aboratory, as before, the visual stimulus was presented on a Tektronix 608 CRT. Photor eceptors facing the CRT received about 5 ·104photons per second at maximum intensity, a value in between t he light intensities seen by the ﬁrst ﬂy in the experiments just before and just after sunset (grey and black symbols in ﬁgure 4a-d). Two experiments were done indoor, one in which the picture on the monitor consisted of vertical bars with a c ontrast pattern measured in a horizontal scan of the outdoor scene (ﬁlled triangles), th e other a high contrast square wave pattern with contrast=1, and spatial wavelength=12.5◦(ﬁlled squares). From ﬁgure 4g we see that the information transmitted by H1 is much lower in the laboratory experiments than in the outdoor experiment, due to the small er stimulated area and the lower light level. Figure 4e shows that the decrease in in formation, as before, is mainly due to a lowering of the total entropy. The noise entro py also decreases (ﬁgure 4f), but not enough to compensate. Somewhat surprisingly, t he experiment with the high contrast pattern indoors leads to a slightly higher cod ing eﬃciency than even the outdoor experiment. 4. Discussion Outdoor illumination can easily be a hundred times brighter than anything displayed on common laboratory equipment, and in the outdoor experime nt stimuli extend over a large fraction of the ﬂy’s full visual ﬁeld rather than bein g conﬁned to a small ﬂat monitor. Both eﬀects are relevant for our experiments, as th e higher brightness leads to higher photoreceptor signal to noise ratios (de Ruyter va n Steveninck and Laughlin, 1996), and as H1’s receptive ﬁeld covers almost a hemisphere (Krapp and Hengstenberg, 1997). In moving from laboratory to outdoor conditions, bot h eﬀects increase the signal to noise ratio of the input available for computation of rigi d wide ﬁeld motion from the photoreceptor array. The question then is whether the ﬂy’s b rain uses this improvement in input signal quality to produce more accurate estimates o f visual motion, and/or increase its operating range of motion detection. Figure 2 s hows that the range of velocities that are encoded increases markedly when the vis ual input becomes more reliable. If the accuracy of information processing is limited by nois e sources within the nervous system, we should observe a plateau, that is, inform ation transmission should saturate at some deﬁned level of input signal quality. There is some arbitrariness in the choice of the level of input signal quality, however: In prin ciple we can surpass any degreeNeural coding of naturalistic motion stimuli 11 of accuracy of the physical input signal by simply increasin g the light intensity, and at some point the internal randomness of the brain’s component s must become the limiting factor in information processing. However, statements abo ut the magnitude of internal versus external noise in sensory information processing ar e primarily meaningful in the context of reasonable, physiological levels of input signa l quality. Those stimuli that the animal encounters naturally, taken at the high end of their d ynamic range, would meet this criterion. For the case considered here the dynamic ran ge refers to light intensity, size of stimulated visual ﬁeld, and dynamics of motion. The d ata we recorded outdoors show no sign of saturation in information transmission when the input signal quality increases. On the contrary, if we compare the rasters of ﬁgur e 3b and 3c, we see that there is a marked improvement in the timing of spikes, even ov er the highest decade of light intensity (2 ×105to 3×106photons/s at zenith per photoreceptor). This improvement translates into a signiﬁcant gain in informati on transmission, especially at ﬁne time resolution, as shown in ﬁgure 4c. Thus, in computing motion from the array of photoreceptors, the ﬂy’s brain does not suﬀer noticeably fr om information bottlenecks imposed by internal noise, under ecologically relevant con ditions. In our outdoor experiments, the information content of the s pike train varies primarily as a result of a varying total entropy (ﬁgure 4a). T he noise entropy (ﬁgure 4b) appears to be almost constant as a function of light level . One can distinguish two diﬀerent ways to increase information transmission thr ough a channel. The ﬁrst is to encode the same messages more accurately, the second to increase the variety of messages, keeping the accuracy of each individual message t he same. The ﬁrst scheme implies constant total entropy and decreasing noise entrop y, the second an increase in total entropy at constant noise entropy. 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Odour-plume dynamics inﬂuence the brain’s olfactory code. Nature 410,466–470 (2001). Vinje, W. E., and Gallant, J. L. Sparse coding and decorrelat ion in primary visual cortex during natural vision, Science 287,1273–1276 (2000). Warzecha, A.–K., and Egelhaaf, M. On the performance of biol ogical movement detectors and ideal velocity sensors in the context of optomotor course stabili zation, Vis. Neurosci. 15,113–122 (1998). Warzecha, A.–K., and Egelhaaf, M. Neural encoding of visual motion in real-time. In: J.M. Zanker and J. Zeil (eds.) Motion vision. Computational and ecological constraints. Springer, Berlin, Heidelberg,Neural coding of naturalistic motion stimuli 13 New York (2001), pp. 239–278.Neural coding of naturalistic motion stimuli 14 Figure legends Figure 1. Left: Setup used in the outdoor experiments. The ﬂy is in a pla stic tube, head protruding, and immobilized with wax. A small feeding t able is made, from which the ﬂy can drink sugarwater. The part of the setup shown here r otates around the axis indicated at the bottom, by means of a stepper motor. A si lver reference wire makes electrical contact with the body ﬂuid, while a tungste n microelectrode records action potentials extracellularly from H1, a wide-ﬁeld mot ion sensitive neuron in the ﬂy’s lobula plate. The electrode signals are preampliﬁed by a Burr-Brown INA111 integrated instrumentation ampliﬁer, the output of which i s fed through a slip ring system to a second stage ampliﬁer and ﬁlter and digitized by a National Instruments PCMCIA data acquisition card in a laptop computer. The part o f the setup visible in the ﬁgure is mounted on a stepper motor which is driven by co mputer-controlled laboratory built electronics. Right: Occlusion in the left visual ﬁeld of the ﬂy. The dot in centre represents the position of the ﬂy. The animal is loo king in the direction of the arrow and has the same orientation as the ﬂy in the setup at left. The thin mesh bordered by the heavy line represents the excluded part of th e visual ﬁeld of the left eye for a free ﬂying ﬂy (based on Beersma et al. 1977). The heav y mesh represents the overlap of the left eye’s natural visual ﬁeld with those part s of the setup that rotate along with the ﬂy, and therefore do not contribute to a motion signal. The total visual ﬁeld of the left eye is 6.85 sr, or 0 .55·4π. The overlap depicted by the heavy mesh subtends about 1.52 sr, or 22% of the visual ﬁeld of the left ey e. Figure 2. A comparison of responses to constant velocity in a typical l aboratory experiment (closed squares), and in an outdoor setting wher e the ﬂy is rotating (open circles). Average ﬁring rates were computed over the last 0. 5 seconds of a 1 second constant velocity presentation. Figure 3. Responses of the H1 neuron to the same motion trace recorded o utside at diﬀerent times of the day. (a)Short segment of the motion trace executed by the stepper motor with the ﬂy. The full segment of motion lasted 5 seconds, and was derived from video recordings of natural ﬂy ﬂight during a ch ase (see Methods) (b) 50 Spike rasters in response to the motion trace in (a), taken at noon. (c)As(b), but recorded about half an hour before sunset. (d)As(b), but recorded about half an hour after sunset.Neural coding of naturalistic motion stimuli 15 Figure 4. Lefthand column: Information theoretic quantities for the three outdoor experiments whose rasters are shown in ﬁgure 3. The symbol sh adings refer to the diﬀerent conditions of illumination in the experiments. Al l ﬁgures refer to a 30 ms measurement window in which neural ﬁring patterns are deﬁne d at time resolutions, ∆t, of 1, 2, 3, 5, 10, 15, and 30 ms, as given by the abscissae. (a): Total entropy of spike ﬁring patterns. (b): Average noise entropy. (c): Average information transmitted by ﬁring patterns. (d): Coding eﬃciency, deﬁned as the transmitted information di vided by the total entropy. Righthand column: The same quantities as plotted in the left hand column, but now for an experiment outdoors (open symbols), and two exper iments in the laboratory (closed symbols, see text for further description of condit ions). Squares are for a moving square wave pattern of high contrast (C=1), and spatial wave length 12.5◦; triangles are for a moving sample of the natural scene at the location where the outdoor experiments were done. Both these stimulus patterns were generated on th e cathode ray tube in the laboratory.This figure "fig1.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103088v1velocity (°/s)0.1 1 10 100 1000 10000average rate (spikes/s) 050100150200250300 outside laboratory Lewen et al. Figure 2This figure "fig3.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0103088v1zenith photon flux (photons/s per photoreceptor) time resolution ∆t (ms)1 10coding efficiency 0.00.20.40.60.81.01 10noise entropy (bits) 0246810 1 10information (bits) 0123456a b hgfe dc Lewen et al Fig 41 10information (bits) 01234563×102 2×105 3×106 time resolution ∆t (ms)1 10coding efficiency 0.00.20.40.60.81.01 10noise entropy (bits) 02468101 10total entropy (bits) 0246810location and stimulus pattern 1 10total entropy (bits) 0246810outside lab, bar pattern lab, sampled scene
arXiv:physics/0103089 28 Mar 2001 MOTION, UNIVERSALITY OF VELOCITIES, MASSES IN WAVE UNIVERSE. TRANSITIVE STATES (RESONAN/G4BES) - MASS SPECTRUM Chechelnitsky A.M. Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia E’mail: ach@thsun1.jinr.ru ABSTRACT Wave Universe Concept (WU Concept) opens new wide possibilities for the effective description of Elementar Objects of Matter (EOM) hierarchy, in particular, of particles, resonances mass spectrum of subatomic (and HEP) physics. The special attention to analysis and precise description of wide and important set - transitive states (resonances) of EOM is payed. Its are obtained sufficiently precise representations for mass values, cross relations between masses of wide set objects of particle physics - metastable resonances - (fast moving) transitive states - in terms of representations of Wave Universe Concept (WU Concept). Wide set of observed in experiments effects and connected with its resonances (including - Darmstadt effect, ABC effect,etc.) may be effectively interpreted in WU Concept and described with use of mass formula - as manifestation of rapidly moving, physically distinguished transitive states (resonances). DISCRETENESS, COMMENSURABILITY, QUANTIZATION of WAVE DYNAMIC SYSTEMS (WDS) According to ideas of Wave Universe Concept (WU Concept) [Chechelnitsky, 1980-1998], (any) arbitrary real objects of micro (atoms, particles) and megaworld (astronomical systems) represent principally - the wave dynamic systems (WDS). In that case the following assertion is valid. Proposition # Internal structure, geometry, dynamics, physics of WDS are essentially connected with observated effects of discreteness, commensurability, quantization of its dynamical parameters. # That, first of all, relates to discreteness, commensurability, quantization of two sets conjugated values (parameters) ∗ Sectorial velocities (circulations) L=LN[s], ∗ Keplerian (orbital) velocities v=vN[s]. # Nature "prefers" to manifest (it's activity) at some dynamically, physically distinguished (with most probability observed) values of dynamical parameters, - first of all, at elite (dominant) values of (sectiorial and keplerian) velocities. These special states are the most simple, easily detectable ones - even at preliminary heuristic analysis of discreteness and commansurability. By its - usually and first of all - having physical intuition researchers "come across" in their search investigations. TRANSITIVE STATES (RESONANCES) Motion Factor Motion of transitive state (resonance) may naturally arise in framework of following simple intuitive consideration. Let some (being a stationary at rest) state - stable particle, for instance, π - meson with (table) mass M, moves with high ("relativistic") velocity v. Is that moving object (dynamic system, wave configuration) - the same particle ? Or it represents arbitary (some another) quasistable state π∧∧ ? Situation gets out from indeterminancy and doub ts, if we take into account the following important consequence of WU Concept - the existance of physycally distnguished vN[s] elite (dominant) velocities and connected with its phenomenon of discreteness, commensurability of elite velocities . Reality of these circumstances signifies, that particles by virtue of fundamental laws of nature "prefer" (with most probability) to move (only) with specific, physycally distinguished elite (dominant) velocities v=vN[s]. In such case moving mass configuration, indeed, represents some (quasistable), will be say, the transitive Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 2 (moving) state (resonance) π∧∧ with fixed energy, momentum (impulse) and, probable, M∧∧ mass. "Complex" ("Compaund") States (Resonances). Reconstruction of states Notion of transitive may be extended at more wide class of objects. Let some ("complex") state (resonance) disintegrates into a few moving (stationary) states. Using conservation laws (of energy, etc.) it may be restored, reconstructed the initial "complex" (occasionally, slowly moving or resting) state (resonance). For this initial state it may be saved "transitive" term (transitive state, resonance), taking into account the way of it's reconstruction - by scuttering, moving decay products, with essential regarding for its velocities. Transitive States (Resonances). Mass Formulae Using the standard representations of (relativistic) kinematics it is not difficult to receive the representation for the transitive state (resonance) mass in the form M = M#(1+z)1/2 = M#(1+β2)1/2, z= β2, β=v/c, where c - light velocity. In fact, taking into account the admissible spectrum of v=vN[s] elite (dominant) velocities, this mass representation converts in following explicit form MN[s] = M#(1+z)1/2 = M# (1+β2)1/2, M ⇒ MN[s], β ⇒ βN[s] ⇒ vN[s]/c. Decay Processes. Resonance "Reconstruction". Binary Decay The simplest situation - resonance decay into two identifical with equal m0 masses and equa l P=mv - momenta (impulses) (m-mass, v-vellocity) - gives a possibility to obtain the simplest representation for the mass of disintegreting resonance M = M#(1+z)1/2 = M#(1+β2) 1/2 with M# = 2m0 Of course, similar representation is justify when taking into account all circumstances of decay, that correspond to conservation of energy law. Expanding of this approach to more composite cases of decay, in general, is not difficult. Principally it remains the central moment - accounting velocities of moving particles - of decay products must be close to physically distinguished v=vN[s] elite (dominant) velocities. Binary Collisions In collaiders experiments where collision of identical particles takes places, evidently, it may be expected the appea rance of transitive states (resonances) in the indicated above sense (as generating by collading, moving particles). And then, as in the case of reconstruction by moving components in mass formula, it may be used as M# the value M# = 2m0, where m0 - (table, at rest) mass of collading particles. Transitive Resonances. Many-Particles Decays Very wide class of observing in the subatomic world objects (states) is connected with the follwing typical picture. The short-living state (resonance) with the effective mass M decays to some more stable - let speek - stationary states, having the table masses (of rest) mo,i. If at process of decaing its move with velocities vi (momenta Pi=mo,ivi), than by force of the standard (relativistic) kinematics it is valid the representation for, let speek, effective mass mi of each component of decay m i = mo, i(1+zi)1/2=mo, i(1+vi2/c2)1/2, where zi = βi2 =vi2/c2, βi = vi/c, c - the light velocity. Than, by force of mass - energy retaining low, effective mass of initial decaing state (resonance) is equal M = Σ m i = Σmo, i(1+zi)1/2 By force of close connection such mass with movement of decay products with none-zero velocities (moreover, this mass is wholly born by movement), let name initial state the transitive resonance. Binary decays The most simple and often the decay is, when two identical particles (n=2) with identical velocities are obtained. Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 3 In such case for mass of transitive resonance it is valid the representation M = 2mo(1+z)1/2 In connection with that it is represntating the possibility of observation of following transitive resonances. # Di-proton resonances (at mo = mp - proton mass) M = 2mp(1+z)1/2 # Di-pion resonances (at mo = mπ - pion mass) M = 2mπ(1+z)1/2 # Di-muon resonances (at mo= mµ - muon mass) M = 2mµ(1+z)1/2 # Di-electron resonances (at mo= me - electron mass) M = 2me(1+z)1/2 General Case: Transitive States (Resonances) Mass Spectrum of Transitive States (Resonances) The central idea of following examination may be breafly formulated in form of the Proposition # Mass Spectrum of transitive states (resonances) is characterized by preferable, physically distinguished values of mass - it is discrete, (not continuous), commensurable, quantized. # This discrete mass spectrum is generated by discrete spectrum of preferable, physically distinguished - elite (most brightly - dominant) velocities - by the Universal spectrum of velocities (of Universe - micro- and megaworld) vN[s]. This assertion, may be seeming unusual, extraordinary for the ultimate standard theory of particles, representes as evident and natural in the framework of base ideas of Wave Universe Concept. VELOCITIES HIERARCHY AND UNIVERSALITY Hierarchy and Spectrum of Elite Velocities. The Fundamental wave equation [Chechelnitsky, 1980], described of Solar system (similarly to the atom system), separates the spectrum of physically distinguished, stationary - elite - orbits, corresponding to mean quantum numbers N, including the spectrum of permissible elite velocities vN. It is the follow representation for the physically distinguished - elite dominant velocities vN in G[s] Shells of wave dynamical (in particular, astronomical) systems [Chechelnitsky, 1986] vN[s] = C∗[s](2π)1/2/N, s=...,-2,-1,0,1,2,... C∗[s] = (1/χs-1)⋅C∗[1]. Here C∗[1] = 154.3864 km⋅s-1 is the calculated value of sound velocity of wave dynamic system (WDS) in the G[1] Shell, that was made valid by observations, χ - the Fundamental parameter of hierarchy - Chechelnitsky Number χ = 3.66(6) [Chechelnitsky, 1980 - 1986], s - the countable parameter of Shells, N - (Mega)Quantum numbers of elite states, a) Close to NDom = 8; 11; 13; (15.5)16; (19,5); (21,5) 22,5 - for the strong elite (dominant) states (orbits); b) Close to N - Integer, Semi-Integer - for the week elite (recessive) states (orbits). In the wave structure of the Solar System for planetary orbits of Mercury (ME), Venus (V), Earth (E), Mars (MA), we have, in particular, N = (2πa/a∗)1/2 (a - semi-major axes of planetary orbits, a∗[1]=8R/G7E - semi-major axis of TR∗[1] - Transsphere, R/G7E - radius of Sun) [Chechelnitsky, 1986] N = 8.083; 11.050; 12.993; 16.038, close to integer N = 8; 11; 13; 16. Taking into account Ceres (CE) orbit and transponated in G[1] (from G[2]) planetary orbits of Uranus - (U), Neptune - (NE), Pluto - (P), it can be received the general representation for observational dominant N TR∗ ME TR V E (U) MA (NE) CE (P) N= (2π)1/2=2.5066 8.083 (2π)1/2χ=9.191 11.050 12.993 15.512 16.038 19.431 21.614 22.235 Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 4 It may be show, that N = N∗ = (2π)1/2=2.5066 (critical - transspheric value) and NTR=χ(2π)1/2 ≅ 9.191 also are physically distinguished (dominant) N values [Chechelnitsky, 1986]. Extended Representation It is possible, in principle, to examine the following substitution 1/N → ς / N# or N → N#/ς and to extend formula for elite velocities vN[s] = C∗[s](2π)1/2(ς / N#), s=...,-2,-1,0,1,2,... ς , N# - integer. In this case, for instance, the previouns condition N - semi - integer will be indicated (for the set of integer numbers) by the condition ς =2, N# - integer, and thus, - the substitution N → N#/2. General Dichotomy Very close (to discussed above) variant of description of physically distinguished states may be possible with using of effective approximation, proposing by the General Dichotomy Law [Chechelnitsky, 1992]. Connected with it compact representation for the N quantum numbers have the explicit form Nν = Nν=0·2ν/2, Nν=0 = 6.5037 that is depended from countable parameter ν = k/2, k=0,1,2,3,... It follows to, in particular, exponential, (power) dependen ce for a semi-major axes aν[s] = aν=0[s] 2ν, aν=0[s] = a∗[s] (Νν=0)2/2π, In the some sense - this is the expansion and gene ralization to all WDS of Universe of the well-known Titius-Bode Law for the planetary orbits. Such idealazing model representation - the General Dychotomy Law (GDL) - gives approximate, but easy observed description of the set of distinguished (dominant) orbits. Universal Spectrum of Elite Velocities in the Universe. Megaworld and Microworld (Quasars and Particles). Proposition. The spectrum of physically distinguished elite velocities vN[s] and quan tum numbers N of arbitrary wave dynamic systems (WDS) has the some universal peculiarity. It is practically identical - universal (invariant) for all known observed systems of Universe (of megaworld and microworld ). In particular, velocities spectrum of experimentally well investigated Solar and satellite systems practically coincides for observed planetary and satellite - dominant orbits, corresponding to some (dominant) values of quantum numbers NDom. Thus it may be expected, that spectrum of elite (dominant - planetary) velosities of the Solar system (well identificated by observations) may be effectivelly used as quite representive - internal (endogenic) - spectrum of physically distinguiched, well observed - elite (dominant) velocities, for example, of far astronomical systems of Universe [Chechelnitsky, 1986, 1997] and of wave dynamic systems (WDS) - elementary objects of subatomic physics. Quantization of Circulation and Velocity. We once more repit in the compact form the important conclusion which was obtained in the monograph (Chechelnitsky, 1980) and repeatedly underlined afterwards. Proposition (Quantization of Velocities). In the frames of Wave Universe Concept and Universal wave dynamics # The fundamental properties of discreteness, quantization of wave dynamic systems (WDS) - objects both mega and microworld - are connected not only with discreteness, quantization of i ) Kinetic momentum (angular momentum) Km= mva, ii ) And momentum (impuls) P = mv (as that is discrabed in well known formalism of quantum mechanics), # But - on the fundamental level - are connected with discreteness, quantization of v) Sectorial velocity (circulation) Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 5 L = Km/m = va, (L∗ = ξd- = ξh-/m), ξ - nondimensial coefficient, vv) And (Keplerian) velocity v = P/m. vvv) Together with the relating to its sizes (lengths) - a - semi - major axes of orbits and T - periods (frequencies). Universality of observed, physically distinguished velocities From the point of view of experimental investigations of real systems of Universe the Law of Universality of Elite (Dominant) velocities may be briefly formulated as follows Proposition (Universality of Elite - Dominant Velocities in Universe) # Detectable in experiments and observations velocities of real systems of Universe - from objects of microworld (subatomic physics) to objects of megaworld - astronomical systems - with the most probability belong to the Universal Spectrum of elite (dominant) velocities of Universe. # This Universal Spectrum of Velocities in the sufficient approximation may be represented in the form: vN[s] = C∗[s](2π)1/2/N, s=...,-2,-1,0,1,2,... C∗[s] = (1/χs-1)⋅C∗[1]. General Gomological Series of Sound Velocities Once more let pay our attention to the hierarchy of sound velocities, that is definded by the recurrence relation C∗[s] = (1/χs-1)⋅C∗[1] s=...,-2,-1,0,1,2,... In view of its special important significance and possibility of following generalizations we will name it "The General Gomological Series (GGS) of sound velocities". By the quality of generative member in that series essentially it is used, for instance, the C∗[1]=154.3864 km⋅s-1 - value of sound velocity in G[1] Shell of WDS. As a matter of fact, this is primary source (eponim) of that series. Of course, in the role of primary source any member of that series may be used. Testimony (Evidence) for that is only most knowlege reliability of that value - its experimental definiteness (determination). THEORY, OBSERVATIONS, EXPERIMENTS Two problems will lie in field of our attention below. # If are known, fixed in observations and experiments any facts, that prove argue a reality of existence of theory effects of velocities discreteness, commensurability, quantization? # How much effective, in frame of theory (WU Concept), is the description of mass spectrum of transitive resonances objects, that are close connected with consequant decay to rapidly moving components? Phenomenon of Velocities Discreteness Effects of velocities discreteness in experiments of subatomic phisics, apparently, appeared long ago, but, indeed, conceptually its were not "observed" till now. This is the situation, which is typical for science. Results of experiments, at first, must be comrehended in frame of any theoretical representations, of arbitrary conceptual expectations in order to that facts, properly, will be taken into consideration. Otherwise its remain unnoticed and sink in array of suppress by its volume information. It is very important, that only on the base of some expectation any successful experiments may be constructed. This theme is interesting by itself and we hope, may be, to return to it afterwards (later or subsequently). Let us point out (indicate) only several facts and investigations of last time. Observations in Space Quantizations of Velocities and Redshifts of Astronomical Systems Information about existence of distinguished velocities spectrum most brightly, evidently, (as that often occur in the history of science) for a long time enters from area of study of megasystem - beside from close - Solar system [Chechelnitsky, 1980-1998], but from distant - galaxies, quasars [Burbidge, 1967, 1968; Tifft and Cocke, 1984; Arp et al, 1990; Chechelnitsky, 1997]. In fact, namely in the world of astronomical systems, frequantly, important phenomena in particular descriptive, unmuddy form are fixed. The question is about observations of preferable velocities of not only celestial bodies, but also of plasma, that is high speed particles flows inside astronomical systems. The last is evident from the fact, that the Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 6 discreteness phenomena are connected with velocities, that essentially exceed all conceivable admissible limits of velocities of large celestial bodies. By such high (subluminal) velocitites only motions of plasma, highenergetic particles may be characterized. So, even only on the base of similar facts and its comrehend in the frame of WU Concept we can regard, that real objects, components of its decay "prefer" to move with some physical distinguished - elite (dominant) velocities [Chechelnitsky, 1997]. Particles and Quasars It is interesting to point that above mentioned representation z=v2/c2 describes (in megaworld) the redshifts of quasars and galaxies [Chechelnitsky, 1997]. It may be shown - this is not accidental coincidence. Experiments on the Earth But, it is clear, of course, that it is interesting to detect the effects of velocities discreteness, commensurability, quantization not so far - in Space, but - in immediate nearness, at a short distance - at the Earth, in physical laboratories. Already now, even without of specially oriented, purposeful experiments it may be suggested, that in physical experiments effects were fixed, which may be interpreted as phenomena of velocities discreteness, commensurability. Undoubtedly, this topic deserves of the special investigation in history of science (physics). Effects of Discreteness, Commensurability in Decay Reactions "Well forgotten" old. Radioactive decay. Still the pioneers - investigators of radioactivity remarked a set of interesting peculiarities, by which the radioactive decay was characterized, in particular, it was remarked [Dorfman, 1979]: "It was shown, that all α - particles, thrown by one radioactive radiation, have identical run length and identical for this radiative velocity [Reserford, 1905] ". In particular, Reserford [Reserford, 1972, p.77] remarked: "We see, that issuing velocity of α-particles of differ radioactive matters lay in enough na rrow interval, between 1.59⋅109 and 2.25⋅109 cm/s". (P.75): "All initial issuing velocities of products of radioactive elements lay between 1.59⋅109 and 2.25⋅109 cm/s, that is the maximal issuing velocity is only in 1.44 times larger than the minimal velocity." This Reseford observation finds an interesting comment in this discussed approach of WU Concept. The observating limits of velocities correspond to the dominant values in Shell G[-4], and really are close to the characteristic value 21/2 =1.414 of such velocities in frames of Generalized Di /G6Bhotomy [Chechelnitsky, 1997]. There are many another important results of interesting. Such brilliant experiments passed as unremarked by theory, proved to be out of mainstream of fundamental representations of standard science. Evidently, generally accepted representations possess by special selection. We discovered for ourselves the results of such buried experiments, becouse looked for namely these effects. One way or another, the effects of velocitites discreteness, commensurability, quantization must appear itselves in more wide circle of occurrences. "Unremarked" New Let us point results only one experiment conducted in Dubna [Avdeychikov, Nikitin, 1987,1988]: "...In range of low kinematic energy of the Ef - fragment the endow of source with limit velocity β1=0.02c dominates, in range of high energy - with β2=0.08 c, where c - the light velocity. The pointed values β1 and β2 are characteristic for all z - fragments and energies of a beam." This observations of experimentators are extraordinary and a'priori - far not evident. Really, why the whole set of fragments, that essentially differ by charges and energies, must have the same velocities of issuing? From the formed standard ("probable") representations such conclusion does not follow. But in frame of (WU Concept) representations about universality of velocities spectrum - of its discreteness, commensurability, quantization, presence of physically distinguished states - such effect is quite expected. Moreover, it is really correct from the point of view of theory, because of the observated velocities correspond to the theoretically calculated values vV[-4] = 0.0774c = 23210 km/s, vV[-3] = 0.0211c = 6330 km/s, Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 7 vν=1.5[-4] = 0.0782c = 23448 km/s, v ν=1.5[-3] = 0.0213c = 6395 km/s. It is not difficult to show also, that the observating velocities concern as β2/β1=4=22, that is its belong for all that to the velocities set of the Generalized Dihotomy. It is interesting to mark, that the detected by modern methods physically distinguished velocity β2=0.08c corresponds to the velocity, that Reserford observed as close to the upper limit of velocities at α - decay. Another New It is interesting point, that information about the same velocity v ≤ 0.02 c, generating the e+ e- resonances, is mentioned in [Koinig, 1993]. Another information [Pokotilovsky, 1993]: "...Data testify that mass center of probable decaying e+e- system move in center-of- mass of collision ions with little velocity, not exceeding 0.03 - 0.05 c." Latent Possibilities of RPP (Review of Particle Physics) Meanwhile, wide possibilities and reason (cause) for reflections and investigations the most known compendium of experimental data of Particle Physics (RPP) allows, especially when in it the data concerning to momenta (impulses) of decay modes of particles were ublished. Lying at a surface and detecting at purposeful search, the effect of P - momenta (impulses) discreteness and commensurability (together with masses discreteness) naturally leads to discovery of v - velocities discreteness, commensurability. Investigations of Gareev's Group Recently, to the indicated by the Wave Universe Concept universal effects of (sectorial and Keplerian) velocities discreteness, commensurability (in micro - and megaworld) Gareev pays special attention [Gareev et al, 1996]. Possessing by developed physical intuition and conducting wide work with using of RPP experimental data, Gareev and his co-workers made convinced in validity of expectations of WU Concept for objects of subatomic world too. Indeed, in a wide array of particles decays the effect of velocities discreteness, commensurability is observed, exists, brightly manifests. Information (in RPP) about experimentally observed P mass momenta (impulses) and connected with its v velocities opens the possibilities to calculate the potentially virtual masses spectrum of resonances by semi - empirical way. Correlation of computations and experimentally known data is impressionable. In any case, such coinciding is a challenge to the standard theory and a stimulus for further purposeful investigations. What is further? But the heuristic analysis exhausts itself even as the following questions arize: # From where these physically distinguished velocities? # What must we do without RPP by the hand - i.e. without information about empirical values of velocities (of particles decays)? # Why these velocities, but not another? # If exist for its (velocities) any theoretical representations? Answers to these questions the Wave Universe Concept gives, in particular, by the Proposition (Theorem) about Universality of (physically preferable) elite (dominant) velocities spectrum - for real systems of Universe (micro - and megaworld). Integer Commensurability Simplest integer commensurability of observed in experiments (in RPP) velocities vi /vj = Nj / Ni N i, N j - Integer is entirely evident in frame of WU Concept and directly follows from the principal representation for elite velocities vN[s] = C∗[s](2π)1/2/N, N - Integer (semi - Integer) At the Begining of Way If the history experience learns to something, then it is relevant to note, that with the comprehension (by a wide circle of physicists - professionals) of real existence of velocities discreteness, commensurability, quantization effects, the subatomic physics fixes itself at the stage, which corresponds (roughly) to the time of Roy and Ovenden pub lication [Roy and Ovenden, 1954], concerning world of megasystems. In that paper at wide material the presence of commensurability in motions of Solar system celestial Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 8 bodies is convincingly stated, although separate phenomena was known for a long time [see review at Chechelnitsky, 1980]. In front of that we yet have much work in theoretical conceiveness and fundamental comprehension of these phenomena. Transitive States (Resonances) Spectrum of Masses. Theory and Observations We shall bring only some results of calculation and comparision with experimental data. Its concern of Transitive states (Resonances) spectrum, connecting with dominant velocities only by some G[s] Shells. Mentioned data correspond to states, reconstructed by binary decays. Transitive (T)G[-5] Shell Transitive States (Resonances). Spectrum of Masses In Table 1 are cited the data of theory and comparison its with avialable experimental information for three families of transitive resonances # Di - electron Family. Set of transitive resonanes of this family is generated by mass M# = 2me = 1.022 Mev/c2, where me = 0,511 Mev/c2 - electron mass. Observed in experiments values of resonances masses are taken from [RPP - Review of Particles Physics; Pokotilovsky, 1993; Ganz et al., 1996 and review Gareev et al., 1996, 1997 (E4-97-183)]. # Di-pion Family. Masses spectrum of transitive resonances is generated by mass M# = 2mπ = 279.14 Mev/c2, where mπ = 139,56995 Mev/c2 - mass of π± - meson. Experimental data in this range of masses are not known (to us). Calculation is adduced for the orientation of experimentators. # Di-proton Family. Masses spectrum of transitive states (resonances) is generated by mass M# = 2mp = 1876.5446 Mev/c2, where mp = 938.27231 Mev/c2- proton mass. In experiment sufficient developed spectrum of masses is observed. Experimental data are taken from [RPP; Troyan et al., 1991; Troyan, Pechonov, 1993; Tatischeff, 1990,1994, 1997; Edogorov, 1991; Andreev, 1987; Gareev et al, 1996]. Transitive (T)G[-6] Shell Comparision of theory and experiments for Di-electron,Di-pion, Di-proton cites in Table 2. Experimental data # For the Di-electron family are taken from [Ganz et al., 1996; Pokotilovsky, 1993; Gareev et al., 1997]. # For the Di-pion family are taken from [Troyan, 1993; Troyan et al., 1991, 1996; Codino, Plouin, 1994; Gareev, 1996,1997], (m = 447.49 Mev/c2) - from [Troyan et al., 1997]. # For the Di-proton family - from RPP, and also - from [Ball et al., 1994; Ohashi et al., 1987; Tatischeff et al., 1990,1994, 1997; Gareev et al., 1996]. As cause for reflections also comparision of theoretical spectrum with wide class of resonances from RPP is adduced. Transitive States (Resonances) and Multi - Particle Decays It is possible the analysis and comparison with experiment of transitive states (resonances), which decay, in general case, in several different particles with mi masses. With this it is used the indicated above general representation for the M mass of transitive state (resonance) M = Σ m i = Σmo, i(1+zi)1/2 zi = βi2, βi = vi/c, vi ⇒ vN[s], c - the light velocity. Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 9 DISCUSSION Estimating the situation, which fully formed in connection with known early interpretation of masses, it is inevitably come to conclusion, that the theory is situated at the very origin of way. It is fully possible, that discussed above approach will be stimulate more effective advance on that way. So two directions represent as actual # Investigations of conformities of elite velocities universality in Universe - in micro - and megaworld, # Investigations of fundamental conformities of resonances and dynamical spectrum origin. With respect to resonances spectrum even available by this time new data permit to suggest the following. ∗ Detectable in experiments states (resonances) are not phantoms, fancies, ∗ To its real physics and wave dynamics correspond, ∗ Its place, status, role in order of another, more known states may be comprehended in frame of WU Concept. Wide set of observed in experiments effects and connected with its resonances (including - Darmstadt effect [see review Pokotilovsky, 1993], ABC effect [see review Codina, Plouin, 1994], effects discussed by Gareev [see review Gareev, Kazacha, 1996; Gareev et al., 1997]) can effectively interpreted in WU Concept and described with use of mass formula - as manifestation of rapidly moving, physically distinguished transitive states (resonances). Purposeful experiments, stimulated by theory, can brighten up many important details of Universality of elite velocities in the Universe (including - in microworld), and also can broaden the spectrum of observed resonances. There are the serious bases to regard, that results of such experiments will be foreseeing and succesful. The Wave Universe Concept gives such bases. REFERENCES Andreev V.P. et al. - Z. Phys., v.A327, p.363, (1987). Avdeychikov V.V., Nikitin V.A. et al, Preprint JINR P1-87-509, (1987). Avdeychikov V.V., Nikitin V.A. et al, Journal of Nuclear Physics, v.48, i.6(12), pp. 1736-1747, (1988). Arp H., Bi H.G., Chu Y., Zhu X. Periodicity of Quasars Redshifts, Astron. & Astrophys., v.239, p.33, (1990). Ball J. et al., Phys. Rev. Lett., v.B320, p.206, (1994). Burbidge G.R. Astrophys. J., 147, p.851, (1967); 154, L41, (1968). Canz R. et al., Phys. Lett., B 389, 4 (1996). 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Transitive States (Resonances) – Mass Spectrum. 11 TABLE 1 MASS SPECTRUM: TRANSITIVE STATES - (T)G[-5] Shell State Quantum Number N Redshift z=ββ2 ((ββ=v/c) Mass M= 2me(1+z)1/2 2me=1.022 [Mev/c2] Mass (Exp) M [Mev/c2] Mass M= 2mππ(1+z)1/2 2mππ=279.14 [Mev/c2] Mass (Exp) M [Mev/c2] Mass M= 2mp(1+z)1/2 2mp= 1876.5446 [Mev/c2] Mass (Exp) M NN [Mev/c2] Mass (Exp) M __ NN (RPP) [Mev/c2] Mass (Exp) M (RPP) [Mev/c2] TR∗∗ 2.5066 1.57 ME 8.083 0.151 1.0964 1.1 299.46 2013.2 2017 ± 3 2016 2008 ± 3 2022 ± 6 Ξ(2030) Σ(2030) f2(2010) 2007 ± 10 TR 9.191 0.116 1.0796 1.077 294.88 1982.3 1980 ± 2 Σ(2000) 1975 ± 1 V 11.050 0.0806 1.0623 1.062 290.17 1950.7 1956 ± 3 1955 ± 2 1949 ± 10 1943 ± 5 X(1950) Ξ(1950) E 12.993 0.0583 1.0513 287.16 1930.47 1932 ± 3 1930 ± 2 Σ(1940) (U) 15.512 0.0409 1.0426 284.79 1914.53 1918 ± 3 1916 ± 2 1920 1919 ± 3 Σ(1915) MA 16.038 0.0383 1.0413 1.043 284.43 1912.1 X(1910) (NE) 19.431 0.0261 1.0352 282.75 1900.8 1902 CE 21.614 0.0211 1.0327 282.06 1896.2 1898 ± 1 1897 ± 1 1897 ± 1 (P) 22.235 0.0199 1.0321 281.9 1895.1 1892 Λ(1890) Σ(1880) Chechelnitsky A.M. Motion, Universality of Velocities, Masses in Wave Universe. Transitive States (Resonances) – Mass Spectrum. 12 TABLE 2 MASS SPECTRUM: TRANSITIVE STATES - (T)G[-6] Shell State Quantum Number N Redshift z=ββ2 ((ββ=v/c) Mass M= 2me(1+z)1/2 2me=1.022 [Mev/c2] Mass (Exp) M [Mev/c2] Mass M= 2mππ(1+z)1/2 2mππ=279.14 [Mev/c2] Mass (Exp) M [Mev/c2] Mass M= 2mp(1+z)1/2 2mp= 1876.5446 [Mev/c2] Mass (Exp) M NN [Mev/c2] Mass (Exp) M__ NN (RPP) [Mev/c2] Mass (Exp) M (RPP) [Mev/c2] TR∗∗ 2.5066 21.1 ME 8.083 2.02 1.776 1.782 485.09 470± 7 3261.0 X (3250) TR 9.191 1.57 1.638 1.662 447.49 447 3008.3 Σ (3000) V 11.050 1.08 1.473 1.496 402.58 397; 400 2706.4 2735 2710±20 E 12.993 0.784 1.365 372.83 2506.4 ≈ 2500 fb (2510) Ξ (2500) (U) 15.512 0.550 1.272 347.52 354; 350±10 2336.2 2350 2380 f2 (2340) Λ (2350) Λ (2325) MA 16.038 0.514 1.257 1.250 343.46 2308.9 2307± 6 f2 (2300) f4 (2300) (NE) 19.431 0.350 1.187 324.33 2180.3 2194 2172± 5 2180± 10 f2 (2150) CE 21.614 0.283 1.157 316.15 2125.5 2122 Ξ (2120) Σ (2100) (P) 22.235 0.268 1.150 314.32 313± 3 2113.1 2106± 2 2110± 10 2112.4 π2 (2100) Λ (2110)
arXiv:physics/0103090v1 [physics.flu-dyn] 28 Mar 2001Local law-of-the-wall in complex topography: a conﬁrmation from wind tunnel experiments S. Besio, A. Mazzino and C.F. Ratto INFM - National Institute for the Physics of Matter, Department of Physics, Genova University, Genova (Italy). March 24, 2011 Abstract It is well known that in a neutrally-stratiﬁed turbulent ﬂow in a deep constant- stress layer above a ﬂat surface, the variation of the mean ve locity with respect to the distance from the surface obeys the logarithmic law (t he so-called “law-of- the-wall”). More recently, the same logarithmic law has bee n found also in the presence of non ﬂat surfaces. It governs the dynamics of the m ean velocity (i.e. all the smaller scales are averaged out) and involves renormali zed eﬀective parameters. Recent numerical simulations analyzed by the authors of the present Letter show that a more intrinsic logarithmic shape actually takes plac e also at smaller scales. Such a generalized law-of-the-wall involves eﬀective para meters smoothly depending on the position along the underlying topography. Here, we pr esent wind tunnel experimental evidence conﬁrming and corroborating this ne w-found property. New results and their physical interpretation are also present ed and discussed. PACS: 83.10.Ji – 47.27.Nz – 92.60.Fm Boundary layer ﬂows, Ne ar wall turbulence In the realm of boundary layer ﬂows over complex topography, much eﬀort has been devoted in the last few years to investigate both the detaile d form of the surface pressure perturbation arising from the interaction between the shea r ﬂow and the underlying to- pography (see, e.g., Refs. [1, 2, 3]) and its link with the eﬀe ctive parameters describing the large (asymptotic) scale dynamics (see, e.g., Refs. [4, 5]). The latter regime is selected by observing the ﬂow far enough from the surface and, further more, considering solely the mean velocity. It is thus clear that with this approach all in formation on the dynamics at smaller scale becomes completely lost. 1Unlike what happens for the large-scale (asymptotic) dynam ics, the description and un- derstanding of statistical properties of ﬂows at ‘intermed iate’ scales (a regime which we refer to as “pre-asymptotic”, following Ref. [6]) seems str ongly inadequate. Such regime actually attracts much attention in various applicative do mains ranging from wind en- gineering (e.g., for the safe design and siting of buildings ), environmental sciences (e.g., for the simulation of air pollution dispersion) and wind ene rgy exploitation (e.g., for the selection of areas of enhanced wind speed for the economic si ting of wind turbines). This almost unexplored regime is the main concern of the pres ent Letter. A ﬁrst step in the understanding of the pre-asymptotic dynamics has been d one by the present authors in a very recent work [6], where the analysis of simulations o f Navier-Stokes ﬂow ﬁelds [4] over two-dimensional sinusoidal topographies has been performed. More precisely, in the case-studies considered, topography takes a sinusoida l modulation of wavelength λ (along the x−direction, for the sake of simplicity) and amplitude H, its surface having an uniform roughness z0(with z0<< H ). Here, the dominant process governing the dynamics is the interaction between the shear ﬂow and the und erlying topography, the eﬀect of which gives rise to a surface pressure perturbation [2]. Such perturbation has a depth of the order of Hand a downwind phase shift with respect to the topography. Th e latter is the cause of a net force on the ﬂow, acting in the oppo site direction of the ﬂow itself: thus, an enhanced (with respect to the case of ﬂat ter rain) transfer of momentum towards the surface takes place. Far enough from the surface, the averaged (over the periodic ity box of size λ) ﬂow will ‘see’ an ‘eﬀective ﬂat surface’ over which the ‘basic’ logar ithmic law (the well known “law-of-the-wall” relative to ﬂows over ﬂat terrain [7]) is restored but now with larger (again with respect to the ﬂat case) eﬀective parameters ueﬀ ⋆, andzeﬀ 0[4], on account of the enhanced ﬂux of momentum towards the surface originated by t he aforesaid shear-ﬂow – topography interaction. In Ref. [6], we pointed out for the ﬁrst time – as far as we know – that at least in the analyzed WM93 data-set [4], a generalized law-of-the-wall , is observed: U(x, z) =ueﬀ ⋆(x) kln/parenleftBiggz zeﬀ 0(x)/parenrightBigg for z > H (1) where Uis the velocity ﬁeld, xis the horizontal position, zis the height above the terrain, andkis the von K´ arm` an constant which we will take as 0 .4. Notice that the eﬀective parameters, ueﬀ ⋆(x) and zeﬀ 0(x), show a dependence on xat scales of the order of λ(i.e. the ﬂow ‘sees’ some details of the topography and not only its tot al cumulative eﬀects). This is precisely the pre-asymptotic regime already deﬁned in Re f. [6]. In the present Letter, our main goal will be to provide a ﬁrst e xperimental assessment conﬁrming and corroborating the scenario outlined in Ref. [ 6]. In fact, the main trouble of numerical simulations of Navier–Stokes equations is tha t the impact on the results of 2the closure schemes, through which small scale dynamics is a ccounted for, cannot be fully controlled [4]. An experimental conﬁrmation is thus desira ble. To start our analysis, we brieﬂy describe the experimental s et-up relative to the wind tunnel experiment performed by Gong et alin Ref. [8]. Details on the description of the wind tunnel facility and the basic data acquisition and a nalysis system are given also in Ref. [9]. The experiment was conducted in the AES (Atm ospheric Environment Service, Toronto, Canada) meteorological wind tunnel, whi ch has a working volume of 2.44m×1.83m×18.29m(w×h×l). The wave model consisted of sixteen sinusoidal waves with wavelength λ∼610mmand through-to-crest height H∼96.5mmand was placed with its leading edge at distance d∼6.1mdownstream from a honeycomb located at the downstream end of the contraction region. The topogra phy can be thus considered as a fraction of the ideal topography described by: h(x, y) =Hsin2/parenleftbiggπx λ/parenrightbigg (2) where yis the perpendicular-to- x-axis direction coordinate. Two surface roughnesses were considered, corresponding to the natural foam surface (hereafter “smooth case”) and to a carpet cover (“rough case ”), respectively. For the smooth case, velocity proﬁle measurements gave z0∼0.03mm, while for the rough case z0∼0.40mm. The ﬂow was neutrally stratiﬁed and can be considered as a pe rturbation to a stationary horizontally homogeneous inﬁnitely deep un idirectional constant-stress- layer ﬂow above a plane surface of uniform roughness, z0. Thus, this basic ﬂow should have a logarithmic mean velocity proﬁle, U(z) = (u∗/κ) ln(z/z0). The values of u∗were ∼0.43m/sand∼0.62m/sfor the smooth and the rough case, respectively. The free- stream velocity, U0, at approximately 1 mabove the ﬂoor of the tunnel, was set to about 10m/sduring the measurements both in the smooth and in the rough ca se. The bound- ary layer height, hB, was evaluated to be ∼600mm. The rotation of the ﬂow with the height, produced in the numerical simulations [4] by the Cor iolis force, is obviously not present in the wind tunnel experiment and thus the ﬂow is para llel to the x-axis at all elevations. Measurements taken over the crests along the hills showed th at the ﬂow reached an almost periodic state quite rapidly, after the 3rd or 4th wav e. Thus, the perturbed velocity proﬁles, U(x, z), were measured at selected downstream locations between t he 11th and 12th wave crests and a very good agreement between the proﬁle s over these two crests was conﬁrmed. This topography can be thus considered as a goo d approximation to a two-dimensional topography whose shape is described by Eq. (2) In order to compare the numerical simulations analyzed in Re f. [6] with the results from the wind tunnel experiments here shortly described, we used the same approach as by Finardi et al. [10] and Canepa et al. [11]. Accordingly, noticing that in both cases here 3considered U0∼10m/s, we have kept the speeds (including the friction velocities ,u∗) unchanged, while the wind tunnel lengths (and times) have be en multiplied by λW/λG∼ 1000/0.6096∼1640, where λWandλGare the wavelengths in the Wood numerical simulations and in the Gong experiment, respectively. With this change of scale, hB∼ 1000mandλ∼1000min both cases, while the roughness lengths become ∼0.05m (smooth case) and ∼0.66m(rough case), to be compared with the value of ∼0.16mof the numerical simulations. The hill height, H, becomes ∼158m, to be compared with the values 20 m, 100mand 250 min the Wood numerical experiments. The ﬁrst point to emphasize is that logarithmic laws describ ed by (1) are evident also in 1 10 100 1000z (m)024681012U(z) (m/s)(a) 1 10 100 1000z (m)024681012U(z) (m/s)(b) 1 10 100 1000z (m)024681012U(z) (m/s)(c) 1 10 100 1000z (m)024681012U(z) (m/s)(d) Figure 1: The local wind speed proﬁles U(z) from the wind tunnel experiment [8] are plotted (solid lines) as a function of zfor four diﬀerent positions ( xin Eqs. (1) and (2)) along the hill, corresponding to (a) x= 0, (b) x=λ/4, (c) x=λ/2 and (d) x= 3λ/4. The dashed lines represent the unperturbed proﬁle. The do t-dashed lines represent the logarithmic law (1), with parameters ueﬀ ⋆(x) and zeﬀ 0(x) obtained by least- square ﬁts performed inside the scaling regions. The values of these eﬀective parameters are given in the text. the wind tunnel experiments. This can be easily seen in Fig. 1 (the analogous of Fig. 1 in Ref. [6]), where typical behaviours for the horizontal wind speed proﬁle U(z) (see Eq. (1); for the sake of brevity, the dependence on the x-coordinate is omitted in the notation 4from now on) as a function of zare presented in lin-log coordinates for the rough case and for four values of the x-coordinate corresponding to: (a) x= 0 (i.e. h= 0), (b) x=λ/4 (i.e. h=H/2 upwind), (c) x=λ/2 (i.e. h=H) and (d) x= 3λ/4 (i.e. h=H/2 downwind), respectively. Similar behaviours have been fou nd (but not reported here for the sake of brevity) also for the smooth case. From this ﬁgure , clean logarithmic region of the type described by Eq. (1) are evident and both ueﬀ ⋆(x) and zeﬀ 0(x) can be measured by least-square ﬁts. Speciﬁcally, for the four above position s along the hill, we have obtained the following values of ueﬀ ⋆(x) and zeﬀ 0(x): 1.73m/s, 0.053m(forh= 0); 1 .38m/s, 0.028m(forh=H/2 upwind); 0 .95m/s, 0.007m(forh=H) and 1 .27m/s, 0.022m (forh=H/2 downwind), respectively. Such values can be compared with those in the absence of any hill: u⋆∼0.62m/sandz0∼0.66m. −0.1 0.4 0.9x/λ−1.0−0.50.00.51.01.52.0ueff(x)/u−1 smooth −0.1 0.1 0.4 0.7 0.9x/λ0.02.55.07.510.0ln(z0eff(x)/z0) smooth −0.1 0.1 0.3 0.5 0.7 0.9 1.1x/λ0.01.02.03.0ueff(x)/u−1 rough −0.1 0.1 0.3 0.5 0.7 0.9 1.1x/λ1.02.03.04.05.06.0ln(z0eff(x)/z0) rough Figure 2: The measured (circles) eﬀective parameters ueﬀ ⋆(x)/u⋆−1 (on the left) and ln(zeﬀ 0(x)/z0) (on the right) as a function of the ratio x/λalong the axis of the hill in the smooth case (above) and in the rough case (below). The contin uous lines represent the sinusoidal law best-ﬁtting the experimental data. The results of the least-square ﬁts are summarized in Fig. 2 w here both proﬁles ueﬀ ⋆(x)/u⋆−1 (on the left) and ln( zeﬀ 0(x)/z0) (on the right) are shown as a function of x/λfor both the smooth (above) and the rough (below) case (diﬀer ent scales in the or- 5dinates have been adopted). Notice that both ueﬀ ⋆(x)/u⋆−1 and ln( zeﬀ 0(x)/z0) have been ﬁtted with the analytical expression (2), relative to the to pographic proﬁle, but with a shift of λ/2 (i.e. x/mapsto→x+λ/2 in (2)). More precisely, we suggest the expression: y(x) =/angbracketlefty/angbracketright −Ay[h(x)/H−1/2] (3) where y(x) stays for either ueﬀ ⋆(x)/u⋆−1 or ln ( zeﬀ 0(x)/z0),/angbracketlefty/angbracketrightis the average value of y(x) in the interval (0 , λ) and h(x)≡h(x, y) is the topographic shape given in Eq. (2). It should be also stressed that we have considered ln( zeﬀ 0(x)/z0) instead of the simpler ratio zeﬀ 0(x)/z0, because the former parameter is more similar to the topogra phy shape than the second one. It is now interesting to put together the new results here obt ained with those of Ref. [6]. This allows to investigate at which degree of accuracy one ca n express the behaviours of quantities like the average values, /angbracketlefty/angbracketright, and amplitudes, Ay, of these sinusoidal shapes solely in terms of simple geometrical parameters. Simple co nsiderations suggest to look at the ratio H/λ: this is indeed a rough measure of the hill slope. The values of /angbracketlefty/angbracketrightare reported in Fig. 3 for both the smooth and the rough case. T he 0.0 0.1 0.2 0.3 0.4H/λ−0.50.00.51.01.52.0<ueff(x)/u>−1 Wood wt−s wt−r 0.0 0.1 0.1 0.2 0.2 0.2 0.3 0.4 0.4H/λ−2.0−1.00.01.02.03.04.05.06.0<ln(z0eff(x)/z0)> Wood wt−s wt−r Figure 3: The mean values /angbracketleftueﬀ ⋆(x)/u⋆/angbracketright−1 and/angbracketleftln(zeﬀ 0(x)/z0)/angbracketrightin Eq. (3) versus H/λ. Stars are relative to the Wood and Mason numerical simulations ( u⋆∼0.44m/s,z0∼0.16m); diamonds are relative to the smooth case ( u⋆∼0.43m/s,z0∼0.05m) and squares are relative to the rough case ( u⋆∼0.62m/s,z0∼0.66m). Dashed lines represent the linear curve best-ﬁtting the Wood and Mason numerical data. dashed lines (a linear ﬁt in H/λ) are obtained considering only the results of numerical simulations, while the values from the wind tunnel experime nts are reported with their error bars. These monotonic behaviours are expected on acco unt of the increasing of the (total) transfer of momentum towards the surface arisin g for increasing slopes. The amplitudes Ayare shown in Fig. 4. The dashed lines are a parabolic ﬁt in H/λand, as for /angbracketlefty/angbracketright, they have been obtained by only considering the results of t he numerical simulations. 6The values from the wind tunnel experiments are again presen ted in the same ﬁgure. Curves relative to the amplitudes reach a maximum for H/λ∼0.20, after that start to decrease. We can argue that two diﬀerent mechanisms exist an d act in competition. The physical key role is played by curvature eﬀects [12], alread y invoked in Ref. [6] to explain the presence of minima (maxima) located above the hill top (v alley) for both ueﬀ ⋆(x) and lnzeﬀ 0(x). To be more speciﬁc, let us consider the two opposite limits H/λ≪1 and H/λ≫1, from which we can easily isolate the two competing mechani sms. Concerning the former limit, we have gentle slopes and it is well known th at in this case the ﬂow closely follows the surface contour. Streamlines are (weak ly) curved and, as pointed out in Ref. [12], energy is transferred towards the large scale c omponents above the hill tops, while it blows towards the smaller scales above the valleys. The quantity ueﬀ2 ⋆(x), that is a measure of the energy of turbulence, is thus smaller on th e hill top than above the valley. Let us increase (just a little bit) H/λ. The ﬂow again closely follows the surface contour but streamlines are now more curved. As pointed out i n Ref. [12], energy transfer thus increases and, as an immediate consequence, the same ha ppens for the diﬀerence between the maximum and the minimum of ueﬀ ⋆(x). But this means an augmentation of its modulation amplitude. In the second limit H/λ≫1, a further important eﬀect arises due to trapping regions placed on the downstream hill slopes. It is in fact well known (see, e.g., [13]) that, for surface slopes large enough, the ﬂow is not able to follow the contour surface and sepa- rates. In this case, in between two hill crests, the ﬂow is ess entially trapped and, roughly speaking, streamlines are expunged in the wake region. The d ynamical consequence is that the ﬂow streamlines are weakly modulated, and this also happens for the shape of ueﬀ ⋆(x). If we now decrease the ratio H/λ, trapping eﬀects reduce and this means that the wake can penetrate more deeply in the valley, with the conseq uent increasing of curvature eﬀects and thus of the ueﬀ ⋆(x) amplitude. From the inspection of these two limits, it is thus clear that a maximum in the amplitude should be attained for a certain ﬁnite value of H/λ, i.e. when the two competing mecha- nisms are balanced. Being the maximum (minimum) of ln zeﬀ 0(x) directly related to the presence of the maxi- mum (minimum) of ueﬀ ⋆(x) (see Ref. [6] for the discussion of this point) the argument ations above presented hold also for ln zeﬀ 0(x). Comparing the values of both amplitudes and mean values of th e eﬀective parameters extrapolated from the numerical simulations (i.e. from the linear ﬁts in Figs. 3 and 4) and those from the wind tunnel experiments, we notice that, for t he smooth case, experiments are always compatible (within the error bars) with the numer ical simulations. This is not always the case for the rough case. We remark that u∗andz0(relative to the ﬂat terrain) are closer to the WM93 case studies in the smooth cas e than in the rough case. This suggest that the expression of both amplitudes and mean values solely in terms of 7geometrical quantities like the ratio H/λis a reasonable approximation for small variations of the ‘bare’ parameters u∗andz0. When the range of variability of the latter two parameters increases, an explicit dependence on them has to be taken into account. 0.00 0.10 0.20 0.30 0.40H/λ0.00.51.01.5Aueff (x)/u−1 Wood wt−s wt−r 0.00 0.10 0.20 0.30 0.40H/λ0.02.04.06.0Aln(z0eff (x)/z0)Wood wt−s wt−r Figure 4: The values of the amplitude Ayin Eq. (3) relative to y=ueﬀ ⋆(x)/u⋆−1 and y= ln(zeﬀ 0(x)/z0) versus H/λ. Stars are relative to the Wood and Mason numerical simulations ( u⋆∼0.44m/s,z0∼0.16m), diamonds are relative to the smooth case (u⋆∼0.43m/s,z0∼0.05m) and squares are relative to the rough case ( u⋆∼0.62m/s, z0∼0.66m). For a better evaluation and understanding of the dependence of/angbracketlefty/angbracketrightandAyonH/λ, z0andu∗, the analysis of more numerical and wind tunnel experiments , and possibly in nature, is necessary. Nevertheless, the wind tunnel data he re considered give a strong conﬁrmation of the existence of a pre-asymptotic regime cha racterized by a generalized law-of-the-wall given by Eq. (1) and pointed out for the ﬁrst time in Ref. [6]. Thus, this phenomenon appears as a real physical property and not a spur ious feature produced by some of the approximations (e.g. parameterizations of smal l-scale, unresolved dynamics) used to solve the Navier-Stokes equations. Acknowledgements We are particularly grateful to P.A. Taylor for providing us with his data-set relative to the wind tunnel experiments as well as many useful comments and discussions. Helpful discussions and suggestions by E. Fed orovich, D. Mironov, G. Solari, F. Tampieri and S. Zilitinkevich are also acknowledged. References [1] P.A. Taylor, Model prediction of neutrally stratiﬁed pl anetary boundary layer ﬂow over ridges, Q.J.R. Meteorol. Soc., 107, 111-120 (1981). 8[2] S.E. Belcher, T.M.J. Newley and J.C.R. Hunt, The drag on a n undulating surface induced by the ﬂow of a turbulent boundary layer, J. Fluid Mec h., 249, 557-596 (1993). [3] S. Emeis, Pressure drag of obstacles in the atmospheric b oundary layer, J. Appl. Meteorol., 29, 461-476 (1990). [4] N. Wood and P. Mason, The pressure force induced by neutra l, turbulent ﬂow over hills. Q.J.R. Meteorol. Soc., 119, 1233-1267 (1993). [5] D. Xu and P.A. Taylor, Boundary-Layer Parameterization of Drag over Small Scale Topography Q.J.R. Meteorol. Soc., 121, 433-443 (1995). [6] S. Besio, A. Mazzino and C.F. Ratto, Local log-law of the w all: numerical evidences and reasons, Phys. Lett. A, 275, 152-158 (2000). [7] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics , (MIT Press, Cambridge, Mass., 1975). [8] W. Gong, P.A. Taylor and A. Dornbrack, Turbulent boundar y-layer ﬂow over ﬁxed aerodynamically rough two-dimensional sinusoidal waves, Bound. Layer Meteorol., 312, 1-37 (1996). [9] M. Shokr and H.W. Teunissen, Use of hot-wire anemometry i n the AES boundary- layer wind tunnel with particular reference to ﬂow over hill models, Res. Rep. MSRB 88-9, 4905 Duﬀerin Street, Downswiew, Ontario, Canada. [10] S. Finardi, G. Brusasca, M.G. Morselli, F. Trombetti an d F. Tampieri, Boundary- layer ﬂow over analytical two-dimensional hills: a systema tic comparison of diﬀerent models with wind tunnel data, Bound. Layer Meteorol., 63, 25 9-291 (1993). [11] E. Canepa, E. Georgieva, A. Mazzino and C.F. Ratto, Comp arison between the results of a new version of the AVACTA II atmospheric diﬀusio n model and tracer experiments, Il Nuovo Cimento C, 20, 461 (1997). [12] A.A. Townsend, The structure of turbulent shear ﬂow (Ca mbridge University Press, Cambridge, 1980). [13] L.M. Milne–Thomson, Theoretical hydrodynamics (MacM illan & Co, London 1968). 9

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