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arXiv:physics/0008240v1 [physics.gen-ph] 30 Aug 2000Absolutely secure classical cryptography Arindam Mitra V.I.P Enclave, M - 403, Calcutta- 700059. India. Abstract Fundamental problem of cryptography is to create more absolutely secure bits than the initially shared secret bit s between two legitimate users. It is widely believed that thi s task can only be accomplished by quantum cryptosystems. We present a simple classical cryptosystem, which can fulﬁl l this objective. 1Cryptography, the art of secure communication has been deve loped from the dawn of human civilization, but it has been mathematically t reated by Shan- non [1]. At present, we have diﬀerent classical cryptosyste ms whose merits and demerits are discussed below. Vernam cipher [2]: It is proven secure [1] but it can not produ ce more absolutely secure bits than the shared secret bits. Due to th is diﬃculty, it can not be used for widespread secure communication. Data encryption standard [3] and public key distribution sy stem [4]: These are widely used cryptosystems because they can produc e more com- putationally secure bits than the shared secret bits. The problem is that it’s computational security is not proved. The assumption o f computational security has become extremely weak after the discovery of fa st quantum al- gorithms (see ref. 16) To solve the above problems of classical cryptosystem, quan tum cryp- tography [5-9] has been developed over the last two decades. Conceptually quantum cryptography is elegant and many undiscovered poss ibilities might store in it. In the last few years work on its security has been remarkably progressed [10-15], however work is yet not ﬁnished. On quantum cryptography the most widely used conjectures [1 6]are: 1. Completely quantum channel based cryptosystem is imposs ible (existing quantum cryptosystem requires classical channel to operat e). 2. Uncon- ditionally secure quantum bit commitment is impossible. 3. By classical means, it is impossible to create more absolutely secure bit s than the shared secret bits. Recently alternative quantum cryptosystem has been develo ped [17-21] by the present author; which can operate solely on quantum ch annel (both entangled and unentangled type) and can provide unconditio nally secure quantum bit commitment. The same coding and decoding techni que has also been applied for classical cryptosystem incorporatin g noise [18]. In that paper [18], we cast doubt on the third conjecture. But the pro blem of our 2noise based classical cryptosystem is that it is slow and com plex. Here we present a fast and extremely simple system without using noi se. Only we need the concept of ”pseudo-bits” introduced in that paper. Operational procedure: Always in the string of random bits, there are real and pseudo-bits. Real bits contain the message and pseu do-bits are to mislead eavesdropper. Sender encodes the sequence of real b its on to the ﬁxed real bit positions and encodes the sequence of pseudo-b its on to the ﬁxed pseudo-bit positions. It thus forms the entire string w hich is transmit- ted. The ﬁxed positions of real and pseudo-bits are initiall y secretly shared between sender and receiver. Therefore, receiver can decod e the real bits from real bit positions. Obviously he/she ignores the pseud o-bits. For the second message, sender uses new sequences of real and pseudo- bits but the position of real and pseudo-bits are same. So aga in receiver decodes the message from the same real bit positions. In this way inﬁnite number of messages can be coded and decoded. Notice that init ially shared secret positions of real and pseudo-bits are repeatedly use d. That’s why, in some sense, secrecy is being ampliﬁed. Let us illustrate the procedure.  P R R R P R P R P P P R P R R P .... 0b1b1b11b11b11 0 0 b10b1b11.... 1b2b2b20b20b20 1 1 b21b2b20.... 1b3b3b31b31b31 0 0 b30b3b30.... 0b4b4b41b40b41 0 1 b41b4b40.... . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . .... 1bnbnbn1bn0bn0 0 1 bn0bnbn1.... ≡ S M1 M2 M3 M4 . . . Mn  In the above block, the ﬁrst row represents the sequence ”S”, which is initially secretly shared. In that sequence, ”R” and ”P” den ote the position of real and pseudo- bits respectively The next rows represen t the encoded sequences for the messages sequence : M1, M2, M3, M4, ...., M n. In these en- coded sequences biare the real bits - 1s and 0s for i-th message. Obviously 3the sequences of real bits change from message to message. Si milarly pseudo- bits are also changing from message to message. Only positio n of real and pseudo-bits are unchanged. Condition for absolute security: Shannon’s condition for a bsolute security [1] is that the probability of getting the message from ciphe r-text is same with the probability of getting the message without cipher- text. That is, eavesdropper has to guess for absolutely secure system. In o ur system, for a particular sequence of events (bits), if the probability of real events ( prealbits) becomes equal to the probability of pseudo-events ( ppseudo −bits) then eaves- dropper has to guess. As all the sequences are independent so eavesdropper has to guess for all sequences. Therefore, condition for abs olute security can be written as: 1. ppseudo −bits≥prealbits. 2.p1=p0, the probability of 1s and 0s are same. 3. All encoded sequences should be statistically i ndependent. Speed of communication: If we take ppseudo −bits=prealbits and share 100 bits, then we could communicate with half of the speed of digi tal communi- cation as long as we wish. Perhaps no cryptosystems oﬀer such speed. Though this is an extremely simple cryptosystem but it was un discovered since cryptographic power of pseudo-bits was not realized b efore. It should be mentioned that the system is purely classical and can not be u sed to achieve other quantum cryptographic tasks such as cheating free Bel l’s inequality test [19] and quantum bit commitment encoding [20]. Entire area o f quantum cryptography can not be encroached by classical cryptograp hy. References [1] Shannon, C. E. Communication theory of secrecy systems. Bell syst. Technical Jour .28, 657-715 (1949). [2] Vernam, G. S J. Amer. Inst. Electr. Engrs 45, 109-115, 1926. [3] Beker, J. and Piper, F., 1982, Cipher systems: the protec tion of com- munications (London: Northwood publications). 4[4] Hellman, E. M. The mathematics of public-key cryptograp hy.Sci. Amer. August, 1979. [5] Wiesner, S. Congugate coding, Signact News, 15, 78-88, 1983, ( The manuscript was written around 1970). [6] Bennett, C. H. & Brassard, G. Quantum cryptography: Publ ic key dis- tribution and coin tossing. In proc. of IEEE int. conf. on computers, system and signal processing 175-179 ( India, N.Y., 1984). [7] Ekert, A. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.67, 661-663 (1991). [8] Bennett, C. H. Brassard. G. & Mermin. N. D. Quantum crypto graphy without Bell’s theorem. Phys. Rev. Lett .68, 557-559 (1992). [9] Bennett, C. H. Quantum cryptography using any two nonort hogonal states. Phys. Rev. Lett .68, 3121-3124 (1992). [10] Biham, E. & Mor, T. Security of quantum cryptography aga inst collec- tive attack, Phys. Rev. Lett. 78, 2256-2259 (1997). [11] Biham, E. & Mor, T. Bounds on information and the securit y of quan- tum cryptography. Phys. Rev. Lett .79, 4034-4037 (1997). [12] Deutsch, D. et al, Quantum privacy ampliﬁcation and the security of quantum cryptography over noisy channels. Phys. Rev. Lett .77, 2818- 2821 (1996). [13] Mayers, D. Unconditional security in quantum cryptogr aphy. Preprint quant-ph/9802025. [14] Lo, -K. H. & Chau, H. F. Unconditional security of quantu m key distri- bution over arbitrarily long distance. Science .283, 2050 (1999). [15] Brassard, G., Lutkenhaus, N., Mor, T. & Sanders, C. B. Se curity aspect of practical quantum cryptography. Preprint quant-ph/991 1054. [16] Bennett, C. H. & Divincenzo. D Nature ,404, 247,2000. 5[17] Mitra. A, Complete quantum cryptography, Preprint, 5t h version, quant-ph/9812087. [18] Mitra. A, Completely secure practical cryptography. q uant-ph/ 9912074. [19] Mitra. A, Completely entangled based communication wi th security. physics/0007074. [20] Mitra. A, Unconditionally secure quantum bit commitme nt is simply possible. physics/0007089. [21] Mitra. A, Entangled vs unentangled alternative quantu m cryptography. physics/0007090. 6
arXiv:physics/0008241v1 [physics.chem-ph] 30 Aug 2000Rotation in liquid4He: Lessons from a toy model Kevin K. Lehmann Department of Chemistry, Princeton University, Princeton NJ 08544 USA (June 10, 2000) Abstract This paper presents an analysis of a model problem, consisti ng of two inter- acting rigid rings, for the rotation of molecules in liquid4He. Due to Bose symmetry, the excitation of the rotor corresponding to a rin g of N helium atoms is restricted to states with integer multiples of N qua nta of angular momentum. This minimal model shares many of the same feature s of the rotational spectra that have been observed for molecules in nanodroplets of ≈103−104helium atoms. In particular, this model predicts, for the ﬁr st time, the very large enhancement of the centrifugal distort ion constants that have been observed experimentally. It also illustrates the diﬀerent eﬀects of increasing rotational velocity by increases in angular mom entum quantum number or by increasing the rotational constant of the molec ular rotor. It is found that ﬁxed node, diﬀusion Monte Carlo and a hydrodyna mic model provide upper and lower bounds on the size of the eﬀective rot ational constant of the molecular rotor when coupled to the helium. 1The spectroscopy of atoms and molecules dissolved in helium nanodroplets is a topic of intense current interest [1–3]. One particular, almost uni que feature of this spectroscopic host is that even heavy and very anisotropic molecules and co mplexes give spectra with rotationally resolved structure [4]. This spectral struct ure typically corresponds to thermal equilibrium, with T≈0.38K, and has the same symmetry as that of the same species in th e gas phase [5,6]. The rotational constants, however, are gen erally reduced by a factor of up to four or ﬁve, while the centrifugal distortion constants a re four orders of magnitude larger than for the gas phase [7,6]. These large changes clearly reﬂ ect dynamical coupling between the molecular rotation and helium motion. At present, there are at least four diﬀerent models proposed for the increased eﬀective moments of inertia, at l east two of which have reported quantitative agreement with experiment [8–11]. The large o bserved distortion constants have not yet been quantitatively explained, and the most careful attempt to date to calculate them (for OCS in helium) gave an estimate ≈30 times smaller than the experimental value [7] The highly quantum many body dynamics of this condensed phas e system has made it diﬃcult to achieve a qualitative understanding of the obser ved eﬀects. In cases like these, simple models can provide insight, especially if the lesson s learned can be tested against more computationally demanding simulations that seek, how ever, to provide a ﬁrst principles treatment of the properties of the system of interest. In thi s paper, one very simple model system will be explored that seeks to model the coupling of a m olecular rotor to a ﬁrst solvation shell of helium. The existing models for the reduc ed rotational constants agree that most of the observed eﬀect comes from motion of helium in the ﬁrst solvation shell. Some of the qualitative features of this model were discusse d previously [7], but quanitative details were not persued in that work. The ‘toy’ model considered consists of a planar rotor couple d to a symmetric planar ring ofNhelium atoms. This model problem can be solved exactly, and c an reproduce the size of the observed reductions in the rotational constant A ND the size of the centrifugal distortion constants. This is the ﬁrst time, to the authors k nowledge, that the large eﬀective distortion constants of molecules in liquid helium has been reproduced. Further, this model 2clearly resolves a confusion about the signof the centrifugal distortion constant. Based upon the expected decreased following of the helium with inc reasing rotational angular velocity [12,13], one can argue that the rotational spacing should increase faster than for a rigid rotor, i.e., that the eﬀective centrifugal distortion constant should benegative , in conﬂict with experimental observations. The present model demonst rates, however, that opposite behavior is expected when the rotational velocity of the rot or is increased by increasing the rotational quantum number (where an increased angular anis otropy and following of the helium is predicted) or when the rotational constant of the i solated rotor is increased (where decreased angular anisotropy and following of the helium is predicted). The present model, therefore, rationalizes both the observed depenence of the increased moments of inertia on the rotational constant of the isolated molecule and the o bserved centrifugal distortion constants. I. THE TOY MODEL We will consider a highly abstracted model for rotation of a m olecule in liquid helium. The molecule will be treated as a rigid, planar rotor with mom ent of inertia I1. The ori- entation of the molecule is given by θ1The liquid helium is treated as a ring of Nhelium atoms that forms another rigid, planar rotor with moment of i nertiaI2and with orientation given byθ2. Because of the Bose symmetry of the helium, he helium rotor c an only be excited to states with N¯hunits of angular momentum. The lowest order symmetry allowe d coupling between the molecule and the helium ring is given by a potential Vcos [N(θ1−θ2)]. Any coupling spectral components that are not multiples of Nwill lead to mixing of states that are not allowed by Bose symmetry, which is forbidden in q uantum mechanics. The Hamiltonian is given by: H=−¯h2 2I1∂2 ∂θ2 1−¯h2 2I2∂2 ∂θ2 2+VcosN(θ1−θ2) (1) We deﬁneB1,2=¯h2 2I1,2, the rotational constants for the uncoupled rotors. We can s eparate the aboveHby introducing the two new coordinates: 3¯θ=I1θ1+I2θ2 I1+I2θ=θ1−θ2 (2) in which we have: H=Hr+Hv=−¯h2 2(I1+I2)∂2 ∂¯θ12+/bracketleftBigg −¯h2 2/parenleftbigg1 I1+1 I2/parenrightbigg∂2 ∂θ2+Vcos(Nθ)/bracketrightBigg (3) ¯θis the variable conjugate to the total angular momentum; θis a vibrational coordinate. We deﬁneBrigid=¯h2 2(I1+I2)andBrel=¯h2 2/parenleftBig 1 I1+1 I2/parenrightBig . The eigenstates of Hseparate into a product: ψ(¯θ,θ) =eiJ¯θψv(θ) (4) Jis the quantum number for total angular momentum. It would ap pear from the separable Hthat the energy could be written as an uncoupled sum of a rigid rotor energy, BrigidJ2( notJ(J+1) because we have a planar rotor), and a ‘vibrational’ ener gy that is independent ofJ. However, the energies are notsimply additive, due to the fact that the boundary condition for θisJ-dependent. When θ2alone is changed by any multiple of 2 π/N,ψmust be unchanged. However, a change of −2π/Ninθ2results in a change of −(2π/N)(I2/(I1+I2) in¯θand +2π/Ninθ. Thus, the Bose symmetry of the helium ring is satisﬁed by tak ing as the boundary condition for ψv: ψv/parenleftbigg θ+2π N/parenrightbigg = exp/parenleftbigg2πi NI2 I1+I2J/parenrightbigg ψv(θ) (5) As a result, the ‘vibrational energies’ and eigenfunctions are a function of the total angular momentum quantum number, J. Note that the boundary condition is periodic in J, with period given by N(I1+I2)/I2. TheJdependence of the boundary condition of the vibrational fun ction is rather unfa- miliar in molecular physics. This dependence can be removed by a Unitary transformation of the wavefunction: ψ′ v(θ) = exp/parenleftbigg −iI2J I1+I2θ/parenrightbigg ψv(θ) (6) 4The boundary condition on the transformed function is ψ′ rmv/parenleftBig θ+2π N/parenrightBig =ψ′ rmv(θ). The transformed hamiltonian, H′, is given by: H′=1 2I1(J−L)2+1 2I2L2+Vcos (Nθ) (7) whereJ=i¯h∂ ∂¯θis the operator for total angular momentum, and L=i¯h∂ ∂θis the angular momentum of the helium ring relative to a frame moving with th e rotor. This form of the Hamiltonian closely resembles those widely used in trea tment of weakly bound com- plexes [14]. Expansion of the ( J−L)2gives a Coriolis term that couples the overall rotation to the vibrational motion, which makes the nonseperability of these motions evident. This form, however, hides the periodicity in Jof this the coupling. We will now consider the two limiting cases. The ﬁrst to consi der is that where \|V\| ≫ N2Brel. In this case, the potential can be considered harmonic arou ndθ= 2πk/N,k = 0...N−1 whenVis negative (and shifted by π/NforVpositive) with harmonic frequency ν=1 2π/radicalbigg 2N2\|V\|/parenleftBig 1 I1+1 I2/parenrightBig . The wavefunction will decay to nearly zero at the maxima of the potential, and changes in the phase of the periodic bound ary condition at this point (which happens with changes in J) will not signiﬁcantly aﬀect the energy. In this limit, the total energy is E(J,v) =BrigidJ2+hν(v + 1/2), and we have a rigid rotor spectrum with eﬀective moment of inertia I1+I2. While there are Nequivalent minima, Bose symmetry assures that only one linear combination of the states local ized in each well (the totally symmetric combination for J= 0) is allowed, and thus there are no small tunneling splitti ngs, even in the high barrier limit. The total angular momentum is partitioned between the two rotors in proportion to their moments of inertia, i.e.< J1>= ¯hJ·I1/(I1+I2) and <J2>= ¯hJ·I2/(I1+I2). We will now consider the opposite, or uncoupled rotor, limit . The eigenenergies in this case are trivially E(m1,m2) =B1m2 1+B2m2 2with eigenfunctions ψ= exp (im1θ1+im2θ2). m1can be any integer, while m2=Nk, wherekis any integer. Introducing the total angular momentum quantum number J=m1+m2, we haveE(J,m 2) =B1(J−m2)2+B2m2 2. The lowest state for each Jhas quantum numbers m1=J−Nkandm2=Nk, wherekis the 5nearest integer to B1J/N(B1+B2). Treating the quantum numbers as continuous, we have E=BrigidJ2,e.g., the same as for rigid rotation of the rotors. However when we restrictJ to integer values, for J≤N(B1+B2)/(2B1), the energy spacing will be exactly that of a rigid rotor with rotational constant B1. In general, as a function of J, the uncoupled ground state solutions follow the rigid rotor spectrum in B1, but with a series of equally spaced curve crossings when the lowest energy m2value increases by NasJis increased by one quantum. These crossings allow the total energy to oscillat e around that predicted for a rigid rotor with moment of inertia I1+I2. II. NUMERICAL RESULTS Having handled the limiting cases, we can now turn our attent ion to the far more in- teresting question, which is how does the energy and eigenst ate properties change as Vis continuously varied between these limits. We note that chan ging the sign of Vis equivalent to translation of the solution by ∆ θ=π, and thus we will only consider positive values ofVexplicitly. We also note that the eigenstates are not change d, and the eigenenergies scale linearly if B1,B2, andVare multiplied by a constant factor. As a result, we will takeB2= 1 to normalize the energy scale. The solutions for ﬁnite val ues ofVwere cal- culated using the uncoupled basis and the form of Hgiven in Eq. 1 with ﬁxed values of m1+m2=Jandm2=Nk. For each value of J, the matrix representation for His a tridiagonal matrix, with diagonal elements given by the ene rgies for the uncoupled limit, and with oﬀ-diagonal elements given by V/2. Numerical calculations were done using a ﬁnite basis withk=−15,−14,...15. UsingB1=B2andN= 8, we have calculated the lowest eigenvalues of HforJ= 0,1,2 and used these, by ﬁtting to the expression E(v,J) =E0(v) +BeﬀJ2−DeﬀJ4, to determine BeﬀandDeﬀ. Figure 1 shows the value of Beﬀas a function of V(both in units of B2). It can be seen that Beﬀvaries smoothly from B1toBrigidwith increasing V, and is reaches a value half way between these limits for V≈N2B2. 6In order to rationalize this observation, we will now consid er a Quantum Hydrodynamic treatment for the rotation [15]. Let the ground state densit y beρ(θ) =\|ψv(θ)\|2. Now let the molecule classically rotate with angular velocity ω. To ﬁrst order in ω,ρwill not change (i.e. we will have adiabatic following of the helium density for cl assical inﬁnitesimal rotation of the molecule). However, the vibrational wavefunction, ψvwill no longer be real, but instead will have an angle-dependent phase factor whose gradient wi ll give a hydrodynamic velocity. Solving the equation of continuity: d dθ(ρv) =−dρ dt=ωrdρ dθ(8) whereris the radius of the helium ring, gives solutions of the form: v(θ) =ωr·/parenleftBigg 1−C ρ(θ)/parenrightBigg (9) whereCis an integration constant. We determine Cby minimizing the kinetic energy averaged over θ. This gives: C=2π /integraltext2π 0ρ−1(10) and a kinetic energy: ∆Ek=1 2I2ω2·/parenleftBigg 1−4π2 /integraltext2π 0ρ−1dθ/parenrightBigg (11) In the case of a uniform density, ρ= (2π)−1and ∆Ek= 0. As the density gets more anisotropic, the integral becomes larger and ∆ Ekbecomes larger, approaching the value for rigid rotation of the helium ring when ρhas a node in its angular range. We deﬁne the hydrodynamic contribution to the increase in the moment of inertia of the heavy rotor due to partial rotation of the light rotor by ∆ Ek=1 2∆Ihω2. It is interesting to note that for the above lowest energy value of C, we have/integraltext2π 0v(θ)dθ= 0.(i.e.that the solution is ‘irrotational’) and that the net angular momentum induced i n the helium is ∆ Ihω. The lowest energy solution of the three dimensional Quantum hyd rodynamic model satiﬁes these conditions as well [10,16]. 7The hydrodynamic model can be tested against the exact quant um solutions. Deﬁne ∆Ieﬀas the eﬀective moment of inertia for rotation (as calculate d fromBeﬀ) minus the moment of inertia for the molecular rotor. ∆ Ieﬀwill grow from 0 for uncoupled rotors toI2as the coupling approachs the rigid coupling limit of high V. In the hydrodynamic model, ∆Ieﬀ= ∆Ih. Figure 2 shows a plot that compares ∆ Ieﬀand ∆Ihas a function of V. Each has been normalized by I2. They are found to be in qualitative agreement for the full range ofV, though the exact quantum solution is systematically below the hydrodynamic prediction. We note, however, that for the assumed paramete rs, the speed of the molecular rotor is equal to that of the helium rotor, while the hydrodyn amic treatment assumed a classical, inﬁnitesimal rotation of the molecular rotor. The size of ∆ Ieﬀis determined by the degree of anisotropy of the ground state density in the vibrational displacement coordinate θ. IfI1is decreased at ﬁxed I2andV, the eﬀective mass forθ, which is ( I−1 1+I−1 2)−1will also decrease, which will decrease the anisotropy produced by V. Fig 3 shows how the normalized ∆ Ieﬀand ∆Ihvary as the molecular rotational constant, B1, changes from 0 to 2 B2. This calculation was done for V= 100, close to the value corresponding to maximum diﬀerence of ∆ Ieﬀand ∆IhforB1=B2. This plot demonstrates that the hydrodynamic prediction be comes exact in the limit that B1→0,i.e., in the case that the assumption of inﬁnitesimal rotational velocity of the molecule holds. However, it substantially overestimates t he increase eﬀective moment of inertia when B1≥B2. This decrease in the increase moment of inertia with increa sing rotational constant of the heavy rotor is the eﬀect previous ly interpreted as the breakdown of adiabatic following in the literature on the rotational s pectrum of molecules in liquid Helium [12,10,13]. Figure 4 shows a plot of Deﬀas a function of VforB1=B2= 1.Deﬀ= 0 is zero in both limits, and has a maximum value near the value of Vat whichBeﬀis changing most rapidly. It is interesting to explicitly point out that this Deﬀvalue arises entirely from changes in the angular anisotropy of the helium density with J, as the model does not allow for an increase in the radial distance of the helium, which has previously be en considered [7]. Further, the 8peak value of Deﬀ≈1.8·10−3B1is in remarkably good agreement with the ratio of Deﬀ to the gas phase molecular rotational constant observed for a number of molecules in liquid helium. For example, for OCS this ratio is found to be 2 ·10−3[7], while for HCCCN, the same ratio was found to be 1 ·10−3[17]. We can gain further insight by examining the rotational ener gy systematically as a func- tion ofJ. Figure 5 shows the rotational excitation energy ( E(0,J)−E(0,0)) divided by J2as a function of J. The calculations were done with V= 100. The rotational excitation energy approaches that of the BrigidJ2for highJ. Further, it reaches this value for Jequal to multiples of N(I1+I2)/I2, which matches the periodicity of the boundary conditions f or ψv.Jvalues that lead to the same boundary conditions for ψvwill diﬀer in energy only by the eigenvalues of Hr, and thus it follows from Eq. 2, that of a rigid rotor with rota tional constantBrigid. For the ﬁrst half of each period in J,ψvis found to increase in its anisotropy, and therefore the energy increases, as Jis increased (See Fig. 6). This can be understood when one considers the fact that for J=N(I1+I2)/(2I2), the boundary condition is that ψv(2π/N) =−ψv(0), i.e. the wavefunction will be real but have Nnodes in the interval [0,2π]. Classically, the molecular rotor is characterized by its ro tational angular velocity, ω= 2B1J. However, we see that the quantum treatment of the two couple d rotors gives opposite results when ωis increased by increasing either B1orJ. For increases in B1, the ‘degree of following’ of the light rotor decreases for ﬁxed potentia l coupling, as seems intuitively reasonable. However, for increases in J, the anisotropy of the potential and thus the ‘degree of following’ initially increases, and thus so does the eﬀec tive moment of inertia of the coupled system. This behavior continues until one passes through a r esonance condition where the helium can be excited by transfer of Nquantum of angular momentum from molecular rotor to the helium. This resonance condition is missing fro m the classical treatment of the coupling between the rotors, where the angular velocity of t he molecular rotor is treated as a ﬁxed quantity, ω, which is one of the parameters of the problem. 9III. NODAL PROPERTIES OF SOLUTIONS It is possible to calculate the rotational excitation energ ies of clusters of helium around a molecule by use of the Fixed Frame Diﬀusion Monte Carlo (FFD MC) method [12]. As in most DMC methods, this method should yield (except for sta tistical ﬂuctuations) a upper bound on the true energy, ﬁnding the optimal wavefunct ion consistent with the nodal properties that are imposed on the wavefunction by construc tion. In the case of FFDMC, the nodal planes are determined by the free rotor rotational wavefunction for the molecule alone, i.e. that the sign of the wavefunction (which is taken to be real) for any point in conﬁguration space is the same as that of the rotor wavefunct ion at the same Euler angles. We can examine the exact solutions of our toy problem to gain i nsight into the accuracy of the nodal planes assumed in FFDMC. The wavefunctions we ha ve considered up to now are complex, but because of time reversal symmetry, the solu tions withJand−Jrotational quantum numbers must be degenerate. Symmetric combination of these solutions just gives the Real part of Jsolution, and the antisymmetric combination the Imaginary part. The real part is given by: ψR J(θ1,θ2) = cos(Jθ1)Re(ψ′(θ2−θ1))−sin(Jθ2)Im(ψ′(θ2−θ1)) (12) where ψ′(∆θ) =/summationdisplay kcJ kexp(ikN∆θ) (13) andcJ kare the eigenvector coeﬃcients obtained from diagonalizat ion of the real Hamiltonian matrix in the uncoupled basis. Examination of the numerical solutions reveals that for J < N ,Re(ψ′) has no nodes, while Im(ψ′) has nodes at ∆ θequal to integer multiples of π/N. Thus, ifIm(ψ′) = 0, then the solution would satisfy the FFDMC nodal propert ies exactly. However, for ﬁnite Im(ψ′), the nodal surfaces, rather than being on the planes θ1=constant, are modulated N times per cycle along the θ1=constant line. For V= 80 andB1=B2= 1, the maximum value of Im(ψ′) is about 4% of Re(ψ′), and growing approximately linearly for low J. 10In order to test the quantiative implications of this error i n the nodal properties, impor- tance sampled DMC calculations have been done for the presen t two rotor problem. The explicit DMC algorithm given by Reynolds et al.[18] was used with minor change [19]. The guiding function, ψT, which determines the nodes, was selected as cos( Jθ1)ψ0 v(θ1−θ2), where ψ0 vis the real, positive deﬁnite eigenstate for the J= 0 problem. The rotational constant, BDMCis deﬁned as the DMC estimated energy for J= 1 less the exact ground state eigenen- ergy forJ= 0, and will be (except for sampling and ﬁnite time step bias) an upper bound on the true Bvalue calculated earlier. The points plotted in ﬁgure 1 are t he calculated values ofBDMCwith the estimated 2 σerror estimates. It is seen that the ﬁxed node DMC estimates of Beﬀare excellent for low values of V, but underestimate the contribution of the Helium ring to the eﬀective moment of inertia as it is coupled more strongly to the rotor. IV. RELATIONSHIP WITH A MORE REALISTIC MODEL In a series of insightful lectures, Anthony Leggett analyze d the properties of the ground state ofNHelium atoms conﬁned to an annulus of radius Rand spacing d≪R[15]. The walls of the annulus are allowed to classically rotate wi th angular velocity ω. While not stated explicitly, the walls of the annulus couple to the helium via a time dependent potential, which is static in the rotating frame. As such, ou r rotating diatomic molecule can be considered as a special case of the problem treated by L eggett. If one transforms to the rotating frame, the quantum hamiltonian is the same as fo r the static ( ω= 0) problem. However, the boundary condition for the wavefunction in thi s frame is given by [15][Eq. (2.10)]: Ψ′ 0(θ1,θ2...θ j+ 2π...θ N) = exp/parenleftBig 2πimR2ω/¯h/parenrightBig Ψ′ 0(θ1,θ2...θ j...θ N) (14) In making a comparison to the results of the toy model, we note that for this system I2= NmR2(the classical moment of inertia for the helium) and J=ω(I1+I2)/¯h. Substitution shows that the phase factor in Eq. 14 is identical to that deri ved above for Eq. 5. Note, 11however, that Eq. 14 refers to moving one helium atom by 2 π, while Eq. 5 refers to motion of allNhelium atoms by 2 π/N. Motion of all Nhelium atoms by 2 πwill result in a phase factor of 2πiNmR2ω/¯h= 2πiI2J/(I1+I2) in both treatments. Leggett considered the change in helium energy produced by r otation of the walls. Let E0be the ground state energy for the static problem, and E′ 0(ω) the ground state energy in the rotating frame. The ground state energy in the laborat ory frame is given by [15][Eq. (2.12)]: Elab=E0+1 2I2ω2−[E′ 0(ω)−E0] (15) For the ground state of Bosons, we further have that E′ 0(ω)≥E0, with equality only when ωequals integer multiples of ω0= ¯h/mR2since the nodeless state has the lowest possible energy. At ω=kω0, the helium rigidly rotates with the walls. This agrees exac tly with the numerical results of the toy model, as shown in Figure 5. I n making comparisons with this model, one should remember that Elabdoes not include the kinetic energy of the walls (rotor). Thus the more general treatment of Leggett support s one of the central insights of the toy model, that the large eﬀective distortion constants for molecular rotors in helium is a consequence of an increased helium following of the rotor with increasing angular veloc ity, which in turn is a direct consequence of the ωdependence of the single-valuedness boundary condition in the rotating frame. The moment of inertia for the ground state of the helium can be deﬁned by: I=/parenleftBiggd2Elab dω2/parenrightBigg ω→0=I2−/parenleftBiggd2E′ 0(ω) dω2/parenrightBigg ω→0(16) Leggett deﬁned the “normal fraction” of the helium by the rat ioI/I2, which is equal to unity ifE′ 0(ω) is independent of ωasω→0. This will occur if the wavefunction has ‘nontrivial’ nodal planes, since the phase of the wavefunction can be chan ged discontinuously at a node without cost of energy. Nodal plans associated with ove rlap of particles, however, are ‘trivial’ in that the phase relationship on each side of the n ode is determined by the exchange symmetry of the wavefunction, and thus cannot be used to matc h the boundary conditions 12without extra cost of energy. In our toy problem, when Vis very large, the vibrational wavefunction becomes localized, introducing near nodes at the maxima of the potential, and as a result the ground state is described by a near unity norma l fraction; we have what Leggett refers to as a ‘normal solid’. Conversely, as the unc oupled limit is approached, the helium ring does not contribute to the kinetic energy of the l owest rotational states and we haveI→0, and we have zero normal fraction (i.e. the helium has unity superﬂuid fraction). Following Leggett’s deﬁnition, one ﬁnds that th e normal fraction is given by ∆Ieﬀ/I2. Thus, Figure 2 can thus be interpreted as the normal ﬂuid fra ction for the ground state as a function of the strength of the potential coupling . Leggett’s analysis is based upon a classical treatment of the motion of the walls, which impli esI1≫I2, in which limit the hydrodynamic model exactly predicts the normal ﬂuid fracti on. V. ACKNOWLEDGEMENT This work was supported by the National Science Foundation a nd the Air Force Oﬃce of Scientiﬁc Research. 13REFERENCES [1] J. P. Toennies and A. F. Vilesov, Annual Reviews of Physic al Chemistry 49, 1 (1998). [2] S. Grebenev et al., Physica B 280, 65 (2000). [3] K. K. Lehmann and G. Scoles, Science 279, 2065 (1998). [4] M. Hartmann, R. E. Miller, J. P. Toennies, and A. F. Vileso v, Science 272, 1631 (1996). [5] M. Hartmann, R. E. Miller, J. P. Toennies, and A. F. Vileso v, Physical Review Letters 95, 1566 (1995). [6] C. Callegari et al. (unpublished). [7] S. Grebenev et al., Journal of Chemical Physics 112, 4485 (2000). [8] S. Grebenev, J. P. Toennies, and A. F. Vilesov, Science 279, 2083 (1998). [9] Y. Kwon and K. B. Whaley, Physical Review Letters 83, 4108 (1999). [10] C. Callegari et al., Physical Review Letters 83, 5058 (1999). [11] E. B. Gordon and A. F. Shestakov, The Cluster Approach to Description of Atoms and Molecules Isolated by Helium, paper presented at the IV. Wor kshop on quantum Fluid Clusters, Ringberg Schloss, June, 2000. [12] E. Lee, D. Farrelly, and K. B. Whaley, Physical Review Le tters83, 3812 (1999). [13] A. Conjusteau et al., Journal of Chemical Physics 113, (2000), to be published. [14] J. M. Hutson, in Advances in Molecular Vibrations and Collision Dynamics , edited by J. M. Bowman and M. A. Ratner (JAI Press Inc., Greenwich, Co nnecticut, 1991), Vol. 1A, pp. 1–45. [15] A. J. Leggett, Physica Fennica 8, 125 (1973). [16] C. Callegari et al., Physical Review Letters 43, 1848 (2000). 14[17] C. Callegari et al. (unpublished). [18] P. J. Reynolds, D. M. Ceperley, B. J. Alder, and J. Willia m A. Lester, Journal of Chemical Physics 77, 5593 (1982). [19] It was found that when Walkers moved very close to a node, they became trapped, due to the large size of the attempted steps introduced by the dri ft term combined with the ‘detailed balance’ correction selection criteria. When th ey also had large negative values of the local energy, these trapped walkers then grew in weigh t, leading to unphysical large negative values for the DMC estimate for the energy. Th is problem, which is part of the ﬁnite time step bias, was eliminated by killing walkers t hat failed the detail balance selection instead of keeping them at their previous locatio n. 15FIGURES 0.50.60.70.80.91.0 0 100 200 300 400 500Effective Rotational constant for two rotors. B 1 = B 2 = 1., N=8.B eff VB 1 B rigid FIG. 1. Eﬀective Rotational Constant for two coupled rotors as a function of the interaction potential strength. B1=B2= 1, and the second rotor can only be excited to states with mul tiples of 8 quanta. The individual points are the Beﬀective values calclated by a ﬁxed node, Diﬀusion Monte Carlo calculation. The error bars on these points are t he estimated 2 σsampling error. 160.00000.20000.40000.60000.80001.0000 0 100 200 300 400 500Change in Effective moment of inertia for two rotors. B 1=B 2 = 1, N=8∆I / ∆I rigid V∆I eff∆I hydro FIG. 2. Increase in eﬀective moment of inertia of molecule, ∆ eﬀdue to coupling to rotor made of 8 helium atoms. ∆ Ihydrois the same quantity estimated by the hydrodynamic model. Bo th are calculated as a function of the potential coupling strength , and the results normalized to the rigid rotor moment of inertia of the 8 helium rotor 170.10.20.30.40.50.60.70.80.9 0 0.5 1 1.5 2Change in moments of inertia with molecular rotational constant∆I eff / ∆I rigid B 1∆I hydro ∆I eff FIG. 3. Same as Figure 2, except as a function of the rotationa l constant of the molecule, normalized to the rotational constant of the 8 helium rotor. 180.00000.00050.00100.00150.0020 0 100 200 300 400 500Effective Distortion Constant for two rotors. B 1 = B 2 = 1, N = 8D eff V FIG. 4. The eﬀective centrifugal distortion constant, Deﬀ, for molecule coupled to ring of 8 helium atoms as a function of the strength of the coupling, V. Both DeﬀandVare normalized to the rotational constants of the molecule and 8 helium rotor, which are taken as equal. 190.450.500.550.600.650.700.750.80 0 20 40 60 80 100Rotational Excitation Energy / J2 for two rotors. B 1 = B 2 = 1, N = 8 ∆E / J2 JB rigid FIG. 5. The rotational excitation energy, ∆ E, divided by J2as a function of the total rotational angular momentum quantum number, J. Calculated with B1=B2= 1 and V= 100. With rigid following of the helium, the plotted quantity should equal Brigid, which is indicated in the ﬁgure. 200.000.100.200.300.400.500.600.70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Changes in wavefunction with rotational excitation\| ψ(θ) \| θJ = 0 J = 8 FIG. 6. The absolute value of the vibrational wavefunction a s a function of the relative ori- entation between molecule and 8 helium rotor. This quantity is plotted for angular momentum J= 0,1...8. 21
arXiv:physics/0008242 30 Aug 2000STATUS OF THE RFD LINAC STRUCTURE DEVELOPMENT 1 D.A. Swenson, Linac Systems, 1208 Marigold Dr. NE, Albuquerque, NM 87122 1 Work supported by the National Institute of Mental Health (NIMH) and the National Cancer Institute (NCI).Abstract The Proof-of-Principle (POP) prototype of the Rf- Focused Drift tube (RFD) linac structure is currently under test at Linac Systems, after years of delay due to a variety of technical problems. A discussion of these technical problems and their solutions will be presented. The status of these tests will be reported. Plans for future development of this linac structure will be revealed. Potential uses of this linac structure for a variety of scientific, medical, and industrial applications will be described, including: proton linac injectors for proton synchrotrons, compact proton linacs for PET isotope production, epithermal neutron sources for the BNCT application, energy boosters for proton therapy, compact portable neutron sources for thermal neutron radiography, and pulsed cold neutron sources for cold neutron physics and related applications. 1 STATUS OF THE POP PROTOTYPE The POP prototype[4,6,9] came into operation on June 19, 2000 in the Linac Systems laboratory in Waxahachie, TX. We had about 9 mA of 25-kV beam entering the RFQ and about 9 mA of beam transmitted through the RFQ and RFD linac structures. That beam impinged on a 1.25-MeV-thick absorber foil and Faraday cup assembly. The beam that passed through the absorber foil and into the Faraday cup (the accelerated beam) showed the expected threshold for both rf fields and injection energy. Because we suffer from an inadequate amount of rf power (180 kW) and are operating very close to the linac excitation threshold, the accelerated beam corresponded to only about 4% of the transmitted beam. Nevertheless, we now have conditions that produce a steady 0.3-mA beam of protons at 1.636-MeV for hours at a time. The initial operation was sporadic as a result of several problems (in addition to the near threshold operation), which have now been rectified. We have always had a “breathing” phenomenon, which we attributed to some mechanical vibrations in the drift tube stems. A description of the solution to this problem is given below. We often witness a 10% decline in RFQ fields after 15 seconds of operation. A description of the solution to this problem is also given below. We are confident that, with adequate rf power and the improvements that are identified below, the performance of the RFD linac structure will come up to our expectations.2 BRIEF DESCRIPTION OF THE RFD LINAC STRUCTURE The RFD linac structure[1-4] resembles a drift tube linac (DTL) with radio frequency quadrupole (RFQ) focusing incorporated into each drift tube. The RFD drift tubes comprise two separate electrodes, operating at different electrical potentials as excited by the TM010 rf fields, each supporting two fingers pointing inwards towards the opposite end of the drift tube forming a four-finger geometry that produces an rf quadrupole field along the axis. The particles traveling along the axis traverse two distinct regions, namely, the gaps between the drift tubes where the acceleration takes place, and the regions inside the drift tubes where the rf quadrupole focusing takes place. This new structure could become the structure of choice to follow RFQ linacs in many scientific, medical, and industrial applications. 3 TECHNICAL PROBLEMS AND SOLUTIONS Successful operation of the POP was delayed for more than a year by a host of minor problems, each of which have now been overcome. Descriptions of these problems and the solutions that were employed are presented below: RFQ Alignment: The RFQ is assembled from four pieces of machined copper, namely two major pieces (top and bottom) and two minor pieces (sides), each representing one vane of the four-vane structure. The mechanical precision of these pieces was not what it should or could have been. The resulting assembly was difficult to excite in the quadrupole mode. Measuring the actual geometry and placing shims between pieces allowed the quadrupole mode to be excited. Four-rod, dipole-mode detuners, of the type developed at Los Alamos, were incorporated in the upstream end of the structure for additional stability. In the future, a tooling fixture will be used to support the major and minor pieces during machining to achieve the desired precision. RF Power Tubes: The rf power system employed fifteen YU-141 Planar Triodes (PSI/Eimac); one in the intermediate power amplifier, IPA-1, two in IPA-2 and 12 in the final power amplifier (FPA). We started with a full complement of tubes, but quickly suffered 6 or 7 partial failures, making it impossible to reach the rated power of the system (240 kW at 600 MHz). Because of fabrication problems at Eimac, it took more than a yearto get back to a full complement of good tubes and the rated power output. Multipactoring: The twin bladed RFD stem geometry is prone to multipactor. Initially, we were unable to break through the multipactor barrier. In that situation, it is impossible to optimize the rf power system and coaxial drive line. Our first attempt to overcome the multipactoring was to coat the RFD drift tubes and stems with carbon black from an acetylene torch. This worked and allowed us to optimize the rf power system. Next, we decided to try a cleaner and more robust cure for multipactoring, namely vapor deposition of a thin layer of titanium. This also worked and allowed us to excite the RFD structure to high field levels. Some changes in the RFD stem geometry, to reduce the area of parallel surfaces, should reduce the potential for multipactor. Cavity Q: At this point, it was obvious that the Q of the system was less than expected, resulting in a power requirement that was beyond the capabilities of the rf power system. We launched a search for the cause of the depressed Q. Several causes were under suspicion, namely the quality of the copper plating, the spring-ring joints to the end walls, the end walls, and the drift tube stems. Because of the very short tank (350 mm), the two end walls and associated rf joints have an unusually large negative effect on the cavity Q. Because of the very low injection energy (0.8 MeV), the number of drift tubes and associated stems for the short tank are unusually large, which has a negative effect on the cavity Q. In order to separate the possible causes of the depressed Q, we removed the 12 drift tubes and measured the Q of TM010 and TE111 modes in the empty tank. The former, which is effected by bo th the conductivity of the copper and the end seals, was only 38% of theoretical. The latter, which is not effected by end seal conductivity, was 68% of theoretical. We took this to mean that both the copper plating and the end seals were part of the problem. End Seals: We decided to re-machine the tank and end walls to replace the spring ring and o-ring seals with Helicoflex copper seals, providing both rf and vacuum seals. One end of the tank was then modified to employ a custom flexed-fin rf joint backed by an o-ring vacuum seal. Future designs will employ that configuration. Copper Plating: We decided to have the RFD tank and end plates stripped of their original copper plating and re-plated by a copper plating company that had done work for other proton linac projects. In the end, though, the cavity Q is still only 60% of what it should be. Stem Power: A closer analysis of the RFD stem losses revealed that the stem losses were unnecessarily high and that slight changes in the geometry would reduce the stem losses by 50%. This observation, of course, was too late for the POP and we had to live with the higher stem losses. In the future, the modified stem geometry will be used.Rf Drive Line: In a short rf drive line (approximately one wavelength) with no circulator, the length of the line has a large effect on the response of the rf system to the reflected power associated with cavity filling and cavity arcs. To provide some degree of adjustment, we installed and additional length of 3-1/8” coaxial line fitted with a sliding stub tuner. This helped to determine the optimum rf drive line length for the system. Breathing: The RFD tank was quite sensitive to mechanical impact. Throughout this work, we were plagued with a “breathing” phenomenon, which we assumed was due to mechanical vibrations of the drift tube stems, which in turn caused a periodic oscillation in the resonant frequency and rf field levels in the structure. The driving force for this vibration appeared to be rf field, duty factor, and pulse rate dependent. This mechanical vibration was in the vicinity of 20 Hz. Variation of the pulse repetition rate in the vicinity of 20 Hz, caused the observable effects to vary significantly. Recently, we cured this effect by placing an insulating spacer between the two blades of the RFD stems at about 2/3 the distance from the outer wall to the drift tube. We plan to braze a ceramic spacer, at about that location, in all future RFD drift tubes. Fading: We have noticed a “fading” phenomenon in the RFQ fields. We can get more fields in the RFQ after it has been off for several minutes. This field level fades by about 10% in the next 15 seconds. Recently, we have determined that this is due to a thermal distortion of the resonant coupler that couples the RFD fields to the RFQ. This effect has been mitigated by optimizing the tuning of the resonant coupler during operation. In future designs, the resonant coupler will be stiffer and more intimately coupled to the cooled structures, thus eliminating this thermal distortion effect. We have learned a lot in the process of solving these problems. These technical problems can be avoided in the future. 4 MODIFICATIONS TO REDUCE THE RF POWER REQUIREMENT Even with solutions to all of these problems, the rf system could not produce enough po wer to excite the structure to the threshold for proton acceleration. At this point, it would have been expensive to increase the power of the rf system. Instead, we searched for ways to reduce the power requirement of the linac structure. First, we modified the RFD tank to reduce the design gradient from 7.72 MV/m to 5.90 MV/m. Twelve cells at this reduced gradient resulted in a shorter tank (by 22.86 mm) with reduced drift tube spacing. This required drilling 12 new holes in the keel of the tank for mounting the drift tubes. In order to drill these new holes in virgin metal, the longitudinal plane of the drift tube stems (and RFD lenses) had to be rotated 7.2 /G81 about the axis of the tank. This modification reduced the beamenergy from 2.50 MeV to 2.00 MeV and the required rf power by 40%. Unfortunately, this did not reduce the required rf power enough to allow operation with the power available from the rf system Next, we modified the RFD tank to reduce the number of drift tubes from 12 to 9. This shortened the tank by 93.48 mm and reduced the required power by another 19%. This reduced the beam energy of the structure from 2.00 MeV to 1.636 MeV. It was in this configuration that we saw the first accelerated beam. In summary, the energy and intensity of the beam from the POP was less than expected because of the changes that we had to make to fit within the available rf power. Several design and fabrication flaws raised the rf power requirement above the original estimate, namely, the Q of the cavity (18,600) never got to where it should have been (30,000), the RFD stem design was not optimum, tank modifications left 12 additional drift tube mounting holes to be plugged, the very short tank (less than one diameter) accentuated the end wall and end joint losses, and the very low average beam energy accentuated the drift tube stem losses. Future RFD linac designs will employ a number of improvements that will rectify these problems. 5 POTENTIAL USES FOR THE RFD LINAC STRUCTURE We expect the RFD linac structure to form the basis of a new family of compact, economical, and reliable linac systems serving a whole host of scientific, medical, and industrial applications. The principal medical applications include the production of short-lived radio- isotopes for the positron-based diagnostic procedures (PET and SPECT), the production of epithermal neutron beams for BNCT, and accelerated proton beams for injection into proton synchrotrons to produce the energies required for proton therapy. We also propose an S-Band version of the structure to serve as the 10-70- MeV portion of a 200-MeV booster linac for the proton therapy applications. A modest scientific application includes the production of pulsed cold neutrons for cold neutron physics and related applications[8]. The principal industrial and military applications include the production of intense thermal neutron beams for Thermal Neutron Analysis (TNA), Thermal Neutron Radiography (TNR), and Nondestructive Testing (NDT). High duty factor RFD linac systems could produce nanosecond bursts of fast neutrons to support Pulsed Fast Neutron Analysis (PFNA). 6 PLANS FOR CONTINUED DEVELOPMENT OF THE RFD LINAC Further development of RFD-based linac systems is dependent on further financial support or development contracts. We have mature designs for all componentsof two different linac systems, which address two different medical applications, namely isotope production for the PET application, and neutron production for the BNCT application. These systems, with minor modification, could be used to satisfy other scientific, medical, and industrial applications. The PET unit[4], for example, is based on a compact, 12-MeV proton linac with an peak proton beam current of 10 mA and an average proton beam current of 120 µA. This unit, with minor modifications, could be used for production of other isotopes, as injectors for proton synchrotrons, and as injectors for high intensity linear accelerator for energy or materials related applications. The BNCT unit[5,7], on the other hand, is based on a compact 2.5-MeV proton linac with a peak and average proton beam current of 10 mA. This unit, with minor modifications, could be used for thermal neutron analysis (TNA), neutron activation analysis (NAA), non- destructive testing (NDT), thermal neutron radiography (TNR), explosive detection, and gem irradiation. The basic principles of the RFD linac structure are now proven, our designs are mature, and we are ready to accept contracts for the development of RFD-based linac systems for practical applications. 7 ACKNOWLEDGEMENTS The people who played a significant role in the development of the RFD linac structure and the POP prototype are: Frank Guy and Ken Crandall (accelerator physics), Joel Starling (mechanical engineering and commissioning), Jim Potter (rf power), John Lenz (thermal calculations), and Sylvia Revell (radiation safety). Jerry Duggan (Univ. of North Texas) facilitated the titanium coating of the drift tube stems. REFERENCES 1. D.A. Swenson, “RF-Focused Drift-Tube Linac Structure”, LINAC’94, Tsukuba, 1994. 2. D.A. Swenson, Crandall, Guy, Lenz, Ringwall, & Walling, “Development of the RFD Linac Structure”, PAC’95, Dallas, 1995. 3. D.A. Swenson, F.W. Guy, K.R. Crandall, “Merits of the RFD Linac Structure for Proton and Light-Ion Acceleration Systems”, EPAC’96, Sitges, 1996. 4. D.A. Swenson, K.R. Crandall, F.W. Guy, J.M. Potter, T.A. Topolski, “Prototype of the RFD Linac Structure”, LINAC’96, CERN, Geneva, 1996. 5. D.A. Swenson, “CW RFD Linacs for the BNCT Application”, CAARI’96, Denton, 1996. 6. D.A. Swenson, K.R. Crandall, F.W. Guy, J.W. Lenz, W.J. Starling, “First Performance of the RFD Linac Structure”, LINAC’98, Chicago, 1998. 7. D.A. Swenson, “Compact, Inexpensive, Epithermal Neutron Source for BNCT”, CAARI’98, Den., 1998. 8. R.C. Lanza, “Small Accelerator-Based Pulsed Cold Neutron Sources”, CAARI’98, Denton, 1998. 9. D.A. Swenson, F.W. Guy, and W.J. Starling, “Commissioning the 2.5-MeV RFD Linac Prototype”, PAC’99, New York, 1999.
arXiv:physics/0008243v1 [physics.acc-ph] 30 Aug 2000SLAC–PUB–8589 August 2000 Simulation of the Beam-Beam Eﬀects in e+e−Storage Rings with a Method of Reducing the Region of Mesh∗ Yunhai Cai, Alex W. Chao and Stephan I. Tzenov Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Toshi Tajima University of Texas at Austin, Austin, TX 78712 and Lawrence Livermore National Laboratory, Livermore, CA 94551 Abstract A highly accurate self-consistent particle code to simulat e the beam-beam col- lision in e+e−storage rings has been developed. It adopts a method of solvi ng the Poisson equation with an open boundary. The method consi sts of two steps: assigning the potential on a ﬁnite boundary using the Green’ s function, and then solving the potential inside the boundary with a fast Poisso n solver. Since the solution of the Poisson’s equation is unique, our solution i s exactly the same as the one obtained by simply using the Green’s function. The me thod allows us to select much smaller region of mesh and therefore increase th e resolution of the solver. The better resolution makes more accurate the calcu lation of the dynam- ics in the core of the beams. The luminosity simulated with th is method agrees quantitatively with the measurement for the PEP-II B-facto ry ring in the linear and nonlinear beam current regimes, demonstrating its pred ictive capability in detail. Submitted to Physical Review Special Topics: Accelerators and Beams ∗Work supported by Department of Energy contract DE–AC03–76 SF00515.1 Introduction The beam-beam interaction is one of the most important limit ing factors determining the luminosity of storage colliders. It has been studied extens ively by theoretical analysis [1], experimental measurements [2], and computer simulations [ 3]. Historically, due to the com- plexity of the interaction, many approximations, such as st rong-weak [4] or soft-Gaussian [5], have been introduced in order to simulate the interaction in a reasonable computing time. The self-consistent simulation of the beam-beam interacti on by solving the Poisson equation with a boundary condition has been proposed ﬁrst to investig ate the round beams [6] and then the ﬂat beams [7]. To enhance the accuracy and to reduce t he computational over- head, an algorithm (and a code) of the so-called δfmethod that can handle strong-strong interactions has been introduced [8]. Another self-consis tent approach to the beam-beam interaction is to use the Green’s function directly [9] or in directly [10]. In the present paper we will develop a method that takes advan tage from both self- consistent approaches: a smaller region of mesh from the met hod of using the Green’s func- tion and a faster solver for the interior. In order to develop a highly accurate predictive code at the luminosity saturation region, it is necessary to have a fully self-consistent treatment of ﬁeld-particle interaction at collision. Since we are inter ested in simulating the Asymmetric e+e−Storage Collider PEP-II [11], which needs to maximize the lu minosity and thus the beam current, it is even more crucial that the beam-beam inte raction in the large current regime be treated accurately. In a self-consistent simulation of the beam-beam interacti on in storage rings, the beam distributions have to be evolved dynamically during collis ion with the opposing beam to- gether with the propagation in the rings. During collision, the beam distributions are used at each time sequence to compute the force that acts on the opp osing beam. Since positrons and electrons are ultra-relativistic part icles in high energy storage rings, the beam-beam force is transverse and acts only on the opposi ng beam. Hence, given a beam distribution, we can divide the distribution longitud inally into several slices and then solve for the two-dimensional force for each slice. Self-co nsistency is achieved by introducing many-body particles in the ﬁeld that in turn constitute char ge-current, the strategy of the particle-in-cell (PIC) procedure (for example, Ref. [12]) . In this paper, for simplicity, we use only a single longitudinal slice for a bunch, ignoring any be am-beam eﬀects encompassing over the length of the bunch. 2 Method In modern colliders, beams are focused strongly at the inter action point to achieve high luminosity. As a result the transverse dimension of the beam is much smaller than the dimension of the beam pipe at the collision point. Therefore , the open boundary condition is a good approximation for calculating the transverse beam -beam force. 22.1 Green’s Function Given a charge density ρc(x, y), which is normalized to the total charge /integraldisplay dxdyρ c(x, y) =Ne, (2.1) where Nis the total number of particles, the electric potential φ(x, y) satisﬁes the Poisson equation /parenleftigg∂2 ∂x2+∂2 ∂y2/parenrightigg φ(x, y) =−2πρc(x, y) (2.2) withxandybeing the transverse coordinates. The solution of the Poiss on equation can be expressed as φ(x, y) =/integraldisplay dx′dy′G(x−x′, y−y′)ρc(x′, y′), (2.3) where Gis the Green’s function which satisﬁes the equation /parenleftigg∂2 ∂x2+∂2 ∂y2/parenrightigg G(x−x′, y−y′) =−2πδ(x−x′)δ(y−y′). (2.4) In the case of open boundary condition, namely the boundary i s far away so that its con- tribution to the potential can be ignored, one has the well-k nown explicit solution for the Green’s function: G(x−x′, y−y′) =−1 2ln/bracketleftig (x−x′)2+ (y−y′)2/bracketrightig . (2.5) This explicit solution can be used directly to compute the po tential. The main problem of this approach is that it is slow to calculate the logarithm an d the number of computations is proportional to the square of the number of macro particle sN2 p. One can reduce Npby introducing a two-dimensional mesh to smooth out the charge distribution [9]. Or to further improve the computing speed, one can map the solution onto th e space of spectrum by the Fast Fourier Transformation (FFT) and then calculate the po tential [10]. 2.2 Reduce the Region of Mesh Another alternative approach is to solve the Poisson equati on with a boundary condition [7], because the region (20 µm×450µm for PEP-II) occupied by the beam is much smaller than the boundary deﬁned by the beam pipe (2 cm radius) at the c ollision point. In order to achieve required resolution, a few mesh points per σof the beam are needed, otherwise the size of mesh is too large for numerical computation. However, it is unnecessary to cover the entire area with mesh inside the beam pipe since the area is mostly empty. We choose a smaller and ﬁnite area of the mesh, which is large enough to cover the whole beam, and by carefully selecting th e potential on the boundary, we can obtain the accurate solution inside the boundary. 3We denote by φ1the solution (2.3) of the Poisson equation. Let φ2be the solution obtained by solving the Poisson equation in a two-dimension al area Swith the potential prescribed on a closed one-dimensional Lbounding the area S φ2(x, y) =/integraldisplay Sdx′dy′G(x−x′, y−y′)ρc(x′, y′), (2.6) where ( x, y)∈L. By deﬁnition, we have φ1=φ2on the boundary L. Let U=φ1−φ2and use the ﬁrst identity of Green’s theorem [13] in two dimensio ns /integraldisplay S/bracketleftig U∇2U+ (∇U)2/bracketrightig dxdy=/contintegraldisplay LU∂U ∂ndl, (2.7) where dlis a line element of Lwith a unit outward normal n. Since U= 0 on Land∇2U= 0 inside L, we have /integraldisplay S(∇U)2dxdy= 0, (2.8) implying that Uis a constant inside L. We can set U= 0, which is consistent with the value on the boundary. Hence φ1=φ2. The two solutions are identical. 3 Field Solver We adopt the PIC technique to calculate the ﬁelds induced by t he charge (and current) of the beams self-consistently. The charge distribution of a beam is represented by macro particles. These macro particles are treated as single elec tron or positron dynamically. In order to compute the ﬁeld acting on the particles of the oppos ing beam, we ﬁrst deposit their charges onto the gird points of a two-dimensional rect angular mesh. We denote by Hx the horizontal distance between two adjacent grid points an d byHythe distance in vertical direction. 3.1 Charge Assignment We choose the method of the triangular-shaped cloud [15] as o ur scheme for the charge assignment onto the grid. On a two-dimensional grid, associ ated with each macro particle, nine nearest points are assigned with non-vanishing weight s as illustrated in Fig. 1. We use “0” to denote the ﬁrst, “+” as the second, and “-” as the third n earest lines. The weights are quadratic polynomials of the fractional dis tance, rx=δx/H x, to the nearest line w0 x=3 4−r2 x, w+ x=1 2/parenleftbigg1 4+rx+r2 x/parenrightbigg , (3.1) 4+ 0 -rx ry + 0 - Figure 1: Scheme of charge assignment. w− x=1 2/parenleftbigg1 4−rx+r2 x/parenrightbigg . The coeﬃcients are chosen such that the transition at the mid dle of the grid is continuous and smooth, and w0 x+w+ x+w− x= 1 which is required by the conservation of charge. In order to retain these properties, the weights of the two-dimensio nal grid are simply a product of two one-dimensional weights. For example, w00=w0 xw0 yorw+−=w+ xw− y. 3.2 Poisson Solver It is crucial to solve the Poisson equation fast enough (with in a second on a computer workstation) for the beam-beam simulation, because the rad iation damping time is about 5000 turns and several damping times are needed to reach an eq uilibrium distribution. For the reason of the computing speed, we follow Krishnagopal [7 ] and choose the method of cyclic reduction and FFT [14]. A ﬁve-point diﬀerence scheme is used to approximate the two-dimensional Laplacian operator φi−1,j+φi+1,j−2φi,j H2x+φi,j−1+φi,j+1−2φi,j H2y=−2πρci,j, (3.2) where iandjare the horizontal and vertical indices that label the grid p oints on the mesh. Truncation errors are of the order of H2 xandH2 y. It is worthwhile to mention that, if we use the same number of mesh points per σin both transverse directions in the case of beam aspect ratio 30:1, the truncation errors in the horizontal p lane are dominant. To minimize 5the errors in our simulation, we select three times more mesh points per σin horizontal direction compared to the vertical one. 3.3 Field The ﬁeld /vectorE=−∇φis computed on the two dimensional grid, using a six-point di ﬀerence scheme Exi,j=−1 12Hx[(φi+1,j+1−φi−1,j+1) + 4(φi+1,j−φi−1,j) + (φi+1,j−1−φi−1,j−1)],(3.3) Eyi,j=−1 12Hy[(φi+1,j+1−φi+1,j−1) + 4(φi,j+1−φi,j−1) + (φi−1,j+1−φi−1,j−1)].(3.4) The ﬁeld oﬀ the grid is computed with the same smoothing schem e used in the charge assign- ment to ensure the conservation of the momentum. The ﬁelds ExandEyare interpolated between the grid points. They are calculated by using the wei ghted summation of the ﬁelds at the nine nearest points with exactly the same weights used as the charge is assigned. 4 Track Particles The motion of a particle is described by its canonical coordi nates zT= (x, Px, y, P y), (4.1) where PxandPyare particle momenta normalized by the design momentum p0. 4.1 One-Turn Map When synchrotron radiation is turned oﬀ, a matrix is used to d escribe the linear motion in the lattice zn+1=M·zn, (4.2) where Mis a 4 ×4 symplectic matrix which can be partitioned into blocks of 2 ×2 matrices when the linear coupling is ignored M=/parenleftigg Mx0 0My/parenrightigg . (4.3) HereMx, and Myare 2 ×2 symplectic matrices. The matrix Mxis expressed with the Courant-Snyder parameters βx,αx, and γxat the collision point Mx=/parenleftigg cos(2πνx) +αxsin(2πνx)βxsin(2πνx) −γxsin(2πνx) cos(2 πνx)−αxsin(2πνx)/parenrightigg , (4.4) where νxis the horizontal tune. A similar expression is applied in th e vertical plane. 64.2 Damping and Synchrotron Radiation Following Hirata [16], we apply the radiation damping and qu antum excitation in the nor- malized coordinates, since it is easily generalized to incl ude the linear coupling. The motion of a particle in the normalized coordinate is described by a r otation matrix Rx=/parenleftigg cos(2πνx) sin(2 πνx) −sin(2πνx) cos(2 πνx)/parenrightigg , (4.5) which is obtained by performing the similarity transformat ion Rx=A−1 x·Mx·Ax, (4.6) where Ax= √βx0 −αx√ βx1√ βx , A−1 x= 1√ βx0 αx√ βx√βx . (4.7) When synchrotron radiation is switched on, we simply replac e the rotation matrix Rx with following map in the normalized coordinates ¯ xand¯Px /parenleftigg ¯x ¯Px/parenrightigg =e−1 τxRx/parenleftigg ¯x ¯Px/parenrightigg +/radicalig ǫx(1−e−2 τx)/parenleftigg η¯x η¯Px/parenrightigg , (4.8) where η¯xandη¯pxare Gaussian random variables normalized to unity, τxis the damping time in unit of number of turns and ǫxis the equilibrium emittance. In the vertical plane, a similar map is applied. 4.3 Beam-Beam Kick Assuming particles are ultra-relativistic and the collisi on is head-on, the kick on a particle by the opposing beam is given by the Lorenz force δPx=−2e E0Ex, (4.9) δPy=−2e E0Ey, (4.10) where ExandEyare the horizontal and vertical components of the electric ﬁ eld evaluated at the position of the particle. They are computed with the Po isson solver as outlined in the previous section each time two slices of the beam pass eac h other. And the half of the transverse force is the magnetic force due the beam moving at the speed of light. The energy of the particle, E0=cp0, appearing in the denominator of the above expressions come s from the normalization of the canonical momenta PxandPyand the use of the s-coordinate, s=ct, as the “time” variable. 7−2 −1 0 1 2 x 10−3−8−6−4−202468x 10−5 X(meter)Px(rad) −1 −0.5 0 0.5 1 x 10−4−1.5−1−0.500.511.5x 10−4 Y(meter)Py(rad) Figure 2: The beam-beam kick by a ﬂat Gaussian beam with aspect ratio 30 :1 near X axis and Y axis. The dash-dotted curve is the case when φ= 0is assigned as the boundary condition. The long-dashed curve is the kick when inhomogeneous bounda ry condition is used. The short-dashed curve is the kick produced by the Erskine-Bass etti formula [17]. A typical beam-beam kick experienced by a particle near the a xis is shown in Fig. 2 with the PEP-II parameters, which are tabulated in the next s ection. As expected based on the derivation in section 2.2, the kick resulted from solvin g the Poisson equation with the inhomogeneous boundary condition agrees well with the anal ytic solution. In addition, the agreement demonstrates that the scheme of the charge deposi tion works well, the mesh is dense enough and the number of macro particles is large enoug h. The number of macro particles used to represent the distribu tion of the beam is 10240. The area of the mesh is 8 σx×24σyand there are 15 grid points per σxand 5 per σy. There are about 15 macro particles per cell within 3 σof the beam. These parameters are chosen to minimize truncation errors and maximize resolution. The 256×256 mesh is also the maximum allowed by a computer workstation to complete a typi cal job within a reasonable time. The discrepancy between the solution with the homogeneous b oundary condition, φ= 0, and the analytic one worsen as the beam aspect ratio becomes l arger because the actual change of the potential on the horizontal boundaries become s larger. 85 Simulation of PEP-II: Validation An object-oriented C++ class library has been written to sim ulate the beam-beam interac- tion using the method outlined in the previous sections. In t he library, the beam and the Poisson solver are all independent objects that can be const ructed by the user. For example, there is no limitation on how many beam objects are allowed in the simulation and the beams can have diﬀerent parameters as an instance of the beam class . These features provide us with great ﬂexibility to study various phenomena of the beam -beam interaction. We will carry out the simulation of beam-beam interaction wi th the current operating parameters of the PEP-II so that the results of the simulatio n can be compared with the known experimental observations. As a goal of this study, af ter a proper benchmarking of the code against the experiment, we shall be able to make predict ions on parameter dependence and show how to improve the luminosity performance of the col lider. 5.1 PEP-II Operating Parameters Parameter Description LER(e+) HER(e-) E(Gev) Beam energy 3.1 9.0 β∗ x(cm) Beta X at the IP 50.0 50.0 β∗ y(cm) Beta Y at the IP 1.25 1.25 τt(turn) Transverse damping time 9740 5014 ǫx(nm-rad) Emittance X 24.0 48.0 ǫy(nm-rad) Emittance Y 1.50 1.50 νx X tune 0.649 0.569 νy Y tune 0.564 0.639 Table 5.1: Parameters for the beam-beam simulation The parameters used in the simulation are tabulated in Tab. 5 .1. The vertical β∗ yis lowered to 1.25cm [18] from the design value 1.5cm [11]. The h orizontal emittance 24nm-rad in the Low Energy Ring (LER) is half of the design value 48nm-r ad because the wiggler is turned oﬀ to increase the luminosity. The damping time, 9740 turns, in the LER is a factor of two larger than the one in the High Energy Ring (HER) becaus e of the change of the wigglers made during the construction of the machine. The de gradation of luminosity from the increase of the damping time was found then to be about 10% based on the beam-beam simulation. The tunes are split and are determined experime ntally to optimize the peak luminosity. 5.2 Procedure of simulation The distribution of the beam is represented as a collection o f macro particles that are dy- namically tracked. The procedure to obtain equilibrium dis tributions of the two colliding beams is as follows 9•initialize the four-dimensional Gaussian distribution ac cording to the parameters of the lattice at the collision point and the emittance of the beam. Distributions of two beams are independent and diﬀerent. •iterate a loop with three damping times •propagate each beam through corresponding lattice using on e-turn map with syn- chrotron radiation. •cast the particle distributions onto the grid as the charge d istribution with weighting and smoothing. •solve for the potential on the grid with the Poisson solver. •compute the ﬁeld on the grid. •calculate the beam-beam kick to the particles of the other be am with the ﬁeld at the position of the particles. The ﬁeld oﬀ the grid is interpolat ed with the same weighting and smoothing used in the charge deposition. •save data such as beam size, beam centroid and luminosity. •end of the loop. •save the ﬁnal distributions. We vary the beam intensity with a ﬁxed beam current ratio: I+:I−= 2:1, which is close to the ratio for the PEP-II operation. At each beam current, w e compute the equilibrium distributions. 5.3 Beam-Beam Limit Given equilibrium distributions that are close enough to th e Gaussian, we can introduce the beam-beam parameters ξ± x=reN∓β± x 2πγ±σ∓x(σ∓x+σ∓y), ξ± y=reN∓β± y 2πγ±σ∓y(σ∓x+σ∓y), (5.1) where reis the classical electron radius, γis the energy of the beam in unit of the rest energy, and Nis total number of the charge in the bunch. Here the superscri pt “+” denotes quantities corresponding to the positron and “-” quantitie s corresponding to the electron. The results of the simulation are shown in Fig. 3. The beam-be am tune shifts for the electron beam are low because of the large beam-beam blowup o f the positron beam. At this operating point, the positron is the weaker beam. When I+= 1200mA and I−= 600mA, which is the near the maximum allowed currents when the beams are in collision, the positron beam sizes are σ+ x= 260 µm and σ+ y= 7µm. 5.4 Luminosity Given the two beam distributions, ρ+andρ−, the luminosity can be written as 100 500 1000 150000.020.040.060.080.10.120.14Positron I−(mA)ξx+ 0 500 1000 150000.020.040.060.080.10.12Positron I−(mA)ξy+ 050010001500200025000.0050.010.0150.020.025Electron I+(mA)ξx− 050010001500200025000.0050.010.0150.020.025Electron I+(mA)ξy− Figure 3: The beam-beam tune shifts as a function of beam currents. Num ber of bunches, nb= 554, is used for the total beam currents. The revolution frequen cyf0= 136 .312kHz . L=nbf0N+N−∞/integraldisplay −∞∞/integraldisplay −∞ρ+(x, y)ρ−(x, y)dxdy, (5.2) where nbis the number of the colliding bunches, f0is the revolution frequency, and N+, N− are the number of charges in each position and electron bunch , respectively. Since the distribution ρis normalized to unity /integraldisplay dxdyρ (x, y) = 1 (5.3) and proportional to the charge density ρc, we evaluate the overlapping integral by a summa- tion over ρ+ cρ− con the mesh. If we assume the distributions are Gaussian, the overlapping integral can be carried out 11L=nbf0N+N− 2πΣxΣy, (5.4) where Σ x=/radicalig σ+ x2+σ− x2and Σ y=/radicalig σ+ y2+σ− y2. Two methods agree within a few percents. The mesh method gives a higher luminosity than the Gaussian o ne. We always use the mesh method, since it can be applied to broad classes of distribut ion. 0 500 1000 1500 2000 2500 3000 350000.511.522.533.5x 1033 I+(mA)Luminosity(cm−2s−1)nb = 829 nb = 665 nb = 554 nb = 415 PEP−II operating point Figure 4: Luminosity as a function of the beam current. The labels are t he number of the colliding bunches. Figure 4 shows the luminosity of the beams with 415 colliding bunches, which are spaced with every 8 RF buckets and 10% of the gap. The luminosity is be am-beam limited. It also shows that the optimum number of bunches is between 544 and 66 5 and the luminosity is about 2.3 ×1033cm−2s−1given I+= 1200mA. These results quantitatively agree with the experimental observations in the routine operation of t he PEP-II. For example, the peak luminosity of the PEP-II is 1.95 ×1033cm−2s−1withI+= 1170mA, I−= 700mA, andnb= 665 during the period of June, 2000. The fact that the lumino sity value in the simulation is higher than the observation could be explaine d by the hour-glass eﬀect which is ignored in the simulation. 12For a ﬁxed number of bunches, say 554, the simulation shows a m aximum luminosity, which is also seen daily in the control room of the PEP-II. Fro m the simulation, we see that the reason for the peaked luminosity is the rapid growth of σ+ yonce the peak current is passed. In addition, the simulation predicts that we can reach the de sign luminosity 3 ×1033cm−2s−1 by running 829 bunches at the beam current of I+= 1600mA and I−= 800mA. This predic- tion has not been realized yet at this time. Currently, the to tal positron current is probably limited below 1200mA by the electron-cloud instability [19 ]. Once this limitation is removed, we expect to reach the design luminosity with 829 bunches. There is no particle loss outside the area (8 σx×24σy) covered by the mesh in the ﬁrst 15 data points. Beyond the 15th points, particle loss is almost about 1%. 5.5 Damping Time Historically, the damping time is typically not considered to be an important parameter for the beam-beam eﬀects. So we make an attempt to reduce the damp ing time artiﬁcially for the LER to speed up the computation. The result is shown in Fig . 5 The only diﬀerence of the parameters used in two simulations is the damping time in the LER, which is indicated as the labels in the ﬁgure. Indeed, at the low current, the diﬀerence of the luminosity is rather small, which is consistent with t he simulation performed when the change of the wiggler was made. But the diﬀerence grows large r, as the current increases. At the peak luminosity for the PEP-II operation, I+= 1200mA, the diﬀerence is about 40%, which is signiﬁcant. This result shows for the ﬁrst time that the damping time is a r ather important parameter for the computation of the peak luminosity at high beam curre nts. Secondly, it points a way to improve the peak luminosity of the PEP-II without the incr ease of the beam currents, namely to install another wiggler in the LER to reduce the dam ping time to the original design value. 6 Discussion We have developed a hybrid method of solving the potential wi th an open boundary by using Green’s function to ﬁx the potential on a ﬁnite boundar y and then to solve the Poisson equation for the potential inside the boundary. The method i s applied to the simulation of strong-strong interaction of beam-beam eﬀects in PEP-II. T he preliminary results of this simulation show a very good quantitative agreement with the experimental observations. Given the simplicity of the two-dimensional model used, the achievement is surprising and remarkable. We have demonstrated that the present code has a highly reliable predictive capability of realistic beam-beam interaction. To further benchmark the code, we need to extend the simulation to include the ﬁnite length of the bunc h and compare the simulation results directly to the controlled experiments. This method is quite general. It can be applied to the problem of space charge in three dimensions. It can also be used in the beam-beam interaction of a linear collider. Finally, 130200 400 600 800 1000 1200 1400 1600 1800 200000.511.522.533.5x 1033 I+(mA)Luminosity(cm−2s−1)τ+ = 5014 turns τ+ = 9740 turns Figure 5: Luminosity aﬀected by the damping time with 554 bunches. it can be applied to any boundary condition to reduce the regi on of the mesh if Green’s function is known. Acknowledgments We would like to thank John Irwin, John Seeman and Ron Ruth for their continuous support and encouragement. We would like also to thank Franz-Josef D ecker, Miguel Furman, Sam Heifets, Albert Hoﬀmann, Witold Kozanecki, Michiko Minty, Robert Siemann, Mike Sullivan, Robert Warnock, Uli Wienands and Yiton Yan for the helpful di scussions. Especially, we would like to thank Srinvas Krishnagopal for many explanati ons of the PIC method during his visit at SLAC. One of the authors (TT) is supported in part by DOE contract W-7405- Eng.48 and DOE grant DE-FG03-96ER40954. 14References [1] See for example A. W. Chao, P. Bambade and W. T. Weng, “Nonl inear Beam-Beam Resonances,” Proceedings edited by J.M Jowett, M. Month and S. Turner, Sardinia (1985). [2] See for example J. T. Seeman, “Observations of the Beam-B eam Interaction,” Proceed- ings edited by J.M Jowett, M. Month and S. Turner, Sardinia (1 985). [3] See for example S. Myers, “Review of Beam-Beam Simulatio n,” Proceedings edited by J.M Jowett, M. Month and S. Turner, Sardinia (1985). [4] K. Hirata, H. Moshammer and F. Ruggiero, “A Symplectic Be am-Beam Interaction with Energy Change,” Particle Accelerator 40205 (1993). [5] M.A. Furman “Beam-Beam Simulations With the Gaussian Co de TRS,” SLAC-AP-119, LBNL-42669, January 1999. [6] S. Krishnagopal and R. Siemann, “Coherent Beam-Beam Int eractions in Electron- Positron Colliders,” Phys. Rev. Lett., 67, 2461(1991). [7] S. Krishnagopal, “Luminosity-Limiting Coherent Pheno mena in Electron-Positron Col- liders,” Phys. Rev. Lett., 76, 235(1996). [8] J. K. Koga and T. Tajima, “Particle Diﬀusion from the Beam -Beam Interaction in Syn- chrotron Colliders,” Phys. Rev. Lett., 72, 2025(1994), J. K. Koga and T. Tajima, J. Comput. Phys., “The delta-f Algorithm for Beam Dynamics,” 116, 314-329(1995), and J.K.Koga and T. Tajima, “Collective Eﬀects of Beam-Beam Int eraction in a Synchrotron Collider”, AIP Proceedings on SSC Accelerator Physics, eds . Y.T.Yan, J.P.Naples, M.J.Syphers (AIP, NY,1995), p.215. [9] E. B. Anderson, T.I Banks, J.T. Rogers, “ODYSSEUS: Descr iption of Results from a Strong-Strong Beam-Beam Simulation For Storage Rings,’ Pr oceedings of Particle Ac- celerator Conference, New York, 1999. [10] K. Ohmi, “Simulation of the Beam-Beam Eﬀect in KEKB,” Pr oceedings of International Workshop on Performance Improvement of Electron-Positron Collider Particle Factories, September, (1999). [11] “PEP-II: An Asymmetric B Factory”, Conceptual Design R eport, SLAC-418, June 1993. [12] T. Tajima, “Computational Plasma Physics,” Addison-W esley, Reading, Mass., (1989). [13] J. D. Jackson, “Classical Electrodynamics,” Chapter 1 , John Wiley & Sons, Inc. (1962). [14] J.P. Christiansen and R. W. Hockney, “DELSQPHI, a Two-D imensional Poisson-Solver Program,” Computer Physics Communications 2 (1971) 139-15 5. 15[15] R. W. Hockney and J.W. Eastwood, “Computer Simulation U sing Particles,” Chapter 5, Bristol and Philadelphia, (1988) [16] K. Hirata and F. Ruggiero, “Treatment of Radiation in El ectron Storage Rings,” LEP Notes 661, August 8, (1988). [17] M. Bassetti and G. Erskine, CERN ISR TH/80-06 (1980). [18] Y. Nosochkov, et al, “Upgrade of the PEP-II Low Beta Optics,” SLAC-PUB-8481, Ju ne, (2000). [19] M. Izawa, Y. Sato, T. Toyomasu, Phys. Rev. Lett. 74, 5044 (1995). 16
arXiv:physics/0008244v1 [physics.ed-ph] 31 Aug 2000Compactiﬁcation of the moduli space in symplectization and hidden symmetries of its boundary Gang Liu July 26, 2013 1 Introduction The purpose of this paper and the forth coming [L1], [L3] is to lay down a foundation for a sequence of papers concerning the moduli sp ace of connecting pseudo-holomorphic maps in the symplectization of a compac t contact manifold and their applications. In this paper, we will establish the comapctiﬁcation of the moduli space of the pseudo-holomorphic maps in the sympl ectization and ex- hibit some new phenomenon concerning bubbling and the ”hidd en” symmetries of the boundary of the comapctiﬁcation. Combining with the i ndex formula, which will be proved in [L3], we will show in [L3] that the virt ual co-dimension of the boundary components of the moduli space with at least o ne bubble is at least two, while the virtual co-dimension of the boundary co mponents of broken connecting maps of two elements is one. In [L1], we will show t hat these virtual co-dimensions can be realized in the corresponding virtual moduli cycles. In a sequence of forth coming papers, we will give some of possib le applications. In particular, we will deﬁne various versions of index homol ogy for a contact manifold, relative index homology for a symplectic manifol d with contact type boundary, as well as their multiplicative structures in the se holmologies. These multiplicative structures can be thought as analogies of th e usual quantum prod- uct and pants product in quantum cohomology and Floer cohomo logy. We will also investigate the implication of these homologies to Wei nstein conjecture. It is well-known that a family of pseudo-holomorphic maps in the symplec- tization of a compact contact manifold may develop bubbles. Since in the sym- plectization the symplectic form is exact, each top bubble n ecessarily has non- removable singularity at inﬁnity, and along the end at inﬁni ty, the bubble is convergent to some closed orbit of the Reeb ﬁeld of the contac t manifold. This makes the behavior of the boundary components of the compact iﬁcation of the moduli space here very much look like the one of the broken con necting orbits in the usual Floer homology. In particular, it is believed that the co-dimension of the boundary components even coming from bubbling should be one in general. We will show in this paper and [L3] that in the case of the modul i space the 1pseudo-holomorphic maps connecting at least two closed orb its at the two ends of the symplectization,at least virtually, this belief is n ot true. Our starting point is the the following new phenomenon conce rning the bubbling of connecting pseudo-holomorphic maps. Observe t hat each time when a family of pseudo-holomorphic maps connecting two closed o rbits splits into a family of broken connecting maps or develops a bubble, ther e is not only a splitting of the domain but also a splitting of the target at s ame time. Therefore theR-symmetry of the target splits into a two-dimensional or mul ti-dimensional symmetries during the bubbling or splitting. Moreover, the rates of these two types of degeneration of the domain and target are independe nt to each other in general. In fact, the maximum principal implies that in the s implest case when such a family of connecting maps develops only one bubble, th e image of the bubble lies on a new component on the ”left” of the original on e, and there is also a new principal component on the left of the original pri ncipal component. Note that the new ”left” principal componet may be just a triv ial connecting map. However, the limit map itself is still stable. This last kind of degeneration plays a rather special role. Therefore, unlike the usual Gro mov-Floer theory in symplectic case, the bubbling here, splits the domain int o three components and the target into two. Note that this phenomenon can only ha ppen when the pseudo-holomorphic maps involved connect at least two clos ed orbits lying on the two ends of the symplectization. Now using the fact that both symmetry groups of a connecting m ap and a bubble with non-removable singularity are three dimensio nal, it is easy to see that in term of the dimensions of symmetries, bubbling ha s co-dimension three, while the splitting of connecting maps into broken on es is co-dimension two. This seems to suggest a rather diﬀerent picture on the bo undary behavior of the moduli space in the symplectization, which is not only disprove what was believed before but also bring us to a situation of dilemm a. Namely, the situation here is even better than the one in the usual Gromov -Floer theory in symplectic case. One the the main purpose of this paper and [L1], [L3] is to reso lve this dilemma. In this paper, we will give some key ingradients of t he solution of the dilemma. The main body of this paper is devoted to to deﬁne the notion of stable maps in the symplectization and to use them to establi sh the compactness of the moduli space of such maps. It turns out situation here i s diﬀerent from the usual Gromov-Floer theory. There are various new phenom enons, which have to be put into consideration in order to to formulate the notion of stable maps and various related notions. In symplectic geometry, one of the key ingredients to prove t he compactness of the moduli space of stable maps is the bubbling process. It consists of three parts: the Uhlenbeck-Sacks rescaling scheme, removable si ngularity lemma and the analysis concerning the behavior of the ”connecting tub es”. In the case of the symplectization of a contact manifold, we have mentione d above that there is a new phenomenon in the bubbling process. However, as far a s the proof goes, there are still the correspondoing three parts there. The ﬁrst and most important part of the bubbling was established by Hofer in [H ]. He discovered 2the phenomenon of bubbling in the symplecitization with bub ble with non- removable singularity. Since the top bubble in the symplect ization always has non-removable singularity, the corresponding second part of the bubbling here is about the asymptotic behavior of a bubble approaching to i ts non-removable singularities. In particular, it is important to know that a long the end, a bubble with non-removable singularity approaches to some closed o rbit with an expo- nential decay rate. In the case that the contact manifold is t hree dimensional, the desired exponential decay estimate was obtained by Hofe r, Wysocki and Zehnder in [HWZ]. It seem that the third part of the bubbling i n the contact case, especially, the part concerning the behavior of the co nnecting tubes along the non-compact R-direction was not addressed before. We emphasize that in oder to get the desired compactiﬁcation without introducin g unstable trivial connecting maps, it is crucial to know that the ”connecting” tube along the non-compact R-direction behaves essentially like the trivial connectin g map at C0sense. Most analytic part of this paper is aimed to establish the second and the third part of the bubbling process. Once the above bubbling process is established, the main diﬃ culty to es- tablish the compactiﬁcation of the moduli space is more conc eptual rather than technical. In fact what we need here is a a right deﬁnition of s table maps in the contact case, which should incorporate those symmetry s plitting mentioned above as well as ”hidden” symmetries in each component of the target (See Sec. 3). In particular, according to the consideration in Sec 3, b ecause of these ”hid- den” symmetries, one should count the R-symmetry of each component of the target as many times as the number of the connected component s of the domain lying in the component of the target. This will lead to a somew hat ”strange” deﬁnition of the quivalence of stable maps in the contact cas e. In symplectic geometry, historically, the compactness the orem for pseudo- holomorphic maps or connecting ( J,H)-maps was ﬁrst proved by Gromov and Floer [G, F] by adding certain degenerate maps, called cuspi dal maps. Later a smaller compactiﬁcation was found by using stable maps, whi ch plays a impor- tant role for the recent development in symplectic geometry (see, for example, [LiT], [FO] and [LT]). Technically, there is not much diﬃcul ty to pass from the cuspidal map compactiﬁcation to stable map one. The key is to carefully keep track all marked points naturally introduced in the bubblin g, then to study the deformation of the domain equipped with these marked points in a proper mod- uli space of curves. In the same vein, the key to get a right com pactiﬁcation in our case is ﬁrst to understand the two crucial points menti oned in last para- graph, then to keep track carefully all marked points and mar ked lines in the domain, marked sections in the target naturally appeared in the bubbling and to study the deformation of such a structure. Once the desire d compactiﬁcation is established, our main result about the virtual co-dimens ion of the boundary will be a consequence of the compactness theorem, the index f ormula proved in [L3] and a direct dimension counting argument. As mentioned above, it has been believed that bubbling for ps eudo-holomorphic maps in symplectization is a co-dimension one phenomenon. T his has been considered as a major diﬃculty to establish various ”simple ” and ”elementary” 3constructions, such as Floer homology and G-W invariants, i n contact geome- try. A very interesting and much more advanced construction were proposed by Eliashberg, Hofer and Givental under the name contact hom ology or contact Floer homology (see [E]). On the other hand, the work of this paper and [L1], [L3] sugges t a rather diﬀerent picture on the boundary behavior of the moduli spac e in contact ge- ometry. This opens the door to construct those ”simple” cons tructions, such as Floer homology and G-W invariants in contact geometry, whic h have essentially same algebraic structures as the ones in symplectic geometr y. It also makes it possible to generalize various important constructions in symplectic geometry. We now brieﬂy mention some of these possible applications, w hich are outlined in the Sec. 5. The ﬁrst application is to deﬁne an analogy of Floer homology in contact geometry. To distinguish our construction with the one in [E ], which is under the name contact homology or contact Floer homology, we call our construc- tion index homology. The most natural way to do this is to use t he closed orbits of the Reeb ﬁeld to generating a chain group and to coun t the pseudo- holomorphic maps connecting two closed orbits to deﬁne the b oundary map. As mentioned above, the co-dimension of the component of bro ken connect- ing pseudo-holomorphic maps is one, and bubbling is a co-dim ension two phe- nomenon. Therefore, we are in the exactly the same situation as the usual Floer homology, and the desired index homology can be established as an invariant of the contact structure. Once this is done, one can also constr uct G-W invariants and use them to deﬁne ring structure and the action of the usua l homology of the contact manifold on the index homology, which are the a nalogies in the usual quantum cohomology. To see that the index homology so d eﬁned is not always trivial, we introduce Bott-type index homology as a c omputational tool. Using the Bott-type homology, we can compute the index homol ogy for a con- tact manifold, which appears as a regular zero locus of some l ocal Hamiltonian function which generates a S1Hamiltonian action. It turns out that the index homology of the contact manifold in this case is just the inﬁn ite copies of the usual homology of its symplectic quotient indexed by the per iods of the closed orbits. Of course the non-vanishing of index homology impli es the Weinstein conjecture. Therefore, as a corollary, we proved the Weinst ein conjecture in above case. It is also possible to to use the moduli space diﬀerently to de ﬁne various versions of index homology. In particular, in Sec 5, we will o utline how to deﬁne an additive quantum homology of a contact manifolds and rela tive quantum homology of a symplectic manifold with contact type boundar y. There are also some other important constructions that can b e generalized. For example, the relative G-W invariant and its gluing formu la can be estab- lished in general, which was developed by Li-Ruan in [LiR] be fore with an extra assumption on the existence of some local S1Hamiltonian action. This paper is organized as follows. In Sec. 2, we will collect and prove some basic facts about the ﬁrst part of bubbling. Almost all of statements there are well-known d ue to the work 4of Hofer and his collaborators. However, for the completene ss, we give details of the proof for most of these statements. Besides several te chnical lemmas in section, the most important thing in section is the introduc tion of Hofer’s energy function, which leads to the important notion of ﬁnite energ e plane in [H]. In Sec. 3, we formulate the notion of stable maps in the symple ctization and the weak-topology of the moduli space of such maps. We the n proved the compactness of the moduli space and the statement concernin g the co-dimension of its boundary, modulo the statement concerning the expone ntial decay of a bubble approaching its non-removable singularity and the s tatement concerning the behavior of the ”connecting tube”. Both of these stateme nts are proved in Sec. 4. The last section, Sec 5, is an outline of some possible applic ations. the detail of these applications will appear in forth coming papers. Acknowledgment : The author is very grateful to Professor G. Tian for valu- able and inspiring discussions, for his help on various aspe cts of the project and for his encouragement. 2 Bubbling Let (M2n+1,ξ) be a contact manifold. This means that ξis a generic 2 n- dimensional subbundle of TM. A contact form λ=λξassociated to ξis a 1-form such that λ∧(dλ)n/\e}atio\slash= 0 andξ=kerλ. The 2-form dλis non-degenerate when restricted to ξand has a 1-dimensional kernel at each tangent space of M. We denote by ηthe line bundle generated by ker(dλ). It has a canonic sectionXλdeﬁned by requiring that λ(Xλ) = 1. Since ξ∩η={0}, we have TM=ξ⊕RXλ.Letπ:TM→ξbe the projection to the ﬁrst summand. •Symplectization : The symplectization of ( M2n+1,ξ,λ) is deﬁned as follows. Let/tildewiderMbeM×Requipped with the exact symplectic form ω=d(er·λ), whereris the coordinate for the R-factor. Since dλis symplectic along ξ, there exists adλ-compatible almost complex structure Jdeﬁned onξ. In fact, the set of all such J’s is contractible. We extend Jto anr-invariant almost complex structure ˜Jby requiring: ˜J(∂ ∂r) =Xλ,˜J(Xλ) =−∂ ∂r,and˜J=J alongξ. •Equation for ˜J-holomorphic curves in /tildewiderM Let ˜u= (u,a) : Σ =S1×R→/tildewiderMbe a˜J-holomorphic map where u: Σ→M anda: Σ→R. Then we have ˜J(˜u)◦d˜u=d˜u◦i, (⋆) 5whereiis the standard complex structure on Σ ,i.e.i(∂ ∂s) =∂ ∂tandi(∂ ∂t) =−∂ ∂s. Here (s,t) is the cylindrical coordinate of R×S1. Equation (⋆)2 is equivalent to the following equations: /braceleftbiggπ(u)du+J(u)π(u)du◦i= 0 (1) (u∗λ)◦i=da (2) Equation (1) is equivalent to : π(u)(∂u ∂s) +J(u)π(u)(∂u ∂t) = 0. (1′) Lemma 2.1 ∆a=∂2a ∂s2+∂2a ∂t2≥0if˜uis˜J-holomorphic. proof: It follows from (2) that u∗(dλ) = −d(da◦i) =d(−∂a ∂tds+∂a ∂sdt) = (∂2a ∂t2+∂2 ∂s2)ds∧dt. Now u∗(dλ) =dλ(π(∂u ∂s),π(∂u ∂t))ds∧dt =dλ(π(∂u ∂s),J·π(∂u ∂s))ds∧dt =gJ(π(∂u ∂s),π(∂u ∂s))ds∧dt, wheregJis the Riemannian metric deﬁned on ξassociated with dλandJ. Therefore, ∆a=\|π(∂u ∂s)\|2 gJ≥0. QED •Energy Letφ∈C∞(R,[1 2,1]),φ′≥0.For any ˜J-holomorphic curve ˜ u, Itsφ-energy is deﬁned as follows: Eφ(˜u) =/integraldisplay /integraldisplay R1×S1˜u∗d(φλ) and its energy E(˜u) = sup φEφ(˜u). 6Let Eλ(˜u) =/integraldisplay /integraldisplay R1×S1˜u∗d(λ) Note that: ˜u∗(d(φλ)) = ˜u∗(dφ∧λ+φdλ) =φ′(u)da∧u∗λ+φ(a)u∗(dλ) ={φ′(a){∂a ∂sλ(∂u ∂t)−∂a ∂tλ(∂u ∂s)}+φ(a)dλ(∂u ∂s,∂u ∂t)}ds∧dt =1 2{φ′(a){(∂a ∂s)2+ (∂a ∂t)2+λ(∂u ∂s)2+λ(∂u ∂t)2} +φ(a){\|π(∂u ∂s)\|2+\|π(∂u ∂t)\|2}}ds∧dt. This implies that E(˜u)≥0. Note: the above local expression ˜ u∗(d(φλ)) is valid for any conformal coor- dinate. Example Letx:S1→Mbe a closed orbit of Reeb ﬁeld Xλof periodc=/integraltext S1λ( ˙x(t))dt. We get a trivial ˜J- holomorphic map ˜ u(s,t) = (u,a) = (X(t),c·s).Then Eφ(˜u) =1 2/integraldisplay R1×S1φ′(a){(∂a ∂s)2+λ(∂u ∂t)2}ds∧dt =c2/integraldisplay∞ −∞φ′(c·s)ds =c{φ(∞)−φ(−∞)} =1 2c. •Bubbling : Lemma 2.2 Let(X,d)be a complete metric space and φ:X→R+= [0,∞) be a continuous function. Given x∈Xandǫ>0, there exists x′∈Xandǫ′>0 such that (1)ǫ′≤ǫ,φ(x′)ǫ′≥φ(x)·ǫ; (2)d(x,x′)≤2ǫ; (3)2φ(x′)≥φ(y)for ally∈Xsuch thatd(y,x′)≤ǫ′. The proof is elementary. See [H-V]. Proposition 2.1 Let˜un= (un,an) :R1×S1(orC)→/tildewiderMbe a sequence of ˜J- holomorphic maps such that (i) there exists a constant c>0such thatE(˜un)< c; (ii) for each ˜un, there exists a xn∈R1×S1such that \|dun(xn)\| → ∞.Then the sequence {˜un}∞ n=1will bubble oﬀ at {xn}∞ n=1a bubble ˜v:C→/tildewiderM, which is ˜J-holomorphic such that 7(a)\|d˜v(0)\|= 1; (b)\|d˜v(y)\| ≤2for anyy∈C;and (c)E(˜v)<c. Proof : Apply Lemma 2.2 to the case that ( X,d) =R1×S1,φ=\|d˜un\|,x=xn, and ǫ=ǫn, where {ǫn}∞ n=1is a sequence such that dn·ǫn=\|d˜un(xn)\| ·ǫn→ ∞.We may assume that (a)\|d˜un(xn)\| ·ǫn→ ∞; (b) 2\|d˜un(xn)\|>\|d˜un(y)\|for anyy∈Dǫn(xn); (c) ˜un(xn)∈M× {0}after a translation in /tildewiderM. FixR>0, deﬁne ˜vn,R:DR→/tildewiderMto be ˜vn,R(x) = ˜un(xn+x dn). Note that when nis large enough, dn·ǫn> R. Hence,xn+x dn∈Dǫn(xn) forx∈DR. This implies that (i)\|d˜vn,R(0)\|=1 dn\|d˜un(xn)\|= 1; (ii)\|d˜vn,R(x)\|=1 dn\|d˜un(xn+x dn)\| ≤2dn dn= 2 for any x∈DR; (iii) ˜vn,R(0)∈M× {0}. Now the standard elliptic estimation implies that ˜ vn,RisC∞-convergent to a ˜J-holomorphic map ˜ vR:DR→/tildewiderMafter taking a subsequence of ˜ vn,R. LetRn→ ∞, and taking a diagonal subsequence, ˜ vn,R nisC∞-convergent to a˜J-holomorphic map ˜ v=∪R˜vR:C→/tildewiderMsuch that (a) \|d˜v(0)\|= 1; (b) \|d˜v(x)\| ≤2,x∈C; (c)E(˜v)<c; (d) ˜v(0)∈M× {0}. QED The following lemma will be used to prove that the bubbling wi ll stop after ﬁnite steps. Lemma 2.3 Fixc >0. LetVbe the collection of all ˜J-holomorphic maps ˜v:C→/tildewiderMsatisfying the properties (a)-(c) in the previous proposit ion. Then there exists a constant ǫ>0such that for any ˜v∈V,/integraltext C˜v∗(dλ)>ǫ. Proof : If not, there would exist ˜ vn:C→/tildewiderMof˜J-holomorphic maps such that (a) \|d˜vn(0)\|= 1; (b) \|d˜v(x)\| ≤2,x∈C; (c)E(˜vn)< c; and (d) ˜vn(0)∈M× {0} afterR-translation in /tildewiderM; (e) lim n→∞/integraltext C˜v∗ n(dλ) = 0. Now (a)-(d) implies that ˜ vis locallyC∞-convergent to a ˜J-holomorphic map ˜v:C→/tildewiderMwith same properties of (a)-(d). Now /integraldisplay C˜v∗(dλ) = lim R→∞/integraldisplay DR˜v∗(dλ) = lim R→∞lim n→∞/integraldisplay DR˜v∗ n(dλ) = 0. It follows from Lemma 2.5 that ˜ vis a constant map. This contradicts to (a). QED 8Lemma 2.4 Let˜u:R1×S1→˜Mbe a˜J-holomorphic map such that E(˜u)<∞ and/integraltext R1×S1˜u∗(dλ) = 0.Then either ˜ucomes from a closed orbit of Xλas in Example 1, or ˜uis a constant map. Lemma 2.5 Let˜u:C→˜Mbe a˜J- holomorphic map such that E(˜u)<∞, and/integraltext C˜u∗(dλ) = 0, then ˜uis a constant map. Proof of Lemma 2.4 0 =/integraldisplay R1×S1˜u∗(dλ) =1 2/integraldisplay R1×S1{\|π(∂u ∂s)\|2+\|π(∂u ∂t)\|2}ds∧dt =⇒π(∂u ∂s) =π(∂u ∂t) = 0 =⇒uis tangent to RXλ. This implies that u=x◦f,wheref:R1×R1→Randx=x(t) is the solution ofdx dt=Xλ(x(t)).Here we treat uas a function deﬁned on R1×R1which is periodic in the second variable. Therefore,/braceleftbigg∂u ∂s= ˙x∂f ∂s=∂f ∂sXλ(f) ∂u ∂t= ˙x∂f ∂t=∂f ∂tXλ(f). This implies that λ(∂u ∂s) =fsandλ(∂u ∂t) =ft. Now the equation λ◦du= −da◦iis equivalent to/braceleftbigg λ(∂u ∂s) =−at λ(∂u ∂t) =as. We have /braceleftbiggas=ft at=−fs. That isF=a+fiis holomorphic on C. Note that here we treat aas a function onR×Rwhich is periodic on the second variable. Therefor, F′=∂F ∂zis also holomorphic. Now\|dF\|2=\|∂F ∂z\|2=\|∂a ∂z\|2+\|∂f ∂z\|2.If\|da\|is bounded, then \|df\|is also bounded. The holomorphic funtion F′deﬁned on Chas bounded norm, hence, is a constant. This implies that F=c·z+d= (c1+c2i)·(s+ti)+d1+d2i= (c1s−c2t+d1)+(c1t+c2s+d2)i. Hencef= (c1t+c2s+d2) anda(s,t) =c1s−c2t+d1. Sineceais periodic in t:a(s,t+ 1) =a(s,t).This implies that a=c1·s+d1. We claim that in this case x(t) is a closed orbit of Xλ, Suppose this is not true,fhas to be periodic in t:f(s,t+ 1) =f(s,t).This implies that f=c2·s+d1.Now But /braceleftbiggas=ft= 0 at=−fs=−c2 9implies that c1=c2= 0,and henceFis constant. Therefore, we may assume that \|da\|, hence \|d˜u\|is not bounded. Then the bubbling process described before is applicable to this cas e and will produce a bubble ˜v:C→/tildewiderMwith the properties that (a) E(˜v)<∞; (b) ˜v∗(dλ) = 0; (c) \|d˜v(0)\|= 1; (d) \|d˜v\|is bounded. But we will prove in Lemma 5 that (a) and (b) imply that ˜ vis a constant map. This contradicts with (c). QED Proof of Lemma 2.5 As in Lemma 2.4, we have u=x◦fandF=a+fiis holomorphic. If \|df\|and hence \|d˜u\|is unbounded, as above, we would have a bubble ˜ v= (v,a) with the properties (a)-(d) above. Then (b) implies that v=x1◦f1andF1=a1+f1iare holomorphic. Now (d) implies that \|da1\|and hence \|df1\|is bounded. Therefore F′ 1=∂ ∂zF1has bounded norm. Hence F1= (c1s−c2t+d1) + (c1t+c2s+d2)i and /braceleftbigga1=c1s−c2t+d1 f1=c1t+c2s+d2. Now ˜v∗(d(φ·λ)) =1 2φ′(c1s−c2t+d1){(∂a1 ∂s)2+ (∂a1 ∂t)2+ (∂f1 ∂s)2+ (∂f1 ∂t)2} =φ′(c1s−c2t+d1){c2 1+c2 2}. Ifc1orc2/\e}atio\slash= 0 (sayc1>0), then Eφ(˜v∗) = (c2 1+c2 2)/integraldisplay∞ −∞/integraldisplay∞ −∞φ′(c1s−c2t+d1)dsdt =c2 1+c2 2 c1/integraldisplay∞ −∞{φ(+∞)−φ(−∞)}dt= +∞ Hencec1=c2≡0 and ˜v=constant. But this contradicts with (c). Therefore, \|df\|and hence \|da\|is bounded. Hence F′has constant norm. We getF=cz+dagain. As above E(˜u)<∞implies that c= 0. QED Proposition 2.2 Let˜u:R1×S1(orC)→/tildewiderMbe a˜J-holomorphic map such thatE(˜u)<∞.Then there exists a c>0such that \|d˜u(x)\|<c. Proof : E(˜u)<c=⇒/integraltext R1×S1˜u∗(dλ)<c′>0. If\|d˜u\|is not uniformly bounded, then there exists a sequence xn= (sn,tn) withsn→ ±∞ such that \|d˜u(xn)\| → ∞.This will produce a bubble ˜ vwith the 10properties (a)-(d) as in Lemma 2.4. Note that (b) ˜ v∗(dλ) = 0 follows from the fact that the s-coordinate of xn= (sn,tn) tends to ±∞. As in Lemma 2.4, this leads to a contradiction. The proof for the case that ˜ u:C→˜Mis the same. QED Now assume that the contact 1-form λis generic so that 1 is not an eigen value of the Poincare ˘ are returning map at any closed orbit o fXλ.This implies that the set of unparameterized closed orbits of Xλare discrete. Proposition 2.3 Let˜u:R1×S1→/tildewiderMbe a˜J-holomorphic map with E(˜u)<∞ and˜u/\e}atio\slash=constant map. Then lims→∞˜u(s,t), when being projected to Mis either a closed orbit of Xλor a constant map. Assuming the ﬁrst case happens, then ˜u(s,t)is convergent to two closed orbits x±asymptotically with a exponential decay rate. Proof : The proof for the part concerning the exponential decay of t he last statement is given in Sec. 4. By proposition 2.2, there exists a C >0 such that \|d˜u\|< C.For any ﬁxed L >0, we deﬁne ˜ vn,L= ˜u(s+n,t) : [−L,L]×S1→/tildewiderM. Then ˜vn,LisC∞- convergent to ˜ vLafter taking a subsequence and ˜ v= ˜u(s+n,t) is locally C∞- convergent to ˜ v∞:R1×S1→/tildewiderMsuch thatE(˜v∞)<∞and/integraltext R1×S1˜v∗(dλ) = 0. Hence ˜v∞= constant map or ˜ v∞(s,t) = (x(c·t+d1),c·s+d2) withdx dt=Xλ(x) andc=/integraltext S1x∗λdt.Note that in the later case, a(˜vn(s,t))→ ±∞ asn→ ∞. We may assume that c>0, and hence a(s+n,t)→+∞asn→ ∞. Assume the second case happens. Applying the same argument t o the neg- ative end of R1×S1, we get lim n→∞p◦˜u(s−n,t)\|[−L,L]×S1=x−(c−t+d−) for some closed orbits x−ofXλof periodc−,or a constant map. Here pis the projection /tildewiderM→M.Assume again that it is not the constant map. Now/integraldisplay R1×S1˜u∗(dλ) = lim n→∞/integraldisplay {n}×S1˜u∗(λ)−/integraldisplay {−n}×S1˜u∗(λ) =c+−c−. If ˜u(s+ni,t)\|[−L,L]×S1,ni→ ∞ is any other convergent sequence, then the limit must also be a closed orbit of period c+. Letx′be the closed orbit. Under the assumption that λis generic, there are only ﬁnite c-period closed orbits of Xλ. Ifx/\e}atio\slash=x′, we can ﬁnd ( si,ti)∈R1×S1withsi→+∞and ˜u(si,ti)/\e}atio\slash∈a small neighborhood of the set of c-closed orbits. Then ˜ u(s+si,t)\|(−L,L)×S1is C∞-convergent to a constant map. This implies that/integraltext R1×S1˜u∗(dλ) =−c−and leads to a contradiction. Therefore, we conclude that xandx′are the same as unparameterized curves. Then it t is easy to see that as parametrized curve the re is also only one limit lim s→±∞p◦˜u(s,t) =x±(c±t+d±). In the case of the above limits are closed orbits, lim s→±∞a(s,t) =±∞. This can be seen easily from the explicit expression of the limit o f local convergence of the sequence ˜ vn,Lintroduced at the beginning of the proof. QED 113 Compactness LetM±be the two ends of ˜M=M×R. We consider subset of all ﬁnite energy ˜J-holomorphic maps whose two ends asymptotically approxima te to two closed orbits inM±. More precisely, given two parametrized closed orbits x±:S1→ M±≃M, let{x}±be the set of all such parametrized closed orbits diﬀer from x±byS1actions. Deﬁne /tildewiderM(x−,x+,˜J) ={˜u\|˜u:R1×S1→˜M,¯∂˜J= 0,lim s→±∞u(s,t) =x′ ±(t),x′ ±∈ {x±}}. There is an obvious 3-dimensional symmetry group acting on t he moduli space. The actions are induced from the R- translations on the target ˜MandR1×S1- action on the domain R1×S1. Note that the eﬀect of the two types of actions induced from R-actions on the target and the domain are never identical unl ess they act on the trivial ˜ u= ˜u(s,t) = (x±(t),s). LetM(x−,x+,˜J) =/tildewiderM(x−,x+;˜J)/R2×S1. •Energy : Given ˜u∈/tildewiderM(x−,x+;˜J), /integraldisplay R1×S1˜u∗(dλ) =/integraldisplay ∂(R1×S1)˜u∗(λ) =/integraldisplay S1x∗ +(λ)−/integraldisplay S1x∗ −(λ) =c+−c−, wherec±are the periods of x±. Lemma 3.1 Givenu∈/tildewiderM(˜x−,˜x+;˜J), then/integraltext R1×S1˜u∗(dλ)≥0and equality holds if and only if ˜x−= ˜x+and˜u(s,t) =x±(t). We will call such ˜ utrivial map. Therefore, if x−/\e}atio\slash=x+,M(x−,x+;˜J) does not contain trivial map and the R2-action is free. Given ˜u∈/tildewiderM(x−,x+;˜J), Eφ(˜u) =/integraldisplay R1×S1˜u∗(d(φλ)) =/integraldisplay ∂(R1×S1)˜u∗(φλ) =/integraldisplay S1φ(a(+∞))x∗ +(λ)−/integraldisplay S1φ(a(−∞))x∗ −(λ) =c+−1 2c−≥c+−1 2c+=1 2c+>0 Compactiﬁcation of M(x−,x+;˜J): •Stable ˜J-map connecting x−andx+: There are two diﬀerent ways to deﬁne this notion. One is saver but gives less information. We start with this saver one ﬁrst. The Remark 3. 2 in this section will tell us how to modify the deﬁnition here to get the more in formative one. 12Domain Σ = Σ ˜uof a stable ˜J-map ˜uconnecting x−andx+can be written as Σ = ∪iΣpi∪jΣbj,i= 1,···,P, andj= 1,···,B,of the union of domains Σ p of its principal components and Σ bof its bubble components. Each Σ por Σ b is holomorphically equivalent to S2. As a curve, Σ is semi- stable. This means that the worst singularity of Σ is double point singularity. The components of Σ form a connected tree. There are two particular marked point s−∞on Σ p1and +∞on Σ pP.on each Σ pi, there are double points di,−anddi,+such that Σ piand Σpi+1are jointed together in Σ at the double point d=di,+=di+1,−. Therefore, the domain of principal component forms a chain. These joint double points di=di,+=di+1,−are divided into two classes according to the asymptotic behavior of uwhenuapproaches di. We will use IPto denote the set of those indicesisuch thatuapproximates to some closed orbit xpiwhen it approaches di,while for the other i∈P\IP, ˜uis well deﬁned at di,+=di+1,−.Similarly, for all other double points of Σ ,we will make such a distinction. For each of the double point which is treated as inﬁnity of an end, we wi ll introduce a ﬁxS1-parameterization at the inﬁnity of the end. We will include this as part of the structure of Σ. This can be done, for example, by id entify a small neighborhood Uof some double point dwith two copies of R+×S1and using the S1-parameterization on each of R+×S1to give the desired S1-parameterization. For the later application, we mention the following ”canoni cal” way to give the S1-parameterization for the double point on each of top bubble s in the bubble tree. For each of such bubble, we ﬁrst add a marked point y, then choose another marked point z along the circle of of the radius 1 cent ered aty. The ray connecting yandzand started at ygives the required parameterization at inﬁnity. We remark that it is only the parameterization itse lf is included in the structure of Σ, not the other things used to deﬁne it. Therefo re the dimension of the symmetry group of a top bubble is three. Note that our deﬁnition of the domain of a stble map is similar to the one used in the usual Gromov-Floer theory. However if we rest ricted to the compactiﬁcation of M(x−,x+;˜J), then the domains of its elements subject to further restrictions. Although it does not eﬀect construct ions in this paper and the subsequent forth coming papers in any essential way, the se furth restrictions simplify the possible intersection pattern of domains and m ake the situation here is diﬀerent from the corresponding case in the Gromov-Floer theory. We refer the readers to Remark 3.2 of this section on this. The target Uof ˜uis a union U=∪i∈IP˜Mpiwith each ˜Mpi≃˜M. Note that here we have somewhat abused the notation as it may happe n that on each ˜Mpi, there may exist more than one ˜ upi’s. Each ˜Mhas two ends ˜Mpi,±and we identify ˜Mpi,+with ˜MPi+1,−. On each Mpi,±, there is a particular closed orbitxpi,±associated to each index pi,i∈IPand possibly some other closed orbitsxpi,bj,l,i= 1,···,P. Herebj,lare indices of the double points on bubble component Σ pi,bj. Here we have relabeled bubble component Σ bkbefore as Σpi,bj, wherepiis principal component on which the bubble Σ bklies. Note thatx−lies on the negative end of the ﬁrst ˜Mpi’s andx+lies on the positive end of the last ˜Mpi’s,pi∈IP. 13The stable map ˜ u=∪P i=1˜upi∪B j=1˜upi,bjsuch that (i) ˜upi: Σpi−{double points } →˜Mφ(pi)and ˜ubj: Σpi,bj\{double points } → ˜Mφ(pi)are˜J- holomorphic. Here φ(pi) is a function from the set of indices {1,···,P}toIP, which is the identity map when being restricted to IP. (ii) Along each end near the double point di∈Σpi, ˜upiis convergent expo- nentially to some parametrized periodic orbit xpi, ifi∈IP. Otherwise, ˜ upiis well-deﬁned at diand ˜uhas an ordinary double point at di. Similarly at each double point on bubble components or double point on princip al components other than these di’s, ˜ueither asymptotically approximates to a closed orbit x or it extends smoothly across these double points. Note that in the case that ˜uasymptotically approximates to a parameterization closed orbit along some double point, the S1-parameterization (covering) of the closed orbit is given b y theS1-parameterization of the end. (iii) On each ˜Mpi,i∈IP, there is an R1- action ofr-translation. We require that the isotropy subgroup of the components of ˜ uin˜MPiis not the entire R1. This implies that each ˜Mpicontains at least one bubble components if the principle component of ˜ uin˜Mpiis a trivial component. Here a trivial principal component of ˜ uin˜Mpiis the ˜J-holomorphic map ˜ upi:R1×S1→˜Mpisuch that ˜upi(s,t) = (x(ct),cs+d) for some periodic orbit xinM. Clearly the isotropy group of such an ˜ upiisR1itself. (iv) Each constant bubble component is stable in the sense th at it contains at least three double points. (v) ˜uconnectsx−andx+, meaning that it connects some x′ −∈ {x−}and x′ +∈ {x+}. Note that similar to the stable maps used in the usual Floer ho mology, there are two kinds of trivial components, and the trivial princip al components play similar role as closed orbits of a Hamiltonian system regard ed as trivial principal components of a stable ( J,H)-map in Floer homology. The reason to rule out this kind of components can be seen as follows. Example Let ˜u:S1×R1→˜Mbe a˜J- holomorphic map connecting two closed orbits x−andx+. Assume that ˜ uis not trivial. Hence the eﬀect of the R-actions on ˜ u induced by the R-actions on ˜Mis diﬀerent from those induced by the R-actions on the domain S1×R1. Deﬁne ˜un(s,t) = ˜u(s+n,t). Then {˜un}∞ n=0is locally C∞-convergent to ˜ u∞= ˜u∞,0∪˜u∞,1where ˜u∞,0= ˜uand ˜u∞,1:R1×S1→˜M1 with ˜u∞,1(s,t) = (x+(ct),cs+d).Iterating this process, we can produce any number of trivial principal components as limit. Note that this example also indicates that even though the en ergyE(˜u) is a constant for any ˜ u∈/tildewiderM(x−,x+;J), it is not preserved when passing to the limit. We will see that for the element ˜ uin the moduli space ˜ u∈/tildewiderM(x−,x+;J) of stable connecting maps, the energy is uniformly bounded. However, without the assumption of stability, there is no such a bound as above example shows. On the other hand, the quantity/integraltext R1×S1˜u∗dλis obviously preserved under the limit process. •Compactness 14Let M(x−,x+;J) ={[˜u]\|˜uis a stable ˜J-map connecting x−andx+;E(˜u) is ﬁnite. }. Here [˜u] is the equivalent class of ˜ u. The deﬁnition here needs some explanation. In the usual quan tum ho- mology and Floer homology in sympletic geometry, to form the moduli space M(x−,x+;J) and its compactiﬁcation, one needs to ﬁx a relative homotop y class, which is represented by the elements in these moduli s paces. Therefore, there are options here. One is to follow the usual deﬁnition, which is saver but less informative. We will leave the rutin formulation of thi s saver deﬁnition to our reader. On the other hand, the Remark 3.2 together with th e next two lemmas imply that. One still can prove the compactness witho ut restricting to a particular relative homotopy class. Deﬁnition 3.1 Two stable ˜J-map ˜u1and˜u2connecting x−,x+are said to be equivalent if there exists an equivalence φ: Σ1→Σ2of their domains and an equivalence ψ:˜U1→˜U2of the liftings of their targets such that u1=ψ−1◦u2◦φ, whereφis a homomorphism of Σ1andΣ2such that it is bi-holomorphic along each of their components and preserves the variable t∈S1along the chain of principal components and preserves the S1-parameterizations at inﬁnity along those ends of bubble components or principal components app roaching to some closed orbits, and ψis induced from R-translations on each component of the targetU, in the sense explained in the following. Here ˜U1and˜U2are certain ﬁnite liftings determined by the connected components of th e bubble tree in each of the components of U’s. Note that the components of the domain Σ of a stable map ˜ uforms a con- nected tree. If we ﬁx a component ˜Mi=˜Mof the target U, and collect those components of the domain in the bubble tree above whose image s stay in ˜Mi, the components may not be connected anymore. We will associa ted to each of a connected components Σ i,j,j= 1,···,Ji, of the domain in ˜Mi, a˜Mi,j=˜M. We collect all of these ˜Mi,jtogether with same ends as before as the lifting of Umen- tioned above. Then the ψdeﬁned above is just induced by the R-translations on each component of ˜U’s. In particular, it follows from this deﬁnition that on each component of the target of a stable map, there are as many dimensions of R1-actions as the number of the connected components of the dom ain in the component of the target. There is a special case that the above deﬁnition on connected component of the domain in a component of the target is not applicable. A ccording to above deﬁnition, if a component of the target contains a triv ial connecting map, the domain of this map clearly is an isolated component in the domain of the orignal map inside the component of the target. For the obvio us reason that one can not assign an extra R1-symmetry of the target associated the the trivial map. However, as we will prove in this section that on the comp onent of the target, there exists at least one non-trivial connected com ponent of the domain. 15We simply deﬁne a connected component in this case as the unio n of the non- trivial one with the domain of the trivial map. In the case tha t there are several non-trivial connected components together with se veral trivial maps in one component of the target, we consider all possible combin ations of them and consider this as part of the data in the deﬁnition of stable ma p. Another way to deal with this particular case is to only count the R1-symmetry of the domain for each of such stable component. With this interpretation, we note that with respect to the sy mmetry group so deﬁned, the isotropy group of a stable map is always ﬁnite, which is important for deﬁning the virtual moduli cycles in [L1] and will be prov ed there. Theorem 3.1 M(x−,x+;J)is compact and Hausdorﬀ with respect to the C∞- weak topology, which is a compactiﬁcation of M(x−,x+;J). Proof : The Hausdorﬀness follows from the stability of elements in ¯M(x−,x+;J). The proof of the corresponding theorem in [LT] can be easily a dapted here. We refer our readers to the proof there. Lemma 3.2 There exists a constant N=N(x−,x+)such that for any ˜u∈ [˜u]∈M(x−,x+;J), the number of components of uis less than N. Proof : By using local convergence, one can easily show that there ex ists a ﬁxed ǫ >0, such that for any non-trivial bubble ˜ ub,Eλ(˜ub)> ǫ.Same conclusion holds for non-trivial principal component ˜ up:S1×R1\ {doublepoints } →˜M. Note that here we used x−/\e}atio\slash=x+. Since Eλ(˜u) = Σ b,pEλ(˜ub) +Eλ(˜up) =/integraldisplay S1x∗ +λ−/integraldisplay S1x∗ −λ=c+−c− is ﬁxed, we only need to prove that the number of trivial princ ipal components and trivial bubble components is uniformly bounded. Now for each trivial principal component, there exists some non-trivial bub- ble components lying on the same target component. Therefor e, the number of such components is less than or equal to the number of non-t rivial bubble components, which is bounded. Finally, it is easy to see indu ctively that the number of trivial bubble components can be uniformly bounde d by the number of non-trivial bubbles and principal components. Lemma 3.3 Given ˜u∈[˜u]∈M(x−,x+;J),ifxis a closed orbit such that it is an intermediate end of some component of ˜u. Then/integraltextx∗λ </integraltextx∗ +λ=c+. Therefore, there are only ﬁnite such closed orbits. 16Proof : x+⊂Mp,+is the only closed orbit lying on the positive end of ˜Mp, where ˜Mpis the most right (positive) components of the target of ˜ u. Then /integraldisplay x∗ +λ−Σi/integraldisplay x∗ −,iλ=/integraldisplay u\|˜Mpdλ>0, wherex−,iis one of the closed orbits on ˜Mp,−appeared as a non-trivial end of ˜u. The conclusion follows by induction. To see that there are only ﬁnite such intermediate x, we use Remark 3.2. It follows from the remark there that/integraltext x∗λis bounded below by ǫ >0 of the lower bound of the Eλ-energy of non-trival bubbles. QED It follows from this argument that the number of double point s of a compo- nent of a stable map appeared in the compactiﬁcation is also b ounded. Because of these lemma, the proof of the theorem essentially can be reduced to the case that [˜ ui]∞ i=1has only one component and we need to show that such a sequence has a weak-limit [˜ u∞]∈/tildewiderM(x−,x+;J). Note that the above two lemmas together implies that the ener gy ofE(˜u) is uniformly bounded for any ˜ u∈/tildewiderM(x−,x+;J). •Stabilization of the target and its local deformation The target of ˜ uof a stable map is a union U=∪i∈IP˜Mpi. Each ˜Mpi≃˜M= M×Rwith a R1-symmetry coming from the r-translations along the second factorR. To stabilize U, we add a marked point zion the second factor of ˜Mpi to remove theses symmetries. Given ( ˜Mi,zi) =M×(R,zi),i= 1,2 andτ∈R+ of a deformation (gluing) parameter, we can form Uτ= (˜M1,z1)#τ(˜M2,z2) of the local deformation of U= (˜M1,z1)∪(˜M2,z2) with respect to the parameter τby the obvious gluing construction along the second factor o f˜M1and˜M2, namely, cutting oﬀ r1>1 τof˜M1andr2<−1 τand gluing back the remaining parts. Then Uτ≃M×Rwith two marked points z1andz2onR. Similarly, ifU=∪i∈IP˜M0iandτ= (τ1,···,τγ−1)∈(R+)γ−1,γ= #(IP), withγmarked pointszi∈˜Mpi,i= 1,···,γ,we can form Uτ=˜Mp1#τ1˜Mp2#···#τγ−1˜Mpγ with marked points z1,···,zγon it. Another way to think this is to treat the marked point zias a marked section M× {zi}in/tildewiderMi. A cylinder ˜M=M×(R;−∞,+∞,z1,···,zn) with two end points −∞and +∞andndistinct marked points z1,···,znis said to be stable of type Mif n≥1. let Mn(M) be the collection of all such stable cylinders of nmarked points of type M. Then it has an obvious compactiﬁcation ¯Mn(M) =Mn(M)/coproductdisplay l+m=n l, m≥1Ml(M)× M m(M)). The topology of ¯Mn(M) near the boundary points is described by the local deformation (gluing) above. 17•Weak Convergence •Stabilization of a semi-stable curve and its local deformat ion: Domain Σ of a stable map ˜ uis only a semi-stable curve. Therefore, there may exist some non-trivial bubble components or principal c omponents whose domains contain only one or two double points. We can stabili ze these unstable components by adding minimal number of marked points y= (y1,···,ym) to get a stable curve (Σ ,y).In particular, for each top (hence, unstable ) bubble, the symmetry group is three dimensional because of extra str ucture of the S1- parametrization at inﬁnity along its end. To stabilize such a component we introduce an arbitary marked point y1ﬁrst. Then the S1-parametrization at inﬁnity together with the marked point determine a marked ra y connecting y1 toθ= 0 atS1at inﬁnity an obvious way. We add the second marked point y2 on the marked ray with distance of 1 to y1to get the desired stablization. Let (Σα,y) the local deformation of (Σ ,y) in the moduli space of stable curves, where αis the collection of deformation parameters associated wit h double points of Σ.Note that the moduli space of stable maps used here is not the u susal Degline-Mumford compactiﬁcation but an obvious modiﬁcati on of the moduli space of stable ( J,H)-maps used in [LT]. Here for each ordinary double point of ˜u, we associate it with a complex gluing parameter and for each double point corresponding to an end approaching to closed orbit, we asso ciate a positive real gluing parameter. To see this more concretely, we consider the following examp le. Example Consider the semi-stable curve Σ = P∪d1=d2Bwith principal component P=R×S1and bubble Bjoint at the double point d. Assume that the double point corresponds to the two end on PandB. Not that on Pthere are other two marked points corresponding to −∞and + ∞inR×S1. The moduli space of such semi-stable map is 1-demensional due to the choices of S1-parametrization at d. To stable such a Σ, we only need to stabilize B, which is described above. Associated to the double point, there is a one dimensional local deformation of Σ with respect to a gluing parameter α∈R+. By letting Σ vary in above 1-dimension moduli space, the deformation gi ves the elements in the moduli space M0,4, which form a neighbourhood the the 1-dimensional moduli space. Therefore, the above 1-dimensional moduli sp ace can be thought as part of the boundary of M0,4. Of course, this not the usual Degline-Mumford compactiﬁcation of M0,4. On the other hand, give any sequence Σ i∈ M 0,4with Σi= (R×S1;−∞,+∞,yi,ziwithyi→zias in the bubbling below, the process there will give a limit of the sequence in the above 1-dimensi onal moduli space. •Deﬁntion of Weak Convergence : Given [˜ui]∞ i=1∈M(x−,x+;˜J), we say that [˜ u] is weakly C∞-convergent to [˜u∞]∈M(x−,x+;˜J) if there exist ˜ ui∈[˜ui] and ˜u∞∈[˜u∞] such that (i) After adding some marked points yito Σ i, (Σ i,yi) is convergent to the minimal stabilization of (Σ ∞,y∞) in the moduli space of stable curves, where Σiand Σ ∞are the domains of ˜ uiand ˜u∞. Note that the number of marked pointsyi’s is same as the number of marked points y∞.Therefore, when iis 18large enough, there exists an αisuch that (Σ i,yi) is equivalent to (Σ ∞,αi,y∞) of the deformation of (Σ ∞,y∞) with respect to the gluing parameter αi. Letφi: (Σ∞,αi,y∞)→(Σi,yi) be the equivalence map. (ii)LetUi=∪j∈IPi˜Mpi,jandU∞=∪j∈IP∞˜Mp∞,be the targets of ˜ ui∈ [˜ui]∞ i=1and ˜u∞∈[˜u∞]. We stabilize U∞by adding minimal number of marked pointsz∞and require that after adding same number of makred points zito Ui, (Ui,zi) is convergent to ( U∞,z∞) in the space of ¯Mn(M). Herenis the number of marked points of z∞. Therefore, there exists τisuch that (Ui,zi) is equivalent to (( U∞)τi,z∞). Letψi: (Ui,zi)→((U∞)τi,z∞) be the equivalence map. Note that for ui closed tou∞, the gluing parameter αiof the domain (Σ i,yi)≡((Σ∞)αi,y∞) is not compleletely independent of the gluing paprameter τiof the targe (( U∞)τi,z∞) since along these ends where [ u∞] approaches closed orbit αi= 0⇐⇒τi= 0. However, when αi/\e}atio\slash= 0, hence τi/\e}atio\slash= 0, they are essentially independent each other. (iii) Given a compact set K⊂Σ∞\ {double points }, the compact image ˜u∞(K)⊂U∞\ {end ofU∞}. Hence for ilarge enough, ˜ u∞(K)⊂(U∞)τi≡Ui. Therefore,ψ−1 i◦u∞is well-deﬁned on Kand it maps KintoUi. On the other hand, for large i,K⊂(Σ∞)αiandφi(K)⊂Σi, and ˜ui◦φi:K→Ui. We require that (a) ˜ ui◦φi\|KisC0-close to (ψ−1 i◦˜u∞)\|Kwheniis large enough, hence,ψi◦ui◦φi\|K:K→U∞is well-deﬁned. (b) for any compact subset K⊂Σ∞\ {double points },ψi◦˜ui◦φi\|KisC∞-convergent to ˜ u∞\|K. Note thatEλ(˜ui) =Eλ(˜u∞) =c+−c−is ﬁxed. This together with the two statements of Sec. 4. imply that the projections of the im ages of ˜uito the contact manifold MisC0-close to the projection of the image ˜ u∞, and that near a closed orbit xwithλ-periodcas an asymptotic end of ˜ u∞, along the non-compact R-direction of ˜M, ˜uiis essential same as the function c·s, when iis large enough. We start with a detailed description on the case that the sequ ence [˜ui] only develops one bubble, as it already exhibits all of the main po ints of the general case. The following lemma plays an important role both in the proof of this theorem and in the later formal dimension counting of the bou ndary of the modui space of ˜J- holomorphic maps. Lemma 3.4 If{[˜ui]}∞ i=0∈ M(x−,x+;˜J)develops only one bubble at its limit [u∞], then the target U∞contains at least two elements and the image of bubble is not in the right most component. This implies that the doma inΣ∞ofu∞ contains at least three components. Proof : Let ˜ui∈[˜ui], ˜ui= (ui,ai) : Σ i=R1×S1→/tildewiderM, whereui:R1×S1→M andai:R1×S1→R. By assumption, there exists bubble point yi∈Σi such that \|d˜ui(yi)\| → ∞ . First assume that yistays in a compact set of Σi=R1×S1, henceyi→y∞∈R1×S1asi→ ∞ after taking a sub- sequence. We claim that \|ai(yi)\|is not bounded and ai(yi) tends to −∞. Otherwise, assume that \|ai(yi)\|< C. Thenai(yi)→a∞,y∈R.Now the 19domain of Σ ihas two marked points yiandwi, wherewiis a point on the circle centered at yiof radius1 \|d˜ui(yi)\|. Note that here we can make an arbitary choice forwion the circle. See the remark after on how to make ”correct” choice. These two marked points yi,witogether with −∞,+∞onS2= Σ i∪ {−∞,+∞}have moduli, and we can identify the domain (Σ i,−∞,+∞,yi,wi) with (Σ i,−∞,∞,d)#αi(S2,0,1,d′). Here (Σ i;−∞,∞,d)#αi(S2; 0,1,d′) is ob- tained from (Σ i;−∞,∞,d)/logicalortext d=d′(S2; 0,1,d′) by gluing at dwith some deforma- tion parameter αi∈R+withαi→0 asi→ ∞,and (Σ i;−∞,∞,d)/logicalortext d=d′(S2; 0,1,d′) is one of the elements in the 1-dimensional moduli space ment ioned in the pre- vious example. In particular, along the end d, there is a S1-parametrization. Intuitively, what we did here is to conformally enlarg a small disc of Σ inearyi, bringingyi,wiinto standard points 0,1 in standard disc. Now (Σ i#αiS2;−∞,∞,0,1) has four marked points −∞,∞,0,1, and (Σi#αiS2;−∞,∞,0,1)≃(Σi;−∞,∞,yi,wi). Letφi: (Σ i#αiS2;−∞,∞,0,1)→(Σi;−∞,∞,yi,wi) be the identiﬁcation map. LetDRbe the half shpere glued with a ﬁnite cylinder S1×[0;R] along its boundary. We still use DRto denote its obvious conformal image in ( S2; 0,1,d′)⊂ (Σi;−∞,∞,d)/logicalortext d=d′(S2; 0,1,d′) centered at 0, and DR,ithe corresponding im- age in Σ i#αiS2wheniis large enough. Deﬁne /tildewideVi,R= (˜ui◦φi)\|DR,i. Then as we did before for bubbling, ˜ vi,R→ ˜v∞,R:DR→/tildewiderMand ˜vi= ˜ui◦φiis locallyC∞-convergent to ˜ v∞=∪R˜v∞,R: D∞=D2∪(R+×S1)→/tildewiderM. That is {˜ui}produce a bubble at yi. The domain of ˜v∞is the complex plane but thought as half sphere with a half inﬁ nite cylinder attached. Since ˜ v∞is˜J-holomorphic and E(˜v∞)<∞,\|D˜v∞\|is uniformly bounded. As before, lim s→+∞v∞(s,t) =x(t) of some periodic orbit along its cylindrical end. Now ﬁx ǫ >0, and consider ˜ ui,ǫ= ˜ui\|Σi\Dǫ(yi).By our assumption that there is only one bubble we conclude that for any ﬁxedǫ>0, \|d˜ui,ǫ\|< C ǫfor anyi.We may assume that lim i˜ui(0,0) exists at the begining andyi/\e}atio\slash= (0,0).Then the same argument as before implies that ˜ ui,ǫisC∞- convergent to ˜ u∞,ǫ:R1×S1\Dǫ(y∞)→/tildewiderM.Here we used that \|yi\|is bounded and henceyi→y∞∈R1×S1.By lettingǫ→0, we get ˜ui\|Σi\{yi}is locallyC∞- convergent to ˜ u∞\|R1×S1\{y∞}.Identifying Dǫ(y∞)− {y∞}(⊂R1×S1− {yi}) withR+×S1,then lim s→∞u∞(s,t) =x′(t) of a closed orbit. Let ˜v∞= (v∞,b∞). Then lim s→∞b∞(s,t) = +∞. Otherwise, since b∞(s,t)∼ cs+dwithc=/integraltext S1x∗λ/\e}atio\slash= 0, we have lim s→∞b∞(s,t)→ −∞.But since ∆b∞≥0, this contradicts to the maximal principle for sub-harmon ic functions. Therefore,b∞(s,t)∼cs+dwithc=/integraltext S1x∗λ>0.The induced orientation of ˜v∞onxis the same as the one given by λ.By our assumption that there is only one bubble, if we set ˜ u∞= (u∞,a∞), then lim s→∞a∞(s,t) = + ∞, lims→∞u∞(s,t) =x′(t) =x(t).Here, (s,t)∈R+×S1=Dǫ(y∞)\ {y∞}. This implies that the induced orientation on x′(t) =x(t) form ˜u∞is also the same as the one given by λ.However, ˜u∞∪˜v∞is the weak limit of ˜ uiand x(t) =x′(t) is the limit of some corresponding curves xiin ˜ui. Clearly, the 20induced orientations of xiobtained from the two sides of ˜ uiare opposite to each other. This is a contradiction. We remark that one can also get an alternative proof of above s tatement by using gluing in [LT] and maximal principle instead of usin g this orientation consideration. This proves that \|ai(yi)\|is not bounded under the assumption that \|yi\|is bounded. In the case that yi→ ±∞ , (R1×S1;yi,0,−∞,∞) tends to a bound- ary point of moduli space ¯M0,4. If, say,yi→ ∞, let (S2;−∞,y∞,d)/logicalortext d=d′(S2,d′,0,+∞) be the limit curve. Then ( R1×S1;−∞,yi,0,∞)≃(S2#αiS2;−∞,y∞,0,∞) for someαi∈C∗. Now inS2#αiS2,y∞plays the same role yiin Σ ibut it stays away from the two ends. The above argument is still applicabl e except that at the limit, the domain has one more splitting. Therefore,ai(yi)→ ±∞ . Ifai(yi)→+∞, after shifting by −ai(yi) to the target of ˜uiand deﬁne ˜ wi= (ui,ai−ai(yi)), the above argument is still ap- plicable to ˜ wi,and we get bubble at y∞, still denoted by ˜ v∞= (v∞,b∞). In particular, lim s→∞b∞(s,t) = + ∞.Therefore, we get a bubble as before but with target /tildewiderM′lying on the right of /tildewiderMwith the end of the bubble approach- ing a closed orbit lying on the right end of /tildewiderM′. As before, the orientation consideration and maximum principal rule out this possibil ity. Therefore, ai(yi)→ −∞ . Of course, we still can deﬁne ˜ wiby the same formula above. Arguing as before, we conclude that we still g et a bubble from {˜wi}∞ i=1, still denoted by ˜ v∞,D∞=D2∪(R1×S1)→/tildewiderM′. But the target /tildewiderM′lying on the left end of /tildewiderM, and lim s→∞v∞(s,t) =x(t) in the right end of/tildewiderM′. For simplicity, assume that there is no further splitting o f the target. (This follows form our assumption that there is only one bubb le if we also count ”connecting bubbles”.) Then as before, ˜ wi\|Σi−{yi}is locallyC∞- convergent to ˜w∞\|R1×S1−{y∞}(again assume ﬁrst that \|yi\|<cand use deformation as before to deal with general case), and along the end D(y∞)− {y∞} ≃R1×S1, lim s→∞˜w∞(s,t) =x′(t) =x(t)∈/tildewiderM−=/tildewiderM′ +. To see that there is at least one more component of the domain i n the limit of [˜ui], we note that each uiconnectsx−∈/tildewiderM−tox+∈/tildewiderM+, therefore, there exists sisuch that ˜ui\|(−∞,si]×S1lies on the ”left” of ˜ ui(yi). Now ˜ui,−R= ˜ui\|(si−R,si)×S1isC∞-convergent to ˜ u∞,−Rafter identifying ( si−R,si)×S1 with ( −R,0)×S1.We get ˜u∞,−∞=∪R˜u∞,−R: (−∞,0)×S1→/tildewiderM′. We only need to show that for some R, ˜u∞,−Ris not constant map. However, since ˜ ui asymptotically approximates to x−with exponential decay as stends to −∞. More precisely, we have \|ai(s,t)−(cs+di)\|<e−kisfor someki>0. Note that c=−/integraltext S1x∗ −λ/\e}atio\slash= 0 is the same for all i. Therefore we can replace ˜ u1(s,t) by ˜ui(s+si,t) with some very negative sisuch that \|d˜ui\|[0,R]×S1\|>ǫ> 0 for some ﬁxedǫ. Then the above limit ˜ u∞,−∞is not constant. This proves the lemma. Note that the component ˜ u∞,−∞of the limit could come from a closed orbit, i.e. it is a trivial principal compo nents. However, in this case, there is a bubble component lying in the same com ponent of the target. 21We remark that in the general case with multi-bubbling, the s ame proof above proves that each of bubbles lie on some new component of targets which lie on the left of the original ˜M. Moreover, there is at least one more principal component lying on the the new ”left” component. In particul ar, in the ”new” component of the target, where the ﬁrst top bubble lies on, th ere are at least two connected components of the domain of the limit. QED Remark 3.1 Some remark on the special role played by the marked point (yi,wi) in the above lemma and some related issue is in order. Recall t hatyiis the point where \|du(yi)\| → ∞ andwiis the point lying on the cirle of radius1 \|du(yi)\|mea- sured in the standard metric on R×S1. In the process of bubbling we bring (yi,wi)into the standard point (0,1)inS2. On one hand, the point wiwill be used to determine the side of bubbling at each stage, on the ot her, it will also determine two marked lines on the two ends of Σ∞joint at the double point of Σ∞. Since the two components v∞andu∞of the limit approach to a closed orbit{x}, these two marked lines will specify the base point 1∈S1and hence we get a particular paremetrized x:S1→Min{x}. However, in the bubbling one can make arbitary choices for withe the cirle. This implies that in the com- pactiﬁcation below, we can use ﬁxed parametrized closed orb its as asymptotic limit to which bubble components, and hence the adjecent pri ncipal components, approach along their parametrized ends. On the otherhand, t he diﬀerent S1- paramerizations associated to each of such ends contributs an one dimensional moduli to the domain of the limit stable map. An equivalent wa y to think about this is to ﬁx an S1-parametrization for each of such ends of the limit curve. Then we can not ﬁx the parametrizations of limit closed orbit s anymore. Note that in the case of splitting of principal components, a s the maked lines are already ﬁxed a priori, clearly, all elements in {x}may appear in the limit. Remark 3.2 The proof of this lemma can be used the deduce some furth re- strictions on the possible domains of stable maps, which app eared in the com- pactiﬁction of M(x−,x+,J)(meaning as a limit of some sequence of elements inM(x−,x+,J)). There are two general requirements. The ﬁrst one is that th e maximum principal for the a-component of the stable map must hold (as well as the closed related orientation consideration should be inc orperated). The second is that there is no loop in the set of components of a domain. Ap plying these two requirements to the case that there are two connected com ponents of a sta- ble map lying same component of the target with one ordinary d ouble point joint the two components of the domain, one conclude that each conn ected component has at least one end lying on the positive end of the component of the target. Starting from this, inductively one can prove that in the rig htmost (positive) component of the target, there are at least two closed orbits on the positive end of the component of the target, which appeared as the asympto tic limits of the 22stable map. However, the orintation consideration as in the proof of the last lemma ( or the maximum principal plus gluing), implies that t his is impossible. Therefore, we conclude that all double points of a stable map in the com- pactiﬁcation are ends. The same consideration also implies the following simple picture one the structure of the components of a stable map ap peared in the compactiﬁcation. Starting from the leftmost component who se ”left” asympo- totic end is x−, there exist one and only one end of this component, along which the component approaches to a closed orbit x1on the positive end M+,1 of component of the target. It is easy to see that all the other ends of the compo- nents must lie on the negative end of the component of the targ et. In this case, since we are already in the leftmost component, this is impos sible. However, this can happen in general case and we will use this to do induction in a moment. If the next adjecent component lying on the adjecent compone nt of the target, we are in the same position as before and we can inductively go further. We now show that this must be the case. Otherwise, the new compon ent still stay in the same component of the target, then the induced orientati on onx1from the two adjecent components are the same, which is a contradicti on. We conclude that there is a chain of components ( should be called princip al components ), each lying on diﬀerent but adjecent components of the target and each connect- ing two closed orbits on the two diﬀerent ends of the componen t of the target. As mentioned above, for each of the principal components, al l the other ends (if there are any) must lie on the negative end of the component of the target by maximum principle and gluing. To get a complete picture, we need to know the behavior of thos e adjecent components to those negative ends of, say, a typical princip al component. The orientation consideration implies that each of such compon ents must lie in the left adjecnt component to the component of the target, on whi ch the principal component lies. The maximum principle and gluing implies th at the end at which the principal component and the new adjecent componen t joint together is the only positive end for the new component. Now we are in th e position of induction and we get a very simple structure on the compone nts of a stable map which appears as a limit map. Namely, each component of a s table limit connecting map has only one positive end and possibly many ne gative ends with- out any ordinary double points. Starting form the (only) rig htmost end x+, all components of the stable map form a tree pointed to negative a-direction. It follows from this that for each intermediate closed orbit x, which appears as an end of the limit stable map connecting x−andx+,/integraltext x∗λis bounded above by/integraltext x∗ +λand bounded below by the minimum of/integraltext x∗ −λandǫ, the lower bound of theEλ-energy of non-trival bubbles. This is used in the proof of Le mma 3.3. To prove the compactness in general, as in the usual Gromov-W itten theory or Floer homology, there are three steps (i) formation of all bubbles which lie on the top of the bubble tree; (ii) local convergence of the se quence of {˜ui}∞ i=0 along the base, including splitting or degeneration of prin cipal components; (iii) formation of the intermediate bubbles and related ”ze ro bubbling” along connecting necks. Most of analytic part of the proof for thes e are the analogy 23to the symplectic case, except the two statments concerning the exponential decay of a bubble along its non-removable singularity and an d the behavior of ”connecting neck” along the non-compact R-direction, detailed in Sec.4. We will only outline the those parts whose proof are similar to t he symplectic case. To do the step (i), we proceed inductively as in the usual symp lectic case. The proof of the above lemma serves as the staring point of the induction. During the formtion of the ﬁrst bubble, the domain of ˜ uiis deformed into (Σi;y1 i,w1 i), wherey1 i,w1 iare the maked points denoted by yi,wiin the pre- vious lemma. But we think Σ iasR1×S1with a small disc centered at yi removed, then gluing back a portion of a cylinder, [0 ,Ri]×S1with a half sphere attached. In this model of (Σ i;y1 i,w1 i), the maked points y1 i,w1 ibecomes the standard points 0 ,1 in the half sphere. Here Ri=1 αandαiis the deformation parameter in the Lemma before. The target ˜Moriginally has three marked sections −∞,+∞,0. We introduce a new marked section z1 i=ai(yi), whereai is the second factor of ˜ ui. As proved above, z1 i<0 and \|z1 i−0\| → ∞ asi→ ∞. We then check that if \|d˜ui\|measured in the induced metric in the new deformed domain is uniformly bounded. Assume that is it not. Since the injective radius of these new domains are bounded below, we c an repeat the process before to produce second bubble by introducing new m arked points y2 i,w2 iin the domain and marked section z2 iin the taget which play the same role asy1 i,w1 iandz2 iin the formation of the ﬁrst bubble. As each bubble has a minimal amount of Eλ-energy bounded below, this process will stop after ﬁnite steps. We end up with a deformed new domain (Σ i;y1 i,w1 i,y2 i,w2 i,···yk i,wk i) of ˜ui. As above, we think it as R1×S1withk-small disc centered at yj i,j= 1,···k removed, then gluing back a portion of a cylinder, [0 ,Rj i]×S1with a half sphere attached. As before the maked points yj i,wj iin (Σ i;yj i,wj i;j= 1,···,k), becomes the standard points 0 ,1 in thesek-half spheres. The target ˜Mof ˜ui now has marked points −∞,+∞,0,zj i,j= 1,···,k. Now\|d˜ui\|measured in the induced metric in the new deformed domain is uniformly bounded. Let Dj i,R,j= 1,···,k,be one of the khalf spheres centered at 0 =yj iwith a portion of a cylinder of length Rattached in the deformed domain Σ iandDi,Rbe their union. We will use Bi,Rto denote the subset of Σiobtained by removing a small disc around each of those yj iwhich produces a ”top” bubble, and then gluing back a cylinder of length R. Then for any ﬁxed R, ˜ui\|Di,RisC∞-convergent. By letting R→ ∞, we obtained all top bubbles. On the other hand, by restricting ˜ uito part ofBi,Rof, say, length Rand shiftng the target with a suitable constant, we get the local convergence along the ”base” after letting Rtend to inﬁnity. Note that in the local convergence of the base, the domain may splitting further into broken con necting maps. It is possible that only one of the two ends of some component of s uch a broken connecting map approaches to a cloed orbit, the other is just a double point. Note also that during the process of these local convergenes and bubbling, the target also gets split into severl components. For example i n the case that each of the distances between these kmarked sections zj i,j= 1,···,ktends to inﬁnity, the target of the limit has at least k+ 1 components. This essentially 24ﬁnishes the ﬁrst two steps (i) and (ii). It may happen that for some of Bi,R, the limt of the local convergence is only a constant map. In oder to obtain a meaningful limit alon g the ”base”, one has to show it is possible to get a sequence of consective n on-trivial limit connecting x−andx+. The key point to prove this is to observe that one can have isoperemetric inequality and monetonicity lemma for e ach ˜uiprojecting to ξin a small neighbourhood of each point of ˜Mas in the usual symplectic case. Now since each ˜ uiconnectsx−andx+, and approaches to some of closed orbits along the ends of the ”base”, its image projecting to ξis not very small. This implies that the non-trivial limit of above local convergen ce can be obtained. The of the analogy argument in symplectic case is used to prod uce intermediate bubbles, which can be found in [L?]. We refer the readers to th e detail there there, which can be easily adapted here. To do the step(iii), we deﬁne the potential ”connecting bubb le”Ci,R= Σi\Di,R∪Bi,Rfor ﬁxedR. Each componet Ck i,RofCi,Ris a sphere with several small discs removed and cylinder attached, and conn ects the components ofDi,RandBi,R. We may asssume that lim R/mapsto→∞limi/mapsto→∞Eλ(˜ui\|Ci)/\e}atio\slash= 0. Then we get those intermediate connecting bubbles by local conve rgence of ˜ui\|Ck i,R withR→ ∞.As mentioned above isoperemetric inequality and monotonic ity lemma for ˜ uiprojecting to ξcan be used to produce non-trivial connecting bubbles. After this is done, we have lim R/mapsto→∞limi/mapsto→∞Eλ(˜ui\|Ti) = 0, where Ti= Σ\ (Bi,R∪Di,R∪Ci), i.e. there is no Eλ-energy loss any more. We have got the full limit of the sequence ˜ uialong the compact direction. This is the projection of the sequence to the contact manifold Mis already weakly convergent to the projection of the limit map so far obtained. To get the full limit along the non-compact bfR-direction, we observe that since there is no Eλ-energy loss anymore, given any two of ends of any of above three parts, if presumely they should joint together in the d omain acording the above convergence scheme, but they apporach to two closed or bits which lie on diﬀerent ends of the target ( maybe in the diﬀerent component of the target also), then the two closed orbits are the same, and we get triv ial connecting map between them ( maybe passing through several components of the target) as part of the limit. Note that only in the case there is alread y some non-trivail component lying on some component of the target, we may have t o introduce this kind of trivial connecting maps in the component in orde r to get a connected stable map. Therefore, the limit map so obtained is really a s table map deﬁned before. Finally, we note that in the next section we will prove that wh enRandilarge enough, each component Tk iofTi, whose domain is equivalent to [ −Rk i,+Rk i]× S1, is exponetially close to the trivial ˜J-holomrphic map coming from some closed orbit xwhenTk iapproaches to x. QED •Virtual co-dimension of the boundary of M(x−,x+;J): 25Theorem 3.2 The virtual co-dimension of the boundary components of M(x−,x+;J) is at least one. In fact, the co-dimension of the stratum of br oken connecting ˜J-holomorphic maps of two elements is one , and co-dimension o f any stratum whose elements contain bubble component is at least two. proof The proof of this theorem depends on the index formula, which will be proved in [L3]. Let [˜u] be a typical element in the stratum. It is suﬃcient to consid er the follow two cases: (i) The domain Σ of uis Σ1∪Σ2joint together at one of the ends of Σ 1 and Σ 2. Each Σ i,i= 1,2 isS2with two marked points −∞and + ∞treated as ends, and we identify Σ i\ {end}withS1×Rto give two marked lines on Σ. The target Uof ˜uis˜M1∪˜M2joint at one of their ends. ˜ u1connects a closed orbit x−,1on˜M−,1and another closed orbit xon˜M+,1=˜M−,2, and ˜u2 connects the closed orbit xand another closed orbit x+,2on˜M+,2. Note that x/\e}atio\slash=x−,1/\e}atio\slash=x+,2,andx/\e}atio\slash=x+,2.There are ﬁve dimensional symmetries for each element [˜ u] is the above stratum, two dimension coming from the R1-translations on each factor of the target and two dimension alR1- translations on each factor of the domain together with an S1-action on the domain. We will slice out the S1-action ﬁrst. Let ˜M(x−,{x},{x+};J) be the moduli space of marametrized broken connecting ˜J-maps of two elements as above. But we ﬁx a parametrized x−and alowxandx+vary in their equivalent classes. The dimension of the symmetry group of the moduli sp ace is 4. It follows from the index formula in [L3] that the dimension of ˜M(x−,{x},{x+};J) is same as the dimension of ˜M(x−,{x+};J) plus one, due to the one dimensional possible choices of the element x∈ {x}. Now a direct dimension counting on the symmetries shows that in this case the codimension the the boundary component of M(x−,x+;J) is one. (ii) The second case corresponds to the case that there is onl y only one bubbling as described in Lemma 3.4. There are two diﬀerent su bcases: (1) both of the principal components are non-trivial; (2) the ”new” p rincipal components is trivial. In the case (1), along the princial component, as parametrized map, there are three diﬀerent possible parametrized closed orbi ts as asympotic limit along ends, but is the case (2), there are only two of such clos ed orbits. On the other hand, the dimension of the symmetry group of the two components lying in the ”left” component of the target ( not counting the S1-action) is 6 in the case (1) and 5 in the case (2). Note that in the case (1) th ere are two connected components in the ”new’ components of the target, while in the case (2) there is only one according to our convention introduced before. Again index formula in [L3] together with a direct dimension counting ar gument gives the desire conclusion in this case. QED 264 Exponential Decay Estimate We have proved a version of compactness theorem for the modul i space of stable ˜J-holomorphic maps in last section. The result is not quite co mpleted for its own ppurpose as well as for later applications. As we have sho wn before that a sequence of ˜J-holomorphic maps may develop bubbles and split into broken connecting ˜J-holomorphic maps. Unlike the usual Gromov-Floer theory, t hese bubbles always have unremovable singularities. We showed b efore that along the ends of singularities, the bubbles approach to some clos ed orbits. For the purpose of moduli cycles in [L1], it is important to know the r ate of the ˜J- holomorphic maps approach to closed orbits either along the ir ends or along the ends of the singularities of the bubbles. One of the main r esults of this section is to prove that the rate of the approximation is expo nential. When dimM= 3,this is proved by Hofer, Wysocki and Zehnder in [HZW]. When Mhas anS1-symmetry, this is proved by Li-Ruan in [LiR]. We remark that the extra assumption of [LiR] considerably simpliﬁed the an alysis here. On the other hand, the general case, even in dim M= 3, the argument in [HZW] is quite involved. It turns out that the method of [HZW], suitab ly modiﬁed, can be extended to the general case. We will carry out this genera lization somewhere else. In this section, we will give a more abstract and a simpl er proof. To motivate the second main result of this section, we note th at one of the important ingredients of the proof of compactness of the mod uli space in the usual Gromov-Floer theory is an explicit description about the behavior of the ”connecting neck” near bubble point. In our case, it is necessary to know that the behavior of the ”c onnecting necks” near the ”connecting” closed orbit when a family of ˜J-holomorphic maps develop, say, a bubble approaching to the ”connecting” clos ed orbit, or split into a broken ˜J- holomorphic maps of two elements joints at the closed orbit . More precisely, if ˜vi= ˜ui\|[−li,li]×S1: [−li,li]×S1→˜M=M×R is the ”neck” part of ˜ uisuch that the M-projection viis close to the closed orbit x(t) withc=/integraltext S1x∗λ.We claim that ˜ viis essentially the same as the trivial map (s,t)→(x(t),cs)∈M×Rrestricted to [ −li,li]×S1. In particular, the length ofR1- projective of ˜ vidiﬀers from 2 c·liby at most a ﬁxed small constant. Note that wheni→ ∞,li→ ∞.It turns out that this statement plays an important role in the compactness theorem. Recall that we have require d that in the deﬁnition of stable map, there is no unstable trivial connec ting maps appeared as components. The justiﬁcation of this is based on the above statement. Letx(t) be a closed orbit. ˜ u= (u,a),˜w= (w,b) are two ˜J-holomorphic con- necting maps: R1×S1→˜Msuch that lim s→+∞u(s,t) =x(t) = lim s→−∞w(s,t). Assume that lim s→−∞u(s,t) =x−(t) and lim s→+∞w(s,t) =x+(t).Let ˜v∗ i= (v∗ i,f∗ i) :R1×S1→˜Mbe a sequence of ˜J- holomorphic maps connecting x−(t) andx+(t) and locally convergent to ˜ u∪˜w. Hence, lim s→+∞u(s,t) = lims→−∞w(s,t) =x(t) of some closed orbit. Note that the target ˜Mof ˜uand ˜wshould be thought as two diﬀerent spaces joint together at th eir ends. We 27will only prove our results for this particular case. It is ea sy to see that the cor- responding results for the case that ˜ v∗produces only one bubble can be proved in an exactly the same way and the result for the general case c an be obtained by a simple combination of these two cases. The assumption that ˜ v∗ iis locallyC∞-convergent to ˜ u∪˜wimplies that there existni,j∈R,mi,j∈R,j= 1,2 such that ˜ v∗ n(s+ni,1,t) + (0,mi,1) isC∞- convergent to ˜ u(s,t) andv∗ i(s+ni,2,t) = (0,mi,2) isC∞-convergent to ˜ v(s,t) for any compact subset of R1×S1. Now both {˜u(s+n,t)}∞ n=0and{˜v(s−n,t)}∞ n=0are locallyC∞-convergent to the trivial ˜J-holomorphic map ( s,t)→(x(t),cs), after translations in ˜M. We conclude that ∃Nsuch that for any given ǫ >0, whens > N ,\|Dα{u(s,t)− x(t)}\|<ǫ=ǫαandS <−N,\|Dα{w(s,t)−x(t)}\|<ǫ=ǫαfor any \|α\| ≥0, and that\|Dα{a(s,t)−c˙s}\|<ǫ=ǫα,\|Dα{b(s,t)−c˙s}\|<ǫ=ǫαfor any \|α\| ≥1. We now deﬁne ˜ vi(s,t) = ˜v∗ i(s+ni,1+ni,2 2,t).by the assumption on local convergence of ˜ v∗ i,ni,1→ −∞ andni,2→+∞. Letli=1 2{(ni,2−ni,1−2N}. Thenli→+∞.Thenvi(−li,t) =v∗ i(N+ni,1,t)→u(N,t) andvi(lt,t) = v∗ i(−N+ni,2,t)→w(−N,t). Lemma 4.1 Wheniis large enough, for any s∈(−li,li),\|Dα{vi(s,t)− x(t)}\|<2ǫ,\|α\| ≥0and\|Dα{fi(s,t)−cs}\|<2ǫ,\|α\| ≥1. proof Since the proof of the two statements are similar, we will onl y prove the ﬁrst one. Assume that the ﬁrst statement is not true. then the re exists a sequence (si,ti)∈(−li,li)×S1,i→ ∞, such that \|Dα{vi(si,ti)−x(ti)}\|> 2ǫ.If\|si−(−li)\|or\|si−li\|are bounded, say \|si−(−li)\|is bounded, then vi(si+s,t),s∈(−δ,δ) isC∞-convergent to u(N+s,t) for some N> Nand s∈(−δ,δ), which implies that \|Dα{vi(si,t)−x(t)}\|<2ǫ wheniis large enough. This is a contradiction. Hence we may assume that both\|si−(−li)\|and\|si−li\| → ∞. Then ˜vi(si+s,t) is stillC∞-convergent for any ( s,t)∈[−R,R]×S1, with ﬁxedR. LetR→ ∞ and patch all the local limit together, we get a ˜J- holomor- phic map ˜v∞:R1×S1→˜MwithEλ(˜v∞) = 0.This implies that v∞(s,t) =x(t). Therefore, \|Dα(vi(si,t)−x(t))\|<ǫwhenilarge enough. This is a contradiction again. QED To state one of our main results, we deﬁne ˜vi,+(s,t) = (vi(s−li,t),fi(s−li,t)−f(−li,0) +a(N,0)) and ˜vi,−(s,t) = (vi(−s+li,t),fi(−s+li,t)−f(li,0) +b(−N,0)). 28Then ˜vi,+(0,0)→(u(N,0),a(N,0)),and ˜vi,−(0,0)→(w(−N,0),b(−N,0)). •Local Coordinate near x(t) : Theλ-period ofx(t) is/integraltext S1x∗λdt=c. We havedx dt=c˙Xλ(α(t)).By rescaling the parameter ( s,t), we may assume that c= 1. Letτbe the minimal period ofx(t), i.e.τ >0 is the minimal number such that x(t+τ) =x(t).Under this assumption, given any point z=x(t),t∈[0,τ), we assign its θ-coordinate θ=θ(z) =t.For simplicity, we will assume further that τ= 1.Henceθ∈S1= R/Z, andx(θ) =x(t),θ∈S1is the simple closed orbit. Choose a global basis {e1,···,e2n}for the symplectic bundle ( ξ,dλ)\|x(θ)such that the map y= Σyiei(x(θ))∈ξ→(θ,y1,···,y2n)∈(S1×R2n,ω0) gives rise a isomorphism between the two trivial symplectic bundles (ξ,dλ) and (S1×R2n,ω0) overS1.The local coordinate of Mnearx(θ) is deﬁne by ( y,θ)→ expx(θ)Σyiei, wherey= (y1,···,y2n)∈R2n, θ∈S1.The exponential map is taken with respect to the Riemanian metric g˜J. Note that we may assume that J\|ξ\|S1corresponds to J0under above iomorphism of the two symplectic bundles overS1={x(θ)}. LetUbe a small tube neighborhood of xinM. With the above coordinate (y,θ), then at any point z∈U, TzM=R{∂ ∂θ} ⊕R{∂ ∂y1,···,∂ ∂y2n}=RXλ⊕ξz. Since aty= 0,ξ\|y=0=R{∂ ∂y1,···,∂ ∂yn}y=0, the projection dπy:TzM→ R{∂ ∂y1,···,∂ ∂y2n}zwhen restricted to ξz, is an isomorphism, when \|y\|is small enough. Here z= (y,θ). We may assume that any z∈Uhas this property. Then we can ﬁnd ei=ei(z) such that dπy(ei) =∂ ∂yi. Sincedπy(∂ ∂θ) = 0, ∂ ∂θ/\e}atio\slash∈ξz. Hence R{∂ ∂θ} ⊕ξz=TzM. For the application later, we need to compare eiwith∂ ∂yiandXλwith∂ ∂θ. Letei= Σ2n i=1αi,j(z)∂ ∂yi+αi,0(z)∂ ∂θ, Xλ= Σ2n i=1Xi(z)∂ ∂yi+X0(z)∂ ∂θ.Hereαi,j andXiare functions deﬁned on R2n×S1={(y,θ)}.Fixi, sinceei(0,θ) =∂ ∂yi, ei(y,θ)−∂ ∂yi=ei(y,θ)−ei(0,θ) = (∂ ∂θ,∂ ∂y1,···,∂ ∂y2n){/integraldisplay1 0d dτ αi,0(θ,τy) αi,1(θ,τy) ... αi,2n(θ,τy) dτ} = (∂ ∂θ,∂ ∂y1,···,∂ ∂y2n)(/integraldisplay dαi(θ,τy)dτ) y1 ... v2n . 29Heredαi(θ,y) = [∂αi,j ∂yk(θ,y)] is the (2 n+ 1)×(2n+ 1) matrix where the (j,k)th element is∂αi,j ∂yk(θ,y).Similarly, Xλ(y,θ)−∂ ∂θ= (∂ ∂θ,∂ ∂y1,···,∂ ∂y2n)/integraldisplay1 0dx(θ,τy)dτ y1 ... y2n , wheredx(θ,y) is the (2n+1)×(2n+1) matrix whose ( j,k) element is∂Xj ∂yk(θ,y). Not that both matrices dαianddxhas uniformly bounded norm for ( y,θ)∈U. This proves Lemma 4.2 For any (y,θ)∈U,∃constantCsuch that \|ei(y,θ)−∂ ∂yi\|<C˙\|y\|,\|Xλ(y,θ)−∂ ∂θ\|<C˙\|y\|. In the (y,θ,a)-coordinate for U×R⊂˜M, we write ˜u(s,t) = (u(s,t),a(s,t)) andu(s,t) = (yu(s,t),θu(s,t)). If there is no confusion, we will simply ommit the subscript uinyuandθu. Similarly, we write w(s,t) = (yw(s,t),θw(s,t)) andvi(s,t) = (yvi(s,t)),θvi(s,t) in the (y,θ)-coordinate. Lemma 4.3 Letπ=πξ:TM=R{Xλ} ⊕ξ→ξbe the projection. Given any v∈TM, if π(v) = Σ2n i=1ci∂ ∂yi+c0∂ ∂θ= Σ2n i=1diei, thenci=di,i= 1,···,2n. Proof π(v) =v−λ(v)Xλ.Let Xλ= Σ2n i=0Xi∂ ∂yi+X0∂ ∂θ and v= Σ2n i=1vi∂ ∂yi+v0∂ ∂θ. Then π(v) = Σ2n i=1(vi−λ(v)·Xi)∂ ∂yi+ (v0−λ(v)X0)∂ ∂θ = Σ2n i=1(vi−λ(v)·Xi)ei+ (v0−λ(v)X0)∂ ∂θ + Σ2n i=1(vi−λ(v)Xi·(∂ ∂yi−ei). Now sinceπy(∂ ∂yi−ei) =∂ ∂yi−∂ ∂yi= 0, ∂ ∂yi−ei∈kerπy=R{∂ ∂θ}. 30Therefore Σ2n i=1(vi−λ(v)Xi)(∂ ∂yi−ei)∈R{∂ ∂θ} and π(v) = Σ2n i=1(vi−λ(v)Xi)ei,mod(R{∂ ∂θ}). Butπ(v),ei∈ξand∂ ∂θ/\e}atio\slash∈ξ.This implies that π(v) = Σ2n i=1(vi−λ(v)Xi)ei. QED •Equation in the local coordinate : We only write the equation for ˜ u. Same expression is also applicable to ˜ w and ˜vi.That ˜uis˜J- holomorphic is equivalent to:   as =λ(ut) ( a) at =−λ(us) ( b) π(u)◦du◦i=J(u)π(u)◦du (c) LetM(y,θ) be the 2n×2nmatrix for the dλ- compatible almost complex structureJ(y,θ) with respect ot the basis {e1,···,e2n}. We will assume that M(y,θ) =J0, the standard constant complex structure on R2n. That isJ0(ei) = ei+nandJ0(ei+n) =−ei,1≤i≤n.As pointed out in [HZW], the proof of the statements below for general Mcan be reduced to this case. For our purpose of this paper, we can even assume that this is really true as we ca n make choice of J. The eqation (c) is equivalent to π(us) +J(u)π(ut) = 0.In local coordinate we have π(us) = Σ2n i=1{(yi)s−λ(us)Xi}ei π(ut) = Σ2n i=1{(yi)y−λ(ut)Xi}ei. Hence, (ys−λ(us)Y) +M(yt−λ(ut)Y) = 0. Equivalently, ys+Myt+ (at−as·M)·Y= 0. Herey= y1 ... y2n andY= X1 ... X2n ,andM=J0. We have shown that Y(y,θ) ={/integraldisplay1 0dY(τy,θ)dτ} y1 ... y2n  anddY(y,θ) is the 2n×2nmatrix whose ( j,k)-element is∂Xj ∂yk. 31Denote/integraltext1 0dY(τy,θ)dτbyDY(y,θ).Then ys+Myt+{(at−asM)·DY} ·y= 0. Denote {at−asM} ·DY(y(s,t),θ(s,t)) byS(s,t).We deﬁneS∞=−J0· dY(0,t). Lemma 4.4 Whens>N ,\|S(s,t)−S∞(s,t)\|<C·ǫand\|Ss(s,t)\|<C·ǫfor the givenǫand some constant C.Same conclusion for wandviwhens<−N ors∈(−li,li)respectively. Proof : We only prove the statement for u. Whens>N , \|Ds{u(s,t)−x(t)}\|=\|Ds{(y(s,t),θ(s,t))−(0,t)}\| =\|Ds(y(s,t),θ(s,t))\|<ǫ. Note that in the ( y,θ)-coordinate, x(t) = (0,t) sincec= 1.Similarly, when s>N , \|Ds{at(s,t)−∂ ∂t(cs)}\|=\|Dsat(s,t)\|<ǫ and \|Ds{as(s,t)−∂ ∂s(cs)}\|=\|Dsas(s,t)\|<ǫ. This implies that \|DsS(s,t)\|< C·ǫfor some constant Cdepending only on \|\|DY(y,0)\|\|C1onU. Whens>N ,\|at(s,t)\|=\|Dt(a(s,t)−cs)\|<ǫwithc= 1 and \|as(s,t)−1\|= \|Ds(a(s,t)−cs)\|<ǫ,we have \|(y(s,t),θ(s,t))−(0,t)\|<ǫ. This implies that \|S(s,t)− {−J0DY(0,t)}\|<ǫ. But −J0DY(0,t) = −J0/integraldisplay1 odY(0,t)dτ =−J0dY(0,t) =S∞(t). QED Lemma 4.5 S∞(t)is a2n×2nsymmetric metric and all the eigenvalues of the self-adjoint elliptic operator A∞:L2 1(S1,R2n)→L2(S1,R2n)deﬁned by A∞:z→ −J0dz dt−S∞·z, are non-zero. 32Proof : Let Ψ tbe the ﬂow of Xλ.Hence /braceleftbiggdΨt(z) dt=Xλ(Ψt(x)) ( ∗) Ψ0(z) =z,∀z∈M. Ifz0= (0,0) in (y,θ)-coordinate then Ψ t(z0) = Ψ t(0,0) = (0,t) =x(t). Hencez0= Ψ1(z0) is a ﬁxed point of Ψ 1. Note that the ﬂow Ψ tpreserves the decomposition TM=R{Xλ} ⊕ξ, and that along x(t) = (0,t),Xλ=∂ ∂θand ξ=R{∂ ∂y1,···,∂ ∂y2n}.Diﬀerentiating eqaution () above, we get dDΨt dt=DXλ(Ψt)◦DΨt.(∗∗) Now given v,w∈ξ(0,0)⊂T(0,0)M, since Ψ tpreservesdλ=ω, We have ω(J(Ψt)∗(v),J(Ψt)∗(w0)) = ω((Ψt)∗(v),(Ψt)∗(w)) =ω(v,w). Diﬀerentiating this, we get ω(J(d dtDΨt)(v),JDΨt(w)) +ω(JDΨt(v),J(d dtDΨt)(w)) = 0. Here we used that Jis constant along ξ\|x(t).Use equation (), we get ω(JDX λ(Ψt)◦DΨt(v),JDΨt(w)) +ω(JDΨt(v),JDX λ(Ψt)◦DΨt(v)) = 0. Letvt=DΨt(v),wt=DΨt(w).Then gJ(JDX λ(Ψt)(vt),wt) =gJ(vt,JDX λ(Ψt)(wt)). Lett= 1, then Ψ t(z) =zfor anyz= (0,θ).It is easy to see that DXλ(0,t) =/parenleftbigg dY(0,t) 0 0 1/parenrightbigg , JDX λ(0,0)(v1) =JdY(0,0)(v1), and JDX λ(0,0)(w1) =JdY(0,0)(w1). This implies that S∞=−J0dY(0,0) is symmetric. Then general case can be proved by a coordinate change on t.Therefore,A∞=−J0d dt−S∞:L2 1(S1,R2n)→ L2(S1,Rn) is a self-adjoint elliptic operator. We want to show that 0 i s not an eigenvalue of A∞.Given 0 /\e}atio\slash=z∈L2 1(S1,R2n),A∞(z) = 0 is equivalent to dz dt=J0S∞(t)z=dY(0,1)z, (∗ ∗ ∗) 33withz(t+ 1) =z(t).As before let z0= (0,0).Then Ψ t(z0) = (0,t) in (y,θ)- coordinate. We write DΨt(z0) =/parenleftbigg R(t) 1 0 1/parenrightbigg with respect to the basis {∂ ∂y1,···,∂ ∂y2n,∂ ∂θ}. The equation () implies dR(t) dt=dY(0,t)·R(t). (∗ ∗ ∗∗). Ifw(t)/\e}atio\slash= 0 is a solution of (), then w(t+ 1) =w(t). Deﬁne ˜w(t) =R(t)·w(0), then () implies d˜w dt=dY(0,t) ˜w(t). Since ˜w(0) =R(0)w(0) =w(0),we have ˜w(1) =w(1) =w(0), i.e.w(0) is an eigenvalue of R(1) with eigenvalue 1. This implies that dΨ1(z0) has an eigenvector of eigenvalue 1 along ξz0.Conversely, if vis an eigenvector of R(1) with eigenvalue 1, then w(t) =R(t)·vsolves (*) with w(1) =R(1)·v=v= R(0)·v=w(0).Therefore, we get an eigenvector of A∞of eigenvalue 0. It follows from this and previous lemma that Lemma 4.6 There exists a constant δ >0, such that when Nandilarge enough, for ˜uand˜w, withs>N ors<−Nrespectively, /bardbl(−J0d dt−S(s,t))·z/bardbl ≥2δ/bardblz/bardbl,∀z∈L2 1(S1,R2n). For˜viwiths∈(−li,li), same conclusion holds. We denote −J0d dt−S(s,t) byA(s) :L2 1(S1;R2n)→L2(S1;R2n).Note that \|A(s)−A∗(s)\|=\|S−S∗\| ≤ \|S−S∞\|+\|S∗−S∗ ∞\|<c·ǫ,whens>N ors<−N for ˜uor ˜w, orS∈(−li,li) for ˜vi. We now establish the exponential decay estimate for the y-components of ˜u, ˜wand ˜vi. We will use y=y(s) =y(s,−)∈L2 1(S1;R2n) to denote the y-components of ˜ u, ˜wor ˜vi. Letg(s) =1 2<y(s),y(s)>. Lemma 4.7 WhenNandilarge enough, for s > N ors <−Nfor˜uor˜w, and fors∈(−li,li)for˜vi, we have g′′(s)≥δ2g(s). 34Proof : g′(s) =<y′(s),y(s)>. g′′(s) =<ys,ys>+<(ys)′,y(s)> =<A·y,A·y>+<∂ ∂s(−J0dy dt−S·y),y(s)> =<A·y,A·y>+<ys,A∗y>−<Ssy,y> = 2/bardblA·y/bardbl2+<A·y,(A∗−A)·y>−<Ssy,y> ≥2/bardblAy/bardbl2−Cǫ/bardblAy/bardbl · /bardbly/bardbl −Cǫ/bardbly/bardbl2 =/bardblAy/bardbl(2/bardblAy/bardbl −Cǫ/bardbly/bardbl)−Cǫ/bardbly/bardbl2 ≥δ/bardbly/bardbl2(2δ−Cǫ−Cǫ δ) ≥δ2/bardbly/bardbl2=δ2g(s). Here we use the fact that Candδare uniformly bounded for all sandǫcan be made as small as possible by the suitable choice of sin the lemma. QED For ˜uand ˜w, sinces∈[N,+∞) ors∈(−∞,−N], andg(s)→0 ass→ ±∞, the above lemma together with the usual elliptic estimate a pplied to each [si,si+ 1]×S1implies that Lemma 4.8 They-component y(s,t)of˜usatisﬁes: /bardbly(s)/bardbl2 L2≤ /bardbly(N)/bardbl2 L2·e−δ(s−N), s>N. Moreover, there exists a constant C=Cα,with\|α\| ≥0, such that \|Dαy(s,t)\|<C α·e−δ(s−N). Similar conclusion holds for ˜w. To get corresponding estimate for ˜ vi, we note that since lim s→+∞yu(s,t) = 0 = limyw(s,t),we may assume that /bardblyu(N)/bardblL2=/bardblyw(−N)/bardblL2.This implies that/bardblyvi(−li)/bardblL2is very close to /bardblyvi(li)/bardblL2, whenilarge enough. For simplic- ity, we may assume that c+=g(li) =/bardblyvi(li)/bardbl2=/bardblyvi(−li)/bardbl2=g(−li) =c−. Letc=c+=c−, and denote libyl. Deﬁneh(s) =a·(e−δs+eδs),with a=c e−δl+eδl.Thenh(−l) =h(l) =c, andh′′(s) =δ2h(s). Deﬁnef=g−h. Thenf′′(s)≥δ2·f(s),fors∈(−l,l) andf(−l) = f(l) = 0.The maximal principle implies that f(s)≤0,s∈(−l,l).Henceg(s)≤ c·(e−δs+eδs) e−δl+eδl. Now deﬁne g+(s) =g(s−l), andg−(s) =g(l−s),s∈(0,l).Then g+(s)≤c·e−δ(s−l)+eδ(s−l) e−δl+eδl ≤2·ce−δs·eδl eδl= 2g+(0)·e−δs = 2c+e−δs, s ∈[0,l]. 35Similarly,gs(s)≤2c−e−δs. Note that since c+,c−are close to /bardblyu(N)/bardbl2 L2and/bardblyw(−N)/bardbl2 L2which are ﬁxed, we get exponential decay of g+(s) andg−(s). For the general case when c+/\e}atio\slash=c−, we haveg+(s)≤2(c++c−+ǫ)·e−δsfor some ﬁxed small ǫwheni large enough. Deﬁne ˜vi,+= ˜vi(s−li,t) and ˜vi,−(s,t) = ˜vi(li−s,t), and lety+,y−be the corresponding y-components. We have Lemma 4.9 Whenilarge enough, \|y±(s,t)\|2 L2≤2(/bardblyu(N)/bardbl2 L2+/bardblyw(−N)/bardbl2 L2+ǫ)·e−δs, s∈(0,li). Moreover, ∃C=Cα,\|α\| ≥0such that \|Dαy±(s,t)\|<C·e−δs, s ∈(0,li). We now study the behavior of the ( a,θ)-component of ˜ u,˜wand ˜vi. We have shown before that when s > N ors <−N, foruandw, and s∈(−li,li) forvi,\|D(u(s,t)−x(t))\|<ǫ.Since \|Dy(s,t)\|<ǫ, this implies that \|∂tθ−1\|=\|∂tθ−∂tx(t)\|< ǫand\|∂sθ\|=\|∂sθ−∂s(x(t))\|< ǫ.LetP:U⊂ M→R1×S1={(a,θ)}be the projection of the ( a,y,θ)-coordinate chart U to (a,θ)-coordinate chart R1×S1given by (a,y,θ)→(a,θ).Thenu=P ◦˜u, w=P ◦˜wandvi=P ◦˜viare local diﬀeomorphisms from [ −N,+∞)×S1, (−∞,−N]×S1and (−li,li)×S1toR1×S1.Since \|∂sa(s,t)−1\|=\|∂s(a(s,t)−cs)\|<ǫ \|∂ta(s,t)\|<ǫfor these values of sin the above range \|a(s,t)−a(s0,t0)\| ≥ \|a(s,t)−a(s0,t)\| − \|a(s0,t)−a(s0,t0)\| ≥1 2\|s−s0\| −ǫ. This implies that u,wandviare proper.. Hence they are covering maps from open cylinders ( N,+∞)×S1,(−∞,−N) or (−li,li)×S1to their images inR×S1.Assume that the degree of the covering is m. Letπm:R×S1→R×S1be the standard m-fold covering induced from the corresponding covering of S1toS1. Writeu,wandvias (a,θ).We will study viﬁrst. We will only derive the equation for vi= (a,θ).The same formula is also applicable for uandw. Letq0=vi(−li,0) = (a0,θ0) and ˜q0= (a0,˜θ0)∈π−1 m(q0) with ˜θ0∈[0,1) being the smallest of such ˜θ0.DeﬁneVito be the unique lifting of visending (−li,0) to ˜q0. We drop the subscript of Vifrom now on. Then V: (−li,li)× S1→R×S1is an embedding. Note that the length of the image of a-projection ofV({li}×S1) andV({li})×S1) is less than ǫ.Hence the image of VinR×S1 is almost a standard cylinder of the form [0 ,L]×S1.We want to prove that \|L−2li\|is uniformly bounded and tends to zero when iandNtends to inﬁnity. 36Sinceπmpreservesa-length, this also implies the corresponding statement for vi. To this end, deﬁne the complex structure i=i(s,t) on the image of Vby the identiﬁcation: i(s,t) =dV(s,t)◦i◦ {dV(s,t)}−1:TV(s,t)(R1×S1)→TV(s,t)(R1×S1). ThenVis (i,i)-holomorphic, i.e. dV◦i=i(V)·dV. Equivalently, ∂V ∂s+i(s,t)∂V ∂t=∂V ∂s+i(V)∂V ∂t= 0. Switch tovi,+orvi,−and consider the corresponding vi,+,vi,−andV+,V−and associatedi(s,t).By abusing our notations, we will still use i(s,t) to denote the complex structure in these cases. Lemma 4.10 Fors∈(−li,li), there exists a constant C=Cαindependent of i, such that \|Dα(i(s,t)−i)\|<C α·e−δs. Proof : Sinceπmis a local diﬀeomorphism, if we deﬁne I=I(s,t) :Tv(s,t)(R1× S1)→Tv(s,t)(R1×S1) by the formula: dv(s,t)◦i◦ {dv(s,t)}−1, theni(s,t) = dπ−1 m◦I(s,t)◦dπm. Therefore, we only need to prove the corresponding state- ment forI(s,t). Nowvis (i,I)- holomorphic, i.e. dv(s,t)◦i=I(s,t)◦dv(s,t). In terms of the basis∂v ∂s(s,t),∂v ∂t(s,t), I(s,t)(∂v ∂s,∂v ∂t) = (∂v ∂s,∂v ∂t)·/parenleftbigg 0−1 1 0/parenrightbigg . We need to ﬁnd the expressions for∂v ∂sand∂v ∂tin terms of (∂ ∂a,∂ ∂θ). Sublemma (∂v ∂s,∂v ∂t) = (∂ ∂a,∂ ∂θ)·/braceleftbigg/parenleftbiggas−at atas/parenrightbigg +O(e−δs)/bracerightbigg . Proof: ∂v ∂s=dP ◦d˜vi(∂ ∂s) =dP(as∂ ∂a+{(vi)s−λ(vi)sXλ}+λ(vi)sXλ). Let (vi)s−λ(vi)Xλ(vi)s= Σ2n k=1ck∂ ∂yk+c0∂ ∂θ.Then (vi)s−λ(vi)Xλ(vi)s= Σ2n k=1ckek = Σ2n k=1ck∂ ∂yk+ Σ2n k=1ck·(ek−∂ ∂yk). 37Nowckis uniformly bounded and \|ek(u(s,t)−∂ ∂yk)\|<C·\|y(s,t)\|<C·e−δs. Similarly, λ(vi)sXλ=λ(vi)s(Xλ−∂ ∂θ) +λ(vi)·∂ ∂θ, and \|Xλ(u(s,t))−∂ ∂θ)\|<c· \|y(s,t)\|<c·e−δs. This implies that /braceleftbigg∂v ∂s=as∂ ∂a+λ{(vi)s}∂ ∂θ+O(e−δs) ∂v ∂t=at∂ ∂a+λ{(vi)t}∂ ∂θ+O(e−δs). Nowλ{(vi)s}=−atandλ(vi)t=as.The conclusion follows. QED Let A=/parenleftbiggas−at atas/parenrightbigg ,andO=O(e−δs). Then in terms of the basis (∂ ∂s,∂ ∂t): I(s,t) = (A+O)·J0(A+O)−1. SinceAJ0=J0A,I(s,t) =J0+O(e−δs). This proves the lemma for α= 0. The general case with \|α\| ≥1 can be proved similarly. QED Still work with ˜ vi,+and the corresponding V. NowV: (0,li)×S1→R1×S1. ByR-translation, we may assume that V(0,0) = (0,θ0).Note thatθ0→0 when i→ ∞.Consider the unique lifting of Vfrom the universal covering (0 ,li)×R1 of (0,li)×S1to the universal covering R1×R1ofR1×S1,which sends (0 ,0) to (0,θ0), 0≤θ0≤1.We still denote it by V.Then (V−Id) : (0,li)×R1→ R1×R1, and since both VandIdcommutes with deck transformations induced byθ→θ+ 1,,V−Idis periodic on the second factor of (0 ,li)×R1of period 1. Let Φ = V−Id: (0,li)×S1→R2.Then ∂Φ ∂s=∂V ∂s−∂(Id) ∂s=−{i+O(e−s)}∂V ∂t−i∂(Id) ∂t. Since∂V ∂tis bounded, we have ∂Φ ∂s+i∂Φ ∂t+O(e−δs) = 0. Proposition 4.1 \|Φ(s,t)\|< Cfor alls∈(0,li), whereCis bounded by the initial value of O(e−δs),\|Φ(0)\|and\|∂ ∂tΦ\|. All of them tend to zero as iandN tends to inﬁnity. 38Proof: Letφ(s) =/integraltext S1Φ(s,t)dt.Then dφ ds=−/integraldisplay S1O(e−δs)dt=f(s)(=O(e−δs)). Henceφ(s) =φ(0)+/integraltexts 0f(τ)dτ.If\|f(s)\|<d·e−δs,s∈(0,li), then \|/integraltexts 0f(τ)dτ\|< d δ. Now let Ψ( s,t) = Φ(s,t)−φ(s).Then /integraldisplay S1Ψ(s,t)dt=φ(s)−φ(s) = 0. LetC1= max \|∂ ∂tΦ(s,t)\|= max \|∂ ∂tΨ(s,t)\|. Clearly \|Ψ(s,t)\|<2C1.Hence \|Φ(s,t)\|<\|Ψ(θ)(s,t)\|+\|φ(s)\|<\|φ(0)\|+d δ+ 2C1. QED We reamrk that this proposition is the precise statement we m entioned before on the behavior of the ”connecting neck” along the non-compa cta- direction, which is used in the previous section to justify why it is poss ible to get the compactiﬁcation of the moduli space without introducing th e unstable trivial connecting maps. For ˜uand ˜w, we get more. We only prove the result for ˜ u. Deﬁneu,Uand Φ =U−Id: (N,−∞)×S1→R2as above. We have ∂Φ ∂s+i∂Φ ∂t+O(e−δ(s−N)) = 0. LetO(e−δ(s−N)=f(s,t).We identify the image R2of Φ andfwithC. Then the standard complex structure i=/parenleftbigg 0−1 1 0/parenrightbigg onR2is identiﬁed with the multiplication by imaginary number i.Fixs, let Φ(s,t) = Σ n∈Zφn(s)eint f(s,t) = Σ n∈Zfn(s)eint be the Fourier expansion of Φ( s,−) andf(s,−). Thenφ′ n−nφn+fn= 0,n∈Z. Note that \|fn(s)\|< C·e−δ(s−N). In particular, when n= 0, φ′ 0(0) =f0(s). Hence φ0(s) =φ0(N) +/integraldisplays Nf0 =φ0(N) +/integraldisplay∞ Nf0−/integraldisplay∞ sf0 =s0−/integraldisplay∞ sf0, 39wheres0=φ0(N) +/integraltext∞ Nf0is a constant. Now \|/integraldisplay∞ sf0(s)ds\|<C·/integraldisplay∞ se−δ(τ−N)dτ <C δe−δ(s−N). Hence \|φ0(s)−s0\|< C1·e−δ(s−N). Now let Ψ( s,t) = Φ(s,t)−φ0(s) and r(s,t) =f(s,t)−f0(s).Then ∂Ψ ∂s+i∂Ψ ∂t+r(s,t) = 0. Note <i∂Ψ ∂t,i∂Ψ ∂t>=<Σn/\e}atio\slash=0nφn(s)·eint,Σn/\e}atio\slash=0nφn(s)·eins>≥Σn/\e}atio\slash=0φ2 n(s) =<Ψ,Ψ>. Deﬁneg(s) =1 2<Ψ(s),Ψ(s)>.Then g′′(s) =<Ψ′(s),Ψ′(s)>+<Ψ′′(s),Ψ(s)> = 2<iΨt,iΨt>+<r,r> +<iΨt,r>+<r,iΨt>−<rs,Ψ> ≥2/bardblΨ/bardbl(/bardblΨ/bardbl − /bardblr/bardbl − /bardblrs/bardbl). Note that both \|r(s,t)\|and\|rs(s,t)\| ≤C·e−δ(s−N). Leth(s) =/bardblr(s,t)/bardbl+ /bardblrs(s,t)/bardbl ≤2C·e−δ(s−N).Then, if /bardblΨ/bardbl(s)>2h(s), we have g′(s)≥2/bardblΨ/bardbl(/bardblΨ/bardbl −1 2/bardblΨ/bardbl)≥g(s). Now the set P={s\| /bardblΨ/bardbl(s)>2h(s)}is open and is a countable union of (si,si+1) such that /bardblΨ/bardbl(si) = 2h(si) and /bardblΨ/bardbl(si+1) = 2h(si+1).Assume thatδ >1, then the argument before to prove Lemma ??implies that, for s∈(si,si+1),g(s)≤4·C·e−δ(s−N).On the other hand, if s/\e}atio\slash∈P, then /bardblΨ/bardbl(s)≤2h(s) = 4C·e−δ(s−N). We conclude that g(s)≤C1·e−δ(s−N).As before, applying elliptic estimate to get higher order estimate, we get Proposition 4.2 Let˜u(s,t) = (u(s,t),a(s,t))be a ˜J-holomorphic map such thatlims→∞u(s,t) =x(t)of a closed orbit of λ-periodc. Letu(s,t) = (y(s,t),θ(s,t)) in the local (y,θ)-coordinate near x(t).Then there exist positive constants N,C= Cαandδsuch that \|Dαy(s,t)\|< C α·e−δs, \|Dα{(a(s,t),θ(s,t))−(cs+d1,ct+d2)}\|< C α·e−δs for some suitable constants d1∈Randd2∈(0,τ), whereτis the minimal period ofx(t). 405 Some possible applications The following are some immediate possible applications. He re we will only brieﬂy indicate the reasons for these applications and refe r the reader to the forth coming papers on each of these topics. •(A) Index homology in contact geometry : We have already outlined the index homology in contact geome try by using the moduli space of connecting pseudo-holomorphic maps. No te that the moduli space of connecting maps used here has an one dimensional sym metry ofS1- rotations. At the same time, their asymptotic ends of closed orbits also have the S1-symmetry. It is possible to remove the symmetry by using the connecting ˜J=˜Jt,t∈S1holomorphic maps with tdependent ˜J. This will lead to a special Bott-type index homology, even the contact structure is gen eric. •(B) Additive quantum homology in contact geometry : The index homology we deﬁned is an analogy of the usual Floer h omology in symplectic geometry. We now outline a quantum homology in co ntact geometry by a diﬀerent way to use the moduli space. LetMa⊂˜M=M×R1be the section M× {a}in˜M. Given any sin- gular chain αinM=Maconsider the moduli space ˜M(x−,x+;α,M a)/S1, which is a subset of ˜M(x−,x+)/S1whose element usatisﬁes the condition thatu(z0)∈α(∈Ma). Now consider another marked point z1inulying on a ﬁxed marked line and deﬁne the obvious evaluation map eα=eα;a,b,z1: ∪x−,x+˜M(x−,x+;α,M a)/S1→Mb. Note that each element uwith the two marked points without any non-compact symmetry anymore. The intuition here is that by letting abeing very negative, and bvery pos- itive, we ﬂow the singular chain αlying almost in the negative end to get a collection of singular chains almost in the positive end. We now deﬁne additive quantum homology by deﬁning the chain c omplex generated by singular chains αinMwith the boundary D(α) =∂(α)±eα, where ∂(α) is just the usual boundary map of singular homology. By the p roperty of the moduli space established in this paper and [L1], we have D2= 0. One can show that the homology so deﬁned is independent of the choice s involved and is an invariant of M. •(C) Gromov-Witten invariants in contact geometry and ring structure in the index cohomology : Whence the index homology is deﬁned, we can deﬁne G-W invaria nt in exact the same fashion as the G-W invariant in the usual quant um homology and Floer homology. Namely given closed orbits x−= (x1,−,···,xk,−) and x+= (x1,+,···,xl,+), we deﬁne G-W invariant Ψ k,l(x−,x+) by counting J- homomorphic map uin˜Mfrom the domain S2withk+lpunctures such that alongknegative ends uapproaches to x−and alonglpositive ends uapproaches tox+. As in the usual GW-invariant in quantum and Floer homology, o ne can show that the invariant so deﬁned at chain level descends to the ho mology. By using the invariant Ψ 2,1, one can deﬁne a ring structure in the index 41cohomology, which can be thought as a quantum product for the contact man- ifold. One can also extend to deﬁnition of G-W invariants by introdu cing another set of marked points z= (z1,···,zn),zi∈S2, and require that u(zi)∈Ciof some prescribed cycles in ˜M. Using the special case of three marked point invariants with only onez,we get an action of H∗(M) on the index homology, i.e. the index homology is a module over H∗(M). There are obvious generalization of theses constructions, such as higher genus G-W invariants, coupling with gravity and so on. Note that the product structure should be thought as an essen tial part of the structure of these index homologies as there are many cases w here the additive index homology is inﬁnitely generated. •(D) Relative quantum homology : Give a compact symplectic manifold ( P,ω) with a contact boundary, let the boundary be Mwith the compatible contact structure λ=iXω, whereXis the contact ﬁeld i.e. LXω=ω.We now glue ˜MtoPalong the boundary. By using a suitable choice of φmentioned before, we get a new symplectic manifold with a cylindrical end. The chain complex of the quantum homology of a symplectic man ifoldP with contact boundary Mis generated by the pair ( α,β) whereαis a singular chain inPandβis a singular chain in ˜M. The boundary operator D= (D1,D2). HereD2(β) is deﬁned same as the one in (A) above. D1(α) =∂(α)±eα. The deﬁnition of eαhere is also similar to the one in (B). But we use the moduli spa ce ofJ-holomorphic maps, the domain of whose elements is Cbeing treated as a half sphere with an half inﬁnite cylinder attached, to ﬂow th e singular chain α inPto get a collection of singular chains eαinM. Again we have D2= 0. Now there is an obvious embedding of the chain complex of the quan tum homology ofMdeﬁned in (B) to the chain complex we just deﬁned. We deﬁne the chain complex of the relative quantum homology as the quotient of t his pair. Note that unlike (B) above, in the case that the contact bound aryMofP is concave, we may not be able to get a desired uniform energy b ound. In this case we need some extra assumption such as ωis exact. Note that there are some obvious algebraic constructions re lated to these chain complexes, such as the induced long exact sequences re lated these three homologies and Mayer-Vietoris sequence of these homologie s. More general, assume that we can decompose a compact symplectic manifold i n sequence of increasing symplectic sub-manifolds with (convex) contac t type boundaries, we can associate the sequence a ﬁltration of chain complexes de ﬁned above. Then there is a associated spectral sequence associated to the ﬁl tration. It is an interesting question to study further these algebri ac constructions to incorperate the multiplicative structures and to study t heir relation to the quantum homology of a symplectic manifold. It seems that thi s will give a new way to compute quantum homology of a symplectic manifold. •(E) Bott-type index homology, S1-invariant contact manifold and Weinstein conjecture : 42We have assumed so far that the contact form λis generic so that the set of closed orbits is discrete. We can relax this condition by o nly requiring that λis of Bott-type. Then the set of closed orbits decomposes int o an union of diﬀerent components, each being a manifold. Note that the period of any element in a component is the same by Stokes theorem. In the sy mplectic case, in this situation, Ruan and Tian developed a Bott-type Floer homology. One can develop a similar construction in this case. As remarked inA, we have two diﬀerent versions of the Bott-type homology. One of our motivation to consider Bott-type index homology i s to answer the question that if the index homology so deﬁned is always tr ivial. By using the Bott-type index homology, one can compute the in dex ho- mology when the contact manifold appears as a regular zero lo cus of a local Hamiltonian function on some symplectic manifold, which ge nerates a local S1- action. For simplicity, let ( P,ω) be a compact symplectic manifold with a S1Hamil- tonian action generated by a Hamiltonian function H. Assume that ais a regular value ofH. LetPa=H−1(a) andPa=Pa/S1. Under some assumption, Pa is a contact manifold whose contact structure is speciﬁed by ω. In fact the con- tact structure on Pacan be chosen to be S1-invariant. We deﬁne a S1-invariant contact form as follows. Note that P(a−ǫ,a+ǫ)is aS1bundle over P(a−ǫ,a+ǫ), whereP(a−ǫ,a+ǫ)=H−1((a−ǫ,a+ǫ)) andP(a−ǫ,a+ǫ)=H−1((a−ǫ,a+ǫ))/S1. Chose a connection. We can lift any vector ﬁeld X, which is transversal to Pa inP(a−ǫ,a+ǫ)to anS1-equivariant vector ﬁeld ˜X. We deﬁne the S1-invariant contact form λ=i˜Xω. By adjusting X, we may assume that λ(XH) = 1. That isλthe connection 1-form for the S1-bundle. Hence, λis a contact form if the curvaturedλis positive. Now the set of closed orbits of the contact manif old (Pa,λ) of period 1 is just Paand the images of these closed orbits foliated Pa itself. All other components of the set of closed orbits are j ust copies of this one according to diﬀerent periods. We are in the situation of Bott-type index homology. The chain complex of Bott-type homology is genera ted by singular chains in some components of the set of closed orbits and the b oundary map is the combination of the usual boundary map for singular hom ology together with a ”connecting” map by using the connecting J-holomorphic maps between two components of the set of closed orbits to ﬂow the singular chain. In our case, due to the extra S1-symmetry in the moduli space of J-connecting maps, the second part, the part of the ”connecting” map, of the boun dary map has no contribution. Hence, the Bott-type index homology is jus t inﬁnitely many copies of the usual homology of the symplectic quotient Pa. In view of the invariance of Bott-type index homology, this also compute t he index homology for the contact structure. In particular, we proved the non- vanishing of index homology in this case. As a corollary, we proved Weinstein conjecture for this case . It would be interesting to study the relationship of the prod uct structure in the contact manifold Pawith the quantum homology of its quotient, the symplectic manifold Pa. 43•(F) Gluing formula for G-W invariants : Give a compact symplectic manifold ( P,ω), assume that there is a contact type hypersurface M⊂Psuch thatMcutsPinto two pieces P−andP+with the common boundary M. As in (D), we can glue ˜Mto each ofP−andP+to form two non-compact symplectic manifolds P−andP+with cylindrical ends. As in [LR], we can prove a gluing formula for G-W invariants, w hich relates the G-W invariants of Pwith the G-W invariants in P+,P−and˜M. The idea is the following: One ﬁrst collect all J-holomorphic map uinP+,P−or˜Mwith the property thatuapproaches to some of closed orbits lying on the ends of P+,P−or˜M along its punctures, then select among them those ucan be glued along those closed orbits. Note that unlike in [LR], we do not require any local S1Hamiltonian action. •(G) Low dimensional contact manifold A special feature of a three dimensional compact manifold is that it always has a contact structure. Hence the index homology and additive quantum homology is well-de ﬁned associated to the contact structure. It would be very interesting to inves tigate if the invariants we deﬁned here are actually topological invariants. There a re various diﬀerent forms of this type of questions. In view of the work of Taubes o n the relationship of the SW-invariants and GW-invariants, one may hope to get s imilar results for contact 3-fold and symplectic four manifold with contact ty pe boundary. Our result should serve as one of the basis to formulate this type of results. We make the following ﬁnal remark. As we mentioned before, on e of the main results of this paper and [L3] is about the virtual co-dimens ion of the boundary of the moduli space, which is the foundation of the applicati ons outlined in this section. This result is the consequence of the compactness t heorem proved in this paper and the index formula, which will be proved in [L3] . To obtain the result, the index formula we need here is diﬀerent from the us ual one appeared in Bott-type Floer homology due to the extra dimension of the asymptotic R1- motion of a connecting pseudo-holomorphic maps along the en ds (closed orbits). On the other hand, the main body of this paper, the proof of the compact- ness theorem, is independent of the desired index formula. I n fact, the new phenomenon appeared in the bubbling described in Lemma 3.4 a nd the Deﬁni- tion 4.1 on equivalence of stable maps concerning how to coun t symmetries in target already opens the door for various possible applicat ions. References [EH] Y. Eliashberg, Invariants in contact topology, ICM 1998 Vol II (1998), pp. 327-338. [FO] Fukaya and Ono, Arnold conjecture and Gromov-Witten in variants, Topol- ogy(1999). [F] A. Floer, Symplectic ﬁxed points and holomorphic sphere s,Comm. Math. Phys. bf 120(1989), pp. 575-611. 44[G] M. Gromov, Pseudo holomorphic curves in symplectic mani folds, Invent. Math. 82(1985), pp. 307-347. [H] H. Hofer, Pseudo holomorphic curves in symplectization s with applications to Weinstein conjecture in dimension three. Invent. Math. 114(1993), pp. 515-563. [HWZ] H. Hofer, K. Wysocki, E. Zehnder, Holomorphic curves i n symplectiza- tions I: Asympotics. Ann. I. H. P. Analyse Non Lineaire 13(1996), pp. 337-379. [LiR] A. Li and Y. Ruan, Symplectic surgery and Gromov-Witte n invariants of Calabi-Yau 3-folds I, Preprint (1998). [LiT] J. Li and G. Tian, Virtual moduli cycles and GW-invaria nts of gen- eral symplectic manifolds, Proceedings of 1st IP conference at UC, Irvine (1996). [L1] G.Liu, Virtual Moduli cycles in the symplectization, In preperation. [L3] G.Liu, Fredholm theory of the linearized ¯∂-operator and additivity of the index formula, Preprint. [LT] G. Liu and G. Tian, Floer homology and Arnold conjecture ,JDG 49 (1998),pp. 1-74. [RT] Y. Ruan and G. Tian, Bott-type symplectic Floer cohomol ogy and its multiplication structures, preprint (1994). [T] C. Taubes, The Seiberg-Witten invariants and symplecti c forms, Math. Res. Letters 1( 1994) pp. 809-822. 45
arXiv:physics/0008245v1 [physics.chem-ph] 31 Aug 2000The Approach to Ergodicity in Monte Carlo Simulations J. P. Neirotti and David L. Freeman Department of Chemistry, University of Rhode Island 51 Lower College Road, Kingston, RI 02881-0809 and J. D. Doll Department of Chemistry, Brown University Providence, RI 02912 (November 24, 2013) Abstract The approach to the ergodic limit in Monte Carlo simulations is studied using both analytic and numerical methods. With the help of a stochastic model, a metric is deﬁned that enables the examination of a simulation in both the ergodic and non-ergo dic regimes. In the non-ergodic regime, the model implies how the simulation is expected to approach ergodic behavior analytically, and the analytically inferred decay law of the metric allows the mon itoring of the onset of ergodic behavior. The metric is related to previously deﬁned measures develop ed for molecular dynamics simulations, and the metric enables the comparison of the relative eﬃcien cies of diﬀerent Monte Carlo schemes. Applications to Lennard-Jones 13-particle clusters are sh own to match the model for Metropolis, J-walking and parallel tempering based approaches. The rel ative eﬃciencies of these three Monte Carlo approaches are compared, and the decay law is shown to b e useful in determining needed high temperature parameters in parallel tempering and J-walkin g studies of atomic clusters. PACS numbers: 05.10.Ln, 02.70.Lq I. INTRODUCTION A goal of Monte Carlo (MC) simulations in statistical mechanics [1] is the calculation of ensemble mean values of thermodynamic quantities. Ensemble mean values are multidimensional integrals over conﬁguration space /an}bracketle{tU/an}bracketri}ht=/integraldisplay dxP(x)U(x), (1) where P(x) is the probability of ﬁnding a system in the state deﬁned by x, and the functional form of P(x) de- pends on the ensemble investigated. MC simulations usu- ally generate a sampling of conﬁguration space {xk}K k=1by the use of a stochastic process with stationary probability P(xk). The quantity Uevaluated at xkis the output of the simulation U(xk) =Uk, and its arithmetic mean value U, in principle, must approach the ensemble mean value. [1] In this paper we refer to the set of conﬁgurations gener- ated in a Monte Carlo simulation as a time sequence, and we study the behavior of these temporal sequences {Uk} and their arithmetic mean, to understand better how MC simulations approach ergodic behavior. It is important to emphasize that there are two time variables to consider. The time variable klabels the separate conﬁgurations gen- erated in a Monte Carlo walk. Variations of properties withkprovide information about the short-time behavior of a MC simulation. The time variable Klabels the totallength of the MC walk, and variations of computed prop- erties with Kprovide information about the convergence of the simulation on a long time scale. Given an inﬁnite time, the stochastic walker in a MC simulation visits every allowed point in conﬁguration space. [2] Ergodic behavior is reached when the length of the walk is suﬃciently long to sample conﬁguration space appropriately. [3] In practice, this does not mean that the space has been densely covered but that every region with non-negligible probability has been reached. In such a case we can say that the simulation is eﬀectively ergodic or that it has reached the ergodic limit. For a ﬁnite walk, in the event of broken ergodicity [4], phase space is eﬀectively disconnected. The diﬀerent dis- connected regions (called components) are separated by barriers of zero eﬀective probability. If a stochastic walk er starts its walk in one of these regions, it may not cross the barriers within the time of the simulation. If the simula- tion length is increased, some barriers may become acces- sible for the walker and phase space is better sampled. We can conclude that a time τexists such that, for simulation lengths shorter than τ, the walker becomes trapped in one of the phase space components. For simulation lengths much larger than τ, phase space is eﬀectively covered by the walker. In this study we imagine a system having more than one time scale τ1≪τ2≪. . .≪τΛ. In a Monte Carlo sim- ulation each scale comes from stochastic processes with 1 PREPRINTdiﬀerent correlation times. [5] A precise deﬁnition of the correlation times for Monte Carlo processes is given in Section III, but for the moment we can think of these cor- relation times as identical to physical time scales of the system under study. To understand these time scales more fully, it is useful to focus on an example. Prototypical of systems having such disparate time scales are atomic and molecular clusters. Typical cluster potential surfaces ha ve many local minima separated by signiﬁcant energy barri- ers. [6–8] The local minima can be grouped into basins of similar energies, with each basin separated from other basins again by energy barriers. At short Monte Carlo times a cluster system executes small amplitude oscilla- tions about one of its potential minima. We can think of these vibrational time scales as the shortest time scales that deﬁne a cluster system. As the simulation time is extended the system eventually hops between diﬀerent lo- cal minima within the same basin. The time scale for the ﬁrst hops between local minima can be considered the next shortest time scale for the simulation. At still longer Monte Carlo times, the system hops between dif- ferent energy basins deﬁning yet another time scale for the simulation. This grouping of time scales continues until the longest time scale for a given system is reached. At Monte Carlo times that are long compared to this longest time scale, the simulation is ergodic. Consider a system with several time scales as mentioned above. If the length of the simulation is smaller than the smallest correlation time, the walker may become trapped in an eﬀectively disconnected region and the sampling of phase space is incomplete. By increasing the time, the memory of the initial condition in the sampling decreases as the walker crosses to other previously unreachable re- gions. These oscillations and hoppings can be modeled by a superposition of stochastic processes with diﬀerent cor- relation times. These processes with non-zero correlation times are known as colored noise processes (as opposed to zero correlation time white noise processes). [5] From the study of the autocorrelation functions of a stochas- tic model deﬁned using these colored noise processes, we can verify that, at a ﬁxed run length K, there exist two diﬀerent groups of processes; those that contribute to the autocorrelation function with terms that decay like 1 /k (called diﬀusive processes), and those that contribute to the autocorrelation function with terms that decay slower than 1 /k(called non-diﬀusive processes). When the time of the simulation is increased, some non-diﬀusive processe s at shorter run lengths, start to contribute to the autocor- relation function like diﬀusive processes. After the walk length reaches the largest correlation time τΛ, all processes contribute to the autocorrelation function with terms that decay like 1 /k. At this point, the simulation is at the dif- fusive regime and eﬀective ergodicity has been reached. A principal goal of this work is to investigate the way in which the MC output {Uk}reaches the diﬀusive limit (i.e. the ergodic limit) by studying the properties of autocorre- lation functions under changes of scale in time, K→bK withb >1. By time scaling it is possible to infer the decaylaw of the non-diﬀusive contributions with respect to the total simulation time K. The functional dependence of the non-diﬀusive contributions on the parameter bthat is used to scale Kis determined empirically. We have found the decay law so determined to be a particularly valuable method of concluding when a simulation can be considered ergodic. Unlike previous studies [3,9–11] that only have investigated the behavior of certain autocorrelation func - tions in the ergodic regime, by focusing on the approach to ergodic behavior we have a more careful monitor of the onset of ergodicity. Once the non-diﬀusive contributions have decayed to a point where they are too small to be distinguished from zero to within the ﬂuctuations of the calculation, we can say that the ergodic limit has been reached. The autocorrelation functions we use to measure the approach to the ergodic limit are based on one of the probes of ergodicity developed by Thirumalai and co- workers [3,9–11], and is often called the energy metric . The energy metric has been proposed as an alternative to other techniques [3] (like the study of the Lyapunov expo- nents [12]) for the study of ergodic properties in molecular dynamics (MD) simulations. The metric has been used to study the relative eﬃciency of MC simulation methods as well. [13] The MC metric as used in the current work can easily be extended from the energy to other scalar observables of the system. We present two key issues in this paper. First, from the knowledge of the decay law of the non-diﬀusive contribu- tions to the MC metric, we infer how long a simulation must be to be considered eﬀectively ergodic. Second, once the ergodic limit is reached, we can compare the results from diﬀerent numerical algorithms to measure relative eﬃciencies. Because the outcomes of MC simulations are noisy, we have found it useful to separate diﬀusive and non-diﬀusive terms in the MC metric with a Fourier anal- ysis so that we can neglect the high frequency compo- nents of the noise. This technique has given reproducible results. To test the match between the stochastic model and ac- tual Monte Carlo simulations, we examine the approach to ergodic behavior in simulations of Lennard-Jones clus- ters. Recently [14,15] we have studied the thermodynamic properties of Lennard-Jones clusters as a function of tem- perature using both J-walking [16] and parallel temper- ing methods. [17–19] Both simulation techniques require an initial high temperature that must be ergodic when Metropolis Monte Carlo methods [20] are used. If the Metropolis method does not give ergodic results at the initial high temperature, systematic errors propagate to the lower temperatures in J-walking and parallel temper- ing simulations, and the results can be ﬂawed or mean- ingless. In most Monte Carlo simulations of clusters at ﬁnite temperatures, [21,22] the clusters are deﬁned by en- closing the atoms within a constraining potential about the center of mass of the system. The constraining po- tential is necessary because clusters at ﬁnite temperature s have ﬁnite vapor pressures, and the association of any one 2 The Approach to Ergodicity in Monte Carlo Simulationsatom with the cluster can be ill-deﬁned. From experience [14,15,23] we have found that if the radius of the con- straining potential and the initial high temperature are not both carefully chosen, it can be diﬃcult to attain er- godicity with Metropolis methods. A key concern then is the choice of constraining radius and the choice of initial temperature. We verify the stochastic model by investi- gating Monte Carlo simulation results as a function of the temperature and the size of the constraining potential. The contents of the remainder of this paper are as fol- lows. In Section II we motivate the studies that follow by examining numerally the behavior of a set of Monte Carlo simulations of a 13-particle Lennard-Jones cluster. This cluster system is used to illustrate the results through- out this paper. In Section III we introduce the stochastic model based on a continuous time sequence. In Section IV we extend the model to discrete time sequences char- acteristic of actual Monte Carlo simulations. In Section V we test the discrete stochastic model with applications to Lennard-Jones clusters and in Section VI we summarize our conclusions. Many of the key derivations needed for the developments are found in two appendices. II. AN EXAMPLE CALCULATION Before discussing the major developments of this work, it is useful to understand the nature of the problem we are attempting to solve by examining some numerical re- sults on a prototypical system. We take the 13-particle Lennard-Jones cluster deﬁned by the potential function V(x) = 4εN/summationdisplay i=2i−1/summationdisplay j=1/bracketleftBigg/parenleftbiggσ rij/parenrightbigg12 −/parenleftbiggσ rij/parenrightbigg6/bracketrightBigg +N/summationdisplay i=1VC(/vector xi, Rc) (2) where εandσare the standard Lennard-Jones energy and length parameters, Nis the number of particles in the cluster (13 in the present case), rijis the distance between particles iandj rij=\|/vector xi−/vector xj\|, (3) andVCis the constraining potential discussed in Sec. I VC(/vector xi, Rc) =  0/vextendsingle/vextendsingle/vextendsingle/vector xi−/vectorXc/vextendsingle/vextendsingle/vextendsingle< R c ∞Rc</vextendsingle/vextendsingle/vextendsingle/vector xi−/vectorXc/vextendsingle/vextendsingle/vextendsingle, (4) where /vectorXcis the coordinate of the center of mass of the cluster and Rcis the radius of the constraining sphere. The 13-particle Lennard-Jones cluster has a complex po- tential surface with many minima separated by signiﬁcant energy barriers, [6–8] and ergodicity problems associated with the simulation of properties of this system are well- known. [16] We now consider a Metropolis MC simula- tion of the average potential energy of the system in thecanonical ensemble at temperature kBT/ε= 0.393(kBis the Boltzmann constant). This average potential energy Vkis deﬁned by Vk=1 kk/summationdisplay k′=1Vk′ (5) and is displayed in the upper panel of Fig. 1 as a func- tion of the walk length kfor 20 independent simulations each initialized from a random conﬁguration. Over the maximum time scale Kof the walks, it apparent that the potential energy averaged over each independent walk has not converged to the same result. Such unreproducible behavior is symptomatic of a simulation not yet at the ergodic limit. 0 1000 2000 3000 4000 5000 k020406080dk−30−28−26−24−22−20−18Vk\| Fig. 1: The upper panel shows the “time evolution” of Vk(in units of ε) forM= 20 independent experiments. The lower panel shows dk(in units of ε2) vs.kfor the experiments of the upper panel. Rchas been set to 4 σandkBT/ε= 0.393. At least two basins with diﬀerent energies are present. Clearly, dkgoes to a constant when kis increased within the total time scale of the simulation. At the ergodic limit (i.e. for the maximum walk length Kgreater than that included in Fig. 1) the averages dis- J. P. Neirotti, D. L. Freeman. and J. D. Doll 3played in the upper panel of Fig. 1 must approach the same value for each walker. Using related ideas developed elsewhere, [3,9,10] the extent to which the walks approach the same limit can be measured in terms of a metric dk deﬁned by dk=2 M(M−1)M/summationdisplay i=2i−1/summationdisplay j=1/bracketleftBig V(i) k−V(j) k/bracketrightBig2 , (6) In Eq. (6) Mrepresents the number of independent walks, andV(i) kis the average potential energy computed in walk iat MC time k. The metric measures the energy ﬂuctu- ations in the walk as a function of the walk length. For an ergodic simulation, the metric must decay to zero. For the 20 simulations of the 13-particle Lennard-Jones clus- ter, the metric as a function of kis plotted in the lower panel of Fig. 1. Rather than asymptotically approaching zero, over the short length of the walk displayed here, dk has decayed to a constant, and as discussed later in this paper, over the time scale of this simulation, dkcan be qualitatively represented by the function dk=AK k+BK (7) where AKandBKare coeﬃcients that are dependent on the total walk length K. As Kis increased to a time where the walk is ergodic, BKmust decay to zero. Major goals of this work are to understand how BKdecays and to use the discovered decay law to determine the onset of ergodic behavior. Our approach is to introduce ﬁrst a continuous stochastic model of a simulation followed by a discrete model more clearly linked to actual MC studies. III. STOCHASTIC MODEL We have discussed in the introduction how the output of MC simulations can be considered to be a combination of stochastic processes with diﬀerent time scales, and how the contributions to autocorrelation function from these processes can vary when the length of the simulation is enlarged. Here we present a continuous time model for the stochastic processes that occur in a simulation. Even though a MC simulation occurs in a discrete time (each MC point represents a time unit), we ﬁnd that the con- tinuous model helps to understand better the ideas used in the modeling of the MC output. In this section the ensemble mean value is used to ﬁnd the expression for the autocorrelation functions of the model. Although in actual numerical calculations the en- semble mean is replaced by a mean over a ﬁnite number of independent experiments, the results obtained here give information about the limit of an inﬁnite sample. The stationary process used to sample space is a stochastic process. We assume the output of the MC simu- lation can be modeled by a linear superposition of stochas- tic processes with diﬀerent correlation times τℓ≥0,U(t) =Uc+/radicalbig Γ0ξ(t) +Λ/summationdisplay ℓ=1/radicalbig Γℓgℓ(t/τℓ), (8) where Ucis a constant, the random variable ξ(t) represents white noise processes with zero correlation time ( τ0= 0), and the {gℓ(t/τℓ)}are stochastic processes with correla- tion times τℓ>0.ξ(t) and gℓ(t/τℓ) have units of the square root of time, and Γ 0and Γ ℓare constants with units of U2/t. IfUis chosen to be the the x-coordinate of a particle, Γ 0and Γ ℓhave units of a diﬀusion constant. Consequently we refer to these constants as generalized diﬀusion coeﬃcients. The white noise process has the fol- lowing properties [5] /an}bracketle{tξ(t)/an}bracketri}ht= 0 (9) /an}bracketle{tξ(t)ξ(t′)/an}bracketri}ht=δ(t−t′), (10) and the remaining colored noise processes are assumed to satisfy /an}bracketle{tgℓ(t/τℓ)/an}bracketri}ht= 0 (11) /an}bracketle{tgℓ(t/τℓ)gℓ(t′/τℓ)/an}bracketri}ht=1 τℓfℓ/parenleftbigg\|t−t′\| τℓ/parenrightbigg (12) so that they represent processes with a memory fℓ. Even though correlations between processes with diﬀerent cor- relation times may be non-zero, we assume the processes to be independent, i.e. /an}bracketle{tgℓ(t/τℓ)gℓ′(t′/τℓ′)/an}bracketri}ht=/an}bracketle{tgℓ(t/τℓ)/an}bracketri}ht/an}bracketle{tgℓ′(t′/τℓ′)/an}bracketri}ht = 0∀ℓ/ne}ationslash=ℓ′(13) /an}bracketle{tξ(t)gℓ(t′/τℓ)/an}bracketri}ht=/an}bracketle{tξ(t)/an}bracketri}ht/an}bracketle{tgℓ(t′/τℓ)/an}bracketri}ht = 0∀tandt′. (14) The memory function is assumed to be a continuous func- tion that depends only on the distance between tandt′ disregarding the time origin (stationary condition). The memory function represents the correlation between two times of the process gℓ. In our model we impose the con- dition t τℓfℓ/parenleftbiggt τℓ/parenrightbigg </integraldisplayt 0dt′1 τℓfℓ/parenleftbiggt′ τℓ/parenrightbigg <∞. (15) The scope and implications of the leftmost inequality are explored in Appendix A. In Appendix A we also examine the conditions fℓmust satisfy in order to yield contribu- tions to autocorrelation function that decay more weakly than 1 /t. We now assume that this inequality can be taken as a bound to possible maxima of fℓappearing at t >0. The rightmost inequality enables us to assume fℓ is normalized /integraldisplay∞ −∞dt1 τℓfℓ/parenleftbigg\|t\| τℓ/parenrightbigg = 1. (16) 4 The Approach to Ergodicity in Monte Carlo SimulationsWe have identiﬁed here the time scale τℓwith the correla- tion time of the stochastic process gℓ. This identiﬁcation is valid if /integraldisplay∞ −∞dt\|t\| τℓfℓ/parenleftbigg\|t\| τℓ/parenrightbigg =τℓ, (17) which implies that the behavior of fℓat large tmust be O/parenleftbig t−(2+ǫ)/parenrightbig , or smaller. In addition, by the properties of the ensemble mean value, we have that for all real λ 0≤/angbracketleftBig [gℓ(t/τℓ) +λgℓ(t′/τℓ)]2/angbracketrightBig ≤/angbracketleftbig gℓ(t/τℓ)2/angbracketrightbig + 2λ/an}bracketle{tgℓ(t/τℓ)gℓ(t′/τℓ)/an}bracketri}ht+λ2/angbracketleftbig gℓ(t′/τℓ)2/angbracketrightbig ≤1 τℓ/braceleftbigg fℓ(0) + 2 λfℓ/parenleftbigg\|t−t′\| τℓ/parenrightbigg +λ2fℓ(0)/bracerightbigg . (18) Equation (18) must be true for all λ. Therefore, the dis- criminant of the polynomial in λmust be non-positive 4/bracketleftbigg fℓ/parenleftbigg\|t−t′\| τℓ/parenrightbigg −fℓ(0)/bracketrightbigg/bracketleftbigg fℓ/parenleftbigg\|t−t′\| τℓ/parenrightbigg +fℓ(0)/bracketrightbigg ≤0. (19) Consequently, fℓ(0) = max {fℓ(x)∀x≥0}. Other proper- ties of fℓare studied in Appendix A. The ensemble mean value /an}bracketle{tU/an}bracketri}htis time independent. The ensemble mean value of the noise processes is zero. There- fore,Ucmust be equal to /an}bracketle{tU/an}bracketri}ht. Processes deﬁned by Eq.(8) have two diﬀerent components, uncorrelated white noise and correlated processes with correlation time τℓ. Because the goal of the simulation is the calculation of the ensemble mean /an}bracketle{tU/an}bracketri}htby the analysis of the time series, we study the behavior of the temporal mean U(t) U(t) =1 t/integraldisplayt 0dt′U(t′) =/an}bracketle{tU/an}bracketri}ht+1 t/radicalbig Γ0W(t) +1 tΛ/summationdisplay ℓ=1/radicalbig ΓℓGℓ(t/τℓ),(20) where W(t) is a Wiener process, [5] W(t) =/integraldisplayt 0dt′ξ(t′) (21) /an}bracketle{tW(t)/an}bracketri}ht= 0 (22) /an}bracketle{tW(t)W(t′)/an}bracketri}ht=t<, (23) witht<= min( t, t′), and Gℓ(t/τℓ) =/integraltextt 0dt′gℓ(t′/τℓ). Equation (20) implies that the evolution of the tem- poral mean U(t) has the same structure as U, with an uncorrelated term and terms with tailed correlation func- tions. The autocorrelation function of the process Uat times tandt′is deﬁned by κ(t, t′) =/angbracketleftbig/parenleftbig U(t)− /an}bracketle{tU/an}bracketri}ht/parenrightbig/parenleftbig U(t′)− /an}bracketle{tU/an}bracketri}ht/parenrightbig/angbracketrightbig =Γ0 tt′/an}bracketle{tW(t)W(t′)/an}bracketri}ht+1 tt′Λ/summationdisplay ℓ=1Γℓ/an}bracketle{tGℓ(t/τℓ)Gℓ(t′/τℓ)/an}bracketri}ht, (24) where we have used Eqs. (13) and (14) to neglect terms involvi ng processes with diﬀerent correlation times. Because we have assumed fℓis a continuous function, fℓreaches its maximum and minimum value within any closed interval considered. The ℓth non-diﬀusive contribution to κ(t, t′) 1 tt′/an}bracketle{tGℓ(t/τℓ)Gℓ(t′/τℓ)/an}bracketri}ht=1 tt′/integraldisplayt 0dt1/integraldisplayt′ 0dt21 τℓfℓ/parenleftbigg\|t1−t2\| τℓ/parenrightbigg , (25) is bounded 1 t<t>/integraldisplayt< 0dt1/integraldisplayt> 0dt21 τℓfℓ/parenleftbiggtmin τℓ/parenrightbigg ≤1 tt′/an}bracketle{tGℓ(t/τℓ)Gℓ(t′/τℓ)/an}bracketri}ht ≤1 t<t>/integraldisplayt< 0dt1/integraldisplayt> 0dt21 τℓfℓ(0) 1 τℓfℓ/parenleftbiggtmin τℓ/parenrightbigg ≤1 tt′/an}bracketle{tGℓ(t/τℓ)Gℓ(t′/τℓ)/an}bracketri}ht ≤1 τℓfℓ(0), (26) where t>= max( t, t′), and tminis the time at which fℓ reaches its minimum value in the closed interval [0 , t>]. There exists a t∗ ℓ(t>)∈[0, tmin] [24] such that, 1 tt′/an}bracketle{tGℓ(t/τℓ)Gℓ(t′/τℓ)/an}bracketri}ht=1 τℓfℓ/parenleftbiggt∗ ℓ(t>) τℓ/parenrightbigg . (27) Using Eqs. (23) and (27) in (24), we ﬁnd thatκ(t, t′) =Γ0 t>+Λ/summationdisplay ℓ=1Γℓ τℓfℓ/parenleftbiggt∗ ℓ(t>) τℓ/parenrightbigg . (28) For all times shorter than τ1the autocorrelation function is the sum of diﬀusive contributions (proportional to 1 /t) plusnon-diﬀusive contributions . These contributions im- plicitly depend on t>through t∗ ℓ(t>). We assume that fℓ satisﬁes the conditions stated in Appendix A, so that the J. P. Neirotti, D. L. Freeman. and J. D. Doll 5dependence of fℓontisweaker than 1/t(for total time scales shorter than τℓ; see Appendix A). We next consider the behavior of Eq. (28) for time scales greater than τ1. Under the scale change t→bt such that τ1≪bt>≪τ2, the contributions to the cor- relation function from the process with correlation time τ1can be considered diﬀusive [in other words, by virtue of Eqs. (10) and (12), f1/τ1has become a delta func- tion]. With bt>≪τ2, the other processes preserve their old properties. Then, the autocorrelation function can be expressed κ(bt, bt′) =Γ0+ Γ1 bt>+Λ/summationdisplay ℓ=2Γℓ τℓfℓ/parenleftbiggt∗ bℓ(t>) τbℓ/parenrightbigg .(29) The complete derivation of Eq. (29) can be found in Ap- pendix B. For a times larger than the correlation time τΛ, all contributions to the autocorrelation function are diﬀu - sive, the simulation can be considered ergodic, the sam- pling complete, and the temporal mean is equal to the ensemble mean within O(1/t) mean square ﬂuctuations. IV. DISCRETE TIME SEQUENCES AND THE MC METRIC Monte Carlo simulations generate discrete sequences Uk of values of the quantity under study. Additionally, in ac- tual calculations the ensemble of sequences is represented by a ﬁnite rather than an inﬁnite set. In this section, the model developed in the previous section is extended to ﬁnite sets of discrete sequences. We express the Mse- quences/braceleftBig U(m) k/bracerightBigK k=1, where the label ( m) ranges from 1 toM. The exact ensemble mean value /an}bracketle{tU/an}bracketri}htcan be ob- tained in the limit that Mbecomes inﬁnite. In analogy with the model developed in Section III, each output is assumed to have the form U(m) k=/an}bracketle{tU/an}bracketri}ht+/radicalbig Γ0ξ(m) k+Λ/summationdisplay ℓ=1/radicalbig Γℓg(m) ℓ;k/τℓ,(30) where /an}bracketle{tξ(m) k/an}bracketri}ht= 0 (31) /an}bracketle{tξ(m) kξ(n) k′/an}bracketri}ht=δm,nδk,k′ (32)/an}bracketle{tg(m) ℓ;k/τℓ/an}bracketri}ht= 0 (33) /an}bracketle{tg(m) ℓ;k/τℓg(n) ℓ′;k′/τℓ′/an}bracketri}ht=δm,nδℓ,ℓ′fℓ/parenleftbigg\|k−k′\| τℓ/parenrightbigg .(34) The true ensemble average /an}bracketle{tU/an}bracketri}htdoes not depend on the index m. In the discrete case we deﬁne a metric dk=2 M(M−1)M/summationdisplay i=2i−1/summationdisplay j=1/bracketleftBig U(i) k−U(j) k/bracketrightBig2 , (35) where the bars represent the temporal mean value U(m) k=1 kk/summationdisplay k′=1U(m) k′ =/an}bracketle{tU/an}bracketri}ht+√Γ0 kW(m) k+Λ/summationdisplay ℓ=1√Γℓ kG(m) ℓ;k/τℓ,(36) with W(m) k=k/summationdisplay k′=1ξ(m) k′ (37) G(m) ℓ;k/τℓ=k/summationdisplay k′=1g(m) ℓ;k′/τℓ. (38) Observe that in the present case, our ﬁnite sample of the inﬁnite ensemble is the set of outcomes from Mindepen- dent numerical experiments. The metric we have deﬁned in Eq. (35) can be contrasted with alternative metrics [3,9,10] previously deﬁned for molecular dynamics simula- tions. These alternative metrics examine the ﬂuctuations of two simulations initialized from diﬀerent components of conﬁguration space averaged with respect to all the par- ticles in the system. The metric we use in this work is de- termined using an average with respect to Mindependent simulations that represent a subset of the full ensemble. Using the model presented in Eq. (30), we now develop a way to predict the behavior of the MC simulation in the non-ergodic and the ergodic regimes. We ﬁrst con- sider the case that the total simulation time Kis larger than the ﬁrst correlation time τ1but shorter than τ2, i.e. τ1≪K≪τ2. The expression for dkis given by dk=2 M(M−1)M/summationdisplay i=2i−1/summationdisplay j=1/bracketleftBig/parenleftBig U(i) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig −/parenleftBig U(j) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig/bracketrightBig2 =2 MM/summationdisplay i=1/parenleftBig U(i) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig2 −4 M(M−1)M/summationdisplay i=2i−1/summationdisplay j=1/parenleftBig U(i) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig/parenleftBig U(j) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig = 2 Γ 01 MM/summationdisplay i=1/parenleftBigg W(i) k k/parenrightBigg2 + 2 Γ 11 MM/summationdisplay i=1 G(i) 1;k/τ1 k 2 + 2Λ/summationdisplay ℓ=2Γℓ1 MM/summationdisplay i=1 G(i) ℓ;k/τℓ k 2 6 The Approach to Ergodicity in Monte Carlo Simulations+ 4Λ/summationdisplay ℓ=1/radicalbig Γ0Γℓ1 MM/summationdisplay i=1W(i) kG(i) ℓ;k/τℓ k2+ 4Λ/summationdisplay ℓ=2ℓ−1/summationdisplay ℓ′=1/radicalbig ΓℓΓℓ′1 MM/summationdisplay i=1G(i) ℓ;k/τℓG(i) ℓ′;k/τℓ′ k2 −4 M(M−1)M/summationdisplay i=2i−1/summationdisplay j=1/parenleftBig U(i) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig/parenleftBig U(j) k− /an}bracketle{tU/an}bracketri}ht/parenrightBig . (39) If the number of experiments Mis suﬃciently large, we can neglect terms involving process es with diﬀerent correlation times, and products of sequences belonging to diﬀerent expe riments. Under these assumptions we obtain dk= 2Γ0 k1 MM/summationdisplay i=1W(i) k2 k+ 2Γ1 k1 MM/summationdisplay i=1G(i) 1;k/τ12 k+ 2Λ/summationdisplay ℓ=2Γℓ1 MM/summationdisplay i=1 G(i) ℓ;k/τℓ k 2 . (40) Equation (40) preserves the form of Eq. (28). To make this statement explicit, let us rewrite Eq. (40) as dk= 2Γk k+ 2 Υ k, (41) where Γk= Γ01 MM/summationdisplay i=1W(i) k2 k+ Γ11 MM/summationdisplay i=1G(i) 1;k/τ12 k(42) Υk=Λ/summationdisplay ℓ=2Γℓ1 MM/summationdisplay i=1 G(i) ℓ;k/τℓ k 2 . (43) In Appendix B we present a study of the way non-diﬀusive contribution become diﬀusive under time scale changes. If Mis suﬃciently large and τ1≪K≪τ2, by virtue of Ap- pendix B, Γ kmust roughly be a constant. By roughly a constant we mean a constant Cplus some rapidly ﬂuctuat- ing function ζk, with the following properties: a) /an}bracketle{tζk/an}bracketri}ht= 0 and b) \|C\| ≫max k=1,2,...,K(\|ζk\|). Then Γk≃ΓK+ζk (44) IfKis enlarged, we expect to have a larger value of Γ K. Υkis a quantity related to the memory functions fℓwith correlation times τℓ≫K. In the continuous time model, the colored noise processes contribute to the autocorre- lation function with terms proportional to fℓ(t∗ ℓ(t>)/τℓ), which are weakly dependent on t(see Appendix A). We can expect Υ kto be weakly dependent on k, and for se- quences of length Kand for Msuﬃciently large, we con- sider this quantity roughly to be a constant Υk≃ΥK+βk. (45) where βkrepresents additional random noise. Then, for a given length k≤K, the MC metric dkcan be approxi- mated by dk= 2ΓK k+ 2 Υ K+γk, (46) where γk= 2(ζk/k+βk) represents remaining stochas- tic noise from both contributions. In this approximation,ΓKand Υ Kare the quantities that carry the long time dependence. Short time features appear in the 1 /kde- pendence and in the remaining noise γk. If the sequences considered are increased in size by a factor of b, such that τλ−1≪K≪τλ≪bKfor a given 1 ≤λ≤Λ, Γ K(ΥK) is increased (decreased) (see Appendix B). Then, dbk= 2ΓbK bk+ 2 Υ bK+γbk, (47) where Υ bKmust go to zero and Γ bKmust approach a constant when bis increased. By virtue of the expected behavior of the non-diﬀusive contributions (see Appendix A), we propose the following expression for Υ bK ΥbK= Υ Kφ(b), (48) where φ(b) is a decreasing function of b. Moreover, Υ bK is a sum of non-diﬀusive contributions. As presented in Appendix A, each non-diﬀusive contribution to the auto- correlation function has a relative variation smaller than the relative variation of the diﬀusive contribution, namel y 1−1/b. If this inequality is applicable to the sum of non- diﬀusive contributions, we have that 1−1 b>1−ΥbK ΥK(49) 1−1 b>1−φ(b) (50) 1< b φ(b), (51) for all b >1. Then, φmust be either φ(b) =b−υ(52) or φ(b) =1 ηln(b) + 1, (53) with 0 < υ < 1 and 0 < η≤1. Equation (53) can be thought as the limit of Eq. (52) when the exponent goes to zero. We know of no a priori argument to justify Eq. (48). However, as is discussed in Section V, our numerical J. P. Neirotti, D. L. Freeman. and J. D. Doll 7experience has shown Eq. (48) to be obeyed in all cases we have examined. Our goal is to develop a criterion to decide when the simulation can be considered ergodic. From the previ- ous considerations it is clear that the ergodic limit is reached when Υ Kis indistinguishable from zero. The output from a MC simulation is usually noisy. There-fore,γkcan not be neglected. A useful way to separate diﬀusive and non-diﬀusive contributions and to eliminate the stochastic noise from Eq. (46), is to perform a Fourier analysis of the function kdk. Let us deﬁne the frequencies ωn= (2π/K)n, with n= 0,1, . . ., K −1. The discrete Fourier transform of the function kdkis the signal YK(ωn) YK(ωn) =/hatwidestkdk(ωn) =1 KK/summationdisplay k=1exp(−iωnk)kdk (54) =2 KK/summationdisplay k=1exp(−iωnk) ΓK+2 KK/summationdisplay k=1kexp(−iωnk) ΥK+/hatwidestkγk(ωn) = 2δn,0ΓK+{δn,0(K+ 1) + (1 −δn,0)(1 + icot(ωn/2))}ΥK+/hatwidestkγk(ωn) = 2δn,0ΓK+ (K δn,0+ 1) Υ K+i(1−δn,0) cot(ωn/2)Υ K+/hatwidestkγk(ωn) (55) In general,/hatwidestkγk(ωn) is negligible except at high frequen- cies. For small positive values of the frequency we can make the approximation cot( ωn/2)≃2/ωn. From this approximation we have Im(YK(ωn))≃2 ωnΥK. (56) The real part of Eq. (55) for positive frequencies is Re(YK(ωn)) = Υ K. (57) Even though simpler than Eq. (56), we have found Eq. (57) is more sensitive to the deviations of dkfrom the ap- proximation Eq. (46). Therefore, the data obtained from the real part is of poorer quality than the data obtained from the imaginary part. Equation (56) implies that for a given simulation length K, the contributions to the MC metric from the non- diﬀusive process can be determined from a simple rela- tionship involving the Fourier transform of the function kdkat low frequencies. By increasing the length of the runKby a factor of b, it is possible to observe the depen- dence of Υ bKonbK. V. APPLICATIONS The concepts developed in the previous sections are suf- ﬁciently general to be applied to any kind of MC sim- ulation. We devote the present section to the applica- tion of the developments of this paper to the study of the Lennard-Jones 13-particle cluster in the canonical ensem- ble. This system has been introduced previously in Sec. II. Some thermodynamic properties of clusters as a func- tion of temperature exhibit rapid changes that are remi- niscent of similar changes that occur for the same proper- ties in bulk systems at phase transitions. In a bulk system a phase transition occurs at a single temperature. Forclusters the rapid changes in thermodynamic properties occur over a ﬁnite temperature interval. To distinguish the temperature range where thermodynamic properties change rapidly in clusters from a true phase transition, we follow Berry et al. [25] and refer to such changes in physical properties as associated with a phase change. A common property that has been found to be useful in mon- itoring these phase change intervals of temperature is the heat capacity at constant volume [26] CV(T) =1 kBT2/angbracketleftbig (V− /an}bracketle{tV/an}bracketri}htT)2/angbracketrightbig T+3 2NkB,(58) where /an}bracketle{t·/an}bracketri}htTrepresents the classical canonical mean value. In this work we consider the bare Metropolis (Met), [20] J-walking (Jw), [16] and parallel tempering (PT) [17–19] approaches to Monte Carlo simulations. The free variable of all these methods is the reduced temperature kBT/ε. In PT and Jw simulations, the highest temperature used (Th) must be suﬃciently large to ensure that Met is er- godic. [16] From experience simulating a variety of sys- tems, we have found that Thmust also be lower than a temperature Tbwhere cluster evaporation events become frequent. It is useful to think of Tbas the cluster ana- logue of a boiling temperature. We have found that Met is unable to sample the boiling phase change region for clusters ergodically, using total time scales accessible t o current simulations. For the results that follow, U(m) kis chosen to be repre- sented by the potential energy of the system. In general U(m) kcan be any scalar property of the system. We deﬁne a pass to represent a set of single particle MC moves taken sequentially over the 13 particles in the cluster. We take U(m) kto be the potential energy at the k′th pass, in the m′th experiment. Using Eq. (55) we can write YK(0) = 2 Γ K+ (K+ 1)Υ K. (59) In the non-ergodic regime, YK(0) grows with K, while in the ergodic regime, the signal YK(0) approaches a con- stant. 8 The Approach to Ergodicity in Monte Carlo SimulationsWe begin by displaying results obtained for a calcula- tion that has not attained ergodicity over the time scale of the simulation. We examine the 13-particle Lennard- Jones cluster with the Met algorithm setting Rc= 4σat a temperature of kBTh/ε= 0.393. The temperature is chosen to be that typically used as the initial high tem- perature in Jw and PT studies of LJ 13. By choosing a large constraining radius, the evaporation events are so frequent at the chosen temperature that attaining ergodic- ity proves to be quite diﬃcult. We demonstrate the eﬀect of reducing the constraining radius shortly. 0 2000 4000 6000 8000 10000 K−30−25−20−15−10UK050000100000150000200000YK (0) \| Fig. 2: The upper panel is the signal YK(0) (in units of ε2) vs. KforRc= 4σatkBTh/ε= 0.393. from M= 40 independent ex- periments, of LJ 13. The length of the simulation is 104MC passes. The lower panel shows the “time evolution” of UK(in units of ε) for 15 independent experiments. At least three basins with d iﬀerent energies are present. Clearly, the simulation at this scale of time, is not ergodic. The number of replicas used in the calculation is M= 40, andK= 104. The upper panel of Fig. 2 shows the sig- nalYK(0) [evaluated using Eq. (54)], which grows along the entire simulation. This is the behavior expected in the non-ergodic regime. In the lower panel we can seethe “time evolution” of the temporal mean values of 15 experiments. 0 1 2 3 4 5 6 log2 (b)0.02.04.06.08.0ϒK / ϒbKRc = 4.0σ Rc = 3.0σ0.02.04.06.08.010.012.0ϒbKRc = 4.0σ Rc = 3.0σ Rc = 2.5σ Rc = 2.0σ Fig. 3: Upper panel: Υ bK(in units of ε2) as a function of log2(b) forRc= 4σ, 3σ, 2.5σ, and 2 σ. For the two larger radii the full line is the best ﬁt to the data points, according to Eq. (48) with φdeﬁned in Eq. (53). The lower panel shows the linear behavior of Υ K/ΥbK vs. log2(b), for Rc= 4σand 3 σ.Khas been set to 104. There are three sets of curves, each of which is indicative of sampling of at least three diﬀerent energy basins. At low values of Kthe curves in the lower panel diﬀer signif- icantly. At K≃4 000 the high energy basin curves begin to decrease in energy. For a value of Klarger than the data displayed in Fig. 2, the curves can be expected to coalesce with the low energy basin curves. It is clear that forK≤10 000, the simulation is not ergodic. In PT and Jw studies it is essential that the initial high temperature walk be ergodic. Ergodicity can be attained for LJ 13by reducing the radius of the constraining poten- tial so that evaporation events are rare. We now present a study of Υ Kas a function of Kfor several values of Rc. To determine Υ K, we have calculated the Fourier transform function YK(ωn) using Eq. (54) at a series of frequencies ωn= 2πn/K where nhas ranged from 1 to J. P. Neirotti, D. L. Freeman. and J. D. Doll 9min(√ 12bK/20π,100). This range of frequencies ensures the linear approximation used in Eq. (55) is valid while including suﬃcient numbers of points for accuracy. [27] Using Eq. (56), we have calculated the slope of the imag- inary part of 1 /YK(ωn) as a function of ωn, for these fre- quencies. The data points appearing in Fig. 3 are the mean value over twenty independent calculations of the slope of 1 /YK(ωn). 0 1 2 3 4 5 6 log2 (b)0.000.250.500.751.001.251.50ϒbKRc = 2.5σ error in ϒ(Rc = 2.5σ) Rc = 2.0σ error in ϒ(Rc = 2.0σ) Fig. 4: ΥbK(in units of ε2) and its error vs. log2(b) for Rc= 2.5σand 2 σ, with K= 104. When Υ bKis on the order of its own error, the simulation can be considered ergodic. For Rc= 2σ the simulation becomes ergodic at log2(b)≃4 (bK≃16×104). For Rc= 2.5σa longer simulation is needed to reach ergodicity. Starting from random conﬁgurations, we have per- formed 5 ×104Met passes at kBTh/ε= 0.393. After this warmup process, we have created sequences of sizes bK= 104, 2×104, 4×104,. . ., 64×104. The results are presented in Fig. 3 for Rc= 4σ, 3σ, 2.5σ, and 2 σ. The upper panel shows Υ bKas a function of log2(b), for ﬁxed K= 104. We have chosen to present the data using base 2 logarithms for clarity (each increase by 1 unit of log2(b) represents a factor of 2 scale increase). All the data de- crease with increasing b, but only Rc= 2σandRc= 2.5σ appear to vanish to within the error bars over the time scale of the current simulation. In the lower panel we present Υ K/ΥbKas a function of log2(b) forRc= 4 and3σ. The decay law suggested in Eq. (48) with φgiven by Eq. (53) is satisﬁed for both radii. 0 1 2 3 4 5 log2 (b)0.01.02.03.04.05.06.0log2 (ϒK / ϒbK )PT, υ = 0.93(3) Met, υ = 0.94(2)0.00.51.01.52.0ϒbKPT Met Fig. 5: The upper panel shows the decay behavior of Υ bK(in units of ε2) as a function of log2(b) for PT and Met, at the tem- perature of the melting peak of the heat capacity, kBTm/ε= 0.282. From Eq. (52), we plot log2(ΥK/ΥbK) vs. log2(b), to extract the value of the exponent υ(the slope of the linear ﬁt). We have found υ= 0.93±0.03 for PT, and υ= 0.94±0.02 for Met. The straight lines are the best linear ﬁts of the data points. We have stated that the simulation can be considered eﬀectively ergodic when Υ Kis indistinguishable from zero. In Fig. 4 we have plotted Υ bKand its statistical error as a function of log2(b) forRc= 2.5 and 2 σ. For Rc= 2σthe crossing point of Υ bKand its error is at bK≃16×104. ForRc= 2.5σthe crossing point is at a bK > 64×104. We can conclude that for kBTh/ε= 0.393 and Rc= 2σ the simulation can be considered eﬀectively ergodic after 16×104Met passes. Once a constraining radius is chosen, PT and Jw sim- ulations require the highest temperature Thbe chosen so that Met is ergodic. For a given Rc, the extent of ergodic- ity can be tested using the same metric that has been used for determining the optimum value of Rc, but by varying 10 The Approach to Ergodicity in Monte Carlo Simulationsthe temperature. For the parameters kBTh/ε= 0.393 and Rc= 2σthe simulation is ergodic even at very short se- quence lengths. We have found that for kBTh/ε <0.393 the simulations are not ergodic. To be sure that the pa- rameters are appropriate, we have performed a short PT simulation (104passes, ten PT passes consists of nine Met passes plus an exchange attempt) with 40 equally spaced temperatures in the range kBT/ε=[0.028,0.393] in order to obtain a ﬁrst estimate of the position of the melting and boiling temperature regions. The boiling peak in the speciﬁc heat appears to be located at a higher tempera- ture than kBT/ε= 0.393. Moreover, the value of CVat kBT/ε= 0.393 is about one-half the value of CVat the temperature of the melting peak kBTm/ε= 0.282. From these results we feel it is safe to choose Rc= 2σand kBTh/ε= 0.393 for the calculations that follow. We now illustrate the convergence characteristics of Υ K when we increase the total time scale of the calculation by a factor b. We illustrate this behavior using a PT simula- tion of LJ 13, and we focus on results at the temperature of the melting peak in the heat capacity ( kBTm/ε= 0.282). We choose this temperature, because from experience [14,15,23] we know the statistical ﬂuctuations are large at the melting heat capacity maximum. The large statistical ﬂuctuations make it possible to emphasize the behavior of Υ K. We have run the PT simulation at 40 equally spaced temperatures in the range kBT/ε= [0.028,0.393]. The initial warmup time has been set to 104Met passes, followed by 2 ×104PT passes. Following the warm-up period, we perform simulations of 105, 2×105, 4×105, 8×105, 16×105, and 32 ×105PT passes. In each case the initial conﬁguration has been taken to be the last conﬁg- uration of the previous run. The output of the simulation are sequences of the potential energy. Υ Khas been deter- mined in the same way as in the calculation of the high temperature parameters (presented in Fig. 3 and Fig. 4). The data points appearing in the upper panel of Fig. 5 are the mean value over twenty independent calculations of the slope of 1 /YK(ωn). In the lower panel of Fig. 5 we have plotted log2(ΥK/ΥbK) as a function of log2(b), where K= 104andb= 1,2,4, . . .,32. The slope of the linear ﬁt is the exponent υ, according to Eq. (52). At the temperature of the melting peak, υ= 0.93±0.03. It is of interest to perform a similar study of the behav- ior of Υ Kas a function of the time scaling for an Met cal- culation. We have taken the ﬁnal conﬁguration of the PT simulation at kBTm/ε= 0.282 as an initial conﬁguration, and we have performed a simple Met simulation at that melting temperature. A graph of Υ bKand log2(ΥK/ΥbK) as a function of log2(b) for Met is also presented in Fig. 5. From the upper panel of Fig. 5, it is evident that Met results are not ergodic within the same scaled time as the PT results. It is also evident that the power law exponent for both Met and PT are not distinguishable. Similar studies of the power law using the Jw method also give the same exponent. Neither an increase in the number of temperatures nor changing the distribution of temper- atures in both Jw and PT simulations has any eﬀect onthe calculated exponent. 0.0 0.1 0.2 0.3 0.4 kB T / ε010203040Γ / ΓPTJw Met Fig. 6: Comparison of the Met and Jw diﬀusion coeﬃcients with the PT diﬀusion coeﬃcient as a function of the reduced te m- perature. The dashed line represents equivalence between m ethods. By using the results to compare the relative eﬃciencies of Met, Jw and PT simulations for the LJ 13system. We have found that PT and Jw simulations can be considered ergodic if the run length is on the order of 2 ×105passes, while Met simulations that are initialized from conﬁgu- rations generated from an ergodic PT study are ergodic when the total run length consists of 2 ×106passes or more. In order to compare approaches, we have calculated Γ as a function of the reduced temperature, for the three methods. The comparison of diﬀusion coeﬃcients from diﬀerent algorithms has also been used by Andricioaei and Straub [13]. The comparison of Jw and Met with PT is presented in Fig. 6. The Jw and PT simulations are found to have comparable eﬃciencies using Γ as a measure for all calculated temperatures. At intermediate temperatures, Met is signiﬁcantly less eﬃcient. We have chosen to trun- cate the Jw study at kBT/ε= 0.12. For temperatures below kBT/ε= 0.12, Jw simulations require signiﬁcant ef- fort, because a large set of external distributions must be generated. Because at temperatures below kBT/ε= 0.12 J. P. Neirotti, D. L. Freeman. and J. D. Doll 11LJ13is dominated by structures close to the lowest energy icosahedral isomer, we expect the Jw and PT methods to have similar eﬃciencies (as measured by Γ) for all tem- peratures. VI. CONCLUSIONS In this paper we have presented a study of the approach to the ergodic limit in MC simulations. In all the cases examined, the behavior of the MC metric dkcan be ap- proximated by Eq. (46), and the behavior of Υ bKsatisﬁes Eq. (48). Because the exponent υis smaller than one for all the cases studied, the dependence of the non-diﬀusive contributions on dkis weaker (in the sense of Appendix A) than the diﬀusive contributions. The assumption on which we have built the stochastic model have been veri- ﬁed numerically for a system having a suﬃciently complex potential surface to be viewed as prototypical of a large set of many-particle systems. The MC metric used in this work appears to be a valu- able tool to study the ergodicity properties of MC simu- lations. The non-ergodic components of the MC metric enable the prediction of the minimum length a MC simu- lation must have in order to be considered ergodic. The comparison of Γ from diﬀerent algorithms gives a reason- able estimate of their relative eﬃciencies. From the study of the melting region of 13 particle clus- ters, we have found that the exponent υdepends both on the method used and the nature of the potential energy function. We have performed calculations, not discussed in this work, where the functional form of the potential energy is modiﬁed. These studies have shown υto be dependent on the details of the potential. We have not found the exponent υto be a strong function of method. Although PT and Met have signiﬁcantly diﬀerent eﬃcien- cies as measured by their relative diﬀusion coeﬃcients, υis nearly the same in the two methods. The diﬀerence in the decay of Υ Kappears to be dominated by the coeﬃcient in Eqs.(48) and (52) rather than the exponent. As discussed in the text, parallel tempering and J- walking studies of many-particle systems must have an initial high temperature component that is chosen so that a Met simulation is known to be ergodic. For cluster sim- ulations that require an external constraining potential t o deﬁne the cluster, the radius of the constraining poten- tial must be carefully chosen in order to achieve ergodic results. We have found the metric and associated decay laws developed in this work to be a particularly valuable method of choosing these initial parameters in both par- allel tempering and J-walking simulations. We also remark that the metric introduced here may be a more sensitive probe of ergodicity than may be required in some applications. For example in previous J-walking studies [26] of the 13-particle Lennard-Jones cluster, the heat capacity curve determined with a constraining radius of 4σis nearly indistinguishable from the curve obtained with a constraining radius of 2 σ. From the results of thiswork, we know the initial high temperature walk is not ergodic when a constraining radius of 4 σis used. It is striking that the non-ergodicity as measured by the en- ergy metric is not apparent in the heat capacity curve. We have constructed a metric based on an ensemble of MC trajectories. By using an ensemble we attempt to cover suﬃcient portions of space so that all components are accessible. In practice only a ﬁnite subset of a full ensemble can be included, and it is always possible that components of space are missed. In such a case Υ Kmay decay to zero numerically within the subspace, and the be- havior may give misleading evidence that the simulation is ergodic. Because components of space may be missed in any ﬁnite simulation, it is impossible to guarantee ergod- icity. It is hoped by using a suﬃciently large ensemble of trajectories to deﬁne the metric, the possibility of missin g components is minimized. ACKNOWLEDGMENTS We would like to thank Dr. O. Osenda for helpful com- ments. This work has been supported in part by the National Science Foundation under grant numbers CHE- 9714970 and CDA-9724347. This research has been sup- ported in part by the Phillips Laboratory, Air Force Mate- rial Command, USAF, through the use of the MHPCC un- der cooperative agreement number F29601-93-0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as nec- essarily representing the oﬃcial policies or endorsements , either expressed or implied, of Phillips Laboratory or the U.S. Government. APPENDIX A: WEAK DEPENDENCE OF THE NON-DIFFUSIVE CONTRIBUTIONS We have considered two overall time scales for a MC simulation. Properties calculated at short times (labeled kin the discrete case) provide information about each step of the MC process, and properties averaged over the total simulation time (labeled Kin the discrete case) give in- formation about the approach to ergodic behavior. When Kis suﬃciently short we have both diﬀusive and non- diﬀusive contributions as a function of k. In this Ap- pendix we explain the relative time dependence of the diﬀusive and non-diﬀusive contributions to the autocorre- lation function. It has been assumed that the autocorrelation function Eq. (28) can be expressed as the sum of diﬀusive terms plus non-diﬀusive terms, i.e. κ(t, t′) =κd(t, t′) +Λ/summationdisplay ℓ=λ+1κnd , ℓ(t, t′), (A1) where 12 The Approach to Ergodicity in Monte Carlo Simulationsκd(t, t′) =Γ0+ Γ1+ Γ2+. . .+ Γλ t>(A2) κnd , ℓ(t, t′) =Γℓ τℓfℓ/parenleftbiggt∗ ℓ τℓ/parenrightbigg . (A3) Increasing the time variables by a factor b >1, such thatτλ≪bt>≪τλ+1, with λ≥1, we can study the relative variations of each contribution to the correlatio n function, diﬀusive and non-diﬀusive (labeled by ℓ > λ ). In this Appendix we only consider values of bsuch that the transformation t→btdoes not increase the time scale beyond the local correlation time. In Appendix B values ofbare considered that do cross such time scales. By relative variations we mean ∆d(t, t′;b) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleκd(bt, bt′)−κd(t, t′) κd(t, t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A4) ∆nd , ℓ(t, t′;b) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleκnd , ℓ(bt, bt′)−κnd , ℓ(t, t′) κnd , ℓ(t, t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(A5) The relative variation of each non-diﬀusive contribution i s ∆nd , ℓ(t, t′;b) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−1 b2/integraltextbt 0dt1/integraltextbt′ 0dt2fℓ/parenleftBig \|t1−t2\| τℓ/parenrightBig /integraltextt 0dt1/integraltextt′ 0dt2fℓ/parenleftBig \|t1−t2\| τℓ/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (A6) whereas the relative variation of the diﬀusive contributio n is ∆d(t, t′;b) = 1−1 b, (A7) If ∆ d(t, t′;b)>∆nd , ℓ(t, t′;b) for all pair of times tand t′and for all b >1 such that bt>≪τℓ, we say that thenon-diﬀusive contributions are weaker than the diﬀusive contribution in their dependence on t. We explore, in the remainder of this appendix, the properties fℓmust have in order that the inequality ∆ d(t, t′;b)>∆nd , ℓ(t, t′;b) is satisﬁed. Lemma: If the function Hℓ(t;τ) Hℓ(t;τ) =/integraldisplayt 0dt′fℓ/parenleftbiggt′ τ/parenrightbigg >0 (A8) satisﬁes the inequality Hℓ(t;τ)> t f ℓ/parenleftbiggt τ/parenrightbigg ∀tandτ , (A9) then, Hℓ(t;τ) is an increasing function of τ. Demonstration: Forℓandtﬁxed, the function Hℓ(t, τ) evaluated in τ′is Hℓ(t;τ′) =/integraldisplayt 0dt′fℓ/parenleftbiggt′ τ′/parenrightbigg (A10) =/integraldisplayt 0dt′fℓ/parenleftbiggτt′ τ′τ/parenrightbigg (A11) =τ′ τ/integraldisplayτt/τ′ 0du f ℓ/parenleftBigu τ/parenrightBig (A12) =τ′ τHℓ(τt/τ′;τ), (A13) then, for ∆ τ >0 Hℓ(t;τ+ ∆τ)−Hℓ(t;τ) ∆τ=1 ∆τ/braceleftBigg τ+ ∆τ τ/integraldisplayτt/(τ+∆τ) 0dt′fℓ/parenleftbiggt′ τ/parenrightbigg −/integraldisplayt 0dt′fℓ/parenleftbiggt′ τ/parenrightbigg/bracerightBigg (A14) =1 ∆τ/braceleftBigg ∆τ τ/integraldisplayτt/(τ+∆τ) 0dt′fℓ/parenleftbiggt′ τ/parenrightbigg −/integraldisplayt τt/(τ+∆τ)dt′fℓ/parenleftbiggt′ τ/parenrightbigg/bracerightBigg (A15) =1 ∆τ/braceleftBigg ∆τ τ/integraldisplayτt/(τ+∆τ) 0dt′fℓ/parenleftbiggt′ τ/parenrightbigg −t∆τ τ+ ∆τfℓ/parenleftbiggt∗ τ/parenrightbigg/bracerightBigg , (A16) where t∗∈[tτ/(τ+ ∆τ), t]. In the limit ∆ τ→0, and by virtue of the continuity of fℓ, the derivative takes the form ∂Hℓ(t;τ) ∂τ=1 τ/braceleftbigg Hℓ(t;τ)−tfℓ/parenleftbiggt τ/parenrightbigg/bracerightbigg . (A17) Then, ∂Hℓ(t;τ)/∂τ > 0, and Hℓ(t;τ) is an increasing function of τ.✷ Here we have presented the two ﬁrst conditions fℓmust have, namely Eqs. (A8) and (A9). From Eq. (19) fℓ(0) is a global maximum, and the memory functions must have a positive peak at zero. The area below that peakmust be suﬃciently large to satisfy Eq. (A8). Moreover, fℓ(0) must be suﬃciently large to satisfy Eq. (A9), even at points where fℓ(t/τ) is a local maximum. Then, to satisfy this Lemma, we need a memory function with a suﬃciently large global maximum at t= 0. Corollary: Suppose Hℓ(t;τℓ)> tf ℓ(t/τℓ). If b >1, then 0 <∆nd , ℓ(t, t′;b)<1 for all pair of times tandt′. Demonstration: Under the change of scale in time t→bt,κnd , ℓ(t, t′) can be written J. P. Neirotti, D. L. Freeman. and J. D. Doll 13κnd , ℓ(bt, bt′) =1 b2tt′/integraldisplaybt 0dt1/integraldisplaybt′ 0dt21 τℓfℓ/parenleftbigg\|t1−t2\| τℓ/parenrightbigg (A18) =1 tt′/integraldisplayt 0dt1/integraldisplayt′ 0dt21 τℓfℓ/parenleftbiggb\|t1−t2\| τℓ/parenrightbigg , (A19) then, the quotient κnd , ℓ(bt, bt′)/κnd , ℓ(t, t′) is κnd , ℓ(bt, bt′) κnd , ℓ(t, t′)=/integraltextt 0dt1/braceleftBig/integraltextt1 0dt fℓ/parenleftBig t τℓ/b/parenrightBig +/integraltextt>−t1 0dt fℓ/parenleftBig t τℓ/b/parenrightBig/bracerightBig /integraltextt 0dt1/braceleftBig/integraltextt1 0dt fℓ/parenleftBig t τℓ/parenrightBig +/integraltextt>−t1 0dt fℓ/parenleftBig t τℓ/parenrightBig/bracerightBig (A20) =/integraltextt 0dt1{Hℓ(t1;τℓ/b) +Hℓ(t>−t1;τℓ/b)} /integraltextt 0dt1{Hℓ(t1;τℓ) +Hℓ(t>−t1;τℓ)}. (A21) By Eq. (A8), Hℓ(t;τ)>0∀tandτ. By the Lemma the numerator is smaller than the denomina- tor. Then 0 < κ nd , ℓ(bt, bt′)/κnd , ℓ(t, t′)<1 and then, 0<∆nd , ℓ(t, t′;b)<1.✷ Theorem: Suppose that b >1 is such that τℓ−1≪ bt>≪τℓ,Hℓ(t;τℓ)> tf ℓ(t/τℓ), and all fℓsatisfy the Lip- schitz condition [28] (for all closed interval Aexists a real positive number Cℓsuch that \|fℓ(x)−fℓ(y)\| ≤Cℓ\|x−y\| (A22) for all xandyinA). Then ∆ nd , ℓ(t, t′;b)<∆d(t, t′;b) if and only if fℓis non-negative in the interval [0 , t>).Demonstration: If ∆ nd , ℓ(t, t′;b)<∆d(t, t′;b), then 1−1 b>1−1 b2/integraltextbt 0dt1/integraltextbt′ 0dt2fℓ/parenleftBig \|t1−t2\| τℓ/parenrightBig /integraltextt 0dt1/integraltextt′ 0dt2fℓ/parenleftBig \|t1−t2\| τℓ/parenrightBig (A23) 1<1 b/integraltextbt 0dt1/integraltextbt′ 0dt2fℓ/parenleftBig \|t1−t2\| τℓ/parenrightBig /integraltextt 0dt1/integraltextt′ 0dt2fℓ/parenleftBig \|t1−t2\| τℓ/parenrightBig (A24) where the operations to reach Eq. (A24) are valid by using Corollary . Then 0</integraldisplaybt 0dt1/integraldisplaybt′ 0dt21 bfℓ/parenleftbigg\|t1−t2\| τℓ/parenrightbigg −/integraldisplayt 0dt1/integraldisplayt′ 0dt2fℓ/parenleftbigg\|t1−t2\| τℓ/parenrightbigg (A25) 0</integraldisplayt 0dt1/integraldisplayt′ 0dt2/braceleftbigg b fℓ/parenleftbiggb\|t1−t2\| τℓ/parenrightbigg −fℓ/parenleftbigg\|t1−t2\| τℓ/parenrightbigg/bracerightbigg (A26) 0</integraldisplayt< 0dt1/braceleftbigg/integraldisplayt1 0dt2/bracketleftbigg b fℓ/parenleftbiggb(t1−t2) τℓ/parenrightbigg −fℓ/parenleftbiggt1−t2 τℓ/parenrightbigg/bracketrightbigg +/integraldisplayt> t1dt2/bracketleftbigg b fℓ/parenleftbiggb(t2−t1) τℓ/parenrightbigg −fℓ/parenleftbiggt2−t1 τℓ/parenrightbigg/bracketrightbigg/bracerightbigg (A27) 0</integraldisplayt< 0dt1/braceleftbigg/integraldisplayt1 0dt/bracketleftbigg b fℓ/parenleftbiggbt τℓ/parenrightbigg −fℓ/parenleftbiggt τℓ/parenrightbigg/bracketrightbigg +/integraldisplayt>−t1 0dt/bracketleftbigg b fℓ/parenleftbiggbt τℓ/parenrightbigg −fℓ/parenleftbiggt τℓ/parenrightbigg/bracketrightbigg/bracerightbigg (A28) 0</integraldisplayt< 0dt1/braceleftBigg/integraldisplaybt1 0dt fℓ/parenleftbiggt τℓ/parenrightbigg −/integraldisplayt1 0dt fℓ/parenleftbiggt τℓ/parenrightbigg +/integraldisplayb(t>−t1) 0dt fℓ/parenleftbiggt τℓ/parenrightbigg −/integraldisplayt>−t1 0dt fℓ/parenleftbiggt τℓ/parenrightbigg/bracerightBigg (A29) 0</integraldisplayt< 0dt1/braceleftBigg/integraldisplaybt1 t1dt fℓ/parenleftbiggt τℓ/parenrightbigg +/integraldisplayb(t>−t1) t>−t1dt fℓ/parenleftbiggt τℓ/parenrightbigg/bracerightBigg . (A30) Using the intermediate value theorem, [24] we have /integraldisplaybt tdt′fℓ/parenleftbiggt′ τℓ/parenrightbigg = (b−1)t fℓ/parenleftbiggt∗(t) τℓ/parenrightbigg (A31) = (b−1)t fℓ/parenleftbiggt τℓ/parenrightbigg + (b−1)t/bracketleftbigg fℓ/parenleftbiggt∗(t) τℓ/parenrightbigg −fℓ/parenleftbiggt τℓ/parenrightbigg/bracketrightbigg , (A32) where t∗(t)∈[t, bt]. Let be t∗ α(t) and t∗ β(t) the values at which the intermediate value theorem is satis ﬁed, in the intervals [ t, bt] and [ t>−t, b(t>−t)] respectively 14 The Approach to Ergodicity in Monte Carlo Simulations(b−1)t fℓ/parenleftbiggt∗ α(t) τℓ/parenrightbigg =/integraldisplaybt tdt′fℓ/parenleftbiggt′ τℓ/parenrightbigg (A33) (b−1)(t>−t)fℓ/parenleftbiggt∗ β(t) τℓ/parenrightbigg =/integraldisplayb(t>−t) t>−tdt′fℓ/parenleftbiggt′ τℓ/parenrightbigg , (A34) then, the remainder can be written as Rℓ(t<, t>;b) =/integraldisplayt< 0dt/braceleftbigg t/bracketleftbigg fℓ/parenleftbiggt∗ α(t) τℓ/parenrightbigg −fℓ/parenleftbiggt τℓ/parenrightbigg/bracketrightbigg + (t>−t)/bracketleftbigg fℓ/parenleftbiggt∗ β(t) τℓ/parenrightbigg −fℓ/parenleftbiggt>−t τℓ/parenrightbigg/bracketrightbigg/bracerightbigg . (A35) By the Lipschitz condition, we have that Rℓ(t<, t>;b)≤/integraldisplayt< 0dt/braceleftbigg t/vextendsingle/vextendsingle/vextendsingle/vextendsinglefℓ/parenleftbiggt∗ α(t) τℓ/parenrightbigg −fℓ/parenleftbiggt τℓ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+ (t>−t)/vextendsingle/vextendsingle/vextendsingle/vextendsinglefℓ/parenleftbiggt∗ β(t) τℓ/parenrightbigg −fℓ/parenleftbiggt>−t τℓ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg (A36) </integraldisplayt< 0dt/braceleftbigg t Cℓ/vextendsingle/vextendsingle/vextendsingle/vextendsinglet∗ α(t)−t τℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+ (t>−t)Cℓ/vextendsingle/vextendsingle/vextendsingle/vextendsinglet∗ β(t)−(t>−t) τℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg (A37) <Cℓ τℓ/integraldisplayt< 0dt{t\|bt−t\|+ (t>−t)\|b(t>−t)−(t>−t)\|} (A38) <Cℓ τℓ(b−1)/integraldisplayt< 0dt/bracketleftbig t2+ (t>−t)2/bracketrightbig (A39) <Cℓ τℓ(b−1)/parenleftbigg2 3t3 <+t<t>(t>−t<)/parenrightbigg (A40) <2 3t3 >Cℓ τℓ(b−1), (A41) where Cℓis a suitable positive real constant. Using Eqs. (A32) and (A 35) in Eq. (A30) we have 0</integraldisplayt< 0dt(b−1)/braceleftbigg t fℓ/parenleftbiggt τℓ/parenrightbigg + (t>−t)fℓ/parenleftbiggt>−t τℓ/parenrightbigg/bracerightbigg + (b−1)Rℓ(t<, t>;b) (A42) 0</integraldisplayt< 0dt t f ℓ/parenleftbiggt τℓ/parenrightbigg +/integraldisplayt> t>−t<dt t f ℓ/parenleftbiggt τℓ/parenrightbigg +Rℓ(t<, t>;b) (A43) 0</integraldisplayt< 0dt t f ℓ/parenleftbiggt τℓ/parenrightbigg +/integraldisplayt> 0dt t f ℓ/parenleftbiggt τℓ/parenrightbigg −/integraldisplayt>−t< 0dt t f ℓ/parenleftbiggt τℓ/parenrightbigg +Rℓ(t<, t>;b) (A44) 0< Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<) +2 3t3 >Cℓ τℓ(b−1), (A45) where Fℓ(t) =/integraldisplayt 0dt′t′fℓ/parenleftbiggt′ τℓ/parenrightbigg , (A46) is a continuous and diﬀerentiable function of t. The inequality (A45) holds for any b >1. Suppose that Fℓ(t<) + Fℓ(t>)−Fℓ(t>−t<)<0. Then, if bis such that b= 1 +3 2Lτℓ t3>Cℓ\|Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<)\|, (A47) where L >2, we have that 0< Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<) +1 L\|Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<)\| (A48) 0<L−1 L[Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<)] (A49) in contradiction with the hypothesis that Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<) is negative. Then J. P. Neirotti, D. L. Freeman. and J. D. Doll 150≤Fℓ(t<) +Fℓ(t>)−Fℓ(t>−t<). (A50) Let us deﬁne the function ∆Fℓ(t) =Fℓ(t)−Fℓ(t>−t), (A51) where t∈(0, t>). The right derivative at t= 0 of ∆ Fℓ(t) is lim ∆t→0+∆Fℓ(∆t)−∆Fℓ(0) ∆t= lim ∆t→0+Fℓ(∆t)−Fℓ(0) + Fℓ(t>)−Fℓ(t>−∆t) ∆t(A52) = lim ∆t→0+1 ∆t/braceleftBigg/integraldisplay∆t 0dt t f ℓ/parenleftbiggt τℓ/parenrightbigg +/integraldisplayt> t>−∆tdt t f ℓ/parenleftbiggt τℓ/parenrightbigg/bracerightBigg (A53) = lim ∆t→0+1 ∆t/braceleftbigg ∆t t∗ 1fℓ/parenleftbiggt∗ 1 τℓ/parenrightbigg + ∆t t∗ 2fℓ/parenleftbiggt∗ 2 τℓ/parenrightbigg/bracerightbigg (A54) where t∗ 1∈[0,∆t] and t∗ 2∈[t>−∆t, t>]. Thus ∂∆Fℓ(t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle t→0+=t>fℓ/parenleftbiggt> τℓ/parenrightbigg . (A55) If the right derivative at 0 of ∆ Fℓ(t) is negative, ∆ Fℓ(t) approaches −Fℓ(t>) from below, when t→0. There exists a time 0 <˜t < t >, such that 0 > Fℓ(t>) + ∆Fℓ(˜t), in con- tradiction with Eq. (A50). Then, fℓmust be non-negative fort∈(0, t>). By the property Eq. (19) fℓ(0) must be positive. This proves that ∆ nd ,1(t, t′;b)<∆d(t, t′;b)⇒ fℓ(t)≥0 for 0 ≤t < t >. To demonstrate that if fℓis pos- itive yields ∆ nd ,1(t, t′;b)<∆d(t, t′;b) (i.e. the converse), follow the argument backwards, from Eq. (A30). ✷ In conclusion, if the memory functions are positive, sat- isfy the Lipschitz condition, and satisfy the condition Eqs . (A8) and (A9), the non-diﬀusive contributions are more weakly dependent on time than 1 /t. The results of the present appendix are valid in the limit of a complete ensemble. In our numerical experiments only partial samples of the ensemble can be considered. The memory functions that appear in our numerical cal- culations come from partial mean values of the product ofdiscontinuous functions (every noise process is a discon- tinuous function). These memory functions are discon- tinuous. The behavior of the non-diﬀusive contributions observed in our numerical experiments is in agreement with these analytic (inﬁnite ensemble limit) results. We can infer that there might be a version of the theorem applied to discontinuous memory functions, but we have been unable to develop such a theorem. APPENDIX B: CONSEQUENCES OF THE TIME SCALE CHANGE IN THE NON-DIFFUSIVE CONTRIBUTIONS In this appendix we show the behavior of the func- tionf1when its correlation time is changed according to τ1→τb1=τ1/b, with b≫1; i.e. when the total simu- lation time is scaled to exceed the correlation time of the ﬁrst colored noise process. We multiply the time variables by a number b, such that τ1≪bt>≪τ2. We have that the g1process contributes to the autocorrelation function with 1 b2tt′/an}bracketle{tG1(bt/τℓ)G1(bt′/τℓ)/an}bracketri}ht=1 b2tt′/integraldisplaybt< 0dt1/integraldisplaybt> 0dt21 τ1f1/parenleftbigg\|t1−t2\| τ1/parenrightbigg (B1) =1 btt′/integraldisplayt< 0dt′ 1/integraldisplayt> 0dt′ 21 τb1f1/parenleftbigg\|t′ 1−t′ 2\| τb1/parenrightbigg (B2) where t′=t/bandτb1=τ1/b. We want to compute this contribution both within the neighb orhood t1=t2as well as outside such a region. To do so, we can split the integral in Eq . (B2) in three parts 1 b2tt′/an}bracketle{tG1(bt/τℓ)G1(bt′/τℓ)/an}bracketri}ht=I1+I2+I3 (B3) where I1=1 btt′/integraldisplayt< 0dt1/integraldisplaymax(0 ,t1−ǫ/2) 0dt21 τb1f1/parenleftbiggt1−t2 τb1/parenrightbigg (B4) I2=1 btt′/integraldisplayt< 0dt1/integraldisplaymin(t>,t1+ǫ/2) max(0 ,t1−ǫ/2)dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg (B5) I3=1 btt′/integraldisplayt< 0dt1/integraldisplayt> min(t>,t1+ǫ/2)dt21 τb1f1/parenleftbiggt2−t1 τb1/parenrightbigg (B6) 16 The Approach to Ergodicity in Monte Carlo Simulationswitht<> ǫ > 0 (observe that the only integral involving t1=t2isI2). Consider I1. Ift1< ǫ/2 the inner integral is zero. Therefore, t1must be bigger than ǫ/2 and I1=1 bt<t>/integraldisplayt< ǫ/2dt1/integraldisplayt1−ǫ/2 0dt21 τb1f1/parenleftbiggt1−t2 τb1/parenrightbigg , (B7) which, by virtue of the continuity of f1, can be bounded as follows 1 bt<t>/integraldisplayt< ǫ/2dt1b τ1/parenleftBig t1−ǫ 2/parenrightBig f1/parenleftbiggbtmin τ1/parenrightbigg ≤I1≤1 bt<t>/integraldisplayt< ǫ/2dt1b τ1/parenleftBig t1−ǫ 2/parenrightBig f1/parenleftbiggbtmax τ1/parenrightbigg 1 2(t<−ǫ/2)2 t<t>1 τ1f1/parenleftbiggbtmin τ1/parenrightbigg ≤I1≤1 2(t<−ǫ/2)2 t<t>1 τ1f1/parenleftbiggbtmax τ1/parenrightbigg (B8) where tmax(tmin) is the time in the interval [ ǫ/2, t<] at which the function f1reaches its maximum (minimum) value. Because f1is continuous, there exists t∗ 1∈[tmin, tmax] at which I1=1 2(t<−ǫ/2)2 t<t>1 τ1f1/parenleftbiggbt∗ 1 τ1/parenrightbigg . (B9) Consider now I3. Ift1+ǫ/2> t>, the inner integral is zero. Therefore, 0 < t1<min(t<, t>−ǫ/2) and I3=1 btt′/integraldisplaymin(t<,t>−ǫ/2) 0dt1/integraldisplayt> t1+ǫ/2dt21 τb1f1/parenleftbiggt2−t1 τb1/parenrightbigg =min(t<, t>−ǫ/2) t<t>/bracketleftbigg t>−ǫ 2−1 2min(t<, t>−ǫ/2)/bracketrightbigg1 τ1f1/parenleftbiggbt∗ 3 τ1/parenrightbigg , (B10) where t∗ 3∈[tmin, tmax], and now tmax(tmin) is the time in [ ǫ/2, t>] at which the function f1reaches its maximum (minimum) value. − (∋ / 2 − t1) 0 t1 2t1 ∋ / 2 + t1f1 (\|t1 − t2 \| / τb1 ) / τb1 Fig. 7: The area under the curve represents the ﬁrst integral in Eq. ( B11). The darker piece is half of the integral in the interval [−t1, t1], the lighter is half of the integral in [ −ǫ/2, ǫ/2]. Let us consider now I2. First observe that for the integral in t1, if 0 ≤t1≤ǫ/2, max(0 , t1−ǫ/2) = 0 and min(t>, t1+ǫ/2) =t1+ǫ/2. Ifǫ/2≤t1≤t<then max(0 , t1−ǫ/2) =t1−ǫ/2. Then I2=1 bt<t>/braceleftBigg/integraldisplayǫ/2 0dt1/integraldisplayt1+ǫ/2 0dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg +/integraldisplayt< ǫ/2dt1/integraldisplaymin(t>,t1+ǫ/2) t1−ǫ/2dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg/bracerightBigg . (B11) J. P. Neirotti, D. L. Freeman. and J. D. Doll 17The integral in t2between 0 and t1+ǫ/2 can be evaluated with the help of Fig. 7 /integraldisplayt1+ǫ/2 0dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg =1 2/integraldisplayǫ/2 −ǫ/2dt1 τb1f1/parenleftbigg\|t\| τb1/parenrightbigg +1 2/integraldisplayt1 −t1dt1 τb1f1/parenleftbigg\|t\| τb1/parenrightbigg . (B12) The second integral in t1can be separated in two parts; the ﬁrst for ǫ/2≤t1≤min(t<, t>−ǫ/2) and the second for min(t<, t>−ǫ/2)≤t1≤t<. Ift>−t<< ǫ/2 the second term is zero. Then /integraldisplayt< ǫ/2dt1/integraldisplaymin(t>,t1+ǫ/2) t1−ǫ/2dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg =/integraldisplaymin(t<,t>−ǫ/2) ǫ/2dt1/integraldisplaymin(t>,t1+ǫ/2) t1−ǫ/2dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg + Θ/parenleftBigǫ 2+t<−t>/parenrightBig/integraldisplayt< t>−ǫ/2dt1/integraldisplaymin(t>,t1+ǫ/2) t1−ǫ/2dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg , (B13) where Θ is the step function. If t1≤min(t<, t>−ǫ/2) then min( t>, t1+ǫ/2) =t1+ǫ/2. The last integral in t2can be rearranged in the same way as Eq. (B12). Then /integraldisplayt< ǫ/2dt1/integraldisplaymin(t>,t1+ǫ/2) t1−ǫ/2dt21 τb1f1/parenleftbigg\|t1−t2\| τb1/parenrightbigg =/integraldisplaymin(t<,t>−ǫ/2) ǫ/2dt1/integraldisplayǫ/2 −ǫ/2dt1 τb1f1/parenleftbigg\|t\| τb1/parenrightbigg + 1 2Θ/parenleftBigǫ 2+t<−t>/parenrightBig/integraldisplayt< t>−ǫ/2dt1/bracketleftBigg/integraldisplayǫ/2 −ǫ/2dt1 τb1f1/parenleftbigg\|t\| τb1/parenrightbigg +/integraldisplayt>−t1 t1−t>dt1 τb1f1/parenleftbigg\|t\| τb1/parenrightbigg/bracketrightBigg , (B14) We can observe that the correlation time τb1goes to zero when bis increased. The function (1 /τ1)f1(bt/τ1) becomes negligible outside a neighborhood of t= 0 [observe Eqs. (B9) and (B10)]. Equation (16) holds, then, ifbis suﬃciently large, (1 /τb1)f1(t/τb1) can be considered a delta function. The integrals I1andI3become zero, and the integrals involving t= 0 in the expression of I2converge to one. I2becomes I2=1 bt<t>/braceleftBig min/parenleftBig t<, t>−ǫ 2/parenrightBig + Θ/parenleftBigǫ 2+t<−t>/parenrightBig/parenleftBigǫ 2+t<−t>/parenrightBig/bracerightBig =1 bt>, (B15) which is a diﬀusive contribution to the autocorrelation function. The autocorrelation function becomes then κ(bt, bt′) =Γ0+ Γ1 bt>+Λ/summationdisplay ℓ=2Γℓ τℓfℓ/parenleftbiggt∗ bℓ(t>) τbℓ/parenrightbigg .(B16) The same argument can be used when bis such that τ2≪bt>≪τ3. After such changes in the time scale, the diﬀusion coeﬃcient Γ = Γ 0+ Γ1is enlarged, and the non- diﬀusive contributions are reduced. There is an ultimate scale change, such that τΛ≪bt>. Beyond this maximum time scale the process can be considered diﬀusive. [1] J. P. Valleau and S. G. Whittington, A guide to Monte Carlo for Statistical Mechanics: 1 Highways, in Statisti- cal Mechanics, Part A: Equilibrium Techniques , Modern Theoretical Chemistry Series, Vol. 5, Chap. 4, B. Berne Ed. (Plenum, New York, 1976). [2] W. W. Wood and F. R. Parker, J. Chem. Phys. 27, 720 (1957). [3] D. Thirumalai, R. D. Mountain, and T. R. Kirpatrick, Phys. Rev. A 39, 3563 (1989).[4] R. G. Palmer, Adv. Phys. 31, 669 (1982). [5] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer- Verlag. Berlin, Heidelberg, New York, Tokyo, 1983). [6] D. L. Freeman and J. D. Doll, Annu. Rev. Phys. Chem. 47, 43 (1996). [7] R. M. Lynden-Bell and D. J. Wales, J. Chem. Phys. 101, 1460 (1994). [8] J. P. K. Doye, D. J. Wales, and M. A. Miller, J. Chem. Phys.109, 8143 (1998). [9] R. D. Mountain and D. Thirumalai, J. Chem. Phys. 93, 6975 (1989). [10] D. Thirumalai and R. D. Mountain, Phys. Rev. A 42, 4574 (1990). [11] J. E. Straub and D. Thirumalai, Proc. Nat. Acad. Sci. USA90, 809 (1993). [12] A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983). [13] I. Andricioaei and J. E. Straub, J. Chem. Phys. 107, 9117 (1997). [14] J. P. Neirotti, F. Calvo, D. L. Freeman and J. D. Doll, J. Chem. Phys. 112, 10340 (2000). [15] F. Calvo, J. P. Neirotti, D. L. Freeman and J. D. Doll, J. Chem. Phys. 112, 10350 (2000). [16] D. D. Frantz, D. L. Freeman, and J. D. Doll, J. Chem. Phys.93, 2769 (1990). 18 The Approach to Ergodicity in Monte Carlo Simulations[17] E. Marinari and G. Parissi, Europhys. Lett. 19, 451 (1992). [18] C. J. Geyer and E. A. Thompson, J. Am. Stat. Assoc. 90, 909 (1995). [19] M. Falcioni and M. W. Deem, J. Chem. Phys. 110, 1754 (1999). [20] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). [21] J. K. Lee, J. A. Barker, and F. F. Abraham, J. Chem. Phys.58, 3166 (1973). [22] P. Labastie and R. L. Whetten, Phys. Rev. Lett. 65, 1567 (1990).[23] J. P. Neirotti, D. L. Freeman, and J. D. Doll, J. Chem. Phys.112, 3990 (2000). [24] M. Spivak, Calculus (Publish or Perish, 3ed., 1994). [25] R. S. Berry, T. L. Beck, H. L. Davis, and J. Jellinek, Adv. Chem. Phys. 70B, 75 (1988). [26] D. D. Frantz, J. Chem. Phys. 102, 3747 (1995). [27] Assuming the tolerable error to be on the order of 1%, we set 0.01≃ \|[cot(ωn/2)−2/ωn]/(2/ωn)\|=ω2 n/12 +O(ω4 n). Then nmax=bK√ 12/20π. [28] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (Dover, New York, 1970). J. P. Neirotti, D. L. Freeman. and J. D. Doll 19
arXiv:physics/0008246v1 [physics.atom-ph] 31 Aug 2000Wave packet evolution approach to ionization of hydrogen mo lecular ion by fast electrons Vladislav V. Serov, Vladimir L. Derbov Chair of Theoretical and Nuclear Physics, Saratov State Uni versity, 83 Astrakhanskaya, Saratov 410026, Russia Boghos B. Joulakian Institut de Physique, Laboratoire de Physique Mol´ eculair e et des Collisions, Universit´ e de Metz, Technopˆ ole 2000, 1 Rue Arargo, 57078 Metz Cedex 3, France Sergue I. Vinitsky Laboratory of Theoretical Physics, Joint Institute for Nuc lear Research, Dubna 141980, Moscow Region, Russia (January 16, 2014) The multiply diﬀerential cross section of the ionization of hydrogen molecular ion by fast electron impact is calculated by a direct approach, which involves th e reduction of the initial 6D Schr¨ odinger equation to a 3D evolution problem followed by the modeling o f the wave packet dynamics. This approach avoids the use of stationary Coulomb two-centre fu nctions of the continuous spectrum of the ejected electron which demands cumbersome calculation s. The results obtained, after veriﬁcation of the procedure in the case atomic hydrogen, reveal interes ting mechanisms in the case of small scattering angles. PACS number(s): 34.80.Dp I. INTRODUCTION New experimental methods, particularly, based on the multiple coincidence detection technique [1–3] stimulate the interest to fundamental theoretical studies of the dis- sociative ionization of diatomic molecules by electron im- pact. In this context the molecular hydrogen ion can be considered as the basic system in which the removal of the unique electron causes dissociation. Substantial theoret - ical analysis of the dissociative ionization of H+ 2by fast electrons was recently carried out in [4]. As mentioned in [4], the crucial point of calculating the cross-section of such processes is that no closed exact analytical wave functions of the continuum states exist. In [4] the ﬁnal- state wave function of the ejected electron was found by taking a product of two approximate functions that take into account the two scattering centers. To improve the calculation it seems straightforward to obtain these func- tions with the exact numerical solutions of the two-center continuum problem. However, this approach involves a cumbersome procedure of calculating multi-dimensional integrals of the functions presented numerically that re- quires huge computer facilities and may cause additional computational problems. It seems reasonable to search for direct computational approaches, in which the basis of exact two-center continuum wave functions is not in- volved. Note that the potential advantage of such meth- ods is that they could be generalized over a wider class of two-center systems starting from the molecular hydrogen ion as a test object. In the present paper we develop a direct approach to the ionization of hydrogen molecular ion by fast electrons that involves the reduction of the ini-tial 6D Schr¨ odinger equation to a 3D evolution problem followed by modeling of the wave packet dynamics. Originally we intended to treat the incoming elec- tron classically, its trajectory being approximated by a straight line with the deﬂection neglected. The bound electron was to be treated quantum mechanically. Pre- liminary calculations at the impact parameter ρ= 10 a.u. has shown, ﬁrst, that the probability of the emission of the electron having the energy of 50 eV is extremely small, and, second, that the direction of the electron emission is orthogonal to that of the incoming electron motion, that contradicts the results of [4]. This means that the main contribution to the small-angle scattering comes from the central collisions with the bound electron in the region of its localization. Generally, the classical ly estimated deﬂection angle of 1ofor scattered electron cor- responds to the impact parameter of the order of 1 a.u., so that the trajectory passes through the molecule and the classical treatment of the incoming electron is not valid. Here we develop and apply a direct approach to the calculation of the angular distribution of scattered and ejected electrons that involves the reduction of the ini- tial 6D Schr¨ odinger equation to a 3D evolution problem followed by modeling of the wave packet dynamics. The approach does not make use of the basis of stationary Coulomb two-center functions of the continuous spec- trum for the ejected electron, whose proper choice is a crucial point of other model calculations. Our approach can be considered as the linearized version of the phase function method [5,6] for the multi-dimensional scatter- ing problem. The evolution problem is solved using the 1method based on the split-step technique [7] with com- plex scaling, recently proposed by some of us and tested in paraxial optics [8]. In the present paper the method as a whole is also tested using the well known problem of electron scattering by hydrogen atom [9]. II. BASIC EQUATIONS We start from the 6D stationary Schr¨ odinger equation which describes two electrons in the ﬁeld of two ﬁxed protons /bracketleftbigg H0(r)−1 2∇2 R+V(r,R)/bracketrightbigg Ψ(r,R) =EΨ(r,R),(1) FIG. 1. Coordinate frame where ris the radius-vector of the electron initially bound in H+ 2and ﬁnally ejected, Ris the radius-vector of the impact electron, ˆH0=−1 2∇2 r+U(r) is Hamiltonian of ejected electron in the ﬁeld of two protons, V(r,R) = U(R) +Uint(r,R) is the interaction between the impact electron and molecular ion, U(r) =−1/r1−1/r2is the attractive potential between the ejected (scattered) elec - tron and the protons, r1=\|r−r1p\|,r2=\|r−r2p\|, ripis the radius-vector of the i-th proton, Uint(r,R) = 1/\|r−R\|is the repulsive potential of interaction between the electrons. The origin of the coordinate frame is cho- sen in the center of symmetry of the molecular ion with theZaxis directed along the momentum of the incident electron. For the scattering problem solved here the energy of the system may be presented as E=k2 i/2 +E0, where −E0is the ionization potential, kiis the momentum of the incident electron. Let us seek the solution of Eq.(1) in the form Ψ( r,X,Y,Z ) =ψ(r,R⊥,Z)exp(ikiZ). Un- der the condition that ( k2 e+k2 ⊥−2E0)/k2 i<<1 one can neglect the second derivative of ψwith respect to Z. As a result we get the evolution-like equation for the envelope functionψ(r,R⊥,Z) iki∂ψ(r,R⊥,Z) ∂Z= /braceleftbigg ˆH0(r)−1 2∇2 R⊥−E0+V(r,R)/bracerightbigg ψ(r,R⊥,Z).(2)Neglecting the large-angle scattering one can write the initial condition for ψas ψ(r,R⊥,−∞) =ψ0(r). (3) To solve the 5D Schr¨ odinger evolution equation(2) we use Fourier transformation with respect to the variable R⊥ ψ(r,R⊥,Z) =1 2π/integraldisplay ψk⊥(r,Z)exp(ik⊥R⊥)dR⊥.(4) Then Eq.(2) takes the form iki∂ψk⊥(r,Z) ∂Z=/braceleftbigg ˆH0(r) +/parenleftbiggk2 ⊥ 2−E0/parenrightbigg/bracerightbigg ψk⊥(r,Z) +1 (2π)2/integraldisplay Vk⊥k′ ⊥(r,Z)ψk′ ⊥(r,Z)dk′ ⊥,(5) where Vk⊥k′ ⊥(r,Z) =/integraldisplay exp(−i(k⊥−k′ ⊥)R⊥)V(r,R⊥,Z)dR⊥ (6) is the Fourier transform of the interaction potential V(r,R⊥,Z). Further simpliﬁcation of the problem is possible if the amplitude of the incident wave is much greater than that of the scattered wave. In this case one can put ψk⊥(r,Z) =δ(k⊥)ψ0(r) (7) in the integral term of Eq.(5). As a result we get the inhomogeneous equation iki∂ψk⊥(r,Z) ∂Z=/braceleftbigg ˆH0(r) +/parenleftbiggk2 ⊥ 2−E0/parenrightbigg/bracerightbigg ψk⊥(r,Z) +1 (2π)2Vk⊥(r,Z)ψ0(r), (8) whereVk⊥(r,Z) =Vk⊥0(r,Z), with the initial condition ψk⊥(r,−∞) = 0. To calculate the integral with respect to transverse variables in the expression for Vk⊥(r,Z) it is easier to start from the known integral /integraldisplay exp(−ikR)1 RdR=4π k2=4π k2 Z+k2 ⊥. (9) Carrying out the inverse Fourier transformation /integraldisplay∞ −∞exp(ikZZ)dkZ k2 Z+k2 ⊥=π k⊥e−k⊥\|Z\|, (10) one gets Vk⊥(r,Z) =2π k⊥e−k⊥\|Z−z\|−ik⊥r⊥ −2π k⊥/bracketleftBig e−k⊥\|Z−dZ\|−ik⊥d⊥+e−k⊥\|Z+dZ\|+ik⊥d⊥/bracketrightBig .(11) 2Herek⊥=kisinθsis the transverse momentum compo- nent of the scattered electron, θsis the scattering angle, ±dare the positions of the nuclei with respect to the center of symmetry. Note that terms in square brackets determine the elastic scattering of the incident electron by the nuclei. Due to the exponential decrease of the source term with\|Z\|the integration may be actually carried out within a certain ﬁnite interval ( −Zmax,Zmax). Hence the zero initial condition should be imposed at the point −Zmax. Note that the approximation (7) is actually equivalent to the ﬁrst Born approximation [9]. Multiply Eq.(8) by the complex conjugate function of the continuous spec- trum of ˆH0and integrate over all r. Then ikidCk⊥(ke,Z) dZ=/braceleftbiggk2 e 2+k2 ⊥ 2−E0/bracerightbigg Ck⊥(ke,Z) +1 (2π)2/integraldisplay ψ∗(ke,r)Vk⊥(r,Z)ψ0(r)dr, (12) whereCk⊥(ke,Z) =/integraltext ψ∗(ke,r)ψk⊥(r,Z)dris the proba- bility density amplitude for the transition of the initiall y bound electron into the state with the momentum ke. Let us substitute Ck⊥(ke,Z) =˜Ck⊥(ke,Z)exp(ikZZ), wherekZis the increment of the longitudinal component of the momentum of the impact electron determined by the relation kZ=−1 ki/parenleftbiggk2 e 2+k2 ⊥ 2−E0/parenrightbigg . (13) This relation is actually equivalent to the energy conser- vation law written neglecting the terms of the order of k2 Z. The substitution yields ikid˜Ck⊥(ke,Z) dZ= 1 (2π)2exp(−ikZZ)/integraldisplay ψ∗(ke,r)Vk⊥(r,Z)ψ0(r)dr,(14) and ˜Ck⊥(ke,∞) = 1 iki(2π)2/angb∇acketlefteiksRψ(ke,r)\|V(r,R)\|eikiRψ0(r)/angb∇acket∇ight, (15) where ks=ki−Kis the momentum of the scattered electron, K= (−kX,−kY,−kZ) is the momentum trans- fer. Provided that the ejected electron has the momentum ke, the asymptotic form of the solution of Eq. (1) for the wave function of the scattered electron when R→ ∞ is Ψas ke(R) = exp(ikiZ) +exp(iksR) Rfke(θs,φs).(16)The scattering diﬀerential cross-section(DCS) can be then expressed as σke(θs,φs) =keks ki\|fke(θs,φs)\|2, (17) On the other hand, the asymptotic form of the wave function resulting from the solution of Eq.(8) under the conditionZ→ ∞ can be presented as Ψas ke(R) = exp(ikiZ)+ exp(ikiZ)/integraldisplay ˜Ck⊥(ke,∞)exp(ik⊥R⊥+ikZZ)dk⊥.(18) Making use of the fact that the integrand has a stationary point we ﬁnally get Ψas ke(R) =eikiZ+ 1 Zei/parenleftBig ki−k2e/2−E0 ki/parenrightBig Z+iki 2ZR2 ⊥(−2πiki)˜Ck0 ⊥(ke,∞),(19) where k0 ⊥=kisinθs(cosφs,sinφs),R⊥=Rsinθs,Z= Rcosθs. The expression (19) agrees with (16) within the accuracy of the order of θ2 sif we set fke(θs,φs) =−2πiki˜Ck0 ⊥(ke,∞) =−1 2π/angb∇acketlefteiksRψ(ke,r)\|V(r,R)\|eikiRψ0(r)/angb∇acket∇ight. (20) The latter expression is similar to the formula for fke(θs,φs) derived in [9] using the ﬁrst Born approxi- mation. III. CALCULATION OF THE ANGULAR DISTRIBUTION The asymptotic expression of the radial part of the wave function corresponding to the continuous spectrum ofˆH0can be written as ψas E(r,t) =1/radicalbig υ(r)rexp(−iEt+i/integraldisplayr υ(r′)dr′) (21) wheret=Z/kiis the evolution variable, υ(r) =/radicalbig 2(E−Uas(r)),E=k2 e 2+k2 ⊥ 2−E0,Uas(r) =−Z′/r, Z′= 2 is charge of two protons. In the asymptotic limit one can take only the radial component of the momen- tum of the ejected electron into account. Then, accord- ing to [10], the expression for calculating the amplitude A(k,θ,φ ) takes the form Ak⊥(ke,θe,φe) = 1√ 2π/integraldisplayt1 t0dt′j(ψk⊥(r,θ,φ,t′),ψas E(r,t′))/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=rmax,(22) where 3j(Ψ,Φ) =i 2/braceleftbigg Ψr2∂Φ∗ ∂r−Φ∗r2∂Ψ ∂r/bracerightbigg , (23) is the ﬂux introduced in [10], t0=−Zmax/kiandt1>> Zmax/ki. The approximate relation (22) becomes exact whent1→+∞and simultaneously rmax→+∞. The amplitudes deﬁned by (22) are related with the coeﬃcients introduced in Eq.(15) by \|Ak⊥(ke,θe,φe)\|2=ke/vextendsingle/vextendsingle/vextendsingle˜Ck⊥(ke,∞)/vextendsingle/vextendsingle/vextendsingle2 (24) Using (17),(20) and (24) we get the ﬁnal expression for the diﬀerential cross-section σke(θs,φs) = (2π)2kski\|Ak⊥(ke,θe,φe)\|2. (25) In the region where r>r maxwe made use of the com- plex scaling technique [11] to suppress the non-physical reﬂection from the grid boundary. IV. NUMERICAL SCHEME The inhomogeneous Schr¨ odinger equation to be solved can be written as i∂Ψ(r,t) ∂t=ˆH0(r)Ψ(r,t) +F(r,t), (26) The solution of Eq. (26) to within the second-order terms in ∆tcan be expressed as the following sequence of equa- tions: Ψ0 l = Ψ(t)−i∆t 2F(r,t); (27) (1 +i∆t 2ˆH0)Ψ1 l= (1−i∆t 2ˆH0)Ψ0 l; (28) Ψ(t+ ∆t) = Ψ1 l−i∆t 2F(r,t+ ∆t). (29) The key step of the procedure is Eq. (28) which de- ﬁnes nothing but Cranck-Nicholson scheme. To solve this equation we make use of the partial coordinate splitting (PCS). A ﬁnite-diﬀerence scheme is applied for the radial variablerand the polar angle θ. Fast Fourier transform (FFT) is used for the azimuthal angle φ. In the spherical coordinate system, the z-axis of which is directed along the symmetry axis of the molecule (and not along the velocity of the impact electron) and after the substitution Ψ = ψ/rand the Fourier transformation of Eq.(28) the terms ˆH0Ψ0,1 lentering this equation turn into ˆH0(r,η,m )ψ(r,η,m ) = −1 2/bracketleftbigg∂2 ∂r2+1 r2/parenleftbigg∂ ∂η(1−η2)∂ ∂η−m2 1−η2/parenrightbigg/bracketrightbigg ψ(r,η,m ) +U(r,η)ψ(r,η,m ), (30) whereη= cosθ,mis the asimuthal quantum number.Finite-diﬀerence approximation∂2 ∂r2≃Dr i2i1and ∂ ∂η(1−η2)∂ ∂η−m2 1−η2≃Dη,m j2j1of the diﬀerential opera- tors entering Eq.(28) yields Msets of linear equations, each set being of the order L×N, whereM, L andN are the numbers of grid points in φ,ηandr, respectively. Direct solution of each set of equations requires NL2op- erations at each step in t. The FFT that should be per- formed twice, ﬁrst, when proceeding from (27) to (28), and second, from (28) to (29), requires NLM log2Mex- tra operations. To reduce the number of operation we propose a double-cycle split-step scheme. In case when ˆH0can be presented as a sum ˆH0=ˆH1+ˆH2, this scheme can be formulated as follows ψ1=ψ(t); (I+i∆t 4ˆH1)ψ2= (I−i∆t 4ˆH1)ψ1; (I+i∆t 4ˆH2)ψ3= (I−i∆t 4ˆH2)ψ2; (I+i∆t 4ˆH2)ψ4= (I−i∆t 4ˆH2)ψ3; (I+i∆t 4ˆH1)ψ5= (I−i∆t 4ˆH1)ψ4; ψ(t+ ∆t) =ψ5, which to within the second-order terms in ∆ tcorresponds to the initial Cranck-Nicholson scheme, IandH1,2is square matrixes, ( I)i2i1j2j1=δi2i1δj2j1. Now the problem is how to split the Hamiltonian ˆH0 into two parts. Formal separation of radial and angular parts leads to diﬃculties associated with the singularity of the angular part. Due to this singularity the scheme appears to be conditionally stable with severe limitations imposed on the step ∆ t. Practically this version of the splitting scheme is applicable only if the grid in ris rough enough. To remove this limitation we propose a partial coor- dinate splitting scheme. Its principal idea is that in the vicinity ofr= 0 it is preferable not to split oﬀ the angu- lar part at all. To implement this idea we introduce the r-dependent weight function p(r) which is supposed to diminish in the vicinity of r= 0 and deﬁne the discrete approximation of the operators ˆH1,2in the following way (ˆHm 1)i2i1j2j1=−1 2ˆDr i2i1δj2j1+Uas(ri) +p(ri1)/bracketleftBigg −1 2ˆDη,m j2j1 r2 i1δi2i1+U2(ri1,ηj1)/bracketrightBigg ; (31) (ˆHm 2)i2i1j2j1= (1−p(ri1))/bracketleftBigg −1 2ˆDη,m j2j1 r2 i1δi2i1+U2(ri1,ηj1)/bracketrightBigg , (32) hereUas(r)+U2(r,η) =U(r,η). It is reasonable to choose p(r) as a cubic polynomial 4p(r) =  2/bracketleftBig r−ra ap/bracketrightBig3 −3/bracketleftBig r−ra ap/bracketrightBig2 + 1, ra<r<r a+ap; 1, r ≤ra; 0, r ≥ra+ap; whererais the radius of the vicinity of r= 0 where the splitting is absent, apis the width of the area of partial splitting. Such a polynomial satisﬁes the condition of smooth connection at the boundaries that separate the region of partial splitting from the regions of full split- ting, on one hand, and of no splitting at all, on another hand. V. NUMERICAL CALCULATIONS AND RESULTS The method was tested using the well-studied exam- ple of the impact ionization of atomic hydrogen. We compared our results with those given by the well-knownexpression obtained in the ﬁrst Born approximation [12]. Good agreement was demonstrated in the energy interval of interest Eefrom 1 to 3 a.u., Eebeing the energy of the ejected electron. FIG. 2. The multi-fold diﬀerential cross section (MDCS) of the ionization of H+ 2versus the ejection angle θeand ejection energy Eeforθd= 135o 045901351802252703153600.0000.0050.0100.0150.0200.0250.030MDCS(a.u.) θe (deg.)θd=139.2o 045901351802252703153600.000.010.020.030.040.050.060.070.08MDCS(a.u.) θe (deg.)θd=49.2o (a) (b) 045901351802252703153600.000.010.020.030.04MDCS(a.u.) θe (deg.)θd=0o 045901351802252703153600.000.010.020.030.040.05MDCS(a.u.) θe (deg.)θd=90o (c) (d) FIG. 3. The multi-fold diﬀerential cross section (MDCS) of t he ionization of H+ 2versus the ejection angle θefor diﬀerent angles θd: a)θd= 139 .2othat corresponds to d/bardblK; b)θd= 49.2othat corresponds to d⊥K; c)θd= 0o; d)θd= 90o. The energy of the ejected electron Ee= 1.85 a.u.=50.3 eV. 5Our numerical studies concerning the molecular hydro- gen ion focused on the variation of the multi-fold diﬀer- ential cross section (MDCS) concerning a coincidence de- tection of the two emerging electrons and one of the pro- tons with the ejection angle θeat diﬀerent orientations of the molecular axis, provided that the scattering angle is small. The examples of our results illustrated by Figs.2-5 are obtained under the following conditions: the momen- tum of the impact electron ki=12.13 a.u. ( Ei≃2000 eV); the angle of scattering θs= 1o. The impact and ejected electron trajectories and the molecular axis are supposed to lie in one plane. The latter restriction is not imposed by the method as such, it is just an ex- ample. Generally, one gets full information about the ejected electron, i.e., the dependence of MDCS from Ee, θeandφe, after each run of the code at given values of the impact energy, scattering angle and molecular axis orientation. In Fig.2 demonstrates the energy-angle dis- tribution, extracted from the data getting in result of one run of the code. In the planar geometry the orientation of the molecular ion is determined by a single angle θdbe- tween the impact direction and the internuclear axis. We remind that the momentum transfer vector was deﬁned above as K=ki−ks. In Figs. 3 we present the par- ticular cases of the dependence of MDCS upon θewhen internuclear axis is a)parallel to the momentum transfer; b)perpendicular to the momentum transfer; c)parallel to the impact electron direction ki; d)perpendicular to the impact electron direction ki. As it could be expected basing on the elementary symmetry considerations, the ﬁrst two plots are symmetric with respect to the direction of the momentum transfer that corresponds to the angle θe= 319.2o. Since this symmetry is not assumed a pri- oriin the procedure, this may be considered as one more evidence in favour of the validity of the results demon- strated. FIG. 4. The multi-fold diﬀerential cross section (MDCS) of the ionization of H+ 2versus the ejection angle θeand molec- ular angle θd. The energy of the ejected electron Ee= 1.85 a.u.=50.3 eV. The recoil momentum Qrecoil =K−ketransmitted to the target has its minimum for keparallel to K. In this case all the momentum is transferred to the ejectedelectron and the probability of the ionization is maximal. This is conﬁrmed around θe= 319.2oon ﬁgures 3(a) and 3(b) where the inter-nuclear axis is respectively perpen- dicular and parallel to K. So this is a good veriﬁcation for our calculation. Now, for the situation where keis anti- parallel to K, the recoil momentum Qrecoil is maximal and the probability of the ionization is maximal. This is also visible for θe= 139.2o. Now for the directions of the internuclear axis other than θd= 139.2o(where dis parallel to K) orθd= 49.2o(where dis perpendicular to K) the target does not respect the above analysis. This is due to the fact that the diatomic target behaves as an atomic target only for these two angles. The other situ- ations present interference patterns the minima of which move when θdchanges. Fig.4 shows MDCS versus the ejection angle θeand in- ternuclear angle θd. As one can see, this dependence has rather a complex behaviour. 01530456075901051201351501651800.000.010.020.030.040.050.060.070.08MDCS(a.u.) θd (deg.)Ee=1.85 a.u. FIG. 5. The multi-fold diﬀerential cross section of the ion- ization of H+ 2as a function of the angle θdbetween the im- pact direction and the internuclear axis for ﬁxed ejection a ngle θe= 319 .2o. The energy of the ejected electron is Ee= 1.85 a.u. To conﬁrm the above dependence we show in Fig.5 a section of Fig.4 for ﬁxed ejection angle θe= 319.2o which corresponds to the case when the ejected electron direction is parallel to the momentum transfer vector. It presents a variation of the MDCS with respect to the di- rection of the inter-nuclear axis. It can be clearly seen that the maximal value of MDCS is achieved when the in- ternuclear axis is perpendicular to the momentum trans- fer direction that correspond to θd= 49.2o. This result agrees with the hypothesis formulated in [13]. VI. CONCLUSION We have developed a procedure which determines the multiply diﬀerential cross section of the (e,2e) ionizatio n of hydrogen molecular ion by fast electron impact, us- ing a direct approach which reduces the problem to a 63D evolution problem solved numerically. Our method avoids the cumbersome stationary perturbative calcula- tions, and opens the way for near future applications to the (e,2e) ionization of more complex atomic and molec- ular targets. ACKNOWLEDGMENTS Authors would like to thank Dr. A.V. Selin for use- ful discussions. V.V.S and S.I.V. thanks to RFBR for supporting by grants No-00-01-00617, No-00-02-16337. [1] Y.D. Wang, J.H. McGuire, and R.D. Rivarola, Phys. Rev. A40, 3673 (1989). [2] S.E. Corchs, R.D. Rivarola, J.H. McGuire, and Y.D. Wang, Phys. Rev. A 47, 201 (1993). [3] S.E. Corchs, R.D. Rivarola, J.H. McGuire, and Y.D. Wang, Phys. Scr. 50, 469 (1994).[4] B. Joulakian, J. Hassen, R. Rivarola, and A. Motassim, Phys. Rev. A 54, 1473 (1996). [5] V.V. Babikov, Phase function method in quantum me- chanics (Nauka, Moscow, 1968) (in Russian). [6] F. Calogero, Variable phase approach to potential scat- tering, (Academic, New York, 1967). [7] G.I. Marchuk, in Partial Diﬀerential Equation. II. SYNSPADE-1970 (Academic, New York, 1971). [8] V.V. Serov, A.I. Bychenkov, V.L. Derbov, and S.I. Vinit- sky. Numerical scheme with external complex scaling for 2D Schr¨ odinger equation in paraxial optics. Proc. SPIE 4002, 10 (1999). [9] N.F. Mott and H.S.W. Massey, The theory of atomic function, (Clarendon, Oxford, 1965). [10] A.M. Ermolaev, I.V. Puzynin, A.V. Selin, and S.I. Vinit - sky, Phys. Rev. A 60, 4831 (1999). [11] C.W. McCurdy and C.K. Stroud, Computer Phys. Com- mun.63, 323 (1991). [12] L.D. Landau and E.M. Lifshitz. Quantum Mechanics: Non-Relativistic Theory. (Pergamon, London, 1958), p. 458. [13] A. Bugacov, B. Piraux, M. Pont, and R. Shakeshaft, Phys. Rev. A 51, 4877 (1995). 7
arXiv:physics/0008247v1 [physics.chem-ph] 31 Aug 2000Optimal Control of Molecular Motion Expressed Through Quantum Fluid Dynamics Bijoy K Dey∗, Herschel Rabitz Department of Chemistry, Princeton University, Princeton , New Jersey and Attila Askar Department of Mathematics, Koc University, Istanbul, Turk ey ∗Present address : Department of Chemistry, Chemical Physics Theory Group, University of Toronto, Toronto, Canada 1Abstract A quantum ﬂuid dynamic(QFD) control formulation is present ed for optimally manipulating atomic and molecular systems. In QFD the contr ol quantum system is ex- pressed in terms of the probability density ρand the quantum current j. This choice of variables is motivated by the generally expected slowly var ying spatial-temporal depen- dence of the ﬂuid dynamical variables. The QFD approach is il lustrated for manipulation of the ground electronic state dynamics of HCl induced by an e xternal electric ﬁeld. I. INTRODUCTION Manipulating the outcome of quantum dynamics phenomena by a properly tailored external control is a topic of increasing activity [1-15]. P roblems where the external con- trol is electromagnetic have received the most attention, a lthough other applications arise as well. Various implementations of quantum control have be en experimentally realized [16-25]. A variety of control strategies have been suggeste d, and optimal control the- ory(OCT) provides the most general framework for acheiving ﬁeld designs. Such designs will generally require further reﬁnement in the laboratory through learning techniques to overcome design uncertainities [17-20]. A basic diﬃculty i n attaining the control designs is the computational eﬀort called for in solving the time-de pendent Schroedinger equa- tion, often repeatedly in an iterative fashion. This paper i ntroduces the quantum ﬂuid dynamic(QFD) control formulation to simplify this task. Ca lculations have shown [28, 29] that QFD is capable of being much more eﬃcient than conven tional methods(e.g., FFT propagation), and this savings should carry over to the c ontrol design task. This paper will show how OCT can be combined with QFD. The theoretical basis for dynamic control [2-13] is to creat e non-stationary states of one’s choice, by optimally designing the control ﬁeld. Typi cally, the problem is posed as seeking an optimal ﬁeld to drive a quantum wave packet to a des ired target at a chosen 2time t=T. In the traditional approach [3-10] to quantum opti mal control an objective design functional ¯Jis deﬁned, which depends on the system wave function, a wave f unction like Lagrange multiplier, and the external ﬁeld. Minimizat ion of the objective functional leads to the identiﬁcation of external ﬁeld(s) capable of de livering a speciﬁc outcome. This process requires solving for the complex oscillatory w ave function and the similarly behaved Lagrange multiplier, and due care is needed for thei r proper representation in a suitable basis for capturing their behaviour. Often the rap idly varying spatio-temporal behaviour of these functions necesitates the use of many unk nowns in the basis. This paper explores an alternative formulation for OCT to de sign the electric ﬁeld. The formulation is based on the ﬂuid dynamic view point of qua ntum mechanics [26- 29], which by-passes the typically oscillatory nature of th e wave function to exploit the generally smooth behaviour of the real density and the quant um current variables. Recent illustrations have demonstrated the smooth spatial and tem poral nature of the variables and the ability to discretize them on a relatively small numb er of grid points [28, 29]. As background in section 2 we give a brief summary of the QFD fo rmulation of quantum mechanics. Section 3 presents OCT within the framework of QF D for designing an electric ﬁeld to meet a speciﬁc objective. Section 4 applies the OCT-Q FD formulation for the manipulation of HCl. Section 5 concludes the paper. II. QUANTUM FLUID DYNAMICS The treatment below considers a single particle of reduced m ass m, but the QFD formulation has an immediate extension to many parti cles. The time-dependent Schr¨ odinger equation is given by [−¯h2 2m∇2+V+Vext]Ψ(x, t) =i¯h∂ ∂tΨ(x, t) (1) where V typically conﬁnes the particle in a locale and Vextis the control taken here as −µ(x)·E(t) with E(t) being the electric ﬁeld and µ(x) the dipole moment. Substituting Ψ(x, t) =A(x, t)eiS(x,t)/¯h, where A and S are real functions, into Eq.(1) and separating 3the real and imaginary parts one easily obtains two equation s. The imaginary part yields the continuity equation ∂ρ ∂t+∇ ·(ρv) = 0 (2) and the real part the following equation for the phase S ∂S ∂t+∇S· ∇S 2m+Veff= 0 (3) where Veff=V+Vext+VqwithVq=−¯h2 2m∇2ρ1/2 ρ1/2=−¯h2 2m[∇2lnρ1/2+ (∇lnρ1/2)2],ρ=\|Ψ\|2 andv=∇S m. Equation (3) has the form of the classical Hamilton-Jacobi e quation with an extra ’quantum potential’ term Vq. This equation can be transformed into one for the evolution of the velocity vector vby taking the gradient to give ∂ ∂tv=−(v· ∇)v−1 m∇(Veff) (4) Deﬁning the quantum current as j(x, t) =−¯h mIm[Ψ∗(x, t)∇Ψ(x, t)] =ρ(x, t)v(x, t), one readily obtains the equation of motion for jby substitution of∂ρ ∂tand∂ ∂tvfrom Eqs.(2) and (4) as ∂ ∂tj=−v(∇ ·j)−(j· ∇)v−ρ m∇Veff (5) Eqs.(2) and (3) or (2) and (5) describe the motion of a quantum particle within the QFD formulation of quantum mechanics. The motion of a quantum pa rticle is governed by the current vector jand the density ρin Eqs.(2) and (5). Although the QFD equations resemble those of classical ﬂuid dynamics, their quantum id entity prevails due to the presence of the potential Vqwhich has no classical analogue. Equivalently, the QFD equations may be viewed as those of a “classical” ﬂuid with a h ighly non-linear constitutive law prescribed by Veff. Various Eulerian or Lagrangian means can be exploited to so lve the QFD equations [28,29], and available ﬂuid dynamics code s may be adopted to treat 4these equations [30]. The essential simplifying feature of the QFD equations is that ρ andjorρand S are often slowly varying, which is evident from quantum dynamics calculations [28,29], thereby permitting relatively coar se gridding. Despite the non-linear nature of the QFD equations, the general smoothness of ρandjobserved lead to signiﬁcant computational savings [28,29]. III. CONTROL EXPRESSED WITHIN QUANTUM FLUID DYNAMICS Quantum OCT seeks the design of an external ﬁeld to fulﬁll a pa rticular dynamical objective. This section will provide the working equations for OCT-QFD to design an optimal electric ﬁeld that drives a quantum wave packet to a d esired objective at the target time t=T. The OCT-QFD formulation could be expressed in the usual way in terms of the Schr¨ odinger equation where QFD would only act a s a solution procedure. Here we will present a general approach by writing OCT direct ly in terms of QFD. As an example the control of a non-rotating diatomic molecule w ill be used as a simple illustration of the concepts. The treatment of a fully gener al target expectation value ΘT=<Ψ(T)\|Θ\|Ψ(T)>may be considered with QFD, but here we will only treat the common case where the operator Θ( x) is only position x dependent. Then the goal is to steer Θ T ΘT=/integraldisplayxr xlΘ(x)ρ(x, T)dx (6) as close as possible to the desired value Θd. The active spatial control interval is taken as xl≤x≤xrover the time 0 ≤t≤Tthat the control process occurs. We desire to minimize the cost fu nctional Jcost=Jtarget+Jfield where JtargetandJfieldare given by Jtarget=1 2ωx(ΘT−Θd)2and J field=1 2ωe/integraldisplayT 0E2(t)dt (7) withωeandωxbeing the positive weights balancing the signiﬁcance of the two terms. The second term represents the penalty due to the ﬂuence of th e external ﬁeld. The 5minimization of Jcostwith respect to E(t) must be subject to the satisfaction of th e equations of motion for ρandjin Eqs.(2) and (5). We may fulﬁll this constraint by introducing the unconstrained cost functional as ¯J=Jcost−/integraldisplayT 0/integraldisplayxr xlλ1(x, t)[∂ρ(x, t) ∂t+∂j(x, t) ∂x]dxdt (8) −/integraldisplayT 0/integraldisplayxr xlλ2(x, t)[∂j(x, t) ∂t+∂ ∂x(j2 ρ) +ρ m∂ ∂x(V+Vq+Vext)]dxdt where λ1(x, t) and λ2(x, t) are Lagrange’s multiplier functions. An optimal solution satisﬁes δ¯J= 0, which is assured by setting each of the functional derivatives with respect to λ1,λ2,ρ,jand E to zero. The ﬁrst two, i.e., the functional derivatives with respect to λ1andλ2regenerate the QFD equations in Eq.(2) and (5). The three others are obtained in the forms : ∂λ2 ∂t+∂ ∂x(λ2vλ) +S1[ρ, j, λ 2] = 0 (9) ∂λ1 ∂t+∂ ∂x(λ1vλ)−λ2∂ ∂x(V+Vq(λ2) +Vext) +S2[ρ, j, λ 2] = 0 (10) and δ¯J δE(t)=/integraldisplayxr xlλ2(x, t)ρ(x, t)∂ ∂xµ(x)dx+ωeE(t) = 0 (11) where S1= 2j ρ∂λ2 ∂x(12) S2=−λ2 m∂ ∂x(Vq(ρ)−Vq(λ2))−j2 ρ2∂λ2 ∂x(13) −¯h2 4m2ρ1/2∂2 ∂x2[1 ρ1/2∂ ∂x(λ2ρ)] +¯h2 4m2ρ3/2∂2 ∂x2ρ1/2∂ ∂x(λ2ρ) and 6Vq=−¯h2 2m∇2λ1/2 2 λ1/2 2=−¯h2 2m[∇2lnλ1/2 2+ (∇lnλ1/2 2)2] (14) The corresponding ﬁnal conditions are ωx[ΘT−Θd]Θ(x)−λ1(x, T) = 0 (15) and λ2(x, T) = 0 (16) Several other constraint expressions can be obtained by usi ng equivalent forms of the continuity and dynamical equations. The form presented abo ve is used in the subsequent numerical calculations. An alternative form in multi-dime nsions symmetric between the QFD and Lagrange multiplier functions is presented in the Ap pendix. The equations (9) and (10) for λ2andλ1respectively ressemble that of ρandjwith the only diﬀerence being the extra source terms S1andS2. The source terms depend on ρ andj.vλin the above equations is the ’velocity’ associated with the Lagrange’s multiplier and is given as vλ=λ1 λ2. There are now two diﬀerent quantum potential terms, one of which is a function of ρ(x, t) and the other is a function of λ2(x, t). In this formalism the evolution of λ1(x, t) takes place by Vq(λ2) as well as the diﬀerence of the two types of quantum potential. In obtaining the above equations we ha ve standardly assumed no variation of either ρ(x,0) or j(x,0). Thus, we start from the initial value of ρ(x,0) and j(x,0) to solve Eqs.(2) and (5). Eqs.(9) and (10) can be solve d for λ2(x, t) and λ1(x, t) by integrating backward from time T using λ1(x, T) and λ2(x, T) given in Eqs.(15) and (16) respectively. The equations (2), (5), (9) and (10) are n on-linear thereby calling for iteration to solve(cf., the algorithm in Section 4). Finall y the desired control electric ﬁeld is given from Eq.(11) as E(t) =−1 ωe/integraldisplayxr xlλ2(x, t)ρ(x, t)∂ ∂xµ(x)dx (17) 74 APPLICATION TO HCL The OCT-QFD formulation will be applied to manipulating the vibrational motion of HCl on the ground electronic state. The initial density ρ(x,0) = \|Ψ(x)\|2was obtained from solving for the vibrational state from the equation −¯h2 m∂2Ψ(x) ∂x2+V(x)Ψ(x) =EΨ(x) (18) using the Fourier grid Hamiltonian method [31,32] where m is the reduced mass of the HCl molecule and V(x) is the truncated polynomial presented by Olgilvie [33] [V(x) =  a1(2x−xe x+xe)2[1 +/summationtext9 i=2ai(2x−xe x+xe)i−1]−b1for x < 4 A[1−tanh(x−4)]3/2for 4≤x≤6.5 0for x ≥6.5] (19) where xe=2.4086 a.u. is the equilibrium bond length of HCl. The param eters in a.u. entering the potential function are a1= 0.961914, a2=−1.362999, a3= 0.86675, a4= −0.49804, a5= 0.1727, a6= 0.2687, a7=−1.977,a8= 2.78,a9= 4.89,b1= 0.169695 andA=−4.85×10−2. Since Ψ( x) is a stationary real function we have zero initial ﬂux j(x,0)=0. The initial ρ(x,0) is nearly a Gaussian packet centered around xe. The dipole function for HCl is given by [34] µ(x) =c1[g(x) +c2g2(x) +c3g3(x)] (20) where g(x) = 1 −tanh(β(x−xd)) and the parameters are c1= 0.279,c2=−0.905, c3= 1.029,β= 0.687 and xd= 2.555. The following steps were carried out for imple- mentation of the present OCT-QFD algorithm : Step 1: Make an initial guess for the electric ﬁeld E(t), which was ze ro in the present calculations. Step 2: Solve the coupled equations, viz., Eq.(2) and (5) for ρ(x, t)and j(x,t) respectively starting from ρ(x,0)and j(x,0). The solution was 8achieved here by using the Flux-corrected transport(FCT) a lgorithm [35] modiﬁed for the purpose of solving the QFD equations [28 ]. In doing so, we adopt the Eulerian numerical scheme. Step 3: Evaluate the ﬁnal value for λ1(x, T)given by Eq.(15) and set λ2(x, T)=0 by Eq.(16). Step 4: Solve Eqs.(9) and (10) for λ2(x, t)andλ1(x, t), respectively, by back- ward propagation using the same method as in step 2. Equation s (9) and (10) have source terms which depend on ρ(x, t)and j(x,t) calculated from step 2. Step 5: Calculate the diﬀerence between the left and right sides of E q.(16) for use in the conjugate gradient method [36] and calculate Jcostfrom Eq.(7). Step 6: Iterate steps 2 to step 6 until acceptable convergence is met . The spatial range of the calculation was 0 ≤x≤12 a.u., and the time interval was 0≤t≤Twith T=2000 a.u. The total number of spatial mesh points is 64 which gives ∆x= 0.1875 a.u. Similarly, the total number of time steps was 2048, which corresponds to ∆t= 0.9765 a.u. No special eﬀort was made to optimize the grid point s, as the purpose here is to demonstrate the QFD-OCT formulation. The weight ωein Eq.(7) was taken as 1 2, and ωx= 1000. The target operator was Θ = xand Θd= 3.0a.u.. Figure 1 shows the control ﬁeld in atomic units. The slightly non-zero values of the ﬁeld at the beginning and end could be arrested by placing add itional costs if desired. This pulse excites several vibrational states(not shown he re) mainly by a sequence of single quantum transitions. Figure 2 shows the average dist ance< x > as a function of time. The desired control value of < x > =3.0 a.u. at T is obtained through oscillatory motion of the packet. The packet is distorted in shape(not sh own) while approximately retaining its original variance during the evolution. Duri ng the optimization process the total integrated probability density remained at unity up t o a deviation of 10−5. The iteration algorithm takes 10 steps to achieve the results sh own here at 2 CPU mins. on 9an IRIX Silicon Graphics Machine(Release 6.1). Within nume rical precision the results were the same as obtained by solving the original Schr¨ oding er equation. 5. CONCLUSION This paper presents a new QFD based approach for carrying out the optimal design of control ﬁelds with an illustration for the maniputation of t he HCl molecule. Our previous work [28] shows the typical smooth and monotonic behaviour o f the ﬂuid dynamical variables, viz., S and v as opposed to the typical oscillatio ns in the wave functions where the hamiltonian was time independent. In the present case wh ere the system is driven with an optimal time-dependent external ﬁeld we have calcul ated the spatial dependence of j,ρ, S and Ψ at t=T shown in Fig.3. The ﬂuid dynamical variables(F ig.3 curves (a), (b) and (c)) used in the present method are relatively slowly varying spatial functions compared to the wave function(Fig.3, curve(d)) which appar ently enhances the eﬃciency and the numerical saving of the present approach to controll ing dynamics. Although the illustration was for one dimension the QFD tech nique is directly extend- able to higher dimensions, and a QFD wave packet calculation in four dimension has already been performed [28]. The alternating direction met hod can eﬀectively be used with QFD for high dimensions. Comparison with FFT propagati on has been performed for two dimensional systems [29], showing that QFD is capabl e of providing a considerable increase in eﬃciency(i.e., by a factor of 10 or more). Regard less of the dimension, the key advantage of OCT-QFD arises from the expected smooth nature of QFD variables. A spe- cial circumstance will arise if the control “exactly” leads to a bound state with nodes that fully separates one spatial region from another. In practic e placing a lower limit on the density of the order of the machine precision overcomes such diﬃculties. Future studies need to explore the full capabilities of the computational s avings aﬀorded by OCT-QFD. 10ACKNOWLEDGEMENT BD thanks Drs.Jair Botina and Tak-San Ho for useful discussi ons. The authors ac- knowledge support from the NSF and DOD. APPENDIX Two additional forms for the cost functional and associated initial/ﬁnal conditions The forms here are presented for reference as an alternative QFD approach. They have the advantages of simplicity and of giving equations for the Lagrange multiplier in the same form as the dynamical equations. The formalism for deri ving the quations is through the Euler equations corresponding to the minimization of I=/integraldisplay V/integraldisplayT t=0F(f, ft,∇f,∇2f)dtdV (A.1) Here V denotes the volume in coordinate space. The correspon ding Euler equations and conditions on time and space are ∂F ∂f−∂(∂F ∂ft) ∂t− ∇ · (∂F ∂∇f) +∇2(∂F ∂∇2f) = 0 (A.2) Initial condition: f(x,0) =f0(x); Final condition: (∂F ∂ft)\|t=T=fT(x) (A.3) Boundary conditions on dV: f(x, t) =fB(x, t) orn·[∂F ∂∇f− ∇ · (∂F ∂∇f)] = 0; (A.4) n·∇(f) =gB(x, t) or∂F ∂∇2f= 0 (A.5) Starting with the continuity and energy conservation equat ions given in Eqs.(2) and (3) in the text, we rewrite them as At+∇A·∇S m+A∇2S 2m= 0 (A.6) ASt+A∇S· ∇S 2m+V A−¯h2 2m∇2A= 0 (A.7) The use of the dynamical equations above in the cost function al in Eq.(8) becomes 11J=1 2ωx(ΘT−Θd)2+ωe/integraldisplayT t=0E2(t)dt −/integraldisplay V/integraldisplayT t=0[[λ1(At+∇A· ∇S m+A∇2S)/2m] ( A.8) +[λ2(ASt+A∇S· ∇S 2m+V A−µE(t)A−¯h2 2m∇2A)]]dtdV where Θ T=/integraltext VΘ(x)A2(x, T)dV. The corresponding Euler equations are obtained from the formulas in ( A2) for arbitrary variations of A, S, λ1,λ2andE(t) as At+v· ∇A=−A∇ ·v/2 St+v· ∇S/2 =−V+µE(t) +¯h2 2m∇2A/A λ1t+v· ∇λ1=−[λ1∇.v 2+¯h2 2mλ2(∇2A/A− ∇2λ2)/λ2)] ( A.9) λ2t+v· ∇λ2= [λ2∇.v 2+1 2mλ1(∇2A/A− ∇2λ1)/λ1)] ωeE(t) +/integraldisplay Vλ2µAdV = 0 Following the formulas given in Eq.(A3) to (A5), the corresp onding initial and ﬁnal con- ditions become A(x,0) =A0(x);S(x,0) =S0(x); λ1(x, T) + 2ωx(ΘT−Θd)A(x, T)Θ(x);λ2(x, T) = 0 (A.10) The ﬁrst two formulas in A.9 are equivalent to the Schr¨ oding er equation. They can be transformed into various QFD forms in terms of ρ,vandjas in Eqs.(2) to (5) in the main text. The third and fourth equations in A.9 are the ba sic equations for the Lagrange multiplier functions. They are in the same ﬂux cons ervation form as the QFD equations. Indeed, the third equation multiplied by λ1can be rearranged in the form of mass conservation for Λ 1=λ2 1as 12Λ1t+∇ ·(Λ1v) =−[¯h2 mλ1λ2(∇2A/A− ∇2λ2/λ2)] The above derivation also can be obtained starting with the u sual Schr¨ odinger equation and its complex conjugate. Following this approach the cost functional below assures that the external ﬁeld is real J=1 2ωx(ΘT−Θd)2+ωe/integraldisplayT t=0E2(t)dt −/integraldisplay V/integraldisplayT t=0[λ∗[iΨt+¯h2 2m∇2Ψ−VΨ−µE(t)Ψ] +λ[−iΨ∗ t+¯h2 2m∇2Ψ∗−VΨ∗−µE(t)Ψ∗]]dtdV With the substitution Ψ = Aexp(iS), the cost functional reduces to the one in Eq.(A.8) withλ=λ1+iλ2. REFERENCES 1. S. A. Rice, Science, 258, 412 (1992) 2. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985) 3. A. P. Peire, M. A. Dahleh and H. Rabitz, Phys. Rev. A 37, 4950 (1988) 4. D. J. Tannor, R. Kosloﬀ and S. A. Rice, J. Chem. Phys., 85, 5805 (1986) 5. R. Demiralp and H. Rabitz, Phys. Rev. A 47, 809 (1993) 6. J. Botina and H. Rabitz, J. Chem. Phys. 104, 4031 (1996) 7. 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The optimal control ﬁeld in a.u. shown as a function of time. 16/2 /: /1 /2 /: /2 /2 /: /3 /2 /: /4 /2 /: /5 /2 /: /6 /2 /: /7 /2 /: /8 /2 /: /9 /3/0 /5/0/0 /1/0/0/0 /1/5/0/0 /2/0/0/0/< x /> /( t /)/( a/:u/: /)t /( a/:u/: /) FIG. 2. The expectation value < x > shown as a function of time in a.u. The target value is < x > =3.0 a.u. at T=2000 a.u. 17/0 /0 /: /5 /1 /1 /: /5 /2 /2 /: /5 /3/0 /1 /2 /3 /4 /5/ /( x /)x /( a/:u/: /) /( a /)/BnZr /0 /: /0/5 /0 /0 /: /0/5 /0 /: /1 /0 /: /1/5 /0 /: /2/0 /1 /2 /3 /4 /5j /( x /)x /( a/:u/: /) /( b /)/1 /: /4 /1 /: /4/5 /1 /: /5 /1 /: /5/5 /1 /: /6 /1 /: /6/5 /1 /: /7 /1 /: /7/5/0 /1 /2 /3 /4 /5S /( x /)x /( a/:u/: /) /( c /)/BnZr /1 /: /6 /BnZr /1 /: /4 /BnZr /1 /: /2 /BnZr /1 /BnZr /0 /: /8 /BnZr /0 /: /6 /BnZr /0 /: /4 /BnZr /0 /: /2 /0 /0 /: /2 /0 /: /4 /0 /: /6/0 /1 /2 /3 /4 /5/ /( x /)x /( a/:u/: /) /( d /) FIG. 3. Fluid dynamical variables, viz., ρ(x)(curve (a); dotted lines for the initial density and solid lines for the ﬁnal density), j(x)(curve (b)), S(x) (curve (c)) shown as a function of x corresponding to t=T. Curve (d) shows the wave function(Ψ( x))(solid lines for real part and dotted lines for imaginary part) as a function of x correspon ding to t=T. 18
arXiv:physics/0008248v1 [physics.atom-ph] 31 Aug 2000Measurement of persistence in 1-D diﬀusion Glenn P. Wong, Ross W. Mair, and Ronald L. Walsworth Harvard-Smithsonian Center for Astrophysics, 60 Garden St ., Cambridge, MA 02138 David G. Cory Department of Nuclear Engineering, Massachusetts Institu te of Technology, Cambridge, MA 02139 (September 26, 2013) Using a novel NMR scheme we observed persistence in 1-D gas diﬀusion. Analytical approximations and numerical simulations have shown that for an initially random array of spins undergoing diﬀusion, the probability p(t) that the average spin orientation in a given region has not changed si gn (i.e., “persists”) up to time tfollows a power law t−θ, where θdepends on the dimensionality of the system. The large nuclear spin polarization of laser-polarized129Xe gas allowed us both to prepare an initial “quasi-random” 1-D array of spi n orientations and then to perform real-time NMR imaging to monitor the spin diﬀusion. Our measurements are consistent with theoretical and numerical predictions of θ≈0.12. We also observed ﬁnite size eﬀects for long time gas diﬀusion. 02.50.-r,05.40.-a,05.70.Ln,82.20.Mj,76.60.Pc The dynamics of non-equilibrium systems is a ﬁeld of great current interest, including such topics as phase ordering in binary alloys, uniaxial ferromagnets, and nematic liquid crystals, as well as coarsening of soap froth and diﬀusion of inhomogeneous ﬂuids ( e.g.[1]). The evolving spatio-temporal structures in these non- equilibrium systems depend crucially on the history of the system’s evolution and are not completely charac- terized by simple measures such as two-time correlation functions, which do not give information on the entire system history. Therefore, an important problem in the study of non-equilibrium dynamics is the development of simple and easily measurable quantities that give non- trivial information about the history of the system’s evo- lution. The recently identiﬁed phenomenon of “persis- tence” is such a quantity: it characterizes the statistics o f ﬁrst passage events in spatially extended non-equilibrium systems [2–15]. Practically, persistence may be impor- tant in determining what fraction of a system has reached a threshold condition as a function of time; for example, in certain chemical reactions or disinfectant procedures. Consider a non-equilibrium scalar ﬁeld φ(x, t) ﬂuctuat- ing in space and time according to some dynamics (e.g., a random array of interdiﬀusing spins). Persistence is the probability p(t) that at a ﬁxed point in space the quantity sgn[ φ(x, t)− /an}bracketle{tφ(x, t)/an}bracketri}ht] has not changed sign up to time t. It has been found that this probability de- cays as a power law p(t)∼t−θ, where the persistence exponent θis generally nontrivial. This exponent de- pends both on the system dimensionality and the preva-TABLE I. A sample of reported persistence exponents. All values except those indicated are derived from numerical si m- ulations; (∗) denotes exact analytical results, (†) experimental measurements, and (‡) the result reported here. Dim. Diﬀusion Ising q-Potts 1 0.12, 0.118‡3/8∗, 0.35 −1 8+2 π2/bracketleftBig cos−1/parenleftBig (2−q)√ 2q/parenrightBig/bracketrightBig2 ∗ 2 0.19 0.22, 0.19†0.86, 0.88†(large q) 3 0.24 0.26 refs [3–5] [2,11] [15]†[2,12] [14]† lent dynamics, and is diﬃcult to determine analytically due to the non-Markovian nature of the phenomena. Al- though θhas been calculated – largely using numerical techniques – for such systems as simple diﬀusion [3–5], the Ising model [2,8,11], and the more generalized q-state Potts model [2,12], few measurements of persistence have been performed (see Table I). In particular, “breath ﬁg- ures” [13], 2-D soap froth [14], and twisted nematic liquid crystals [15] are the only systems for which experimental results have been reported. In this paper we present the ﬁrst measurement of per- sistence in a system undergoing diﬀusion. Our experi- ment is also the ﬁrst to observe persistence in one dimen- sion (1-D). We employed a novel NMR technique to cre- ate a “quasi-random” initial spatial variation in the spin orientation of a sample of laser-polarized129Xe gas. Sub- sequent 1-D NMR imaging, repeated at diﬀerent times, allowed us to monitor the temporal evolution of the en- semble and observe persistence from the fraction of 1-D regions in the sample that did not change their spin ori- entation as a function of time. Using a simple theory (the “independent interval approximation”) and numeri- cal simulations, both Majumdar et al. [3] and Derrida et al.[4] independently found that θ≈0.121 for 1-D diﬀu- sion. Newman and Toroczkai [5] found θ≈0.125 in 1-D using an analytic expression for the diﬀusion persistence exponent. Our measurements are consistent with these calculations. Recently, laser-polarized noble gas NMR has found wide application in both the physical and biomedical sci- ences. Examples include fundamental symmetry tests [16], probing the structure of porous media [17], and imaging of the lung gas space [18]. These varied investi- gations, as well as the experiment reported here, exploit 1α RF gradienti gi mπ/2 gcrusherβj n Encode Imageτj Delay Store FIG. 1. NMR pulse sequence used to encode a 1-D “quasi-random” pattern on the average spin orientation of laser-polarized129Xe gas. Temporal evolution of the mag- netization pattern is monitored with nrepetitions of a 1-D FLASH imaging routine. For example, with m= 8 encod- ing RF pulse/gradient pairs, the encoding pulse angles αi= [30◦, 35◦, 37◦, 41◦, 45◦, 50◦, 63.5◦, and 90◦] while the gradi- ent amplitudes giwere chosen randomly. The imaging pulse angle βjwas ﬁxed at 8◦and the diﬀusion times τjwere var- ied from 2.4 ms up to ∼2 seconds. The encoding, crusher, pre-image crusher, and imaging wind and rewind gradients were pulsed for 1, 20, 3, 2, and 2.56 ms, respectively. The maximum gradient available was 6.7 G/cm. special features of laser-polarized noble gas: the large nuclear spin polarization ( ∼10%) that can be achieved with optical pumping techniques; the long-lived nuclear spin polarization of the spin-1/2 noble gases129Xe and 3He; and rapid gas-phase diﬀusion. We performed laser-polarization of xenon gas using spin-exchange optical pumping [19]. We ﬁlled a coated cylindrical glass cell [20] ( ∼9 cm long, 2 cm I.D.) with approximately 3 bar of xenon gas isotopically enriched to 90%129Xe, 400 torr of N 2gas, and a small amount of Rb metal. We heated the sealed cell to ∼100◦C to create a signiﬁcant Rb vapor using a resistively-heated oven situated in the fringe ﬁeld (0.01 Tesla) of a high ﬁeld magnet. Optical pumping on the Rb D1 line was achieved with 15 W of circularly-polarized 795 nm light (FWHM ∼3 nm) from a ﬁber-coupled laser diode ar- ray. After 20 minutes the129Xe gas was routinely nu- clear spin-polarized to 1% by spin-exchange collisions with the Rb vapor. We next cooled the cell to room temperature in a water bath – eﬀectively condensing the Rb vapor – and placed the cell inside a homemade RF solenoid coil (2.5 cm diameter, 15 cm long, Q∼900) cen- tered in a 4.7 T horizontal bore magnet (GE Omega/CSI spectrometer/imager) with129Xe Larmor frequency = 55.345 MHz. To allow the gas temperature to reach equi- librium, we left the cell in place for 20 minutes before starting the persistence measurements. Under these con- ditions the129Xe polarization decay time constant ( T1) was in excess of 3 hours, with a129Xe diﬀusion coeﬃcient of 0.0198 cm2/s [21]. The NMR pulse sequence we used to observe per- sistence in laser-polarized129Xe gas diﬀusion is shown schematically in Fig. 1. The initial portion of the pulse sequence encodes a 1-D “quasi-random” pattern on the average spin orientation of the laser-polarized129Xe gassample. The pattern is quasi-random in that there must be a minimum length scale to the induced variations in the129Xe magnetization (typically 500 µm) for there to be suﬃcient NMR signal for useful imaging. Neverthe- less, at longer length scales the induced pattern must be random enough that persistence behavior can be ex- pected. Ideally, /an}bracketle{tφ(x,0)φ(x′,0)/an}bracketri}ht=δ(x−x′); however, calculations indicate that it is suﬃcient for the initial condition correlator to decrease faster than \|x−x′\|−1[3]. The quasi-random patterning pulse sequence employs “cumulative k-space encoding.” Recall that one can de- scribe an NMR experiment in terms of a reciprocal or k- space formalism [22], where kis the wave number charac- terizing a magnetization modulation or “grating” created by RF and magnetic ﬁeld gradient pulses. One can rep- resent a spatial magnetization distribution along a ﬁxed axis by a combination of three basis functions: sin( kx) for variations in the longitudinal magnetization Mz, and e±ikxfor positive and negative “helices” of transverse magnetization Mx,y. RF pulses eﬀectively mix the com- ponents of the magnetization (with amplitudes deter- mined by the ﬂip angle) [23] and gradient pulses change thekvalues of the transverse magnetization [22]. By using mpairs of varying RF and random-strength gra- dient pulses in rapid succession it is possible to create a complex and near-random spatial magnetization dis- tribution; i.e., a large number of gratings with diﬀerent k-values and amplitudes are superposed. As Nelson and coworkers showed [24], the maximum number Nmax(m) ofkvalues one can expect from mpairs of RF and gra- dient pulses is given by Nmax(m) =1 4(3m−2m−1). (1) We found that six to eight RF/gradient pulse pairs (m= 6–8) were optimal for the desired quasi-random 1-D patterning of the129Xe spin orientation. m < 6 resulted in a pattern that was not suﬃciently random, while m >8 signiﬁcantly reduced the signal-to-noise ra- tio (SNR) of the NMR images. The requirement of m≥6 is supported by numerical calculations in which we mod- eled the NMR encoding sequence and simulated the sub- sequent gas diﬀusion using a ﬁnite diﬀerence ﬁrst-order forward Euler scheme [4,25]: we found persistence be- havior (i.e., p(t)∼t−θ) only when m≥6. Furthermore, we acquired 512 (time domain) data points for each im- age. The number of data points per image was limited by the available NMR signal (i.e., the129Xe polariza- tion), the necessity of rapid data acquisition to avoid ex- cessive diﬀusion during the imaging sequence itself, and the maximum imaging gradient strength available. Since there is a one-to-one mapping between the time domain andk-space, we could discern at most 512 magnetiza- tion gratings with diﬀerent kvalues. For m= 6,7,8, Nmax(m) = 179 ,543,and 1636, respectively. Hence, 7 or more RF/gradient pulses maximally covered the avail- 2-4-3-2-1012340123 -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 Position [cm]Time [sec]Xe spin orientation129 FIG. 2. Typical set of 1-D images of129Xe spin orienta- tion from a single experimental run. For this example, 8 encoding RF pulse/gradient pairs were used to create an ini- tial quasi-random pattern of129Xe spin orientation on length scales≥500µm. 32 images were acquired at logarithmically increasing times. Contours from every other image are over- layed on the surface plot, which includes all 32 images. ablek-space and produced as random a 1-D spatial dis- tribution of the129Xe magnetization as was detectable (given the constraints of ﬁnite sample length and min- imum length scale of variations ∼500µm). The fact that our simulations (and data, see below) yield persis- tence behavior for m= 6 indicates some robustness with respect to initial conditions. After the pattern encoding part of the NMR pulse se- quence, a π/2 RF pulse “stores” the quasi-random mag- netization distribution along the longitudinal ( z) direc- tion while a subsequent strong (crusher) gradient pulse dephases any remaining transverse magnetization. The quasi-random magnetization distribution then evolves with time due to diﬀusion and is monitored by a series of 1-D FLASH (Fast Low Angle SHot) NMR images [26] (see Fig. 1). We used a ﬁeld of view (FOV) of 31.5 cm with 0.6 mm resolution, which thus divided the 9 cm cell into about 150 discernible spatial regions. We typically employed 8◦excitation RF ﬂip angles and acquired 32 1-D images spaced logarithmically in time from ∼3 ms to 5 s for a single experimental run. An example of the images acquired in one such run are shown in Fig. 2. We derived spin orientations (aligned or anti-aligned to the main magnetic ﬁeld) from the phase information con- tained in the time-domain NMR image data and spa- tial positions from the frequency information [27]. Each experimental run thus provided a record of the129Xe gas spin orientation as a function of position and time proceeding from the initial quasi-random pattern to the equilibrium condition of homogeneous (near-zero) polar- ization. To measure persistence, we noted the sign of the129Xe spin orientation in each spatial region (i.e., in each 1-D0.01 0.1 1.0 100.50.60.70.80.91.0 t [sec]p(t) FIG. 3. A log-log plot of p(t), the fraction of spin orienta- tion regions that had not changed sign up to a time t, rep- resenting the sum of ∼30 diﬀerent experimental runs. The solid line is a weighted linear least-squares ﬁt to the data f or 0.1 s< t < 1 s, and yields θ= 0.118±0.008. Error bars are derived from the number of pixels with amplitudes close to the image noise level and are shown when they exceed the plot symbol diameter. image pixel) and counted how many remained unchanged as a function of time. We equated the probability p(t) with the fraction of pixels that had not changed sign up to time t. We chose t= 0 to coincide with the ﬁrst im- age and assigned the time index for each image to be the start time of the imaging RF pulse. Images with SNR<40 were excluded from the data to minimize un- certainty in pixel sign changes. We conducted about 30 experiments with image SNR >40, each with a unique set of randomly chosen encoding gradients {gi}. We em- ployed two averaging schemes to combine the results from diﬀerent experimental runs. In the ﬁrst method, we used a linear least-squares ﬁt of log[ p(t)] vs. log[ t] for each run, resulting in a distribution of power law exponents with a weighted mean θ= 0.119±0.048. With our numerical simulations of cumulative k-space-encoded initial condi- tions, we found that this averaging scheme results in a gaussian distribution of exponents with a mean value θ≈ 0.12 in agreement with previous calculations for 1-D dif- fusion [3–5] and our experimental results. In the second averaging scheme, we combined the data from all exper- imental runs; hence p(t) represented the fraction of total pixels from all experiments that had not changed sign up to time t. We found p(t)∼t−θwithθ= 0.118±0.008 for 0.1 s < t < 1 s. Figure 3 shows a log-log plot of p(t) vs.twhen the data is averaged using this method. The observed deviations from power law behavior for t <0.1 s and t >1 s are explained by resolution and ﬁ- nite size eﬀects, respectively. At short times persistence is not observed because129Xe atoms have not yet dif- fused across a single spin orientation region δx≈500µm. The relevant diﬀusion time is ( δx)2/(2DXe)≈0.1 s. At long times, the pattern of129Xe spin orientation becomes 30.01 0.1 1.0 101.010 0.1 t [sec]average domain size [cm] FIG. 4. The average spin orientation domain size Las a function of time t, derived from all experimental runs. For 0.1 s< t < 1 s,L∼tαwhere α= 0.45±0.02 (solid line). The dotted line shows the expected L∼t1/2behavior for an inﬁnite system. The error in Lis shown where it exceeds the plot symbol size. The ﬁnite size limit on Lis evident in the four late-time points ( △), which were taken from the only two runs with suﬃcient SNR at long times. ordered on length scales comparable to the sample dimen- sion, thus curtailing the rate of sign-changing. Both the short and long time deviations are also seen in Fig. 4, where the average length Lof spin orientation domains from all experimental runs is plotted against time. For 0.1 s< t < 1 s, our data are in reasonable agreement with the expected power law L∼t1/2for diﬀusion. How- ever, at short times Lis near the limit to image resolu- tion while at longer times Lgrows more rapidly as it approaches the dimension of the sample cell. In conclusion, we experimentally measured a persis- tence exponent θ≈0.12 for 1-D diﬀusion, consistent with analytical and numerical studies. We performed the mea- surement using a novel NMR scheme with laser-polarized 129Xe gas which allowed us to both encode a “quasi- random” spatial pattern of spin orientation and monitor its evolution over several seconds. We also observed the eﬀect of ﬁnite sample size for long time diﬀusion. In fu- ture work the experimental technique employed in this study may allow measurements of persistence in 2 and 3-D diﬀusion, in heterogeneous systems (e.g., porous me- dia) infused with noble gas, and in ‘patterns’ [28]. The authors thank Satya Majumdar, Michael Cressi- mano, and Lukasz Zielinski for useful discussions. This work was supported by NSF Grant No. CTS-9980194, NASA Grant No. NAG9-1166, and the Smithsonian In- stitution Scholarly Studies Program. [1] A. J. Bray, Adv. Phys. 32, 357 (1994).[2] B. Derrida, V. Hakim, and V. Pasquier, Phys. Rev. Lett. 75, 751 (1995). [3] S. N. Majumdar, C. Sire, A. J. Bray, and S. J. Cornell, Phys. Rev. Lett. 77, 2867 (1996). [4] B. Derrida, V. Hakim, and R. Zeitak, Phys. Rev. Lett. 77, 2871 (1996). [5] T. J. Newman and Z. Toroczkai, Phys. Rev. E 58, R2685 (1998). [6] B. P. Lee and A. D. Rutenberg, Phys. Rev. Lett. 79, 4842 (1997). [7] J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray, and C. Sire, Phys. Rev. E 56, 2702 (1997). [8] S. Jain, Phys. Rev. E 60, R2445 (1999). [9] C. Sire, S. N. Majumdar, and A. R¨ udinger, Phys. Rev. E60, 1258 (2000). [10] V. M. Kendon, M. E. Cates, and J.-C. Desplat, Phys. Rev. E 61, 4029 (2000). [11] S. N. Majumdar and C. Sire, Phys. Rev. Lett. 77, 1420 (1996). [12] B. Derrida, P. M. C. de Oliveira, and D. Stauﬀer, Physica A224, 604 (1996). [13] M. Marcos-Martin, D. Beysens, J. P. Bouchaud, C. Godr` eche, and I. Yekutieli, Physica A 214, 396 (1995). [14] W. Y. Tam, R. Zeitak, K. Y. Szeto, and J. Stavans, Phys. Rev. Lett. 78, 1588 (1997). [15] B. Yurke, A. N. Pargellis, S. N. Majumdar, and C. Sire, Phys. Rev. E 56, R40 (1997). [16] D. Bear, T. E. Chupp, K. Cooper, S. DeDeo, M. Rosen- berry, R. E. Stoner, and R. L. Walsworth, Phys. Rev. A 57, 5006 (1998). [17] R. W. Mair, G. P. Wong, D. Hoﬀmann, M. D. H¨ urlimann, S. Patz, L. M. Schwartz, and R. L. Walsworth, Phys. Rev. Lett.83, 3324 (1999). [18] M. S. Albert, G. D. Cates, B. Driehuys, W. Happer, B. Saam, C. S. Springer, Jr., and A. Wishnia, Nature 370, 199 (1994). [19] T. G. Walker and W. Happer, Rev. Mod. Phys. 69, 629 (1997). [20] We used a wall coating of octadecyltrichlorosilane (OT S) to reduce Xe-wall interactions and hence increase longi- tundinal relaxation times. [21] R. W. Mair, D. G. Cory, S. Peled, C.-H. Tseng, S. Patz, and R. L. Walsworth, J. Mag. Res. 135, 478 (1998). [22] A. Sodickson and D. G. Cory, Prog. Nucl. Magn. Res. Spec.33, 77 (1998). [23] J. Hennig, J. Mag. Res. 78, 397 (1988). [24] R. I. Nelson, Y. Maguire, D. F. Caputo, G. Leu, Y. Kang, M. Pravia, D. Tuch, Y. S. Weinstein, and D. G. Cory, Concepts. in. Magn. Res. 10, 331 (1998). [25] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge Univer- sity Press, Cambridge, U.K., 1988). [26] A. Haase, J. Frahm, D. Matthaei, W. H¨ anicke, and K.-D. Merboldt, J. Mag. Res. 67, 258 (1986). [27] C. B. Ahn and Z. H. Cho, IEEE Trans. Med. Imag. MI-6, 32 (1987). [28] S. N. Majumdar, Curr. Sci. 77, 370 (1999). 4
arXiv:physics/0009001v1 [physics.atom-ph] 1 Sep 2000Comment on “Degenerate Wannier The- ory for Multiple Ionization” Recent Letter of Pattard and Rost [1] suggests new threshold laws for the processes of break up of a par- ticle into several charged fragments, which, regretfully, is incorrect. Previous studies, starting from the classica l paper by Wannier [2] derived fragmentation cross section σin a power-law form σ(ǫ→0)∼ǫµ, (1) where ǫis the energy excess above the break up thresh- old. The primary task of the theory is to evaluate the threshold index µfor diﬀerent multiparticle systems, a comprehensive bibliography could be found in Ref. [3]. Pattard and Rost [1,4] suggest the threshold law of novel functional form, namely, with an extra logarithmic ǫ- dependence: σ(ǫ→0)∼ǫµ\|lnǫ\|−ν. (2) According to the originally idea of Wannier [2], for the Coulomb interaction between the fragments the thresh- old law is deﬁned by the motion along the potential ridge as the constituent particles ﬂy apart. This motion is un- stable; survival on the ridge corresponds to the system fragmentation, whereas sliding from it means that the break up is not achieved. The instability of the motion is characterized by a set of Lyapunov exponents λi. The threshold index µis expressed via sum of all Lyapunov exponents [3,4]. In the present Comment we discuss the key result obtained by Pattard and Rost in [1], and re- peated in [4], namely, the novel threshold law (2). The standard approach to derivation of threshold laws relies on the analysis of classical trajectories in the vici n- ity of the potential ridge where the equations of motion arelinearized over coordinates transversal to the ridge. The system is described by a set of coupled linear equa- tion that are diﬀerential over the time variable t. The time-dependence of the coeﬃcients is eliminated by intro- ducing the eﬀective time τ. The solutions of these equa- tionsq(τ) depend on the eﬀective time as exp( λiτ) where the set of λieigenvalues is found by solving the charac- teristic equation. The threshold index µis expressed via λi, see for details Ref. [2,3] and bibliography therein. Pattard and Rost [1,4] essentially retain the traditional framework, but argue that the situation is changed when the Lyapunov exponents happen to be degenerate . They refer to the standard mathematical textbook [5] saying that “If neigenvalues are degenerate, the general solution contains additional terms τkexp(λτ), (k < n )”. Further they relate the logarithm in the alleged coordinate time- dependence, q(τ) =τkexp(λτ)≡exp(λτ+klnτ) ( k < n),(3) to the logarithmic factor in the energy-dependence (2).This reasoning is invalid. Consider harmonic vibra- tions of a system with many degrees of freedom (for ex- ample, a polyatomic molecule) around an equilibrium po- sition. Some eigenfrequencies might be degenerate due to symmetry reasons, or accidentally. The argumentation by Pattard and Rost fully applies to this case; however, as universally known, the logarithmic solutions (3) in fact do not emerge [see Eq. (23.6) in Ref. [6] and subsequent discussion]. The reason is clear: in the harmonic ap- proximation all the normal modes are fully decoupled and hence a character of the time-dependence in each mode does not depend on whether some other degenerate mode exists or not. This argument is directly related to the motion in the vicinity of the potential ridge. The problem is described by the Hamiltonian that is quadratic both in transversal coordinates and momenta, similar to the harmonic ap- proximation for a polyatomic molecule. The only diﬀer- ence is that the equilibrium is unstable and some eigen- frequencies are complex-valued [3]. Obviously this fact does not inﬂuence the functional form of solutions of the same equations of motion which still reveal no logarithmic terms. The degenerate Lyapunov exponents are related to the diﬀerent modes which are fully decoupled , there- fore the logarithmic solutions (3), being mathematically feasible for general linear diﬀerential equations, do not emerge in the physical applications concerned. As a summary, the threshold law is given by Eq. (1) while the modiﬁcation (2) is never valid. This work has been supported by the Australian Re- search Council. V. N. O. acknowledges the hospitality of the staﬀ of the School of Physics of UNSW where this work has been carried out. M. Yu. Kuchiev and V. N. Ostrovsky School of Physics, University of New South Wales, Syd- ney 2052, Australia PACS numbers: 32.80.Fb, 34.80.Dp, 34.80.Kw, 31.15.Gy [1] T. Pattard and J. M. Rost, Phys. Rev. Lett. 80, 5081 (1998). [2] G. H. Wannier, Phys. Rev. 90, 817 (1953). [3] M. Yu. Kuchiev and V. N. Ostrovsky, Phys. Rev. A 80, 5081 (1998). [4] T. Pattard and J. M. Rost, Phys. Rev. Lett. 81, 2618 (1998). [5] M. R. Spiegel, Advanced Mathematics for Engineers and Scientists (McGraw-Hill, New York, 1980). [6] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press, Oxford , New York, 1976). 1
arXiv:physics/0009002v1 [physics.bio-ph] 1 Sep 2000Statistically Signiﬁcant Strings are Related to Regulatory Elements in the Promoter Regions of Saccharomyces cerevisiae Rui Hu1,2, Bin Wang1 1Institute of Theoretical Physics, Academia Sinica P. O. Box 2735, Beijing 10080, China 2Department of Mordern Physics, University of Science and Technology of China, Anhui, 230027, China Abstract Finding out statistically signiﬁcant words in DNA and prote in sequences forms the basis for many genetic studies. By applying the maximal entropy pr inciple, we give one systematic way to study the nonrandom occurrence of words in DNA or prote in sequences. Through comparison with experimental results, it was shown that pat terns of regulatory binding sites in Saccharomyces cerevisiae (yeast) genomes tend to occur signiﬁcantly in the promoter regions. We studied two correlated gene family of yeast. The method successfully extracts the binding sites variﬁed by experiments in each family. Man y putative regulatory sites in the upstream regions are proposed. The study also suggested that some regulatory sites are active in both directions, while others show directional pr eference. 1 Introduction. It is attractive, but not unexpected, that DNA and protein se quences deviate remarkably from random sequences [1]. According to information theory, ran dom sequences carry minimal infor- mation (maximal entropy) [2], while the total information o f life is assumed to be in DNA and protein sequences. As a result, investigation on the non-ra ndomness of DNA and amino acid sequences would be the focus of Bioinformatics. To ﬁnd out nonrandom occurrence of words (short strings) in n on-coding DNA sequences is interesting because a large portion of regulatory elemen ts of eukaryotes usually are words of 1limit length in the non-coding sequences (for example, abou t 10 bases, while the core part is about 5 bases [3]). subjected to functional constraints, th e patterns of regulatory elements are expected to deviate from random occurrence. In this paper, by applying the Maximal Entropy Principle (ME P), we develop one way to investigate the nonrandom occurrence of words in DNA sequen ces. Each word is given one signiﬁcance index which quantiﬁes the nonrandomness occur rence of the word. The method is then applied to study the promoter regions of Saccharomyces cerevisiae (yeast). [4] We compare the theoretical result with experiments in two ways. In the ﬁ rst way, the promoter database of yeast(SCPD) [5] was analysed. It was found that, statistically, o verrepresented words are more easily encountered in the database. The second way is to stud y the promoters of coregulated gene families. The experimentally found binding sites were successfully extracted, and more putative binding sites are suggested. In the following the method will be developed in details, and in the third section the method will be applied to study the promoter regions of yeast. 2 Treat the nonrandomness of DNA sequences via Maximal En- tropy Principle. The idea comes from a simple observation. Take a long DNA sequ ence as an example. Given only the (normalized) frequencies of A,C,G,T(PA,PC,PG,PT) , one would expected that the frequencies 2-tuples have the form P0c1c2=Pc1×Pc2 (1) Herec1andc2are one of the four bases A,C,G,T . Comparison between the measured frequency Pc1c2and the expected value P0c1c2reveals the statistical signiﬁcance of c1c2in the sequence. To generalize the above idea, one encounters the problem to p redict the frequencies of k+1- tuples from the frequencies of k-tuples when k >1. A reasonable deﬁnation can then be used to evaluate the statistical signiﬁcance of words longer tha n two bases. The following is an attemption to answer this problem. In the treatment, when the compo- sition of a k-tuple is concerned, the word will be written as c1c2· · ·ck−1ck. However when only the length kof the word is relevant, it will be given in the form of wk. A combinatory form may also be used. For example, wkc(cwk) is the word obtained by adding a letter cto the right (left) of wk. The measured and expected frequencies of wkin the sequence will be written as PwkandP0 wk, respectively. 2There are a total of 4kk-tuples. For prediction the Maximal Entropy Principle (MEP ) is a prefered choice. According to modern genetics, the driv ing force for nucleotide sequence evolution is, on one hand, random mutations of bases that max imize the entropy, and, on the other hand, the natural selection which subjects the max imization of entropy to certain constraints. Therefore, DNA sequence analysis shows intri nsic correlation to the MEP. One brief introduction (which is necessary for our use) to MEP wi ll be given below. More details can be found in e.g. [6]. Suppose that {Pi,i= 0,1,2,· · ·}is a discrete distribution. An information entropy can be deﬁned on it [2]: S=/summationdisplay iPilnPi. (2) Usually {Pi}satisﬁes some constraints: Fj({Pi}) = 0, j = 1,2,...,M. (3) HereMis the number of constraints. Deﬁne a target function: H=S+M/summationdisplay j=1λjFj({Pi}), (4) λjbeing Largrange factors. MEP states that the distribution m inimizing the target function H is the most reasonable distribution satisfying constraint s (3). This, however, does not state that {Pi}is the only distribution satisfying (3). The MEP now can be applied to study the problem raised above. T he entropy function here is: S=/summationdisplay iP0 wk+1(i)lnP0 wk+1(i), where iis a index used to distinguish k-tuples from each other. (In o rder to get the index of a word, the following maps were used: Ato 0,Cto 1,Gto 2, and Tto 3. The original word is thus mapped to a string containing only 0,1,2 and 3. The strin g is then considered as quaternary number. After being transformed to decimal, the number is us ed as the index of the word.) Constraints in the present problem is: Pwk(i)=/summationdisplay cP0 wk(i)c, Pwk(i)=/summationdisplay cP0 cwk(i), i= 0,1,2,· · ·,4k−1. (5) 3P0 wk+1is the frequency needs to be predicted and Pwkis the frequency already known. There is a total of 2 ×4kconstraints. It is possible that these constraints are line arly related, so that the number of eﬀective constraints is smaller than 2 ×4k. This, however, does not alternate the result. The solution can be obtained: P0c1c2···ck+1=Pc1c2···ck×Pc2c3···ck+1 Pc2c3···ck. (6) When k=1, the solution reduces to the intuitively result, eq .(1). The above treatment is from k-tuples to k+ 1-tuples. As a generic scheme, the MEP can also be applied to predict the frequencies of k+ 2-tuples, k+ 3-tuples and so on, based on the frequency of k-tuples. Actually, one can get the result by re peatedly applying eq. (6). For example: P0c1c2···ck+1ck+2=P0c1c2···ck+1×P0c2c3···ck+2 Pc2c3···ck+1 =Pc1c2···ck×Pc2c3···ck+1×Pc3c4···ck+2 Pc2c3···ck×Pc3c4···ck+1. (7) Thus, when one refers to the expected frequency of a certain w ord of length k, the knowledge that the prediction is based on must be pointed out. With the frequencies of longer words, one can always obtain t he frequencies of shorter ones. On the other hand, the expected frequencies of longer words, eq.(6), is predicted from the frequencies of shorter words, with no more information adde d. Therefore, the deviation of the measured frequencies from the expected ones gives new in formation emerges only in the frequencies of the longer words. In order to use this part of i nformation, we refer to the following signiﬁcance index Iwk=Pwk−P0 wk/radicalbig P0wk. (8) The indexs of k-tuples form a vector of 4kdimension. It should be pointed out that the simple solution eq.(6) resu lts from the constraints, eq.(5). Although there are many ways to write down the prediction [7, 8], the Maximal Entropy Principle ensure that, submitted to these constraints, the solution e q.(6) is the best one. However, one can consider more constraints. Expect for the continuous wo rds, spaced patterns can also be involved in the above statistical treatment [8]. As an examp le, consider the spaced word c1- c2, where c1andc2are certain bases and the base between them is not relevant. O ne more constraint Pc1−c2=/summationdisplay cPc1cc2 4can be added to the frequencies of 3-tuples, and the statisti cal signiﬁcance of the spaced words can also be evaluated. The MEP, as a general framework, is sti ll applicable, but there will be no simple explicit solution as eq.(6). 3 The relationship between regulatory elements and statist i- cally signiﬁcant words in the yeastpromoter regions. With the accumulation of huge amount of genome sequences, an alysis of the regulatory regions becomes urgent, because they govern the regulation of gene e xpression. Finding out the regula- tory sites in Eukaryotes genomes is especially diﬃcult, lar gely because of their strong variance. This, however, gives the chance for statistical methods to p lay an important role in binding sites prediction. The regulatory elements are functionally constrained and a re often shared by many genes. As a result, the sites are expected to be signiﬁcantly repres ented. Based on this belief, the method developed above is expected to be applicable in ﬁndin g regulatory sites in the promoter regions of yeast. we employ two ways to check this point. In the ﬁrst way, as just an illustration of the eﬀectiveness o f the MEP treatment, a data set including all the promoters of yeast will be used to perform the statistical evaluation. The promoter regions refer, according to Zhang [3], to the upstr eam region of 500 bases long. From the sequence set the word frequencies are obtained and Iwk,k= 2,· · ·,8, are calculated according to eq.(6) and eq.(8). (to obtain Iwk,P0 wkis predicted based on the frequency of k-1-tuples.) For comparison the index Iwk,k= 2,· · ·,8, of words in the coding regions (CDSs) of yeastwere also calculated. To compare the signiﬁcance index of words with experimental ly veriﬁed regulatory elements, a strongly statistically characterized method was pursued . The promoter database of yeast collected by Zhou et al. [5] was used as targets. One word is ca lled to hit the target if it covers a known regulatory element or part of the element. In this way, each word will be checked against all the elements in the database. We want to see if the total hi ts of words show correlation with the signiﬁcance index. Fig.1 shows the ratio of the average hits of words whose signi ﬁcance index are larger than a certain cutoﬀ (5.0, 3.0, or 2.0) to the average hits of all th ek-tuples. Some properties of signiﬁcance index in the promoter regions are revealed. Fir st, for all the cutoﬀ value shown in ﬁg.1, the ratios are always larger than 1. Second, when the wo rds are longer than 4 bases, the average hits increase with the increase of cutoﬀ. Furthermo re, the ratio also increases with the 5increase of word length. As a comparison, Fig.1 shows that th e ratio of hits does not depend on signiﬁcance index in the CDS regions. To see the dependence of hits on signiﬁcance index further, w ords are divided into groups according to their signiﬁcance index values. In each group t he hits were averaged. See table 1, and Fig.2 which is based upon the data in Table.1 but shown as a more audio-visual illustration. The dependence of hits on signiﬁcance index shown in Fig.1 is seen again. Furthermore, the average hits are not the monotonic function of signiﬁcance i ndex in the promoter regions. For words with both positively and negatively large signiﬁcanc e index in the promoter regions, the average hits are larger than those of words whose signiﬁcanc e index is around zero. Again no dependence of average hits on signiﬁcance index in CDS regio ns is observed in Table.1 and Fig.2. That words with large negative signiﬁcant index in the promoter regions also show higher aﬃnity to binding sites deserves more consideration. One ac count is that although some regula- tory elements, such as those involved in the expression of ho usekeeping genes, are expected to be overrepresented since large amount of the genes are needed, others that control the expression of some essential but restrictedly needed genes, are expect ed to be underrepresented to avoid inappropriate translation. However, more convincing expl aination exists: if a word, e.g., wA, has high positive index, then some of wC,wG,wT are expected to have negative index. This can be seen from the following example. While the index of TATAT i s 16.3, that of TATAA is -12.2. Actually, both have much high counts in the sequences and bot h are variance of binding site of the same transcriptional factor. For universally existing regulatory elements, as expected , the signiﬁcance index in the pro- moter regions are much high. One example is the poly(A/T) str etches. As given above, the signiﬁcance index of TATAT is 16.3. Also the signiﬁcance ind ex of TATATAT, 8.1, is high. As another example, the signiﬁcance index of the core of CAAT-b ox, CAAT, is 8.95. However, in order to develop an algorithm for regulatory elements pre diction, more subtle consideration must be involved. First, genes are needed to be classiﬁed int o families to improve the composi- tional bias of the sequences. Furthermore, more complicate d usage of the information given by signiﬁcance index should be considered, because, accordin g to eq.(7), the expected frequency of k-mers can be deﬁned in k−1 ways, i.e., based on the frequency of 1 ,2,· · ·,k−1-mers, respec- tively. For each deﬁnition the signiﬁcance index can be obta ined. On considering the statistical property of words in the sequences, each of these indexes wou ld give useful information. We choose two coregulated gene family to further test our metho d. The coregulated genes of yeastmetabolism have been widely studied, and these datasets pro - vide ideal material to test the methods for binding sites pre diction. Two families of coregulated 6genes, GCN and TUP, were shown in table 2. Detailed informati on on them can be found in [9]. For each family, the frequencies of 6-tuples in the promoter regions were ﬁrst collected. The expected frequencies them were predicted in ﬁve ways, which are based on the frequencies of bases, 2-tuples, 3-tuples, 4-tuples, 5-tuples, respectiv ely. In stead of Iwk, a simpler signiﬁcance index Pwk/P0 wkwas used. In our study only the single strand of promoter sequ ences is consid- ered. This is diﬀerent from that of [9]. They count the number of each words in both strands. In this way there are only 2080 distinct oligonucleotides, w hile the number in ours is 46= 4096. Table 3 shows the words that possess no less than 3 among the 5 s igniﬁcance index larger than 3. There are 13 such words for GCN family, and 23 for TUP family. I n table 3 several words tend to cluster together to form a longer pattern. Generally spea king, the clusters can be expanded by involving words with slightly lower signiﬁcance. In both families, 6-tuples corresponding to regulatory bin ding sites found by experimental analysis are observed in table 3. See the ﬁrst cluster of word s for GCN family and the ﬁrst and the second clusters for TUP family. Most of these words also s how high statistical signiﬁcance in the analysis of [9]. Some words predicted by [9] but not var iﬁed by experiments are also observed in table 3 (signiﬁcant words shared by [9] and the pr esent analysis are shown in bold in table 3). However, our analysis also found many signiﬁcant w ords which do not show as highly signiﬁcant scores according to [9]. Two clusters of words for TUP families is noteworthy (see the ﬁrst and the second clus- ters in table 3). The ﬁrst cluster includes GTGGGG, AGGGGC, A CGGGC, TGGGGT, and GGGGTA, and the second cluster involves TACCCC, ACCCCG, CCC CGC, and CCCCAC. be- tween them GTGGGG and CCCCAC, GGGGTA and TACCCC are reverse c omplements. The two clusters both correspond to the binding sites of transcr iption factor Mig1p (Zn ﬁnger), but seen from diﬀerent strands. This may imply that the binding s ites of Mig1p are active in both orientation. This property, however, was not found for the b inding sites of Gcn4p (see table 3). For example, when the cutoﬀ of signiﬁcance index is reduc ed to 1.3 (now 46 words satisfy the creterion), the cluster of TGACTC and GACTCA expands to i nvolve another 4 members: CGATGA, GATGAC, ATGACT, and GTGACT; while only one of invers e complements of them, GAGTCA, also has 3 index larger than 1.3. Among the 46 words, i t can only be clustered with another words GGAGTC. Thus, the binding sites of Gcn4p seem t o be active preferencially in one direction. Among the available methods of binding sites prediction, ou rs is similar to that of [9] in that both work by deﬁning expected frequencies of words. the diﬀe rence is that our method deﬁnes the expected frequences on the statistical stproperties of the sequences themselves, while [9] 7more or less heuristically deﬁnes the word frequencies of wh ole non-coding sequences as the expected value. It is thus expected that our method is more pr ecise and gives more unbiased result. An alternative method developed by Li et al [10]. gives more s ubtle consideration on the statistical feature of DNA sequences. In their model, the se quence is considered as a text without interwords dilimiters. They apply maximal likelihood cons ideration to recover the words, which they consider as possible binding site condidates. But the c omputation is far more complex to get meaningful result. More methods to detect unknown elements within funtionally related sequences are availible (for a review, see [11]), most of which, such as the consensus [12] and the Gibbs sampler [13], are based upon well diﬁned biological models. The type of sig nals that can be detected are generally limited; it is diﬃcult for them to detect multiple signals. But these methods are able to detect much larger patterns with high precision. The pres ent method can be used to detect multiple elements, but the pattern it can ﬁnd is short. It is also a widely explored problem in biology to compare the noncoding and coding regions of DNA sequences [14, 15, 16]. The MEP treatment gives one syste matic way to study the statistical diﬀerences between coding and noncoding regions. In table. 1 it is shown that signiﬁcance index in CDS regions distribute much more stretchy than that of the promoter regions. The contrast keeps for all the word lengths we studied (up to 8 bases). This reveals that CDS regions are in a more nonrandom state. Two factors may help to interpret t his phenomenon. First, the mutation rate of CDS regions is much lower than that of the pro moter regions [15]. Secondly, the code usage in CDS region is universal and deﬁnite, while i n the promoter regions the length of regulatory elements diﬀer from each other and the regulat ory elements may diﬀer strongly from the consensus sequences. ACKNOWLEDGMENTS We are grateful to professor Bai-lin Hao and Wei-mou Zheng fo r stimulating discussions. We also thank Guo-yi Chen for helps on computation. 8References [1] C.-K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F . Sciortino, M. Simons, H.E. Stanley, Nature 356 (1992) 168. [2] A.E. Shannon, Bell System Tech. J. 27 (1948) 379. [3] M.Q. Zhang, comput. Chem. 23 (1999) 233. [4] A. Goﬀeau, B.G. Barrell, H. Bussey, R.W. Davis, B. Dujon, H. Feldmann, F. Galibert, J.D. Hoheisel, C. Jacq, M. Johnston, E.J. Louis, H.W. Mewes, Y. Mu rakami, P. Philippsen, H. Tettelin, S.G. Oliver, Science 274 (1996) 546. [5] J. Zhu, M.Q. Zhang, Bioinformatics 15 (1999) 607. [6] J. Honerkamp, Statistical Physics: an Advanced Approac h with Application, Springer, Berlin, 1998. [7] G.J. Phillips, J. Arnold, R. Ivarie, Nucl. Acids. Res. 15 (1987) 2611. [8] P.A. Pevzer, M.Y. Borodovsky, A.A. Mironov, J. Biomol. S truct. Dynam. 6 (1989) 1013. [9] J.V. Helden, B. Andre, J. Collado-Vides, J. Mol. Biol. 28 1 (1998) 827. [10] H.J. Bussemaker, H. Li, and E.D. Siggia, Preprint. [11] J.W. Fickett, A. G. Hatzigeorgiou, Eukaryotic Promote r Recognition in Genome Research, Cold Spring Harbor Laboratory Press, 1997. [12] G.Z. Hertz, G. W. Hartzell, G.D. Stormo, Comput. Appl. B iosci. 6 (1990) 81. [13] C.E. Lawrence, S.F. Altschul, M.S. Boguski, J.S. Liu, A .F. Neuwald, J.C. Wootton Science 262 (1993) 208. [14] C. Burge, A. Campbell, S. Karlin, Proc. Natl, Acad. Sci. USA 89 (1992) 1358. [15] W.-H. Li, Molecular Evolution, Sinauer Associates, Ca nada, 1997. [16] R.N. Mantegna, S.V. Buldyrev, A.L. Goldberger, C.-K. P eng, M. Simons, H.E. Stanley, Phys. Rev. Lett. 73 (1994) 3169. [17] A. G. Hinnebusch, General and pathway-speciﬁc regulat ory mechanisms controlling the synthesis of amino acid biosynthetic enzymes in Saccharomyces cerevisiae , in: E.W. Jones, J.R. Pringle, J.R. Broach (Eds), The molecular and Cellular Biology of the Yeast Saccha- romyces : Gene Expression, pp. 319-414, Sold Spring Harbor Laborato ry Press, Cold Spring Harbor, NY, 1992. [18] J.L. Derisi, V.R. Iyer, P.O. Brown, Science, 278 (1997) 680. 9110 2 3 4 5 6 7 8H/H0 string lengthP5.0 P3.0 P2.0 C4.0 C2.0 Figure 1: The ratio of average hits ( H) of words above certain cutoﬀ of signiﬁcance in- dex to the average hits ( H0) of all the words of same length. The H0(word length) are 405(2) ,94.4(3),21.7(4),4.92(5),1.10(6),0.241(7) ,0.0528(8). 0.511.522.53 -10 -5 0 5 10hits SSIpromoter CDS Figure 2: The dependence of average hits of 6-tuples on their average signiﬁcance index. The data in this ﬁgure are shown as a more audio-visual illustrat ion of the 6-tuple data in Table 1. 10Table 1: The dependence of average hits on the signiﬁcance in dexIw=Pw−P0w√ P0w.The values shown in the hits volume are averaged over the hits of the poin ts (words) included in the signiﬁcance index range shown in the Iwcolummn. pentemer hexmer promoter CDS promoter CDS Iw points hits Iw points hits Iw points hits Iw points hits -15,-9 9 7.33 -29,-12 16 5.50 -11,-6 10 2.10 -15,-9 12 1.17 -9,-7 12 4.75 -12,-10 12 4.33 -6,-4 15 1.40 -9,-7 15 0.60 -7,-5 16 4.00 -10,-9 12 5.67 -4,-3 28 1.79 -7,-6 30 1.40 -5,-4 22 3.50 -9,-8 14 5,14 -3,-2 161 1.04 -6,-5 95 0.78 -4,-3 29 4.31 -8,-7 23 4.87 -2,-1 716 0.976 -5,-4 102 0.98 -3,-2 91 4.02 -7,-6 40 5.38 -1,0 1137 0.997 -4,-3 202 1.15 -2,-1 158 4.03 -6,-5 44 4.34 0,1 1169 1.05 -3,-2 334 1.10 -1,0 182 4.46 -5,-4 46 5.41 1,2 593 1.27 -2,-1 563 1.07 0,1 198 5.02 -4,-3 63 5.27 2,3 178 1.51 -1,0 738 1.09 1,2 134 5.58 -3,-2 74 5.34 3,4 52 1.28 0,1 750 1.04 2,3 77 5.25 -2,-1 79 4.52 4,5 19 1.68 1,2 570 1.09 3,4 47 6.55 -1,0 94 5.53 5,6 11 2.18 2,3 345 1.24 4,6 21 7.09 0,1 101 4.58 6,13 12 3.17 3,4 117 1.07 6,8 13 6.15 1,2 83 4.16 4,5 94 1.29 8,10 10 7.40 2,3 59 5.47 5,6 64 1.17 10,19 9 10.22 3,4 60 4.60 6,7 21 1.23 4,5 48 4.70 7,9 18 1.00 5,6 38 4.08 9,19 16 1.44 6,7 27 4.19 7,8 25 4.80 8,9 19 6.84 9,10 13 5.46 10,12 12 4.42 12,29 19 5.21 11Table 2: The coregulated gene family GCN and TUP, and criteri on for them being clustered. Family Genes Shared regulatory property References GCN ARG1,ARG3,ARG4,ARG8,ARO3,ARO4, ARO7,CPA1,CPA2,GLN1,HIS1,HIS2, HIS3,HIS4,HIS5,HOM2,HOM3,HOM6, ILV1,ILV2,ILV5,LEU1,LEU2,LEU3, LEU4,LYS1,LYS2,LYS5,LYS9,MES1, MET14,MET3,MET6,TRP2,TRP3, TRP4,TRP5,THR1General amino acid contral; genes activated by Gcn4p.Hinnebusch [17] TUP FSP2,YNR073C,YOL157C,HXT15,SUC2, YNR071C,YDR533C,YEL070W,RNR2, YER067W,CWP1,YGR243W,YDR043C, YER096W,HXT6,YLR327C,YJL171C, YGR138C,HXT4,GSY1,YOR389W, MAL31,YML131W,RCK1All genes which are both derepressed by a facter larger than 4 when TUP1 is deleted, and induced by a factor larger than during the diauxic shiftDeRisi et al. [18] 12Table 3: Highly overrepresented words in promoter regions o f GCN and TUP family.For each family, the 6-tuples with no less than 3 among the 5 signiﬁcan ce index larger than 3 are indicated. The words also appear in table 2 of [9] as signiﬁcant patterns are highlighted in bold. Words are clustered according to their similarity. sig(i) is the value of Pw6/P0w6withP0w6being the frequency of 6-tuple w6predicted based on the frequencied of i-tuples. analysis result on 6-tuples sites previously characterized Family Sequences counts sig(1) sig(2) sig(3) sig(4) sig(5) Consensus binding factors GCN TGACTC 29 4.47 4.61 4.16 2.93 1.39 RRTGACTCTTT Gcn4p GACTCA 21 3.28 3.36 3.25 3.24 1.39 (bZip) CCGGTT 12 3.18 3.38 3.47 2.07 1.50 CCGGGT 6 2.77 3.27 3.29 3.01 1.70 - - GGGCGC 5 4.02 3.10 2.93 3.66 1.68 CAGCAG 16 4.35 3.45 3.12 1.99 1.69 - - CAGCGG 12 5.61 4.95 4.63 2.28 1.55 CCGCTG 12 4.99 4.60 3.51 2.18 1.36 - - CCCCCC 7 3.71 3.88 3.15 2.10 1.84 CCTGCC 10 3.75 3.15 3.22 1.94 1.55 - - GTGCCA 14 3.76 3.35 3.06 2.23 1.37 GGTGGT 10 3.26 3.73 3.14 2.19 1.53 - - TUP GTGGGG 9 6.67 5.23 3.76 3.27 1.47 AGGGGC 10 6.77 3.90 3.65 2.64 1.64 KANWWWWATSYGGGGW Mig1p ACGGGC 7 4.49 3.57 3.23 2.62 1.97 (Zn ﬁnger) TGGGGT 9 4.10 3.21 3.14 3.39 1.37 GGGGTA 10 4.39 4.29 3.52 2.89 1.58 TACCCC 16 5.67 5.73 4.22 2.52 1.32 ACCCCG 11 6.34 5.24 5.15 3.27 1.39 Complement of Mig1p CCCCGC 8 7.37 4.68 3.60 2.31 1.33 KANWWWWATSYGGGGW (Zn ﬁnger) CCCCAC 12 6.55 5.05 3.49 2.27 1.36 AGGAGG 11 4.66 3.79 3.12 1.70 1.44 - - GGTGGT 9 4.10 4.27 3.41 2.21 1.31 CTCGAG 8 3.15 4.00 4.42 2.22 1.17 - - TCGAGG 9 3.75 3.88 4.38 2.15 1.73 GCGGAG 7 4.74 4.07 3.20 1.84 1.35 - - CGGAGA 10 4.02 4.17 3.05 1.97 1.69 CTGCTA 10 2.42 3.23 4.28 3.21 1.90 GTGCCT 17 6.95 6.81 4.86 3.34 1.71 - - TGCCAC 10 3.74 3.38 3.02 1.74 1.51 GCGCCG 4 4.10 3.12 3.13 3.23 2.67 GCAACG 9 3.43 2.88 3.12 3.13 1.37 - - GCACGG 8 5.13 4.66 3.12 2.58 1.66 CAGTGG 8 3.33 3.48 3.01 1.90 1.61 - - CGCGAT 7 2.76 3.48 4.12 3.68 2.083 - - 13
arXiv:physics/0009003v1 [physics.bio-ph] 1 Sep 2000Noise Delays Bifurcation in a Positively Coupled Neural Cir cuit Boris Gutkin1, Tim Hely2, Juergen Jost2,3 1. Unite de Neurosciences Integratives et Computationalle s, INAF, CNRS, Avenue de la Terrasse, 91198, Gif-sur-Yvette, Cedex, France. Email: Bo ris.Gutkin@iaf.cnrs-gif.fr 2. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501. 3. Max Pl anck Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany . Abstract We report a noise induced delay of bifurcation in a simple pul se-coupled neural circuit. We study the behavior of two neural oscillat ors, each individ- ually governed by saddle-node dynamics, with reciprocal ex citatory synaptic connections. In the deterministic circuit, the synaptic cu rrent amplitude acts as a control parameter to move the circuit from a mono-stable regime through a bifurcation into a bistable regime. In this regime stable s ustained anti-phase oscillations in both neurons coexist with a stable rest stat e. We introduce a small amount of random current into both neurons to model pos sible ran- domly arriving synaptic inputs. We ﬁnd that such random nois e delays the onset of bistability, even though in decoupled neurons nois e tends to advance bifurcations and the circuit has only excitatory coupling. We show that the delay is dependent on the level of noise and suggest that a cur ious stochastic “anti-resonance” is present. PACS numbers: 87.10.+e,87.18.Bb,87.18.Sn,87.19.La 1I. INTRODUCTION The eﬀects of random currents on the ﬁring behavior of real an d model neurons have received a considerable amount of attention in neurobiolog y and physics literature [1–8]. Several experimental results indicate that in vivo neural s pike trains seem to be excessively noisy, with interspike interval distribution showing 1/f s pectra [9]. However, other in vitro experiments have shown that noisy stimuli can produce highl y reliable ﬁring with the neu- ron locking onto large range variations of the noise [2,10]. A number of theoretical studies have attempted to reconcile such seemingly disparate resul ts by studying the dynamics of neural networks with additive noise, showing that high vari ance ﬁring behavior can arise in networks of threshold elements [12]. At the same time additi ve noise in oscillating networks of more realistic neurons destabilizes synchronous and pha se locked behavior, producing complicated spatiotemporal patterns [13]. These simulati on results have appeared in the context of a body of literature that has delved into the eﬀect s of noise on the response of ex- citable and oscillatory non-linear dynamical systems. In p articular a number of investigators have considered what happens to single neurons and circuits of neurons when noise perturbs periodically modulated input signals. Experimental work h as identiﬁed noise induced signal ampliﬁcation and resonance in a number of preparations e.g. [14]. Theoretical analyses have successfully explained such ﬁndings employing the languag e of stochastic resonance devel- oped originally for general multi-stable dynamical system s. There the enhancement of the subthreshold stimuli and encoding of stimulus structure ha d a non-linear relationship with the noise amplitude, resulting in a signal-to-noise ratio r elationship with a pronounced peak. Noise eﬀects have also been studied in the context of indigen ous oscillations in neural mod- els, focusing on the so-called “autonomous stochastic reso nance” [5]. For example, a recent report by Lee shows noise induced coherence resonance in a Ho dgkin-Huxley model, with pre-cursors of the sub-critical Hopf bifurcation revealed by the action of random currents [6]. In this sense noise “advanced” the bifurcation. Simila r eﬀects have also been found in a generic saddle-node driven oscillator where noise advanc es the onset of oscillations and 2upregulates the mean frequency [15,16] Although pulse coupled or synaptically coupled neural netw orks have received much recent attention with regard to their dynamics [17,18] and c omputational power [19], we believe that this Letter is the ﬁrst attempt to look in detail at the eﬀects of noise on the onset of synaptically sustained ﬁring in such networks. Tha t is, circuits of intrinsically quiescent neurons where activity occurs purely due to the re current synaptic interactions. To our knowledge, almost all eﬀorts to study the interplay of noise and neural oscillators report noise induced increase in ﬁring and advancement of bifurcations, e.g. [15]. In this light our ﬁnding is rather intriguing since we observe a nois e induced delay of bifurcation in a purely positively coupled circuit of neural oscillator s. We also observe a phenomenon that may be termed “stochastic anti-resonance”, since the d elay of bifurcation depends non- linearly on the noise level. Our analysis of this system lead s us to conclude that the relative width of the attractor basins for the quiescent and persiste nt ﬁring states is the key factor in determining whether stochastic resonance has a delaying , neutral, or advancing eﬀect on the bifurcation. Below we summarize the dynamics of the spiking neuron used in this circuit (the θ- neuron), and analyze the case of two coupled cells in the regi mes of weak and strong excita- tory coupling in both noise free and noisy simulations. Sinc e we believe that the phenomenon we observe is generic for circuits of recurrently coupled sp iking neurons, ﬁrst we describe the stochastic anti-resonance phenomenon observed in this simple circuit. Figure 1A, upper trace shows the ﬁring patterns of two cells w hose spiking behavior results from mutual excitatory synapses. The cells are init ially quiescent (they are not intrinsically spiking) and their activity results from an i nitial external input to one cell. Activity can be terminated by small levels of noise (Figure 1 A, middle trace), whilst increased noise levels cause intermittent ﬁring (Figure 1A, lower tra ce). Figure 1B plots the probability (M1) of observing ﬁring in the last 200 msecs of a 2000 msec run ove r an ensemble of 1000 sample paths. The x-axis plots the strength of the synaptic c oupling ( gs). In the noise free circuit, g∗ sis the critical value of coupling above which sustained ﬁrin g occurs (i.e. M1=0 for 3gs< g∗ s,M1=1 for gs≥g∗ s). Since the synaptically sustained ﬁring apears with a non- zero frequency, we suspect that the bifurcation here is of a sub-c ritical Hopf type. At small noise levels (Figure 1B, traces 1,2), increasing the noise amplit ude progressively shifts the curves ofM1to the right with respect to the noise-free case. This behavi or is surprising as addition of small amounts of noise for a single autonomously spiking θ-neuron induces the opposite eﬀect - noise advanced bifurcation (see [21]). The eﬀect has been described in a generic saddle-node oscillator in [15]. Above a critical noise valu e, the onset of sustained ﬁring is advanced back to the left (Figure 1B, traces 3 and higher). Th us the bifurcation is delayed for low noise amplitudes and advanced with higher noise. Fig ure 1C shows that there is a non-linear relationship between the amount of injected noi se and the ﬁring probability. Here we plot the value of gs=g2/3 sat which continuous sustained ﬁring is observed in 2/3 of the sample paths, the same points are marked on the probability p lots in Figure 1B. Adding small amounts of noise moves the probability curves to the ri ght. This can be viewed as a probabilistic signature of a delay of the bifurcation. As t he noise amplitude grows, the bifurcation is delayed further, and the test point g2/3 soccurs at higher gsvalues. As the noise is increased further, noise ﬂuctuations are strong enough t o induce intermittent ﬁring. Both the probability curves and the location of the test point the n move back to the left towards the noise-free value. If we consider the sustained ﬁring as s ignal (perturbed by noise), this resembles stochastic resonance, however here the net eﬀect of noise is “negative”. It should be noted that this eﬀect is not restricted to the dyn amics of the θ-neuron. All aspects of noise induced delay of bifurcation seen above als o occur in a circuit where each of the cells is modeled with a more complicated conductance b ased, Hodgkin-Huxley model for a pyramidal neuron [11] (simulations not shown). This is not surprising, since this model can be readily reduced to the θ-neuron which we now describe. 4II. THE θ-NEURON Theθ-neuron model developed by Ermentrout and Gutkin [20,21] is derived from the observation that wide class of neuronal models of cortical n eurons, based on the electro- physiological model of Hodgkin and Huxley show a saddle-nod e type bifurcation at a critical parameter value. This parameter determines the dynamical b ehavior of the solutions of the corresponding system of ordinary diﬀerential equations. G eneral dynamical systems theory tells us that the qualitative behavior in some neighborhood of the bifurcation point (which may be quite large as it extends up to the next bifurcation or o ther dynamic transition) is governed by the reduction of the system to the center manifol d. In the present case of the saddle-node bifurcation which is the simplest bifurcation type, this leads to the following diﬀerential equation dx dt=λ+x2. (1) Here, the bifurcation parameter λis considered as the input to the neuron while xrecords its activity. Obviously, a solution to this equation tends t o inﬁnity in ﬁnite time. This is considered as a spiking event, and the initial values are the n reset to −∞. In order to have a model that does not exhibit such formal singularities, one introduces a phase variable θ that is 2 π-periodic via x= tan(θ 2). (2) θis then a variable with domain the unit circle S1, and a spike now corresponds to a period ofθ. Spikes are no longer represented by transitions through in ﬁnity, but by changes of some discrete topological invariant. The original diﬀerential equation is then transformed into dθ dt= (1−cos(θ)) + (1 + cos( θ))λ. (3) Due to the nonlinearity of the transformation from xtoθ, the input λis no longer additive. In fact, it is easy to show that (1 + cosθ) is the phase resetting function for the model [20]. As before, the bifurcation occurs at λ= 0. There, we have precisely one rest point, namely 5θ= 0 which is degenerate. In any case, the sensitivity to the in putλis highest at θ= 0 and lowest at θ=πwhich according to the derivation of our equation is conside red as the spike point. When λis positive, the equation does not have any rest point. In thi s case, θ continues to increase all the time, and the neuron is perpetu ally ﬁring. When λis negative, however, there are two rest points, a stable one denoted by θrand an unstable one θt> θr. Ifθis larger than θtit increases until it completes a period and comes to rest at θr+ 2π which is identiﬁed with θras we are working on the unit circle S1. Thus, if the phase is above the threshold value θt, a spike occurs and the neuron returns to rest. So far, we have tacitly assumed that the input λis constant. We now consider the situation where the input can be decomposed as λ=β+ση, (4) where βis a constant term, the so-called bias, while ηis (white) noise and σits intensity. In this case, suﬃciently strong noise can occasionally push the phase θbeyond the thresh- old value θtcausing intermittent ﬁring (Figure 1C). Equation 3 now beco mes a canonical stochastic saddle-node oscillator which has been studied i n Rappel & Wooten and Gutkin & Ermentrout [21]. III. COUPLED NEURONS We now consider the situation where we have two neurons (dist inguished by subscripts i= 1,2). The dynamics then takes place on the product of two circle s, i.e. on a two- dimensional torus T, represented by the square [ −π, π]×[−π, π] in the plane, with periodic boundary identiﬁcations. We ﬁrst consider the simple case o f two uncoupled, noise-free neurons ( σ1=σ2= 0) with the same bias β. Their dynamics are independent. In the phase plot shown in Figure 2(i) the diagonal is always an inva riant curve, corresponding to synchronized activity of the two neurons. If β >0, both neurons continue to ﬁre, although their phase diﬀerence, if not 0 initially, is not constant, d ue to the nonlinearity of the 6diﬀerential equation governing it. If β= 0, (0 ,0) is a degenerate rest point (Figure 2(ii)). The two curves θ1=θ2= 0 are homoclinic orbits and all ﬂow lines eventually termin ate at this ﬁxed point. One or both neurons will spike before return ing to rest if their initial phase is between 0 and π. Ifβ <0 (Figure 2(iii)), we have four ﬁxed points - the attractor ( θ1=θ2=θr), the repeller ( θ1=θ2=θt), and the two saddles where one of the neurons has its phase at θr (rest) and the other one at θt(threshold). Some special heteroclinic orbits are given by the straight lines where one of the two neurons stays at θtwhile the other one moves from the threshold to the rest value, spiking if its initial phase was above threshold. All other ﬂow lines terminate at the attractor. We now add an interaction t ermsigsto the input of neuron i.siis considered as the synaptic input from neuron jto neuron i(i/negationslash=j) and gsis the synaptic intensity. (One could also study the case of a singl e neuron ifor which sirepresents synaptic self-coupling, but here we are interested in the ca se of two coupled neurons). A precise equation for sican be derived from electrophysiological models, however f or our qualitative study we only need the characteristic features that it stays bounded between 0 and 1. Typically, it is peaked near the spike of neuron j, i.e. where θj=π. With this interaction term, the equation for neuron ithen becomes dθi dt= (1−cos(θi)) + (1 + cos( θi))(β+gssi+ση). (5) Since sirepresents the input that neuron ireceives from neuron j,sishould essentially be considered as a function of the phase θjofj. Once more, we ﬁrst consider the situation without noise, i.e. σ= 0 (although our ﬁnal aim is to understand the eﬀect of noise on the dynamic behavior of the coupled neurons). We also assu me that we are in the excitable region, i.e. β <0.gsis assumed to be positive (excitatory coupling), and so the coupling counteracts the eﬀect of the bias to a certain exten t, a crucial diﬀerence being, however, that the synaptic input to each neuron is time depen dent, in contrast to the constant bias. If gsis suﬃciently small, the qualitative situation does not cha nge compared to the case without coupling, i.e. gs= 0. We still have a heteroclinic orbit from the saddle 7(θ1=θt, θ2=θr) to the attractor ( θr, θr), although θ2does not stay constant anymore along that orbit, but increases ﬁrst a little due to the input from n euron 1 before it descends again to the rest value. (Figure 2(iv)). (Of course, we also get suc h an orbit with the roles of the two neurons reversed; in fact, the dynamical picture is a lways invariant under reﬂection across the diagonal, i.e.under exchanging the two neurons. ) Ifgsreaches some critical value g∗ s, however, the heteroclinic orbit starting at ( θt, θr) does not terminate anymore at the attractor, and the value of the phase of neuron 2 is increased so much by the synaptic interaction that it reaches the other saddle ( θr, θt) (Figure 2v). Besides two heteroclinic orbits that go from the repeller to the two saddles as before, all other orbits still terminate at the attractor ( θr, θr), for gs=g∗ s. Ifgsis increased beyond g∗ s, however, the heteroclinic orbit between the two saddles mutates into a stable attracto r (Figure 2(vi)). It corresponds to sustained asynchronous ﬁring of the two neurons. In fact, if the phase diﬀerence between the two neurons is too small, the dynamics converges towards the double rest point (except in some region in the vicinity of the node), and both neurons s top ﬁring. This is caused by the fact that when the two neurons are close to synchrony, nei ther cell is sensitive enough to its synaptic input to maintain ﬁring (an eﬀective refract ory period). Conversely, if they are out of synchrony, a single spike can induce the second neu ron to ﬁre at a time when the ﬁrst one is close to rest, and sensitive to synaptic input its elf. If gsis only slightly above the critical value, the basin of attraction of that limit cyc le will still be relatively small, but as gsis increased further, the basin grows in size until eventual ly it is larger than the basin of attraction of the double rest point. On the basis of t he preceding analysis, it is now straightforward to predict the eﬀect of noise. If gsis only slightly above the critical value g∗ s, a small amount of noise is more likely to kick the dynamics ou t of the narrow basin of attraction of the asynchronous limit cycle and into the large basin of the double rest point than vice versa. In eﬀect, a small noise level incr eases the critical parameter value required for the qualitative transition to sustained async hronous ﬁring. A larger amount of noise, however, has the potential to move the dynamics from t he rest point into the basin of attraction of the asynchronous limit cycle. Once in that b asin, the neurons will ﬁre. 8Thus, for large noise in that regime, one will observe that th e neurons will ﬁre, perhaps with some intermissions spent near the double rest point. So , a larger value of noise will cause intermittent periods of sustained ﬁring of the two neu rons even at somewhat smaller values of gs. In eﬀect, it decreases the value of the critical parameter. Thus, we observe a genuinely nonlinear eﬀect of the noise level σ(Figure 1E). For values of the coupling gs that are substantially larger than the critical value g∗ s, even small amounts of noise have a good chance of perturbing the dynamics out of the attractin g vicinity of the double rest point into the attracting region of the asynchronous limit c ycle. This will further enhance the sustained asynchronous ﬁring pattern of the two neurons . IV. CONCLUSIONS In this work we report a new and unusual eﬀect of noise in a simp le neural circuit. When the sustained oscillations in the circuit are induced by rec urrent excitatory coupling, small noise levels can exert a strong inﬂuence on the circuit dynam ics, often abolishing the ﬁring. The probability of observing sustained ﬁring has been used t o characterize the transition from quiescent to oscillatory behavior. Figure 1B clearly s hows that in this system, noise delays this transition. Noise induced delay of bifurcation can therefore occur in a completely positively coupled circuit. The same noise has the exact opp osite eﬀect of advancing the bifurcation when it is applied to a single autonomously ﬁrin g neuron. The paradoxical eﬀect of noise in this circuit can be understood by considering the structure of its phase plane - and in particular the width of the attractor basins for the susta ined antiphase oscillations. When the width of the attractor basin is small, small levels of noi se can perturb the system into the larger basin of the stable quiescent state. However, transi tions in the opposite direction from the rest-state to a sustained ﬁring state can only occur when noise ﬂuctuations reach a critical value. Above this value, transitions into the ﬁring state be gin to counteract transitions into the quiescent state. Alternatively, as the coupling streng th increases, the basin of attraction for the sustained ﬁring solution grows at the expense of the q uiescent state. The negative 9(bifurcation-delaying) eﬀect of the noise is then eliminat ed. In this system low levels of noise eﬀectively act as a switch to turn oﬀ otherwise continuous ﬁr ing behavior. Alternatively, low levels of noise ensure that sustained ﬁring can only take place above a critical coupling threshold. In this way, small amounts of noise may in fact hel p to reduce overall noise levels by eliminating the formation of spurious attractors. It has yet to be determined whether this eﬀective noise-induced control mechanism can be obser ved in large ensembles of coupled neurons. Funding was provided by National Science Foundation Bioinf ormatics Postdoctoral Fel- lowship (B.S.G.) and the Santa Fe Institute (T.H. and J.J.) T he authors thank Cosma Shalizi for helpful discussions. 10REFERENCES [1] J. P. Segundo, O.D. Martinez, K. Pakdaman, M. Stiber, and F. Vibert, J. Noise in sensory and synaptic coding - a survey of its history and a sum mary of its conclusions. Biophysical Journal , 66(2), 1994. [2] Z.F. Mainen and T.J. Sejnowski. Reliability of spike tim ing in neocortical neurons. Science , 268(5216):1503–1506, 1995. [3] J. J. Collins, C.C. Chow, and P. Grigg. Noise-mediated en hancements and decrements in human tactile sensation. Physical Review E , 56(1):923–926, 1997. [4] D.R. Chialvo, A. Longtin, and J. MullerGerking. Stochas tic resonance in models of neuronal ensembles. Physical Review E. , 55(2):1798–1808, 1997. [5] A. Longtin. Autonomous stochastic resonance in burstin g neurons. Physical Review E. , 55(1):868–786, 1997. [6] S. Lee, A. Neiman, and S. Kim. Coherence resonance in a Hod gkin-Huxley neuron. Physical Review E , 57(3):3292–3297, 1998. [7] R Rodriguez and H.C. Tuckwell. Noisy spiking neurons and networks: useful approxi- mations for ﬁring probabilities and global behavior. Biosystems , 48(1-3):187–194, 1998. [8] D.J. Mar, C.C. Chow, W. Gerstner, R.W. Adams, and J.J. Col lins. Noise shaping in populations of coupled model neurons. Proc.Natl.Acad.Sci, USA , 96(18):10450–10455, 1999. [9] M. Usher, M. Stemmler, and Z. Olami. Dynamic pattern-for mation leads to 1/f noise in neural populations. Physical Review Letters , 74(2):326–329, 1995. [10] D.S. Reich, J.D. Victor, B.W. Knight, T. Ozaki, and E. Ka plan. Response variability and timing precision of neuronal spike trains in vivo. Journal of Neurophysiology , 77:2836– 2841, 1997. 11[11] R. Traub, M.A. Whittington, I.M. Standford, J.G.R. Jeﬀ reys. A mechanism for gener- ation of long-range oscillations in the cortex Nature , 282:621–624, 1996. [12] C. van Vreeswijk and H. Sompolinksy. Chaotic balanced s tate in a model of cortical circuits. Neural Computation , 10(6):1321–1371, 1998. [13] D. Golomb and Y. Amitai. Propagating neuronal discharg es in neocortical slices: Com- putational and experimental study. Journal of Neurophysiology , 78(3):1199–1211, 1997. [14] J. J. Collins, T.T. Imhoﬀ, and P. Grigg. Noise-enhanced tactile sensation. Nature , 383(6603):770, 1996. [15] W.J. Rappel and S.H. Strogatz. Stochastic resonance in an autonomous system with a nonuniform limit- cycle. Physical Review E , 50(4):3249–3250, 1994. [16] A. S. Pikovsky and J. Kurths. Coherence resonance in a no ise-driven excitable system. Physical Review Letters , 78(5):775–778, 1997. [17] W. Maass. Networks of spiking neurons: The third genera tion of neural network models. Neural Networks , 10(9):1659–1671, 1997. [18] J.K. Lin, K. Pawelzik, U. Ernst, and T.J. Sejnowski. Irr egular synchronous activity in stochastically-coupled networks of integrate-and-ﬁre ne urons. Network-Computation in Neural Systems , 9(3):333–344, 1998. [19] W. Maass. Bounds for the computational power and learni ng complexity of analog neural nets. SIAM Journal on Computing , 26(3):708–732, 1997. [20] G.B. Ermentrout. Type i membranes, phase resetting cur ves, and synchrony. Neural Computation , 8(5):979–1001, 1996. [21] B.S. Gutkin and B. Ermentrout. Dynamics of membrane exc itability determine inter- spike interval variability: A link between spike generatio n mechanisms and cortical spike train statistics. Neural Computation , 10(5):1047–1065, 1998. 12FIGURES 0.20.40.60.811.21.41.61.8 60 80 100 120 140 160 180 200 0.20.40.60.811.21.41.61.8 60 80 100 120 140 160 180 200 0.20.40.60.811.21.41.61.8 100 150 200 250 300 350 400 450 500M1 8 9 10 11 12 gs0.140.120.10.080.060.040.02s=0gs=8.625 01 12 3 450 7.588.599.510gs 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14sData #2gs2/3 13FIG. 1. Asynchronous synaptically sustained oscillations in a positively coupled 2-cell circuit. A. Upper trace: the sustained ﬁring in the noise free circuit . Middle trace: the sustained ﬁring can be terminated by the action of small amplitude noise. Low er trace: larger amplitude noise induces an intermittent ﬁring pattern. Here the noise injec ted into the two neurons is completely correlated, but the results are qualitatively identical fo r uncorrelated noises. B. Increasing noise delays sustained ﬁring for low noise levels (traces 1,2) and advances ﬁring for higher noise levels (traces 3,4,5), the horizontal dashed line and the numbers m ark the test points g2/3 s. C. Addition of noise has a non-linear eﬀect on sustained ﬁring in this cou pled circuit. Here we plot the location of the test points g2/3 sSee text for further details. 14/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /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 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1cell2cell1+ +t q rq1q2 -ppp -p qrqt qrqtqrqt qtqrqr qtqt qq t tq0 0spikeb(i) >0, g =0 b b 0qr 0b (vi) <0, g >gb eventsynaptic(ii) =0, g =0 s s (iii) <0, g =0 s (iv) <0, 0<g <gb ss(v) <0, g =gssssqtqr FIG. 2. Diﬀerent states of the network for various values of t he intrinsic excitability of the cells, β, and the coupling strength, gs. Axes plot the phase ( θ1,θ2) of each cell. See text for full details. 15
arXiv:physics/0009004v1 [physics.atom-ph] 1 Sep 2000The beryllium atom and beryllium positive ion in strong magn etic ﬁelds M. V. Ivanov †and P. Schmelcher Theoretische Chemie, Physikalisch–Chemisches Institut, Universit¨ at Heidelberg, INF 229, D-69120 Heidelberg, Fed eral Republic of Germany †Permanent address: Institute of Precambrian Geology and Ge ochronology, Russian Academy of Sciences, Nab. Makarova 2, St. Petersburg 199034, Russia (February 9, 2008) The ground and a few excited states of the beryllium atom in external uniform magnetic ﬁelds are calculated by means of our 2D mesh Hartree-Fock method for ﬁeld strengths rang- ing from zero up to 2 .35·109T. With changing ﬁeld strength the ground state of the Be atom undergoes three transitions involving four diﬀerent electronic conﬁgurations which be long to three groups with diﬀerent spin projections Sz= 0,−1,−2. For weak ﬁelds the ground state conﬁguration arises from the 1s22s2,Sz= 0 conﬁguration. With increasing ﬁeld strength the ground state evolves into the two Sz=−1 conﬁgurations 1s22s2p−1and 1 s22p−13d−2, followed by the fully spin po- larised Sz=−2 conﬁguration 1 s2p−13d−24f−3. The latter conﬁguration forms the ground state of the beryllium atom in the high ﬁeld regime γ >4.567. The analogous calculations for the Be+ion provide the sequence of the three following ground state conﬁgurations: 1 s22sand 1s22p−1(Sz=−1/2) and 1 s2p−13d−2(Sz=−3/2). I. INTRODUCTION The behaviour and properties of atoms in strong mag- netic ﬁelds is a subject of increasing interest. This is motivated by the astrophysical discovery of strong ﬁelds on white dwarfs and neutron stars [1–3]. On the other hand the competition of the diamagnetic and Coulomb interaction causes a rich variety of complex properties which are, of course, also of interest on their own. For a long time the investigations in the literature fo- cused on the hydrogen atom (for a list of references see, for example, [4–7]). As a result of the corresponding in- vestigations the absorption features of certain magnetic white dwarfs could be understood in detail and a mod- elling of their atmospheres was possible (see ref. [8] for a review up to 1994 and [9] for more recent references). De- tailed spectroscopic calculations were carried out recent lyfor the helium atom in strong magnetic ﬁelds [10]. These calculations allow to identify spectra of other, namely helium-rich objects, including the prominent white dwarf GD229 [11]. Recently a number of new magnetic white dwarfs have been found whose spectra are still unex- plained (see, e.g., Reimers et al [12] in the course of the Hamburg ESO survey). Investigations on the electronic structure in the pres- ence of a magnetic ﬁeld appear to be quite complicated due to the intricate geometry of this quantum prob- lem. For the hydrogen atom the impact of the com- peting Coulomb and diamagnetic interaction is particu- larly evident and pronounced in the intermediate regime for which the magnetic and Coulomb forces are compa- rable. For diﬀerent electronic degrees of excitation of the atom the intermediate regime is met for diﬀerent absolute values of the ﬁeld strength. For the ground state this regime corresponds to ﬁeld strengths around γ= 1 (for the magnetic ﬁeld strength as well as for other physical values we use atomic units and, in partic- ular,γ=B/B 0,B0corresponds to the magnetic ﬁeld strengthB0= ¯hc/ea2 0= 2.3505·105T). Both early [13,14] and more recent works [4,15] on the hydrogen atom have used diﬀerent approaches for relatively weak ﬁelds (the Coulomb force prevails over the magnetic force) and for very strong ﬁelds (the Coulomb force can be considered as weak in comparison with the magnetic forces which is the so-called adiabatic regime). A powerful method to obtain comprehensive results on low-lying energy levels of the hydrogen atom in particular in the intermediate regime is provided by mesh methods [5]. For atoms with several electrons there are two decisive factors which en- rich the possible changes in the electronic structure with varying ﬁeld strength compared to the one-electron sys- tem. First we have a third competing interaction which is the electron-electron repulsion and second the diﬀer- 1ent electrons feel very diﬀerent Coulomb forces, i.e. pos- sess diﬀerent one particle energies, and consequently the regime of the intermediate ﬁeld strengths appears to be the sum of the intermediate regimes for the separate elec- trons. Opposite to the hydrogen atom the wavefunctions of the multi-electron atoms change their symmetries with increasing ﬁeld strength. It is well known that the sin- glet zero-ﬁeld ground state of the helium atom (1 s2in the Hartree-Fock language) is replaced in the high-ﬁeld regime by the triplet fully spin polarised conﬁguration 1s2p−1. For atoms with more than two electrons the evolution of the ground state within the whole range of ﬁeld strengths 0 ≤γ <+∞includes multiple intermedi- ate conﬁgurations besides the zero-ﬁeld ground state and the ground state corresponding to the high ﬁeld limit. In view of the above there is a need for further quantum mechanical investigations and data on atoms with more than two electrons in order to understand their electronic structure in strong magnetic ﬁelds. Our calculations al- lowed us to obtain the ﬁrst conclusive results on the series of ground state conﬁgurations for the Li [16] and C [17] atoms. These results are substantially diﬀerent from pre- viously published ones [18]. The ground state electronic conﬁgurations of the beryllium atom for 0 ≤γ <+∞ were not investigated so far. A previous work on the beryllium atom [19] focused on problems associated with the symmetries of the Hartree-Fock wavefunction of the low-ﬁeld ground state 1 s22s2of this atom. For strong ﬁelds the 1 s22s2state represents a highly excited state and the electronic ground state conﬁguration of Be is, so far, not investigated. In the current paper we present results of our fully nu- merical 2D Hartree-Fock mesh calculations of the beryl- lium atom and Be+ion in magnetic ﬁelds and obtain for the ﬁrst time conclusive results on the structure and en- ergy of the ground state conﬁgurations of these systems for arbitrary ﬁeld strengths. II. METHOD The computational method applied in the current work coincides with the method described in our works [5,19–22] and applied afterwards in [16,17,23,24]. Wesolve the electronic Schr¨ odinger equation for the beryl- lium atom in a magnetic ﬁeld under the assumption of an inﬁnitely heavy nucleus in the (unrestricted) Hartree- Fock (HF) approximation. The solution is established in the cylindrical coordinate system ( ρ,φ,z ) with thez-axis oriented along the magnetic ﬁeld. We prescribe to each electron a deﬁnite value of the magnetic quantum num- bermµ. Each one-electron wave function Ψ µdepends on the variables φand (ρ,z) Ψµ(ρ,φ,z ) = (2π)−1/2e−imµφψµ(z,ρ) (1) whereµindicates the numbering of the electrons. The re- sulting partial diﬀerential equations for ψµ(z,ρ) and the formulae for the Coulomb and exchange potentials have been presented in ref. [21]. These equations as well as the Poisson equations for inter-electronic Coulomb and exchange potentials are solved by means of the fully nu- merical mesh method described in refs. [5,21]. The ﬁnite- diﬀerence solution of the Poisson equations on sets of nodes coinciding with those of the Hartree-Fock equa- tions turns out to be possible due to a special form of uniform meshes used in the present calculations and in refs. [16,17,19]. Details and discussion on these meshes are presented in ref. [25]. Our mesh approach is ﬂexible enough to yield precise results for arbitrary ﬁeld strengths. Some minor decrease of the precision appears for electronic conﬁgurations with big diﬀerences in the spatial distribution of the electroni c density for diﬀerent electrons. This results in big diﬀer- ences with respect to the spatial extension of the density distribution for diﬀerent electrons. This situation is mor e typical for the electronic conﬁgurations which do not rep- resent the ground state at the corresponding ﬁelds (e.g. 1s22s2at very strong ﬁelds or 1 s2p−13d−24f−3in the weak ﬁeld regime). The precision of our results depends, of course, on the number of mesh nodes and can be al- ways improved in calculations with denser meshes. Most of the present calculations are carried out on sequences of meshes with the maximal number of nodes being 80 ×80. Along with the numerical solution of the Schr¨ odinger equation the key element for solving the problem of the ground state electronic conﬁgurations is a proper choice of the conﬁgurations, which could potentially be the ground state ones. An example of solving this problem is presented in [17]. In that work we have developed a 2strategy which enables one to reduce the set of possi- ble ground state conﬁgurations which are then subject to a following numerical investigation. This removes the risk of missing some ground state conﬁgurations due to the limited possibilities of performing numerical investi - gations. With increasing number of electrons the number of conﬁgurations which cannot a priori be excluded from becoming the ground state increase rapidly. A compre- hensive numerical investigation of all these conﬁguration s is, in general, not feasible. The above-mentioned strat- egy to exclude certain conﬁgurations is therefore highly desirable. It is based on a combination of qualitative theoretical arguments and numerical calculations of the energies of electronic conﬁgurations. As a ﬁrst step the set of electronic conﬁgurations has to be separated into several groups according to their spin projections Sz. The following considerations have to be carried out in each subset separately, and are certainly more transpar- ent by starting with the limit of inﬁnite strong ﬁelds and analysing the electronic conﬁgurations with decreasing ﬁeld strengths. The qualitative theoretical consideratio ns mentioned above are based on the geometry of the spa- tial part of the wavefunction and enable one to determine the ground state for the high-ﬁeld limit as well as several candidates for the ground state conﬁguration with de- creasing ﬁeld strength. The numerical calculations then enable us to decide which of these candidates becomes the actual ﬁrst intermediate ground state and yields the transition ﬁeld strength. The knowledge of the ﬁrst inter- mediate ground state allows us to repeat the qualitative considerations for the second intermediate ground state to obtain a list of candidates which is then investigated by means of numerical calculations. Repeating this pro- cedure one can determine the full sequence of the ground state conﬁgurations for each subset Szand ﬁnally the sequence of ground state conﬁgurations for the physical system. III. GROUND STATE ELECTRONIC CONFIGURATIONS FOR γ= 0AND γ→ ∞ In this section we provide some helpful qualitative con- siderations on the problem of the atomic multi-electron ground states particularly in the limit of strong magneticﬁelds. For the case γ= 0 the ground state conﬁguration of the beryllium atom can be characterised as 1 s22s2. This no- tation has a literal meaning when considering the atom in the framework of the restricted Hartree-Fock approach. The latter is an approximation of limited quality in de- scribing the beryllium atom as it was shown in many fully correlated calculations both for the ﬁeld-free Be atom [26,27] and for its polarizabilities in electric ﬁelds [28, 29]. It was pointed out in these works that the Be atom is a strongly correlated system and that the HF ground state wavefunction (i. e. the spherically symmetric 1 s22s2) is not a very accurate zeroth-order wavefunction, especially for calculations of electric polarizabilities. This is due to a signiﬁcant contribution of the 1 s22p2conﬁguration to the ground state wave function. The latter conﬁguration is evidently a non-spherical one. This fact is in agree- ment with results of ref. [19] where the fully numerical 2D unrestricted Hartree-Fock approach provides the 2 s2 shell stretched along the zaxis even for γ= 0. In terms of spherical functions it is natural to describe this geom- etry of the 2 s2shell as a mixture of 2 sand 2p0functions. We remark that the s,p,d...orbital notation both for γ= 0 andγ/negationslash= 0 is based on the behaviour of the wave functions in the vicinity of the origin and on the topol- ogy of the nodal surfaces, but does not imply any detailed geometry or certain values of the orbital moment l. It is evident that the ﬁeld-free ground state of the beryllium atom remains the ground state only for rela- tively weak ﬁelds. The set of one-electron wave functions constituting the HF ground state for the opposite case of extremely strong magnetic ﬁelds can be determined as follows. The nuclear attraction energies and HF poten- tials (which determine the motion along zaxis) are small for largeγin comparison to the interaction energies with the magnetic ﬁeld (which determines the motion perpen- dicular to the magnetic ﬁeld and is responsible for the Landau zonal structure of the spectrum). Thus, all the one-electron wavefunctions must correspond to the low- est Landau zones, i.e. the magnetic quantum numbers mµare not postive for all the electrons mµ≤0, and the system must be fully spin-polarised, i.e. szµ=−1 2. For the Coulomb central ﬁeld the one-electron levels form (as B→ ∞) quasi 1D Coulomb series with the binding en- 3ergyǫB=1 2n2zfornz>0 andǫB→ ∞ fornz= 0, wherenzis the number of nodal surfaces of the wave function with respect to the zaxis. The binding energy of a separate electron has the form ǫB= (m+\|m\|+ 2sz+ 1)γ/2−ǫ (2) whereǫis the energy of the electron. When considering the case γ→ ∞ it is evident, that the wave functions with nz= 0 have to be chosen for the ground state conﬁguration. Furthermore starting with the energetically lowest one particle level the electrons occupy according to the above arguments orbitals with increasing absolute value of the magnetic quantum num- bermµ. Consequently the ground state of the beryllium atom must be given by the fully spin-polarised conﬁgu- ration 1s2p−13d−24f−3. In our notation of the electronic conﬁgurations we assume in the following that all paired electrons, like for example the 1 s2part of a conﬁguration, are of course in a spin up and spin down orbital, respec- tively, whereas all unpaired electrons possess a negative projection of the spin onto the magnetic ﬁeld direction. On a qualitative level the conﬁguration 1 s2p−13d−24f−3 is not very diﬀerent from similar electronic conﬁgurations for other atoms (see ref. [24]). This is a manifestation of the simpliﬁcation of the picture of atomic properties in the limitγ→ ∞ where a linear sequence of electronic conﬁgurations replaces the periodic table of elements of the ﬁeld-free case. The problem of the conﬁgurations of the ground state for the intermediate ﬁeld region cannot be solved without doing explicit calculations combined with some qualita- tive considerations in order to extract the relevant con- ﬁgurations. IV. GROUND STATE ELECTRONIC CONFIGURATIONS FOR ARBITRARY FIELD STRENGTHS In order to determine the ground state electronic con- ﬁgurations of the beryllium atom we employ here the strategy introduced in ref. [17] where the carbon atom has been investigated. First of all, we divide the possible ground state conﬁgurations into three groups accordingto their total spin projection Sz: theSz= 0 group (low- ﬁeld ground state conﬁgurations), the intermediate group Sz=−1 and theSz=−2 group (the high-ﬁeld ground state conﬁgurations). This grouping is required for the following qualitative considerations which are based on the geometry of the spatial parts of the one-electron wave functions. In the course of this discussion it is expedi- ent to treat localground states for each Szsubset (i.e. the lowest states with a certain Szvalue) along with the global ground state of the atom as a whole. For each value of the magnetic ﬁeld strength one of these local ground states is the global ground state of the atom. According to the general arguments presented in the previous section we know that the ground state conﬁgu- ration of the beryllium atom in the high ﬁeld limit must be the fully spin-polarised state 1 s2p−13d−24f−3. The question of the ground state conﬁgurations at interme- diate ﬁelds cannot be solved without performing explicit electronic structure calculations. On the other hand, the a priori set of possible intermediate ground state con- ﬁgurations increases enormously with increasing number of electrons and is rather large already for the beryl- lium atom. Some qualitative considerations are there- fore needed in order to exclude certain conﬁgurations as possible ground state conﬁgurations thereby reducing the number of candidates for which explicit calculations have to be performed. As mentioned in the previous section the optimal strategy hereby consists of the repeated pro- cedure of determining neighbouring ground state conﬁg- urations with increasing (or decreasing) magnetic ﬁeld strength using both qualitative arguments as well as the results of the calculations for concrete conﬁgurations. The total energies for the considered states and par- ticularly of those states which become the global ground state of the atom for some regime of the ﬁeld strength are illustrated in ﬁgure 1. In the following paragraphs we describe our sequence of selection procedure and calcu- lations for the candidates of the electronic ground state conﬁgurations. Due to the simplicity of the ground state electronic conﬁgurations of atoms in the limit γ→ ∞ it is natural to start the consideration for γ/negationslash= 0 with the high-ﬁeld ground state and then consider other possible candidates in question for the electronic ground state for Sz=−2 4(see ﬁgure 1) with decreasing ﬁeld strength . The con- sideration of the high-ﬁeld (i.e. the fully spin-polarised ) regime was carried out in ref. [24] and for this case (i.e. Sz=−2 for beryllium) we repeat this consideration in more detail. In particular, we have found in ref. [24] that the beryllium atom, opposite to the carbon and heavier elements has only one fully spin-polarised ground state conﬁguration. All the one electron wave functions of the high-ﬁeld ground state 1 s2p−13d−24f−3possess no nodal surfaces crossing the z-axis and occupy the energetically lowest orbitals with magnetic quantum numbers ranging from m= 0 down to m=−3. The 4f−3orbital possesses the smallest binding energy of all orbitals constituting the high-ﬁeld ground state. Its binding energy decreases rapidly with decreasing ﬁeld strength. Thus, we can expect that the ﬁrst crossover of ground state conﬁg- urations happens due to a change of the 4 f−3orbital into one possessing a higher binding energy at the cor- responding lowered ﬁeld strength. One may think that the ﬁrst transition while decreasing the magnetic ﬁeld strength will involve a transition from an orbital possess- ingnz= 0 to one for nz= 1. The energetically lowest available one particle state with nz= 1 is the 2 p0or- bital. Another possible orbital into which the 4 f−3wave function could evolve is the 2 sstate. For the hydro- gen atom or hydrogen-like ions in a magnetic ﬁeld the 2p0is stronger bound than the 2 sorbital. On the other hand, owing to the electron screening in multi-electron atoms in ﬁeld-free space the 2 sorbital tends to be more tightly bound than the 2 p0orbital. Thus, two states i.e. 1s2p02p−13d−2and 1s2s2p−13d−2are candidates for be- coming the ground state in the Sz=−2 set when we lower the ﬁeld strength coming from the high ﬁeld sit- uation. The numerical calculations show that the ﬁrst crossover of the Sz=−2 subset takes place between the 1s2p−13d−24f−3and 1s2p02p−13d−2conﬁgurations (ﬁg- ure 1). On the other hand, the calculations show that even earlier (i.e. at higher magnetic ﬁeld strengths) the global ground state acquires the total spin Sz=−1 due to a crossover of the energy curve of the 1 s2p−13d−24f−3 conﬁguration with that of the conﬁguration 1 s22p−13d−2 (which is the local ground state for the Sz=−1 subset in the high-ﬁeld limit). For the ﬁelds below this pointγ= 4.567 the ground state electronic conﬁgurations of the beryllium atom belong to the subset Sz=−1. This means that the beryllium atom has only one fully spin po- larised ground state conﬁguration (as mentioned above). The electronic conﬁgurations 1s22p−13d−2and 1s2p−13d−24f−3diﬀer by the replace- ment of the spin down 4 f−3orbital through the spin up 1 s orbital and according to the argumentation presented in the previous section the 1 s22p−13d−2represents the local ground state conﬁguration for the subset Sz=−1 in the limitγ→ ∞. Analogous arguments to that presented in the previous paragraph provide the conclusion, that in the process of decreasing ﬁeld strength the 1 s22p−13d−2 ground state electronic conﬁguration can be replaced ei- ther by the 1 s22s2p−1or by the 1 s22p02p−1conﬁgura- tion. The numerical calculations show, that the curve E1s22s2p−1(γ) intersects the curve E1s22p−13d−2(γ) at a higher magnetic ﬁeld ( γ= 0.957) thanE1s22p02p−1(γ) crossesE1s22p−13d−2(γ). The diﬀerence with respect to the order of the local ground state conﬁgurations in the subsetsSz=−2 andSz=−1 stems from the diﬀer- ence in the magnetic ﬁeld strengths characteristic for the crossovers in these subsets. At moderate ﬁeld strengths (Sz=−1) the inﬂuence of the Coulomb ﬁelds of the nucleus and electrons prevails over the inﬂuence of the magnetic ﬁeld and make the energy of the 2 sorbital lower than that of the 2 p0orbital. On the other hand, at stronger ﬁelds characteristic for the subset Sz=−2 the energies of these orbitals are governed mostly by the magnetic ﬁeld and, in result, the energy of the 2 p0orbital becomes lower than the energy of the 2 sorbital. From our simple qualitative considerations we can conclude, that the conﬁguration 1 s22s2p−1is the local ground state conﬁguration of the subset Sz=−1 for the weak ﬁeld case, i.e. for γ→0. Indeed, when we construct such a conﬁguration, the ﬁrst three electrons go to orbitals 1 sand 2sforming the 1 s22sconﬁguration withSz=−1/2. The fourth electron must then have the same spin as the 2 sorbital electron to obtain the total spin valueSz=−1. Thus, the lowest orbital which it can occupy is the 2 p−1. Therefore, there are two local ground state conﬁgurations in the subset Sz=−1 and they both represent the global ground state for some ranges of the magnetic ﬁeld strengths. 5The necessary considerations for the subset Sz= 0 are quite simple. At γ= 0 and, evidently, for very weak ﬁelds the ground state of the beryllium atom has the con- ﬁguration 1 s22s2. We can expect, that when increasing the magnetic ﬁeld strength, the next lowest state with Sz= 0 will be the 1 s22s2p−1conﬁguration with oppo- site directions of the spins of the 2 sand 2p−1electrons. But both contributions, the Zeeman spin term and the electronic exchange make the energy of this state higher than the energy of the state 1 s22s2p−1with the parallel orientation of the spins of the 2 sand 2p−1electrons (i.e. Sz=−1) considered above. The calculated energies for these states are presented in ﬁgure 1. Thus, the beryl- lium atom has one ground state electronic conﬁguration 1s22s2with the total spin z-projection Sz= 0. This state is the global ground state for the magnetic ﬁeld strengths between γ= 0 andγ= 0.0412. Above this point the ground state conﬁguration is 1 s22s2p−1with Sz=−1. Summarising the results on the ground state conﬁg- urations of the beryllium atom we can state that de- pending on the magnetic ﬁeld strength this atom has four diﬀerent electronic ground state conﬁgurations. For 0≤γ <0.0412 the ground state conﬁguration coin- cides with the ﬁeld-free ground state conﬁguration 1 s22s2 which has zero values for the total magnetic quantum numberMand spin projection Sz. The following are two ground state conﬁgurations with Sz=−1: 1s22s2p−1 (M=−1) for 0.0412< γ < 0.957 and 1s22p−13d−2 (M=−3) for 0.957< γ < 4.567. Forγ > 4.567 the ground state conﬁguration is 1 s2p−13d−24f−3with Sz=−2 andM=−6. The complete results of the investigations of the sequence of the ground state conﬁg- urations of the Be atom are presented in table I which contains the critical values of γat which the crossovers of diﬀerent ground state conﬁgurations take place. The next aim of this section is the corresponding inves- tigation of the ground state conﬁgurations of the ion Be+. The ﬁeld-free ground state of this ion corresponds to the 1s22sconﬁguration ( Sz=−1/2 andM= 0). In the op- posite case γ→ ∞ the ground state is obviously given by the 1s2p−13d−2conﬁguration ( Sz=−3/2 andM=−3). Thus, we need to investigate only two diﬀerent subsets of electronic ground state conﬁgurations: Sz=−1/2 andSz=−3/2. The energy curves which are necessary for this investigation are presented in ﬁgure 2. The subset Sz=−1/2 contains only two possible ground state con- ﬁgurations 1 s22sand 1s22p−1. The latter is the local ground state conﬁguration for this subset in the limit γ→ ∞. The curves E1s22s(γ) andE1s22p−1(γ) intersect atγ= 0.3185 and above this point E1s22p−1<E1s22s. In the subsetSz=−3/2 we have to consider the conﬁgura- tions 1s2p02p−1and 1s2s2p−1along with the high-ﬁeld ground state conﬁguration 1 s2p−13d−2. But the numeri- cal calculations show that the energies of both 1 s2p02p−1 and 1s2s2p−1lie above the energy of the 1 s2p−13d−2con- ﬁguration at the intersection point ( γ= 4.501) between E1s2p−13d−2(γ) andE1s22p−1(γ). Thus, the ion Be+has three diﬀerent electronic ground state conﬁgurations in external magnetic ﬁelds: for 0 ≤γ <0.3185 it is 1 s22s (Sz=−1/2 andM= 0), then for 0 .38483< γ < 4.501 it is 1s22p−1(Sz=−1/2 andM=−1) and for all the values γ >4.501 the ground state conﬁguration is 1s2p−13d−2(Sz=−3/2 andM=−3). These results are summarised in table II. The set of the electronic ground state conﬁgurations for the Be+ion appears to be qualitatively the same as for the lithium atom [16]. The ﬁeld strengths for the corresponding transition points are roughly two times higher for the Be+ion than for the Li atom. V. SELECTED QUANTITATIVE ASPECTS In tables III and IV we present the total energies of the four ground state electronic conﬁgurations of the beryl- lium atom and the three ground state electronic conﬁg- urations of the ion Be+, respectively. These data cover a very broad range of the ﬁeld strengths from γ= 0 and very weak magnetic ﬁelds starting with γ= 0.001 up to extremely strong ﬁelds γ= 10000. The latter value of the ﬁeld strength can be considered as a rough limit of applicability of the non-relativistic quantum equations t o the problem (see below). The corresponding data on the Be+ion can be found in tables II and IV. So far there exist three works which should be men- tioned in the context of the problem of the beryllium atom in strong magnetic ﬁelds. Ref. [19] deals with the 1s22s2state of this atom in ﬁelds 0 ≤γ≤1000 and 6ref. [24] investigates the ground state energies of atoms with nuclear charge number Z≤10 in the high-ﬁeld, i.e. fully spin polarised regime. Both these works contain calculations carried out by the method used in the cur- rent work and do not represent a basis for comparison. The comparison of our results with an adiabatic Hartree- Fock calculation of atoms with Z≤10 [30] is presented in [24] and we can brieﬂy summarise this comparison for two values of the magnetic ﬁeld strengths: for B12= 0.1 (i.e.B= 0.1×1012G) our result is E=−0.89833keV whereas ref. [30] yields E=−0.846keV; for B12= 5 (i.e. B= 5×1012G) our result is E=−3.61033keV, whereas ref. [30] yields E=−3.5840keV. This comparison allows us to draw the conclusion of a relatively low precision of the adiabatic approximation for multi-electron atoms even for relatively high magnetic ﬁelds. In ﬁgure 3 we present the ionization energy Eionof the beryllium atom depending on the magnetic ﬁeld strength. This continuous dependence is divided into six parts cor- responding to diﬀerent pairs of the ground state conﬁg- urations of the Be atom and Be+ion involved into the ionization energy. The ﬁve vertical dotted lines in ﬁgure 3 mark the boundaries of these sections. The alteration of the sections of growing and decreasing ionization energy originates from diﬀerent dependencies of the total ener- gies of the Be and Be+on the magnetic ﬁeld strength for diﬀerent pairs of the ground state conﬁgurations of these two systems. One can see the sharp decrease of the ion- ization energy between the crossovers (4) and (5). This behaviour is due to the fact that Eionis determined in this section by the rapidly decreasing total energy of the state 1s2p−13d−2of the Be+ion (ﬁgure 2) and by the energy of the Be atom in the state 1 s22p−13d−2which is very weakly dependent on the ﬁeld strength (ﬁgure 1). Another remarkable feature of the curve Eion(γ) is its behaviour in the range of ﬁeld strengths between the transitions (2) and (3). The ionization energy in this re- gion contains a very shallow maximum and in the whole section it is almost independent on the magnetic ﬁeld. Thus, the ionization energy is stationary in this regime of ﬁeld strengths γ= 0.3−0.5 a.u. typical for many magnetic white dwarfs [8]. The above-discussed properties are based on the be- haviour of the total energy of the Be atom and Be+ion.On the other hand, the behaviour of the wavefunctions and many intrinsic characteristics of atoms in external magnetic ﬁelds are associated not with the total energy, but with the binding energies of separate electrons (2) and the total binding energy of the system EB=N/summationdisplay µ=1(mµ+\|mµ\|+ 2szµ+ 1)γ/2−E (3) whereNis the number of electrons. The binding en- ergies of the ground state electronic conﬁgurations of Be and Be+depending on the magnetic ﬁeld strength are presented in ﬁgures 4 and 5. These dependencies at very strong magnetic ﬁelds may illustrate our consid- erations of the previous sections. One can see in ﬁg- ure 4 that the high-ﬁeld ground state 1 s2p−13d−24f−3 is not the most tightly bound state of the beryllium atom. For all the values of the magnetic ﬁelds consid- ered in this paper its binding energy is lower than that of states 1s22s2p−1and 1s22p−13d−2and forγ <100 it is lower than EB1s22s2. The latter circumstance can be easily explained by the fact that the 1 s22s2conﬁgura- tion contains two tightly bound orbitals 1 swhereas the 1s2p−13d−24f−3possess only one such orbital. However, with increasing magnetic ﬁeld strengths the contribution of the group 2 p−13d−24f−3to the binding energy turns out to be larger than that of the 1 s2s2group. Analo- gously we can expect EB1s2p−13d−24f−3>EB1s22s2p−1at some very large ﬁelds γ >10000. On the other hand, it is evident that the state 1 s2p−13d−24f−3must be less bound than 1 s22p−13d−2because both these con- ﬁgurations are constructed of orbitals with binding en- ergies, logarithmically increasing as γ→ ∞ , but the 1s22p−13d−2contains an additional 1 sorbital, which is more tightly bound than 4 f−3at arbitrary ﬁeld strengths. The plot for the Be+ion (ﬁgure 5) illustrates the same features and one can see in this ﬁgure nearly parallel curvesEB1s22p−1(γ) andEB1s2p−13d−2(γ) in the strong ﬁeld regime. Figures 6 and 7 allow us to add some details to the considerations of the previous section. These ﬁgures present spatial distributions of the total electronic dens i- ties for the ground state conﬁgurations of the beryllium atom and its positive ion, respectively. These pictures allow us to gain insights into the geometry of the dis- tribution of the electronic density in space and in par- 7ticular its dependence on the magnetic quantum num- ber and the total spin. The ﬁrst pictures in these ﬁg- ures present the distribution of the electronic density for the ground state for γ= 0. The following pictures show the distributions of the electronic densities for val- ues of the ﬁeld strength which mark the boundaries of the regimes of ﬁeld strengths belonging to the diﬀerent ground state conﬁgurations. For the high-ﬁeld ground states we present the distribution of the electronic densit y at the crossover ﬁeld strength and at γ= 500. For each conﬁguration the eﬀect of the increasing ﬁeld strength consists in compressing the electronic distribution to- wards thezaxis. However the crossovers of ground state conﬁgurations involve the opposite eﬀect due to the fact that these crossovers are associated with an increase of the total magnetic quantum number M. In the ﬁrst rows of ﬁgures 6 and 7 one can see a dense core of 1s2electrons inside the bold solid line contour and a diﬀuse distribution of 2 selectrons outside this core. The prolate shape of the bold solid line contour in the ﬁrst plot of the ﬁgure 6 (1 s22s2,γ= 0) reﬂects the non-spherical distribution of the 2 selectrons in our calculations or the admixture of the 1 s22p2 0conﬁguration to the 1s22s2one from the point of view of the multi- conﬁgurational approach [26–29]. Some additional issues concerning the results presented above have to be discussed. First, our HF results do not include the eﬀects of correlation. To take into ac- count the latter would require a multi-conﬁgurational ap- proach which goes beyond the scope of the present paper. We, however, do not expect that the correlation energy changes our main conclusions like, for example, the fact of the crossovers with respect to the diﬀerent ground states conﬁgurations. With increasing ﬁeld strength the eﬀec- tive one particle picture should be an increasingly better description of the wave function and the percentage of the correlation energy should therefore decrease (see ref. [23] for an investigation on this subject). Two other im- portant issues are relativistic eﬀects and eﬀects due to the ﬁnite nuclear mass. Both these points are basically important for very high magnetic ﬁeld strengths and they have been discussed in ref. [24]. For the systems Be and Be+and for most of the ﬁeld strengths considered here these eﬀects result in minor corrections to the energy.VI. SUMMARY AND CONCLUSIONS We have applied our 2D mesh Hartree-Fock method to the magnetised neutral beryllium atom and beryllium positive ion. The method is ﬂexible enough to yield pre- cise results for arbitrary ﬁeld strengths and our calcula- tions for the ground and several excited states are per- formed for magnetic ﬁeld strengths ranging from zero up to 2.3505·109T (γ= 10000). Our considerations focused on the ground states and their crossovers with increasing ﬁeld strength. The ground state of the beryllium atom undergoes three transitions involving four diﬀerent elec- tronic conﬁgurations. For weak ﬁelds up to γ= 0.0412 the ground state arises from the ﬁeld-free ground state conﬁguration 1 s22s2with the total spin z-projection Sz= 0. With increasing strength of the ﬁeld two diﬀer- ent electronic conﬁgurations with Sz=−1 consequently become the ground state: 1 s22s2p−1and 1s22p−13d−2. Atγ= 4.567 the last crossover of the ground state con- ﬁgurations takes place and for γ >4.567 the ground state wavefunction is represented by the high-ﬁeld-limit fully spin polarised conﬁguration 1 s2p−13d−24f−3,Sz=−2. For the ion Be+we obtain three diﬀerent ground state conﬁgurations possessing two values of the spin projec- tion. For ﬁelds below γ= 0.3185 the ground state electronic conﬁguration has the spin projection Sz= −1/2, magnetic quantum number M= 0 and qualita- tively coincides with the zero-ﬁeld ground state conﬁg- uration 1s22s. Between γ= 0.3185 andγ= 4.501 the ground state is represented by another conﬁguration with Sz=−1/2, i.e. 1s22p−1(M=−1). Above the point γ= 4.501 the fully spin polarised high-ﬁeld-limit conﬁg- uration 1s2p−13d−2(Sz=−3/2) is the actual ground state of the Be+ion. Thus, the sequence of electronic ground state conﬁgurations for the Be+ion is similar to the sequence for the Li atom [16]. We present detailed tables of energies of the ground state conﬁgurations for Be and Be+. For Be and Be+we have presented also the binding energies of the ground state conﬁgurations dependent on the magnetic ﬁeld strength and maps of electronic densi- ties for these conﬁgurations. For the Be atom we present its ionization energy dependent on the ﬁeld strength. Our investigation represents the ﬁrst conclusive study 8of the ground state of the beryllium atom and Be+ion for arbitrary ﬁeld strengths. For the Be atom we have obtained a new sequence of electronic conﬁgurations with increasing ﬁeld strength. This sequence does not coincide with any such sequences obtained previously for other atoms and ions and could not be predicted even quali- tatively without detailed calculations. Putting together what we currently know about ground states of atomic systems in strong magnetic ﬁelds we can conclude that the H, He, Li, Be, C, He+, Li+and Be+ground states have been identiﬁed. For other atoms and multiple series of ions the question about the ground state conﬁgurations is still open. [1] J. P. Ostriker and F. D. A. Hartwick, Astrophys. J. 153, 797 (1968). [2] J. Tr¨ umper, W. Pietsch, C. Reppin, W. Voges, R. Stauben, and E. Kendziorra, Astrophys. J. 219, L105 (1978). [3] J. D. Landstreet, in Cosmical Magnetism , edited by D. Lynden-Bell (Kluwer, Boston, 1994), p.55. [4] W. R¨ osner, G. Wunner, H. Herold, and H. Ruder, J. Phys. B 17, 29 (1984). [5] M. V. Ivanov, J. Phys. B 21, 447 (1988). [6] H. Friedrich and D. Wintgen, Phys.Rep. 183, 37 (1989). [7] Yu. P. Kravchenko, M. A. Liberman, and B. Johansson, Phys.Rev.Lett. 77, 619 (1996). [8] H. Ruder, G. Wunner, H. Herold and F. Geyer, Atoms in Strong Magnetic Fields , Springer-Verlag 1994. [9]Atoms and Molecules in Strong External Fields , edited by P. Schmelcher and W. Schweizer, Plenum Press New York and London (1998) [10] W. Becken, P. Schmelcher, and F.K. Diakonos, J. Phys.B32, 1557 (1999). [11] S. Jordan, P. Schmelcher, W. Becken, and W. Schweizer, Astr.&Astrophys. 336, 33 (1998). [12] D. Reimers, S. Jordan, V. Beckmann, N. Christlieb, L. Wisotzki, Astr.& Astrophys. 337, L13 (1998) [13] R. H. Garstang, Rep. Prog. Phys. 40, 105 (1977). [14] J. Simola and J Virtamo, J. Phys. B 11, 3309 (1978). [15] H. Friedrich, Phys. Rev. A 26, 1827 (1982). [16] M. V. Ivanov and P. Schmelcher, Phys. Rev. A 57, 3793 (1998). [17] M. V. Ivanov and P. Schmelcher, Phys. Rev. A 60, 3558 (1999). [18] M. D. Jones, G. Ortiz, and D. M. Ceperley, Phys. Rev. A.54, 219 (1996). [19] M. V. Ivanov, Phys. Lett. A 239, 72 (1998). [20] M. V. Ivanov, Optics and Spectroscopy 70, 148 (1991). [21] M. V. Ivanov, J. Phys. B 27, 4513 (1994). [22] M. V. Ivanov, USSR Comput. Math. & Math. Phys. 26, 140 (1986). [23] P. Schmelcher, M. V. Ivanov and W. Becken, Phys. Rev. A59, 3424 (1999). [24] M. V. Ivanov and P. Schmelcher, Phys. Rev. A 61, 022505 (2000). [25] M. V. Ivanov and P. Schmelcher, Advances in Quantum Chemistry, to be published. [26] K. J. Miller, and K. Ruedenberg, J. Chem. Phys. 48, 3450 (1968). [27] J. S. Sims and S. Hagstrom, Phys. Rev. A 4, 908 (1971). [28] G. H. F. Diercksen and A. J. Sadlej, Chem. Phys. 77, 429 (1983). [29] G. Maroulis and A. J. Thakkar, J. Phys. B: At. Mol. Opt. Phys.21, 3819 (1988). [30] D. Neuhauser, S. E. Koonin, and K. Langanke, Phys. Rev. A 33, 2084 (1986); 36, 4163 (1987). 9Figure Captions Figure 1. The total energies (in atomic units) of the states of the beryllium atom as functions of the magnetic ﬁeld strength considered for the determination of the ground state electronic conﬁgurations. The ﬁeld strength is given in units of γ= (B B0),B0= ¯hc/ea2 0= 2.3505·105T. Figure 2. The total energies (in atomic units) of the states of the beryllium positive ion as functions of the magnetic ﬁeld strength considered for the determination of the ground state electronic conﬁgurations. The ﬁeld strength is given in units of γ= (B B0),B0= ¯hc/ea2 0= 2.3505·105T. Figure 3. Be atom ground state ionization energy EI. Transition points are marked by broken vertical lines. The sequence of the transitions are (from left to right): 1. Be: 1 s22s2− →1s22s2p−1; 2. Be+: 1s22s− → 1s22p−1; 3. Be: 1s22s2p−1−→1s22p−13d−2. 4. Be+: 1s22p−1− →1s2p−13d−2; 5. Be: 1 s22p−13d−2− → 1s2p−13d−24f−3. Crossovers (4) and (5) take place at relatively close values of γand are not resolved in the ﬁgure. Figure 4. The binding energies (in atomic units) of the ground state electronic conﬁgurations of the Be atom depending on the magnetic ﬁeld strength. The ﬁeld strength is given in units of γ= (B B0),B0= ¯hc/ea2 0= 2.3505·105T. Figure 5. The binding energies (in atomic units) of the ground state electronic conﬁgurations of the Be+ ion depending on the magnetic ﬁeld strength. The ﬁeld strength is given in units of γ= (B B0),B0= ¯hc/ea2 0= 2.3505·105T. Figure 6. Contour plots of the total electronic den- sities for the ground state of the beryllium atom. For neighbouring lines the densities are diﬀerent by a factor of 2. The coordinates z,xas well as the corresponding ﬁeld strengths are given in atomic units. Each row presents plots for a ground state conﬁguration at its lower (left) and upper (right) intersection points. Rows: 1. 1 s22s2: γ= 0 andγ= 0.0412; 2. 1s22s2p−1:γ= 0.0412 and γ= 0.957; 3. 1s22p−13d−2:γ= 0.957 andγ= 4.567; 4.1s2p−13d−24f−3:γ= 4.567 andγ= 500. Figure 7. Contour plots of the total electronic den- sities for the ground state of the beryllium positive ion. For neighbouring lines the densities are diﬀerent by a factor of 2. The coordinates z,xas well as the corre- sponding ﬁeld strengths are given in atomic units. Each row presents plots for a ground state conﬁguration at its lower (left) and upper (right) intersection points. Rows: 1. 1s22s:γ= 0 andγ= 0.3185; 2. 1s22p−1:γ= 0.3185 andγ= 4.501; 3. 1s2p−13d−2:γ= 4.501 andγ= 500. 10TABLE I. The Hartree-Fock ground state conﬁgurations of the beryllium atom in external magnetic ﬁelds. The conﬁguratio ns presented in the table are the ground state conﬁgurations fo rγmin≤γ≤γmax. Atomic units are used. no. γmin γmax The ground state conﬁguration M S z E(γmin) 1 0 0.0412 1 s22s20 0 −14.57336 2 0.0412 0.957 1 s22s2p−1 −1 −1 −14.57098 3 0.957 4.567 1 s22p−13d−2 −3 −1 −15.13756 4 4.567 ∞ 1s2p−13d−24f−3 −6 −2 −15.91660 TABLE II. The Hartree-Fock ground state conﬁgurations of th e Be+ion in external magnetic ﬁelds. The conﬁgurations presented in the table are the ground state conﬁgurations fo rγmin≤γ≤γmax. Atomic units are used. no. γmin γmax The ground state conﬁguration M S z E(γmin) 1 0 0.3185 1 s22s 0 −1/2 −14.27747 2 0.3185 4.501 1 s22p−1 −1 −1/2 −14.38602 3 4.501 ∞ 1s2p−13d−2 −3 −3/2 −15.01775 11TABLE III. The total energies of the ground state conﬁgurati ons of the beryllium atom depending on the magnetic ﬁeld strength. The ﬁgures in parentheses are the labels of the gro und state conﬁgurations provided in the ﬁrst column of table I. Atomic units are used. γ E (1) E(2) E(3) E(4) 0.000 −14.57336 −14.51206 −14.19023 −9.44321 0.001 −14.57336 −14.51357 −14.1928 −9.4483 0.002 −14.57335 −14.51507 −14.1952 −9.4532 0.005 −14.57332 −14.51953 −14.2025 −9.4675 0.01 −14.57322 −14.52690 −14.2142 −9.4903 0.02 −14.57279 −14.54138 −14.2361 −9.5331 0.03 −14.57209 −14.55553 −14.2566 −9.5735 0.04 −14.57111 −14.56933 −14.27587 −9.6121 0.05 −14.56986 −14.58281 −14.29437 −9.6493 0.07 −14.56657 −14.60879 −14.32933 −9.7198 0.1 −14.55971 −14.64548 −14.3780 −9.82 0.12 −14.55395 −14.66851 −14.40838 −9.8805 0.15 −14.54367 −14.70108 −14.45145 −9.9692 0.2 −14.52261 −14.75065 −14.51761 −10.1081 0.3 −14.46861 −14.83520 −14.63369 −10.36220 0.3185 −14.84905 0.4 −14.40279 −14.90464 −14.73396 −10.59384 0.5 −14.32860 −14.96264 −14.82272 −10.80901 0.6 −14.24832 −15.01171 −14.90262 −11.01121 0.7 −14.16352 −15.05368 −14.97542 −11.20281 0.8 −14.07526 −15.08989 −15.04232 −11.38545 0.9 −13.98431 −15.12138 −15.10422 −11.56045 1. −13.89120 −15.14899 −15.16178 −11.72880 1.2 −13.69990 −15.19498 −15.26583 −12.04863 1.5 −13.40329 −15.24757 −15.39926 −12.49432 2. −12.88908 −15.30815 −15.57496 −13.16961 3. −11.79811 −15.36376 −15.79985 −14.35016 4. −10.633617 −15.34275 −15.90161 −15.38050 4.501 −15.91626 5. −9.40602 −15.25183 −15.91027 −16.30690 7. −6.79760 −14.89530 −15.71644 −17.94005 8. −5.43095 −14.64516 −15.53623 −18.67389 10. −2.5988 −14.03046 −15.04644 −20.01753 12. +0 .34064 −13.29115 −14.41743 −21.23057 15. +4 .9055 −12.00063 −13.27286 −22.86513 20. +12 .8201 −9.49118 −10.97100 −25.23250 30. +29 .3964 −3.59324 −5.40704 −29.11102 40. +46 .5935 +3 .04026 +0 .95677 −32.28415 50. +64 .186 +10 .1472 +7 .83395 −35.00768 100. +155 .286 +49 .4177 +46 .25962 −45.10519 200. +343 .899 +135 .659 +131 .4188 −58.08264 500. +924 .20 +411 .830 +405 .7027 −80.67357 121000. +1905 .14 +888 .70 +880 .706 −102.75480 2000. +3881 .5 +1860 .40 +1850 .052 −129.9790 5000. +4813 .56 +4799 .35 −175.2704 10000. +9770 .37 +9752 .24 −217.695 13TABLE IV. The total energies of the ground state conﬁguratio ns of the Be+ion depending on the magnetic ﬁeld strength. The ﬁgures in parentheses are the labels of the ground state c onﬁgurations provided in the ﬁrst column of table II. Atomic units are used. γ E (1) E(2) E(3) 0.000 −14.27747 −14.13093 −9.41056 0.001 −14.27797 −14.13195 −9.41358 0.002 −14.27846 −14.13294 −9.41657 0.005 −14.27995 −14.13593 −9.42551 0.01 −14.28241 −14.14087 −9.44028 0.02 −14.28725 −14.15066 −9.46939 0.03 −14.29198 −14.16030 −9.49791 0.04 −14.29659 −14.16980 −9.52587 0.0412 −14.29714 0.05 −14.30111 −14.17916 −9.55332 0.07 −14.30981 −14.19746 −9.60670 0.1 −14.32207 −14.22390 −9.68356 0.12 −14.32972 −14.24088 −9.73294 0.15 −14.34047 −14.26542 −9.80463 0.2 −14.35648 −14.30406 −9.91878 0.3 −14.38212 −14.37402 −10.13144 0.4 −14.40046 −14.43599 −10.32817 0.5 −14.41282 −14.49163 −10.51259 0.6 −14.42022 −14.54210 −10.68705 0.7 −14.42350 −14.58821 −10.85323 0.8 −14.42335 −14.63059 −11.01236 0.9 −14.42029 −14.66971 −11.16540 0.957 −14.69069 1. −14.41478 −14.70591 −11.31312 1.2 −14.39782 −14.77070 −11.59490 1.5 −14.36143 −14.85169 −11.98978 2. −14.28225 −14.95181 −12.59206 3. −14.08247 −15.05201 −13.65352 4. −13.83797 −15.05004 −14.58615 4.567 −15.07310 5. −13.55019 −14.96820 −15.42817 7. −12.85647 −14.61928 −16.91814 8. −12.45821 −14.37080 −17.58931 10. −11.57652 −13.75773 −18.820184 12. −10.59993 −13.01900 −19.93310 15. −8.99386 −11.72840 −21.43461 20. −6.03364 −9.217910 −23.612005 30. +0 .59244 −3.31723 −27.18373 40. +7 .81557 +3 .31895 −30.10832 50. +15 .4261 +10 .42836 −32.61959 100. +56 .5516 +49 .70820 −41.93414 200. +145 .1649 +135 .95916 −53.90638 14500. +425 .471 +412 .14745 −74.73619 1000. +906 .37 +889 .0264 −95.07513 2000. +1883 .08 +1860 .7100 −120.11947 5000. +4844 .6 +4814 .005 −161.7052 10000. +9809 .3 +9770 .643 −200.5709 1510−210−1100101 γ−17−16−15−14−13−12EFigure 1 1s22s2 1s22s2p−1 1s22p−13d−2 1s2p−13d−24f−3 1s2p02p−13d−2 1s2s2p−13d−2 1s22p02p−1 1s22s2p−1, Sz=010−210−1100101 γ−16−15−14−13EFigure 2 1s22s 1s22p−1 1s2p−13d−2 1s2p−12p0 1s2s2p−110−310−210−1100101102103104 γ0.20.30.50.7123571020EionFigure 3 1 2 3 4 510−310−210−1100101102103104 γ1030100300EBFigure 4 1s22s2 1s22s2p−1 1s22p−13d−2 1s2p−13d−24f−310−310−210−1100101102103104 γ1030100300EBFigure 5 1s22s 1s22p−1 1s2p−13d−2/G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23 /G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23 /G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23 /G91/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23 /G91/G16/G23/G16/G21/G19/G21/G23 /G41/G76/G74 /G88/G85/G72/G3/G25/G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G16/G23 /G16/G21 /G19 /G21 /G23 /G91/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23 /G91/G16/G23/G16/G21/G19/G21/G23/G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23/G93 /G16/G23 /G16/G21 /G19 /G21 /G23/G16/G23/G16/G21/G19/G21/G23 /G41/G76/G74 /G88/G85/G72/G3/G26
arXiv:physics/0009005 1 Sep 2000Beam-based Feedback Simulations for the NLC Linac* L. Hendrickson#, N. Phinney, P. Raimondi, T. Raubenheimer, A. Seryi, P. Tenenbaum SLAC, P.O. Box 4349, Stanford CA 94309 (USA) Abstract Extensive beam-based feedback systems are planned a s an integral part of the Next Linear Collider (NLC) control system. Wakefield effects are a significant influence on the feedback design, imposing both architectural and algorithmic constraints. Studies are in progress to assure the optimal selection of devices and to refine and confirm the algorithms for the system design. We show the results of initial simulations, along with evaluations of system response for various conditions of ground motion and other operational disturbances. 1 INTRODUCTION The NLC design specifies a sequence of beam-based trajectory feedback systems along the main linacs. This feedback is intended to operate on the average properties of a pulse train of 95 bunches at a repe tition rate of 120 Hz. The measurements are beam position monitor (BPM) readings for a single pulse which may be averaged over a number of bunches. The correctio ns are primarily applied with fast dipole magnets but other devices were also studied. The feedback is designed to damp low frequency trajectory errors up to about 5 Hz. The system is modeled after the generalized feedbac k developed at the SLC [1] with improvements to provi de more optimal response for the NLC. Simulations of the NLC linac feedback systems have been performed in a Matlab environment [2] using th e LIAR program [3] for tracking and beam calculations . Beam tests have also been done on the SLAC linac to confirm the simulation results [4]. 2 MULTI-CASCADE SIMULATIONS In the main linacs of both the SLC and NLC design, a series of feedbacks are used to keep the beam traje ctory well centered in the quadrupoles and structures. To avoid overcorrection and ringing in response to an incoming disturbance, these systems need to communicate beam information. In the SLC, a simple one-to-one "cascade" system between loops was implemented. Each feedback controlled a number of beam states which were typically fitted beam positi ons and angles at a single point. On each beam pulse, these _______________________ \\*Work supported by the U.S. Dept. of Energy under co ntract DE- AC03-76SF00515 # Email: ljh@slac.stanford.edu states were recalculated and the information passed to the next downstream loop before a correction was implemented. The downstream feedback would then use transport matrices to calculate the correspondi ng local states and subtract these from the current measurement to determine the residual beam motion t o be corrected. The beam transport from the adjacent upstream feedback loop was calculated adaptively fr om the natural beam jitter. In the presence of wakefields, the beam transport i s non-linear and depends on the location where the perturbation occurs and distance it propagates. Thi s means that the simple SLC single-cascade system is inadequate. In the NLC multi-cascade scheme, each feedback receives information from all of the upstr eam loops. The beam transport is an overconstrained lea st squares fit matrix, which converts many upstream states to the fitted downstream location. The NLC simulations were done with five feedback systems distributed along the linac. Figure 1 shows the response of the last of these loops to a step distu rbance early in the linac. The raw state is the measured b eam orbit in the linac which can be fixed perfectly in simulation, when accelerator or modeling imperfecti ons are not included. This simulation simply shows that the multi-cascade algorithm is correct. The response se en on the BPMs of the last feedback loop is identical to the design response for a single feedback loop, even th ough five loops respond. The cascade-adjusted state is the 0 5 10 15 20 25 30 35-5-4-3-2-1012x 10-5 Time, pulsesActuator kick Figure 1: Multi-cascade simulation results. The low er plot (+) shows the real beam motion seen by the las t NLC linac feedback following a perturbation. The upper plot (O) shows the cascade-adjusted state, wh ich is the portion of the oscillation that this feedbac k corrects. residual beam motion to be corrected by this feedba ck. Intuitively one might expect that the downstream lo ops would have nothing to do. However with the non-line ar beam transport, the downstream feedbacks must make small corrections to achieve optimal system respons e. 3 DEVICE CONFIGURATION STUDIES Another topic of study is the optimal number, type and placement of feedback devices for the NLC linac. In the SLC, the BPMs and correction dipoles were grouped together over a distance of at most 200 m because of bandwidth and connectivity constraints. The present NLC configuration includes 5 feedback syste ms distributed along the linac. Each feedback has 2 se ts of 4 dipole correctors (two vertical and two horizonta l). The first set of correctors is at the beginning of the section and the second set is located midway to the next feedback. Four sets of BPM measurements are used pe r loop, with 4 monitors in each set. Figures 2, 3 and 4 show the response to a step disturbance in the midd le of the first loop with this configuration. 0 2 4 6 8 10 12105051015 S position (km)BPM Y measurement (microns) Figure 2: BPM orbit along the linac after feedback responds to a step function. The feedback corrector s are located at the position of the vertical lines and t he BPMs are marked with (+). 0 2 4 6 8 10 120.40.60.811.21.41.61.822.22.4x 10-7 S position (km)Normalized emittance Figure 3: Simulation of emittance growth in the lin ac following a simple step disturbance, with (solid) a nd without (dashed) feedback. When a bunch passes off-axis through the accelerati ng structures, the wakefields from the head of the bun ch deflect the particles which follow causing a "tilt" or y-z correlation along the bunch. Local feedback correct s the bunch centroid but does not remove the tilt. Th e additional sets of BPMs and correctors help find a solution which minimizes both centroid offset and t ilt. Figure 4 shows the offset of individual slices alon g the length of the bunch for the same simulation. The be am profile remains fairly flat for this device configu ration. 400 300 200 100 0 100 200 300 400 1.5 1 0.5 0 0.5 1 1.5 Feedback corrected Y position, um 400 300 200 100 0 100 200 300 400 1.5 1 0.5 0 0.5 1 1.5 Z position of the slice, um Uncorrected Y position, um Figure 4: Offset of slices along the bunch at the e nd of the linac. The top figure shows the profile with NL C feedback on (+). The lower figure (O) shows the uncorrected bunch profile. Most of the simulations have been done using dipole magnets as the correction devices. Other devices su ch as crab cavities or structure movers have also been studied to determine if they better correct the dis tortion of the bunches and the emittance dilution. Figure 5 shows the response to a perturbation for a system i n which structure movers are used for one set of correctors in each feedback. The effectiveness of various alternatives is still under study. 0 2 4 6 8 10 12-8-6-4-20246 S position (km)BPM Y measurement (microns) Figure 5: BPM orbit in the linac after a step respo nse. Both dipole correctors and structure movers are use d in this feedback configuration. 4 GROUND MOTION STUDIES The critical test for a feedback system is performa nce in the presence of ground motion and accelerator errors. Simulations have been done using both the A TL ground motion model of Shiltsev [6] and the more complete model developed by Seryi [7]. Initial stud ies evaluated the RMS beam jitter and emittance growth after several seconds of ground motion changes, wit h feedback at 120 Hz rate. The ATL ground motion model was used to compare the performance of different feedback configuration s. The simulations used 30 minutes of ATL-like ground motion with a coefficient of 5.0e-7 µm2/m/sec, a typical value for the SLAC site. The BPM resolution was 0.1 µm and results from 100 random seeds were averaged. For the proposed NLC configuration, the emittance growth was less than 6%. Table 1 lists the results showing significantly larger emittance growth if le ss correctors or BPMs are used. # Feedback Loops # BPMs per loop # Cors per loop per plane Emittance Growth (%) 0 (off) 0 0 104 5 16 2 31 5 8 4 21 5 16 4 5.7 Table 1: Emittance Growth for different feedback device configurations. Additional correctors and BP Ms are effective in reducing emittance dilution. Figure 6 shows how the emittance grows along the linac with the NLC feedback configuration. The emittance increases at the location of the correcto r dipoles which create dispersion. However, the resul ting emittance growth is only 5.7% compared with 104% without feedback, as shown in figure 7. 0 2 4 6 8 10 1201234567 S position (km)Vertical emittance growth, % Figure 6: Simulation of vertical emittance along th e NLC linac growth after 30 minutes of ATL ground motion, with feedback. Vertical lines mark the loca tion of feedback corrector dipoles. 0 2 4 6 8 10 12-20020406080100120 S position (km)Vertical emittance growth, % Figure 7: Simulation of vertical emittance growth a fter 30 minutes of ATL ground motion without feedback. 5 FUTURE PLANS Further studies are needed to optimize the performa nce of the NLC feedback systems in the presence of grou nd motion and other errors. Integrated simulations wit h feedback, ground motion, cultural noise, and beam- based alignment algorithms are planned. These more complete simulations will be used to estimate the operational control requirements for the NLC and th e beam jitter and emittance dilution expected during operation. Another topic of study is the feasibility of adapti ve or semi-invasive methods for measuring and updating the feedback model of beam transport in the linac. Because the planned feedback systems are distribute d over large distances, accumulated energy or focusin g errors can cause the actual model to differ signifi cantly from the theoretical model. Present simulations ass ume that the model is updated by calibration measuremen ts but adaptive methods would perform better. REFERENCES [1] L. Hendrickson, et al., "Generalized Fast Feed back System in the SLC," ICALEPCS, Tsukuba, Japan, SLAC- PUB-5683 (1991). [2] P. Tenenbaum, et al., "Use of Simulation Progr ams for the Modeling of the Next Linear Collider,'' PAC, Ne w York, New York (1999). [3] R. Assmann, et al., "The Computer program LIAR for the simulation and modeling of high performance linacs, " PAC, Vancouver, Canada (1997). [4] L. Hendrickson, et al., "Beam-based Feedback T esting and Simulations for the SLC Linac," LINAC, Monterey , California (2000). [5] T. Himel, et al., "Adaptive Cascaded Beam-Base d Feedback at the SLC," PAC, Washington, D.C., SLAC-P UB- 6125 (1993). [6] V. Shiltsev, "Space-Time Ground Diffusion: The ATL Law for Accelerators," Proceedings IWAA-93, 352 (19 95). [7] A Seryi, et al., "Simulation Studies of the NL C with Improved Ground Motion Models," LINAC, Monterey, California (2000).
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/D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/B8 /CB/C8/C5/B8 /C3/C4/C5/B8/CV/D6/D3/D9/D4/B9/DA /CT/D0/D3 /CX/D8 /DD /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CP/D2/CS /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CQ /DD /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /BA /CC/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2 /B4 /D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D4/D9/D0/D7/CT/B5 /CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/D7 /CU/D3/D9/D2/CS /CP/D2/CS /D8/CW/CT /D3/D2 /D8/D6/CX/B9/CQ/D9/D8/CX/D3/D2 /D3/CU /C3/CT/D6/D6/B9/D0/CT/D2/D7/CX/D2/CV /D8/D3 /D7/D3/D0/CX/D8/D3/D2 /CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/D7 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/CS/BA /CF /CT /D7/CW/D3 /DB /CT/CS /D8/CW/CP/D8/D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /CW/CX/D6/D4/B9/CU/D6/CT/CT 2π /B9 /CP/D2/CSπ /B9 /D7/D3/D0/CX/D8/D3/D2/D7 /CP/D2/CS /CW/CX/D6/D4 /CT/CS /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/D7/DB/CX/D8/CW /DA /CP/D6/CX/CP/CQ/D0/CT /D7/D5/D9/CP/D6/CT /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CP/D6/CT /D7/D8/CP/CQ/D0/CT /CP/CV/CP/CX/D2/D7/D8/D0/CP/D7/CT/D6 /D2/D3/CX/D7/CT /DB/CX/D8/CW /D2/D3 /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D1/CX/D2/CX/D1/CP/D0 /D2/CT /CT/D7/D7/CP/D6/DD /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CS/CT/D4/D8/CW /CX/D2/B9/D8/D6/D3 /CS/D9 /CT/CS /CQ /DD /CP/CQ/D7/D3/D6/CQ /CT/D6/BA /BT/D2 /CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CS /D7/CT/D0/CU/B9/D7/D8/CP/D6/D8/CX/D2/CV /CP/CQ/CX/D0/CX/D8 /DD/D3/CU /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CP/D6/CT /CT/D7/D8/CX/D1/CP/D8/CT/CS/B8 /D8/D3 /D3/BA/BE/BE /C5/D3 /CS/CT/D0/BU/CP/D7/CT/CS /D3/D2 /D8/CW/CT /D7/D0/D3 /DB/D0/DD /DA /CP/D6/DD/CX/D2/CV /CT/D2 /DA /CT/D0/D3/D4 /CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /AS/CT/D0/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT a(t) /B8 /DB /CT /D3/D2/D7/CX/CS/CT/D6 /CP /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /D7/DD/D7/D8/CT/D1 /CX/D2 /D0/D9/CS/CX/D2/CV /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CV/CP/CX/D2α /B8 /D0/CX/D2/CT/CP/D6 /D0/D3/D7/D7 γ /B8 /CB/C8/C5β /B8 /C3/CT/D6/D6/B9/D0/CT/D2/D7/B9/CX/D2/CS/D9 /CT/CS /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /DB/CX/D8/CW /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D2/B9/D8/CT/D2/D7/CX/D8 /DD 1/σ /B8 /CV/D6/D3/D9/D4/B9/DA /CT/D0/D3 /CX/D8 /DD /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2D /CP/D2/CS /CQ/CP/D2/CS/DB/CX/CS/D8/CW /D0/CX/D1/CX/D8/CX/D2/CV /CT/D0/CT/D1/CT/D2 /D8/DB/CX/D8/CW /D8/D6/CP/D2/D7/D1/CX/D7/D7/CX/D3/D2 /CQ/CP/D2/CS 1/tf /BA /CC /D3 /CP /D3/D9/D2 /D8 /CU/D3/D6 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /DB/CX/D8/CW/D8/CW/CT /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /DB/CW/CX/D0/CT /D7/D8/CP /DD/CX/D2/CV /CX/D2 /CU/D6/CP/D1/CT /D3/CU /CP/D2/CP/D0/DD/D8/CX /CP/D4/D4/D6/D3/CP /CW /DB /CT/CP/CS/D3/D4/D8/CT/CS /CP /D8 /DB /D3/B9/D0/CT/DA /CT/D0 /D1/D3 /CS/CT/D0 /D3/CU /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/BA /CC/CW/CX/D7 /CP/D7/D7/D9/D1/D4/B9/D8/CX/D3/D2 /CX/D7 /DA /CP/D0/CX/CS /CU/D3/D6 /D5/D9/CP/D2 /D8/D9/D1/B9 /D3/D2/AS/D2/CT/CS /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/D8/D6/D9 /D8/D9/D6/CT/D7 /D9/D8/CX/D0/CX/DE/CT/CS /CX/D2 /D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /CU/D7/B9/D0/CP/D7/CT/D6/D7/BA/CF/CW/CT/D2 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D1 /D9 /CW /D7/CW/D3/D6/D8/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /CS/CT/D4/CW/CP/D7/CX/D2/CV /D8/CX/D1/CT /CX/D2/CP/CQ/D7/D3/D6/CQ /CT/D6/B8tp<<t coh /CP/D2/CS /D8/CW/CT /AS/CT/D0/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD \|a(t)\|2/CX/D7 /D2/D3/D8 /CT/D2/D3/D9/CV/CW /CU/D3/D6 /D8/CW/CT /CB/D8/CP/D6/CZ/CT/AR/CT /D8 /D1/CP/D2/CX/CU/CT/D7/D8/CP/D8/CX/D3/D2/B8 /D8/CW/CT /D4/D9/D0/D7/CT/B9/D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CQ /CT/DD/D7 /D8/CW/CT /BU/D0/D3 /CW/CT/D5/D9/CP/D8/CX/D3/D2/D7 /CJ/BD/BE℄/BM du dt= (∆ −dφ dt)v+qaw,dv dt=−(∆−dφ dt)u,dw dt=−qau, /B4/BD/B5/DB/CW/CT/D6/CTu(t) /B8v(t) /CP/D2/CSw(t) /CP/D6/CT /D8/CW/CT /D7/D0/D3 /DB/D0/DD /DA /CP/D6/DD/CX/D2/CV /CT/D2 /DA /CT/D0/D3/D4 /CT/D7 /D3/CU /D8/CW/CT /D4 /D3/D0/CP/D6/B9/CX/DE/CP/D8/CX/D3/D2 /D5/D9/CP/CS/D6/CP/D8/D9/D6/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D2/CS /D8/CW/CT /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CS/CX/AR/CT/D6/CT/D2 /CT/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8 q=d/¯h /B8d= 0.28×e /BV/D3/D9/D0/D3/D1 /CQ/D2/D1 /CX/D7 /D8/CW/CT /CS/CX/D4 /D3/D0/CT /D1/D3/D1/CT/D2 /D8/D9/D1/B8 e /CX/D7 /D8/CW/CT /CT/D0/CT/B9/D1/CT/D2 /D8/CP/D6/DD /CW/CP/D6/CV/CT /B4/CX/D2 /D3/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /DB /CT /D9/D7/CT/CS /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /BZ/CP/BT/D7/BB/BT/D0/BT/D7/CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DDEa= 50µJ/cm2/CP/D2/CStcoh= 50fs /B5/B8∆ /CX/D7 /D8/CW/CT/D1/CX/D7/D1/CP/D8 /CW /CQ /CT/D8 /DB /CT/CT/D2 /D3/D4/D8/CX /CP/D0 /D6/CT/D7/D3/D2/CP/D2 /CT /CP/D2/CS /D4/D9/D0/D7/CT /CP/D6/D6/CX/CT/D6 /CU/D6/CT/D5/D9/CT/D2 /DD /B8φ /CX/D7 /D8/CW/CT /CX/D2/B9/D7/D8/CP/D2 /D8 /AS/CT/D0/CS /D4/CW/CP/D7/CT/BA /BT/D2 /CX/D2/CX/D8/CX/CP/D0 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 γa= 2π×N×d×2ω×za× tcoh/(c¯h) = 0.01 /B4ω /CX/D7 /D8/CW/CT /AS/CT/D0/CS /CU/D6/CT/D5/D9/CT/D2 /DD/B5 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CP/D6/D6/CX/CT/D6/D7 /CS/CT/D2/D7/CX/D8 /DD N=γaEa/(¯h×ω×za) = 2 ×1018cm−3/CP/D2/CS /D8/CW/CT /D8/CW/CX /CZ/D2/CT/D7/D7 /D3/CU /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/CP/CQ/D7/D3/D6/CQ /CT/D6za= 10nm /BA/CC/CW/CT /D0/CP/D7/CT/D6 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/CS /C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV/CT/D5/D9/CP/D8/CX/D3/D2 /CJ/BE℄/BA /CC/CW/CT/D2 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /D0/CP/D7/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7/BM ∂a(z,t) ∂z=/bracketleftigg α−γ+iθ+δ∂ ∂t+ (t2 f+iD)∂2 ∂t2+σ−iβ η2\|a\|2/bracketrightigg a+ /bracketleftigg2πNz aωd cu−2πNz ad cdv dt/bracketrightigg , /B4/BE/B5/DB/CW/CT/D6/CTz /CX/D7 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CP /DA/CX/D8 /DD /D0/CT/D2/CV/D8/CW/B8/CX/BA/CT/BA /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D8/CW/CT /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4/B8 c /CX/D7 /D8/CW/CT /D0/CX/CV/CW /D8 /DA /CT/D0/D3 /CX/D8 /DD /B8θ /CP/D2/CSδ /CP/D6/CT/D8/CW/CT /D4/CW/CP/D7/CT /CP/D2/CS /D8/CX/D1/CT /AS/CT/D0/CS /CS/CT/D0/CP /DD /CP/CU/D8/CT/D6 /D8/CW/CT /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /CF /CT/D2/CT/CV/D0/CT /D8/CT/CS /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/AR/CT /D8/D7 /CX/D2 /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/BA /C4/CP/D8/CT/D6 /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D3/D2/D0/DD/BF/D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D4/D9/D0/D7/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2/D7/B8 /DB/CW/CX /CW /CP/D0/D0/D3 /DB/D7 /D8/D3 /CT/D0/CX/D1/CX/D2/CP/D8/CT /D8/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/D2 z /BA /CF /CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/CW/CT /D8/CX/D1/CT/D7 /D8/D3tf /CP/D2/CS /D8/CW/CT /AS/CT/D0/CS /D8/D3q×tf /BAβ /CP/D2/CSσ /CP/D6/CT/D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D32(q×tf)2/(n×c×ε0) = 5 ×10−12cm2/W /B8 /DB/CW/CT/D6/CTn /CX/D7 /D8/CW/CT/CX/D2/CS/CT/DC /D3/CU /D6/CT/CU/D6/CP /D8/CX/DA/CX/D8 /DD /B8 ε0 /CX/D7 /D8/CW/CT /D4 /CT/D6/D1/CX/D8/D8/CX/DA/CX/D8 /DD /CP/D2/CStf= 2.5fs /B4/CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT/D0/CP/D7/CT/D6/B5/BA /CF/CX/D8/CW /D8/CW/CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7 σ= 0.14 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/D4/CP/D6/CP/D1/CT/D8/CT/D6 /D3/CU /C3/CT/D6/D6/B9/D0/CT/D2/D7 /CX/D2/CS/D9 /CT/CS /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D3/CU107W /CP/D2/CS30µm/D7/D4 /D3/D8 /D7/CX/DE/CT /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/BA /BW/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CB/C8/C5 /D4/CP/D6/CP/D1/CT/D8/CT/D6 β /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 0.26 /CU/D3/D61mm /D8/CW/CX /CZ /CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT /D6/DD/D7/D8/CP/D0/BA /B4/C6/D3/D8/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /CP /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7/D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /BY/CX/CV/D7/BA /BE /CP/D2/CS /BG/B5/BA /BT /CS/CS/CX/D8/CX/D3/D2/CP/D0/D0/DD /B8 /DB /CT /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS /CP/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6η /B8 /DB/CW/CX /CW /CX/D7 /CV/D3 /DA /CT/D6/D2/CT/CS /CQ /DD /BD/B5 /D8/CW/CT /D6/CP/D8/CX/D3 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D7/CX/DE/CT /D3/CU/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D1/D3 /CS/CT /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CP/D2/CS /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /D3/D6 /CQ /DD /BE/B5/D8/CW/CT /D6/CT/AT/CT /D8/CX/DA/CX/D8 /DD /D3/CU /D8/CW/CT /D9/D4/D4 /CT/D6 /D7/D9/D6/CU/CP /CT /D3/CU /D8/CW/CT /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CS/CT/DA/CX /CT/BA/BY /D3/D6/D1/CP/D0/D0/DD /B8 /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CUη /D1/CT/CP/D2/D7 /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D6/CT/D0/CP/D8/CX/DA /CT /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU/CB/C8/C5/B8 /C3/CT/D6/D6/B9/D0/CT/D2/D7/B9/CX/D2/CS/D9 /CT/CS /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D3/D6 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CV/CP/CX/D2 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8/D8/D3 /D8/CW/CT /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/BA/BT/D7 /DB /CT /D7/CW/CP/D0/D0 /D7/CT/CT /D0/CP/D8/CT/D6/B8 /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CQ /DD /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /CX/D7 /CP/D2/CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /CU/CP /D8/D3/D6 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CS /D8/CW /D9/D7 /D7/CW/D3/D9/D0/CS /CQ /CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8/BA/CC/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /DB /CP /DD /D8/D3 /CS/D3 /D8/CW/CX/D7 /CX/D7 /D8/D3 /D9/D7/CT /CP /D5/D9/CP/D7/CX/B9/D8 /DB /D3 /D0/CT/DA /CT/D0 /D1/D3 /CS/CT/D0 /CU/D3/D6 /CP /D8/CX/DA /CT/D1/CT/CS/CX/D9/D1/BA /BT/CU/D8/CT/D6 /D7/D3/D1/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CT/CS /CQ /DD /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT/CT/D2/CT/D6/CV/DDE /CX/D2 /D8/CW/CT /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D3/D2/CS/CX/D8/CX/D3/D2 /DB /CT /CW/CP /DA /CTα=Pαmax P+τE/η2+1/Tr /B8 /DB/CW/CT/D6/CT αmax /CX/D7 /D8/CW/CT /CV/CP/CX/D2 /CU/D3/D6 /D8/CW/CT /CU/D9/D0/D0 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CX/D2 /DA /CT/D6/D7/CX/D3/D2/B8Tr /CX/D7 /D8/CW/CT /CV/CP/CX/D2 /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2/D8/CX/D1/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CP /DA/CX/D8 /DD /D4 /CT/D6/CX/D3 /CSTcav /B8P=σ14×Tcav×Ip/(h×νp) /CX/D7 /D8/CW/CT/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8νp /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /CU/D6/CT/D5/D9/CT/D2 /DD /B8σ14 /CX/D7 /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D6/D3/D7/D7 /D7/CT /D8/CX/D3/D2 /D3/CU /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8Ip /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8τ= 6.25×10−4/CX/D7/D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CX/D2 /DA /CT/D6/D7/CT /CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/BA/C4/CP/D8/CT/D6 /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D7/CX/D1/D4/D0/CX/CU/DD/CX/D2/CV /D6/CT/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /D3/D9/D6 /D1/D3 /CS/CT/D0/CP/CX/D1/CT/CS /D8/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /B4/BD/B8 /BE/B5/BA/BF /C8/D9/D0/D7/CT /D3/CU /D8/CW/CT /D7/CT/D0/CU/B9/CX/D2/CS/D9 /CT/CS /D8/D6/CP/D2/D7/D4/CP/D6/CT/D2 /DD /CX/D2 /D8/CW/CT/CP/CQ/D7/CT/D2 /CT /D3/CU /C3/C4/C5/CF /CT /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CP/D7/CT /BW /BP/BC /CP/D2/CS /D6/CT/D7/D8/D6/CX /D8 /D3/D9/D6/D7/CT/D0/DA /CT/D7 /D8/D3 /D8/CW/CT /CP/D7/CT /D3/CU /CW/CX/D6/D4/B9/CU/D6/CT/CT/D7/D3/D0/D9/D8/CX/D3/D2/D7/BA /BT/CU/D8/CT/D6 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BD/B5/B8 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BE/B5/D6/CT/CP/CS/D7/BM ∂a(z,t) ∂z=/bracketleftigg α−γ+iθ+δ∂ ∂t+ (1 +iD)∂2 ∂t2+σ−iβ η2\|a\|2/bracketrightigg a− γa tcohsin(ψ(z,t)), /B4/BF/B5/BG/DB/CW/CT/D6/CTψ(z,t) =t/integraltext −∞a(z,t′)dt′ /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT /B4/D2/D3/D8/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /AS/CT/D0/CS /CP/D2/CS/D8/CX/D1/CT /CP/D6/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CW/CT/D6/CT/B5/BA /CD/D2/CS/CT/D6 /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D3/D2/CS/CX/D8/CX/D3/D2 /B4/D8/CW/CT/D4/D9/D0/D7/CT /CT/D2 /DA /CT/D0/D3/D4 /CT /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/D2 /DE /B5/B8 /CP/D2 /CX/D2 /D8/CT/CV/D6/D3/B9/CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /BX/D5/BA /B4/BF/B5 /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7/D8/D3 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2  (α−γ+iθ)d dt+δd2 dt2+ (1 +iD)d3 dt3+σ−iβ η2/parenleftiggdψ(t) dt/parenrightigg2d dt ψ(t)− γa tcohsin(ψ(t)) = 0. /B4/BG/B5/C1/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D0/CP/D7/CX/D2/CV /CU/CP /D8/D3/D6/D7 /DB /CT /CW/CP /DA /CT /CP /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C3/D0/CT/CX/D2/B9/BZ/D3/D6/CS/D3/D2/B3/D7 /D8 /DD/D4 /CT /DB/CX/D8/CW2π /B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CU/D3/D6/D1 a(t) =a0sech(t/tp) /B8 /DB/CW/CT/D6/CTa0 /CX/D7 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8 tp /CX/D7 /D8/CW/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CJ/BD/BE℄/BA/BU/D9/D8 /D8/CW/CX/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /CS/D3 /CT/D7 /D2/D3/D8 /D7/CP/D8/CX/D7/CU/DD /D8/CW/CT /CU/D9/D0/D0 /BX/D5/BA /B4/BG/B5 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /C3/C4/C5/B4σ= 0 /B5/BA /BY /D9/D6/D8/CW/CT/D6 /DB /CT /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /BX/D5/BA /B4/BG/B5 /D2/CT/CV/D0/CT /D8/CX/D2/CV /D8/CW/CT /C3/C4/C5/B8 /CB/C8/C5 /CP/D2/CS/BZ/CE/BW /B4σ=β=D=θ= 0 /B5/BA/CC/CW/CT /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 ψ(t) =x /B8dψ(t)/dt=y(x) /B4/AG/D7/D5/D9/CP/D6/CT/B9/CP/D1/D4/D0/CX/D8/D9/CS/CT/AH /D6/CT/D4/D6/CT/B9/D7/CT/D2 /D8/CP/D8/CX/D3/D2/B5 /D6/CT/CS/D9 /CT/D7 /D8/CW/CT /D8/CW/CX/D6/CS/B9/D3/D6/CS/CT/D6 /BX/D5/BA /B4/BG/B5 /D8/D3 /D8/CW/CT /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D3/D2/CT/BM  /parenleftiggd2y dx2/parenrightigg y+/parenleftiggdy dx/parenrightigg2 +δdy dx+ (α−γ) y−γa tcohsin(x) = 0. /B4/BH/B5/CF /CT /D7/D3/D0/DA /CT/CS /D8/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /D2 /D9/D1/CT/D6/CX /CP/D0/D0/DD /CP/D2/CS /CU/D3/D9/D2/CS /CP /D2/D3/D2/D7/D3/D0/CX/D8/D3/D2 /B4/CX/BA/CT/BA /D2/D3/D2/B9/D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS/B5 2π /B9/D7/D3/D0/D9/D8/CX/D3/D2/D7 /CU/D3/D6 /CX/D8 /B4/D7/CT/CT /BY/CX/CV/BA /BD/B8 /DB/CW/CT/D6/CT /D2 /D9/D1/CT/D6/CX /CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7/D7/CW/D3 /DB/D2 /CX/D2 /D3/D1/D4/CP/D6/CX/D7/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D4/D9/D0/D7/CT/B5/BA /CC/CW/CX/D7 /D0/CT/CP/D6/D0/DD /CX/D2/CS/CX /CP/D8/CT/D7 /D3/D2/D7/D3/D1/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CU/CP /D8/D3/D6/D7 /DB/CW/CX /CW /CP/D6/CT /D2/CT /CT/D7/D7/CP/D6/DD /CU/D3/D6 /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2 /CU/D3/D6/D1/CP/B9/D8/CX/D3/D2 /CP/D2/CS /DB/CW/CX /CW /CP/D6/CT /CP/CQ/D7/CT/D2 /D8 /CX/D2 /BX/D5/BA /B4/BH/B5/BA/C0/D3 /DB /CT/DA /CT/D6/B8 2π /B9 /D2/D3/D2/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /DB /D3/D6/D8/CW /D1/D3/D6/CT /CS/CT/D8/CP/CX/D0/CT/CS /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2 /CU/D3/D6/CX/D8 /D1/CX/CV/CW /D8 /CQ /CT /D6/CT/D0/CT/DA /CP/D2 /D8 /D1/CT /CW/CP/D2/CX/D7/D1 /CU/D3/D6 /D9/D0/D8/CX/D1/CP/D8/CT/D0/DD /D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CX/D2 /D6/CT/CP/D0/D0/CP/D7/CT/D6/D7/BA /BT/D7 /CX/D7 /CZ/D2/D3 /DB/D2 /CJ/BG℄/B8 /D8/CW/CT /D1/CP/CX/D2 /D1/CT /CW/CP/D2/CX/D7/D1 /D3/CU /CS/CT/D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CU/D7/B9/D4/D9/D0/D7/CT/D7 /CX/D7/D8/CW/CT /D2/D3/CX/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CP/D7 /D6/CT/D7/D9/D0/D8 /D3/CU /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /CP/CQ/D7/D3/D6/CQ /CT/D6/BA /C1/D2 /D8/CW/CT /CP/D7/CT /D3/CU 2π /B9/D4/D9/D0/D7/CT /CU/D3/D6/D1/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CA/CP/CQ/CX /AT/D3/D4/D4/CX/D2/CV /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /D7/D9/D4/D4/D6/CT/D7/D7/CT/D7/D8/CW/CT /D2/D3/CX/D7/CT /CQ /CT/CW/CX/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /D8/CP/CX/D0/B8 /D8/CW /D9/D7 /D7/D8/CP/CQ/CX/D0/CX/DE/CX/D2/CV /CU/D7/B9/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CJ/BD/BC℄/BA/CC /D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /D4/D9/D0/D7/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /BX/D5/BA /B4/BH/B5 /CP/D2/CP/D0/DD/D8/CX /CP/D0/D0/DD /B8 /DB /CT /D9/D7/CT/CS /CP/CW/CP/D6/D1/D3/D2/CX /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BM ψ(x) =a1sin(x/2) +a2sin(x) +... /BA /CA/CT/D8/CP/CX/D2/CX/D2/CV/D3/D2/D0/DD /D8/CW/CT /AS/D6/D7/D8 /D8/CT/D6/D1/B8 /CX/D2 /D8/CW/CT /AG/D7/D5/D9/CP/D6/CT/B9/CP/D1/D4/D0/CX/D8/D9/CS/CT/AH /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/B8 /DB /CT /CP/D6/D6/CX/DA /CT /D8/D3/D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2a1= 2/radicalig 2(α−γ) /B8δ=γa/(2(α−γ)tcoh) /B8tp= 2/a1 /B8 /DB/CW/CX /CW /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /AG/D8/CX/D1/CT/B9/CP/D1/D4/D0/CX/D8/D9/CS/CT/AH /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/BA/CC/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT /CP/D2/CP/D0/D3/CV/D9/CT/D7 /D8/D3 /D8/CW/CP/D8 /D3/D2/CT/D7 /CU/D3/D62π/BH/D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D7/D3/D0/D9/D8/CX/D3/D2/B8 /CT/DC /CT/D4/D8 /CP/D2 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2 /CP/D6/CX/D7/CX/D2/CV /CQ /CT/D8 /DB /CT/CT/D2 /D4/D9/D0/D7/CT/CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D2/CS /D0/CP/D7/CX/D2/CV /CU/CP /D8/D3/D6/D7α /CP/D2/CSγ /BA/BY/CX/CV/BA /BE /B4 /D9/D6/DA /CT/D7 /BD /CP/D2/CS /BE/B5 /D4/D6/CT/D7/CT/D2 /D8/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8 /DB /D3 /D4/CW /DD/D7/CX /CP/D0/D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /BX/D5/BA /B4/BH/B5/BA /C7/D2/CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /D4/D6/D3 /DA/CX/CS/CT/D7 /D7/D9/CQ/B9/BD/BC/CU/D7 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D7/D8/CP/D6/D8/CX/D2/CV /CU/D6/D3/D1 /D7/D3/D1/CT /D1/CX/D2/CX/D1/CP/D0 /D4/D9/D1/D4/BA /BT/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /CU/CT/CP/D8/D9/D6/CT/D3/CU /D8/CW/CX/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT /CS/CX/AR/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2 /CP/D2/CS /D0/CX/D2/CT/CP/D6/D0/D3/D7/D7α−γ >0 /B8 /DB/CW/CX /CW /CX/D1/D4 /D3/D7/CT/D7 /CP /D6/CT/D5/D9/CX/D6/CT/D1/CT/D2 /D8 /D3/D2 /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /D1/CX/D2/CX/D1/CP/D0 /D7/CP/D8/D9/D6/CP/CQ/D0/CT/D0/D3/D7/D7γa /D2/CT /CT/D7/D7/CP/D6/DD /CU/D3/D6 /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/B8 /CX/BA/CT/BA /D8/CW/CT /D1/CX/D2/CX/D1/CP/D0 /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CS/CT/D4/D8/CW /D3/CU /CP/CQ/D7/D3/D6/CQ /CT/D6 /D8/CW/CP/D8 /D3/D2/AS/D2/CT/D7 /D8/CW/CT /D6/CT/CV/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /D0/CP/D7/CT/D6/D2/D3/CX/D7/CT/BA /BT/D7 /CX/D8 /DB /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /CJ/BE℄/B8 /D8/CW/CT /D4/D9/D0/D7/CT /CX/D7 /D7/D8/CP/CQ/D0/CT /CX/CU /D8/CW/CT /D2/CT/D8/B9/CV/CP/CX/D2 /D3/D9/D8/D7/CX/CS/CT /D4/D9/D0/D7/CT/CX/D7 /D2/CT/CV/CP/D8/CX/DA /CT /D8/CW/CP/D8 /D4/D6/D3 /CS/D9 /CT/D7 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2α−γ−γa<0 /B8 /CX/BA/CT/BAγa> α−γ /BA/CC/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /D8/CW/CT /D1/CX/D2/CX/D1/CP/D0 /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CS/CT/D4/D8/CW /D3/D2 /D8/CW/CT /D4/D9/D1/D4 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2/BY/CX/CV/BA /BF /CU/D3/D6 /D8 /DB /D3 /D4/CW /DD/D7/CX /CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /BX/D5/BA /B4/BH/B5 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /BY/CX/CV/BA /BE /B4/D0/D3 /DB /CT/D6 /D9/D6/DA /CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /DB/CX/D8/CW /D0/CP/D6/CV/CT/D6 /CS/D9/D6/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CS/CP/D7/CW/CT/CS /D9/D6/DA /CT /CS/CT/D4/CX /D8/D7/D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CW/CP/D8 /CW/CX/D2/CV /D7/CW/D3 /DB/D7 /CP /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /DE/D3/D2/CT/B5/BA /BT/D7/D3/D2/CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /CP/CV/CP/CX/D2/D7/D8 /D0/CP/D7/CT/D6 /D2/D3/CX/D7/CT /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D3/D2/D0/DD /CU/D3/D6 /D8/CW/CT/D7/D3/D0/D9/D8/CX/D3/D2 /DB/CX/D8/CW /D0/D3/D2/CV/CT/D6 /CS/D9/D6/CP/D8/CX/D3/D2/BA /CC/CW/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D6/CP/D2/CV/CT /DB/CX/CS/CT/D2/D7 /CX/D2γa /B8 /CW/D3 /DB /CT/DA /CT/D6 /CP/D8/D8/CW/CT /D3/D7/D8 /D3/CU /D4/D9/D1/D4 /CV/D6/D3 /DB/D8/CW/B8 /DB/CW/CX /CW /CX/D7 /D8/CW/CT /D3/CQ /DA/CX/D3/D9/D7 /CS/CX/D7/CP/CS/DA /CP/D2 /D8/CP/CV/CT /D3/CU /D8/CW/CX/D7 /D6/CT/CV/CX/D1/CT/BA/C6/D3 /DB /DB /CT /CX/D2 /D8/D6/D3 /CS/D9 /CT /CX/D2 /D8/D3 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D8/CW/CT /CB/C8/C5 /CP/D2/CS /D8/CW/CT /BZ/CE/BW /D8/CT/D6/D1/D7/B8 /D8/CW/CP/D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP /D6/CT/CP/D0 /CU/CT/D1 /D8/D3/D7/CT /D3/D2/CS /D0/CP/D7/CT/D6/D7/BA /BT/D7/D7/D9/D1/CX/D2/CV /CP /CW/CX/D6/D4/B9/CU/D6/CT/CT /B4/CX/BA/CT/BA /D4/D9/D6/CT/D6/CT/CP/D0/B5 /D2/CP/D8/D9/D6/CT /D3/CU /D4 /D3/D7/D7/CX/CQ/D0/CT /D7/D3/D0/D9/D8/CX/D3/D2 /DB /CT /CP/D2 /D6/CT/CS/D9 /CT /BX/D5/BA /B4/BG/B5 /D8/D3 /D8/CW/CT /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6/CT/D5/D9/CP/D8/CX/D3/D2/BM /bracketleftigg δdy(x) dx+β η2Dy(x)2+ (α−γ−θ D)/bracketrightigg y(x)−γa tcohsinx= 0. /B4/BI/B5/CF/CX/D8/CW /D8/CW/CT /CP/CQ /D3 /DA /CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CW/CP/D6/D1/D3/D2/CX /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /DB /CT /CW/CP /DA /CT /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/D7/D3/D0/D9/D8/CX/D3/D2/BMθ= 3β×a2 1/(4η2)+D(α−γ) /B8δ= 4γa/(tcoh×a2 1) /B8a1= 2/radicalig 3(α−γ) /B8 tp= 2/a1 /CU/D3/D62π /B9/D7/D5/D9/CP/D6/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2/BA /CC/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /BY/CX/CV/BA/BE /CQ /DD /D9/D6/DA /CT/D7 /BD/B3 /CP/D2/CS /BE/B3/BA /CC/CW/CT /D7/D8/CP/CQ/D0/CT /CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D2/D3/CX/D7/CT /D7/D3/D0/D9/D8/CX/D3/D2 /CW/CP/D7 /CP /D7/D0/CX/CV/CW /D8/D0/DD/D0/D3/D2/CV/CT/D6 /CS/D9/D6/CP/D8/CX/D3/D2 /D8/CW/CP/D2 /D8/CW/CT /D9/D2/D7/D8/CP/CQ/D0/CT /D3/D2/CT/BA/BY /D3/D6/D1/CP/D0/D0/DD /B8 /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CP/D7/CT/D6 /D4/CP/D6/D8 /D3/CU /D1/CP/D7/D8/CT/D6/CT/D5/D9/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /D7/CP/D1/CT /D8/CX/D1/CT /D7/CP/D8/CX/D7/AS/CT/D7 /D8/CW/CT /BU/D0/D3 /CW /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CC/CW/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD/D3/CU /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D6/CT/D7/D9/D0/D8/D7 /CU/D6/D3/D1 /D8/CW/CT /D7/CT/D0/CU/B9/CX/D2/CS/D9 /CT/CS /D8/D6/CP/D2/D7/D4/CP/D6/CT/D2 /DD /CX/D2 /CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /DB/CW/CT/D2/D8/CW/CT /D4/D9/D0/D7/CT /D4/D6/D3/D4/CP/CV/CP/D8/CT/D7 /CX/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT /D2/CT/D8/B9/CV/CP/CX/D2 /CQ/D9/D8 /D2/D3/CX/D7/CT /CX/D7/D7/D9/D4/D4/D6/CT/D7/D7/CT/CS /CS/D9/CT /D8/D3 /CA/CP/CQ/CX /AT/D3/D4/D4/CX/D2/CV /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2/BA /CF /CT /CS/D3 /D2/D3/D8/CP/D2/CP/D0/DD/DE/CT /D8/CW/CT /CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CJ/BD/BF℄ /D3/CU /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D7/CX/D2 /CT /CX/D8 /CW/CP/D7 /CQ /CT/CT/D2/D8/CT/D7/D8/CX/AS/CT/CS /CS/CX/D6/CT /D8/D0/DD /CQ /DD /D8/CW/CT /D2 /D9/D1/CT/D6/CX /CP/D0 /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /CX/D2 /CJ/BD/BC℄/BA/CC/CW /D9/D7 /DB /CT /CP/D2 /D3/D2 /D0/D9/CS/CT/B8 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /D2/D3/D8 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/B9/D8/D3/D2 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /C3/CT/D6/D6/B9/D0/CT/D2/D7/B9/CX/D2/CS/D9 /CT/CS /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2/BA /BU/D9/D8 /D8/CW/CT/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D2/D3/D2/D7/D3/D0/CX/D8/D3/D2 2π /B9/D4/D9/D0/D7/CT /D8/CP/CZ /CT/D7 /D4/D0/CP /CT/B8 /DB/CW/CX /CW /CX/D1/D4 /D3/D7/CT/D7 /CP /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2 /D3/D2/BI/D1/CX/D2/CX/D1/CP/D0 /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CS/CT/D4/D8/CW /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /DB/CX/D8/CW /D7/D9/CQ/D7/CT/D5/D9/CT/D2 /D8 /CV/D6/D3 /DB/D8/CW /D3/CU /CV/CT/D2/B9/CT/D6/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /C1/D2 /D8/CW/CT /D2/CT/DC/D8 /D7/CT /D8/CX/D3/D2 /DB /CT /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D3/CU /C3/C4/C5/BA/BG /BV/D3/CW/CT/D6/CT/D2 /D8 2π /B9/D7/D3/D0/CX/D8/D3/D2 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /C3/C4/C5/CC/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /C3/C4/C5 /CX/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT /D8/CT/D6/D1σ\|a\|2/CX/D2 /BX/D5/BA /B4/BF/B5/BA /C1/D2 /D8/CW/CX/D7 /CP/D7/CT /D8/CW/CT/D6/CT /CX/D7 /CP2π /B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BF/B5/B8 /CW/D3 /DB /CT/DA /CT/D6 /CU/D3/D6 /CP /D7/D8/D6/CX /D8 /D6/CT/D0/CP/D8/CX/D3/D2/CQ /CT/D8 /DB /CT/CT/D2σ /CP/D2/CSη /B8 /D7/D3 /D8/CW/CP/D8σ=η2 2 /BA /CC/CW/CT /D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CW/CP/D7 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BM a0=2 tp,tp=1√γ−α,δ=γa tcoh(γ−α). /B4/BJ/B5/CC/CW/CT/D6/CT /CP/D6/CT /D8 /DB /D3 /CS/CX/D7/D8/CX/D2 /D8 /CU/CT/CP/D8/D9/D6/CT/D7 /D3/CU2π /B9/D7/D3/D0/CX/D8/D3/D2 /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/BM /BD/B5 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 α−γ <0 /CX/D7 /D7/CP/D8/CX/D7/AS/CT/CS /CP/D2/CS/B8 /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /B8 /D8/CW/CT/D6/CT /CX/D7 /D2/D3 /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D1/CX/D2/CX/D1/CP/D0/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CS/CT/D4/D8/CW /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/BN /BE/B5 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/CX/D7 /D4/D6/CT /CX/D7/CT/D0/DD /D8/CW/CT /D7/CP/D1/CT /CP/D7 /CU/D3/D6 /D8/CW/CT /CP/D7/CT /D3/CU /D4/D9/D6/CT /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D1/D3 /CS/CT/D0/D3 /CZ/CX/D2/CV /CJ/BE℄/B8 /D8/CW/CP/D8 /D7/D9/CV/CV/CT/D7/D8/D7 /D8/CW/CP/D8 /D8/CW/CT /C3/C4/C5 /CX/D7 /D8/CW/CT /D1/CP/CX/D2 /D1/CT /CW/CP/D2/CX/D7/D1 /CS/CT/D8/CT/D6/D1/CX/D2/CX/D2/CV/D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/BA /CC/CW/CX/D7 /D3/D2 /D0/D9/D7/CX/D3/D2 /D3/D6/D6/D3/CQ /D3/D6/CP/D8/CT/D7 /DB/CX/D8/CW /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /D6/CT/CU/BA /CJ/BD/BD℄/BA/CC/CW/CT /CP /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /CS/CT/D8/CT/D6/D1/CX/D2/CT/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/CT/D0/CP /DDδ /CP/D2/CS /CX/D1/D4 /D3/D7/CT/D7/CP /D6/CT/D7/D8/D6/CX /D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT/B8 /CX/BA /CT/BA /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/CP/D2/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT/BA /C8/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /C3/C4/C5 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BE/CQ /DD /CS/D3/D8/D8/CT/CS /D9/D6/DA /CT /BF/BA /BT/D7 /CX/D7 /D7/CT/CT/D2/B8 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D1 /D9 /CW /D7/CW/D3/D6/D8/CT/D6 /D8/CW/CP/D2 /CU/D3/D6/D8/CW/CT /CP/D7/CT /DB/CX/D8/CW /D2/D3 /C3/C4/C5/B8 /CT/D7/D4 /CT /CX/CP/D0/D0/DD /CU/D3/D6 /D8/CW/CT /D7/D1/CP/D0/D0 /D4/D9/D1/D4/BA /BT/D7 /D8/CW/CT /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D3/CU /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /CX/D7 /CX/D2 /D6/CT/CP/D7/CT/CS /B4η /CP/D4/D4/D6/D3/CP /CW/CT/D7 /BD/B8 /D9/D6/DA /CT /BD /CX/D2 /BY/CX/CV/BA /BG/B5/D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D6/CT/CS/D9 /CT/CS /CS/D3 /DB/D2 /D8/D3 /D8/CW/CT /D0/CX/D1/CX/D8 /D3/CU /D8/CW/CT /DA /CP/D0/CX/CS/CX/D8 /DD /D3/CU /D8/CW/CT /D7/D0/D3 /DB/D0/DD/DA /CP/D6/DD/CX/D2/CV /CT/D2 /DA /CT/D0/D3/D4 /CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BA/BT/D2 /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /D3/CU /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /C3/CT/D6/D6/B9/D0/CT/D2/D7/B9/CX/D2/CS/D9 /CT/CS /CP/CQ/D7/D3/D6/CQ /CT/D6 /CP/D2/CS /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /CX/D7 /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM /CP /D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /D7/CP/D8/CX/D7/AS/CT/D7 /CQ /D3/D8/CW /D4/D9/D6/CT /D0/CP/D7/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /BU/D0/D3 /CW /CT/D5/D9/CP/D8/CX/D3/D2/D7/B8/CQ/D9/D8 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CS/CX/D7 /D6/CX/D1/CX/D2/CP/D8/CT/D7 /D8/CW/CT /D7/D4 /CT /CX/CP/D0 /CP/D7/CT/D7 /D3/CUnπ /B9/D7/D5/D9/CP/D6/CT/CS/D4/D9/D0/D7/CT/D7/B8 /CX/D2 /D4/CP/D6/D8/CX /D9/D0/CP/D6 2π /B9/D4/D9/D0/D7/CT/D7/B8 /DB/CW/CX /CW /CP/D6/CT /D4/D6/D3 /DA/CX/CS/CT/CS /CQ /DDσ=η2/2 /D6/CT/D0/CP/D8/CX/D3/D2/BA/C4/CT/D8 /D9/D7 /D7/D8/D9/CS/DD /CP/D2 /CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /D8/CW/CT /D0/CP/D7/CT/D6 /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/D8/D3/D2/B8/DB/CW/CX /CW /DB /CT /D7/CW/D3 /DB /CT/CS /CQ /CT/CU/D3/D6/CT /D8/D3 /CQ /CT /DA /CT/D6/DD /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /CU/CP /D8/D3/D6 /CX/D2 /CU/D7/B9/D0/CP/D7/CT/D6/D7 /CJ/BD/BF℄/BA /CF /CT/D9/D7/CT/CS /CP/D2 /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2/D0/CT/D7/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B8 /DB/CW/CX /CW /CP/D7/D7/D9/D1/CT/D7 /CP/D2 /D9/D2 /CW/CP/D2/CV/CT/CS /CU/D3/D6/D1 /D3/CU/D7/D3/D0/D9/D8/CX/D3/D2 /CP/D2/CSz /B9/CS/CT/D4 /CT/D2/CS/CT/D2 /CT /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /CC/CW/CT /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D4/D9/D0/D7/CT /CT/D2 /DA /CT/D0/D3/D4 /CT /CX/D2 /BX/D5/BA /B4/BF/B5 /DB/CX/D8/CW /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D2 /D8/D3 /D8/CW/CT /D8/CX/D1/CT /D7/CT/D6/CX/CT/D7 /DD/CX/CT/D0/CS/D7/BM da0 dz= 2(α−γ)η2t2 p−η2+ 4σ η2t3p,dtp dz= 4η2−2σ a0η2t2p,/BJθ=2(D+ 4β η2) a0t3p,δ= 2γatp tcoha0. /B4/BK/B5/BX/D5/D7/BA /B4/BK/B5 /DB /CT/D6/CT /CS/CT/D6/CX/DA /CT/CS /CU/D3/D6 /CP /CW/CX/D6/D4/B9/CU/D6/CT/CT /D7/D3/D0/D9/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2D= −2β/η2/CT/DC/CP /D8/D0/DD /D3/D1/D4 /CT/D2/D7/CP/D8/CX/D2/CV /CU/D3/D6 /CB/C8/C5/BA /CC /D3 /CQ /CT /D7/CT/D0/CU/B9 /D3/D2/D7/CX/D7/D8/CT/D2 /D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /B4/BK/B5/D7/CW/D3/D9/D0/CS /CQ /CT /D3/D1/D4/D0/CT/D8/CT/CS /CQ /DD /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CP/D1/B9/D4/D0/CX/D8/D9/CS/CT /CP/D6/CX/D7/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /BU/D0/D3 /CW /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /BT/CU/D8/CT/D6 /D7/D3/D1/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /DB /CT /CW/CP /DA /CT/D8/CW/CT /CT/DC/D4/D0/CX /CX/D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7/B8 /DB/CW/CX /CW /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /BA /CC/CW/CT /D4/D9/D0/D7/CT /CX/D7/D7/D8/CP/CQ/D0/CT /CX/CU /D8/CW/CT /C2/CP /D3/CQ /CT/CP/D2 /D3/CU /D8/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D7/CX/CS/CT/D7 /D3/CU /AS/D6/D7/D8 /D8 /DB /D3 /BX/D5/D7/BA /B4/BK/B5 /CW/CP/D7 /D3/D2/D0/DD/D2/D3/D2/B9/D4 /D3/D7/CX/D8/CX/DA /CT /CT/CX/CV/CT/D2 /DA /CP/D0/D9/CT/D7/BA /CC/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /CP/D1/D4/D0/CX/D8/D9/CS/CT /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /CS/CT /CP /DD −4(γ−α)2<0 /CX/D7 /D7/CP/D8/CX/D7/AS/CT/CS /CP/D9/D8/D3/D1/CP/D8/CX /CP/D0/D0/DD /BA /BY /D3/D6σ=η2/2 /D8/CW/CT /D4/D9/D0/D7/CT /D4 /D3/D7/D7/CT/D7/D7/CT/D7 /CP/D1/CP/D6/CV/CX/D2/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/BA /C0/D3 /DB /CT/DA /CT/D6 /CU/D3/D6 /CP/D2 /DDσ<η2/2/D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/D2/CS/CX/D8/CX/D3/D2 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D7/CP/D8/CX/D7/AS/CT/CS/BA/CC/CW /D9/D7/B8 /DB /CT /CW/CP /DA /CT /CP/D2/CP/D0/DD/DE/CT/CS /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D7 /D3/CU2π /B9/D7/D3/D0/CX/D8/D3/D2/D7 /CV/CT/D2/CT/D6/CP/D8/CT/CS /CX/D2/CU/D7 /C3/C4/C5 /D0/CP/D7/CT/D6/D7 /DB/CX/D8/CW /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/CQ /CT/D6/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CT/D5/D9/CP/D8/CX/D3/D2/D7/CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CX/D7 /D4/CW /DD/D7/CX /CP/D0 /D7/CX/D8/D9/CP/D8/CX/D3/D2 /CP/D0/D0/D3 /DB /DD /CT/D8 /CP/D2/D3/D8/CW/CT/D6 /D8 /DD/D4 /CT /D3/CU /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA/BH /BV/D3/CW/CT/D6/CT/D2 /D8π /B9/D7/D3/D0/CX/D8/D3/D2 /CP/D2/CS /CW/CX/D6/D4 /CT/CS /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/D7/CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /C3/C4/C5/BT/D7 /D3/D2/CT /CP/D2 /D7/CT/CT /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D4/CP/D6/D8 /D3/CU /D3/D9/D6 /DB /D3/D6/CZ/B8 /D8/CW/CT /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CX/D7/D8/CW/CT /D7/D3/D0/CX/D8/D3/D2 /CU/D3/D6 /D8/CW/CT /CQ /D3/D8/CW /D0/CP/D7/CT/D6 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /BU/D0/D3 /CW/CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /C6/D3 /DB /DB /CT /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D3/D1/D4/D0/CT/DC /CP/D2/D7/CP/D8/DE /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /CP /D4/D9/D0/D7/CT /DB/CX/D8/CW /CW/CX/D6/D4ζ /B8a(t) =a0sech(t tp)1−iζ/BA /C1/D8 /CX/D7 /CZ/D2/D3 /DB/D2 /CJ/BD/BE℄/B8 /D8/CW/CP/D8 /D8/CW/CT /BX/D5/D7/BA /B4/BD/B5 /CW/CP /DA /CT /CP/D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CX/D2 /CU/D3/D6/D1 /D3/CU /CW/CX/D6/D4/B9/CU/D6/CT/CTπ /B9/D4/D9/D0/D7/CT /D3/D6 /D7/CT /CW /B9/D7/CW/CP/D4 /CT/CS /CW/CX/D6/D4 /CT/CS/D7/D3/D0/D9/D8/CX/D3/D2/B8 /DB/CW/CT/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CW/D3/D0/CS/BM u(t) =u0sech(t tp), v(t) =v0sech(t tp), w(t) = tanh(t tp), dφ(t) dt=ζ tptanh(t tp),/DB/CW/CT/D6/CT a0=√ 1+ζ2 tp, u0=−1√ 1+ζ2, v0=ζ√ 1+ζ2./BT /CW/CX/D6/D4/B9/CU/D6/CT/CT /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /CPπ /B9/D7/D3/D0/CX/D8/D3/D2/B8 /DB/CW/CX /CW /CX/D7 /D3/CQ /DA/CX/D3/D9/D7/D0/DD /D9/D2/D7/D8/CP/CQ/D0/CT /CX/D2 /D8/CW/CT/CP/CQ/D7/D3/D6/CQ /CT/D6 /D7/CX/D2 /CT /D8/CW/CT /CU/D9/D0/D0 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CX/D2 /DA /CT/D6/D7/CX/D3/D2 /CQ /CT/CW/CX/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /D8/CP/CX/D0 /CP/D1/D4/D0/CX/AS/CT/D7/D8/CW/CT /D2/D3/CX/D7/CT/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CP/D2/D3/D8/CW/CT/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CU/CP /D8/D3/D6/D7 /CX/D2 /C3/C4/C5/B9/D0/CP/D7/CT/D6 /CP/D2 /D7/D8/CP/CQ/CX/D0/CX/DE/CT /D8/CW/CT/D4/D9/D0/D7/CT /CP/D2/CS /D8/CW/CX/D7 /D6/CT/D5/D9/CX/D6/CT/D7 /CP /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2/BA/C8 /CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /CW/CX/D6/D4/B9/CU/D6/CT/CTπ /B9/D7/D3/D0/CX/D8/D3/D2 /CX/D2 /C3/C4/C5/B9/D0/CP/D7/CT/D6 /CP/D6/CT/BM a0=1 tp, tp=1√γ−α, D=−β 2η2, θ=β(γ−α) 2η2, σ= 2η2. /B4/BL/B5/BK/CC/CW/CX/D7 /CX/D7 /DA /CT/D6/DD /D7/CX/D1/CX/D0/CP/D6 /D8/D32π /B9/D7/D3/D0/D9/D8/CX/D3/D2 /B4/BJ/B5/B8 /CW/D3 /DB /CT/DA /CT/D6 /DB/CX/D8/CW /D7/D3/D1/CT /CS/CX/AR/CT/D6/CT/D2 /CT/D7/BA /BT/D7/D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D8/CW/CTπ /B9/D7/D3/D0/CX/D8/D3/D2 /CX/D7 /D8 /DB /D3 /D8/CX/D1/CT/D7 /D7/D1/CP/D0/D0/CT/D6 /D8/CW/CP/D2 /CU/D3/D6 /D8/CW/CT2π /B9/D7/D3/D0/CX/D8/D3/D2/B8 /CP /C3/C4/C5/B9/D4/CP/D6/CP/D1/CT/D8/CT/D6 /CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D4/D6/D3 /CS/D9 /CT /D8/CW/CT /D7/CP/D1/CT /CT/AR/CT /D8 /CP/D2/CS /D7/D9/D4/D4 /D3/D6/D8/D7/D3/D0/CX/D8/D3/D2 /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D7/CW/D3/D9/D0/CS /CQ /CT /CX/D2 /D6/CT/CP/D7/CT/CS /CU/D3/D9/D6 /D8/CX/D1/CT/D7 /CP/D2/CS /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D7/CW/D3/D9/D0/CS/CQ /CT /CS/CT /D6/CT/CP/D7/CT/CS /CP /D3/D6/CS/CX/D2/CV/D0/DD /BA/BV/D9/D6/DA /CT /BG /CX/D2 /BY/CX/CV/BA /BE /CS/CT/D4/CX /D8/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CU/D3/D6π /AL/D7/D3/D0/CX/D8/D3/D2/BA /BT/D7 /CX/D7 /D7/CT/CT/D2/B8 /D8/CW/CT/D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /D7/D0/CX/CV/CW /D8/D0/DD /CS/CX/AR/CT/D6/D7 /CU/D6/D3/D1 /D8/CW/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CU/D3/D62π /B9/D7/D3/D0/CX/D8/D3/D2 /B4 /D9/D6/DA /CT /BE/B5 /CP/D2/CS/CX/D7 /D7/CW/D3/D6/D8/CT/D6 /CX/D2 /D8/CW/CT /D6/CT/CV/CX/D3/D2 /D3/CU /D7/D1/CP/D0/D0η /B4 /D9/D6/DA /CT /BE /CX/D2 /BY/CX/CV/BA /BG/B5/BA/BT/D7 /DB /CP/D7 /D7/CP/CX/CS /CQ /CT/CU/D3/D6/CT/B8 /CPπ /B9/D7/D3/D0/CX/D8/D3/D2 /CX/D2 /DA /CT/D6/D8/D7 /D8/CW/CT /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CS/CX/AR/CT/D6/CT/D2 /CT /CX/D2 /D8/CW/CT/CP/CQ/D7/D3/D6/CQ /CT/D6 /D8/CW/CP/D8 /CP/D9/D7/CT/D7 /D8/CW/CT /D2/D3/CX/D7/CT /CP/D1/D4/D0/CX/AS /CP/D8/CX/D3/D2 /CQ /CT/CW/CX/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /D8/CP/CX/D0/BA /C0/CT/D2 /CT/B8 /D8/CW/CT/D0/CP/D7/CT/D6 /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /CX/CU /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2α+γa−γ <0 /CX/D7 /D7/CP/D8/CX/D7/AS/CT/CS/BA/CC/CW/CX/D7 /CS/CT/AS/D2/CT/D7 /D8/CW/CT /D1/CP/DC/CX/D1/CP/D0 /CX/D2/CX/D8/CX/CP/D0 /D0/D3/D7/D7 /CX/D2 /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /B4/CS/D3/D8/D8/CT/CS /D9/D6/DA /CT /CX/D2 /BY/CX/CV/BA /BF/B5/BA/C1/D8 /CX/D7 /D7/CT/CT/D2/B8 /D8/CW/CP/D8 /D8/CW/CT /D1/CP/DC/CX/D1/CP/D0 /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CS/CT/D4/D8/CW /CT/DC /CT/CT/CS/D7 /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /B4/CS/CP/D7/CW/CT/CS /D9/D6/DA /CT/B5 /CP/D2/CS/B8 /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /B8 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CTπ /B9/D7/D3/D0/CX/D8/D3/D2 /CX/D7 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arXiv:physics/0009007v1 [physics.flu-dyn] 2 Sep 2000Breaking of vortex lines - a new mechanism of collapse in hydr odynamics E.A.Kuznetsov†and V.P.Ruban∗ L.D.Landau Institute for Theoretical Physics, 2 Kosygin st r., 117334 Moscow, Russia. A new mechanism of the collapse in hydrodynamics is suggested, due to breaking of continuously distributed vor - tex lines. Collapse results in formation of the point singu- larities of the vorticity ﬁeld \|Ω\|. At the collapse point, the value of the vorticity blows up as ( t0−t)−1wheret0is a collapse time. The spatial structure of the collapsing dist ri- bution approaches a pancake form: contraction occurs by the lawl1∼(t0−t)3/2along the ”soft” direction, the characteris- tic scales vanish like l2∼(t0−t)1/2along two other (”hard”) directions. This scenario of the collapse is shown to take pl ace in the integrable three-dimensional hydrodynamics with th e Hamiltonian H=/integraltext \|Ω\|dr. Most numerical studies of col- lapse in the Euler equation are in a good agreement with the proposed theory. PACS: 47.15.Ki, 47.32.Cc I. INTRODUCTION Collapse in hydrodynamics of an ideal incompressible ﬂuid, as a process of singularity formation in a ﬁnite time, is one of the central problems in the theory of developed hydrody- namic turbulence. The classical examples of such type spec- tra are the Phillips spectrum for water-wind waves [2] and th e Kadomtsev-Petviashvili spectrum for acoustic turbulence [3]. In the ﬁrst case white caps – wages of water surface – play a role of singularities, and in the second case these are densi ty breaks (or shocks). The question about collapse in hydrodynamics is an old problem. For example, in 1981 P.Saﬀman [4] considered col- lapse as one of the most important problems in hydrodynam- ics (see also papers [6] and references therein), probably a l- ready L.Richardson and A.N.Kolmogorov understood an im- portance of this problem. In spite of so long a history of the question, there is no deep understanding of the nature of a collapse in hydrodynamics, even though there are many numerical simulations testifying to the collapse existenc e col- lected by now. As far as theory is concerned, essential resul ts are absent, and, moreover, there is no common agreement about collapse as a subject for incompressible hydrodynami cs (see, e.g., Sec. 7.8 of the book of U.Frisch [5] and reference s therein). In the theory, the only appreciable exclusion is t he work of V.E.Zakharov, 1988 [7] (more detailed publication h as appeared in 1999 [8]), where the consistent theory of collap se for two anti-parallel vortex ﬁlaments of small thickness wa s developed in quasi-two-dimensional approximation, when a ﬂow is almost two-dimensional with a slow dependence with respect to the third coordinate (see also the paper [9]). A signiﬁcant progress in studying hydrodynamic collapse has been achieved in numerical simulations of the Euler equatio n. Many numerical experiments testify that the value of the vor - ticity\|Ω\|becomes inﬁnite in isolated points in a ﬁnite time. As follows from the papers of Kerr [10], Grauer, Marlianiand Germashewsky [11], Pelz [12], Boratav and Pelz [13], \|Ω\| grows at the collapse point like ( t0−t)−1, wheret0is a collapse time. According to [10], [13], a spatial scale of the collaps - ing distribution contracts as ( t0−t)1/2. In the recent paper by Kerr [17], an anisotropy of the collapsing region has been reported. The data processing gave two scales, one of them being contracted as the root l1∼(t0−t)1/2, and another one asl2∼t0−t. It should be noted that the initial ﬂow either possessed a deﬁnite symmetry or it was close to a symmetric ﬂow in most numerical simulations. As a result, several sin- gularities arose simultaneously. For example, the evoluti on of two anti-parallel vortex tubes was studied in [10]. The col- lapse here is caused by the Crow instability [14] leading at t he nonlinear stage to the vortex reconnection. Symmetry of the ﬂow makes the collapse to happen in two symmetric points. In the present paper, we suggest a new mechanism of the singularity formation connected with breaking of continuo usly distributed vortex lines. This mechanism is not related to a ny symmetry of the initial vorticity. The collapse itself is po ssible in one separate point. Probably, just this type of collapse h as been observed in the recent numerical experiment [15]. The mechanism suggested can be naturally incorporated into the classical catastrophe theory [16]. From this point of view, collapse can be considered as caustic formation for a solenoidal ﬁeld. It is not so easy to understand, how a collap se arises in the Eulerian description. First, this is connecte d with a hidden symmetry of the Euler equation, i.e., the relabelin g symmetry (for more details see the reviews [18], [19]). This symmetry generates the conservation law for the Lagrangian invariant – the so-called Cauchy invariant, which is expres sed through the velocity curl and the Jacoby matrix of a map- ping from the Eulerian variables to the Lagrangian ones, and by this reason this invariant occurs very nonlocal in terms o f the velocity ﬁeld. On the one hand, the Cauchy invariant is known as invariant which characterizes the property of froz en- ness of vortex lines into a ﬂuid. On the other hand, all known conservation laws for vorticity, such as the Kelvin’s and Er - tel’s theorems, conservation of the topological Hopf invar iant, being a measure of the ﬂow knottiness, are a simple conse- quence of the Cauchy invariant constancy. The frozenness of vortex lines means that ﬂuid particles are pasted to a given vortex line and never leave it. A destruction of frozenness is possible only due to viscosity, i.e., beyond ideal hydrod y- namics. Therefore, as a next natural step in the vortex mo- tion description, a mixed Lagrangian-Eulerian descriptio n has been introduced, where the main object is a vortex line [20], [21]. Each vortex line in this description is labeled by a two - dimensional Lagrangian marker, while the third coordinate serves as a parameter determining the curve. This represen- tation, which we called as the vortex line representation, i s a key-point in the description of hydrodynamic collapse whic h can be considered as a process of a caustic formation for the solenoidal vorticity ﬁeld. The paper is organized as follows: in Sec.II we introduce 1the vortex line representation and explain its meaning. In Sec.III we consider the 3D integrable hydrodynamic model introduced in our previous paper [20]. The Hamiltonian of this model is unusual, it is expressed through the absolute value of the vorticity \|Ω\| H=/integraldisplay \|Ω\|dr. (1.1) The given model can be integrated by means of combina- tion of the vortex line representation and the inverse scat- tering transform. By applying the vortex line representa- tion, the Hamiltonian is decomposed into a sum of Hamiltoni- ans of non-interacting vortex lines. Dynamics of each vorte x line is described by the integrable one-dimensional Landau - Lifshitz equation for a Heisenberg ferromagnet or by its gau ge- equivalent - the nonlinear Schroedinger equation. Thus, th e integrable hydrodynamics represents a hydrodynamics of fr ee vortex lines. As for hydrodynamics of free particles – hy- drodynamics of dust with a null pressure (see, e.g. [22]), fo r the 3D integrable hydrodynamics typical singularities are also caustics. For hydrodynamics of dust, density turns into inﬁ n- ity at caustics. Unlike the dust density, being a scalar char ac- teristics, vorticity is a vector solinoidal ﬁeld. Therefor e, the latter imposes some restrictions on a spatial structure nea r singularity. As it is shown in Sec.IV, the singularity struc - ture, being very anisotropic, turns into a pancake form. The spatial collapsing distribution at t→t0leads to quasi-two- dimensional. Along the ”soft” direction a more rapid com- pression takes place as l1∼(t0−t)3/2, and along two other– ”hard” – directions l2∼(t0−t)1/2. At the collapse time, the vorticity vector lies in the pancake plane and its value \|Ω\|blows up like ( t0−t)−1. This behavior corresponds to the general situation. The degenerated case is considered i n Sect. V, where we consider collapse for topologically nontr iv- ial axi-symmetric distribution of vorticity in the form of t he so-called Hopf mapping when any two vortex lines are linked once with each other. In this case two eigen-values of the Ja- cobi matrix vanish simultaneously in the collapse point. As its sequence, vorticity occurs to have more strong singular ity: \|Ω\| ∼(t0−t)−2. And we conclude in Sec. VI with discussion of numerical experiments on collapse observation in the Eul er equation and their correspondence to the proposed theory. II. VORTEX LINES REPRESENTATION OF HYDRODYNAMICS Let us consider the equations of the hydrodynamic type ∂Ω ∂t= curl/bracketleftBig curlδH δΩ×Ω/bracketrightBig , (2.1) where H{Ω}is the Hamiltonian of a system, Ω(r,t) = curlp(r,t) is the generalized vorticity, pis the canonical mo- mentum. The vector ﬁeld v= curl(δH/δΩ) (2.2) is nothing else but the ﬂuid velocity. By the deﬁnition divv= 0, i.e., we deal with an incompressible ﬂuid. If theHamiltonian coincides with the kinetic energy of the ﬂuid H=/integraldisplay p2 2dr=/integraldisplay /integraldisplay Ω(r1)·Ω(r2) 8π\|r1−r2\|dr1dr2 then the expression (2.2) yields the usual relation Ω= curl v between velocity vand vorticity Ω, while Eq.(2.1) transforms into the Euler equation for vorticity ∂Ω ∂t= curl[ v×Ω],divv= 0. The important property of the equation (2.1) is the frozenne ss of vorticity into the substance, i.e. all Lagrangian ﬂuid pa r- ticles att>0 remain at their own vortex line. After that, it is natural to introduce a mixed Lagrangian-Eulerian descri p- tion when each vortex line is labeled by its own Lagrangian markerν, which lies in a ﬁxed two-dimensional manifold N, while a parameter salong the line has a meaning of an Eu- lerian variable. In such vortex line representation vortic ity is expressed as follows [20] Ω(r,t) =/integraldisplay NΩ0(ν)d2ν/integraldisplay δ(r−R(s,ν,t))∂R ∂sds. (2.3) Here, the closed curve r=R(s,ν,t) corresponds to each vor- tex lineν, so that Rsis its tangent vector. The quantity Ω0(ν), with Rsbeing its tangent vector. The ﬁxed function Ω0(ν) is the strength of vortex loop. However, without loss of generality, this function can be put equal to the unity. This can be achieved by both an appropriate re-deﬁnition of label s νand changing the vortex orientation to the opposite one for those lines from the manifold N, for which Ω 0(ν)<0. There- fore, in the next section we will omit the multiplier Ω 0(ν) in front ofd2νin the corresponding formulae. The generalization of Eq. (2.3) to the case of arbitrary topology of the vortex lines is served by the formula: Ω(r,t) =/integraldisplay δ(r−R(a,t))(Ω0(a)∇a)R(a,t)d3a,(2.4) where Ω0(a) is the Cauchy invariant, characterizing the frozenness property. The div aΩ0(a) = 0 condition guar- antees automatically incompressibility for the ﬁeld Ω(r,t): divΩ(r) = 0. In the expression (2.4), the vector b= (Ω0(a)∇a)R(a,t) is a tangent vector to the vortex line at the point r=R(a,t). (2.5) In the representation (2.3), an arc-length of line of the ini tial ﬁeldΩ0(a) can serve as the parameter s. After integrating (2.4) over a-variables, the vector Ω(r,t) is expressed through the Jacobian Jof the mapping (2.5) J= det\|\|∂R/∂a\|\|and the Cauchy invariant Ω0(a): Ω(R) =1 J(Ω0(a)∇a)R(a). (2.6) It is important to emphasize, that in this expression the Ja- cobian is not to be equal to the unity: J/ne}ationslash= 1. Nevertheless, it does not contradict to the incompressibility condition for the ﬂuid. 2As it was shown in [20,21], the equations of motion for vortex lines can be obtained directly from the equation of frozenness (2.1) [{(Ω0(a)∇a)R(a,t)} × {Rt(a,t)−v(R(a,t),t)}] = 0.(2.7) This equation describes the transverse dynamics of vortex line: obviously any motion along a curve leaves the curve un- changed. In particular, this helps to understand why there a re no restrictions imposed on the value of the Jacobian J. Let us recall that in a purely Lagrangian description of an incom - pressible ﬂuid Jacobian is equal to the unity identically. B ut accordingly to the equation (2.7), the motion of Lagrangian particles along vortex lines is excluded from the mapping (2 .5) for the mixed Lagrangian-Eulerian description. It is exact ly the reason why the Jacobian is not necessarily equal to unity now. This point is principal and will be instrumental below in explaining how collapse is possible in the hydrodynamic systems (2.1). As it was shown in [20,21], Eq.(2.7) can be written in the Hamiltonian form [{(Ω0(a)∇a)R(a,t)} ×Rt(a)] =δH{Ω{R}} δR(a)/vextendsingle/vextendsingle/vextendsingle Ω0.(2.8) This equation describes a motion of vortex lines in systems with an arbitrary Hamiltonian that depends on Rthrough theΩ(r,t) only. It is useful also to keep in mind that the expressions for such important characteristics of the system as its momentu m P=/integraltext pdrand angular momentum M=/integraltext [r×p]dr, being transformed by integration by parts to a form, where, instea d ofp, the vorticity Ωis employed, and being then rewritten in terms of vortex lines, have the form P∼1 2/integraldisplay [r×Ω]dr=/integraldisplay Nd2ν1 2/integraldisplay [R×Rs]ds (2.9) M∼1 3/integraldisplay [r×[r×Ω]]dr=/integraldisplay Nd2ν1 3/integraldisplay [R×[R×Rs]]ds (2.10) The∼-sign in these relations means that equalities take place up to integrals over surface with inﬁnitely large radius. He nce one can see that the momentum and the angular momentum are composed of momenta and angular momenta of each vor- tex line, the momentum of a closed line being equal to the oriented area of a surface tightened on the vortex loop. It is easily to verify that uniform shift R0ofRdoes not change the momentum, while the angular momentum is sub- jected to the well known transformation M→M′= [R0×P] +M. (2.11) III. INTEGRABLE HYDRODYNAMICS In this and two next sections, we will show how and why collapse is possible in 3D integrable hydrodynamics. Thismodel was introduced in our previous paper [20]. The Hamil- tonian of this model is expressed through the absolute value ofΩ(r,t) H=/integraldisplay \|Ω(r)\|dr, (3.1) and the equation of motion coincides with the frozenness equation (2.1) with velocity v= curl/vector τ where/vector τ= (Ω/Ω) is the unit tangent vector along the vortex line. Assuming all the lines closed, choosing the labeling b y such a way so that Ω 0(ν) = 1, and substituting the represen- tation (2.3) into (3.1), it is easy to see that the Hamiltonia n is decomposed as a sum of Hamiltonians for the vortex lines 1: H{R}=/integraldisplay d2ν/integraldisplay/vextendsingle/vextendsingle/vextendsingle∂R ∂s/vextendsingle/vextendsingle/vextendsingleds. (3.2) Here, the integral over sis the length of the vortex line with the indexν. The equation of motion for the vector R(ν,s), in accordance with the Eq. (2.8), is local in these variables – it doesn’t contain an interaction with other vortices: [Rs×Rt] = [/vector τ×[/vector τ×/vector τs]]. (3.3) By this reason, not only the total energy, momentum, and angular momentum are conserved, but also the corresponding geometrical invariants for each vortex loop: its length H(ν) =/integraldisplay \|Rs(ν)\|ds, the oriented area spanned on the vortex loop which coincides with its momentumits momentum P(ν) =1 2/integraldisplay [R(ν)×Rs(ν)]ds, and its angular momentum M(ν) =1 3/integraldisplay [R(ν)×[R(ν)×Rs(ν)]]ds. It is important to pay attention to the following fact: The equation (3.3) is invariant with respect to changes s→˜s(s,t). Therefore, it can be solved for Rtup to a shift along the vortex line – the transformation leaves the vorticity Ωunchanged. This means that to ﬁnd the vorticity Ωit is enough to have one solution of the equation \|Rs\|Rt= [/vector τ×/vector τs] +βRs (3.4) which follows from Eq.(3.3) for some choice β. This leads to an equation for /vector τas a function of the ﬁlament length l 1It is worth to notice that this property is common for all systems with the Hamiltonians of the type H=/integraltext F(τ,(τ∇)τ,(τ∇)2τ,...)\|Ω\|dr. To explain the idea of col- lapse of vortex lines, we have chosen the simplest example (3.1), which has a physical meaning. 3(dl=\|Rs\|ds) and time t(by choosing a new value β= 0), which reduces to the integrable one-dimensional (1D) Landa u- Lifshits equation for a Heisenberg ferromagnet ∂/vector τ ∂t=/bracketleftbigg /vector τ×∂2/vector τ ∂l2/bracketrightbigg . (3.5) This equation, in its turn, is gauge equivalent to the 1D non- linear Shr¨ odinger equation [23] iψt+ψll+ (1/2)\|ψ\|2ψ= 0 (3.6) and, for instance, can be reduced to the NLSE by means of the Hasimoto transformation [24] ψ(l,t) =κ(l,t)·exp/bracketleftbigg i/integraldisplayl χ(˜l,t)d˜l/bracketrightbigg , whereκ(l,t) is a curvature and χ(l,t) the line torsion. The system under consideration has direct relation to hy- drodynamics. As it is known [25], [26], the local induction approximation for a thin vortex ﬁlament, under assumption of smallness of the ﬁlament width to the characteristic long i- tudinal scale, leads to the Hamiltonian (3.2), but only for a single separate line. The essence of this approximation is i n in replacing the logarithmic interaction law by a delta-funct ional one. When the widths of the ﬁlaments are small comparable with distances between them, in the same approximation, the Hamiltonian of vortex lines transforms into the sum of the Hamiltonians of independent vortex loops, yielding in a ”co n- tinuous” limit the Hamiltonian (3.1). By such a way, we have the model of 3D integrable hydro- dynamics of free vortex ﬁlaments. In this model, each vortex is a nonlinear object with its own internal dynamics. As we will see later, already in the framework of this model, a sing u- larity formation is possible. Singularities appear in this model as a result of intersection of vortex lines that is analogous to the phenomenon of wave breaking in gas-dynamics. A. Stationary vortices Let us consider now the simplest solution of Eq.(3.3), i.e., a stationary propagation of a closed vortex line: Rt=V≡ const. In this case the velocity Vis determined from solution of the equation [Rs×V] = [/vector τ×[/vector τ×/vector τs]]. (3.7) It is easily to check that this equation follows from the vari - ational principle δ(H(ν)−V·P(ν)) = 0, (3.8) i.e., any solution of (3.7) represents a stationary point of the Hamiltonian for a ﬁxed momentum P(ν). The equation (3.7) can be simply integrated, being rewritten in terms of the bi- normal band the curvature κof the line as follows [τ×V] =κ[τ×b], (3.9) that gives V=κb. (3.10)A constant value of the velocity Vin this expression implies constancy of the curvature κ, i.e. the vortex line must be a ring of radius r= 1/κand V= 1/r. (3.11) The direction of the ring motion is perpendicular to its plan e. It is interesting to note that the exact answer to the velocit y of a thin (with width d≪r) vortex ring in ideal hydrodynamics ( [27]) coincides with Eq.(3.10) up to the logarithmic accu- racy that just diﬀers the considering model from the Euler equation. Stationary solutions (3.10) in the form of rings are remark- able within this model, because they are stable, moreover, they are stable in the Lyapunov’ sense. Remind, that momen- tumPof a closed vortex line is its oriented surface spanned on the loop: P=Sn, whereSis the surface value, nits normal. Inasmuch as the Hamiltonian of a vortex loop coincides with its length, a max - imum of the momentum, or, that is the same as a maximum of surfaceSis obviously attained, for ﬁxed length, at the per- fect circle. Just this proves stability of the vortex ring so lution (3.10) in the Lyapunov sense. IV. COLLAPSE The solution (3.10), (3.11) enables us to construct the sim- plest mappings R=R(ν,s,t). Let all vortex lines be circle-shaped and oriented in the same direction, for instance, along z-axis. We will see fur- ther that collapse in our model is a purely local phenomenon. Therefore, it is suﬃcient to consider some vortex tube (whic h can be imagined as a torus) to ﬁnd a mapping. Let vortex rings be distributed continuously inside the tube. We label each vortex line by the two-dimensional parameter ν, which values coincide with coordinate of some cross section of the tube att= 0. We will use the ring arc-length as longitudinal parameters(ds=rdφ, whereφis the polar angle around z-axis). Then, with the help of (3.10), the desired mapping can be written as follows R=R0(ν) +r(ν)cosφex+r(ν)sinφey+V(ν)tez.(4.1) In this formula ex,y,zare unit vectors along the corresponding axes. It can be easily veriﬁed for this mapping that the Jacobian is a linear function of time J=∂(X,Y,Z ) ∂(ν1,ν2,s))=J0(ν,s) +A(ν,s)t. (4.2) HereA(ν,s) is a coeﬃcient linearly dependent on the veloc- ity derivatives with respect to νandJ0the initial value of Jacobian. Dependence J(4.2) on time means that for every ﬁxed pair νandsthere exists such a moment of time t=˜t(ν,s) (t>0, ort <0), when Jacobian is equal to zero: J(ν,s,t) = 0. Denote ast0the minimal value of t=˜t(ν,s) att >0. And let this minimum be attained at some point a=a0(here we 4denote a point ( ν1,ν2,s) asa). It is evident that at t=t0 /parenleftbig ∂˜t/∂a/parenrightbig \|a=a0= 0 or ∇aJ(a,t)\|a=a0= 0, (4.3) since ∇aJ(a,t)\|a=a0+∂J(a,t) ∂t∂˜t ∂a/vextendsingle/vextendsingle/vextendsingle/vextendsingle a=a0= 0. It is clear also that at t=t0the tensor of second derivatives ofJagainst a, 2γij=∂2J ∂ai∂aj, will be positive deﬁnite at the point a=a0. Hence, it is easy to deﬁne a behavior of the Jacobian in a small vicinity of a=a0. Expansion of Jnear this point (in a typical situation) att→t0is as follows J(a,t) =α(t0−t) +γij∆ai∆aj+..., (4.4) where α=−∂J(a,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=t0,a=a0>0,∆a=a−a0. Those are the leading contributions to the Jacobian expansi on 2. Geometrically, the above expansion corresponds to a suﬃ- ciently simple picture. The (hyper-) surface J=J(a,t) de- forms with time in such a manner, that its minimum reaches the (hyper-) plane J= 0 att=t0, where two surfaces touch each other. Obviously, for smooth mappings in a typical case , this touching takes place in one separate point. In a degener - ated situation, touching is possible in a few points simulta ne- ously, or even at a curve. A case, when two eigenvalues of the Jacoby matrix ˆJtend to zero simultaneously at the collapse point, will be regarded also as degenerated (such an exam- ple will be considered in the next section). In that case one should keep the next terms in the Jacobian expansion along the corresponding directions. We would like to repeat that all these cases can not be considered as typical ones. In accordance with Eq. (2.6), the equality J= 0 at the singular point means the formation of a singularity for the vorticity at the moment t=t0: Ω(r,t) =Ω0(ν)Rs α(t0−t) +γij∆ai∆aj. (4.5) It is important, that the numerator in this fraction – the tangent vector of a vortex line – does not vanish at the point a0due to its geometrical meaning. Therefore, the vorticity at the singular point blows up as ( t0−t)−1, and the characteristic size of the collapsing distribution in a-coordinates decreases as√t0−t. The above type of collapse arises as a result of vortex line breaking when one vortex overtakes another. For ﬂows of the 2For instance, the term containing mixed time-space deriva- tives, in the general case, is a small correction to the ﬁrst t erm in (4.4).general type (without symmetries) a singularity must arise at the ﬁrst time always in one separate point. As we will see further, for the given type of collapse , the dependence (4.5) for Ω(r,t), derived for the particular initial distribution, is actually the general answer, which can be a p- plied not only to the integrable hydrodynamics but also to the whole family (2.1) of hydrodynamic systems, of course, under the condition that they admit such (quasi-) inertial regime of collapse. What necessary conditions should satis fy the Hamiltonian of some particular system, in order to ex- hibit such regime? Now the answer to this question has been unknown that provides a wide ﬁeld for future investigations , both theoretical and numerical. A. Non-stationary vortices Let us consider now the integrable hydrodynamics in a more general case when closed vortex lines are not circles. In this situation, it is possible to map each vortex contour t o some vortex ring. It is then natural to introduce the (mean) direction n, as well as the mean area S=πr2 0, with the help of the expression for momentum of vortex line, P=nS.The position R0of the ring center changes linearly in time. A cor- responding mean velocity of the motion of closed line must be directed along the momentum, in order to satisfy the conser- vation law for angular momentum. The mean velocity value V0, generally speaking, is a function of those fundamental in- tegrals of motion, which are independent on the origin choic e – the Hamiltonian (i.e. the length L), the momentum, and the projection on the vector nof the angular momentum. It is clear also that increasing the contour sizes in λtimes should result in decreasing the velocity in λtimes, so that V0=8 π2P L3·U/parenleftbigg 16π2P2 L4,(M·P) L5/parenrightbigg . (4.6) Explicit dependence of the U(ξ,η)-function has been unknown now. Its ﬁrst argument can be considered as a measure of a ”crumpleness” of the vortex line and changes in the limits 0≤ξ≤1, and the second argument determines a measure of a spirality of the line. In a more or less reasonable approxim a- tion, which doesn’t lead to excessive errors, one can suppos e thatU(ξ,η)∼U(1,0) = 1. After introducing the mean characteristics, the mapping R(ν,s,t) can be represented as the sum: R(ν,s,t) =˜R(ν,s,t) +δr(ν,s,t), (4.7) where the mean value ˜R(ν,s,t) is given by the relation ˜R(ν,s,t) =R0+r0cosφ′e′ x+r0sinφ′e′ y, ˙R0=V0n. (4.8) Here angular parameter φ′= 2π(s/L) is proportional to the arc-lengths, and unit vectors e′x,e′ylie in the plane, per- pendicular to the local z′-axis, directed along the n. The relationship between r0andV0is given by the Eq. (4.6). The vector function δr(ν,s,t) describes deviations (generally speaking, not small) from the mean value ˜R(ν,s,t). The separation of the mean and oscillatory motions, intro- duced by means of (4.7), (4.8), (4.6) for each vortex contour , 5shows that the mapping R=R(ν,s,t) at each ﬁxed value a= (ν,s) is a linear function of time with nonlinear oscil- lations, which are described by the Landau-Lifshits equati on (3.5) or its gauge equivalent (3.6). The linear mean depen- dence reﬂects the fact that the model under consideration is a model of free vortices. The collapse thus arises as a result of a running of one vortex into another. Due to a continuous distribution of vortex lines, their ”density” increases in ﬁnitely in some point. A similar situation takes place in the model for large scale structure formation in the Universe studied by V.I.Arnold, Ya.B.Zeldovich and S.F.Shandarin [22]. In the basement of this model, the suggestion lies about initial dust-like dis tribu- tion of masses, when their behavior can be described by the zero-pressure hydrodynamic equations ρt+ divρv= 0, (4.9) dv dt=vt+ (v∇)v= 0. (4.10) The integration of this system in the Lagrangian variables gives that i) all ﬂuid particles being free move with a consta nt velocity r=a+v(a)t, (4.11) and ii) the density ρis expressed through the initial value and the Jacobian of the mapping (4.11) as follows ρ(r,t) =ρ0(a) J. (4.12) In the framework of this model, appearance of large scale structures is connected with the breaking phenomenon re- sulting in density singularities, due to the Jacobian vanis hing for the mapping (4.11). In a typical situation, these struc- tures have a pancake shape and can be considered as pcurlo- galaxies. The formula, analogous to the Eq. (4.12), takes place, as we have seen above, for the vorticity Ω– see the Eq. (2.6). However, there is a diﬀerence from Eq. (4.12), connected with the vector nature of the Ωﬁeld. In the given task we will be interested in a geometric struc- ture of a the singularity at t→t0, butt < t 0, i.e., in some sense, at the initial stage of collapse, but not at the develo ped stage, which has the meaning for astrophysics applications , but remains rather unclear for the incompressible hydrody- namics, when viscosity is necessary to be taken into account at small scales. B. Structure of the singularity Let us consider in more details the structure of the col- lapsing region in a typical situation. First, it is clear fro m the above discussion that the vorticity distribution near s in- gularity will be given by the former expression (4.5). Secon d, the main features of the singularity will be determined by the Jacobian, that is the denominator of (4.5). Numerator (Ω0(a)∇a)R( tangent vector to the lines) can be taken at the point a=a0,t=t0and considered as a constant. According to Eq. (4.5) the Jacobian expansion contains positive deﬁnite symmetric matrix γijof its second deriva- tives taken at the point a=a0,t=t0. Att<t 0, this matrixis assumed to be non-degenerate: all its eigenvalues are pos i- tive, and the matrix itself can be diagonalized. Hence it fol - lows immediately that compression along all principal axis es ina-space will have the same law: la∼√t0−t. Therefore, near the singular point, vorticity Ωwill have the self-similar asymptotics Ω(r,t) =Ω0(ν)Rs (t0−t)(α+γijηiηj), (4.13) whereη= ∆a/√t0−tare self-similar variables in a-space. However, the equation (4.13) does not mean, that compres- sion in r-space will be the same. When the Jacobian takes zero value, one of the eigenvalues of the Jacobi matrix becomes equal to zero. This eigenvalue (denote it as λ1), as it easily to see, coincides with the Jaco- bian (4.4) at a small vicinity of the collapse point up to the almost constant multiplier λ2·λ3. Represent now the Jacoby matrix ˆJas a decomposition over eigenvectors of direct ( ˆJ\|ψ(n)>=λn\|ψ(n)>) and conjugated (<˜ψ(n)\|ˆJ=λn<˜ψ(n)\|) spectral problems Jik≡∂xk ∂ai=3/summationdisplay n=1λnψ(n) i˜ψ(n) k. (4.14) Here two sets of eigenvectors of direct and conjugated prob- lems are mutually orthogonal <˜ψ(n)\|ψ(m)>=δnm. In a small vicinity of the collapse point eigenvalues λ2,3can be considered as constants, while λ1≡J λ2λ3= (λ2λ3)−1[α(t0−t) +γijaiaj]. Here, for simplicity, we have placed the origin at the point a=a0. As for the eigenvectors, they also can be considered constant. Let us decompose the vectors xand∇ain Eq. (4.14) through the corresponding bases, denoting their appropria te projections as XnandAn: Xn=<x\|ψ(n)>,∂ ∂An=<˜ψ(n)\|∇a>. In this case the vector ais written in terms of Anas follows aα=/summationdisplay nψ(n) α\|˜ψ(n)\|2An. As a result, Eq. (4.14) can be rewritten in the form ∂X1 ∂A1=τ+ Γ mnAmAn, (4.15) ∂X2 ∂A2=λ2,∂X3 ∂A3=λ3. (4.16) Here, the matrix Γmn=γαβ(λ2λ3)−1ψ(n) αψ(m) β\|˜ψ(n)\|2\|˜ψ(m)\|2, and parameter τ=α(t0−t)(λ2λ3)−1is assumed to be small. It follows from here immediately that size reduction along t he second (X2) and the third ( X3) directions is the same as in 6the auxiliary a-space, i.e., ∼√τ, but along the ”soft” direc- tionX1behaves like τ3/2. Respectively, in terms of new self- similar variables ξ1=X1/τ3/2,ξ2=X2/τ1/2,ξ3=X3/τ1/2, integration of the system gives for ξ2andξ3a linear depen- dence onη, while forξ1it a cubic dependence ξ1= (1 + Γ ijηiηj)η1+1 2Γ1iηiη2 1+1 3Γ11η3 1, i,j= 2,3 (4.17) ξ2=λ2η2, ξ3=λ3η3.(4.18) Together with (4.13), the relations (4.17) and (4.18) deter - mine implicit dependence Ω(r,t). The presence of two diﬀer- ent self-similarities shows, that the spatial vorticity di stribu- tion becomes strongly ﬂattened in the ﬁrst direction, takin g a pancake form as t→t0. The direction of the ﬁeld Ωcan be found from the incom- pressibility condition div Ω= 0. It is easy to see that in the leading order (as t→t0) the gradient of Jis determined by the soft direction e1: ∇J≈τ−3/2∂J ∂ξ1e1. Contributions from two other directions are small in the pa- rameterτ. Hence, it follows that vector lines of the ﬁeld Ωlie in the pancake plane that is in agreement with transversality of mo - tion of vortex lines (compare with Eq.(2.7)). V. EXAMPLE OF COLLAPSE IN THE DEGENERATED CASE In the previous section we have considered collapse for a non-degenerated situation, when the only one eigenvalue of the Jacoby matrix tends to zero at the touching point. Now we shall consider an example of collapse, when two eigenval- ues of the Jacoby matrix vanish simultaneously at the col- lapse point. We will examine an initial vorticity distribut ion with the nontrivial topology of vortex lines with linking nu m- berN= 1. This special distribution is the so-called Hopf mapping. There are several ways how to construct the cor- responding ﬁeld Ω. We will keep here the approach of the paper [28]. Following to that work, let us represent the ﬁeld Ωthrough then-ﬁeld ( n2= 1) Ωα(r) =1 32ǫαβγ(n·[∂βn×∂γn]), (5.1) where the n-ﬁeld is supposed to be a smooth function of coor- dinates tending to the constant value eat the inﬁnity r→ ∞. It is easy to check that in accordance with Eq.(5.1) each pointn=n0on the unit sphere S2deﬁnes a closed vortex line . Indeed, parameterization of the unit vector nthrough the spherical angles θandϕallows to write down the ﬁeld Ω as follows Ω=1 16[∇ϕ× ∇cosθ]), (5.2) so that the variables ϕand cosθplay the role of the Clebsch variables. Thus, each vortex line coincides in this case wit h the intersection of two surfaces ϕ=const and cosθ=const,i.e. a closed vortex line is the pcurlotype in R3of a point on the sphere S2. By the deﬁnition, the Hopf degree Nof mapping R3→ S2is called an integer number of linkages between arbitrary two vortex lines – pcurlotypes of two poin ts on the unit sphere. The Hopf mapping (with N= 1) is given by the next relation: (n·σ) =U(e·σ)U†, U=1 +i(a·σ) 1−i(a·σ), (5.3) whereσare the Pauli matrices. Expressing the vector nfrom here and substituting the result into Eq.(5.1), one can get (see [29]) Ω0(a) =e(1−a2) + 2a(e·a) + 2[e×a] (1 +a2)3. (5.4) As it was shown in [29], all the ﬂux lines of this ﬁeld are circles, and each line is linked once with another line. By th is reason, the singularity formation is inevitable. It is worth to note that the ﬁeld (5.4) has no singular points in the whole space and its absolute value depends only on the absolute value \|a\|. The unit vector /vector τis deﬁned everywhere: /vector τ(a) =e(1−a2) + 2a(e·a) + 2[e×a] (1 +a2). (5.5) The velocity of each ring is connected with the binormal b(ν) and the radius r(ν) by the relation V(ν) =b(ν)/r(ν). The radiir(ν) of rings and their orientations b(ν) are integrals of motion. Only positions of centers of rings can change, and this motion occurs with a constant velocity for each ring. In the given problem, instead of the variables νands, it is convenient to use directly the variables a, in which the mapping R(a,t) is written as R(a,t) =a+tV(a), (5.6) where the velocity V(a) is expressed through the unit tangent vector/vector τby means of the relation V(a) = [/vector τ(a)×(/vector τ(a)∇a)/vector τ(a)]. (5.7) Hence one can ﬁnd the expression for the Jacobian J(a,t) = det/vextenddouble/vextenddouble/vextenddoubleˆI+t/parenleftbigg ∂V(a) ∂a/parenrightbigg/vextenddouble/vextenddouble/vextenddouble. (5.8) It should be noted that the velocity of each ring is constant along the vortex line. Therefore the determinant of the matr ix ∂V(a)/∂a, which is the coeﬃcient in front of t3in the previous expression, is equal to zero identically. As a result, Jhas a quadratic dependence on time tonly: J= 1 +tc1(a) +t2c2(a). Singularity formation corresponds to the case J→0 at some point. The collapse moment of time t0will be determined by the coeﬃcients c1andc2so thatt0will be a minimal positive root of the equation minaJ(a,t0)≡Jmin(t0) = 0. 7Calculations by means of Eq-s (5.5), (5.7) and (5.8) lead to the following expressions: V(a) =−2 (1 +a2)2/parenleftbig [e×a](1−a2) + 2a(e·a)−2ea2/parenrightbig ,(5.9) J(a,t) =(1 +a2)3−8ta3(1 +a2) + 4t2(1 +a2 3−a2 1−a2 2) (1 +a2)3, (5.10) wherea3is the projection of the vector ato the axis e. Analyzing the latter expression one can show that at t<1 a minimum of the Jacobian is attained on the symmetry axis, i.e., ata1= 0,a2= 0. In this case J\|axis=(1−a2 3)2+ 4(t−a3)2 (1 +a2 3)2, (5.11) hence it becomes clear that the singularity takes place at t0= 1, a3= 1, and the Jacobian tends to zero with the quadratic asymptotics. Thus, in this example \|Ω\|max∼(t0−t)−2. It can be easily veriﬁed also that the Jacoby matrix has two zero eigenvalues at the touching point which eigenvectors l ie in the plane orthogonal to e. In the collapse vicinity the ﬁeldΩis directed along the vector e. Compression along this direction is linear in time l3∼(t0−t), but in the perpendicular plane it is more fast, with the law l1,2∼(t0−t)3/2. As a result, the singularity structure occurs strongly stretched along the anisotropy axis. CONCLUDING REMARKS Considering the hydrodynamic model with the Hamilto- nianH=/integraltext \|Ω\|dr, we have arrived at the conclusion that each vortex line in the given system moves independently of other lines. Just this property makes possible to form singu - larity in a ﬁnite time for the generalized vorticity Ω(r,t) from smooth initial data. A typical singularity of this kind look s like an inﬁnite condensation of the vortex lines near some point. Thus, the collapse in the integrable hydrodynamics has purely inertial origin. If one will assume that this type of collapse is possible also in the Euler hydrodynamics, then t he asymptotics of the vorticity near the singularity point (th e point of vortex ”overturning”) will be given by Eq-s (4.5) or (4.13) in a non-degenerated situation. That is already clea r from the general consideration. Namely, the curl of the velo c- ity will blow up as ( t0−t)−1. Exactly this dependence for vorticity near a singular point has been observed in practic ally all numerical simulations of the Euler equation, including the above cited: [6], [10,17], [11], [13], [12]. However, not all the simulations are regarded to numerical integration of the Euler equations for continuous distribu tions of vorticity. The ﬁrst numerical experiments [6] relate to i n- vestigation of collapse for two anti-parallel vortex ﬁlame nts which, as it was shown by Crow [14], are linearly unstable with respect to transverse perturbations ( about the develo p- ment of this direction see also [12]). The theory of collapse fortwo thin vortex ﬁlaments, as the nonlinear stage of the Crow instability, was developed by V.E.Zakharov [7], [8] (see al so [9]). The conclusions of this theory are in a good agreement with the numerical experiments up to the distances compara- ble with a core size of ﬁlaments. This theory predicts decrea s- ing distance between vortex ﬁlaments as√t0−t. For smaller distances the cores of vortices loose their round shape. The y become ﬂat, and the process of attraction between vortices becomes more slow [30], [31]. The same tendency was ob- served also in the numerical experiments of Kerr [10], the most advanced (in our opinion) for the problem of recon- nection, and where, unlike [6], collapse of two anti-parall el vortices was simulated with continuously distributed vortic- ity. Besides a natural contraction of the minimal distance b e- tween the distributed vortices, these simulations have sho wn, for the ﬁrst time, the formation of two singularities in two symmetric points. While approaching a collapse moment in time, the explosive growth of the maximal value of vorticity was observed with the law ( t0−t)−1. According to the recent publications of Kerr [17], the analysis of numerical data ga ve two distinguished scales, one scale being contracted as the square root: l1∼(t0−t)1/2, and another as the ﬁrst power of time:l2∼t0−t. In the work of Grauer, Marliani and Germaschewsky [11], the successfull attempt was undertaken to observe collapse for the initial condition not possessing a low symmetry. The initial vorticity was concentrated in the vicinity of a cyli nder, and was modulated over the angle in such a way so that the simplest symmetries were absent. In the present time this experiment has the best spatio-temporal resolution. In thi s simulation appearance of a separate collapsing region was o b- served with the vorticity growth at the center as ( t0−t)−1. Thus, the results of the numerical simulations for ideal hy- drodynamics go along with the concept of collapse as a proces s of caustic formation for the solenoidal ﬁeld. There is an agr ee- ment for behavior of the vorticity maximum. Concerning the spatial structure of the collapsing domain, we can make only a qualitative agreement. The results of the papers [30], [31 ] (where some contraction of the vortex core was observed at the initial stage of reconnection neglecting viscosity) an d also the results of Kerr seem to support our theory. We would like to pay attention to the numerical results of the paper [3 3] where for short-time dynamics for the initial conditions in the form of the Taylor-Green vortex and for random initial con- ditions it was observed formation of thin vortex layers with high vorticity that also support our predictions. All these results allow one to say that the scenario pre- sented here looks like very plausible. We would like to repeat once more, that if such scenario takes place then behavior of the vorticity near of singular point is deﬁned by Eq-s (4.5) or (4.13). The structure of this domain should be highly anisotropic: in one of two direction s perpendicular to the vorticity, there is more rapid contrac tion (∼τ3/2) than with respect to other directions ( ∼τ1/2). The spatial distribution becomes close to the two-dimensional one familiar to a tangential discontinuity. The velocity of ﬂow in this region can be approximated with a good accuracy by the linear dependence v⊥∼ΩmaxX1, i.e. the ﬂow looks like a shear ﬂow. The given type of col- 8lapse, according to the classiﬁcation of [32], belongs to a w eak collapse: the energy captured into singularity (with accou nt of the viscosity ν) tends to zero as ν→0. It is interesting to note that the dissipation rate ∼/integraltext Ω2drfrom the collapsing region also vanishes when ν→0. One should note once more that unlike the model of free vortices considered here, in the Euler hydrodynamics vorte x lines interact pairly in accordance with the Hamiltonian HEuler=1 8π/integraldisplay /integraldisplay Rs(ν,s)·Rξ(µ,ξ) \|R(ν,s)−R(µ,ξ)\|d2νdsd2µdξ. (5.12) From the viewpoint of investigation of the collapse problem it is very important that the interaction function (i.e. the Gr een function for the Laplace operator) has the singularity for e qual arguments R(ν,s)→R(µ,ξ). If this singularity would be ab- sent, i.e. if the interaction function would be completely r eg- ular, then anyinitial vortex line distribution equivalent in the sense of iso-vorticity to a smooth ﬁeld Ω0(a) , even very singu- lar distribution, would produce a suﬃciently smooth veloci ty ﬁeldv(r). In a smooth velocity ﬁeld, an initial singularity of the generalized vorticity could not disappear in subsequ ent moment of time, it could be only transported and deformated by the ﬂuid ﬂow. As far as the equations of motion for an in- viscid substance are time-reversible, it follows from this , that the formation of singularity from a smooth initial data also would not possible. So, the existence and possible types of collapse of vortex lines, in the systems with the quadratic o n ΩHamiltonians, depend on the asymptotics of the interaction functionG(r1,r2) atr1→r2. So, for a better understanding of the collapse problem in hydrodynamics, it is reasonable t o investigate such systems, for which the interaction functi on has the asymptotics G∼ \|r1−r2\|−q, and the exponent qis not necessary equal to the unity. What is the inﬂuence of viscosity on the structure of col- lapsing region? How does the given type of collapse have eﬀect on turbulent spectra? That are only few of the most important questions, which need investigation. It would be interesting to verify numerically our hypothesis about pos si- bility of (quasi-)inertial collapse in ideal hydrodynamic s, both in Eulerian variables and in the vortex line representation . ACKNOWLEDGMENTS The discussions with V.E.Zakharov, R.Z.Sagdeev, V.V.Le- bedev, G.E.Falkovich, A.Tsinober, I.Goldhirsch, N.Zabus ky, J.Herring, R.Kerr, R.Pelz of many questions concerned in th e given article were very useful. The authors are grateful to a ll of them. E.K. thanks to P.L.Sulem for pointing out the pa- per [33] to the attention. This work was supported by RFBR (grant 00-01-00929), by Program of Support of the Leading Scientiﬁc Schools (grant 00-15-96007). V.R. thanks also th e Fund of Landau Postdoc Scholarship (KFA, Forschungszen- trum, Juelich, Germany) for support. E.K. wishes to thank the Nice Observatory, where the paper was completed, for its hospitality and ﬁnancial support through the Landau-CNRS agreement.†e-mail: kuznetso@itp.ac.ru ∗e-mail: ruban@itp.ac.ru [1] A.N.Kolmogorov, Doklady AN SSSR 30, 9 (1941) (in Russian). [2] O.M.Phillips, J. Fluid Mech. 4, 426 (1958). [3] B.B.Kadomtsev and V.I. Petviashvili, Doklady AN SSSR (Soviet Physics Uspekhi) 208, 794 (1973) (in Russian). [4] P.G.Saﬀman, J. Fluid Mech., 106, 49 (1981). [5] U.Frisch, ”Turbulence. The legacy of A.N.Kolmogorov”, Cambridge Univ. Press (1995) . [6] A.Pumir and E.D.Siggia, Collapsing solutions in the 3-D Euler equations , in: ”Topological Fluid Mechanics”, Eds. H.K.Moﬀatt and A.Tsinober, Cambridge Univ. Press, Cambridge, 469 (1990); Phys. Fluids A, 4, 1472 (1992). [7] V.E.Zakharov, Uspekhi Fizicheskikh Nauk (Soviet Physics Uspekhi), 155, 529 (1988) (in Russian). [8] V.E.Zakharov, in: Lecture Notes in Physics, Nonlin- ear MHD Waves and Turbulence , ed. T.Passot and P.L.Sulem, Springer, Berlin, 369-385 (1999). [9] R.Klein, A.Maida and K.Damodaran, J. Fluid Mech., 228, 201 (1995). [10] R.M.Kerr, Phys. Fluids A, 4, 2845 (1993). [11] R.Grauer, C.Marliani and K.Germaschewski, Phys. Rev.Lett., 80, 4177 (1998). [12] R.B.Pelz, Phys. Rev. E, 55, 1617 (1997). [13] O.N.Boratav and R.B.Pelz, Phys. Fluids, 6, 2757 (1994). [14] S.C.Crow, Amer. Inst. Aeronaut. Astronaut. J., 8, 2172- 2179 (1970). [15] Ya.G.Sinai, (private communication), (1999). [16] V.I.Arnold, Theory of Catastrophe , Moscow, Zna- nie,(1981); Mathematical Methods of Classical Mechan- ics, 2nd edition (Springer-Verlag, New York, 1989). [17] R.M.Kerr, Trends in Math., Birkhauser Verlag Basel / Switzerland, 41-48 (1999). [18] R.Salmon, Ann. Rev. Fluid Mech., 20, 225 (1988). [19] V.E.Zakharov and E.A.Kuznetsov Phys. Usp. 40, 1087 (1997). [20] E.A.Kuznetsov and V.P.Ruban, JETP Letters, 67, 1076 (1998). [21] E.A.Kuznetsov and V.P.Ruban, Phys. Rev. E 61, 831 (2000). [22] V.I.Arnold, S.F.Shandarin and Ya.B.Zeldovich, Geohy s. Astrophys. Fluid Dynam. 20, 111 (1982). [23] V.E.Zakharov and L.A.Takhtajan, Teor Mat. Fiz (Theor. Math. Phys.), 38, 26 (1979). [24] R.Hasimoto, J. Fluid Mech., 51, 477 (1972). [25] L.S.Da Rios, Rend. Circ. Mat. Palerno, 22, 117 (1906). [26] R.Betchov, J. Fluid Mech., 22, 471 (1965). [27] H.Lamb. Hydrodynamics , 6th edition, Dover, New York. [28] E.A.Kuznetsov, A.V.Mikhailov, Phys. Lett. 77 A 37 (1980). [29] A.M.Kamchatnov, ZhETF (JETP), 82, 117 (1982). [30] M.V.Melander and F.Hussain, Phys. Fluids A, 1, 633-636 (1989). [31] M.J.Shelley, D.J.Meiron and S.A.Orszag, J. Fluid Mech ., 246, 613 (1993). [32] V.E.Zakharov and E.A.Kuznetsov, ZhETF, 91, 1310 (1986) [Sov. Phys. JETP, 64, 773 (1986)]. [33] M.E.Brachet, M.Meneguzzi, A.Vincent, H.Politano and P.L.Sulem, Phys. Fluids, A 4, 2845 (1992). 9
arXiv:physics/0009008v1 [physics.bio-ph] 3 Sep 2000One way to characterize the compact structures of lattice protein model∗ Bin Wang1, Zu-Guo Yu2,1 1Institute of Theoretical Physics, Chinese Academy of Scien ces, P.O. Box 2735, Beijing 100080, P. R. China. 2Department of Mathematics, Xiangtan Universiy, Hunan 411105, P.R. China February 2, 2008 Abstract On the study of protein folding, our understanding about the protein structures is limited. In this paper we ﬁnd one way to characterize the compact struc tures of lattice protein model. A quantity called Partnum is given to each compact structure. The Partnum is compared with the concept Designability of protein structures emerged recently. It is shown that the highly designable structures have, on average, an atypical number of local degree of freedom. The statistical property of Partnum and its dependence on se quence length is also studied. 1 Introduction The study of protein folding is fundamental on both theory an d application. In order to tackle protein folding problem physically, it is important to pay m uch attention to concrete proteins and consider the details of interactions, such as for medica l purpose. But there are also “global views” that should be noticed. For example, The possible con ﬁgurations of folded proteins are enormous, while that can be observed in living form is rather limited. These protein structures generally can be described as belonging to a limit number of f amilies. In each family, ignoring the details, the proteins possess similar overall conforma tions, and in many cases the structures show regular forms or approximate symmetry.[1, 2, 3, 4, 5, 6] Another example is that single domain proteins was observed only within a certain range of s equence length: the number of amino acid residues in single domain proteins seldom exceed s 200. Larger proteins usually fold into multi-domains native states.[6] With the accumulation of knowledge about the structures and functions of proteins, it was found that many proteins of similar structures pursue compl ete diﬀerent functions, while pro- teins with diﬀerent tertiary structures may perform simila r functions. These suggested that to ∗This project was supported partly by Chinese Natural Scienc e Foundation. 1understand the protein folding problem physically, one sho uld ﬁrst get to know the properties of protein structures.[7] Based on the the concepts from the physics of spin glass, study shows that to fold eﬃciently, proteins require a specially shaped energy landscape resembling a fun- nel. A heteropolymer with a completely random sequence gene rically possess a rugged energy landscape without a funnel.[8, 9] Goldstein et al[10, 11] have worked on optimizing energy func- tions for protein structure prediction. They found that som e structures are more optimizable than others, i.e., there exist structures for which the funn eled energy landscape can be obtained within a wide range of interaction parameters, while for som e other structures the parameters for fast folding are much more restricted. The funneled land scape theory argued that the inter- actions in the folded structure must act in concert more eﬀec tively than expected in the most random cases.[12] Accordingly, compared with most other st ructures, the superiority of highly optimizable structure should be that its geometric arrange ment permit more sequences to reach the concert interaction states. Other studies on the thermodynamic of lattice protein model s also support the above idea.[13, 14, 15, 16] In the lattice HP models, a protein is represented by a self avoiding chain of beads placed on a discrete lattice with two types of beads: the Pola r (P) and the Hydrophobic (H). A sequence is speciﬁed by a choice of monomer type at each posi tion on the chain {xi}. Where xicould be either H- or P-type, and iis a monomer index. A structure is speciﬁed by a set of coordinates for all the monomers {ri}. The energy has the form: H=/summationdisplay i<jExixj△(ri−rj) where △(ri−rj) = 1 when riandrjare adjoining lattice sites while they are not adjacent alon g the sequence, and △(ri−rj) = 0 in other cases. Interaction parameter Exixjdiﬀer according to the contact type HH, HP, or PP. Given the interaction param eters, it is possible to ﬁnd out the ground state structure(s) of each sequence. Study shows that structures diﬀer markedly in their tendency to be chosen by sequences as their unique gr ound states. The number of sequences which choice the structure as unique ground state is called the Designability of this structure. It was argued that only highly designable st ructures are thermodynamically stable and stable against mutation, and thus can be chosen by nature to fulﬁll the duty of life.[13] Though interaction parameters used may diﬀer str ongly in diﬀerent studies, the mostly designed structures do not depend strongly on the detail of i nteractions.[13, 15, 16] From above discussion we see that it should be essential to in vestigate the protein folding problem from structural point of view. To see the problem mor e clearly, we take square lattice HP model as an example. The total number of the most compact st ructures of 36 beads chain is 57337.[14] Consider 36 beads homopolymer with interaction parameter Exixj=E0<0. All the 57337 structures give the same energy when one such homop olymer fold onto each of them. Therefore the folded energy can not be used to distinguish th e compact structures from each other. The essential here is that of discrimination, or char acterization: give ways to tell how and why structures diﬀer from each other. Nature’s way to bre ak the symmetry is to replace homopolymer with heteropolymer. From this point of view, th e success of lattice protein model 2is that it help to reveal this secret of nature. Studies focusing on the properties of protein structures is still lack,[17] in spite of some recent elaborations in this direction.[18, 19, 20] In this a rticle we present one way to break the symmetry, to distinguish the compact structures of lattice model without explicitly considering concrete interaction form. However, since only compact str uctures are considered here, an loose constraint is actually set on interactions: interact ions under which compact structures are preferred as ground energy states. The method gives a num ber called partition number (Partnum ) to each compact structure during a simple process. The Part nums of structures diﬀer strongly, so giving one way to distinguish them from ea ch other. In the following section we will give the detail of the method , and compare the Partnum with designability. The statistical properties of Partnum s are discussed in section II. The last section is for some remarks. 2 The deﬁnition and interpretation of Partnum It is easy to ﬁnd out all the compact structures of certain cha in length with computer.[21] Take 9 beads chain as an example. The search is self avoiding and re stricted to the 3 ×3 square lattice shown in Fig.1(A), and the resulting structures should not b e related by rotation or reﬂection symmetry. As a result, there are only three starting points, (0,0), (0,1) and (1 ,1), for the search of structures. To ﬁnd the structures start at (0 ,0), the ﬁrst step is to go to (1 ,0). This is the only choice, because (0 ,1) is a symmetric point of (1 ,0). We give all the structures following this step a number p1=ln(1). Now go to the next site. There are two possible choices: ( 2,0) or (1,1). Since the walk is self avoiding and restricted to the 3 ×3 lattices, the walk following certain choice may fail to extend to 9 beads length. The choice that wi ll reach to 9 beads length is called acceptable. Suppose that both (2 ,0) and (1 ,1) are acceptable. Then each compact structure which will be generated following (0 ,0)−→(1,0)−→(2,0) or (0 ,0)−→(1,0)−→(1,1) is given a number ln(1/2). Generally speaking, restricting to 3 ×3 lattice and beginning at a starting point, there are totally 8 steps to ﬁnish a self avoi ding walk. Each step is given one number according to the following rule: if the i-th step has totally Cacceptable choices not being symmetrically related, then the step is given a number called partnum of i-th step pi=ln(1/C). For 2D square lattice, the largest possible choice C0is 3. Adding all the 8 numbers and then dividing the sum by 8, we get t he Partnum P1. Here the structure is actually oriented . The consideration of orien ted walk is reasonable in the case of protein structures, because the native protein structure w ould become unstable if the sequence is reversed, and also protein in life are produced successiv ely from one end to another. However, if one consider the start and end reversal of the walk as a symm etric operation, then one oriented walk and its reverse together correspond to a structure that is not related with the direction. In the follows, oriented walk andnon-oriented structures are used to distinguish the two diﬀerent ways of viewing structure, and the Partnums corresponding t o them are denoted as P1 and P2, respectively. However, when it is no need to distinguish t hem, simply structure is used and the Partnum is denoted as P. For the non-oriented structure, the Partnum can be deﬁne as : 3P2 =P1(1) + P1(2), where P1(1) is the Partnum of one of two oriented walks and P1(2) is that of its reverse. The Partnums of structures of other chain length can be obtai ned similarly. Since the original motivation of developing the Partnums of structures is to account for the diﬀerence of Designability of structures, in Fig.2 we give t he plot of Designability against Part- num of orientd structures on 5 ×5 lattice (the interaction parameters for calculating Desi gnability is the same as used in Ref. [13]). There is not strict correspo ndence between Designability and Partnum. However the linear ﬁt of the data revealed that Desi gnability tends to increase with the increase of Partnum (see Fig.2). The same thing happens f or other sequence length. In the case of 6 ×6 lattice, the structure with highest Designability[13, 15 ] possess the second largest Partnum ( P2). According to Fig.1(B), an oriented walk corresponds to one p ath from the root to the top leave of the hierarchical tree. The value of Partnum of the st ructure is determined by the frequency of the path being disturbed by branches. If the pat h of a walk meet with fewer branches, the Partnum would be larger. This can be compared w ith the conclusion in Ref. [16]. In Ref. [16] a simple version of HP model of protein is employe d. A walk is reduced to a string of 0s and 1s, which represent the surface and core site respecti vely, as the backbone is traced. Each walk is therefore associated with a point in a high dimension al space. Sequences are represented by strings of their hydrophobicity and thus can be mapped int o the same space. It was found that walks far away from other walks in the high dimensional s pace are highly designable and thermodynamically stable. For this reason, highly designa ble structures are called atypical in Ref. [16]. Here the structures with large Partnum can also be called atypical (atypical average local freedom) since these structures correspond to paths o n the hierarchical tree with fewer branches. In an analog to the suggestion that nature selected out only h ighly designable structures, we assume that there exists a random process which selects out o nly the structures with the largest Partnum. It is interesting to see what this assumption will r esult in. For concise we assume a critical Partnum Pc, so that only a small portion of oriented walks for which P1> Pccan be selected out. Two oriented walks are called n-level similar if their ﬁrst n−1 steps are along the same path, and they branched at the n-th steps. Suppose s1is among the structures with the most highest Partnum satisfying P1(s1)> Pc. This means that there are few branches along the path of s1. As a result, it is diﬃcult to ﬁnd walks which show high level similarity to s1. But if there do exist such walks, these walks should have hig h possibility to be selected out. For example, if s2isN−1 level similar with s1,Nbeing the chain length, thenP1(s2) =P1(s1)> Pc. More generally, let n12being the similarity level between s2ands1. We know that P1(s2) =P1(s1)−(1−n12 N−1ln(C0)),C0= 3 being the maximal possible choices per step during the search of structures. According to this e xpression, the more similar s2is to s1, the more possible it is to be selected out. Assuming that s3is another walks with P1(s3)> Pc, but it is dissimilar to s1. From above discussion we know that there are two families, all the membe rs of which are selected out. Within each family, the similarity level of two walks is much higher thann13, while any two walks from 4diﬀerent families are dissimilar from each other, and the si milarity level is n13. We thus come to the conclusion that the selected walks belong to separate families. walks within each family are similar, while walks belonging to diﬀerent families are dissimilar. For the non-oriented structures, there is no the convenienc e of the hierarchical tree to discuss their properties. But it is believable that the above result be kept once similarity between structures is properly deﬁned. This is the case for the class iﬁcation of real protein structures, where more or less arbitrary criteria[1, 2, 3, 22, 23, 24, 25] are used to deﬁne the similarity between protein structures and to classify structure into f amilies, superfamilies , folds, and so on. 3 The statistical properties of Partnums Natural single domain proteins exist only within a limit ran ge of sequence length. By both theoretical and numerical studies it is showed in Ref. [26] t hat the stability of folded sequences against mutation decrease with the increase of chain length . In that follows the dependence of the statistical properties of Partnum on chain length will b e discussed. We will show how some structural properties are determined by general statistic al principle. The density distribution of P2 are shown in Fig.4. Things are similar for P1. In both cases, visually the distribution becomes more and more norm al. Actually it will be shown that the distribution is Gauss distribution in the long chain lim it. As the ﬁrst step, however, let’s much generally, assume that the Partnums of chain length Ncan be described by a density distribution function, F(P,v1,v2...),vibeing the moment of i-th orders. It is easy to get the average v1=< P > and variance v2=△P. The results of both oriented walk and non-oriented structures are shown in Fig.3. Fig.3 shows that < P > (both < P1>and< P2>) decrease with the increase of chain length. However, from the deﬁnition of Partnum we know that < P1>(< P2>) can not be smaller than −ln3 (−2ln3). So, for either oriented walks or non-oriented structure s, there must existδ, so that limN−→∞< P > =δ. A similar argument applies to △P, where limN−→∞△P=ǫ,ǫ≥0. It is known that the total number of compact structures Mincrease exponentially with the increase of chain length N:M(N)∼(Cav)N,Cav< C0= 3 being the average number possible choices per step for the walks. This gives one way to estimate the value of Cavusing the knowledge of M(N). Fig.5 show the ﬁt of the data M(N) tof(N) = ( lnCav)N+b. The result is Cav= 1.397. Viewing this value of Cavas the value in long chain limit, we get that δ=ln(1/Cav) =−0.3343 for oriented walks, a reasonable estimation (see Fig.3 ). It should be noticed that Cavget this way is much larger that given by mean ﬁeld considerat ion,[21, 27] where Cav=C0/e= 1.1. According to this Cav,δ=−0.099 for non-oriented walks. From Fig.3 5we know that this is a value too large to be the long chain limit of< P1>. So it seems that the mean ﬁeld treatment does not apply to the two dimensional protein model. With the help of central limit theorem, we can argue that the d ensity distribution is Gauss distribution in long chain limit, and ǫ= 0. See follows. In the space of compact structures, the Partnum Pof certain structure is the average of the partnums piof all the Tsteps. For oriented walks Tequals to the chain length subtracted by 1, and for non-oriented structures this value should be do ubled further. Now divide the T partnums into ( T)/ngroups (suppose T/nis an integer). In each group the nmembers are chosen randomly within the total Tnumbers. For each group we deﬁne a new random variable qk=/summationtext ipi/n,kbeing the group index. Since the members in each group are cho sen randomly form the total Tnumbers, the T/nnewly deﬁned random variable should have the same average and variance when n−→ ∞ . At the same time, since P=/summationtext kqk (T−1)/n, applying the central limit theorem,[28] we know that Pis a Gaussian random variable, and δP−→0 when T/n−→ ∞ . From the above discussion we know that, according to Partnum s, statistically all the compact structures become indistinguishable in long chain limit. R ecalling the selection rule assumed above, we know that it becomes increasingly diﬃcult to selec t outatypical structures when chain length increases. These results show some connection to the work of Ejtihadi et al..[20] With a purely geometrical approach, they were able to reduce large ly the candidates of structures that can be chosen as the ground states of sequences. They found th at for the case of HP protein model the number of ground state candidates grows only as N2,Nbeing the sequence length. While, as pointed out above, the total number of compact stru ctures increase exponentially with the increase of N. So it becomes increasingly diﬃcult to ﬁnd the ground state c andidates. This is in accordance with the statistical property of Partnum. For fulﬁlling biology functions, proteins should possess s ome properties, for example fast folding, thermodynamically stable and stable against muta tion.[12, 13, 26, 29, 30] It was pos- tulated that with the increase of sequence length, the folde d structures become more and more diﬃcult to possess these properties.[26] Based on the study of Partnum, we propose that this property of proteins is determined by the statistical prope rties of protein structures, the detail of interaction having weak inﬂuence. 4 Conclusion Remarks Protein structures seem to be a very special class among all t he possible folded conﬁgurations of polypeptide chain. We now know something about howspecial it is, but little on whyit be so. Ways of characterizing folded structures, from whatever po int of view, will help to deepen our understanding about protein structures. In this paper, the study on Partnum itself is interesting, and more interesting when compared with the dynamic and ther modynamic study of proteins. The concept of Partnum is simple and can only be applied to lat tice model. But the study on it reveals that it is possible to investigate protein structur es with no consideration of interaction detail. 6ACKNOWLEDGMENTS The authors would like to give thanks to Proﬀ. Wei-Mou Zheng a nd Proﬀ. Bai-Lin Hao for stimulating discussion. We also thank Mr. Guo-Yi Chen for ma ny helps on computation. References [1] M. Levitt and C. Chothia, Nature (London) 261, 552 (1976). [2] L. A. Orengo, D. T. Jones and J. M. Thornton, Nature (Londo n)372, 631 (1994). [3] Z. X. Wang, Proteins 26, 186 (1996). [4] J. S. Richardson, Proc. Natl. Acad. Sci. USA 73, 2619 (1976). [5] J. S. Richardson, Adv. Protein Chem. 34, 167 (1981). [6]Introduction to protein structure (Laslow Publishing, New York, 1996), edited by, C. Branden a nd J. Tooze. [7] S. Govindarajan and R. A. Goldstein, Proc. Natl. Acad. Sc i. USA 93, 3341 (1996). [8] J. D. Bryngelson, J. Onuchic, N. Socci and P. G. Wolynes, P rotein Struct. Funct. Genet. 21, 167 (1995). [9] P. E. Leopold, M. Montal and J. Onuchic, Proc. Natl. Acad. Sci. USA 98, 8721 (1992). [10] R. A. Goldstein, Z. A. Luthey-Schulten and P. G. Wolynes , Proc. Natl. Acad. Sci. USA 78, 4818 (1992). [11] R. A. Goldstein, Z. A. Luthey-Schulten and P. G. Wolynes , Proc. Natl. Acad. Sci. USA 78, 9029 (1992). [12] J. D. Bryngelson and P. G. Wolynes, Proc. Natl. Acad. Sci . USA 84, 7524 (1987). [13] H. Li, R. Helling, C. Tang and N. S. Wingreen, Science 273, 666 (1996). [14] K. A. Dill, Biochemistry 24(1985) 1501; H. S. Chan and K. A. Dill, Macromlecular 22, 4559 (1989). [15] C. Micheletti, J. R. Banarvar, A. Martin and F. Seno, Phy . Rev. Lett. 80, 5683 (1998). [16] H. Li, C. Tang and N. S. Wingreen, Proc. Natl. Acad. Sci. U SA95, 4987 (1998). [17] P. G. Wolynes, Proc. Natl. Acad. Sci. USA 93, 14249 (1996). [18] S. Govindarajan and R. A. Goldstein, Biopolymers 36, 43 (1995). [19] C. Micheletti, J. R. Banavar, A. Maritan and F. Seno, Phy s. Rev. Lett. 82, 3372 (1999). [20] M. R. Ejtehadi, N. Hamedani and V. Shahrezaei, Phys. Rev . Lett. 82, 4723 (1999). [21] V. S. Pande, C. Joerg, A. Y. Grosberg, and T. Tanaka, J. Ph ys. A27, 6231 (1994). [22] C. Chothia, Nature 357, 543 (1992). [23] T. L. Blundell and M. S. Johnson, Protein Sci. 2, 887 (1993). [24] N. N. Alexandrov and N. Go, Protein Sci 3, 866 (1994). [25] A. G. Murzin, S. E. Bremmer, T. Hubbard and C. Chothia, J. Mol. Biol. 247, 536 (1995). [26] H. J. Bussemaker, D. Thirumalai and J. K. Bhattacharjee , Phys. Rev. Lett. 79, 3530 (1997). [27] P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, New York, 1953). [28] See, for example, A mordern course in statistical Physics (University of Texas Press, Austin, 1980), edited by. L. E. Reichl. [29] J. D. Bryngelson and P. G. Walynes, Proc. Natl. Acad. Sci . USA 84, 7524 (1987). [30] M. Vendruscolo, A. Maritan and J. R. Banavar, Phys. Rev. Lett.20, 3967 (1997). 70 1 2 3 4 5 6 7 8 0 1 2012 (0,0)(1,1) (A) (B)(2,1) (1,2) (1,0) Figure 1: (A):The 3 ×3 square lattices used to ﬁnd out all the compact structures o f 9 beads chain. The bold curve is an oriented walk start at (0,0) and en d at (0,2). The arrows show that instead of walking along the bold curve, one can ﬁnd other str uctures in the direction of the arrows. (B): The oriented walks and their branching pattern during the search of them. Note that only some points show branching on the tree. Others are t runcated because they can not extend to 9 beads length due to the restriction of lattice siz e and self avoiding. The number at the right of the ﬁgure show the steps of the search. 020000400006000080000100000120000140000 -0.9 -0.7 -0.5 -0.3designability partnum Figure 2: Points: Designability against Partnum of non-ori ented compact structures of chain length 25. Line: the curve of f(x) =ax+bwitha= 488809 and b= 300344. The correlation coeﬃcient is r= 0.447, with totally 621 data points. 8-0.7-0.65-0.6-0.55-0.5-0.45-0.4-0.35-0.3-0.25-0.2 20304050average of Partnum ( A )chain length0.0020.0040.0060.0080.010.0120.014 20304050variance of Partnum ( B )chain length Figure 3: (A): The dependence of the average of Partnums < P > on chain length. The upper line-points curve is for the oriented walks, and the lower li ne-points curve is for the non-oriented structures. The upper and lower doted straight lines < P > =−0.3343 and < P > =−0.6686 are the estimated long chain limit of < P1>and< P2>, respectively (see text). (B): The dependence of the variance of Partnums on chain length. 901234567 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3density distribution PartnumP2-49 P2-42 P2-30 P2-20 Figure 4: The density distributions of Partnums of non-orie nted structures under various chain length. The number “49” in “P2-49”, for example, is the chain length. The distribution curves are shown in step curve style. 678910111213141516 25 30 35 40 45 50lnM chain length Figure 5: Logarithm of the total number of oriented walks ver sus the chain length. The line is the ﬁt using lnM=ln(Cav)×N+b, with Cav= 1.3969 and b=−0.9489. The correlation coeﬃcient is r= 0.99. 10
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/D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CU/CT/D1/B9/D8/D3/D7/CT /D3/D2/CS /B4/D7/D3/D0/CX/D8/D3/D2/B5 /CP/D2/CS /D4/CX /D3/D7/CT /D3/D2/CS /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/BA /CF /CT /D7/CW/CP/D0/D0 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D8/CW/CP/D8/D7/D9 /CW /CP /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CX/D7 /CP /D3/D1/D4/CP/D2/CX/CT/CS /CQ /DD /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /CW /DD/D7/D8/CT/D6/CT/D7/CX/D7 /D4/CW/CT/D2/D3/D1/B9/CT/D2/CP/BA /BU/CP/D7/CT/CS /D3/D2 /D7/D3/D0/CX/D8/D3/D2 /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /D8/CW/CT/D3/D6/DD /B8 /D8/CW/CT /D2 /D9/D1/CT/D6/CX /CP/D0 /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D8 /DB /D3/CS/CX/AR/CT/D6/CT/D2 /D8 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /D4 /CT/D6/CU/D3/D6/D1/CT/CS/BA /CC/CW/CT /AS/D6/D7/D8 /D3/D2/CT /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /B4/CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3/D6 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/B5/B8/DB/CW/CT/D2 /CU/D3/D6 /CT/DA /CT/D6/DD /D2/CT/DB /DA /CP/D0/D9/CT /D3/CU /D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D8/CW/CT /D0/CP/D7/CT/D6 /CX/D7 /D8/D9/D6/D2/CT/CS /D3/D2 /CP/CU/D6/CT/D7/CW/BA/CC/CW/CT /D7/CT /D3/D2/CS /D7/CX/D8/D9/CP/D8/CX/D3/D2 /CX/D7 /CU/D3/D6 /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /CS/D9/D6/B9/CX/D2/CV /CP /D7/CX/D2/CV/D0/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D7/CT/D7/D7/CX/D3/D2/BA /CF /CT /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /CP/D0/D7/D3 /D8/CW/CP/D8 /D8/CW/CT /D8/CW/CX/D6/CS /D3/D6/CS/CT/D6/CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CS/CT/D7/D8/CP/CQ/CX/D0/CX/DE/CT/D7 /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2 /CP/D2/CS /D4/D6/D3 /CS/D9 /CT/D7 /CP /D7/D8/D6/D3/D2/CV /CU/D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8 /D3/CU/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /CV/CP/CX/D2 /CT/D2 /D8/CT/D6/BA/BE/BE /C5/D3 /CS/CT/D0/CF /CT /CP/D2/CP/D0/DD/DE/CT/CS /D8/CW/CT /AS/CT/D0/CS /CT/DA /D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /D0/CP/D7/CT/D6 /D7/DD/D7/D8/CT/D1 /D3/D2 /D8/CP/CX/D2/CX/D2/CV /D7/CP/D8/B9/D9/D6/CP/CQ/D0/CT /D5/D9/CP/D7/CX/B9/CU/D3/D9/D6 /D0/CT/DA /CT/D0 /CV/CP/CX/D2 /D6/DD/D7/D8/CP/D0/B8 /CU/D6/CT/D5/D9/CT/D2 /DD /AS/D0/D8/CT/D6/B8 /BZ/CE/BW /CP/D2/CS /CB/C8/C5 /CT/D0/CT/D1/CT/D2 /D8/D7/B8/D8 /DB /D3 /D0/CT/DA /CT/D0 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /CP/D2/CS /D0/CX/D2/CT/CP/D6 /D0/D3/D7/D7/BA /CF /CT /CP/D7/D7/D9/D1/CT/CS/B8 /D8/CW/CP/D8 /CX/D2 /D2/D3/D2 /D3/CW/CT/D6/CT/D2 /D8/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /D8/CW/CT /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /CP /D8/CX/D3/D2 /CP/D2 /CQ /CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /D3/D4 /CT/D6/CP/D8/D3/D6 −γexp(−E/E s) 1+∂/∂t /CJ/BK℄/B8 /DB/CW/CT/D6/CT γ /CX/D7 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /D0/D3/D7/D7/B8E=t/integraltext −∞\|a(t′)\|2dt′ /CX/D7/D8/CW/CT /CT/D2/CT/D6/CV/DD /AT/D9/CT/D2 /DD /D4/CP/D7/D7/CT/CS /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D8/D3 /D1/D3/D1/CT/D2 /D8 t /B8t /CX/D7 /D8/CW/CT /D0/D3 /CP/D0/D8/CX/D1/CT/B8 Es /CX/D7 /D8/CW/CT /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /AT/D9/CT/D2 /DD/BN /D8/CW/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /D8/CW/CT /CS/CT/D2/D3/D1/B9/CX/D2/CP/D8/D3/D6 /CX/D2 /D8/D3 /D8/CW/CT /D8/CX/D1/CT /D7/CT/D6/CX/CT/D7 /CX/D2∂ ∂t /CX/D7 /CU/D9/D6/D8/CW/CT/D6 /D7/D9/D4/D4 /D3/D7/CT/CS/BA /CC/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D7/CS/D9/CT /D8/D3 /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /BA /CC/CW/CT /D6/CP/D8/CX/D3 /D3/CU /D8/CW/CT /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /D8/D3 /D8/CW/CT/CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/CX/CT/D7 τ /D9/D2/CS/CT/D6 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D5/D9/CP/D0 /D6/D3/D7/D7 /D7/CT /D8/CX/D3/D2/D7 /D3/CU/D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D1/D3 /CS/CT /CP/D8 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D1/D3 /CS/D9/D0/CP/D8/D3/D6 /CP/D2/CS /CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT /CP /D8/CX/DA /CT/D1/CT/CS/CX/D9/D1 /CX/D71.25×10−4/CU/D3/D6Es= 100 µJ/cm2/B4 /D3/D1/D4/CP/D6/CT /D8/CW/CX/D7 /DB/CX/D8/CW /D8/CW/CT /AS/CV/D9/D6/CT/CU/D3/D6 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /D6/CT/D4 /D3/D6/D8/CT/CS /CX/D2 /CJ/BF℄/B5/BA/CC/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /AS/CT/D0/CS /CP /D3/CQ /CT/DD/D7 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D3/D4 /CT/D6/CP/D8/D3/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BM ∂a(z, t) ∂z=/bracketleftBigg αexp(−τE)−γexp(−E) 1 +∂ ∂t+/bracketleftBigg1 1 +∂ ∂t−1/bracketrightBigg −l/bracketrightBigg a(z, t) +/bracketleftBigg ik2∂2 ∂t2+k3∂3 ∂t3−iβ\|a(z, t)\|2/bracketrightBigg a(z, t) /B4/BD/B5/DB /CT/D6/CTz /CX/D7 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CP /DA/CX/D8 /DD /D0/CT/D2/CV/D8/CW/B8 /D3/D6/D8/D6/CP/D2/D7/CX/D8 /D2 /D9/D1 /CQ /CT/D6/B8 α /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2/B8 l /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D0/D3/D7/D7/B8k2,3 /CP/D6/CT /D8/CW/CT/D7/CT /D3/D2/CS/B9 /CP/D2/CS /D8/CW/CX/D6/CS/B9/D3/D6/CS/CT/D6 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8/D7/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8 /D8/CW/CT /CT/D2/CT/D6/CV/DD /AT/D9/B9/CT/D2 /DD /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3Es /BA /CC/CW/CT /D8/CX/D1/CT /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW /D3/CU/D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 1/ta /BA /CF/CX/D8/CW /D8/CW/CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /CB/C8/C5 /D3 /CTꜶ /CX/CT/D2 /D8 β /CX/D72πn2LEs λnta /B8/DB/CW/CT/D6/CT L /CX/D7 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D6/DD/D7/D8/CP/D0/B8 n /CP/D2/CSn2 /CP/D6/CT /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CP/D2/CS /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3 /CTꜶ/B9 /CX/CT/D2 /D8/D7 /D3/CU /D6/CT/CU/D6/CP /D8/CX/DA/CX/D8 /DD /B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8 λ /CX/D7 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/BA /CC/CW/CT /D8/CT/D6/D1/CX/D2 /D7/D5/D9/CP/D6/CT /D4/CP/D6/CT/D2 /D8/CW/CT/D7/CX/D7 /D7/D8/CP/D2/CS/D7 /CU/D3/D6 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /AS/D0/D8/CT/D6/CX/D2/CV/BA /CC/CW/CT /D8/D6/CP/D2/D7/D1/CX/D7/D7/CX/D3/D2/CQ/CP/D2/CS /D3/CU /D8/CW/CT /AS/D0/D8/CT/D6 /CP/D2/CS /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CQ/CP/D2/CS /D3/CU /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /DB /CT/D6/CT /CP/D7/D7/D9/D1/CT/CS/D8/D3 /D3/CX/D2 /CX/CS/CT/BA /BT/D2 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BD/B5 /D9/D4 /D8/D3 /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /CX/D2∂/∂t /D6/CT/D7/D9/D0/D8/D7 /CX/D2/D2/D3/D2/D0/CX/D2/CT/CP/D6 /C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV /CT/D5/D9/CP/D8/CX/D3/D2 /CJ/BG/B8 /BK℄/B8 /DB/CW/CX /CW /DB /CT /CS/D3 /D2/D3/D8 /DB/D6/CX/D8/CT /D3/D9/D8 /CW/CT/D6/CT/CS/D9/CT /D8/D3 /CX/D8/D7 /D3/D1/D4/D0/CT/DC/CX/D8 /DD /BA/CB/CX/D2 /CT /CP/D2 /CT/DC/CP /D8 /CV/CT/D2/CT/D6/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BD/B5 /CX/D7 /D9/D2/CZ/D2/D3 /DB/D2/B8 /DB /CT /D7/D3/D9/CV/CW /D8 /CU/D3/D6 /D8/CW/CT/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CU/D3/D6/D1a(z, t) =a0exp[iω(t−zδ)+iϕz] cosh[( t−zδ)/tp]1+iΨ /B8/DB/CW/CT/D6/CT a0 /CX/D7 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8 tp /CX/D7 /D8/CW/CT /DB/CX/CS/D8/CW/B8 ω /CX/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /D1/CX/D7/D1/CP/D8 /CW /CU/D6/D3/D1/AS/D0/D8/CT/D6 /CQ/CP/D2/CS /CT/D2 /D8/CT/D6/B8 Ψ /CX/D7 /D8/CW/CT /CW/CX/D6/D4/B8 ϕ /CP/D2/CSδ /CP/D6/CT /D8/CW/CT /D4/CW/CP/D7/CT /CP/D2/CS /D8/CX/D1/CT /CS/CT/D0/CP /DD/D7/CP/CU/D8/CT/D6 /D8/CW/CT /CU/D9/D0/D0 /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS /D8/D6/CX/D4/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /C1/D2 /D8/CW/CT /CU/D6/CP/D1/CT /D3/CU /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2/D0/CT/D7/D7/BF/CP/D4/D4/D6/D3/CP /CW /CJ/BL℄/B8 /D8/CW/CT /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CX/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /BX/D5/BA /B4/BD/B5 /DB/CX/D8/CW /CU/D3/D0/D0/D3 /DB/CX/D2/CV/CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D2t /DD/CX/CT/D0/CS/D7 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D7/CT/D8 /D3/CU /D3/D6/CS/CX/D2/CP/D6/DD /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CU/D3/D6/CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BM da0 dz=a0 αexp/parenleftBig −τa2 0 v/parenrightBig + (1−ω2−v2)γexp/parenleftBig −a2 0 v/parenrightBig −l+k2Ψv2− ω2−v2−3k3ωv2Ψ , dω dz=/parenleftBig Ψa2 0ω2+a2 0ω−2v2ω−2v2ωΨ2−Ψa2 0/parenrightBig γexp/parenleftBigg −a2 0 v/parenrightBigg − 2v2ωΨ2−2v2ω−Ψa2 0ταexp/parenleftBigg −τa2 0 v/parenrightBigg −3k3v4Ψ/parenleftBig Ψ2+ 1/parenrightBig , dv dz=1 2v2/parenleftBig a4 0v2+ 2v4Ψ2−a4 0+a4 0ω2+ 2a2 0v2−4a2 0ωv2Ψ−4v4/parenrightBig × /B4/BE/B5 γexp/parenleftBigg −a2 0 v/parenrightBigg −a4 0τ2α 2v2exp/parenleftBigg −τa2 0 v/parenrightBigg + 3k2v2Ψ +v2Ψ2− 2v2−9k3ωΨv2, dΨ dz=/parenleftBig 4a2 0ωv2+a4 0Ψ + 4a2 0ωv2Ψ2−2v4Ψ−Ψa4 0ω2−a4 0ω−2v4Ψ3/parenrightBig × γ v2exp/parenleftBigg −a2 0 v/parenrightBigg +Ψαa4 0τ2 v2exp/parenleftBigg −τa2 0 v/parenrightBigg −2a2 0β−2v2Ψ− 4k2v2Ψ2−2v2Ψ3−4k2v2+ 12k3ωv2/parenleftBig Ψ2+ 1/parenrightBig ,/DB/CW/CT/D6/CT v= 1/tp /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D4/D9/D0/D7/CT /DB/CX/CS/D8/CW/B8 /CP/D2/CS /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CX/D1/CT /CP/D2/CS/D4/CW/CP/D7/CT /CS/CT/D0/CP /DD/D7/BM δ= [a2 0(ω2+v2−1)−2v2ωΨ)]γexp/parenleftBig −a2 0 v/parenrightBig −2ωΨv2−2k2ωv2+ k3v2(5v2+ 3ω2−3v2Ψ2)−a2 0ταexp/parenleftBig −τa2 0 v/parenrightBig , ϕ= [a2 0ω(ω2+v2−1) +v2Ψ (v2−2ω2)]γexp/parenleftBig −a2 0 v/parenrightBig +k3v2ω× (2v2+ 2ω2−3v2Ψ2) +k2v2(v2−ω2) + Ψv4+βa2 0v2−2ω2Ψv2− ωταa2 0exp/parenleftBig −τa2 0 v/parenrightBig ./BT/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /D4/D9/D0/D7/CT /DB/CX/CS/D8/CW /CX/D7 /D1 /D9 /CW /D7/CW/D3/D6/D8/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /CP /DA/CX/D8 /DD /D4 /CT/D6/CX/D3 /CS Tcav /B8/D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CV/CP/CX/D2 /CT/DA /D3/D0/D9/D8/CX/D3/D2 /CX/D7 /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM dα dz= (αm−α)P−2ατa2 0 v−α Tg, /B4/BF/B5/DB/CW/CT/D6/CT P=σTcavIp hυ /CX/D7 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8 σ /CX/D7 /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D6/D3/D7/D7/B9/D7/CT /D8/CX/D3/D2 /CP/D8 /D8/CW/CT /D4/D9/D1/D4 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/B8 hυ /CX/D7 /D8/CW/CT /CT/D2/CT/D6/CV/DD /D3/CU /D4/D9/D1/D4 /D4/CW/D3/D8/D3/D2/B8 αm/CX/D7 /D8/CW/CT /D1/CP/DC/CX/D1/CP/D0 /CV/CP/CX/D2 /CP/D8 /D8/CW/CT /CU/D9/D0/D0 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CX/D2 /DA /CT/D6/D7/CX/D3/D2/BA P= 10−4 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7/D8/D3 /D8/CW/CT /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6 /D3/CU /BD/CF /CU/D3/D6100µm /CS/CX/CP/D1/CT/D8/CT/D6 /D3/CU /D4/D9/D1/D4 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/B4/CB/BX/CB/BT/C5/B3/D7/B5 /CU/D3/D6 /CU/CT/D1 /D8/D3/D7/CT /D3/D2/CS /D8/D3 /D2/CP/D2/D3/D7/CT /D3/D2/CS/D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CX/D2 /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6/D7/AH/B8 /C1/BX/BX/BX /C2/BA /CB/CT/D0/CT /D8/CT/CS /CC /D3/D4/CX /D7 /CX/D2 /C9/D9/CP/D2 /D8/BA/BX/D0/CT /D8/D6/BA/B8 /BE/B8 /BG/BF/BH/B9/BG/BH/BD /B4/BD/BL/BL/BI/B5/BA/BG/BA /BY/BA /CG/BA /C3/GC/D6/D8/D2/CT/D6/B8 /C1/BA /BW/BA /C2/D9/D2/CV/B8 /CP/D2/CS /CD/BA /C3/CT/D0/D0/CT/D6/B8 /AH/CB/D3/D0/CX/D8/D3/D2 /D1/D3 /CS/CT/B9/D0/D3 /CZ/CX/D2/CV /DB/CX/D8/CW/D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/D7/AH/B8 /C1/BX/BX/BX /C2/BA /CB/CT/D0/CT /D8/CT/CS /CC /D3/D4/CX /D7 /CX/D2 /C9/D9/CP/D2 /D8/BA /BX/D0/CT /D8/D6/BA/B8 /BE/B8 /BH/BG/BC/B9/BH/BH/BH/B4/BD/BL/BL/BI/B5/BA/BH/BA /CB/BA /CA/BA /BU/D3/D0/D8/D3/D2/B8 /CA/BA /BT/BA /C2/CT/D2/CZ/D7/B8 /BV/BA /C6/BA /BX/D0/CZ/CX/D2 /D8/D3/D2/B8 /BZ/BA /CB/D9 /CW/CP/B8 /AH/C8/D9/D0/D7/CT/B9/D6/CT/D7/D3/D0/DA /CT/CS/D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D3/CU /D7/D9/CQ/CW/CP/D6/D1/D3/D2/CX /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CX/D2 /CP /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /CC/CX/BM/D7/CP/D4/D4/CW/CX/D6/CT /D0/CP/D7/CT/D6/AH/B8 /C2/BA /C7/D4/D8/BA /CB/D3 /BA /BT/D1/BA /BU/B8 /BD/BI/B8 /BF/BF/BL/B9/BF/BG/BG /B4/BD/BL/BL/BL/B5/BA/BI/BA /CE/BA /C4/BA /C3/CP/D0/CP/D7/CW/D2/CX/CZ /D3 /DA/B8 /C1/BA /BZ/BA /C8 /D3/D0/D3 /DD/CZ /D3/B8 /CE/BA /C8 /BA /C5/CX/CZ/CW/CP/CX/D0/D3 /DA/B8 /BW/BA /DA /D3/D2 /CS/CT/D6 /C4/CX/D2/CS/CT/B8/AH/CA/CT/CV/D9/D0/CP/D6/B8 /D5/D9/CP/D7/CX/B9/D4 /CT/D6/CX/D3 /CS/CX /CP/D2/CS /CW/CP/D3/D8/CX /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2 /DB /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /C3/CT/D6/D6/B9/D0/CT/D2/D7/D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /D0/CP/D7/CT/D6/D7/AH/B8 /C2/BA /C7/D4/D8/BA /CB/D3 /BA /BT/D1/BA /BU/B8 /BD/BG/B8 /BE/BI/BL/BD/B9/BE/BI/BL/BH /B4/BD/BL/BL/BJ/B5/BA/BJ/BA /BV/BA /C0/GU/D2/D2/CX/D2/CV/CT/D6/B8 /CA/BA /C8 /CP/D7 /CW/D3/D8/D8/CP/B8 /BY/BA /C5/D3/D6/CX/CT/D6/B9/BZ/CT/D2/D3/D9/CS/B8 /C5/BA /C5/D3/D7/CT/D6/B8 /CP/D2/CS /CD/BA/C3/CT/D0/D0/CT/D6/B8 /AH/C9/B9/D7/DB/CX/D8 /CW/CX/D2/CV /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D0/CX/D1/CX/D8/D7 /D3/CU /D3/D2 /D8/CX/D2 /D9/D3/D9/D7/B9/DB /CP /DA /CT /D4/CP/D7/D7/CX/DA /CT /D1/D3 /CS/CT /D0/D3 /CZ/B9/CX/D2/CV/AH/B8 /C2/BA /C7/D4/D8/BA /CB/D3 /BA /BT/D1/BA /BU/B8 /BD/BI/B8 /BG/BI/B9/BH/BI /B4/BD/BL/BL/BL/B5/BA/BK/BA /CE/BA /C4/BA /C3/CP/D0/CP/D7/CW/D2/CX/CZ /D3 /DA/B8 /BW/BA /C7/BA /C3/D6/CX/D1/CT/D6/B8 /C1/BA /BZ/BA /C8 /D3/D0/D3 /DD/CZ /D3/B8 /CE/BA/C8 /BA /C5/CX/CZ/CW/CP/CX/D0/D3 /DA/B8 /AH/CD/D0/B9/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CX/D2 /DB /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6 /DB/CX/D8/CW /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/CP/D8/D9/D6/CP/CQ/D0/CT/CP/CQ/D7/D3/D6/CQ /CT/D6 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D0/CX/D2/CT/DB/CX/CS/D8/CW /CT/D2/CW/CP/D2 /CT/D1/CT/D2 /D8/AH/B8 /C7/D4/D8/CX /D7/BV/D3/D1/D1 /D9/D2/BA/B8 /BD/BH/BL/B8 /BE/BF/BJ/B9/BE/BG/BE /B4/BD/BL/BL/BL/B5/BA/BL/BA /BT/BA /C5/BA /CB/CT/D6/CV/CT/CT/DA/B8 /BX/BA /CE/BA /CE /CP/D2/CX/D2/B8 /BY/BA /CF/BA /CF/CX/D7/CT/B8 /AH/CB/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /D4/CP/D7/D7/CX/DA /CT/D0/DD /D1/D3 /CS/B9/CT/D0/D3 /CZ /CT/CS /D0/CP/D7/CT/D6/D7 /DB/CX/D8/CW /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/D7/AH/B8 /C7/D4/D8/CX /D7 /BV/D3/D1/D1 /D9/D2/BA/B8 /BD/BG/BC/B8 /BI/BD/B9/BI/BG/B4/BD/BL/BL/BJ/B5/BA/BK-20 0101001000 dtp 43 321 BA xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx/BY/CX/CV/D9/D6/CT /BD/BM /C8/D9/D0/D7/CT /DB/CX/CS/D8/CW tp /DA /CT/D6/D7/D9/D7 /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8 k2 /B4/D8/CX/D1/CT /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 ta /B5/BA /CA/CT/CV/CX/D3/D2 /BT/B8 /D4/CX /D3/D7/CT /D3/D2/CS /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/B8 /D6/CT/CV/CX/D3/D2 /BU/B8 /CP/D9/D8/D3 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D6/CT/CV/CX/D1/CT /CU/D3/D6/D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D9/D6/DA /CT /BF/BAP= 5×10−4/B8αm= 0.5 /B8ta= 2.5fs /B8Tg= 3µs /B8 Tcav= 10ns /B8l= 0.01 /B8γ= 0.05 /B8k3= 0 /BAβ= 0.001 /B4/BD/B5/B80.01 /B4/BE/B5/B80.05 /B4/BF/B5/B8 0.1 /B4/BG/B5/BA/BLxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx -20 0 20-60-3003060 dψ B 31 12 2, 3, 4/BY/CX/CV/D9/D6/CT /BE/BM /BV/CW/CX/D6/D4 Ψ /DA /CT/D6/D7/D9/D7 /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8 k2 /BA /CC/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D8/D3 /BY/CX/CV/BA /BD/BA /BV/D9/D6/DA /CT /BF /CP/D2/CS /BG /CT/DC/CX/D7/D8 /D9/D2/CS/CT/D6 k2≥ −3 /B8−1 /B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA/BD/BC200 400123 zI/BY/CX/CV/D9/D6/CT /BF/BM /C8 /CT/CP/CZ /CX/D2 /D8/CT/D2/D7/CX/D8 /DD I /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3Est−1 a /DA /CT/D6/D7/D9/D7 /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4 /D2 /D9/D1 /CQ /CT/D6 z /BA k2=−20 /B8k3= 0 /B8P= 5×10−4/B8β= 0.05 /BA/BD/BD0.0001 0.0002101001000 1 2 finish of trajectory PtpA 2start of trajectory 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 2/BY/CX/CV/D9/D6/CT /BG/BM /C8/D9/D0/D7/CT /DB/CX/CS/D8/CW tp /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD P /BA /BT/D6/D6/D3 /DB /BD /B9 /D7/DB/CX/D8 /CW /CQ /CT/B9/D8 /DB /CT/CT/D2 /D4/D7/B9 /CP/D2/CS /CU/D7/B9/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /B4/D8/D6/CX/CP/D2/CV/D0/CT/D7/B5/BA k2=−10 /B8k3= 0 /B8β= 0.1 /B8 /D3/D8/CW/CT/D6/D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /BY/CX/CV/BA /BD/BA/BD/BE-80 -60 -40 -20 0 20101001000 dtp xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxA/BY/CX/CV/D9/D6/CT /BH/BM /C8/D9/D0/D7/CT /DB/CX/CS/D8/CW tp /DA /CT/D6/D7/D9/D7 /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8 k2 /BA /C8 /CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D8/D3 /D9/D6/DA /CT /BE /CX/D2 /BY/CX/CV/BA/BD/BA/BD/BF-20 0 20-0.10.00.1 1, 2, 3, 44 3221 BBB xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx B dωxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx/BY/CX/CV/D9/D6/CT /BI/BM /BY /D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8ω /DA /CT/D6/D7/D9/D7 /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8 k2 /BA /C8 /CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS /D8/D3 /D9/D6/DA /CT /BF /CX/D2 /BY/CX/CV/BA /BD/BAk3= 0 /B4/BD/B5/B8−5 /B4/BE/B5/B8−10 /B4/BF/B5/B810 /B4/BG/B5/BA /CC/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /BY/CX/CV/D7/BA /BD/B8 /BE/BA/BD/BG
arXiv:physics/0009010v1 [physics.gen-ph] 3 Sep 2000PSYCHOPHYSICAL QUANTUM–STRING WELTDRAMA Denis V. Juriev ul.Miklukho-Maklaya 20-180 Moscow 117437 Russia (e-mail: denis@juriev.msk.ru) The goal of this article is to formulate hypothesis of an exis tence of the universal quantum string ﬁeld for all interactive phenomena and to cla rify its practical value. I. Quantum string field theory and interactive phenomena [1] 1.1. Experimental detection of interactive phenomena. Let us consider a natural, behavioral, social or economical system S. It will be described by a set {ϕ}of quntities, which characterize it at any moment of time t(so that ϕ=ϕt). One may suppose that the evolution of the system is described by a diﬀerential equation ˙ϕ= Φ(ϕ) and look for the explicit form of the function Φ from the exper imental data on the system S. However, the function Φ may depend on time, it means that the re are some hidden parameters, which control the system Sand its evolution is of the form ˙ϕ= Φ(ϕ, u), where uare such parameters of unknown nature. One may suspect that s uch parameters are chosen in a way to minimize some goal function K, which may be an integrodiﬀerential functional of ϕt: K=K([ϕτ]τ≤t) (such integrodiﬀerential dependence will be brieﬂy notate d asK=K([ϕ]) below). More generally, the parameters umay be divided on parts u= (u1, . . . , u n) and each partuihas its own goal function Ki. However, this hypothesis may be conﬁrmed by the experiment very rarely. In the most cases the choice of parameters uwill seem accidental or even random. Nevertheless, one may suspe ct that the controls uiareinteractive , it means that they are the couplings of the pure controls u◦ iwith theunknown or incompletely known feedbacks: ui=ui(u◦ i,[ϕ]) Typeset by AMS-TEXand each pure control has its own goal function Ki. Thus, it is suspected that the system Srealizes an interactive game . There are several ways to deﬁne the pure controls u◦ i. One of them is the integrodiﬀerential ﬁltration of the cont rolsui: u◦ i=Fi([ui],[ϕ]). To verify the formulated hypothesis and to ﬁnd the explicit f orm of the convenient ﬁltrations Fiand goal functions Kione should use the theory of interactive games, which supplies us by the predictions of the game, and compare the predictions with the real history of the game for any considered FiandKiand choose such ﬁltrations and goal functions, which describe the reality better. One m ay suspect that the dependence of uionϕis purely diﬀerential for simplicity or to introduce the so- called intention ﬁelds , which allow to consider any interactive game as diﬀerentia l. Moreover, one may suppose that ui=ui(u◦ i, ϕ) and apply the elaborated procedures of a posteriori analysis and predictions to the system. In many cases this simple algorithm eﬀectively unravels the hidden interactivity of a complex system. However, more sophisticated psychophy sical procedures exist. Below we shall consider the complex systems S, which have been yet represented as the n-person interactive games by the procedure described above . 1.2. Functional analysis of interactive phenomena. To perform an analysis of the interactive control let us note that often for the n-person interactive game the interactive controls ui=ui(u◦ i,[ϕ]) may be represented in the form ui=ui(u◦ i,[ϕ];εi), where the dependence of the interactive controls on the argu ments u◦ i, [ϕ] andεiis known but the ε-parameters εiare the unknown or incompletely known functions ofu◦ i, [ε]. Such representation is very useful in the theory of intera ctive games and is called the ε-representation . One may regard ε-parameters as new magnitudes, which characterize the syst em, and apply the algorithm of the unraveling of interactivity t o them. Note that ε- parameters are of an existential nature depending as on the s tatesϕof the system Sas on the controls. Theε-parameters are useful for the functional analysis of the in teractive controls described below. First of all, let us consider new integrodiﬀerential ﬁltrat ionsVα: v◦ α=Vα([ε],[ϕ]), where ε= (ε1, . . . , ε n). Second, we shall suppose that the ε-parameters are ex- pressed via the new controls v◦ α, which will be called desires: εi=εi(v◦ 1, . . . , v◦ m,[ϕ]) and the least have the goal functions Lα. The procedure of unraveling of interac- tivity speciﬁes as the ﬁltrations Vαas the goal functions Lα. 21.3. SD-transform and SD-pairs. The interesting feature of the proposed de- scription (which will be called the S-picture ) of an interactive system Sis that it contains as the real (usually personal) subjects with the pu re controls uias the impersonal desires vα. The least are interpreted as certain perturbations of the ﬁrst so the subjects act in the system by the interactive cont rolsuiwhereas the desires are hidden in their actions. One is able to construct the dual picture (the D-picture ), where the desires act in the system Sinteractively and the pure controls of the real subjects are hidden in their actions. Precisely, the evolution of the system is g overned by the equations ˙ϕ=˜Φ(ϕ, v), where v= (v1, . . . , v m) are the ε-represented interactive desires: vα=vα(v◦ α,[ϕ]; ˜εα) and the ε-parameters ˜ εare the unknown or incompletely known functions of the states [ ϕ] and the pure controls u◦ i. D-picture is convenient for a description of systems Swith a variable number of acting persons. Addition of a new person does not make any i nﬂuence on the evolution equations, a subsidiary term to the ε-parameters should be added only. The transition from the S-picture to the D-picture is called theSD-transform . TheSD-pair is deﬁned by the evolution equations in the system Sof the form ˙ϕ= Φ(ϕ, u) =˜Φ(ϕ, v), where u= (u1, . . . , u n),v= (v1, . . . , v m), ui=ui(u◦ i,[ϕ];εi) vα=vα(v◦ α,[ϕ]; ˜εα) and the ε-parameters ε= (ε1, . . . , ε n) and ˜ ε= (˜ε1, . . . ,˜εm) are the unknown or incompletely known functions of [ ϕ] and v◦= (v◦ 1, . . . , v◦ m) oru◦= (u◦ 1, . . . , u◦ n), respectively. Note that the S-picture and the D-picture may be regarded as c omplementary in the N.Bohr sense. Both descriptions of the system Scan not be applied to it simultaneously during its analysis, however, they are comp atible and the structure of SD-pair is a manifestation of their compatibility. 1.4. The second quantization of desires. Intuitively it is reasonable to con- sider systems with a variable number of desires. It can be don e via the second quantization. To perform the second quantization of desires let us mention that they are deﬁned as the integrodiﬀerential functionals of ϕandεvia the integrodiﬀerential ﬁltrations. So one is able to deﬁne the linear space Hof all ﬁltrations (regarded as classical ﬁelds) and a submanifold Mof the dual H∗so that His naturally identiﬁed with a subspace of the linear space O(M) of smooth functions on M. The quantized ﬁelds of desires are certain operators in the space O(M) (one is able to regard them as unbounded operators in its certain Hilbert completion). Th e creation/annihilation 3operators are constructed from the operators of multiplica tion on an element of H⊂ O(M) and their conjugates. To deﬁne the quantum dynamics one should separate the quick a nd slow time. Quick time is used to make a ﬁltration and the dynamics is real ized in slow time. Such dynamics may have a Hamiltonian form being governed by a quantum Hamil- tonian, which is usually diﬀerential operator in O(M). IfMcoincides with the whole H∗then the quadratic part of a Hamiltonian de- scribes a propagator of the quantum desire whereas the highe st terms correspond to the vertex structure of self-interaction of the quantum ﬁ eld. If the submani- foldMis nonlinear, the extraction of propagators and interactio n vertices is not straightforward. 1.5. Quantum string ﬁeld theoretic structure of the second q uantization of desires. First of all, let us mark that the functions ϕ(τ) and ε(τ) may be regarded formally as an open string. The target space is a pro duct of the spaces of states and ε-parameters. Second, let us consider a classical counterpart of the evolu tion of the integrodif- ferential ﬁltration. It is natural to suspect that such evol ution is local in time, i.e. ﬁltrations do not enlarge their support (as a time interval) during their evolution. For instance, if the integrodiﬀerential ﬁltration depends on the values of ϕ(τ),ε(τ) forτ∈[t0−t1, t0−t2] at the ﬁxed moment t0, it will depend on the same values forτ∈[t−t1, t−t2] at other moments t > t 0. This supposition provides the reparametrization invariance of the classical evolution. Hence, it is reasonable to think that the quantum evolution is also reparametrization invariant. Reparametrization invariance allows to apply the quantum s tring ﬁeld theoretic models to the second quantization of desires. For instance, one may use the string ﬁeld actions constructed from the closed string vertices (n ote that the phase space for an open string coincides with the conﬁguration space of a closed string) or string ﬁeld theoretic nonperturbative actions. In the least case t he theoretic presence of additional “vacua” (minimums of the string ﬁeld action) as w ell as their structure is very interesting. II. Psychophysical quantum–string Weltdrama 2.1. Psychophysical quantum string ﬁelds as quantized inte ntion ﬁelds. Note that one may assume that ε-parameters are history-independent functions of states ϕand desires vafter an introduction of an explicitely time-dependent classical string ﬁeld Ξ: ε=ε(ϕ, v◦; Ξ). If it is so, the classical string ﬁeld is just an intention ﬁel d and the obtained quan- tum string ﬁeld may be regarded as a quantized intention ﬁeld . The formalism of intention ﬁelds and their second quantization was describe d in [2]. An interpretation of quantum string ﬁelds as quantized inte ntion ﬁelds allows to avoid an introduction of a new additional concept and clariﬁ es the relation between desires and intentions. It should be marked that an eﬀective way to manipulate the int ention ﬁelds is to visualize them. Concrete procedures of visualization of in tention ﬁelds by use of the so-called overcolors were discussed by the author many times. Such visualization s, which identify intention ﬁelds with “latent lights” , preserve the algebraic structure 4of the intention ﬁelds. Some practical applications of this scheme were described in [3]. 2.2. Hypothesis of the universal psychophysical quantum st ring ﬁeld. In the article [4] it was supposed that allintention ﬁelds have the common dynamical nature and some discussion of this nature was performed. Therefore, it is reasonable to believe that all quantum stri ng ﬁelds, which appear in interactive phenomena, can be derived from one universal quantum string ﬁeld, which will be called the universal psychophysical quantum string ﬁeld . Precisely it may mean that the string ﬁeld algebras of concrete quantum st ring ﬁelds are certain subalgebras of the string ﬁeld algebra of this universal psy chophysical quantum string ﬁeld. If the tactical aspects [5] are also taken into account, the u niversal psychophysical quantum string ﬁeld would describe the dramatic structure o f the Universe and would represent the least as psychophysical quantum–string Weltdrama . Its existence and explication of its nature will allow to use the simple and more than inexpensive experimental data, obtained from analysi s of interactive computer games (especially, various perception games), to more soph isticated, less accessible and more important interactive phenomena. References [1] Juriev D., Experimental detection of interactive phenomen a and their analysis: math.GM/0003001; New mathematical methods for psychophys ical ﬁltering of experi- mental data and their processing: math.GM/0005275; Quantum string ﬁeld theory and psychophysics: physics/0008058. [2] Juriev D., Interactive games and representation theory: math .FA/9803020; Interactive games and representation theory. II. A second quantization: ma th.RT/9808098. [3] Juriev D., Droems: experimental mathematics, informatics a nd inﬁnite-dimensional ge- ometry. Report RCMPI-96/05+(1996) [e-version: cs.HC/9809119]. [4] Juriev D., On the dynamics of physical interactive informat ion systems: mp arc/97-158. [5] Juriev D., Tactical games & behavioral self-organization: math.GM/9911147; Tactics, dialectics, representation theory: math.GM/0001032; Theory o f transpersonal tactics, in preparation. 5
arXiv:physics/0009011v1 [physics.atm-clus] 3 Sep 2000ORIENTATIONAL MELTING OF TWO-SHELL CARBON NANOPARTICLES: MOLECULAR DYNAMICS STUDY. Yu. E. Lozovik, A. M. Popov∗ Institute of Spectroscopy, Russian Academy of Science, 142 190, Troitsk, Moscow region, Russia The energetic characteristics of two-shell carbon nanopar ticles (”onions”) with diﬀerent shapes of second shell are calculated. The barriers of relative rot ation of shells are found to be surprisingly small therefore free relative rotation of shells can take pl ace at room temperature. The intershell orientational melting of the nanoparticle is studied by mol ecular dynamics. The parameters of Arrhenius formula for jump rotational intershell diﬀusion are calculated. The rotation of shells can be observed beginning from temperature 70 K. I. INTRODUCTION The discovery of fullerenes [1] and the elaboration of metho d of their production in arc discharge [2] gives rise the interest to another carbon nanostructures produced in arc d ischarge, in particular, nanoparticles with shell structu re [3]. A set of works is devoted to their structure and energeti cs [4]– [12]. Nevertheless, an attention has not yet been given to thermodynamical properties of carbon nanoparticl es with shell structure. The melting of single cluster can essentially diﬀer from phase transitions in macroscopic sy stems [13]- [19]. Particularly, the melting of a mesoscopic cluster with shell structure can manifest itself as an hiera rchy of rearrangements with breaking intershell orientati onal order and then breaking shell structure and order in particl es positions inside shells. E.g., in 2D mesoscopic clusters with Coulomb [13]- [16], screened Coulomb [17], logarithmi c [18] and dipole [19] interaction between particles the orientational melting (breaking the orientational order b etween the shells) precedes melting inside the shells. The van der Waals interaction between atoms of neighbour she lls in carbon nanoparticles is considerably weaker than chemical bonds between atoms inside the shell. So it is n aturally that these nanoparticles are possible candidates for orientational melting [4]. The possibility of orientat ional melting of long two-shell carbon nanotube was discuss ed [20]. The orientational melting in carbon nanotube bundle w as also theoretically studied [21]. The orientational melting can be considered as a two stage ph enomenon. At low temperatures the relative orien- tations of shells are freezed. The intershell reorientatio ns begin with increasing of temperature. For low temperatur e these reorientations occur as jumps between ﬁxed relative s hell orientations corresponding to minima of nanoparticle energy (jump rotational diﬀusion). For high temperature fr ee rotation of shells take place. In the present paper the zero temperature energetic characteristics of two-she ll carbon nanoparticle C60@C240are calculated. The ob- tained values for barriers of relative rotations of shells a re small enough to free rotation of shells take place at room temperature. The orientational melting of this nanopartic le is studied here by molecular dynamics technique. II. SIMULATION DETAILS The following reasons have determined our choice of nanopar ticle shells. The TEM images shows that the inner shell of carbon nanoparticle can have a size that is close to t he size of fullerene C 60[22,23]. The fullerene C 60withIh symmetry is the smallest fullerene without adjacent pentag ons in its structure. Fullerenes smaller than C 60can not be directly extracted by the use of any solvent from soot, obt ained in arc discharge (see, for example, [24,25]). To explain this fact it was proposed that atoms of fullerenes wh ich belong to two adjacent pentagons can have chemical bonds with neighbor fullerenes in soot [27]. For example, ch emical bonds between all neighbour fullerenes are present in solid C 36[29]. Therefore we consider C 60as the smallest inner shell where the absence of chemical bon ds between shells is very probable (it is a necessary condition for exis tence of relative rotation of shells). Used single and doubl e bonds lengths of C 60are 1.391 ˚Aand 1.455 ˚A, respectively [30]. We accept the fullerene C 240withIhsymmetry as outer shell of nanoparticle. This model gives the distance b etween shells in agreement with experiment [23] being close to the distance between graphite planes. Besides the f ullerene C 240withIhsymmetry have greater binding energy than fullerenes C 240with other structures [6]. Several sets of geometric parame ters corresponding to diﬀerent shapes of fullerene C 240obtained by ab initio calculations of minima of binding energy [4,7,8] are used. D iﬀerent shell ∗Corresponding author. Fax: +7-095-334-0886; e-mail: popo v@isan.troitsk.ru 1shapesB,C,DandEwere found by optimization of all independent geometric par ameters of fullerene C 240withIh symmetry. The shapes BandDcorresponding to global and local minima found by York et al[7] that are close to sphere and truncated icosahedron respectively. The shape Ecorresponds to the single minimum found by Osawa [4]. It is intermediate between shapes BandD. The shape Cis rather close to shapes E. It corresponds to the minimum found by Scuceria [8]. The shape Ais obtained by optimization of less number of independent ge ometric parameters so that all atoms of this shape are arranged on the sphere [7]. We describe the interaction between atoms of neighbour shel ls by Lennard-Jones potential U= 4ǫ((σ/r12)−(σ/r)6) with parameters ǫ= 28 K and σ= 3.4˚A. These parameters were used for the simulation of solid C 60[31]. The interaction between atoms inside shells we describe by Born potential: U=α−β 260/summationdisplay i,j=1((ui−uj)rij \|rij\|)2+β 260/summationdisplay i,j=1(ui−uj)2(1) where ui,ujare displacements of atoms from equilibrium positions, rijare distances between atoms. We take α= 1.14·103N/m andβ= 1.24·102N/m. Born potential with these values of force constants giv es adequate internal vibrational spectrum of C 60[32]. Born potential is correct only near the bottom of poten tial well. Nevertheless we believe that this potential is adequate for our simulation b ecause we use it at temperatures that are one-two order of magnitude less than the temperature of fullerene destructi on. The orientational melting of nanoparticle C60@C240with shape Dof C240we studied by molecular dynamics technique. The simulations are performed in microcanonica l ensemble. The equations of motion were integrated using the leap frog algorithm. We used the integration step τ= 6.1·10−16s (about one hundred steps for period of atoms vibration inside shells). Initially the system has be en brought to the equilibrium during 300-500 ps that is about 30–50 librations of shells. The average ﬂuctuations of the t otal energy and temperature of the system fall and ﬂatten out during this period. Then the system was studied during 10 0 ps. The average ﬂuctuations of the total energy of the system were within 0.3 % and the average ﬂuctuations of te mperature were within 1.3 %. The angular velocities of shells change rather slowly. Therefore all investigated quantities were averaged over 34-46 diﬀerent realizations of the systems at the same temperature but with diﬀerent energi es accounting for relative rotation of shells (i.e. with diﬀerent random angular velocities of shells). III. RESULTS AND DISCUSSION A. Ground state energetics The global and local minima of total nanoparticle energy are found by optimization of three angles of their relative orientation. The total energy of nanoparticle includes the energy of interaction between shells and energy of shell deformation. We describe the relative orientations corres ponding to minima of total energy in terms of three angles αz,αyandαxof subsequent rotations of ﬁrst shell around axes OZ, OY and O X of coordinate system. The centers of both shells coincide with the center of coordinate system . The angles αz,αyandαxwere measured from the initial orientation shown on Fig. 1. Due to the high Ihsymmetry of shells the number of any equivalent minima (global or local) is 60. Such equivalent minima correspond t o diﬀerent relative orientations of shells. The energies of interaction between shells and angles of one of the orient ations corresponding to global and local minima of total energy of nanoparticle are listed in Table 1. The energies of interaction between shells calculated here are slightly less than 16.9 [9], 18.57 [10] and 20.3 [9] meV/atom obtained using another representations of van der Waals interaction and are about three times less than estimation 65.3 meV/atom for graphite [33]. Note, that the e nergy of total interaction between shells is not maximal for perfect sphere in comparison with other shapes of C 240contrary to the assumption of Lu and Yang [10]. We observed that the angles of orientations corresponding t o global and local minima are determined by the shape of second shell. For the shapes C,DandEof C240the initial relative orientation of shells (where symmetry axis of shells coincide) corresponds to global minima of total nano particle energy (note, that all these shapes of C 240are close to the truncated icosahedron). Several global minima for sh apeDare shown on Fig. 2a. One type of local minima is found for these shapes of C 240. For the shape B(which is close to sphere) orientations with coinciding sym metry axes correspond only to local minima (see Fig 2b). No minima corre spond to such orientations for the ”spherical” shape A. For ”spherical” shape of C 240two types of local minima are found. The diﬀerences ∆ Elocin total nanoparticle energies between global and local minima are very small and a lso determined by the shape of second shell (see Table 2). These diﬀerences decrease with decreasing the average d eviation<∆Ri2>=<\|Ri2−< R i2>\|of second shell from perfect sphere, where Ri2is the distance between an atom of second shell and the center of nanoparticle. The 2diﬀerences ∆ Elocalso decrease when the average distance between shells h=<R i2>−<R i1>approaches to the distancermincorresponding to the minimum in pair interatomic potential . This fact can be explained as follows: the change of distance d12between two atoms of neighbour shells causes the less change of interaction energy between these atoms for the distances d12corresponding to the bottom of interatomic potential well i n comparison with the distancesd12corresponding to the walls of this well. The calculated energies of shell deformation are presented in Table 2. The inﬂuence of shell deformation on the barriers of relative rotation of shells is studied as an exam ple for barriers B5of shell rotation around ﬁfth order axes of symmetry. (Barriers B5was calculated for the relative orientation where symmetry axes of shells have the same directions). The comparison of barriers B5calculated with taking into account shell deformation and w ithout it gives the diﬀerence less than 1 % for all ﬁve shapes of C 240investigated here (Note that the barrier B5calculated here for the shapeEof C240is 12 % less than that obtained by Osawa used the tandem of mole cular orbital and molecular mechanics calculations [4]). Therefore the shell deformat ions are disregarded here in calculation of barriers of rela tive rotation of shells, i.e. lengths of bonds angles between bon ds inside shells are supposed to be ﬁxed during intershell rotation. Note, that opposite situation take place e.g. for clusters with logarithmic interaction between particles [ 18]. In this case the interparticle interactions inside shell an d between shells are the same and therefore the considering of shells deformation is necessary in calculation of barrie rs for rotation. The relative displacement of the centres of symmetry of shells causes an increase in intershell interac tion energy. Therefore the common center of symmetry of both shells also supposed to be ﬁxed during rotation. The barriers of relative rotation of shells in the nanoparti cles under consideration are calculated for relative orien - tations corresponding to global minima of total nanopartic le energies. It is found that the obtained values of barriers for rotation are surprisingly small (see Table 2). Magnitudes of these barriers are very sensiti ve to the shape of C 240 and decrease when <∆Ri2>→0 andh→rmin(analogously to the diﬀerences ∆ Elocin interaction energies between global and local minima). Moreover, these barriers are only several times greater than barriers Bain dependencies of interaction energy between only one atom of the second shell and the whole ﬁrst shell vs. angle of rotat ion. For example, for the nanoparticle with shape Dof C240the barrier for rotation around ﬁfth order axis is 158.8 K. Si mul- taneously the maximal barrier among the barriers Bafor diﬀerent atoms of the second shell is 21.6 K. The detailed analysis shows that maxima of barriers Bafor individual atoms in the same shell corresponds to diﬀerent angles of rotation and so the dependence of total energy on angle of rot ation is essentially smoothed (see Fig. 3). Note, that the using of spherical shape of C 240leads to signiﬁcant underestimation of barriers for rotati on. The radii of shells of nanoparticle C60@C240are very close to radii of shells of (5,5)@(10,10) two-shell carbon nanotube. It is interest that barriers for relative rotatio n of shells per one atom calculated here for all considered nanoparticles are order of magnitude less than appropriate barrier in (5,5)@(10,10) two-shell carbon nanotube calcu- lated by Kwon and Tomanek [20]. B. Molecular dynamics simulation We have investigated by molecular dynamics technique the an gular velocity autocorrelation function of shells, the spectrum of shell librations, the frequency of shell reorie ntations, distributions of Eiler angles of relative orient ations of shells and heat capacity of nanoparticle. The dependence of total energy on temperature is used to calc ulate the heat capacity of nanoparticle. In investigated temperature region 30 −150 K the heat capacity per one degree of freedom has no diﬀere nce from the heat capacity of harmonic oscillator system within the accuracy of calcul ation that is less than 5 %. Only three degrees of freedom accounted for relative orientation of shells. Therefore as was to be expected there is not any peculiarities in the dependency of heat capacity on temperature and the orientat ional melting of two-shell carbon nanoparticle has a crossover behavior: the free rotation of shells observed in few realizations of the system at temperature 70 K and in a half realizations of the system at temperature 140 K. The dependence of shells reorientation frequency νvs. temperature Tis shown on Fig. 4. The jump orientational intershell diﬀusion takes place where kT≪Bre,Breis an eﬀective energy barrier of reorientation. The reorien tation frequencyνfor jump orientational intershell diﬀusion we interpolate at temperatures 30 −100 K by the Arrhenius formula (thick line on Fig. 4): ν= Ω0exp/parenleftbigg −Bre kT/parenrightbigg , (2) where Ω 0is a frequency multiplier. The ﬁtting by least square techni que givesBre= 167 ±22 Kelvin degrees and Ω0= 540 ±180 ns−1. The using of shorter temperature range T= 30−75 K for interpolation is found to have only a slight inﬂuence on calculated parameters Breand Ω 0. 3The exponential increase of reorientation frequency νends at temperatures 100 −150 and this shows the beginning of free rotation of shells. It can be shown that the reorienta tion frequency νat temperature T≫Brecan be estimated by the expression ν=n 2π/radicalBigg 3kT(I1+I2) I1I2(3) wherenis an average number of reorientations over the period of rel ative shell rotation ( n≈5),I1andI2are moments of inertia of 1-st and 2-nd shells respectively. The depende nce of reorientation frequency on temperature deﬁned by Eq. (3) is shown on Fig. 4 by thin line. The prominent smooth of distributions of Eiler angles of rel ative orientations of shells (Fig. 5), the disappearance of maxima in the angular velocity autocorrelation function of shells (Fig. 6) and in the spectrum of shell librations (Fig. 7) conﬁrm that the free rotation of shells determines t he thermodynamical behaviour of the nanoparticle at temperatures greater than 140 . Thus it is found that process of orientational melting for th e two-shell nanoparticle occurs at temperatures that are at least 10 times less than the temperature of total melti ng. Analogeously orientational melting can occur also in many-shell nanoparticles and short many shell nanotubes [3 4]. As we have shown the barriers for rotation are very sensitive to the shape of shells. Therefore, the realizatio n of possible rotational melting in many-shell nanoparticl es is determined by their shape. The nanoparticles obtained in arc discharge are faceted in shape [3,35]. However, they change their shape to almost spherical one when they are subjected to very strong electron irradiation in a high-resolution electron microscope [23,33,36]. The accu rateab initio calculation of geometric parameters of large shells are necessary for performance of theoretical studie s of possible orientational melting of many-shell nanopart icles. Nevertheless, the theory does not provide accurate coordin ates. Some works predict that many-shell nanoparticles are faceted [6,9] and some that they are spherical [10,12]. T he calculations also shown that the faceted nanoparticles transform to spherical under high temperature [9,11]. Ther efore the barriers for rotation may decrease withincreasing of temperature due to change of shell structure. The carbon nanoparticles with shell structure are not the si ngle example of diﬀerent types of atom interaction inside shell and between shells. A two-shell spherical nano particle from MoS 2was produced [12]. We believe that orientational melting can also take place in nanoparticles from this material. The orientational melting in a single nanoparticle may be re vealed by IR or Raman study of the temperature depen- dence of width of spectral lines. The last must have Arrheniu s-like contribution in reorientational phase (analogousl y to the behavior in plastic crystals, see, e.g., [37] and refe rences herein). Moreover this study can give the estimation of reorientational barriers. Besides NMR line narrowing ca n be observed in reorientational phase. ACKNOWLEDGEMENTS This work was supported by grants of Russian Foundation of Ba sic Researches, Programs ”Fullerenes and Atomic Clusters” and ”Surface and Atomic Structures”. [1] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Sma lley, Nature 318 (1985) 162. [2] W. Kratschmer, L.D. Lamb, K. Fostiroupolos, D.R. Huﬀman , Nature 347 (1990) 354. [3] S. Iijima, J. Crystal Growth, 50 (1980) 675. [4] M. Yoshida, E. Osawa, Ful. 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Hare, H.W. Kroto, R. Taylor, D.R.M . Walton, Nature 353 (1991) 147. [31] A. Cheng, M.L. Klein, J. Phys. Chem. 95 (1991) 6750. [32] Q. Jiang, H. Xia, Z. Zhang, D. Tian, Chem. Phys. Lett. 191 (1991) 197. [33] D. Ugarte, Europhys. Lett. 22 (1993) 45. [34] Yu.E. Lozovik, to be published. [35] D. Ugarte, Chem. Phys. Lett. 198 (1992) 596. [36] D. Ugarte, Chem. Phys. Lett. 207 (1993) 473. [37] G.N. Zhizhin, Yu.E. Lozovik, M.A. Moskalova, A. Usmano v, Soviet Physics-Doklady 15 (1970) 36. 5Table 1. The energies Eintof interaction between shells of nanoparticle and one of the relative orientations of shells corre- sponding to the global and local minima of total nanoparticl e energy;αz,αyandαxare the angles of subsequent rotations of inner shell from initial orientation around ax es OZ, OY and OX respectively. ShapeEint,αz αzαx (meV/atom) (in radians) (in radians) (in radians) A 15.034 0.0819 0.1452 0.0540 A 15.033 0.2495 0.8128 -0.0081 A 15.032 0.6283 0.4634 0.0 B 15.124 0.6283 0.4634 0.0 B 15.101 0.0 0.0 0.0 C 15.180 0.0 0.0 0.0 C 15.098 0.6283 0.4634 0.0 D 13.819 0.0 0.0 0.0 D 13.777 0.6283 0.4634 0.0 E 15.166 0.0 0.0 0.0 E 15.061 0.6283 0.4634 0.0 6Table 2. The characteristics of second shell shape: the average devi ation of second shell from perfect sphere <∆Ri2> and the diﬀerence between average intershell distance hand the distance rmincorresponding to the minimum in pair interparticle potential l=h−rmin; the diﬀerences ∆ Elocin total energies of nanoparticle between global and local minima; the minimal and average barriers for rotation Bmin,Bav±∆Bavrespectively, where the barrier Bav is averaged over all directions of rotation axis and ∆ Bavis its dispersion; the average energies of shell deformatio n Ed1±∆Ed1andEd2±∆Ed2for ﬁrst and second shells respectively, where the energies Ed1andEd2are averaged over all relative orientations of shells and ∆ Ed1and ∆Ed2are their dispersions. Shape<∆Ri2> l ∆ElocBminBav±∆BavEd1±∆Ed1Ed2±∆Ed2 (˚A) ( ˚A) (oK) (oK) (oK) (oK) (oK) A -0.245 0.0 3.2; 5.5 19.0 20.5 ±0.8 2.09 ±0.02 34.56 ±0.12 B -0.258 0.057 76.7 82.9 122.1 ±12.1 1.62 ±0.07 29.98 ±0.50 C -0.289 0.152 287.4 349.3 363.1 ±8.8 2.17 ±0.26 18.19 ±0.42 D -0.119 0.244 144.4 160.3 177.3 ±9.6 3.75 ±0.20 34.40 ±0.55 E -0.299 0.147 368.3 441.2 459.9 ±12.9 4.58 ±0.44 13.78 ±0.38 7Captions for illustrations. Fig. 1. The fragments of two shells (shape Dof second shell) at their initial orientations. OX, OY and OZ are axes of coordinate system. One ﬁvefold axis of each shell is a ligned with the axis OZ. One of the closest to axis OZ atoms of ﬁrst and second shells (shown by black circles) lie i n plane OXZ. This ﬁxes the orientation of axes OX and OY. Fig. 2. The dependencies of binding energies for interaction betwe en shells of nanoparticle on their relative orientation. αzandαyare the angles of subsequent rotations of inner shell from in itial orientation around axes Z and Y respectively. The angle of rotation around axis X is ﬁxed eq ual to zero. a) shape D of second shell; b) shape B of second shell; Fig. 3. Interaction energies between ﬁrst shell of nanoparticle an d groups of atoms of second shell with shape D vs.angleαzof rotation of inner shell from initial orientation around a xis Z. An each group include all atoms with the same dependencies of interaction energy Eabetween this atom and the ﬁrst shell on angle of rotation. The curves corresponding to all 25 groups of atoms with diﬀerent depend enciesEafor individual atom are shown by thin lines (23 groups from 10 atoms and 2 groups from 5 atoms). The depend ence of total interaction energy between shells on angleαzis shown by bold line. All energies are measured from their mi nima. Fig. 4 The dependence of shells reorientation frequency νon temperature Tin Kelvin degrees. The interpolation by the Arrhenius formula at T <B reis shown by thick line. The estimation at T >B reis shown by thin line. Fig. 5 The distributions of Eiler angles θ,ψandφof relative orientations of shells at temperatures 21 K, 36 K and 140 K are shown by dotted lines, thin lines and thick lines respectively; a) the distribution of angle φ; b) the distribution of angle θ; c) the distribution of angle ψ. Fig. 6 The angular velocity of autocorrelation function of the ﬁrs t shell at temperatures 21 K, 36 K and 140 K are shown by dotted lines, thin lines and thick lines respect ively. Fig. 7 The spectrum of shell librations at temperatures 21 K, 36 K an d 140 K are shown by dotted lines, thin lines and thick lines respectively. 8/G3D /G3B/G3C /G32/G13/G11/G17 /G13/G11/G1B /G14/G11/G15/G13/G18/G13/G14/G13/G13/G14/G18/G13/G15/G13/G13/G15/G18/G13 /G5D/G28/G51/G48/G55/G4A/G5C/G03/G03/G0B /G52/G2E/G0C /G44/G03/G03/G03/G0B/G55/G44/G47/G4C/G44/G51/G0C/G13 /G18/G13 /G14/G13/G13 /G14/G18/G13/G13/G18/G13/G14/G13/G13/G14/G18/G13/G15/G13/G13/G49/G55/G48/G54/G58/G48/G51/G46/G5C/G03/G03/G0B/G2A/G2B/G5D/G0C /G03/G37/G48/G50/G53/G48/G55/G44/G57/G58/G55/G48/G03/G0B/G52/G2E/G0C/G13 /G14 /G15 /G16/G49/G13 /G14 /G15 /G16/G54/G13 /G14 /G15 /G16/G5C/G13 /G14/G13 /G15/G13 /G16/G13 /G17/G13 /G18/G13/G10/G14/G11/G13/G10/G13/G11/G18/G13/G11/G13/G13/G11/G18/G14/G11/G13/G44/G51/G4A/G58/G4F/G44/G55/G03/G59/G48/G4F/G52/G46/G4C/G57/G5C/G03/G44/G58/G57/G52/G46/G52/G55/G55/G48/G4F/G44/G57/G4C/G52/G51/G03/G49/G58/G51/G46/G57/G4C/G52/G51 /G57/G4C/G50/G48/G03/G0B/G53/G56/G0C/G13/G11/G15 /G13/G11/G17 /G13/G11/G19 /G13/G11/G1B/G13/G11/G13/G13/G11/G17/G13/G11/G1B/G14/G11/G15/G4F/G4C/G45/G55/G52/G51/G03/G56/G53/G48/G46/G57/G55/G58/G50 /G49/G55/G48/G54/G58/G48/G51/G46/G5C/G03/G0B/G53/G56/G10/G14/G0C
T est of Cosmic Spatial Isotrop y for P olarized ElectronsUsing a Rotatable T orsion BalanceLi/-Shing Hou /, W ei/-T ou Ni and Y u/-Ch uM /. L iCen ter for Gra vitation and CosmologyDepartmen to f P h ysics/, National Tsing Hua Univ ersit yHsinc h u/, T aiw an /3/0/0/5/5/, Republic of ChinaAbstractT o test the cosmic spatial isotrop y /,w e use a rotatable torsion balance carrying a transv ersely spin/-p olarized ferrimagnetic Dy/6 F e/2/3 mass/. With a rotation p erio d of one hour/, the p erio d of anisotrop y signalis reduced from one sidereal da yb y ab out /2/4 times/, and hence the /1//f noise is greatly reduced/. Ourpresen t exp erimen tal results constrain the cosmic anisotrop y Hamiltanian H /= C/1 //1 /+ C/2 //2 /+ C/3 //3 /( //3is in the axis of earth rotation/) to /( C /2/1 /+ C /2/2 /) /1 /= /2/=/( /1 /: /8 / /5 /: /3/) / /1/0 /; /2/1eV and j C/3 j /=/( /1 /: /2 / /3 /: /5/) / /1/0 /; /1/9eV/. This impro v es the previous limits on /( C/1 /;;C/2 /)b y /1/2/0 times and C/3 b y a factor of /8/0/0/.PCAS n um b er/(s/) /: /0/4/./8/0/./-y /, /1/1/./3/0/.Cp/, /9/8/./8/0/./-kEinstein equiv alence principle /(EEP/) is the cor/-nerstone of metric theories of gra vit y and go v ernsthe microscopic and macroscopic structures in exter/-nal gra vitational /elds/. In metric theories /(includinggeneral relativit y/)/, the EEP guaran tees lo cal Loren tzin v ariance /(LLI/)/. Ho w ev er/, in the spirit of Mac h/, theinertial and related prop erties should b e determinedb y the distribution of matter in the cosmos/. T ot e s tthis/, Hughes/-Drev er t yp e exp erimen ts ha v e b een p er/-formed o v er last /4/0 y ears with increasing precision onthe anomalous atomic energy lev el splittings of Li /[/1/-/3/]/, Be /[/4/-/6/] and Hg /[/7/-/9/]/. With the adv en t of the con/-cept of sp on taneous brok en symmetry of v acuum andthe disco v ery of the quadrup ole anisotrop y in cosmicmicro w a v eb a c kground radiation /[/1/0/, /1/1/]/, the test ofcosmic isotrop y at the fundamen tal la w lev el and theenhancemen t of precision of the Hughes/-Drev er t yp eexp erimen ts b ecome ev en more signi/can t/.Ref/. /[/1/-/9/] are mainly Hughes/-Drev er t yp e exp eri/-men ts on n uclei/. Phillips w ork ed on a Hughes/-Drev ert yp e exp erimen t on electron since /1/9/6/5 /[/1/2/]/. He useda cry ogenic torsion p endulum carrying a transv erselyp olarized magnet with sup erconducting shields/. In/1/9/8/7 /[/1/3/]/, he set a stringen t upp er limit of /8 /: /5 / /1/0 /; /1/8eV for the energy splitting of electron spin/-states/. Inour lab oratory w eh a v e used a ro om/-temp erature tor/-sion balance with a magnetically/-comp ensated DyF e/3p olarized/-mass to impro v e the limit/. Our cum ulatedresults set a limit of /2 /: /9/6 / /1/0 /; /1/8eV /[/1/4/-/1/6/]/. Berglundet al/. /[/9/] ha v e used the relativ e frequency of Hg andCs magnetometers to monitor the p oten tial energylev el v ariations due to spatial anisotrop y and giv esan upp er limit of /1 /: /7 / /1/0 /; /1/8eV for electron/. F orall of these exp erimen ts/, the signals detected ha v ep erio d of one sidereal da y /(/2/3 hr /5/6 min /4 sec/)/. T a/-ble /1 lists the corresp onding limits giv en b yv ariousexp erimen ts/.F or the analysis of cosmic anisotrop y for elec/-trons/, w e use the follo wing Hamiltonian/:H /= C/1 //1 /+ C/2 //2 /+ C/3 //3 /(/1/)in the celestial frame of reference/. This includes thefollo wing t w o cases/: /(i/) Hcosmic /= g/ / n with C/1 /=gn/1 /, C/2 /= gn/2 /,C/3 /= gn/3 as considered in /[/1/4/-/1/6/]/;;here C/'s are constan ts/, and /(ii/) Hcosmic /= g/ / v T able /1/: Hughes/-Drev er t yp es exp erimen ts usingelectron spins/. /E/? /=/2 /( C /2/1 /+ C /2/2 /) /1 /= /2and /Ek /=/2 jC/3 j are the energy lev el splittings parallel and trans/-v erse to the earth rotation axis resp ectiv ely /.O f a l lthe previous exp erimen ts/, only /[/4/] giv es constrain tson /Ek /. Reference /E/? /Ek /(/1/0 /; /1/8eV/) /(/1/0 /; /1/8eV/) Phillips/(/1/9/8/7/) /[/1/3/] / /8 /: /5 N/.A/. Wineland et al/. /(/1/9/9/1/) /[/4/] / /5/5/0 / /7/8/0 Chen et al/. /(/1/9/9/2/) /[/1/4/] / /7 /: /3 N/.A/. W ang et al/. /(/1/9/9/2/)/[/1/5/] / /3 /: /8/7 N/.A/. Chang et al/. /(/1/9/9/5/) /[/1/6/] / /2 /: /9/6 N/.A/. Berglund et al/. /(/1/9/9/5/) /[/9/] / /1 /: /7 N/.A/. This w ork / /0 /: /0/5/7 / /0 /: /9/7 with C/1 /= gv/1 /, C/2 /= gv/2 /, C/3 /= gv/3 as consideredin /[/1/2/,/1/3/,/1/7/,/1 /8/]/;; in this case/, since v is largely /,t h ev elo cit y of our solar system through the cosmic pre/-ferred frame/, to a /rst appro ximation/, C/'s are alsoconstan ts/. F or con v enience/, w e use the celestial equa/-torial co ordinate system with the earth rotation axisas z/-axis and the direction of the spring equino xa sthe p ositiv e x/-direction /(Fig/. /1/)/. The righ t ascension/ of our lab oratory is measured east w ard along thecelestial equator from the spring equino x /(//) to its in/-tersection with lab oratory/'s hour circle/. Declination/ is the geographical latitude/. F or our lab oratory /, /is /2/4 o/4/7 /0/4/3 /0/0and the longitude is /1/2/0 o/5/9 /0/5/8 /0/0/.F or asusp ended electron p olarized/-b o dy with its net spinaxis p oin ting in a horizon tal direction rotated coun/-terclo c kwise from east direction b y / /, the torque from/(/1/) is/~ / /= n /~C / /</~ //> /;; /(/2/)where n is the n um b er of p olarized electrons and /</~ //> is the a v erage p olarization v ector/. When w esusp end this p olarized/-b o dy with a /bre/, the torqueon the /bre is/ve rt /= /1 /2 n j /</~ //> j /[ C/1 /(/1 /+ sin / /) cos /( / /+ / /)/; C/1 /(/1 /; sin / /)c o s /( / /; / /)/+ C/2 /(/1 /+ sin / /) sin /( / /+ / /)/; C/2 /(/1 /; sin / /) sin /( / /; / /) /; /2 C/3 cos / cos / /] /: /(/3/)PΩ ω EYZ(C )3 (C )1 (C )2ZZ X// αδθYLPE P Y L L X EXFigure /1/: The celestial equatorial frame /(XE /,YE /,ZE /)/, the lab oratory frame /(XL /,YL /,ZL /) and the ro/-tatable table frame /(XP /,YP /,ZP /)/.When the torsion p endulum is rotating with angularfrequency /! /,/ /= /!t /+ //0 /;;/ /=/ t /+ //0 /;; /(/4/)with / the earth/-rotation angular frequency /, //0 theinitial / angle and //0 the initial righ t ascension/.The equilibrium angular p osition of the /bre is//= K with K the torque constan t of the loaded /bre/.In our exp erimen t/, w e measure this angle p ositionc hange to giv e constrain ts on C/1 /, C/2 and C/3 /. F orC/1 and C/2 /,w e use the earth rotation to p erform /1/2/-sidereal/-hr o/set substraction/, and hence the accu/-racy in determining them is higher/. W e notice thatthe signals of C/1 /, C/2 and C/3 are at di/eren t frequen/-cies /\| / /+ /! /,/ /; /! and /! /.F or C/3 /, the constrain tcomes only from the frequency of the rotating ta/-ble/, and could not b e observ ed from a non/-rotatingexp erimen t/. This is the new information from therotating torsion/-balance exp erimen t/, although its ac/-curacy will not b e so precise as those for C/1 and C/2 /.The measuremen ts c heme will b e presen t e di nt h emeasuremen t pro cedure follo wing our description ofv arious parts of the exp erimen tal setup /(Fig/. /2/)/. The p olarize db o dy /\|T o obtain large net spins forincreasing the p ossibilit y of detecting an anisotrop ysignal while still a v oiding magnetic in teraction/, spin/-p olarized b o dy of Dy/6 F e/2/3 w as used in the previousexp erimen ts /[/1/9/]/. Dy/-F e comp ounds are ferrimag/-netic at ro om temp erature/. The e/ectiv e orderingof the iron lattice and dysprosium lattice ha v e di/er/-en t temp erature dep endence b ecause the strengthsof exc hange in teractions are di/eren t/. Near the com/-p ensation temp erature/, the magnetic momen ts of t w olattices comp ensate eac h other mostly /.D y /+/+/+hasL /= /5 and S /=/5 /= /2/. Half of the Dy magnetizationcomes from orbital angular momen tum/, the other halffrom spin/. Most of the iron magnetization comesfrom spin/. So there is a net spin /(and net total an/-gular momen tum/) remaining/.T om a k e samples/, Dy/6 F e/2/3 w as syn thesized b ymelting stoic hiometric quan tities of metallic iron andmetallic dysprosium/. The Dy/6 F e/2/3 ingots w ere crushed/,pressed in to a cylindrical alumin um cup/, and magne/-tized along a tran v erse direction/. The magnetic mo/-men t of a small sample w as measured as a functionof temp erature from /3/0/0 K to /4/./2 K using an RFSQUID measuremen t system/. W e compare measure/-men t data with mo del calculations to conclude thatthere is at least /0/./4 net p olarized electron p er atomof Dy/6 F e/2/3 /. The net ferrimagnetic magnetization w asshielded b yt w oh a l v es of pure iron casing/, a thin alu/-min um spacer and a set of t w o /tting / /-metal cups/.The a v erage magnetization after shielding is /2/./5/7 mG/(/4 / M/)/.Our Dy/6 F e/2/3 sample has diameter /1/6/./0 mm/, heigh t/1/9/./6 mm/, mass /2/8/./9/7 g/, and n um be r /( n j /</~ //> j /)o fnet p olarized electrons /8 /: /9/5 / /1/0 /2/2/. The magneticallyshielded p olarized/-b o dy has a dimension of /2/2 mm / / /2/6 mm heigh t with a mass /6/8/./3 g/.T orsion b alanc e and r otatable table /\| As in Fig/. /2/,the torsion balance is h ung from the magnetic damp erusing a /2/5 / m General Electric tungsten /b er/. Themagnetic damp er is h ung from top of the c ham be rhousing the torsion balance using a /7/5 / m tungsten/b er/. The p erio d of the torsion balance is measuredto b e /1/4/4/./3/4 sec/. The momen t of inertia relativ et othe cen tral v ertical axis of the p endulum set is /3/2/./3/2g / cm /2/. Hence/, the torsion constan t K is /6 /: /1/2 / /1/0 /; /2dyne / cm//rad/.TableDamping Fiber Rotatable LaserVacuum Chamber MotorMagnet Coils Computer Coil-damping FeedbackMagnetic Feedback µ− ChambermetalHelmholtz Polarized Mass ScrewsTiltFigure /2/: Sc hematic exp erimen tal set/-up/.F or the angle detection/, w e set up an optical lev elmade of a /6/3/3 nm laser dio de/, a b eamsplitter/, a mir/-ror on the torsion balance/, a /3/0 cm fo cal/-length cylin/-drical lens and a /3/4/5/6/-pixel F airc hild linear CCD with/7 / mp i t c h /(/1 pixel/)/. /1 pixel di/erence in the CCDdetection corresp onds to /1/1/./7 / rad de/ ection of thetorsion balance and an amplitude of this size amoun tsto /4 /: /9/8 / /1/0 /; /1/8eV in the C /'s/. The torsional motion ofthe p endulum is damp ed b y the D /(Deriv ativ e/) feed/-bac k of the coil damping system without c hangingthe equilibrium p osition/.The torsion p endulum with its housing is moun tedon a rotatable table /xed to a Hub er Mo del /4/4/0 Go/-niometer/. The angle p ositioning repro ducibilit yi sb etter than /2 arcsec and the absolute angle deviationis less than / /1/0 arcsec/. The torsion balance togetherwith the rotatable table and goniometer is moun tedwith /4 adjustable screws on the optical table inside the v acuum c ham b er/. The four/-phase stepping motorfor rotating the table is outside the v acuum c ham be rand connected to the table b y an acrylic ro d/. A /0/./0/2/ rad resoultion biaxial tiltmeter is attac hed to thetable to moin tor the tilt/.Magnetic /eld c omp ensation and temp er atur ec on/- tr ol /\|W e use three pairs of square Helmholtz coils/(/1/./2 m for eac h side/) to comp ensate the earth mag/-netic /eld/. A / /-metal c ham b er with atten uation fac/-tor larger than /3/0 is placed inside the p endulum hous/-ing to magnetically shield the p olarized/-b o dy /. Themagnetic /eld is measured /3/0 cm b elo w the exp er/-imen tc ham be r b y a /3/-axis magnetometer to mak esure the region to b e o ccupied b y / /-metal c ham be rand the p olarized b o dy to b e less than /2 mG b eforew e set up the torsion balance/. The /3/-axis magne/-tometer signals are fed bac kt o c o n trol the curren tsof /3 pairs of Helmholtz coils with a precision b etterthan /0/./1 mG rms/.One thermometer is attac hed to the middle partof the alumin um tub e housing the /b er/. The otherfour are placed outside the w all of the v acuum c ham/-b er/. The c ham b er temp erature is con trolled throughan air conditioner and four radian t heater outside thec ham be r w h i c h are feedbac k con trolled b y these /v ethermometers through a p ersonal computer/. T em/-p erature v ariation during /2/-da y data run is b elo w/2 /0mK p eak to p eak for the thermometer on the tub e/.Me asur ement pr o c e dur e /\|E a c h complete datarun consists of /4 con tiguous p erio ds/. Eac h p erio dlasts for /1/2 sidereal hours /(/1/1 hr /5/8 min /2 sec/)/. In the/rst p erio d w e rotate the torsion balance clo c kwise orcoun terclcokwise with /1 hr p erio d for /1/1 turns/, andthen stop the torsion balance for /5/8 min /2 sec to pre/-pare for the second p erio d/. In the second p erio d/, w erep eat with opp osite rotation/. In the third /(fourth/)pe r i od /, w e rep eat with the same sense of rotationas in the second /(/rst/) p erio d/. The torsion balanceangular p osition F /( t /) is measured/. F /( t /) is basicallyequal to the equilibrium p osition //= K plus devia/-tion and noise/. The signal part of F /( t /)/, //= K /, giv esv alues of C/1 /, C/2 /, C/3 /, the deviation and noise giv esuncertain t y /. Let T b e /1/2 sidereal hours/. Adding t w odata sets with same rotating direction for F usingeqs/. /(/3/) and /(/4/)/, w e can eliminate C/1 and C/2 andestimate C/3 /;; substracting/, w e can eliminate C/3 andestimate C/1 and C/2 /.F or /0 / t / /1/2 siderial hours/,in the case the rotation is coun terclo c kwise in the/rst p erio d/, de/ne F/+ /( t /)/= F /( t /) /; F /( t /+/3 T /)a n dF/; /( t /)/= F /( t /+/2 T /) /; F /( t /+ T /)/;; in the other case/,de/ne F/+ /( t /)/= F /( t /+/2 T /) /; F /( t /+ T /) and F/; /( t /)/=F /( t /) /; F /( t /+/3 T /)/. W e form the follo wing com bina/-tions to separate signals with di/eren t frequencies/:f/1 /( t /) /= f /(/1 /+ sin / /) F/+ /( t /)/+/( /1 /; sin / /) F/; /( t /) g /= /(/4 sin / /)/= n j /</~ //> jf C/1 cos /[/(/ /+ j /! j /) t /+ //0 /+ //0 /]/+ C/2 sin /[/(/ /+ j /! j /) t /+ //0 /+ //0 /] g /;; /(/5/)f/2 /( t /) /= f /(/1 /; sin / /) F/+ /( t /)/+/( /1/+s i n / /) F/; /( t /) g /= /(/4 sin / /)/= n j /</~ //> jf C/1 cos /[/(/ /;j /! j /) t /+ //0 /; //0 /]/+ C/2 sin /[/(/ /;j /! j /) t /+ //0 /; //0 /] g /;; /(/6/)f/3 /( t /) /= f F /( t /)/+ F /( t /+/3 T /) g /= /(/2 cos / /)/= /; n j /</~ //> j C/3 cos/( j /! j t /+ //0 /) /;; /(/7/)f/4 /( t /) /= f F /( t /+ T /)/+ F /( t /+/2 T /) g /= /(/2 cos / /)/= /; n j /</~ //> j C/3 cos/( /;j /! j t /+ //0 /) /: /(/8/)A nalysis and r esults /\|F rom the FFT analysis ofthe linear/-drift/-reduced CCD residuals of f/1 /( t /)/, f/2 /( t /)/,f/3 /( t /) and f/4 /( t /)/, w e obtain t w o estimates of the C/1 /, C/2and C/3 /. Fig/. /3 sho ws a t ypical data set for f/1 /( t /) andits F ourier sp ectra/. In this case/, w e strat rotatingthe torsion p endulum set at /2/2/:/5/8/:/0/7/, F ebruary /1/6/,/1/9/9/9 coun terclo c kwise for spin initially in the w estdirection /( //0 /= /1/8/0/)/. Because of starting transien ts/,w e discard the /rst hour data and use the in terv alt /= /1 hr to t /=/1 /0 /: /5/9/9 hr /(/1/0 cycles for angular fre/-quency / /+ /! /) for F ourier analysis/. A t t /= /1 hr/, theinitial righ t ascension //0 is /2/7/9 /: /6 o/. As w ec a n s e ein Fig/. /3/(b/)/, the /9th and /1/0th harmonics are higherthan the neigh b oring harmonics/. This is due to thecon tribution of uncancelled residue of one/-hour rota/-tion p erio d /(frequency /= /0 /: /2/7/7/8 mHz/) and is clear inFig/. /3/(c/) when w e use ten/-hour data for F ourier anal/-ysis/. In Fig/. /3/(d/)/, w e substract this one/-hour p erio dresidue from Fig/. /3/(b/)/. The cos/[/(/ /+ /! /) t /+ //0 /+ //0 /]a m /-plitude is no w /; /0 /: /0/0/1/8 pixel and sin/[/(/ /+ /! /) t /+ //0 /+ //0 /]amplitude /; /0 /: /0/0/2/8 pixel/, corresp onding to n j /</~ //> jC/1 /=/0 /: /0/0/3/1 pixel and n j /</~ //> j C/2 /= /; /0 /: /0/0/1/3pixel/. An estimate of uncertain t y is obtained b ya v/-eraging the t w on e i g h b oring FFT amplitudes withthis amplitude/;; this giv es an uncertain t y of /0/./0/0/3/1pixel/. Con v erting to the estimate of C/1 and C/2 /,w eha v e/( C /2/1 /+ C /2/2 /) /1 /= /2/=/( /1 /: /6/5 / /1 /: /5/5/) / /1/0 /; /2/0eV/. Theaccum ulation of /1/6 da ys of data giv es /1/6 sets of thesen um b e r/( /8 s e t sf o r f/1 /( t /)a n d/8s e t sf o r f/2 /( t /)/)/. Thew eigh ted a v erage for /( C /2/1 /+ C /2/2 /) /1 /= /2is /(/1 /: /8 / /5 /: /3/) / /1/0 /; /2/1eV/.F or the determination of C/1 and C/2 /, the e/ects with /1/-hr rotation p erio d are largely cancelled out inF/+ and F/; /. The uncancelled residues in f/1 and f/2can b e substracted as explained in the last paragraph/.Ho w ev er/, for determination of C/3 /, the e/ects withrotation p erio d need to b e mo delled in order to b eable separate from the C/3 signals/. The tilt e/ect ismo delled/. Other e/ects are put in to systematic error/.The later data ha v em uc h less tilt e/ect/. With these /8da ys of data/, C/3 is determined to b e /(/1 /: /2 / /3 /: /5/) / /1/0 /; /1/9eV/.A b etter adjusted system for the tilt of the rotat/-able torsion balance is exp ected to reduce the noise/.One order/-of/-magnitude impro v emen t will reac ht h esensitivt y to prob e the macroscopic evidences of spin/-rotation coupling Hef f /= /; /~/ / /~S on earth /[/2/0/]/. Thisnon/-inertial e/ect is calculated to b e equiv alen tt o aC/3 of /2 /: /4 / /1/0 /; /2/0eV/.Our constrain t on the Loren tz and CPT violationparameters /~b e/? and /~b eZ of Bluhm and Kostelec ky is/~b e/? /[/= /( C /2/1 /+ C /2/2 /) /1 /= /2/] / /3 / /1/0 /; /2/9GeV and /~b eZ /(/= C/3 /) //5 / /1/0 /; /2/8GeV /[/2/1/]/.W e thank Sheau/-shi P an/, W an/-Sun Tse and Hsien/-Chi Y eh for their help and encouragemen t/. W e alsothank the National Science Council of the Republicof China for supp orting this w ork/.References/[/1/] V/. W/. Hughes et al/. /,P h ys/. Rev/. Lett/. /4 /, /3/4/2/(/1/9/6/0/)/;; V/. Beltran/-Lop ez et al/. /, Bull/. Am/. Ph ys/.So c/. /6 /, /4/2/4 /(/1/9/6/1/)/./[/2/] R/. W/. P /. Drev er/, Phil/. Mag/. /6 /, /6/8/3 /(/1/9/6/2/)/./[/3/] J/. E/. Ellena et al/. /, IEEE T rans/. on Meas/./, IM/-/3/6 /, /1/7/5 /(/1/9/8/7/)/./[/4/] D/. J/. Wineland et al/. /,P h ys/. Rev/. A /5 /, /8/2/1/(/1/9/7/2/)/./[/5/] J/. D/. Prestage et al/. /,P h ys/. Rev/. Lett/, /5/4 /, /2/3/8/7/(/1/9/8/5/)/./[/6/] D/. J/. Wineland et al/. /,P h ys/. Rev/. Lett/. /2/3 /, /1/7/3/5/(/1/9/9/1/)/./[/7/] S/. K/. Lamoreaux et al/. /,P h ys/. Rev/. A /3/9 /, /1/0/8/2/(/1/9/9/0/)/.Figure /3/: /(a/) A t ypical data set for f/1 /( t /)/. /(b/) F ouriersp ectrum F/1 /( / /) for f/1 /( t /) from t /=/1 h r t o t /=/1 /0 /: /5/9/9hr/. The /rst hour data is abandoned b ecause ofstarting transien ts/. The time in terv al for F ouriertransform is exactly /1/0 cycles for angular frequency/ /+ j /! j /. /(c/) F ourier sp ectrum of /1/0 hr in terv al/. TheF ourier comp onen t with one/-hour p erio d is conspicu/-ous/. When this is substracted/, the F ourier sp ectrumin /(b/) is corrected to /(d/)/. /[ /8 /] B /.J /.V enema et al/. /,P h ys/. Rev/. Lett/. /6/8 /, /1/3/5/(/1/9/9/2/)/./[/9/] C/. J/. Berglund et al/. /,P h ys/. Rev/. Lett/. /7/5 /, /1/8/7/9/(/1/9/9/5/)/./[/1/0/] G/. F/. Smo ot et al/. /, Astroph ys/. J/. /3/9/6 /, L/1 /(/1/9/9/2/)/./[/1/1/] C/. M/. Guti / errez et al/. /, Astroph ys/. J/. /5/2/9 /,/4 /7/(/2/0/0/0/)/;; and references therein/./[/1/2/] P /. R/. Phillips/, Ph ys/. Rev/. /1/3/9B /, /4/9/1 /(/1/9/6/5/)/;;P /. R/. Phillips and D/. W o olum/, Nuo v o Cimen to/6/4B /, /2/8 /(/1/9/6/9/)/./[/1/3/] P /. R/. Phillips/, Ph ys/. Rev/. Lett/. /5/9 /, /1/7/8/4 /(/1/9/8/7/)/./[/1/4/] S/./-C/. Chen et al/. /, Pr o c e e dings of the Sixth Mar/-c el Gr ossmann Me eting on Gener al R elativity /,eds/. H/. Sato and T/. Nak am ura /(W orld Scien ti/c/,/1/9/9/2/) p/. /1/6/2/5/./[/1/5/] S/./-L/. W ang et al/. /,M o d /. P h ys/. Lett/. A /8 /, /3/7/1/5/(/1/9/9/3/)/./[/1/6/] F/./-L/. Chang et al/. /, /"Impro v ed exp erimen tal limiton the cosmological spatial anisotrop y for p olar/-ized electrons/"/, pp/. /2/1/-/2/9 in International Work/-shop on Gr avitation and Cosmolo gy /(Tsing HuaU/./, Hsinc h u/, /1/9/9/5/)/./[/1/7/] L/. Sto dolsky /,P h ys/. Rev/. Lett/. /3/4 /, /1/1/0 /(/1/9/7/5/)/./[/1/8/] H/. Nielson and I/. Picek/, Nucl/. Ph ys/. /2/1/1B /, /2/6/9/(/1/9/8/3/)/./[/1/9/] W/./-T/. Ni/, in Pr o c e e dings of the F ourth Mar c elGr ossmann Me eting on Gener al R elativity /, ed/.b y R/. Ru/ni /(Elsevier Science Publishers B/. V/./,/1/9/8/6/) p/. /1/3/3/5/;; R/. C/. Ritter et al/. /,P h ys/. Rev/.D /4/2 /, /9/7/7 /(/1/9/9/0/)/;; T/. C/. P /.C h ui and W/./-T/. Ni/,Ph ys/. Rev/. Lett/. /7/1 /, /3/2/4/7 /(/1/9/9/3/)/./[/2/0/] W/./-T/. Ni/, Bull/. Am/. Ph us/. So c/. /2/9 /, /7/5/1 /(/1/9/8/4/)/;;B/. Mashhon/, Ph ys/. Rev/. Lett/. /6/1 /, /2/6/3/9 /(/1/9/8/8/)/;;F/. W/. Hehl and W/./-T/. Ni/, Ph ys/. Rev/. D /4/2 /, /2/0/4/0/(/1/9/9/0/)/./[/2/1/] R/. Bluhm and V/. A/. Kostelec ky /,P h ys/. Rev/. Lett/./8/4 /, /1/3/8/1 /(/2/0/0/0/)/;; R/. Bluhm et al/. /,P h ys/. Rev/.Lett/. /7/9 /, /1/4/3/2 /(/1/9/9/7/)/;; and references therein/.
Mo dern Ph ysics Letters A/,fcW orld Scien ti/c Publishing Compan yR OT A T ABLE/-TORSION/-BALANCE EQUIV ALENCE PRINCIPLEEXPERIMENT F OR THE SPIN/-POLARIZED HoF e/3LI/-SHING HOU and WEI/-TOU NICenter for Gr avitation and Cosmolo gy/, Dep artment of Physics/, National Tsing Hua University/,Hsinchu/, T aiwan /3/0/0/5/5/, R epublic of ChinaReceiv ed /(receiv ed date/)Revised /(revised date/)W e use a rotatable torsion balance to p erform an equiv alence principle test on a mag/-netically shielded spin/-p olarize d bod y o f H o F e/3 /. With a rotation p erio d of one hour/, thep erio d of p ossible signal is reduced from one solar da yb y /2/4 times/, and hence the /1//f noiseis greatly reduced/. Our presen t exp erimen tal results giv es a limit /(/0 /: /2/5 / /1 /: /2/6/) / /1/0 /; /9onthe E/ otv/ os parameter / of equiv alence of the p olarized b o dy compared with unp olarize daluminium/-brass cylinders in the solar gra vitiona l /eld/, and a limit /(/0 /: /3/4 / /0 /: /5/2/) / /1/0 /; /9in the earth gra vitional /elds/. This impro v es the previous limit on p olarized b o dies b ya factor of /4/5 for solar /eld and b y a factor of /1/1 for earth /eld/./1/. In tro ductionThe Einstein Equiv alence Principle /(EEP/) is the cornerstone of the gra vitationalcoupling of matter and non/-gra vitational /elds in general relativit y and metric the/-ories of gra vit y /. P ossible deviation from equiv alence w ould giv e clue to the mi/-croscopic origin of gra vit y or some new fundamen tal forces/. Mass and spin /(orhelicit y in the case of zero mass/) are the t w o indep enden ti n v arian ts c haracteriz/-ing irreducible represen tations of the P oincar / e group/. Both electro w eak and strongin teractions are strongly p olarization/-dep enden t/. A general parametric mo del/, / /- gframew ork for electromagnetically coupled particles/, has b een analyzed to sho w thatWEP I /(Galileo/'s w eak equiv alence principle/) do es not imply EEP /, but WEP I plusthe univ ersalit y of free fall rotation /(WEP I I/) do es imply EEP in this framew ork/. /1The nonmetric theory /, obtained in this in v estigation/, whic h serv es as a coun terex/-ample to Sc hi/ /'s conjecture /2/(WEP I implies EEP /./) giv es gra vit y/-induced rotationof linearly p olarized ligh t/, and astroph ysical observ ations on long/-range electro/-magnetic w a v e propagations are suggested for tests/. /3The nonmetric part of thistheory is em b o died in the axion in teraction in the string theories/. /4Using p olar/-ization observ ations of radio galaxies/, signi/can t limits on the strength of cou/-pling are obtained/. /5F uture observ ations ma yg i v e a decisiv e test of this nonmetriccosmological electromagnetic/-w a v e propagation with relev ance to the Cop ernicanPrinciple/. /6Th us/, analysis in the deep relationships among equiv alence principles/1/2 L/./-S/. Hou and W/./-T/. Nileads to testable cosmological implications/. And this in v olv es p olarization/. F ormatter/, exp erimen ts on the macroscopic spin/-p olarized b o dies are sensitiv et o o l s t oprob e the spin/-dep enden t e/ects in gra vitation/. /7 /;; /8These t w o reasons motiv ated usto do equiv alence principle exp erimen t for p olarized b o dies/.In the new general relativit yo f H a y ashi and Shirafuji/, /9the coupling with anan tisymmetric /eld leads to a univ ersal spin/-spin in teraction/. F rom gauging a sub/-group of the Loren tz group/, Naik and Pradhan/, /1/0prop osed a similair in teraction/.In connecting with P /(parit y/) and T /(time rev ersal/) nonin v ariance/, Leitner andOkub o/, /1/1and Hari Dass /1/2suggested the follo wing t yp e of spin/-gra vit yi n teraction/,Hint /= f /( r /)/^ r / / /(/1/)where /^ r is the unit v ector from the massiv e b o dy to the particle with spin / h / /.I nan e/ort to solv e the strong PT problem/, axion theories with similiar monop ole/-dip ole in tereaction w ere prop osed/. /1/3 /;; /1/4Axion and other pseudoscalar Goldstoneb osons are p ossible candidates for dark matter/. T o searc h for this dark matter/, itis imp ortan t to determine the form of in teraction in the lab oratory /.T h i s c a n b eexplored exp erimen tally b y gra vitation/-t yp e exp erimen t on macroscopic p olarizedb o dies using E/ otv/ os/-t yp e exp erimen ts/, /7 /;; /8or SQUID measuremen ts on p olarizableb o dies with suitable sources/. /1/5 /;; /1/6Recen tly /,w eh a v e set up a rotatable torsion balance to test the cosmic spatialisotrop y for p olarized electrons using spin/-p olarized ferrimagnetic Dy/6 F e/2/3 mass/.Our curren t results impro v e the previous limits b y more than t w o orders of magnitude/. /1/7Using this setup and c hanging the pan/-set/, w eh a v e adapted it to a test of equiv a/-lence principle for p olarized HoF e/3 /. In this pap er/, w e presen t the exp erimen tu s i n gthis rotatable torsion balance to prob e the p ossible mass/-spin /(monop ole/-dip ole/)in teraction of a HoF e/3 p olarized b o dy with b oth the sun and the earth to test theequiv alence principle/. In previous in v estigations/, w eh a v e used a /xed torsion bal/-ance susp ended from a /7/5 cm long /b er to p erform an equiv alence principle test onspin/-p olarized b o dy of Dy/6 F e/2/3 /. The equiv alence of this p olarized b o dy comparedwith unp olarized alumin/-brass cylinders is go o d to /7 / /1/0 /; /8in the solar gra vitional/eld /1/8/. Similar result is obtained later using a /1/6/1/./5 cm long /bre/. /1/9T o prob e thespin/-mass in tereaction of p olarized/-b o dies with earth/, w eh a v e used a b eam balance/(Metter HK/1/0/0/0 Single/-P an Mass Comparator/) to compare the mass of a magneti/-cally shielded spin/-p olarized b o dy of Dy/6 F e/2/3 with an unp olarized set of referencemasses/. The equiv alence of spin/-up and spin/-do wn p ositions is go o d to /1 part in/1/0 /; /8in the earth gra vitational /eld/. /2/0 /;; /2/1F or a rotatable torsion balance/, b oth the sun and the earth act as dynamicsource/, and the p erio d of signal w as reduced from one da y to ab out one hour /(thep erio d of rotating table/) to reduce the /1//f noise/. F rom the earth/'s gra vitational/eld/, the p ossible EP violation torque on the /b er is/ Eve rt /= ml / a/? sin //: /(/2/)R otatable/-T orsion/-Balanc e Equivalenc e Principle Exp eriment /:/:/: /3Here / a/? /= /E gE /? is the p ossible acceleration from the violation of equiv alenceprinciple/;; gE /? /= gE sin / /=/1 /: /6/7 cm//sec /2is the gra vitional acceleration pro jectedon the pan set plane with / /(/= /2/4 /: /8 o/) the declination latitude of our lab oratory/;;/E /=/( mI /; mG /) /=mG is the E/ otv/ os parameter for the earth gra vitational /eld/. mIis the inertial mass of the test b o dy /,mG is the gra vitational mass of the test b o dy /,and l is the length from the cen ter of pan set to test b o dy /. / /= /!t /+ //0 is the anglebe t w een the direction from the cen ter of the pan to the p olarized/-b o dy with thesouth direction/. /! is the angular v elo cit y of the rotating table/.F or the sun/, w e de/ne the direction from the sun to the earth as X/-axis directionand earth/'s rotation direction as Z/-axis direction/. The p ossible violation torque onthe /b er is/ Sve rt /=/( /1 /= /2/) ml /S gS /[/(/1 /+ sin / /) cos /( / /+ / /) /; /(/1 /; sin / /) cos /( / /; / /)/] /(/3/)Here gS /=/0 /: /5/9 cm//sec /2is the earth/'s gra vitational acceleration to w ard the sun/;;/S /=/( mI /; mG /) /=mG is the E/ otv/ os parameter for the sun/'s gra vitational /eld/./ /=/ t /+ //0 is the angle of lab oratory rotated due to earth motion from Y axis/.The equilibrium angular p osition of the /b er is //= K where K is the torsionconstan t of /b er with load/. W e measure this angle/-p osition c hange to giv e constrain ton /E and /S /.A n a ylsis of our presen t results of /v e /2/-da y runs/, giv es a limit of/S /=/( /0 /: /2/5 / /1 /: /2/6/) / /1/0 /; /9/,a n d /E /=/( /0 /: /3/4 / /0 /: /5/2/) / /1/0 /; /9/.In Sec/. /2/, w e describ e our exp erimen tal setup/. In Sec/. /3/, w e describ e our mea/-suremen ts c heme and measuremen t pro cedure/. In Sec/. /4/, w e presen t our results andanalysis/. In Sec/. /5/, w e conclude this pap er/./2/. Exp erimen tal SetupThe sc heme of our exp erimen tal setup is sho wn in Fig/. /1/. The v arious parts aredescrib ed b elo w/./2/./1/. The p olarize db o dyT om a k e a p olarized/-b o dy with a net spin but without net magnetic momen t/, w eneed b oth the orbital angular momen tum con tribution and spin con tribution ofmagnetic momen ts so that these con tributions cancel eac h other/, with a net totalspin remaining/. The e/ectiv e ordering of the iron lattice and holmium lattice ha v edi/eren t temp erature dep endence b ecause the strengths of exc hange in teractionsare di/eren t/. Near the comp ensation temp erature/, the magnetic momen ts of t w olattices comp ensate eac h other mostly /.The holmium ion has a large magnetic /eld at n ucleus and /1/6/5Ho is /1/0/0 /%abundan t with a large n uclear momen t/4 /. /1 /7 /N /.A t ro om temp erature the fractionp /=/1 /: /7 / /1/0 /; /3of an electron/-p olarize d holmium atom has n uclear p olarization/. /2/2T om a k e the sample/, the HoF e/3 ingots w ere crushed/, pressed in to a cylindricalaluminium cup and magnetized along the axial direction/. The magnetic /eld w asshielded b yt w o halv es of pure iron casing/, a thin aluminium spacer and a /tting/4 L/./-S/. Hou and W/./-T/. Ni MirrorDamping Fiber Rotatable LaserVacuum Chamber MotorMagnet Coils Computer Coil-damping FeedbackMagnetic Feedback µ− ChambermetalHelmholtz ScrewsTiltTableE.P. Test Pan Set Polarized BodyFig/. /1/. Sc hematic exp erimen tal set/-up with the pan/-set con/gurati on sho wn on the righ to f t h ediagram/./ /-metal cup/. The mass of HoF e/3 is /6/./8/5 g and the total mass including shieldingmaterials is /2/3/./8/1 g/. In HoF e/3 /, there is at least /0/./4 p olarized electrons p er atom/. Thetotal mass of p olarized electrons is /1 /: /9 / /1/0 /; /5g/, and the total mass of p olarizedn uclei is /1 /: /7 / /1/0 /; /3g/. Of our p olarized b o dy /, /0/./8 ppm of total mass consists ofp olarized electron/, and /7/1 ppm of total mass consists of p olarized n uclei/./2/./2/. The p an setThe pan set consists of an aluminium triangle plate /(side length/: /5/./7/1 cm/) withthree test b o dies ep o xied underneath its three corners/, and a mirror holder plusa mirror /(Fig/. /1/)/. One of the test b o dies is the p olarized b o dy men tioned ab o v ewith p olarization in the v ertical direction /(spin up/)/. The other t w o test b o dies arecylinders made of brass and aluminium with matc hing masses/. The total mass ofthis pan set in /9/3/./1/7 g/. The total momen t of inertial loaded on the /b er relativ et othe cen ter of the axis is /8/1/2/./2/5 g/-cm /2/. The triangle plate and the t w o unp olarizedtest b o dies are the ones used in the exp erimen t of reference /1/8/. The p olarized b o dyis used in the exp erimen t of reference /1/9/./2/./3/. T orsion b alanc eAs in Fig/. /1/, the torsion balance is h ung from the magnetic damp er using a /2/5R otatable/-T orsion/-Balanc e Equivalenc e Principle Exp eriment /:/:/: /5/ m General Electric tungsten /b er/. The magnetic damp er is h ung from top of thec ham b er housing the torsion balance using a /7/5 / m tungsten /b er/. The p erio d ofthe torsion balance is measured to b e /6/5/4/./4 sec/. The momen t of inertia relativ et othe cen tral v ertical axis of the p endulum set is /8/1/2/./2/5 g / cm /2/. Hence/, the torsionconstan t K is /7 /: /2/8 / /1/0 /; /2dyne / cm//rad/./2/./4/. Optic al lever and dete ction systemThe laser dio de ligh t with w a v elength /6/3/3 nm shines on the mirror of the torsionbalance/. After re/ ected from the torsion balance mirror/, the ligh t is de/ ected fromthe b eamsplitter and is fo cussed on a linear CCD arra y detector b y a cylindrical lenswith cylindrical fo cal length /3/0 cm/. When the torsion balance is turned b y an angle/ /, the CCD will detect a displacemen t of the ligh ts p o t w h i c h amoun ts to /2 f/ /.I nthis exp erimen t/, the ph ysical quan tit yt h a tw e measured is the displacemen to f t h esp ot giv en b y the CCD readout/. The F airc hild linear CCD w e used has /3/4/5/6 pixelswith a pitc ho f /7 / m /(/1 pixel/)/. Eac h pixel is equiv alen t to /1/1/./7 / rad de/ ection of themirror/. Since the torsion constan ti s K /=/7 /: /2/8 / /1/0 /; /2dyne / cm//rad/, /1 / rad anglec hange amoun ts to a torque c hange of /6 /: /1/0 / /1/0 /; /8dyne / cm/, and / /S /=/1 /: /3/7 / /1/0 /; /9or / /E /=/4 /: /4/8 / /1/0 /; /1/0/./2/./5/. Magnetic /eld c omp ensation and thermal shieldW e use three pairs of square Helmholtz Coils /(/1/./2 m for eac h side/) to comp ensate theearth magnetic /eld/. The magnetic /eld is measured /3/0 cm b elo w the exp erimen tc ham be r b y a /3/-axis magnetometer to mak e sure the region to b e o ccupied b y / /-metal c ham b er and the p olarized b o dy to b e less than /1 mG b efore w es e tu pthe torsion balance/. The /3/-axis magnetometer signals are fed bac kt oc o n trol thecurren ts of /3 pairs of Helmholtz coils with a precision b etter than /0/./1 mG rms/. Thesample is shielded in a / /-metal c ham b er with atten uation factor /3/0 or more/.The whole torsion balance is placed in a v acuum c ham b er to reduce the temp era/-ture / uctuation/. One thermometer is attatc hed on the middle part of the alumin umtub e housing the /b er/. The other four are placed outside the w all of the v acuumc ham b er/. The c ham b er temp erature is con trolled through an air conditioner andfour radian t heaters whic h are con trolled b y these /v e thermometers through a p er/-sonal computer/. Under the feedbac kc o n trol the temp erature v ariation is b elo w/2 /0mK p eak to p eak for the thermometer on the tub e during t w o/-da y data run/./2/./6/. R otatable tableThe torsion p endulum with its housing is moun ted on a rotatable table /xed to aHub er Mo del /4/4/0 Goniometer/. The angle p ositioning repro ducibili t y i sb e t t e rt h a n /2arcsec and the absolute angle deviation is less than / /1/0 arcsec/. The torsion balancetogether with the rotatable table and goniometer is moun ted with /4 adjustablescrews on the optical table inside the v acuum c ham b er/. The four/-phase steppingmotor for rotating the table is outside the v acuum c ham b er and connected to the/6 L/./-S/. Hou and W/./-T/. Nitable b y an acrylic ro d/. A /0/./0/2 / rad resoultion biaxial tiltmeter is attac hed to thetable to moin tor the tilt/.In our previous exp erimen t /1/8/,w e used a traditional /xed torsion balance/. F or atraditional /xed torsion balance/, it tak es one da y to complete one p erio d of rotation/.With a rotation p erio d of one hour/, the p erio d of p ossible violation signal is reduced from one da yb y /2/4 times/, and hence the /1//f noise is greatly reduced/./2/./7/. Coil/-damping systemThe coil/-damping system mainly consists of t w o p erp endicular coils/, the electronicphase b o x/, D//A card/, a function generator/, and a computer/. The computer DCoutput v oltage /( VD /) is mixed with the function generator output/. After b eingphase/-shifted b y the phase b o x/, the outputs are supplied to t w o p erp endicular coils/(/1/0/0/0 turns eac h/) to generate A C curren ts with phase di/erence //= /2o r/3 //= /2f o rpro ducing a rotating magnetic /eld/. Rotation of magnetic /eld induces an eddy curren t on the p endulum damping mass/. The in teraction of induced curren tw i t hthe coil magnetic /eld giv es an external torque on the p endulum/. The amplitudeof torque is prop ortional to the square of VD /.W ec o n trol the rotational motion ofthe torsion balance b yt h eP /.I/.D/. feedbac k equation/:/ /= Im // /= /; K/ /; P/ /; D /_/ /; I Z/ /(/4/)Here / is the torque/, K is the torsion constan t/. P /, I and D are constan ts cor/-resp onding to prop ortional/, in tegrativ e and deriv ativ e /(damping/) feedbac kc o e / /-cien ts/. Although w eh a v e three parameters/, w e only use nonzero D v alue to dampthe oscillation without c hanging the equilibrium p osition of p endulum/. The setsof coils and damping mass is /1/5 cm b elo w the p olarized b o dy and isolated b yt h e/ /-metal c ham b er shield with atten uation factor /3/0 or more/./3/. Measuremen tS c hemeEac h complete data run consists of /4 con tiguous p erio ds/. Eac h p erio d lasts for /1/2hours/. In the /rst p erio d w e rotate the torsion balance coun terclo c kwise with /1hr p erio d for /1/1 turns/, and then stop the torsion balance to prepare for the second p erio d/. In the second p erio d/, w e rep eat with opp osite rotation/. In the third /(fourth/)p erio d/, w e rep eat with the same sense of rotation as in the second /(/rst/) p erio d/.The torsion balance angular p osition F /( t /) is measured/. F /( t /) is basically equal tothe equilibrium p osition //= K plus deviation and noise/. The signal part of F /( t /)/,//= K /, giv es v alues of /E /, /S /, the deviation and noise giv es uncertain t y /. Let T be/1/2 hours/. Adding t w o data sets with same rotating direction for F using eqs/. /(/2/)and /(/3/)/, w e can eliminate / Sve rt and estimate /E /;; substracting/, w e can eliminate/ Eve rt and estimate /S /.F or /0 / t / /1/2 hours/, de/ne F/+ /( t /)/= F /( t /) /; F /( t /+/3 T /)a n dF/; /( t /)/= F /( t /+/2 T /) /; F /( t /+ T /)/. W e form the follo wing com binations to separateR otatable/-T orsion/-Balanc e Equivalenc e Principle Exp eriment /:/:/: /7signals with di/eren t frequencies/:f/1 /( t /) /= f /(/1 /+ sin / /) F/+ /( t /)/+ /( /1 /; sin / /) F/; /( t /) g /= /(/4 sin / /)/= /( ml /S gS /=K /) cos/[/(/ /+ j /! j /) t /+ //0 /+ //0 /] /;; /(/5/)f/2 /( t /) /= f /(/1 /; sin / /) F/+ /( t /)/+ /( /1 /+s i n / /) F/; /( t /) g /= /(/4 sin / /)/= /( ml /S gS /=K /) cos/[/( j /! j/; / /) t /+ //0 /; //0 /] /;; /(/6/)f/3 /( t /) /= f F /( t /)/+ F /( t /+/3 T /) g /= /(/2 cos / /)/= /( ml /E gE /=K /)s i n /( j /! j t /+ //0 /) /;; /(/7/)f/4 /( t /) /= f F /( t /+ T /)/+ F /( t /+/2 T /) g /= /(/2 cos / /)/= /; /( ml /E gE /=K /)s i n /( j /! j t /+ //0 /) /: /(/8/)When w e start or stop the rotation/, the torsion balance swings out of the CCDdetection range/. The p erio d of the torsional motion of the torsion balance is /6/5/4sec/. T om a k e the angle p osition of the torsion balance tractable /(within the CCDdetection/)/, w e /rst rotate the torsion balance with half sp eed /(i/.e/./, with /2/-hr rotationp erio d/) for /3/2/7 sec /(half torsion p erio d/) and then con tin ue with full rotational sp eed/.A t the end of /3/2/7 sec/, the torsion balance/'s angle p osition en ters in to the CCDdetection range/, but with a maxim un acceleration/;; at this time/, when the table isset to full rotational sp eed/, the relativ e acceleration of the torsional balance b ecomeszero and the CCD signals sta y near the starting p osition/. The same metho d is usedwhen w e stop the table/./4/. Analysis and ResultsF rom the FFT analysis of the linear/-drift/-reduced CCD residuals of f/1 /( t /)/, f/2 /( t /)/,f/3 /( t /) and f/4 /( t /)/, w e obtain t w o estimates of the /S and /E /. Fig/. /2 sho ws a t ypicaldata set for f/1 /( t /) and its F ourier sp ectrum/. In this case/, w e start rotating the torsionp endulum set at /0/9/:/3/0/:/0/0/, April /2/3/, /2/0/0/0 for p olarized b o dy initially in the southdirection /( //0 /= /0/)/. Because of starting transien ts/, w e discard the /rst hour dataand use the in terv al t /= /1 hr to t /=/1 /0 /: /6/0 hr /(/1/0 cycles for angular frequency / /+ /! /)for F ourier analysis/. A t t /= /1 hr/, the initial righ t ascension //0 is /6/7 /: /5 o/. In Fig/. /2/(b/)/,w e sho wt h e F ourier sp ectrum of f/1 /(t/)/. F rom the /1/0th harmonics/, w e estimate/S /.T h e c o s /( / /+ /! /) t amplitude is /; /0 /: /1/1 / rad and sin /(/ /+ /! /) t amplitude is /; /2 /: /0/9/ rad/. An estimate of uncertain t y is obtained b ya v eraging the t w o neigh b oring totalFFT amplitudes with this amplitude/;; this giv es an uncertain t yo f /2 /. /0 /6 / rad/. Theaccum ulation of /1/0 da ys of data giv es /1/0 sets of these n um b ers /( /5 sets for f/1 /( t /)a n d/5 sets for f/2 /( t /)/)/. The w eigh ted a v erage for /S is /(/0 /: /2/5 / /1 /: /2/6/) / /1/0 /; /9/(T able /1/)/.F or the determination of /S /, the e/ects with /1/-hr rotation p erio d are largelycancelled out in F/+ and F/; /.H o w ev er/, for determination of /E /, the e/ects with thep erio d of rotation need to b e mo delled in order to b e able to separate them from the /E signals/.W e mo del the tilt e/ect as follo ws /:/8 L/./-S/. Hou and W/./-T/. Ni /GC0/GC4 /GC8 /GC3/GC0/GC4 /GC3 /GC3/GC0/GC8 /GC3/GC3/GC8/GC3/GC4/GC3 /GC3/GC4/GC8 /GC3 /GE7/GDC /GE0 /GD8 /GBB/GC4 /GC4 /GFB /G05 /GBC/GC3 /GC4 /GC5/GC6 /GC7/GC8 /GC9 /GCA/GCB /GCC /GC4 /GC3 /GC4 /GC4/G0B/G44/G0C /GD9 /GE2/GE8/GE5/GDC /GD8 /GE5 /GE6 /GE3 /GD8 /GD6 /GE7 /GE5/GE8/GE0 /GC3/GC5/GC7/GC9/GCB/GC4/GC3 /GD9/GE5 /GD8 /GE4 /GE8/GD8 /GE1 /GD6 /GEC /GBB /G08/G01 /GFC /G07 /GCD /GC3/GC1 /GC5 /GCB/GCC/GC7 /G00 /GDB /G0D /GBC/GC3 /GC4 /GC3 /GC5 /GC3/GC6 /GC3 /GC7 /GC3/GC8 /GC3 /GC9 /GC3 /GCA/GC3 /GCB/GC3 /GCC/GC3 /GC4/GC3/GC3/G0B/G45/G0C µ /G55/G44 /G47µ/G55/G44 /G47Fig/. /2/. /(a/) A t ypical data set for f/1 /( t /)/. /(b/) F ourier sp ectrum for f/1 /( t /) from t /=/1 h r t o t /=/1 /0 /: /6/0/0hr/. The /rst hour data is abandone d b ecause of starting transien ts/. The time in terv al for F ouriertransform is exactly /1/0 cycles for angular frequency / /+ j /! j /.T able /1/: F ourier amplitudes of f/1 and f/2 with resp ectiv e frequencies /( /! /+/ /) a n d/( /! /; / /) of /v e runs to test the equiv alence principle/. Run n um be r cosine sine T otaland F unction amplitude amplitude amplitude Uncertain t yTime /(/2/0/0/0/) /( / rad/) /( / rad/) /( / rad/) /( / rad/) /1 f/1 /( t /) /1/./3/4 /2/./3/4 /2/./7/0 /3/./2/5April /1/7/-/1/9 f/2 /( t /) /1/./8/7 /2/./5/0 /3/./1/2 /3/./3/0/2 f/1 /( t /) /; /0 /: /3/0 /4/./1/5 /4/./1/6 /4/./9/8April /2/0/-/2/2 f/2 /( t /) /; /2 /: /2/8 /; /2 /: /9/6 /3/./7/4 /4/./4/0/3 f/1 /( t /) /; /0 /: /2/3 /2/./4/9 /2/./5/0 /2/./2/7April /2/3/-/2/5 f/2 /( t /) /; /0 /: /7/8 /; /2 /: /3/3 /2/./4/5 /2/./2/8/4 f/1 /( t /) /; /0 /: /1/1 /; /2 /: /0/9 /2/./1/1 /2/./0/6June /2/5/-/2/7 f/2 /( t /) /0/./4/1 /2/./2/2 /2/./2/6 /2/./1/3/5 f/1 /( t /) /; /1 /: /4/7 /; /2 /: /6/7 /3/./0/5 /3/./0/1June /2/8/-/3/0 f/2 /( t /) /1/./9/9 /1/./4/5 /2/./4/6 /2/./9/0 W eigh ted a v erage /0/./1/8 /0/./5/2 /0/./5/5 /0/./9/2 R otatable/-T orsion/-Balanc e Equivalenc e Principle Exp eriment /:/:/: /9T able /2/: P arameter /tting for the determination of the E/ otv/ os parameter /E /. Fitted Cf Sf Ctilt Stilt Fitting P arameterF unction /( / rad/) /( / rad/) /( / rad/) /( / rad/) a b / /E /3/9/./7/8 /4/./5/6 /1/6/5/./3 /1/3/9/./2/3/0/./4/2 /3/./5/1 /1/4/7/./9 /1/0/4/./4f/3 /1/4/./0/4 /-/2/./5/7 /9/5/./7 /3/4/./8 /0/./1/5/1 /0/./0/7/4 /0/./6/1 /0/./7/7/1/6/./1/5 /0/./8/2 /8/7 /4/3/./5 / /0/./0/0/2 / /0/./0/0/1/8 / /1/./2/2 / /1/./2/3/1/3/./8/1 /3/./7/4 /6/0/./8 /4/2/./7/3/8/./8/4 /; /3 /: /8/2 /1/6/5/./3 /; /1/3/9 /: /2/3/1/./2/4 /; /2 /: /7/8 /1/4/7/./9 /1/0/4/./4f/4 /1/5/./2/3 /3/./0/2 /9/5/./7 /; /3/4 /: /8 /0/./1/4/9 /0/./0/7/1 /0/./6/4 /0/./7/4/1/6/./2/3 /; /1 /: /0/8 /8/7 /; /4/3 /: /5 / /0/./0/0/2 / /0/./0/0/1/7 / /1/./0/4 / /1/./0/8/1/2/./6/4 /; /2 /: /8/6 /6/0/./8 /; /4/2 /: /7 Av erage of the E/ otv/ os parameter /E /=/0 /: /7/5 / /1 /: /1/6 / rad/. /CfSf //= /a b/; b a //CtiltStilt //+ ///E //(/9/)where /( Cf /;;Sf /) and /( Ctilt /;;Stilt /) are cosine and sine amplitudes of the f/3 or f/4 CCDdata and the tiltmeter resp ectiv el y /, a and b are the tilt/-e/ect parameters/, /E isthe E/ otv/ os parameter/, and / is an extra parameter for comparison and consistencyc hec k/. The /4 parameters a/, b/, / /, and /E of this simple mo del are to b e determinedfrom the least square /tting of /v e sets of data for f/3 and /v e sets of data for f/4 /.The t w o determinations are listed in T able /2/. Systematic errors due to temp erature/,magnetic /eld and gra vit y/-gradien t are small/. The t w o determinations are consisten twith eac h other/. The a v erage of the t w o determinations of /E is /E /=/( /0 /: /3/4 / /0 /: /5/2/) //1/0 /; /9/./5/. ConclusionF or our spin/-p olarized HoF e/3 of /2/3/./8/1 g/, the equiv alence with resp ect to unp olarizedaluminium/-brass masses is /S /=/( /0 /: /2/5 / /1 /: /2/6/) / /1/0 /; /9in the solar gra vitational /eldand /E /=/( /0 /: /3/4 / /0 /: /5/2/) / /1/0 /; /9in the earth gra vitational /eld/. This result indicatesthat to /(/0 /: /3/1 / /1 /: /5/8/) / /1/0 /; /3///( /0 /: /4/3 / /0 /: /6/5/) / /1/0 /; /3/, the p olarized electron falls withthe same rate as unp olarized b o dies in the solar // earth gra vitational /eld/;; and thatto /(/0 /: /3/5 / /1 /: /7/6/) / /1/0 /; /5///( /0 /: /4/8 / /0 /: /7/3/) / /1/0 /; /5the p olarized n uclei falls with the samerate as unp olarized b o dies in the solar // earth gra vitational /eld/. This impro v esour previous results b y /4/5///1/1 times for p olarized electron /1/8 /;; /2/0and b y /6/5/0///8/0 timesfor p olarized n uclei /1/8 /;; /2/0 /;; /2/2for the solar//earth gra vitational /eld/./1/0 L/./-S/. Hou and W/./-T/. NiAc kno wledgmen tsW e thank the National Science Council of the Republic of China for supp orting thisw ork under in part con tract Nos/. NSC /8/9/-/2/1/1/2/-M/-/0/0/7/-/0/4 /1/.References/1/. W/./-T/. Ni/, Bull/. A m/. Phys/. So c/. /1/9 /, /6/5/5 /(/1/9/7/4/)/;; Phys/. R ev/. 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arXiv:physics/0009014v1 [physics.gen-ph] 4 Sep 2000ALARMING OXYGEN DEPLETION CAUSED BY HYDROGEN COMBUSTION AND FUEL CELLS AND THEIR RESOLUTION BY MAGNEGASTM Ruggero Maria Santilli R&D Director, USMagnegas, Inc. 13100 Belcher Road, Largo, FL 33773, U.S.A. Tel.: +1-727-507 9520, Fax: +1-727-507 8261, E-mail: ibr@g te.net Contributed paper, International Hydrogen Energy Forum 2000, Munich, Germany, September 11-15, 2000 Abstract We recall that hydrogen combustion does resolve the environ men- tal problems of fossil fuels due to excessive emission of car cinogenic substances and carbon dioxide. However, hydrogen combusti on im- plies the permanent removal from our atmosphere of directly usable oxygen, a serious environmental problem called oxygen depl etion, since the combustion turns oxygen into water whose separation to r estore the original oxygen is prohibitive due to cost. We then show t hat a conceivable global use of hydrogen in complete replacemen t of fos- sil fuels would imply the permanent removal from our atmosph ere of 2.8875×107metric tons O 2/day. Fuel cells are brieﬂy discussed to point out similarly serious environmental problems, again , for large uses. We propose the possibility of resolving these problem s by up- grading hydrogen to the new combustible fuel called magnega stm, 0whose chemical structure is composed by the new chemical spe cies of magnecules, whose energy content and other features are b eyond the descriptive capacities of quantum chemistry. In fact, m agnegas contains up to 50% hydrogen, while having combustion exhaus t with: 1) a positive oxygen balance (releasing more oxygen in the ex haust than that used in the combustion); 2) no appreciable carcino genic or toxic substances; 3) considerably reduced carbon dioxide a s compared to fossil fuels; 4) considerably reduced nitrogen oxides; a nd 5) gen- eral reduction of pollutants in the exhaust up to 96% of curre nt EPA standards. We also discuss the possibility of further reduc ing carbon dioxide via suitable disposable sponges in the exhaust syst em, as well as the further reduction of nitrogen oxides with more eﬃcien t engine cooling and other means. The analysis therefore indicates t hat mag- negas combustion exhaust already is dramatically below EPA stan- dards, while the achievement of a completely clean exhaust i s within technological reach. Therefore, magnegas appears to be an e xcellent upgrading of hydrogen, both, for direct combustion and for u se in fuel cells. We ﬁnally indicate that one of the best applicati ons of the new technology is that of processing crude oil in the magnega s reac- tors, by yielding a fuel dramatically cleaner than gasoline , at a cost smaller than that via reﬁneries. In conclusion, crude oil, h ydrogen and fuel cells remain indeed fully admissible in this new era of envi- ronmental concern, provided that they are treated via a basi cally new technology whose quantitative study requires a new chemist ry, called hadronic chemistry [1-5]. 1As is well known, gasoline combustion requires atmospheric oxygen which is then turned into CO 2and various HydroCarbon (HC). In turn, CO 2is recycled by plants via the known reaction, H 2O + CO 2+ (hν)→O2+ (-(CH 2O)-), which restores oxygen in the atmosphere. Essentially this was the scenario at the beginning of the 20-th century. The same s cenario at the beginning of the 21-st century is dramatically diﬀerent, be cause forests have rapidly diminished while we have reached the following unre assuring daily consumption of crude oil: 74.18 million of barrel per day = (74.18 million barrels/24h) ×(55 gallons/barrel) = 4.08×109gallons/24h = 1.54×1013cc/24h (using 4 quarts/gallon and 946 cc/quart) (1) = (4.08 ×109gallons) ×(4 qrt./gallon)x(946 cc/qrt.)/day = 1.5438 ×1013cc/day = (1.5438 ×1013cc/day) ×(0.7028 grams/cc) = 1.0850 ×1013grams octane/day = (1.0850 ×1013grams)/(114.23 grams/mole) = 9.4984 ×1010moles n-octane/day, (see, e.g., http://www.eia.doe.gov/emeu/international /energy.html) where we have replaced, for simplicity, crude oil with a straight c hain of n-octanes CH3-(CH 2)6-CH3with the known density of 0.7028 g/cc at 20oC. It should be indicated that data (1) do not include the additional larg e use of natural gas and coals, which would bring the daily combustion of all f ossil fuel to the equivalent of about 120 million barrels of crude oil per day. The primary environmental problems caused by the above disp roportion- ate consumption of fossil fuel per day are the following: 1) Excessive emission of carcinogenic and other toxic subst ances in the combustion exhaust. It is well known by experts that gasolin e combustion releases in our atmosphere the largest percentage of carcin ogenic and other toxic substances as compared to any other source. The terms ” atmospheric pollution” are an euphemism for very toxic breathing. 2) Excessive release of carbon dioxide. It is evident that, u nder the very large daily combustion (1), plants cannot recycle the entir e production of CO2, thus resulting in an alarming increase of CO 2in our atmosphere, an occurrence known as green house eﬀect. In fact, by using the k nown reaction 2C8H18+ (25/2)O 2→8CO 2+ 9H 2O, we have the following alarming daily production of CO 2from fossil fuel combustion: (9.4984×1010moles C 8H18)×(8/1)/day = 7.5987 ×1011moles CO 2/day = (7.5987 ×1011moles)×(0.044 kg/mole)/day (2) = 3.3434 ×1010kg/day = (3.3434 ×1010kg/day)/(1000 kg/metric ton) = 3.3434 ×107metric tons/day. It is evident that plants cannot possibly recycle such a disp roportionate amount of daily production of CO 2. This has implied a considerable increase of CO 2in our atmosphere which can be measured by any person serious ly interested in the environment via the mere purchase of a CO 2meter, and then compare current readings of CO 2with standard values on record, e.g., the percentage of CO 2in our atmosphere at sea level in 1950 was 0.033% ± 0.01% (see, e.g., Encyclopedia Britannica of that period). Along these lines, in our laboratory in Florida we measure a thirty fold increas e of CO 2in our atmosphere over the indicated standard. We assume the reade r is aware of recent TV reports of small areas in the North Pole containing liquid water, an occurrence which has never been observed before. Increasin gly catastrophic climactic events are known to everybody. 3) Excessive removal of directly usable oxygen from our atmo sphere, an environmental problem of fossil fuel combustion, which is l esser known than the green house eﬀect, even among environmentalists, but po tentially more serious. The problem is called oxygen depletion, and refers to the diﬀerence between the oxygen needed for the combustion less that expel led in the ex- haust. By using again the reaction C 8H18+(25/2)O 2→8CO 2+ 9H 2O and data (2), it is easy to obtain the following additionally ala rming daily use of oxygen for the combustion of fossil fuel, (9.4984×1010moles octane/day) ×(12.5 moles O 2/1 mole octane) = 1.1873 ×1012moles of O 2/day = (1.1873 ×1012moles of O 2)×(0.032 kg/mole O 2) (3) = 3.7994 ×1010kg O 2/day = 3.7994 ×107metric tons/day. 3Again, this large volume of oxygen is turned by the combustio n into CO 2of which only an unknown part is recycled by plants into usable o xygen. Thus, the actual and permanent oxygen depletion caused by fossil f uel combustion in our planet is currently unknown. However, it should be ind icated that the very existence of the green house eﬀect is unquestionable ev idence of oxygen depletion, because we are dealing precisely with the quanti ty of CO 2which has not been re-converted into O 2by plants. Oxygen depletion is today measurable by any person seriousl y interested in the environment via the mere purchase of an oxygen meter, m easure the local percentage of oxygen, and then compare the result to st andards on record, e.g., the oxygen percentage in our atmosphere at sea level in 1950 was 20.946% ±0.02% (see, e.g., Encyclopedia Britannica of that period). Along these lines, in our laboratory in Florida we measure a l ocal oxygen depletion of 3%-5%. Evidently, bigger oxygen depletions ar e expected for densely populated areas, such as Manhattan, London, and Tok yo, or at high elevation. We assume the reader is aware of the recent decisi on by U.S. airlines to lower the altitude of their ﬂights despite the ev ident increase of cost. This decision has been apparently motivated by oxygen depletion, e.g., fainting spells due to insuﬃcient oxygen suﬀered by passeng ers during ﬂights at previous higher altitudes. The purpose of this note is to indicate that, whether used for direct com- bustion or in fuel cells, hydrogen combustion does not relea se carcinogenic and carbon dioxide in the exhaust, but causes an alarming oxy gen depletion which is considerably bigger than that caused by fossil fuel combustion for the same energy. This depletion is due to to the fact that gaso line combus- tion turns oxygen into CO 2part of which is recycled by plants into O 2, while hydrogen combustion turns atmospheric oxygen into H 2O. This process per- manently removes oxygen from our planet in a directly usable form due to the excessive cost of water separation to restore the origin al oxygen balance. By assuming, for simplicity, that gasoline is solely compos ed of one octane C8H18, thus ignoring other isomers, the combustion of one mole of H 2gives 68.32 Kcal, while the combustion of one mole of octane produc es 1,302.7 Kcal. Thus, we need 19.07 = 1302.7/68.32 moles of H 2to produce the same energy of one mole of octane. In turn, the combustion of 19.07 moles of H 2requires 9.535 moles of O2, while the combustion of one mole of octane requires 12.5 mol es of O 2. Therefore, on grounds of the same energy release, the combus tion of hydrogen 4requires less oxygen than gasoline (about 76% of the oxygen c onsumed by the octane). The alarming oxygen depletion occurs, again, because of the fact that the combustion of hydrogen turns oxygen into water, by therefor e permanently removing usable oxygen from our planet. When used in modest a mounts, the combustion of hydrogen constitutes no appreciable envi ronmental prob- lem. However, when used in large amounts, the combustion of h ydrogen is potentially catastrophic on environmental grounds, becau se oxygen is the foundation of life. At the limit, a global use of hydrogen as fuel in complete repl acement of fossil fuels would render our planet uninhabitable in a shor t period of time. In fact, such a vast use of hydrogen would imply the permanent removal from our atmosphere of 76% of the oxygen currently consumed t o burn fossil fuels, i.e., from Eqs. (2) and (3), we have the following perm anent oxygen depletion due to global hydrogen combustion: 76% oxygen used for fossil fuel combustion (4) = 2.8875 ×107metric tons O 2depleted/day, which would imply the termination of any life on Earth within a few months. Predictably, the above feature of hydrogen combustion has a larmed envi- ronmental groups, labor unions, and other concerned people . As an illustra- tion, calculations show that, in the event all fuels in Manha ttan were replaced by hydrogen, the local oxygen depletion would cause heart fa ilures, with ev- ident large ﬁnancial liabilities and legal implications fo r hydrogen suppliers. An inspection of fuel cells reveals essentially the same sce nario. If hydrogen is used as fuel we have the above indicated oxygen depletion. If, instead, we use fossil fuels in fuel cells, we are back to essentially the original problems caused by fossil fuel combustions. The main open issue creat ed by the above scenario is: since pure hydrogen is potentially catastroph ic on a large scale use whether as direct fuel or in fuel cells, how can hydrogen b e upgraded to a form avoiding the oxygen depletion? It is easy to see that th is question does not admit an industrially and environmentally accepta ble answer via the use of conventional gases. For instance, the addition of CO to H 2in a 50-50 mixture would leave the oxygen depletion unchanged. I n fact, each of the two reactions, H 2+ (1/2) O 2→H2and CO + (1/2) O 2→CO2, requires 1/2 mole of O 2.Therefore, the 50-50 mixture of H 2and CO would also require 51/2 mole of O 2, exactly as it is the case for the pure H 2. After studying the above problems for years, the only answer known to this author is that of upgrading hydrogen into a new combusti ble gas, called magnegastm[1] (international patents pending), which is produced as a by- product in the recycling of liquid waste (such as automotive antifreeze and oil waste, city and farm sewage, etc.) or the processing of ca rbon-rich liquids (such as crude oil, etc.). The new technology, called Plasma ArcFlowtm(in- ternational patents pending), is essentially based on ﬂowi ng liquids through a submerged electric arc with at least one carbon electrode. The arc essen- tially decomposes the liquid molecules into a plasma at 7,00 0oF composed of mostly ionized H, O and C atoms, plus solid precipitates. T he technology then controls the recombination of H, O and C into a combustib le gas with a new chemical species, tentatively called magneculestm[2], which is currently under study. A ﬁrst peculiarity of magnegastmnonexistent in other gases, is that, following numerous tests in analytic laboratories, its che mical structure can- not be identiﬁed via conventional Gas Chromatographic Mass Spectromet- ric (GC-MS) measurements, since it results to be constitute d by large clus- ters (all the way to 1,000 a.m.u. in molecular weight) which r emain com- pletely unidentiﬁed by the MS. The chemical structure of mag negas is equally unidentifyable via InfraRed Detectors (IRD), because the n ew clusters com- posing magnegastmhave no IR signature at all, thus suggesting a bond of non-valence type (because these large clusters cannot poss ibly be all sym- metric). Moreover, the IR signature of conventional molecu les such as CO and CO 2result to be mutated with the appearance of new peaks, which e v- idently indicate new internal bonds. These features establ ish that magnegas has an energy content considerably bigger than that predict ed by quantum chemistry, since it can store energy in three diﬀerent level s: magnecules, molecules, and new internal molecular bonds. As a result, th e combustion of conventional fuels can be conceived as a singlet rocket ﬁrin g, while the com- bustion of ,magnegas can be referred to the burning of a multi -stage rocket, with intriguing new features. In vies of the above occurrenc es, quantitative scientiﬁc studies of magnegas are, therefore, beyond the ca pabilities of quan- tum chemistry. A broader theory suitable for scientiﬁc stud ies of the new chemical species and the combustion of the new gas has been de veloped by R. M. Santilli and D. D. Shillady under the name of hadronic ch emistry [3, 4] (see also papers [5]). 6Scans of the same sample of magnegas at diﬀerent times shows d iﬀerent magnecules, a phenomenon called magnecule mutation. The eﬀ ect is ex- pected to be due to collisions among magnecules, resulting f ragmentations due to their large size, and their subsequent recombination s with other frag- ments. This results in macroscopic changes of the MS peaks fo r the same gas under the same GC-MS test, only conducted at diﬀerent tim es. These mutations have identiﬁed the presence in the clusters of ind ividual atoms of H, O and C, plus ordinary molecules H 2, CO, and O 2[2, 3]. The estimated conventional composition of magnegas consists of about 40% -45% hydrogen, 55%-60% carbon monoxide, the rest being composed by traces o f oxygen and carbon dioxide. Evidently, small traces of light HC are poss ible in ppm, but no heavy HC is possible in magnegas since the gas is created at 7,000oF of the electric arc, as conﬁrmed by the lack of activation of cat alytic converters during the combustion. As a working hypothesis in the absenc e of a more accurate knowledge, it is conjectured that the very intense magnetic ﬁelds in the microscopic vicinity of 1,000-3,000 DC Amps of the subme rged electric arc (which can be as high at 1014Gauss at distances of 10−8cm) cause a polarization of the orbits of at least the valence electrons from a spherical into a toroidal conﬁguration, resulting in strong magnetic ﬁelds estimated to be 1,415 times nuclear magnetic ﬁelds [2, 5a]. It is then ex pected that strongly polarized individual atoms and molecules bond tog ether like little magnets, resulting in clusters which are stable at ordinary conditions. Since the new bonds do not appear to be of valence type (or any of its v ariations), they can only be of electric, magnetic, or electromagnetic n ature. The new clusters are called magnecules because of the dominance of m agnetic over other eﬀects in their creation, while electric eﬀects are ge nerally unstable, and often repulsive (as it is the case of ions). Besides direct calculations [2, 5a], the magnetic polariza tion of the atoms and molecules constituting magnegas is further supported b y a number of in- direct eﬀects, such as the capability of magnegastmto stick to instruments walls, called magnecule adhesion. As an illustration, foll owing the removal of magnegas from a GC-MS and its conventional ﬂushing, the ba ckground preserves all the anomalous peaks of magnegas. This occurre nce can only be interpreted numerically via adhesion due to induced magnet ic polarization, and not via electrostatic, coordination, and other eﬀects. Mutatis mutan- dae, stable clusters can only exist under a suﬃciently stron g attractive force, which must be numerically identiﬁed for a model to have suﬃci ent depth. 7Among all possible non-valence bonds, the magnetic attract ion among polar- ized valence orbits is the only model available at this writi ng with a concrete attractive bond, while all other models lack such an identiﬁ cation (as it is the case for electric eﬀects, coordination eﬀects, co-vale nce, etc.). Due to the implications here at stake, the study of alternative str ucture of the new clusters in magnegas is warmly recommended, provided that, again, the at- tractive force creating the clusters is speciﬁcally and num erically identiﬁed, and models based on pure nomenclatures are avoided. Even though the chemical structure of magnegas escapes curr ent quan- tum chemical knowledge, its combustion exhaust has a conven tional chemical structure, because the exhaust temperature is beyond the Cu rie point of mag- necules. As a result, all magnecules and other anomalies are removed by the combustion. Following numerous tests, including various c onversions of au- tomobiles to run on magnegas, we have the following combusti on exhaust of magnegas measured before the catalytic converter, in perce ntages: Water vapor : 65%-70% Oxygen : 9.5%-10.5% Carbon dioxide : 6%-8% Carbon monoxide : 0.00%-0.01% Hydrocarbons : minus 2 to minus 5 ppm Rest atmospheric(5) As one can see, the upgrading of hydrogen into magnegas: 1) tu rns the oxygen depletion caused by hydrogen combustion into a posit ive oxygen bal- ance (more oxygen in the exhaust than that used for the combus tion) 2) emits no carcinogenic or toxic substance in the exhaust; and 3) implies a signiﬁcant reduction of carbon dioxide emission over fossi l fuels. In particu- lar, magnegas exhaust meets the most stringent governmenta l requirements without a catalytic converter while having a positive oxyge n balance. Pre- liminary magnegas exhaust measurements have been recently conducted at the EPA Certiﬁed, Vehicle Certiﬁcation Laboratory Liphard t & Associates of Long Island, New York, via the Varied Test Procedure (VTP) as per Reg- ulation 40-CFR, Part 86 on a Honda Civic Natural Gas Vehicle V IN number 1HGEN1649WL000160, produced in 1998 (and purchased new in 1 999) to 8operate with Compressed Natural Gas (CNG). This car was conv erted by US- Magnegas, Inc., Largo, Florida, to operate on Compressed Ma gneGas (CMG) via: 1) the replacement of CNG with CMG; 2) the disabling of th e oxygen sensor (because magnegas has 20 times more oxygen in the exha ust than nat- ural gas); and 3) installing a multiple spark system (to impr ove combustion); while leaving the rest of the car unchanged, including its co mputer. The tests consisted of the conventional EPA routine for Regu lation 40- CFR, Part 89, consisting of three separate and sequential te sts conducted on a computerized dynamometer, the ﬁrst and the third tests usi ng the car at its maximal possible capability to simulate an up-hill trav el at 60 mph, while the second test consists in simulating normal city driving o f the car. Three corresponding bags with the exhaust residues are collected , jointly with a fourth bag containing atmospheric contaminants. The ﬁnal m easurements expressed in grams/mile are given by the average of the measu rements on the three EPA test bags, less the measurements of atmospheri c pollutants in the fourth bag. The results of the above preliminary tests on magnegas exhaust are: Hydrocarbons : 0.026 gram/mile = 93.6% reduction of the EPA standard of 0.41 gram/mile Carbon monoxide : 0.262 grams/mile = 92.6% reduction of the EPA standard of 3.40 grams/mile Nitrogen oxides : 0.281 gram/mile = 29.7% reduction of the EPA standard of 0.4 gm/mile Carbon dioxide : 235 grams/mile - there is no EPA standard on CO 2at this moment Oxygen : not measured because not requested in Regulation 40-CFR, Part 86(6) The following comments are important for an appraisal of the above re- sults: 1) Magnegas does not contain heavy HC since it is created at 7, 000oF. Therefore, the measured HC is expected to be due, at least in p art, to com- bustion of oil, either originating from magnegas compressi on pumps (thus contaminating the gas), or from engine oil. New tests are und er way in 9which magnegas is ﬁltered after compression, and all oils of fossil fuels origin are replaced with synthetic oils. 2) Carbon monoxide is fuel for magnegas (while being a combus tion prod- uct for gasoline). Therefore, any presence of CO in the exhau st is evidence of insuﬃcient combustion. 3) The great majority of measurements (6) originate from the ﬁrst and third parts of the test at extreme performance, because, dur ing ordinary city traﬃc, magnegas exhaust is essentially pollutant free, as s hown in Figure 1. 4) Nitrogen oxides are not due, in general, to the fuel (wheth er magnegas or other fuel), but to the temperature of the engine, thus bei ng an indication of the quality of its cooling system. Therefore, for each giv en fuel, including magnegas, NOx’s can be decreased by improving the cooling sy stem and other means. 5) Measurements (6) do not refer to the best possible combust ion of mag- negas, but only to the combustion of magnegas in a vehicle who se carburetion was developed for natural gas. Alternatively, the test was p rimarily intended to prove the interchangeability of magnegas with natural ga s without any ma- jor automotive changes, while keeping essentially the same performance and consumption. The measurements under combustion speciﬁcal ly conceived for magnegas are under way, and will be released in the near fu ture. The main diﬀerence in the latter tests is a considerable reducti on in the emis- sion of carbon dioxide for certain technical reasons relate d to the magnegas combustion. In Figure 1, the ﬁrst three diagrams illustrate the very low c ombustion emission of magnegas in city driving, by keeping in mind that most of mea- surements (6) are due to the heavy duty, hill climbing part of the EPA test. Even though 29.7% of EPA standard, the fourth diagram on nitr ogen oxides is an indication of insuﬃcient cooling of the engine. The bot tom diagram indicates the simulated speed of the car versus time, where ﬂ at tracts simu- late idle portions at traﬃc lights. By keeping in mind: 1) the lack of (heavy) hydrocarbon in magnegas (because produced at 7,000oF of the electric arc); 2) the expectation of no appreciable carbon dioxide in the ma gnegas exhaust under proper combustion (because CO is fuel for magnegas); 3 ) the possi- ble further reduction of carbon dioxide via disposable spon ges placed in the exhaust systems; 4) the decrease of nitrogen oxides with a mo re eﬃcient en- gine cooling and other improvements; and 5) the positive oxy gen balance of magnegas (not measured in the test because not included in cu rrent EPA 10Figure 1: An illustration of the city part of the EPA test acco rding to Reg- ulation 40-CFR, Part 86, conducted at the Vehicle Certiﬁcat ion Laboratory Liphardt & Associates of Long Island, New York on a Honda Civi c Natural Gas Vehicle converted to magnegas. 11regulations); the measurements depicted in this diagram in dicate that the achievement of a truly clean fuel is indeed within technolog ical reach. We should also indicate considerable research eﬀorts under way to further reduce the CO 2content via suitable cartridges of disposable chemical spo nges placed in the exhaust system. Admittedly, these catalytic m eans generally implies the creation of acids harmful to the human skin, if re leased in the environment. However, the ongoing research aims at the chem ical and/or technological resolution of these problems. Additional re search is under way via liqueﬁed magnegas obtained via catalytic or convention al liquefaction, which is expected to have an anomalous energy content with re spect to other liquid fuels, and an expected, consequential decrease of po llutants. As a result of these eﬀorts, the achievement of an exhaust essent ially free of CO 2 appears to be within technological reach. As a comparison for measurements (6), a similar (but diﬀeren t) Honda car running on indolene (a version of gasoline) without aﬀec ting performance was tested in the same laboratory with the same EPA procedure , resulting in the following data: Hydrocarbons : 0.234 gram/mile = 900% of magnegas emission Carbon monoxide : 1.965 gram/mile = 750% of magnegas emission Nitrogen oxides : 0.247 gram/mile = 86% of magnegas emission (7) Carbon dioxide : 458.655 grams/mile = 195% of magnegas emission which illustrates the environmental superiority of magneg as over gasoline. The improvement of emission by magnegas over the above data a re evident. Other features favoring the upgrading of pure hydrogen into magnegas are (international patents pending): 1) magnegas is cost competitiveness with respect to fossil f uels (since it is produced as a byproduct of an income-producing recycling ); 2) magnegas increases the energy content from about 300 BTU/ cf for hydrogen to about 800-900 BTU/cf (due to the new means of ener gy storage); 3) magnegas is more readily availability anywhere desired ( since easily transportable PlasmaArcFlow reactors as big as a desk produ ce up to 1,500 cf of magnegas per hour, i.e, a production in one hour suﬃcien t for about three hours city travel by a compact car); 124) magnegas admits easier liquefaction, e.g., via Fischer- Tropsch cat- alytic synthesis or conventional liquefaction (due to attr actions between mag- necules); 5) magnegas has a better penetration through membranes (due to mea- sured decreases of average molecular sizes of magnetically polarized conven- tional molecules); 6) magnegas can be used for any conventional fuel applicatio n, including metal cutting, cooking, automotive use, etc. 7) Magnegas can be used in fuel cells, by preserving its envir onmental advantages. Above all, the magnegastmtechnology appears to permit an ultimate merger of crude oil and hydrogen technologies. One of the bes t liquids usable in the PlasmaArcFlowtmreactors is crude oil, which is then turned into a fuel much cleaner than gasoline (plus usable heat and solid p recipitates) at a cost visibly smaller than that that via huge reﬁneries. The fuel produced by the above new processing of crude oil is 40%-45% hydrogen. In conclusion, crude oil, hydrogen, and fuel cells remain in deed fully ad- missible in this new era of environmental concern, provided that they are treated via a basically new technology whose quantitative s tudy requires a new chemistry, hadronic chemistry [1-5]. Acknowledgments The author would like to thank D. D. Shillady, Chemistry Depa rtment, Vir- ginia Commonwealth University, U.S.A., and A. K. Aringazin , Department of Theoretical Physics, Karaganda State University, Kazak stan. Particu- lar thanks are also due to all member of USMagnegas, Inc., for invaluable assistance without which this paper could not have seen the l ight of the day. 13References [1] http://www.magnegas.com [2] R. M. Santilli, Hadronic Journal 21, 789 (1998). [3] R. M. Santilli and D. D. Shillady, Ab Initio Hadronic Chemistry , Hadronic Press, Florida (2000). [4] R. M. Santilli and D. D. Shillady, International Journal of Hydrogen Energy 24, 943 (1999), and 25, 173 (2000). [5] (a) M. G. Kucherenko and A. K. Aringazin, Hadronic Journa l21, 895 (1998). (b) M. G. Kucherenko and A. K. Aringazin, Hadronic Jo urnal 23, 1 (2000), e-print: physics/0001056 . (c) A. K. Aringazin, Hadronic Journal 23, 57 (2000), e-print: physics/0001057 . 14
arXiv:physics/0009015v1 [physics.gen-ph] 4 Sep 2000Perfectly secure cipher system. Arindam Mitra Lakurdhi, Tikarhat Road, Burdwan. 713102. India. Abstract We present a perfectly secure cipher system based on the concept of fake bits which has never been used in either clas- sical or quantum cryptography. 1Cryptography, the art of secure communication has been deve loped since the dawn of human civilization, but it has been mathematically t reated by Shan- non [1]. At present, we have diﬀerent classical cryptosyste ms whose merits and demerits are discussed below. Vernam cipher [2]: It is proven secure [1] but it can not produ ce more absolutely secure bits than the shared secret bits. Due to th is diﬃculty, it has not become popular, however it is still routinely used in diplomatic se- cure communication. Data encryption standard [3] and public key distribution sy stem [4]: These are widely used cryptosystems because they can produc e more com- putationally secure bits than the shared secret bits. The problem is that its computational security is not proved. The assumption of com putational se- curity has now become weak after the discovery of fast quantu m algorithms (see ref. 16) To solve the above problems of classical cryptosystem, quan tum cryp- tography [5-9] has been developed over the last two decades. Conceptually quantum cryptography is elegant and many undiscovered poss ibilities might store in it. In the last few years work on its security has been remarkably progressed [10-14], however work is yet not ﬁnished. Recently it is revealed [15] that all practical quantum cryp tographic systems are insecure. Regarding quantum cryptosystems, the popular conjectures are: 1. Completely quantum channel based cryptosystem is imposs ible [16] (exist- ing quantum cryptosystem requires classical channel to ope rate). 2. Uncon- ditionally secure quantum bit commitment is impossible [16 ]. 3. By classical means, it is impossible to create more absolutely secure bit s than the shared secret bits. Recently alternative quantum cryptosystem has been develo ped [17-20] by the present author; which can operate solely on quantum ch annel (both entangled and unentangled type)[17,18] and can provide unc onditionally se- 2cure quantum bit commitment [19]. Here we shall see that thir d conjecture is also not true. Operational procedure : For two party protocol, the problem of Vernam ciper (popularly called one time pad) [2] is that two users ha ve to meet at regular interval to exchange the key material. We observe th at key material can be simply transmitted without compromising security. In the presented cipher system, always in the string of rando m bits, there are real and pseudo-bits (fake bits). Real bits contain key m aterial and pseudo-bits are to mislead eavesdropper. Sender encodes th e sequence of real bits on to the ﬁxed real bit positions and encodes the sequenc e of pseudo- bits on to the ﬁxed pseudo-bit positions. It thus forms the en tire encoded sequence. which is transmitted. The ﬁxed positions of real a nd pseudo-bits are initially secretly shared between sender and receiver. Therefore, receiver can decode the real bits (the ﬁrst key) from real bit position s. Obviously he/she ignores the pseudo-bits. For the second encoded sequence, sender uses new sequence of real and pseudo-bits but the position of real and pseudo-bits are sam e. So again receiver decodes the second key from the same real bit positi ons. In this way inﬁnite number of keys can be coded and decoded. Notice th at initially shared secret positions of real and pseudo-bits are repeate dly used. That’s why, in some sense, secrecy is being ampliﬁed. Let us illustr ate the procedure.  P R R R P R P R P P P R P R R P .... 0b1b1b11b11b11 0 0 b10b1b11.... 1b2b2b20b20b20 1 1 b21b2b20.... 1b3b3b31b31b31 0 0 b30b3b30.... 0b4b4b41b40b41 0 1 b41b4b40.... . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . .... 1bnbnbn1bn0bn0 0 1 bn0bnbn1.... ≡ Ss Se1 Se2 Se3 Se4 . . . Sen  3In the above block, the ﬁrst row represents the sequence Ss, which is ini- tially secretly shared. In that sequence, ”R” and ”P” denote the position of real and pseudo- bits respectively. The next rows represent the encoded se- quences : Se1, Se2, Se3, Se4, ...., S en. In these encoding, biare the real bits for i-th real string of bits. Other bits are pseudo-bits. Obviou sly the sequences of real bits always form new real keys. Similarly sequences o f pseudo-bits always form new pseudo-keys. But positions of real and pseud o-bits are un- changed. As receiver ignores pseudo-bits and pseudo-keys, so the decoded strings of real bits (keys) will look like:  R R R R R R R R .... b1b1b1b1b1b1b1b1.... b2b2b2b2b2b2b2b2.... b3b3b3b3b3b3b3b3.... b4b4b4b4b4b4b4b4.... . . . . . . . . .... . . . . . . . . .... . . . . . . . . .... bnbnbnbnbnbnbnbn.... ≡ Ss K1 K2 K3 K4 . . . Kn  HereK1, K2, K3, K4....., K Nare independent keys. Condition for absolute security: Shannon’s condition for a bsolute security [1] is that eavesdropper has to depend on guess for absolutel y secure system. In our system, for a particular encoded sequence of events (b its), if the prob- ability of real events ( prealbits) becomes equal to the probability of pseudo- events ( ppseudo −bits) then eavesdropper has to guess. Since all the encoded sequences are independent so eavesdropper has to guess all s equences. There- fore, condition for absolute security can be written as: 1. ppseudo −bits≥prealbits. 2. All encoded sequences should be statistically independe nt. That is, any encoded sequence should not have pseudo randomness. Speed of communication: If we take ppseudo −bits=prealbits and share 100 bits, then message can be communicated with 1/4 speed of digi tal commu- nication (data rate will reduce a factor of 1/2 due to key prod uction and 4another factor of 1/2 due to message encoding) as long as we wi sh. If the key (Ki) itself carries meaningful message, then speed of secure co mmuni- cation will be just half of the speed of digital communicatio n. Perhaps no cryptosystems oﬀer such speed. The above art of key exchange is mainly based on the idea of pse udo-bits, which was ﬁrst introduced in our noised based cryptosystem[ 21]. But that system will be slow and complicated. In contrast, this syste m will be fast and simple. Note that, noise has never been a threat to the securi ty of any classi- cal cryptographic protocol ( rather it can be helpful [21] to achive security). This is the main advantage of classical cryptographic proto col over quantum key distribution protocols, where noise indeed a threat to t he security. It should be mentioned that the classical cipher system can not achieve other quantum cryptographic tasks such as cheating free Bell’s in equality test [18] and quantum bit commitment encoding [19]. Indeed classical cryptography can not be encroach entire area of quantum cryptography. References [1] Shannon, C. E. Communication theory of secrecy systems. Bell syst. Technical Jour .28, 657-715 (1949). [2] Vernam, G. S J. Amer. Inst. Electr. Engrs 45, 109-115, 1926. [3] Beker, J. and Piper, F., 1982, Cipher systems: the protec tion of com- munications (London: Northwood publications). [4] Hellman, E. M. The mathematics of public-key cryptograp hy.Sci. Amer. August, 1979. [5] Wiesner, S. Congugate coding, Signact News, 15, 78-88, 1983, ( The manuscript was written around 1970). [6] Bennett, C. H. & Brassard, G. Quantum cryptography: Publ ic key dis- tribution and coin tossing. In proc. of IEEE int. conf. on computers, system and signal processing 175-179 ( India, N.Y., 1984). [7] Ekert, A. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.67, 661-663 (1991). 5[8] Bennett, C. H. Brassard. G. & Mermin. N. D. Quantum crypto graphy without Bell’s theorem. Phys. Rev. Lett .68, 557-559 (1992). [9] Bennett, C. H. Quantum cryptography using any two nonort hogonal states. Phys. Rev. Lett .68, 3121-3124 (1992). [10] Biham, E. & Mor, T. Security of quantum cryptography aga inst collec- tive attack, Phys. Rev. Lett. 78, 2256-2259 (1997). [11] Biham, E. & Mor, T. Bounds on information and the securit y of quan- tum cryptography. Phys. Rev. Lett .79, 4034-4037 (1997). [12] Deutsch, D. et al, Quantum privacy ampliﬁcation and the security of quantum cryptography over noisy channels. Phys. Rev. Lett .77, 2818- 2821 (1996). [13] Mayers, D. Unconditional security in quantum cryptogr aphy. Preprint quant-ph/9802025. [14] Lo, -K. H. & Chau, H. F. Unconditional security of quantu m key distri- bution over arbitrarily long distance. Science .283, 2050 (1999). [15] Brassard, G., Lutkenhaus, N., Mor, T. & Sanders, C. B. Se curity aspect of practical quantum cryptography. Preprint quant-ph/991 1054. [16] Bennett, C. H. & Divincenzo. D Nature ,404, 247,2000. [17] Mitra. A, Complete quantum cryptography, Preprint, 5t h version, quant-ph/9812087. [18] Mitra. A, Completely entangled based communication wi th security. physics/0007074. [19] Mitra. A, Unconditionally secure quantum bit commitme nt is simply possible. physics/0007089. [20] Mitra. A, Entangled vs unentangled alternative quantu m cryptography. physics/0007090. [21] Mitra. A, Completely secure practical cryptography. q uant-ph/ 9912074. 6
Second law versus variation principles by W.D. Bauer Abstract: We apply Pontrjagin's extremum principle of control theory to mechanics and transfer it to equilibrium thermodynamics in order to test it as an ansatz.This approach allows to derive the Boltzmann distribution as result of a variation problem. Furthermore, a principle of extremal ent ropy can be derived and -last not least - the second law in a modified form. Contrary to the Clausius version of second law, the derivation can predict secon d law violations if potential fields are included into consideration. This is illustrated by a experimental example from literature. Therefore, a purely math ematical approach of thermodynamics, which derives the direction of irreversibilities from the inverse variational problem built around the thermody namic formalism, can contradict the Clausius second law for some special cases. 1. Introduction The big success of the second law in thermodynamics relies on the fact that it predicts the direction of the known irreversible processes correctly. The inherent problem with it is that it is based on experience. Therefore, due to axiomatic character of s econd law, the question arises incidentally, whether the second law is an overgeneralisation. On the other hand, unconsciously and without any notice, other basic concepts are used in order to explain the direction of irreversibilities in thermodynamics. This can be the case if a chemist speaks about that his reaction is "driven by enthalpy". Landau and Lifshitz [1] obtain the direction of electro-thermodynamic irreversibilities by the application of variational princ iples on potentials. Because variational principles are included in the mathematics of a physical problem, the question arises, wheth er the second law as additional physical principle becomes obsolete if this purely mathematic aspect is included into consideratio n . This article checks the consistence and equivalence of the second law against approaches using the variational principles appli ed to potentials. We will present a field-dependent derivation of equilibrium thermodynamics from classical mechanics of a many particle system. Many elements and steps of this derivation are well known [2][3]. The new point is that we use the information about theHamiltonian from the extremum principle of Pontrjagin[4] gained from classical mechanics and and transfer it intothermodynamic as extremum principle of the potentials. Therefrom, the Boltzmann distribution, an extremum entropy principleand a modified second law is derived for special examples. 2. The Hamiltonian as minimizing function It is well known that the equations of motion are a solution of the Lagrange variational problem. The solution is obtained, if the functional is an extremum, where x(t 0)=x0and x(t1)=x1 are start and end point of the path. It is not so well known in physics [4] that the variational problem can be regarded as a special case of a problem of control theory if we substitute (t) , where u(t) is the control function to be optimized. The functional for a control theory problem be with and .The Hamiltonian is defined to The adjunct system is defined to Acc. to the Pontrjagin theorem the optimum control function u(t) can be determined by looking for the extremum of the Hamiltonian, i.e. 1 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmIf we differentiate for t the solution is exactly the Euler-Lagrange equation because of and as defined above. The extremum of the Hamiltonian is a minimum because Huu > 0 . Similarly, as shown by Landau and Lifshitz [1], the Lagrangian of electrostatics can be varied with respect to electric coordin ates E or D. If only one Maxwell equation is given the other can be reconstructed by the variation formalism applied to any thermodynamic potential. These results could be embedded in a more general mathematical framework [6] which derives generalrelativity including all sub-theories using a Lagrange energy approach developed to second order. 3. The transition from mechanics to thermodynamics As shown in the last section the mechanic Hamiltonian (or inner energy) goes to an extremum with respect to if the solution of the equation of motion of the many particle system is inserted into the Hamiltonian. Non-stationary variations of the solution deviate from the extremum. Such variations relative to a stable stationary state can exist physically if the Hamiltonian vari es in time if the system goes into the equilibrium by phase transitions which change the number of degrees of freedom or constraintsbetween the particles. Then, H tot changes in time during non-equilibrium until equilibrium is reached. This interpretation, however, holds only for stable states (or equilibrium thermodynamics), because the time t is not contained in the description as independent variable there. It is different for dynamic processes (or equilibrium thermodynamics[7,8]) wher e t is contained explicitely . In equilibrium, due to energy conservation, the Hamiltonian is constant. Then, the time mean of the Hamiltonian is identical to the Hamiltonian as well. If the assumption is founded that the ergodic hypothesis can be applied to a many particle system, the mean total energy in t ime of all particles is identical to the ensemble average or in mathematic language(using the definition T as measuring time interv al) where is the Hamilton energy of a single particle = p2/(2m) + and represents the mean inner potential between the particles due to the real fluid behaviour. H( , ) = + U( ) is the total energy formula of a single particle in the field U( ) and W( , ) is the probability function of the ensemble. The probability function is normed acc. to As shown in the last section the stationary mechanic Hamiltonian sits in an extremum with respect to all particle velocity coordinates. If we transfer this feature from mechanics into thermodynamic notation as an ansatz analogously, then at every spa ce cell the Hamilton density function representing the sum of all particles of each energy should go to an extremum as well. The Legendre transform of this Hamilton function is the Lagrange function with H= +U( ) and . From the Euler-Lagrange equation we obtain the corresponding differential equation The solution of this differential equation is the Boltzmann distribution. 2 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmwith introduced in order to get dimensionless units. It holds W0=1 due to (8). Contrary to mechanics, however, the second variation is exactly zero. Therefore, neither a minimum neither a maximum Hamiltonian exists with respect to the chosen independent coordinates here. Now, the conventional thermodynamic descriptioncan be obtained by a coordinate transformation of the Hamiltonian ensemble average (or inner energy) H tot . Proof: We take the total differential of the ensemble average for and r , where . Then it follows using the definitions of unit area:=A and V := volume. We discuss each partial derivatives of (15): 1) We change the coordinate of the first derivative by inserting the Boltzmann distribution W( , ):= exp[- /(kT( ))] in (14). If the partial derivative of (14) is taken respectively to , the potential term U( ri) cancels to zero due to (13) . Therefore, we can write We identify the specific Boltzmann entropy density as new independent coordinate. We note here that due to eq.(16) the variational formalism from eq.(10) ff. can be applied as well either to Hamiltonian energy either for entropy density if H= +U( ) is replaced trivially by . Therefore, the extremum entropy principle can be regarded to be derived here from the extremum principle of the Hamiltonian. If we apply information theory additionally we obtain the Boltzmann distribution as maximum entropy distribution of all possible Shannon entropy distributions [9]. We see however, that the maximum entropy principle contains less information than the extremum principle of the Hamiltonian because spatial information of the potential U( ) is lost during partial differentiation of (14) with respect to . As it will shown later, this can lead to contradictions. In thermodynamics this weakness is compensated by the additional demand that the materi al equation have to be designed such that second law is always fulfilled [10]. 2) Due to the coordinate transformation the second partial derivative of (15) can be written We denote P* as the global total pressure. It is a fictive value without any physical empirical relevance. It is constant over the volume in equilibrium in the field and characterizes mathematically the global coupling over the volume. Due to H( , ) = + U( ), P* can be evaluated as 3 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmP( ) can be identified as the measureable pressure. The second term represents the additional hydrostatic or barometric pressure contribution due to the outer field. 3) The third partial derivative of (15) is We denote µ* as global total potential[5]. It is constant over the volume in equilibrium and characterizes mathematically the global coupling over the volume similarly to the total pressure. Due to the norm of W( , ), comp.eq. (8), and H( , ) = + U( ) this can be evaluated to with µi( ) as the chemical potential of a substance without field. The above space dependent formalism of thermodynamics allows to deal generally with static and space dependent thermodynamic problems in potential fields like hydrostatic or barometric pressure and coincides with other known formulationsof space dependent equilibrium thermodynamics [11]. The thermodynamic equilibrium conditions can be found by the following control problem: The mean Hamilton energy H(S,V,n i) can be regarded acc. to section 2 to to be controled by the control vector u=(S(r),V(r),ni(r)) and should go to an extremum.We have to optimize The solution of this problem is acc.to the line of section 2. In the end we get the equations which are the conditions of equilibrium because of T= H/ S, P*=- H/ V and µi=H /ni. The second variation of the Hamiltonian (or inner energy) with respect to the new coordinates system depends from the choice o f the material, the fields and similarly from the inner potentials Uij of the material acc. to the variational point of view discussed here. We will prove in the next section that it is not an absolute minimum generally which would be necessary to obtain Clausius's version of the second to be valid generally. 4. The second law as consequence of the extremum principle of the Hamiltonian The minimum principle of potentials is derived in many textbooks of thermodynamics from second law [12]. In this section we will reverse this procedure and derive the second law from the extremum behaviour of the Hamiltonian. Acc. to section 2 and 3the second derivative of the Hamiltonian can be obtained principally from the mathematics of the problem and gives independentinformation about the direction of the extremum of the potentials. This information is independent from any additional empirica l physical information like second law. Therefore, the question has to be discussed whether both approaches are equivalent. It will be shown in the following that both approaches make the same prediction for thermodynamic standard cases where irreversibilities obey dH<0. However, if potential fields are included into consideration the variational approach can predictsecond law violations for special cases. 4.1. Thermodynamic standard case - no fields present As an example we discuss the behaviour of a cycle of a simple fluid which may condensate. We imagine a periodic cycle which includes a reversible part of the closed path over the points 1-> 2-> 3 . The path 3-> 1 is irreversible. During this phase th e fluid has contact with the (only) heat of environment at constant temperature T. We start at the cycle at point 1 of the dewline of a phase diagram of a simple substance like water, comp. fig.1 , 2 and 3. The path 1->2 is an isentropic expansion (dS1-> 2= 0). At point 2 the volume is separated into two parts 2' and 2´´ by closing a tap. One part 2´contains vapour only, the other2´´ liquid and vapour. Then both volumes are recompressed isentropically back to t he initial total volume (dS2-> 3= 0). At point 3 the tap is reopened, the system is set into contact with the heat bath of environment and the cycle is closed 3->1 irreversible at constant volume. Because the inner energy H is a potential it holds 4 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmAcc. to the second law it should hold due to minimizing of H during irreversible transitions. This follows as well from the 2nd variation of the Hamiltonian with respect of pi. Therefrom follows using dH = -PdV + TdS , dS1->2 = dS2->3 = 0 and From follows then This result is in accordance with the standard versions of the second law. 4.2 Violation of 2nd law acc. to the extremum principle if fields are present The inner energies H' and H" of a capacitive loaded thermodynamic system are defined [1]by where we used the definitions := susceptibility, E:= electric field, P:= electric polarisation. The dielectric constant of vacuum is omitted in all formulas for convenience. The same formulas can be written in differentials Regarding the 2nd derivative with respect to the electric variables P or E of both these potentials we see in formulas (28) that for constant homogeneous dielectrics the potential H'(V,S,ni,P) approaches a minimum in the extremal state of thermodynamic equilibrium and H''(V,S,ni,E) approaches a maximum in the equilibrium state, if > 0, dS=0 and dV=0. Therefore, because the second variations of H with respect of the fields are identical to the second derivative of H with respect to the field coordin ate [1], the following unequalities hold for irreversible changes of state Due to the Legendre transformation formalism analogous expressions on eq.(28) and (29) hold for free enthalpy, i.e. and In words, these equations (28) - (31) can be interpreted as "electric Chatelier-Braun-principle", which states for simple dielectrics that they tend to discharge themself. In order to show the contradiction between second law and the "electric le Chatelier-Braun principle" we regard an isothermalelectric cycle with an irreversible path into a maximum of G ´´ at constant field E, comp. fig.4 and 5: Because G'' is a potential, we have for a closed cycle over three points(1->2->3->1) According to the extremum principle 23G''irrev > 0 holds. Because of the isofield ( E2=E3) irreversible change of state (2->3) we have also . This zero expression is added to the second formula of (32) and we can write 5 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmRegarding the sign of this integral we see that the orientation of this cycle is reversed compared to the usual hysteresis of a ferroelectric substance. Because the cycle proceeds isothermically (with only one heat reservoir) the Clausius statement of sec ond law is violated. The proof is as follows: Due to energy conservation and because H" is a potential holds. Therefrom, because the first term is negative, it follows that the net heat exchange T dS has a positive sign. This impl ies dS>0 because T=constant . This means that the cycle takes heat from the environment and gives off electrical work under isothermal conditions which is contrary to the Clausius formulation of the 2nd law. Example: A concrete physical model containing this contradiction to the 2nd law can be found in polymer chemistry. Wirtz and Fuller [13] [14] investigated electrically induced sol-gel phase transitions of polymer solutions. To explain their experiments they used aFlory-Huggins model [15] extended by an electric interaction term. Their model describes the qualitative behaviour of suchsolutions correctly, however, they did not note the inconsistence regarding to Clausius version of second law. This can be shown by proceeding a isothermal closed cycle, which is the electric analog of a Serogodsky or a van Platen cycle o f binary mixtures discussed recently [16]. This closed splitted cycle is proceeded using a dielectric sol-gel mixture in a capaci tor like polystyrene in cyclohexane (upper critical point solution) or p-chlorostyrene in ethylcarbitol (lower critical point solut ion). The composition of the solution is separated periodically by a demixing phase transitions induced by switching off the field. A fter the separation of both phases by splitting into two volumes they are remixed again (irreversible phase !) after openig theseparating tap in a strong field. The cycle is started in the 2- phase region at zero field at the points 1, comp. fig.4 ,5 and 6, where the volume is splitted by closing the tap separating both phases. Then a strong electric field E is applied. So we reach the points 2' and 2'' representing different phases of the solution in both compartments. Then we open the tap and let mix the solutions in both compartments. During themixing (2->3) the electric field is kept constant by decharging the capacitor during the decline of the dieleletric constant, c f. fig.7 . Then, in the phase diagram fig. 6 and as well in fig.4 + 5, the mixed solution is at the phase separation line at point 3. In the last step of the cycle the capacitor is discharged completely and the system goes back into the 2-phase area to point 1 and demixes.According to the theory (cf. Eq. 31) Now, we define S:=V ´/V to be the splitting factor of the total volume V , V' and V'' are the volumes of the compartments each where V:=V ´+V". We write the difference of the free enthalpy using the definition or G ´´ in (30) assuming := -1 to be dependent from and independent from E for zeroth order The right side of the first line represents the stored linear combined field energy of the separated volume parts (points 2' an d 2") at point 2, the second line stands for the field energy difference (1->3) of both the connected compartments containing of two volume parts of the coexisting phases ´ and ". In the first line S, , ' and " are constant, in the second line S, , ' and " are dependent from E in 2-phase area. For the system investigated by Wirtz et al., the Flory free-energy density approach of an incompressible dilute monodisperse polymer solution is useful. The "ansatz"[13,14] is where N:= polymerisation number, := total dielectric constant, p := dielectric constant of the polymer solution,m:= dielectric constant of the monomer solution, :=1/(kT) with k:=Boltzmann number and :=Flory parameter. Therefrom, the chemical potential µ and osmotic pressure follow to The phase equilibrium is determined by the equations 6 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmThe first equation describes the chemical potential to be equal in both phases. The second equation is the Maxwell construction applied to the chemical potential. The solution of this system of equations can be done numerically or by deriving a parameterrepresentation [13,14]. Results are shown qualitatively in fig.6 . 5.Conclusion The analogies between mechanics and thermodynamics- shown in tab.1- suggest that the direction of irreversibilities can be understood from a variation principle applied to inner energy analogously to the extremum principle of Pontrjagin applied tomechanics. Therefrom, it can be derived the Boltzmann distribution, an extremum entropy principle and the second law in a modified form which allows second law violations if fields are included into consideration. This result becomes understandable if we note that the reverse problem of variational calculus (i.e. the reconstruction of a variation functional from a function or differential equation) cannot be solved uniquely, because the solutions can differ by a integration constant. These integration constants can contain other variables to be varied. Therefore, the prediction of the di rection of an irreversibility based on the evaluation of the potential or based only on entropy can be different, because the entropy as derivative of a potential has lost information. The lost integration constants, however, influence the direction of the irrever sibility in the state space acc. to the variational calculation applied to all possible coordinates of the Hamiltonian. Acc. to a purely mathematical variational approach of thermodynamics all empirical information about a system is in the potential describing the material behaviour. Therefrom, the application of all possible variation principles allows to determ ine the directions of the irreversible processes. Contradictions to second law between due to "strange" material behaviour areexcluded by this approach a priori. Instead of this Clausius's version of second law can be violated under defined conditions. Therefore, model systems showing this theoretical contradiction between second law and second variation of inner energy could be interesting for experimental research. References: 1) L.D. Landau, E.M.Lifshitz Elektrodynamik der Kontinua , §18 Akademie Verlag , Berlin 1990, 5 Auflage 2) M. Leontovich Introduction à la thermodynamique Physique statistique Edition Mir, Moscou, 1983 transduction francaise1986 3) R.L. Stratonovich Nonlinear Nonequilibrium Thermodynamics I, Springer Berlin, 1992 4) Bronstein-Semendjajew Taschenbuch der Mathematik, Harri Deutsch, Frankfurt, 1984 5) The term total potential stems from van der Waals and Kohnstam see for example: V.Freise Chemische Thermodynamik BI Taschenbuch 1973 (in German) 6) V.Benci, D.Fortunato, Foundations of Physics 28,No.2, 1998, p. 333 -352 A new variational principle for the fundamental equations of classiccal physics 7) L. Onsager, S. Machlup, Phys. Rev. 91, No.6, 1953, p. 1505 8) I. Gyarmati Non-equilibrium Thermodynamics Springer, Berlin, 1970 9) F. Topsoe Informationstheorie B.G.Teubner Stuttgart 1974 (in German 10) W. Muschik, H. Ehrentraut, J. Non-Equilib. Thermodyn. 21, 1996, p. 175-192 11) J.U. Keller Thermodynamik der irreversiblen Prozesse, 7 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmde Gruyter, Berlin, 1977 (in German) 12) Stephan K., Mayinger F. Thermodynamik Bd.II Springer Verlag Berlin, New York 1988 13) D. Wirtz, G.G. Fuller, Phys.Rev.Lett.71 (1993) 2236 14) D. Wirtz, K. Berend, G.G. Fuller Macromolecules 25 (1992) 7234 15) J. Des Cloizeaux ,G. Jannink Polymers in Solution Oxford University Press, Oxford 1987 16) W.D. Bauer, W Muschik, J. Non-Equilib. Thermodyn. 23 (1998), p.141-158 17)P. Debye, K. J. Kleboth, Chem.Phys.42 (1965) 3155 Tab.1: analogous features between thermodynamics and mechanics mechanics thermodynamics time mean or least action functional ensemble average Hamilton energy inner energy non-extremal state of functional non-equilibrium state Legendre transformations, i.e. L, H Legendre transformations, i.e. U, H, F, G Pontrjagin's extremum principle extremum principle of potentials second variation of the Hamiltonian "second law" Captions: Fig.1: Cycle with periodic irreversibility of a simple fluid: 1-2 isentropic expansion, 2 splitting the volume, 2-3 separate isentropic compression of both compartement, 3 opening the tap, 3-1 irreversible closing of the cycle and contact with heat bath of environment at T=constant, comp. fig.2 and 3 and text 8 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmfig.2: Cycle with periodic irreversibility in the T-S phase diagram of a simple fluid 1-2 isentropic expansion, 2 splitting the volume, 2-3 separate isentropic compression of both compartement, 3 opening the tap, 3-1 irreversible closing of the cycle and contact with heat bath of environment at T=constant, comp. fig.1 and 3 and text fig.3: Pressure P vs. total volume V of a cycle of a simple fluid with periodic irreversibility 1-2 isentropic expansion, 2 splitting the volume, 2-3 separate isentropic compression of both compartement, 3 opening the tap, 3-1 irreversible closing of the cycle and contact with heat bath of environment at T=constant, comp. fig.2 and 3 and text 9 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmfig 4: Isothermic isobaric electric cycle with a diluted polymer solution as dielectric 1) voltage U=0: system in 2-phase region 2) both volumes separated, rise of voltage from zero to U=const.: each volume compartment in 1-phase region 3 ) voltage U=const.: opening the tap and returning to the phase separation line by remixing back to section 4.2 or section 4.2 Example fig.5: Isothermal electric cycle in capacitor with electrically induced phase transitions; charge Q versus voltage U plotted; 1 starting at 2 phase area line with zero field, 1-2 applying a field with tap closed, 2 opening the tap, 2-3 discharging and remixing in field, 3 returning to starting point 1 by discharging the capacitor; a negative work area is predicted according to Gibbs thermodynamics contrary to the second law back to section 4.2 or section 4.2 Example 10 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmfig.6: vs. volume fraction (with T:=temperature) phase diagram of a polymer solution with and without electric field E according to [13,14]; plot shows modified Flory-parameter versus volume fraction of polymers; points 1: E=0, 2-phases, both points 1 at the phase separation line; points 2: E=const., both points 2 of the splitted volume in 1 phase area; point 3: E=const., after opening the tap: point 3 returns exactly at the phase separation line; more information about the construction of this phase diagrams, see references [14,15] back to section 4.2 Example or section 4.2 end of Example 11 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htmfig.7: Dielectric constant versus volume fraction of polymers in a dilute solution; points 1, 2 and 3 refer to points in fig.1-3. According to the theory [13,14,15] d2/d2>0 holds near the critical point. Therefore ( ) has to turn to the left and the dielectric constant has to decline during remixing 2->3. Observations at a similar system[17] support this prediction back to section 4.2 Example 12 of 12 04.09.00 16:20file:///C\|/Eigene Dateien/FP/HTML/FP.htm
arXiv:physics/0009017v1 [physics.optics] 5 Sep 2000On the sensitivity of wave channeling of X-ray beam to the shape of interface channels. T.A.Bobrova, L.I.Ognev1 Abstract The using of microdiﬀraction of X-ray radiation for analysi s of the structure of material specimens with submicron resolution becomes very promisin g investigation method [1]. One of the methods for obtaining of submicron beams of hard X-ray radiation is formation in a narrow channel of dielectrical resonator [1, 2]. In the pres ent work the eﬀect of transmission of X-ray through narrow submicron rough channels was investig ated by numerical simulation with account for diﬀraction and decay of coherency. It was found t hat transmission can be strongly decreased for channels with periodic deformations. The eﬀe cts of roughnes were explained with the statistical theory of X-ray scattering in rough transit ional layer. The wave mode attenuation coeﬃcients βscale as β∼1/d3(dis channel width) and proportional to roughness amplitude σ. Possible explanation of observed anomalous energy depend ence of transmission through thin Cr/C/Cr channel was given. The sensitivity of transmission of dielectrical channel to the presence of roughness and deformation with large space period was inv estigated. PACS 41.50 61.10 61.10.E 78.70.C 1Nuclear Fusion Institute, Russian Research Center ”Kurcha tov Institute”, Moscow, 123182, Russia E-mail: ognev@nﬁ.kiae.ru 1The using of microdiﬀraction of X-ray radiation for analysi s of the structure of material specimens with submicron resolution becomes very promisin g investigation method [1]. One of the methods for obtaining of submicron beams of hard X- ray radiation is formation in a narrow channel of dielectrical resonator [1, 2]. Monito ring of X-ray beam by capture into a narrow dielectric channel is used in waveguide X-ray l aser physics [3, 4], production of thin X-ray probe beams [2] and other applications [5] due t o the eﬀect of total external reﬂection. In this work we consider the role of diﬀraction that can be imp ortant for narrow beams especielly when roughness is high. Scattering from su rfaces with high roughness needs special approach because small perturbation methods fail [6]. Theoretical model X-ray scattering on rough surfaces is usually investigated within the well known Andronov-Leontovich approach [7] but for very small angles of incidence the model of ”parabolic equation” for slowly varying scalar amplitudes of electrical ﬁeld vector A(x, z) should be used. Within the model scattering and absorption d o not disappear at small grazing angle limit that results from Andronov-Leontovich approach [7]. In this case large angle scattering is neglected so ∂2A(x, z)/∂z2≪k·∂A(x, z)/∂z and because the beam is narrow ∂2A(x, z)/∂z2≪∂2A(x, z)/∂x2, where zandxare coordinates along and across the channel. The considera tion will be re- stricted here to 2-dimensional channels (gaps) although th e same approach can be applied to capillaries. The assumption results in ”parabolic equat ion” of quazioptics [8]: 22ik∂A ∂z= ∆ ⊥A+k2ε−ε0 ε0A (1) A(x, z= 0) = A0(x), where k=√ε0ω c. (In this case ε0is dielectrical permittance of air, ε1- dielectrical per- mittance of glass.) The evolution of the channeled X-ray bea m was calculated by direct integration of the ”parabolic” equation [9]. The dielectri c permitance on the rough bound- ary with the random shape x=ξ(z) was presented as ε(x, z) =ε1+ (ε0−ε1)H(x−ξ(z)) where H(x) is a step function. The distribution of roughness heights is assumed to be normal. It is known from re sults of [7] that at grazing incidence the eﬀect of scatering is very small. So special su rfaces are needed to observe scattering eﬀects in the gap interface at reasonable distan ce. In the calculations we used roughness amplitude up to 400 ˚A. The reﬂection of X-ray beam on very rough surfaces (up to 1500 ˚A) of silicon was observed in[10]. The results of direct simu lation of scattering with the model rough surface by integration of equation (1) c alculated for X-ray energy E= 10keV, width of the channel d= 0.5µm,σ= 400 ˚Aand correlation length of roughness zcorr= 5µmaveraged over 40 realizations are shown on Fig.1 as normaliz ed to initial value total intensity of the beam rtot, incoherent part rinc, where ri=/integraltext∞ −∞Ii(x)dx/ /integraltextd/2 −d/2I0(x)dx. Initial angles of incidence of plane wave were ϑ= 0; 3·10−4and 6·10−4rad (Fresnel angle ϑF= 3·10−3rad). The atomic scattering factors used in the calculations were taken from [11]. 300.20.40.60.811.2 0 5000 10000 15000 20000ri µm1 1′22′ 33′ Fig.1. Evolution of the total integral normalizied intensity of th e beam rtotand normalized incoherent part rpart=rinc/rtotfor diﬀerent incidence angles ϑ.ϑ= 0,rtot(curve 1), rpart(curve 1′);ϑF/10 (curves 2 and 2′);ϑF/5 (curve 3 and 3′). The main result of direct simulation is that the loss of coher ency comes along with attunuation of the beam and in the transmitted beam the coher ent part prevails [8]. Analytical results for transmission of coherent part of X-r ay can be obtained with statistical averaging of equation (1) using Tatarsky metho d (see [12]) as it was made in [8] by generalization of the method for stratiﬁed media. T he same generalization of the method to include stratiﬁed media was used in the case of e lectron channeling in single crystals [13]. The method results in additional atte nuation of coherent part of the amplitude < A > due to ”scattering potential” W(x). W(x) = (−ik/4)/integraldisplay∞ −∞< δε′(x, z)δε′(x, z′)> dz′ As it was shown in [8] ”scattering potential” can be expessed as 4W(x)≈ −k 4(ε0−ε1)2 π(ε0)2/integraldisplay∞ −∞dz′/integraldisplay0 −∞exp(−ξ2)dξ/integraldisplay −R(z′)ξ (1−R2(z′))1/2 0exp(−η2)dη ·exp(−x2 σ2) (2) with clear dependence on vertical coordinate xwhere R(z) is the autocorrelation coeﬃ- cient, σis dispersion of ξ(z) distribution. The decay of coherency for particular wave modes can be descr ibed with attenuation coeﬃcients βl. Attenuation coeﬃcients can be found as overlap integrals βl=−k 2/integraldisplay ϕl∗(x)[Im(χ(x)) +W(x)]ϕl(x)dx, where eigenfunctions ϕj(x) are solutions of equations ∆⊥ϕj(x) =k[2kjz−kRe(χ(x))]ϕj(x). Statistically avaraged refraction and absorption are acco unted for by normalized term χ(x, z) = (< ε(x)>−ε0)/ε0. It can be shown for lower channeled modes that incoherent sca ttering attenuation coeﬃ- cient is proportional to σ(see discussion above about dependence of W(x) onσ) βscatter∼k2(ε0−ε1)2σ/integraldisplay∞ −∞dz′/integraldisplay0 −∞exp(−ξ2/2)dξ/integraldisplay −R(z′)ξ (1−R2(z′))1/2 0exp(−η/22)dη. The proportionality of losses of beam intensity to roughnes s amplitude σunder su- permall gliding angles was obtained also in the numerical si mulation results ([14], Fig. 5). Results The dependence of attenuation coeﬃcients βof X-ray beam on the channel width d between silicon plates for three lower modes were shown in[1 5] and demonsrate β∼1/d3 5dependence. Such dependence accounts for decreasing of diﬀ ractional eﬀects with beam width∼λ/d2and the eﬀective portion of the beam that interacts with the s urface ∼σ/d. When lead plates were taken into consideration instead of si licon the value of attenuation coeﬃcients became 1.5 times greater. Increasing of βwith decreasing of energy is stronger than∼1/Ethat can be accounted for by incresing of diﬀraction along wi th increasing of optical density of channel walls. Recently published experiments with Cr/C/Cr channel with l ength L= 3mmand width d= 1620 ˚Aof carbon layer [2] had shown nonmonotonous energy dependen ce of transmision for ’0’ wave mode (Fig.2, rombs). As it was suppo sed [2] roughness of the interfaces couldnot exceed ∼10˚A. 00.20.40.60.81 10 12 14 16 18 20 22 24 26T E, keV12 3 333 333 Fig.2. Calculated dependence of basic ’0’ wave mode transmission Tin Cr/C/Cr channel on X-ray energy. L= 3mm,d= 1620 ˚A. Defor- mation amplitude a= 120 ˚A, period Λ = 100 µm(curve 1), 500 µm (2), 1000 µm(3).σ= 0˚A. Experimental points of W. Jark et al [2] are shown by rombs. Direct numerical simulation of the transmission of X-ray be am with equation (1) was 6developed to investigate the dependence of ’0’ and ’1’ modes transmission on roughness amplitude. The account for roughness decreases transmissi on of the basic mode with E= 17keVby 1.3 % for σ= 10˚Aand by 5 % for σ= 20˚A(see Fig.3 ) that cannot explain prominent depression of experimental results on Fi g.2. 00.20.40.60.81 0 5 10 15 20T roughness σ,˚A′0′ ′1′ Fig.3 Dependence of transmission of 17 keV X-ray beam in the chan- nel Cr/C/Cr width C layer d= 1620 ˚Afor modes ’0’ and ’1’ on roughness σ,zcorr= 5µm. For the expanation of anomalous dependence of 17keV radiati on basic mode trans- mision through Cr/C/Cr channels periodic deformation of th e layers were taken into acount. The results are shown on Fig.2 for deformation ampli tudea= 120 ˚Aand periods Λ = 100 µm(curve 1), 500 µm(2) and 1000 µm(3). The dependence of transmission on deformation period Λ for E= 17keV a = 120 ˚A and without roughness (the eﬀect of roughness was not import ant; see Fig.3 above) is shown on Fig.4. Several resonanses can be recognised in shor t Λ region. So the results shown on Fig.2 and Fig.4 are similar to the complicated eﬀect s of strong wave function transformation of channeled electrons in superlattices [9 ]. 7Thus the depression of the transmission for E= 17keVon Fig.2 observed in [2] can be result of the periodic corrugation of Cr/C interface and w ave mode interference. To clear out the mechanism of decay of x-ray beam in thin ﬁlm wa veguide with periodic perpurbations both decay of total intensity and basic mode ” 0” intensity on the distance was investigated for diﬀerent periods Λ. In the case of small scale perturbations (Λ ≤45µm) basic mode intensity decreases nearly the same as the whole beam. And in the case of resonant p erturbation Λ = 45 µm the intensity of basic mode is subjected to strong oscillati ons with the period Λ /2, de- creasing at the distance z= 3000 ˚Ato 0.03 part of the initial value. In the nonresonance case Λ = 40 µmthe basic mode ”0” oscilations are substantial only near the entrance to the carbon channel. Intensity at the distance z= 3000 ˚Aon exit of the channel decreases to 0.6 of the initial value. For the period (Λ = 1000 µm) the dependence of total intensity and basic mode ”0” intensity are shown on Fig.5 with curves 2 (points) and 2′(solid). Curves 1 and 1′corre- spond to the direct channel. Pulsations on the curve 1′are due to calcuiation uncertainties. From the Fig.5 it is seen that in the case of large scale pertur bations the decreasing of total intensity slightly diﬀeres from the streight channel . But decreasing of basic mode having the oscilation manner with the period Λ /2, may reach nearly 0.1 of the initial value. That is why the inﬂuence of large scale perturbations must result in substantial increasing of angular spread of the beam at the exit of the cha nnel. Discussion The investigations developed show strong inﬂuence of defor mations of the channel on the transmission of x-ray channeled beams. Small scale rand om perturbations of the sur- face with the roughness amplitude up to 20 ˚Ado not eﬀect considerably the transmission 8of X-rays in comparison with diﬀraction eﬀects that determi ne the decay of intensity in the channel in the case. The X-ray transmission is the most se nsitive to the resonant periodical perturbations of the channel corresponding to t he pendulum oscillations of modes ”0” and ”1”. In this case nearly complete dempening of t he beam due to transfer of basic mode ”0” to upper modes which decay rapidly[8, 15]. I n the case of large periods of deformations of the channel eﬀective transfer of the beam to the higher modes takes place but it do not succed in substancial change of total inte nsity. It is worth noting that the eﬀect of abnormal energy dependence of transmission of t he beam through Cr/C/Cr channel that was observed in [2] dissapeared after the techn ology of production of X-ray waveguides was improved [16] that can serve as the conﬁrmati on of the results of the present work. The results of the present work can be used for creation of new type of tunable X-ray ﬁlters for formation of thin beams of synchrotron X-ray radi ation. References [1]C. Riekel , Report Progress Phys., 63(2000), 232. [2]W. Jark, S. Di Fonzo, G. Soullie, A. Cedola, S. Lagomarsino , J. Alloys and Compounds, 286(1999), 9-13. [3]S.V. Kukhlevsky, G. Lubkovics, K. Negrea, L. Kozma , Pure Appl. Opt., 6 (1999), 97. [4]S.V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, Zs. Kozma, A. Rit- ucci, L. Palladino, A. Reale, G. Tomassetti , X-ray spectrometry, 29(2000), 0000. 9[5]V.L. Kantsyrev, R. Bruch, M. Bailey, A. Shlaptseva , Applied Phys. Lett., 66, n.26 (1995), 3567. [6]S.S. Fanchenko, A.A. Nefedov , phys. stat. solidi (b), 212/1 (1999), R3. [7]A.V. Vinogradov, N.N. Zorev, I.V. Kozhevnikov, I.G. Yakushki n, Sov. Phys. JETP, 62(1985), 1225. [8]T.A. Bobrova, L.I. Ognev , JEPT Letters, 69(1999), 734. [9]T.A. Bobrova, L.I. Ognev , phys. stat. sol. (b), 203/2 (1997), R11. [10]K. Tsuji, T. Yamada, H. Hirokava , J. Applied Phys., 78(1995), 969. [11]B.L. Henke, E.M. Gullikson, J.C. Davis , Atomic Data and Nuclear Data Tables, 54, no. 2 (1993), 181-342. (Excess http://www-cxro.lbl.gov/optical constants/). [12]V. Hol ´y, K.T.Gabrielyan , phys. stat. sol. (b), 140(1987), 39. [13]L.I. Ognev , Radiation Eﬀects and Defects in Solids, 25(1993), 81. [14]T.A. Bobrova, L.I. Ognev , Preprint IAE-6051/11, Moscow, 1997 (in Russian; English translation can be obtained from http://xxx.itep. ru/abs/physics/9807033). [15]L.I. Ognev , Technical Phys. Lett., 26(2000), 67-69. [16]W. Jark , private communication. 1000.20.40.60.81 0 50 100 150 200 250 300 350 400 450 500T Λ, µm Fig. 4 The dependence of basic ’0’ wave mode transmission TofE= 17keVX-ray beam in Cr/C/Cr d= 1620 ˚Achannel on the defor- mation period Λ. σ= 0˚A,L= 3mm,a= 120 ˚A. 1100.20.40.60.81 0 500 1000 1500 2000 2500 3000T z, µm1′1 2′2 Λ = 1000 µm↑ ↓ւ ր Fig. 5 The evolution of total intensity (curves 1 and 2, point) and b asic mode ”0” (curves 1′and 2′) of X-ray beam with energy E= 17keV in the channel Cr/C/Cr, d= 1620 ˚Afor direct channel (1 and 1′) and under deformations with the period Λ = 1000 µm(2 and 2′). σ= 0˚A,L= 3mm,a= 120 ˚A. The initial beam corresponds to the basic waveguide mode. 12
arXiv:physics/0009018v1 [physics.atom-ph] 5 Sep 2000Two-time Green function method in quantum electrodynamics of high-Z few-electron atoms V. M. Shabaev Department of Physics, St. Petersburg State University, Oulianovskaya Street 1, Petrodvorets, 198904 St. Petersbu rg, Russia The two-time Green function method in quantum electrodynam ics of high-Z few-electron atoms is described in detail. This method provides a simple proced ure for deriving formal expressions for the energy shift of a single level and for the energies and wav e functions of degenerate and quasi- degenerate states. It also allows one to derive formal expre ssions for the transition and scattering amplitudes. Application of the method to resonance scatter ing processes yields a systematic theory for the spectral line shape. The practical ability of the met hod is demonstrated by deriving the formal expressions for the QED and interelectronic-interaction c orrections to energy levels and transition and scattering amplitudes in one-, two-, and three-electro n atoms. The resonance scattering of a photon by a one-electron atom is also considered. PACS number(s): 12.20.-m, 12.20.Ds, 31.30. Jv 1Contents I Introduction 3 II Energy levels of atomic systems 3 A 2N-time Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 B Two-time Green function and its analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 7 C Energy shift of a single level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 D Perturbation theory for degenerate and quasidegenerate l evels . . . . . . . . . . . . . . . . . . . . . 12 E Practical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1 Zeroth order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 One-electron atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Atom with one electron over closed shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Two-electron atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Two-photon exchange diagrams for the ground state of a heli umlike atom . . . . . . . . . . . . . 21 6 Qusidegenerate states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7 Nuclear recoil corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 III Transition probabilities and cross sections of scatter ing processes 30 A Photon emission by an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 B Transition probability in a one-electron atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 Zeroth order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 QED corrections of ﬁrst order in α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 C Radiative recombination of an electron with an atom . . . . . . . . . . . . . . . . . . . . . . . . . . 37 D Radiative recombination of an electron with a high- Zhydrogenlike atom . . . . . . . . . . . . . . . 40 1 Zeroth order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 Interelectronic-interaction corrections of ﬁrst order i n 1/Z . . . . . . . . . . . . . . . . . . . . . . 41 E Photon scattering on an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 F Resonance scattering: Spectral line shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 G Resonance photon scattering on a one-electron atom . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 IV Conclusion 49 APPENDIXES 50 A 50 B 50 C 52 D 53 E 53 2I. INTRODUCTION A great progress in experimental investigations of high-Z f ew-electron systems (see, e.g., [1]) stimulated theorists to perform accurate QED calculations for these systems. The calculations of the QED and interelectronic-interaction corrections in high-Z few-electron systems are convenient ly divided in two stages. The ﬁrst stage consists in deriving formal expressions for these corrections from the ﬁrst prin ciples of QED. The second one consists in numerical evaluations of these expressions. The present paper will be completely focused on the ﬁrst stage. As to the numerical evalutions of the QED corrections, they are recently review ed in [2–6]. Historically, the ﬁrst method suitable for derivation of th e formal expressions for the energy shift of a bound state level was formulated by Gell-Mann, Low, and Sucher [7,8]. Th is method is based on introducing an adiabatically damped factor, exp ( −λ\|t\|), in the interaction Hamiltonian and expressing the energy shift in terms of so-called adiabaticSλmatrix elements. Due to its simple formulation, the Gell-Ma nn–Low–Sucher formula for the energy shift gained wide spreading in the literature related to high-Z fe w-electron systems [9–16]. However, the practical use of th is method showed that it has several serious drawbacks. One of t hem consists in strong complication of the derivation of the formal expressions for so-called reducible diagrams. By the reducible diagrams we mean here the diagram s in which an intermediate state energy of the atom coincides wit h the reference state energy (this terminology is quite natural since it can be considered as an extension of the deﬁn itions introduced by Dyson [17] and by Bethe and Salpeter [18] to high-Z few-electron atoms). As to irreducuble diagrams, i.e. the diagrams in which the intermediate state energies diﬀer from the reference state energy, the de rivation of the formal expressions for them can easily be reduced to the usual ( λ= 0)S-matrix elements in every method, including the Gell-Mann– Low–Sucher method as well (see, e.g., [11,15]), and, therefore, causes no proble m. Another serious drawback of the Gell-Mann–Low–Sucher method consists in the fact that this method needs special st udying the renormalization procedure since the adibatic Sλ-matrix is suﬀered from the ultraviolet divergences. Due to noncovariantness of the adiabatically damped factor, the ultraviolet divergences can not be removed from Sλifλ/ne}ationslash= 0. However, from the physical point of view one may expect that these divergenes cancel each other in the expres sion for the energy shift and, therefore, may be disregarded in the calculation of the energy shift of a single level. But f or the case of degenerate levels, this problem becomes very urgent since we can not expect that the standard renorma lization procedure makes the secular operator to be ﬁnite in the ultraviolet limit [10,12]. In addition, we shou ld note that at present there is no formalism based on the Gell-Mann–Low–Sucher approach which would provide treatm ent of quasidegenerate levels. To date, no formalism in the framework of this approach was developed for calculatio n of the transition or scattering amplitudes. The same problems refer to the evolution operator method dev eloped in [19–23]. Another way to formulate the perturbation theory for high-Z few-electron systems consists in using Green’s func- tions. These functions contain the complete information ab out the energy levels and the transition and scattering amplitudes. The renormalization problem does not appear in this way since Green’s functions can be renormalized from the very beginning (see, e.g., [24]). To date, various v ersions of the Green function formalism were developed which diﬀer from each other by methods of extracting the phys ical information (the energy levels and the transition and scattering amplitudes) from Green’s functions. One of t hese methods was worked out in [25–29] and was success- fully employed in many practical calculations [30–41]. Sin ce one of the key elements of this methods consists in using two-time Green’s functions, in what follows we will call it a s the two-time Green function (TTGF) method. This method, which provided solving all the problems appeared in the other formalisms indicated above, will be considered in detail in the present paper. As to other versions of the Green function method [12,43–51, 16], a detailed discussion of them would be beyond the scope of the present paper. We note only that some of these met hods are also based on employing two-time Green function’s but yield other forms of the perturbation theory . So, in [12,44–47] the two-time Green functions were used for constructing quasipotential equations for high-Z few- electron systems. It corresponds to the perturbation theor y in the Brillouin-Wigner form while the method of Refs. [25–29] yields the perturbation theory in the Rayleigh-Schr¨ oding er form. Various versions of the Bethe-Salpeter equation deri ved from the 2 N-time Green function formalism for high-Z few-electron systems can be found in [12,48]. In [49,50] the perturbation theory in the Rayleigh-Schr¨ odinger form is constructed in the case of a one-electron system where the pr oblem of relative electron times is absent. The relativistic unit system ( ¯ h=c= 1 ) and the Heaviside charge unit ( α=e2 4π,e<0) are used in the paper. II. ENERGY LEVELS OF ATOMIC SYSTEMS In this section we formulate the perturbation theory for cal culation of the energy levels in high-Z few-electron atoms. In these systems the number of the electrons, which we denote byN, is much smaller than the nuclear charge numberZ. It follows that the interaction of the electrons with each o ther and with the quantized electromagnetic 3ﬁeld is much smaller (by factors 1 /Zandα, respectively) than the interaction of the electrons with t he Coulomb ﬁeld of the nucleus. Therefore, it is natural to assume that in zer oth approximation the electrons interact only with the Coulomb ﬁeld of the nucleus and obey the Dirac equation (−iα·∇+βm+VC(x))ψn(x) =εnψn(x). (2.1) The interaction of the electrons with each other and with the quantized electromagnetic ﬁeld is accounted by pertur- bation theory. In this way we get the quantum electrodynamic s in the Furry picture. It should be noted that we could start also with the Dirac equation with an eﬀective potentia lVeﬀ(x) which describes approximately the interaction with the other electrons. In this case the interaction with t he potential δV(x) =VC(x)−Veﬀ(x) must be accounted for perturbatively. Using the eﬀective potential provides an extention of the theory to many-electron atoms where, for instance, a local version of the Hartree-Fock potential can be used as Veﬀ(x). However, for simplicity, in what follows we will assume that in zeroth approximation the elec trons interact only with the Coulomb ﬁeld of the nucleus. In the present paper we will mainly consider the perturbatio n theory with the standard QED vacuum. The transition to the formalism in which the role of the vacuum is played by cl osed shells is realized by replacing i0 with −i0 in the electron propagator denominators corresponding to the clo sed shells. Before to formulate the perturbation theory for calculatio ns of the interelectronic interaction and radiative correc - tions to the energy levels, we consider standard equations o f the Green function approach in quantum electrodynamics. A. 2N-time Green function It can be shown that the complete information about the energ y levels of an N-electron atom is contained in Green’s function deﬁned as G(x′ 1,...x′ N;x1,...x N) =/an}b∇acketle{t0\|Tψ(x′ 1)···ψ(x′ N)ψ(xN)···ψ(x1)\|0/an}b∇acket∇i}ht, (2.2) whereψ(x) is the electron-positron ﬁeld operator in the Heisenberg r epresentation, ψ(x) =ψ†γ0, andTis the time- ordered product operator. The basic equations of the quantu m electrodynamics in the Heisenberg representation are summarized in Appendix A. The equation (2.2) presents a stan dard deﬁnition of 2 N-time Green’s function which is a fundamental object of the quantum electrodynamics. It can be shown (see, e.g., [24,52]) that in the interaction representation the Green function is given by G(x′ 1,...x′ N;x1,...x N) =/an}b∇acketle{t0\|Tψin(x′ 1)···ψin(x′ N)ψin(xN)···ψin(x1)exp{−i/integraltext d4zHI(z)}\|0/an}b∇acket∇i}ht /an}b∇acketle{t0\|Texp{−i/integraltext d4zHI(z)}\|0/an}b∇acket∇i}ht(2.3) =/braceleftBig∞/summationdisplay m=0(−i)m m!/integraldisplay d4y1···d4ym/an}b∇acketle{t0\|Tψin(x′ 1)···ψin(x′ N)ψin(xN)···ψin(x1) ×HI(y1)···H I(ym)\|0/an}b∇acket∇i}ht/bracerightBig/braceleftBig∞/summationdisplay l=0(−i)l l!/integraldisplay d4z1···d4zl/an}b∇acketle{t0\|HI(z1)···H I(zl)\|0/an}b∇acket∇i}ht/bracerightBig−1 (2.4) where HI(x) =e 2[ψin(x)γµ,ψin(x)]Aµ in(x)−δm 2[ψin(x),ψin(x)] (2.5) is the interaction Hamiltonian. Commutators in equation (2 .5) refer to operators only. The ﬁrst term in (2.5) describes the interaction of the electron-positron ﬁeld with the quan tized electromagnetic ﬁeld and the second one is the mass renormalization counterterm. We consider here that the int eraction of the electrons with the Coulomb ﬁeld of the nucleus is included into the unperturbed Hamiltonian (the F urry picture). However, there is also an alternative method to get the Furry picture. In this method the interacti on with the Coulomb ﬁeld of the nucleus is included in the interaction Hamiltonion and the Furry picture is obtain ed by summing inﬁnite sequences of Feynman diagrams describing the interaction of electrons with the Coulomb po tential. As a result of this summation, the free-electron propagators are replaced by the bound-electron propagator s. This method is very convenient for studying processes involving continuum-electron states and will be used in the section concerning the radiative recombination process. The Green function Gis constructed by perturbation theory according to equatio n (2.4). This is carried out with the aid of Wick’s theorem (see, e.g., [24]). According to thi s theorem the time-ordered product of ﬁeld operators is equal to the sum of normal-ordered products with all possibl e contractions between the operators 4T(ABCD ···) =N(ABCD ···) +N(AaBaCD···) +N(AaBCaD···) + all possible contractions , (2.6) whereNis the normal-ordered product operator and the superscript s denote the contraction between the correspond- ing operators. The contraction between neighbouring opera tors is deﬁned by AaBa=T(AB)−N(AB) =/an}b∇acketle{t0\|T(AB)\|0/an}b∇acket∇i}ht. (2.7) If the contracted operators are boson’s operators they can b e put one next to another. If the contracted operators are fermion’s operators they also can be put one next to another b ut in this case the expression must be multiplied with the parity of the permutation of fermionic operators. Since in the Green function the vacuum expectation value is calculated, only the term with all contracted operators sur vives on the right-hand side of equation (2.6). In contrast to the free-electron QED, in the Furry picture the time-orde red product of two fermions operators must be deﬁned also for the equal-time case to produce the correct vacuum po larization terms. As was noticed in [53], the deﬁnition T[A(t)B(t)] =1 2A(t)B(t)−1 2B(t)A(t) (2.8) provides the following simple rule for dealing with the inte raction operator. It can be written as HI(x) =eψin(x)γµψin(x)Aµ in(x)−δmψin(x)ψin(x) (2.9) and then Wick’s theorem is applied with contractions betwee n all operators, including equal-time operators. We note that the problem of the deﬁnition of the time-ordered produc t of fermion operators at equal times does not appear at all if the alternative method for producing the Furry pict ure discussed above is employed. The contractions between the electron-positron ﬁelds and b etween the photon ﬁelds lead to the following propagators /an}b∇acketle{t0\|Tψin(x)ψin(y)\|0/an}b∇acket∇i}ht=i 2π/integraldisplay∞ −∞dω/summationdisplay nψn(x)ψn(y) ω−εn(1−i0)exp[−iω(x0−y0)] (2.10) and /an}b∇acketle{t0\|TAµ in(x)Aν in(y)\|0/an}b∇acket∇i}ht=−igµν/integraldisplayd4k (2π)4exp [−ik·(x−y)] k2+i0. (2.11) Here the Feynman gauge is considered. In equation (2.10) the labelnruns over all bound and continuum states. The denominator in equation (2.3) describes unobservable v acuum-vacuum transitions and, as one can show (see, e.g., [24]), it cancels disconnected vacuum-vacuum subdia grams from the nominator. Therefore, we can simply omit all diagrams containing disconnected vacuum-vacuum subdi agrams in the nominator and replace the denominator by 1. In practical calculations of the Green function it is conven ient to work with the Fourier transform with respect to time variables G((p′0 1,x′ 1),...,(p′0 N,x′ N); (p0 1,x1),...,(p0 N,xN)) = (2π)−2N/integraldisplay∞ −∞dx0 1···dx0 Ndx′0 1···dx′0 N ×exp(ip′0 1x′0 1+···+ip′0 Nx′0 N−ip0 1x0 1− ··· −ip0 Nx0 N) ×G(x′ 1,...,x′ N;x1,...,x N). (2.12) For the Green function G((p′0 1,x′ 1),...,(p′0 N,x′ N); (p0 1,x1),...,(p0 N,xN)) the following Feynman rules can be derived: (1) External electron line x y✛i 2πS(ω,x,y), where S(ω,x,y) =/summationdisplay nψn(x)ψn(y) ω−εn(1−i0), (2.13) 5ψn(x) are solutions of the Dirac equation (2.1). (2) Internal electron line x y✛i 2π/integraltext∞ −∞dω S(ω,x,y) . (3) Disconnected electron line x y✛i 2πS(ω,x,y)δ(ω−ω′). (4) Internal photon line ✂✁✂✁✂✁✂✁✂✁✄
arXiv:physics/0009019v1 [physics.data-an] 5 Sep 20001 A Generalization of the Maximum Noise Fraction Transform Christopher Gordon Abstract —A generalization of the maximum noise fraction (MNF) transform is proposed. Powers of each band are included as new bands before the MNF transform is per- formed. The generalized MNF (GMNF) is shown to per- form better than the MNF on a time dependent airborne electromagnetic (AEM) data ﬁltering problem.1 Keywords — Maximum noise fraction transform, noise ﬁltering, time dependent airborne electromagnetic data. I. Introduction THE maximum noise fraction (MNF) transform was in- troduced by Green et al.[1]. It is similar to the principle component transform [2] in that it consists of a linear trans - form of the original data. However, the MNF transform orders the bands in terms of noise fraction. One application of the MNF transform is noise ﬁltering of multivariate data [1]. The data is MNF transformed, the high noise fraction bands are ﬁltered and then the reverse transform is performed. We show an example where the MNF noise removal adds artiﬁcial features due to the nonlinear relationship betwe en the diﬀerent variables of the data. A polynomial generaliza - tion of the MNF is introduced which removes this problem. In Section II we summarize the MNF procedure. The problem data set is introduced in Section III and the MNF is applied to it. In Section IV, the generalized MNF trans- form is explained and applied. The conclusions are given in Section V. II. The Maximum Noise Fraction (MNF) Transform In this section we deﬁne the MNF transform and list some of its properties. For further details the reader is referre d to Green et al. [1] and Switzer and Green [3]. A good re- view is also given by Nielsen [4]. A reformulation of the MNF transform as the noise-adjusted principle component (NAPC) transform was given by Lee et al.[5]. An eﬃcient method of computing the MNF transform is given by Roger [6]. Let Zi(x), i= 1, . . . , p be a multivariate data set with pbands and with xgiving the position of the sample. The means of Zi(x)are assumed to be zero. The data can always be made to approximately satisfy this assumption by subtracting the sample means. An additive noise model is assumed: Z(x) =S(x) +N(x) C. Gordon is with the School of Computer Science and Mathematics at the University of Portsmouth in the UK. E- mail:christopher.gordon@port.ac.uk 1Copyright (c) 2000 Institute of Electrical and Electron- ics Engineers. Reprinted from [IEEE Transactions on Geo- science and Remote Sensing, Jan 01, 2000, v38, n1 p2, 608]. This material is posted here with permission of the IEEE. Internal or personal use of this material is permitted. How- ever, permission to reprint/republish this material for ad - vertising or promotional purposes or for creating new col- lective works for resale or redistribution must be obtained from the IEEE by sending a blank email message to pubs- permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.where ZT(x) ={Z1(x), . . . , Z p(x)}is the corrupted signal and S(x)andN(x)are the uncorrelated signal and noise compo- nents of Z(x). The covariance matrices are related by: Cov{Z(x)}= Σ = Σ S+ ΣN where ΣNandΣSare the noise and signal covariance matri- ces. The noise fraction of the ith band is deﬁned as Var{Ni(x)}/Var{Zi(x)}. The maximum noise fraction transform (MNF) results in a newpband uncorrelated data set which is a linear transform of the original data: Y(x) =ATZ(x). The linear transform coeﬃcients, A, are found by solving the eigenvalue equation: AΣNΣ−1= ΛA (1) where Λis a diagonal matrix of the eigenvalues, λi. The noise fraction in Yi(x)is given by λi. By convention the λi are ordered so that λ1≥λ2≥. . .≥λp. Thus the MNF trans- formed data will be arranged in bands of decreasing noise fraction. The proportion of the noise variance described by the ﬁrst rMNF bands is given by /summationtextr i=1λi/summationtextp i=1λi. The eigenvectors are normed so that ATΣAis equal to an identity matrix. The advantages of the MNF transform over the PC trans- form are that it is invariant to linear transforms on the data and the MNF transformed bands are ordered by noise frac- tion. The high noise fraction bands can be ﬁltered and then the transform reversed. This can lead to an improvement in the ﬁltering results because the high noise fraction bands shou ld contain less signal that might be distorted by the ﬁltering. Examples of this approach have been given by Green et al. [1], Nielsen and Larsen [7] and Lee et al.[5]. An extreme version of MNF ﬁltering is based on excluding the eﬀects of the ﬁrst rcomponents. That is ris chosen so as to include only bands with high enough noise ratios. This can be achieved by: Z∗(x) = (A−1)TRATZ(x) (2) where Z∗(x)is the ﬁltered data and Ris an identity matrix with the ﬁrst rdiagonal elements set to zero. Thus elimi- nating the eﬀect of one or more of the MNF bands produces a ﬁltered data set which is a linear transform of the original data. This MNF based ﬁlter uses interband correlation to remove noise. In order to use Equation (1) to compute A,ΣNhas to be known. Nielsen and Larsen [7] have given four diﬀerent ways of estimating N(x). They all rely on the data being spatially correlated. A simple method for computing N(x) is by N(x) =Z(x)−Z(x+δ) (3)2 where δis an appropriately determined step length. We are eﬀectively assuming S(x) =S(x+δ). To the extent that this is not true, the estimate of N(x)is in error. When this method of noise estimation is used, the MNF transform is equivalent to the min / max autocorrelation factor transform [3]. III. Airborne Electromagnetic Data We test the MNF ﬁltering methodology on a ﬂight line produced by SPECTREM’s time dependent airborne elec- tromagnetic (AEM) system. Background information on this AEM system has been explained by Leggatt [8]. A multiband image can be formed by consecutive ﬂight lines but usually each ﬂight line is examined separately. Fig. 1 shows a ﬂight line of data, consisting of the seven windowed AEM X band spectra. All seven bands are dis- played stacked above each other. The amplitude of a band at a particular point is proportional to the vertical distan ce of the spectrum from its corresponding zero amplitude ref- erence (dotted) line. Neighbouring points along a line are responses from neighbouring points on the ground. The higher band numbers are associated with greater under- ground depths. Ore bodies are often associated with small features in the higher bands. Analysis can be made easier by ﬁltering the spectra. Because this data set has substantial interband correlation, the MNF ﬁltering methodology can be used. Fig. 2 (b) shows the MNF ﬁltering of the spectra in Fig. 1. Only the last three bands (i.e. 5, 6 and 7) and a portion of the ﬂight line are shown. The noise was estimated by taking the diﬀerence in neighboring pixels, as in Equation (3). The data were ﬁltered by excluding the ﬁrst two MNF bands which accounted for approximately 86% of the noise fraction. Although the noise has been reduced, spurious features have been added, indicated by ‘S’. Excluding only the last MNF component does not signiﬁcantly reduce the magnitude of the spurious features and does almost no noise reduction. As seen in Equation (2), the MNF ﬁltered data is com- posed from a linear function of the original data. Fig. 3 shows a plot of Z1(x)against RZ1(x), where RZ1(x)is the diﬀerence between Z1(x)and a least squares regression of Z1(x)based on all the other bands. The clear pattern of the residuals plotted in Fig. 3 is evidence that the relationshi p between Z1(x)and the other bands is not linear. Similar patterned residuals were found for residual plots based on the other bands. In the next section we show how the linear assumption can be relaxed. IV. The Generalized Maximum Noise Fraction Transform From the discussion in the previous section it appears that using a linear ﬁlter is too restrictive for this data set . Gnanadesikan [9] proposed a generalization of the principl e component transform. Powers of the original bands were appended to the data set as new bands. For example, pnew bands can be created by appending the square of each band to the original data set. Thus each generalized principle component would be a polynomial, as opposed to linear, function of all the bands in the original data set. The same procedure can be applied to generalize the MNF transform. More formally, a new data set, Z′(x), can be created by appending up to qpowers of the original data set: Z′(x) = {Z1(x), Z2(x), . . . , Z p(x), Z2 1(x), Z2 2(x), . . . , Z2 p(x), . . . , Z1(x)q, . . . , Zq p(x)}. We are assuming that the Zi(x)have zero means. Cross terms, such as Z1(x)Z2(x)can also be appended. The rest of the methodology remains unchanged.From Equation (2), each band of the generalized MNF (GMNF) ﬁltered data can be seen to be, Z∗ i(x) =p/summationdisplay j=1q/summationdisplay k=1Fi,j+(k−1)pZk j(x) where Fi,j+(k−1)pis the element in row iand column j+(k−1)p of the ﬁlter matrix: F= (A−1)TRAT Thus the GMNF transform leads to a polynomial ﬁlter. To apply the GMNF ﬁlter to the data in Fig. 1, the GMNF transform was applied with powers of up to order 6 for each band appended to the original data. Cross terms were found to make little diﬀerence to the result and so were not included. The ﬁrst 15 of the 42 GMNF components, contributing approximately 80% of the noise fraction, were eliminated. Fig. 2(c) shows the GMNF ﬁltered AEM data. A com- parison with the MNF ﬁltered data (Fig. 2(b)) shows that for GMNF ﬁltered data , the noise reduction is greater and spurious features are much less evident. V. Conclusion We have proposed a generalized maximum noise fraction transform (GMNF) that is a polynomial as opposed to linear transform. The GMNF was applied to ﬁltering a test AEM data set. It was found to remove more noise while adding less artiﬁcial features than the MNF based ﬁlter. Implementing the GMNF is a simple extension of the MNF implementation. Software written for the MNF trans- form can be be used for the GMNF transform without any modiﬁcation.3 References [1] A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A tran s- formation for ordering multispectral data in terms of image qual- ity with implications for noise removal,” IEEE Transactions on Geoscience and Remote Sensing , vol. 26, no. 1, pp. 65–74, 1988. [2] Rafael C. Gonzalez and Richard E. Woods, Digital Image Pro- cessing , Addison-Wesley publishing company, 1992. [3] P. Switzer and A. Green, “Min/max autocorrelation facto rs for multivariate spatial imagery,” Tech. Rep. 6, Department of Statis- tics, Standford University, 1984. [4] Alan Aasbjerg Nielsen, Analysis of Regularly and Irregularly Sampled Spatial, Multivariate, and Multi-temporal Data , Ph.D. thesis, Institute of Mathematical Modelling. University o f Den- mark, 1994. [5] J. B. Lee, A. S. Woodyatt, and M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-ad justed principal components transform,” IEEE Transactions on Geo- science and Remote Sensing , vol. 28, no. 3, pp. 295–304, 1990. [6] R. E. Roger, “A faster way to compute the noise-adjusted p rinci- pal components transform matrix.,” IEEE Transactions on Geo- science and Remote Sensing , vol. 32, no. 6, 1994. [7] Alan Aasbjerg Nielsen and R. Larsen, “Restoration of GER IS data using the maximum noise fractions transform,” in Proceed- ings form the First International Airborne Remote Sensing C on- ference and Exhibition, Volume II , Strasbourg, France, 1994, pp. 557–568. [8] Peter Bethune Leggatt, Some Algorithms and Code for the Com- putation of the Step Response Secondary EMF Signal for the SPECTREM AEM System , Ph.D. thesis, University of the Wit- watersrand, Johannesburg, South Africa, 1996. [9] R. Gnanadesikan and M. B. Wilk, “Data analytic methods in mul- tivariate statistical analysis,” in Multivariate Analysis II , P. R. Krishnaiah, Ed. 1969, pp. 593–638, Academic Press. New York , U. S. A. 0 500 1000 1500 SampleBand1 1 2 3 4 5 6 7 Fig. 1 Unfiltered AEM data. Bands 1 to 7 are shown. The band number of each spectrum is labelled to the left of the spectrum. The dotted line of each spectrum marks the zero amplitude for that spectrum.4 0 500 1000 (c)Band5 5 6 7GMNF0 500 1000 (b)Band5 5 6 7MNF S SS S0 500 1000 (a)Band5 5 6 7RAW Fig. 2 A comparison of the MNF and GMNF filtering methods. Only a portion of the flight line for bands 5, 6 and 7 is shown for each figure. The sample number is displayed on the horizontal axis of each subplot. (a) Unfiltered AEM data. (b) MNF filtered AEM data. The ‘S’ symbols mark parts of the data where spurious features have been introduced by the MNF filtering. (c) GMNF filtered AEM data.0 100 200 300 400 500 Band 1-80-4004080Band 1 residual Fig. 3 A plot of the residual of a linear regression of band 1 based on bands 2 to 7, versus band 1 values.
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/CT/D0/D3/D4 /CT /CS/D9/D6/CX/D2/CV /D3/D2/CT /D6/D3/D9/D2/CS/D8/D6/CX/D4 /CX/D7/D7/D1/CP/D0/D0/B8 /D8/CW/CT /D0/CP/D7/CT/D6 /CS/DD/D2/CP/D1/CX /D7 /CP/D2 /CQ /CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /CP /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV /D8 /DD/D4 /CT /CJ/BD/BC/B8/BD/BL℄/BM a(k,t) k=/braceleftBig α−l+ (1 +id)∂2/∂t2+ (σ−iβ)\|a\|2/bracerightBig a(k,t). /B4/BD/B5/C0/CT/D6/CTa(k,t) /CX/D7 /D8/CW/CT /CT/D0/CT /D8/D6/CX /AS/CT/D0/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8 k /CX/D7 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4 /D2 /D9/D1 /CQ /CT/D6/B8 t /CX/D7 /D8/CW/CT /D0/D3 /CP/D0 /D8/CX/D1/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CV/CP/CX/D2 /CQ/CP/D2/CS/DB/CX/CS/D8/CW tg /DB/CW/CX /CW /CU/D3/D6/CC/CX/BM/D7/CP/D4/D4/CW/CX/D6/CT /CX/D7 /CP/CQ /D3/D9/D8 2.5fs /B8α /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8/B8l /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6/D0/D3/D7/D7/B8 /CP/D2/CSd /CX/D7 /D8/CW/CT /BZ/BW/BW /D3 /CTꜶ /CX/CT/D2 /D8 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3t2 g /BA /CC/CW/CT /CU/CP /D8/D3/D6 /CQ /CT/CU/D3/D6/CT /D8/CW/CT/D7/CT /D3/D2/CS /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/B8(1 +id) /B8 /D8/CP/CZ /CT/D7 /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /BZ/BW/BW /CP/D2/CS /D8/CW/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW/D0/CX/D1/CX/D8/CX/D2/CV /CT/AR/CT /D8 /D3/CU /D8/CW/CT /CP /DA/CX/D8 /DD /BA /BY /D3/D6 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /DB /CT /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8 /D8/CW/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW /D3/CU/D8/CW/CT /CV/CP/CX/D2 /D1/CT/CS/CX/D9/D1 /D4/D0/CP /DD/D7 /D8/CW/CT /CS/D3/D1/CX/D2/CP/D2 /D8 /D6/D3/D0/CT/BA /CC/CW/CT /D0/CP/D7/D8 /D7/D9/D1/D1/CP/D2/CS /D3/D2/D7/CX/D7/D8/D7 /D3/CU /D8 /DB /D3/D4/CP/D6/D8/D7/B8 /CP /CU/CP/D7/D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /B4/C3/CT/D6/D6 /D0/CT/D2/D7/CX/D2/CV/B5 /D8/CT/D6/D1σ\|a\|2/CP/D2/CS /CP /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/D8/CT/D6/D1 −iβ\|a\|2/BA /BY /D3/D6 /CQ /D3/D8/CW /D8/CT/D6/D1/D7 /DB /CT /CP/D7/D7/D9/D1/CT/CS /D7/CP/D8/D9/D6/CP/D8/CX/D2/CV /CQ /CT/CW/CP /DA/CX/D3/D6 /CP /D3/D6/CS/CX/D2/CV/D8/D3σ=σ0(1−1 2σ0\|a0\|2) /CP/D2/CSβ=β0(1−1 2β0\|a0\|2) /B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8 /DB/CW/CT/D6/CT a0 /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /D4 /CT/CP/CZ /CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D2/CSβ0=2πzn2 λn= 1.7×10−12cm2/W /CX/D7 /D8/CW/CT/D9/D2/D7/CP/D8/D9/D6/CP/D8/CT/CS /CB/C8/C5 /D3 /CTꜶ /CX/CT/D2 /D8 /CU/D3/D6 /CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT /D6/DD/D7/D8/CP/D0/D0/BA /CC/CW/D6/D3/D9/CV/CW/D3/D9/D8 /D8/CW/CT /D4/CP/D4 /CT/D6/CQ /D3/D8/CW \|a0\|2/CP/D2/CS\|a\|2/CP/D6/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D35.9×1011W/cm2/B8 /D8/CW/CT/D2β0 /CP/D2/CSσ0 /DB/CX/D0/D0/BH/CQ /CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D31.7×10−12cm2/W /BA /CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DDσ0 /D4/D0/CP /DD/D7 /D8/CW/CT /D6/D3/D0/CT /D3/CU /CP/D2/CX/D2 /DA /CT/D6/D7/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D3/CU /D8/CW/CT /CU/CP/D7/D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /B4/C3/CT/D6/D6/B9/D0/CT/D2/D7/CX/D2/CV/B5/BA /CC/CW/CT /D0/CP/D6/CV/CT/D6σ0/D8/CW/CT /D0/CP/D6/CV/CT/D6 /CX/D7 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT /D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/BA/CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7n2 /BP1.3×10−16cm2/W /CP/D2/CSn= 1.76 /CP/D6/CT /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6/CP/D2/CS /D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/DA /CT /CX/D2/CS/CT/DC/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8z= 3mm /CX/D7 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /C3/CT/D6/D6/D1/CT/CS/CX/D9/D1/B8 /CP/D2/CSλ= 800nm /CX/D7 /D8/CW/CT /CT/D2 /D8/CT/D6 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/BA /CC/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CUσ0 /CP/D2/CQ /CT /D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD /D8/CW/CT /CP /DA/CX/D8 /DD /CP/D0/CX/CV/D2/D1/CT/D2 /D8 /CP/D2/CS /CU/D3/D6 /D8 /DD/D4/CX /CP/D0 /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT/D0/D3 /CZ /CT/CS/D0/CP/D7/CT/D6/D7σ0 /CX/D710−10−10−12cm2/W /CJ/BE/BC℄/BA /CC/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D3/CU /CB/C8/C5 /CX/D7 /CS/D9/CT/CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT /D8/D3 /D8/CW/CT /D2/CT/DC/D8/B9/CW/CX/CV/CW/CT/D6 /D3/D6/CS/CT/D6 /D8/CT/D6/D1 /CX/D2 /D8/CW/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D6/CT/CU/D6/CP /D8/CX/DA /CT/CX/D2/CS/CT/DC/B8 /D8/CW/CTn4 /B9 /D8/CT/D6/D1/BA /CC/CW/CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D8/CT/D6/D1 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2/D8/CW/CT /D4 /CT/CP/CZ /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D6/CP/D8/CW/CT/D6 /D8/CW/CT /CX/D2/D7/D8/CP/D2 /D8/CP/D2/CT/D3/D9/D7 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CX/D7 /D2/CT /CT/D7/D7/CP/D6/DD /CU/D3/D6 /D7/D3/D0/DA/CX/D2/CV/BX/D5/BA /B4/BD/B5 /CP/D2/CP/D0/DD/D8/CX /CP/D0/D0/DD /BA/CC/CW/CT /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8/B8 /CS/D9/D6/CX/D2/CV /D3/D2/CT /D6/D3/D9/D2/CS/D8/D6/CX/D4/B8 /CW/CP/D2/CV/CT/D7 /CS/D9/CT /D8/D3 /CS/CT/D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2/CQ /DD /D8/CW/CT /D0/CP/D7/CT/D6 /D4/D9/D0/D7/CT/B8 /CV/CP/CX/D2 /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /DB/CX/D8/CW /CP /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D8/CX/D1/CTT31 /B43µs /CU/D3/D6/CC/CX/BM/D7/CP/D4/D4/CW/CX/D6/CT/B5/B8 /CP/D2/CS /D8/CW/CT /D4/D9/D1/D4/CX/D2/CV /D4/D6/D3 /CT/D7/D7/BM dα dt=σ14(αm−α)Ip/hνp−σ32α\|a\|2/hν−α/T31. /B4/BE/B5/C0/CT/D6/CTσ14= 10−19cm2/CP/D2/CSσ32= 3 ×10−19cm2/CP/D6/CT /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2/CP/D2/CS /CT/D1/CX/D7/D7/CX/D3/D2 /D6/D3/D7/D7/B9/D7/CT /D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8αm /CX/D7 /D8/CW/CT/D9/D2/D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8/B8νp /CP/D2/CSν /CP/D6/CT /D8/CW/CT /D4/D9/D1/D4 /CP/D2/CS /D0/CP/D7/CT/D6 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7/B8/D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /B8 /CP/D2/CSIp /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /BA /BY /D6/D3/D1 /D8/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CV/CP/CX/D2 /CT/DA /D3/B9/D0/D9/D8/CX/D3/D2 /D3/D2/CT /CP/D2 /CS/CT/D6/CX/DA /CT /CP /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CV/CP/CX/D2 /CP/CU/D8/CT/D6 /D8/CW/CT /CZ/B7/BD /B9/D8/CW /D6/D3/D9/D2/CS/D8/D6/CX/D4/B4/D0/CT/CU/D8/B9/CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD/B5 /CP/D2/CS /D8/CW/CT /CZ /B9/D8/CW /D6/D3/D9/D2/CS/D8/D6/CX/D4/B4/D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /CT/D5/D9/CP/D8/CX/D3/D2/B5/BM α′=αexp(−2τ\|a\|2tp−Tcav T31−U)+αmU (U+Tcav/T31)[1−exp(−Tcav T31−U)], /B4/BF/B5/DB/CW/CT/D6/CTtp /CX/D7 /D8/CW/CT /D0/CP/D7/CT/D6 /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3tg /CP/D2/CSTcav /B410ns /B5 /CX/D7/D8/CW/CT /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS/D8/D6/CX/D4 /D8/CX/D1/CT/BA /CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /B8τ−1/CX/D7 /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD/AT/D9/CT/D2 /DDEg /B40.82J/cm2/B5 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3λntg 2πzn2 /B41.5×10−3J/cm2/B5 /CU/D3/D6 /DB/CW/CX /CW/DB /CT /D3/CQ/D8/CP/CX/D2 /CP /DA /CP/D0/D9/CT /D3/CU555 /D8/CW/CP/D8 /DB /CP/D7 /D9/D7/CT/CS /CX/D2 /D3/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7/BA U=σ14Tcav hνpIp /CX/D7 /CP/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D4/D9/D1/D4 /D4/CP/D6/CP/D1/CT/D8/CT/D6/B8 /DB/CW/CX /CW /DB /CT /CW/D3/D7/CT /D0/D3/D7/CT /D8/D34×10−4/B4/CU/D3/D6 /CP /D4/D9/D1/D4/D0/CP/D7/CT/D6 /D7/D4 /D3/D8 /D7/CX/DE/CT /D3/CU100µm /D8/CW/CX/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP/CQ /D3/D9/D8 4W /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/B5/BA /CF/CW/CT/D2/CP/D2/CP/D0/DD/DE/CX/D2/CV /D7/D8/CP/D8/CX/D3/D2/CP/D6/DD /D6/CT/CV/CX/D1/CT/D7 /DB/CW/CT/D6/CT /D8/CW/CT /D4/D9/D0/D7/CT/D7 /D6/CT/D4/D6/D3 /CS/D9 /CT /CP/CU/D8/CT/D6 /CT/CP /CW /D6/D3/D9/D2/CS/D8/D6/CX/D4/CP /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8 /CX/D7 /D9/D7/CT/CS /D8/CW/CP/D8 /CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D7/CT/D8/D8/CX/D2/CVα=α′/CX/D2 /BX/D5/BA /B4 /BF /B5/BA /C1/D2 /D8/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D2/D3/D2/D7/D8/CP/D8/CX/D3/D2/CP/D6/DD /D6/CT/CV/CX/D1/CT/D7 /D8/CW/CT /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8 /CW/CP/D2/CV/CT/D7 /CU/D6/D3/D1 /D6/D3/D9/D2/CS/D8/D6/CX/D4 /D8/D3 /D6/D3/D9/D2/CS/D8/D6/CX/D4 /CP/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /BX/D5/BA/B4 /BF /B5/BA /BT/D7 /CX/D7 /CZ/D2/D3 /DB/D2/B8/D7/CT/CT /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT /CJ/BD/BE℄/B8 /BX/D5/BA /B4/BD/B5 /CW/CP/D7 /CP /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D3/D6/D1/BIa(t) =a0exp(iφ)/cosh1+iψ(t/tp), /B4/BG/B5/DB/CW/CT/D6/CTψ /CX/D7 /D8/CW/CT /CW/CX/D6/D4 /D8/CT/D6/D1/B8 /CP/D2/CSφ /CX/D7 /D8/CW/CT /D3/D2/D7/D8/CP/D2 /D8 /D4/CW/CP/D7/CT /CP /D9/D1 /D9/D0/CP/D8/CT/CS /CX/D2/D3/D2/CT /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS/D8/D6/CX/D4/BA /CC /D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /DB /CT /D9/D7/CT/CS /CP /D7/D3/B9 /CP/D0/D0/CT/CS/CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2/D0/CT/D7/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CJ/BE/BD℄/B8 /DB/CW/CX /CW /CP/D0/D0/D3 /DB/D7 /D3/D2/CT /D8/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /CS/CT/D4 /CT/D2/B9/CS/CT/D2 /CT /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D2 /D8/CW/CT /D6/D3/D9/D2/CS/D8/D6/CX/D4 /D2 /D9/D1 /CQ /CT/D6k /BA /BT/CU/D8/CT/D6 /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV/D8/CW/CT /CP/D2/D7/CP/D8/DE /B4/BG/B5 /CX/D2 /D8/D3 /BX/D5/BA /B4/BD/B5 /CP/D2/CS /CT/DC/D4/CP/D2/CS/CX/D2/CV /D3/CU /D3/CQ/D8/CP/CX/D2/CT/CS /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D2 /D7/CT/D6/CX/CT/D7 /D3/CU t /D9/D4 /D8/D3 /D8/CW/CX/D6/CS /D3/D6/CS/CT/D6/B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /D8/CW/D6/CT/CT /D3/D6/CS/CX/D2/CP/D6/DD /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D7 /D3 /CT/CU/B9/AS /CX/CT/D2 /D8/D7 /D3/CU /CT/DC/D4/CP/D2/D7/CX/D3/D2/BA /CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/DD/D7/D8/CT/D1 /CQ /DD /D8/CW/CT /CU/D3/D6/DB /CP/D6/CS/BX/D9/D0/CT/D6 /D1/CT/D8/CW/D3 /CS /D6/CT/D0/CP/D8/CT/D7 /D8/CW/CT /D9/D2/CZ/D2/D3 /DB/D2 /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2tp /B8 /CW/CX/D6/D4 /D4/CP/D6/CP/D1/CT/D8/CT/D6ψ /B8 /CP/D2/CS/D4 /CT/CP/CZ /CP/D1/D4/D0/CX/D8/D9/CS/CTa0 /CP/CU/D8/CT/D6 /D8/CW/CTk+1 /B9/D8/CW /D6/D3/D9/D2/CS/D8/D6/CX/D4 /B4/D0/CT/CU/D8/B9/CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/B8/D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B5 /D8/D3 /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/CU/D8/CT/D6 /D8/CW/CTk /B9/D8/CW /D8/D6/CP/D2/D7/CX/D8 /B4/D6/CX/CV/CW /D8 /CW/CP/D2/CS/D7/CX/CS/CT /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7/B5/BM tp′=tp+4−7dψ−3ψ2+ (φψ−2σa2 0−ψβa2 0)t2 p 2t2 p, /B4/BH/B5 ψ′= (1 −2σa2 0)ψ+ (φ−βa2 0)ψ2+φ−3βa2 0−5d+ 3ψ+ 5dψ2+ 3ψ3 t2p, /B4/BI/B5 a0′=a0[1 +dψ−1 + (α−l−σa2 0)t2 p t2 p]. /B4/BJ/B5/BT /CU/D3/D9/D6/D8/CW /CT/D5/D9/CP/D8/CX/D3/D2 /DD/CX/CT/D0/CS/D7 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CT/D0/CP /DD /B8 φ=βa2 0+d+ψ t2p. 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/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D8/CW/CX/D6/CS/B9/D3/D6/CS/CT/D6 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8/B8 /DB/CW/CX /CW /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS t3 g /BA /CC /D3 /D7/D3/D0/DA /CT /D8/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/DB /CT /D2/D3 /DB /D1/CP/CZ /CT /D8/CW/CT /CP/D2/D7/CP/D8/DE a(k,t) =a0sech1+iψ[(t−ϑk)/tp]ei[φk+ω(t−ϑk)], /B4/BL/B5/DB/CW/CT/D6/CTϑ /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/CT/D0/CP /DD /CP/CU/D8/CT/D6 /D3/D2/CT /CU/D9/D0/D0 /D6/D3/D9/D2/CS /D8/D6/CX/D4 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /D0/D3 /CP/D0/D8/CX/D1/CT /CU/D6/CP/D1/CT/B8 /CP/D2/CSω /CX/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /CS/CT/D8/D9/D2/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /CT/D2 /D8/CT/D6 /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /D8/CW/CT/CQ/CP/D2/CS/DB/CX/CS/D8/CW/B9/D0/CX/D1/CX/D8/CX/D2/CV /CT/D0/CT/D1/CT/D2 /D8 /CJ/BE/BE℄/BA /CC/CW/CT/D7/CT /D8 /DB /D3 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CQ /CT /D3/D1/CT/D2/CT /CT/D7/D7/CP/D6/DD /CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /CT/AR/CT /D8 /D3/CU /D8/CW/CX/D6/CS/B9/D3/D6/CS/CT/D6 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2/BA /BT/CU/D8/CT/D6/CX/D2/D7/CT/D6/D8/CX/D2/CV /D8/CW/CT /CP/D2/D7/CP/D8/DE /B4/BL/B5 /CX/D2 /D8/D3 /D8/CW/CT /D1/D3 /CS/CX/AS/CT/CS /D0/CP/D7/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/B5/B8 /DB /CT /D2/D3 /DB /D3/CQ/D8/CP/CX/D2/D7/CX/DC /CX/D8/CT/D6/CP/D8/CX/DA /CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /CC/CW/CT /AS/D6/D7/D8 /D8/CW/D6/CT/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D3/D1/D4/D6/CX/D7/CT/CS /D3/CU /BX/D5/D7/BA /B4/BH/B9/BJ/B5 /D7/D9/D4/D4/D0/CT/D1/CT/D2 /D8/CT/CS /CQ /DD /D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D8/CT/D6/D1/D7 −ω(21d3ψ+ϑψt2 p)−dψω2t2 p−d3ω3ψt2 p, ω(3ϑ+θψ2−[25 + 9ψ2]d3/t2 p)−ω2(6d−5ψ −dψ2+φt2 p−σa2 0t2 p)−ω3(10d3−d3ψ2+ϑt2 p)− dω4t2 p−d3ω5t2 p, 3d3ωψ−ω2t2 p,/BL/D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /CC /D3 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CT/D0/CP /DD /CJ/BX/D5/BA/B4/BK/B5℄ /D8/CW/CT /D8/CT/D6/D1 ω(3d3ω/t2 p−ϑ) +dω2+d3ω3/CW/CP/D7 /D8/D3 /CQ /CT /CP/CS/CS/CT/CS/BA /CC/CW/CT /D8 /DB /D3 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7/CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /CS/CT/D0/CP /DD /D4 /CT/D6 /D6/D3/D9/D2/CS/D8/D6/CX/D4ϑ /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /DD /CS/CT/D8/D9/D2/CX/D2/CVω /CP/D6/CT ϑ=ω(2−4dψ+ 2d2ψ2) +ω2d3(9dψ2−ψ) + 9ω3d2 3ψ2−8d3ψ(α−l+σa2 0) −ψ+dψ2+ 3d3ωψ2,/B4/BD/BC/B5/CP/D2/CS ω′=ω+8d3ψ−t2 p[2ω+ϑψ−2dωψ −3d3ψω2] t4p, /B4/BD/BD/B5/D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /BT/D7 /CX/D7 /D3/CQ /DA/CX/D3/D9/D7 /CU/D6/D3/D1 /BX/D5/D7/BA /B4/BD/BC/B5 /CP/D2/CS /B4/BD/BD/B5/B8 /CP /D2/D3/D2/DE/CT/D6/D3 /D8/CW/CX/D6/CS/B9/D3/D6/CS/CT/D6 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D8/CT/D6/D1 /CV/CX/DA /CT/D7 /D6/CX/D7/CT /D8/D3 /CP /D4/D9/D0/D7/CT /CV/D6/D3/D9/D4 /CS/CT/D0/CP /DD /B4/DB/CX/D8/CW /D6/CT/D7/D4 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/D3/B8 /CE/BA /C8 /BA /C5/CX/CZ/CW/CP/CX/D0/D3 /DA/B8/AG/C6/CT/DB /D4/D6/CX/D2 /CX/D4/D0/CT /D3/CU /CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT/D7 /CX/D2 /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6/D7 /DB/CX/D8/CW /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT/B9/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CP/D2/CS /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/AH/B8 /C9/D9/CP/D2 /D8/D9/D1 /BX/D0/CT /D8/D6/BA/B8 /BE/BI/B8 /BE/BF/BI /B4/BD/BL/BL/BI/B5/BA/BD/BE700 750 800 850 900spectral intensity (arb. units) wavelength (nm)0 5 10 15A-O modulator trigger signal Ti:S laser pulse train time (µs)(a) (b)/BY/CX/CV/D9/D6/CT /BD/BM /B4 /CP /B5 /C8/D9/D0/D7/CT /D8/D6/CP/CX/D2 /CU/D6/D3/D1 /CP /D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /CC/CX/BM/D7/CP/D4/D4/CW/CX/D6/CT /D0/CP/D7/CT/D6 /CX/D2 /CP /D7/CT/D0/CU /C9/B9/D7/DB/CX/D8 /CW/CT/CS /D6/CT/CV/CX/D1/CT /BT /B4/D9/D4/D4 /CT/D6 /D9/D6/DA /CT/B5 /CP/D2/CS /D8/CW/CT /CP /D3/D9/D7/D8/D3/B9/D3/D4/D8/CX /CS/D6/CX/DA /CT/D6 /D7/CX/CV/D2/CP/D0 /D9/D7/CT/CS/CX/D2 /D8/CW/CT /CP/D7/CT /DB/CW/CT/D2 /D8/CW/CT /D4/D9/D1/D4 /D0/CP/D7/CT/D6 /DB /CP/D7 /D1/D3 /CS/D9/D0/CP/D8/CT/CS/BA /C7/D2 /D8/CW/CT /D8/CX/D1/CT /D7 /CP/D0/CT /D7/CW/D3 /DB/D2/CW/CT/D6/CT /D8/CW/CT/D6/CT /DB /CP/D7 /D2/D3 /D3/CQ/D7/CT/D6/DA /CP/CQ/D0/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CC/CX/BM/D7/CP/D4/D4/CW/CX/D6/CT /D0/CP/D7/CT/D6 /D4/D9/D0/D7/CT/D8/D6/CP/CX/D2 /DB/CX/D8/CW /D8/CW/CT /D1/D3 /CS/D9/D0/CP/D8/CT/CS /CP/D2/CS /D9/D2/D1/D3 /CS/D9/D0/CP/D8/CT/CS /D4/D9/D1/D4 /D0/CP/D7/CT/D6/BA /B4 /CQ /B5 /C4/CP/D7/CT/D6 /D7/D4 /CT /D8/D6/D9/D1/D3/CQ/D7/CT/D6/DA /CT/CS /CS/D9/D6/CX/D2/CV /D6/CT/CV/CX/D1/CT /BT /BA/BD/BF-1 0 1 2770nmintensity (arb. units) time (µs)-1 0 1 2-1 0 1 2830nm 820nm 810nm 800nm 780nm 750nm 730nmintensity (arb. units) time (µs)time (µs)/BY/CX/CV/D9/D6/CT /BE/BM /CC /CT/D1/D4 /D3/D6/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /CP/D6/CX/D3/D9/D7 /D7/D4 /CT /D8/D6/CP/D0 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3/D8/CW/CT /D4 /CT/CP/CZ /D3/CU /D8/CW/CT /C9/B9/D7/DB/CX/D8 /CW/CT/CS /CT/D2 /DA /CT/D0/D3/D4 /CT /CP/D2/CS /D8/CW/CT /D1/D3 /CS/CT/D0/D3 /CZ /CT/CS /D4/D9/D0/D7/CT /D8/D6/CP/CX/D2 /B4/CQ /D3/D8/D8/D3/D1 /D9/D6/DA /CT/B5/BA/BD/BG760 780 800 820 840 860 880 900spectral intensity (arb. units) wavelength(nm)-4 -2 0 4 6detector signal (arb. units) time (µ s)(a) (b) 760 780 800 820 840 860 880 900spectral intensity (arb. units) wavelength(nm)- - 2 8 10(b)/BY/CX/CV/D9/D6/CT /BF/BM /B4 /CP /B5 /C8/D9/D0/D7/CT /D8/D6/CP/CX/D2 /CP/D2/CS /B4 /CQ /B5 /D7/D4 /CT /D8/D6/D9/D1 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 /D6/CT/CV/CX/D1/CT /BU /BA/BD/BH-1.0 -0.5 0.0 0.5 1.0 1.5λ= 827 nmλ= 810 nmspectral intensity (arb. units) time (µs)- - 1.5/BY/CX/CV/D9/D6/CT /BG/BM /CC /CT/D1/D4 /D3/D6/CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D7/D4 /CT /D8/D6/CP/D0 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D8810nm /CP/D2/CS827nm/CX/D2 /D6/CT/CV/CX/D1/CT /BU /CP/D2/CS /D8/CW/CT /D1/D3 /CS/CT/D0/D3 /CZ /CT/CS /D4/D9/D0/D7/CT /D8/D6/CP/CX/D2 /B4/CQ /D3/D8/D8/D3/D1 /D9/D6/DA /CT/B5/BA/BD/BI0 2 4 6 8 101101001000 pulse duration tp (a) 2 1 SPM parameter 0A(b) SPM parameter 00 2 4 6 8 10-1.0-0.50.00.51.0chirp parameter ψ A2 1( ) β(b) β0-- xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Cxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx C xxxxxxxxxxxx Bxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Cxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx C xxxxxxxx B/BY/CX/CV/D9/D6/CT /BH/BM /CB/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /B4 /CP /B5 /CP/D2/CS /CW/CX/D6/D4 /B4 /CQ /B5 /DA /CT/D6/D7/D9/D7 /D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS/CB/C8/C5 /D3 /CTꜶ /CX/CT/D2 /D8β0 /CU/D3/D6 /CP /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /BZ/BW/BW /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CUd= 0 /B4 /D9/D6/DA /CT /BD/B5 /CP/D2/CS d=−10 /B4 /D9/D6/DA /CT /BE/B5/BA /CC/CW/CT /D3/D8/CW/CT/D6 /D0/CP/D7/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CTσ0= 1 /B8l= 0.05 /B8αm = 0.5 /B8U= 4×10−4/B8z= 0.3cm /B8λ= 800nm /B8 /CP/D2/CSTcav= 10ns /BA /CC/CW/CT/CW/CP/D8 /CW/CT/CS /D6/CT/CV/CX/D3/D2/D7 /CS/CT/D7 /D6/CX/CQ /CT /DA /CP/D6/CX/D3/D9/D7 /D2/D3/D2/D7/D8/CP/D8/CX/D3/D2/CP/D6/DD /D4/D9/D0/D7/CT /D6/CT/CV/CX/D1/CT/D7/B8 /D7/CT/CT /D8/CT/DC/D8/BA/BD/BJ-30 -15 01101001000pulse duration tp(b) (a) d31 A for1B for23 2 2 1 d-200 -100 0-4-3-2-1012chirp parameter ψ 1 2 3321 - -(b) - ----- xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Dxxxxxxxxxxxxxxxxxxxxxxxxx Cxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx DA xxxxxxxxxxxxxxxxxxxxxxxx C/BY/CX/CV/D9/D6/CT /BI/BM /CB/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /B4 /CP /B5 /CP/D2/CS /CW/CX/D6/D4 /B4 /CQ /B5 /DA /CT/D6/D7/D9/D7 /D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS/BZ/BW/BW /D3 /CTꜶ /CX/CT/D2 /D8d /CU/D3/D6β0= 1 /CP/D2/CSσ0= 10 /B4 /D9/D6/DA /CT /BD/B5/B8σ0= 1 /B4 /D9/D6/DA /CT /BE/B5/B8σ0= 0.1/B4 /D9/D6/DA /CT /BF/B5/BA /CC/CW/CT /D3/D8/CW/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT /D8/CW/CT /D7/CP/D1/CT /CP/D7 /CX/D2 /BY/CX/CV/BA /BH/BA/BD/BK1E-30.01 10 0T, µs0.0 0.2 0.4 0.60.11 1E-91E-4 \|a0\|2\|a0\|2\|a0\|2\|a0\|2 200 150 100 50 0 00.11 T, µsT, µsT, µs 0.60.5 0.40.3 0.20.1 /BY/CX/CV/D9/D6/CT /BJ/BM /CC/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D4/D9/D0/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD\|a0\|2/B4/D0/D3/CV /D7 /CP/D0/CT/B5 /DA /CT/D6/D7/D9/D7 /CV/D0/D3/CQ/CP/D0 /D8/CX/D1/CT T /BA /CC/CW/CT /D0/CP/D7/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CU/D3/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D9/D6/DA /CT/D7 /CP/D6/CT /CW/D3/D7/CT/D2 /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BA /D9/D6/DA /CT/CP /BMU= 4×10−4/B8β0= 0 /B8d= 0 /B8σ0= 1 /D9/D6/DA /CT /CQ /BMU= 5×10−4/B8β0= 0 /B8 d=−10 /B8σ0= 1 /D9/D6/DA /CT /BMU= 4×10−4/B8β0= 1 /B8d= 0 /B8σ0= 1 /D9/D6/DA /CT /CS /BM U= 6×10−4/B8β0= 1 /B8d=−20 /B8σ0= 0.1/BD/BL-0.10-0.050.00frequency detuning ω(b)(a) 0 20 T(µs)010203040chirp parameter ψ T(µs)20 0 -0.10-0.050.00 --(b)( ) 0 20 T(µs)T(µs)20 0/BY/CX/CV/D9/D6/CT /BK/BM /C8/D9/D0/D7/CT /CW/CX/D6/D4ψ /B4 /CP /B5 /CP/D2/CS /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CU/D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8ω /B4 /CQ /B5 /DA /CT/D6/D7/D9/D7 /CV/D0/D3/CQ/CP/D0/D8/CX/D1/CTT /CU/D3/D6 /D8/CW/CT /D0/CP/D7/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D9/D7/CT/CS /CX/D2 /BY/CX/CV/BA /BJ/B8 /D9/D6/DA /CT /BW /B8 /CP/D2/CS /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS/D4/CP/D6/CP/D1/CT/D8/CT/D6d3=−30 /BA/BE/BC750800850 (nm) 50100150 0(ns)/BY/CX/CV/D9/D6/CT /BL/BM /BV/CP/D0 /D9/D0/CP/D8/CT/CS /D4/D9/D0/D7/CT /D7/D4 /CT /D8/D6/D9/D1 /CP/D8 /CS/CX/AR/CT/D6/CT/D2 /D8 /D8/CX/D1/CT/D7 /D2/CT/CP/D6 /D8/CW/CT /D4 /CT/CP/CZ /D3/CU /D8/CW/CT/C9/B9/D7/DB/CX/D8 /CW/CT/CS /D4/D9/D0/D7/CT /CU/D3/D6 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /BY/CX/CV/BA /BK/BA/BE/BD
arXiv:physics/0009021v1 [physics.gen-ph] 5 Sep 2000EFUAZ FT-96-27 Las Construcciones de Dirac y de Majorana (Agenda para Estudiantes)∗ Valeri V. Dvoeglazov Escuela de F´ ısica, Universidad Aut´ onoma de Zacatecas Antonio Doval´ ı Jaime s/n, Zacatecas 98068, ZAC., M´ exico Correo electronico: VALERI@CANTERA.REDUAZ.MX (5 de julio de 1996) Resumen El presente trabajo es una breve revisi´ on del desarollo rec iente de la mec´ anica cu´ antica relativista en el espacio de representaci´ on (1 /2,0)⊕(0,1/2). Est´ a dirigida a estudiantes de licenciatura, pero tambi´ en puede ser ´ util a estudiantes de posgrado en f´ ısica de part´ ıculas y teor´ ıa cu´ antica de campos. The present work is a brief review of the recent development o f the relativistic quantum mechanics in the (1 /2)⊕(0,1/2) representation space. It is directed for B. Sc. students of Mexican Universities, but can be also useful for students graduated in particle physics and quantum ﬁeld theory. PACS: 03.65.Pm, 11.30.Cp, 11.30.Er Typeset using REVT EX ∗Enviado a “Investigaci´ on Cientiﬁca”. 1La construcci´ on de Dirac [1] para fermiones es conocida por todos los f´ ısicos. Es muy ´ util porque nos permite describir fen´ omenos con part´ ıculas cargadas, las que en nuestra disposici´ on en los aceleradores. Se ha dado cons iderable atenci´ on desde los primeros principios a esta construcci´ on en art´ ıculos precedentes para los estu- diantes [2,3]. En los ´ ultimos sesenta a˜ nos los f´ ısicos pr estaron poca atenci´ on a la construcci´ on de Majorana [4] para part´ ıculas neutras tal es como neutrino y fot´ on; ex- isten tres art´ ıculos [5,6] de cierta antig¨ uedad que hacen referencia a estos problemas. Adem´ as, los modelos diferentes del modelo del electr´ on de Dirac han sido propuestos por Pauli, Weisskopf y Markov [7–9]. Recientemente, gracia s a muchas discrepancias entre experimento y el modelo est´ andar de Weinberg, Salam y Glashow en f´ ısica del neutrino, ademas a unas causalidades, rigurosa considerac i´ on del espacio de repre- sentaci´ on (j,0)⊕(0,j) basada en la teor´ ıa del grupo de Lorentz ha sido empezada po r D. V. Ahluwalia [10–12], G. Ziino, A. O. Barut [13], y por V. V. Dvoeglazov [14–17]. Continuando quisiera aclarar, si el lector quiere entender nuestras ideas, ´ el tiene que acostumbrarse a pensar que la construcci´ on de Dirac es l a verdad, s´ olo la verdad perono toda la verdad . Los postulados que usamos son los siguientes: •La relaci´ on dispersional relativista E2/c2−p2=m2c2es valida para los estados de part´ ıculas observables; ces la velocidad de la luz. •Los 2-espinores derechos e izquierdos en el espacio ( j,0)⊕(0,j), dondejes el esp´ ın de part´ ıcula, se transforman de acuerdo con reglas d e Wigner [18,19]; ϕ son los par´ ametros del boost dado, (j,0) :φR(pµ) = ΛR(pµ←◦pµ)φR(◦pµ) = exp(+ J·ϕ)φR(◦pµ), (1a) (0,j) :φL(pµ) = ΛL(pµ←◦pµ)φL(◦pµ) = exp(−J·ϕ)φL(◦pµ), (1b) que son relacionadas con el postulado (1), vease Ryder [20]. •En el sistema de referencia donde el momento lineal de la part ´ ıcula es igual a 2cero tenemos la relaci´ on de Ryder-Burgard [20,10,12,14,1 5,17] entre los espinores izquierdos y derechos. Esta relaci´ on se basa en la declarac i´ on [20]: “ When a particle is at rest, one cannot deﬁne its spin as either left- or right-handed, so φR(0) =φL(0). Vease mas adelante para la relaci´ on generalizada y su d iscusi´ on. Como lo notiﬁc´ o D. V. Ahluwalia [12] aparte del bispinor de Dirac Ψ( pµ) = column (φR(pµ), φL(pµ)) es posible considerar los bispinores auto/contr-auto co n- jugados desde el principio porque sabemos [20] que Θ [j]φ∗ Lse transforma como los espinores derechos y Θ [j]φ∗ R, como los espinores izquierdos, Θ [j]es el operador de re- versi´ on de tiempo para el esp´ ın j. Entonces, podemos deﬁnir cuadriespinores λ(pµ)≡/parenleftbigg/parenleftBig ζλΘ[j]/parenrightBig φ∗ L(pµ) φL(pµ)/parenrightbigg , ρ(pµ)≡/parenleftBigg φR(pµ)/parenleftBig ζρΘ[j]/parenrightBig∗φ∗ R(pµ)/parenrightBigg , (2) dondeζλyζρson los factores de fase que se ﬁjan por las condiciones de aut o/contr- auto conjugaci´ on de carga el´ ectrica (o, sus generalizaci ones para los esp´ ınes enteros [12, ec.(12)]) Sc [j]λS,A(pµ) =±λS,A(pµ). (3) Los bispinores tienen propiedades extra˜ nas: a) no son func iones propias del operador de helicidad en el espacio ( j,0)⊕(0,j); b) no son funciones propias del operador de paridad, que en la representaci´ on de Weyl tiene la forma Ss [j]=eiϑs [j]/parenleftbigg0 11j 11j0/parenrightbigg ; c) los estados que describen son estados bi-ortonormales en sentido matem´ atico; d) los estados noson estados propios del operador hamiltoniano de Dirac. Los bispinores pueden satisfacer dos tipos de ecuaciones. Las primeras han sido dadas por D. V. Ahluwalia para espinores λS,A[12] /parenleftbigg−11 ζλexp (J·ϕ) Θ[j]Ξ[j]exp (J·ϕ) ζλexp (−J·ϕ)Ξ−1 [j]Θ[j]exp (−J·ϕ) −11/parenrightbigg λ(pµ) = 0,(4) y por V. V. Dvoeglazov para espinores ρS,A[17] 3/parenleftbigg−11 ζ∗ ρexp (J·ϕ)Ξ−1 [j]Θ[j]exp (J·ϕ) ζ∗ ρexp (−J·ϕ) Θ[j]Ξ[j]exp (−J·ϕ) −11/parenrightbigg ρ(pµ) = 0.(5) La deducci´ on de las ecuaciones ha sido presentada en ref. [1 2,2] y est´ a basada en la relaci´ on de Ryder-Burgard de la siguiente forma: /bracketleftBig φh L,R(◦pµ)/bracketrightBig∗=Ξ[j]φh L,R(◦pµ), (6) donde Ξ [j]es la matriz que conecta el bispinor de momento cero con su com plejo conjugado. Esta ecuaci´ on no se puede representar en la form a covariante [Γµνpµpν+ mΓµpµ−2m211]λ= 0 porque, como mencion´ o Ahluwalia, “it turns out that Γµνand Γµdo not transfrom as Poincar` e tensors”. Adem´ as, despues de aplicar el sistema para calculos analiticos en computadora MATHEMATICA 2.2 se puede convencer que aparte de las soluciones que satisfacen el postulado (1) , existen soluciones de la ecuaci´ on para j= 1/2 [12, ec. (31)] con la relaci´ on dispersional p0=−2m±/radicalbig p2+m2, a las cuales tenemos que proponer la interpretaci´ on. El otr o camino para encontrar las ecuaciones de los bispinores auto/contr-auto conjugad os es el que usa la forma de la relaci´ on de Ryder-Burgard siguiente: /bracketleftBig φh L(◦pµ)/bracketrightBig∗= (−1)1/2−he−i(θ1+θ2)Θ[1/2]φ−h L(◦pµ), (7) que relaciona los espinores de helicidad opuesta. Obtenemo s el conjunto de ecuaciones covariantes: iγµ∂µλS(x)−mρA(x) = 0, (8a) iγµ∂µρA(x)−mλS(x) = 0, (8b) para los estados con la energ´ ıa positiva, y iγµ∂µλA(x) +mρS(x) = 0, (9a) iγµ∂µρS(x) +mλA(x) = 0, (9b) para los estados con energ´ ıa negativa. Estas ecuaciones se pueden escribir en la forma 8- dimensional y ellas tienen las relaciones dispercionales p0=±/radicalbig p2+m2´ unicamente. 4La interpetaci´ on f´ ısica de las ecuaciones (8a-9b) es pare cida a la interpretaci´ on de los art´ ıculos de Barut y Ziino [13]. Por ejemplo, el pricipal re sultado es que los estados auto/contr-auto conjugados de carga el´ ectrica aunque son neutras en el sentido de las interacciones electromagn´ eticas no son neutras con re specto de carga quiral, el nuevo tipo del operador de carga que anticomuta con el operad or de carga el´ ectrica. El primero quien lo predijo hace mucho tiempo fue R. E. Marsha k, y el operador de carga quiral ha sido conectado con las transformaciones d e norma cos α±iγ5sinα. Analizando la forma de los bispinores (2) podemos deducir tr es tipos m´ as de interacci´ on de norma. Quisiera mencionar que las ecuaci´ ones del tipo (8 a-9b) fueron presentadas por primera vez por M. Markov [8,9], el autor de conocida idea de friedmones. El fue tambi´ en el primero qui´ en expres´ o la idea que el problema d e jerarqu´ ıa (del espectro de leptones, en otras palabras) tiene que ser resuelta en base a l an´ alisis de las soluciones de “la ecuaci´ on de Dirac” con ocho componentes y sus interac ciones con el vector potencial (v´ ease tamb´ en Barut [21]). La conexi´ on de la ec uaci´ on de Dirac con ocho componentes con la construcci´ on de Majorana ha sido estudi ada por Tokuoka [6]. Finalmente, queremos recordar que diferentes tipos de las t eor´ ıas invariantes con re- specto al grupo de Lorentz extendido (que toman en cuenta las operaciones de simetr´ ıas discretas), las teor´ ıas que recibieron el nombre del tipo d e Bargmann, Wightman y Wigner [10], han sido consideradas por varios autores [19,2 3,22,24,25,27]. Nuestras construcciones pertencen a este tipo. Adem´ as, como se indi caron [24–27] en apli- caci´ on a neutrino, aunque la part´ ıcula puede tener masa ce ro (de la ecuaci´ on de Klein-Gordon), sin embargo la ecuaci´ on de primer orden par ecida a la ecuaci´ on de Dirac puede contener el t´ ermino con alg´ un par´ ametro (‘ma sa variable’ como lo llamo el profesor Fushchich) de la dimensi´ on de masa. Por ejemplo , [iγµ∂µ+m(1±γ5)]ψ(xµ) = 0. (10) Otros tipos de ecuaciones no tienen la forma covariante expl icita. En base a los trabajos que mencion´ e en esta peque˜ na revisi´ on podemos delinear 5nuestros planes en el futuro pr´ oximo: •Por consideraci´ on a las transformaciones de norma m´ as gen erales para los es- pinoresλyρdeducir el modelo de Weinberg, Salam y Glashow. •En base de las ideas de Dowker [28], Evans y Vigier [29] entend er qu´ e es esp´ ın. •Deducir el espectro de leptones en base de la desarollo matem ´ atico de las ideas de Markov [9] y Barut [21]. •Investigar la estructura matem´ atica del espacio de Fock en base a las ideas de Dirac [30]. •En base de la consideraci´ on de los espinores en el sistema de referencia con el momento lineal cero entender que es color , el n´ umero cu´ antico de los quarks y gluones en cromodin´ amica cu´ antica y por qu´ e los estados ‘de color ’ no pueden ser observables directamente. Deducir la teor´ ıa de norma b asada en el grupo SUc(3), la cromodin´ amica cu´ antica. •Incluir la gravitaci´ on para uniﬁcar todas las interacci´ o nes. Como conclusi´ on de mis tres art´ ıculos en “Investigaci´ on Cient´ ıﬁca” sobre la mec´ anica cu´ antica relativista: aunque el modelo est´ and ar de Glashow, Weinberg y Salam y otros modelos de norma fueron muy utiles en prediccio nes fenomenologicas, ellos todav´ ıa no pueden ser considerados como base para una teor´ ıa completa. Los campos cu´ anticos con 2(2 j+ 1) componentes construidos en base de los postulados de teor´ ıa de relatividad y el principio de causalidad contien en ciertas simetr´ ıas que junto con los postulados de los modelos de norma pueden ser la base p ara la construcci´ on de una teor´ ıa uniﬁcada. Agradezco mucho a los doctores Dharam V. Ahluwalia y Anatoly F. Pashkov por su apoyo y paciente introducci´ on en estas materias. Agradezc o el apoyo en la ortograf´ ıa espa˜ nola del Sr. Jes´ us Alberto C´ azares. 6REFERENCIAS [1] P. A. M. Dirac, Proc. Roy. Soc. (London) A 117(1928) 610; ibid 118(1928) 351. [2] V. V. Dvoeglazov, De Dirac a Maxwell: Un Camino con Grupo de Lorentz. Investigaci´ on Cient´ ıﬁca 1, No. 10 (1997) 23-29. [3] V. V. Dvoeglazov, Interacci´ on ‘Oscilador’ de Part´ ıculas Relativistas. Investigaci´ on Cient´ ıﬁca 2, No. 1 (1999) 5-15. [4] E. Majorana, Nuovo Cim. 14(1937) 171. [5] J. A. McLennan, Phys. Rev. 106(1957) 821; K. M. Case, Phys. Rev. 107(1957) 307. [6] Z. Tokuoka, Prog. Theor. Phys. 37(1967) 581. [7] W. Pauli y V. Weisskopf, Helv. Phys. Acta 7(1934) 709. [8] M. Markov, ZhETF 7(1937) 579, 603. [9] M. A. Markov, Nucl. Phys. 55(1964) 130. [10] D. V. Ahluwalia, T. Goldman y M. B. Johnson, Phys. Lett. B 316(1993) 102. [11] D. V. Ahluwalia, M. B. Johnson and T. Goldman, Mod. Phys. Lett. A 9(1994) 439; Acta Phys. Polon. B 25(1994) 1267. [12] D. V. Ahluwalia, Int. J. Mod. Phys. A 11(1996) 1855. [13] A. O. Barut y G. Ziino, Mod. Phys. Lett. A 8(1993) 1011; G. Ziino, Int. J. Mod. Phys. A11(1996) 2081. [14] V. V. Dvoeglazov, Rev. Mex. Fis. Suppl. 41(1995) 159. [15] V. V. Dvoeglazov, Hadronic J. Suppl. 10(1995) 349; Nuovo Cim. 111B (1996) 483. [16] V. V. Dvoeglazov, Int. J. Theor. Phys. 34(1995) 2467. [17] V. V. Dvoeglazov, Nuovo Cim. 108A (1995) 1467. [18] E. P. Wigner, Ann. Math. 40(1939) 149. [19] E. P. Wigner, en “Group theoretical concepts and methods in elementary part icle physics – Lectures of the Istanbul Summer School of Theoretical Phys ics, 1962”. Ed. F. G¨ ursey (Gordon & Breach, 1964). [20] L. H. Ryder, Quantum Field Theory. (Cambridge University Press, 1985). [21] A. O. Barut, Phys. Lett. 73B (1978) 310; Phys. Rev. Lett. 42(1979) 1251. [22] I. M. Gelfand y M. L. Tsetlin, Sov. Phys. JETP 4(1957) 947; G. A. Sokolik, ibid 6(1958) 1170. [23] B. P. Nigam y L. L. Foldy, Phys. Rev. 102(1956) 1410. [24] N. D. Sen Gupta, Nucl. Phys. B 4(1967) 147. [25] V. I. Fushchich, Nucl. Phys. B 21(1970) 321; Lett. Nuovo Cim. 4(1972) 344. [26] M. T. Simon, Lett. Nuovo Cim. 2(1971) 616. [27] T. S. Santhanam y A. R. Tekumala, Lett. Nuovo Cim. 3(1972) 190. 7[28] J. S. Dowker y Y. P. Dowker, Proc. Roy. Soc. A 294(1966) 175; J. S. Dowker, ibid 297 (1967) 351. [29] M. W. Evans y J.-P. Vigier, Enigmatic Photon. Vols. 1-3. El tercer volumen con S. Roy y S. Jeﬀers. (Kluwer, 1994-96). [30] P. A. M. Dirac, Proc. Roy. Soc. A 322(1971) 435; ibid A 328(1972) 1. 8
Highlights of AAC 2000 Workshop J.B. Rosenzweig, Department of Physics and Astronomy, University of California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095 Abstract The Advanced Accelerator Concepts 2000 (AAC2K) Workshop was held in Santa Fe in June, 2000, and included a wide array of conceptual and theoretical advances at the frontier of accelerator physics. This paper reviews the highlights of the workshop, with subjects ranging from acceleration using lasers, plasmas and microstructures, to the beam physics of muon colliders. Particular emphasis is given to the topics which arerelevant to research at existing linear accelerator facilities, and the effect of this research on the capabilities of such facilities. 1 INTRODUCTION The field of advanced accelerators, which is in search of new and revolutionary technology to allow progress in high energy physics experimentation, holds approxi- mately biannual meetings. The latest in this series, Advanced Accelerator Concepts 2000, was held in June, 2000 in Santa Fe, New Mexico, and was hosted by Los Alamos National Laboratory, which also hosted the original workshop in the series[1]. The advanced accelerator field initially emerged in the early 1980’s as a response to the need to scale linac technology to TeV-class colliders. It was realized at that time, and it is still true today, that some dramatic change in the physical and technological paradigms of charged acceleration are necessary in order to build a compact and affordable version of such a collider. The search for such alternate technologies has now matured into a vigorous sub-field of accelerator physics. The field of advanced accelerator concepts (AAC) has renewed sense of purpose in light of both the failure of the SSC, and in the considerable experimental progress made in the field. In this paper, we review the conceptual structure of the AAC field, and mark its progress as of AAC2K. It is important to keep in mind that, even with dramatic proof- of-principle experimental results, that AAC has yet to produce a working accelerator in the sense that the linac community would recognize one. Nevertheless, the AAC field has already made a strong impact on the linac field, as AAC efforts have pushed the state of the art in linac- based experiments. AAC work has thus stimulated great progress in electron beam sources (notably the rf photo- injector), and in ultra-low emittance, sub-picosecond beam measurements. These techniques have already found their way into the more conventional fields of linear colliders and free-electron lasers. It can be expected that AAC work will next introduce non-conventional technologies intothe rf linac and related high energy physics fields, in the form of novel radiation and particle sources, and lenses. 2 AAC: PHYSICAL PRINCIPLES In order to review the progress made in AAC that was evident in the workshop, we first introduce the major categories of advanced accelerator schemes. • Wake-field accelerators (WFA): This type of accelerator uses what might be best termed a novel approach to creation of high frequency rf power, using a tightly bunched beam pulse or train of pulses which traverses a high impedance environment. The accelerating beam may traverse the same environment (collinear WFA) or a nearby structure (a two-beam accelerator, or TBA). The WFA addressed the problems of power creation and distribution at high frequency and high fields. The highest impedance environment one may use is a plasma (PWFA[2]) of high density n0, in which the power radiated by the beam in its wake (generalized Cerenkov radiation) is in the form of electrostatic or electromagnetic plasma waves. On the more conv- entional side, dielectric and metallic structures have been studied for WFA use; the CLIC[3] linear collider is essentially a TBA based on metallic structures. • Direct laser acceleration: Since lasers are known to be a cheap and efficient source of electromagnetic radiation at extremely high field and power levels, theyare very attractive for AAC applications. For relativistic beam particles, however, acceleration is only made possible by bending particle trajectories (inverse free-electron laser, or IFEL[4]), or by intro- ducing non-vacuum boundary or impedance conditions.These conditions can be similar to rf linac structures, or deformed into a planar geometry[5,6], or use the inverse Cerenkov effect, or ICA[7]. The obstacles to realization of such schemes center on the problem of scaling the accelerating wave down in size by four orders of magnitude(!) from present rf linac tech- nology. On the other hand, laser-based techniques have the tide of history on their side; all technologies have been pushed towards miniaturization in recent years. • Plasma accelerators: This classification clearly over- laps with the previous two, but is often discussed separately in the context of intense laser-plasma interaction. Plasma waves can be excited by lasers, just as with electron beams in the case of the PWFA, when the laser pulse is very short compared to the plasma wavelength λπpenr=/0 (laser wake-field accelerator, or LWFA[8]) and modulated at the plasma wavelength (plasma beatwave accelerator, or PBWA[9]). In plasmas, since the medium is already ionized, structure breakdown does not limit the field amplitude, and acceleration rates of several GeV/m have been reported[8,9]. Plasma accelerators have sev-eral interesting byproducts, notably plasma lenses[10], which may have application in linear collider final foci, and ultra-high brightness particle sources[11]. 3 PROGRESS REPORTED AT AAC2K 3.1 Wake-field Acceleration Progress in WFA was reported in a number of significant areas at AAC2K. Investigators at the facility most dedicated to WFA research, the Argonne Wake-field Accelerator (AWA), showed experimental results in the areas of multiple-pulse excitation[12] which verified the linear theory of Cerenkov wake-fields. In addition, the AWA performed initial acceleration experiments using the so-called step-up transformer, a form TBA in which the rfpower excited by the drive beam wake in one dielectric tube is transferred to another tube of higher impedance, in which a test beam is accelerated[13]. The investigation of dielectric wakes (coherent Cerenkov emission) has been extended at the AWA to an experiment designed to observe a signal similar to that expected by an EM shower induced by the interaction of a high energy neutrino with lunar matter[14], as in a newly proposed scheme of ultra-high energy neutrino detection. The E-157 collaboration has reported acceleration using the PWFA mechanism in the so-called “blow-out”[2] regime at the SLAC FFTB. The 3.3 nC, 2 psec rms, 30 GeV beam is injected into a 1.4 m long, very uniform plasma of density near 10 14 cm-3. Much of the beam is decelerated, with the tail being accelerated by the wake- field. This beam was observed by a time-resolved imaging and energy measurement system. The analysis of the data taken in this experiment has proven difficult due to the presence of strong transverse kicks which mimic energy changes in the spectrometer. The data is, despite these problems, consistent with the computational models of the PWFA in this nonlinear regime, with wakes in the range of several hundred MeV/m measured[15]. As the acceleration gradient in wake-field accelerators has long been recognized to scale strongly with the bunch length (with σz−2), several AAC photoinjector labs have developed bunch compressors. Pulse compression with such high currents and low emittances has inherent physics interest, due to such poorly understood processes as coherent synchrotron radiation and related emittance growth. Also measurement of such pulses in the sub-psec regime presents serious challenges, giving rise to methods beyond the limits of streak camera resolution[16] which rely on coherent transition radiation. The performance of compressor systems at the FNAL A0 photoinjector[16], the UCLA Neptune lab[17], and the Univ. of Tokyo[18], were all reported at AAC2K. The 5 nC, 2 psec bunches produced at A0 were used by a UCLA team to drive wake- fields in the blow-out regime which also reached several hundred MeV/m, nearly stopping the 17 MeV beam in 8 cm of a 7 1013× cm-3plasma[19]. The availability of 50 GeV electron beams which can be compressed to less than 30 µm rms pulse length at the end of the SLC arcs at SLAC gave rise to the suggestion by Katsouleas that a dense plasma of 5 m length placed after the arcs could double the energy of the SLC collisions[20]. This suggestion, termed the wake-field “after-burner”, will undoubtedly spur further investigation. 3.2 Laser Acceleration Experiments on direct laser acceleration which use structures, despite having been generally proposed for over a decade, have just begun. The daunting challenges (aperture, timing) of scaling the acceleration wavelength into the infrared have proven to be considerable. Progress in a 1 µm(!) wave-length planar-geometry exper- iment termed LEAP[21] at Stanford was reported at AAC2K, yet without definitive proof of acceleration . Acceleration, however, has been reported for years in non-structure-based (IFEL, ICA) experiments at the BNL ATF using 10 µm light. The next generation of laser acceleration experiment, in the form of a scheme termed STELLA, has just begun to produce impressive results asreported at AAC2K. STELLA (STaged ELectron Laser Acceleration[22]) is a two-stage system based on a 10 µm IFEL. The first stage microbunches the several psec (mm)ATF beam into the 10 µm IFEL period. This beam is then injected at the correct phase for acceleration into thesecond stage , where the captured beam is accelerated with small phase and energy spread. Some results of this remarkable experiment are shown in Fig. 1, which dis- plays the momentum spectrometer images of a beam before interaction, after the bunching of the beam in the first stage, and after the capture and acceleration of the beam in the second IFEL stage. This robust result thus shows the first precise manipulation of accelerating electron beams in waves with only 31 fsec period. Figure 1. False-color spectrometer images from STELLA experiment at BNL ATF (a) without IFEL, (b) after initial stage IFEL bunching, and (c) after correctly phased second IFEL stage (courtesy Ilan Ben-zvi). 3.3 Plasma Accelerators, Lenses and Sources In the past few AAC workshops, great leaps forward into multi-GeV/m acceleration was reported in laser-driven plasma accelerators. No such discoveries were reported at AAC2K, which found other plasma-based experiments, including PWFA, in the spotlight. In the context of PWFA, an important milestone in beam-plasma interaction was reported by the E-150collaboration at SLAC. In this experiment, positron beams at the FFTB were injected into short plasmas, withthe measured focusing of the beam from 8 µm to 4 µm rms width[10,23]. This promising result has resounding implications for use of plasmas for focusing andaccelerating positrons in future colliders. The highest level of attention at AAC2K in the area of laser-plasma acceleration was given to the production of high-brightness electron beams by the breaking of large amplitude, laser-driven plasma waves. Pioneering work onthis effect performed at the Univ. of Michigan (UM) was reviewed by Umstadter[24]. Further work using a more intense laser at LBNL was also presented, in which the extracted high brightness fsec electron pulses were measured to have energies in excess of 25 MeV[25]. Further control over the injection dynamics of these electron micropulses is expected when the wave-breaking is dictated by the collision of the a second laser pulse[25], or caused by a sharp density transition, as proposed in the context of the PWFA by Suk, et al.[26]. Figure 2. Radiographic film spectrum of protons emitted by PW laser interaction with solid target, showing peak near maximum energy (courtesy T. Cowan). 3.4 Heavy Particle Acceleration While the AAC Workshop traditionally is focused on acceleration and manipulation of electrons and positrons, at AAC2K there was much discussion of advances in heavy particle accelerator physics. The recent emphasis of laser-plasma based sources of electrons has been recently paralleled by the unexpected discovery of ion beam generation from multi-TW to PW,sub-psec laser-solid interactions. At AAC2K, results from the UM group (in the ten TW regime), as well as from LLNL (PW class laser) were reported. In the lower power experiments, it has been shown that the energetic (several MeV) ions are generated on the upstream sided of the thin-foil targets. In the LLNL experiments[27], on the other hand, the accelerated ions (>30 MeV) are found to be derived from the back side of the target, ejected along the surface normal. In both experiments, the emitted ionsshowed small transverse phase space extent. Even more remarkably, in the LLNL case the energy spectrum showed a notable peak near the maximum observed energy — the emitted ions formed a true narrow band (and initially sub-picosecond) beam. While such observation ofsuch unprecedentedly high brightness ion beam sources capture the imagination, it is not yet clear how one captures such a beam into a more conventional linac to use it in applications. The recent emergence of muon colliders , and the related concept of neutrino factories, was recognized in the working groups at AAC2K. There are many challenging research problems which need to be investigated before such projects are undertaken, such as ionization cooling, and manipulation of beam phase spaces under conditions where the angular momentum of the beam is nonzero[28]. It was apparent from the presentations at AAC2K he muon/neutrino factory field is still in its youth, and that with the onset of active experiments to prove the relevant principles, that this proposed high-energy physics instrument will grow in importance in the near future. 4 REFERENCES 1. Laser Acceleration of Particles, AIP Conf. Proc. 91 (AIP, New York, 1982)2. J.B.Rosenzweig, et al., Phys.Rev.A 44, R6189 (1991) 3. “The CLIC Study of a Multi-TeV e± Linear Collider”, Jean-Pierre Delahaye, et al. Proc. 1999 Part. Accel. Conf. 250 (IEEE, New York,1999)4. R. Palmer, J. Appl. Phys., 43 3014 (1972). 5. Y.Huang, R.L.Byer , Appl.Phys. Lett. 69 2175 (1996) 6. J.B. Rosenzweig, A. Murokh, and C. Pellegrini, Phys. Rev. Lett. 74, 2467 (1995). 7. W.D. Kimura, et al., Phys. Rev. Lett . 74, 546 (1995 ) 8. K. Nakajima, et al., Phys. Rev. Lett. 74, 4428 (1995) 9. C.E. Clayton, et al., Phys. Rev. Lett. 70, 37 (1993) 10. P. Chen, et al. Phys. Rev. D 40, 923-926 (1989) 11. D.Umstadter et al., Phys. Rev. Lett. 76, 2073 (1996) 12. J.G. Power, et al., Phys. Rev. E, 60, 6061 (1999) . 13. P. Zou, et al., Rev. Sci. Inst. 71, 2301 (2000). 14. P. Gorham, et al., LANL e-print hep- ex/ 0004007 . 15. P. Muggli, these proceedings.16. M. Fitch, et al. . Proc. 1999 Part. Accel. Conf. 2181 (IEEE, New York,1999).17. S. Anderson, et al., to appear in Proc. AAC2K (AIP) 18. M. Uesaka, et al., to appear in Proc. AAC2K (AIP) 19. N. Barov, et al., to appear in Proc. AAC2K (AIP) 20. T. Katsouleas, to appear in Proc. AAC2K (AIP) 21. E. Colby, et al., to appear in Proc. AAC2K (AIP) 22. W.D Kimura, et al., to appear in Proc. AAC2K (AIP) 23. J. Ng, et al., to appear in Proc. AAC2K (AIP) 24. Xiaofang Wang, et al. PRL , 84,5324 (2000) 25. W. Leemans, these proceedings. 26. H. Suk, et al., to appear in Proc. AAC2K (AIP) 27. T. Cowan, to appear in Physics of High Brightness Beams (World Sci., Singapore, 2000). 28. G.Penn, J.S.Wurtele, Phys.Rev.Lett . 85, 764 (2000)
1Experimental Evidence of Near-field Superluminally Propagating Electromagnetic Fields William D. Walker Royal Institute of Technology, KTH-Visby Department of Electrical Engineering Cram /G01rgatan 3, S-621 57 Visby, Sweden bill@visby.kth.se 1 Introduction A simple experiment is presented which indicates that electromagnetic fields propagate superluminally in the near-field next to an oscillating electric dipole source.A high frequency 437MHz, 2 watt sinusoidal electrical signal is transmitted from adipole antenna to a parallel near-field dipole detecting antenna. The phase differencebetween the two antenna signals is monitored with an oscilloscope as the distancebetween the antennas is increased. Analysis of the phase vs distance curve indicatesthat superluminal transverse electric field waves (phase and group) are generatedapproximately one-quarter wavelength outside the source and propagate toward andaway from the source. Upon creation, the transverse waves travel with infinite speed.The outgoing transverse waves reduce to the speed of light after they propagate aboutone wavelength away from the source. The inward propagating transverse fieldsrapidly reduce to the speed of light and then rapidly increase to infinite speed as theytravel into the source. The results are shown to be consistent with standardelectrodynamic theory. Theoretical analysis of an oscillating electric dipole reveals that the longitudinal component of the electric field and the transverse magnetic field are generated at thesource and propagate away from the source. Upon creation, the waves travel withinfinite speed and then rapidly reduce to the speed of light after they propagate aboutone wavelength away from the source. It is noted that the special theory of relativitypredicts that from a moving reference frame superluminal signals can propagatebackward in time. Arguments against the superluminal wave interpretation presentedin this paper are reviewed and shown to be invalid. Because of the similarity of thegoverning partial differential equations, two other physical systems (magnetic dipoleand a gravitationally radiating oscillating mass) are noted to have similarsuperluminal near-field theoretical results. 2 Theoretical expectations from electromagnetic theory 2.1 Electromagnetic theoretical solution of oscillating electric dipole Numerous textbooks present solutions of the electromagnetic (EM) fields generated by an oscillating electric dipole. The resultant electrical and magnetic fieldcomponents for an oscillating electric dipole are known to be [1, 2]: /G01/G02 /G01/G02wtkri or ekrirCosE/G01/G01 /G02 12)( 3/G01/G02/G03 /G04 (1) /G01/G02wtkri oekrikrrSinE/G01/G01 /G01 /G022 314)( /G01/G02/G03 /G04 /G01 (2) /G01/G02wtkrieikrrSinH/G01/G01 /G01 /G0224)( /G01/G03 /G05/G04 /G02 (3) Figure 1: Spherical co-ordinate system used to analyze electric dipole and resulting EM field solutionsErz xH /G01 E /G02/G01 y/G02 /G03r2Alternatively the electric dipole solution can be expressed as a superposition of sinusoidal waves which propagate at the speed of light. Using the identity: )()()(tkrSinitkrCos etkri/G01 /G01/G03/G01 /G02 /G01 /G03/G01 and extracting the imaginary part of the solution yields: tkrCoskrtkrSinrCosE or )(2)( 3 (4) )()()( )( 4)( 2 3tkrCoskrtkrSinkrtkrSin rSinE o /G01 (5) )()()( 4)( 2tkrCostkrSinkr rSinH /G02 (6) It should be noted that all of the above solutions are only valid for distances (r) much greater than the dipole length (d o). In the region next to the source (r ~ d o), the source cannot be modelled as a sinusoid: tSin /G05. Instead it must be modelled as a sinusoid inside a Dirac delta function: tSindro /G01 /G02 /G01 . The solution of this hyper-near-field problem can be calculated using the Liénard-Wiechert potentials [3, 4, 5] 2.2 Analysis of instantaneous phase speed and group speed It is noted from the above analysis that the field solutions of the electric dipole can be written as a sum of sinusoidal waves, which travel away from the dipole sourceat the speed of light. Even if the waves are generated by unique physical mechanisms,only the superposition of the waves is observable at any point in space. These wavecomponents in effect form a new wave which may have different properties than theoriginal components. Only the longitudinal and transverse wave components are realsince they can be decoupled by proper configuration of a measurement antenna. Thefollowing analysis derives general relations that are used to determine theinstantaneous phase and group speed vs distance graphs for the longitudinal andtransverse field components. 2.2.1 Derivation of phase speed relation In this section a mathematical relation is derived which enables the instantaneous phase speed of a wave to be determined from its phase vs distance curve. Given a propagating wave of the form: Sin(kr- t) the instantaneous wave phase speed (c ph = r/t) is the propagation speed of a point of constant phase ( t) on the wave. Solving this relation for time ( t and inserting it into the phase speed relation yields: c ph = /G09r/G0AIn the limit and using the relation ( = cok, where k is a far-field constant) the instantaneous phase speed becomes [6]: (7) Alternatively this phase speed relation can be derived from the known relation: cph = /k. Solving the phase ( kr) for (k) and inserting it in the phase speed equation yields: c ph = /G09r/G0AIn the limit this becomes Eqn. 7. Since = co k (where k is a far-field constant) the phase speed becomes: c ph = (co k)/(r) = co / [/G09(kr)].rkcrco ph 3Inserting the relation: k = 2 yields: c ph = /G0Dco / [/G09(rel)]. In the limit the instantaneous phase speed relation becomes: elo ph rcc 360 (8) where the electrical length is: r el = r//G0AA more rigorous derivation of this relation can be found in a previous paper by the author [3, 4, 7]. The above relation (ref. Eq. 8) indicates that the instantaneous phase speed is inversely proportional to the slope ofthe phase vs distance curve. Note that zero slope on this curve would indicate aninfinite instantaneous phase speed. 2.2.2 Derivation of group speed relation In this section a mathematical relation is derived that enables the instantaneous group speed of a wave to be determined given its phase vs distance curve. The groupspeed is known to be the speed at which wave energy and information travel. It can becalculated by considering two Fourier components of a wave group which form an amplitude modulated signal: ) () (22 11 rktSinrktSin /G01 /G02 /G01 /G01 /G01 )() ( krtSinkrtSin /G01 /G02 /G01 /G02 /G03 /G01 /G01 in which: = ( /G03 /G04/G0Dk = (k1-k2)/2, =( /G03 /G04/G0Dand k = (k 1+k2)/2. The instantaneous group speed (c g = /k) is the propagation speed of a point of constant phase ( k r) of the modulation component of the modulated wave. Solving the phase relation for ( k)and inserting it into the group speed relation yields: cg = /(r) = [( /G09r )]-1. In the limit and using the relation ( = cok) the instantaneous group speed becomes [6]: 12121/G01 /G01 /G09 /G09 krc rc og (9) A more rigorous derivation of this relation can be found in a previous paper by the author [3, 4, 7]. The above relation can also be made a function of (kr) by multiplyingthe numerator and the denominator by (k) and using the relation (k = w/c o) yielding: cg = co [ /G09kr]-1. In the limit this becomes: c g = co [(d/d dd/G09kr)]-1 = co [ (d/d /G09dd(kr) dd/G09kr)]-1. Using the relation ( cok) the instantaneous group speed becomes: c g = co [kr (d/d /G09kr/G09dd(kr) dd/G09kr)]-1. Using the relation for the electrical wavelength (r el = r/ kr/(/G0Dthe group speed becomes: cg = 2 co [rel (d/d/G09rel/G09dd(rel) dd/G09rel)]-1 \| /G02in rad. In conclusion the instan- taneous group speed becomes: (10) The above relation (ref. Eq. 10) indicates that the instantaneous group speed is inversely proportional to both the curvature and the slope of the phase vs distancecurve. Note that if the denominator of the above equation (ref. Eq. 10) is zero, thegroup speed will be infinite. Also note that if the curvature is zero, the group speedequation (ref. Eq. 10) will be the same as the phase speed equation (ref. Eq. 8). /G09 /G0A el elelo g rrrcc 22360/G01in deg in deg42.2.3 Radial electric field (E r) Applying the above phase and group speed relations (ref. Eq: 7, 9) to the radial electrical field (E r) component (ref. Eq. 1 or 4) yields the following results [7]: 3 11 31kr krTankr kr/G01 /G02 /G01 /G03 /G01/G01/G02/G03 (11) okro kroph c krc krcc 1212)(11 /G03/G03 /G01/G01/G02 /G02/G04/G04 /G05/G06 /G07/G07 /G08/G09/G0A /G03 (12) okrph kro g cc krkrkrcc 1 14 222 3)()(3)(1 /G03/G03 /G01/G01/G02 /G02/G0A/G0A/G03 (13) Figure 2: E r Phase vs kr Figure 3: E r cph vs kr Figure 4: E r cg vs kr 2.2.4 Transverse electric field (E /G02) Applying the above phase and group speed relations (ref. Eq. 7, 9) to the transverse electrical field (E /G02) component (ref. Eq. 2 or 5) yields the following results [7]: /G01/G02 /G01/G02 /G01/G02/G04/G04 /G05/G06 /G07/G07 /G08/G09 /G02 /G01/G01/G01 /G03/G01 4 22 1 11 krkrkrCoskr /G02 (14) /G01/G02 /G01/G02 /G01/G02 /G01/G02/G04/G04 /G05/G06 /G07/G07 /G08/G09 /G02 /G01/G02 /G01/G034 24 2 21 krkrkrkrccoph (15) 8 6 4 224 2 )()()(7)(6)()(1 krkrkrkrkrkrcco g/G02 /G01 /G02 /G01/G02 /G01/G03 (16) Figure 5: E /G01 phase ( /G02) vs kr Figure 6: E /G01 cph vs kr Figure 7: E /G01 cg vs kr0 2 4 6 8 10kr012345678910Chp /G01CoEr Cphvs kr coEr /G030 2 4 6 8 10kr0.811.21.41.61.8Cg /G01CoEr Cgvs kr coEr /G030 2 4 6 8 10kr0100200300400500600/G01 /G02geD /G03Er Phase /G01 /G01 /G02vs kr co Er /G03/G01 90 deg 0 2 4 6 8 10kr-1000100200300400500600/G01 /G02geD /G03E /G01Phase /G01 /G01 /G02vs kr co /G03E /G02180 deg 0 2 4 6 8 10kr0 -1 -2 -3 -4 -512345Cg /G01CoE /G02Cgvs kr co /G03E /G02 -co 0 2 4 6 8 10kr0 -1 -2 -3 -4 -512345Chp /G01CoE /G01Cphvs kr co /G03E /G02 -co52.2.5 Transverse magnetic field (H /G01) Applying the above phase and group speed relations (ref. Eq. 7, 9) to the transverse magnetic field (H /G01) component (ref. Eq. 3 or 6) yields the following results [7]: /G01/G01 /G02/G03 /G04/G04 /G05/G06 /G07/G08/G08 /G09/G01 21 )(1krkrCoskr /G01 (17) /G04/G04 /G05/G06 /G07/G07 /G08/G09/G0A /G032)(11krccoph(18) 4 222 )()(3)(1 krkrkrcco g/G0A/G0A/G03 ( 19) Figure 8: H /G02 phase ( /G02) vs kr Figure 9: H /G02 cph vs kr Figure 10: H /G02 cg vs kr 2.2.6 Animated field plots In this section, animated contour plots are presented which show how the longitudinal and transverse electric fields propagate. A cosinusoidal dipole source isused and the resultant fields are assumed to be a vectoral sum of all the wavecomponents. The resultant field magnitude and phase are then inserted into a propagating cosine wave: Mag Cos( t + ph) and plotted at different moments in time. The plots are generated using Mathematica Ver. 3 software. The code generates 24 plots evenly spaced within a specified analysis period. Several of the resultant framesare shown below. The vertical dipole source is located in the center of the plots. Theframes shown below (ref. Fig. 11) are animated plots of the longitudinal electric field.They clearly show that the waves are generated at the source and propagate awayfrom the source. Tt2317 Tt2318 Tt2319 Tt23200 2 4 6 8 10 kr-900100200300400500600/G01 /G02geD /G03H /G01Phase /G01 /G02 /G02vs kr co /G03H /G01180 deg 0 2 4 6 8 10kr0.811.21.41.61.8Cg /G01CoH /G02Cgvs kr co /G03H /G01 0 2 4 6 8 10kr012345678910Chp /G01CoH /G02Cphvs kr co /G03H /G01 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 /G066Mathematica code used to generate animations Eth=MagEthCos[wt+PhEth]; MagEth=po/4/Pi/eoSqrt[(1-(kr)^2)^2+(kr)^2]/r^3Sin[th]; PhEth=-kr+ArcCos[(1-(kr)^2)/Sqrt[1-(kr)^2+(kr)^4]]; Er=MagErCos[wt+PhEr];MagEr=po/2/Pi/eoSqrt[1+(kr)^2]/r^3Cos[th];PhEr=-kr+ArcTan[kr];L=1;k:=23.14159/L;c=310^8;w=23.141159c/L;T=L/c;po=1.610^(-19);eo=8.8510^(-12); r=Sqrt[x^2+y^2]; Animate[ContourPlot[Er/(110^(-7)),{x,-Pi/4,Pi/4},{y,-Pi/4,Pi/4},PlotPoints->100],{t,0,1T},ContourShading->False,Contours->{-.9,-.7,-.5,-.3,-.1,.1,.3,.5,.7,.9}] Figure 11: E r near-field wave animation plots The frames shown below (ref. Fig. 12) are animated plots of the transverse electrical field. The plots clearly show that the waves are created outside the source andpropagate toward and away from the source. Tt231 Tt232 Tt234 Tt235 Figure 12: E /G02 near-field wave animation plots The frames shown below (ref. Fig. 13) are animated plots of the longitudinal and transverse electrical fields vectorially added together (vector plot). The vertical dipolesource is located in the middle of the left-hand side of the plot Tt237 Tt2311 Tt2315 Tt2316-0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 -0.75 -0.5 -0.25 00.25 0.5 0.75-0.75-0.5-0.2500.250.50.75 /G06 /G067Additional Mathematica code used to generate vector plot animations (Add this code to the previous code used above) Ex=ErSin[th]+EthCos[th]; Ey=ErCos[th]-EthSin[th];r=Sqrt[x^2+y^2];th=ArcCos[y/(Sqrt[x^2+y^2])];<<Calculus`VectorAnalysis`<<Graphics`PlotField` Ett={Ex,Ey}; Etmag=Sqrt[Ex^2+Ey^2];Animate[PlotVectorField[Ett/Etmag,{x,0,0.5},{y,-0.25,0.25}],{t,0,T}] Figure 13: Animated plot of Er and E /G02 vectorially added together A more detailed plot of the total electric field can be obtained by using the fact that a line element crossed with an electric field is zero. The resulting partial differentialequation can be solved yielding [1, 2]: Resultant Equation: 0 /G0A /G02 /G02 dSinrCdrSinrC r (20) Solution: ConsttkrTankrCosSin kr /G0A/G01 1 2 211 (21) A contour plot of this solution yields the plot below (ref. Fig.14). Tt231 Tt232 Tt233 Tt234 Mathematica code used to generate E field contour plot L= 1;c=310^8; f= c/L;T=1/f;w=2N[Pi]f;k= w/c; fn= Sqrt[1/ (kr)^2+ 1] Cos [Th]^2* Cos[wt-kr+ArcTan[kr]] ; r = Sqrt[x^2 + y^2] ;Th = ArcTan[y/x] ; Animate[ContourPlot[fn, {x, 0.01,.5}, (y, -.4,.4), PlotPoints-> 100] , {t, 0, T} , ContourShading->False, Contours->{-. 9, -.7, -.5, -.3, -.1, .1, .3, .5, .7, .9}, AspectRatio->3/2] Figure 14: Animated plot of E field in near-field0.10.20.30.40.5-0.4-0.200.20.4 0.10.20.30.40.5-0.4-0.200.20.4 0.10.20.30.40.5-0.4-0.200.20.4 0.10.20.30.40.5-0.4-0.200.20.4 8Further away from the source the plot of the electric field becomes: Tt3230 Tt3233 Figure 15: Plot of E field in far-field Note that careful inspection of the plot reveals that the wavelength of the transverse electric field in the near-field (a) is larger than the wavelength in the far-field (b). The phase speed (c ph) is known to be a function of wavelength ( ) and frequency (f): cph = f. Solving the relation for (f), which is constant both in the near-field and far- field, yields: f = Cph near/near = Cph far/far. Solving this for Cph near yields: Cph near = Cphfar (near / far). Since near > far the phase speed of the transverse electric field is larger than the speed of light. (c ph > co). 2.2.7 Interpretation of theoretical results The above theoretical results suggest that longitudinal electric field waves and transverse magnetic field waves are generated at the dipole source and propagateaway. Upon creation, the waves (phase and group) travel with infinite speed and thenrapidly reduce to the speed of light after they propagate about one wavelength awayfrom the source. In addition, transverse electric field waves (phase and group) aregenerated approximately one-quarter wavelength outside the source and propagatetoward and away from the source. Upon creation, the transverse waves travel withinfinite speed. The outgoing transverse waves reduce to the speed of light after theypropagate about one wavelength away from the source. The inward propagatingtransverse fields rapidly reduce to the speed of light and then rapidly increase toinfinite speed as they travel into the source. In addition, the above results show thatthe transverse electrical field waves are generated about 90 degrees out of phase withrespect to the longitudinal waves. In the near-field the outward propagatinglongitudinal waves and the inward propagating transverse waves combine together toform a type of oscillating standing wave. Note that unlike a typical standing wave thethe outward and inward waves are completley different types of waves (longitudinalvs transverse) and can be separated by proper orientation of a detecting antenna. Inaddition, it should also be noted that both the phase and group waves are not confinedto one side of the speed of light boundary and propagate at speeds above and belowthe speed of light in specific regions from the source. The mechanism by which the electromagntetic near-field waves become superluminal can be understood by noting that the field componets can be consideredrectangular vector components of the total field (ref. Fig. 16). For example, the vectordiagram for the longitudinal electric field is (ref. Eq. 4): 00.5 11.5 22.53-2-1012 ab Transverse Field (E /G02) 00.5 11.5 22.53-2-10129 Figure 16: Vector diagram for longitudinal electric field From this vector diagram it can be seen that the phase of the longitudinal electric field is: - kr. Also it can be seen that angle: ArcTan[kr]. Combining these relations yields phase relation Eq. 11: ArcTan[kr] – kr. Note that for small (kr r << ) the angle bisector: B = 1 Sin(kr) kr has about the same length as vector (kr). Therefore when the (kr) is small the two vector components add together to form a longitudinal electric field vector which has nearly zero phase. Note that the anglebisector approximation is valid for several values of (kr) when (kr) is small. Thisresult can also be seen by Taylor expanding the phase relation for small (kr) yielding: = kr – [kr + (kr) 3/3 + O(kr)5] = (kr)3/3 + O(kr)5, where kr = r/c. These results show that very near the dipole source the phase of the longitudinal electric field is zero, causing both the phase speed and the group speed to be infinite (ref Eq. 7, 9). Inthe near-field the phase increases to (kr) 3/3, causing the phase speed to be: c o/(kr)2 (ref. Eq. 7) and the group speed to be: co/(kr)2/3 (ref. Eq. 9). In the far-field the phase becomes: kr, causing both the phase speed and the group speed to be equal to the speed of light (ref Eq. 7, 9). The other components of the electromagnetic field (E /G02 /G01) can also be analysed in the same way yielding similar results. 3 Experimental results 3.1 Description of experiment A simple experimental setup (ref. Fig. 17) has been developed to qualitatively verify the transverse electric field phase vs distance plot predicted from standard EMtheory (ref. Fig. 5). Figure 17: Experimental setupCh2 Ch1 Oscilloscope437 MHz 2 Watt Transmitterx Tx Dipole Ant /G01t T Rx Dipole Ant/G03 /G01/G04 kr ErkrB/G04 kr Erkr kr small i.e. r << 10The experiment setup consists of a high frequency UHF FM transmitter (Hamtronics model no. TA451)1 which generates a 437MHz (68.65cm wavelength), 2 watt sinusoidal electrical signal. The output of the transmitter is connected with a RG58coaxial cable to a vertical dipole antenna designed for the carrier frequency (modelno. RA3126) 2. The output of the transmitter is also connected to channel 1 of the input of a high frequency 500MHz digital oscilloscope (model no. HP54615B). Thetransmitter output, cable, antenna, and oscilloscope input all have 50 Ohm impedancein order to minimize reflections. A second identical receiver dipole antenna isconnected to channel 2 of the high frequency oscilloscope and the antenna ispositioned parallel to the vertical transmitting antenna. The sinusoidal signals fromthe two antennas are monitored with the oscilloscope, triggered to channel 1. Thephase difference between the signals is measured using the oscilloscope measurementcursors as the antennas are moved apart from 5 cm to 70 cm in increments of 5 cm(measurements made with a ruler). The oscilloscope calculates the phase from the measured time delay ( t) and the measured wave period (T): deg/G09t)/T. The phase vs distance data is analyzed using HPVEE (Ver. 4.01) PC software. The data is then curvefit with a 3rd order polynomial and the data is superimposed to visually verify the accuracy of the curvefit. The phase speed vs distance curve and the groupspeed vs distance curve are then generated by differentiating the resultant curvefitequation with respect to space and using the transformation relations (ref. Eq. 8, 10). 1 Ref. Internet site: www.hamtronics.com 2 Ref. Internet site: www.elfa.se - part no. 78-069-95113.2 Experimental results The following graph (ref. Fig. 19) is a plot of the phase vs distance data [ref. Fig. 18] taken during one experiment. The phase and group speed graphs weregenerated by curvefitting the experimental data and inserting the curvefit equationinto the phase and group speed transformations: (ref. Eq. 8, 10). The first data point isnot real and was added to improve the polynomial curvefit. The curvefit yielded thefollowing polynomial: ph = (132.2) + (-262.5)r el + (838.9)r el2 + (-353.4)r el3 Data # r el (cm) Ph (Deg) 0 0 140.0 1 5 111.7 2 10 102.4 3 15 108.64 20 121.05 25 136.66 30 155.2 7 35 170.7 8 40 195.59 45 211.0 10 50 245.211 55 282.4 12 60 316.6 13 65 325.914 70 366.2 Figure 18: Phase data vs r el Figure 19: Curvefit of phase vs distance data (r el) Figure 20 Figure 21 Calculated phase speed (c ph/co) vs distance (r el) graph Calculated group speed (c g/co) vs distance (r el) graph It should be noted that these experimental results are only qualitative due to EMreflections from nearby walls and objects. Quantitative measurements can only beattained in an anechoic chamber. The experiment has been repeated several times indifferent parts of a 4 x 4m (area) x 2m (height) room at different angular orientationsto the walls and the phase vs distance curve always appears the same within 10%. It is also observed that changing the scope input impedance from 50 Ohms to 1M Ohm input impedance does not noticeably affect the phase vs distance curve. Since noeffect is observed it is concluded that the Tx antenna to Rx antenna variablecapacitance combined with the scope input impedance (thereby forming a high passfilter) is not the cause of the phase change. Experimentally it is observed that the electrical field near the source (less than 0.6 ) is at least an order of magnitudeFirst data point not real. Point added to improve curvefit 0 0.2 0.4 0.6 0.8 1 r /G01 /G01100150200250300350hp /G01geD /G02 00.1 0.2 0.3 0.4 0.5 0.6 r /G01 /G01-15-10-505101520hpC /G03oC -2000.1 0.2 0.3 0.4 0.5 0.6 r /G01 /G01-15-10-505101520gC /G03oC12greater than electric field several wavelengths away from the source, which may be reflected. It is concluded that the observed field near the source is predominantly dueto near-field effects thereby making the observed results qualitatively reliable. Theexperimental results (ref. Fig. 19, 20, 21) are qualitatively similar to the electricdipole solution presented (ref. Fig. 5, 6, 7). Differences between experiment and thetheory presented can be attributed to EM reflections and also to the fact that thetheoretical model for a real dipole antenna is somewhat different from the simpleelectric dipole solution presented. 3.3 Interpretation of experimental results Analysis of the experimentally derived phase vs distance curve (ref. Fig. 19) indicates that the phase vs distance curve generated from the experimental data is verysimilar to the curve predicted from electric dipole theory (ref. Fig. 5). Performing theexperiment in an anacroic chamber and improving theoretical model for the dipoleantenna should yield a better match between theory and experiment. The phase speed(ref. Fig. 20, 6) and group speed (ref. Fig. 21, 7) vs distance curves do not matchtheory as well as the phase vs distance curves (ref. Fig. 19, 5). This difference is dueto the fact that small errors in the experimental data and curvefit become magnifiedafter differentiating the data, which is required by the phase and group speedtransformation relations (ref. Eq. 8, 10). Although the experimental results are not asaccurate as they can be, it can be qualitatively seen from the results that transverseelectric field waves (phase and group) are generated approximately one-quarterwavelength outside the source and propagate toward and away from the source(ref. Fig. 20, 21). Upon creation, the transverse waves travel with infinite speed. Theoutgoing transverse waves reduce to the speed of light after they propagate about onewavelength away from the source. Note that infinite phase speed and group speed isexpected since the phase vs distance curve (ref. Fig. 19) has a minimum at about onequarter wavelength distance from the source (ref. Eq. 8, 10). 4 Discussion 4.1 Arguments against superluminal interpretation 4.1.1 Superluminal illusion due to velocity dependent field cancellation An argument against the superluminal interpretation presented in this paper has appeared in the literature [8, 9, 10] in which it is argued that the electrical field can bedecomposed into 3 terms: a retarded coulomb field component, a retarded velocitydependent component, and a retarded acceleration component. /G09/G0A /G0D /G0A/G02/G02 /G02 rr r oedtd cre dtd cr reqE22 2 2 21 4 (22) Where (r´) denotes the retarded distance, (q) charge, (t) time, (c) speed of light, and (er´) unit vector from retarded position to field point. Note, retardation refers to the time it takes for the field to propagate a distance (r) at the speed of light. In the near-field the first two terms dominate and the velocity component partially cancels theeffects of the retardation causing the field to be nearly instantaneous in the near-field.In the far-field the retarded acceleration component dominates. The above equationcan be used to analyze the electrical field produced by an oscillating dipole.According to Feynman [8] the electric field for the electric dipole becomes:13(23) where )/(/crtpcrcrtpp /G0A Given the retarded dipole moment {p= Sin(kr- t)}, and using the relation (kr = r/c) the longitudinal component of the electric field becomes: tkrCoskrtkrSin rCospcrp rCosE o or /G09/G0A )( 2)( 2)( 3 3 (24) The above solution is equivalent to the longitudinal electrical field solution presented at the beginning of the paper (ref. Eq. 1, 4), which yields the superluminal phase andgroup speed curves (ref. Fig. 3, 4). This result shows that the solution is the sum oftwo retarded longitudinal vectorial components resulting in a superluminallongitudinal electrical field. The first sinusoidal term is due to the retarded coulombfield and the second cosinusoidal term is due to the retarded velocity dependentcomponent. Opponents would argue that the two retarded longitudinal componentsare real and that they combine together to produce a superluminal illusion. Theinterpretation presented in this paper is that the resultant longitudinal electrical field isa superposition of all the retarded wave components. At each point in space the wavecomponents vectorially add together to form a new wave that propagatessuperluminally as it is created and reduces to the speed of light after it has propagatedapproximately one wavelength from the source. In addition, the transverse component of the electric field can also be determined using (ref. Eq. 23) yielding: /G09/G0A p cpcrp rSinE o 2 31 4)( /G01 tkrCoskrtkrSinkrtkrSin rSin o 2 34)( (25) The solution is also equivalent to the longitudinal electrical field solution presented at the beginning of the paper (ref. Eq. 2, 5), which yields the superluminal phase andgroup speed curves (ref. Fig. 6, 7). Opponents would also argue that the transverseelectrical field is composed of three retarded sinusoidal propagating transverse waveswhich combine together to form a superluminal illusion. It is argued in this paper thatthese waves add vectorially as they propagate from the source and form a new type ofwave that has different properties from its components. An interesting proof of this isthat it is known (by using the pointing vector) that only the far-field transverse components (E /G02, H /G01) radiate, and that the near-field components form a type of standing wave and do not radiate [2]. But it can be seen from the above solutions (ref. Eq. 24, 25) that all of the components of the electric field propagate away fromthe dipole source and none of the components propagate toward the source which isrequired for the near-field propagating fields to not radiate from the source. Using theinterpretation presented in this paper it can be understood that in the near-field, theoutward propagating longitudinal waves and the inward propagating transverse wavescombine together in the near-field and form a type of oscillating standing wave. In thefar-field only the outward propagating transverse waves radiate. /G09 /G0A rrcrtpcrrrpprE o)/(1)(3 41 2 2* * 3 144.1.2 Superluminal illusion due to presence of standing waves It is also suggested by some authors that the near-field of an electrical dipole consists of an electrical field which grows and collapses synchronized with theoscillation of the electric dipole, resulting in a type of standing wave. Since standingwaves are thought to be the addition of transmitted and reflected waves the resultantfield may yield phase shifts unrelated to how the fields propagate, thereby refuting theresults presented in this paper. It is argued that the theory and experimental resultspresented in this paper suggest that the oscillating electrical dipole generates anelectrical field composed of two types of waves: longitudinal and transverse. Thelongitudinal electrical wave is generated at the source and propagates superluminallyaway from the source. The transverse electrical wave is generated about one quarterwavelength outside the source and propagates superluminally toward and away fromthe source. The apparent growth and collapse of the near-field electrical field is due tothe fact that the waves are produced 90 degrees out of phase. In the near-field theoutward propagating longitudinal waves and the inward propagating transverse wavescombine together to form a type of oscillating standing wave. Note that unlike atypical standing wave the outward and inward waves are completely different types ofwaves (longitudinal vs transverse) and can be separated by proper orientation of adetecting antenna. 4.2 Causality issue Although superluminal phase speeds are known to exist in other physical systems (eg. EM wave propagation in the ionosphere [11] ), group speeds exceeding the speedof light are not known to exist. In Einstein’s 1907 paper [12] he indicated thatalthough relativity does not prohibit the existence of superluminal signals (groupspeed > c) relativity does predict that superluminal signals can be seen by a movingobserver to travel backward in time. Einstein concluded that a superluminal signal (w)propagating a known distance (l) would be seen by a moving observer (v) to have crossed the distance in time: t = [(1-w v/c 2)/(w-v)] l . If the signal speed is hyperluminal: w > (c2)/v then the signal would be seen by the moving observer to travel backward in time ( t becomes negative). The result can also be derived from the relativistic equation for time: t´= [t-(v/c2)x], since t = l/w and x = l then: t´ = (l/w – v l/c2) < 0 for time reversal. Solving for (w) yields: w > (c2)/v. This effect can also be intuitively understood by using a spacetime diagram, with the moving coordinates (xc´, t´) superimposed on the reference frame of a stationaryobserver (xc, t). [13] Figure 22. Spacetime diagram showing a mechanism for time reversalA/G01/G01 cct ct´ x´ xw15The (x´) and (t´) axes are at angles /G09withrespect to the (x) and (t) axes, where = ArcTan(v/c). If a signal is transmitted superluminally (with respect to the stationary reference frame) from the origin to point (A), then the signal speed is: ct/x < Tan( ), but ct/x = c/w, therefore c/w < v/c. Solving this relation for (w) yields: w > c2/v. Although ( t) with respect to the stationary reference frame is positive, ( t) with respect to the moving reference frame is negative, indicating that from themoving reference frame the signal will be seen to travel backward in time. This is commonly referred to as a violation of causality where effect precedes cause.Although special relativity does not forbid that signals can travel faster than the speedof light, it does predict that if signals travel hyperluminally (w > c 2/v), the signal would be seen by a moving observer to travel backward in time. From the theorypresented in this paper, it is seen that all of the waves generated by an oscillatingelectric dipole travel with infinite speed at their point of creation and travel superluminally within a limited region of space (~0.1 ). It should be noted that this region of space can be very large for low frequencies (frequencies less than 30MHz yield: 0.1 > 1m). Therefore, it is concluded that according to relativity theory a moving observer can see these superluminally propagating waves propagating backward in time provided w > c 2/v. It should be noted that the moving reference frame can travel subluminally. 4.3 Speed of information propagation and detection Although the speed of information propagation (group speed (c g) ) may be superluminal, the speed of information propagation and detection may be less. If asinusoidally modulated signal propagates with a group speed (c g) and the sinusoidal modulation (Period T = 1/f) propagates a distance (d) in time (t), detection of thesignal may require several cycles (nT) of the signal in order to decode theinformation. The speed of information detection (c inf) can then be modelled: cinf = d / (t+nT). Since d = c g t, then c inf = (cg t) / (t+nT). In the far-field the propagation time (t) can be much larger than the number of cycles (nT) needed todecode the signal, therefore: c inf = cg. In the near-field the propagation time (t) can be much smaller than the number of cycles (nT) needed to decode the signal, therefore:c inf = cg t / (nT). This result shows that depending on the number of cycles required to detect the signal, the speed of information propagation and detection may besignificantly less than the group speed in the near-field. It is known from Fouriertheory that several cycles of a sinusoid are required for the information (frequency) tobe determined. Therefore, if information detection is based on Fourier decompositionof the signal, the speed of information transmission and detection may be significantlyless than the group speed. It is also known from information theory that only twopoints of the modulated sinusoid signal are required to determine its frequency,amplitude and phase. If the signal noise is small, these points can be very closetogether (nT < t) and a sinusoidal curvefit can be performed to detect the signal. If information detection is based on this method, the speed of information detection maybe only slightly less than the group speed. Note that applying this effect to the electricdipole will not eliminate the infinities in the phase and group speed curves; it willonly reduce the width of the superluminal regions.164.4 Magnetic dipole and oscillating gravitational mass Two other physical systems are noted to generate similar superluminal waves. Mathematical analysis of a magnetic dipole and a gravitationally radiating oscillatingmass [3, 4, 5] reveals that they are governed by the same partial differential equationas the electric dipole. For the magnetic dipole, the only difference is that electric andmagnetic fields are reversed. Consequently all of the analysis presented in this paperalso applies to this system, and therefore similar superluminal wave propagation nearthe source is also predicted from theory. For a vibrating gravitational mass, the difference is that electric (E) and magnetic (B) fields are replaced by analogs: the electric (G) and magnetic (P) component of thegravitational field [14]. In addition, a second mass vibrating with opposite phase isrequired to conserve momentum thereby making the source a quadrapole. But veryclose to the source, the effect of the second mass is negligible and can be neglected inthe analysis. Consequently superluminal wave propagation is also predicted next tothe source. Further away from the source the fields tend to cancel. Evidence of infinitegravitational phase speed at zero frequency has been observed by a few researchers bynoting the high stability of the earth’s orbit about the sun [15, 16]. Light from the sunis not observed to be collinear with the sun’s gravitational force. Astronomical studiesindicate that the earth’s acceleration is toward the gravitational center of the sun eventhough it is moving around the sun, whereas light from the sun is observed to beaberated. If the gravitational force between the sun and the earth were aberated thengravitational forces tangential to the earth’s orbit would result, causing the earth tospiral away from the sun, due to conservation of angular momentum. Currentastronomical observations estimate the phase speed of gravity to be greater than2x10 10c. Arguments against the superluminal interpretation have appeared in the literature [9, 10] 5 Conclusion A simple experiment has been presented which shows that an oscillating electric dipole generates superluminal transverse electric field waves (phase and group) aboutone quarter wavelength outside the dipole source and that the waves travelsuperluminally toward and away from the source. The results have been shown to beconsistent with electromagnetic theory. Arguments against this superluminalinterpretation have been reviewed and shown to be deficient. Relativistic analysisindicates that from a moving observer’s perspective, the superluminal signalsgenerated by a stationary electric dipole can be seen to travel backward in time. Dueto the mathematical similarity, two other physical systems are noted to have similarsuperluminal results: radiating magnetic dipole and oscillating gravitational mass.17References 1 W. Panofsky, M. Philips, Classical electricity and magnetism , Ch. 14, Addison-Wesley, (1962). 2 P. Lorrain, D. Corson, Electromagnetic fields and waves , W. H. Freeman and Company, Ch. 14, (1970). 3 W. Walker, ‘Superluminal propagation speed of longitudinally oscillating electrical fields’ , Conference on causality and locality in modern physics , Kluwer Acad, (1998). 4 W. Walker, J. Dual, ‘Phase speed of longitudinally oscillating gravitational fields’ , Edoardo Amaldi conference on gravitational waves , World Scientific, (1997). Also ref. elect. archive: http://xxx.lanl.gov/abs/gr-qc/9706082 5 W. Walker, Gravitational interaction studies , ETH Dissertation No. 12289, Zürich, Switzerland, (1997). 6 M. Born and E. Wolf, Principles of Optics , 6th Ed., Pergamon Press, pp. 15-23, (1980). 7 W. Walker, International Workshop “Lorentz Group, CPT an Neutrinos”, Zacatecas, Mexico, June 23-26, (1999) to be published in Conference proceedings, World Scientific. Also ref. elect archive: http://xxx.lanl.gov/abs/physics/0001063 8 R. P. Feynman, Feynman lectures in physics, Addison-Wesley Pub. Co., Vol. 2, Ch. 21, (1964). 9 S. Carlip, Aberration and the speed of gravity, Phys. Lett. A267 (2000) 81-87, Also ref elect archive: http://xxx.lanl.gov/abs/gr-qc/9909087 10 M. Ibison, H. Puthoff, S. Little, The speed of gravity revisited, ref. elect archive: http:// xxx.lanl.gov/abs/physics/9910050 11 F. Crawford, Waves: berkeley physics course , McGraw-Hill, Vol 3, pp. 170, 340-342, (1968). 12 A. Einstein, Die vom relativitätsprinzipgeforderte trägheit der energie, Ann. Phys., 23, 371-384, (1907). Also reference: A. Miller, Albert Einstein’s special theory of relativity, Springer-Verlag New York, (1998). 13 P. Nahin, Time machines, AIP New York, pp. 329-335, (1993). 14 R. Forward, General relativity for the experimentalist, Proceedings of the IRE, 49, May (1961). 15 P. S. Laplace, Mechanique, English translation, Chelsea Publ., New York, pp.1799-1825, (1966). 16 T. Van Flanderrn, The speed of gravity – what the experiments say, Phys. Lett. A 250, (1998).
arXiv:physics/0009024v1 [physics.atom-ph] 6 Sep 2000Local Momenta and a Three-Body Gauge Michael Jay Schillaci, Francis Marion University, Florence, SC 29501 March 3, 2008 Abstract In recent years researchers have attempted to improve the co ntinuum state three-body wavefunc- tion for three, mutually interacting Coulomb particles by i ncluding, so called, local momentum eﬀects, which depend upon the logarithmic gradient of the continuum , two-body Coulomb waves. Using the exact three-body wavefunction in the region where two of the three particles remain close, a revised description of these local momenta, is attained and predict s that a quantum-mechanical impulse may develop in the reaction zone, causing like-sign–charged pa rticles to decrease their radial separation and opposite-sign–charged particles to increase their radial separation. The consequences of these predictions are investigated through both quantum and semi-classical t echniques where the total energy of a two- body continuum Coulomb system in the presence of a third, mut ually interacting body are analyzed. Numerical calculations conﬁrm that while ignoring these lo cal eﬀects for light-ion–atom processes, may be appropriate, three-body eﬀects may dominate in the react ion zone for heavy-ion–atom processes. The techniques developed here are then applied to explain the ob served asymmetry in the data collected by Wiese, et. al. ,[PRL 79, 4982] on the correlated breakup of three massive, Coulom b interacting particles. The results are attributed to a genuine three-body eﬀect whi ch rearranges particles in the reaction zone, while retaining the appropriate asymptotic behavior. Prel iminary calculations show a deviation of less than 10% between the predicted and observed asymmetries. Th e strength of these results is then used to argue that the local momenta, herein developed, be treate d as a formal gauge constraint for three- body interactions. This hypothesis is investigated and it i s shown that a real-valued, position-dependent phase is added to the wavefunction. A semi-classical analys is of the proposed three-body gauge, reveals that while genuine three-body eﬀects may arise in the reacti on zone, the asymptotic form of the rele- vant two-body Hamiltonian remains unchanged for relative e nergies greater than ∼1eV for all atomic species. Further analysis shows however that one may detect asymptotic variations in the scattering amplitudes for massive systems at energies ∼µeV. These results provide convincing theoretical and physical evidence for the success of many current experimen ts and indicate that more experimentation with near-threshold, massive three-body systems is needed . 1 Introduction The ”three-body” problem is as old as the study of Physics its elf. After Sir Isaac Newton showed (in his Principia ) that it was possible to infer the orbits of two, mutually interacting bodies using only the laws of mathematics, mankind has endeavored to derive an analyti c description of the motion of three, mutually interacting particles. Three-body interactions abound in natural processes as div erse as stellar evolution and thin-ﬁlm growth and occur over a very wide range of energies. For the present d iscussion, continuum atomic scattering will be considered for three mutually interacting charged p articles interacting via the inﬁnite range Coulomb potential. Calculation of local distortion eﬀects herein d erived will be carried out over representative energies of between 3 −54.4 eV.1 1These energies were chosen because of the particular releva nce to the results oﬀered in [6] on electron or positron scatt ering from hydrogen. 1In Section 2, the traditional Jacobi coordinates are introd uced and key historical results are given. While the notation used here is not substantially diﬀerent, minor errors have been corrected and additional properties and unique results have been added. A revised description of the so called local momenta, ﬁrst pr esented in [1] is oﬀered in Section 3. While the local momenta derived here also depend upon the logarith mic gradient of a continuum state, Coulomb wave - here referred to as the local distortion - the coordina te dependence is such that variations of the momenta can not be ignored in an a priori manner. The development continues in Section 4, where a detailed discussion of the mathematical and physical prope rties of the local momenta is oﬀered. Appendix A shows that the local distortion can be expressed as a damped oscillator function that is analytic over all regions, thus improving the ability to assess the possible c ontributions of these local eﬀects in both the inner (reaction-zone), and the outer (asymptotic) regions of the three-body scattering event.2This analysis reveals that while local distortion eﬀects generally “fall-oﬀ” in t he asymptotic regions, they may alter the outcome of a scattering event by rearranging particles in the reacti on-zone. Appendix B demonstrates however that one may incorporate local eﬀects while retaining the approp riate asymptotic form for the continuum state, three-body wavefunction. From this a new interpretation of the three-body scattering event emerges, wherein a two-body continuum pair acquires a local momentum by scatt ering oﬀ of an exact interaction potential. Section 5 illustrates that generalized local momentum eﬀec ts may be used to explain the observed asym- metry in the data collected in the triple-coincidence detec tion experiment of Wiese et. al., [2]. The experiment considered three large, very nearly equal mass, mutually in teracting, charged particles. With the appropriate total center-of-mass energy and reduced mass parameters ch osen to reﬂect those used in the actual experi- ment, the predicted local eﬀects are shown to be large enough to reproduce the observed asymmetry. This result is taken as motivation to hypothesize that the genera lized local momentum be treated as a possible gauge transformation for three-body interactions. In Sec. 6 this hypothesis is formalized and a semi-classical analysis reveals that while a real-valued, position depend ent phase is added to the two-body continuum state wavefunction, the asymptotic form of the relevant two-body Hamiltonian remains unchanged. 2 Notation The traditional Jacobi coordinates, here denoted by ( /vector rν, /vector ρν) along with their respective, congugate momenta, (/vectorkν, /vector qν),where ν=α, β, γ will be used to indicate a particular channel representatio n. These coordinates are particularly well suited for the study of the motions of t hree mutually interacting particles because they locate the conventional3reduced masses of the system. These coordinates are shown in Figure 1 and the “alpha-channel” representation will be used unless otherw ise speciﬁed. The reduced masses located by the ( /vector rα, /vector ρα) coordinates are given by µα≡mβmγ mβ+mγ(1a) Mα≡mα(mβ+mγ) mα+mβ+mγ, (1b) respectively and the other masses are deﬁned cyclically. In addition, there are two relationships that the Jacobi coo rdinates and their conjugate momenta obey that will be of use in the current development. These are, /vector rα+/vector rβ+/vector rγ= 0 (2a) /vectorkα µα+/vectorkβ µβ+/vectorkγ µγ= 0. (2b) 2While there is no rigorous convention for the quantiﬁcation of these regions, the “reaction zone” is here taken to be the region wherein r <10a0. 3While there is some debate in the literature, as noted in [3], over the appropriate reduced masses to use in the Classical treatment of the three-body problem, the possible renormal ization of mass in the Quantum treatment makes these argumen ts irrelevant here. 2While (2a) can be “seen” in Figure 1, (2b) is more subtle and is a statement of the “relative-velocity conservation” for the three-body system. These relationsh ips follow from the orthogonality of the Jacobi coordinates and are easily veriﬁed using the transformatio n matrices for the coordinates, /parenleftbigg /vector ρβ /vector rβ/parenrightbigg =/parenleftigg −µβ mγ−µα Mβ 1−µα mγ/parenrightigg/parenleftbigg /vector ρα /vector rα/parenrightbigg , (3a) /parenleftbigg/vector ργ /vector rγ/parenrightbigg =/parenleftigg −µγ mβµα Mγ −1−µα mβ/parenrightigg/parenleftbigg/vector ρα /vector rα/parenrightbigg , (3b) and for the momenta, /parenleftbigg/vector qβ /vectorkβ/parenrightbigg =/parenleftigg −µα mγ−1 µβ Mα−µβ mγ/parenrightigg/parenleftbigg/vector qα /vectorkα/parenrightbigg , (4a) /parenleftbigg/vector qγ /vectorkγ/parenrightbigg =/parenleftigg −µα mβ1 −µγ Mα−µγ mβ/parenrightigg/parenleftbigg/vector qα /vectorkα/parenrightbigg . (4b) A further consequence of these transformations is that the t hree-body, Coulomb potential may be ex- pressed as follows4: Vα C(/vector rα, /vector ρα) =e2 4πǫ0 ZβZγ rα+ZαZγ/vextendsingle/vextendsingle/vextendsingle/vector rα−µα mγ/vector ρα/vextendsingle/vextendsingle/vextendsingle /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright rβ+ZαZβ/vextendsingle/vextendsingle/vextendsingle/vector rα+µα mβ/vector ρα/vextendsingle/vextendsingle/vextendsingle /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright rγ , (5) where the Zν(ν=α, β, γ ) are the appropriate charge signs. Moreover, the orthogona lity of the Jacobi coordinates can be used to show that, the three-body Schr¨ od inger equation may be written in a channel- independent way. i.e., (Hν 0+Vν C)Ψ/vectorkν,/vector qν(/vector rν, /vector ρν) =EtΨ/vectorkν,/vector qν(/vector rν, /vector ρν), (6) where Hν 0=−¯h2 2µν/vector∇2 /vector rν−¯h2 2Mν/vector∇2 /vector ρν, (7) andVν C(/vector rν, /vector ρν) has been used, with supressed coordinate dependence, to in dicate the full three-body Coulomb potential of the ν-channel. Here Etis the total center of mass energy of three-body system. To date, the most successful and widely used approximate sol ution to the three-body Schr¨ odinger equation is the paradigm “3C” wavefunction proposed by Redmond[4],[ 5] and rigorously derived and tested by Brauner et. al. [6]. The solution is valid in the asymptotic region where all particles are far apart. Traditionally one denotes this region with, Ω 0, and the solution is given by, ΨRED(Ω0) /vectorkν,/vector qν(/vector rν, /vector ρν) =ei(/vectorkν·/vector rν+/vector qν·/vector ρν)/productdisplay ν=α,β,γe−iηkνlnζkν, (8) where for instance, ηkα≡ZβZγ/parenleftbigge2 4πǫ0¯h/parenrightbiggµα ¯hkα(9) 4While it is conventional to use “atomic units” wherein, ¯ h=e= 1, all physical constants are retained so that the interest ed reader may verify explicit numerical results cited later in the text without normalization. 3is the atomic Sommerfeld parameter. The hyper-parabolic co ordinate, ζkν≡kνrν+/vectorkν·/vector rν, (10) has been introduced for notational simplicity only, and wit h the above deﬁnition, the wavefunction (8) satisﬁes all incoming boundary conditions in the region Ω 0. The overwhelming opinion in the literature is that all valid three-body wavefunctions must match smoothl y with this solution in the region Ω 0. The logarithmic phase factors in the Redmond solution are pr esentphysically because of the inﬁnite nature of the Coulomb potential, and arise mathematically as the leading term in the asymptotic expansion of the conﬂuent hypergeometric function. i.e.,[7] Cηkν(ζkν) = Nηkν1F1(iηkν,1,−iζkν) (11a) Cηkν(ζkν) = Γ[1 −iηkν]e−ηkνπ 2∞/summationdisplay n=0(iηkν)n (1)n(−iζkν)n n!(11b) Cηkν(ζkν) = lim rν→∞e−iηkνlnζkν. (11c) These “C-functions” provide one of the representations for theexact solution to the two-body scattering problem. That is, if one writes the two-body wavefunction as ΨTwo−Body /vectorkν(/vector rν) =ei/vectorkν·/vector rνCηkν(ζkν), (12) then substitution of this form into the two-body Schr¨ oding er equation shows that /bracketleftbigg −¯h2 2µα/vector∇2 /vector rα−i¯h µα/vectorkα·/vector∇/vector rα+ZβZγe2 4πǫ01 rα/bracketrightbigg Cηkα(ζkα) = 0. (13) While the very intuitive solution (8) has been used with grea t success by researchers to model both atom- ion [6],[9] and photo-ionization processes [10], a more rob ust solution has been sought in recent years[1],[11]. Particularly, a solution is sought that may be extended into the so called “interior regions,” where at least two of the three particles remain close to one another. It is s igniﬁcant to note that all of the successes of the 3C wavefunction have been achieved for light-ion - atom systems, where the asymptotic form in Ω 0has been shown to be generally adequate. What is sought however i s a more precise accounting of the intimacies of the three-body interaction for arbitrary masses and in al l regions of the scattering space. 3 Origin of the Local Momenta Because an exact solution to the atomic three-body problem d oes not exist, the best hope in achieving an improved wavefunction has been to improve the approximatio n schemes used. Generally, these approximation schemes fall into two categories: •Approximating the Kinetic Terms of the Hamiltonian (i.e., t he Eikinol approximation) •Approximating the Potential Terms of the Hamiltonian While these techniques would seem to be mutually exclusive, the problem is that the inseparable nature of the three-body Coulomb potential makes the choice of the kin ematic description impossible. Research has thus continued along a fragile path and to account for approx imation and/or distortion eﬀects, two well established interpretations have emerged: •Introduce a Local Momentum , which depends upon the radial separation of two of the three particles through the logarithmic gradient of the continuum two-body wavefunction, and attribute the distortion to a velocity-dependent, auxilary potential. cf., [1] 4•Introduce an Eﬀective Charge , which depends upon the radial separation of two of the three particles through the logarithmic gradient of the continuum two-body wavefunction, and attribute the distortion to dynamical screening of charges. cf., [8] The common dependence upon the two-body solution in these tw o interpretations is clear and in both the dependence is derived in a rigorous, but a posteriori way, to satisfy the relevant boundary conditions in the asymptotic regions. The natural question is if this de pendence can be achieved in an a priori way, utilizing only physical and mathematical intuition. To approach an answer to this question, it is important to kno w that there does exist an exact solution to the continuum three-body Schr¨ odinger equation for the c ase when two of the particles remain close together(or equivalently if one of the particles is inﬁnite ly massive). The solution depends critically upon the form of the three-body Coulomb potential in the region Ω α, where two of the particles, βandγare close together, while particle αis inﬁnitely far away from both of them. That is, one may write a description of the region Ω αsuccinctly as follows: Ωα: lim rα→ ∞,limrα ρα→0 Clearly the region Ω αmatches smoothly with the region Ω 0, and to investigate the behavior of the three-body Coulomb potential in this region, one may write Vα C(/vector rα, /vector ρα) =ZβZγe2 4πǫ01 rα+Zαe2 4πǫ01 ρα×∞/summationdisplay L′=0ZL′/parenleftbiggrα ρα/parenrightbiggL′ PL′(ˆrα·ˆρα), r α≪ρα, (14) where ZL′=Zβ/parenleftbigg −µα mβ/parenrightbiggL′ +Zγ/parenleftbiggµα mγ/parenrightbiggL′ , and the PL′(ˆrα·ˆρα) are the Legendre polynomials of the ﬁrst kind. Now in the region Ω α,the only surviving term in the expansion is for L′= 0.Hence the potential has the following asymptotic form: Vα(Ωα) C (/vector rα, /vector ρα) =ZβZγe2 4πǫ01 rα+Zα(Zβ+Zγ)e2 4πǫ01 ρα. (15) The second term in this expansion is often referred to as the “ reduced charge potential,” and the resulting form of the three-body wavefunction is separable. e.g., the three-body Schr¨ odinger equation takes the asymptotic form, /bracketleftbigg −¯h2 2µα/vector∇2 /vector rα+ZβZγe2 4πǫ01 rα−¯h2 2Mα/vector∇2 /vector ρα+Zα(Zβ+Zγ)e2 4πǫ01 ρα/bracketrightbigg ΨΩα /vectorkα,/vector qα(/vector rα, /vector ρα) =EtΨΩα /vectorkα,/vector qα(/vector rα, /vector ρα).(16) Because the termZα(Zβ+Zγ)e2 4πǫ0ραcouples to the kinetic term −¯h2 2Mα/vector∇2 /vector ρα, the three-body Hamiltonian naturally separates and if one assumes that the three-body wavefuncti on has the standard plane wave form, ΨΩα /vectorkα,/vector qα(/vector rα, /vector ρα) =ei(/vectorkα·/vector rα+/vector qν·/vector ρν)Φ/vectorkα,/vector qα(/vector rα, /vector ρα), (17) where the total energy of the three-body system may be writte n as Et=¯h2k2 α 2µα+¯h2q2 α 2Mα>0, (18) for continuum scattering, then substitution into the asymp totic three-body Schr¨ odinger equation yields the following equation for Φ /vectorkα,/vector qα(/vector rα, /vector ρα) in the region Ω α: /bracketleftbigg −¯h2 2µα/vector∇2 /vector rα+ZβZγe2 4πǫ01 rα−¯h2 2Mα/vector∇2 /vector ρα+Zα(Zβ+Zγ)e2 4πǫ01 ρα/bracketrightbigg Φ/vectorkα,/vector qα(/vector rα, /vector ρα) = 0. (19) 5Therefore the exact solution in the region Ω αis given by, Ψ(Ωα) /vectorkα,/vector qα(/vector rα, /vector ρα) =ei(/vectorkα·/vector rα+/vector qν·/vector ρν)Cηkα(ζkα)Cηqα(ζqα). (20) Here the function Cηqα(ζqα) is a reduced charge continuum state Coulomb wave, which sat isﬁes /bracketleftbigg −¯h2 2Mα/vector∇2 /vector ρα−i¯h Mα/vector qα·/vector∇/vector ρα+Zα(Zβ+Zγ)e2 4πǫ01 ρα/bracketrightbigg Cηqα(ζqα) = 0, (21) and ηq≡Zα(Zβ+Zγ)/bracketleftbigge2 4πǫ0¯h/bracketrightbiggMα ¯hqα. While equation (20) constitutes a rigorous solution it is no t a valid solution because it does not match smoothly with the result asserted by Redmond. (cf., equatio n (8) above.) To see this note that the asymptotic form of (20) in Ω 0would be given by Ψ(Ω0) /vectorkα,/vector qα(/vector rα, /vector ρα) =ei(/vectorkα·/vector rα+/vector qα·/vector ρα)e−iηkαlnζkαe−iηqαlnζqα (22) Clearly, the twologarithmic phases present in this solution can not match sm oothly with the three present in the Redmond solution. Hence (20) does not constitute a val id solution in the asymptotic region Ω 0. In addition, note that as a general wavefunction in the regio n Ω0, (20) may not be a good choice simply because when the two charges ZβandZγare of equal and opposite charge, as in the paradigm case of electron scattering from hydrogen, the solution in /vector ραreduces to that of a free particle. i.e., Ψ(Ωα,Zβ=−Zγ) /vectorkα,/vector qα(/vector rα, /vector ρα) =ei(/vectorkα·/vector rα+/vector qν·/vector ρν)Cηkα(ζkα). (23) These reasons make it clear that in order to extend the 3C wave function into the the asymptotic region, Ωα(or into the interior regions for that matter) a diﬀerent app roach is warranted. The current approach, refered to here as “kinematic coupling,” involves the intro duction of an exact interaction potential given by: Vα I(/vector rα, /vector ρα)≡e2 4πǫ0 ZαZγ/vextendsingle/vextendsingle/vextendsingle/vector rα−µα mγ/vector ρα/vextendsingle/vextendsingle/vextendsingle+ZαZβ/vextendsingle/vextendsingle/vextendsingle/vector rα+µα mβ/vector ρα/vextendsingle/vextendsingle/vextendsingle−Zα(Zβ+Zγ) ρα , (24) Hence the three-body Schr¨ odinger equation takes the exact form: /bracketleftbigg −¯h2 2µα/vector∇2 /vector rα+ZβZγe2 4πǫ01 rα−¯h2 2Mα/vector∇2 /vector ρα+Zα(Zβ+Zγ)e2 4πǫ01 ρα+Vα I(/vector rα, /vector ρα)/bracketrightbigg Ψ/vectorkα,/vector qα(/vector rα, /vector ρα) =EtΨ/vectorkα,/vector qα(/vector rα, /vector ρα). (25) To remain completely general, one then asserts that the tota l center of mass energy may be written in the form Et=Eµα+EMα+EI, (26) where Eµα≡¯h@2k2 α 2µα>0 and EMα≡¯h2q2 α 2Mα>0 are the energies associated with the continuum state, two- body clusters, µαandMαrespectively, and the term EIaccounts for the remaining energy of the three-body interaction. Though the deﬁnition (26) is nonstandard, it r eﬂects the fact that because the three-body potential must remain inseparable in a completely general s olution, so too must the total energy. Due to the fact that the inseparable portion of the three-bod y Coulomb potential is now contained within the interaction potential, one can assume that the three-bo dy wavefunction takes the form, Ψ/vectorkα,/vector qα(/vector rα, /vector ρα) =ei(/vectorkα·/vector rα+/vector qα·/vector ρα)Cηkα(ζkα)Cηqα(ζqα)χ/vectorkα,/vector qα(/vector rα, /vector ρα), (27) 6where χ/vectorkα,/vector qα(/vector rα, /vector ρα) is an unknown function, and Cηkα(ζkα) and Cηqα(ζqα) are the known Coulomb waves deﬁned in equations (13) and (21) respectively. That is, the ansatz (27) incorporates the exact solution in the region Ω α(cf., equation (20)) explicitly. Substitution of the form (27) into the three-body Schr¨ odin ger equation (25) yields the following result: Cηqα(ζqα)χ/vectorkα,/vector qα(/vector rα, /vector ρα)/parenleftig −¯h2 2µα/vector∇2 /vector rα−i¯h µα/vectorkα·/vector∇/vector rα+ZβZγe2 4πǫ01 rα/parenrightig Cηkα(ζkα) +Cηkα(ζkα)χ/vectorkα,/vector qα(/vector rα, /vector ρα)/parenleftig −¯h2 2Mα/vector∇2 /vector ρα−i¯h Mα/vector qα·/vector∇/vector ρα+Zα(Zβ+Zγ)e2 4πǫ01 ρα/parenrightig Cηqα(ζqα) −Cηkα(ζkα)Cηqα(ζqα)/parenleftig −¯h2 2µα/vector∇2 /vector rα−¯h2 2Mα/vector∇2 /vector ρα−i¯h µα/vectorkα·/vector∇/vector rα−i¯h Mα/vector qα·/vector∇/vector ρα/parenrightig χ/vectorkα,/vector qα(/vector rα, /vector ρα) (28) −¯h2 µαCηqα(ζqα)/vector∇/vector rαCηkα(ζkα)·/vector∇/vector rαχ/vectorkα,/vector qα(/vector rα, /vector ρα)−¯h2 MαCηkα(ζkα)/vector∇/vector rαCηqα(ζqα)·/vector∇/vector ραχ/vectorkα,/vector qα(/vector rα, /vector ρα) +VI α(/vector rα, /vector ρα)Cηkα(ζkα)Cηqα(ζqα)χ/vectorkα,/vector qα(/vector rα, /vector ρα) =EICηkα(ζkα)Cηqα(ζqα)χ/vectorkα,/vector qα(/vector rα, /vector ρα), where the exponential terms have been cancelled from both si des of the equation and the following vector identities have been employed: /vector∇(AB) =B/vector∇A+A/vector∇B, and /vector∇2(AB) =B/vector∇2A+A/vector∇2B+ 2/vector∇A·/vector∇B. Using equations (13) and (21) in conjunction with the deﬁnit ion (26), one ﬁnds that the ﬁrst two lines in this unwieldy expression are identically zero. Then afte r dividing by the product Cηkα(ζkα)Cηqα(ζqα), the following equation for the unknown function, χ/vectorkα,/vector qα(/vector rα, /vector ρα) is derived: /bracketleftbigg −¯h2 2µα/vector∇2 /vector rα−i¯h µα/vectorK(/vector rα)·/vector∇/vector rα−¯h2 2Mα/vector∇2 /vector ρα−i¯h Mα/vectorQ(/vector ρα)·/vector∇/vector ρα+VI α(/vector rα, /vector ρα)/bracketrightbigg χ/vectorkα,/vector qα(/vector rα, /vector ρα) =EIχ/vectorkα,/vector qα(/vector rα, /vector ρα), (29) is found, where /vectorK(/vector rα)≡/vectorkα−i/vector∇/vector rαCηkα(ζkα) Cηkα(ζkα), (30a) /vectorQ(/vector ρα)≡/vector qα−i/vector∇/vector ραCηqα(ζqα) Cηqα(ζqα), (30b) are the proposed position-dependent local momenta for this kinematic coupling model. The principle diﬀerence in this development is that the loca l momenta arise purely from the physical structure of the inseparable three-body Schr¨ odinger equa tion, and that their mathematical form implies the existence of position dependent momenta. Furthermore, in t he current development, the local momenta are predicted in a symmetric fashion so that both “legs” of the th ree-body interaction experience distortions which depend upon the congugate coordinates. That is, /vector rαis congugate to /vectorkαand/vector ραis congugate to /vector qα. This means that the distortion that two of the particles experien ce does not depend explicitly upon the distance to the third particle as in [1], but is rather wholly attribut able to a genuine three-body eﬀect wherein a very intuitive description of the three-body scattering event a rises: The two reduced mass clusters µαandMα, initially described by the two-body waves, Cηkα(ζkα) andCηqα(ζqα) respectively, scatter oﬀ of the interaction potential VI(/vector rα, /vector ρα) and acquire a local momenta. The subsequent motion of these clusters is then of course dic tated by the function χ/vectorkα,/vector qα(/vector rα, /vector ρα) which must satisfy (29). While one may argue that solving equation (29) is a greater ta sk than solving the original Schhr¨ odinger equation, a solution that is consistent with the 3C wavefunc tion in the region Ω 0may be established. The details of this solution are not important to the present dis cussion and are relegated to the appendices. (See appendix B.) What will be of great importance is the nature of the predicted local momenta, (30). 74 A Generalized Local Momentum Thelocal momenta introduced in this kinematically coupled model depend upon the congugate coordinate. Because of this variations in the momenta will generally con tribute to the solutions. To understand how and where these contributions will be important, a generalized “local distortion” term, /vectorD/vector p(/vector r) is introduced, where /vector pand/vector rmay be either /vectorkνor/vector qνand/vector rνor/vector ρν, respectively. Hence a general position-dependent moment um may be deﬁned as follows: /vectorP(/vector r) =/vector p−/vectorD/vector p(/vector r), (31) where the exact form of the local distortion is given by5: /vectorD/vector p(/vector r)≡i/vector∇rlnCηp=iηpp1F1[(1 +iηp),2,−iζp] 1F1(iηp,1,−iζp)(ˆp+ ˆr) (32a) /vectorD/vector p(/vector r) =/braceleftiggiηpp(ˆp+ ˆr) , r= 0 ηp r/bracketleftig 1−e−i(ζp+2δηp(ζp))/bracketrightig (ˆp+ˆr) ˆp·(ˆp+ˆr), r >0. (32b) Here ηp=Z/parenleftbigge2 4πǫ0¯h/parenrightbiggµ ¯hp=Z/parenleftbigge2 4πǫ0¯h/parenrightbigg/radicalbiggµ 2E, (33) Cηp(ζp) satisﬁes /parenleftbigg −¯h2 2µ/vector∇2 /vector r−i¯h µ/vector p·/vector∇/vector r+Ze2 4πǫ01 r/parenrightbigg Cηp(ζp) = 0, (34) andδηp(ζp) is areal-valued, position dependent phase, deﬁned by tan/bracketleftbig δηp(ζp)/bracketrightbig =ℑ[1F1(iηp,1,−iζp)] ℜ[1F1(iηp,1,−iζp)]. (35) While the general form (32a) has been used by many researcher s, the form (32b) is unique and has been introduced to help illucidate the physical signiﬁcance of t he local momentum. e.g., one may write: ( r∝negationslash= 0) /vectorD/vector p(/vector r) =Z/parenleftbigge2 4πǫ0¯h/parenrightbigg/radicalbiggµ 2Ea∗ ηp(ζp) r(ˆp+ ˆr), (36) where Zandµare the relevant product charge andreduced mass of the pair described by Cηp(ζp), and a∗ ηp(ζp)≡/bracketleftig 1−ei(ζp+2δηp(ζp))/bracketrightig (1 + ˆp·ˆr). (37) The form (36) shows that the local momentum is essentially a1 rdamped-oscillator function, and that it is analytic everywhere except possibly at requal to zero. Interestingly, it can also be seen that the loc al momentum has equal radiation (in the ˆ rdirection) and induction (in the ˆ pdirection) components. This is indicative of the possibility of a tensor force in a Classica l treatment or of a nonconserved current density in a Quantum treatment of the three-body interaction. Perha ps more importantly, one sees that the local distortion depends upon not only the radial separation of th e constituent particles, but also upon both the relative energy and reduced mass of the system. In Figure 2 the realpart of the distortion experienced by a electron-proton con tinuum pair has been plotted with various relative energies and an arbitrary sca ttering angle of θ= 0. One sees immediately that 5See Appendix A for details concerning the derivation of this form. 8the distortion “falls oﬀ” very quickly with increasing radi al separation and even more dramatically with increased relative energy. To see how the distortion depends upon the reduced mass of the system, observe that in Figure 3, the real part of the distortion experienced by an electron-electron pair and a electron-proton pair have been plotted. Notice that the distortion increases accordingly with incr eased mass, and that the sign of the distortion is diﬀerent for the cases of attraction and repulsion. This very interesting point will be discussed in greater det ail below, but for now note that in addition to the important physical properties of the local distortion, there are also important mathematical properties. The most important of these is that because the local distort ion is essentially an oscillatory function, the existence of turning and stationary (i.e., maxima and minim a) points may yield much insight. In Figure 4 one sees that the stationary points6occur quite often, so that the local distortion will very often be identically zero! Moreover, these zeroes will be ve ry dense on a macroscopic scale and so may greatly alter the topology of a scattering event. This can be seen even more dramatically in Figure 5, where three-dimensional and contour plots of the local distortio n have been shown. Though there is no analytic form for determining the zeroes o f the conﬂuent hypergeometric function, the stationary points of the local distortion may be found by ﬁnding the nonzero roots to (37). Hence one ends up solving the following transcendental equation: 1 r=p 2[nπ−δp(/vector r)](1−cosθ), (38) where nis an integer. Note that if δp(/vector r) were a constant, (38) would be the equation of a conic sectio n with an eccentricity of 1 - parabolas. Hence one sees that the stat ionary points of the local distortion lie (very nearly) along the classically forbidden “trajectories” fo r particles with positive energy! While this may seem to be no more than coincidence, δp(/vector r) is in fact very nearly constant between the stationary poin ts (see Fig. 6), and one may infer that these “zero-distortion-trajecto ries” are in fact the dominant contributions in a path-integral approach to the quantum, three-body scatter ing problem. Indeed, from this point of view, one may argue that they must be the dominant contributions for li ght-ion–atom processes in the asymptotic regions, where great success has been achieved while ignori ng local distortion eﬀects. What is not clear however is how these local eﬀects will contribute in the so ca lled “interior regions” or for heavy-ion–atom processes. 5 Kinematic Rearrangement Above it was noted that the local distortion had diﬀerent sig ns for the attractive and repulsive cases. The real importance of this observation is that if one imagines a three-body system composed of a tightly bound continuum state pair, with an initial momentum /vector p, and a third, mutually interacting particle, then upon break-up the continuum pair will acquire a local momentum /vectorP(/vector r). Hence the continuum pair will experience a momentum change (or impulse ) given by /vectorI(/vector r)≡ ℜ/vectorP(/vector r)−/vector p=−ℜ/vectorD/vector p(/vector r). (39) Evidently one ﬁnds, (for r∝negationslash= 0) /vectorIattractive = +\|Z\|/parenleftbigge2 4πǫ0¯h/parenrightbigg/radicalbiggµ 2E/bracketleftbig 1−cos(ζp+ 2δηp)/bracketrightbig ˆp·(ˆp+ ˆr)1 r(ˆp+ ˆr) (40a) /vectorIrepulsive =−\|Z\|/parenleftbigge2 4πǫ0¯h/parenrightbigg/radicalbiggµ 2E/bracketleftbig 1−cos(ζp+ 2δηp)/bracketrightbig ˆp·(ˆp+ ˆr)1 r(ˆp+ ˆr). (40b) The equations (40) imply that the two particles that are init ially attracted to one another, within the continuum pair described by Cηp(ζp), will experience an impulse that will tend to increase thei r radial 6That is, the points where the zeroes of the real and imaginary parts of the local distortion coincide. 9separation upon breakup and conversely, that two particles that are initially repelled from one another will experience an impulse that will tend to decrease their radia l separation. In other words, at small values of the radial separation, opposite(same) sign-charged parti cles will tend to repel(attract) one another due to the local distortion eﬀects! To illustrate this point, the realpart of the distortion for several diﬀerent continuum pairs have been calculated and the results are shown in Figure 7. Observe tha t even though all of these distortion terms decrease with increasing radial separation, the magnitude of the distortion in the interior regions depends critically upon the system being studied. As was shown above in Figure 3, the distortion eﬀects increases dramatically with an increase in the reduced mass. Speciﬁca lly, note that for the case of a proton–anti-proton ((e) if Fig. 7) or proton-proton ((f) in Fig. 7), the range has been shortened accordingly, to show that the distortion in the reaction zone is dramatically changed. In deed, for the case of the proton–anti-proton, the magnitude of the local distortion eﬀect may be large enou gh and in a direction such that a kinematic rearrangement of the particles may occur. In addition, beca use of the dependence upon the inverse of the square-root of the relative energy, these eﬀects will be eve n more pronounced at lower relative energies. This kind of phenomena can be observed in the data obtained by Wiese et. al. , [2] on the breakup of the excited ion ( H+ 3)∗. This highly unstable ion of hydrogen decays in a two step pro cess, as follows: (H+ 3)∗−→H∗∗ 2+H+ i→H+ f+H−+H+ i. The data (see Figure 8) show a slight asymmetry with small inc reases in energy (from 6 .5 eV to 7 .5 eV) which can be accounted for in a direct manner by considering t he local distortion eﬀects herein derived. To see this note that during the nearly co-linear breakup of t he excited H+ 3ion the “initial”7proton H+ i is emitted in the forward direction and the doubly excited H∗∗ 2ion is emitted in the backward direction. The subsequent breakup of this ion occurs such that a “ﬁnal” p roton H+ fand the nearly equally weighted H−ion are formed.[2] At this point the H−is preferentially associated with H+ f. However, after the breakup, the continuum pair ( H+ f, H−) will acquire a local momentum ,/vectorP(/vector r), which has a component in the ˆrdirection. Hence this gained momentum will act to increase t he radial separation and will push the H−over the Coulomb Saddle, so that it H−will be preferentially associated with the “initial” proto n,H+ i. Now, because the two protons are in fact indistinguishable, one will observe an asymmetry between “low” and “high” energy scattering events. To help visualize this process, a schematic diagram (see Fig. 9.) has been developed. Further analysis shows that, as mentioned above, because th e local distortion depends upon1√ E, as the energy is increased, a proportionately smaller number of pa rticles will experience an impulse that is large enough to alter their ﬁnal state distribution. Indeed, duri ng the breakup of the H+ 3ion, the magnitude of the local distortion eﬀects experienced by the ( H+ f, H−) continuum pair while in the reaction zone are many orders of magnitude larger than the eﬀects experienced by an electron-proton continuum pair. (see Fig. 10) With this understanding, one may conclude in a straightforw ard manner, that the observed asymmetry can be attributed to local distortion eﬀects of the three-body s ystem. Moreover, the degree of asymmetry may be predicted as follows: /radicalbigg Ehigh Elow∝amplitude of rearrangment at low energy amplitude of rearrangement at high energy. (41) Therefore, with the probabilistic interpretation of the wa vefunction, one may write, /radicalbigg Ehigh Elow≡/radicaligg Plow Phigh(42) where PhighandPloware the probabilities of rearrangement for the high and low e nergy states, respectively. That is, using the actual data (See Fig. 8.) the ratio of the nu mber of rearranged particles to the total 7“Initial” and “ﬁnal” are used here as in [2] to indicate the or der in which the particles were detected during triple coinc idence measurements. 10number of detected particles may be calculated. Doing this o ne obtains the following results: /radicalbigg Ehigh Elow=/radicalbigg 7.5 eV 6.5 eV= 1.1 and Plow Phigh=/radicalig 31 72/radicalig 14 47= 1.2 The absolute error between these two calculations is ≈8.7% and illustrates that the generalized local momentum, herein developed may in fact be the actual momentu m of the ( H−, H+ f) continuum pair during breakup. Indeed, one may consider revaluating the importan ce of the generalized local momenta. Instead of treating it as a mere mathematical nicety, one may place it on ﬁrm physical ground by viewing it is a formal gauge condition for three-body interactions. 6 Towards a Three-Body Gauge If one grants that the mathematical and computational evide nce gathered here, is suﬃcient to guide further experimental and theoretical investigations by consideri ng the generalized local momentum as a formal three-body gauge condition, then one can construct a three- body gauge transformation. i.e., one may write: /vector p−→/vectorP(/vector r) =/vector p−/vectorD/vectorP(/vector r), (43) when working with three-body systems, in much the same way th at one would write, /vector p− →/vectorP(/vector r) =/vector p−e¯h/vectorA(/vector r), (44) when working with particles in an external magnetic ﬁeld. (i .e., the Coulomb Gauge.) The subtlety here is that while the transformation, /vectorA(/vector r)−→/vectorA(/vector r) +/vector∇/vector rΩ(/vector r) preserves the form of the magnetic ﬁeld, /vectorB(/vector r) =/vector∇/vector r×/vectorA(/vector r), and provides for the global gauge symmetry of the electromagnetic interaction, the condition (43) will c ontribute a real-valued, position-dependent phase to the relevant wavefunction.8Thus any proposed three-body interaction mechanism would r equire a local gauge symmetry .9 To see how a position-dependent phase arises, one may consid er a situation similar to that discussed in Sec. 5. That is, consider three-body system consisting of a c ontinuum pair described by the wavefunction, Ψ/vector p(/vector r) =e−i¯h/vector p·/vector rCηp(ζp) together with a third, mutually interacting particle. The n, upon breakup, the pair will acquire a local momentum and according to (43), the wave function will transform as follows: Ψ/vector p(/vector r)−→Ψ/vectorP(/vector r)(/vector r) =e−i[/vectorP(/vector r)·/vector r]Cηp(ζp) =e−i/vector p·/vector re+i/vectorD/vector p(/vector r)·/vector rCηp(ζp). (45) Using (32b), one then ﬁnds that10, (for r∝negationslash= 0) /vectorD/vector p(/vector r)·/vector r=a0ηp/bracketleftbig 1−cos(ζp+ 2δηp)/bracketrightbig +ia0ηpsin(ζp+ 2δηp). (46) 8It is interesting to note however that, /vector∇/vector r×/vectorD/vector p(/vector r) =i/vector∇/vector r×/vector∇/vector rlnCηp(ζp)≡/vector0. 9For those readers that are unfamiliar with the terms global andlocalin this context, note that a global gauge transformation is one that preserves the modulus of the appropriate wavefun ction. eg., Ψ( /vector r)→e−iaΨ(/vector r). On the other hand, a local gauge transformation generally does not and may take the form, Ψ( /vector r)→ef(/vector r)Ψ(/vector r). 10Note that the “extra” a0arises here because of the scale used in the plots. i.e., one w rites/vector r→a0rˆr. 11Therefore the transformed continuum two-body wavefunctio n takes the following form: Ψ/vectorP(/vector r)(/vector r) =ei/vector p·/vector reia0ηp[1−cos(ζp+2δηp)]e−a0ηpsin(ζp+2δηp)Cηp(ζp), (47) which shows the explicit form for the position dependent pha se. i.e., one may write, Sηp(ζp)≡ia0ηp/bracketleftbig 1−cos(ζp+ 2δηp)/bracketrightbig −a0ηpsin(ζp+ 2δηp), (48) and see that the transformation is indeed indicative of a glo bal gauge symmetry. eg., Ψ/vectorP(/vector r)(/vector r) =eSηp(ζp)Ψ/vector p(/vector r) (49) It is of course the absolute square of the wavefunction that i s truly important for predicting whether or not this phase will signiﬁcantly eﬀect the experimental ﬁnding s. To this end, one may take the absolute square of (47) and show that the imaginary -part of the local distortion contributes a real-valued, position-dependent phase. i.e., \|Ψ/vector p(/vector r)\|2=e−2a0ηpsin(ζp+2δηp)/vextendsingle/vextendsingleCηp(ζp)/vextendsingle/vextendsingle2=e2ℜSηp(ζp)/vextendsingle/vextendsingleCηp(ζp)/vextendsingle/vextendsingle2. (50) The exceedingly small magnitude of the term 2 a0ηpsin(ζp+2δηp) for the relative energy and reduced masses of the systems considered in current research ﬁndings, make s it clear that one may write e−2a0ηpsin(ζp+2δηp)− →1. (51) While this reinforces the fact that the proposed framework w ill leave the asymptotic description of the three- body scattering event unaltered, (as was established in App endix B) it does not address the extent to which the proposed gauge transformation, (43) will alter the desc ription in the reaction zone. To begin an investigation of the expected behavior in the rea ction zone, one may construct a semi-classical expectation value for the total energy of the continuum pair ,Cηp(ζp). To do this recall that the expectation value of the semi-classical Hamiltonian for this system (be fore breakup) would be, ∝angb∇acketleftH0(/vector p,/vector r)∝angb∇acket∇ight=¯h2p2 2µ+Ze2 4πǫ01 r. (52) where p=√2µE ¯his the momentum of the continuum pair before breakup. During breakup, the pair will acquire a local momentum, so that the expectation value of th e semi-classical Hamiltonian for the system would then become, /angbracketleftig H(/vectorP(/vector r),/vector r)/angbracketrightig =¯h2/parenleftig k2+\|/vectorD/vector p(/vector r)\|2/parenrightig 2µ+Ze2 4πǫ01 r. (53) The eﬀects of the proposed gauge transformation may then be o bserved by plotting the expectation values and varying the relative energy, E(see Fig. 11), the scattering angle, θ(see Fig. 12), and the reduced mass, µ(see Fig. 13). All of these show undeniably that while the asy mptotic form remains unchanged with respect to variations of all kinematic parameters, gen uine three-body distortion eﬀects may arise in the reaction zone. Note speciﬁcally that the variation of th e relative energy with the scattering reinforces the interpretation oﬀered in Sec. 5. Speciﬁcally, the magni tude of the distortion eﬀects for large scattering angles (corresponding to the nearly colinear breakup of the continuum pair) in the reaction zone may cause two opposite-sign-charged particles to increase their rad ial separation, and appear to repel one another. Conversely, two like-sign-charged particles would decrea se their radial separation, and appear to attract one another. As a last exercise, one may ask if the position-dependent pha se introduced by the three-body gauge formalism could be measured. To answer this question one may observe that e−2ηpa0sin(ζp+2δp)∼=e−2ηpa0=e−√ 2za0e2 4πǫ0¯h√µ eE, (54) 12where (33) has been used for ηp, andEis measured in electron-volts. As mentioned above, this con tribution approaches unity for all systems of physical interest in the realm of current experiments in atomic physics. One can however venture outside of this realm and ask what ene rgy and/or reduced mass is needed to obtain a measurable deviation from unity. If one uses the heaviest p urely atomic species, either a proton-proton (p−p) or a proton–anti-proton ( p−¯p) continuum pair, and assumes that a deviation of one part in a million can be measured, then the energy scale needed is on the order o f 1µeV! The variation of the term e−2a0ηp at these energies is shown in Figure 14, and illustrates that one may detect a change in the absolute square of the wavefucntion, thus altering the relevant scattering amplitude. Moreover, the exceedingly small energy scale required to detect these asymptotic distortion eﬀect s in either electron(or positron) scattering from hydrogen or in electron-electron or electron-positron ion ization processes provides a precise understanding for the success achieved in these areas while ignoring local momentum eﬀects. 7 Conclusion In the above it has been shown that a generalized, position-d ependent local momenta, which depends upon the congugate coordinate through the logarithmic gradient of a continuum state Coulomb wave, may provide evidence for the manifestation of genuine three-body disto rtion eﬀects in the reaction zone. The form of this local momenta was derived from a consideration of the exact t hree-body wavefunction, for the case when two of the three, mutually interacting, particles are far ap art, and it indicates that the eﬀects do not depend explicitly upon the location of the third particle. For this reason, the eﬀects may be viewed as a distortion of the initial two-body continuum state wavefunction of the two remaining particles. This interpretation was adopted, and it was shown that the local distortion eﬀects co uld be used to provided a rigorous, physical description of the observed asymmetry in the data obtained b y Wiese et. al., [2] on the breakup of three massive Coulomb particles. The degree of this asymmetry was then predicted with an error of less than 10%, and it was also shown that while the distortion eﬀects we re large in the reaction zone, the asymptotic form of the relevant two-body interaction may be retained by treating the local momentum acquisition as a three-body gauge constraint. Furthermore, the evidence f or detecting asymptotic variations in the µeV range, as presented in Fig. 14 suggest that more experimenta tion be focused on these low energy, heavy- ion–atom processes. Indeed, these experiments may yield ne w insight into a mechanism by which electrical forces may contribute to the fusion process! That is, the qua ntum-mechanical–impulse interpretation oﬀered here shows that two like-charged particles may in fact attra ct one another due to local distortion eﬀects in the reaction zone. In addition to these ﬁndings and predictions, one may learn m uch by noting that by adopting the proposed three-body gauge transformation, one ﬁnds that an electron -proton continuum pair exhibited a very small amount of distortion in the reaction zone. This result provi des a rigorous explanation of why the paradigm 3C wavefunction works so well for light-ion–atom scatterin g [6],[9]. Moreover, the general framework shows that an electron-electron continuum pair would experience distortion eﬀects of lesser magnitude, due to its greatly decreased reduced mass. These results can again be u sed to explain why Qiu et. al.,[10] achieved amazing success in modeling electron-electron photo-ioni zation processes, while ignoring local momentum eﬀects. Indeed, if one recalls the path-integral interpret ation suggested in Sec. 5, then the success of these ﬁndings for light-ion–atom processes may be attributed to t he fact that the leading contribution to the relevant cross sections are the “paths” along which the loca l distortion eﬀects are identically zero. While no analytic form exists for calculation of these roots, one can compile a table for use in numerical calculations and construct a solution that better reﬂects the physical na ture of the three-body interaction. References [1] E. O. Alt and A. M. Mukhamedzhanov, Phys. Rev. A 47, 2004(1993). [2] L. Wiese et. al. , PRL 79, 4982 (1997). 13[3] P.P Fiziev, T. Y. Fizieva, Few-Body Systems 271 (1987). [4] P. J. Redmond, ca. 1972,unpublished (as referenced by Ro senberg). [5] L. Rosenberg, Phys. Rev. D 8, 1833(1972). 56, 370(1997). [6] M. Brauner et. al. , J. Phys. B 22, 2265(1989). [7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Ser ies, and Products . Academic Press, London (1965). [8] J. Berakdar, J. S. Briggs, Phys. Rev. A [9] S. Jones and D. Madison, Phys. Rev. A 55444 (1997). [10] Y. Qiu et. al. , Phys. Rev. A 57, R1489 (1998). [11] M. Lieber and A. M. Mukhamedzhanov, Phys. Rev. A 54, 3078(1996). [12] A. Engelns et. al, , J. Phys. B (30), L811(1997). A The Logarithmic Derivative of the 1F1(iηp,1;−1ζp)Function The logarithmic derivative of the conﬂuent hypergeometric function is deﬁned here as follows: /vectorD/vector p(/vector r)≡i/vector∇rln1F1(iηp,1;−iζp) =i/vector∇r1F1(iηp,1;−iζp) 1F1(iηp,1;−iζp) (55) =iηpp1F1[(iηp+ 1),2;−iζp] 1F1(iηp,1;−iζp)(ˆp+ ˆr). Then using the well known recursion relationship,(See refe rence [7] for instance.) z b1F1[(a+ 1),(b+ 1), z] =1F1[(a+ 1), b, z]−1F1(a, b, z), (56) one may write11 /vectorD/vector p(/vector r) = iηpp[1F1[(iηp+ 1),1;−iζp]−1F1(iηp,1;−iζp)] ζp1F1(iηp,1;−iζp)(ˆp+ ˆr), ζ p>0. (57) Further simpliﬁcation of the logarithmic derivative is obt ained by use of the Kummer relation[7] for the conﬂuent hypergeometric equation. e.g., 1F1(a, b, z) =ez1F1[(b−a), b,−z]. (58) With this relation one ﬁnds that the distortion may be rewrit ten as follows: /vectorD/vector p(/vector r) = iηpp/bracketleftbig e−iζp1F1(−iηp,1;iζp)−1F1(iηp,1;−iζp)/bracketrightbig ζp1F1(iηp,1;−iζp)(ˆp+ ˆr) (59a) =ηp rˆp·(ˆp+ ˆr)/bracketleftbigg 1−e−iζp1F1(−iηp,1;iζp) 1F1(iηp,1;−iζp)/bracketrightbigg (ˆp+ ˆr), r > 0. (59b) 11Because of the division by ζpin equation (57), the region of validity for the logarithmic derivative is limited to ζp>0 only as indicated. One can however ﬁnd the actual value at ζp= 0.c.f.equation (62). 14At this point note that the ratio of conﬂuent hypergeometric equations in equation (59b) is in fact a very special case because the function on top is the complex conju gate of the function on the bottom! i.e., 1F1(−iηp,1;iζp) =1F1†(iηp,1;−iζp), (60) so that one may deﬁne a real-valued, position-dependent phase, tan/bracketleftbig δηp(ζp)/bracketrightbig =ℑ[1F1(iηp,1;−iζp)] ℜ[1F1(iηp,1;−iζp)]. (61) The logarithmic derivative may then be written as follows: /vectorD/vector p(/vector r) =/braceleftiggiηpp(ˆp+ ˆr), r= 0 ηp r/bracketleftig 1−e−i(ζp+2δηp(ζp))/bracketrightig (ˆp+ˆr) ˆp·(ˆp+ˆr), r > 0, (62) and one sees that in this form, the local distortion is analyt ic everywhere, except possibly at r= 0. This result is very signiﬁcant, because it allows one to investig ate the nature of the local momenta well inside of the interior regions of a scattering event in a rigorous ma nner. Only in this way can one determine the relative importance of these local eﬀects. B An Alternative Three-Body Wavefunction To solve equation (28) for the unknown function χ/vectorkα,/vector qα(/vector rα, /vector ρα) one normally assumes that the interaction energy (cf, equation (26)) satisﬁes EI= 0. This technique was developed by Popalilios as referenced in [6], and asserts that the total center of mass energy of the three-body system is partitioned among the two reduced mass clusters, µαandMα. Hence one may write /bracketleftbigg −¯h2 2µα/vector∇2 /vector rα−i¯h µα/vectorK(/vector rα)·/vector∇/vector rα−¯h2 2Mα/vector∇2 /vector ρα−i¯h Mα/vectorQ(/vector ρα)·/vector∇/vector ρα+VI α(/vector rα, /vector ρα)/bracketrightbigg χ/vectorkα,/vector qα(/vector rα, /vector ρα) = 0.(63) and see that χ/vectorkα,/vector qα(/vector rα, /vector ρα) must be built to incorporate each term in the interaction po tential. To accomplish this, assume that χ/vectorkα,/vector qα(/vector rα, /vector ρα) is given by χ/vectorkα,/vector qα(/vector rα, /vector ρα) =C−ηqα(ζqα)fβ(/vector rα, /vector ρα)fγ(/vector rα, /vector ρα), (64) where C−ηqα(ζqα) satisﬁes /bracketleftbigg −¯h2 2Mα/vector∇2 /vector ρα−i¯h Mα/vector qα·/vector∇/vector ρα−Zα(Zβ+Zγ)e2 4πǫ01 ρα/bracketrightbigg Cηqα(ζqα) = 0. (65) Because the function C−ηqα(ζqα) is a known function, which exactly incorporates the intera ction term −Zα(Zβ+Zγ)e2 4πǫ0ρα, this substitution results in the following (exact) couple d equations for the functions fν(/vector rν, /vector ρν):(ν= β, γ) /bracketleftig −¯h2 2µα/vector∇2 /vector rα−¯h2 2Mα/vector∇2 /vector ρα−i¯h µα/vectorKγ α(/vector rα;/vector rγ, /vector ργ)·/vector∇/vector rα−i¯h Mα/vectorQγ α(/vector ρα,/vector rγ, /vector ργ)·/vector∇/vector ρα+V(γ) ν,eﬀ(/vector rβ, /vector ρα)/bracketrightig fβ(/vector rβ, /vector ρβ) = 0, (66a) /bracketleftig −¯h2 2µα/vector∇2 /vector rα−¯h2 2Mα/vector∇2 /vector ρα−i¯h µα/vectorKβ α(/vector rα;/vector rβ, /vector ρβ)·/vector∇/vector rα−i¯h Mα/vectorQβ α(/vector ρα,/vector rβ, /vector ρβ)·/vector∇/vector ρα+V(β) γ,eﬀ(/vector rγ, /vector ρα)/bracketrightig fγ(/vector rγ, /vector ργ) = 0, (66b) where (for ǫ∝negationslash=ν, with ǫ=β, γ) /vectorKǫ α(/vector rα;/vector rǫ, /vector ρǫ) = /vectorkα−i/vector∇/vector rαlnCηkα(ζkα)−i 2/vector∇/vector rαlnfǫ(/vector rǫ, /vector ρǫ), (67a) /vectorQǫ α(/vector ρα;/vector rǫ, /vector ρǫ) = /vector qα−i/vector∇/vector ρα/bracketleftbig lnCηqα(ζqα)C−ηqα(ζqα)/bracketrightbig −i 2/vector∇/vector ραlnfǫ(/vector rǫ, /vector ρǫ), (67b) 15and12 V(ǫ) ν,eﬀ(/vector rν, /vector ρα) =ZαZǫe2 4πǫ01 rν×/braceleftigg 1 +4πǫ0η2 qα ZαZǫe2Mα/parenleftbiggrν ρα/parenrightbigg/bracketleftig 1−e−i(ζqα+2δηqα(ζqα))/bracketrightig/bracketleftig 1−e−i(ζqα+2δ−ηqα(ζqα))/bracketrightig/bracerightigg . (68) While these equations may seem even more complicated than th e original equation, there are many important aspects to note: •The eﬀective potential reduces to the Coulomb potential for theν-channel, in the asymptotic regions Ω0and Ω α. i.e., V(ǫ)(Ω0,Ωα) ν,eﬀ(/vector rν, /vector ρα) =ZαZǫe2 4πǫ01 rν(69) •The second term in /vectorQǫ α(/vector ρα,/vector rǫ, /vector ρǫ) vanishes in the asymptotic regions because, /bracketleftbig Cηqα(ζqα)C−ηqα(ζqα)/bracketrightbig limρα→∞−→e−iηqαlnζqαe+iηqαlnζqα= 1. (70) (Indeed, this cancellation of the logarithmic phases was “b uilt in” to the solution!) •The ﬁrst two terms in /vectorKǫ α(/vector rα,/vector rǫ, /vector ρǫ) may be identiﬁed as a position-dependent, local momentum. i.e., /vectorKα(/vector rα) =/vectorkα−i/vector∇/vector rαlnCηkα(ζkα). •The solutions are coupled in a completely symmetric way by th e terms −i 2/vector∇/vector rαlnfν(/vector rǫ, /vector ρǫ) and −i 2/vector∇/vector ραlnfν(/vector rǫ, /vector ρǫ), so that a numerical solution of these equations would be ma nifestly less computa- tionally intensive. Furthermore, because the equations (66) are exact, they wou ld be more reliable in ab initio calculations. Approximate solutions that are valid through second order, may be achieved in a direct manner following the techniques outlined in [12] and may be written in the form of distorted and coupled Coulomb waves. e.g., fν(/vector rν, /vector ρν) =CηKǫν(ζKǫν)≡Γ(1−iηKǫν)e−πηKǫν 21F1/parenleftbig iηKǫν,1,−iζKǫν/parenrightbig , (71) where the conventional, two-body normalization procedure has been employed so that that incoming wave has unit magnitude. The complete three-body wavefunction may then be written in the form: ΨKC5C /vectorkα,/vector qα(/vector rα, /vector ρα) =ei(/vectorkα·/vector rα+/vector qα·/vector ρα)Cηkα(ζkα)Cηqα(ζqα)C−ηqα(ζqα)CηKγ β(ζKγ β)CηKβ γ(ζKβ γ) (72) a product of ﬁve, kinematically coupled Coulomb waves. To se e that this solution is valid, ﬁrst note that one can show in a straightforward manner that the distorted C oulomb waves have the following asymptotic form: C(Ω0) ηKǫν(ζKǫν)→Cηkν(ζkν) (73) Hence, due to the relationship (70), the asymptotic form in Ω 0will be identical to that of the Redmond, 3C wavefunction. 12Note that the general form of the local distortion given in (3 2b) has been used to derive this form. 16The Jacobi Coordinates /vector rα /vector rβ/vector rγ /vector ρα /vector ρβ/vector ργ mαmβ mγ Figure 1: The Jacobi coordinates are most often used to study many-body kinematics, because any orthogonal pair, ( /vector rν, /vector ρν), ν=α, β, γ may be used. See equations (1) for the deﬁnition of the reduce d masses. EDependence of the Local Distortion 0 5 10 15 20-3-2-10 54.4105 r(a0)ℜD/vector p(/vector r) (1 p)eV eV eV Figure 2: The real part of the local distortion experienced b y an electron-proton continuum pair with relative energies as indicated and a scattering angle of θ= 4◦. 17µandZDependence of the Local Distortion 5 10 15 20-2.5-2-1.5-0.500.51 r(a0)ℜD/vector p(/vector r) (1 p) e−¯e e−p Figure 3: The real part of the local distortion experienced b y both electron-electron and electron-proton continuum pairs, with values as in Fig. ??. Note that the sign of the distortion is diﬀerent for the attr active (e−, p+) and repulsive ( e−, e−) cases. Stationary Points of the Local Distortion 0 5 10 15 20-2-1.5-1-0.500.5 r(a0)D/vector p(/vector r) (1 p) ℜ ℑ Figure 4: The local distortion experienced by an electron-p roton continuum pair with a relative energy of 10 eV, and a scattering angle of θ= 16◦. Note that when the zeroes of realandimaginary part of the distortion coincide, the distortion contribution will be i dentically zero. 18Topology of the Local Distortion 0 5 10 15 20045901351800 -1 0 5 10 15ℜD/vector p(/vector r) (1 p) r(a0)r(a0)θ (a) 0 5 10 15 2004590135180 -20 r(a0)θℜD (b) Figure 5: Shown here are (a) three-dimensional and (b) conto ur plots of the realpart of the local distortion experienced by an electron-proton continuum pair with a rel ative energy of 10 eV. 19Local Distortion Phase Eﬀects 0 5 10 15 20 25 30-1.5-1-0.500.511.5 2010 r(a0)δηp(ζp) θ= 20◦ eV eV Figure 6: Shown here is the position dependent-phase, δηp(ζp). Note that it is very nearly constant over one log-cycle and that it is nearly independent of the relative e nergy of the electron-proton continuum pair. 20Figure 7: Shown are the local distortion eﬀects experienced by (a) electron-positron, (b) electron-electron, (c) electron-antiproton , (d) positron-proton, (e) proton -antiproton, and (f) proton-proton, continuum pairs withE= 10 eV and θ= 16◦. While these eﬀects become negligible asymptotically, the eﬀects become more pronounced in the interior regions; the range has been short ened in (e) and (f) to show the dramatic change due to the increase in the reduced mass of the pair 21WYJ Dalitz Plot Et= 6.5 eV,Plow=/radicalig 31 72 (a) WYJ Dalitz Plot Et= 7.5 eV,Phigh=/radicalig 14 47 (b) Figure 8: These original data, obtained directly from the au thors, show that as the total center-of-mass energy of the three-body system, Et, is increased, a proportionately smaller number of particl es undergoes kinematic rearrangement. Shown also are the predicted prob ability ratios for this rearrangement for the (a) Et= 6.5 eV and (b) Et= 7.5 eV, triple-coincidence events. 22Explanation of Three-Body Distortion Eﬀects 12345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123123456789012345678901234567890121234567890123456789012345678901212312345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123123456789012345678901234567890121234567890123456789012345678901212312345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123123456789012345678901234567890121234567890123456789012345678901212312345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123123456789012345678901234567890121234567890123456789012345678901212312345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123123456789012345678901234567890121234567890123456789012345678901212312345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123123456789012345678901234567890121234567890123456789012345678901212312345678901234567890123456789012123456789012345678901234567890121231234567890123456789012345678901212345678901234567890123456789012123(H+ 3)∗ H∗∗ 2H∗∗ 2 H+ iH+ i H+ fH+ f H−H−H−H− H+ H+ H+H+/vector p /vector r/vector r /vectorP(/vector r)/vectorP(/vector r) Coulomb Saddle Low Energy High Energy(a) (b) (c) (d) Figure 9: (a) During the breakup of the H+ 3ion the H+ iis emitted in the forward direction and the H∗∗ 2is emitted in the backward direction. (b) The subsequent break up of this ion occurs such that the H+ fandH− ions are formed, and the H−is preferentially associated with H+ f. (c) The continuum pair, ( H+ f, H−) acquires a local momentum, /vectorP(/vector r), pushing the H−over the Coulomb Saddle, so that the H−will be associated with theH+ i. (d) Because the two protons are in fact indistinguishable, one will observe an asymmetry between “low” and “high” energy scattering events. 23Figure 10: During the breakup of the H+ 3ion, the magnitude of the local distortion eﬀects experienc ed by the (H+ f, H−) continuum pair while in the reaction zone are many orders of magnitude larger than the eﬀects experienced by either the ( e,¯e) or (e, e) continuum pairs. Here θ= 176◦andE= 1.5 eV for comparison with the experiment. 24Three-Body Gauge Variation with E 0 5 10 15 20-200204060 r(a0)ERelative EnergyHH0 (a) 0 2 4 6 8 10-20-1001020 r(a0)ERelative EnergyH0 H (b) 0 2 4 6 8 10-200204060 r(a0)ERelative EnergyH0 H (c) Figure 11: The eﬀects of the three-body gauge transformatio n are shown to depend critically upon the relative energy of the continuum pair. Observe that as the re lative energy of an electron-proton continuum pair, with a scattering angle of θ= 0◦, is increased from (a) E= 3 eV to (b) E= 10 eV to (c) E= 54.4 eV, the eﬀects in the reaction zone nearly vanish. 25Three-Body Gauge Variation with θ 0 2 4 6 8 10-20-1001020 r(a0)ERelative EnergyH0 H (a) 0 5 10 15 20-200204060 r(a0)ERelative EnergyH0 H (b) 0 5 10 15 20-200204060 r(a0)ERelative EnergyH0 H (c) Figure 12: The eﬀects of the three-body gauge transformatio n are shown to depend critically upon the scattering angle of the continuum pair. Observe that as the s cattering angle of an electron-proton continuum pair, with a relative energy of E= 10 eV, is increased from (a) θ= 0◦to (b) θ= 75◦to (c) θ= 150◦, the eﬀects in the reaction zone become more pronounced. 26Three-Body Gauge Variation with µ 0 2 4 6 8 10-20-1001020 r(a0)ERelative EnergyH0 H (a) 0 5 10 15 20-200204060 r(a0)ERelative EnergyHH0 (b) 0 2 4 6 8 10-20-1001020 r(a0)ERelative EnergyH0 H (c) Figure 13: The eﬀects of the three-body gauge transformatio n are shown to depend critically upon the reduced mass of the continuum pair. Observe that as the reduc ed mass of the continuum pair, with a relative energy of E= 10 eV and a scattering angle of θ= 0◦, is increased from (a) µ= 1µepto (b) µ= 10µepto (c) µ= 20µep, the eﬀects in the reaction are dramatically altered. 27Asymptotic Phase Variation 00.2 0.4 0.6 0.8 10.99980.999911.00011.0002e−2a0ηp E(µeV)p−p p−¯p (a) 00.2 0.4 0.6 0.8 10.99980.999911.00011.0002e−2a0ηp E(neV)e−p ¯e−p (b) 00.2 0.4 0.6 0.8 10.9940.9960.99811.0021.0041.006e−2a0ηp E(peV)e−e e−¯e (c) Figure 14: Shown is the variation of the phase achieved by the three-body gauge transformation (see text equation (43)) for (a) a proton-proton or proton–anti-prot on, (b) an electron-proton or positron-proton and (c) electron-electron or electron-positron continuum pai r. Observe that as the reduced mass of the system is decreased accordingly, the corresponding energy decrea se is suﬃcient to detect a variation of one part in a million. 28/BW/CX/D7/D8/D3/D6/D8/CX/D3/D2 /BX/AB/CT /D8/D7 /BW/D9/D6/CX/D2/CV /B4 /C0 /A0/BF /B5 /A3/BU/D6/CT/CP/CZ/D9/D4 0 5 10 15 20-2000-1500-1000-5000/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /D4/B5/C4/D3 /CP/D0 /BW/CX/D7/D8/D3/D6/D8/CX/D3/D2 /BX/AB/CT /D8/D7 /CU/D3/D6 /CE /CP/D6/CX/D3/D9/D7 /BV/D3/D2 /D8/CX/D2 /D9/D9/D1 /C8 /CP/CX/D6/D7 0 5 10 15 20-1.25-1-0.75-0.5-0.2500.250.5/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /BP/D4 /B5 /B4/CP/B5 /B4 /CT /A0 /AM /CT /B5 0 5 10 15 20-0.4-0.200.20.4/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /BP/D4 /B5 /B4/CQ/B5 /B4 /CT /A0 /CT /B5 0 5 10 15 20-2-1.5-1-0.500.51/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /BP/D4 /B5 /B4 /B5 /B4 /CT /A0 /D4 /B5 0 5 10 15 20-0.4-0.200.20.4/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /BP/D4 /B5 /B4/CS/B5 /B4 /AM /CT /A0 /AM /D4 /B5 0 0.05 0.1 0.15 0.2-300-200-1000100/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /BP/D4 /B5 /B4/CT/B5 /B4 /D4 /A0 /AM /D4 /B5 0.5 1 1.5 200.20.40.6/C8/CB/CU/D6/CP/CV /D6/CT/D4/D0/CP /CT/D1/CT/D2 /D8/D7/D6 /B4 /CP/BC /B5/BO /BW /DI /D4/B4 /DI /D6 /B5 /B4/BD /BP/D4 /B5 /B4/CU /B5 /B4 /D4 /A0 /D4 /B5
DESIGN, FABRICATION AND MEASUREMENT OF THE FIRST ROUNDED DAMPED DETUNED ACCELERATOR STRUCTURE (RDDS1)* J.W. Wang, C. Adolphsen, G.B. Bowden, D.L. Burke, J. Cornuelle, V.A. Dolgashev, W.B. Fowkes, R.K. Jobe, R.M. Jones, K. Ko, N. Kroll, Z.Li, R.J. Loewen, D. McCormick, R.H. Miller, C.K. Ng, C. Pearson, T.O. Raubenhemer, R. Reed, M. Ross, R.D. Ruth, T. Smith, G. Stupakov, SLAC, Stanford, CA, USA T. Higo, Y. Funahashi, Y. Higashi, T. Higo, N. Hitomi, T. Suzuki, K. Takata, T. Takatomi, N. Toge, Y. Watanabe, KEK, Tsukuba, JAPAN Abstract As a joint effort in the JLC/NLC research program, we have developed a new type of damped detuned acceleratorstructure with optimized round-shaped cavities (RDDS).This paper discusses some important R&D aspects of thefirst structure in this series (RDDS1). The design aspectscovered are the cell design with sub-MHz precision,HOM detuning, coupling and damping technique andwakefield simulation. The fabrication issues covered areultra-precision cell machining with micron accuracy,assembly and diffusion bonding technologies tosatisfactorily meet bookshelf, straightness and cellrotational alignment requirements. The measurementsdescribed are the RF properties of single cavities andcomplete accelerator section, as well as wakefields fromthe ASSET tests at SLAC. Finally, future improvementsare also discussed. 1 INTRODUCTION An active collaboration between KEK and SLAC on the development of X-band accelerating structures beganmore than 10 years ago. As either joint or parallel efforts,seven JLC/NLC prototypes Detuned (DS) or DampedDetuned Structures (DDS) have been built [1] withseveral goals in mind. These include testing of long-rangewakefield suppression, studying high gradient limits,accelerating beams in NLCTA and improving acceleratorefficiency. This last goal has been the main focus behindthe latest the Damped Detuned Structure with Round-shaped cavities or RDDS1 (see Figure. 1). Like itspredecessors, it is 1.8 m long and consists of 206 cells. Toimprove the rf efficiency, the structure cells were madewith a rounded shape compared to the flat disk shape usedin past structures. This change increases the cell shuntimpedance by 14%, allowing operation at lower inputpower to achieve the same average gradient. 2 STRUCTURE DESIGN 2.1 Cell profile optimisation and accurate dimension calculation The disc-loaded waweguide cross section for DS, DDS types and RDDS type disks are shown in Figure 2. ___________________ *Work supported by U.S. Department of Energy, contract DE-AC03-76F00515 and part of Japan-US CollaborationProgram In High Energy Physics Research. Figure1: RDDS1 section installed on a strongback. Properly chosen, three arcs in the RDDS design optimized the shunt impedance, Q value and minimized the ratio ofpeak surface to average accelerating electrical field. Also,the modified HOM coupling slots reduced the leakage ofthe fundamental mode by a factor of two. Compared toDDS, the overall improvement in shunt impedance forRDDS1 is about 19%. A newly developed finite element parallel-processingcode Omega3P was used for the design calculations [2].To verify its accuracy, five sets of test disks werecarefully measured mechanically with a CMM machineand electrically with a network analyzer. Excellentagreement was obtained between the experimental dataand extrapolated quadratic finite element results. A finalcell dimensional table was created with a frequencyaccuracy of better than 1 MHz. DS&DDS RDDS Figure 2: Cell shapes for different disc loaded waveguides. 2.2 Design of Accelerator Couplers The design goal for the input and output fundamentalcoupler and the high order mode couplers is to provide agood match for the rf signals travelling through each ofthem. Time domain MAFIA and finite-element OmegaT3codes were used to simulate these structures, and usuallysome cold test models were fabricated for confirmation. Figure 3 shows the cross-sectional view of the input end of RDDS1 structure. Figure 3: Cutaway view of input end of RDDS1. 2.3 Further effort to minimize the long-range transverse wakefield Space limitations prevented coupling higher order modes to the damping manifolds from the last fewcavities. From the ASSET test of an earlier structure(DDS3), it was realised that the pair of fundamentaloutput ports appears to couple to these modes, at least forone polarisation. To exploit this in RDDS1, foursymmetric output ports were used that were oriented inthe ±45ο plane because of space constraints (see Figure 4). Computer simulation and microwave measurementsshowed that when properly terminated, these last severalcavities had Q’s as low as 300 [3]. Besides this change indamping, the width of the dipole mode distribution waswidened from 10.1 % to 11.3 % to enhance the effect ofdetuning. Figure 4: Cutaway view of the output end of RDDS1. 3 FABRICATION 3.1 Disk Fabrication After being cut from bar-shaped Class-I OFC copper material, each disk was rough machined to have ~ 40 µm remaining copper for all critical dimensions [4]. The finaldiamond turning was performed on an ultra-high-precision lathe under controlled temperature of 20 ° ±0.3 οC. It was confirmed that the cell profile tolerance of ±1µm was satisfactorily met. A surface finish of 50nm and a flatness better than 0.5 µm assured success in stack alignment and diffusion bonding. 3.2 Microwave QC and Feed forward Extensive single and multi-disk rf measurements were made to verify the fundamental and dipole modefrequencies [5]. For single disk QC, each disk wassandwiched between two flat plates. Frequencies deviation of four modes (fundamental zero and π mode, π mode of first dipole band and zero mode of second dipole band) were measured as shown in Figure 5. The deviationof any of the four frequencies is within 0.6 MHz rms. Figure 5: Four frequencies measured in single-disk QC versus cell number. Systematic errors in the fundamental frequency werecompensated by introducing small corrections forsubsequent cells. We set the tolerance of 5 degrees in theintegrated phase advance at any point along the structure.It should be noted that this criterion is equivalent to a 0.1MHz systematic frequency error. The beginning disks were machined to the design dimensions and a series ofmulti-disk microwave measurements were made. Basedon this frequency information, a feed-forward correctionin "b" (cavity radius) was applied to the disks in laterfabrication in order to correct the integrated phase. Figure6 shows the 2 π/3 mode frequencies for six consecutive disks, 2b feed-forward correction and the integrated Figure 6: Fundamental mode measurement from stack QC. phase shift (less than 3° finally). With this amount of feed-forward correction, the frequency distribution of the first dipole mode was changed from of the design -3-2-10123 0 50 100 150 200Accelerating mode frequency2b offset [micron] Frequency [MHz] Integrated Phase Slip [degree] Disk number2b_offsetIntegrated Phase SlipMeasured Frequency-2-1.5-1-0.500.511.52 0 50 100 150 200Single-disk RF-QCdel_sf00 del_sf0pi del_sf1pi del_sf20Frequency Deviation [MHz] Disk numberdistribution by less than 1MHz. This will not degrade the designed decoherence for dipole modes [6]. 3.3 Structure Assembly The disks were manually stacked on a stainless steel V-block, inclined at 60 ° from horizontal. The disk misalignment was measured by running two capacitivegap sensors along the V-block. An autocollimator wasused for monitoring the perpendicularly of the disks. Thefinished stack was pre-diffusion bonded with 600kg axialforce in a furnace at 180 °C for a day. The section was then orientated vertically and fully diffusion bonded at850 ° - 890°C for four hours. Afterward, waveguide components, cooling and vacuum parts were brazed in ahydrogen furnace. Final measurement showed that thecell-to-cell misalignment was better than ±1 µm and the bookshelving was within 50 µrad. There was a gentle bow of 200µm that was straightened after final assembly on the strongback. 4 LESSONS LEARNED Differential expansion of the endplate supports relative to outer most copper cells lead to a flaring of the ends ofthe structure during bonding. Likewise, the relativeexpansion of a stainless-steel torodial vacuum manifoldenlarged a few cells near the center of the structure whereit was brazed on. Although much of the flaring wasremoved by symmetrically squeezing the ends of thestructure, the schedule for installing the structure inASSET did not allow time to attempt to correct the centercells (a scheme has since been worked out). Even with thecenter fundamental frequency errors, which are of theorder of 12 MHz, it was decided that much could still belearned from the ASSET measurements. 5 ASSET Measurements The structure was installed in ASSET facility in Sector2 of the SLAC Linac on May 24 and removed on August4. [7] As in previous wakefield measurements, positronsextracted from the South Damping Ring served as thedrive bunch and electrons extracted from the NorthDamping Ring served as the witness bunch. Data weretaken between PEP II fills, which required streamliningthe magnet and timing setup procedure. Parasitic single-beam measurements were also made of the manifolddipole signals versus beam position. The wakefield results are plotted in Figure 7. Overallthere is good agreement between the data and predictionwith the frequency errors, which were estimated frombead-pull measurement data of phase shift of thefundamental mode. The wakefield is likely dominated bya few modes in the region of the cell errors. This issupported by the fact that the wakefield phase shows asmooth variation relative to a single frequency oscillation(15.094 GHz). Additional measurements were taken in a search forhigher dipole band contributions in the wakefield dipregion where they should be readily apparent. AlthoughFnothing obvious was found in this region, 26 GHz wakefields are seen at shorter times once the 15.1 GHzcomponents are removed. This wakefield is likely frommodes in the third dipole band. Analysis of higher bandcontributions is still ongoing. Figure 7: ASSET wakefield data (vertical = dots, horizontal = crosses) and prediction with and withoutcentral frequency errors. For the single beam studies, the manifold dipole signals were either measured with a spectrum analyzer or withdowmmixers and digitizers. In one set of measurementsthe spectra from the nine RDDS1 ports that had cableconnections were recorded. Each measurement was madewith the beam roughly centred (based on minimizing thedipole power), and then offset by 0.5 mm in X and Y. Theshape of the spectrum with the beam offset in X is in fairagreement with predictions from the same model used tocompute the wakefield. However, several narrowmonopole-like signals were seen in the data and are notunderstood. There is a 10% X-Y power coupling of the14.2 to 14.6 GHz signals that may be related to the cellellipticity observed near the upstream end of the structure.Finally, the high end of the spectrum falls off smoothlyindicating that the fundamental output ports are dampingthe uncoupled modes as planned. The wakefield data includes measurements made in both the horizontal and vertical plane: they agree well in bothamplitude and phase. For this test, the upstream verticalHOM ports, which couple to the horizontal dipole modes,were blanked off to see if any X-Y differences would beobserved. The similarity of the results suggest that theupstream ports have little effect as expected since littlepower propagates upstream. Therefore, these ports can beremoved in future designs. 7 REFERENCES [1] International Study Group Progress Report, SLACReport-559, KEK Report 2000-7, April 2000.[2] Z. Li et al, TUA04, this conference. [3] R. Jones et al, TUA09, this conference. [4] N. Hitomi et al, TUA01, this conference. [5] T. Higo et al, TUA02, this conference. [6] R. Jones et al, TUA19, this conference. [7] C. Adolphsen et al. NLC Newsletter, Vol.1, No. 2,August 200010-210-1100101102 0 2 4 6 8 10 12 14 16Wake Amplitude (V/pC/m/mm) SQRT[Time(ns)]
arXiv:physics/0009026v1 [physics.atom-ph] 6 Sep 2000Stable, Strongly Attractive, Two-State Mixture of Lithium Fermions in an Optical Trap K. M. O’Hara, M. E. Gehm, S. R. Granade, S. Bali, and J. E. Thoma s Physics Department, Duke University, Durham, North Caroli na 27708-0305 (February 2, 2008) We use an all-optical trap to conﬁne a strongly attractive tw o-state mixture of lithium fermions. By measuring the rate of evaporation from the trap, we determin e the eﬀective elastic scattering cross section 4 πa2to show that the magnitude of the scattering length \|a\|is very large, in agreement with predictions. We show that the mixture is stable against inel astic decay provided that a small bias magnetic ﬁeld is applied. For this system, the s-wave intera ction is widely tunable at low magnetic ﬁeld, and can be turned on and oﬀ rapidly via a Raman πpulse. Hence, this mixture is well suited for fundamental studies of an interacting Fermi gas. PACS numbers: 32.80.Pj Copyright 2000 by the American Physical Society Trapped, ultracold atomic vapors oﬀer exciting new opportunities for fundamental studies of an interacting Fermi gas in which the temperature, density and interac- tion strength can be independently controlled. Recently, a degenerate gas of fermionic40K has been produced by using a two-state mixture to enable s-wave scattering and evaporation in a magnetic trap [1]. By removing one species, the properties of the noninteracting degenerate gas were measured, demonstrating that the momentum distribution and the total energy obey Fermi-Dirac statis- tics [1]. However, the properties of interacting two-state fermionic vapors have not been explored experimentally. Theoretical treatments of an interacting Fermi gas have focused extensively on6Li [2–9]. Certain two- state6Li mixtures are predicted to be strongly attractive, i.e., they have anomalously large and negative scattering lengths [10] arising from a near-zero energy resonance in the triplet state [11]. It has been predicted that these strongly attractive mixtures can undergo a transition to a superﬂuid state at a relatively high transition tempera- ture [2,4]. In addition, the two-state eﬀective interactio n potential is widely tunable in a magnetic ﬁeld, permit- ting systematic studies of fundamental phenomena such as collective oscillations for both the normal and super- ﬂuid phases [3,5,6], as well as new tests of superconduc- tivity theory [4]. Unfortunately, magnetically trappable mixtures in6Li with large s-wave scattering lengths are not stable, since there are correspondingly large spin-exchange and dipo- lar decay rates [2,7,10]. Hence, the methods employed to study degenerate40K are not applicable. For this reason, we developed an ultrastable CO 2laser trap to conﬁne a stable mixture of the two lowest6Li hyperﬁne states [12]. However, attaining a large and negative scattering length in this mixture requires high magnetic ﬁelds B≥800 G to exploit either a Feshbach resonance or the triplet scat- tering length [7,10]. In this Letter, we show that there exists another sta- ble hyperﬁne state mixture in6Li which has the following unique properties. First, we predict that the scattering length ais large, negative, and widely tunable at low magnetic ﬁeld B. By monitoring the rate of evaporationfrom the CO 2laser trap at a ﬁxed well depth, we measure \|a\|= 540+210 −100a0atB= 8.3 G. This result conﬁrms for the ﬁrst time that very large scattering lengths exist in 6Li mixtures. The predicted scattering length is −490a0 atB= 8.3 G, consistent with our observations, and is expected to increase to −1615a0asB→0. Second, we ﬁnd that this system is stable against spin exchange collisions provided that B/ne}ationslash= 0. In addition, the dipolar decay rate is predicted to be very small [13], consistent with our observations. Finally, in the experiments, a Ra- manπpulse is employed to abruptly create an interacting mixture from a noninteracting one, a desirable feature for studies of many-body quantum dynamics [14]. -3-2-10123 E/ahf 200 150 100 50 0 B(Gauss)3 2 1456)(m ) 2 / 3(+ ) 2 / 1(+ ) 2 / 1(− ) 2 / 3(− ) 2 / 1(− ) 2 / 1(+F = 1/2F = 3/2 FIG. 1.6Li hyperﬁne states, labeled \|1/an}bracketri}htto\|6/an}bracketri}htin order of increasing energy in a magnetic ﬁeld. The magnetic quantum number of each state is denoted by m. The hyperﬁne constant ahf= 152 .1 MHz. Fig. 1 shows the hyperﬁne states for6Li labelled \|1/an}bracketri}ht − \|6/an}bracketri}ht, in order of increasing energy in a magnetic ﬁeld. At low ﬁeld, the states \|1/an}bracketri}htand\|2/an}bracketri}htcorrespond to the \|F= 1/2, m/an}bracketri}htstates, while states \|3/an}bracketri}htthrough \|6/an}bracketri}htcorrespond to states \|F= 3/2, m/an}bracketri}ht. At nonzero magnetic ﬁeld, only the magnetic quantum number mis conserved. The subject of this paper is the \|3/an}bracketri}ht − \|1/an}bracketri}htmixture. 1-1400-1200-1000-800-600-400Scattering Length ( a0) 0.010.1110100 Magnetic Field (Gauss) FIG. 2. Magnetic ﬁeld dependence of the scattering length a31for a mixture of the \|3/an}bracketri}htand\|1/an}bracketri}hthyperﬁne states of6Li. Fig. 2 shows the scattering length a31for the \|3/an}bracketri}ht − \|1/an}bracketri}ht mixture as a function of magnetic bias ﬁeld B. We es- timate a31(B) by using the asymptotic boundary con- dition (ABC) approximation [10]. This calculation in- corporates the singlet and triplet scattering lengths [11] , aS= 45.5±2.5a0andaT=−2160±250a0, and a bound- ary radius which we take to be R= 40a0[10]. The scat- tering length varies from −1620a0(≃3aT/4 asB→0) to−480a0atB= 10 G. The results of our approximate calculation for B= 0 to B= 200 G are conﬁrmed within 10% by van Abeelen and Verhaar [13] using a coupled channel calculation which includes the uncertainties in the potentials. At higher ﬁelds, near 800 G, we believe the scattering length exhibits a Feshbach resonance (not shown). Above this resonance, the scattering length ap- proaches the triplet scattering length of −2160a0. The\|3/an}bracketri}ht − \|1/an}bracketri}htmixture is stable against spin-exchange collisions provided that a small bias magnetic ﬁeld is applied. Spin-exchange inelastic collisions conserve the two-particle total magnetic quantum number MT, where MT=−1 for the \|{3,1}/an}bracketri}htstate. Note that {,}denotes the antisymmetric two-particle spin state, as required for s-wave scattering which dominates at low tempera- tures. There are no lower-lying antisymmetric states with MT=−1. Hence, exothermic collisions are precluded. The only other states with MT=−1 are \|{4,2}/an}bracketri}htand \|{5,3}/an}bracketri}ht. Without an adequate bias magnetic ﬁeld, transi- tions to these states lead to population in level \|4/an}bracketri}ht. Then, exothermic \|{3,4}/an}bracketri}ht → \|{ 3,2}/an}bracketri}htand\|{4,1}/an}bracketri}ht → \|{ 1,2}/an}bracketri}ht collisions can occur. With an adequate bias magnetic ﬁeld, the energy of states \|{4,2}/an}bracketri}htand\|{5,3}/an}bracketri}htcan be in- creased relative to that of state \|{3,1}/an}bracketri}htby more than the maximum relative kinetic energy, i.e., twice the well depth during evaporative cooling. By energy conserva- tion, spin-exchange transfer is then suppressed. In this case, the inelastic rate is limited to magnetic dipole- dipole (dipolar) interactions which contain a rank 2 rela- tive coordinate operator of even parity [2]. Since parity is conserved and p −wave→p−wave scattering is frozenout at low temperature, the dominant dipolar process is a small s →d rate in which \|{3,1}/an}bracketri}ht → \|{ 1,2}/an}bracketri}ht[13]. In the experiments, the CO 2laser trap is initially loaded from a magneto-optical trap (MOT) [12]. At the end of the loading period, the MOT laser beams are tuned near resonance and the intensity is lowered to de- crease the temperature. Then, optical pumping is used to empty the F= 3/2 state to produce a 50-50 mixture of the \|1/an}bracketri}ht − \|2/an}bracketri}htstates. These states are noninteracting at low magnetic ﬁeld, i.e, the scattering amplitude van- ishes as a result of an accidental cancellation [10]. With a CO 2laser trap depth of 330 µK, up to 4 ×105atoms are conﬁned in the lowest-lying hyperﬁne states at an initial temperature between 100 and 200 µK. A bias magnetic ﬁeld of 8.3 G is applied to split the two-particle energy states by ≃16 MHz. This is twice the maximum attain- able energy at our largest well depth of 400 µK = 8 MHz. After a delay of 0.5 second relative to the loading phase, a pair of optical ﬁelds is pulsed on to induce a Raman πpulse. This pulse transfers the population in state \|2/an}bracketri}ht to state \|3/an}bracketri}htin two microseconds, initiating evaporative cooling in the resulting \|3/an}bracketri}ht − \|1/an}bracketri}htmixture. The optical ﬁelds are detuned from resonance with the D2 transition by≃700 MHz to suppress optical pumping. If the Ra- man pulse is not applied, the trapped atoms remain in the noninteracting \|1/an}bracketri}ht−\|2/an}bracketri}htmixture and exhibit a purely exponential decay with a time constant ≃300 seconds. An acousto-optic modulator (A/O) in front of the CO 2 laser controls the laser intensity, which is reduced to yiel d a shallow trap depth of 100 µK. By using a shallow well, we avoid the problem that the elastic cross section be- comes independent of the scattering length at high en- ergy, as described below. In addition, the shallow well greatly reduces the number of loaded atoms and makes the sample optically thin, simplifying calibration of the number of trapped atoms. To determine the trap pa- rameters, the laser power is modulated and parametric resonances [15] are observed at drive frequencies of 2 νfor three diﬀerent trap oscillation frequencies ν: At 100 µK well depth, νx= 2.4 kHz νy= 1.8 kHz and νz= 100 Hz, where the trap laser beam propagates along z. Using the measured total power as a constraint, we obtain the trap intensity 1 /e2radii, wx= 50µm and wy= 67µm, and the axial intensity 1 /e2length, zf≃1.13 mm, where zf is consistent with the expected Rayleigh length within 15%. The number of atoms in the trap N(t) is estimated using a calibrated photomultiplier. The detection sys- tem monitors the ﬂuorescence induced by pulsed, retrore- ﬂected, σ±probe and repumper beams which are strongly saturating ( I/Isat= 26 for the strongest transition). To simplify calibration, only the isotropic component of the ﬂuorescence angular distribution is measured: The collecting lens is placed at the magic angle [16] of 55◦ (P2(cosθ) = 0) with respect to the propagation direction of the probe beams. The net eﬃciency of the detection system is determined using laser light of known power. The primary uncertainty in the calibration arises from 2the excited state population fraction, which we estimate lies between 1/4 and 1/2. 5 4 3 2 1Number of Atoms (104) 20 15 105 0 Time (sec)5 4 3 2 1 1.0 0.5 0.0 FIG. 3. Number of trapped atoms versus time for evapo- ration of a \|3/an}bracketri}ht − \|1/an}bracketri}htmixture of6Li at a ﬁxed well depth of 100µK. The solid curve shows the s-wave Boltzmann equa- tion ﬁt for a scattering length of \|a31\|= 540 a0. Inset: 0-1 second. Fig. 3 shows the number of trapped atoms N(t) mea- sured for the \|3/an}bracketri}ht−\|1/an}bracketri}htmixture at a well depth U0= 100 µK and a bias ﬁeld of 8.3 G as a function of time between 5 ms and 20 seconds after evaporation is initiated. For times beyond 50 seconds (not shown), the evaporation stagnates, and we observe an exponential decay of the cooled \|3/an}bracketri}ht − \|1/an}bracketri}htmixture with a time constant of 370 sec- onds over a period of a few hundred seconds. The error bars are the standard deviation of the mean of ten com- plete runs through the entire time sequence. A model based on the s-wave Boltzmann equation [17] is used to predict N(t) for comparison to the experiments. This equation is modiﬁed to include the density of states for a gaussian potential well [12] and to include the en- ergy dependence of the elastic cross section. Assuming a short range potential and a symmetric (s-wave) spatial state, the cross section takes the form σ(k) =8πa2 31 1 +k2a2 31, (1) where ¯ hkis the relative momentum. For k\|a31\|<<1, the cross section is maximized. When k\|a31\|>>1, the cross section approaches the unitarity limit 8 π/k2which is independent of a31. Note that k\|a31\|= 1 corresponds to a relative kinetic energy of ǫ= ¯h2/(2µ a2 31), where µ= M/2 is the reduced mass. For \|a31\|= 500 a0,ǫ= 115 µK. For a two-state mixture of fermions, the eﬀective cross section is reduced from that of Eq. 1 by a factor of 2 since pairs of colliding atoms are in an antisymmetric hyper- ﬁne state with a probability 1/2. This eﬀective cross section is used in a Boltzmann collision integral for each statei= 1,3. A decay term −Ni(t)/τwithτ= 370 secis added to account for the measured trap lifetime. A detailed description of our coupled Boltzmann equation model will be published elsewhere. The coupled s-wave Boltzmann equations for the two states are numerically integrated to determine N(t) using the well parameters as ﬁxed inputs. From the calibrated photomultiplier signal, assuming that 1/3 of the atoms are in the excited state, we obtain an initial total number N0= 44,000. For this case, the initial collision rate in Hz is estimated to be 1 /(2πτc)≃N0Mσ0ν3/(kBT), where ν3=νxνyνz,σ0= 8πa2 31, and Mis the6Li mass. Assuming \|a31\|= 500 a0,τc= 30 ms. Hence, for t >0.3 seconds, when on average 10 collisions have occurred, the sample should be thermalized as assumed in the theory. The best ﬁt to the data starting with 22,000 atoms in each state is shown as the solid curve in Fig. 3. The χ2 per degree of freedom for this ﬁt is 1.4 and is found to be very sensitive to the initial temperature T0of the atoms in the optical trap. ¿From the ﬁt, we ﬁnd T0= 46µK, which is less than the well depth. We believe that this low temperature is a consequence of the MOT gradient magnet, which is turned oﬀ after the MOT laser beams. The eﬀective well depth of the optical trap is therefore reduced until the gradient is fully oﬀ, allowing hotter atoms to escape before the Raman pulse is applied to create the \|3/an}bracketri}ht − \|1/an}bracketri}htmixture. The ﬁt is most sensitive to data for t >0.5 second, where the thermal approxima- tion is expected to be valid. From the ﬁt, we obtain the scattering length \|a31\|= 540 ±25a0, which is within 10% of the predictions of Fig. 2. The quoted error corresponds to a change of 1 in the total χ2. We determine the systematic errors in a31due to the uncertainties in the calibration and in the population im- balance as follows. The data is ﬁt for an initial num- ber of atoms N0of 58,000 and 29,000, corresponding to an excited state fraction of 1/4 and 1/2. This yields \|a31\|= 440 ±20a0and\|a31\|= 750 ±42a0, respectively. Note that for the larger scattering lengths, the cross sec- tion given by Eq. 1 approaches the unitarity limit and the error increases. We assume that the initial popula- tion imbalance for states \|3/an}bracketri}htand\|1/an}bracketri}htis comparable to that of states \|2/an}bracketri}htand\|1/an}bracketri}htin the optically pumped MOT. To estimate the latter population imbalance, we use state- selective Raman πpulses to excite \|2/an}bracketri}ht → \|3/an}bracketri}htor\|1/an}bracketri}ht → \|6/an}bracketri}ht transitions in the MOT. Probe-induced ﬂuorescence sig- nals from states \|3/an}bracketri}htor\|6/an}bracketri}htshow that the initial \|1/an}bracketri}htand \|2/an}bracketri}htpopulations are equal within 10%. Note that resid- ual population in state \|2/an}bracketri}htis expected to be stable and weakly interacting, since we estimate \|a32\|<30a0for 0≤B≤50 G using the ABC method, and a12≃0 [10]. Using the parameters for the ﬁt shown in Fig. 3, but changing the initial mixture from 50-50 to 60-40, we ﬁnd a slight increase in the ﬁtted scattering length from 540 a0 to 563 a0. Thus, the uncertainty in the calibration of the number of atoms produces the dominant uncertainty and \|a31\|= 540+210 −100a0. To demonstrate that evaporative cooling is occuring, rather than just trap loss, we have also measured the 3ﬁnal temperature of the mixture using release and recap- ture [18] from the CO 2laser trap. We obtain 9 .8±1µK, which is within 10 % of the ﬁnal temperature of 8 .7µK predicted by the Boltzmann equation model. An excel- lent ﬁt to the data is obtained for the ﬁnal temperature, which describes a thermal distribution. However, the ini- tial temperature is not so readily measured, as it is non- thermal before evaporation is initiated, and is rapidly changing during evaporation, unlike the ﬁnal tempera- ture, which stagnates. Good ﬁts to the evaporation data are obtained neglect- ing inelastic collisions, suggesting that the dipolar rate for the \|3/an}bracketri}ht − \|1/an}bracketri}htmixture is small, in contrast to the scat- tering length. A limit on the dipolar loss rate for the \|3/an}bracketri}ht − \|1/an}bracketri}htmixture can be estimated from the τ= 370 sec- ond lifetime of the mixture after evaporation stagnates. For equal populations in both states, dipolar decay re- sults in an initial loss rate ˙ n=−Gn2/4, where Gis the dipolar rate constant and nis the total density. To ob- tain a high density, the trap is loaded at a well depth of 330 µK and the temperature of the atoms is reduced by evaporation to T≃30±1µK. The number of atoms remaining in each state after evaporation is estimated to beN= 6.5±2.2×104, where the uncertainty is in the calibration. We cannot rule out the possibility that one state is depleted on a long time scale, since we do not directly measure the individual state populations. How- ever, we believe that, after evaporation stagnates in the deep well, a \|3/an}bracketri}ht − \|1/an}bracketri}htmixture remains, since subsequent reduction of the well depth yields ﬁnal temperatures con- sistent with evaporative cooling. Note that the mixture ratio is not critical: An 80-20 mixture yields an initial loss rate ˙ n=−0.16Gn2,≃2/3 that of a 50/50 mixture. For a ﬁxed 330 µK trap depth, ν3= 2.6±0.3 kHz3, and the phase space density for one state in the harmonic approximation is then ρph=N/(kBT/hν)3= 7×10−4. This corresponds to a maximum total density of n= 2ρph/λ3 B= 6.4×1011/cm3, where λB≡h/√2πM k BT. Since the exponential decay time of the \|3/an}bracketri}ht−\|1/an}bracketri}htmixture is similar to that obtained in the noninteracting \|1/an}bracketri}ht− \|2/an}bracketri}ht mixture, we assume the loss is dominated by background gas collisions. Thus, we must have Gn/4<<1/τ, which yields G << 2×10−14cm3/sec. This result is consis- tent with the value G≃2×10−15cm3/sec predicted for the dipolar rate constant at 30 µK by van Abeelen and Verhaar [13]. Future experiments will employ continuous evapora- tion by slowly reducing the well depth [19]. In this case, very large scattering lengths can be obtained at low tem- peratures and small well depths by using a reduced bias magnetic ﬁeld B. By adiabatically recompressing the well, experiments can be carried out with the precooled atoms in a deep trap to obtain high density as well. In such experiments, the ﬁnal low temperature limits the number of atoms in the high energy tail of the energy dis- tribution, exponentially suppressing spin-exchange coll i- sions for B/ne}ationslash= 0. For example, if a total of 3 ×105atoms were contained in our trap at a well depth of 400 µK, theFermi temperature TF= 7µK and the Fermi density is 4×1013/cm3. At a temperature of T= 0.1TF= 0.7µK, a bias ﬁeld of B= 0.16 G would split the two-particle hyperﬁne states by kBTF+ 12kBT, suppressing the spin exchange rate by exp( −12), and giving a31≃ −1200a0. Alternatively, as shown in Fig. 2, large a31can be ob- tained at moderate B≃300 G. In conclusion, we have observed that an optically trapped \|3/an}bracketri}ht − \|1/an}bracketri}htmixture of6Li atoms has a very large scattering length at low magnetic ﬁeld. This mixture is stable against spin-exchange collisions provided that a small bias magnetic ﬁeld is applied. The evaporation curves measured for this mixture are in good agreement with a model based on an s-wave Boltzmann equation which neglects inelastic processes. We have predicted that the scattering interactions are strongly attractive and widely tunable at low magnetic ﬁeld. If the param- eters described above for deep wells can be attained, the system will be close to the threshold for superﬂuidity [2] and ideal for investigating frequency shifts and damp- ing in collective oscillations [3,5]. Further, since s-wav e interactions can be turned on and oﬀ in a few microsec- onds, this system is well suited for studies of many-body quantum dynamics. This research has been supported by the Army Re- search Oﬃce and the National Science Foundation. [1] B. DeMarco, and D. S. Jin, Science 285, 1703 (1999). [2] H. T. C. Stoof, et al., Phys. Rev. Lett. 76, 10 (1996); See also, M. Houbiers et al., Phys. Rev. A 56, 4864 (1997). [3] L. Vichi and S. Stringari, Phys. Rev. A 60, 4734 (1999). [4] R. Combescot, Phys. Rev Lett. 83, 3766 (1999). [5] G. M. Bruun, and C. W. Clark, Phys. Rev. Lett. 83, 5415 (1999). [6] G. M. Bruun and C. W. Clark, cond-mat/9906392. [7] M. Houbiers and H. T. C. Stoof, Phys. Rev. A 59, 1556 (1999). [8] G. Bruun, et al., Eur. Phys. J. D 7, 433 (1999). [9] M. Houbiers and H. T. C. Stoof, cond-mat/9808171. [10] M. Houbiers, et al., Phys. Rev. A 57, R1497 (1998). [11] E. R. I. Abraham, et al., Phys. Rev A 55, R3299 (1997). [12] K. M. O’Hara, et al., Phys. Rev. Lett. 82, 4204 (1999). [13] We are indebted to F. A. van Abeelen and B. J. Verhaar who calculated the inelastic \|{3,1}/an}bracketri}ht → \|{ 1,2}/an}bracketri}htdipolar rate and conﬁrmed our calculations of the magnetic ﬁeld dependence of a31. [14] P. T¨ orm¨ a and P. Zoller, Phys. Rev. Lett. 85, 487 (2000). [15] S. Friebel, et al., Phys. Rev. A 57, R20 (1998). [16]Atomic, Molecular, and Optical Physics Handbook , ed. G. W. Drake, (AIP Press, New York, 1996), p. 176. [17] O. J. Luiten, et al., Phys. Rev. A 53, 381 (1996). [18] S. Chu, et al., Phys. Rev. Lett. 55, 48 (1985). [19] C. S. Adams, et al., Phys. Rev. Lett. 74, 3577 (1995). 4
-180-135-90-4504590135180 Longitude, E-90-60-300306090Latitude, NGPS global network Fig/. /1/: V ariations in soft X/-ra y emission at /0/./1/{/0/./8 nm /(a/)/; /> /1MeV proton/ ux /(b/)/; and /> /5 MeV /(c/) measured at geostationary orbit of the GOES/-/1/0satellite /(/1/3/5 /W /) with /5/-min time resolution/; and of the H/-comp onen t of thegeomagnetic /eld record at station Irkutsk /(/5/2/./2 N/; /1/0/4/./3 E /(d/)/, and Dst /(e/)during the magnetic storm of April /6/, /2/0/0/0/. Mean v alues of the relativ e densit yof phase slips P /( t /) for PRN/0/2 /(f /)/, and PRN/0/7 /(g/)/. Horizon tal lines sho w thethreshold for the selected n um b er of coincidences P /3/.1E+11E+21E+31E+4> 1 MeV protons 1E-11E+01E+11E+2> 5 Mev protons -300-1500150300H(t), nT6 April, 2000 SSC Irkutsk -300-1500Dst, nTGOES 10, 1350 W 1E-71E-61E-51E-4longe X - ray 0510P(t), % 0:004:008:0012:0016:0020:000:00 Time, UT051015P(t), %a b c d e f gPRN02 PRN07 P3P3Dayside DaysideFig/. /2/: V ariations in soft X/-ra y emission at /0/./1/{/0/./8 nm /(a/)/; /> /1MeV proton/ ux /(b/)/; and /> /5 MeV /(c/) measured at geostationary orbit of the GOES/-/1/0satellite /(/1/3/5 /W /) with /5/-min time resolution/; and of the H/-comp onen t of thegeomagnetic /eld record at station Irkutsk /(/5/2/./2 N/; /1/0/4/./3 E /(d/)/, and Dst /(e/)during the magnetic storm of April /6/, /2/0/0/0/. Mean v alues of the relativ e densit yof phase slips P /( t /) for PRN/0/2 /(f /)/, and PRN/0/7 /(g/)/. Horizon tal lines sho w thethreshold for the selected n um b er of coincidences P /3/.1E+31E+41E+5> 1 MeV protons 1E+31E+41E+5> 5 MeV protons -4000400H(t), nTJuly 15, 2000 Irkutsk -3000300Dst, nTGOES 10, 1350 W SSC 1E-21E-11E+0long X - ray 0.02.04.0P(t), % 0:004:008:0012:0016:0020:000:00 Time, UT0.03.06.0P(t), %a b c d e f gPRN04 PRN24P3 P3Fig/. /3/: Same as in Fig/. /2a/, but for the magnetic storm of July /1/5/,/2/0/0/0/. Meanv alues of the relativ e densit y of phase slips P /( t /) for PRN/0/2 /(g/) and PRN/2/4 /(h/)/.1E+21E+31E+4> 1 Mev protons 1E+11E+21E+31E+4> 5 MeV protons -1000100H(t), nTJuly 14, 2000 Irkutsk1E-21E-11E+01E+1 long X - ray GOES 08, 740 W -1000100Dst, nT 0.01.02.0P(t), % 0:004:008:0012:0016:0020:000:00 Time, UT0.01.02.0P(t), %a b c d e f g P3P3PRN15 PRN21Fig/. /4/: Same as in Fig/. /2a/, but for the p o w erful solar / are of July /1/4/, /2/0/0/0/. Meanv alues of the relativ e densit y of phase slips P /( t /) for ORN/1/5 /(g/) and PRN/2/1 /(h/)/.Time, UT0.05.010.0M(t) Time, UT0.01.02.03.04.05.0P(t), % Time, LT0.001.252.50 M(t) 06121824 Time, LT0.000.100.20P(t), %Time, UT0.05.010.0M(t) Time, UT0.05.010.015.0P(t), % Time, LT0.001.252.50 M(t) 06121824 Time, LT0.000.501.001.502.00P(t), %a e b f c g d hJuly 29, 1999 April 6, 2000 PRN07 PRN07SSC S(t) S(t)S(t)S(t) P3 P3 P3 P3DaysideFig/. /5/: Mean v alues of the observ ation densit y M /( t /)/, the densit y of phase slipsS /( t /)/, and of the relativ e densit y of phase slips P /( t /)/: for individual GPS satellitesas a function of univ ersal time UT /(a/, b/, e/, f /)/. The same v alues a v eraged o v er allobserv ed satellites and smo othed with a time windo w of /2 hours but as a functionof lo cal time L T /(c/, d/, g/, h/)/, Left /- magnetically quiet da y of July /2/9/, /1/9/9/9/; righ t/- ma jor magnetic storm of April /6/, /2/0/0/0/. The SSC time /1/6/:/4/2 UT is sho wn inpanels e and f b y a v ertical bar/. Horizon tal lines corresp ond to the threshold forthe selected n um b er of coincidences P /3/.0:00 6:00 12:00 18:00 0:00 Time, LT0.00.51.01.52.0P(t), % 06.04.2000 15.07.2000 14.07.2000 09.01.2000 29.07.1999Fig/. /6/: Lo cal time L T dep endence of the relativ e mean densit y of phase slips P /( t /)obtained b y a v eraging the data from all GPS satellites observ ed sim ultaneouslyat the GPS stations net w ork at elev ations less than /3/0 /. The P /( t /)/-dep endenciesare smo othed with a time windo w of /2 hours/. The resp ectiv e dates are sho wn atthe P /( t /)/-curv es/.0.01.02.03.04.05.06.0M(t) 0.00.51.01.52.0P(t), % 0.01.02.03.04.05.06.0 M(t) 6789101112 Time, UT0.00.51.01.52.0P(t), %0.05.010.0M(t) 0.01.02.03.04.0P(t), % 0.05.010.0 M(t) 101214161820 Time, UT0.01.02.03.04.05.06.0P(t), %a e b f c g d hJuly 14, 2000 July 15, 2000 PRN04 PRN15SSC S(t) S(t)S(t) S(t) PRN24 PRN2110:03 UT P3 P3P3 P3Fig/. /7/: Mean v alues of the observ ation densit y M /( t /)/, the densit y of phase slipsS /( t /)/, and of the relativ e densit y of phase slips P /( t /) for individual GPS satellites asa function of univ ersal time UT/. Left /- solar / are of July /1/4/, /2/0/0/0/; righ t /- magneticstorm of July /1/5/, /2/0/0/0/. The onset times of the solar / are /(/1/0/:/0/3 UT/) and themagnetic storm SSC time /(/1/4/:/4/0 UT/) are sho wn b y a v ertical bar/. Horizon tallines corresp ond to the threshold for the selected n um b er of coincidences P /3/. Theresp ectiv e satellite n um b ers are sho wn at the M /( t /)/-curv es/.arXiv:physics/0009027v1 [physics.geo-ph] 6 Sep 20001 MAGNETOSPHERIC DISTURBANCES, AND THE GPS OPERATION Afraimovich E. L., and O. S. Lesyuta Institute of Solar-Terrestrial Physics SD RAS, p. o. box 4026, Irkutsk, 664033, Russia fax: +7 3952 462557; e-mail: afra@iszf.irk.ru Short title: MAGNETOSPHERIC DISTURBANCES, AND THE GPS OPERATION2 Abstract. We have detected a strong dependence of the relative density of phase slips in the GPS navigation system which are most likely to have been caus ed by the processes occurring in the neighborhood of the GPS satellites, or in the plasmasphere, on the disturbance level of the Earth’s magnetosphere during major magnetic storms and powerful so lar ﬂares. The study is based on using Internet-available selected dat a from the global GPS network, with the simultaneously handled number of receiving stations rangi ng from 160 to 323. The analysis used ﬁve days from the period 1999–2000, with the daily mean values of the geomagnetic ﬁeld disturbance index Dst from -4 to -70 nT. During strong magnetic storms, the relative density of phas e slips exceeds the one for magnetically quiet days by one-two orders of magnitude as a minimum, and re aches a few and (for some of the GPS satellites) even ten percent of the total density of obse rvations, which may be unacceptable when tackling important navigation problems. Furthermore, the level of phase slips for the GPS satellites located on the sunward side of the Earth was by a factor of 3-5 l arger compared with the opposite side of the Earth. For large isolated magnetic storms, a clearly pronounced eﬀ ects of abrupt increase in the density of phase slips was also observed to occur immediately after a su dden storm commencement, SSC. A similar eﬀect was also detected during a power solar ﬂare of class X5.7 on July 14, 2000.3 1. Introduction The satellite navigation GPS system has become a powerful fa ctor of scientiﬁc and technological progress worldwide, and enjoys wide use in a great variety of human activity. In this connection, much attention is given to continuous perfection of the GPS syste m and to the widening of the scope of its application for solving the navigation problems themselve s, as well as for developing higher-precision systems for time and accuracy determinations. Even greater capabilities are expected in the near future through the combined use of the GPS with a similar Russian sys tem (GLONASS). Recently the GPS system has also gained wide-spread accepta nce in research in the ﬁeld of geodynamics, in the physics of the Earth’s atmosphere, iono sphere and plasmasphere, etc. [ Davies and Hartman , 1997; Klobuchar , 1997]. Investigations of this kind are not only of purely sc ientiﬁc interest but are also important for perfection of the GPS system itself. T o address these problems, a global network of receiving GPS stations was set up, which consisted, by Aug ust 2000, of no less than 732 points, the data from which are placed on the Internet. Using two-frequency multichannel receivers of the global n avigation GPS system, at almost any point on the globe and at any time simultaneously at two coher ently-coupled frequencies f1= 1575 .42 MHz and f2= 1227 .60 MHz, highly accurate measurements of the group and phase d elays are being underway along the line of sight (LOS) between the receiver o n the ground and the transmitters on-board the GPS system satellites which are in the zone of re ception. These data, converted to values of total electron content (T EC), are of considerable current use in the study of the regular ionosphere and of disturbances of na tural and technogenic origins (solar eclipses, ﬂares, earthquakes, volcanoes, strong thunderstorms, aur oral heating; nuclear explosions, chemical explosion events, launches of rockets). We do not cite here t he relevant references for reasons of space, which account for hundreds of publications to date. The study of deep, fast variations in TEC caused by a strong sc attering of satellite signals from intense small-scale irregularities of the ionospheric F2-layer at equatorial and polar latitudes has a special place among ionospheric investigations based on us ing satellite (including GPS) signals [ Aarons et al., 1996, 1997; Klobuchar , 1997; Pi et al ., 1997; Aarons and Lin , 1999]. The interest to this problem as regards the practical implementation is explained by the fact that as a result of such a scattering, the signal undergoes deep amplitude fadings, which leads to a ph ase slip at the GPS working frequencies. To achieve a more eﬀective detection of disturbances in the n earterrestrial space environment, researchers at the ISTP SD RAS have developed a new technolog y of a global detector GLOBDET, and a relevant software which makes it possible to automate t he acquisition, ﬁltering and pretreatment process of the GPS data received via the Internet [ Afraimovich , 2000b]. This technology is being used to detect, on a global and regional scales, ionospheric eﬀec ts of strong magnetic storms [ Afraimovich et al., 2000a], solar ﬂares [ Afraimovich , 2000b], solar eclipses [ Afraimovich et al ., 1998], launches of rockets [Afraimovich et al ., 2000c], earthquakes, etc. However, the data from the GPS system that are available from the Internet in the RINEX-format4 contain also information about failures of phase delay meas urements [ Hofmann-Wellenhof et al ., 1992]. In TEC measurements, this information is usually used to ass ess data quality, and in solving problems of restoring lost or distorted information about the TEC. In this paper we have used an earlier GLOBDET technology in a global analysis of the relative dens ity of phase slips in the GPS system during disturbances of the near-terrestrial space environ ment. It was found that not only can these disturbances (caused by c orresponding processes in the “Sun-Earth” system) be detected in TEC measurements, but th ey also inﬂuence the functioning of the GPS system itself. The experimental geometry and general information about th e data base used are presented in Section 2. The determination of the relative density of phas e slips, and the method of processing the data available from the Internet are brieﬂy outlined in Sect ion 3. Section 4 describes the results obtained for magnetically quiet and disturbed conditions. Results a re discussed in Section 5. 2. Experimental geometry and general information about the data base used This study is based on using the data from a global network of G PS receiving stations available from the Internet. For a number of reasons, slightly diﬀering set s of GPS stations were chosen for the various events under investigation; however, the experimental geo metry for all events was virtually identical. Figure 1 illustrates the geometry of the global GPS array tha t was used in our analysis of the phase slips in the GPS system. Dots show the location of GPS station s (totalling 732 stations by August 2000). The analysis used a set of stations (from 160 to 323) wi th a relatively even distribution across the globe. For reasons of space, we do not give here the statio ns coordinates. This information may be obtained from http://lox.ucsd.edu/cgi-bin/allCoords.c gi?. As is evident from Figure 1, the set of stations, which we sele cted out of the part of the global GPS network available to us, covers rather densely North Ame rica and Europe; Asia has much poorer coverage. The number of GPS stations in the Paciﬁc and Atlant ic oceans is even smaller. However, such coverage over the globe is already presently suﬃcient for a g lobal detection of disturbances with spatial accumulation unavailable before. Thus, in the western hemi sphere, the corresponding number of stations can, already today, reach at least 500, and the number of beam s to the satellites no less than 2000-3000. The analysis involved ﬁve days of the period 1999-2000, with daily mean values of the geomagnetic ﬁeld disturbance index /angbracketleftDst/angbracketrightranging from -4 to -70 nT and /angbracketleftKp/angbracketrightfrom 2.38 to 6.38 . These events are summarized in Table 1. Figures 3, 4, 5 presents the measured variations of a soft X-r ay 0.1-0.8 nm (a); >1 MeV proton ﬂux (b), and >5 MeV proton ﬂux (c) that were measured at geostationary orbi t by the GOES-10 stations (135◦W) with 5-min time resolution; H-component of the geomagneti c ﬁeld at station Irkutsk (52 .2◦N; 104.3◦E–d), and Dst (e) during major magnetic storms (on April 4, and July 15, 2000), and at the time5 of a powerful solar ﬂare of July 14, 2000 (a correlative analy sis of the data is made in Sections 4 and 5). The statistic of the data used in this paper for each of the day s under examination is characterized by the information in Table 1 about the number of stations use dm, the number of passes n, during which a given satellite was observed at some of the stations, and about a total number Σ lof 30-s observations. The total amount of data exceeds 11 .3×10630-s observations. 3. The method of processing the data from the Internet A concise account of the data handling procedure is in order. Primary data series represent the values of TEC, elevations and azimuths of the beam connectin g the receiver with the satellite; these values are calculated using our developed program CONVTEC w hich converts standard RINEX-ﬁles received via the Internet (http://lox.ucsd.edu/cgi-bin/ dbSimpleDailyDataBrowser.cgi). Besides, these ﬁles contain also information about phase sl ip measurements reﬂected in a special code Loss of Lock Indicator LLI (W. Gurther, unpublished report, 1993, available from International GPS Service for Geodynamics Central Bureav at http://igscb.jp l.nasa.gov:80/igscb/data/format/rinex2.txt). This code identiﬁes phase slips caused by phase losses at one or simultaneously two working frequencies f1= 1575 .42 MHz or f2= 1227 .60 MHz. This study uses the LLI code only in recording any phas e slip, no matter what the type of code. The TEC series that are identi ﬁed from the diﬀerence of phase L1 and L2 are then used only to conﬁrm the phase diﬀerence slip event . Thus, as a result of a pretreatment of the RINEX-ﬁles, we have the number of phase slips within a single selected time interval dT=5 min, as well as the corresponding number of observations t hat is required for normalizing the data. Our choice of such an inte rval was dictated by the need to reduce the amount of the data analyzed without decreasing the time reso lution that is required for the analysis (a standard time step for the RINEX-ﬁles equal to 30 s would requ ire a larger memory capacity). These data for each of the GPS satellites were then averaged f or all the stations selected in order to infer the mean density of observations M(t) and the mean density of phase slips S(t). In the middle of the observed satellite pass, the density of observations M(t) averages 10 ±1 (30-s counts); at the beginning and end of the pass it can decrease because the time intervals of observation of a given satellite at elevations larger than that speciﬁed do not coincide at di ﬀerent stations. Subsequently, we calculated the mean relative density of phase slips P(t)=S(t)/M(t), %. Furthermore, the daily mean value of the relative number of phase slips /angbracketleftP/angbracketrightthat was averaged over all GPS satellites and stations was us eful for our analysis. Slips of phase measurements can be caused by reception condi tions for the signal in the neighborhood of the receiver (interference from thunderstorms, radioin terferences), which is particularly pronounced at low elevations θ. To exclude the inﬂuence of the signal reception conditions , in this paper we used only observations with satellite elevations θlarger than 30◦. Besides, in the analysis we made use of the known scheme of coincidences, and recorded - as phase slips f or each point in time - only those which were observed simultaneously at more than s = 3 sites.6 To identity a part of coincident phase slips for a single time interval of analysis dT= 5 s from a total number of slips, we determined every time the threshol d value of the relative density of phase slips which were recorded for a given satellite at more than 3 stati ons simultaneously: P3 = 1/n, where n is the number of stations. These data were also veriﬁed using the series of 30-s counts; for reasons of space, we do not give here the relevant data. Another possible reasons for the phase slips, as has been poi nted out in the Introduction, is due to deep, fast changes in TEC because of a strong scattering of satellite signals from intense small-scale irregularities of the ionospheric F2-layer at equatorial and polar latitudes [ Aarons et al ., 1996, 1997; Klobuchar , 1997; Pi et al ., 1997; Aarons and Lin , 1999]. However, since we are using a global averaging of the number of phase slips for all beams and stations, as a co nsequence of the uneven distribution of stations the proportion of mid-latitude stations of North A merica and, to a lesser extent, of Europe is predominant (see above). Besides, to exclude situations as sociated with scintillations in the ionosphere, in this paper we used only the stations located in the mid-lat itude zone 30◦–60◦N of the northern hemisphere. Consequently, it is most likely that our detect ed phase slips were not caused by a strong scattering of satellite signals from small-scale irregula rities of the ionospheric F2-layer. 4. Results derived from analyzing the relative density of ph ase slips 4.1. Magnetically quiet days Figure 5 (left) plots the dependencies of the mean values of t he observation density M(t), the density of phase slips S(t), and of the relative density of phase slips P(t) for PRN07 as a function of universal time UT (a, b). The same values, averaged over all o bserved satellites but as a function of local time LT, are plotted in panels (c, d). Horizontal lines show the threshold for the chosen number of coincidences ( P3). The local time for each GPS station was calculated, based on the value of its geographic longitude. As would be expected the mean observation density M(t) for a single satellite exhibits a diurnal variation that is determined by the satellite’s orbit, and v aries over the range from 0 to 8. When averaged over all satellites, the value of M(t) shows almost no essential minima but decreases to the value 1.7, on average. With the scale of Figure 5 selected, th eS(t)-dependence for July 29, 1999 merges with the axis of abscissa. As is evident from Figure 5b, 5d, the phase slips on a magnetic ally quiet day have a sporadic character, and are not time-coincident at diﬀerent station s (i.e. they are most likely to correspond to the signal reception conditions at separate stations). As a con sequence, the value of the relative density of phase slips P(t) for PRN07 (b) and for al satellites in general (d) does not ex ceed the required threshold Psats= 3. The daily mean value of the relative density of phase slip s/angbracketleftP/angbracketright, averaged over all GPS satellites and stations, was 0.005 % for the magnetically qu iet day of July 29, 1999 (the ﬁrst line in the7 Table 1). Figure 6 plots the local time (LT) dependence of the relative mean density of phase slips P(t) obtained by averaging the data from all GPS satellites obser ved simultaneously at the GPS stations network at elevations larger than 30◦. The P(t)-dependencies are smoothed with a time window of 2 hours. The respective dates are shown at the P(t)- curves. Wi th the scale of Figure 6 selected, the P(t)-dependence for July 29, 1999 (the non-smoothed dependenc e in Figure 5d) virtually merges with the abscissa axis. For the other magnetically quiet day of January 9, 2000, howe ver, the mean value of /angbracketleftP/angbracketrightwas already an order of magnitude larger, 0.06 % (the second line in the Ta ble 1). For the diurnal P(t)-dependence on January 9, 2000, one can point out the irregularity of the m ean density of phase slips as a function of local time LT. 4.2. Magnetic storms of April 6 and July 15, 2000 A totally diﬀerent picture was observed on April 6, 2000 duri ng a strong magnetic storm with a well-deﬁned sudden commencement. Figure 2 presents the var iations in soft X-ray emission (a); >1 MeV proton ﬂux (b), >5 MeV (c) measured at geostationary orbit of the GOES-10 sate llite (135◦W) with 5-min time resolution; H-component of the geomagnetic ﬁeld at station Irkutsk (52 .2◦N; 104.3◦E – d) and Dst (e). Figure 5 (right) plots the mean values of the observation den sityM(t), the density of phase slips S(t), and of the relative density of phase slips P(t) for PRN07 as a function of universal time UT (e, f). The SSC time 16:42 UT is shown in panels e) and f) by a vertical b ar. In this case, with the purpose of achieving a clearer detecti on of the eﬀect of the magnetic storm SSC inﬂuence on the P(t)-dependence, we chose only those GPS stations which were on the dayside of the Earth at the SSC time. The same values, averaged over all o bserved satellites but as a function of local time LT, are plotted in panels (g, h). Horizontal lines show the threshold for the chosen number of coincidences ( P3). The local time LT dependence of the relative density of phase slips P(t), obtained by averaging the data from all GPS satellites, is presented in Figure 6. First of all, it should be noted that the relative density of p hase slips P(t) exceeds that for magnetically quiet days by one (when compared with January 9 , 2000) or even two (when compared with July 29, 1999) orders of magnitude, and reaches a few and (for some of the GPS satellites) even ten percent of the total observation density (Figure 5f). The me an value of /angbracketleftP/angbracketrightfor this storm is 0.53 % (the third line in the Table 1), which is by a factor of 100 larger th an that of /angbracketleftP/angbracketrightfor July 29, 1000, and by a factor of 10 larger than that for January 9, 2000. It was also found that the averaged (over all satellites) lev el of phase slips for the GPS satellites on the subsolar side of the Earth is by a factor of 3-5 larger than that on the opposite side of the Earth (Figure 6).8 Besides, noteworthy is a well-deﬁned eﬀect of an abrupt incr ease in the density of phase slips that occurred after a sudden storm commencement SSC (Figure 5f). To synchronize this eﬀect with the data illustrating magnetic ﬁeld and energetic particle ﬂux vari ations at geostationary orbit, Figure 2 presents theP(t)-dependencies for PRN02 (g) and PRN07 (h). A similar result conﬁrming all of the above-mentioned featu res of the April 6, 2000 storm was also obtained for the other magnetic storm of July 15, 2000 (see th e measurements at geostationary orbit and at magnetic observatory Irkutsk in Figure 3, and the loca l time LT dependence of the relative mean density of phase slips P(t) obtained by averaging the data from all GPS satellites, in F igure 6). The mean value of /angbracketleftP/angbracketrightfor this storm is 0.33 % (the ﬁfth line in the Table 1), which is also in appreciable excess of the level of phase slips for magnetica lly quiet days. The eﬀect of an abrupt increase in the density of phase slips i mmediately after a sudden storm commencement SSC is clearly pronounced for this storm as wel l. Figure 7 (left) plots the dependencies of the mean observation density M(t), the density of phase slips S(t) and of the relative density of phase slips P(t) for PRN04 (e, f) and PRN24 (g, h) as a function of universal ti me UT. The SSC time (14:40 UT) is shown by a vertical bar. It is evident from Figur e 7 that these satellites were observed without phase slips coinciding at more than 3 stations, at le ast as early as 3 hours before the SSC onset; signiﬁcant phase slips are noticed thereafter, however. 4.3. Powerful solar ﬂare of July 14, 2000 The fact that disturbances in the Earth’s immediate interpl anetary environment aﬀect the operation of the GPS system was also conﬁrmed during a powerful class X5 .7 solar ﬂare of July 14, 2000. Figure 4 presents the variations in and soft X-ray emission ( a);>1 MeV proton ﬂux (b); >5 MeV proton ﬂux (c) measured at geostationary orbit of the GOE S-8 satellite (74◦W) with 5-min time resolution and H-component of the geomagnetic ﬁeld at stati on Irkutsk (52 .2◦N; 104.3◦E) (d) and Dst (e). The mean value of Dst for that day did not exceed -4 nT; therefo re, the mean level of phase slips was substantially below that for magnetically quiet days. T he mean value of /angbracketleftP/angbracketrightfor this storm is 0.077 % (the fourth line in the Table 1), which exceeds only slightl y the level of phase slips for magnetically quiet days. However, the averaged (over all satellites) lev el of phase slips for GPS satellites on the subsolar side of the Earth was also larger than that on the opp osite side of the Earth (Figure 6). Of the greatest interest is the eﬀect of an abrupt increase in th e phase slip density immediately after the start of the solar ﬂare which is clearly seen in the Figure 7 de pendencies of the mean relative density of phase slips P(t) for PRN15 (a,b) and PRN21 (c,d). The onset time of the ﬂare (1 0:03 UT) is shown by a vertical bar. It is evident from Figure 5 that these satel lites were observed without phase slips coinciding at more than 2 stations at least as early as 3 hours before the ﬂare onset, and only after that did phase slips set in. A comparison of this dependence with the data from GOES-8 (74◦W) presented in Figure 4 suggest9 the conclusion that such a response was caused by a signiﬁcan t step-like increase in ﬂux density of high-energy protons rather than by a short-duration increa se in X-ray emission ﬂux. 5. Discussion and Conclusions Our intention was to investigate how magnetospheric distur bances accompanying solar ﬂares and magnetic storms aﬀect the operation of the GPS system. A deta iled analysis of the factors responsible for the phase slips in the GPS system is a highly diﬃcult task, and is beyond the scope of this paper. The main results of this study may be summarized as follows: 1. Our recorded phase slips are most likely to be caused by the processes in the neighborhood of the GPS satellites or in the plasmasphere rather than by a strong scattering of satellite signals from small-scale ionospheric F2-layer irregularities at equatorial and polar latitudes [ Aarons et al ., 1996, 1997;Klobuchar , 1997; Pi et al ., 1997; Aarons and Lin , 1999]. We have restricted our analysis to mid-latitudes, however. Phase scintilaltions in the cited references were recorded from the excess of a maximum value of the time derivative of TEC of a given threshold. To estimate t his value requires measuring TEC without a phase loss at both frequencies. If a phase slip is re corded, then it is no longer possible to determine TEC for this point in time. Therefore, the data o n phase scintillations reported in [Aarons et al ., 1996, 1997; Klobuchar , 1997; Pi et al ., 1997; Aarons and Lin , 1999] provide no statistic of phase slips in the GPS system. The authors of the cited references point out that phase scin tillations are most frequently manifested in the dawn hours of local time LT. According to ou r data, a maximum phase slip density on magnetically disturbed days corresponds to 12–1 4 LT. 2. We have detected a strong dependence of the relative densi ty of phase slips in the navigation system GPS on the disturbance level of the Earth’s magnetosp here during major magnetic storms and powerful solar ﬂares. The phase slips that are caused by t he processes occurring near the satellites can be elicited by break-down of the microsystem electronics due to high-energy particles [Stanley and Curtis , 1976], by anomalous surface charging of the satellite [ Degtyarev et al ., 1989], as well as by a change in antenna impedance in the presence of the time-varying plasma environment. There may also be possible eﬀects of signal propagation in th e plasmasphere, including those caused by a strong scattering from plasmaspheric irregular ities extended along the magnetic ﬁeld [Jacobson et al ., 1996]. However, the eﬀect of high-energy protons is most probable, because under geomagneticly quiet conditions the GPS satellite resides for about half the time in regions of the space environment where solar cosmic rays freely penetrate. Thus, the residen ce time of the satellite in regions where solar protons with energies 30 MeV (L 5) and 1 MeV (L 6.5) is, re spectively, about10 8 and 5 hours per orbit. During geomagnetic disturbances ( \|Dst\|>100 nT) the boundary of penetration of solar cosmic- ray protons into the magnetosp here is shifted to smaller L-shells, and the entire GPS orbit becomes accessible for protons with Ep>1 MeV. To ascertain the physical mechanisms for phase slips, it is n ecessary to detail the character of phase slips for the GPS satellites depending on their location wit h respect to a particular conﬁguration of the Earth’s magnetosphere with due regard for the dynamic changes of this conﬁguration during the development of magnetospheric disturbances, and for da ta on the magnetic ﬁeld, electric ﬁelds and particle ﬂuxes both at GPS satellite orbits and over the e ntire range of distances in the “Sun-Earth” system. 3. During major magnetic storms the relative density of phas e slips exceeds that for magnetically quiet days at least by one or two orders of magnitude, and reac hes a few and (for some of the GPS satellites) even ten percent of the total observation densi ty, which can become unaccepTable when solving some important navigation problems. Of course, pha se measurements are more sensitive to equipment failures and to interferences in the GPS “satel lite-receiver” channel when compared with group delay measurements which are directly used in sol ving navigation problems. Therefore, it is necessary to have a monitoring of the errors of determin ing the coordinates of stationary sites of the global GPS network, based on the data in the RINEX-form at available from the Internet, and to analyze these series in conjunction with the data on th e conditions of the near-terrestrial space environment. 4. The level of phase slips for the GPS satellites on the sunwa rd side of the Earth is by a factor of 3-5 times than that on the opposite side of the Earth. This diﬀere nce can be caused by the fact that the physical conditions in the immediate environment of the satellites are substantially diﬀerent on the dayside and nightside. 5. For strong isolated magnetic storms, we detected also a cl early pronounced eﬀect of an abrupt increase in phase slip density immediately after a sudden st orm commencement SSC. 6. A similar eﬀect was also detected at the time of a powerful s olar ﬂare of class X5.7 on July 14, 2000. Hence, not only can the disturbances in the near-terrestria l space environment (caused by corresponding processes in the “Sun-Earth” system) be dete cted in TEC measurements by processing the GPS data, as has now been demonstrated in a large number of studies, but they also aﬀect the operation of the navigation system GPS itself. This means th at to develop methods and techniques with the purpose of improving the GPS system operation relia bility under disturbed conditions requires invoking the whole of the research tools of solar-t errestrial physics - from models to various special-purpose facilities for space monitoring of the pro cesses in the “Sun-Earth” system.11 We are aware that this study has revealed only the key average d patterns of this inﬂuence, and we hope that it would give impetus to a wide variety of more detai led investigations. Acknowledgments. Authors are grateful to A. V. Mikhalev, V. V. Koshelev, V. V. Yevstaﬁev, G. V. Popov and A. V. Tashilin for their encou raging interest in this study and active participations in discussions. We are also indeb ted to S. A. Nechaev for the data from magnetic observatory Irkutsk made available to us. Thanks a re also due V. G. Mikhalkovsky for his assistance in preparing the English version of the T EX-manuscript. This work was done with support under RFBR grant of leading scientiﬁc schools o f the Russian Federation No. 00-15-98509 and Russian Foundation for Basic Research (gra nt 99-05-64753), as well as RF Minvuz Grant 1999; supervisor B. O. Vugmeister.12 References Aarons, J., M. Mendillo, E. Kudeki, and R. Yantosca, GPS phas e ﬂuctuations in the equatorial region during the MISETA 1994 campain, J. Geophys. Res. ,101, 26851–26862, 1996. Aarons, J., M. Mendillo, and R. Yantosca, GPS phase ﬂuctuati ons in the equatorial region during sunspot minimum, Radio Sci. ,32, 1535–1550, 1997. Aarons, J., and B. Lin, Development of high latitude phase ﬂu ctuations during the January 10, April 10-11, and May 15, 1997 magnetic storms, J. Atmos. and Sol.-Terr. Phys. ,61, 309–327, 1999. Afraimovich, E. L., K. S. Palamartchouk, N. P. Perevalova, a nd V. V. Chernukhov, Ionospheric eﬀects of the solar eclipse of March 9, 1997, as deduced from G PS data, Geophys. Res. Lett.,25, 465–468, 1998. Afraimovich, E. L., E. A. Kosogorov, L. A. Leonovich, K. S. Pa lamarchouk, N. P. Perevalova, and O. M. Pirog, Determining parameters of large-scale trav eling ionospheric disturbances of auroral origin using GPS-arrays. J. Atmos. and Sol.-Terr. Phys. ,62, 553–565, 2000. Afraimovich, E. L., GPS global detection of the ionospheric response to solar ﬂares. Radio Sci., – accepted. Afraimovich, E. L., E. A. Kosogorov, K. S. Palamarchouk, N. P . Perevalova, and A. V. Plotnikov, The use of GPS arrays in detecting the ionosp heric response during rocket launchings – accepted to special issue of ‘Earth, Pla nets, and Space’: Proceedings of International Symposium on GPS: Application to Earth Sci ences and Interaction with Other Space Geodetic Techniques, 1999. Davies, K., G. K. Hartmann, Studying the ionosphere with the Global Positioning System, Radio Sci. ,32, 1695–1703, 1997. Degtyarev, V. I., G. V. Popov, O. S. Grafodatsky, and Sh. N. Is lyaev, Satellite electrization in circular orbit (of 20000 km altitude), Issledovania po geomagnetizmy, aeronomii i ﬁzike solnca ,85, 15–26, 1989. Hofmann-Wellenhof, B., H. Lichtenegger, and J. Collins, Global Positioning System: Theory and Practice , Springer-Verlag Wien, New York, 1992. Jacobson, A. R., G. Hoogeveen, R. C. Carlos, G. Wu, B. G. Fejer , and M. C. Kelley, Observations13 of inner plasmasphere irregularities with a satellite-bea con radio-interferometer array, J. Geophys. Res. ,101, 19665-1982, 1996. Klobuchar, J. A., Real-time ionospheric science: The new re ality,Radio Sci. ,32, 1943–1952, 1997. Pi, X., A. J. Mannucci, U. J. Lindgwister, and C. M. Ho, Monito ring of global ionospheric irregularities using the woldwide GPS network, Geophys. Res. Lett. ,24, 2283–2286, 1997. Stanley, B. Curtis, JEEE Transactions on Nuclear Science, 23, 1355–1361, 1976. E. L. Afraimovich, Institute of Solar-Terrestrial Physics SD RAS, p. o. box 4026, Irkutsk, 664033, Russia, fax: +7 3952 462557; e-mail: afra@iszf.irk.ru ReceivedarXiv:physics/0009027v1 [physics.geo-ph] 6 Sep 2000Table 1: Statistics of experiments N DateDay number m n/summationtextl ×106/angbracketleftP/angbracketright, %/angbracketleftDst/angbracketright, nT /angbracketleftKp/angbracketright 129.07.1999 210 160 2300 1.86 0.006 -17.9 2.38 29.01.2000 009 323 4293 2.13 0.07 -4.79 – 36.04.2000 097 243 2520 1.94 0.86 -67.42 4.5 414.07.2000 196 310 5029 3.22 0.19 -4 4.25 515.07.2000 197 306 5042 3.1 0.49 -58.38 6.38
arXiv:physics/0009028v1 [physics.data-an] 7 Sep 2000Technical Report No. 2005, Department of Statistics, Unive rsity of Toronto Slice Sampling Radford M. Neal Department of Statistics and Department of Computer Scienc e University of Toronto, Toronto, Ontario, Canada http://www.cs.utoronto.ca/ ∼radford/ radford@stat.utoronto.ca 29 August 2000 Abstract. Markov chain sampling methods that automatically adapt to c haracteristics of the distribution being sampled can be constructed by expl oiting the principle that one can sample from a distribution by sampling uniformly from th e region under the plot of its density function. A Markov chain that converges to this u niform distribution can be constructed by alternating uniform sampling in the vertica l direction with uniform sampling from the horizontal ‘slice’ deﬁned by the current vertical p osition, or more generally, with some update that leaves the uniform distribution over this s lice invariant. Variations on such ‘slice sampling’ methods are easily implemented for un ivariate distributions, and can be used to sample from a multivariate distribution by updati ng each variable in turn. This approach is often easier to implement than Gibbs sampling, a nd more eﬃcient than simple Metropolis updates, due to the ability of slice sampling to a daptively choose the magnitude of changes made. It is therefore attractive for routine and a utomated use. Slice sampling methods that update all variables simultaneously are also p ossible. These methods can adaptively choose the magnitudes of changes made to each var iable, based on the local properties of the density function. More ambitiously, such methods could potentially allow the sampling to adapt to dependencies between variables by c onstructing local quadratic approximations. Another approach is to improve sampling eﬃ ciency by suppressing random walks. This can be done using ‘overrelaxed’ versions of univ ariate slice sampling procedures, or by using ‘reﬂective’ multivariate slice sampling method s, which bounce oﬀ the edges of the slice. Keywords: Markov chain Monte Carlo, adaptive methods, Gibbs sampling , Metropolis algorithm, overrelaxation, dynamical methods. 1 Introduction Markov chain methods such as Gibbs sampling (Gelfand and Smi th 1990) and the Metropolis algorithm (Metropolis, et al 1953, Hastings 1970) can be used to sample from many of the complex, multivariate distributions encountered in st atistics. However, to implement Gibbs sampling, one may need to devise methods for sampling f rom non-standard univariate 1distributions, and to use the Metropolis algorithm, one mus t ﬁnd an appropriate ‘proposal’ distribution that will lead to eﬃcient sampling. The need fo r such special tailoring limits the routine use of these methods, and inhibits the development o f software that automatically constructs Markov chain samplers from model speciﬁcations . Furthermore, many common Markov chain samplers are ineﬃcient, due to a combination of two ﬂaws. First, they may try to make changes that are not well adapted to the local prop erties of the density function, with the result that changes must be made in small steps. Seco nd, these small steps take the form of a random walk, in which about n2such steps are needed in order to move a distance that could be traversed in only nsteps if these steps moved consistently in one direction. In this paper, I describe a class of ‘slice sampling’ methods that can be applied to a wide variety of distributions. Simple forms of univariate s lice sampling are an alternative to Gibbs sampling that avoids the need to sample from non-sta ndard distributions. These slice sampling methods can adaptively change the scale of ch anges made, which makes them easier to tune than Metropolis methods, and also avoids problems that arise when the appropriate scale of changes varys over the distributio n. More complex slice sampling methods can adapt to the dependencies between variables, al lowing larger changes than would be possible with Gibbs sampling or simple Metropolis m ethods. Slice sampling methods that improve sampling by suppressing random walks c an also be constructed. Slice sampling originates with the observation that one can sample from a univariate distribution by sampling points uniformly from the region u nder the curve of its density function, and then looking only at the horizontal coordinat es of the sample points. A Markov chain that converges to this uniform distribution ca n be constructed by alternately sampling uniformly from the vertical interval deﬁned by the density at the current point, and from the union of intervals that constitutes the horizon tal ‘slice’ though the plot of the density function that this vertical position deﬁnes. If thi s last step is still diﬃcult, one may substitute some other update that leaves the uniform distri bution over the current slice invariant. To sample from a multivariate distribution, suc h single-variable slice sampling updates can be applied to each variable in turn. The details o f these single-variable slice sampling methods are described in Section 4. One can also apply the slice sampling approach to a multivari ate distribution directly, as described in Section 5, by sampling uniformly under the mu ltidimensional plot of its density function. As for a univariate distribution, this ca n be done by alternately sampling uniformly from the vertical interval from zero up to the dens ity at the current point, and then uniformly from the slice deﬁned by this vertical positi on. When the slice is high- dimensional, how to eﬃciently sample from it is less obvious than for single-variable slice sampling, but one gains the possibility of sampling in a way t hat respects the dependencies between variables. I show how, in the context of slice sampli ng, the way changes are proposed can be adapted to respect these dependencies, base d on local information about the density function. In particular, local quadratic appro ximations could be constructed, as have been used very successfully for optimization proble ms. Adaptive slice sampling appears to be simpler than a somewhat analogous scheme propo sed for the Metropolis algorithm (Mira 1998, Chapter 5; Tierney and Mira 1999; Gree n and Mira 1999). However, 2further research will be needed to fully exploit the adaptiv e capabilities of multivariate slice sampling. One might instead accept that dependencies between variabl es will lead to the distribu- tion being explored in small steps, but try at least to avoid e xploring the distribution by an ineﬃcient random walk, which is what happens when simple f orms of the Metropolis algorithm are used. The beneﬁts of random walk suppression a re analysed theoretically in some simple contexts by Diaconis, Holmes, and Neal (in press ). Large gains in sampling eﬃciency can be obtained in practice when random walks are su ppressed using the Hybrid Monte Carlo or other dynamical methods (Duane, Kennedy, Pen dleton, and Roweth 1987; Horowitz 1991; Neal 1994; and for reviews from a more statist ical perspective, Neal 1993, 1996), or by using an overrelaxation method (Adler 1981; Bar one and Frigessi 1990; Green and Han 1992; Neal 1998). Dynamical and overrelaxation meth ods are not always easy to apply, however. Use of Markov chain samplers that avoid ra ndom walks would be as- sisted by the development of methods that require less speci al programming and parameter tuning. Two approaches to random walk suppression based on slice sam pling are discussed in this paper. In Section 6, I show how one can implement an overrelax ed version of the single- variable slice sampling scheme. This may provide the beneﬁt s of Adler’s (1981) Gaussian overrelaxation method for more general distributions. In S ection 7, I describe slice sampling analogues of dynamical methods, which move around a multi-v ariable slice using a stepping procedure that proceeds consistently in one direction whil e reﬂecting oﬀ the slice boundaries. Although these more elaborate slice sampling methods requi re more tuning than the single- variable slice sampling schemes, they may still be easier to apply than alternative methods that avoid random walks. To illustrate the advantages of the adaptive nature of slice sampling, I show in Section 8 how it can help avoid disaster when sampling from a distribut ion that is typical of priors for hierarchical Bayesian models. Simple Metropolis metho ds can give the wrong answer for this problem, while providing little indication that an ything is amiss. This paper concludes (in Section 9) with a discussion of the m erits of the various slice sampling methods in comparison with other Markov chain meth ods, and of their suitability for routine and automated use. Below, I set the stage by discu ssing general-purpose Markov chain methods that are currently in wide use. Readers who are quite familiar with Markov chain sampling and are eager to get to the main idea can skip im mediately to Section 3. 2 General-purpose Markov chain sampling methods Applications of Markov chain sampling in statistics often i nvolve sampling from many dis- tributions. In Bayesian applications, we must sample from t he posterior distribution for the parameters of a model given certain data. Diﬀerent datas ets will produce diﬀerent pos- terior distributions, which may diﬀer in important charact eristics such as diﬀuseness and multimodality. Furthermore, we will often wish to consider a variety of models. For routine use of Markov chain methods, it is important to minimize the a mount of eﬀort that the data analyst must spend in order to sample from all these dist ributions. Ideally, a Markov 3chain sampler would be constructed automatically for each m odel and dataset. The Markov chain method most commonly used in statistics is G ibbs sampling, popu- larized by Gelfand and Smith (1990). Suppose that we wish to s ample from a distribution overnstate variables (eg, model parameters), written as x= (x1,... ,x n), with probabil- ity density p(x). Gibbs sampling proceeds by sampling in succession from th e conditional distributions for each xigiven the current values of the other xjforj∝ne}ationslash=i, with conditional densities written as p(xi\|{xj}j/negationslash=i). Repetition of this procedure deﬁnes a Markov chain which leaves the desired distribution invariant, and which in many circumstances is ergodic (eg, when p(x)>0 for all x). Running the Gibbs sampler for a suﬃciently long time will then produce a sample of values for xfrom close to the desired distribution, from which the expectations of quantities of interest (eg, posterior mean s of parameters) can be estimated. Gibbs sampling can be done only if we know how to sample from al l the required con- ditional distributions. These sometimes have standard for ms for which eﬃcient sampling methods have been developed, but there are many models for wh ich sampling from these conditional distributions requires the development of cus tom algorithms, or is infeasible in practice (eg, for multilayer perceptron networks (Neal 199 6)). Note, however, that once methods for sampling from these conditional distributions have been found, no further tuning parameters need be set in order to produce the ﬁnal Mar kov chain sampler. The routine use of Gibbs sampling has been assisted by the dev elopment of Adaptive Rejection Sampling (ARS) (Gilks and Wild 1992; Gilks 1992), which can be used to eﬃ- ciently sample from any conditional distribution whose den sity function is log concave, given only the ability to compute some function, fi(xi), that is proportional to the conditional density, p(xi\|{xj}j/negationslash=i) (the ability to also compute the derivative, f′ i(xi), is helpful, but not essential). This method has been used for some time by the BUGS software (Thomas, Spiegelhalter, and Gilks 1992) to automatically generate M arkov chain samplers from model speciﬁcations. The ﬁrst step in applying ARS is to ﬁnd points on each side of the mode of the conditional distribution (one of which can be the curren t point). This will in general require a search, which will in turn require the choice of som e length scale for an initial step. However, the burden of setting this scale parameter is lessened by the fact that a good value for it can be chosen ‘retrospectively’, based on p ast iterations of the Markov chain, without invalidating the results (since the setting of this parameter aﬀects only the computation time, not the distribution sampled from). The Adaptive Rejection Metropolis Sampling (ARMS) method ( Gilks, Best, and Tan 1995) generalizes ARS to conditional distributions whose d ensity functions may not be log-concave. However, when the density is not log-concave, ARMS does not produce a new point drawn independently from the conditional distrib ution, but merely updates the current point in a fashion that leaves this distribution inv ariant. Applying ARMS to sample from the conditional distribution of each variable in succe ssion will result in an equilibrium distribution that is exactly correct, but when some conditi onal distributions are not log- concave, it may take longer to approach this equilibrium tha n would be the case if true Gibbs sampling were used. Also, when a conditional distribution i s not log-concave, the points used to set up the initial approximation to it must not be chos en with reference to past iterations, as this could result in the wrong distribution b eing sampled (Gilks, Neal, Best, 4and Tan 1997). The initial approximation must be chosen base d only on prior knowledge (including any preliminary Markov chain sampling runs), an d on the current values of the other variables. Unlike ARS, neither the current value of th e variable being updated, nor any statistics collected from previous updates (eg, the typ ical scale of changes) can be used. This hinders routine use of the method. Another general way of constructing a Markov chain sampler i s to perform Metropolis updates (Metropolis, et al1953, Hastings 1970), either for each variable in turn, as wi th Gibbs sampling, or for all variables simultaneously. A Metr opolis update starts with the random selection of a ‘candidate’ state, drawn from a ‘propo sal’ distribution. The candidate state is then accepted or rejected as the new state of the Mark ov chain, based on the ratio of the probability densities of the candidate state and the c urrent state. If the candidate state is rejected, the new state is the same as the old state. A simple ‘random-walk’ Metropolis scheme can be constructe d based on a symmetric proposal distribution (eg, Gaussian) that is centred on the current state. All variables could be updated simultaneously in such a scheme, or alterna tively, one variable could be updated at a time. In either case, a scale parameter is requir ed for each variable to ﬁx the width of the proposal distribution in that dimension. For th e method to be valid, these scale parameters must not be set on the basis of past iteratio ns, but rather only on the basis of prior knowledge (including preliminary runs), and the current values of variables that are not being changed in the present update. Choosing to o large a value for the scale of a proposal distribution will result in a high rejection ra te, while choosing too small a value will result in ineﬃcient exploration via a random walk with unnecessarily small steps. Furthermore, the appropriate scale for Metropolis proposa ls may vary from one part of the distribution to another, in which case no single value wi ll produce acceptable results. Selecting a scale at random from some range can sometimes all eviate these problems, but at a large cost in wasted eﬀort when the scale selected is inap propriate. It is tempting to tune the Metropolis proposal distribution based on the rejection rate in past iterations of the Markov chain, but such ‘retrospectiv e tuning’ is not valid in general, since it can disturb the stationary distribution to which th e process converges (as was also the case for ARMS). Fixing the proposal distribution based o n a preliminary run is allowed, but if the original proposal distribution was not good, such a preliminary run may not have sampled from the whole distribution, and hence may be a bad gu ide for tuning. We therefore see that although Gibbs sampling and Metropoli s methods have been used to do much useful work, there is a need for better methods, tha t can be routinely applied in a wider variety of situations. One of my objectives in this paper is to ﬁnd variations on slice sampling that can be used to sample from any continuous distribution, given only the ability to evaluate a ‘black-box’ function that is proporti onal to its density, and in some cases, to also evaluate the gradient of the log of this functi on. For many distributions, these new methods will not sample more eﬃciently than true Gi bbs sampling or a well- designed Metropolis scheme, but the slice sampling methods will often requiring less eﬀort to implement and tune. For some distributions, however, sli ce sampling can be much more eﬃcient, because it can adaptively choose a scale for change s appropriate for the region of the distribution currently being sampled. Slice samplers t hat adapt in more elaborate ways, 5or that are designed to suppress random walks, can potential ly be much faster than simple Metropolis methods or Gibbs sampling. 3 The idea of slice sampling Suppose we wish to sample from a distribution for a variable, x, taking values in some subset ofℜn, with density function proportional to some function f(x). We can do this by sampling uniformly from the n+1 dimensional region that lies under the plot of f(x). This idea can be formalized by introducing an auxiliary real variable, y, and deﬁning a joint distribution overxandythat is uniform over the region U={(x,y) : 0 < y < f (x)}below the curve or surface deﬁned by f(x). That is, the joint density for ( x,y) is p(x,y) =/braceleftBigg 1/Z if 0< y < f (x) 0 otherwise(1) where Z=/integraltextf(x)dx. The marginal density for xis then p(x) =/integraldisplayf(x) 0(1/Z)dy=f(x)/Z (2) as desired. To sample for x, we can sample jointly for ( x,y), and then ignore y. Generating independent points drawn uniformly from Umay not be easy, so we might in- stead deﬁne a Markov chain that will converge to this uniform distribution. Gibbs sampling is one possibility: We sample alternately from the conditio nal distribution for ygiven the current x— which is uniform over the interval (0 ,f(x)) — and from the conditional distri- bution for xgiven the current y— which is uniform over the region S={x:y < f(x)}, which I call the ‘slice’ deﬁned by y. Generating an independent point drawn uniformly from Smay still be diﬃcult, in which case we can substitute some upd ate for xthat leaves the uniform distribution over Sinvariant. Similar auxiliary variable methods been used in the past. Hi gdon (1996) has interpreted the standard Metropolis algorithm in these terms. The highl y successful Markov chain algorithm for the Ising model due to Swendsen and Wang (1987) can also be seen as an auxiliary variable method, which has been generalized by Ed wards and Sokal (1988). In their scheme, the density (or probability mass) function is proportional to a product of k functions: p(x)∝f1(x)···fk(x). They introduce kauxiliary variables, y1,... ,y k, and deﬁne a joint distribution for ( x,y1,... ,y k) which is uniform over the region where 0 < yi< fi(x) fori= 1,... ,k . Gibbs sampling, or some related Markov chain procedure, ca n then be used to sample for ( x,y1,... ,y k), much as described above for the case of a single auxiliary variable. Applications of such methods to image analysis ha ve been discussed by Besag and Green (1993) and by Higdon (1996). Mira and Tierney (in press) have shown that these auxiliary v ariable methods, with one or with many auxiliary variables, are uniformly ergodic under certain conditions. Roberts and Rosenthal (1999) have shown that these methods are geometri cally ergodic under weaker conditions, and have also found some quantitative converge nce bounds. These results all assume that the sampler generates a new value for xthat is uniformly drawn from S, independently of the old value, which is often diﬃcult in pra ctice. 6Concurrently with the work reported here1, Damien, Wakeﬁeld, and Walker (in press) have viewed methods based on multiple auxiliary variables a s a general approach to con- structing Markov chain samplers for Bayesian inference pro blems. They illustrate how one can often decompose f(x) into a product of kfactors for which the intersection of the sets {x:yi< fi(x)}is easy to compute. This leads to an easily implemented sampl er, but convergence is slowed by the presence of many auxiliary vari ables. For example, for a model ofki.i.d. data points, one simple approach (similar to some exa mples of Damien, et al) is to have a factor (and auxiliary variable) for each data point, w ith the product of these factors being the likelihood. (Suppose for simplicity that the prio r is uniform, and so needn’t be represented in the posterior density.) For many models, ﬁnd ing{x:yi< fi(x)}will be easy to compute when fiis the likelihood from one data point. However, if this appro ach is applied to ndata points that are modeled as coming from a Gaussian distri bution with mean µand variance 1, it is easy to see that after the yiare chosen, the allowable range for µwill have width of order 1 /n. Since the width of the posterior distribution for µwill be of order 1 /√n, and since the posterior will be explored by a random walk, th e convergence time will be of order n. Gibbs sampling would, of course, converge in a single itera tion when there is only one parameter, and the slice sampling meth ods of this paper would also converge very rapidly for this problem, for any n. Using a large number of auxiliary variables is a costly way to avoid diﬃcult computations. I therefore am concerned in this paper with methods based on s lice sampling with a single auxiliary variable. So that these methods will be practical for a wide range of problems, they often use updates for xthat do not produce a point drawn independently from the slice, S, but merely change xin some fashion that leaves the uniform distribution over S invariant. This allows the methods to be used for any continu ous distribution, provided only that we can compute some function, f(x), that is proportional to the density. 4 Single-variable slice sampling methods Slice sampling is simplest when only one (real-valued) vari able is being updated. This will of course be the case when the distribution of interest is uni variate, but more typically, the single-variable slice sampling methods of this section will be used to sample from a multivariate distribution for x= (x1,... ,x n) by sampling repeatedly for each variable in turn. To update xi, we must be able to compute a function, fi(xi), that is proportional to p(xi\|{xj}j/negationslash=i), where{xj}j/negationslash=iare the values of the other variables. Often, the joint distr i- bution for x1,... ,x nwill be deﬁned by some function, f(x1,... ,x n), that is proportional to the joint density, in which case we can simply take fi(xi) =f(... ,x i,...), where the variables other than xiare ﬁxed to their current values. To simplify notation, I will here write the single real varia ble being updated as x(with subscripts denoting diﬀerent such points, not components o fx). I will write f(x) for the 1An earlier version of this paper, under the title “Markov cha in Monte Carlo methods based on ‘slicing’ the density function” was issued in November 1997 as Technic al Report 9722, Department of Statistics, University of Toronto (available from my web page). It conta ins essentially all the material in this paper with the exception of Sections 5 and 8. The ﬁrst application o f the methods developed here, by Frey (1997), predates this technical report, referencing it as being ‘in preparation’. 7function proportional to the probability density of x. The single-variable slice sampling methods discussed here replace the current value, x0, with a new value, x1, found by a three-step procedure: a)Draw a real value, y, uniformly from (0 ,f(x0)), thereby deﬁning a horizontal ‘slice’: S={x:y < f(x)}. Note that x0is always within S. b)Find an interval, I= (L,R), around x0that contains at least a big part of the slice. c)Draw the new point, x1, from the part of the slice within this interval (ie, from S∩I). Step(a)picks a value for the auxiliary variable that is characteris tic of slice sampling. Note that there is no need to retain this auxiliary variable f rom one iteration of the Markov chain to the next, since its old value is forgotten at this poi nt anyway. In practice, it is often safer to compute g(x) = log( f(x)) rather than f(x) itself, in order to avoid pos- sible problems with ﬂoating-point underﬂow. One can then us e the auxiliary variable z= log( y) =g(x0)−e, where eis exponentially distributed with mean one, and deﬁne the slice by S={x:z < g(x)}. Steps (b)and(c)can potentially be implemented in several ways, which must o f course be such that the resulting Markov chain leaves the distribut ion deﬁned by f(x) invariant. Figure 1 illustrates one generally-applicable method, in w hich the interval is found by ‘stepping out’, and the new point is drawn with a ‘shrinkage’ procedure. Figure 2 illustrates an alternative ‘doubling’ procedure for ﬁnding the interva l. These and some other variations are described in detail next, followed by a proof that the res ulting transitions leave the correct distribution invariant. I then describe some short cuts that are possible when the distribution is unimodal. 4.1 Finding an appropriate interval After a value for the auxiliary variable has been drawn, deﬁn ing the slice S, the next task is to ﬁnd an interval I= (L,R), containing the current point, x0, from which the new point, x1, will be drawn. We would like this interval to contain as much of the slice as is feasible, so as to allow the new point to diﬀer as much as possi ble from the old point, but we would also like to avoid intervals that are much larger tha n the slice, as this will make the subsequent sampling step less eﬃcient. Several schemes for ﬁnding an interval are possible: 1) Ideally, we would set L= inf( S) and R= sup( S). That is, we would set Ito the smallest interval that contains the whole of S. This may not be feasible, however. 2) If the range of xis bounded, we might simply let Ibe that range. However, this may not be good if the slice is typically much smaller than the ran ge. 3) Given an estimate, w, for the scale of S, we can randomly pick an initial interval of sizew, containing x0, and then perhaps expand it by a ‘stepping out’ procedure. 4) Similarly, we can randomly pick an initial interval of siz ew, and then expand it by a ‘doubling’ procedure. 8wyf(x )0 (c)(a) (b) 0x 0 x x x 1 0 Figure 1: A single-variable slice sampling update using the stepping-out and shrinkage procedures. A new point, x1, is selected to follow the current point, x0, in three steps. (a) A vertical level, y, is drawn uniformly from (0 ,f(x0)), and used to deﬁne a horizontal ‘slice’, indicated in bold. (b)An interval of width wis randomly positioned around x0, and then expanded in steps of size wuntil both ends are outside the slice. (c)A new point, x1, is found by picking uniformly from the interval until a point in side the slice is found. Points picked that are outside the slice are used to shrink the inter val. (b)(a) Figure 2: The doubling procedure. In (a), the initial interval is doubled twice, until both ends are outside the slice. In (b), where the start state is diﬀerent, no doubling is done. 9For each scheme, we must also be able to ﬁnd the set Aof acceptable successor states, deﬁned as follows: A={x:x∈S∩IandP(Select I\|At state x) =P(Select I\|At state x0)}(3) That is, Ais the set of states from which we would be as likely to choose t he interval I as we were to choose this Ifrom the current state. When we subsequently sample from within I(see section 4.2), we will ensure that the state chosen is in A, a fact which will be used in the proof of correctness in section 4.3. Clearly, for schemes (1) and (2), A=S. For scheme (3), we will arrange that A=S∩I. Things are not so simple for scheme (4), for which a special test of whether a state is in Amay be necessary. Scheme (1), in which Iis set to the smallest interval containing S, will be feasible when all solutions of f(x) =ycan be found analytically, or by an eﬃcient and robust numeri cal method, but one cannot expect this in general. Often, even th e number of disjoint intervals making up Swill be hard to determine. Scheme (2) is certainly easy to implement when the range of xis bounded, and one can of course always arrange this by applying a suitable transfo rmation. However, if the slice is usually much smaller than the full range, the subsequent s ampling (see section 4.2) will be ineﬃcient. This scheme has been used by Frey (1997). The ‘stepping out’ procedure (scheme (3) above) is appropri ate for any distribution, provided that some rough estimate, w, for the typical width of the slice is available. The manner in which an interval is found by stepping out is illust rated in Figure 1(b) and the procedure is given in detail in Figure 3. The size of the inter val found can be limited to mw, for some speciﬁed integer m, or the interval can be allowed to grow to any size (ie, mcan be set to inﬁnity), in which case the procedure can be simp liﬁed in an obvious way (eliminating all references to JandK). Note that the random positioning of the initial interval and the random apportioning of the maximum number o f steps minto a limit on going to the left and a limit on going to the right are essentia l for correctness, as they ensure that the ﬁnal interval could equally well have been produced from any point within S∩I. Ifmis set to one in the stepping out procedure, the interval will always be of size w, and there will be no need to evaluate fat its endpoints. This saves some computation time, but is undesirable if wmight be much too small. The ‘doubling’ procedure (scheme (4) above) can expand the i nterval faster than the stepping out procedure, and hence may be more eﬃcient when th e estimated size of the slice ( w) turns out to be too small. This procedure is illustrated in F igure 2, and given in detail in Figure 4. Doubling produces a sequence of interval s, each twice the size of the previous one, until an interval is found with both ends outsi de the slice, or a predetermined limit is reached. Note that when the interval is doubled the t wo sides are not expanded equally. Instead just one side is expanded, chosen at random (irrespective of whether that side is already outside the slice). This is essential to the c orrectness of the method, since it produces a ﬁnal interval that could have been obtained from p oints other than the current one. The set Aof acceptable next states is restricted to those for which th e same interval could have been produced, and is in general not all of S∩I. This complicates the subsequent sampling somewhat, as described below. 10Input: f= function proportional to the density x0= the current point y= the vertical level deﬁning the slice w= estimate of the typical size of a slice m= integer limiting the size of a slice to mw Output: ( L,R) = the interval found U∼Uniform(0 ,1) L←x0−w∗U R←L+w V∼Uniform(0 ,1) J←Floor ( m∗V) K←(m−1)−J repeat while J >0 and y < f(L): L←L−w J←J−1 repeat while K >0 and y < f(R): R←R+w K←K−1 Figure 3: The ‘stepping out’ procedure for ﬁnding an interva l around the current point. The notation U∼Uniform(0 ,1) indicates that Uis set to a number randomly drawn from the uniform distribution on (0 ,1). Input: f= function proportional to the density x0= the current point y= the vertical level deﬁning the slice w= estimate of the typical size of a slice p= integer limiting the size of a slice to 2pw Output: ( L,R) = the interval found U∼Uniform(0 ,1) L←x0−w∗U R←L+w K←p repeat while K >0 and{y < f(L) ory < f(R)}: V∼Uniform (0 ,1) ifV <1/2 then L←L−(R−L) elseR←R+ (R−L) K←K−1 Figure 4: The ‘doubling’ procedure for ﬁnding an interval ar ound the current point. Note that it is possible to save some computation in second and lat er iterations of the loop, since only one of f(L) and f(R) will have changed from the previous iteration. 114.2 Sampling from the part of the slice within the interval Once an interval, I= (L,R), has been found containing the current point, x0, the ﬁnal step of the single-variable slice sampling procedure is to rando mly draw a new point, x1, from within this interval. This point must lie within the set Aof points acceptable as the next state of the Markov chain, deﬁned in equation (3). Two methods could be used to sample from I: i)Repeatedly sample uniformly from Iuntil a point is drawn that lies within A. ii)Repeatedly sample uniformly from an interval that is initia lly equal to I, and which shrinks each time a point is drawn that is not in A, until a point within Ais found. Method (i)could potentially be very ineﬃcient, if ever Aturns out to be a tiny portion of I. The shrinkage of the interval in method (ii)ensures that the expected number of points drawn will not be too large, making this a more appropriate me thod for general use. The shrinkage procedure is shown in detail in Figure 5. Note t hat each rejected point is used to shrink the interval in such a way that the current poin t remains within it. Since the current point is always within A, the interval used always contains acceptable points, ensuring that the procedure will terminate. If the interval was found by scheme (1), (2), or (3), the set Ais simply S∩I. However, if the doubling procedure (scheme (4)) was used, Amay be a smaller subset of S∩I. This is illustrated in Figure 2. In 2(a), an interval is found by do ubling an initial interval until both ends are outside the slice. A diﬀerent starting point is considered in 2(b), one which might have been drawn from the interval found in 2(a). The dou bling procedure terminates earlier starting from here, so this point is not in A. (Note that Ais here deﬁned conditional on the alignment of the initial interval.) The Accept ( x1) predicate in Figure 6 tests whether a point in S∩Iis inAwhen the doubling procedure (scheme (4)) was used. This procedure wo rks backward through the intervals that the doubling procedure would pass through to arrive at Iwhen starting from the new point, checking that none of them have both ends outsi de the slice, which would lead to earlier termination of the doubling procedure. (Not e that one needn’t check this explicitly until the intervals diﬀer from those followed fr om the current point, a condition tracked with the variable Din the procedure.) If the distribution is known to be unimoda l, this test can be omitted, as discussed in section 4.4. 4.3 Correctness of single-variable slice sampling To show that single-variable slice sampling is a correct pro cedure, we must show that each update leaves the desired distribution invariant. To g uarantee convergence to this distribution, the resulting Markov chain must also ergodic . This is not always true, but it is in those situations (such as when f(x)>0 for all x) for which one can easily show that Gibbs sampling is ergodic. I will not discuss the more diﬃcul t situations here. To show invariance, we suppose that the initial state, x0, is distributed according to f(x). In step (a)of single-variable slice sampling, a value for yis drawn uniformly from (0 ,f(x)). 12Input: f= function proportional to the density x0= the current point y= the vertical level deﬁning the slice w= estimate of the typical size of a slice (L,R) = the interval to sample from Output: x1= the new point ¯L←L,¯R←R repeat: U∼Uniform (0 ,1) x1←¯L+U∗(¯R−¯L) ify < f(x1) and Accept ( x1) then exit loop ifx1< x0then¯L←x1else¯R←x1 Figure 5: The ‘shrinkage’ procedure for sampling from the in terval. The notation Accept ( x1) represents a test for whether a point, x1, that is within S∩Iis an accept- able next state. If scheme (1), (2), or (3) was used for constr ucting the interval, all points within S∩Iare acceptable. If the doubling procedure (scheme (4)) was u sed, the point must pass the test of Figure 6, below. Input: f= function proportional to the density x0= the current point x1= the possible next point y= the vertical level deﬁning the slice w= estimate of the typical size of a slice (L,R) = the interval found by the doubling procedure Output: whether or not x1is an acceptable next state ˆL←L,ˆR←R D←false repeat while ˆR−ˆL >1.1∗w: M←(ˆL+ˆR)/2 if{x0< Mandx1≥M}or{x0≥Mandx1< M}thenD←true ifx1< MthenˆR←MelseˆL←M ifDandy≥f(ˆL) and y≥f(ˆR) then The new point is not acceptable The new point is acceptable if not rejected in the loop above Figure 6: The test for whether a new point, x1, that is within S∩Iis an acceptable next state, when the interval was found by the ‘doubling’ procedu re. The multiplication by 1.1 in the ‘while’ condition guards against possible round-oﬀ e rror. 13The joint distribution for x0andywill therefore be as in equation (1). If the subsequent steps update x0tox1in a manner that leaves this joint distribution invariant, t hen when we subsequently discard y, the resulting distribution for x1will be the marginal of this joint distribution, which is the same as that deﬁned by f(x), as desired. We therefore need only show that the selection of x1to follow x0in steps (b)and(c)of the single-variable slice sampling procedure leaves the jo int distribution of xandyinvariant. Since these steps do not change y, this is the same as leaving the conditional distribution fo r xgiven yinvariant, and this conditional distribution is uniform ov erS={x:y < f(x)}, the slice deﬁned by y. We can show invariance of this distribution by showing that the updates satisfy detailed balance, which for a uniform distr ibution reduces to showing that the probability density for x1to be selected as the next state, given that the current state isx0, is the same as the probability density for x0to be the next state, given that x1is the current state, for any states x0andx1within S. In the process of picking a new state, various intermediate c hoices are made randomly. When the interval is found by the stepping out procedure of Fi gure 3, the alignment of the initial interval is randomly chosen, as is the division of th e maximum number of intervals into those used to extend to the left and those used to extend t o the right. For the doubling procedure of Figure 4, the alignment of the initial interval is random and the decisions whether to extend to the right or to the left are also made rand omly. When sampling is done using the shrinkage procedure of Figure 5, zero or more r ejected points will be chosen before the ﬁnal point. Let rdenote these intermediate random choices. I will prove that detailed balance holds for the entire procedure by showing t he following stronger result: P(next state = x1,intermediate choices = r\|current state = x0) =P(next state = x0,intermediate choices = π(r)\|current state = x1) (4) where π(r) is some one-to-one mapping that has Jacobian one (with rega rd to the real- valued variables), which may depend on x0andx1. Integrating over all possible values for rthen gives the desired result. In detail, the mapping πused is as follows. First, if the interval Iis found by the stepping out or doubling procedure, an intermediate value, U, will be generated by the procedure of Figure 3 or 4, and used to deﬁne the initial interval. We deﬁne πso that it maps the value U0chosen when the state is x0to the following U1when the state is x1: U1= Frac ( U0+ (x1−x0)/w) (5) where Frac ( x) =x−Floor ( x) is the fractional part of x. This mapping associates values that produce the same alignment of the initial interval. Not e also that it has Jacobian one. If the stepping out procedure is used, a value for Jis also generated, uniformly from the set{0, ... , m−1}. The mapping πassociates the J0found when the state is x0with the following J1when the state is x1: J1=J0+ (x1/w−U1)−(x0/w−U0) (6) Here, ( x1/w−U1)−(x0/w−U0) is an integer giving the number of steps (of size w) from the left end of the interval containing x0to the left end of the interval containing x1. This 14is the amount by which we must adjust J0in order to ensures that if the interval found starting from x0grows to its maximum size, the associated interval found sta rting from x1will be identical. Similarly, if the doubling procedure of F igure 4 is used, the sequence of random decisions as to which side of the interval to expand is mapped by πto the sequence of decisions that would cause the interval expandi ng from x1to become identical to the interval expanding from x0when the latter ﬁrst includes x1, and to remain identical through further expansions. Note in this respect that there is at most one way that an given ﬁnal interval can be obtained by successive doublings from a given initial interval, and that the alignment of the initial intervals by the associ ation of U0withU1ensures that doubling starting from x1can indeed lead to the same interval as found from x0. Finally, to complete the deﬁnition, πmaps the sequence of rejected points used to shrink the inter val found from x0(see Figure 5) to the same sequence of points when x1is the start state. It remains to show that with this deﬁnition of π, equation (4) does indeed hold, for all points x0andx1, and all possible intermediate values r. The equation certainly holds when both sides are zero, so we needn’t consider situations where movement between x0andx1 is impossible (in conjunction with the given intermediate v alues). Consider ﬁrst the probability (density) for producing the i ntermediate values that deﬁne the interval I. For the stepping out and doubling procedures, the values U0andU1(related byπ) that are generated from x0andx1will certainly have the same probability density, since Uis drawn from a uniform distribution. Similarly, for the ste pping out procedure, the values J0andJ1are drawn from a uniform distribution over {0, ... , m−1}, and hence have the same probability as long as J0andJ1are both in this set, which will be true whenever movement between x0andx1is possible. For the doubling procedure, a sequence of decisions as to which side to extend is made, with all seque nces of a given length having the same probability. Here also, the sequences associated by πwill have the same probability, provided the same number of doublings are done starting from x0as from x1. This need not be true in general, but if the sequence from x1is shorter, the test of Figure 6 will eliminate x1as a possible successor to x0, and if the sequence from x0is shorter, x1will not be a possible successor because it will be outside the interval Ifound from x0. Both sides of equation 4 will therefore be zero in this situation. Note next that the intervals found by any of the schemes of sec tion 4.1 will be the same for x0as for x1, when the intermediate values chosen are related by π, assuming a transition fromx0tox1is possible. For the stepping out procedure, the maximum ext ent of the intervals will be the same because of the relationships betw eenU0andU1and between J0 andJ1. Furthermore, the actual intervals found by stepping out (l imited by the maximum) must also be the same whenever a transition between x0andx1is possible, since if the interval starting from x0has reached x1, expansion of both intervals will continue in the same direction until the outside of the slice or the maximum i s reached, and likewise in the other direction. Similarly, the mapping πis deﬁned to be such that if the interval found by the doubling procedure starting from x0includes x1, the same interval would be found fromx1, provided the process was not terminated earlier (by both en ds being outside the slice), in which case x1is not a possible successor (as it would be rejected by the pro cedure of Figure 6). Note also that since the set Ais determined by I(for any start state), it too 15will be the same for x0as for x1. If we sample from this Iby simple rejection (scheme (i)in section 4.2), the state chosen will be uniformly distributed over A, so the probability of picking x0will be the same as that of picking x1. If we instead use the shrinkage procedure (scheme (ii), in Figure 5), we need to consider as intermediate values the sequence of reje cted points that were used to narrow the interval (recall that under πthis sequence is the same for x0as for x1). The probability density for the ﬁrst of these is clearly the same for both starting points, since Iis the same. As the interval shrinks, it remains the same for b othx0andx1, since the rejection decisions (based on A) are the same, and since we need consider only the case where the same end of the interval is moved to the rejected poi nt (as otherwise a transition between x0andx1in conjunction with these intermediate values would be impo ssible). The probability densities for later rejected points, and for th e ﬁnal accepted state, are therefore also the same. This completes the proof. Various seemingly reasonable mod iﬁcations — such as changing the doubling procedure of Figure 4 to not expand the interval on a side that is already outside the slice — would undermine the argument of the proof , and hence cannot be used. However, some shortcuts are allowed when the distribution i s unimodal, as discussed next. 4.4 Shortcuts for unimodal distributions Certain shortcuts can be used when the conditional distribu tion for the variable being updated is known to be unimodal, or more generally, when the s lice,S, is known to consist of a single interval. For some values of the auxiliary variab le,Smay be a single interval even when the distribution is multimodal, but the eﬀort requ ired to conﬁrm this probably exceeds the gain from using the shortcuts, so I will refer onl y to the unimodal case here. Two shortcuts apply when the ‘doubling’ procedure is used to ﬁnd the interval. First, for a unimodal distribution, the acceptance test in Figure 6 can be omitted, since it will always indicate that the new point is acceptable. To see this, note t hat the procedure rejects a point when one of the intervals found by doubling from that st arting point has both ends outside the slice, but does not contain the current point. Si nce both the current point and the new point are inside the slice, this is impossible if the s lice consists of only one interval. Second, the interval found by the doubling procedure can som etimes be shrunk at the outset. The side chosen for extension when the interval doub les will sometimes be outside the slice already. When the distribution is known to be unimo dal, it is not possible for such an extension to contain any points within the slice. Accordi ngly, before sampling is begun, the endpoints of the interval can be set to the ﬁrst point in ea ch direction that was found to lie outside the slice. This may reduce the number of points generated, while having no eﬀect on the distribution of the point ﬁnally chosen. Finally, if the distribution is known to be unimodal andno limit is imposed on the size of the interval found (ie, mandpin Figures 3 and 4 are inﬁnite), the estimate, w, for the typical size of a slice can be set on the basis of past iteratio ns. One could, for example, keep a running average of the distance between the old and new poin ts in past iterations, and use this (or some suitable multiple) as the estimate w. This is valid because the distribution 16of the new point does not depend on win this situation, even though winﬂuences how eﬃciently this new point is found. Indeed, when the distribu tion is known to be unimodal, one can use any method at all for ﬁnding an interval that conta ins the current point and has both ends outside the slice, as any such interval will lea d to the new point ﬁnally chosen being drawn uniformly from the slice. 5 Multivariate slice sampling methods Rather than sample from a distribution for x= (x1,... ,x n) by applying one of the single- variable slice sampling procedures described above to each xiin turn, we might try instead to apply the idea of slice sampling directly to the multivari ate distribution. I will start by describing a straightforward generalization of the single -variable methods to multivariate distributions, and then describe a more sophisticated meth od, which can potentially allow for adaptation to the local dependencies between variables . 5.1 Multivariate slice sampling with hyperrectangles We can generalize the single-variable slice sampling metho ds of Section 4 to methods for performing multivariate updates by replacing the interval I= (L,R) by an axis-aligned hyperrectangle H={x:Li< xi< R ifor all i= 1,... ,n}. Here, LiandRideﬁne the extent of the hyperrectangle along the axis for variable xi. The procedure for ﬁnding the next state, x1= (x1,1,... ,x 1,n), from the current state, x0= (x0,1,... ,x 0,n), parallels the single-variable procedure: a)Draw a real value, y, uniformly from (0 ,f(x0)), to deﬁne the slice S={x:y < f(x)}. b)Find a hyperrectangle, H= (L1,R1)×···× (Ln,Rn), around x0, which preferably contains at least a big part of the slice. c)Draw the new point, x1, from the part of the slice within this hyperrectangle (ie, f rom S∩H). It would perhaps be ideal for step (b) to set Hto the smallest hyperrectangle containing S, but this is unlikely to be feasible. An easy option when all t he variables have bounded range is to set Hto be the entire space, but this will often be rather ineﬃcien t, since Sis likely to be much smaller. In practice, we must usually be content to ﬁnd an Hthat contains the current point, x0, but probably not all of S, using width parameters, wi, for the dimensions of Halong each axis. If we know nothing about the relative scales of diﬀeren t variables, we might set all the wito a single scale parameter, w. The simplest way of ﬁnding His to randomly position a hyperrectangle with these dimensions, uniformly over posi tions that lead to Hcontaining x0. This generalizes the random positioning of the initial int ervalIfor the single-variable slice sampling methods. The stepping out and doubling proce dures used with single-variable slice sampling do not generalize so easily, however. The goa l of ﬁnding an interval whose endpoints are outside the slice would generalize to ﬁnding a hyperrectangle all of whose vertices are outside the slice, but since an ndimensional hyperrectangle has 2nvertices, 170x (b)(a) 0x1 1x x Figure 7: Multivariate slice sampling with hyperrectangle s. The heavy line outlines the slice, containing the current point, x0. The large square is the initial hyperrectangle. In (a), the hyperrectangle is shrunk in all directions when the point dr awn is outside the slice, until a new point, x1, inside the slice is found. In (b), the hyperrectangle is shr unk along only one axis, determined from the gradient and the current dimensio ns of the hyperrectangle. The dashed lines are contours of the density function, indicati ng the direction of the gradient. we would certainly not want to test for this when nis large. The stepping out procedure seems to be too time consuming in any case, since one would nee d to step out in each of the ndirections. The doubling procedure does generalize approp riately, and one could decide to stop doubling when a randomly-drawn point picked u niformly from the current hyperrectangle is outside the slice. This idea is worth expl oring, but here I will consider only the simplest scheme, which is to use the randomly positioned hyperrectangle without any expansion, though it is then crucial that the winot be much smaller than they should be. The shrinkage procedure of Figure 5 generalizes easily to mu ltiple dimensions — the hyperrectangle can simply be shrunk independently along ea ch axis. Combining this with simple random positioning of H, one gets the multivariate slice sampling method shown in Figure 7(a), and given in detail in Figure 8. The validity of t his method can be proven in the same way as was done for single-variable slice sampling i n Section 4.3. Although this simple multivariate slice sampling method is easily implemented, and will often be reasonably eﬃcient, in one respect it works less wel l than applying single-variable slice sampling to each variable in turn. When each variable i s updated separately, the 18Input: f= function proportional to the density x0= the current point, of dimension n wi= scale estimates for each variable, i= 1,... ,n Output: x1= the new point Step (a): Find the value of ythat deﬁnes the slice. y∼Uniform(0 , f(x0)) Step (b): Randomly position the hyperrectangle H= (L1,R1)×···× (Ln,Rn). Fori= 1 to n: Ui∼Uniform (0 ,1) Li←x0,i−wi∗Ui Ri←Li+wi Step (c): Sample from H, shrinking when points are rejected. Repeat: Fori= 1 to n: Ui∼Uniform (0 ,1) x1,i←Li+Ui∗(Ri−Li) ify < f(x1) then exit loop Fori= 1 to n: ifx1,i< x0,ithenLi←x1,ielseRi←x1,i Figure 8: A simple multivariate slice sampling procedure, w ith randomly positioned hyper- rectangle, and shrinkage in all directions. 19interval for that variable will be shrunk only as far as neede d in order to obtain a new value within the slice. The amount of shrinkage can be diﬀere nt for diﬀerent variables. In contrast, the procedure of Figure 8 shrinks all dimensions o f the hyperrectangle until a point inside the slice is found, even though the probability densi ty may not vary rapidly in some of these dimensions, making shrinkage in these directions u nnecessary (and undesirable). One way to try to avoid this problem is illustrated in Figure 7 (b). Rather than shrink all dimensions of the hyperrectangle when the last point chosen was outside the slice, we can instead shrink along only one axis, basing the choice on the g radient of log f(x), evaluated at the last point. Speciﬁcally, only the axis corresponding to variable xiis shrink, where i maximizes the following product: (Ri−Li)\|Gi\| (7) where Gis the gradient of log f(x) at the last point chosen. By multiplying the magnitude of component iof the gradient by the width of the hyperrectangle in this dir ection, we get an estimate of the amount by which log f(x) changes along axis i. The axis for which this change is thought to be largest is likely to be the best one to s hrink in order to eliminate points outside the slice. Unfortunately, if this decision w ere based as well on whether the sign of the gradient indicates that log f(x) is increasing or decreasing as we move toward the current point, x0, the shrinkage decision might be diﬀerent if we were to shrin k from the ﬁnal accepted point, x1, which would invalidate the method (unless we somehow avoid ed or rejected such points). Many more elaborate schemes along these lines are possible. For instance, we might shrink along all axes for which the product (7) is greater than some t hreshold. A good scheme might preserve the ability of single-variable slice sampli ng to adapt diﬀerently for diﬀerent variables, while keeping the advantages that simultaneous updates may sometimes have (eg, in producing an ergodic chain when there are tight dependenc ies between variables). More ambitiously, we might hope that a multivariate slice sa mpler could adapt to the dependencies between variables, not just to their diﬀerent scales. This will require that we go beyond axis-aligned hyperrectangles, as is done in the ne xt section. 5.2 A framework for adaptive multivariate slice sampling We would like a more general framework by which trial points o utside the slice that were previously rejected can be used to guide the selection of fut ure trial points. In contrast to schemes based on hyperrectangles, we would like future tr ial points to potentially come from distributions that take account of the dependencies be tween variables. The scheme I present here achieves this by laying down a trail of ‘crumbs’ that guide the selection of future trial points, leading eventually to a point inside the slice . A crumb can be anything — eg, a discrete value, a real number, a vector, a hyperrectangle — but the method is perhaps most easily visualized when crumbs are points in the state sp ace being sampled from. As with the previous slice sampling schemes, we start by chos ing a value yuniformly between zero and f(x0), where x0is the current point. A crumb, c1, is then draw at random from some distribution with density (or probability mass) function g1(c;x0, y). 20Note that this distribution may depend on both the current po int,x0, and on the value of ythat deﬁnes the slice. A ﬁrst trial point, x∗ 1, is then drawn from the distribution with density h1(x∗;y, c1) =g1(c1;x∗, y)/Z1(y, c1), where Z1(y, c1) =/integraltextg1(c1;x∗, y)dx∗is the appropriate normalizing constant. One can view x∗ 1as being drawn from a pseudo-posterior distribution, based on a uniform prior, and the “data” that t he ﬁrst crumb was c1. Ifx∗ 1 is inside the slice, we set the new point, x1, tox∗ 1, and are ﬁnished. Otherwise, a second crumb, c2, is drawn from some distribution g2(c;x0, y, c 1, x∗ 1), which may depend on the previous crumb and the previous trial point, as well as x0andy. The second trial point is then drawn from the pseudo-posterior distribution based on the “data” c1andc2— that is,x∗ 2is drawn from h2(x∗;y, c1, x∗ 1, c2) = g1(c1;x∗, y)g2(c2;x∗, y, c 1, x∗ 1)/Z2(y, c1, x∗ 1, c2) (8) where Z2(y, c1, x∗ 1, c2) =/integraltextg1(c1;x∗, y)g2(c2;x∗, y, c 1, x∗ 1)dx∗. Ifx∗ 2is inside the slice, it becomes the new state. Otherwise, we draw a third crumb, from a distribution that may depend on the current state, the value deﬁning the slice, the previous crumbs, and the previous trial points, generate a third trial point using th is and the previous crumbs, and so forth until a trial point lying within the slice is found. To show that this procedure leaves the distribution with den sityf(x)/Zinvariant, it suﬃces to show that it separately satisﬁes detailed balance with respect to transitions that occur in conjunction with any given number of crumbs being dr awn. In the case, for instance, of transitions involving two crumbs, we can show t his by showing the stronger property that for any x∗ 1that is not in the slice deﬁned by yand any x∗ 2that is in this slice, the following will hold: P(x0)P(y, c1, x∗ 1, c2, x∗ 2\|x0) = P(x∗ 2)P(y, c1, x∗ 1, c2, x0\|x∗ 2) (9) Here, P(x0) and P(x∗ 2) are the probability densities for the current point and the point that will become the new point (which are proportional to f(x)). The conditional probabilities above are the densities for the given sequence of values bein g chosen during the procedure, given that the current point is the one conditioned on. The le ft side of equation (9) can be written as follows: P(x0)·P(y\|x0)·P(c1\|x0, y)·P(x∗ 1\|y, c1)·P(c2\|x0, y, c 1, x∗ 1)·P(x∗ 2\|y, c1, x∗ 1, c2) = [f(x0)/Z]·[1/f(x0)]·g1(c1;x0, y)·[g1(c1;x∗ 1, y)/Z1(y, c1)] ·g2(c2;x0, y, c 1, x∗ 1)·[g1(c1;x∗ 2, y)g2(c2;x∗ 2, y, c 1, x∗ 1)/Z2(y, c1, x∗ 1, c2)] The right side is P(x∗ 2)·P(y\|x∗ 2)·P(c1\|x∗ 2, y)·P(x∗ 1\|y, c1)·P(c2\|x∗ 2, y, c 1, x∗ 1)·P(x0\|y, c1, x∗ 1, c2) = [f(x∗ 2)/Z]·[1/f(x∗ 2)]·g1(c1;x∗ 2, y)·[g1(c1;x∗ 1, y)/Z1(y, c1)] ·g2(c2;x∗ 2, y, c 1, x∗ 1)·[g1(c1;x0, y)g2(c2;x0, y, c 1, x∗ 1)/Z2(y, c1, x∗ 1, c2)] These are readily seen to be equal, as is true in general for tr ansitions involving any number of crumbs. 21The hyperrectangle methods of Section 5.1 can be viewed in th is framework. The ran- domly placed initial hyperrectangle is the ﬁrst crumb. The ﬁ rst trial point is chosen from those points that could produce this initial hyperrectangl e, which is simply the set of points within the hyperrectangle. The second and later crumbs are t he shrunken hyperrectangles. Conditional on the current point, the previous crumb (ie, th e previous hyperrectangle), and the previous trial point, these have degenerate distrib utions, concentrated on a single hyperrectangle. The possible corresponding trial points a re the points within the shrunken hyperrectangle. By using diﬀerent sorts of crumbs, and diﬀerent distributio ns for them, a huge variety of methods could be constructed within this framework. I wil l here only brieﬂy discuss methods in which the crumbs are points in the state space, and have multivariate Gaussian distributions. The distributions of the trial points given the crumbs will then also be multivariate Gaussians. In the simplest method of this sort, every giis Gaussian with mean x0and covariance matrix σ2I, for some ﬁxed σ2. The distribution, hi, forx∗ iwill then be Gaussian with mean ¯ ci= (c1+...+ci)/iand covariance matrix ( σ2/i)I. As more and more trial points are generated, they will come from narrower and narrower dis tributions, which will be concentrated closer and closer to the current point (since ¯ ciwill approach x0). This is analogous to shrinkage in the hyperrectangle method. In pra ctice, it would probably be desirable to let σ2 idecrease with i(perhaps exponentially), so that the trial points would be forced closer to x0more quickly. Alternatively, one might look at f(x∗ i−1)/yin order to estimate what value for σiwould produce a distribution for the next trial point, x∗ i, that is likely to lie within the slice. More generally, gicould be a multivariate Gaussian with mean x0and some covariance matrix Σ i, which may depend on the value of y, the previous crumbs, and the previous trial points. In particular, Σ icould depend on the gradients of f(x∗ j) forj < i, which provide information on what Gaussian distribution would be a good lo cal approximation to f(x). The distribution, hi, for trial point x∗ iwill then have covariance Σ∗ i= [Σ−1 1+···+ Σ−1 i]−1 and mean ¯ ci= Σ∗ i[Σ−1 1c1+···+ Σ−1 ici]. When xis of only moderate dimensionality, explicitly performing operations involving these covariance matrices would be tolerable, and a wide var iety of ways for producing Σiwould be feasible. For higher-dimensional problems, such o perations would need to be avoided, as is done in an optimization context with the con jugate gradient and other related methods. Further research is therefore needed in or der to fully exploit the potential of this promising framework for adaptation, and to compare i t with methods based on the ‘delayed rejection’ (also called ‘splitting rejection’) f ramework of Tierney and Mira (Mira 1998, Chapter 5; Tierney and Mira 1999). 6 Overrelaxed slice sampling When the updates used do not account for the dependencies bet ween variables, many up- dates will be needed to move from one part of the distribution to another. Sampling 22eﬃciency can be improved in this context by suppressing the r andom walk behaviour char- acteristic of simple schemes such as Gibbs sampling. One way of achieving this is by using ‘overrelaxed’ updates. Like Gibbs sampling, overrelaxati on methods update each variable in turn, but rather than drawing a new value for a variable fro m its conditional distribution independently of the current value, the new value is instead chosen to be on the opposite side of the mode from the current value. In Adler’s (1981) sch eme, applicable when the conditional distributions are Gaussian, the new value for v ariable iis x′ i=µi+α(xi−µi) +σi(1−α2)1/2n (10) where µiandσiare the conditional mean and standard deviation of variable i,nis a Gaus- sian random variate with mean zero and variance one, and αis a parameter slightly greater than−1. This method is analysed and discussed by Barone and Friges si (1990) and by Green and Han (1992), though these discussions fail in some r espects to properly eluci- date the true beneﬁts and limitations of overrelaxation (Ne al 1998). The crucial ability of overrelaxation to (sometimes) suppress random walks is ill ustrated for a bivariate Gaussian distribution in Figure 9. Various attempts have been made to produce overrelaxation s chemes that can be used when the conditional distributions are not Gaussian. I have reviewed several such schemes, and introduced one of my own (Neal 1998). The concept of overr elaxation seems to apply only when the conditional distributions are unimodal, so we may assume that this is usually the case, though we would like the method to at least remain va lid (ie, leave the desired distribution invariant) even if this assumption turns out t o be false. To obtain the full beneﬁts of overrelaxation, it is important that almost ever y update be overrelaxed, with few or no ‘rejections’ that leave the state unchanged, as suc h rejections re-introduce an undesirable random walk aspect to the motion through state s pace (Neal 1998). In this section, I will show how overrelaxation can be done us ing slice sampling. Many schemes for overrelaxed slice sampling are possible, but I w ill describe only one in detail, based on the stepping out procedure and on bisection. This sc heme is illustrated in Fig- ure 10, and given in detail in Figure 11. To begin, we apply the stepping out procedure of Figure 3 to ﬁn d an interval around the current point. Normally, we would apply this procedure w ith the maximum size of the interval ( m) set to inﬁnity, or to some large value, since a proper overre laxation operation requires that the entire slice be found, but the scheme remai ns valid for any m. If the stepping out procedure found an interval around the sl ice that is bigger than the initial interval, then the two outermost steps will serve to locate the endpoints of the slice to within an interval of size w. (Here, we assume that the slice consists of a single interva l, as it will if the distribution is unimodal.) We then locate the e ndpoints more precisely using a bisection procedure. For each endpoint, we test whether the mid-point of the interval within which it is located is inside or outside the slice, and shrink this interval appropriately to narrow the location of the endpoint. This is repeated atimes, after which each endpoint will be known to lie within an interval of size 2−aw. If the stepping out procedure found that the initial interva l (of size w) already had both ends outside the slice, then before doing any bisection, we n arrow this interval, by shrinking 23-10+1 -1 0 +1-10+1 -1 0 +1 Gibbs Sampling Adler’s Method, α=−0.98 Figure 9: Gibbs sampling and Adler’s overrelaxation method applied to a bivariate Gaussian with correlation 0.998 (whose one-standard-deviation contour is plotted). The top left shows the progress of 40 Gibbs sampling itera- tions (each consisting of one update for each variable). The top right shows 40 overrelaxed iterations, with α=−0.98. The close-up on the right shows how successive overrelaxed updates operate to avoid a random walk.0+1 0 +1 it in half repeatedly until its mid-point is within the slice . We then use bisection as above to locate the endpoints to within an interval of size 2−aw. Once the locations of the endpoints have been narrowed down, we can approximate the entire slice by the interval ( ˆL,ˆR), formed from the outer bounds on the endpoint locations. To do an overrelaxed update, we ﬂip from the current point, x0, to a new point, x1, that is the same distance as the current point from the middle of this interval, but on the opposite side. That is, we let x1=ˆL+ˆR 2−/parenleftBigg x0−ˆL+ˆR 2/parenrightBigg =ˆL+ˆR−x0 (11) We must sometimes reject this candidate point, in which case the new point is the same as the current point. First of all, we must reject x1if it lies outside the interval, ( ¯L,¯R), that had 24(b)(a) (c) ˆLˆL+ˆR 2ˆR Figure 10: Overrelaxation using the stepping out procedure and bisection. In (a), an interval with both ends outside the slice is found by steppin g out from the current point, as was illustrated in Figure 1(b). In (b), the endpoints of the slice are located more accurately using bisection. In (c), a candidate point is found by ﬂipping through the point half -way between the approximations to the endpoints. In this case, t he candidate point is accepted, since it is within the slice, and within the orginal interval (prior to bisection). been found prior to bisection, since the interval found from x1would then be diﬀerent, and detailed balance would not hold. However, this situation ca nnot arise when the distribution is unimodal. Secondly, we must reject x1if it lies outside the slice. This can easily happen for a multimodal distribution, and can happen even for a unim odal distribution when the endpoints of the slice have not been located exactly. Howeve r, the probability of rejection for a unimodal distribution can be reduced to as low a level as desired, at moderate cost, by locating the endpoints more precisely using more iterati ons of bisection. The correctness of this procedure can be seen using argument s similar to those of sec- tion 4.3. The interval before bisection can be found by the do ubling procedure instead of stepping out, provided the point found is rejected if it fa ils the acceptance test of Fig- ure 6. However, rejection for this reason will not occur in th e presumably typical case of a unimodal distribution. One could use many methods other than bisection to narrow dow n the locations of the endpoints before overrelaxing. If the derivative of f(x) can easily be calculated, one could use Newton iteration, whose rapid convergence would often a llow the endpoints to be cal- culated to machine precision in a few iterations. For unimod al distributions, such exact calculations would eliminate the possibility of rejection , and would also make the ﬁnal re- sult be independent of the way the interval containing the sl ice was found, thereby allowing use of retrospective methods for tuning the procedure for ﬁn ding this interval. To obtain a full sampling scheme, overrelaxed updates of thi s sort would be applied to each variable in turn, in a ﬁxed order, for a number of cycles, after which a normal slice 25Input: f= function proportional to the density x0= the current point y= the vertical level deﬁning the slice w= estimate of the typical size of a slice a= integer limiting endpoint accuracy to 2−aw (L,R) = interval found by the stepping out procedure Output: x1= the new point ¯L←L,¯R←R ¯w←w, ¯a←a When the interval is only of size w, the following section will narrow it until the mid-point is inside the slice (or the accuracy limi t is reached). ifR−L <1.1∗wthen repeat: M←(¯L+¯R)/2 if ¯a= 0 or y < f (M) then exit loop ifx0> Mthen¯L←Melse¯R←M ¯a←¯a−1 ¯w←¯w /2 Endpoint locations are now reﬁned by bisection, to the speci ﬁed accuracy. ˆL←¯L,ˆR←¯R repeat while ¯ a >0: ¯a←¯a−1 ¯w←¯w /2 ify≥f(ˆL+ ¯w) then ˆL←ˆL+ ¯w ify≥f(ˆR−¯w) then ˆR←ˆR−¯w A candidate point is found by ﬂipping from the current point t o the opposite side of (ˆL,ˆR). It is then tested for acceptability. x1←ˆL+ˆR−x0 ifx1<¯Lorx1>¯Rory≥f(x1) then x1←x0 Figure 11: The overrelaxation procedure using bisection. I t is assumed that the interval (L,R) was found by the stepping out procedure, with a stepsize of w. 26sampling update would be done. Alternatively, each update c ould be done normally with some small probability. A Markov chain consisting solely of overrelaxed updates might not be ergodic, and might in any case suppress random walks for to o long. The frequency of normal updates is a tuning parameter, analogous to the choic e ofαin Adler’s overrelaxation method, and would ideally be set so that the Markov chain move s systematically, rather than in a random walk, for long enough that it traverses a dist ance comparable to the largest dimension of the multivariate distribution, but fo r no longer than this. To keep from doing a random walk for around ksteps, one would do every k’th update normally, and also arrange for the rejection rate for the overrelaxed u pdates to be less than 1 /k. 7 Reﬂective slice sampling Multivariate slice sampling methods can also be designed to suppress random walks. In this section I describe methods that ‘reﬂect’ oﬀ the boundar ies of the slice. Such movement with reﬂection can be seen as a specialization to uniform dis tributions of the Hamiltonian dynamics that forms the basis for Hybrid Monte Carlo (Duane, et al1987). As before, suppose we wish to sample from a distribution over ℜn, deﬁned by a function f(x) that is proportional to the probability density, and which we here assume is diﬀeren- tiable. We must be able to calculate both f(x) and its gradient (or equivalently, the value and gradient of log f(x)). In each iteration of the Markov chain, we will draw a value for an auxiliary variable, y, uniformly from (0 ,f(x)), thereby deﬁning an n-dimensional slice S={x:y < f(x)}. We will also introduce nadditional ‘momentum’ variables, written as a vector p, which serve to indicate the current direction and speed of m otion through state space. At the start of each iteration, we pick a value fo rp, independently of x, from some rotationally symmetric distribution, typically Gaus sian with mean zero and identity covariance matrix. Once yandphave been drawn, we repeatedly update xby stepping in the direction of p. After some predetermined number of steps, we take the ﬁnal v alue of xas our new state (provided it is acceptable). In each step, we try to set x′=x+wp, for some scale parameter wthat determines the average step size. However, if the resul tingx′is outside the slice S (ie,y≥f(x′)), we must somehow bring try to bring it back inside. The sche mes considered here all do this by some form of reﬂection, but diﬀer in the exa ct procedure used. Ideally, we would reﬂect from the exact point at which moveme nt in the direction of p ﬁrst takes us outside the slice. This reﬂection operation mo diﬁes p, after which motion continues in the new direction, until we again encounter the boundary of the slice. When we hit the boundary at a point where the gradient of f(x) isg, reﬂection will change pas follows: p′=p−2gp·g \|g\|2(12) This ideal reﬂection scheme is illustrated for a two-dimens ional slice in Figure 12. Using the fact that the reﬂection transformation above has Jacobian o ne and is its own inverse, one can show that movement with reﬂection for some pre-determined d uration leaves invariant the joint distribution of x(uniform within the slice) and p(rotationally symmetric, independent 27Figure 12: Moving around a two-dimensional slice by reﬂecti on from the exact boundaries. w \|p\| (a)w \|p\| (b) Figure 13: Reﬂection from an inside point. The trajectories here go in steps of size w\|p\|, starting from the top right, until a point outside the slice i s reached, when a reﬂection is attempted based on the inner contour shown. In (a), the reﬂection is is sucessful; in (b), it must be rejected, since the reverse trajectory would not reﬂ ect at this point. Figure 14: Reﬂection from outside points. Starting from the left, two reﬂections based on outside contours lead back inside the slice after the next st ep. The step after the third reﬂection is still outside the slice, so further reﬂections must be done. In this case, the trajectory eventually returns to the slice, and its endpoin t would therefore be accepted. 28ofx), so this way of sampling is valid, with no need for an accepta nce test. One can also see from the ﬁgure how such motion can proceed consistently in on e direction (until the end of the slice is reached), rather than in a random walk. Ideal reﬂection is diﬃcult to implement, however, as it requ ires precise calculation of where the current path intersects the boundary of the slice. Finding this point analytically might sometimes be possible, or we might try to solve for it nu merically, but if the slice is not known to be convex, it may be diﬃcult even to determine w ith certainty that an intersection point that has been found is in fact the ﬁrst one that would be encountered. Rather than attempt such exact calculations, we can instead employ one of two approximate schemes, based on ‘inside’ or ‘outside’ reﬂection, althoug h the trajectories these schemes produce must sometimes be rejected. When stepping from xtox′=x+wptakes us outside the slice, we can try to reﬂect from the last inside point, x, instead of from the exact point where the path intersects the boundary, using the gradient of f(x) at this inside point. The process is illustrated in Figure 13. However, for this method to be valid, we must check that the reverse trajectory would also reﬂect at this point, by verifying that a step in th e direction opposite to our new heading would take us outside the slice. If this is not so, we m ust either reject the entire trajectory of which this reﬂection step forms a part, or alte rnatively, set pandxso that we retrace the path taken to this point (starting at the inside p oint where the reﬂection failed). Alternatively, when we step outside the slice, we can try to r eﬂect from the outside point, x′, based on the gradient at that point. A trajectory with sever al such reﬂections is shown in Figure 14. After performing a pre-determined number of st eps, we accept the trajectory if the ﬁnal point is inside the slice. Note that for this metho d to be valid, one must reﬂect whenever the current point is outside the slice, even if this leads one away from the slice rather than toward it. This will sometimes result in the traj ectory never returning to the slice, and hence being rejected, but other times, as in the ﬁg ure, it does return eventually. Many variations on these procedures are possible. Above, it was assumed that values foryandpare are randomly drawn at the beginning of a trajectory, and t hen kept the same for many steps (apart from the changes to pwhen reﬂections occur). When using inside reﬂection, we might instead pick a new value for ymore often, perhaps before every step, and we might also partially update p, as is done in Horowitz’s (1991) variation on Hybrid Monte Carlo. When using outside reﬂection, the accep tance rate can be increased by terminating the trajectory when either some pre-set maxi mum number of steps have been taken, orsome pre-set number of steps have ended inside the slice. Whe n termination occurs for the latter reason, the ﬁnal point will necessaril y be inside the slice, and the trajectory will therefore be accepted. 8 A Demonstration To illustrate the beneﬁts stemming from the adaptive nature of slice sampling, I show here how it can help avoid a disastrous scenario, in which a seriou sly wrong answer is obtained without any obvious indication that something is amiss. 29The task is to sample from a distribution for ten real-valued variables, vandx1tox9. The marginal distribution of vis Gaussian with mean zero and standard deviation 3. Conditi onal on a given value of v, the other variables, x1tox9, are independent, with the conditional distribution for each being Gaussian with mean zero and vari anceev. The resulting shape resembles a ten-dimensional funnel, with small values for vat its narrow end, and large values for vat its wide end. Such a distribution is typical of priors for c omponents of Bayesian hierarchical models — x1tox9might, for example, be random eﬀects for nine subjects, with vbeing the log of the variance of these random eﬀects. If the da ta happens to be largely uninformative, the problem of sampling from the p osterior will be similar to that of sampling from the prior, so this test is relevant to actual Bayesian inference problems. It is of course possible to sample from this distribution dir ectly, by simply sampling for v, and then sampling for each of x1tox9given this value for v, thereby obtaining independent points from exactly the correct distribution. And in any cas e, we already know the correct marginal distribution for v, which will be the main focus below. For this test, however, we will pretend that we don’t already know the answer, and the n compare what we would conclude using various Markov chain methods to what we know i s actually correct. Figure 15 shows the results of trying to sample from this dist ribution using Metropolis methods. The upper plot shows 2000 iterations of a run in whic h each iteration consists of 10000 multivariate Metropolis updates (ie, 20 million Me tropolis updates altogether). The proposal distribution used was a spherical Gaussian cen tred on the current state, with standard deviation of one for each of the ten variables. The i nitial state had v= 0 and all xi= 1, which is a typical magnitude for the xigiven that v= 0. The points plotted are the value of vat each iteration. Dotted lines are shown at v=±7.5. The results of this run are grossly incorrect. We know that th e marginal distribution for vis Gaussian with mean zero and standard deviation 3. One woul d expect that out of 2000 points from this distribution, on average 95.6 (4.8%) shoul d be less than -5, but none of the points sampled by the multivariate Metropolis method are in this region. Moreover, there is little in the plot to indicate that anything is wrong. In an actual application, the results of a run such as this could easily be accepted as being correct , with serious consequences. The source of the problem is the low probability of accepting a proposal when in a state where vis small. When vis−4, for example, the standard deviation of the xiconditional on this value for vis 0.135. The chances that a multivariate Metropolis propos al in which eachxihas standard deviation one will produce values for all the xithat are within this range of zero is about 0 .1359≈1.5×10−8. The proposal will include a change to vas well as the xi, so this calculation does not give the exact acceptance prob ability, but it does indicate that when vis small, the acceptance probability can become very small, and the chain will remain in the same state for a very long time. Since the Markov chain leaves the correct distribution invariant, it follows that the chain w ill only very rarely move from a large value of v(which happens to be where this run was started) to a small val ue for v— indeed, this never occurred in the actual run. Once one suspects a problem of this sort, signs of it can be see n in the plot. In particular, starting at iteration 1198, the value of vstays at around−3.3 for 25 iterations (ie, for 30Multivariate Metropolis updates, standard deviation 1 0 500 1000 1500 2000-10 -5 0 5 10··· ··· ··· ···· · ··· ·· ·· ·· ··· ··· ··· ···· ···· ··· · ·· ···· ··· ······ ····· ··· ·· ·· ·· ····· ··· · · · ···· · ····· ·· · ··· ···· ··· ···· ······ ·· ··· ······ · · · · ···· ··· · ····· ·· ·· ·· · ·· ·· ·· ····· · ·· ····· ······ ·· ·· ·· ······ ······ ······ ··· ·· ·· ···· ··· ·· ···· · ········ ··· ·· ·· ·· ··· ·· ···· ··· ··· ······ ···· ·· ·· ···· ···· ··· ·· ···· ·· ··· ···· · ······· ···· ··· ·· · · · ··· · ··· ··· ··· ··· ·· · · ·· ····· ·· ······ ·· ··· ···· ···· ··· ·· · ······ ·· ·· ·· ·· ···· · ··· ·· ·· ····· ···· ····· ·· ······ · ·· ····· · ··· ··· ···· ·· ···· ····· · ···· ·· ··· ···· ·· ··· ·· · ······ ·· · ··· ··· ····· ·· ··· ····· ···· ···· ·· ······· ····· · ·· · ··· ··· ··· ······ ·· ·· ···· · ·· ·· ···· ·· ···· ··· · ·· ·· ···· ·· ·· ·· ····· · ·· ····· ···· · ···· 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· · · ·· · · · ···· ···· ··· ··· · ·· ···· · · · · · · · ·· · · · · · · · · · ···· ·· ·· ·· ·· ···· ·· ····· · · ····· ··· ··· ····· ··· ·· ··· ·· ·· ·· · ·· · ·· · · · ······ · ·· ··· ·· ·· ···· ···· ··· ·· ·· ·· ··· ···· ··· ··· ······ ···· ·· ······ ······ ····· · ······· ··· ·· ···· · ··· ·· · · · ······ ···· · · · ··· · ···· ·· ··· ···· ··· ···· ··· ··· · · ···· ··· ··· ··· · ··· ····· ····· ··· ···· ··· ·· ···· ··· ·· ··· · ·· ··· ······· ········· ··· ··· ·· ···· ··· · ······· ··· ·· ·· ······ ···· ·· · ····· ··· ········ ··· ······ ·· ·· ·· ··· ······· ·· ·· ·· · ···· ···· ······ · ···· ··· ···· · ······ ··· · · ···· · ··· ··· · · ·· ······ ··· · ··· ·· ····· · ···· ·· ····· ·· ·· ···· · ···· ·· ···· ··· ······ · ···· · ·· ·· ···· ·· ···· ···· ··· ·· ··· ······ ·· ·· ······· ·· · ···· ·· ··· ···· ···· · ·· ···· ·· · ·· ··· ·· ·· ····· ·· ··· ···· ··· ··· Single-variable Metropolis updates, standard deviation 1 0 500 1000 1500 2000-10 -5 0 5 10···· ·· · ···· ··· ···· ··· ··· ·· ······ ·· ·· ·· ···· ··· ·· ········ ···· ·· ······ ···· ··· ·· · · ···· ··· ·· ···· · ···· ·· ·· · ···· · ···· ·· ·· ····· · ··· ·· ··· ···· ·· ··· ·· · ·· ·· ·· ····· ··· ·· ···· ···· ··· · ··· ····· ·· ·· · ·· ···· ··· ··· ·· ··· ··· ········· · ·· ···· ··· · ···· ···· ······· ···· ··· ··· · ···· ··· ·· ·· ···· · ··· ·· ······· ·· ·· ···· · ··· ····· ··· ·· ····· ·· · ·· ·· ···· ··· ··· · ··· ·· ····· ····· ····· ··· ··········· ······ ·· · ·· · · ··· · ···· ·· ··· ·· ··· ·· ····· ·· ·· ·· ·· ·· ·· ···· ··· · ······ ···· ·· · ···· ·· ···· ··· ·· ······· ··· ·· ·· ···· ·· ······ · ·· ···· ·· · ··· ·· ·· · ···· ······ ··· ·· ··· ·· ·· ·· · ··· ··· ·· ·· · ·· ·· ····· ····· ·· ·· ······ ····· ··· · ·· ···· ·· ·· ·· ··· ······ ····· ·· ····· ··· · ·· · ·· ···· ······ ·· ··· · ·· · ····· ·· ···· · ··· ··· ···· ·· ··· ·· ·· ··· · ···· ···· ···· ·· · ···· ·· ······ ·· ·· ·· ····· ··· ·· ·· ·· ·· ·· ··· · ··· · ··· ····· · ···· · ·· · ··· ··· ·· ···· ····· ··· ·· ···· ·· · · ·· ··· ····· 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·········· ···· ··· · ···· ·· ··· ·· ·· ···· ···· ·· ·· ··· ·· ··· ··· ·· ··· ···· · ·· ···· ··· ·· ·· ··· ······ ·· ·· ······· ··· ········ ·· · ·· ·· ··· · · ··· ·· ····· ·· · ·· ··· ··· ····· ··· ·· ·· ·· ··· ··· ··· ······ ·· ·· ·· ·· ···· ··· ····· ··· ··· ··· ··· ··· ·· ·· ······· · · ·· ·· ·· ·· ··· ·· ···· ·· ···· ·· ·· · ·· ·· ·· ···· ···· ·· ··· ·· · ··· ·· ····· · ·· ··· ··· ··· · ·· ··· ·· ·· ··· ··· ·· ·· ·· · ··· · ·· ·· · ··· · ·· ·· ··· ·· ·· · · ··· ····· ···· ···· ··· ·· ··· ····· ·· ··· ··· ·· ·· · ······ · · ··· ·· ····· · ··· ········ ··· ··· · ···· ·· ······· ···· ·· · ··· · ·· ··· · ··· ··· ······· ····· ·· ·· ·· ··· ··· ··· ··· Multivariate Metropolis updates, random standard deviati on 0 500 1000 1500 2000-10 -5 0 5 10·· · ·· ····· ······ · ····· · ···· ·· ····· ········ ·· ·· ·· · ······ · ···· ···· ··· ····· ·· ···· ··· ·· ·· ·· · ·· ··· ··· · ····· ··· ·· ·· ······ ·· ···· ···· ···· ·· · ·· ··· ···· · ··· ··· ·· ··· ·· ··· ······· ········ ·· ··· ···· ··· ·· 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··· ··· ··· ····· · ·· ····· · ··· ·· ······ ·· ·· ··· ···· ·· ·· ·· ·· ··· ······ ······ ······· ·· ·· ·· ·· ······ ······ · ··· ··· ···· · ······· ·· ·· · ···· · ··· · ······ ···· · ··· ······· · ··· ·· ·· · ·· · ··· ······ ··· ······· ·· ········· ········· ·· ···· ······· ····· ···· · ··· ···· ···· · · ··· ···· · ·· ····· ·· ··· ·· ·· ···· ············ ·· ·· · ··· ·· ·· ····· ···· · Figure 15: Sampling from the funnel distribution with Metro polis methods. 31250,000 Metropolis updates). However, there are no obvious occurrences of this sort in the ﬁrst 1000 iterations, so the problem would not be apparent ev en to a suspicious user if only half as many iterations had been done. Running several chain s from diﬀerent starting states might have revealed the problem, but when sampling from more complex distributions, it is diﬃcult to be sure that an appropriate variety of starting states has been tried. The middle plot in Figure 15 shows the results of sampling fro m the funnel distribution using single-variable Metropolis updates, applied to each variable in sequence. The proposal distribution was a Gaussian centred on the current value, wi th standard deviation one. Each iteration for this run consisted of 1300 updates for eac h variable in turn, which take approximately as long as 10000 multivariate Metropolis upd ates (with the program and machine used). As before, the plot shows the value of vafter each of 2000 such iterations. The results using single-variable Metropolis updates are n ot as grossly wrong as those obtained using multivariate Metropolis updates. Small val ues for vare obtained in the expected proportion. The previous problem of very low accep tance rates when vis small is avoided because even when the standard deviation for one of t hexigiven vis much smaller than the proposal standard deviation, proposals to change a single xiare still accepted occasionally (eg, when v=−9, the standard deviation of the xiis 0.011, and about one proposal in 100 is accepted). However, large values for vare sampled poorly in this run. About 0.6% of the values should be greater than 7.5 (which is marked by a dotted line), but no such values are seen in the ﬁrst half of the run (1000 iterations, 1.3 million upda tes for each variable). Around iteration 1200, the chain moves to large values of vand stays there for 17 iterations (22100 updates for each variable). This number of points above 7.5 i s not too far from the expected number in 2000 iterations, which is 12.4, so in this sense the run produced approximately the right answer. However, it is clear that this was largely a matter of luck. Movement to large values of vis rare, because once such a value for vis reached, the chain is likely to stay at a large value for vfor a long time. In this case, the problem is not a high rejecti on rate, but rather slow exploration of the space in small steps . For example, the standard deviation of the xiwhen vis 7.5 is 42.5. Exploring a range of plus or minus twice this by a random walk with steps of size around one takes about (4 ×42.5)2= 28900 updates of each variable. While exploring this range, substantial amo unts of time will be spent with values for the xithat are not compatible with smaller values of v. (This problem is not as severe in the previous run, because the multivariate propos als take larger steps, since they change all variables at once.) We might try to avoid the problems sampling for both large and small values of vby picking the proposal standard deviation at random, from a wi de range. The lower plot in Figure 15 shows the results when using multivariate Metropo lis proposals in which the log base 10 of the proposal standard deviation is chosen uniform ly from the interval ( −3,3). Large values for vare sampled fairly well, but small values for vare still a problem, though the results are not as bad as for multivariate Metropolis wit h the proposal standard deviation ﬁxed at one. Increasing the range of proposal standard devia tions to even more than six orders of magnitude might ﬁx the problem, but at an even gr eater cost in wasted computation when the random choice is inappropriate. 32Single-variable slice sampling, initial width of 1 0 500 1000 1500 2000-10 -5 0 5 10····· ·· ·· · ···· ···· · ·· ··· ·· ····· ·· ··· ··· ··· · · ···· ·· ·· ··· ·· ···· ·· ····· ·· ······· ······ ·· ··· ·· ··· · ····· ···· ·· ·· ·· · ··· ··· ·· ····· ··· · ·· ·· · · ·· ·· ···· · ··· ···· ·· ·· ··· ··· ···· ·· ····· ····· ··· · ··· ·· ··· ···· ··· · ·· · ···· ·· ·· ··· ·· ·· ······ · ·· ··· · ·· · ····· · ··· ·· ·· ··· ··· ····· ··· ·· ··· · · ·· ··· ·· · ··· ·· · ·· ·· · ······ · · ······ ·· · ··· · ···· ···· · ····· ··· · ···· ·· ···· · ·· ····· ······· ·· ····· ···· ··· ······ · ··· ·· ···· · ·· ···· · ··· ··· ······· ··· ···· ······ ·· ···· ··· ··· ····· ··· ·· ·· ··· ·· ·· · ·· ·· ······ ·· ·· ··· ·· ·· ··· ··· · ··· ··· ···· ···· ··· ···· ·· ·· ·· ·· ··· ·· ·· · ·· ··· ·· ·· ·· ·· ··· · ····· ·· ·· ··· ··· ··· ······· ··· ·· · ···· ·· ···· ··· · ··· ····· ···· ·· ···· ·· ·· ··· ···· ···· ··· ·· ··· ·· ···· ·· ··· ·· ·· ·· · ·· ··· · ····· ···· ··· ·· · ··· ··· ·· ··· ·· · ··· ·· ···· ··· ···· ··· ·· ···· · ·· ·· · ··· ··· ·· ·· ···· ···· ··· ·· ·· ·· ··· ·· ·· ·· ··· ·· ·· ·· ·· ···· ·· ·· · ··· ····· ·· ·· ···· ··· · · ····· ·· · ·· ··· ···· ···· · ··· ···· ·· ·· ···· ·· ·· ·· · ···· ·· ·· ··· ·· ··· ········ ·· · ··· ···· ······· ·· ·· · · ···· ···· ····· · ··· ·· ··· ··· ·· ··· ··· ······ · ·· ··· · ·· ···· · ···· · ·· ·· ··· ·· · ·· ···· ····· · ···· ······ ··· ··· ·· ··· ·· ·· ·· ·· · ···· ·· ··· · ····· · ···· ·· ·· ·· ··· ·· ··· ··· ······ ·· ···· ···· · ·· ·· ·· ······ ····· · ·· ··· ···· · ·· ···· ··· ·· ·· ···· · ·· ··· ········ · ···· ·· · ···· ·· ·· ··· · ·· ··· ···· ·· ···· ·· ········· ·· ···· · ····· ······· ···· ······ ·· ·· ·· ····· ·· ·· ··· · ···· ··· ···· ··· ··· ···· ····· ·· · · ·· ·· ··· · ···· · ·· ·· · ·· ···· ······ ····· 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·· ··· ··· · ·· ·· · Multivariate slice sampling, initial width of 1 0 500 1000 1500 2000-10 -5 0 5 10· ··· · ·· ····· ···· ···· ···· ···· ··· ·· · ··· ··· ··· · · ·· ····· · ··· ·· ·· ··· ·· ·· ·· ·· · ··· ··· ·· ·· ······ ··· ··· ···· ·· · ········· ·· ·· ···· · ··· · ·· ····· ·· ····· ····· ·· ···· ·· ·· ·· · ··· ·· ··· ··· ··· · ··· ·· ·· ···· ··· · ··· ··· ·· ·· ·· ··· ··· ·· ·· ···· ······ ····· ·· ···· ·· ···· ··· ·· · ···· ·· ··· ··· · ···· ··· ····· ····· ·· · ······· · ······· · ····· · · ····· ·· · ····· ·· ·· ·· ···· ··· ··· · ··· · ··· ·· ··· ···· · ·· ··· ·· ···· · ···· ·· · ····· ··· ·· ············ · ···· ·· ··· ········ · ·· ·· ··· · ···· ······ ······· ·· ·· ·· ·· ·········· ··· ··· ·· ···· ·· ··· ··· ···· ·· ·· ··· ······· ··· ···· ·· · · ··· ··· ··· · ··· · ······ ·· · ·· ·· ··· ····· ·· ·· · · ··· ···· ·· ··· ······ · ·· ····· ··· ··· ·· ·· · ·· ·· · ······ ···· ·· ·· ·· · ····· ··· ·· ·· ··· · ··· ······ ····· ········· ·· ·· · ··· · ·· ·· · ··· ·· · ··· ···· 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·· · ··· · ······· ·· ·· ··· · ·· ·· ·· ·· ·· ·· · ···· ··· ··· ······· ··· ····· ··· ·· ··· ··· ·· ·· · ····· ······· ··· · · ··· ······· ···· ··· ·· ···· ··· ····· ··· ·· ······ ··· ··· ·· ········ · ··· · ···· ····· ·· ······· ·· ·· ······ ·· ·· ····· ·· ··· · ·· ···· ···· ·· ·· ··· ··· ··· · ······· ····· ···· · · · ······ · ······ ·· ······· ·· ·· ·········· ···· ·· ···· ·· ······ ······ ··· ···· ·· ···· ··· · ·· ··· ·· ··· ·· ··· ·· ··· ···· ·· ·· ·· ··· ·· ····· · ····· ·· · ··· ·· ·· ··· ···· · ··· ······ ·· ··· ···· ···· ·· ·· ···· · ···· ··· ···· ···· ··· ··· ···· ·· · ·· ······ ···· · ······ · ··· ·· ·· · ·· ·· ·· ····· ·· ···· · ·· ·· ··· ·· ···· ··· ··· ··· · · ·· · ··· ·· · ·· ····· · ·· ·· ··· ······ ·· ··· ····· · · ······ · ····· ·· ···· ·· ··· · ··· ··· ···· ····· · Figure 16: Sampling from the funnel distribution with slice sampling methods. Figure 16 shows the results of trying to sample from the funne l distribution using slice sampling methods. In the upper plot, single-variable slice sampling was used, with an initial interval of size one, expanded by the stepping-out procedur e (Figure 3) until both ends are outside the slice, and then sampled from with the shrinkage p rocedure (Figure 5). Each of the 2000 iterations done consisted of 120 such updates for each variable in turn, which takes approximately the same amount of time as the Metropoli s methods of Figure 15. The average number of evaluations of ffor these slice sampling updates was 12 .7, but a few updates required more than a hundred evaluations. The results with single-variable slice sampling are quite g ood. Small values of vare perhaps sampled slightly less well than with single-variab le Metropolis updates (Figure 15, middle plot), but the diﬀerence is not large. Large values of vare sampled better than with any of the Metropolis methods. The lower plot in Figure 16 shows the results when using the mu ltivariate slice sampling procedure of Figure 8, with an initial hyperrectangle with s ides of length one. Sampling is good out to values of vof about±7.5, but the extreme tails are not sampled well. Recall that this procedure does not expand the initial hyperrectangle, which explains the poor sampling 33for large values of v. The problem with small values of vis probably due to shrinkage being done for all dimensions of the hyperrectangle, with the resu lt that changes to vare very small when vis small (since changes to the ximust then be small). As mentioned brieﬂy in Section 5.1, ways to improve the method in both respects co uld be explored. 9 Discussion As seen in this paper, the idea of slice sampling can be used to produce many Markov chain sampling schemes. In Figure 17, I attempt to summarize the characteristics of these schemes, and of some competing approaches for sampling from general distributions on continuous state spaces. The table separates single-variable methods that update ea ch variable in turn from multi- variate methods that update all variables at once. Single-v ariable methods may be preferred Derivatives How critical Retrospective Can suppress needed? is tuning? tuning allowed? random walks? Single-variable methods ARS/ARMS No (but helpful) Low/Medium If log concave No Single-variable No Medium No No Metropolis Single-variable No Low If unimodal No slice sampling Overrelaxed No (but helpful) Low If unimodal and Yes slice sampling endpoints exact Multivariate methods Multivariate No Medium-High No No Metropolis Dynamical methods Yes High No Yes Slice sampling with No Low–Medium No No hyperrectangles Slice sampling with Possibly Low–Medium No No Gaussian crumbs helpful Reﬂective Yes Medium–High No Yes slice sampling Figure 17: Characteristics of some general-purpose Markov chain sampling methods. 34when the coordinate system used is such that one expects many of the variables to be almost independent, allowing these variables to change by a large a mount even when the other vari- ables are ﬁxed. Also, for some models, recomputing the proba bility density after a change to one variable may be much faster than recomputing it after a change to all variables. On the other hand, if there are strong dependencies between var iables, using single-variable updates may lead to slow convergence, or even a lack of ergodi city — though for high- dimensional problems with strong dependencies, simple-mi nded multivariate methods will also be quite slow. The ﬁrst column in the table indicates whether the method req uires that derivatives of the (unnormalized) probability density be computable. Deriva tives are needed by dynamical methods and reﬂective slice sampling, which limits their ap plicability. Adaptive rejection sampling (Gilks and Wild 1992; Gilks 1992) and overrelaxed s lice sampling can take ad- vantage of derivatives, but can operate without such inform ation with only a moderate loss of eﬃciency — eg, when no derivatives are available, overrel axed slice sampling can use bisection rather than Newton iteration to ﬁnd the endpoints of the slice. The second and third columns indicate how critical it is that tuning parameters be set to good values, and whether or under what conditions ‘retros pective tuning’ is allowed — that is, whether parameters of the method can be set based on information from past iterations. Adaptive rejection sampling (ARS) for log conc ave distributions is very good in these respects — a parameter is needed for the size of the ﬁr st step taken in search of a point on the other size of the mode, but subsequent steps can be made larger (eg, by doubling), so the eﬀect of a poor initial step is not too serio us; furthermore, it is allowable to set this size parameter based on the stepsize that was foun d to be necessary in previous iterations. Parameter tuning is more of a problem when ARMS ( Gilks, Best, and Tan 1995) is used for distributions not known to be log concave — a poor choice of parameters may have worse eﬀects, and retrospective tuning is not allow ed (Gilks, Neal, Best, and Tan 1997). Tuning is also a problem for single-variable and mult ivariate Metropolis methods — proposing changes that are too small leads to an ineﬃcient ra ndom walk, while proposing changes that are too large leads to frequent rejections. Usi ng too small a stepsize with a dynamical method is not quite as bad, since movement is not in a random walk, but too large a stepsize is disastrous, since the dynamical simulat ion becomes unstable, and very few changes are accepted. For Metropolis and dynamical meth ods, the stepsize parameter must not be set retrospectively. Single-variable slice sampling and overrelaxed slice samp ling oﬀer advantages over other methods in these respects. Whereas ARS/ARMS allows retrosp ective tuning only for log concave distributions, this is allowed for these slice samp ling methods when they are ap- plied to any unimodal distribution (provided the interval i s expanded to the whole slice, and endpoints for overrelaxation are computed exactly). Fu rthermore, the tuning is less critical for slice sampling than for the other methods (apar t from ARS), as discussed further below. For reﬂective slice sampling, however, tuning is at l east moderately critical, though perhaps less so than for dynamical methods, and retrospecti ve tuning is not allowed. Tun- ing for multivariate slice sampling using hyperrectangles is less critical than for multivariate Metropolis methods, but as was seen in the demonstration of S ection 8, tuning can be more 35critical for multivariate slice sampling than for single-v ariable slice sampling. The ﬁnal column indicates whether the method can potentiall y suppress random walk behaviour. This is important when sampling from a distribut ion with high dependencies between variables, as in such a situation, exploration of th e distribution may have to proceed in small steps, and the diﬀerence in eﬃciency between diﬀusi ve and systematic exploration of the distribution can be very large (as is typical, for exam ple, with neural network models (Neal 1996)). Another way of exploring the diﬀerences between these metho ds is to see how well they work in various circumstances. The most favourable situati on is when our prior knowledge lets us choose good tuning parameters for all the methods (eg , the width of a Metropolis proposal distribution or of the initial interval for slice s ampling). A Metropolis algorithm with a simple proposal distribution will then move about the distribution fairly eﬃciently (although in a random walk), and will have low overhead, sinc e it requires evaluation of f(x) at only a single new point in each iteration. Single-variab le slice sampling will be comparably eﬃcient, however, provided we stick with the int erval chosen initially (ie, we set m= 1 in the stepping out procedure of Figure 3). There will then be no need to evaluate f(x) at the boundaries of the interval, and if the ﬁrst point chos en from this interval is within the slice, only a single evaluation of f(x) will be done. If this point is outside the slice, further evaluations will be required, but this ineﬃc iency corresponds to the possibility of rejection with the Metropolis algorithm. This situation is similar for multivariate slice sampling with an initial hyperrectangle that is not expande d. Metropolis and slice sampling methods should therefore perform quite similarly. However , slice sampling will work better if it turns out that we mistakenly chose too large a width for t he Metropolis proposal distribution and the initial slice sampling interval. This error will lead to a high rejection rate for the Metropolis algorithm, but the sampling procedu res of Figures 5 and 8 use the rejected points to shrink the interval, which is much more eﬃ cient when the initial interval was too large. As seen in the demonstration of Section 8, the advantage of sl ice sampling over Metropolis methods can be quite dramatic if we don’t know enough to choos e a good tuning parameter, or if no single value of the tuning parameter is appropriate f or the entire distribution. Another possibility is that we know that the conditional dis tributions are log concave, but we do not know how wide they are. Adaptive Rejection Sampl ing (ARS) then works very well, because its width parameter can be retrospective ly tuned, based on previous iterations. Single-variable slice sampling will also work well, since in this situation it can also be tuned retrospectively (provided no limit is set on th e size of the interval). However, ARS does true Gibbs sampling, whereas the slice sampling upd ates do not produce points that are independent of the previous point. Such dependency is probably a disadvantage (unless deliberately directed to useful ends, as in overrel axation), so ARS is probably better than single-variable slice sampling in this context. Suppose, however, that we know only that the conditional dis tributions are unimodal, but not necessarily log concave. We would then need to use ARM S rather than ARS, and would not be able to tune it retrospectively, whereas we can s till use single-variable slice 36sampling with retrospective tuning. This will likely not be as good as true Gibbs sampling, however, which we should prefer if the conditional distribu tion happens to be one that can be eﬃciently sampled from. In particular, if slice sampl ing is used to sample from a heavy-tailed distribution, it may move only infrequently b etween the tails and the central region, since this transition can occur only when we move to a point under the curve of f(x) that is as low as the region under the tails, but whose horizon tal position is in the central region. However, there appears to be no general purpose sche me that avoids problems in this situation. Finally, consider a situation where we do not know that the co nditional distributions are unimodal, and have only a rough idea of an appropriate width p arameter for a proposal distribution or initial slice sampling interval. Single-v ariable slice sampling copes fairly well with this uncertainty. If the initial interval is too small i t can be expanded as needed, either by stepping out or by doubling (which is better will depend on whether the faster expansion of doubling is worth the extra overhead from the acceptance t est of Figure 6). If instead the initial interval is too big, it will be shrunk eﬃciently by th e procedure of Figure 5. We might try to achieve similar robustness with the Metropolis algor ithm by doing several updates for each variable, using proposal distributions with a rang e of widths. For example, if wis our best guess at an appropriate width, we might do updates wi th widths of w/4,w/2,w, 2w, and 4 w. This may ensure that an appropriate proposal distribution is used some of the time, but it is unattractive for two reasons. First, the limi ts of the range (eg, from w/4 to 4w) must be set a priori . Second, for this approach to be valid, we must continue thro ugh the original sequence of widths even after it is clear that we have gone past the appropriate one. These problems are not present with slice sampling. In any of these situations, we might prefer to use a method tha t can suppress random walks. Dynamical methods such as Hybrid Monte Carlo (Duane, et al1987) do this well for a wide range of distributions; reﬂective slice sampling may also work for a wide range of distributions, but preliminary indications are that is l ess eﬃcient than Hybrid Monte Carlo, when both are tuned optimally. Overrelaxation is som etimes beneﬁcial, but not always (whether it is or not depends on the types of correlati on present). For problems where overrelaxation is helpful, overrelaxed slice sampli ng may often be the best approach to suppressing random walks. If the conditional distributi ons are unimodal, it oﬀers the possibility of retrospective tuning. It does not require co mputation of derivatives. For some models, the fact that overrelaxation updates one variable a t a time will permit computa- tional saving, in comparison with the simultaneous updates for dynamical and reﬂective methods. Multivariate slice sampling using hyperrectangles does no t appear to oﬀer much, if any, advantage over single-variable slice sampling, except for the uncommon situation were it is known that the coordinate system used is especially bad (a nd hence updating variables singly will be particularly ineﬃcient). However, the more g eneral framework for multivariate slice sampling based on ‘crumbs’ that was outlined in Sectio n 5.2 oﬀers the possibility of adapting not just to the scales of the variables, but also to t he dependencies between them. The beneﬁts of such methods can only be determined afte r further research, but huge increases in eﬃciency would seem conceivable, if one is to judge from the analogous 37comparison of minimization by simple steepest descent vers us more sophisticated quasi- Newton or conjugate gradient methods. The practical utility of the slice sampling methods describ ed here will ultimately be determined by experience on a variety of applications. Some such applications will involve tailor-made sampling schemes for particular models — for in stance, Frey (1997) used slice sampling successfully to sample for latent variables in a ne ural network. Slice sampling is also particularly suitable for use in automatically genera ted samplers, and is now used in some situations by the WinBUGS system (Lunn, et al2000). Readers can try out slice sampling methods for themselves, on a variety of Bayesian mo dels, using the “software for ﬂexible Bayesian modeling” that is available from my web pag e. This software (version of 2000-08-21) implements most of the methods discussed in thi s paper. Acknowledgements I thank Brendan Frey, Gareth Roberts, Jeﬀrey Rosenthal, and David MacKay for helpful discussions. This research was supported by the Natural Sci ences and Engineering Research Council of Canada and by the Institute for Robotics and Intel ligent Systems. References Adler, S. L. 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arXiv:physics/0009029v1 [physics.comp-ph] 8 Sep 2000Time-Symmetric ADI and Causal Reconnection: Stable Numeri cal Techniques for Hyperbolic Systems on Moving Grids. Miguel Alcubierre and Bernard F. Schutz Department of Physics and Astronomy, University of Wales, College of Cardiﬀ, P.O. Box 913, Cardiﬀ CF1 3YB, UK. Moving grids are of interest in the numerical solution of hyd rodynamical problems and in nu- merical relativity. We show that conventional integration methods for the simple wave equation in one and more than one dimension exhibit a number of instabili ties on moving grids. We introduce two techniques, which we call causal reconnection andtime-symmetric ADI , which together allow integration of the wave equation with absolute local stabil ity in any number of dimensions on grids that may move very much faster than the wave speed and that can even accelerate. These methods allow very long time-steps, are fully second-order accurat e, and oﬀer the computational eﬃciency of operator-splitting. We develop causal reconnection ﬁrst in the one-dimensional case: we ﬁnd that a conventional implicit integration scheme that is unconditionally stabl e as long as the speed of the grid is smaller than that of the waves nevertheless turns unstable whenever the grid speed increases beyond this value. We introduce a notion of local stability for diﬀerence equations with variable coeﬃcients. We show that, by “reconnecting” the computational molecule at each time-step in such a way as to ensure that its members at diﬀerent time-steps are within one anoth er’s causal domains, one eliminates the instability, even if the grid accelerates. This permits ver y long time-steps on rapidly moving grids. The method extends in a straightforward and eﬃcient way to mo re than one dimension. However, in more than one dimension, it is very desirable to u se operator-splitting techniques to reduce the computational demands of implicit methods, an d we ﬁnd that standard schemes for integrating the wave equation — Lees’ First and Second Al ternating Direction Implicit (ADI) methods — go unstable for quite small grid velocities. Lees’ ﬁrst method, which is only ﬁrst-order accurate on a shifting grid, has mild but nevertheless signi ﬁcant instabilities. Lees’ second method, which is second-order accurate, is very unstable. By adopting a systematic approach to the design of ADI scheme s, we develop a new ADI method that cures the instability for all velocities in any directi on up to the wave speed. This scheme is uniquely deﬁned by a simple physical principle: the ADI diﬀe rence equations should be invariant under time-inversion. (The wave equation itself and the ful l implicit diﬀerence equations satisfy this criterion, but neither of Lees’ methods do.) This new time-s ymmetric ADI scheme is, as a bonus, second-order accurate. It is thus far more eﬃcient than a ful l implicit scheme, just as stable, and just as accurate. By implementing causal reconnection of the computational m olecules, we extend the time- symmetric ADI scheme to arrive at a scheme that is second orde r accurate, computationally ef- ﬁcient and unconditionally locally stable for all grid spee ds and long time-steps. We have tested the method by integrating the wave equation on a rotating grid, w here it remains stable even when the grid speed at the edge is 15 times the wave speed. Because our m ethods are based on simple phys- ical principles, they should generalize in a straightforwa rd way to many other hyperbolic systems. We discuss brieﬂy their application to general relativity a nd their potential generalization to ﬂuid dynamics. I. INTRODUCTION. In the numerical study of wave phenomena it is often necessar y to use a reference frame that is moving with respect to the medium in which the waves propagate. This could be the c ase, for example, when studying the waves generated by a moving source, where it may prove convenient to use a refe rence frame attached to this source. In some cases, one may even need to use a frame that moves faster than the wave s themselves, as in the case of a supersonic ﬂow. In general relativity, especially in black-hole problems, one may have to use a grid that shifts rapidly, even faster than light. All these problems arise in more than one spatial dimension, where computational eﬃciency may make stringent demands on the algorithm. It is a common experienc e to ﬁnd that standard algorithms seem to go unstable in realistic problems. In this paper, by studying the simple wave equation, we show that the consistent application of two fundamental physical principles — causality and time-r eversal-invariance — produces remarkably stable, eﬃcient 1and accurate integration methods. These principles can eas ily be applied to more complex physical systems, where we would expect similar beneﬁts. Our principal motivation for studying these techniques is t he development of suitable algorithms for the numerical simulation of moving, interacting black holes. Relativist s have long acknowledged the importance of using shifting grids in some problems, but to our knowledge there has been no systematic study of the eﬀects of such shifts on the stability of numerical algorithms. In the next two paragrap hs we develop this motivation. Readers not concerned with numerical relativity may skip these without loss of con tinuity. Let us consider the requirements that black-hole problems w ill make of our algorithms. Within the context of the 3 + 1 formalism of General Relativity ( [1], [2]), it would seem to be desirable to develop methods on a quasi- rectangular 3-dimensional grid, so that no special coordin ate features would prevent one from studying quite general problems. If we imagine a picture in which a black hole moves “ through” such a grid, much the way a star would if it were interacting with another, then some requirements be come clear: 1. Grid points will move from outside to inside the horizon, b ut the grid as a whole should not be sucked in. This may require an inner boundary to the grid, say on a margin ally trapped surface, and this boundary will have to move at faster than the speed of light. Grid points may cross this boundary and be forgotten, at least temporarily, but others will emerge on the other side of the b oundary. 2. Grid points that so emerge will then move from inside to out side the horizon as the hole passes over them; this will require grids that shift faster than light. This is ines capable unless one ties the grid to the hole as it moves. 3. If two black holes begin in orbit around one another, then i t may be desirable to adopt a grid that rotates with respect to inﬁnity, in which the holes move relatively slowl y at ﬁrst. In such a grid one would expect that one could take long time-steps without losing accuracy, since n ot much happens initially. One therefore would like to be free of the Courant condition on time-steps, i.e.one wants to use implicit methods. 4. Integrating the equations of general relativity on a grid with reasonable resolution will tax the capacity of the best available computers for some time to come. Full implici t schemes are very time-consuming in more than one spatial dimension, because they require the inversion o f huge sparse matrices. Alternating-direction-implicit (ADI) schemes reduce this burden enormously by turning the i ntegration into a succession of one-dimensional implicit integrations, so an ADI scheme that can cope with gr id shifts is very desirable. In this paper we show that it should be possible to develop sta ble methods that satisfy the last three requirements above: ADI schemes that are absolutely stable and computati onally eﬃcient even on grids that shift at many times the speed of light. As a bonus, our ADI methods preserve the se cond-order accuracy of the full implicit equations. The ﬁrst requirement, that of dealing with an inner boundary that moves faster than light, is closely related to these techniques and will be addressed elsewhere. Having these requirements in mind, we have studied the eﬀect s that the use of a moving reference frame has on the ﬁnite diﬀerence approximation to the simple wave equati on, centering our attention particularly in the stability properties. The wave equation is the simplest system, so the instabilities we ﬁnd in the standard ADI methods should certainly also be present when they are applied to more reali stic physical systems. Of course, the wave equation is much simpler than other systems, so it is possible that metho ds that stabilize its integration will not extend to other systems. However, the principles that we ﬁnd here are of such a fundamental physical nature that it seems certain that they should be applied wherever possible. Other kinds o f instabilities may of course arise in complex systems, especially those directly due to nonlinearity, but we feel t hat moving-grid instabilities are likely to be cured by the methods we describe here. We shall conclude this introductory section by summarizing the approach and results of the following sections. In the second section we develop the mathematical framework of shifting grids. Then in Section III we study the one-dimensional wave equation. We ﬁnd simple implicit ﬁnit e diﬀerence schemes that are locally stable for any speed up to that of the waves, even when the grid is accelerating as w ell as moving. When formulated on a grid that is moving, and even accelerating, it is not immediately obviou s how one deﬁnes stability: solutions of the diﬀerential equation do not have simple harmonic time-dependence in thi s frame. We ﬁnd that a satisfactory criterion for local stability of these simple schemes is that no solutions of the diﬀerence equations should grow faster anywhere on the grid than local solutions of the diﬀerential equation. However, as soon as the reference frame moves faster than the wave speed, these schemes become highly unstable. We trace the origin of this instability to the fact that the co mputational molecules no longer represent in an adequate way the causal relationships between the grid points. We ﬁnd that by modifying the molecules so that they link 2a given point on one time-slice with one on the next one that is within the ﬁrst point’s cone of characteristics (its forward “light cone”), one can restore stability. We discus s one algorithm for doing this in Appendix B. We call this causal reconnection. It is important to note that this has a minimal impact on the in tegration scheme: for implicit schemes, the matrix that must be solved for the s olution at a given time-step is constructed only from the relations between grid-points at that time-step, while cau sal reconnection aﬀects only the relations between points o n diﬀerent time-steps. Thus, it can be incorporated into the p art of the algorithm that constructs the “inhomogeneous terms” that generate the right-hand-side of the implicit ma trix equation. For the 1-dimensional wave equation, the extra work involved in seeking out causally related grid poi nts can be signiﬁcant, but it becomes a smaller proportion of the overhead in more than one dimension, and for complicat ed systems of equations, such as one has in general relativity, the overhead will be a negligible fraction of th e total work per time-step. We have tested causal reconnecti on and found it to be stable even on grids moving at many times the speed of the waves. It is also insensitive to the acceleration of the grid. We then move our attention in Section IV to operator-splitting ADI methods [4], which are computationally eﬃcient ways of implementing implicit schemes in more than one dimen sion. We ﬁnd it helpful to derive ADI schemes from a more systematic point of view than one usually ﬁnds in exposi tions of this technique. The goal is to add extra terms to a set of diﬀerence equations that (i)do not change its accuracy, but (ii)replace the large sparse non-tridiagonal matrix which has to be solved in implicit schemes with a matri x that is a simple product of tridiagonal matrices of the 1-D implicit form for each dimension, which are easy to so lve. The extra terms are related to the “left-over” terms that appear as the diﬀerence between the true operator and it s factored replacement acting on the data values on the ﬁnal time-step. These ﬁnal-time-step terms must be elimina ted. They are in eﬀect replaced by similar terms from earlier time-steps, which replacement makes no diﬀerence w hen ∆t→0, but which removes them from the matrix inversion and allows them to be included as part of the inhomo geneous terms in the matrix solution. Then the new equations will be a valid approximation to the diﬀerential e quation but can be solved by a succession of (rapid) 1-D tridiagonal matrix solutions. When subjected to the same stability analysis as we devised f or causal reconnection, the standard ADI methods show instabilities even when the reference frame moves very slowly. The instability is most marked in Lees’ second method, in which the extra terms added in are of second order a nd therefore do not degrade the accuracy of the full implicit scheme. The instability is also present, albeit mo re weakly, in Lees’ ﬁrst scheme, which is only ﬁrst-order accurate. We trace these instabilities to the fact that the extra terms added in either of the standard methods break the time-reversal invariance exhibited by the original diﬀere ntial equation and by the full implicit diﬀerence equations . Demanding that the extra terms be time-symmetric uniquely d etermines an ADI scheme that is essentially a hybrid of Lees’ ﬁrst and second methods. This time-symmetric ADI meth od turns out to be fully stable for all grid shifts up to the wave speed. Although not built in as a requirement, the ne w method also turns out to be second-order accurate. The method can then be extended to grid speeds larger than the wave speed by a direct generalization of the causal reconnection approach developed for the one-dimensional c ase. We demonstrate this by performing an integration on a rotating grid whose edge moves faster than the wave speed. In Appendix A, we derive the wave equation in the acceleratin g coordinate system using the eﬃcient tensorial techniques of relativity. In Appendix B, we discuss one meth od of implementing causal reconnection. II. THE WAVE EQUATION ON A MOVING GRID. The wave equation is a good testing ground for any new algorit hms for hyperbolic systems. The equations governing many wave systems can be reduced to the standard wave equatio n, and its cone of characteristics has the causal structure of space-time. We shall use it to test methods for i ntegrating hyperbolic systems on moving grids. We consider the wave equation in an arbitrary number of spati al dimensions n, ∇2φ−1 c2∂2φ ∂t2= 0. (II.1) written in a standard inertial coordinate system denoted by (t,ξi). We are interested in ﬁnding a ﬁnite diﬀerence approximation to this equation using a grid of points that moves with an arbitrary non-uniform speed. Moreover, we will assume th at the speed of each grid point can change with time. In order to represent this situation, we need to introduce a sec ond coordinate system ( t,xi) that will be comoving with the grid. We introduce these coordinates in the continuous c ase by a transformation of the form 3xi=xi(t, ξk). (II.2) We have not changed the time coordinate, so we assume that the identiﬁcation of surfaces of constant time does not change. This is thus not the usual Lorentz transformatio n of special relativity, so there is no reason for the form of the wave equation to remain invariant. This will have the i mplication that, in ﬁnite diﬀerences, the time interval betweent= const slices will be constant, independent of position. Fo r problems in general relativity, this is somewhat of a restriction, but we do not feel it is a serious one. If the c ausal relations are properly taken into account, then a spatial dependence in the lapse function ought not to change our physical conclusions. In Appendix A we show that the wave equation takes the followi ng form in the new coordinates ( t,xi): (gi k−βiβk)∂2φ ∂xi∂xk+2βi c∂2φ ∂xi∂t−Γi∂φ ∂xi−1 c2∂2φ ∂t2= 0. (II.3) The following quantities derived from the coordinate trans formation appear in the last equation: βi:=−1 c∂xi ∂t, (II.4) gi j:=n/summationdisplay l=1∂ξl ∂xi∂ξl ∂xj, (II.5) gi j:=n/summationdisplay l=1∂xi ∂ξl∂xj ∂ξl, (II.6) g:= det(gij), (II.7) Γi:=−1√g/braceleftbigg∂ ∂t/parenleftbig√gβi/parenrightbig +∂ ∂xj/bracketleftbig√g/parenleftbig gij−βiβj/parenrightbig/bracketrightbig/bracerightbigg . (II.8) Each of these quantities has a physical interpretation, whi ch we now explain. Readers familiar with these ideas may skip to the next section. Theshift vector βigives the relationship between the two coordinate systems o n nearby surfaces of constant time.1 Let the line of constant {ξi}have coordinates {xi t}at the lower hypersurface and {xi t+dt}at the upper hypersurface. From the deﬁnition of the shift vector in Equation II.4, it is clear that xi(ξj,t+dt)≈xi(ξj,t)−cβidt. (II.9) As we illustrate in Figure 1, if one starts at any given point a t timet, then by time t+dtthe{xi}coordinates will have shifted by an amount equal to the shift vector times cdtrelative to the {ξi}(inertial) coordinates. The shift vectorβiwill in general be a function of both {xi t}and t. We now introduce the spatial metric tensor gi j, which we have deﬁned in Equation II.5. Its name comes from th e fact that the distance dlbetween two points whose coordinates diﬀer by dξlin the original coordinates and by dxi in the shifting coordinates is given by the Pythagorean Theo rem: dl2=n/summationdisplay l=1dξldξl=n/summationdisplay i,j=1gi jdxidxj(II.10) That this gives Equation II.5 for gijis readily seen by substituting the following transformati on fromdξltodxi into the ﬁrst version of the Pythagorean Theorem: dξl=n/summationdisplay i=1∂ξl ∂xidxi. (II.11) 1This is just the standard deﬁnition of the shift vector in the 3+1 formalism of Numerical Relativity [1]. 4✟✟✟✟✟✟ ✟✟✟✟✟✟✟✟✟✟✟✟ ✟✟✟✟✟✟ ✻✻ ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ ✁✁✁✁✁ ✕✁✁✁ ✕ ✛ tt+dt xi txi t xi t+dtcβidtlineof constant/braceleftbig xi/bracerightbig lineof constant/braceleftbig ξi/bracerightbig FIG. 1. Shift vector βi. The next tensor that appears in the general form of the wave eq uation is the inverse metric tensor given by Equation II.6. This is the matrix inverse of the metric tenso r, n/summationdisplay j=1gijgjk=δi k, (II.12) as can easily be seen by substituting Equations II.5 and II.6 into the above. The ﬁnal quantity we need is Γi, a measure of the acceleration of the shifting coordinates w ith respect to the old ones, given by Equation II.8. We will leave the full derivati on of Γito Appendix A, but to illustrate our interpretation of it as an acceleration term, we shall explicitly evaluate i t in the case where the new coordinates are obtained from the inertial ones by a simple shift independent of position. Then the shift vector βiis only a function of time, and the spatial metric gi jis just the unit matrix: ∂βi ∂xj= 0, g i j=δi j. (II.13) It is not diﬃcult to see that in this case the coeﬃcients Γireduce to: Γi=−1 cdβi dt. (II.14) Since the shift vector gives the speed of the {xi}coordinates, the last expression implies that the Γicoeﬃcients are essentially the acceleration. Notice that if there is no acceleration, the only essential d iﬀerence from the normal wave equation is the transport term/vectorβ· ∇˙φ, which arises as well in hydrodynamical problems. We will se e that the local stability properties of the algorithms we study are determined mainly by βi, not Γi, which is one reason we expect our analysis to have much wider applicability than just to problems involving the wav e equation. Having derived the form of the wave equation in our new coordi nates, we now establish a grid for formulating diﬀerence equations in these coordinates. By assumption, w e take the time-interval ∆ tbetween successive surfaces of constant time to be uniform (independent of position) and co nstant (the same for any pair of surfaces). We take each grid point to have a ﬁxed spatial coordinate position xi, and for convenience we take the spacing between grid points ∆xito be uniform in each coordinate direction. As seen in the ine rtial frame, the grid deforms itself as in Figure 2. The corresponding picture in the xi-coordinate frame looks much more regular (Figure 3). 5❈❈❈❈❈❈❈❈❈❈❈❈ ❈❈❈ ❖❈❈❈ ❖ ✇✇✇ ✻✻ ✇✇✇ ✄✄✄✄✄✄✄✄✄✄✄✄ ✄✄✄ ✗✄✄✄ ✗ ✇✇✇ ✂✂✂✂✂✂✂✂✂✂✂✂ ✂ ✂✂✂ ✂ ✍✂✂✂ ✂ ✍ ✇✇✇ ✁✁✁✁✁✁✁✁✁✁✁✁ ✁✁✁ ✕✁✁✁ ✕ ✇✇✇ ✲✻t ξi FIG. 2. Grid in original coordinates, showing true distance s. ✻ ✻ ✻ ✻ ✻✻ ✻ ✻ ✻ ✻ ✇ ✇ ✇ ✇ ✇✇ ✇ ✇ ✇ ✇ ✲✻t xi∆t∆x FIG. 3. Grid in new coordinates. III. THE ONE DIMENSIONAL CASE. A. Finite diﬀerence approximation. The one-dimensional wave equation allows us to study shifti ng grids in a relatively simple fashion. The added complication of extra dimensions will be treated in the next section. In one spatial dimension, the metric, shift, and accelerati on coeﬃcients reduce to scalar functions: g11(x,t) =g(x,t), β1(x,t) =β(x,t), Γ1(x,t) = Γ(x,t).  (III.1) Because the metric scales the squares of the coordinate distances (Equation II.10), it is conveni ent to deﬁne the linear scale function s(x,t) by s(x,t) :=/radicalbig g(x,t) =∂ξ ∂x, (III.2) so that the spatial proper distance is given by dξ=s(x,t)dx. (III.3) 6② ② ②② ② ②② ② ② ✲✻t x∆x ∆t i−1 i i + 1j−1jj+ 1 FIG. 4. Computational molecule. Using this expression, Equation II.3 becomes: (1 s2−β2)∂2φ ∂x2+2β c∂2φ ∂x∂t−Γ∂φ ∂x−1 c2∂2φ ∂t2= 0. (III.4) For the ﬁnite diﬀerence approximation to this equation we em ploy the usual notation: φj i:=φ(i∆x,j∆t). (III.5) We deﬁne the ﬁrst and second centered spatial diﬀerences as: δxφj i:=φj i+1−φj i−1, δ2 xφj i:=φj i+1−2φj i+φj i−1.  (III.6) It is important to note that with the last deﬁnitions ( δx)2/ne}ationslash=δ2 x. We can also deﬁne analogous diﬀerences for the time direction. We now write the ﬁnite diﬀerence approximation to the diﬀere ntial operators that appear in Equation III.4 using the computational molecule shown in Figure 4. We have: /parenleftbig ∂2 tφ/parenrightbigj i=φj+1 i−2φj i+φj−1 i (∆t)2+Ett, (III.7) whereEttis the truncation error whose principal part is: Ett=−(∆t)2 12/parenleftbig ∂4 tφ/parenrightbigj i+.... (III.8) Similarly, for the mixed derivative in space and time we ﬁnd: (∂x∂tφ)j i=δxφj+1 i−δxφj−1 i 4 ∆x∆t+Ext, (III.9) with: Ext=−1 6/bracketleftig (∆x)2/parenleftbig ∂3 x∂tφ/parenrightbigj i+ (∆t)2/parenleftbig ∂x∂3 tφ/parenrightbigj i/bracketrightig +.... (III.10) For the second derivative in the xdirection we use an implicit approximation of the following form: /parenleftbig ∂2 xφ/parenrightbigj i=θ1 2/bracketleftigg δ2 xφj+1 i (∆x)2+δ2 xφj−1 i (∆x)2/bracketrightigg + (1−θ1)/bracketleftigg δ2 xφj i (∆x)2/bracketrightigg +Exx, (III.11) 7whereθ1is an arbitrary parameter that gives the weight of the implic it terms. If θ1= 0 the approximation is explicit, while if θ1= 1 all the weight is given to the initial and ﬁnal time-steps o f the molecule in the ﬁgure. Note that the last equation is symmetric in time. The error for thi sxderivative is: Exx=−(∆x)2 12/parenleftbig ∂4 xφ/parenrightbigj i−θ1(∆t)2 2/parenleftbig ∂2 x∂2 tφ/parenrightbigj i+... . (III.12) Finally, for the ﬁrst derivative in xwe take: (∂xφ)j i=θ2 2/bracketleftigg δxφj+1 i 2 ∆x+δxφj−1 i 2 ∆x/bracketrightigg + (1−θ2)/bracketleftigg δxφj i 2 ∆x/bracketrightigg +Ex, (III.13) where we have used again an implicit approximation with a diﬀ erent parameter θ2. The truncation error Exis: Ex=−/bracketleftigg (∆x)2 6/parenleftbig ∂3 xφ/parenrightbigj i+θ2(∆t)2 2/parenleftbig ∂x∂2 tφ/parenrightbigj i/bracketrightigg +.... (III.14) We can now write down a second order ﬁnite diﬀerence approxim ation to Equation III.4: ρ2/parenleftbigg1 s2−β2/parenrightbigg/braceleftbiggθ1 2/bracketleftig δ2 xφj+1 i+δ2 xφj−1 i/bracketrightig + (1−θ1)/bracketleftig δ2 xφj i/bracketrightig/bracerightbigg +ρβ 2/bracketleftig δxφj+1 i−δxφj−1 i/bracketrightig −/bracketleftig φj+1 i−2φj i+φj−1 i/bracketrightig −ρ(c∆t) 2Γ/braceleftbiggθ2 2/bracketleftig δxφj+1 i+δxφj−1 i/bracketrightig + (1−θ2)/bracketleftig δxφj i/bracketrightig/bracerightbigg = 0, (III.15) whereρis the ‘Courant parameter’ [3] given by: ρ:=c∆t ∆x. (III.16) The coeﬃcients {s,β, Γ}appearing in Equation III.15 should be evaluated at the poin t (i,j) that corresponds to the center of the molecule. To arrive at the ﬁnal form of the diﬀerence equation we multip lied it through by (∆ t)2. This means that the overall truncation error is now EIII.15=O/bracketleftig (∆x)2(∆t)2/bracketrightig +O/bracketleftig (∆t)4/bracketrightig . (III.17) Equation III.15 is well studied in the particular case when β= Γ = 0 and s= 1 [4]. It is important to note that, because we use centered diﬀerences in the transport te rm, the above ﬁnite diﬀerence approximation will be implicit whenever the shift vector is diﬀerent from zero, ev en whenθ1=θ2= 0. Therefore the use of implicit approximations for the spatial derivatives does not add any extra numerical diﬃculty. We shall need to know how much numerical work is involved in us ing the implicit scheme. Suppose there are N spatial grid points. Then Equation III.15 is to be solved for theNvalues {φj+1 i, i= 1,...,N }at the ﬁnal time-step. The equation for index irelates three such values, at points {i−1,i, i+ 1}. The system of equations therefore has the matrix form ˆQxφj+1=f(φj, φj−1), (III.18) where ˆQxis a tridiagonal N×Nmatrix, and the inhomogeneous term fis constructed from ﬁeld values at the ﬁrst two time-steps. Solving a tridiagonal matrix involves O(N) operations. Since we also need O(N) operations for the solution of an explicit scheme, we see that the use of an impli cit method in one dimension will increase the number of operations per time-step by at most a multiplier, independe nt of the number of grid points. Against this, the implicit scheme for certain choices of θ1andθ2can, on a ﬁxed grid, take much larger time-steps, limited onl y by accuracy considerations. In the next section, we shall show that this property of the implicit scheme in one dimension can, with suitable modiﬁcations, be extended to grids that shift essentially arbitrarily fast. 8B. Local stability: deﬁnition and analysis of the implicit s cheme on a shifting grid. It is well known [4] that the implicit approximation to the wa ve equation can be made unconditionally stable in the case when β= Γ = 0 and s= 1 by using an implicit parameter θ1≥1/2. We are interested in studying under what conditions this property is preserved in the case of a shifting grid. The shifting grid introduces a major diﬃculty: the coeﬃcients in the equation generally depend o n both position and time. This complicates the deﬁnition of stability. This diﬃculty means that an analytic stability analysis mus t belocal: we will actually only consider the stability of the diﬀerence equation obtained from Equation III.15 by, at each point ( x,t), taking the coeﬃcients to be constant, with the values corresponding to that point. We feel that thi s is not a very restrictive assumption, since in practice instabilities usually appear as local phenomena [4], with t he fastest growing modes having wavelengths comparable to the grid spacing. Moreover, if the coeﬃcients in the diﬀeren ce equation are not practically constant over a few grid points, then we are probably not approximating the original diﬀerential equation adequately anyway. We will start then by considering the nature of the solutions of the diﬀerential equation in a very small region around the point ( x,t) . As usual, we look for a solution of the form: φ(x,t) =eıαteıkx. (III.19) Substituting this in Equation III.4 gives the following “di spersion relation” for α: α±=kβc±c/bracketleftbiggk2 s2+ikΓ/bracketrightbigg1/2 . (III.20) The general solution for a wavenumber kis φ(x,t) =eıkx/bracketleftbig Z+eıα+t+Z−eıα−t/bracketrightbig . (III.21) Clearly, if Γ /ne}ationslash= 0, then one of the independent solutions will grow with time , but the other one will decay because of what we shall call the analytic boundedness condition : /vextendsingle/vextendsingleeıα+teıα−t/vextendsingle/vextendsingle= 1. (III.22) This does not mean that the system is physically unstable, bu t only that in an accelerating coordinate system (Γ /ne}ationslash= 0) the wave equation does not have purely sinusoidal solutions . One can understand this intuitively in the following way: Consider a sinusoidal solution in a static coordinate syste m. From the point of view of an accelerating observer, the frequency of this solution will be changing with time (he wil l be seeing more and more crests per unit time). This change in frequency will have important local eﬀects. As a cr est approaches our accelerated observer, he will see the wave function rising faster than an observer moving at th e same speed, but not accelerating, would. Hence the appearance of locally growing modes in our analysis. Simila rly, after a crest is reached, the accelerated observer will see the value of the wave function falling faster than a unifo rmly moving observer would. It is not diﬃcult to see that the diﬀerence between the growth rate in the ﬁrst case and the decay rate in the other, as seen by our two observers, will be the same. This is the origin of the analytic boundedne ss condition given above. The presence of the growing modes is crucial for our local sta bility analysis. Since the solutions of the diﬀerential equation can grow with time, we can not ask the solutions of th e ﬁnite diﬀerence approximation not to do so. What we are entitled to ask is for the numerical solutions not to gr ow faster than the corresponding normal modes of the diﬀerential equation. Our stability criterion is, therefo re:a diﬀerence equation is locally stable if every solution for a given wavenumber kis bounded in time by a solution of the diﬀerential equation f or the same k. Bearing this in mind, we now proceed to an analogous analysis of the solutions of the ﬁnite diﬀerence scheme. We look for a local stability condition around the point ( n,m) by making the substitution: φm n= (ψ)mei k n∆x. (III.23) Substituting the last expression in the ﬁnite diﬀerence app roximation (Equation III.15) we ﬁnd a quadratic equation inψof the form: A(ψ)2+Bψ+C= 0, (III.24) with coeﬃcients given by: 9A=/braceleftbigg θ1ρ2/parenleftbigg1 s2−β2/parenrightbigg/bracketleftbigg cos(k∆x)−1/bracketrightbigg −1/bracerightbigg +iρsin (k∆x)/bracketleftbigg β−θ2 2(c∆t)Γ/bracketrightbigg , (III.25) B=/braceleftbigg 2 (1−θ1)ρ2/parenleftbigg1 s2−β2/parenrightbigg/bracketleftbigg cos(k∆x)−1/bracketrightbigg + 2/bracerightbigg −iρ(1−θ2)(c∆t)Γ sin (k∆x), (III.26) C=/braceleftbigg θ1ρ2/parenleftbigg1 s2−β2/parenrightbigg/bracketleftbigg cos(k∆x)−1/bracketrightbigg −1/bracerightbigg −iρsin (k∆x)/bracketleftbigg β+θ2 2(c∆t)Γ/bracketrightbigg . (III.27) The two roots of this equation are: ψ±=−B±/parenleftbig B2−4AC/parenrightbig1/2 2A, (III.28) and the general solution of the diﬀerence equation is φm n=eıkn∆x[Z+(ψ+)m+Z−(ψ−)m]. (III.29) It is not diﬃcult to see that the coeﬃcients AandChave the property: \|A\|2=\|C\|2−2θ2ρ2β(c∆t) Γ sin (k∆x), (III.30) which implies: \|ψ+ψ−\|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleC A/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ne}ationslash= 1. (III.31) This contrasts with the diﬀerential case, Equation III.22, where the product of the magnitudes of the two funda- mental solutions was 1. Since the ratio \|C/A\|depends on the value of kin Equation III.30, there will always exist wavenumbers for which the product \|ψ+ψ−\|exceeds 1. This would seem to be undesirable from the point of view of stability, but we can eliminate it as a potential problem by s etting from now on θ2= 0. (III.32) This means that we will use an implicit approximation only fo r the second spatial derivatives ( θ1/ne}ationslash= 0), and not for the ﬁrst spatial derivatives ( θ2= 0). Since from now on we will have only one θparameter, we will change notation now and deﬁne θ:=θ1. The solutions of the diﬀerence equation now satisfy: \|ψ+ψ−\|= 1. (III.33) Next we introduce the ampliﬁcation measure M: M:= max k/parenleftig \|ψ+\|2,\|ψ−\|2/parenrightig , (III.34) and analogously for the solutions of the diﬀerential equati on. The ampliﬁcation measure bounds the growth in the magnitude of any normal mode in one time-step. Our local stab ility condition is then equivalent to MNum≤MAna, (III.35) whereMNumandMAnalare the ampliﬁcation measures for the ﬁnite diﬀerence appro ximation and the diﬀerential equation respectively. 10FIG. 5. Stability on a static grid. In the left-hand ﬁgure, we treat the explicit scheme, where we ﬁnd, as expected, that instability sets in for Courant parameter ρ/s > 1. On the right, we see that a fully implicit scheme ( θ= 0.5) is stable for all time-steps, again as expected. When all the parameters are free to take any value, Equation I II.35 is very complicated, and it is then diﬃcult to ﬁnd its consequences analytically. We shall therefore stud y this equation numerically, in order to ﬁnd regions of the parameter space in which the ﬁnite diﬀerence scheme is stabl e. First let us consider the case of a static grid, β= 0,Γ = 0. This case has, of course, been studied analytically [4] , and it is known that if θ<1/2 the generalized Courant stability condition is: /parenleftigρ s/parenrightig2 ≤1 (1−2θ), (III.36) while ifθ≥1/2 the scheme is absolutely stable. In Figure 5 we show graphs o f both the numerical ampliﬁcation measure (solid line) and the one corresponding to the diﬀere ntial equation (dotted line). We have only plotted the functions for k∆x=πbecause this turns out to be the worst case. The ﬁrst graph sho ws how for θ= 0 the scheme is stable for values of the Courant parameter ρ/ssmaller than one. However, when this parameter takes values slightly larger than one, the numerical ampliﬁcation measu re begins to grow very fast. In the second case we see that forθ= 1/2 the scheme is locally stable for all values of the Courant pa rameter, in agreement with the known stability condition given above. The next group of graphs (Figure 6) shows the eﬀect of a unifor m shift. In both graphs we have assumed that there is no acceleration (Γ = 0), and we have taken θ= 1/2 in order to avoid any instability of the type seen in Figure 5 . The ﬁrst of these shows that the scheme remains locally stabl e for all values of the Courant parameter, even when the grid shift speed sβis very close to 1. However, in the second graph we see that, as soon assβbecomes larger than 1, the scheme turns unstable for allvalues ofρ. In this last case there is no stable choice of time-step. Thi s is a very important property: The ﬁnite diﬀerence scheme becomes unconditionally unstab le whenever the shift is faster than the speed of the waves. Finally, in Figures 7 and 8 we consider the eﬀects of an accele rating grid for the particular case when: θ= 1/2, sβ= 1/2, ands2Γ ∆x= 1 . In Figure 7 we show the behavior of the ampliﬁcation measu re for the ﬁnite-diﬀerence equation and the diﬀerential equation for two diﬀerent norm al modes (two values of k). As we expect, the ampliﬁcation measure corresponding to the diﬀerential equation, MAnais no longer 1. For the ﬁrst graph we have k∆x= 1 and for the second k∆x= 2. For the smaller wave number (larger wavelength) the ampl iﬁcation measures for the diﬀerential and ﬁnite-diﬀerence cases are relatively clos e to each other. As the wavenumber increases, the ﬁnite- diﬀerence ampliﬁcation measure falls further below that of the diﬀerential equation, so that the ﬁnite-diﬀerence scheme remains stable (although less accurate). Figure 8 sh ows a surface plot of ( MAna−MNum) in the region: 11FIG. 6. Stability on a uniformly shifting grid. The ﬁgure on t he left has a grid speed 0 .9 times the wave speed. On the right the grid moves at 1 .1 times the wave speed. In both cases we have set θ= 1/2 and Γ = 0 (no acceleration). ρ/s∈(0,2) k∆x∈(0,π), We clearly see how ( MAna−MNum)≥0 in the whole region. Since k∆x=πcorresponds to the smallest wavelength that can be represented on the grid ( λ= 2 ∆x), we ﬁnd that the ﬁnite-diﬀerence scheme will be stable for a ll modes. We have searched through other values of Γ , and we have found t hat, although the details of the graphs change, the qualitative behavior is preserved. The acceleration parameter Γthus seems to have no important eﬀect on the stability of the scheme. In summary, our stability analysis shows that the ﬁnite diﬀe rence scheme given by Equation III.15 will be locally stable for all values of the Courant parameter ρif the following conditions are satisﬁed: •θ1≥1/2, •θ2= 0, • \|sβ\|<1, •Γirrelevant.  (III.37) The limit on βis inconvenient in many problems, where it is desirable to ha ve grids shifting faster than the wave speed. We turn now to a method for removing this restriction. C. Causal reconnection of the computational molecules. 1. Causality problem. The causal structure of a grid shifting faster than the wave s peed is particularly clearly illustrated in the original (ξ,t) coordinates. In Figure 9 awe see how, for a very large shift, the individual grid points move faster than the 12FIG. 7. Stability on an accelerating grid. For two diﬀerent m odes, the ﬁnite-diﬀerence ampliﬁcation measure (solid cur ve) lies below that of the diﬀerential equation (dotted curve). This means the ﬁnite-diﬀerence scheme is stable, at least fo r these modes. FIG. 8. Stability on an accelerating grid. Here we show a surf ace plot of the diﬀerence between the analytical and numeric al ampliﬁcation measures. This diﬀerence is always positive, which means that the ﬁnite-diﬀerence scheme is stable in the whole region. 13waves, that is, they move outside the light-cone.2Since the diﬀerential equation propagates data along this c one, it seems plausible that the instability found in the previous s ection arises in the fact that the diﬀerence scheme attempts to determine the solution at points on the ﬁnal time-step usi ng data that are outside the past light-cone of these points. This suggests that we should not build the computational mol ecules from grid points with ﬁxed index labels, but instead use those points that have the closest causal relationship (Figure 9 b). We shall now proceed to show analytically how such a reconnection can stabilize the sche me. In order to build this causal molecule let us consider then an individual grid point at the last time level. We look for that grid point in the previous time level that is closest to it in the causal sense. Having found this point, we repeat the procedure to ﬁnd the closest causally connected p oint in the ﬁrst time level. In Appendix B we give a simple algorithm for ﬁnding these points in an integration o f the wave equation. The algorithm adapts easily to other linear hyperbolic systems. We shall return in a later p aper to its generalization to nonlinear equations, and in particular to the case of shocks in hydrodynamics. First we c onsider the general constraints on the time-step that causal reconnection imposes, and then address the issue of h ow much extra computational eﬀort causal reconnection may involve. 2. The causal reconnection condition. Since we permit the grid to move with an arbitrary non-unifor m speed, there is no reason that these causally con- nected points should be in a straight line in either the origi nal inertial reference frame ( ξ,t) or in the moving reference frame (x,t). In the moving coordinate system ( x,t) the relationship among these three points may generically look something like that shown in Figure 10. Accordingly, we introduce a new local coordinate system ( x′,t′) adapted to the three given points. In order to do this, it is convenient to introduce the interpolating secon d order polynomial that can be obtained from these three points: P(t) =A(t−t0)2 2+B(t−t0) +x0, (III.38) wheret0is the time at the central point of the molecule, xt0the position of that point, and: A:=/parenleftigg xt0+∆t−2xt0+xt0−∆t (∆t)2/parenrightigg , B :=/parenleftbiggxt0+∆t−xt0−∆t 2 ∆t/parenrightbigg . (III.39) We deﬁne the new local coordinate system adapted to the causa l molecule by: x′:= (x−xt0)−P(t), t′:=t.  (III.40) It can easily be seen that this new coordinate system ( x′,t′) moves with respect to the old one ( x,t) with a speed Bat timet=t0, and with a constant acceleration A. Since in general the value of the coeﬃcients AandBwill change from molecule to molecule, the above change of variab les must be repeated for each molecule. We assume that this can be done in a smooth manner; this may not be possible in a nonlinear system, if the characteristics depend on the solution. In the primed coordinate system, the diﬀerential equation h as exactly the same form as before (see Equation III.4), except for the following substitutions: β−→β+A(t−t0) +B c, Γ−→Γ−A c2. (III.41) 2From here on we will adopt the language of relativity and refe r to the characteristic cone of the hyperbolic equation as th e ‘light-cone’. 14Light coneLight cone t t(a) Non-causal molecule. (b) Causal molecule. FIG. 9. Causal computational molecule. In both ﬁgures, the d ashed lines represent the light-cone, and the thick solid li nes show the computational molecule. Figure ( a) shows the usual molecule that follows the motion of the grid points. Figure ( b) shows the reconnected molecule, where we pay attention to th e causal structure instead. 15✇✇✇ ✚✚✚✚✚✚✚✚✡✡✡✡✡✡ ✡ t0−∆tt0t0+ ∆t xt0−∆t xt0xt0+∆t✲✻t x FIG. 10. Causally connected grid points. In the same way, the ﬁnite diﬀerence approximation will have the same form as before (Equation III.15), except for the substitutions: β−→β+B c, Γ−→Γ−A c2, (III.42) where the term with ( t−t0) has disappeared because in this case the coeﬃcients should be evaluated at the center of the molecule where t=t0. Since the original ﬁnite diﬀerence approximation could be m ade stable as long as Equation III.37 was satisﬁed (and θ≥1/2), the analogous condition for a ﬁnite-diﬀerence scheme ad apted to the new local coordinates takes the form: 1 s2−/parenleftbigg β+B c/parenrightbigg2 ≥0. (III.43) We will say that the three given points form a proper causal molecule when the last condition is satisﬁed. In order to ﬁnd when this happens, we will start by deﬁning the eﬀective numerical light-cone of the point xt0as the region between the lines: x±(t) :=xt0+/parenleftbigg −β±1 s/parenrightbigg c(t−t0) +1 2Γc2(t−t0)2. (III.44) This numerical light-cone will coincide with the exact ligh t-cone when: ∂β ∂x= 0∂Γ ∂x=∂Γ ∂t= 0. (III.45) We will also deﬁne the axis of the numerical light-cone as the line: xa(t) :=xt0−βc(t−t0) +1 2Γc2(t−t0)2. (III.46) We will now show that if xt0+∆tandxt0−∆tare inside the numerical light-cone of xt0then the three points will form a proper causal molecule. From the deﬁnition of the nume rical light-cone we see that if xt0+∆tandxt0−∆tare inside it then xt0+∆t=xa(∆t) +D+ xt0−∆t=xa(−∆t) +D−, (III.47) with D+,D−∈/bracketleftbigg −c∆t s,c∆t s/bracketrightbigg . (III.48) The coeﬃcients AandBwill then be given by 16B=−βc+D+−D− 2 (∆t), A = Γc2+D++D− (∆t)2, (III.49) which in turn means /parenleftbigg β+B c/parenrightbigg ∈/bracketleftbigg −1 s,1 s/bracketrightbigg ,/parenleftbigg Γ−A c2/parenrightbigg ∈/bracketleftbigg −2 s(c∆t),2 s(c∆t)/bracketrightbigg . (III.50) From this it is easy to see that condition III.43 is indeed sat isﬁed, that is, the three points do form a proper causal molecule. Moreover, the absolute value of the acceleration in the new local coordinates will be bounded, and even though this doesn’t aﬀect the stability of the ﬁnite diﬀeren ce scheme, it does improve its accuracy. In order to be able to form proper causal molecules everywher e, we must guarantee that two logically distinct conditions hold. If we call the central point at t0as the “parent” and the points at t0±∆tthe “children”, then every parent must have two children and every child must have a pare nt: 1.Every parent must have two children. There must always be at least one grid point in the upper and lo wer time levels inside the numerical light-cone of any given point in the middle time level. This can be guaranteed if we ask for the distance between grid points to be smaller than th e spread of the smallest light-cone, that is, ∆x≤2c∆t max (s), (III.51) which implies 2ρmin/parenleftbigg1 s/parenrightbigg ≥1. (III.52) (Without loss of generality, we assume in this section that ∆ tand henceρare positive.) 2.Every child must have a parent. All the grid points in the upper and lower time levels must be i nside the numerical light-cone of at least one point in the middle time level. This requires that the distance between the axes of the numerical light-cones of two consecutive grid po ints must be smaller than the spread of the minimum light-cone. Let us therefore consider two consecutive grid points x1andx2=x1+ ∆x. The distance between the axis of their light-cones at the next time level is given by: d+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(xa)2(∆t)−(xa)1(∆t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (III.53) Using the deﬁnition of xawe ﬁnd that: d+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆x−c∆t/bracketleftbigg β(x2)−β(x1)/bracketrightbigg +1 2(c∆t)2/bracketleftbigg Γ (x2)−Γ (x1)/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (III.54) In the same way we ﬁnd that the distance between the axis of the light-cones at the previous time level is: d−=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆x+c∆t/bracketleftbigg β(x2)−β(x1)/bracketrightbigg +1 2(c∆t)2/bracketleftbigg Γ (x2)−Γ (x1)/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (III.55) The maximum of these two is: d=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆x+c∆t/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ(x2)−β(x1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+1 2(c∆t)2/bracketleftbigg Γ (x2)−Γ (x1)/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (III.56) Let us assume now that both βand Γ are continuous functions. Then we can expand them in a Ta ylor series around the point: 17¯x:=x1+x2 2. (III.57) We then ﬁnd that, to second order in ∆ x: d=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆x+c∆t/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂β ∂x∆x/vextendsingle/vextendsingle/vextendsingle/vextendsingle+1 2(c∆t)2/bracketleftbigg∂Γ ∂x∆x/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (III.58) where the derivatives are evaluated at the point ¯ x. The condition that the maximum value of this distance should be smaller than the spread of the minimum light-cone is now: max(d)≤2c∆t max(s), (III.59) Using now the expression for d, we can rewrite this as: 2ρmin/parenleftbigg1 s/parenrightbigg ≥/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 +ρ∆xmax/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂β ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle+1 2ρ2(∆x)2max/bracketleftbigg∂Γ ∂x/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (III.60) This is the “no orphans” condition, that every child point sh ould have a parent. Since we want this to be true for all grid points, it must hold for all x. We call equations III.52 and III.60 the causal reconnection conditions: when they are satisﬁed, one is guaranteed that causal molecules can be formed everywhere. It is clear that if the derivatives of βand Γ are too large, it will be impossible to satisfy the secon d of the causal reconnection conditions, Equation III.60 (no orphans). Th is can be avoided if we require that βand Γ change very little from one grid point to another: ∆xmax/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂β ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1 ∆ xmax/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂Γ ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1 (III.61) As we mentioned before, if this is not the case our ﬁnite diﬀer ence approximation is unlikely to be good anyway, and a more reﬁned grid spacing should be used. Another interesting feature of condition III.60 is the fact that, whenever Γ is not uniform, there will always be a value ofρlarge enough for the condition to be violated. These sets an upper bound on the time-step, which can be understood if we examine the eﬀects of a non-uniform accel eration on two adjacent numerical light-cones. If the acceleration increases with x, the numerical light-cones will eventually converge and pa ss through each other at a large enough time, even if they were diverging initially. Si milarly, if the acceleration decreases with x, the numerical light-cones will eventually diverge, even if they intersec t each other for a while. Clearly these situations do not aris e in the exact (diﬀerential) case because they would break the causal structure of the solutions. We must therefore conclude that the numerical light-cones will not approxima te the real light-cones properly when we have a time-step large enough for these eﬀects to occur. However, if Γ is such t hat III.61 holds, then the upper limit on ∆ twill be very large indeed, much larger than the Courant limit, and so large that the accuracy of the integration must be breaking down anyway. Moreover, for any given time-step con dition III.60 can always be satisﬁed for a small enough grid spacing ∆ x. 3. Numerical overheads of causal reconnection. Notice that causal reconnection does not change the fundame ntal structure of the diﬀerence equation, since it does not aﬀect the relations between points at the ﬁnal time-step . Therefore, even with causal reconnection, the equation will have the form ˆQxφj+1=f′(φj, φj−1), (III.62) wheref′is a diﬀerent function, which reﬂects the fact that causal re connection identiﬁes diﬀerent points at time-steps jandj−1 to use to generate the points at the ﬁnal time-step. Therefo re, any algorithms that are used without causal reconnection for the solution of this tridiagonal system of equations can be used equally well with causal reconnection . 18There will, of course, be an overhead associated with the sea rch for causally related grid points. In a one-dimensional problem with Ngrid points, this will require only O(N) operations, since once the causal molecule of one grid poin t has been constructed, the causal molecule of its neighbor wi ll, by continuity, usually diﬀer by at most one spatial shift at any time level. In more than one dimension, the search shou ld still scale linearly with the number of grid points, since again by continuity the causal molecule of any point ca n usually be guessed from that of any of its neighbors. We have found that for the one-dimensional wave equation, th e implementation of causal reconnection given in Appendix B can multiply the computation time by something li ke a factor of two. But for a more realistic problem, such as general relativity, where there are many dependent v ariables per grid point, the overhead of searching for the causal structure will be no diﬀerent than for the simple wave equation, so it will represent a small percentage of the overall computing time. D. Numerical examples of causal reconnection. As an example of the methods that we have developed in the last sections, we will consider a grid that is oscillating in the original coordinates ( ξ,t). The scale and shift functions are given by: s(x,t) = 1, β (x,t) =Acos(ωt), (III.63) from which we deduce Γ =Aωsin (ωt). (III.64) This grid turns out to give a very good illustration of all the properties we have mentioned so far. In the calculations we have taken c= 1. In Figure 11 we show two calculations using the ﬁnite diﬀeren ce approximation given by Equation III.15. In these examples we have not taken into consideration the causal str ucture. The ﬁgures show the evolution of a Gaussian wave packet that was originally at rest at the center of the gr id. In both cases we have taken ω= 6, and we show the situation after 35 time-steps (3.5 periods of oscillati on of the grid). In the ﬁrst graph A= 1: the maximum shift equals the speed of the waves, but for al l the rest of the time the shift is less than 1. At the end of the calculation there is no e vidence of any instabilities. In fact, we have integrated this for a very large number of time-steps with the same resul t: no instabilities appear. For the second graph we have taken A= 1.1: the maximum shift is now slightly larger than the speed of t he waves, although even here the grid spends most of its time at speeds l ess than 1. By the end of the calculation an instability has started to form. It exhibits the characteristic feature of ﬁnite-diﬀerence instabilities, that the shortest wavel engths are the most unstable. In Figure 12 we compare the direct, non-causal, approach and the causal approach for a larger shift amplitude. We use the same initial Gaussian wave packet as before and take A= 1.3,ω= 6. Here the instability appears very fast in the direct approach (after only 25 time-steps). With caus al reconnection, however, the calculation remains stable. We have carried out the same calculation for many more time-s teps, and also for larger values of A(up toA= 15), and the results are the same: no instabilities develop in the causal approach. Therefore causal reconnection of the computational molecu les seems to cure all the local instabilities on grids that move faster than the waves. We will see that in more than one di mension it will also guarantee local stability for rapid shifts, but only after we cure a further instability th at arises in operator-splitting methods for small velociti es. IV. THE MULTI-DIMENSIONAL CASE. A. How to design an ADI scheme for a hyperbolic system. 1. Fully implicit scheme. We shall begin our discussion of stable integration schemes in more than one dimension by introducing ADI schemes in a way that makes our time-symmetric ADI method emerge natu rally, and which makes it clear how to generalize it to other hyperbolic systems in a straightforward manner. ADI is basically a device for implementing an im- plicit integration scheme in many dimensions without the en ormous computational overheads that the direct implicit 19FIG. 11. Oscillating grid, non-causal approach. 20FIG. 12. Oscillating grid, causal reconnection. 21scheme would involve. We begin our discussion, therefore, w ith an examination of the direct implicit scheme and its computational demands. We shall concentrate on two dimensi ons, but the generalization to more is straightforward. The general wave equation (Equation II.3) in two dimensions is /bracketleftig gx x−(βx)2/bracketrightig∂2φ ∂x2+ 2 (gx y−βxβy)∂2φ ∂x∂y+/bracketleftig gy y−(βy)2/bracketrightig∂2φ ∂y2 +2βx c∂2φ ∂x∂t+2βy c∂2φ ∂y∂t−Γx∂φ ∂x−Γy∂φ ∂y−1 c2∂2φ ∂t2= 0. (IV.1) In the ﬁnite diﬀerence approximations to this equation, we w ill use the notation φj:=φj ix,iy; (IV.2) that is, we will suppress the spatial indices and write them o nly when the possibility of confusion arises. The ﬁnite diﬀerence approximations to all the diﬀerential operators that appear in Equation IV.1 have the same form as in the one dimensional case, except for a new term that did not exist before: (∂x∂yφ)j=δxδyφ 4 (∆x)2+Exy, (IV.3) where the spatial diﬀerences are deﬁned in the same way as bef ore, and the truncation error is: Exy=−1 6(∆x)2/bracketleftig/parenleftbig ∂3 x∂yφ/parenrightbigj+/parenleftbig ∂x∂3 yφ/parenrightbigj/bracketrightig +.... (IV.4) As we learned to do in the one-dimensional case, we will use on ly explicit approximations for the ﬁrst spatial derivatives. We can then write our second-order implicit ﬁn ite diﬀerence approximation to Equation IV.1 in the form: ρ2/bracketleftig gx x−(βx)2/bracketrightig/bracketleftbiggθ 2/parenleftbig δ2 xφj+1+δ2 xφj−1/parenrightbig + (1−θ)/parenleftbig δ2 xφj/parenrightbig/bracketrightbigg +ρ2/bracketleftig gy y−(βy)2/bracketrightig/bracketleftbiggθ 2/parenleftbig δ2 yφj+1+δ2 yφj−1/parenrightbig + (1−θ)/parenleftbig δ2 yφj/parenrightbig/bracketrightbigg +ρ2 2(gx y−βxβy)/parenleftbig δxδyφj/parenrightbig +ρβx 2/parenleftbig δxφj+1−δxφj−1/parenrightbig +ρβy 2/parenleftbig δyφj+1−δyφj−1/parenrightbig −ρ(c∆t) 2Γx/parenleftbig δxφj/parenrightbig −ρ(c∆t) 2Γy/parenleftbig δyφj/parenrightbig −/parenleftbig φj+1−2φj+φj−1/parenrightbig = 0. (IV.5) In this equation we have assumed for convenience that the spa tial increment is the same in both directions, and we have deﬁned the Courant parameter in the same way as before. A s in the one dimensional case, to arrive at the last expression we have multiplied through by (∆ t)2. The truncation error is therefore of order: EIV.5=O/bracketleftig (∆x)2(∆t)2/bracketrightig +O/bracketleftig (∆t)4/bracketrightig . (IV.6) Equation IV.5 is the most direct ﬁnite diﬀerence approximat ion to the original diﬀerential equation in 2 dimensions. We call it the “fully implicit” scheme. As in the one-dimensi onal case, it takes the form of a matrix equation ˆQ2φj+1=f(φj, φj−1). (IV.7) However, as is well-known, the numerical solution of this eq uation is considerably more time-consuming than in the one dimensional case, and the computational demands increa se very rapidly with the number of dimensions. This is due to the fact that, if we have Ngrid points in each of nspatial directions, the matrix ˆQnwill haveNnrows and columns. Most importantly, this matrix will notbe tridiagonal: it may be possible to arrange that the neares t neighbors in, say, the x-direction of any point should occupy adjacent columns, but those in other directions will be far away in another part of the matrix. The matrix will still b e sparse, but the number of operations involved in solving it may be very large indeed, in the worst case involvi ng of order N3noperations at each time-step. Even if a well-designed relaxation method is used, the number of oper ations will in general increase faster than Nn. ADI schemes oﬀer a systematic way around this problem, usual ly aﬀording considerable savings in computational eﬀort with, as we will show, no sacriﬁce in accuracy. However , while the fully implicit scheme may be expected to be as stable against grid shifts in ndimensions as in one, this is not true of ADI schemes, and we sh all have to be careful to design a stable one. 222. Designing ADI schemes: how to make the operator factoriza ble. Alternating Direction Implicit (ADI) methods [4] reduce th e numerical work involved in an n-dimensional problem by modifying the ﬁnite-diﬀerence scheme in such a way as to re place the original large sparse matrix ˆQnby one that can be factored into a product of tridiagonal matrices relat ed to ˆQxfor each spatial direction. If we assume that we have the same number Nof grid points in all directions, we will have to invert a seri es ofNn−1tridiagonal matrices of sizeN×Nfor each spatial dimension. This means that we will need only O(nNn) operations to solve the system. We see then that the number of operations for the ADI scheme wi ll scale with the number of grid points in the same way as it does for an explicit method. The reason that one can contemplate replacing the original o perator ˆQnwith a diﬀerent one is that the fully implicit ﬁnite-diﬀerence equation is only an approximation to the di ﬀerential equation, so if we modify it by adding extra high-order terms that are of the same order as those neglecte d in the original approximation, the accuracy of the scheme will not be aﬀected. If we can then choose these extra t erms to change the operator acting on the function at the last time level φj+1into a factorizable one, we will have speeded up the solution by a huge amount. For our two-dimensional wave equation, the operator acting onφj+1is (see Equation IV.5): ˆQ2:=−1 +ρ 2(βxδx+βyδy) +ρ2θ 2/braceleftig/bracketleftig gx x−(βx)2/bracketrightig δ2 x+/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightig . (IV.8) We want to add high-order terms to this expression to transfo rm it into a product of one-dimensional operators of the form of the similar term we had in the one-dimensional cas e (Equation III.15 with θ2= 0 )3: ˆQ′ 2=ˆQxˆQy :=−/braceleftbigg 1−ρβx 2δx−ρ2θ 2/bracketleftig gx x−(βx)2/bracketrightig δ2 x/bracerightbigg ×/braceleftbigg 1−ρβy 2δy−ρ2θ 2/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightbigg . (IV.9) Let us deﬁne ˆSto be the diﬀerence between these operators: ˆS:=ˆQ′ 2−ˆQ2. (IV.10) Then we can rearrange the fully implicit equation (Equation IV.7) to read ˆQ′ 2φj+1=f(φj, φj−1) +ˆSφj+1. (IV.11) Now, this is not directly any help, since although we have the factorizable operator ˆQ′ 2on the left-hand-side, we have unknown terms in φj+1on the right. However, let us consider the following related equation: ˆQ′ 2φj+1=f(φj, φj−1) +ˆSφj. (IV.12) This equation isin a form that can be solved easily, since the unknown φj+1appears only with the operator ˆQ′. Moreover, since in the limit ∆ t→0 we haveφj→φj+1, in that limit Equation IV.12 approaches Equation IV.11, which is the original fully implicit equation. This fully im plicit approximation is itself only a valid approximation to the original diﬀerential equation in the same limit, so it follows that Equation IV.12 also approximates the diﬀer- ential equation in that limit. (It need not be as good an appro ximation, of course: the error terms of the original Equation IV.11 may be smaller than those introduced by the ch ange to Equation IV.12. We will address this point below.) There is nothing unique about changing ˆSφj+1toˆSφjto make the equation factorizable. One could change ˆSφj to any combination of terms that limits to ˆSφj+1as ∆t→0. The diﬀerent ADI methods make diﬀerent choices of 3This form of the factorized operator is not unique, there are many diﬀerent operators that one could choose instead of ˆQxˆQy. See for example [5] and [6]. 23these terms. We shall see that the standard choices produce e quations that are very unstable when the grid shifts, but that by imposing the simple physical requirement of time -reversibility one gets a uniquely deﬁned ADI scheme that is stable and just as accurate as the original fully impl icit method. It will be helpful to write out explicitly what the operator ˆSdeﬁned in Equation IV.10 is: ˆS=−/bracketleftbiggρ2 4βxδx(βyδy) +ρ3θ 4/bracketleftig βxδx/braceleftig/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightig +/bracketleftig gx x−(βx)2/bracketrightig δ2 x(βyδy)/bracketrightig +ρ4θ2 4/bracketleftig gx x−(βx)2/bracketrightig δ2 x/braceleftig/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightig/bracketrightbigg . (IV.13) It is important to note here that the expression for the opera torˆSincludes terms in which the diﬀerence operators in thexdirection act on the functions gyyandβy, because these functions in general depend on both xandy. Had we deﬁned ˆQ′ 2asˆQyˆQxwe would have had a diﬀerent expression for ˆS, because, whenever the coeﬃcients in the ﬁnite diﬀerence approximation change with position, th e operators ˆQxandˆQydo not commute. 3. ADI schemes old and new. We ﬁrst cast the two original, and still standard, ADI method s for the wave equation, Lees’ ﬁrst and second methods, into the notation we have used above. They serve to i llustrate how our approach to ADI methods works on a familiar method, and we will subsequently analyze the st ability of these schemes. Then we will introduce the scheme that will turn out to be stable for shifting grids, the time-symmetric scheme. a. Lees’ ﬁrst method. The most straightforward approach is that introduced by Lee s in 1962 ( [7], [8]) for the case of the ordinary wave equation on a ﬁxed grid. It is convenient to describe it in terms of the extra terms that one adds to the left-hand-side of Equation IV.5 to produce a factoriz able equation: ˆS/parenleftbig φj+1−φj−1/parenrightbig . (IV.14) This eﬀectively produces the equation ˆQ′ 2φj+1=f(φj, φj−1) +ˆSφj−1. (IV.15) This is a simple change from Equation IV.12. It is clear that as ∆ t→0 the extra term (IV.14) will vanish, and we will recover the o riginal diﬀerential equation. However, we will see below that the extra terms do not vanish a s fast as the errors in Equation IV.5, which are given in Equation IV.6. Lees’ ﬁrst method is only ﬁrst-order accur ate on a shifting grid. It is known that Lees’ ﬁrst method is absolutely stable on a static grid. However, it will turn o ut to be subject to weak but signiﬁcant instabilities when used on shifting grids. b. Lees’ second method. Another way to modify the equation is to add instead a second- time-diﬀerence term. This is known as Lees’ second method: ˆS/parenleftbig φj+1−2φj+φj−1/parenrightbig . (IV.16) Here again we recover the original diﬀerential equation in t he limit ∆t→0. The result is in fact a linear combination of Equations (IV.12) and (IV.15). We will see below that this method does not sacriﬁce accuracy: the introduced terms are of the same order as the original truncation error o n shifting grids. Moreover, it is absolutely stable on a static grid. However, our stability analysis will reveal t hat this method shows strong instabilities when used on shifting grids. c. The time-symmetric ADI method. A more general approach would be to separate the operator ˆSinto diﬀerent pieces and use a ﬁrst time diﬀerence with some of them, and a se cond time diﬀerence with the rest. We will call this a mixed ADI method. One can try many diﬀerent mixed methods, b ut there is one that is natural from the point of view of the original diﬀerential equation. This is the one we call the time-symmetric ADI method. The original diﬀerential equation, Equation IV.1, has, in c ommon with all fundamental physics diﬀerential equations, the property of time-reversal invariance. In this case, the equation is invariant if we make the replacements t−→ −tandβi−→ −βi. (IV.17) 24The fully implicit diﬀerence equation, Equation IV.5, also has this property, since the approximations used for the tim e- derivatives are centered diﬀerences: they do not bias the di rection of time. In our case, time-reversal is implemented by the exchange of the time-step indices j+ 1 andj−1. However, replacing the fully implicit operator ˆQ2with ˆQ′ 2breaks this invariance, because this change modiﬁes the way thatφj+1enters the equation without automatically modifying the φj−1terms in a symmetrical way. If we look at the deﬁnition of the operator ˆSin Equation IV.13, we see that it is not itself invariant: it c ontains terms both linear and quadratic in βi. Therefore, since Lees’ ﬁrst and second schemes both add ter ms in which ˆS operates on an expression with a deﬁnite time-symmetry, nei ther scheme is time-reversal invariant. What we need to do is to separate ˆSinto parts ˆSeandˆSothat are even and odd with respect to βiand then to allow them to act, respectively, on even and odd extra terms. That is, we add to E quation IV.5 the term ˆSe/parenleftbig φj+1−2φj+φj−1/parenrightbig +ˆSo/parenleftbig φj+1−φj−1/parenrightbig , (IV.18) where the even and odd parts of the operator ˆSare deﬁned by ˆSe:=−ρ2 4βxδx(βyδy)−ρ4θ2 4/bracketleftig gx x−(βx)2/bracketrightig δ2 x/braceleftig/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightig , (IV.19) ˆSo:=−ρ3θ 4/bracketleftig βxδx/braceleftig/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightig +/bracketleftig gx x−(βx)2/bracketrightig δ2 x(βyδy)/bracketrightig . (IV.20) This eﬀectively ensures that we apply the same modiﬁcation t o theφj−1terms as to the φj+1terms in producing a factorizable equation that limits to the fully implicit eq uation as the time-step goes to zero. We will see that, by so preserving the time-symmetry of the original equation, w e have also produced a method that is just as accurate as the fully implicit method and, perhaps more importantly, is unconditionally locally stable on grids shifting at any speed up to the wave speed. 4. Intermediate values and the implementation of ADI scheme s. Whichever ADI method we choose to use, we will always produce an equation of the form: ˆQ′ 2φj+1=ˆAφj+ˆBφj−1, (IV.21) where ˆAandˆBare spatial ﬁnite diﬀerence operators whose speciﬁc form wi ll depend on the method chosen. Looking at the deﬁnition of ˆQ′ 2in Equation IV.9, we see that the last equation can be decompo sed into a system of two coupled equations in the following way: /braceleftbigg 1−ρβy 2δy−ρ2θ 2/bracketleftig gy y−(βy)2/bracketrightig δ2 y/bracerightbigg φj+1:=φ∗j+1, (IV.22) /braceleftbigg 1−ρβx 2δx−ρ2θ 2/bracketleftig gx x−(βx)2/bracketrightig δ2 x/bracerightbigg φ∗j+1=ˆAφj+ˆBφj−1, (IV.23) where the ﬁrst equation deﬁnes the so-called intermediate value φ∗j+1. These two equations give the simplest ADI split of the ﬁnite d iﬀerence approximation. In order to solve the system, one ﬁrst solves the second equation for φ∗j+1using values of φin the previous time levels. This operation involves solving a tridiagonal system of equations for each ﬁxed valu e of they-index. One then solves for φj+1using the ﬁrst equation, again solving only tridiagonal equations. In the general case of an ndimensional problem, this procedure will take us to a system of nequations and n−1 intermediate values. Each equation employs an operator ac ting only in one of the spatial directions. It is important to realize that the splitting of Equation IV. 21 given above is by no means unique. One may ﬁnd many diﬀerent splittings of the same equation, and some may p rove to be more computationally eﬃcient than the one we have given above. However, the diﬀerences will only be in t he algebra (and in roundoﬀ errors): diﬀerent splittings are only diﬀerent ways of writing the same ADI scheme. 255. Accuracy of ADI methods. The diﬀerent methods of forming ˆAandˆBwill diﬀer in general in their accuracy and stability. To ﬁnd the accuracy of the diﬀerent ADI schemes on shifting grids, we start by con sidering Lees’ ﬁrst method. In this case we must add to the left hand side of Equation IV.5 the following term: ˆS/parenleftbig φj+1−φj−1/parenrightbig =−ρ2 4βxδx/bracketleftbig βyδy/parenleftbig φj+1−φj−1/parenrightbig/bracketrightbig −ρ3θ 4/braceleftig βxδx/bracketleftig/parenleftig gy y−(βy)2/parenrightig δ2 y/parenleftbig φj+1−φj−1/parenrightbig/bracketrightig +/parenleftig gx x−(βx)2/parenrightig δ2 x/bracketleftbig βyδy/parenleftbig φj+1−φj−1/parenrightbig/bracketrightbig/bracerightig −ρ4θ2 4/parenleftig gx x−(βx)2/parenrightig δ2 x/bracketleftig/parenleftig gy y−(βy)2/parenrightig δ2 y/parenleftbig φj+1−φj−1/parenrightbig/bracketrightig . (IV.24) The order of this term is found by replacing diﬀerences with d erivatives: ˆS/parenleftbig φj+1−φj−1/parenrightbig ≈ −2c2(∆t)3βx∂ ∂x/bracketleftbigg βy∂2φ ∂y∂t/bracketrightbigg −θc3(∆t)4/braceleftbigg βx∂ ∂x/bracketleftbigg/parenleftig gy y−(βy)2/parenrightig∂3φ ∂y2∂t/bracketrightbigg +/parenleftig gx x−(βx)2/parenrightig∂2 ∂x2/bracketleftbigg βy∂2φ ∂y∂t/bracketrightbigg/bracerightbigg −θ2 2c4(∆t)5/parenleftig gx x−(βx)2/parenrightig∂2 ∂x2/bracketleftbigg/parenleftig gy y−(βy)2/parenrightig∂3φ ∂y2∂t/bracketrightbigg =O/parenleftig (∆t)3/parenrightig +O/parenleftig (∆t)4/parenrightig +O/parenleftig (∆t)5/parenrightig . (IV.25) From this we can see that the principal part of the truncation error introduced by the new terms is of order (∆ t)3, which is in fact one order less than the original accuracy of E quation IV.5. Using Lees’ ﬁrst ADI decomposition reduces the accuracy of the original scheme. This is only tru e, however, when we consider a moving grid. From the last expression it is clear that for a ﬁxed grid ( βx=βy= 0), the truncation error introduced by this method will only be of order (∆ t)5, as is well known. When we do the same analysis for the case of the ADI scheme base d onLees’ second method , we ﬁnd that the principal part of the truncation error introduced by the new terms is of order (∆ t)4for a shifting grid. The accuracy of the original equation is therefore preserved. In princip le, one would therefore prefer Lees’ second method. However , we will see below that the second method is far more unstable t han the ﬁrst when the grid shifts, so its higher accuracy is of limited usefulness. For the time-symmetric ADI scheme , the principal part of the truncation error introduced by th e extra terms is again of order (∆ t)4. This method is thus as accurate as the fully implicit one. We will see that it is also stable. B. Local stability analysis. We turn now to the all-important question of the stability of the ADI schemes that we have described in the last section. In the same way as in the one dimensional case, we wil l start by studying the nature of the solutions of the diﬀerential equation (Equation IV.1), and we will again con sider the solutions in a very small region around the point/parenleftbig xi,t/parenrightbig , assuming that the coeﬃcients remain constant in this regio n. Moreover, for simplicity we will assume that we can take the f unctionsgyyandβyout of the diﬀerence operators in the expression for ˆS(Equation IV.20). Again, if these functions change rapidly from one grid point to the next, the accuracy of the ﬁnite-diﬀerence scheme on this grid is pr obably poor anyway. 1. Solutions of the diﬀerential equation. Following the same procedure as in the one-dimensional case , we will look for solutions of the diﬀerential equation Equation IV.1 that take the form: 26φ(x,t) =eı α(/vectork)teı/vectork·/vector x, (IV.26) where/vectorkis the 2-dimensional wave vector. We denote its components b yki, and deﬁne the associated covector (one-form) components kiby ki=gijkj. (IV.27) Substituting into Equation IV.1 and solving for α, we ﬁnd the dispersion relation: α±=c/parenleftbig kjβj/parenrightbig ±c/bracketleftbig/parenleftbig kjkj/parenrightbig +i/parenleftbig kjΓj/parenrightbig/bracketrightbig1/2. (IV.28) This is a simple generalization of Equation III.20. Again we have the analytic boundedness condition /vextendsingle/vextendsingleeıα+teıα−t/vextendsingle/vextendsingle2= 1. (IV.29) Therefore, if one of the solutions is growing, the other is dy ing at the same rate. 2. Local stability of the diﬀerence equations. We will now proceed to the local stability analysis of the diﬀ erent numerical approximations. We look for numerical solutions to the ﬁnite diﬀerence approximations of the foll owing form: φm nxny=ψmei(kxnx+kyny) ∆x, (IV.30) where we have used our simplifying assumption that ∆ y= ∆x. By substituting this equation into any of the ﬁnite diﬀerence approximations, we shall always obtain a quadrat ic equation for ψof the form: Aφ2+Bψ+C= 0. (IV.31) The coeﬃcients in this equation will depend on the particula r approximation used. We call the two solutions of this equationψ±. As in the one-dimensional case, we deﬁne the numerical ampli ﬁcation measure: MNum:= max /vectork/parenleftig \|ψ+\|2,\|ψ−\|2/parenrightig , (IV.32) and we take our local stability condition to be: MNum≤MAna. (IV.33) One general consideration applies to all diﬀerence schemes . It is not diﬃcult to see that the analytic boundedness condition Equation IV.29 will also hold in the ﬁnite diﬀeren ce case when the following condition on the coeﬃcients in Equation IV.31 is satisﬁed: \|A\|=\|C\|. (IV.34) a. Lees’ ﬁrst scheme. Lees showed that his methods are stable for all time-steps if θ≥1/2, for a static grid. In the shifting case, the coeﬃcients of the quadratic equation for Lees’ ﬁrst method are: A=/braceleftig 1−iρβxsin(kx∆x)−ρ2θ/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1]/bracerightig ×/braceleftig 1−iρβysin (ky∆x)−ρ2θ/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig , (IV.35) B=−/parenleftig 2ρ2(1−θ)/braceleftig/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1] +/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig −2ρ2(gxy−βxβy) sin (kx∆x) sin (ky∆x) −iρ2∆x[Γxsin (kx∆x) + Γysin (ky∆x)] + 2/parenrightbig , (IV.36) 27C=−/parenleftig/braceleftig 1−iρβxsin(kx∆x)−ρ2θ/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1]/bracerightig ×/braceleftig 1−iρβysin (ky∆x)−ρ2θ/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig + 2ρ2θ/braceleftig/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1] +/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig −2/parenrightig . (IV.37) It is clear that these coeﬃcients do not satisfy the boundedn ess condition Equation IV.34. We shall test the stability condition given by Equation IV.3 3 on Lees’ ﬁrst method by numerically calculating the ampliﬁcation measure from the roots of Equation IV.31. For s implicity, we consider the case where: gxx=gyy= 1, gxy= 0. Γi= 0.  (IV.38) We do not believe this restricts the generality of our conclu sions: from our analysis of the one-dimensional case, the restriction on Γishould not cause a problem, and the particular values of the m etric tensor are unlikely to have a determining eﬀect on stability. The results of our local stability analysis appear in Figure 13, where we show the following region of the shift vector space: βx, βy∈(0,1.2). Since/vextendsingle/vextendsingle/vextendsingle/vectorβ/vextendsingle/vextendsingle/vextendsingle= 1 corresponds to a grid shifting with a speed c, the region considered in the graphs will include grids that shift faster than the waves. We have considered 50 ×50 uniformly spaced values of the shift vector inside this region. For a given point in the shift vector space, we ﬁnd the maximum value of the quantity: R:=MNum MAna, using 10 ×10 diﬀerent values of the wave vector /vectork: kx, ky∈(0,2π), and, for each wave vector, 100 diﬀerent values of the Courant parameter ρin the interval (0 ,10). Having found Rmaxwe plot its value on a logarithmic scale. In the graphs, value s of log10(Rmax) smaller than or equal to 0 ( Rmax≤1) are represented by clear regions, and values larger than 2 (Rmax≥100) by the darkest regions. It is not diﬃcult to see that the clear regions will c orrespond to values of the shift vector for which the ﬁnite diﬀerence scheme is locally stable (at least for ρ∈(0,10)), and dark regions to values of the shift that give rise to instabilities. The darker the region, the more violent the i nstabilities. Finally, the solid arc in the graphs marks the end of the light-cone. It is important to note that the presence of a dark region does not mean that for the given value of ! pvecβ the scheme will be unstable for all ρ∈(0,10), but only that we must expect instabilities for at least s ome values of the ρin that interval. In the upper graph in Figure 13, we show the case where θ= 1/4. The ﬁnite diﬀerence scheme is unstable for at least some value of ρat all values of the shift vector. This instability becomes m uch stronger whenever one of the components of the shift vector is greater than the speed of th e waves. We recall that even in the one-dimensional case, the implicit scheme for θ= 1/4 is only conditionally stable, so the behavior here is no sur prise. This ﬁgure also shows how the introduction of an operator splitting has broken the rotational symmetry of our problem: it is no longer the light-cone which is the important feature, but th e rectangular region in which the light-cone is inscribed. The lower graph corresponds to the case θ= 1/2, which is unconditionally stable in the one-dimensional c ase. In two dimensions, the scheme is still locally stable for value s of the shift vector along the direction of the coordinate axis, just as the 1-D scheme was. However, instabilities app ear for speeds in other directions. These instabilities are weak compared to those for speeds faster than the wave speed, but their presence will nevertheless be signiﬁcant, as we will show in the examples of numerical integrations that w e give below. We have looked at larger values of the parameter θ, but the situation doesn’t improve beyond θ= 1/2. Lees’ ﬁrst scheme is therefore not very useful for grid speeds that are n ot aligned with the coordinate axis. 28FIG. 13. Stability for a method of Lees’ ﬁrst type. 29b. Lees’ second scheme. We next turn to Lees’ second method, for which the coeﬃcients of the quadratic equation are: A=/braceleftig 1−iρβxsin (kx∆x)−ρ2θ/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1]/bracerightig ×/braceleftig 1−iρβysin (ky∆x)−ρ2θ/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig , (IV.39) B=−/parenleftig 2ρ2(1−θ)/braceleftig/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1] +/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig −2ρ2gxysin (kx∆x) sin (ky∆x)−iρ2∆x[Γxsin (kx∆x) + Γysin (ky∆x)] + 2iρ3θ/braceleftig βx/bracketleftig gyy−(βy)2/bracketrightig sin (kx∆x) [cos(ky∆x)−1] +βy/bracketleftig gxx−(βx)2/bracketrightig sin (ky∆x) [cos(kx∆x)−1]/bracerightig + 2ρ4θ2/bracketleftig gxx−(βx)2/bracketrightig/bracketleftig gyy−(βy)2/bracketrightig [cos(kx∆x)−1][cos(ky∆x)−1] + 2/parenrightig , (IV.40) C=−/parenleftig ρ2θ/braceleftig/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1] +/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig −iρ[βxsin (kx∆x) +βysin (ky∆x)] +ρ2βxβysin (kx∆x) sin (ky∆x) −iρ3θ/braceleftig βx/bracketleftig gyy−(βy)2/bracketrightig sin(kx∆x) [cos(ky∆x)−1] +βy/bracketleftig gxx−(βx)2/bracketrightig sin (ky∆x) [cos(kx∆x)−1]/bracerightig −ρ4θ2/bracketleftig gxx−(βx)2/bracketrightig/bracketleftig gyy−(βy)2/bracketrightig [cos(kx∆x)−1] [cos(ky∆x)−1] −1/parenrightig . (IV.41) Again, these do not satisfy the boundedness condition Equat ion IV.34. In Figure 14 we again portray the cases for θ= 1/4 andθ= 1/2 . The situation is even worse than before: the instabilitie s in Lees’ second scheme grow faster than for the ﬁrst scheme, and even for θ >1/2 there is only a very small region of stability just around th e origin. Clearly, this scheme will not be practical for any moving gri d. c. The time-symmetric scheme. We have found that both standard ADI methods become unstable when the reference frame is moving. Neither satisﬁes the symmetry co ndition Equation IV.34. Now we look in the same way at the time-symmetric scheme. The fact that this scheme does in deed satisfy Equation IV.34 can readily be seen from the form that the coeﬃcients of the quadratic equation take i n this case: A=/braceleftig 1−iρβxsin(kx∆x)−ρ2θ/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1]/bracerightig ×/braceleftig 1−iρβysin (ky∆x)−ρ2θ/bracketleftig gyy−(βy)2/bracketrightig ([cos(ky∆x)−1]/bracerightig , (IV.42) B=−/parenleftig 2ρ2(1−θ)/braceleftig/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1] +/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig −2ρ2gxysin (kx∆x) sin (ky∆x)−iρ2∆x[Γxsin(kx∆x) + Γysin (ky∆x)] + 2ρ4θ2/bracketleftig gxx−(βx)2/bracketrightig/bracketleftig gyy−(βy)2/bracketrightig [cos(kx∆x)−1][cos(ky∆x)−1] + 2/parenrightig , (IV.43) C=/braceleftig 1 +iρβxsin(kx∆x)−ρ2θ/bracketleftig gxx−(βx)2/bracketrightig [cos(kx∆x)−1]/bracerightig ×/braceleftig 1 +iρβysin (ky∆x)−ρ2θ/bracketleftig gyy−(βy)2/bracketrightig [cos(ky∆x)−1]/bracerightig . (IV.44) Figure 15 shows the local stability analysis for this scheme , where again we show what happens for θ= 1/4 and θ= 1/2. 30FIG. 14. Stability for a method of Lees’ second type. 31FIG. 15. Stability for the time-symmetric scheme. 32For the ﬁrst case, the situation is no better than before: the scheme is unstable for practically every value of the shift vector. However, when we set θ= 1/2, the value that gave absolute stability in the one-dimensi onal case, the scheme becomes locally stable for every value of the shift vector inside the rectangular re gion that inscribes the light-cone . We ﬁnd that this stability is maintained for larger values of θ. What we see here is eﬀectively a ‘light-cone stability condi tion’, except for the fact that instead of a cone we now have a rectangle, as a consequence of the fact that the ADI spl itting breaks the rotational symmetry of the problem. In the general case, this local stability condition can be ex pressed in the following way: gi i−/parenleftbig βi/parenrightbig2≥0 (no sum) , (IV.45) whereimay refer to any spatial direction. The time-symmetric scheme has also the important property t hat in the stable region the numerical solutions will always be non-dissipative (at least in the non-acceleratin g case), that is: max/parenleftbiggMNum MAna/parenrightbigg =min/parenleftbiggMNum MAna/parenrightbigg = 1 This can be easily proved from the fact that this scheme satis ﬁes condition IV.34: If the scheme has a dissipative so- lution with MNum< M Ana, then condition IV.34 together with the analytic boundedne ss condition (Equation IV.29) implies that it must also have an unstable solution with MNum> M Ana. With the time-symmetric scheme we have then found what we are looking for: an absolutely locally stable, second- order accurate ADI decomposition for the ﬁnite diﬀerence ap proximation to the wave equation in a reference frame moving at any speed up to the wave speed in any direction. This scheme can be easily generalized to any number of spatial dimensions, as can be seen from its deﬁnition in the l ast section. The restriction to frames moving slower than the wave speed i s expected, of course. To remove it, we now deﬁne causal reconnection for the 2-dimensional case, using the t ime-symmetric ADI scheme as our starting point. C. Causal reconnection in 2 dimensions. In the last section we found that the time-symmetric ADI sche me had stability properties superior to both the schemes of Less’ ﬁrst and Lees’ second types. However, even f or the time-symmetric scheme, instabilities appear as soon as one of the components of the grid speed becomes larger than the speed of the waves. In the one dimensional case, we saw that this instability could be avoided if we used a computational molecule based on the causal structure of the wave equation, and not in the motion of the individual g rid points. We now want to generalize this approach to the two dimensional case. We will again look for a computational molecule that guarant ees that the light-cone is properly represented in the immediate vicinity of the central point. For the moment we wi ll assume that we have already found the points that form such a molecule. We then introduce the local coordinate system {xi′,t′}adapted to the causal molecule as a direct generalization of the one dimensional case: xi′:=/parenleftbig xi−xi t0/parenrightbig −Pi(t), t′:=t,  (IV.46) where {xi t0,t0}are the coordinates of the central point of the molecule, and where Pi(t) =Ai(t−t0)2 2+Bi(t−t0) +xi 0, (IV.47) Ai:=/parenleftigg xi t0+∆t−2xi t0+xi t0−∆t (∆t)2/parenrightigg , (IV.48) Bi:=/parenleftigg xi t0+∆t−xi t0−∆t 2 ∆t/parenrightigg . (IV.49) In the new coordinate system, the wave equation has the same f orm as before, except for the substitutions: 33βi−→βi+Ai(t−t0) +Bi c, Γi−→Γi−Ai c2. (IV.50) Since in the ﬁnite diﬀerence approximation the coeﬃcients s hould be evaluated at the center of the molecule, the above expressions will reduce to: βi− →βi+Bi c, Γi−→Γi−Ai c2. (IV.51) We know that the original ﬁnite diﬀerence approximation was locally stable as long as Equation IV.45 was satisﬁed. This implies that the approach based on the reconnected mole cule will be stable if gi i−/parenleftbigg βi+Bi c/parenrightbigg2 ≥0 ∀i. (IV.52) As in the one-dimensional case, we will say that the given thr ee points form a proper causal molecule if the last condition is satisﬁed. We will now deﬁne a generalization to two dimensions of the co ncept of eﬀective numerical light-cone. We do this by deﬁning ﬁrst the axis of this numerical light-cone as the l ine: xi a(t) :=xi t0−βic(t−t0) +1 2Γic2(t−t0)2. (IV.53) The numerical light-cone will then be deﬁned by taking at eac h time the region covered by a rectangle that is centered at the axis and has sides: 2c∆t/parenleftbig gi i/parenrightbig1/2. (IV.54) With this deﬁnition, the numerical “light-cone” is really a prism and not a cone. It is not diﬃcult to prove now that if the points xi t0−∆tandxi t0+∆tare inside the numerical light-cone of xi t0, then we will have: /parenleftbigg βi+Bi c/parenrightbigg ∈/bracketleftigg −/parenleftbig gi i/parenrightbig1/2,/parenleftbig gi i/parenrightbig1/2/bracketrightigg , (IV.55) /parenleftbigg Γi−Ai c2/parenrightbigg ∈/bracketleftigg −2/parenleftbig gi i/parenrightbig1/2 (c∆t),2/parenleftbig gi i/parenrightbig1/2 (c∆t)/bracketrightigg . (IV.56) This means that the three points do form a proper causal molec ule, and also that the acceleration in the new local coordinates will be bounded. As in the one-dimensional case, we can guarantee that proper causal molecules can be formed everywhere if we ask for two conditions: 1.Every parent has at least two children. There must always be at least one grid point in the upper and lo wer time levels inside the numerical light-cones of all points i n the middle time level. It is easy to see that this will require: 2ρmin/parenleftbig gi i/parenrightbig1/2≥1 ∀i. (IV.57) 2.Every child has a parent. All the grid points in the upper and lower time levels must be i nside the numerical light-cone of at least one point in the middle time level. We g uarantee this by asking that the light-cones of the points in the middle time level should cover completely the u pper and lower time levels, in other words that the union of the intersections of these light-cones with both th e upper and lower levels should be the entire grid. Let us consider a square of nearest neighbours in the middle t ime level. We want their numerical light-cones to cover the whole quadrilateral area deﬁned by the points wher e the axis of those light-cones intersect the adjacent time levels. A suﬃcient condition for this to happen is to ask for the sides of this quadrilateral to be smaller than the spread of the smallest light-cone divided by√ 2 (this factor arises from the fact that the diagonals of a square are√ 2 times larger than its sides). Following now the same procedure as before, we can show that t his condition takes the form: 342√ 2ρmin/parenleftbig gi i/parenrightbig1/2≥/braceleftigg 1 + 2 max ( d11,d22) +/bracketleftbigg max (d11,d12)/bracketrightbigg2 +/bracketleftbigg max(d21,d22)/bracketrightbigg2/bracerightigg1/2 ∀i. (IV.58) where the quantities djkare deﬁned as: djk=ρ/parenleftigg ∆x/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂βj ∂xk/vextendsingle/vextendsingle/vextendsingle/vextendsingle+(∆x)2 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2βj ∂x1∂x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightigg +ρ2(∆x)2 2/parenleftbigg∂Γj ∂xk/parenrightbigg (IV.59) and the maximum should be taken over all values of xi. As in the one-dimensional case, the last condition is valid only to second order in ∆ x. Conditions IV.57 and IV.58 are the causal reconnection conditions in the two dimensional case. They will guarantee that proper causal molecules can always be formed. D. Numerical examples. To test the ﬁnite diﬀerence methods that we have developed, w e will consider two diﬀerent situations: a grid moving with a uniform speed, and a grid rotating with constant angul ar velocity. 1. Uniformly moving grid. We will ﬁrst study the case of the grid moving with uniform spe ed, in order to show the advantages of the time- symmetric scheme. If the grid is moving with velocity /vector v= (vx,vy), it is not diﬃcult to see that: gi j=δi j (IV.60) βi=vi/c,and (IV.61) Γx= Γy= 0. (IV.62) Using these values for the coeﬃcients, we have studied the nu merical solution to the wave equation for a number of examples, comparing the three diﬀerent ADI methods develop ed earlier. The ﬁrst set of graphs (Figure 16) show the result of one such calculation for a scheme of Lees’ ﬁrst type . In the graphs we show the grid region [(0 ,10)×(0,10)], and we calculate the evolution of a Gaussian wave packet orig inally at rest at the point (7 ,7). For simplicity, we have imposed reﬂecting boundaries. We have taken a time-step suc h thatρ= 1, which means that we are well beyond the Courant limit.4The evolution is followed using a grid with a speed given by: /vector v= (1 2,1 2),\|/vector v\|= 0.707<1, where we have taken c= 1. We see how an instability is beginning to grow even though the grid is moving slower than the wave speed. This is precisely in accordance with the results of our local stab ility analysis. This instability grows slowly, as expected . Nevertheless, it is clear that its presence is unacceptable in a calculation of any duration. If we use a method of Lee’s second type, the instability takes longer to develop, but once it appears it grows very fast, much faster that with Lees’ ﬁrst method. The fact that the instabilities in ge neral take longer to appear with Lees’ second method can be traced to the particular wave modes that are involved. As we can see in the graphs, the instabilities in Lees’ ﬁrst method are associated with relatively long wavelength s (several grid points), and since these modes are already present in the initial data, they start growing right away. I n Lees’ second method, however, the instabilities turn out 4In andimensional problem, the Courant limit for the stability of an explicit scheme is ρ= 1/√n. 35FIG. 16. Uniform shift vector: Lees’ ﬁrst scheme. 36to be associated with very short wavelengths (one or two grid points), which do not contribute signiﬁcantly to the initial data. These means that, even though the instabiliti es are more violent with this method, it will take a long time for the unstable modes to grow to the scale of the real sol ution. In the next set of graphs (Figure 17) we have applied the time- symmetric scheme to the same problem. The instability has completely disappeared. This is again in ag reement with our previous conclusions, and shows the superiority of this method. We have performed similar calculations for many diﬀerent va lues of the grid speed and we have found the essentially similar results. The time-symmetric scheme remains stable as long as the grid moves slower than the waves, while the other schemes present instabilities for quite small gri d velocities. 2. Rotating grid. In order to show the advantage of causal reconnection of the c omputational molecules when the grid moves very fast, we will consider now an example with a grid rotating wit h the constant angular velocity ω. It is not diﬃcult to show that: gi j=δi j (IV.63) βx=−ω cy, βy=ω cx, (IV.64) Γx=−ω2 c2x, Γy=−ω2 c2y. (IV.65) To test for local stability when using causal reconnection, we take ω= 0.25 in units in which c= 1 and the grid extends over the range [( −5,5)×(−5,5)]. This means that the centers of the edges of the grid will be moving faster than the wave speed, wi th a linear velocity of 1 .25, while the corners will be going even faster. The next graphs show the results of a time-symmetric calcula tion, ﬁrst using a “direct” calculation (ﬁxed compu- tational molecule) and second using causal reconnection. A gain we use a time-step such that ρ= 1. In Figure 18, we show the evolution of a Gaussian wave packet originally at re st at the center of the grid, using the direct approach. We see how after 32 time-steps an instability has appeared cl ose to the boundaries. Only ﬁve time-steps later, this instability has grown so large that the original wave is no lo nger visible (the scale is automatically adjusted to displa y the largest value of the function). Figure 19 shows the same calculation using causal reconnect ion. The instability is not present. In fact, we have done the same calculation with much larger values of the angu lar velocity (up to ω= 3.0, where the edge is travelling at 15 times the wave speed), and the scheme remains locally st able. These examples demonstrate dramatically that time-symmet ric ADI can be married with causal reconnection, and that together the two techniques provide a robust diﬀerence approximation to the wave equation on a moving grid. These methods are stable, oﬀer all the computational advant ages of ADI schemes, and remain second order accurate in ∆xand ∆t. A comment on how to enforce causal reconnection at the bounda ries seems in order here. In all our examples we have taken the practical approach of setting the value of the wave function to zero whenever a complete causal molecule cannot be formed. This can happen not only at the boundaries, but also at inner points close to the boundaries for large enough grid speeds. The philosophy behind this approa ch is simple: If the causal molecule is incomplete, then we would need information from outside the grid to evolve the wave function at that place. If we impose the condition that no information can come from the outside, then we must ta ke the value of the wave function as zero at that point. This requirement can be relaxed somewhat be using an o utgoing wave boundary condition whenever we can still ﬁnd a causally related point in the previous time level , but not before that. At places where one can’t even ﬁnd a causally related point in the previous time level, the only legitimate thing one can do is to set the value of the wave function to zero. 37FIG. 17. Uniform shift vector: time-symmetric scheme. 38FIG. 18. Rotating grid: non-causal approach. 39FIG. 19. Rotating grid: causal reconnection. 40V. CONCLUSIONS. The wave equation we have studied here is a prototype for more complex equations of mathematical physics, such as the Einstein ﬁeld equations. In fact, many hyperbolic syste ms in mathematical physics can be formulated in terms of the wave operator. One would expect the instabilities we hav e found here to be generic: any numerical approximation to a hyperbolic system on a shifting grid should exhibit them . Only experience will show us just how well our cures for these generic instabilities transfer to more interesting equations. However, the instabilities we have described he re are cured by the application of two clear physical principles, causality and time-reﬂection invariance. It s eems clear that it would be asking for trouble notto incorporate these principles into the design of algorithms for the numer ical integration of any fundamental physical equations. We have, of course, studied in detail only one second-order d iﬀerential equation in one and two dimensions. The restriction to two dimensions is not important. The physica l principles involved do not depend on the number of dimensions, and the savings obtained by using an ADI scheme i nstead of a fully-implicit formulation increase rapidly with the number of dimensions. In many physical systems, it i s advantageous to formulate the equations of the theory as ﬁrst-order diﬀerential equations. This is true in hydrod ynamics and in many studies of general relativity. The general principles of causality and time-reﬂection invari ance extend in a simple way to such systems. Time-symmetric ADI should prove relatively straightforward to apply to mor e complicated systems of equations, provided the original diﬀerential equations embody time symmetry. Causality may be less straightforward in nonlinear equatio ns, where the structure of the characteristic cone will depend on the solution, and so the exact causal relationship s between time-levels cannot be decided independently of solving the equations. However, causal reconnection is imp lemented via an inequality: one requires that grid points should be within the characteristic cones of their relatives at the previous time-step. In most cases, one would hope that the inequality can be assured simply by extrapolation f rom the behavior of the characteristic cones on the known time-steps. An important area for the application of the techniques we ha ve developed here would be numerical ﬂuid dynamics, where the study of wave phenomena in supersonic ﬂows is a natu ral place to expect causality problems, and where the interest in three-dimensional problems makes ADI essen tial in many cases. In some restricted situations, it may be straightforward to apply these techniques; for example, a neutron star moving supersonically through a grid in general relativity will, i f treated in the standard way, use acausal computational molecules. Using causal reconnection, adapted to the chara cteristics of the ﬂuid problem, should prevent instabiliti es of the type we have found here. But the application of our techniques to more general proble ms in ﬂuid dynamics will not be automatic. Causal reconnection will have to be generalized to deal with hydrod ynamic shocks. At a shock, the regular causal structure of the ﬂuid breaks down. This does not mean that causal reconnec tion cannot be implemented there. On the contrary, the fact that a causal algorithm is constantly mapping the st ructure of the characteristics means that it can be programmed automatically to locate and to identify shocks. The idea of correctly representing the causal structure of t he original diﬀerential equation is not new, existing methods for handling shocks and related transport problems , such as upwind diﬀerencing [9] and Godunov methods [10], are already based on the local structure of the charact eristics of the ﬂuid. These ideas have also been introduced in the numerical study of steady supersonic ﬂows, where the d irection of ﬂow behaves like a time coordinate and the equations become hyperbolic. Integration methods have bee n developed that use retarded diﬀerences in the upstream direction to maintain stability [11], [12]. All these metho ds diﬀer from causal reconnection in the fact that they keep using only the nearest neighbours to build the computationa l molecules. We have recently become aware, however, of a paper by E. Seidel and Wai-Mo Suen that introduces an idea they call “causal diﬀerencing”, that is very similar to our causal reconnection [13]. Fluid dynamics also presents special challenges to time-sy mmetric ADI. The usual equations of inviscid gas dynamics are time-symmetric, but the presence of viscosity or shocks introduces a fundamental irreversibility into the problem . We hope to treat the ﬂuid dynamic problem in a future paper. We are conﬁdent, however, that the present techniques will g eneralize easily to problems in numerical general relativity, such as that of the motion of black holes through ﬁxed grids. Causal reconnection should allow the equations to remain stable and causal. Moreover, the computational ad vantages oﬀered by ADI schemes, of permitting stable large time-steps (provided the physical situation allows s uch steps to remain accurate) while avoiding time-consumin g sparse-matrix solutions, can be obtained without sacriﬁci ng accuracy or stability. It is hard now to imagine any situation in numerical integrations of the vacuum ﬁeld equa tions of general relativity where one would use implicit methods without employing time-symmetric ADI. 41. APPENDIX A. DERIVATION OF WAVE EQUATION ON A SHIFTING GRID. In this appendix we will sketch the derivation of Equation II .3 by making use of elegant tensorial techniques. There are many alternative approaches, of course, and a reader unf amiliar with tensors can obtain the same result in a straightforward, but rather long, way simply by making the f ollowing general change of variables in the original wave equation from the physical (inertial) coordinates {ξµ}to the computational coordinates {xα}: xi=xi(ξµ), x0=ξ0, (A.1) with the associated change of derivatives ∂ ∂ξi=∂xk ∂ξi∂ ∂xk,∂ ∂ξ0=∂xk ∂ξ0∂ ∂xk+∂ ∂x0. (A.2) The functions that we have identiﬁed as the shift vector βi, the spatial metric gi jand the Γicoeﬃcients in Equations (II.4), (II.5) and (II.8) come out as part of the al gebra. A reader who wants an introduction to the use of tensors in mathematical physics is invited to consult refer ence [14]. We will start from the expression of the wave equation in a gen eral coordinate system: ✷2φ= (γµνφ;µ);ν= 0, (A.3) where the semicolon stands for covariant derivative. Using the explicit expression for the covariant derivatives, the last equation takes the form: γµν∂2φ ∂xµ∂xν−Γλ∂φ ∂xλ= 0, (A.4) where the coeﬃcients Γλare deﬁned in terms of the Christoﬀel symbols as: Γλ:=γµνΓλ µν. (A.5) Our ﬁrst task is, then, to ﬁnd the inverse metric γµν. The metric in the new coordinates is given by γµν=∂ξα ∂xµ∂ξβ ∂xνηαβ, (A.6) withηαβthe Minkowski metric tensor: η00=−1, ηi i= 1 (no sum) , ηµν= 0 µ/ne}ationslash=ν.  (A.7) We now note that for a line of constant {ξi}we have 0 =dξi dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle {ξi}=∂ξi ∂xjdxj dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle {ξi}+∂ξi ∂t, which implies ∂ξi ∂t=−∂ξi ∂xjdxj dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle {ξi}. (A.8) Using now the deﬁnition of the shift vector (Equation II.4) a nd writing x0=ctwe ﬁnd: ∂ξi ∂x0=βj∂ξi ∂xj. (A.9) This is an important relation, and we will use it to rewrite th e metric coeﬃcients given by Equation A.6. For the mixed components in space and time of γµνwe ﬁnd: 42γ0i=γi0=n/summationdisplay l=1∂ξl ∂xi∂ξl ∂x0 =βjn/summationdisplay l=1∂ξl ∂xi∂ξl ∂xj, and ﬁnally: γ0i=γi0=γi jβj=gi jβj. (A.10) In a similar way we can ﬁnd the coeﬃcient γ00: γ00=−/parenleftbigg∂ξ0 ∂x0/parenrightbigg2 +n/summationdisplay l=1/parenleftbigg∂ξl ∂x0/parenrightbigg2 =−1 +n/summationdisplay l=1βj∂ξl ∂xjβi∂ξl ∂xi, and from this we obtain: γ00=−1 +gi jβiβj. (A.11) We will adopt the convention that the indices of the shift vec tor can be raised and lowered by using only the spatial metric: βi=gi jβj, βi=gi jβj, (A.12) wheregi jare the coeﬃcients of the inverse of the spatial metric matri xgi j. The coeﬃcients of γµνcan now be written as: γµν= (−1 +βiβi)βk βjgj k . (A.13) Using the last expression it is not diﬃcult to see that the coe ﬃcients of the inverse metric γµνwill be given by: γµν= −1βk βj(gj k−βjβk) . (A.14) Having found γµν, we will now look for an expression for the coeﬃcients Γi. Since the original coordinates {ξα} deﬁne an inertial reference frame, the Christoﬀel symbols c an be expressed in terms of their transformation to the general coordinates: Γλ µν=∂xλ ∂ξα∂2ξα ∂xµ∂xν. (A.15) From the last expression it is easy to see that: Γ0 µν= 0, (A.16) which in turn means: Γ0= 0. (A.17) On the other hand, from the general expression for the Christ oﬀel symbols: Γλ µν:=1 2γλα/bracketleftbigg∂γµα ∂xν+∂γνα ∂xµ−∂γµν ∂xα/bracketrightbigg , (A.18) 43it is not diﬃcult to show that: Γi:=γµνΓi µν=−1√g/braceleftbigg∂ ∂t/parenleftbig√gβi/parenrightbig +∂ ∂xj/bracketleftbig√g/parenleftbig gij−βiβj/parenrightbig/bracketrightbig/bracerightbigg . (A.19) Using the previous results, we can ﬁnally rewrite Equation A .4 in the following way: (gi k−βiβk)∂2φ ∂xi∂xk+2βi c∂2φ ∂xi∂t−Γi∂φ ∂xi−1 c2∂2φ ∂t2= 0. (A.20) This is the ﬁnal form of the wave equation in the coordinate sy stem adapted to the motion of the grid. . APPENDIX B. IMPLEMENTATION OF CAUSAL RECONNECTION. In this appendix, we discuss one algorithm that determines t he positions of the points that form the causal compu- tational molecules. We will consider the case of an arbitrar y number of spatial dimensions n. The particular cases of one and two dimensions can then be found in a straightforwa rd way. There are many diﬀerent ways of ﬁnding the closest causally c onnected grid points. For example, if it is possible to ﬁnd the transformation of coordinates that takes us back t o the original inertial reference frame ξi=ξi/parenleftbig xj,t/parenrightbig , (B.1) then we could use the fact that in that reference frame the cau sal structure is particularly simple: we would simply select those points in the diﬀerent time levels that have the closest values of/braceleftbig ξi/bracerightbig . This method, however, will only be useful in a few special cases. Indeed, in the general case i t may prove almost impossible to ﬁnd the functional relation (B.1). With this in mind, we have developed a method that can be appli ed in the general case. Let us then assume that we are given a point in the last time level {t0+ ∆t}with position xi t0+∆t. Our aim is to ﬁnd points xi t0andxi t0−∆t in the two previous time levels in such a way as to guarantee th at a proper causal molecule will be formed. We have already seen that this will happen if both xi t0−∆tandxi t0+∆tare inside the numerical light-cone of xi t0. Our algorithm to ﬁnd the proper causal molecules linking the grids at time-steps t0+ ∆t,t0, andt0−∆tassumes that the causal reconnection condition holds. (If it doesn’ t, then remedial action, changing ∆ tor ∆xi, is required.) Our procedure is the following. 1. Choose some point xi t0+∆t. The center of its causal molecule will be that grid point yi t0for which the following function reaches a minimum: f1/parenleftbig yi/parenrightbig :=N/summationdisplay i=1/braceleftbigg/bracketleftbigg yi−βi/parenleftbig yj,t0/parenrightbig ∆t+1 2Γi/parenleftbig yj,t0/parenrightbig (∆t)2/bracketrightbigg −xi t0+∆t/bracerightbigg2 . (B.2) This minimum can easily be found by standard multi-dimensio nal search techniques. Once the minimum is found, we have our best approximation to the exact inverse coordina te transformation: the point xiis approximately at the same spatial location as yiin the original inertial frame. 2. Once we have found the appropriate yi t0, we look in the third time level {t0−∆t}for the completion of the causal molecule, the point zi t0−∆tthat minimizes the function: f2/parenleftbig zi/parenrightbig :=N/summationdisplay i=1/braceleftbigg/bracketleftbigg yi t0+βi/parenleftig yj t0,t/parenrightig ∆t+1 2Γi/parenleftig yj t0,t/parenrightig (∆t)2/bracketrightbigg −zi/bracerightbigg2 . (B.3) This is much easier than the previous step because we already have the point yi t0, so we don’t have to calculate again the functions βand Γ . In fact, minimizing f2is equivalent to ﬁnding the point ziin the third time level that is closest to the axis of the light cone of yi t0in the original inertial frame. We do not have to go through the grid again to ﬁnd this point. 443. So far we have constructed only one molecule. One needs to r epeat the above steps for all points at time-step t0+ ∆t, but of course the best guess for a causal molecule for any gri d point is simply to translate the molecule found for its neighbor. This will occasionally fail, but onl y by one grid point. So the minimization steps will require a computing eﬀort that is only proportional to the nu mber of grid points. It is not diﬃcult to prove that, whenever the causal reconnec tion conditions hold, this algorithm does indeed produce a proper causal molecule. For reasons indicated above, the c omputational eﬀort is proportional to the number of grid points. In complex problems, such as general relativity, th is is likely to be a very small overhead. . ACKNOWLEDGEMENTS. We want to thank G.D. Allen for many useful discussions and co mments. One of the authors (M. Alcubierre) also thanks the ‘Universidad Nacional Aut´ onoma de M´ exico’ for ﬁnancial support. [1] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation , W.H. Freeman and Co., U.S.A., 1973. [2] J.W. York, ‘Kinematics and Dynamics of General Relativi ty’ in: Sources of Gravitational Radiation , ed. L.L. Smarr, pp. 83-126, Cambridge University Press, U.S.A., 1979. [3] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vette rling, Numerical recipes: The Art of Scientiﬁc Computing , Cambridge University Press, U.S.A., 1989. [4] R.D. Richtmyer and K.W. Morton, Diﬀerence Methods for Initial-Value Problems , 2nd. ed., Interscience, U.S.A, 1967. [5] G. Strang, SIAM J. Num. Anal. ,5, p. 506 (1968). [6] G.I. Marchuck, Methods of Numerical Mathematics , 2nd. Ed., Springer-Verlag, U.S.A., 1982. [7] M. Lees, J. Soc. ind. appl. Math. ,10, p. 610 (1962). [8] G. Fairweather and A.R. Mitchell, J. Inst. Maths. Applics. ,1, p. 309 (1965). [9] G.E. Farsythe and W.R. Wasow, Finite-Diﬀerence Methods for Partial Diﬀerential Equatio ns, John Wiley and Sons, U.S.A., 1967. [10] M. Holt, Numerical Methods in Fluid Dynamics , Springer-Verlag, U.S.A., 1977. [11] E.M. Murman and J.D. Cole, AIAA Journal ,9, p. 114, (1971). [12] A. Jameson, Comm. Pure Appl. Math. ,27, p. 283 (1974). [13] E. Seidel and Wai-Mo Suen, Phys. Rev. Lett. ,69No. 13, p. 1845 (1992). [14] B.F. Schutz, Geometrical Methods of Mathematical Physics , Cambridge University Press, Cambridge, UK, 1980. 45
arXiv:physics/0009030v1 [physics.optics] 8 Sep 2000The inﬂuence of a metallic sheet on an evanescent mode atomic mirror J. B. Kirk∗, C. R. Bennett†, M. Babiker†andS. Al-Awﬁ†† ∗Department of Physics, University of Essex, Colchester, Es sex CO4 3SQ, UK †Department of Physics, University of York, Heslington, Yor k, YO10 5DD, UK ††Department of Physics, King Abdulaziz University, P O Box 34 4, Medina, Saudi-Arabia Abstract A theory of evanescent mode atomic mirrors utilising a metal lic sheet on a dielectric substrate is described. The emphasis here is on the role of th e metallic sheet and on the evaluation of atomic trajectories using the ﬁeld-dipol e orientation picture. At low intensity, the atomic reﬂection process is controlled by tw o separate mechanisms both of which are modiﬁed by the presence of the metallic sheet and inﬂuenced by the use of the ﬁeld-dipole orientation picture. The ﬁrst mechanism in volves the spontaneous force which accelerates the atom parallel to the sheet plane. The s econd mechanism involves the combined dipole plus van der Waals force which acts to rep el the atom from the surface, decelerating its motion until it attains an instan taneous halt before changing direction away from the surface at an appropriate turning po int in the trajectory. Various quantitative features arising from varying the con trollable parameters of the system, including screening eﬀects as well as desirable enh ancement eﬀects, are pointed out and discussed. PACS numbers: 42.50.Vk, 32.70.Jz, 32.80.Pj, 32.80.Lg 1One of the primary aims, yet to be fully accomplished in the ra pidly developing ﬁeld of atom optics is the standardisation of the atomic mirror as a routine atom optical element. The idea of an atomic mirror, a device capable of reﬂ ecting atoms, was ﬁrst put forward by Cooke and Hill in 1982 [1]. Since then, a nu mber of studies, both theoretical and experimental, have sought to explore v arious aspects of atomic mirrors, based on diﬀerent mechanisms of atom reﬂection fro m planar surfaces [2-19]. In particular, the evanescent mode atomic mirror is suitable f or neutral atoms undergoing electric dipole transitions [14,18,19]. The mechanism in t his type of mirror makes use of light evanescing into the vacuum region outside the plana r surface of a dielectric when laser light is internally reﬂected. The evanescent lig ht sets up a repulsive dipole potential which acts on any neutral atom possessing a transi tion frequency at near resonance with, and blue-detuned from, the laser frequency . The limiting factors of such a mirror stem primarily from heating eﬀects and also fro m ﬂuctuations due to the light which become negligible at low intensities. A device o perating at low intensities was thus needed. This was the motivation behind the studies w hich highlight the role of a thin metallic layer deposited on the surface in providin g an enhancement of the evanescent component [18,19]. The purpose of this article is to report a theory of evanescen t mode atomic mirrors with a metallic sheet, exploring the nature of the enhanceme nt, determining the general features of the mirror and evaluating the trajectories for a typical set of parameters. It turns out that a quantitative theoretical analysis deman ds the implementation of a number of steps. First, a mode normalisation procedure in te rms of intensity is needed for the incident light mode responsible for the generation o f the evanescent component. Secondly, the jump conditions should be implemented involv ing the surface current arising from the ﬁnite two-dimensional conductivity of the metallic sheet. Thirdly, the attractive force between the atom and the dielectric in the p resence of the metallic sheet should be included in the dynamics to determine the tra jectories. Finally, use 2should made of the ﬁeld-dipole orientation picture for eval uating the radiation forces as well as the van der Waals-type atom-surface force. The rol e of the ﬁeld-dipole orientation picture is to determine an average value for the local orientation of the electric dipole moment vector in the presence of the evanesc ent ﬁeld. We show here that a programme incorporating the above theoretical featu res permits the atomic trajectories to be determined by direct solution of the equa tion of motion, leading either to a reﬂection of the atom oﬀ the mirror or a collision with it, in a manner dependent on the chosen set of parameters. Existing treatments determ ining trajectories are primarily based on Monte Carlo techniques. The basic elements comprising the atomic mirror are shown sc hematically in Fig. 1. Here a metallic sheet in the form of an inﬁnitesimally thin la yer is deposited on the planar surface of a dielectric substrate (or glass prism). L ight of frequency ωincident from within the dielectric is internally reﬂected at the int erface between the dielectric substrate and the metallic sheet. This creates a ﬁeld in the v acuum region which is decaying with distance from the metallic sheet and is propag ating along the surface. A neutral atom possessing a transition frequency ω0< ωand which is moving in the plane of incidence would be subject to the repulsive dipo le force plus an attractive atom-surface force and will also experience a light pressur e force parallel to the surface. As we show below the combined inﬂuence of these forces can be m ade to control the reﬂection process. The relevant electric ﬁeld vector at frequency ωcan be written in terms of incident (I), reﬂected (R) and evanescent (1) parts as follows E(,r, t) =/braceleftBig/parenleftBig EI(k/bardbl,r, t) +ER(k/bardbl,r, t)/parenrightBig θ(−z) +E1(k/bardbl,r, t)θ(z)/bracerightBig a+h.c. (1) where θis the unit step function and the ﬁelds are given by EI(k/bardbl,r, t) =AI/parenleftBigg 1,0,−k/bardbl kz2/parenrightBigg eikz2zei(k/bardbl.r/bardbl−ωt)(2) 3ER(k/bardbl,r, t) =AR/parenleftBigg 1,0,k/bardbl kz2/parenrightBigg e−ikz2zei(k/bardbl.r/bardbl−ωt)(3) E1(k/bardbl,r, t) =B/parenleftBigg 1,0,ik/bardbl kz1/parenrightBigg e−kz1zei(k/bardbl.r/bardbl−ωt)(4) Herek/bardblis the wavevector parallel to the surface. Its magnitude k/bardblis given by c2k2 /bardbl= ω2ǫ2sin2φwhere φis the angle of incidence. The three quantities between the b rackets in each of Eqs.(2) to (4) stand for the vector components para llel tok/bardbl, perpendicular to it on the surface plane and along the z-direction, respect ively. kz1andkz2(both real) are deﬁned by k2 z1=k2 /bardbl−ǫ1ω2 c2>0;k2 z2=ǫ2ω2 c2−k2 /bardbl>0 (5) Finally, AI, ARandBare ﬁeld amplitude factors, to be determined. The notation is such that parameters associated with the substrate are la belled by the subscript 2, while for the outer region (vacuum) the label is 1. Both die lectric functions ǫ1 andǫ2are assumed to be frequency-independent and we take ǫ1= 1, as appropriate for vacuum. The position vector is written as r= (r/bardbl, z) in terms of an in-plane position vector r/bardbland a z-coordinate relative to the metallic sheet. The role o f the metallic sheet is primarily to provide a two-dimensional ch arge density nsand, so, an electrical conductivity inse2/m∗(ω+iγ) where m∗andeare the electronic eﬀective mass and charge and γ≪ωaccounts for metallic plasma loss eﬀects. The metallic sheet only enters the formalism via the electromagnetic bou ndary conditions involving the tangential components of the magnetic ﬁelds correspond ing to Eqs.(2) to (4), which can be calculated using Maxwell’s equation H=−(iǫ0c2/ω)∇×E. Application of the ﬁrst boundary condition, namely the continuity of the tange ntial component of the electric ﬁeld vector at z= 0, yields AI+AR=B (6) The second electromagnetic boundary condition is that the t angential component of the magnetic ﬁeld vector experiences a discontinuity at z= 0 arising from the surface 4current induced by the in-plane component of the electric ﬁe ld at the metallic sheet. We have H/bardbl(0−)−H/bardbl(0+) =inse2 m∗(ω+iγ)E/bardbl(0) (7) where 0 ±are the limits as ξ→0 of (0 ±ξ). Application of this boundary condition leads to a second relation connecting the ﬁeld amplitudes ǫ2 kz2(AI−AR) +iǫ1 kz1B=inse2 ǫ0m∗ω(ω+iγ)(8) Elimination of ARbetween Eqs.(6) and (8) yields straightforwardly B= 2AI/bracketleftBigg ikz2 ǫ2/parenleftBiggnse2 ǫ0m∗ω(ω+iγ)−ǫ1 kz1/parenrightBigg + 1/bracketrightBigg−1 (9) The next step is to determine the value for the amplitude AI. This is the amplitude of the incident ﬁeld in the unbounded bulk of material 2 and th e value of AIis such that the Hamiltonian HIreduces to the canonical form HI=ǫ0 2/integraldisplay dr/braceleftBigg ǫ2E2 I+1 (ǫ0c)2H2 I/bracerightBigg =1 2¯hω/parenleftBig aa†+a†a/parenrightBig (10) Thus we ﬁnd A2 I=¯hk2 z2c2 2V ǫ0ǫ2 2ω(11) where Vis the (large) volume of material 2. The average radiation force acting on an atom of transition f requency ω0moving in the vacuum region at velocity vis given by the well known expression [20,22] F(r,v) = 2¯h/braceleftBiggΓΩ2 Rk/bardbl−1 2∆∇Ω2 R ∆2+ 2Ω2 R+ Γ2/bracerightBigg =Fs+Fd (12) where Fscorresponds to the ﬁrst term, identiﬁed as the spontaneous f orce along the wave propagation direction k/bardbl, andFdcorresponds to the second term, identiﬁed as the dipole force; ∆ is the dynamic detuning given by ∆ = ∆ 0−k/bardbl.v (13) with ∆ 0=ω−ω0the static detuning of the light from the atomic resonance. T he force given in Eq.(12) is applicable in the low intensity lim it for which the saturation 5parameter Ssatisﬁes the inequality S= 2Ω2 R/(∆2+ Γ2)<1, where Ω R(0) is the Rabi frequency, deﬁned below, evaluated at z= 0, and Γ is the spontaneous emission rate. At the high intensity regime, corresponding to S >> 1 one needs to adopt the dressed atom approach [21]. The low intensity regime is the correct r egime in the context of the atomic mirrors considered here for which it can readily b e veriﬁed that S <1, so that the dynamics based on the average radiation forces give n by Eq.(12) is applicable. Of the two average radiation forces, it is easy to see that Fdcan act as a repulsive force provided that the detuning ∆ 0is positive (blue detuning). Note that v, the velocity of the atom, is in the plane of incidence as in Fig. 1, i.e. has o nly two components, a z-component and a component parallel to k/bardbl. The Rabi frequency Ω Rentering Eq.(12) is deﬁned for the electric dipole µin the evanescent ﬁeld E1(k/bardbl,r, t) ΩR=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleαµ.E1(k/bardbl,r, t) ¯h/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(14) where αis a complex amplitude factor such that a→αin the classical electromagnetic ﬁeld limit. It is in fact related to the intensity Iof the incident beam by the well known relation [23] \|α\|2=IV ¯hcω(15) It can be seen from Eq.(4) that the evanescent ﬁeld possesses two vector components. However, once the evanescent ﬁeld has been set up, the averag e atomic dipole moment vector at any given point aligns itself along, and follows th e oscillations of, the local evanescent electric ﬁeld vector. The appropriate Rabi freq uency in this ﬁeld-dipole orientation picture is thus given by [22] ¯hΩR(r) =\|α\|µE1(r) (16) where µandE1are the magnitudes of these vectors. Using Eqs.(4) and (9) we can write the square of the Rabi frequency in the limit γ→0 in the following form, emphasising the dependence on the angle of incidence φ, which is related to k/bardblthrough 6c2k2 /bardbl=ω2ǫ2sinφ, Ω2 R(ω, φ, z ) =4\|α\|2µ2k2 z2c2e−2kz1z 2¯hV ǫ0ǫ2 2ω 1 +k2 z2ǫ2 1 k2 z1ǫ2 2/parenleftBiggΛ2kz1d ω2−1/parenrightBigg2 −1 (1 +k2 /bardbl/k2 z1) (17) where Λ is a convenient scaling frequency deﬁned by Λ2=nse2 m∗ǫ0ǫ1d(18) withda convenient scaling length. Note the presence of the second term in the ex- pression between the last pair of brackets in Eq.(17) which i s proportional to k2 /bardbl/k2 z1. This represents the contribution from the z-component of E1, and since kz1can be very small (when the angle of incidence is close to the total i nternal reﬂection angle), this could lead to an enhancement, in contrast to the contrib ution from the in-plane component (corresponding to the ﬁrst term in the last pair of bracket in Eq.(17)) which does not exhibit this feature. Figure 2 displays the eﬀects of varying the sheet electron de nsitynson the value of the squared Rabi frequency Ω2 R(ω, φ,0), i.e. evaluated at the surface z= 0. This quantity provides an indication of the eﬀectiveness of the structure as an atomic mirror. The curves correspond to diﬀerent values of angle of incidence φ. There are two features worthy of note here. First, for a given φthe variation of Ω2 Rwith increasing nsis practically ﬂat at the value corresponding to ns= 0 (absence of the metallic sheet) up to a point where it changes rapidly, increasing to a maximu m and then dropping sharply to a relatively small value at a higher range of ns. This feature at large ns signiﬁes screening eﬀects. The peak of Ω2 Rcorresponds to the value of nssatisfying the condition kz1d=ω2/Λ2, which is similar to (but not the same as) the dispersion relation of the surface mode in this system which is [24-26] kz1d/parenleftbigg ǫ2 ǫ1+/bracketleftBig 1−ω2 c2k2 z1d2(ǫ2−ǫ1)/bracketrightBig1/2/parenrightbigg=ω2 Λ2(19) This surface mode, in turn, diﬀers from the usual surface pla smon mode arising on a semi-inﬁnite metallic half space. At an angle of incidence a pproximately equal to φ0for 7total internal reﬂection, the Rabi frequency exhibits a res onance which is at least two orders of magnitude larger than the value corresponding to t he absence of the metallic sheet. At φ=φ0(not shown in the Fig. 2) the squared Rabi frequency Ω2 Rwould be in the form of a delta function. In principle, then, pronounc ed repulsion eﬀects would be expected in conditions corresponding to the region of the resonance. The real mirror action is only partially inﬂuenced by the beh aviour of the Rabi fre- quency; dynamical eﬀects are, of course, controlled by the f orces acting on the atomic centre of mass. The atom experiences besides the average rad iation force given by Eq.(12), an attractive force due to interaction with the vac uum ﬁelds which are con- strained by the presence of the surface. At distances from th e surface large compared to a reduced transition wavelength λ0/2πthe atom-surface force takes the Casimir-Polder form. At distances smaller than λ0/2πthe force assumes the van der Waals form. As we show shortly, the important region for the atomic mirrors considered here, is, in fact, the van der Waals regime for which the force can be writt en as Fvw(z) =−∂Uvw ∂z(20) where Uvwis the image potential. As is well known, the van der Waals pot ential arises as the position-dependent change in the radiative self ener gy due to the presence of the surface, but it depends on the orientation of the electri c dipole moment relative to the surface. It should be emphasised that the evanescent ﬁ eld has no other role to play in the determination of the van der Waals potential, exc ept that it is responsible for the dipole orientation. The average electric dipole ali gns itself parallel to ˆ e1, a unit vector in the direction of the local evanescent electric ﬁel d so that E1(r) =ˆ e1E1(r). In the ﬁeld-dipole orientation picture this state of aﬀairs ap plies at every point along the atom trajectory. The leading contribution to the van der Waa ls potential arises from the interaction of the electric dipole with its image and we t herefore have Uvw(z) =−µ2 32πǫ0z3/braceleftBig 3(ˆ e1.ˆ z)(˜ˆ e1.ˆ z)−ˆ e1˜.ˆ e1/bracerightBig (21) 8where the unit vector ˜ˆ e1is the image of ˆ e1. The atomic reﬂection process is controlled by two separate m echanisms. Firstly, the spontaneous force Fsacts on the atom in the direction of k/bardbland, secondly, the combined forceFd+Fvwacts to repel the atom from the surface, decelerating its mot ion along the z-axis towards the surface, attaining an instantaneous hal t before changing direction away from the surface at an appropriate turning point in the t rajectory. This behaviour can be seen more clearly by examining the corresponding repu lsive potential UT(v, z) = Uvw(z)+Ud(v, z) which is shown in Fig. 3 for a typical set of parameters. Note that by controlling the average dipole direction the ﬁeld-dipole o rientation picture inﬂuences the variations of both UvwandUd. The trajectory of the atom of mass Mapproaching a mirror for a given set up is obtainable by solving the equation of motion Md2r dt2=Fs+Fd+Fvw−Mgˆ z (22) subject to given initial conditions. Figure 4 displays typi cal trajectories in the plane of incidence. The parameters are such that the spontaneous r ate Γ is taken to be the free space value Γ 0. This is in fact a very good approximation in the trajectory r egion which is suﬃciently far from the metallic sheet and the subst rate. The static detuning is taken to be ∆ 0= 5×102Γ0. Finally the intensity of the light is assumed to be I= 2.0×104W m−2. It is straightforward to verify that for this light intensi ty and for the detuning value used here, the saturation parameter S= 2Ω2 R(0)/(∆2+ Γ2 0)≈0.23 which conforms with the low intensity regime. In view of Fig. 4 one concludes that the structure operates as an atomic mirror in three of the cases displayed, while for the fourth case the traject ory of the atom terminates with a collision at the surface. An approximate guide to the c ondition leading to a collision with the surface is to compare the maximum height Umaxof the potential in Fig. 3 to the initial kinetic energy Mv2 z(0)/2. For vz(0)>/radicalBig 2Umax/Ma collision 9occurs. This interpretation indeed conforms with the resul ts of the type shown in Fig. 3. Note that reﬂected atom trajectories are, in general, asy mmetric with respect to the turning point. This is a consequence of the action of Fswhich in the present example where the light and the atom are incident on the same side (lef t side) of the z-axis, accelerates the atom to the right along the surface. In conclusion, we have explored the inﬂuence of adding a meta llic sheet to the usual evanescent mode atomic mirror set up. The theory presented p rovides information about the range of metallic sheet densities and the angle of i ncidence at which the Rabi frequency exhibits pronounced enhancement eﬀects. We have also adopted the ﬁeld-dipole orientation picture in which the average atomi c dipole oscillates along the direction of the electric ﬁeld of the evanescent mode at ever y point in the trajectory. We have seen that this step which allowed us to identify the dire ction of the average dipole moment vector as that of the local electric ﬁeld vector at eve ry point in the trajectory, has important consequences for the evaluation of the forces and, hence, the dynamics of the atom. Our results show that enhancement arises at an an gle of incidence close to that of the total internal reﬂection condition, but requi res a relatively high sheet density. Acknowledgement: JBK and CRB are grateful to the EPSRC for ﬁnancial support. Th is work has been carried out under the EPSRC grant number GR/M16313 10References 1. V I Cooke and R K Hill, Opt. Commun. 43, 258 (1982) 2. W Seifert, C S Adams, V I Balykin, C Heine, Yu Ovchinikov and J Mlynek, Phys. Rev.A49, 3814 (1994) 3. G I Opat, S J Wark and A Cimmino, Appl. Phys. B54, 396 (1992) 4. S Tan and D F Walls, J. Phys. II (France) 4, 1879 (1994) 5. P Ryytty, M Kaivola and C G Aminoﬀ, Europhys. Lett. 36, 343 (1996) 6. R J Wilson, B Holst and W Allison, Rev. Sci. Instrum 70, 2960 (1999) 7. D C Lau, A I Sidrov, G I Opat, R J McLean, W J Rowlands and P Hann aford, Eur. Phys. J D 5, 193 (1999) 8. R Cote, B Segev and M G Raisen, Phys. Rev. A58, 3999 (1998) 9. L Cognet, V Savalli, G Zs K Horvath, D Holleville, R Marani, N Westbrook, C I Westbrook and A Aspect, Phys. Rev. Lett. 81, 5044 (1998) 10. H Gauck, M Hartl, D Schneble, H Schnitzler, T Pfau and J Mly nek, Phys. Rev. Lett.81, 5298 (1998) 11. L Santos and L Rose, Phys. Rev. A58, 2407 (1998) 12. N Friedman, R Ozeri and N Davidson, J. Opt. Soc. Am. 15, 1749 (1998) 13. P Szriftgiser, D Guery-Odelin, P Desbiolles, J Dalibard , M Arndt and A Steane, Acta Phys. Pol. 93, 197 (1998) 14. J P Dowling and J Gea- Banacloche, Adv. Atom. Mol. Opt. Phy .37, 1 (1997) 15. A Landragin, J Y Courtois, G Labeyrie, N Vansteenkiste, C I Westbrook and A Aspect, Phys. Rev. Lett. 77, 1464 (1996) 1116. N Vansteenkiste, A Landragin, G. Labeyrie, R Kaiser, C I W estbrook and A Aspect, Ann. Phys.-Paris 20, 595 (1995) 17. S M Tan and D F Walls, Phys. Rev. A50, 1561 (1994) 18. S Feron, J Reinhardt, S Lebouteux, O Gocreix, J Baudon, M D ucloy, J Robert, C Miniatura, S N Chormaic, H Haberland and V Lorent, Opt. Comm un.102, 83 (1993) 19. T Esslinger, M Weidenm¨ uller, A Hammerich and T W H¨ anch, Opt. Lett. 18, 450 (1993) 20. J P Gordon and A Ashkin, Phys. Rev. A21, 1606 (1980) 21. J Dalibard and C Cohen-Tannoudji, J. Phys. B18, 1661 (1985) 22. M Babiker and S Al-Awﬁ, Opt. Commun. 168, 145 (1999); L Allen, M Babiker, W K Lai and V E Lembessis, Phys. Rev. A54, 4259 (1996) 23. R Loudon The Quantum Theory of Light , 2nd Edn. (Oxford, 1995) 24. F Stern, Phys. Rev. Lett. 18, 546 (1967) 25. A L Fetter, Ann. Phys. (NY) 81, 367 (1973) 26. C R Bennett, J B Kirk, and M Babiker, to be submitted. 12Figure Captions Figure 1 Schematic arrangement of the elements comprising an evanes cent mode atomic mirror with a metallic sheet. The plane of incidence contains the in ternally reﬂected light beam as well as a typical atomic trajectory. Here both the ato m and the light are initially propagating on the same side of the vertical (z-ax is). Figure 2 Variation with areal density ns(in units of nsilver s= 5.573×1018m−2) of Ω2 R, the squared evanescent mode Rabi frequency (in arbitrary units), evalu ated just outside the metallic sheet in the limit of small γ. The diﬀerent curves correspond to diﬀerent angles of incidence φ= 42.00o(dashes); φ= 47.00o(dash-dots); φ= 52.00o(dash-double dots). The inset (solid curve) shows the variation for φ= 41.37o, which is very close to the angle φ0for total internal reﬂection. The parameters are such that ǫ2= 2.298, corresponding to an angle of total internal reﬂection φ0= 41.27o; Γ0= 6.128×106s−1; ∆0= 5.0×102Γ0and the intensity of the light is taken as I= 2.0×104Wm−2for the Rb resonance at the transition wavelength λ0= 780nm. The scaling length dis taken asd= 15 nm. Figure 3 Variation with z(in units of d= 15nm) of the combined static dipole potential U= Ud+Uvwacting on a Rb atom (in units of ¯ hΓ0/2). The curves correspond to a ﬁxed value of ns= 215 nsilver s, but diﬀerent values of angle of incidence φ. They are as follows: φ= 42.00o(dashes); φ= 45.00o(dash-dots); φ= 47.00o(dash-double dots); φ= 41.37o (solid curve). The dotted curve shows the variation of the va n der Waals potential. In 13the evaluation of the potentials, the direction of the dipol e moment vector conforms with the ﬁeld-dipole orientation picture. The parameters a re the same as those used in the evaluation of Fig. 2. Figure 4 Atomic trajectories of a Rb atom in the atomic mirror arrange ment shown in Fig.1 with the x-z plane as the plane of incidence. The angle of incidenc e is ﬁxed at φ= 42.000and the metallic sheet corresponds to ns= 215 nsilver s, as in Fig. 3. In all cases the initial position of the atom is at the point ( x= 0, z= 100 d), where d= 15nm. The diﬀerent trajectories correspond to the same initial condition for t he horizontal component of the velocity v/bardbl(0) = 0 .054 ms−1, but diﬀer in their initial z-component of velocity vz(0). They are as follows: vz(0) = 0 .1ms−1(solid curve); vz(0) = 0 .4ms−1(dashed curve); vz(0) = 0 .7ms−1(dash-dotted curve); vz(0) = 1 .1ms−1(dash-double dotted curve). The parameters are the same as those in Figs. 2 and 3. 14Figure 1 φε2 Insz r \|\|µ µ ε1Figure 2 ns/nssilver1 10 100 1000 ΩR2 (Arbitrary Units) 0.00.10.20.30.40.50.60.7 ns/nssilver1 10 100 1000 ΩR2 (Arbitrary Units) 0123456Figure 3 z / d0 20 40 60 80 1002U / /G104Γ0 -50-30-101030507090Figure 4 x / d0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0z / d 020406080100
DESIGN OF THE JLC/NLC RDDS STRUCTURE USING PARALLEL EIGENSOLVER OMEGA3P * Z. Li, N.T. Folwell, K. Ko, R.J. Loewen, E.W. Lundahl, B. McCandless, R.H. Miller, R.D. Ruth, M.D. Starkey, Y. Sun, J.W. Wang, SLAC; T. Higo, KEK *Work supported by the DOE, contract DE-AC03-76SF00515.Abstract The complexity of the Round Damped Detuned Structue (RDDS) for the JLC/NLC main linac is driven bythe considerations of rf efficiency and dipole wakefieldsuppression. As a time and cost saving measure for theJLC/NLC, the dimensions of the 3D RDDS cell are beingdetermined through computer modeling to withinfabrication precision so that no tuning may be neededonce the structures are assembled. The tolerances on thefrequency errors for the RDDS structure are about oneMHz for the fundamental mode and a few MHz for thedipole modes. At the X-band frequency, these correspondto errors of a micron level on the major cell dimensions.Such a level of resolution requires highly accurate fieldsolvers and vast amount of computer resources. A parallelfinite-element eigensolver Omega3P was developed atSLAC that runs on massively parallel computers such asthe Cray T3E at NERSC. The code was applied in thedesign of the RDDS cell dimensions that are accurate towithin fabrication precision. We will present thenumerical approach of using these codes to determine theRDDS dimensions and compare the numerical predictionswith the cold test measurements on RDDS prototypes thatare diamond-turned using these dimensions. 1 INTRODUCTION The 1 TeV JLC/NLC[1,2] collider consists of ten thousand X-band accelerator structures. Each structure ismade up of 206 cells with dimensions tailored from cell tocell to detune the dipole modes frequencies to suppressthe dipole wakefields. The cells are optimized to have around profile for higher rf efficiency. With the manifoldsand slots added to damp the wakefields, the structure cellsbecome fully 3 dimensional and complex, as shown inFig. 1. A time and cost saving approach for the structuredesign is to determine the 3D RDDS dimensions throughcomputer modeling and to machine the cells using highprecision diamond turning machines. With the advancedmodeling and machining tools, the structure can bedesigned and manufactured with a greater accuracy suchthat no tuning will be need once the structure isassembled. The frequency tolerance on the fundamentalmode of the X-band structure is about 1~MHz in order tomaintain a better than 98% acceleration efficiency in thestructure. At the frequency of 11.424~GHz, this tolerance corresponds to one micron in the major dimensions of thecell. To model the RDDS with such an accuracy, vastcomputer resources and accurate field solvers arerequired. With the newly developed finite-elementparallel code Omega3P running on massive parallelsupercomputers, such an accuracy is readily achievable.In this paper, we will present the numerical design of theRDDS structure using the Omega3P code. We willcompare the numerical results with the rf measurementson the high precision machined cells. 2 RDDS STRUCTURE The round damped-detuned structure (RDDS)[3] was designed to suppress wakefields in the JLC/NLC linacs.The RDDS consists of 206 cells connected via slotopenings to four pumping manifolds that run the length ofan accelerator section. The dimensions of the cells arechosen such that the deflecting modes are detuned in aprescribed manner such that the wakefields are decoheredand reduced in magnitude. The coupling slots provide theconduit by which the wakefields within the structure canescape out to the manifold to be absorbed externally,thereby further decreasing the wakefields. Thefundamental frequency of the RDDS needs to bedetermined within 1~part of 10,000, which corresponds toan accuracy in the 3D cell dimensions of one micron. Thiscomplex shape can only be modeled accurately byconformal meshes on unstructured grids. 3 OMEGA3P Omega3P[4,5], a thrust effort in the DOE Grand Challenge, is a new 3D eigensolver designed to modelFigure 1. Round damped-detuned structure: cell profileoptimized for rf efficiency, dipole modes detuned anddamped to suppress long-range wakefield. large, complex RF cavities with unprecedented accuracy and speed by employing advanced eigensolver algorithmson massively parallel supercomputing platforms. The codeuses quadratic finite element formulations for fieldinterpolation, uses unstructured mesh to model complexgeometries. The code takes advantage of existing parallellibraries such as ParMETIS[6] for mesh partitioning andload balancing, and AZTEC[7] for scalable matrix-vectoroperations. The unstructured mesh is generated usingexisting packages such as SIMAIL[8] and CUBIT[9]. Themesh generator can take geometry inputs from solidmodels obtained with popular CAD tools such asSolidEdge and EMS. Omega3P is written in C++programming language and currently runs onsupercomputing platforms such as SGI T3E, IBM SP2,and Linux clusters. The code also runs on single processorworkstations. With the parallel code Omega3P, the computational domain is divided into smaller subdomains via domaindecomposition with each subdomain fitted into one of theprocessors for computation. This allows to model largecomplex problems with much higher grid resolutionswithin reasonable run time, far beyond the computationalresource that can be provided by single processorworkstations. It provides a tool that is essential formodeling complex rf components such as the RDDSstructure. 3.1 Omega3P convergence The convergence of Omega3P on modeling the RDDS1 cell is as shown in Fig. 2. The color-coded mesh showsthe domain decomposition for the 16-processor parallelcomputation. The mesh was generated using CUBIT. Thefrequency converges as the fourth power of the mesh size.To reach a better than 1 MHz accuracy for the RDDS cell,the run time is less than 20 min on 16 processors. 3.2 Comparison with cold test cells Three test cells were calculated using Omega3P on NERSC SGI/CRAY T3E with about 3 million degrees offreedom. The dimensions of the test cells were from therough machine table. These cells were diamond tuning machined. The cell profiles were measured using a highprecision mechanical CMM machine. The profile errorand skin depth effect were corrected on the measuredfrequencies in order to compare with the numericalresults. Table 1. Comparison of numerical result using Omega3P and the cold test measurement on diamond turningmachined cells. Cell Numerical(MHz) Meas.(MHz) Diff.(MHz) 001 11420.57 11420.3 0.27 102 11420.35 11420.4 -0.05 203 11420.09 11419.7 0.39 Omega3P results agree with the cold test measurementwithin half a MHz. The dimensions determined by usingOmega3P has reached an accuracy well within themanufacturing tolerance. These benchmark results give usthe confidence of determining the RDDS dimensionsnumerically using Omega3P. As a result, the number ofcold test cells for the RDDS1 structure was reduced,minimizing the cost and turn around time of structureR&D. 4 DETERMINE RDDS1 STRUCTURE MACHINE TABLE USING OMEGA3P RESULTS The RDDS1 structure machine table was generated based on the numerical results of the Omega3Pcalculation. The dimensions of seven cells along thestructure were calculated using Omega3P, running onNERSC SGI/CRAY T3E, with up to 3 million degrees offreedom for the numerical model. The third-order Splinefunction interpolation was used to obtain the dimensionsof the rest of the cells. Since the code models only perfectconducting materials, the skin depth effect of 0.7 MHz forcopper was corrected for the final machine table. 11.42411.424211.424411.424611.424811.42511.4252 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 dX^4Frequency (GHz) 2.9 M D.O.F.1.8 M D.O.F.0.38 M D.O.F. Figure 2. Paralell modeling of the RDDS1 cell. Left) domain decomposition, each color assigned to oneprocessor; right) convergence of the Omega3P eigensolver, converges as the fourth power of the meshsize. -2-1.5-1-0.500.511.52 0 50 100 150 200Single-disk RF-QCdel_sf00 del_sf0pi del_sf1pi del_sf20Frequency Deviation [MHz] Disk number Figure 3. Single cell RF QC of the RDDS1 cells. Shown are the deviations of the fundamental and dipolemodes at zero and PI phase advances.The 206 RDDS1 structure were diamond tuning machined[10]. The frequency of zero and PI phaseadvance of the fundamental and dipole modes weremeasured for each disk. The deviations of the frequenciesfrom a smooth curve fitting of the corresponding modeare as shown in Fig. 3. The rms values for all of the fourmodes are better than 0.5 MHz. We are confident at thispoint that both the numerical predictions and highprecision machining of the 3D RDDS cell dimensionshave achieved an accuracy better than 10 -4 in frequency or 1 micron in major cell dimensions, which is well withinthe NLC design tolerance. 5 RF PULSE HEATING Damping slots perturb current of accelerating mode, which causes additional wall loss and reduces the Q of theaccelerating mode. This wall loss is concentrated in asmall region around the slot opening, as shown in Fig. 4.The power density, if high, can cause large temperaturerise during the rf pulse. It is shown in ref[11] that atemperature rise of 120 0C can result in structural damage on the copper surface. It is important to evaluate andminimize, if necessary, this pulse heating. Since the highpower is distributed within a thin region, a good meshresolution is need to calculate the power densityaccurately, to which Omega3P is an ideal tool. Thetemperature rise on the copper surface can be estimatedusing the following formula[12] where R s=1/σδ is the surface resistance, Hwall is the magnetic field on the wall (proportional to the wallcurrent), T pulse is the rf pulse length, ρ is the density of copper, c is the specific heat of copper, and k is the thermal conductivity. At an average gradient of 70 MV/m, the rf pulse heating temperature ris for the RDDS1 structure is from25 0C to 550C depending on the location along the structure as shown in Fig. 4. As a comparison, the temperature risefor the detuned RDS temperature is about 14 0C. The high temperature rise at the end of the structure is due to dipoledetuning which may be reduced in the future design. 6 SUMMARY The RDDS1 X-band structure is the first structure that was optimized and designed using parallel finite elementcode Omega3P. The Omega3P results were comparedwith cold test measurements. The accuracy of Omega3Ppredictions on the RDDS cell dimensions is better than10 -4, or 1 MHz at X-band frequency. The final machine table of the RDDS1 structure was determined numericallyusing Omega3P. This approach becomes possible onlywith advanced field solver and supercomputing resources.Omega3P is shown to be a powerful and essential tool formodeling complex rf components with unprecedented accuracy and speed. REFERENCES [1] JLC Design Study, KEK Report 97-001, 1997. [2] Zeroth-order design report for the Next Linear Collider, LBNL-PUB-5424, SLAC Report 474,UCRL-ID-124161, 1996. [3] J. Wang, et al, Design Fabrication and Measurement of the First Rounded Damped Detuned Accelerator Structure (RDDS1), this proceedings. [4] K. Ko, High Performance Computing in Accelerator Physics, this proccedings, WE201. [5] N.T. Folwell, et al , SLAC Parallel Electromagnetic Code Development & Applications, this proceedings, THA17. [6] George Karypis, et al , “ParMETIS: Parallel Graph Partitioning and Sparse Matrix Ordering Library: Version 2.0,” University of Minnesota, September24, 1998. [7] Aztec User's Guide, SAND95-1559, 1995.[8] SIMAIL user's manual, Simulog 1995.[9] CUBIT user's manual, SAND94-1100, or http://endo.sandia.gov/cubit . [10] T. Higo, et al, Meeting Tight Frequency Requirement of Rounded Damped Detuned Structure, this proceedings, THA02. [11] D.P. Pritzkau, Possible High Power Limitations From RF Pulse Heating, SLAC-PUB-8013. [12] G. Bowden, RF Surface Heating Stresses, NLC ME Note, No. 8-96 Rev 0, August 23, 1996.Figure 4 Pulse heating of the RDDS1 structure. Top) wall loss power distribution, clear enhancement aroundthe wide slot; bottom) temperature rise along thestructure. 2 pulse Sw a l ltTR Hckπρ∆=
arXiv:physics/0009032v1 [physics.data-an] 8 Sep 2000Information Theory and Learning: A Physical Approach Ilya Mark Nemenman A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS NOVEMBER 2000c∝circleco√yrtCopyright 2008 by Ilya Mark Nemenman. All rights reserved.Abstract We try to establish a uniﬁed information theoretic approach to learning and to ex- plore some of its applications. First, we deﬁne predictive information as the mutual information between the past and the future of a time series, discuss its behav- ior as a function of the length of the series, and explain how o ther quantities of interest studied previously in learning theory—as well as i n dynamical systems and statistical mechanics—emerge from this universally de ﬁnable concept. We then prove that predictive information provides the unique measure for the com- plexity of dynamics underlying the time series and show that there ar e classes of models characterized by power–law growth of the predictive information that are qualitatively more complex than any of the systems that have been investigated before. Further, we investigate numerically the learning o f a nonparametric prob- ability density, which is an example of a problem with power– law complexity, and show that the proper Bayesian formulation of this proble m provides for the ‘Occam’ factors that punish overly complex models and thus a llow one to learn not only a solution within a speciﬁc model class, but also the class itself using the data only and with very few a priori assumptions. We study a possib leinformation theoretic method that regularizes the learning of an undersampled discrete v ari- able, and show that learning in such a setup goes through stag es of very different complexities. Finally, we discuss how all of these ideas may be useful in various problems in physics, statistics, and, most importantly, bi ology. iiiAcknowledgements Most importantly, I thank my family and my dearest friends; f or if not for their wisdom, knowledge, love, and support I would never be who I am now. And I thank Bill Bialek, who is not just a perfect advisor, but a fri end to me. This thesis would not be what it is now if not for many, too many to name them all, people who mentored life and physics to me. So, in or der of their ap- pearance in my life I am grateful to Leonid Demikhovsky, Mikh ail Polozov, Al- bert Minkevich, Valentin Rusak, people of the Department of Theoretical Physics at Belarusian State University, Betty Young, people of Sant a Clara University, Ger- ald Fisher, Ronald Adler, people of the Physics and Astronom y Department at San Francisco State University, Anatoly Spitkovsky, Alexa nder Silbergleit, Grav- ity Probe B theory group, Curtis Callan, Vipul Periwal, the l ate Howard Stone, Alexander Polyakov, Olexei Motrunich, Akakii Melikidze, S tanislav Boldyrev, Sergei Gukov, Andrei Mikhailov, Timur Shutenko, people of t he Department of Physics at Princeton University, Naftali Tishby, Gonzalo G arcia de Polavieja Em- bid, Methods in Computational Neuroscience course at Marin e Biological Labo- ratory, Rob de Ruyter van Steveninck, Adrienne Fairhall, Jo nathan Miller, Dmitry Rinberg, people of NEC Research Institute, and many others. Thank you all! ivCollaborators This thesis is based on the work done in collaboration with Wi lliam Bialek, Naf- tali Tishby, Adrienne Fairhall, and Jonathan Miller. In par ticular, Chapters 2 and 3 largely follow the papers by Bialek, Nemenman, and Tish by (2000), and Bialek and Nemenman (2000) respectively, and Chapter 4 is a p art of the work in progress by Bialek, Fairhall, Miller, and Nemenman. vContents Abstract iii Acknowledgements iv Collaborators v 1 Introduction: what do we know? 2 2 Predictability, Complexity, and Learning 6 2.1 Why study predictability? . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 A curious observation . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Learning and predictability . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 A test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Learning a parameterized distribution . . . . . . . . . . . . . 28 2.4.3 Learning a parameterized process . . . . . . . . . . . . . . . 36 2.4.4 Taming the ﬂuctuations: ﬁnite dVCcase . . . . . . . . . . . . 38 2.4.5 Taming the ﬂuctuations: the role of the prior . . . . . . . . . 43 2.4.6 Beyond ﬁnite parameterization: general considerati ons . . . 48 2.4.7 Beyond ﬁnite parameterization: example . . . . . . . . . . . 51 vi2.5Ipredas a measure of complexity . . . . . . . . . . . . . . . . . . . . . 56 2.5.1 Complexity of statistical models . . . . . . . . . . . . . . . . 5 7 2.5.2 Complexity of dynamical systems . . . . . . . . . . . . . . . 60 2.5.3 A unique measure of complexity? . . . . . . . . . . . . . . . 62 2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3 Learning continuous distributions: Simulations with ﬁeld theoretic priors 77 3.1 Occam factors in statistics . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3 Learning with the correct prior . . . . . . . . . . . . . . . . . . . . . 82 3.4 Learning with ‘wrong’ priors . . . . . . . . . . . . . . . . . . . . . . 8 4 3.5 Selecting the smoothness scale . . . . . . . . . . . . . . . . . . . . . 88 3.6 Can the ‘wrong’ prior help? . . . . . . . . . . . . . . . . . . . . . . . 90 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4 Learning discrete variables: Information–theoretic regularization 95 4.1 The general paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Discrete variables: a need for special attention . . . . . . . . . . . . 97 4.3 Toy model: theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.4 Toy model: numerical analysis . . . . . . . . . . . . . . . . . . . . . 1 05 4.4.1 Learning with the correct prior . . . . . . . . . . . . . . . . . 10 6 4.4.2 Learning with ‘wrong’ priors . . . . . . . . . . . . . . . . . . 107 4.4.3 Selecting λwith the help of the data . . . . . . . . . . . . . . 109 4.5 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 vii5 Conclusion: what have we achieved? 114 A Appendix 116 A.1 Summary of nonparametric learning . . . . . . . . . . . . . . . . . . 116 A.2 Correlation function evaluation . . . . . . . . . . . . . . . . . . . . . 121 Bibliography 123 viiiList of Figures 2.1 Calculating entropy of spin words. . . . . . . . . . . . . . . . . . . . 11 2.2 Entropy as a function of the word length . . . . . . . . . . . . . . . . 12 2.3 Subextensive part of the entropy as a function of the word length. . 13 3.1Qclfound for different Natl= 0.2. . . . . . . . . . . . . . . . . . . . 83 3.2Λas a function of Nandl. . . . . . . . . . . . . . . . . . . . . . . . 84 3.3Λas a function of Nandla. . . . . . . . . . . . . . . . . . . . . . . . 85 3.4Λas a function of N,ηaandla. . . . . . . . . . . . . . . . . . . . . . 87 3.5 Smoothness scale selection by the data . . . . . . . . . . . . . . . . . 89 3.6 Learning with the data induced smoothness scale. . . . . . . . . . . 91 3.7 Learning speed comparison . . . . . . . . . . . . . . . . . . . . . . . 92 4.1 The reference and a random distributions . . . . . . . . . . . . . . . 106 4.2Λas a function of λ,K, andN. . . . . . . . . . . . . . . . . . . . . . 107 4.3Λas a function of λ,λa,Q∗ aandN. . . . . . . . . . . . . . . . . . . . 108 4.4λ∗for various ensembles of target distributions. . . . . . . . . . . . 110 4.5 Learning with an adaptive λ∗. . . . . . . . . . . . . . . . . . . . . . . 111 A.1 Integration contours. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 22 ix1 “All of the books in the world contain no more information tha n is broadcast as video in a single large American city in a single year. Not all bits have equal value.” Carl Sagan1 “My interest is in the future because I am going to spend the re st of my life there.” Charles F. Kettering “That is what learning is. You suddenly understand somethin g you’ve understood all your life, but in a new way.” Doris Lessing “Learning is not compulsory ...Neither is survival.” W. Edwards Deming “Where is the knowledge we’ve lost in information?” T. S. Eliot “What most experimenters take for granted before they begin their experiments is inﬁnitely more interesting than any results to which thei r experiments lead.” Norbert Wiener 1All quotations shown on this page can be found at the electron ic archive http://www.starlingtech.com/quotes/Chapter 1 Introduction: what do we know? We hope that while reading this work our readers will unsurpr isingly realize that they actually are learning something. However, what may com e as a surprise is that they learn a lot more than they think: while reading this very sentence the photoreceptors in the eyes estimate the mean intensity of th e ambient light and adapt to it; the auditory cortex monitors the surroundings a nd warns if a visi- tor knocks on the door. The reader skips the endings of some lo ng, complicated words because he has already guessed what is coming; he then n otices peculiar- ities in the stylistics of the text and soon learns to disting uish sentences written late at night. And then, ﬁnally, there is the “true” learning of the thoughts that the authors try to convey in their writing. Learning is everywhere around and inside us, and it is absolu tely essential for our second–to–second survival. In fact, because of its u tmost importance and omnipresence each one of us has a well developed personal, un ique intuition on what “learning” means, and how it works. One might think th at such enor- mous experience would come in handy when studying learning f rom a scientiﬁc perspective, but the situation is quite the opposite: it is e xtremely difﬁcult to build a theory that unites the enormous spectrum of possible learning problems. 23 Intuition built up for the case of learning to play a musical i nstrument may be totally useless (and even destructive) for studying, for ex ample, how we learn our ﬁrst language, or master mathematical concepts. A multi tude of ideas and approaches, each treating its speciﬁc problem and having on ly a slight relation to another, is indeed what we see in learning science now. In fact, there even is no such thing as the “learning theory.” There is statistical learning theory, which builds probabilistic bounds on our a bility to estimate the parameters of models that describe some observations, and i ts formalism seems completely disjoint from the designs of psychological and p hysiological exper- iments that study learning in humans and animals. Then there is the Minimal Description Length paradigm, which states that the shorter is the code for a set of samples, the better is the knowledge of the structure insi de the samples; it is not clear how to connect these ideas to numerous learning cur ves deﬁned in spe- ciﬁc contexts of neural networks. Then there are ideas that s ince the speed or (conversely) the difﬁculty of learning is related intuitiv ely to the complexity of the studied problem, learning and complexity should be stud ied together; this opens the Pandora box of different approaches to complexity (later in this work we list over a dozen of deﬁnitions of this quantity!) and does not even come close to quantifying learning and complexity of, say, some simple geometric concept. We can continue this list, but the point is clear. We believe t hat speciﬁc learning scenarios, however interesting and practical they may be, a re not going to bring any more insight to our current understanding of learning (a nd, for that matter, complexity). What we need at this stage is not another exampl e—there are too many of them to comprehend already—but a unifying, generali zing theory. What do we expect from such a theory? We want it to be physical i n its spirit.4 That is, it must explain and unify all accumulated knowledge of the subject (and thus necessarily have an element of a review), but this expla nation should bring a new level of understanding to the old problems, a level from which all the problems appear as different realizations of one general ph enomenon. However, explaining old data is just a half of a good theory. Using new t ools we must also be able to ask and answer meaningful new questions, thus the t heory should be constructive enough to serve as a kernel for development. We build our presentation to address all of these questions. In Chapter 2 we introduce a version of the theory of learning and complexity which is built on information theory and the notion of predictability. After ﬁnishing the construc- tion, we extensively analyze the literature to show that mos t of prior knowledge of the subject is subsumed in our more general approach. Then we try to show that the ideas do not only explain the old results but can be us ed to study new problems as well. For this, we discuss a broad spectrum of pos sible applica- tions to physics, to computer science, and to biology, and th en single out two examples for a detailed analysis. In Chapter 3 we study appli cations of our ideas to the learning of nonparametric continuous probability de nsities, and we show how complexity penalizing Occam factors work in this case. T hen in Chapter 4 we turn to the seemingly easier problem of learning a probabi lity distribution of a discrete variable, and we study how regularization based o nly on information theory makes learning possible in the undersampled regime. One may argue that the examples we discuss are not enough to cl aim for cer- tain that our theory indeed is constructive. We hope to resol ve these fears in the nearest future by studying other possible applications tha t we mention through- out our work. However, we want to stress here explicitly that we believe that the5 theory itself is complete, the deﬁnitions that we make are se nsible and unique, and the conclusions are general and universal.Chapter 2 Predictability, Complexity, and Learning 2.1 Why study predictability? There is an obvious interest in having practical algorithms for predicting the fu- ture, and there is a correspondingly large literature on the problem of time series extrapolation.1But prediction is both more and less than extrapolation: we m ight be able to predict, for example, the chance of rain in the comi ng week even if we cannot extrapolate the trajectory of temperature ﬂuctua tions. In the spirit of its thermodynamic origins, information theory (Shannon 19 48) characterizes the potentialities and limitations of all possible prediction algorithms, as well as uni- fying the analysis of extrapolation with the more general no tion of predictability. Speciﬁcally, we can deﬁne a quantity—the predictive information —that measures 1The classic papers are by Kolmogoroff (1939, 1941) and Wiene r (1949), who essentially solved all the extrapolation problems that could be solved by linea r methods. Our understanding of pre- dictability was changed by developments in dynamical syste ms, which showed that apparently random (chaotic) time series could arise from simple determ inistic rules, and this led to vigorous exploration of nonlinear extrapolation algorithms (Abarb anel et al. 1993). For a review com- paring different approaches, see the conference proceedin gs edited by Weigend and Gershenfeld (1994). 62.1. Why study predictability? 7 how much our observations of the past can tell us about the fut ure. The predictive information characterizes the world we are observing, and w e shall see that this characterization is close to our intuition about the comple xity of the underlying dynamics. Prediction is one of the fundamental problems in neural comp utation. Much of what we admire in expert human performance is predictive i n character—the point guard who passes the basketball to a place where his tea mmate will arrive in a split second, the chess master who knows how moves made no w will inﬂu- ence the end game two hours hence, the investor who buys a stoc k in anticipation that it will grow in the year to come. More generally, we gathe r sensory informa- tion not for its own sake but in the hope that this information will guide our actions (including our verbal actions). But acting takes ti me, and sense data can guide us only to the extent that those data inform us about the state of the world at the time of our actions, so the only components of the incom ing data that have a chance of being useful are those that are predictive. Put bl untly, nonpredictive information is useless to the organism , and it therefore makes sense to isolate the predictive information. It will turn out that most of the inf ormation we collect over a long period of time is nonpredictive, so that isolatin g the predictive infor- mation must go a long way toward separating out those feature s of the sensory world that are relevant for behavior. One of the most important examples of prediction is the pheno menon of gen- eralization in learning. Learning is formalized as ﬁnding a model that explains or describes a set of observations, but again this is useful p recisely (and only) be- cause we expect this model will continue to be valid: in the la nguage of learning theory [see, for example, Vapnik (1998)] an animal can gain s elective advantage2.1. Why study predictability? 8 not from its performance on the training data but only from it s performance at generalization. Generalizing—and not “overﬁtting” the tr aining data—is pre- cisely the problem of isolating those features of the data th at have predictive value (see also Bialek and Tishby, in preparation). Further , we know that the success of generalization hinges on controlling the comple xity of the models that we are willing to consider as possibilities. Finally, learn ing a model to describe a data set can be seen as an encoding of those data, as emphasiz ed by Rissanen (1989), and the quality of this encoding can be measured usin g the ideas of in- formation theory. Thus the exploration of learning problems should provide us with explicit links among the concepts of entropy, predictabili ty, and complexity . The notion of complexity arises not only in learning theory, but also in several other contexts. Some physical systems exhibit more complex dynamics than oth- ers (turbulent vs. laminar ﬂows in ﬂuids), and some systems e volve toward more complex states than others (spin glasses vs. ferromagnets) . The problem of char- acterizing complexity in physical systems has a substantia l literature of its own [for an overview see Bennett (1990)]. In this context severa l authors have con- sidered complexity measures based on entropy or mutual info rmation, although as far as we know no clear connections have been drawn among th e measures of complexity that arise in learning theory and those that aris e in dynamical systems and statistical mechanics. An essential difﬁculty in quantifying complexity is to dist inguish complexity from randomness. A true random string cannot be compressed a nd hence re- quires a long description; it thus is complex in the sense deﬁ ned by Kolmogorov (1965, Li and Vit´ anyi 1993, Vit´ anyi and Li 2000), yet the ph ysical process that2.1. Why study predictability? 9 generates this string may have a very simple description. Bo th in statistical me- chanics and in learning theory our intuitive notions of comp lexity correspond to the statements about complexity of the underlying process, and not directly to the description length or Kolmogorov complexity. Our central result is that the predictive information provides a general measure of complexity which includes as special cases some relevant concepts from learning theory and from dynamical systems. While the work on the comp lexity of mod- els in learning theory rests speciﬁcally on the idea that one is trying to infer a model from data, the predictive information is a property of the data (or, more precisely, of an ensemble of data) itself without reference to a speciﬁc class of underlying models. If the data are generated by a process in a known class but with unknown parameters, then we can calculate the predicti ve information ex- plicitly and show that this information diverges logarithmically with the size of the data set we have observed ; the coefﬁcient of this divergence counts the number of parameters in the model, or more precisely the effective dim ension of the model class, and this provides a link to known results of Rissanen a nd others. But our approach also allows us to quantify the complexity of proces ses that fall outside the ﬁnite dimensional models of conventional learning theo ry, and we show that these more complex processes are characterized by a power–law rat her than a logarithmic divergence of the predictive information . By analogy with the analysis of critical phenomena in statis tical physics, the separation of logarithmic from power–law divergences, tog ether with the mea- surement of coefﬁcients and exponents for these divergence s, allows us to deﬁne “universality classes” for the complexity of data streams. The power–law or non- parametric class of processes may be crucial in real world le arning tasks, where2.2. A curious observation 10 the effective number of parameters becomes so large that asy mptotic results for ﬁnitely parameterizable models are inaccessible in practi ce. There is empirical evidence that simple physical systems can generate dynamic s in this complexity class, and there are hints that language also may fall in this class. Finally, we argue that the divergent components of the predictive information pro - vide a unique measure of complexity that is consistent with certain simple require- ments. This argument is in the spirit of Shannon’s original d erivation of entropy as the unique measure of available information. We believe t hat this uniqueness argument provides a conclusive answer to the question of how one should quan- tify the complexity of a process generating a time series. With the evident cost of lengthening our discussion, we have tried to give a self–contained presentation that develops our point of vie w, uses simple exam- ples to connect with known results, and then generalizes and goes beyond these results.2Even in cases where at least the qualitative form of our resul ts is known from previous work, we believe that our point of view elucida tes some issues that may have been less the focus of earlier studies. Last but not least, we explore the possibilities for connecting our theoretical discussi on with the experimental characterization of learning and complexity in neural syst ems. 2.2 A curious observation Before starting the systematic analysis of the problem, we w ant to motivate our discussion further by presenting results of some simple num erical experiments. 2Some of the basic ideas presented here, together with some co nnections to earlier work, can be found in brief preliminary reports (Bialek 1995; Bialek a nd Tishby 1999). The central results of the present work, however, were at best conjectures in these preliminary accounts.2.2. A curious observation 11 7W1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 W1W0W1W9W0 . . . . . . . . .W = 0 0 0 0 W = 0 0 0 1 W = 0 0 1 0 W = 1 1 1 1 . . .0 1 2 15 Figure 2.1: Calculating entropy of words of length 4in a chain of 17spins. For this chain, n(W0) =n(W1) =n(W3) =n(W7) =n(W12) =n(W14) = 2 ,n(W8) = n(W9) = 1 , and all other frequencies are zero. Thus, S(4)≈2.95 bits . Suppose we have a 1-dimensional chain of Ising spins with the Hamiltonian given by H=−/summationdisplay i,jJijσiσj, (2.1) where the matrix Jijis not necessarily tridiagonal (that is, long range interac - tions are also allowed). One may identify spins pointing upw ards with 1and downwards with 0, and then a spin chain is equivalent to some sequence of bi- nary digits. This sequence consists of (overlapping) words ofNdigits each, Wk, k= 0,1···2N−1. Even though there are 2Nsuch words total, they appear with very different frequencies n(Wk)in the spin chain [see Fig. (2.1) for details]. If the number of spins is large, then counting these frequencies pr ovides a good empir- ical estimate to PN(Wk), the probability distribution of different words of length N. Then one can calculate the entropy S(N)of this probability distribution by the usual formula S(N) =−2N−1/summationdisplay k=0PN(Wk) log2PN(Wk) (bits). (2.2) Since entropy is an extensive property, S(N)is asymptotically proportional to Nfor any spin chain. Choosing a different set of couplings Jijmay change the coefﬁcient of proportionality (and ﬁnding this coefﬁcient is usually the goal of2.2. A curious observation 12 5 10 15 20 250510152025 N Sfixed J variable J, short range interactions variable J’s, long range decaying interactions Figure 2.2: Entropy as a function of the word length for spin c hains with different interactions. Notice that all lines start from S(N) = log22 = 1 since at the values of the coupling we investigated the correlation length is mu ch smaller than the chain length ( 1·109spins). statistical mechanics) but the linearity is never challeng ed. We investigated this in three different spin chains of one bi llion spins each (the temperature is always kBT= 1). For the ﬁrst chain, only Ji,i+1= 1was nonzero, and its value was the same for all i’s. The second chain was also generated using the nearest neighbor interactions, but the value of the coup ling was reinitialized every 400,000 spins by taking a random number from a Gaussian distribution with a zero mean and a unit variance. In the third case, we agai n reinitialized at the same frequency, but now interactions were long–range d, and the variance of coupling constants decreased with the distance between t he spins as ∝an}bracketle{tJ2 ij∝an}bracketri}ht= 1/(i−j)2. We plotted S(N)for all these cases in Fig. (2.2), and, of course, the asymptotically linear behavior seems to be evident—the ext ensive entropy shows2.2. A curious observation 13 5 10 15 20 2501234567 S1=const S1=const1+1/2 log N S1=const1+const2 N0.5 N S1fixed J variable J, short range interactions variable J’s, long range decaying interactions fits Figure 2.3: Subextensive part of the entropy as a function of the word length. no qualitative distinction between the three cases we consi der. However, the situation changes drastically if we remove the asymptotic linear contribution and plot only the sublinear component S1(N)of the entropy. As we see in Fig. (2.3), the three investigated chains then exhibi tqualitatively different features: for the ﬁrst one, S1is constant; for the second one, it is logarithmic; and, for the third one, it clearly shows a power–law behavior. What is the signiﬁcance of this observation? Of course, the d ifferences must be related to the ways we chose J’s for the simulations. In the ﬁrst case, Jis ﬁxed, and there is not much one can learn from observing the sp in chain. For the second chain, Jchanges, and the statistics of the spin–words is different i n different parts of the sequence. By looking at this statisti cs, one can thus esti- mate coupling at the current position. Finally, in the third case there are many coupling constants that can be learned. In principle, as Nincreases one becomes2.3. Fundamentals 14 sensitive to correlations caused by interactions over larg er and larger distances, and, since the variance of the couplings decays with the dist ance, interactions of longer range do not interfere with learning short–scale pro perties. So, intuitively, the qualitatively different behavior of S1(N)for the three plotted cases is due to a different character of learning tasks involved in underst anding the spin chains. Much of this Chapter can be seen as expanding on and quantifyi ng this intuitive observation.3 2.3 Fundamentals The problem of prediction comes in various forms, as noted ab ove. Information theory allows us to treat the different notions of predictio n on the same footing. The ﬁrst step is to recognize that all predictions are probab ilistic—even if we can predict the temperature at noon tomorrow, we should provide error bars or conﬁ- dence limits on our prediction. The next step is to remember t hat, even before we look at the data, we know that certain futures are more likely than others, and we can summarize this knowledge by a prior probability distrib ution for the future. Our observations on the past lead us to a new, more tightly con centrated distri- bution, the distribution of futures conditional on the past data. Different kinds of predictions are different slices through or averages ove r this conditional distri- bution, but information theory quantiﬁes the “concentrati on” of the distribution without making any commitment as to which averages will be mo st 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 3Note again that we are dealing here with subextensive proper ties of systems. These are the properties that are ignored in most problems in statistical mechanics.2.3. Fundamentals 15 in saying something about the future, so we want to know about the datax(t) that will be observed in the time interval 0< t < T′; let these future data be calledxfuture . In the absence of any other knowledge, futures are drawn fro m the probability distribution P(xfuture), while observations of particular past data xpast tell us that futures will be drawn from the conditional distr ibutionP(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 p ast 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 (2.3) =−∝an}bracketle{tlog2P(xfuture)∝an}bracketri}ht − ∝an}bracketle{tlog2P(xpast)∝an}bracketri}ht −[−∝an}bracketle{tlog2P(xfuture,xpast)∝an}bracketri}ht], (2.4) where ∝an}bracketle{t···∝an}bracketri}ht denotes an average over the joint distribution of the past an d the future,P(xfuture,xpast). Each of the terms in Eq. (2.4) is an entropy. Since we are inter ested in pre- dictability or generalization, which are associated with s ome features of the signal persisting forever, we may assume stationarity or invarian ce under time transla- tions. Then the entropy of the past data depends only on the du ration 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 durati onT+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′). (2.5)2.3. Fundamentals 16 In the same way that the entropy of a gas at ﬁxed density is prop ortional to the volume, the entropy of a time series (asymptotically) is proportional to its duration, so that limT→∞S(T)/T=S0; entropy is an extensive quantity. But from Eq. (2.5) any extensive component of the entropy cancels in t he computation of the predictive information: predictability is a deviation from extensivity . If we write S(T) =S0T+S1(T), then Eq. (2.5) tells us that the predictive information is related only to 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 lim T→∞S(T) T=S0 (2.6) exists, and for this to be true we must also have lim T→∞S1(T) T= 0. (2.7) Thus the nonextensive terms in the entropy must be subextensive, that is they must grow with Tless rapidly than a linear function. Taken together, these f acts guarantee that the predictive information is positive and s ubextensive. 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 e ntire future, Ipred(T) = lim T′→∞Ipred(T,T′) =S1(T). (2.8) If we have been observing a time series for a (long) time T, then the total amount of data we have taken in is measured by the entropy S(T), and at large Tthis is given approximately by S0T. But the predictive information that we2.3. Fundamentals 17 have gathered cannot grow linearly with time, even if we are m aking 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. (2.9) In this precise sense, most of what we observe is irrelevant t o the problem of predicting the future. We can think of Eq. (2.9) as a law of dim inishing returns: although we collect data in proportion to our observation ti meT, a smaller and smaller fraction of this information is useful in the proble m of prediction. Note that these diminishing returns are not due to a limited lifet ime, since we calculate the predictive information assuming that we have a future ex tending forward to inﬁnity. Now consider the case where time is measured in discrete step s, so that we have seenNtime points x1,x2,···,xN. How much have we learned about the un- derlying pattern in these data? The more we know, the more eff ectively we can predict the next data point xN+1and hence the fewer bits we will need to describe the deviation of this data point from our prediction: our acc umulated knowledge about the time series is measured by the degree to which we can compress the description of new observations. On average, the length of t he code word re- quired to describe the point xN+1, given that we have seen the previous Npoints, is given by ℓ(N) =−∝an}bracketle{tlog2P(xN+1\|x1,x2,···,xN)∝an}bracketri}htbits, (2.10) where the expectation value is taken over the joint distribu tion of all the N+ 12.3. Fundamentals 18 points,P(x1,x2,···,xN,xN+1). It is easy to see that ℓ(N) =S(N+ 1)−S(N)≈∂S(N) ∂N. (2.11) As we observe for longer times, we learn more and this word len gth decreases. It is natural to deﬁne a learning curve that measures this impro vement. Usually we deﬁne learning curves by measuring the frequency or costs of errors; here the cost is that our encoding of the point xN+1is longer than it could be if we had perfect knowledge. This ideal encoding has a length which we can ﬁnd b y imagining that we observe the time series for an inﬁnitely long time, ℓideal= limN→∞ℓ(N), but this is just another way of deﬁning the extensive compone nt of the entropy S0. Thus we can deﬁne a learning curve Λ(N)≡ℓ(N)−ℓideal (2.12) =S(N+ 1)−S(N)− S0 =S1(N+ 1)−S1(N) ≈∂S1(N) ∂N=∂Ipred(N) ∂N, (2.13) and we see once again that the extensive component of the entr opy cancels. It is well known that the problems of prediction and compress ion are related, and what we have done here is to illustrate one aspect of this c onnection. Specif- ically, if we ask how much one segment of a time series can tell us about the future, the answer is contained in the subextensive behavio r of the entropy. If we ask how much we are learning about the structure of the time se ries, then the nat- ural and universally deﬁned learning curve is related again to the subextensive entropy: the learning curve is the derivative of the predict ive information. This universal learning curve is connected to the more conve ntional learning2.3. Fundamentals 19 curves in speciﬁc contexts. As an example (cf. Section 2.4.1 ), consider ﬁtting a set of data points {xn,yn}with some class of functions y=f(x;α),where the αare unknown parameters that need to be learned; we also allow for some Gaussian noise in our observation of the yn. Here the natural learning curve is the evolution ofχ2for generalization as a function of the number of examples. W ithin the approximations discussed below, it is straightforward to s how that as Nbecomes large, ∝an}bracketle{tχ2(N)∝an}bracketri}ht=1 σ2∝an}bracketle{t[y−f(x;α)]2∝an}bracketri}ht →2 ln2 Λ(N) + 1, (2.14) whereσ2is the variance of the noise. Thus a more conventional measur e of per- formance at learning a function is equal to the universal lea rning curve deﬁned purely by information theoretic criteria. In other words, i f a learning curve is measured in the right units, then its integral represents th e amount of the useful information accumulated. Since one would expect any learni ng curve to decrease to zero eventually, we again obtain the ‘law of diminishing r eturns’. Different quantities related to the subextensive entropy h ave been discussed in several contexts. For example, the code length ℓ(N)has been deﬁned as a learning curve in the speciﬁc case of neural networks (Opper and Haussler 1995) and has been termed the “thermodynamic dive” (Crutchﬁeld an d Shalizi 1998) and “Nthorder block entropy” (Grassberger 1986). Mutual informati on between all of the past and all of the future (both semi–inﬁnite) is kn own also as the “ex- cess entropy,” “effective measure complexity,” “stored in formation,” and so on [see Shalizi and Crutchﬁeld (1999) and references therein, as well as the discus- sion below]. If the data allow a description by a model with a ﬁ nite number of parameters, then mutual information between the data and th e parameters is of2.3. Fundamentals 20 interest, and this is also the predictive information about all of the future; some special cases of this problem have been discussed by Opper an d Haussler (1995) and by Herschkowitz and Nadal (1999). What is important is th atthe predictive information or subextensive entropy is related to all these quantities , and that it can be deﬁned for any process without a reference to a class of model s. It is this universality that we ﬁnd appealing, and this universality is strongest if we focus on the limit of long observation times. Qualitatively, in this regime ( T→ ∞ ) we expect the predictive information to behave in one of three different w ays: it may either 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, limT→∞Ipred(T) = constant, means that no matter how long we observe we gain only a ﬁnite amount of information abo ut the future. This situation prevails, for example, when the dynamics are too regular: for a purely periodic system, complete prediction is possible on ce we know the phase, and if we sample the data at discrete times this is a ﬁnite amou nt of information; longer period orbits intuitively are more complex and also h ave largerIpred, but this doesn’t change the limiting behavior limT→∞Ipred(T) =constant. Alternatively, the predictive information can be small whe n the dynamics are irregular but the best predictions are controlled only b y the immediate past, so that the correlation times of the observable data are ﬁnit e [see, for example, Crutchﬁeld and Feldman (1997) and the ﬁxed short–range inte ractions plot on Fig. (2.3)]. 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 can always write2.3. Fundamentals 21 the joint distribution of the Ndata points as a product, P(x1,x2,···,xN) =P(x1)P(x2\|x1)P(x3\|x2,x1)···. (2.15) For Markov processes, what we observe at tndepends only on events at the pre- vious time step tn−1, so that P(xn\|{x1≤i≤n−1}) =P(xn\|xn−1), (2.16) and hence the predictive information reduces to Ipred=/angbracketleftBigg log2/bracketleftBiggP(xn\|xn−1) P(xn)/bracketrightBigg/angbracketrightBigg . (2.17) The maximum possible predictive information in this case is the entropy of the distribution of states at one time step, which in turn is boun ded by the logarithm of the number of accessible states. To approach this bound th e system must main- tain memory for a long time, since the predictive informatio n 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)diverges at large T. In physi- cal systems we know that there are critical points where corr elation times become inﬁnite, so that optimal predictions will be inﬂuenced by ev ents in the arbitrarily distant past. Under these conditions the predictive inform ation can grow with- out bound as Tbecomes large; for many systems the divergence is logarithm ic, Ipred(T→ ∞)∝lnT, as for the variable Jij, short range Ising model of Figs. (2.2, 2.3). Long range correlation also are important in a time ser ies where we can learn some underlying rules. It will turn out that when the set of po ssible rules can be described by a ﬁnite number of parameters, the predictive in formation again di- verges logarithmically, and the coefﬁcient of this diverge nce counts the number2.4. Learning and predictability 22 of parameters. Finally, a faster growth is also possible, so thatIpred(T→ ∞)∝Tα, as for the variable Jijlong range Ising model, and we shall see that this behavior emerges from, for example, nonparametric learning problem s. 2.4 Learning and predictability Learning is of interest precisely in those situations where correlations or associ- ations persist over long periods of time. In the usual theore tical models, there is some rule underlying the observable data, and this rule is valid forever; ex- amples seen at one time inform us about the rule, and this info rmation can be used to make predictions or generalizations. The predictiv e information quanti- ﬁes the average generalization power of examples, and we sha ll see that there is a direct connection between the predictive information and the complexity of the possible underlying rules. 2.4.1 A test case Let us begin with a simple example already mentioned above. W e observe two streams of data xandy, or equivalently a stream of pairs (x1,y1),(x2,y2),···, (xN,yN). Assume that we know in advance that the x’s are drawn independently and at random from some distribution P(x), while the y’s are noisy versions of some function acting on x, yn=f(xn;α) +ηn, (2.18)2.4. Learning and predictability 23 wheref(x;α)is a class of functions parameterized by α, andηnis some noise which for simplicity we will assume is Gaussian with some kno wn standard de- viationσ. We can even start with a very simple case, where the function class is just a linear combination of some basis functions, so that f(x;α) =K/summationdisplay µ=1αµφµ(x). (2.19) The usual problem is to estimate, from Npairs{xi,yi}, the values of the param- etersα; in favorable cases such as this we might even be able to ﬁnd an effective regression formula. We are interested in evaluating the pre dictive information, which means that we need to know the entropy S(N). We go through the calcu- lation in some detail because it provides a model for the more general case. To evaluate the entropy S(N)we ﬁrst construct the probability distribution P(x1,y1,x2,y2,···,xN,yN). The same set of rules apply to the whole data stream, which here means that the same parameters αapply for all pairs {xi,yi}, but these parameters are chosen at random from a distribution P(α)at the start of the stream. Thus we write P(x1,y1,x2,y2,···,xN,yN) =/integraldisplay dKαP(x1,y1,x2,y2,···,xN,yN\|α)P(α), (2.20) and now we need to construct the conditional distributions f or ﬁxed α. By hy- pothesis each xis chosen independently, and once we ﬁx αeachyiis correlated only with the corresponding xi, so that we have P(x1,y1,x2,y2,···,xN,yN\|α) =N/productdisplay i=1[P(xi)P(yi\|xi;α)]. (2.21) Further, with the simple assumptions above about the class o f functions and2.4. Learning and predictability 24 Gaussian noise, the conditional distribution of yihas the form P(yi\|xi;α) =1√ 2πσ2exp −1 2σ2 yi−K/summationdisplay µ=1αµφµ(xi) 2 . (2.22) Putting all these factors together, P(x1,y1,x2,y2,···,xN,yN) =/bracketleftBiggN/productdisplay i=1P(xi)/bracketrightBigg/parenleftBigg1√ 2πσ2/parenrightBiggN/integraldisplay dKαP(α) exp/bracketleftBigg −1 2σ2N/summationdisplay i=1y2 i/bracketrightBigg ×exp −N 2K/summationdisplay µ,ν=1Aµν({xi})αµαν+NK/summationdisplay µ=1Bµ({xi,yi})αµ ,(2.23) where Aµν({xi}) =1 σ2NN/summationdisplay i=1φµ(xi)φν(xi),and (2.24) Bµ({xi,yi}) =1 σ2NN/summationdisplay i=1yiφµ(xi). (2.25) Our placement of the factors of Nmeans that both AµνandBµare of order unity asN→ ∞ . These quantities are empirical averages over the samples {xi,yi}, and if theφµare well behaved we expect that these empirical means conver ge to expectation values for most realizations of the series {xi}: lim N→∞Aµν({xi}) =A∞ µν=1 σ2/integraldisplay dxP(x)φµ(x)φν(x), (2.26) lim N→∞Bµ({xi,yi}) =B∞ µ=K/summationdisplay ν=1A∞ µν¯αν, (2.27) where ¯αare the parameters that actually gave rise to the data stream {xi,yi}. In fact we can make the same argument about the terms in/summationtexty2 i, lim N→∞N/summationdisplay i=1y2 i=Nσ2 K/summationdisplay µ,ν=1¯αµA∞ µν¯αν+ 1 . (2.28)2.4. Learning and predictability 25 Conditions for this convergence of empirical means to expec tation values are at the heart of learning theory. Our approach here is ﬁrst to ass ume that this conver- gence works, then to examine the consequences for the predic tive information, and ﬁnally to address the conditions for and implications of this convergence breaking down. Putting the different factors together, we obtain P(x1,y1,x2,y2,···,xN,yN) /tildewider→/bracketleftBiggN/productdisplay i=1P(xi)/bracketrightBigg/parenleftBigg1√ 2πσ2/parenrightBiggN/integraldisplay dKαP(α) exp [−NEN(α;{xi,yi})], (2.29) where the effective “energy” per sample is given by EN(α;{xi,yi}) =1 2+1 2K/summationdisplay µ,ν=1(αµ−¯αµ)A∞ µν(αν−¯αν). (2.30) Here we use the symbol /tildewider→to indicate that we not only take the limit of large N, but also neglect the ﬂuctuations. Note that in this approxim ation the dependence on the sample points themselves is hidden in the deﬁnition of ¯αas being the parameters that generated the samples. The integral that we need to do in Eq. (2.29) involves an expon ential with a large factor Nin the exponent; the free energy FNis of order unity as N→ ∞ . This suggests that we evaluate the integral by a saddle point or steepest descent approximation [similar analyses were performed by Clarke a nd Barron (1990), by MacKay (1992), and by Balasubramanian (1997)]: /integraldisplay dKαP(α) exp [−NEN(α;{xi,yi})]≈ P(αcl) ×exp/bracketleftbigg −NEN(αcl;{xi,yi})−K 2lnN 2π−1 2ln detFN+···/bracketrightbigg ,(2.31)2.4. Learning and predictability 26 where αclis the “classical” value of αdetermined by the extremal conditions ∂EN(α;{xi,yi}) ∂αµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleα=αcl= 0, (2.32) the matrix FNconsists of the second derivatives of EN, FN=∂2EN(α;{xi,yi}) ∂αµ∂αν/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleα=αcl, (2.33) and···denotes terms that vanish as N→ ∞ . If we formulate the problem of estimating the parameters αfrom the samples {xi,yi}, then asN→ ∞ the matrix NFNis the Fisher information matrix (Cover and Thomas 1991); th e eigenvectors of this matrix give the principal axes for the error ellipsoi d in parameter space, and the (inverse) eigenvalues give the variances of paramet er estimates along each of these directions. The classical αcldiffers from ¯αonly in terms of order 1/N; we neglect this difference and further simplify the calcul ation of leading terms asNbecomes large. After a little more algebra, then, we ﬁnd the p robabil- ity distribution we have been looking for: P(x1,y1,x2,y2,···,xN,yN) /tildewider→/bracketleftBiggN/productdisplay i=1P(xi)/bracketrightBigg1 ZAP(¯α) exp/bracketleftbigg −N 2ln(2πeσ2)−K 2lnN+···/bracketrightbigg ,(2.34) where the normalization constant ZA=/radicalBig (2π)KdetA∞. (2.35) Again we note that the sample points {xi,yi}are hidden in the value of ¯αthat gave rise to these points.4 4We emphasize again that there are two approximations leadin g to Eq. (2.34). First, we have replaced empirical means by expectation values, neglectin g ﬂuctuations associated with the par- ticular set of sample points {xi, yi}. Second, we have evaluated the average over parameters in a2.4. Learning and predictability 27 To evaluate the entropy S(N)we need to compute the expectation value of the (negative) logarithm of the probability distribution in Eq . (2.34); there are three terms. One is constant, so averaging is trivial. The second t erm depends only on thexi, and because these are chosen independently from the distri butionP(x) the average again is easy to evaluate. The third term involve s¯α, and we need to average this over the joint distribution P(x1,y1,x2,y2,···,xN,yN). As above, we can evaluate this average in steps: ﬁrst we choose a value of t he parameters ¯α, then we average over the samples given these parameters, and ﬁnally we average over parameters. But because ¯αis deﬁned as the parameters that generate the samples, this stepwise procedure simpliﬁes enormously. Th e end result is that S(N) =N/bracketleftbigg Sx+1 2log2(2πeσ2)/bracketrightbigg +K 2log2N+Sα+∝an}bracketle{tlog2ZA∝an}bracketri}htα+···, (2.36) where ∝an}bracketle{t···∝an}bracketri}ht αmeans averaging over parameters, Sxis the entropy of the distribu- tion ofx, Sx=−/integraldisplay dxP(x) log2P(x), (2.37) and similarly for the entropy of the distribution of paramet ers, Sα=−/integraldisplay dKαP(α) log2P(α). (2.38) saddle point approximation. At least under some condition, both of these approximations would become increasingly accurate as N→ ∞ , so that this approach should yield the asymptotic be- havior of the distribution and hence the subextensive entro py at large N. Although we give a more detailed analysis below, it is worth noting here how thi ngs can go wrong. The two ap- proximations are independent, and we could imagine that ﬂuc tuations are important but saddle point integration still works, for example. Controlling th e ﬂuctuations turns out to be exactly the question of whether our ﬁnite parameterization captures th e true dimensionality of the class of models, as discussed in the classic work of Vapnik, Chervone nkis, and others [see Vapnik (1998) for a review]. The saddle point approximation can break down because the saddle point becomes unstable or because multiple saddle points become importan t. It will turn out that instability is exponentially improbable as N→ ∞ , while multiple saddle points are a real problem in certain classes of models, again when counting parameters doesn’t r eally measure the complexity of the model class.2.4. Learning and predictability 28 The different terms in the entropy Eq. (2.36) have a straight forward interpre- tation. First we see that the extensive term in the entropy, S0=Sx+1 2log2(2πeσ2), (2.39) reﬂects contributions from the random choice of xand from the Gaussian noise iny; these extensive terms are independent of the variations in parameters α, and these would be the only terms if the parameters were not va rying (that is, if there were nothing to learn). There also is a term which reﬂ ects the entropy of variations in the parameters themselves, Sα. This entropy is not invariant with respect to coordinate transformations in the paramete r space, but the term ∝an}bracketle{tlog2ZA∝an}bracketri}htαcompensates for this noninvariance. Finally, and most inte restingly for our purposes, the subextensive piece of the entropy is domin ated by a logarith- mic divergence, S1(N)→K 2log2N(bits). (2.40) The coefﬁcient of this divergence counts the number of param eters independent of the coordinate system that we choose in the parameter spac e. Furthermore, this result does not depend on the set of basis functions {φµ(x)}. This is a hint that the result in Eq. (2.40) is more universal than our simpl e example. 2.4.2 Learning a parameterized distribution The problem discussed above is an example of supervised lear ning: we are given examples of how the points xnmap intoyn, and from these examples we are to in- duce the association or functional relation between xandy. An alternative view is that pair of points (x,y)should be viewed as a vector /vector x, and what we are learning is the distribution of this vector. The problem of learning a distribution usually2.4. Learning and predictability 29 is called unsupervised learning, but in this case supervise d learning formally is a special case of unsupervised learning; if we admit that all t he functional relations or associations that we are trying to learn have an element of noise or stochastic- ity, then this connection between supervised and unsupervi sed problems is quite general. Suppose a series of random vector variables {/vector xi}are drawn independently from the same probability distribution Q(/vector x\|α), and this distribution depends on a (potentially inﬁnite dimensional) vector of parameters α. As above, the param- eters are unknown, and before the series starts they are chos en randomly from a distribution P(α). With no constraints on the densities P(α)orQ(/vector x\|α)it is impossible to derive any regression formulas for parameter estimation, but one can still calculate the leading terms in the entropy of the da ta series and thus the predictive information. We begin with the deﬁnition of entropy S(N)≡S[{/vector xi}] =−/integraldisplay d/vector x1···d/vector xNP(/vector x1,/vector x2,···,/vector xN) log2P(/vector x1,/vector x2,···,/vector xN).(2.41) By analogy with Eq. (2.20) we then write P(/vector x1,/vector x2,···,/vector xN) =/integraldisplay dKαP(α)N/productdisplay i=1Q(/vector xi\|α). (2.42) Next, combining the last two equations and rearranging the o rder of integration, we can rewrite S(N)as S(N) = −/integraldisplay dK¯αP(¯α)  /integraldisplay d/vector x1···d/vector xNN/productdisplay j=1Q(/vector xj\|¯α) log2P({/vector xi})  .(2.43) Eq. (2.43) allows an easy interpretation. There is the ‘true ’ set of parameters ¯αthat gave rise to the data sequence /vector x1···/vector xNwith the probability/producttextN j=1Q(/vector xj\|¯α).2.4. Learning and predictability 30 We need to average log2P(/vector x1···/vector xN)ﬁrst over all possible realizations of the data keeping the true parameters ﬁxed, and then over the paramete rs¯αthemselves. With this interpretation in mind, the joint probability den sity, the logarithm of which is being averaged, can be rewritten in the following us eful way: P(/vector x1,···,/vector xN) =N/productdisplay j=1Q(/vector xj\|¯α)/integraldisplay dKαP(α)N/productdisplay i=1/bracketleftBiggQ(/vector xi\|α) Q(/vector xi\|¯α)/bracketrightBigg =N/productdisplay j=1Q(/vector xj\|¯α)/integraldisplay dKαP(α) exp [−NEN(α;{/vector xi})], (2.44) EN(α;{/vector xi}) = −1 NN/summationdisplay i=1ln/bracketleftBiggQ(/vector xi\|α) Q(/vector xi\|¯α)/bracketrightBigg . (2.45) Since, by our interpretation, ¯αare the true parameters that gave rise to the partic- ular data {/vector xi}, we may expect empirical means to converge to expectation va lues, so that EN(α;{/vector xi}) =−/integraldisplay dDxQ(x\|¯α) ln/bracketleftBiggQ(/vector x\|α) Q(/vector x\|¯α)/bracketrightBigg −ψ(α,¯α;{xi}), (2.46) whereψ→0asN→ ∞ ; here we neglect ψ, and return to this term below. The ﬁrst term on the right hand side of Eq. (2.46) is the Kullba ck–Leibler di- vergence,DKL(¯α\|\|α), between the true distribution characterized by parameter s ¯αand the possible distribution characterized by α. Thus at large Nwe have P(/vector x1,/vector x2,···,/vector xN)/tildewider→N/productdisplay j=1Q(/vector xj\|¯α)/integraldisplay dKαP(α) exp [−NDKL(¯α\|\|α)], (2.47) where again the notation /tildewider→reminds us that we are not only taking the limit of largeNbut also making another approximation in neglecting ﬂuctua tions. By the same arguments as above we can proceed (formally) to comp ute the entropy of this distribution, and we ﬁnd S(N)≈ S 0·N+S(a) 1(N), (2.48)2.4. Learning and predictability 31 S0=/integraldisplay dKαP(α)/bracketleftbigg −/integraldisplay dDxQ(/vector x\|α) log2Q(/vector x\|α)/bracketrightbigg ,and (2.49) S(a) 1(N) = −/integraldisplay dK¯αP(¯α) log2/bracketleftbigg/integraldisplay dKαP(α)e−NDKL(¯α\|\|α)/bracketrightbigg . (2.50) HereS(a) 1is an approximation to S1that neglects ﬂuctuations ψ. This is the same as the annealed approximation in the statistical mechanics of disordered systems, as has been used widely in the study of supervised learning pr oblems (Seung et al. 1992). Thus we can identify the data sequence /vector x1···/vector xNwith the disorder, EN(α;{/vector xi})with the energy of the quenched system, and DKL(¯α\|\|α)with its an- nealed analogue. The extensive term S0, Eq. (2.49), is the average entropy of a distribution in our family of possible distributions, generalizing the res ult of Eq. (2.39). The subextensive terms in the entropy are controlled by the Ndependence of the partition function Z(¯α;N) =/integraldisplay dKαP(α) exp [−NDKL(¯α\|\|α)], (2.51) andS1(N) =−∝an}bracketle{tlog2Z(¯α;N)∝an}bracketri}ht¯αis analogous to the free energy. Since what is important in this integral is the Kullback–Leibler (KL) div ergence between dif- ferent distributions, it is natural to ask about the density of models that are KL divergence Daway from the target ¯α, ρ(D;¯α) =/integraldisplay dKαP(α)δ[D−DKL(¯α\|\|α)]; (2.52) note that this density could be very different for different targets. The density of divergences is normalized because the original distribu tion over parameter space,P(α), is normalized, /integraldisplay dDρ(D;¯α) =/integraldisplay dKαP(α) = 1. (2.53)2.4. Learning and predictability 32 Finally, the partition function takes the simple form Z(¯α;N) =/integraldisplay dDρ(D;¯α) exp[−ND]. (2.54) We recall that in statistical mechanics the partition funct ion is given by Z(β) =/integraldisplay dEρ(E) exp[−βE], (2.55) whereρ(E)is the density of states that have energy E, andβis the inverse tem- perature. Thus the subextensive entropy in our learning pro blem is analogous to a system in which energy corresponds to the Kullback–Leible r divergence rela- tive to the target model, and temperature is inverse to the nu mber of examples. As we increase the length Nof the time series we have observed, we “cool” the system and hence probe models which approach the target; the dynamics of this approach is determined by the density of low energy states, t hat is the behavior ofρ(D;¯α)asD→0. The structure of the partition function is determined by a co mpetition be- tween the (Boltzmann) exponential term, which favors model s with small D, and the density term, which favors values of Dthat can be achieved by the largest possible number of models. Because there (typically) are ma ny parameters, there are very few models with D→0. This picture of competition between the Boltz- mann factor and a density of states has been emphasized in pre vious work on supervised learning (Haussler et al. 1996). The behavior of the density of states, ρ(D;¯α), at smallDis related to the more intuitive notion of dimensionality. In a parameterized fam ily of distributions, the Kullback–Leibler divergence between two distributions wi th nearby parameters2.4. Learning and predictability 33 is approximately a quadratic form, DKL(¯α\|\|α)≈1 2/summationdisplay µν(¯αµ−αµ)Fµν(¯αν−αν) +···, (2.56) where Fis the Fisher information matrix. Intuitively, if we have a r easonable parameterization of the distributions, then similar distr ibutions will be nearby in parameter space, and more importantly points that are far apart in parameter space will never correspond to similar distributions; Clar ke and Barron (1990) refer to this condition as the parameterization forming a “s ound” family of dis- tributions. If this condition is obeyed, then we can approxi mate the low Dlimit of the density ρ(D;¯α): ρ(D;¯α) =/integraldisplay dKαP(α)δ[D−DKL(¯α\|\|α)] ≈/integraldisplay dKαP(α)δ/bracketleftBigg D−1 2/summationdisplay µν(¯αµ−αµ)Fµν(¯αν−αν)/bracketrightBigg =/integraldisplay dKαP(¯α+U ·ξ)δ/bracketleftBigg D−1 2/summationdisplay µΛµξ2 µ/bracketrightBigg , (2.57) where Uis a matrix that diagonalizes F, (UT· F · U )µν= Λµδµν. (2.58) The delta function restricts the components of ξin Eq. (2.57) to be of order√ Dor less, and so if P(α)is smooth we can make a perturbation expansion. After some algebra the leading term becomes ρ(D→0;¯α)≈ P(¯α)2πK/2 Γ(K/2)(detF)−1/2D(K−2)/2. (2.59) Here, as before, Kis the dimensionality of the parameter vector. Computing th e partition function from Eq. (2.54), we ﬁnd Z(¯α;N→ ∞)≈f(¯α)·Γ(K/2) NK/2, (2.60)2.4. Learning and predictability 34 wheref(¯α)is some function of the target parameter values. Finally, th is allows us to evaluate the subextensive entropy, from Eqs. (2.50, 2. 51): S(a) 1(N) = −/integraldisplay dK¯αP(¯α) log2Z(¯α;N) (2.61) →K 2log2N+···(bits), (2.62) where ···are ﬁnite as N→ ∞ . Thus, general K–parameter model classes have the same subextensive entropy as for the simplest example co nsidered in the pre- vious section. To the leading order, this result is independ ent even of the prior distribution P(α)on the parameter space, so that the predictive information seems to count the number of parameters under some very gener al conditions [cf. Fig. (2.3) for a numerical example of the logarithmic be havior]. Although Eq. (2.62) is true under a wide range of conditions, this cannot be the whole story. Much of modern learning theory is concerned with the fact that counting parameters is not quite enough to characteriz e the complexity of a model class; the naive dimension of the parameter space Kshould be viewed in conjunction with the Vapnik–Chervonenkis (VC) dimensio ndVC(also known as the pseudodimension) and the phase space dimension d. The phase space di- mension is deﬁned in the usual way through the scaling of volu mes in the model space (see, for example, Opper 1994). On the other hand, dVCmeasures not vol- umes, but capacity of the model class, and its deﬁnition is a b it trickier: for a set of binary (indicator) functions F(/vector x,α), VC dimension is deﬁned as the maximal number of vectors /vector x1···/vector xdVCthat can be classiﬁed into two different classes in all2dVCpossible ways using this set of functions. Similarly, for re al–valued func- tionsF(/vector x,α)one can ﬁrst deﬁne a complete set of indicators using step fun ctions, θ[F(/vector x,α)−β], and then the VC dimension of this set is the VC dimension of th e2.4. Learning and predictability 35 real–valued functions (Vapnik 1998). Separation of a vecto r in all possible ways is called shattering, and hence another name for the VC dimensi on—the shattering dimension. BothdanddVCcan differ from the number of parameters in several ways. One possibility is that dVCis inﬁnite when the number of parameters is ﬁnite, a prob- lem discussed below. Another possibility is that the determ inant of Fis zero, and hencedVCanddare both smaller than the number of parameters because we have adopted a redundant description. It is possible that th is sort of degeneracy occurs over a ﬁnite fraction but not all of the parameter spac e, and this is one way to generate an effective fractional dimensionality. One ca n imagine multifractal models such that the effective dimensionality varies conti nuously over the pa- rameter space, but it is not obvious where this would be relev ant. Finally, models withd<d VC<∞are also possible [see, for example, Opper (1994)], and this list probably is not exhaustive. The calculation above, Eq. (2.59), lets us actually deﬁne the phase space dimen- sion through the exponent in the small DKLbehavior of the model density, ρ(D→0;¯α)∝D(d−2)/2, (2.63) and thendappears in place of Kas the coefﬁcient of the log divergence in S1(N) (Clarke and Barron 1990, Opper 1994). However, this simple c onclusion can fail in two ways. First, it can happen that a macroscopic weight ge ts accumulated at some nonzero value of DKL, so that the small DKLbehavior is irrelevant for the largeNasymptotics. Second, the ﬂuctuations neglected here may be uncon- trollably large, so that the asymptotics are never reached. Since controllability of ﬂuctuations is a function of dVC(see Vapnik 1998 and later in this paper), we may2.4. Learning and predictability 36 summarize this in the following way. Provided that the small DKLbehavior of the density function is the relevant one, the coefﬁcient of the l ogarithmic divergence ofIpredmeasures the phase space or the scaling dimension dand nothing else. This asymptote is valid, however, only for N≫dVC. It is still an open question whether the two pathologies that can violate this asymptoti c behavior are related. 2.4.3 Learning a parameterized process Consider a process where samples are not independent, and ou r task is to learn their joint distribution Q(/vector x1,···,/vector xN\|α). Again, αis an unknown parameter vector which is chosen randomly at the beginning of the series. If αis aKdimensional vector, then one still tries to learn just Knumbers and there are still Nexamples, even if there are correlations. Therefore, although such pr oblems are much more general than those considered above, it is reasonable to exp ect that the predictive information is still measured by (K/2) log2Nprovided that some conditions are met. One might suppose that conditions for simple results on the p redictive in- formation are very strong, for example that the distributio nQis a ﬁnite order Markov model. In fact all we really need are the following two conditions: S[{/vector xi}\|α]≡ −/integraldisplay dN/vector xQ({/vector xi}\|α) log2Q({/vector xi}\|α) →NS0+S∗ 0;S∗ 0=O(1), (2.64) DKL[Q({/vector xi}\|¯α)\|\|Q({/vector xi}\|α)]→NDKL(¯α\|\|α) +o(N). (2.65) Here the quantities S0,S∗ 0, andDKL(¯α\|\|α)are deﬁned by taking limits N→ ∞ in both equations. The ﬁrst of the constraints limits deviat ions from extensivity to be of order unity, so that if αis known there are no long range correlations2.4. Learning and predictability 37 in the data—all of the long range predictability is associat ed with learning the parameters.5The second constraint, Eq. (2.65), is a less restrictive one , and it ensures that the “energy” of our statistical system is an ext ensive quantity. With these conditions it is straightforward to show that the results of the pre- vious subsection carry over virtually unchanged. With the s ame cautious state- ments about ﬂuctuations and the distinction between K,d, anddVC, one arrives at the result: S(N) = S0·N+S(a) 1(N), (2.66) S(a) 1(N) =K 2log2N+···(bits), (2.67) where ···stands for terms of order one. Note again that for the results Eq. (2.67) to be valid, the process considered is not required to be a ﬁni te order Markov pro- cess. Memory of all previous outcomes may be kept, provided t hat the accumu- lated memory does not contribute a divergent term to the sube xtensive entropy. It is interesting to ask what happens if the condition in Eq. ( 2.64) is vio- lated, so that there are long range correlations even in the c onditional distribution Q(/vector x1,···,/vector xN\|α). Suppose, for example, that S∗ 0= (K∗/2) log2N. Then the subex- tensive entropy becomes S(a) 1(N) =K+K∗ 2log2N+···(bits). (2.68) We see the that the subextensive entropy makes no distinctio n between predicta- bility that comes from unknown parameters and predictabili ty that comes from intrinsic correlations in the data; in this sense, two model s with the same K+K∗ 5Suppose that we observe a Gaussian stochastic process and we try to learn the power spec- trum. If the class of possible spectra includes ratios of pol ynomials in the frequency (rational spectra) then this condition is met. On the other hand, if the class of possible spectra includes 1/f noise, then the condition may not be met. For more on long rang e correlations, see below.2.4. Learning and predictability 38 are equivalent. This, actually, must be so. As an example, consider a chain of Ising spins with long range interactions in one dimension. T his system can order (magnetize) and exhibit long range correlations, and so the predictive informa- tion will diverge at the transition to ordering. In one view, there is no global pa- rameter analogous to α, just the long range interactions. On the other hand, there are regimes in which we can approximate the effect of these in teractions by say- ing that all the spins experience a mean ﬁeld which is constan t across the whole length of the system, and then formally we can think of the pre dictive informa- tion as being carried by the mean ﬁeld itself. In fact there ar e situations in which this is not just an approximation, but an exact statement. Th us we can trade a de- scription in terms of long range interactions ( K∗∝ne}ationslash= 0, butK= 0) for one in which there are unknown parameters describing the system but give n these parameters there are no long range correlations ( K∝ne}ationslash= 0, K∗= 0). The two descriptions are equivalent, and this is captured by the subextensive entrop y.6 2.4.4 Taming the ﬂuctuations: ﬁnite dVCcase The preceding calculations of the subextensive entropy S1are worthless unless we prove that the ﬂuctuations ψare controllable. In this subsection we are going to discuss when and if this, indeed, happens. We limit the dis cussion to the anal- ysis of ﬂuctuations in the case of ﬁnding a probability densi ty (Section 2.4.2); the case of learning a process (Section 2.4.3) is very similar. Clarke and Barron (1990) solved essentially the same proble m. They did not make a separation into the annealed and the ﬂuctuation term, and the quantity 6There are a number of interesting questions about how the coe fﬁcients in the diverging pre- dictive information relate to the usual critical exponents , and we hope to return to this problem in a later paper.2.4. Learning and predictability 39 they were interested in was a bit different from ours, but, in terpreting loosely, they proved that, modulo some reasonable technical assumpt ions on differen- tiability of functions in question, the ﬂuctuation term alw ays approaches zero. However, they did not investigate the speed of this approach , and we believe that, by doing so, they missed some important qualitative di stinctions between different problems that can arise due to a difference betwee ndanddVC. In or- der to illuminate these distinctions, we here go through the trouble of analyzing ﬂuctuations all over again. Returning to Eqs. (2.44, 2.46) and the deﬁnition of entropy, we can write the entropyS(N)exactly as S(N) = −/integraldisplay dK¯αP(¯α)/integraldisplayN/productdisplay j=1[d/vector xjQ(/vector xj\|¯α)] ×log2/bracketleftBiggN/productdisplay i=1Q(/vector xi\|¯α)/integraldisplay dKαP(α) e−NDKL(¯α\|\|α)+Nψ(α,¯α;{/vector xi})/bracketrightBigg .(2.69) This expression can be decomposed into the terms identiﬁed a bove, plus a new contribution to the subextensive entropy that comes from th e ﬂuctuations alone, S(f) 1(N): S(N) = S0·N+S(a) 1(N) +S(f) 1(N), (2.70) S(f) 1=−/integraldisplay dK¯αP(¯α)N/productdisplay j=1[d/vector xjQ(/vector xj\|¯α)] ×log2/bracketleftBigg/integraldisplaydKαP(α) Z(¯α;N)e−NDKL(¯α\|\|α)+Nψ(α,¯α;{/vector xi})/bracketrightBigg , (2.71) whereψis deﬁned as in Eq. (2.46), and Zas in Eq. (2.51). Some loose but useful bounds can be established. First, the p redictive infor- mation is a positive (semideﬁnite) quantity, and so the ﬂuct uation term may not be smaller than the value of −S(a) 1as calculated in Eqs. (2.62, 2.67). Second, since2.4. Learning and predictability 40 ﬂuctuations make it more difﬁcult to generalize from sample s, the predictive in- formation should always be reduced by ﬂuctuations, so that S(f)is negative. This last statement corresponds to the fact that for the statisti cal mechanics of disor- dered systems, the annealed free energy always is less than t he average quenched free energy, and may be proven rigorously by applying Jensen ’s inequality to the (concave) logarithm function in Eq. (2.71); essentially th e same argument was given by Opper and Haussler (1995). A related Jensen’s inequ ality argument al- lows us to show that the total S1(N)is bounded, S1(N)≤N/integraldisplay dKα/integraldisplay dK¯αP(α)P(¯α)DKL(¯α\|\|α) ≡ ∝an}bracketle{tDKL(¯α\|\|α)∝an}bracketri}ht¯α,α, (2.72) so that if we have a class of models (and a prior P(α)) such that the average Kullback–Leibler divergence among pairs of models is ﬁnite , then the subexten- sive entropy is necessarily properly deﬁned. Note that ∝an}bracketle{tDKL(¯α\|\|α)∝an}bracketri}ht¯α,αincludes S0as one of its terms, so that usually S0andS1are well– or ill–deﬁned together. Tighter bounds require nontrivial assumptions about the cl asses of distribu- tions considered. The ﬂuctuation term would be zero if ψwere zero, and ψis the difference between an expectation value (KL divergence) an d the corresponding empirical mean. There is a broad literature that deals with t his type of difference (see, for example, Vapnik 1998). We start with the case when the pseudo-dimension ( dVC) of the set of proba- bility densities {Q(/vector x\|α)}is ﬁnite. Then for any reasonable function F(/vector x;β), de- viations of the empirical mean from the expectation value ca n be bounded by probabilistic bounds of the form P/braceleftBigg sup β/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 N/summationtext jF(/vector xj;β)−/integraltextd/vector xQ(/vector x\|¯α)F(/vector x;β) L[F]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ǫ/bracerightBigg2.4. Learning and predictability 41 <M(ǫ,N,d VC)e−cNǫ2, (2.73) wherecandL[F]depend on the details of the particular bound used. Typicall y, cis a constant of order one, and L[F]is either some moment of For the range of its variation. In our case, Fis the log–ratio of two densities, so that L[F] may be assumed bounded for almost all βwithout loss of generality in view of Eq. (2.72). In addition, M(ǫ,N,d VC)is ﬁnite at zero, grows at most subexpo- nentially in its ﬁrst two arguments, and depends exponentia lly ondVC. Bounds of this form may have different names in different contexts: Glivenko–Cantelli, Vapnik–Chervonenkis, Hoeffding, Chernoff, ...; for revie w see Vapnik (1998) and the references therein. To start the proof of ﬁniteness of S(f) 1in this case, we ﬁrst show that only the region α≈¯αis important when calculating the inner integral in Eq. (2.7 1). This statement is equivalent to saying that at large values o fα−¯αthe KL di- vergence almost always dominates the ﬂuctuation term, that is, the contribution of sequences of {/vector xi}with atypically large ﬂuctuations is negligible (atypical ity is deﬁned as ψ≥δ, whereδis some small constant independent of N). Since the ﬂuctuations decrease as 1/√ N[see Eq. (2.73)], and DKLis of order one, this is plausible. To show this, we bound the logarithm in Eq. (2.7 1) byNtimes the supremum value of ψ. Then we realize that the averaging over ¯αand{/vector xi}is equivalent to integration over all possible values of the ﬂu ctuations. The worst case density of the ﬂuctuations may be estimated by differen tiating Eq. (2.73) with respect to ǫ(this brings down an extra factor of Nǫ). Thus the worst case contribution of these atypical sequences is S(f),atypical 1 ∼/integraldisplay∞ δdǫN2ǫ2M(ǫ)e−cNǫ2∼e−cNδ2≪1 for largeN. (2.74)2.4. Learning and predictability 42 This bound lets us focus our attention on the region α≈¯α. We expand the exponent of the integrand of Eq. (2.71) around this point and perform a simple Gaussian integration. In principle, large ﬂuctuations mig ht lead to an instability (positive or zero curvature) at the saddle point, but this is atypical and therefore is accounted for already. Curvatures at the saddle points of both numerator and denominator are of the same order, and throwing away unimpor tant additive and multiplicative constants of order unity, we obtain the f ollowing result for the contribution of typical sequences: S(f),typical 1 ∼/integraldisplay dK¯αP(¯α)dN/vector x/productdisplay jQ(/vector xj\|¯α)N(BA−1B) ; (2.75) Bµ=1 N/summationdisplay i∂logQ(/vector xi\|¯α) ∂¯αµ,∝an}bracketle{tB∝an}bracketri}ht/vector x= 0 ; (A)µν=1 N/summationdisplay i∂2logQ(/vector xi\|¯α) ∂¯αµ∂¯αν,∝an}bracketle{tA∝an}bracketri}ht/vector x=F. Here∝an}bracketle{t···∝an}bracketri}ht/vector xmeans an averaging with respect to all /vector xi’s keeping ¯αconstant. One immediately recognizes that BandAare, respectively, ﬁrst and second deriva- tives of the empirical KL divergence that was in the exponent of the inner integral in Eq. (2.71). We are dealing now with typical cases. Therefore, large devi ations of Afrom Fare not allowed, and we may bound Eq. (2.75) by replacing A−1withF−1(1+δ), whereδagain is independent of N. Now we have to average a bunch of products like ∂logQ(/vector xi\|¯α) ∂¯αµ(F−1)µν∂logQ(/vector xj\|¯α) ∂¯αν(2.76) over all/vector xi’s. Only the terms with i = j survive the averaging. There are K2Nsuch terms, each contributing of order N−1. This means that the total contribution of2.4. Learning and predictability 43 the typical ﬂuctuations is bounded by a number of order one an d does not grow withN. This concludes the proof of controllability of ﬂuctuation s fordVC<∞. 2.4.5 Taming the ﬂuctuations: the role of the prior One may notice that we never used the speciﬁc form of M(ǫ,N,d VC), which is the only thing dependent on the precise value of the dimensio n. Actually, a more thorough look at the proof shows that we do not even need the st rict uniform convergence enforced by the Glivenko–Cantelli bound. With some modiﬁcations the proof should still hold if there exist some a priori improbable values of αand ¯αthat lead to violation of the bound. That is, if the prior P(α)has sufﬁciently narrow support, then we may still expect ﬂuctuations to be un important even for VC–inﬁnite problems. To see this, consider two examples. A variable xis distributed according to the following probability density functions: Q(x\|α) =1√ 2πexp/bracketleftbigg −1 2(x−α)2/bracketrightbigg , x∈(−∞; +∞) ; (2.77) Q(x\|α) =exp (−sinαx) /integraltext2π 0dxexp (−sinαx), x∈[0; 2π). (2.78) Learning the parameter in the ﬁrst case is a dVC= 1 problem, while in the sec- ond casedVC=∞. In the ﬁrst example, as we have shown above, one may construct a uniform bound on ﬂuctuations irrespective of th e prior P(α). The second one does not allow this. Indeed, suppose that the prio r is uniform in a box0< α < α max, and zero elsewhere, with αmaxrather large. Then for not too many sample points N, the data would be better ﬁtted not by some value in the vicinity of the actual parameter, but by some much larger val ue, for which almost all data points are at the crests of −sinαx. Adding a new data point would not2.4. Learning and predictability 44 help, until that best, but wrong, parameter estimate is less thanαmax.7So the ﬂuctuations are large, and the predictive information is sm all in this case. Even- tually, however, data points would overwhelm the box size, a nd the best esti- mate ofαwould swiftly approach the actual value. At this point the ar gument of Clarke and Barron (1990) would become applicable, and the le ading behavior of the subextensive entropy would converge to its asymptotic v alue of (1/2) logN. On the other hand, there is no uniform bound on the value of Nfor which this convergence will occur—it is guaranteed only for N≫dVC, which is never true ifdVC=∞. For some sufﬁciently wide priors this asymptotically corr ect behav- ior would be never reached in practice. Further, if we imagin e a thermodynamic limit where the box size and the number of samples both become large, then by analogy with problems in supervised learning (Seung et al. 1 992, Haussler et al. 1996) we expect that there can be sudden changes in performan ce as a function of the number of examples. The arguments of Clarke and Barron ca nnot encompass these phase transitions or “aha!” phenomena. Following the intuition inferred from this example, we can n ow proceed with a more formal analysis. As the above argument about the small ness of ﬂuctua- tions in the ﬁnite dVCcase paralleled the discussion of the Empirical Risk Mini- mization (ERM) approach (Vapnik 1998), this present argume nt closely resembles some statements of the Structural Risk Minimization (SRM) t heory (Vapnik 1998), which deals with the case of dVC=∞or, equivalently, N/d VC<1. While ERM solves the problem of uniform non–Bayesian learning, there seems to be a general 7Interestingly, since for the model Eq. (2.78) KL divergence is bounded from below and above , forαmax→ ∞ the weight in ρ(D;¯α)at small DKLvanishes, and a ﬁnite weight accumulates at some nonzero value of D. Thus, even putting the ﬂuctuations aside, the asymptotic b ehavior based on the phase space dimension is invalidated, as mentio ned above.2.4. Learning and predictability 45 agreement that SRM theory is a solution to the problem of lear ning with a prior. However, to our knowledge, no explicit identiﬁcation of why this is so has been done, so we try to do it here. Suppose that, as in the above example, admissible solutions of a learning problem belong to some subset C1of the whole K–dimensional parameter space C. Suppose also that for any ﬁnite C1the VC dimension of the correspond- ing learning problem, dVC(C1), is ﬁnite, but dVC(C) =∞. In SRM theory a nested set of such subspaces C1⊂C2⊂C3⊂ ··· is called a structure Cif C=/uniontextCn. EachCnis known as a structure element . Since the subsets are nested, dVC(C1)≤dVC(C2)≤dVC(C3)≤ ··· . We know that these are the large VC di- mensions and, therefore, parameters that belong to the larg e structure elements Cn, n→ ∞ , that are responsible for large ﬂuctuations. But in view of E q. (2.53), for any properly deﬁned prior P(α), very large values of αare a priori improb- able. Thus the ﬁght between the prior and the data may result i n an effective cutoffn∗, so that all Cn,n > n∗,contribute little to S(f) 1, and the ﬂuctuations are controlled. Indeed, let’s form a structure by assigning all α’s for which −logP(α) + max log P ≤nto the element Cn(nis not necessarily integer). This imposes an a priori probability ν(n)on the elements themselves. Now we can bound the internal integral in Eq. (2.71) by replacing ψ(α,¯α,{/vector xi})withψn(¯α,{/vector xi})—its maximal value on the smallest element Cnthat includes α. If the logarithm of the a priori probability ν(n)falls off faster than Nψn(¯α,{/vector xi})increases as ngrows, then one can select a particular n∗, for which the integral over all Cn, n > n∗,is smaller than any predeﬁned δ. Effectively n∗then serves as a cutoff. Note that, since ﬂuctuations enter multiplied by N,n∗(N)is a nondecreasing function. If it2.4. Learning and predictability 46 grows in a way such that dVC(Cn∗)is sublinear in N(∼N/logNsufﬁces), then M(ǫ,N,d VC)is still subexponential, and we can use the proofs of the prec eding section to show that the ﬂuctuations are controllable. The o nly difference that occurs is that the contribution of typical ﬂuctuations is do minated by a saddle point near αcl, which solves the equation ∂ ∂αµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleαcl[−logP(α) +ND(¯α\|\|α)] = 0. (2.79) If¯αis only in very large structure elements that contribute lit tle to the internal integral of Eq. (2.71), then αclmay be quite far from ¯α. That is, the best estimate of¯αmay be imprecise at any ﬁnite N. This is particularly important in the case of nonparametric learning (see Sections 2.4.7, 3.5). In ﬁnite dimensional cases similar to the above example, eve ryCn, n<∞,has ﬁnite VC dimension dVC, and this dimension is bounded from above by the phase space dimension d. The magnitude of ﬂuctuations depends mostly on dVC. There- fore, beyond some n∗(N)for whichdVC(Cn∗) =d, the ﬂuctuations will practically stop growing. This means that any proper prior P, however slowly decreasing at inﬁnities, is enough to impose a ﬁnite cutoff and render ﬂuct uations ﬁnite. This is in complete agreement with Clark and Barron—but prior-de pendent. We want to emphasize again that, in general, ﬂuctuations are controlled only if two related, but not equivalent, assumptions are true. Fi rst, for any ﬁnite N there has to be a ﬁnite cutoff n∗(N). This means that P(α)is narrow enough to deﬁne a valid structure. Second, for the ﬂuctuations with inCn∗to be small, dVC(Cn∗(N))must grow sublinearly in N.8In this case the number of samples 8Actually, the n∗-dependence of the factors similar to L[F], deﬁned above, may require a dif- ferent, yet slower, growth [see Vapnik (1998) for details]. But this is outside the scope of this discussion.2.4. Learning and predictability 47 eventually outgrows the current VC dimension by an arbitrar ily large factor, and determination of parameters is possible to any precision. B oth of these conditions are well known in SRM theory (Vapnik 1998). In the classical SRM theory, only selection of the law n∗=n∗(N)is a part of the problem, and the structure is usually assumed to be given . Ideally, this law is selected by minimizing the expected error of learning, wh ich consists of uncer- tainties due to the limited set of allowed solutions ( n∗<∞) and due to the ﬂuc- tuations within this set. These uncertainties behave oppos itely asn∗increases. If calculating the expected error is difﬁcult, people may be co ntent with even pre- selecting the law n∗=n∗(N), and then every law for which the VC dimension grows sublinearly does the job—better or worse—just as we ha ve shown above. In our current treatment the structure andthe law of the VC dimension growth are both a result of the prior. If the prior is appropriate, then s o are the structure and the law. If not, then learning with this prior is impossible. On general grounds, we know that when the prior correctly embodies the a priori kn owledge, it re- sults in the fastest average learning possible. Therefore w e are guaranteed that, on average, the law n∗=n∗(N)is optimal if this law is imposed by the prior (see Sections 3.4, 3.5 for more on this). Summarizing, we note that while much of learning theory has p roperly fo- cused on problems with ﬁnite VC dimension, it might be that th e conventional scenario in which the number of examples eventually overwhe lms the number of parameters or dimensions is too weak to deal with many real world prob- lems. Certainly in the present context there is not only a qua ntitative, but also a qualitative difference between reaching the asymptotic re gime in just a few mea- surements, or in many millions of them. Finitely parameteri zable models with2.4. Learning and predictability 48 ﬁnite or inﬁnite dVCfall in essentially different universality classes with respect to the predictive information. 2.4.6 Beyond ﬁnite parameterization: general considerati ons The previous sections have considered learning from time se ries where the un- derlying class of possible models is described with a ﬁnite n umber of parameters. If the number of parameters is not ﬁnite then in principle it i s impossible to learn anything unless there is some appropriate regularization o f the problem. If we let the number of parameters stay ﬁnite but become large, the n there is more to be learned and correspondingly the predictive information grows in proportion to this number, as in Eq. (2.62). On the other hand, if the numb er of parameters becomes inﬁnite without regularization, then the predicti ve information should go to zero since nothing can be learned. We should be able to se e this happen in a regularized problem as the regularization weakens: event ually the regulariza- tion would be insufﬁcient and the predictive information wo uld vanish. The only way this can happen is if the subextensive term in the entropy grows more and more rapidly with Nas we weaken the regularization, until ﬁnally it becomes extensive at the point where learning becomes impossible. M ore precisely, if this scenario for the breakdown of learning is to work, there must be situations in which the predictive information grows with Nmore rapidly than the logarith- mic behavior found in the case of ﬁnite parameterization. Subextensive terms in the entropy are controlled by the dens ity of models as function of their Kullback–Leibler divergence to the targe t model. If the models have ﬁnite VC and phase space dimensions then this density va nishes for small2.4. Learning and predictability 49 divergences as ρ∼D(d−2)/2. Phenomenologically, if we let the number of param- eters increase, the density vanishes more and more rapidly. We can imagine that beyond the class of ﬁnitely parameterizable problems there is a class of regular- ized inﬁnite dimensional problems in which the density ρ(D→0)vanishes more rapidly than any power of D. As an example, we could have ρ(D→0)≈Aexp/bracketleftbigg −B Dµ/bracketrightbigg , µ> 0; (2.80) that is, an essential singularity at D= 0. For simplicity we assume that the con- stantsAandBcan depend on the target model, but that the nature of the esse ntial singularity ( µ) is the same everywhere. Before providing an explicit examp le, let us explore the consequences of this behavior. From Eq. (2.54) above, we can write the partition function as Z(¯α;N) =/integraldisplay dDρ(D;¯α) exp[−ND] ≈A(¯α)/integraldisplay dDexp/bracketleftBigg −B(¯α) Dµ−ND/bracketrightBigg ≈˜A(¯α) exp/bracketleftBigg −1 2µ+ 2 µ+ 1lnN−C(¯α)Nµ/(µ+1)/bracketrightBigg , (2.81) where in the last step we use a saddle point or steepest descen t approximation which is accurate at large N, and the coefﬁcients are ˜A(¯α) =A(¯α)/parenleftBigg2πµ1/(µ+1) µ+ 1/parenrightBigg1/2 ·[B(¯α)]1/(2µ+2)(2.82) C(¯α) = [B(¯α)]1/(µ+1)/parenleftBigg1 µµ/(µ+1)+µ1/(µ+1)/parenrightBigg . (2.83) Finally we can use Eqs. (2.61, 2.81) to compute the subextens ive term in the en- tropy, keeping only the dominant term at large N, S(a) 1(N)→1 ln 2∝an}bracketle{tC(¯α)∝an}bracketri}ht¯αNµ/(µ+1)(bits), (2.84)2.4. Learning and predictability 50 where ∝an}bracketle{t···∝an}bracketri}ht ¯αdenotes an average over all the target models. This behavior of the ﬁrst subextensive term is qualitativel y different from ev- erything we have observed so far. A power law divergence is mu ch stronger than a logarithmic one. Therefore, a lot more predictive informa tion is accumulated in an “inﬁnite parameter” (or nonparametric) system; the sy stem is much richer and more complex, both intuitively and quantitatively. Subextensive entropy also grows as a power law in a ﬁnitely pa rameterizable system with a growing number of parameters [compare to the sp in chain with decaying interactions on Fig. (2.3)]. For example, suppose that we approximate the distribution of a random variable by a histogram with Kbins, and we let K grow with the quantity of available samples as K∼Nν. Equation (2.62) sug- gests that in a K–parameter system, the Nthsample point contributes ∼K/2N bits to the subextensive entropy. If Kchanges as mentioned, the Nthexample then carries ∼Nν−1bits. Summing this up over all samples, we ﬁnd S(a) 1∼Nν, and if we let ν=µ/(µ+ 1) we obtain Eq. (2.84). Note that the growth of the number of parameters is slower than N(ν=µ/(µ+ 1)<1), which makes sense both intuitively and within the framework of the above SRM ﬂu ctuation analysis. Indeed, this growing number of parameters is nothing but exp anding structure elements, and dVCincreasing with them, dVC(Cn∗(N))≡dVC(N). Therefore, sub- linear growth is needed for the ﬂuctuation control. Power law growth of the predictive information illustrates the point made earlier about the transition from learning more to ﬁnally le arning nothing as the class of investigated models becomes more complex. As µincreases, the problem2.4. Learning and predictability 51 becomes richer and more complex, and this is expressed in the stronger diver- gence of the ﬁrst subextensive term of the entropy; for ﬁxed l argeN, the predic- tive information increases with µ. However, if µ→ ∞ the problem is too complex for learning—in our model example the number of bins grows in proportion to the number of samples, which means that we are trying to ﬁnd to o much detail in the underlying distribution. As a result, the subextensi ve term becomes exten- sive and stops contributing to predictive information. Thu s, at least to the leading order, predictability is lost, as promised. 2.4.7 Beyond ﬁnite parameterization: example The discussion in the previous section suggests that we shou ld look for power– law behavior of the predictive information in learning prob lems where rather than learning ever more precise values for a ﬁxed set of param eters, we learn a progressively more detailed description—effectively in creasing the number of parameters—as we collect more data. One example of such a pro blem is learn- ing the distribution Q(x)for a continuous variable x, but rather than writing a parametric form of Q(x)we assume only that this function itself is chosen from some distribution that enforces a degree of smoothness. The re are some natu- ral connections of this problem to the methods of quantum ﬁel d theory (Bialek, Callan, and Strong 1996) which we can exploit to give a comple te calculation of the predictive information, at least for a class of smoothne ss constraints. We writeQ(x) = (1/l0) exp[−φ(x)]so that positivity of the distribution is au- tomatic, and then smoothness may be expressed by saying that the ‘energy’ (or action) associated with a function φ(x)is related to an integral over its derivatives,2.4. Learning and predictability 52 like the strain energy in a stretched string. The simplest po ssibility following this line of ideas is that the distribution of functions is given b y P[φ(x)] =1 Zexp −l 2/integraldisplay dx/parenleftBigg∂φ ∂x/parenrightBigg2 δ/bracketleftbigg1 l0/integraldisplay dxe−φ(x)−1/bracketrightbigg , (2.85) where Zis the normalization constant for P[φ], the delta function insures that each distribution Q(x)is normalized, and lsets a scale for smoothness. If dis- tributions are chosen from this distribution, then the opti mal Bayesian estimate ofQ(x)from a set of samples x1,x2,···,xNconverges to the correct answer, and the distribution at ﬁnite Nis nonsingular, so that the regularization provided by this prior is strong enough to prevent the development of s ingular peaks at the location of observed data points (Bialek, Callan, and St rong 1996)9. Further developments of the theory, including alternative choices ofP[φ(x)], have been given by Periwal (1997, 1998), Holy (1997), and Aida (1998). We chose the origi- nal formulation for our analysis because our goal here is to b e illustrative rather than exhaustive. From the discussion above we know that the predictive inform ation is related to the density of Kullback–Leibler divergences, and that th e power–law behavior we are looking for comes from an essential singularity in thi s density function. To illustrate this point, we calculate the predictive infor mation using the density, even though an easier direct way exists. WithQ(x) = (1/l0) exp[−φ(x)], we can write the KL divergence as DKL[¯φ(x)\|\|φ(x)] =1 l0/integraldisplay dxexp[−¯φ(x)][φ(x)−¯φ(x)]. (2.86) 9We caution the reader that our discussion in this section is l ess self–contained than in other sections. Since the crucial steps exactly parallel those in the earlier work, here we just give refer- ences. To compensate for this, we compiled a summary of the or iginal results by Bialek et al. in the Appendix A.1.2.4. Learning and predictability 53 We want to compute the density, ρ(D;¯φ) =/integraldisplay [dφ(x)]P[φ(x)]δ/parenleftBig D−DKL[¯φ(x)\|\|φ(x)]/parenrightBig (2.87) =M/integraldisplay [dφ(x)]P[φ(x)]δ/parenleftBig MD−MD KL[¯φ(x)\|\|φ(x)]/parenrightBig , (2.88) where we introduce a factor Mwhich we will allow to become large so that we can focus our attention on the interesting limit D→0. To compute this inte- gral over all functions φ(x), we introduce a Fourier representation for the delta function, and then rearrange the terms: ρ(D;¯φ) =M/integraldisplaydz 2πexp(izMD )/integraldisplay [dφ(x)]P[φ(x)] exp( −izMD KL)(2.89) =M/integraldisplaydz 2πexp/parenleftbigg izMD +izM l0/integraldisplay dx¯φ(x) exp[−¯φ(x)]/parenrightbigg ×/integraldisplay [dφ(x)]P[φ(x)] exp/parenleftbigg −izM l0/integraldisplay dxφ(x) exp[−¯φ(x)]/parenrightbigg .(2.90) The inner integral over the functions φ(x)is exactly the integral which was evalu- ated in the original discussion of this problem (Bialek, Cal lan and Strong 1996); in the limit that zMis large we can use a saddle point approximation, and standar d ﬁeld theoretic methods allow us to compute the ﬂuctuations a round the saddle point. The result is that [cf. Eqs. (A.6)–(A.8)] /integraldisplay [dφ(x)]P[φ(x)] exp/parenleftbigg −izM l0/integraldisplay dxφ(x) exp[−¯φ(x)]/parenrightbigg = exp/parenleftbigg −izM l0/integraldisplay dxφcl(x) exp[−¯φ(x)]−Seﬀ[φcl(x);zM]/parenrightbigg ,(2.91) Seﬀ[φcl;zM] =l 2/integraldisplay dx/parenleftBigg∂φcl ∂x/parenrightBigg2 +1 2/parenleftbiggizM ll0/parenrightbigg1/2/integraldisplay dxexp[−φcl(x)/2],(2.92) l∂2 xφcl(x) +izM l0exp[−φcl(x)] =izM l0exp[−¯φ(x)]. (2.93) Now we can do the integral over z, again by a saddle point method. The two saddle point approximations are both valid in the limit that D→0andMD3/2→2.4. Learning and predictability 54 ∞; we are interested precisely in the ﬁrst limit, and we are fre e to setMas we wish, so this gives us a good approximation for ρ(D→0;¯φ). Also, since Mis arbitrarily large, φcl(x) =¯φ(x). This results in ρ(D→0;¯φ) =A[¯φ(x)]D−3/2exp/parenleftBigg −B[¯φ(x)] D/parenrightBigg , (2.94) A[¯φ(x)] =1√16πll0exp/bracketleftBigg −l 2/integraldisplay dx/parenleftBig ∂x¯φ/parenrightBig2/bracketrightBigg/integraldisplay dxexp[−¯φ(x)/2](2.95) B[¯φ(x)] =1 16ll0/parenleftbigg/integraldisplay dxexp[−¯φ(x)/2]/parenrightbigg2 . (2.96) Except for the factor of D−3/2, this is exactly the sort of essential singularity that we considered in the previous section, with µ= 1. TheD−3/2prefactor does not change the leading large Nbehavior of the predictive information, and we ﬁnd that S(a) 1(N)∼1 2 ln2√ll0/angbracketleftBigg/integraldisplay dxexp[−¯φ(x)/2]/angbracketrightBigg ¯φN1/2, (2.97) where ∝an}bracketle{t···∝an}bracketri}ht ¯φdenotes an average over the target distributions ¯φ(x)weighted once again by P[¯φ(x)]. Notice that if xis unbounded then the average in Eq. (2.97) is infrared divergent; if instead we let the variable xrange from 0toLthen this average should be dominated by the uniform distribution. Re placing the average by its value at this point, we obtain the approximate result S(a) 1(N)∼1 2 ln2√ N/parenleftbiggL l/parenrightbigg1/2 bits. (2.98) To understand the result in Eq. (2.98), we recall that this ﬁe ld theoretic ap- proach is more or less equivalent to an adaptive binning proc edure in which we divide the range of xinto bins of local size/radicalBig l/NQ(x)(Bialek, Callan, and Strong 1996, see also Appendix A.1). From this point of view, Eq. (2. 98) makes perfect sense: the predictive information is directly proportiona l to the number of bins2.4. Learning and predictability 55 that can be put in the range of x. This also is in direct accord with a comment from the previous subsection that power law behavior of predicti ve information arises from the number of parameters in the problem depending on the number of sam- ples. More importantly, since learning a distribution cons isting of ∼/radicalBig NL/l bins is, certainly, a dVC∼/radicalBig NL/l problem, we can refer back to our discussion of ﬂuc- tuations in prior controlled learning scenarios (Section 2 .4.5) to infer that ﬂuctua- tions pose no threat to this nonparametric learning setup. One thing which remains troubling is that the predictive inf ormation depends on the arbitrary parameter ldescribing the scale of smoothness in the distribu- tion. In the original work it was proposed that one should int egrate over possible values ofl(Bialek, Callan and Strong 1996). Numerical simulations de monstrate that this parameter can be learned from the data itself (see C hapter 3), but per- haps even more interesting is a formulation of the problem by Periwal (1997, 1998) which recovers complete coordinate invariance by eff ectively allowing lto vary withx. In this case the whole theory has no length scale, and there a lso is no need to conﬁne the variable xto a box (here of size L). We expect that this co- ordinate invariant approach will lead to a universal coefﬁc ient multiplying√ N in the analog of Eq. (2.98), but this remains to be shown. In summary, the ﬁeld theoretic approach to learning a smooth distribution in one dimension provides us with a concrete, calculable exa mple of a learning problem with power–law growth of the predictive informatio n. The scenario is exactly as suggested in the previous section, where the dens ity of KL divergences develops an essential singularity. Heuristic considerati ons (Bialek, Callan, and2.5.Ipredas a measure of complexity 56 Strong 1996; Aida 1999) suggest that different smoothness p enalties [for exam- ple, replacing the kinetic term in the prior, Eq. (2.85), by (∂η xφ)2] and generaliza- tions to higher dimensional problems ( dim/vector x=ζ) will have sensible effects on the predictive information S1(N)∼Nζ/2η. (2.99) This shows a power–law growth. Smoother functions have smal ler powers (less to learn) and higher dimensional problems have larger power s (more to learn)— but real calculations for these cases remain challenging. 2.5Ipredas a measure of complexity The problem of quantifying complexity is very old. Solomono ff (1964), Kol- mogorov (1965), and Chaitin (1975) investigated a mathemat ically rigorous no- tion of complexity that measures (roughly) the minimum leng th of a computer program that simulates the observed time series [see also Li and Vit´ anyi (1993)]. Unfortunately there is no algorithm that can calculate the K olmogorov complex- ity of any data set. In addition, algorithmic or Kolmogorov c omplexity is closely related to the Shannon entropy, which means that it measures something closer to our intuitive concept of randomness than to the intuitive co ncept of complexity [as discussed, for example, by Bennett (1990)]. These probl ems have fueled con- tinued research along two different paths, representing tw o major motivations for deﬁning complexity. First, one would like to make precis e an impression that some systems—such as life on earth or a turbulent ﬂuid ﬂow—ev olve toward a state of higher complexity, and one would like to be able to cl assify those states. Second, in choosing among different models that describe an experiment, one2.5.Ipredas a measure of complexity 57 wants to quantify a preference for simpler explanations or, equivalently, provide a penalty for complex models that can be weighed against the m ore conventional goodness of ﬁt criteria. We bring our readers up to date with s ome developments in both of these directions, and then discuss the role of pred ictive information as a measure of complexity. This also gives us an opportunity to discuss more carefully the relation of our results to previous work. 2.5.1 Complexity of statistical models The construction of complexity penalties for model selecti on is a statistics prob- lem. As far as we know, however, the ﬁrst discussions of compl exity in this con- text belong to philosophical literature. Even leaving asid e the early contributions of William of Occam on the need for simplicity, Hume on the pro blem of in- duction, and Popper on falsiﬁability, Kemeney (1953) sugge sted explicitly that it would be possible to create a model selection criterion that balances goodness of ﬁt versus complexity. Wallace and Burton (1968) hinted that this balance may re- sult in the model with “the briefest recording of all attribu te information.” Even though he probably had a somewhat different motivation, Aka ike (1974a, 1974b) made the ﬁrst quantitative step along these lines. His ad hoc complexity term was independent of the number of data points and was proportiona l to the number of free independent parameters in the model. These ideas were rediscovered and developed systematicall y by Rissanen in a series of papers starting from 1978. He has emphasized stron gly (Rissanen 1984, 1986, 1987) that ﬁtting a model to data represents an encodin g of those data, or2.5.Ipredas a measure of complexity 58 predicting future data, and that in searching for an efﬁcien t code we need to mea- sure not only the number of bits required to describe the devi ations of the data from the model’s predictions (goodness of ﬁt), but also the n umber of bits re- quired to specify the parameters of the model (complexity). This speciﬁcation has to be done to a precision supported by the data.10Rissanen (1984) and Clarke and Barron (1990) in full generality were able to prove that t he optimal encod- ing of a model requires a code with length asymptotically pro portional to the number of independent parameters and logarithmically depe ndent on the num- ber of data points we have observed. The minimal amount of spa ce one needs to encode a data string (minimum description length or MDL) w ithin a certain assumed model class was termed by Rissanen stochastic complexity, and in recent work he refers to the piece of the stochastic complexity requ ired for coding the parameters as the model complexity (Rissanen 1996). This approach was further strengthened by a recent result (Vit´ anyi and Li 2000) that a n estimation of param- eters using the MDL principle is equivalent to Bayesian para meter estimations with a “universal” prior (Li and Vit´ anyi 1993). There should be a close connection between Rissanen’s ideas of encoding the data stream and the subextensive entropy. We are accustomed to the idea that the average length of a code word for symbols drawn from a distrib utionPis given by the entropy of that distribution; thus it is tempting to sa y that an encoding of a stream x1,x2,···,xNwill require an amount of space equal to the entropy of the joint distribution P(x1,x2,···,xN). The situation here is a bit more subtle, because the usual proofs of equivalence between code length and entropy rely 10Within this framework Akaike’s suggestion can be seen as cod ing the model to (suboptimal) ﬁxed precision.2.5.Ipredas a measure of complexity 59 on notions of typicality and asymptotics as we try to encode s equences of many symbols; here we already have Nsymbols and so it doesn’t really make sense to talk about a stream of streams. One can argue, however, that a typical sequences are not truly random within a considered distribution since their coding by the methods optimized for the distribution is not optimal. So at ypical sequences are better considered as typical ones coming from a different di stribution [a point also made by Grassberger (1986)]. This allows us to identify properties of an observed (long) string with the properties of the distribut ion it comes from, as was done by Vit´ anyi and Li (2000). If we accept this identiﬁc ation of entropy with code length, then Rissanen’s stochastic complexity sh ould be the entropy of the distribution P(x1,x2,···,xN). As emphasized by Balasubramanian (1996), the entropy of the joint distribu- tion ofNpoints can be decomposed into pieces that represent noise or errors in the model’s local predictions—an extensive entropy—and th e space required to encode the model itself, which must be the subextensive entr opy. Since in the usual formulation all long–term predictions are associate d with the continued validity of the model parameters, the dominant component of the subextensive entropy must be this parameter coding, or model complexity i n Rissanen’s ter- minology. Thus the subextensive entropy should be the model complexity, and in simple cases where we can describe the data by a K–parameter model both quantities are equal to (K/2) log2Nbits to the leading order. The fact that the subextensive entropy or predictive inform ation agrees with Rissanen’s model complexity suggests that Ipredprovides a reasonable measure of complexity in learning problems. On the other hand, this a greement might2.5.Ipredas a measure of complexity 60 lead the reader to wonder if all we have done is to rewrite the r esults of Ris- sanen et al. in a different notation. To calm these fears we re call again that our approach distinguishes inﬁnite VC problems from ﬁnite ones and treats nonpara- metric cases as well. Indeed, the predictive information is deﬁned without ref- erence to the idea that we are learning a model, and thus we can make a link to physical aspects of the problem. 2.5.2 Complexity of dynamical systems There is a strong prejudice that the complexity of physical s ystems should be measured by quantities that are at least related to more conv entional thermody- namic quantities (temperature, entropy, ...), since this is the only way one will be able to calculate complexity within the framework of statis tical mechanics. Most proposals deﬁne complexity as an entropy–like quantity, bu t an entropy of some unusual ensemble. For example, Lloyd and Pagels (1988) iden tiﬁed complexity asthermodynamic depth , the entropy of the state sequences that lead to the cur- rent state. The idea is clearly in the same spirit as the measu rement of predictive information, but this depth measure does not completely dis card the extensive component of the entropy (Crutchﬁeld and Shalizi 1999) and t hus fails to resolve the essential difﬁculty in constructing complexity measur es for physical systems: distinguishing genuine complexity from randomness (entro py), the complexity should be zero both for purely regular and for purely random s ystems. New deﬁnitions of complexity that try to satisfy these crite ria (Lopez–Ruiz et al. 1995, Gell–Mann and Lloyd 1996, Shiner et al. 1999, Sol e and Luque 1999, Adami and Cerf 2000) and criticisms of these proposals (Crut chﬁeld et al. 1999,2.5.Ipredas a measure of complexity 61 Feldman and Crutchﬁeld 1998, Sole and Luque 1999) continue t o emerge even now. Aside from the obvious problems of not actually elimina ting the extensive component for all or a part of the parameter space or not expre ssing complexity as an average over a physical ensemble, the critiques often a re based on a clever argument ﬁrst mentioned explicitly by Feldman and Crutchﬁe ld (1998). In an at- tempt to create a universal measure, the constructions can b e made over–universal : many proposed complexity measures depend only on the entrop y density S0and thus are functions only of disorder—not a desired feature. I n addition, many of these and other deﬁnitions are ﬂawed because they fail to dis tinguish among the richness of classes beyond some very simple ones. In a series of papers, Crutchﬁeld and coworkers identiﬁed statistical complexity with the entropy of causal states, which in turn are deﬁned as all those microstates (or histories) that have the same conditional distribution of futures (Crutchﬁeld and Young 1989, Shalizi and Crutchﬁeld 1999). The causal sta tes provide an op- timal description of a system’s dynamics in the sense that th ese states make as good a prediction as the histories themselves. Statistical complexity is very simi- lar to predictive information, but Shalizi and Crutchﬁeld ( 1999) deﬁne a quantity which is even closer to the spirit of our discussion: their excess entropy is exactly the mutual information between the semi–inﬁnite past and fu ture. Unfortunately, by focusing on cases in which the past and future are inﬁnite b ut the excess en- tropy is ﬁnite, their discussion is limited to systems for wh ich (in our language) Ipred(T→ ∞) = constant . In our view, Grassberger (1986) has made the clearest and the most appealing deﬁnitions. He emphasized that the slow approach of the entr opy to its extensive limit is a sign of complexity, and has proposed several funct ions to analyze this2.5.Ipredas a measure of complexity 62 slow approach. His effective measure complexity is the subextensive entropy term of an inﬁnite data sample. Unlike Crutchﬁeld et al., he allows t his measure to grow to inﬁnity. As an example, for low dimensional dynamical sys tems, the effec- tive measure complexity is ﬁnite whether the system exhibit s periodic or chaotic behavior, but at the bifurcation point that marks the onset o f chaos, it diverges logarithmically. More interestingly, Grassberger also no tes that simulations of speciﬁc cellular automaton models that are capable of unive rsal computation in- dicate that these systems can exhibit an even stronger, powe r–law, divergence. Grassberger (1986) also introduces the true measure complexity , which is the minimal information one needs to extract from the past in ord er to provide opti- mal prediction. This quantity is exactly the statistical co mplexity of Crutchﬁeld et al., and the two approaches are actually much closer than t hey seem. The re- lation between the true and the effective measure complexit ies, or between the statistical complexity and the excess entropy, closely par allels the idea of extract- ing or compressing relevant information (Tishby et al. 1999 , Bialek and Tishby, in preparation), as discussed below. 2.5.3 A unique measure of complexity? We recall that entropy provides a measure of information tha t is unique in sat- isfying certain plausible constraints (Shannon 1948). It w ould be attractive if we could prove a similar uniqueness theorem for the predictive information, or any part of it, as a measure of the complexity or richness of a time dependent signal x(0< t < T )drawn from a distribution P[x(t)]. Before proceeding with such an argument we have to ask, however, whether we want to attach measures of2.5.Ipredas a measure of complexity 63 complexity to a particular signal x(t), or whether we are interested in measures (like the entropy itself) which constitute an average over t he ensemble P[x(t)]. In most cases, including the learning problems discussed ab ove, it is clear that we want to measure complexity of the dynamics underlyin g the signal, or equivalently the complexity of a model that might be used to d escribe the signal. This is very different from trying to deﬁne the complexity of a single realization, because there can be atypical data streams. Either we must tr eat atypicality ex- plicitly, arguing that atypical data streams from one sourc e should be viewed as typical streams from another source, as discussed by Vit´ an yi and Li (2000), or we have to look at average quantities. Grassberger (1986) in pa rticular has argued that our visual intuition about the complexity of spatial pa tterns is an ensemble concept, even if the ensemble is only implicit [see also Tong in the discussion ses- sion of Rissanen (1987)]. So we shall search for measures of c omplexity that are averages over the distribution P[x(t)]. Once we focus on average quantities, we can start by adopting Shannon’s postulates as constraints on a measure of complexity: if the re areNequally likely signals, then the measure should be monotonic in N; if the signal is decompos- able into statistically independent parts then the measure should be additive with respect to this decomposition; and if the signal can be descr ibed as a leaf on a tree of statistically independent decisions then the measu re should be a weighted sum of the measures at each branching point. We believe that t hese constraints are as plausible for complexity measures as for information measures, and it is well known from Shannon’s original work that this set of cons traints leaves the entropy as the only possibility. Since we are discussing a ti me dependent signal, this entropy depends on the duration of our sample, S(T). We know of course2.5.Ipredas a measure of complexity 64 that this cannot be the end of the discussion, because we need to distinguish be- tween 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 in- variant under transformations of x. It seems reasonable to ask that complexity be a function of the process we are observing and not of the coo rdinate system in which we choose to record our observations. The examples a bove show us, however, that it is not the whole function S(T)which depends on the coordinate system for x;11it is only the extensive component of the entropy that has thi s non- invariance. This can be seen more generally by noting that su bextensive terms in the entropy contribute to the mutual information among diff erent segments of the data stream (including the predictive information deﬁn ed here), while the ex- tensive entropy cannot; mutual information is coordinate i nvariant, so all of the noninvariance must reside in the extensive term. Thus, any m easure complexity that is coordinate invariant must discard the extensive com ponent of the entropy. The fact that extensive entropy cannot contribute to comple xity is discussed widely in the physics literature (Bennett 1990), as our shor t review above shows. To statisticians and computer scientists, who are used to Ko lmogorov’s ideas, this is less obvious. However, Rissanen (1986, 1987) also talks a bout “noise” and “use- ful information” in a data sequence, which is similar to spli tting entropy into its extensive and the subextensive parts. His “model complexit y,” aside from not be- ing an average as required above, is essentially equal to the subextensive entropy. Similarly, Whittle [in the discussion of Rissanen (1987)] t alks about separating the 11Here we consider instantaneous transformations of x, not ﬁltering or other transformations that mix points at different times.2.5.Ipredas a measure of complexity 65 predictive part of the data from the rest. If we continue along these lines, we can think about the asymp totic 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 te rms? In the con- text of learning a parameterized model, most of the terms in t his series depend in detail on our prior distribution in parameter space, whic h might seem odd for a measure of complexity. More generally, if we consider tran sformations 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 under such temporally local transformations.12So if we insist that measures of complexity be invariant not only under instantaneous coordinate transformations, but also under tempo- rally local transformations, then we can discard both the ex tensive and the ﬁnite subextensive terms in the entropy, leaving only the diverge nt subextensive terms as a possible measure of complexity. An interesting example of these arguments is provided by the statistical me- chanics of polymers. It is conventional to make models of pol ymers as random walks on a lattice, with various interactions or self avoida nce constraints among different elements of the polymer chain. If we count the numb erNof walks withNsteps, we ﬁnd that N(N)∼ANγzN(de Gennes 1979). Now the entropy is the logarithm of the number of states, and so there is an ext ensive entropy S0= log2z, a constant subextensive entropy log2A, and a divergent subextensive 12Throughout this discussion we assume that the signal xat one point in time is ﬁnite dimen- sional. There are subtleties if we allow xto represent the conﬁguration of a spatially inﬁnite system.2.5.Ipredas a measure of complexity 66 termS1(N)→γlog2N. Of these three terms, only the divergent subextensive term (related to the critical exponent γ) is universal, that is independent of the detailed structure of the lattice. Thus, as in our general ar gument, it is only the divergent subextensive terms in the entropy that are invari ant to changes in our description of the local, small scale dynamics. We can recast the invariance arguments in a slightly differe nt form using the relative entropy. We recall that entropy is deﬁned cleanly o nly for discrete pro- cesses, and that in the continuum there are ambiguities. We w ould like to write the continuum generalization of the entropy of a process x(t)distributed accord- ing toP[x(t)]as Scont=−/integraldisplay Dx(t)P[x(t)] log2P[x(t)], (2.100) but this is not well deﬁned because we are taking the logarith m of a dimensionful quantity. Shannon gave the solution to this problem: we use a s a measure of in- formation the relative entropy or KL divergence between the distribution P[x(t)] and some reference distribution Q[x(t)], Srel=−/integraldisplay Dx(t)P[x(t)] log2/parenleftBiggP[x(t)] Q[x(t)]/parenrightBigg , (2.101) which is invariant under changes of our coordinate system on the space of sig- nals. The cost of this invariance is that we have introduced a n arbitrary dis- tributionQ[x(t)], and so really we have a family of measures. We can ﬁnd a unique complexity measure within this family by imposing in variance principles as above, but in this language we must make our measure invari ant to different choices of the reference distribution Q[x(t)]. The reference distribution Q[x(t)]embodies our expectations for the signal x(t); in particular, Srelmeasures the extra space needed to encode signals drawn2.5.Ipredas a measure of complexity 67 from the distribution P[x(t)]if we use coding strategies that are optimized for Q[x(t)]. Ifx(t)is a written text, two readers who expect different numbers o f spelling errors will have different Qs, but to the extent that spelling errors can be corrected by reference to the immediate neighboring lett ers we insist that any measure of complexity be invariant to these differences in Q. On the other hand, readers who differ in their expectations about the global su bject of the text may well disagree about the richness of a newspaper article. Thi s suggests that com- plexity is a component of the relative entropy that is invari ant under some class of local translations and misspellings. Suppose that we leave aside global expectations, and constr uct our reference distribution Q[x(t)]by allowing only for short ranged interactions—certain let ters tend to follow one another, letters form words, and so on, but we bound the range over which these rules are applied. Models of this class cann ot embody the full structure of most interesting time series (including langu age), but in the present context we are not asking for this. On the contrary, we are loo king for a measure that is invariant to differences in this short ranged struct ure. In the terminology of ﬁeld theory or statistical mechanics, we are constructin g our reference distribu- tionQ[x(t)]from local operators. Because we are considering a one dimen sional signal (the one dimension being time), distributions const ructed from local op- erators cannot have any phase transitions as a function of pa rameters; again it is important that the signal xat one point in time is ﬁnite dimensional. The absence of critical points means that the entropy of these distribut ions (or their contri- bution to the relative entropy) consists of an extensive ter m (proportional to the time window T) plus a constant subextensive term, plus terms that vanish a sT becomes large. Thus, if we choose different reference distr ibutions within the2.6. Discussion 68 class constructible from local operators, we can change the extensive component of the relative entropy, and we can change constant subexten sive terms, but the divergent subextensive terms are invariant. To summarize, the usual constraints on information measure s in the contin- uum produce a family of allowable complexity measures, the r elative entropy to an arbitrary reference distribution. If we insist that al l observers who choose reference distributions constructed from local operators arrive at the same mea- sure of complexity, or if we follow the ﬁrst line of arguments presented above, then this measure must be the divergent subextensive compon ent of the entropy or, equivalently, the predictive information. We have seen that this component is connected to learning, quantifying the amount that can be learned about dy- namics that generate the signal, and to measures of complexi ty that have arisen in statistics and in dynamical systems theory. 2.6 Discussion We have presented predictive information as a characterization of data streams. In the context of learning, predictive information is relat ed directly to general- ization. More generally, the structure or order in a time ser ies or a sequence is related almost by deﬁnition to the fact that there is predict ability along the se- quence. The predictive information measures the amount of s uch structure, but doesn’t exhibit the structure in a concrete form. Having col lected a data stream of duration T, what are the features of these data that carry the predictiv e infor- mationIpred(T)? From Eq. (2.9) we know that most of what we have seen over2.6. Discussion 69 the timeTmust be irrelevant to the problem of prediction, so that the p redic- tive information is a small fraction of the total informatio n; can we separate these predictive bits from the vast amount of nonpredictive data? The problem of separating predictive from nonpredictive in formation is a spe- cial case of the problem discussed recently (Tishby et al. 19 99, Bialek and Tishby, in preparation): given some data x, how do we compress our description of x while preserving as much information as possible about some other variable y? Here we identify x=xpastas the past data and y=xfuture as the future. When we compressxpastinto some reduced description ˆxpastwe keep a certain amount of information about the past, I(ˆxpast;xpast), and we also preserve a certain amount of information about the future, I(ˆxpast;xfuture). There is no single correct com- pressionxpast→ˆxpast; instead there is a one parameter family of strategies which trace out an optimal curve in the plane deﬁned by these two mut ual informations, I(ˆxpast;xfuture)vs.I(ˆxpast;xpast). The predictive information preserved by compression must b e less than the total, so that I(ˆxpast;xfuture)≤Ipred(T). Generically no compression can preserve all of the predictive information so that the inequality wil l be strict, but there are interesting special cases where equality can be achieved. I f prediction proceeds by learning a model with a ﬁnite number of parameters, we migh t have a re- gression formula that speciﬁes the best estimate of the para meters given the past data; using the regression formula compresses the data but p reserves all of the predictive power. In cases like this (more generally, if the re exist sufﬁcient statis- tics for the prediction problem) we can ask for the minimal se t ofˆxpastsuch that I(ˆxpast;xfuture) =Ipred(T). The entropy of this minimal ˆxpastis the true measure complexity deﬁned by Grassberger (1986) or the statistical complexity deﬁned by2.6. Discussion 70 Crutchﬁeld and Young (1989) [in the framework of the causal s tates theory a very similar comment was made recently by Shalizi and Crutchﬁeld (2000)]. In the context of statistical mechanics, long range correla tions are charac- terized by computing the correlation functions of order par ameters, which are coarse–grained functions of the system’s microscopic vari ables. When we know something about the nature of the order parameter (e. g., whe ther it is a vector or a scalar), then general principles allow a fairly complet e classiﬁcation and de- scription of long range ordering and the nature of the critic al points at which this order can appear or change. On the other hand, deﬁning th e order param- eter itself remains something of an art. For a ferromagnet, t he order parameter is obtained by local averaging of the microscopic spins, whi le for an antiferro- magnet one must average the staggered magnetization to capt ure the fact that the ordering involves an alternation from site to site, and s o on. Since the order parameter carries all the information that contributes to l ong range correlations in space and time, it might be possible to deﬁne order paramet ers more generally as those variables that provide the most efﬁcient compressi on of the predictive information, and this should be especially interesting for complex or disordered systems where the nature of the order is not obvious intuitiv ely; a ﬁrst try in this direction was made by Bruder (1998). At critical points the p redictive information will diverge with the size of the system, and the coefﬁcients of these divergences should be related to the standard scaling dimensions of the o rder parameters, but the details of this connection need to be worked out. If we compress or extract the predictive information from a t ime series we are in effect discovering “features” that capture the nature of the ordering in time.2.6. Discussion 71 Learning itself can be seen as an example of this, where we dis cover the parame- ters of an underlying model by trying to compress the informa tion that one sam- ple ofNpoints provides about the next, and in this way we address dir ectly the problem of generalization (Bialek and Tishby, in preparati on). The fact that (as mentioned above) nonpredictive information is useless to t he organism suggests that one crucial goal of neural information processing is to separate predictive in- formation from the background. Perhaps rather than providi ng an efﬁcient rep- resentation of the current state of the world—as suggested b y Attneave (1954), Barlow (1959, 1961), and others (Atick 1992)—the nervous sy stem provides an ef- ﬁcient representation of the predictive information.13It should be possible to test this directly by studying the encoding of reasonably natura l signals and asking if the information which neural responses provide about the fu ture of the input is close to the limit set by the statistics of the input itself, g iven that the neuron only captures a certain number of bits about the past. Thus we migh t ask if, under natural stimulus conditions, a motion sensitive visual neu ron captures features of the motion trajectory that allow for optimal prediction o r extrapolation of that 13If, as seems likely, the stream of data reaching our senses ha s diverging predictive information then the space required to write down our description grows a nd grows as we observe the world for longer periods of time. In particular, if we can observe f or a very long time then the amount that we know about the future will exceed, by an arbitrarily l arge factor, the amount that we know about the present. Thus representing the predictive in formation may require many more neurons than would be required to represent the current data . If we imagine that the goal of primary sensory cortex is to represent the current state of t he sensory world, then it is difﬁcult to understand why these cortices have so many more neurons th an they have sensory inputs. In the extreme case, the region of primary visual cortex devote d to inputs from the fovea has nearly 30,000 neurons for each photoreceptor cell in the retina (Ha wken and Parker 1991); although much remains to be learned about these cells, it is difﬁcult t o imagine that the activity of so many neurons constitutes an efﬁcient representation of the curr ent sensory inputs. But if we live in a world where the predictive information in the movies reachi ng our retina diverges, it is perfectly possible that an efﬁcient representation of the predictive information available to us at one instant requires thousands of times more space than an efﬁcient repr esentation of the image currently falling on our retina.2.6. Discussion 72 trajectory; by using information theoretic measures we bot h test the “efﬁcient representation” hypothesis directly and avoid arbitrary a ssumptions about the metric for errors in prediction. For more complex signals su ch as communication sounds, even identifying the features that capture the pred ictive information is an interesting problem. It is natural to ask if these ideas about predictive informat ion could be used to analyze experiments on learning in animals or humans. We hav e emphasized the problem of learning probability distributions or probabil istic models rather than learning deterministic functions, associations or rules. It is known that the ner- vous system adapts to the statistics of its inputs, and this a daptation is evident in the responses of single neurons (Smirnakis et al. 1996, Br enner et al. 2000); these experiments provide a simple example of the system lea rning a parameter- ized distribution. When making saccadic eye movements, hum an subjects alter their distribution of reaction times in relation to the rela tive probabilities of dif- ferent targets, as if they had learned an estimate of the rele vant likelihood ratios (Carpenter and Williams 1995). Humans also can learn to disc riminate almost optimally between random sequences (fair coin tosses) and s equences that are correlated or anticorrelated according to a Markov process ; this learning can be accomplished from examples alone, with no other feedback (L opes and Oden 1987). Acquisition of language may require learning the joi nt distribution of suc- cessive phonemes, syllables, or words, and there is direct e vidence for learning of conditional probabilities from artiﬁcial sound sequenc es, both by infants and by adults (Saffran et al. 1996; 1999). These examples, which are not exhaustive, indicate that the nervous system can learn an appropriate pr obabilistic model,14 14As emphasized above, many other learning problems, includi ng learning a function from2.6. Discussion 73 and this offers the opportunity to analyze the dynamics of th is learning using in- formation theoretic methods: What is the entropy of Nsuccessive reaction times following a switch to a new set of relative probabilities in t he saccade experi- ment? How much information does a single reaction time provi de about the relevant probabilities? Following the arguments above, su ch analysis could lead to a measurement of the universal learning curve Λ(N). The learning curve Λ(N)exhibited by a human observer is limited by the pre- dictive information in the time series of stimulus trials it self. Comparing Λ(N)to this limit deﬁnes an efﬁciency of learning in the spirit of th e discussion by Barlow (1983); while it is known that the nervous system can make efﬁ cient use of avail- able information in signal processing tasks [cf. Chapter 4 o f Rieke et al. (1997)], it is not known whether the brain is an efﬁcient learning machin e in the analogous sense. Given our classiﬁcation of learning tasks by their co mplexity, it would be natural to ask if the efﬁciency of learning were a critical fu nction of task complex- ity: perhaps we can even identify a limit beyond which efﬁcie nt learning fails, indicating a limit to the complexity of the internal model us ed by the brain dur- ing a class of learning tasks. We believe that our theoretica l discussion here at least frames a clear question about the complexity of intern al models, even if for the present we can only speculate about the outcome of such ex periments. An important result of our analysis is the characterization of time series or learning problems beyond the class of ﬁnitely parameteriza ble models, that is the class with power–law divergent predictive information. Qu alitatively this class is more complex than anyparametric model, no matter how many parameters there noisy examples, can be seen as the learning of a probabilisti c model. Thus we expect that this description applies to a much wider range of biological lear ning tasks.2.6. Discussion 74 may be, because of the more rapid asymptotic growth of Ipred(N). On the other hand, with a ﬁnite number of observations N, the actual amount of predictive in- formation in such a nonparametric problem may be smaller than in a model with a large but ﬁnite number of parameters. Speciﬁcally, if we ha ve two models, one withIpred(N)∼ANνand one with Kparameters so that Ipred(N)∼(K/2) log2N, the inﬁnite parameter model has less predictive informatio n for allNsmaller than some critical value Nc∼/bracketleftbiggK 2Aνlog2/parenleftbiggK 2A/parenrightbigg/bracketrightbigg1/ν . (2.102) In the regime N≪Nc, it is possible to achieve more efﬁcient prediction by tryin g to learn the (asymptotically) more complex model, as we illu strate concretely in numerical simulations of the density estimation problem, S ection 3.6. Even if there are a ﬁnite number of parameters—such as the ﬁnite numb er of synapses in a small volume of the brain—this number may be so large that we always have N≪Nc, so that it may be more effective to think of the many paramete r model as approximating a continuous or nonparametric one. It is tempting to suggest that the regime N <<N cis the relevant one for much of biology. If we consider, for example, 10 mm2of inferotemporal cortex devoted to object recognition (Logothetis and Sheinberg 1996), the number of synapses is K∼5×109. On the other hand, object recognition depends on foveation , and we move our eyes roughly three times per second throughout pe rhaps 15 years of waking life during which we master the art of object recogn ition. This limits us to at most N∼109examples. Remembering that we must have ν <1, even with large values of AEq. (2.102) suggests that we operate with N < Nc. One can make similar arguments about very different brains, suc h as the mushroom2.6. Discussion 75 bodies in insects (Capaldi, Robinson and Fahrbach 1999). If this identiﬁcation of biological learning with the regime N << N cis correct, then the success of learning in animals must depend on strategies that implemen t sensible priors over the space of possible models. There is one clear empirical hint that humans can make effect ive use of models that are beyond ﬁnite parameterization (in the sense that pr edictive information diverges as a power–law), and this comes from studies of lang uage. Long ago, Shannon (1951) used the knowledge of native speakers to plac e bounds on the entropy of written English, and his strategy made explicit u se of predictability. Shannon showed N–letter sequences to native speakers (readers), asked them to guess the next letter, and recorded how many guesses were req uired before they got the right answer. Thus each letter in the text is turned in to a number, and the entropy of the distribution of these numbers is an upper boun d on the conditional entropyℓ(N)[cf. Eq. (2.10)]. Shannon himself thought that the converge nce as Nbecomes large was rather quick, and quoted an estimate of the extensive en- tropy per letter S0. Many years later, Hilberg (1990) reanalyzed Shannon’s dat a and found that the approach to extensivity in fact was very sl ow: certainly there is evidence for a large component S1(N)∝N1/2, and this may even dominate the extensive component for accessible N. Ebeling and P¨ oschel (1994; see also P¨ oschel, Ebeling, and Ros´ e 1995) studied the statistics o f letter sequences in long texts (like Moby Dick ) and found the same strong subextensive component. It would be attractive to repeat Shannon’s experiments with a s lightly different de- sign that emphasizes the detection of subextensive terms at largeN.15 15Associated with the slow approach to extensivity is a large m utual information between words or characters separated by long distances, and severa l groups have found that this mu- tual information declines as a power law. Cover and King (197 8) criticize such observations by2.6. Discussion 76 In summary, we believe that our analysis of predictive infor mation solves the problem of measuring the complexity of time series. This ana lysis uniﬁes ideas from learning theory, coding theory, dynamical systems, an d statistical mechan- ics. In particular we have focused attention on a class of pro cesses that are qual- itatively more complex than those treated in conventional l earning theory, and there are several reasons to think that this class includes m any examples of rele- vance to biology and cognition. noting that such behavior is impossible in Markov chains of a rbitrary order. While it is possi- ble that existing mutual information data have not reached a symptotia, the criticism of Cover and King misses the possibility that language is nota Markov process. Of course it cannot be Markovian if it has a power–law divergence in the predictive information.Chapter 3 Learning continuous distributions: Simulations with ﬁeld theoretic priors 3.1 Occam factors in statistics As we have discussed extensively in the preceding Chapter, o ne of the central problems in learning is to balance ‘goodness of ﬁt’ criteria against the complexity of models. An important development in the Bayesian approac h thus was the re- alization that there does not need to be any extra penalty for model complexity: if we compute the total probability that data are generated b y a model, there is a factor from the volume in parameter space—the “Occam factor ”—that discrimi- nates against more complex models (MacKay 1992, Balasubram anian 1997). This works remarkably well for systems with a ﬁnite number of para meters and cre- ates a complexity “razor” (named after “Occam’s razor”) tha t is equivalent to the model complexity of the celebrated Minimal Description Length (MDL) princip le (Rissanen 1989, 1996). It is not clear, however, what happen s if we leave this ﬁnite 773.1. Occam factors in statistics 78 dimensional setting and consider nonparametric problems s uch as the estimation of a smooth probability density. As we have emphasized, the behavior of the predictive inform ation, Eq. (2.8), is controlled by the density of models, and therefore the pre dictive information is closely related to the Occam factor. Since the density and consequently Ipred are well deﬁned for the ﬁnite parameter as well as for the nonp arametric cases (cf. Sections 2.4.2 and 2.4.6) one can hope that a nonparamet ric analogue of the Occam factor exists and can do its job of punishing complexit y. However, since in these two cases the densities of models are very different , the Occam factor details certainly must be different too. The 1996 formulation of nonparametric learning by Bialek, C allan, and Strong, which we have summarized in Appendix A.1 and investigated fu rther in Sec- tion 2.4.7, may serve as a good example in which to study inﬁni te dimensional Occam factors. In this Bayesian quantum ﬁeld theory formula tion, standard ﬁeld theory methods may be used not only to ﬁnd a nowhere singular e stimate of a continuous density, but also to compute the inﬁnite dimensi onal analog of the Occam factor, at least asymptotically for large numbers of s amples. This factor, which we also call the ﬂuctuation determinant, is the second term of the effective Hamiltonian Eqs. (A.7, 2.92) R=1 2/parenleftbiggN ll0/parenrightbigg1/2/integraldisplay dxexp[−φcl(x)/2]. (3.1) Intuitively, smaller values of lallow more rapidly varying and thus more com- plex [as measured by the predictive information, Eqs. (2.97 , 2.98)] estimates of the density. Correspondingly, the inﬁnite dimensional Occam f actor is bigger and3.2. The algorithm 79 thus exponentially punishes more complex models. As Bialek et al. have spec- ulated,l, the only free parameter of their theory, can be determined b y a ﬁght between the log–likelihood goodness of ﬁt and the Occam fact or to provide for the shortest total description (highest probability) of th e data, much like in the ﬁnite parameter MDL theory. However, their proposed scalin g forl∗(the best value of the parameter) as a function of N,l∗∼N1/3seems to be over–universal and requires further analysis. There are more questions not clearly answered either by the o riginal work, or its further developments (Periwal 1997, 1998, Holy 1997, Aida 1999). Can this method be implemented in practice? Can we really use the inﬁn ite dimensional Occam factor to balance against the goodness of ﬁt? How does t he algorithm’s performance compare to the absolute bounds set by the predic tive information? What happens if the learning problem is strongly atypical of the prior distribu- tion? And what is the role of the Occam factor in this case? To answer all of these questions we turn to numerical simulat ions. 3.2 The algorithm To simplify the algorithm, maximize the speed of simulation s, and shorten our presentation, we do the numerical analysis only in the frame work of the original paper (Bialek, Callan, and Strong 1996, see also Appendix A. 1 and Section 2.4.7). This may seem too speciﬁc, but we believe that our results are very general and will hold for the alternative formulations of Periwal (1997 , 1998) and Holy (1997) since the mechanisms of regularization and complexity cont rol are everywhere the same.3.2. The algorithm 80 Due to our most recent developments (cf. Section 2.4.7) and t he speciﬁc ques- tions we ask, we need to modify the original setup slightly be fore proceeding. First of all, we will investigate the performance of the meth od in many different learning problems, some of them not characteristic of the pr ior Eq. (A.5). For these purposes we will take densities at random from an ‘actu al’ a priori distri- bution that minimally generalizes Eq. (A.5), P[φ(x)] =1 Zexp −l2ηa−1 a 2/integraldisplay dx/parenleftBigg∂ηaφ ∂xηa/parenrightBigg2 δ/bracketleftbigg1 l0/integraldisplay dxe−φ(x)−1/bracketrightbigg . (3.2) Hereηa>1/2to ensure UV convergence, Zis the normalization constant, and theδ–function enforces normalization of Q. We refer to laandηaas the smoothness scale and the exponent , respectively, and the subscript astands for ‘actual’. We will use non–subscripted ηandlto indicate the parameters the algorithm uses, that is, the learning machine’s own a priori expectations, and th enηa=η≡1and la=lreduces to the original formulation of Bialek et al. The other modiﬁcation we make relates to the problem of the in frared diver- gence of the predictive information, Eq. (2.97), or, equiva lently, to the nonuniform convergence of the estimate Qest(x)to the target P(x), Eq. (A.11). To cure this we can put the system in the box of size L, just like we did in Section 2.4.7. Also, we realize that the variance of ﬂuctuations between the targ et and the estimate (Bialek et al. 1996) is just an ad hoc measure of performance of the learning ma- chine. The universal learning curve Λ(N), Eq. (2.13), is a much better choice. For a proper Bayesian learning with the prior Eq. (A.5), using Eqs . (2.51, 2.98), we write Λ(N)≡/angbracketleftBig ∝an}bracketle{tDKL[P(x)\|\|Qest(x)]∝an}bracketri}ht{xi}/angbracketrightBig(0)∼1 4/radicalBigg L lN, (3.3) where ∝an}bracketle{t···∝an}bracketri}ht(0)means an average over the prior, and the log 2 factor is omitted because we choose to measure entropies in nats (that is, use n atural logarithms)3.2. The algorithm 81 in this Chapter. Note that the coefﬁcient in front of the squa re root is probably meaningless since it is calculated here only to the zeroth or der (see Section 2.4.7). After these modiﬁcations, the algorithm to implement the th eory is rather sim- ple. We need to solve the second order differential equation [cf. Eq. (A.8)] l∂2 xφcl(x) +N l0exp [−φcl(x)] =N/summationdisplay j=1δ(x−xj). (3.4) Normalization of Qclﬁxes one integration constant, and the other is set by a peri- odicity constraint for φcl, φ(x) =φ(x+L), (3.5) which is due to xbeing in a box. The resulting boundary value problem is solve d by a standard ‘relaxation’ (or Newton) method of iterative i mprovements to a guessed solution (see, for example, Press et al. 1988). The t arget precision is al- ways 10−5, which is smaller than the smallest DKL∼10−4we intend to investi- gate. It turns out that the method converges regardless of th e initial guess for all lup to∼5. However, convergence is not uniform in land, asl→0, the number of iterations required to reach the same precision grows alm ost quadratically in 1/l. The independent variable x∈[0,L]is discretized in equal steps to ensure stability of the method. We expect the estimate distributio n to vary over a local length scale [Bialek et al. 1996, cf. Eq. (A.9)] ξ(x)∼[l/NQ est(x)]1/2≈[l/NP(x)]1/2. (3.6) Empirically we see that, for small l, the maximal value of the target distribu- tionP(x)grows approximately as ∼l−1/2. This means that for Figs. (3.1–3.4), whereN≤105andl≥0.05, we are safe with 104grid points. Similarly, for Figs. (3.5, 3.6), we need 105discretization steps because N= 106andl= 0.01are present there.3.3. Learning with the correct prior 82 Since the prior we use, Eq. (3.2), is UV convergent, we can gen erate random probability densities from it by replacing φwith its Fourier series and truncating the latter at some sufﬁciently high wavenumber kc, φ(x) =kc/summationdisplay k=0/bracketleftBigg Akcos2πkx L+Bksin2πkx L/bracketrightBigg . (3.7) Then Eq. (3.2) enforces the amplitudes of the k’th mode to be distributed normally around zero with the standard deviation ∝an}bracketle{tA2 k∝an}bracketri}ht1/2=∝an}bracketle{tB2 k∝an}bracketri}ht1/2=21/2 lη−1/2/parenleftbiggL 2πk/parenrightbiggη , k= 1,2,···. (3.8) In addition, the amplitude of the zeroth mode, A0, is always set by the normal- ization constraint. For the same sets of ﬁgures, it is enough to havekc= 1000 and 5000 respectively, and then we should see very little effects ass ociated with the ﬁniteness of kc. Coded in such a way, the simulations are extremely computati onally inten- sive. Therefore, the Monte Carlo averages given here are onl y over 500 runs, and ﬂuctuation determinants are calculated according to Eq . (3.1), but not using numerical path integration. 3.3 Learning with the correct prior As an example of the algorithm’s performance, Fig. (3.1) sho ws one particular learning run for η=ηa= 1 andl=la= 0.2. We see that singularities and overﬁtting are absent even for Nas low as 10. Moreover, the approach of Qcl(x) to the actual distribution P(x)is remarkably fast: for N= 10 , they are similar; forN= 1000 , very close; for N= 100000 , one needs to look carefully to see the difference between the two.3.3. Learning with the correct prior 83 0 0.2 0.4 0.6 0.8 100.511.522.533.5 x Q cl(x), P(x)Fit for 10 samples Fit for 1000 samples Fit for 100000 samples Actual distribution Figure 3.1: Qclfound for different Natl= 0.2. The next question on our list is the behavior of the learning c urves. For the sameηandl= 0.4,0.2,0.05, these are shown on Fig. (3.2). One sees that the exponents are extremely close to the expected 1/2, and the ratios of the prefactors are within the errors from the predicted scaling ∼1/√ l. All of this means that the proposed algorithm for ﬁnding densities not only works a s predicted, but is, at most, a constant factor away from being optimal in usin g the predictive information of the sample set. Note also that the data points approach their asymptotic reg imes very differ- ently for different values of l: the bigger lis, the lower the data start compared to their respective ﬁts. This is explainable in view of the fa ct that smoother dis- tributions usually vary over smaller ranges. For example, f orl= 0.4the target distribution P(x)usually takes values from ∼0.5to∼2. On the other hand, for3.4. Learning with ‘wrong’ priors 84 10110210310410510−310−210−1100Λ N l=0.4, data and best fit l=0.2, data and best fit l=0.05, data and best fit Figure 3.2: Λas a function of Nandl. The best ﬁts are: for l= 0.4,Λ = (0.54± 0.07)N−0.483±0.014; forl= 0.2,Λ = (0.83±0.08)N−0.493±0.09; forl= 0.05,Λ = (1.64±0.16)N−0.507±0.09. not too large Nthe estimates are also smooth and close to being uniform. The re- fore, the KL divergence usually comes out small in this case. Thus thel= 0.4data starts so low not because we manage to learn the distribution extremely well for N= 10 , but because almost any guess is as good as any other at this le vel of detail. 3.4 Learning with ‘wrong’ priors We stress ﬁrst that there is no such thing as a wrong prior . If one admits the possibility of a prior being wrong, then that prior does not e ncode all of our a priori knowledge! It does make sense, however, to ask what h appens if the distribution we are trying to learn is an extreme outlier in t he prior P[φ(x)]. One3.4. Learning with ‘wrong’ priors 85 10110210310410510−310−210−1100Λ N la=0.2, data and best fit la=0.4, data and best fit la=0.05, data and best fit variable la, data and best fit Figure 3.3: Λas a function of Nandla. Best ﬁts are: for la= 0.4,Λ = (0.56± 0.08)N−0.477±0.015; forla= 0.05,Λ = (1.90±0.16)N−0.502±0.008; for variable la, Λ = (1.28±0.13)N−0.498±0.016. In all cases we learn with l= 0.2. way to generate such an example is to choose a typical functio n from a different priorP′[φ(x)], and this is what we mean by ‘learning with an incorrect prior .’ To study this we learn using η= 1andl, but we choose our target distributions from the prior Eq. (3.2) with different values of ηaandla. If the prior is wrong in the above sense, and the learning proc ess is as usual, Eqs. (A.3, A.6–A.8), then we still expect the asymptotic beh avior, Eq. (3.3), to hold. Indeed, once φclbecomes close to −logP, it takes the same time to discover that the distribution’s features at the current relevant scale ξ(N)are as expected, too big, or almost absent. Thus only the prefactors of Λshould change, and those must increase because there is an obvious advantage in havin g the right prior. We illustrate these points in Figs. (3.3, 3.4).3.4. Learning with ‘wrong’ priors 86 Figure (3.3) shows the learning curve for distributions gen erated with the ‘ac- tual’ smoothness scale la= 0.4,0.05and studied using the ‘learning’ smoothness scalel= 0.2(we show the case la=l= 0.2again as a reference). The Λ∼1/√ N behavior is seen unmistakably. The prefactors are a bit larg er (unfortunately, in- signiﬁcantly) than the corresponding ones from Fig. (3.2), so we may expect that the ‘right’l, indeed, provides better learning (see Section 3.5 for a det ailed dis- cussion). Finally, the approach to the asymptotes again is d ifferent for the differ- ent examples considered, but it is still explainable by the a rgument we used for Fig. (3.2). To generate outliers that are even more uncommon than the one s just dis- cussed one may want to distort the xaxis (use different parameterization), and this results in a variable smoothness scale la(x). As an example, Fig. (3.3) shows the learning curve for la= 0.2distributions that have been rescaled according tox→x−0.9 sin(2πx/L). For the rescaled variable, la(x)varies from 0.02to 0.38. Two separate straight line ﬁts—through the ﬁrst ﬁve (shown ) and the last four points—are possible for this data. Each of the ﬁts separ ately agrees with the prediction, but their prefactors are different. This is pro bably just a numerical artifact because 1000 Fourier modes used here feel like much less in some places due to the rescaling, and this shows up at large enough N. Alternatively, this may be an indication that a detailed analysis of the reparame terization invariant formulation (Periwal 1997, 1998) is needed. Finally, Fig. (3.4) illustrates learning when not only l, but alsoηis ‘wrong’ in the sense deﬁned above. We illustrate this for ηa= 2,0.8,0.6,0(remember that onlyηa>0.5removes UV divergences). Again, the inverse square root dec ay ofΛshould be observed, and this is evident for ηa= 2. Theηa= 0.8,0.6,03.4. Learning with ‘wrong’ priors 87 10110210310410510−410−310−210−1100101Λ Nηa=1, la=0.2, data, best fit ηa=2, la=0.1, data, best fit ηa=0.8, la=0.1, data, best fit ηa=0.6, la=0.1, data, one run ηa=0, la=0.12, data, one run Figure 3.4: Λas a function of N,ηaandla. Best ﬁts: for ηa= 2,la= 0.1,Λ = (0.40±0.05)N−0.493±0.013, forηa= 0.8,la= 0.1,Λ = (1.06±0.08)N−0.355±0.008. l= 0.2for all graphs, but the one with ηa= 0, for which l= 0.1. cases are different: even for Nas high as 105the estimate of the distribution is far from the target, thus the asymptotic regime is not reache d. This is a crucial observation for our subsequent analysis of the smoothness s cale determination from the data (Section 3.5). Remarkably, Λ(both averaged and in the single runs shown) is monotonic, so even in the cases of qualitatively less smooth distributions there is still no overﬁtting . On the other hand, Λis well above the asymptote for η= 2and smallN, which means that initially too many details are expected an d wrongfully introduced into the estimate, but then they are a lmost immediately (N∼300) eliminated by the data.3.5. Selecting the smoothness scale 88 3.5 Selecting the smoothness scale Results presented in the last Figures already suggest that O ccam factors should work in this inﬁnite dimensional case, and that it indeed is p ossible to see this in numerical simulations. The competition between the data an d the Occam factor is equivalent to minimizing the expression [cf. Eq. (A.6, A. 7)] H∗[φcl;{xi};l] =/integraldisplay dxl 2(∂xφcl)2+N/summationdisplay j=1φcl(xj) +1 2/radicalBigg N ll0/integraldisplay dxexp [−φcl(x)/2],(3.9) which makes the total probability of the data maximal, and th us the length nee- ded to code it minimal. How does the smoothness scale l∗that minimizes H∗ behave? The data term [second in Eq. (3.9)] on average is equa l toNDKL(P\|\|Qcl), and it can be small compared to the other terms for very regula rP(x). Then only the kinetic and the ﬂuctuation terms matter, and l∗∼N1/3, as was obtained by Bialek, Callan, and Strong (1996). For less regular distrib utionsP(x)[cf. graphs forηa= 0,0.6,0.8on Fig. (3.4)], this is not true. Indeed, for η= 1,Qcl(x)approx- imates large scale features of P(x)very well, but details at scales smaller than ∼/radicalBig l/NL are not present in it. If P(x)is taken from the prior, Eq. (3.2), charac- terized by some ηaandla, then according to Eq. (3.8) the contribution of these details falls off with the wave number kas∼(L/la)ηa−1/2k−ηa. Thus the expected data term is NDKL(P\|\|Qcl)∼N/integraldisplay∞ √ NL/l/parenleftbiggL la/parenrightbigg2ηa−1 k−2ηa=N/parenleftbiggL la/parenrightbigg2ηa−1/parenleftbiggNL l/parenrightbigg−ηa+1/2 ,(3.10) and this is not necessarily small. For ηa<1.5it actually dominates the kinetic term and competes with the Occam factor to establish the rele vant smoothness scale. Summarizing, l∗∼N1/3, ηa≥1.5 (3.11)3.5. Selecting the smoothness scale 89 10210410610−310−210−1100 l* Nηa=1, la=0.2 ηa=0.8, la=0.1 ηa=1, variable la, mean 0.12 ηa=2, la=0.1 Figure 3.5: Smoothness scale selection by the data. The line s that go off the axis for smallNsymbolize that H∗monotonically decreases as l→ ∞ . l∗∼N(ηa−1)/ηa,0.5<ηa<1.5. (3.12) There are two noteworthy things about Eq. (3.12). First, for ηa=η= 1,l∗sta- bilizes at some constant value, which we expect to be equal to la. Second, even ifηa∝ne}ationslash=η, butηa<1.5, then Eqs. (3.3, 3.12) ensure that Λ∼N1/2ηa−1, and this asymptotic behavior will be reached almost immediately, un like in theηa= 0,0.6 examples from Fig. (3.4). This performance is, at most, a con stant factor away from the limits set by heuristic calculations of predictive information, Eq. (2.99), with the ‘right’ priors ηa=η∝ne}ationslash= 1—a remarkable result! We present simulations relevant to these predictions in Fig s. (3.5, 3.6). Unlike in the previous Figures, these results are not averaged due t o extreme computa- tional costs, so all our further claims, which are inherentl y statistical, have to be3.6. Can the ‘wrong’ prior help? 90 taken cautiously. On the other hand, selecting l∗and observing the effects asso- ciated with it in single runs has some practical advantages s ince then we are able to ensure the best possible learning for any given realizati on of the data, not just on average. Figure (3.5) shows single learning runs for various ηaandla. In addition, to keep the Figure readable, we do not show runs with ηa= 0.6,0.7,1.2,1.5,3, and ηa→ ∞ , which is a ﬁnitely parameterizable distribution. All of th ese display a good agreement with the predicted scalings, Eq. (3.11, 3.12 ). Figure (3.6) shows the KL divergence between the target dist ribution and its classical estimate calculated at l∗; the average of this divergence over the samples and the prior is the learning curve. Again, the predictions a re clearly fulﬁlled. Note that for all ηawith exception of ηa=η= 1 there is indeed a qualitative advantage in using the data induced smoothness scale. To ill ustrate this more clearly and ease the comparison we replotted some of the curv es with adaptive l side by side with their ﬁxed lanalogues on Fig. (3.7). 3.6 Can the ‘wrong’ prior help? The last four Figures have illustrated some aspects of learn ing with ‘wrong’ pri- ors. However, more importantly, all of our results may be con sidered as belong- ing to the ‘wrong prior’ class. Indeed, the actual probabili ty distributions we used were not nonparametric continuous functions with smoo thness constraints, but were composed of kcFourier modes and thus had exactly 2kcparameters. Usually it would take well over 2kcsamples to even start to close down on the3.6. Can the ‘wrong’ prior help? 91 10210410610−510−410−310−210−1100 DKL( P\| Qcl) Nηa=1, la=0.2 ηa=0.8, la=0.1 ηa=1, variable la, mean 0.12 ηa=2, la=0.1 Figure 3.6: Learning with the data induced smoothness scale . actual value of the parameters, and many more to get accurate results. How- ever, using the wrong continuous parameterization [ φ(x)] we were able to obtain good ﬁts for as low as 1000 samples [cf. Figs. (3.1)] with the help of the prior Eq. (A.5). Moreover, learning happened continuously and mo notonically with- out huge chaotic jumps of overﬁtting that necessarily accom pany any brute force parameter estimation method at low N. So, for some cases, a seemingly more com- plex model is actually easier to learn! We can summarize this by stating that, when data are scarce an d the param- eters are abundant, one gains even by using the regularizing powers of wrong priors. The priors select some large scale features that are the most important to learn ﬁrst and ﬁll in the details as more data become availabl e. If the global fea- tures are dominant (arguably, this is generic), one actuall y wins in the learning speed [cf. Figs. (3.2, 3.3, 3.6)]. If, however, small scale d etails are as important,3.6. Can the ‘wrong’ prior help? 92 10210410610−510−410−310−210−1100 DKL( P\| Qcl) Nηa=0.8, la=0.1, l= l* ηa=0.8, la=0.1, l=0.2 ηa=2, la=0.1, l= l* ηa=2, la=0.1, l=0.2 Figure 3.7: Comparison of learning speeds for the same data s ets with different a priori assumptions. In all runs we learn using the model wit hη= 1, andlis either predeﬁned, or set by the Occam factor. then one is at least guaranteed to avoid overﬁtting [cf. Fig. (3.4)]. Thus we can argue that our numerical experiments support the Occam–like claim we made in Section 2.6: if two models provide equally go od ﬁts to data, the simpler one should always be used . In particular, using Eq. (2.102) we see that for our simulations, the nonparametric QFT model is simpler (as characterized by the predictive information) than a ﬁnite dimensional one for N < Nc∼(kclogkc)2. We operate in this regime in all of our simulations, and so we m ust learn faster and with less overﬁtting if we use the wrong parameterizatio n. Note, that these results are very much in the spirit of our whole program: not o nly is the value of l∗selected that simpliﬁes the description of the data, but the continuous param- eterization itself serves the same purpose. This is an unexp ectedly neat general- ization of the MDL (Rissanen 1989) principle to nonparametr ic cases.3.7. Discussion 93 3.7 Discussion The ﬁeld theoretic approach to density estimation in princi ple not only regular- izes the learning process but also allows the self–consiste nt selection of smooth- ness criteria through an inﬁnite dimensional version of the Occam factors. We have shown numerically that this works, even more clearly th an was conjectured. Forηa<1.5,Qestand the learning curve Λtruly become properties of the data and not of the Bayesian prior used for learning: one can set a l earning machine to work atη= 1and be sure that this does not bias the estimates in any excess ive way. If we can extend these results to include ηa>1.5and combine this work with the reparameterization invariant formulation of Peri wal (1997, 1998), this should give a complete theory of Bayesian learning for one di mensional distri- butions, and this theory has no arbitrary parameters. In add ition, if this theory properly treats the limit ηa→ ∞ , we should be able to see how the well–studied ﬁnite dimensional Occam razors and the MDL principle arise f rom a more gen- eral nonparametric formulation. These results also have some biological implications. Firs t of all, it may be that this smoothness scale adaptation mechanism is partly r esponsible for a com- monly known effect: children learn faster than adults. More seriously, and more closely connected to the models discussed here is the learni ng and development of smooth ‘maps’ in the nervous system (see, for example, Knu dsen et al. 1987). These maps become much less susceptible to the sensory input s as time passes, and this may be interpreted in terms of stiffening of the smoo thness constraint. Indeed, starting from scratch, it is very easy to drift the sm oothness scale to such3.7. Discussion 94 large values that susceptibility of the learning machine (i n other words, an ani- mal) to the new data will be extremely small. Second, if our conclusions are correct, then a learning mach ine that is pro- grammed to solve problems at η= 1 caneasily solve more complex problems with anyηa,1.5> ηa>0.5, by performing a simple averaging over the smoothness scales. At worst, this procedure may lead to a constant multi plicative drop in performance. By analogy, we may expect that, once an animal i s able to treat a problem that falls in any power–law class with respect to the predictive informa- tion, then it is able to treat any problem that provides more p redictive informa- tion with only a multiplicative overhead. Since, as we have a lready discussed (Hilberg 1990, Ebeling and P¨ oschel 1994, P¨ oschel, Ebelin g, and Ros´ e 1995), hu- mans can solve power–law problems, it is encouraging to know that there is no learning task in this world that is, in principle, too difﬁcu lt for us (our lifetime is the only limiting factor). More seriously, if it is, indeed, possible to construct a complete theory of one (and, possibly, higher) dimensional learning, where both the smoothness scale and the exponent can be self–consisten tly determined, then the questions we asked in Section 2.6 (like “what models do hu mans use for learn- ing?”) may lose their meaning—any model is almost as good as a ny other, and it is very difﬁcult to look for possible multiplicative differ ences between them. Sur- prisingly, these questions are meaningful if such a theory c annot be constructed. In this case, as we saw in Eqs. (3.11, 3.12) and on Figs. (3.5, 3 .6), a complex learn- ing machine cannot effectively adapt to simple tasks. This a gain accords with our subjective experience that it is sometime very hard to ﬁn d a simple solution when expecting a complex one. It should be possible to devise an experiment that would quantify this failure to solve simple problems in complex contexts.Chapter 4 Learning discrete variables: Information–theoretic regularization 4.1 The general paradigm In Chapters 2 and 3 of this work we discussed what we consider t o be some of the most interesting problems in modern statistics and le arning theory. We deﬁned predictive information and complexity, studied the learning of nonpara- metric distributions, and showed an example of how Occam fac tors generalized to inﬁnite dimensional problems. There is one tantalizing q uestion that followed us through this whole discussion: many problems require a pr ior to regularize learning, but then how can one make sure that the estimates an d the conclusions are inferred from the data, rather then being imposed by some a priori knowledge that the data do not support? The results in the inﬁnite dimen sional generaliza- tion of the Occam factors seemed especially interesting in t his respect—estimates can become almost insensitive even to the qualitative choic e of prior, at least in a broad range. This conclusion, together with Periwal’s (19 97, 1998) work, are the only results known to us that aim towards constructing a r eparameterization 954.1. The general paradigm 96 and prior invariant (or, better yet, ignorant) theory of non parametric Bayesian inference. Even though we have concentrated on nonparametric problems , similar difﬁ- culties also exist in seemingly simpler, parametric cases. The choice of the prior for ﬁnite parameter learning scenarios is still a hot topic, and various alternatives are proposed that make some universal theoretical sense wit hin the framework of information theory, MDL theory, etc. Examples include un iversal priors (Ris- sanen 1983, Lee and Vit´ anyi 1993) or Jeffreys’ priors (Clar ke and Barron 1994, Balasubramanian 1997). We think, however, that the prior really should embody a prio ri knowledge, and thus we cannot agree with the use of universal, globally d eﬁnable choices. On the other hand, there is an obvious appeal for approaches b ased on more general theoretical principles. For example, the problems of nonuniform conver- gence of the estimate to the target, Eq. (A.11), and of the inf rared divergence of predictive information, Eq. (2.97), are easily resolved if one turns to reparameter- ization invariant priors (Periwal 1997, 1998). These have a clear theoretical edge over non–invariant ones since they estimate a true density, that is, a function that transforms like a density under reparameterizations of the independent variable. As we tried to emphasize in this work, learning is informatio n accrual, and, therefore, it is a part of Shannon’s information theory. Thu s a general theoretical principle can be to construct priors solely from informatio n–theoretic quantities like entropy (or self–information), Kullback–Leibler div ergence (or relative in- formation), various mutual informations, etc., and with no other constraints. In addition, since information–related quantities are forme d from log–probabilities it makes sense to include them exponentially into the priors , which are, after4.2. Discrete variables: a need for special attention 97 all, also probabilities. In Sections 4.3 and 4.4 we will pres ent a simple example that illustrates the use of this general principle—regular ization with information– theoretic quantities—and show that it is possible and advan tageous to learn with such priors. 4.2 Discrete variables: a need for special attention When learning probability densities of continuous variabl es, except for the very simple cases N >> d VC, priors are used to balance the quality of ﬁts to the data against the complexity of solutions (cf. Section 2.4.5 and C hapter 3); this smooth- ing of data prevents overﬁtting. It is easy to construct smoo thing priors for con- tinuous variables: continuity implies a metric, so localit y is deﬁned, and then one just needs to punish distributions that exhibit large va riations over small dis- tances. Such a ‘smoothness’ prior will work as a regularizer . The case when a variable is discrete, and the (discrete) metr ic is impossible to deﬁne, presents a problem. Usually this case is not consid ered interesting, because the law of large numbers guarantees that, at large N, the frequencies of events estimate probabilities well. However, if the numb er of examples N is smaller or comparable to the number of possible outcomes K, then statistical ﬂuctuations are large. This situation is not as uncommon as o ne might hope. For example, it is possible that syntactic structures in a langu age can be inferred from statistical arguments alone (see, for example, Pereira et a l. 1993, Manning and Sch ¨ utze 1999). Estimating probabilities of occurrence of a few thousand common words is rather easy. It is even feasible to construct condit ional distributions of nouns given verbs (Pereira et al. 1993). However, it is total ly unrealistic to expect4.2. Discrete variables: a need for special attention 98 to build an accurate probability distribution of whole sent ences from the raw data without some a priori knowledge. Similar problems appear in molecular biology. For example, gene expression is governed by promoter regions in DNA molecules. These regi ons are usually thought to be constructed from two ﬁve base pair long blocks ( there are 45pos- sible different realizations of these) that appear anywher e inside ‘junk’ genetic material, which is about a hundred bases in length. If one tri es to ﬁnd the mean- ingful 5 + 5 structures by statistical methods (see, for example, van He lden et al. 1998), then a probability distribution over 45∗45∼106possibilities must be constructed. Many experiments are done on yeast, and their g enome is only a few millions of base pairs long. So getting a full statistics is, at best, problematic. Even worse, if one tries to look for a possibility of promoter s of a different length, then the problem becomes totally hopeless. To solve these and similar problems, one needs smoothing pri ors that regu- larize ﬂuctuations. However, now there is no notion of local ity to create them. It is not at all evident how to impose smoothness conditions, or whether these conditions will speed up learning in any way. It is clear thou gh that any smooth- ing must be global—we cannot talk about local changes, but gl obal variability is well deﬁned. Recall that variability can be measured non– metrically by en- tropies, and these have a very special meaning in informatio n theory—they are the unique measure of information (Shannon 1948). Therefor e, learning a dis- crete variable may turn out to be an excellent example of the g eneral paradigm introduced above. In the next Sections we present a toy model for learning a disc rete variable and show that it is possible to regularize and speed up such le arning with the4.3. Toy model: theory 99 help of information–theoretic priors. In general, discret e calculus is much less developed than its continuous analogue (precisely for the r eason that the notion of locality is absent), so analytical results can usually be achieved only for very simple problems, and our example is like that. Nonetheless, this case is of interest as it solves some real world problems and, more importantly, because it develops techniques that later will be used in more complicated tasks ; the work on those is already in progress. 4.3 Toy model: theory Consider the following ‘real world’ problem. A US Census Bur eau needs a pre- liminary report on statistics of people in Trenton, NJ based on the Census–2000 data. Unfortunately, only a few thousand households have ﬁl led in their Cen- sus cards, and this is clearly not enough to sample adequatel y many classes x, x= 1···K(we also call them possible outcomes orbins) into which the peo- ple are classiﬁed (ethnicity, marital status, educational level, size of the house- hold, etc.) Suppose now that Newark, NJ, perceived to have a s imilar population statistics, has been quick to organize door–to–door counti ng of people, so that a good sampling of Newark’s population, Q∗(x), is available. Can the Census Bureau statisticians use these data to answer their questio ns about the Trenton people? An obvious solution would be to take a weighted avera ge of the (un- dersampled) Trenton counts and the (well sampled) Newark pr obabilities. But how should the weights be set in order to ensure that the Trent on estimate is just smoothed and not unfairly biased by the Newark data? We can offer a solution to the problem by choosing an a prior pr obability4.3. Toy model: theory 100 density for Q(x), the Trenton distribution, that is biased towards the refer ence Newark distribution Q∗(x). This may be done in the following form P[Q(x),λ] =1 ZQ(λ)exp [−λD(Q∗\|\|Q)]δ/parenleftBiggK/summationdisplay x=1Q(x)−1/parenrightBigg P(λ), (4.1) where P(λ)is some a priori normalized density for the inverse temperat ure–like parameter λ,ZQ(λ)≡/integraltext[dQ(x)]P[Q(x)\|\|λ]is the normalization of the a priori dis- tribution of Q(x)conditional on λ, andDis some measure of distance between the two distributions, Q∗andQ. If we want to stick to our paradigm of using information–theoretic quantities only, then we do not have much freedom in se- lectingDsince the natural distance between any two distributions in information theory is the familiar Kullback-Leibler divergence, D(Q∗\|\|Q)≡DKL(Q∗\|\|Q) =K/summationdisplay x=1Q∗(x) logQ∗(x) Q(x). (4.2) In the language of coding theory, this choice of Dmeans that we optimize our coding strategies for Q, but we want Q’s such that the data coming from the refer- ence distribution can be coded compactly also. We could have chosen to measure distances in the opposite direction and switch arguments in the KL divergence. Then the best coding for Q∗is ﬁxed, and we look for Qthat is still coded well. We chose Eq. (4.2) over the other choice because this allows an e xact solution. Now, similarly to Bialek et al. (1996, see also Appendix A.1) , we apply the Bayes formula to get the probability density for Q(x)andλgiven the data {xi} P[Q(x),λ\|{xi}] =P[{xi}\|Q(x),λ]P[Q(x),λ] P({xi}) =P[Q(x),λ]/producttextN i=1Q(xi) /integraltext[dQ(x)]dλP[Q(x),λ]/producttextN i=1Q(xi). (4.3) The least square estimator of Q(x)is then, as usual, Qest(x\|{xi}) =/integraldisplay dλ[dQ(x)]Q(x)P[Q(x),λ\|{xi}]4.3. Toy model: theory 101 =∝an}bracketle{tQ(x)Q(x1)Q(x2)···Q(xN)∝an}bracketri}ht(Q,λ) ∝an}bracketle{tQ(x1)Q(x2)···Q(xN)∝an}bracketri}ht(Q,λ), (4.4) where ∝an}bracketle{t···∝an}bracketri}ht(Q,λ)stands for averaging over Qandλwith respect to the prior; simi- larly,∝an}bracketle{t···∝an}bracketri}ht(Q)means averaging only over P[Q\|λ]. Ifλwas ﬁxed, then the averaging in Eq. (4.4) would have one integ ral less—a simpler problem. However, varying it may allow the Occam fac tor that arises from volumes in the Q-space to ﬁnd some λ∗that creates the shortest (thus the most probable) description of the data (cf. Section 3.5). By achieving the optimal balance between the ‘goodness of ﬁt’ and closeness to the ref erence, this may resolve the problem of a possible erroneous bias towards Q∗. As mentioned above, the solution of this model is rather simp le. We write n(x) for the data count in the bin x,/summationtext xn(x) =N. Then, leaving aside the λintegral for a while, we have (see Appendix A.2): ∝an}bracketle{tQ(x1)···Q(xN)∝an}bracketri}ht(Q)=/integraldisplay [dQ]e−λDKL(Q∗\|\|Q) ZQ(λ)δ(/summationdisplay xQ(x)−1)/productdisplay xQ(x)n(x)(4.5) =eS[Q∗] ZQ(λ)/integraldisplay [dQ]δ(/summationdisplay xQ(x)−1)/productdisplay xQ(x)n(x)+λQ∗(x)(4.6) =eS[Q∗] ZQ(λ)/producttextK x=1Γ/parenleftBig n(x) +λQ∗(x) + 1/parenrightBig Γ(λ+N+K), (4.7) whereS[Q∗]is just the entropy of the reference distribution. ZQ(λ)is given by the same integral, Eq. (4.5, 4.7), but with n(x) =N= 0. Therefore, if we now integrate overλusing the steepest descent technique, then the most probabl e value of the inverse temperature is determined by N/summationdisplay j=11 λ∗+K+j−1−K/summationdisplay x=1n(x)/summationdisplay j=1Q∗(x) λ∗Q∗(x) +j= 0. (4.8) Numerically (cf. Section 4.4) this equation has one nontriv ial solution for large N, so that the Occam factor seems to work again. Unfortunately , however, we4.3. Toy model: theory 102 were unable to obtain satisfying analytical results with ex ception of some trivial asymptotic limits. If, on the other hand, we keep λﬁxed, then [see Eq. (4.4)] Qest(x\|{xi};λ) =n(x) +λQ∗(x) + 1 N+λ+K(4.9) The simplicity of this result is intriguing. Our initial sug gestion to average the actual undersampled data with the well sampled smoothing di stribution turns out to have deep roots in Bayesian inference with the prior Eq . (4.1)! Note the presence of ‘ +1’ in Eqs. (4.7, 4.9). Due to this term the estimated distribution is a smoothed version of the counts n(x)even ifλ→0. In this the- ory, no bin has an estimated probability of zero, so that DKL(Q∗\|\|Q)always is well deﬁned, and observing the next sample in any bin never is a completely unexpected event. This summand, which has so many desirable consequences, appears because we deﬁne the prior, Eq. (4.1), on the space of Q’s. Changing the variables from probabilities to likelihoods, −φ(x) = logQ(x), creates a Jacobian which effectively adds one count in every bin. Maximal likel ihood estimation of parameters does not have this feature, and this is yet anothe r argument in favor of the Bayesian formulation. Even though this toy model has an exact solution, it still is i nstructive to per- form a saddle point (large N) analysis in the hope that some useful knowledge reusable in more complex settings will be gained. With the us ual change of vari- ables, Q(x) = e−φ(x), Q∗(x) = e−φ∗(x), (4.10) leaving the λintegral aside again, and replacing the δ-function by its Fourier representation, we get the following expression for the cor relation functions: ∝an}bracketle{tQ(x1)···Q(xN)∝an}bracketri}ht(Q)=/integraldisplay[dφ(x)] ZQ(λ)dµ 2πe−H[φ(x),iµ,λ,φ∗(x)]−/summationtext x[n(x)+1]φ(x),(4.11)4.3. Toy model: theory 103 H[φ(x),iµ,λ,φ∗(x)] =λ/summationdisplay xQ∗(x)[φ(x)−φ∗(x)] +iµ(/summationdisplay xe−φ(x)−1).(4.12) Calculating up to one–loop corrections using the steepest d escent techniques, this Hamiltonian results in ∝an}bracketle{tQ(x1)···Q(xN)∝an}bracketri}ht(Q)= e−Heﬀ[φcl(x),λ,φ∗(x)]−/summationtext x[n(x)+1]φcl(x), (4.13) Heﬀ[φcl(x),λ,φ∗(x)] =H[φcl(x),iµ=N+λ+K,λ,φ∗(x)] +K 2log/parenleftbigg 1 +N λ/parenrightbigg +1 2/summationdisplay x(φ∗(x)−φcl(x)),(4.14) Qcl(x)≡e−φcl(x)=n(x) +λQ∗(x) + 1 N+λ+K(4.15) Again, just like in Eqs. (3.1, A.7), the ﬂuctuation determin ant [the last two terms in Eq. (4.14)] has a different λdependence than the data and the prior terms. This suggests, even more clearly than the exact result Eq. (4 .7), that competition between them will select a nontrivial most probable value of λ∗. Note thatQest, Eq. (4.9), is equal to Qcl. A priori one should not expect them to be similar even for large N. Indeed, the matrix of second derivatives in the saddle point calculation has eigenvalues ∼n(x), and these are small for bins with small counts. So, in principle, one could expect large discrepanc ies between the exact and the classical solutions. The fact that they arethe same inspires a hope that the saddle point analysis may remain useful for other, more comp lex, problems. Finally, we want to illustrate yet another important aspect of this toy model. What is the predictive information for this system? In gener al, it is difﬁcult to calculate, so we consider two very simple limits: λ≪N, K ≪N, and 1≪N≪ λ,K≪λ. The ﬁrst case closely parallels calculations of Section 2. 4.2 and yields S1(N)≈K 2logN. (4.16)4.3. Toy model: theory 104 In the second case, somewhat lengthy calculations lead to S(N) =NS[Q∗] +S1(N), lim N→∞,N/λ→0S1≈0, (4.17) whereS[Q∗]is again the entropy of the reference distribution. These re sults are expected: for small λand largeNthis problem is just learning a K–parameter distribution, so Eq. (2.50) should hold. On the other hand, f or extremely large λ, the estimate converges to the reference distribution regar dless of the data, so the problem is effectively zero dimensional. If we do estimates at some large, but ﬁxed, λ, then we should see a crossover from the zero to the Kparameter regime.1For generic distributions the crossover will be smooth, since each bin starts to add its (1/2) logNto the predictive in- formation independently when the estimate for that bin swit ches from the ref- erence value to the count, i. e., when n(x)> λQ∗(x)[cf. Eq. (4.9)]. A smooth crossover from zero to the logarithm is possible only throug h a faster than loga- rithmic growth for some range of N. Indeed, from the discussion in Section 2.4.6, we know that continuous addition of extra degrees of freedom is a sign of the power–law growth of predictive information. It is remarkab le that a discrete sys- tem as simple as this toy model can exhibit such a rich behavio r that, so far, has been associated with only very complex nonparametric model s; we will see a numerical demonstration of this in the next section. 1If we average over λ, andλ∗(N)/Nstarts large and drifts to below 1asNincreases, then we will still observe the crossover. This behavior of λ∗is possible since at low Nmost of the weight should go to the smoothing term, while at large Nthe actual counts are to be trusted.4.4. Toy model: numerical analysis 105 4.4 Toy model: numerical analysis If we want to be able to observe all of the features described i n the previous sec- tion, thenK, the number of bins, should be large enough to allow for a prom inent prior–dominated (scarce data) region, but it also should be small enough so that we can generate enough samples and observe a cross–over to th e data–driven regime in all bins. The choice of K= 75,100,125reasonably satisﬁes both condi- tions and will be used in all of our simulations. The next important question is the generation of random dist ributions from the prior, Eq. (4.1). Due to the δ-function normalization constraint this is a com- plicated task. However, in the limit of large λ,DKL[Q∗\|\|Q]typically is small and can be approximated by the χ2distance lim DKL→0DKL[Q∗(x)\|\|Q(x)] =1 2K/summationdisplay x=1[Q(x)−Q∗(x)]2 Q∗(x). (4.18) Then the prior, Eq. (4.1), becomes a multi–variable normal d ensity, and genera- tion of random distributions is easy2. The choice of λis motivated by the same arguments as the choice of K. The asymptotic regime is reached for λK > N , while simulations are time limited byN∼105. Therefore, we must use λof up to about 1000 . On the other hand we want to see as much of the prior–enforced learning as possi ble, so we choose to work close to this limit: λ= 300,500,1000 . One may be concerned that these large values will produce random distributions that are alm ost identical to the reference, which would make some of our results less interes ting. Fig. (4.1) shows 2The KL divergence tends to the χ2measure from below. Thus replacing DKLwithχ2pro- duces a slightly narrower prior; this difference becomes sm aller as λgrows. In principle, this can produce dramatic changes in statistics, but for our choices of independent parameters this turns out to be almost irrelevant.4.4. Toy model: numerical analysis 106 00.0050.010.0150.020.0250.030.03500.0050.010.0150.020.0250.030.035 Q* Qdata Q= Q* line Figure 4.1: Comparison between the reference distribution and a typical random one generated with λ= 500 , andK= 100 . Each point corresponds to the values/parenleftBig Q∗(x),Q(x)/parenrightBig in one of the bins x. that these fears are misplaced. The reference and the random distributions are similar (as we want them to be), but not excessively so. 4.4.1 Learning with the correct prior Fig. (4.2) shows the Ndependence of the universal learning curve Λ[averaged KL divergence, Eqs. (2.13, 3.3)], for various combinations ofλandK. All of the behavior predicted in Section 4.3 is observed clearly. The l earning curves start out ﬂat (predictive information and the effective number of parameters are zero) and soon enter a continuous series of transitions that add mo re and more degrees of freedom. Finally the behavior enforced by Eq. (2.62), Λ(N) =K 2N, (4.19) is reached. Notice that, in agreement with the claim that the number of active4.4. Toy model: numerical analysis 107 10110210310410510−410−310−210−1100 NΛ K =100, λ = 500 K =100, λ = 300 K =100, λ = 1000 K =75, λ = 500 K =125, λ = 500 Figure 4.2: Λas a function of λ,K, andNforλ=λa. parameters depends on the comparison between the counts and the value of λ, the asymptotic regime is reached at earlier Nfor smaller values of the parame- ter. The agreement with this asymptotic behavior is so remar kably good that we chose not to quote any ﬁts: all of them are within the expected errors. In the pre–asymptotic regime, learning curves with larger λstart lower since there the random distributions are very close to the referen ce and are estimated much better by it. Similarly, curves with smaller Kalso start lower because now there are fewer degrees of freedom and, therefore, fewer way s to get a larger KL divergence. 4.4.2 Learning with ‘wrong’ priors As we did for nonparametric distributions in Section 3.4, we now want to in- vestigate the performance of the learning algorithm on atyp ical data sequences.4.4. Toy model: numerical analysis 108 10110210310410510−410−310−210−1100 NΛ λa = λ= 500 λa = 500, λ = 0 λa = 300, λ = 500 λa = 1000, λ = 500 λa = λ =500, Q* ≠ Qa* Figure 4.3: Λas a function of λ,λa,Q∗ aandN. For all curves K= 100 . That is, as before, we will differentiate two sets of paramet ers:λandQ∗, which encode the expectations of the learning machine, and λaandQ∗ a, which together with Eq. (4.1) describe the ensemble of atypical target dist ributions. Simulations related to this question are summarized in Fig. (4.3). Here w e have shown learn- ing with different combinations of λ∝ne}ationslash=λa, and the case λ=λa= 500,Q∗=Q∗ ais plotted as a reference. Comparing the curves with the corres ponding ones from the previous Figure, we clearly see that, even though learni ng with an ‘incorrect’ prior is possible, there are discrepancies: convergence to the asymptotic limit is different, and the curves start slightly higher then their ‘ correct’ counterparts. Another interesting example shown is when λis ‘correct’, but the reference distribution itself is totally wrong. We see that this type o f mistake is much more costly: even for Nas large as 105the inﬂuence of the wrong reference distribution is still strong enough to compromise fast learning.4.4. Toy model: numerical analysis 109 Note the curve with λa= 500 ,λ= 0. This is a case of ‘trivial’ learning, when the only regularization is due to the Jacobian of the Q→φtransformation [the ‘+1’ term in Eq. (4.9)]. If not for it, the estimate would be extre mely overﬁtted and would have zeros, and the Λwould explode. On the other hand, with the ’+1’ correction, Λstarts out from a constant value, which is the KL divergence between the target and the uniform distribution. 4.4.3 Selecting λwith the help of the data Having shown how the learning machine performs on expected a nd unexpected data samples, we are now in a position to ask if the Occam facto rs can select the right regularization parameter λas in the case of ﬁnite dimensional models (MacKay 1992, Balasubramanian 1997) and nonparametric mod els (Bialek, Cal- lan, and Strong 1996, and Section 3.5 of the present work). Th is question has an afﬁrmative answer, and Fig. (4.5) shows λ∗, the value of λthat maximizes the correlation function Eq. (4.7) and minimizes the expone nt in Eq. (4.13). We show the results averaged over many runs, even though this fo rm of presentation is questionable because λ∗ﬂuctuates a lot in different trials. These ﬂuctuations explain the kinks on the three lower curves. For Nto the left of the kinks, there are many realizations of the data, for which there is no best v alue forλ, and the correlation functions are maximized at λ∗→ ∞ . The kinks appear due to the numerically imposed ﬁnite cutoff on possible values of t he parameter. Apart than this, the rest of the learning curves’ behavior is as hop ed. For ensembles generated at λa= 500 and studied at Q∗=Q∗ a,λ∗turns out to be very close to 500. IfQ∗∝ne}ationslash=Q∗ a,λ∗drifts to smaller values, letting the data, not the referenc e,4.4. Toy model: numerical analysis 110 10110210310410510010210410610810101012 Nλλa = 500, Q = Qa* λa = 500, Q* ≠ Qa* λa = 0 λa = ∞, Q* = Qa* Figure 4.4: λ∗for various ensembles of target distributions. control the estimate. The same happens for λa= 0for anyQ∗since at this value of the parameter the target is, again, far from the reference . Finally, we examine the case where the ensemble of the distri butions is qual- itatively closer to the reference than the prior, Eq. (4.1), allows (this corresponds toη∝ne}ationslash=ηafor nonparametric learning). This can be achieved by having a higher power of the KL divergence in the exponent of the prior, but su ch a choice is very difﬁcult for numerical simulations. So we take an easier, bu t less illustrative ex- ample ofλa=∞; that is, the target distribution is exactly equal to the ref erence. Here, not surprisingly, λ∗quickly becomes very large. Again, it often exceeds the numerical cutoff, so the average line shown should not be tak en to literally. Finishing up this section, in Fig. (4.4) we show the learning curves calculated atλ=λ∗for all the cases considered above. Comparing to the corresp onding learning curves in Fig. (4.3), we deduce that learning with a n adaptive λis much4.5. Further work 111 10110210310410510−610−510−410−310−210−1100 NΛ λa = 500, Q* = Qa* λa = 500, Q* ≠ Qa* λa = 0 λa = ∞, Q* = Qa* Figure 4.5: Learning with an adaptive λ∗. faster than with a ﬁxed wrong one, and the example with λa=∞is particularly demonstrative3. The only learning curve which starts off slightly worse tha n its ﬁxedλanalogue is that for λa= 500, Q∗=Q∗ a. Even though it improves very quickly, this once again proves the common knowledge that fo r small sample sizes nothing beats learning with the ‘correct’ prior. 4.5 Further work The toy example we have investigated resolves the questions it was meant to answer. Yet it is just a toy example, and most of its value lies in the extension to more difﬁcult problems. The most straightforward, but very interesting developmen t is one that has 3This remarkable performance for λa=∞is achieved with the upper cutoff on λ∗, and it is possible that Λcan fall off even faster without the cutoff.4.5. Further work 112 been already mentioned in passing. Reversing the direction of the KL divergence in Eq. (4.1) and choosing a uniform reference distribution Q∗, we obtain a prior that favors distributions with larger entropies, i. e., the ﬂattest and the most regu- lar distributions. Finding the least variable distributio n compatible with the data is certainly in the spirit of our work and deserves an investi gation. This idea may be made more sophisticated when the independen t variable is a vector,/vector x={x,y}. If the distribution Q(/vector x)is expected to be smooth, but the cardinalities of xandy(KxandKyrespectively) are large, then only the marginal distributions Q(x) =/summationtext yQ(x,y)andQ(y) =/summationtext xQ(x,y)are sampled well for Kx andKy≪N∼KxKy. To smooth the data we might want to weight the a priori probability of Q(x,y)by the entropy S[Q(x,y)]. However, this choice, though valid, is not the best. Indeed, the entropy can be small becau se the marginals are very narrow. But in the limit we are interested in, the margin als are well deﬁned by the data and do not require separate smoothing. So it is not the entropy of a distribution, but its value with respect to the entropy of th e marginals that should enter a regularizing prior. This is the mutual information I(x,y)betweenxand y, and it is, once again, a meaningful information–theoretic quantity. A more ambitious but very appealing direction is to combine t hese methods with the relevant information extraction ideas of Tishby et al. (1999) and Bialek and Tishby (in preparation), discussed brieﬂy in Section 2. 6. Recall, that these authors proposed to compress (that is, to smooth) the variab lexintoˆxso that the mutual information I(ˆx,y)remains as close to I(x,y), as possible. There is a one parameter family of solutions to the problem, and this param eter measures the relative importance of compression (smoothing) and preser vation of the informa- tion (ﬁt to the data). In many practical applications, choos ing the right value of4.5. Further work 113 this parameter is a problem. We can view the result of Tishby e t al. as a classical solution to a problem with some (yet to be deﬁned) prior. One c an realistically expect that the value of the parameter will, once again, be se t by the Occam factor. If this theory succeeds, we can use these results to further a dvance the the- ory of learning a nonparametric variable. Derivatives of de nsities do not have any special meaning in the framework of information theory. Using them, as in Eq. (A.5) or its reparameterization invariant version (Per iwal 1997, 1998), is thus not a preferred regularization. Building priors that inclu de terms with deriva- tives of many different orders and ﬁxing coefﬁcients of the t erms by requiring that the estimate does not depend on the UV details of the data (rounding or truncation)4may work and have a meaning in QFT, but this is an approach alie n to information theory. Similarly, preferring distributio ns that are close to their ﬁltered version and averaging over the ﬁlters afterwards5has the same prob- lem. What we mean by smoothness in any formulation, includin g nonparametric continuous ones, is that the independent, possibly continu ous, variable xcan be successfully coded in some ˆxof ﬁnite and small cardinality such that this com- pacted version explains the data almost as well as the origin al one. For example, in the ﬁnite parameter case, ˆx=α. Developing a theory for smoothing through compression would be a great achievement in itself, and with in this formalism we get the added advantage that the right balance between the goodness of ﬁt and the compression will again be determined by the Occam fac tor. This is obviously only a start to an extensive program of gene ralizations, and we hope make a signiﬁcant progress along these lines in the ne ar future. 4This Wilsonian renormalization group approach was suggest ed by V . Periwal. 5This idea is by W. Bialek.Chapter 5 Conclusion: what have we achieved? Let us summarize our achievements and compare them to the pro mises we made and the desires we expressed in the beginning of this work. As promised, we built a uniform and universally valid approach to learning b y using information theory and treating learning as an ability to predict. For th is we deﬁned a new quantity, the predictive information, and the study of it re vealed that it not only measures the information relevant to prediction of a time se ries, but also deﬁnes uniquely the complexity of the process that generated the se ries. Statistical me- chanics gave an insight on how learning is always annealing i n the model space, and then we could illuminate numerous connections to statis tical learning and coding theories and catch some omissions in those. Summariz ing, we indeed delivered on the promise of a coherent re–treatment of the ol d knowledge. Then we went further and showed that conventional ﬁnite para meter models are not the only possible scenario. We investigated nonpara metric and (under- sampled) discrete learning to show that the capacity contro l mechanisms work much better then one might have thought. We showed that the ’i nformation the- ory only’ approach to learning can be made self consistent an d does not need any supplemental help to survive. All of these efforts constitu te an attempt to build 114115 up some new ﬂesh around the core of ideas that are the focal poi nt of our atten- tion. This ﬂesh may seem too thin, and it certainly is—a lot mo re has to be done before one can ﬁnally say: “The End!” However, one has to star t somewhere, and we did.Appendix A Appendix A.1 Summary of nonparametric learning The main problem of statistics—inferring distributions fr om a ﬁnite data set— has a wide variety of possible practical applications. Usua lly, based on some a priori considerations, an observer has some ﬁnite-dimensi onal model for the dis- tribution being studied. Then the problem reduces to an esti mation of a ﬁnite number of parameters from a large data set, and this is relati vely well-studied (see, for example, Vapnik 1998, Balasubramanian 1997, and S ections 2.4.1–2.4.5 of the current work). Unfortunately, reducing the problem t o a ﬁnite number of parameters heavily biases the outcome of statistical infer ence: the true distribu- tion may not even be in the chosen family. Thus lately it has be come popular to look for nonparametric solutions to the problem of learning distributions (recent reviews are Dey et al. 1998 and Lemm 1999). As discussed in Sec tion 2.4.5, non- parametric estimations necessarily are prior dependent, i . e., Bayesian. Therefore, the result of the inference is a probability distribution of probability distributions, which becomes more concentrated as the number of samples inc reases. Even though the result depends on the prior, the prior may be very m ildly restrictive 116A.1. Summary of nonparametric learning 117 (say only some smoothness constraints are assumed), and the n the bias is less than in any ﬁnite parameter setting. Recently Bialek, Callan, and Strong presented an elegant fo rmulation that casts nonparametric Bayesian learning in the familiar sett ing of statistical me- chanics or, equivalently, Euclidean Quantum Field Theory ( QFT) (Bialek, Callan, and Strong 1996). This approach and some alternative formul ations were further developed by Periwal (1997, 1998), Holy (1997), and Aida (19 98). In the present work, we have utilized heavily the techniques and results of Bialek et al. and ex- panded or corrected some of their conclusions. To make our pr esentation more self contained, we here present a brief overview of the theor y augmented with some comments of our own. Following the original reference, if Ni. i. d. samples {xi},i= 1...N, are ob- served, then the probability that a particular density Q(x)gave rise to these data is given by the application of the Bayes formula P[Q(x)\|{xi}] =P[{xi}\|Q(x)]P[Q(x)] P({xi})=P[Q(x)]/producttextN i=1Q(xi) /integraltext[dQ(x)]P[Q(x)]/producttextN i=1Q(xi), (A.1) where P[Q(x)]encodes our a priori expectations of Q. Specifying this prior on a space of functions deﬁnes a QFT, and the optimal least square s estimator is then given by a ratio of correlation functions Qest(x\|{xi}) =/integraldisplay [dQ(x)]Q(x)P[Q(x\|{xi})] (A.2) =∝an}bracketle{tQ(x)Q(x1)Q(x2)...Q(xN)∝an}bracketri}ht(Q) ∝an}bracketle{tQ(x1)Q(x2)...Q(xN)∝an}bracketri}ht(Q), (A.3) where ∝an}bracketle{t...∝an}bracketri}ht(Q)means averaging with respect to the prior. Since Q(x)≥0, it is convenient to deﬁne an unconstrained ﬁeld φ(x) Q(x)≡1 l0e−φ(x), (A.4)A.1. Summary of nonparametric learning 118 Unlike another possible choice, Q(x) = [φ(x)]2, (Holy 1997) this deﬁnition puts Q andφin one–to–one correspondence. The next step is to select a prior that regularizes the inﬁnit e number of degrees of freedom and allows learning. We require that when we estim ate the distribu- tionQ(x)the answer must be everywhere ﬁnite. Also we want the prior P[φ]to make sense as a continuous theory, so that the statistics of φ(x)on large scales are not affected, for example, by discretization or round–o ff errors in xon small scales. This implies that we should look for a renormalizati on group invariant prior (the ﬁrst steps along this direction were performed in Aida 1998)1. Simpler, but almost equally satisfying, is any ultraviolet (UV) conv ergent prior. For xin one dimension, a minimal choice that is the easiest for theor etical (and, acciden- tally, numerical) analysis is P[φ(x)] =1 Zexp −l 2/integraldisplay dx/parenleftBigg∂φ ∂x/parenrightBigg2 δ/bracketleftbigg1 l0/integraldisplay dxe−φ(x)−1/bracketrightbigg , (A.5) where Zis the normalization constant, and the δ-function enforces normalization ofQ. The coefﬁcient ldeﬁnes a scale below which variations in φare considered to be too rapid, thus we refer to las the smoothness scale . By making this scale local, one may also achieve reparameterization invariance of lear ned results (Periwal 1997, 1998). The resulting ﬁeld theory was solved by Bialek et al. up to one –loop correc- tions in the limit of large Nusing standard semiclassical techniques: ∝an}bracketle{tQ(x1)···Q(xN)∝an}bracketri}ht(Q)≈1 lN 0exp/parenleftBig −Heﬀ[φcl(x);{xi};l]−N/summationdisplay j=1φcl(xj)/parenrightBig ,(A.6) 1As noted by V . Periwal in private communication, one may hope to construct a complete the- ory of nonparametric learning in many dimensions by choosin g a renormalization group compli- ant prior. That is, the prior’s (many) parameters have to be d eﬁned to change with the renormal- ization group ﬂow in such a way that the resulting correlatio n functions do not depend on the cutoff scale, which is in its turn due to round–off, discreti zation, or ﬁltering.A.1. Summary of nonparametric learning 119 Heﬀ[φcl(x);{xi};l] =l 2/integraldisplay dx(∂xφcl)2+1 2/parenleftbiggN ll0/parenrightbigg1/2/integraldisplay dxe−φcl(x)/2,(A.7) l∂2 xφcl(x) +N l0exp[−φcl(x)] =N/summationdisplay j=1δ(x−xj), (A.8) whereφclis the ‘classical’ solution to the ﬁeld theory. In the effect ive Hamiltonian [Eq. (A.7)], the ﬁrst term is due to the value of the prior at φcl, while the second one is the inﬁnite dimensional determinant arising from one –loop integration over ﬂuctuations around the classical solution. Calculati ng this determinant is the most technically involved step in the solution, and this can be done using a standard van Vleck technique (see, for example, Chapter 7 o f Coleman 1988). This term is a direct analog of Occam factors that appear in ﬁn itely parameter- izable models (MacKay 1992, Balasubramanian 1997) and allo w one to build a complexity penalizing razor. Using the WKB method, the authors have shown that the solutio ns [both the classical approximation Qcl= (1/l0) exp(−φcl), and the optimal least squares es- timatorQest, Eq. (A.3)] are non–singular even at ﬁnite Nand are essentially self consistent averagings of ﬂuctuations (samples) over regio ns of a (local) size ξ∼[l/NQ cl(x)]1/2. (A.9) It was assumed implicitly that the target distribution P(x)being learned varies negligibly over this length scale, and then the WKB method ca n be used again to show that the ﬂuctuations in the estimate, ψ(x)≡φ(x)−[−logP(x)]behave at largeNas ∝an}bracketle{tψ(x)∝an}bracketri}ht=l NP(x)∂2 xlogP(x) +···, (A.10) /angbracketleftBig [δψ(x)]2/angbracketrightBig =1 41/radicalBig NP(x)l+···. (A.11)A.1. Summary of nonparametric learning 120 IfP(x)is not smooth enough, then in both of these equations one must replace Pby its version smoothed over the local smoothness scale ξ(for more on this see Section 3.5). Note also that the variance of the ﬂuctuati ons is not uniformly small; this is a direct result of parameterization dependen ce of the prior, Eq. (A.5) (Periwal 1998). One of the most interesting conjectures of the Bialek et al. ( 1996) paper is that the Occam factor (the ﬂuctuation determinant) is enough to c onstruct a complex- ity razor just as in the ﬁnite parameter case. Indeed, one may impose an a priori distribution on land average over it after the correlation function, Eq. (A.6 ), is found. The kinetic and the ﬂuctuation determinant terms of t he effective Hamil- tonian, Eq. (A.7), have opposite dependences on l, so at large Nthe average should be dominated by some l∗, for which Heﬀis minimal. The data itself should select the best smoothness scale consistent with the ﬁnite p arameter Minimal De- scription Length paradigm of Rissanen (1983, 1989) and the O ccam’s complexity razor of MacKay (1992) and Balasubramanian (1997). With the same implicit as- sumption of a very smooth P(x)the authors have concluded that l∗∼N1/3. (A.12) IfP(x)is not smooth enough, a different dependence of l∗onNshould be ex- pected (cf. Section 3.5). This approach has a few shortcomings, most of them arising fr om reparame- terization noninvariance and omission of clear identiﬁcat ion of the above–men- tioned smooth target assumption. Some of these problems are discussed in our present work, and some have been analyzed by Periwal (1997, 1 998).A.2. Correlation function evaluation 121 A.2 Correlation function evaluation In the step from Eq. (4.6) to Eq. (4.7) we have performed the in tegral of the fol- lowing form I=/integraldisplay1 0dt1dt2···dtKK/productdisplay j=1tzj jδ/parenleftBigg 1−K/summationdisplay k=1tk/parenrightBigg , (A.13) wheretjwasQ(x),zjwasn(x) +λQ∗(x), and the limit of integration for each variable is 1because probabilities are normalized to one. This integral may be calculated in a straightforward manner by integrating out e ach variable in turn. This creates a product of B–functions, which can be simply reduced to the ﬁnal result, Eq. (A.17). However, this ease of integration is a co nsequence of the sim- plicity of our toy model. Some other models currently under i nvestigation (see Section 4.5) involve similar integrals, but they are sufﬁci ently different to prohibit easy exact calculations. Keeping these possible applicati ons in mind, we want to show a trickier integration method of Eq. (A.13) that may be m ore useful in other problems. First note that due to the δ–function, which enforces the normalization, only tj’s less than or equal to 1matter. So we may equally well replace the upper limits of all integrals in Eq. (A.13) by +∞. Then we can replace the delta function by its Fourier representation, shift the contour of integrati on by a small εto the right [the contours and the directions of the integrations are sho wn in Fig. (A.1)], and exchange the order of the integrals I=/integraldisplay∞ 0dt1dt2···dtKK/productdisplay j=1tzj j/integraldisplay C1dµ 2πieµ(1−/summationtextK k=1tk)(A.14) = lim ε→0/integraldisplay C2dµ 2πieµK/productdisplay j=1/bracketleftbigg/integraldisplay∞ 0dtjtzj je−µtj/bracketrightbigg . (A.15) Now, since we shifted the contour, Reµ > 0. Therefore each of the internalA.2. Correlation function evaluation 122 Re µC1Imµ C C2 3 ε0 Figure A.1: Integration contours. integrals in Eq. (A.15) is a Γfunction: I=K/productdisplay j=1Γ(zj+ 1)×lim ε→0/integraldisplay C2dµ 2πieµ /producttext jµzj+1(A.16) In more complex cases we probably would have stopped here, tr adingKreal integrals of Eq. (A.13) for one contour integration in the co mplex plane—this is why this method may be of use. However, our toy model is very easy, so we can proceed further and do the µintegration. Remembering that εis small, we can now use the Jordan’s lemma and bend the integration conto urC2intoC3. Then using a well known formula for the integral representat ion of the inverse Γ–function [Gradshtein and Ryzhik 1965, Eq. (8.315)] we get I=/integraldisplay1 0dt1dt2···dtKK/productdisplay j=1tzjδ/parenleftBigg 1−K/summationdisplay k=1tk/parenrightBigg =/producttextK j=1Γ(zj+ 1) Γ(/summationtextK j=1zj+K)(A.17) Note that, in the case of Eq. 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arXiv:physics/0009033v1 [physics.data-an] 9 Sep 2000Bayesian Blocks: Divide and Conquer, MCMC, and Cell Coalescence Approaches Jeﬀrey D. Scargle NASA-Ames Research Center MS 245-3, Moﬀett Field, CA1 From the 19th International Workshop on Bayesian Inference and Maximum Entropy Methods (MaxEnt ’99), August 2-6, 1999, Boi se State University, Boise, Idaho, USA. This conference proce edings will be published (Josh Rychert, Editor). Key Words: Poisson process, changepoint detection, time series, imag e process- ing, Voronoi tessellation Abstract. Identiﬁcation of local structure in intensive data – such as time series, im- ages, and higher dimensional processes – is an important pro blem in astronomy. Since the data are typically generated by an inhomogeneous Poisso n process, an appropriate model is one that partitions the data space into cells, each o f which is described by a homogeneous (constant event rate) Poisson process. It is ke y that the sizes and loca- tions of the cells are determined by the data, and are not pred eﬁned or even constrained to be evenly spaced. For one-dimensional time series, the me thod amounts to Bayesian changepoint detection. Three approaches to solving the mul tiple changepoint problem are sketched, based on: (1) divide and conquer with single ch angepoints, (2) maximum posterior for the number of changepoints, and (3) cell coale scence. The last method starts from the Voronoi tessellation of the data, and thus sh ould easily generalize to spaces of higher dimension. 1)Email: jeﬀrey@sunshine.arc.nasa.govI INTRODUCTION: THE DATA ANALYSIS PROBLEM Developments in detector technology for high energy astrop hysics2have led to observation systems capable of reporting accurate arrival times for individual pho- tons. These times, while not binned, are quantized in micros econd-scale units I like to call “ticks” – since they are in fact generated by the ticki ng of the computer clock on-board the spacecraft. In the approximation where the tic ks are short compared to time scales of interest, we very accurately have a Poisson process. Note that, depending on the nature of the variability of the process, di ﬀerent mathematical models apply, as indicated in the table below. Nature of Mathematical Variability Model Process Constant Homogeneous Poisson Deterministic Inhomogeneous Poisson Random Doubly Stochastic Poisson (Cox Process) A number of mathematical references [20,36,1,2,40] descri be the nature of spa- tial Poisson processes. Time series are the 1 Dspecial case, consisting of streams of numbers representing photon detection times. In this conte xt, the Poisson distribu- tion is so accurate that it is hardly an approximation. Stati sticians seem to have a hard time understanding this point. I believe the culprit is the fact that the Poisson process is usually taught as the limit of a binomial process o r the like. Technically, our data do comprise a ﬁnite Bernoulli lattice [40], since typically more than one event cannot be recorded at a given tick. (See [8] for an inter esting discussion of a lattice theory of quantum ﬁelds.) But the rates in units of ev ents3per tick are so 2)The term high energy astrophysics is used loosely for both the study of astronomical objects which produce and emit large amounts of energy, and for obser vations of radiation consisting of high-energy photons, e.g.X-rays and gamma-rays. Often the two meanings coincide. 3)We use the term event for photon detections, or other data points. 2low that the probability of such multiple events is very low. In practice, the most signiﬁcant imperfection is a departur e, not from the as- sumed distribution, but from the assumption of independenc e of the process at diﬀerent times. Photon detectors always have a ﬁnite dead time – an interval after the detection of one event during which another photon canno t be recorded, either because the detector mechanism itself is temporarily paral yzed, or the data system is too busy processing the event. This yields interdependen ces in the detection of photons close together in time. The discrete nature of photon counting data is most often con sidered a challenge. Many analysts ﬁrst bin their data and then apply standard met hods. Further, the notion is rife that not only are bins necessary, but that t hey must be large and contain enough events to produce a “statistically signi ﬁcant sample.” This is wrong and wasteful. Analysis can be carried out directly on t he event data. It is my belief that the discreteness and utter simplicity of the f undamental event – a photon was detected or it wasn’t – are a big advantage. This si mplicity and the fact that the observational errors accurately follow a dist ribution with essentially no free parameters4mean that the posterior marginalization can be carried out exactly, at least for some of the nuisance parameters [see Eq s. (4) and 5]. Happily, the simplicity of the basic events allows almost im mediate generalization of one-dimensional results to data spaces of higher dimensi on. Simple examples are cases where the photons, in addition to being timed, are also tagged with spectral and spatial information – i.e.energies of photons and their location on the sky. Processes that sample the density of events in spaces of vari ous dimension (denoted D) include: •time series ( D= 1) •other sequential data ( D= 1;e.g., genetic sequences) •images ( D= 2) •time sequences of images ( D= 3) •time sequences of spectra ( D= 2) •points in parameter spaces (any D) The corresponding data types have in common that they compri se sets of points in some well deﬁned space. Since the underlying process deﬁnes the intensity of some physical quantity over the space, we call these intensive data . The actual data can consist of a list of points, or the number of points in pre- selected bins ( e.g. image pixels), or other data modes. A problem of broad interest is the dete ction of local5structures in such data. Pulses, or other time-domain struc tures in time 4)The Poisson rate parameter is pretty closely nailed at the lo cal event rate. 5)I use the term localto distinguish features of limited extent from global features, such as periodic signals extending over the whole of the data space. 3series data, features in images and clusters in parameter sp aces of various dimension (classiﬁcation) are examples. For one-dimensional time series, the method of Bayesian Blocks approximates the signal as a piecewise constant Poisson process [35]. Genera lizing this representation to a space, denoted S, of arbitrary dimension, the basic problem can be phrased as the following relatively straightforward Bayesian Maximum a posteriori (MAP) problem: Consider all partitions of Sinto subvolumes, or cells. Model the event rate within each cell as a homogeneous Poisso n Process. Among all such partitioned models, which is the most likely? We determine the parameters in order of how fundamental they are to the nature of the model. The piecewise constant Poisson nature of the mo del is assumed, so the most fundamental parameter is Ncp, the number of changepoints. This is deter- mined by marginalizing all of the other parameters – the loca tions of the change- points and the Poisson rates between them. Then the location s of the changepoints are determined. Finally, the block event rates are determin ed; trivially, the MAP value is just the ratio of N, the number of events in the cell, to V, its volume. The resulting evidence in favor of this model for a single cel l, called the marginal likelihood , and deﬁned below in Eq. (4), is a simple function of VandN[below, Eqs. (4) and (5), and also see [35]]. For binned data and other data modes, and for other distributions than the Poisson one assumed here, the l ikelihood is similarly an explicit function of the same two parameters. We assume the c ells are independent, so the total likelihood is the product of the likelihoods of t he cells (see Section III). II EXACT BAYES FACTORS For the multiple changepoint problem we need to evaluate the posterior proba- bility for piecewise constant models, given the data. An imp ortant simpliﬁcation is that the marginal likelihoods and posterior probabilities for such models factor into the product of the same quantities for each independent segm ent of the model. We refer to these segments as blocks in 1Dcontexts, and as cellsin higher dimensions. This section gives the computation of the posterior for a sin gle cell, which can then be used in various ways – such as the evaluation of Bayes facto rs for comparison of two or more models. The general form of Bayes factor comparin g two models M1 andM2, given data D{c.f.[15], eq. (6.4)}is: Bayes factor( M2;M1) =p(D\|M2) p(D\|M1)=/integraltextp(λ2\|M2)p(D\|λ2, M2)dλ2/integraltextp(λ1\|M1)p(D\|λ1, M1)dλ1(1) 4where p(λ\|M) is the prior on λandp(D\|λ, M) is the likelihood. Given the discussion above, it is easy to compute the appropriate factor for a Pois son model with rate parameter λ(units: events per unit volume) for a set of events in some blo ck or cell. The Poisson Likelihood, obtained by multiplying likelihoo ds for individual sample bins, is L(N\|λ, V) =e−V λλN(2) The usual Poisson factorial does not appear because the numb er of counts in a tick is 0 or 1. Marginalizing λusing the conjugate Poisson prior ( [15]; Section 2.7, p. 49-50) p(λ) =βα Γ(α)e−βλλα−1, (3) the contribution to the Bayes factor for a cell – often called themarginal likelihood – is P(M\|D) =/integraldisplay∞ 0p(λ)L(N\|λ, V)dλ=βα Γ(α)Γ(N+α) (V+β)N+α. (4) Based on a prior assigning a uniform distribution to the prob ability of occurrence of a single event, the marginal likelihood P(M\|D) =Γ(N+ 1)Γ( V−N+ 1) Γ(V+ 2), (5) was derived in [35]. A function of the same two suﬃcient stati stics, NandV, it behaves similarly to the marginal likelihood in Eq. (4), but was found less accurate in simulation studies such as the one discussed below. III THREE APPROACHES TO THE MULTIPLE CHANGEPOINT PROBLEM We now have to evaluate the above marginal likelihood for eac h segment of a piecewise homogeneous Poisson model consisting of success ive, independent6blocks of data, and cobble the results together. The next section de scribes three ways to do this. The ﬁrst two seem eﬀective in 1D, but have problemati cal extensions to higher dimensions. The third was inspired by its simplicity in 2Dand 3D. All three methods are demonstrated on the same synthetic 1 Ddata. A An Iterative Approach: Top Down 6)That is, due to a fundamental property of the Poisson distrib ution, random variables corre- sponding to counts in successive blocks (or diﬀerent cells i n general) are independent. This fact should not be confused with independence of the actual Poiss on rates, which in general does not hold. 5As detailed in [35], the formulas above allow easy computati on of the Bayes factor comparing (1) the two-block model in which the interv al is segmented into two parts at a changepoint, with (2) a single Poisson model fo r the whole interval. The decision whether to keep an interval unsegmented or to di vide it into two subintervals is based on comparison of the corresponding ma rginal likelihoods. Let Φ(N, V) stand for the marginal likelihood corresponding to a Poiss on model for a volume of size Vcontaining Nevents, such as one of the two functions given above. Then an interval should be broken in two if Φ(Ni, Vi)Φ(Ni+1, Vi+1)−Φ(N, V)>0, (6) where the putative changepoint divides the interval of size Vinto two parts, of sizeViandVi+1=V=Vi, containing NiandNi+1=N−Nievents, respectively. This criterion is easily computed as a function of the locati on of the changepoint. The interval is then segmented at the point that maximizes th e expression in Eq. (6). This divide and conquer scheme is applied ﬁrst to the whole data interval, and then iteratively to any subintervals found at the previous s tep. When this criterion favors segmentation of no further intervals, computation h alts. Figure 1 shows the results for synthetic data generated by a p iecewise constant Poisson process, between eleven known changepoints. I used a modiﬁed form of the Blocks function popularized by David Donoho, Iain Johnstone, and t heWaveLab project [3] as a sample function with many discontinuities t hat can be detected using wavelet methods (see Fig. 1 in [27]). The original func tion has blocks of negative amplitude, which will not do as Poisson rates. Henc e I added a constant, 31 2, to the classic Blocks function, and then generated points o beying the Poisson distribution. The three panels show three divide and conquer iterations. I t can be seen that the various changepoint locations are found rather accurat ely, allbeit with the single changepoint algorithm. In the early steps the posterior has multiple peaks at the various changepoints, but only the highest one is used at eac h step. There is a tendency for changepoints determined early in the process to be less accurate. These errors are not corrected as the iteration proceeds, bu t the algorithm could be modiﬁed to do so. B Maximum Posterior for NcpviaMCMC It is clear that the above method is not rigorous, in that it do es not solve for all of the changepoints simultaneously. It is relatively st raightforward, however, to remedy this with a more rigorous, but also more computationa lly intensive scheme. If there are multiple changepoints, say Ncpin number, the full posterior is just the product of the posteriors of all the subintervals. The margi nalization of all the locations of the changepoints requires evaluating the Ncp-dimensional integral7of the posterior over all values of the changepoint locations. 7)In practice, this is a ﬁnite sum, e.g.over a bin index or an event index. 6Results obtained in this way, using a simple Markov Chain Mon te Carlo (MCMC) method, are surprisingly good. Figure 2 shows the block repr esentation of the same data as in Figure 1, obtained from a simple MCMC evaluation of the marginal likelihood as a function of the number of changepoints. The m aximum posterior was at 16 changepoints (17 blocks), compared to the correct v alue of 11. The “extra” blocks are narrow, and while they do not look pretty, they will not much aﬀect derived quantities (such as pulse widths, rise times, etc). The good performance with only a few iterations may be due to a combination of several factors: •the well behaved nature of the data ◦low dynamic range of the signal ◦well behaved backgrounds ◦exact Poisson statistics for the observational uncertaint ies •a simple, exact Likelihood; only one parameter •the fact that the posterior does not have to be computed accur ately to distin- guish one value of Ncpfrom another •the useful characteristics of the ﬁnal model are not that sen sitive to Ncp Nevertheless, when the number of changepoints and the numbe r of data points are large, the computation is quite long. Since there is little g lobal communication, in the sense that the location of a change point in one part of t he observation interval aﬀects that of one elsewhere, breaking the data up i nto smaller intervals is an eﬀective way to speed up the overall computation. C Cell Coalescence (Bottom Up) Based on preliminary tests, a new algorithm may be a signiﬁca nt improvement over either of the above approaches. It begins with a ﬁne-gra ined segmentation, namely the Voronoi tessellation of the data points, and merg es these many cells to form fewer, larger ones. The outline of the algorithm is simp le: 7Bayesian Cell Coalescence (0) Initial Segmentation: Voronoi Tessellation of the Even ts (1) Compute Bayes Factor for Merging Each Pair of Adjacent Ce lls (2) Identify Largest Bayes Factor (at i) (3) If Largest Bayes Factor is <1, HALT! (4) Otherwise Merge Pair of Cells with Largest Bayes Factor: •Ni←Ni+Ni+1 •Vi←Vi+Vi+1 •Delete Cell i+ 1 (5) Go to 1 While there is no rigorous justiﬁcation for this procedure, one has the sense that it should come reasonably close to obtaining the optimal sol ution. At each step in the iteration the local event rate in a cell isN V, the number of events in the cell divided by its volume. The rate estimates for the initial cel ls,1 V, are reasonable estimates of the ﬁne-grain, local event rate if these cells a re assigned roughly their surrounding volume, extending approximately halfway to ne ighboring points. An obvious choice for the initial partition is thus the Voronoi tessellation [40,19] of the data points. The Voronoi cell for a data point consists of all the space closer to that point than to any other data point. The Voronoi tessellation of a set of points on an interval (1D ) is trivial: it is simply the set of intervals spanned by pairs of midpoints b etween successive data points. Many algorithms exist for computing Voronoi te ssellations in higher dimensions [21,13,32,11], allowing one to address problem s such as cluster detection in parameter spaces, and identiﬁcation of structures in ima ges. Since the marginal posteriors discussed above are valid for piecewise constan t Poisson data in any dimension, the methods demonstrated here for 1 Dshould apply in general. The decision whether to merge two cells or to halt, Step (3), i s based on com- parison of the Bayes factors. Using the same notation as abov e, in Eq. (6), cells i andi+ 1 are merged if Φ(Ni+Ni+1, Vi+Vi+1)−Φ(Ni, Vi)Φ(Ni+1, Vi+1)>0 (7) and kept separate otherwise. When this criterion favors the merging of no further cells, computation halts. 8Figure 3 demonstrates the application of this algorithm to t he same synthetic data as in the previous two examples. The top panel depicts th e state part way into the iterations, but is not far from the initial Voronoi t essellation, one Voronoi cell for each data point. Successive panels show the evoluti on of the representation as the cells merge into fewer, larger ones. In the last panel, the halting criterion mentioned above has just terminated the iteration. It can be seen that most of the changepoints are accurately detected. Several though are m issed. IV RELATED WORK It is well known that Bayesian methods are well-suited to ﬁnd ing changepoints (e.g.[30,34]; see also Appendix C of [16]). In [42] methods simila r to those de- scribed here are used to ﬁnd changepoints in binned data, to a n accuracy better than the bin size. A number of recent publication are relevan t to this problem [37,17,18,42,6,34,38,39,31,7] and [22–29]. More recentl y, Raftery and colleagues have developed similar methods, mainly for 2 Dproblems. In [5] the Voronoi sites are ﬁducial markers for the cells, not the tessellation deﬁn ed by the individual data points. Other approaches are described in [14,4,10,41,33] . Bayesian methods are also useful for segmentation of autoregressive models, wit h applications to speech processing [9]. After this article was completed, I became aware that NASA’s Chandra X-ray Observatory, is using an unbinned source detection techniq ue much like cell coales- cence, and starting from the same Voronoi tessellation [12] . Source cells are merged with a percolation process based on a sample estimate of the density distributi on. V ACKNOWLEDGEMENTS I am greatful to Alanna Connors, Seth Digel, Tom Loredo, and J ay Norris for helpful comments and encouragement. This work is partly fun ded by the NASA Applied Information Systems Research Program. REFERENCES 1. Andersen, P. K., Borgan, O., Gill, R. D., and Keiding, N. (1 993),Statistical Models Based on Counting Processes , Springer Series in Statistics, Springer-Verlag: New York 2. Barndorﬀ-Nielsen, O. E., Kendall, W. S., and van Lieshout , M. N. M. (1999), Stochastic Geometry: Likelihood and Computation , Chapman & Hall/CRC: Boca Raton 3. Buckheit, J. B. and Donoho, D. L. (1995), Wavelab and Repro ducible Research, in Wavelets and Statistics , pp. 53-81, Springer-Verlag: Berlin. http://stat.Stanford.EDU:80/~wavelab/ 94. Byers, S. D., and Raftery, A. E. (1998) “Nearest Neighbor C lutter Removal for Estimating Features in Spatial Point Processes,” J. American Statistical Association , June 1998. Also (1996) Technical Report No. 305, Dept. of Sta tistics, University of Washington. 5. Byers, S. D., and Raftery, A. E. 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B., Stern, H. S. and Rubin, D. B. (1995) Bayesian Data Analysis , Chapman & Hall, New York 16. Gregory, P. C. and Loredo, T. J. (1992) “A New Method for th e Detection of a Periodic Signal of Unknown Shape and Period,” Astrophysical Journal ,398, pp. 146- 168. 17. Gustafsson, F. (1996) “The marginalized likelihood rat io test for detecting changes,” IEEE Transactions on Automatic Control ,41, pp. 66-78. 18. Gustafsson, F. (1999) “A Change Detection and Segmentat ion Toolbox for Matlab” 19. Klein, R. (1989), Concrete and Abstract Voronoi Diagrams , Lecture Notes in Com- puter Science No. 400, Springer-Verlag: Berlin 20. Kutoyants, Yu. A. (1998), Statistical Inference for Spatial Poisson Processes , Lecture Notes in Statistics No. 134, Springer-Verlag: Berlin 21. Noltemeier, H./ ed. (1988), Computational Geometry and its Applications , Lecture 10Notes in Computer Science No. 333, Springer-Verlag: Berlin 22. Ogden, T. notes: Change-points in Poisson Data http://www.stat.sc.edu/~ogden/slides/temp.html . 23. Ogden, T. and Lynch, J. D. (1998) “Bayesian analysis of Ch ange-point models,” Bayesian Inference in Wavelet Based Models , P. Muller and B. Vidokovic eds., Springer-Verlag: New York 24. Ogden, T. and Cheng, C. (1997) “Testing for abrupt jumps w ith wavelets,” Proceed- ings of the 1997 Conference on the Interface of Statistics an d Computer Science ,56, pp. 293-302. 25. Ogden, T. and Wes, R. W. (1998) “Continuous-time Estimat ion of a Change-point in a Poisson Process,” Journal of Statistical Computation and Simulation ,56, pp. 293- 302. 26. Ogden, T. (1996) “Wavelets in Bayesian Change-point Ana lysis,” American Statis- tical Association, 1996 Proceedings of the Section on Bayes ian Statistical Science , pp. 164-169. 27. Ogden, T. and Parzen, E. (1996) “Data Dependent Wavelet T hresholding in Non- parametric Regression with Change-point Applications,” Computational Statistics and Data Analysis ,22, pp. 53-702. 28. Ogden, T. and Parzen, E. (1996) “Change-point Approach t o Data Analytic Wavelet Thresholding,” Statistics and Computing ,6, pp. 93-99. 29. Ogden, T. and Sugiura, N. (1994) “Testing Change-points with Linear Trend,” Com- munications in Statistics B: Simulation and Computation ,23, pp. 287-322. 30.`O Ruanaidh, J. J. K. and Fitzgerald, W. J. (1996) Numerical Bayesian Methods Applied to Signal Processing , Springer: New York 31. Pettitt, A. N. (1979) “A non-parametric approach to the c hange-point problem,” Journal of Applied Statistics ,28, pp. 126-135. 32. Preparata, F. P., and Sahmos, M. I. (1985), Computational Geometry: An Introduc- tion, Springer-Verlag: New York 33. Raftery, A. E. (1993) “Change Point and Change Curve Mode ling in Stochastic Processes and Spatial Statistics,” Technical Report No. 25 3, Dept. of Statistics, University of Washington. 34. Raftery, A. E. and Akman, V. E. (1986) “Bayesian analysis of a Poisson process with a change-point,” Biometrika ,73, pp. 85-89. 35. Scargle, J. D. (1998) “Studies in Astronomical Time Seri es Analysis. V. Bayesian Blocks, A New Method to Analyze Structure in Photon Counting Data,” Astrophys- ical Journal ,504, pp. 405-418. 36. Snyder, D. L. and Miller, M. I. (1991), Random Point Processes in Time and Space , Second Edition, Springer-Verlag: New York 37. Stark, J. A., Fitzgerald, W. J. and Hladky, S. B. (1997) “M ultiple-order Markov Chain Monte Carlo Sampling Methods with Application to a Cha ngepoint Model” 38. Stephens, D. A. (1994) “Bayesian retrospective multipl e-changepoint identiﬁcation,” Journal of Applied Statistics ,43, pp. 159. 39. Stephens, D. A. and Smith, A. F. M. (1992) “Bayesian edge- detection in images via changepoint methods” 40. Stoyan, D. Kendall, W. S. and Mecke, J. (1995) Stochastic Geometry and its Appli- 11cations . 2nd edition, John Wiley & Sons: New York 41. Walsh, D. and Raftery, A. E. (1999) “Detecting Mines in Mi neﬁelds with Linear Characteristics,” Technical Report No. 359, Dept. of Stati stics, University of Wash- ington. 42. Webster, R. and Ogden, R. T. (1998) Continuous-time Esti mation of a Change-point in a Poisson Process 12FIGURE 1. Block representations based on the divide and conquer algor ithm. The piecewise homogeneous Poisson data were generated from a modiﬁcation of Donoho’s Blocks function. The actual changepoints used to generate the data are shown as ve rtical lines at the bottom, and the changepoints determined on the ﬁrst three steps of the itera tion are dashed lines. 13FIGURE 2. Block representations based on the MCMC algorithm. The piec ewise homogeneous Poisson data were generated from a modiﬁcation of Donoho’s B locks function. The actual change- points used to generate the data are shown as vertical dashed lines. Spurious, narrow blocks have been emphasized ( e.g.the one at about t= 0.76. 14FIGURE 3. Block representations using the Cell Coalescence algorith m. The top panel shows the overly ﬁne segmentation in the early stages of the iterat ion. In successive panels the process is converging toward a coarser representation. The bottom p anel shows the ﬁrst stage at which the Bayes factor contraindicates merging of all of the remai ning blocks. The actual changepoints used to generate the data are shown as vertical dotted lines. 15
arXiv:physics/0009034v1 [physics.gen-ph] 10 Sep 2000Has superluminal light propagation been observed?∗ Yuan-Zhong Zhang† CCAST (World Laboratory), P.O. Box 8730, Beijing, China Institute of Theoretical Physics, Chinese Academy of Scien ces, P.O. Box 2735, Beijing, China‡ It says in the report1by Wang et al. that a negative group velocity u=−c/310is obtained and that a pulse advancement shift 62-ns is mea- sured. The authors claim that the negative group velocity is associated with superluminal light propagation and that the pulse adva ncement is not at odds with causality or special relativity. However , it is shown here that their conclusions above are not true. Furthermore , I give some suggestion concerning a re-deﬁnition of group-veloci ty and a new explanation in special relativity of causality. The velocity of u =−(c/310)ˆk0is subluminal but not superluminal (the term “superluminal” is usually understood as such a lig ht propagation with phase, group, and energy velocities all exceeding the vacuu m speed of light2). It is well-known that the 4-dimensional interval for a signal i n special relativity is given by ds2=c2dt2−/parenleftBig dx2+dy2+dz2/parenrightBig =dt2/parenleftBig c2−u2/parenrightBig , (1) as seen in the inertial frame K, where u2=u·u=/parenleftBiggdx dt/parenrightBigg2 +/parenleftBiggdy dt/parenrightBigg2 +/parenleftBiggdz dt/parenrightBigg2 (2) withubeing the velocity of the signal. According to special relat ivity,ds2is an invariant under Lorentz transformations, i.e. ds2=ds′2. This means dt′2/parenleftBig c2−u′2/parenrightBig =dt2/parenleftBig c2−u2/parenrightBig , (3) where the quantities with a prime stand for the ones as seen in other inertial frame K′. This shows that both u2andu′2are all bigger, or all less, than c2. Explicitly, \|u\|< cleads to \|u′\|< c. Similarly, \|u\|> cleads to \|u′\|> c. For the case in the report by Wang et al., the velocity is found to be (i n terms of vector symbol) u=−(c/310)ˆk0withˆk0being the unit vector of the incident direction (see below) and hence \|u\|< cas seen in the laboratory frame. So that we have \|u′\|< c, i.e. the velocity of the pulse would also be smaller than the vacuum speed of light c, as seen in any of other inertial frames. Therefore, the nega tive velocity obtained by Wang et al. is simply subluminal but not superluminal. ∗The project supported partially by the Ministry of Science a nd Technology of China under Grant No. 95-Yu-34 and National Natural Science Foundation of China under Grant Nos. 19745008 and 19835040 †Email: yzhang@itp.ac.cn ‡Mailing address 1New suggestion concerning re-deﬁnition of negative veloci ty.Now I want to give a new explanation of the so-called “negative” g roup velocity. By deﬁnition the group velocity of a light pulse propagating in a dispersive linear medium is given by3 u=∂ω ∂k=c n+νdn dνˆk, (4) where ˆk≡k/\|k\|is the unit vector of the direction of phase velocity (or wave vector), kis wave vector, ν=ω/2πis frequency, and n=n(ν) is the optical refractive index of the medium. For a normal medium we have d n/dν >0 and so that \|u\|< c. But for anomalous dispersive linear media in where d n/dν <0 , one arrives at the fol- lowing two situations: (i) For 1 > n(ν) +νdn/dν≥0, we have \|u\|> c; (ii) For n(ν) +νdn/dν <0, one gets u=−c \|n+νdn dν\|ˆk. (5) In case of a light pulse propagating vertically towards a sur face of dispersive medium from vacuum, the incident direction ˆk0is usually deﬁned as to be pos- itive. The wave vector (or phase velocity) of the pulse in the medium is usually assumed to have a positive direction (i.e. ˆk=ˆk0) while the group velocity uthen has a negative direction4. Ifurepresents the velocity of an actual information, then the deﬁnition of negative group-velocity must give vio lation of causality. Thus it is needed to modify the usual deﬁnition of phase and gr oup velocities. Here it must be emphasized that the negative sign “ −” in Eq. (5) simply indicates the directions of the group-velocity uand wave-vector kare opposite each other, but not say which one should be negative. In fact t here is no reason to identify the direction of kin the medium with the incident one ˆk0. Contrarily, it should be more reasonable to suppose the group-velocity uhas the same direction to that of the incident light signal, while the wave-vector k( and hence phase velocity) then has a negative direction, i.e., ˆk=−ˆk0. By use of the new deﬁnition, we never meet any problem concerning violation of causality in case where the group-velocity udoes represent a velocity of an actual information. Now come back to the case of u=−c/310 in terms of the symbol by Wang et al.. Note that the negative velocity is not directly measu red but calculated by Wang et al. from their measured refraction index by use of t he deﬁnition: u=c/ngwithng≡n(ν) +νdn(ν)/dν. In other words, the negative sign “ −” forng<0 is just deﬁned by them. Contrarily, according to the presen t new deﬁnition, the group-velocity calculated from their measu red refraction index should be positive, and less than c. The observed 62-ns advancement shift must be violation of ca usality. Another result in the report1by Wang et al. is the 62-ns advancement shift (see Fig. 4 of Ref. 1). They claim that it is not at odds with causali ty. They argue that it is a result of the wave nature of light and that no actua l information, or signal, is trasmitted1,5. However it must be pointed out that the authors 2make confusion of the direct observation with theoretical p rediction. At ﬁrst it is emphasized that the 62-ns shift is a directly measured d atum but not a theoretical prediction. Secondly it is needed to clear whet her the observed 62- ns shift is an actual information. If not, one must face the qu estion: Can you measure a non-actual information in a laboratory? In fact, i t is not possible for any experimental device to record a non-actual signal. In ot her words, what a device records is certainly an actual information. So that t he curves A and B in Fig. 4 in the report1are just the records of actual information. The curves A and B show that the actual signal B is advanced for 62-ns in tim e compared to the the actual signal A. If A were the source of B, then the 62-n s advancement would certainly be violation of causality. This conclusion is independent of any theoretical prediction concerning phase, group, or other k ind of velocity. Owing to any actual signal should not violate causality, then the c urve B could be connected causally not with the curve A but only with a measur ement error. The 3.7-µs full-width at half-maximum of the probe pulse means the pul se spatial extension of more than 1-km much larger than the 6-cm length o f the atomic cell. On the other hand, the curve B is only translated in time but almost not changed in shape compared to the curve A. So that possible sou rces of the 62-ns translation would be a systematic error, or a pulse-reshapi ng phenomenon such as the ampliﬁcation of the pulsefront and reduction of its tail . In order to determine ﬁnally the source of the advancement shift, it is needed to pe rform further similar measurements in diﬀerent experiment conditions, such as di ﬀerent probe pulses, diﬀerent cell lengths, and so on. New suggestion for explanation of causality in special rela tivity. For an anomalous dispersive medium with 1 > n(ν) +νdn/dν≥0, group-velocity is superluminal (i.e. u > c) in laboratory frame in which the medium is at rest. Eq. (3) gives that the group-velocity in any of inertial fram es is still superluminal (i.e.u′> c). Let t1andt2be the instants at which the light signal arrives at points 1 and 2, respectively, in the medium. Due to the fact of △t=t2−t1>0, no causality would be violated in the laboratory frame. But b y making use of Lorentz transformations, one always ﬁnd such an inertial fr ame, e.g. the frame K′, in which we have △t′=t′ 2−t′ 1<0. This is just the so-called violation of causality as seen within K′. However it must be addressed that △t′and△tare coordinate time intervals but not proper ones. It is well-known that a co ordinate time interval is related to the deﬁnition of simultaneity and thus is not di rectly observable6. In special relativity, therefore, all of physical observatio ns must be used to compare with such quantities which are invariant under Lorentz tran sformations, while the only exception is just the explanation of causality abov e. Here I suggest to explain causality by means of proper time interval in stead o f coordinate one. To do it, let the signal come back to the point 1 after it reaches t he point 2. In this case we have △τ=t3−t1>0 where t3is the instant at which the signal returns to the point 1. Due to t1andt3are readings of the same standard clock at rest at the point 1, so that △τis just a proper time interval to be positive in all of inertial frames. Using the new deﬁnition one could arrive at the conclusion: The superluminal light propagation (i.e. u > c ) is not at odds with both causality 3and special relativity. References [1] Wang, L. J., Kuzmich, A. & Dogariu, A. Gain-assisted supe rluminal light propa- gation. Nature 406, 277– 279 (2000). [2] Chiao, R. Y. Superluminal (but causal) propagation of wa ve packets in transparent media with inverted atomic populations. Phy. Rev. A 48, R34–R37 (1993). [3] Landau, L. D. & Lifshitz, E. M. Electrodynamics of Continuous Media (Pergamon, Oxford, 1960). [4] Brillouin, L. Wave Propagation and Group velocity (Academic, New York, 1960). [5] Marangos, J. Faster than a speeding photon. Nature 406, 243–244 (2000). [6] Zhang, Y. Z. Special Relativity and its Experimental Foundations (World Scientiﬁc, 1997). 4
arXiv:physics/0009035v1 [physics.atm-clus] 10 Sep 20001 Binding Energies and Scattering Observables in the4He3Atomic System∗ A. K. Motovilova, W. Sandhasb, S. A. Soﬁanosc, and E. A. Kolganovaad aJINR, 141980 Dubna, Moscow Region, Russia bPI Universit¨ at Bonn, Endenicher Allee 11-13, D-53115 Bonn , Germany cPhysics Department, UNISA, P.O.Box 392, Pretoria 0003, Sou th Africa dIAMS, Academia Sinica, PO Box 23-166, Taipei, Taiwan 10764, R.O.C. The4He3system is investigated using a hard-core version of the Fadd eev diﬀerential equations and realistic4He–4He interactions. We calculate the binding energies of the 4He trimer but concentrate in particular on scattering obser vables. The atom-diatom scattering lengths are calculated as well as the atom-diato m phase shifts for center of mass energies up to 2 .45mK. There is a great number of experimental and theoretical stud ies of the4He three-atomic system (see, e.g., [1–9] and references cited therein). Mos t of the theoretical investigations consist merely in computing the bound states, while scatter ing processes found compar- atively little attention. In Ref. [10] the characteristics of the He–He 2scattering at zero energy were studied. The recombination rate of the reaction (1 + 1 + 1 →2 + 1) was esti- mated in [11]. The phase shifts of the He–He 2elastic scattering and breakup amplitudes at ultra-low energies have been calculated for the ﬁrst time just recently [12] but only for the comparatively old HFD-B potential by Aziz et al. [13]. In the present paper we extend the investigations of Ref. [12 ]. There, the formalism, which consists of a hard-core version of the Faddeev diﬀeren tial equations, has been described in detail. As in [12] we use the ﬁnite-diﬀerence ap proximation of the two- dimensional partial-wave Faddeev equations. We consider o nly the case of zero total angular momentum and take /planckover2pi12/m= 12.12K˚A2. In this work, we employ grids of the dimension 500–800 in both the hyperradius and hyperangle, w hile the cutoﬀ hyperradius is chosen to be up to 1000 ˚A. As compared to [12] we use in the present work the reﬁned He–He interatomic potentials LM2M2 of Ref. [14], and TTY of R ef. [15]. Our numerical methods have also been substantially improved, which allow ed us to deal with considerably larger grids. Furthermore, due to better computing facilit ies, we could take into account more partial waves. Although we have performed detailed calculations of the4He3binding energies, the main goal of this work was to perform calculations for the sca ttering of a helium atom oﬀ ∗This work was supported by the Deutsche Forschungsgemeinsc haft, Russian Foundation for Basic Research, and National Research Foundation of South Africa . Contribution to Proceedings of 16th International Confere nce on Few-Body Problems in Physics, 6-10 March 2000, Taipei, Taiwan. LANL e-print physics/0009035.2 Table 1 Dimer energies ǫd,4He−4He diatom scattering lengths ℓ(2) sc, trimer ground-state energies E(0) t, trimer excited-state energies E(1) t, and4He atom –4He dimer scattering lengths ℓsc for the potentials used. Potential ǫd(mK) ℓ(2) sc(˚A)lmaxE(0) t(K) E(1) t(mK) ℓsc(˚A) 0−0.0942 −2.45 168 HFD-B −1.68541 88 .50 2−0.1277 −2.71 138 4−0.1325 −2.74 135 0−0.0891 −2.02 168 LM2M2 −1.30348 100 .23 2−0.1213 −2.25 134 4−0.1259 −2.28 131 0−0.0890 −2.02 168 TTY −1.30962 100 .01 2−0.1212 −2.25 134 4−0.1258 −2.28 131 a helium dimer at ultra-low energies. Our results for the tri mer binding energies and4He– atom4He–dimer scattering lengths for the potentials employed ar e presented in Table 1. We also put in this table the corresponding dimer binding ene rgies together with the4He– 4He atomic scattering lengths. It should be noted that the mai n contribution to the trimer binding energies stems from the l= 0 and l= 2 partial-wave components, the latter being about 30%, and is approximately the same for all potentials u sed. The contribution from thel= 4 partial wave is of the order of 3-4% (cf. [5]). We notice tha t that our results for the ground-state energy E(0) tof the trimer for lmax= 4 are in a perfect agreement with the corresponding values obtained in the most advanced calc ulations [4,7–9]. The same also holds true for the excited-state energy E(1) t. Our results for E(1) tare in quite a good agreement with those of Refs. [8,9]. There are not many results in the literature concerning the H e–He 2scattering length. Apart from our previous result [12], there is that of Ref. [10 ] ofℓsc= 195 ˚A, obtained within a zero-energy scattering calculation based on a sepa rable approximation of the oldest Aziz et al. potential HFDHE2, and a more recent one obtained by Blume and Greene [16] via a Monte Carlo hyperspherical calculation wi th the LM2M2 potential. The latter result of ℓsc= 126 ˚A is in good agreement with our result of 131 ±5˚A (see Table 1). Within the accuracy of our calculations, the scatt ering lengths provided by the LM2M2 and TTY potentials, like the energies of the excited st ate, are exactly the same. It should be mentioned that in this case also the two-body bin ding energies and scattering lengths are almost indentical. We have also calculated the4He–atom4He–dimer scattering shifts for the HFD-B, LM2M2 and TTY potentials for center of mass energies up to 2 .45mK. After transfor- mation to the laboratory system the phase shifts for the thes e potentials turn out to be practically the same, especially those for LM2M2 and TTY. Th us, in Fig.1 we only plot the phase-shift results obtained for the HFD-B and TTY poten tials. Note that for the phase shifts we use the normalization required by the Levins on theorem. Inclusion of the l= 4 partial-wave channel only adds about 0.5% to the phase shi fts obtained for l= 0 andl= 2. This is the reason why the corresponding curve for lmax= 4 is not depicted in Fig.1.3 260280300320340360 0 1 2 3 4 5TTY HFD-B //0 /, degreesElab /, mK Figure 1.4He atom –4He dimer scattering shifts for the HFD-B and TTY potentials. The lower curve corresponds to the case where lmax= 0 while for the upper lmax= 2. A detailed exposition of the material presented, including tables for the phase shifts, is given in an extended paper [17]. REFERENCES 1. S. Grebenev, J. P. Toennies, and A. F. Vilesov, Science 279, 2083 (1998). 2. F. Luo, C. F. Giese, and W. R. Gentry, J. Chem. Phys. 104, 1151 (1996). 3. Th. Cornelius and W. Gl¨ ockle, J. Chem. Phys. 85, 3906 (1986). 4. R. N. Barnett and K. B. Whaley, Phys. Rev. A 47, 4082 (1993). 5. J. Carbonell, C. Gignoux, and S. P. Merkuriev, Few–Body Sy stems15, 15 (1993). 6. B. D. Esry, C. D. Lin, and C. H. Greene, Phys. Rev. A 54, 394 (1996). 7. M. Lewerenz, J. Chem. Phys. 106, 4596 (1997). 8. E. Nielsen, D. V. Fedorov, and A. S. Jensen, J. Phys. B 31, 4085 (1998). 9. V. Roudnev and S. Yakovlev, LANL e-print physics/9910030 . 10. S. Nakaichi-Maeda and T. K. Lim, Phys. Rev A 28, 692 (1983). 11. P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, Phy s. Rev. Lett. 77, 2921 (1996). 12. E. A. Kolganova, A. K. Motovilov, and S. A. Soﬁanos, J. Phy s. B.31, 1279 (1998). 13. R. A. Aziz, F. R. W. McCourt, and C. C. K. Wong, Mol. Phys. 61, 1487 (1987). 14. R. A. Aziz and M. J. Slaman, J. Chem. Phys. 94, 8047 (1991). 15. K. T. Tang, J. P. Toennies, and C. L. Yiu, Phys. Rev. Lett. 74, 1546 (1995). 16. D. Blume and C. H. Green, J. Chem. Phys., to appear. 17. A. K. Motovilov, W. Sandhas, S. A. Soﬁanos, and E. A. Kolga nova, LANL e-print physics/9910016.
EXOTIC ION-BEAMS, TARGETS AND SOURCES J.A. Lettry, CERN, Geneva, Switzerland Abstract Exotic beams of short-lived radioisotopes are produced in nuclear reactions such as thermal neutron induced fission, target or projectile fragmentation and fusion reactions. For a given radioactive ion beam (RIB), different production modes are in competition. For each of them the cross section, the intensity of the projectile beam and the target thickness define an upper production rate. The final yield relies on the optimisation of the ion- source, which should be fast and highly efficient in view of the limited production cross section, and on obtaining a minimum diffusion time out of the target matrix or fragment catcher to reduce decay losses. Eventually, either chemical or isobaric selectivity is needed to confine unwanted elements near to the production site. These considerations are discussed for pulsed or dc- driven RIB facilities and the solutions to some of the technical challenges will be illustrated by examples of currently produced near-drip-line elements. 1 INTRODUCTION Over the last 30 years, RIB facilities based on Isotope On Line (ISOL) and in-flight fragment separators [1-4] played a major role in very different research domains such as astrophysics, atomic physics, nuclear physics, solid state physics and nuclear medicine. Nuclei far from stability (so called exotic nuclei) are produced in spallation, fragmentation, fission and fusion nuclear reactions between the primary beam particles and the target nucleus. RIB facilities consist of four parts. The driver produces a primary beam of particles and directs them to the target where the nuclear reactions take place. This is followed in the case of ISOL by the ion source and the separator which ionise and select the isotopes and direct them to the physics experiment. The drivers at the origin of the nuclear reaction are based on light or heavy ion accelerators and on various actinide fission processes. The fission of uranium produced via high-energy protons or thermal neutrons is now being complemented by fast neutron induced fission [5] and even photofission [6]. Existing RIB facilities can be classified according to the typical time span between the nuclear reaction and the delivery of the radioisotope to the experimental set- up into three complementary categories: the /G50s-class regrouping the in-flight fragment separators; the ms-class where fragments are caught and thermalized in a noble gas beam stopper; and the s-class where fragments or fission products are stopped or produced at rest in solids,diffuse out of the target material and eventually get ionised in the ion source. Ion sources will be described is section 2 and targets in section 3. 2 ION-SOURCES ALONG MENDELEEV’S TABLE RIB ion-sources should be fast to limit the decay losses of short-lived isotopes, efficient as the available amount of material is limited by the production cross section, and chemically selective to reduce isobar contaminants and confine unwanted species as close as possible to the production area. In this section, a selection of single charge ion sources is briefly presented according to their chemical properties, all of them designed to work close to the high-radiation target area. 2.1 Alkalis, alkaline earth and rare earth A feature common to alkalis, alkaline earth and rare earth elements is their low ionisation potential (between 3.9 and 6 eV). This property, shared by some diatomic molecules, allows a very efficient ionisation on a high temperature metallic surface. The ionisation of an atom bouncing from a high-temperature metallic surface was observed for the first time in 1923. Its efficiency is described as a function of the difference between the energy required to remove an electron from its surface (its work function) and the ionisation potential of the atom. Since then, high-temperature cavities were designed to optimise the ion fraction caught by the extraction field. A commonly used geometry consists of a simple refractory metal (niobium, tantalum, tungsten or rhenium) tube heated by a dc-current. 2.2 Metals Plasma ion sources are based on a controlled electron beam of a few hundred eV which generates a plasma in a low pressure of noble gas in a high-temperature cavity with a low magnetic field. The Forced Electron Beam Arc Discharge (FEBIAD) sources [6-7] efficiently ionise all elements which do not have strong chemical reactions with the ion source materials. Their intrinsic low chemical selectivity is improved via thermo- chromatographic methods. The material and the temperature of the transfer line between the target and the ion source is chosen according to the elements to be condensed. The Resonant Ionisation Laser Ion Source (RILIS) [8,9] is based on the stepwise excitation of 2 or 3 atomic transitions leading either to auto-ionising states ordirectly to the continuum. The interaction between the laser beams and the atoms takes place in a high- temperature metallic cavity. While drifting towards the extraction hole, the RILIS ions are confined by a radial potential well originating from the electron layer at the surface of the hot metal. The RILIS elegantly solves the selectivity issue with the notable exception for the alkalis, which are ionised in the high-temperature cavity. It is a proven technique for the efficient production of radioisotopes of 19 elements. In principle an individual ionisation scheme can be developed for 75% of the elements [10]. The RILIS atoms are ionised within the few ns laser pulse, but the different path lengths of drift towards extraction leads to a RILIS ion bunch of typically a few tens of /G50s duration. The micro-gating technique consists in deflecting the surface-ionised ions from the experimental set-up to a beam dump between two RILIS ion bunches. The selectivity towards surface- ionised isobars can be increased up to a factor of 10 via micro gating [11]. A very elegant solution to the selective ionisation of refractory elements is the Ion Guide Laser Ion Source (IGLIS) [12] at Leuven: recoiling refractory metals fragments are stopped in a noble gas catcher, thermalized and neutralized, and eventually ionised via resonant laser ionization. 2.3 Halogens Halogens have high electron affinities, therefore, a halogen atom hitting a low work function material has a tendency to pick an electron from the surface and to be released as a negative ion. The LaB6 ISOLDE surface ion source [13] efficiently produced negatively charged beams of chlorine, bromine, iodine and astatine. Due to its reactivity, fluorine beams could not be produced this way but rather as a molecular side band (AlF+) in a plasma ion source [14] A dedicated Cs sputter ion sources was tested off-line for fluorine production [15]. 2.4 Gaseous elements Despite their high ionisation potentials, noble gases, nitrogen and oxygen are very efficiently ionised in ECR ion sources. Besides the standard high-charge-state ECR, a new generation of ECR designed for single-charge ions was developed [16]. Compact, equipped with permanent magnets and fed with the commercial 2.45 MHz RF generators. They have efficiencies of the order of 90% for 1+ argon. FEBIAD ion sources equipped with a cooled Cu- transfer line are very selective, the only contaminants being the higher charge state of the heavier noble gases. Their efficiency is of the order of 50% for the heavier gases drops to typically 1% for the lighter.3 SELECTED DRIVERS, TARGETS AND ION BEAMS This section is dedicated to the presentation of examples of targets and drivers dedicated to the production of radioisotopes belonging to specific regions of the chart of nuclides. As more powerful drivers are planned, targets must be designed to cope with higher power densities. The challenge is to dissipate the deposited heat in a target that must be kept at high temperature for maximum diffusion speed. In the cases presented, the radioisotopes are stopped or produced at rest. The release of radioisotopes from thick targets is governed by diffusion out of the target matrix, folded with the usually faster effusion from the surface of the target material to the ion source [17]. A release function gives the probability density for an atom produced at t=0 to be released at a given time [18]. It is the normalised and decay-loss corrected equivalent to the time dependence of the radioisotope current. The sharp rise of the first few ms is usually associated with the effusion process, while the slower tail gives information on the diffusion process. As release curves are often measured with pulsed beams, the induced temperature shock affects the release during the heat transport transient time of typically 1s. The last part of the release function is representative of the diffusion process. Neutron-rich isotopes are mainly produced in the fission of actinides to benefit from the neutron excess. Fission can be induced by high-energy charged ions, thermal or fast neutrons and even photons [19]. The distribution of the fission fragments in the chart of nuclides depends on the energy of the primary particle, and is widest for high-energy primary particles. In this case, fragmentation and spallation reactions contribute to the production of both light respectively heavy radioisotopes, on top of the fission products. Among the highest cross-sections, the fission of 235U via thermal neutrons is characterised by its well-known double peak. On the other side of the valley of stability, isotopes close to the proton drip line have to be produced by spallation of a slightly heavier element, or by fragmentation. 3.1 A target for a high flux reactor Thermal neutron induced fission is, thanks to its very high cross section, a very powerful tool for the production of fission fragments as demonstrated for decades by the OSIRIS facility in Studsvik. The next generation of such facilities was proposed at Grenoble ( PIAFE project at the ILL high flux reactor [20]) and Munich (Munich Accelerator for Fission Fragments MAFF [21] currently under construction). One of the objectives of MAAF is the production of superheavy elements via fusion reaction between two different neutron-rich fission products. This would be realised by implanting neutron-rich fission fragments on line on asubstrate, to constantly regenerate a thin very neutron rich target and, simultaneously, accelerate other fission fragments at high energy onto this target [22]. The 235U primary target consists of 1g uranium carbide in a porous graphite matrix. It is confined in a rhenium vessel directly connected to the ion source (surface, plasma or laser). The lifetime of the target and ion source system has to exceed the cycle of the reactor of 52 days. One of the preferred ion sources to match this challenge is the RILIS, as all complex parts are out of the highly active region except for the high-temperature metallic cavity. The neutron flux of maximum 3 /G751011 n/cm2s can be adjusted via the position of the target at the edge of the reactor. The fission power heats up the target to a nominal temperature above 2000 /G71C. A higher fission rate density can be achieved than with charged particle induced fission schemes where energy losses contribute to the heating. The expected fission rate is 1014 s-1. 3.2 Dc vs. pulsed proton beam drivers The ISOLDE facility driver used to be a 600MeV synchrocyclotron (SC) for 2 decades. Following the move of the facility to CERN’s Proton Synchrotron Booster (PSB) [23,24], all targets developed for the dc- beam of the SC were tested with low frequency proton pulses at the PSB. Similar targets are currently very successful at TRIUMF [25] with one order of magnitude higher dc-proton current. The effects of the increase of energy density of 3 orders of magnitude between the almost continuous SC and the pulsed PSB proton beam are briefly described in this section. Molten metal targets: The thermal shock generated by a pressure wave during the 2.4 /G50s proton pulse was sufficient to break the welds of the target container, to generate vapour pressure bursts, and to splash the molten metal into the ion-source [26]. Corrosion and even cavitation-like attacks were observed on a Ta-container for molten lead. The transfer lines between the containers and the ion-sources were equipped with temperature controlled baffle systems, also designed to condense the excess of vapours. This intermediate temperature control allows in some cases to reach higher temperature. The power of a 1 GeV proton beam at an average current of 1 /G50A heats up a lead target unit up to 800 /G71C and sets the power dissipation limit of the present design, which can be improved by cooling. Metal foil targets: The rapid release from metal targets benefits from the pulsed production of the radioisotopes, which allows additional noise suppression by collecting data only after the proton pulses, according to the release and to the specific half-life. On the other hand, the temperature increase following the proton pulse can reach 600 /G71C and occurs in metals that are often beyond their elasticity domain. This leads to the destruction of the target container and rapid sintering of the target material. Defocusing the proton beam gives acontrol over the maximum temperature increase and measurements show the expected increase of the rapidly released isotopes induced by the thermal shock. The synchronous production of isotopes with half-lives of a few ms duration reduces dramatically the data acquisition time fraction. As an example, for 14Be (4.35 ms) the background is reduced by a factor 200 for a data acquisition time of more than 12 half-lives. The pulsed generation of radioisotopes in a target without (or with a very reduced) thermal shock is possible for fission products if the fission is generated with pulsed neutrons. A uranium carbide target placed in the vicinity of a tantalum cylinder bombarded with high- energy pulsed protons corresponds to this description and is presently under test at ISOLDE. The principle of the conversion has been demonstrated at Orsay (section 3.3), Gatchina (ISIS) [27] and of course in spallation neutron sources. 3.3 Molten Uranium target Fast neutrons resulting from the interaction of deuterium with a light converter (beryllium, carbon or liquid lithium) do induce 238U fission. One of the aims of the first proposal by Nolen [5] was to remove the charged particle energy losses from the high temperature uranium target. This concept was partly tested within the PARRNe project [28,29]. The primary targets called neutron converters were built out of low Z material but as demonstrated by the spallation sources, high Z materials including uranium can be envisaged. The RIB production of uranium carbide and molten uranium targets coupled to graphite and beryllium converters were compared at deuteron energies between 15 and 150 MeV. The net gain of neutron production using Be converters was reduced by geometrical effects as the C- converter can be placed closer to the high-temperature target. While the fast release from uranium carbide matrix is well-known, the first measurement from molten uranium confirmed the usually slow (tens of seconds) release behaviour common to all molten metal targets. The challenging container for molten uranium at high temperature was made out of sintered yttria which kept 200g uranium at temperatures up to 1700 /G71C [30] for a few days. Observations of the container after the run showed no damage but a blackening of the yttria (due to oxygen losses). These tests will be used as a benchmark for the simulation codes necessary to optimise the numerous parameters such as deuteron energy, converter and target thickness and geometry. With this technique applied in SPIRAL-II, a target of typically 3 kg 238U would be necessary to achieve of the order of 1014 fissions/s.3.4 Very thin Ta-foil target for neutron halo nuclides The half-lives of the neutron halo nuclides 11Li (8.7ms) and 14Be (4.34ms) are comparable to the typical effusion time from a standard tantalum foil target. Their diffusion time constants are even orders of magnitude larger. Therefore, the target geometry and the diffusion thickness must be optimised to reduce the large decay losses of four orders of magnitude. Instead of the 130 g/cm2 20 /G50m Ta-foils contained in a 20 cm long 2 cm diameter oven only 10 g/cm2 2 /G50m Ta-foils were deposited in a u-shaped support and oriented along the beam axis inside the oven or target container [31]. This container was connected to a surface ion-source consisting of a 3 mm-diameter 30 mm long tungsten cavity. While lithium was ionised on the 2400 /G71C surface, the RILIS laser beams ionised the beryllium isotopes. The specific yield from 11Li increased by a factor of more than 20 and its absolute yield reached 7000 ions/ /G50C (1.4 GeV protons). The best yield of 12Be (23.6 ms) obtained increased by an order of magnitude, and doubled for 14Be. This successful example demonstrates that a target has possibly to be designed for each or a few specific isotopes. 3.5 Radiation-cooled high-power targets Radiation-cooled targets or very refractory materials were first proposed by Bennet et al. [32] within the RIST project. The RIST target consists of a succession of diffusion-bonded 25 /G50m Ta-spacers and discs. A hole of decreasing diameter is cut in the centre to distribute the power along the axis. The target is designed to work with 100 /G50A pulsed beam of 800 MeV protons. The total electrical power that could be dissipated by radiation at 2400 /G71C in an off-line test reached 30 kW. The heavy ion beam from GANIL foreseen for the SPIRAL project, has a total power of 6kW. The short range of heavy ions in matter sets stringent constraints on the target/catcher material. The shape of the graphite target developed for noble gas isotopes is a succession of increasing diameter 0.5 mm thick disks held on a central rod which can also be used as heating resistor. The heavy-ion beam circles around the axis. The target is contained in a water-cooled box and the gaps between two adjacent disks are designed to increase the heat- radiating surface (2500 /G71C). In addition they allow a rapid effusion towards the ECR ion source. The micrometer-size graphite structure is optimum for fast diffusion. During the test of the first prototype, the sublimation temperature of graphite was reached and resulted in a hole located at the Bragg peak. A heat transfer code including heat radiation, was developed to define the final geometry of a target which was then able to dissipate the power.A complementary approach based on conductive cooling was tested at TRIUMF, where a water-cooled set of diffusion-bonded disks of molybdenum representing a medium temperature target was submitted to a 100 /G50A beam of 500 MeV protons. The heat transfer calculations showed small discrepancies with respect to the recorded temperatures [33]. 3.6 Target for alkali suppression The on-line production of neutron-rich copper isotopes is very efficient with high-energy protons impinging on a thorium (or uranium) target. Unfortunately, the yield of rubidium isobars is four orders of magnitude higher. A target unit was therefore designed to purify the copper signal from its rubidium isobars. The micro-gating technique around the typically 30 /G50s laser ionised Cu bunches contributes to the selectivity by a factor of 4. Thanks to their low ionisation potential, rubidium isotopes will be in a positive charge state already inside the target oven. The dc heated high-temperature tantalum oven generates an electric field, which directs rubidium ions opposite to the RILIS cavity (while the neutral copper atoms are insensitive to this electric field). At this point, a small aperture extraction system collects them. The selectivity gain measured on a Ta-foil target is 5 for rubidium and caesium, and it rapidly drops for elements with higher ionisation potentials. The last, but not least, way to reduce the production of rubidium is to generate the fission via fast neutrons produced in a converter placed close to the target oven. The expected cross-section ratios contribute to the selectivity up to an order of magnitude. These suppression systems are in addition to the selectivity provided by the resolving power of the mass separator. 3.7 Ion beam manipulation The characteristics of radioactive ion beams such as emittance, energy spread or charge state sometimes have to be matched to the physics set-up or to the acceptance of the following beam optics element. For this purposes an increasing number of facilities make use of ion traps and gas cells. An ion trap acting as buncher and used to improve the beam emittance is installed in the REX- ISOLDE LINAC [34] which requires a q/m larger than 0.25. Its charge state breeding relies on the trapping and bunching of the ions prior to the injection into an Electron Beam Ion Source (EBIS). In this trap, up to 107 ions are accumulated and bunched, and their emittance is reduced by on e order of magnitude to match the acceptance and the time structure of the EBIS. At the Ion Guide Isotope Separator Leuven (IGISOL) facility, laser ionised refractory metal fragments are transported via a noble gas flow. The skimmer electrode used to extract the singly charged fragments induces an energy straggling which vanished when the skimmer was replaced by a Sextupole Ion Guide (SPIG).The Rare Isotope Separator (RIA) includes, beside the standard ISOL and in-flight separation, a new production scheme. In this variant, the fragments (possibly produced on a liquid Li target) will be stopped in a noble gas cell where, after thermalization, the singly charged ions are transported by weak electric fields towards the ion beam transport system. Preliminary tests showed that the release time from such a gas cell is well below 1 s [35] depending on the thickness of the gas cell (in this case length /G75 pressure). This part of the RIA project aims at combining the ion beam quality from an ISOL technique and the short release delays of a fragment separator. 4 CONCLUSION The standard complementary methods used in RIB facilities are now completed with an intermediate one, namely the stopping of fragments in gas cells as in the RIA project. The increase of the total beam power available at TRIUMF, or expected for the SIRIUS project, has triggered a large effort in the development of high power targets for high-energy proton driver schemes. The targets developed for moderate intensities or dc. beams could be adapted for beam intensities up to 20 /G50A and for the today’s pulsed drivers. The limitation on the maximum power which can be deposited in a small target volume, can be overcome partially for the production of fission products, by the introduction of a converter producing fast neutrons. However, fission in reactor-based facilities like MAFF sets a challenging goal of 1014 fissions/s. A large R&D effort from the whole RIB community is visible, as witnessed by the RIA project and by the creation of study groups on targets, traps and ion- sources, triggered by the EURISOL facility project. Obviously, with the expected intensities, not only the target area, but in a less critical manner the beam transport, traps and experimental set-ups will have to be designed with regard of their contamination when handling the very intense beams foreseeable in the near future. REFERENCES [1]H. Grunder, Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [2]B. Jonson, Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [3] I. Tanihata, ENAM 98 AIP Conf. Proc. 455, 943 (1998). [4]H. Ravn, Radioactive ion-beam projects based on the two accelerator or ISOL principle, Phil. Trans. R. Soc. Lond. A 365, 1955 (1998). [5]J. Nolen, RNB-3 Conf. Proc. D. Morissey Ed., Editions Frontières, Gif sur Yvette. 111, (1993). [6]R. Kirchner and E. Roeckl, Nucl. Instrum. and Meth. 133, 187 (1976).[7]R. Kirchner, K. H. Burkard, W. Hüller and O. Klepper, Nucl. Instrum. and Meth. 186, 295 (1981). [8]V.I. Mishin et al., Nucl. Instrum. and Meth. B 73,550 (1993). [9]A. E. Barzakh, et al., Nucl. Instrum. and Meth. B 126, 85 (1997). [10]U. Koester et al., Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [11]J. Lettry et al., Rev. Sci. Instrum., 69, 761 (1997). [12]Y. Kudriavtsev et al., Nucl. Instrum. and Meth. B 114, 350 (1996). [13]B. Vosicki et al., Nucl. Instrum. and Meth. 186, 307 (1981). [14]R. Welton et al., Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [15]G. Alton, et al., Nucl. Instrum. and Meth. B 142, 578 (1998). [16]A. Villari et al., Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [17]J.R.R. Bennett, Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [18]J. Lettry et al., Nucl. Instrum. and Meth. B 126 (1997) 130. [19]Y. Oganesian, Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [20]PIAFE project report, ed. By U. Köster and J.-A. Pinston, ISN Grenoble, 1998. [21]D. Habs et al., Nucl. Phys. A616 (1997). http://www.ha.physik.uni-muenchen.de/maff/ [22]U. Köster, AIP Conf. Proc. 475, 269, (1998). [23]B. Allardyce and R. Billinge, CERN PS/DL 89-37. [24]B. Allardyce et al., CERN PS/92-46 (PA). [25]R. Laxdal et al., These proceedings. [26]J. Lettry et al., Nucl. Instrum. and Meth. B 126 171, (1997). [27]J.Nolen and V. Panteleev, communication to the TWIST workshop GANIL Caen 2000. [28]S. Kandri-Rody et al., Nucl. Instrum. and Meth. B160, 1, (2000). [29]F. Clapier et. al., Phys. Rev. Special topics, Accelerators and Beams, 1, 013501 (1988). [30]O. Bajeat, communication to the TWIST workshop GANIL Caen 2000. [31]J.R.J. Bennett et al., Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [32]J.R.J. Bennett, Nucl. Instrum. and Meth. B 126, 105 (1997). [33]W. Talbert et al., Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A. [34]O. Kester et al., these proceedings. [35]G.Savard et al., Proc. of the RNB–5 Conf. Divonne 2000, to be published in Nucl. Phys. A.
arXiv:physics/0009037v1 [physics.class-ph] 11 Sep 2000Inﬂuence of guiding magnetic ﬁeld on emission of stimulated photons in generators utilizing periodic slow-wave struct ures. V. G. Baryshevsky, K. G. Batrakov Institute of Nuclear Problems 220050, Minsk, Republic of Be larus (January 11, 2014) Abstract Eﬀect of guiding magnetic ﬁeld on evolution of stimulated em ission is con- sidered. It is shown that the transverse dynamics of electro ns contributes to generation process and that contribution decreases with th e magnetic ﬁeld grows. The equation of generation for stimulated radiation of electron beam passing over periodic medium in magnetic ﬁeld of arbitrary v alue is derived. The critical value of guiding ﬁeld for which the transverse d ynamics of elec- tron don’t contribute to emission is determined. It is shown , that transverse dynamics of electron modiﬁes the boundary conditions. It fo llows from the derived generation equation that transverse dynamics yiel ds to ∼25% in- crease of the increment magnitude in high gain regime. In the limit of small signal the generation gain twice increases when transverse dynamics evolves. Obtained results are valid for every FEL system, which use th e mechanism of waves slowing in the slow-wave structures for generation. I. INTRODUCTION The various types of devices utilizing the electron beam int eraction with electromagnetic ﬁelds in slow-wave system (Cherenkov [1], Smith-Purcell [3 ], quasi-Cherenkov [2], transition [4] radiation mechanisms) was considered in the past.The gr eat number of researches in the area of microwave electronics resulted in the traveling wav e tube (TWT), backward-wave oscillator (BWO), orotrons and so on. Theory of such generat ors, as a rule, considers the one-dimentional (longitudinal) dynamics of electrons in t he ﬁeld of an electromagnetic wave. On the other hand in our previous works devoted to quasi-Cher enkov FEL the stimulated emission of unmagnetized electron beam with three - dimensi onal dynamics was studied ( [5], [6], [7]). In our works [8], [9] the guiding ﬁeld was con sidered as strong and the electron beam dynamics as one-dimensional. It was shown tha t quasi-Cherenkov volume FEL (VFEL) can produce radiation with lower current density (especially in X-ray spectrum region of frequencies) in comparison with ordinary FELs. Ho wever, multiple scattering of electrons in the medium destructs the coherence of radiatio n process. The surface scheme of the quasi-Cherenkov VFEL can be used for multiple scatter ing reducing ( [8], [9]). In that case an electron beam moves over a periodic medium at a di stance d≤u 4πcγλ(λ is the radiation wavelength, γ= 1//radicalbigg 1−u2 c2is the electron Lorentz factor) and radiation 1is formed along the whole electron trajectory in vacuum with out multiple scattering. In these works ( [8], [9]) the generation of the surface paramet ric FEL was considered for electron beam placed in strong longitudinal guiding magnet ic ﬁeld. So, only the contribution of the one-dimensional (longitudinal) dynamics of electro ns to stimulated radiation was considered. This is true if inequalityeH(0) mγc∆≫1 is satisﬁed ( H(0)is the magnitude of magnetic ﬁeld, ∆ is the detuning from synchronism condition ). In opposite caseeH(0) mγc∆≤1 it is necessary to take into account the transverse motion of electrons. The ∆ increases with the interaction length L∗decrease ( L∗is the length of interaction between the electron beam and radiation) and with the current density increase. T herefore the transverse motion contributes to stimulated quasi-Cherenkov emission in the case of high current or small interaction length (eH(0) mγc≤max{2πc L∗; (ωω2 b/γ5)1/3}, where ωb= 4πe2ne/meis the Langmuir frequency). In this paper we present the analysis of volume FEL (VFEL) ope ration in the periodic slow-wave structures including the eﬀect of the ﬁnite guidi ng magnetic ﬁeld value on stimu- lated emission. So we take into account the contribution of t ransverse electron beam motion to generation process. Using the linearized perturbation ﬁ eld approximation we derive the boundary conditions, dispersion relations and generation equation. All results received be- low relate to Compton regime, when the amplitudes of excited longitudinal Langmuir waves are small. Raman regime was considered in our previos work [1 0] in two limiting cases 1)when guiding magnetic ﬁeld is absent; 2)when guiding ﬁeld is strong. II. THE PROBLEM STATEMENT The considered system is represented on Figure 1. An electro n beam located at height h over the spatial periodic structure is moving parallel to st ructure surface. δis the beam thick- ness in the direction normal to the surface. The axis xis chosen along the direction of elec- tron motion, the axis zis normal to the periodic structure surface. Dynamical diﬀr action on the periodic structure forms the 3-dimensional volume dist ribution feedback which provides generation regime. The set of reciprocal lattice vectors τn={2πn1/d1; 2πn2/d2; 2πn3/d3} deﬁning diﬀraction process can be directed at an arbitrary a ngle relative to the particle ve- locity and to the surface, ( diare translation periods of periodic structure, niare intergers). The interaction between the electron beam and the grating pr oduces an emission spectrum. The emission frequency is deﬁned by the spatial period and by the volume geometry (by the direction of the reciprocal vectors for example). The perio dic structure perfoms two basic functions. Firstly it slows down the phase velocity of an ele ctromagnetic wave that enables conditions for coherent radiation. Secondly, due to 3-dime nsional distributed feedback, the periodic structure is an eﬀective volume resonator,which g ives the possibility for an oscillator regime realization. The emitted photons bunch the electron beam. This bunching leads to greater emission, which leads to more bunching.Three-dime nsional Bragg distributed feed- back keeps emission in the interaction region. The regime of an oscillator is realized as the result of these processes. Dependence of the emitted wav elength on system geometry 2provides smooth frequency tuning. For deriving the generat ion equation it is necessary to obtain the dispersion equations in all regions and to use the boundary conditions on the surfaces of the electron beam and surfaces of the slow-wave s tructure. III. DISPERSION EQUATIONS In most of previous works concerning slow-wave FELs the elec tron beam is considered as magnetized. Therefore only longitudinal dynamics of ele ctron beam was taken into ac- count. The magnetic ﬁeld is used for electron beam guiding ov er slow-wave structure surface. However, the transverse motion of electron still can contri bute to the process of stimulated radiation. The contribution of transverse degrees of freed om depends on: 1) parameters of an electron beam such as the energy of electrons, current den sity, the velocity spread of an electron beam; 2) the parameters of emitted radiation such a s photon wavelength and the ﬁeld amplitudes; 3) the parameters of the electrodynamical structure such as photoabsop- tion length, interaction length of electron beam with emitt ed radiation and the binding of an electron beam with eigenmodes of an electrodynamic system; 4) the magnitude of guiding ﬁeld. The slow electromagnetic wave which is in synhronism with th e electron beam produces modulation of the density and current density. This leads to development of instability. The stimulated radiation is the result of this instability. Let us consider the inﬂuence of guiding magnetic ﬁeld on the stimulated radiation. The velocity and radius-vector of an electron can be presented as: vα(t) =u+δvα(t),rα(t) =r0α+ut+δrα(t). Here the pertubations δvα(t) and δrα(t) are results of electron interaction with an electromagnet ic wave. In the linear ﬁeld approximation the current density can be writte n as j(z, kx, ky, ω) =e/summationdisplay α{u[δ(z−z0α) exp(−ikxx0α−ikyy0α)(−ikxδxα(ω−kxu)− ikyδyα(ω−kxu))− (3.1) ∂ ∂zδ(z−z0α)δzα(ω−kxu) exp(−ikxx0α−ikyy0α)] + δvα(ω−kxu)δ(z−z) exp(−ikxx0α−ikyy0α)}, where xα(t) = x0α+ut+δxα(t),yα(t) = y0α+ut+δyα(t),zα(t) = z0α+ δzα(t) are the radius vectors of an electrons in a beam and {δxα, δzα, δvα}(ω) =/integraldisplay dtexp(iωt){δxα, δyα, δzα, δvα}(t),j(z, kx, ky, ω) =/integraldisplay dxdyexp(−ikxx−ikyy)j(z, x, y, ω ). Dynamics of electron in the ﬁeld of electromagnetic wave is d escribed by equation dδvα(t) dt−e mγc[δvα(t)H0] =e mγ{E(rα(t), t) +1 c[uH(rα(t), t)]− (3.2) u c2(uE(rα(t), t))}. The distinction from the schemes studied earlier ( [8], [9]) is in considering the term with guiding magnetic ﬁeld H0. The Fourier transformation of (3.2) gives 3δvα(ω)−e mγc[δvα(ω)H0] =ie mγω/integraldisplaydk′ xdk′ y (2π)2exp(ik′ xx0α+ik′ yy0α) {E(z0α, k′ x, k′ y, ω−k′ xu) +1 c[uH(z0α, k′ x, k′ y, ω+k′ xu)]− (3.3) u c2(uE(z0α, k′ x, k′ y, ω+k′ xu))};δrα(ω) =i ωδvα(ω) Decomposing (3.3) by components it can be received δvxα(ω) =ie mγ3ω/integraldisplaydk′ xdk′ y (2π)2exp(ik′ xx0α+ik′ yy0α)Ex(z0α, k′ x, k′ y, ω+k′ xu) δvyα(ω) =1 ω2−/parenleftig eH(0) mγc/parenrightig2/integraldisplaydk′ xdk′ y (2π)2exp(ik′ xx0α+ik′ yy0α) /braceleftbigg −e2H(0) m2γ2c/braceleftbiggω ω+k′xuEz−iu ω+k′xu∂Ex ∂z/bracerightbigg +ieω mγ/parenleftbiggω ω+k′xuEy+k′ yu ω+k′xuEx/parenrightbigg/bracerightbigg δvzα(ω) =1 ω2−/parenleftig eH(0) mγc/parenrightig2/integraldisplaydk′ xdk′ y (2π)2exp(ik′ xx0α+ik′ yy0α) /braceleftbiggie mγω/braceleftbiggω ω+k′ xuEz−iu ω+k′ xu∂Ex ∂z/bracerightbigg +e2H(0) m2γ2c/parenleftbiggω ω+k′ xuEy+k′ yu ω+k′ xuEx/parenrightbigg/bracerightbigg If electrons in the beam are distributed as n=nef(zα) it can be derived from (3.1,3.3) δj(z, kx, ky, ω) =ω2 L 4πf(z)/braceleftbiggiω γ3(ω−ku)2Exex+ −eH(0) mγ2c/braceleftbigω−kxu ωEz−iu ω∂Ex ∂z/bracerightbig +i(ω−ku) γ/parenleftig ω−kxu ωEy+kyu ωEx/parenrightig (ω−ku)2−/parenleftig eH(0) mγc/parenrightig2ey+ i γ(ω−ku)/braceleftbigω−kxu ωEz−iu ω∂Ex ∂z/bracerightbig +eH(0) mγ2c/parenleftig ω−kxu ωEy+kyu ωEx/parenrightig (ω−ku)2−/parenleftig eH(0) mγc/parenrightig2ez  + (3.4) uky ω−kuω2 L 4πf(z)−eH(0) mγ2c/braceleftbigω−kxu ωEz−iu ω∂Ex ∂z/bracerightbig +i(ω−ku) γ/parenleftig ω−kxu ωEy+kyu ωEx/parenrightig (ω−ku)2−/parenleftig eH(0) mγc/parenrightig2− iu ω−kuω2 L 4π∂ ∂z f(z)i γ(ω−ku)/braceleftbigω−kxu ωEz−iu ω∂Ex ∂z/bracerightbig +eH(0) mγ2c/parenleftig ω−kxu ωEy+kyu ωEx/parenrightig (ω−ku)2−/parenleftig eH(0) mγc/parenrightig2  Current density contains terms with Cherenkov and cyclotro n resonances. We shall study the Compton regime of Cherenkov instability. The terms corr esponding to second order resonances give maximal contributions in that case. Below w e use this fact for separation of wave polarisations. The dispersion equation in the region ﬁlled with electron be am is deﬁned by equating of the determinant of the system to zero. (k2c2−ω2)E−c2k(kE) =−4πiωδj(k, ω), (3.5) 4Hereδj(k, ω) is derived from (3.4) ( δj(z, kx, ky, ω)∼δj(k, ω) exp(ikzz). It is considered in this case that f(z) = 1 in the region with electron beam). Let us discuss some features of this dispersion equation. In general case of an arbitrary guiding magnetic ﬁeld it has six roots kza(kx, ky, ω),a= 1÷6. For the case of strong guiding magnetic ﬁeld when the condition ω−ku<<eH(0) mγcthere exist four roots kbz=±/radicalbigg ω2 c2−k2 \|\|when the wave polarisation is normal to uandk (3.6) kbz=±/radicaligg/parenleftbiggω2 c2−k2 \|\|/parenrightbigg /parenleftbigg 1−ω2 L γ3(ω−kxu)2/parenrightbigg when the wave polarisation is in the plane of uandk The ﬁst two roots correspond to electromagnetic waves which don’t interact with the electron beam. The last two roots correspond to waves which a re result of electromag- netic wave with electron beam interactions.In particular c ase when kz= 0, these two wave degenerate to longitudinal slow and fast Langmuir wave s with the dispersion equa- tion 1 −ω2 L γ3(ω−kxu)2= 0. In the opposite case of low guiding ﬁeld, when inequality ω−ku>>eH(0) mγcis satisﬁed there exist four roots of (3.5) kbz=±/radicaligg ω2 c2−k2 \|\|−ω2 L γwhen the wave polarisation is normal to uandk kbz=±/radicaligg ω2 c2−k2 \|\|−ω2 L γwhen the wave polarisation is in the plane of uandk and the Langmuir waves polarized parallel to wavevector k. So in the region ﬁlled by beam (h < z < h +δ) the ﬁeld can be written as /summationdisplay {in}/bracketleftig e(i) bna(n) bexp(−ikbnzz) exp{i(k\|\|+τn\|\|)r\|\|}+e(i) bnb(n) bexp(ikbnzz) exp{i(k\|\|+τn\|\|)r\|\|}/bracketrightig (3.7) In vacuum regions 1 ( z > h +δ), 3 (0 < z < h ) and 5 ( z <−D) the electromagnetic ﬁeld is a set of transverse polarized plane waves 5/summationdisplay {in}e(i) 1na(n) 1exp(−iknzz) exp{i(k\|\|+τn\|\|)r\|\|}in region 1 which is over the electron beam /summationdisplay {in}/bracketleftig e(i) 3na(n) 3exp(−iknzz) exp{i(k\|\|+τn\|\|)r\|\|}+e(i) 3nb(n) 3exp(iknzz) exp{i(k\|\|+τn\|\|)r\|\|}/bracketrightig in the gap between the electron beam and slow-wave structure/summationdisplay {in}e(i) 5na(n) 5exp(iknzz) exp{i(k\|\|+τn\|\|)r\|\|}in region 5 which is under the slow-wave structure (3.8) Writing ﬁelds in region 1 and 5 we use the lack of the incident w aves. The electromagnetic ﬁeld in the slow wave structure ( −D < z < 0) can be written as a sum of Bloch functions /summationdisplay αfαEα(r) =fαexp{ik(α)r}uα(r), (3.9) whereuα(r) satisﬁes to conditions uα(r+di) =uα(r) anddiis arbitrary translation vector of spatially periodic slow-wave structure. IV. THE BOUNDARY CONDITIONS To derive the generation conditions it is necessary to write the equations for ﬁeld coef- ﬁcient in (3.7,3.8,3.9). These equations are produced by ut ilizing the boundary conditions on the surfaces. If the surface currents and surface charges are not excited on the bound- ary, then we shall use the conditions of transverse magnetic and electric ﬁeld continuity on the boundary.In general case, as will be shown below, the i nduced surface currents and charges exist at the electron beam surfaces. For deﬁning of t his currents and deriving of corresponding boundary conditions the consideration of se lf-consistent problem of electron beam-radiation interaction should be performed. To produc e the boundary condition for tangential component of magnetic ﬁeld we use (3.4) and Maxwe ll equation rotH=4π cδj (4.1) By integrating left and right hand sides (4.1) in narrow regi on near the electron beam surface and using (3.4), it can be derived the following boundary con ditions  Hy+u cω2 L γf(z)∆/braceleftbig∆ ωEz−iu ω∂Ex ∂z/bracerightbig −ia0/parenleftig ∆ ωEy+kyu ωEx/parenrightig ∆D0  zb= 0 (4.2) [Hx] = 0 Here the new symbols are introduced ∆ = ω−ku,D0= (ω−ku)2−/parenleftbiggeH(0) mγc/parenrightbigg2 ,a0=eH(0) mγc. As can be seen from (4.2) the tangential component of magneti c ﬁeld which is normal to electron beam velocity udon’t conserve on the electron beam surfaces. The component of 6magnetic ﬁeld parallel to velocity is conserved. The noncon serving of Hyon the electron beam density discontinuity is caused by arising of surface c urrent directed along the electron velocity vector u. The following limit cases of boundary conditions (4.2) exi st. 1) the limit of strong guiding magnetic ﬁeld ∆ << a 0, theycomponent of magnetic ﬁeld is conserved. That is result of transverse dynamics lac k in strong longitudinal magnetic ﬁeld; 2) the opposite limit of weak guiding ﬁeld ∆ >> a 0, in this case the boundary condition has the form: /bracketleftigg Hy+u cω2 L γf(z)∆ ωEz−iu ω∂Ex ∂z ∆2/bracketrightigg zb= 0, (4.3) and electron beam gives the resonant contribution to bounda ry condition (4.3) in the region of Cherenkov synhronism. V. THE GENERATION EQUATIONS The scheme of a surface VFEL is shown in Fig 1. There his the distance between an electron beam and a target surface, δis a transverse size of an electron beam. The electromagnetic ﬁeld excited in this system has the foll owing form: 1)z > h +δ texp{−ikz(h+δ)}exp{ikr}+/summationdisplay imiexp{ikir} (5.1) 2)h < z < h +δ aexp{ikbr}+bexp{ik(−) br}+ (5.2)/summationdisplay imiexp{ikir} 3) 0< z < h cexp{ikr}+dexp{−ik(−)r}+/summationdisplay imiexp{ikir} (5.3) 4)−D < z < 0 /summationdisplay αfαexp{ik(α)r}uα(r) (5.4) 5)z <−D /summationdisplay igiexp{ik(−) ir} (5.5) where k= (k⊥;kz),k(−)= (k⊥;−kz);kz=/radicalig ω2/c2−k2 ⊥;ki= (k⊥+τi⊥, kiz),k(−) i= (k⊥+τi⊥,−kiz),kiz=/radicalbig ω2/c2−(k⊥+τi⊥)2, the wave vectors {ki}and{k(−) i}correspond 7to electromagnetic waves escaping from the system (if kizis real) and evanescent waves (if kiz is imaginary). {Fα= exp {ik(α)r}uα(r)}are Bloch waves ( α= 1, ..., n) excited in the target, {uα(r)}are periodical functions: uα(r+lm) =uα(r), where lmare the translation vector of the periodic structure, kbz=kz/radicaligg 1 +ω2 La2 0 γ3∆2D0,kb= (k⊥;kbz),k(−) b= (k⊥;−kbz) are the wave vector corresponding to an electromagnetic waves in th e electron beam. kbandk(−) b are produced as the solution of dispersion equation for elec tromagnetic waves in the beam, ω2 b= 4πnb/meis the Langmuir frequency of electron beam. We assume that only the wave with wave vectors kandk(−)are under the Cherenkov synhronism conditions with the particles. Therefore the el ectron beam does not aﬀect the diﬀracted waves with the wave vectors ki=k+τiifτi/ne}ationslash= 0.a,b,{mi},c,d,t,{fα},{gi} are the coeﬃcients deﬁned from boundary conditions on the su rfaces of discontinuity.Using equations (4.3) the following system for these coeﬃcients c an be written: f=aexp(ikbzH) +bexp(−ikbzH) +βω2 L γukzη ωD0{aexp(ikbzH)−bexp(−ikbzH)} f=s{aexp(ikbzH)−bexp(−ikbzH)} aexp(ikbzh) +bexp(−ikbzh) +βω2 L γukzη ωD0{aexp(ikbzh)−bexp(−ikbzh)}= cexp(ikzh) +dexp(−ikzh) s{aexp(ikbzh)−bexp(−ikbzh)}=cexp(ikzh)−dexp(−ikzh) ... where s=kbz kz/braceleftig 1 +ω2 La2 0 γ3∆2D0/bracerightig, η =ckbz ω/braceleftig 1 +ω2 La2 0 γ3∆2D0/bracerightig(5.6) The conditions on the beam boundaries are written in (5.6), t he dots ...denote remaining boundary conditions on the surfaces of slow wave system. Res olving (5.6) it can be produced the following equality d= exp( −2αh)[1−s2+ 2η1]exp(αbδ)−exp(−αbδ) (s+ 1)2exp(αbδ)−(s−1)2exp(−αbδ)a, (5.7) where η1=βω2 Luck2 z γω2D0 Hereα=kz/i, α b=kbz/i. Let us note that roots of equation d= 0 gives the eigenstate of ”cold” waveguide without an electron beam. Therefore the generation equation for the system ”electron beam + slow-wave system” looks like −exp(−2αh)/braceleftbiggω2 L γ3∆2+ω2 L γ3D0/bracerightbiggexp(αbδ)−exp(−αbδ) (s+ 1)2exp(αbδ)−(s−1)2exp(−αbδ)= (5.8) N(k,k1, ...,kn, ω) In (5.8) the function N(k,k1, ...,kn, ω) describes the ”cold” slow-wave system. It is easy to see distinction between lasing in cases with low and stron g guiding ﬁeld from (5.8). For strong guiding ﬁeld (5.8) has form 8−exp(−2αh)ω2 L γ3∆2exp(αbδ)−exp(−αbδ) (s+ 1)2exp(αbδ)−(s−1)2exp(−αbδ)=N(k,k1, ...,kn, ω) (5.9) and for the slow magnetic ﬁeld −exp(−2αh)2ω2 L γ3∆2exp(αbδ)−exp(−αbδ) (s+ 1)2exp(αbδ)−(s−1)2exp(−αbδ)=N(k,k1, ...,kn, ω) (5.10) The terms related with electron beam diﬀer in two times. It is result of transverse dynamics lack in the case of (5.9). In the case of slow guiding magnetic ﬁeld (5.10) the transverse motion and longitudinal motion give the same contribution t o generation process. VI. INCREMENTS OF QUASI-CHERENKOV INSTABILITY The slow electromagnetic wave produces the modulation in th e electron current and this modulation forms the coherent quasi-Cherenkov radiation w hich acts on the electron beam again. As the result the emission increases during the proce ss of ”electron beam - radiation” interaction in the slow-wave system. Dynamics of this proce ss can be described by the increment of instability. Received generation equation (5 .8) will be used for calculation of increment. Expanding the equation (5.8) in vicinity of Cherenkov reson ance and in vicinity of the eigenmode of slow-wave system the generation equation can b e written in the form −A/braceleftbigg1 ν2+1 ν2−a2 0/bracerightbigg =N0+∂N ∂νν+∂2N ∂ν2ν2+... (6.1) where A= exp( −2αh)ω2 L γ3ω2,a0=eH(0) mγcω. The dependence of increment on the value of magnetic ﬁeld can be studied using (6.1). Let discuss the phy sical meaning of the terms in right hand side of (6.1). As was shown in [7], the N0is proportional to absorption losses of slow-wave system ( ∼χ′′ 0).∂N ∂νis equal to zero at the point of root degeneration. In the case of great photoabsoption losses ( \|N0\| ≫ \|∂N ∂νν\|) the dissipative instability developes. At the root degeneration point dependence of increment on current density changes ν∼j1/(2+s), where sis the number of degenerated modes.The dependence of increm ent on the magnitude of guiding ﬁeld is presented on (Figure 2, Figure 3) for curre nt density j= 10A/cm2(Figure 2) and j= 100 A/cm2(Figure 3). The following parameters were taken: ω∼5·1011s−1, u∼1.4·1010cm/s. It follows from (Figure 2, Figure 3) that for the magnitude o f magnetic H0= 3 KGs only longitudinal motions of electron contributes to stimulated emission if the current density j= 10A/cm2. Ifj= 100 A/cm2, the critical magnitude of the guiding ﬁeld isH0≈6 KGs. If guiding ﬁeld less these values, the transverse dyna mics contributes to emission also. 9VII. CONCLUSIONS This paper presents analysis of the guiding magnetic ﬁeld in ﬂuence on the quasi- Cherenkov stimulated radiation. It is shown that the increm ent is maximal in the case without magnetic ﬁeld. However, for guiding of the electron beam over the surface of the slow-wave system the magnetic ﬁeld has to be strong enough to oppose to Coulomb repul- sion of electron beam. The following simple estimation for g uiding ﬁeld can be used: 1) the deviation from the Cherenkov synchronism ∆ must be less the w idth of stimulated emission line \|δ∆\| ∼ωδvx/u+ωδvz/γu≤max{πu/L int;βων}, (7.1) where δvxandδvzare velocity pertubations caused by Coulomb repulsion, 2) t he ampli- tude of oscillation in crossed ﬁelds can be less then the gap w idthh:mc2E eH2< h. If period of transverse electron oscillations in guiding magn etic ﬁeld less then interaction time (ωHL/v≥1), then (7.1) can be writen as ωcE/γ2uH≤max{πu/L int;βων}. Take into account that E∼2πI ul, 2πωcI/ (u2lH)≤max{πu/L int;βων},2πmc2I eH2ulγ2< hwhere lis the width of an electron beam along the yaxis,Iis the beam current.So, if the inequalities max{eH(0) mγc; 2πωcI/ (γ2u2lH)}<max{πu/L int;βων}are fulﬁlled, then the transverse dy- namics of electrons contributes to generation process. As w as shown above the transverse dynamics contributes to stimulated emission for magnitude of guiding magnetic ﬁeld ≤few KGs. The transverse electron dynamics can increase the increme nt on 25 percent in the high gain exponentional regime as can be seen from (Figure 2, Figure 3). In the slow gain regime the magnitude of the gain can increase in two times due to transverse dynamics. 10REFERENCES [1] P.A. Cherenkov. DAN SSSR. 2, 451 (1934). [2] V.G.Baryshevsky and I.D.Feranchuk, J. Physique. 44, 913 (1983). [3] S.J. Smith, E.M. Purcell, Phys. Rev. 92, 1069 (1953). [4] M.A. Piestrup, R.L.Finman, IEEE J. Quant. Electron. 19(357). [5] V.G.Baryshevsky and I.D.Feranchuk, Phys.Letters. 102A ,103 (1984). [6] V.G.Baryshevsky, K.G.Batrakov and I.Ya.Dubovskaya, J .Phys. D 24,1250 (1991). [7] V.G.Baryshevsky, K.G.Batrakov and I.Ya.Dubovskaya, P hys. Stat.Sol. 169 b , 235 (1992). [8] V.G.Baryshevsky, K.G.Batrakov and I.Ya.Dubovskaya, N IM,341A , 274 (1994). [9] V.G.Baryshevsky, K.G.Batrakov, I.Ya.Dubovskaya, S.S ytova, NIM, 358A , 508 (1995). [10] V.G.Baryshevsky, K.G.Batrakov. 21thInternational FEL99 Conference Contributions http://www.desy.de/fel99/contributions/T05/M0-P-10. pdf, DESY, Hamburg, Ger- many 23,28 Aug. (1999). 11FIGURES FIG. 1. Electron beam is moving over the periodic structure 0.02 0.04 0.06 0.08 0.1a0-0.048-0.046-0.044-0.042n 12FIG. 2. Dependence of dimensionless instability increment νon the magnitude of guiding ﬁeld (a0=eH0 mcγω). The current density of electron beam is j= 10A/cm2. 0.2 0.4 0.6 0.8 1a0 -0.105-0.095-0.09n FIG. 3. Dependence of dimensionless instability increment νon the magnitude of guiding ﬁeld (a0=eH0 mcγω). The current density of electron beam is j= 100 A/cm2. 1300.05 0.1 0.15 0.2 0.25 0.351015202530 FIG. 4. The contour plot for instability increment.The absc issa corresponds to a0, the ordinate corresponds to current density j(A/cm2). 14
arXiv:physics/0009038v1 [physics.atom-ph] 11 Sep 2000Relativistic recoil corrections to the atomic energy level s V. M. Shabaev Department of Physics, St. Petersburg State University, Oulianovskaya Street 1, Petrodvorets, St. Petersburg 1985 04, Russia The quantum electrodynamic theory of the nuclear recoil eﬀe ct in atoms to all orders in αZ and to ﬁrst order in m/M is considered. The complete αZ-dependence formulas for the relativistic recoil corrections to the atomic energy levels are derived i n a simple way. The results of numerical calculations of the recoil eﬀect to all orders in αZare presented for hydrogenlike and lithiumlike atoms. These results are compared with analytical results o btained to lowest orders in αZ. It is shown that even for hydrogen the numerical calculations to a ll orders in αZprovide most precise theoretical predictions for the relativistic recoil corre ction of ﬁrst order in m/M. I. INTRODUCTION In the non-relativistic quantum mechanics the nuclear reco il eﬀect for a hydrogenlike atom is easily taken into account by using the reduced mass µ=mM/(m+M) instead of the electron mass m(Mis the nuclear mass). It means that to account for the nuclear recoil eﬀect to ﬁrst ord er inm/M we must simply replace the binding energy EbyE(1−m/M). Let us consider now a relativistic hydrogenlike atom. In the inﬁnite nucleus mass approximation a hydrogenlike atom is described by the Dirac equation (¯ h=c= 1) (−iα·∇+βm+VC(x))ψ(x) =εψ(x), (1.1) whereVCis the Coulomb potential of the nucleus. For the point-nucle us case, analytical solution of this equation yields the well known formula for the energy of a bound state: εnj=mc2 /radicalbigg 1 +(αZ)2 [n−(j+1/2)+√ (j+1/2)2−(αZ)2]2, (1.2) wherenis the principal quantum number and jis the total angular momentum of the electron. The main probl em we will discuss in this paper is the following: what is the rec oil correction to this formula? It is known that to the lowest order in αZthe relativistic recoil correction to the energy levels can be derived from the Breit equation. Such a derivation was made by Breit a nd Brown in 1948 [1] (see also [2]). They found that the relativistic recoil correction to the lowest order inαZconsists of two terms. The ﬁrst term reduces the ﬁne structure splitting by the factor (1 −m/M). The second term does not aﬀect the ﬁne structure splitting and is equal to−(αZ)4m2/(8Mn4). Calculations of the recoil eﬀect to higher orders in αZdemand using QED beyond the Breit approximation. In quantum electrodynamics a two-body syst em is generally treated by the Bethe-Salpeter method [3] or by one of versions of the quasipotential method propos ed ﬁrst by Logunov and Tavkhelidze [4]. In Ref. [5] (see also [6]), using the Bethe-Salpeter equation, Salpete r calculated the recoil correction of order ( αZ)5m2/Mto the energy levels of a hydrogenlike atom. This correction gives a contribution of 359 kHz to the 2 s- 2p1/2splitting in hydrogen. The current uncertainties of the Lamb and isotopi c shift measurements are much smaller than this value (see, e.g., [7]) and, therefore, calculations of the recoil corrections of higher orders in αZare required. In addition, for the last decade a great progress was made in high precision me asurements of the Lamb shifts in high-Z few-electron ions [8–10]. In these systems, the parameter αZis not small and, therefore, calculations of the relativist ic recoil corrections to all orders in αZare needed. II. RELATIVISTIC FORMULA FOR THE RECOIL CORRECTION First attempts to derive formulas for the relativistic reco il corrections to all orders in αZwere undertaken in [11,12]. As a result of these attempts, only a part of the desired expre ssions was found in [12] (see Ref. [13] for details). The completeαZ-dependence formula for the relativistic recoil eﬀect in th e case of a hydrogenlike atom was derived in [14]. 1The derivation of [14] was based on using a quasipotential eq uation in which the heavy particle is put on the mass shell [15,16]. According to [14], the relativistic recoil c orrection to the energy of a state ais the sum of a lower-order term ∆ELand a higher-order term ∆ EH: ∆E= ∆EL+ ∆EH, (2.1) ∆EL=1 2M/angb∇acketlefta\|/parenleftBig p2−αZ r/parenleftBig α+(α·r)r r2/parenrightBig ·p/parenrightBig \|a/angb∇acket∇ight, (2.2) ∆EH=i 2πM/integraldisplay∞ −∞dω/angb∇acketlefta\|/parenleftBig D(ω)−[p,VC] ω+i0/parenrightBig ×G(ω+εa)/parenleftBig D(ω) +[p,VC] ω+i0/parenrightBig \|a/angb∇acket∇ight. (2.3) Here \|a/angb∇acket∇ightis the unperturbed state of the Dirac electron in the Coulomb ﬁeldVC(r) =−αZ/r,p=−i∇is the momentum operator, G(ω) = [ω−H(1−i0)]−1is the relativistic Coulomb-Green function, H=α·p+βm+VC, αl(l= 1,2,3) are the Dirac matrices, εais the unperturbed Dirac-Coulomb energy, Dm(ω) =−4παZα lDlm(ω), (2.4) Dlm(ω) is the transverse part of the photon propagator in the Coulo mb gauge. In the coordinate representation it is Dik(ω,r) =−1 4π/braceleftBigexp (i\|ω\|r) rδik+∇i∇k(exp (i\|ω\|r)−1) ω2r/bracerightBig . (2.5) The scalar product is implicit in the equation (2.3). In Refs . [17,18], the formulas (2.1)-(2.3) were rederived by other methods and in [17] it was noticed that ∆ Ecan be written in the following compact form: ∆E=i 2πM/integraldisplay∞ −∞dω/angb∇acketlefta\|[p−D(ω)]G(ω+εa)[p−D(ω)]\|a/angb∇acket∇ight. (2.6) However, the representation (2.1)-(2.3) is more convenien t for practical calculations. The term ∆ ELcan easily be calculated by using the virial relations for th e Dirac equation [19,20]. Such a calculation gives [14] ∆EL=m2−ε2 a 2M. (2.7) This simple formula contains all the recoil corrections wit hin the (αZ)4m2/Mapproximation. The term ∆ EHtaken to the lowest order in αZgives the Salpeter correction [5]. Evaluation of this term t o all orders in αZwill be discussed below. The complete αZ-dependence formulas for the nuclear recoil corrections in highZfew-electron atoms were derived in Ref. [21]. As it follows from these formulas, within the ( αZ)4m2/Mapproximation the nuclear recoil corrections can be obtained by averaging the operator H(L) M=1 2M/summationdisplay s,s′/parenleftBig ps·ps′−αZ rs/parenleftBig αs+(αs·rs)rs r2s/parenrightBig ·ps′/parenrightBig (2.8) with the Dirac wave functions. This operator can also be used for relativistic calculations of the nuclear recoil eﬀect in neutral atoms. An independent derivation of this operato r was done in [22]. The operator (2.8) was employed in [23] to calculate the ( αZ)4m2/Mcorrections to the energy levels of two- and three-electron multicharged ions. III. SIMPLE APPROACH TO THE RECOIL EFFECT IN ATOMS As was shown in [13], to include the relativistic recoil corr ections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron ﬁeld i nteracting with the quantized electromagnetic ﬁeld and with the Coulomb ﬁeld of the nucleus VC, taken in the Coulomb gauge, the following term HM=1 2M/integraldisplay dxψ†(x)(−i∇x)ψ(x)/integraldisplay dyψ†(y)(−i∇y)ψ(y) −eZ M/integraldisplay dxψ†(x)(−i∇x)ψ(x)A(0) +e2Z2 2MA2(0). (3.1) 2This operator acts only on the electron-positron and electr omagnetic ﬁeld variables. The normal ordered form of HM taken in the interaction representation must be added to the interaction Hamiltonian. It gives additional elements to the Feynman rules for the Green function. In the Furry pict ure, in addition to the standard Feynman rules in the energy representation (see [24,13]), the following vertec es and lines appear (we assume that the Coulomb gauge is used) 1.Coulomb contribution . An additional line (”Coulomb-recoil” line) appears to be q q q q q q q q q q q q q q qs sω x yi 2πδkl M/integraltext∞ −∞dω. This line joins two vertices each of which corresponds to AAAA q q q q q q q q q qsxω2 ω1ω3 -K −2πiγ0δ(ω1−ω2−ω3)/integraltext dxpk, where p=−i∇xandk= 1,2,3. 2.One-transverse-photon contribution . An additional vertex on an electron line appears to be AAAA sxω2 ω1ω3 -K −2πiγ0δ(ω1−ω2−ω3)eZ M/integraltextdxpk, The transverse photon line attached to this vertex (at the po intx) is sω x yi 2π/integraltext∞ −∞dωD kl(ω,y). At the point ythis line is to be attached to an usual vertex in which we have −2πieγ0αl2πδ(ω1−ω2−ω3)/integraltextdy, whereαl(l= 1,2,3) are the usual Dirac matrices. 3.Two-transverse-photon contribution . An additional line (”two-transverse-photon-recoil” line ) appears to be s x yω i 2πe2Z2 M/integraltext∞ −∞dωD il(ω,x)Dlk(ω,y). 3This line joins usual vertices (see the previous item). Let as apply this formalism to the case of a single level ain a one-electron atom. To ﬁnd the Coulomb nuclear recoil correction we have to calculate the contribution of t he diagram shown in Fig. 1. A simple calculation of this diagram yields (see Ref. [13] for details) ∆EC=1 Mi 2π/integraldisplay∞ −∞dω/summationdisplay n/angb∇acketlefta\|pi\|n/angb∇acket∇ight/angb∇acketleftn\|pi\|a/angb∇acket∇ight ω−εn(1−i0). (3.2) The one-transverse-photon nuclear recoil correction corr esponds to the diagrams shown in Fig. 2. One easily obtains ∆Etr(1)=4παZ Mi 2π/integraldisplay∞ −∞dω/summationdisplay n/braceleftBigg /angb∇acketlefta\|pi\|n/angb∇acket∇ight/angb∇acketleftn\|αkDik(εa−ω)\|a/angb∇acket∇ight ω−εn(1−i0) +/angb∇acketlefta\|αkDik(εa−ω)\|n/angb∇acket∇ight/angb∇acketleftn\|pi\|a/angb∇acket∇ight ω−εn(1−i0)/bracerightBigg . (3.3) The two-transverse-photon nuclear recoil correction is de ﬁned by the diagram shown in Fig. 3. We ﬁnd ∆Etr(2)=(4παZ)2 Mi 2π/integraldisplay∞ −∞dω/summationdisplay n ×/angb∇acketlefta\|αiDil(εa−ω)\|n/angb∇acket∇ight/angb∇acketleftn\|αkDlk(εa−ω)\|a/angb∇acket∇ight ω−εn(1−i0). (3.4) The sum of the contributions (3.2)-(3.4) is ∆E=1 Mi 2π/integraldisplay∞ −∞dω/angb∇acketlefta\|(pi+ 4παZα lDli(ω)) ×G(ω+εa)(pi+ 4παZα mDmi(ω))\|a/angb∇acket∇ight. (3.5) This exactly coincides with formula (2.6). Consider now a high- Ztwo-electron atom. For simplicity, we will assume that the u nperturbed wave function is a one-determinant function u(x1,x2) =1√ 2/summationdisplay P(−1)PψPa(x1)ψPb(x2). (3.6) The nuclear recoil correction is the sum of the one-electron and two-electron contributions. The one-electron contri- bution is the sum of the expressions (3.5) for the aandbstates. The two-electron contributions are deﬁned by the diagrams shown in Figs. 4-6. A simple calculation of these di agrams yields ∆E(int)=1 M/summationdisplay P(−1)P/angb∇acketleftPa\|pi+ 4παZα lDli(εPa−εa)\|a/angb∇acket∇ight ×/angb∇acketleftPb\|pi+ 4παZα mDmi(εPb−εb)\|b/angb∇acket∇ight. (3.7) The formula (3.7) was ﬁrst derived by the quasipotential met hod in [21]. IV. NUMERICAL RESULTS A. Hydrogenlike atoms According to equations (2.1)-(2.3) the recoil correction i s the sum of the low-order and higher-order terms. The low-order term ∆ ELis given by equation (2.7). The higher order term ∆ EHwas calculated to all orders in αZin [25–27]. The results of these calculations expressed in ter ms of the function P(αZ) deﬁned as 4∆EH=m2 M(αZ)5 πn3P(αZ) (4.1) are presented in Table 1. To the lowest order in αZthe function P(αZ) is given by Salpeter’s expressions: P(1s) S(αZ) =−2 3log (αZ)−8 32.984129 +14 3log 2 +62 9, (4.2) P(2s) S(αZ) =−2 3log (αZ)−8 32.811769 +187 18, (4.3) P(2p1 2) S =P(2p3 2) S =8 30.030017 −7 18. (4.4) Comparing the function P(αZ) from Table 1 with the lowest order contributions (4.2)-(4. 4) shows that for high Z the completeαZ-dependence results diﬀer considerably from Salpeter’s on es. In the case of hydrogen, the diﬀerence ∆ P=P−PSamounts to -0.01616(3), -0.01617(5), and 0.00772 for the 1 s, 2s, and 2p1/2states, respectively. Table 2 displays the relativistic re coil corrections, beyond the Salpeter ones, to the hydrogen energy levels. These values include also the corre sponding correction from the low-order term (2.7) which is calculated by ∆′E(1s) L= 0, (4.5) ∆′E(2s) L= ∆′E(2p1/2) L =(αZ)6 642 [3 +/radicalbig 1−(αZ)2] [1 +/radicalbig 1−(αZ)2]3m2 M. (4.6) The results of Refs. [25,27] which are exact in αZare compared with the related corrections obtained to the lo west order inαZ. In [28,29] it was found that the ( αZ)6log (αZ)m2/Mcorrections cancel each other. The ( αZ)6m2/M correction was derived in [18] for s-states and in [30] for p-states. The ( αZ)7log2(αZ)m2/Mcorrection was recently evaluated in Refs. [31,32]. The uncertainty of the calculat ion based on the expansion in αZis deﬁned by uncalculated terms of order ( αZ)7m2/Mand is expected to be about 1 kHz for the 1 sstate. It follows that the results of the completeαZ-dependence calculations are in a good agreement with the re sults obtained to lowest orders in αZbut are of much higher accuracy. As it follows from Ref. [13], the formulas (2.1)- (2.3) will i ncorporate partially the nuclear size corrections to the recoil eﬀect if VC(r) is taken to be the potential of an extended nucleus. In parti cular, this replacement allows one to account for the nuclear size corrections to the Coulomb part of the recoil eﬀect. In Ref. [33], where the calculations of the recoil eﬀect for extended nuclei were performed, it was f ound that, in the case of hydrogen, the leading relativistic nuclear size correction to the Coulomb low-order part is com parable with the total value of the ( αZ)6m2/Mcorrection but is cancelled by the nuclear size correction to the Coulom b higher-order part. One of the main goals of the calculations of Refs. [25,26,33] was to evaluate the nuclear recoil correction for highly charged ions. In the case of the ground state of hydrogenlike uranium these calculations yield -0.51 eV for the point nucleus case [25] and -0.46 eV for the extended nucleus case [ 33]. This correction is big enough to be included in the current theoretical prediction for the 1s Lamb shift in h ydrogenlike uranium [34] but is small compared with the present experimental uncertainty which amounts to 13 eV [10 ]. However, a much higher precision was obtained in experiments with heavy lithiumlike ions [8,9]. In this conn ection in Refs. [25,26] the nuclear recoil corrections for lithiumlike ions were calculated as well. B. Lithiumlike ions In lithiumlike ions, in addition to the one-electron contri butions, we must evaluate the two-electron contributions. In the case of one electron over the (1 s)2shell the total two-electron contribution to the zeroth ord er in 1/Zis given by the expression ∆Eint=−1 M/summationdisplay εn=ε1s/angb∇acketlefta\|p−D(εa−εn)\|n/angb∇acket∇ight/angb∇acketleftn\|p−D(εa−εn)\|a/angb∇acket∇ight, (4.7) where Dis deﬁned by equation (2.4). Calculation of this term causes no problem [25,26]. For the 2 p1/2and 2p3/2 states, the results of this calculation expressed in terms o f the function Q(αZ) deﬁned by ∆Eint=−29 38m2 M(αZ)2Q(αZ) (4.8) 5are presented in Table 3. For the s-states the two-electron contribution is equal zero. To the lowest orders in αZthe functionQ(αZ) is given by [23] Q(2p1/2) L(αZ) = 1 + (αZ)2/parenleftBig −29 48+ log9 8/parenrightBig , (4.9) Q(2p3/2) L (αZ) = 1 + (αZ)2/parenleftBig −13 48+1 2log27 32/parenrightBig . (4.10) The expressions (4.9)-(4.10) serve as a good approximation for theQ(αZ) function even for very high Z. For lowZ, in addition to the corrections considered here, the Coulom b interelectronic interaction eﬀect on the non- relativistic nuclear recoil correction must be taken into a ccount. It contributes on the level of order (1 /Z)(αZ)2m2/M. To date, the highest precision in experiments with heavy ion s was obtained for the 2 p3/2−2stransition in lithiumlike bismuth [9]. The transition energy measured in this experim ent amounts to (2788 .14±0.04) eV. In [8] the energy of the 2p1/2−2stransition in lithiumlike uranium was measured to be (280 .59±0.10) eV. In both cases the recoil correction amounts to -0.07 eV and, therefore, is comparable with the ex perimental uncertainty. At present, the uncertainty of the theoretical predictions for these transition energi es is deﬁned by uncalculated contributions of second order i n α(see Refs. [34,35]). When calculations of these contributi ons are completed, it will be possible to probe the recoil eﬀect in high-Z few-electron systems. This will provide a un ique possibility for testing the quantum electrodynamics in the region of strong coupling ( αZ∼1) beyond the external ﬁeld approximation since in calculat ions of all other QED corrections in heavy ions the nucleus is considered only as a stationary source of the classical electromagnetic ﬁeld. V. CONCLUSION In this paper the relativistic theory of the recoil eﬀect in a toms is considered. It is shown that the complete αZ- dependence calculation of the recoil correction provides t he highest precision even in the case of hydrogen. The recoil corrections to the energy levels of highly charged ions cont ribute on the level of the present experimental accuracy. It provides good perspectives for testing the quantum electro dynamics in the region of strong coupling ( αZ∼1) beyond the external ﬁeld approximation. ACKNOWLEDGMENTS The author wants to express his thanks to A.N. Artemyev, T. Be ier, G. Plunien, G. Soﬀ, and V.A. Yerokhin for stimulating collaboration. Valuable conversations wi th S.G. Karshenboim, P.J. Mohr, and A.S. Yelkhovsky are gratefully acknowledged. 6[1] G. Breit, G.E. Brown: Phys. Rev. 74, 1278 (1948) [2] K. Bechert, J. Meixner: Ann. Phys., Lpz. 22, 525 (1935) [3] E.E. Salpeter and H.A. Bethe: Phys. Rev. 84, 1232 (1951) [4] A.A. Logunov and A.N. Tavkhelidze: Nuovo Cimento 29, 380 (1963) [5] E.E. Salpeter: Phys. Rev. 87, 328 (1952) [6] H.A. Bethe, E.E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957) [7] F. Biraben and T. W. H¨ ansch: this volume [8] J. Schweppe, A. Belkacem, L. Blumenfeld, N. Claytor, B. F einberg, H. Gould, V.E. Kostroun, L. Levy, S. Misawa, J.R. Mowat, M.H. Prior: Phys. Rev. Lett. 66, 1434 (1991) [9] P. Beiersdorfer, A. Osterheld, J. Scoﬁeld, J.R. Crespo L opez-Urrutia, K. Widmann: Phys. Rev. Lett. 80, 3022 (1998) [10] T. St¨ ohlker: this volume [11] L.N. Labzowsky: In: Papers at 17th All-Union Symposium on Spectroscopy (Astrosovet, Moscow, 1972), Part 2, pp. 89- 93 [12] M.A. Braun: Sov. Phys. JETP 37, 211 (1973) [13] V.M. Shabaev: Phys. Rev. A 57, 59 (1998) [14] V.M. Shabaev: Theor. Math. Phys. 63, 588 (1985); In: Papers at First Soviet-British Symposium o n Spectroscopy of Multicharged Ions (Academy of Sciences, Troitsk, 1986), pp . 238-240 [15] F. Gross: Phys. 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Khriplovich, A.I. Milstein, A.S. Yelkh ovsky: Phys. Lett. A 181, 172 (1993) [30] E.A. Golosov, I.B. Khriplovich, A.I. Milstein, A.S. Ye lkhovsky: JETP 80, 208 (1995) [31] K. Pachucki, S. Karshenboim: Phys. Rev. A 60, 2792 (1999) [32] K. Melnikov, A. Yelkhovsky: Phys. Lett. B 458, 143 (1999) [33] V.M. Shabaev, A.N. Artemyev, T. Beier, G. Plunien, V.A. Yerokhin, G. Soﬀ: Phys. Rev. A 57, 4235 (1998); Phys. Scr. T 80, 493 (1999) [34] V.M. Shabaev, A.N. Artemyev, V.A. Yerokhin: Phys. Scr. T86, 7 (2000) [35] V.A. Yerokhin, A.N. Artemyev, V.M. Shabaev, M.M. Sysak , O.M. Zherebtsov, G. Soﬀ: to be published 7rrrrrrrrrru u FIG. 1. Coulomb nuclear recoil diagram. u u a b FIG. 2. One-transverse-photon nuclear recoil diagrams. u FIG. 3. Two-transverse-photon nuclear recoil diagram. 8r r r r r r r r r ru u FIG. 4. Two-electron Coulomb nuclear recoil diagram. u a bu FIG. 5. Two-electron one-transverse-photon nuclear recoi l diagrams. u FIG. 6. Two-electron two-transverse-photon nuclear recoi l diagram. 9TABLE I. The results of the numerical calculation of the func tionP(αZ) for low-lying states of hydrogenlike atoms. Z 1s 2s 2p1/2 2p3/2 1 5.42990(3) 6.15483(5) -0.30112 -0.3013(4) 5 4.3033(4) 5.0335(2) -0.2692 -0.2724(1) 10 3.7950(1) 4.5383(1) -0.2277 -0.2379 20 3.2940(1) 4.0825 -0.1393 -0.1726 30 3.0437(1) 3.9037 -0.0421 -0.1107 40 2.9268(1) 3.8900 0.0685 -0.0517 50 2.9137(1) 4.0228(1) 0.2000 0.0050 60 3.0061(2) 4.3248(2) 0.3655 0.0597 70 3.2334(4) 4.8656(5) 0.5894 0.1125 80 3.672(1) 5.807(2) 0.9214(2) 0.1638 90 4.519(8) 7.557(9) 1.481(1) 0.2138 100 6.4(1) 11.4(2) 2.63(2) 0.2625 TABLE II. The values of the relativistic recoil correction t o hydrogen energy levels beyond the Salpeter contribution, in kHz. The values given in the second and third rows include the (αZ)6m2/Mcontribution and all the contributions of higher orders in αZ. In the last row the sum of the ( αZ)6m2/Mand (αZ)7log2(αZ)m2/Mcontributions is given. State 1s 2s 2p1/2 To all orders in αZ, Ref. [25] -7.1(9) -0.73(6) 0.59 To all orders in αZ, Ref. [27] -7.16(1) -0.737(3) 0.587 (αZ)6m2/M, Refs. [18,30] -7.4 -0.77 0.58 (αZ)7log2(αZ)m2/M, Refs. [31,32] -0.4 -0.05 The sum of the low-order terms -7.8 -0.82 TABLE III. The results of the numerical calculation of the fu nction Q(αZ) for low-lying states of lithiumlike ions. Z (1s)22p1/2 (1s)22p3/2 10 0.99741 0.99810 20 0.98959 0.99239 30 0.97645 0.98281 40 0.95776 0.96926 50 0.93313 0.95165 60 0.90195 0.92988 70 0.86320 0.90390 80 0.81529 0.87362 90 0.75570 0.83896 100 0.68041 0.79951 10
NUMERICAL STUDIES ON LOCALLY DAMPED STRUCTURES* Z. Li, N.T. Folwell and T.O. Raubenheimer, SLAC, Stanford University, Stanford, CA 94039, USA *Work supported by the DOE, contract DE-AC03-76SF00515.Abstract In the JLC/NLC X-band linear collider, it is essential to reduce the long-range dipole wakefields in the acceleratorstructure to prevent beam break up (BBU) and emittancedegradation. The two methods of reducing the long-rangewakefields are detuning and damping. Detuning reducesthe wakefields rapidly as the dipole modes de-cohere but,with a finite number of modes, the wakefield will growagain as the modes re-cohere. In contrast, dampingsuppresses the wakefields at a longer distance. There aretwo principal damping schemes: synchronous dampingusing HOM manifolds such as that used in the RDDS1structure and local damping similar to that used in theCLIC structure. In a locally damped scheme, one canobtain almost any Q value, however, the damping canhave significant effects on the accelerating mode. In thispaper, we present a medium local-damping scheme wherethe wakefields are controlled to meet the BBUrequirement while minimizing the degradations of thefundamental rf parameters. We will address the loaddesign and pulse heating issues associated with themedium damping scheme. 1 INTRODUCTION The Next Linear Collider (JLC/NLC) [1,2] will provide a luminosity of 1034cm-2sec-1 or higher at a center of mass of 1 TeV. To obtain this luminosity, the NLC linacs mustaccelerate bunch trains with lengths of 266 ns and currentsthat are over half an ampere while preserving the verysmall beam emittances. Dipole wakefields in theaccelerator structures can cause beam emittancedegradation and beam-break-up. It is essential in thecollider design to reduce the long-range dipole wakefieldsto prevent the beam breakup instability (BBU). The twomethods of reducing the long-range wakefields aredetuning and damping of the dipole modes. Detuning canreduce the wakefields rapidly by causing the modes to de-cohere however it is less effective at longer time scalesbecause the finite number of modes can re-cohere. Incontrast, damping can be used to control the wakefields ata longer distance. There are two primary damping schemes for traveling wave structures: the synchronous coupling scheme as usedin the RDDS1[3] design and the local damping schemeused in the CLIC[4] structure design. In the local dampingscheme, each cell is coupled to four radial wave-guidesvia coupling irises. Heavy damping of the dipole modecan be achieved with the proper design of the irisopenings however this can have significant effects on the fundament mode and cause reductions of Q 0 and the shunt impedance. As was demonstrated in the CLIC 15 GHz teststructure [4] where the dipole Q 1 was about 20, the Q0 reduction of the fundamental mode, due to the dampingirises, can be as high as 20%. This reduces the rfefficiency and may lead to unwanted effects such aspulsed heating. For the NLC X-band linacs, we propose a medium damping scheme with a Q 1 of a few hundred as means of solving the rf efficiency and the pulse heating limitations.In this paper, we address some of the design issues forlocally damped structures at X-band. We will also discussthe design of compact loads for the medium dampedstructures to minimize the physical size associated withthe local damping and thereby reduce the cost. 2 LOCALLY DAMPED STRUCTURE The numerical studies in this paper will be based on a structure design with 1500 phase advance per cell. The idea of going to a higher phase advance is to reduce thegroup velocity while keeping a large structure aperture[5];this may reduce the damage due to rf breakdown [6]without increasing the wakefields. The structureparameters are shown in Table 1. Table 1. 1500 X-band structure parameters F = 11.424 GHz Tf = 127 ns φ = 1500/cell /G116 = 0.544 Lcell = 0.0109343 m F1,center = 15.21 GHz Lstruct = 1.4 m Gave = 6.22 MV/m/(MW)1/2 Ncell = 128 aw /c94 = 4.74 mm For high efficiency, the structure will use rounded cell profiles, which have 12% higher Q0 and shunt impedances than that in the standard disk-loaded waveguide structures.In the locally damped structure, the dipole wakefieldgenerated by the beam is coupled out via four higher-order-mode (HOM) couplers, illustrated in Fig. 1. Thecoupler wave-guide propagates only the dipole modefrequencies and is cut-off at the fundamental modefrequency. The coupler is narrower in height than the celllength and is positioned off the symmetry plane of the cellin the axial direction so that modes which haveTE(M)111-like symmetry can also be damped. The damping properties of the cell are determined by the coupling iris to the damping waveguide. To determinethe damping requirements for the structure, numericalsimulations using LIAR[7] were performed to study theBBU as a function of the dipole damping. The wakefields were calculated from the uncoupled frequencies and kick factors for the structure parameters in Table 1. Thissimple model must be updated in the future with awakefield calculated from the coupled mode frequenciesand kick factors however it provides a good estimate ofthe expected performance. The tracking was done for twodifferent beam configurations: 95 bunches having a 2.8 nsbunch spacing and 1.0x10 10e/bunch; 190 bunches having a 1.4 ns bunch spacing and 0.75x1010e/bunch. In both cases, the train length was 266 ns and the initial oscillation was1 σy ≈ 3µm at 10 GeV with βy ≈ 5.7 m in standard NLC linac. In addition, both were tracked using 1 slice perbunch – this eliminates the effect of the initial energyspread and the short-range wakefield which will reducethe BBU. The results are listed in Table 2 and indicatethat a Q 1 of less than 750 is acceptable for the X-band design. With a Q1 of over 1000, the emittance of the beam starts to deteriorate quickly. Table 2. Results of tracking a 1 σy oscillation with (95 bunches of 1.0x1010e / 190 bunches of 0.75x1010e). Q1 ∆ε / ε (axis) [%]∆ε / ε (centroid) [%]Wsum (rms) [V/pC/m/mm]Wsum (std dev) [V/pC/m/mm] 250 47 / 47 0.4 / 0.2 0.48 / 0.26 0.06 / 0.06 500 47 / 47 1.0 / 0.5 0.52 / 0.38 0.08 / 0.09 750 50 / 48 2.7 / 1.5 0.62 / 0.51 0.12 / 0.12 1000 64 / 55 13 / 6.9 0.94 / 0.77 0.28 / 0.24 The damping Q1 of the dipole mode was studied using MAFIA [8]. The HOM coupler waveguide dimensions forthe model are 12 mm by 3 mm. The coupling iris opensthe full height of the HOM waveguide but the width of theiris is adjusted to obtain the required Q 1 values while the thickness was kept at 0.6mm. The Q of the dipole mode isvery sensitive to the iris opening as is shown in Fig. 2. Itis also clear from the plot that very low Q’s can beobtained with the local damping scheme. However, the coupling iris also perturbs the fundamental mode, degrading the fundamental Q 0 and the shunt impedance. The degradation is almost inverselyproportional to the dipole Q 1. The cause of the Q0 reduction is the higher wall-loss in a small region aroundthe iris opening resulted from the current concentrationdue to the iris perturbation. This wall-loss power may cause excessive local pulsed heating, which must beminimized to avoid damage from the thermal stresses. 3 PULSED HEATING The Q0 reduction of the fundamental mode is due to the perturbation of the HOM coupling iris to the fundamentalmode current. The power-loss distribution of the mediumdamped structure was calculated using Omega3P[9]running on NERSC T3E and SP2 supercomputers. Asnap-shot of the wall-loss is shown in Fig. 3. The red spotin a small region around the HOM coupler indicateshigher power density which can cause an excessivetemperature rise during the rf pulse. A temperature rise of120 0C has been shown [10] to be detrimental to the copper surface. Thus, it is important to design the structure tomaintain the heating at a safe level. Knowing the powerdensity from the Omega3P simulation, the temperaturerise can be calculated using the formula given in[10,11,12] where Rs=1/σδ is the surface resistance (0.0279 Ω at 11.424 GHz for copper), Hwall is the magnetic field on the wall (proportional to the wall current), Tpulse is the rf pulse length, ρ is the density of copper (8900 kg/m3), c is the specific heat of copper (385.39 J/kg0C), and k is the thermal conductivity (380 W/0C-m). The temperature rise scaled to a gradient of 70 MV/m is shown in Table 3. Thetemperature rise depends on both the dipole Q 1 and the iris geometry. Smaller iris openings have the least2 pulse Sw a l ltTR Hckπρ∆= Figure 3. Wall loss distributionFigure 1. Locally damped structure cell with four damping waveguides; ¼ of the geometry is shown. Figure 2. Dipole Q 1 and shunt impedance reduction at different iris openings. perturbation on the fundamental mode. Rounding the iris corners can further reduce the ∆T. Table 3. Temperature rise versus iris geometry Iris geometry Dipole Q1 ∆T 3mmx6mm square 150 127 3mmx6mm rounded 430 80 3mmx5mm square 255 71 The design strategy is to damp the dipole mode only as much as needed to prevent BBU thereby minimizing theloss in Q 0 and shunt impedance. This approach leads to a medium damping with a Q1 of a few hundred. We chose a Q1 of 500 for the current medium damped structure design. With such a scheme, it is expected that the rfheating temperature rise will be acceptable. 4 HOM LOAD FOR MDS The loads for the medium damped structure should be simple and compact in geometry. The ideal load would bean adiabatically tapered load which has the advantages ofa broad bandwidth and an insensitivity to the materialvariation. However, the adiabatically tapered load is long;at the X-band dipole frequency, the load would be roughly50 mm. The disk diameter with such loads would be atleast 155 mm which becomes cumbersome as shown inFig. 4. We took an approach of using quasi-adiabatic quasi- resonant tapers for the loads. Using either MAGO ( ε= 9.2(1-j0.13)) or AlN+SiC ( ε=25(1-j0.35)) lossy materials, the load can be designed to be shorter than 20 mm. Withthese loads, the cell can have diameter of 85 mm. Forexample, the cell geometry with the MAGO load is shownin Fig. 4. As anticipated, the compact load design has a finite bandwidth which is shown in Fig. 5. Full cell simulations,which include the cell, the HOM waveguide and the loads,suggested that a reflection better than 0.1 is adequate forthe medium damped structure. The bandwidth of the loadwithin this reflection limit is shown to be about 1.0 GHz.To cover the whole bandwidth of the dipole modes in thestructure, a few different load designs are needed alongstructure. Simulations show the load is also effective at damping the higher band frequencies (Fig. 5). 5 VACUUM AND STRUCTURE BPM Vacuum manifolds can be easily incorporated to the design. A 6mmx10mm manifold, shown in Fig. 4, cuttingthrough the damping waveguide provides additionallongitudinal vacuum conductance with little effect on theload design. Unlike the RDDS type structures, the structure BPM for the locally damped structure is not readily available. Thepresent thought is to utilize probes in the dampingwaveguides to monitor the dipole signals. A few suchprobes are needed to determine the alignment of thestructure. Another option is to couple the vacuummanifold weakly to the rf in the damping waveguide andpick up the signals in the manifold for beam positionanalysis. Further studies are needed on these issues. ACKNOWLEGEMENT The authors would like to thank G. Bowden, R. Ruth, P. Wilson and R. Miller for helpful discussions on pulseheating of accelerator structures. REFERENCES [1] JLC Design Study, KEK Report 97-001, 1997. [2] Zeroth-order design report for the Next Linear Collider, LBNL-PUB-5424, SLAC Report 474,UCRL-ID-124161, 1996. [3] J. Wang, et al, this proceedings, TUA03. [4] CLIC structure, this proceedings, TUA17. [5] Z. Li, T.O. Raubenheimer, in preparation (2000).[6] C. Adolphsen, C. Nantista, these proceedings. TUE02. [7] R. Assmann, et al , LIAR, SLAC/AP-103, April, 1997. [8] MAFIA User ’s Guide, CST GmbH, Darmstadt, Germany. [9] K. Ko, High Performance Computing in Accelerator Physics, this proceedings, WE201. [10]D.P. Pritzkau, Possible High Power Limitations From RF Pulse Heating, SLAC-PUB-8013. [11] Z.D. Farkas and Perry Wilson, Parameter for RDS Accelerating Structure with a/ /G108=0.171 and 0.180, August 2, 1999. [12] G. Bowden, RF Surface Heating Stresses, NLC ME Note, No. 8-96 Rev 0, August 23, 1996.Figure 4. left) adiabatically tapered load, cell OD>155mm; right) compact load using quasi-adiabatic quasi-resonantapproach, fits in cell OD of 85 mm.OD=155 bvacuum manifoldFigure 5. Load reflection at first and third dipoleband frequencies.
arXiv:physics/0009040v1 [physics.space-ph] 13 Sep 2000Mirrormodes: Nonmaxwelliandistributions M. Gedalin and Yu. E. Lyubarsky Ben-Gurion University, Beer-Sheva 84105, Israel M. Balikhin ACSE, University of Shefﬁeld, Mappin Street, Shefﬁeld S1 3J D, United Kingdom R.J. Strangeway and C.T. Russell IGPP/UCLA, 405 Hilgard Ave., Los Angeles, CA 90095-1567 (Dated: November 24, 2013) We perform direct analysis of mirror mode instabilities fro m the general dielectric tensor for several model distributions,inthe longwavelength limit. The growthrat e at the instabilitythreshold depends on thederivative of the distribution for zero parallel energy. The maximum gr owth rate is always ∼k/bardblvT/bardbland the instability is of nonresonant kind. The instability growth rate and its d ependence on the propagation angle depend on the shape of the ion andelectrondistributionfunctions. PACS numbers: I. INTRODUCTION Numerousobservationsof waves in the the Earth magnetoshea th,as well as at other planets have stimulated studies of lon g- wavelength and low-frequencymodes in high βmagnetized plasmas. It has been theoretically shown that th e features of low- frequency waves in hot plasmas differ signiﬁcantly from tho se in cool plasmas, even in the limit corresponding to the usu al magnetohydrodynamicwaves [1]. These ﬁndings have been sub sequently proven by direct comparison with observations [2 ]. However,particularinteresttothelow-frequencymodesin hotplasmasisexplainedbyobservationsofthemirrormodes ,which werefoundinplanetarymagnetosheaths[3,4,5,6],intheso larwind[7],incometarycomas[8,9],andinthewakeofIo[10 ,11]. Thesemodesarenonpropagatingzerofrequencymodes(somet imesconsideredasthekineticcounterpartofthehydrodyna mical entropymode),whichareexpectedtogrowin ananisotropicp lasmawith sufﬁcientlyhigh β⊥/β/bardbl(see,e.g.,Hasegawa[12]). Usual high amplitudes of observed mirror modes show that the y easily achieve the nonlinear regime. At the same time, in several cases low-amplitude magnetic ﬁeld structures with the same properties were observed which may mean that the lin ear and nonlinear mirror mode features are generically related . Yet we do not know so far what makes these modes so ubiquitous andwhatdeterminestheirnonlinearamplitudes. The early explanationofthe mirrorinstability [12] is base d onthe simple pictureof the adiabatic responseof the aniso tropic pressureofmagnetizedparticles. Numericalanalysesofth emirrorinstabilityinbi-Maxwellianplasmas[13,14,15]h aveshown that the maximum of the growth rate occursat k⊥ρi∼1(where ρiis the ion thermal gyroradius),which was interpreted as an indicationonthekineticnatureoftheinstability. At the same time, Southwoodand Kivelson[16] proposeda new e xplanationof the instability mechanismas a resonantone, where the presence of a group of the resonant particles (with v/bardbl= 0) plays the destructive role in the mode excitation: the growth rate of the instability is claimed to be inversely pro portional to the number of the resonant particles. This expl anation wasfurtherreiteratedwithsomemodiﬁcationsbyPantellin iandSchwartz[17]andPokhotelovetal.[18],andusedbyKiv elson and Southwood [19] for the explanation of the nonlinear satu ration mechanism. The analysis of Southwood and Kivelson [1 6] is done in the regime where the phase velocity of the perturba tion is much less than the parallel thermal velocity, in othe r words, γ≪k/bardblvTi/bardbl, and therefore, is directly applied only at the very thresho ld of the instability. At the same time, numerical calculations[15]showthatmostimportanteventsoccurint herange γ∼k/bardblvTi/bardbl,whichisnotcoveredinthepreviousanalytical studies. Thepreviousanalyticalandnumericalconsiderationsofth elinearregimeofthemirrorinstability,eveninthelongwa velength limit,are,asarule,restrictedtotheusageofthebi-Maxwe lliandistribution. At thesametimeparticledistribution sincollision- lessplasmamaysubstantiallydifferfromtheMaxwellian. F orexample,duetotheionheatingmechanismattheshock(see ,e.g., Sckopkeet al. [20]), the magnetosheathion distributionsm ay well deviatefrom the bi-Maxwellian. It is thereforeof in terest to studythedependenceofthe instabilityonthe shapeoftheio nandelectrondistributions. Yet anotherargumentin favor of the analysis of otherdistri butionsfis that thereis no goodanalytical approximations for the dielectric tensor forthe Maxwellianplasma in the range \|ω\|/k/bardblvT/bardbl∼1, which forcedresearchersto considermoreconvenient asymptotics. It is, however, possible to ﬁnd the shapes of th e distribution which allow closed analytical presentation of the dielectrictensorin thewholerangeofphasevelocitiesand makethestudyofthe instabilityphysicsmoretransparent. Inthepresentpaperwestudyindetailthedependenceofthem irrorinstabilityontheshapeoftheionandelectrondistri butions, using model distribution functions which allow direct expl icit analytical calculation of the dielectric tensor. We es tablish the2 genericrelationof themirrorinstability withthe oscilla torymodeswhentheLandaudampingis absentandstudythetra nsition ofdampingmodesto the unstableregime. We also proposeanap proximationwhichis usefulfortheanalyticaltreatmentof the instabilityinthemost importantrange γ∼k/bardblvTi/bardblingeneralcase. The paper is organized as follows. In section II we derive the general dispersion relation in the longwavelength for arbi - trary distribution function. In sections III-IV we apply th e general analysis to three different distributions. In sec tion V we derive the instability conditionand the growth rate at the t hresholdfor arbitrarydistribution. In section VI we devel opa useful approximationfortheanalysisofthebi-Maxwellian-kindd istributionsin theregionofthe maximumgrowthrate. II. DISPERSIONRELATIONINTHE LONGWAVELENGTHLIMIT In what follows we will be interested in the longwavelength l imit where ω≪ΩandkvT≪Ω, while maintaining the phase velocity ﬁnite 0< ω/k < ∞. The last inequality means that the phase velocity does not t end to zero in all propagation angle range but it certainly may vanish for particular set of parameters. For simplicity we assume that both ions and elec trons are Maxwellian in the perpendicular direction, so that ∝angbracketleftv2 ⊥∝angbracketright= 2v2 T⊥and∝angbracketleftv4 ⊥∝angbracketright= 8v4 T⊥. We also denote ∝angbracketleftv2 /bardbl∝angbracketright=v2 T/bardbland β/bardbl,⊥= 2v2 T/bardbl,⊥ω2 p/c2Ω2for each species (subscript istands for ions and subscript efor electrons). Let us introduce the refraction index vector N=kc/ω, such that N= (N⊥,0, N/bardbl) =N(sinθ,0,cosθ). With all this the components of the dispersionmatrix Dij=N2δij−NiNj−ǫijtakethefollowingform(seeAppendixB): D11=N2 /bardbl/parenleftbig 1−1 2(β/bardbl−β⊥)/parenrightbig −1−ω2 pi Ω2 i−ω2 pe Ω2e, (1) D12= 0, (2) D13=−N/bardblN⊥/parenleftbig 1−1 2(β/bardbl−β⊥)/parenrightbig , (3) D22=N2/parenleftbig 1−1 2cos2θ(β/bardbl−β⊥) + sin2θβ⊥ (4) −sin2θ(riβi⊥¯χi+reβe⊥¯χe/parenrightbig −1−ω2 pi Ω2 i−ω2 pe Ω2e, D23=−iω2 pitanθ Ωiω(ri¯χi−re¯χe), (5) D33=N2 ⊥/parenleftbig 1−1 2(β/bardbl−β⊥)/parenrightbig −1−ω2 piβi/bardbl k2 /bardblv2 Ti/bardbl/parenleftbigg¯χi βi/bardbl+¯χe βe/bardbl/parenrightbigg (6) +ω2 pitan2θ Ω2 iri¯χi+ω2 petan2θ Ω2ere¯χe, where β/bardbl=βi/bardbl+βe/bardbl,β⊥=βi⊥+βe⊥,ri=βi⊥/βi/bardbl,re=βe⊥/βe/bardbl,and ¯χ=v2 T/bardbl/integraldisplay (u−v/bardbl)−1∂f ∂v/bardbldv/bardbl. (7) Theintegrationin(7)istakenalongthepathbelowthesingu larityv/bardbl=u. Inwhatfollowsweshallalsoassumethat ω2 pi/Ω2 i≫1 and neglect unity relative to this large parameter (which co rresponds to the assumption vA≪c, where vA=cΩi/ωpiis the Alfven velocity). In what follows we also neglect ω2 pe/Ω2 e= (ω2 pi/Ω2 i)(me/mi). In the above derivation we used ω2 pi/Ωi= −ω2 pe/Ωeinthe quasineutralelectronprotonplasma(thisisnotcorr ectif anyadmixtureofotherchargedparticlesispresent). In the limit ω/Ωi→0(andω/kﬁnite) the dispersion relation D= det ∝bardblDij∝bardbl= 0splits into two ones. One describes the purelytransverseAlfvenwave (the wave electric ﬁeld vecto rin the kB0plane, the wave magneticﬁeld vectorperpendicularto theexternalmagneticﬁeld)with thedispersion ω2=k2v2 Acos2θ/parenleftbig 1−1 2(β/bardbl−β⊥)/parenrightbig . (8) Inthiswavetheabsolutevalueofthemagneticﬁeld doesnotc hange,butthemagneticﬁeldrotates. Theseconddispersionrelationreads Ψ(Z) =/bracketleftbig 2−cos2θ(β/bardbl−β⊥) + 2 sin2θβ⊥−2 sin2θ(riβi⊥¯χi+reβe⊥¯χe) −Z2βi/bardblcos2θ/bracketrightbig/bracketleftbigg¯χi βi/bardbl+¯χe βe/bardbl/bracketrightbigg + sin2θ[ri¯χi−re¯χe]2= 0,(9)3 where we introduced Z=ω/k/bardblvTi/bardblfor convenience( ωis complex,in general, so that Z=W+iG), and ri,e=βi,e⊥/βi,e/bardbl. Eq.(9)describesellipticallypolarizedwaveswithallthr eecomponentsofthewaveelectricﬁeldpresent,sothatinge neralthere existsanonzerocomponentofthewavemagneticﬁeld Bz=N⊥Eyinthedirectionoftheexternalmagneticﬁeld. Thesewaves notonlyrotatethemagneticﬁeld butchangeitsmagnitudeas well. The functions ¯χplay the crucial role in the subsequent analysis. They are de ﬁned by the integral containing the distribution function f(v/bardbl)and cannot be explicitly calculated without particular cho ice of these distributions. It is common to choose f as Maxwellian. In this case ¯χis well-known and tabulated but has good asymptotic expansi ons only for \|Z\| ≪1or\|Z\| ≫1 (for electrons Zshould be substituted by Z(me/mi)(vTi/bardbl/vTe/bardbl)). This actually restricts possible analytical considerat ions of the mirror instability only with the range \|Z\| ≪1. Yet, numerical analyses show that the most important event s occur in the vicinity of \|Z\| ∼1which is unavailable to direct theoretical analysis when Ma xwellian is chosen. On the other hand, there are vague indications that the qualitative features of long waves (instabilities) in the high βmore or less sensibly depend on the lowest moments of the distribution function (provided i t is sufﬁciently “normal”: smooth, no beams, no holes, etc.) . It thereforemakessense toinvestigatethe dispersionrelati onsforasuitablychosenmodeldistributionsothat ¯χcanbe calculated and analyzed in the range \|Z\| ∼1. In what follows we shall use three different distributions for these purposes. The waterbag distribution f= Θ(v2 0−v2 /bardbl)/2v0will be used for study of the behavior of longwavelength mode s and their dependence on the plasma parameters in the absence of Landau damping. Here Θ(x) = 1ifx >0andΘ(x) = 0ifx <0. The hard-bell distribution f= 3(v2 0−v2 /bardbl)Θ(v2 0−v2 /bardbl)/4v3 0willallowtoincludetheLandaudampingeffects,andtheLor entz-likedistribution f= (2v3 0/π)(v2 0+v2 /bardbl)−2removes the upper limit on the particle velocities. The four distributions (including Maxwellian f= (2πv2 T/bardbl)−1/2exp(−v2 /bardbl/2v2 T/bardbl)) mentionedinthispaperareshowninFigure1. III. WATERBAG The waterbag distribution f= Θ(v2 0−v2 /bardbl)/2v0is somewhat peculiar since the Landau damping is absent. The analysis of this distribution allows to establish the generic relation of the instability to nondamping propagating modes. It is ea sy to ﬁnd thatin thiscase ¯χi=1 3−Z2, ¯χe=1 3−Z2µR(10) where µ=me/mi≈1/2000,R=βi/bardbl/βe/bardbl, andv2 T/bardbl=v2 0/3. In the limit Z= 0one has d≡¯χ(Z= 0) = 1 /3. It is worth noting that for the Maxwellian distribution d= 1. In this section we use for electronsthe approximationof th e massless bi-Maxwellian (instead of above waterbag,which is used onl y for ions), for which ¯χe= 1. The resulting dispersion relation (9) is a third orderequationwith respect to Z2with real coefﬁcients. Althoughthis equationcan be analyz eddirectly and even solvedanalytically,graphicalrepresentationofthe root sis muchmoreconvenient. Figure 2 shows the mode with the highest phase velocity (fast mode) for the case when βi/bardbl=βi⊥=βe/bardbl=βe⊥= 0.1and massless bi-Maxwellian electrons. It it worth noting that i ons arenotisotropic since they are Maxwellian in the perpendicular directionandwaterbagintheparalleldirection. Thephase velocityofthefastmodeiswellabove k/bardblvTi/bardblsothatithasnothingto dowiththemirrorinstability. We donotconsiderthismodei ntherestofthepaper. We donotconsidertheAlfvenmodeeith er. The remaining two low-phase velocity modes are shown in Figu re 3 together with ω=k/bardblv0i(solid line). The upper curve is above the resonant region having \|ω\|>\|k/bardblv/bardbl\|for all ions. The lower mode is inside the resonant region and would damp if therewerenonzero ∂f/∂v /bardbl. Figure 4 shows the same two modes but in the case βi/bardbl=βi⊥=βe/bardbl=βe⊥= 0.5. In both cases the naive instability condition K=β⊥/β/bardbl−2−2/β⊥>0 (11) is not fulﬁlled, although in the second case Kis closer to the threshold just because of the larger β⊥. There is no much differencein thebehaviorofthe two modesforthese two case s, excepta little strongerdecreaseof the phasevelocities towards theperpendicularpropagationregimein thehigher β⊥case. Figure 5 shows the behavior of the two modes in the anisotropi c case βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5(so that K= 1), and bi-Maxwellian electrons. The lower mode now remains pur ely propagating mode for smaller angles (diamonds) but turn s into an aperiodic instability for larger angles of propagat ion(stars). The obviousconclusion from Figure 5 is that the unstable modehasitspropagationcounterpartforthesmallerangles ofpropagation. Therelativegrowthrate G=γ/k/bardblvTi/bardbl∼1islarge inthe wholerangeofinstability,sothatthe approximation G≪1[16]is notapplicable. It is of interest to compare this case with the massless water bag electrons ¯χe= 1/3. The corresponding curves in Figure 6 showthatthereisnoinstabilityinthiscasedespitethe fac tthatthethreshold(11)isexceeded. Thus, the analysis of the waterbag distribution already sho ws that (a) there is, in general, the propagatingcounterpar t of the mirror instability if Landau damping is absent, (b) the inst ability threshold and growth rate are sensitive to the detai ls of the4 distribution and not only to the second moment, and (c) the in stability is aperiodic, that is, in the unstable range W= 0and G >0. Itcanbeshownthatthelast featureisgenerallyvalidunle ssthedistributionfunctionisverypeculiar(seeAppendix C). IV. HARD-BELLANDLORENTZDISTRIBUTIONS ThewaterbagdistributiondoesnotallowLandaudampingsin ce∂f/∂v /bardbl= 0everywhere. Inordertogetridofthisrestriction we consider the hard-belldistribution f= 3(v2 0−v2 /bardbl)Θ(v2 0−v2 /bardbl)/4v3 0, which has nonzeroderivativebut is is compact ( f= 0 for\|v/bardbl\|> v0. Inthiscase ¯χi=3 5/bracketleftBigg 1 +Z 4√ 5ln(√ 5−W)2+G2 (√ 5 +W)2+G2 +iZ 2√ 5/parenleftBigg arctan√ 5−W G+ arctan√ 5 +W G/parenrightBigg/bracketrightBigg ,(12) where Z=W+iG,WandGbeingreal, G >0,andv2 0= 5v2 T/bardbl. Thecorresponding d= ¯χ(Z= 0) = 3 /5. Thecorresponding expressionfor ¯χeisobtainedfrom(12)bysubstitution Z→Z√µR. Inordertoanalyzenon-compactdistributionstooweshallc onsidertheLorentzdistribution f= (2v3 0/π)(v2 0+v2 /bardbl)−2. Inthis case ¯χi=16iZ (1 +Z2)3+3i i−Z−2Z (i−Z)3+3iZ (i−Z)2, (13) withv2 0=v2 T/bardblandd= 3. Again, ¯χeisobtainedbysubstitution Z→Z√µR. We shall also compare the results for these distributions wi th the bi-Maxwellian. In this case there is no compact analyt ical expressionfor χandwe usedirectnumericalcalculation. In what followswe are interestedonlyin the unstable region . The subparticlemodeis expectedto be stronglydampedin th e propagationrange. The“superparticle”modeisnotdampedi nthehard-bellcaseandalmostnotdampedintheLorentzcase . As the ﬁrst set of parameters for the unstable regime we choos eβi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, and massless bi- Maxwellian electrons ¯χe= 1. Figure 7 shows the growth rates for the three distributions . The highest growth rate is for the Lorentzian,the lowest is forthe waterbag. Figure8 showsth e same growthratesas in Figure 7 but normalizedon kvTi/bardblwhich allowstocomparegrowthratesofthemodeswiththesamewave number kanddifferentanglesofpropagation. Itisseenthatthe maximumgrowthratesisachievedapproximatelyatthesamea ngleofpropagation ≈60◦foralldistributions,butthethreshold anglemovestowardsmorequasiparallelregimesfordistrib utionswithstrongertails(MaxwellianandLorentzian). Figure 9 shows the dependence of the growth rate on β⊥whenK= 1andβi⊥/βe/bardbl=βi/bardbl/βe/bardbl= 1remain constant. Both curves correspond to the waterbag ions and massless bi-Maxw ellian electrons. Diamonds stand for the same parameters as in Figure5,crossescorrespondto βi⊥= 1andβi/bardbl= 0.25. Theinstabilityisstrongerforhigher β⊥. In the previousanalysiswe always used the approximationof massless bi-Maxwelliandistributioncorrespondingto χe= 1. Figures 10 and 11 show the growth rate of the instability when the electron distributions are chosen in the same form as the ion distributions. One can see that the waterbag distributi ons become stable, while the growth rate in the case of Lorent zian drasticallyincreases. Theratioofthemaximumgrowthrate sshowninFigures8and11roughlycorrespondsto de=χe(Z= 0) whichshowsthatthemaximumgrowthratesigniﬁcantlyonele ctrons(see sectionsV andVI). Forothercombinationsofionandelectrondistributionsth eratiosmaybeevengreaterasisseeninFigure12,wherediam onds correspond to waterbag ions and massless bi-Maxwellian ele ctrons, while circles correspond to waterbag ions and Loren tz electrons. The βparametersarethesame forbothcases. V. NEAR THETHRESHOLD It is possible to obtain general results just above the thres hold of the instability, where Z=iG→+0. Forf=f(v2)it is easyto ﬁnd χ=/integraldisplay1 iG−v/bardbl∂f ∂v/bardbldv/bardbl =−/integraldisplayv/bardbl v2 /bardbl+G2∂f ∂v/bardbldv/bardbl=−/integraldisplayd f dEdv/bardbl+G/integraldisplayG v2 /bardbl+G2d f dEdv/bardbl =−/integraldisplayd f dEdv/bardbl+πGd f dE\|v/bardbl=0=d−κG,(14)5 where E=v2 /bardbl/2is the energy(onthe unit mass). Substituting this into (9) a ndneglectingall termsof the order Z2and higher, onehas G=−A/B, (15) A= [2−cos2θ(β/bardbl−β⊥) + 2 sin2θβ⊥−2 sin2θ(riβi⊥di +reβe⊥de)]/parenleftbiggdi βi/bardbl+de βe/bardbl/parenrightbigg + sin2θ(ridi−rede)2, (16) B=−κi βi/bardbl[2−cos2θ(β/bardbl−β⊥) + 2 sin2θβ⊥−2 sin2θ(riβi⊥di +reβe⊥de)] + 2 sin2θriβi⊥κi/parenleftbiggdi βi/bardbl+de βe/bardbl/parenrightbigg −2 sin2θκi(ridi−rede), (17) wherewe neglected κe∼κi/radicalbig me/mi. The instability thresholdfor given θis foundfromthe condition G= 0, that is, A= 0, whichgives 2 +β⊥−β/bardbl+ sin2θ[β/bardbl+β⊥−2(riβi⊥di+reβe⊥de) + (ridi−rede)2/(di/βi/bardbl+de/βe/bardbl)] = 0 .(18) Since0≤sin2θ≤1,the globalinstabilitycriterionreads(intheassumption that2 +β⊥> β/bardbl): 2(riβi⊥di+reβe⊥de)−2−2β⊥−(ridi−rede)2/parenleftbiggdi βi/bardbl+de βe/bardbl/parenrightbigg−1 >0. (19) It is instructive to consider several simple cases. We can ne glect completely the electron contribution by putting de= 0, whichgivesthe instabilitycriterionintheform dβ⊥ β/bardbl>2(1 +1 β⊥), (20) andforthebi-Maxwelliandistribution, d= 1,reducestothe naivemirrorinstabilitycriterion. Ontheotherhand,when re=ri=β⊥/β/bardblandde=di=d, onegets dβ⊥ β/bardbl>1 +1 β⊥. (21) This condition is harder for more compact distributions ( d= 1/3for waterbag and d= 3/5for hard-bell) and softer for distributions with long tails ( d= 1for Maxwellian and d= 3for Lorentzian). The global instability condition (19) can be writtenina moresymmetricformasfollows: (ridi+rede)2+2(β2 i⊥+β2 e⊥)dedi βi/bardblβe/bardbl −2(1 + β⊥)/parenleftbiggde βe/bardbl+di βi/bardbl/parenrightbigg >0(22) whichemphasizesthesymmetricroleofionsandelectronsin theinstabilityonset(cf. Pokhotelovetal. [18]). Indeed, near the threshold γ/k/bardblvT/bardbl≪1and the response of both electrons and ions is adiabatic, tha t is, their inertia does not play anyrole. In these circumstancesthe mass of the part icle is of notimportance. Their role in the response to the pa rallel electric ﬁeld is, however, antisymmetric because of the dif ferent signs of the charge: the adiabatic response is obtain ed from eEz−(1/n)(dp/dz ) =eEz−ik/bardblp/n= 0. The parallel response plays the crucial role in the instabi lity development. As is knownthe instability occursbecause of the breakdownof the local frozen-inconditionand efﬁcient drag of particles ou t of the ﬁeld enhancementinto theﬁeld depletionregion[16, 17, 18] . Thus, whenthemagneticﬁeld is perturbed, Bz=B0+δBz, the perturbationofthe densityofthespecies sis δns n0s=δBz B0+δn(ext) s n0s, (23) where δn(ext) sis due to the motion along the ﬁeld lines. In the adiabatic reg imeγ/k/bardblvT/bardbl≪1this change can be consideredas a quasistatic responseto the effectivepotential φeff=φ+µsδBz/qs, where φis theelectrostatic potential, µs=∝angbracketleftv2 ⊥∝angbracketrights/2B0is6 the averagemagneticmoment, and qsis the chargeof the species. The density response to this eff ectivepotential can be found fromthereducedVlasovequation ∂fs ∂t+v/bardbl∂fs ∂z=qs∂φeff ∂z∂fs ∂v/bardbl, (24) whichfor ∂/∂t=γand∂/∂z=ik/bardblgives δn(ext) s n0s=φeff/integraldisplayik/bardbl γ+ik/bardblv/bardbl∂fs ∂v/bardbldv/bardbl. (25) Itis easytosee thatin theadiabaticregimenearthethresho ldoftheinstability, γ→0,thisexpressionreducestothefollowing δn(ext) s n0s=−qsφeff 4πn0sq2sr2 D, (26) where rDis the Debye length calculated with the parallel distributi on function. It is easy to see that r2 D=v2 T/bardbl/ω2 pd, where d= ¯χ(Z= 0). The electrostatic potential φcan be excluded using the quasineutrality condition δne=δni, which eventually gives δn n=δBz B0/bracketleftbigg 1−Te⊥+Ti⊥ 4πe2n0(r2 De+r2 Di)/bracketrightbigg , (27) wherewehavetakenintoaccountthat µ=T⊥/B0. Eq.(27)showsthatsmallerDebyelengths rD(larger d)resultinthestronger drag of the particles into the weak ﬁeld region, that reducin g the kinetic pressure response to the magnetic ﬁeld enhance ment and supporting instability. Therefore, stronger Debye scr eening (larger d) would lower the instability threshold, in agreement withthe foundfromrigorouscalculations. From(15)–(17)itiseasilyseenthatthegrowthrateisinver selyproportionalto κi=−π(d f/dE)\|v/bardbl=0,andnottothenumber of particles with v/bardbl= 0(cf. Southwood and Kivelson [16]). The latter is correct for the bi-Maxwellian distribution since (d f/dE)∝fin this case. For other distributions this relation may well be wrong. For example, for the waterbag distribution (d f/dE)\|v/bardbl=0= 0and higher order terms should be retained to investigate the behavior near the threshold. It is easy to see from(14) that in this case ¯χ=d−αG2, where α=−/integraltext v−2 /bardbl(d f/dE)dv/bardblis well-deﬁned. The dispersionrelation (9) becomes than a ﬁrst orderequationfor G2, which hasone positivesolution near the threshold. It is cl ear that in this case the growthrate isdeterminedbythewholedistributionandnotonlybythebe haviorin v/bardbl= 0. VI. HYDRODYNAMICALREGIME The previous analysis shows that maximum Zis always of the order of unity or larger, which means that ion s no longer respondadiabaticallytothemagneticﬁeldenhancementsan dtheirinertiabeginstoplayanimportantrole. Thisalsome ansthat it is thermal particles of the ion distribution body with v∼vTi/bardblwhich are mainly responsible for the instability developme nt andnotthegroupofresonantparticleswith v/bardbl= 0. Figure10showsthatforsomedistributionstheinstabilit y maybe veryfast so thattheelectroninertiashouldbetakenintoaccount. The previous analysis gives a clue to the treatment of the ins tability in the range of maximum growth rates, where G/greaterorsimilar1. Let us assume that the distribution function is such that v/bardblf(v/bardbl)has a sharp maximum at some vm∼vT/bardbl. An example of a distributionof thiskindis theMaxwellian fi= (1/√ 2πvTi/bardbl)exp(−v2 /bardbl/2v2 Ti/bardbl)forwhichtherewas nogoodapproximationfor ¯χintherange \|Z\| ∼1so far. Fortheaperiodicmirrorinstabilitywith Z=iG,G >0,onehas ¯χ=/integraldisplay1 iG−v/bardbl∂f ∂v/bardbldv/bardbl=−/integraldisplay1 G2+v2 /bardblv/bardbl∂f ∂v/bardbldv/bardbl. (28) Forvm∼1/lessorsimilarG(vmis normalized on vT/bardbl) the function (G2+v2 /bardbl)−1varies slowly in the vicinity of the maximum of v/bardbl(∂f/∂v /bardbl),so thatonemayapproximate ¯χ=−1 G2+v2m/integraldisplay v/bardbl∂f ∂v/bardbldv/bardbl=1 G2+v2m. (29) Figure 13 shows the comparison of the numerically found ¯χfor the Maxwellian distribution ( v2 m= 2) andZ=iG, G > 0 with the approximation (29). The approximation proves to be very good for G≥1and is only by the factor 2 smaller at7 G→0. Figure 14 shows similar comparison for Lorentzian. Now the maximum growth rate can be obtained by substituting ¯χi= 1/(G2+v2 mi)in (9). If Gis expectedto be high, so that G2Rµ∼1, as it occursfor the Lorentzian e−idistributionsin Figure10,theelectroninertiashouldbealsotakenintoacc ountbysubstituting ¯χe= 1/(G2Rµ+v2 me). If,however,thegrowth ratesare relativelymodest(asin othercases studiedin the presentpaper),the electronsstill respondadiabatically and¯χe=de. Inthelastcase(9)turnsintoathirdorderequationwithres pectto G2. Findingthemaximumgrowthratefromthisequationisa technicalproblem. Weshallstopforawhileatthephysicals enseoftheaboveapproximation. Thedependenceofthemaxim um growth rate on vmindicates that the particles with high velocities v∼vT/bardblare taking part in the process. This is related to the dynamic redistribution (closely related to the dynamic Debye screening): if the potential changes quickly the low v elocity particlesdonot haveenoughtime to changetheir positionan d leave the ﬁeld enhancements. Thisredistributionis descr ibedby thesameEq.(25)butnow γ/k/bardblvT/bardbl∼1. Highvelocityparticlescanleavetheseregionsandreduce thekineticpressureresponse but their contributionrapidlydecreaseswith the velocity since their numberdecreases. The increaseof redistributi onefﬁciency and the decrease of the number of screeners with the velocity increase ﬁnds its manifestation in that the main contributi on belongsto the particlesat the maximumof v/bardblf(v/bardbl). Sincethe redistributionplaysthe destabilizingrole,it canbe expectedthat the smaller is vmthe higher is the growth rate. This can be seen already from Fi gure 8 where the growth rate for Lorentzian ions,v2 m= 0.5, is larger than the growth rate for the Maxwellian, v2 m= 2(with the same massless Maxwellian electrons). Figure15 showsthe comparisonof the growthratesobtainedw ith the proposedapproximationfor several v2 m= 2(diamonds), 1.5 (crosses), 1 (triangles), 0.5 (circles), and massless M axwellian electrons. The parameters chosen are βi⊥=βe⊥= 0.5, βi/bardbl=βe/bardbl. Asexpectedthedecreaseof vmresultsin theincreaseofthe maximumgrowthrate. Finally, Figure 16 shows the comparison of the growth rates o btained directly and with the above approximation for Maxwellian (diamondsand crosses) and Lorentzian (triangl es and circles), for the same parameter set. The agreement is quite satisfactory. VII. CONCLUSIONS We havederivedthemost generaldispersionrelationforlon gwavelengthmodesin hotplasmas. We havederivedthe genera l mirror instability condition for arbitrary ion and electro n distributions and growth rate of the instability near the t hreshold. The instabilitythresholddependsnotonly onthe plasma spe ciesβbut also onanotherintegralcharacteristicof the distribu tion function d=/integraltext v−1 /bardbl(∂f/∂v /bardbl)dv/bardblforbothspecies. Larger dcorrespondstosmallerDebyelength. SmallerDebyelength, inturn, corresponds to stronger response of the density to the pertu rbations of the potential, which allows stronger density de pletions in the regions of the magnetic ﬁeld enhancements. Therefore , the kinetic pressure response to the magnetic pressure bui ldup weakens. Hence,thelargeris dthelowerisinstabilitythreshold. Thenear-the-threshol dgrowthrateisinverselyproportionalto ∂f/∂E,where E=v2 /bardbl/2istheparallelenergy. The mirror instability is always aperiodic and (γ/k/bardblvTi/bardbl)max∼1(and sometimes substantially greater). Maximum growth rates are normallydeterminedby vmisuch that v/bardbl∂fi/∂v/bardblhas a sharp maximumin v/bardbl=vmi, andde(if the instability is very strong vmetakestheplaceof de). Thisisrelatedtothedynamicredistributioninwhichthe thermalparticlesparticipate. Growth rates are higher for distributions with tails and lower for c ompact distributions (those, for which f= 0if\|v/bardbl\|> v0, where v0 is some upper limit). For noncompact distributions the maxi mum growth rate is larger for smaller vm, which corresponds to the weaker dynamic screening of the parallel electric ﬁeld. For the distributions analyzed in this paper the behavior of dand vmcorrelates( dincreaseswhen vmdecreases)since all these aresingle-parameterdistribut ions. Formoregeneraldistributions the behavior of deandvmmay be uncorrelated. It is also worth noting that it not, in ge neral, any speciﬁc group of particles which are responsibleforthe instabilitydevelopment. Com pare,forexample,two similar distributions(velocitynor malizedon the thermal velocity vT/bardbl):f1= (2/π)(1 + v2 /bardbl)−2withd= 3andv2 m= 0.5, andf2= (√ 2/π)(1 + v4 /bardbl)−1withd= 1and v2 m= 1. While the behaviorof the two is similar for v/bardbl= 0andv/bardbl→ ∞(the onlydifferenceis the factor√ 2), the ﬁrst one is expectedto bemoreunstablebecauseofthethreetimesstron gerDebyescreening. At the same timethe behaviorofthe seco nd distributionnearthethresholdshouldbeclosetothatofth eMaxwellian, d= 1,despitetheverydifferentsuprathermaltailsand (d f/dE)\|v/bardbl=0. We have also proposed a useful approximation for the dielect ric function in the range G/k/bardblvTi/bardbl/greaterorsimilar1for distributions with sharpmaximaof v/bardbl(∂f/∂v /bardbl)(Maxwellianas oneof such distributions). Thisapproximat ionprovesto be quite satisfactoryfor Maxwelliantypedistributionsandallowsto studyanalytic allytheinstabilitybehaviorin themaximumgrowthrateran ge. Acknowledgments FiguresaremadeusingMatlab.8 APPENDIXA: GENERALEXPRESSIONS We start withthe generalexpressionforthedielectrictens orin thefollowingform: ǫij=δij+/summationdisplay λij, (A1) wherethesummationisonthe speciesand λij=−ω2 p ω2δij+ηij. (A2) Theexpressionfor ηijis well-known(see,e.g.,Hasegawa[12]): ηij=−/summationdisplay nω2 p ω2/integraldisplay v⊥dv⊥dv/bardbl/parenleftbiggnΩ v⊥∂f0 ∂v⊥+k/bardbl∂f0 ∂v/bardbl/parenrightbiggΠij nΩ−ζ, (A3) where ζ=ω−k/bardblv/bardbl,and Πij= (n2Ω2/k2 ⊥)J2 ni(v⊥nΩ/k⊥)JnJ′ n(v/bardblnΩ/k⊥)J2 n −i(v⊥nΩ/k⊥)JnJ′ n v2 ⊥J′ n2−iv⊥v/bardblJnJ′ n (v/bardblnΩ/k⊥)J2 n iv⊥v/bardblJnJ′ n v2 /bardblJ2 n . (A4) HereJn=Jn(x),x=k⊥ρ=k⊥v⊥/Ω,andJ′ n=dJn/dx. Fortheanalysisin thelow-frequencyrange ω/Ω≪1let uswrite ηij=η(0) ij+η(n/negationslash=0) ij, (A5) andexpand 1 nΩ−ζ=1 nΩ/parenleftbigg 1 +ζ nΩ+ζ2 n2Ω2+···/parenrightbigg . Letalso f0=f1(v⊥)f2(v2 /bardbl), anddenote ∝angbracketleft. . .∝angbracketright=/integraltext (. . .)fdvj, where j=⊥,∝bardbl. Onehas η(0) ij=ω2 p ω2/integraldisplay v⊥dv⊥dv/bardblk/bardbl ζ∂f0 ∂v/bardbl× 0 0 0v2 ⊥J′ 02−iv⊥v/bardblJ0J′ 0 0iv⊥v/bardblJ0J′ 0 v2 /bardblJ2 0  (A6) and η(n/negationslash=0) ij =−/summationdisplayω2 p ω2/integraldisplay v⊥dv⊥dv/bardbl/parenleftbigg1 v⊥∂f0 ∂v⊥+k/bardbl nΩ∂f0 ∂v/bardbl/parenrightbigg ×/parenleftbigg 1 +ζ nΩ+ζ2 n2Ω2/parenrightbigg Πij.(A7) Now,upto Ω−2oneobtains η(n/negationslash=0) 11=−/summationdisplay nω2 p ω2/bracketleftBigg ∝angbracketleftJ2 n∂ ∂v⊥∝angbracketrightn2Ω2+ω2+k2 /bardbl∝angbracketleftv2 /bardbl∝angbracketright k2 ⊥+k2 /bardbl k2 ⊥∝angbracketleftv⊥J2 n∝angbracketright/bracketrightBigg , (A8) η(n/negationslash=0) 12=−i/summationdisplay nω2 p ω2ω k⊥∝angbracketleftv⊥JnJ′ n∂ ∂v⊥∝angbracketright, (A9) η(n/negationslash=0) 13=/summationdisplay nω2 p ω2k/bardbl k⊥/bracketleftbigg ∝angbracketleftv⊥J2 n∝angbracketright+∝angbracketleftv2 /bardbl∝angbracketright∝angbracketleftJ2 n∂∝angbracketright ∂v⊥/bracketrightbigg , (A10) η(n/negationslash=0) 22=−/summationdisplay nω2 p ω2/bracketleftBigg ∝angbracketleftv2 ⊥J′ n2∂ ∂v⊥∝angbracketright/parenleftBigg 1 +ω2+k2 /bardbl∝angbracketleftv2 /bardbl∝angbracketright n2Ω2/parenrightBigg +k2 /bardbl n2Ω2∝angbracketleftv3 ⊥J′ n2∝angbracketright/bracketrightBigg , (A11)9 η(n/negationslash=0) 23=−i/summationdisplay nω2 pk/bardbl ωn2Ω2/bracketleftbigg ∝angbracketleftv2 ⊥JnJ′ n∝angbracketright+ 2∝angbracketleftv2 /bardbl∝angbracketright∝angbracketleftv⊥JnJ′ n∂ ∂v⊥∝angbracketright/bracketrightbigg , (A12) η(n/negationslash=0) 33=−/summationdisplay nω2 p ω2∝angbracketleftv2 /bardbl∝angbracketright∝angbracketleftJ2 n∂ ∂v⊥∝angbracketright, (A13) and η(0) 22=ω2 p ω2k/bardbl∝angbracketleftv3 ⊥J′ 02∝angbracketrightχ, (A14) η(0) 23=−iω2 p ω∝angbracketleftv2 ⊥J0J′ 0∝angbracketrightχ, (A15) η(0) 33=ω2 p ω2∝angbracketleftv⊥J2 0∝angbracketright/parenleftbigg 1 +ω2 k/bardblχ/parenrightbigg . (A16) where χ=∝angbracketleft1 ζ∂ ∂v/bardbl∝angbracketright. (A17) UsinginEqs.(A8)-(A13)thefollowingsummationrules /summationdisplay n/negationslash=0J2 n= 1−J2 0, (A18) /summationdisplay n/negationslash=0n2J2 n=x2 2=k2 ⊥v2 ⊥ 2Ω2, (A19) /summationdisplay n/negationslash=0JnJ′ n=−J0J′ 0, (A20) /summationdisplay n/negationslash=0J′ n2=1 2−J′ 02, (A21) one obtains eventually the following general expression fo rλijin the limit of ω, k/bardblv/bardbl≪Ωwhen expanded up to the second orderin ζ/Ω: λ11=ω2 p k2 ⊥/parenleftBigg 1 +k2 /bardbl∝angbracketleftv2 /bardbl∝angbracketright ω2/parenrightBigg ∝angbracketleftJ2 0∂ ∂v⊥∝angbracketright −ω2 p ω2k2 /bardbl k2 ⊥∝angbracketleftv⊥(1−J2 0)∝angbracketright, (A22) λ12=iω2 p ωk⊥∝angbracketleftv⊥J0J′ 0∂ ∂v⊥∝angbracketright, (A23) λ13=ω2 p ω2k/bardbl k⊥/bracketleftbigg ∝angbracketleftv⊥(1−J2 0)∝angbracketright − ∝angbracketleftv2 /bardbl∝angbracketright∝angbracketleftJ2 0∂ ∂v⊥∝angbracketright/bracketrightbigg , (A24) λ22=ω2 p ω2∝angbracketleftv2 ⊥J′ 02∂ ∂v⊥∝angbracketright −ω2 p ω2/summationdisplay n/bracketleftBigg ω2+k2 /bardbl∝angbracketleftv2 /bardbl∝angbracketright n2Ω2∝angbracketleftv2 ⊥J′ n2∂ ∂v⊥∝angbracketright (A25) +k2 /bardbl n2Ω2∝angbracketleftv3 ⊥J′ n2∝angbracketright/bracketrightBigg +ω2 p ω2k/bardbl∝angbracketleftv3 ⊥J′ 02∝angbracketrightχ, λ23=−i/summationdisplay nω2 pk/bardbl ωn2Ω2/bracketleftbigg ∝angbracketleftv2 ⊥JnJ′ n∝angbracketright+ 2∝angbracketleftv2 /bardbl∝angbracketright∝angbracketleftv⊥JnJ′ n∂ ∂v⊥∝angbracketright/bracketrightbigg −iω2 p ω∝angbracketleftv2 ⊥J0J′ 0∝angbracketrightχ, (A26) λ33=−ω2 p ω2/bracketleftbigg 1− ∝angbracketleftv2 /bardbl∝angbracketright∝angbracketleftJ2 0∂ ∂v⊥∝angbracketright/bracketrightbigg +ω2 p ω2∝angbracketleftv⊥J2 0∝angbracketright/parenleftbigg 1 +ω2 k/bardblχ/parenrightbigg . (A27) It ispossibletogetridoftheseriesin(A25)–(A27)using[2 1] Φ =/summationdisplay n/negationslash=0J2 n n2=4 π/integraldisplayπ/2 0t2J0(2xcost)dt−π2 6J2 0, (A28)10 Ψ =/summationdisplay n/negationslash=0JnJ′ n n2=1 2dΦ dx, (A29) Υ =/summationdisplay n/negationslash=0(J′ n)2 n2=1 2/bracketleftbiggd2Φ dx2+1 xdΦ dx+ 2Φ−2 x(1−J2 0)/bracketrightbigg . (A30) Thismaybeusefulforcalculationsintheregime k⊥v⊥/Ω∼1. Thegeneraldispersionrelationis obtainedfromthe determ inantdet\|D\|= 0,where Dij=N2δij−NiNj−ǫij,that is, D11=N2 /bardbl−1−/summationdisplay λ11, (A31) D12=−/summationdisplay λ12, (A32) D13=−N/bardblN⊥−/summationdisplay λ13, (A33) D22=N2−1−/summationdisplay λ22, (A34) D23=−/summationdisplay λ23, (A35) D33=N2 ⊥−1−/summationdisplay λ33. (A36) Thepolarizationshouldbe foundfromtheequations DijEj= 0. (A37) Eq. (A37)providestheratio of the electric ﬁeld components . In orderto translate that into the magneticpolarizationo nehas to usethe relation B=k×E/ω. Inordertoﬁndthedensityperturbationsonehastousethe c urrentconservationasfollows δρ=k·j/ω, (A38) where ji=−iω 4πλijEj, (A39) so thatonehaseventually δρ=−i 4πkiλijEj. (A40) Furthersimpliﬁcationsarepossibleinthe longwavelength limit. APPENDIXB: LONGWAVELENGTHAPPROXIMATION In this appendix we provide general expressions for the diel ectric tensor in the longwavelength limit k⊥v⊥/Ω≪1, where J±1=±k⊥v⊥/2Ω,J0= 1−k2 ⊥v2 ⊥/2Ω2, andhigherorderBessel functionsmaybeneglected. Inthis limit onehas λ11=ω2 p Ω2+1 2N2 /bardbl(β/bardbl−β⊥), (B1) λ12=iω2 p ωΩ, (B2) λ13=−1 2N/bardblN⊥(β/bardbl−β⊥), (B3) λ22=ω2 p Ω2+1 2N2 /bardbl(β/bardbl−β⊥) (B4) −N2 ⊥β⊥+N2 ⊥β⊥ 4∝angbracketleftv4 ⊥∝angbracketright ∝angbracketleftv2 ⊥∝angbracketrightχ], λ23=iβ⊥tanθΩ 2ωc2χ, (B5) λ33=1 2N2 ⊥(β/bardbl−β⊥) + (ω2 p k2 /bardbl−β⊥tan2θc2 2)χ, (B6)11 where N=kc/ω,N⊥=k⊥c/ω=Nsinθ,N/bardbl=lk/bardblc/ω=Ncosθ,β/bardbl= 2ω2 p∝angbracketleftv2 /bardbl∝angbracketright/c2Ω2,β⊥=ω2 p∝angbracketleftv2 ⊥∝angbracketright/c2Ω2, and∝angbracketleft. . .∝angbracketright denotes usual averaging over the distribution. Here also ζ=u−v/bardbl, where u=ω/k/bardbl. The last term in (B6) is given for completeness. In the limit used in this paper, ω/Ω→0andω/kﬁnite, it should be neglected. Throughout the paper we also assume ωpi≫Ωi. APPENDIXC:APERIODICNATUREOFTHE MIRRORINSTABILITY In order to show that the mirror instability is aperiodic we a nalyze the behavior of the roots of (9) when the βparameters are changed. In the waterbag case the transition from the sta ble to the unstable regimes occurs when W= 0, G= 0and for the mode whose phase velocity is less than the highest partic le velocity, ReZ < v /bardbl,max(“subparticle” mode), that is, in the resonant region. In the general case, where Landau damping i s nonzero, in the resonant range every propagatingwave havi ng W∝negationslash= 0has also nonzero damping rate G < 0(we assume that there are no other kinetic instabilities in t he mirror-stable region). By continuously changing the plasma parameters (e .g., the anisotropy ratio β⊥/β/bardbl) we can bring the system into the unstable regime. Assuming continuousdependence of WandGon the plasma parameters we see that it is impossible for the “subparticle” mode with W∝negationslash= 0to transform into the unstable mode, since G <0and cannot be made positive continuously. Thus,theonlywayto dothat istogothrough W= 0,G= 0. Let us nowconsiderthe vicinityof the transitionto the inst ability, \|Z\| ≪1. In the most generalway, expanding ¯χin powers ofZonegets: ¯χ=/integraldisplay1 Z−v/bardbl∂f ∂v⊥dv/bardbl =−1 v/bardbl∂f ∂v⊥dv/bardbl+Z/integraldisplay1 Z−v/bardbl1 v/bardbl∂f ∂v⊥dv/bardbl →d+iκZ,(C1) provided (∂f/∂v /bardbl)\|v/bardbl=0∝negationslash= 0. The quantities dandκare deﬁned in (14). It is easy to see that (9) is the ﬁrst order e quation foriZ(with real coefﬁcients) in the lowest order on \|Z\| ≪1, which means that there is a simple (one and only one) aperiod ic rootinthevicinityof Z= 0. Suchaperiodicsolutionscannotbeconvertedintonon-ape riodiconesbycontinuouschangeofthe plasmaparameters,forthesame reasonasabove. Therefore, theunstablesolutionsmust beaperiodic. The function Ψ(Z), deﬁned in (9), is an analytical function of Z=W+iGand a continuous function of its parameters βandθ. Let us consider how Zmoves from the lower half-plane (stable regime) to the upper half-plane (unstable regime) with the change of βandθ=const. The transition to instability occurs, in general, in the vicinity of Z= 0where Ψ(Z) = Ψ(0) + ( dΨ/dZ)\|Z=0Z=A+BZ(see sec. V). In the transition point A= 0. Using (15)–(17) it is easy to show that in the transition point B >0(provided κ >0, this condition being violated if ∂f/∂E>0atv/bardbl= 0, corresponding to the regime of two-hump instability), so that in the vicinity of the tran sition point A >0corresponds to the stable regime, while A <0 correspondstotheinstability. Becauseofthecontinuity, inthe wholeinstabilityrange A <0. Let us show now that (9) always has a solution Z=iG,G >0in the unstable range. Indeed, Ψ(0)<0as is shown above. Ontheotherhand,if G→ ∞onehas χ→1/G2andΨ(∞)>0. Thismeansthatthereexists G >0suchthat Ψ(G) = 0. In the absence of kinetic instabilities, in the stable regim e all roots of (9) with nonzero Ware either in the lower half-plane (Landaudampingornonpropagation)orattherealaxis(if (∂f/∂v /bardbl)\|v/bardbl=W= 0). Intheﬁrstcasenorootcancrosstherealaxis except at W= 0, when the parameters are changed continuously to bring the s ystem in the unstable regime. As can be seen from(15)–(17)thereis onlyonerootcrossingthereal axisa t thispoint,provided ∂f/∂E<0. Therefore,thereisonlyoneroot inthe upperhalf-planeandit ispurelyimaginary. If∂f/∂E= 0there are two or morerootsin the vicinityof Z= 0(dependingonthe behaviorof f) but onlyone is positive, G >0. Since in this case the analytical continuation through Z= 0into the lower half-plane is straightforward (no pole at v/bardbl= 0)otherrootscorrespondtodampingsolutions,andthereisa gainonlyonerootinthe upperhalf-plane. Finally, let us consider the case where there are roots with G= 0andW=W0∝negationslash= 0. 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[14] McKean, M.E.,Winske, D.,and Gary, S.P.,Mirror and ion cyclotron anisotropy instabilitiesin the magnetosheath, J. Geophys. Res., 97, 19,421, 1992. [15] Gary, S.P.,Fuselier,S.A.,and Anderson, B.J.,Ion ani sotropy instabilitiesinthe magnetosheath, J.Geophys. Re s.,98, 1481, 1993. [16] Southwood, D.J,andKivelson, M.G.,Mirror instabilit y,1, Physicalmechanism of linear stability,J. Geophys. Re s.,98, 9181, 1993. [17] Pantellini, F.G.E., and Schwartz, S.J., Electron temp erature effects in the linear proton mirror instability, J. Geophys. Res., 100, 3539, 1995. [18] Pokhotelov, O.A., Balikhin, M.A., Alleyne, H.S.C.K., Onishchenko, O.G., Mirror instability with ﬁnite electron temperature effects, J. Geophys. Res.,105, 2393, 2000. [19] Kivelson,M.G.,andSouthwood, D.J.,Mirrorinstabili ty,2,Themechanism ofnonlinear saturation,J.Geophys. Re s.,101, 17,365, 1996. 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[21] Prudnikov, A.P.,Brychkov, Yu.A.,and Marichev, O.I., Integrals and series, New York: Gordon andBreach Science Publishers,1988.13 -5 -4 -3 -2 -1 0 1 2 3 4 500.10.20.30.40.50.60.7 v/vT/bardblf(v/bardbl) FIG.1: Waterbag (solidline), hardbell (dashed), Lorentz ( dotted), andMaxwellian (dash-dotted) distributions. 0 10 20 30 40 50 60 70 80 90020406080100120140 θω/k/bardblvTi/bardbl FIG. 2: Phase velocity of the fast mode as a function of propag ation angle for the case of the waterbag distribution with βi/bardbl=βi⊥=βe/bardbl= βe⊥= 0.1and massless bi-Maxwellianelectrons.14 0 10 20 30 40 50 60 70 80 901.71.751.81.851.91.952 θω/k/bardblvTi/bardbl FIG. 3: Phase velocity (diamonds) of the two low-velocity mo des as a function of propagation angle for the case of the wate rbag distribution withβi/bardbl=βi⊥=βe/bardbl=βe⊥= 0.1and massless bi-Maxwellian electrons. The solidline is ω=√aik/bardblvTi/bardbl=k/bardblv0i. 0 10 20 30 40 50 60 70 80 901.651.71.751.81.851.91.952 θω/k/bardblvTi/bardbl FIG. 4: Phase velocity (diamonds) of the two low-velocity mo des as a function of propagation angle for the case of the wate rbag distribution withβi/bardbl=βi⊥=βe/bardbl=βe⊥= 0.5and massless bi-Maxwellian electrons. The solidline is ω=√aik/bardblvTi/bardbl=k/bardblv0i. 0 10 20 30 40 50 60 70 80 900.511.522.53 θω/k/bardblvTi/bardbl,γ/k/bardblvTi/bardbl frequency W growthrate G FIG.5: Behavior ofthetwolow-velocitymodes as afunctiono fpropagation angle forthecase ofthewaterbag distributio nwith βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5(sothat K= 1),andmasslessbi-Maxwellianelectrons. Diamondsmarkthe modesintherangewheretheirfrequencies are purelyreal, stars show the growth rateof the aperiodic i nstability.15 0 10 20 30 40 50 60 70 80 900.511.522.53 θω/k/bardblvTi/bardbl FIG.6: Behavior ofthetwolow-velocitymodes as afunctiono fpropagation angle forthecase ofthewaterbag distributio nwith βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5(sothat K= 1), andwaterbag electrons, de= 1/3. There isno instability. 10 20 30 40 50 60 70 80 9000.511.522.53 θγ/k/bardblvTi/bardbl FIG. 7: Growth rates for the mirror instability in the case of βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, and massless bi-Maxwellian electrons de= 1,and four different distributions: waterbag (diamonds), h ard-bell (crosses), Lorentz (circles),and bi-Maxwellian (triangles). 0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.8 θγ/kv Ti/bardbl FIG. 8: Growth rates for the mirror instability in the case of βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, and massless bi-Maxwellian electrons de= 1,andfourdifferentdistributions: waterbag(diamonds),h ard-bell(crosses),Lorentz(circles),andbi-Maxwellian (triangles),normalized onkvTi/bardbl.16 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.80.91 θγ/kv Ti/bardbl FIG. 9: Dependence of the growth rate on β⊥withK= 1for waterbag ions and massless bi-Maxwellian electrons: di amonds correspond to βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, crosses correspond to βi/bardbl=βe/bardbl= 0.25,βi⊥=βe⊥= 1. 10 20 30 40 50 60 70 80 90051015 θγ/k/bardblvTi/bardbl FIG.10: Growthratesforthemirrorinstabilityinthecaseo fβi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5,andthreedifferentcombinations: hard-bell ions and electrons (crosses), Lorentz ions and electrons (c ircles), andbi-Maxwellian ions and (massive) electrons (t riangles). 10 20 30 40 50 60 70 80 9000.511.522.53 θγ/kv Ti/bardbl FIG.11: Same as inFigure 10 butnormalized on kvTi/bardbl.17 10 20 30 40 50 60 70 80 9000.511.522.53 θγ/kv Ti/bardbl FIG. 12: Growth rates for the mirror instability in the case o fβi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, waterbag ions and two different electron distributions: massive bi-Maxwellian(diamonds) and Lore ntz (circles). 0 1 2 3 4 5 6 7 8 9 1000.10.20.30.40.50.60.70.80.91 Gχ FIG.13: Approximationof χ(G)fortheMaxwelliandistribution. Thenumericallycalculat edχ(G)(solidline)iscomparedto χ= (G2+2)−1 (crosses). 0 1 2 3 4 5 6 7 8 9 1000.511.522.53 Gχ FIG. 14: Approximation of χ(G)for the Lorentzian distribution. The numerically calculat edχ(G)(solid line) is compared to χ= (G2+ 0.5)−1(crosses).18 0 10 20 30 40 50 60 70 80 9000.511.5 θγ/kv Ti/bardbl FIG. 15: Growth rates for the mirror instability in the case o fβi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, calculated with the approximation ¯χi= 1/(G2+v2 m),for several v2 m= 2(diamonds), 1.5(crosses), 1(triangles), and0.5(circles ). The electrons are massless Maxwellian. 0 10 20 30 40 50 60 70 80 9000.511.5 θγ/kv Ti/bardbl FIG. 16: Comparison of the growth rates for the mirror instab ility in the case of βi/bardbl=βe/bardbl= 0.1,βi⊥=βe⊥= 0.5, calculated directly and with the approximation ¯χi= 1/(G2+v2 m), for Maxwellian (diamonds and crosses, respectively) and L orentzian (triangles and circles, respectively). The electrons are massless Maxwellian.
arXiv:physics/0009041v1 [physics.flu-dyn] 13 Sep 2000Charged-Surface Instability Development in Liquid Helium; Exact Solutions N. M. Zubarev Institute of Electrophysics, Ural Division, Russian Acade my of Sciences, 106 Amundsena Street, 620016 Ekaterinburg, Russia e-mail: nick@ami.uran.ru The nonlinear dynamics of charged-surface instability dev elopment was investigated for liquid helium far above the critical point. It is found th at, if the surface charge completely screens the ﬁeld above the surface, the equation s of three-dimensional (3D) potential motion of a ﬂuid are reduced to the well-known equa tions describing the 3D Laplacian growth process. The integrability of these equat ions in 2D geometry allows the analytic description of the free-surface evolution up t o the formation of cuspidal singularities at the surface. It is known [1] that the ﬂat electron-charged surface of liqu id helium is unstable if the electric ﬁeld strength above ( E+) and inside ( E−) the ﬂuid satisfy inequality E+2+E−2>Ec2= 8π√gαρ, wheregis the free fall acceleration, αis the surface tension coeﬃcient, and ρis the ﬂuid density. An analysis of the critical behavior of the system s uggests that, depending on the dimensionless parameter S= (E−2−E+2)/Ec2, the nonlinearity leads either to the saturation of linear instability or, conversely, to the explosive incr ease in amplitude. The ﬁrst situation may result in the formation of a stationary perturbed surfac e relief (hexagons [2] and many- electron dimples [3]) in liquid helium. The use of a perturba tion theory, with the surface slope as a small parameter, allowed the detailed analytic study of such structures in the critical region [4,5]. In the second case, the small-angle approxima tion fails. The cinematographic study by V.P. Volodin, M.S. Khaikin, and V.S. Edelman [6] has demonstrated that the development of surface instability leads to the formation o f dimples and their sharpening in a ﬁnite time. A substantial nonlinearity of this processes c alls for a theoretical model that is free from the requirement for smallness of surface perturba tions and adequately describes the formation dynamics of a singular surface proﬁle in liquid he lium. This work demonstrates that such a model can be developed if the condition E−≫E+is fulﬁlled, i.e., the ﬁeld above liquid helium is fully screened by the surface electro n charge, and if the electric ﬁeld far exceeds its critical value, i.e., E−≫Ec. Let us consider the potential motion of an ideal ﬂuid (liquid helium), in a region bounded by the free surface z=η(x,y,t). We assume that the characteristic scale λof surface perturbations is much smaller than the ﬂuid depth. We also as sume that αE−2 −≪λ≪E−2/(gρ), so that the capillary and gravity eﬀects can be ignored. The e lectric-ﬁeld potential ϕ(x,y,z,t ) in the medium and the ﬂuid velocity potential Φ( x,y,z,t ) satisfy Laplace equations ∇2ϕ= 0,∇2Φ = 0, (1) 1which should be solved jointly with the conditions at the sur face 8πρΦt+ 4πρ(∇Φ)2+ (∇ϕ)2=E−2, z =η(x,y,t), (2) ηt= Φz− ∇⊥η· ∇⊥Φ, z =η(x,y,t), (3) ϕ= 0, z =η(x,y,t), (4) and conditions at inﬁnity ϕ→ −zE−, z → −∞, (5) Φ→0, z → −∞. (6) Let us pass to the dimensionless variables, taking λas a length unit, E−as a unit of electric ﬁeld strength, and λE−1 −(4πρ)1/2as a time unit. It is convenient to rewrite the equations of motion of free surface z=η(x,y,t) in the implicit form (not containing the η function explicitly). Let us introduce the perturbed poten tial ˜ϕ=ϕ+zdecaying at inﬁnity. One has at the boundary: ˜ ϕ\|z=η=η. It is then straightforward to obtain the following relationships: ηt=˜ϕt 1−˜ϕz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=η,∇⊥η=∇⊥˜ϕ 1−˜ϕz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=η, which allow one to eliminate the ηfunction from Eq. (3). The dynamic and kinematic boundary conditions (2) and (3) then take the form Φt−˜ϕz=−(∇Φ)2/2−(∇˜ϕ)2/2, z =η(x,y,t), (7) ˜ϕt−Φz=−∇˜ϕ· ∇Φ, z =η(x,y,t). (8) Let now introduce a pair of auxiliary potentials: φ(±)(x,y,z,t ) = (˜ϕ±Φ)/2. With these potentials, the ﬂuid surface shape can be deﬁned b y relationship η= (φ(+)+φ(−))/vextendsingle/vextendsingle/vextendsingle z=η, (9) while equations of motion (1)–(6) are reduced to the followi ng symmetric form: ∇2φ(±)= 0, (10) φ(±) t=±φ(±) z∓(∇φ(±))2, z =η(x,y,t), (11) φ(±)→0, z → −∞, (12) where boundary conditions (11) are obtained by combining Eq s. (7) and (8) with plus and minus sign, respectively. It is seen that the equations of motion are split into two syst ems of equations for the potentialsφ(+)andφ(−), which are implicitly related by equation for surface shape (9). An essential point is that these equations are compatible wi th eitherφ(−)= 0 orφ(+)= 0 condition. One can readily see that the ﬁrst condition corr esponds to those solutions 2whose amplitude increases with time, while the second condi tion corresponds to the decaying solutions that are of no interest to us. Thus, an analysis of the equations of motion of a charged surf ace of liquid helium reveals a solution increasing with tand corresponding to the φ(−)= 0 condition or, what is the same, to theϕ+z= Φ condition (the stability of this branch of solutions is pr oved below). The functional relation between the potentials can be used to el iminate the velocity potential Φ from initial Eqs. (1)–(6). In the moving system of coordinat es{x′,y′,z′}={x,y,z−t}, one has ∇2ϕ= 0, (13) η′ t=∂nϕ/radicalBig 1 + (∇⊥η′)2, z′=η′(x′,y′,t). (14) ϕ= 0, z′=η′(x′,y′,t) (15) ϕ→ −z′, z′→ −∞, (16) whereη′(x′,y′,t) =η−t, and∂ndenotes the normal derivative. These equations explicitly describe the motion of a free charged surface z′=η′(x′,y′,t). They coincide with the equations for the so-called Laplacian growth process, i.e. , the phase boundary movement with velocity directly proportional to the normal derivati ve of a certain scalar ﬁeld ( ϕin our case). Depending on the system, this may be the temperatu re (Stefan problem in the quasi-stationary limit), the electrostatic potential (el ectrolytic deposition), the pressure (ﬂow through a porous medium), etc. Note that the boundary movement described by Eqs. (13)–(16) is invariably directed inward from the surface. Let η′be a single-valued function of variables x′andy′at zero time t= 0. Then the inequality η′(x′,y′,t)≤η′(x′,y′,0) holds for t>0. In the initial notations, η(x,y,t)≤η(x,y,0) +t (17) for anyxandy. This condition can be used to prove the stability of the asce nding branch to the small perturbations of potential φ(−). Clearly, the boundary motion at small φ(−)values is entirely controlled by the potential φ(+)[one should set φ(−)= 0 in Eq. (9)] and, hence, obeys Eqs. (13)–(16). The evolution of the φ(−)potential is described by Eqs. (10)–(12), with the following simple boundary condition in the linear a pproximation: φ(−) t=−φ(−) z, z =η(x,y,t). Let the potential distribution at zero time t= 0 be determined by the expression φ(−)\|t=0=φ0(x,y,z ), whereφ0is a harmonic function at z≤η(x,y,0) decaying at z→ −∞ . It is then straight- forward to show that the time dynamics of the φ(−)potential is given by φ(−)=φ0(x,y,z−t). This implies that the singularities of the φ(−)function will drift in the zdirection, so that they will occur only in the z > η (x,y,0) +tregion. Taking into account inequality (17), 3one ﬁnds that the singularities always move away from the bou ndaryz=η(x,y,t) of liquid helium. Consequently, the perturbation φ(−)will relax to zero, as we wished to prove. Let us now turn to the analysis of the dynamics of surface inst ability development in liquid helium. In the 2D case (all quantities are taken to be i ndependent of the yvariable), system of Eqs. (13)–(16) is reduced to the well-known Laplac ian growth equation (see, e.g., [7] and references therein): Im(f∗ tfw) = 1, ϕ = 0. In this expression, f=x′+iz′is a complex function analytical in the lower half-plane of the complex variable w=ψ−iϕand satisfying condition f→watw→ψ−i∞. Note that theψfunction is a harmonic conjugate to ϕ, while the condition ψ= const deﬁnes the electric ﬁeld lines in a medium. The Laplacian growth equati on is integrable in the sense that it allows for the inﬁnite number of partial solutions: f(w) =w−it−iN/summationdisplay n=1anln (w−wn(t)) +i/parenleftBiggN/summationdisplay n=1an/parenrightBigg ln (w−˜w(t)), whereanare complex constants and Im( wn)>0. The last term is added in order that the condition η→0 be fulﬁlled at \|x\| → ∞ . One can put Im( ˜ w)≫Im(wn); in this case, the inﬂuence of this term on the surface evolution can be igno red. The functions wn(t) are deﬁned by the following set of transcendental equations [7] : wn+it+iN/summationdisplay m=1a∗ mln (wn−w∗ m) =Cn, whereCnare arbitrary complex constants. Let us consider the simplest ( N= 1) solution to the Laplacian growth equation: f(w) =w−it+iln(w−ir(t)), r (t)−lnr(t) = 1 +tc−t, (18) wheretcis a real constant and the real function r(t)≥1. The shape of a solitary perturbation corresponding to Eqs. (18) is given parametrically by expre ssions z(ψ,t) = ln/radicalBig ψ2+r2(t), x (ψ,t) =ψ−arctan (ψ/r(t)). (19) This solution exists only during a ﬁnite time period and culm inates in the formation, at time t=tc, of a singularity in the form of a ﬁrst-kind cusp at the ﬂuid su rface. Indeed, setting r=r(tc) = 1 in Eq. (19), one obtains 2z=\|3x\|2/3 in the leading order near the singular point (see also [8]). N ote that the electric ﬁeld turns to inﬁnity at the cusp: ∂nϕ∼x−1 ψ/vextendsingle/vextendsingle/vextendsingle ψ=0∼1/√tc−t. The surface velocity also becomes inﬁnite in a ﬁnite time: ηt=zt\|ψ=0∼1/√tc−t. 4It is worth noting that the singular solution in the leading o rder is also true when the ﬁeld above the surface is screened incompletely. The point is tha t the requirement that the ﬁeld above the surface be small compared to the ﬁeld in ﬂuid is natu rally satisﬁed in the vicinity of the singularity. Let now discuss the inﬂuence of the capillary eﬀects. One can readily estimate the surface and electrostatic pressures near the surface: αR−1∼αρ1/2E−1 −(tc−t)−1,(∂nϕ)2∼λρ1/2E−(tc−t)−1. Insofar as we assumed that λ≫αE−2 −, the capillary forces cannot compete with the elec- trostatic ones, so that there is no need to take into account t he surface forces at the stage of cusp formation. In summary, we succeeded in ﬁnding a broad class of exact solu tions to the equations of motion of a charged surface of liquid helium. It is remarka ble that the solutions ob- tained are not constrained by the condition for the smallnes s of surface perturbations: the model suggested describes the free-surface instability de velopment up to the formation of the singularities (cusps) similar to those observed in the e xperiment [6]. I am grateful to E.A. Kuznetsov for stimulating discussions . This work was supported by the Russian Foundation for Basic Research (project no. 00-0 2-17428) and the INTAS (grant no. 99-1068). 1. L.P. Gorkov and D.M. Chernikova, Sov. Phys. Dokl. 21, 328 (1976). 2. M. Wanner and P. Leiderer. Phys. Rev. Lett. 42, 315 (1979). 3. A.A. Levchenko, E. Teske, G.V. Kolmakov, et al., JETP Lett. 65, 572 (1997). 4. V.B. Shikin and P. Leiderer, JETP Lett. 32, 572 (1980). 5. V.I. Melnikov and S.V. Meshkov, Sov. Phys. JETP 54, 505 (1981); 55, 1099 (1982). 6. V.P. Volodin, M.S. Khaikin, and V.S. Edelman, JETP Lett. 26, 543 (1977). 7. M.B. Mineev-Weinstein and S.P. Dawson, Phys. Rev. E 50, R24 (1994). 8. S.D. Howison, SIAM J. Appl. Math. 46, 20 (1986). 5
1 RELATIVISTIC GEOMETRY AND QUANTUM ELECTRODYNAMICS Gustavo González-Martín Departamento de Física, Universidad Simón Bolívar, Apartado 89000, Caracas 1080-A, Venezuela. Web page URL http:\\prof.usb.ve\ggonzalm\ Excitations of a relativistic geometry were used to represent the theory of quantum electrodynamics. The connection excitations and the frame excitations reduce, respectively,to the electromagnetic field operator and electron field operator. Because of the inherentgeometric algebraic structure these operators obey the standard commutation rules of QED.If we work with excitations, we need to use statistical theory when considering the evolutionof microscopic subsystems. The use of classical statistics, in particular techniques ofirreversible thermodynamics, determine that the probability of absorption or emission of ageometric excitation is a function of the classical energy density. Emission and absorption ofgeometric excitations imply discrete changes of certain physical variables, but with aprobability determined by its wave energy density. Hence, this geometric theory does notcontradict the fundamental aspects of quantum theory.SB/F/272-992 Introduction A geometrical unified theory of gravitation and electromagnetism, which leads to equations of relativistic quantum mechanics, was shown to imply the quantization of electric charge and magnetic flux [1], providinga plausible explanation for the fractional quantum Hall effect. Furthermore the theory leads to a geometricalmodel for the process of field quantization[2], implying the existence of fermionic and bosonic operators. Weshall indicate the latter paper by I. Here we discuss whether the theory gives quantum electrodynamics (QED)including its probabilistic interpretation. A basic feature of the proposed theory is non-linearity. A solution cannot be obtained by the addition of two or more solutions and therefore it is not possible to build exact solutions from small subsystems. Neverthe-less, it is possible to study its local linearized equations which represent excitations that evolve approximatelyunder the influence of effects inherited from the nonlinear equation. Geometric Relations Product of Jacobi Operators. The product αβ has been taken in I as the Clifford product. Other candidates for the ring product were discussed in that paper. The chosen product presents difficulties when relating the geometry to standardQED. Since the fiber bundle of frames, E, is a principal bundle and its vertical tangent bundle TE has for fiber a Lie algebra structure inheritted from the group, it would be more natural that the chosen product be closedin the algebra so that the result be also valued in the Lie algebra. Geometrically we should specialize the ringproduct to be the Lie subproduct. Then the product αβ is zero whenever its gradation is even.The bracket survives only when the commutator is nonzero because its gradation is odd and corresponds to the commuta-tor. In other words the ring product obeys []αβ α β•=, for odd grade , ( 1) αβ•=0 for even grade , ( 2) with this product, the bracket defined in I becomes {} {}αβ βαVW WV,,= , ( 3) which is the anticommutator, for the matter Jacobi operator fields. The fiber bundle of connections, W, is an affine bundle and the ring product associated to the fiber of TW is commutative. Therefore, the bracket is the commutator for the connection Jacobi operator fields Commutation Relations If we take Schwingwer ’s action principle as described in I, we obtain the quantization relations by requiring that the quantum operators Ψ be Jacobi operators, () (){}Ψxy,δΨ=0 , ( 4) () (){} () ΨΠxy x y,,µµδ=− . ( 5) As discussed in previous articles, the Jacobi vectors Vs represent fluctuations of sections of fiber bundles E. The corresponding jet prolongation jV of an extension of Vs is a vector field on JE which acts as an operator on functions on JE. A Jacobi field, as a vertical vector on the section s(M) may be considered as a displacement of s. Similarly, its jet prolongation may be considered as a displacement of js, in other words a variation, ()ds dVsj jj λ= . ( 6) This linear action of the Jacobi fields on the background sections allow us to associate them with quantum theory objects. We may consider the background section as the state 〉 of the physical system and the Jacobi fields as physical linear operators Ψ that naturally act on the states. In this condition the background section3 remains fixed and the Jacobi operators obey the first order linearized equations. (Heisenberg picture). A given physical matter section may be expressed in terms of reference frame sections. This reference frames are also physical systems . Hence they may evolve under the action of the same group that acts on thephysical sections. This represents the known equivalence of the active and passive views of evolution. We mayconsider the physical system evolving in a fixed frame or, equivalently, we may consider the system fixed in anevolving reference frame. We may choose the reference frame so that the substratum section does not evolve. In this condition the substratum section remains fixed and the Jacobi operators obey the first order linearized equations of motion.(Heisenberg picture). The Jacobi vector fields and their jet prolongations transform under the adjoint representation of the group that transforms the sections. We may use these transformations to a new reference frame where the Jacobioperators do not evolve. In this condition, the time dependence of the Jacobi operators may be eliminated andthe substratum sections obey equations of motion, indicating that the state is time dependent (Schroedingerpicture). Geometric Electrodynamics Reduction of the electromagnetic sector In order to get the standard QED from the geometric theory, we have to reduce the structure group to one of the 3 U(1) subgroups in the SU(2) electromagnetic sector. The corresponding u(1) component of the fluc-tuation of the generalized connection is the electromagnetic potential A. Similarly, an electromagnetic fluctua- tion of the matter section determines a fluctuation of the generalized current. The standard QED electriccurrent is the electromagnetic sector component of this current fluctuation, () []() () () jJ J J J== = =1 423 1 4123 5 1 45123 1 40tr1κκκδ κκκ κ κκκκ κ tr , tr tr . ( 7) It was shown in [3], that the standard electric current in quantum theory is related to the κ0 component of the generalized current, ()1 40 1 40tr tr~κκ κ γµµ µJe e== ΨΨ , ( 8) which is the electric current for a particle with a charge equal to one quantum in the geometric units. It should be clear that an exact solution of the proposed problem is not possible with this technique, The reason for this is the presence of the nonlinearity of the self interaction in the equations. We must considerapproximate solutions. In QED the interaction is imposed on special fields called “free fields ”. Here we have to discuss which frame and connection sections correspond to “free fields ”. The field equation, DJ∗=Ω4πα , ( 9) reduces when the connection has only a κ0 electromagnetic component to dd A j∗=4πα . ( 10) It is natural to consider a free connection field as one that satisfies the previous equation with j=0. A solution to the free equation provides a background connection section that can be used with the proposedtechnique. The equation of motion κµ µ∇=e0 , ( 11) presents the difficulty with the self interaction. It is not possible to set the connection to zero because it will automatically eliminate the self energy and the mass parameter in accordance with our theory. The simplestapproximation is then to assume that all self interaction effects, to first order, are concentrated in the singlemass parameter. The fluctuation equation is, κ∂µ µem e=+ interactions . ( 12)4 It is natural to consider a free frame field as one that satisfies the previous equation without the interac- tion term. A solution to this free equation provides a background frame section that can be used with ourtechnique. The linearized fluctuation equations, with the interaction term, determine the fluctuations of theconnection and frame fields. Quantum Electrodynamics We shall consider that the cause of the fluctuation is the interaction that couples the two free systems. The two background sections are not a solution to the interacting system, they are a reference for the method ofapproximation. The solution is represented by two sections that deviate (fluctuate) from the background sec-tions. In other words, there is a fluctuation operator that represents the interaction. This situation correspondsto an interaction picture in QED. By choosing an appropiate reference frame, the background sections havethe “free motion ” and the fluctuation operators represent the dynamics. We assumed that the only effect of the self interaction in the free motion of the fields is the mass parameter m. The remaining effects of the total physical interaction, associated with the flutuation fields, corresponds to an effective net interaction energy, ~ee mHκµ µΓ−= . ( 13) The first part of H is the total physical interaction. The meaning of the second part is that massive self energy is not included in the fluctuation and this method (or QED) is not adequate for calculation of mass. It is known that the Lagrangian for the first variation of the Lagrange equations is the second variation of the Lagrangian. For both free fields, the corresponding second variation of the Lagrangian gives the freeMaxwell and Dirac fields Lagrangian. For the interaction fields the second variation gives () () δδ Γ δ κ γ γµ µµ µ2 1 41 40LJ AA== =tr tr ΨΨ ΨΨ , ( 14) which is the standard interaction Hamiltonian in QED. We have obtained a framework of linear operators (the prolongations of Jacobi fields) acting on states of a Banach space (sections forming a Banach space) where the operators obey the bracket commutation rela-tions, commutators for A and anticommutators for Ψ. Furthermore, the geometric Lagrangian of the theory reduces to a Lagrangian in terms of operators which is the standard Lagrangian for QED). From this point we can proceed using the standard language and notation of QED for any calculation we wish to carry in this approximation of the geometric theory. The result of the calculation should be interpretedphysically. This is our next task, to give a statistical interpretation to fluctuations in the geometrical theory. Statistical interpretation The physical significance of the local sections in the geometrical theory is the representation of the influ- ence of the total universe of matter and interactions. The geometry of the theory, including the notion ofcorpuscles is determined by the large number of sources in the universe. Within this theory and interpretation,it is not appropriate to consider the sections to be associated with a single particle or excitation. Rather, theyare associated to the global non-linear geometric effects of bulk matter and radiation. This is not the customary situation in which physical theories are set. It is usual to postulate fundamental laws between elementary microscopic objects (particles). In order to translate the global universal geometry tothe usual microscopic physics, since we associate particles to geometric excitations, we shall stablish what wecall a “many excitation microscopic regime ”, distinguished from the situation described in the previous para- graph. Statistics enter in our approach in a manner different than the usual one, where fundamental microscopic physical laws are postulated between idealized elementary particles and statistical analysis enters because ofthe difficulties that arise when combining particles to form complex systems. Here, the fundamental laws arepostulated geometrically among all matter and radiation in the system ( the physical universe) and statisticalanalysis arises because of the difficulties and approximations in splitting the non-linear system into elemen-tary microscopic linear fluctuation subsystems. In this holistic approach, the results associated to excitationshave the statistical character of quantum theory. We consider that we may work in two different regimes of the geometrical theory. One is the geometrical regime where there are exact non-linear equations between the local frame sections, representing matter func-5 tions, and the bundle connection. The other is the many excitations microscopic regime where we have linear- ized approximate equations between the variations of the frame and connection sections, representing particlesand fields. In the many-excitations regime, the non-linear local effects of the interaction are replaced by a linear local effects for which the systems are seen, approximately, as a collection of excitations of a background solution.The number of excitations is naturally very large and the cross interactions among them preclude an exacttreatment for a single excitation. Instead it is necessary to treat the excitations as one among a large ensembleof excitations leading to a convenient and necessary use of statistical theory. The geometric excitations form statistical ensembles of population density n i . It is not possible to follow the evolution of any single one because of the previous arguments. It is absolutely necessary to use statistics indescribing the evolution of the excitations. The situation is similar to that of chemical reactions, where mol-ecules are statistically created or annihilated. There are adequate classical statistical techniques for describing these processes. In particular we may use the theory of irreversible thermodynamics [4] to calculate the rates of reaction between different geometricexcitations. The process is described by the flux density /G4A characterizing the flow of excitations between two systems. /G4A=dn dt . ( 15) It is also necessary to introduce a driving force function, /G46, which is called affinity and represents differ- ences of thermodynamic intensive parameters. In case of equilibrium between two different subsystems, boththe affinity and the flux are zero. The identification of the affinity is done by considering the rate of production of entropy s, ds dts ndn dtkk==∂ ∂/G46/G4A/G6B k , ( 16) from which the affinity associated to any given excitation is /G46k kks uu nT==∂ ∂∂ ∂µ , ( 17) where u is the energy, T is the temperature and µ is the excitation potential, similar to the chemical potential. The statistical flux is a function of the affinity and we see that the statistics of reaction of geometric excitations should depend on the classical geometric energy of the excitations. If the excitations are oscilla-tory, the energy depends on the excitation amplitude. Then the probability of occurrence of a single reactionevent is determined by the excitation potential which in turn depends on the excitation amplitude. These is thesignificance of the probabilistic interpretation of quantum fields. In some cases, for linear Markoffian systems (systems whose future is determined by their present and not their past) the flux is proportional to the affinity ∆∆/G4A/G46/G46=J∂ ∂0 , ( 18) and the calculation of rates is simplified, indicating explicitly the dependence of the flux on the different of excitation potentials. Applications In order to illustrate this statistical character of the linearized regime, we discuss the wave corpuscular duality of light and matter, typically demonstrated in Young ’s double slit interference experiment, within the concepts of the geometric theory. There are excitations that correspond to multiple particles and lead toSchrödinger’s “entangled states ”. There has been a revolution in the experimental preparation of multi article entanglements [5,6,7,8,9]. In particular, consider a two-particle interferometer illustrated in figure 1. At thecenter there is a source of decaying particles, with a vertical extension d. Two collimating screens, each with a6 pair of holes, offer alternative paths to a given pair of particles that are finally detected in two detecting screens . A physical particle (f.e. a photon) is an excitation of the corresponding background field. In geometric terms, a Jacobi vector Ψ, associated to a variation of a section e or ω, represents the particle. As indicated in the previous section, it is not possible to follow the evolution of a single excitation Ψ. The statistical approach is to treat the radiation as a thermodynamic reservoir of excitations . This approach was used in blackbodyradiation[10,11,12] and was the origin of Planck ’s quantum theory. In recent years, a similar idea lead to introduction of stochastic quantization [13,14], which has been shown to be equivalent to path integralquantization. Our approach is different, for example we do not introduce evolution along a fictitious timedirection as done in stochastic quantization. We rely on the existence of a global nonlinear geometry whichmakes a practical necessity the statistical treatment of the linearized equations describing the evolution ofmicroscopical subsytems. The atoms in the screen are an ensemble of systems in contact with the reservoir of radiation. The equilib- rium of the total system, radiation and screen, is determined by the equality of the excitation potentials associ-ated to the geometric excitations forming the radiation. When there is no equilibrium there is a flow of excita-tions from the radiation to the screen. The techniques of irreversible thermodynamics relate this excitationflux density to the affinity, which depends on the differences of the associated excitation potentials. We alsoassume that we are dealing with a Markoffian system. This approximation has been used successfully in thequantum theory of damping [15,16] for laser systems. In order to calculate the excitation potential we need to express the energy in terms of the number of excitations. If the electromagnetic excitation field is expressed, as usual in terms of harmonic oscillators at each point, the oscillator displacement (excitation or variation of the electromagnetic field) is a thermody-namic extensive parameter, x. Associated to it there is an intensive parameter F that gives the energy change, du Fdx= . ( 19) Fig. 1 y zxA BB’ A’PP’7 For linear harmonic oscillators the energy is related to the square of the displacement amplitude X UK X X=1 2✝ , ( 20) where K is a constant As indicated previously, the Jacobi vector Ψ may be decomposed in its Fourier components. The ampli- tudes become operators. Then the operator, Na a=✝ . ( 21) has discrete eigenvalues which determine the energy of the field. It should be clear that because of the commu- tation relations of the operators a, the energy of the harmonic oscillators corresponding to the excitations is quantized, with the same results of the quantum theory. The energy of the excitations has quanta of value ν, in geometric units where /GDF equals 1. The radiation field has an amplitude which is modulated throughout certain space regions because of the interference pattern produced by the waves. This means that the average energy isnot homogeneously distributed but concentrated in certain regions. We should consider a local excitation po-tential defined within domains of volume, determined by a correlation distance where the density of energy u may taken as constant. Then the excitation potential is µ∂ ∂=u n , ( 22) where n is the eigenvalue of N. In the standard Young ´s experiment the waves arriving at a point in the screen have the well known phase difference φ that allow us to write for the resultant amplitude, ΨΨ Ψ=+− 00eeiiφφ . ( 23) When the amplitude Ψ0 is expressed in terms of the Fourier components a, the energy of the excitation leads to the expression for the potential, µεφεπ λα ==   44cos cos sinl . ( 24) In our case we have to consider an m-particle wave. To clarify the situation and avoid confusion with the concept of particle, we define an m-corpuscle excitation as the tensor product representation of m fundamental (1-corpuscle) representations.The Jacobi excitation field amplitude is then an m-product representation. In this language, we consider the case of a 2-corpuscle excitation experiment, in particular the case of two Young’s experiments side by side. We not only have a correlation betweeen alternative paths through A and B for the right corpuscle but also correlations among paths A or B and A ’ or B’ for the 2-corpuscle excitation formed by the product of the two 1-corpuscle excitations. If a particle decays at height x above the center line, a particle may be detected in the right screen at height y and another in the left screen at height z. The phase difference ϕ for the right side particle has a contribution from the screen angle φπ λPly r= , ( 25) and another from the source angle, φπ λPlx r= , ( 26) leading to, for small y an z, ()φπθ λPxy=+ , ( 27) where θ is the angle subtended by the two holes. A similar expression is obtained for the left side particle, ()φπθ λ′=+Pxz . ( 28)8 The amplitude of the Jacobi excitation, which is a product representation of the two excitations, is () () Ψ∝+  +   −+ ∫a ddx x y x z dd cos cosπθ λπθ λ 22 ( 29) If d is much smaller than λ/θ the integral gives the product of two Young ’s patterns as it should. If, on the contrary, d is much larger than λ/θ, this integral gives, () Ψ∝−  azy2cosπθ λ . ( 30) We obtain for the excitation potential () µπθ λ∝−    coszy2 . ( 31) The probability for the simultaneous detection of particles at P an P ’ is given by this excitation potential. There are regions of high and low probabilities because of the interference of the excitation amplitudes. Thesecases just illustrated for 1-particle and 2-particle interferometers give a general characteristic of section exci-tation in the geometrical theory. There are no exact equations for physical situations with a single excitation.Any real physical situation in the microscopic linear regime of the theory requires the use of statistics. Theoutcome of microscopic transition experiments depends on the excitation potentials of the systems. The re-quirement of statistics necessarily leads to the probability interpretation of quantum theory. If, within a par-ticular experimental setup, we can physically distinguish between two excitation states, there no room to applystatistics and there follows the absence of the interference pattern. This is the content of Feynman ’s statement about experimentally indistinguishable alternatives [17]. Fundamentally the quantum statistics are the classi-cal statistics of section excitations in this physical geometry. The objections raised by Einstein to the probabi-listic interpretation are resolved automatically [18,19]. Conclusion Excitations of a physical geometry were used to represent the theory of quantum electrodynamics. The connection excitations and the frame excitations reduce, respectively, to the electromagnetic field operator andelectron field operator. Because of the inherent geometric algebraic structure this operators obey the standardcommutation rules of QED. Use of harmonic analysis introduces creation and annihilation operators for theassociated excitation waves. The energy of the connection excitations is /GDF ν. The nonlinear geometric equations apply to the total universe of matter and radiation. If we work with excitations, this implies the need to use statistical theory when considering the evolution of microscopic sub-systems. The use of classical statistics, in particular techniques of irreversible thermodynamics, determine thatthe probability of absorption or emission of a geometric excitation is a function of the classical energy density,which determines the excitation (chemical) potential. Emission and absorption of geometric excitations imply discrete changes of certain physical variables, but with a probability determined by its wave energy density. Hence, this geometric theory does not contradict thefundamental aspects of quantum theory. On the contrary, it offers a geometrical representation for the exist-ence of discrete quanta of energy, spin, electric charge and magnetic flux. References 1G. Gonz ález-Martín, Gen. Rel. Grav. 23, 827 (1991); G. González-Martín, Physical Geometry, (Universidad Simón Bolívar, Caracas) (2000), Electronic copy posted at http:\\prof.usb.ve\ggonzalm\invstg\book.htm 2G. Gonz ález-Martín, Gen. Rel. Grav. 24, 501 (1992). 3G. Gonz ález-Martín, Phys. Rev. D 35, 1225 (1987). 4H. B. Callen, Thermodynamics, (J. Wiley & Sons, New York), p. 289 (1960).5M. A. Horne, A. Zeilinger, in Proc. Symp. on Foundations of Modern Physics, P. Lahti, P. Mittelstaedt, eds.,9 (World Science, Singapore), p. 435 (1985). 6C. O. Alley, Y. H. Shih, Proc. 2nd Int. Symp. on Foudations of Quantum Mechanics in the Light of New Technology, M. Namiki et al, eds. (Phys. Soc. Japan, Tokyio), p. 47 (1986). 7M. A. Horne, A. Shimony, A Zeilinger, Phys. Rev. Lett. 62, 2209 (1989). 8R. Ghosh, L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).9Y. H. Shih, C. O. Alley, Phys Rev. Lett. 61, 2921 (1988). 10M. Planck, Verh. Dtsch. Phys. Gesellschaft, 2, 202 (1900). 11S. Bose, Z. Physik 26, 178 (1924). 12A. Einstein, Preuss. Ak. der Wissenschaft, Phys. Math. Klasse, Sitzungsberichte, p, 18 (1925). 13G. Parisi, Y. S Wu, Sc. Sinica, 24, 483 (1981). 14P. H. Damgaard, H. Huffel, Physics Reports 152, 229 (1987). 15W. H. Louisell, L. R. Walker, Phys. Rev. 137B, 204 (1965). 16W: H. Louisell, J. H. Marburger, J. Quantum Electron. QE-3, 348 (1967). 17R. P. Feynman, The Feyman Lectures on Physics, Quantum Mechanics. (Addison Wesley, Reading), p.3-7 (1965). 18A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935). 19N. Bohr, Phys. Rev. 48, 696 (1936).
arXiv:physics/0009043v1 [physics.class-ph] 13 Sep 2000From Superluminal Velocity To Time Machines? G. Nimtz∗, A.A. Stahlhofen†, and A. Haibel∗ February 2, 2008 Various experiments have shown superluminal group and sign al velocities recently. Ex- periments were essentials carried out with microwave tunne lling [1], with frustrated total internal reﬂection [2], and with gain-assisted anomalous d ispersion [3]. According to text books a superluminal signal velocity violates Einstein cau sality implying that cause and eﬀect can be changed and time machines known from science ﬁct ion could be constructed. This naive analysis, however, assumes a signal to be a point i n the time dimension ne- glecting its ﬁnite duration. A signal is not presented by a po int nor by its front, but by its total length. On the other hand a signal energy is ﬁnite th us its frequency band is limited, the latter is a fundamental physical property in co nsequence of ﬁeld quantiza- tion with quantum hν. All superluminal experiments have been carried out with ra ther narrow frequency bands. The narrow band width is a condition sine qua non to avoid pulse reshaping of the signal due to the dispersion relation of the tunnelling barrier or of the excited gas, respectively [4]. In consequence of the nar row frequency band width the time duration of the signal is long so that causality is prese rved. However, superluminal signal velocity shortens the otherwise luminal time span be tween cause and eﬀect. Can a signal travel faster than light? If this happens, would it really violate the principle of causality stating that cause precedes eﬀect [5, 6]? The latter stateme nt has been widely assumed as a matter of fact. It has been shown according to the theory of special r elativity that a signal velocity faster than light allows to change the past. The line of arguments ho w to manipulate the past in this case is illustrated in Fig. 1 [6, 7]. There are two frames of refere nce displayed. In the ﬁrst one at the time t = 0 lottery numbers are presented, whereas at t = -10 ps the co unters were closed. Mary (A) sends the lottery numbers to her friend Susan (B) with a signal velo city twice the velocity of light. Susan ∗Universit¨ at zu K¨ oln, II. Physikalisches Institut, Z¨ ulp icher Str. 77, D–50937 K¨ oln, Germany †Universit¨ at Koblenz, Institut f¨ ur Physik, Rheinau 1, D–5 6075 Koblenz, Germanymoving in the second inertial system at a relative speed of 0. 75c, sends the data back at an even faster speed of 4c, which arrives in the ﬁrst system at t = -50 ps, thus in time to deliver the correct lottery numbers before the counters close at t = -10 ps. v =4c´t xt x´ ´ LB Av=2c -50psx=ctv =0.75cr Figure 1: Coordinates of two observers A (0,0) and B with O(x, t) and O’(x’,t’) moving with a relative velocity of 0.75c. The distance L between A and B is 0.1 m. A has available a signal velocity v = 2c and B v’= 4c. Taking into consideration a ﬁnite signal dura tion, the lottery fraud is impossible as shown in Fig. 3. (The numbers in the example are chosen accord ing to [6].) The time shift of a point on the time coordinate into the past i s given by the relation [6]: tA=−L c(c−c/N−c/N′+vr/NN′) (c−vr/N), (1) where Lis the transmission length of the signal, vris the relative velocity of the two inertial systems A and B, and Nc, N’c are the signal velocities in A and B, respect ively. N and N’ are numbers assumed to be >1. This is an example often encountered in the literature suppo sed to show that a superluminal signal velocity results in negative times and allows to manipulate the past. We show now that this simple model is not correct. First we are going to recall the basic properties of a signal. Microwave pulses [1, 8] and quite recently light pulses [3] of frequency νand bandwidth ∆ νhave been shown to travel at a velocity much faster than light. The pulses in the two experiments correspond to s ignals used nowadays in telephone as well as in inter-computer communication. Frequency band limitation of signals, the basis of the sampling theorem [9], is a backbone of digital communicatio n technology discussed in detail in the literature (e.g. in Encyclopedia Britannica), but scarcel y addressed to in textbooks of physics. Theﬁnite energy content of a signal actually implies the freque ncy band limitation [4]. This fundamental physical property is in consequence of the energy of any freq uency components of a signal to be nhν where nis a whole number, hthe Planck constant, and νthe frequency. A pulse represents an amplitude modulated (AM) signal on a ca rrier frequency. The carrier frequency is in charge of the receivers address and the half-width of th e pulse represents the number of digits, i.e. the information. In the case of modern ﬁber optics the re lative frequency band width is 10−3 approximately, in the superluminal microwave experiments the band width was 10−1and in the optical experiment mentioned above it was less than 10−9. Due to the narrow frequency bands there was no signiﬁcant pulse reshaping neither in the microwave tunn elling experiment nor in the gain-assisted light propagation experiment. The superluminal signals ar e shown together with the luminal reference signals in Fig. 2. 00.10.20.30.40.50.60.70.80.91 -2-101234Intensity (normalized) Time [ns](1)(2) Figure 2: Display of the superluminal gain–assisted optica l pulses (left [3]) and tunnelled microwave (right [8]). The pulses are normalized and compared with the air born or the wave-guided signals. The measured velocities have been –310c and 4.7c, respectiv ely. Thus in both experiments the signal travelled at a superlumi nal velocity, e.g. with 4.7c [8] or with –310c [3], respectively. Nevertheless, the principle of ca usality has not been violated in both experi- ments. In the example with the lottery data the signal was assumed to be a point on the time coordinate. However, a signal has a ﬁnite duration as the pulse sketched a long the time coordinate in Fig. 3. (In the experiments in question 7.5 µs and 5 ns, see Fig. 2.) Any information like a word has a ﬁnite extension on the time coordinate. In the two cited superlumi nal experiments the superluminal time shift compared with the pulse length is about 30% in the micro wave experiment with the velocity 4.7c and about 1 % in the light experiment with the velocity –310c. Due to the signal’s ﬁnite duration of 200 ps the information is obtained only at positive times u nder the assumptions as illustrated inv =4c´t xt x´ ´ LB A+150psv=2c -50psx=ctv =0.75cr Figure 3: In contrast to Fig. 1 the pulse has a ﬁnite duration o f 200 ps. This data is used for a clear demonstration of the eﬀect. (In both experiments, the pulse length is extremely long compared with measured time shift in consequence of the superluminal sign al velocity as shown in Fig. 2.) Fig. 3. The same holds a fortiori for the two discussed experi ments. The ﬁnite duration of a signal is the reason that a superluminal velocity does not violate the principle of causality. On the other hand a shorter signal corresponds to a broader frequency band. In consequence of the dispersion relation of either a tunnelling barrier or of an excited atomic gas wit h an extremely narrow frequency regime of anomalous dispersion strong pulse reshaping would occur . Summing up, the principle of causality has not been violated by the experiments with superluminal s ignal velocities, but amazing the time span between cause and eﬀect has been reduced compared with l uminal propagation. References [1] A. Enders and G. Nimtz, J. Phys. I France 2,1693-1698 (1992) [2] J. J. Carey et al., Phys. Rev. Lett. 84, 1431-1434 (2000) [3] L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406,277-279 (2000) [4] G. Nimtz, Eur. Phys. J. B 7,523-525 (1999) [5] J. Marangos, Nature 406,243-244 (2000) [6] P. Mittelstaedt, Eur. Phys. J. B 13,353-355 (2000) [7] R. Sexl and H. Schmidt, Raum-Zeit-Relativit¨ at (vieweg studium, Braunschweig, 1978)[8] G. Nimtz, A. Enders, and H. Spieker, J. Phys. I France 4,565-570 (1994) [9] C. E. Shannon, Bell Syst. Tech. J. 27, 379-423 (1948); C. E. Shannon, Proc. IRE 37, 10-21 (1949) Acknowledgment We gratefully acknowledge helpful discussions with P. Mitt elstaedt.
arXiv:physics/0009044v1 [physics.gen-ph] 13 Sep 2000Universal Tunnelling Time in photonic Barriers A. Haibel and G. Nimtz Universit¨ at zu K¨ oln, II. Phys. Institut, Z¨ ulpicher Str.77, D-50937 K¨ oln, Germany Tunnelling transit time for a frustrated total internal reﬂ ection (FTIR) in a double–prism experiment was measured using microwave r adiation. We have found that the transit time is of the same order of magn itude as the corresponding transit time measured either in an unders ized wave- guide (evanescent modes) or in a photonic lattice. Moreover we have established that in all such experiments the tunnelling tra nsit time is approximately equal to the reciprocal (1 /f) of the corresponding fre- quency of radiation. Previous photonic tunnelling transit time experiments hav e been carried out using elec- tromagnetic radiation both at microwave and optical freque ncies. Such experiments were stimulated by the formal analogy between the classical Helm holtz wave equation and the quantum–mechanical Schr¨ odinger equation. The correspon ding tunnelling transit time data for e.g. electrons are not yet available. In our Letter we are considering the tunnelling transit time for opaque photonic barriers [1, 2, 3, 4]. We suggest that in general the transit or delay time i s approximately equal to the reciprocal frequency 1 /fof the corresponding radiation and that it is independent of the type or shape of the actual barrier. The transit time or gr oup delay time is deﬁned asτgr=x vgr, were xis the tunnelling distance and vgr=dω dk. This deﬁnition agrees with that introduced by Eisenbud and Wigner who put τϕ=dϕ dω=x dω/dk[5]. The tunnelling transit time or just tunnelling time for short, is measured a s the time interval between2 the respective times of arrival of the signal’s envelope at t he two ends of the tunnelling length x. We are not suggesting here that this is equivalent to the mea sure of the signal velocity within the barrier. In order to justify this hypothesis of a universal tunnellin g time we have carefully analyzed our own experimental results and those of others. Three diﬀe rent types of photonic barrier have been used. Investigations carried out in such experime nts as shown in Fig.1. b c a FIG. 1: Three types of the photonic barrier. a) A double–pris m, b) a photonic lattice of dielectric layers, c) an undersized waveguide In Fig.1a the tunnelling eﬀect occurs between two prisms (fr ustrated total internal re- ﬂection or FTIR) [6, 7, 8], in Fig.1b tunnelling is modelled b y the forbidden band of an one–dimensional photonic lattice [2, 3, 9], and in Fig.1c tu nnelling occurs in the under- sized section of the waveguide [1]. Since in one dimension th e Helmholtz and Schr¨ odinger equations are similar, it is suggested that the three kinds o f barrier can be used to model the one–dimensional process of wave mechanical tunnelling [10, 11]. Let us start by presenting some new data on the double–prism e xperiment. For n1> n 2 and an angle of incidence θi> θ c:= arcsin n2/n1the incoming beam penetrates into the second medium and travels for some distance along the interf ace before being scattered back into the ﬁrst medium (see Fig.2); here n1andn2are respectively the refractive indices of the ﬁrst prism and of the air. If a second prism with n3=n1=nis used to probe the “evanescent” component of the wave, the total reﬂe ction becomes “frustrated” and photonic tunnelling across the air gap takes place. It is indicated in Fig.2 that the barrier traversal time of th e double-prism, or what we3 call here the tunnelling time can be split into two component sttunnel=t/bardbl+t⊥, one along the surface due to the Goos-H¨ anchen shift D, and another part perpendicular to the surface [12]. The measured tunnelling time represents the g roup or phase time delay as explained earlier. The ﬁrst component is related to a non-ev anescent wave characterized t t dDθ FIG. 2: The tunnelling time of the double-prism experiment c onsists of two components. t/bardblfor the Goos–H¨ anchen shift Dparallel to the prism’s surface and t⊥for crossing the gap in the direction perpendicular to the two surfaces of the gap. by the real wavenumber k/bardbl:=k0nsinθiwhile the second one k⊥:=i k0/radicalBig n2sin2θi−1 is related to the evanescent mode traversing the gap between the two prisms. ( k0= 2π/λ0, λ0is the corresponding vacuum wavelength, and nthe refractive index of both prisms.) The Goos-H¨ anchen shift Dis a sensitive function of the gap width d, the frequency of radiation and its polarization, the beam width and the angle θiof the incoming beam [7, 13, 14]. With increasing air gap the shift reaches a constant asymptotic value D= dϕ/dk /bardbl[7, 13], where ϕis the phase shift of the reﬂected or transmitted beam. We have performed a double-prism experiment with two prisms of perspex, obtained from a 400 mm cube by a diagonal cut. The corresponding refractive index of n= 1.605 gives a total reﬂection angle of θc= 38.68◦. Microwave radiation at f= 8.45GHz generated by a 2K25 klystron was fed to a parabolic dish antenna which tr ansmitted a near parallel beam to the prisms. (Beam spread was less than 2◦). In order to appoint the tunnelling time we measured the time f or a signal travelling the closed and the opened prism. The transmission time through t he opened prism is faster4 than through the closed prism. Considering the modiﬁcation s of the path length the tunnelling time was determinated from the diﬀerence of both times. The signal was then picked up by a microwave horn antenna and f ed ampliﬁed to an oscilloscope (HP 54825A). Due to the Goos–H¨ anchen shift (s ee Fig.2) the position of the beam’s maximum had to be found by scanning the reﬂected and tr ansmitted beams. It was found that the signal had a Gaussian-like shape, its half -width being 8 ns [17]. Since the total propagation time (antenna–prism–antenna) is longer than the signal half– width, it is safe to assume that the transmitter, the prism, a nd the detector are well decoupled since there is no danger of the circuit components being coupled by a standing wave building up. The experimental set–up permits asymptot ic measurements. The tunnelling time was measured at the frequency of 8.345GH z (vacuum wavelength λ0 = 36 mm) using a TE-polarized beam. The beam diameter was 190m m and the angle of incidence was chosen to be θi= 45◦. We tested whether all beam components were parallel and whet her the angle of incidence was within the regime of total reﬂection by measuring at two d iﬀerent frequencies the transmission as a function of the gap between the two prisms. The measured transmission was 0.73dB/mm at 8.345GHz and 0.93dB/mm for 9.72GHz respect ively compared to the theoretical values of 0.76dB/mm and 0.94dB/mm (see Fig.3). This agreement between the theoretical (as quoted for k⊥in [11]) and experimental results indicates that our method of measuring FTIR is very sensitive, provided the bou ndary conditions are well deﬁned. The tunnelling time was measured in the regime of constant as ymptotic Goos–H¨ anchen shiftD, where in our case (see experimental parameters given above )D= 31mm. The timetGH=t/bardblfor the Goos–H¨ anchen shift can be obtained from [7] by writi ng: tGH≡t/bardbl=D nsin(θi) c(1) ForD= 31mm, n = 1.605 and θi= 45◦we obtain from (1) tGH= 117ps. Actually this value equals the measured Goos–H¨ anchen time for the to tal reﬂected beam in the absence of the second prism. (The measured time was obtained by properly taking into5 0 10 20 30 40 50 Air Gap [mm]−40−30−20−100Transmission [dB]8.345GHz, 0.73dB/mm 9.72GHz, 0.93dB/mm FIG. 3: Transmission vs Air Gap measured at two diﬀerent freq uencies. consideration the beam’s path in the prism.) As mentioned be fore for a transmitted beam the total tunnelling time is the sum of the two components t/bardblandt⊥. Surprisingly the measured total tunnelling time proved to b e equal to t/bardblalone. Since our accuracy of time measurement was ±10 ps, this means that t⊥≤10ps or at least t⊥=0. Thus it would appear that the measured total tunnelling time depends mostly on the Goos- H¨ anchen shift and hence is approximately equal to the Goos– H¨ anchen time tGH. This result is compatible with some theoretical investigations bearing in mind the imaginary wavenumber k⊥of the evanescent mode in the gap [15]. For large gaps where the transit time does not depend on dthe theoretical value for the FTIR-tunnelling time is 82 ps, using the model of Ghatak and B anerjee [16]. This value is quite near to the measured value of 117 ±10 ps. It is now quite interesting to note that the reciprocal of the carrier frequency, 1 /f= 120ps, gives approximately the same value for the time interval as the measured tunnelling time. This r esult is also in agreement with the theoretical model of Ghatak and Banerjee, being val id over a wide range of frequencies and at all angles of incidence, except in the vic inity of the critical angle θc and for θi>80◦(grazing incidence) (see Fig.4). This relationship we have obtained for FTIR seems to be a univ ersal property of many tunnelling processes. Some previously obtained experimental results are collect ed in Table I; they all seems6 2e+09 4e+09 6e+09 8e+09 1e+10Carrier Frequency [Hz] 405060708090 Angle of Incidence [Degrees]02e-104e-106e-108e-101e-09Tunnelling Time [s] FIG. 4: Tunnelling time calculated using the reciprocal of t he carrier frequency. The model of Ghatak and Banerjee [16] (dashed lines) is in quite a good agreement with the above calculations over a wide frequency range and for most a ngles of incidence (expect the critical angle θc= 38.68◦and the grazing incidence (angles beyond 80◦) ). to conﬁrm the suggested universal property that the tunnell ing time is approximately equal to the reciprocal of the carrier frequency. Some devia tions may have arisen from two experimental short comings: the studied barriers have n ot been suﬃciently opaque or some tunnelling experiments were too diﬃcult to perform. Our experimental data obtained from FTIR using microwave ra diation show that the ﬁnite tunnelling time is largely dependent on the interference eﬀ ects at the entrance boundary of the barrier. In the case of FTIR it is the time equivalent to the Goos-H¨ anchen shift as was ﬁrst pointed out by Stahlhofen [15]. We have also checked using a waveguide at a microwave frequen cy of 8.85GHz, whether a similar behaviour applies in the case of a photonic lattice type shown in Fig.1b. The measured group delay time τreﬂ= 75±5ps of the reﬂected beam was found to be the same as the time measured for traversing the barrier, τtrans= 74±5ps, or what we have called the tunnelling time. Once again there is no indicatio n that the evanescent mode7 Photonic Barrier Reference Tunnelling Time Reciprocal Frequency Double–Prism FTIR this paper 117 ps 120 ps Carey et al. [6] ≈1 ps 3ps Balcou/Dutriaux [7] 40 fs 11.3 fs Mugnai et al. [8] 134 ps 100 ps Photonic Lattice Steinberg et al. [2] 1.47 fs 2.3 fs Spielmann et al. [3] 2.7 fs 2.7 fs Nimtz et al. [9] 81 ps 115 ps Undersized Waveguide Enders/Nimtz [1] 130 ps 115 ps TABLE I: Results of tunnelling time measurements using thre e diﬀerent types of photonic barrier and performed at quite diﬀerent frequencies. spend any time inside the barrier, similary to FTIR [17]. Hartman [18] calculated the tunnelling time (phase time del ay) of Gaussian wave packets for one-dimensional barriers based on the time dependent Sc hr¨ odinger equation. It is interesting to note that his theoretical wave-mechanical r esults are also in agreement with the photonic experiments for diﬀerent barrier lengths [19] . All experimental measurements of the tunnelling time are in agreement with the theoret- ical calculations and indicate a universal tunnelling time in the case of opaque barriers. Both the measured ﬁnite total tunnelling time and the time de lay of the reﬂected beam are associated with the front of the barrier and closely corr elate with the reciprocal of frequency of the corresponding radiation. We gratefully acknowledge helpful discussions with P. Mitt elstaedt, A. Stahlhofen, R.– M. Vetter, and the Referee who made us familiar with the calcu lations of Ghatak and Banerjee. G.N. likes to thank X. Chen and P. Lindsay for stimu lating discussions during his short stay at Queen Mary and Southﬁeld College London.8 REFERENCES [1] A. Enders and G. Nimtz, J. Phys.I, France 2, 1693 (1992) [2] A. Steinberg, P. Kwiat, and R. Chiao, Phys. Rev. Letters 71, 708 (1993) [3] Ch. Spielmann, R. Szip¨ ocs, A. Stingle, and F. Kraus, Phy s. Rev. Letters 73, 2308 (1994) [4] A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, Phys. Rev . E48, 1453 (1994) [5] Merzbacher, E., Quantum Mechanics , 2nd ed., John Wiley & Sons, New York (1970) [6] J. J. Carey, J. Zawadzka, D. Jaroszynski, and K. Wynne, Ph ys. Rev. Letters, 84, 1431 (2000) [7] Ph. Balcou and L. Dutriaux, Phys. Rev. Letters 78, 851 (1997) [8] D. Mugnai, A. Ranfagni, and L. Ronchi, Phys. Letters A 247, 281 (1998) [9] G. Nimtz, A. Enders, and H. Spieker, J. Phys. I., France 4, 565 (1994) [10] A. Sommerfeld, ’Vorlesungen ¨ uber Theoretische Physi k’, Band IV, Optik, Dieterich- sche Verlagsbuchhandlung (1950) [11] Feynman, R. P.,Leighton, R. B. and Sands, M., ’The Feyma n Lectures on Physics’, Addison–Wesley Publishing Company, II33–12 (1964) [12] Artmann, K., Ann. Phys. 287-102 (1948) [13] Cowan, J. J. and Anicin, B., J. Opt. Soc. Am. 671307-1314 (1977) [14] Horowitz, B. R. and Tamir, T., J. Opt. Soc. Am. 61586-594 (1971) [15] A. A. Stahlhofen, Phys. Rev. A 62, 12112 (2000) [16] A. Ghatak and S. Banerjee, Appl. Opt. 28, 1960 (1989) [17] G. Nimtz, Eur. Phys. J. B 7, 523 (1999) [18] Th. Hartman, J Appl. Phys. 33, 3427 (1962) [19] A. Enders and G. Nimtz, Phys. Rev. E 48, 632 (1993)
1 RELATIVISTIC GEOMETRY AND WEAK INTERACTIONS Gustavo González-Martín Departamento de Física, Universidad Simón Bolívar, Apartado 89000, Caracas 1080-A, Venezuela. Web page URL http:\\prof.usb.ve\ggonzalm\ Geometric interactions in a new relativistic geometric unified theory include interactions other than gravitation and electromagnetism. In a low energy limit one of these interactions leads essentially to a Fermi type theory of weak interactions.SB/F/271-992 Introduction. A non linear geometric theory that unifies gravitation and electromagnetism offers the possibility of representing other interactions by a sector of the connection [1]. It has been shown that new electromagnetic consequence of thetheory lead to quanta of electric charge and magnetic flux, providing a plausible explanation to the fractional quan-tum Hall effect [2], [3]. On the other hand we also have considered an approximation to this geometric non linear theory [4] where the microscopic physical objects (geometric particles) are realized as linear geometric excitations,geometrically described in a jet bundle formalism shown to lead to the standard quantum field theory techniques.These geometric excitations are essentially perturbations around a non linear geometric background space solution,where the excitations may be considered to evolve with time. In this framework, a geometric particle is acted upon bythe background connection and is never really free except in absolute empty background space (zero backgroundcurvature). The background space carries the universal inertial properties which should be consistent with the ideasof Mach [5] and Einstein [6] that assign fundamental importance of far-away matter in determining the inertialproperties of local matter. We may interpret the geometric excitations as geometric particles and the background asthe particle vacuum. This is a generalization of what is normally done in quantum field theory when particles areinterpreted as vacuum excitations. The vacuum is replaced by a geometric symmetric curved background space andcurrent solution which we call the substratum. The field equation of the theory admits a constant connection andcurrent substratum solution [7], [8]. The role that Clifford algebras play in the geometrical structure of the theory provides a link to the non classical interactions theories. The structure group of the physical geometry is the simple group of automorphisms of theClifford algebra. The combinations of the fundamental geometric excitations corresponding to the holonomy sub-groups of the connection display an interesting SU(3) /G31SU(2)/G31U(1) combinatorial symmetry [9]. The standard model, that shows this symmetry, has had many successes in describing weak and strong interactions that lead to a generalacceptance of the model. Nevertheless, nowadays, this model may be considered an effective theory of some othermore general theory. History has shown us that in many cases, progress in physics is attained by the evolution orreplacement of models, that provide partial fits to experimental data, by more general ones. The consideration ofgeometrical strings to represent physical concepts indicates a present trend in this direction. Therefore, in the questof unification, the lack of an a established relation between geometry and the standard model should not deter us frominvestigating the possible physical interpretations of the odd sector of the connection which precisely holds the key toits relation with the standard model. This sector certainly does not represent simply the classical interactions and, inparticular, may help clarify a possible relation to the standard model. As a matter of fact, the physical geometryindicates there is complementary approach to particles and their interactions related to another model [10]. As firsttask we consider here low energy aspects of weak interactions. The holonomy groups of the connection may be used geometrically to classify the interactions contained in theory. The subgroup chain SL(4,R) ⊃ Sp(4,R) ⊃ SL(2,C) characterizes a chain of subinteractions with reducing sectors of non classical interactions In this article we shall mainly concern ourselves with Sp(4,R) and SL(2,C) connections A section is related by charts (coordinates) to elements of the group, SL(4,R), which are matrices that form a frame of general SL(4,R) column spinors. The connection is a sl(4,R) 1-form that acts naturally on the frame e (sections). The field equation, which relates the derivatives of the curvature to a current source J is Dk J J∗∗ ∗==Ω 4πα , ( 1) Je u eµα αµκ=~/G24 /G24 , ( 2) in terms of the frame e, an orthonormal set of the algebra κ, the correlation in spinor spaces and an space-time tetrad u. The coupling constant is 4πα, where α, is the fine structure constant. This relation introduces a fundamental length, as we shall see later.The field equation implies a conservation law for the geometric current, which determines a generalized Dirac equa-tion in terms of the frames. This equation, for the even an odd parts of a frame f reduces, under certain restrictions [11], to κ∂κµ µµ µff m f+− − −==Γ , ( 3)3 κ∂κµ µµ µff m f−− + +==Γ , ( 4) implying that a frame for a massive corpuscle must have odd and even parts. In our case fm−=⇒0 =0 . ( 5) Therefore, for an even frame we have, multiplying by κ0, σ∂µ µf+=0 , ( 6) which is the equation normally associated with a neutrino field. A fluctuation of f+ on the fixed background obeys also the last equation.We also have suggested that particles may be represented by excitations on a geometric background. In particular, theelectron and neutrino, at fixed states, correspond to matrices with only one non zero column which form an algebraicrepresentation of the group. Geometric Weak Interaction. If the geometric theory has anything to do with the weak interactions, it should be possible to represent electronneutrino interactions within these geometric ideas. From the discussion of previous articles, we consider that anelectroweak interaction may be related to the action of the Sp(4,R) holonomy group. The total interaction field shouldbe represented by a sp(4,R) connection Γ. The total matter current should be associated to a Sp(4,R) frame f present- ing both the electron field e and the neutrino field ν. to avoid confusion, in this article we use the symbol f for a general section reserving e for electron sections At a point, the total frame f of the interacting e, ν is related to an element of the group Sp(4,R), a subspace of the geometric algebra R3,1, =R(4). The frame f may be decomposed into fields associated to the particles e, ν by means of the addition operation within the algebra. These e, ν fields are not necessarily frames because addition does not preserve the group subspace Sp(4,R) and geometrically , e ν are sections of a fiber bundle with space-time as base and the Clifford algebra as fiber. The source current J in the theory is Jf ff f==~κκ ,( 7 ) where f is the frame a section associated to the total field of the electron and the neutrino and κ represents the orthonormal set. For the Sp(4,R) group the correlation reduces to conjugation.Due to the properties that a neutral particle field should have, we consider that the effect produced by ν should be small relative to the effect of e (there are no electromagnetic or massive effects). Hence we may assume that, in the composite system, ν is a perturbation of the order of the structure constant α, the only physical constant in the theory. fe=+αν . ( 8) Then the current becomes, () () () Je e e e e e=+ + = + + +αν κ αν κ α κν νκ α νκν2 . ( 9) The intermediate terms may be considered as a perturbation of order α to a background electronic matter frame. The perturbation current may be written, by splitting e into its even and odd parts and noticing that ν has only even part () () [] Je e−= + + +++ κα η ξ κ κ νν κ η κ ξ00 . ( 10) As usual in particle theory, we neglect gravitation, which is an even SL(2,C) connection. If we look for effects not imputable to gravitation, it is logical to center our attention on the odd part of the perturbation current as a candidatefor the interaction current ()αα η κ ν ν κ ηµµ µj=+ . ( 11)4 It should be noted that the neutrino ν automatically associates itself, by Clifford addition, with the even part η of the electron. This corresponds to the Weinberg Salam association of the left handed components as a doublet, with thesame Lorentz transformation properties.If we apply perturbation theory to the field equations, we expand the connection Γ in terms of the coupling constant α. We have, JJ J J=+ + +012 2 αα /G4C , ( 12) ΓΓ Γ=+ + +012 2 αΓ α /G4C . ( 13) The background equation and the first varied equation, which is second order in α, have the following structure, ()DJ∗=ΩE E4πα , ( 14) ()δπ α δDJ∗=Ω4 . ( 15) If we let, the first order terms be, Jj1= , ( 16) Γ1=W , ( 17) we obtain for the variation, the linear equation, ()απ αdd W L W j∗+=42( 18) where L is a linear first order differential operator determined by the background. This equation may be solved in principle using its Green ’s function /G47. The solution in terms of components with respect to a base Ea in the algebra is () ( ) W dx x x jxi ij j µ µννπα=′ −′′∫4 /G47 . ( 19) It is well known that the second variation of a Lagrangian serves as Lagrangian for the first varied Euler equations. Therefore, the ΓJ term in the covariant derivative present in the Lagrangian proposed in [1] provides an interaction coupling term that may be taken as part of the Lagrangian for the process in discussion. When the ΓJ term is taken in energy units, considering that the Lagrangian has an overall multiplier, it should lead to the interaction energy for theprocess. The second variation (or differential) in a taylor expansion of the energy U, corresponds to the Hessian of U. () () Ux UU xxU xxxxii jiij001 22 ,δ∂ ∂δ∂ ∂∂δδ =+ + + /G4C ( 20 ) The ΓJ term gives the interaction Lagrangian for the background and for the perturbations, respectively, []() /G4CEE=− + ≈−1 41 2trJJ j AEE E Eµ µµµµ µ ΓΓ , ( 21) () /G4C=-αµµµ µ21 41 2trWj jW+ . ( 22) It is clear that this interaction is carried by Γ or W. Nevertheless, we wish to obtain a current -current interaction to compare with other theories at low energies. Substitution of eq. (19) in the last equation gives the correspondingaction which, for clarity, we indicate by, ()[] () ( ) /G57 /G47 =− ′−′+ ′∫231 4παµνµνtrdxdxiji l l i jl xxE E E Ejx jx , ( 23)5 representing a current-current interaction Hamiltonian, with a coupling constant derived from the fine structure constant. This new expression may be interpreted as a weak interaction Fermi Lagrangian. The associated couplingconstant is of the same order as the standard weak interaction coupling constant, up to terms in the Green ’s functions /G47 . Relation with Fermi’s Theory The action, in terms of elements of the algebra and the trace, corresponds to the scalar product. For the case of the isotropic homogeneous constant solution [12] the Greens function is a multiple of the unit matrix with respect to thealgebra components. We assume here that for a class of solutions the Green ’s function have this property. Then for this class of solutions, the last equation may be written as () ( ) ( ) /G57 /G47 =− ′−′′ ∫431 4παµνµνtrdxdx x x j x j x , ( 24) where the j are matrices. The current defines an equivalent even current j-, the κ0 component, by inserting κ0κ0 and commuting, as follows, () () jµµµ µ µηκ κ κ ν νκ κ κ η κ η σ ν ν σ η=+= +00 00 0 ✝✝ , ( 25) j−=ησ ν✝ + h.c. . ( 26) Each 2×2 block component of this 4 dimensional real current matrix is an even matrix which may be represented as a complex number, by the known isomorphism between the complex algebra and a subalgebra of 2 ×2 matrices. Thus, we may use, instead of the generalized spinors forming the frame f, the standard spinor pairs, η1 η2 and ν1, ν2 which form the matrices corresponding to the even part, respectively, of the electron and neutrino. The first term of j-, which corresponds to a non hermitian term, is ησνησν ησν ησν ησνµµµ µµ✝✝✝ ✝✝/G24B A=  1 12 1 1 22 2 . ( 27) In this manner, if we use the standard quantum mechanics notation, using Weyl ’s representation, each component is of the form () jenµµ µγγ η σ ν=+ =1 215ΨΨ✝ , (28) which may be recognized as the standard non hermitian Fermi weak interaction current for the electron neutrino system. We shall indicate by jF .this current plus its hermitian conjugate. As discussed in the previous publications, within this theory particles would be represented by excitations of frames.These fluctuations are matrices that correspond to a single representation of the subgroup in question. This meansthat for each pair of spinors of a spinor frame, only one is active for a particular fluctuation matrix. In other words, fora fluctuation, only one of the components of the matrix in eq. (27) is non zero and we may omit the indices on thespinors. Now we may evaluate the trace of the currents in the expression for /G57/G20, ()() () ( ) tr tr tr trjj j j j j j j===−− − − − −κκ κ κ00 0 0 ✝✝ . ( 29) We see from the equations that there are various terms contributing to /G57. In particular there is one term of the form,6 ()() ( ) () ()() () ()() ()tr† h.c. h.c. h.c. h.c.ησ ν η σ ν ησ ν η σ νµν µν µν✝✝ ✝✝xx xxjj xx xx+   ′′+     == +× ′′ +0 000 0000 , ( 30) where the right side is in complex number notation instead of 2 ×2 even matrices. This term combines with its hermitian conjugate. It should be noticed that in going from the real 4 ×4 to the 2 ×2 complex matrix realizations the 1/4 in front of the trace changes to 1/2.We have the result that the expression for /G57 has a term of the form, () ( ) ( ) /G57 /G47 =− ′−′′∫23πα µνµνdxdx xxj x jx00✝ . ( 31) If we further assume simplifications in the Green ’s function, which in particular are met by the fluctuations around a constant solution, () ()∗∗++ =dd m m Jgg δω δω δω δ παδδα δα ρρ δα δα /G24/G24/G24 /G24/G24 22 2 422 , ( 32) which is a Yukawa equation because of the constant background ω. In general, we may expect that the Green ’s function has a Dirac time δ function that allows integration in t’. The spatial part of the Green ’s function should provide an equivalent range for the interaction. For example, if we assume a constant solution the equation reduces,for a point source, to the radial equation, and the Green ’s function is () /G47=− −′−−′1 4πµe rrrr ( 33) If we further assume that the currents vary slightly in the small region of integration so that j(x’) approximately equals j(x), we obtain an approximation for the Hamiltonian contained in the geometric theory. () () () ()/G57w 00 =− • ′ = −•−′∞ ∫∫ ∫ ∫1 2 23 002 04 0 3 2α πα µµπ dxj x j x dr re d dxj x j xr ✝ ✝Ω . . ( 34) This expression fixes the value of a weak interaction constant G in terms of the fine structure constant α and a characteristic length of the Green ’s function /G47. The value of G, determined from results of experiments like the muon decay is G=1.16639 ×10-5 GeV-2. Comparing the theoretical value with the experimental value, we may assign a mass to the value of ω, −   =− 2 23 2πα µϕcosG . ( 35) ()µπ α== =−2 2 54409234123212mGg . M e v . , ( 36) which corresponds to a mass close to the η particle. From results of previous articles, in particular for the homogeneous isotropic constant solution, using the samenotation, the value of the range R w 2 may be obtained theoretically in terms of a geometric fundamental unit of length, designated as g.7 Rmw2 2221 41 1650769401116161095755 == = =ω.. g2 , ( 37) The value of the geometric fundamental unit g would then be calibrated to 1 06905811 Gevg−=. ( 38) 1 285743 10 028574314 g= c m= f..×−( 39) We recognize that the general perturbation for the interaction of the electron and the neutrino fields includes eqs. (28, 34) which essentially are the current and Lagrangian assumed in Fermi ’s theory [13, 14] of weak interactions [15, 16] of leptons. Fermi ’s theory is contained, as a limit within the unified theory of connections and frames discussed in these series of papers. On the other hand, the full theory is accepted there are certainly new effects and implicationsto consider, which should be determined without making the phenomenological considerations made in this sectionto display the relation of our theory to low energy weak interactions. In particular, we should not expect that electroweak theory is related to another coupling “constant”, Because of the non linearity of the theory, it is not correct to assume that if we subtract from a full solution, a partial electromagneticsolution, we get another solution. The same thing applies to strong nuclear forces. Strong and nuclear interactionswere historically introduced to account for physical phenomena not explained by electromagnetic and gravitationalfields. We may say that nuclear effects are residual, in the sense that they theoretically correspond to the residue ofsubtracting a solution to linear equation from a non linear equation Conclusion. It was shown that the general perturbation technique for the interaction of the electron and neutrino fields lead to eqs.(28,34) which are essentially the current and Lagrangian assumed in Fermi ’s theory of weak interactions of leptons. Fermi’s theory is contained, as a low energy limit within the geometric unified theory of connections and frames discussed in these series of papers. It is also clear that if the full theory is accepted there are certainly new effects. The geometric fundamental unit of length was calibrated in terms of the accepted value for weak interaction coupling constant calculated from muon decay. From this knowledge, a value for the range of the W was estimated which although far from other theoretical estimates may be considered within the experimental “ballpark”, without present day theoretical assumptions. Of course, we must consider in the future high energy applications which may shed somelight into the relation of this geometry with the standard model [17,18]. 1 G. Gonz ález-Martín, Gen. Rel. and Grav. 22, 481 (1990); G. Gonz ález-Martín, Physical Geometry, (Universidad Simón Bolívar, Caracas) (2000), Electronic copy posted at http:\\prof.usb.ve\ggonzalm\invstg\book.htm 2 G. Gonz ález-Martín, Gen. Rel. and Grav. 23, 827 (1991). 3 G. Gonz ález-Martín, Charge to Magnetic Flux Ratios in the FQHE , Universidad Sim ón Bolívar Report, SB/F/273-99 (1999). 4 G. Gonz ález-Martín, Gen. Rel. Grav. 24, 501 (1992); See related publications . 5 E. Mach, The Science of Mechanics, 5th English ed. (Open Court, LaSalle), ch. 1 (1947). 6 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p.55 (1956). 7 G. Gonz ález-Martín, Fundamental Lengths in a Geometric Unified Theory, preprint 96a (1997). 8 G. Gonz ález-Martín, p/e Geometric Mass Ratio , Universidad Sim ón Bolívar Report, SB/F/274-99 (1999). 9 G. Gonz ález-Martín, The Importance of Symmetric Spaces for the Geometric Classification of Particles and Interactions , to be published as Universidad Sim ón Bolívar Report SB/F/279-00 (2000). 10W. T. Grandy, Found. of Phys. 23, 439 (1993). 11G. Gonz ález-Martín, Phys. Rev. D 35, 1225 (1987).8 12G. Gonz ález-Martín, Fundamental Lengths in a Geometric Unified Theory, USB preprint 96a (1996). 13E, Fermi, Z. Physik 88, 161 (1934). 14E. Fermi, N. Cimento, 11 1 (1934). 15R. Feyman and M. Gell-Mann, Phys. Rev, 109, 193 (1958). 16E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109, 1860 (1958). 17S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). 18A. Salam, in Elementary Particle Theory, ed.N. Swartholm (Almquist and Wissell, Stockholm) (1968).
arXiv:physics/0009046v1 [physics.space-ph] 13 Sep 2000Average observed properties of the Earth’s quasi-perpendi cular and quasi-parallel bow shock A. Czaykowska, T. M. Bauer, R. A. Treumann, and W. Baumjohann Max-Planck-Institut f¨ ur extraterrestrische Physik, Gar ching, Germany acz@mpe.mpg.de, thb@mpe.mpg.de, tre@mpe.mpg.de, bj@mpe .mpg.de ABSTRACT We present a statistical analysis of 132 dayside (LT 0700-17 00) bow shock crossings of the AMPTE/IRM spacecraft. We perform a superposed epoch a nalysis of plasma and magnetic ﬁeld parameters as well as of low frequency magneti c power spectra some min- utes upstream and downstream of the bow shock by dividing the events into categories depending on the angle θBnbetween bow shock normal and interplanetary magnetic ﬁeld and on the plasma- β, i.e., the ratio of plasma to magnetic pressure. On average, the proton temperature is nearly isotropic downstream of th e quasi-parallel bow shock (θBn<45◦) and it is clearly anisotropic with Tp⊥/Tp/bardbl≈1.5 downstream of the quasi- perpendicular bow shock ( θBn>45◦). In the foreshock upstream of the quasi-parallel bow shock, the power of magnetic ﬂuctuations is roughly 1 ord er of magnitude larger (δB∼4nT for frequencies 0.01–0.04 Hz) than upstream of the quasi -perpendicular bow shock. There is no signiﬁcant diﬀerence in the magnetic p ower spectra upstream and downstream of the quasi-parallel bow shock, only at the b ow shock itself magnetic power is enhanced by a factor of 4. This enhancement may be due to an ampliﬁcation of convecting upstream waves or due to wave generation at the shock interface. On the contrary, downstream of the quasi-perpendicular bow shock the magnetic wave activity is considerably higher than upstream. Downstream of the qua si-perpendicular low- β bow shock we ﬁnd a dominance of the left-hand polarized compo nent at frequencies just below the ion cyclotron frequency with amplitudes of ab out 3 nT. These waves are identiﬁed as ion cyclotron waves which grow in a low- βregime due to the proton temperature anisotropy. We ﬁnd a strong correlation of this anisotropy with the inten- sity of the left-hand polarized component. Downstream of so me nearly perpendicular (θBn≈90◦) high- βcrossings mirror waves are identiﬁed. However, there are al so cases where the conditions for mirror modes are met downstream of t he nearly perpendicular shock, but no mirror waves are observed. Subject headings: Earth: bow shock – Earth: magnetosheath– 2 – 1. Introduction Since the beginning of the space age, the Earth’s bow shock is of particular interest because it serves as a unique laboratory for the study of shock waves i n collisionless plasmas. Most of our understanding of structure, dynamics, and dissipation pro cesses of such shocks has come from in situ spacecraft measurements crossing the bow shock. Early observations of waves and particles upstream of the bow shock can be found in the special issue of t heJournal of Geophysical Research , 86, pp. 4317–4536 [1981]. A collection of observational and th eoretical work on the bow shock is contained in Stone & Tsurutani [1985] and Tsurutani & Stone [1985]. Reviews focusing on the dissipation processes taking place at the Earth’s bow shock have been given by, e.g., Kennel et al . [1985] and, more recently, by Omidi [1995]. Plasma wave observations across the bow shock in the high frequency range have been reviewed by Gurnett [1985]. Schwartz et al . [1996] have reviewed results concerning low frequency waves in the magnetosheat h region behind the bow shock. Many of the previous measurements have demonstrated that a l arge variety of nonthermal particles is generated at the bow shock. While nonthermal el ectrons can act as a source for high frequency waves, nonthermal ions can be responsible for low frequency waves. Electrons and ions reﬂected at the shock stream sunward along the interplaneta ry magnetic ﬁeld, thus forming the electron and ion foreshock, respectively. Structure, dynamics, and dissipation processes of the bow s hock vary considerably depending on the angle θBnbetween the upstream magnetic ﬁeld and the shock normal, on t he plasma β, i.e., the ratio of plasma to magnetic pressure in the upstream regi on, and on the Mach numbers MA orMms, i.e., the ratios of the solar wind velocity along the shock n ormal to the upstream Alfv´ en or magnetosonic speed. For quasi-perpendicular shocks wit hθBn>45◦, the main transition from the solar wind to the magnetosheath is accomplished at a shar p ramp. In contrast, quasi-parallel shocks with θBn<45◦consist of large-amplitude pulsations extending into the f oreshock region. For larger Mach numbers this pulsating structure continuously re-forms by virtue of collisions between convecting upstream waves and the shock [ Burgess , 1989] or due to an instability at the interface between solar wind and heated downstream plasma [ Winske et al ., 1990]. Two-ﬂuid theories of shocks have indicated the presence of a critical magnetosonic Mach num- ber,M∗, above which ion reﬂection is required to provide the necess ary dissipation. However, it has been demonstrated by observations [ Greenstadt & Mellott , 1987; Sckopke et al ., 1990] that ion reﬂection occurs also below M∗and that the distinction between subcritical ( Mms< M∗) and supercritical ( Mms> M∗) shocks is not sharp. Whereas the ramp of quasi-perpendicul ar shocks at high Mach numbers is preceded by a foot and followed by an ov ershoot, these features are less prominent at low Mach numbers [ Mellott & Livesey , 1987]. While quasi-parallel shocks are steady at low Alfv´ en Mach numbers, MA≤2.3, they become unsteady for higher Mach numbers, where they continuously re-form [ Krauss-Varban & Omidi , 1991]. An important role in the dissipation process is played by ion s reﬂected at the bow shock. At quasi-parallel shocks they can escape from the shock into th e foreshock region and drive ion beam– 3 – instabilities. These instabilities may excite large-ampl itude waves observed in the foreshock region, e.g., by Le & Russell [1992] and Blanco-Cano & Schwartz [1995]. At quasi-perpendicular shocks the reﬂected ions gyrate back to the shock and enter the downstre am region, where their presence leads to a strong perpendicular temperature anisotropy [ Sckopke et al ., 1983]. This anisotropy leads to the generation of ion cyclotron and mirror waves [e.g., Price et al. , 1986; Gary et al. , 1993]. These waves have been observed in the Earth’s magnetosheath, e.g. , bySckopke et al. [1990] and Anderson et al. [1993,1994]. Closer to the magnetopause the mechanism of ﬁe ld line draping leads to the formation of anisotropic ion distributions and the formati on of a plasma depletion layer. Waves in this environment have also been described by Anderson et al. [1993,1994]. Large-amplitude mirror waves have been observed in planetary magnetospheres, e.g. , byBavassano-Cattaneo et al. [1998] in Saturn’s magnetosphere, where the ion temperature aniso tropies are due to both shock heating and ﬁeld line draping. In the present study we investigate the average behavior of p lasma and magnetic ﬁeld param- eters including the low frequency magnetic wave power as mea sured by AMPTE/IRM during a fairly large number of bow shock crossings. We show that, as e xpected, quasi-perpendicular and quasi-parallel bow shocks behave diﬀerently even in their a verage properties. This diﬀerence has been quantiﬁed in our investigation. Section 2 provides a sh ort description of the available data. It is followed by Section 3 which compares the properties of q uasi-perpendicular and quasi-parallel shocks. In Section 4 low and high- βbow shock crossings are compared for the quasi-perpendicul ar shock, and it is outlined why a classiﬁcation by βis preferred to a classiﬁcation by Mach number. Finally, Section 5 presents concluding remarks. 2. Data description The present analysis uses data from the AMPTE/IRM satellite . From the periods when the apogee of AMPTE/IRM was on the Earth’s dayside (August – Dece mber 1984 and August 1985 – January 1986), we have selected all crossings of the satelli te through the Earth’s bow shock in the local time interval 0700 – 1700, whenever there was a reasona ble amount of data measured on both sides of the bow shock, i.e., at least 2min upstream and 4 min d ownstream. Altogether this gives 132 events, with some events belonging to multiple crossing s due to the fast movement of the bow shock relative to the slowly moving satellite. Due to the sat ellite’s orbital parameters, all crossings occurred at low latitudes, i.e., in the interval ±30◦from the ecliptic plane. We analyze the data from the triaxial ﬂuxgate magnetometer described by L¨ uhr et al. [1985] which gives the magnetic ﬁeld vector at a rate of 32 samples per second. In addition, we use t he plasma moments calculated from the three-dimensional particle distribution functions me asured once every spacecraft revolution (∼4.3 s) by the plasma instrument [ Paschmann et al. , 1985]. In Fig. 1 we show the locations of the individual bow shock cro ssings rotated into the ecliptic along meridians. Cases where the angle θBnis less than 45◦, i.e., quasi-parallel events, and cases where the angle θBnis greater than 45◦, i.e., quasi-perpendicular events, are shown in addition– 4 – Fig. 1.— GSM-Positions of the 132 AMPTE/IRM bow shock crossi ngs rotated to the ecliptic plane along meridians. Coordinates are given in Earth radii (RE). Quasi-perpendicular crossings are marked with a square, quasi-parallel crossings with a tr iangle. The solid curve represents the best ﬁt hyperbola of Fairﬁeld [1971]. to the best ﬁt hyperbola of Fairﬁeld [1971] using data from the Imp 1 to 4 and Explorer 33 and 35 spacecraft. It is found that most of the AMPTE/IRM bow shoc k crossings occurred closer to the Earth than Fairﬁeld’s average bow shock. Since we analyz e only bow shock crossings on the dayside, the best ﬁt hyperbola derived from our data is not re liable at the ﬂanks. However, the distance of the subsolar point is well deﬁned. We ﬁnd a value o f 12.3 RE, which is more than 2 RE– 5 – closer to the Earth than the value of 14.6 REfound by Fairﬁeld. In his study, Formisano [1979] analyzed 1500 bow shock crossings. He normalized the observ ed distance Robsof these crossings to an average value of the solar wind dynamical pressure using Rnorm=Robs/parenleftbiggnobsv2 obs n0v2 0/parenrightbigg1/6 (1) with a typical value of the solar wind speed v0= 450 km/s and particle density n0= 9.4 cm−3. He derived a distance of the subsolar point of 11.9 RE. Applying the same normalization to the AMPTE/IRM data, we ﬁnd a value of 11.7 RE, which is in good agreement with the result of Formisano [1979]. This indicates that the diﬀerence of the distance of the subsolar point between Fairﬁeld’s and our study is due to diﬀerent average solar win d dynamical pressure. We interpret this ﬁnding as a solar cycle eﬀect since the AMPTE/IRM data ar e obtained close to solar activity minimum, whereas Fairﬁeld’s data set is from the years 1964- 1968, when solar activity increased from minimum to maximum. In solar minimum the Earth is hit mor e frequently by high speed solar wind streams than in solar maximum. The high speed sola r wind has, although less dense, a higher dynamical pressure than the slow solar wind. Hence, the solar wind dynamical pressure is usually higher on average during solar minimum than durin g solar maximum [ Fairﬁeld , 1979]. In addition, during solar minimum, the heliospheric plasma sheet described by Winterhalter et al. [1994] is fairly ﬂat, i.e., near the ecliptic plane. With its very high densities it can enhance the solar wind pressure although the solar wind velocity is only around 350 km/s. In a more recent study, Peredo et al. [1995] investigated 1392 bow shock crossings from 17 spacecraft during the years 1963–1979, i.e., one and a half s olar cycles. They found a dependence on the Alfv´ en Mach number MA. With the average Alfv´ en Mach number MA= 5.6±2.9 of the AMPTE/IRM data set the distance of the subsolar point should be in the range of 14.0-14.9 RE. Performing a normalization with the average values of n0= 7.8 cm−3andv0= 454 km/s used by Peredo et al. [1995], the distance of the subsolar point of the AMPTE/IRM d ata set is 12.1 RE. The results of Peredo et al. [1995] are thus not in agreement with our results and those of Formisano [1979]. Peredo et al. [1995] explain this disagreement with the fact that the stud y ofFormisano [1979] is biased by the dominance of the high latitude HEOS 2 b ow shock crossing. However, our data are low-latitude and agree well with the results of Formisano [1979]. In principle, the bow shock position depends on the magnetopause position and on t he standoﬀ distance between the magnetopause and the bow shock. Whereas the magnetopause po sition depends only on the solar wind dynamical pressure, the standoﬀ distance at a given Mac h number depends on the polytropic index γ(Spreiter et al., 1966). Since our data were sampled at typical solar minimum c onditions, the polytropic index might be diﬀerent than in other phases o f the solar cycle. This might contribute to the discrepancy.– 6 – 3. Comparison of quasi-perpendicular and quasi-parallel b ow shock crossings We divided the 132 events into 92 quasi-perpendicular ( θBn>45◦) and 40 quasi-parallel (θBn<45◦) cases and compared the average behavior of plasma and magne tic ﬁeld parameters and low frequency magnetic ﬂuctuations of the two groups. Actually, for the quasi-parallel bow shock crossings θBnvaries substantially with time in the dynamic foreshock region. For these events the angle θBnhad to be averaged over a time interval of about 20 s further upstream to identify them with quasi-pa rallel shock crossings. The high level of ﬂuctuations in the region upstream of the quasi-parallel bow shock is well known [e.g., Hoppe et al., 1981, Greenstadt et al., 1995]. In order to obtain the average behavior of plasma and magneti c ﬁeld parameter at the bow shock, one would ideally need average spatial proﬁles of the se parameters. However, with just one satellite and in a region with strong plasma ﬂows and strong m otions of the region itself, it is not unambiguously possible to translate the time proﬁles into s patial proﬁles. Therefore we perform a superposed epoch analysis by averaging time proﬁles cente red on the bow shock crossing time and consider the result as an approximation for the average s patial behavior. The time series are aligned on the keytime with the upstream always preceding, i .e., for outbound crossings the time sequence had to be reversed. The keytime, i.e., the bow shock crossing time, is identiﬁed with the steepest drop in the proton velocity. This drop is well deﬁned for the quasi-perpendicu lar cases and corresponds, of course, to the shock ramp. Due to the large-amplitude pulsations in the foreshock, the keytime cannot as easily be found in the quasi-parallel cases. We therefore applied, as an additional criterion for the quasi- parallel events, that no solar wind-like plasma is allowed t o be visible in the downstream region. As noted in Section 1, quasi-parallel shocks consist of larg e-amplitude pulsations associated with a sequence of partial transitions from solar wind-like to mag netosheath-like plasma and vice versa. Thus our deﬁnition of the keytime implies that the keytime of quasi-parallel shocks corresponds to the downstream end of this pulsating transition region. We use data from 2 min upstream to 4 min downstream for the anal ysis of the plasma and magnetic ﬁeld parameters and data from 3 min upstream to 9 min downstream for the low frequency ﬂuctuations, although not for all events such long time proﬁ les are available. One has to be aware that superposed epoch analysis can mask sm all scale structures, in par- ticular if they are not visible in all events, like magnetic f oot and overshoot structures. These and other features can be smeared out due to diﬀerent bow shoc k velocities with respect to the satellite for diﬀerent crossings. This eﬀect becomes worse the farther away from the keytime the data are averaged. Nevertheless, superposed epoch analysi s is useful to reveal the average plasma and magnetic ﬁeld parameters and the typical features in the vicinity of the key-structure as has been shown by, e.g., Paschmann et al. [1993], Phan et al. [1994], and Bauer et al. [1997] in there studies at the magnetopause.– 7 – Fig. 2.— Superposed-epoch analysis of plasma and magnetic ﬁ eld parameters from 2 min upstream to 4 min downstream of the bow shock (BS) of 92 quasi-perpendi cular (left) and 40 quasi-parallel (right) bow shock crossings. Shown are from top to bottom the magnitude Bof the magnetic ﬁeld, the magnitude vpof the proton velocity, the proton velocity vnparallel to the bow shock normal vector, and the root mean square amplitude δB2of magnetic ﬂuctuations. 3.1. Plasma and magnetic ﬁeld parameters The time series adjusted in the way described above are super imposed by averaging 10-s bins. The averages are performed geometrically in order to reduce the dominance of cases with large dynamical ranges. The result of the superposition is shown i n Fig. 2 and Fig. 3. The averages of the downstream and upstream values, excluding the 4 bins clo sest to the bow shock on both sides, are given in Table 1, together with the ratios of the downstre am to the upstream values. The top panel of Fig. 2 shows the magnitude Bof the magnetic ﬁeld. It increases steeply by a factor of 3 at the quasi-perpendicular bow shock and gradua lly by a factor of 1.8 at the quasi- parallel bow shock. The proton bulk velocity vp, shown in the next panel, decreases to somewhat less than half of its solar wind value for the quasi-perpendi cular bow shock and to slightly more than half for the quasi-parallel bow shock. The third panel s hows the proton velocity parallel to the shock normal vector. The latter is calculated from the Fa irﬁeld bow shock model [ Fairﬁeld ,– 8 – Table 1: Averages of plasma and magnetic ﬁeld parameters Upstream Downstream Ratio q-⊥9.4±0.1 28 .5±0.8 3 .0±0.1 B[nT] q-/bardbl10.9±0.8 19 .6±1.1 1 .8±0.2 q-⊥ 436±6 198 ±7 0 .45±0.03 vp[km/s] q-/bardbl332±16 171 ±17 0 .52±0.08 q-⊥ − 401±9−122±7 0 .30±0.02 vn[km/s] q-/bardbl − 283±18−96±16 0 .34±0.08 q-⊥0.6±0.1 2 .3±0.5 4 .0±1.4 δB2[nT] q-/bardbl1.9±0.3 3 .2±0.5 1 .7±0.5 q-⊥7.4±0.1 20 .1±0.4 2 .7±0.1 N∗ e[cm−3] q-/bardbl7.8±0.3 14 .4±1.1 1 .8±0.2 q-⊥0.15±0.01 0 .38±0.03 2 .6±0.3 N2p[10−2cm−3] q-/bardbl4.0±0.5 2.6 ±0.5 0 .7±0.2 q-⊥27.5±0.4 71 .3±0.7 2 .6±0.1 Te[104K] q-/bardbl 43±1 73 ±3 1 .7±0.1 q-⊥ 1.51±0.06 Tp⊥/Tp/bardblq-/bardbl— 1.05±0.03— Table 1: Averages of 92 quasi-perpendicular (q- ⊥) and 40 quasi-parallel (q- /bardbl) bow shock crossings from 120 to 20 seconds upstream and 20 to 240 seconds downstre am: the magnitude Bof the magnetic ﬁeld, the magnitude vpof the proton velocity, the proton velocity vnparallel to the bow shock normal vector, the magnetic ﬂuctuations δB2, the corrected electron density N∗ e, the density of energetic protons N2p, the electron temperature Te, and the proton temperature anisotropy Tp⊥/Tp/bardbl. Ratios of the downstream to the upstream values are given in the last column.– 9 – Fig. 3.— Superposed-epoch analysis of plasma parameters fr om 2 min upstream to 4 min down- stream of the bow shock (BS) of 92 quasi-perpendicular (left ) and 40 quasi-parallel (right) bow shock crossings. Shown are from top to bottom the corrected e lectron density N∗ e, the density of energetic protons N2p, the electron temperature Te, and the proton temperature anisotropy Tp⊥/Tp/bardbl. 1971]. For both categories vndecreases by more than the magnitude of the proton velocity vp, indicating that the plasma is deﬂected away from the bow shoc k normal direction in order to ﬂow around the magnetopause. In the last panel we show the root me an square amplitude δB2of the high resolution magnetic ﬁeld measurements during one spin period: δB2=/bracketleftBigg3/summationdisplay k=11 nn/summationdisplay i=1(Bk,i−¯Bk)2/bracketrightBigg1/2 (2) where i= 1,...,n counts the measurements during one spin period, and Bk,k= 1,2,3 denote the magnetic ﬁeld vector components. ¯Bkis the average of Bktaken during one spin period. At the keytime, δB2is approximately half the jump of the magnitude of the magnet ic ﬁeld. Further upstream and downstream it is a measure of wave activity with Doppler-shifted periods shorter than the spin period of about 4.3 s. Although the magnetic ﬁeld inc reases only gradually at the quasi- parallel bow shock, δB2is strongly enhanced at the bow shock crossing time and has a m aximum– 10 – immediately downstream of the shock. This jump in δB2, as a parameter measured independently of the velocity, is a conﬁrmation that the selection of the cr ossing times with the help of the velocity jump (Section 2) is reasonable. In the foreshock region upst ream of the quasi-parallel bow shock, δB2is considerably higher than in the solar wind regime upstrea m of the quasi-perpendicular bow shock. The ﬁrst panel of Fig. 3 shows the electron density N∗ e, approximately corrected for the low- energy cut-oﬀ of the plasma instrument [ Sckopke et al. , 1990]. Like the magnetic ﬁeld, the density rises sharply at the quasi-perpendicular bow shock, wherea s it increases gradually at the quasi- parallel bow shock. The second panel shows N2p, the proton density in the energy interval 8-40 keV.N2pincreases by a factor of 2.6 at the quasi-perpendicular bow s hock. In the foreshock region upstream of the quasi-parallel bow shock N2pis an order of magnitude higher than upstream of the quasi-perpendicular bow shock and decreases by a factor of a bout 0.7 in the downstream region. In the next panel the electron temperature Teis shown. Downstream of the quasi-parallel bow shock crossing it has about the same value as downstream of th e quasi-perpendicular bow shock. However, upstream of the quasi-parallel bow shock Teis about a factor of 1.6 higher than upstream of the quasi-perpendicular bow shock. The reason for this is again the foreshock region where the solar wind kinetic energy is already partly transformed into thermal energy. The last panel shows the proton temperature anisotropy Tp⊥/Tp/bardbl. The plasma instrument did not resolve the cold, supersonic distributions of the solar wind ions. The calcul ated proton densities and temperatures are therefore not reliable in the solar wind regime. Hence, t he proton temperature anisotropy cannot be determined in the upstream region and is therefore set to 1 . Whereas the proton temperature anisotropy downstream of the quasi-parallel bow shock is in signiﬁcant, there is a strong proton temperature anisotropy, Tp⊥/Tp/bardbl>1.4, downstream of the quasi-perpendicular bow shock, with a maximum value of more than 2 immediately behind the shock. C omparing the downstream values of the electron and proton temperatures (not shown), we ﬁnd Tp≈4.3×106K≈6Tefor the quasi-parallel bow shock and Tp≈2.9×106K≈4Tefor the quasi-perpendicular bow shock. Whereas the electron temperature is slightly anisotropic, Te⊥/Te/bardbl≈0.9, upstream and downstream of quasi-perpendicular shocks, no signiﬁcant anisotropy i s observed at quasi-parallel shocks. 3.2. Low frequency magnetic ﬂuctuations In order to analyze the low frequency magnetic ﬂuctuations w e perform a spectral analysis of the magnetic ﬁeld using a cosine-bell ﬁlter [see, e.g., Bauer et al. , 1995]. The Fourier transform is taken over a time interval of 4 min. Fig. 4 and Fig. 5 show the resulting power spectra of the compressive and the right- and left-hand polarized mode s, respectively. In each graph the center time (-3,-1,1,3,5,7) of the transformed time interv al is given in minutes relative to crossing time. A cross marks the proton cyclotron frequency fcp=eB/2πmpwithmpthe proton mass. The solar wind (SW) spectrum of Fig. 4, 3 min upstream of the qu asi-perpendicular bow shock, shows a structureless decrease to higher frequencies follo wing the power law S∼f−1.3. The– 11 – Fig. 4.— Superposed-epoch analysis of magnetic spectra of 9 2 quasi-perpendicular (left) and 40 quasi-parallel bow shock crossings (right) from 3 min upstr eam to 1 min downstream. Solid line: compressive component, dashed line: left hand polarized co mponent, dotted line: right hand po- larized component. The cross with the horizontal error bar m arks the proton cyclotron frequency. The acronyms SW, BS, SH mean solar wind, bow shock, and magnet osheath, respectively. compressive mode lies below the transverse modes that repre sent Alfv´ en waves frequently observed in the interplanetary medium. They are thought to have their origin in the vicinity of the Sun– 12 – Fig. 5.— Superposed-epoch analysis of magnetic spectra of 9 2 quasi-perpendicular (left) and 40 quasi-parallel bow shock crossings (right) from 3 min to 7 mi n downstream. Same format as Fig. 4. [Belcher & Davis , 1971]. The spectrum upstream of the quasi-parallel bow sho ck has much more power than that upstream of the quasi-perpendicular bow sho ck. It also has a diﬀerent structure: For lower frequencies it shows a ﬂatter decrease ( S∼f−0.5), while for higher frequencies it decreases more steeply ( S∼f−2.0). The kink in the spectrum lies below the proton cyclotron fr equency.– 13 – The next power spectra (-1, SW/BS) contain magnetic ﬁeld dat a from the upstream region and the bow shock itself. At the quasi-perpendicular bow sho ck the compressive mode at low frequencies is one order of magnitude higher compared to the transverse modes and follows a power law of S∼f−2.1. This represents simply the spectrum of the jump of the magne tic ﬁeld across the shock ﬁltered with a cosine bell function. The spectra of the transverse modes are a little higher than 2 min earlier and the decrease with frequency is not cons tant any more. At the quasi-parallel bow shock the increase of the magnetic ﬁeld is not visible, wh ich is not surprising since the magnetic ﬁeld increases only gradually. Level and structure of the sp ectrum are similar to those calculated 2 min earlier. The spectra (1, SW/BS) contain magnetic ﬁeld data from again the bow shock itself and from the magnetosheath just downstream of the bow shock. At the qu asi-perpendicular bow shock the compressive mode behaves similar to the spectrum 2 min earli er, whereas the spectral power of the transverse modes is higher than 2 min earlier. Just below the proton cyclotron frequency ﬁrst indications of a plateau are visible. At the quasi-parallel bow shock wave activity is signiﬁcantly enhanced by a factor of 4 compared to the upstream spectra. Ag ain all three modes behave similar. The proton cyclotron frequencies increase according to the magnetic ﬁeld increase by a factor of 3 at the quasi-perpendicular bow shock and by a factor of 2 at th e quasi-parallel bow shock. Figure 5 shows that for both categories of shock crossings th e spectra do not change much in the interval 2 to 8 min downstream of the bow shock. Below fcpthe compressive and the right-hand polarized modes downstream of the quasi-perpendicular bow shock follow a power law S∼f−1.1, whereas the spectral energy of the left-hand polarized mode is clearly enhanced in the frequency interval from about 0.1 Hz to about the proton cyclotron freq uency. This fact is investigated more carefully in Section 4. Downstream of the quasi-parallel bo w shock the spectral energy is again higher than downstream of the quasi-perpendicular bow shoc k. For f < f cpit follows the power lawS∼f−0.8. 3min after the crossing the spectral energy is higher than i n the 2 later spectra but already lower than directly at the bow shock. For both catego ries the spectral energy decreases steeply ( S∼f−2.6) forf > f cp. 3.3. Discussion The processes in the bow shock transition region itself cann ot be described in terms of mag- netohydrodynamics (MHD). However, if one considers the bow shock as an inﬁnitesimally thin discontinuity one can derive from the MHD equations the Rank ine-Hugoniot jump conditions (see, e.g.,Siscoe , 1983), which are relations for the conditions in the plasma upstream and downstream of the discontinuity under the assumption of time independe nce. For mass continuity the jump condition is n2vn2−n1vn1= 0 (3)– 14 – where the subscripts 1 and 2 denote the upstream and downstre am values of the corresponding parameters, respectively. According to Eq. (3) the product of the downstream-to-upstream ratio of the electron density Neand the normal proton velocity vn, which is equivalent to the plasma bulk speed normal to the discontinuity, must be unity. For th e quasi-parallel events this product is observed to be about 0.6. This fact is not surprising, since t he considered upstream time proﬁles are not taken from the quiet solar wind regime but from the dyn amic foreshock region, which is highly time dependent. For the quasi-perpendicular events this product is observed to be about 0.8, which means that the jump condition Eq. (3) is not fully s atisﬁed. This could be explained by the fact that we have not measured the exact electron densi ty, but only the density of electrons with energies between 15 eV and 30 keV. Electrons with higher energies are negligible since already in the energy range of 1.8 - 30 keV the upstream density is of th e order of 10−5cm−3. However, particularly in the cold solar wind, electrons with energie s below the instrument cut-oﬀ contribute an essential part to the total electron density. Therefore w e use the corrected electron density N∗ e. The correction is calculated with the assumption of a Maxwel lian distribution of the measured temperature. However, as has been measured, e.g., recently by the Wind spacecraft [Fig. 4 of Lin, 1997], the quiet solar wind ﬂow cannot be described by a singl e Maxwellian distribution over its whole energy range. Therefore the electron density could ea sily be overestimated by the correction, especially in the cold solar wind regime where the low energy electrons are more important than in the warmer magnetosheath regime. In order to satisfy the jum p condition Eq. (3) we can estimate, that the value of the corrected electron density upstream of the quasi-perpendicular bow shock is about 20% too high if we assume that the error in the downstrea m value is not of importance. Other factors for the apparent deviation from mass continui ty could be the unknown bow shock motion, which changes the downstream plasma velocity by a hi gher percentage than the upstream plasma velocity, and the uncertainty of the shock normal vec tor. For an exactly parallel shock wave one derives from the Ranki ne-Hugoniot jump condition that the magnetic ﬁeld remains constant in magnitude and directi on and one is left with the equations for a purely hydrodynamic shock wave. In the case of an exactl y perpendicular shock wave the jump conditions give the relation n2 n1=B2 B1=v1 v2(4) For the limit of high Mach numbers, i.e., the Alfv´ en Mach num berMA→ ∞ and the sonic Mach number Ms→ ∞, Eq. (4) has a value of 4 for γ= 5/3. Since we do not observe the extreme cases of exactly perpendicular and exactly parallel geomet ry but quasi-perpendicular and quasi- parallel shock wave crossings with ﬁnite Mach numbers the nu mbers given above are not reached. For the quasi-perpendicular bow shock the ratio Eq. (4) is ab out 3 for the average magnetic ﬁeld B and proton normal velocity vn, and somewhat less for the corrected electron density N∗ eaccording to the above mentioned probable overestimation of this para meter in the solar wind regime. For the quasi-parallel bow shock the change in the magnitude of t he magnetic ﬁeld is signiﬁcantly lower than for the quasi-perpendicular bow shock, which is consis tent with theory.– 15 – In their study of subcritical quasi-perpendicular shocks, Thomsen et al. , [1985] found that the downstream electron temperature is nearly isotropic, Te⊥2/Te/bardbl2≈0.9 , and that the downstream- to-upstream ratio Te⊥2/Te⊥1of the perpendicular temperature is approximately equal to the ratio B2/B1of the magnetic ﬁeld strength. From the latter result they co ncluded that the net heating is adiabatic, although Te⊥/Bis not constant. Obtaining averages, Te⊥2/Te/bardbl2≈0.9,Te⊥2/Te⊥1≈2.6, andB2/B1≈3.0, we can conﬁrm these results for our data set of (subcritica l and supercritical) quasi-perpendicular shocks. Moreover, isotropy of the dow nstream electron temperature and adia- batic net heating is also found for our data set of quasi-para llel shocks, for which we obtain averages Te⊥2/Te/bardbl2≈1.0,Te⊥2/Te⊥1≈1.7, and B2/B1≈1.8. An interesting question of shock physics is how the dissipat ed bulk ﬂow energy of the solar wind is partitioned amongst ion and electron heating. For the ave rage proton-to-electron temperature ratio we found Tp/Te≈6 downstream of the quasi-parallel bow shock and Tp/Te≈4 downstream of the quasi-perpendicular bow shock. Hence, for quasi-paral lel shocks proton heating is more favored with respect to electron heating than for quasi-perpendicu lar shocks. Let us turn to the low frequency waves of Fig. 4 and Fig. 5. At qu asi-parallel bow shocks the wave power observed 3 min upstream of the keytime is much h igher than the power of the interplanetary Alfv´ en waves observed upstream of quasi-p erpendicular shocks. This enhanced power reﬂects upstream waves generated in the foreshock. Observa tions of upstream waves have recently been reviewed by Greenstadt et al. [1995] and Russell & Farris [1995]. The nonlinear steepening of the shock leads to whistler precursors phase standing in the shock frame. The interaction between ions reﬂected at the shock and the incoming solar wind can dri ve ion beam instabilities. These are probably the source of large-amplitude waves observed at pe riods around 30 s. Finally, there are upstream propagating whistlers with frequencies around 1 H z, which seem to be generated directly at the shock. The most striking feature in the average spectr um of upstream waves observed 3 min before the keytime is the kink at 0 .04Hz≈fcp/3. The average power measured in the ﬂat portion 0.01–0.04 Hz of the spectrum corresponds to a mea n square amplitude δB1≈4nTor δB1/B≈0.4. Large-amplitude waves observed in this frequency range [ Le & Russell , 1992; Blanco- Cano & Schwartz , 1995] have been interpreted as upstream propagating magne tosonic waves excited by the right-hand resonant ion beam instability, upstream p ropagating Alfv´ en/ion cyclotron waves excited by the left-hand resonant ion beam instability, or d ownstream propagating magnetosonic waves excited by the non-resonant instability. Whereas the upstream propagating magnetosonic waves should be left-hand polarized in the shock frame (and a lso in the spacecraft frame), the other two wave types should be right-hand polarized. This mi ght explain why none of the two circular polarizations dominates in our average spectra. M oreover, the compressional component is comparable to the two transverse components. This shows t hat the waves propagate at oblique angles to the magnetic ﬁeld. For oblique propagation, low fr equency waves have only a small helicity [Gary, 1986]. Thus they are rather linearly than circularly polar ized. The power spectra presented by Le & Russell [1992] exhibit clear peaks at f≈fcp/3. Looking into the spectra of individual time intervals, we ﬁnd that so metimes the IRM data exhibit similar– 16 – spectral peaks. However, most of the individual spectra do n ot have clear peaks, but are rather ﬂat in the range 0.01–0.04 Hz like the average spectrum of Fig. 4. The steep decrease of the power above about f≈fcp/3 is common to our spectra and those reported previously. In f act, the maximum growth rate of the ion beam generated waves is expected for fr equencies below the proton cyclotron frequency [e.g., Scholer et al. , 1997]. In Fig. 2 we saw that magnetic ﬂuctuations above 0.23 Hz in the foreshock have root mean square amplitudes δB2≈1.5 nT. This is comparable to typical amplitudes of the upstrea m propa- gating whistlers. In individual time intervals these narro w-band waves lead to clear spectral peaks at frequencies around 1 Hz. However, since the frequency var ies from event to event, no such peak appears in the average spectra. At the keytime of quasi-parallel bow shocks we observed a cle ar enhancement of the wave power. This enhancement can either be due to an ampliﬁcation of the upstream waves or due to wave generation at the shock interface. Wave generation a t the shock due to the interface instability has been found in hybrid simulations of Winske et al. [1990] and Scholer et al. [1997]. This instability is driven by the interaction between the in coming solar wind ions and the heated downstream plasma at the shock interface. Ampliﬁcation of u pstream waves has been predicted byMcKenzie & Westphal [1969] who analyzed the transmission of MHD waves across a fa st shock. They found that the amplitude of Alfv´ en waves increases by a factor of 3. For compressional waves the ampliﬁcation can even be stronger. However, the hybrid s imulations of Krauss-Varban [1995] show that the transmission of waves across the shock is compl icated by mode conversion. The proton temperature anisotropy downstream of the quasi- perpendicular bow shock serves as a source of free energy. According to both observations an d simulations this kind of free energy drives two modes of low frequency waves under the plasma cond itions in the magnetosheath: the ion cyclotron wave and the mirror mode (see e.g., Sckopke et al. , 1990, Hubert et al. , 1989 and Anderson et al. , 1994 for observations, Price et al. , 1986, and Gary et al. , 1993 for simulations, and Schwartz et al. 1996 for a review). Which of these waves grow under which conditions is investig ated in Section 4, where we divide the crossings of quasi-perpendicular shocks into cases wit h low and high upstream β, respectively. It turned out that for this classiﬁcation by βthe diﬀerences become somewhat clearer than for a classiﬁcation by upstream Mach number. The critical Mach nu mber M∗above which ion reﬂection is required to provide the necessary dissipation is strongl y dependent on the plasma- β[Edmiston & Kennel , 1984]. We have calculated the ratio Mms/M∗for our shock crossings and have found that all subcritical shocks are low- β, i.e., that the classiﬁcation low- βversus high- βis more or less identical to the classiﬁcation subcritical versus supercr itical. The reason for this is that M∗→1 for β≫1. As the excitation of mirror and ion cyclotron waves depend s onβ, the results of Section 4 should be interpreted as the eﬀect of the plasma- βand not as an eﬀect of subcritical or supercritical Mach numbers. For the quasi-parallel bow shocks, we could not investigate the diﬀerence between subcritical– 17 – and supercritical shocks, because no subcritical quasi-pa rallel shock was identiﬁed in the data set. Trying higher thresholds for the division into low and high M ach numbers, we did not ﬁnd any qualitative diﬀerences. In this context it should be noted t hat only one of the cases in our data set has an Alfv´ en Mach number in the range MA≤2.3, for which quasi-parallel shocks are steady according to the hybrid simulations of Krauss-Varban & Omidi [1991]. 4. Comparison of quasi-perpendicular low- βand high- βbow shock crossings In order to reveal the origin of the left-hand polarized comp onent in the power spectra down- stream of the quasi-perpendicular bow shock (Fig. 5) we divi de the crossings into classes with low (<0.5) and high ( >1.0) upstream βand compare these two classes. There are 20 low- βand 47 high- βcases. The crossings with 0 .5≤β≤1.0 are not included in this analysis in order to emphasize the diﬀerences between the low- βand high- βregimes. Fig. 6.— Superposed-epoch analysis of the plasma parameter β, the proton temperature anisotropy Tp⊥/Tp/bardbl, and the mirror instability criterion from 2min upstream to 4 min downstream of the quasi- perpendicular bow shock (BS) on the left for 20 low- β, on the right of 47 high- βcrossings. 4.1. Plasma and magnetic ﬁeld parameters In Fig. 6 we show some interesting diﬀerences between the low -βand high- βcategories. Of course the plasma parameter βdiﬀers essentially. In the upstream region βis derived by setting the proton density to the corrected electron density and the pro ton temperature to 10−5K, the long– 18 – term average of the proton temperature, since proton distri bution functions are not well measured in the cold solar wind (Section 3.1). The classiﬁcation for t he low- βand high- βcategories is derived from the estimated plasma βin the upstream region. The ﬁrst panel of Fig. 6 shows that the same classiﬁcation could be obtained using the plasma- βof the downstream region with the limits shifted to larger values. The most striking diﬀerenc es between the low- βand high- βbow shock are shown in the next two panels, i.e., the proton tempe rature anisotropy Tp⊥/Tp/bardbland the mirror wave instability criterion. Both parameters are aga in determined only in the downstream region. The instability criterion for almost perpendicula r propagation of the mirror mode in its general form [ Hasegawa , 1969] is given by −1 +/summationdisplay jβj⊥/parenleftbiggβj⊥ βj/bardbl−1/parenrightbigg >0. (5) The subscript jdenotes the particle species ( j=e,pfor electrons and protons, respectively). Downstream of the quasi-perpendicular low- βbow shock the proton temperature anisotropy is very high, Tp⊥/Tp/bardbl≈2.5, immediately behind the shock and remains high, Tp⊥/Tp/bardbl>2, throughout the whole magnetosheath interval investigated. Downstrea m of the quasi-perpendicular high- βbow shock the proton temperature anisotropy is also signiﬁcant , but lower compared to the low- βbow shock, i.e., Tp⊥/Tp/bardbl≈1.8 just behind the bow shock and Tp⊥/Tp/bardbl≈1.3 further downstream. The mirror instability criterion is only marginally satisﬁ ed immediately downstream of the quasi- perpendicular low- βbow shock and is not satisﬁed at later times, since the instab ility criterion does not only depend on the particle temperature anisotropy but also on the absolute value of β. Downstream of the quasi-perpendicular high- βbow shock the mirror instability criterion is satisﬁed in the entire interval of 4min behind the shock. Ext remely high values of the left-hand side of Eq. (5) are occasionally observed immediately behin d the shock. 4.2. Low frequency magnetic ﬂuctuations Fig. 7 shows the magnetic power spectra downstream of the qua si-perpendicular low- β(left) and high- β(right) bow shock, 3, 5, and 7 min after the crossing time. Now it becomes obvious that the left-hand polarized mode dominates only behind the quasi-perpendicular bow shock with low-βin a frequency interval below the proton cyclotron frequenc y. 5 and 7 min downstream, the left-hand polarized mode has up to one order of magnitude more power spectral density than the compressive and the right-hand polarized components. B elow the frequency range where the left-hand polarized component dominates in the low- βcases, the spectrum downstream of the quasi- perpendicular high- βbow shock shows a weaker gradient than downstream of the low- βbow shock. In addition, the compressive mode lies on the same level as th e transverse modes and occasionally at somewhat higher levels, whereas 5 and 7 min downstream of t he low- βbow shock the compressive mode lies clearly below the transverse modes.Due to the high er magnetic ﬁeld the proton cyclotron frequency is a factor of about 2.3 higher downstream of the lo w-βbow shock ( B≈45 nT) than– 19 – Fig. 7.— Superposed-epoch analysis of magnetic power spect ra downstream of 20 quasi- perpendicular low- βand 47 quasi-perpendicular high- βbow shock crossings. Same format as Fig. 5. downstream of the high- βbow shock ( B≈23 nT). At this point, let us collect some numbers for typical wave am plitudes. For that purpose we use the root mean square amplitude δB2given in Eq. (2), which has already been shown in Fig. 2 and can be obtained by integrating the power spectra for frequen cies above 0.23 Hz, and we use the root mean square amplitude δB1of ﬂuctuations in the frequency range 0.01–0.04 Hz. The rang e 0.01–– 20 – 0.04 Hz has been chosen, because it corresponds to the ﬂat por tion of the spectrum observed 3min upstream of the keytime of quasi-parallel bow shocks. These upstream waves have δB1≈4 nT or δB1/B≈0.4. In contrast, the Alfv´ en waves seen in the solar wind upstr eam of quasi-perpendicular bow shocks have δB1≈0.8 nT or δB1/B≈0.09. At the keytime of quasi-parallel bow shocks we observed a clear enhancement of the wave power, which leads t oδB≈8 nT or δB/B≈0.5. In this section we saw that the characteristics of the wave ac tivity downstream of quasi- perpendicular shocks depend on β. For low β, the magnetic ﬂuctuations above 0.23 Hz are domi- nated by left-hand polarized ﬂuctuations with δB2≈3nT or δB2/B≈0.08, which will be inter- preted as ion cyclotron waves. For high β, typical amplitudes are δB2≈1.5 nT or δB2/B≈0.08. 4.3. Discussion Sckopke et al. [1990] have performed a case study of low- βsubcritical bow shock crossings, using the AMPTE/IRM data of September 5, and November 2, 1984 . These events are also included in our quasi-perpendicular low- βdata set: There are 8 events from September 5 and 3 events from November 2, 1984. Sckopke et al. [1990] identiﬁed the dominating left-hand polarized component with the ion cyclotron wave which can be generated by a proton temperature anisotropy (e.g.,Hasegawa , 1975). The growth rate of the ion cyclotron wave is positive when the instability criterion is satisﬁed. This holds for the resonance with pro tons when Tp⊥ Tp/bardbl>fcp fcp−f. (6) The AMPTE/IRM plasma instrument did not resolve ion masses, therefore all ions are assumed to be protons. Figure 8 shows the ratio of the left-hand polarized to the rig ht-hand polarized component in the frequency band 0.3–0 .8fcpfor 32 2-min intervals from 4 to 8 min downstream of the quasi- perpendicular low- βbow shock as a function of the proton temperature anisotropy . There is a clear correlation found between these two ratios with a corr elation coeﬃcient of 0.8. This shows that the wave intensity more than 4 min downstream of the quas i-perpendicular bow shock depends strongly on the local temperature anisotropy. The temporal evolution of the correlation coeﬃcients, calc ulated for 2-min intervals down- stream of the quasi-perpendicular low- βbow shock, is shown in Fig. 9. The value of the correlation coeﬃcient 11 minutes downstream is not reliable since only a limited data set extends so far down- stream. Although the temperature anisotropy is highest imm ediately downstream of the bow shock, the best correlation is found around 5min downstream. This s hows that the ion cyclotron waves need a certain time to develop in the moving plasma. Since downstream of the quasi-perpendicular high- βbow shock the mirror instability criterion is satisﬁed on average, we have looked more carefully for thi s highly compressive non-propagating– 21 – 00510152025 1 2 3 4 5 Tp / TpIII (l) / I (r) ⊥ Fig. 8.— Ratio of the left-hand polarized to the right hand po larized component I(l)/I(r) against the proton temperature anisotropy Tp⊥/Tp/bardblot 32 2-min intervals from 4 to 8 min downstream of the quasi-perpendicular low- βbow shock. 0-0.200.20.40.60.8 2 4 6 8 10 12 Minutes after crossingCorrelation coefficient Fig. 9.— Temporal evolution of the correlation coeﬃcient of the ratios I(l)/I(r) and Tp⊥/Tp/bardbl downstream of the quasi-perpendicular low- βbow shock. mode. The fact that the compressive mode has a higher power sp ectral density downstream of the high- βthan downstream of the low- βshock might indicate the existence of mirror modes. We therefore perform a superposed epoch analysis for 9 cases fo r which the mirror instability criterion– 22 – Frequency (Hz)Magnetic Power Spectrum (nT2/Hz) Fig. 10.— Superposed-epoch analysis of magnetic spectra 7 m in downstream of 9 quasi- perpendicular high- βbow shock crossings of which the mirror instability criteri on is well satisﬁed. is particularly well satisﬁed, 7min downstream of the high- βbow shock. The resulting spectrum is shown in Fig. 10. In this spectrum we ﬁnd a compressive mode slightly dominating at some frequencies well below the proton cyclotron frequency. Contrary to the correlation of the intensity of the left-han d polarized ion cyclotron waves to the proton temperature anisotropy for quasi-perpendicula r low- βcases, our investigation does not reveal a clear correlation of the intensity of the compressi ve mode to any parameter for the quasi- perpendicular high- βcases. One reason for this could be that the highly compressi ve mirror mode, which is expected to exist under the observed high- βconditions is a purely growing mode with frequencies being pure Doppler-shifted frequencies. Cons equently, the waves do not appear in a ﬁxed frequency interval and can be smeared out in the superpo sition. Therefore we have looked into the individual spectra of intervals 5, 7, and 9 min downstrea m of the quasi-perpendicular high- β bow shock when the mirror criterion is fulﬁlled. This is the c ase in 34 of the quasi-perpendicular high-βevents (72 %). Only in 4 cases (12 % of the quasi-perpendicula r high- βevents where the mirror criterion is fulﬁlled) mirror waves can clearly be id entiﬁed in the magnetosheath and are visible in several consecutive spectra. These events and th e means of identiﬁcation of the mirror modes are described in Czaykowska et al. [1998]. The 4 events have in common that the angle θBnis larger than 80◦. But there are also almost perpendicular high- βevents with large values of the left-hand side of Eq. (5) where no indication for mirror w aves is visible. Thus a more complex– 23 – dependence on diﬀerent parameters seems to determine the gr owth of the mirror wave. For 14 of the high-βevents (41 %) where the mirror instability criterion is fulﬁ lled, the compressive component is at least slightly dominating at several frequencies. Sever al of these events have an angle θBn<80◦. In addition, 2 high- βevents show ion cyclotron waves in the consecutive spectra t aken 5, 7, and 9 min downstream. It is well known [ Price et al. , 1986; Gary et al. , 1993] that the growths of the ion cyclotron and mirror waves are competing processes. In t he magnetosheath plasma the crucial parameters for this competition are the α-particle concentration and the plasma- β. However, in our data set we have found many events in the high- βregime where none of the two wave modes can be identiﬁed although the proton temperature anisotrop y is high. This seems to indicate that the dominance of the growing mode does not persist long enoug h to be visible in one spectrum. The energy of the growing mode might be transferred to other m odes by nonlinear eﬀects. Statistical studies of measurements in the magnetosheath s uggest that a relation of the form Tp⊥ Tp/bardbl−1 =S (βp/bardbl)α(7) exists between the proton temperature anisotropy and the ra tioβp/bardblof ﬁeld-aligned proton pressure and magnetic pressure. Analyzing AMPTE/CCE data, Anderson et al. [1994] determined S= 0.85 andα= 0.48 and Fuselier et al. [1994] determined S= 0.83 and α= 0.58. Using AMPTE/IRM data,Phan et al. [1994] obtained S= 0.58 and α= 0.53. We performed a similar analysis on our data set of quasi-perpendicular bow shock crossings. For th is analysis we computed 2-min averages of all measurements of Tp⊥/Tp/bardblandβp/bardbltaken between the keytime and 8min downstream of the keytime. We ﬁnd a reasonable ﬁt to Eq. (7) with S= 0.43±0.03 and α= 0.58±0.05, which is not too diﬀerent from the result of Phan et al. [1994]. A relation of the form of Eq. (7) is regarded as the consequenc e of the combined action of ion cyclotron and mirror waves, which grow due to the tempera ture anisotropy and reduce this anisotropy by means of pitch angle scattering. The growth of the waves depends on Tp⊥/Tp/bardbland βp/bardbland it is expected that the anisotropy is reduced until the gr owth rate, γm, of the most unstable wave falls below some threshold. In fact, Anderson et al. [1994] showed that Eq. (7) with their values of Sandαcorresponds approximately to the threshold γm/2πfcp= 0.01. Hence, the validity of Eq. (7) indicates that the magnetosheath plasma reaches a state near marginal stability of the waves driven by the temperature anisotropy. As noted above, we found that data obtained less than 8min downstream of quasi-perpendicular bow shocks sat isfy a relation of the form Eq. (7) with values of Sandαthat are not too diﬀerent from those determined by Phan et al. [1994] for the entire magnetosheath. This indicates that the state nea r marginal stability is already reached close to the shock. Figure 7 shows that the largest amplitudes of the ion cyclotr on waves at quasi-perpendicular lowβshocks are observed on average about 5min downstream of the k eytime (see also Fig. 9). The bow shock moves relative to the spacecraft at speeds of 10 –100 km/s. Taking a typical speed of 30 km/s, we can translate 5min to a downstream distance of 9 000 km. According to Fig. 2, the plasma velocity, vpn, normal to quasi-perpendicular shock is 120 km/s on average . Thus the plasma– 24 – needs about 75 s to ﬂow 9000 km downstream. Since the ion cyclo tron waves are convected with the plasma while they are growing, these 75 s can serve as a rou gh estimate for the time τthat the waves need to reach their maximum amplitudes and saturate. I n terms of gyro-periods, we have τ∼75s≈50/fcp. Moreover, we ﬁnd that on the same time scale τthe temperature anisotropy is reduced from about 2.5 immediately downstream of the low βshock to about 2.1 (Fig. 6) and that the ion cyclotron waves typically reach amplitudes of δB/B∼0.07. These results can be compared with two-dimensional hybrid s imulations of McKean et al. [1994]. These authors examined a plasma with βp/bardbl= 1 and Tp⊥/Tp/bardbl= 3. Under these conditions the ion cyclotron mode is found to be the dominant mode and the waves saturate after τ≈5/fcp and reach amplitudes of δB/B∼0.15. The proton temperature anisotropy is reduced on the same time scale τfrom 3 to about 1.8. For our data set of low βbow shocks Tp⊥/Tp/bardblis on average 2.5 andβp/bardblis on average 0.5 immediately downstream of the shock. Since these values are considerably lower than the initial values used by McKean et al. [1994], the plasma simulated by McKean et al. [1994] is initially much farther away from the state of margi nal stability. Thus it is not surprising that the waves grow faster, reach larger amplitudes and ther efore lead to a stronger reduction of the anisotropy by means of pitch angle scattering. McKean et al. [1994] also examined a plasma with βp/bardbl= 4 and Tp⊥/Tp/bardbl= 3. Under these conditions the ion cyclotron mode dominates for low α-particle concentration, whereas the mirror mode dominates for high α-particle concentration. The waves saturate after τ≈10/fcpand reach amplitudes of δB/B∼0.2. The proton temperature anisotropy is reduced on the same t ime scale τfrom 3 to about 1.5. This can be compared with data obtained at the quasi-perpendicular high βshock. For our data set of high- βbow shocks Tp⊥/Tp/bardblis on average 1.8 and βp/bardblis on average 5 immediately downstream of the shock. Again, the plasma sim ulated by McKean et al. [1994] is initially much farther away from the state of marginal stabi lity. Fig. 6 shows that a reduction of the anisotropy to 1.3 is observed 30 s downstream of the keyti me. This can again be translated to a downstream distance and used to estimate the time span that passes while the plasma travels this distance. This estimate gives 8s ≈2.5/fcp. Finally, it should be noted that the mirror waves analyzed by Czaykowska et al. [1998] have amplitudes of δB/B∼0.2, which is comparable to those found in the simulations of McKean et al. [1994]. 5. Conclusions We have analyzed the plasma and magnetic ﬁeld parameters as w ell as low frequency magnetic ﬂuctuations at 132 dayside AMPTE/IRM bow shock crossings. T he average distance of the subsolar point, which results from the coordinates of the investigat ed bow shock crossings, is considerably smaller than in other studies, even when normalized to the av erage solar wind dynamical pressure. A reason for this discrepancy might be a variation of the poly tropic index with the solar cycle since our observations are performed during typical solar m inimum conditions. The position of the Earth’s bow shock still seems to be a matter of discussion.– 25 – A superposed epoch analysis has been carried out by averagin g particle and magnetic ﬁeld data as well as low frequency magnetic spectra upstream and downs tream of the bow shock. We have performed this analysis by dividing the events into diﬀeren t categories, i.e., quasi-perpendicular and quasi-parallel events as well as quasi-perpendicular l ow-βand high- βevents. The particle and magnetic ﬁeld data show that upstream of the quasi-parallel bow shock, in the foreshock region, the plasma is already heated compar ed to the undisturbed solar wind. Moreover, there are more energetic protons in the foreshock region, and the magnetic ﬁeld is highly variable. Downstream of the quasi-perpendicular bow shock , a proton temperature anisotropy is found, which is higher on average downstream of the quasi-pe rpendicular low- βthan downstream of the quasi-perpendicular high- βbow shock. Concerning the low frequency magnetic ﬂuctuations we ﬁnd th at upstream of the quasi- perpendicular bow shock the solar wind spectrum is undistur bed with transverse Alfv´ en waves surpassing the compressive spectral component. Upstream o f the quasi-parallel bow shock largely enhanced wave activity is detected in the turbulent foresho ck region. These upstream waves are convected downstream, experiencing an enhancement at the b ow shock itself. Downstream of the quasi-perpendicular bow shock the observed proton tempera ture anisotropy leads to the generation of left-hand polarized ion cyclotron waves under low- βconditions and in some cases to the gen- eration of mirror waves under high- βconditions. A clear correlation has been observed between the intensity of the left-hand polarized component of the ma gnetic power spectrum relative to the right-hand polarized component and the proton temperature anisotropy. On the other hand, we could not ﬁnd a simple correlation between the intensity of t he compressive component and any single plasma or magnetic ﬁeld parameter. In cases where mir ror waves are obviously observable mostly three conditions are fulﬁlled: the plasma- βis high, the mirror instability criterion is satisﬁed and the angle θBnis large, i.e., θBn/greaterorsimilar80◦. 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This preprint was prepared with the AAS L ATEX macros v5.0.
arXiv:physics/0009047v1 [physics.ins-det] 13 Sep 2000The Longitudinal Polarimeter at HERA M. Beckmanna, A. Borissovb, S. Brauksiepea, F. Burkarta, H. Fischera, J. Franza, F.H. Heinsiusa, K. K¨ onigsmanna, W. Lorenzonb,1, F.M. Mendena, A. Mostb, S. Rudnitskyb, C. Schilla, J. Seiberta, A. Simona aFakult¨ at f¨ ur Physik, Universit¨ at Freiburg, 79104 Freib urg, Germany bRandall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120 Abstract The design, construction and operation of a Compton back-sc attering laser po- larimeter at the HERA storage ring at DESY are described. The device measures the longitudinal polarization of the electron beam between the spin rotators at the HERMES experiment with a fractional systematic uncertaint y of 1.6 %. A measure- ment of the beam polarization to an absolute statistical pre cision of 0.01 requires typically one minute when the device is operated in the multi -photon mode. The polarimeter also measures the polarization of each individ ual electron bunch to an absolute statistical precision of 0.06 in approximately ﬁv e minutes. It was found that colliding and non-colliding bunches can have substantiall y diﬀerent polarizations. This information is important to the collider experiments H 1 and ZEUS for their future longitudinally polarized electron program because those experiments use the colliding bunches only. Key words: Polarized Compton scattering; Electron polarimetry PACS: 29.20.Dh; 29.27.Bd; 29.27.Fh; 29.27.Hj 1 Introduction In high-energy storage rings, electron (positron) beams ca n become trans- versely polarized through the emission of synchrotron radi ation [1]. This pro- cess involves a small asymmetry in the spin-ﬂip amplitudes, which enhances 1Corresponding author; email: lorenzon@umich.edu Preprint submitted to Elsevier Preprint 2 February 2008the population of the spin state antiparallel (parallel) to the magnetic bending ﬁeld. The polarization develops in time according to P(t) =P∞/parenleftBig 1−e−t/τ/parenrightBig , (1) where the asymptotic polarization P∞and the time constant τare character- istics of the ring conditions. In the absence of depolarizin g eﬀects, the max- imum polarization theoretically achievable is Pth= 0.924, and the rise-time constant, which depends on the bending radius of the storage ring and the beam energy, is τth= 37 min for the HERA storage ring operated at an energy Ee= 27.5 GeV. Depolarizing eﬀects can however substantially reduce the m aximum achievable polarization. These intricate eﬀects cannot generally be p recisely controlled, making it necessary to continuously measure the beam polari zation. The depo- larizing eﬀects also aﬀect the actual rise-time, which scal es with P∞according to τ=P∞/parenleftbiggτth Pth/parenrightbigg . (2) Thus for a typical beam polarization of 0.55, the rise-time i s about 22 min. This article describes a polarization monitor at the HERA el ectron ring at DESY, which is based on Compton scattering of circularly pol arized photons from an intense pulsed laser beam. This method for measuring the polariza- tion of stored electron beams2was suggested more than 30 years ago [2], and has been employed at many laboratories [3] to measure the tra nsverse polar- ization. Compton scattering has also been employed at linea r accelerators [4] to measure the longitudinal polarization. In recent years, the NIKHEF [5] and MIT-Bates [6] laboratories have developed Compton polarim eters to monitor the longitudinal beam polarization in their storage rings. At DESY, a Comp- ton polarimeter [7] had been constructed in 1992 to measure t he transverse polarization of the electron beam in the HERA West section. T his Trans- verse Polarimeter measured the electron beam polarization with an initial fractional systematic uncertainty of 9%, which has subsequ ently been im- proved to 3.4% [8]. The Longitudinal Compton Polarimeter was added to obtain an independent and more precise measurement of the beam polarization at HER A, with very diﬀerent systematic uncertainties and the capability to me asure individual bunch polarizations. It was commissioned during fall 1996, and provides a 2electron beams refer to both, electron and positron beams fo r the remainder of this article 2measurement of the longitudinal beam polarization in the Ea st section of HERA between the spin rotators [9] at the HERMES experiment [ 10]. 2 Polarized Compton Scattering The cross section for Compton scattering of circularly pola rized photons oﬀ longitudinally polarized electrons can be written [11] in t he laboratory frame as dσ dEγ=dσ0 dEγ[1−PλPeAz(Eγ)], (3) where dσ0/dE γis the unpolarized cross section, Eγis the energy of the back- scattered Compton photons, Pλis the circular polarization of the incident photons for the two helicity states λ=±1,Peis the longitudinal polarization of the electron beam, and Az(Eγ) is the longitudinal asymmetry function, which is shown in Fig. 1 for a 2.33 eV photon scattered oﬀ a 27.5 GeV electron. -0.4-0.200.20.40.60.8 0 2 4 6 8 10 12 14 Eγ (GeV)Az Fig. 1. The longitudinal asymmetry function Azversus the energy Eγof the back-scattered Compton photons for the case of a 2.33 eV phot on incident on a 27.5 GeV electron. The total unpolarized cross section is 377 mb, and the diﬀere ntial cross sec- tion is peaked at the maximum energy ( Eγ,max= 13.6 GeV) of the back- scattered Compton photons, hereafter called the Compton ed ge. The longitu- dinal asymmetry function has a maximum of about 0.60 at the Co mpton edge. As the energy of the Compton photons decreases, Azdecreases rapidly and becomes negative below 9.1 GeV, corresponding to scatterin g angles smaller than 90◦in the electron rest frame, and returns to zero at Eγ= 0. Due to the enormous kinematic boost from the electron beam (the Lor entz factor is Ee/me≈5.4·104), most back-scattered Compton photons are contained in a narrow cone centered around the initial direction of the str uck electrons. This is very advantageous to a polarization monitor because it al lows the detector to be far away (many tens of meters) from the interaction regi on. However, the 3spatial distribution of the Compton photons on the detector surface is given not only by the Compton kinematics but also by the electron be am optics. Therefore, if the interaction point is chosen at a position w here the divergence of the electron beam is small, the transverse size of the phot on detector can be suﬃciently small to accommodate only moderate separations of the Compton photons and the electron beam, and thus meet the spatial cons traints given by the HERA electron ring. 3 Apparatus A schematic overview of the Longitudinal Polarimeter arran gement is shown in Fig. 2. A circularly polarized photon beam from a pulsed la ser is focused on the HERA electron beam. The laser-electron interaction p oint is located between the two bending magnets BH39 and BH90 at 39 m and 90 m fr om the HERMES target, respectively. A calorimeter measures th e energy of the back-scattered Compton photons for each laser pulse. If the electron beam is longitudinally polarized, the energy distributions of t he Compton photons diﬀer for left-handed and right-handed photon helicities. BH90 HERMES targetCompton photons calorimeter BH39.13 m 39 m 16 m 38 m HERMES experiment laser - electron. HERA electron beamlaser beam interaction point Fig. 2. Schematic overview of the Longitudinal Polarimeter in the HERA East sec- tion. 3.1 Laser and Optics A frequency-doubled, pulsed Nd:YAG laser [12] operated at 5 32 nm, corre- sponding to a photon energy of Eλ= 2.33 eV, is used for the measurements. The laser produces 3 ns long pulses of linearly polarized lig ht and can be operated with a continuously variable repetition rate from single shot up to 100 Hz, and pulse energies from 1 to 250 mJ. The laser is synchr onized with the electron bunches in the HERA ring, and triggered at close to 100 Hz. The timing and the intensity of each laser pulse are measured by t wo photo diodes as shown in Fig. 3. To minimize pulse-to-pulse intensity ﬂuc tuations, the laser is operated at a ﬁxed energy of 100 mJ per pulse. The intensity of the laser 4pulses can be controlled by passing the laser beam through a r otatable half wave plate and a ﬁxed Glan-Thompson prism. The linearly polarized laser light is converted to a circula rly polarized beam by passing it through an electrically reversible birefringen t cell, known as a Pock- els cell. The voltage on the Pockels cell [13] is adjusted to p roduce a quarter wave phase shift which is reversed each pulse. The degree of c ircular polariza- tion\|Pλ\|of the laser beam is larger than 0.999 for each of the two helic ity states λ=±1, and is checked regularly with a polarization analyzer [14 ] consisting of a rotatable half wave plate, a Glan-Thompson prism, and a p hoto diode. Before entering the laser transport system, the beam diamet er is expanded by a factor of four by a set of plano-concave and plano-convex le nses [15]. This beam expander reduces the divergence of the laser beam to all ow it to traverse the 80 m long light path to the laser-electron interaction re gion and also to reduce the resulting waist at the interaction point. In addi tion, it minimizes the sensitivity to variations in the laser beam divergence, and it reduces the energy density of the laser beam to prevent damage to the opti cal components in the light path. /0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1mirrors"laser intensity"PIN diode polarization analyzer"laser timing"half wave beam shutter plateprism half waveplate PIN diodeentrance window 1:4 beam expander Nd:YAG laserbeam dump laser transport PIN diodebeam dump prismGlan-ThompsonPockels cell system/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 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 "AB 1"filter (OD 3.0) Glan- Thompson filter (OD 2.4)neutral densityneutral density HV Fig. 3. Arrangement of the optical system in the laser room. The laser beam is guided by six remotely controlled mirrors [ 16] through a total of 72 m of stainless steel vacuum pipe, and focused with a lens doublet [17] on the HERA electron beam, as shown in Fig. 4. The mirrors are arr anged in three phase-compensated pairs to maintain the polarization of th e photon beam close to 100%. Behind each mirror, a video camera is installe d to monitor the laser beam position. The laser beam enters the storage ring v acuum through a 1 cm thick fused silica window [18] and is brought into colli sion with the electron beam at a vertical angle of 8.7 mrad. The window was m ounted with Helicoﬂex gaskets [19] to minimize stress such that it has ne gligible optical retardation. 5At the interaction point, the laser spot has a diameter of app roximately 0.5 mm, and the transverse size of the electron beam is σx≈0.6 mm hor- izontally and σy≈0.2 mm vertically. Each electron bunch is approximately 11 mm long (corresponding to 37 ps), i.e. about one hundred ti mes shorter than the laser pulse. After passing through the interaction poin t, the laser beam exits the storage ring vacuum system through an identical va cuum window and enters a second polarization analyzer which also monito rs the position and the intensity of the laser light. Nd:YAG laser screenlens doubletlaser roomcable shaft screen polarization analyzermirror M 3 interaction pointlaser - electron HERA exit windowstand3.3 m shutter 2.5 m10.6 m 5.6 m8.4 mbeam & optical system pump mirror M 2mirror M 4 Compton photonscalorimeter HERA electron beam electronswindow screen HERA tunnel, section East Rightmirror M 1 mirrors M 5/6entrance 47.2 m 6.3 mHERA entrance window Fig. 4. Layout of the Longitudinal Polarimeter in the HERA Ea st section. 3.2 Laser −Electron Interaction Region The location of the laser-electron interaction region was c hosen to optimize the rate of the back-scattered Compton photons versus the ba ckground rate, and to minimize changes to the electron ring vacuum system. M aximizing the Compton rate means that the crossing angle between the laser beam and the electron beam should be as small as possible, and the horizon tal widths of the electron and laser beams should both be small. In additio n, the transverse spatial distribution of the back-scattered Compton photon s due to the size and divergence of the electron beam had to be minimized, since th e back-scattered photons have to travel about 54 m to the calorimeter. The laser-electron interaction point is located in the East Right HERA tunnel section, 13 m downstream of the ﬁrst dipole magnet BH39, whic h bends the beam by 0.54 mrad (Fig. 2). This is enough to prevent a large fr action of the bremsstrahlung generated by the residual gas in the long straight vacuum section upstream of BH39, and by the HERMES gas target in part icular, from reaching the calorimeter. On the other hand, it is littl e enough that it rotates the spin by only 1.9◦. The corresponding reduction of the measured longitudinal beam polarization is negligible (0.06%). 6The scattered electrons and photons travel with the unscatt ered electron beam until the electrons are deﬂected by the dipole magnet BH90, w hich has a bending radius of 1262 m and deﬂects the beam by 2.7 mrad. A col limator is installed in the beam line 6 m downstream of BH90 to further re duce possible bremsstrahlung contributions from the HERMES target. In or der to minimize changes to the electron ring vacuum system, the calorimeter position was cho- sen 16 m downstream of BH90. This puts strict constraints on t he transverse size of the calorimeter, since the electron beam and the cent er of the back- scattered Compton photon distribution are separated by onl y 42 mm at the chosen position. 3.3 The Compton Photon Detector The detector assembly is mounted on a remotely controlled ta ble that can be moved vertically and horizontally. A light tight aluminum b ox contains the electromagnetic shower detector shown schematically in Fi g. 5. The detector is positioned very close to the electron beam pipe during nor mal operation. Therefore, the lateral face of the box near the beam pipe is ma de of a 3 mm thick tungsten plate to protect the detector against soft sy nchrotron radiation emerging from the beam pipe. photons 2NaBi(WO ) crystal2 x 6 mm lead plates 4 photomultiplieraluminized mylarCompton Fig. 5. Schematic layout of the NaBi(WO 4)2crystal calorimeter. The front of the detector is positioned 21 mm downstream of a c opper vac- uum window in the HERA beam tube. This window is 2 mm thick and 3 4 mm in diameter. The Compton photons enter the detector through a set of two 6 mm (2 ×1.1 radiation lengths) thick lead plates, which serves as an eﬀ ec- tive shield against the intense synchrotron radiation gene rated by the dipole magnet BH90. The electromagnetic calorimeter consists of f our optically iso- lated NaBi(WO 4)2crystals. Each crystal is 20 cm long (19 radiation lengths), 22 mm wide and 22 mm high, arranged in a 2 ×2 array, as displayed in Fig. 5. The crystal material has a high index of refraction ( n= 2.15), and is very radiation-hard (7 ·107rad) [20,21] and compact (Moli` ere radius 2.38 cm). The Compton photons generate an electromagnetic shower in the l ead preshower 7and the crystals. The charged particles of the electromagne tic shower pro- duce ˇCerenkov light, which is detected by one photomultiplier tu be [22] for each crystal. The sharing of the shower between the four calo rimeter blocks allows a sub-millimeter alignment of the NaBi(WO 4)2array with respect to the Compton photon beam. 3.4 Trigger and Electronics The event trigger is provided by a pulser at a rate of approxim ately 200 Hz. The laser is ﬁred by only every second pulse, allowing a backg round event to be recorded following each Compton event. Each laser pulse i s synchronized with the HERA bunch clock, which is provided by a bunch trigge r module (BTM) [23]. The BTM is also used to select a speciﬁc electron b unch in a sequence determined by a programmable Digital Signal Proce ssor (DSP) [24]. Four consecutive events are recorded for each selected bunc h: for each of the two light helicity states, one background event and one even t where the laser was ﬁred. The program of the DSP further provides the option o f scanning any subset of the beam bunches in any sequence. From the recorded single bunch data, one can extract the polarization of a single bunch, the average beam polarization of all the bunches, or the polarization of any s et of bunches, e.g. only colliding or non-colliding bunches. The trigger also a llows for pedestal or gain monitoring events during empty HERA beam bunches. The H ERMES gain monitoring system [10] monitors the response of the Com pton photon detector by sending laser light pulses through glass ﬁbers t hat are coupled to the front faces of the NaBi(WO 4)2crystals. The signals from the four photomultiplier tubes are digitiz ed by a charge sen- sitive ADC [25], and transferred by the DSP to the HERMES data acquisition system, which is described in detail elsewhere [10]. Also th e signals from the various photodiodes are recorded. For each Compton event, t he timing of the laser pulse measured by the “laser timing” PIN diode (see Fig . 3) is recorded relative to the bunch timing by a TDC [26]. 4 Polarimeter Operation Normal operation of the Longitudinal Polarimeter requires an optimum over- lap of the laser and electron beams in both space and time to ma ximize the back-scattered Compton rate. The spatial overlap is achiev ed by steering the laser beam horizontally through the interaction point with mirror M4 (see Fig. 4). The timing of the laser pulse is set with respect to th e electron bunches by adjusting the laser trigger delay for maximum luminosity . 8The luminosity is monitored continuously by the polarimete r control sys- tem [27]. If the luminosity drops below a speciﬁed value, the procedures de- scribed above are executed automatically to re-optimize it . This online feed- back system also ensures that the calorimeter remains cente red on the back- scattered Compton photon distribution, and that the startu p and shutdown of the laser system are executed automatically. Therefore, under normal con- ditions, polarization measurements are performed without intervention during HERA operation. The detector can be operated in two diﬀerent modes, the singl e-photon and the multi-photon mode. In contrast to the single-photon mode, i n which the energy of each individual Compton photon is analyzed, in the multi- photon mode one measures the total energy deposited in the detector by ma ny Compton photons per laser pulse interaction with an electron bunch. The multi-photon mode was chosen as the standard mode of operation to provide h igh statistics single bunch measurements in real time, and to overwhelm the bremsstrahlung backgrounds originating from the residual vacuum pressure in the straight section between the two dipole magnets BH39 and BH90. The sin gle-photon mode is used for test and diagnosis purposes only. 4.1 Single-Photon Mode The advantages of running in single-photon mode would be two fold. The asym- metries are large, up to 0.60 at the Compton edge (see Fig. 1), and the energy spectra can be compared to the Compton cross sections. Opera tion of the Longitudinal Polarimeter in this mode is possible if the las er pulse intensity is drastically reduced. However, the resolution for single-p hoton events is rather poor, as shown in Fig. 6, because most of the generated ˇCerenkov light is trapped in the crystals and does not reach the photomultipli er tubes due to the high index of refraction of the crystals and the 3 mm air ga p between the crystals and the photomultiplier tubes. While Compton spec tra can be pro- duced, and the beam polarization can be extracted, this is no t a feasible mode of operation with the 100 Hz laser, since a measurement of the beam polar- ization with an absolute statistical accuracy of 0.01 takes about 2.5 hours. In comparison, such a measurement takes only one minute in the m ulti-photon mode. In the single-photon mode, the asymmetry can be written as As(Eγ) =(dσ/dE γ)1 2−(dσ/dE γ)3 2 (dσ/dE γ)1 2+ (dσ/dE γ)3 2=PcPeAz(Eγ), (4) where ( dσ/dE γ)1 2and (dσ/dEγ)3 2are the cross sections for the electron-photon 90255075100125150175200225 0 10 20 30 photon energy (GeV)events spin 1/2 0255075100125150175200225 0 10 20 30 photon energy (GeV)spin 3/2 Fig. 6. Energy spectra collected in single-photon mode for t he spin-1 2and spin-3 2 conﬁgurations at a beam polarization of 0.51. The solid line is the result of a simu- lation [28] for a Compton (bremsstrahlung) rate of 0.02 (0.0 6) per bunch. conﬁgurations where the incident spins are antiparallel an d parallel, respec- tively, and Pc=1 2(\|P+1\|+\|P−1\|) is the average circular light polarization. The electron beam polarization is determined by ﬁtting the e nergy spectra for the two spin conﬁgurations using a simulation (solid line in Fig. 6) that in- cludes the response function and resolution of the detector , and realistic back- ground conditions [28] above a Compton photon energy of 4 GeV . Whereas the simulation represents the data well above 4 GeV, it consi derably underes- timates the background at lower energies. 4.2 Multi-Photon Mode The operation of the Longitudinal Polarimeter in multi-pho ton mode has the advantage of being eﬀectively independent of bremsstrahlu ng background in the HERA storage ring. A large number of Compton photons is pr oduced each time a laser pulse interacts with an electron bunch. The se photons are detected together by the calorimeter, which measures their energy sums I1 2 andI3 2for the spin-1 2and spin-3 2electron-photon conﬁgurations, respectively. In the multi-photon mode, an energy asymmetry is formed as Am=I1 2−I3 2 I1 2+I3 2=PcPeAp, (5) where Apis the analyzing power of the process. Under the assumption t hat the photomultiplier signals are linear over the full single-ph oton to multi-photon operating range, Apis given by the integrals over the energy weighted cross sections for the spin-1 2and spin-3 2conﬁgurations and PcPe= 1, multiplied by the single-photon relative response function r(Eγ) =S(Eγ)/Eγ(where Sis 10the digitized ADC signal) of the detector. The analyzing can be written as Ap=Σ1 2−Σ3 2 Σ1 2+ Σ 3 2, (6) with Σ i=Eγ,max/integraldisplay Eγ,min(dσ/dE γ)iEγr(Eγ)dEγ, i=1 2,3 2. Assuming a linear energy response of the detector, the analy zing power has a value of 0.1838 for Eλ= 2.33 eV and Ee= 27.5 GeV. The energy-weighted Compton cross sections for the two spin conﬁgurations are sh own in Fig. 7. Their asymmetry is largest for photon energies close to the C ompton edge (Eγ,max). For small photon energies the two distributions are nearl y identical. This has the advantage that the analyzing power is not very se nsitive to the detector energy threshold Eγ,min. 020040060080010001200 0 2 4 6 8 10 12 14 Eγ (GeV)Eγ dσ/dEγ (mb). spin 3/2spin 1/2 Fig. 7. Energy-weighted cross sections for the spin-1 2(solid curve) and spin-3 2(dashed curve) conﬁgurations. 4.3 Polarization Determination In order to determine the single-photon relative response f unction r(Eγ) of the NaBi(WO 4)2calorimeter, test beam measurements were performed at DESY and CERN, covering the entire energy range of the back-scatt ered Compton photons, as shown in Fig. 8. The resulting analyzing power wa s found to be 0.1929±0.0017. Note that only the precision of the measurement of the calorimeter response is important, and not its deviation fr om linearity. However, to apply the result for the relative response funct ion obtained from the test beam measurements to Eq. (6), it was necessary to sho w that the photomultiplier response is linear over the full single-ph oton to multi-photon operating range. This was veriﬁed using the gain monitoring system. The ul- timate test was then to show that the measurement of the beam p olarization 110.80.850.90.9511.051.11.151.2 0 5 10 15 20 25 30 beam energy (GeV)rel. response function r(E) Fig. 8. Relative calorimeter response function, normalize d to unity at 5 GeV, as determined in the DESY T22 and CERN X5 test beams. The band rep resents the systematic uncertainty. was not aﬀected by changing from single-photon to multi-pho ton mode. This was demonstrated by attenuating the laser beam intensity ov er three orders of magnitude using a rotatable half wave plate and a ﬁxed Glan-T hompson prism while increasing the photomultiplier high-voltage such th at the digitized sig- nal of the photomultiplier remained constant. The test was p erformed during stable electron beam conditions while monitoring the beam p olarization with the Transverse Polarimeter by observing the ratio of the pol arization of the two polarimeters. As shown in Fig. 9, the ratio was constant o ver the entire range. 0.50.60.70.80.911.11.21.31.41.5 10-11 10 102103 number of Compton photonspolarization ratio Fig. 9. Ratio of longitudinal to transverse electron beam po larization as a function of the average number of Compton photons detected in the calo rimeter. Once the calorimeter response was understood in the single- photon and multi- photon modes, the longitudinal beam polarization was deter mined by evalu- ating the calorimeter signals for every bunch individually . Although the laser is triggered by a precise electronic signal that is synchron ized with the HERA bunch timing, the time of the resulting light pulse ﬂuctuate s within ±1.5 ns. Because of this ﬂuctuation and the ﬁnite crossing angle, the 37 ps long electron bunches interact with varying parts of the 3 ns long laser pul ses. As mentioned earlier, the timing of each laser pulse is recorded relative to the trigger signal. The calorimeter signal reﬂects the temporal proﬁle of the la ser pulses if it is 12plotted versus the relative trigger time, as shown in Fig. 10 . With a ﬁt to this distribution, the calorimeter response is corrected for th is variation. 0100020003000400050006000 0 1 2 3 4 5 relative trigger time (ns)calorimeter signal (a.u.) Fig. 10. Temporal proﬁle of the laser pulses as sampled by an e lectron bunch. The solid line through the distribution is a ﬁt which is used to co rrect the calorimeter response. Switching between the two light helicity states results in t he two energy distri- butions for the corrected calorimeter signals I1 2andI3 2, displayed in Fig. 11 for an individual bunch. The longitudinal polarization of each electron bunch is determined from the asymmetry of the means of these two energ y distributions divided by the analyzing power and the measured circular lig ht polarization (Eq. (5)). This calculation is provided every minute. The lo ngitudinal beam polarization is ﬁnally computed as the mean of the individua l bunch polariza- tions weighted by the corresponding time-averaged bunch cu rrents. 050100150200250300350 0 1000 2000 3000 4000 5000spin 3/2spin 1/2 I 1/2 , 3/2 (a.u.)counts Fig. 11. Spectra collected in multi-photon mode for the spin -1 2(solid histogram) and spin-3 2(dashed histogram) conﬁgurations for a speciﬁc electron bu nch with a beam polarization of 0.59. 4.4ˇCerenkov Light Attenuation A large number of Compton photons can be produced per laser pu lse when the polarimeter is operated in the multi-photon mode, rangi ng from a few photons to many thousand. During normal operating mode in wh ich about 131000 back-scattered Compton photons are produced at the beg inning of a ﬁll, approximately 250 times more energy is deposited in the calo rimeter than the highest energies (bremsstrahlung) deposited in the single -photon mode, since the average energy deposited per Compton photon is 6.8 GeV. I n order to attenuate the ˇCerenkov light to protect the photomultiplier tubes from sa tu- ration, a remotely controlled movable, perforated nickel f oil could be inserted into the 3 mm air gap between the NaBi(WO 4)2crystals and the photomulti- plier tubes. This was initially the standard mode of operati on. Even though the NaBi(WO 4)2crystals are 19 radiation lengths long, there is a small amount of longitudinal shower leakage into the photo multiplier tubes. Unfortunately, the corresponding shower particles genera te a large signal in the photomultiplier tubes, which introduces a substantial non-linearity in the energy response. The longitudinal shower leakage signal de rives mostly from the highest energy Compton photons and hence has a large anal yzing power. When the ˇCerenkov light produced in the NaBi(WO 4)2crystals was attenu- ated by the nickel foil, the shower leakage signal dominated the signal in the photomultiplier tubes. This altered the response function and increased the analyzing power of the detector by about 25%. The polarimeter has been operated without any light attenua tors since early 1999. This was also the case for the test beam calibrations. T he gain in the photomultiplier tubes has to be reduced in the multi-photon mode by about a factor of 200. As described in Section 4.3, it was veriﬁed tha t the photomulti- plier tubes are linear over this large range in gain. To addre ss concerns about long-term stability, linearity, and radiation damage, a tu ngsten/scintillator sampling calorimeter, similar to the one employed in the Tra nsverse Polarime- ter [7] but without position sensitivity, is moved in the Com pton photon beam periodically. It acts as an independent device to check the b eam polarization measurement and is otherwise not exposed to bremsstrahlung and direct syn- chrotron radiation. 5 Polarimeter Performance Since early 1997 the Longitudinal Polarimeter has routinel y measured the HERA electron beam polarization for the HERMES experiment. Typical elec- tron beam ﬁlls last from eight to twelve hours, starting with a ramped injection current of 40 to 45 mA and ending usually with a controlled bea m dump when the current reaches about 10 mA. Fig. 12 shows an example of po larization measurements as a function of time for three consecutive ﬁll s. Each data point represents a one minute measurement of the longitudinal pol arization with an absolute statistical accuracy of 0.01. The time structure i n the ﬁrst ﬁll dis- played in Fig. 12 is the result of tuning eﬀorts by the HERA ope rators. In the 141997 and 1998 running periods the electron beam helicity was reversed every few months; since then it has been reversed once a month. 00.10.20.30.40.50.60.7 0 5 10 15 20 25 30 35 40 45 50 time (h)electron polarization Fig. 12. Longitudinal beam polarization versus time for thr ee consecutive ﬁlls. 5.1 Single Bunch Measurements The Longitudinal Polarimeter measures the polarization of individual bunches, as shown in Fig. 13. Each data point represents a measurement lasting twenty minutes with an absolute statistical accuracy of 0.03. Not a ll electron bunches collide with proton bunches in HERA, and it was found that the colliding and non-colliding electron bunches can have diﬀerent polariza tion values. This is believed to be caused by beam-beam interactions between the electron and proton beams and the associated tune shifts. Comparison of t he polarization of the 174 colliding and the 15 non-colliding bunches is a use ful tool for tuning the accelerator to optimize polarization. This informatio n is shown in Fig. 14 and is provided in real time to the HERA control room every min ute, with an absolute statistical precision of 0.01 (0.04) for the col liding (non-colliding) bunches. 00.10.20.30.40.50.60.7 0 25 50 75 100 125 150 175 200 bunch numberbunch polarization non-colliding bunchescolliding bunches Fig. 13. Polarization of the individual beam bunches, as mea sured by the Longitu- dinal Polarimeter. Analyzing individual electron bunches is as of yet unique to the Longitudinal Polarimeter. An upgrade of the data acquisition system [29] of the Transverse 15Polarimeter at HERA, which is in progress, will also have thi s important feature. This detailed polarization information about the electron beam will be crucial for the collider experiments H1 and ZEUS since the y are preparing to measure spin observables after the luminosity upgrade in 2001. Whereas the HERMES experiment is sensitive to the average beam polar ization of all the bunches, the collider experiments are sensitive to the c olliding bunches only. 00.10.20.30.40.50.60.7 20 22 24 26 28 30 32 time (h)electron polarization non-colliding bunchescolliding bunches Fig. 14. Longitudinal polarization of the colliding and non -colliding beam bunches versus time, for the second beam ﬁll in Fig. 12. 5.2 Systematic Uncertainties Various studies have been performed to investigate the syst ematic uncertainty associated with the polarization measurements by the Longi tudinal Polarime- ter operated in the multi-photon mode. Since the polarizati on of the electron beam is obtained from a measurement of an asymmetry Am(see Eq. (5)), po- tential sources of false asymmetries were investigated. Ot her studies quantiﬁed the precision with which the analyzing power Apand the circular light polar- ization Pcat the interaction point are determined. The various contri butions to the total systematic uncertainty are summarized in Table 1, and apply to the polarimeter operating conditions without optical ﬁlte rs in the calorimeter, i.e. since early 1999. The largest contribution to the overall systematic uncerta inty originates from the determination of the analyzing power, which depends str ongly on the ex- act shape of the relative detector response function r(Eγ) (see Fig. 8). The energy calibration of the detector in the test beams takes in to account most sources that can lead to a non-linear response of the detecto r including the crystals, and the signal generated in the photomultiplier t ubes by the longi- tudinal shower leakage (see section 4.4). It does not accoun t for the response of the photomultiplier tubes in the multi-photon mode, but i t accounts for the low-energy cut-oﬀ from the lead absorber, and the limite d size of the calorimeter. The analyzing power was determined to a precis ion of 1 .2 % that 16was calculated by propagating the systematic uncertainty o f the relative re- sponse function (0 .9 %) shown in Fig. 8, and by including the uncertainty of the transition from single-photon to multi-photon mode ( 0.8 %) shown in Fig. 9. The long-term stability of the detector response function i s checked by mon- itoring the sources that can produce a time-dependent non-l inear detector response. The linearity of the photomultiplier tubes is che cked continuously with the gain monitoring system over the full multi-photon o perating range and is found to deviate by less than 0 .4 % over the annual running period. The eﬀect on the analyzing power is of the same size. The annua l radiation dose deposited in the crystals was determined to be about ten times below the level of damage. Instead of considering all contributio ns separately, the overall systematic uncertainty can also be estimated by per iodically perform- ing measurements with the sampling calorimeter to compare t he polarization measurements of the two detectors. Based on this comparison , the systematic uncertainty associated with the long-term instability of t he analyzing power is 0.5 %. Possible false asymmetries introduced by gain-mismatched photomultiplier tubes have been considered. An iterative method based on dat a from scan- ning the Compton photon beam across the detector front face i s used to gain- match the detector elements within an accuracy of about 5%. T he gains of the photomultiplier tubes are monitored continuously and f ound to change by approximately 20 −30 % during a beam ﬁll, returning to their initial val- ues between ﬁlls. These short-term drifts diﬀer by only a few percent for the four photomultiplier tubes and therefore have no net eﬀect o n the polariza- tion measurement, since the Longitudinal Polarimeter does not depend on an absolute energy calibration. Long-term deviations of the r elative gains during the annual running were found to aﬀect the beam polarization measurement by less than 0 .3 %. The circular polarization of the laser light can be determin ed very precisely [30] immediately following the Pockels cell in the laser room. Ho wever, it can in principle be diﬀerent at the interaction point, given the fa ct that the laser beam has to be passed through windows and lenses and be reﬂect ed from mir- rors before it interacts with the electron beam. To estimate this uncertainty, the laser beam polarization was measured after the storage r ing vacuum win- dow with the identical analyzer that is normally mounted in t he laser room. The two vacuum windows were removed, then each window was mou nted sep- arately, and ﬁnally the ring vacuum was re-established, whi le measuring the laser beam polarization after each step. Based on these meas urements, a sys- tematic uncertainty of 0 .2 % was assigned to the circular polarization of the laser light at the interaction point. 17The measurement may also be aﬀected by changes in the phase sp ace of the laser beam at the interaction point due to an imperfectly ali gned Pockels cell. Horizontal or vertical shifts of the laser beam can occur whe n the voltage across the cell is changed, resulting in a helicity-depende nt luminosity and hence a false energy asymmetry. To quantify extraneous heli city-dependent beam shift eﬀects in the system, we performed two tests. Firs t, a half wave plate was temporarily mounted immediately following the Po ckels cell. Except for the expected change of sign in the measurement of the elec tron beam polar- ization, no change in the magnitude was observed within the 0 .3 % precision of the test. However, this test does not account for non-optima l laser and electron beam overlap. This is important since the sensitivity to a he licity-dependent laser beam shift increases with decreasing overlap. Theref ore a second test was performed by changing the overlap of the two beams within the limits of the normal operating conditions. This showed that the imp act on the en- ergy asymmetry is at most 0 .3 %. Combining the two values leads to a total contribution of 0 .4 %. The position and size of the Compton photon beam incident on t he calorime- ter is determined by the electron beam orbit conditions at th e interaction point. During normal HERA luminosity operation, variation s of the size and divergence of the electron beam are so small that the impact o n the calorime- ter response is negligible. However, a change of the positio n or slope of the electron beam at the interaction point can result in a shift o f the Compton photon distribution away from the center of the calorimeter . In these cases, the online feedback system of the polarimeter automaticall y repositions the calorimeter center on the Compton photon beam to better than 1 mm preci- sion. By scanning the Compton photon beam across the calorim eter front face, it has been determined that within the relevant operating ra nge, the eﬀect on the measurement of the beam polarization is less than 0 .6 %. To estimate the eﬀect of slow beam drifts during a ﬁll, the slope of the electr on beam was moved over the maximal observed range while keeping the Comp ton photon distribution centered on the calorimeter. No inﬂuence on th e polarization mea- surement was observed within the 0 .5 % accuracy of the study. Combining the uncertainties of the two tests leads to a total contribution of at most 0 .8 %. The various contributions to the systematic uncertainties of the Longitudi- nal Polarimeter have been considered separately and added i n quadrature to a total uncertainty of 1 .6 % (see Table 1). Those systematic uncertainties of the two HERA electron beam polarimeters that relate to stabi lity and re- producibility (not absolute scale) can be further studied b y comparing their performances over an extended period of time. Non-statisti cal ﬂuctuations in the ratio of their results over the 1999-2000 running period s correspond to a relative systematic stability of σ= 1.6 %, which is compatible with the quadratic sum of contributions estimated from the two instr uments. 18Table 1 The various contributions to the fractional systematic unc ertainty of the longitudi- nal electron beam polarization Pe. Source of systematic uncertainty ∆ Pe/Pe Analyzing power ±1.2% Analyzing power long-term instability ±0.5% Gain mismatching ±0.3% Laser light polarization ±0.2% Pockels Cell misalignment ±0.4% Electron beam instability ±0.8% Total ±1.6% 6 Summary We have designed and constructed a Compton back-scattering laser polarime- ter which routinely measures the longitudinal polarizatio n of the HERA elec- tron beam for the HERMES experiment. The Longitudinal Polar imeter de- termines the beam polarization with an absolute statistica l precision of 0.01 per minute and a fractional systematic uncertainty of 1 .6 %. The polarimeter also measures the polarization of individual electron bunc hes, a feature that is currently not available to the Transverse Polarimeter. I t was found that the individual bunches can each have a signiﬁcantly diﬀerent po larization. This observation can be further analyzed if one groups the bunche s into colliding and non-colliding bunches. The variation and the time evolu tion of the polar- ization of the individual bunches and of classes of bunches p rovide important additional information for achieving high beam polarizati on at HERA. 7 Acknowledgments We would like to thank R. Fastner and the crew of the machine sh op from the University of Freiburg for the design and construction o f the optical trans- port system, and G. Braun for his help in designing some of the electronic components. We thank M. Spengos and E. Steﬀens for their help in the ini- tial planning of the project, N. Meyners for his design of the safety interlock system, and H.D. Bremer for his help in the design and constru ction of the calorimeter and the calorimeter table. We also thank T. Behn ke, A. Miller, and P. Sch¨ uler for many useful discussions and their commen ts to this article. We are grateful to E. Belz, O. H¨ ausser, R. Henderson, M. Ruh, M. Woods, and R. Zurm¨ uhle for their help and advice. We acknowledge th e DESY man- 19agement for its support, and the DESY staﬀ for the signiﬁcant eﬀort in the planning, design, and construction of the Longitudinal Pol arimeter. We es- pecially acknowledge the eﬀorts of the HERA machine group to deliver high beam polarization. This work was supported by the German Bun desminis- terium f¨ ur Bildung, Wissenschaft, Forschung und Technolo gie, and the US National Science Foundation. References [1] A.A. Sokolov and I.M. Ternov, Sov. Phys. Doklady 8 (1964) 1203. [2] V.N. Baier and V.A. Khoze, Sov. J. Nucl. Phys. 9 (1969) 238 . [3] D.P. Barber, Proc. of the 12th International Symposium o n High-Energy Spin Physics, edited by C.W. de Jager et al., Amsterdam, The N etherlands, World Scientiﬁc (1996) 98. The laboratories include: VEPP- 3 and VEPP-4 at Novosibirsk, SPEAR at SLAC, CESR at Cornell, DORIS, PETRA an d HERA at DESY, and LEP at CERN. [4] M. Woods et al., hep-ex/9611005; SLAC-PUB-7319 (1996); and C. Cavata, Proc. of the 11th International Symposium on High-Energy Sp in Physics, edited by K.J. Heller and S.L. Smith, Bloomington, USA, IAP Conf. Pr oc. 343 (1995) 250. [5] I. Passchier et al., Nucl. Instr. Meth. A 414 (1998) 446. [6] W.A. Franklin et al., Progress in Part. and Nucl. Phys. 44 (2000) 61. [7] D.P. Barber et al., Nucl. Instr. Meth. A 329 (1993) 79. [8] K. Ackerstaﬀ et al., Phys. Lett. B 464 (1999) 123. [9] J. Buon and K. Steﬀen, Nucl. Instr. Meth. A 245 (1986) 248. [10] K. Ackerstaﬀ et al., Nucl. Instr. Meth. A 417 (1998) 230. [11] F.W. Lipps and H.A. Tolhoek, Physica 20 (1954) 85 and 395 . [12] Model Inﬁnity 40-100 from Coherent Laser Group. [13] Pockels cell with 21 mm clear aperture from Gs¨ anger Com pany in Germany, now Linos Photonics. [14] F. Burkart, Wissenschaftliche Arbeit, Univ. Freiburg (1996). [15] The beam expander consists of two fused silica lenses fr om CVI Laser Corporation: a plano-concave lens of type PLCC-25.4-77.3- UV (−150 mm focal length) and a plano-convex lens of type PLCX-50.8-309.1-UV (+600 mm focal length). 20[16] Mirrors M1 to M4 are 4-inch diameter fused silica mirror s of type Y2-4050- 45UNP-37 and mirrors M5 and M6 are 2-inch diameter fused sili ca mirrors of type Y2-2037-45UNP-37. All mirrors are from CVI Laser Cor poration. The vacuum compatible mounts and their remote control system ar e from OWIS GmbH, Germany. [17] Fused Silica lens doublet of type PLCC-95.0-206.0-C-5 32 (−400 mm focal length) and PLCX-95.0-206.0-C-532 (+400 mm focal length) f rom CVI Laser Corporation. [18] Fused Silica windows of type PP 1537 UV-5320-0 from CVI L aser Corporation. [19] Aluminum gaskets of type HNV 200 from Helicoﬂex. [20] G.I. Britvich et al., Nucl. Instr. Meth. A 321 (1992) 64. [21] A.V. Antipov et al., Nucl. Instr. Meth. A 327 (1993) 346. [22] Hamamatsu photomultiplier tubes model R4125 MOD with 1 5 mm active diameter. [23] The bunch trigger module is developed by the MKI group at DESY. [24] Motorola 96002 DSP. [25] Fastbus Analog to Digital Converter model 1881M from Le Croy Research Systems. [26] The Time to Digital Converter consists of a Fast Encodin g Time to Charge Converter model 4303 from LeCroy Research Systems followed by the Fastbus ADC model 1881M also from LeCroy Research Systems. [27] S. Brauksiepe, Proc. of the 12th International Symposi um on High-Energy Spin Physics, edited by C.W. de Jager et al., Amsterdam, The Nethe rlands, World Scientiﬁc (1996) 771. [28] C. Pascaud and F. Zomer, private communications. [29] V. Andreev et al., DESY PRC-99-04: Proposal for an upgra de of the HERA polarimeters for HERA 2000, Dec 1998. [30] M. Beckmann, Ph.D. thesis, Univ. Freiburg (2000). 21
arXiv:physics/0009048v1 [physics.bio-ph] 13 Sep 2000Balanced Branching in Transcription Termination K. J. Harrington and R. B. Laughlin Department of Physics, Stanford University, Stanford, Cal ifornia 94305 S. Liang NASA Ames Research Center, Moﬀett Field, California 94035 (January 12, 2014) The theory of stochastic transcription termination based o n free-energy competition1requires two or more reaction rates to be delicately balanced over a wide r ange of physical conditions. A large body of work on glasses and large molecules suggests that thi s should be impossible in such a large system in the absence of a new organizing principle of matter . We review the experimental literature of termination and ﬁnd no evidence for such a principle but ma ny troubling inconsistencies, most notably anomalous memory eﬀects. These suggest that termin ation has a deterministic component and may conceivably be not stochastic at all. We ﬁnd that a key experiment by Wilson and von Hippel2allegedly refuting deterministic termination was an incor rectly analyzed regulatory eﬀect of Mg2+binding. PACS numbers: 87.15.-v, 82.20.Pm, 61.43.Fs I. INTRODUCTION The branching ratio of the termination process in gene transcription is balanced. In the case most thoroughly studied, ρ-independent termination in procaryotes, con- ventional gel experiments performed in vitro ﬁnd a frac- tionPof elongating RNA polymerase reading through the termination sequence with \|ln(1/P−1)\|<4 essen- tially always, even though Pis diﬀerent for diﬀerent ter- minators and can be made to exhibit order-1 changes by perturbing the environment. This eﬀect is astonish- ing from the standpoint of microscopic physics because a stochastic decision to read through or not requires a com- petition of transition rates - quantities of inverse time - that must be nearly equal for the branching to be bal- anced. RNA polymerase, however, is more the size of a glass simulation than a small molecule and thus possesses a broad spectrum of natural time scales spanning many decades. Without some physical reason for a particu- lar scale to be preferred, rate competition ought to have been severely unbalanced, meaning that one event occurs essentially always and the other never. Balanced branch- ing in termination has been implicated in transcription regulation in a few cases,3but its broader signiﬁcance, especially its robustness, is still a mystery. In this paper we examine the experimental facts rele- vant to the physical nature of termination with the goal of determining what, if any, principle selects the time scale for stochastic rate balance. Our conclusion is both surprising and unsettling. We ﬁnd no evidence for such a principle, but glaring weaknesses in the case for stochas- ticity and a large body of unexplained experimental re- sults pointing to a termination decision that is partially deterministic. In light of the inaccessability of systems this large to ab-initio computation we conclude that tran- scription termination is a fundamentally unsolved prob-lem in mesoscopic physics and an ideal target for the emerging techniques of nanoscience. II. TERMINATION EFFICIENCY The simplest termination sequences are the ρ- independent terminators of procaryotes, which are capa- ble of causing polymerase to terminate in vitro in the ab- sence of the ρprotein factor. A representative sampling of these is reproduced in Table I. This diﬀers from lists that have appeared in the literature before4,5by having been rechecked against the fully-sequenced genome6and expunged of “theoretical” terminators identiﬁed only by computer search. They conform for the most part to the motif of a palindrome of typically 10 base pairs followed by a short poly-T stretch, although there is tremendous variety in the length and composition of the palindrome, variation in the length of the poly-T stretch, and occa- sional extension of the palindrome to include the poly- T stretch. This enormous variability contrasts with the simplicity of stop codons, which terminate protein syn- thesis by ribosomes and have no other function. ρ-independent terminators are characterized by “eﬃ- ciencies”, i.e., the fraction of assayed transcripts that t er- minate. These rarely take on extreme values close to 1 or 0 when measured in vitro . In cases where a measurement in vivo exists as well the latter is usually larger7and is occasionally unmeasurably close to 1. Balanced termina- tion eﬃciency is commonly observed in vivo as well, how- ever. Table II shows results from a particularly careful study7in vitro in which termination probabilities in E. colifor wild-type terminators, mutant terminators, phage terminators,8and terminators from S. Boydii were mea- sured under identical conditions. Despite the great vari- ety of these sequences the termination eﬃciency runs only 1Sequence4,5Name Address6± Reference CGTTAATCCGCAAATAACGTAAAAACCCGC TTCGGCGGGTTTTT TTATGGGGGGA rpoC t 4187152 + RNA polymerase operon57 CAGTTTCACCTGATTTACGTAAAAACCCGC TTCGGCGGGTTTTT GCTTTTGGAGG M1-RNA 3267812 - RNA of RNase P58 CGTACCCCAGCCACATTAAAAAAGCTCGC TTCGGCGAGCTTTTT GCTTTTCTGCG sup 0695610 - supBC tRNA operon59 ACACTAATCGAACCCGGCTCAAAGACCCGC TGCGGCGGGT TTTTTTGTCTGTAAT 1260102 - Nucleotide synthesis60 AGTAATCTGAAGCAACGTAAAAAAACCCGCC CCGGCGGGTTTTTTT ATACCCGTA L17 3437202 - Ribosomal RNA operon61 TCTCGCTTTGATGTAACAAAAAACCCCGCC CCGGCGGGGTTTTTTGTTA TCTGCT rpm 3808820 - Ribosome rpm operon62 GAGTAAGGTTGCCATTTGCCCTCCGC TGCGGCGGGGGGC TTTTAACCGGGCAGGA t2 3306624 -Polynucleotide phosphorylase63 CGATTGCCTTGTGAAGCCGGAGCGG GAGACTGCTCCGGC TTTTTAGTATCTATTC deo t 4619189 + deo operon64 CGTAAAGAAATCAGATACCCGCCCGC CTAATGAGCGGGC TTTTTTTTGAACAAAA trp a 1321015 - tryptophan synthesis65 GCGCAGTTAATCCCACAGCCGCCAG TTCCGCTGGCGGC ATTTTAACTTTCTTTAA trp t 1314395 - tryptophan synthesis66 AAATCAGGCTGATGGCTGGTGACT TTTTAGTCACCAGCC TTTTTGCGCTGTAAGG rplL t 4178530 +Ribosomal proteins L7/L1267 AGGAAACACAGAAAAAAGCCCGCAC CTGACAGTGCGGGCTTTTTT TTTCGACCAA thr a 0000263 + threonine operon68 AGCACGCAGTCAAACAAAAAACCCGCGC CATTGCGCGGGTTTTTT TATGCCCGAA leu a 0083564 - leucine synthesis69 CCCGTTGATCACCCATTCCCAGCCCCTC AATCGAGGGGCT TTTTTTTGCCCAGGC pyrBI a 4469985 - pyrimidine synthesis70 ACACGATTCCAAAACCCCGCCGG CGCAAACCGGGCGGGGTTTT TCGTTTAAGCAC ilvB a 3850449 - ilvB operon71 GAAACGGAAAACAGCGCCTGAAAGCCTCC CAGTGGAGGCTTT TTTTGTATGCGCG pheS a 1797160 -Phenylalanyl-tRNA synthetase72 CTTAACGAACTAAGACCCCCG CACCGAAAGGTCCGGGGGT TTTTTTTGACCTTAA ilvGEDA a 3948053 + ilvGEDA operon73 CCGCCCCTGCCAGAAATCATCCTTA GCGAAACGTAAGGAT TTTTTTTATCTGAAA rrnC t 3944645 + Ribosomal RNA operon74 CATCAAATAAAACAAAAGGC TCAGTCGGAAGACTG GGCCTTTTGTTTTAT CTGTT rrnD t 3421006 + Ribosomal RNA operon75 TCCGCCACTTATTAAGAAGCCTCGAG TTAACGCTCGAGG TTTTTTTTCGTCTGTA rrnF (G) t 0228998 + Ribosomal RNA operon76 GCATCGCCAATGTAAATCCGGCCCGCC TATGGCGGGCCG TTTTGTATGGAAACCA frdB t 4376529 - Fumarate reductase77 TGAATATTTTAGCCGCCCCAGTCA GTAATGACTGGGGCG TTTTTTATTGGGCGAA spot42-RNA 4047542 + spot42 RNA78 ATTCAGTAAGCAGAAAGTCAAAAGCCTCCG ACCGGAGGCTTTTGACT ATTACTCA tonB t 1309824 + Membrane protein79 AGAAACAGCAAACAATCCAAAACGCCGC GTTCAGCGGCGTTTT TTCTGCTTTTCT glnS T 0707159 +Glutaminyl-tRNA synthetase80 CTGGCATAAGCCAGTTGAAAGAGGGAG CTAGTCTCCCTCTTT TCGTTTCAACGCC rplT t 1797371 - Ribosome protein L2081 GCATCGCCAATGTAAATCCGGCCCGCC TATGGCGGGCCG TTTTGTATGGAAACCA ampC a 4376529 - β-lactamase82 TGCGAAGACGAACAATAAGGCC TCCCAAATCGGG GGGCCTTTTTTATTGATAACA phe a 2735697 + Phenylalanine operon83 ACGCATGAGAAAGCCCCCGGAAG ATCACCTTCCGGGGGCTTT TTTATTGCGCGGT hisG a 2088121 + ATP synthesis84 CATCAAATAAAACGAAAGGC TCAGTCGAAAGACTG GGCCTTTCGTTTTAT CTGTT rrnB t 14169333 + Ribosomal RNA operon GGCATCAAATTAAGCAGAAGGCCATCC TGACGGATGGCCTT TTTGCGTTTCTACA rrnB t 24169493 + Ribosomal RNA operon AATTAATGTGAG TTAGCTCAC TCATTAGGCACCCCAGGCTTTACACTTTATGCTT lacI tII 0365588 - Lactose synthesis85 CTTTTTGGCGGAGGGCG TTGCGCTTCTCCGCC CAACCTATTTTTACGCGGCGGTG uvrD a 3995538 + DNA helicase II86 TABLE I. ρ-independent terminators in E. coli taken primarily from Brendel et al.4These are oriented in the reading direction and are aligned at the poly-T stretch. The palindr ome is underlined. The beginning and end of the selected sequ ences have no absolute meaning but simply follow the convention of d’Aubenton et al.5The address identiﬁes the location in the standard E. coli genome6of the left-most nucleotide in the table. Sequence Name % T GGCTCAGTC GAAAGACTGGGCC TTTCGTTTT AAT rrnB t 184±1 TCAAAAGCCTCCG ACCGGAGGCT TTTGACTATTA tonB t 19±1 CCAGCCCGC CTAATGAGCGGGCT TTTTTTTGAAC trp a 71±2 CCAGCCCGC CTAATGAGCGGGCT TTTGCAAGGTT trp a 1419 2±1 CCAGCCCGC CTAATAAGCGGGC TTTTTTTT GAAC trp a L126 65±4 CCAGCCCGC CTAATAAGCGG ACTTTTTTTT GAAC trp a L153 8±4 CTGGCTCACC TTCGGGTGGGCC TTTCTGCG TTTA T7T e88±2 GGCTCACCTTCACGGGTG AGCCTTTCTTCG TTCX T3T e14±2 GGCCTGCTGGTAATCGCAGGCC TTTTTATTT GGG tR2 49±4 AAACCACCGTT GGTTAGCGGTGG TTTTTTGTTTG RNA I 73±4 TABLE II. Termination eﬃciencies measured in vitro .7 The ﬁrst 3 terminators are native to E. coli . These are fol- lowed by 3 mutants, 3 phage terminators,8and one from S. Boydii . Far-right underlined sequences are termination zones.Sequence Name % T GTTAATAACAGGCCTGC TGGTAATCGCAGGCCT TTTTATT tR2 40 GTTAATAACAGGGGACG TGGTAATCCGTCCCC TTTTTATT tR2-6 56 TAATAACAGGCCTGGC TGGTAATCGCCAGGCCT TTTTATT tR2-11 54 CCGGGTTAATAACAGGCCTGC TTCGGCAGGCCT TTTTATT tR2-12 69 CGGGTTATTAACAGGCCTC TGGTAATCGAGGC TTTTTATT tR2-13 11 ATAACAGGGGACG TGGTAATCGCCAGCAGGCC TTTTTATT tR2-14 20 GTTAATAAAAGGCCTGC TGGTAATCGCAGGCCTTTTTATT tR2-16 36 GGTTCTTCTCGGCCTGC TGGTAATCGCAGGCC TTTTTATT tR2-17 67 TABLE III. Termination eﬃciencies for modiﬁed versions of the phage λterminator tR2.11 from 2% to 88%. Many other researchers report similar values for terminators in E. coli and other bacteria,9including artiﬁcially altered terminators.10 2Sequence Name rpo+ rpo203 GCAACCGCTGGGG AATTCCCCAG TTTTCA trpC 301 0 20 AACCGCTGGCCGG GATCGGCCAG TTTTCA trpC 302 8 35 CAGCCGCCAG TTCCGCTGGCGGCT TTTAA trp t 25 45 ACCAGCCCGC CTAATGAGCGGGCT TTTGC trp a 1419 3 35 CAGCCCGC CTAATGAGCGGGCTG TTTTTT trp a 135 65 80 TABLE IV. Termination eﬃciences for wild-type E. coli polymerase (rpo+) and mutant polymerase (rpo203).12trp t is native to the genome. The rest are either mutants or syn- thetic. The results in Tables III and IV show balanced ter- mination for modiﬁed versions of the phage terminator tR211and for mutant polymerase.12This also makes order-1 changes to the eﬃciencies themselves. Simi- lar eﬀects were reported by other researchers9,13with diﬀerent mutant polymerases. Modiﬁcations up to 20 base pairs upstream and downstream of the terminator cause large changes to the eﬃciency without causing it to unbalance.7Thus balanced termination eﬃciency is the norm rather than the exception. III. LARGE MOLECULES AND GLASSES Large systems are qualitatively diﬀerent from small ones.14The speciﬁc heat of all non-crystalline matter in macroscopic quantities - including biological matter - is proportional to Tat low temperatures.15This behavior is fundamentally incompatible with the linear vibration of the atoms around sites, and is caused by collective quan- tum tunneling of atoms between energetically equivalent “frustrated” conﬁgurations.16It contrasts sharply with theT3behavior of crystals with small unit cells. Glasses also exhibit stretched-exponential time dependence in re- sponse to perturbations, i.e., of the form exp( −Atβ) with β <1, indicating a broad spectrum of decay rates rather than just one. They also exhibit memory eﬀects, such as “remanence” in spin glasses17or the well-known failure of ordinary silica to crystallize without annealing. This be- havior is universal and robust. All non-crystalline macro- scopic matter exhibits hysteresis, metastability, a broad spectrum of relaxation times, and memory. How large a system must be before it can exhibit such behavior is not known, as the relevant experiments are diﬃcult to perform except on macroscopic samples, but there are many indications that even medium-sized proteins have glass-like properties. Crystals of myo- globin, a protein with a molecular weight of only 17,000, have linear speciﬁc heats at low temperatures18and exhibit stretched-exponential response to photodissoci- ation pulses.19Denatured proteins refold on a variety of time scales ranging from nanoseconds to seconds,20 and amino acids sequences chosen at random will not fold at all.21Permanent misfolding of proteins with molecular weights of only 30,000 has been implicated in prion diseases.22Many enzymes exhibit hysteresis in theircatalytic rates.23,24The activity of cholesterol oxidase ofBrevibacterium sp. , a protein with molecular weight 53,000, was recently shown by ﬂuorescence correlation techniques to have a memory eﬀect persisting about 1 second under normal conditions at room temperature.25 Other notable examples include wheat germ hexokinase (mol. wt. 50,00026) with a half-life of 2 minutes,27rat liver glucokinase (mol. wt. 52,00028) at 1 minute,23and yeast hexokinase (mol. wt. 50,000) at 1-2 minutes.29 Thus RNA polymerase complexes, which have a molec- ular weight of 379,000 and are comparable in size to the largest computer simulations of glasses ever performed, are good candidates for systems that exhibit glassy be- havior. Glassiness in enzymes is not always easy to observe. The mnemonic eﬀect in yeast hexokinase occurs when it is preincubated with MgATP and free Mg2+and the re- action is started with glucose, or preincubated with glu- cose and free Mg2+and started with MgATP, but notif the enzyme is preincubated with glucose and metal-free ATP and then started with Mg2+.23Mnemonic behav- ior can be destroyed by “desensitizing” the enzyme with contaminants.26Time scales can depend on enzyme, sub- strate, product, activator and eﬀector ligand concentra- tions as well as pH, buﬀers, and temperature.23,29,30Be- fore hysteresis and memory eﬀects were recognized, early investigators generally adjusted such reaction condition s until the “improper” behavior was eliminated.23 IV. POLYMERASE STATES While the size of RNA polymerase makes it plausible to expect glassy behavior on purely theoretical grounds, several direct lines of evidence indicate that the enzyme exhibits a spectrum of multiconformational, mnemonic and hysteretic behavior: 1. Polymerase has a catalytic mode distinct from RNA synthesis, as it can cleave the RNA transcript through hydrolysis (rather than pyrophosphorol- ysis, the reverse reaction of RNA synthesis),31 with the cleavage reaction requiring Mg2+,31being template-dependent,32changing the polymerase footprint size,33and stimulated either by GreA and GreB proteins34,35or by high pH (8.5-10.0).36 The last eﬀect was discovered serendipitously, go- ing unobserved for decades because assay condi- tions were being optimized to maximize elongation rates, which occur at lower pH values (7.8-8.237).36 2. RNA polymerase mobilities in non-denaturing elec- trophoresis gels show signiﬁcant and discontinu- ous variance while bearing nearly identical tran- scripts or identical length transcripts with diﬀerent sequences.38These mobility variances are still ob- served if the RNA transcript is ﬁrst removed by ribonuclease digestion.39 33. RNA polymerase ternary complexes vary greatly in their stability and mode of binding to DNA (ionic or non-ionic) in a template-dependent man- ner. Some complexes are stable against very high salt concentrations ([K+] = 1 M), while others (speciﬁcally those proximal to an upstream palin- drome sequence) are salt-sensitive (completely dis- sociating in concentrations as low as 20 mM K+). However, the salt-sensitive complexes are stabilized by millimolar concentrations of Mg2+.40 4. The size of the RNA polymerase footprint on the DNA template measured by ribonuclease digestion is signiﬁcantly altered even at adjacent template positions, suggesting that the enzyme assumes dif- ferent conformations during elongation.41 5. Guanosine tetraphosphate (ppGpp) inhibits the rate of elongation on natural DNA templates but not on synthetic dinucleotide polymer templates, and does not inhibit elongation by competing with NTP binding, but by enhancing pausing. It must therefore bind to polymerase and modify its behav- ior at an unrelated regulatory site in an allosteric manner, rather than interfering with the substrate binding site.42 6. The stability of a stalled elongation complex de- pends on whether the polymerase arrives at the stall site via synthesis or pyrophosphorolysis.43 7. Termination eﬃciencies are aﬀected by transcribed upstream sequences and untranscribed downstream sequences adjacent to the terminator.44 8. Stalling elongating polymerase complexes (via nu- cleotide starvation) and then restarting them by nucleotide addition perturbs pausing patterns 50- 60 base pairs downstream.45 9. An elongating polymerase’s Michaelis constants KS for NTPs vary over 500-fold for diﬀerent DNA tem- plate positions,46and for diﬀerent templates,47al- though these eﬀects are not observed for synthetic dinucleotide polymer templates.47 10. The rate of misincorporation at a single site for which the correct NTP is absent is signiﬁ- cantly diﬀerent before and after isolation of ternary complexes.48 11. Stalled polymerase gradually “arrests” (i.e., is in- capable of elongating when supplied with NTPs), with the approximate half-time for arrest estimated at 5 minutes40and 10 minutes49for diﬀerent DNA templates. The polymerase can continue elongat- ing if reactivated by pyrophosphorolysis.40 12. Even after undergoing arrest, crosslinking exper- iments show that the internal structure of poly- merase gradually changes over the course of the next hour.4913. Observations of single elongating RNA polymerase molecules show that it has two elongation modes with diﬀerent intrinsic transcription rates and propensities to pause and arrest.50 The possibility of metastability - through shape mem- ory or the conditional attachment of factors - is di- rectly relevant to the rate-balance conundrum because it provides a simple alternative to balanced stochastic branching that requires no physical miracles. If, for example, the polymerase possessed a small number of metastable conﬁgurational states and terminated deter- ministically depending on which state it was in, then balanced branching would be a simple, automatic con- sequence of scrambling the state populations. V. THERMAL ACTIVATION The idea that polymerase memory is potentially rel- evant to expression regulation is not new.47It is im- plicit in the work of Goliger et al51and Telesnitsky and Chamberlin44and even explicitly speculated by the latter in print. However, because of the experimental evidence supporting the stochastic model of termination1and the widespread belief - unjustiﬁed, in our view - that pro- teins equilibrate rapidly, this suggestion generated litt le enthusiasm. A key experiment supporting the stochastic model by Wilson and von Hippel2is both historically im- portant and typical, so it is appropriate that we consider it carefully. Wilson and von Hippel promoted and stalled RNA polymerase 8 base pairs upstream of the tR2 termina- tor hairpin of phage λin vitro , thermally equilibrated at temperature T, and then launched it forward by adding NTP. The results are reproduced in Fig. 1a. Termina- tion occurred at sites 7, 8, and 9 base pairs downstream of the beginning of the poly-T stretch (cf. Table II) with probabilities P7=N7/N,P8=N8/NandP9=N9/N. The data were originally reported as a semilogarithmic plot of 1 /ˆP−1 against temperature, where ˆP7=N7/N, ˆP8=N8/(N−N7) and ˆP9=N9/(N−N7−N8). They concluded that all three branching probabilities ˆPwere thermally activated and had distinctly diﬀerent activa- tion energies. However, it is clear from Fig. 1a that this conclusion is false. The three probabilities Pare essen- tially the same function and are well characterized by the sumP=P7+P8+P9, also plotted in Fig. 1a. This is shown more explicitly in Fig. 1b, where the ratios P7/P, P8/P, and P9/Pare plotted against temperature. The ﬂatness of these curves shows that the branching ratios among the three sites are essentially constant and in- dependent of temperature within the error bars of the experiment. Note that these fractions are also all of or- der 1. Thus the alleged spread in activation energies was an artifact of the plotting procedure. Let us now consider the temperature dependence. It may be seen from Fig. 1a that Psaturates to 1 at 80 ◦C, the temperature at which Wilson and von Hippel 4/0 /0/./2 /0/./4 /0/./6 /0/./8 /1/1/0 /2/0 /3/0 /4/0 /5/0 /6/0 /7/0 /8/0 PT /( /C/) a/)/3 /3 /3 /3 /3 /3/+ /+ /+ /+ /+ /+/2 /2 /2 /2 /2 /2/ / / / / //0/./1 /0/./4 /0/./5/1/0 /2/0 /3/0 /4/0 /5/0 /6/0 /7/0 /8/0 fT /( /C/) b/)/+/+ /+ /+ /+ /+ /2 /2/2/2/2/2// / / / //0 /1/1/0 /2/0 /3/0 /4/0 /5/0 /6/0 /7/0 /8/0 PT /( /C/) c/)/3 /3 /3 /3 /3 /3/0 /1/0 /1/0 /2/0 /3/0 /4/0 P/[MgCl/2 /] /(mM/) d/) /3/3/3/3 FIG. 1. a) Temperature dependence of termination proba- bility Pfor phage λterminator tR2 reported by Wilson and von Hippel.2+,✷, and×denote the probabilities to termi- nate 7, 8, and 9 nucleotides downstream from the beginning of the poly-T stretch. The sum is shown as ✸. b) +, ✷, and ×above divided by ✸to make a branching fraction f. c) Comparison of ionization model Eq. (1) with ✸from a). The ionization energy has been ﬁt to ǫ0= 0.7eV(16 kcals/mole) and the quantity n/M3/2adjusted to make the curves match at 30◦C. d) Prediction of Eqn. (1) for dependence on Mg2+ concentration compared with data of Reynolds et al.7 report that the polymerase “will not elongate”, i.e., has stopped working properly. This suggests that the eﬀect has something to do with the overall mechanical integrity of the enzyme rather than the termination process alone.Guided by this observation we note that the activated be- havior identiﬁed by Wilson and von Hippel is actually the formula for conventional monomolecular chemical equi- librium. The probability for a particle of mass Mwith a binding energy of E0to be ionized oﬀ the polymerase is P=1 1 +ZeE0/kBTn λ3 th(λth=/radicalBigg 2π¯h2 MkBT),(1) where nis the concentration of this component and Z is the change to the internal partition function that re- sults from binding. If one makes the approximation that λthis a slowly-varying function of temperature and can thus be taken to be constant then this reduces to the for- mula with which Wilson and von Hippel ﬁt their data.2 That it works may be seen in Fig. 1c, where we plot the total termination probability from experiment against Eq. (1) with E0= 0.7eVandZadjusted to match ex- periment at T = 30◦C. Thus reinterpreting this eﬀect as an ionization equilibrium, we may account for the high- temperature intercept and weak temperature dependence seen in Fig. 1b in the following way: In addition to the ionization state the polymerase possesses an internal con- ﬁgurational memory with a number of states of order 10. These code for termination at sites 7, 8 or 9. In the equili- bration step, the polymerase molecules come to thermal equilibrium and a fraction Pof them become ionized. All of these terminate at one of the three sites when launched. The rest read through. A candidate for the ionizable component is an Mg2+ ion. In their studies of the eﬀects of ion concentrations on termination eﬃciency, Reynolds et al7discovered that Mg2+has the strange and unique eﬀect of increasing ter- mination eﬃciency to 100% for all terminators studied when reduced below 1 mM. The Mg2+concentration in the experiments shown in Fig. 1d was 10 mM.2Extrap- olating at T = 30◦C53using Eq. (1) we obtain, with no adjustable parameters, the ﬁt to the [MgCl 2] dependence found by Reynolds et al7shown in Fig. 1d. The quality of this ﬁt suggests that Mg2+has a special function in regulating transcription, and that the temperature de- pendence in Fig. 1a is simply a thermal binding relation for this ion. This is corroborated by the recent structural studies of Zhang et al,54who report that polymerase crys- tallized out of 10 mM solution of MgCl 2has a Mg2+ion bound at what appears to be the catalytic site of the enzyme. There is evidence for more termination channels other than the ionization of Mg2+. In Fig. 2 we reproduce re- sults of Reynolds et al7showing that terminator eﬃcien- cies tend to saturate at large Mg2+concentration to val- ues other than zero. The saturation values are balanced, and there is an evident tendency of them to cluster. Both eﬀects are consistent with the polymerase executing an instruction at the terminator to read through condition- ally, even when the ionizable component is bound, if its memory is appropriately set. There is obviously not 5/0/./2 /0/./3 /0/./6 /0/./7/0 /5 /1/0 /1/5 /2/0 /2/5 /3/0 /3/5 /4/0 P/[MgCl/2 /] /(mM/) qqq q qq qq qqqq qqqq qqqq qqqq qqq q qqqq qqq q qqq q FIG. 2. Termination eﬃciency as function of [MgCl 2] for 10 terminators, as reported by Reynolds et al.7The terminators are, top to bottom at the right edge, RNA I ,T7Te,rrnB T1 , trp a L126 ,trp a,tR2,T3Te,P14,tonB t , andtrp a L153 . enough data here to draw such a conclusion, however. We note that Reynolds et al7also found order-1 eﬀects on the termination eﬃciency from Cl−and K+, although with the opposite sign. The function of these ions is not yet known. VI. ANTITERMINATION What experiments can detect internal memory? In general, one would look for cases in which polymerase acts diﬀerently under apparently identical conditions, suggesting an internal control mechanism of some kind. Such thinking motivates the following hypothetical ex- periment: one constructs a template with promoter P followed by two identical terminators and ﬂanking DNA sequences in succession. If termination is stochastic, the n the branching ratio at T 2will be the same as that at T 1. If termination is deterministic and hysteretic, then the branching ratios will be diﬀerent, depending on details. A passive termination at T 1would result in no termi- nation at T 2, since the polymerase that reads through has been “polarized”, i.e., selected for the memory set- ting that codes for read-through. An active termination at T 1would reprogram the memory there and cause a termination probability at T 2diﬀerent from that of T 1 but not necessarily zero. Variations of this design, e.g., adding more terminators, combining diﬀerent termina- tors, changing their order, etc., could, in principle, an- swer more sophisticated questions, such as whether and how polymerase is reprogrammed in active read-through and whether non-equilibrium eﬀects are important. A few such experiments have already been performed on DNA templates containing antiterminators (sequences upstream of terminators that reduce termination eﬃcien- cies) and are thus less general than one would like, but they strongly support the idea of polymerase memory. There is indirect evidence in the case of N-antitermina- tion of phage λ, the case most studied, that the memory is a physical attachment of the transcribed mRNA to theSequence T7Te trp a AATTGTGAGCGGATAACAATT TCACACAGGAAACAGGGAA 61 99 AATTGTGAGCGGATAACAATT TCACACAGGAAACAGAA.. 51 52 AATTGTGAGCGGATAACAATT TCACACAGGAA... 73 99 AATTGTGAGCGGATAACAATT TCACGGAA... 45 99 AATTGTGAGCGGATAACAATT TCAGGAA... 71 99 AATTGTGAGCGGATAACAATT TCGGAA... 75 66 AATTGTGAGCGGATAGGAA... 88 75 No Antiterminator 99 80 TABLE V. Sequences and corresponding termination probabilities at downstream T7Te andtrp afor modiﬁed lac antiterminators reported by Telesnitsky and Chamberlin.44 Sequence oop t rpoC t AAATCTGATAATTT TGCCAATGTTGTACGGAATTC 37 22 AAATCTGATAATTT TGCCAATGTTGGGAATTC ... 45 17 AAATCTGATAATTT TGCCAATGTTGGAATTC ... 31 19 AAATCTGATAATTT TGCCAATGGAATTC ... 29 16 AAATCTGATAATTT TGCCGGAATTC ... 25 18 AAATCTGATAATTT GGAATTC... 17 20 AAATCTGATAATT GGAATTC... 15 22 AAATCTGATAAT GGAATTC... 11 20 AAATCTGATAAGGAATTC... 19 21 AAATCGGAATTC ... 20 16 TABLE VI. Antiterminator sequences constructed by Goliger et al51from a promoter from phage 82, together with the readthrough probabilities in vitro for downstream termi- nators oop tandrpoC t . Note that these terminators are not in series. The underlined sequence on the right is the EcoRI linker. polymerase to form a loop.55There is also evidence that it is not true generally.44 In 1989 Telesnitsky and Chamberlin44reported mem- ory eﬀects associated with the lacantiterminator found just downstream of the Ptacpromoter in E. coli . Their key result is reproduced in Table V. Insertion of lac353 nucleotides upstream of the terminator makes diﬀerent order-1 modiﬁcations to the termination eﬃciences of T7Te phage and trp a. The antiterminator contains a palindrome, and the antitermination eﬀect is sensitive to modiﬁcations of the downstream 15-base-pair sequence. 3 copies of T7Te placed in tandem downstream of lac showed that the antitermination eﬀect is partially re- membered through multiple terminators: the eﬃciencies were 44%, 60%, and 90%, but without the antiterminator they were 90%, >90%, and >90%. In another experiment in vitro reported in 1989, Goliger et al51found that the E. coli terminator rpoC tand phage terminators oop t andt82were strongly antiterminated by a sequence they constructed ac- cidentally. Their key result is reproduced in Ta- ble VI. A phage 82 promoter was fused onto a sequence containing either rpoC t alone or oop t 6Sequence Name GAGCGCGGCGGGTTCA GGATGAAC GGCAATGCTGCTCATTAGC putL GCGTGGTCA AGGATGAC TGTCAATGGTGCACGATAAAAACCCA putR TABLE VII. Antitermination sequences putL andputR from the Hong Kong phage HK022.56 andrpoC t in tandem using the EcoRI linker sequence GGAATTC . This resulted in unexpected antitermination in vitro of both terminators, but of diﬀerent sizes that depended sensitively on the insertion point. The read- through eﬀects in the tandem experiments were unfortu- nately poorly documented. One can see from Table V that the phage terminator responded more strongly in this experiment than did rpoC t . However, the reverse was the case in another experiment in which the antiter- minator was a portion of the 6S RNA gene downstream of a phage λpR′promoter, and in which factor NusA was present. As a control, this latter experiment was rerun with the phage terminator t82, which terminated at greater than 98% in all cases, seemingly immune to antitermination. King et al52reported in 1996 that the putL and putRantitermination sequences of the Hong Kong phage HK022,56shown in Table VII, caused downstream readthrough of a triple terminator consisting of tR′from phage λfollowed by the strong E. coli ribosome operon terminators rrn B t 1andrrn B t 2. This eﬀect was sensi- tive to the choice of promoter. When putLwas inserted between the Ptacpromoter and the triple terminator 284 nucleotides downstream and studied in vivo the termina- tion probability was 50%. Substituting the phage λPL promoter for Ptacunder the same conditions resulted in complete readthrough (though with wide error bars). When this experiment was repeated in vitro the antiter- mination eﬀect was found to be smaller and to persist through all three terminators. The read-through proba- bilities at tR′were 34% and 31% for promotion by P L andPtac, respectively, but 57% and 27% for rrnB t 1and 76% and 40% for rrnB t 2. This result is incompatible with statistical termination, for both the antitermina- tion eﬀect itself and the changes resulting from switching promoters are order-1 eﬀects that do not add. They also reported that reduced Mg2+concentration destroys the antitermination eﬀect. VII. CONCLUSION In summary we ﬁnd that the theory of stochastic ter- mination, which requires natural selection to engineer a physical miracle of balanced rates, is ﬂawed, but that there is ample evidence of a sophisticated and as-yet poorly understood regulatory system in RNA polymerase involving hysteresis, metastability, and long-term conﬁg - urational memory, all robust phenomena in inanimate matter. On this basis we predict that branching ratios ofidentical terminators in series will diﬀer by order-1 amounts very generally - speciﬁcally in the absence of looping. We propose that the confusion surrounding the existence of polymerase memory is symptomatic of the larger problem that measurement of physical activ- ity on the length and time scales appropriate to life has thus far been impossible, and that overcoming this prob- lem should be one of the high-priority goals of modern nanoscience. This work was supported primarily by NASA Collab- orative Agreement NCC 2-794. SL would like to ac- knowledge informative discussions with Brian Ring. RBL wishes to express special thanks to the organizers of the Keystone conferences, particularly R. Craig, for encour- aging interdisciplinary science. We also wish to thank the Institute for Complex Adaptive Matter at Los Alamos, the Brown-Botstein group at Stanford for numerous stim- ulating discussions, and H. Frauenfelder, J. W. Roberts, D. 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Single-bubble sonoluminescence: Shape stability analysis of collapse dynamics in a semianalytical approach Vladislav A. Bogoyavlenskiy * Low Temperature Physics Department, Moscow State University, 119899 Moscow, Russia ~Received 29 December 1999; revised manuscript received 21 April 2000 ! This paper theoretically analyzes the hydrodynamic shape stability problem for sonoluminescing bubbles. We present a semianalytical approach to describe the evolution of shape perturbations in the strongly nonlinearregime of violent collapse. The proposed approximation estimating the damping rate produced by liquidviscosity is used to elucidate the inﬂuence of the collapse phase on subsequent evolution of the Rayleigh-Taylor instability. We demonstrate that time derivatives of shape perturbations grow signiﬁcantly as the bubbleradius vanishes, forming the dominant contribution to destabilization during the ensuing bounce phase. By thiseffect the Rayleigh-Taylor instability can be enhanced drastically, yielding a viable explanation of the upperthreshold of driving pressure experimentally observed by Barber et al. @Phys. Rev. Lett. 72, 1380 ~1994!#. PACS number ~s!: 47.20. 2k, 78.60.Mq I. INTRODUCTION In the last decade, a signiﬁcant impulse to theoretical and experimental studies of the bubble collapse problem wasgiven by the discovery of the single-bubble sonolumines-cence phenomenon @1–4#. If a gas bubble in water is sub- jected to a periodic spherical sound wave of ultrasonic fre- quency, the acoustic energy can be concentrated by over 12 orders of magnitude in very small volume. During the rar-efaction part of the acoustic cycle the bubble absorbs energyfrom the sound wave, and the subsequent compressional por-tion of the sound ﬁeld causes the collapse; the resulting ex-citation and heating of the gas inside the bubble may lead toUV-light emission of picosecond duration. One of the re-markable features of sonoluminescing bubbles observed byPutterman and co-workers is high sensitivity of the lightemission to experimental conditions such as forcing pres-sure, ambient bubble radius, water temperature, and type ofgas mixture @4–8#. Optical measurements reveal different dynamic regimes of bubble behavior, and stable sonolumi-nescence is found only in a narrow range of external param-eters. Particularly puzzling is the dependence on the ampli- tude of the forcing pressure where an upper threshold effectwas reported @5#. Usually, the emission of light takes place when the amplitude of the sound wave exceeds the edge ofsonoluminescence; if the sound intensity is increased further,beyond a threshold, the light is quenched. In order to describe the nontrivial experimental results, Brenneret al.introduced the concept that the observed upper threshold marks the onset of shape instabilities on the bubblesurface @9,10#. On the basis of linear hydrodynamic analysis, they argued that the strongest destabilization develops whenthe bubble radius reaches its minimum. The acceleration ofcompressed gas into the surrounding liquid is enormous, mo-tivating the Rayleigh-Taylor instability that causes exponen-tial growth of shape perturbations on time scales of less than 10 29s@10#. This theory, supported by theoretical @11–13 #and experimental @14,15 #investigations, was nethertheless criticized by Putterman and co-workers as to its background;they claimed that under experimental conditions the liquidviscosity would quench the shape perturbations, so it is nec-essary to ﬁnd some mechanism other than the Rayleigh-Taylor instability that results in the quenching of sonolumi-nescence @7,8#~different points of view on the problem are published in Refs. @16,17 #!. The posed discrepancies were recently examined by numerical simulations of the full hy-drodynamic model considering the viscous nonlocal effects@18,19 #. Although these studies have demonstrated a satisfac- tory agreement between the exact hydrodynamics and its ap-proximation @9,10#, further theoretical clariﬁcations still seem desirable. In the present work, we propose a semianalytical ap- proach to clarify the shape stability problem for sonolumi-nescing bubbles. Our goal is formulated as a detailed inves-tigation of shape perturbations in the region of the violentcollapse preceding the intensive development of theRayleigh-Taylor instability. For this purpose, analytical so-lution of the Rayleigh-Plesset equation modeling the liquidviscosity @20#is used to derive the perturbation dynamics as a single relation ~distortion amplitude vs bubble radius !ap- propriate for subsequent theoretical analysis. We demon- strate that time derivatives of the shape perturbations cangrow drastically as the bubble collapses, giving the dominantcontribution to posterior evolution of the Rayleigh-Taylorinstability during the shocklike bounce. This allows us toelucidate the destabilization mechanism leading to the upperthreshold effect @5#, and also to estimate the inﬂuence of liquid viscosity on the shape stability. The paper is organizedas follows. In Sec. II, we propose and justify the analyticalapproximation of the bubble dynamics for the violent col-lapse phase. The subject of Sec. III is the derivation of theperturbation dynamics for collapsing bubbles. In Sec. IV, weanalyze evolution of the shape perturbations during the col-lapse, and then give a phenomenological description of theRayleigh-Taylor instability development leading to thequenching of sonoluminescence. Finally, Sec. V formulates asummary of the results obtained. Electronic address: bogoyavlenskiy@usa.netPHYSICAL REVIEW E AUGUST 2000 VOLUME 62, NUMBER 2 PRE 62 1063-651X/2000/62 ~2!/2158 ~10!/$15.00 2158 ©2000 The American Physical SocietyII. BUBBLE DYNAMICS A. The Rayleigh-Plesset equation Since Lord Rayleigh treated the collapse of an empty cav- ity in an inviscid liquid @21#, much reﬁnement has been done in the theory of bubble dynamics @22#. The main step was the introduction of liquid viscosity, surface tension, and variableexternal driving pressure by Plesset @23#. Following this for- mulation, the motion of the bubble wall R(t) obeys the rela- tion~named the Rayleigh-Plesset equation ! rSR¨R13 2R˙2D14mR˙ R12s R1S11R cd dtD~P01Pa2Pg! 50. ~1! Here overdots denote time derivatives; r,m, and sare the density, shear viscosity, and surface tension coefﬁcient of the liquid, respectively; cis the sound speed in the liquid; P0 5const is the ambient hydrostatic pressure; Pais the driving acoustic pressure; and Pgis the gas pressure inside the bubble. For sonoluminescing bubbles, the external sound ﬁeldParepresents a spatially homogeneous, standing wave: Pa52Pa0sin2pvt, ~2! wherePa0is the amplitude and vis the frequency of the acoustic ﬁeld. A key aspect of modeling the Rayleigh-Plesset dynamics is the speciﬁcation of the internal pressure Pg. The problem consists in the complexity of the thermoﬂuid mechanicalprocesses such as heat transport at the bubble-liquid interfaceand formation of shock waves inside the bubble @24–29 #.I t should be mentioned, however, that in an early paper Trilling@24#concluded that these shock waves would not signiﬁ- cantly affect the pressure variation at the bubble wall, whichis the primary determinant of the radial motion; as also dem-onstrated by Prosperetti et al. @25,26 #, at moderate pressure amplitudes the temperature variations of the liquid near thebubble are negligible. As a consequence, at conditions rel- evant to sonoluminescing bubbles the gas pressure P gcan be considered to obey the van der Waals process equation, giv-ing a rather precise resemblance between theoretical curves R(t) and experimentally obtained data @30#: P g5P0SR032h3 R32h3Dk . ~3! HereR0is the ambient bubble radius, his the collective hard core van der Waals radius, and kis the effective polytropic exponent varying from 1 ~the isothermal condition !to the ratio of speciﬁc heats g~the adiabatic condition !. In Fig. 1, we present a typical example of the dynamics R(t) simulated for an air bubble in water; the values of pa- rameters in Eqs. ~1!–~3!are chosen to satisfy an experimen- tal regime where sonoluminescence is observed @31#. For one acoustic period T537.7 ms, one can resolve three distinct stages of the bubble dynamics. During the ﬁrst part of the acoustic cycle, 0 ,t(ms)<17.5, the bubble radius expands from its ambient value R054.5mm, to the maximum Rmax 547.1 mm. Then the collapse phase, 17.5 ,t(ms)<21.8,takes place; at the end of the compression, a sharp peak of UV light is emitted as the bubble radius approaches the mini- mumRmin50.56 mm. After the collapse, there is the stage of weak secondary oscillations, 21.8 ,t(ms)<T; during this phase, the bubble dissipates the energy accumulated from thesound ﬁeld by viscous damping, and its radius approaches the ambient value R 0by the beginning of the next acoustic cycle. Among the three stages of the bubble dynamics, we focus on the collapse phase responsible for signiﬁcant accumula-tion of sound energy. In order to discuss this region in detail,we rewrite Eq. ~1!as 2 rR¨R5~Pvel1Pext1Psur!2~Pvis1Pgas!, ~4! where Pvel[3rR˙2 2,Pext[P01Pa1RP˙a c,Psur[2s R,~5! Pvis[24mR˙ R,Pgas[Pg1RP˙g c. ~6! In this representation, we separate the terms in Eq. ~1!that either accelerate ( Pvel,Pext,Psur) or decelerate ( Pvis,Pgas) the bubble wall motion. The overall picture demonstratingthe contributions of the accelerating and decelerating termsfor the discussed Rayleigh-Plesset dynamics ~Fig. 1 !is sum- marized by Fig. 2: ~a!shows the dependence R(t) during the collapse phase; in plots ~b!and~c!, we present the evolution of pressures P vel,Pext,Psur,Pvis, and Pgas.A s FIG. 1. Dynamics Rvstfor an air bubble in water during one acoustic period T. Numerical simulation of the Rayleigh-Plesset equation corresponds to an experimental regime where sonolumi- nescence is observed: R054.5mm,P051 atm,Pa051.325 atm, andv526.5 kHz. The material constants are r51 g/cm3,m 50.01 g/cm s, s573 g/s2,c51481 m/s, R0/h58.5, and k 51.4.PRE 62 2159 SINGLE-BUBBLE SONOLUMINESCENCE: SHAP E...illustrated by the auxilary lines, the collapse region can be conditionally subdivided into three intervals: ~i!the weak collapse, ~ii!the violent collapse, and ~iii!the bounce; within these intervals, the Pext,Pvel, andPgaspressure terms, re- spectively, are dominant in Eq. ~4!. B. The violent collapse phase Since our goal is the investigation of the bubble shape stability during the violent collapse phase, we need to for- mulate an adequate approximation of the dynamics R(t)i n this region. From Fig. 2, the term Pvelgives the dominant contribution to the bubble wall acceleration so the dynamics R(t) principally follows from the classic Rayleigh equation @21#: R¨R13 2R˙250. ~7! To derive the next approximation, we take into consideration the viscosity term Pvis, which dominates among the decel- erating pressures until the shocklike bounce emerges @Fig. 2~c!#. As a result, we introduce the following simpliﬁcation of the Rayleigh-Plesset equation ~see Appendix A !: R¨R13 2R˙214m rR˙ R50, ~8! with initial conditions R$t5ti%5Ri,R˙$t5ti%52Vi. ~9! Hereti,Ri, andViare the initial time, bubble radius, and bubble wall velocity, respectively, related to the beginning of the violent collapse ( Pvel;Pext); to improve the ﬁt, the ve- locityVishould slightly exceed its actual value 2dR/dt$t 5ti%, as discussed in Appendix A.The introduced approximation of the Rayleigh-Plesset dy- namics @Eqs.~8!and~9!#is integrable, giving the analytical dependence between the bubble radius and time @20#: 4m rRi2~t2ti!51 4~12R˜2!2a 3~12R˜3/2!1a2 2~12R˜! 2a3~12R˜1/2!2a4lna1R˜1/2 a11, ~10! whereR˜[R/Riis the dimensionless bubble radius and ais the parameter of liquid viscosity deﬁned by the relation a[rRiVi 8m21. ~11! In the case of a.0, Eq. ~10!describes the dynamics of viscous collapse, R˙52S8m rD11aR˜21/2 R, ~12! which satisﬁes the Rayleigh scaling law as the bubble radius vanishes: R}~tC2t!2/5,tC5const. ~13! C. The bounce region As the gas inside the bubble is compressed to the hard core radius R!Rmin’h, the violent collapse phase is halted abruptly, and the shocklike bounce emerges. During thisvery short region ~as shown by Fig. 2, it lasts approximately 10 210s) the bubble wall velocity falls from supersonic speeds down to zero, releasing the energy stored in the com-pressed gas through emission of sound waves. The corre- FIG. 2. Collapse phase @17.5,t(ms)<21.8#of the Rayleigh-Plesset equation simulated with the same parameters as in Fig. 1; ~a!bubble dynamics Rvst;~b!log10P/P0vstfor accelerating pressures Pvel,Pext, andPsur;~c!log10P/P0vstfor decelerating pressures Pvisand Pgas. Left plots present the regions of weak and violent collapses; right plots, the bounce region in expanded time scale.2160 PRE 62 VLADISLAV A. BOGOYAVLENSKIYsponding dynamics is governed almost exclusively by the extended gas pressure Pgas, which gives the dominant con- tribution to the Rayleigh-Plesset equation: rR¨R5Pgas5S11R cd dtDPg, ~14! wherePgobeys Eq. ~3!. As pointed out by Lo ¨fstedtet al. @30#, this relation yields satisfactory description of the bubble dynamics R(t) for a time interval tin the vicinity of the instant of collapse, where the value of tis estimated as t’Rmin ~2dR/dt!max;h c. ~15! III. STABILITY EQUATIONS A. General formulation Let us consider an initially spherical bubble immersed in an inﬁnite viscous liquid. In order to study the problem ofshape stability, we assume a ﬂuctuation ﬁeld that perturbsthe bubble-liquid interface @32#. This ﬁeld of perturbations is represented by spherical Legendre polynomials as Rˆ ~t,u,w!5R~t!1( n52‘ an~t!Yn~u,w!. ~16! HereR(t) andRˆ(t,u,w) are the undistorted and distorted bubble radii, respectively ( uandware parameters of the spherical coordinate system whose origin is at the center of the bubble !; functions Yn(u,w) are the spherical harmonics of degree n5(2 ,..., ‘); the distortion amplitudes an(t) are considered to be small, uan(t)u!R(t). The classic example of a surface instability is the growth of perturbations on aplane interface separating a light liquid from a heavier oneinto which it is being uniformly accelerated; this is generallyknown as the Rayleigh-Taylor instability @33#. The instabilities that arise on the surface of an acousti- cally driven bubble are accompanied by effects related to thespherical geometry @34–38 #. For the simplest case of inviscid liquids, the perturbation dynamics was derived by Plesset@34#: a¨ n13R˙ Ra˙n1~n21!S2R¨ R1~n11!~n12!s rR3Dan50. ~17! The inﬂuence of liquid viscosity, being neglected in Plesset’s derivation, was taken into account by Prosperetti @39#. The intrinsic difﬁculty of this consideration is that viscousstresses produce vorticity of the liquid in neighborhood ofthe bubble wall; this vorticity spreads by both convective anddiffusive processes and the problem becomes strongly non-local:a¨ n1S3R˙ R22~n21!~n11!~n12!m rR2Da˙n 1~n21!S2R¨ R1~n11!~n12!~s12mR˙! rR3 Dan 2n~n11!R˙ R2E R‘Rn rnS12R3 r3DT~r,t!dr 1n~n11!~n12!m rR2T~R,t! 50. ~18! Here the ﬁeld T5T(r,t), the toroidal component of the liq- uid vorticity, obeys the diffusion equation ]T ]t1R˙R2] ]rST r2D5m rS]2T ]r22n~n11!T r2D ~19! and the boundary condition at the bubble wall T~R,t!12 RE R‘ T~r,t!Rn rndr 52 n11S~n12!a˙n2~n21!R˙ RanD.~20! This exact formulation of the viscous problem is too com- plex for detailed analysis, although some cases of the fullnumerical integration of Eqs. ~18!–~20!were recently re- ported @18,19 #. In order to make the model local and more appropriate for analytical investigation, we apply several rea-sonable simpliﬁcations as follows. ~i!The surface tension can be excluded since s!2mR˙for collapsing bubbles in water ~this inequality, equivalent to Psur!Pvis, is illustrated by Fig. 2 in the previous section !.~ii!The integral in Eq. ~18! does not contain viscous terms and, therefore, only results in tiny increments to coefﬁcients a˙nandan, in comparison with 3R˙/Rand2R¨/R, respectively. ~iii!As an issue of sim- pliﬁcation, one needs to approximate the viscous dampingrate caused by the vorticity ﬁeld. For this purpose, aboundary-layer type model was proposed by Prosperetti @40# and examined by Brener and co-workers @9,10#. According to this approach, considerable vorticity is localized within a small boundary layer of thickness daround the bubble; then the integral in Eq. ~20!is written as 2 dT(R,t)/R,s ot h e vorticity at the bubble wall follows from the expression T~R,t!52 ~n11!~112d/R!S~n12!a˙n2~n21!R˙ RanD, ~21! where the value of the parameter dis given by d5minHAm rv,R 2nJ. ~22!PRE 62 2161 SINGLE-BUBBLE SONOLUMINESCENCE: SHAP E...Although the boundary-layer approximation is based on rather questionable assumptions ~a relevant discussion is published in Refs. @16,17 #!, its application leads to satisfac- tory estimations of the viscous damping ~see@18,19 #and Appendix B !. In this work, we consider the limiting case of a thin layer related to maximal viscous dissipation, implying the vorticity ﬁeld as given in Eq. ~21!with d!0. By our three considerations ~i!–~iii!, the system of Eqs. ~18!–~20!is reduced, yielding the following relation for the perturbation dynamics: a¨n1S3R˙ R12m~n12!~2n11! rR2Da˙n 1~n21!S2R¨ R12mR˙~n12! rR3Dan 50. ~23! B. Perturbation dynamics of the violent collapse In order to analyze the shape stability problem in the vio- lent collapse region, we combine the dynamical system an 5an(R,t) andt5t(R)@Eqs. ~23!and~10!#to derive the single relation an5an(R). Since Eq. ~10!gives the bubble dynamics inverted @t(R) instead of R(t)#, we need to apply the formulas of conversion R˙5Sdt dRD21 , ~24! a˙n5Sdt dRD21 an8, ~25! R¨52Sd2t dR2DSdt dRD23 , ~26! a¨n5Sdt dRD22 an92Sd2t dR2DSdt dRD23 an8, ~27! where primes denote the radial derivatives: an8[dan/dRand an9[d2an/dR2. The expressions for dt/dRandd2t/dR2can be obtained from the Rdifferentiation of Eq. ~10!: dt dR52Sr 8mDR 11aR˜21/2, ~28! d2t dR252Sr 8mD11~3a/2!R˜21/2 ~11aR˜21/2!2. ~29! Finally, the substitution of Eqs. ~28!and~29!into Eqs. ~24!– ~27!and then into Eq. ~23!gives the following differential relation for the perturbation dynamics ~for details, see Ap- pendix C !:~11aR˜21/2!d2an d~lnR˜!21S11a 2R˜21/22~n12!~2n11! 4D 3dan d~lnR˜!1~n21!S3a 2R˜21/22~n22! 4Dan 50. ~30! IV. RESULTS AND DISCUSSION A. Stability analysis of the violent collapse 1. Theoretical investigation In the previous section, we have derived Eq. ~30!, which governs the perturbation dynamics in the violent collapseregion. Before we proceed to numerical simulations of thisdifferential relation, some of its asymptotic properties can bepointed out theoretically. ~i!Let us take the inviscid limit aR˜21/2@1; in this case, Eq.~30!is reduced to d2an d~lnR˜!211 2dan d~lnR˜!13~n21! 2an50. ~31! Then, ﬁnding a solution as an5exp(jlnR˜)5R˜j(jis an un- known constant !, one obtains j56iA24~n21! 421 4. ~32! As a result, the family of solutions is represented as an5AR˜21/4sin@~v0lnR˜!1w0#, ~33! whereAandw0are parameters determined by initial condi- tions, and v0[A24(n21)21/4. By Eq. ~33!, the distortions anoscillate on the logarithmic scale of R˜; the amplitude of the oscillations slightly increases as the bubble collapses,obeying the relation max uanu}R˜21/4. ~34! ~ii!In the opposite case of high viscosity aR˜21/2!1, Eq. ~30!is transformed to d2an d~lnR˜!21S12~n12!~2n11! 4Ddan d~lnR˜! 2~n21!~n22! 4an50. ~35! Repeating the same procedure to ﬁnd a solution, an5R˜j, leads to j561 2AS~n12!~2n11! 421D2 1~n21!~n22! 11 2S~n12!~2n11! 421D, ~36!2162 PRE 62 VLADISLAV A. BOGOYAVLENSKIYwhich allows us to obtain the following approximation for the roots, due to the inequality ( n21)(n22)!@(n12)(2n 11)/421#2: j1’~n12!~2n11! 421,j2’2~n21!~n22! ~n12!~2n11!24. ~37! As a result, the family of solutions is an5AR˜j11BR˜j2, ~38! where j1andj2are deﬁned by Eq. ~37!, and the parameters AandBare determined by initial conditions. This functional dependence means that the distortions anobey a power-law behavior of R˜, giving the following scaling as the bubble radius vanishes: an}R˜j2. ~39! However, the absolute values of the negative root j2are small for low modes n~e.g., uj2urises from 0 to 0.2 as n varies from 2 to 6 !, so the growth of perturbations by Eq. ~39!is rather weak, as in the inviscid limit @Eq.~34!#. 2. Numerical simulations To give the overall picture for the perturbation dynamics an(R˜), results of the numerical simulations of Eq. ~30!are presented in the range 0.01 <R˜<1~in logarithmic scale of the dimensionless bubble radius R˜) covering all the compres- sion stages of sonoluminescing bubbles. In Fig. 3, we illustrate the evolution of the quadrupole modea2for various values of the viscosity parameter a; the initial conditions for the simulation are chosen as a25a20andda2/dR˜50a tR˜51. The calculated curves a2(log10R˜) dem- onstrate a nonlinear oscillating behavior, where the succes- sive increase of liquid viscosity ~decrease of a) results in a monotonic damping of the distortion amplitude. However,this viscous damping yields a substantial contribution to the perturbation dynamics only if a,10; this inequality is not valid for sonoluminescing bubbles ~where a;100 due to @41#!, so the inﬂuence of viscosity on the shape stability can be considered as negligible in the violent collapse region. The perturbation dynamics an(log10R˜) for different modesn52 ,..., 6a n d ﬁxed viscosity parameter a5100 is summarized by Fig. 4; the initial conditions at R˜51 are the same as in Fig. 3. As one can see by comparing Figs. 3 and 4, the high harmonics n>3 qualitatively resemble the dy- namics of the quadrupole mode a2: the obtained curves an(log10R˜) oscillate with a slight increase of amplitude as the bubble radius R˜diminishes, obeying Eq. ~34!. B. The Rayleigh-Taylor instability The stability analysis of the violent collapse region pre- sented above has shown that the growth of the perturbationamplitude is rather weak @Eq.~34!#, resulting in an insigniﬁ- cant contribution to the shape destabilization. This raises thefollowing problem: does it mean that the inﬂuence of thecollapse phase on subsequent development of the Rayleigh-Taylor instability is inﬁnitesimal? In order to answer thequestion posed, let us study the transition from the violentcollapse to the bounce in detail. When the bounce phaseemerges ~see Sec. IIC and Fig. 2 !, the bubble dynamics transforms abruptly from collapsing @Eq.~10!#to shock- FIG. 3. Perturbation dynamics for quadrupole mode a2/a20vs log10R˜at various values of viscosity parameter a51, 2, 4, 10, 100, and ‘~shown for each curve !in the violent collapse region. Initial conditions are a25a20,da2/dR˜50a sR˜51. FIG. 4. Perturbation dynamics an/an0vs log10R˜for modes n 52 ,...,6 ~shown for each curve !in the violent collapse region. Parameter of liquid viscosity ais ﬁxed, a5100. Initial conditions arean5an05constn,dan/dR˜50a sR˜51.PRE 62 2163 SINGLE-BUBBLE SONOLUMINESCENCE: SHAP E...like@Eq.~14!#. Denoting the moment of the transformation t5t, we write the initial conditions for the posterior evolu- tion of perturbations as an$t5t%5an,a˙n$t5t%5a˙n, ~40! whereananda˙nare the value and its time derivative of the distortion an(t), corresponding to the end of the violent col- lapse region. From Figs. 3 and 4, the absolute values of perturbations anare comparable to an0.T oﬁ n d a˙n, we use the relation a˙n51 RiSdR dtDSdan dR˜Dast!t,R˜!Rmin Ri.~41! The dynamics of the radial derivatives dan/dR˜vs log10R˜ for modes n52 ,..., 6i s presented in Fig. 5 ~obtained by the R˜differentiation of Fig. 4 !. The curves dan/dR˜(log10R˜) are characterized by strongly nonlinear oscillations with therapid increase of amplitude as the bubble radius vanishes: the values of udan/dR˜uachieve 500 an0for compression ratio R˜ of the order of 1022. For sonoluminescing bubbles, the co- efﬁcient of dan/dR˜in Eq. ~41!exceeds 108s21@42#,s ot h e absolute values of a˙ncan reach 1011an0s21by the ﬁnal stage of the violent collapse. As a consequence, on the time scale t;10210s@Eq.~15!#the following inequality is valid: ua˙nut@an. ~42! This relation means that the dominant contribution to the Rayleigh-Taylor instability stems from the time derivatives of the distortions a˙n.The crucial role of the derivative terms a˙non the shape destabilization is elucidated by Fig. 6. We present evolution of the quadrupole mode a2(t) during the bounce phase t P@0,t#~the moment t50 corresponds to the end of the vio- lent collapse phase !. These are results for an air bubble in water: the ambient bubble radius R054.5mm, ambient hy- drostatic pressure P051 atm, and driving frequency v 526.5 kHz are ﬁxed; the amplitude of the forcing pressure Pa0is varied in the range 1.2 <Pa0(atm) <1.4. We assume the bubble dynamics R(t) to obey Eq. ~10!during the violent collapse and then Eq. ~14!during the bounce phase; the pa- rametersa2anda˙2are calculated with the use of Figs. 4 and 5. Two plots are composed: ~a!the initial time derivative a˙2 is ignored and ~b!the term a˙2is taken into consideration @Eq.~41!#. The ﬁgure shows the signiﬁcant effect of the ini- tial time derivatives: at the end of the bounce interval t5t, the values of the quadrupole mode a2with identical values ofPa0differ by more than an order of magnitude. The differ- ence increases with the forcing pressure amplitude: for Pa0 51.4 atm the ratio a2$a˙2Þ0%/a2$a˙250%exceeds 30. The results obtained allow us to discuss the development of the Rayleigh-Taylor instability quantitatively. The distor- tion value of the distortion a20relevant to the beginning of the collapse phase can be estimated as a microscopic ﬂuc- tuation formed by a random displacement of magnitude ;1n m ~several diameters of the water molecule !. Then the crucial perturbation of the initially spherical bubble a2cr ;Rmin’0.6mm is related to the increase of a2by a factor of;600 during the bounce. From Fig. 6 ~a!, the curves a2(t) FIG. 5. Dynamics of radial derivatives dan/dR˜vs log10R˜for modesn52 ,..., 6 ~shown for each curve !in the violent collapse region. Parameter of liquid viscosity ais ﬁxed, a5100. Initial conditions are the same as in Fig. 4. FIG. 6. Perturbation dynamics for quadrupole mode a2/a20vs t/tdemonstrating intensive development of the Rayleigh-Taylor instability in the bounce region; initial derivatives a˙2are either ignored ~a!or taken into consideration ~b!. These are results for various values of acoustic ﬁeld amplitude Pa0(atm) 51.20, 1.25, 1.30, 1.35, and 1.40 ~shown for each curve !. Ambient bubble radius R0, ambient hydrostatic pressure P0, and driving frequency vare ﬁxed:R054.5mm,P051 atm, and v526.5 kHz.2164 PRE 62 VLADISLAV A. BOGOYAVLENSKIYcannot achieve this threshold even as Pa051.4 atm where no sonoluminescence was observed @8#. In contrast to the plot ~a!, the distortions a2(t) shown in Fig. 6 ~b!increase by three orders of magnitude, so that should result in almost full de- struction of the initial bubble sphericity at Pa0>1.35 atm, i.e., in the region where the upper threshold of the driving amplitude Pa0was experimentally reported @5#. C. Coexistence of different instability mechanisms Although the main reason for the shape destabilization of sonoluminescing bubbles consists in the strongest develop-ment of the Rayleigh-Taylor instability in the instant of col-lapse, some additional mechanisms also coexist, such as theparametric and afterbounce instabilities distinguished byBrenner and co-workers @9,10#. The ﬁrst arises due to the accumulation of perturbations from sphericity over many os-cillation periods, similar to Faraday waves. The secondgrows during the rapid afterbounces @secondary weak oscil- lations ~Fig. 1 !#that bubbles execute after the point of mini- mal radius. The increments of the destabilization mecha-nisms to the Rayleigh-Taylor instability lead to a verycomplex structure for the stability boundary @8–10,17–19 #. Since a detailed quantitative analysis of these increments interms of our semianalytical approach seems rather difﬁcult,we propose a phenomenological description as follows. From our studies, the dominant contribution to the Rayleigh-Taylor instability comes from the time derivatives of the surface distortion a˙ nat the end of the violent collapse phase. As shown in Fig. 5, the derivative terms a˙n(R˜) oscil- late extremely nonlinear as the bubble radius R˜diminishes. As a consequence, an inﬁnitesimal shift in the initial condi- tionsan0[an$R˜51%anda˙n0[a˙n$R˜51%related to the begin- ning of the collapse may lead to drastic change of the abso- lute values of a˙n~by several orders of magnitude !or even to sign inversion. These initial conditions an0anda˙n0are formed during the oscillation period Tbetween the collapse mo- mentsand,therefore,arestrictlyinﬂuencedbytheparametricand afterbounce destabilization mechanisms. V. SUMMARY ~i!We have shown that the general Rayleigh-Plesset equation governing the dynamics of sonoluminescingbubbles allows an analytically integrable approximation~which takes into account the liquid viscosity term !in the violent collapse region. ~ii!Based on the boundary-layer approach, we have de- rived a single differential relation ~distortion amplitude vs bubble radius !for the perturbation dynamics during the vio- lent collapse. Theoretical and numerical investigations revealstrongly nonlinear oscillations of the distortion amplitudes~weakly dependent on the liquid viscosity !as the bubble ra- dius vanishes. ~iii!We have estimated the contribution of the violent collapse phase to the posterior intensive development of theRayleigh-Taylor instability, and then have elucidated the up-per threshold effect, discussing the increments from paramet-ric and afterbounce destabilization. ACKNOWLEDGMENTS I would like to thank Dr. Natasha Chernova and Dr. Maxim Lobanov for stimulating discussions and helpfulcomments. APPENDIX A: DYNAMICS OF VISCOUS COLLAPSE The proposed simpliﬁcation of the general Rayleigh- Plesset dynamics @Eqs.~8!and~9!#may need additional jus- tiﬁcation, since the external pressure Pextexceeds ~or is comparable with !the viscous term Pvisduring the violent collapse region ~Fig. 2 !. Despite the fact that it seems desir- able to include Pextin the approximation proposed @Eq.~8!#, we have nevertheless ignored the external pressure term inour consideration for the following reasons. ~i!The function P ext(R) is approximately constant during the whole collapse so its increment to the bubble dynamics R(t) can be compensated by an insigniﬁcant increase of the initial velocity Viin Eq. ~9!, in contrast to the strongly non- linear behavior of the viscous contribution Pvisas the bubble approaches the minimum. ~ii!The viscous pressure Pvisis the only term in the Rayleigh-Plesset equation responsible for the dissipation ofsound energy until the bounce emerges and, therefore, its consideration is preferable to that of P ext. ~iii!The approximation introduced allows us to obtain an analytical solution of the bubble dynamics @Eq.~10!#; that yields the opportunity for subsequent detailed analysis of theshape stability problem. APPENDIX B: BOUNDARY-LAYER APPROXIMATION The boundary-layer approximation was criticized by Putterman and Roberts @16#since, as they claimed, it under- estimates the viscous damping rate produced by the vorticity ﬁeld. They argued that the thickness dof the boundary layer can drastically exceed the value estimated by Eq. ~22!, espe- cially when the bubble reaches its minimum, yielding en-hanced dissipation within the layer. Further investigations ofthe problem @18,19 #stimulated by this criticism have re- vealed that the actual thickness dis several times greater than the assumed one, but the discrepancies between the ex- act model and its dapproximation are nevertheless insigniﬁ- cant. The solution of the seeming paradox follows from simple analysis of Eq. ~21!: increase of the parameter dre- sults in a monotonic decrease of T(R,t), i.e., the viscous dissipation caused by the liquid vorticity decreases as theboundary layer is enhanced. In other words, consideration ofthe vorticity localized cannot underestimate, but rather over-estimates, the inﬂuence of liquid viscosity. The reported con-ditions related to slight underestimation of the viscous damp-ing rate @18,19 #are easily explained as follows. The criterion by which one can determine if the dmodel overestimates ~or underestimates !the viscosity effect states that the sign of T(R,t) is the same as ~or opposite to !the sign of *R‘T(r,t)/rndr. The function T(r,t) usually demonstratesPRE 62 2165 SINGLE-BUBBLE SONOLUMINESCENCE: SHAP E...the oscillating behavior of rwith diminishing amplitude @18#, so in most cases the signs are the same and, therefore, theboundary-layer type approximation is adequate. Some rarecases when the equality of the signs is broken, relevant to theobserved discrepancies between the models, correspond tothe afterbounce phase of the bubble dynamics @18#. Since in this paper we are focused on the collapse and on the bounce, the application of the dmodel to our stability analysis seems rather reasonable. APPENDIX C: DERIVATION OF PERTURBATION DYNAMICS The substitution of dt/dRandd2t/dR2from Eqs. ~28! and~29!into Eqs. ~24!–~27!yields R˙52S8m rD11aR˜21/2 R, ~C1! a˙n52S8m rD11aR˜21/2 Ran8, ~C2! R¨52S8m rD2~11aR˜21/2!@11~3a/2!R˜21/2# R3,~C3!a¨n5S8m rD2~11aR˜21/2!2 R2an9 2S8m rD2~11aR˜21/2!@11~3a/2!R˜21/2# R3an8. ~C4! Then, the combination of Eqs. ~C1!–~C4!with Eq. ~23!gives the following: R2~11aR˜21/2!an91RS213a 2R˜21/22~n12!~2n11! 4Dan8 1~n21!S3a 2R˜21/22~n22! 4Dan50. ~C5! Finally, the variable replacement R$lnR˜as dan dR51 Rdan d~lnR˜!, ~C6! d2an dR251 R2Sd2an d~lnR˜!22dan d~lnR˜!D ~C7! allows us to obtain Eq. ~30!. @1#D.F. Gaitan and L.A. Crum, in Frontiers in Nonlinear Acous- tics, edited by M. Hamilton and D.T. Blackstock ~Elsevier, New York, 1990 !, p. 459; D. F. Gaitan, Ph.D. thesis, Univer- sity of Mississippi, 1990 ~unpublished !. @2#D.F. Gaitan, L.A. Crum, R.A. Roy, and C.C. Church, J. Acoust. Soc. Am. 91, 3166 ~1992!. @3#B.P. Barber and S.J. Putterman, Nature ~London !352, 318 ~1991!. @4#R. Hiller, K. Weninger, S.J. Putterman, and B.P. Barber, Sci- ence266, 248 ~1994!. @5#B.P. Barber, C.C. Wu, R. Lo ¨fstedt, P.H. Roberts, and S.J. Putterman, Phys. Rev. Lett. 72, 1380 ~1994!. @6#B.P. Barber, K. Weninger, R. Lo ¨fstedt, and S.J. Putterman, Phys. Rev. Lett. 74, 5276 ~1995!. @7#R. Lo¨fstedt, K. Weninger, S. Putterman, and B.P. Barber, Phys. Rev. E 51, 4400 ~1995!. @8#B.P. Barber, R.A. Hiller, R. Lo ¨fstedt, S.J. Putterman, and K.R. Weninger, Phys. Rep. 281,6 5~1997!. @9#M.P. Brenner, D. Lohse, and T.F. Dupont, Phys. Rev. Lett. 75, 954~1995!. @10#S. Hilgenfeldt, D. Lohse, and M.P. Brenner, Phys. Fluids 8, 2808 ~1996!;9, 2462 ~E!~1997!. @11#M.P. Brenner, D. Lohse, D. Oxtoby, and T.F. Dupont, Phys. Rev. Lett. 76, 1158 ~1996!. @12#L. Kondic, C. Yuan, and C.K. Chan, Phys. Rev. E 57, R32 ~1998!. @13#S. Hilgenfeldt, D. Lohse, and W.C. Moss, Phys. Rev. Lett. 80, 1332 ~1998!. @14#R.G. Holt and D.F. Gaitan, Phys. Rev. Lett. 77, 3791 ~1996!. @15#D.F. Gaitan and R.G. Holt, Phys. Rev. E 59, 5495 ~1999!.@16#S.J. Putterman and P.H. Roberts, Phys. Rev. Lett. 80, 3666 ~1998!. @17#M.P. Brenner, T. Dupont, S. Hilgenfeldt, and D. Lohse, Phys. Rev. Lett. 80, 3668 ~1998!. @18#C.C. Wu and P.H. Roberts, Phys. Lett. A 250, 131 ~1998!; Phys. Fluids 10, 3227 ~1998!. @19#Y. Hao and A. Prosperetti, Phys. Fluids 11, 1309 ~1999!;A . Prosperetti and Y. Hao, Philos. Trans. R. Soc. London, Ser. A357, 203 ~1999!. @20#V.A. Bogoyavlenskiy, Phys. Rev. E 60, 504 ~1999!. @21#Lord Rayleigh, Philos. Mag. 34,9 4~1917!. @22#For a recent review, see S. Hilgenfeldt, M.P. Brenner, S. Grossmann, and D. Lohse, J. Fluid Mech. 365, 171 ~1998!, and references therein. @23#M.S. Plesset, J. Appl. Mech. 16, 277 ~1949!. @24#L. Trilling, J. Appl. Phys. 23,1 4~1952!. @25#A. Prosperetti, J. Fluid Mech. 222, 587 ~1991!. @26#V. Kamath, A. Prosperetti, and F. Egolfopoulos, J. Acoust. Soc. Am. 94, 248 ~1993!. @27#C.C. Wu and P.H. Roberts, Phys. Rev. Lett. 70, 3424 ~1993!; Proc. R. Soc. London, Ser. A 445, 323 ~1994!. @28#L. Kondic, J.I. Gersten, and C. Yuan, Phys. Rev. E 52, 4976 ~1995!. @29#V.Q. Vuong, A.J. Szeri, and D.A. Young, Phys. Fluids 11,1 0 ~1999!. @30#R. Lo¨fstedt, B.P. Barber, and S.J. Putterman, Phys. Fluids A 5, 2911 ~1993!. @31#R. Hiller, S.J. Putterman, and B.P. Barber, Phys. Rev. Lett. 69, 1182 ~1992!; B.P. Barber and S.J. Putterman, ibid.69, 3839 ~1992!.2166 PRE 62 VLADISLAV A. BOGOYAVLENSKIY@32#H. Lamb, Hydrodynamics , 6th ed. ~Dover Publications, New York, 1945 !. @33#Lord Rayleigh, Proc. London Math. Soc. XIV, 170 ~1883!; G.I. Taylor, Proc. R. Soc. London, Ser. A 201, 192 ~1950!. @34#M.S. Plesset, J. Appl. Phys. 25,9 6~1954!. @35#M.S. Plesset and T.P. Mitchell, Q. Appl. Math. 13, 419 ~1956!. @36#G. Birkhoff, Q. Appl. Math. 12, 306 ~1954!;13, 451 ~1956!. @37#A.I. Eller and L.A. Crum, J. Acoust. Soc. Am. 47, 762 ~1970!. @38#H.W. Strube, Acustica 25, 289 ~1971!. @39#A. Prosperetti, Q. Appl. Math. 34, 339 ~1977!; Ph.D. thesis, California Institute of Technology, Pasadena, California, 1974. @40#A. Prosperetti, Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat.,Rend.62, 196 ~1977!. @41#The beginning of the violent collapse region is deﬁned by the condition Pvel.Pext, yielding the estimation Vi’2(P0/r)1/2 ;20 m/s. Since for sonoluminescing bubbles the initial radius Ri;40mm, corresponding values of the liquid viscosity pa- rameter acalculated from Eq. ~11!are of the order of 100. @42#Sonoluminescence is observed when the bubble wall velocity achieves supersonic speeds, ( dR/dt)max5Mc, wherec’1.5 3103m/s is the sound speed in water and M;2–4 is the Mach number. Since corresponding values of Riare approxi- mately 40 mm, the expression ( dR/dt)/Riis estimated as 4M3107s21;108s21.PRE 62 2167 SINGLE-BUBBLE SONOLUMINESCENCE: SHAP E...
arXiv:physics/0009050v1 [physics.flu-dyn] 14 Sep 2000Formation of Root Singularities on the Free Surface of a Conducting Fluid in an Electric Field N. M. Zubarev Institute of Electrophysics, Ural Division, Russian Acade my of Sciences, 106 Amundsena Street, 620016 Ekaterinburg, Russia e-mail: nick@ami.uran.ru Abstract The formation of singularities on a free surface of a conduct ing ideal ﬂuid in a strong electric ﬁeld is considered. It is found that the nonlinear e quations of two-dimensional ﬂuid motion can be solved in the small-angle approximation. This enables us to show that for almost arbitrary initial conditions the surface cu rvature becomes inﬁnite in a ﬁnite time. Electrohydrodynamic instability of a free surface of a cond ucting ﬂuid in an external electric ﬁeld [1,2] plays an essential role in a general prob lem of the electric strength. The interaction of strong electric ﬁeld with induced charges at the surface of the ﬂuid (liquid metal for applications) leads to the avalanche-like growth of surface perturbations and, as a consequence, to the formation of regions with high energy co ncentration which destruction can be accompanied by intensive emissive processes. In this Letter we will show that the nonlinear equations of mo tion of a conducting ﬂuid can be eﬀectively solved in the approximation of small pertu rbations of the boundary. This allows us to study the nonlinear dynamics of the electrohydr odynamic instability and, in particular, the most physically meaningful singular solut ions. Let us consider an irrotational motion of a conducting ideal ﬂuid with a free surface, z=η(x,y,t), that occupies the region −∞<z≤η(x,y,t), in an external uniform electric ﬁeldE. We will assume the inﬂuence of gravitational and capillary forces to be negligibly small, which corresponds to the condition E2≫8π√gαρ, wheregis the acceleration of gravity, αis the surface tension coeﬃcient, and ρis the mass density. The potential of the electric ﬁeld ϕsatisﬁes the Laplace equation, ∆ϕ= 0, with the following boundary conditions, ϕ→ −Ez, z → ∞, ϕ= 0, z =η. 1The velocity potential Φ satisﬁes the incompressibility eq uation ∆Φ = 0, which one should solve together with the dynamic and kinemat ic relations on the free surface, ∂Φ ∂t+(∇Φ)2 2=(∇ϕ)2 8πρ+F(t), z =η, ∂η ∂t=∂Φ ∂z− ∇η· ∇Φ, z =η, whereFis some function of variable t, and the boundary condition Φ→0, z → −∞. The quantities η(x,y,t)ψ(x,y,t) = Φ\|z=ηare canonically conjugated, so that the equations of motion take the Hamiltonian form [3], ∂ψ ∂t=−δH δη,∂η ∂t=δH δψ, where the Hamiltonian H=/integraldisplay z≤η(∇Φ)2 2d3r−/integraldisplay z≥η(∇ϕ)2 8πρd3r coincides with the total energy of a system. With the help of t he Green formula it can be rewritten as the surface integral, H=/integraldisplay s/bracketleftBiggψ 2∂Φ ∂n+Eη 8πρ∂˜ϕ ∂n/bracketrightBigg ds, where ˜ϕ=ϕ+Ezis the perturbation of the electric ﬁeld potential; dsis the surface diﬀerential. Let us assume \|∇η\| ≪1, which corresponds to the approximation of small surface a ngles. In such a case we can expand the integrand in a power series of c anonical variables ηandψ. Restricting ourselves to quadratic and cubic terms we ﬁnd af ter scale transformations t→tE−1(4πρ)1/2, ψ→ψE/(4πρ)1/2, H→HE2/(4πρ) the following expression for the Hamiltonian, H=1 2/integraldisplay/bracketleftBig ψˆkψ+η/parenleftBig (∇ψ)2−(ˆkψ)2/parenrightBig/bracketrightBig d2r −1 2/integraldisplay/bracketleftBig ηˆkη−η/parenleftBig (∇η)2−(ˆkη)2/parenrightBig/bracketrightBig d2r. 2Hereˆkis the integral operator with the diﬀerence kernel, whose Fo urier transform is the modulus of the wave vector, ˆkf=−1 2π+∞/integraldisplay −∞+∞/integraldisplay −∞f(x′,y′) [(x′−x)2+ (y′−y)2]3/2dx′dy′. The equations of motion, corresponding to this Hamiltonian , take the following form, ψt−ˆkη=1 2/bracketleftBig (ˆkψ)2−(∇ψ)2+ (ˆkη)2−(∇η)2/bracketrightBig +ˆk(ηˆkη) +∇(η∇η), (1) ηt−ˆkψ=−ˆk(ηˆkψ)− ∇(η∇ψ). (2) Subtraction of Eqs. (2) and (1) gives in the linear approxima tion the relaxation equation (ψ−η)t=−ˆk(ψ−η), whence it follows that we can set ψ=ηin the nonlinear terms of Eqs. (1) and (2), which allows us to simplify the equations of motion. Actually, add ing Eqs. (1) and (2) we obtain an equation for a new function f= (ψ+η)/2, ft−ˆkf=1 2(ˆkf)2−1 2(∇f)2, (3) which corresponds to the consideration of the growing branc h of the solutions. As f=ηin the linear approximation, Eq. (3) governs the behavior of th e elevation η. First we consider the one-dimensional case when function fdepends only on x(andt) and the integral operator ˆkcan be expressed in terms of the Hilbert transform ˆH, ˆk=−∂ ∂xˆH, ˆHf=1 πP+∞/integraldisplay −∞f(x′) x′−xdx′, where P denotes the principal value of the integral. As a resu lt, Eq. (3) can be rewritten as ft+ˆHfx=1 2(ˆHfx)2−1 2(fx)2. (4) It should be noted that if one introduces a new function ˜f=ˆHf, then Eq. (4) transforms into the equation proposed in Ref. [4] for the description of the nonlinear stages of the Kelvin-Helmholtz instability. For further consideration it is convenient to introduce a fu nction, analytically extendable into the upper half-plane of the complex variable x, v=1 2(1−iˆH)fx. Then Eq. (4) takes the form Re(vt+ivx+ 2vvx) = 0, 3that is, the investigation of integro-diﬀerential equatio n (4) amounts to the analysis of the partial diﬀerential equation vt+ivx+ 2vvx= 0, (5) which describes the wave breaking in the complex plane. Let u s study this process in analogy with [5,6], where a similar problem was considered. Eq. (5) c an be solved by the standard method of characteristics, v=Q(x′), (6) x=x′+it+ 2Q(x′)t. (7) where the function Qis deﬁned from initial conditions. It is clear that in order t o obtain an explicit form of the solution we must resolve Eq. (7) with res pect tox′. A mapping x→x′, deﬁned by Eq. (7), will be ambiguous if ∂x/∂x′= 0 in some point, i.e. 1 + 2Qx′t= 0. (8) Solution of (8) gives a trajectory x′=x′(t) on the complex plane x′. Then the motion of the branch points of the function vis deﬁned by an expression x(t) =x′(t) +it+ 2Q(x′(t))t. At some moment t0when the branch point touches the real axis, the analiticity ofv(x,t) at the upper half-plane of variable xbreaks, and a singularity appears in the solution of Eq. (4). Let us consider the solution behavior close to the singulari ty. Expansion of (6) and (7) at a small vicinity of x=x(t0) up to the leading orders gives v=Q0−δx′/(2t0), δx=iδt+ 2Q0δt+Q′′t0(δx′)2, whereQ0=Q(x′(t0)),Q′′=Qx′x′(x′(t0)),δx=x−x(t0),δx′=x′−x′(t0), andδt=t−t0. Eliminating δx′from these equations, we ﬁnd that close to singularity vxcan be represented in the self-similar form ( δx∼δt), vx=−/bracketleftBig 16Q′′t3 0(δx−iδt−2Q0δt)/bracketrightBig−1/2. As Re(v) =η/2 in the linear approximation, we have at t=t0 ηxx∼ \|δx\|−1/2, that is the surface curvature becomes inﬁnite in a ﬁnite time . It should be mentioned that such a behavior of the charged surface is similar to the behav ior of a free surface of an ideal ﬂuid in the absence of external forces [5,6], though the sing ularities are of a diﬀerent nature (in the latter case the singularity formation is connected w ith inertial forces). 4Let us show that the solutions corresponding to the root sing ularity regime are consistent with the applicability condition of the truncated equation (3). LetQ(x′) be a rational function with one pole in the lower half-plane, Q(x′) =−is 2(x′+iA)2, (9) which corresponds to the spatially localized one-dimensio nal perturbation of the surface (s>0 andA>0). The characteristic surface angles are thought to be smal l,γ≈s/A2≪1. It is clear from the symmetries of (9) that the most rapid bran ch point touches the real axis atx= 0. Then the critical moment t0can be found directly from Eqs. (7) and (8). Expansion of t0with respect to the small parameter γgives t0≈A/bracketleftBig 1−3(γ/4)1/3/bracketrightBig . (10) Taking into account that the evolution of the surface pertur bation can be described by an approximate formula η(x,t) =s(A−t) (A−t)2+x2, we have for the dynamics of the characteristic angles γ(t)≈s (A−t)2. Then, substituting the expression for t0(10) into this formula, we ﬁnd that at the moment of the singularity formation with the required accuracy γ(t0)∼γ1/3, that is, the angles remain small and the root singularities a re consistent with our assumption about small surface angles. In conclusion, we would like to consider the more general cas e where the weak dependence of all quantities from the spatial variable yis taken into account. One can ﬁnd that if the condition \|kx\| ≪ \|ky\|holds for the characteristic wave numbers, then the evoluti on of the ﬂuid surface is described by an equation [vt+ivx+ 2vvx]x=−ivyy/2, which extends Eq. (5) to the two-dimensional case. An interesting group of particular solutions of this equati on can be found with the help of substitution v(x,y,t) =w(z,t), where z=x−i 2(y−y0)2 t. The equation for wlooks like wt+iwz+ 2wwz=−w/(2t). 5It is integrable by the method of characteristics, so that we can study the analyticity violation similarly to the one-dimensional case. Considering a motio n of branch points in the complex plane of the variable zwe ﬁnd that a singularity arises at some moment t0<0 at the point y0along they-axis. Close to the singular point at the critical moment t=t0we get ηxx\|δy=0∼ \|δx\|−1/2, η xx\|δx=0∼ \|δy\|−1. This means that in the examined quasi-two-dimensional case the second derivative of the surface proﬁle becomes inﬁnite at a single isolated point. Thus, the consideration of the behavior of a conducting ﬂuid surface in a strong electric ﬁeld shows that the nonlinearity determines the tendency fo r the formation of singularities of the root character, corresponding to the surface points w ith inﬁnite curvature. We can assume that such weak singularities serve as the origin of th e more powerful singularities observed in the experiments [7,8]. I would like to thank A.M. Iskoldsky and N.B. Volkov for helpf ul discussions, and E.A. Kuznetsov for attracting my attention to Refs. [5,6]. This w ork was supported by Russian Foundation for Basic Research, Grant No. 97–02–16177. References 1. L. Tonks, Phys. Rev. 48 (1935) 562. 2. Ya.I. Frenkel, Zh. Teh. Fiz. 6 (1936) 347. 3. V.E. Zakharov, J. Appl. Mech. Tech. Phys. 2 (1968) 190. 4. S.K. Zhdanov and B.A. Trubnikov, Sov. Phys. JETP 67 (1988) 1575. 5. E.A. Kuznetsov, M.D. Spector, and V.E. Zakharov, Phys. Le tt. A 182 (1993) 387. 6. E.A. Kuznetsov, M.D. Spector, and V.E. Zakharov, Phys. Re v. E 49 (1994) 1283. 7. M.D. Gabovich and V.Ya. Poritsky, JETP Lett. 33, (1981) 30 4. 8. A.V. Batrakov, S.A. Popov, and D.I. Proskurovsky, Tech. P hys. Lett. 19 (1993) 627. 6
1 THE ALPHA CONSTANT FROM RELATIVISTIC GROUPS. Gustavo González-Martín Departamento de Física, Universidad Simón Bolívar, Apartado 89000, Caracas 1080-A, Venezuela Web page at http:\\prof.usb.ve\ggonzalm\ The value of the alpha constant, known to be equal to an algebraic expression in terms of π and entire numbers related to certain group volumes, is derived from the relativitistic structure group of a geometric unified theory , its subgroups and corresponding quotients.SB/F/277-002 1. Introduction. It is known that the fine structure constant α is essentially equal to an algebraic expression in terms of π and entire numbers that arises from the quotient of the volume of certain groups [1,2]. This expression may also be obtained,using different physical arguments, starting from the structure group of a unified geometric theory [3]. Using the geometric fact, in this theory, that the tangent space to space time is the image of a Minkowsian subspace of thegeometric algebra, the invariant measure in the symmetric space, defined by the associated groups is transported tospace time. 2. A Geometric Measure. The current J is a 3-form on M valued in the Clifford algebra A. It is constructed starting from a vector field on the symmetric space K. This space is G/G+ where G is the simple group whose action produces the automorphisms of A and G+ is the even subgroup, relative to the orthonormal base of the algebra. The vector field is the image, under the Clifford injection κ of a vector field in space time M. This injection allows us to define J as the pullback form of a 3-form in K. The integration of the current in a three dimensional boundary of a region R in M is equivalent to the integration of this 3-form pulled back from a geometric form in a three dimensional boundary of the image of the region in the symmetric space G/G+. The latter form is defined by the existence of a geometric invariant measure in G/G+. The constant coefficient of this invariant measure may be calculated in the particular case where the fiber bundle is flat and the field equation reduces to the linear equation equivalent to electromagnetism. This relation defines a geometric interpretation for the coupling constant of the geometric unified theory: “The coupling constant is the constant coefficient of J introduced by the invariant measure in the symmetric space G/G+”. 2.1. Symmetric Space K. As indicated before, the group G is SL(4,R) and the even subgroup G+ is SL1(2,C). The symmetric space K is a non compact real form of the complex symmetric space corresponding to the complex extension of the non compact SU(2,2) and its quotients. The corresponding series of symmetric spaces coincides with the series characterized by thegroup SO(4,2) as shown in appendix B. In particular we can identify the quotients with the same character, +4, inorder to write the series of spaces in the following form, RSO SO SOSL R SL C SOKSO SO SO≡×≈×≅≅ ≈×(,) () ()(, ) (, ) ()() () ()42 424 226 42/G4C . ( 2.1) These quotients include the non compact Riemannian Hermitian R and the non compact pseudo-Riemannian non Hermitian K symmetric spaces. Since some of these groups and quotients are non compact we shall use the normal- ized invariant measure µN calculated from a known measure, as usually done when working with non compact groups. For compact groups the integral of the invariant measure over the group parameter space gives the group volume. In general, the normalized measure gives only the functional structure of the volume element, in otherwords, the invariant measure up to a multiplicative constant. The center of G, which is not discrete, contains a generating element κ5 whose square is -1. We shall designate by 2J the restriction of ad( κ5) to the tangent space TKk. This space, that has for base the 8 matrices κα, κβκ5, is the proper subspace corresponding to the eigenvalue -1 of the operator J2, or, () [][] Jx y x y xy2 51 455 5 5λ λλ λλ λλ λ λ λλ λκκ κκ κ κκ κ κκ κ+= + = −−,, . ( 2.2) The endomorphism J defines an almost complex structure over K. In addition, using the Killing metric, in the3 Clifford representation, the complex structure preserves the pseudo Riemannian (Minkowskian) metric. Furthermore the torsion S vanishes, ()[][ ][ ] [ ]Sab ab JJ ab JaJ b J aJ b,, , , ,=+ + − = 0 . ( 2.3) In this form, the conditions for J to be an integrable complex structure, invariant by G are met and the space K is a non Hermitian complex symmetric space. 2.2. Realization of K as a unit polydisc D4(K). The bilinear complex metric in K is invariant under SO(4,C) and does not have a definite signature. Using Weyl ’s unitary trick on the Minkowskian coordinates xλ, yλ of the symmetric space K, its complex structure is related to the complex structure of R. The generators of the quotient K are 2 compact and 6 non compact instead of the 8 non compact generators of the quotient R. Both quotients have the matricial structure, Kxy xy xyxy xy xy=                    * 00 11 2233 44 55 . ( 2.4) where the lower right submatrix is xy xyxx xy yx yy44 551 2 1 1  =+• • •+ •   . ( 2.5) The conditions imposed by the associated groups SL(4,R) and SO(4,2) over the corresponding coordinates on these spaces, expressed by the scalar product in this submatrix, are related respectively by the Minkowskian and Euclidianmetrics. Define the six complex coordinates ta on R that relate this space to the complex space C6, where R is inmersed, that is, txi y aaa a=+ ≤ ≤ 05 . ( 2.6) These coordinates may be expressed in terms of the four corresponding coordinates uα on K by recognizing the scalar product in eq. ( 2.5), δη κ κ µµνµν µνµν µν tt uu I↔− = ≤ ≤() 03 , ( 2.7) and if we introduce new coordinates t on K, tumm= , ( 2.8) ti u00= , ( 2.9) we find the same conditions on the coordinates t on K that exist on the coordinates t on R. The conditions over the coordinates t4 and t5 allow us to reduce to C5 the complex space where the realization of K is immersed. If we introduce the four complex projective coordinates zµ, we obtain the realization,4 zt ti tµµ µ =−≤≤45 03 . ( 2.10) If we indicate the transposed by z’ the conditions on these coordinates are those of the unit polydisc, D4⊂D5, defined by (){}D K z C zz zz zznn=∈ + ′−′>′< ;, 12 0 12 . ( 2.11) In this manner, the complex coordinates define a holomorphic diffeomorphism h of K onto the interior of a bounded symmetric domain D. The bounded realization of the space K is the unit polydisc D4(K). This realization D4(K) corresponds to the bounded realization of the space R, the unit polydisc D4(R) by a change in coordinates. Although the interior of D4 is not compact we can apply the existing mathematical techniques of the classical bounded domains to study the space K, in particular we can find normalized invariant measures for the spaces K and R. 2.3. Invariant Measure on the Polydisc. A geometric measure on the space K, an 8 dimensional hyperboloid H8, arises from a measure on C5 in a manner similar as the measure on the Euclidian spheres is obtained from a measure on Rn. In order to evaluate this measure, it is convenient to use the immersion, i:D4→D5 defined on the intersection of the D5 and the plane z5=0. Since D5 is a homogeneous space under the action of the compact group SO(5) ×SO(2), using this group and SO(5,2) we may obtain the measure on the quotient R, [4], which is equal the measure on K. In order to construct these measures it is convenient to define certain domains related to D5, [5]. Silov ’s boundary, the generalization of the circle as the boundary of the 1 dimensional complex disk, is established by the Fourier transfor-mation on the symmetric space D n. It is the characteristic space of Dn, in other words it allows us to characterize the holomorphic functions on Dn by their value on this boundary. It is defined by (){}Q K xe x R xxni n== ∈ ′=≤ ≤ ξθ πθ;,, 10 . ( 2.12) Poisson’s kernel Pn(z,ξ) over Dn× Qn is defined as the Euclidian invariant measure on the characteristic space Qn. This kernel has the value ()() ( ) () ()Pzzz zz VQ z znn nn ,ξ ξξ=+′−′ ×− −′122 2 , ( 2.13) determined by Hua [5]. The actual construction of the measure over Q4, due to Wyler, is indicated in the appendix A. The expression for the harmonic functions over Dn is ()() ( ) ϕξ ϕ ξ ξzP z dn Qn=∫, , ( 2.14) which, for the case of the disc, reduces to a solution the Dirichlet problem using Poisson ’s integral formula that gives the harmonic function knowing its value on circle boundary, ()()() ϕθ ϕ θ θπ zP z d=∫, 02 . ( 2.15) The Poisson kernel defines a normalized form µN over the characteristic space because5 ()Pz dn Qn,ξξ∫ =1 . ( 2.16) As indicated above, the measure of interest over the polydisc D4, representative of the hyperboloid K, is obtained from the complex space C5 where it is immersed. There are injective mappings, MKD Dhi 484 5κ→ →  →  , ( 2.17) that correspond to Clifford ’s mapping κ and the holomorphic mapping h. The immersion i:D4→D5 allows us to pull back the Euclidian measure on the characteristic space Q5, boundary of D5, to Q4 and then to corresponding bound- aries in K and M. The form µN on the image of M, defines, in this way, a geometric form on space time, ()[] [ ] hM h MNN/G6Fκµ µ κ∗=44 ** . ( 2.18) 3. Value of the Geometric Coefficient. We may associate a physical current 3-form J to the standard unnormalized volume form µ. Similarly, in a natural form, we may associate another geometric current 3-form Jg to the normalized geometric form µNg with the same factor or constant coefficient αg. In addition, since the physical current form should not be associated to a normalized volume form, this geometric current 3-form should be defined by additionally multiplying by the volume of a charac- teristic space determined by the physical solutions. Let us define Jg on the boundary of a region R in M, ()( ) []() ( ) [] ()JV Rh RV R h R =V R JgN g g g≡× = × ×∗∗∂κ µ ∂ ∂ α κ µ ∂ ∂α/G6F/G6F . ( 3.1) The constant coefficient in this equation may be identified, once and for all, using any solution. If we assume staticity conditions and spherical symmetry that allow a decomposition of space time M4 in two orthogonal subspaces, spatial spheres S2 and the supplementary space time M2, the forms decompose in two components and it becomes easier to calculate the constant coefficient, ()[] [] [ ] hM h M h SNg Ng Ng/G6Fκ µ κµ κµ∗ ∗∗ ∗∗=∧42 2 . ( 3.2) The geometric charge Qg, given by the induced geometric measure, is the geometric calculation of interest. It repre- sents physically the integral of the geometric current αj, obtained by absorbing the constant α in the current, over a spatial hypersurface σ/G71M that contains the S2 subspace. Let us restrict the problem to the special particular case of pure electromagnetism in a flat space time M. All solutions of this restricted problem may be found as a sum of fundamental solutions that correspond to the Green ’s function for the electromagnetic field. The Green ’s function determines the field of a point source which always corresponds to a spherically symmetric static field relative to an observer at rest with the source. Thus the restricted problem reduces to Coulomb ’s problem in flat three dimensional space. This is precisely the situation where the α constant, or equivalently Coulomb ’s constant, is introduced. The spherically symmetric harmonic potential solutions are determined, using Poisson ’s integral formula, by its value on a boundary sphere. We see that the characteristic boundary space, where integration is performed to find solutions of the restricted physical problem, is the sphere S2 in R3. Geometrically, in accordance with our theory, the form hµNg should be restricted to its component over the image of a sphere, determined by the Clifford mapping κ, in the space K. Therefore, the characteristic space is κ(S2). We have then the volume of this characteristic space,6 ()()VS h hNg SNg Sκµ κ µ π κ2 2228 == =∗∗ ∗∫∫ . ( 3.3) The volume of κ(S2), indicated in the previous equation, is twice the volume of S2 due to the 2-1 homomorphism between standard spinors and vectors determined by its homomorphic groups SU(2) and SO(3). Hence the geometric coefficient α is ()αα κ π α==∗g g VS28 . ( 3.4) The value of this constant coefficient is obtained substituting, in the last equation, the value of Wyler ’s coefficient calculated in appendix A, []απ ππ ππα M p432 655 41 4 31 4 23 22 59 161201 13703608245=×  =  ==! . , ( 3.5) which is equal to the experimental physical value of the alpha constant. The field equation may be written geometri- cally, DJ Jg∗∗ ∗==Ω44πα π . ( 3.6) 4. Conclusions. The coupling constant of the geometric unified theory may be calculated from the volumes of certain symmetric spaces related to the structure group of the theory and its subgroups. There is no additional arbitrary constant to beused in the theory. Appendix A. In what follows, we indicate Wyler ’s calculation of the value of the constant coefficient of the measure on Q4, Silov’s boundary of D4. This measure is obtained by constructing Poisson ’s measure invariant under general coordinate transformations in D5 by the group of analytic mappings of D5 onto itself which is SO(5,2). The calculation is based on the following proposition: The isotropy subgroup at the origin, SO(5) ×SO(2), acts tran- sitively over Q5 and Poisson ’s kernel Pn(z,ξ), harmonic on Dn, represents an invariant measure of the action of SO(5)×SO(2) over Q5. Although P4(x,ξ) represents a measure µ on Q4, it is not adequate in this case because we need the measure defined by the Euclidian measure in one dimension higher, C5, the one induced from Q5. Since P5(z,ξ) is an invariant Euclidian measure over Q5 we construct the induced measure by the immersion i:Q4→Q5 which is equal to the kernel P4(z,ξ) up to constant coefficient factor. Functionally both measures are equivalent. The invariant normalized form over the characteristic space is []()() ()[] ()µξ ξξµ NQP z dz VQQ VQ5 55 55 5== =,,Π , ( A.1)7 () µξ ξN QQPz d 555 1 ∫∫==, , ( A.2) where µ represents an invariant form, not normalized, over Q5 defined by these equations. The action of SO(5,2) over the Hermitian structure of D5 defines the Bergman metric on this space. The group SO(5,2), of coordinate transforma- tions, acts on D5 and consequently on Q5. The measure defined by the Poisson kernel is not invariant under SO(5,2), but is related to the invariant metric measure by []()()[] µξ ξ µNg ggQP z j dj VQQ5 55 55==, , ( A.3) in terms of jg, which denotes the complex Jacobian or determinant of the Jacobian matrix JC of the mapping z→G(z) where G=SO(5,2). To find jg we use a relation, given by Hua, between the Bergman kernel and the volume density over the domain D5. The Bergman kernel may be written as ()()() ()B VD z z z zJz VDnC 55 22 51 12= ×+′−′=det . ( A.4) The real Bergman metric h is defined by the invariant bilinear form () () ()dsdzJ z J z dz VDCC 2 5=′′ . ( A.5) Thus, the value of the complex Jacobian of the transformation z→G(z) is () () () ()() jJ J h V DgC R== = =−det det det1 21 451 4 , ( A.6) obtaining by substitution in eq. ( A.3) the metric measure, []()()() ()[] µξ ξ µNg gQP z j dVD VQQ5 555 551 4 ==, . ( A.7) To obtain the Wyler measure, in Q4, it is necessary to reduce the action of the isotropy group I(5,2) to the isotropy subgroup I(4,2), []()() ()()[] ()[][] [] [] µµ α µNg gQVD VQVI VIQQ Q45 544 41 442 52== ×, , . ( A.8) The inverse of the measure of the isotropy groups quotient is () ()[] () ()[]()[] ()[]()()VS O S O VS O S OVS O VS OVS52 425 422 3452 5232× ×== = =ππ Γ . ( A.9)8 Under this reduction the coefficient of Poisson ’s kernel over Q4, the constant coefficient of the normalized measure µN in eq. ( A.8), defines the coefficient of the measure over D4, []()() ()()[] ()[]()() ()() () ()()αgQVD VQVI VIVD VS O VQ VS O45 55 51 41 442 524 5==× ×, , . ( A.10) The indicated volumes are known. The volume of the polydisc is ()Dnnn n=−π 21!( A.11) and the volume of Silov ’s boundary is the inverse of the coefficient in the Poisson kernel, ()()Qn nn =+221 2π Γ . ( A.12) In particular we have, ()D55 425=×π ! , ( A.13) ()Q5332 3=π . ( A.14) Substitution of theses expressions in equation ( A.10) gives Wyler ’s coefficient of the induced invariant measure, []αππ gQ42 655 43 22 51 4 =×  ! . ( A.15) Appendix B Here we indicate the relationship between the spaces R and K, as components of a series of symmetric spaces charac- terized by the group SO(4,2). The Cartan Killing metric for a group quotient space, G’/H, is taken as the metric in the subspace of the algebra of G’, complementary to the algebra of H. The exponentiation of this subspace is a globally symmetric space [4] because any point and its neighborhood can be translated to any other point by a group operation. In this way it is possible to show that the metric is invariant. Since both the group G’ and the subgroup H are related to compact groups by means of involutive automorphisms, there are different quotients related among each other by two involutions that we shall indicate as σ and τ. The possibilities for both involutions are exactly the same available for the involutive automorphisms of the complex extension of the algebra. Hence, the classification of the symmetric spaces is determined applying a pair of theseinvolutions, taken from the indicated set, that commute and exhaust the possibilities. The simultaneous eigenvalues of this pair serve to describe the Lie algebra A, AA A A A=⊕⊕⊕++ +− −+ −− . ( B.3) The involutive automorphism τ serves to select a compact subgroup, starting from a compact group G9 () GA A++ + − +=⊕τexp ( B.4) and the compact symmetric space with definite metric ()()expiA AG G+− −− +⊕= τ . ( B.5) The other automorphism σ serves to convert the compact subgroup to a non compact subgroup ()() GA A A i A Gt++ + − + + + − + +=⊕  → ⊕ = exp expσ τσ( B.6) and to convert the symmetric space with definite metric in one with indefinite metric ()() ()()G GiA A iA i AG G+++ −− +− −− +=⊕  → ⊕ = τσσ τσ exp exp . ( B.7) The symmetric space G/G+ has a negative definite metric derived from the Cartan Killing metric. The dual space G/G+, where G+ is the maximal compact subgroup, has an equal but positive definite metric, derived from the Cartan Killing metric restricted to the complementary subspace of A. Both spaces are, therefore, Riemannian spaces. There is a theorem that says that the non compact irreducible Hermitian symmetric spaces are exactly the manifolds G/H where G is a connected non compact simple group with center { I} and H is a maximal compact subgroup of G with a non discrete center [6]. There is a standard notation for the classification of Riemannian spaces, indicating the Cartan subspace (A,B …), the involution type (I,II,III) and the dimensions that characterize the groups (2n,p,q). There are other real forms of the complex group GC between the two extremes G y G* and therefore there is a series of symmetric spaces that are real forms of the complex extension of the quotient, GGG GCC C + +  = , ( B.8) that fall between the two extreme Riemannian spaces. These intermediate spaces have an indefinite metric and are considered pseudo Riemannian spaces. The different real forms within the series corresponding to a complex sym-metric space are classified by their characters. The character of a real form is defined as the trace of the canonicalform of the metric. This integer, corresponds to the difference in the number of compact and non compact generators.The series may be characterized by its non compact end group. The series of quotient spaces related to the A 3 Cartan space are of interest. In particular we choose the involutive automorphism τ of type AIII(p=2,q=2) that determines a seven dimensional compact subgroup G+. We obtain, in this manner, a series of eight dimensional spaces, characterized by the non compact group SU(2,2), corresponding to the Riemannian space G/G+ and its dual G/G+, SU SU SU USU SL C SOSU SL C SO SL ,R SL C SOSU SU SU U() () () ()( ) (, ) ()(,) (, ) (,) () (, ) ()(,) () () ()4 22 14 2222 21 1 4 2222 22 1⊗⊗≈⊗≈⊗≈ ≈⊗≈⊗⊗ . ( B.9) Due to the isomorphism of the spaces A3 and D3 we have the isomorphic series, characterized by the non compact group SO(4,2), corresponding to the Riemannian coset space G/G+ and its dual G*/G+, with involution τ of the type BDI(p=4,q=2),10 SO SO SOSO SO SOSO SO SO SO SO SOSO SO SO() () ()(,) (,) ()(,) (,) (,) (,) (,) ()(,) () ()6 4251 31 242 31 11 33 31 242 42⊗≈⊗≈⊗ ≈⊗≈⊗ . ( B.10) The characters of the real forms of both isomorphic series are -8, -4, 0, +4, +8. 1A. Wyler, Acad, Sci. Paris, Comtes Rendus, 269A, 743 (1969). 2A. Wyler, Acad, Sci. Paris, Comtes Rendus, 271A, 180 (1971). 3G. Gonz ález-Martín, Gen. Rel. Grav. 26, 1177 (1994); G. Gonz ález-Martín, Physical Geometry, (Universidad Simón Bolívar, Caracas) (2000), Electronic copy posted at http:\\prof.usb.ve\ggonzalm\invstg\book.htm ; See also related publications . 4R. Gilmore, Lie Groups, Lie Algebras and some of their applications (John Wiley and Sons, New York), ch. 9 (1974). 5L. K. Hua, Harmonic Analysis in the Classical Domains (Science Press, Peking 1958), translated by l Ebner, A Korányi (American Mathematical Soc., Providence) (1963). 6S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York) (1962).
1 A PROGRAM FOR THE GEOMETRIC CLASSIFICATION OF PARTICLES AND RELATIVISTIC INTERACTIONS Gustavo González-Martín Departamento de Física, Universidad Simón Bolívar, Apartado 89000, Caracas 1080-A, Venezuela. Web page at http:\\prof.usb.ve\ggonzalm\ Geometric relativistic interactions in a new geometric unified theory are classified using the dynamic holonomy groups of the connection. Physical meaning may be given to these interactions if the frame excitations represent particles. These excitations have algebraic and topological quantum numbers. The proton, electron and neutrinomay be associated to the frame excitations of the three dynamical holonomy subgroups. In particular, theproton excitation has a dual mathematical structure of a triplet of subexcitations. Hadronic, leptonic andgravitational interactions correspond to the same subgroups. The background geometry determines non trivialfiber bundles where excitations live, introducing topological quantum numbers that classify families ofexcitations. From these particles, the only stable ones, it may be possible, as suggested by Barut, to build therest of the particles. The combinations of the three fundamental excitations display SU(3) /G31SU(2)/G31U(1) symmetries.SB/F/279-002 1. Introduction. The group of the geometric space-time structure of special relativity is fundamental to the field theories of elementary particles, which are representations of this group. In contrast the geometry of general relativity hasnot played such a fundamental role. Nevertheless, our geometric relativistic unified theory of gravitation and electromagnetism may have non trivial applications for particle theory [1,2]. Here we discuss this questionusing groupal and geometric features and ideas, avoiding unnecessary aspects that may present obstacles inunderstanding the physical implications of the geometry for physical particles. The study of groups that act onthe geometric structures of a physical theory and their related symmetric spaces may display essential physicalfeatures without actually solving the equations of the theory. We hope to show the relation of this particularnew approach in the treatment of elementary particles and their interactions, a geometric group realization asfiber bundle sections, with the standard model [3,4]. It is well known that the holonomy groups of a connection in a fiber bundle may be used to classify its possible connections. It is interesting to use this method to gain insight on the types of physical interactionsrepresented by the connection in the unified geometric theory. In QFT certain particles are given masses by a Higgs mechanism which relies on certain symmetries that a vacuum solution may possess. This appears to assign to the vacuum a not completely passive role. We may tryto assign to the vacuum an even more active role. This is accomplished by recognizing that the particle vacuumis a geometric space with physical meaning, related to a unified non linear geometric theory of physics. Weconsider an approximation, to the geometric non linear theory, where the microscopic physical objets arerealized as linear geometric excitations around a non linear substratum geometric space. This is consistentwith the QFT interpretation of particles as vacuum excitations. We interpret the excitation as particles and thesubstratum as the particle vacuum. With this definition, a particle is acted upon by the substratum and is never really free except in absolute empty space (zero substratum curvature). The substratum space carries the universal inertial properties. A freeparticle is an idealization. In a fundamental level, if we accept physics in the substratum we are assuming, inpart, a (Parmenides) holistic principle which should be consistent with the ideas of Mach [5] and Einstein [6]that assign fundamental importance of far-away matter in determining the inertial properties of local matter.On the contrary, if we assume only particles we are assuming a (Democritus) atomistic principle. This impliesthere is no physics in the substratum. Restriction to holonomy subgroups has also implications for the equations of motion of matter, since within the theory particles are represented by frame excitations (sections of a bundle). We may expect that the geomet-ric association of frames to holonomy subgroups may naturally classify particles and associate them to interac-tions. Hopes to accomplish this objectives are supported on results of recent previous work [7], [8], [9] which we shall summarize as follows. Constants associated to the substratum connection appear as constant mass parameters in the excitation equations and play a fundamental role. As in General Relativity, the covariant equations may be referred toframes (coordinates) that must be related to observers determined by the physical experiment in question.Otherwise theoretical results remain indeterminate. The freedom to choose an arbitrary reference frame by agroup transformation generates a class of equivalent solutions, linking relativistic interactions, represented by one particular reference frame e r. Any excitation must be associated to a definite substratum. An arbitrary observation of an excitation prop- erty depends on both the excitation and the substratum, but the physical observer must be the same for bothexcitation and substratum. We may use the freedom to select the reference frame, to refer the excitation to thephysical frame defined by its own substratum, which satisfies the nonlinear equation, DJbb∗∗=Ω4πα . ( 1.1) Then the substratum is referred to itself and the substratum matter frame eb, referred to er becomes the group identity I. Actually this generalizes comoving coordinates (coordinates adapted to dust matter geode- sics) [10]. We adopt coordinates adapted to local substratum matter frames (the only non arbitrary frame is3 itself). Free matter shows no self acceleration, self action. In its own reference frame these effects actually disappear. Only constant self energy terms, determined by the nonlinearity of the substratum, make sense andshould be the origin of the constant mass parameter. At a small distance λ, characteristic of free excitations, the connection and frame of the substratum appears symmetric. Mathematically we should say that the substratum is a locally symmetric space [11] or hyperbolicmanifold [12]. We recognize the necessary condition that locally the substratum be a bundle that locally ad-mits a maximal set of Killing vectors of the space time symmetry with null connection Lie derivative [13].This means that there are space time Killing coordinates such that the connection is constant non zero in theregion of particle interest. (A flat connection is too strong assumption). It is clear from the definition of excitation that a free particle (excitation) is a representation of the struc- ture group of the theory and consequently an algebraic element. A representation (and therefore a particle) ischaracterized by the eigenvalues of the Casimir operators. The state of a representation (particle) are charac-terized by the eigenvalues of Cartan subspace basis operators. This provides a set of algebraic quantum num-bers to the excitation. Of course, we must somehow choose the respective representations associated to theseparticles. It has been indicated that the physical particles are representations of holonomy subgroups of SL(4,R)induced from the subgroup SL(2,C), realized as functions on the coset spaces. In fact, it has been shown thatnew electromagnetic consequences of this theory leads to quantization of electric charge and magnetic flux[14], providing a plausible explanation for the fractional quantum Hall effect. One important issue is: How do we calculate mass in a consistent manner? Mass arises from constants with inverse length dimension corresponding to a symmetric substratum solution of the non linear field equation.These constants appear in the linear excitation equations for a fermionic frame fluctuation δe and a bosonic connection fluctuation W=δΓ, which are of the general form, in the defining representation, ()() /G50δκ ∂δδµ µee m e=+ = /G4C , ( 1.2) ∗∗ ∗−+=dd W W jµπ α24 /G4C , ( 1.3) where m is the fermionic mass parameter given by () () me e J==−1 41 1 4 tr trκµ µµ µ ΓΓ ( 1.4) and µ is the bosonic mass parameter given by, µνν 21 4=trΓΓ , ( 1.5) in terms of the constants of the substratum solution. If we consider geometric excitations on a substratum, these expressions may be expanded as a perturbation around the substratum in terms of a small parameter ε, characterizing the excitation, indicating that the zeroth order term, the bare mass, is given entirely by the substratum current and connection, with corrections depend-ing on the excitation self interaction. As indicated in previous work [15], these corrections correspond to ageometric quantum field theory. In this manner, the particle bare mass parameters are not determined from the linear particle equation themselves, but rather from the holistic inertial non linear substratum solution. The existence of a constantconnection solution [16] for these asymptotic fields (free particle substratum) provides a fundamental dis-tance, due to non linearity, and gives a mechanism to calculate mass ratios in terms of volumes of respectivesymmetric coset spaces. The extreme case of absolute empty space that implies zero substratum curvature selfinteraction, is only mathematically possible in this theory for a universe without matter, because of thenonlinearity of the substratum field equations. Nevertheless, the association of a mass to one of these geometric distances remains arbitrary because this association actually calibrates a geometric scale of distance in terms of a physical mass scale. Since mass isinverse length, we may associate a “standard physical mass ” to the “standard geometric length ”. After the association we may attempt to calculate a ratio of masses. For example, if we select the mass of the protonexcitation to calibrate the geometric unit, we may calculate the mass of the electron excitation. The result ofthis calculation, using the symmetric space quotients of holonomy groups and the constant substratum solutionis the ratio [17]4 () ()m mVK VCm mG HR Rp e== = ≈ 6 183611815π . , ( 1.6) which essentially agrees with the experimental value and may lend credibility to this geometric theory. 2. Geometric Classification of the Connection. In order to classify the relativistic interactions we look for dynamical holonomy groups H of the associated connection. It is clear that H must be a subgroup of the structure group of the theory, SL(4,R), chosen to be the simple group of automorphisms of the space time Clifford algebra. From the geometric meaning of the algebra, the elements ±κa are associated to triads of opposite orienta- tion. In principle, both sets with opposite signs may be used as part of the orthonormal set κα of the algebra. The arbitrary sign is determined by a standard relation of products of Pauli matrices, in terms of Hodge dualityε ijk and Clifford (complex) duality i. We should adhere to the same convention in the choice of sign for the ±κa. The standard orientation in space time [ u0,u1,u2,u3] induces an orientation in the geometric algebra which should be used to define the Clifford geometric duality κ0κ1κ2κ3. This four dimensional duality operation de- noted by κ5, and Hodge duality should relate the matrices representing the Pauli matrices within the algebra, preserving the standard relation. Mathematically, κκκκ κ σσσσ κκκκκκ0123 5 0123 010203=≡ = =i , ( 2.1) σκ κii=0. ( 2.2) The group SL(4,R) is generated by exponentiation of the 15 traceless matrices of a base in the Clifford algebra. For physical reasons we want that the relativistic interactions be associated to a dynamic evolution of the matter sources. The dynamical effects are produced in the theory by the action of the group, in particularaccelerations should be produced by generators equivalent to the Lorentz boosts as seen by an observer. There-fore, we require that the boost generators, κ0κα, be present in a connection identifiable with a dynamical interaction as seen by the observer associated to the Minkowski subspace generated vectorially by κα. Because of the nature of the source current, Je u eµα αµκ=~/G24 /G24 , ( 2.3) which corresponds to the adjoint action of the group on the algebra, it should be clear that all generators of the form κ[ακβ] of SL(4,R) should be present in the dynamical holonomy group of a physical interaction connection. If we designate the even subgroup, generated by κ[ακβ], by L, the previous discussion means that LHG⊆⊆ . ( 2.4) Furthermore, H must be simple. The different possibilities may be obtained from the knowledge of the sub- groups of G. The possible simple subgroups are as follows: 1. The 10 dimensional group P generated by κα, κ[βκγ]. This group is isomorphic to the groups generated by, κ[ακβκγ], κ[ακβ] and by κ[aκbκc], κ[aκb], κ5 and in fact to any subgroup generated by a linear combina- tion of these three generators; 2. The 6 dimensional subgroup L, corresponding to the even generators of the algebra, κ[ακβ]. This group is isomorphic to the subgroups generated by κa, κ[aκb], and by κ0κ[aκb], κ[aκb] and in fact to any subgroup generated by a linear combination of these three generators; 3. The group G itself. The P subgroup is Sp(4,R), as may be verified by explicitly showing that the generators satisfy the simplectic requirement [18]. This group is known to be homomorphic to SO(3,2), a De Sitter group. The L subgroup is Sp(2,C), isomorphic to SL(2,C). In addition there are only two simple compact subgroups of G, non dynami- cal, generated by κ[aκb] and κ0, κ5, κ1κ2κ3, apart from the unidimensional subgroups. Then we have only three possible dynamical holonomy groups: L, P, or G. For each case we have an equiva- lence class of connections and a possible physical interaction within the theory. Other holonomy groups are notdynamical, in the sense that they do not produce a geometrical accelerating action on matter, as determined by5 an observer boost. This is the case of a U(1) holonomy subgroup which may represent electromagnetism but does not provide, by its direct action, a geometric dynamics (force) on charged matter. The dynamics requiresa separate Lorenz force postulate. Since L ⊂P⊂G, this scheme allows us to classify interactions geometrically in increasing order of complex- ity. This group chain has a symmetry because there is no unique way of identifying L/G5FG and L/G5FP. The coset G/L represents how many equivalent L subgroups are in G. There is also an equivalence relation R among the non compact generators of G, all of them equivalent to a boost generator or space-time external symmetry. The subspace obtained as the quotient of G/L by this relation is the internal symmetry group of L/G5FG, GL RSU/() =2 . ( 2.5) Similarly the coset P/L gives, as the internal symmetry of L/G5FP, the group PL RU/()=1 . ( 2.6) The total internal symmetry of the chain L/G5FP/G5FG is the product of the two groups SU(2) /G31U(1) which coin- cides with the symmetry group of the weak interactions. There is no geometrical reason to identify the struc- ture group of the theory with the symmetry group because they are different geometrical concepts. At the endwe shall come back to discuss the physical significance of this symmetry. 3. Subexcitations Corresponding to Subgroups. If we restrict the group to either the P or L subgroups, the corresponding frame matrices (subframes) are elements of the subgroup. The total frame f, which is an element a of the algebra, decomposes into even and odd parts in accordance with the general Clifford algebra decomposition, aa a=++−κ0 . ( 3.1) From previous results, the equations of motion for the even f+ and odd f- parts of f are under certain restrictions, κ∂κµ µµ µff m f+− − −==Γ , ( 3.2) κ∂κµ µµ µff m f−− + +==Γ , ( 3.3) implying that a frame for a massive particle must have odd and even parts. In our case, if we set f- equal to zero we obtain also that m is zero. Therefore, for an L-subframe we have multiplying by κ0 , σ∂µ µf+=0 , ( 3.4) the equation of the neutrino as discussed before. If an excitation corresponds to a representation of a subgroup with specific quantum numbers, it may be associated to only one of the spinor columns of the frame, the one with the corresponding quantum numbers.Accordingly, we restrict the fluctuations of frames to matrices that have only one column in each of the twoparts of the frame, the even η and the odd ξ. ηη η=  11 120 0/G24 /G24 , ( 3.5) ξξ ξ=  11 120 0/G24 /G24 . ( 3.6) We now restrict to the even simple subgroup SL(2,C), homomorphic the Lorentz group. As shown in previ-6 ous work, the η, ξ parts have inequivalent transformations under this group, ′=ηηl , ( 3.7) ′=ξξl* . ( 3.8) These columns are spinor representations of the group. We may form a four dimensional (Dirac) spinor by adjoining the two spinors, where the components η, ξ are two complex 2-spinors. We may combine the 2 columns into a single column Dirac 4-spinor, ψξ ξ η η=      11 12 11 12/G24 /G24 /G24 /G24 . ( 3.9) We now show that the even and odd parts of a frame are related to the left and right handed components of the field. We calculate the left handed and right handed components, and obtain, omitting the indices, () () ψ γψ γξ ηηL=+ =+  =  1 25 1 25110 , ( 3.10) () () ψγ ψ γξ ηξ R=− =−  =  1 25 1 25110 . ( 3.11) We have that the left handed component is equivalent to the η field which in turn is defined in terms of the even f+ field. Similarly, we see that the right handed component is equivalent to the ξ field and consequently to the odd f- field. Therefore, an even frame corresponds to a left handed particle, as should be for a neutrino. The Lorentz frame excitation has neutrino like properties. It is necessary to point out that, since the mass in thisequation is zero, Wigner ’s little group, which preserves the momentum k, is ISO(2) instead of SO(3) corre- sponding to massive particles. The induced representations of the neutrino excitation are sections over thelight 3-hypercone valued in representations of ISO(2) and are characterized by the value of helicity. A general Sp(4,R) excitation has a κ0 generator, which is an electromagnetic generator. We expect that this excitation should represent a particle with one quantum of charge and one quantum of spin. The component ofthe current along this generator coincides with the standard electric current [19] in quantum mechanics. Inaddition, it was shown that the general perturbation technique for the interaction of the electron and neutrinofields lead, essentially, to the current and Lagrangian assumed in Fermi ’s theory of weak interactions [20] of leptons. This lead to the conjecture that a Sp(4,R) excitation may represent an electron. With this in mind, thevalue of the ratio of the proton to electron masses was derived from the definition of mass in terms of energy asindicated in a previous section, using the properties of the structure group and its subgroups and respectivecosets in the theory. It is possible to define the eigenvalue of the odd operator in the geometric algebra as a quantum number called oddity. Furthermore, transformations by the subgroup SL(2,C) leave invariant the even and odd sub-spaces of a frame. Since these eigenspaces correspond to the left handed and right handed parts, or equivalentto the helicity eigenvalues, it is clear that, within a Sp(4,R) frame, e L may be assigned a νL partner and may be considered a doublet, under another group, while eR has no νR partner and may be considered a singlet. This indicates a possible link with the standard model of weak interactions. 4. The General Material Excitation. In general, the equation of motion for matter, 20κκµ µα µαµ∇+∇ =eu e/G24 /G24 , ( 4.1)7 applies to the three classes of dynamical frames, according to the three dynamical holonomy groups. The three equations, together with the non linear field equation should have a substratum state solution. Then we mayassociate an excitation to each class of frames around the substratum solution. This excitations are elements ofthe Lie algebra of the structure group of the theory. The maximally commuting subspace of the Lie algebra sl(4,R), generated by the chosen regular element, [21] is a three dimensional Cartan subalgebra, which is spanned by the generators, X11 2=κκ , ( 4.2) X20 1 2 3=κκκκ , ( 4.3) X30 3=κκ . ( 4.4) It is clear that X1, and X2 are compact generators and therefore have imaginary eigenvalues. Because of the way they were constructed, they should be associated, respectively, to the z-component of angular momentum and the electric charge. Both of them may be diagonalized simultaneously in terms of their imaginary eigen-values. As shown in figure 1 the 4 members of the fundamental representation form a tetrahedron in the three dimensional Cartan space. They represent the combination of the two spin states and the two charge states ofthe associated particle. For example, charge spin flux negative charge with spin up negative charge with spin down positive charge with spin up positive charge with spin down−+− −−++++ +−−−↑ −↓ +↑ +↓111 111111 111f fff The fundamental representation f of SL(4,R), which may be indicated by f+↑, f+↓, f-↑, f-↓ groups together two excitation states of positive charge with two excitation states of negative charge. The presence of oppositecharges in a representation forces us to make a clarification. To avoid confusion we should restrict the term(charge) conjugation to indicate the original Dirac operation to relate states of opposite charge. There is also another fundamental representation f ∼ dual to f and of the same dimensions, with all signs reversed, which is inequivalent to the original one and represented by the inverted tetrahedron in the Cartansubspace. One of the two representations is arbitrarily assigned to represent the physical excitation. The math-ematics of the representation algebra indicates that a state may also be described as composed of 3 members ofits dual representation. In turn, the dual representation may also be similarly described as composed of 3members of its dual, which is the original representation. This mutual mathematical decomposition is reminis-cent of the idea of “nuclear democracy ” proposed in the 1960 ’s [22] but restricted to dual representations. To avoid confusion we restrict the use of the term duality to relate these inequivalent representations. In standard particle language the antiparticle is the dual particle which implies also the conjugate particle. Nevertheless, this assignment of a physical particle/antiparticle pair to the fundamental representation and itsinequivalent dual is not a mathematical implication of standard particle theory, only a physical assumption.Since our fundamental representation includes opposite charges it is not appropriate to consider the dualrepresentation as antiparticles. We may, as well, simply say that the antiparticle is related to the conjugateparticle. The dual particle is just a necessary dual mathematical structure. We have then for a particular state p of the fundamental representation, () ( ) p f f f f qqq↑ +↑ +↑ +↓ −↑ +↑ +↓ −↑≡= ≡~~~ , ( 4.5) in terms of the dual states q. Similarly for a particular q state, () ()qq q q p p p+↑ +↑ +↓ −↓ +↑ +↑ −↓==~~~ . ( 4.6)8 It does not follow that the q necessarily are states of a different physical excitation, only that the q form a dual triplet mathematical representation of the p representing the same excitation. This allows a different physical interpretation for these mathematical constructions. It should be noted that all p, q have electric charge equal to the geometric unit, electron charge ±e. Since these excitations have particle properties, there is a dual mathematical representation of the physical excitation (particle). We may raise the following question?What happens if we physically identify p with the proton, which mathematically may be expressed as 3 q, interpreted as quarks? In our theory there is no need to assign fractional charges to quarks. In accordance withthe “restricted nuclear democracy ” we may assume that the quark states, in turn, may be mathematically expressed as 3 protons p, which may justify the large experimental mass of these states. The fundamental representation of Sp(4,R) which may be displayed as a tetrahedron projected square in a two dimensional Cartan space is e -↑, e-↓, e+↑, e+↓. Its dual Sp(4,R) representation, obtained by reversing all signs is mathematically the same e+↓, e+↑ e-↓, e-↑. Similarly the SL(2,C) representation ν↑, ν↓ which may be displayed in a one dimensional Cartan space is mathematically its own SL(2,C) dual. Therefore, the only one of the three excitations with a mathematically inequivalent dual structure is p. Since the Sp(4,R) and SL(2,C) subgroups may be imbedded in SL(4,R), the corresponding Cartan spaces of Sp(4,R) and SL(2,C) may also be imbeddedin the Cartan subspace of SL(4,R). The plane subspace Q=-1 has charges of one definite sign and is a repre- sentation of Sp(4,R). Another plane, Q=1, containing opposite charges is another representation of Sp(4,R) that completes the SL(4,R) Cartan subspace (anti-particle e +). States q+↑, q+↓ may be considered as an Sp(4,R) antimatter representation different from e-↑, e-↓, which is present inside the proton implying that there is no need to look for antimatter elsewhere. The analogous relations between the mathematical q+ and e- structures Figure 1+1, +1, +1-1, +1, -1-1, -1, +1 +1, -1, -19 as Sp(4,R) representations may explain the parallelism of hadron and lepton families. Nevertheless, they are different because the q are acted upon by the complete G connection whereas the e are acted only by the P connection. It may be seen that the electron state e-↓ is complementary to q+↑ q+↓ q-↑ occupying the quartet of states of a SL(4,R) representation, reflecting the neutrality of matter composed of p, and e. The fundamental (1, ±1, ±1) SL(4,R) states define a proton charge sign. Since matter is neutral the fundamental (-1, ±1, ±1) Sp(4,R) states, imbedded in the proton Cartan space as subspace Q=-1, define the electron charge sign, opposed to the proton sign. The space SL(4,R)/SU(2) /G31SU(2) is a 9-dimension Riemannian space of the non compact type. There are 9 boost generators Bm a. The rotation SU(2), indicated by S, contained in SO(4) acts on the m index and the electromagnetic SU(2), indicated by Q, contained in SO(4) acts on the a index, QBS Bba an nm bm=′ . ( 4.7) Similarly the subspace Sp(4,R)/SU(2) /G31U(1) is a 6-dimensional Riemannian space of the non compact type. The complementary subspace within SL(4,R)/SU(2) /G31SU(2) is 3-dimensional. There is a triple infinity of these subspaces within the total space, reflecting the triple infinity of groups P in G, depending on the choice of an electromagnetic generator among the three possible ones in SU(2). In Sp(4,R)/SU(2) /G31U(1), with a fixed electromagnetic generator, we have a vector in the odd sector, repre- senting a momentum k associated to an excitation e. Since we have this situation in SL(4,R)/SU(2) /G31SU(2) for each generator in SU(2), we have, in effect, 3 momenta, ki, that may characterize an excitation p. We must consider excitations characterized by 3 momenta, ki, which may be interpreted as three subexcitations. The 3 momenta, ki, of the three subexcitations characterize the protonic frame excitation p, but neither those of e nor those of ν. Mathematically what we have is a system characterized by 3 momenta, ki, that may be scattered into another system of 3 momenta, kj’. Could this be interpreted as a 3-point particle? It should be expected that a scattering analysis of excitations must include three momenta in some δ functions that appear in the scattering results. Experimentally, at high energies, this should appear as a collision with a system ofthree point like particles. Again, this could be interpreted as three partons (quarks) inside the proton. In anycase, it is clear that a point like proton excitation is not predicted by our theory. 5. Physical Interpretation in Terms of Particles and Relativistic inter-actions. We have found properties of the geometric excitations which are particle like. We take the position that this is no coincidence but indicates a geometrical structure for physics. The source current J depends on a frame. To each holonomy group we may associate a class of frames thus giving three classes of matter. As already discussed in section 3, the corresponding L-frame represents a zero mass, neutral, spin ½, left handed geometric excitation, which obeys eq. ( 3.4), and has the particle properties of the neutrino. For the P-connection, the corresponding P-frame represents a massive, opposite charge -1, spin ½, geomet- ric point like excitation which has the properties of the electron [23]. For the G-connection the corresponding G-frame should be a massive, charge 1, spin ½, 3-point geometric excitation, with a bare mass of 1836.12 times the electron mass which we conjecture is the proton. In this manner, we have associated to each of the three holonomy groups, one of the only three known stable particles. For the L-connection it is not difficult to recognize that the interaction is gravitation, from the discussion in previous chapters and the work of Carmelli [24]. Similarly, it also was shown in previous work that electro-magnetism (without dynamics) is associated to one of the SU(2) generators and that the physical Fermi weakinteraction is related to the odd sector of a P-connection. We propose here that the P/L generators may be interpreted as an electroweak interaction and the G/P as strong nuclear interaction. Then the three dynamic holonomy classes of connections may correspond to threeclasses of relativistic interactions as follows: 1. The L-connection describes gravitational interaction; 2. The P-connection describes coupled gravitational and electroweak interactions; 3. The G-connection describes coupled gravitational, electroweak and strong interactions.10 The L-frames obey equations that may be obtained from the general equations of motion when the frame e has only the even part e+. From the field equation it is seen that a P-frame generates a P-connection and that a G-frame generates a G-connection. From this classification it follows, in agreement with the physical interactions that: 1. All (matter) frames self interact gravitationally;2.L-frames self interact only gravitationally (uncharged matter); 3.G-frames (hadrons) are the only frames that self interact strongly (hadronic matter); 4.G-frames self interact through all three interactions; 5.G-frames (hadronic matter) and P-frames (leptonic matter) self interact electroweakly and gravita- tionally; 6.P-frames self interact gravitationallly an electroweakly but not strongly (leptonic matter). 6. Particle Families. The framework of this theory is compatible with a phenomenological classification of particles in a manner similar to what is normally done with the standard symmetry groups. First we should notice that the theory suggests naturally three stable ground particles ( ν, e, p). In fact, if we consider the possibility of different levels of excited states, each particle may generate a class of unstableparticles or resonances. In particular, since the equations for each of the three particle classes are the same, differing only in the subgroup that applies, it may be expected that there is some relation among corresponding levels of excitationsfor each class, forming families. The ratio of the mass of the hadron in the ground level family (proton) to the mass of the lepton in the ground level family (electron) has been calculated from the ratio of volumes of coset spaces based on theexistence of a trivial non zero constant solution for the substratum connection. This trivial solution is therepresentative of a class of equivalent solutions generated by the action of the structure group. Now consider only topological properties, independent of the connection, of the space of complete solutions (substratum plus excitation solutions). An incoming scattering solution is a jet bundle local section, over aworld tube in the space time base manifold, that describes the evolution of the solution in terms a time likeparameter τ from past infinity to some finite time t. Similarly, an outgoing solution is a local section from time t to future infinity. The local sections in the bundle represent classes of solutions relative to local observers. Scattering solutions at infinity are asymptotically free excitation solutions around a substratum. The substrata(incoming and outgoing) are equivalent to each other and to the constant trivial solution if we choose observerframes adapted to the substrata. Since the equations are of hyperbolic type, we should provide initial conditions on an initial 3 dimensional hypersurface at past infinity /G49 /G2D. The sections of interest are the three fundamental excitations. We require that all incoming solutions, at the past infinity hypersurface /G49/G2D, reduce to a free excitation around the trivial sub- stratum solution at the two dimensional spatial infinity subspace /G49/G2D(∞). Since the incoming solution substrata are equivalent to the trivial substratum solution at spatial infinity /G20/G49/G2D(∞), we may treat this spatial infinity as a single point, thus realizing a single point compactification of /G49/G2D, so the initial hypersurface /G49/G2D is homeomor- phic to S3. All incoming solutions on /G49/G2D are classified by the functions over S3. The same requirements may be applied to the outgoing remote future solutions and, in fact, to any solution along an intermediate 3 dimen-sional hypersurface, a section of the world tube. Thus, the final hypersurface at future infinity /G49 /G2B is also homeo- morphic to S3. The incoming and outgoing substratum local sections over /G49/G2D and /G49/G2B must be pasted together in some common region around the present t, by the transition functions of the bundle. All generators of the group produce a transformation to a different, but equivalent under the group, expression for the solution. Ifthe holonomy group of the solution is not the whole group, there is a reference frame that reduces the structuregroup to the particular holonomy subgroup. But in general for arbitrary observer, there are solutions formallygenerated by SL(4,R). Since this transition region, the “equator”, has the topology of S 3×R, the transition functions ϕ define a mapping, at the τ=0 hypersurface, ϕ:( , )SS L R34→ , ( 6.1) which is classified by the third homotopy group [25] of the structure group SL(4,R) or the respective holonomy subgroup. There are some solutions not deformable to the trivial solution by a homeomorphism because ϕ11 represents the twisting of local pieces of the bundle when glued together. To determine π3(G), we realize that the exponential map from the maximally non compact subalgebra of sl(4,R) is a diffeomorphism [26] to the non compact Riemannian coset subspace which is contractile. We havea short exact sequence in the general homotopy sequence [27], () () () () /G4C/G4Bππ π π43 3 3GHHGGHip ∆∗∗ ∗→ →  → ) , ( 6.2) {} () () {} 0033→ →  → ∗ππHG , ( 6.3) which implies that the intermediate mapping is an isomorphism and () ()ππ33GH= ( 6.4) where H is the maximal compact subgroup. We also know that there is an isomorphism between the homotopy groups of a group and its covering group, except for the first homotopy group [28]. For the homotopy group ofSL(4,R) we get () () ()() ()() () ππ π π33 3 342 2 2 2SL R SU SU SU SU Z Z(, ) ()=⊗ = ⊗ = ⊗ . ( 6.5) Similarly, we have for the homotopy groups of the other two holonomy groups, () ()() () ππ π33 342 2Sp R R SU SU Z(, ) ()=⊗ = = , ( 6.6) () ( )ππ3322SL C SU Z(, ) ()== . ( 6.7) These scattering solutions are characterized by topological quantum integral numbers n, for the three groups, and m only for group G, called winding numbers. In all cases the scattering solutions are characterized by one topological winding number n and in particular the hadronic scattering solutions have an additional topologi- cal winding number m. This result implies that there are solutions ωn, en that are not homotopically equivalent to ω0, e0. All p, e, ν excitation solutions with a given n may be associated among themselves because of the isomor- phism of the homotopy groups Z. This is an equivalence relation. Two p, e, ν excitation with the same n are in the same topological class determined by the substratum. Each topological class, characterized by the topologi-cal quantum number n defines a physical class, a family of particles with the same winding number, which is respected by transitions the same as the algebraic quantum numbers s, q, f. For example, the association of a solution of each type to any value of n may be part of a general scheme of relations labeled by n as follows, lel l hphp hel e012 012 01 2=== === ==,,,µτ ννννννµ µ/G4C /G4C /G4C, ( 6.8) among the excitation levels of the electron (proper leptons), the excitation levels of the neutrinos (other neutri- nos) and the excitation levels of the proton (hadrons). 7. Relation with Barut’s Model. Since in our geometric theory the only constant that enters is α, the ratio of the masses of the excited states and the fundamental state should depend on this constant. It is possible that the geometric theory may providean explanation for the interesting approximate relation of the masses of the leptons and the electron, mm nlel =+  ∑13 24 0α . ( 7.9) This equation was proposed by Barut [29] with n interpreted as the number of νν pairs. Although this12 interpretation is also characterized by the group Z, it is not clear whether both interpretations are compatible. In any case, the possibility exists that there be excited states which may be interpreted as a particle com- posed of p, e and ν . As a matter of fact Barut has suggested that the muon should be considered as an electromagnetic excitation of the electron. Although the details should be different, because Barut ’s approach is only electromagnetic in nature and ours is a unified interaction, we may conjecture also that the muon is anexcited electron state with a leptonic winding number n=1. Similarly the τ would be an excitation with wind- ing number n=2. Having a possible scheme for the geometric origin of the muon, it is possible that the process of building other particles as suggested by Barut [30,31,32] may be accomplished in terms of the stable articles, theelectron, the proton and the neutrino together with the muon. In order to adapt our geometric model to Barut ’s model we establish that the quantum numbers of an excitation correspond to the quantum numbers of its stable components ( p, e, ν). In this manner, following Barut’s ideas, we define the charge of an excitation as the number of protons minus the number of electrons contained in the excitation, QN Npe=− . ( 7.10) Similarly, the barionic number of an excitation is the number of protons in the excitation, BNp≡ ( 7.11) and the leptonic number of an excitation is the number of electrons and neutrino components, LN Ne≡+ν . ( 7.12) Strangeness of an excitation is the number of excited electrons (winding number 1), or muons µ in the excitation, −≡SNµ ( 7.13) and charm is the number of muonic neutrinos, ()CN=νµ . ( 7.14) The isotopic spin depends on the number of stable particles in the following manner, 23INNNpe≡−+ν . ( 7.15) From these definitions it is possible to derive the Gell-Mann Nishijima formula, () QI BS=+ +31 2 . ( 7.16) and define strong hypercharge YBS≡+ . ( 7.17) In additon to these Barut definitions, it is also possible to define weak hypercharge for a P-excitation in terms of leptonic number and the oddity Ο of the representative algebraic element, ()yL≡− +Ο . ( 7.18) The hadronic states contain protons. In figure 2 we show the barionic octet JP=1 2+ and in figure 4 the barionic decuplet JP=3 2+. The meson states are bound states of two leptons ll as indicated in the mesonic octect JP=0+ in Figure 3. Furthermore, the geometric theory allows a discussion of the approximate quantum interaction between two of quark excitations forming a two 3-velocity system (Quarkonium?). We speculate that the non relativisticeffective potential should be similar to the one in QCD because there are similarities of the mathematics of thetheories. If this were the case, (this has to be shown) the non relativistic effective potential would be [33]13 ()ra rkr=− + ( 7.19) and then we may fit the experimental Ψ excited levels accordingly as is done in QCD. The full frame excitation has proton like properties. Of course, within the geometric theory the quark excitations are not the fundamental building blocks of matter. They are only approximate subexcitations thatare useful and necessary in describing a series of hadronic excitations. 8. Relation with the Standard Model. We have replaced the space time related structure group, fundamental to the theory, by using SL(4,R) instead of SO(3,1). On top of this geometry, as is done in the standard model over the geometry of standardspecial relativity, it is possible to add a related structure to help in the understanding of the physical particles.It is known that heavy nuclei may be partially understood by using groups SU(N), U(N), O(N) to associateprotons and neutrons in a dynamic symmetric or supersymmetric manner [34]. This is essentially the use ofgroup theory to study the combinations of the two building blocks, protons and neutrons, assumed to formnuclei. In the same manner we may use certain groups to describe the combinations of the three fundamental geometrical building blocks introduced by the physical geometry, protons, electrons and neutrinos, assumingthey form other particles. We are interested in how to combine G, P and L excitations. This is accomplished by attaching L excita-()Σ0pµνννµ−()Λ0pµνµ−p ()Σ−− −peνµνµ ()Ξ−− −pµµννµµ()np e−ν ()Ξ0peµµνµ−−+()Σ++ −peµ Barionic octet()Σ0pµνννµ−()Λ0pµνµ− Figure 214 tions to P excitations and then to G excitations. This combination may not be done uniquely, rather it depends on the identification of a subgroup H with a subspace in the group fiber bundle space G. Any non compact generator is equivalent to a boost, by the adjoint action of a compact generator. Thus, the compact generatorsgenerate a symmetry of the combination of excitations. In particular, the identification of L within P is not unique. The group L may be expressed as the principal bundle (L,3B,S), where the fiber S is the SU(2)R associated to rotations and 3B is the three dimensional boost symmetric space, 32 2BSL C SUR=(, ) ()( 8.1) The group P may be expressed as the principal bundle ( P,6B,S/G31U(1)) where 6B is the six dimensional double boost symmetric space, 6 4 21BSp R SU URQ=⊗(, ) () ()( 8.2) The choice of the boost sector of L (even generators), inside P, depends on the action of the U(1) group in the fiber of P, generated by κ0, a rotation within the double boost sector in P. []κκ κ κ01 0 12,= . ( 8.3) Mathematically this corresponds to the adjoint action of the only compact generator in the odd quotient P/L. Similarly, the identification of P inside G is not unique either. The group G may be expressed as the bundle Pseudo scalar mesonic octet()ηννµµ++− + −ee1 6()Ke0−+µ ()K++µνµ ()K−−µνµ ()Ke0+−µ() πνννµµ−−e ()πνννµµ++e() πνν0ee+−+ ()ηνν µµ +−+− + −ee 21 6() πνν0 1 2−+−ee Figure 315 (G,9B,S/G31Q) where Q is the SU(2)Q associated to charge and 9B is the nine dimensional triple boost symmetric space, 9 4 22BSL R SU SURQ=⊗(, ) () ()( 8.4) The choice of the boost sectors of L and P sector inside G depends on the action of the SU(2) in the fiber of G, generated by the compact electromagnetic generators. Mathematically this corresponds to the action of allcompact generators in the quotient G/L. The total symmetry of first identifying L/G71P and then ( L/G71P)/G71G is the product of the two groups. Therefore, there is a symmetry action of a group equal to SU(2) /G31U(1), but different to the subgroups of G , on the identification of the subgroups in the chain L/G71P/G71G. There is an induced symmetry on the combination of L and P subexcitations on G excitations corresponding to this SU(2) /G31U(1) symmetry. The action of this combinatorial group may be interpreted as the determination of the possible combinations of L and P excitations to give flavor to states of G and P excitations and thus related to the weak interactions. This SU(2) F group relates electron equivalent P-excitation states or neutrino equivalent L-excitation states. Clearly at sufficiently high energies the mass of any excitation kinematically appears very small and its effectson results are small deviations from those of a zero mass excitation that always corresponds to an excitation ofthe even subgroup or L-excitation. For this reason, at high energies, the even part (left handed part) of a P- excitation may be related to an even L-excitation by an SU(2) transformation. Both corresponding physical leptonic excitations, the even part e + or e0 and ν, may be considered members of a doublet, labeled by oddity 0 or weak hypercharge -1, while the odd part e- or e1 is a singlet labeled by oddity 1 or weak hypercharge -2. This weak interaction association of leptons into hypercharged states has the approximate symmetry SU(2)F. InBarionic decuplet()Ω−− − − +peµµµννµµ()Ξ∗− − −pµµννµµ()∆−− −pe eννµµ()∆0pe−νµ ()∆+p ()∆++ +peνµ ()Ξ∗− − +0peµµν()Σ∗+ + −peµ ()Σ∗− − −peµννµ()Σ∗−0pµνννµ()Σ∗−0pµνννµ Figure 416 addition, there is an undetermined orientation of L1 inside G that depends of the action of SU(2) /G31U(1). Under this approach, the standard physical electromagnetic potential A, the connection of U+(1) in SL1(2,C), has an orientation angle within the chiral SU(2) /G31U(1). Since our description may be made in terms of the proton representation or, alternately, in terms of its dual quark representation as indicated in section 4, we could obtain complementary dual realizations of the rela- tions among different experimental results. These are really different perceptions or pictures of the same physicalreality of matter. The proton or G-excitation has a quark structure: it behaves as if formed by three points. Using the quark representation to build other particles, we may consider that in G the 3 quark states span a 2 dimensional Cartan A 2 subspace. There is a combinatorial unitary symmetry, SU(3), related to this Cartan subspace, that may be interpreted as the color of the combinations of states of G and thus related to strong interactions. States of G-excitations display an SU(3)C symmetry. In this manner, the physical geometry determines an approximate combinatorial internal symmetry charac- terized by the combinatorial symmetry group SC. SS U S U UCC F=⊗⊗() () ()32 1 . ( 8.5) If the probability of combination is propagated infinitesimally by a connection, we have the elements of a high energy quasi standard model. At high energies we may expect the appearance of resonances with high masses.Of course, in order to get the electromagnetic potential at low energy, the chiral part of the phenomenologicalcombinatorial internal symmetry should be broken. The standard model is built from certain general features of experimental and theoretical results. In par- ticular the SU(3) transformation among a triplet of components, the quark-parton model [35,36], and theSU(2) L transformation of the chiral parts of a doublet, the electroweak theory [37,38], are the starting points of the model. Since these features are present in the physical geometry, as shown here and in sections 3 and 4,we have the real possibility of building an approximate “quasi standard model ” on top of our geometry. The combinatorial groups actually relate discrete asymptotic, in or out, states in a scattering theory of excitations.It is possible, in an approximate way, to “gauge” these groups to obtain an approximate dynamic theory. Theoretically, the infinitesimal evolution of physical systems or dynamics is determined by the generators ofSL(4,R) in accordance with the equations of the theory, but the scattering solutions should show symmetriesrelated to the combinatorial group. In this manner, this approximate construction should share certain features and results with the standard model while there would be differences in some other features. Due to the vastness of particle physics experi-mental results, it is not clear, at this time, how these differences would compare with experiments, in particu-lar because the geometrical ideas may introduce a rearrangement of experimental results. This actually repre-sents a program to be carried out. It is true that the standard model has led to success, but this does not implythere is no better way of reordering the physical results. Summary. The geometric unified theory discussed represents relativistic interactions that may be classified into three classes by the dynamical holonomy groups of the possible physical connections. The three classes correspondto gravitation, electro-weak and strong physical interactions. Solutions to the three corresponding frame excitation equations representing matter, show algebraic and topological quantum numbers. Using them and previously calculated mass ratios, we are able to identify threeclasses of fundamental excitations, corresponding to the three stable physical particles (neutrino, electron andproton). The algebraic numbers correspond to electric charge, spin and magnetic flux. The leptonic and had-ronic topological quantum winding numbers correspond to higher levels of excitation, defining families foreach integer. The interactions felt by these excitations conform with the general classification scheme of par-ticles into neutrinos, leptons and hadrons. The combinations of the three particles display an SU(3) /G31SU(2)/G31U(1) symmetry. A program to establish the relation of the theory with the standard model of particles was outlined. 1 G. Gonz ález-Martín, Phys. Rev. D35, 1225 (1987); G. González-Martín, Physical Geometry, (Universidad Simón Bolívar, Caracas) (2000), Electronic copy posted at http:\\prof.usb.ve\ggonzalm\invstg\book.htm 2 G. Gonz ález-Martín, Gen. Rel. Grav. 22, 481 (1990); See related public ations. 3 S Weinberg, The Quantum Theory of Fields, (Cambridge Univ. Press, Cambridge) V. 2, 384, (1996).17 4 J. C. Taylor, Gauge Theories of Weak Interactions (Cambridge Univ. Press, Cambridge), ch. 6 (1976). 5 E. Mach, The Science of Mechanics, 5th English ed. (Open Court, LaSalle), ch. 1 (1947). 6 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p.55 (1956). 7 G. Gonz ález-Martín, p/e Geometric Mass Ratio , Universidad Sim ón Bolívar Report, SB/F/274-99 (1999). 8 G. Gonz ález-Martín, Physical Geometry and Weak Interactions , Universidad Sim ón Bolívar Report, SB/F/271-99 (1999). 9 G. Gonz ález-Martín, Charge to Magnetic Flux Ratios in the FQHE , Universidad Sim ón Bolívar Report, SB/F/273-99 (1999). 10W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco), p. 715 (1973). 11E. Cartan, Assoc. Avanc. Sc. Lyon, p. 53 (1926).12J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (Springer-Verlag, New York) (1994)13A. Trautman, Geometrical Aspects of Gauge Configurations, preprint IFT/4/81, Acta Phys. Austriaca, Supl. (1981). 14G. Gonz ález-Martín, Gen. Rel. Grav. 23, 827 (1991); G. González-Martín, Physical Geometry, (Universidad Simón Bolívar, Caracas) (2000), Electronic copy posted at http:\\prof.usb.ve\ggonzalm\invstg\book.htm 15G. Gonz ález-Martín, Gen. Rel. Grav. 24, 501 (1992). 16G. Gonz ález-Martín, Fundamental Lengths in a Geometric Unified Theory, USB preprint, 96a (1996). 17See ref. 9. 18R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (John Wiley and Sons, New York), p.188 (1974). 19See ref. 1 20See ref. 8. 21S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York) p. 137 (1962).22G. F. Chew, S. C. Frautschi, Phys. Rev. Lett. 7, 394 (1961). 23G. Gonz ález-Martín, in Strings, Membranes and QED, Proc. of LASSF, Eds. C. Aragone, A.Restuccia, S. Salamó (Ed. Equinoccio, Caracas) p, 97 (1989) 24M. Carmelli, Ann: Phys. 71, 603 (1972). 25R. E. Marshak, Conceptual Foundations of Modern Particle Physics, (World Scientific, Singapore) ch. 10 (1993). 26S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York) p. 130, 137, 214 (1962). 27G. W. Whitehead, Elements of Homotoy Theory (Springer Verlag, New York) (1978).28F. H. Croom, Basic Concepts in Algebraic Topology, (Springer Verlag, New York) (1978).29A. O. Barut, Phys. Rev. Lett. B 42, 1251 (1979). 30A. O. Barut, in Lecture notes in Physics, 94, (Springer, Berlin) (1979). 31A. O. Barut, Physics Reports, 172, 1 (1989). 32W. T. Grandy, Found. of Phys. 23, 439 (1993). 33E. Eichen, K. Gottfried, T. Kinoshita, K. Lane, T. Yan, Phys. Rev. D21, 203 (1980). 34A. Arima, F. Iachello, Phys. Rev. Lett., 25, 1069 (1974). 35M. Gell-Mann, Phys. Lett. 8, 214 (1964). 36G. Zweig, CERN Report 8182/Th. 401 (unpublished) (1964).37S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). 38A. Salam, in Elementary Particle Theory, ed.N. Swartholm (Almquist and Wissell, Stockholm) (1968).
arXiv:physics/0009053v1 [physics.atom-ph] 14 Sep 2000Interaction dynamics between atoms in optical lattices J. Piilo,1K.-A. Suominen,1,2,3and K. Berg-Sørensen4 1Helsinki Institute of Physics, PL 9, FIN-00014 Helsingin yl iopisto, Finland 2Department of Applied Physics, University of Turku, FIN-20 014 Turun yliopisto, Finland 3Ørsted Laboratory, Universitetsparken 5, DK-2100 Copenha gen Ø, Denmark 4Nordita, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark We have simulated the dynamics of interacting atoms in optic al lattices. The periodic lattice struc- ture is produced with laser beams detuned a few linewidths to the red from the atomic transition. The atoms localize in lattice sites as a steady state is reach ed during the cooling, but they can still to some extent move between the sites. Our Monte Carlo Wave Fu nction simulations show that in such situations the interactions can lead to evaporative co oling of the atoms as the hotter ones are ejected from the lattice. 32.80.Pj, 34.50.Rk, 42.50.Vk, 03.65.-w Neutral atoms can be cooled and trapped in light- induced optical lattices [1]. By controlling the laser ligh t one can adjust the properties of the lattices in order to study e.g. the quantum nature of atomic motion in a pe- riodic structure [2], including the analogues to the behav- ior of electrons in periodic solid state lattices [3]. Ideas regarding the possibility to use optical lattices in atom optics and quantum computation have also emerged re- cently [4–7]. In experiments the trapped gas density is typically very low, providing at best a ﬁlling ratio of 10 % for the near red detuned lattices [1]. Thus it is normally a very good approximation to ignore that the atoms in- teract with each other. In magneto-optical traps for cold atoms the interaction processes are highly inelastic and limit the numbers and temperatures achievable for the atomic gas as densities increase to about 1011atoms/cm3 [8]. By using Bose-Einstein condensates or combining lattices and other types of optical traps it is becoming possible to obtain ﬁlling ratios close to unity and even higher [9]. Applications such as quantum computing re- quire atoms to interact in order to perform quantum log- ical operations [5,6]. Controlled interaction studies could be performed e.g. by superimposing two optical lattices, which can be moved in respect to each other [6]. This, however, does not answer the question of what happens in a basic lattice conﬁguration when the ﬁlling ratio increases, especially when the interactions become an unseparable part of the cooling process, including the localization of atoms at lattice sites. For low densities the atom cloud reaches a thermal equilibrium state, and based on the studies in magneto-optical traps one would expect that inelastic interactions increase the temperature of this equilibrium state as the gas density increases [10]. We have performed Monte Carlo simulations of two atoms in a lattice. They show that (for the parameters of our study) an equilib- rium is not easily obtained. Instead, one observes an evaporative cooling process . The main atomic interaction process in lattices is thattwo cold atoms get close enough to each other to form a long–range quasimolecule [8]. Compared to single atoms, such a molecule interacts diﬀerently with the surrounding laser light, and this interaction depends strongly on the interatomic distance. Previously the atomic interactions in lattices have usually been modelled by assuming ﬁxed positions for both atoms and calculating how the atomic energy levels are shifted by the interaction [11–14]. Such static models ignore the dynamical nature of the inelas- tic processes. But to allow the atoms to move makes the problem complicated and computationally tedious. We present in this Letter a study of interactions in a lattice between moving atoms. Once the dynamical processes are understood, they can be used as input for macro- scopic theories. The distribution of atoms in an optical lattice depends on the choice of laser ﬁeld conﬁguration and the atomic level structure. The laser ﬁeld should have a spatially changing polarization, and the atom needs at least two Zeeman sublevels in the lower energy state, and a diﬀer- ent angular momentum in the upper energy state. The interaction between the laser ﬁeld and an atom gives rise to periodic light-induced potentials for atoms in the Zee- man states of their internal ground states. A single atom moving in such a lattice will undergo Sisyphus cooling because of optical pumping from one ground state to an- other, in a manner that favors the reduction of kinetic energy between the rapid optical pumping cycles [15]. This cooling eﬀect takes place rapidly in lattices created by lasers which are tuned only a few linewidths below the atomic transition (red-detuned bright lattices). Af- ter cooling, the atoms are to a large extent localized in these potential wells. We have chosen as a basis for our studies the simplest atomic transition for a red-detuned laser ﬁeld, i.e., a sys- tem with a lower state angular momentum Jg= 1/2 and an upper state angular momentum Je= 3/2. We de- note the ﬁrst state as the ground state \|g±1/2/angbracketright, the index referring to the quantum number mfor the eigenvalue 1of the zprojection of the angular momentum operator, Jz. Similarly, we denote the second state as the excited state, with eigenstates \|e±3/2/angbracketrightand\|e±1/2/angbracketright. The reso- nance frequency of the transition is ω0. In the numerical calculations we have used the atomic properties of Cs. The laser ﬁeld has periodicity in one dimension, and consists of two linearly polarized counter-propagating beams, with orthogonal linear polarization and frequency ω. For this conﬁguration, the combined laser ﬁeld is E(z, t) =E0(exeikz−ieye−ikz)e−iωt+c.c., (1) where E0is the amplitude and kis the wavenumber. When the interactions become important, the atomic cloud is still relatively dilute so that only two atoms at a time are involved, and the dipole-dipole interaction (DDI) dominates the process. We calculate the two-atom DDI potentials following the procedure described in Ap- pendix A of Ref. [16]. We consider two atoms interact- ing with the laser ﬁeld, coupled to a reservoir, namely the vacuum electric ﬁeld. The system Hamiltonian reads (after rotating wave approximation) Hs=/summationdisplay α=1,2p2 α 2M−¯hδPe,α+V, (2) where the sum over αis over the two atoms, δis the detuning δ=ω−ω0,M= 133 a.u. is the Cs atom mass, andPe,α=/summationtext3/2 m=−3/2\|em/angbracketrightα α/angbracketleftem\|. The potential Vgives the interaction with the laser ﬁeld. The strength of this interaction is given by the Rabi frequency Ω = 2 dE0/¯h where dis the dipole moment of the transition. The system interacts with the reservoir through a dipole coupling between the atoms and the vacuum modes. As we want an expression for the DDI poten- tial, we concentrate our eﬀort on the calculations leading to an expression like Eq. (25) of Ref. [16], which origi- nates from the commutator between the system density operator, ρ, and the DDI potential, Vdip. Also, we con- centrate on spontaneous terms, i.e., terms with vanishing average photon number. Let us introduce the operators Sα +,q=m=1/2/summationdisplay m=−1/2CGq m\|em+q/angbracketrightα α/angbracketleftgm\|, (3) where CGq mare the appropriate Clebsch-Gordan coeﬃ- cients and qis the polarization label in spherical basis. Furthermore, we use a description in terms of a center of mass coordinate Zand a relative coordinate r=r2−r1 (with coordinate along the quantization axis z=z2−z1). With these coordinates, the interaction potential with the laser ﬁeld reads V=−i¯hΩ√ 2sinkZcoskz 2S+,++i¯hΩ√ 2coskZsinkz 2∆S+,+ +¯hΩ√ 2coskZcoskz 2S+,−+¯hΩ√ 2sinkZsinkz 2∆S+,− +h.c., (4)where S+,q=S1 +,q+S2 +,q, and ∆ S+,q=S1 +,q−S2 +,q. In order to calculate the DDI term, we look at the Hamiltonian part of the damping terms in the equation of motion for the system density operator ρ. After ma- nipulations similar to those presented in appendix A of Ref. [16], and using arguments from Ref. [17] to evalu- ate integrals of Bessel functions multiplied with principa l value functions, we ﬁnd the DDI potentials between the two atoms. In the following we look only at atoms on the axis of the laser ﬁeld, i.e., a one-dimensional situa- tion, and in this case, the DDI potential reduces to Vaxis dip=3 8¯hΓ/braceleftbigg1 3cosq0r q0r+ 2/bracketleftbiggsinq0r (q0r)2+cosq0r (q0r)3/bracketrightbigg/bracerightbigg × (S++S−++S+−S−−−2S+0S−0). (5) Here, Γ is the atomic linewidth, q0is the resonant wavenumber q0=ω0/c, and S+qS−q′≡/parenleftbig S1 +,qS2 −,q′+S2 +,qS1 −,q′/parenrightbig . (6) Numerical simulation of the motion of atoms in the lat- tice ﬁeld in one dimension only, using the Monte Carlo Wave Function (MCWF) method [18], is computation- ally very demanding [19]. In order to perform two-atom studies, which require even in one dimension at least two translational degrees of freedom, we have ﬁxed one atom in position, and let the other one move freely. This ﬁxes the relation between the lattice coordinates and the rela- tive interatomic coordinate. Thus an inelastic interactio n process will not change the kinetic energy for both atoms, but we use the relative kinetic energy as an estimate for the kinetic energy change per atom. (We express en- ergy and momentum in recoil units: Er= ¯h2k2/2Mand pr= ¯hkrespectively). We have formulated the problem in the two-atom basis, which leads to a system of 36 internal states. In studies for magneto-optical traps one tends to use the molecu- lar frame, where the atom-atom interactions have been included to the molecular potential structure [20]. How- ever, the quantum jump processes needed for the Monte Carlo method are easier to describe in the atomic basis. One aspect of the simulations is that we do not use the adiabatic elimination of the excited states [21], which is typically employed in order to simplify the equations for atomic motion. For simplicity we neglect Doppler cool- ing. In the molecular frame the system of two interacting atoms is excited resonantly to a molecular state with an attractive interatomic potential (see Fig. 1). This leads to the acceleration of the relative motion of the atoms, until the process terminates with spontaneous decay. We use these attractive potentials for the verbal description of the process but it must be emphasised that they do not directly appear in the two-atom basis. The kinetic energy change due to the attractive potentials also com- plicates greatly the numerical simulations by demanding 2larger momentum and ﬁner spatial grid than in the single atom Sisyphus cooling simulations [22]. We use the laser parameters δ=−3Γ and Ω = 1 .5Γ corresponding to a lattice modulation depth of U0= 584Er. These parameters correspond to a lattice where the atoms move from one lattice site to another on a timescale that is comparable to the timescale of a har- monic oscillation within one of the lattice potential wells . In our selected system the atomic interactions are too weak to really destroy the lattice, so the actual case of interest is the one where the atoms need to be simulta- neously at the same lattice site. In the MCWF method an approximation for the two- atom steady state density matrix is obtained as an en- semble average of diﬀerent wave function histories, for which the spontaneous emission occurs as probabilistic quantum jumps [18]. These quantum jumps (both atoms in our case have six decay channels) occur according to probabilities weighted by the appropriate Clebsch- Gordan coeﬃcients of the decay channels. There are various ways how to calculate the results by ensemble averaging. We take the ensemble average of single his- tory time averages in the steady state time domain [23]. Thus we obtain the kinetic energy per atom, and the spa- tial and momentum probability distributions for various occupation densities ( ρo) of the lattice. A comparison between the number of atoms having gained large kinetic energy via interactions and the total number of interaction processes show (see Table I) that basically every interaction process produces hot atoms in our chosen parameter range. This leads to an evaporative cooling process in the optical lattice: those atoms which are able to move from one well to the other and which have larger kinetic energy than localized atoms leave the trap. A crucial ingredient in the interaction process in- creasing the kinetic energy by a large amount and leading to evaporation is that a large fraction of the population has to enter the attractive molecular excited state during the interaction process. This fraction in turn depends on the relative velocity between the interacting atoms when they reach the resonance point for the attractive molec- ular states. The relative velocity in turn depends on the lattice depth. In our simulations the surroundings is still favorable so that the relative velocity between atoms is low enough to keep the excitation probability high when atoms approach each other and cross the molecular res- onance point. The number of attractive molecular states is ﬁve (with two degenerate ones) and the resonant excitation to these potentials takes place at diﬀerent interatomic distances (see Fig. 1) [20]. If the atoms do not get a large increase in kinetic energy at the ﬁrst resonance they reach, there are still other resonances left. A comparison with semiclas- sical (SC) excitation and survival calculations suggests that the potential which becomes resonant ﬁrst when the atoms approach each other has the dominant role in theinteraction process. When calculating the steady state kinetic energy per atom (Table I), we use two critical wavenumbers kc. Wavefunction histories which at some time point have gained larger total kinetic energy than given by kcare ne- glected in ensemble averaging (considered lost from the lattice). The smallest value of kcwe use [24] is more than two times larger than the semiclassical critical value ksc cgiven in Ref. [15]. The denser the lattice is initially, the larger is the number of interaction processes and the more eﬀective is the evaporative cooling process. This can be seen in the results for kinetic energy per atom using kc= 40 (see Table I). The kinetic energy decreases when the initial density of the lattice increases. Results withkc= 70 include atoms that are lost from the lat- tice, and the value of kinetic energy is slightly above the sparse lattice (non-interacting case) result. The momentum distribution in Fig. 2 shows the eﬀect of the evaporative cooling process clearly. Due to the interactions between atoms part of the population has shifted to the region of large k(wings in Fig. 2) and does not localize back to the lattice because the atoms are above the recapture range. Thus the central peak of the momentum distribution corresponding to atoms localized at lattice sites has a 13% narrower FWHM for an initially dense lattice compared to the non-interacting case. We have shown that at high-density, red-detuned (a few linewidths) optical lattices, atomic interactions should lead to the ejection of the hotter atoms from the lattice. This is because (a) atoms may move from one lattice site to another even in the steady state for Sisy- phus cooling, and (b) because the molecular interaction is strong enough to give to each clearly interacting atom pair almost always enough energy to escape from the lat- tice. Earlier simulations for roughly the same laser (and Cs) parameters in magneto-optical traps have indicated that the dominating eﬀect would be a clear broadening of the atomic momentum distribution, i.e., radiative heat- ing [10]. In both situations high-momentum atoms are produced, but in magneto-optical traps the production is independent of initial relative momentum, whereas in optical lattices atoms with higher momentum are favored (in both cases the fast atoms get involved in more close encounters than the slow ones, but in lattices working in our selected parameter region this fact becomes en- hanced). J.P. and K.-A.S. acknowledge the Academy of Finland (project 43336), NorFA, and Nordita for ﬁnancial sup- port, and the Finnish Center for Scientiﬁc Computing (CSC) for computing resources. J.P. acknowledges sup- port from the National Graduate School on Modern Op- tics and Photonics. 3[1] P. S. Jessen and I. H. Deutsch, Adv. At. Mol. Opt. Phys. 37, 95 (1996); D. R. Meacher, Contemp. Phys. 39, 329 (1998); S. Rolston, Phys. World 11(10), 27 (1998); L. Guidoni and P. Verkerk, J. Opt. B 1, R23 (1999). [2] Y. Castin and J. Dalibard, Europhys. Lett. 14, 761 (1991). [3] G. Birkl et al., Phys. Rev. Lett. 75, 2823 (1995); M. Weidem¨ uller et al.,ibid75, 4583 (1995); Q. Niu et al., ibid76, 4504 (1996); M. Ben Dahan et al.,ibid76, 4508 (1996); S. R. Wilkinson et al.,ibid76, 4512 (1996). [4] C. S. Adams et al., Phys. Rep. 240, 143 (1994). [5] G. K. Brennen et al., Phys. Rev. Lett. 82, 1060 (1999); G. K. Brennen et al., Phys. Rev. A 61, 062309 (2000); I. H. Deutsch et al., quant-ph/0003022. [6] D. Jaksch et al., Phys. Rev. Lett. 82, 1975 (1999); H.-J. Briegel et al., J. Mod. Opt. 47, 415 (2000). [7] A. Hemmerich, Phys. Rev. A 60, 943 (1999). [8] K.-A. Suominen, J. Phys. B 29, 5981 (1996); J. Weiner et al., Rev. Mod. Phys. 71, 1 (1999). [9] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998); D.-I. Choi and Q. Niu, ibid.82, 2022 (1999); M. T. DePue et al.,ibid82, 2262 (1999). [10] M. J. Holland et al., Phys. Rev. Lett. 72, 2367 (1994); Phys. Rev. A 50, 1513 (1994). [11] E. V. Goldstein et al., Phys. Rev. A 53, 2604 (1996). [12] C. Boisseau and J. Vigu´ e, Opt. Commun. 127, 251 (1996). [13] A. M. Guzm´ an and P. Meystre, Phys. Rev. A 57, 1139 (1998). [14] C. Menotti and H. Ritsch, Phys. Rev. A 60, R2653 (1999); Appl. Phys. B 69, 311 (1999). [15] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B6, 2023 (1989); P. J. Ungar et al.,ibid.6, 2058 (1989). [16] G. Lenz and P. Meystre, Phys. Rev. A 48, 3365 (1993). [17] P. R. Berman, Phys. Rev. A 55, 4466 (1997). [18] J. Dalibard et al., Phys. Rev. Lett. 68, 580 (1992); K. Mølmer et al., J. Opt. Soc. Am. B 10, 524 (1993). [19] We use 32 processors of SGI Origin 2000 machine having 128 MIPS R12000 processors. The total memory taken by a simulation is 14 Gb. Generating a single history requires 6 hours of CPU time. [20] P. S. Julienne and J. Vigu´ e, Phys. Rev. A 44, 4464 (1991). [21] K. I. Petsas et al., Eur. Phys. J. D 6, 29 (1999) and references therein. [22] Due to the large kinetic energies involved, the spatial step size ∆ zhas to be smaller than in single atom simulations since the momentum space size Lk= 2π/∆z. We use the split operator Fourier method to solve the non–Hermitian time–dependent Schr¨ odinger equation. The density of the lattice is controlled by changing the length Lzof the nu- merical grid. [23] K. Mølmer and Y. Castin, Quantum Semiclass. Opt. 8, 49 (1996). [24] The semiclassical value ksc c(from Ref. [15]) gives the point where the cooling force has its maximum value, but the cooling is still eﬀective above that point. Our values ofkcgives us a criterion for neglecting energetic histories from the ensemble (i.e., atoms lost from the lattice).TABLE I. Escaped atoms and kinetic energies. The num- ber of MC histories which have been neglected ( Nout) in the ensemble averaging due to escape from the lattice and steady state kinetic energies per atom ( < E k>) for various occu- pation densities ( ρo) of the lattice. Two diﬀerent critical wavenumbers kchave been used. Nint totgives an estimate for the total number of atom–atom interaction processes based on single atom MC collision rate calculations by monitoring the quantum ﬂux at a mean atomic separation given by ρo. The total number of MC histories for each simulation is 128. The absolute values of the standard deviation for the kineti c energies are given in parentheses. ρo(%) Nint tot Nout Nout < E k> < E k> kc= 40 kc= 70 kc= 40 kc= 70 25.0 39 38 26 61(6) 110(18) 20.0 25 27 19 69(5) 99(12) 16.7 19 19 11 80(6) 103(12) 14.3 16 19 12 80(7) 104(12) no interactions 0 0 0 91(8) 91(8) 0.2 0.4 0.6 0.8 1 1.2−15000−10000−50000 z (λ/2π)U (Er)1u 0− g 2u 1g 0u+ FIG. 1. The shifted ground state and the attractive ex- cited state [labeled by Hund’s case (c) notation] molecular potentials for δ=−3.0Γ. The interatomic distance zis ex- pressed in terms of laser wavelength. −60 −40 −20 0 20 40 60 p (pr )\|Ψ\| 225 % no interactions FIG. 2. The steady state momentum probability distribu- tions for a densely populated ( ρo= 25%) lattice and for the non–interacting atoms case (see text). All of the MC histori es are included. Here δ=−3.0Γ, and Ω = 1 .5Γ. 4
/G36/G23 /G20/G21/G22 /G37 /G37 /G38 /G38-5.0-2.50.02.55.0 0 10 20 30 40 50 60-6.0-4.0-2.00.02.04.0a) I2 I1Heater current (mA) Choi et al., Fig. 2 a&bb) ∆V1∆V22V1 2V2 Time (sec)Voltmeter signal ( µV)0.01 0.11.001.011.021.031.041.051.06 c) Choi et al., Fig. 2c TEP vs freq. SAC/SDC freq. (Hz)0 2 4 6 8 10 12 14 16-0.25-0.20-0.15-0.10-0.050.00 S (µV/K) T (K)-2.0-1.00.01.02.03.04.05.0 Choi et al., Fig. 3Sc Sa19.8 19.9 20.0 20.1 20.2 20.3 20.4 0 5 10 15 20 25-15-10-5051015202530 a) B (T)∆V2∆V3∆V1 1 µVvoltmeter signal ( µV) B (T) Magnetothermopower α-(BEDT-TTF)2KHg(SCN)4 Choi et al., Fig. 4 a&b b)S (µV/K)-30-25-20-15-10 -5 0 5 10 15 20 25 30-8.0-6.0-4.0-2.00.02.04.06.0 0 5 10 15 20 25 30-6.0-5.0-4.0-3.0-2.0-1.00.01.02.0 Choi et al., Fig. 5Normal Field Direction Reversed Field Directiona) T=1.4 KSxy (µV/K) Nernst Signal α-(BEDT-TTF)2KHg(SCN)4 b) B (T)Nernst effect ( µV/K) B (T)0 5 10 15 20 25 30-0.8-0.6-0.4-0.20.00.20.40.60.00.10.20.30.40.5 MTEP of Chromel-Au(0.07%Fe)(S(H)-S(0))/S(0) B (T) Choi et al., Fig. 6b)a) MTEP of Au 1.4 K 4.8 K 6.8 KS(H)-S(0) ( µV/K)1Low-frequency method for magnetothermopower and Ner nst effect measurements on single crystal samples at low tempe ratures and high magnetic fields E. S. Choi Research Institute for Basic Sciences, Ewha Womans University, Seoul 120-750, Korea and National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310 J. S. Brooks and J. S. Qualls* National High Magnetic Field Laboratory and Physics Department, Florida State University, Tallahassee, Florida 32310 Y. S. Song Texas Center for Superconductivity, University of Houston, Houston, TX 77204-5932 ABSRACT: We describe an AC method for the measurement of the longitudinal (S xx) and transverse (Sxy, i.e. Nernst) thermopower of mm-size single crystal samples a t low temperatures (T<1 K) and high magnetic fields (B>30 T). A low-frequency (33 mHz) heating method is used to increase the resolution, and to determine the temperature gradient reliably in high magnetic fields. Samples are mounted between two thermal blocks which are heated by a sinusoidal frequency f0 with a π/2 phase difference. The phase difference between two heater currents gives a temperature gradient at 2 f0. The corresponding thermopower and Nernst effect signals are extracted by using a digita l signal processing method due. An important component of the method involves a superconducting link, YBa 2Cu3O7+δ (YBCO) , which is mounted in parallel with sample to remove the bac kground magnetothermopower of the lead wires. The method is demonstrated fo r the quasi two- dimensional organic conductor α-(BEDT-TTF) 2KHg(SCN) 4, which exhibits a complex, magnetic field dependent ground state above 22.5 T at low temperatures. Draft (Thursday, September 14, 2000; 5:38:32 PM) Corresponding Author: Prof. James Brooks, Physics NHMFL/Physics 1800 E. Paul Dirac Dr. Tallahassee FL 32310 USA brooks@magnet.fsu.edu Phone: 1-850-644-2836 (-5038 fax)2I. INTRODUCTION The application of a thermal gradient ( ∆T) across a conducting material leads to a corresponding potential difference, or thermo-electric power (TEP , or thermopower). Thermopower measurements yield information about both thermodynamic proper ties and the transport properties of carriers. Advantages of TEP include the z ero-current nature of the measurement, and its sensitivity to band structure, especially in the case of anisotropic (low dimensional Fermi surface) materials. Following Mott and Jones, the thermopower1 of a metal may be expressed as FEEB dEEd dEEvd dEEnd eTkS= + + = ))(ln)(ln)(ln(32 22τ π (1) where n(E) is the density of the states, v(E) is an average charge velocity, and τ (E) is the carrier scattering relaxation time. As we will show in the present application, the derivative of n(E) at the Fermi energy will lead to large oscillations in sy stems where Landau quantization of the electronic energy levels occurs at hig h magnetic fields2. Hence oscillations in TEP associated with the de Haas van Alphen effect can be observed. In a typical experimental setup, similar to that shown in Fig. 1, a sample is connected between two thermal platforms. A temperature differenc e is applied by heating one of the platforms, and ∆T is measured either between the platforms, or at points on the sample. The electric potential difference ∆V is measured with contact leads on the sample. In general, the apparatus is in weak thermal contact wit h a reference bath at a variable temperature T. As with other such measurements on small samples (e.g. specifi c3heat), the effects of the addenda, and the magnetic field dependence of the sensors, cannot be negelected. Previous magneto-thermopower (hereafter MTEP) m easurements at low temperatures and in high magnetic fields have addressed expe rimental issues such as the magnetothermopower of lead wires3,4. By using well-studied elemental metal wires of copper or gold, and high Tc superconductors (where S = 0 for T << T c and B << B c2), these background contributions may be sorted out. For long, thin samples (mm size samples with 10:1 to 100:1 aspect ratios), an AC technique has been used to measure MTEP for wide range of temperature5. But these techniques cannot be easily adapted to small single crystals with 1:1 aspect ratios, as in the case of the quasi-two dimensional “ET” organic conductors (see results section), where an accurate determination of ∆T in high magnetic fields becomes difficult. Resel et al. introduced a MTEP measurement technique up to 17 Tesla and down to 3K where chromel-constantan thermocouples are used as voltage and ∆T leads simultaneously6. Their technique (as does ours) includes the alternate heating method ("seesaw heating") to increase the m easurement accuracy. However, there are limits in their methods in the case of smal l samples, since the large absolute TEP of chromel wire can introduce substantial background signa l, and the application of a thermocouple junction directly to a small sample, ca n cause complications. The technique to be introduced in this paper utilizes a stable, alter nating heating method at very low frequency. Here the lead wires are in-situ calibrated using a YBa2Cu3O7+δ high Tc superconductor sample as a reference. When combined with digital signal processing methods, our procedure leads to enhanced resolution a nd accuracy for the MTEP of small samples, with direct application to high magnetic field meas urements.4II. EXPERIMENTAL TECHNIQUE A. Measurement setup Fig. 1 shows the schematic diagram of the MTEP and the Nernst effect measurement holder in a top-view, where the magnetic field is applied normal to the plane of the figure. The apparatus is held in a 10 mm diameter cylindrical copper holder that is sealed with a copper cap (with a threaded grease seal). The copper holder can be maintained at any temperature T between 300 K and 0.5 K in a standard cryogenic, high-field dewar arrangement. The integrity of the seal is checked by a small jump in an applied temperature gradient of the apparatus when the encapsulated air i n the holder condenses out below 80 K. Since 3He exchange gas is used to cool the holder, superfluid leaks do not present a problem. Samples are mounted between two quartz blocks ( 2.9 x 2.4 x 1.0 mm3), A and B, with the ends attached using Apiezon N grease7. Electrical contacts to the samples are made by 12.5 µm gold wires using silver (or carbon) paste. The electrical connection between the lead wires and the external wires is kept i n an isothermal condition by thermally anchoring them to the copper holder. Chromel-Au(F e0.07%) thermocouple wires7, used to measure ∆T between the quartz blocks, are attached to the quartz blocks using Stycast 2850 GT epoxy8. To minimize the experimental inaccuracy, which may result from the difference of temperature between the quartz block and the sample, the temperature gradient was produced by heating quartz bl ocks at a low frequency of order 33 mHz. Two chip resistor heaters (220 Ω RuO2 miniature surface- mount resistors) are attached by Stycast 2850 GT epoxy8 to the edge of quartz blocks to enhance the homogeneous heat conduction.5 Two sinusoidal currents are applied by Keithley 220 programmable DC current sources9 with same frequency f0 but with a π/2 phase difference. The relation between the two heater currents and ∆T for ideal system (perfect heat conduction from the heater to the quartz block) is the following: )2sin()(0 0 1 t f ItI π = )2cos() 2 / 2sin()(0 0 0 0 2 t f I t f ItI π ππ =+ = (2) × = −( ∝ ∆ ))2 (2sin(/ /) )(02 02 22 1 tf CRICRIItTp p π where I1, I2 are heater currents, R is heater resistance, Cp is the heat capacity of the quartz block and t is the time. The validity of last expression in Eq. (2), i .e. the assumption of equal power for both heaters, can be checked by the Fourier analysis of the voltage signal of thermocouple wires. If the values of R and Cp are not identical, ∆T will oscillate with frequency of 2 f0 and f0 as follows; )2cos())2 (2sin()2cos()()2sin()(0 0 0 0 t f tf At f At f AtT πδ π πδ π − = +− ∝ ∆ (3) By the spectrum analysis of ∆T through the Fourier transform, two peaks will appear at f=f0 and f=2f0. One can estimate the contribution of the non-identical heat transfe r (~δ/A) from the ratio of amplitudes of peaks. For our holder, the value was found to be about 0.02. Even the contribution from the non-identical heat transfer is substant ial, it can be dealt with by an appropriate analysis of the signal, as disc ussed below. For the present case, we consider only the dominant 2 f0 contribution. Since ∆T oscillates with 2 f0 frequency, corresponding TEP and the Nernst voltage will also oscil late with the same frequency. This second harmonic detection has an advantage in reducing el ectrical cross- talk which may arise from single harmonic generation in the heaters.6 Figs. 2a and 2b show the applied heater currents and the corresponding thermoelectric potential ∆V1 and the thermocouple emf ∆V2 as a function of the time. (∆V1 and ∆V2 refer to voltage leads 1 and 2 as shown in Fig. 1.) ∆V2 , the measured emf of the Chromel-Au(Fe0.07%) thermocouple, is related to ∆T by ∆V2 = - ∆T ×SCh-AuFe + Voffset, where SCh-AuFe is the TEP of Chromel-Au(Fe0.07%) thermocouples7 and Voffset is the offset voltage which lies in the range of 0.1 ~ 0.2 µV at low temperature . ∆V1 and ∆V2 were measured with Keithley 2182 nanovoltmeters9. There is a slight phase difference (~ 14 degrees or about 0.6 s delay) between ∆T and ∆V2 due to the non-ideal heat conduction. However, since the signals are digitally averaged over 2 ½ periods (see below) to obtain the amplitudes of ∆T and ∆V2, the phase difference does not enter into the final TEP value. When ∆T is small compared to the measurement temperature T, the absolute TEP ( Sxx) and Nernst voltage ( Sxy) can be expressed as : ),(),(/ ),(),(),(),(),(),(/ ),(),(),(),(),(),(),( 21211 BTPBTSBTVBTVBTSBTSBTPBTSBTVBTVBTSBTPBTTBTVBTS xy AuFeChxyAu xx AuFeChAu xx xx × =+ × =+ ×∆= −− (4) where SAu is the absolute thermopower of Au and P is either +1 or –1 depending on the phase difference between V1(V3) and V2. SAu can be determined from the measurement of another sample (YBCO in this paper) whose value is known at a certa in temperature and magnetic field. For the Nernst voltage, it is assumed that sample alignment is ideal so that there is no contribution from Sxx. When the misalignment is substantial, Sxy can be7obtained from the difference in the Nernst voltage for two magnet ic field sweeps with opposite polarities. The amplitude of oscillation ( V1 and V2 in Fig. 2) can be determined from the discrete values (∆Vi) by the following formula10 : V H V H Vx i x j= − × ×σ µ22 8 3 2 [ { ( ( ))}] /∆ ∆ (5)-1 where σx2 is the variance, µx is the mean value and H (xi ) is the Hanning window defined by )}#2cos(1 {5 . 0)(elementstheofixHiπ−×= for i = 0, 1, 2, ….., n-1. (5)-2 The Hanning window was used to separate AC signal from DC signa l (Voffset), and hence the signal was compensated for the windowing effect by a multipl icative constant (8/3 in this case). Finally the root mean square value was used to extra ct an amplitude of the AC signal. Because this method does not discriminate oscillations with different frequencies (for example, dominant 2 f0 signal and f0 signal from non-identical heat transfer), this method has an advantage that the non-identical heat transfer term c an be also considered. However, if there is a substantial low frequency noise, one should use digital bandpass filters or FFT analysis to obtain the amplitudes. We note that there have been previous AC TEP measurements using t he 2f0 mode by Kettler et al.11, where limitations due to the heat capacity of the material and the characteristic times of thermal relaxation were identified. To overcome these problems, the excitation frequency 2 f0 should be as small as possible. In our measurement setup, 2 f08was chosen to be 67 mHz, i. e., the oscillation period of the heater cur rents is 30 seconds, and the corresponding oscillation period of ∆T is 15 seconds. Our method for determining the frequency range where the AC and ideal DC methods coincide is shown in Fig. 2c. A suitable excitation frequency range for the apparatus in Fig. 1 i s for 2f0 below 100 mHz. Above 100 mHz, the AC TEP increases as a function of frequency due t o various thermal relaxation rates which are characteristic of the apparatus. B. Measurements in magnetic field The difficulties for the MTEP measurement comes from the fie ld dependence of SAu(B) and SCh-AuFe(B). To avoid the problem of SCh-AuFe (B), we exploit the high reproducibility of ∆T for corresponding constant amplitudes of the heater currents. V2 changes very little with time during a measurement for fixed temperature (typically less than 1 mK over a 20 minute period of measurement). Correspondingly, the change of ∆T is also very small for magnetic field sweeps , with less than 1% deviation in ∆T as compared with the zero field value . The change of ∆T will come from the field dependence of specific heat of the quartz block and the magnetoresistance of the heater resistor. The former is negligible for the quartz block and the latter can be calibrated at each magneti c field. Although the magnetoresistance ( MR = R(B)/R(B=0)) of the chip resistor used in this measurement is very small ( for 30 T : ~ -1.5% at 4K, ~ -1.4% at 1.1K, ~ -2.0 % at 0.7 K ), we also included the MR effect in the determination of ∆T (T, B). Once ∆T(T, B) is determined, SAu(B) can be easily measured using a YBCO sample as a referen ce below its critical field, where SYBCO(B) = 0. Therefore the MTEP and Nernst voltage can be written as9),( ),() 0,() 0,(),(),(),(),( ),() 0,() 0,(),(),( 2321 BTP BTMRBTSBTVBTVBTSBTSBTP BTMRBTSBTVBTVBTS xy AuFeChxyAu xx AuFeChxx × ×===+ × ×=== −− (6) where ),( ),() 0,() 0,(),(0),( 21BTP BTMRBTSBTVBTVBTSYBCOxx AuFeChAu − −× ×==−= (7) for T < Tc and B < Bc of YBCO. III. EXPERIMENTAL RESULTS To demonstrate the techniques described here, we consider the MTEP a nd the Nernst effect of organic conductor α-(BEDT-TTF) 2KHg(SCN) 4 . The material α-(BEDT- TTF)2KHg(SCN) 4 is a well known quasi two-dimensional organic conductor12 which shows metal - density wave transition around T = 8 K, and which has a magnetoresistance anomaly at 22.5 T (below 8 K) where there is a magnetic field i nduced change in the electronic structure. The typical size of the sample is about 1.4×0.6×0.26 mm3 , with a plate-like morphology. The sample is mounted so that the temperature gradient is in the plane of the conducting layers (a or c-axis), with the field perpendicul ar to the conducting layers (b-axis). A polycrystalline YBCO sample, of comparable dimensions, was used for in-situ calibration. The sample holder was attached to the probe of 3He cryostat and a 3010T resistive magnet at the National High Magnetic Field Labor atory was used for the high field measurements. Fig. 3 shows zero field TEP results for the crystallographic a- and c-axis, i.e. ∆T is parallel with a- and c-axis respectively. As mentioned in the introduction, the TEP is sensitive to the anisotropy of the band structure (and therefore the Fermi surface), hence it depends on the direction of ∆T with respect to the crystallographic axes. Structure in the TEP, due to the opening of a partial gap, is clearly seen ar ound 8K for both axes. The sum of the amount of jump is about 1 µV/K, which is in reasonable agreement with the heat capacity measurement results13. For MTEP and the Nernst effect measurement, the magnetic fiel d is swept very slowly (0.042 T/min) for fixed temperatures. Fig 4 (a) shows the ra w data of ∆V1, ∆V2 and ∆V3 for T = 0.7 K. When the heater power is about 220 µW, ∆T is 0.085K at zero field and it is assumed to decrease to 0.083 K at B = 27 T due to the negative MR of the chip resistor. The oscillation of V1 and V3 is huge and the change of polarity ( P(T, B) of Eq. (4) and(6))can be also seen from the raw data.. The derived MTE P data are shown in Fig. 4 (b). Each MTEP data point is obtained using Eq. (5) - (7) by a veraging over ~ 100 raw data points. Fig. 5 shows the Nernst voltage ( Sxy) for positive and negative magnetic field sweep and the corresponding Nernst effect at T = 1.4K. For this c ase, the xy sample electrode alignment is quite good and quantum oscillations can be clea rly seen, even though the final results are obtained by subtracting Sxy (B<0) from Sxy (B>0), and dividing by 2. The corresponding period of quantum oscillations can be obtained from the Fourier11transform, and it is found to be ~ 670 T, which agrees with the value m easured from the magnetoresistance and magnetization experiments14. Finally, we describe the MTEP of the thermocouple (Chromel-AuFe) and electrical (Au) leads used in the present study. This is done under the assumption that ∆T depends only on the heater power (taking into account the negative MR), and assuming that the magnetic field effect on the heat capacity of quartz and stycast epoxy is negligible. Then, for a fixed heater power ( ∆T) and temperature T, ∆V4 on the YBCO sample and ∆V2 on the quartz blocks are measured as a function of magnetic field. SAu(T, B) can then be determined directly, and from Eq. (7), we can determi ne SCh-AuFe(T, B). Fig. 6 shows the results at different temperatures as a funct ion of the magnetic field. For the MTEP of SCh-AuFe, the overall behavior is quite similar to the previous determinations15 in which the MTEP was measured by a different method up to B = 14 T. ACKNOWLEDGEMENTS This work was supported in part by NSF-DMR95-10427 and DMR99-71474. The work was carried out at the National High Magnetic Field Laboratory , supported by a contractual agreement between the State of Florida and the NSF t hrough NSF-DMR-95- 27035. E. S. Choi was financially supported by the KOSEF postdoctoral fel lowship program.12REFERENCES * Present address: Physics Dept., Wake Forest University, Winston-Salem, NC 27109. 1 See Mott and Jones in D. K. C. MacDonald, Thermoelectricity : An Introduction to the Principles (John Wiley & Sons, 1962). 2 F. J. Blatt, P. A. Schroeder and C. L. Folies, Thermoelectric Power of Metals (Plenum, New York, 1976). 3 W. Kang, S. T. Hannahs, L. Y. Ching, R. Upasani and P. M. Chaikin, Phys. Re v. B 45, 13566 (1992). 4 H.-C. Ri, R. Gross, F. Gollnik, A. Beck, and R. P. Huebener, P. Wagner and H . Adrian, Phys. Rev. B 50, 3312 (1994). 5 W. H. Kettler, R. Wernhardt and M. Rosenberg, Rev. Sci. Insrum. 57, 3053 (1986). 6 R. Resel, E. Gratz, A. T. Burkov, T. Nakama, M. Higa and K. Yagasake , Rev. Sci. Instrum. 67, 190 (1996). 7 The Apiezon N grease and Chromel-Au(Fe0.07%) thermocouple wires (0.127 mm diameter) were purchased from LakeShore Cryotronics, Inc. , 575 McCo rkle Blvd., Westerville OH 43082. 8Emerson and Cuming, 46 Manning Road, Billerica, MA 01821. 9Keithley Instruments, Inc., 28775 Aurora Road, Cleveland, OH 44139. 10 Labview Analysis VI Reference Manual , The National Instruments Co. (1998), National Instruments Corporation 11500 N Mopac Expwy , Austin, TX 78759-3504.1311 See Kettler et al., in L. L. Sparks and R. L. Powell, Temperature, Its Measurement and Control in Science and Industry (Instrument Society of America, Pittsburgh, 1972) Vol. IV. Pt. 3, p. 1569. 12 T. Ishiguro, G. Saito and K. Yamaji, Organic Superconductors , Second Edition, (Springer, Berlin 1998). 13 A. Kovalev, H. Mueller and M. V. Kartsovnik, JETP 86, 578 (1998). 14 J. Wosnitza, Fermi Surfaces of Low-Dimensional Organic Metals and Superconductors (Springer-Verlag, Berlin, 1996). 15 H. H. Sample, L. J. Neuringer and L. G. Rubin, Rev. Sci. Instrum. 45, 64 (1974).14Figure captions Fig. 1. Diagram of the measurement holder (the outer diameter of the cylindrical copper holder is 10 mm). A : Cu heat sink, B : quartz blocks, C : heaters. 1 : thermopower leads of sample, 2 : Chromel-Au(Fe0.07%) thermocouples for ∆T leads, 3: Nernst voltage leads of sample, 4 : thermopower leads of reference YBCO sample. Fig. 2. a) Heater currents and b) ∆V1(∆V2) as a function of the time. The period of the heating cycle is 30 seconds and the corresponding periods of oscillation of temperature gradient and thermopower signal are 15 seconds. c) S AC/SDC vs. frequency method used to determine the optimum frequency range where S AC/SDC /G167/G3/G20/G3/G73/G82/G85/G3/G87/G75/G72/G3/G55/G40/G51/G3/G80/G72/G68/G86/G88/G85/G72/G80/G72/G81/G87/G86/G17 Fig. 3. Zero-field thermopower results of α-(BEDT-TTF) 2KHg(SCN) 4 as a function of temperature. A gap opens in the quasi-one dimensional part of the Ferm i surface near 8 K, which is seen as a peak in the a-axis data. Filled circles : ∆T \|\| a-axis, open circles : ∆T \|\| c-axis. Fig. 4. Magnetothermopower. (a) ∆V1, ∆V2 and ∆V3 curves under magnetic field for α- (BEDT-TTF) 2KHg(SCN) 4 at T = 0.7 K. (b) Derived magnetothermopower results. Note the narrow range of field in (a) , which corresponds to only a few qu antum oscillations in (b). Fig. 5. Nernst effect. a) Sxy signal of α-(BEDT-TTF) 2KHg(SCN) 4 for normal and reversed field sweeps at T=1.4 K. The large asymmetry, even in the raw data, indicates a15significant xy component of the thermopower. b) Corresponding Nernst volt age (= (Sxy(B>0)-Sxy(B<0))/2) . Fig. 6. a) ( S(B)-S(B=0)) of Au and b) ( S(B)-S(B=0))/S(B=0) of Chromel-Au(Fe0.07%) thermocouples.
arXiv:physics/0009055v1 [physics.class-ph] 15 Sep 2000Apparent Superluminal Behavior in Wave Propagation A. D. Jackson∗, A. Lande†, and B. Lautrup∗ December 13, 2013 Abstract The apparent superluminal propagation of electromagnetic signals seen in recent experiments is shown to be the result of simple and robust properties of relativistic ﬁeld equations. Althoug h the wave front of a signal passing through a classically forbidden re gion can never move faster than light, an attenuated replica of the si gnal is reproduced “instantaneously” on the other side of the barri er. The reconstructed signal, causally connected to the forerunne r rather than the bulk of the input signal, appears to move through the barr ier faster than light. Recent experimental reports of the propagation of electrom agnetic signals with velocities larger than cin dispersive media [1], in wave guides [2], and in electronic circuits [3] have once again focused attentio n on a subject of long-standing interest. The immediate and wide-spread hos tility which such reports seem to generate among theorists is an understandab le consequence of a ﬁrm belief in the consistency of electromagnetism and re lativity. In order to dispel such concerns at the outset, we distinguish b etween “true” and “apparent” superluminal phenomena. Consider a right-m oving pulse with a wave front located at x0att= 0. A true superluminal phenomenon would permit the observation of some signal at positions x >x 0+ct. True superluminality is not predicted by either Maxwell’s equat ions or by the manifestly covariant Klein-Gordon equation which we shall consider here. Indeed, most recent experimental papers are careful to emph asize their con- sistency with Maxwell’s equations and, hence, do not claim t he observation of ∗The Niels Bohr Institute, University of Copenhagen, Blegda msvej 17, 2100 Copen- hagen,Denmark †Institute for Theoretical Physics, University of Groninge n, Nijenborgh 4, 9747AG Groningen, The Netherlands 1true superluminal eﬀects. Rather, these experiments have d emonstrated the existence of “apparent” superluminal phenomena taking pla ce well behind the light front. In the case of microwave experiments, obser vations generi- cally involve pulses which seem to traverse a “classically f orbidden” region instantaneously. While these results illuminate an intere sting and potentially useful eﬀect, such transmission occurs well behind the ligh t front and does not challenge the wisdom and authority of Maxwell and Einste in. As we shall see below, apparent superluminality is extremely general a nd to be expected. Papers by Sommerfeld and Brillouin [6] represent some of the earliest and most beautiful investigations of the question of superlumi nality. Their con- cern was with unbounded, dispersive media. There were at the time abundant examples of anomalous dispersion, i.e. substances for whic h phase and group velocities were both larger than c. Since the group velocity was then be- lieved to be identical to the velocity of energy propagation , Sommerfeld and Brillouin understandably found the question of superlumin al propagation to be of importance. Their strategy was to write the requisite p ropagator us- ing Laplace transforms and a suitable analytic form for the p hase velocity, v(ω) =ω/k. The fact that the singularities of v(ω) were restricted to the lower halfωplane was then suﬃcient to prove that the signal was necessar - ily zero ahead of the light front and that the light front alwa ys moves with a velocity of c. While the eﬀorts of Sommerfeld and Brillouin would seem to have settled the issue of true superluminal propagation d eﬁnitively, the situation is somewhat more subtle. Their work conclusively demonstrated that Maxwell’s equations preclude superluminal propagati on for media with a causal form of v(ω). It did not, however, extend to a proof that the singu- larities ofv(ω) must lie in the lower half plane. In simple electron resonan ce models of dielectrics, such behavior follows from the absor ptive nature of the material [5]. We are not, however, aware of any completely ge neral proof of material causality. The present paper addresses the current issue of apparent su perluminality. In order to avoid the diﬃcult issue of modeling v(ω), we will restrict our atten- tion to propagation in two-dimensional wave guides with con strictions. For slow variations in the shape of the constriction, Maxwell’s equations reduce to a one-dimensional Klein-Gordon equation, in which the no n-uniformities can be modeled through a suitable potential. A side beneﬁt of this replace- ment is that the strict impossibility of true superluminal p ropagation is easily demonstrated. We then consider the propagation of a wave for m with a pre- cise front initially to the left of a potential barrier locat ed in the interval 0≤x≤b. The barrier is presumed to be high relative to the dominant wave numbers contained in the pulse but is otherwise arbitra ry. Our results 2for the incoming and transmitted waves can then be expressed simply: The incoming wave moves with a uniform velocity of c= 1. The transmitted waveψ(x,t) (forx > b) is attenuated as an obvious consequence of barrier penetration, and its amplitude is proportional to the deriv ative of the initial pulse evaluated at the point ( x−ct−b). The additional displacement, b, suggests instantaneous transmission of the pulse through t he barrier and is the source of the apparent superluminality observed empiri cally. The fact that the transmitted pulse is an image of the derivative of th e original pulse and not the pulse itself is an elementary consequence of the f act that trans- mission amplitudes generally vanish in the limit ω→0. When the signal is a low-frequency modulation of a carrier wave, as is the case i n many exper- imental investigations, the envelopes of the incident and t ransmitted waves are identical. Some of the topics treated here have been discussed elsewher e both analyt- ically and numerically [4, 3]. Our intention is to emphasize the generality and extreme simplicity of this phenomenon. The Model: We consider a scalar wave, Ψ, moving in a two-dimensional wave guide according to the Klein-Gordon equation ∇2Ψ =∂2Ψ ∂t2. (1) The wave guide is inﬁnite in the x-direction and extends from 0 ≤z≤h(x) in thez-direction. We assume that Ψ vanishes at the transverse boun ding surfaces. If his a slowly varying function of x, we can approximate Ψ as the productψ(x,t) sin (πz/h). Neglecting derivatives of h, eqn.(1) reduces to a one-dimensional equation: −∂2ψ ∂x2+V(x)ψ=−∂2ψ ∂t2. (2) The eﬀective potential is determined by the width of the wave guide so that V(x) =π2/h(x)2for the lowest transverse mode. For simplicity, we assume thath(x) is large except in the vicinity of the constriction, so that V(x) can be modeled as non-zero only in the region 0 ≤x≤b. We seek solutions to eqn.(2) which describe the motion of an i nitial pulse, ψ(x,0) =f(x), which has arbitrary shape but satisﬁes the following two conditions. First, the pulse has a well-deﬁned wave front, x0, initially to the left of the barrier, V(x), i.e.,ψ(x,0)/negationslash= 0 only when x≤x0<0. Second, att= 0 the pulse moves uniformly to the right with a velocity of 1, so that 3∂ψ/∂t =−∂ψ/∂x att= 0. For any given potential, this problem can be solved with the aid of the corresponding Green’s function. This model has been considered in detail in ref. [7], where it was shown in generality that the transmitted wave is given by ψ(x,t) =/integraldisplayx0 −∞/tildewideT(x−t−x′)f(x′)dx′, (3) whereT(x) is the retarded transmission kernel, which may be expresse d in terms of its Fourier transform T(ω) /tildewideT(u) =/integraldisplay∞ −∞T(ω)eiωudω 2π. (4) The physical interpretation of T(ω) as a transmission amplitude is elemen- tary: An incoming plane wave exp ( iωx), incident on the potential barrier from the left, leads to a transmitted wave T(ω) exp (iωx). Since \|T(ω)\| →1 for\|ω\| → ∞ , the integration contour in eqn.(4) can be closed in the uppe r halfω-plane forx>0. IfT(ω) is free of singularities in the upper half plane, it follows that /tildewideT(x−t−x′) = 0 forx>x′+tand thus that ψ(x,t) is strictly zero forx>x 0+t. Nothing precedes the light front. A Special Case: The authors of ref. [7] considered the special case where V(x) =m2is a positive constant and found T(ω) =4ωκ Dei(κ−ω)b(5) withκ=i/radicalbig m2−(ω+iǫ)2and D= (ω+κ)2−(ω−κ)2e2iκb. (6) The singularities of T(ω) are due to the zeros of Din theω-plane. Given the form ofκ, the zeros of Dare conﬁned to the lower half plane, and T(ω) is indeed analytic in the upper half plane. As expected, this mo del precludes genuine superluminal propagation. The analytic propertie s ofT(ω) are, of course, dictated by those of V(ω). The general proof that any given potential will lead to such analyticity is more challenging.1All real, local, and bounded 1A general proof can be constructed along the following lines . Write the transmission amplitude as a linear integral equation of the form ψ=ϕ+/integraltext G0V ψ, whereG0is a suitable free propagator. Singularities in ψarise when singularities of the integrand pinch the integration contour. Analyticity of V(ω) in the upper half plane then ensures the desired analyticity properties of T(ω). 4potentials which vanish suﬃciently rapidly as x→ ∞ are expected to respect these analyticity conditions, and the absence of true super luminality is thus to be expected for all physically sensible choices of V. Apparent Superluminal Behaviour: Conﬁdent that our Klein-Gordon model is free of genuine acausal propagation, we turn to appa rent superlumi- nal phenomena. Consider a strong barrier (with mb>> 1) and imagine that the initial wave form, ψ(x,0), is dominated by low frequency components for which \|ω\|<< m . In this case, the form of ψ(x,t) is both simple and intuitive. Speciﬁcally, we need only consider ω≈0 for which κ≈im. In this domain, the transmission amplitude can be approximate d as T(ω)≈ −ω4i me−iωbe−mb. (7) We shall see shortly that this form of the transmission ampli tude is quite general. Using eqn.(7), we see that eqn.(4) reduces to /tildewideT(x−t−x′) =−4 me−mb∂ ∂uδ(u−b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle u=x−t−x′(8) Thus, we ﬁnd that ψ(x,t)≈ −4 me−bmf′(x−t−b). (9) When a pulse dominated by low frequencies components imping es on a strongly repulsive barrier, the transmitted wave is a stron gly attenuated replica of the derivative of the original pulse. The transmi tted pulse ap- pears to traverse the region of the potential barrier in zero time. This is the apparent superluminal phenomenon observed empiricall y. It occurs well behind the light front of the original signal and is not an ind ication of true superluminal propagation. Rather, it is an interference ph enomenon which is in no sense acausal. There is an evident inconsistency between the present assum ption thatψ(x,0) is dominated by low frequency components and our initial ass umption that the signal has a well-deﬁned light front (which necessarily implies the pres- ence of high frequency components). The consideration of si gnals which are the product of, e.g., a gaussian pulse (with clear low-frequ ency dominance) and a step function to impose the light front makes it clear th at the eﬀects of this inconsistency can be made arbitrarily small [7]. 5Carrier Waves: Experiments frequently involve a modulated carrier wave, f(x) = eiω0xF(x), (10) whereF(x) provides a slowly-varying modulation of the carrier. Inse rting this into eqn.(9), the transmitted signal becomes ψ(x,t) =−4 me−bm(iω0F(u) +F′(u))eiω0u/vextendsingle/vextendsingle u=x−t−b(11) Since the Fourier transform of F(x) is presumed to have support only for frequencies \|ω\| ≪ω0, the second term may be ignored. We conclude that the envelope of the pulse, \|ψ(x,t)\|, is unaltered by the transmission. Again, the argument of the right side of eqn.(11) suggests that tran smission of the envelope through the barrier is instantaneous. Generality of the Results: The various factors contributing to the ap- proximate form, eqn.(7), for T(ω) are all of general origin. The factor exp (−iωb) represents the phase diﬀerence between the free plane wave , exp (iωx), at the boundaries of the region of non-zero potential. It w ill al- ways appear. Similarly, the linear vanishing of the transmi ssion amplitude as ω→0 is a feature common to all potentials which do not have a zero -energy bound state.2 The ﬁnal barrier penetration factor is also familiar and is e xpected when- ever there is strong attenuation. Consider the transmissio n amplitude for a strongly repulsive but otherwise arbitrary potential usin g the WKB approx- imation.3The resulting transmission amplitude is readily calculate d, and shows that the factor exp( −mb) is replaced by exp/bracketleftbigg −/integraldisplayb 0/radicalbig V(x)dx/bracketrightbigg . (12) 2Consider scattering from an arbitrary potential which is ze ro except in the interval 0≤x≤bwith an interior solution ϕ(x). Join this interior solution to the right-moving plane wave, exp iω(x−b), atx=bwith the usual requirement of continuity of the wave function and its ﬁrst derivative. In the limit ω→0,ϕ(b) = 1 andϕ′(b) = 0. Similarly, join the interior solution to the linear combination Aexp(iωx) +Bexp(−iωx) atx= 0. Ifϕ′(0)/negationslash= 0, the coeﬃcients AandBwill diverge like 1 /ω. The transmission amplitude, T(ω) =e−iωb/A, will vanish linearly with ωunlessϕ′(0) is zero. The condition that ϕ′(0) =ϕ′(b) = 0 is precisely the condition that the potential should sup port a zero- energy bound state. 3We consider a potential which is strongly repulsive for all 0 ≤x≤band zero elsewhere. Hence, it is appropriate to match the plane wave solutions di rectly to the WKB wave function. 6A localized repulsive barrier of suﬃcient strength will tra nsmit an instan- taneous image of the derivative of the incoming signal accor ding to eqn.(9) independent of the details of both the potential barrier and the pulse. Ap- parent superluminal behavior is a robust phenomenon. The Time Delay: The above results can also be expressed as a time delay of the pulse, τ, deﬁned as the diﬀerence between the time actually required for transmission across the barrier less the time required f or a free wave to travel the same distance. In the case of the square barrier an d a low frequency pulse,τ=−b. Negative values of τcorrespond to apparent superluminal propagation. For a modulated carrier wave, eqn.(10), one ma y expandT(ω) about the carrier frequency, ω0, and obtain T(ω)≈ \|T(ω0)\|eiΦ(ω0)ei(ω−ω0)Φ′(ω0), (13) where Φ(ω) is the phase of T(ω). The second exponential factor gives rise to the time delay, τ(ω0) = Φ′(ω0), through the Fourier exponential in eqn.(4). Familiar results from quantum mechanics for purely repulsi ve potentials re- mind us that Φ( ω) is less than 0 for all ωand that Φ(0) = Φ( ∞) = 0.4The time delay is necessarily negative for suﬃciently small ω0and apparent su- perluminal eﬀects can be observed for all repulsive potenti als. The time delay changes sign for some value of ω0comparable to the height of the potential barrier, and it approaches zero from above as ω0tends to inﬁnity. Apparent superluminality is a very general phenomenon. Conclusions: Using the model of a Klein-Gordon equation with a poten- tial, we have presented a simple description of the apparent superluminal phenomena seen in wave guides: Low frequency waves seem to tr averse such barriers in zero time. Strongly repulsive barriers always t ransmit an attenu- ated image of the spatial derivative of the incident signal. When the signal consists of a modulated carrier wave, the envelope of this wa ve is transmitted unaltered. We have attempted to demonstrate that the phenom ena described here are both extremely general and non-controversial. The ir experimental observation does not challenge received wisdom and in no sen se compromises our conﬁdence in general notions of causality. It will be int eresting to see whether this interesting and general consequence of wave th eory will have practical applications. 4This second result is a consequence of Levinson’s theorem wh ich relates the asymptotic behavior of the phase shift to the number of bound states supp orted by the potential, n, through Φ(0) −Φ(∞) =nπ. There are no bound states for purely repulsive potentials. 7References [1] Wang, L. J., Kuzmich, A. & Dogariu, A., Nature 406(2000) 277; and references therein. [2] D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84(2000) 4830; and references therein. [3] Mitchell, M. W., and Chiao, R. Y., American Journal of Phy sics66 (1998) 14. [4] Emig, T., Phys. Rev. E54(1996) 5780. [5] Jackson, J. D., Classical Electrodynamics ,(Wiley, New York, 1999), p. 310. [6] A. Sommerfeld, Ann. Physik 44(1914) 177; L. Brillouin, Ann. Physik 44 (1914) 203. See also, Stratton, J.A., Electromagnetic Theory , (McGraw- Hill, New York, 1941). [7] J. M. Deutch and F. E. Low, Ann. of Physics 228(1993) 184. 8
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/B4 /CX/D7 /D8/CW/CT /D3/D1/D4/D0/CT/DC /D3/D2/CY/D9/CV/CP/D8/CT/CS /D8/CT/D6/D1/B5/BM >/D6/CT/D7/D8/CP/D6/D8/BM >/DB/CX/D8/CW/B4/D4/D0/D3/D8/D7/B5/BM >/DB/CX/D8/CW/B4/BW/BX/D8/D3/D3/D0/D7/B5/BM >/BT/C1 /BM/BP /BT/BC/B6/CT/DC/D4/B4/C1/B6/B4/D3/D1/CT/CV/CP/B6/D8/B7/CZ/B6/DC/B5/B5/B7 /BN AI:=A0e(I(ωt+kx))+cc/BE/CC/CW/CT/D2 /D8/CW/CT /D6/CT/AT/CT /D8/CT/CS /DB /CP /DA /CT /B4/D2/D3/D6/D1/CP/D0 /CX/D2 /CX/CS/CT/D2 /CT /CP/D2/CS /CU/D9/D0/D0 /D6/CT/AT/CT /D8/CX/DA/CX/D8 /DD /CP/D6/CT /D7/D9/D4/B9/D4 /D3/D7/CT/CS/B5 /CX/D7/BM >/BT/CA /BM/BP /BT/BC/B6/CT/DC/D4/B4/C1/B6/B4/D3/D1/CT/CV/CP/B6/D8/B9/CZ/B6/DC/B7/C8/CX/B5/B5/B7 /BN AR:=A0e(I(ωt−kx+π))+cc/DB/CW/CT/D6/CTπ /CX/D7 /D8/CW/CT /D4/CW/CP/D7/CT /D7/CW/CX/CU/D8 /CS/D9/CT /D8/D3 /D6/CT/AT/CT /D8/CX/D3/D2/BA /BT/D2 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2/CX/D2 /CX/CS/CT/D2 /D8 /CP/D2/CS /D6/CT/AT/CT /D8/CT/CS /DB /CP /DA /CT/D7 /D6/CT/D7/D9/D0/D8/D7 /CX/D2 > /D3/D2/DA/CT/D6/D8/B4/BT/C1/B7/BT/CA/B8/D8/D6/CX/CV/B5/BM >/CT/DC/D4/CP/D2/CS/B4/B1/B5/BM >/CU/CP /D8/D3/D6/B4/B1/B5/BN −2A0sin(ωt) sin(kx) + 2IA0cos(ωt) sin(kx) + 2cc/CC/CW/CX/D7 /CX/D7 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /D7/D8/CP/D2/CS/CX/D2/CV /DB /CP /DA /CT/BM >/CP/D2/CX/D1/CP/D8/CT/B4/D7/CX/D2/B4/DC/B5/B6/D7/CX/D2/B4/D8/B5/B8 /DC/BP/BC/BA/BA/BE/B6/C8/CX/B8/D8/BP/BC/BA/BA/BE/B6/C8/CX/B8 /CP/DC/CT/D7/BP/CQ/D3/DC/CT/CS/B8/CK >/D8/CX/D8/D0/CT/BP/CO/CB/D8/CP/D2/CS/CX/D2/CV /DB/CP/DA/CT/CO/B8 /D3/D0/D3/D6/BP/D6/CT/CS/B8/CU/D6/CP/D1/CT/D7/BP/BG/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/B1/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BD/B8 /D4/CP/CV/CT /BG/BC/C6/D3/D8/CT/B8 /D8/CW/CP/D8 /CP /DB /CP /DA /CT /D2/D3 /CS/CT /D0/CX/CT/D7 /D3/D2 /CP /D7/D9/D6/CU/CP /CT /B4/D4 /D3/CX/D2 /D8 /DC /BP/BC/B5/BA /CC/CW/CT /D7/CX/D1/CX/D0/CP/D6 /D7/CX/D8/D9/CP/D8/CX/D3/D2/D8/CP/CZ /CT/D7 /D4/D0/CP /CT /CX/D2 /D8/CW/CT /D0/CP/D7/CT/D6 /D6/CT/D7/D3/D2/CP/D8/D3/D6/BA /BU/D9/D8 /D8/CW/CT /D0/CP/D7/CT/D6 /D6/CT/D7/D3/D2/CP/D8/D3/D6 /D3/D2/D7/CX/D7/D8/D7 /D3/CU /D8 /DB /D3 /B4/D3/D6/D1/D3/D6/CT/B5 /D1/CX/D6/D6/D3/D6/D7 /CP/D2/CS /D8/CW/CT /D7/D8/CP/D2/CS/CX/D2/CV /DB /CP /DA /CT /CX/D7 /CU/D3/D6/D1/CT/CS /CS/D9/CT /D8/D3 /D6/CT/AT/CT /D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT /CT/CP /CW/D1/CX/D6/D6/D3/D6/BA /CB/D3/B8 /D8/CW/CT /DB /CP /DA /CT /CX/D2 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CX/D7 /D8/CW/CT /D7/D8/CP/D2/CS/CX/D2/CV /DB /CP /DA /CT /DB/CX/D8/CW /D8/CW/CT /D2/D3 /CS/CT/D7/D4/D0/CP /CT/CS /D3/D2 /D8/CW/CT /D1/CX/D6/D6/D3/D6/D7/BA /CB/D9 /CW /DB /CP /DA /CT/D7 /CP/D6/CT /CP/D0/D0/CT/CS /CP/D7 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D0/CP/D7/CT/D6 /D1/D3 /CS/CT/D7 /BA/CC/CW/CT /D0/CP/D7/CT/D6 /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CP/D2 /D3/D2 /D8/CP/CX/D2 /CP /D0/D3/D8 /D3/CU /D1/D3 /CS/CT/D7 /DB/CX/D8/CW /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7/B4/CQ/D9/D8 /CX/D8/D7 /D2/D3 /CS/CT/D7 /CW/CP /DA /CT /D8/D3 /D0/CX/CT /D3/D2 /D8/CW/CT /D1/CX/D6/D6/D3/D6/D7/AX/B5 /CP/D2/CS /D8/CW/CT/D7/CT /D1/D3 /CS/CT/D7 /CP/D2 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/BA /C4/CT/D8/D7/D9/D4/D4 /D3/D7/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D1/D3 /CS/CT/D7 /CP/D6/CT /D2 /D9/D1 /CQ /CT/D6/CT/CS /CQ /DD /D8/CW/CT /CX/D2/CS/CT/DC /D1 /BA /C1/D2 /CU/CP /D8/B8/DB /CT /CW/CP /DA /CT /C5 /CW/CP/D6/D1/D3/D2/CX /D3/D7 /CX/D0/D0/CP/D8/D3/D6/D7 /DB/CX/D8/CW /D8/CW/CT /D4/CW/CP/D7/CT /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /DD /CS/CX/AR/CT/D6/CT/D2 /CT/D7 /CSφ/CP/D2/CS /CSω /B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D0/DD /BA /C4/CT/D8 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D1/D3 /CS/CT/D7 /CX/D7 /BT/BC /BA >/D1/D3 /CS/CT /BM/BP /BD/BB/BE/B6/BT/BC/B6/CT/DC/D4/B4/C1/B6/B4/D4/CW/CX/BC/B7/D1/B6/CS/D4/CW/CX/B5/B7/C1/B6/B4/D3/D1/CT/CV/CP/BC/B7/D1/B6/CS/D3/D1/CT/CV/CP /B5/B6/D8/B5 \/B7 /BN /AZ /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D8/CW/CT /D1/D3/CS/CT /D2/D9/D1/CQ/CT/D6/CT/CS /CQ/DD /CX/D2/CS/CT/DC /D1 mode :=1 2A0e(I(φ0+mdφ)+I(ω0+mdω)t)+cc/C0/CT/D6/CTφ /BC /CP/D2/CSω /BC /CP/D6/CT /D8/CW/CT /D4/CW/CP/D7/CT /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /D8/CW/CT /CT/D2 /D8/D6/CP/D0 /D1/D3 /CS/CT/B8 /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS/CX/D2/CV/D0/DD /BA /CC/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D7/CT /D1/D3 /CS/CT/D7 /D4/D6/D3 /CS/D9 /CT/D7 /D8/CW/CT /DB /CP /DA /CT /D4/CP /CZ /CT/D8/BM >/D4/CP /CZ /CT/D8 /BM/BP /D7/D9/D1/B4/D1/D3 /CS/CT/B9 /B8/D1/BP/B9/B4/C5/B9/BD/B5/BB/BE/BA/BA/B4/C5/B9/BD/B5/BB/BE/B5 \/B7 /BN/AZ/CX/D2/D8/CT/D6/CU/CT/D6/CT/D2 /CT /D3/CU /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D1/D3/CS/CT/D7 /DB/CX/D8/CW /D3/D2/D7/D8/CP/D2/D8 /D4/CW/CP/D7/CT/D7 packet :=1 2A0e(I(φ0+(1/2M+1/2)dφ)+I(ω0+(1/2M+1/2)dω)t) e(Idφ)e(Idωt)−1 −1 2A0e(I(φ0+(−1/2M+1/2)dφ)+I(ω0+(−1/2M+1/2)dω)t) e(Idφ)e(Idωt)−1+cc/BF/C6/D3 /DB/B8 /DB /CT /CP/D2 /CT/DC/D8/D6/CP /D8/D7 /D8/CW/CT /D8/CT/D6/D1 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT /CU/CP/D7/D8 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /CT/D2 /D8/D6/CP/D0/B4Ꜽ /CP/D6/D6/CX/CT/D6Ꜽ/B5 /CU/D6/CT/D5/D9/CT/D2 /DDω /BC /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/BA /CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D6/CT/D7/D9/D0/D8/CX/D7 /D8/CW/CT /D4/CP /CZ /CT/D8/B3/D7 /CT/D2 /DA /CT/D0/D3/D4 /CT /B4/CX/D8/D7 /D7/D0/D3 /DB/D0/DD /DA /CP/D6/DD/CX/D2/CV /CP/D1/D4/D0/CX/D8/D9/CS/CT/B5/BM >/CT/D2 /DA /CT/D0/D3/D4 /CT /BM/BP /CT/DC/D4/CP/D2/CS/B4/B4/D4/CP /CZ /CT/D8 /B9 /B5/BB/CT/DC/D4/B4/C1/B6/B4/D8/B6/D3/D1/CT/CV/CP/BC/B7/D4/CW/CX/BC/B5/B5/B5/BN/AZ /D7/D0/D3/DB/D0/DD /DA/CP/D6/DD/CX/D2/CV /CT/D2/DA/CT/D0/D3/D4/CT /D3/CU /D8/CW/CT /DB/CP/DA/CT /D4/CP /CZ/CT/D8 envelope :=1 2A0e(1/2IdφM)e(1/2Idφ)e(1/2ItdωM)e(1/2Idωt) e(Idφ)e(Idωt)−1 −1 2A0e(−1/2IdφM)e(1/2Idφ)e(−1/2ItdωM)e(1/2Idωt) e(Idφ)e(Idωt)−1/C1/D8 /CX/D7 /D3/CQ /DA/CX/D3/D9/D7/D0/DD /B8 /D8/CW/CP/D8 /D8/CW/CX/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CP/D2 /CQ /CT /D3/D2 /DA /CT/D6/D8/CT/CS /CX/D2 /D8/D3 /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CU/D3/D6/D1/BM >/CT/D2 /DA /CT/D0/D3/D4 /CT /BM/BP /BD/BB/BE/B6/BT/BC/B6/D7/CX/D2/CW/B4/BD/BB/BE/B6/C1/B6/C5/B6/B4/CS/D4/CW/CX/B7/D8/B6/CS/D3/D1/CT/CV/CP/B5/B5/BB \/D7/CX/D2/CW/B4/BD/BB/BE/B6/C1/B6/B4/CS/D4/CW/CX/B7/D8/B6/CS/D3/D1/CT/CV/CP/B5/B5/BN envelope :=1 2A0sin(1 2M(dφ+dωt)) sin(1 2dφ+1 2dωt)/CC/CW/CT /D7/D5/D9/CP/D6/CT /D3/CU /D8/CW/CT /CT/D2 /DA /CT/D0/D3/D4 /CT/B3/D7 /CP/D1/D4/D0/CX/D8/D9/CS/CT /B4/CX/BA /CT/BA /CP /AS/CT/D0/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD/B5 /CX/D7 /CS/CT/D4/CX /D8/CT/CS/CX/D2 /D8/CW/CT /D2/CT/DC/D8 /AS/CV/D9/D6/CT /CU/D3/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /C5 /BA >/D7/D9/CQ/D7/B4/DF/C5/BP/BH/B8/BT/BC/BP/BD/B8/CS/D4/CW/CX/BP/BC/BA/BD/B8/CS/D3 /D1/CT/CV/CP /BP/BC/BA/BD /DH/B8/BE/B6 /CT/D2/DA/CT /D0/D3/D4/CT /CM/BE/B5 /BM >/D7/D9/CQ/D7/B4/DF/C5/BP/BE/BC/B8/BT/BC/BP/BD/B8/CS/D4/CW/CX/BP/BC/BA/BD/B8 /CS/D3/D1/CT /CV/CP/BP/BC /BA/BD/DH/B8 /BE/B6/CT/D2 /DA/CT/D0/D3 /D4/CT/CM /BE/B5/BM >/D4/D0/D3/D8/B4/DF/B1/B8/B1/B1/DH/B8/D8/BP/B9/BK/BC/BA/BA/BK/BC/B8/CP/DC/CT/D7/BP/CQ /D3 /DC/CT/CS/B8 /D8/CX/D8/D0/CT/BP/CO/D6/CT/D7/D9/D0/D8 /D3/CU /D1/D3 /CS/CT/D7 /CX/D2 /D8/CT/D6/B9/CU/CT/D6/CT/D2 /CT/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/B8 /D4/CP/CV/CT /BG/BD/C7/D2/CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2 /CT /D3/CU /D1/D3 /CS/CT/D7 /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT/D7/BA /CC/CW/CT /CX/D2 /D8/CT/D6/DA /CP/D0 /CQ /CT/D8 /DB /CT/CT/D2 /D4/D9/D0/D7/CT/D7 /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /BEπ /BB /CSω /BA /CC/CW/CT /CV/D6/D3 /DB/D8/CW /D3/CU/C5 /CS/CT /D6/CT/CP/D7/CT/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /BEπ /BB /C5/B6/CSω /CP/D2/CS /D8/D3 /CX/D2 /D6/CT/CP/D7/CT/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD M2∗A02/BA /CC/CW/CT /D0/CP/D7/D8 /CX/D7 /D8/CW/CT /D3/D2/D7/CT/D5/D9/CT/D2 /CT /D3/CU /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D6/CT/D0/CP/D8/CX/D3/D2/BM >/AZ /D1/CP/DC/CX/D1/CP/D0 /CU/CX/CT/D0/CS /CX/D2/D8/CT/D2/D7/CX/D8/DD >/B4/D0/CX/D1/CX/D8/B4/D7/CX/D2/B4/C5/BB/BE/B6/DC/B5/BB/D7/CX/D2/B4/DC/BB/BE/B5 /B8/DC/BP/BC /B5/B6/BT/BC /B5/CM/BE/BN M2A02/C1/D2 /D8/CW/CX/D7 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D2/CT/CX/CV/CW /CQ /D3/D6/CX/D2/CV /D1/D3 /CS/CT/D7 /CX/D7 /D3/D2/B9/D7/D8/CP/D2 /D8/BA /CB/D9 /CW /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CP/D9/D7/CT/D7 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D7/CW/D3/D6/D8 /CP/D2/CS /CX/D2 /D8/CT/D2/D7/CT /D4/D9/D0/D7/CT/D7/BA/BU/D9/D8 /CX/D2 /D8/CW/CT /D6/CT/CP/D0/CX/D8 /DD /B8 /D8/CW/CT /D0/CP/D7/CT/D6 /D1/D3 /CS/CT/D7 /CP/D6/CT /D2/D3/D8 /D0/D3 /CZ /CT/CS/B8 /CX/BA /CT/BA /D8/CW/CT /D1/D3 /CS/CT/D7 /CP/D6/CT /D8/CW/CT/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /D8/CW/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CP/D2/CS /CP /CX/CS/CT/D2 /D8/CP/D0 /D4/CW/CP/D7/CT/D7/BA /C1/D2 /D8/CW/CX/D7 /CP/D7/CT/BM >/C5 /BM/BP /BE/BC/BM/AZ /BE/BC /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D1/D3/CS/CT/D7 >/BT/BC /BM/BP /BD/BM/AZ /D3/D2/D7/D8/CP/D2/D8 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D1/D3/CS/CT >/CS/D3/D1/CT/CV/CP /BM/BP /BC/BA/BD/BM/AZ /D3/D2/D7/D8/CP/D2/D8 /CU/D6/CT/D5/D9/CT/D2 /DD /CS/CX/CU/CU/CT/D6/CT/D2 /CT >/D4/CW/CX/BC /BM/BP /BC/BM/AZ /D4/CW/CP/D7/CT /D3/CU /D8/CW/CT /CT/D2/D8/D6/CP/D0 /D1/D3/CS/CT >/D3/D1/CT/CV/CP/BC /BM/BP /BD/BM/AZ /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /CT/D2/D8/D6/CP/D0 /D1/D3/CS/CT >/D1/D3/CS/CT /BM/BP /BC/BM >/CU/D3/D6 /D1 /CU/D6/D3/D1 /B9/B4/C5/B9/BD/B5/BB/BE /D8/D3 /B4/C5/B9/BD/B5/BB/BE /CS/D3/BM >/CS/CX/CT /BM/BP /D6/CP/D2/CS/B4/BI/B5/BM/BG>/CS/D4/CW/CX /BM/BP /CS/CX/CT/B4/B5/BM/AZ /CP /CX/CS/CT/D2/D8/CP/D0 /D4/CW/CP/D7/CT /CS/CX/CU/CU/CT/D6/CT/D2 /CT /CQ/CT/D8/DB/CT/CT/D2 /D1/D3/CS/CT/D7 >/D1/D3/CS/CT /BM/BP /D1/D3/CS/CT/B7/BT/BC/B6 /D3/D7/B4/D4/CW/CX/BC/B7/D1/B6/CS/D4/CW/CX/B7/B4 /D3/D1/CT/CV /CP/BC/B7/D1 /B6/CS/D3/D1 /CT/CV/CP /B5/B6/D8/B5 /BM >/D3/CS/BM >/D4/D0/D3/D8/B4/D1/D3/CS/CT/B8/D8/BP/B9/BK/BC/BA/BA/BK/BC/B8 /CP/DC/CT/D7/BP/CQ/D3/DC/CT/CS/B8 /CK >/D8/CX/D8/D0/CT/BP/CO/D6/CT/D7/D9/D0/D8 /D3/CU /D1/D3/CS/CT/D7 /CX/D2/D8/CT/D6/CU/CT/D6/CT/D2 /CT/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BF/B8 /D4/CP/CV/CT /BG/BE/CC/CW /D9/D7/B8 /D8/CW/CT /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2 /CT /D3/CU /D8/CW/CT /D9/D2/D0/D3 /CZ /CT/CS /D1/D3 /CS/CT/D7 /D4/D6/D3 /CS/D9 /CT/D7 /D8/CW/CT /CX/D6/D6/CT/CV/D9/D0/CP/D6 /AS/CT/D0/CS/CQ /CT/CP/D8/CX/D2/CV/D7/B8 /CX/BA /CT/BA /D8/CW/CT /D2/D3/CX/D7/CT /D7/D4/CX/CZ /CT/D7 /DB/CX/D8/CW /CP /CS/D9/D6/CP/D8/CX/D3/D2 /DI /BD/BB/B4/C5/B6/CSω /B5 /BA/CF/CW/CP/D8 /CP/D6/CT /D8/CW/CT /D1/CT/D8/CW/D3 /CS/D7 /CU/D3/D6 /D8/CW/CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV/BR /BY/CX/D6/D7/D8/D0/DD /B8 /D0/CT/D8 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT/D7/CX/D1/D4/D0/CT/D7/D8 /D1/D3 /CS/CT/D0 /D3/CU /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D4 /CT/D6/CX/D3 /CS/CX /CP/D0/CU/D3/D6 /CT/BA >/AZ /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D4 /CT/D6/CX/D3 /CS/CX /CP/D0 /CU/D3/D6 /CT /B4/CS/CT/D0/D8/CP /CP/D2/CS /D4/CW/CX /CP/D6/CT /D8/CW/CT/CU/D6/CT/D5/D9/CT/D2 /DD /CP/D2/CS /D4/CW/CP/D7/CT /D3/CU /D8/CW/CT /CU/D3/D6 /CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D0/DD/B5/CS/CX/CU/CU/CN/CT/D5/D9/CP/D8/CX/D3/D2/BM/BP/CS/CX/CU/CU/B4/DD/B4/D8/B5/B8/D8/B0 /BE/B5/B7/D3 /D1/CT/CV/CP /CM/BE/B6/DD /B4/D8/B5/BP /D3/D7/B4 /CS/CT/D0/D8 /CP/B6/D8/B7 /D4/CW/CX/B5 /BN >/CS/D7/D3/D0/DA/CT/B4/DF/CS/CX/CU/CU/CN/CT/D5/D9/CP/D8/CX/D3/D2/B8 /DD/B4/BC/B5/BP/BD/B8 /BW/B4/DD/B5/B4/BC/B5/BP/BC/DH/B8/DD/B4/D8/B5/B5/BM >/D3/D7 /CX/D0/D0/BD /BM/BP /D3/D1/CQ/CX/D2/CT/B4/B1/B5/BN diﬀ /CNequation := (∂2 ∂t2y(t)) +ω2y(t) = cos(δt+φ) oscill1 := y(t) = (−2ωcos(δt+φ) +ωcos(−ωt+φ) +ωcos(ωt+φ)−2 cos(ωt)ω3+ 2ωcos(ωt)δ2 −δcos(−ωt+φ) +δcos(ωt+φ)) /(−2ω3+ 2ωδ2) >/CP/D2/CX/D1/CP/D8/CT/B4/D7/D9/CQ/D7/B4/DF/D3/D1/CT/CV/CP/BP/BD/B8/CS/CT/D0/D8/CP/BP/BC/BA/BD /DH/B8/D7/D9/CQ/D7/B4/D3/D7 /CX/D0/D0/BD/B8/DD/B4/D8/B5/B5/B5/B8 \/D8/BP/BC/BA/BA/BD/BC/BC/B8/D4/CW/CX/BP/BC/BA/BA/BE/B6/C8/CX/B8 /D3/D0/D3/D6/BP/D6/CT/CS/B8/D7/D8 /DD/D0/CT/BP/D4 /D3/CX/D2 /D8/B8 /CP/DC/CT/D7/BP/CQ /D3 /DC/CT/CS/B8/CU/D6/CP/D1/CT/D7/BP/BG/B5/BM/CS/CX/D7/D4/D0/CP/DD/B4/B1/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BG/B8 /D4/CP/CV/CT /BG/BF/CF /CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT /CP/D9/D7/CT/D7 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /D8/CW/CT /CP/CS/CS/CX/B9/D8/CX/D3/D2/CP/D0 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7/BM ω /B7δ /CP/D2/CSω /B9δ /BA /C1/CUδ /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D6/D1/D3 /CS/CT /CX/D2 /D8/CT/D6/DA /CP/D0/B8 /D8/CW/CT/CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D3/CU /D1/D3 /CS/CT /D4/D0/CP /DD/D7 /CP /D6/D3/D0/CT /D3/CU /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2 /CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT /CU/D3/D6/D8/CW/CT /D2/CT/CX/CV/CW /CQ /D3/D6/CX/D2/CV /D1/D3 /CS/CT/D7/BA /C4/CT/D8 /D3/D2/D7/CX/CS/CT/D6 /D7/D9 /CW /D6/CT/D7/D3/D2/CP/D2 /D8 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT/D3/CU /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2 /CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT/BM >/AZ /D6/CT/D7/D3/D2/CP/D2 /CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT >/CS/CX/CU/CU/CN/CT/D5/D9/CP/D8/CX/D3/D2/BM/BP/CS/CX/CU/CU/B4/DD/B4/D8/B5/B8/D8 /B0/BE/B5/B7 /D3/D1/CT/CV /CP/CM/BE/B6 /DD/B4/D8/B5 /BP/CK > /D3/D7/B4/D3/D1/CT/CV/CP/B6/D8/B7/D4/CW/CX/B5/BN >/CS/D7/D3/D0/DA/CT/B4/DF/CS/CX/CU/CU/CN/CT/D5/D9/CP/D8/CX/D3/D2/B8/DD/B4/BC/B5 /BP/BD/B8/BW /B4/DD/B5/B4 /BC/B5/BP/BC /DH/B8/DD/B4 /D8/B5/B5/BM >/D3/D7 /CX/D0/D0/BE /BM/BP /D3/D1/CQ/CX/D2/CT/B4/B1/B5/BN diﬀ /CNequation := (∂2 ∂t2y(t)) +ω2y(t) = cos(ωt+φ) oscill2 := y(t) = 1 4cos(ωt+φ) + 2ωtsin(ωt+φ)−cos(−ωt+φ) + 4 cos(ωt)ω2 ω2/BH/CC/CW/CT /D8/CT/D6/D1/B8 /DB/CW/CX /CW /CX/D7 /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3 /D8 /B4Ꜽ/D7/CT /D9/D0/CP/D6Ꜽ /D8/CT/D6/D1/B5/B8 /CT/D5/D9/CP/D0/CX/DE/CT/D7 /D8/CW/CT /D4/CW/CP/D7/CT/D3/CU /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT /D4/CW/CP/D7/CT /D3/CU /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT/BA /C1/D8 /CX/D7 /D8/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /D1/D3 /CS/CT/D0/D3/CU /CP /D7/D3/B9 /CP/D0/D0/CT/CS /CP /D8/CX/DA/CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /BA /C0/CT/D6/CT /D8/CW/CT /D6/D3/D0/CT /D3/CU /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT /CP/D2 /CQ /CT/D4/D0/CP /DD /CT/CS /CQ /DD /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/D6 /D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/D3/D6/BA /C5/CP/CX/D2 /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6/D8/CW/CX/D7 /D1/D3 /CS/D9/D0/CP/D8/D3/D6 /CX/D7 /D8/CW/CT /CT/D5/D9/CP/D0/CX/D8 /DD /D3/CU /D8/CW/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D6/D1/D3 /CS/CT/CU/D6/CT/D5/D9/CT/D2 /DD /CX/D2 /D8/CT/D6/DA /CP/D0 /D8/CW/CP/D8 /CP/D9/D7/CT/D7 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2 /D8 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D1/D3 /CS/CT/D7 /CP/D2/CS/B8/CP/D7 /D3/D2/D7/CT/D5/D9/CT/D2 /CT/B8 /D8/CW/CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV/BA/CC/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D1/CT /CW/CP/D2/CX/D7/D1/B8 /CP /D4 /CP/D7/D7/CX/DA/CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /B8 /CX/D7 /D4/D6/D3 /CS/D9 /CT/CS /CQ /DD /D8/CW/CT /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CU /D1/D3 /CS/CT/D7 /DB/CX/D8/CW /CP/D2 /D3/D4/D8/CX /CP/D0 /D1/CT/CS/CX/D9/D1/BA /CB/D9 /CW /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CP/D2/CQ /CT /CP/D9/D7/CT/CS /CQ /DD /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2/B8 /D7/CT/D0/CU/B9/CU/D3 /D9/D7/CX/D2/CV /CT/D8 /BA /B4/D7/CT/CT /CU/D9/D6/D8/CW/CT/D6 /D4/CP/D6/D8/D7 /D3/CU/DB /D3/D6/CZ/D7/CW/CT/CT/D8/B5/BA /C6/D3 /DB /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /D1/D3 /CS/CT/D0 /D3/CU /D8/CW/CT /D4/CP/D7/D7/CX/DA /CT /D1/D3 /CS/CT/D0/D3 /CZ/CX/D2/CV/BA /C4/CT/D8 /D7/D9/D4/D4 /D3/D7/CT/B8 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CP/D6/CT /D8 /DB /D3 /D1/D3 /CS/CT/D7/B8 /DB/CW/CX /CW /D3/D7 /CX/D0/D0/CP/D8/CT /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8/D4/CW/CP/D7/CT/D7 /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CX/D2 /D8/CW/CT /D9/CQ/CX /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D1/CT/CS/CX/D9/D1/BM >/AZ /D8/DB/D3 /D3/D7 /CX/D0/D0/CP/D8/CX/D2/CV /D1/D3/CS/CT/D7 /DB/CX/D8/CW /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3/D9/D4/D0/CX/D2/CV >/D3/D1/CT/CV/CP /BM/BP /BD/BM >/CS/CT/D0/D8/CP /BM/BP /BC/BA/BD/BM >/CT/BD/BM/BP/CS/CX/CU/CU/B4/CS/CX/CU/CU/B4/DD/B4/D8/B5/B8/D8/B5/B8/D8/B5/B7/D3 /D1/CT/CV/CP /CM/BE/B6/DD /B4/D8/B5/BP /B9/B4/DD/B4 /D8/B5/CM/BF /B7/DE/B4 /D8/B5/CM/BF /B5/BM >/CT/BE/BM/BP/CS/CX/CU/CU/B4/CS/CX/CU/CU/B4/DD/B4/D8/B5/B8/D8/B5/B8/D8/B5/B7 /B4/D3/D1/CT /CV/CP/B7/CS /CT/D0/D8/CP /B5/CM/BE/B6 /DE/B4/D8/B5 /BP/CK >/B9/B4/DD/B4/D8/B5/CM/BF/B7/DE/B4/D8/B5/CM/BF/B5/BM >/D7/DD/D7 /BM/BP /CJ/CT/BD/B8 /CT/BE℄/BM >/BW/BX/D4/D0/D3/D8/BF/CS/B4/D7/DD/D7/B8/CJ/DD/B4/D8/B5/B8/DE/B4/D8/B5℄/B8/D8/BP/BC/BA/BA/BD/BC/BC/B8/CJ/CJ/DD/B4/BC/B5/BP/B9/BD/B8/DE/B4/BC/B5/BP/BD/B8 \/BW/B4/DD/B5/B4/BC/B5/BP/BC/B8/BW/B4/DE/B5/B4/BC/B5/BP/BC℄℄/B8/D7/D8/CT/D4/D7/CX/DE/CT/BP/BD/B8/D7 /CT/D2/CT/BP/CJ/DD/B4/D8/B5/B8/DE/B4/D8/B5/B8/D8℄/B8/CP/DC/CT/D7/BP/CQ /D3 /DC/CT/CS/B8 \/D0/CX/D2/CT /D3/D0/D3/D6/BP/BU/C4/BT /BV/C3/B8/D3/D6/CX/CT/D2 /D8/CP/D8/CX /D3/D2/BP/CJ/B9/BI/BC/B8/BJ/BC℄/B8 /D8/CX/D8/D0/CT/BP/CO/D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV/CO/B5/BN sys:= [(∂2 ∂t2y(t))+y(t) =−y(t)3−z(t)3,(∂2 ∂t2z(t))+1.21 z(t) =−y(t)3−z(t)3]/CB/CT/CT /BY/CX/CV/D9/D6/CT /BH/B8 /D4/CP/CV/CT /BG/BG/CF /CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/CW/CP/D7/CT/D7 /CP/D6/CT /D0/D3 /CZ /CT/CS/CS/D9/CT /D8/D3 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D8/CW/CP/D8 /D4/D6/D3 /CS/D9 /CT/D7 /D8/CW/CT /D7/DD/D2 /CW/D6/D3/D2/D3/D9/D7 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/BA/BT/D7 /D3/D2 /D0/D9/D7/CX/D3/D2/B8 /DB /CT /D2/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CX/D7 /D6/CT/D7/D9/D0/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /CX/D2/B9/D8/CT/D6/CU/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D7/D8/CP/D2/CS/CX/D2/CV /DB /CP /DA /CT/D7 /DB/CX/D8/CW /D3/D2/D7/D8/CP/D2 /D8 /CP/D2/CS /D0/D3 /CZ /CT/CS /D4/CW/CP/D7/CT/D7/BA /CB/D9 /CW/CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2 /CT /CU/D3/D6/D1/D7 /CP /D8/D6/CP/CX/D2 /D3/CU /D8/CW/CT /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT/D7/BA /CC/CW/CT /D1/CT /CW/CP/D2/CX/D7/D1/D7 /D3/CU /D8/CW/CT/D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CP/D6/CT /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW /CU/D6/CT/D5/D9/CT/D2 /DD /B8 /DB/CW/CX /CW /CX/D7 /CT/D5/D9/CP/D0/D8/D3 /CX/D2 /D8/CT/D6/D1/D3 /CS/CT /CU/D6/CT/D5/D9/CT/D2 /DD /CX/D2 /D8/CT/D6/DA /CP/D0/B8 /D3/D6 /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT /D3/D4/D8/CX /CP/D0/D1/CT/CS/CX/D9/D1/BA/CC/CW/CT /D1/CT/D8/CW/D3 /CS /D3/CU /D8/CW/CT /D4/CP/D7/D7/CX/DA /CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CX/D7 /DA /CT/D6/DD /CP/D8/D8/D6/CP /D8/CX/DA /CT /CS/D9/CT /D8/D3 /CX/D8/D7/D6/CT/D0/CP/D8/CX/DA /CT /D7/CX/D1/D4/D0/CX /CX/D8 /DD /BA /C4/CP/D8/CT/D6 /D3/D2 /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D3/D2/D0/DD /D8/CW/CX/D7 /D1/CT/D8/CW/D3 /CS/BA /BU/D9/D8 /AS/D6/D7/D8/D0/DD/DB /CT /CW/CP /DA /CT /D8/D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D1/D3/D6/CT /D6/CT/CP/D0/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/BA/BI/BF /BU/CP/D7/CX /D1/D3 /CS/CT/D0/CC/CW/CT /D1/D3 /CS/CT/D0/D7 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT /D0/CP/D7/CT/D6 /AS/CT/D0/CS /CT/DA /D3/D0/D9/D8/CX/D3/D2 /CP/D6/CT /CQ/CP/D7/CT/CS /D9/D7/D9/CP/D0/D0/DD /D3/D2 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /D7/CT/D1/CX/B9 /D0/CP/D7/D7/CX /CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BA /C1/D2 /D8/CW/CT /CU/D6/CP/D1/CT/DB /D3/D6/CZ /D3/CU /D8/CW/CX/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D8/CW/CT /AS/CT/D0/CS /D3/CQ /CT/DD/D7 /D8/CW/CT /D0/CP/D7/D7/CX /CP/D0 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /D1/CT/CS/CX/D9/D1 /CT/DA /D3/D0/D9/D8/CX/D3/D2 /CW/CP/D7/D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1/B9/D1/CT /CW/CP/D2/CX /CP/D0 /CW/CP/D6/CP /D8/CT/D6/BA /C0/CT/D6/CT /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/DB/CX/D8/CW/D3/D9/D8 /D3/D2 /D6/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CT/CS/CX/D9/D1 /CT/DA /D3/D0/D9/D8/CX/D3/D2/BA/CC/CW/CT /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D0/CX/CV/CW /D8 /DB /CP /DA /CT /CP/D2 /CQ /CT /DB/D6/D3/D8/CT /CP/D7/BM >/D6/CT/D7/D8/CP/D6/D8/BM >/DB/CX/D8/CW/B4/C8/BW/BX/D8/D3/D3/D0/D7/B8/CS /CW/CP/D2/CV/CT/B5/BM >/D1/CP/DC/DB/CT/D0/D0/CN/CT/D5 /BM/BP /CS/CX/CU/CU/B4/BX/B4/DE/B8/D8/B5/B8/DE/B8/DE/B5/B9/CS/CX/CU/CU/B4/BX/B4/DE/B8/D8/B5 /B8/D8/B8/D8 /B5/BB /CM/BE/BP /CK >/BG/B6/C8/CX/B6/CS/CX/CU/CU/B4/C8/B4/D8/B5/B8/D8/B8/D8/B5/BB /CM/BE/BN maxwell /CNeq:= (∂2 ∂z2E(z, t))−∂2 ∂t2E(z, t) c2= 4π(∂2 ∂t2P(t)) c2/DB/CW/CT/D6/CT /BX/B4/DE/B8/D8/B5 /CX/D7 /D8/CW/CT /AS/CT/D0/CS /D7/D8/D6/CT/D2/CV/D8/CW/B8 /C8/B4/D8/B5 /CX/D7 /D8/CW/CT /D1/CT/CS/CX/D9/D1 /D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2/B8 /DE /CX/D7/D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 /D8 /CX/D7 /D8/CW/CT /D8/CX/D1/CT/B8 /CX/D7 /D8/CW/CT /D0/CX/CV/CW /D8 /DA /CT/D0/D3 /CX/D8 /DD /BA /CC/CW/CT /CW/CP/D2/CV/CT/D3/CU /D8/CW/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /DE /AL> /DE/B6 /B8 /D8 /B9 /DE/BB /AL> /D8/B6 /D4/D6/D3 /CS/D9 /CT/D7 >/D1/CP /D6/D3/B4/DE/D7/BP/CO/DE/B6/CO/B8/D8/D7/BP/CO/D8/B6/CO/B5/BM >/D8/D6 /BM/BP /DF/DE /BP /DE/D7/B8 /D8 /BP /D8/D7 /B7 /DE/D7/BB /DH/BN >/D1/CP/DC/DB/CT/D0/D0/CN/D1 /BM/BP /CS /CW/CP/D2/CV/CT/B4/D8/D6/B8/D1/CP/DC/DB/CT/D0/D0/CN/CT/D5/B8/CJ/DE/D7/B8/D8 /D7℄/B8/D7 /CX/D1/D4 /D0/CX/CU/DD /B5/BN tr:={z=z∗, t=t∗+z∗ c} maxwell /CNm:=(∂2 ∂z∗2E(z∗,t∗, c))c−2 (∂2 ∂z∗∂t∗E(z∗,t∗, c)) c= 4π(∂2 ∂t∗2P(z∗,t∗, c)) c2/CF /CT /CS/D3 /CT/D7 /D2/D3/D8 /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /CT/AR/CT /D8/D7 /D3/D2/D2/CT /D8/CT/CS /DB/CX/D8/CW /CP /DB /CP /DA /CT /D4/D6/D3/D4/CP/CV/CP/B9/D8/CX/D3/D2 /CX/D2 /D8/CW/CX/D2 /D1/CT/CS/CX/D9/D1 /D0/CP /DD /CT/D6/B8 /D8/CW/CP/D8 /CP/D0/D0/D3 /DB/D7 /D8/D3 /CT/D0/CX/D1/CX/D2/CP/D8/CT /D8/CW/CT /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /CS/CT/D6/CX/DA /CP/B9/D8/CX/D3/D2 /D3/D2 /DE/B6 /BA /CC/CW/CT/D2 >/CX/D2/D8/B4/D3/D4/B4/BE/B8/CT/DC/D4/CP/D2/CS/B4/D0/CW/D7/B4/D1/CP/DC/DB/CT/D0 /D0/CN/D1/B5 /B5/B5/B8/D8 /D7/B5/B9/CK >/CX/D2/D8/B4/D6/CW/D7/B4/D1/CP/DC/DB/CT/D0/D0/CN/D1/B5/B8/D8/D7/B5/BM/AZ/CX /D2/D8/CT/CV /D6/CP/D8/CX /D3/D2/CK >/D3/CU /CQ/D3/D8/CW /D7/CX/CS/CT/D7 /D3/CU /DB/CP/DA/CT /CT/D5/D9/CP/D8/CX/D3/D2 >/D2/D9/D1/CT/D6/B4/B1/B5/BM >/DB/CP/DA/CT/CN/BD /BM/BP /CT/DC/D4/CP/D2/CS/B4/B1/BB/B4/B9/BE/B5/B5/BN wave /CN1:= (∂ ∂z∗E(z∗,t∗, c))c+ 2π(∂ ∂t∗P(z∗,t∗, c))/CB/D3/B8 /DB /CT /D6/CT/CS/D9 /CT/CS /D8/CW/CT /D3/D6/CS/CT/D6 /D3/CU /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/BA /CC/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D3/CU /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /D7/CW/D3/D6/D8/CT/D2/CT/CS /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/BM >/D8/D6 /BM/BP /DF/DE/D7 /BP /DE/B8 /D8/D7 /BP /D8 /B9 /DE/BB /DH/BN >/DB/CP/DA/CT/CN/BE /BM/BP /CS /CW/CP/D2/CV/CT/B4/D8/D6/B8/DB/CP/DA/CT/CN/BD/B8/CJ/DE/B8/D8℄/B8/D7/CX/D1 /D4/D0/CX/CU /DD/B5/BN tr:={z∗=z,t∗=t−z c}/BJwave /CN2:= (∂ ∂zE(z, t, c ))c+ (∂ ∂tE(z, t, c )) + 2π(∂ ∂tP(z, t, c ))/C6/CT/DC/D8 /D7/D8/CT/D4 /CX/D7 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /D7/D0/D3 /DB/D0/DD/B9/DA /CP/D6/DD/CX/D2/CV /CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BA/CF /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /AS/CT/D0/CS /CT/D2 /DA /CT/D0/D3/D4 /CTρ /B4/DE/B8/D8/B5 /CP/D2/CS /D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2 /C8/B4/D8/B5 /B8 /DB/CW/CX /CW /CP/D6/CT /AS/D0/D0/CT/CS/CQ /DD /D8/CW/CT /CU/CP/D7/D8 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW /CU/D6/CT/D5/D9/CT/D2 /DDω /B4 /CZ /CX/D7 /D8/CW/CT /DB /CP /DA /CT /D2 /D9/D1 /CQ /CT/D6/B8 /C6 /CX/D7 /D8/CW/CT /D3/D2 /CT/D2 /D8/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP/D8/D3/D1/D7/B8 /CS /CX/D7 /D8/CW/CT /D1/CT/CS/CX/D9/D1 /D0/CT/D2/CV/D8/CW/B5/BM >/CC/CW/CT/D8/CP/BP/D3/D1/CT/CV/CP/B6/D8/B9/CZ/B6/DE/BN/AZ /D4/CW/CP/D7/CT >/BX /BM/BP /D6/CW/D3/B4/DE/B8/D8/B5/B6 /D3/D7/B4/CC/CW/CT/D8/CP/B5/BN/AZ /CU/CX/CT/D0/CS >/C8 /BM/BP /C6/B6/CS/B6/B4/CP/B4/D8/B5/B6 /D3/D7/B4/CC/CW/CT/D8/CP/B5/B9/CK >/CQ/B4/D8/B5/B6/D7/CX/D2/B4/CC/CW/CT/D8/CP/B5/B5/BN/AZ /D4/D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2/CK >/B4/CP /CP/D2/CS /CQ /CP/D6/CT /D8/CW/CT /D5/D9/CP/CS/D6/CP/D8/D9/D6/CT /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/B5 Q=ωt−kz E:=ρ(z, t) cos(Q) P:=Nd(a(t) cos(Q)−b(t) sin(Q))/CC/CW/CT/D2 /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2 /CQ /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CP/D7/BM >/CC/CW/CT/D8/CP/BM/BP/D3/D1/CT/CV/CP/B6/D8/B9/CZ/B6/DE/BM >/CS/CX/CU/CU/B4/BX/B8/DE/B5/B7/CS/CX/CU/CU/B4/BX/B8/D8/B5/BB /B7/BE/B6/C8 /CX/B6/CS/CX /CU/CU/B4/C8 /B8/D8/B5/BB /BM > /D3/D1/CQ/CX/D2/CT/B4/B1/B8/D8/D6/CX/CV/B5/BM > /D3/D0/D0/CT /D8/B4/B1/B8 /D3/D7/B4/CC/CW/CT/D8/CP/B5/B5/BM >/DB/CP/DA/CT/CN/BF /BM/BP /D3/D0/D0/CT /D8/B4/B1/B8/D7/CX/D2/B4/CC/CW/CT/D8/CP/B5/B5/BM >/CT/D5/CN/CU/CX/CT/D0/CS /BM/BP /D3/D4/B4/BD/B8/CT/DC/D4/CP/D2/CS/B4 /D3/CT/CU/CU/B4/DB/CP/DA/CT/CN/BF/B8 /D3/D7/B4 /CC/CW/CT/D8 /CP/B5/B5 /B5/B5/B7/CK >/D3/D4/B4/BF/B8/CT/DC/D4/CP/D2/CS/B4 /D3/CT/CU/CU/B4/DB/CP/DA/CT/CN/BF/B8 /D3/D7/B4/CC /CW /CT/D8/CP/B5/B5/B5/B5/BP/CK >/B9/D3/D4/B4/BE/B8/CT/DC/D4/CP/D2/CS/B4 /D3/CT/CU/CU/B4/DB/CP/DA/CT/CN/BF/B8 /D3/D7/B4 /CC/CW/CT/D8 /CP/B5/B5/B5 /B5/B9/CK >/D3/D4/B4/BG/B8/CT/DC/D4/CP/D2/CS/B4 /D3/CT/CU/CU/B4/DB/CP/DA/CT /CN/BF/B8 /D3/D7/B4/CC/CW/CT/D8/CP/B5/B5/B5/B5/BN/AZ /CU/CX/CT/D0/CS /CT/D5/D9/CP/D8/CX/D3/D2 >/CS/CX/D7/D4/CN /D3/D2/CS/CX/D8/CX/D3/D2/D7 /BM/BP /C1/D2/D8/B4 /D3/CT/CU/CU/B4/DB/CP/DA/CT/CN/BF/B8/D7/CX/D2/B4/CC/CW/CT/D8/CP /B5/B5/B8 /D8/B5/CK >/BP/BC/BN/AZ /CS/CX/D7/D4/CT/D6/D7/CX/D3/D2 /D3/D2/CS/CX/D8/CX/D3/D2 eq /CNﬁeld:= (∂ ∂zρ(z, t)) +∂ ∂tρ(z, t) c=−2πNd (∂ ∂ta(t)) c+ 2πNd b(t)ω c disp /CNconditions := /integraldisplayρ(z, t)kc−2πNd (∂ ∂tb(t))−ρ(z, t)ω−2πNd a(t)ω cdt= 0/CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D6/CT/D7/D9/D0/D8 /CX/D7 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /D3/CU /D8/CW/CT /D7/CW/D3/D6/D8/CT/D2/CT/CS /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT/CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3/D2/CS/CX/D8/CX/D3/D2/BA /CC/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /D8/CW/CT /AS/CT/D0/CS /CT/D5/D9/CP/D8/CX/D3/D2 /B4/D1/CP/D8/CW/CT/D6/CX/CP/D0/D4/CP/D6/D8/B5 /DB/CX/D0/D0 /CQ /CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D3/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CP/D4/D4/D0/CX /CP/D8/CX/D3/D2/D7 /B4/D7/CT/CT /CQ /CT/D0/D3 /DB/B5/BA/BK/BG /C6/D3/D2/D0/CX/D2/CT/CP/D6 /CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BM /D3/D2/D7/D8/D6/D9 /B9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT/CS/CX/D6/CT /D8 /C0/CX/D6/D3/D8/CP/B3/D7 /D1/CT/D8/CW/D3 /CS/CC/CW/CT /CB/CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT/DB /CT/CP/CZ /D2/D3/D2/D0/CX/D2/CT/CP/D6 /DB /CP /DA /CT/D7/B8 /CX/D2 /D8/CW/CT /D4/CP/D6/D8/CX /D9/D0/CP/D6/B8 /D8/CW/CT /D0/CP/D7/CT/D6 /D4/D9/D0/D7/CT /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2 /CX/D2 /AS/CQ /CT/D6/D7/BA/C1/D2 /D8/CW/CT /D0/CP/D7/D8 /CP/D7/CT/B8 /CP /D4/D9/D0/D7/CT /CP/D2 /D4/D6/D3/D4/CP/CV/CP/D8/CT /DB/CX/D8/CW/D3/D9/D8 /CS/CT /CP /DD/CX/D2/CV /D3 /DA /CT/D6 /D0/CP/D6/CV/CT /CS/CX/D7/D8/CP/D2 /CT/CS/D9/CT /D8/D3 /CQ/CP/D0/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /CU/CP /D8/D3/D6/D7/BM /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /B4/CB/C8/C5/B5 /CP/D2/CS /CV/D6/D3/D9/D4/CS/CT/D0/CP /DD /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /B4/BZ/BW/BW/B5/BA /CC/CW/CT/D7/CT /D4/D9/D0/D7/CT/D7 /CP/D6/CT /D2/CP/D1/CT/CS /CP/D7 /D3/D4/D8/CX /CP/D0 /D7/D3/D0/CX/D8/D3/D2/D7 /CJ /C5/BA /C2/BA/BT /CQ/D0/D3/DB/CX/D8/DE/B8 /C0/BA /CB/CT /CV/D9/D6/B8 Ꜽ/CB/D3/D0/CX/D8/D3/D2/D7 /CP/D2/CS /D8/CW/CT /C1/D2/DA/CT/D6/D7/CT /CB /CP/D8/D8/CT/D6/CX/D2/CV /CC /D6 /CP/D2/D7/CU/D3/D6/D1Ꜽ/B8 /CB/C1/BT/C5/C8/CW/CX/D0/CP/CS/CT/D0/D4/CW/CX/CP/B8 /BD/BL/BK/BC ℄/BA /CC/CW/CT /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CQ /CT/DD/D7 /D8/D3 /D8/CW/CT /D2/CT/DC/D8 /D1/CP/D7/D8/CT/D6/CT/D5/D9/CP/D8/CX/D3/D2/BM I(∂ ∂zρ) =k2(∂2 ∂t2ρ) +β\|ρ\|2ρ/DB/CW/CX /CW /CX/D7 /D8/CW/CT /D3/D2/D7/CT/D5/D9/CT/D2 /CT /D3/CU /CT /D5/CN/AS/CT/D0/CS /CU/D6/D3/D1 /D4/D6/CT/DA/CX/D3/D9/D7 /D4/CP/D6/D8 /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU/D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D8/D3 /D0/D3 /CP/D0 /D8/CX/D1/CT /D8/AL> /D8/B9/DE/BB /BA /CC/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D7/CX/CS/CT /D8/CT/D6/D1/D7 /CS/CT/D7 /D6/CX/CQ /CT /BZ/BW/BW/B4/DB/CX/D8/CW /D3 /CTꜶ /CX/CT/D2 /D8k2 /B5 /CP/D2/CS /CB/C8/C5 /B4/DB/CX/D8/CW /D3 /CTꜶ /CX/CT/D2 /D8β /B5/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D0/DD /BA/C1/D8 /CX/D7 /DA /CT/D6/DD /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D8/D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /CT/DC/CP /D8 /D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6/CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CC/CW/CT/D6/CT /CP/D6/CT /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CP/D2/CS /CS/CX/D6/CT /D8 /D1/CT/D8/CW/D3 /CS/D7 /D8/D3 /D3/CQ/D8/CP/CX/D2 /D7/D9 /CW /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA/C7/D2/CT /D3/CU /D8/CW/CT /CS/CX/D6/CT /D8 /D1/CT/D8/CW/D3 /CS/D7 /CX/D7 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /C0/CX/D6/D3/D8/CP/B3/D7 /D1/CT/D8/CW/D3 /CS/BA /CC/CW/CT /D1/CP/CX/D2 /D7/D8/CT/D4/D7/D3/CU /D8/CW/CX/D7 /D1/CT/D8/CW/D3 /CS /CP/D6/CT/BM /BD/B5 /D8/CW/CT /D7/CT/D0/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D9/CX/D8/CP/CQ/D0/CT /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /CX/D2/D7/D8/CT/CP/CS /D3/CU/D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2ρ /B4/D7/CT/CT /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/B5/B8 /D8/CW/CP/D8 /CP/D0/D0/D3 /DB/D7 /D8/D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /CQ/CX/D0/CX/D2/CT/CP/D6/CU/D3/D6/D1 /D3/CU /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/BN /BE/B5 /D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D3/D6/D1/CP/D0 /D7/CT/D6/CX/CT/D7 /D3/CU/D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /D8/CW/CT/D3/D6/DD /CU/D3/D6 /D8/CW/CX/D7 /CQ/CX/D0/CX/D2/CT/CP/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BA /C1/D2 /D8/CW/CT /CP/D7/CT /D3/CU /D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2/D7/D8/CW/CT/D7/CT /D7/CT/D6/CX/CT/D7 /CP/D6/CT /D8/CT/D6/D1/CX/D2/CP/D8/CT/CS/BA/CC/CW/CT /D9/D7/CT/CU/D9/D0 /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CB/CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 ρ /B4/DE/B8/D8/B5/BP/BZ/B4/DE/B8/D8/B5/BB/BY/B4/DE/B8/D8/B5/BA /C4/CT/D8 /D7/D9/D4/D4 /D3/D7/CT /D8/CW/CP/D8 /BY /CX/D7 /D8/CW/CT /D6/CT/CP/D0 /CU/D9/D2 /D8/CX/D3/D2/BA /C1/D8 /D7/CW/D3/D9/D0/CS /CQ /CT/D2/D3/D8/CT/CS /D8/CW/CP/D8 /DB /CT /CP/D2 /D1/CP/CZ /CT /CP/D2 /DD /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /CP/CQ /D3/D9/D8ρ /D8/D3 /D7/CP/D8/CX/D7/CU/DD /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2/D7/BD/B5 /CP/D2/CS /BE/B5/BA /C0/CX/D6/D3/D8/CP /D4/D6/D3/D4 /D3/D7/CT/CS /D8/D3 /CX/D2 /D8/D6/D3 /CS/D9 /CT /CP /D2/CT/DB /BW/B9/D3/D4 /CT/D6 /CP/D8/D3/D6 /CX/D2 /CU/D3/D0/D0/D3 /DB/CX/D2/CV /DB /CP /DD/BM DzmDtnab= ((∂ ∂z)−(∂ ∂z1))m((∂ ∂t)−(∂ ∂t1))na(z, t) b(z1, t1) [IDz+k2Dt2]GF= 0 k2Dt2FF−βGG∗= 0 /B4/BD/B5/CC/CW/CT /CU/D9/D2 /D8/CX/D3/D2/D7 /BZ /CP/D2/CS /BY /CP/D2 /CQ /CT /CT/DC/D4/CP/D2/CS/CT/CS /CX/D2 /D8/D3 /D8/CW/CT /D7/CT/D6/CX/CT/D7 /D3/CU /D8/CW/CT /CU/D3/D6/D1/CP/D0/D4/CP/D6/CP/D1/CT/D8/CT/D6ε /BM G=εG1+ε3G3+ε5G5;F= 1 +ε2F2+ε4F4+ε6F6/BL/CB/D9/CQ/D7/D8/CX/D8/D9/D8/CT /BZ /CP/D2/CS /BY /CX/D2 /D8/D3 /BX/D5/BA /B4/BD/B5 /CP/D2/CS /D8/D6/CT/CP/D8 /D8/CW/CT /D8/CT/D6/D1/D7 /DB/CX/D8/CW /D4 /D3 /DB /CT/D6/D7 /D3/CUε/CP/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8/B8 /D8/D3 /CV/CT/D8 /D8/CW/CT /CX/D2/AS/D2/CX/D8/CT /D7/CT/D8 /D3/CU /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD/BZ/BD/B8 /BZ/BF/B8 /BA/BA/BA/BN /BY/BE/B8 /BY/BG/B8 /BA/BA/BA /BA /CC/CW/CT/D7/CT /CU/D3/D6/D1/CP/D0 /D7/CT/D6/CX/CT/D7 /CP/D6/CT /D8/CT/D6/D1/CX/D2/CP/D8/CT/CS /D3/D2/D0/DD /CX/D2 /D8/CW/CT /CP/D7/CT/DB/CW/CT/D2 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CW/CP/D7 /CT/DC/CP /D8 /C6 /B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2/BA /BY /D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B8 /D8/CW/CT /D7/CT/D8/D3/CU /AS/D6/D7/D8 /D7/CX/DC /CS/CX/AR/CT/D6/CP/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CX/D2 /D3/D9/D6 /CP/D7/CT /CX/D7/BM I(∂ ∂zG1) +k2(∂2 ∂t2G1) = 0 ; 2k2(∂2 ∂t2F2)−βG1 G1 ∗= 0 ;I(∂ ∂zG3) +k2(∂2 ∂t2G3) + [IDz+k2Dt2]G1 F2 = 0 ; 2k2(∂2 ∂t2F4) +Dt2F2F2 −β(G3 G1 ∗+G1 G3 ∗) = 0 ;I(∂ ∂zG5) +k2(∂2 ∂t2G5) + [IDz+k2Dt2] (G3 F2 +G1 F4 ) = 0 ; 2k2(∂2 ∂t2F6)+k2Dt2(F4 F2 +F2 F4 )−β(G5 G1 ∗+G3 G3 ∗+G1 G5 ∗) = 0/BY /D3/D6 /D7/CP/CZ /CT /D3/CU /D8/CW/CT /D7/CX/D1/D4/D0/CX/AS /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /CT/D6/DD /D9/D1 /CQ /CT/D6/D7/D3/D1/CT /D1/CP/D2/CX/D4/D9/D0/CP/D8/CX/D3/D2/D7 /DB /CT/CX/D2 /D8/D6/D3 /CS/D9 /CT /D8/CW/CT /D4/D6/D3 /CT/CS/D9/D6/CT /CU/D3/D6 /D3/D4 /CT/D6/CP/D8/D3/D6 DtmDzn/B8 /DB/CW/CX /CW /CP /D8/D7 /D3/D2 /D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2/D7 /CP/CP/D2/CS /CQ /BA /CC/CW/CT /D0/CP/D7/D8/D7 /CP/D6/CT /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/D7 /B4/D3/D6 /D0/CX/D2/CT/CP/D6 /D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/D7/B5/CX/D2 /D8/CW/CT /CU/D3/D6/D1eη/B8 /DB/CW/CT/D6/CTη(z, t) /CX/D7 /D0/CX/D2/CT/CP/D6 /CU/D9/D2 /D8/CX/D3/D2/BA >/D6/CT/D7/D8/CP/D6/D8/BM >/DB/CX/D8/CW/B4/D4/D0/D3/D8/D7/B5/BM/BG/BA/BD /C8/D6/D3 /CT/CS/D9/D6/CT DtmDzn >/BW/D8/CN/BW/DE /BM/BP /D4/D6/D3 /B4/CP/B8/CQ/B8/D1/B8/D2/B5 /D0/D3 /CP/D0 /CB/D9/D1/D1/CP/B8/CZ/B8/D6/B8/D6/CT/D7/D9/D0/D8/BM >/CB/D9/D1/D1/CP /BM/BP /BC/BM >/CX/CU /B4/D2/BQ/BD/B5 /CP/D2/CS /B4/D1/BO/BQ/BC/B5 /D8/CW/CT/D2 >/CU/D3/D6 /CZ /CU/D6/D3/D1 /BD /D8/D3 /D2/B9/BD /CS/D3 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D2/B8/CZ/B5/B6/B4/B9/BD/B5/CM/B4/D2/B9 /CZ/B7/D1/B5 /B6/CK >/CS/CT/D6/B4 /CS/CT/D6/B4/CQ/B8/DE/B8/B4/D2/B9/CZ/B5/B5/B8/D8/B8/D1/B5/B6/CS/CT/D6/B4/CP/B8 /DE/B8/CZ/B5 /B7/CK >/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D2/B8/CZ/B5/B6/B4/B9/BD/B5/CM/B4/D2/B9/CZ/B5/B6/CS/CT /D6/B4/CQ/B8 /DE/B8/B4/D2 /B9/CZ/B5/B5 /B6/CK >/CS/CT/D6/B4 /CS/CT/D6/B4/CP/B8/DE/B8/CZ/B5/B8/D8/B8/D1/B5 >/D3/CS/BM >/CU/CX/BM >/CX/CU /B4/D2/BQ/BD/B5 /CP/D2/CS /B4/D1/BQ/BD/B5 /D8/CW/CT/D2 >/CU/D3/D6 /D6 /CU/D6/D3/D1 /BD /D8/D3 /B4/D1/B9/BD/B5 /CS/D3 >/CU/D3/D6 /CZ /CU/D6/D3/D1 /BD /D8/D3 /B4/D2/B9/BD/B5 /CS/D3 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D1/B8/D6/B5/B6/CK >/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D2/B8/CZ/B5/B6/B4/B9/BD/B5/CM/B4/D2/B9/CZ/B7/D1/B9/D6 /B5/B6/CK >/CS/CT/D6/B4 /CS/CT/D6/B4/CQ/B8/DE/B8/B4/D2/B9/CZ/B5/B5/B8/D8/B8/B4/D1/B9/D6/B5/B5/B6/CK >/CS/CT/D6/B4 /CS/CT/D6/B4/CP/B8/DE/B8/CZ/B5/B8/D8/B8/D6/B5/BN >/D3/CS/BM >/D3/CS/BM >/CU/CX/BM >/CX/CU /B4/D1/BQ/BD/B5 /CP/D2/CS /B4/D2/BO/BQ/BC/B5 /D8/CW/CT/D2 >/CU/D3/D6 /D6 /CU/D6/D3/D1 /BD /D8/D3 /B4/D1/B9/BD/B5 /CS/D3/BD/BC>/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D1/B8/D6/B5/B6/B4/B9/BD/B5/CM/B4/D1 /B9/D6/B7/D2 /B5/B6/CK >/CS/CT/D6/B4 /CS/CT/D6/B4/CQ/B8/DE/B8/D2/B5/B8/D8/B8/B4/D1/B9/D6/B5/B5/B6/CS/CT/D6/B4/CP/B8/D8 /B8/D6/B5/B7 /CK >/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D1/B8/D6/B5/B6/B4/B9/BD/B5/CM/B4/D1/B9/D6/B5/B6/CS/CT/D6 /B4/CQ/B8/D8 /B8/B4/D1/B9 /D6/B5/B5/B6 /CK >/CS/CT/D6/B4 /CS/CT/D6/B4/CP/B8/DE/B8/D2/B5/B8/D8/B8/D6/B5/BN >/D3/CS/BN >/CU/CX/BM >/CX/CU /B4/D1/BO/BQ/BC/B5 /CP/D2/CS /B4/D2/BO/BQ/BC/B5 /D8/CW/CT/D2 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/B4/B9/BD/B5/CM/B4/D1/B7/D2/B5/B6/CS/CT/D6/B4/CS/CT/D6/B4/CQ/B8/DE /B8/D2/B5/B8 /D8/B8/D1/B5 /B6/CP/B7 /CK >/B4/B9/BD/B5/CM/D1/B6/CS/CT/D6/B4/CP/B8/DE/B8/D2/B5/B6/CS/CT/D6/B4/CQ/B8/D8 /B8/D1/B5/B7 /B4/B9/BD/B5 /CM/D2/B6/CS /CT/D6/B4/CP /B8/D8/B8/D1 /B5/B6/CK >/CS/CT/D6/B4/CQ/B8/DE/B8/D2/B5/B7/CS/CT/D6/B4/CS/CT/D6/B4/CP/B8/DE/B8/D2/B5 /B8/D8/B8/D1 /B5/B6/CQ/BN >/CU/CX/BM >/CX/CU /B4/D1/BP/BC/B5 /CP/D2/CS /B4/D2/BQ/BD/B5 /D8/CW/CT/D2 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/B4/B9/BD/B5/CM/B4/D2/B5/B6/CS/CT/D6/B4/CQ/B8/DE/B8/D2/B5/B6/CP/B7 /CS/CT/D6/B4 /CP/B8/DE/B8 /D2/B5/B6 /CQ/BN >/CU/D3/D6 /CZ /CU/D6/D3/D1 /BD /D8/D3 /B4/D2/B9/BD/B5 /CS/D3 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D2/B8/CZ/B5/B6/B4/B9/BD/B5/CM/B4 /D2/B9/CZ/B5 /B6/CK >/CS/CT/D6/B4/CQ/B8/DE/B8/B4/D2/B9/CZ/B5/B5/B6/CS/CT/D6/B4/CP/B8/DE/B8/CZ/B5/BN >/D3/CS/BM >/CU/CX/BM >/CX/CU /B4/D1/BP/BC/B5 /CP/D2/CS /B4/D2/BP/BD/B5 /D8/CW/CT/D2 >/CB/D9/D1/D1/CP /BM/BP /CS/CT/D6/B4/CP/B8/DE/B8/BD/B5/B6/CQ/B9/CS/CT/D6/B4/CQ/B8/DE/B8/BD/B5/B6/CP/BM >/CU/CX/BM >/CX/CU /B4/D2/BP/BC/B5 /CP/D2/CS /B4/D1/BQ/BD/B5 /D8/CW/CT/D2 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/B4/B9/BD/B5/CM/B4/D1/B5/B6/CS/CT/D6/B4/CQ/B8/D8/B8/D1/B5 /B6/CP/B7/CS /CT/D6/B4/CP /B8/D8/B8/D1 /B5/B6/CQ /BN >/CU/D3/D6 /D6 /CU/D6/D3/D1 /BD /D8/D3 /B4/D1/B9/BD/B5 /CS/D3 >/CB/D9/D1/D1/CP /BM/BP /CB/D9/D1/D1/CP/B7/CQ/CX/D2/D3/D1/CX/CP/D0/B4/D1/B8/D6/B5/B6/B4/B9/BD/B5/CM/B4/D1 /B9/D6/B5/B6 /CK >/CS/CT/D6/B4/CQ/B8/D8/B8/B4/D1/B9/D6/B5/B5/B6/CS/CT/D6/B4/CP/B8/D8/B8/D6/B5/BN >/D3/CS/BM >/CU/CX/BM >/CX/CU /B4/D2/BP/BC/B5 /CP/D2/CS /B4/D1/BP/BD/B5 /D8/CW/CT/D2 >/CB/D9/D1/D1/CP /BM/BP /CS/CT/D6/B4/CP/B8/D8/B8/BD/B5/B6/CQ/B9/CS/CT/D6/B4/CQ/B8/D8/B8/BD/B5/B6/CP/BM >/CU/CX/BM >/CX/CU /B4/D2/BP/BC/B5 /CP/D2/CS /B4/D1/BP/BC/B5 /D8/CW/CT/D2 >/CB/D9/D1/D1/CP /BM/BP /CP/B6/CQ >/CU/CX/BM >/D6/CT/D7/D9/D0/D8 /BM/BP /D3/D1/CQ/CX/D2/CT/B4/CB/D9/D1/D1/CP/B8/CT/DC/D4/B5/BM >/CT/D2/CS/BM/CC/CW/CT /D2/CT/DC/D8 /D4/D6/D3 /CT/CS/D9/D6/CT /DB/CX/D0/D0 /CQ /CT /D9/D7/CT/CS /CU/D3/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /D3/CUeη/B4/D3/D6 /D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /CT/DC/D4 /D3/D2/CT/D2 /D8/D7/B5 /D3/D2 /D8 /D3/D6 /DE /DB/CX/D8/CW /CU/D9/D6/D8/CW/CT/D6 /D7/CX/D1/D4/D0/CX/AS /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CQ/B9/D8/CP/CX/D2/CT/CS /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/BA/BG/BA/BE /C8/D6/D3 /CT/CS/D9/D6/CT der >/CS/CT/D6 /BM/BP /D4/D6/D3 /B4/CU/B8/DC/B8/D1/B5 >/D0/D3 /CP/D0 /CS/CX/CU/BY/B8/CX/B8/CU/D9/D2 /D8/CX/D3/D2/BM >/D7/D9/CQ/D7/B4/CT/D8/CP/BD/BP/CT/D8/CP/BD/B4/DC/B5/B8/CT/D8/CP/BD/D7/BP/CT/D8/CP/BD /D7/B4/DC/B5 /B8 >/CT/D8/CP/BE/BP/CT/D8/CP/BE/B4/DC/B5/B8/CT/D8/CP/BE/D7/BP/CT/D8/CP/BE/D7/B4 /DC/B5/B8/CU /B5/BM >/CS/CX/CU/BY /BM/BP /CS/CX/CU/CU/B4/B1/B8/DC/B0/D1/B5/BM/BD/BD>/CX/CU /B4/DC/BP/D8/B5 /D8/CW/CT/D2 >/CU/D9/D2 /D8/CX/D3/D2 /BM/BP 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>/CS/CX/CU/CU/B4/CQ/BD/B8/DC/B5/BP/BC/B8/CS/CX/CU/CU/B4/CQ/BE/B8/DC/B5/BP/BC/B8/CS/CX /CU/CU/B4/CP /BD/D7/B8/DC /B5/BP/BC/B8 >/CS/CX/CU/CU/B4/CP/BE/D7/B8/DC/B5/BP/BC/B8/CS/CX/CU/CU/B4/CQ/BD/D7/B8/DC/B5 /BP/BC/B8/CS /CX/CU/CU/B4 /CQ/BE/D7/B8 /DC/B5/BP/BC /DH/B8/B1/B5 >/CT/D0/D7/CT > /D3/D1/CQ/CX/D2/CT/B4/B1/B5 >/CU/CX/BM > /D3/D0/D0/CT /D8/B4/B1/B8/CT/DC/D4/B5/BM >/CT/D2/CS/BM/CC/CW/CT /D2/CT/DC/D8 /D4/D6/D3 /CT/CS/D9/D6/CT /CX/D7 /D9/D7/CT/CS /D8/D3 /CP/D0 /D9/D0/CP/D8/CT /CP/D2 /CX/D2 /D8/CT/CV/D6/CP/D0 /D3/CUeη/B4/D3/D6 /D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU/CT/DC/D4 /D3/D2/CT/D2 /D8/D7/B5 /D3/D2 /D8 /D3/D6 /DE /DB/CX/D8/CW /CU/D9/D6/D8/CW/CT/D6 /D7/CX/D1/D4/D0/CX/AS /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/BA/BG/BA/BF /C8/D6/D3 /CT/CS/D9/D6/CT Integr >/CX/D2/D8/CT/CV/D6 /BM/BP /D4/D6/D3 /B4/CU/B8/DC/B8/D1/B5 >/D0/D3 /CP/D0 /CX/D2/D8/BY/B8/CX/B8/CV/BD/B8/CV/BD/D7/B8/CV/BE/B8/CV/BE/D7/B8/CU/D9/D2 /D8/CX /D3/D2/BM >/CX/D2/D8/BY /BM/BP /D7/D9/CQ/D7/B4/CT/D8/CP/BD/BP/CV/BD/B6/DC/B8/CT/D8/CP/BD/D7/BP/CV/BD/D7/B6/DC/B8/CT /D8/CP/BE/BP /CV/BE/B6/DC /B8/CK >/CT/D8/CP/BE/D7/BP/CV/BE/D7/B6/DC/B8/CU/B5/BM >/CU/D3/D6 /CX /CU/D6/D3/D1 /BD /D8/D3 /D1 /CS/D3 >/CX/D2/D8/BY /BM/BP /CX/D2/D8/B4/CX/D2/D8/BY/B8/DC/B5/BN >/D3/CS/BM >/CX/CU /B4/DC/BP/D8/B5 /D8/CW/CT/D2 >/DC /BM/BP /D8/BN /CV/BD /BM/BP /CQ/BD/BN /CV/BD/D7 /BM/BP /CQ/BD/D7/BN /CV/BE /BM/BP /CQ/BE/BN /CV/BE/D7 /BM/BP /CQ/BE/D7/BN >/CT/D0/D7/CT >/DC /BM/BP /DE/BN /CV/BD /BM/BP /CP/BD/BN /CV/BD/D7 /BM/BP /CP/BD/D7/BN /CV/BE /BM/BP /CP/BE/BN /CV/BE/D7 /BM/BP /CP/BE/D7/BN >/CU/CX/BM >/CX/D2/D8/BY/BN > /D3/D0/D0/CT /D8/B4/B1/B8/CT/DC/D4/B5/BM >/D7/D9/CQ/D7/B4/CQ/BD/B6/D8/BP/CT/D8/CP/BD/B8/CQ/BD/D7/B6/D8/BP/CT/D8/CP/BD/D7/B8/CQ /BE/B6/D8/BP /CT/D8/CP/BE /B8/CK >/CQ/BE/D7/B6/D8/BP/CT/D8/CP/BE/D7/B8/CP/BD/B6/DE/BP/CT/D8/CP/BD/B8/CP/BD/D7 /B6/DE/BP/CT /D8/CP/BD/D7 /B8/CP/BE/B6 /DE/BP/CT/D8 /CP/BE/B8/CK >/CP/BE/D7/B6/DE/BP/CT/D8/CP/BE/D7/B8/B1/B5/BN /CT/D2/CS/BM/C6/D3 /DB /D8/D6/DD /D8/D3 /D3/CQ/D8/CP/CX/D2 /CP /AS/D6/D7/D8/B9/D3/D6 /CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /CU/D3/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BA >/D1/CP /D6/D3/B4/BZ/D7/BP/CO/BZ/B6/CO/B8/BZ/BD/D7/BP/CO/BZ/BD/B6/CO/B8/BZ/BF /D7/BP/CO/BZ /BF/B6/CO/B8 /BZ/BH/D7/BP /CO/BZ/BH/B6 /CO/B8/CK >/CP/D7/BP/CO/CP/B6/CO/B8/CQ/D7/BP/CO/CQ/B6/CO/B8/CT/D8/CP/BC/D7 /BP /CO/CT/D8/CP/BC/B6/CO/B5/BM >/BZ/BD /BM/BP /CT/DC/D4/B4/CT/D8/CP/BD/B5/BM/AZ /D7/D9 /CT/D7/D7/CU/D9/D0 /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2/AX/BD/BE>/C1/B6/CS/CT/D6/B4/B1/B8/DE/B8/BD/B5/B7/CZ/CN/BE/B6/CS/CT/D6/B4/B1/B8/D8/B8/BE/B5/BM /AZ/CU/CX/D6 /D7/D8 /CU/D6/D3/D1 /D8/CW/CT >/CT/D5/D9/CP/D8/CX/D3/D2/D7 /D7/CT/D8 >/CU/CP /D8/D3/D6/B4/B1/B5/BN eη1(Ia1+b12k /CN2) >/AZ/CP/D7 /D6/CT/D7/D9/D0/D8 /DB/CT /CP/D2 /CU/CX/D2/CS /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B0/CP/BD/B0 >/CP/BD /BM/BP /C1/B6/CZ/CN/BE/B6/CQ/BD/CM/BE/BM >/CP/BD/D7 /BM/BP /B9/C1/B6/CZ/CN/BE/B6/CQ/BD/D7/CM/BE/BM >/BZ/BD/D7 /BM/BP /CT/DC/D4/B4/CT/D8/CP/BD/D7/B5/BM/AZ /D3/D2/CY/D9/CV/CP/D8/CT/CS /D8/D3 /B0/BZ/BD/B0 >/BZ/BD/BZ/BD/D7 /BM/BP /D3/D1/CQ/CX/D2/CT/B4/BZ/BD/B6/BZ/BD/D7/B5/BM >/BY/BE /BM/BP /CQ/CT/D8/CP/BB/B4/BE/B6/CZ/CN/BE/B5/B6/CX/D2/D8/CT/CV/D6/B4/B1/B8/D8/B8/BE/B5 /BN/AZ/BY/BE /CU/D6/D3/D1 /D8/CW/CT /D7/CT /D3/D2/CS >/CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CT/D8 F2:=1 2βe(η1+eta1s) k /CN2(b1+b1s)2 >/AZ /BU/D9/D8 /D8/CW/CT /D2/CT/DC/D8 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CT/D8 /D6/CT/D7/D9/D0/D8/D7 /CX/D2 >/C1/CN/BW/DE/CN/BZ/BD/CN/BY/BE /BM/BP /C1/B6/CU/CP /D8/D3/D6/B4/BW/D8/CN/BW/DE/B4/BZ/BD/B8/BY/BE/B8/BC/B8/BD/B5/B5/BM >/CS/CN/BW/D8/BE/CN/BZ/BD/CN/BY/BE /BM/BP /CZ/CN/BE/B6/CU/CP /D8/D3/D6/B4/BW/D8/CN/BW/DE/B4/BZ/BD/B8/BY/BE/B8/BE/B8 /BC/B5/B5/BM >/CU/CP /D8/D3/D6/B4/C1/CN/BW/DE/CN/BZ/BD/CN/BY/BE/B7/CS/CN/BW/D8/BE/CN/BZ/BD/CN /BY/BE/B5/BN >/BW/D8/CN/BW/DE/B4/BY/BE/B8/BY/BE/B8/BE/B8/BC/B5/BN 0 0/CC /D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /DB /CT /D9/D7/CT /D8/CW/CT /D8/D6/CX/DA/CX/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/D7/BM >/CT/D8/CP/BD/BP/CP/BD/B6/DE/B7/CQ/BD/B6/D8/B7/CT/D8/CP/BD/BC/BM >/CT/D8/CP/BE/BP/CP/BE/B6/DE/B7/CQ/BE/B6/D8/B7/CT/D8/CP/BE/BC/BM >/BW/DE/B6/CT/DC/D4/B4/CT/D8/CP/BD/B5/B6/CT/DC/D4/B4/CT/D8/CP/BE/B5/BP/B4/CP/BD/B9/CP /BE/B5/B6/CT /DC/D4/B4/CT /D8/CP/BD/B7 /CT/D8/CP/BE /B5/BN >/BW/D8/CM/BE/B6/CT/DC/D4/B4/CT/D8/CP/BD/B5/B6/CT/DC/D4/B4/CT/D8/CP/BE/B5/BP/B4/CQ /BD/B9/CQ/BE /B5/CM/BE/B6 /CT/DC/D4/B4 /CT/D8/CP/BD /B7/CT/D8 /CP/BE/B5/BN Dzeη1eη2= (Ik /CN2 b12−a2)e(η1+η2) Dt2eη1eη2= (b1−b2)2e(η1+η2)/BT/D7 /DB /CP/D7 /D7/CW/D3 /DB/D2 /CP/CQ /D3 /DA /CT /CP/BP /B9 /CX/B6k2 /B6b2/B8 /CW/CT/D2 /CT /D8/CW/CT /D0/CP/D7/D8 /D8/CT/D6/D1 /CX/D2 /D8/CW/CX/D6/CS /CT/D5/D9/CP/D8/CX/D3/D2/D3/CU /D7/CT/D8 /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /BC /BA /CB/D3/B8 /DB /CT /CP/D6/CT /D8/D3 /CW/D3 /D3/D7/CT /BZ/BF /BP /BC /D8/D3 /D7/CP/D8/CX/D7/CU/DD /D8/CW/CX/D6/CS /CT/D5/D9/CP/D8/CX/D3/D2/BA/BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT Dt2/B6/BY/BE /CU/D6/D3/D1 /CU/D3/D9/D6/D8/CW /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CT/D8 /CX/D7 /B4β 2(b+bs)2k /CN2 /B5/CM/BE/B6 Dt2/CT/DC/D4/B4η /B7η /CN/D7/B5/BA /BU/D9/D8 /CP /D3/D6/CS/CX/D2/CV /DB/CX/D8/CW /CP/CQ /D3 /DA /CT /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/D7 /D8/CW/CX/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CX/D7/CT/D5/D9/CP/D0 /D8/D3 /DE/CT/D6/D3/BA /CB/D3 /DB /CT /CP/D2 /CW/D3 /D3/D7/CT /BY/BG /BP /BC/BA /CC/CW /D9/D7 /D8/D3 /D7/CP/D8/CX/D7/CU/DD /D3/D8/CW/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB /CT /CP/D2 /CZ /CT/CT/D4 /CX/D2 /D8/CW/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /CU/D9/D2 /D8/CX/D3/D2/D7 /BZ /CP/D2/CS /BY /D3/D2/D0/DD /BZ/BD /B8 /BZ/BF /CP/D2/CS /BY/BE /BA/CB/D3/B8 /D8/CW/CT /CU/D3/D6/D1/CP/D0 /D7/CT/D6/CX/CT/D7 /CP/D6/CT /D8/CT/D6/D1/CX/D2/CP/D8/CT/CS/BA /CB/CX/D2 /CTε /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D4/CP/D6/CP/D1/CT/D8/CT/D6/DB /CT /CP/D2 /D8/CP/CZ /CTε /BP/BD/BA >/D6/CW/D3 /BM/BP /BZ/BD/BB/B4/BD/B7/BY/BE/B5/BN ρ:=eη1 1 +1 2βe(η1+eta1s) k /CN2(b1+b1s)2/BD/BF>/D7/D9/CQ/D7/B4/CT/D8/CP/BD/BP/CP/BD/B6/DE/B7/CQ/BD/B6/D8/B7/CT/D8/CP/BD/BC/B8 /CT/D8/CP/BD /D7/BP/CP/BD /D7/B6/DE/B7 /CQ/BD/D7/B6 /D8/B7/CT/D8 /CP/BD/BC /D7/B8 /D6/CW/D3/B5/BM >/D7/D3/D0/CX/D8/D3/D2 /BM/BP /CT/DC/D4/CP/D2/CS/B4/D7/D9/CQ/D7/B4/DF/CQ/BD/D7/BP/CQ/BD/B8/CQ/CT/D8/CP/BP /BD/B8/CZ/CN /BE/BP/BD/BB /BE/DH/B8 /B1/B5/B5/BN >/AZ/D8/CW/CT /CW/D3/CX /CT /D3/CU /CZ/CN/BE /CP/D2/CS /CQ/CT/D8/CP /CX/D7 /D3/D2/D0/DD /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 >/D3/CU /D8/CW/CT /DA/CP/D0/D9/CT/D7 /CX/D2 /CT/D5/D9/CP/D8/CX/D3/D2 soliton :=e(1/2Ib12z)e(b1t)eη10 1 +1 4(e(b1t))2eη10eeta10s b12 >/AZ /C6/D3/DB /DB/CT /CW/CT /CZ /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CQ/DD >/D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU /D3/D2/CT /CX/D2 /CS/DD/D2/CP/D1/CX /CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 >/C1/B6/CS/CT/D6/B4/D6/CW/D3/B8/DE/B8/BD/B5/BB/D6/CW/D3/B7/CQ/CT/D8/CP/B6/CT/DC /D4/B4/CT/D8 /CP/BD/B7/CT /D8/CP/BD/D7 /B5/BB/CK >/B4/B4/BD/B7/BD/BB/BE/B6/CQ/CT/D8/CP/B6/CT/DC/D4/B4/CT/D8/CP/BD/B7/CT/D8/CP/BD /D7/B5/BB/CK >/B4/CZ/CN/BE/B6/B4/CQ/BD/B7/CQ/BD/D7/B5/CM/BE/B5/B5/B5/CM/BE/B7/CZ/CN/BE/B6/CS /CT/D6/B4/D6 /CW/D3/B8/D8 /B8/BE/B5/BB /D6/CW/D3/BM >/D7/CX/D1/D4/D0/CX/CU/DD/B4/B1/B5/BN 0/BT/D0/D0 /D6/CX/CV/CW /D8/AX /CC/CW/CX/D7 /CX/D7 /D8/CW/CT /CT/DC/CP /D8 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BA /C8/CW /DD/D7/B9/CX /CP/D0/D0/DD /CQ/BD /CW/CP/D7 /CP /D7/CT/D2/D7/CT /D3/CU /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/BA /CB/D3 /CX/D8 /CX/D7 /D6/CT/CP/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/BA/BU/D9/D8 /DB/CW/CP/D8 /CX/D7 /D8/CW/CT /CU/D6/CT/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6η /BD/BC /BR /C4/CT/D8η /BD/BC /CX/D7 /D6/CT/CP/D0 /CP/D2/CSeη10/BP /CQ/BD/BA /CC/CW/CT/D2 >/D7/D9/CQ/D7/B4/DF/CT/DC/D4/B4/CT/D8/CP/BD/BC/B5/BP/BE/B6/CQ/BD/B8 /CT/DC/D4/B4/CT/D8/CP/BD/BC/D7/B5/BP/BE/B6/CQ/BD/DH/B8/D7/D3/D0/CX/D8/D3/D2/B5 /BM >/D7/CX/D1/D4/D0/CX/CU/DD/B4/B1/B5/BN 2e(1/2b1(Izb1+2t))b1 1 +e(2b1t)/CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /AS/D6/D7/D8/B9/D3/D6 /CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /DB/CX/D8/CW /CS/D9/D6/CP/D8/CX/D3/D21 b1 /B8/CP/D1/D4/D0/CX/D8/D9/CS/CT /CQ/BD /CP/D2/CS /D4/CW/CP/D7/CT b12/B6/DE/BB/BE /BM ρ /B4/DE/B8/D8/B5/BP/CQ/BD/B6/D7/CT /CW/B4/CQ/BD/B6/D8/B5/B6/CT/DC/D4/B4/CX/B6 b12/B6/DE/BB/BE/B5 >/D4/D0/D3/D8/B4/D7/D9/CQ/D7/B4/DF/DE/BP/BC/B8/CQ/BD/BP/BD/DH/B8/B1/B5/B8/D8/BP /B9/BH/BA/BA /BH/B8/CP/DC /CT/D7/BP/CQ /D3/DC/CT/CS /B8/CK >/D8/CX/D8/D0/CT/BP/CO/CU/CX/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BI/B8 /D4/CP/CV/CT /BG/BH/BU/D9/D8 /DB/CW/CP/D8 /CX/D7 /CP/CQ /D3/D9/D8 /CS/CX/AR/CT/D6/CT/D2 /D8 /DA /CP/D0/D9/CT/D7 /D3/CUη /BD/BC /BR /CC/CW/CT /CW/D3/CX /CT /D3/CU /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/DA /CP/D0/D9/CT/D7 /D3/CUη /BD/BC /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CX/D2/CV /D3/CU /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /D3/D0/D0/CP/D4/D7/CX/D2/CV /D4/D9/D0/D7/CT/D7/B8 /CX/BA /CT/BA/D4/D9/D0/D7/CT/D7 /DB/CX/D8/CW /D7/CX/D2/CV/D9/D0/CP/D6/CX/D8 /DD /CX/D2 /D8/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /D8/CW/CT/CX/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D2 /DE /BA/CC/CW/CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CW/CT/D6/CT /D4/D6/D3 /CT/CS/D9/D6/CT /CX/D7 /CP /DA /CP/CX/D0/CP/CQ/D0/CT /CU/D3/D6 /D8/CW/CT /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CW/CX/CV/CW/CT/D6/B9/D3/D6/CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2/D7/BA /BY /D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU /BZ/BD /BPeη1/B7eη2 /CP/D9/D7/CT/D7 /D8/CW/CT/D8/CT/D6/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CT/D6/CX/CT/D7 /D3/D2 /AS/CU/D8/CW /CT/D5/D9/CP/D8/CX/D3/D2 /B4/DD /D3/D9 /CP/D2 /D4/D6/D3 /DA /CT /D8/CW/CX/D7 /D7/D8/CP/D8/CT/D1/CT/D2 /D8 /CQ /DD/D9/D7/CX/D2/CV /D3/CU /D8/CW/CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CP/CQ /D3 /DA /CT /D4/D6/D3 /CT/CS/D9/D6/CT/D7/B5/BA /CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2/DE /CP/D2/CS /CX/D7 /CP/D0/D0/CT/CS /CP/D7 /D8/CW/CT /D7/CT /D3/D2/CS /D3/D6 /CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /BA >/BG/B6/CT/DC/D4/B4/B9/C1/B6/DE/BB/BE/B5/B4 /D3/D7/CW/B4/BF/B6/D8/B5/B7/BF/B6 /CT/DC/D4/B4 /B9/BG/B6/C1 /B6/DE/B5/B6 /D3/D7/CW /B4/D8/B5/B5 /BB/CK >/B4 /D3/D7/CW/B4/BG/B6/D8/B5/B7/BG/B6 /D3/D7/CW/B4/D8/B5/B7/BF/B6 /D3/D7 /B4/BG/B6/DE /B5/B5/BM >/AZ /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2 >/D4/D0/D3/D8/BF/CS/B4/CP/CQ/D7/B4/B1/B5/CM/BE/B8/D8/BP/B9/BE/BA/BA/BE/B8/DE/BP/BC/BA /BA/BH/B8 /D3/D0/D3/D6 /BP/D6/CT/CS /B8/CP/DC/CT /D7/BP/CQ /D3/DC/CT/CS /B8/CK >/D8/CX/D8/D0/CT/BP/CO/D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BJ/B8 /D4/CP/CV/CT /BG/BI/BD/BG/C1/D2 /D8/CW/CT /CP/D7/CT /D3/CU /CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D8/CW/CT /D1/CP/CX/D2 /CU/CT/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /CS/CT/D7 /D6/CX/CQ /CT/CS/CP/CQ /D3 /DA /CT /D1/CT/D8/CW/D3 /CS /CX/D7 /D8/CW/CT /D8/CT/D6/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D3/D6/D1/CP/D0 /D7/CT/D6/CX/CT/D7 /CU/D3/D6 /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /D3/D6/CS/CT/D6 /D3/CU/D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/BA /CB/D9 /CW /CQ /CT/CW/CP /DA/CX/D3/D6 /D8/CT/D7/D8/CX/AS/CT/D7 /CP/CQ /D3/D9/D8 /DA /CT/D6/DD /D6/CX /CW /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /D7/D8/D6/D9 /D8/D9/D6/CT/D3/CU /D8/CW/CT /CS/DD/D2/CP/D1/CX /CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/BM /CP/D2 /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /D8/CW/CT /CX/D2/AS/D2/CX/D8/CT/D0/DD /D1/CP/D2 /DD /D2/D3/D2 /D8/D6/CX/DA/CX/CP/D0 /D7/DD/D1/B9/D1/CT/D8/D6/CX/CT/D7 /CP/D2/CS /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB/D7/B8 /C8 /CP/CX/D2/D0/CT/DA /CT /D4/D6/D3/D4 /CT/D6/D8 /DD /CP/D2/CS /CX/D2 /D8/CT/CV/D6/CP/CQ/CX/D0/CX/D8 /DD /CQ /DD /D1/CT/CP/D2/D7/D3/CU /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D7 /CP/D8/D8/CT/D6/CX/D2/CV /D1/CT/D8/CW/D3 /CS/BA /CC/CW/CT /D2/D3/D2/B9/CS/CT /CP /DD/CX/D2/CV /D4/D9/D0/D7/CT/B9/D0/CX/CZ /CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU/D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/CQ/D0/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CT/DA /D3/D0/D9/D8/CX/D3/D2/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CP/D0/D0/CT/CS /CP/D7 /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2/D7 /BA /BU/D9/D8/CP/D7 /DB /CT /DB/CX/D0/D0 /D7/CT/CT /D0/CP/D8/CT/D6/B8 /D7/D3/D1/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CW/CP /DA /CT /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2/B9/D0/CX/CZ /CT /D7/D3/D0/D9/D8/CX/D3/D2/D7/B8/CQ/D9/D8 /CS/D3 /D2/D3/D8 /CQ /CT/D0/D3/D2/CV /D8/D3 /CX/D2 /D8/CT/CV/D6/CP/CQ/D0/CT /D0/CP/D7/D7/BA /BY /D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CU/D3/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV /CT/D5/D9/CP/D8/CX/D3/D2 /B4/D7/CT/CT /D2/CT/DC/D8 /D7/CT /D8/CX/D3/D2/B5 /D8/CW/CT/D6/CT /CX/D7 /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2/B9/D0/CX/CZ /CT /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT/AS/D6/D7/D8 /D3/D6/CS/CT/D6 /D3/CU /D8/CW/CT /C0/CX/D6/D3/D8/CP/B3/D7 /D1/CT/D8/CW/D3 /CS/BA /BU/D9/D8 /D8/CW/CT /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D7/D3/D0/D9/D8/CX/D3/D2 /CS/D3 /CT/D7 /D2/D3/D8/D0/CT/CP/CS /D8/D3 /D8/CW/CT /D8/CT/D6/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CT/D6/CX/CT/D7/BA /CB/D9 /CW /D7/D3/D0/CX/D8/D3/D2/B9/D0/CX/CZ /CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D2/D3/D2/B9/CX/D2 /D8/CT/CV/D6/CP/CQ/D0/CT /CS/DD/D2/CP/D1/CX /CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D6/CT /CP/D0/D0/CT/CS /CP/D7 /D8/CW/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/D7 /BA/BH /C6/D3/D2/D0/CX/D2/CT/CP/D6 /C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV /CT/D5/D9/CP/D8/CX/D3/D2/BM/D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2/C0/CT/D6/CT /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /CP /D7/D3/D0/CX/D8/D3/D2/B9/D0/CX/CZ /CT /D4/D9/D0/D7/CT/B8 /DB/CW/CX /CW /CX/D7 /CV/CT/D2/CT/D6/CP/D8/CT/CS /CX/D2 /D8/CW/CT /D3/D2 /D8/CX/D2 /D9/D3/D9/D7/DB /CP /DA /CT /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6 /CS/D9/CT /D8/D3 /D4 /D3 /DB /CT/D6/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CS/CX/AR/D6/CP /D8/CX/D3/D2/D0/D3/D7/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D8/CW/CT /AS/CT/D0/CS /D7/CT/D0/CU/B9/CU/D3 /D9/D7/CX/D2/CV /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CJ /C0/BA/BT/BA /C0/CP/D9/D7/B8/C2/BA/BZ/BA /BY /D9/CY/CX/D1/D3/D8/D3/B8 /CP/D2/CS /BX/BA/C8/BA /C1/D4/D4 /CT/D2/B8 Ꜽ/BT /D2/CP/D0/DD/D8/CX /D8/CW/CT /D3/D6/DD /D3/CU /CP/CS/CS/CX/D8/CX/DA/CT /D4/D9/D0/D7/CT /CP/D2/CS /C3/CT/D6/D6/D0/CT/D2/D7 /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CVꜼ/B8 /C1/BX/BX/BX /C2/BA /C9/D9/CP/D2/D8/D9/D1 /BX/D0/CT /D8/D6 /D3/D2/CX /D7/B8 /DA/BA /BE/BK/B8 /C6/D3/BA /BD/BC/B8 /D4/D4/BA /BE/BC/BK/BI/B9/BD/BC/BL/BI /B4/BD/BL/BL/BE/B5 ℄/BA /CC/CW/CT /CP /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CP/D8/D9/D6/CP/CQ/D0/CT /D0/D3/D7/D7 /CP/D2 /CQ /CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT /D6/CT/CP/D0 /D9/CQ/CX /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D8/CT/D6/D1/BA /CC/CW/CT /CT/D2/CT/D6/CV/DD /CS/CX/D7/D7/CX/D4/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /D7/D4 /CT /D8/D6/CP/D0 /AS/D0/D8/CT/D6/CX/D2/CV /CP/D2/CQ /CT /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT /D6/CT/CP/D0 /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /D3/D2 /D8 /BA /CC/CW/CT/D2 /CX/D2/D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /CB/C8/C5 /CP/D2/CS /BZ/BW/BW/B8 /D8/CW/CT /CS/DD/D2/CP/D1/CX /CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /CP /CP/D2/CP/D0/D3/CV /D3/CU /D8/CW/CT/CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/B8 /CQ/D9/D8 /DB/CX/D8/CW /D4/D9/D6/CT /D6/CT/CP/D0 /D8/CT/D6/D1/D7/BA ∂ ∂zρ(z, t) /BP /CVρ(z, t) /B7tf2∂2 ∂t2ρ(z, t) /B7σ ρ(z, t)3/C0/CT/D6/CT /CV /CX/D7 /D8/CW/CT /D2/CT/D8/B9/CV/CP/CX/D2 /CX/D2 /D8/CW/CT /D0/CP/D7/CT/D6 /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /CV/CP/CX/D2 /CP/D2/CS /D0/CX/D2/CT/CP/D6/D0/D3/D7/D7 /CX/D2 /D8/CW/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8 /D3/D9/D8/D4/D9/D8 /D0/D3/D7/D7/B8 /CP/D2/CS /CS/CX/AR/D6/CP /D8/CX/D3/D2 /D0/D3/D7/D7/BA /BY /D3/D6 /D7/CP/CZ /CT /D3/CU /D7/CX/D1/D4/D0/CX/B9/AS /CP/D8/CX/D3/D2/B8 /DB /CT /DB/CX/D0/D0 /D7/D9/D4/D4 /D3/D7/CT /D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT /D8/D3 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW/D3/CU /D8/CW/CT /D7/D4 /CT /D8/D6/CP/D0 /AS/D0/D8/CT/D6tf /B4/D0/CT/D8tf /BP /BE/BA/BH /CU/D7/B8 /D8/CW/CP/D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CU/D9/D0/D0 /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/CQ/CP/D2/CS/DB/CX/CS/D8/CW /D3/CU /CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT /D0/CP/D7/CT/D6/B5 /CP/D2/CS /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D4/D9/D0/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D8/D3 /D8/CW/CT/CX/D2 /DA /CT/D6/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D3/CU /D8/CW/CT /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2σ /B4/D8/CW/CT /D8 /DD/D4/CX /CP/D0 /DA /CP/D0/D9/CT/D7 /D3/CUσ /CP/D6/CT /DI10−10/B910−12cm2/BB/CF/B5/BA /CF /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D4/D9/D0/D7/CT /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2/B8 /D8/CW/CT/D2 ∂ ∂zρ(z, t) /BP /BC/BA /CB/D3/B8 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D2 /CQ /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /D8/D3 /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2/BW/D9Ꜷ/D2/CV/B3/D7 /D8 /DD/D4 /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW/D3/D9/D8 /CS/CP/D1/D4/CX/D2/CV/BM >/D6/CT/D7/D8/CP/D6/D8/BM/BD/BH>/DB/CX/D8/CW/B4/D4/D0/D3/D8/D7/B5/BM >/DB/CX/D8/CW/B4/BW/BX/D8/D3/D3/D0/D7/B5/BM >/D3/CS/CT /BM/BP /CS/CX/CU/CU/B4/D6/CW/D3/B4/D8/B5/B8/CO/B0/CO/B4/D8/B8/BE/B5/B5 /B7 /D6/CW/D3/B4/D8/B5/CM/BF /B7 /CV/B6/D6/CW/D3/B4/D8/B5/BN ode:= (∂2 ∂t2ρ(t)) +ρ(t)3+gρ(t)/C1/D8/D7 /CX/D1/D4/D0/CX /CX/D8 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CP/D6/CT/BM >/D7/D3/D0 /BM/BP /CS/D7/D3/D0/DA/CT/B4/D3/CS/CT/BP/BC/B8/D6/CW/D3/B4/D8/B5/B5/BN sol:=−2/integraldisplayρ(t) 1/radicalBig −2 /CNa4−4g /CNa2+ 4 /CNC1d /CNa−t− /CNC2= 0, 2/integraldisplayρ(t) 1/radicalBig −2 /CNa4−4g /CNa2+ 4 /CNC1d /CNa−t− /CNC2= 0/CC/CW/CT /AS/D6/D7/D8 /CX/D2 /D8/CT/CV/D6/CP/D0 /D3/CU /D1/D3/D8/CX/D3/D2 /CX/D7/BM >/D2/D9/D1/CT/D6/B4/CS/CX/CU/CU/B4/D0/CW/D7/B4/D7/D3/D0/CJ/BD℄/B5/B8/D8/B5/B5 /BM >/CX/D2/D8/CN/D1/D3/D8/CX/D3/D2 /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/B4/D3/D4/B4/BD/B8/B1/B5/CM/BE/B9/D3/D4/B4/BE/B8/B1/B5 /CM/BE/B5/BB /BE/B5/BN int /CNmotion := 2 (∂ ∂tρ(t))2+ρ(t)4+ 2gρ(t)2−2 /CNC1/CC/CW/CT/D7/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /D1/D3/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/BM >/D4/D3/D8 /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/D3/D4/B4/BE/B8/CX/D2/D8/CN/D1/D3/D8/CX/D3/D2/B5/B7/D3 /D4/B4/BF/B8 /CX/D2/D8/CN /D1/D3/D8/CX /D3/D2/B5 /B5/BN pot:=ρ(t)4+ 2gρ(t)2/CC/CW/CT /DA /CP/D0/D9/CT /D3/CU /BG/B6/CN/BV/BD /D4/D0/CP /DD/D7 /CP /D6/D3/D0/CT /D3/CU /D8/CW/CT /CU/D9/D0/D0 /CT/D2/CT/D6/CV/DD /D3/CU /D7/DD/D7/D8/CT/D1/BA /CC/CW/CT /CS/CT/D4 /CT/D2/B9/CS/CT/D2 /CT /D3/CU /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D3/D2ρ /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /CV /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /D2/CT/DC/D8 /AS/CV/D9/D6/CT/BM >/D4/D0/D3/D8/BF/CS/B4/D7/D9/CQ/D7/B4/D6/CW/D3/B4/D8/B5/BP/D6/CW/D3/B8/D4/D3/D8 /B5/B8/CV/BP /BC/BA/BC/BH /BA/BA/B9/BC /BA/BD/B8/D6 /CW/D3/BP/B9 /BC/BA/BH /BA/BA/BC/BA /BH/B8/CK >/CP/DC/CT/D7/BP/CQ/D3/DC/CT/CS/B8/D8/CX/D8/D0/CT/BP/CO/C8/D3/D8/CT/D2/D8/CX/CP /D0 /D3/CU /D4/CT/D2/CS/D9/D0/D9/D1/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BK/B8 /D4/CP/CV/CT /BG/BJ/CF /CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CP/D8 /CU/D3/D6 /CV> /BC /D8/CW/CT/D6/CT /CX/D7 /D3/D2/CT /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT /D3/CU /D4 /CT/D2/CS/D9/D0/D9/D1 /CU/D3/D6 ρ /BP /BC /B4/D7/D8/CP/CQ/D0/CT /D7/D8/CP/D8/CT/B5/B8 /CP/D2/CS /CU/D3/D6 /CV< /BC /D8/CW/CT/D6/CT /CP/D6/CT /D8/CW/D6/CT/CT /D3/D2/CT /B4/D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6ρ /BP /BC/CP/D2/CS /D7/D8/CP/CQ/D0/CT /CU/D3/D6ρ /BP /B7/BB/B9√−g /B5 /BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD /B8 /D8/CW/CP/D8 /CX/D2 /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D6/CT/CV/CX/D1/CT/D7 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT/B8 /D8/CW/CP/D8 /CX/D7 /CX/D0/D0/D9/D7/D8/D6/CP/D8/CT/CS /CQ /DD /D8/CW/CT /D4/CW/CP/D7/CT /D4 /D3/D6/D8/D6/CP/CX/D8 /D3/D2 /D8/CW/CT/D4/D0/CP/D2/CT Ꜽ /DD< /AM> /DE/BP/CSρ /BB/CS/D8 Ꜽ /BA >/D7/DD/D7 /BM/BP /D3/D2/DA/CT/D6/D8/D7/DD/D7/B4/D3/CS/CT /BP /BC/B8/CJ ℄/B8/D6/CW/D3/B4/D8/B5/B8/D8/B8/DE/B5/BN >/CS/CU/CX/CT/D0/CS/D4/D0/D3/D8/B4/CJ/CS/CX/CU/CU/B4/D6/CW/D3/B4/D8/B5/B8/D8 /B5/BP/DE/B4 /D8/B5/B8/CK >/CS/CX/CU/CU/B4/DE/B4/D8/B5/B8/D8/B5/BP/B9/D6/CW/D3/B4/D8/B5/CM/BF/B9/D7/D9/CQ/D7/B4 /CV/BP/B9/BC /BA/BD/B8/CV /B5/B6/CK >/D6/CW/D3/B4/D8/B5℄/B8 /CJ/DE/B4/D8/B5/B8/D6/CW/D3/B4/D8/B5℄/B8/D8/BP/B9/BE/BA/BA/BE/B8/D6/CW/D3/BP/B9/BC /BA/BH/BA/BA /BC/BA/BH/B8 /CK >/DE/BP/B9/BC/BA/BD/BA/BA/BC/BA/BD/B8/CP/D6/D6/D3/DB/D7/BP/C4/BT/CA/BZ/BX/B8/CP /DC/CT/D7/BP /CQ/D3/DC/CT /CS/B8/D8/CX /D8/D0/CT/BP /CO/C6/D3/D2 /D0/CX/D2 /CT/CP/D6/CK >/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/CO/B8 /D3/D0/D3/D6/BP/CQ/D0/CP /CZ/B5 /BN sys:= [[YP1=z2,YP2=−z13−gz1],[z1=ρ(t), z2= ∂ ∂tρ(t)],undeﬁned,[]]/BD/BI/CB/CT/CT /BY/CX/CV/D9/D6/CT /BL/B8 /D4/CP/CV/CT /BG/BK/CC/CW/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D3/D7 /CX/D0/B9/D0/CP/D8/CX/D3/D2 /CP/D6/D3/D9/D2/CS /D8/CW/CT /D7/D8/CP/CQ/D0/CT /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT /DB/CX/D8/CW /CX/D2/AS/D2/CX/D8/CT /D4 /CT/D6/CX/D3 /CS/BA /C1/D2 /D8/CW/CX/D7 /CP/D7/CT/D8/CW/CT /CU/D9/D0/D0 /CT/D2/CT/D6/CV/DD /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /BC/BA /CC/CW/CT/D2 /CN/BV/BD /BP /BC /CP/D2/CS /D8/CW/CT /D1/D3/D8/CX/D3/D2 /CQ /CT/CV/CX/D2/D7 /CU/D6/D3/D1ρ/BP /BC /CP/D8 /D8 /BP /B9∞ /BM >/D4/D0/D3/D8/B4/DF/D7/D5/D6/D8/B4/D7/D9/CQ/D7/B4/DF/D6/CW/D3/B4/D8/B5/BP/D6/CW /D3/B8/CV/BP /B9/BC/BA/BD /DH/B8/CK >/B9/D4/D3/D8/B5/B5/BB/BE/B8/B9/D7/D5/D6/D8/B4/D7/D9/CQ/D7/B4/DF/D6/CW/D3/B4/D8 /B5/BP/D6/CW /D3/B8/CV/BP /B9/BC/BA/BD /DH/B8/B9/D4 /D3/D8/B5/B5 /BB/BE/DH /B8/CK >/D6/CW/D3/BP/BC/BA/BA/BC/BA/BG/BH/B8/CP/DC/CT/D7/BP/CQ/D3/DC/CT/CS/B8/D0/CP/CQ /CT/D0/D7/BP /CJ/CO/D6/CW /D3/B4/D8/B5 /CO/B8/CO/CS /D6/CW/D3/B4 /D8/B5/BB /CS/D8/CO℄ /B8/CK > /D3/D0/D3/D6/BP/D6/CT/CS/B8/D2/D9/D1/D4/D3/CX/D2/D8/D7/BP/BE/BC/BC/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BD/BC/B8 /D4/CP/CV/CT /BG/BL/CC/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CU/D6/D3/D1 /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /D3/CU /D1/D3/D8/CX/D3/D2/B4/CX/BA /CT/BA /D4/D9/D0/D7/CT /D1/CP/DC/CX/D1 /D9/D1 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CSρ /BB/CS/D8 /BP /BC /CP/D2/CS /CN/BV/BD /BP /BC /CX/D2 /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0/D3/CU /D1/D3/D8/CX/D3/D2/B5/BA >/D6/CW/D3/BC /BM/BP /D7/D3/D0/DA/CT/B4/CU/CP /D8/D3/D6/B4/D4/D3/D8/B5/BB/D6/CW/D3/B4/D8/B5/CM/BE /BP/BC/B8/D6 /CW/D3/B4/D8 /B5/B5/BN ρ0 :=/radicalBig −2g,−/radicalBig −2g/CC/CW/CT /CT/DC/D4/D0/CX /CX/D8 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D4/D6/D3 /CS/D9 /CT/D7/BM >/CP/D7/D7/D9/D1/CT/B4/CV/BO/BC/B5/BM >/D7/D3/D0/BD/CN/CP /BM/BP /D2/D9/D1/CT/D6/B4/DA/CP/D0/D9/CT/B4/D7/D9/CQ/D7/B4/CN/BV/BD/BP/BC/B8/D0/CW/D7/B4 /D7/D3/D0/CJ /BD℄/B5/B5 /B5/B5/BN >/D7/D3/D0/BD/CN/CQ /BM/BP /D2/D9/D1/CT/D6/B4/DA/CP/D0/D9/CT/B4/D7/D9/CQ/D7/B4/CN/BV/BD/BP/BC/B8/D0/CW/D7 /B4/D7/D3/D0 /CJ/BE℄/B5 /B5/B5/B5 /BN sol1 /CNa:=−ρ(t)/radicalBig −2ρ(t)2−4g˜arctanh(2g˜ √−g˜/radicalBig −2ρ(t)2−4g˜) −t/radicalBig −2ρ(t)4−4g˜ρ(t)2/radicalBig −g˜− /CNC2/radicalBig −2ρ(t)4−4g˜ρ(t)2/radicalBig −g˜ sol1 /CNb:=ρ(t)/radicalBig −2ρ(t)2−4g˜arctanh(2g˜ √−g˜/radicalBig −2ρ(t)2−4g˜) −t/radicalBig −2ρ(t)4−4g˜ρ(t)2/radicalBig −g˜− /CNC2/radicalBig −2ρ(t)4−4g˜ρ(t)2/radicalBig −g˜/C5/CP/CZ /CT /D7/D3/D1/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/BM >/D7/D3/D0/BE/CN/CP /BM/BP /D2/D9/D1/CT/D6/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/CT/DC/D4/CP/D2/CS/B4/D7/D3/D0/BD/CN /CP/BB/D3/D4 /B4/BD/B8/D7 /D3/D0/BD /CN/CP/B5/B5 /B8/CK >/D6/CP/CS/CX /CP/D0/B8/D7/DD/D1/CQ/D3/D0/CX /B5/B5/BN >/D7/D3/D0/BE/CN/CQ /BM/BP /D2/D9/D1/CT/D6/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/CT/DC/D4/CP/D2/CS/B4/D7/D3/D0/BD/CN /CQ/BB/D3/D4 /B4/BD/B8/D7 /D3/D0/BD /CN/CQ/B5/B5 /B8/CK >/D6/CP/CS/CX /CP/D0/B8/D7/DD/D1/CQ/D3/D0/CX /B5/B5/BN sol2 /CNa:= arctanh(2g˜ √−g˜/radicalBig −2ρ(t)2−4g˜) +t/radicalBig −g˜ + /CNC2/radicalBig −g˜ sol2 /CNb:= arctanh(2g˜ √−g˜/radicalBig −2ρ(t)2−4g˜)−t/radicalBig −g˜− /CNC2/radicalBig −g˜/BD/BJ/CF/CW/CT/D2 /CP/D8 /D8/CW/CT /D4/D9/D0/D7/CT /D1/CP/DC/CX/D1 /D9/D1ρ /B4/BC/B5 /BPρmax /B8 /D8/CW/CT /DA /CP/D0/D9/CT /D3/CU /CN/BV/BE /CP/D2 /CQ /CT /CU/D3/D9/D2/CS/CP/D7/BM >/CX/CN/BV/BE/CN/CP /BM/BP /D7/D3/D0/DA/CT/B4/D7/D9/CQ/D7/B4/DF/D8/BP/BC/B8/D6/CW/D3/B4/D8/B5/BP/D6/CW/D3/CN/D1 /CP/DC/DH/B8 /D7/D3/D0/BE /CN/CP/B5 /BP/BC/B8/CN /BV/BE/B5/BN >/CX/CN/BV/BE/CN/CQ /BM/BP /D7/D3/D0/DA/CT/B4/D7/D9/CQ/D7/B4/DF/D8/BP/BC/B8/D6/CW/D3/B4/D8/B5/BP/D6/CW/D3/CN/D1 /CP/DC/DH/B8 /D7/D3/D0/BE /CN/CQ/B5 /BP/BC/B8/CN /BV/BE/B5/BN i /CNC2 /CNa:=−arctanh(2g˜ √−g˜/radicalBig −2rho /CNmax2−4g˜) √−g˜ i /CNC2 /CNb:=arctanh(2g˜ √−g˜/radicalBig −2rho /CNmax2−4g˜) √−g˜/C4/CT/D8 /D8/CP/D2/CW/B4/CX/CN/BV/BE/CN/CP/B6/D7/D5/D6/D8/B4/B9/CV/B5/B5 /BP /D8/CP/D2/CW/B4/B9/CX/CN/BV/BE/CN/CQ/B6/D7/D5/D6/D8/B4/B9/CV/B5 /B5 /BP /B9/BB/B7υ /B8 /D8/CW/CT/D2/CU/D6/D3/D1 /CP/D2 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D8/CP/D2/CV/CT/D2 /D8/D7 /D3/CU /D7/D9/D1 /D3/CU /CP/D6/CV/D9/D1/CT/D2 /D8/D7/BM /D8/CP/D2/CW/B4/CP /B7 /CQ/B5 /BP/D8/CP/D2/CW/B4/CP/B5 /B7 /D8/CP/D2/CW/B4/CQ/B5 /BB/B4/BD /B7 /D8/CP/D2/CW/B4/CP/B5/B6/D8/CP/D2/CW/B4/CQ/B5 /B5/B8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D2 /CQ /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS/CP/D7/BM >/D7/D3/D0/BF/CN/CP /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/D8/CP/D2/CW/B4/D3/D4/B4/BD/B8/D7/D3/D0/BE/CN/CP/B5/B5 /B5/BP/CK >/B4/D8/CP/D2/CW/B4/B9/D3/D4/B4/BE/B8/D7/D3/D0/BE/CN/CP/B5/B5/B7/D9/D4/D7/CX/D0 /D3/D2/B5/BB /CK >/B4/BD/B7/D8/CP/D2/CW/B4/B9/D3/D4/B4/BE/B8/D7/D3/D0/BE/CN/CP/B5/B5/B6/D9/D4/D7 /CX/D0/D3/D2 /B5/BN >/D7/D3/D0/BF/CN/CQ /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/D8/CP/D2/CW/B4/D3/D4/B4/BD/B8/D7/D3/D0/BE/CN/CQ/B5/B5 /B5/BP/CK >/B4/D8/CP/D2/CW/B4/B9/D3/D4/B4/BE/B8/D7/D3/D0/BE/CN/CQ/B5/B5/B7/D9/D4/D7/CX/D0 /D3/D2/B5/BB /CK >/B4/BD/B7/D8/CP/D2/CW/B4/B9/D3/D4/B4/BE/B8/D7/D3/D0/BE/CN/CQ/B5/B5/B6/D9/D4/D7 /CX/D0/D3/D2 /B5/BN sol3 /CNa:= 2g˜ √−g˜/radicalBig −2ρ(t)2−4g˜=−tanh(t√−g˜) +υ 1−tanh(t√−g˜)υ sol3 /CNb:= 2g˜ √−g˜/radicalBig −2ρ(t)2−4g˜=tanh(t√−g˜) +υ 1 + tanh(t√−g˜)υ >/D7/D3/D0/BG/CN/CP /BM/BP /D7/D3/D0/DA/CT/B4/D7/D3/D0/BF/CN/CP/B8 /D6/CW/D3/B4/D8/B5/B5/BN >/D7/D3/D0/BG/CN/CQ /BM/BP /D7/D3/D0/DA/CT/B4/D7/D3/D0/BF/CN/CQ/B8 /D6/CW/D3/B4/D8/B5/B5/BN sol4 /CNa:= 2√2g˜−2g˜υ2e(t√−g˜) −%1 + 1 +υ%1 +υ,−2√2g˜−2g˜υ2e(t√−g˜) −%1 + 1 +υ%1 +υ %1 := (e(t√−g˜))2 sol4 /CNb:= 2√2g˜−2g˜υ2e(t√−g˜) %1−1 +υ%1 +υ,−2√2g˜−2g˜υ2e(t√−g˜) %1−1 +υ%1 +υ %1 := (e(t√−g˜))2/C6/D3 /DB/B8 /DB /CT /CP/D6/CT /D8/D3 /D2/D3/D8/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8ρmax /B9>ρ /BC /B4/D7/CT/CT /CP/CQ /D3 /DA /CT/B5 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7/D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8υ /AL>∞ /BA /CC/CW/CT/D2 /D8/CW/CT /AS/D2/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CP/D6/CT /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /D8/CW/CT /D2/CT/DC/D8/D3/D4 /CT/D6/CP/D8/CX/D3/D2/D7/BM >/D7/D3/D0/CN/CU/CX/D2/CN/BD /BM/BP /D0/CX/D1/CX/D8/B4/D7/D3/D0/BG/CN/CP/CJ/BD℄/B8/D9/D4/D7/CX/D0/D3/D2/BP/CX /D2/CU/CX/D2 /CX/D8/DD/B5 /BN >/D7/D3/D0/CN/CU/CX/D2/CN/BE /BM/BP /D0/CX/D1/CX/D8/B4/D7/D3/D0/BG/CN/CP/CJ/BE℄/B8/D9/D4/D7/CX/D0/D3/D2/BP/CX/D2/CU/CX /D2/CX/D8/DD /B5/BN >/D7/D3/D0/CN/CU/CX/D2/CN/BD /BM/BP /D0/CX/D1/CX/D8/B4/D7/D3/D0/BG/CN/CQ/CJ/BD℄/B8/D9/D4/D7/CX/D0/D3/D2/BP/CX/D2/CU /CX/D2/CX/D8 /DD/B5/BN/BD/BK>/D7/D3/D0/CN/CU/CX/D2/CN/BE /BM/BP /D0/CX/D1/CX/D8/B4/D7/D3/D0/BG/CN/CQ/CJ/BE℄/B8/D9/D4/D7/CX/D0/D3/D2/BP/CX/D2 /CU/CX/D2/CX /D8/DD/B5 /BN sol /CNﬁn /CN1:= 2√−2g˜e(t√−g˜) e(2t√−g˜)+ 1 sol /CNﬁn /CN2:=−2√−2g˜e(t√−g˜) e(2t√−g˜)+ 1 sol /CNﬁn /CN1:= 2√−2g˜e(t√−g˜) e(2t√−g˜)+ 1 sol /CNﬁn /CN2:=−2√−2g˜e(t√−g˜) e(2t√−g˜)+ 1/CC/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT /D6/D3 /D3/D8/D7 /D7/CP/D8/CX/D7/CU/DD /D8/D3 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /D6/CT/D7/D9/D0/D8 /CX/D7 /D8/CW/CT/D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /D4/D9/D0/D7/CT /DB/CX/D8/CW /D7/CT /CW /B9 /D7/CW/CP/D4 /CT /CT/D2 /DA /CT/D0/D3/D4 /CT/B8 /DB/CW/CX /CW /CW/CP/D7 /D8/CW/CT /CS/D9/D6/CP/D8/CX/D3/D2 /BD/BB√−g /CP/D2/CS /D8/CW/CT /D4 /CT/CP/CZ /CP/D1/D4/D0/CX/D8/D9/CS/CTρ /BC /BP√−2g /BA /CC/CW/CT /D4/D9/D0/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D4/D6/D3/AS/D0/CT /CX/D7/D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /D2/CT/DC/D8 /AS/CV/D9/D6/CT /B4σ /BP10−11cm2/BB/CF/B8ι /CX/D7 /D8/CW/CT /D8/CX/D1/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3tf /B5/BM >/CP/D2/CX/D1/CP/D8/CT/B4/CT/DA/CP/D0/CU/B4/D7/D9/CQ/D7/B4/D8/BP/CX/D3/D8/CP/BB /B4/BE/BA/BH /CT/B9/BD/BH /B5/B8/D7/D3 /D0/CN/CU/CX /D2/CN/BD/B5 /CM/BE/B6 /BD/BC/BC/B5 /B8 /CK >/CX/D3/D8/CP/BP/B9/BD/CT/B9/BD/BF/BA/BA/BD/CT/B9/BD/BF/B8/CV/BP/B9/BC/BA/BC/BH /BA/BA/B9/BC /BA/BC/BD/B8 /CU/D6/CP/D1 /CT/D7/BP/BH /BC/B8/CK >/CP/DC/CT/D7/BP/CQ/D3/DC/CT/CS/B8 /D3/D0/D3/D6/BP/D6/CT/CS/B8/D0/CP/CQ/CT/D0 /D7/BP/CJ/CO /D8/CX/D1/CT /B8 /CU/D7/CO/B8/CO/D6/CW/D3/CM/BE/B8 /CK >/BZ/CF/BB /D1/CM/BE/CO℄/B8/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CT/D2/DA/CT/D0/D3/D4/CT/CO/B8/CU/D6/CP/D1/CT/D7/BP/BG/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/B1/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BD/BD/B8 /D4/CP/CV/CT /BH/BC/CB/D3/B8 /D8/CW/CT/D6/CT /CX/D7 /D8/CW/CT /D0/D3/D7/CT /CP/D2/CP/D0/D3/CV/DD /CQ /CT/D8 /DB /CT/CT/D2 /CP/D2 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CX/D2/D8/CW/CT /D0/CP/D7/CT/D6 /DB/CX/D8/CW /D4 /D3 /DB /CT/D6/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CP/D2 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6/D4 /CT/D2/CS/D9/D0/D9/D1/BA /CC/CW/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /CW/CP/D7 /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6 /D3/CU /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/BA /BU/D9/D8 /CX/D2 /D8/CW/CT /D2/CT/DC/D8 /D7/CT /D8/CX/D3/D2 /DB /CT /CP/D6/CT /CV/D3/CX/D2/CV /D8/D3 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D7/D3/D1/CT /CP/D2/CP/D0/D3/CV/CX/CT/D7/D3/CU /D8/CW/CT /CW/CX/CV/CW/B9/D3/D6/CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /CQ /CT/CW/CP /DA/CX/D3/D6 /B4/D0/CP/D7/CT/D6 /CQ/D6/CT/CT/DE/CT/D6/D7/B5/BA/BI /BT/D9/D8/D3 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/D7 /CX/D2 /D8/CW/CT /D0/CP/D7/CT/D6/CC/CW/CT /D1/CP/CX/D2 /CT/AR/D3/D6/D8/D7 /CX/D2 /D8/CW/CT /CS/CT/D7/CX/CV/D2 /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D6/D2 /CU/CT/D1 /D8/D3/D7/CT /D3/D2/CS /D0/CP/D7/CT/D6/D7 /CP/CX/D1 /D8/D3 /D8/CW/CT/D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D9/D0/D8/D6/CP /D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /BY /D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CP/D7 /CX/D8 /DB /CP/D7/D7/CW/D3 /DB/D2 /CX/D2 /CJ /BT/BA /C5/BA /CB/CT/D6 /CV/CT /CT/DA/B8 /BX/BA /CE/BA /CE /CP/D2/CX/D2/B8 /BY/BA /CF/BA /CF/CX/D7/CT/B8 /CB/D8/CP/CQ/CX/D0/CX/D8/DD /D3/CU /D4 /CP/D7/D7/CX/DA/CT/D0/DD/D1/D3 /CS/CT/B9/D0/D3 /CZ/CT /CS /D0/CP/D7/CT/D6/D7 /DB/CX/D8/CW /CU/CP/D7/D8 /D7/CP/D8/D9/D6 /CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/D7/B8 /C7/D4/D8/BA /BV/D3/D1/D1/D9/D2/BA/B8 /BD/BG/BC/B8 /BI/BD/B4/BD/BL/BL/BJ/B5 ℄/B8 /D8/CW/CT /D4/D9/D0/D7/CT /CS/CT/D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2/D7 /CP/D6/CT /DA /CT/D6/DD /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /CU/CP /D8/D3/D6/D7 /D0/CX/D1/CX/D8/CX/D2/CV /D8/CW/CT/D3/D4 /CT/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CP/D7/CT/D6 /DB/CX/D8/CW /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /B4/D8/CW/CT /CQ/D0/CT/CP /CW/CX/D2/CV /D3/CU /CS/CX/AR/D6/CP /D8/CX/D3/D2/D0/D3/D7/D7 /CS/D9/CT /D8/D3 /C3/CT/D6/D6/B9/D0/CT/D2/D7/CX/D2/CV /CX/D7 /D8/CW/CT /CT/DC/CP/D1/D4/D0/CT /D3/CU /D7/D9 /CW /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /D7/CT/CT /D4/D6/CT/B9/DA/CX/D3/D9/D7 /D4/CP/D6/D8/B5/BA /CC/CW/CT /CS/CT/D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CP/D7/CT/D6 /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /CP/D2 /D4/D6/D3 /CS/D9 /CT /CX/D8/D7/CS/CT/D7/D8/D6/D9 /D8/CX/D3/D2 /D3/D6 /D6/CT/CV/D9/D0/CP/D6/BB/D2/D3/D2/D6/CT/CV/D9/D0/CP/D6 /CP/D9/D8/D3 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/BA /CC/CW/CT /D0/CP/D7/D8 /CP/D2 /CQ /CT /D3/D2/D7/CX/CS/B9/CT/D6/CT/CS /CP/D7 /D8/CW/CT /CP/D2/CP/D0/D3/CV /D3/CU /CP /CW/CX/CV/CW/B9/D3/D6/CS/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /CU/D3/D6/D1/CP/D8/CX/D3/D2 /B4/D7/CT/CT /D7/CT /D8/CX/D3/D2 /BE/B5/B8 /DB/CW/CX /CW /CX/D7/D6/CT/D1/CX/D2/CX/D7 /CT/D2 /D8 /D3/CU /D8/CW/CT /CQ/D6/CT/CT/DE/CT/D6/D7 /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CS/DD/D2/CP/D1/CX /CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/BA/CF /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /D0/CP/D7/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/B8 /DB/CW/CX /CW /CY/D3/CX/D2/D7 /D8/CW/CT /C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV /CP/D2/CS /CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CC/CW/CT /D1/CP/CX/D2 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CU/CP /D8/D3/D6/D7 /D2/D3 /DB /CP/D6/CT/BD/BL/CB/C8/C5 /CP/D2/CS /D4 /D3 /DB /CT/D6/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/BA /BZ/BW/BW /CP/D2/CS /D7/D4 /CT /D8/D6/CP/D0 /AS/D0/D8/CT/D6/CX/D2/CV /DB/CX/D0/D0/CQ /CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2/B8 /D8/D3 /D3/BA >/D6/CT/D7/D8/CP/D6/D8/BM >/DB/CX/D8/CW/B4/BW/BX/D8/D3/D3/D0/D7/B5/BM >/DB/CX/D8/CW/B4/D4/D0/D3/D8/D7/B5/BM >/D1/CP/D7/D8/CT/D6/CN/BD /BM/BP /CS/CX/CU/CU/B4/D6/CW/D3/B4/DE/B8/D8/B5/B8/DE/B5 /BP/CK >/CP/D0/D4/CW/CP/B6/D6/CW/D3/B4/DE/B8/D8/B5/B9/CV/CP/D1/D1/CP/B6/D6/CW/D3/B4/DE /B8/D8/B5/B7 /C1/B6/D4/CW /CX/B6/D6/CW /D3/B4/DE/B8 /D8/B5/B7/CK >/D8/CU/CM/BE/B6/CS/CX/CU/CU/B4/D6/CW/D3/B4/DE/B8/D8/B5/B8/CO/B0/CO/B4/D8/B8/BE /B5/B5/B7/CK >/C1/B6/CZ/CN/BE/B6/CS/CX/CU/CU/B4/D6/CW/D3/B4/DE/B8/D8/B5/B8/CO/B0/CO/B4/D8/B8 /BE/B5/B5/B7 /D7/CX/CV/D1 /CP/B6/D6/CW /D3/B4/DE/B8 /D8/B5/CM/BE /B6/CK > /D3/D2/CY/D9/CV/CP/D8/CT/B4/D6/CW/D3/B4/DE/B8/D8/B5/B5/B9/CK >/C1/B6/CQ/CT/D8/CP/B6/D6/CW/D3/B4/DE/B8/D8/B5/CM/BE/B6 /D3/D2/CY/D9/CV/CP/D8 /CT/B4/D6/CW /D3/B4/DE/B8 /D8/B5/B5/BN master /CN1:=∂ ∂zρ(z, t) =αρ(z, t)−γρ(z, t) +Iφρ(z, t) +tf2(∂2 ∂t2ρ(z, t)) +Ik2(∂2 ∂t2ρ(z, t)) +σρ(z, t)2ρ(z, t)−Iβρ(z, t)2ρ(z, t)/C0/CT/D6/CTφ /CX/D7 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CT/D0/CP /DD /D3/D2 /D8/CW/CT /CU/D9/D0/D0 /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS /D8/D6/CX/D4/B8α /CP/D2/CSγ /CP/D6/CT /D8/CW/CT/CV/CP/CX/D2 /CP/D2/CS /D0/D3/D7/D7 /D3 /CTꜶ /CX/CT/D2 /D8/D7/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D0/DD /BA /CC/CW/CT /CV/CT/D2/CT/D6/CP/D0 /CT/DC/CP /D8 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CX/D7/CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D2/D3/D8 /CZ/D2/D3 /DB/D2/B8 /CQ/D9/D8 /D8/CW/CT/D6/CT /CX/D7 /D8/CW/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/CU/D3/D6/D1/BM >/CU/BD /BM/BP /B4/DE/B8/D8/B5/B9/BQ/D6/CW/D3/BC/B4/DE/B5/B6/D7/CT /CW/B4/D8/B6/D8/CP/D9/B4/DE/B5 /B5/CM/B4/BD /B7/C1/B6/D4 /D7/CX/B4/DE /B5/B5/BN f1:= (z, t)→ρ0(z) sech(tτ(z))(1+Iψ(z))/C0/CT/D6/CTρ /BC /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8 τ /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D4/D9/D0/D7/CT /DB/CX/CS/D8/CW/B8ψ /CX/D7 /D8/CW/CT /CW/CX/D6/D4/BA/CC/CW/CX/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CQ /CT/DD/D7 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D2∂ ∂zρ /BP /BC/B8/CX/BA/CT/BA /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT /D3/D2/D7/D8/CP/D2 /D8/BA /CC /D3 /CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /D2/D3/D2/D7/D8/CP/D8/CX/D3/D2/CP/D6/DD /D4/D9/D0/D7/CT/D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2/B8 /DB /CT /DB/CX/D0/D0 /D9/D7/CT /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2/D0/CT/D7/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BM /D8/CW/CT /CW/CP/D2/CV/CT/D7 /D3/CU/D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CS/D3 /D2/D3/D8 /CP/D9/D7/CT /D8/CW/CT /CP/CQ /CT/D6/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /CU/D3/D6/D1/BA /C6/CT/DC/D8/D7/D8/CT/D4 /CX/D7 /D8/CW/CT /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU /CU/BD /CX/D2 /D8/D3 /D1/CP/D7/D8/CT/D6/CN/BD /DB/CX/D8/CW /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D2 /D8/B9 /D7/CT/D6/CX/CT/D7/BA /CC/CW/CT /D3 /CTꜶ /CX/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D4/D6/D3 /CS/D9 /CT /D8/CW/CT /D7/CT/D8 /D3/CU /C7/BW/BX /CU/D3/D6 /D8/CW/CT/CT/DA /CP/D0/D9/CP/D8/CX/D2/CV /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA >/CP/D7/D7/D9/D1/CT/B4/D8/CP/D9/B4/DE/B5/B8/D6/CT/CP/D0/B5/BM >/CP/D7/D7/D9/D1/CT/B4/D8/B8/D6/CT/CP/D0/B5/BM >/CT/DC/D4/CP/D2/CS/B4/D0/CW/D7/B4/D7/D9/CQ/D7/B4/D6/CW/D3/B4/DE/B8/D8/B5/BP/CU /BD/B4/DE/B8 /D8/B5/B8/D1 /CP/D7/D8/CT /D6/CN/BD/B5 /B5/B9/CK >/D6/CW/D7/B4/D7/D9/CQ/D7/B4/D6/CW/D3/B4/DE/B8/D8/B5/BP/CU/BD/B4/DE/B8/D8/B5/B8 /D1/CP/D7/D8 /CT/D6/CN/BD /B5/B5/B5/BM >/CT/D5 /BM/BP/CK >/D7/D9/CQ/D7/B4/CK >/DF/CP/D0/D4/CW/CP/BP/CP/D0/D4/CW/CP/B4/DE/B5/B8/CS/CX/CU/CU/B4/D6/CW/D3/BC/B4 /DE/B5/B8/DE /B5/BP/CP/B8 /CK >/CS/CX/CU/CU/B4/D8/CP/D9/B4/DE/B5/B8/DE/B5/BP/CQ/B8/CS/CX/CU/CU/B4/D4/D7/CX/B4 /DE/B5/B8/DE /B5/BP /DH > /D3/D2/DA/CT/D6/D8/B4/D7/CT/D6/CX/CT/D7/B4/B1/B8/D8/BP/BC/B8/BF/B5/B8/D4/D3 /D0/DD/D2/D3 /D1/B5/B5/BM >/CP/D7/D7/D9/D1/CT/B4/D6/CW/D3/BC/B4/DE/B5/B8/D6/CT/CP/D0/B5/BM >/CT/D5/BD /BM/BP /CT/DA/CP/D0 /B4 /D3/CT/CU/CU/B4/CT/D5/B8/D8/CM/BE/B5/B5/BM >/CT/D5/BE /BM/BP /CT/DA/CP/D0 /B4 /D3/CT/CU/CU/B4/CT/D5/B8/D8/B5/B5/BM >/CT/D5/BF /BM/BP /CT/DA/CP/D0 /B4 /D3/CT/CU/CU/B4/CT/D5/B8/D8/B8/BC/B5/B5/BM/BE/BC>/CT/D5/BG /BM/BP /D3/CT/CU/CU/B4/CT/D5/BD/B8/C1/B5/BM >/CT/D5/BH /BM/BP /D3/CT/CU/CU/B4/CT/D5/BD/B8/C1/B8/BC/B5/BM >/CT/D5/BI /BM/BP /D3/CT/CU/CU/B4/CT/D5/BF/B8/C1/B5/BM >/CT/D5/BJ /BM/BP /D3/CT/CU/CU/B4/CT/D5/BF/B8/C1/B8/BC/B5/BM >/D7/D3/D0/DA/CT/B4/DF/CT/D5/BG/BP/BC/B8/CT/D5/BH/BP/BC/B8/CT/D5/BI/BP/BC/B8/CT /D5/BJ/BP/BC /DH/B8/DF/CP /B8/CQ/B8 /B8/D4/CW/CX /DH/B5/BM >/D7/DD/D7 /BM/BP /CS/CX/CU/CU/B4/D6/CW/D3/BC/B4/DE/B5/B8/DE/B5/BP/D7/D9/CQ/D7/B4/B1/B8/CP/B5 /B8/CK >/CS/CX/CU/CU/B4/D8/CP/D9/B4/DE/B5/B8/DE/B5/BP/D7/D9/CQ/D7/B4/B1/B8/CQ/B5/B8/CS /CX/CU/CU/B4 /D4/D7/CX/B4 /DE/B5/B8/DE /B5/BP/D7/D9 /CQ/D7 /B4/B1/B8 /B5/BM/CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/DD/D7/D8/CT/D1 /D7/DD/D7 /CW/CP /DA /CT /D8/D3 /CQ /CT /D7/D9/D4/D4/D0/CT/D1/CT/D2 /D8/CT/CS /DB/CX/D8/CW /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /CV/CP/CX/D2/CT/DA /D3/D0/D9/D8/CX/D3/D2/BM >/D7/DD/D7 /BM/BP /DF/B1/B8/CS/CX/CU/CU/B4/CP/D0/D4/CW/CP/B4/DE/B5/B8/DE/B5/BP/CK >/CP/D0/D4/CW/CP/B4/DE/B5/B6/CT/DC/D4/B4/B9/BE/B6/DC/CX/B6/D6/CW/D3/BC/B4/DE/B5 /CM/BE/BB/D8 /CP/D9/B4/DE /B5/B9/BD/BB /CC/D6/B9/C8 /D9/D1/D4/B5 /B7/CK >/C8/D9/D1/D4/B6/CP/D0/D4/CW/CP/D1/DC/B6/B4/BD/B9/CT/DC/D4/B4/B9/BD/BB/CC/D6/B9 /C8/D9/D1/D4 /B5/B5/BB/B4 /C8/D9/D1/D4 /B7/BD/BB/CC /D6/B5/B9/CP /D0/D4/CW /CP/B4/DE/B5 /DH/BN sys:=  ∂ ∂z˜ψ(z˜) =−4k2τ(z˜)2−2tf2ψ(z˜)τ(z˜)2 −4k2ψ(z˜)2τ(z˜)2−2βρ0(z˜)2 −2βρ0(z˜)2ψ(z˜)2−2tf2ψ(z˜)3τ(z˜)2, ∂ ∂z˜τ(z˜) =−2tf2τ(z˜)3+σρ0(z˜)2τ(z˜) + 3k2τ(z˜)3ψ(z˜) +βρ0(z˜)2ψ(z˜)τ(z˜) +tf2ψ(z˜)2τ(z˜)3, ∂ ∂z˜ρ0(z˜) = σρ0(z˜)3−γρ0(z˜)−tf2ρ0(z˜)τ(z˜)2+α(z˜)ρ0(z˜) + k2ρ0(z˜)ψ(z˜)τ(z˜)2, ∂ ∂z˜α(z˜) =α(z˜)e(−2ξ ρ0(z˜)2 τ(z˜)−1 Tr−Pump )+ Pump alphamx (1−e(−1 Tr−Pump )) Pump +1 Tr−α(z˜)  /C0/CT/D6/CT /CC /D6 /CX/D7 /D8/CW/CT /CV/CP/CX/D2 /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D8/CX/D1/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CP /DA/CX/D8 /DD /D4 /CT/D6/CX/D3 /CS/B8 /C8/D9/D1/D4/CX/D7 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D4/D9/D1/D4/B8ξ /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /BA/C6/D3 /DB/B8 /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT/B3/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D2/D8/CW/CT /CQ/CP/D7/CX/D7 /D3/CU /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/DD/D7/D8/CT/D1 /D3/CU /C7/BW/BX/BA /CF /CT /D7/D9/D4/D4 /D3/D7/CT /D8/D3 /D7/D3/D0/DA /CT /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /DD/D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /D3/D4 /CT/D6/CP/D8/D3/D6 /BW/BX/D4/D0/D3/D8/BA /C4/CT/D8 /D2/D3/D6/D1/CP/D0/CX/DE/CTσ /CP/D2/CSβ /D8/D3 /BD/B8/BJ/B610−12cm2/BB/CF/B8/D8/CX/D1/CT/D7 /D8/D3tf /B4/BE/B8/BH /CU/D7 /CU/D3/D6 /CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT /D0/CP/D7/CT/D6/B5/B8 /D8/CW/CT/D2ξ /BP /BC/BA/BC/BC/BD/BK/BA /CC/CW/CT /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0/D7/D8/CT/D4 /CX/D7 /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C3/CT/D6/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD/BM σ /BPσ0 /B6/B4/BD /B9ρ02σ0 2 /B5/B8 β /BPβ0 /B6/B4/BD /B9ρ02β0 2 /B5/B8/DB/CW/CT/D6/CTσ0 /CP/D2/CSβ0 /CP/D6/CT /D8/CW/CT /D9/D2/D7/CP/D8/D9/D6/CP/D8/CT/CS /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA >/AZ/D4/D6/D3 /CT/CS/D9/D6/CT /CU/D3/D6 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS/CK/BE/BD>/D7/DD/D7/D8/CT/D1 /D3/CU /C7/BW/BX >/C7/BW/BX/CN/D4/D0/D3/D8 /BM/BP /D4/D6/D3 /B4/CP/D0/D4/CW/CP/D1/DC/B8/CV/CP/D1/B8/D7/CX/CV/D1/CP/BC/B8/CQ/CT /D8/CP/BC/B8 /CC/D6/B8/C8 /D9/D1/D4 /B8/CK >/DC/CX/B8/CS/CX/D7/D4/B8/D8/CV/B8/D2/B5 >/D7/CX/CV/D1/CP /BM/BP /D7/CX/CV/D1/CP/BC/B6/B4/BD /B9 /D7/CX/CV/D1/CP/BC/B6/D6/CW/D3/BC/B4/DE/B5/CM/BE/BB/BE /B5/BM >/CQ/CT/D8/CP /BM/BP /CQ/CT/D8/CP/BC/B6/B4/BD /B9 /CQ/CT/D8/CP/BC/B6/D6/CW/D3/BC/B4/DE/B5/CM/BE/BB/BE /B5/BM >/D7/DD/D7 /BM/BP /CJ/BW/B4/CP/D0/D4/CW/CP/B5/B4/DE/B5 /BP/CK >/CP/D0/D4/CW/CP/B4/DE/B5/B6/CT/DC/D4/B4/B9/BE/B6/DC/CX/B6/D6/CW/D3/BC/B4/DE/B5 /CM/BE/BB/D8 /CP/D9/B4/DE /B5/B9/BD/BB /CC/D6/B9/C8 /D9/D1/D4/B5 /B7/CK >/C8/D9/D1/D4/B6/CP/D0/D4/CW/CP/D1/DC/B6/B4/BD/B9/CT/DC/D4/B4/B9/BD/BB/CK 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/CS/D9/D6/CP/D8/CX/D3/D2tf /BP /BE/BA/BH /CU/D7/BA /CF /CT /DB/CX/D0/D0 /D9/D7/CT /D8/CW/CT /D2/CT/DC/D8 /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7/BM/BD/B5 /D8/CW/CT /D8/CX/D1/CT /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3tf /BN /BE/B5 /D8/CW/CT /AS/CT/D0/CS /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /BD/BB/D5/BBtf /BP /BC/BA/BL/BH/B6 107/CE/BB /D1/BN /BF/B5 /D8/CW/CT /AS/CT/D0/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3ε0 /B6/D2/B6 /BB/B4/BE/B6/D5/B6 t2 f /B5 /B8 /D8/CW/CP/D8 /CX/D7 >/CX/D2/D8/CT/D2/D7/CX/D8/DD/CN/D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/CN/D4/CP /D6/CP/D1/CT /D8/CT/D6 /BM/BP /D4/D6/D3 /B4/B5 >/D0/D3 /CP/D0 /CT/D4/D7/CX/D0/D3/D2/BC/B8/D2/B8 /B8/D5/B8/D8/CV/B8/D4/CP/D6/BM >/CT/D4/D7/CX/D0/D3/D2/BC /BM/BP /BK/BA/BK/BH/CT/B9/BD/BG/BM >/D2 /BM/BP /BF/BA/BF/BE/BM > /BM/BP /BF/CT/BD/BC/BM >/D8/CU /BM/BP /BE/BA/BH/CT/B9/BD/BH/BM >/D5 /BM/BP /D4/CP/D6/CP/D1/CT/D8/CT/D6/CN/D5/B4/B5/BM >/CT/D4/D7/CX/D0/D3/D2/BC/B6/D2/B6 /BB/B4/BE/B6/D5/B6/D8/CU/B5/CM/BE/BM >/CT/D2/CS/BM >/CT/DA/CP/D0/CU/B4/CX/D2/D8/CT/D2/D7/CX/D8/DD/CN/D2/D3/D6/D1/CP/D0/CX/DE/CP/D8 /CX/D3/D2/CN /D4/CP/D6/CP /D1/CT/D8/CT /D6/B4/B5/B5 /BN >/AZ/CC/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2 /CX/D7 /CJ/CF/BB /D1/CM/BE℄ .2000000000 1012/BE/BH/BY /D3/D6 /D7/D9 /CW /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D3/CU /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CU/D3/D6 /CC/CX/BM /D7/CP/D4/B9/D4/CW/CX/D6/CT /B4/D8/CW/CT /CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /BC/BA/BK /C2/BB /D1/CM/BE/B5 /CX/D7ξ /BP /BI/BA/BE/BH/B6 10−4/BA/BT/D7 /CX/D8 /DB/CX/D0/D0 /CQ /CT /D7/CT/CT/D2/B8 /D8/CW/CT /D4/D6/CX/D2 /CX/D4/CP/D0 /CU/CP /D8/D3/D6 /CX/D2 /D3/D9/D6 /D1/D3 /CS/CT/D0 /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD/D3/CU /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D3/CU /CS/CX/AR/D6/CP /D8/CX/D3/D2 /D0/D3/D7/D7 /CX/D2 /D8/CW/CT /D0/CP/D7/CT/D6 /B4/D7/D3/B9 /CP/D0/D0/CT/CS /C3/CT/D6/D6 /B9 /D0/CT/D2/D7 /D1/D3 /CS/CT/D0/D3 /CZ/CX/D2/CV /D4/CP/D6/CP/D1/CT/D8/CT/D6/B5 σ /BP /BC/BA/BD/BG /B4/D4/CP/D6/CP/D1/CT/D8/CT/D6 /D3/CU /D8/CW/CT /CS/CX/AR/D6/CP /D8/CX/D3/D2 /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D7 107/CF /CP/D2/CS /D8/CW/CT /D1/D3 /CS/CT /D6/D3/D7/D7 /D7/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT /BZ/CP/D9/D7/D7/CX/CP/D2 /CQ /CT/CP/D1 /CX/D7 /BF/BCµ /D1/B5/BA /CC/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CB/C8/C5 /CX/D2 /BD /D1/D1 /CP /D8/CX/DA /CT /D6/DD/D7/D8/CP/D0 /CX/D7β /BP /BC/BA/BE/BI/BA/CC/CW/CT /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /CX/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /D3/D2 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/CU/BU/D0/D3 /CW /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CJ /C4/BA /BT /D0 /D0/CT/D2 /CP/D2/CS /C2/BA /C0/BA /BX/CQ /CT/D6/D0/DD/B8 /C7/D4/D8/CX /CP/D0 /CA /CT/D7/D3/D2/CP/D2 /CT /CP/D2/CS /CC/DB/D3/B9/C4 /CT/DA/CT/D0 /BT /D8/D3/D1/D7 /B4/CF/CX/D0/CT/DD/B8 /C6/CT/DB /CH /D3/D6/CZ/B8 /BD/BL/BJ/BH/B5 ℄/BA /C1/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /D8/CW/CT /AS/CT/D0/CS /D4/CW/CP/D7/CT/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CP/D2/CS /D8/D9/D2/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /D1/CT/CS/CX/D9/D1 /D6/CT/D7/D3/D2/CP/D2 /CT /CU/D6/CT/D5/D9/CT/D2 /DD /CP/D2/CS /CX/D2 /D8/CW/CT/CP/CQ/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D8/CX/D1/CT /CX/D2 /D8/CT/D6/DA /CP/D0/B8 /DB/CW/CX /CW /CX/D7 /D3/D1/D4/CP/D6/CP/CQ/D0/CT /DB/CX/D8/CW /D8/CW/CT/D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2 /B4 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/B4/D8/B5 /CP/D2/CS /CQ/B4/D8/B5 /B5/CP/D2/CS /D8/CW/CT /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CS/CX/AR/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /CT/DC /CX/D8/CT/CS /CP/D2/CS /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7 /DB/B4/D8/B5 /D3/CQ /CT/DD/D8/D3 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CT/D5/D9/CP/D8/CX/D3/D2/D7/BM >/CQ/D0/D3 /CW/CN/BD /BM/BP /CS/CX/CU/CU/B4/CQ/B4/D8/B5/B8/D8/B5/BP/D5/B6/D6/CW/D3/B4/D8/B5/B6/DB/B4/D8/B5/BN >/CQ/D0/D3 /CW/CN/BE /BM/BP /CS/CX/CU/CU/B4/CP/B4/D8/B5/B8/D8/B5/BP/BC/BN >/CQ/D0/D3 /CW/CN/BF /BM/BP /CS/CX/CU/CU/B4/DB/B4/D8/B5/B8/D8/B5/BP/B9/D5/B6/D6/CW/D3/B4/D8/B5/B6/CQ/B4 /D8/B5/BN bloch /CN1:=∂ ∂tb(t) =qρ(t) w(t) bloch /CN2:=∂ ∂ta(t) = 0 bloch /CN3:=∂ ∂tw(t) =−qρ(t) b(t)/CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1 /CP/D6/CT >/D7/D3/D0/CN/CQ/D0/D3 /CW /BM/BP/CK >/CS/D7/D3/D0/DA/CT/B4/DF/CQ/D0/D3 /CW/CN/BD/B8/CQ/D0/D3 /CW/CN/BE/B8/CQ /D0/D3 /CW /CN/BF/B8/DB /B4/BC/B5/BP /B9/BD/B8/CQ /B4/BC/B5/BP /BC/B8/CP /B4/BC/B5/BP /BC/DH/B8/CK >/DF/CP/B4/D8/B5/B8/CQ/B4/D8/B5/B8/DB/B4/D8/B5/DH/B5/BM >/D7/D3/D0/CN/CP /BM/BP /D7/D9/CQ/D7/B4/D7/D3/D0/CN/CQ/D0/D3 /CW/B8/CP/B4/D8/B5/B5/BN >/D7/D3/D0/CN/CQ /BM/BP /D7/D9/CQ/D7/B4/D7/D3/D0/CN/CQ/D0/D3 /CW/B8/CQ/B4/D8/B5/B5/BN >/D7/D3/D0/CN/DB /BM/BP /D7/D9/CQ/D7/B4/D7/D3/D0/CN/CQ/D0/D3 /CW/B8/DB/B4/D8/B5/B5/BN sol /CNa:= 0 sol /CNb:=−sin(q/integraldisplayt 0ρ(u)du) sol /CNw:=−cos(q/integraldisplayt 0ρ(u)du)/CC/CW/CT /CP/D6/CV/D9/D1/CT/D2 /D8 /D3/CUsin/cos /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CTψ /BM >/CQ/B4/D8/B5/BM/BP /B9/D7/CX/D2/B4/D4/D7/CX/B4/D8/B5/B5/BN /CP/B4/D8/B5/BM/BP /BC/BN b(t) :=−sin(ψ(t)) a(t) := 0/BE/BI/CC/CW/CT /D0/CX/D2/CT/CP/D6 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT /D0/CP/D7/CT/D6 /DB/CX/D8/CW /D8/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8α /B8 /D0/D3/D7/D7 /D3 /CTꜶ /CX/CT/D2 /D8 /CV/CP/D1 /B8 /CU/D6/CT/D5/D9/CT/D2 /DD /AS/D0/D8/CT/D6 /DB/CX/D8/CW /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW tf /B8 /D0/CP/D7/CT/D6 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CT/D0/CT/D1/CT/D2 /D8/D7 /DB/CX/D8/CW /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8k2 /CX/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT/D8/CT/D6/D1/D7 /CX/D2 /D8/CW/CT /D6/CX/CV/CW /D8 /CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/D7/CT/CT /CB/CT /D8/CX/D3/D2/D7 /BG/B8 /BH/B5/BM >/C4/CP/D7/CT/D6/CN/D0/CX/D2/CT/CP/D6 /BM/BP/CK >/CP/D0/D4/CW/CP/B6/D6/CW/D3/B4/DE/B8/D8/B5/B9/CV/CP/D1/B6/D6/CW/D3/B4/DE/B8/D8 /B5/B7/D8/CU /CM/BE/B6/CS /CX/CU/CU/B4 /D6/CW/D3/B4 /DE/B8/D8/B5 /B8/D8/B8 /D8/B5/B7/CK >/C1/B6/CZ/CN/BE/B6/CS/CX/CU/CU/B4/D6/CW/D3/B4/DE/B8/D8/B5/B8/D8/B8/D8/B5/BN Laser /CNlinear :=αρ(z, t)−gamρ(z, t)+tf2(∂2 ∂t2ρ(z, t))+Ik /CN2(∂2 ∂t2ρ(z, t))/CC/CW/CT /D6/CT/D7/D4 /D3/D2 /CT /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D0/CP/D7/CT/D6 /CU/CP /D8/D3/D6/D7 /CX/D7 >/C4/CP/D7/CT/D6/CN/D2/D3/D2/D0/CX/D2/CT/CP/D6 /BM/BP/CK >/D7/CX/CV/D1/CP/B6/D6/CW/D3/B4/DE/B8/D8/B5/B6 /D3/D2/CY/D9/CV/CP/D8/CT/B4/D6 /CW/D3/B4/DE /B8/D8/B5/B5 /B6/D6/CW/D3 /B4/DE/B8/D8 /B5/B9/CK >/C1/B6/CQ/CT/D8/CP/B6/D6/CW/D3/B4/DE/B8/D8/B5/B6 /D3/D2/CY/D9/CV/CP/D8/CT/B4 /D6/CW/D3/B4 /DE/B8/D8/B5 /B5/B6/D6/CW /D3/B4/DE/B8 /D8/B5/BN Laser /CNnonlinear :=σρ(z, t)2ρ(z, t)−Iβρ(z, t)2ρ(z, t)/C0/CT/D6/CTα /CP/D2/CS /CV/CP/D1 /CP/D6/CT /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /DA /CP/D0/D9/CT/D7/B8 /D8/CW/CP/D8 /D7/D9/D4/D4 /D3/D7/CT/D7 /D8/CW/CT /D2/D3/D6/D1/CP/D0/B9/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /DE /D8/D3 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D3/D4/D8/CX /CP/D0 /D1/CT/CS/CX/D9/D1 /B4/CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CX/D2/D3/D9/D6 /CP/D7/CT/B5/B8 /CV/CP/D1 /CX/D2 /D0/D9/CS/CT/D7 /D2/D3/D8 /D3/D2/D0/DD /D7 /CP/D8/D8/CT/D6/CX/D2/CV /D0/D3/D7/D7 /CX/D2 /D8/D3 /D3/D4/D8/CX /CP/D0 /CT/D0/CT/D1/CT/D2 /D8/D7/B8 /CQ/D9/D8/CP/D0/D7/D3 /D8/CW/CT /D3/D9/D8/D4/D9/D8 /D0/D3/D7/D7 /D3/D2 /D8/CW/CT /D0/CP/D7/CT/D6 /D1/CX/D6/D6/D3/D6/BAσ /CP/D2/CSβ /CW/CP /DA /CT /CS/CX/D1/CT/D2/D7/CX/D3/D2 /D3/CU /D8/CW/CT/CX/D2 /DA /CT/D6/D7/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8 /CX/BA /CT/BA\|ρ\|2/CX/D7 /D8/CW/CT /AS/CT/D0/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B4/DB /CT /CS/D3 /CT/D7 /D2/D3/D8 /DB/D6/CX/D8/CT /D8/CW/CT /CU/CP /D8/D3/D6 ε /BC/B6/D2/B6 /BB/BE /B8 /DB/CW/CX /CW /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 ﬁeld2/B9> /CX/D2 /D8/CT/D2/D7/CX/D8 /DD/B5/BA/BT/D7 /D8/CW/CT /D6/CT/D7/D9/D0/D8/B8 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CX/D2 /D8/CT/CV/D6/D3/B9/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 /AS/CT/D0/CS /CT/DA /D3/D0/D9/B9/D8/CX/D3/D2 /CX/D7 >/D1/CP/D7/D8/CT/D6/CN/BD /BM/BP/CK >/CS/CX/CU/CU/B4/D6/CW/D3/B4/DE/B8/D8/B5/B8/DE/B5/B7/CS/CX/CU/CU/B4/D6/CW/D3/B4 /DE/B8/D8/B5 /B8/D8/B5/BB /BP/CK >/D7/D9/CQ/D7/B4/C6/BP/C6/B6/DE/CN/CP/CQ/D7/B8/B9/BE/B6/C8/CX/B6/C6/B6/CS/B6/CS /CX/CU/CU/B4 /CP/B4/D8/B5 /B8/D8/B5/BB /B7/CK >/BE/B6/C8/CX/B6/C6/B6/CS/B6/CQ/B4/D8/B5/B6/D3/D1/CT/CV/CP/BB /B5/B7/C4/CP/D7 /CT/D6/CN/D0 /CX/D2/CT/CP /D6/B7/C4/CP /D7/CT/D6/CN /D2/D3/D2/D0 /CX/D2/CT /CP/D6/BN >/AZ/D7/CT/CT /CB/CT /D8/CX/D3/D2 /BE master /CN1:= (∂ ∂zρ(z, t)) +∂ ∂tρ(z, t) c=−2πNz /CNabsdsin(ψ(t))ω c +αρ(z, t)−gamρ(z, t) +tf2(∂2 ∂t2ρ(z, t)) +Ik2(∂2 ∂t2ρ(z, t)) +σρ(z, t)2ρ(z, t)−Iβρ(z, t)2ρ(z, t)/C4/CT/D8 /D8/D6/CP/D2/D7/CX/D8 /D8/D3 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/BM >/CP/D7/D7/D9/D1/CT/B4/D5/B8/D6/CT/CP/D0/B5/BM >/D1/CP/D7/D8/CT/D6/CN/BE /BM/BP /CT/DC/D4/CP/D2/CS/B4/D7/D9/CQ/D7/B4/D6/CW/D3/B4/DE/B8/D8/B5/BP/CK >/CS/CX/CU/CU/B4/D4/D7/CX/B4/D8/B5/B8/D8/B5/BB/D5/B8/D1/CP/D7/D8/CT/D6/CN/BD/B5 /B5/BN master /CN2:=∂2 ∂t2ψ(t) q˜c=−2πNz /CNabsdsin(ψ(t))ω c+α(∂ ∂tψ(t)) q˜/BE/BJ−gam(∂ ∂tψ(t)) q˜+tf2(∂3 ∂t3ψ(t)) q˜ +Ik2(∂3 ∂t3ψ(t)) q˜+σ(∂ ∂tψ(t))2∂ ∂tψ(t) q˜3 −Iβ(∂ ∂tψ(t))2∂ ∂tψ(t) q˜3/CF /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D3/D2/D0/DD /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /AS/CT/D0/CS /D7/D8/CP/D8/CT/D7/B8 /CX/BA /CT/BA∂ ∂zρ /B4/DE/B8 /D8/B5 /BP /BC/B8 /CP/D2/CS /D8/D3/CX/D2 /D8/D6/D3 /CS/D9 /CT /D8/CW/CT /D8/CX/D1/CT /CS/CT/D0/CP /DD /D3/D2 /D8/CW/CT /CP /DA/CX/D8 /DD /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4δ /BA >/D1/CP/D7/D8/CT/D6/CN/BF /BM/BP /D6/CW/D7/B4/D1/CP/D7/D8/CT/D6/CN/BE/B5/B7/CS/CT/D0/D8/CP/B6/CS/CX/CU/CU/B4/D4 /D7/CX/B4/D8 /B5/BB/D5/B8 /CO/B0/CO /B4/D8/B8/BE /B5/B5/BN master /CN3:=−2πNz /CNabsdsin(ψ(t))ω c+α(∂ ∂tψ(t)) q˜ −gam(∂ ∂tψ(t)) q˜+tf2(∂3 ∂t3ψ(t)) q˜ +Ik2(∂3 ∂t3ψ(t)) q˜+σ(∂ ∂tψ(t))2∂ ∂tψ(t) q˜3 −Iβ(∂ ∂tψ(t))2∂ ∂tψ(t) q˜3+δ(∂2 ∂t2ψ(t)) q˜/C4/CT/D8 /CX/D2 /D8/D6/D3 /CS/D9 /CT /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 λ /B8 /DB/CW/CX /CW /CP/D2 /CQ /CT /BD/B5 /D6/CP/D8/CX/D3 /D3/CU /D1/D3 /CS/CT /D6/D3/D7/D7/B9/D7/CT /D8/CX/D3/D2 /D3/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /D8/D3 /D3/D2/CT /D3/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /D3/D6 /BE/B5 /D3 /CTꜶ /CX/CT/D2 /D8/D3/CU /D6/CT/CU/D6/CP /D8/CX/DA/CX/D8 /DD /B4/CU/D3/D6 /AS/CT/D0/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT/B5 /D3/CU /D1 /D9/D0/D8/CX/D0/CP /DD /CT/D6 /D1/CX/D6/D6/D3/D6 /D3/D2 /D8/CW/CT /D7/CT/D1/CX /D3/D2/CS/D9 /B9/D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /B4/D7/D3/B9 /CP/D0/D0/CT/CS /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D1/CX/D6/D6/D3/D6 /B9 /CB/BX/CB/BT/C5/B5/BA/CC/CW/CT/D2 >/D1/CP/D7/D8/CT/D6/CN/BG /BM/BP/CK >/D7/D9/CQ/D7/B4/DF/D7/CX/CV/D1/CP/BP/D7/CX/CV/D1/CP/BB/D0/CP/D1/CQ/CS/CP/CM/BE /B8/CQ/CT/D8 /CP/BP/CQ/CT /D8/CP/BB/D0 /CP/D1/CQ/CS /CP/CM/BE/DH /B8/D1/CP /D7/D8/CT/D6 /CN/BF/B5/BN master /CN4:=−2πNz /CNabsdsin(ψ(t))ω c+α(∂ ∂tψ(t)) q˜ −gam(∂ ∂tψ(t)) q˜+tf2(∂3 ∂t3ψ(t)) q˜ +Ik2(∂3 ∂t3ψ(t)) q˜+σ(∂ ∂tψ(t))2∂ ∂tψ(t) λ2q˜3 −Iβ(∂ ∂tψ(t))2∂ ∂tψ(t) λ2q˜3+δ(∂2 ∂t2ψ(t)) q˜/C6/D3 /DB /D8/D6/CP/D2/D7/CX/D8 /D8/D3 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /B3/D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /B9 /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT/B3 /CP/D2/CS /CT/D0/CX/D1/CX/B9/D2/CP/D8/CT /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /BZ/BW/BW /CP/D2/CS /CB/C8/C5/BA /CC/CW/CT/D7/CT /D7/D9/D4/D4 /D3/D7/CX/D8/CX/D3/D2/D7 /D0/CT/CP/CS /D8/D3 /D8/CW/CT/D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /C7/BW/BX/BA /CC/CW/CP/D8 /CX/D7 >/D1/CP/D7/D8/CT/D6/CN/BH /BM/BP/CK > /D3/D0/D0/CT /D8/B4/CT/DC/D4/CP/D2/CS/B4/D7/D9/CQ/D7/B4/DF/CZ/CN/BE/BP/BC /B8/CQ/CT/D8 /CP/BP/BC/B8 /D7/CX/CV/D1 /CP/BP/BC/DH /B8/CK >/D1/CP/D7/D8/CT/D6/CN/BG/B5/BB/B4/B9/BE/B6/C8/CX/B6/C6/B6/CS/B6/D3/D1/CT/CV/CP /B6/D8/CU/B6 /DE/CN/CP/CQ /D7/BB /B5 /B5/B8/CS/CX /CU/CU/B4/D4 /D7/CX/B4 /D8/B5/B8/D8 /B5/B5/BN >/AZ /D6/CT/CS/D9 /CT/CS /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CT/CU/CU/CX /CX/CT/D2/D8/D7/BE/BK>/D7/D9/CQ/BD /BM/BP /CP/BD/BP/B9/BD/BB/BE/B6 /BB/B4/C8/CX/B6/C6/B6/CS/B6/D3/D1/CT/CV/CP/B6/D8 /CU/B6/D5/B6 /DE/CN/CP/CQ /D7/B5/BM >/D7/D9/CQ/BE /BM/BP /CP/BE/BP/B9/BD/BB/BE/B6 /B6/CS/CT/D0/D8/CP/BB/B4/C8/CX/B6/C6/B6/CS/B6/D3/D1/CT/CV /CP/B6/D8/CU /B6/D5/B6/DE /CN/CP/CQ /D7/B5/BM >/D7/D9/CQ/BF /BM/BP/CK >/CP/BF/BP/B9/BD/BB/BE/B6 /B6/CP/D0/D4/CW/CP/BB/B4/C8/CX/B6/C6/B6/CS/B6/D3/D1 /CT/CV/CP/B6 /D8/CU/B6/D5 /B6/DE/CN/CP /CQ/D7/B5/B7 /CK >/BD/BB/BE/B6 /B6/CV/CP/D1/BB/B4/C8/CX/B6/C6/B6/CS/B6/D3/D1/CT/CV/CP/B6/D8/CU /B6/D5/B6/DE /CN/CP/CQ/D7 /B5/BM >/D1/CP/D7/D8/CT/D6/CN/BI /BM/BP /CP/BD/B6/CS/CX/CU/CU/B4/D4/D7/CX/B4/D8/B5/B8/CO/B0/CO/B4/D8/B8/BF/B5/B5/B7 /CK >/CP/BE/B6/CS/CX/CU/CU/B4/D4/D7/CX/B4/D8/B5/B8/CO/B0/CO/B4/D8/B8/BE/B5/B5/B7/CP /BF/B6/CS/CX /CU/CU/B4/D4 /D7/CX/B4/D8 /B5/B8/D8/B5 /B7/D7/CX/D2 /B4/D4/D7 /CX/B4/D8/B5 /B5/BN >/CS/D7/D3/D0/DA/CT/B4/D1/CP/D7/D8/CT/D6/CN/BI/BP/BC/B8/D4/D7/CX/B4/D8/B5/B5 /BN /AZ /D8/D6/DD /D8/D3 /D7/D3/D0/DA/CT >/D1/CP/D7/D8/CT/D6/CN/BI /CP/D2/CS /CU/CX/D2/CS /CP /DA/CT/D6/DD /D9/D7/CT/CU/D9/D0 /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT /DA/CP/D6/CX/CP/CQ/D0/CT/D7 >/D1/CP/D7/D8/CT/D6/CN/BJ /BM/BP/CK >/CT/DC/D4/CP/D2/CS/B4/D2/D9/D1/CT/D6/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7 /B4/DF/CP/BD /BP/D6/CW/D7 /B4/D7/D9/CQ /BD/B5/B8/CP /BE/BP/D6/CW /D7/B4/D7 /D9/CQ/BE/B5 /B8/CK >/CP/BF/BP/D6/CW/D7/B4/D7/D9/CQ/BF/B5/DH/B8/CP/BD/B6/B4/CS/CX/CU/CU/B4/D6/CW/D3 /B4/D4/D7/CX /B5/B8/D4/D7 /CX/B8/D4/D7 /CX/B5/B6/D6 /CW/D3/B4/D4 /D7/CX/B5 /CM/BE/B7/CK >/CS/CX/CU/CU/B4/D6/CW/D3/B4/D4/D7/CX/B5/B8/D4/D7/CX/B5/CM/BE/B6/D6/CW/D3/B4/D4 /D7/CX/B5/B5 /B7/CP/BE/B6 /CS/CX/CU/CU /B4/D6/CW/D3 /B4/D4/D7/CX /B5/B8/D4 /D7/CX/B5/B6 /CK >/D6/CW/D3/B4/D4/D7/CX/B5/B7/CP/BF/B6/D6/CW/D3/B4/D4/D7/CX/B5/B7/D7/CX/D2/B4/D4 /D7/CX/B5/B5 /B5/B5/BB/B4 /B9 /B5/B5 /BN >/AZ /D9/D7/CT /D8/CW/CT /CU/D3/D9/D2/CS/CT/CS /CW/CP/D2/CV/CT /D8/D3 /D6/CT/CS/D9 /CT /D8/CW/CT /D3/D6/CS/CT/D6 /D3/CU /C7/BW/BX master /CN5:= (−1 2cα πNdω tf z /CNabs q˜+1 2cgam πNdω tf z /CNabs q˜) (∂ ∂tψ(t)) +sin(ψ(t)) tf−1 2tfc(∂3 ∂t3ψ(t)) πNdω z /CNabs q˜−1 2cδ(∂2 ∂t2ψ(t)) πNdω tf z /CNabs q˜ master /CN6:=a1(∂3 ∂t3ψ(t)) +a2(∂2 ∂t2ψ(t)) +a3(∂ ∂tψ(t)) + sin(ψ(t)) ψ(t) = /CNa&where {a1((∂2 ∂ /CNa2 /CNb( /CNa)) /CNb( /CNa)2 + (∂ ∂ /CNa /CNb( /CNa))2/CNb( /CNa)) +a2(∂ ∂ /CNa /CNb( /CNa)) /CNb( /CNa) +a3 /CNb( /CNa) + sin( /CNa) = 0},{ /CNb( /CNa) =∂ ∂tψ(t), /CNa=ψ(t)}, /braceleftBigg ψ(t) = /CNa, t=/integraldisplay1/CNb( /CNa)d /CNa+ /CNC1/bracerightBigg  master /CN7:= (∂2 ∂ψ2ρ(ψ))ρ(ψ)2+ (∂ ∂ψρ(ψ))2ρ(ψ) +δ(∂ ∂ψρ(ψ))ρ(ψ) +ρ(ψ)α−ρ(ψ)gam −2sin(ψ)πNdω tf q˜z /CNabs c/BT/D7 /CX/D8 /CX/D7 /CZ/D2/D3 /DB/D2 /B4/D7/CT/CT/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /C4/BA /BT /D0 /D0/CT/D2 /CP/D2/CS /C2/BA /C0/BA /BX/CQ /CT/D6/D0/DD/B8 /C7/D4/D8/CX /CP/D0/CA /CT/D7/D3/D2/CP/D2 /CT /CP/D2/CS /CC/DB/D3/B9/C4 /CT/DA/CT/D0 /BT /D8/D3/D1/D7 /B4/CF/CX/D0/CT/DD/B8 /C6/CT/DB /CH /D3/D6/CZ/B8 /BD/BL/BJ/BH/B5/B5/B8 /D8/CW/CT /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2/D3/CU /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT/D0/DD /D7/CW/D3/D6/D8 /D0/CP/D7/CT/D6 /D4/D9/D0/D7/CT /CX/D2 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /CP/D9/D7/CT/D7 /D8/CW/CT /CT/AR/CT /D8 /D3/CU/D8/CW/CT /D7/CT/D0/CU/B9/CX/D2/CS/D9 /CT/CS /D8/D6/CP/D2/D7/D4/CP/D6/CT/D2 /DD /B8 /DB/CW/CT/D2 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D3 /CT/D7 /D2/D3/D8 /D7/D9/AR/CT/D6 /D8/CW/CT /CS/CT /CP /DD /CP/D2/CS/D8/CW/CT/D6/CT /CX/D7 /D2/D3/D8 /CP /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D7/CW/CP/D4 /CT /CX/D2 /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/BA /CC/CW/CX/D7 /CT/AR/CT /D8/BE/BL/CX/D7 /D8/CW/CT /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /DB/CX/D8/CW /D8/CW/CT /CP/D8/D3/D1/D7 /CP/D2/CS /CX/D7/CS/CT/D7 /D6/CX/CQ /CT/CS /D3/D2 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/CU /BU/D0/D3 /CW /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /C1/D2 /D8/CW/CT /CQ /CT/CV/CX/D2/D2/CX/D2/CV /DB /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6/D8/CW/CT /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D9/D0/D8/D6/CP/B9/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2/DB/CX/D8/CW /CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /CQ/D9/D8 /DB/CX/D8/CW/D3/D9/D8 /CP/D2 /DD /D0/CP/D7/CX/D2/CV /CU/CP /D8/D3/D6/D7/BA /C1/D2 /D8/CW/CX/D7 /CP/D7/CT/B8 /D8/CW/CT /D1/D3 /CS/CX/AS/CT/CS/D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /D1/CP/D7/D8/CT/D6/CN/BJ /D3/D2 /D8/CP/CX/D2/D7 /D3/D2/D0/DD /D8 /DB /D3 /D8/CT/D6/D1/D7/BM /D8/CW/CT /D8/CT/D6/D1 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT/D4/D9/D0/D7/CT /D8/CX/D1/CT /CS/CT/D0/CP /DD /B4 /D9 /BP1 δ /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /DA /CT/D0/D3 /CX/D8 /DD /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT/B5 /CP/D2/CS /D8/CW/CT /D8/CT/D6/D1 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D6/CT/D7/D4 /D3/D2/D7/CT/BA /C6/D3/D8/CT/B8 /D8/CW/CP/D8ρ /CX/D7 /D8/CW/CT /AS/CT/D0/CS/CP/D1/D4/D0/CX/D8/D9/CS/CT /D1 /D9/D0/D8/CX/D4/D0/CX/CT/CS /CQ /DD /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D5 /B8ψ /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT/BA >/D3/CS/CT/BD /BM/BP /D9/B6/D6/CW/D3/B4/D4/D7/CX/B5/B6/CS/CX/CU/CU/B4/D6/CW/D3/B4/D4/D7/CX/B5/B8/D4 /D7/CX/B5/B7 /D4/B6/D7/CX /D2/B4/D4/D7 /CX/B5/BN ode1:=uρ(ψ) (∂ ∂ψρ(ψ)) +psin(ψ)/CC/CW/CT /D2/CP/D8/D9/D6/CP/D0 /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /CX/D7ρ /B4/BC/B5 /BP /BC/B8 /D8/CW/CP/D8 /CX/D7 /CQ /CT/CU/D3/D6/CT /D4/D9/D0/D7/CT /CU/D6/D3/D2 /D8 /CX/D8/D7/CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D2/CS /D7/D5/D9/CP/D6/CT /CP/D6/CT /BC/BA /CC/CW/CT/D2 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D6/CT >/D7/D3/D0 /BM/BP /CS/D7/D3/D0/DA/CT/B4/DF/D3/CS/CT/BD/B8/D6/CW/D3/B4/BC/B5/BP/BC/DH/B8/D6/CW/D3/B4 /D4/D7/CX/B5 /B5/BN sol:=ρ(ψ) =/radicalBig u(2 cos(ψ)p−2p) u, ρ(ψ) =−/radicalBig u(2 cos(ψ)p−2p) u/BT/D7 /D3/D2/CT /CP/D2 /D7/CT/CT/B8 /D3/D2/D0/DD /D3/D2/CT /D7/D3/D0/D9/D8/CX/D3/D2 /B4 /D9< /BC/B5 /CX/D7 /D4/CW /DD/D7/CX /CP/D0/B8 /CQ /CT /CP/D9/D7/CT /D3/CU /D8/CW/CT/CP/D1/D4/D0/CX/D8/D9/CS/CT /CX/D7 /D8/D3 /CQ /CT /D6/CT/CP/D0/BA/CC/CW/CT /CS/CX/D7/D8/CX/D2 /D8 /D4/D9/D0/D7/CT /DA /CT/D0/D3 /CX/D8 /DD /CS/CT/AS/D2/CT/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT/BA /CC/CW/CX/D7 /CU/CP /D8 /CX/D7 /CX/D0/D0/D9/D7/B9/D8/D6/CP/D8/CT/CS /CQ /DD /D8/CW/CT /D2/CT/DC/D8 /AS/CV/D9/D6/CT/BM >/CP/D2/CX/D1/CP/D8/CT/B4/CT/DA/CP/D0/CU/B4/CP/CQ/D7/B4/D7/D9/CQ/D7/B4/D4/BP/BH /CT/B9/BG/B8 /CK >/D7/D9/CQ/D7/B4/D7/D3/D0/CJ/BE℄/B8/D6/CW/D3/B4/D4/D7/CX/B5/B5/B5/BB/BD/BC/B5 /B5/B8/D4/D7 /CX/BP/BC/BA /BA/BE/B6/C8 /CX/B8/CK >/D9/BP/B9/BD/CT/B9/BG/BA/BA/B9/BC/BA/BF/CT/B9/BH/B8/CU/D6/CP/D1/CT/D7/BP/BD/BC /BC/B8/CP/DC /CT/D7/BP/CQ /D3/DC/CT/CS /B8 /D3/D0 /D3/D6/BP/D6 /CT/CS/B8 /CK >/D0/CP/CQ/CT/D0/D7/BP/CJ/CO/D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT/CO/B8/CO/DD/B8 /C5/CE/BB /D1/CO℄/B8/D8/CX/D8/D0/CT/BP/CK >/CO/C8/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /DA/CT/D6/D7/D9/D7 /CX/D8/D7 /D7/D5/D9/CP/D6/CT/CO/B8/CU/D6/CP/D1/CT/D7/BP/BG/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/B1/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BD/BJ/B8 /D4/CP/CV/CT /BH/BI/CC/CW/CT /AS/CV/D9/D6/CT /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /BEπ /BA /CB/D9 /CW/D4/D9/D0/D7/CT /CX/D7 /D2/CP/D1/CT/CS /CP/D7 Ꜽ/BEπ /B9 /D4/D9/D0/D7/CTꜼ /CP/D2/CS /D8/CW/CT /D4/D6/D3 /CT/D7/D7 /D3/CU /CX/D8/D7 /CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2 /D3/CU /D8/CW/CT /D7/CT/D0/CU/B9/CX/D2/CS/D9 /CT/CS /D8/D6/CP/D2/D7/D4/CP/D6/CT/D2 /DD /BA/C5/D3/D6/CT /D2/CP/D8/D9/D6/CP/D0 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/D8/D3/D2/D7 /CX/D2 /D3/D9/D6 /CP/D7/CT /CX/D7 /D4/D6/D3/B9/CS/D9 /CT/CS /CQ /DD /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 Ꜽ/D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT /B9 /D8/CX/D1/CTꜼ/BA /CC/CW/CT/D2 /D8/CW/CT/D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/D7/CT/CT /D1/CP/D7/D8/CT/D6/CN/BG /B5 /CP/D2 /CQ /CT /DB/D6/D3/D8/CT /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM >/D3/CS/CT/BE /BM/BP /CS/CX/CU/CU/B4/D4/D7/CX/B4/D8/B5/B8/D8/B0/BE/B5/BP/B9/CP/B6/D7/CX/D2/B4/D4/D7 /CX/B4/D8/B5 /B5/BN >/AZ/CP/D2/CP/D0/D3/CV /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6 /C3/D0/CT/CX/D2/B9/BZ/D3/D6/CS/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2 ode2:=∂2 ∂t2ψ(t) =−asin(ψ(t))/C0/CT/D6/CT /CP/BP/D4/BB/D9 /BA /CC/CW/CX/D7 /CX/D7 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/B8 /DB/CW/CX /CW /CX/D7 /CP/D2/CP/D0/D3/CV /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D4 /CT/D2/CS/D9/D0/D9/D1 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /B4/D8/CW/CT /CP/D2/CV/D0/CT /DA /CP/D6/CX/CP/CQ/D0/CT /DC /CX/D2 /D3/D9/D6 /CP/D7/CT /CX/D7 /D8/CW/CT /D9/D0/D8/D6/CP/B9/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT/D7/D5/D9/CP/D6/CT/B5 /CP/D2/CS /D8/CW/CT /CP/D2/CV/D0/CT /CX/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /CT/CV/CX/D2/D2/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /D9/D4/D4 /CT/D6 /D4 /D3/CX/D2 /D8 /CX/CU /D9< /BC/B4/D7/CT/CT /CP/CQ /D3 /DA /CT/B5/BA /CC/CW/CX/D7 /D7/CX/D8/D9/CP/D8/CX/D3/D2 /BT /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D2/CT/DC/D8 /AS/CV/D9/D6/CT/BA/BF/BC/CB/CT/CT /BY/CX/CV/D9/D6/CT /BD/BL/B8 /D4/CP/CV/CT /BH/BJ/CF /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /CP /D1/D3/D6/CT /D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /D7/CX/D8/D9/CP/D8/CX/D3/D2 /BU /B8 /DB/CW/CT/D2 /D8/CW/CT /D6/CT/CU/CT/D6/CT/D2 /CT/D4 /D3/CX/D2 /D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D0/D3 /DB /CT/D6 /D4 /D3/CX/D2 /D8 /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/B8 /CP/D2/CS /D2/CT/DB /CP/D6/CV/D9/D1/CT/D2 /D8 /DC/B4/D8/B5/CX/D7 /CT/D5/D9/CP/D0 /D8/D3π /D1/CX/D2 /D9/D7 /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT/BA /CC/CW/CT/D2 >/D3/CS/CT/BE /BM/BP /CS/CX/CU/CU/B4/DC/B4/D8/B5/B8/D8/B0/BE/B5/BP/CP/B6/D7/CX/D2/B4/DC/B4/D8/B5/B5 /BN ode2:=∂2 ∂t2x(t) =asin(x(t))/C0/CT/D6/CT /CP /CX/D7 /D2/CT/CV/CP/D8/CX/DA /CT /DA /CP/D0/D9/CT /B4/D7/CT/CT /CP/CQ /D3 /DA /CT/B5/BA /CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /DB /CT/D0/D0/CZ/D2/D3 /DB/D2/BA >/D7/D3/D0/BD /BM/BP /CS/D7/D3/D0/DA/CT/B4/D3/CS/CT/BE/B8/DC/B4/D8/B5/B5/BN sol1:=/integraldisplayx(t) 1/radicalBig −2acos( /CNa) + /CNC1d /CNa−t− /CNC2= 0, −/integraldisplayx(t) 1/radicalBig −2acos( /CNa) + /CNC1d /CNa−t− /CNC2= 0/C5/CP/CZ /CT /CP/D2 /CT/DC/D4/D0/CX /CX/D8 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/BM >/D7/D3/D0/BE/CN/CP /BM/BP /DA/CP/D0/D9/CT/B4/D0/CW/D7/B4/D7/D3/D0/BD/CJ/BD℄/B5/B5/BN >/D7/D3/D0/BE/CN/CQ /BM/BP /DA/CP/D0/D9/CT/B4/D0/CW/D7/B4/D7/D3/D0/BD/CJ/BE℄/B5/B5/BN/CC/CW/CT /D6/CT/D7/D9/D0/D8 /CX/D7 /CT/DC/D4/D6/CT/D7/D7/CT/CS /D8/CW/D6/D3/D9/CV/CW /CT/D0/D0/CX/D4/D8/CX /CX/D2 /D8/CT/CV/D6/CP/D0/D7/BA /CC/CW/CT /D7/CX/D1/D4/D0/CX/AS /CP/D8/CX/D3/D2 /D3/CU/D8/CW/CT /D6/CP/CS/CX /CP/D0/D7 /D4/D6/D3 /CS/D9 /CT/D7 /D8/CW/CT /D2/CT/DC/D8 /D6/CT/D7/D9/D0/D8/BM >/D7/D3/D0/BF/CN/CP /BM/BP/CK >/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D3/D0/BE/CN/CP/B7/D8/B7/CN/BV/BE/B8/D6/CP/CS/CX /CP/D0/B8 /D7/DD/D1/CQ /D3/D0/CX /B5/BP/D8/B7 /CN/BV/BE/BN >/D7/D3/D0/BF/CN/CQ /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D3/D0/BE/CN/CQ/B7/D8/B7/CN/BV/BE/B8/D6/CP/CS/CX /CP/D0/B8/D7 /DD/D1/CQ/D3 /D0/CX /B5/BP/D8/B7 /CN/BV/BE/BN sol3 /CNa:=−2IEllipticF(sin(1 2x(t)),2/radicalBig a(− /CNC1+ 2a) − /CNC1+ 2a) √− /CNC1+ 2a=t+ /CNC2 sol3 /CNb:= 2IEllipticF(sin(1 2x(t)),2/radicalBig a(− /CNC1+ 2a) − /CNC1+ 2a) √− /CNC1+ 2a=t+ /CNC2/C6/D3 /DB /DB /CT /CS/CT/AS/D2/CT /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/D7/BA /C4/CT/D8 /D7/D9/D4/D4 /D3/D7/CT/B8 /D8/CW/CP/D8 /DC/B4/BC/B5 /BP /BC/B8 /CS/DC/B4/BC/B5/BB/CS/D8/BPα /B8 /DB/CW/CT/D6/CTα /CX/D7 /D8/CW/CT /D7/D3/D1/CT /D4 /D3/D7/CX/D8/CX/DA /CT /DA /CP/D0/D9/CT/BA /CC/CW/CT/D2 >/CX/CN/BV/BD /BM/BP/CK >/D7/D3/D0/DA/CT/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/DF/CS/CX/CU/CU/B4 /DC/B4/D8/B5 /B8/D8/B5/BP /CP/D0/D4/CW /CP/B8/DC/B4 /D8/B5/BP/BC /CK >/DH/B8/CT/DC/D4/CP/D2/CS/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/CS/CX/CU/CU/B4/D0/CW/D7 /B4/D7/D3/D0 /BF/CN/CP/B5 /B8/D8/B5/B5 /B5/B5/B5/BP /BD/B8/CN/BV /BD/B5/BN >/CX/CN/BV/BD /BM/BP/CK >/D7/D3/D0/DA/CT/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/DF/CS/CX/CU/CU/B4 /DC/B4/D8/B5 /B8/D8/B5/BP /CP/D0/D4/CW /CP/B8/DC/B4 /D8/B5/BP/BC /CK >/DH/B8/CT/DC/D4/CP/D2/CS/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/CS/CX/CU/CU/B4/D0/CW/D7 /B4/D7/D3/D0 /BF/CN/CQ/B5 /B8/D8/B5/B5 /B5/B5/B5/BP /BD/B8/CN/BV /BD/B5/BN i /CNC1:= 2a+α2 i /CNC1:= 2a+α2/BF/BD/CC/CW/CT /D7/CT /D3/D2/CS /D3/D2/D7/D8/CP/D2 /D8 /D3/CU /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM >/CX/CN/BV/BE /BM/BP/CK >/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/DF/CN/BV/BD/BP/CX/CN/BV/BD/B8/DC/B4 /D8/B5/BP/BC /DH/B8/D0/CW /D7/B4/D7/D3 /D0/BF/CN/CP /B5/B5/B5/BP /BV/BE/BN >/CX/CN/BV/BE /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/DF/CN/BV/BD/BP/CX/CN/BV/BD/B8/DC/B4 /D8/B5/BP/BC /DH/B8/D0/CW /D7/B4/D7/D3 /D0/BF/CN /CQ/B5/B5/B5 /BP/BV/BE/BN i /CNC2:= 0 = C2 i /CNC2:= 0 = C2/CC/CW/CT /CX/D1/D4/D0/CX /CX/D8 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 >/D7/D3/D0/BG/CN/CP /BM/BP/CK >/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/CN/BV/BD/BP/CX/CN/BV/BD/B8/D0/CW/D7 /B4/D7/D3/D0 /BF/CN/CP/B5 /B5/B8/D6/CP /CS/CX /CP /D0/B8/D7/DD /D1/CQ/D3 /D0/CX /B5 /CK >/BP/D8/B7/D0/CW/D7/B4/CX/CN/BV/BE/B5/BN >/D7/D3/D0/BG/CN/CQ /BM/BP/CK >/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/CN/BV/BD/BP/CX/CN/BV/BD/B8/D0/CW/D7 /B4/D7/D3/D0 /BF/CN/CQ/B5 /B5/B8/D6/CP /CS/CX /CP /D0/B8/D7/DD /D1/CQ/D3 /D0/CX /B5 /CK >/BP/D8/B7/D0/CW/D7/B4/CX/CN/BV/BE/B5/BN sol4 /CNa:=−2EllipticF(sin(1 2x(t)),2√−a α) α=t sol4 /CNb:= 2EllipticF(sin(1 2x(t)),2√−a α) α=t/C4/CT/D8 /D3/D2/D7/CX/CS/CT/D6 /CP /D7/D4 /CT /CX/CP/D0 /D7/CX/D8/D9/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D2 /BE/B6√−p u α /BP /BD/BM >/D7/D3/D0/BH/CN/CP /BM/BP/CK >/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D0/CW/D7/B4/D7/D9/CQ/D7/B4/BE/B6/D7/D5/D6/D8/B4 /B9/CP/B5/BB /CP/D0/D4/CW /CP/BP/BD/B8 /D7/D3/D0/BG /CN/CP/B5/B5 /B5/CK >/BP/D6/CW/D7/B4/D7/D3/D0/BG/CN/CP/B5/B8/DC/B4/D8/B5/B5/BN >/D7/D3/D0/BH/CN/CQ /BM/BP/CK >/D7/D3/D0/DA/CT/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D0/CW/D7/B4/D7/D9/CQ/D7/B4/BE/B6 /D7/D5/D6/D8 /B4/B9/CP/B5 /BB/CP/D0/D4 /CW/CP/BP/BD /B8/D7/D3/D0 /BG/CN/CQ /B5/B5/B5/CK >/BP/D6/CW/D7/B4/D7/D3/D0/BG/CN/CQ/B5/B8/DC/B4/D8/B5/B5/BN sol5 /CNa:=−2 arcsin(tanh(1 2tα)) sol5 /CNb:= 2 arcsin(tanh(1 2tα)) >/CX/CU /D7/CX/CV/D2/B4/D7/D3/D0/BH/CN/CP/B5 /BO /BC /D8/CW/CT/D2 /AZ /DB/CT /CW/D3/D3/D7/CT/CK >/D3/D2/D0/DD /D3/D2/CT /D6/D3/D3/D8 >/D7/D3/D0 /BM/BP /D7/D3/D0/BH/CN/CP >/CT/D0/D7/CT >/D7/D3/D0 /BM/BP /D7/D3/D0/BH/CN/CQ >/CU/CX/BM/CC/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT /D3/D2 /D8/CW/CT /D8/CX/D1/CT /CX/D7/BM >/D4/D9/D0/D7/CT/CN/D7/D5/D9/CP/D6/CT /BM/BP/CK >/D7/D9/CQ/D7/B4/CP/D0/D4/CW/CP/BP/D7/D3/D0/DA/CT/B4/BE/B6/D7/D5/D6/D8/B4/B9/D4 /BB/D9/B5/BB /CP/D0/D4/CW /CP/BP/BD/B8 /CP/D0/D4/CW /CP/B5/B8/C8 /CX/B9/D7 /D3/D0/B5/BN /CK >/CP/D2/CX/D1/CP/D8/CT/B4/CT/DA/CP/D0/CU/B4/D7/D9/CQ/D7/B4/D4/BP/BH/CT/B9/BG /B8/D4/D9/D0 /D7/CT/CN/D7 /D5/D9/CP/D6 /CT/B5/B5/B8 /CK >/D8/BP/B9/BD/BC/BA/BA/BD/BC/B8/D9/BP/B9/BD/CT/B9/BD/BA/BA/B9/BC/BA/BF/CT/B9/BH /B8/CP/DC/CT /D7/BP/CQ/D3 /DC/CT/CS/B8 /D3/D0/D3 /D6/BP/D6/CT /CS/B8/CK >/D0/CP/CQ/CT/D0/D7/BP/CJ/CO/D8/CX/D1/CT/B8 /D8/BB/D8/CU/CO/B8/CO/D4/D7/CX/CO℄/B8/CK >/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT /DA/CT/D6/D7/D9/D7 /D8/CX/D1/CT/CO/B8/CU/D6/CP/D1/CT/D7/BP/BG/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/B1/B5/BN/BF/BE>/AZ/D8/CX/D1/CT /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CU pulse /CNsquare :=π+ 2 arcsin(tanh( t/radicalbigg −p u))/CB/CT/CT /BY/CX/CV/D9/D6/CT /BD/BL/B8 /D4/CP/CV/CT /BH/BK/CC/CW/CT /D4/D9/D0/D7/CT /D4/D6/D3/AS/D0/CT /CX/D7/BM >/CP/D7/D7/D9/D1/CT/B4 /D3/D7/CW/B4/D8/B6/D7/D5/D6/D8/B4/B9/D4/BB/D9/B5/B5/B8 /D4/D3/D7/CX /D8/CX/DA/CT /B5/BM >/CU/CX/CT/D0/CS /BM/BP /DA/CP/D0/D9/CT/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4 /D3/D2/DA/CT/D6/D8/B4/CS/CX/CU/CU /CK >/B4 /D4/D9/D0/D7/CT/CN/D7/D5/D9/CP/D6/CT/B8/D8/B5/B8/D7/CX/D2 /D3/D7/B5/B5/B5/BN ﬁeld:= 2/radicalBigg −p˜ u˜ cosh(t˜/radicalBigg −p˜ u˜)/CC/CW/CX/D7 /CX/D7 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/D8/D3/D2 /DB/CX/D8/CW /CS/D9/D6/CP/D8/CX/D3/D2tp /BP/radicalBig−u p /CP/D2/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT2 qtp >/CP/D2/CX/D1/CP/D8/CT/B4/CT/DA/CP/D0/CU/B4/D7/D9/CQ/D7/B4/D4/BP/BH/CT/B9/BG/B8 /CU/CX/CT/D0 /CS/B5/BB/BD /BC/B5/B8/D8 /BP/B9/BD/BC /BA/BA/BD/BC /B8/CK >/D9/BP/B9/BD/CT/B9/BD/BA/BA/B9/BC/BA/BF/CT/B9/BF/B8/CP/DC/CT/D7/BP/CQ/D3/DC/CT /CS/B8 /D3 /D0/D3/D6/BP /D6/CT/CS/B8 /D0/CP/CQ/CT /D0/D7/BP/CK >/CJ/CO/D8/CX/D1/CT/B8/D8/BB/D8/CU/CO/B8/CO/D6/CW/D3/B8 /C5/CE/BB /D1/CO℄/B8/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CT/D2/DA/CT/D0/D3/D4/CT/CO/B8/CK >/CU/D6/CP/D1/CT/D7/BP/BG/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/B1/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/BC/B8 /D4/CP/CV/CT /BH/BL/CF /CT /CW/CP /DA /CT /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/D8/D3/D2 /DB/CX/D8/CW /D7/CT /CW /B9 /D7/CW/CP/D4 /CT /D4/D6/D3/AS/D0/CT/CX/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D4/D6/D3/D4/CP/CV/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D1/CT/CS/CX/CP /CS/CT/D7 /D6/CX/CQ /CT/CS /D3/D2 /D8/CW/CT/CQ/CP/D7/CX/D7 /D3/CU /BU/D0/D3 /CW /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CC/CW/CX/D7 /D4/D9/D0/D7/CT /D1/CP /DD /CQ /CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CX/D2 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /B3/AS/CT/D0/CS/B9 /D7/D5/D9/CP/D6/CT/B3 /D3/D6 /B3/AS/CT/D0/CS /B9 /D8/CX/D1/CT/B3/BA /CC/CW/CT /AS/D6/D7/D8 /CX/D7 /CU/D3/D6/D1/CP/D0/D0/DD /DA /CT/D6/DD /D7/CX/D1/D4/D0/CT/B8 /CQ/D9/D8 /D8/CW/CT /D7/CT /D3/D2/CS/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /CX/D7 /D4/CW /DD/D7/CX /CP/D0/D0/DD /D1/D3/D6/CT /D3/CQ /DA/CX/D3/D9/D7 /CP/D2/CS /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D1/D3 /CS/CT/D0 /D3/CU/D2/D3/D2/D0/CX/D2/CT/CP/D6 /D4 /CT/D2/CS/D9/D0/D9/D1/BA/C6/D3 /DB /DB /CT /D6/CT/D8/D9/D6/D2 /D8/D3 /D8/CW/CT /D0/CP/D7/CT/D6 /D1/D3 /CS/CT/D0 /B4 /D1/CP/D7/D8/CT/D6/CN/BJ /B5/BA /CC/CW/CX/D7 /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6 /D2/D3/D2/CP/D9/D8/D3/D2/D3/D1/D3/D9/D7 /C7/BW/BX /CP/D2 /CQ /CT /D7/D3/D0/DA /CT/CS /D2 /D9/D1/CT/D6/CX /CP/D0/D0/DD /B4 /D4/DE /BP /BE/B6π /B6/C6/B6d2/B6/B4 ω /B6/D8/CU /B5/B6/DE/CN/CP/CQ/D7/BB/B4 /B6/CW/B5 /BP /CV/CP/D1/CN/CP/CQ/D7 /B6 /D8/CU /BB /D8 /D3/CW /B8 /DB /CT /D7/D9/D4/D4 /D3/D7/CT/CS /CV/CP/D1/CN/CP/CQ/D7 /BP /BC/BA/BC/BD/B5 >/CS/CT /BM/BP /D7/D9/CQ/D7/B4/DF/CK >/CP/D0/D4/CW/CP/BP/BC/BA/BD/B8/CK >/CV/CP/D1/BP/BC/BA/BC/BG/B8/CK >/CS/CT/D0/D8/CP/BP/BC/BA/BC/BC/BG/BE/B8/CK >/D4/DE/BP/BH/CT/B9/BG/DH/B8/CK >/D7/D9/CQ/D7/B4/DF/D3/D4/B4/BI/B8/D1/CP/D7/D8/CT/D6/CN/BJ/B5/BP/CK >/B9/D4/DE/B6/D7/CX/D2/B4/D4/D7/CX/B5/B8/D6/CW/D3/B4/D4/D7/CX/B5/BP/D6/CW/D3/B4 /D4/D7/CX/B5 /BB/BD/BC/DH /B8/D1/CP/D7 /D8/CT/D6/CN /BJ/B5/B5/BM >/AZ/CW/CT/D6/CT /BD/BC /CX/D7 /D8/CW/CT /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CT /D8/CX/D1/CT /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/CK >/D8/D3 /D8/CU /B4/CX/BA/CT/BA /D8/CW/CT /CU/CX/CT/D0/CS /CX/D7 /D1/CT/D7/D9/D6/CT/CS /CX/D2/CK >/BD/BB/B4/D5/B6/D8/CU/B5 /CJ/C5/CE/BB /D1℄ /B9 /D9/D2/CX/D8/D7/B5 >/CU/CX/CV /BM/BP/CK >/BW/BX/D4/D0/D3/D8/B4/CJ/CS/CT/BP/BC℄/B8/D6/CW/D3/B4/D4/D7/CX/B5/B8/D4/D7/CX /BP/BC/BA/BC /BD/BA/BA/BD /BA/BL/BK/BH /B6/C8/CX/B8 /CK >/CJ/CJ/D6/CW/D3/B4/C8/CX/B5/BP/BC/BA/BJ/BI/B6/BD/BC/B8/BW/B4/D6/CW/D3/B5/B4/C8 /CX/B5/BP/BD /CT/B9/BD/BH ℄℄/B8/CK >/D6/CW/D3/BP/BC/BA/BA/BC/BA/BJ/BI/B6/BD/BC/B8/D7/D8/CT/D4/D7/CX/DE/CT/BP/BC/BA /BC/BC/BD/B8 /D0/CX/D2/CT /D3/D0/D3 /D6/BP/CV/D6 /CT/CT/D2/B5 /BM >/CS/CX/D7/D4/D0/CP/DD/B4/CU/CX/CV/B8/D0/CP/CQ/CT/D0/D7/BP/CJ/CO/D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT/CO/B8/CO/D6/CW/D3/B8 /C5/CE/BB /D1/CO℄/B8/CK/BF/BF>/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /DA/CT/D6/D7/D9/D7 /CX/D8/D7 /D7/D5/D9/CP/D6/CT/CO/B8/DA/CX/CT/DB/BP/BC/BA/BA/BJ/BA/BI/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/BD/B8 /D4/CP/CV/CT /BI/BC/C4/CT/D8 /D3/D1/D4/CP/D6/CT /D8/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /DB/CX/D8/CW /D8/CW/CT /D7/CT /CW /B9/D7/CW/CP/D4 /CT /D4/D6/D3/AS/D0/CT /B4/CQ/D0/D9/CT /D3/D0/D3/D6/B5 /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/D8/D3/D2/BM >/D4/D0/D3/D8/B4/D7/D9/CQ/D7/B4/CP/D1/BD/BP/BC/BA/BJ/BI/B8/CP/D1/BD/B6/D7/CX/D2 /B4/D4/D7/CX /BB/BE/B5/B6 /BD/BC/B5/B8 /CK >/D4/D7/CX/BP/BC/BA/BA/BE/B6/C8/CX/B8 /D3/D0/D3/D6/BP/CQ/D0/D9/CT/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/CU/CX/CV/B8/B1/B8/D0/CP/CQ/CT/D0/D7/BP/CJ/CO/D4/D9 /D0/D7/CT /D7/D5/D9/CP/D6/CT/CO/B8/CO/D6/CW/D3/B8 /C5/CE/BB /D1/CO℄/B8/CK >/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /DA/CT/D6/D7/D9/D7 /CX/D8/D7 /D7/D5/D9/CP/D6/CT/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/BE/B8 /D4/CP/CV/CT /BI/BD/CF /CT /D3/CQ/D8/CP/CX/D2/CT/CS /BEπ /B9 /D4/D9/D0/D7/CT/B8 /CQ/D9/D8/B8 /CP/D7 /CX/D8 /CP/D2 /D7/CT/CT /CU/D6/D3/D1 /D4/D6/CT/DA/CX/D3/D9/D7 /AS/CV/D9/D6/CT/B8 /D7/D9 /CW/D4/D9/D0/D7/CT /CX/D7 /D2/D3/D8 /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /DB/CX/D8/CW /D7/CT /CW /B9/D7/CW/CP/D4 /CT /D4/D6/D3/AS/D0/CT /B4/DB/CW/CX /CW /CX/D7 /D7/CW/D3 /DB/D2 /CQ /DD /D8/CW/CT/D0/D3 /DB /CT/D6 /D9/D6/DA /CT/B8 /D7/CT/CT /CQ /CT/D0/D3 /DB /CP/CQ /D3/D9/D8 /D3/D2/D2/CT /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /B3/AS/CT/D0/CS /B9 /D7/D5/D9/CP/D6/CT/B3/CP/D2/CS /B3/AS/CT/D0/CS /B9 /D8/CX/D1/CT/B3 /CX/D2 /D8/CW/CX/D7 /CP/D7/CT/B5/BA /CC/CW/CT /AS/CT/D0/CS106/CE/BB /D1 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D2/B9/D7/CX/D8 /DD /CX/D2 /DA /CP /D9/D9/D1 /BD/BA/BF /BZ/CF/BBcm2/B8 /D8/CW/CP/D8 /CX/D7 /D8/CW/CT /D8 /DD/D4/CX /CP/D0 /CX/D2 /D8/D6/CP /CP /DA/CX/D8 /DD /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CU/D3/D6 /D8/CW/CT/D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6/BA /C6/D3 /DB /D1/CP/CZ /CT /CP /D8/D6/DD /CU/D3/D6 /D3/CQ/D8/CP/CX/D2/CX/D2/CV /D3/CU /CP/D2 /CP/D4/D4/D6/D3 /DC/CX/B9/D1/CP/D8/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BA /CF /CT /DB/CX/D0/D0 /D9/D7/CT /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /CU/D3/D6/D1/B8 /DB/CW/CX /CW/CX/D7 /D8 /DD/D4/CX /CP/D0 /CU/D3/D6 /D8/CW/CT /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT /CP/D9/D8/D3 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/B4/CW/CP/D6/D1/D3/D2/CX /CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B5/BM >/AZ/CP/D4/D4/D6/D3/DC/CX/D1/CP/D8/CX/D3/D2 >/CP/D4/D4/D6/D3/DC/CN/D7/D3/D0 /BM/BP /CP/D1/BD/B6/D7/CX/D2/B4/D4/D7/CX/BB/BE/B5/B7/CP/D1/BE/B6/D7/CX/D2/B4/D4/D7/CX/B5/BN >/CU/BJ /BM/BP/CK >/D2/D9/D1/CT/D6/B4 /D3/D1/CQ/CX/D2/CT/B4/CT/DC/D4/CP/D2/CS/B4/D7/D9/CQ/D7/B4 /D6/CW/D3/B4 /D4/D7/CX/B5 /BP/CK >/CP/D4/D4/D6/D3/DC/CN/D7/D3/D0/B8/D1/CP/D7/D8/CT/D6/CN/BJ/B5/B5/B8/D8/D6/CX/CV /B5/B5/BN/AZ /D7/D9/CQ/D7 /D8/CX/D8/D9 /D8/CX/D3/D2 /CK >/D8/D3 /CX/D2/CX/D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 approx /CNsol:=am1sin(1 2ψ) +am2sin(ψ) f7:=−4δam1 am2 sin(1 2ψ)c+ 12δam1 am2 sin(3 2ψ)c + 8am23sin(3ψ)c−2am13sin(1 2ψ)c + 2am13sin(3 2ψ)c−32 sin(ψ)πNdω tf q˜z /CNabs −8am23sin(ψ)c−9am1 am22sin(3 2ψ)c + 17am1 am22sin(5 2ψ)c+ 16αam2sin(ψ)c + 16αam1sin(1 2ψ)c−16gam am2 sin(ψ)c −16gam am1 sin(1 2ψ)c−10am1sin(1 2ψ)am22c + 11am12am2sin(2ψ)c−10am12am2sin(ψ)c + 4δam12sin(ψ)c+ 8δam22sin(2ψ)c/BF/BG/CF /CT /CW/CP /DA /CT /D8/D3 /D3/D0/D0/CT /D8 /D8/CW/CT /D3 /CTꜶ /CX/CT/D2 /D8/D7 /CQ /CT/CU/D3/D6/CT /D7/CX/D2/B4ψ /BB/BE/B5 /CP/D2/CS /D7/CX/D2/B4ψ /B5/BM >/CU/BK /BM/BP /D3/CT/CU/CU/B4/CU/BJ/B8/D7/CX/D2/B4/D4/D7/CX/BB/BE/B5/B5/BN >/CU/BL /BM/BP /D3/CT/CU/CU/B4/CU/BJ/B8/D7/CX/D2/B4/D4/D7/CX/B5/B5/BN f8:=−4δam1 am2c−2am13c+ 16αam1c −16gam am1c−10am1 am22c f9:=−32πNdω tf q˜z /CNabs−8am23c +16αam2c−16gam am2c −10am12am2c+ 4δam12c/C6/D3/D8/CT/B8 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D0/CP/D7/CX/D2/CV /CU/CP /D8/D3/D6/D7 /CP/D4/D4/D6 /D3/DC/CN/D7/D3/D0 /CX/D7 /D8/CW/CT /CT/DC/CP /D8/D7/D3/D0/D9/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D1/BE /BP /BC/B8 /CP/D1/BD /BP /BE/B6/radicalBigpz δ /BA /BU/CT/D0/D3 /DB /DB /CT /DB/CX/D0/D0/D7/D9/D4/D4 /D3/D7/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /D0/D3/D7/CT /D8/D3 /D8/CW/CT /D7/DD/D1/D1/CT/D8/D6/CX /CP/D0 /D7/CW/CP/D4 /CT/B8 /CX/BA/CT/BA /CP/D1/BE /BP /BC/BA /CC/CW/CT/D2 >/CU/BD/BC /BM/BP/CK >/CT/DC/D4/CP/D2/CS/B4/CU/CP /D8/D3/D6/B4/D7/D9/CQ/D7/B4/CP/D1/BE/BP/BC/B8/CU /BK/B5/B5/BB /B4/B9/BE/B6 /B6/CP/D1 /BD/B5/B5/BN >/CU/BD/BD /BM/BP /D7/D9/CQ/D7/B4/CP/D1/BE/BP/BC/B8/CU/BL/B5/BN >/D7/DD/D1/D1/CT/D8/D6/CX /CP/D0/CN/D7/D3/D0/CN/BD /BM/BP/CK >/CP/D0/D0/DA/CP/D0/D9/CT/D7/B4/D7/D3/D0/DA/CT/B4/DF/CU/BD/BC/BP/BC/B8/CU/BD/BD /BP/BC/DH/B8 /DF/CP/D1/BD /B8/CS/CT/D0 /D8/CP/DH/B5 /B5/BN f10:=am12−8α+ 8gam f11:=−32πNdω tf q˜z /CNabs+ 4δam12c symmetrical /CNsol /CN1:={am1= 2/radicalBig 2α−2gam, δ=−πNdω tf q˜z /CNabs c(−α+gam)}, {am1=−2/radicalBig 2α−2gam, δ=−πNdω tf q˜z /CNabs c(−α+gam)}/CB/D3/B8 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /CX/D7 /CS/CT/AS/D2/CT/CS /CQ /DD /D8/CW/CT /D0/CP/D7/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/B8 /CP/D0/D8/CW/D3/D9/CV/CW /D8/CW/CT/D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8 /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CS/CT/D0/CP /DD /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3/D3/D2/CT /CU/D3/D6 /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /D7/D3/D0/CX/D8/D3/D2 /B4/D7/CT/CT /CP/CQ /D3 /DA /CT/B5/BA /C6/D3 /DB /D6/CT/D8/D9/D6/D2 /D8/D3 /D8/CW/CT /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7/B3/CP/D1/D4/D0/CX/D8/D9/CS/CT /B9 /D8/CX/D1/CT/B3 /CU/D3/D6 /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX /CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BA >/D7/DD/D1/D1/CT/D8/D6/CX /CP/D0/CN/D7/D3/D0/CN/BE /BM/BP/CK >/CS/D7/D3/D0/DA/CT/B4/CS/CX/CU/CU/B4/D4/D7/CX/B4/D8/B5/B8/D8/B5/B9/D7/D9/CQ/D7 /B4/DF/D4/D7 /CX/BP/D4/D7 /CX/B4/D8/B5 /B8/CP/D1/BE /BP/BC/DH/B8 /CK >/CP/D4/D4/D6/D3/DC/CN/D7/D3/D0/B5/B8/D4/D7/CX/B4/D8/B5/B5/BN symmetrical /CNsol /CN2:= ψ(t˜) = 2 arctan(2e(1/2t˜am1+1/2 /CNC1 am1 ) 1 +e(t˜am1+ /CNC1 am1 ),−e(t˜am1+ /CNC1 am1 )+ 1 1 +e(t˜am1+ /CNC1 am1 ))/CC/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /AS/CT/D0/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT /CX/D7/BM >/D7/DD/D1/D1/CT/D8/D6/CX /CP/D0/CN/D7/D3/D0/CN/BF/BM/BP/CK >/D7/CX/D1/D4/D0/CX/CU/DD/B4/CS/CX/CU/CU/B4/D7/D9/CQ/D7/B4/D7/DD/D1/D1/CT/D8/D6 /CX /CP/D0 /CN/D7/D3/D0 /CN/BE/B8/D4 /D7/CX/B4/D8 /B5/B5/B8/D8 /B5/B5/BN symmetrical /CNsol /CN3:= 2am1e(1/2am1(t˜+ /CNC1)) 1 +e(am1(t˜+ /CNC1))/BF/BH/CC/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /CX/D7ρ /B4/BC/B5 /BP /CP/D1/BD /BA /CC/CW/CT/D2 >/CX/D2/CN/BV /BM/BP/CK >/D7/D3/D0/DA/CT/B4/D7/D9/CQ/D7/B4/D8/BP/BC/B8/D7/DD/D1/D1/CT/D8/D6/CX /CP/D0 /CN/D7/D3/D0 /CN/BF/B5/BP /CP/D1/BD/B8 /CN/BV/BD/B5 /BN in /CNC:= 0,0/BT/D2/CS /AS/D2/CP/D0/D0/DD /DB /CT /CW/CP /DA /CT/BM >/D7/DD/D1/D1/CT/D8/D6/CX /CP/D0/CN/D7/D3/D0 /BM/BP/CK >/D7/D9/CQ/D7/B4/CN/BV/BD/BP/BC/B8/D7/DD/D1/D1/CT/D8/D6/CX /CP/D0/CN/D7/D3/D0 /CN/BF/B5/BN symmetrical /CNsol:= 2am1e(1/2t˜am1) 1 +e(t˜am1)/C6/CP/D8/D9/D6/CP/D0/D0/DD /B8 /D8/CW/CX/D7 /CX/D7 /CP /D7/CT /CW /B9 /D7/CW/CP/D4 /CT /D4/D9/D0/D7/CT /DB/CX/D8/CW /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D1/BD /BP /BE/B6/radicalBig 2 (α−gam) /BB/D5/BB/D8/CU /CP/D2/CS /D8/CW/CT /CS/D9/D6/CP/D8/CX/D3/D2 /D8/D4 /BP /BE/BB/B4 /CP/D1/BD/B6/D5 /B5 /BP /D8/CU /BB/radicalBig 2 (α−gam) /BA/CC/CW/CT /D4/D9/D0/D7/CT /D4/D6/D3/AS/D0/CT /CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /D8/CW/CT /D2/CT/DC/D8 /AS/CV/D9/D6/CT/BM >/D4/D0/D3/D8/BF/CS/B4/D7/D9/CQ/D7/B4/CV/CP/D1/BP/BC/BA/BC/BG/B8/BE/B6/D7/D5/D6 /D8/B4/BE/B6 /B4/CP/D0/D4 /CW/CP/B9/CV /CP/D1/B5/B5 /B6/CK >/D7/CT /CW/B4/D8/B6/D7/D5/D6/D8/B4/BE/B6/B4/CP/D0/D4/CW/CP/B9/CV/CP/D1/B5/B5 /BB/BE/BA/BH /CT/B9/BD/BH /B5/B6/BD/BC /B5/B8/CK >/D8/BP/B9/BD/CT/B9/BD/BF/BA/BA/BD/CT/B9/BD/BF/B8/CP/D0/D4/CW/CP/BP/BC/BA/BC/BG /BA/BA/BC/BA /BD/B8/CP/DC /CT/D7/BP/CQ /D3/DC/CT/CS /B8/CK >/D3/D6/CX/CT/D2/D8/CP/D8/CX/D3/D2/BP/CJ/BE/BL/BC/B8/BJ/BC℄/B8/D0/CP/CQ/CT/D0 /D7/BP/CJ/CO /D8/B8/D7/CO /B8/CO/CP/D0 /D4/CW/CP/CO /B8/CO/D6/CW /D3/CO℄ /B8/CK >/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /B4/C5/CE/BB /D1/B5 /DA/CT/D6/D7/D9/D7 /D8/CX/D1/CT/CO/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/BF/B8 /D4/CP/CV/CT /BI/BE/C6/D3 /DB /DB /CT /CP/D2 /CU/D3/D9/D2/CS /D8/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/D2 /D8/CW/CT /D6/CX/D8/CX /CP/D0/D0/CP/D7/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/B8 /D8/CW/CP/D8 /CX/D7 /D8/CW/CT /D4/D9/D1/D4/BA /BY /D3/D6 /D8/CW/CX/D7 /CP/CX/D1/B8 /DB /CT /CW/CP /DA /CT /D8/D3 /CT/DC/D4/D6/CT/D7/D7 /D8/CW/CT /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8 /CU/D6/D3/D1 /D8/CW/CT /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /BA /CF /CT /DB/CX/D0/D0 /D7/D9/D4/D4 /D3/D7/CT/B8 /D8/CW/CP/D8 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/D3/D4 /CT/D6/CP/D8/CT/D7 /CP/D7 /D8/CW/CT /CU/D3/D9/D6 /D0/CT/DA /CT/D0 /D7 /CW/CT/D1/CT/BA /C1/D2 /D8/CW/CX/D7 /CP/D7/CT /D8/CW/CT /D7/D8/CT/CP/CS/DD/B9/D7/D8/CP/D8/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS/CV/CP/CX/D2 /CX/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /CP/D7 /CU/D3/D0/D0/D3 /DB/D7 /CJ /C2/BA /C0/CT/D6/D6/D1/CP/D2/D2/B8 /BU/BA /CF/CX/D0/CW/CT/D0/D1/CX/B8 Ꜽ/C4 /CP/D7/CT/D6 /CU/D9/D6 /CD/D0/D8/D6 /CP/CZ/D9/D6/DE/CT/C4/CX /CW/D8/CX/D1/D4/D9/D0/D7/CTꜼ/B8 /BT /CZ/CP/CS/CT/D1/CX/CT/B9/CE /CT/D6/D0/CP/CV/B8 /BU/CT/D6/D0/CX/D2 /B4/BD/BL/BK/BG/B5 ℄/BM >/CP/D0/D4/CW/CP/BP/C8/D9/D1/D4/B6/CP/D0/D4/CW/CP/D1/DC/BB/B4/C8/D9/D1/D4/B7/D8 /CP/D9/B6/BX /D2/CT/D6/CV /DD/B7/BD/BB /CC/D6/B5/BN α=Pump alphamx Pump +τEnergy +1 Tr/C0/CT/D6/CT /C8/D9/D1/D4 /BPσ /CN /CP/CQ/B6/CC /B6/C1/D4/BB/CW /B6ν /CX/D7 /D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8σ/CN /CP/CQ /CX/D7 /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D6/D3/D7/D7 /D7/CT /D8/CX/D3/D2 /CP/D8 /D8/CW/CT /D4/D9/D1/D4 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/B8 /CC /CX/D7 /D8/CW/CT /CP /DA/CX/D8 /DD/D4 /CT/D6/CX/D3 /CS/B8 /C1/D4 /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8 /CW /B6ν /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /D4/CW/D3/D8/D3/D2 /CT/D2/CT/D6/CV/DD /B8 /CP/D0/D4/CW/CP/D1/DC/CX/D7 /D8/CW/CT /D1/CP/DC/CX/D1/CP/D0 /CV/CP/CX/D2/B8 /BX/D2/CT/D6 /CV/DD /CX/D7 /D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /B8 /CC /D6 /CX/D7 /D8/CW/CT /CV/CP/CX/D2/D6/CT /D3 /DA /CT/D6/DD /D8/CX/D1/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /CC /B4/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CC /D6 /BP /BF/BC/BC /CU/D3/D6 /CC/CX/BM /D7/CP/D4/D4/CW/CX/D6/CT/D0/CP/D7/CT/D6 /DB/CX/D8/CW /CP /DA/CX/D8 /DD /D4 /CT/D6/CX/D3 /CS /BD/BC /D2/D7/B5/BA/BY /D3/D6 /CW/CP/D6/D1/D3/D2/CX /CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BM >/BX/D2/CT/D6/CV/DD/BP/BE/B6/CP/D1/BD/CM/BE/B6/D8/D4/BM/AZ/D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD >/CU/BD/BE /BM/BP /C8/D9/D1/D4/B6/CP/D0/D4/CW/CP/D1/DC/BB/B4/C8/D9/D1/D4/B7/D8/CP/D9/B6/BX/D2/CT/D6 /CV/DD/B7/BD /BB/CC/D6/B5 /B9/CP/D0/D4 /CW/CP/BM >/CU/BD/BF /BM/BP/CK >/D2/D9/D1/CT/D6/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/CK >/DF/CP/D1/BD/BP/BE/B6/D7/D5/D6/D8/B4/BE/B6/B4/CP/D0/D4/CW/CP/B9/CV/CP/D1/B5/B5 /B8/D8/D4/BP /BD/BB/D7/D5 /D6/D8/B4/BE /B6/B4/CP/D0 /D4/CW/CP/B9 /CV/CP/D1 /B5/B5/CK >/DH/B8/D7/D9/CQ/D7/B4/BX/D2/CT/D6/CV/DD/BP/BE/B6/CP/D1/BD/CM/BE/B6/D8/D4/B8/CU /BD/BE/B5/B5 /B5/B5/BM >/CP/D0/D4/CW/CP/CN/D7/D3/D0 /BM/BP /D7/D3/D0/DA/CT/B4/CU/BD/BF/BP/BC/B8/CP/D0/D4/CW/CP/B5/BM /AZ/D7/D3/D0/D9/D8/CX/D3/D2 /CU/D3/D6 /CK/BF/BI>/D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2/CC/CW/CT /CS/CT/D4 /CT/D2/CS/CP/D2 /CT /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /B4/D8 /DB /D3 /D4/CW /DD/D7/CX /CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D8/D3 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD/B5 /DA /CT/D6/D7/D9/D7 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D4/D9/D1/D4 /D3 /CTꜶ /CX/CT/D2 /D8 /CX/D7 >/CU/CX/CV /BM/BP /D4/D0/D3/D8/B4/DF/CK >/CA/CT/B4/CT/DA/CP/D0/CU/B4/D7/D9/CQ/D7/B4/D0/CP/D1/CQ/CS/CP/BP/BD/B8/D7/D9/CQ /D7/B4/CK >/DF/CP/D0/D4/CW/CP/D1/DC/BP/BC/BA/BD/B8/CC/D6/BP/BF/BC/BC/B8/D8/CP/D9/BP/BI/BA /BE/BH/CT/B9 /BG/BB/D0/CP /D1/CQ/CS/CP /CM/BE/B8/CV /CP/D1/BP/BC /BA/BC/BD /CK >/DH/B8/D7/D9/CQ/D7/B4/CP/D0/D4/CW/CP/BP/CP/D0/D4/CW/CP/CN/D7/D3/D0/CJ/BD℄/B8 /BE/BA/BH/BB /D7/D5/D6/D8 /B4/BE/B6/B4 /CP/D0/D4/CW /CP/B9/CV/CP /D1/B5/B5 /B5/B5/B5/B5 /B5/B8/CK >/CA/CT/B4/CT/DA/CP/D0/CU/B4/D7/D9/CQ/D7/B4/D0/CP/D1/CQ/CS/CP/BP/BD/B8/D7/D9/CQ /D7/B4/CK >/DF/CP/D0/D4/CW/CP/D1/DC/BP/BC/BA/BD/B8/CC/D6/BP/BF/BC/BC/B8/D8/CP/D9/BP/BI/BA /BE/BH/CT/B9 /BG/BB/D0/CP /D1/CQ/CS/CP /CM/BE/B8/CV /CP/D1/BP/BC /BA/BC/BD /CK >/DH/B8/D7/D9/CQ/D7/B4/CP/D0/D4/CW/CP/BP/CP/D0/D4/CW/CP/CN/D7/D3/D0/CJ/BF℄/B8 /BE/BA/BH/BB /D7/D5/D6/D8 /B4/BE/B6/B4 /CP/D0/D4/CW /CP/B9/CV/CP /D1/B5/B5 /B5/B5/B5/B5 /B5/DH/B8/CK >/C8/D9/D1/D4/BP/BC/BA/BC/BC/BC/BH/BA/BA/BC/BA/BC/BC/BH/B8/CP/DC/CT/D7/BP/CQ/D3 /DC/CT/CS/B8 /D0/CP/CQ/CT /D0/D7/BP/CJ /CO/C8/D9/D1 /D4/B8 /CP/BA/D9/BA/CO/B8 /CO/D8/D4/B8/CK >/CU/D7/CO℄/B8/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /DA/CT/D6/D7/D9/D7 /D4/D9/D1/D4/CO/B8 /D3/D0/D3/D6/BP/D6/CT/CS/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/CU/CX/CV/B8/DA/CX/CT/DB/BP/BH/BA/BA/BG/BC/B5/BN/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/BG/B8 /D4/CP/CV/CT /BI/BF/BT/D7 /DB /CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CT /D4/D9/D1/D4 /CV/D6/D3 /DB/D8/CW /CS/CT /D6/CT/CP/D7/CT/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /D9/D4 /D8/D3 /D7/D9/CQ/B9/BD/BC/CU/D7 /CS/CX/CP/D4/CP/D7/D3/D2/BA/CC/CW/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /CX/D2 /D8/CW/CT /D0/CP/D7/CT/D6 /DB/CX/D8/CW/D7/CT/D0/CU/B9/CU/D3 /D9/D7/CX/D2/CV /CX/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /CX/D2 /CB/CT /D8/CX/D3/D2 /BG/BA /C6/D3 /DB /DB /CT /DB/CX/D0/D0 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/B8 /D8/CW/CP/D8 /D8/CW/CT/CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CU/CT/CP/D8/D9/D6/CT/D7 /D3/CU /D8/CW/CT /D0/CP/D7/CX/D2/CV /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /CQ /D3/D8/CW /CU/CP /D8/D3/D6/D7 /CX/D7 /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD/D3/CU /D8/CW/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /B4 /D3/D1/D4/CP/D6/CT /DB/CX/D8/CW /CP/CQ /D3 /DA /CT /CS/CX/D7 /D9/D7/D7/CT/CS /D7/CX/D8/D9/CP/D8/CX/D3/D2/B5/BA /CF /CT/DB/CX/D0/D0 /AS/D2/CS /D7/D9 /CW /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA >/CP/D7/D7/D9/D1/CT/B4/D8/B8/D6/CT/CP/D0/B5/BM >/CP/D7/D7/D9/D1/CT/B4/D8/D4/B8/D6/CT/CP/D0/B5/BM >/CP/BC /BM/BP /BE/BB/D8/D4/BM /AZ /CC/CW/CX/D7 /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT >/D7/D3/D0/BM/BP/CX/D2/D8/B4/CP/BC/BB /D3/D7/CW/B4/D8/BB/D8/D4/B5/B8/D8/B5 /BM/AZ/CC/CW /CX/D7 /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D5/D9/CP/D6/CT /D4/D7/CX >/D7/D9/CQ/D7/B4/DF/CS/CX/D7/D4/BP/BC/B8/CQ/CT/D8/CP/BP/BC/B8/D4/D7/CX/B4/D8/B5 /BP/D7/D3/D0 /B8/CZ/CN/BE /BP/BC/B8/CQ /CT/D8/CP/BP /BC/DH/B8/D1 /CP/D7/D8 /CT/D6/CN/BG /B5/BM >/CT/DC/D4/CP/D2/CS/B4/B1/B5/BM >/D2/D9/D1/CT/D6/B4/B1/B5/BM >/CT/D5/BD /BM/BP /CT/DC/D4/CP/D2/CS/B4/B1/BB/B4/BG/B6/CT/DC/D4/B4/D8/BB/D8/D4/B5/B5/B5/BN eq1:=−2πNz /CNabsdωtp˜3q˜3λ2+ 2πNz /CNabsdωtp˜3q˜3λ2(e(t˜ tp˜))4 +αctp˜2q˜2λ2+ 2αctp˜2q˜2λ2(e(t˜ tp˜))2+αctp˜2q˜2λ2(e(t˜ tp˜))4 −gamctp˜2q˜2λ2−2gamctp˜2q˜2λ2(e(t˜ tp˜))2 −gamctp˜2q˜2λ2(e(t˜ tp˜))4+tf2cq˜2λ2 −6tf2cq˜2λ2(e(t˜ tp˜))2+tf2(e(t˜ tp˜))4cq˜2λ2 +16σ(e(t˜ tp˜))2c+δctp˜q˜2λ2−δctp˜q˜2λ2(e(t˜ tp˜))4/BV/D3/D0/D0/CT /D8 /D8/CW/CT /D8/CT/D6/D1/D7 /DB/CX/D8/CW /CT/D5/D9/CP/D0 /CS/CT/CV/D6/CT/CT/D7 /D3/CU /CT/DC/D4/B4/D8/BB/D8/D4/B5 /BA /BT/D7 /D6/CT/D7/D9/D0/D8 /DB /CT /D3/CQ/D8/CP/CX/D2/D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D2/CS /D7/DD/D7/D8/CT/D1 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA >/CT/BD /BM/BP /D3/CT/CU/CU/B4/CT/D5/BD/B8/CT/DC/D4/B4/D8/BB/D8/D4/B5/CM/BG/B5/BN >/CT/BE /BM/BP /D3/CT/CU/CU/B4/CT/D5/BD/B8/CT/DC/D4/B4/D8/BB/D8/D4/B5/CM/BE/B5/BN >/CT/BF /BM/BP /CT/DC/D4/CP/D2/CS/B4/CT/D5/BD/B9/CT/BD/B6/CT/DC/D4/B4/D8/BB/D8/D4/B5/CM/BG/B9 /CT/BE/B6/CT /DC/D4/B4/D8 /BB/D8/D4/B5 /CM/BE/B5/BN/BF/BJe1:= 2πNz /CNabsdωtp˜3q˜3λ2+αctp˜2q˜2λ2−gamctp˜2q˜2λ2+tf2cq˜2λ2 −δctp˜q˜2λ2 e2:= 2αctp˜2q˜2λ2−2gamctp˜2q˜2λ2−6tf2cq˜2λ2+ 16σc e3:=−2πNz /CNabsdωtp˜3q˜3λ2+αctp˜2q˜2λ2−gamctp˜2q˜2λ2+tf2cq˜2λ2 +δctp˜q˜2λ2 >/CT/BG /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/CT/BD/B9/CT/BF/B5/BN >/CT/BH /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/CT/BD/B7/CT/BF/B5/BN >/CT/BI /BM/BP /D7/CX/D1/D4/D0/CX/CU/DD/B4/CT/BE/B9/CT/BH/B5/BN e4:= 4πNz /CNabsdωtp˜3q˜3λ2−2δctp˜q˜2λ2 e5:= 2αctp˜2q˜2λ2−2gamctp˜2q˜2λ2+ 2tf2cq˜2λ2 e6:=−8tf2cq˜2λ2+ 16σc >/CP/D0/D0/DA/CP/D0/D9/CT/D7/B4/D7/D3/D0/DA/CT/B4/DF/CT/BG/BP/BC/B8/CT/BH/BP/BC /B8/CT/BI/BP /BC/DH/B8/CK >/DF/D8/D4/B8/CS/CT/D0/D8/CP/B8/D7/CX/CV/D1/CP/DH/B5/B5/BN {δ= 2tf2πNz /CNabsdωq˜ c(−α+gam),tp˜ =tf√−α+gam, σ=1 2tf2q˜2λ2}, {δ= 2tf2πNz /CNabsdωq˜ c(−α+gam),tp˜ =−tf√−α+gam, σ=1 2tf2q˜2λ2}/CF /CT /D7/CT/CT /D8/CW/CT /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CS/CX/AR/CT/D6/CT/D2 /CT/D7 /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CX/D8/D9/CP/D8/CX/D3/D2/BM /BD/B5 /D8/CW/CT/D6/CT /CX/D7/D8/CW/CT /D4/D9/D0/D7/CT /DB/CX/D8/CW /D7/CT /CW /B9/D7/CW/CP/D4 /CT /B4/D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/B5/BN /BE/B5 /D8/CW/CT /D4/D9/D0/D7/CT /CT/DC/CX/D7/D8/D7/B8 /DB/CW/CT/D2α < γ /B8/CX/BA /CT/BA /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D0/D3/D7/D7 /CT/DC /CT/CT/CS/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2/BA /CC/CW/CX/D7 /CX/D7 /CP/D2 /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CS/CT/D1/CP/D2/CS/D8/D3 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS /CQ/D6/CT/CP/CZ/D7 /D8/CW/CT /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D0/D3/D7/D7 /D3 /CTꜶ /CX/CT/D2 /D8 /CX/D2/D8/CW/CT /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6/BN /BF/B5 /D8/CW/CT /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /CT/DC/CX/D7/D8/D7 /D3/D2/D0/DD /CU/D3/D6 /D8/CW/CT /CS/CT/AS/D2/CT/CS/DA /CP/D0/D9/CT /D3/CUσ /B8 /DB/CW/CX /CW /CP/D2 /CQ /CT /CW/CP/D2/CV/CT/CS /CU/D3/D6 /D8/CW/CT /AS/DC/CT/CS /CP/CQ/D7/D3/D6/CQ /CT/D6 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CQ /DD /D8/CW/CT/DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CUλ /B8 /CX/BA /CT/BA /CQ /DD /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D1/D3 /CS/CT /D6/D3/D7/D7 /D7/CT /D8/CX/D3/D2 /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CP/D2/CS/CX/D2 /CP/CQ/D7/D3/D6/CQ /CT/D6 /D3/D6 /CQ /DD /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D1/CX/D6/D6/D3/D6 /D6/CT/AT/CT /D8/CX/DA/CX/D8 /DD /BA /C6/D3/D8/CT/B8 /D8/CW/CP/D8 /D8/CW/CT/D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /CS/CT/AS/D2/CT/CS /CQ /DD /D8/CW/CT /CU/D3/D6/D1 /D9/D0/CP/B8 /DB/CW/CX /CW /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D3/D2/CT /CU/D3/D6 /C3/CT/D6/D6/B9/D0/CT/D2/D7/D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /B4/D7/CT/CT /CB/CT /D8/CX/D3/D2 /BG/B5/BA /C4/CT/D8 /CU/D3/D9/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D7 /D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2/D3/CU /D4/D9/D1/D4/BA >/BX/D2/CT/D6/CV/DD/BP/BE/B6/CP/BC/CM/BE/B6/D8/D4/BM/AZ/CT/D2/CT/D6/CV/DD /D3/CU /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 >/CU/BD/BG /BM/BP/CK >/D2/D9/D1/CT/D6/B4/D7/CX/D1/D4/D0/CX/CU/DD/B4/D7/D9/CQ/D7/B4/D8/D4/BP/BD/BB/D7 /D5/D6/D8/B4 /CV/CP/D1/B9 /CP/D0/D4/CW /CP/B5/B8/CK >/D7/D9/CQ/D7/B4/BX/D2/CT/D6/CV/DD/BP/BE/B6/CP/BC/CM/BE/B6/D8/D4/B8/CU/BD/BE/B5 /B5/B5/B5/BM >/CP/D0/D4/CW/CP/CN/D7/D3/D0/BE /BM/BP /D7/D3/D0/DA/CT/B4/CU/BD/BG/BP/BC/B8/CP/D0/D4/CW/CP/B5/BM/AZ/D7/CP/D8/D9/D6/CP/D8 /CT/CS /CV/CP/CX/D2 >/CU/CX/CV/BE /BM/BP /D4/D0/D3/D8/B4/CK >/CA/CT/B4/CT/DA/CP/D0/CU/B4/D7/D9/CQ/D7/B4/D0/CP/D1/CQ/CS/CP/BP/BC/BA/BH/B8/D7 /D9/CQ/D7/B4 /CK >/DF/CP/D0/D4/CW/CP/D1/DC/BP/BC/BA/BD/B8/CC/D6/BP/BF/BC/BC/B8/D8/CP/D9/BP/BI/BA /BE/BH/CT/B9 /BG/BB/D0/CP /D1/CQ/CS/CP /CM/BE/B8/CV /CP/D1/BP/BC /BA/BC/BD /CK >/DH/B8/D7/D9/CQ/D7/B4/CP/D0/D4/CW/CP/BP/CP/D0/D4/CW/CP/CN/D7/D3/D0/BE/CJ/BE℄ /B8/BE/BA/BH /BB/D7/D5/D6 /D8/B4/CV/CP /D1/B9/CP/D0 /D4/CW/CP/B5 /B5/B5/B5 /B5/B5/B8/CK >/C8/D9/D1/D4/BP/BC/BA/BC/BC/BC/BH/BA/BA/BC/BA/BC/BC/BH/B8/CP/DC/CT/D7/BP/CQ/D3 /DC/CT/CS/B8 /D0/CP/CQ/CT /D0/D7/BP/CJ /CO/C8/D9/D1 /D4/B8 /CP/BA/D9/BA/CO/B8 /CO/D8/D4/B8/CK >/CU/D7/CO℄/B8/D8/CX/D8/D0/CT/BP/CO/C8/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /DA/CT/D6/D7/D9/D7 /D4/D9/D1/D4/CO/B8 /D3/D0/D3/D6/BP/CQ/D0/D9/CT/B5/BM >/CS/CX/D7/D4/D0/CP/DD/B4/CU/CX/CV/B8/CU/CX/CV/BE/B8/DA/CX/CT/DB/BP/BH/BA/BA/BE/BC/B5 /BN/BF/BK/CB/CT/CT /BY/CX/CV/D9/D6/CT /BE/BH/B8 /D4/CP/CV/CT /BI/BG/CC/CW /D9/D7/B8 /D8/CW/CT /C3/CT/D6/D6/B9/D0/CT/D2/D7/CX/D2/CV /B4/D0/D3 /DB /CT/D6 /D9/D6/DA /CT/B5 /CP/D0/D0/D3 /DB/D7 /D8/D3 /D6/CT/CS/D9 /CT /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/CP/D2/CS /D8/D3 /CV/CT/D2/CT/D6/CP/D8/CT /D8/CW/CT /D7/D9/CQ/B9/BD/BC /CU/D7 /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/BA/C1/D2 /D8/CW/CT /D3/D2 /D0/D9/D7/CX/D3/D2 /B8 /DB /CT /CW/CP/CS /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/CS/B8 /D8/CW/CP/D8 /CP/D2 /CY/D3/CX/D2 /D8 /CP /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CP/D7/CX/D2/CV/CU/CP /D8/D3/D6/D7 /CP/D2/CS /D3/CW/CT/D6/CT/D2 /D8 /CP/CQ/D7/D3/D6/CQ /CT/D6 /D3/CQ /CY/CT /D8/CX/DA /CT/D7 /D8/D3 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CW/CT/D6/CT/D2 /D8/D7/D3/D0/CX/D8/D3/D2/BA /CC/CW/CT /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CX/D2 /D8/CW/CX/D7 /CP/D7/CT /CW/CP/D7 /BEπ /B9/D7/D5/D9/CP/D6/CT/B8 /CQ/D9/D8 /CX/D8 /CX/D7 /D2/D3/D8 /D7/CT /CW /B9/D7/CW/CP/D4 /CT /D4/D9/D0/D7/CT /B4/D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/B5/BA /CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D6/CT/D4/D0/CP /CT/CS /DB/CX/D8/CW/CX/D2 /CX/D2 /D8/CT/D6/DA 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/CP/D0/D0/B9/D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT/B8 Ꜽ/CW/CP/D2/CS /CU/D6/CT/CTꜼ /CU/CT/D1 /D8/D3/D7/CT /D3/D2/CS /D0/CP/D7/CT/D6/D7/BA/BK /BV/D3/D2 /D0/D9/D7/CX/D3/D2/CC/CW/CT /D4 /D3 /DB /CT/D6/CU/D9/D0 /D3/D1/D4/D9/D8/CP/D8/CX/D3/D2 /CP/CQ/CX/D0/CX/D8/CX/CT/D7 /D3/CU /C5/CP/D4/D0/CT /CE /CP/D0/D0/D3 /DB /CT/CS /D8/D3 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D8/CW/CT/CQ/CP/D7/CX /D3/D2 /CT/D4/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D6/D2 /CU/CT/D1 /D8/D3/D7/CT /D3/D2/CS /D8/CT /CW/D2/D3/D0/D3/CV/DD /BA /CC/CW/CT /CP/D4/D4/D0/CX /CP/D8/CX/D3/D2/D3/CU /D8/CW/CT/D7/CT /D3/D2 /CT/D4/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CP/D2/CS /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CS/D9/CT/D8/D3 /D3/CW/CT/D6/CT/D2 /D8 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6 /D0/CT/CP/CS/CT/CS /D8/D3 /D8/CW/CT /D2/CT/DB /D7 /CX/CT/D2 /D8/CX/AS /D6/CT/D7/D9/D0/D8/D7 /B4/D7/CT/CT/B8/CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /C2/BA /C2/CP/D7/CP/D4 /CP/D6 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±1.6.5.4.3.2.1.0 Standing wave xStanding wave x Standing wave xStanding wave x/BG/BC050100150200 ±60 ±40 ±20 0 20 40 60 80 tresult of modes interference/BG/BD/BG/BEt t .50 0 -.50 ±1.100.80.60.40.20.02.1. 0 ±1.±2.100.80.60.40.20.02.1. 0 ±1.±2. 100.80.60.40.20.0.50 0 -.50 ±1.100.80.60.40.20.0 t t/BG/BF±1±0.8±0.6±0.4±0.200.20.40.60.81y(t)01 z(t)20406080100 tmode locking/BG/BG00.20.40.60.81 ±4 ±2 0 2 4 tfirst-order soliton/BG/BH/BG/BI/BG/BJ/BG/BK±0.04±0.0200.020.04 drho(t)/dt 0.1 0.2 0.3 0.4 rho(t)/BG/BLPulse envelope rho^2, GW/cm^2 time, fsPulse envelope rho^2, GW/cm^2 time, fsPulse envelope rho^2, GW/cm^2 time, fs Pulse envelope rho^2, GW/cm^2 time, fs2.001.801.601.401.201.00 .80.60.40.20 -.1e±124.3.2.1. -.1e±127.6.5.4.3.2.1. -.1e±1210. 8.6.4.2. -.1e±12 /BH/BC/BH/BD/BH/BE/BH/BF/BH/BG0.050.10.150.20.25 0.4 0.6 0.8 1 1.2 1.4 1.6/BH/BHy, MV/cm pulse squarePulse amplitude versus its square y, MV/cm pulse squarePulse amplitude versus its square 2.502.001.501.00 .50 06.5.4.3.2.1.0.70.60.50.40.30.20.10 06.5.4.3.2.1.0.50.40.30.20.10 06.5.4.3.2.1.0.40.30.20.10 06.5.4.3.2.1.0 y, MV/cm pulse squarePulse amplitude versus its square y, MV/cm pulse squarePulse amplitude versus its square /BH/BI/BH/BJpsi time, t/tfPulse square versus timepsi time, t/tfPulse square versus time psi time, t/tfPulse square versus time psi time, t/tfPulse square versus time 6.0000000005.0000000004.0000000003.0000000002.0000000001.000000000 10.5.0 ±5. ±10.5.004.504.003.503.002.502.00 1.5 10.5.0 ±5. ±10.4.504.003.503.002.50 2. 10.5.0 ±5. ±10.4.003.503.002.50 2. 10.5.0 ±5. ±10. /BH/BKrho, MV/cm time, t/tfPulse envelope rho, MV/cm time, t/tfPulse envelope rho, MV/cm time, t/tfPulse envelope rho, MV/cm time, t/tfPulse envelope .250.200.150.100 .50e±1 10.5.0 ±5. ±10..240e±1.220e±1.200e±1.180e±1.160e±1 .14e±1 10.5.0 ±5. ±10..170e±1.160e±1.150e±1.140e±1 .13e±1 10.5.0 ±5. ±10..1400e±1.1350e±1.1300e±1.1250e±1 .12e±1 10.5.0 ±5. ±10. /BH/BL/BI/BC/BI/BD/BI/BE6810121416182022242628303234363840 tp, fs 0.001 0.002 0.003 0.004 0.005 Pump, a.u.Pulse duration versus pump/BI/BF68101214161820 tp, fs 0.001 0.002 0.003 0.004 0.005 Pump, a.u.Pulse duration versus pump/BI/BG
1 SONOLUMINESCENCE: NATURE’S SMALLEST BLACKBODY G. Vazquez, C. Camara, S. Putterman, K. Weninger Physics Department, University of California, Los Angeles, CA 90095 The Spectrum of the light emitted by a sonoluminescing bubble is extremely well fit by the spectrum of a blackbody. Furthermore the radius of emission can be smaller than the wavelength of the light. Consequences, for theories of sonoluminescence are discussed. The phenomenon of sonoluminescence occurs when a trapped bubble of gas in water is driven into high amplitude pulsations by a strong sound wave. Key parameters that characterize the bubble and sound field are the amplitude of the sound aPand the ambient radius of the bubble 0R. This is the radius when 0 =aP . During the rarefaction portion of the cycle of sound the bubble expands from 0R to mR the maximum radius that is about 010R. During the ensuing compression the bubble catastrophically collapses right through 0R on its way down to its collapsed radius cR that is determined by the van der Waals hard core of the particular gas molecule [for Helium 100R Rc= ]. As R approaches cR the input acoustic energy density concentrates by 12 orders of magnitude [11] and leads to the emission of a broad band flash of light whose width is delineated in picoseconds. Plasma processes have been invoked by a number of researchers as the basis for explaining the light emitting mechanism [5, 12-17]. Atoms that are neutral at low 2 temperatures ‘T’ become ionized as T increases inside the collapsed bubble. In the resulting plasma charged particles emit and absorb light through various processes. Electrons moving with the thermal velocity can collide with neutral atoms or with ions to radiate light as they accelerate [thermal Bremsstrahlung]. Light propagating in the plasma can be absorbed by the inverse of these processes as well as by ionizing collisions with atoms. If the distance that light propagates from generation to re- absorption γl is small compared to the size of the plasma then the medium is opaque and the spectrum will be that of a blackbody and radiation is from the surface [1]. For large values of γl the body becomes transparent. Radiation is emitted form the volume of the bubble, so therefore the spectrum yields information about the underling collisional processes. For hydrogen at 6000K and a density of 2.3x1022 molecules/cm3, m lr10≈ as calculated from formulas for the inverse of electron-neutral Bremsstrahlung. This number is much greater than the mµ35. radius of the collapsed bubble [18] and therefore also greater than R e. Similar calculations have led researchers to claim that the sonoluminescing medium is a transparent plasma [12-17,19]. Nevertheless, we observe the blackbody behavior shown in Figure 1. Reconciliation of theory and experiment must be sought in the fact that the contents of the SL bubble are very dense, being compressed to the van der Waal’s hard core [5,11]. Saha’s equation for the degree of ionization and the standard Bremsstrahlung formulas [including rl] are calculated in the dilute gas limit where behavior is dominated by binary collisions. This could be the opposite limit to the one that describes SL [17]. A mechanism where a ‘cool’ blackbody 3 spectrum can mask an arbitrarily high interior temperature [and shorter rl] is pre- heating due to thermal radiation transport that is set off by a shock wave [1]. Another paradox of the blackbody interpretation of SL relates to the density of states. The derivation of Planck’s formula assumes that many modes [i.e. many wavelengths of light] fit into the hot blackbody, but the SL hot spot is smaller than the wavelength of light that we measure. Perhaps those processes that lead to a small rl also lead to a large index of refraction [and therefore a smaller λ] inside the hot spot. In this model light would be released as rl suddenly increased in the cooling hot spot. The existence of such a switching process inside the bubble is suggested by the plot in Figure 2B which shows that the ratio of the ionization temperature to the blackbody temperature is the same value for all noble gas bubbles. That noble gas bubbles are accurately described by Planck’s formula is shown in Figure 2A. Measurements of the flash width as a function of color also support a switching mechanism. It is found that between 200nm and 700nm the flash width varies by less than 20% [being longer in the ultra-violet! {18}]. Within the absolute timing resolution of about 40ps it appears that the spectrum is blackbody from start to finish. Such behavior is also found to exist in plasmas that are generated at the focus of a strong laser in water [20]. A clue to the transition from opaque to transparent behavior is suggested by Figure 3 that displays the SL spectrum of a cloud of cavitating argon bubbles driven by sound at 1MHz [21]. The spectrum of light generated at this higher acoustic frequency is not well described by Planck’s formula, but is characteristic of formulas for thermal 4 Bremsstrahlung[1]. The radii, cR, of the 1MHz bubbles [21] at the moment of collapse are so small [ ≈cR 50nm] that perhaps these bubbles are transparent to light. Thus for SL the transition from opaque [blackbody] to transparent takes place at around 100nm. The robust behavior of SL even at the nanoscale has been pushed to 11MHz where scaling arguments suggest that nmcR 10≈ . The spectral densities at 1MHz and 11MHz are similar [Figure 3]. The transduction of sound into light by a collapsing bubble is robust down to the nanoscale. Although radiation is emitted from a surface whose radius is smaller than, the wavelength of light and the photon matter mean free path, the observed spectrum can match that of a blackbody for [ nm nm 800 200 <<λ and meRµ1.> ]. Whether blackbody behavior extends out into the Rayleigh-Jeans limit of large λ is a question for future experiments. As an equilibrated blackbody radiates only from its surface the extent to which the bubble’s contents are stressed may be unrepresented in the observed spectrum. The spatial correlations and mode counting dilemma that would characterize such a small blackbody could constitute a new regime for the application of statistical optics. The inability to reconcile the long photon mean free path with the smallness of the hot spot suggests new physics in the modeling of SL. REFERENCES 1. Ya.B.Zeldovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic, Phenomena, 2vols [Academic Press, NY 1966,1967]. 5 2. S. Glasstone, R. Lovberg, Controlled Thermonuclear Reactions, [Van Nostrand Reinhold 1960]. 3. R.Hiller, S. Putterman, B.P. Barber, Phys. Rev. Lett. 69, 1182 (1992). 4. R. Hiller, S. Putterman, Phys. Rev. Lett. 75, 3549 (1995); 77,2345 (E). 5. B.P. Barber, R.A. Hiller, R. Löfstedt, S.J. Putterman, and K.R. Weninger, Phys. Reports, 281, 67 (1997). 6. Hiller, R.A., Putterman, S.J., and Weninger, K.R., Phys. Rev. Lett. 80, 1090 (1998). 7. T. Matula et al Phys. Rev. Lett. 75,2602 (1995). 8. All data in the paper is acquired from bubbles acoustically driven in sealed cylindrical resonators constructed with quartz walls and stainless steel end-caps [5]. Spectra were measured with light falling directly on the input slit (through order sorting filters) of a spectrometer (Acton 308i) read out by an intensified CCD (Princeton Inst. IMAX) and are fully radiance calibrated against commercially available QTH and D 2 lamps. Spectra in figures 1 and 3 acquired through commercial grade quartz (GM assoc. # 214) whereas fig2 is acquired in cells constructed of ‘suprasil’. No spectra are corrected for transmission of water or quartz. For our commercial grade quartz, there is absorption for wavelengths below 300 nm that rises to 25 % at 200nm. The suprasil has constant transmission for all wavelengths above 200nm. We attribute the bump in the data at 550 nm and the dip at 360 nm to documented errors in the manufacturer-supplied calibration of our lamps (see fig 75 reference 5). 6 9. Spectra reported here have the same spectral density and detailed shape as reported in previous papers. But in the course of recalibrating the system we find that the scale for the ‘y’ axis, namely spectral radiance, is generally lower, being down by about a factor of 12 compared to 4,6. We have verified the new data calibrated against various lamp standards with photon counting through bandpass filters. Previously quoted values of photons per flash remain unchanged. We believe that the mistake in scaling the ‘y’ axis is greater than can be accounted for by resonator variability, drive level, and thermal drift [see discussion in ref.5]. The corrected value of radiance plus our ability to measure flash width and bubble size combine to make possible the quantitative comparisons to blackbody radiation proposed here. 10. J. Maddox, Nature 361,397 (1993). 11. B.P. Barber, and P.J. Putterman, Nature, 352, 318 (1991). 12. C.C.Wu and P.H. Roberts, Phys. Rev Lett. 70, 3424 (1993). 13. W. C. Moss, D. B. Clarke, and D.A. Young, Science, 276, 1398 (1997). 14. S. Putterman, Scientific American, 272 (2), 46 (1995). 15. D. Hammer and L. Fromhold, Phys. Rev Lett. 85, 1326 (2000). 16. S.Hilgenfeldt, S. Grossman, D. Lohse Nature 398, 402 (1999). 17. S.J. Putterman and K.R. Weninger Ann. Rev. Fluid Mech. 32, 445 (2000). 7 18. G. Vazquez, et al, preprint, present the data that determines an ambient radius of 3.5microns and the time resolved spectrum of a hydrogen bubble. The collapse radius is derived as the van der Waals hard core corresponding to the measured R 0 , being roughly R 0/10. 19. R. Hiller, PhD Thesis 1995. 20. R.J. Thomas, D.X. Hammer, G.D. Noojin, D.J. Stolarski, B.A. Rockwell and W.P. Roach, “Time-Resolved Spectroscopy of laser induced breakdown in water”, Proc. Laser Tissue Interaction VII, SPIE, 2681 , 402 (1996). O. Baghdassarian, B. Tabbert, G.A. Williams [preprint] report blackbody like spectral densities from laser generated plasmas in water and from the ensuing cavitation collapse. 21. K. Weninger et al, preprint, present the details of the experimental technique for exciting and measuring a cloud of sonoluminescing bubbles at 1MHz, and 11MHz. 22. Research supported by DARPA and the NSF. FIGURE CAPTIONS Figure 1: Spectrum of Sonoluminescence from a hydrogen bubble in water (23C) driven at 33 kHz. The hydrogen is dissolved into the water at a partial pressure of 5 torr. Data acquired with 24 nm FWHM resolution. The solid line is a fit to a blackbody at 6230 K. Using the measured flash width of 110 ps, this fit requires emission from a surface with radius of 0.22 microns. The dashed line is a bremsstrahlung fit with a temperature of 15000 K. 8 Figure 2: A) Spectrum of Sonoluminescence from bubbles of helium (150 torr) and xenon (3 torr) in water (23C) driven at 42 kHz. Resolution is 12 nm FWHM. The solid lines are blackbody fits at 8000K (xenon) and 20400K (helium). Using measured flash widths of 100 ps (helium) and 200 ps (xenon) gives emission from a surface of radius 0.1 microns (helium) and 0.4 microns (xenon). Ambient radii measured with light scattering techniques [5] are about 5.5 micron (xenon) and 4.5 micron (average value for helium) from which we estimate cR (= 6.70R (xenon), = 8.90R (helium)) as 0.7 micron (xenon) and 0.5 micron (helium). [Note that a 150Torr He bubble is not in diffusive equilibrium [5].] The dashed line is a bremsstrahlung spectrum at infinite temperature. B) Ratio of ionization potential ( χ) of the gas used to make sonoluminescence to the temperature of the blackbody (T BB) best fitting the observed spectrum for each gas. This ratio is plotted vs. the ionization potential of the gas (k is Boltzman’s constant). Gases plotted include the 5 light noble gases as well as hydrogen (χ=15.5 eV for H 2). Figure 3: Spectrum of Sonoluminescence from a cloud of cavitation bubbles in water driven at 1 MHz given per cm3 of cloud volume. The water is saturated with argon and maintained at C018 . Resolution is 12 nm FWHM. The solid line is a bremsstrahlung fit with temperature 85,000 K and the dashed lines are blackbody curves with temperatures of 9900 K and 15,000K. The open circles are the measured spectrum (60 nm FWWM) of xenon bubbles driven in water (11C) with sound at 11 MHz and offset vertically by an arbitrary amount.
arXiv:physics/0009058v1 [physics.plasm-ph] 15 Sep 2000Transport control by coherent zonal ﬂows in the core/edge tr ansitional regime K. Hallatschek, D. Biskamp Max-Planck Institut f¨ ur Plasmaphysik, EURATOM-IPP Assoc iation, D-85748 Garching, Germany 3D Braginskii turbulence simulations show that the energy ﬂux in the core/edge transition region of a tokamak is strong ly modulated – locally and on average – by radially propagat- ing, nearly coherent sinusoidal or solitary zonal ﬂows. The ﬂows are geodesic acoustic modes (GAM), which are primar- ily driven by the Stringer-Winsor term. The ﬂow amplitude together with the average anomalous transport sensitively de- pend on the GAM frequency and on the magnetic curvature acting on the ﬂows, which could be inﬂuenced in a real toka- mak, e.g., by shaping the plasma cross section. The local modulation of the turbulence by the ﬂows and the excitation of the ﬂows are due to wave-kinetic eﬀects, which have been studied for the ﬁrst time in a turbulence simulation. a. Introduction — It is now commonly believed, that the transport in the tokamak core is controlled by zonal ﬂows [1–3]. In the plasma edge, the ﬂows were not stud- ied thoroughly yet, but they tend to be weak [4] (al- though they can possibly completely quench the tur- bulence [5]). The zonal ﬂows in the core and at the edge were found to be incoherent, random ﬂuctuations [2,6,7]. In contrast to this, in the transitional regime be- tween core and edge, strong radially coherent sinusoidal or solitary zonal ﬂows are ubiquitous according to the numerical simulations described below. The zonal ﬂows in the transitional regime are essentially geodesic acous- tic modes [8], i.e., the poloidal rotation is coupled to an (m,n) = (1,0) pressure perturbation by the inhomoge- neous magnetic ﬁeld, which results in a restoring force and hence an oscillation. Transcending the present mod- els of the zonal ﬂow generation based purely on Reynolds stress [1], the major part of the ﬂow energy in the tran- sitional regime is apparently generated by the Stringer- Winsor term [9], i.e., the torque on the plasma column caused by the interaction of pressure inhomogeneities with the inhomogeneous magnetic ﬁeld. An order of mag- nitude variation of the shear ﬂow level and average trans- port has been observed upon a sole modiﬁcation of the curvature terms acting on the ﬂows, at ﬁxed curvature terms acting on the turbulence. The pressure inhomo- geneities are driven by anomalous transport modulations, which can be understood by a drift-wave model for the wave-kinetic turbulence response to the ﬂows. Some pre- dictions of the model have been veriﬁed subsequently in numerical experiments. Zonal ﬂows in the transitional regime are GAMs— The numerical turbulence simulations have been carried out using the three dimensional electrostatic drift Bra- ginskii equations with isothermal electrons, including th e ion temperature ﬂuctuations with the associated polar- ization drift eﬀects, the resistive (non-adiabatic) paral lel electron response and the parallel sound waves (a subsetof the equations of Ref. [5]). The nondimensional param- eters have been varied around a reference parameter set resembling the transitional core/edge regime: αd= 0.6, ǫn= 0.08,q= 3.1,τ= 1,ηi= 3, ˆs= 1. The compu- tational domain is a ﬂux tube winding around the toka- mak for three poloidal connection lengths. The radial and poloidal domain width is 50 LRB, with the resistive ballooning scale length, LRB. For a deﬁnition of these pa- rameters and units see Ref. [5]. The parameters are con- sistent with the physical parameters R= 3 m,a= 1.5 m, Ln= 12 cm,n= 3.5×1019m−3,Zeﬀ= 4,B0= 3.5 T, andT= 200 eV, LRB= 3.6 mm,ρs= 0.82 mm. At these parameters the ITG mode turbulence is the domi- nant cause of the heat ﬂux [4], and the restriction of the domain to a ﬂux tube is justiﬁed [7]. Viewed as a function of radius and time, the ﬂux sur- face averaged poloidal E×Bﬂows [ﬁg. 1 (a)] start as an irregular pattern of radially propagating independent wavelets and merge into a radially coherent standing wave later on. The ﬁnal standing wave pattern consists of GAM oscillations. They can be described by suitable poloidal Fourier components of the vorticity and pressure evolution equations (neglecting the parallel sound wave), ∂t∝angbracketleftvE∝angbracketright − ∝angbracketleftsv∝angbracketright=−C1∝angbracketleftpsinθ∝angbracketright (1) ∂t∝angbracketleftpsinθ∝angbracketright − ∝angbracketleftspsinθ∝angbracketright=C2ǫn 33 + 5τ 1 +τ∝angbracketleftvEsin2θ∝angbracketright,(2) with∝angbracketleft.∝angbracketrightdenoting the ﬂux surface average, the pressure ﬂuctuations p=n+τ 1+τTi, the poloidal ﬂow velocity vE=∂xφ, and the source terms sv/spof ﬂow/pressure due to Reynolds stress/anomalous transport, respec- tively. For easier reference, the two diﬀerent curvature terms in the equations have been adorned with the factors C1andC2, which are both 1 in the turbulence equations for a low aspect ratio circular tokamak. The curvature termC1is the Stringer-Winsor term, the term C2repre- sents the up-down asymmetric compression of the plasma due to the poloidal rotation. For all parameters used in the numerical simulations, the zonal ﬂows oscillate in the stationary state with a frequency within 5% of the eigen- frequency of Eqs. (1,2) without source terms and with the approximation ∝angbracketleftvEsin2θ∝angbracketright ≈ ∝angbracketleftvE∝angbracketright/2, ω=/radicalbigg C1C2ǫn 63 + 5τ 1 +τ. (3) In physical units this is equal to/radicalbig (6 + 10τ)/(3 + 3τ)cs/R∼cs/R. Since the parallel sound frequency cs/(qR) is much lower than the GAM frequency, the neglect of the parallel sound wave 1is justiﬁed. The energy balance equation of the GAM oscillations is according to (1,2) ∂t1 2/bracketleftbigg ∝angbracketleftvE∝angbracketright2+C1 ω∝angbracketleftpsinθ∝angbracketright2/bracketrightbigg = ∝angbracketleftvE∝angbracketright∝angbracketleftsv∝angbracketright+C1 ω∝angbracketleftpsinθ∝angbracketright∝angbracketleftspsinθ∝angbracketright (4) The average contributions to the GAM energy from the Reynolds stress term ∝angbracketleftsv∝angbracketrightand the Stringer-Winsor term ∝angbracketleftspsinθ∝angbracketrighthave been listed in table I for varying turbulence parameters. In the transitional regime with its strong co- herent zonal ﬂows most of the ﬂow energy is generated by the Stringer-Winsor term, while with decreasing temper- ature toward the very edge the Stringer-Winsor energy input eventually becomes negative indicating a braking force on the ﬂows, while simultaneously we get weak in- coherent ﬂows. One is tempted to attribute the decrease of the zonal ﬂows towards the very edge to the Stringer- Winsor term. Indeed, eliminating the ﬂow source term ∝angbracketleftspsinθ∝angbracketrightdue to the anomalous transport in the numer- ical simulations leads to relatively strong coherent ﬂows even for parameters in the very edge. With the natural drive of the GAM being apparently the Stringer-Winsor term, it has been found that altering the amplitude of the curvature terms C1,C2acting on the ﬂows but keeping the curvature terms acting on the tur- bulence modes ﬁxed, changes the ﬂow amplitudes and the anomalous transport by one order of magnitude. Empir- ically, the ﬂow level rises with increasing C1and decreas- ingω. One reason is, that the higher C1/ω∼/radicalbig C1/C2 is, the higher is the contribution of the anomalous trans- port source term spto the ﬂow energy (4). At constant ratioC1/ωthe ﬂows are still somewhat increasing for decreasing ω. This is understood, since with increasing oscillation period the ﬂows have more time to inﬂuence the turbulence resulting in an increased sp. The cause of the pressure perturbations driving the ﬂows are local modulations of the anomalous transport (the usual Stringer spin-up mechanism is ineﬀective due to the relatively long sound transit time in the con- sidered regime). Plots of the radial pressure transport ∝angbracketleftvrp∝angbracketright=Q(r,t) and its up-down antisymmetric compo- nent∝angbracketleftvrpsinθ∝angbracketright=U(r,t) are shown in ﬁg. 1 (b) and (c). (Note that the pressure source term is the divergence of the anomalous pressure ﬂux, i.e., sp=−∂r(vrp), and ∝angbracketleftspsinθ∝angbracketright=−∂rU.) In the initial phase of ﬂow genera- tion,Udevelops dipoles around the ﬂows [ﬁg. 1 (d)], in whichUhas always the same sign as the local shear- ing rate. These dipolar transport structures generate the pressure up-down asymmetries driving the GAMs. As soon as suﬃciently strong ﬂows exist, the anoma- lous transport Qdevelops a striking peaking at the radii of positive ﬂow resembling transport fronts propagating with the ﬂow “waves”. In addition to the dipole struc- turesUdevelops a unipolar part component, whose signdepends on the propagation direction of the correspond- ing ﬂow. These up-down antisymmetric transport fronts can be viewed as avalanches running outward on the lower half of the torus ( θ <0) and inward on the up- per half. The unipolar up-down asymmetries have been found to be responsible for the setup of the ﬂow pattern. If in a numerical experiment the GAMs are initially set to zero outside of one ﬂow peak, the turbulence is still capable of moving this ﬂow into the original direction, until a new standing wave pattern has been formed. The necessary radial GAM energy ﬂow due to the turbulence has been found to be primarily caused by the unipolar up-down asymmetries. Similar events happen, if a numerical sim- ulation is started with a GAM pattern with the wrong wave-length from a simulation run with diﬀerent GAM parameters. In that case, strong unipolar up-down trans- port asymmetries develop, which enforce the equilibrium ﬂow propagation velocity, i.e., ﬂow wavelength. The described peculiar modulation of the transport by the ﬂows is absent for weak diamagnetic drift and vanish- ing gyro radius, such as in the resistive ballooning regime [4]. Instead, the shear ﬂows simply weaken the turbu- lence. Apart from this, there is a tendency to ﬂatten pressure gradients, i.e., to eliminate the pressure ﬂuctua - tions associated with the GAM, resulting in the observed braking force. The wave-kinetic eﬀects The dependence of the transport modulation on drift eﬀects and the gyro ra- dius suggests a simpliﬁed drift wave model containing only the radial mode coupling due to the polarization drift, eliminating two ﬂuid and curvature eﬀects. Since in the numerical studies the mode wavelengths are not small compared to the zonal ﬂow scales, we refrain from a geometrical optics approach [1] and instead move back to the linearized adiabatic drift wave equation, Dt(1−ρ2 s∆⊥)φ+αd∂yφ= 0, (5) withDt=∂t+vE(x,t)∂y. We study the impact of the polarization drift term −Dtρ2 s∆⊥nup to ﬁrst order in ρ2 s, without assumptions on the ratio of ﬂow vs. turbulence scales. The 0-th order time evolution (with yin Fourier space) is φ0(x,t) =φi(x)ψ(x,t) (6) with initial amplitude φi(x) and the ﬂow-and-drift in- duced phase factor ψ(x,t) = exp[ −ikyξ], ξ=/integraldisplayt 0[vE(x,τ) +αd]dτ.(7) Inserting (6) into (5) results in the ﬁrst order correction φ1due toρ2 s φ1(x,t)ψ∗(x,t) =−ikyαdρ2 s/integraldisplayt 0ψ∗(x,τ)∆⊥(φiψ(x,τ))dτ+ 2+ρ2 s/bracketleftbig ψ∗(x,τ)∂2 x(φiψ(x,τ))/bracketrightbigτ=t τ=0, (8) in which [f(...,τ)]τ=t τ=0≡f(...,t)−f(...,0). From this, the change in turbulence intensity can be computed to ﬁrst order, δ\|φ\|2= 2Re(φ∗ 1φ0) = 2kyαdρ2 s/integraldisplayt 0∂x[kx(x,τ)\|φi\|2]dτ −2ρ2 s[k2 x(t)−k2 x(0)]\|φi\|2, (9) with a suitable local kx(x,τ)≡ −kyξ′(x,τ) + Im(φ′ i/φi). The term involving the diamagnetic drift velocity αdcor- responds to the advection of mode intensity in response to the shearing distortion, while the other term is analo- gous to the adiabatic compression of a wave ﬁeld. If the initial φihas no radial structure ( kx= 0) and the shear ﬂows do not change with time, we obtain due to radial mode advection δ\|φ\|2 advection =−ρ2 sαd\|φi\|2t2k2 yv′′ e, (10) which explains the empirical peaking in turbulence inten- sity at the locations of positive ﬂows. To verify the pres- ence of the advection term, a complementary scenario has been simulated, with the turbulence initially conﬁned to a small region and a constant linear shear ﬂow vE∝x enforced upon it. This results in a motion of the turbu- lence maxima towards increasing vE, which conﬁrms the presence of the advection term (see ﬁg. 2). In another nu- merical experiment, a stationary bell shaped ﬂow proﬁle has been superposed on an initially radially homogeneous turbulence ﬁeld, which has been prepared with the zonal ﬂows switched oﬀ. After the turbulence rises transiently at the ﬂow maximum, it drops to a level below the initial one. This corroborates that the turbulence ampliﬁcation at the ﬂow maxima is not due to an increase of drive at the ﬂow maxima [10], but due to a transient wave-kinetic concentration of ﬂuctuation energy. The numerically observed radial dipole layers of up- down antisymmetric transport apparently result from the compressional terms in (9) acting on the mode structure enforced by the magnetic shear, namely kx∼kyˆsθ. (11) Hence, with ˆ s= 1 a shear ﬂow with positive v′ ereduces k2 xon the upper side of the tokamak ( θ >0) increasing the mode amplitude there, while the mode is attenuated for (θ <0). The dependence of the shear ﬂow action on the magnetic shear has been veriﬁed in a numerical experiment with ˆ s=−1, in which the modes are am- pliﬁed forθv′ e<0. Moreover, a “swinging through” eﬀect hast been observed, i.e., the turbulence intensity decreases again, when k2 xrises afterkxhas gone through zero. Last, we consider the unipolar up-down transport asymmetries induced by a propagating positive shear ﬂow(which is accompanied by a general transport peaking due to the advective eﬀect). With vE(x,t) =V(x−νt) (propagation velocity ν) we obtain from deﬁnition (7) ξ′(x,t) =−ν−1V(x−νt), i.e. kx(x,t)2= (kx0−kyξ′)2=k2 y(ˆsθ+ν−1V(x−νt))2. (12) A reduction of k2 xand corresponding ampliﬁcation of the turbulence modes via eq. (9) occurs for ˆ sθνV < 0. The sign of this eﬀect of a moving ﬂow has been conﬁrmed numerically for positive and negative shear. Conclusions — GAMs, oscillating zonal ﬂows, have been found to be the main mechanism controlling the tur- bulence level in the transitional core/edge regime. Quasi- stationary zonal ﬂows are unimportant in the simula- tions, because of the strong restoring force due to the pressure imbalance generated by the magnetic ﬁeld in- homogeneity, which cannot be short-circuited along the magnetic ﬁeld lines due to the long parallel sound transit time in the transitional regime. Primarily, the ﬂows are not driven by Reynolds stress but by the pressure asym- metries on a ﬂux surface generated by modulations of the anomalous transport. These modulations in turn are caused by the ﬂows, on one hand by the radial advec- tion of turbulence energy concentrating the turbulence in locations of increased ﬂow in electron diamagnetic di- rection, and on the other hand by adiabatic compression eﬀects on the wave ﬁeld which together with the mag- netic shear result in up-down asymmetries of the trans- port. The peculiar nature of the drive mechanism leads to propagating peaked ﬂow structures acompanied by pro- nounced transport fronts, which sometimes come close to propagating solitons. We have studied for the ﬁrst time the action of wave-kinetic eﬀects in a numerical simu- lation. Similar results should be expected for the core, except that the polarization due to the ion larmor ra- dius has to be replaced by the neoclassical polarization due to the banana width [11]. I.e., we expect the wave- kinetic transport modulation in the core to be signiﬁ- cantly stronger than in in the edge. GAMs on the other hand are less important in the core due to the shorter parallel connection length ( q∼1). On one hand, the GAM drive eﬃcency depends on the nature of the turbulence, in that it depends on the pres- ence of ﬁnite ρseﬀects. Absence of these, such as in the resistive ballooning regime leads to a strong damp- ing of the GAM due to the anomalous diﬀusion erod- ing the pressure ﬂuctuations connected with the GAM. However, apart from the turbulence the zonal ﬂow am- plitude is inﬂuenced strongly by the linear properties of the GAM itself, such as its frequency or the torque ex- certed on a pressure ﬂuctuation. Manipulation of both of them can lead to a reduction of the transport by up to an order of magnitude. Consequentially, any discussion of the anomalous transport focusing on the inﬂuence of the magnetic geometry on the turbulence drive falls short of an essential factor, if the inﬂuence of the magnetic geom- 3etry on the ﬂows is neglected. Since the GAM properties could be inﬂuenced in a real tokamak by, e.g., shaping the plasma column, the coherent ﬂows should be taken into consideration to further reduce the transport in advanced tokamaks. Moreover, the clear signature of the radially coherent ﬂows in the numerical simulations makes them an interesting target of experimental investigation, e.g. , by means of microwave reﬂectometry. This work has been performed under the auspices of the Center for Interdisciplinary Plasma Science, a joint initiative by the Max-Planck-Institutes for Plasma Physics and for Extraterrestrial Physics. [1] P. H. Diamond, M. N. Rosenbluth et al., 17th IAEA Fu- sion Energy Conference, IAEA-CN-69/TH3/1 (1998) [2] T. S. Hahm et al., Phys. Plasmas 6, 922 (1999) [3] P. W. Terry, Rev. Mod. Phys. 72, 109 (2000) [4] A. Zeiler et al., Phys. Plasmas 5, 2654 (1998) [5] B. N. Rogers et al., Phys. Rev. Lett 81, 4396 (1998) [6] Z. Lin et al., Phys. Plasmas 7, 1857 (2000) [7] K. Hallatschek, Phys. Rev. Lett. 84, 5145 (2000) [8] S. V. Novakovskii et al., Phys. Plasmas 4, 4272 (1997) [9] A. B. Hassam et al., Phys. Plasmas 1, 337 (1994) [10] K. L. Sidikman et al., Phys. Plasmas 1, 1142 (1994) [11] M. N. Rosenbluth et al., Phys. Rev. Lett. 80, 724 (1997) (a) (b) (c) (d) 4FIG. 1. Time evolution of (a) poloidal E×Bﬂow pro- ﬁle, (b) pressure ﬂux proﬁle /angbracketleftvrp/angbracketright, (c) up-down asymmetric pressure ﬂux /angbracketleftvrpsinθ/angbracketrightfor the standard parameters; (d) in- stantaneous proﬁles at t= 80 of the ﬂow (above), pressure ﬂux (solid, below), and the up-down asymmetric pressure ﬂux (dashed). Note the dipole layers of up-down asymmetric pres - sure ﬂux around the ﬂows, e.g., at x=−7. The unipolar component of the ﬂux asymmetries can be observed in ﬁgure (c): depending on their propagation direction the traces lo ok either bright or dark. FIG. 2. Time evolution of heat ﬂux proﬁle demonstrating turbulence movement in responce to a linear shear ﬂow with diﬀerent sign for the lower and upper half of the plots. TABLE I. Mean GAM energy production due to Reynolds stress and Stringer-Winsor eﬀect, ﬂow intensity /angbracketleftv2 E/angbracketrightand anomalous pressure transport /angbracketleftpvr/angbracketrightfor varying parameters. The units are the turbulence units described in reference [5 ]. In these units, the diamagnetic velocity is equal to αd. The parameters are consistent with the physical reference para m- eters given in the text, except for an altered temperature Ti=Te=T. For αd= 0 the ions have been assumed cold. In the line marked with an ∗the Stringer-Winsor drive has been switched oﬀ, eliminating its braking force in the balloonin g regime. αdηiT(eV) /angbracketleftpvr/angbracketright /angbracketleftv2 E/angbracketrightReynolds drive Stringer drive 0− 0 0.27 0.04 1 .4 −1.0 0∗− 0 0.25 0.2 3.3 −26 0.2 1 100 0.27 0.03 2.0 −1.0 0.2 3 100 0.87 0.23 1.5 3.6 0.6 3 300 0.15 0.9 −0.1 10.0 1.1 3 550 0.24 0.9 16.0 33.4 5
arXiv:physics/0009059v1 [physics.chem-ph] 16 Sep 2000Eﬃciency of diﬀerent numerical methods for solving Redﬁeld equations Ivan Kondov, Ulrich Kleinekath¨ ofer, and Michael Schreibe r Institut f¨ ur Physik, Technische Universit¨ at, D-09107 Ch emnitz, Germany Abstract The numerical eﬃciency of diﬀerent schemes for solving the L iouville- von Neumann equation within multilevel Redﬁeld theory has b een studied. Among the tested algorithms are the well-known Runge-Kutta scheme in two diﬀerent implementations as well as methods especially dev eloped for time propagation: the Short Iterative Arnoldi, Chebyshev and Ne wtonian propa- gators. In addition, an implementation of a symplectic inte grator has been studied. For a simple example of a two-center electron trans fer system we dis- cuss some aspects of the eﬃciency of these methods to integra te the equations of motion. Overall for time-independent potentials the New tonian method is recommended. For time-dependent potentials implementati ons of the Runge- Kutta algorithm are very eﬃcient. PACS: 02.60.Cb, 31.70.Hq, 34.70.+e Keywords: Liouville-von Neumann equation, Redﬁeld theory , electron trans- fer, Runge-Kutta, Chebyshev Typeset using REVT EX 1I. INTRODUCTION Besides classical and semi-classical descriptions of diss ipative molecular systems several quantum theories exist which fully account for the quantum e ﬀects in dissipative dynamics. Among the latter are the reduced density matrix (RDM) formal ism [1–5] and the path integral methods [6, 7]. Here we concentrate on the Redﬁeld t heory [8, 9], in which one has to solve a master equation for the RDM. It is obtained by pe rforming a second order perturbation treatment in the system-bath coupling as well as restricting the calculation to the Markovian limit. With this approach the quantum dynamic s of an “open” system, e.g., the exchange of energy and phases with the surroundings mode led as a heat bath, can be described. The unidirectional energy ﬂow into the environm ent is called dissipation. Within this theory it is possible e.g. to simulate the dissipative s hort-time population dynamics usually detected by modern ultrafast spectroscopies. Because the Redﬁeld theory is a Markovian theory the time evo lution of the RDM is governed by equations containing no memory kernel. In the or iginal Redﬁeld theory [8,9] the secular approximation was performed. In this approxima tion it is assumed that every element of the RDM in the energy representation is coupled on ly to those elements that oscillate at the same frequency. In the present study we do no t perform this additional approximation. For larger systems numerical implementations for solving t he Redﬁeld equation are nu- merically very demanding and therefore require that one ﬁnd s the most appropriate way to perform the time evolution of the RDM. Straightforward on e can construct and directly diagonalize the Liouville superoperator. For processes wi th a time-independent Hamilto- nian the rates, i.e. the characteristic inverse times of an e xponential decay of the occupation probability of the excited states, can be obtained in this wa y. In such an approach a huge number of ﬂoating point operations will be involved and the o verall computational eﬀort will scale as N6where Nis the size of the basis of vibronic states. Furthermore the d irect diagonalization can be numerically unstable, but neverthe less has been successfully used (see e.g. [10]). Another strategy suggests solving N2ordinary diﬀerential equations and requires products between the Liouville superoperator and the RDM wh ich scale as N4(see e.g. [11]). This is also numerically demanding for larger systems. Assu ming a bilinear system-bath cou- pling the numerical eﬀort can be reduced considerably by rew riting the Redﬁeld equation in such a form that only matrix-matrix multiplications are n eeded [12] rather than applying a superoperator onto the RDM. Hence, a computational time sc aling of N3and a storage requirement of N2is achieved. In the present paper our numerical studies will be based on this approach. If one wants to reduce the scaling of the numerical eﬀort with increasing number of basis functions even more one has to go to stochastic wave fun ction methods [13–16]. They prescribe certain recipes to unravel the Redﬁeld equation a nd to substitute the RDM by a set of wave functions which evolve partially stochasticall y in time. The method will have the typical scaling of the well developed and optimized wave function propagators, i.e. N2. It has been applied, for example, to electron transfer syste ms [17–19] and shown to give accurate results. Direct solutions and the stochastic wave packet simulations have already been compared numerically [20, 21]. All these ﬁrst studies w ere restricted to dissipation operators with Lindblad form [22]. Breuer et al. [23] showed that the stochastic wave function 2approach can also be applied to Redﬁeld master equations wit hout the secular approximation and for non-Markovian quantum dissipation. In particular f or complex systems with a large number of levels its practical application is very adv antageous. Between the direct and the stochastic methods to solve the Redﬁeld equation the accuracy diﬀers especially because direct RDM integrators are numerically “exact” whi le the stochastic wave function simulation methods have statistical error. The scope of thi s work are small and medium size problems. Therefore we compare the diﬀerent numerical algo rithms for a direct integration of the Redﬁeld equation. But when using stochastic methods f or density matrix propagation one has to solve Schr¨ odinger-type equations with a non-Her mitian Hamiltonian. For that purpose the same algorithms as investigated here can be used . In this sense the present study is also of importance for solving Redﬁeld equations by means of stochastic methods. Here we use diﬀerent numerical schemes to solve the Liouvill e-von Neumann equation. The performance of the well-known Runge-Kutta (RK) scheme i s studied in two diﬀerent implementations: as given in the Numerical Recipes [24] and by the Numerical Algorithms Group [25]. Compared to these general-purpose solvers are more sp ecial algorithms which have been applied previously to the time evolution of wave pa ckets and density matrices. For density matrices these are the Short-Iterative-Arnoldi (S IA) propagator [12,26], the short- time Chebyshev polynomial (CP) propagator [11], and the New tonian polynomial (NP) propagator [27, 28]. The latter propagator is also used as a r eference method because of its high accuracy. In addition, a symplectic integrator (SI ), which was originally developed for solving classical equations of motion and extended to wa ve packet [29,30] and density matrix propagation [31], is tested. Besides the propagation algorithm one has to determine the a ppropriate representation in which to calculate the elements of the RDM and the Liouvill e superoperator. The choice depends strongly on the type of physical problem that one con siders. Coordinate (grid) rep- resentation has an advantage when dealing with complicated potentials, e.g. non-bonding potentials. For the problem of dissociation dynamics in the condensed phase the grid repre- sentation has been applied based on a Lindblad-type master e quation [32]. Other examples using grids are works of Gao [33] and Berman et al. [27]. Conve nient treatment of electron transfer dynamics is done in state representation, because one can model the system using a set of harmonic diabatic potentials [34]. Other authors [1 2,35, 36] choose the adiabatic eigenstates of the whole system as a basis set to treat simila r problems. A comparative anal- ysis of the beneﬁts and drawbacks of the diabatic and adiabat ic representation in Redﬁeld theory will be given elsewhere [37]. From the viewpoint of nu merical eﬃciency we focus on both representations in the present article. The paper is organized as follows. In the next section we will make a short introduction to the model system and a discussion on the versions of the Red ﬁeld equation and the numerical scaling that they exhibit. In Section III the meth ods for propagation used in this work are brieﬂy reviewed. In Section IV we compare the eﬃcien cy of several propagators in solving a simple electron transfer problem with multiple le vels. A summary is given in the last section. 3II. THE REDFIELD EQUATION AND ITS SCALING PROPERTIES In the RDM theory the full system is divided into a relevant sy stem part and a heat bath. Therefore the total Hamiltonian consists of three ter ms – the system part HS, the bath part HBand the system-bath interaction HSB: H=HS+HB+HSB. (2.1) The separation into system and bath allows one to formulate t he system dynamics, described by the Liouville-von-Neumann equation, in terms of the degrees of freedom of the relevant system. In this way one loses the mostly unimpor tant knowledge of the bath dynamics but gains a great reduction in the size of the proble m. Such a reduction together with a second order perturbative treatment of the system-ba th interaction HSBand the Markov approximation leads to the Redﬁeld equation [1,2,8, 9]: ˙ρ=−i ¯h[HS, ρ] +Rρ=Lρ. (2.2) In this equation ρdenotes the reduced density operator and Rthe Redﬁeld tensor. If one assumes bilinear system-bath coupling with system part Kand bath part Φ HSB=KΦ (2.3) one can take advantage of the following decomposition [2,12 ]: ˙ρ=−i ¯h[HS, ρ] +1 ¯h2{[Λρ, K] +/bracketleftBig K, ρΛ†/bracketrightBig }. (2.4) Here the system part Kof the system-bath interaction and Λ together hold equivale nt information as the Redﬁeld tensor R. The Λ operator can be written in the form Λ =K∞/integraldisplay 0dτ/an}bracketle{tΦ(τ)Φ(0)/an}bracketri}hteiωτ=KC(ω). (2.5) The two-time correlation function of the bath operator C(τ) =/an}bracketle{tΦ(τ)Φ(0)/an}bracketri}htand its Fourier-Laplace image C(ω) can be relatively arbitrarily deﬁned and depend on a micro- scopic model of the environment. Diﬀerent classical and qua ntum bath models exist. Here we take a quantum bath [26], i.e. a large collection of harmon ic oscillators in equilibrium, that is characterized by a Bose-Einstein distribution and a spectral density J(ω): C(ω) = 2π/bracketleftbigg 1 +/parenleftBig e¯hω/k BT−1/parenrightBig−1/bracketrightbigg [J(ω)−J(−ω)]. (2.6) Now we look for bases that span the degrees of freedom of the re levant system. Let us consider atomic or molecular centers mat which the electronic states \|m/an}bracketri}htof the system are localized. Their potential energy surfaces (PESs) will be approximated by harmonic oscillator potentials, displaced along the reaction coord inate qof the system. As such a coordinate one can, for example, choose a normal mode of the r elevant system part which is supposed to be strongly coupled to the electronic states. The centers are coupled to each 4other with constant coupling vmn. For example two such coupled centers are sketched in Fig. 1. The coupled surfaces \|1/an}bracketri}htand\|2/an}bracketri}htare assumed to describe excited electronic states. The electron transfer takes place after an excitation of the system from its ground state \|g/an}bracketri}ht. Using this microscopic concept we deﬁne Kas the system’s coordinate operator K=q=/summationdisplay m(2ωmM/¯h)−1/2/parenleftBig a† m+am/parenrightBig \|m/an}bracketri}ht/an}bracketle{tm\| (2.7) where amanda† mare the boson operators for the normal modes at the center m,Mis the reduced mass and ωmare the eigenfrequencies of the oscillators. In the same pic ture the Hamiltonian of the relevant system reads HS=/summationdisplay mn{δmn/bracketleftbigg Um+/parenleftbigg a† mam+1 2/parenrightbigg ¯hωm/bracketrightbigg + (1−δmn)vmn}\|m/an}bracketri}ht/an}bracketle{tn\|. (2.8) From this point on we consider two possible state representa tions in order to calculate the matrix elements of ρand the operators HS,Kand Λ. The diabatic (local) basis is a direct product of the eigenstates \|M/an}bracketri}htof the harmonic oscillators and the relevant electronic sta tes \|m/an}bracketri}ht(Fig. 1, left panel). The intercenter coupling vmngives rise to oﬀ-diagonal elements of the Hamiltonian matrix /an}bracketle{tmM\|HS\|nN/an}bracketri}ht=/bracketleftbigg Um+/parenleftbigg M+1 2/parenrightbigg ¯hωm/bracketrightbigg δmnδMN+ (1−δmn)vmn/an}bracketle{tmM\|nN/an}bracketri}ht, (2.9) where Umare the energies of the minima of the diabatic PESs. Other imp ortant properties of the diabatic representation are the equidistance in the l evel structure and the diagonal form of the system-bath interaction operator HSB. This determines the tridiagonal band form of the operator K. When neglecting the inﬂuence of the intercenter coupling on the dissipation a very eﬃ- cient numerical algorithm can be derived [34,38]. Of course , for strong intercenter couplings the populations /an}bracketle{tmM\|ρ\|mM/an}bracketri}htat long times deviate from their expected equilibrium value s. But even for very small couplings there are cases in which the population does not converge to its equilibrium value [37]. Therefore this neglect of the inﬂuence of the intercenter cou- pling on the dissipation has to be handled with care. On the ot her hand this approximation makes the extension of the present electron transfer model t o many modes conceptually much easier [17,18]. The matrix elements of the operators in the dissipative part of Eq. (2.4) read /an}bracketle{tmM\|K\|nN/an}bracketri}ht= (2ωmM/¯h)−1/2δmn/parenleftBig δM+1,N√ M+ 1 + δM−1,N√ M/parenrightBig (2.10) and /an}bracketle{tmM\|Λ\|nN/an}bracketri}ht=/an}bracketle{tmM\|K\|nN/an}bracketri}htC(ωmMnN ). (2.11) In Eq. (2.11) ωmMnN denote the transition frequencies of the system ¯hωmMnN =/an}bracketle{tmM\|HS\|mM/an}bracketri}ht − /an}bracketle{tnN\|HS\|nN/an}bracketri}ht. (2.12) Since the system can emit or absorb only at the eigenfrequenc ies of the system oscillators ωmthe spectral density of the bath J(ω) is eﬀectively reduced to a few discrete values 5J(ω) =/summationtext mγmδ(ω−ωm). The advantage of this approach lies in the scaling behavio r with the number of basis functions. As shown in Fig. 2 it scales lik eN2.3where Nis the number of basis functions. This is far better than the scaling witho ut neglecting the inﬂuence of the intercenter coupling on the dissipation as described below . If the system Hamiltonian HSis diagonalized it is possible to use its eigenstates as a bas is (Fig. 1, right panel), in which to calculate the elements of t he operators in Eq. (2.4). Of course there will be no longer any convenient structure in Kor Λ, so that the full matrix- matrix multiplications are inevitable. For this reason the computation of Lρ(t) scales as N3, where Nis the number of eigenstates of HS. There appears to be a minimal number N0below which the diagonalization of HSfails or the completeness relation for \|mM/an}bracketri}htis violated. Nevertheless, the beneﬁt of this choice is the exa ct treatment of the system-bath interaction. Denoting the unitary transformation that dia gonalizes HSbyUone has ǫ=U†HSU (2.13) where ǫis a diagonal matrix containing the eigenvalues. In this way it is straightforward to obtain the matrices for ρandK. Equation (2.6) for C(ω) still holds but a new deﬁnition of the spectral density J(ω) is necessary because of the non-equidistant adiabatic eig enstates. The bath absorbs over a large region of frequencies and this i s characterized in the model byJ(ω). One needs the full frequency dependence of J(ω) which we take to be of Ohmic form with exponential cut-oﬀ: J(ω) =ηΘ(ω)ωe−ω/ω c. (2.14) Here Θ denotes the step function and ωcthe cut-oﬀ frequency. In this study all system oscillators have the same frequency ω1(see Table I) and the cut-oﬀ frequency ωcis set to be equal to ω1. The normalization prefactor ηis determined such that ∞/integraldisplay 0dωJ(ω) =γ1. (2.15) Equation (2.14) together with Eq. (2.6) yields the correlat ion function in adiabatic repre- sentation. The introduced representations allow us to consider the num erical eﬀort for a single computation of the right hand side of Eq. (2.4), i.e. of Lρ(t). In the diabatic representation its computation can be approached using two diﬀerent algori thms. It is possible to per- form matrix-matrix multiplication only on those elements o fKand Λ which have nonzero contributions to the elements of Lρ(t) (Fig. 2, solid line). This is advantageous because of the tridiagonal form of Kin diabatic representation and shows the best scaling prope rties, namely N2.3. In the same representation but performing the full matrix- matrix multiplica- tions in Eq. (2.4) (Fig. 2, dashed line) the scaling behavior is slightly worse than the same operation in adiabatic representation (Fig. 2, dotted line ). This is due to the non-diagonal Hamiltonian HSin the former case that makes the computing of the coherent te rm (see Eq. (2.2)) more expensive. Below we will take the full matrix -matrix multiplication to eval- uateLρ(t) in both representations. We do this to concentrate on the va rious propagation schemes, not the unequal representations. Nevertheless th ere are performance changes in the diﬀerent representations because of the disparate basis fu nctions and forms of the operators in these basis functions. 6III. THE DIFFERENT PROPAGATION SCHEMES A. Runge-Kutta method The RK algorithm is a well-known tool for solving ordinary di ﬀerential equations. Thus, this method can be successfully applied to solve a set of ordi nary diﬀerential equations for the matrix elements of Eq. (2.4). It is based on a few terms of t he Taylor series expansion. In the present work we use the FORTRAN77 implementation as gi ven in the Numerical Recipes [24] which is a ﬁfth-order Cash-Karp RK algorithm and will be denoted as RK-NR. As alternative the RK subroutine from the Numerical Algorithms Group [25] which is based on RKSUITE [39] was tested with a 4(5) pair. It will be referre d to as RK-NAG. Both RK- NAG and RK-NR involve terms of ﬁfth order and use a prespeciﬁe d tolerance τas an input parameter for the time step control. The tolerance τand the accuracy of the calculation are not always simply proportional. Usually decreasing τresults in longer CPU times. In our previous work [40] a time step control mechanism diﬀer ent from those used in RK- NAG and RK-NR was tested. Discretizing the time derivative i n Eq. (2.2) and requiring /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ(ti+1)−ρ(ti) ∆t+Lρ(ti)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< τ (3.1) one only has to call the propagation subroutine once and to st ore the previous RDM. In addition one has to calculate the action of the Liouville sup eroperator Lonto the RDM but the numerical eﬀort for this is small compared to a call of the propagation subroutine. It was shown in Ref. [40] that this time step control is the most eﬃci ent for propagation with the coherent terms in Eq. (2.2) only but disadvantageous for pro blems with dissipation. This is the reason why we do not include this algorithm in the present study. B. Short Iterative Arnoldi propagator The SIA propagator [12,26] is a generalized version of the Sh ort Iterative Lanczos propa- gator [41] to non-Hermitian operators. With the Short Itera tive Lanczos algorithm the wave function can be propagated by approximating the time evolut ion operator in Krylov space, which is generated by consecutive multiplications of the Ha miltonian on the wave function. In analogy the Krylov space within the SIA method is construc ted by recursive applications of the Liouville superoperator onto the RDM ρn=Lnρ(t). In this way it is tailored for the RDM at every moment in time. The Liouville superoperator, de noted by lin Krylov space, has Hessenberg form L ≈V lVT, (3.2) where the orthogonal transformation matrix Vis constructed iteratively using the so-called Lanczos procedure [12]. The Krylov representation lcan be easily diagonalized to Lwith the help of a transformation matrix S: eLt≈V SeLtS−1VT. (3.3) Since the diagonalization is performed in the Krylov space t he numerical eﬀort depends on its dimension which can be chosen small in practice. Havin g thus derived a diagonal operator eLtthe calculation of ρ(t) is straightforward. 7C. Symplectic integrator The SIs were originally developed for solving classical equ ations of motion [42]. The time evolution of a classical Hamiltonian system can be viewed as a canonical transformation and SIs are sequences of canonical transformations. Recent ly it was shown that the time evolution of wave packets [29,30] and density matrices [31] can also be performed using SIs. In order to rewrite the Redﬁeld equations in the form of coupl ed canonical variables that are analogous to classical equations of motion one deﬁnes th e functions [31] Q(t) =ρ(t), (3.4) P(t) = ˙ρ(t), (3.5) the operator W=−1 ¯h2L2, (3.6) and the Hamiltonian function G(Q, P) =1 2[PTP+QTWQ]. (3.7) Doing so one obtains equations of motion analogous to the cla ssical ones d dtP(t) =−∂G(Q, P) ∂Q=−WQ(t), (3.8) d dtQ(t) =∂G(Q, P) ∂P=P(t). (3.9) Rewriting this into the SI algorithm of order myields [31] Pi=Pi−1+bi∆t ¯h3L2Qi−1 (3.10) Qi=Qi−1+ai∆t ¯hPi (3.11) fori= 1, . . ., m . Diﬀerent sets of coeﬃcients {ai}and{bi}are given in the literature. Here we choose the McLachlan-Atela fourth-order method [43]. Th e coeﬃcients for this method are listed in Ref. [44]. A comparison of the McLachlan-Atela fourth-order method with the McLachlan-Atela third-order method [43] and Ruth’s third- order method [42] has been given elsewhere [31]. D. Newton polynomial scheme Another way to solve Eq. (2.2) is by a polynomial expansion of the time-evolution opera- tor. Such methods are well established and approved for wave -function propagation [28,41]. Recently the Faber [45] and NP [27] algorithms have been appl ied to propagate density ma- trices and it has been shown that they behave very similarly [ 45]. The main idea of the NP method is the representation of the Liouville superoperato r by a polynomial interpolation 8eLt≈ P Np−1(L)≡Np−1/summationdisplay n=0anρn=Np−1/summationdisplay n=0ann−1/productdisplay j=0(L −λj) (3.12) of order Npwhere the ρnare computed recursively and anare the n-th divided diﬀerences. The interpolation points λjcan be chosen to form a rectangular area in the complex plane (see Fig. 3) which contains all eigenvalues of L. This interpolation scheme is uniform, i.e., the accuracy in energy space is approximately the same in the who le spectral range of L. This is in contrast to schemes such as the SIA propagator which are no nuniform approximations. A consequence of this property is the very high accuracy which can be achieved with uniform propagators. This is why we take a high-order NP expansion as reference solution. Since the quality of the approximation of the time evolution operator is equivalent to a scalar function with the same interpolation points λj, one can, before performing the actual calculation, check the accuracy on a scalar function. For the calculation with the NP propagator we set the truncation limit of the expansion to 10−15, i.e., the sum in Eq. (3.12) is truncated when the residuum fulﬁlls an\|\|ρn\|\|<10−15[28]. E. Chebyshev polynomial scheme As a last contribution to the present study we will examine th e CP propagator. Re- cently it was studied by Guo et al. [11] for density matrices. The Liouville superoperator is approximated by a series of CPs Tk(x). Generally the CPs diverge for non-real arguments. For propagators of the kind e−iHtit has been shown [28] that the CPs may tolerate some imaginary part in the eigenvalues of H. The stability region has the form of an ellipse with a center at the origin and a very small half-axis in imagi nary directions [28]. In con- trast, the eigenvalues of the Liouville superoperator are s pread over the negative real half of the complex plane and symmetrically with respect to the re al axis (see Fig. 3). The real components for the system that we consider are one order of ma gnitude smaller than the imaginary components. This is why we make the expansion alon g the imaginary axis and use an expression similar to that already applied to wave fun ction propagation [41]: eLt≈eL+∆tNp−1/summationdisplay n=0(2−δn0)Jn(L−∆t)Tn(˜L). (3.13) Here the expansion coeﬃcients Jnare the Bessel functions of the ﬁrst kind, and ˜Lis the appropriately scaled Liouville superoperator: ˜L= (L −L+)/L−, where L−andL+are the half span and the middle point of the spectrum of L. Since the spectrum is symmetric with respect to the real axis, L+= 0. The time evolution of ρis given by ρ(t+ ∆t)≈Np−1/summationdisplay n=0(2−δn0)Jn(L−∆t)˜ρn. (3.14) The Chebyshev vectors ˜ ρnare generated by means of a recurrence procedure: ˜ρn= 2˜L˜ρn−1+ ˜ρn−2,˜ρ0=ρ(t) and ˜ ρ1=˜L˜ρ0. (3.15) 9For the CP and NP methods one has to adjust the values of the spe ctral parameters L− andL+. One can obtain some knowledge about the spectrum of Lby an approximate diag- onalization, e.g. by Krylov subspace methods. For instance , Fig. 3 shows an approximate spectrum of Lappropriately scaled so that all eigenvalues lie within the rectangle formed by the Newtonian interpolation points. IV. PERFORMANCE OF PROPAGATION METHODS The aim of this section is to compare the diﬀerent numerical m ethods described above for propagating the RDM in time. The calculations were perfo rmed for both RK methods with diﬀerent tolerance parameters τand for the SI as well as the NP, CP, SIA propagators with diﬀerent timesteps. The number of expansion terms Npin NP and CP propagators is 170 and 64, respectively. The dimension of the Krylov space f or the SIA method was set to 12 because smaller as well as larger values are less eﬃcien t for the example studied here. All computations were made on Pentium III 550 MHz personal co mputers with intensive use of BLAS and LAPACK libraries. The code was compiled using the PGF90 Fortran compiler [46]. For estimation of the computational error of all methods the NP algorithm with 210 terms was chosen as a benchmark. In this work we consider only two centers m= 1,2 which is the minimal model to describe the main physics of an electron transfer reaction. A basis si ze of 16 levels per center satisﬁes the completeness relation and presents no diﬃculties durin g the diagonalization of HS. The electronic coupling was v12= 0.1 eV. We choose γ1=γ2= 1.57863 ×10−2eV˚A−2and M= 20mpwhere mpis the proton mass. The temperature T= 298 K is used. In Table I the parameters for the system oscillators are given. The process that is simulated involves the following scenar io. A Gaussian wave packet is prepared as initial state by a vertical transition from th e lowest vibrational level of the ground electronic state \|g/an}bracketri}htto the ﬁrst (upper) center \|1/an}bracketri}ht: ρ1M1N(t= 0) = /an}bracketle{t1M\|g0/an}bracketri}ht/an}bracketle{tg0\|1N/an}bracketri}ht. (4.1) The energy distribution of the occupied eigenstates by the w ave packet depends on the displacement q0 g−q0 1between the PESs of \|g/an}bracketri}htand\|1/an}bracketri}ht. During the pulse the two excited electronic states \|1/an}bracketri}htand\|2/an}bracketri}htare assumed to be decoupled. In this way one can simulate the absorption of electromagnetic radiation from a pulse wi th vanishing width. Right after the pulse is over, the wave packet starts moving on the excite d PESs and spreading. The relevant system part begins losing energy to the bath and dep hasing. The population on the upper center starts decaying. When the damping is not too strong, as for the model parameter studied here, a damped oscillation of the populat ion between the two excited PESs can be seen. We assume no coupling to the ground state aft er the pulse. After a certain time the system reaches its equilibrium state. In all cases the RDM was propagated for a total time period of 3 ×105a.u. which is suﬃcient for complete relaxation to equilibrium. It was com pared to the RDM ρrefevaluated by the NP algorithm at the same points in time. The relative er rorε(t) of each method at a certain moment in time thas been estimated using a formula similar to that proposed f or wave functions by Leforestier et al. [41]: 10ε(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−Tr (ρ(t)ρref(t)) Tr(ρ2 ref(t))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (4.2) As the error εwe deﬁne the maximum value of ε(t) over the total propagation time. For more details we refer to our previous paper [40]. Other error meas ures (see for example [11,47]) can be used as well but they will have the same qualitative beh avior. As an index for the numerical eﬀort two possibilities were ex plored. The ﬁrst one is a direct measurement of the CPU time of the total propagation ( Fig. 4). It may look quite diﬀerent on other computer architectures or even on the same architecture but under changed operation conditions. An evidence of the performance (Fig. 4) will be expressed by means of CPU time versus the error ε. Another approach to describe the numerical eﬀort has been pr oposed [11] and is called eﬃciency factor. It is deﬁned as the ratio between the timest ep ∆tand the number of operations Lρ(t) within this timestep. Because of the deﬁnition it is a machi ne independent quantity. The larger the eﬃciency factor, the better the per formance of the algorithm. Because the RK algorithms propagate with variable timestep we cannot directly use the deﬁnition of the eﬃciency factor. Instead we deﬁne a quantit yαas the total number Ncof Lρ(t)-evaluations divided by the total time for the propagation : α=Nc/(Ns∆t). (4.3) HereNsdenotes the total number of timesteps. The inverse of αwill have the meaning of an eﬃciency factor for an averaged timestep ∆t. We should point out that Ncdoes not take into account the eﬀort for summation of the diﬀerent contrib utions. In particular in the case of the NP method the summation of the diﬀerent terms in th e polynomial expansion Eq. (3.12) can be non-negligible. This can be seen in the diﬀe rent relative performance of the propagators shown in Figs. 4 and 5. We consider both the CP U time and the quantity αas measures of the numerical eﬀort. Contributions from the algorithm to calculate Lρ(t) also inﬂuence the CPU time. As discussed above, in all computations represented in Figs. 4 and 5 the full matrix-matrix multiplications in Eq. (2.4) were performed. The performan ce of the CP, NP, SI and SIA methods is only inﬂuenced very little by the choice between d iabatic and adiabatic repre- sentation. Both RK implementations are less eﬃcient in the a diabatic than in the diabatic representation, though the RK-NAG scheme has still the best performance besides the NP algorithm. The RK-NR scheme has an advantage for computatio n in diabatic rather than in adiabatic representation especially for medium precisi on requirements. In that range the performance curves of the RK methods exhibit a shoulder for t he adiabatic case which seems to result from a numerical artifact. Because the error of the SIA algorithm is not uniformly distr ibuted in energy space [48] we could expect some diﬀerence in its performance in diabati c and adiabatic representation. But because the coupling v12chosen here is not very large, the eigenstates of the coupled system are just slightly disordered (see Fig. 1, right plot) and hence the performance of the SIA algorithm is almost not changed. The uniformity, stability and high accuracy of the CP propag ator for wave functions is well known [41, 47, 48]. The CP approach to density matrix pro pagation was introduced by Guo and Chen [11]. Using a damped harmonic oscillator as mo del system and starting 11from a pure state they established that the relative error ca n reach the machine precision limits (10−15) for suﬃciently small stepsize. However, for the system of c oupled harmonic potentials studied here and using an initial RDM with non-ze ro oﬀ-diagonal elements the error saturates at ε≈10−8(see Fig. 4). It was not possible to decrease this saturation limit of εneither by increasing the order of the CP nor by decreasing th e timestep. This saturation limit seems to depend strongly on the imaginary p art of the eigenvalues of L. For large timesteps the CP method loses its stability and one nee ds to estimate the eﬃciency range of Np, ∆tandL−. Turning oﬀ the dissipation we could reach much higher accur acy with the CP propagator as expected. The SI is easy to implement. The expansion coeﬃcients are ﬁxe d and can be taken from literature. At the same time the ﬁxed coeﬃcients seem to limit the accuracy. For not too high accuracy the performance of the SI is as good as that o f the other propagators in adiabatic representation. In diabatic representation its performance is a little worse. But we were not able to achieve very high accuracy with this metho d. This might be due to the special version, the fourth order McLachlan-Atela method, which we chose. As already highlighted [45] the NP scheme is very stable for a rbitrary spectral properties ofL. The only restriction is that the spectrum must be conﬁned wi thin the area formed by the interpolation points. In our investigation the NP pro pagator performs with a good accuracy for timesteps of 1500 a.u. ( Np= 170) which is 10 times larger than the step size of the CP scheme. Higher order expansions might be even more e ﬃcient but the numerical implementation gets tricky and easily unstable. For timest eps of 100 a.u. and Np= 50 the NP algorithm is already numerically exact but computati onal very expensive (see the arrows in Fig. 4 and Fig. 5). For problems with time-dependen t Hamiltonians (e.g. non- stationary external ﬁelds with relatively small amplitude ) the RK and SIA methods will be more eﬃcient with small timestep. At the end we should point out that there exists no ultimate me thod to determine the performance of a certain numerical approach which could be v alid for diﬀerent platforms. Tuning and optimization features are generally not portabl e and this may cause even diﬀerent scaling behavior and hence a diﬀerent method of preference. That is the reason why the generality of the results is limited to similar computation platforms and even to systems with similar properties of the corresponding Liouville sup eroperator. But on the other hand this study can give hints on the performance of the diﬀerent a lgorithms in general. V. SUMMARY In the present work an estimation of the numerical eﬃciency o f several methods for density matrix propagation has been given. The example of el ectron transfer in a two-center system has been used for this purpose. A speciﬁc measure of th e numerical eﬀort has been introduced in order to compare methods with ﬁxed timestep an d such ones with timestep control (RK). Besides the method of reference (NP) the RK-NA G approach shows best performance for both cases of adiabatic and diabatic repres entation. The advantage of the SIA propagator is that the accuracy improves with decreasin g the timestep in all cases we investigated. That is not the case with the CP propagator whi ch exhibits a saturation of accuracy and is therefore not convenient for very small time steps. The easy-to-implement SI 12gives reasonable performance for not too high accuracy. The present SI seems to be limited in accuracy due to the ﬁxed coeﬃcients. The present studies were restricted to state bases. Of cours e similar calculations can be done on a grid which is especially useful for complicated or u nbound potentials. In these cases another propagator, the split operator [49], should b e taken into account. This operator has the advantage that its performance does not (directly) d epend on the spectral range of the Hamiltonian or Liouville operator. So it may perform ver y well for problems with a large spectral range although it cannot be applied to operat ors which have mixed terms in coordinate and momentum operators. The use of the mapped F ourier method [50] may reduce the number of grid points signiﬁcantly and ﬁrst wave p acket propagations with this method have been done [51,52]. Recently the multi-conﬁgura tion time-dependent Hartree method has been established to treat density matrix operato rs [53]. This method might be favorable for multi-dimensional systems. The presented methods can be used for various applications i n the ﬁeld of dissipative molecular dynamics in condensed phases, where the RDM appro ach provides a good way of describing processes in systems with one or more degrees o f freedom. This includes the electron transfer processes mentioned in the introduction as well as exciton transfer processes [54]. 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Phys. 112, 10718 (2000). [54] T. Renger and V. May, Phys. Rev. Lett. 78, 3406 (1997). [55] C. J. Bardeen, J. Che, K. R. Wilson, V. V. Yakovlev, V. A. A pkarian, C. C. Martens, R. Zado, B. Kohler, and M. Messina, J. Chem. Phys. 106, 8486 (1997). 15TABLES TABLE I. Parameters of the system oscillators used for the co mputations. Center \|m/an}bracketri}ht Um, eV q0 m,˚A ωm, eV \|0/an}bracketri}ht ≡ \|g/an}bracketri}ht − 0.60 0.000 0.1 \|1/an}bracketri}ht 0.25 0.125 0.1 \|2/an}bracketri}ht 0.05 0.363 0.1 16FIGURES \|g〉 q\|1〉 \|2〉 \|g〉\|e〉 q FIG. 1. Potential energy surfaces (PESs) for a model electro n transfer system. Diabatic PESs are plotted on the left side, and the PES of the adiabatic exci ted state \|e/an}bracketri}hton the right side. 20 30 40 50 Number of basis states0246CPU time FIG. 2. Scaling behavior of the product Lρ(t). Solid line – tridiagonal form of Kin diabatic representation, dotted line – adiabatic representation, d ashed line – diabatic representation with full matrix-matrix multiplications. The CPU time is scaled so that it is equal to 1 for N= 50 in diabatic representation. 17−0.25 −0.15 −0.05 0.05 Real−2−1012Imaginary FIG. 3. Scaled spectrum L/L−of the Liouville superoperator for the model of electron tra nsfer. Approximate eigenvalues obtained in Krylov subspace are pl otted as dots. Open squares denote the interpolation points λjfor the NP scheme. 10−1310−910−510−1 ε0250050007500CPU, [s]RK−NR RK−NAG SIA CP NP SI 10−1410−1010−610−2 ε FIG. 4. Numerical performance of diﬀerent numerical propag ators. Results obtained in the diabatic (adiabatic) representation are shown on the left ( right) plot. The arrows represent the numerical performance for the NP propagator with 50 terms an d timestep 100 a.u. 1810−1310−910−510−1 ε00.511.5αRK−NR RK−NAG SIA CP NP SI 10−1410−1010−610−2 ε FIG. 5. Numerical performance of diﬀerent numerical propag ators. The numerical eﬀort αis deﬁned in Section IV. Results obtained in the diabatic (adia batic) representation are shown on the left (right) panel. The arrows represent the numerical perf ormance for the NP propagator with 50 terms and timestep 100 a.u. 19
arXiv:physics/0009060v1 [physics.chem-ph] 17 Sep 2000LETTER TO THE EDITOR Convergence improvement for coupled cluster calculations N S Mosyagin †§, E Eliav ‡and U Kaldor ‡ †Petersburg Nuclear Physics Institute, Gatchina, St.-Petersburg district 188350, Russia ‡School of Chemistry, Tel Aviv University, Tel Aviv 69978, Is rael Abstract. Convergence problems in coupled-cluster iterations are di scussed, and a new iteration scheme is proposed. Whereas the Jacobi metho d inverts only the diagonal part of the large matrix of equation coeﬃcients, we invert a matrix which also includes a relatively small number of oﬀ-diagonal coeﬃcien ts, selected according to the excitation amplitudes undergoing the largest change in the coupled cluster iteration. A test case shows that the new IPM (inversion of partial matri x) method gives much better convergence than the straightforward Jacobi-type s cheme or such well-known convergence aids as the reduced linear equations or direct i nversion in iterative subspace methods. PACS numbers: 31.15.Dv, 31.15.-p, 31.25.-v, 31.15.Ar Submitted to: J. Phys. B: At. Mol. Opt. Phys. §E-mail for correspondence: Mosyagin@lnpi.spb.su; http:/ /www.qchem.pnpi.spb.ruLetter to the Editor 2 The coupled cluster (CC) method is widely used in electronic structure calculations. The CC theory has been described in many reviews (see, e.g., [ 1, 2, 3, 4, 5]), and will not be presented here. The basic equation for the CC method is the Bloch equation ΩHΩ =HΩ, (1) where His the Hamiltonian and Ωis the wave operator. The resulting equations have the general algebraic form Ai+N/summationdisplay j=1B(t)ijtj= 0, i = 1,2, . . ., N , (2) where tjare the cluster or excitation amplitudes to be determined, Nis the number of the unknown amplitudes, Ais a vector and B(t) is a square matrix which in general depends upon t. For simplicity, we consider the case when Bdoes not depend upon t, Ai+N/summationdisplay j=1Bijtj= 0, i = 1,2, . . ., N. (3) The generalization for the case of B(t) is straightforward (and implemented in the relativistic CC code employed for the test examples below). The direct solution of equations (3) using the Gauss elimination method is feasibl e only for systems with a few thousand cluster amplitudes at most, whereas problems enco untered in our relativistic CC may involve millions of such amplitudes. A Jacobi-type it erative method is usually applied to solve these equations. Using the fact that Bis normally a diagonally dominant matrix, the method involves direct inversion of the diagona l part DofB. The system (3) is rewritten in the form ti=−(D−1)ii[Ai+N/summationdisplay j=1(B−D)ijtj], i = 1,2, . . ., N (4) and is solved iteratively. The coupled cluster calculations are often beset by converg ence diﬃculties. This is particularly true for multireference CC methods, such as th e Fock-space approach [2, 3]. Several methods for improving convergence have been propos ed; the most commonly used are the reduced linear equations (RLE) [6] and direct in version in the iterative subspace (DIIS) [7, 8] approaches. These help in some, but no t all, cases. Most severe convergence problems may be traced to the existence of intru der states. While increasing the model (or P) space improves the quality of the calculation by including a larger part of the correlation, it also increases the probability of enc ountering intruder states and getting no valid results at all. New methods for improving co nvergence are therefore highly desirable. One such method is presented in this Lette r. The problem may be illustrated by an example taken from recen t work [9], where ground and excited state energies of Hg and its ions were calc ulated by the relativistic coupled cluster method. The 5 d10ground state of the Hg2+ion served as the reference state, and the Fock-space CC scheme was Hg2+[(0)sector] →Hg+[(1)sector] →Hg[(2)sector] , (5)Letter to the Editor 3 with electrons added in the 6 sand 6porbitals, designated as valence particles. While the calculations in [9] were relativistic, the nonrelativi stic notation will be employed for brevity. The model space in the (1) sector, with one valen ce particle, consisted of determinants with 5 d106s1and 5d106p1conﬁgurations. Adding the 7 s, 7p, and 6 d orbitals to the list of valence particles would yield more st ate energies in the (1) sector, as well as better description of the 6 s16p1states[9] in the (2) sector, corresponding to neutral Hg. Unfortunately, adding the 5 d107s1, 5d107p1, and 5 d106d1conﬁgurations to the model space leads to divergence of the CC iterations (4 ) in the (1) sector. Analysis shows that the divergence is caused by the 5 d96s16p1, 5d96s2and other intruder states from the complementary Qspace, which are close in energy to certain Pstates (5d107p1, 5d106d1, and others). The diagonal elements Biiof the matrix Bcorrespond to diﬀerences between the total energies of the PandQdeterminants connected by the tiexcitations. Some of these elements will be very small in thi s case, leading to large elements in D−1. Small changes in tamplitudes on the right hand side of equations (4) will therefore cause large changes in the amplitudes on the l eft hand side, leading to divergence. We propose to overcome this problem by replacing the Dmatrix by D′which includes, in addition to the diagonal elements of B, those nondiagonal Belements which are large in comparison with corresponding diagonal elemen ts. The calculation of all B matrix elements is impractical, and a selection procedure f or nondiagonal elements to be included in D′is described below. This new matrix is constructed so that it s matrix elements, D′ ij, are equal to or approximate the Bijmatrix elements both for i=jand fori, j∈I, where Iis some small subset of the amplitudes. The other nondiagona lD′ ij matrix elements ( i/negationslash∈Iorj/negationslash∈I) are set to zero. The method involves the inversion of the partial matrix (IPM) D′. A modiﬁed form of the system of equations (4), ti=−N/summationdisplay k=1(D′−1)ik[Ak+N/summationdisplay j=1(B−D′)kjtj], i = 1,2, . . ., N (6) is obtained and solved iteratively. Equations (6) can be div ided into two sets, ti=−/summationdisplay k∈I(D′−1)ik[Ak+N/summationdisplay j=1(B−D)kjtj−/summationdisplay j∈I(D′−D)kjtj], i ∈I,(7) ti=−(D′−1)ii[Ai+N/summationdisplay j=1(B−D)ijtj−(D′−D)iiti], i /negationslash∈I,(8) where the second part is similar to equations (4). The size of the subset Imust be kept small, so that the calculation (the most time- consuming step), storage and manipulation of the non-zero o ﬀ-diagonal D′elements remains feasible. Careful selection of the amplitudes to be included in Iis therefore of paramount importance. The algorithm followed here start s with calculating the ti amplitudes by the standard iteration scheme (4). The amplit udes which have undergone the largest changes are included in I, the corresponding D′ ijmatrix elements are evaluated, and the tiamplitudes in Iare recalculated by equations (7). The dimensionLetter to the Editor 4 Mof the Isubset was kept at 1000, which makes the calculation and mani pulation of D′feasible. Optimal algorithms for determining M, selecting excitations to be included inI, and calculating the D′matrix will be studied in the future. It should be noted that the system (6) is equivalent to the sta ndard equations (4) in the limit M= 0; in the limit M=N, scheme (6) converges in one iteration, if one takes D′ ij=Bij, ti=−N/summationdisplay k=1(B−1)ikAk, i = 1,2, . . ., N. (9) Formally, one can always achieve convergence of the iterati ons (6) by increasing M. The IPM method proposed here may be combined with other proce dures for accelerating convergence, such as the reduced linear equations [6] and di rect inversion in the iterative subspace [7, 8] methods. This has not been done in the present application, and will be tried in the future. It should be mentioned that the identi ﬁcation of the resulting high-lying levels may require careful analysis of the tamplitudes, particularly if some of the latter are large, indicating large contributions of Qconﬁgurations. Finally, it should be noted that the IPM scheme described above may be reg arded as adopting the Gershgorn-Shavitt Akperturbation theory approach rather than that of A0[10]. The diﬀerent iteration schemes were tested for the 33-elect ron relativistic Fock- space CC calculation with single and double cluster amplitu des of Hg+levels in the (spdfg) basis from [9] in the framework of the Dirac-Coulomb Hamilt onian. Two model spaces were used, one consisting of determinants with 5d106s1and 5 d106p1 conﬁgurations, the other including in addition the 5 d107s1, 5d107p1, and 5 d106d1 conﬁgurations. All iterations involved 1:1 damping (the in put amplitudes for iteration n+ 1 were taken as the average of input and output amplitudes of iteration n). The IPM scheme is compared with the standard scheme (4) and with t he RLE [6] and DIIS [7, 8] methods in tables 1 and 2. The RLE and DIIS methods used t he output of the last ﬁve iterations to form the new input vector. All meth ods led to convergence for the small model space (table 1). The RLE, DIIS, and IPM sch emes were about equally eﬀective in reducing the number of iterations requi red. The large model space (table 2) shows markedly diﬀerent behavior for the diﬀerent methods. Straightforward iteration by the Jacobi-type method blows up almost immedia tely; the large excitation amplitudes may be traced to the intruder states mentioned ab ove. The RLE and DIIS schemes exhibit better behavior, but could not achieve conv ergence even after several hundred iterations. Only the IPM approach proposed in this L etter led to convergence (in the 29th iteration), showing the potential of the method . Acknowledgments This work was supported by INTAS grant No 96–1266. N M thanks t he Russian Foundation for Basic Research (grant No 99–03–33249). Work at TAU was supported by the Israel Science Foundation. The authors are grateful f or valuable discussions with A.V. Titov.Letter to the Editor 5 References [1] Bartlett R J 1989 J. Phys. Chem. 931697 [2] Mukherjee D and Pal S 1989 Advances in Quantum Chemistry vol 20 (Academic Press) 291 [3] Kaldor U 1991 Theor. Chim. Acta 80427 [4] Paldus J 1992 Methods in Computational Molecular Physics ed S Wilson and G H F Diercksen (New York: Plenum Press) p 99 [5] Bartlett R J 1995 Modern Electronic Structure Theory ed D R Yarkony vol 2 (Singapore: World Scientiﬁc) p 1047 [6] Purvis III G D and Bartlett R J 1981 J. Chem. Phys. 751284 [7] Pulay P 1980 Chem. Phys. Lett. 73393 [8] Pulay P 1982 J. Comp. Chem. 3556 [9] Mosyagin N S, Eliav E, Titov A V and Kaldor U 2000 J. Phys. B 33667 [10] Gershgorn Z and Shavitt I 1968 Int. J. Quantum Chem. 2751Letter to the Editor 6 Tables and table captions Table 1. The largest change in the single and double cluster amplitud es (Nmax i=1\|t(n+1) i −t(n) i\|) at iteration n. The changes are obtained by equations (4) in the RCC calculations with the Jacobi-type, RLE, DIIS and IPM iteration schemes. The model space consists of determinants with 5 d106s1and 5 d106p1conﬁgurations. The convergence threshold is 10−6. Iteration Jacobi RLE DIIS IPM 0 1 .62·10−11.62·10−11.62·10−11.62·10−1 3 2 .99·10−22.99·10−22.59·10−21.30·10−2 6 1 .39·10−21.41·10−31.10·10−31.38·10−3 9 6 .65·10−36.81·10−41.21·10−42.01·10−4 12 3 .19·10−31.69·10−51.12·10−53.60·10−5 15 1 .54·10−38.42·10−65.15·10−67.01·10−6 18 7 .53·10−49.47·10−72.43·10−61.43·10−6 21 3 .72·10−4convergence 1 .15·10−6convergence 24 1 .86·10−4convergence 27 9 .46·10−5 30 4 .87·10−5 33 2 .53·10−5 36 1 .34·10−5 39 7 .11·10−6 42 3 .82·10−6 45 2 .07·10−6 48 1 .13·10−6 convergenceLetter to the Editor 7 Table 2. Same as Table 1, except that the model space is larger, consis ting of determinants with 5 d106s1, 5d106p1, 5d107s1, 5d107p1, and 5 d106d1conﬁgurations. Iteration Jacobi RLE DIIS IPM 0 2 .14·10−12.14·10−12.14·10−12.14·10−1 3 7 .54·10−17.54·10−16.93·10−19.38·10−2 6 2 .61 5 .18·10−11.22·10−11.21·10−2 9 9 .31 1 .82 5 .75·10−21.25·10−3 12 3 .42·1012.73·10−13.20·10−23.19·10−4 15 1 .24·1025.65·10−12.90·10−27.39·10−5 18 4 .45·1021.80·1012.87·10−21.61·10−5 21 1 .54·1032.77·10−13.06·10−21.04·10−5 24 5 .00·1031.01 2 .55·10−29.55·10−6 27 1 .49·1041.46·10−12.19·10−23.64·10−6 30 3 .93·1045.51·10−12.23·10−2convergence 33 9 .08·1042.40·10−11.93·10−2 36 1 .88·1051.73·10−11.99·10−2 39 3 .61·1054.91·10−11.36·10−2 42 6 .70·1052.95·10−11.62·10−2 45 1 .22·1066.04·10−11.44·10−2 48 2 .22·1065.00·10−11.45·10−2 divergence no convergence no convergence
arXiv:physics/0009061v1 [physics.plasm-ph] 17 Sep 2000SHAFRANOV’S VIRIAL THEOREM AND MAGNETIC PLASMA CONFINEMENT Ludvig Faddeev♯‡, Lisa Freyhult⋆, Antti J. Niemi⋆and Peter Rajan⋆ ♯St.Petersburg Branch of Steklov Mathematical Institute Russian Academy of Sciences, Fontanka 27 , St.Petersburg, R ussia§ ⋆Department of Theoretical Physics, Uppsala University P.O. Box 803, S-75108, Uppsala, Sweden ‡Helsinki Institute of Physics P.O. Box 9, FIN-00014 University of Helsinki, Finland Shafranov’s virial theorem implies that nontrivial magnet ohydrodynamical equilib- rium conﬁgurations must be supported by externally supplie d currents. Here we extend the virial theorem to ﬁeld theory, where it relates to Derric k’s scaling argument on soliton stability. We then employ virial arguments to investigate a realistic ﬁeld theory model of a two-component plasma, and conclude that stable localiz ed solitons can exist in the bulk of a ﬁnite density plasma. These solitons entail a nontr ivial electric ﬁeld which implies that purely magnetohydrodynamical arguments are i nsuﬃcient for describing stable, nontrivial structures within the bulk of a plasma. ♯Supported by grants RFFR 99-01-00101 and INTAS 9606 ⋆Supported by NFR Grant F-AA/FU 06821-308 §Permanent address E-mail: FADDEEV@PDMI.RAS.RU, FREYHULT@TEORFYS.UU.SE, NIEMI@TE ORFYS.UU.SE, RAJAN@TEORFYS.UU.SEIdeal single ﬂuid magnetohydrodynamics obeys an integral r elation which is known as Shafranov’s virial theorem [1]. It implies that an ideal m agnetohydrodynamical sys- tem can not support nontrivial localized structures. Inste ad any nontrivial equilibrium conﬁguration must be maintained by externally supplied cur rents, a guiding principle in the design of contemporary magnetic fusion devices. Ideal magnetohydrodynamics is supposedly adequate for des cribing the ground state equilibrium geometry of a plasma. It also provides a startin g point for a weak coupling Bolzmannian transport theory [1]. But as an eﬀective mean ﬁe ld theory it lacks the kind of detailed microscopic information which is needed to properly account for the electromagnetic interactions between the charged particl es within the plasma. For this, ideal magnetohydrodynamics should be replaced by an approp riate classical ﬁeld theory of charged particles. With a ﬁrm microscopic basis and estab lished set of rules for systematic computations, a ﬁeld theory model can provide a r igorous basis for describing thermal ﬂuctuations and dynamical eﬀects, including trans port phenomena and issues related to plasma stability and conﬁnement. Here we extend Shafranov’s virial theorem to classical ﬁeld theory where it yields a variant of Derrick’s scaling argument, widely employed to i nspect soliton stability. We then apply virial arguments to a realistic ﬁeld theory model of plasma. In accordance with ideal magnetohydrodynamics, we conclude that the ﬁeld theory does not support localized self-conﬁned plasma conﬁgurations in isolation , in an otherwise empty space. But when we inspect the ﬁnite density bulk properties of the ﬁ eld theoretical plasma, we ﬁnd that the virial theorem does allow for the existence of st able solitons. These solitons describe extended collective excitations of charged parti cles in the otherwise uniform ﬁnite density environment. Our results are consistent with a recent proposal [2], that a ﬁnite density ﬁeld theoretical plasma supports stable kno tted solitons [3]. Indeed, we expect that these solitons can be employed to describe a vari ety of observed phenomena. For example coronal loops that are present in solar photosph ere are natural candidates. The properties of these solitons may also become attractive in fusion experiments, where their stability might help in the design of particularly sta ble magnetic geometries [1]. Shafranov’s virial theorem follows from the properties of t he magnetohydrodynam- ical energy-momentum tensor Tµνin the ideal single ﬂuid approximation. Its spatial components are [1] Tij=ρvivj+/parenleftbigg p+1 2B2/parenrightbigg δij−BiBj(1) while the purely temporal component coincides with the inte rnal energy density, T00=1 2ρv2+1 2B2+p γ−1(2) Hereγis the ratio of speciﬁc heats. The ﬂuid variables are the mass densityρ, the (bulk) ﬂuid velocity viand the pressure p, andBiis the magnetic ﬁeld in natural 1units withµ0= 1. The plasma evolves according to the Navier-Stokes equat ion which follows when we equate the divergence of the energy-momentu m tensor with external dissipative forces. These dissipative forces are present w henever the plasma is in motion, but cease when the plasma reaches a stable magnetostatic equ ilibrium conﬁguration that minimizes the internal energy E=/integraldisplay d3x T00(3) Shafranov’s virial theorem follows when we subject (3) to a s cale transformation of the spatial coordinates xi→λxiwithλa constant. For the magnetic ﬁeld we select Bi(x)→λ2Bi(λx), as customary in Maxwell’s theory. But for the pressure pideal magnetohydrodynamics does not supply enough information t o determine its behaviour under a scale transformation. For this we assume that the pre ssure is subject to the standard thermodynamic scaling relation of a thermally iso lated gas, pVγ=constant (4) This implies that under a scaling p(x)→λ3γp(λx). If we assume that the value λ= 1 corresponds to an actual minimum energy conﬁguration of the energy (3), when viewed as a function of λthe energy (3) then has an extremum at λ= 1. Consequently 0 =δE(λ) δλ \|λ=1=−/integraldisplay d3x/parenleftbigg 3p+1 2B2/parenrightbigg ≡ −/integraldisplay d3x Ti i (5) The magnetic contribution to the pressure is manifestly pos itive deﬁnite. Furthermore, (collisionless) kinetic theory relates the pressure pto the kinetic energy of the individual particles, which is similarly a positive deﬁnite quantity. The integrand in (5) is then pos- itive deﬁnite, and we conclude that under the present assump tions non-trivial localized equilibrium conﬁgurations do not exist in ideal magnetohyd rodynamics [1]. We have formulated our derivation of Shafranov’s virial the orem so that it relates to Derrick’s scaling argument in classical ﬁeld theory [4]. Fo r this, we consider a generic three-dimensional Hamiltonian ﬁeld theory model with clas sical action S=/integraldisplay dtd3xL(ψ) =/integraldisplay dtd3x{πα˙ϕα−H[π,ϕ]} (6) The ﬁeldsψα∼(πα,ϕα) are canonical conjugates with Poisson bracket {πα(x),ϕβ(y)}=δαβ(x−y) (7) Notice that the time derivative in (6) acts asymmetrically. But in the sequel it will be useful to consider symmetrized quantities, and for this we g eneralize the time derivative term by a canonical transformation into /integraldisplay dtd3x πα˙ϕα→/integraldisplay dtd3x[a˙παϕα+ (1−a)πα˙ϕα] (8) 2whereaparametrizes the canonical transformation. We assume that the Hamiltonian His a functional of the ﬁelds ψ= (π,ϕ) and their ﬁrst derivatives only, with no explicit dependence on the sp ace coordinates xiand time t≡x0. We then obtain the energy-momentum tensor directly from No ether’s theorem: Since there is no explicit dependence on xµ ∂L ∂xµ=δL δψα∂µψα+δL δ∂νψα∂µ∂νψα (9) and by employing the equations of motion we identify the comp onents of the energy- momentum tensor with the ensuing four conserved currents Tµ ν=δL δ∂µψα∂νψα−δµ νL (10) In general (10) fails to be symmetric. But in the following we ﬁnd it useful to consider symmetrized quantities, and for this we can re-deﬁne Tµ ν→Tµ ν+∂ρXρµ ν (11) whereXρµν=−Xµρνhas no eﬀect on the dynamics. (Note that if the theory fails to be Lorentz invariant, in general there will be no symmetry be tween the momentum ﬂux T0iand the energy ﬂux Ti0.) We are interested in a scale transformation of the spatial co ordinatesxi→λxi, which sends ψα(x)→λDαψα(λx). HereDαis the scale dimension of the ﬁeld ψα. By considering an inﬁnitesimal transformation with λ= 1 +ǫwe ﬁnd for the generator δS of the scale transformation δSπα=xi∂iπα+Dα ππα (12) δSϕα=xi∂iϕα+Dϕ αϕα(13) For the energy density this yields δST0 0=/braceleftBigg −Ti i+∂i(xiT0 0) +/summationdisplay αDα/bracketleftBiggδT00 δψαψα+δT00 δ∂kψα∂kψα/bracketrightBigg/bracerightBigg (14) In general the scale dimensions can be arbitrary, and there i s noa priori relation between the diﬀerent Dα. But if the scale transformation is a canonical transformat ion it must preserve Poisson brackets, which implies that the scale dim ensions of a canonical pair are subject to Dα π+Dϕ α= 3 (15) The generator δC Sof such a canonical scale transformation can be computed fro m Noether’s theorem, and by properly selecting the value of ain (8) we arrive at the symmetrized form δC S=/integraldisplay d3x xiT0 i (16) 3We are interested in a reﬁnement of ideal magnetohydrodynam ics, a microscopic ﬁeld theory model of a two-component plasma with negatively charged electrons ( e) and positively charged ions ( i) and classical (ﬁrst-order) Lagrangian [2], L=Ek∂tAk+i 2(ψ∗ e∂tψe−∂tψ∗ eψe+ψ∗ i∂tψi−∂tψ∗ iψi)−1 2E2 k−1 2B2 k −1 2m\|(∂k+ieAk)ψe\|2−1 2M\|(∂k−ieAk)ψi\|2+A0(∂kEk−eψe∗ψe+eψi∗ψi) (17) Hereψeandψiare (complex) non-relativistic Hartree-type ﬁelds that de scribe electrons and ions with masses mandMand electric charges ±erespectively, together with their electromagnetic interactions. Note that we have realized M axwell’s theory canonically so that the electric ﬁeld Eiand spatial gauge ﬁeld Aiform a canonical pair, with the temporalA0a Lagrange multiplier that enforces Gauss’ law. Since the ti me derivative appears linearly in the charged ﬁelds, the action (17) admit s a proper Hamiltonian interpretation with ψ∗ e,ithe canonical conjugates of ψe,i. Notice that for deﬁniteness we have chosen both charged ﬁelds to be commuting. This shoul d be adequate in the Bolzmannian limit, relevant in conventional plasma scenar ios where the temperature is suﬃciently high so that bound states (hydrogen atom) are pre vented but not high enough for relativistic corrections to become important. Notice t hat we have also introduced an appropriate symmetrization of the form (8) in the time deriv ative terms of the charged ﬁelds. Finally, besides the terms that we have displayed in ( 17) we implicitely assume the presence of chemical potential terms that ensure overal l charge neutrality. However, fr the present purposes such terms are redundant and willeit her remain implicit, or will be enforced by appropriate boundary conditions. We propose that the advantage of (17) over ideal magnetohydr odynamics is, that (17) provides a ﬁrm microscopic basis for systematically co mputing various properties of a plasma. For example an appropriate version of the equati on of state (4) can be derived from (17). In particular, (17) yields immediately t he standard electromagnetic many-body Schr¨ odinger equation for a gas of electrons and i ons. The energy-momentum tensor Tµνcan be computed directly from (10). After we introduce an appropriate symmetrization which ensures man ifest gauge invariance, we ﬁnd for the energy density T0 0=1 2µ{sin2α\|Dkψe\|2+ cos2α\|D∗ kψi\|2}+E2 2+B2 2−A0(∂iEi+e[ψ∗ iψi−ψ∗ eψe]) (18) whereDk=∂k+ieAkandµ=msin2α=Mcos2αis the reduced mass. For the spatial components of the energy-momentum tensor we ﬁnd sim ilarly, with the help of the equations of motion Ti k=EiEk+BiBk−1 2µ/braceleftbigg sin2α[(Diψe)∗(Dkψe) + (Dkψe)∗(Diψe)] 4+ cos2α[(D∗ iψi)∗(D∗ kψi) + (D∗ kψi)∗(D∗ kψi)]/bracerightbigg −δi kL (19) Finally, for the generator of the canonical scale transform ation we get δC S=/integraldisplay d3x xkT0 k=/integraldisplay d3x xk/bracketleftbigg EiFki+i 2{ψ∗ eDkψe−D∗ kψ∗ eψe+ψ∗ iD∗ kψi−Dkψ∗ iψi}/bracketrightbigg (20) It yields the following gauge covariantized version of (12) , (13), δC SEk=xi∂iEk+ 2Ek+xk(∂iEi+e[ψ∗ iψi−ψ∗ eψe]) (21) δC SAk=xi∂iAk+Ak−∂k(xiAi) (22) δC Sψe,i=xi∂iψe,i+3 2ψe,i±iexiAiψe,ixi∂iψ∗ e,i+3 2ψ∗ e,i (23) In particular, for each of the canonical variable ( ψe,i,ψ∗ e,i) the scale dimension is 3 /2 so that the canonical scale generator commutes with the number operators for the charged particles {δC S, Ne,i}=δC S/integraldisplay d3x ψ∗ e,iψe,i= 0 (24) We now proceed to inspect the consequences of Shafranov’s vi rial arguments. For this we remind that a static minimum energy conﬁguration mus t be a stationary point of the energy (3), (18) under any localvariation of the ﬁelds. Since the scale transformation (12), (13) is a non-local variation it does not need to leave t he energy intact, unless it also preserves the pertinent boundary conditions. To determine these boundary conditions, we consider the plasma in two diﬀerent physical environment s: In the ﬁrst scenario we have an isolated, localized plasma co nﬁguration in an other- wise empty space, with a deﬁnite number of charged particles Ne+Ni=/integraldisplay d3x(ψ∗ eψe+ψ∗ iψi) (25) Since the canonical scale generator commutes with the indiv idual number operators (24), the ensuing variation of the ﬁelds is consistent with the bou ndary condition that the number of particles remains intact. By a direct computation we then ﬁnd for a static stationary point of the energy, 0 =δC S/integraldisplay d3x T0 0=−/integraldisplay d3x Ti i =−/integraldisplay d3x/parenleftBigg −1 µ{sin2α\|Dkψe\|2+ cos2α\|D∗ kψi\|2} −E2 2−B2 2/parenrightBigg (26) Since the trace of the spatial stress tensor is a sum of positi ve deﬁnite terms, in analogy with Shafranov’s virial theorem in ideal magnetohydrodyna mics (5) we conclude that there can not be any nontrivial stationary points. This mean s that in an otherwise 5empty space an initially localized plasma conﬁguration can not be conﬁned solely by its internal electromagnetic interactions. additional inter actions such as gravity must be present. Otherwise the canonical scale transformation dil utes the plasma by expanding its volume while keeping the number of the charged particles intact, until the collective behaviour of the plasma becomes replaced by an individual-p article behaviour of the charged constituents. The second physical scenario of interest to us describes the bulk properties of a plasma: We are interested in an initially localized plasma c onﬁguration, located within the bulk of a ﬁnite density plasma background. In this case th e relevant boundary condition on the charged ﬁelds states, that at large distanc es their densities approach a non-vanishing constant value ρ0which is the density of the uniform background plasma, \|ψe,i\|2r→∞−→ρ2 0 (27) The canonical scale transformation assigns a non-trivial s cale dimension to the charged ﬁelds. Consequently it can not leave the asymptotic particl e density intact, and fails to be consistent with the boundary condition (27) unless ρ0= 0. Instead of the canonical version of the scale transformation, we need to employ a non- canonical version of (12), (13) where the scale dimensions of the charged ﬁelds vanish, Dψ= 0. When we perform the ensuing variation of the ﬁelds in the energy density (18) , instead of (26) we ﬁnd δS/integraldisplay d3x T0 0=−/integraldisplay d3x/bracketleftBiggE2 2+B2 2−sin2α\|Dkψe\|2−cos2α\|D∗ kψi\|2/bracketrightBigg (28) Now the integrand acquires both positive and negative contr ibutions, which implies that a virial argument can not exclude the existence of stable ﬁni te energy solitons. Indeed, in [2] it has been argued that stable knotted solitons are prese nt. These solitons are formed within the bulk of the plasma, in an environment with an asymp totically constant back- ground density. A physical example of such an environment is the solar photosphere, the solitons are natural candidates for describing stable coro nal loops. Another, somewhat more hypothetical example could be the ball lightning, in th e background of Earth’s atmosphere. Such solitons could also become relevant in ide ntifying particularly stable plasma conﬁgurations in fusion experiments, when the plasm a is kept at ﬁnite density by the boundaries of an appropriate vessel. We shall now proceed to demonstrate, that the virial theorem (28) is also consistent with an appropriate canonical scale tranformation. For thi s we ﬁrst notice that ex- cluding the kinetic terms, the Lagrangian (17) coincides wi th that of relativistic scalar electrodynamics with two ﬂavors of scalar ﬁelds, L=\|(∂µ+iAµ)φ1\|2+\|(∂µ−iAµ)φ2\|2−V(φ)−1 4F2 µν (29) Here we have included a Higgs potential V(φ), to ensure a non-vanishing asymptotic value for the charged ﬁelds. For example, we can choose V(φ)∝(φ2 1+φ2 2−ρ2 0)2. The 6Hamiltonian version of (29) is L=π∗ 1∂0φ1+π1∂0φ∗ 1+π∗ 2∂0φ2+π2∂0φ∗ 2+Ei∂0Ai− \|(∂k+iAk)φ1\|2−\|(∂k−iAk)φ2\|2 −π∗ 1π1−π∗ 2π2−V(φ)−E2 2−B2 2−A0(∂iEi+iπ∗ 1φ1−iπ1φ∗ 1−iπ∗ 2φ2+iπ2φ∗ 2) (30) Notice that now the charged ﬁelds are canonically independe nt variables, a consequence of Lorentz invariance. The energy-momentum tensor can be co mputed directly from (10). With a proper symmetrization it becomes fully symmetr ic, as it should since the theory is Lorentz invariant. For the energy density we ﬁnd T0 0=\|Dkφ1\|2+\|D∗ kφ2\|2+E2 2+B2 2+π∗ 1π1+π∗ 2π2+V(φ) −A0{∂iEi+i(π∗ 1φ1−π1φ∗ 1)−i(π∗ 2φ2−π2φ∗ 2)} (31) The momentum ﬂux is T0 k=EiFki+π∗ 1Dkφ1+π1D∗ kφ∗ 1+π∗ 2D∗ kφ2+π2Dkφ∗ 2 (32) so that instead of (23), we ﬁnd that the canonical scale dimen sions of the charged scalar ﬁelds now vanish. As a consequence the canonical scale trans formation is consistent with the relevant boundary condition that in the r→ ∞ limit the system approaches a Higgs vacuumφ2 1+φ2 2→ρ2 0. This means that the canonical scale transformation must no w leave the energy of a static stationary point intact which le ads to the following virial theorem 0 =δC S/integraldisplay d3xT0 0=−/integraldisplay d3x Ti i =−/integraldisplay d3x/bracketleftBiggE2 2+B2 2+ 3(π∗ 1π1+π∗ 2π2)− \|Dkφ1\|2− \|D∗ kφ2\|2/bracketrightBigg (33) Since the contribution to the pressure from the charged ﬁeld s is negative, the virial theorem can not exclude stable static solitons. Finally, we note that even though the static sectors of the tw o theories (17) and (29) are very similar, these theories actually have a quite diﬀer ent physical content: In the relativistic case we may consistently set π1=π2=Ei=A0= 0 in the static equations of motion. This reduces the energy density to a functional fo rm which is manifestly magnetohydrodynamical (2), E=/integraldisplay d3x/bracketleftBigg \|Dkφ1\|2+\|D∗ kφ2\|2+V(φ) +B2 2/bracketrightBigg (34) But since the canonical scaling dimensions of the charged ﬁe lds now vanish, the virial theorem does not exclude the existence of purely magnetic so litons. On the other hand, 7if in the non-relativistic case (17) we set the electric ﬁeld to vanish, the equations of motion become inconsistent unless the electron and ion char ge densities are everywhere identical. This leads to a contradiction whenever the Hopf i nvariant is nontrivial [2], [3]. Hence solitons with a nontrivial Hopf invariant are nec essarily accompanied with a nontrivial electric ﬁeld. In particular, this means that t heir properties can not be consistently inspected by pure magnetohydrodynamics. In conclusion, we have extended Shafranov’s virial theorem from ideal magnetohy- drodynamics to classical ﬁeld theory and related it with Der rick’s scaling argument. We then employed the virial theorem to inspect soliton stabili ty in a realistic ﬁeld theory model of a two component plasma. In line with ideal magnetohy drodynamics, a scaling argument reveals that the ﬁeld theory model does not support stable isolated solitons in an otherwise empty space. But the virial theorem does allow f or the existence of stable solitons within the bulk of the plasma. These solitons are ac companied by a nontriv- ial electric ﬁeld, hence they can not be probed by magnetohyd rodynamics alone. We suggest that these solitons are relevant in describing coro nal loops in solar photosphere, maybe even ball lightning in Earth’s atmosphere. They might also become useful in the design of particularly stable magnetic fusion geometries. We thank A. Bondeson, R. Jackiw, S. Nasir, A. Polychronakos a nd G. Semenoﬀ for discussions. References [1] J. P. Freidberg, Ideal Magnetohydrodynamics Plenum Press, New York and London 1987; D. Biskamp, Nonlinear Magnetohydrodynamics Cambridge University Press, Cambridge 1993 [2] L. Faddeev and A. J. Niemi, physics/0003083 (Physical Re view Letters, to appear) [3] L. Faddeev and A.J. Niemi, Nature 387(1997) 58 [4] S. Coleman, Aspects of Symmetry - Selected Erice Lectures of Sidney Cole man(Cam- bridge Univ. Press) 1985 8
1/5 LANL archive reference : physics/0009062 Two exact derivations of the mass/energy relationship, E=mc2 Eric Baird (eric_baird@compuserve.com ) The E=mc2 relationship is not unique to special relativity. Einstein published one exact derivation from special relativity and two approximate derivations that used general extension s to Newtonian mechanics , and an exact derivation is also possible if we use a “first-order ” Doppler equation instead of special relativity's "relativistic Doppler" formula. We present two sample derivations based on different non -transverse Doppler relationships, and briefly look at the two diverging systems of physics that result. 1. INTRODUCTION The E=mc2 relationship between mass and energy was first made explicit in a short piece by Einstein (“On the Origin of Inertia” [1]) which was written as a postscript to the famous 1905 “Electrodynamics” paper, [2] and which presented t he E=mc2 result as a consequence of the mathematical relationships that had appeared in the earlier piece. W. L. Fadner has also unearthed and discussed a number of contemporary pieces that either came close to deriving E=mc2, or presented similar equations without fully exploring the consequences or claiming the result to be general. [3] Further pieces by Einstein in the 19 40s presented the E=mc2 relationship as a result that could be derived (approximately) from general principles without making specific reference to special relativity, [4][5] and we can also obtain an exact derivation by applying relativistic principles to a different shift formula to the one used by special relativity. Various authors have reworked the E=mc2 result since. [6][7][8][9][10] What appears to be missing from the literature is a direct comparison between derivations based on these two different shift formulae, and this is what we present here. 2. ASSUMPTIONS AND DEPENDENCIES Used by both derivations: • Momentum of particle, mom = mv • Momentum of light -pressure, mom = E / 2c • Energy of light taken as being proportional to its frequency • Newton's law of the conservation of momentum Used by first derivation only: • the "emitter -theory" non -transverse frequency -shift formula, Used by second derivation only: • Special relativity’s non -transverse frequency -shift formula, The variable v is taken to be recession velocity throughout.   −=cvc frequencyfrequency' 22 cv vccvcvc frequencyfrequency' −×   +=+−= 1 Newton, Optiks 3:1:Qu30: “Are not gross Bodies and Light convertible into one another, and may not Bodies receive much of their Activity from the particles of Light which enter their Composition? For all fix'd Bodies being heated e mit Light so long as they continue sufficiently hot, and Light mutually stops in Bodies as often as its Rays strike upon their Parts, as we shew'd above. ... The changing of Bodies into Light, and Light into Bodies, is very conformable to the Course of Nature, which seems delighted with Transmutations. ... And among such various and strange transmutations, why may not Nature change Bodies into Light, and Light into Bodies?” “Two exact derivations of the mass/energy relationship, E=mc2" Eric Baird 18th September 2000 2/5 physics/0009062 3. E=mc², FROM A FIRST -ORDER DOPPLER FREQUENCY -SHIFT FORMULA: Non-radiating particle A particle travels forwards at v m/s inside a nominally -stationary box. When the particle hits the rear wall of the box, it gives the box a forward momentum of mv. We will set the mass of the particle to be extremely small compared to the mass of the box, so that the change in box velocity caused by the impact is arbitrarily small. Radiating particle The particle emits a burst of energy E before it hits the box, as two plane waves each with energy E/2 in the particle ’s frame , one traveling "forwards" and one traveling "backwards" along the particle's path. The speed of the particle is not affected by the emission, because the reaction forces of both plane -waves (in the particle’s own frame) are equal and opposite. However, in the frame of the box, the two waves do not have the same energy or momentum. The frequency of the forward -aimed wave is blueshifted by freq' / freq = (c-(-v)) / c , the frequency of the rearward -aimed wave is redshifted by freq' / freq = (c-v) / c , and the energies of the two waves as they impact on the box are altered by the same amount. If the individual momentum of each wave is given by E / 2c, the combined forward momentum of the two waves striking the opposite sides of the box, is calculated as: E [ (c+v) - (c-v) ] / 2c2 = Ev/c² , so (in the box frame) the two pulses of light emitted by the particle carry Ev/c² of total forward momentum. If the particle and its emitted light have the same total overall momentum before and after emission takes pl ace, then after emission the forward momentum of the particle must be reduced by the same amount, Ev/c² . Since our “spent” particle has reduced momentum but an unchanged velocity, we can apply mass = momentum/velocity to calculate a reduced value for the mass of the particle after emission. mass 1 - mass 2 = (mom 1 - mom 2) / v mass LOST = (Ev/c² ) / v mass LOST = E/c² In order to lose an amount of inertial mass m, a particle with a fixed velocity has to lose an amount of energy (measured in its own frame) equal to E = mc² If we want to try this derivation for pairs of plane -waves aligned at angles other than 0 ° and 180 °, we need to know the exact aberration angle -changes associated with our first -order Doppler equation. This derivation will be attempted in a future paper. “Two exact derivations of the mass/energy relationship, E=mc2" Eric Baird 18th September 2000 3/5 physics/0009062 4. E=mc², FROM SPECIAL RELATIVITY’S FREQUENCY -SHIFT FORMULA: Non-radiating particle A particle travels forwards at v m/s inside a nominally -stationary box. When the particle hits the rear wall of the box, it gives the box a forward momentum of mv. We will set the mass of the particle to be extremely small compared to the mass of the box, so that the change in box velocity caused by the impact is arbitrarily small. Radiating particle The partic le emits a burst of energy E before it hits the box, as two plane waves each with energy E/2 in the particle ’s frame, one traveling "forwards" and one traveling "backwards" along the particle's path. The speed of the particle is not affected by the emission, because the reaction forces of both plane -waves (in the particle’s own frame) are equal and opposite. However, in the frame of the box, the two waves do not have the same energy or momentum. The frequency of the forward -aimed wave is blueshifted by freq'/freq = c / (c+(-v)) × √(1 - v²/c²) , the frequency of the rearward -aimed wave is redshifted by freq'/freq = c / (c+v) × √(1 - v²/c²) , and the energies of the two waves as they strike the box are altered by the same amount. If the individual momentum of each wave is given by E / 2c, the combined forward momentum of the two waves striking the opposite sides of the box, is calculated as: E/2c × [ c/(c-v) - c/(c+v) ] × √(1 - v²/c²) = [ Ev/c ² / (1 - v²/c²) ] × √(1 - v²/c²) = Ev/c ² / √(1 - v²/c²) , so (in th e box frame) the two pulses of light emitted by the particle carry Ev/c² / √(1 - v²/c² ) of momentum, and if total momentum is conserved, the act of emission must reduce the particle’s momentum by this amount. Since our “spent” particle has reduced momentum but an unchanged velocity, we can apply mass = momentum /velocity to calculate a reduced value for the mass of the particle after emission. mass 1 - mass 2 = (mom 1 - mom 2) / v mass LOST = (Ev/c²)/v / √(1 - v²/c²) mass LOST = E/c² / √(1 - v²/c²) We can now rearrange this so that the " E" term is on the left. In order to lose an amount of inertial mass m, a particle with a fixed velocity v has to lose an amount of energy equal to E = mc ² / √(1 - v²/c²) Since the total (Lorentz -dilated) mass of this particle under SR [11] is mTOTAL = mREST / √(1 - v²/c²) , we can get rid of the Lorentz term by either: • making v=0 and m=rest mass , (so that v²/c²=0 , and the Lorentz term disappears) or • making v>0 and using m as Lorentz -dilated mass (in which3 case the tw o Lorentz terms cancel). Either way, we then get the more general relationship E = mc² This is essentially Einstein’s 1905 derivation, for f=0. The concept of Lorentz mass -dilation is now considered by some people to be “old -fashioned”, but is still arguably valid for a ”historical” derivation. “Two exact derivations of the mass/energy relationship, E=mc2" Eric Baird 18th September 2000 4/5 physics/0009062 5. COMPARISON OF THE TWO DERIVATIONS Although both derivations produce the same nominal result, the resulting physics is different in each case. A First derivation (non -standard) With the first -order Doppler formula used in Section 3, the wavelength of the forward - emitted wave does not become infinite in the lab frame until the forward velocity of the particle is also infinite. This inability of the particle to catch up with and overtake its own wavefront even at nominally “superluminal” velocities means that a model based around this derivation does not use a single fixed speed of light – the speed of a signal leaving the “moving” particle’s nose must be different to the speed of a signal emitted by a similar particle that is stationary in the laboratory frame. If we take the root product of the object’s apparent ageing rates when viewed at 0° and 180° to its path, we get an averaged rate of apparent timeflow for the object of √(1 - v2/c2), which is the counterpart of the Lorentz time - dilation effect under special relativity. Emitter -theory and “dragged -light” models should show this behaviour. In contrast to special relativity, these variable -lightspeed models allow speeds up to and beyond cBACKGROUND , with the “imaginary” Lorentz result for v>c BACKGROUND being a consequence of the breakdown of direct observations of a superluminally -receding object. Emitter -theories based on flat spacetime are known not to work. [12][13][14] B Second derivation (special relativity) With the so -called “relativistic Doppler“ equation (second derivation) the energy of the forward -emitted wave becomes infinite in the background frame as v tends to c, and the requirement that no more energy can be taken out of the experiment than was put in dictates that the amount of energy required to give our particle a velocity v also tends towards infinity as v tends to c. This still leaves the question of whether the “c” in question refers to cBACKGROU ND or to cPARTICLE – these can have different values in a dragged -light model, where v=cPARTICLE represents an unattainable upper limit, with the particle and its forward -emitted light only having the same nominal speed when both have an infinite velocity in the lab frame. Under special relativity, these issues are avoided by declaring that spacetime is wholly undistorted by relative velocities between physical particles (“spacetime is flat”). The forward -emitted wave traveling at cPARTICLE is then also traveling at cBACKGROUND , and the upper limit to the particle’s speed has a definite finite velocity in the laboratory frame. There is again a calibrated Lorentz reduction in the apparent rate of timeflow for a moving object, with the rate becoming “imaginary” for v>c, but the infinite “energy barrier ” at v=c arguably makes this a moot point . 6. CONCLUSIONS Because Einstein’s “inertia” paper was published shortly after the electrodynamics paper, and presented the E=mc2 relationship as a consequence of the equations in the earlier piece, it is perhaps natural to suppose that the E=mc2 relationship would not have been discovered without special relativity. However, E=mc2 is also a consequence of the first -order Doppler equations associated with Newtonian ballistic -photon models, and, given Newton’s published hypothesis about the interconvertibility of matter and light, [15] the significance of the relationship could have been recognised much earlier. Fadner’s research [3] shows that some c ontemporary papers did include near -derivations, or derivations that were presented as limited special cases. If we consider the three most fundamental non-transverse Doppler equations (the two basic first -order equations and special relativity’s intermediate “relativistic Doppler” equation), only one of the three has not been shown to give us an exact derivation of E=mc2 – and of the two derivations given, special relativity’s is not the shortest. Although special relativity deserves credit for providing a credible theoretical platform that allowed Einstein to derive and publish the result, a verification of E=mc2 is not necessarily a verification of special relativity. “Two exact derivations of the mass/energy relationship, E=mc2" Eric Baird 18th September 2000 5/5 physics/0009062 REFERENCES [1] A. Einstein, “Does the Inertia of a Body depend on its Energy -Content?” (1905), translated and reprinted in The Principle of Relativity (Dover, NY, 1952) pp.67 -71. [2] A. Einstein, “On the Electrodynamics of Moving Bodies" (1905), translated and reprinted in The Principle of Relativity (Dover, NY, 1952) pp.35 -65. [3] W.L. Fadner "Did Einstein really discover “ E=mc² ?," Am.J.Phys. 56 (2) 114 -122 (February 1988) . [4] A. Einste in, “ E=mc2 ,” Science Illustrated (1946), reprinted in Out of My Later Years , (Philosophical Library, NY, 1950 ). [5] A. Einstein, “An Elementary Derivation of the Equivalence of Mass and Energy” (1946), reprinted in Out of My Later Years , (Philosophical Library, NY, 1950 ). [6] Mitchell J. Feigenbaum and N David Mermin, "E=mc²," Am.J.Phys. 56 (1) 18 -21 (January 1988) . [7] Fritz Rohrlich, "An elementary derivation of E=mc² ," Am.J.Phys. 58 (4) 348 -349 (April 1990) . [8] Lawrence Ruby and Robert E. Reynolds, "Comment on ‘An elementary derivation of E=mc²’ … ," Am.J.Phys. 59 (8) 756 (August 199 1) . [9] Fritz Rohrlich, "A response to ‘Comment on “An elementary derivation of E=mc²” … ’ ,” Am.J.Phys. 59 (8) 757 (August 1991) . [10] Ø Grøn, “A modification of Einstein’s first deduction of the inertia -energy relationship,” Eur.J.Phys. 8 24-26 (1987) . [11] T.R. Sandin, "In defense of relativistic mass”, Am.J.Phys. 59 (11) 1032 -1036 (November 1991) . [12] W. deSitter, "A proof of the constancy of the velocity of light," Kon. Acad. van Weten. 15 (2) 1297 -1298 (1913). [13] W. deSitter, "On the constan cy of the velocity of light," Kon. Acad. van Weten. 16 (1) 395 -396 (1913). [14] R. S. Shankland "Conversations with Albert Einstein," Am. J. Phys. 31 (1) 47 -57, section I (1963). [15] Isaac Newton, Optiks Book Three, Part 1, Qu. 30 .
arXiv:physics/0009063v1 [physics.bio-ph] 18 Sep 2000A minimum principle in mRNA editing of Physarum ? L. Frappatac, A. Sciarrinob, P. Sorbaa aLaboratoire d’Annecy-le-Vieux de Physique Th´ eorique LAP TH CNRS, UMR 5108, associ´ ee ` a l’Universit´ e de Savoie BP 110, F-74941 Annecy-le-Vieux Cedex, France bDipartimento di Scienze Fisiche, Universit` a di Napoli “Fe derico II” and I.N.F.N., Sezione di Napoli Complesso Universitario di Monte S. Angelo, Via Cintia, I-8 0126 Napoli, Italy cMember of the Institut Universitaire de France Abstract mRNA editing in three sequences of Physarum polycephalum is analyzed. Once ﬁxed the edited peptide chain, the nature of the inserted nucleot ides and the position of the insertion sites are explained by introducing a minimum prin ciple in the framework of the crystal basis model of the genetic code introduced by the aut hors. PACS number: 87.10.+e, 02.10.-v LAPTH-812/00 physics/0009063 September 2000 1One of the basic dogmas of molecular genetics states that the information contained in DNA ﬂows faithfully, via the mRNA intermediate molecule, in to the production of proteins. In late eighties it has been discovered that the information contained in DNA is not always found unmodiﬁed in the RNA products. It has been demonstrate d that in several organisms (kinetoplastid protozoa, mitochondria or chloroplasts of plants) some yet unknown biochemical machinery alters the sequence of the ﬁnal transcriptions pr oduct. This process is called RNA editing . Mainly three diﬀerent types of mRNA editing have been ident iﬁed: insertion of urydine (U); substitution of U by cytidine (C) and insertion of C. We c onsider in the following the mRNA editing in Physarum polycephalum, ﬁrstly discovered i n 1991 by R. Mahendran, M. Spottswood and D. Miller [1]. We have analyzed 151 single ins ertions (144 C and 7 U) in three published sequences of Physarum p. [1,2,3], remarking that the same amino acid chain could have been obtained by insertion of C in a site diﬀerent from th e observed one or by insertion of a nucleotide diﬀerent from C or U. Here we show that the nature of the inserted nucleotide and the position of the insertion site can be explained by introd ucing a minimum principle in the framework of the crystal basis model of the genetic code intr oduced in Ref. [4]. In this model any nucleotide sequence is characterized as an element of a v ector space. A simple function is deﬁned on this space, which takes in the observed conﬁgurati on a value smaller than the value in the alternative conﬁgurations. The mRNA editing in Physarum polycephalum has been extensiv ely studied and it presents the peculiar feature to be characterized mainly by C inserti on. We concentrate our analysis only on this biological species due to the high statistics of available data. Main feature of the RNA editing in Physarum p. is that in about 80 % of the cases the insertion occurs in the third position of the codon, the insertion sites are non rand om and in about 68 % of the cases the C is inserted after a purine-pyrimidine dinucleotide. M oreover no rule for the location of the editing sites has been determined. We have analyzed thre e published sequences of mRNA editing in portion of the ATP-9 Mitochondrial, of mRNA of Cyt ochrome c and b of Physarum polycephalum [1, 2, 3], showing respectively, 54 insertion s of a single C, 62 insertions (59 single C, 1 single U) and 40 insertions (31 single C, 6 single U). We do not consider presently multiple insertions and substitutions as the statistics is too low to derive any signiﬁcant conclusion. In the whole of the analyzed sequences we have remarked: 1. the presence of at least 22 alternative insertion sites fo r C (15 % of the cases, see Table 2), which would produce the same ﬁnal amino acids, so not alterin g the protein biosynthesis. For example, at the insertion site 9 of Ref. [1], the (observe d) sequence is ACC TTA (Thr Leu), while the (unobserved) sequence with alternative ins ertion site may be ACT C TA. 2. in at least 108 (resp. 98 and 63) of the 144 single C insertio ns (75 %, resp. 68 % and 44 % of the cases, see Table 3), the same ﬁnal amino acid may hav e been obtained by a single U (resp. A and G) insertion. Note that in writing Table 3, when the insertion site is ambiguous, i.e. when the inserted C is next to another C, so metimes a shift has been performed. Moreover, we have to consider the two cases GCC UCU →GCU ACU – site 16′– and CUU AAA→UUA AAR – site 21* – where C insertion is replaced by an A or an R ( R = A, G) insertion together with a shift of the insertion site. A simi lar analysis has been performed for the single U insertions. This implies two natural questions: 1) why the insertion sit es are the observed ones and not the other ones ? 2) why the C insertion is largely preferred ? In physics when a phenomenon occurs in one ﬁxed way between ma ny possible choices, one assumes that some minimum principle has to be satisﬁed. The s implest example is the straight 2path of light (in absence of strong gravitation ﬁelds), corr esponding to the shortest path between two point in euclidean geometry (the so-called geodesics). Can we think of the existence of a sort of minimum principle to explain mRNA editing and/or other process in DNA ? There ar e several technical and conceptual diﬃculties in this way of t ackling the problem. One should give a mathematical modelisation of RNA and identify the seq uence by a possibly discrete set of variables. Deﬁning a topological metric space depending on discrete variables and introducing on it a variation principle is a hard mathematical problem. M oreover we do not have a priori any theoretical guidelines, such as the Hamiltonian and/or Lagrangian formalisms, so we must have some good empirical grounding to begin with. In the pres ent note, as a ﬁrst step, we look for a simple function which would take the smallest valu e in the observed conﬁguration of insertion sites and single C insertion, with respect to th e conﬁguration with insertion in alternative sites and/or with a single U, G, A insertion. The starting point for a mathematical modelisation of DNA or mRNA is the crystal basis model of the genetic code [4] where the nucleotides are assig ned to the 4-dim irreducible fun- damental representation (1 /2,1/2) ofUq→0(sl(2)⊕sl(2)) and any sequence of Nnucleotides to the N-fold tensor product of (1 /2,1/2) (for codons, see [4] or Table 4 of [5], here reported in Table 1 for completeness). Then we make the assumption tha t the location sites for the insertion of a nucleotide should minimize the following fun ction for the mRNA or cDNA A0= exp/bracketleftBigg −/summationdisplay k4α Ck H+ 4β Ck V+ 2γJk 3,H/bracketrightBigg (1) where the sum in kis over all the codons in the edited sequence, Ck H(Ck V) and Jk 3,H, are the values of the Casimir operator and of the third component of the generator of the H-sl(2) (V- sl(2)), see [4], in the irreducible representation to which th ek-th codon belongs, see Table 1. Let us recall that the value of the Casimir operator on a state in an irreducible representation (IR) labelled by ( JH, JV) is CH(JH, JV) =JH(JH+ 1) and CV(JH, JV) =JV(JV+ 1) (2) In (1) the simpliﬁed assumption that the dependence of A0on the irreducible representation to which the codon belongs is given only by the values of the Casi mir operators has been made. The parameters α, β, γ are constants, depending on the biological species. The minimum of A0has to be computed in the whole set of conﬁgurations satisfyi ng to the constraints: i) the starting point should be the mtDNA an d ii) the ﬁnal peptide chain should not be modiﬁed. It is obvious that the global minimiza tion of expression (1) is ensured ifA0takes the smallest value locally, i.e. in the neighborhood o f each insertion site. The choice of the function A0is rather arbitrary; one of the reasons is that the chosen exp ression is computationally quite easily tractable. If the paramete rsα, β, γ are strictly positive with γ/6> β > α , the minimization of (1) explains the observed conﬁguratio ns in all cases, except for the cases 12, 33, 45 and 41* where there is equality and the cases 18* and 51* where the minimization is not satisﬁed (see Table 2). In order to deal with the remaining cases and to take into acco unt the observed fact that the dinucleotide preceding the insertion site is predomina ntly a purine-pyrimidine, we add to the exponent of the function A0an ”interaction term” which is equivalent to multiply (1) by the function A1where A1= exp/bracketleftBigg/summationdisplay i−4ω1j(i) 3,V·j(i−1) 3,V+ 4ω2j(i) 3,V·j(i−2) 3,V/bracketrightBigg (3) 3The sum in iis over the insertion sites and j(i−n) 3,Vis the value of the third component of the generator of V-sl(2) of the n-th nucleotide preceding the inserted nucleotide C (i.e. +1 /2 for C, U and −1/2 for G, A) and ω1, ω2are constants, depending on the biological species. In the case where the insertion site cannot be unambiguously de termined, i.e. when the inserted nucleotide is next to a nucleotide of the same type, (3) shoul d be computed in the conﬁguration which minimizes the value of A1. Ifω1> ω 2>0 and ω1>12αthe minimization of the function A=A0A1explains all the observed positions for C insertions, see Ta ble 2. It is reasonable, but not taken into account in (1), to argue that t he insertion sites and the nature of the inserted nucleotides also depend on the content of the particular sequence. Moreover A might be considered as the ﬁrst terms of a development, next t erms involving representations corresponding to more than one codon, the nature of the nucle otides following the insertion site, etc. These further terms may play a role in a more reﬁned analysis. An analysis of the 7 single U insertions shows that in 6 cases – sites 22, 10′, 18′, 22′, 24′, 26′– (resp. 3 cases – sites 10′, 18′, 24′–) the replacement U →C (resp. U →R) gives the same amino acid. In 4 of these cases the minimization of eq. (1 ) should prefer the insertion of C, giving rise to UUU →UUC, site 22; CUU →CUC, sites 10′, 18′; ACU →ACC, site 24′, while in sites 22′, 26′UUA is more preferred than CUA. This may explain why the U inse rtions are so rare compared with the C insertions. Also in this case f urther terms in Amay help for a more reﬁned analysis of the preferred conﬁguration of the i nsertions. In conclusion our eﬀective model does not explain why and where mRNA editing occurs in Physarum polycephalum, but it seems to be able to determine t he location sites and the nature of inserted nucleotides, once ﬁxed the amino acid chain. Ind eed in 110 of the 114 sites in which the insertion of C or U, and in all the cases where also an inser tion of purine can produce the same amino acid, the observed mRNA editing makes use of the nu cleotide C or U which does minimize A. It is natural to argue that mRNA editing in other biological organism in which the U insertion is preferred may be governed by an analogous e xpression of eqs. (1)-(3) with diﬀerent values of the constants. It is our program to invest igate the RNA editing in other biological species. Acknowledgements : It is a pleasure to thank D.L. Miller for pointing us the refe rences on RNA editing. Partially supported by MURST (Italy) and MAE (F rance) in the framework of french-italian collaboration Galileo. References [1] R. Mahendran, M.R. Spottswood, D.L. Miller, RNA editing by cytidine insertion in mito- chondria of Physarum polycephalum , Nature 349, 434 (1991). [2] J.M. Gott, L.M. Visomirski, J.L. Hunter, Substitutional and Inseertional RNA Editing by the Cytochrome c Oxidase Subunit 1 mRNA of Physarum polyceph alum, J. Biol. Chem. 268, 25483 (1993). [3] S.S. Wang, R. Mahendran, D.L. Miller, Editing of Cytochrome b mRNA in Physarum polycephalum , J. Biol. Chem. 274, 2725 (1999). [4] L. Frappat, A. Sciarrino, P. Sorba, A crystal base for the genetic code , Phys. Lett. A 250, 214 (1998) (physics/9801027). [5] L. Frappat, A. Sciarrino, P. Sorba, Symmetry and codon usage correlations in the genetic code, Phys. Lett. A 259, 339 (1999) (physics/9812041). 4Table 1: The eukariotic code. The upper label denotes diﬀere nt irreducible representations. codon a.a. JHJVJ3,HJ3,Vcodon a.a. JHJVJ3,HJ3,V CCC Pro3 23 23 23 2UCC Ser3 23 21 23 2 CCU Pro (1 23 2)1 1 23 2UCU Ser (1 23 2)1−1 23 2 CCG Pro (3 21 2)1 3 21 2UCG Ser (3 21 2)1 1 21 2 CCA Pro (1 21 2)1 1 21 2UCA Ser (1 21 2)1−1 21 2 CUC Leu (1 23 2)2 1 23 2UUC Phe3 23 2−1 23 2 CUU Leu (1 23 2)2−1 23 2UUU Phe3 23 2−3 23 2 CUG Leu (1 21 2)3 1 21 2UUG Leu (3 21 2)1−1 21 2 CUA Leu (1 21 2)3−1 21 2UUA Leu (3 21 2)1−3 21 2 CGC Arg (3 21 2)2 3 21 2UGC Cys (3 21 2)2 1 21 2 CGU Arg (1 21 2)2 1 21 2UGU Cys (1 21 2)2−1 21 2 CGG Arg (3 21 2)2 3 2−1 2UGG Trp (3 21 2)2 1 2−1 2 CGA Arg (1 21 2)2 1 2−1 2UGA Ter (1 21 2)2−1 2−1 2 CAC His (1 21 2)4 1 21 2UAC Tyr (3 21 2)2−1 21 2 CAU His (1 21 2)4−1 21 2UAU Tyr (3 21 2)2−3 21 2 CAG Gln (1 21 2)4 1 2−1 2UAG Ter (3 21 2)2−1 2−1 2 CAA Gln (1 21 2)4−1 2−1 2UAA Ter (3 21 2)2−3 2−1 2 GCC Ala3 23 23 21 2ACC Thr3 23 21 21 2 GCU Ala (1 23 2)1 1 21 2ACU Thr (1 23 2)1−1 21 2 GCG Ala (3 21 2)1 3 2−1 2ACG Thr (3 21 2)1 1 2−1 2 GCA Ala (1 21 2)1 1 2−1 2ACA Thr (1 21 2)1−1 2−1 2 GUC Val (1 23 2)2 1 21 2AUC Ile3 23 2−1 21 2 GUU Val (1 23 2)2−1 21 2AUU Ile3 23 2−3 21 2 GUG Val (1 21 2)3 1 2−1 2AUG Met (3 21 2)1−1 2−1 2 GUA Val (1 21 2)3−1 2−1 2AUA Ile (3 21 2)1−3 2−1 2 GGC Gly3 23 23 2−1 2AGC Ser3 23 21 2−1 2 GGU Gly (1 23 2)1 1 2−1 2AGU Ser (1 23 2)1−1 2−1 2 GGG Gly3 23 23 2−3 2AGG Arg3 23 21 2−3 2 GGA Gly (1 23 2)1 1 2−3 2AGA Arg (1 23 2)1−1 2−3 2 GAC Asp (1 23 2)2 1 2−1 2AAC Asn3 23 2−1 2−1 2 GAU Asp (1 23 2)2−1 2−1 2AAU Asn3 23 2−3 2−1 2 GAG Glu (1 23 2)2 1 2−3 2AAG Lys3 23 2−1 2−3 2 GAA Glu (1 23 2)2−1 2−3 2AAA Lys3 23 2−3 2−3 2 5Table 2: From the left: the a.a., the C insertion site, the cod ons coding for the a.a., the dinucleotide preceding C; the shift with respect to the obse rved site of the alternative insertion site, the new codons, the dinucleotide preceding C in the alt ernative site. Ref. to ﬁg. 3 of [1], ﬁg. 2 of [2] (with an asterisk ), ﬁg. 2 of [3] (with a prime′). a.a. site codons dinucl. shift codons dinucl. Thr, Leu 9, 24, 55 ACC, UUA AC + 1 ACU, CUA CU Ile, Leu 23, 30* AUC, UUG AU +1 AUU, CUG UU Ala, Phe 32 GCC, UUU GC +3 GCU, UUC UU Val, Phe 33, 45, 41* GUC, UUU GU +3 GUU, UUC UU Ser, Arg 34 UCC, AGA UC +1 UCA, CGA CA Asn, Phe 12 AAU, UUC UU −3AAC, UUU AA Ile, Leu 49, 48, 20′AUC, UUA AU +1 AUU, CUA UU Ala, Leu 5′GCC, UUA GC +1 GCU, CUA CU Ser, Phe 43, 13′UCC, UUU UC +3 UCU, UUC UU Thr, Arg 3* ACC, AGA AC +1 ACA, CGA CA Ser, Leu 18* AGU, CUG GU −1AGC, UUG AG Val, Leu 23, 40 GUC, UUA GU +1 GUU, CUA UU His, Leu 51* CAU, CUA AU −1CAC, UUA CA 6Table 3: From the left: the a.a., the codon created by C insert ion, the alternative codon created by alternative insertion, the site with reference t o ﬁg. 3 of [1], ﬁg. 2 of [2] (with an asterisk ), ﬁg. 2 of [3] (with a prime′). Here X = U, A, G and R = A, G. a.a. codon alt. codon site Asn AAC AAU 35, 4′ Thr ACC ACX 5, 7, 9, 10, 21, 24, 26, 36, 3, 4, 5, 12, 20, 26* 33, 35, 39, 49, 50, 55, 62, 15′, 39′ Ser AGC AGU 1, 36, 34′ Ile AUC AUU, AUA 1, 4, 13, 15, 17, 18, 20, 23, 38, 46, 49, 50, 51, 6 7, 9, 16, 17, 19, 24, 27, 30, 34, 38 48, 54, 57, 58, 60, 61, 20′, 32′, 36′, 37′ His CAC CAU 44* Pro CCC CCX 17′ Arg CGA AGA 30 Leu CUA UUA 31, 40, 8, 51, 6′ Leu CUG UUG 18* Leu CUC CUX 22 Leu CUU UUR 3, 13, 21, 47, 8′, 39′ Asp GAC GAU 54 Ala GCC GCX 25, 27, 29, 32, 37, 10, 13, 28, 53, 5′, 16′, 27′, 30′ Val GUC GUX 2, 6, 11, 14, 33, 42, 45, 23, 40, 41, 56, 9′, 21′, 25′ Tyr UAC UAU 43 Ser UCC UCX 34, 42, 2′, 12′, 13′ Phe UUC UUU 12, 52, 45* 7
arXiv:physics/0009064v1 [physics.data-an] 18 Sep 2000Conﬁdence intervals for the parameter of Poisson distribution in presence of background S.I. Bityukova,∗N.V. Krasnikovb aDivision of Experimental Physics, Institute for High Energ y Physics, Protvino, Moscow Region, Russia bDivision of Quantum Field Theory, Institute for Nuclear Res earch RAS, Moscow, Russia Abstract A numerical procedure is developed for construction of conﬁ dence intervals for parameter of Poisson distribution for signal in the presence of backgr ound which has Poisson distribution with known value of parameter. Keywords: statistics, conﬁdence intervals, Poisson distribution, G amma distribution, sample. I. INTRODUCTION In paper [1] the uniﬁed approach to the construction of conﬁd ence intervals and conﬁ- dence limits for a signal with a background presence, in part icular for Poisson distributions, has been proposed. The method is widely used for the presenta tion of physical results [2] though a number of investigators criticize this approach [3 ] In present paper we propose a simple method for construction of conﬁdence intervals for parameter of Poisson distribution for signal in the presenc e of background which has Poisson distribution with known value of parameter. This method is b ased on the statement [4] that the true value of parameter of the Poisson distribution in th e case of observed number of ∗Corresponding author Email addresses: bityukov@mx.ihep.su, Serguei.Bitioukov@cern.ch 1events ˆ xhas a Gamma distribution. In contrast to the approach propos ed in [1], the width of conﬁdence intervals in the case of ˆ x= 0 is independent on the value of the parameter of the background distribution. In Section 2 the method of construction of conﬁdence interva ls for parameter of Poisson distribution for signal in the presence of background which has Poisson distribution with known value of parameter is described. The results of conﬁde nce intervals construction and their comparison with the results of uniﬁed approach are als o given in the Section 2. The main results of this note are formulated in the Conclusion. II. THE METHOD OF CONSTRUCTION OF CONFIDENCE INTERVALS Assume that in the experiment with the ﬁxed integral luminos ity (i.e. a process under study may be considered as a homogeneous process during give n time) the ˆ xevents of some Poisson process were observed. It means that we have an exper imental estimation ˆλ(ˆx) of the parameter λof Poisson distribution. We have to construct a conﬁdence in terval (ˆλ1(ˆx),ˆλ2(ˆx)), covering the true value of the parameter λof the distribution under study with conﬁdence level 1 −α, where αis a signiﬁcance level. It is known from the theory of statistics [5], that the mean value of a sample of data is an un biased estimation of mean of distribution under study. In our case the sample consists of one observation ˆ x. For the discrete Poisson distribution the mean coincides with the e stimation of parameter value, i.e.ˆλ= ˆxin our case. As it is shown in ref [4] the true value of paramete rλhas Gamma distribution Γ 1,ˆx+1, where the scale parameter is equal to 1 and the shape paramet er is equal to ˆx+ 1, i.e. P(λ\|ˆx) =λˆx ˆx!e−λ. (2.1) Let us consider the Poisson distribution with two component s: signal component with a parameter λsand background component with a parameter λb, where λbis known. To construct conﬁdence intervals for parameter λsof signal in the case of observed value ˆ xwe must ﬁnd the distribution P(λs\|ˆx). 2At ﬁrst let us consider the simplest case ˆ x= ˆs+ˆb= 1. Here ˆ sis a number of signal events and ˆbis a number of background events among observed ˆ xevents. Theˆbcan be equal to 0 and to 1. We know that the ˆbis equal to 0 with probability p0=P(ˆb= 0) =λ0 b 0!e−λb=e−λb (2.2) and the ˆbis equal to 1 with probability p1=P(ˆb= 1) =λ1 b 1!e−λb=λbe−λb. (2.3) Correspondingly, P(ˆb= 0\|ˆx= 1) = P(ˆs= 1\|ˆx= 1) =p0 p0+p1andP(ˆb= 1\|ˆx= 1) = P(ˆs= 0\|ˆx= 1) =p1 p0+p1. It means that distribution of P(λs\|ˆx= 1) is equal to sum of distributions P(ˆs= 1\|ˆx= 1)Γ 1,2+P(ˆs= 0\|ˆx= 1)Γ 1,1=p0 p0+p1Γ1,2+p1 p0+p1Γ1,1, (2.4) where Γ 1,1is Gamma distribution with probability density P(λs\|ˆs= 0) = e−λsand Γ 1,2is Gamma distribution with probability density P(λs\|ˆs= 1) = λse−λs. As a result we have P(λs\|ˆx= 1) =λs+λb 1 +λbe−λs. (2.5) Using formula (2.5) for P(λs\|ˆx= 1) we construct the shortest conﬁdence interval of any conﬁdence level in a trivial way [4]. In this manner we can construct the distribution of P(λs\|ˆx) for any values of ˆ xandλb. We have obtained the formula P(λs\|ˆx) =(λs+λb)ˆx ˆx!ˆx/summationdisplay i=0λi b i!e−λs. (2.6) The numerical results for the conﬁdence intervals and for co mparison the results of paper [1] are presented in Table 1 and Table 2. It should be noted that in our approach the dependence of the w idth of conﬁdence intervals for parameter λson the value of λbin the case ˆ x= 0 is absent. For ˆ x= 0 the method proposed in ref. [6] also gives a 90% upper limit indep endent of λb. 3III. CONCLUSION In this note the construction of classical conﬁdence interv als for the parameter λsof Pois- son distribution for the signal in the presence of backgroun d with known value of parameter λbis proposed. The results of numerical construction are pres ented. Acknowledgments We are grateful to V.A. Matveev, V.F. Obraztsov and Fred Jame s for the interest to this work and for valuable comments. We wish to thank S.S. Bit yukov, A.V. Dorokhov, V.A. Litvine and V.N. Susoikin for useful discussions. This work has been supported by RFFI grant 99-02-16956 and grant INTAS-CERN 377. 4REFERENCES [1] Feldman, G.J. and R.D. Cousins, Uniﬁed approach to the cl assical statistical analysis of small signal, Phys.Rev. D573873-3889 (1998). [2] Groom, D.E. et al., Review of particle physics, Eur.Phys.J. C 15 (2000) 198-199. [3] as an example, Zech, G., Classical and Bayesian Conﬁdenc e Limits, in: F. James, L. Lyons, and Y. Perrin (Eds.), Proc. of 1st Workshop on Conﬁd ence Limits, CERN 2000-005, Geneva, Switzerland, (2000) 141-154. [4] Bityukov S.I, N.V. Krasnikov and V.A. Taperechkina, On t he conﬁdence interval for the parameter of Poisson distribution. e-print arXiv: physics /0008082, August 2000. [5] as an example, Handbook of Probability Theory and Mathematical Statistic s (in Russian), V.S. Korolyuk (Ed.), (Kiev, ”Naukova Dumka”, 1978) [6] Roy B.P. and M.B. Woodroofe, Phys.Rev. D59(1999) 053009. 5TABLES TABLE I. 90% C.L. intervals for the Poisson signal mean λs, for total events observed ˆ x, for known mean background λbranging from 0 to 4. A comparison between results of ref.[1] a nd results from present note. ˆx\λb0.0 ref.[1] 0.0 1.0 ref.[1] 1.0 2.0 ref.[1] 2.0 3.0 ref.[1] 3. 0 4.0 ref.[1] 4.0 00.00, 2.44 0.00, 2.30 0.00, 1.61 0.00, 2.30 0.00, 1.26 0.00, 2 .30 0.00, 1.08 0.00, 2.30 0.00, 1.01 0.00, 2.30 10.11, 4.36 0.09, 3.93 0.00, 3.36 0.00, 3.27 0.00, 2.53 0.00, 3 .00 0.00, 1.88 0.00, 2.84 0.00, 1.39 0.00, 2.74 20.53, 5.91 0.44, 5.48 0.00, 4.91 0.00, 4.44 0.00, 3.91 0.00, 3 .88 0.00, 3.04 0.00, 3.53 0.00, 2.33 0.00, 3.29 31.10, 7.42 0.93, 6.94 0.10, 6.42 0.00, 5.71 0.00, 5.42 0.00, 4 .93 0.00, 4.42 0.00, 4.36 0.00, 3.53 0.00, 3.97 41.47, 8.60 1.51, 8.36 0.74, 7.60 0.51, 7.29 0.00, 6.60 0.00, 6 .09 0.00, 5.60 0.00, 5.35 0.00, 4.60 0.00, 4.78 51.84, 9.99 2.12, 9.71 1.25, 8.99 1.15, 8.73 0.43, 7.99 0.20, 7 .47 0.00, 6.99 0.00, 6.44 0.00, 5.99 0.00, 5.72 62.21,11.47 2.78,11.05 1.61,10.47 1.79,10.07 1.08, 9.47 0. 83, 9.01 0.15, 8.47 0.00, 7.60 0.00, 7.47 0.00, 6.76 73.56,12.53 3.47,12.38 2.56,11.53 2.47,11.38 1.59,10.53 1 .49,10.37 0.89, 9.53 0.57, 9.20 0.00, 8.53 0.00, 7.88 83.96,13.99 4.16,13.65 2.96,12.99 3.18,12.68 2.14,11.99 2 .20,11.69 1.51,10.99 1.21,10.60 0.66, 9.99 0.34, 9.33 94.36,15.30 4.91,14.95 3.36,14.30 3.91,13.96 2.53,13.30 2 .90,12.94 1.88,12.30 1.92,11.94 1.33,11.30 0.97,10.81 105.50,16.50 5.64,16.21 4.50,15.50 4.66,15.22 3.50,14.50 3 .66,14.22 2.63,13.50 2.64,13.21 1.94,12.50 1.67,12.16 2013.55,28.52 13.50,28.33 12.55,27.52 12.53,27.34 11.55,2 6.52 11.53,26.34 10.55,25.52 10.53,25.34 9.55,24.52 9.53 ,24.34 TABLE II. 90% C.L. intervals for the Poisson signal mean λs, for total events observed ˆ x, for known mean background λbranging from 6 to 15. A comparison between results of ref.[1] and results from present note. ˆx\λb6.0 ref.[1] 6.0 8.0 ref.[1] 8.0 10.0 ref.[1] 10.0 12.0 ref.[1 ] 12.0 15.0 ref.[1] 15.0 00.00, 0.97 0.00, 2.30 0.00, 0.94 0.00, 2.30 0.00, 0.93 0.00, 2 .30 0.00, 0.92 0.00, 2.30 0.00, 0.92 0.00, 2.30 10.00, 1.14 0.00, 2.63 0.00, 1.07 0.00, 2.56 0.00, 1.03 0.00, 2 .51 0.00, 1.00 0.00, 2.48 0.00, 0.98 0.00, 2.45 20.00, 1.57 0.00, 3.01 0.00, 1.27 0.00, 2.85 0.00, 1.15 0.00, 2 .75 0.00, 1.09 0.00, 2.68 0.00, 1.05 0.00, 2.61 30.00, 2.14 0.00, 3.48 0.00, 1.49 0.00, 3.20 0.00, 1.29 0.00, 3 .02 0.00, 1.21 0.00, 2.91 0.00, 1.14 0.00, 2.78 40.00, 2.83 0.00, 4.04 0.00, 1.98 0.00, 3.61 0.00, 1.57 0.00, 3 .34 0.00, 1.37 0.00, 3.16 0.00, 1.24 0.00, 2.98 50.00, 4.07 0.00, 4.71 0.00, 2.60 0.00, 4.10 0.00, 1.85 0.00, 3 .72 0.00, 1.58 0.00, 3.46 0.00, 1.32 0.00, 3.20 60.00, 5.47 0.00, 5.49 0.00, 3.73 0.00, 4.67 0.00, 2.40 0.00, 4 .15 0.00, 1.86 0.00, 3.80 0.00, 1.47 0.00, 3.46 70.00, 6.53 0.00, 6.38 0.00, 4.58 0.00, 5.34 0.00, 3.26 0.00, 4 .65 0.00, 2.23 0.00, 4.19 0.00, 1.69 0.00, 3.74 80.00, 7.99 0.00, 7.35 0.00, 5.99 0.00, 6.10 0.00, 4.22 0.00, 5 .23 0.00, 2.83 0.00, 4.64 0.00, 1.95 0.00, 4.06 90.00, 9.30 0.00, 8.41 0.00, 7.30 0.00, 6.95 0.00, 5.30 0.00, 5 .89 0.00, 3.93 0.00, 5.15 0.00, 2.45 0.00, 4.42 100.22,10.50 0.02, 9.53 0.00, 8.50 0.00, 7.88 0.00, 6.50 0.00, 6.63 0.00, 4.71 0.00, 5.73 0.00, 3.00 0.00, 4.83 207.55,22.52 7.53,22.34 5.55,20.52 5.53,20.34 3.55,18.52 3 .55,18.30 2.23,16.52 1.70,16.08 0.00,13.52 0.00,12.31 6
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/CI/D2/CB/CT /CP/D7 /D1/CT/CS/CX/CP/CU/D3/D6 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /D0/CP/D7/CT/D6 /CX/D7 /CP /DA /CT/D6/DD /CQ/D6/D3/CP/CS /CV/CP/CX/D2 /CQ/CP/D2/CS/B8 /DB/CW/CX /CW /CP/D2 /D7/D9/D4/D4 /D3/D6/D8 /D8/CW/CT/CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D3/CU /BD/BE /CU/D7 /D4/D9/D0/D7/CT/D7/BA/C0/D3 /DB /CT/DA /CT/D6/B8 /CW/CX/CV/CW /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8 /CX/D7 /D8/CW/CT /D7/D3/D9/D6 /CT /D3/CU /D7/D3/D1/CT /D7/CW/D3/D6/D8/B9 /D3/D1/CX/D2/CV/D7/BA /BT/D7 /CX/D8 /DB /CP/D7 /D7/CW/D3 /DB/D25/B8 /D8/CW/CT /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /D0/CP/D7/CT/D6/D7 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /CP/DA /CP/D6/CX/CT/D8 /DD /D3/CU /D9/D2/D7/D8/CP/CQ/D0/CT /D6/CT/CV/CX/D1/CT/D7 /D8/CW/CP/D8 /CP/D2 /CS/CT/D7/D8/D6/D3 /DD /D8/CW/CT /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CP/D2/CS /D4/D6/CT/DA /CT/D2 /D8/CU/D6/D3/D1 /D8/CW/CT /D4/D9/D0/D7/CT /D7/CW/D3/D6/D8/CT/D2/CX/D2/CV/BA/C0/CT/D6/CT/B8 /CU/D3/D6 /AS/D6/D7/D8 /D8/CX/D1/CT /D8/D3 /D3/D9/D6 /CZ/D2/D3 /DB/D0/CT/CS/CV/CT /DB /CT /CP/D2/CP/D0/DD/DE/CT /D8/CW/CT /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT /D0/D3 /CZ/B9/CX/D2/CV /CP/CQ/CX/D0/CX/D8/CX/CT/D7 /D3/CU /BV/D62+/BM /CI/D2/CB/CT /D0/CP/D7/CT/D6/BA /CC/CW/CT /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /CI/D2/CB/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7/CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /D8/CW/CP/D8 /CX/D7 /D2/CT /CT/D7/D7/CP/D6/DD /CU/D3/D6 /CP/D2 /D3/D4/D8/CX/D1/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CP/D7/CT/D6 /CS/CT/D7/CX/CV/D2 /CP/D2/CS /CU/D3/D6/CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2 /D3/CU /D7/CT/D0/CU/B9/D7/D8/CP/D6/D8/CX/D2/CV /CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CS /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CX/D2 /BV/D62+/BM/CI/D2/CB/CT /D0/CP/D7/CT/D6/BA/BE /C6/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/D3/D2 /CX/D2 /CP/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/BT/D7 /CX/D8 /CX/D7 /CZ/D2/D3 /DB/D23/B8 /D8/CW/CT /D8 /DB /D3/B9/D4/CW/D3/D8/D3/D2 /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2/B8 /CA/CP/D1/CP/D2 /D7 /CP/D8/D8/CT/D6/CX/D2/CV /CP/D2/CS /CB/D8/CP/D6/CZ/CT/AR/CT /D8 /D7/D8/D6/D3/D2/CV/D0/DD /D3/D2 /D8/D6/CX/CQ/D9/D8/CT /D8/D3 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/DA/CX/D8 /DD /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/D7/BA /CC/CW/CT/D1/CP/CX/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6/B8 /DB/CW/CX /CW /CS/CT/AS/D2/CT/D7 /CP /DA /CP/D0/D9/CT /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8 n2 /B8/CX/D7 /D8/CW/CT /CQ/CP/D2/CS/CV/CP/D4 /CT/D2/CT/D6/CV/DD Eg /BM n2[esu] =K/radicalBig EpG2(¯hω Eg) n0E4g./C0/CT/D6/CTn0 /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CU /D6/CT/CU/D6/CP /D8/CX/D3/D2/BN Ep= 21eV /CU/D3/D6 /D8/CW/CT /D1/D3/D7/D8 /D3/CU/CS/CX/D6/CT /D8 /CV/CP/D4 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/D7/BN K= 1.5·10−8/CX/D7 /CP /D1/CP/D8/CT/D6/CX/CP/D0/B9/CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/D2/D7/D8/CP/D2 /D8/BN G2 /CX/D7 /D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/CX/D2/CV /D3/D2/D0/DD /D3/D2 /D8/CW/CT /D6/CP/D8/CX/D3 /D3/CU /D4/CW/D3/D8/D3/D2 /CT/D2/CT/D6/CV/DD /CP/D2/CS /D8/CW/CT/CT/D2/CT/D6/CV/DD /CV/CP/D4 /D3/CU /D8/CW/CT /D1/CP/D8/CT/D6/CX/CP/D0/BA /BY /D3/D6 /CI/D2/CB/CT Eg= 2.58eV /BA /BY /D3/D6G2 /DB /CT /D9/D7/CT/CS /D8/CW/CT/CU/D3/D0/D0/D3 /DB/CX/D2/CV /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/CV/CP/D4 /D6/CT/CV/CX/D3/D2/BM G2(x) = 0.01373 +0.656·10−14 x2+0.1889·10−37 x6,/DB/CW/CX /CW /DB /CP/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /CS/CP/D8/CP /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /CA/CT/CU4/BA /C1/D2 /CP/CS/CS/CX/D8/CX/D3/D2 /D8/D3/BE/CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CUG2 /D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/B8 /DB /CT /CW/CP /DA /CT /D8/D3 /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2/D3/CUn0 /BA /BY /D3/D6 /CI/D2/CB/CT /D8/CW/CT /D2/CT/DC/D8 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CX/D7 /DA /CP/D0/CX/CS/BM n0= 2.4215 +0.4995·10−7 λ−0.1747·10−13 λ2+0.3429·10−19 λ3/B4/CW/CT/D6/CT λ /CX/D7 /CX/D2 /D1/CT/D8/CT/D6/D7/B5/BA /BT /D6/CT/D7/D9/D0/D8/CX/D2/CV /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CUn2 /D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /CX/D7 /D7/CW/D3 /DB/D2/CX/D2 /BY/CX/CV/BA /BD/BA /CC/CW/CT/D6/CT /CX/D7 /CP /D7/D8/D6/D3/D2/CV /C3/CT/D6/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /D4/D6/D3 /CS/D9 /CX/D2/CV /D7/CT/D0/CU/B9/CU/D3 /D9/D7/CX/D2/CV /CT/AR/CT /D8/BA/CC/CW/CT /DA /CP/D0/D9/CT /D3/CUn2 /CT/DC /CT/CT/CS/D7 /D8/CW/CT /D8 /DD/D4/CX /CP/D0 /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D1/D3/D7/D8 /D3/CU /CS/CX/CT/D0/CT /D8/D6/CX /D6/DD/D7/D8/CP/D0/D7/CQ /DD /D8 /DB /D3 /D3/D6/CS/CT/D6/D7 /D3/CU /D1/CP/CV/D2/CX/D8/D9/CS/CT/BA/C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CX/D7 /CX/D7 /D2/D3/D8 /D7/D9Ꜷ /CX/CT/D2 /D8 /CU/D3/D6 /D8/CW/CT /CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D4/D6/D3/D4 /CT/D6/B9/D8/CX/CT/D7 /D3/CU /CI/D2/CB/CT/BA /BW/CT/D7/D4/CX/D8/CT /D8/CW/CT /CU/CP /D8 /D8/CW/CP/D8 /CI/D2/CB/CT /CW/CP/D7 /D9/CQ/CX /CP/D0 /D0/CP/D8/D8/CX /CT/B8 /D8/CW/CT/D6/CT /CX/D7 /D2/D3/D8 /CT/D2 /D8/CT/D6/D3/CU /CX/D2 /DA /CT/D6/D7/CX/D3/D2 /CX/D2 /D6/DD/D7/D8/CP/D0/BA /BT/D7 /D6/CT/D7/D9/D0/D8/B8 /D8/CW/CT /D1/CT/CS/CX/D9/D1 /D4 /D3/D7/D7/CT/D7/D7/CT/D7 /D8/CW/CT /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7/BA /CC/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3 /CTꜶ /CX/CT/D2 /D8 d= 80 pm/V6/BA/BT/D0/D8/CW/D3/D9/CV/CW /D8/CW/CT/D6/CT /CX/D7 /D2/D3 /CTꜶ /CX/CT/D2 /D8 /CU/D6/CT/D5/D9/CT/D2 /DD /D3/D2 /DA /CT/D6/D7/CX/D3/D2/B8 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CP/D2 /D7/D8/D6/D3/D2/CV/D0/DD /D3/D2 /D8/D6/CX/CQ/D9/D8/CT/D7 /D8/D3 /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/DA/CX/D8 /DD /CS/D9/CT/D8/D3 /D7/D3/B9 /CP/D0/D0/CT/CS /CP/D7 /CP/CS/CT/CS /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /B8 /DB/CW/CX /CW /CP/D2 /CQ /CT /D9/D7/CT/CS /CU/D3/D6 /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV7/BA /CD/D7/D9/CP/D0/D0/DD /B8 /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT /CX/D2/CS/CT/DC /D3/CU /D6/CT/CU/D6/CP /D8/CX/DA/CX/D8 /DD /CW/CP/D7 /D2/D3/D2/B9/C3/CT/D6/D6 /CW/CP/D6/CP /D8/CT/D6/B8 /CQ/D9/D8 /D8/CW/CT /C3/CT/D6/D6/B9/D0/CX/CZ /CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 n=n0+n2I /B4/DB/CW/CT/D6/CT I /CX/D7 /D8/CW/CT /AS/CT/D0/CS/CX/D2 /D8/CT/D2/D7/CX/D8 /DD/B5 /CX/D7 /DA /CP/D0/CX/CS /CX/CU6(∆k 2χ)2 I<<1 /B8 /DB/CW/CT/D6/CT χ=2ωd√ 2ǫ0n2 0n2ωc3 /BN∆k= 2k0−k2ω /BNω /B8k0 /CP/D6/CT /D8/CW/CT /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /CU/D6/CT/D5/D9/CT/D2 /DD/CP/D2/CS /DB /CP /DA /CT /D2 /D9/D1 /CQ /CT/D6/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D0/DD/BN n2ω /B8k2ω /CP/D6/CT /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/DA /CT /CX/D2/CS/CT/DC/CP/D2/CS /DB /CP /DA /CT /D2 /D9/D1 /CQ /CT/D6 /CU/D3/D6 /D8/CW/CT /D7/CT /D3/D2/CS /CW/CP/D6/D1/D3/D2/CX /BA /C7/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /D7/CW/D3 /DB /CT/CS/B8 /D8/CW/CP/D8/D8/CW/CT /C3/CT/D6/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CU/D3/D6 /CI/D2/CB/CT /CX/D7 /DA /CP/D0/CX/CS /D9/D4 /D8/D3 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /D3/CU2.2TW/cm2/CP/D8/BEµ /D1 /CP/D2/CS600GW/cm2/CP/D8 /BFµ /D1/B8 /CP/D2/CS /CX/D8 /CX/D7 /D9/D7/CT/CS /D8/CW/D6/D3/D9/CV/CW/D3/D9/D8 /D4/D6/CT/D7/CT/D2 /D8 /DB /D3/D6/CZ/BA/CC/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CU /D6/CT/CU/D6/CP /D8/CX/D3/D2 /CS/D9/CT /D8/D3 /CP/D7 /CP/CS/CX/D2/CV /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CX/D76/BM n2=4πd2 ǫ0n2 0n2ωcλ∆k./C6/D3/D8/CT/B8 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU /D8/CW/CT /D2/D3/D6/D1/CP/D0 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3/CUn0 /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD/CS/D9/CT /D8/D3 /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CW/CP/D7 /CP /CS/CT/CU/D3 /D9/D7/CX/D2/CV /D2/CP/D8/D9/D6/CT/BA /CC/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT/D3/CUn2 /D3/D2 /D8/CW/CT /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BE/BA /C7/D2/CT /CP/D2 /D7/CT/CT/B8 /D8/CW/CP/D8 /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6/D6/CT/CU/D6/CP /D8/CX/D3/D2 /CX/D7 /DA /CT/D6/DD /D7/D8/D6/D3/D2/CV/BA /BT/D7 /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CT /CY/D3/CX/D2 /D8 /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CX/D6/CS/B9 /CP/D2/CS/D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8/CX/CT/D7/B8 /CP /D2/CT/D8/B9 /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CU /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/D3/D2 /CX/D7 /D7/CW/D3 /DB/D2/CX/D2 /BY/CX/CV/BA /BF/BA /CC/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CX/D7 /CU/D3 /D9/D7/CX/D2/CV /D9/D4 /D8/D3 /BD/BA/BEµ /D1 /CP/D2/CS /CS/CT/CU/D3 /D9/D7/CX/D2/CV /CX/D2 /D1/CX/CS/B9/C1/CA/D6/CP/D2/CV/CT/BA /BT/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /DA /CP/D0/D9/CT /D3/CUn2 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/D8/CW/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/D4/CP/D6/CP/D1/CT/D8/CT/D6 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BH/BA /C1/D8 /CX/D7 /D7/CT/CT/D2/B8 /D8/CW/CP/D8 /D8/CW/CT /C3/CT/D6/D6/B9/D0/CT/D2/D7/CX/D2/CV /CTꜶ /CX/CT/D2 /DD /CX/D2/D8/CW/CX/D7 /CP/D7/CT /CX/D7 /CW/CX/CV/CW/CT/D6 /CQ /DD /D8 /DB /D3 /D3/D6/CS/CT/D6/D7 /D3/CU /D1/CP/CV/D2/CX/D8/D9/CS/CT/BA /C1/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV/D0/DD /B8 /D8/CW/CP/D8 /D8/CW/CT/CS/CT/CU/D3 /D9/D7/CX/D2/CV /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CS/D3 /CT/D7 /D2/D3/D8 /CP/AR/CT /D8 /D8/CW/CT /D4/D6/D3 /CT/D7/D7 /D3/CU /D8/CW/CT /CP /DA/CX/D8 /DD/D3/D4/D8/CX/D1/CX/DE/CP/D8/CX/D3/D2 /CT/D7/D7/CT/D2 /D8/CX/CP/D0/D0/DD/BM /D8/CW/CT /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT /D7/CX/CV/D2 /D3/CUn2 /D3/D2/D0/DD /D7/D0/CX/CV/CW /D8/D0/DD /CP/AR/CT /D8/D7/D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CP/D2/CS /CU/D3/D0/CS/CX/D2/CV /CS/CX/D7/D8/CP/D2 /CT/BA /CC/CW/CT /D3/D4/D8/CX/D1/CX/DE/CT/CS /CP /DA/CX/D8 /DD /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BG/BA/BG /CD/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD/BV/CP/D0 /D9/D0/CP/D8/CT/CS /DA /CP/D0/D9/CT/D7 /D3/CU∂T ∂I /D7/D9/CV/CV/CT/D7/D8 /CP /CV/D3 /D3 /CS /C3/CT/D6/D6/B9/D0/CT/D2/D7 /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /CP/CQ/CX/D0/CX/D8 /DD /D3/CU /BV/D62+/BM/CI/D2/CB/CT /AL /D0/CP/D7/CT/D6/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /D7/D8/D6/D3/D2/CV /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /DB/CW/CX/D0/CT /CQ /CT/CX/D2/CV /D8/CW/CT/D4 /D3/D7/CX/D8/CX/DA /CT /CU/CP /D8/D3/D6 /CU/CP /CX/D0/CX/D8/CP/D8/CX/D2/CV /D7/CT/D0/CU/B9/CU/D3 /D9/D7/CX/D2/CV/B8 /CP/D2/B8 /D3/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /CS/CT/D7/D8/CP/CQ/CX/D0/CX/DE/CT/CP/D2 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD5/BA /C1/D2 /D3/D6/CS/CT/D6/D8/D3 /CT/D7/D8/CX/D1/CP/D8/CT /D8/CW/CT /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CX/D2 /BV/D62+/BM /CI/D2/CB/CT /AL /D0/CP/D7/CT/D6 /DB /CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS/D8/CW/CT /D2 /D9/D1/CT/D6/CX /CP/D0 /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/CU /AT/D9 /D8/D9/CP/D8/CX/D3/D2 /D1/D3 /CS/CT/D0 /CP/D2/CS /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS/D7 /CW/CT/D1/CT /D3/CU /D8/CW/CT /D0/CP/D7/CT/D6 /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/BA /CC/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /DB /CP/D7/BM ∂a ∂z=/parenleftBigg α+ (tf+iD)∂2 ∂t2−iβ\|a\|2−γ 1 +σ\|a\|2/parenrightBigg a,/DB/CW/CT/D6/CT a /CX/D7 /D8/CW/CT /AS/CT/D0/CS/B8z /CX/D7 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/B8 α /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS/CQ /DD /CU/D9/D0/D0 /AS/CT/D0/CS /CT/D2/CT/D6/CV/DD /CV/CP/CX/D2 /D3 /CTꜶ /CX/CT/D2 /D8/B8 tf /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW /D3/CU /D8/CW/CT /D7/D4 /CT /B9/D8/D6/CP/D0 /AS/D0/D8/CT/D6/B8 D /CX/D7 /D8/CW/CT /D2/CT/D8/B9/CV/D6/D3/D9/D4 /CS/CT/D0/CP /DD /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2/B8 β=2πn2zg λn0 /CX/D7 /D8/CW/CT /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6/B8 zg /CX/D7 /D8/CW/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /D0/CT/D2/CV/D8/CW/B8 γ /CX/D7 /D8/CW/CT /CS/CX/AR/D6/CP /D8/CX/D3/D2/D0/D3/D7/D7/B8 /CS/CT/AS/D2/CT/CS /CU/D6/D3/D1 /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D0/CP/D7/CT/D6/B8 σ /BP∂T ∂I /BA/CC/CW/CT /D7/D8/D6/D3/D2/CV /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CP/D2/CS /D7/CT/D0/CU/B9/CU/D3 /D9/D7/CX/D2/CV /CS/D3/D1/CX/D2/CP/D8/CT /D3 /DA /CT/D6 /D8/CW/CT/CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/BA /CC/CW/CT /D0/CP/D7/D8 /CU/CP /D8/D3/D6 /D4/D6/D3 /DA/CX/CS/CT/D7 /D8/CW/CT /D2/CT/CV/CP/D8/CX/DA /CT /CU/CT/CT/CS/CQ/CP /CZ /CX/D2 /D0/CP/D7/CT/D6 /D8/CW/CP/D8/D7/D8/CP/CQ/CX/D0/CX/DE/CT/D7 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1 p/e GEOMETRIC MASS RATIO Gustavo González-Martín Departamento de Física, Universidad Simón Bolívar, Apartado 89000, Caracas 1080-A, Venezuela.Web page URL http:\\prof.usb.ve\ggonzalm\ A previously proposed geometric definition of mass in terms of energy, in a geometrical unified theory, is used to obtain a numerical expression for a ratio of masses of geometrical excitations. The resultant geometric ratio is approximately equal the ratio of the proton to electron physicalmasses.SB/F/274-992 1. Introduction. We have presented a definition of mass [1], within our geometric unified theory of gravitation and other interac- tions, in terms of the concept of self energy of the non linear self interaction in a geometric space, leading to the massterm in Dirac’s equation. The concept of mass plays a fundamental role in relativity as shown by the relation betweenmass and energy and the principle of equivalence between inertial and gravitational mass. It is then of interest to finda method to carry a numerical calculation to test the proposed geometric model. On the other hand we also have considered an approximation to the geometric non linear theory [2] where the microscopic physical objects (geometric particles) are realized as linear geometric excitations, geometrically de-scribed in a jet bundle formalism shown to lead to the standard quantum field theory techniques. These geometricexcitations are essentially perturbations around a non linear geometric background space, where the excitations maybe considered to evolve with time. In this framework, a geometric particle is acted upon by the background and isnever really free except in absolute empty background space (zero background curvature). The background spacecarries the universal inertial properties which should be consistent with the ideas of Mach [3] and Einstein [4] thatassign fundamental importance of far-away matter in determining the inertial properties of local matter. We mayinterpret the geometric excitations as geometric particles and the background as the particle vacuum. This is ageneralization of what is normally done in quantum field theory when particles are interpreted as vacuum excitations.The vacuum is replaced by a geometric symmetric curved background space. In quantum electrodynamis the self energy is a source of problems. The infinite energy of the vacuum state is resolved by subtracting the infinite self energy by using the ordered products of operators. Similarly, the electron massis infinite in quantum electrodynamics. This problem is resolved by subtracting the infinite self energy effect andassuming the experimental value for the mass, in the process of renormalization. These problems do not arise in our treatment. If we consider geometric excitations on a background, the expres- sion defining mass may be expanded in a perturbation series around the background with a finite zeroth order term given entirely by the system self energy in terms of the background current and connection . The higher order terms are corrections depending on the excitation self interaction. As indicated in previous work, these corrections corre-spond to a geometric quantum field theory and may be taken care by the standard quantum field techinques. There isno ad-hoc assumption on our calculation because it is based, as indicated in the first paragraph, in a definition of massgiven in a unified theory long before this calculation was attempted. In this article we limit ourselves to the zeroth order background contribution, which we consider a bare mass, subject to further corrections. The existence of a simple particular solution to the background field equations allowsthe possibility of calculating these geometrical bare masses and obtain mass ratios using ratios of volumes of symmet-ric spaces, obtained from the structure group as cosets with respect to some of its subgroups. It should be noted thatthe structure group G of the theory, SL(4,R), has been used to describe particle properties [5,6,7] in another ap- proach. First we shall review the concept of geometric excitations. The fundamental dynamic process in the theory is the action of the connection on the frame. Since the connection is valued in the Lie algebra of G and the frame is an element of the group G, the dynamics is realized by the action of the group on itself. The principal bundle structure of the group, ( G,K,L), provides a natural geometric interpretation of its action on itself. A particular subgroup L defines a symmetric space K, the coset G/L, the base space of the bundle. The subgroup L, the fiber of the bundle, is the isotropy subgroup of the coset K acts on itself. The complemen- tary coset elements act as translations on the symmetric coset. This geometric interpretation may be transformed into a physical interpretation if we choose L to be homomorphic to the spinor group, SL(2,C), related to the Lorentz group. The action of L is then interpreted as a pseudo rotation (Lorentz transformation) of the external space, the tangent space TM of the physical space time manifold M, defining a metric. The action of the complementary coset K is interpreted as a translation in an internal space, the symmetric coset K itself. There is then a non trivial geometric relation between the internal and external spaces determined by the Clifford algebra structure of the manifold. The space K is the exponentiation of the odd sector of the Clifford algebra and is related to local copies of the tangent space TM and its dual cotangent space TM and may be interpreted as a generalized momentum space. The states of momentum k would correspond to a point of K. It follows that the frame excitations are also acted by the connection and evolve as representations of G. Different observers would measure different relative momenta k for a given excitation. A measurement for each k corresponds to a function in momentum space. An abstract excitation is an equivalence class of these functions, under the relativ-ity group. Since the group space itself carries its own representations, the realization of excitations as representationsdefined on the group space have a fundamental geometric character. The geometric action of the K sector is a trans-3 lation on itself and the functions on K are the observable internal linear representations. The action of the L sector is a Lorentz transformation and sl(2,C) spinors are the observable external representations. For these reasons, we must represent physically observable excitations by classes of spinor valued functions on the symmetric K space . In particular, we realize them on a vector bundle, associated to the principal bundle ( E,M,G), with fiber the sl(2,C) representation valued functions on the symmetric K space. In mathematics, these representations are called represen- tations of SL(4,R) induced from SL(2,C). Essentially, this is, in fact, done in particle physics when consideringrepresentations of the Poincare group. From our geometric point of view, it has been indicated [8] that the three stable particles, proton, electron and neutrino appear to be representations of the only three dynamical subgroups of SL(4,R) induced respectively from theeven subgroups SL 1(2,C) and SL(2,C). This is a generalization of particles as representations of the Poincare group induced from its Lorentz subgroup. Therefore we shall apply the proposed definition of mass to these representations,thus obtaining a physical and mathematically different calculation. It should be noted that there is no contradiction inthis calculation with present physical theories, which may be considered as effective theories derived under certainconditions and limits from other theories. 2. Bare Masses. The definition of the mass parameter m, in terms of a connection ω on the principal fiber bundle (E, M, G) , has been given in the fundamental defining representation of SL(4,R) in terms of 4×4 matrices, but in general, may be written for other representations using the Cartan-Killing metric gC, defined by the trace. The definition of this metric may be extended to the Clifford algebra A, which is an enveloping algebra of both sl(4,R) and sp(4,R). The Clifford algebra A is a representation and a subalgebra of the universal enveloping algebra U of these Lie algebras. The dimension of the vector space carrying the group representations is the trace of the representation of the identity in A. We may write the definition of the mass parameter in any representation of the algebras sl(4,R) and sp(4,R) and thecorresponding representation of the common enveloping Clifford algebra /G44(A) as () ()()mJJ IgJ gI AC CA== ≡1 4trtr trµ µµ µµ µΓΓ Γ /G44 . ( 2.1) It is known that the Cartan metric depends on the representations, but we shall only apply this expression to find ratios within a particular fixed induced representation of the enveloping algebra. If we consider geometric excitations on a background, this mass may be expanded as a perturbation around the background in terms of the only small parameter α, characterizing the excitation, () ()mJ J JO=+ + +1 400 10 012trµ µµ µµ µ αα αΓΓΓ , ( 2.2) indicating that the zeroth order term is given entirely by the background current and connection, with corrections depending on the excitation self interaction. As indicated in previous work [9], these corrections correspond to ageometric quantum field theory. In this article we limit ourselves to the zeroth order term which we consider the baremass of QFT. The structure group G is SL(4,R) and the even subgroup G + is SL1(2,C). The subgroup L is the subgroup of G+ with real determinant in other words, Sl(2,C). There is another subgroup H in the group chain G⊃H⊃L which is Sp(4,R). The corresponding symmetric spaces and their isomorphisms are discussed in appendix B. We are dealing with twoquotients which we shall designate as C and K, KG GSL R SL C SOSO SO SO≡≅⊗≅⊗+(, ) (, ) ()(,) (,) ()4 2233 31 2 , ( 2.3) CH LSp R Sp CSO SO≡≅ ≅(, ) (, )(,) (,)4 232 31 . ( 2.4) These groups have a principal bundle structure over the cosets and themselves carry representations. The geometric action of the K generators are translations on the coset K. The functions on K are the natural internal representations.4 We shall consider, then, the representations of SL(4,R) and Sp(4,R) induced from the subgroups SL1(2,C) and SL(2,C) over the symmetric spaces SL(4,R)/SL1(2,C) and Sp(4,R)/SL(2,C), respectively. The geometric induced representations may be realized as sections of a homogeneous vector bundle ( D, K, /G44[SL1(2,C)], G+) with SL1(2,C) representations /G44 as fiber F over the coset K [10]. To the induced representation of SL(4,R) on D, there corresponds an induced representation of the enveloping Clifford algebra A on D [11]. Furthermore, to the latter also corresponds a representation of the subgroup Sp(4,R) on D. In other words, the vector bundle D carries corresponding representations of A, SL(4,R) and Sp(4,R). These three representations are functions on K valued on representations of SL1(2,C). At each point of the base space M we consider the function space Σ of all sections of the homogeneous vector bundle D. Define a vector bundle S≡ (S, M, Σ, G), associated to the principal bundle E, with fiber the function space Σ of sections of D. The fiber of S is formed by induced representations of G. There is an induced connection acting, as the adjoint representation of G, on the bundle S. The connection ω is represented by matrix operators (Lorentz rotations on L and translations on K). The induced connection ω may be decomposed in terms of a set of basis functions characterized by a parameter k, the generalized spherical functions Yk on the symmetric space [12]. If K were compact, the basis of this function space would be discrete, of infinite dimensions d. The components, relative to this basis would be labeled by an infinite number of discrete indices k. Expressing the Cartan-Killing metric in the induced representation, we formally would have for the mass parameter,in terms of the Γ, J components mdJkk kk kk=′′ ′∑1 4tr ,µ µΓ . ( 2.5 ) Since the spaces under discussion are non compact, the discrete indices k that label the components, become continuous labels and the summation in matrix multiplication becomes integration over the continuous parameter k. In addition, we are working with 4 dimensional matrices and continuous functions on the cosets, and the Cartan-Killing metric in A, expressed by trace and integration, introduces a 4V(K R) dimensionality factor for the common carrier space D, giving, ()() () m VAdkJkk k kd k R=∫∫1 422 2tr ,,Γ . ( 2.6) where V(AR) is a volume characterizing the dimension of the continuous representation of A. We interpret the value of a function at k as the component with respect to the basis functions Y(kx) of the symmetric space K, parametrized by k, as usually done in flat space in terms of a Fourier expansion. We may say that there are as many “translations ” as points in K. It should be noted that these “translations ” do not form the well known Abelian translation group. The G-connection on E induces a SO(3,1)-connection on TM. The combined action of the connections, under the even subgroup G+, leaves the orthonormal set κµ invariant [13], defining a geometric relativistic equivalence relation R in the odd subspace K. Each element of the coset is a group element k that corresponds to an odd moving frame. A class of equivalent moving frames k is represented, up to an SL1(2,C) transformation, by a single rest frame, a point k0. Decomposition of Γ and J is into equivalence classes of state functions Y(kx). The dimension of the induced representation space, or number of classes of state functions Y(kx) (independent bases), is the volume of a subspace KR⊂K of classes (non equivalent) of points. The current and connection components are functions over the coset K. Physically the integral represents summation of the connection × current product, over all inequivalent odd observers, represented by a rest observer. There is a constant solution [14] for the non linear differential equations (see the appendix) that provides a trivial connection ϖ to the principal fiber bundle ( E,M,G). This SL(4,R) valued 1-form on E represents the class of equivalent local connection forms represented by the constant solution. At some particular frame s, that we may take as origin of the coset, the local expression for ϖ is the constant sm d x m Jgg∗=− =−ϖκαα . ( 2.7 ) All points of K or C may be reached by the action of a translation by k, restricted to the corresponding subgroup, from the origin of the coset. As the reference frame changes from s at the origin to sk at point k of the coset, the local connection form changes k s k s k k dk m J k dk k dkg∗∗ − ∗ − − −=+ = − + ≡ +ϖϖ11 1 1Λ , ( 2.8) which corresponds to the equivalence class of constant solutions ϖ. All ksϖ correspond the same constant solution5 class ϖ, seen in the different frames of the coset. In the principal bundle the constant connection ϖ, combines with the constant J to produce a product constant over the coset K, as long as the transformation is orthogonal to J [15]. As indicated before, the mass variation, produced by the last term in the equation due to an arbitrary choice of frame, corresponds to inertial effects. The noninertial effects are due to the first term in the right hand side of the equation which corresponds to the symbol Λ. It is clear that its subtraction from the connection transforms as a connection, because the current J is a tensorial form, and corresponds to the inertial connection. The latter connection is expressed by the last term in the equation and only hasa k dependence. The physical contribution to the bare mass parameter may be calculated in the special frame s giving, () () () gs J gJ m gJ JCC g C∗ •== −ωµ µµ µ Λ , ( 2.9) and defining an invariant expression in terms of Λ, valid for a given representation. We are interested in the induced representations of SL(4,R) and Sp(4,R) corresponding to the same trivial constant solution. In the defining representation of 4 dimensional matrices, the product JJ, JJ e e e e I JJ JJ JJGG HH GG HH•−−••• == − = = ′′=′′114 κκµµ , ( 2.10) is invariant under a SL(4,R) transformation and equal for both the G group and the H subgroup. There is a representation of A on the bundle S corresponding to the induced representation. The invariance (equality) of the product JJ must be valid in any representation of A, although the value of the product may differ from one representation to the other. For the induced function representations valued in the sl(2,C) algebra (Pauli matrices), the product becomes () () () () ()J k k J k k dk F k k F Jkk Jkk d kµα αβµβ µα αβµβκκ κκ κκ κκ/G24 /G24/G24/G24 /G24 /G24/G24/G24,, , ,,12 0 230 21 3 0 12 0 230 2∫ ∫≡= = ′′ ′′ ′′ , ( 2.11) which, must be a constant 4 ×4 matrix, independent of k1 and k3. The expression for the mass becomes, ()() mm VAFkkd kg RKR=∫4tr, . ( 2.12) The integrand is the same F0 constant for both groups, but the range of integration X differs. Integration is on a subspace KR⊂K of relativistic inequivalent points of K for the group G and on a subspace CR⊂C⊂K for the group H. The expressions for the masses corresponding to G-excitations and H-excitations become, ()()() ()() mm VAFkkd kVK VAmFGg RKR Rg R= =∫440tr, tr , ( 2.13) ()()()()()() ()mm VAFkkd km VAFVCmVC VKHg RCg RRGR RR= ==∫440tr, tr . ( 2.14) The bare mass parameters m are related to integration on certain subspaces of the coset spaces, depending proportionally on the volumes of the respective coset spaces. In other words, the ratio of the bare mass parameters for representationsof SL(4,R) and Sp(4,R), induced from the same SL(2,C) representation as functions on cosets K and C ⊂K would be equal to the ratio of the volumes of the respective subspaces.6 3. Volume Ratios. 3.1. Volume of C Space First consider, the volume of the four dimensional symmetric space Sp(4,R)/SL(2,C) which coincides with the quotient SO(3,2)/SO(3,1) as shown in appendix B. In the regular representation, it has the structure [] []Cx x x x x=                     * 0 1 23 4 , ( 3.1) where x4 must be a function of the xµ imposed by the group structure [16], ()xx x41 21=−ηµνµν . ( 3.2) The Euclidian volume element dV(c) given in terms of the forms dxµ(c) varies over the four dimensional coset. The invariant measure dµ(c) is determined by weighing the Euclidian element by a density equal to the inverse of the Jacobian of the transformation generated by a translation in the coset, ()() ()dcdV c Jcµ= , ( 3.3) ddx dx dx dx xxdV xxvµ ηηµνµ µνµν=∧∧∧ −= −0123 11 , ( 3.4) or in R5 with w as the fifth coordinate x4, ()δηµνµνw x x d xd xd xd xd w20 1 2 31 + − ∧∧∧∧ , ( 3.5) leading to ddx dx dx dx wdV wµ==0123 , ( 3.6) where, ηηµνµν µνµνxx xx w→+2 . (3.7) Inspection of this equation allows us to physically interpret the parameter w as a measure of a variation of mass energy throughout the coset C , wx x m=− =−112ηµνµν . (3.8) The presence of the Jacobian means that that we should use coordinates adapted to this symmetric space, polar hyperbolic coordinates, to calculate its invariant volume density, ()Cd wgdV CC=±=−∫∫µ . ( 3.9) The quotient space must be one of the standard four dimensional hyperboloids H4 [17]. We add a superscript α that indicates the number of negative signs in the invariant standard definition of Hn,α,7 () () 122 12 1=+ − =+ =∑∑wx xi in i i αα . ( 3.10) In particular, the space corresponds to the hyperboloid H4,3 , 122222 222=+ −−− =+−wuxyzwuk , ( 3.11) in terms of the coordinates u, k corresponding respectivelly to the energy and to the momentum absolute value, and an overall parameter λ that characterizes the size of a particular four dimensional hyperboloid. We introduce a parametrization in terms of arcs in the symmetric space by defining the hyperbolic coordinates ϕ, θ, β, χ , cosϕ= +x xy22 , ( 3.12) cosθ=z k , ( 3.13) coshβ= −u uk22 , ( 3.14) cosχλ=w . ( 3.15) It should be noted that the hyperbolic arc parameter β is not the relativistic velocity but is related to it by tanhβ==k uv c . ( 3.16) In particular the volume of C is obtained by an integration over this Minkowskian momentum space, where the coordinates x stand for u, k. We split the integration into the angular integration on the compact sphere S2, the radial boost boost β and the energy parameter χ., () ()VC dd d dIC= = =∫∫∫ ∫χχ ψ β β πββπββ π π βsin sinh32 02 04 0 016 316 3Ω sinh2 , ( 3.17) in terms of a boost integral I(β). 3.2. Volume of K space. For the volume of K, the integration is over an 8 dimensional symmetric space. This space G/G+ has a complex structure and is a non Hermitian space. The center of G+, which is not discrete, contains a generating element κ5 whose square is -1. We shall designate by 2J the restriction of ad(κ5) to the space TKk. This space, that has for base the 8 matrices κ α, κ βκ5, is the proper subspace corresponding to the eigen value -1 of the operator J2 , () [][] Jx y x y xy2 51 455 5 5λ λλ λλ λλ λ λ λλ λκκ κκ κ κκ κ κκ κ+= + = −−,, . ( 3.18) The endomorphism J defines an almost complex structure over K. In addition, using the Killing metric, in the Clifford representation, () () ()() ()() g Ja Jb JaJb J a b ab g a b,t r t r t r ,== − = =1 41 42 1 4 , ( 3.19) we have that the complex structure preserves the pseudo Riemannian (Minkowskian) Killing metric. Furthermore the torsion S is zero,8 ()[][ ][ ] [ ]Sab ab JJ ab JaJ b J aJ b,, , , ,=+ + − = 0 . ( 3.20) In this form, the conditions for J to be an integrable complex structure, invariant by G, are met and the space K is a non Hermitian complex symmetric space [18]. It is known that the Hermitian symmetric spaces are classified by certain group quotients. The symmetric space K is a non compact real form of the complex symmetric space corresponding to the complex extension of the noncompact group SU(2,2) and its quotients as shown in appendix B. This space coincides with the quotient SO(3,3)/SO(3,1) /G31SO(2) of the SO(4,2) series. In particular we have the 8 dimensional spaces RSO SO SOSL R SL C SOKSO SO SO≡×≈×≅≅ ≈×(,) () ()(, ) (, ) ()() () ()42 424 226 42/G4C . ( 3.21) which are the five real forms characterized by SO(4,2). The extreme spaces correspond to the two Hermitian spaces, compact and non compact R. Between the extremes we find the three non Hermitian non compact spaces, in particular the space of interest K. In the regular representation these quotients have the matricial structure, Kxy xy xyxy xy xy=                   * * 00 11 2233 44 55 , ( 3.22) where the lower right submatrix determines the conditions, xy xyxx xy yx yy44 551 2 1 1  =+• • •+ •   , ( 3.23) imposed by the corresponding associated groups on the coordinates x4, x5, y4, y5 in higher dimensional spaces (d>8),expressed by the scalar product in this submatrix, in terms of the corresponding 4-vectors x, y and respective metric, related to the Euclidian metric by Weyl ’s unitary trick. As in the previous case, this condition determines a unit symmetric space. Since these conditions are difficult to analyze, it is convenient to find the volume using the complex structure of the manifold. We may introduce complex coordinates zµ on K. In this way we obtain a symmetric bilinear complex metric in the complex four dimensional space K, ()1 42trzz z Cα ααβ αβ κη=− ∈ , ( 3.24) zz ei µµ ψµ= ( 3.25) The group that preserves this symmetric bilinear complex metric is the orthogonal group SO(4,C), There are two standard projections of the complex numbers C on the real numbers R, the real part and the modulus. In a similar manner as the modulus projects the complex plane to the real half line , we define an equivalencerelation S in the points on K by defining equivalent points as points with equal moduli coordinates. This equivalence relation may be expressed by S SSSS=×××1111 , ( 3.26) where S1 is the one dimensional phase sphere. We may define the 4 dimensional quotient Q by QK S= . ( 3.27)9 It is then convenient to parametrize K in accordance with this equivalence relation S. Each of the 4 complex coordinates z has a modulus zand a phase ψ which will be used as parameters. The Haar measure in terms of the Euclidian measure should correspond to integration on a complex 4 dimensional space. We choose the parametersdefining the Euclidian volume element at the identity, () () () () () () () () ()dV I d z I d I d z I d I dz I d I dz I d I=∧∧∧∧ ∧∧∧00 1 1 2233ψψ ψψ , ( 3.28) and translate it to the point k by a coset operation, ()() ()dkdV k Jkµ= . ( 3.29) As before, the Jacobian is a measure of a variation of mass energy throughout the coset K. We calculate the volume integral using the invariant volume density, ()ddz dz dz dz d d d d Jkµψψψψ=∧∧∧∧∧∧∧01230123 . ( 3.30) The Jacobian does not depend on the phases and the volume element of S is separable from dµ defining a complementary volume element corresponding to the quotient Q. The volume of K will then be the product of the volumes of S and Q, as indicated also by eq. ( 3.2.27). It may be seen that Q is a symmetric space, by taking as its representative points those with real coordinates. On the other hand Q must be non compact, otherwise K would also be compact since the relation S is compact. The quotient cannot be a product of a manifold times a discrete set because then K would have disconnected components. The SO(4,R) subgroup of SO(4,C), which acts on this subspace Q of K preserving the bilinear metric in this space, has S3 as orbit. Therefore Q is symmetric, four dimensional, non compact with a three dimensional compact subspace ΣS equal to the Riemannian sphere S3, QK SS=⊃3 . ( 3.31) The quotient Q must, then, be the hyperboloid H4,4, the only one with a S3 spacelike subspace, characterized in five dimensional space by the invariant, 122222 222= −+−−−= −+−wuxyz wuk . ( 3.32) The physical boost parameter β (the one that represents a change in kinetic energy) is the hyperbolic arc parameter along the orbit of a one parameter non compact subgroup. The parametrization is in terms of the hyperbolic coordinates ϕ, θ, ζ, β defined, instead of eqs. (3.14,3.15), by, coshβλ=u , ( 3.33) cosζ= +w wk22 . ( 3.34) We split the integration into the integration on the 4 compact phase spaces S and integration on the 4 dimensional space Q, parametrized by the coordinate moduli. The last integration is further split into integration on the compact 3 spheres S3 and integration on the complementary non compact direction, which corresponds to boosts β, obtaining, () ()() VK gdVdddd VQ VS dd d dK=− =×= ×  ∫ ∫∫∫∫4 0123 3 023 04 002 4ψψψψ ββ ζζ ψβ π ππ sinh sin Ω , ( 3.35)10 ()()() () VK d IK ==∫22 22 4 3 056ππ ββπββ sinh , ( 3.36) in terms of another boost integral I(β). 3.3 Ratio of Geometric Volumes We expect that the ratio of the volumes V of the inequivalent subspaces corresponding to fundamental representations of spin 1/2 should be related to the ratio of the corresponding physical bare masses. As indicated in section 2, there is a subset of points of the symmetric spaces that are relativistic equivalent. We have to eliminate these equivalent pointsby dividing by the equivalence relation R under the boosts of SO(3,1). Equivalent points are related by a Lorentz boost transformation of magnitude β . There are as many equivalent points as the volume of the orbit developed by the parameter β. The respective inequivalent volumes are, for C, ()() ()()() ()VCVC VRI IRC C= = = βπβ βπ16 3 16 3 , ( 3.37) and for K, ()() ()()() ()VKVK VRI IRK K== = βπβ βπ2256 56 . ( 3.38) Although the volumes of the non compact spaces diverge, their ratio taking in consideration the equivalence relation R has a well defined limit, obtaining, () ()VK VCR R=65π . ( 3.39) We actually have shown in this section a theorem that says: The ratio of the volumes of K and C, up to the equivalence relation R under the relativity isotropy subgroup, is finite and has the value 6 π5 . 4. Physical Mass Ratios. It is clear from the discussion in section 1 that, for a constant solution, the masses are proportional to the respective volumes V(KR). The constants of proportionality only depend on the specified inducing SL(2,C) representations. In particular for any two representation induced from the the spin 1/2 representation of L, the respective constants are equal. Armed with the theorem of the previous section, the ratio of masses of G-excitations and H-excitations given by the fundamental representations of the groups G and its subgroup H induced from the spin 1/2 representation of L, for the constant solution, equals the ratio of the volumes of the inequivalent subspaces of the respective cosets G/L1 and H/L. The ratio of bare masses for these geometric excitations has the finite exact value, () ()m mVK VCm mG HR Rp e== = ≈ 6 183611815π . , ( 4.1) which is a very good approximation for the ratio of the experimental physical values for the proton mass and the electron mass. This geometrical expression was previously known but not physically explained [19,20,21]. If we reject the easy explanation that this result is a pure coincidence, we should try to relate the G group to the proton and the H group to the electron. If this were the case the only other dynamical subgroup L=SL(2,C) of G should lead to a similar mass ratio. Previously we have related L to the neutrino. In this case the quotient space is the identity and we get,11 () ()() ()m mVLL VCVI VCm mL HR RR Re== = =/0ν , ( 4.2) which is in accordance with the zero bare mass of the neutrino. There may be small corrections to obtain the physical masses due to excitation contributions. 5. Conclusion. The ratio of geometrical masses mL/mH, given by the volumes of the quotients SL(4,R)/SL1(2,C) and Sp(4,R)/ SL(2,C)) up to the equivalence relation under SL(2,C) is equal, with a discrepancy of 2 ×10-5, to the value of the ratio of the masses of the proton and the electron. It appears that the values of the bare masses of the proton and electron may be calculated as masses of geometric excitations in this unified theory. These values necessarily need correction terms to obtain the physical masses,because of the interactions of the excitations. These corrections, due to the self interaction of the excitations may beexpected to be of the order of α2, equal to the order of the discrepancy. 6. Appendix A. The non linear equations of the theory are applicable to an isolated background physical system interacting with itself. Of course the equations must be expressed in terms of components with respect to an arbitrary reference frame.A reference frame adapted to an arbitrary observer introduces arbitrary fields which do not contain any informationrelated to the physical system in question. Any excitation must be associated to a definite background. An arbitrary observation of an excitation property depends on both the excitation and the background, but the physical observer must be the same for both excitation andbackground. We may use the freedom to select the reference frame, to refer the excitation to the physical framedefined by its own background. We have chosen the current density 3-form J corresponding to the vector, Je u eµα αµκ=~/G24 /G24 , ( 6.1) in terms of the matter spinor frame e and the orthonormal space-time tetrad u . Since we selected that the background be referred to itself, the background matter frame eb, referred to er becomes the group identity I. Actually this generalizes comoving coordinates (coordinates adapted to dust matter geodesics) [22]. We adopt coordinates adapted to local background matter frames (the only non arbitrary frame is itself, as arethe comoving coordinates defined by an isolated fluid). If the frame e becomes the identity I (in the group), the background current density becomes a constant, invariant under the action of the connections, Jd x=καα . ( 6.2) Comparison of an object with itself gives trivial information. For example free matter or an observer are always at rest with themselves, no velocity, no acceleration, no self forces, etc. In its own reference frame these effects actuallydisappear. Only constant self energy terms, determined by the non linearity, makes sense and should be the origin ofthe constant mass. In this reference frame the background field equation, DJbb∗∗=Ω4πα , ( 6.3) admits a constant ω solution. The differential expression for the left side of this equation reduces to an algebraic expression, a triple product of ω, giving, () ()ωω ω ω ωωπ α καα∧∧ −∧ ∧ =∗∗4d x , ( 6.4) which has for a trivial particular solution, ωκα α =−md xg/G24 /G24 , ( 6.5) where mg is a geometric dimensionless constant ()mg=πα313 . ( 6.6)12 In an arbitrary reference system the connection becomes, in terms of the current form J, ()ωκα α =− + = −+−− −e m dx e e de m J e degg11 1. ( 6.7) The trivial connection is essentially proportional to the current, up to an automorphism. It should be noted that in the expression for ω, the term containing the current J is the potential tensorial form Λ, used in chapter 8 in the definition of mass. Its subtraction from ω gives an object, e-1de, that transforms as a connection. 7. Appendix B. The series of quotient spaces related to the A3 Cartan space are of interest. In particular we choose the involutive automorphism of type AIII(p=2,q=2) that determines a seven dimensional compact subgroup G+. We obtain, in this manner, a series of eight dimensional spaces, characterized by the non compact group SU(2,2), corresponding to theRiemannian space G/G + and its dual G/G+, SU SU SU USU SL C SOSU SL C SO SL ,R SL C SOSU SU SU U() () () ()( ) (, ) ()(,) (, ) (,) () (, ) ()(,) () () ()4 22 14 2222 21 1 4 2222 22 1⊗⊗≈⊗≈⊗≈ ≈⊗≈⊗⊗ . ( 7.1) Due to the isomorphism of the spaces A3 and D3 we have the isomorphic series, characterized by the non compact group SO(4,2), corresponding to the Riemannian space G/G+ and its dual G/G+, with involution of the type BDI(p=4,q=2), SO SO SOSO SO SOSO SO SO SO SO SOSO SO SO() () ()(,) (,) ()(,) (,) (,) (,) (,) ()(,) () ()6 4251 31 242 31 11 33 31 242 42⊗≈⊗≈⊗ ≈⊗≈⊗ . ( 7.2) The characters of the real forms of both isomorphic series are -8, -4, 0, +4, +8. Another series of interest are the ones related to the C2. Cartan space. In particular, we choose the involutive automorphism of the type CII(p=2,q=2) that determines a six dimensional compact subgroup. We obtain a series offour dimensional spaces, characterized by the non compact group USp(2,2), corresponding to the Riemannian spaceG/G + and its dual G/G+, USp USp USpUSp Sp CSp R Sp R Sp R Sp R Sp CUSp USp USp() () ()(,) (, )(, ) (, ) (, ) (, ) (, )(,) () ()4 2222 24 22 4 222 22⊗≈≈⊗≈ ≈≈⊗ . ( 7.3) Also, due to the isomorphism of the spaces B2 and C2 we have the isomorphic series, characterized by the non compact group SO(4,1), corresponding to the Riemannian space G/G+ and its dual G*/G+ with involution of the type BDI(p=4,q=1), SO SOSO SOSO SOSO SOSO SO() ()(,) (,)(,) (,)(,) (,)(,) ()5 441 3132 2232 3141 4≈≈≈≈ . ( 7.4) The characters of the real forms of both isomorphic series are -4, -2, 0, +2, +4. 1 G. Gonz ález-Martín, Gen. Rel. Grav. 26, 1177 (1994); See also related publications . 2 G. Gonz ález-Martín, Gen. Rel. Grav. 24, 501 (1992); G. Gonz ález-Martín, Physical Geometry, (Universidad13 Simón Bolívar, Caracas) (2000), Electronic copy posted at http:\\prof.usb.ve\ggonzalm\invstg\book.htm 3 E. Mach, The Science of Mechanics, 5th English ed. (Open Court, LaSalle), ch. 1 (1947). 4 A. Einstein The Meaning of Relativity, 5th ed. (Princeton Univ. Press, Princeton), p.55 (1956). 5 Y. Ne’eman, Dj. Sijacki, Phys. Lett., 157B, 267. 6 Y. Ne’eman, Dj. Sijacki, Phys. Lett., 157B, 275. 7 F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne ’eman, Phys. Rep. 258, 1 (1995). 8 G. G. Gonz ález-Martín, in Strings, Membranes and QED, Proc. of LASSF, Eds. C. Aragone, A. Restuccia, S. Salamó (Ed. Equinoccio, Caracas) p, 97 (1989). 9 G. Gonz ález-Martín, Gen. Rel. Grav. 24, 501 (1992). 10R. Hermann, Lie Groups for Physicists (W. A. Benjamin, New York) (1966).11S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York) p. 90 (1962).12S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York) p. 360 (1962).13G. Gonz ález-Martín, Phys. Rev. D 35, 1225 (1987). 14G. Gonz ález-Martín, Fundamental Lengths in a Geometric Unified Theory, preprint USB 96a (1997). 15G. Gonz ález-Martín, Gen. Rel. Grav. 26, 1177 (1994) 16R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (John Wiley and Sons, New York), ch. 9 (1974). 17J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (Springer-Verlag, New York) (1994).18S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York) p. 130 (1962).19F. Lenz, Phys. Rev. 82, 554 (1951). 20I. J. Good, Phys. Lett. 33A, 383 (1970). 21A. Wyler, Acad, Sci. Paris, Comtes Rendus, 271A, 180 (1971). 22W. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco), p. 715 (1973).
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/CW/CX/CV/CW /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /B8 /DB/CW/CX /CW /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT /CP/D6/D6/CX/CT/D6 /CS/CT/D2/D7/CX/D8 /DD/CP/D2/CS/B8 /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /B8 /D3/D2 /D8/CW/CT /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /B4/D7/CT/CT/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CJ/BI℄/B5/BA /CC/CW/CX/D7 /D4/D6/D3 /CS/D9 /CT/D7 /CP/D7/D8/D6/D3/D2/CV /CT/D2/CT/D6/CV/DD/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D7/CT/D0/CU/B9/D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /B4/CB/C8/C5/B5/B8 /DB/CW/CX /CW /CX/D7 /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0/D8/D3 /D0/D3/D7/D7 /D3 /CTꜶ /CX/CT/D2 /D8/BA /CC/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CU /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0/CX/D8 /DD /B4/C0/CT/D2/D6/DD/B3/D7/CU/CP /D8/D3/D6/B5 /CX/D7 /CP/CQ /D3/D9/D8 /B9/BF÷ /B9/BK /CJ/BJ/B8 /BK℄/BA/CC/CW/CT /CP/CX/D1 /D3/CU /D8/CW/CX/D7 /D7/CT /D8/CX/D3/D2 /CX/D7 /D8/D3 /CS/CT/D7 /D6/CX/CQ /CT /CP/D2 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /CX/D2 /DB/D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /CU/CP/D7/D8 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/D3/D2 /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/CP/D2/CS /CP/D6/D6/CX/CT/D6/D7 /CS/CT/D2/D7/CX/D8 /DD /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CB/C8/C5 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6/BA /BT/D7 /DB /CT /D7/CW/CP/D0/D0/CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D0/CP/D8/CT/D6 /D3/D2/B8 /D8/CW/CT /D0/CP/D7/D8 /CU/CP /D8/D3/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D7/CT/D7/D7/CT/D2 /D8/CX/CP/D0/D0/DD /CP/D2/CS /CP/D2 /D7/D8/CP/CQ/CX/D0/CX/DE/CT /CP/D2 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CP/CV/CP/CX/D2/D7/D8 /CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0/CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /BA /C1/D2 /CP/CS/CS/CX/D8/CX/D3/D2/B8 /CX/D8 /DB/CX/D0/D0 /CQ /CT /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D0/CX/D2/CT/DB/CX/CS/D8/CW /CT/D2/B9/CW/CP/D2 /CT/D1/CT/D2 /D8 /D4/D6/D3 /CS/D9 /CT/D7 /CP /D2/CT/CV/CP/D8/CX/DA /CT /CU/CT/CT/CS/CQ/CP /CZ/B8 /DB/CW/CX /CW /D0/CT/CP/CS/D7 /D8/D3 /D1 /D9/D0/D8/CX/D7/D8/CP/CQ/D0/CT /D3/D4 /CT/D6/B9/CP/D8/CX/D3/D2/BA /C7/D2/CT /D7/CW/D3/D9/D0/CS /D2/D3/D8/CT /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CU/D6/D3/D1 /CP/D2 /CT/DC/CX/D7/D8/CX/D2/CV /D8/CW/CT/D3/D6/DD /CJ/BG℄/B8 /DB/CW/CT/D6/CT/D5/D9/CP/D7/CX/B9/CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /D0/CP/D7/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /DB /CP/D7 /D7/D8/D9/CS/CX/CT/CS/BA/BE/BA/BD /C5/D3 /CS/CT/D0/BU/CP/D7/CT/CS /D3/D2 /D8/CW/CT /D7/CT/D0/CU/B9 /D3/D2/D7/CX/D7/D8/CT/D2 /D8 /AS/CT/D0/CS /D8/CW/CT/D3/D6/DD /CJ/BL℄ /CP/D2/CS /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /CV/CP/CX/D2/B8/D8/CW/CT /D7/CP/D8/D9/D6/CP/CQ/D0/CT /D0/D3/D7/D7 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/B8 /CU/D6/CT/D5/D9/CT/D2 /DD /AS/D0/D8/CT/D6/CX/D2/CV/B8 /D8/CW/CT /BZ/CE/BW /CP/D2/CS /D8/CW/CT/CB/C8/C5 /DB /CT /CP/D6/D6/CX/DA /CT /D8/D3 /D8/CW/CT /D1/CP/D7/D8/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/BM ∂a(k,t) ∂k=α−Γ[exp[ −ǫ Ua] +iχ(exp[−ǫ Ua]−1)](1 +∂ ∂t)−1+ /B4/BD/B5/BE[1 (1 +∂ ∂t)−1]−l+id∂2 ∂t2−iβ\|a(k,t)\|2+iφa(k,t),/DB/CW/CT/D6/CTa(k,t) /CX/D7 /D8/CW/CT /AS/CT/D0/CS/B8k /CX/D7 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8 /D2 /D9/D1 /CQ /CT/D6/B8t /CX/D7 /D8/CW/CT /D0/D3 /CP/D0 /D8/CX/D1/CT/B8α /CX/D7/D8/CW/CT /CS/DD/D2/CP/D1/CX /CP/D0/D0/DD /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2/B8l /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D0/D3/D7/D7/B8Γ /CX/D7 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D7/CP/D8/D9/D6/CP/CQ/D0/CT/D0/D3/D7/D7/B8ε /CX/D7 /D8/CW/CT /CX/D2/D7/D8/CP/D2 /D8 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /AT/D9/DC/B8Ua /CX/D7 /D8/CW/CT /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /AT/D9/DC/B8φ/CX/D7 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CT/D0/CP /DD /CP/CU/D8/CT/D6 /D8/CW/CT /CU/D9/D0/D0 /D6/D3/D9/D2/CS /D8/D6/CX/D4/B8d /CX/D7 /D8/CW/CT /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8/B8 β /CX/D7 /D8/CW/CT/CB/C8/C5 /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CU /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8χ /CX/D7 /D8/CW/CT /C0/CT/D2/D6/DD/B3/D7 /CU/CP /D8/D3/D6/BA /CC/CW/CT /D8/CT/D6/D1 /CX/D2 /D8/CW/CT/D7/D8/D6/CP/CX/CV/CW /D8 /D4/CP/D6/CT/D2 /D8/CW/CT/D7/CT/D7 /DB/CX/D8/CW /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D7/D8/CP/D2/CS/D7 /CU/D3/D6 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /AS/D0/D8/CT/D6/CX/D2/CV/BA/BT/D0/D0 /D8/CX/D1/CT/D7 /CP/D6/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /AS/D0/D8/CT/D6 /CQ/CP/D2/CS/DB/CX/CS/D8/CW tf /CP/D2/CS /D8/CW/CT /CT/D5/D9/CP/D0/CQ/CP/D2/CS/DB/CX/CS/D8/CW/D7 /CU/D3/D6 /D8/CW/CT /D0/D3/D7/D7 /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /DD /AS/D0/D8/CT/D6 /CP/D6/CT /CP/D7/D7/D9/D1/CT/CS /CU/D3/D6 /D8/CW/CT /D7/CP/CZ /CT /D3/CU/D7/CX/D1/D4/D0/CX /CX/D8 /DD /BA /CC/CW/CT /CV/CP/CX/D2 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CX/D7 /D2/CT/CV/D0/CT /D8/CT/CS/BA/BT/D2 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /CT/D5/BA /B4/BD/B5 /CX/D2 /D8/D3 /D8/CW/CT /D7/CT/D6/CX/CT/D7 /CX/D2 /CT/D2/CT/D6/CV/DD /CP/D2/CS /CP /D3/D9/D2 /D8 /CU/D3/D6 /D8/CW/CT/CS/DD/D2/CP/D1/CX /CP/D0 /D0/D3/D7/D7 /CP/D2/CS /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CQ /DD /D8/CW/CT /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD ǫ=t/integraltext t0\|a(k,t′)\|2dt′ /B4t0 /CX/D7 /D8/CW/CT /D8/CX/D1/CT /D1/D3/D1/CT/D2 /D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D4/D9/D0/D7/CT/D4 /CT/CP/CZ/B5 /DD/CX/CT/D0/CS/D7/BM ∂a(k,t) ∂k=α0(1−τǫ+(τǫ)2 2)−l−γ0− /B4/BE/B5 γ0(1 +iχ)(ǫ2 2−ǫ)−(1−γ0(1−ǫ))∂ ∂t+ (1−γ0)∂2 ∂t2+id∂2 ∂t2−iχγ0ǫ∂ ∂t+ iφ−ip\|a(k,t)\|2a(k,t),/DB/CW/CT/D6/CTα0 /CP/D2/CSγ0 /CP/D6/CT /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2 /CP/D2/CS /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /D0/D3/D7/D7 /CP/D8 /D8/CW/CT/D4/D9/D0/D7/CT /D4 /CT/CP/CZ/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /C1/D2 /CT/D5/BA /B4/BE/B5 /D8/CW/CT /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /AT/D9/DC /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /D8/CW/CT/D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /AT/D9/DC /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6Ua /B8τ /CX/D7 /D8/CW/CT /D6/CP/D8/CX/D3 /D3/CU /D8/CW/CT /D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/CT/D2/CT/D6/CV/DD /AT/D9/DC /D8/D3 /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /AT/D9/DC /B4/DB/CX/D8/CW /CP /D3/D9/D2 /D8 /CU/D3/D6 /D8/CW/CT /D1/D3 /CS/CT /D6/D3/D7/D7/B9/D7/CT /D8/CX/D3/D2/D7 /CP/D8 /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /CP/D2/CS /CP/D8 /D8/CW/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B5/B8p=βUa tf /BA/BT /D7/D3/D0/CX/D8/D3/D2/B9/D0/CX/CZ /CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /CT/D5/BA /B4/BE/B5 /CX/D7 /D7/D3/D9/CV/CW /D8 /CX/D2 /D8/CW/CT /CU/D3/D6/D1 a(k,t) =a0sech1+iψ[(t−kδ)/tp] exp[iω(t−kδ)], /B4/BF/B5/DB/CW/CT/D6/CTa0 /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8 δ /CX/D7 /D8/CW/CT /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4 /CS/CT/D0/CP /DD /D7/D3 /D8/CW/CP/D8t0= kδ /B8tp /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/B8ω /CX/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8 /CU/D6/D3/D1 /D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU /D8/CW/CT/D8/D6/CP/D2/D7/D1/CX/D7/D7/CX/D3/D2 /CQ/CP/D2/CS /D3/CU /D8/CW/CT /AS/D0/D8/CT/D6/B8ψ /CX/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CW/CX/D6/D4/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BF/B5 /CX/D2 /D8/D3 /B4/BE/B5 /CV/CX/DA /CT/D7 /D8/CW/CT /D7/CT/D8 /D3/CU /D7/CX/DC /CP/D0/CV/CT/CQ/D6/CP/CX /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CB/D3/D0/D9/D8/CX/D3/D2 /D3/CU/D8/CW/CX/D7 /D7/CT/D8 /CV/CX/DA /CT/D7 /CT/DC/D4/D0/CX /CX/D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D7/CX/DC /D9/D2/CZ/D2/D3 /DB/D2a0 /B8tp /B8ψ /B8ω /B8δ /CP/D2/CSφ /B8/CW/D3 /DB /CT/DA /CT/D6 /D8/D3 /D3 /D0/D9/D1/D7/DD /D8/D3 /DB/D6/CX/D8/CT /D8/CW/CT/D1 /D3/D9/D8 /CW/CT/D6/CT/BA/BF/CC /D3 /D6/CT/D0/CP/D8/CT /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D3/D9/D6 /D1/D3 /CS/CT/D0 /D8/D3 /D8/CW/D3/D7/CT /D3/D2 /D8/D6/D3/D0/D0/CP/CQ/D0/CT /CX/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/DB /CT /CW/CP /DA /CT /D8/D3 /D6/CT /CP/D0 /D9/D0/CP/D8/CT /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /D0/D3/D7/D7 /CP/D8 /D4/D9/D0/D7/CT /D8/CW/CT /D4 /CT/CP/CZγ0 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8/D8/D3 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /D0/D3/D7/D7Γ /BMγ0=e−E/2, /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D0/D3/D7/D7 /D6/CT /D3 /DA /CT/D6/DD/D8/CX/D1/CT /CX/D7 /D1 /D9 /CW /D7/CW/D3/D6/D8/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /CP /DA/CX/D8 /DD /D4 /CT/D6/CX/D3 /CSTcav /CP/D2/CS /D1 /D9 /CW /D0/D3/D2/CV/CT/D6 /D8/CW/CP/D2 /D8/CW/CT/D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/BA /C0/CT/D6/CTE=∞/integraltext −∞\|a\|2dt /CX/D7 /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /BA /BT/D0/D8/CW/D3/D9/CV/CW /CX/D2 /D8/CW/CT/D8 /DD/D4/CX /CP/D0 /CU/CT/D1 /D8/D3/D7/CT /D3/D2/CS /D0/CP/D7/CT/D6/D7 /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /CX/D7 /D1 /D9 /CW /CQ/CX/CV/CV/CT/D6 /D8/CW/CP/D2 /D8/CW/CT/D0/D3/D7/D7 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /B4/CX/D2 /D3/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 τ= 0.0015 /B8 /DB/CW/CX /CW /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3/D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /AT/D9/DC /D3/CU100µJ/cm2/B8 /CC/CX/BM/D7/CP/D4/D4/CW/CX/D6/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CP/D2/CS/D8/CW/CT /CQ /CT/CP/D1 /D6/CP/CS/CX/CX /D3/CU30 /CP/D2/CS106µm /CP/D8 /D8/CW/CT /CP /D8/CX/DA /CT /D6/DD/D7/D8/CP/D0 /CP/D2/CS /CP/D8 /D8/CW/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/B8/D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD/B5/B8 /D3/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /CW/CP /DA /CT /D7/CW/D3 /DB/D2/B8 /D8/CW/CP/D8 /D8/CW/CT /CQ/CP/D0/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D7/CT/D8 /DB /D3 /CU/CP /D8/D3/D6/D7 /D2/D3/D8/CX /CT/CP/CQ/D0/DD /CP/AR/CT /D8/D7 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /CC/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS/CV/CP/CX/D2α0 /CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CP/D7 /CU/D3/D0/D0/D3 /DB/BMα0=αm1−exp(−U) 1−exp(−U−τE)exp(−τE/2) /B8 /DB/CW/CT/D6/CT U /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3σ14Tcav hν /B8hν /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /D4/CW/D3/D8/D3/D2 /CT/D2/CT/D6/CV/DD /B8 σ14 /CX/D7 /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D6/D3/D7/D7 /D7/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8αm /CX/D7 /D8/CW/CT /CV/CP/CX/D2 /CP/D8/CU/D9/D0/D0 /CX/D2 /DA /CT/D6/D7/CX/D3/D2/BA /CC/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CV/CX/DA /CT/D7 /CP/D2 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6 /CS/CT/D8/CT/D6/D1/CX/D2/CX/D2/CV/D9/D2/CZ/D2/D3 /DB/D2 /D7/DD/D7/D8/CT/D1/B3/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6α0 /BA/BE/BA/BE /CD/D0/D8/D6/CP/B9/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D7 /CP/D2/CS /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD/C7/D9/D6 /CP/D2/CP/D0/DD/D7/CX/D7 /D7/CW/D3 /DB /CT/CS/B8 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /CB/C8/C5 /CX/D2 /D8/CW/CT /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B4p/ne}ationslash= 0 /CX/D2 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/D2/D3/D2/D0/CX/D2/CT/CP/D6 /D6/CT/CU/D6/CP /D8/CX/D3/D2 /B4χ/ne}ationslash= 0 /CX/D2 /CT/D5/BA /B4/BE/B5/B5/AJ/D1/CX/DC/CT/D7/AK /D8/CW/CT/D7/CT /D8 /DB /D3 /D7/D8/CP/D8/CT/D7/B8 /D7/D3 /D8/CW/CP/D8 /D8/CW/CT /CW/CX/D6/D4 /D3/D1/D4 /CT/D2/D7/CP/D8/CX/D3/D2 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D3/D2/D0/DD /CU/D3/D6/D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D0/CP/D6/CV/CT /D4/D9/D1/D4 /B4/BY/CX/CV/BA /BD/B8 /D9/D6/DA /CT/D7 /BE/B8 /BF/B5 /CP/D2/CS /CU/D3/D6 /D8/CW/CT /D7/CW/D3/D6/D8/CT/D7/D8 /D4/D9/D0/D7/CT /D8/CW/CT /CW/CX/D6/D4/D6/CT/D1/CP/CX/D2/D7 /D9/D2 /D3/D1/D4 /CT/D2/D7/CP/D8/CT/CS /B4/BY/CX/CV/BA /BF/B8 /D9/D6/DA /CT/D7 /BE/B8 /BF/B5/BA /CC/CW/CT /CU/CT/CP/D8/D9/D6/CT/D7 /CX/D2 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6/D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D8/CW/CT /D0/CX/D2/CT/DB/CX/CS/D8/CW /CT/D2/CW/CP/D2 /CT/D1/CT/D2 /D8 /CP/D6/CT /D8/CW/CT/CQ/D6/D3/CP/CS/CT/D2/CX/D2/CV /D3/CU /D8/CW/CT /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6 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/CT/BA/C6/D3 /DB /DB /CT /CW/CP /DA /CT /D8/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CQ/D8/CP/CX/D2/CT/CS/BA /CC/CW/CT/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /D7/D1/CP/D0/D0 /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /B4/CX/BA/CT/BA /CP/D1/D4/D0/CX/D8/D9/CS/CT/B8/CS/D9/D6/CP/D8/CX/D3/D2/B8 /CU/D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8/B8 /CW/CX/D6/D4 /CP/D2/CS /CT/D2/CT/D6/CV/DD/B5 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /CP/D4 /CT/D6/D8/D9/D6/CQ /CT/CS /D7/D3/D0/D9/D8/CX/D3/D2 /B4/BF/B5 /CX/D2 /D8/D3 /CT/D5/BA /B4/BE/B5 /CP/D2/CS /CT/DC/D4/CP/D2/CS/CX/D2/CV /CX/D8 /CX/D2 /D8/D3 /D8/CW/CT /D8/CX/D1/CT /D7/CT/D6/CX/CT/D7 /DB /CT/CV/CT/D8 /CP/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D7/CT/D8 /CU/D3/D6 /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BM da0 dk=a0[α0−γ0−l−ω2(1−γ0)−υ(1−γ0−dψ)], /B4/BG/B5 dω dk=a2 0[γ0(χ−ω−ψ−χωψ) +α0a2 0ψτ+ 2υω(γψ2+γ0−ψ2−1)], /B4/BH/B5 dυ dk=a4 0(γ0−α0τ2) + 2a2 0γ0υ(χψ−1) + 2υ2(3dψ+ψ2+γ0−γ0ψ−1), /B4/BI/B5 dψ dk=−2a2 0(γ0χ+p+γ0χψ2) +a4 0 υ(γ0χ−γ0ψ+α0τ2χ) + /B4/BJ/B5 2υ(γ0ψ2+ψγ0−2dψ2−ψ2−2d),/DB/CW/CT/D6/CTυ=1 t2p /BA/CC /D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /CT/D2/CT/D6/CV/DD /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2/DB /CT /CU/D3/D0/D0/D3 /DB /CT/CS /D8/CW/CT /D7 /CW/CT/D1/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /CJ/BD/BC℄/BM /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D3/CU /CT/D5/BA /B4/BE/B5 /CP/D2/CS /D7/D9/D1/D1/CX/D2/CV/D9/D4 /DB/CX/D8/CW /CX/D8/D7 /D3/D1/D4/D0/CT/DC /D3/D2/CY/D9/CV/CP/D8/CT /CV/CX/DA /CT/D7 /D8/CW/CT /CT/D2/CT/D6/CV/DD /D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB/BA /BY /D6/D3/D1 /D8/CW/CX/D7/CX/D2 /D8/CT/CV/D6/CP/D0 /D3/CU /D1/D3/D8/CX/D3/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CS/CT /CP /DD /D3/CU /D8/CW/CT /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT/CT/D2/CT/D6/CV/DD /CU/D3/D0/D0/D3 /DB/D7/BM −α0τeτE 2(1 +e−τE) +l+ (1 +e−E)− /B4/BK/B5 2√υ 3(1−γ0)(1 +ψ2+ 3ω2/υ)−2 3γ0a2 0(χψ−1)<0/C0/CT/D6/CT /DB /CT /CP/D7/D7/D9/D1/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D0/D3/D7/D7 /CP/D2/CS /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D3/CQ /CT/DD /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2/B9/D8/CX/CP/D0 /D0/CP /DB/B8 /DB/CW/CX /CW /CX/D7 /D8/CW/CT /CP/D7/CT /CU/D3/D6 /D5/D9/CP/D7/CX/B9/D8 /DB /D3 /D0/CT/DA /CT/D0 /D7/DD/D7/D8/CT/D1 /DB/CX/D8/CW /CP /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D8/CX/D1/CT/D1 /D9 /CW /D0/D3/D2/CV/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/BA /C7/D9/D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CP/D0/DD/D7/CX/D7 /CX/D7 /CP /D1/D3 /CS/CX/AS /CP/D8/CX/D3/D2/D3/CU /D8/CW/CT /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /D0/CP/D7/CT/D6 /D2/D3/CX/D7/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /CJ/BD/BC℄ /DB/CX/D8/CW /CT/D2/CT/D6/CV/DD/D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /D4/D0/CP /DD/CX/D2/CV /D8/CW/CT /D6/D3/D0/CT /D3/CU /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CQ /DD /D8/CW/CT /D2/D3/CX/D7/CT /D3/D2 /D8/CX/D2 /D9/D9/D1/BA/C6/CT/CV/CP/D8/CX/DA /CT /D6/CT/CP/D0 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /C2/CP /D3/CQ /CT/CP/D2 /D3/CU /D8/CW/CT /CT/D5/D7/BA /D7/CT/D8 /B4/BG /B9 /BK/B5 /CX/D7 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /BA /C1/D8 /CX/D7 /D7/CT/CT/D2 /CU/D6/D3/D1 /CT/D5/D7/BA /B4/BG/B5 /CP/D2/CS /B4/BK/B5/B8 /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD/BH/D7/CW/CX/CU/D8/B8 /CW/CX/D6/D4 /CP/D2/CS /D7/D4 /CT /D8/D6/CP/D0 /AS/D0/D8/CT/D6/CX/D2/CV /B4/CU/D3/D6/D8/CW /D8/CT/D6/D1 /CX/D2 /CT/D5/BA /B4/BK/B5/B5 /D7/D8/CP/CQ/CX/D0/CX/DE/CT /D8/CW/CT /D4/D9/D0/D7/CT/CP/CV/CP/CX/D2/D7/D8 /CT/D2/CT/D6/CV/DD /CP/D2/CS /CP/D1/D4/D0/CX/D8/D9/CS/CT /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2/D7/B8 /D8/CW/CT /AJ/D7/D0/D3 /DB/AK/B4/AS/D6/D7/D8 /D8/CT/D6/D1 /CX/D2 /CT/D5/BA /B4/BH/B5/CP/D2/CS /D7/CT /D3/D2/CS /D8/CT/D6/D1 /CX/D2 /CT/D5/BA /B4/BJ/B5/B5 /CP/D2/CS /D8/CW/CT /AJ/CU/CP/D7/D8/AK /B4/AS/D6/D7/D8 /D8/CT/D6/D1 /CX/D2 /CT/D5/BA /B4/BJ/B5/B5 /CB/C8/C5 /D7/D8/CP/CQ/CX/B9/D0/CX/DE/CT /D8/CW/CT /D4/D9/D0/D7/CT /CP/CV/CP/CX/D2/D7/D8 /CU/D6/CT/D5/D9/CT/D2 /DD /CP/D2/CS /CW/CX/D6/D4 /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2/D7/BA /CC/CW/CX/D7 /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2/CX/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /CQ /DD /D2/CT/CV/CP/D8/CX/DA /CT /CU/CT/CT/CS/CQ/CP /CZ /CS/D9/CT /D8/D3 /D4/D9/D0/D7/CT /CW/CX/D6/D4/CX/D2/CV /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8 /CX/D2/D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /AS/D2/CX/D8/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CP/D2/CS /AS/D0/D8/CT/D6 /CQ/CP/D2/CS/DB/CX/CS/D8/CW/D7/BA/BV/D3/D2/CS/CX/D8/CX/D3/D2 /B4/BK/B5 /CX/D7 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /D0/CP/D7/CT/D6 /D2/D3/CX/D7/CT/CV/CT/D2/CT/D6/CP/D8/CT/CS /DB/CX/D8/CW/CX/D2 /D8/CW/CT /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /CP/D2/CS /CX/D8 /D7/CW/D3/D9/D0/CS /CQ /CT /D3/D1/D4/D0/CT/D8/CT/CS /CQ /DD /D8/CW/CT/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CP/D0/DD/D7/CX/D7 /CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D2/D3/CX/D7/CT /CV/CT/D2/CT/D6/CP/D8/CT/CS /CQ /CT/CW/CX/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /DB/CX/D8/CW/CX/D2 /D8/CW/CT/DB/CX/D2/CS/D3 /DB /D3/CU /D8/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT /D2/CT/D8/B9/CV/CP/CX/D2/BA /BT/D7 /DB /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /CJ/BD/BD℄/B8 /D8/CW/CT /D1/CP/CX/D2 /D7/D8/CP/CQ/CX/D0/CX/DE/B9/CX/D2/CV /CU/CP /D8/D3/D6 /CX/D2 /D8/CW/CX/D7 /CP/D7/CT /CX/D7 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CX/D2 /D8/CW/CT /CV/D6/D3/D9/D4 /DA /CT/D0/D3 /CX/D8/CX/CT/D7 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT/CP/D2/CS /D8/CW/CT /D2/D3/CX/D7/CT/BA /CC/CW/CT /CS/CT/D0/CP /DD /D3/CU /D8/CW/CT /D2/D3/CX/D7/CT /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT /D2/CT/D8/B9/CV/CP/CX/D2/DB/CX/D2/CS/D3 /DB /D4/D6/D3 /CS/D9 /CT/D7 /CP/D2 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D0/D3/D7/D7 /CU/D3/D6 /CX/D8 /CP/D2/CS/B8 /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /B8 /D7/D8/CP/CQ/CX/D0/CX/DE/CT/D7 /D8/CW/CT/D4/D9/D0/D7/CT/BA /BT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CX/D7 /D3/D2/CS/CX/D8/CX/D3/D2 /D0/CT/CP/CS/D7 /D8/D3/CT/DA /D3/D0/D9/D8/CX/D3/D2/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /DB/B9 /D2/D3/CX/D7/CT /DB/CX/D8/CW /CX/D2 /D8/CT/D2/D7/CX/D8 /DDN /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D2/D3/CX/D7/CT /D3/D2/D7/CX/D7/D8/D7 /D3/CU /D7/D4 /D3/D2 /D8/CP/D2/CT/D3/D9/D7 /D7/D4/CX/CZ /CT/D7 /DB/CX/D8/CW /CS/D9/D6/CP/D8/CX/D3/D2/D7 /D1 /D9 /CW /D0/D3/D2/CV/CT/D6 /D8/CW/CP/D2tf /BM dN(k,t) dk=/braceleftBigg α0e−τE/2−l−V−δ∂V ∂t/bracerightBigg N(k,t), /B4/BL/B5/DB/CW/CT/D6/CTV= (1 + (e−E−1)e−t/Ta) /CX/D7 /AG /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/AH /D6/CT/CP/D8/CT/CS /CQ /DD /D8/CW/CT /D4/D9/D0/D7/CT/B8/D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /CS/CT/D7 /D6/CX/CQ /CT/D7 /D8/CW/CT /D8/CX/D1/CT /D7/CW/CX/CU/D8 /D3/CU /D8/CW/CT /D2/D3/CX/D7/CT /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /D4/D9/D0/D7/CT/BA/BY /D6/D3/D1 /D8/CW/CX/D7 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CS/CT /CP /DD /D3/CU /D8/CW/CT /D2/D3/CX/D7/CT /CT/D2/CT/D6/CV/DD /D6/CT/D7/D9/D0/D8/D7/BM α0e−τE 2−l−e−E+δ Ta(1−e−E)<0. /B4/BD/BC/B5/CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /CT/D5/BA /B4/BE/B5 /DB/CW/CX /CW /CX/D7 /D7/D8/CP/CQ/D0/CT /CP/CV/CP/CX/D2/D7/D8 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2/D7/B4/CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD/B5 /CP/D6/CT /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/D7/BA /BD /B9 /BF /CQ /DD /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/BA /C1/D8 /D7/CW/D3/D9/D0/CS/CQ /CT /D2/D3/D8/CT/CS/B8 /D8/CW/CP/D8 /D8/CW/CT /D7/D3/D0/CX/D8/D3/D2 /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV /D1/CT /CW/CP/D2/CX/D7/D1 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/D8/BT/D2 /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D0/CX/D2/CT/DB/CX/CS/D8/CW /CT/D2/CW/CP/D2 /CT/D1/CT/D2 /D8 /CX/D7 /D2/D3/D8 /CP /D7/CX/D2/CV/D0/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D1/CT /CW/CP/D2/CX/D7/D1/CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/D7 /D8/CW/CP/D8 /D1/CP /DD /D3/D2 /D8/D6/CX/CQ/D9/D8/CT /D8/D3 /D1/D3 /CS/CT/B9/D0/D3 /CZ/CX/D2/CV/BA /BX/D7/D8/CX/D1/CP/D8/CX/D3/D2/D7 /D7/CW/D3 /DB/D8/CW/CP/D8 /D8/CW/CT /CB/D8/CP/D6/CZ /D7/CW/CX/CU/D8 /D3/CU /D8/CW/CT /CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D7/D8/D6/D3/D2/CV/D0/CP/D7/CT/D6 /AS/CT/D0/CS /CP/D2 /CQ /CT /CP /D7/D8/D6/D3/D2/CV /D4/D9/D0/D7/CT/B9/D7/CW/CP/D4/CX/D2/CV /CU/CP /D8/D3/D6/B8 /D8/D3 /D3 /CJ/BH/B8 /BD/BE℄/BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 /CP/D2/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D9/D8/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CB/D8/CP/D6/CZ /CT/AR/CT /D8 /CS/D9/CT /D8/D3 /CT/DC/D8/CT/D6/D2/CP/D0 /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /DB /CP/D7/D7/D9 /CT/D7/D7/CU/D9/D0/D0/DD /D9/D7/CT/CS /CU/D3/D6 /CP /D8/CX/DA /CT /D1/D3 /CS/CT/B9/D0/D3 /CZ/CX/D2/CV /D3/CU /CS/CX/D3 /CS/CT/B9/D4/D9/D1/D4 /CT/CS /C6/CS/BM /CH /BT /BZ /D0/CP/D7/CT/D6/CJ/BD/BF℄/BA/BF/BA/BD /C5/D3 /CS/CT/D0/C9/D9/CP/CS/D6/CP/D8/CX /CP /CB/D8/CP/D6/CZ /D7/CW/CX/CU/D8∆ω /D8/CW/CP/D8 /CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /CX/D2/AT/D9/CT/D2 /CT /D3/CU /D2/D3/D2/D6/CT/D7/D3/D2/CP/D2 /D8 /D8/D6/CP/D2/B9/D7/CX/D8/CX/D3/D2/D7 /D3/D2 /CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT /CX/D7 /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3 /D8/CW/CT /D4 /D3/D0/CP/D6/CX/DE/CP/CQ/CX/D0/CX/D8 /DD /CS/CX/AR/CT/D6/CT/D2 /CT/BJ/CQ /CT/D8 /DB /CT/CT/D2 /CV/D6/D3/D9/D2/CS /CP/D2/CS /CT/DC /CX/D8/CT/CS /D7/D8/CP/D8/CT/D7 ∆α /CJ/BD/BG℄/BM∆ω=\|∆α\| × \|E\|2/h /B8 /DB/CW/CT/D6/CTE/CX/D7 /D8/CW/CT /AS/CT/D0/CS /D7/D8/D6/CT/D2/CV/D8/CW/BA /CC/CW/CT /D8 /DD/D4/CX /CP/D0 /DA /CP/D0/D9/CT/D7 /D3/CU\|∆α\| /CU/D3/D6 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/D7 /CP/D6/CT /D3/CU/D3/D6/CS/CT/D6 10−19÷10−21cm3/CJ/BD/BH℄/B8 /D8/CW/CP/D8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CB/D8/CP/D6/CZ /D7/CW/CX/CU/D8 /D3 /CTꜶ /CX/CT/D2 /D8 ζ= 8π\|∆α\|/(nch) = (105÷103)/ncm2J−1/B8 /DB/CW/CT/D6/CTn /CX/D7 /D8/CW/CT /CX/D2/CS/CT/DC /D3/CU /D6/CT/CU/D6/CP /D8/CX/DA/B9/CX/D8 /DD /BA /C6/CT/CV/D0/CT /D8/CX/D2/CV /D8/CW/CT /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /D3/CU /CW/CX/CV/CW/CT/D6 /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /D3/CU /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/D2/CS/D9/D2/CS/CT/D6 /DB /CT/CP/CZ /CT/DC /CX/D8/D3/D2/B9/CT/DC /CX/D8/D3/D2/B8 /CT/DC /CX/D8/D3/D2/B9/D4/CW/D3/D2/D3/D2 /CQ /D3/D9/D2/CS /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B8 /D3/D2/CT /CP/D2/CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D0/CP/D7/CT/D6 /AS/CT/D0/CS /CP/D2/CS /D5/D9/CP/D2 /D8/D9/D1/B9 /D3/D2/AS/D2/CT/CS /CP/CQ/D7/D3/D6/CQ /CT/D6/CQ /DD /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/CS /D8 /DB /D3/B9/D0/CT/DA /CT/D0 /D1/D3 /CS/CT/D0 /CJ/BD/BG℄/BA /CC/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/AR/B9/CS/CX/CP/CV/D3/D2/CP/D0 /CT/D0/B9/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD /D1/CP/D8/D6/CX/DC Π /CP/D2/CS /D8/CW/CT /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CS/CX/AR/CT/D6/CT/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /CV/D6/D3/D9/D2/CS/CP/D2/CS /CT/DC /CX/D8/CT/CS /D7/D8/CP/D8/CT/D7 Ξ /D3/CQ /CT/DD /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /D7/CT/D8/BM dΠ dt+ [1 ta−i(ωl−ωa−∆ω)]Π =i h℘Ξ, /B4/BD/BD/B5 dΞ dt+Ξ−Ξ0 T=−4 hIm(Π℘∗),/DB/CW/CT/D6/CTta /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW /D3/CU /D8/CW/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D0/CX/D2/CT/B8ωa /CX/D7 /D8/CW/CT /D6/CT/D7/D3/B9/D2/CP/D2 /CT /CU/D6/CT/D5/D9/CT/D2 /DD /B8ωl /CX/D7 /D8/CW/CT /AS/CT/D0/CS /CP/D6/D6/CX/CT/D6 /CU/D6/CT/D5/D9/CT/D2 /DD /B8℘ /CX/D7 /D8/CW/CT /D1/CP/D8/D6/CX/DC /CT/D0/CT/D1/CT/D2 /D8 /D3/CU/D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2/B8 Ξ0 /CX/D7 /D8/CW/CT /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CS/CX/AR/CT/D6/CT/D2 /CT/BA /CC/CW/CT /CB/D8/CP/D6/CZ /D7/CW/CX/CU/D8/DB/CW/CX /CW /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D3/D2/D0/DD /CX/D2 /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/CS /D8 /DB /D3/B9/D0/CT/DA /CT/D0 /D1/D3 /CS/CT/D0/B8 /CX/D2 /D5/D9/CP/D7/CX/B9/D1/D3/D2/D3 /CW/D6/D3/D1/CP/D8/CX /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B8 /CX/BA /CT/BA /D9/D2/CS/CT/D6 /D2/CT/CV/D0/CT /D8/CX/D2/CV /D8/CW/CT /D6/D3/D7/D7/B9/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CS/CX/AR/CT/D6/CT/D2 /D8/D4/D9/D0/D7/CT /D7/D4 /CT /D8/D6/CP/D0 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/B8 /CX/D7 /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3ς\|a(t)\|2/BA/C1/D2 /D8/CW/CT /D2/D3/D2 /D3/CW/CT/D6/CT/D2 /D8 /CP/D4/D4/D6/D3/CP /CW/B8 /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/D1/D4/D0/CT/DC /AS/CT/D0/CSa(t) /CX/D2/D8/CW/CT /D0/CP/D7/CT/D6 /D7/DD/D7/D8/CT/D1 /D3/D2 /D8/CP/CX/D2/CX/D2/CV /D8/CW/CT /CV/CP/CX/D2 /D1/CT/CS/CX/D9/D1/B8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/B8 /CU/D6/CT/D5/D9/CT/D2 /DD/AS/D0/D8/CT/D6 /CP/D2/CS /CS/CX/D7/D4 /CT/D6/D7/CX/DA /CT /CT/D0/CT/D1/CT/D2 /D8 /D3/CQ /CT/DD/D7 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3/D4 /CT/D6/CP/D8/D3/D6/B3/D7 /CT/D5/D9/CP/D8/CX/D3/D2/BMak+1(t) = ˜Γ˜G˜Dak(t), /DB/CW/CT/D6/CTk /CX/D7 /D8/CW/CT /D6/D3/D9/D2/CS /D8/D6/CX/D4 /D2 /D9/D1 /CQ /CT/D6/B8t /CX/D7 /D8/CW/CT /D0/D3 /CP/D0 /D8/CX/D1/CT/BA /CC/CW/CT /C4/D3/D6/CT/D2/B9/DE/CX/CP/D2 /CV/CP/CX/D2 /CQ/CP/D2/CS /CP /D8/CX/D3/D2 /CX/D7 /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD˜G= exp(αLg 1+Lgtg∂ ∂t) /B8Lg=1 1+i(ωl−ωg)tg/DB/CW/CT/D6/CTωg /CX/D7 /D8/CW/CT /CV/CP/CX/D2 /D6/CT/D7/D3/D2/CP/D2 /CT/B8 tg /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /CV/CP/CX/D2 /CQ/CP/D2/CS/DB/CX/CS/D8/CW /B4/CU/D3/D6 /BV/D6/BM/CU/D3/D6/D7/D8/CT/D6/CX/D8/CT /CW/D3/D7/CT/D2 /CU/D3/D6 /D3/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7/B8 tg= 20fs /B5/B8α /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2/BA ˜D= exp(id∂2 ∂t2) /CX/D7 /D8/CW/CT /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /BZ/CE/BW /CV/D6/D3/D9/D4 /DA /CT/D0/D3 /CX/D8 /DD /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/B8/DB/CW/CT/D6/CTd /CX/D7 /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CP/D1/D3/D9/D2 /D8 /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS /CQ /DD /D8/CW/CT /D4/D6/CX/D7/D1 /D4/CP/CX/D6/BA /BY /D6/D3/D1 /CT/D5/BA/B4/BD/BD/B5 /D3/D2/CT /CS/CT/D6/CX/DA /CT/D7 /CP/D2 /D3/D4 /CT/D6/CP/D8/D3/D6 /CP /D3/D9/D2 /D8/CX/D2/CV /CU/D3/D6 /D8/CW/CT /CT/AR/CT /D8/D7 /D3/CU /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/B9/D8/CX/D3/D2 /CP/D2/CS /D2/D3/D2/D0/CX/D2/CT/CP/D6 /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /CB/D8/CP/D6/CZ/B9/CT/AR/CT /D8 Γ = exp−γLaexp[−Re(La)t/integraltext t0\|a(t′)\|2exp(−t−t′ T)dt′/Ua] 1+Lata∂ ∂t−l /B8 /DB/CW/CT/D6/CT La=1 1+i[ωl−(ωa+µ\|a(t)\|2)]ta /B8Ua /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CT/D2/CT/D6/CV/DD /B8γ /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS/D0/D3/D7/D7 /CP/D8 /D8/CW/CT /D1/D3/D1/CT/D2 /D8t0 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D4/D9/D0/D7/CT /D4 /CT/CP/CZ/B8l /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D0/D3/D7/D7/BA/CC/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D2Γ /CP /D3/D9/D2 /D8/D7 /CU/D3/D6 /D8/CW/CT /D3/D6/CS/CX/D2/CP/D6/DD /D7/D0/D3 /DB /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /DB/CX/D8/CW/D6/CT /D3 /DA /CT/D6/DD /D8/CX/D1/CTT /BA /C0/CT/D6/CT /DB /CT /D9/D7/CT/CS /D8/CW/CT /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /CP/D7 /CX/D2 /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT /D8/CX/D3/D2/B8 /D7/D3/D8/CW/CP/D8 /CU/D3/D6 /C8/CQ/CB /AL /CQ/CP/D7/CT/CS /D1/D3 /CS/D9/D0/CP/D8/D3/D6 /DB/CX/D8/CWUa= 390µJ/cm2/CJ/BD/BI℄µ=ζUa= 13 /BA/BK/BT/D2 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /D0/CP/D7/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D2 /D8/D3 /D7/CT/D6/CX/CT/D7 /D3/D2t /B8 /D0/D3 /CP/D0 /AS/CT/D0/CS /CT/D2/CT/D6/CV/DD /CP/D2/CS/CX/D2 /D8/CT/D2/D7/CX/D8 /DD /B8 /D4/D6/D3 /DA/CX/CS/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D1 /D9 /CW /D7/CW/D3/D6/D8/CT/D6 /D8/CW/CP/D2T /B8 /CV/CX/DA /CT/D7/D8/CW/CT /D0/CP/D7/CT/D6 /CS/DD/D2/CP/D1/CX /CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/CS /C4/CP/D2/CS/CP/D9/B9/BZ/CX/D2/DE/CQ/D9/D6/CV/CT/D5/D9/CP/D8/CX/D3/D2/BM ∂a(k,t) ∂k= [c1+ic2∂ ∂t+ (c3+ic4)∂2 ∂t2+ /B4/BD/BE/B5 (c5+ic6)\|a(k,t)\|2+ (c7+ic8)ǫ+ (c9+ic10)ǫ2 2+ (c11+ic12)ǫ∂ ∂t]a(k,t),/DB/CW/CT/D6/CTǫ=t/integraltext t0\|a(k,t′)\|2dt′/B8c1=αJg−γJa−l /B8c2= 2αΩJ2 g−2γωϑ2J2 a /B8c3= (1−3Ω2)J3 g−γ(1−3ω2ϑ2)J3 aϑ2/B8c4=α(Ω3−3Ω)J3 g−γ(ω3ϑ3−3ωϑ)J3 aϑ2+d /B8 c5=−2γωϑµHa /B8c6=−γµ(1−ω2ϑ2)Ha /B8c7=γJ2 2 /B8c8=−γωϑJ2 a /B8c9= −γJ3 a /B8c10=γωϑJ3 a /B8c11=−γ(1−ϑ2ω2)J3 aϑ /B8c12= 2ωγϑ2J3 a /B8Ja=1 1+ω2ϑ2 /B8 Jg=1 1+Ω2 /B8Ha=1 (1−ω2ϑ2)2+4ω2ϑ2 /B8ω=ωl−ωa /B8Ω =ωl−ωg /B8ϑ=ta/tg /BA /CF /CT/D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CP/D0/D0 /D8/CX/D1/CT/D7 /D8/D3tg /B8 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D8/D3t−1 g /B8 /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8 /D8/D3t2 g /B8 /D8/CW/CT/CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D8/D3Ua tg /BA/BF/BA/BE /CB/D8/CP/D6/CZ /CX/D2/CS/D9 /CT/CS /D1/D3 /CS/CT /D0/D3 /CZ/CX/D2/CV/BT/D7 /D3/D2/CT /CP/D2 /D7/CT/CT/B8 /BX/D5/BA /B4/BD/BE/B5 /D3/D2 /D8/CP/CX/D2/D7 /D8/CW/CT /D8/CT/D6/D1 −2µγHaωϑ\|a(k,t)\|2a(k,t) /B8/DB/CW/CX /CW /CX/D7 /D6/CT/D7/D4 /D3/D2/D7/CX/CQ/D0/CT /CU/D3/D6 /D8/CW/CT /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CP/D8 /D8/CW/CT /CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/B9/CV/CP/D4/CT/DC /CX/D8/CP/D8/CX/D3/D2 /DB/CX/D8/CWω <0 /B4/CX/BA /CT/BA /D6/CT/CS /D7/CW/CX/CU/D8 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /CP/D6/D6/CX/CT/D6 /CU/D6/CT/D5/D9/CT/D2 /DD /CU/D6/D3/D1 /D8/CW/CT/CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT/B5/BA /CC/CW/CT /D7/CW/CX/CU/D8ω /CX/D7 /CS/D9/CT /D8/D3 /CS/CT/D8/D9/D2/CX/D2/CV /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CV/CP/CX/D2 /CP/D2/CS/D0/D3/D7/D7 /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /B4ωg−ωa<0 /B5/BA /CC/CW/CT /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /CP /D8/CX/D3/D2 /CX/D7 /CP/D9/D7/CT/CS/CQ /DD /D8/CW/CT /CX/D2/CS/D9 /CT/CS /AG/D4/D9/D7/CW/CX/D2/CV /D3/D9/D8/AH /D3/CU /D8/CW/CT /CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT /CU/D6/D3/D1 /D8/CW/CT /D6/CT/CS/B9/D7/CW/CX/CU/D8/CT/CS/D4/D9/D0/D7/CT /D7/D4 /CT /D8/D6/D9/D1 /CS/D9/CT /D8/D3 /D4 /D3 /DB /CT/D6/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CQ/D0/D9/CT /CB/D8/CP/D6/CZ /D7/CW/CX/CU/D8/BA /CC/CW/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CX/D7Is= [−2µγHata(ωl−ωa)]−1/BA /C7/D9/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /D7/CW/D3 /DB /CT/CS/B8/D8/CW/CP/D8 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CX/D7 /D0/D3/D7/CT /D8/D3 /D8/CW/CT /D0/CT/DA /CT/D0/B8 /DB/CW/CX /CW /CX/D7 /D8 /DD/D4/CX /CP/D0 /CU/D3/D6 /C3/CT/D6/D6/B9/D0/CT/D2/D7/D1/D3 /CS/CT/B9/D0/D3 /CZ /CT/CS /D0/CP/D7/CT/D6/D7 /CP/D2/CS /CX/D7 /CW/CX/CV/CW /CT/D2/D3/D9/CV/CW /D8/D3 /D4/D6/D3 /DA/CX/CS/CT /D7/CT/D0/CU/B9/D7/D8/CP/D6/D8/CX/D2/CV/BA/BX/D5/BA /B4/BD/BE/B5 /CW/CP/D7 /D5/D9/CP/D7/CX/B9/D7/D3/D0/D9/D8/CX/D3/D2 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/CS/D9/D6/CP/D8/CX/D3/D2/BA /CF/CX/D8/CW /D8/CW/CT/CP/CQ /D3 /DA /CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2/D7/B8 /CP/D2 /D3/D4 /CT/D6/CP/D8/D3/D6 /CU/D3/D6 /D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CU /D8/CW/CT /AS/CT/D0/CS /DB/CX/D8/CW /CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT /CX/D7 /B4/D7/CT/CT /D7/CT /D8/CX/D3/D2 /BF/BA/BD/B5/BM Γ =−Laexp[−ReLat/integraltext −∞\|a(t′)\|2dt′/Ua] 1 +Lata∂ ∂t /B4/BD/BF/B5/BD/BD/DB/CW/CT/D6/CTLa=1 1+iκta\|a(t′)\| /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/CQ/D0/CT /D0/D3/D7/D7/B8κ=√ς /B8ta /CX/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT/CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CQ/CP/D2/CS/DB/CX/CS/D8/CW/B8 t /CX/D7 /D8/CW/CT /D0/D3 /CP/D0 /D8/CX/D1/CT/BA/CA/CT/CP/D0 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6 /D1/CX/D6/D6/D3/D6/D7 /B4/CB/BX/CB/BT/C5/B3/D7/B5 /D3/D4 /CT/D6/CP/D8/CT /CX/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /D7/D8/D6/D3/D2/CV /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CJ/BE/BE℄/B8 /CP/D2/CS /D3/D2/CT /D1/CP /DD /D8/CW/CT/D6/CT/CU/D3/D6/CT /D2/CT/CV/D0/CT /D8 /D8/CW/CT /CS/DD/D2/CP/D1/B9/CX /CP/D0 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D2 /D8/D6/D3 /CS/D9 /CT /CP /D7/CP/D8/D9/D6/CP/D8/CT/CS /CQ /DD /D8/CW/CT /CU/D9/D0/D0 /D4/D9/D0/D7/CT /CT/D2/CT/D6/CV/DD /D0/D3/D7/D7γ /BM γ=Γ 1+ReL a∞/integraltext −∞\|a(t′)\|2dt′/Ua /BA/CC/CW /D9/D7/B8 /D8/CW/CT /CQ/CP/D7/CX /D4/D9/D0/D7/CT /D7/CW/CP/D4/CX/D2/CV /D1/CT /CW/CP/D2/CX/D7/D1 /CW/CT/D6/CT /CX/D7 /D8/CW/CP/D8 /D3/CU /D7/D3/D0/CX/D8/D3/D2 /CU/D3/D6/D1/CP/B9/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /CB/C8/C5 /CQ/CP/D0/CP/D2 /CT/CS /CQ /DD /D8/CW/CT /BZ/CE/BW/B8 /CQ/D9/D8 /D8/CW/CT /CB/C8/C5 /D2/D3 /DB /CW/CP/D7 /D8 /DB /D3 /D3/D1/B9/D4 /D3/D2/CT/D2 /D8/D7/B8 /D9/CQ/CX /CX/D2 /AS/CT/D0/CS /B4/C3/CT/D6/D6/B9/D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B5 /CP/D2/CS /D5/D9/CP/CS/D6/CP/D8/CX /CX/D2 /AS/CT/D0/CS /B4/CB/D8/CP/D6/CZ/B9/D7/CW/CX/CU/D8 /CX/D2/CS/D9 /CT/CS /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/B5/B8 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /CT/D7/D7/CT/D2/B9/D8/CX/CP/D0/D0/DD /D8/D6/CP/D2/D7/CU/D3/D6/D1/CX/D2/CV /D8/CW/CT /D2/CP/D8/D9/D6/CT /CP/D2/CS /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT/BA/BX/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /CT/D5/BA /B4/BD/BD/B5 /CX/D2 /D8/D3 /D8/CW/CT /D4 /D3 /DB /CT/D6 /D7/CT/D6/CX/CT/D7 /CX/D2 /D0/D3 /CP/D0 /D8/CX/D1/CTt /CP/D2/CS /AS/CT/D0/CSa /CP/D2/CS/D6/CT/D8/CP/CX/D2/CX/D2/CV /D8/CW/CT /D8/CT/D6/D1/D7 /D9/D4 /D8/D3 /D8/CW/CT /BE/D2/CS /D3/D6/CS/CT/D6 /DD/CX/CT/D0/CS/D7/BM ∂a(t,k) ∂k= [α−γ−l+iϕ]a(t,k) + /B4/BD/BG/B5 /bracketleftBigg (α−γ+id)∂2 ∂t2/bracketrightBigg a(t,k)∓ /bracketleftBig iγ\|a\|+ (γ−ip)\|a\|2/bracketrightBig a(t,k),/DB/CW/CT/D6/CTk /CX/D7 /D6/D3/D9/D2/CS/B9/D8/D6/CX/D4 /D2 /D9/D1 /CQ /CT/D6/B8α /CX/D7 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2/B8l /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6/D0/D3/D7/D7/B8ϕ /CX/D7 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CT/D0/CP /DD /B8d /CX/D7 /D8/CW/CT /BZ/CE/BW /CP/D1/D3/D9/D2 /D8/B8p /CX/D7 /D3 /CTꜶ /CX/CT/D2 /D8 /D3/CU /C3/CT/D6/D6/B9/D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD β=2πn2z λn /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3t2 aς /B4n2 /CP/D2/CSn /CP/D6/CT /D2/D3/D2/D0/CX/D2/CT/CP/D6 /CP/D2/CS /D0/CX/D2/CT/CP/D6/D6/CT/CU/D6/CP /D8/CX/D3/D2 /CX/D2/CS/CX /CT/D7 /CP/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8z /CX/D7 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/B8λ /CX/D7/D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/B5/B8 /D8/CW/CT /D7/CX/CV/D2/D7 /AG− /AG /CP/D2/CS /AG+ /AH /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CQ /CT/D0/D3 /DB /CP/D2/CS/CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/BA /CC/CW/CT /D7/CP/D8/D9/D6/CP/D8/CT/CS /CV/CP/CX/D2α /CX/D7 /CP/D0 /D9/D0/CP/D8/CT/CS /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT/CV/CP/CX/D2 /CP/D8 /CU/D9/D0/D0 /CX/D2 /DA /CT/D6/D7/CX/D3/D2 /CP/D7 α=αm1−exp(−U) 1−exp(−U−σ∞/integraltext −∞\|a\|2dt)exp(−τ∞/integraltext −∞\|a\|2dt/2) /B8 /DB/CW/CT/D6/CTU /CX/D7 /D8/CW/CT /D4/D9/D1/D4 /D4/CW/D3/D8/D3/D2/AT/D9/DC/B8 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3 /CP /DA/CX/D8 /DD /D4 /CT/D6/CX/D3 /CSTcav /CP/D2/CS /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D6/D3/D7/D7/B9/D7/CT /D8/CX/D3/D2 σ13 /B8τ−1= Ugςta /CX/D7 /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CT/D2/CT/D6/CV/DD /AT/D9/DC /D3/CU /D8/CW/CT /CV/CP/CX/D2 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/BA /C1/D2 /CT/D5/BA /B4/BD/BG/B5 /D8/CW/CT /D8/CX/D1/CT/CP/D2/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CP/D6/CT /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /D8/D3ta /CP/D2/CS /CP/D2/CSt2 aς /B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /CF /CT /CP/D7/D7/D9/D1/CT/CS /CP/D0/D7/D3/D8/CW/CP/D8 /D8/CW/CT /CV/CP/CX/D2 /D0/CX/D2/CT /CW/CP/D7 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE/CX/CP/D2 /D4/D6/D3/AS/D0/CT /CP/D2/CS /CX/D2 /DA /CT/D6/D7/CT /CQ/CP/D2/CS/DB/CX/CS/D8/CW /CX/D7 /CT/D5/D9/CP/D0 /D8/D3/D8/CW/CP/D8 /D3/CU /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/CQ/D7/D3/D6/CQ /CT/D6ta /BA /BY /D3/D6 /BV/D6/BM /CU/D3/D6/D7/D8/CT/D6/CX/D8/CT /D0/CP/D7/CT/D6 /B4ta= 20fs /B5 /DB/CW/CT/D6/CT/CP/D2 /D9/D0/D8/D6/CP/D7/CW/D3/D6/D8 /D4/D9/D0/D7/CT /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /D9/D7/CX/D2/CV /D8/CW/CT /CQ/D0/CT/CP /CW/CX/D2/CV /D3/CU /CT/DC /CX/D8/CP/D8/CX/D3/D2 /D6/CT/D7/D3/D2/CP/D2 /CT /CX/D2/C8/CQ/CB /D5/D9/CP/D2 /D8/D9/D1 /CS/D3/D8/D7 /CW/CP/D7 /CQ /CT/CT/D2 /D6/CT/D4 /D3/D6/D8/CT/CS /CJ/BD/BE℄ /DB/CX/D8/CW /D3/D9/D6 /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7 /CP/D2/CS /BF/B9/D1/D1/D0/D3/D2/CV /D6/DD/D7/D8/CP/D0 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 τ /CP/D2/CSp /CP/D6/CT /BC/BA/BC/BD /CP/D2/CS /BC/BA/BD/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA/C1/D8 /CX/D7 /D7/CT/CT/D2 /CU/D6/D3/D1 /CT/D5/BA /B4/BD/BG/B5/B8 /D8/CW/CP/D8 /D8/CW/CT /CB/D8/CP/D6/CZ /D7/CW/CX/CU/D8 /D3/CU /CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT /DB/CW/CX /CW/D1/CP /DD /CQ /CT /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /CP/D7 /D4/D9/D0/D7/CT/B9/CX/D2/CS/D9 /CT/CS /AG /D4/D9/D7/CW/CX/D2/CV/B9/D3/D9/D8/AH /D3/CU /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /CQ/CP/D2/CS /CU/D3/D6/D1/BD/BE/D8/CW/CT /D4/D9/D0/D7/CT /D7/D4 /CT /D8/D6/CP/D0 /D4/D6/D3/AS/D0/CT /CX/D2 /D8/D6/D3 /CS/D9 /CT/D7 /D2/D3/D2/D0/CX/D2/CT/CP/D6 /B4/D5/D9/CP/CS/D6/CP/D8/CX /B5 /D4/CW/CP/D7/CT /D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D7 /DB /CT/D0/D0 /CP/D7 /CU/CP/D7/D8 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /D0/D3/D7/D7 /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /BA /CC/CW/CT /D7/CX/CV/D2 /D3/CU /CB/C8/C5 /CX/D2/CP/CQ/D7/D3/D6/CQ /CT/D6 /D3/CX/D2 /CX/CS/CT/D7 /DB/CX/D8/CW /D8/CW/CP/D8 /D4/D6/D3 /CS/D9 /CT/CS /CQ /DD /C3/CT/D6/D6/B9/D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1/CU/D3/D6 /D8/CW/CT /CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2 /CP/D2/CS /CX/D7 /D3/D4/D4 /D3/D7/CX/D8/CT /CU/D3/D6 /D8/CW/CT /CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/B9/CV/CP/D4/CT/DC /CX/D8/CP/D8/CX/D3/D2/BA/C1/D2 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2 /CU/D3/D6 /CT/D5/BA /B4/BD/BG/B5 /CX/D7 /D9/D2/CZ/D2/D3 /DB/D2/B8 /D7/D3 /DB /CT /D7/CW/CP/D0/D0 /D3/D2/D7/CX/CS/CT/D6 /D8 /DB /D3/D0/CX/D1/CX/D8/CX/D2/CV /CP/D7/CT/D7 /D3/CU /CT/D5/BA /B4/BD/BG/B5/BM /CX/B5 /DB /CT/CP/CZ /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2 /AS/CT/D0/CS /CP/D2/CS /CX/CX/B5 /D7/D8/D6/D3/D2/CV /CV/CT/D2/CT/D6/CP/D8/CX/D3/D2/AS/CT/D0/CS/BA /C1/D2 /D8/CW/CT /AS/D6/D7/D8 /CP/D7/CT /D3/D2/CT /D1/CP /DD /D2/CT/CV/D0/CT /D8 /D8/CW/CT /D9/CQ/CX /CX/D2 /AS/CT/D0/CS /D8/CT/D6/D1/BM ∂a ∂k= [α−γ−l+iϕ]a+ /B4/BD/BH/B5 /bracketleftBigg (α−γ+id)∂2 ∂t2/bracketrightBigg a∓ iγ\|a\|a./C1/D2 /D8/CW/CT /D7/CT /D3/D2/CS /CP/D7/CT /D3/D2/CT /D1/CP /DD /D2/CT/CV/D0/CT /D8 /D8/CW/CT /D5/D9/CP/CS/D6/CP/D8/CX /CX/D2 /AS/CT/D0/CS /D8/CT/D6/D1/BM ∂a ∂k= [α−γ−l+iϕ]a+ /B4/BD/BI/B5 /bracketleftBigg (α−γ+id)∂2 ∂t2/bracketrightBigg a+ (γ−ip)\|a\|2a/BG/BA/BE /C6/D3/D2/B9/CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /CX/D2 /CP /D7/D3/D0/CX/CS/B9/D7/D8/CP/D8/CT /D0/CP/D7/CT/D6 /DB/CX/D8/CW/D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D7/CP/D8/D9/D6/CP/CQ/D0/CT /CP/CQ/D7/D3/D6/CQ /CT/D6/BX/D5/BA /B4/BD/BH/B5/B8 /DB/CW/CX /CW /CX/D7 /CP/D2/CP/D0/D3/CV/D9/CT/D7 /D8/D3 /D8/CW/CT /BY/CX/D7/CW/CT/D6/B9/CT/D5/D9/CP/D8/CX/D3/D2 /CJ/BE/BF℄/B8 /CW/CP/D7 /CP /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2/D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CU/D3/D6/D1/BM a(t) =a0sech2+iψ(t/tp), /B4/BD/BJ/B5/DB/CW/CT/D6/CT /D8/CW/CT /D4/D9/D0/D7/CT /CP/D1/D4/D0/CX/D8/D9/CS/CTa0 /B8 /D8/CW/CT /CW/CX/D6/D4ψ /CP/D2/CS /D8/CW/CT /CS/D9/D6/CP/D8/CX/D3/D2tp /D6/CT/D0/CP/D8/CT /CP/D7/CU/D3/D0/D0/D3 /DB/D7/BM ψ=5d±/radicalBig 25d2+ 24(α−γ)2 2(γ−α), /B4/BD/BK/B5 tp=/radicalBigg 2(α−γ)−dψ α−γ−l, a0=5ψ[(α−γ)2+d2] γt2p(γ−α),/BD/BF/CC/CW/CT /D7/CX/CV/D2/D7 /AG− /AH /CP/D2/CS /AG+ /AH /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT /D1/CT/CP/D2/CX/D2/CV /CP/D7 /CX/D2 /CT/D5/BA /B4/BD/BG/B5/BA/BX/D5/BA /B4/BD/BI/B5 /CX/D7 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D0/CP/D7/CT/D6 /CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /D7/D3/D0/CX/D8/D3/D2 /CV/CT/D2/CT/D6/CP/D8/CT/CS /CX/D2 /D8/CW/CT/D7/DD/D7/D8/CT/D1 /DB/CX/D8/CW /D8/CW/CT /CV/CP/CX/D2/B8 /D0/D3/D7/D7/B8 /C3/CT/D6/D6/B9/D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CP/D2/CS /BZ/CE/BW /CJ/BE/BG℄/BA /CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /CU/D3/D6/CT/D5/BA /B4/BD/BI/B5 /CX/D7 a(t) =a0sech1+iψ(t/tp), /B4/BD/BL/B5/DB/CX/D8/CW a0=/radicaltp/radicalvertex/radicalvertex/radicalbt(γ−α)(ψ2−2)−3dψ γt2 p, /B4/BE/BC/B5 tp=/radicalBigg (γ−α)(ψ2−1)−2dψ γ−α+l, ψ={3[dp+γ(γ−α)] +/radicalBig 9[dp+γ(γ−α)]2+ 8[p(γ−α)−γd]2}/ 2[p(γ−α)−γd]./C8/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CW/CX/D6/D4 /DA/D7 /BZ/CE/BW /CP/D6/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /BY/CX/CV/D7/BA /BK /CP/D2/CS /BL/BA /CB/D3/D0/CX/CS /D9/D6/DA /CT/D7 /BD /CP/D2/CS /BE /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /CP/D7/CT /D3/CU /DB /CT/CP/CZ /AS/CT/D0/CS /CP/D2/CS /CQ /CT/D0/D3 /DB /CP/D2/CS /CP/CQ /D3 /DA /CT/CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /BW/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /CP/D7/CT /D3/CU/D7/D8/D6/D3/D2/CV /AS/CT/D0/CS/BA/BY /D6/D3/D1 /BY/CX/CV/BA /BL /B4/D7/D3/D0/CX/CS /D0/CX/D2/CT /BD /CP/D2/CS /CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/B5 /D3/D2/CT /D1/CP /DD /D7/CT/CT /D8/CW/CP/D8 /CP/D8 /CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2 /D8/CW/CT /D7/CX/CV/D2/D7 /D3/CU /D8/CW/CT /CW/CX/D6/D4 /D4/D6/D3 /CS/D9 /CT/CS /CQ /DD /CB/D8/CP/D6/CZ/B9/D7/CW/CX/CU/D8 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/CP/D2/CS /C3/CT/D6/D6/B9/D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /CX/D2 /CP /D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /D3/CX/D2 /CX/CS/CT/BA /C1/D2 /D8/CW/CT /AS/D6/D7/D8 /D0/CX/D1/CX/D8/CX/D2/CV /CP/D7/CT/D8/CW/CX/D7 /D6/CT/CS/D9 /CT/D7 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D6/CT/CV/CX/D3/D2 /D3/CU /D2/CT/CV/CP/D8/CX/DA /CT /BZ/CE/BW/BA /BV/CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /CU/CT/CP/D8/D9/D6/CT/D7 /D3/CU /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8 /DD/D4 /CT /B4/BD/BJ/B5 /CP/D6/CT /D8/CW/CP/D8 /CX/B5 /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D0/D3/D7/CT /D8/D3/D8/CW/CT /D1/CX/D2/CX/D1/CP/D0 /D4 /D3/D7/D7/CX/CQ/D0/CTta /B8 /CP/D2/CS /D8/CW/CT /D2/CT /CT/D7/D7/CP/D6/DD /D4/D9/D1/D4 /D6/CP/D8/CT/D7 /CP/D6/CT /D1 /D9 /CW /D0/D3 /DB /CT/D6 /D8/CW/CP/D2/CU/D3/D6 /D4/D9/D6/CT /D5/D9/CP/D7/CX/B9/CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /D7/D3/D0/CX/D8/D3/D2/B8 /D8/CW/CP/D8 /CX/CX/B5 /CU/D9/D0/D0 /CW/CX/D6/D4 /D3/D1/D4 /CT/D2/D7/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3/BZ/CE/BW /CX/D7 /CX/D1/D4 /D3/D7/D7/CX/CQ/D0/CT /CP/D2/CS /D8/CW/CP/D8 /CX/CX/CX/B5 /D8/CW/CT /BZ/CE/BW /D6/CP/D2/CV/CT /DB/CW/CT/D6/CT /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /CT/DC/CX/D7/D8/D7 /CX/D7/D1 /D9 /CW /D2/CP/D6/D6/D3 /DB /CT/D6 /D8/CW/CP/D2 /D8/CW/CP/D8 /D3/D2/CT /CU/D3/D6 /D5/D9/CP/D7/CX/B9/CB /CW/D6/GU /CS/CX/D2/CV/CT/D6 /D7/D3/D0/CX/D8/D3/D2/BA/CB/D3/D0/D9/D8/CX/D3/D2 /B4/BD/BJ/B5 /CX/D7 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CT/D7/D7/CT/D2 /D8/CX/CP/D0/D0/DD /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU /D8/CW/CT /CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4/CT/DC /CX/D8/CP/D8/CX/D3/D2 /B4 /D9/D6/DA /CT/D7 /BE /CX/D2 /BY/CX/CV/D7/BA /BK /CP/D2/CS /BL/B5/BM /D8/CW/CT /CW/CX/D6/D4 /CW/CP/D2/CV/CT/D7 /CX/D8/D7 /D7/CX/CV/D2 /CP /D3/D6/CS/CX/D2/CV /D8/D3/AG/CS/CT/CU/D3 /D9/D7/CX/D2/CV/AH /CP /D8/CX/D3/D2 /D3/CU /CB/C8/C5 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /CP/D2/CS /D6/CT/CP /CW/CT/D7 /CX/D8/D7 /D1/CX/D2/CX/D1 /D9/D1 /B4/CP/D0/D3/D2/CV/DB/CX/D8/CW /D8/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2/B5 /CP/D8 /D7/D3/D1/CT /D4 /D3/D7/CX/D8/CX/DA /CT /BZ/CE/BW/BN /D8/CW/CT /BZ/CE/BW /D6/CP/D2/CV/CT /DB/CW/CT/D6/CT /D8/CW/CT/D7/D3/D0/D9/D8/CX/D3/D2 /CT/DC/CX/D7/D8/D7 /D2/CP/D6/D6/D3 /DB/D7 /CP/D7 /D3/D1/D4/CP/D6/CT /D8/D3 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CP/D7/CT/BA/CC /D3 /CW/CT /CZ /D8/CW/CT /D3/D6/D6/CT /D8/D2/CT/D7/D7 /D3/CU /D8/CW/CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/BG/B5/CQ /DD /CT/D5/BA /B4/BD/BH/B5 /CU/D3/D6 /DB /CT/CP/CZ/B9/AS/CT/D0/CS /D0/CX/D1/CX/D8 /DB /CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/D3/CU /D5/D9/CP/D7/CX/B9/D7/D3/D0/CX/D8/D3/D2 /B4/BD/BJ/B5 /CX/D2 /CT/D5/BA /B4/BD/BH/B5 /D4 /CT/D6/D8/D9/D6/CQ /CT/CS /CQ /DD /D9/CQ/CX /D8/CT/D6/D1/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D3/D2 /D3/CU/B4/BD/BJ/B5 /CX/D2 /D8/D3 /B4/BD/BG/B5 /CP/D2/CS /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D2 /D8/D3 /D8/CW/CT /D8/CX/D1/CT /D7/CT/D6/CX/CT/D7 /DD/CX/CT/D0/CS/D7 /CP/D2 /CP/D0/CV/CT/CQ/D6/CP/CX /CT/D5/D9/CP/D8/CX/D3/D2/D7/D7/CT/D8/BM/BD/BG2(γ−α) +dψ−(γ−α+l)t2 p+γa2 0t2 p= 0, /B4/BE/BD/B5 −6(γ−α)−5dψ+ (γ−α)ψ2−2γa2 0t2 p= 0, −12d+ 4(γ−α)ψ−3dψ2+ (γ−α)ψ3−2a0t2 pγ−2a2 0t2 p(2p+ψγ) = 0./CC/CW/CT /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT /CW/CX/D6/D4 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT/D7/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D4/D6/CT/B9/D7/CT/D2 /D8/CT/CS /CQ /DD /CS/D3/D8/D8/CT/CS /D0/CX/D2/CT/D7 /CX/D2 /BY/CX/CV/D7/BA /BK /CP/D2/CS /BL/BA /C1/D8 /CX/D7 /D7/CT/CT/D2/B8 /D8/CW/CP/D8 /CU/D3/D6 /D8/CW/CT /DB /CT/CP/CZ/B9/AS/CT/D0/CS/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /D8/CW/CT /D9/CQ/CX /D8/CT/D6/D1 /CX/D7 /D6/CT/CP/D0/D0/DD /CY/D9/D7/D8 /CP /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /CP/D2/CS /CW/CP/D2/CV/CT/D7 /D8/CW/CT/D7/D3/D0/D9/D8/CX/D3/D2 /D3/D2/D0/DD /D7/D0/CX/CV/CW /D8/D0/DD /BA /C7/D2/CT /CP/D2 /D7/CT/CT /CP/D0/D7/D3/B8 /D8/CW/CP/D8 /DB/CW/CT/D2 /D8/CW/CT /D7/CX/CV/D2/D7 /D3/CU /CB/C8/C5 /CX/D2 /CP /B9/D8/CX/DA /CT /D1/CT/CS/CX/D9/D1 /CP/D2/CS /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6 /D3/CX/D2 /CX/CS/CT /B4/CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B5/B8 /D8/CW/CT/CS/CT /D6/CT/CP/D7/CT /D3/CU /D4/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS/B8 /DB/CW/CX/D0/CT /CU/D3/D6 /D8/CW/CT /D3/D4/D4 /D3/D7/CX/D8/CT /D7/CX/CV/D2/D7 /B4/CQ /CT/D0/D3 /DB/CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B5 /D8/CW/CT /CS/D9/D6/CP/D8/CX/D3/D2 /CX/D2 /D6/CT/CP/D7/CT/D7/BA/BT/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /D4 /CT/D6/D8/D9/D6/CQ/CP/D8/CX/D3/D2 /D3/CU /CX/D8/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /B4/CP/D1/B9/D4/D0/CX/D8/D9/CS/CT/B8 /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CW/CX/D6/D4/B5 /CP/D2/CS /CT/D2/CT/D6/CV/DD /DB /CP/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CX/D7 /CU/D3/D0/D0/D3 /DB/D7/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/B9/CX/D2/CV /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BD/BJ/B5 /CX/D2 /D8/D3 /CT/D5/BA /B4/BD/BG/B5 /CP/D2/CS /CT/DC/D4/CP/D2/CS/CX/D2/CV /CX/D8 /CX/D2 /D8/D3 /D8/CW/CT /D8/CX/D1/CT /D7/CT/D6/CX/CT/D7 /DB /CT/CP/D6/D6/CX/DA /CT /D8/D3 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D7/CT/D8 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CT/DA /D3/D0/D9/D8/CX/D3/D2/BA /BT/D7 /CQ /CT/CU/D3/D6/CT/B4/CB/CT /BA /BE/BA/BE/B5 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D4/D9/D0/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD 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/BY/D0/D9 /CZ/B8 /BV/BA /C0/GU/D2/D2/CX/D2/CV/CT/D6/B8 /C6/BA /C5/CP/D8/D9/D7 /CW/CT/CZ/B8 /CP/D2/CS /C2/BA /CP/D9/D7 /CS/CT/D6 /BT/D9/B8 /C1/BX/BX/BX /C2/BA /D3/CU/CB/CT/D0/CT /D8/CT/CS /CC /D3/D4/CX /D7 /CX/D2 /C9/D9/CP/D2 /D8/BA /BX/D0/CT /D8/D6/D3/D2/BA/B8 /BE /B4/BD/BL/BL/BI/B5 /BG/BF/BH/BA/CJ/BE/BF℄ /C5/BA /BT/CQ/D0/D3 /DB/CX/D8/DE/B8 /C0/BA /CB/CT/CV/D9/D6/B8 /CB/D3/D0/CX/D8/D3/D2/D7 /CP/D2/CS /CX/D2 /DA /CT/D6/D7/CT /D7 /CP/D8/D8/CT/D6/CX/D2/CV /D4/D6/D3/CQ/D0/CT/D1/B8/B4/C8/CW/CX/D0/CP/CS/CT/D0/D4/CW/CX/CP /BD/BL/BK/BJ/B5/BA/CJ/BE/BG℄ /CE/BA /C4/BA /C3/CP/D0/CP/D7/CW/D2/CX/CZ /D3 /DA/B8 /BW/BA /C7/BA /C3/D6/CX/D1/CT/D6/B8 /C1/BA /BZ/BA /C8 /D3/D0/D3 /DD/CZ /D3/B8 /C2/BA /C7/D4/D8/BA /CB/D3 /BA /BT/D1/BA/B8/BU /BD/BJ /B4/BE/BC/BC/BC/B5 /BH/BD/BL /B4/D7/CT/CT /CP/D0/D7/D3 /CP/D6/CG/CX/DA/BM /D4/CW /DD/D7/CX /D7/BB /BC/BC/BC/BL/BC/BC/BL/B5/BA/BD/BK/BJ /BY/CX/CV/D9/D6/CT /CP/D4/D8/CX/D3/D2/D7/BY/CX/CV/BA /BD/BA /BV/CW/CX/D6/D4ψ /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6 /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /D0/CX/D2/CT/DB/CX/CS/D8/CW /CT/D2/CW/CP/D2 /CT/D1/CT/D2 /D8/CU/CP /D8/D3/D6/D7/BMχ= /BC /B4/BD/B5/B8 /B9/BC/BA/BC/BC/BH /B4/BE/B5/B8 /B9/BC/BA/BC/BH /B4/BF/B5/B8 /B9/BE/BA/BH /B4/BG/B5/BA /BX/DA /CT/D6/DD /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D7/CT/D8 /CW/CP/D7/D8 /DB /D3 /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA /CB/D8/CP/CQ/D0/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CP/D6/CT /D4/D0/D3/D8/D8/CT/CS /CQ /DD /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/BA /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8 /CX/D7/B9/BF/BI/BC /CU/D72/B8Γ = /BC/BA/BC/BH/B8l= /BC/BA/BC/BH/B8am= /BD/BA/BH/B8p= /BF/BA/BY/CX/CV/BA /BE/BA /BY /D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8ω /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6 /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /D0/CX/D2/CT/DB/CX/CS/D8/CW/CT/D2/CW/CP/D2 /CT/D1/CT/D2 /D8 /CU/CP /D8/D3/D6/D7 /CP/D2/CS /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8/D7/BM χ= /BC /B4/BD/B5/B8 /B9/BC/BA/BC/BC/BH /B4/BE/B5/B8 /B9/BC/BA/BC/BH /B4/BF/B5/B8/B9/BE/BA/BH /B4/BG/B8 /BH/B5/BNd= /B9/BF/BI/BC /CU/D72/B4/BD /B9 /BG/B5/B8 /B9/BL/BC /CU/D72/B4/BH/B5/BA /C7/D8/CW/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT /CP/D7 /CX/D2 /BY/CX/CV/BA /BE/BA/CB/D8/CP/CQ/D0/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CP/D6/CT /D4/D0/D3/D8/D8/CT/CS /CQ /DD /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/BA /CC /D3 /CQ /CT/D8/D8/CT/D6 /CX/D0/D0/D9/D7/D8/D6/CP/D8/CT /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6/D3/CU /D8/CW/CT /D9/D6/DA /CT/D7 /D8/CW/CT /D4/D0/D3/D8 /CX/D7 /CS/CX/DA/CX/CS/CT/CS /CX/D2 /D8/D3 /D4/CP/D6/D8/D7 /CP /CP/D2/CS /CQ /BA/BY/CX/CV/BA /BF/BA /C8/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2tp /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6 /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /D0/CX/D2/CT/DB/CX/CS/D8/CW/CT/D2/CW/CP/D2 /CT/D1/CT/D2 /D8 /CU/CP /D8/D3/D6/D7 /CP/D2/CS /BZ/CE/BW /D3 /CTꜶ /CX/CT/D2 /D8/D7/BM χ= /BC /B4/BD/B5/B8 /B9/BC/BA/BC/BC/BH /B4/BE/B5/B8 /B9/BC/BA/BC/BH /B4/BF/B5/B8/B9/BE/BA/BH /B4/BG/B8 /BH/B5/BNd= /B9/BF/BI/BC /CU/D72/B4/BD /B9 /BG/B5/B8 /B9/BL/BC /CU/D72/B4/BH/B5/BA /C7/D8/CW/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT /CP/D7 /CX/D2 /BY/CX/CV/BA/BE/BA /CB/D8/CP/CQ/D0/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CP/D6/CT /D4/D0/D3/D8/D8/CT/CS /CQ /DD /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/BA/BY/CX/CV/BA /BG/BA /CA/CT/CV/CX/D3/D2/D7 /D3/CU /D4/D9/D0/D7/CT /CT/DC/CX/D7/D8/CT/D2 /CT/BA /B4 /BT /B5 /AL /D6/CT/CV/CX/D3/D2 /D3/CU /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /B8 /B4 /BU /B5 /B9 /CP/D9/D8/D3/B9/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /B8 /B4 /BV /B5 /B9 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/CV/CP/CX/D2/D7/D8 /D2/D3/CX/D7/CT/B8 /B4 /BW /B5 /AL /CP/D9/D8/D3/D1/D3 /CS/D9/D0/CP/D8/CX/D3/D2/CP/D0/CP/D2/CS /D2/D3/CX/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/D2 /D8/CW/CT /D4/D0/CP/D2/CT /B4/BZ/CE/BW /AL /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/B5/BAχ= /BC /B4 /CP /B5/B8 /B9/BE/BA/BH /B4 /CQ /B5/BA/BY/CX/CV/BA /BH/BA /BW/D9/D6/CP/D8/CX/D3/D2tp /B4 /CP /B5/B8 /BZ/CE/BW /CP/D1/D3/D9/D2 /D8d /B4 /CQ /B5 /CP/D2/CS /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CU/D6/CT/D5/D9/CT/D2 /DD/D1/CX/D7/D1/CP/D8 /CW /CU/D6/D3/D1 /CT/DC /CX/D8/D3/D2/CX /D6/CT/D7/D3/D2/CP/D2 /CT Ωtg /B4 /B5 /CU/D3/D6 /CW/CX/D6/D4/B9/CU/D6/CT/CT /D7/D3/D0/D9/D8/CX/D3/D2 /DA /CT/D6/D7/D9/D7 /D2/D3/D6/B9/D1/CP/D0/CX/DE/CT/CS /D1/CX/D7/D1/CP/D8 /CW /CQ /CT/D8 /DB /CT/CT/D2 /CV/CP/CX/D2 /CP/D2/CS /CP/CQ/D7/D3/D6/D4/D8/CX/D3/D2 /D6/CT/D7/D3/D2/CP/D2 /CT/D7 (ωa−ωg)×tg /BA χ= /BD/BF /B4/BD/B8 /BF/B8 /BG/B5/B8 /BK /B4/BE/B5/B8Γ = /BC/BA/BC/BH /B4/BD/B8 /BE/B5/B8 /BC/BA/BD /B4/BF/B8 /BG/B5/B8ϑ= /BD /B4/BD/B9/BF/B5/B8 /BF /B4/BG/B5/B8α−r= /BC/BA/BC/BD/BA/BY/CX/CV/BA /BI/BA /BV/CW/CX/D6/D4ψ /B4 /CP /B5 /CP/D2/CS /CS/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D9/D0/D7/CTtp /B4 /CQ /B5 /DA /CT/D6/D7/D9/D7 /BZ/CE/BW /CP/D1/D3/D9/D2 /D8d /BA ϑ= /BD /B4/BD/B5/B8 /BF/BC /B4/BE/B5/B8(ωa−ωg)×tg= /BC/BA/BH /B4/BD/B5/B8 /BC/BA/BE /B4/BE/B5/BNχ= /BD/BF/B8Γ = /BC/BA/BD/B8α−r= /BC/BA/BC/BC/BD/BA/BY/CX/CV/BA /BJ/BA /C6/CT/D8/B9/CV/CP/CX/D2 Σ /CQ /CT/CW/CX/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /D8/CP/CX/D0 /CU/D3/D6 /AS/DA /CT /D7/CT/D0/CT /D8/CT/CS /D2/D3/CX/D7/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 (ωn−ωl)×tg= /BD /B4 /D9/D6/DA /CT/BD/B5/B8 /BC/BA/BH /B4/BE/B5/B8 /BC /B4/BF/B5/B8 /B9/BC/BA/BH /B4/BG/B5/B8 /B9/BD /B4/BH/B5/BAχ= /BD/BF /B4 /CP /B8 /B8 /CS /B5/B8 /BK/B4/CQ/B5/B8Γ = /BC/BA/BC/BH /B4 /CP /B8 /CQ /B5/B8 /BC/BA/BD /B4 /B8 /CS /B5/B8ϑ= /BD /B4 /CP /B8 /CQ /B8 /B5/B8 /BF /B4 /CS /B5/BA/BY/CX/CV/BA /BK/BA /BW/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CTtp /DA /CT/D6/D7/D9/D7 /BZ/CE/BWd /BA /BV/D9/D6/DA /CT /BD /B9 /D4/D9/D0/D7/CT/B4/BD/BJ/B5 /CP/D8 /CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B8 /D9/D6/DA /CT /BD /B9 /D4/D9/D0/D7/CT /B4/BD/BJ/B5 /CP/D8 /CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4/CT/DC /CX/D8/CP/D8/CX/D3/D2/B8 /CS/CP/D7/CW/CT/CS /D9/D6/DA /CT /B9 /D4/D9/D0/D7/CT /B4/BD/BL/B5/B8 /CS/D3/D8/D8/CT/CS /D9/D6/DA /CT/D7 /B9 /D4/D9/D0/D7/CT /B4/BD/BJ/B5 /CP/D7 /D7/D3/D0/D9/D8/CX/D3/D2/CU/D3/D6 /CT/D5/BA /B4/BD/BH/B5 /D4 /CT/D6/D8/D9/D6/CQ /CT/CS /CQ /DD /D9/CQ/CX /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /BA /C8/D9/D1/D4 /D4 /D3 /DB /CT/D6 /CX/D7 /BF/BD/BC /D1 /CF /B4/D7/D3/D0/CX/CS/CP/D2/CS /CS/D3/D8/D8/CT/CS /D9/D6/DA /CT/D7/B5 /CP/D2/CS /BL/BA/BJ /CF /B4/CS/CP/D7/CW/CT/CS /D9/D6/DA /CT/B5/B8 /D4/D9/D1/D4 /CQ /CT/CP/D1 /D6/CP/CS/CX/D9/D7 /CX/D7 /BH/BCµ /D1/B8 αm= /BD/B8Γ = /BC/BA/BD/B8l= /BC/BA/BC/BH/BA/BY/CX/CV/BA /BL/BA /BV/CW/CX/D6/D4ψ /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CT /DA /CT/D6/D7/D9/D7 /BZ/CE/BWd /BA /BV/D9/D6/DA /CT /BD /B9 /D4/D9/D0/D7/CT /B4/BD/BJ/B5 /CP/D8/CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B8 /D9/D6/DA /CT /BD /B9 /D4/D9/D0/D7/CT /B4/BD/BJ/B5 /CP/D8 /CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B8/CS/CP/D7/CW/CT/CS /D9/D6/DA /CT /B9 /D4/D9/D0/D7/CT /B4/BD/BL/B5 /B8 /CS/D3/D8/D8/CT/CS /D9/D6/DA /CT/D7 /B9 /D4/D9/D0/D7/CT /B4/BD/BJ/B5 /CP/D7 /D7/D3/D0/D9/D8/CX/D3/D2 /CU/D3/D6 /CT/D5/BA /B4/BD/BH/B5/D4 /CT/D6/D8/D9/D6/CQ /CT/CS /CQ /DD /D9/CQ/CX /D2/D3/D2/D0/CX/D2/CT/CP/D6/CX/D8 /DD /BA /C7/D8/CW/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT /CP/D7 /CX/D2 /BY/CX/CV/BA /BK/BA/BY/CX/CV/BA /BD/BC/BA /BW/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CT /B4/BE/BE/B5tp /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/BA /BV/D9/D6/DA /CT/BD /B9 /CQ /CT/D0/D3 /DB /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B8 /D9/D6/DA /CT /BD /B9 /CP/CQ /D3 /DA /CT /CQ/CP/D2/CS/B9/CV/CP/D4 /CT/DC /CX/D8/CP/D8/CX/D3/D2/B8 /CS/D3/D8/D8/CT/CS /D9/D6/DA /CT /B9 /D8/CW/CT /CP/D7/CT /DB/CX/D8/CW /D2/D3 /CB/D8/CP/D6/CZ/B9/CT/AR/CT /D8 /CX/D2 /D7/CT/D1/CX /D3/D2/CS/D9 /D8/D3/D6/B8 d= /B9/BD/BH/BA/BI /CU/D72/CP/D2/CS/D3/D8/CW/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D7 /CX/D2 /BY/CX/CV/BA /BK/BA/BD/BL1 2 3-0.0200.020.04 44 3 322 11ψ Pump power, W/BY/CX/CV/D9/D6/CT /BD/BM /BV/CW/CX/D6/D4ψ /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/BE/BC1.0 1.5 2.0-0.020-0.015-0.010-0.0050.000 ω Pump power, W3221 1 23 4-0.3-0.2-0.10.0b a Pump power, W5 5 44/BY/CX/CV/D9/D6/CT /BE/BM /BY /D6/CT/D5/D9/CT/D2 /DD /D7/CW/CX/CU/D8ω /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/BE/BD1.0 1.52.02.5 3.03.54.0303005 55 4 43 322 11 t p, fs Pump power, W/BY/CX/CV/D9/D6/CT /BF/BM /C8/D9/D0/D7/CT /CS/D9/D6/CP/D8/CX/D3/D2tp /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/BE/BE/BY/CX/CV/D9/D6/CT /BG/BM /CA/CT/CV/CX/D3/D2/D7 /D3/CU /D4/D9/D0/D7/CT /CT/DC/CX/D7/D8/CT/D2 /CT/BE/BF0.00 0.25 0.50 0.75-0.08-0.040.000.04 (ωa-ωg)tg(ωa-ωg)tg(ωa-ωg)tga cb Ω4 431 0.00 0.25 0.50 0.75-1.6-1.2-0.8-0.40.0 32 1 2 10.00 0.25 0.50 0.7550100150200 dtp, fs 4 32/BY/CX/CV/D9/D6/CT /BH/BMtp /B4 /CP /B5/B8d /B4 /CQ /B5 /CP/D2/CSΩtg /B4 /B5 /CU/D3/D6 /CW/CX/D6/D4/B9/CU/D6/CT/CT /D7/D3/D0/D9/D8/CX/D3/D2/BE/BG-2 0 2510152025ba tp, fs 2 1-2 0 2-20 ddψ 2 1 2 1/BY/CX/CV/D9/D6/CT /BI/BMψ /B4 /CP /B5 /CP/D2/CStp /B4 /CQ /B5 /DA /CT/D6/D7/D9/D7d/BE/BH0.3 0.4 0.5-20-1004, 5 3120.4 0.6 0.8-40-200 5 43 210.55 0.60 0.650510 53, 4210.3 0.6 0.9-10010 (ωa-ωg)tg Σ x 103Σ x 103Σ x 103Σ x 103 (ωa-ωg)tg(ωa-ωg)tg(ωa-ωg)tg4, 5 3 2 1 dcba/BY/CX/CV/D9/D6/CT /BJ/BM /C6/CT/D8/B9/CV/CP/CX/D2 Σ /CQ /CT/CW/CX/D2/CS /D8/CW/CT /D4/D9/D0/D7/CT /D8/CP/CX/D0/BE/BI50 0 -50 -100050100150200250300350400 d, fs221tp, fs/BY/CX/CV/D9/D6/CT /BK/BM /BW/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CTtp /DA /CT/D6/D7/D9/D7 /BZ/CE/BWd/BE/BJ50 0 -50 -100-10-50510 d, fs22 1ψ/BY/CX/CV/D9/D6/CT /BL/BM /BV/CW/CX/D6/D4ψ /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CT /DA /CT/D6/D7/D9/D7 /BZ/CE/BWd/BE/BK0.0 0.5 1.0 1.5 2.001002003004005006007008009001000 2 1tp, fs Pump power, W/BY/CX/CV/D9/D6/CT /BD/BC/BM /BW/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4/D9/D0/D7/CTtp /DA /CT/D6/D7/D9/D7 /D4/D9/D1/D4 /D4 /D3 /DB /CT/D6/BE/BL
arXiv:physics/0009068v1 [physics.comp-ph] 22 Sep 2000 /CB/D8/D9/CS/DD /C6/D3/D8/CT/D7 /D3/D2 /C6/D9/D1/CT/D6/CX /CP/D0 /CB/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CF /CP /DA /CT /BX/D5/D9/CP/D8/CX/D3/D2/DB/CX/D8/CW /D8/CW/CT /BY/CX/D2/CX/D8/CT /BW/CX/AR/CT/D6/CT/D2 /CT /C5/CT/D8/CW/D3 /CS /BD/BT/D6/D8/D9/D6 /BU/BA /BT /CS/CX/CQ /BE/BW/CT/D4/D8/D3/BA /CS/CT /BY/GL/D7/CX /CP/B8 /CD/D2/CX/DA /CT/D6/D7/CX/CS/CP/CS/CT /BY /CT/CS/CT/D6/CP/D0 /CS/D3 /BV/CT/CP/D6/G9 /B9 /BU/D6/CP/DE/CX/D0/BD/CB/D8/D9/CS/DD /D7/D9/D4/D4 /D3/D6/D8/CT/CS /CQ /DD /D8/CW/CT /C8/C1/BU/C1/BV/BB/BV/C6/C8/D5 /D9/D2/CS/CT/D6/CV/D6/CP/CS/D9/CP/D8/CT /D6/CT/D7/CT/CP/D6 /CW /D4/D6/D3/CV/D6/CP/D1/B8 /BU/D6/CP/DE/CX/D0/BA/BE/CT/B9/D1/CP/CX/D0/BM /CP/CS/CX/CQ/AS/D7/CX /CP/BA/D9/CU /BA/CQ/D6/C8/D6/CT/CU/CP /CT/CC/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CW/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CS/DD/D2/CP/D1/CX /D7 /D3/CU /BV/D0/CP/D7/D7/CX /CP/D0 /BY/CX/CT/D0/CS /CC/CW/CT/D3/D6/CX/CT/D7 /CX/D7 /D9/D6/D6/CT/D2 /D8/D0/DD 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/BA /BA /BA /BA /BA /BA /BA /BA /BD/BE/BF /BX/DC/CP/D1/D4/D0/CT/D7 /BD/BH/BF/BA/BD /CC/CW/CT /BY /D6/CT/CT /CB/D8/D6/CX/D2/CV /B4/BD/BW/B5 /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BD/BH/BF/BA/BE /CC/CW/CT /C5/CT/D1 /CQ/D6/CP/D2/CT /B4/BE/BW/B5 /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BD/BH/BT /CC /CP /DD/D0/D3/D6/B3/D7 /CC/CW/CT/D3/D6/CT/D1 /BD/BJ/BT/BA/BD /BW/CT/AS/D2/CX/D8/CX/D3/D2/D7 /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BD/BJ/BT/BA/BE /CD/D7/CT/CU/D9/D0 /BX/DC/D4/CP/D2/D7/CX/D3/D2/D7 /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BA /BD/BJ/BE/BV/CW/CP/D4/D8/CT/D6 /BD/CC/CW/CT /CF /CP /DA /CT /BX/D5/D9/CP/D8/CX/D3/D2/BD/BA/BD /C1/D2 /D8/D6/D3 /CS/D9 /D8/CX/D3/D2/C8 /CP/D6/D8/CX/CP/D0 /BW/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /BX/D5/D9/CP/D8/CX/D3/D2/D7 /B4/CU/D6/D3/D1 /D2/D3 /DB /D3/D2 /D7/CX/D1/D4/D0/DD /C8/BW/BX/D7 /B5 /CP/D6/CT /CS/CX/DA/CX/CS/CT/CS /CX/D2 /D8/CW/CT /D0/CX/D8/CT/D6/CP/D8/D9/D6/CT /CQ/CP/D7/CX /CP/D0/D0/DD/CX/D2 /D8/CW/D6/CT/CT /CZ/CX/D2/CS/D7/BM /D4/CP/D6/CP/CQ /D3/D0/CX /B8 /CT/D0/D0/CX/D4/D8/CX /CP/D2/CS /CW /DD/D4 /CT/D6/CQ /D3/D0/CX /B4/D8/CW/CT /D6/CX/D8/CT/D6/CX/D3/D2 /D3/CU /D0/CP/D7/D7/CX/AS /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D7/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CX/D2 /CJ/BG /B8 /BV/CW/CP/D4/BA /BK℄/B5/BA /C1/D2 /D8/CW/CX/D7 /DB /D3/D6/CZ /DB /CT /DB/CX/D0/D0 /CQ /CT /CX/D2 /D8/CT/D6/CT/D7/D8/CT/CS /D1/CP/CX/D2/D0/DD /D3/D2 /CW /DD/D4 /CT/D6/CQ /D3/D0/CX /CT/D5/D9/CP/D8/CX/D3/D2/D7/B8/D3/CU /DB/CW/CX /CW /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /D4/CP/D6/CP/CS/CX/CV/D1/BM ∇2u=1 v2∂2u ∂t2, /B4/BD/BA/BD/B5/DB/CW/CT/D6/CT v2=τ λ /CX/D7 /D8/CW/CT /D7/D5/D9/CP/D6/CT /D3/CU /D8/CW/CT /DB /CP /DA /CT /DA /CT/D0/D3 /CX/D8 /DD /CX/D2 /D8/CW/CT /D1/CT/CS/CX/D9/D1/B8 /DB/CW/CX /CW /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU /CP /CU/D6/CT/CT /D7/D8/D6/CX/D2/CV /D3/D9/D0/CS /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 τ /CP/D2/CS /D8/CW/CT /D1/CP/D7/D7 /CS/CT/D2/D7/CX/D8 /DD /D4 /CT/D6 /D0/CT/D2/CV/D8/CW /D9/D2/CX/D8λ /BA/BY /D6/D3/D1 /D8/CW/CT /D7/D8/D6/CX /D8 /D2 /D9/D1/CT/D6/CX /CP/D0 /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB/B8 /D8/CW/CT /CS/CX/D7/D8/CX/D2 /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D7/CT /D0/CP/D7/D7/CT/D7 /D3/CU /C8/BW/BX/D7 /CX/D7/D2/B3/D8 /D3/CU/D1 /D9 /CW /CX/D1/D4 /D3/D6/D8/CP/D2 /CT /CJ/BH℄/BA /CC/CW/CT/D6/CT /CX/D7/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /CP/D2/D3/D8/CW/CT/D6 /D7/D3/D6/D8 /D3/CU /D0/CP/D7/D7/CX/AS /CP/D8/CX/D3/D2 /D3/CU /C8/BW/BX/D7 /DB/CW/CX /CW /CX/D7 /D6/CT/D0/CT/DA /CP/D2 /D8 /CU/D3/D6/D2 /D9/D1/CT/D6/CX /CP/D0 /D4/D9/D6/D4 /D3/D7/CT/D7/BM /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /DA/CP/D0/D9/CT /D4/D6 /D3/CQ/D0/CT/D1/D7 /B4/DB/CW/CX /CW /CX/D2 /D0/D9/CS/CT /D8/CW/CT /CP/D7/CT /D3/CU /D8/CW/CT /CW /DD/D4 /CT/D6/CQ /D3/D0/CX /CT/D5/D9/CP/D8/CX/D3/D2/D7/B5/CP/D2/CS /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2 /D4/D6 /D3/CQ/D0/CT/D1/D7 /B4/DB/CW/CX /CW /CX/D2 /D0/D9/CS/CT/B8 /CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B8 /D4/CP/D6/CP/CQ /D3/D0/CX /CT/D5/D9/CP/D8/CX/D3/D2/D7/B5 /BA /C1/D2 /D8/CW/CX/D7/DB /D3/D6/CZ /DB /CT /DB/CX/D0/D0 /D6/CT/D7/D8/D6/CX /D8 /D3/D9/D6/D7/CT/D0/DA /CT/D7 /D8/D3 /CX/D2/CX/D8/CX/CP/D0 /DA /CP/D0/D9/CT /D4/D6/D3/CQ/D0/CT/D1/D7/BA /CB/CT/CT /D6/CT/CU/CT/D6/CT/D2 /CT /CJ/BH℄ /CU/D3/D6 /CP /CV/D3 /D3 /CS /CX/D2 /D8/D6/D3 /CS/D9 /D8/CX/D3/D2/D8/D3 /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2 /D4/D6/D3/CQ/D0/CT/D1/D7/BA/C1/D2 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/BA/BD/B5 /DB /CT /D3/D9/D0/CS /D7/D8/CX/D0/D0 /CP/CS/CS /CP /CS/CX/D7/D7/CX/D4/CP/D8/CX/DA /CT /D8/CT/D6/D1 /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3 /D8/CW/CT /AS/D6/D7/D8 /D4 /D3 /DB /CT/D6 /D3/CU/D8/CW/CT /D8/CX/D1/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /D3/CUu /B8 /CX/BA/CT/BA/B8 τ∇2u=λ∂2u ∂t2+η∂u ∂t, /B4/BD/BA/BE/B5/DB/CW/CT/D6/CT η /CX/D7 /D8/CW/CT /DA/CX/D7 /D3/D7/CX/D8 /DD /D3 /CTꜶ /CX/CT/D2 /D8/BA/C7/D9/D6 /AS/D6/D7/D8 /D7/D8/CT/D4 /DB/CX/D0/D0 /CQ /CT /D8/D3 /CS/CT/D6/CX/DA /CT /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D6/D3/D1 /CP /D7/CX/D1/D4/D0/CT /D1/CT /CW/CP/D2/CX /CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CU/D6/CT/CT/D6/D3/D4 /CT/BA /BU/CT/CX/D2/CV /D8/CW/CX/D7 /CP /DB /CT/D0/D0 /CZ/D2/D3 /DB/D2 /D4/D6/D3/CQ/D0/CT/D1 /CX/D2 /D0/CP/D7/D7/CX /CP/D0 /D1/CT /CW/CP/D2/CX /D7/B8 /DB /CT /DB/CX/D0/D0 /CV/D3 /D8/CW/D6/D3/D9/CV/CW /D3/D2/D0/DD /D8/CW/CT /D1/CP/CX/D2/D7/D8/CT/D4/D7 /D3/CU /CX/D8 /B4/CU/D3/D6 /CP /D1/D3/D6/CT /D3/D1/D4/D0/CT/D8/CT /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /D3/CU /D8/CW/CT /CU/D6/CT/CT /D7/D8/D6/CX/D2/CV/B8 /D7/CT/CT/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CJ/BI/B8/BV/CW/CP/D4/D7/BA /BK /CP/D2/CS /BL℄/B5/BA /C7/D2 /CT /DB /CT /CP/D6/CT /CS/D3/D2/CT /DB/CX/D8/CW /D8/CW/CT /BD/B9/CS /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/B8 /DB /CT /DB/CX/D0/D0 /D4/D6/D3 /CT/CT/CS /CU/D9/D6/D8/CW/CT/D6 /D8/D3 /D8/CW/CT/BE/B9/CS /CP/D7/CT/B8 /DB/CW/CX /CW /CX/D7/D2/B3/D8 /CP/D7 /CP/CQ/D9/D2/CS/CP/D2 /D8 /CX/D2 /D8/CW/CT /D0/CX/D8/CT/D6/CP/D8/D9/D6/CT /CP/D7 /D8/CW/CT /BD/B9/CS /CP/D7/CT/BA/BD/BA/BE /CF /CP /DA /CT/D7 /CX/D2 /BD/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2 /B4/D8/CW/CT /CU/D6/CT/CT /D7/D8/D6/CX/D2/CV/B5/BD/BA/BE/BA/BD /BX/D5/D9/CP/D8/CX/D3/D2 /D3/CU /C5/D3/D8/CX/D3/D2/BY/CX/CV/D9/D6/CT /BD/BA/BD /CV/CX/DA /CT/D7 /D9/D7 /CP/D2 /CX/CS/CT/CP /D3/CU /CP /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8 dm /DB/CX/D8/CW /D0/CX/D2/CT/CP/D6 /CS/CX/D1/CT/D2/D7/CX/D3/D2 dx /D7/D9/CQ /CY/CT /D8 /D8/D3 /D8/CT/D2/D7/CX/D3/D2/CU/D3/D6 /CT/D7/BA /CF /CT /CP/D6/CT /CX/D2 /D8/CT/D6/CT/D7/D8/CT/CS /D3/D2 /D8/CW/CT /DA /CT/D6/D8/CX /CP/D0 /CS/CX/D7/D4/D0/CP /CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CX/D7 /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8/B8 /D7/D3/B8 /CU/D3/D6 /D8/CW/CX/D7 /CS/CX/D6/CT /D8/CX/D3/D2/B8/DB /CT /D3/D9/D0/CS /DB/D6/CX/D8/CT /D8/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D3/D6 /CT/BM dFu=/vector τ·ˆu\|x+dx−/vector τ·ˆu\|x, /B4/BD/BA/BF/B5/BF/BV/C0/BT/C8/CC/BX/CA /BD/BA /CC/C0/BX /CF /BT /CE/BX /BX/C9/CD/BT /CC/C1/C7/C6 /BG x+dxxß uu+du/BY/CX/CV/D9/D6/CT /BD/BA/BD/BM /CA/CT/D4/D6/CT/D7/CT/D2 /D8/CX/D2/CV /D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 /CU/D3/D6 /CT/D7 /CP /D8/CX/D2/CV /D3/D2 /CP/D2 /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /CT/D0/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D6/D3/D4 /CT/BA/DB/CW/CT/D6/CT /vector τ /CX/D7 /D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 /CP/D2/CS /D8/CW/CT /D9/D2/CX/D8 /DA /CT /D8/D3/D6 ˆu /D6/CT/CU/CT/D6/D7 /D8/D3 /D8/CW/CT /DA /CT/D6/D8/CX /CP/D0 /CS/CX/D6/CT /D8/CX/D3/D2/BA /CF/CX/D8/CW/CX/D2 /D8/CW/CT /CS/D3/D1/CP/CX/D2 /D3/CU/D7/D1/D3 /D3/D8/CW /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D7/D8/D6/CX/D2/CV /B4/CX/BA/CT/BA/B8 /D7/D1/CP/D0/D0 β /B5/B8 /DB /CT /D3/D9/D0/CS /DB/D6/CX/D8/CT/BM τu≡/vector τ·ˆu=τsinβ≈τtanβ=τ∂u ∂x /B4/BD/BA/BG/B5/CF /CT /D2/D3/D8/CX /CT /D2/D3 /DB /D8/CW/CP/D8 /B4/BD/BA/BF/B5 /D3/D9/D0/CS /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7/BM dFu=τu\|x+dx−τu\|x dxdx=∂τu ∂xdx=∂ ∂x/parenleftbigg τ∂u ∂x/parenrightbigg dx/CF /CT /DB/CX/D0/D0 /D2/D3 /DB /D6/CT/D7/D8/D6/CX /D8 /D3/D9/D6/D7/CT/D0/DA /CT/D7 /D8/D3 /D8/CW/CT /CP/D7/CT /D3/CU /D3/D2/D7/D8/CP/D2 /D8 /D8/CT/D2/D7/CX/D3/D2/D7 /CP/D0/D3/D2/CV /D8/CW/CT /D6/D3/D4 /CT/B8 /D7/D3 /D8/CW/CP/D8/BM dFu=τ∂2u ∂x2dx /B4/BD/BA/BH/B5/BX/D5/D9/CP/D8/CX/D2/CV /D8/CW/CX/D7 /DB/CX/D8/CW /C6/CT/DB/D8/D3/D2/B3/D7 /D7/CT /D3/D2/CS /D0/CP /DB dFu=dm∂2u ∂t2=λ∂2u ∂t2dx,/DB/CW/CT/D6/CT λ /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D1/CP/D7/D7 /CS/CT/D2/D7/CX/D8 /DD /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /D8/CW/CT/D2 /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /CP /CU/D6/CT/CT /D7/D8/D6/CX/D2/CV/BM τ∂2u ∂x2=λ∂2u ∂t2 /B4/BD/BA/BI/B5/BD/BA/BE/BA/BE /BX/D2/CT/D6/CV/DD/CC/CW/CT /CZ/CX/D2/CT/D8/CX /CT/D2/CT/D6/CV/DD /D3/D9/D0/CS /CQ /CT /CT/DA /CP/D0/D9/CP/D8/CT/CS /CX/D2 /CP /D7/D8/D6/CP/CX/CV/CW /D8/CU/D3/D6/DB /CP/D6/CS /D1/CP/D2/D2/CT/D6/B8 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D2/CV /D8/CW/CT /CZ/CX/D2/CT/D8/CX /CT/D2/CT/D6/CV/DD/D8/CT/D6/D1 /CU/D3/D6 /CP /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/DA /CT /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8/BM dT=1 2dm·v2 u,/DB/CX/D8/CWdm=λ·dx /CP/D2/CSvu=∂u ∂t /B8 /CX/D2 /D3/D8/CW/CT/D6 /DB /D3/D6/CS/D7/B8 T=/integraldisplay dT=λ 2/integraldisplayl 0/parenleftbigg∂u ∂t/parenrightbigg2 dx /B4/BD/BA/BJ/B5/CC/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD /CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /CP/D0 /D9/D0/CP/D8/CX/D2/CV /D8/CW/CT /DB /D3/D6/CZ /D2/CT /CT/D7/D7/CP/D6/DD /D8/D3 /CQ/D6/CX/D2/CV /D8/CW/CT /D7/D8/D6/CX/D2/CV/CU/D6/D3/D1 /CP /AG/D8/D6/CX/DA/CX/CP/D0/AH /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 u(x,0) = 0 /D8/D3 /D8/CW/CT /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CP/D8 /DB/CW/CX /CW /DB /CT /DB /CP/D2 /D8 /D8/D3 /CT/DA /CP/D0/D9/CP/D8/CT /D8/CW/CT/D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD u(x, t) /BA /CF /CT /DB/CX/D0/D0 /AS/DC /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 u(0, t) =u(l, t) = 0 /B4/D7/D8/D6/CX/D2/CV /D8/CX/CT/CS /CP/D8/D8/CW/CT /CT/D2/CS/D7/B5 /CP/D2/CS /CP/D7 /CP /D3/D2/D7/CT/D5/D9/CT/D2 /CT /D3/CU /D8/CW/CX/D7/B8∂tu(0, t) =∂tu(l, t) = 0 /BA /CC/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3/D8/CW/CT /DB /D3/D6/CZ /D2/CT /CT/D7/D7/CP/D6/DD /D8/D3 /CW/CP/D2/CV/CT /D3/CUδu /D8/CW/CT /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /D3/CU /CP/D2 /CT/D0/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D7/D8/D6/CX/D2/CV /CX/D2 /CP/D2 /CX/D2 /D8/CT/D6/DA /CP/D0 /D3/CU/D8/CX/D1/CTdt /CX/D7/BM δV=−dFu·δu=−dFu·/parenleftbigg∂u ∂t/parenrightbigg dt/BV/C0/BT/C8/CC/BX/CA /BD/BA /CC/C0/BX /CF /BT /CE/BX /BX/C9/CD/BT /CC/C1/C7/C6 /BH/CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/D8/D6/CX/D2/CV /CX/D2 /D8/CW/CX/D7 /D7/CP/D1/CT /CX/D2/D8/CT/D6/DA/CP/D0 /D3/CU /D8/CX/D1/CT /CX/D7/BM dV=−dt·/integraldisplay dFu/parenleftbigg∂u ∂t/parenrightbigg/CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /B4/BD/BA/BH/B5 /CX/D2 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /DB /CT /CV/CT/D8/BM dV=−dt·/integraldisplayl 0τ/parenleftbigg∂2u ∂x2/parenrightbigg /parenleftbigg∂u ∂t/parenrightbigg dx/CF /CT /CP/D6/CT /CW/D3 /DB /CT/DA /CT/D6 /CX/D2 /D8/CT/D6/CT/D7/D8/CT/CS /D3/D2V[u(x, t)] /B8 /D7/D3 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D2/CV /CX/D2 /D8/CX/D1/CT /CP/D2/CS /D9/D7/CX/D2/CV /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/B9/D8/CX/D3/D2/D7 /CP/CQ /D3 /DA /CT/B8 /DB /CT /CW/CP /DA /CT/BM V=/integraldisplay dV=−τ/integraldisplayt 0dt/integraldisplayl 0dx/parenleftbigg∂u ∂t/parenrightbigg /parenleftbigg∂2u ∂x2/parenrightbigg = =−τ/integraldisplayt 0dt/braceleftBigg ∂u ∂t∂u ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsinglel 0−/integraldisplayl 0∂u ∂x∂2u ∂x∂tdx/bracerightBigg = =τ/integraldisplayt 0dt/integraldisplayl 0∂u ∂x∂2u ∂x∂tdx= =τ/integraldisplayt 0dt1 2∂ ∂t/integraldisplayl 0/parenleftbigg∂u ∂x/parenrightbigg2 dx= =τ 2/integraldisplayl 0/parenleftbigg∂u ∂x/parenrightbigg2 dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet 0⇒ V=τ 2/integraldisplayl 0/parenleftbigg∂u ∂x/parenrightbigg2 dx /B4/BD/BA/BK/B5/CF/CX/D8/CW /B4/BD/BA/BJ/B5 /CP/D2/CS /B4/BD/BA/BK/B5 /DB /CT /CW/CP /DA /CT /AS/D2/CP/D0/D0/DD /D8/CW/CT /D8/D3/D8/CP/D0 /CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT /D6/D3/D4 /CT/BM E=λ 2/integraldisplayl 0/parenleftbigg∂u ∂t/parenrightbigg2 dx+τ 2/integraldisplayl 0/parenleftbigg∂u ∂x/parenrightbigg2 dx /B4/BD/BA/BL/B5/C1/D2 /D3/D9/D6 /CP/D4/D4/D0/CX /CP/D8/CX/D3/D2/D7 /DB /CT /DB/CX/D0/D0 /D8/CP/CZ /CTλ=τ /D7/D9 /CW /D8/CW/CP/D8v2= 1 /CP/D2/CS /B4/BD/BA/BL/B5 /CP/D7/D7/D9/D1/CT/D7 /D8/CW/CT /D7/CX/D1/D4/D0/CT /CU/D3/D6/D1/BM E=1 2/integraldisplayl 0/bracketleftBigg/parenleftbigg∂u ∂t/parenrightbigg2 +/parenleftbigg∂u ∂x/parenrightbigg2/bracketrightBigg dx /B4/BD/BA/BD/BC/B5/CC/CW/CX/D7 /CT/D2/CT/D6/CV/DD /CT/D5/D9/CP/D8/CX/D3/D2 /DB/CX/D0/D0 /CQ /CT /DA /CT/D6/DD /D9/D7/CT/CU/D9/D0 /D8/D3 /D8/CT/D7/D8 /D3/D9/D6 /CP/D0/CV/D3/D6/CX/D8/CW/D1/D7 /D8/CW/D6/D3/D9/CV/CW /CP/D2 /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D3/D2/D7/CT/D6/B9/DA /CP/D8/CX/D3/D2 /B4/D3/D6 /CS/CX/D7/D7/CX/D4/CP/D8/CX/D3/D2/B5 /CS/D9/D6/CX/D2/CV /D8/CW/CT /CS/DD/D2/CP/D1/CX /CP/D0 /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA/BD/BA/BF /CF /CP /DA /CT/D7 /CX/D2 /BE/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/D7 /B4/D8/CW/CT /CU/D6/CT/CT /D1/CT/D1 /CQ/D6/CP/D2/CT/B5/BD/BA/BF/BA/BD /BX/D5/D9/CP/D8/CX/D3/D2 /D3/CU /C5/D3/D8/CX/D3/D2/C1/D2 /CX/D8/D7 /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /DA /CT/D6/D7/CX/D3/D2/B8 /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /D3/D9/D0/CS /CQ /CT /CS/CT/D7 /D6/CX/CQ/CX/D2/CV /CP /D1/CT/D1 /CQ/D6/CP/D2/CT/B8 /CP /D0/CX/D5/D9/CX/CS /D7/D9/D6/CU/CP /CT/B8/D3/D6 /D7/D3/D1/CT /AG /D3/CP/D6/D7/CT/B9/CV/D6/CP/CX/D2/CT/CS/AH /AS/CT/D0/CS /CX/D2 /D8/CW/CT /D7/D9/D6/CU/CP /CT /D4/CW /DD/D7/CX /D7/B8 /D8/D3 /CX/D8/CT /CP /CU/CT/DB/BA /C1/D2 /D8/CW/CT /CP/D7/CT /D3/CU /D8/CW/CT /D1/CT/D1 /CQ/D6/CP/D2/CT/D3/D6 /D3/D8/CW/CT/D6 /CT/D0/CP/D7/D8/CX /D7/D9/D6/CU/CP /CT/B8 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CP/D0/D7/D3 /D3/D2/D7/D8/D6/CP/CX/D2/CT/CS /D8/D3 /CQ /CT /D7/D1/CP/D0/D0 /B4/CP/D2/CP/D0/D3/CV/D3/D9/D7/D0/DD /D8/D3 /D8/CW/CT /BD/B9/CS/D7/D8/D6/CX/D2/CV/B5/BA/CF /CT /D2/D3/D8/CX /CT /D2/D3 /DB /D8/CW/CP/D8 /D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /CS/CX/D1/CT/D2/D7/CX/D3/D2 /CU/D3/D6 /CT/D7 /D9/D7 /D8/D3 /CS/CT/AS/D2/CT /D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 /AG/D4 /CT/D6 /D9/D2/CX/D8 /D0/CT/D2/CV/D8/CW/AH/BM f=τ l /B4/BD/BA/BD/BD/B5/BV/C0/BT/C8/CC/BX/CA /BD/BA /CC/C0/BX /CF /BT /CE/BX /BX/C9/CD/BT /CC/C1/C7/C6 /BI x ydydx x x+dxy y+dydm uu+du/BY/CX/CV/D9/D6/CT /BD/BA/BE/BM /CA/CT/D4/D6/CT/D7/CT/D2 /D8/CX/D2/CV /CP /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8 dm /D3/CU /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7 dxdy /BA /CC/CW/CT /CT/D0/CT/D1/CT/D2 /D8 /CX/D7 /D7/D9/CQ /CY/CT /D8 /D8/D3 /D8/CT/D2/D7/CX/D3/D2/CU/D3/D6 /CT/D7 /D3/D2 /CT/CP /CW /D7/CX/CS/CT /B4/CP/D2/CP/D0/D3/CV/D3/D9/D7 /D8/D3 /D8/CW/CT /CQ /D3/D6/CS/CT/D6/D7 /D3/CU /D8/CW/CT /D1/CP/D7/D7 /CT/D0/CT/D1/CT/D2 /D8 /CX/D2 /D8/CW/CT /BD/B9/CS /CP/D7/CT/B5/B8 /CQ /CT/CX/D2/CV /D8/CW/CT/D7/CT/CU/D3/D6 /CT/D7 /D3/D6/D8/CW/D3/CV/D3/D2/CP/D0 /D8/D3 /D8/CW/CT /CP/DC/CT/D7 /D3/CU /D8/CW/CT /D7/CX/CS/CT/D7/BA /C7/D2/D0/DD /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3x /D3/CUu /CX/D7 /CS/D6/CP /DB/D2 /B4/CX/BA/CT/BA/B8 u /CP/D2/CSu+du /CX/D2 /D8/CW/CT /AS/CV/D9/D6/CT /CP/D6/CT /CS/CX/D7/D4/D0/CP /CT/D1/CT/D2 /D8/D7 /D3/CUu(x, y) /CZ /CT/CT/D4/CX/D2/CV y /D3/D2/D7/D8/CP/D2 /D8 /CP/D2/CS /DA /CP/D6/DD/CX/D2/CV x /B5/BA/CC/CW/CX/D7 /CU/D3/D6 /CT /D4 /CT/D6 /D9/D2/CX/D8 /D0/CT/D2/CV/D8/CW /D3/D9/D0/CS /CQ /CT /D9/D2/CS/CT/D6/D7/D8/D3 /D3 /CS /DB/CX/D8/CW /CP /D7/CX/D1/D4/D0/CT /CT/DC/CP/D1/D4/D0/CT/BM /D7/D8/D6/CT/D8 /CW /CP /D8/CP/D4 /CT /D3/CU /DB/CX/CS/D8/CW l /CU/D6/D3/D1 /CX/D8/D7 /CT/DC/D8/D6/CT/D1/CX/D8/CX/CT/D7 /DB/CX/D8/CW /CU/D3/D6 /CTτ /BA /CF /CT /CP/D2/B3/D8 /CP/D7/CZ /D8/CW/CT /CU/D3/D6 /CT /D3/D2 /CP /D4 /D3/CX/D2/D8 /D3/CU /D8/CW/CT /D8/CP/D4 /CT/B8 /CQ/D9/D8 /D3/D2/D0/DD /D3/D2 /D7/D3/D1/CT/CT/D0/CT/D1/CT/D2 /D8 /D3/CU /D7/D3/D1/CT /CS/CT/AS/D2/CX/D8/CT /D0/CT/D2/CV/D8/CW /CP/D2/CS /DB/CX/CS/D8/CW /B4/D3/CU /D3/D9/D6/D7/CT/B8 /D8/CW/CX/D7 /CT/D0/CT/D1/CT/D2 /D8 /D3/D9/D0/CS /CQ /CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0/B8 /D4/D0/CP /DD/CX/D2/CV/D8/CW/CT /D7/CP/D1/CT /D6/D3/D0/CT /D3/CU /CP /D0/CX/D2/CT/CP/D6 /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D0/CT/D1/CT/D2 /D8 /CX/D2 /D8/CW/CT /CP/D7/CT /D3/CU /D8/CW/CT /D7/D8/D6/CX/D2/CV/B5/BA /BY /D3/D6/D1 /D9/D0/CP /B4/BD/BA/BD/BD/B5/B8 /D8/CX/D1/CT/D7 /D8/CW/CT/D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CT/D0/CT/D1/CT/D2 /D8/B8 /D8/CW/CT/D2 /CV/CX/DA /CT/D7 /DD /D3/D9 /D8/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D3/D6 /CT /B4/D8/CT/D2/D7/CX/D3/D2/B5 /D3/D2 /D8/CW/CT /CT/D0/CT/D1/CT/D2 /D8/BA /C1/D2 /D8/CW/CX/D7 /DB /CP /DD /B8 /DB /CT/CT/DC/D8/CT/D2/CS /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/BA/BF/B5 /D8/D3 /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/BM dFu=/bracketleftbigg /vectorfx·ˆu/vextendsingle/vextendsingle/vextendsingle x+dx,ydy−/vectorfx·ˆu/vextendsingle/vextendsingle/vextendsingle x,ydy/bracketrightbigg +/bracketleftbigg /vectorfy·ˆu/vextendsingle/vextendsingle/vextendsingle x,y+dydx−/vectorfy·ˆu/vextendsingle/vextendsingle/vextendsingle x,ydx/bracketrightbigg , /B4/BD/BA/BD/BE/B5/DB/CW/CT/D6/CT /vectorfx·ˆu/vextendsingle/vextendsingle/vextendsingle x,ydy≡fx,u\|x,ydy /CX/D7 /D8/CW/CTx /D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 /CX/D2 /D8/CW/CT /CS/CX/D6/CT /D8/CX/D3/D2 ˆu /CP /D8/CX/D2/CV /D3/D2/D8/CW/CT /D7/CX/CS/CT /CS/CT/AS/D2/CT/CS /CQ /DD /D8/CW/CT /D4 /D3/CX/D2 /D8/D7 (x, y) /CT(x, y+dy) /B8 /CP/D2/CS /D7/D3 /D3/D2/BA /CF /CT /CP/D6/CT /CV/D3/CX/D2/CV /D8/D3 /D7/D9/D4/D4 /D3/D7/CT /D8/CW/CP/D8 /D8/CW/CT/CU/D3/D6 /CT/D7 /D3/D2 /D8/CW/CT /D7/CX/CS/CT/D7 /D3/CU /D8/CW/CT /CT/D0/CT/D1/CT/D2 /D8/D7 /CP/D6/CT /D3/D6/D8/CW/D3/CV/D3/D2/CP/D0 /D8/D3 /D8/CW/CT/CX/D6 /CP/DC/CT/D7/B8 /DB/CW/CX /CW /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CP/D7 /CS/CT /D3/D1/D4 /D3/D7/CX/D2/CV/D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 /CU/D3/D6 /CT /D3/D2dm /CX/D2 /D8/D3 /CU/D3/D9/D6 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/B8 /D3/D2/CT /CU/D3/D6 /CT/CP /CW /D7/CX/CS/CT /B4/D2/D3/D8/CX /CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CP/D8 /DB /CT /CW/CP /DA /CT/CT/AR/CT /D8/CX/DA /CT/D0/DD /D3/D2/D0/DD /D8/DB/D3 /D6/CT/D7/D9/D0/D8/CX/D2/CV /D3/D1/D4 /D3/D2/CT/D2 /D8/D7/B8 /D8/D3 /DB/CX/D8ˆx /CP/D2/CSˆy /B5/BA /BW/D3/CX/D2/CV /D8/CW/CX/D7 /DB /CT /DB /D3/D2/B3/D8 /D2/CT/CT/CS /D8/D3 /CT/D1/D4/CW/CP/D7/CX/DE/CT/D8/CW/CT /D8/CT/D2/D7/CX/D3/D2 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D0/D3/D2/CV x /D3/D6y /B8/vectorfx·ˆu/vextendsingle/vextendsingle/vextendsingle x,y /CQ /CT /D3/D1/CX/D2/CV /D7/CX/D1/D4/D0/DD fu\|x,y /CP/D2/CS /D7/D3 /D3/D2/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/CX/D8 /CX/D7 /D7/D8/CX/D0/D0 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8/B8 /CU/D3/D6 /DB/CW/CP/D8 /DB /CT /D7/CP/CX/CS /CP/CQ /D3 /DA /CT/B8 /D8/D3 /CZ/D2/D3 /DB /DB/CW/CP/D8 /D7/CX/CS/CT /DB /CT /CP/D6/CT /D8/CP/D0/CZ/CX/D2/CV /CP/CQ /D3/D9/D8/BA /CB/D3/B8 /CX/D2 /D8/CW/CT/D6/CT/CV/CX/D1/CT /D3/CU /D7/D1/CP/D0/D0 /DA/CX/CQ/D6/CP/D8/CX/D3/D2/D7 /B4/CX/BA/CT/BA/B8 /D7/D1/CP/D0/D0 /CP/D2/CV/D0/CT/D7 /D3/CU /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2/B5/B8 /DB /CT /D3/D9/D0/CS /AS/D2/CSfu /CP/D2/CP/D0/D3/CV/D3/D9/D7/D0/DD /D8/D3 /D8/CW/CT/D7/D8/D6/CX/D2/CV /CP/D7/CT fudy=f∂u ∂xdy /B4/BD/BA/BD/BF/B5 fudx=f∂u ∂ydx, /B4/BD/BA/BD/BG/B5/DB/CW/CT/D6/CT fudy /CP/D2/CSfudx /CP/D6/CT /D8/CW/CT /D8/CT/D2/D7/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /CS/CX/D6/CT /D8/CX/D3/D2 ˆu /D3/D2 /CP /D7/CX/CS/CT /D3/CU /D0/CT/D2/CV/D8/CW dy /CP/D2/CSdx /CP/D0/D3/D2/CV y/CP/D2/CSx /B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA /CF /CT /CT/D1/D4/CW/CP/D7/CX/DE/CT /D8/CW/CP/D8/B8 /DB/CX/D8/CW /D8/CW/CX/D7 /D2/D3/D8/CP/D8/CX/D3/D2 /D4/D0/D9/D7 /D8/CW/CT /CZ/D2/D3 /DB/D0/CT/CS/CV/CT /D3/CU /D8/CW/CT /D4 /D3/CX/D2/D8 /DB/CW/CT/D6/CT/DB /CT /CP/D6/CT /CV/D3/CX/D2/CV /D8/D3 /CT/DA /CP/D0/D9/CP/D8/CT /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/D7/B8 /DB /CT /CW/CP /DA /CT /CP /D3/D1/D4/D0/CT/D8/CT /D7/D4 /CT /CX/AS /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CX/CS/CT /D3/D2 /DB/CW/CX /CW /D8/CW/CT/D8/CT/D2/D7/CX/D3/D2 /CP /D8/D7 /BD/BA /CF/CX/D8/CW /CP/D0/D0 /D8/CW/CX/D7 /CX/D2 /CW/CP/D2/CS/D7/B8 /BX/D5/BA /B4/BD/BA/BD/BE/B5 /CQ /CT /D3/D1/CT/D7/BM dFu=/bracketleftBig fu\|x+dx,ydy−fu\|x,ydy/bracketrightBig +/bracketleftBig fu\|x,y+dydx−fu\|x,ydx/bracketrightBig/CF/CX/D8/CW /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /CP/D2/CP/D0/D3/CV/D3/D9/D7 /D8/D3 /D8/CW/D3/D7/CT /D3/CU /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT /D8/CX/D3/D2/B8 /DB /CT /CW/CP /DA /CT/BM dFu=fu\|x+dx,y−fu\|x,y dxdxdy+fu\|x,y+dy−fu\|x,y dydydx= /B4/BD/BA/BD/BH/B5/BD/C1/D2/CS/CT/CT/CS/B8 /D3/D2 /CT /D7/D4 /CT /CX/AS/CT/CS /D8/CW/CT /CQ /CT /CV/CX/D2/D2/CX/D2/CV /D3/CU /D8/CW/CT /D7/CX/CS/CT /DB/CX/D8/CW /D8/CW/CT /D4/CP/CX/D6(x, y) /B8 /D7/D4 /CT /CX/CU/DD/CX/D2/CV /D8/CW/CT /D0/CT/D2/CV/D8/CW /DB/CX/D8/CWdx /D3/D6dy /CU/D9/D6/D2/CX/D7/CW/CT/D7/D9/D7 /DB/CX/D8/CW /D8/CW/CT /CS/CX/D6 /CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/CX/CS/CT /CX/D2 /D5/D9/CT/D7/D8/CX/D3/D2/BA /CC/CW/CX/D7 /CX/D7 /D7/D9Ꜷ /CX/CT/D2 /D8 /D8/D3 /D0/D3 /CP/D0/CX/DE/CT /CX/D8/B8 /D7/CX/D2 /CT /D8/CW/CTz /D3 /D3/D6/CS/CX/D2/CP/D8/CT /CX/D7 /D9/D2/CP/D1 /CQ/CX/CV/D9/D3/D9/D7/D0/DD/CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /DA/CX/CPz=u(x, y) /BA/BV/C0/BT/C8/CC/BX/CA /BD/BA /CC/C0/BX /CF /BT /CE/BX /BX/C9/CD/BT /CC/C1/C7/C6 /BJ =∂fu ∂xdxdy+∂fu ∂ydxdy= /B4/BD/BA/BD/BI/B5 =f∂2u ∂x2dxdy+f∂2u ∂y2dxdy /B4/BD/BA/BD/BJ/B5/CF/CX/D8/CW /C6/CT/DB/D8/D3/D2/B3/D7 /D7/CT /D3/D2/CS /D0/CP /DB /DB /CT /D3/CQ/D8/CP/CX/D2/BM f∂2u ∂x2dxdy+f∂2u ∂y2dxdy =dm∂2u ∂t2⇒ f∂2u ∂x2dxdy+f∂2u ∂y2dxdy =σdxdy∂2u ∂t2⇒ f/parenleftbigg∂2u ∂x2+∂2u ∂y2/parenrightbigg =σ∂2u ∂t2⇒ ∂2u ∂x2+∂2u ∂y2=1 v2∂2u ∂t2, /B4/BD/BA/BD/BK/B5/DB/CW/CX /CW /CX/D7 /D8/CW/CT /CS/CT/D7/CX/D6/CT/CS /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/B8 /DB/CX/D8/CWv2=f σ /CP/D2/CSσ /D8/CW/CT /D7/D9/D6/CU/CP /CT /D1/CP/D7/D7 /CS/CT/D2/D7/CX/D8 /DD /BA/BD/BA/BF/BA/BE /BX/D2/CT/D6/CV/DD/CC/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /CT/D2/CT/D6/CV/DD /CX/D7 /CS/D3/D2/CT /CX/D2 /D8/CW/CT /D7/CP/D1/CT /D1/CP/D2/D2/CT/D6 /CP/D7 /D8/CW/CT /BD/CS /CP/D7/CT/BA /CF /CT /DB/CX/D0/D0 /D3/D2/D7/CX/CS/CT/D6/CP /D7/D9/D6/CU/CP /CT z=u(x, y, t) /DB/CX/D8/CW /D7/D9/D4/D4 /D3/D6/D8 /D3/CU /CS/CX/D1/CT/D2/D7/CX/D3/D2 l×l /B8 /D7/D9/CQ /CY/CT /D8 /D8/D3 /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 u\|boundary≡u(0, y, t) =u(l, y, t) =u(x,0, t) =u(x, l, t) = 0 /CP/D2/CS˙u\|boundary≡˙u(0, y, t) = ˙u(l, y, t) = ˙u(x,0, t) = ˙u(x, l, t) = 0 /B8 /DB/CW/CT/D6/CT ˙u≡∂u/∂t /BA /C4/CT/D8 /D9/D7 /CQ /CT/CV/CX/D2 /DB/CX/D8/CW /D8/CW/CT /CZ/CX/D2/CT/D8/CX /D8/CT/D6/D1/BM dT=1 2dm·˙u2=σ 2˙u2dxdy⇒ /B4/BD/BA/BD/BL/B5 T=σ 2/integraldisplayl 0/integraldisplayl 0/parenleftbigg∂u ∂t/parenrightbigg2 dxdy, /B4/BD/BA/BE/BC/B5/DB/CW/CT/D6/CT σ /CX/D7 /D8/CW/CT /D7/D9/D6/CU/CP /CT /D1/CP/D7/D7 /CS/CT/D2/D7/CX/D8 /DD /BA/CC/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /CP/D2 /CP/D2/CP/D0/D3/CV/D3/D9/D7 /DB /CP /DD /D8/D3 /D8/CW/CT /CB/CT /D8/CX/D3/D2 /B4/BD/BA/BE/B5/BM dV=−dt·/integraldisplay dFu/parenleftbigg∂u ∂t/parenrightbigg = =−dt·/integraldisplayl 0/integraldisplayl 0f/parenleftbigg∂2u ∂x2+∂2u ∂y2/parenrightbigg /parenleftbigg∂u ∂t/parenrightbigg dxdy⇒ V=−f/integraldisplayt 0dt/braceleftBigg/integraldisplayl 0dy/integraldisplayl 0dx/parenleftbigg∂2u ∂x2∂u ∂t/parenrightbigg +/integraldisplayl 0dx/integraldisplayl 0dy/parenleftbigg∂2u ∂y2∂u ∂t/parenrightbigg/bracerightBigg = =f/integraldisplayt 0dt/braceleftBigg/integraldisplayl 0dy1 2∂ ∂t/integraldisplayl 0/parenleftbigg∂u ∂x/parenrightbigg2 dx+/integraldisplayl 0dx1 2∂ ∂t/integraldisplayl 0/parenleftbigg∂u ∂y/parenrightbigg2 dy/bracerightBigg = =f 2/integraldisplayt 0dt∂ ∂t/braceleftBigg/integraldisplayl 0/integraldisplayl 0/bracketleftBigg/parenleftbigg∂u ∂x/parenrightbigg2 +/parenleftbigg∂u ∂y/parenrightbigg2/bracketrightBigg dxdy/bracerightBigg = =f 2/integraldisplayl 0/integraldisplayl 0/bracketleftBigg/parenleftbigg∂u ∂x/parenrightbigg2 +/parenleftbigg∂u ∂y/parenrightbigg2/bracketrightBigg dxdy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet 0⇒ V=f 2/integraldisplayl 0/integraldisplayl 0/bracketleftBigg/parenleftbigg∂u ∂x/parenrightbigg2 +/parenleftbigg∂u ∂y/parenrightbigg2/bracketrightBigg dxdy /B4/BD/BA/BE/BD/B5/BV/C0/BT/C8/CC/BX/CA /BD/BA /CC/C0/BX /CF /BT /CE/BX /BX/C9/CD/BT /CC/C1/C7/C6 /BK/CC/CW/CT/D2 /D8/CW/CT /D8/D3/D8/CP/D0 /CT/D2/CT/D6/CV/DD /CU/D3/D6 /D3/D9/D6 /D9/D7/D9/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 f=σ⇒v2= 1 /CX/D7/BM E=1 2/integraldisplayl 0/integraldisplayl 0/bracketleftBigg/parenleftbigg∂u ∂x/parenrightbigg2 +/parenleftbigg∂u ∂y/parenrightbigg2 +/parenleftbigg∂u ∂t/parenrightbigg2/bracketrightBigg dxdy/D3/D6/B8 /CX/D2 /CP /D1/D3/D6/CT /CV/CT/D2/CT/D6/CP/D0 /DB /CP /DD /B4/DB /CT /CP/D6/CT /CV/D3/CX/D2/CV /D8/D3 /D8/CP/CZ /CT /CP/D7 /CV/D6/CP/D2 /D8/CT/CS /D8/CW/CT /D6/CT/D7/D9/D0/D8 /CU/D3/D6 /D1/D3/D6/CT /D8/CW/CP/D2 /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/B5/B8 E=/integraldisplay dn/vector r/bracketleftbigg1 2(/vector∇u)2+1 2˙u2/bracketrightbigg/C7/D2 /CT /CP/CV/CP/CX/D2 /C1 /CT/D1/D4/CW/CP/D7/CX/DE/CT /D8/CW/CP/D8 /D8/CW/CT/D7/CT /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /CU/D3/D6 /D8/CW/CT /DA /CT/D6/CX/AS /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU/D3/D9/D6 /D2 /D9/D1/CT/D6/CX /CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7/BA /CC/CW/CX/D7 /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /CX/D7 /D7/CW/D3 /DB/D2 /CW/CT/D6/CT /D2/D3/D8 /D3/D2/D0/DD /CP/D7 /CP/D2 /CT/DC/CT/D6 /CX/D7/CT/B8 /CQ/D9/D8 /CP/D0/D7/D3 /CQ /CT /CP/D9/D7/CT /C1 /D3/D9/D0/CS/D2/B3/D8 /AS/D2/CS /D8/CW/CT /BE/B9/CS /DA /CT/D6/D7/CX/D3/D2 /CX/D2 /CP/D2 /DD /D8/CT/DC/D8/CQ /D3 /D3/CZ/BA/BV/CW/CP/D4/D8/CT/D6 /BE/BY/CX/D2/CX/D8/CT /BW/CX/AR/CT/D6/CT/D2 /CT/D7/BE/BA/BD /C1/D2 /D8/D6/D3 /CS/D9 /D8/CX/D3/D2/BW/CX/AR/CT/D6/CT/D2 /D8/D0/DD /CU/D6/D3/D1 /D8/CW/CT /D2/D3/D2/B9/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT /CP/D2/CP/D0/DD/D8/CX /CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /C8/BW/BX/D7 /CX/D2 /D8/CW/CT /D3/D2 /D8/CX/D2 /D9/D9/D1 /B4/CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B8/D8/CW/D3/D7/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D8/CW/D6/D3/D9/CV/CW /DA /CP/D6/CX/CP/CQ/D0/CT /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2 /CP/D2/CS /D7/D9/CQ/D7/CT/D5/D9/CT/D2 /D8 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/B5/B8 /D2 /D9/D1/CT/D6/CX /CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CQ/B9/D8/CP/CX/D2/CT/CS /CX/D2 /CP /D3/D1/D4/D9/D8/CT/D6 /CW/CP /DA /CT /D0/CX/D1/CX/D8/CT/CS /D4/D6/CT /CX/D7/CX/D3/D2 /BD/BA /C1/D8 /CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /DB /CP /DD /CX/D2 /DB/CW/CX /CW /D3/D1/D4/D9/D8/CT/D6/D7 /D7/D8/D3/D6/CT /CS/CP/D8/CP/CP/D2/CS /CP/D0/D7/D3 /CQ /CT /CP/D9/D7/CT /D3/CU /D8/CW/CT/CX/D6 /D0/CX/D1/CX/D8/CT/CS /D1/CT/D1/D3/D6/DD /BA /BT/CU/D8/CT/D6 /CP/D0/D0/B8 /CW/D3 /DB /D3/D9/D0/CS /DB /CT /DB/D6/CX/D8/CT /CX/D2 /CS/CT /CX/D1/CP/D0 /D2/D3/D8/CP/D8/CX/D3/D2 /B4/D3/D6/CX/D2 /CP/D2 /DD /D3/D8/CW/CT/D6 /CQ/CP/D7/CT/B5 /CP/D2 /CX/D6/D6/CP/D8/CX/D3/D2/CP/D0 /D2 /D9/D1 /CQ /CT/D6 /D0/CX/CZ /CT√ 2 /D1/CP/CZ/CX/D2/CV /D9/D7/CT /D3/CU /CP /AS/D2/CX/D8/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CS/CX/CV/CX/D8/D7/BR /C1/D2 /D8/CW/CX/D7/DB /D3/D6/CZ /DB /CT /DB /D3/D2/B3/D8 /D7/D8/CX /CZ /DB/CX/D8/CW /D6/CX/CV/D3/D6/D3/D9/D7 /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D8/CW/CT/D3/D6/CT/D1/D7 /D2/D3/D6 /D3/CU /D1/D3/D7/D8 /D3/CU /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS/BA/CC/CW/CT /D6/CT/CU/CT/D6/CT/D2 /CT/D7 /D0/CX/D7/D8/CT/CS /CX/D2 /D8/CW/CT /CT/D2/CS /D7/CW/D3/D9/D0/CS /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CU/D3/D6 /D8/CW/CX/D7 /CT/D2/CS/BA/CC/CW/CT /CT/D2 /D8/D6/CP/D0 /CX/CS/CT/CP /D3/CU /D2 /D9/D1/CT/D6/CX /CP/D0 /D1/CT/D8/CW/D3 /CS/D7 /CX/D7 /D5/D9/CX/D8/CT /D7/CX/D1/D4/D0/CT/BM /D8/D3 /CV/CX/DA /CT /AS/D2/CX/D8/CT /D4/D6/CT /CX/D7/CX/D3/D2 /B4/AG/D8/CW/CT /CS/CX/D7 /D6/CT/D8/CT/AH/B5/D8/D3 /D8/CW/CP/D8 /D3/CQ /CY/CT /D8 /CT/D2/CS/D3 /DB /CT/CS /DB/CX/D8/CW /CX/D2/AS/D2/CX/D8/CT /D4/D6/CT /CX/D7/CX/D3/D2 /B4/AG/D8/CW/CT /D3/D2 /D8/CX/D2 /D9/D9/D1/AH/B5/BA /BU/DD /CS/CX/D7 /D6 /CT/D8/CX/DE/CT /DB /CT /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /D8/D3/D8/D6/CP/D2/D7/CU/D3/D6/D1 /D3/D2 /D8/CX/D2 /D9/D9/D1 /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /D0/CX/CZ /CTx, y, .., z /CX/D2 /D8/D3 /CP /D7/CT/D8 /D3/CU /CS/CX/D7 /D6/CT/D8/CT /DA /CP/D0/D9/CT/D7 {xi},{yi}, ...,{zi} /B8 /DB/CW/CT/D6/CT i /D6/D9/D2/D7 /D3 /DA /CT/D6 /CP /AS/D2/CX/D8/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /DA /CP/D0/D9/CT/D7/B8 /D8/CW /D9/D7 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/CU/D3/D6/D1 /CU/D3/D6 /D8/CW/CT /BX/D5/BA /B4/BE/BA/BF/B5/BM f(x+h)−f(x) h≈g(x)/C6/D3/D8/CX /CT /D8/CW/CP/D8 /D8/CW/CX/D7 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D2 /DA /D3/D0/DA /CT/D7 /D3/D2/D0/DD /CS/CX/AR/CT/D6/CT/D2 /CT/D7 /CP/D7 /DB /CT /D7/CP/CX/CS /CP/CQ /D3 /DA /CT/B8 /CP/D2/CS /D8/D3 /D7/D3/D0/DA /CT /CX/D8 /CX/D2 /CP /D3/D1/D4/D9/D8/CT/D6/DB /CT /D7/CW/CP/D0/D0 /D2/CT/CT/CS /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CX/D8/CT/D6 /CP/D8/CX/DA/CT /D6 /CT/D0/CP/D8/CX/D3/D2 /D3/CQ/D8/CP/CX/D2/CT/CS /CS/CX/D6/CT /D8/D0/DD /CU/D6/D3/D1 /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/BM f(x+h) =hg(x) +f(x)/D3/D6/B8 /CX/D2 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /D2 /D9/D1/CT/D6/CX /CP/D0 /D2/D3/D8/CP/D8/CX/D3/D2/BM fn+1=hgn+fn /B4/BE/BA/BG/B5/CC /CT /CW/D2/CX /CP/D0/D0/DD /B8 /D3/D2 /CT /D4/D6/D3 /DA/CX/CS/CT/CS /CQ /D3/D8/CW /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /B4/CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B8 f0= 0 /B5 /CP/D2/CS /D8/CW/CT /CU/D9/D2 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/CT/D5/D9/CP/D8/CX/D3/D2/B8/DB /CT /D3/D9/D0/CS /CP /CW/CX/CT/DA /CT /D8/CW/CT /CS/CT/D7/CX/D6/CT/CS /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/D2/CS/CX/D8/CX/D3/D2/BA /C7/D2/CT /D4 /D3/D7/D7/CX/CQ/D0/CT zn m /D7/CP/D8/CX/D7/CU/DD/CX/D2/CV /D8/CW/CT /D6/CX/D8/CT/D6/CX/CP /CP/CQ /D3 /DA /CT /CX/D7/BM zn m=eαn∆teiβm∆x/B4/BE/BA/BD/BC/B5/C1/D2/CS/CT/CT/CS/B8 /D2/D3/D8/CX /CT /D8/CW/CP/D8 /CU/D3/D6n= 0 /DB /CT /CW/CP /DA /CT/vextendsingle/vextendsinglez0 m/vextendsingle/vextendsingle= 1 /B8 /CP/D2/CS /DB/CX/D8/CWα /CP/D2/CSβ /CP/D6/CQ/CX/D8/D6/CP/D6/DD /DA /CP/D0/D9/CT/D7 /DB /CT /D7/CP/D8/CX/D7/CU/DD/D8/CW/CT /CP/CQ /D3 /DA /CT /CS/CX/D7 /D9/D7/D7/CX/D3/D2/BA /CF/CX/D8/CW /D8/CW/CX/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/B8 /DB /CT /D3/D9/D0/CS /D2/D3 /DB /DB/D6/CX/D8/CT /D8/CW/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /DA /D3/D2/C6/CT/D9/D1/CP/D2/D2 /CP/D2/CP/D0/DD/D7/CX/D7/BM \|ξn\| ≤1, /B4/BE/BA/BD/BD/B5/BV/C0/BT/C8/CC/BX/CA /BE/BA /BY/C1/C6/C1/CC/BX 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/D8/D3 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /B4/BE/BA/BD/BD/B5/B8 /DB /CT /CV/CT/D8 /D8/CW/CT /D2/CT /CT/D7/D7/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /BA/BE/BA/BG /CC/CW/CT /BV/D3/D9/D6/CP/D2 /D8 /BV/D3/D2/CS/CX/D8/CX/D3/D2/BT/D2/D3/D8/CW/CT/D6 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D3/D2/CS/CX/D8/CX/D3/D2 /D8/CW/CP/D8 /DB /CT /D7/CW/D3/D9/D0/CS /D4/CP /DD /D7/D3/D1/CT /CP/D8/D8/CT/D2 /D8/CX/D3/D2 /CX/D2 /CX/D2/CX/D8/CX/CP/D0 /DA /CP/D0/D9/CT /D4/D6/D3/CQ/D0/CT/D1/D7 /CX/D7 /D6/CT/D0/CP/D8/CT/CS/D8/D3 /D8/CW/CT /D7/D4 /CT /CT /CS /DB/CX/D8/CW /DB/CW/CX /CW /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/D9/D0/CS /D4/D6/D3/D4/CP/CV/CP/D8/CT /CX/D2 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CT/D5/D9/CP/D8/CX/D3/D2/BA /CF /CT /D3/D9/D0/CS /DA/CX/D7/D9/CP/D0/CX/DE/CT/D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /CX/D2 /D8/CW/CT /D7 /CW/CT/D1/CT /D3/CU /BY/CX/CV/D9/D6/CT /B4/BE/BA/BD/B5/BA/C1/D8 /D3/D9/D0/CS /CQ /CT /D7/CW/D3 /DB/D2 /CJ/BH℄ /D8/CW/CP/D8/B8 /CP/D4/D4/D0/DD/CX/D2/CV /DA /D3/D2 /C6/CT/D9/D1/CP/D2/D2/B3/D7 /D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /CW /DD/D4 /CT/D6/CQ /D3/D0/CX /D4/D6/D3/CQ/D0/CT/D1/D7 /DB /CT /CP/D6/D6/CX/DA /CT/CP/D8 /D8/CW/CT /BV/D3/D9/D6 /CP/D2/D8 /D3/D2/CS/CX/D8/CX/D3/D2 /BM /CX/CU /D8/CW/CT /AG/D4/CW /DD/D7/CX /CP/D0/AH /DB /CP /DA /CT /DA /CT/D0/D3 /CX/D8 /DD \|v\| /CX/D2 /CP /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /CV/D6/CT/CP/D8/CT/D6/D8/CW/CP/D2 /D8/CW/CT /AG/CP/D0/CV/D3/D6/CX/D8/CW/D1 /D7/D4 /CT/CT/CS/AH ∆x/∆t /B8 /D8/CW/CT/D2 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /CX/D7 /D7/D8/CP/CQ/D0/CT/BA /CF /CT /CW/CP /DA /CT /D8/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /BV/D3/D9/D6/CP/D2 /D8 /D3/D2/CS/CX/D8/CX/D3/D2/BM \|v\| ≤∆x ∆t /B4/BE/BA/BD/BF/B5/BE/BA/BH /CC/CW/CT /C4 /CT /CP/D4/B9/BY /D6 /D3 /CV /BT/D0/CV/D3/D6/CX/D8/CW/D1 /CP/D2/CS /D8/CW/CT /CF /CP /DA /CT /BX/D5/D9/CP/D8/CX/D3/D2/BV/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /BD/B9/CS /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/D8/D3 /CT/CP/D7/CT /D8/CW/CT /D2/D3/D8/CP/D8/CX/D3/D2/B8 /CU/D6/D3/D1 /D2/D3 /DB /D3/D2 /DB /CT /DB/CX/D0/D0 /D8/CP/CZ /CTv=τ=λ=σ= 1 /B5/BM ∂2u ∂x2=∂2u ∂t2/CD/D7/CX/D2/CV /D8/CW/CT /D7/CT /D3/D2/CS /D3/D6/CS/CT/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /B4/BT/BA/BL/B5 /CU/D6/D3/D1 /BT/D4/D4 /CT/D2/CS/CX/DC /BT /CU/D3/D6 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/D7 /CP/CQ /D3 /DA /CT/B8 /DB /CT /CW/CP /DA /CT/BM un i+1−2un i+un i−1 ∆x2=un+1 i−2un i+un−1 i ∆t2, /B4/BE/BA/BD/BG/B5/D3/D6/B8 /D7/D3/D0/DA/CX/D2/CV /D8/CW/CX/D7 /CP/D0/CV/CT/CQ/D6/CP/CX /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6un+1 i /B8 un+1 i=ρ/parenleftbig un j+1+un j−1/parenrightbig + 2(1−ρ)un j−un−1 j,/DB/CX/D8/CWρ= (∆t/∆x)2/BA/C6/D3/D8/CX /CT /D8/CW/CP/D8 /D8/CW/CX/D7 /CT /D5/D9/CP/D8/CX/D3/D2 /CX/D7 /CT/DC/D4/D0/CX /CX/D8 /CP/D2/CS /CW/CP/D7 /D7/CT /D3/D2/CS /D3/D6 /CS/CT/D6 /D4/D6 /CT /CX/D7/CX/D3/D2 /CQ /D3/D8/CW /CX/D2 /D8/CX/D1/CT /CP/D2/CS /D7/D4 /CP /CT /B4/DB /CT/CS/CX/CS/D2/B3/D8 /DB/D6/CX/D8/CT /D8/CW/CT /CT/D6/D6/D3/D6/D7 O(∆x2,∆t2) /B8 /CQ/D9/D8 /DD /D3/D9 /CP/D2 /CT/CP/D7/CX/D0/DD /D8/D6/CP /CZ /CX/D8 /DB/CW/CT/D2 /DD /D3/D9 /CS/D3 /D8/CW/CT /CP/CQ /D3 /DA /CT /D4/CP/D7/D7/CP/CV/CT/B5/BA/CF /CT /D3/D9/D0/CS /CP/D0/D7/D3 /D3/CQ/D8/CP/CX/D2 /CX/D8 /D8/CP/CZ/CX/D2/CV /D7/CT /D3/D2/CS /D3/D6/CS/CT/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D7 /D3/CU /D8/CW/CT /DB /CP /DA /CT/CT/D5/D9/CP/D8/CX/D3/D2 /B4/D8/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D7 /D3/CU /CP/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/D9/D0/CS /CQ /CT /CU/D3/D9/D2/CS /CX/D2 /CJ/BG/B8 /BV/CP/D4/BA /BK℄/B5/B8 ∂u ∂x±∂u ∂t= 0,/BV/C0/BT/C8/CC/BX/CA /BE/BA /BY/C1/C6/C1/CC/BX /BW/C1/BY/BY/BX/CA/BX/C6/BV/BX/CB /BD/BF/D7/D3/B8 /DB/CX/D8/CW /D8/CW/CT/D7/CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D7/B8 /D8/D3 /AS/D2/CSun+1 m /DB /CT /D2/CT/CT/CSun−1 m /B8un m−1 /CP/D2/CSun m+1 /BA /CC/CW/CT/D2 /D8/CW/CT /D2/CP/D1/CT/D0/CT /CP/D4/B9/CU/D6 /D3 /CV /B8 /D7/CX/D2 /CT/B8 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT /D8/CX/D1/CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/B8 /DB /CT /AG/D0/CT/CP/D4/AH/B8 /CU/D6/D3/D1n−1/D8/D3n+ 1 /B8 /D3 /DA /CT/D6 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /DB/CW/CX /CW /CX/D2 /DA /D3/D0/DA /CT/D7 /D3/D2/D0/DD /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D7 /CP/D8 /D8/CW/CT /CX/D2/D7/D8/CP/D2 /D8 n /BA/C6/D3 /DB /D0/CT/D8 /D9/D7 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CP/D7/CT /D3/CU /D8/CW/CT /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /DB/CX/D8/CW /CS/CX/D7/D7/CX/D4/CP/D8/CX/D3/D2/BM ∂2u ∂x2=∂2u ∂t2+η∂u ∂t,/DB/CW/CT/D6/CT η /CX/D7 /D8/CW/CT /DA/CX/D7 /D3/D7/CX/D8 /DD /D3 /CTꜶ /CX/CT/D2 /D8/BA /CD/D7/CX/D2/CV /CP/D0/D7/D3 /CP /D7/CT /D3/D2/CS /D3/D6/CS/CT/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D2/CT/DB /D8/CT/D6/D1 /DB /CT /CW/CP /DA /CT/BM un i+1−2un i+un i−1 ∆x2=un+1 i−2un i+un−1 i ∆t2+ηun+1 i−un−1 i 2∆t,/CU/D6/D3/D1 /DB/CW/CX /CW /DB /CT /D3/D9/D0/CS /D7/D3/D0/DA /CT /CU/D3/D6 /D8/CW/CT /D8/CT/D6/D1un+1 i /BM un+1 i=/bracketleftbigg 1 +η∆t 2/bracketrightbigg−1/braceleftbigg ρ/parenleftbig un i+1+un i−1/parenrightbig + 2(1−ρ)un i−/bracketleftbigg 1−η∆t 2/bracketrightbigg un−1 i/bracerightbigg/B4/BE/BA/BD/BH/B5/C4/CT/D8/B3/D7 /D3/D2/D7/CX/CS/CT/D6 /D2/D3 /DB /D8/CW/CT /BE/CS /CP/D7/CT /DB/CX/D8/CW /CP /D7/DD/D1/D1/CT/D8/D6/CX /D7/D4/CP /CX/D2/CV /CU/D3/D6 /D8/CW/CT /D8 /DB /D3 /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /B4∆x= ∆y= ∆l /B5 /D4/D0/D9/D7 /CP /DA/CX/D7 /D3/D7/CX/D8 /DD /D8/CT/D6/D1/BM un i+1,j−2un i,j+un i−1,j ∆l2+un i,j+1−2un i,j+un i,j−1 ∆l2=un+1 i,j−2un i,j+un+1 i,j ∆t2+ηun+1 i,j−un−1 i,j 2∆t/CP/D2/CS/B8 /CP/CV/CP/CX/D2 /D7/D3/D0/DA/CX/D2/CV /CU/D3/D6un+1 i,j /B8 un+1 i,j =/bracketleftBig 1 +η∆t 2/bracketrightBig−1/braceleftbig ρ/bracketleftbig un i+1,j+un i−1,j+un i,j+1+un i,j−1−4un i,j/bracketrightbig + 2un i,j−/bracketleftBig 1−η∆t 2/bracketrightBig un−1 i,j/bracerightBig/B4/BE/BA/BD/BI/B5/C1/D8 /D3/D9/D0/CS /CQ /CT /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /D8/CW/CT/D7/CT /CS/CX/AR/CT/D6/CT/D2 /CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D7/CP/D8/CX/D7/CU/DD /D8/CW/CT /DA/D3/D2 /C6/CT/D9/D1/CP/D2/D2 /D6/CX/D8/CT/D6/CX/D3/D2 /DB/CW/CT/D2 ∆x/∆t /D7/CP/D8/CX/D7/CU/DD /D8/CW/CT /BV/D3/D9/D6 /CP/D2/D8 /D6/CX/D8/CT/D6/CX/D3/D2 /CJ/BH℄ /BA /BY /D3/D6 /D8/CW/CT /D7/CP/CZ /CT /D3/CU /D3/D1/D4/D0/CT/D8/CT/D2/CT/D7/D7/B8 /D0/CT/D8/B3/D7 /D2/D3 /DB /D7/CT/CT /CW/D3 /DB /D8/CW/CT/CP/CQ /D3 /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/CX/BA/CT/BA/B8 /D8/CW/CT /BD/B9/CS /CP/D2/CS /BE/B9/CS /CS/CX/D7 /D6/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7/B5 /CW/CP/D2/CV/CT /DB/CW/CT/D2 /DB /CT /CP/CS/CS /CP/D2 /AG/CP/D6/CQ/CX/D8/D6/CP/D6/DD/AH /D8/CT/D6/D1 /BF F(un) /DB/CW/CX /CW /CS/CT/D4 /CT/D2/CS/D7 /D3/D2u(t) /B4/CX/D8 /D3/D9/D0/CS /CQ /CT/B8 /CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B8 /D8/CW/CT /D8/CT/D6/D1δV[φ]/δφ(/vector x) /DB/CW/CX /CW /CP/D6/CX/D7/CT/D7 /CX/D2 /D0/CP/D7/D7/CX /CP/D0 /AS/CT/D0/CS /D8/CW/CT/D3/D6/CX/CT/D7/B8 /DB/CW/CT/D6/CT /D8/CW/CTλφ4/CX/D7 /D8/CW/CT /D4/CP/D6/CP/CS/CX/CV/D1/B5/BA /CC/CW/CT /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/D7/CT /CP/D7/CT/D7/CP/D6/CT/BM ∂2u ∂t2=∂2u ∂x2−η∂u ∂t−F(u) ∂2u ∂t2=∂2u ∂x2+∂2u ∂y2−η∂u ∂t−F(u),/DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT/CX/D6 /CS/CX/D7 /D6/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7 /CP/D6/CT/BM un+1 i=/bracketleftbigg 1 +η∆t 2/bracketrightbigg−1/braceleftbigg ρ/parenleftbig un i+1+un i−1/parenrightbig + 2(1−ρ)un i−/bracketleftbigg 1−η∆t 2/bracketrightbigg un−1 i−∆t2F(un)/bracerightbigg/B4/BE/BA/BD/BJ/B5 un+1 i,j =/bracketleftBig 1 +η∆t 2/bracketrightBig−1/braceleftbig ρ/bracketleftbig un i+1,j+un i−1,j+un i,j+1+un i,j−1−4un i,j/bracketrightbig + 2un i,j−/bracketleftBig 1−η∆t 2/bracketrightBig un−1 i,j−∆t2F(un)/bracerightBig/B4/BE/BA/BD/BK/B5/C1/D8 /D1/CP /DD /CQ /CT /DB /D3/D6/D8/CW /D1/CT/D2 /D8/CX/D3/D2/CX/D2/CV /D8/CW/CP/D8 /DB /CT /D2/CT/CT/CS /D8/D3 /CS/CT/AS/D2/CT /CQ /D3/D8/CW /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /DA /CP/D0/D9/CT/D7 /CP/D2/CS /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D7/D3/D0/DA /CT /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /BY /D3/D6 /D8/CW/CT /CP/D4/D4/D0/CX /CP/D8/CX/D3/D2/D7 /DB/CW/CX /CW /DB/CX/D0/D0 /CQ /CT /D7/CW/D3 /DB/D2 /CX/D2/BV/CW/CP/D4/D8/CT/D6 /BF/B8 /DB /CT /D7/CW/CP/D0/D0 /D9/D7/CT /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D3/D2/CS/CX/D8/CX/D3/D2/D7/BM/BF/C6/D3/D8/CX /CT /D8/CW/CP/D8 /D8/CW/CX/D7 /D8/CT/D6/D1 /D3/D9/D0/CS /D2/D3/D8 /CS/CT/D4 /CT/D2/CS /D3/D2 /D8/CX/D1/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/D7 /D3/CUu(t) /D3/D6 /CP/D2 /DD /D3/D8/CW/CT/D6 /AG/D2/D3/D2/B9/D0/D3 /CP/D0/AH /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT/B8/D8/CW/CP/D8 /CX/D7/B8 /CX/D8 /D7/CW/D3/D9/D0/CS /CQ /CT /CS/CT/AS/D2/CT/CS /D3/D1/D4/D0/CT/D8/CT/D0/DD /CX/D2 /D8/CT/D6/D1/D7 /D3/CUun/BN /D3/D8/CW/CT/D6/DB/CX/D7/CT /DB /CT /D1/CX/CV/CW /D8 /D2/D3/D8 /CQ /CT /CP/CQ/D0/CT /D8/D3 /D7/D3/D0/DA /CT /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 un+1/CT/DC/D4/D0/CX /CX/D8/D0/DD /BA /CB/D4/CP/D8/CX/CP/D0 /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/D7 /CP/D6/CT /D2/D3/D8 /CP /D4/D6/D3/CQ/D0/CT/D1 /D7/CX/D2 /CT /D8/CW/CT/DD /CP/D6/CT /CS/CT/AS/D2/CT/CS /D0/D3 /CP/D0/D0/DD /CX/D2 /D8/CX/D1/CT/BA/BV/C0/BT/C8/CC/BX/CA /BE/BA /BY/C1/C6/C1/CC/BX /BW/C1/BY/BY/BX/CA/BX/C6/BV/BX/CB /BD/BG   u(t)\|boundaries= 0 ∂u(t) ∂t/vextendsingle/vextendsingle/vextendsingle boundaries= 0 u(x, y)\|t=0=Cexp/bracketleftBig −(/vector r−/vector r0)2 2γ/bracketrightBig , /B4/BE/BA/BD/BL/B5/DB/CW/CT/D6/CT /vector r=xˆi+yˆj /B8/vector r0=l 2ˆi+l 2ˆj /B8l /CX/D7 /D8/CW/CT /D0/CP/D8/D8/CX /CT /D0/CT/D2/CV/D8/CW/B8 C /CX/D7 /CP /D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/D2/D7/D8/CP/D2 /D8/B8 /CP/D2/CSγ /CX/D7 /CP/D7/D9Ꜷ /CX/CT/D2 /D8/D0/DD /D7/D1/CP/D0/D0 /D3/D2/D7/D8/CP/D2 /D8 /D7/D9 /CW /D8/CW/CP/D8u(/vector r)→0 /CP/D7/vector r→boundaries /B8 /D8/CW/CP/D8 /CX/D7/B8 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /CX/D7/CP /CV/CP/D9/D7/D7/CX/CP/D2 /D7/D9Ꜷ /CX/CT/D2 /D8/D0/DD /D0/D3 /CP/D0/CX/DE/CT/CS /D8/D3 /D1/CP/CZ /CT u /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /CP/D8 /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/CX/CT/D7/BA/BV/CW/CP/D4/D8/CT/D6 /BF/BX/DC/CP/D1/D4/D0/CT/D7/BF/BA/BD /CC/CW/CT /BY /D6/CT/CT /CB/D8/D6/CX/D2/CV /B4/BD/BW/B5/CC/CW/CT/D7/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT /CT/DC/CT /D9/D8/CT/CS /CX/D2 /CP /C8/BV /D3/CU350MHz /B8 /CU/D3/D6 /D0/CP/D8/D8/CX /CT/D7 /D3/CU /CP/D8 /D1/D3/D7/D8 N= 1000 /BA /CC/CW/CT/CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D8/CX/D1/CT /D0/CP /DD /CX/D2 /D8/CW/CT /D3/D6/CS/CT/D6 /D3/CU /D7/CT /D3/D2/CS/D7/BA/BY/CX/CV/D9/D6/CT /BF/BA/BD /D7/CW/D3 /DB/D7 /D7/D3/D1/CT /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT /D8/CX/D3/D2 /B4/BE/BA/BD/BL/B5 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7/D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /C6/D3/D8/CX /CT /D8/CW/CT /CX/D1/D4/D6/D3 /DA /CT/D1/CT/D2 /D8 /CU/D3/D6 /AS/D2/CT/D6 /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2/D7 /CP/D2/CS /D8/CW/CT /D4/D6/CT /CX/D7/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /AS/D8/D8/CX/D2/CV /CU/D3/D6 η= 1 /BA /CC/CW/CX/D7 /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /D6/CT/D7/D9/D0/D8 /CX/D7 /CT/DC/D4 /CT /D8/CT/CS/B8 /D7/CX/D2 /CT /D8/CW/CT /DA/CX/CQ/D6/CP/D8/CX/D2/CV /D7/D8/D6/CX/D2/CV /D3/D9/D0/CS /CQ /CT /D9/D2/CS/CT/D6/D7/D8/D3 /D3 /CS /CX/D2 /D8/CT/D6/D1/D7/D3/CU /D8/CW/CT /BY /D3/D9/D6/CX/CT/D6 /D7/D4/CP /CT/B8 /DB/CW/CT/D6/CT /CT/CP /CW /D1/D3 /CS/CT /CQ /CT/CW/CP /DA /CT/D7 /CP/D7 /CP /CS/CT /D3/D9/D4/D0/CT/CS /CW/CP/D6/D1/D3/D2/CX /D3/D7 /CX/D0/D0/CP/D8/D3/D6 /DB/CX/D8/CW /CP /CS/CP/D1/D4/CX/D2/CV/CV/CX/DA /CT/D2 /CQ /DDη /BA/BF/BA/BE /CC/CW/CT /C5/CT/D1 /CQ/D6/CP/D2/CT /B4/BE/BW/B5/CC/CW/CT/D7/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT /CP/D0/D7/D3 /CT/DC/CT /D9/D8/CT/CS /CX/D2 /CP /C8/BV /D3/CU350MHz /B8 /CP/D2/CS /CU/D3/D6 /D0/CP/D8/D8/CX /CT/D7 /D3/CUN= 500 /D8/CW/CT/CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D8/CX/D1/CT /D6/CT/CP /CW/CT/CS /CW/CP/D0/CU /CP/D2 /CW/D3/D9/D6/BA/BY/CX/CV/D9/D6/CT /BF/BA/BE /D7/CW/D3 /DB/D7 /D7/D3/D1/CT /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6η= 0 /CP/D2/CSη= 1 /BA /BY /D3/D6 /CP /D2/D3/D2/B9 /D3/D2/D7/CT/D6/DA /CP/D8/CX/DA /CT /D7/DD/D7/D8/CT/D1 /B4η= 1 /B5/B8 /DB /CT/D7/CT/CT /CP/D0/D7/D3 /D8/CW/CT /CT/DC/D4 /D3/D2/CT/D2 /D8/CX/CP/D0 /AS/D8/D8/CX/D2/CV /CU/D3/D6N= 200 /BA/BD/BH/BV/C0/BT/C8/CC/BX/CA /BF/BA /BX/CG/BT/C5/C8/C4/BX/CB /BD/BI 0 1 2 3 4 5 t0.30.350.40.450.50.550.6E N=50 N=100 N=500 N=1000 0 1 2 3 4 500.20.40.60.8 Numerico Ajuste Exponencial ~ exp(−Ax)/BY/CX/CV/D9/D6/CT /BF/BA/BD/BME×t /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /D0/CP/D8/D8/CX /CT /D7/D4/CP /CX/D2/CV/D7 /CP/D2/CSη= 0 /B4/CP/CQ /D3 /DA /CT/B5 /CP/D2/CSE×t /CU/D3/D6N= 1000 /CP/D2/CS η= 1 /B4/CQ /CT/D0/D3 /DB/B5/BA 0 2 4 6 81111.51212.5 N=50 N=200 N=500 0 2 4 6 8051015 Resultado Numerico Ajuste Exponencial/BY/CX/CV/D9/D6/CT /BF/BA/BE/BME×t /CU/D3/D6η= 0 /CP/D2/CS /DA /CP/D6/CX/D3/D9/D7 N /B4/CP/CQ /D3 /DA /CT/B5 /CP/D2/CSE×t /CU/D3/D6η= 1 /CP/D2/CSN= 200 /B4/CQ /CT/D0/D3 /DB/B5/BA/BT/D4/D4 /CT/D2/CS/CX/DC /BT/CC /CP /DD/D0/D3/D6/B3/D7 /CC/CW/CT/D3/D6/CT/D1/BT/BA/BD /BW/CT/AS/D2/CX/D8/CX/D3/D2/D7/CF/CW/CT/D2 /DB /CT /DB /CP/D2 /D8 /D8/D3 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /CP /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D2 /D8/D3 /CP /CS/CX/AR/CT/D6/CT/D2 /CT /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CC /CP /DD/D0/D3/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2/CX/D7 /D3/CU/D8/CT/D2 /D9/D7/CT/CS/BM f(x) = f(x0) +f′(x0)(x−x0) +1 2f′′(x0)(x−x0)2+...= /B4/BT/BA/BD/B5 =∞/summationdisplay n=0f(n)(x0) n!(x−x0)n, /B4/BT/BA/BE/B5/DB/CW/CT/D6/CT x0 /CX/D7 /D8/CW/CT /D4 /D3/CX/D2 /D8 /CP/D6/D3/D9/D2/CS /DB/CW/CX /CW /DB /CT /DB /CP/D2 /D8 /D8/D3 /CT/DC/D4/CP/D2/CS f(x) /BD/BA /BT/D2 /CP/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT /CU/D3/D6/D1 /CU/D3/D6 /D8/CW/CX/D7/CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D7 /CP /CW/CX/CT/DA /CT/CS /CS/D3/CX/D2/CV /CP /D7/CX/D1/D4/D0/CT /DA /CP/D6/CX/CP/CQ/D0/CT /CW/CP/D2/CV/CT x→x+h /CP/D2/CSx0→x /BM f(x+h) =∞/summationdisplay n=0f(n)(x) n!hn/B4/BT/BA/BF/B5/C1/D2 /D8/CW/CT /D2 /D9/D1/CT/D6/CX /CP/D0 /CP/D7/CT /DB /CT /DB/CX/D0/D0 /CQ /CT /CX/D2 /D8/CT/D6/CT/D7/D8/CT/CS /CX/D2 /D8/D6/D9/D2 /CP/D8/CX/D2/CV /D8/CW/CT /D7/CT/D6/CX/CT/D7/B8 /D7/D3 /D8/CW/CP/D8 /DB /CT /AS/D2/CX/D7/CW /DB/CX/D8/CW /CP/AS/D2/CX/D8/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D8/CT/D6/D1/D7/BA /CF /CT /D3/D9/D0/CS /D8/CW/CT/D2 /DB/D6/CX/D8/CT /D8/CW/CX/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CU/D3/D6/D1/BM f(x+h) =m/summationdisplay n=0f(n)(x) n!hn+O(hm+1), /B4/BT/BA/BG/B5/DB/CW/CT/D6/CT O(hm+1) /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D8/D6/D9/D2 /CP/D8/CT/CS /D8/CT/D6/D1/D7 /DB/CW/CX /CW /D4 /D3 /DB /CT/D6/D7 /D3/CUh /CP/D6/CT /CT/D5/D9/CP/D0 /D3/D6 /CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 m+ 1 /B4/D8/CW/CX/D7 /D8/CT/D6/D1 /CX/D7 /CU/D6/CT/D5/D9/CT/D2 /D8/D0/DD /CP/D0/D0/CT/CS /D8/CW/CT /CT/D6/D6 /D3/D6 /D3/CU /D3/D6 /CS/CT/D6 m+ 1 /B5/BA /C6/D3/D8/CX /CT /D8/CW/CP/D8 /D9/D2/CS/CT/D6 /D8/CW/CT /D2 /D9/D1/CT/D6/CX /CP/D0/D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /CX/D8 /CX/D7 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D8/D3 /CZ/D2/D3 /DB /D8/CW/CT /D3/D6/CS/CT/D6 /D3/CUO /CX/D2 /D8/CW/CT /CS/CX/D7 /D6/CT/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D7/CX/D2 /CT /CU/D3/D6h≪1 /B8 /D8/CW/CT/CV/D6/CT/CP/D8/CT/D6 /D8/CW/CT /D3/D6/CS/CT/D6 /D3/CUO /D8/CW/CT /D1/D3/D6/CT /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /DB/CX/D0/D0 /CQ /CT /D8/CW/CT /CT/D6/D6/D3/D6/BA/BT/BA/BE /CD/D7/CT/CU/D9/D0 /BX/DC/D4/CP/D2/D7/CX/D3/D2/D7/CB/D3/D1/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /D8/CW/CP/D8 /DB/CX/D0/D0 /CQ /CT /D9/D7/CT/CS /D8/CW/D6/D3/D9/CV/CW/D3/D9/D8 /D8/CW/CX/D7 /D8/CT/DC/D8 /CP/D6/CT /D7/CW/D3 /DB/D2 /CQ /CT/D0/D3 /DB/BA /BT/D0/D0 /D3/CU /D8/CW/CT/D1 /D3/D9/D0/CS /CQ /CT/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /B4/BT/BA/BG/B5 /CQ /DD /CS/CX/D6/CT /D8/D0/DD /D7/D3/D0/DA/CX/D2/CV /CU/D3/D6 /D8/CW/CT /CS/CT/D7/CX/D6/CT/CS /D8/CT/D6/D1 /D3/D6 /D9/D7/CX/D2/CV /D1/D3/D6/CT /D8/CW/CP/D2 /D3/D2/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D8/D3/AS/D2/CS /CW/CX/CV/CW/CT/D6 /D3/D6/CS/CT/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT/B8 /CP/D2/CS /D8/CW/CT/D2 /D7/D3/D0/DA/CX/D2/CV /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA /BY /D3/D6 /CX/D2/D7/D8/CP/D2 /CT/BM /braceleftbiggf(x+h) =f(x) +f′(x)h+1 2f′′(x)h2+... f(x−h) =f(x)−f′(x)h+1 2f′′(x)h2−... f′(x) =f(x+h)−f(x) h−O(h) /B4/BT/BA/BH/B5/BD/BY /D3/D6x0= 0 /D8/CW/CX/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CX/D7 /CP/D0/D7/D3 /CZ/D2/D3 /DB/D2 /CP/D7 /C5/CP /D0/CP/D9/D6/CX/D2 /CT/DC/D4 /CP/D2/D7/CX/D3/D2 /BA/BD/BJ/BT/C8/C8/BX/C6/BW/C1/CG /BT/BA /CC /BT /CH/C4/C7/CA/B3/CB /CC/C0/BX/C7/CA/BX/C5 /BD/BK f′(x) =f(x+h)−f(x−h) 2h−2O(h2) /B4/BT/BA/BI/B5 f′′(x) =f(x+h)−2f(x) +f(x−h) h2−2O(h2) /B4/BT/BA/BJ/B5 ∂f(x, y) ∂x=f(x+h, y)−f(x, y) h−O(h) /B4/BT/BA/BK/B5 ∂2f(x, y) ∂x2=f(x+h, y)−2f(x, y) +f(x−h, y) h2−2O(h2) /B4/BT/BA/BL/B5/C6/D3/D8/CX /CT /D8/CW/CP/D8 /DB/CW/CT/D2 /DB /CT /CS/CX/DA/CX/CS/CT /CP/D2 /CT/D6/D6/D3/D6 /D3/CU /D3/D6/CS/CT/D6 O(hn) /CQ /DDhr/B8 /CP/D9/D8/D3/D1/CP/D8/CX /CP/D0/D0/DD /D8/CW/CX/D7 /CT/D6/D6/D3/D6 /D8/D9/D6/D2/D7 /D8/D3 /D3/D6/CS/CT/D6 n−r /B8 /CX/BA/CT/BA/B8O(hn)/hr=O(hn−r) /BA/BU/CX/CQ/D0/CX/D3/CV/D6/CP/D4/CW /DD/CJ/BD℄ /C5/BA /BZ/D0/CT/CX/D7/CT/D6 /CP/D2/CS /CA/BA /C7/BA /CA/CP/D1/D3/D7/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /BW /BH/BC /B8 /BE/BG/BG/BD /B4/BD/BL/BL/BG/B5/BA/CJ/BE℄ /BZ/BA /BT/CP/D6/D8/D7 /CP/D2/CS /C2/BA /CB/D1/CX/D8/B8 /C6/D9 /D0/BA /C8/CW /DD/D7/BA /BU /BH/BH/BH /B8 /BF/BH/BH /B4/BD/BL/BL/BL/B5/BN /C8/CW /DD/D7/BA /CA/CT/DA/BA /BW /BI/BD /B8 /BC/BE/BH/BC/BC/BE /B4/BE/BC/BC/BC/B5/BA/CJ/BF℄ /C2/BA /BU/D3/D6/D6/CX/D0/D0 /CP/D2/CS /C5/BA /BZ/D0/CT/CX/D7/CT/D6/B8 /C6/D9 /D0/BA /C8/CW /DD/D7/BA /BU/BG/BK/BF /B8 /BG/BD/BI /B4/BD/BL/BL/BJ/B5/CJ/BG℄ /BZ/BA /BU/BA /BT/D6/CU/CZ /CT/D2/B8 /C0/BA /C2/BA /CF /CT/CQ /CT/D6/B8 /C5/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /C5/CT/D8/CW/D3 /CS/D7 /CU/D3/D6 /C8/CW/DD/D7/CX /CX/D7/D8/D7/B8 /BG/D8/CW /BX/CS/BA/CJ/BH℄ /CF/BA /C0/BA /C8/D6/CT/D7/D7 /CT/D8 /CP/D0/BA/B8 /C6/D9/D1/CT/D6/CX /CP/D0 /CA /CT /CX/D4 /CT/D7/B8 /BE/D2/CS /BX/CS/BA/CJ/BI℄ /C3/BA /CA/BA /CB/DD/D1/D3/D2/B8 /C5/CT /CW/CP/D2/CX /D7 /B8 /BF/D6/CS/BA /BX/CS/BA/CJ/BJ℄ /CF/BA /BV/CW/CT/D2/CT/DD /CT /BW/BA /C3/CX/D2 /CP/CX/CS/B8 /C6/D9/D1/CT/D6/CX /CP/D0 /C5/CP/D8/CW/CT/D1/CP/D8/CX /D7 /CP/D2/CS /BV/D3/D1/D4/D9/D8/CX/D2/CV /B8 /BF/D6/CS /BX/CS/BA/CJ/BK℄ /BT/BA /CA/BA /C5/CX/D8 /CW/CT/D0/D0 /CT /BW/BA /BY/BA /BZ/D6/CXꜶ/D8/CW/D7/B8 /CC/CW/CT /BY/CX/D2/CX/D8/CT /BW/CX/AR/CT/D6 /CT/D2 /CT /C5/CT/D8/CW/D3 /CS /CX/D2 /C8/CP/D6/D8/CX/CP/D0 /BW/CX/AR/CT/D6 /CT/D2/D8/CX/CP/D0 /BX/D5/D9/CP/D8/CX/D3/D2/D7/BD/BL
arXiv:physics/0009069v1 [physics.atom-ph] 23 Sep 2000Lamb Shift in Light Hydrogen-Like Atoms Vladimir G. Ivanov1⋆and Savely G. Karshenboim2,3⋆⋆ 1Pulkovo Observatory, St. Petersburg, Russia 2D. I. Mendeleev Institute for Metrology, St. Petersburg, Ru ssia 3Max-Planck-Institut f¨ ur Quantenoptik, Garching, German y† Abstract. Calculation of higher-order two-loop corrections is now a l imiting factor in development of the bound state QED theory of the Lamb shift in the hydrogen atom and in precision determination of the Rydberg constant . Progress in the study of light hydrogen-like ions of helium and nitrogen can be hel pful to investigate these uncalculated terms experimentally. To do that it is necessa ry to develop a theory of such ions. We present here a theoretical calculation for low energy levels of helium and nitrogen ions. 1 Introduction The Quantum Electrodynamics (QED) theory of simple atoms li ke hydrogen or hydrogen-like ions provides precise predictions for diﬀ erent energy levels [1,2]. Particularly, some accurate results were obtained f or the Lamb shift in the ground state of the hydrogen atom. The accuracy of the QED calcula- tions of the Lamb shift has been limited by unknown higher-or der two-loop corrections and inaccuracy of determination of the proton c harge radius [3]. As far as the proton size is going to be determined very precis ely from a new experiment [4] on the Lamb shift in muonic hydrogen, the only theoretical uncertainty is now due to the two-loop contribution. Improv ement of the the- ory is important to determine the Rydberg constant with high accuracy [5,6] and to test the bound state QED precisely. Since the theory seems not to be able to give now any results on higher- order two-loop corrections ( α2(Zα)6mand higher) we have to look for another way to estimate these terms and so the uncertainty of the hydr ogen Lamb shift theory. An opportunity is to study the problem experim entally, mea- suring the Lamb shift in diﬀerent hydrogen-like ions at not t oo high value of the nuclear charge Z. Only for two such ions the Lamb shift can be available with a high accuracy from experiment at the present time or in the near fu- ture. Namely, these are helium [7,8] and nitrogen [9] ions. T he experimental estimation of higher-order two-loop terms is quite of inter est also because of recent speculation on a great higher-order term [10] (see al so Refs. [11,12]). The advantage of using Z >1 is determined by the scaling behaviour of diﬀerent QED values: ⋆E-mail: ivanovv@gao.spb.ru ⋆⋆E-mail: sek@mpq.mpg.de †The summer address2 Vladimir G. Ivanov and Savely G. Karshenboim •The scaling of the Lamb shift is Z4; •The scaling of the radiative line width of excited states (e. g. of the 2 p and 3sstates) is Z4as well; •The scaling of the unknown higher-order two-loop correctio ns to the Lamb shift is Z6. Thus, relatively imprecise measurements with higher Zcan nevertheless give some quite accurate data on some QED corrections. Our target is to develop a theory for the Lamb shift and the ﬁne structure in these two atomic systems. Eventually we need to determine the 2s−2p1/2 splitting in the helium ion (for comparison with the experim ent [7]), diﬀerence of the Lamb shifts EL(2s)−EL(3s) in4He+(for the project [8]) and the 2p3/2−2sinterval in hydrogen-like nitrogen. The diﬀerence mention ed is necessary [13,1] if one needs to compare the results of the La mb shift ( n= 2) measurement [7] and the 2 s−3sexperiment. Since the uncertainty of the QED calculations is determined for these two ions (He+and N6+) by the higher-order two-loop terms, we are going to reduce the other sources of uncertainty. We present resul ts appropriate to provide an interpretation of the experiments mentioned a s a direct study of the higher-order two loop corrections. The results of the ions experiments should afterwards be useful for the hydrogen atom. 2 Theoretical contributions 2.1 Deﬁnitions and notation The Lamb shift is deﬁned throughout the paper as a deviation f rom an un- perturbed energy level1 E(0)(nlj) =mR/bracketleftig f(nj)−1/bracketrightig −m2 R 2(M+m)/bracketleftig f(nj)−1/bracketrightig2 , (1) where Mandmare the mass of the nucleus and of the electron, and mR stands for the reduced mass. The dimensionless Dirac energy of the electron in the external Coulomb ﬁeld is of the form f(nj) = 1 +(Zα)2 /parenleftig n−j−1 2+/radicalig (j+1 2)2−(Zα)2/parenrightig2 −1 2 . (2) The Lamb shift is mainly a QED eﬀect, perturbed by the inﬂuenc e of the nuclear structure ∆E=∆EQED+∆ENucl. (3) 1We use the relativistic units in which ¯ h=c= 1.Lamb Shift in Light Hydrogen-Like Atoms 3 The QED contribution ∆EQED=∆E∞+∆EM (4) includes one-, two- and three-loop terms calculated within the external-ﬁeld approximation ∆E∞=α(Zα)4 π/parenleftigmR m/parenrightig3m n3/bracketleftbigg F+/parenleftigα π/parenrightig H+/parenleftigα π/parenrightig2 K/bracketrightbigg , (5) and a recoil corrections ∆EM, which is a sum of pure recoil and radiative recoil contributions depending on the nuclear mass M ∆EM=∆ERec+∆ERRC. (6) 2.2 One-loop contributions: self energy of the electron Let us start with the one-loop contribution. The terms of the external-ﬁeld contribution are usually written in the form of an expansion (Zα)4F(Z) =/summationdisplay i,jAij(Zα)ilnj1 (Zα)2. (7) The dominant contribution comes from the one-loop self ener gy of the elec- tron. The known results for some low levels are summarized be low2: FSE ns(Z) =4 3lnm (Zα)2mR−4 3ln/parenleftbig k0(ns)/parenrightbig +10 9 + (Zα)4π/parenleftbigg139 128−1 2ln(2)/parenrightbigg + (Zα)2/parenleftbigg −ln21 (Zα)2 +A61(ns) ln1 (Zα)2+Gns(Z)/parenrightbigg , (8) FSE 2p1/2(Z) =−4 3ln/parenleftbig k0(2p)/parenrightbig −1 6m mR+ (Zα)2/parenleftbigg103 180ln1 (Zα)2 +G2p1/2(Z)/parenrightbigg , (9) FSE 2p3/2(Z) =−4 3ln/parenleftbig k0(2p)/parenrightbig +1 12m mR+ (Zα)2/parenleftbigg29 90ln1 (Zα)2 +G2p3/2(Z)/parenrightbigg , (10) where the state-dependent logarithmic coeﬃcient A61(ns) is known A61(1s) =28 3ln(2) −21 20, 2It is useful to keep somewhere the reduced mass mR.4 Vladimir G. Ivanov and Savely G. Karshenboim A61(2s) =16 3ln(2) +67 30, A61(3s) =−4 ln(3) +28 3ln(2) +6163 1620, the Bethe logarithm is [14] ln/parenleftbig k0(1s)/parenrightbig = 2.984 128 56 . . . , ln/parenleftbig k0(2s)/parenrightbig = 2.811 769 89 . . . , ln/parenleftbig k0(3s)/parenrightbig = 2.767 663 61 . . . , ln/parenleftbig k0(2p)/parenrightbig =−0.030 016 71 . . . and higher-order self-energy terms Gnl(Z) are numerically found in Sect. 2.5. 2.3 One-loop contributions: polarization of vacuum The coeﬃcient of the expansion (7) for free vacuum polarizat ion can be cal- culated in any order of Zαin a closed analytic form [15]. In particular, the result was found in Ref. [15] for the circular states ( l=n−1) with j=l+1/2. Forn= 1,2 one can expand at low Zαand ﬁnd FV P 1s(Z) =−4 15+5π 48(Zα) + (Zα)2/parenleftbigg −2 15ln1 (Zα)2+4 15log(2) −1289 1575/parenrightbigg + (Zα)3/parenleftbigg5π 96ln1 (Zα)2+5π 48ln(2) +23π 288/parenrightbigg , (11) FV P 2p3/2(Z) =−1 70(Zα)2+7π 1024(Zα)3. (12) In the case of other states the result has been known only up to the order (Zα)2[17,16,13]. We present here new results for two other states atn= 2 and for the 3 sstate: FV P 2s(Z) =−4 15+5π 48(Zα) + (Zα)2/parenleftbigg −2 15ln1 (Zα)2−743 900/parenrightbigg + (Zα)3/parenleftbigg5π 96ln1 (Zα)2+5π 24ln(2) +841π 9216/parenrightbigg , (13) FV P 2p1/2(Z) =−9 140(Zα)2+41π 3072(Zα)3, (14) FV P 3s(Z) =−4 15+5π 48(Zα) + (Zα)2/parenleftbigg −2 15ln1 (Zα)2−4 15ln3 2−1139 1575/parenrightbiggLamb Shift in Light Hydrogen-Like Atoms 5 + (Zα)3/parenleftbigg5π 96ln1 (Zα)2+5π 48ln(6) +137π 2592/parenrightbigg . (15) 2.4 Wichmann-Kroll contributions It is not enough to consider the free vacuum polarization. Th e relativistic corrections to the free vacuum polarization in Eqs. (11–14) are of the same order as the so-called Wichmann-Kroll term due to Coulomb eﬀ ects inside the electronic vacuum-polarization loop. To estimate this term we ﬁtted its numerical values from Ref. [18], which are more accurate for some higher Z∼30, by expression [18,17] FWK(Z) = (Zα)2/parenleftbigg19 45−π2 27/parenrightbigg + (Zα)3/parenleftbigg A71ln1 (Zα)2+A70/parenrightbigg .(16) We found that the contribution of A70andA71terms is small enough. The accuracy of the calculation in Ref. [18] is not high at Z= 7 and we performed some ﬁtting of higher Zdata. To make a conservative estimate we ﬁnd two pairs of coeﬃcients which reproduce the result at Z= 30. The results are: A71(ns) =−0.23(2), A 70(ns) = 0 and FWK(7) = 0 .000 139(1) ; A71(ns) = 0 , A 70(ns) =−0.07(1) and FWK(7) = 0 .000 130(2) , where the uncertainty comes from inaccuracy of the numerica l calculations ofFWK(30) = 0 .0020 [18], which is estimated here as a value of a unit in the last digital place. Comparing the results above one can ﬁnd a conservative estimate: FWK(7) = 0 .000 134(6). The value FWK(30) = 0 .0020 is valid for both the 1 sand 2sstates and we use this value for the 3 sstate as well. On this level of accuracy ( δFWK(30)≃0.0001) there is no shift of the 2 plevels [18] and we use a zero value for them. 2.5 Fitting of one-loop self energy contributions We separate from the expression for the self-energy part of t he one-loop correction (7) the function G(Z) =A60+/angbracketlefthigher-order terms /angbracketright. (17) Using numerical values of A60from Refs. [19,20] and ones of G(Z) from Refs. [21,22], we performed several types of ﬁtting for these func tions. We started with ﬁtting (I)with function A60+ (Zα)/parenleftbigg A71ln1 (Zα)2+A70/parenrightbigg , (18)6 Vladimir G. Ivanov and Savely G. Karshenboim minimizing the sum /summationdisplay Z/braceleftigg/tildewideG(Z)−/bracketleftbig A60+ (Zα)/parenleftbig A71ln/parenleftbig 1/(Zα)2/parenrightbig +A70/parenrightbig/bracketrightbig δG(Z)/bracerightigg2 , (19) with respect to A70andA71for 1sstate (where Z= 1. . .5) and to A70 for 2sstate (where Z= 5,10). In the latter case we used the fact that A71(1s) =A71(2s). The statistical error of data /bracketleftig δ/tildewideG(Z)/bracketrightig2 = [δnumG(Z)]2+/bracketleftig π2(Zα)3A(0) 70/bracketrightig2 . (20) contains uncertainty of numerical integrations in Refs. [2 1,22] and of the ﬁt in (17) due to neglecting of higher-order terms of absolute o rderα(Zα)8m (where A(0) 70is a result of preliminary ﬁtting with δ/tildewideG=δnum/tildewideG). To estimate the additional systematic uncertainty which or iginates from the unknown term of order α(Zα)7we studied a sensitivity of the ﬁt (I)to introduction of some perturbation function h(z) /tildewideG(Z) =G(Z) +h(Z), (21) The ﬁnal uncertainty of the ﬁt was calculated as a random sum o f diﬀerences of the ﬁts without function hand with h(Z) from the binomial expansion of the expression (Zα)2/parenleftbigg ln1 (Zα)2+π/parenrightbigg3 (22) for the 1 sand 2sstates and (Zα)2/parenleftbigg ln1 (Zα)2+π/parenrightbigg2 (23) for the 2 p-states and for the diﬀerence G2s−Gns(see below). The logarithm ln/parenleftbig 1/(Zα)2/parenrightbig in the expansion (7) is a large value3at very lowZbut it is quite a smooth function of Zaround Z= 7 (see Table 1). Due to that, we can also use a non-logarithmic ﬁtting functio n G(Z0) +A(Z−Z0) +B 2(Z−Z0)2(24) with smooth behaviour at Z∼7. In particular, we applied the Eq. (24) for numerical data from Ref. [21,22] at Z= 3,4,5 with central value Z0= 4 (ﬁt II) and Z= 5,10,15 with Z0= 10 (ﬁt III). For the 2 sstate we also performed independent ﬁts for G1sand the dif- ference G1s−G2s, ﬁnding G2sas their combination. Values of corresponding 3Note that a natural value for the constant term is about ln/parenleftbig k0(ns)/parenrightbig ∼3).Lamb Shift in Light Hydrogen-Like Atoms 7 Table 1. Function ln/parenleftbig 1/(Zα)2/parenrightbig for small Z Zln/parenleftbig 1/(Zα)2/parenrightbig Zln/parenleftbig 1/(Zα)2/parenrightbig 1 9.840 7 5.949 2 8.454 8 5.682 3 7.643 9 5.446 4 7.068 10 5.235 5 6.622 15 4.424 6 6.257 20 3.849 ﬁts are labeled (IV)(ﬁts for low Z),(V)(Z= 3,4,5 for 1 sandZ= 5,10,15 for the diﬀerence G1s−G2s) and(VI)(both ﬁts for Z= 5,10,15). That can be useful because data on the 1 sstate is more accurate [21,22], and in case of diﬀerence the uncertainty is smaller (cf. Eqs. (22) and (2 3)) and one of higher-order parameters is known ( A71= 0) [23,13]. Only few data are available for 3 s[24] at Z= 10,20,30. . .and we perform two ﬁtting: ﬁt IwithZ= 10,20 and ﬁt IIIwithZ= 10,20,30 at Z0= 20. Diﬀerent ﬁtting functions are plotted in Figs. 1–3 for n= 1, n= 2, G1s−G2sandG2s−G3s. The points with error bars are for numerical values obtained in Refs. [24,21,22].
arXiv:physics/0009070v1 [physics.chem-ph] 24 Sep 2000Nuclear spin conversion in formaldehyde P.L. Chapovsky∗ Institute of Automation and Electrometry, Russian Academy of Sciences, 630090 Novosibirsk, Russia (January 1, 2014) Abstract Theoretical model of the nuclear spin conversion in formald ehyde (H 2CO) has been developed. The conversion is governed by the intram olecular spin- rotation mixing of molecular ortho and para states. The rate of conversion has been found equal γ/P= 1.4·10−4s−1/Torr. Temperature dependence of the spin conversion has been predicted to be weak in the wide t emperature range T= 200 −900 K. 03.65.-w; 31.30.Gs; 33.50.-j; Typeset using REVT EX ∗E-mail: chapovsky@iae.nsk.su 1I. INTRODUCTION It is well known that many symmetrical molecules exist in nat ure only in the form of nuclear spin isomers [1]. The spin isomers demonstrate anom alous stability. For example, the ortho and para isomers of H 2survive almost 1 year at 1 atm and room temperature [2]. Existence of spin isomers is well-understood in the framewo rk of the spin-statistics relation in quantum mechanics. On the other hand, the dynamical part o f the problem, viz., the isomer stability, their conversion rates and responsible c onversion mechanisms are less clear. This is because experimental data on spin isomers are rare du e to substantial diﬃculties in preparation of enriched spin isomer samples. Thus each new e xperimental result in this ﬁeld deserves close attention. In this paper we perform theoretical analysis of the nuclear spin conversion in formalde- hyde (H 2CO). This process was considered theoretically previously [3]. But it is worthwhile to perform new analysis in order to account new important inf ormation. First, the formalde- hyde spin conversion was recently investigated experiment ally and the gas phase conversion rate was measured for the ﬁrst time [4]. Second, formaldehyd e molecular structure and molecular spectroscopic parameters have been determined v ery accurately [5–7]. Third, simple theoretical model of nuclear spin conversion in asym metric tops was developed and tested by the spin conversion in ethylene [8]. All these circ umstances allow to advance signiﬁcantly theory of the spin conversion in formaldehyde . II. DIRECT AND INDIRECT CONVERSION MECHANISMS There are two main mechanisms known for the gas phase spin con version in molecules. The ﬁrst one, direct, consists in the following. In the course of collision inhom ogeneous magnetic ﬁeld produced by collision partner induces direct transitions in the test molecule between spin states. This mechanism is responsible for the h ydrogen isomer conversion by paramagnetic O 2[9]. It is important for us that the hydrogen conversion indu ced by O 2(the rate is equal to 8 ·10−6s−1/Torr [2]) appears to be much slower than the spin conversion in formaldehyde [4], γexp/P= (1.1±0.3)·10−3s−1/Torr. (1) 2Note that we refer here to the formaldehyde ortho-para equil ibration rate. In notations of [4] this rate is equal to k1+k2, where k1andk2are the ortho-to-para and para-to-ortho conversion rates, respectively. Hydrogen is an exceptional molecule due to its anomalously l arge ortho-para level spacing and high symmetry. It is more appropriate for the present dis cussion to consider the direct mechanism in polyatomic molecules. The isomer conversion i n CH 3F induced by O 2was investigated theoretically and experimentally in [10,11] . It was found that this mechanism provides the conversion rate on the order of 10−4−10−5s−1/Torr. These rates refer to the collisions with paramagnetic O 2which have magnetic moment close to Bohr magneton, µB. An experimental arrangement in [4] corresponds to “nonmagn etic” collision partner having a magnetic moment on the order of nuclear magneton, µn, thus by 103times smaller. The conversion rate by direct mechanism depends on magnetic mom ent as µ2[10]. It implies that the conversion by direct process is too slow to be taken into a ccount for the formaldehyde conversion. In fact, the authors [4] had arrived at the same c onclusion. Second mechanism is applicable to molecules in which diﬀere nt spin states can be quan- tum mechanically mixed by weak intramolecular perturbation. Mixing and interruption of this mixing by collisions with the surrounding particles re sult in the isomer conversion. We will refer to this mechanism as quantum relaxation . Quantum relaxation can provide sig- niﬁcantly faster conversion than direct process does for no nmagnetic particles. It can be illustrated by a few examples in which quantum relaxation wa s established to be a leading process. The most studied case is the spin conversion in13CH3F (for the review see [12]). The conversion rate in this molecule was determined as (12 .2±0.6)·10−3s−1/Torr. The conversion rate in ethylene (13CCH 4) is equal to (5 .2±0.8)·10−4s−1/Torr [13]. In this molecule too the quantum relaxation is the leading process [ 8]. Thus one can conclude that for the spin conversion in formaldehyde quantum relaxation is an appropriate mechanism. The formaldehyde molecules have two nuclear spin isomers, o rtho (total spin of the two hydrogen nuclei, I= 1) and para ( I= 0). Each rotational state of the molecule can belong only to ortho, or to para isomers. Thus the molecular states a re divided into two subspaces as it is shown in Fig. 1. Simple physical picture of spin conve rsion by quantum relaxation is given elsewhere, e.g., [12]. Quantitative description of t he process can be performed in the framework of the density matrix formalism. The result of thi s description is as follows [14]. 3One has to split the molecular Hamiltonian into two parts, ˆH=ˆH0+ˆV , (2) where the main part of the Hamiltonian, ˆH0, has pure ortho and para states as the eigen- states; the perturbation ˆVmixes the ortho and para states. If at initial instant the non equi- librium concentration of, say, ortho molecules, δρo(t= 0), was created, the system will relax then exponentially, δρo(t) =δρo(0)e−γt, with the rate γ=/summationdisplay a∈o,a′∈p2ΓF(a\|a′) Γ2+ω2 aa′(Wo(α) +Wp(α′)) ;F(a\|a′)≡/summationdisplay ν∈o,ν′∈p\|Vαα′\|2. (3) where Γ is the decay rate of the oﬀ-diagonal density matrix el ement ραα′(α∈ortho;α′∈ para) assumed here to be equal for all ortho-para level pairs; ωaa′is the gap between the states aanda′;Wo(α) and Wp(α′) are the Boltzmann factors of the corresponding states. The sets of quantum numbers α≡ {a, ν}andα′≡ {a′, ν′}consist of the degenerate quantum numbers ν,ν′and the quantum numbers a,a′which determine the energy of the states. In Eq. (3) and further the ortho states will be denoted by unprim ed characters, but para states by primed characters. For the following it is convenient to i ntroduce the strength of mixing , F(a\|a′). In the deﬁnition of F(a\|a′) in Eq. (3) the summation is made over all degenerate states. The general model (3) was tested comprehensively by convers ion in symmetric tops (13CH3F,12CH3F) and asymmetric top,13CCH 4. It was proven that the ortho-para mixing is performed in these molecules by intramolecular hyperﬁne interactions. Thus, the spin isomer conversion gives an alternative access to very weak i ntramolecular forces which was investigated previously by high resolution spectroscopy, e.g., Laser Stark Spectroscopy [15] and Microwave Fourier Transform Spectroscopy [16]. To avoid confusion we stress that γfrom Eq. (3) gives the equilibration rate in the system if one would measure the concentration of ortho (or pa ra) molecules. The authors [4] introduced the ortho-to-para ( k1) and para-to-ortho ( k2) rates which are equal, respectively, to the ﬁrst and second terms in the expression (3) for γ. Thus γ≡k1+k2. Relation between the formaldehyde ortho and para partition functions (see be low) explains the equality k2= 3k1[4]. 4III. ROTATIONAL STATES OF FORMALDEHYDE The formaldehyde molecule is a prolate, nearly symmetric to p having symmetry group C2v. The characters of the group operations and its irreducible representations are given in the Table 1. The molecular structure and orientation of the m olecular system of coordinates are given in Fig. 2. The formaldehyde is a planar molecule hav ing the following parameters in the ground state rCH= 1.1003±0.0005˚A,rCO= 1.2031±0.0005˚A, and αHCO= 121.62±0.05o[6]. Rotational states of formaldehyde in the ground electronic and vibrational state can be determined with high accuracy using the octic order Hamilto nian of Watson [17,5,7], ˆH0=1 2(B+C)J2+ (A−1 2(B+C))J2 z−∆JJ4−∆JKJ2J2 z−∆KJ4 z +HJJ6+HJKJ4J2 z+HKJJ2J4 z+HKJ6 z +LJJKJ6J2 z+LJKJ4J4 z+LKKJJ2J6 z+LKJ8 z +1 4(B−C)F0−δJJ2F0−δKF2+hJJ4F0+hJKJ2F2+hKF4+lKJF4, (4) whereJ,Jx,Jy, andJzare the molecular angular momentum operator and its project ions on the molecular axes. The B,C, andAare the parameters of a rigid top which characterize the rotation around x,y, andzmolecular axes, respectively (see Fig. 2). The rest of param eters account for the centrifugal distortion eﬀects [17]. In Eq. ( 4) the notation was used Fn≡Jn z(J2 x−J2 y) + (J2 x−J2 y)Jn z. (5) We left in the Hamiltonian (4) only those terms for which mole cular parameters in [5] were not set to zero. It is convenient to diagonalise the Hamiltonian (4) in the Wa ng basis [1], \|α, p > =1√ 2/bracketleftig \|α >+(−1)J+K+p\|α >/bracketrightig ; 0< K≤J, \|α0, p > =1 + (−1)J+p 2\|α0>;K= 0. (6) Herep= 0,1;\|α >are the symmetric-top rotational states; the sets of quantu m numbers areα≡ {J, K, M };\|α >≡ {J,−K, M};α0≡ {J, K= 0, M}where J,K, and Mare the quantum numbers of angular momentum and its projection on th e molecular symmetry axis and on the laboratory quantization axis, respectively. Dep ending on the parity of J,Kand 5p, the states (6) generate 4 diﬀerent irreducible representa tions of the molecular symmetry group C 2v, as it is explained in the Table 1. In the following we will nee d the reduction of the matrix elements of full symmetric operator ˆVin the basis \|α, p > to the matrix elements of symmetric-top states \|α >. This reduction reads < α, p \|V\|α′, p′>=δp,p′/bracketleftig < α\|V\|α′>+(−1)J′+p′< α\|V\|α′>/bracketrightig ;K′>0, < α, p \|V\|α′ 0, p > =δp,p′1 + (−1)J′+p′ √ 2< α\|V\|α′ 0>;K′= 0. (7) The molecular Hamiltonian, ˆH0, is full symmetric (symmetry A 1). Consequently, the matrix elements between the states of diﬀerent symmetry dis appear. Thus diagonalization of the total Hamiltonian in the basis of (6) is reduced to the dia gonalization of four independent submatrices, each for the states of particular symmetry. Th e rotational states of asymmetric top can be expanded over the basis states (6), \|β, p > =/summationdisplay KAK\|α, p >, (8) where AKstands for the expansion coeﬃcients. The summation variabl e,K, is shown explicitly in (8), although AKdepends on other quantum numbers as well. All coeﬃcients in the expansion (8) are real numbers because the Hamiltonia n (4) is symmetric in the basis \|α, p > . Complete description of the asymmetric-top quantum state n eeds indication of all expan- sion coeﬃcients, AK, from (8) which is not practical. There are a few schemes for a bbreviate notations, see, e.g., [18]. We will use here the notations wh ich are somewhat better adopted to the consideration of the spin isomer problem in asymmetri c tops [8]. We will designate the rotational states of asymmetric top by indicating p,Jand prescribing the allowed K values to the eigen states keeping both in ascending order. F or example, the eigen state having p= 0, J= 20, the allowed Kin the expansion (8) equal K= 0,2,4. . .20 and being the third in ascending order will be designated by ( p= 0, J= 20,K= 4). Note the diﬀerence between the two characters KandK. It gives unambiguous notation of rotational states for each of the four species A 1, A2(K-even) and B 1, B2(K-odd). This classiﬁcation becomes exact for a prolate symmetric top for which K=K. Calculation of the level energies and wave functions of mole cular quantum states were performed in the paper numerically. Accuracy of these calcu lations can be estimated by 6comparing with the experimental rotational spectra in the g round state of formaldehyde [5]. This comparison shows that the accuracy of the calculations for most rotational states is in the range of 10 −100 kHz. This should be suﬃcient for the investigation of the spin isomer conversion in formaldehyde. The two equivalent hydrogen nuclei in H 2CO have spin 1/2. It implies that the total wave function (product of spin and spatial wave functions) i s of symmetry B 1. Spin states of the two hydrogen nuclei can be either triplet (ortho, I= 1, symmetry A 1), or singlet (para, I= 0, symmetry B 1). In order to have the total wave function of symmetry B 1, the ortho molecules should have the spatial wave function of symmetry B 1, but the para molecules should have the spatial wave function of symmetry A1. Consequently, in the ground electronic and vibrational state the rotational sta tes A 1and B 1are only positive (even in parity), but the states A 2and B 2are only negative (odd in parity). The ortho states of the formaldehyde molecule can be present ed as \|µ >=\|β, p > \|I= 1, σ >;K −odd. (9) where σis the projection of the nuclear spin Ion the laboratory quantization axis. The para states can be presented as \|µ′>=\|β′, p′>\|I′= 0>;K′−even. (10) The Boltzmann factors Wo(α) and Wp(α′) in Eq. (3) determine the population of the states αandα′in the ortho and para families, ρα=ρoWo(α);ρα′=ρpWp(α′), (11) where ρoandρpare the total densities of ortho and para molecules, respect ively. The partition functions for ortho and para molecules at room tem perature (T=300 K) are found to be equal to Zortho= 2.16·103;Zpara= 721 . (12) In the calculation of these partition functions the energie s of the rotational states were determined numerically using the Hamiltonian (4). The dege neracy over M,σ, as well as the restrictions imposed by the quantum statistics were tak en into account. 7IV. MIXING OF THE ORTHO AND PARA STATES There are two known intramolecular perturbations able to mi x ortho and para states in polyatomic molecules. The ﬁrst one is the spin-spin inter action between the molecular nuclei. This interaction has simple form in formaldehyde an d can be expressed as [1] ˆVSS=P12ˆI(1)ˆI(2)• •T; Tij=δij−3δi,xδj,x;P12=µ2 p/r3I(1)I(2)h , (13) where ˆI(1)andˆI(2)are the spin operators of the hydrogen nuclei 1 and 2; µpis magnetic moment of proton; iandjare the Cartesian indices. The second rank tensor ˆI(1)ˆI(2)acts on spin variables. The second rank tensor Trepresents a spatial part of the spin-spin interaction. One can deduce from the angular momentum algeb ra that all matrix elements of the perturbation ˆVSSbetween ortho and para states of formaldehyde are vanishing . This is because one cannot draw a triangle having the sides 2, 1, an d 0, which are the rank of the tensor ˆI(1)ˆI(2), the total spin of ortho, and para states, respectively. The second intramolecular perturbation which should be con sidered is the spin-rotation coupling between spins of hydrogen nuclei and molecular rot ation. The spin-rotation cou- pling can be presented in general as [19,18,20] ˆVSR≡/summationdisplay nˆV(n) SR=1 2/parenleftigg/summationdisplay nˆI(n)•C(n)•ˆJ+h.c./parenrightigg ;n= 1,2. (14) HereCis the spin-rotation tensor; ˆJis the angular momentum operator. Index nin (14) refers only to the hydrogen nuclei because we are interested now in the perturbation able to mix ortho and para states. Calculation of the spin-rotational tensor, C, is a complicated problem. Further, a few simpliﬁcations will be made. First, we neglect small contri bution to the tensor Cdue to the electric ﬁelds at the position of protons [21]. The remainin g part of the tensor Coriginates from the magnetic ﬁelds produced by the electrical currents in the molecule. One can split the tensor Cinto two terms, C=eC+nC, (15) whereeCandnCare due to the electron and nuclear currents in the molecule, respectively. The electron part,eC, appears as an average over electron state, which has nonzer o value 8only in the second order perturbation theory. Its calculati on needs the knowledge of electron excited states and thus is rather diﬃcult to perform. On the o ther hand, the nuclear part, nC, has nonzero value for the ground electron state and can be ea sily estimated. In the following we will neglect the electron part,eC, and will use the estimation C≃nC. This can be considered as an upper limit for Cbecause molecular electrons are following the rotation of nuclear frame and thus compensate partially the magnetic ﬁeld produced by nuclei. The spin-rotation tensornCcan be presented (in Hz) as [21] C(n)=/summationdisplay k/negationslash=nbk[(rk•Rk)1−rkRk]•B; bk= 2µpqk/c¯hR3 k, (16) whereRkis the radial vector from the proton H(n)to the charge k;rkis the radial from the center of mass to the particle k;qkare the nuclei’ charges; Bis the inverse matrix of inertia moment. Bis a diagonal matrix having the elements Bxx= 68.0 GHz, Byy= 77.67 GHz, andBzz= 563 .9 GHz. Index kruns here over all nuclei in the molecule except the proton n. Using the symmetry operation C 2one can prove the equality of the two matrix elements, < µ\|V(1) SR\|µ′>=< µ\|V(2) SR\|µ′>. Thus for the evaluation of the spin-rotation coupling in formaldehyde it is suﬃcient to calculate one matrix element , e.g., < µ\|V(1) SR\|µ′>. We write this matrix element using an expansion over symmetric-top s tates (8), < µ\|V(1) SR\|µ′>=< I= 1, σ\| /summationdisplay K,K′AKA′ K′< α, p \|V(1) SR\|α′, p′> \|I′= 0> . (17) This expression and (7) reduce the calculation of the spin-r otation matrix elements of asym- metric tops to the calculation of symmetric-top matrix elem ents. Solution for the latter can be found in [22,20,23], which allows to express the strength of mixing in formaldehyde by ˆVSRas FSR(a′\|a) =1 4(2J′+ 1)(2 J+ 1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay K>0,qAK+qA′ KΦ(J, K′+q\|J′, K′) +1 + (−1)J′+p′ √ 2A1A′ 0Φ(J,1\|J′,0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (18) Hereq=±1; In (18) the notation was used 9Φ(J, K\|J′, K′) =/summationdisplay l√ 2l+ 1Cl,q J′l J −K′q K ×  y(J)(−1)l  J′J l 1 1J  +y(J′)  J J′l 1 1J′   , (19) where (: : :) stands for the 3j-symbol; {: : :}stands for the 6j-symbol; y(J) = /radicalig J(J+ 1)(2 J+ 1); Cl,qare the spherical components of the spin-rotation tensor of the rank l(l= 1,2) for the ﬁrst proton calculated in the molecular frame. Cl,qcan be determined using Eq. (16). For the formaldehyde molecular structure fr om the Ref. [6] and bare nuclei’ charges these components are C2,1= 3.39 kHz; C1,1=−3.39 kHz . (20) We stress that these values give an upper limit to the C-tensor. The selection rules for the ortho-para mixing by spin-rotat ion perturbation in formalde- hyde read ∆p= 0; \|∆J\| ≤1. (21) Parity of K′andKis opposite. V. CONVERSION RATES For the calculations of the isomer conversion rate one needs the value of the ortho-para decoherence rate, Γ, see Eq. (3). Experimental determinati on of this parameter can be based on the level-crossing resonances in spin conversion. Such m easurements were performed so far only for the13CH3F spin conversion and gave Γ /P≃2·108s−1/Torr (see the discussion in [12]). Estimation of Γ can be done using the pressure line b roadening data. For polar molecules the line broadening is on the order of ∼108s−1/Torr. Further, we will assume the decoherence rate being equal to Γ/P= 1·108s−1/Torr. (22) The same value of Γ was used in [4]. It is clear from Eq. (3) that only close ortho-para level pair s can contribute signiﬁcantly to the spin conversion. Formaldehyde is a light molecule hav ing large level spacing. The 10average density of levels in the range 0 −1000 cm−1is low, 1 level per each 5 cm−1. As was pointed out already in [3], there are regular and “accidenta l” ortho-para resonances. An example of regular resonances ( p= 0, J,K= 1)−(p′= 0, J,K′= 0) is shown in Fig. 3. The ortho-para gaps in this sequence of states goes rapidly down as∼exp(−0.23J). Analogous phenomenon of collapsing ortho and para states exists also i n ethylene where decrease of gaps is even faster [8]. Calculated spin conversion rates are given in the Table 2. Th e total conversion rate combines contributions from all ortho-para level pair havi ngJup to 40 and \|ω\|<40 GHz. Thus the spin conversion rate in formaldehyde is γ/P= 1.4·10−4s−1/Torr. (23) One can conclude from the data presented in Table 2 that the cl ose sequence of states from Fig. 3 does not contribute signiﬁcantly to the conversi on. The same eﬀect was found in the ethylene conversion [8]. Another observation from the T able 2 is that there are no close ortho-para level pairs in formaldehyde which can be mixed by the spin-rotation coupling. The most important ortho-para level pair which contributes more than 50% to the total rate is the pair (1,18,2)–(1,17,3). It has the energy gap ≃6 GHz. The expansion coeﬃcients, AK, for these states are presented in Fig. 4. One can see from the se data that the wave functions of these states are rather close to the symmetric- top case which would have just one term in the expansion (8). VI. DISCUSSION AND CONCLUSIONS Calculated value of the conversion rate in formaldehyde was found to be almost 10 times smaller than the experimental one [4]. There are two main unc ertainties in the present calculations which both can be deﬁned more accurately by fut ure experiments. The ﬁrst uncertainty comes from the decoherence rate, Γ. This parame ter can be determined by careful study of the pressure broadening of rotational line s in the formaldehyde ground state. The second uncertainty originates from poor knowled ge of the spin-rotation coupling in formaldehyde. The spin-rotation coupling in H 2CO can be investigated using high res- olution spectroscopy methods, e.g., Laser Stark Spectrosc opy [15] and Microwave Fourier 11Transform Spectroscopy [16] which were proven to be eﬃcient for the investigation of hy- perﬁne interactions in molecules. In general, it is diﬃcult to verify the mechanism of spin conv ersion in formaldehyde by comparing single values of the theoretical and experimenta l conversion rates, which depends in theory on a number of parameters and in experiment can be re sulted from a few eﬀects, e.g., chemical reactions. It is more appropriate to compare the dependencies predicted by the theory with the experimental dependencies. First, the m odel predicts rather strong dependence of the conversion rate on the type of buﬀer gas. By varying the collision partner one can change the decoherence rate, Γ, by nearly one order of magnitude. In the same proportion the spin conversion rate should be changed if con version is governed by quantum relaxation. It is alarming that the authors [4] observed ver y small change of γby adding the argon gas up to the pressure of 760 Torr. Another experimental veriﬁcation of the conversion mechan ism could be the investigation of temperature dependence. The theoretical temperature de pendence of the spin conversion rate in formaldehyde is shown in the Fig. 5. The calculation w ere done under an assumption that the decoherence rate, Γ, is temperature independent. S uch assumption is supported by the slow temperature dependence of Γ observed in the case o f spin conversion in13CH3F [12]. The theoretical model predicts rather weak inﬂuence o f the gas temperature on the formaldehyde conversion rate, γ, in the wide range of temperatures, T= 200 −900 K. In conclusion, we have developed theoretical model of the sp in conversion in formalde- hyde. Although this model has the same basic concepts as the m odel developed in [3], the key parameters of the new model are more precise. First of all , it refers to the molecular level energies and the wave functions. We have analysed the relation between the theoretical model of the formaldehyde spin conversion and the experiment [4]. The theoretical model in its present form gives 10 times smaller conversion rate than the rate measured in [4]. Two ty pes of experiments have been proposed which can help to resolve the puzzle. 12REFERENCES [1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon Press, Oxford, 1981). [2] A. Farkas, Orthohydrogen, Parahydrogen and Heavy Hydrogen (Cambridge University Press, London, 1935), p. 215. [3] R. F. Curl, Jr., J. V. V. Kasper, and K. S. Pitzer, J. Chem. P hys.46, 3220 (1967). [4] G. Peters and B. Schramm, Chem. Phys. Lett. 302, 181 (1999). [5] R. Bocquet and et al, J. Mol. Spectrosc. 177, 154 (1996). [6] S. Carter and N. C. Handy, J. Mol. Spectrosc. 179, 65 (1996). [7] H. S. P. Muller, G. Winnewisser, J. Demaison, A. Perrin, a nd A. Valentin, J. Mol. Spectrosc. 200, 143 (2000). [8] P. L. Chapovsky and E. Ilisca, (2000), http://arXiv.org /abs/physics/0008083. [9] E. Wigner, Z. f. Physikal Chemie 23, 28 (1933). [10] P. L. Chapovsky, Chem. Phys. Lett. 254, 1 (1996). [11] B. Nagels, P. Bakker, L. J. F. Hermans, and P. L. Chapovsk y, Chem. Phys. Lett. 294, 387 (1998). [12] P. L. Chapovsky and L. J. F. Hermans, Annu. Rev. Phys. Che m.50, 315 (1999). [13] P. L. Chapovsky, J. Cosl´ eou, F. Herlemont, M. Khelkhal , and J. Legrand, Chem. Phys. Lett.322, 414 (2000). [14] P. L. Chapovsky, Phys. Rev. A 43, 3624 (1991). [15] G. Duxbury, International Reviews in Physical Chemist ry4, 237 (1985). [16] A. Bauder, in Microwave Fourier Transform Spectroscopy , Vol. 20 of Vibrational spectra and structure , edited by J. R. During (Elsivier, Amsterdam, 1993), pp. 157 –188. [17] J. K. G. Watson, in Aspects of quartic and sexic centrifugal eﬀects on rotation al energy levels, Vol. 6 of Vibrational spectra and structure , edited by J. R. During (Elsivier, 13Amsterdam, 1977), pp. 1–89. [18] C. H. Townes and A. L. Shawlow, Microwave Spectroscopy (McGraw-Hill Publ. Comp., New York, 1955), p. 698. [19] G. R. Gunther-Mohr, C. H. Townes, and J. H. Van Vleck, Phy s. Rev. 94, 1191 (1954). [20] E. Ilisca and K. Bahloul, Phys. Rev. A 57, 4296 (1998). [21] K. Bahloul, M. Irac-Astaud, E. Ilisca, and P. L. Chapovs ky, J. Phys. B: At. Mol. Opt. Phys.31, 73 (1998). [22] K. I. Gus’kov, Zh. Eksp. Teor. Fiz. 107, 704 (1995), [JETP. 80, 400-414 (1995)]. [23] K. I. Gus’kov, J. Phys. B: At. Mol. Opt. Phys. 32, 2963 (1999). 14Table 1. The character table for the C 2vsymmetry group and the classiﬁcation of the basis states (6). E C2σvσ′ vK-even K=0K-oddSign A11 1 1 1 p=0J, p-even, – + B21 -1 -1 1 – – p=1 - A21 1 -1 -1 p=1J, p-odd, – - B11 -1 1 -1 – – p=0 + 15Table 2. The most important ortho-para levels and their cont ributions to the spin con- version in formaldehyde. Level pair Energy ω/2π FSR γ/P p′, J′,K′-p, J,K(cm−1) (MHz) (10−2MHz2) (10−5s−1/Torr) 1,18,2–1,17,3 445.87 -6162 6.59 7.56 0,8,2–0,9,1 120.93 9252 0.98 2.38 1,21,4–1,22,3 692.52 -10379 12.5 1.55 0,13,2–0,14,1 253.82 17211 2.45 0.91 1,12,2–1,13,1 221.18 -19305 2.01 0.69 Total rate 14.0 160200400 para orthoVα'α α' α Level energy (cm-1) FIG. 1. Ortho and para states of formaldehyde (H 2CO). The levels are calculated using the molecular parameters from Ref. [7]. Bent lines indicate tra nsitions inside the ortho and para sub- spaces induced by collisions. The level pair most important for the spin conversion in formaldehyde in shown to be mixed by intramolecular perturbation V 17z x y 1 2 FIG. 2. Formaldehyde molecule, H 2CO, and orientation of the molecular system of coordinates. 1801020304050100101102103104105106 Ortho-para gaps (MHz) J FIG. 3. Frequency gaps in the egular sequence of close ortho ( p= 0,J,K= 1) and para p′= 0,J,K′= 0) states in formaldehyde. 190 10 20-0.40.00.40.8 Expansion coefficients, AK K FIG. 4. The expansion coeﬃcients, AK, for the states most important for the spin conversion in H2CO. (o)–ortho state ( 1,J= 17,K= 3); (•)–para state ( 1,J′= 18,K′= 2). 200200 400 600 80010000.00.40.81.21.62.0 Conversion rate (10-4 s-1/Torr) Temperature (K) FIG. 5. Temperature dependence of the total spin conversion rate in formaldehyde. 21
arXiv:physics/0009071v1 [physics.atom-ph] 25 Sep 2000Relativistic corrections to the dipole polarizability of t he ground state of the molecular ion H+ 2 V.I. Korobov Institute of Theoretical Atomic and Molecular Physics Harvard-Smithsonian Center for Astrophysics Harvard, Cambridge, MA 02138 and Joint Institute for Nuclear Research 141980, Dubna, Russia The recently reported precise experimental determination of the dipole polarizability of the H+ 2 molecular ion ground state [P.L. Jacobson, R.A. Komara, W.G . Sturrus, and S.R. Lundeen, Phys. Rev. A 62, 012509 (2000)] reveals a discrepancy between theory and ex periment of about 0 .0007a3 0, which has been attributed to relativistic and QED eﬀects. In present work we analyze an inﬂuence of the relativistic eﬀects on the scalar dipole polarizabil ity of an isolated H+ 2molecular ion. Our conclusion is that it accounts for only 1/5 of the measured di screpancy. I. INTRODUCTION Recent measurements [1,2] of the scalar electric dipole pol arizability of H+ 2molecular ion through the study of H2molecule states with one Rydberg electron stimulated the in troduction of methods [3]– [6] which are able to accurately describe wave functions of molecular ions with t wo heavy nuclei beyond the adiabatic approximation. The accuracy for the dipole polarizability constant ( ∼10−7a3 0) reached in the last work [6] in its turn become a challenge to experiment. The new experimental work [2] subs tantially increases the accuracy of measurements and reveals a discrepancy of about 0 .0007a3 0between theory and experiment, which can not be accounted fo r within purely nonrelativistic approximation. In present work we c onsider relativistic corrections of order α2to the dipole polarizability of the ground state of an isolated H+ 2molecular ion. II. THEORY The nonrelativistic Hamiltonian of the hydrogen molecular ion H+ 2is H0=−1 2M∇2 1−1 2M∇2 2−1 2m∇2+1 R12−1 r1−1 r2, (1) We adopt atomic units ( e= ¯h=m= 1) throughout this paper. The interaction with an external electric ﬁeld (details of the nonrelativistic treatment of the problem can be found in previous papers [3]– [6]) is expressed by Vp=En·d, (2) where d=µrc=/parenleftbigg2M 2M+m+ 2m 2M+m/parenrightbigg /bracketleftbigg r−r1+r2 2/bracketrightbigg is the electric dipole moment of the three particles with res pect to the center of mass of the system. Without loss of generality we assume that n·d=µzc. The Breit α2correction to the nonrelativistic Hamiltonian is describe d by an operator VB=α2/braceleftbigg −p4 8m3+4π 8m2[δ(r1) +δ(r2)] +1 2m2/bracketleftbigg[r1×p] r3 1+[r2×p] r3 2/bracketrightbiggσ 2/bracerightbigg . (3) Then the total Hamiltonian reads, H=H0+VB+Vp. (4) Let us deﬁne the ground state nonrelativistic wave function as follows 1(H0−E0)Ψ0= 0. (5) In the nonrelativistic case the change of energy due to polar izability of molecular ion is expressed by E(2) p=/ang⌊ra⌋ketleftΨ0\|Vp(E0−H0)−1Vp\|Ψ0/ang⌊ra⌋ketright=E2µ2/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright=−1 2α0 sE2, (6) and α0 s=−2µ2/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright. (7) Let us introduce H1=H0+VB, then the scalar dipole polarizability αswith account of relativistic corrections can be rewritten in a form (we assume that VB≈α2H0and ΨB 0= Ψ0+ ΨB) α1 s=−2µ2/ang⌊ra⌋ketleftΨB 0\|zc(E1−H1)−1zc\|ΨB 0/ang⌊ra⌋ketright =−2µ2/ang⌊ra⌋ketleftΨB 0\|zc/bracketleftbig (E0−H0)−1+ (E0−H0)−1(VB− /ang⌊ra⌋ketleftVB/ang⌊ra⌋ketright)(E0−H0)−1+. . ./bracketrightbig zc\|ΨB 0/ang⌊ra⌋ketright =−2µ2/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright −2µ2/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1VB(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright −2µ2/parenleftBig /ang⌊ra⌋ketleftΨB\|zc(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright+/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1zc\|ΨB/ang⌊ra⌋ketright/parenrightBig ,(8) and ΨB= (E0−H0)−1VB\|Ψ0/ang⌊ra⌋ketright. Thus relativistic correction to the scalar dipole polariz ability αsis reduced to evaluation of the following matrix elements ∆αs=−2µ2/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1(VB− /ang⌊ra⌋ketleftVB/ang⌊ra⌋ketright)(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright −2µ2/ang⌊ra⌋ketleftΨ0\|VB(E0−H0)−1zc(E0−H0)−1zc\|Ψ0/ang⌊ra⌋ketright −2µ2/ang⌊ra⌋ketleftΨ0\|zc(E0−H0)−1zc(E0−H0)−1VB\|Ψ0/ang⌊ra⌋ketright.(9) At this point we can note that the spin–orbit term does not con tribute to αssince the magnetic dipole operator has selection rules m′=m±1. III. VARIATIONAL NONRELATIVISTIC WAVE FUNCTION Variational wave function describing the ground state of th e hydrogen molecular ion is taken in a form Ψ0=∞/summationdisplay i=1/bracketleftBig Cicos(νiR12) +Disin(νiR12)/bracketrightBig ×e−αir1−βir2−γiR12+ (r1↔r2).(10) Hereαi,βi,γi, and νiare parameters generated in a quasirandom manner, αi=/floorleftbigg1 2i(i+ 1)√pα/floorrightbigg [(A2−A1) +A1], where ⌊x⌋designates the fractional part of x,pαandqαare some prime numbers, the end points of an interval [ A1, A2] are real variational parameters. Parameters βi,γi, and νiare obtained in a similar way. Details of the method and discussion of various aspects of its application can be foun d in our previous papers [7,8]. The perturbed function Ψ 1=µ(E0−H0)−1rcΨ0is expanded in the similar way Ψ1=∞/summationdisplay i=1r1/bracketleftBig ˆCicos(νiR12) +ˆDisin(νiR12)/bracketrightBig ×e−αir1−βir2−γiR12+ (r1↔r2).(11) Technically evaluation of the matrix elements in Eq. (9) can proceed in the following way. For the matrix element in the ﬁrst line we need to solve one linear equation, ( E0−H0)Ψ1=zcΨ0, and then average operator ( VB− /ang⌊ra⌋ketleftVB/ang⌊ra⌋ketright over Ψ 1. To get rid of singularities in the solutions of implicit equ ations of lines 2 and 3 of Eq. (9) one can solve a sequence of equations from the right to the left in the second line and in the reverse order for the third line. In the latter two cases solution as well as the rhs of the last equati on should be projected onto subspace orthogonal to Ψ 0. 2IV. RESULTS AND CONCLUSIONS In Table I results of numerical calculations are presented. For the zeroth-order approximation a wave function with a basis set of N= 800 has been used that yields the nonrelativistic energy E0=−0.59713906312340(1) , (12) which is in a good agreement with our previous accurate resul t [8]. Here mp= 1836 .152701 meis adopted. As seen from the Table convergence for the relativistic correction is slower due to singular operators encountered in the matri x elements. Nevertheless we can conclude from this Table that the resulting value is ∆αs=−0.00015214(1) . (13) Combining this result with the nonrelativistic value from t he Table I one obtains that the static electric dipole polarizability of H+ 2molecular ion ground state with relativistic α2corrections is to be αs= 3.1685737(1) . (14) We see that thus obtained value for the dipole polarizabilit y does not fully account for present disagreement between theory and experiment (comparison of our results with resul ts of previous calculations and experiment are presented in Table II). Our consideration does not include leading ord er QED corrections but usually they are one order of magnitude smaller than relativistic corrections and have a diﬀerent sign. Thus they could not cover the rest 4/5 of the discrepancy. On the other hand the experimental value for the dipole polar izability has been deduced from the eﬀective model Hamiltonian [11] which is a fully nonrelativistic Hamilton ian and it does not include the retardation Casimir–Polder eﬀect [12] for the Rydberg electron. On the importance of thi s phenomena has been pointed out in a paper of Babb and Spruch [13]. So, our conclusion is that the Casimir–Polder p otential has to be included into the eﬀective Hamiltonian to meet the requirements of the present level of experimenta l accuracy. That will enable to deduce scalar electric dipole polarizability in a more reliable way. This work was supported by the National Science Foundation t hrough a grant for the Institute for Theoretical Atomic and Molecular Physics at Harvard University and Smit hsonian Astrophysical Observatory, which is gratefully acknowledged. [1] P.L. Jacobson, D.S. Fisher, C.W. Fehrenbach, W.G. Sturr us, and S.R. Lundeen, Phys. Rev. A 56, R4361 (1998); 57, 4065(E) (1998). [2] P.L. Jacobson, R.A. Komara, W.G. Sturrus, and S.R. Lunde en, Phys. Rev. A 62012509 (2000). [3] J. Shertzer and C.H. Greene, Phys. Rev. A 58, 1082 (1998). [4] A.K. Bhatia and R.J. Drachman, Phys. Rev. A 59, 205 (1999). [5] R.E. Moss, Phys. Rev. A 58, 4447 (1998). [6] J.M. Taylor, A. Dalgarno, and J.F. Babb, Phys. Rev A 60, R2630 (1999). [7] V.I. Korobov, D. Bakalov, and H.J. Monkhorst, Phys. Rev. A,59, R919 (1999). [8] V.I. Korobov, Phys. Rev. A, 61, 064503 (2000). [9] V.B. Berestetsky, E.M. Lifshitz, and L.P. Pitaevsky, Quantum Electrodynamics , (Oxford, Pergamon, 1982). [10] H. Bethe and E. Salpeter, Quantum Mechanics of One– and Two–Electron Atoms , Springer-Verlag, 1957. [11] W.G. Sturrus, E.A. Hessels, P.W. Arcuni, and S.R. Lunde en, Phys. Rev. A 44, 3032 (1991). [12] H.B.G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948). [13] J.F. Babb and L. Spruch, Phys. Rev. A 50, 3845 (1994). 3α0 s ∆αs N= 400 3.1685962 −1.52065[ −4] N= 600 3.1687252 −1.52140[ −4] N= 800 3.1687258 −1.52137[ −4] TABLE I. Dipole polarizability of H+ 2(0,0). Convergence of the numerical results with the size of bas is set. αs Experiment [2] 3.16796(15) Shertzer and Greene [3] 3.1682(4) Bhatia and Drachman [4] 3.1680 Moss [5] 3.16850 Taylor, Dalgarno, Babb [6] 3.1687256(1) present work nonrelativistic 3.1687258 withα2corrections 3.1685737 TABLE II. Dipole polarizability of H+ 2(0,0). Comparison with other calculations and experiment. 4
arXiv:physics/0009072v1 [physics.atom-ph] 25 Sep 2000Spin Relaxation Resonances Due to the Spin-Axis Interactio n in Dense Rubidium and Cesium Vapor C. J. Erickson, D. Levron∗, W. Happer Joseph Henry Laboratory, Physics Department, Princeton Un iversity, Princeton, New Jersey, 08544 S. Kadlecek, B. Chann, L. W. Anderson, T. G. Walker Department of Physics, University of Wisconsin-Madison, M adison, Wisconsin, 53706 (July 23, 2013) Resonances in the magnetic decoupling curves for the spin relaxation of dense alkali-metal vapors prove that much of t he relaxation is due to the spin-axis interaction in triplet di mers. Initial estimates of the spin-axis coupling coeﬃcients for the dimers (likely accurate to a factor of 2) are \|λ\|/h= 290 MHz for Rb; 2500 MHz for Cs. Hot, dense alkali-metal vapors are used to polarize the nuclear spins of noble gases, especially3He and129Xe, by spin exchange optical pumping [1]. Applications of these “hyperpolarized” noble gases in medical imaging [2] and fundamental physics [3] are of considerable cur- rent interest. Nevertheless, the basic physical limits to the eﬃciency of the spin-exchange process are still not fully understood. Under typical operating conditions of Rb-3He spin exchange polarizers (saturated Rb vapor at 200◦C in 8 amagat of3He) about 45% of the Rb spin depolarization is due to Rb-Rb collisions [4]. Here we re- port conclusive experimental evidence that much of the spin relaxation in dense alkali-metal vapors comes from the spin-axis interaction in triplet dimers. Our measured spin-axis coupling strengths also impact the ﬁeld of ul- tracold collisions, where the spin-axis interaction cause s trap-loss and aﬀects the widths and positions of Feshbach resonances [5,6]. In 1960, Anderson et al. [7] showed that the dominant eﬀect of binary collisions between alkali-metal atoms is exchange of the electron spins. In 1974 Gupta et al. [8] showed that the spin angular momentum of the alkali- metal atoms was freely exchanged with the nuclear spin of the singlet dimers, for example, Rb 2and Cs 2(where nuclear quadrupole interactions cause some spin depo- larization). In 1980, Bhaskar et al. [9] showed that Cs- Cs interactions destroy spin at about 1% of of the spin- exchange rate. The rate coeﬃcient for Cs-induced re- laxation showed little dependence on buﬀer gas pressure or species. Relaxation mechanisms in singlet or triplet dimers were expected to depend strongly on pressure. Consequently, Bhaskar et al. [9] proposed that the relax- ation was due to binary collisions between alkali-metal atoms. They pointed out that although there should be ∗Permanent Address: Nuclear Research Center, Beer Sheva, Israel.a spin-rotation interaction, Vsr=γN·Sin the triplet dimers – similar to that postulated by Bernheim [10] – the coupling coeﬃcient γwas probably too small to ac- count for relaxation in binary collisions. Instead, they suggested that the source of the relaxation was a spin- axis interaction of the classic form [11] Vsa=2λ 3S·(3ζζ−1)·S. (1) HereS=S1+S2is the total electron spin of the col- liding pair, and ζis a unit vector along the direction of the internuclear axis. The spin-axis coupling coeﬃcient λ=λ(R) arises from both the interaction energy of the electrons’ magnetic dipole moments and the spin-orbit interaction in second order [12]. Because Knize’s [13] measurements also showed little dependence of the Rb-induced relaxation on He or N 2 pressure, it came as a surprise when in 1998, Kadlecek, Anderson and Walker [14] discovered that the relaxation could be substantially suppressed by externally applied magnetic ﬁelds of a few thousand Gauss. If the interac- tion were due to pairs of alkali-metal atoms interacting on the triplet potential curve, this ﬁeld dependence sets a lower limit on the correlation time of 40 ps, the in- verse of the electron Larmor frequency at the decoupling ﬁeld and a time much longer than the duration of a bi- nary collision. Similar magnetic decoupling curves for K and Cs vapors were soon observed in our laboratories at Wisconsin and Princeton. Although we still do not understand the relaxation at He or N 2pressures of an atmosphere or more, recent low- pressure experiments leave no doubt that an important part of the spin relaxation in dense alkali-metal vapors comes from the spin-axis interaction (1), acting in triplet dimer molecules – even though the triplet dimer density is no more than 10−6that of the monomers. The key experimental observation is the existence of resonances in the magnetic decoupling curves, resonances which are predicted from the spin-axis interaction (1), and which cannot be produced by the spin-rotation interaction or by spin-1/2 species such as trimer molecules. A representative arrangement for measuring relaxation transients in Rb is shown at the top of Fig. 1. The alkali vapor and N 2buﬀer gas were contained in 1 inch spherical glass bulbs that were heated in an oven placed between the pole faces of a 10 kG electromagnet. We measured 1the number density of Rb atoms using Faraday rotation [15] of a tunable diode laser [16]. 0 10 20 30 40 50 60 70-15-10-50 Time, msln ln I(t) / I( oo)ChopperBeam CombinerMagnet OvenCellλ/4 Plate Polarizing Beamsplitter Cube Photodiode Photodiodeσ Pump Beam π Probe Beam FIG. 1. Top. Experimental apparatus for measuring the spin relaxtion of Rb vapor. Bottom . Representative transient decay curve obtained with the apparatus; at late-times the transient decay is characterized by a single exponential ti me constant. An intense pulse of 7947 ˚A circularly polarized pump laser radiation built up the spin polarization for a short (10 ms) interval. The pump laser was then blocked by the chopper wheel, and photodiodes monitored the trans- mission of a weak linearly polarized probe laser through the cell. The probe beam was resolved into its two circular components, giving positive and negative helic- ity signals V±(t) for sampling times t. The signal ra- tioI(t) =V+(t)/V−(t) greatly suppressed probe-laser noise. The diﬀerence in the attenuation coeﬃcients of the two circular components of the probe laser is pro- portional to the spin polarization P(forP≪1) of the vapor, so ln I(t) =a+bP(t), where a= lnI(∞) and bare constants. Fig. 1 shows a typical decay transient ln lnI(t)/I(∞). For early times, the decay is not expo- nential since the faster eigenmodes [17] of spin relaxation are still contributing to the polarization P. However, at late times there is a time interval, several e-foldings in length, over which ln ln I(t)/I(∞) is nearly a straight line with a slope −γ. Thus, when the polarization has decayed for a suﬃciently long time, it is characterized by a single, exponential late-time decay rate γ. The late-time decay rates of polarized Cs vapor in N 2 gas were measured as described previously [14], usingFaraday rotation of linearly polarized light both to mea- sure the Cs density and to monitor the spin-polarization. Representative measurements of γin87Rb are shown in Fig. 2(a), where one can recognize 3 well-resolved res- onances on the magnetic decoupling curve at magnetic ﬁelds of approximately 900, 1500 and 2100 Gauss. Sim- ilar representative data for133Cs are shown in Fig. 3, where the theoretical curve has 7 poorly resolved reso- nances, which nevertheless provide a good ﬁt to the ex- perimental data. 0 1000 2000 3000 4000 5000 6000-25000-20000-15000-10000-50000500010000150002000025000 Magnetic Field (Gauss)E/h, MHz Anti-Crossings0 1000 2000 3000 4000 5000 6000 80 90 100 110 120 130 140 150 160 170 180 190 Magnetic Field, Gauss Rate, 1/s Resonances due to SPIN-AXIS Interaction λ/h = 290 MHz 1 /τ = 3.36 x 109 s-1 Binary Relaxation and DiffusionQuadrupole RelaxationSpin-Axis FIG. 2. Top. Late-time decay rate of87Rb versus mag- netic ﬁeld B. Resonant enhancements of the decay rate are observed at the ﬁelds B1= 915 G, B2= 1526 G andB3= 2136 G, in agreement with the predicted values BI=A(2I+ 1)/4gSµB. The cell contained 0.061 amagat of N2gas. The cell temperature was 220 C. The solid line is a theoretical ﬁt assuming a ﬁeld-independent contribution to γof 97 sec−1from binary collisions and diﬀusion to the cell walls, a ﬁeld-dependent contribution from quadrupole inte rac- tions in singlet dimers, calculated as outlined in Ref. [8], and a contribution from a spin-axis interaction in triplet dime rs, calculated according to Eq. (4). Bottom . Energy levels for a representative spin space with I= 2. The level anticrossings are responsible for the resonances in the magnetic decoupli ng curves. Also shown in Fig. 2(b) are representative energies of the spin sublevels of the triplet dimer molecules of87Rb for a value of the total nuclear spin quantum number I= 2. The resonances in the relaxation rate occur at the 2same magnetic ﬁelds as anticrossings in the energy level diagrams. The anticrossings are the result of the spin- axis interaction (1), acting in a total spin Hamiltonian for homonuclear triplet dimers of the form H=A 2S·I+gsµBBSz−λ 3S·(3nn−1)·S.(2) HereAis the magnetic-dipole hyperﬁne coupling coeﬃ- cient for the free alkali atoms ( A(87Rb)/h= 3,417.4 MHz andA(133Cs)/h= 2,298.2 MHz). The operator for the total nuclear spin, I=I1+I2is the sum of the nuclear spin operators I1andI2for the two atoms. The operator I·Icommutes with (2), and for homonuclear dimers its eigenvalues I(I+ 1) can have I= 0,1,2, . . .,2Im, where Im=I1=I2is the nuclear spin quantum number of a monomer. The Hamiltonian (2) can therefore be diago- nalized in a spin subspace Iof dimension 3(2 I+1) to ﬁnd eigenfunctions \|i/an}bracketri}ht=\|Ii/an}bracketri}htand eigenvalues Ei=EIi. For λ= 0 there is a curious degeneracy in the spin subspace I/ne}ationslash= 0, with 2 I+ 1 sublevels crossing at the magnetic ﬁeldBI=A(2I+ 1)/4gSµB. The last term in (2) is the spin-axis interaction (1), where we assume that the rota- tional angular momentum Nof the dimers is large enough that we may treat it as a classical vector, pointing along the unit vector n=N/N, and make the replacement (3ζζ−1)→ −(3nn−1)/2. We estimate the relaxation caused by the interaction (2) with the following simple model. Let the total longi- tudinal spin operator of the triplet dimer be Gz=Sz+Iz. When triplet dimers are formed in dense alkali-metal va- pors with low spin polarization, the spin density matrix of the dimers will be very nearly ρ(0) = eβGz/Z, where β≈4P≪1 is the spin-temperature parameter [7]. The partition function is very nearly Z= 3(2 Im+ 1)2. The spin of the dimer at the instant of formation is therefore /an}bracketle{tGz0/an}bracketri}ht= 2(I2 m+Im+ 1)β/3, where we have neglected non-linear terms in the small parameter β. The mean change in the density matrix by the time it breaks up again is ∆ ρ=/an}bracketle{tUρ(0)U−1/an}bracketri}ht −ρ(0), where the time evolution operator is U=e−iHt/¯h, and the angle brackets /an}bracketle{t· · ·/an}bracketri}htdenote an average over all directions of n, and also an average over the probability e−t/τdt/τthat the dimer will break up in the time interval between t andt+dt. The collisional breakup rate of the dimers is 1/τ. The mean change in the dimer spin is there- fore/an}bracketle{t∆Gz/an}bracketri}ht= Tr{Gz∆ρ}=−W/an}bracketle{tGz0/an}bracketri}ht, where the spin- destruction probability is W= 2/summationdisplay ij\|/an}bracketle{ti\|Gz\|j/an}bracketri}ht\|2(ωijτ)2 (2Im+ 1)2[(2Im+ 1)2+ 3][1 + ( ωijτ)2],(3) and the Bohr frequencies are ωij= (Ei−Ej)/¯h. An average over the directions of nis to be understood in (3). Almost all of the spin is carried by monomers of number density [Ak] ( e.g., [Ak]=[Rb] or [Ak]=[Cs]), which have a spin density per unit volume [Ak] β(4I2 m+4Im+3)/12. The spin loss rate per unit volume from triplet dimersof number density [Ak 2] is [Ak 2]/an}bracketle{t∆Gz/an}bracketri}ht/τ, from which one obtains the relaxation equation −(d/dt)lnβ=γ= κ[Ak], where the rate coeﬃcient is κ= 2/bracketleftbigg(2Im+ 1)2+ 3 (2Im+ 1)2+ 2/bracketrightbiggWK τ. (4) The chemical equilibrium coeﬃcient of the triplet dimers isK= [Ak 2]/[Ak]2. We have used theoretical triplet dimer potential curves of Krauss and Stevens [18] to es- timate K(Rb) = 330 ˚A3andK(Cs) = 650 ˚A3at repre- sentative temperatures of 220◦C and 180◦C respectively. 200 100 0Relaxation rate (1/s) 12840 Field (kGauss)Pressure (Torr) 1039100397 FIG. 3. Late-time ﬁeld-dependent decay rate of133Cs versus magnetic ﬁeld and pressure at 180◦C, [Cs]= 6 .6×1014 cm−3. The cell temperature was 177 C. The ﬁts to the data use the reorientation model with negative λ. Non-vanishing values of λlift the degeneracy at the ﬁeldBIand convert the level crossings into anticrossings, while making the corresponding matrix element /an}bracketle{ti\|Gz\|j/an}bracketri}ht non-zero, as is illustrated in Fig. 2. It is these anticross- ings of the energy levels that are responsible for the res- onances, most clearly visible in Fig. 2. Remarkably, the spin-rotation interaction does not break the degeneracy so the spin-axis interaction must be responsible for the relaxation resonances. Since there are 2 Imnonvanish- ing values of Ifor atoms with the nuclear spin quantum number Im, we expect there to be 3 resonaces for87Rb withIm= 3/2, as observed, and 7 resonances, at ﬁelds between 600 G and 3100 G for133Cs with Im= 7/2, which ﬁts well the observed ﬁeld dependence shown in Fig 3. We ﬁt the experimental data to theoretical curves that include a ﬁeld-independent part due to diﬀusion to the cells walls and binary collisions, a ﬁeld-dependent nu- clear quadrupole relaxation in singlet dimers (negligible for Cs), and relaxation due to the spin-axis interaction 3in triplet dimers. We expect a small portion of the re- laxation in triplet dimers to arise from the spin-rotation interaction, and we may estimate the magnitude of this relaxation mechanism by assuming that the spin-rotation interaction of Rb-Rb triplet dimers is the same as that that of Rb-Kr, and that the spin-rotation interaction for Cs-Cs is the same as that for Rb-Xe, both of which are ex- perimentally known from the work of Bouchiat et al.[19]. These estimates should be fairly reliable since the spin- orbit interaction that produces the spin-rotation couplin g [1] varies only slightly from Rb to Kr or Cs to Xe. Using these estimates for γN·S, we ﬁnd that the addition of the spin-rotation interaction causes a few percent change in the predicted relaxation rates. In the model described above, the relaxation due to triplet dimers dependends on two parameters: λand the molecular lifetime τ, assuming the theoretical val- ues for Kare accurate. The molecular lifetime should be inversely proportional to pressure, making it natu- ral to introduce a cross-section σ= (τ[N2]v)−1. Our Rb data is well-described over the limited pressure range from 50 Torr to 300 Torr by the values \|λ\|= 290 MHz, andσ= 290 ˚A2. The latter is somewhat larger than the typical 150 ˚A2breakup cross sections deduced from magnetic decoupling studies of RbKr [19]. The sign of λ cannot be determined from the data. Since there is little contribution to Cs relaxation from singlet molecules, relaxation from Cs-Cs triplet molecule s can be observed over a wider range of pressures. Fig. 3 shows a sampling of our spin-relaxation data for Cs as a function of magnetic ﬁeld at a variety of N 2pressures. In this case the data are well-described by λ=−2.9 GHz, σ= 98 ˚A2,K= 350 ˚A3, or, assuming the op- posite sign of λ,λ= 3.4 GHz, σ= 82 ˚A2,K= 307 ˚A3. The deduced values of Kare about a factor of 2 smaller than predicted from the potential curves (which are believed to be fairly reliable). We have also ﬁt the data with a model that allows for collisional reorienta- tion of the triplet molecules without breakup, making the assumption that a spin-temperature distribution de- scribes the dimer density matrix after reorientation. In this case τin Eq. 3 is replaced by the coherence time τc, andW=Wτ/(τc+ (τ−τc)W) replaces Win Eq. 4. In this case we obtain λ=−2.13 GHz, σc= 81˚A2,σ= 42 ˚A2,K= 584 ˚A3, orλ= 2.71 GHz, σc= 93˚A2,σ= 55 ˚A2,K= 431 ˚A3. We emphasize that single values of λandσcaccurately represent the magnetic ﬁeld depen- dences at 9 pressures (not all shown in Fig 3), covering nearly two decades in pressure. Spin-axis coupling in Cs triplet dimers is also of cur- rent interest for studies of ultracold collisions [5] since it produces loss of spin-polarized atoms in magnetic traps, and aﬀects the widths and positions of Feshbach reso- nances in Cs collisions [6]. The magnitude and sign of λ, as well as its dependence on interatomic separation R were predicted by Mies et al. [12]. Previous analysis [5] led to the conclusion that in order to agree with exper- iment the second-order spin-orbit calculated by Mies etal.needed to be scaled by a factor of SC= 4.0±0.5, recently updated to 3 .2±0.5 [6]. We ﬁnd, by averaging λ(R) over the ro-vibrational levels of the triplet state us- ing the Krauss and Stevens potentials [18], scaling factors ofSC= 12 for Rb and SC= 3.6 for Cs are needed to reproduce our data. In conclusion, the resonances observed in the magnetic decoupling curves of Rb, coupled with the observed pres- sure and ﬁeld dependences of Cs, prove that a major con- tribution to the spin relaxation of hot, dense alkali-metal vapors is caused by the spin-axis interaction in triplet dimers. Knowing this key piece of the physics should help to unravel the still-puzzling independence of the re- laxation rates on He and N 2pressures of an atmosphere or more [14]. Support for this research came from NSF (Wisconsin & Princeton), and NIH (Princeton). [1] T. G. Walker and W. Happer, Rev. Mod. Phys. 69, 629 (1997). [2] M. S. Albert, G. D. Cates, B. Driehuys, W. Happer, B. Saam, C. S. Springer and A. Wishnia, Nature, 370, 199 (1994). [3] T. E. Chupp, R. J. Hoare, R. L. Walsworth, and Bo Wu, Phys. Rev. Lett., 72, 2363 (1994). [4] A. Ben-Amar Baranga, S. Appelt, M. V. Romalis, C. J. Erickson, A. R. Young, G. D. Cates, and W. Happer, Phys. Rev. Lett. 80, 2801 (1998). [5] P. Leo, E. Tiesinga, P. Julienne, D. Walter, S. Kadlecek, and T. Walker, Phys. Rev. Lett. 81, 1389 (1998) [6] P. Leo, C. Williams, and P. Julienne, preprint. [7] L. W. Anderson, F. M. Pipkin and J. C. Baird, Phys. Rev.116, 87 (1959). [8] R. Gupta, W. Happer, G. Moe and W. Park, Phys. Rev. Lett.32574 (1974); Note that in Eq. 7 of this paper, the numerical coeﬃcient 3 /40 should have been 3 /160. [9] N. D. Bhaskar, J. Pietras, J. Camparo, and W. Happer, Phys. Rev. Lett., 44, p. 930 (1980). [10] R. A. Bernheim, J. Chem. Phys. 36, 135 (1962). [11] M. H. Hebb, Phys. Rev. 49, p. 610 (1936); C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, p. 182, Dover Publications, New York (1975). [12] F. H. Mies, C. Williams, P. Julienne, and M. Krauss, J. Res. Natl. Inst. Stand. Technol. 101, 521 (1996). [13] R. Knize, Phys. Rev. A 40, 6219 (1989). [14] S. Kadlecek, L. W. Anderson and T. G. Walker, Phys. Rev. Lett. 80, 5512 (1998). [15] Z. Wu, M. Kitano, W. Happer, M. Hou, and J. Daniels, Applied Optics 25, 4483 (1986) [16] K. B. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys.60, 1098 (1992). [17] W. Happer, Rev. Mod. Phys. 44, 169 (1972). [18] M. Krauss and W. J. Stevens, J. Chem. Phys. 93, 4236 (1990). 4[19] M. A. Bouchiat, J. Brossel, and L. C. Pottier, J. Chem. Phys.56, 3703 (1972). 5
arXiv:physics/0009073v1 [physics.atom-ph] 25 Sep 2000Non-destructive spatial heterodyne imaging of cold atoms S. Kadlecek, J. Sebby, R. Newell, and T. G. Walker Department of Physics, University of Wisconsin-Madison, M adison, Wisconsin, 53706 (February 20, 2014) We demonstrate a new method for non-destructive imag- ing of laser-cooled atoms. This spatial heterodyne techniq ue forms a phase image by interfering a strong carrier laser bea m with a weak probe beam that passes through the cold atom cloud. The ﬁgure of merit equals or exceeds that of phase- contrast imaging, and the technique can be used over a wider range of spatial scales. We show images of a dark spot MOT taken with imaging ﬂuences as low as 61 pJ/cm2at a detuning of 11Γ, resulting in 0.0004 photons scattered per atom. In this paper we demonstrate a new “spatial hetero- dyne” method for non-destructive imaging of trapped atoms. As with other non-destructive techniques, spatial heterodyne imaging minimizes the number of absorbed photons required for an image and is therefore particu- larly useful in applications such as Bose-Einstein Con- densation [1], magnetic trapping, and far-oﬀ-resonance trapping that are particularly sensitive to heating and optical pumping from absorbed photons. Oﬀ resonant, non-destructive imaging of clouds of trapped atoms [2] has been previously demonstrated us- ing several diﬀerent methods, all of which image the phase shift produced by the atoms on a collimated probe laser: dark-ground imaging [3], polarization- rotation imaging [4], and phase-contrast imaging [5]. Non-destructive detection without imaging was recently demonstrated using FM spectroscopy [7]. The most pop- ular of these methods, the phase-contrast technique, uses a small ( ∼10−100µm)π/2 phase mask that is inserted into the imaging laser focus at the Fourier plane of an imaging lens. In the image plane the π/2 phase-shifted laser ﬁeld interferes with the signal ﬁeld produced by the atoms to give an image intensity that is linear with re- spect to the atom-induced phase shift. To implement spatial heterodyne imaging, we used two laser beams: a carrier laser beam which does not pass through the trapped atoms, and a probe beam which is phase shifted as it passes through the atom cloud. The beams are coincident on a CCD camera and straightfor- ward digital image processing techniques use the result- ing interference pattern to reconstruct the phase shift due to the cloud. Spatial heterodyne imaging has several practical ad- vantages for non-destructive imaging. First, there is no need for precision fabrication and alignment of a phase plate. Second, it has a signiﬁcant signal-to-noise advan- tage for low imaging intensities. Third, at high intensitie s it has a larger signal per absorbed photon, allowing the large dynamic range of CCD cameras to be better used. Fourth, the method works over a wide range of spatialscales. Finally, rejection of spurious interference fring es due to various optical elements such as vacuum windows is automatically accomplished. The principle of spatial heterodyne imaging is similar to heterodyne spectroscopy [6], with interference occur- ing in the spatial rather than the temporal domain. As shown in Fig. 1a) a probe beam of intensity Iptravels through a cloud of trapped atoms and accumulates a po- sition dependent phase shift φ(r) due to the index of re- fraction of the atoms. A lens placed in this beam images the atom cloud onto a CCD detector. A carrier beam of intensity Ic, and derived from the same laser as the probe beam interferes with the probe beam at an angle θ. For convenience, we assume equal radii of curvature for the carrier and probe beams. The interference pat- tern on the CCD detector I(r) is a set of straight line fringes whose position is determined by an overall phase shift between the beams χ, and which are distorted by the accumulated phase shift from the atoms: I(r) =Ic+Ip+ 2/radicalbig IcIpcos(χ+ 2πθˆk⊥·r/λ−φ(r)) (1) from which φ(r) can be reconstructed. Here ˆk⊥is a unit vector pointing along the direction of the component of the carrier wavevector kperpendicular to the direction of the probe beam. FIG. 1. Apparatus. The phase shift φis most easily determined in two lim- its:θ≪δ/λ(parallel mode) and θ≫δ/λ(tilted mode), where δis the desired resolution element on the image. In the parallel mode the phase of the interference pat- tern is uniform across the cloud image, and the resulting interference pattern is (with χ=π/2): I(r) =Ic+Ip+ 2/radicalbig IcIpsinφ(r) (2) IfIc=Ip, this is identical to phase contrast imaging. If not, the signal size is increased by a factor of/radicalbig Ic/Ip. 1The spatial variation of the phase shift from the cloud becomes a spatial variation of the intensity at the CCD detector, producing a real image on the detector. In prac- tice the phase shift χbetween the two beams must be stabilized using feedback. For this paper we have implemented spatial hetero- dyne imaging in the tilted mode. In this case a set of high spatial frequency fringes appear and the eﬀect of the atom cloud is to give a spatially varying phase shift to these fringes. The analysis of the fringes then pro- ceeds in a manner highly analogous to lock-in detection: we demodulate the interference pattern to zero spatial frequency and apply a low-pass ﬁlter to the result. FFT techniques make the demodulation and ﬁltering eﬃcient (2 sec for a 784 ×520 pixel camera on a 400 MHz Pen- tium). It is not necessary to stabilize the relative phase between the probe and carrier beams. To demonstrate the method we use an atom cloud with an on-resonant optical thickness of about 15 in a dark spot87Rb MOT [8,9]. The experimental arrange- ment is shown in Fig. 1b. Typically 3 ×107atoms from a MOT are accumulated in the dark state at a den- sity of roughly 5 ×1011cm−3by imaging a 1 mm ob- struction in the MOT repumping laser onto the trap. The atoms in the dark spot are quite sensitive to reso- nant light and hence absorption imaging is diﬃcult. The imaging laser beam is tuned in the range of 2 −11Γ away from the87Rb 5S 1/2(F=1)→5P1/2(F=2) resonance, switched via an acousto-optic modulator, and then split into two beams by a non-polarizing beamsplitter. The probe beam is attenuated by a factor of 1-200 by a neu- tral density ﬁlter before passing through the atom cloud, which is imaged onto a CCD array. An interference ﬁl- ter placed in the Fourier plane of the imaging lens rejects 780 nm ﬂuorescence from the bright state trapped atoms. The carrier passes around the vacuum chamber and is in- cident on the CCD detector tilted at an angle θ≈1 deg. For convenience, we roughly match the radii of curvature of the probe and carrier beams at the CCD, to produce nearly straight fringes. We tilt the fringes at an angle of typically 30◦from the rows of the CCD chip to avoid aliasing. Two competing factors determine the optimum angle θ. As with lock-in detection, it is important to modulate the signal at somewhat higher spatial frequency than the smallest feature to be resolved. The ﬁnite camera pixel size sets an upper limit on the modulation frequency without loss of fringe contrast. We ﬁnd that a fringe spacing of 4-5 pixels is a good compromise between res- olution and fringe contrast. In the parallel mode the full resolution of the camera is acheived. To begin processing we subtract oﬀ reference images of each laser beam. This leaves only the interference term in Eq. 1, which we Fourier transform. The transform contains the phase image information in two sections cen- tered on spatial wavenumbers k0=±2πθ/λ. We shift one of these sections to zero spatial frequency and attenuate the high frequencies with a ﬁlter, typically the GaussianFIG. 2. a) Image of a dark spot MOT taken at a probe de- tuning ∆ = −11Γ and carrier-to-probe intensity ratio r= 20. b) Image optimized for minimum light scattering: ∆ = −11Γ, r= 60 : 1. This image required approximately 0.0004 pho- tons to be scattered per atom. c) Side-on image of a dark sheet trap formed by an 59 µm wire image in the repumping beam. The image of the cloud is approximately 51 µm. The right-hand scale is the phase shift in radians. ﬁlter exp(-(3 k/k0)2). Finally, we take the inverse trans- form whose phase (tan−1(Im/Re)) is φ(r). This proce- dure automatically reduces spurious interference fringes that arise from various optical elements since they are likely to be at the wrong spatial frequency. To compen- sate for slight curvature of the interference fringes, we subtract φ(r) from another image, similarly processed, but taken in the absence of atoms. This also reduces distortion due to spatial inhomogeneities in IcandIp. Figure 2 shows several images φ(r) taken using the above procedure. At a typical line-center optical thick- ness of 3-15 we have successfully imaged the dark spot trap for a variety of detunings and carrier-to-probe inten- sity ratios r. Fig. 2a) shows a typical image with r= 20, ∆ =−11Γ, and about 1 .2×10−3scattered photons per atom. As another example, Fig. 2b) shows an image taken at ∆ = −11Γ and r= 60. The total ﬂuence used to make the image was only 61 pJ/cm2, corresponding to 0.0004 photons scattered per atom. The S/N ratio on a given resolution element is about 10 for this image. 2Depending on the details of the imaging system, ﬁlter- ing of the Fourier transform may limit the spatial res- olution of the ﬁnal image. In our system, with a mag- niﬁcation of 5 and a CCD pixel spacing of 8.8 µm, the resolution is limited to about 20 µm, compared to a the- oretical diﬀraction limit of about 5 µm. Fig. 2 shows an image of a 50 µmwide trap. Depending on the application, the ﬁgure of merit for spatial heterodyne imaging is comparable with or supe- rior to phase constrast imaging. For simplicity, we con- sider here the parallel mode. The intensity pattern is Ic+Ip+2/radicalbig IcIpcos(χ−φ(x)),IcandIpare measured in numbers of photons. For small phase shifts and χ≈π/2, the signal size is approximately 2 η/radicalbig IcIpφwhere ηis the quantum eﬃciency of the detector, typically ∼0.3 for CCD chips in the near infrared. Noise sources include shot noise and other sources of technical noise, b, such as the camera read noise and ﬁnite resolution of the cam- era’s A/D converter. The signal-to-noise ratio is there- fore (S/N)SH=2η/radicalbig IcIpφ/radicalbig η(Ic+Ip) +b2(3) The maximum S/Noccurs for Ic≫Ip, b2/η, giving (S/N)max= 2φ/radicalbig ηIp (4) which shows that there is a minimum number of photons that must be scattered from the atoms to acheive a given S/N. A similar relation holds for phase contrast imaging. A natural ﬁgure of merit for non-destructive imaging is the number of absorbed photons required to attain the desired signal to noise ratio. For optically thick clouds this number is greatly reduced because the probing can be done at large detuning [2]. Thus the shot-noise limited ﬁgure of merit for either technique is S/N A=2φ/radicalbig ηIp αIp≈2∆ Γ/radicalbiggη Ip(5) for ∆≫Γ. When the technical noise bis signiﬁcant, however, spa- tial heterodyne imaging has a better S/N ratio than phase contrast imaging. Figure 3 compares the S/N ra- tio per radian of phase shift for the two techniques. As with any heterodyne method, the interference between the carrier and the signal boosts the signal level at a given probe intensity. Furthermore, for highest quality images it is desirable to maximize signal size and thereby minimize the dis- cretization errors from the A/D converter. In this case the spatial heterodyne method oﬀers a√rperformance enhancement as compared to the phase contrast tech- nique. Fig. 2b) shows an image taken with r= 60, repre- senting 3 bits of increased signal size for ﬁxed absorption. We have demonstrated the spatial heterodyne method for non-destructive imaging of trapped atoms and shownFIG. 3. S/N comparison for phase contrast imaging and spatial heterodyne imaging. The camera read noise is as- sumed to be b= 25 e−, and the quantum eﬃciency is η= 0.5. that it has some advantages over other techniques. Our method is a special case of of a more general class of holographic imaging techniques that could be used with cold atoms. Support for this research came from the NSF. We ac- knowledge helpful communications with D. Jin, W. Ket- terle, and S. Rolston, and assistance from N. Harrison. [1] M. Anderson et al., Science 269, 198(1995); K. Davis et al., Phys. Rev. Lett. 75, 3969 (1995); C. Bradley, C. Sack- ett, and R. Hulet, Phys. Rev. Lett. 78, 985 (1997). [2] W. Ketterle, D.S. Durfee, and D.M. Stamper-Kurn, In ”Bose-Einstein condensation in atomic gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CXL,” edited by M. Inguscio, S. Stringari and C.E. Wieman (IOS Press, Amsterdam, 1999) pp. 67-176. [3] M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 273, 84 (1996). [4] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phs. Rev. Lett.79, 985 (1997). [5] M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, Phs. Rev. Lett.79, 553 (1997). [6] A. Yariv, Optical Electronics (New York: Holt, Rinehart, and Winston) 305 ﬀ. [7] V. Savalli, G. Horvath, P. Featonby, L. Cognet, N. West- brook, C. Westbrook, Opt. Lett. 24, 1552 (1999). [8] W. Ketterle, K. B. Davis, M. A. Joﬀe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). [9] M. H. Anderson, W. Petrich, J. R. Ensher, and E. A. Cornell, Phys. Rev. A 50, R3597 (1994). 30.690.000.60 0.000.78 0.00a b c a) b)
arXiv:physics/0009074v1 [physics.gen-ph] 25 Sep 2000Deﬁnition of the elements of length and time by means of photon D.L. Khokhlov Sumy State University, R.-Korsakov St. 2 Sumy 40007 Ukraine e-mail: khokhlov@cafe.sumy.ua Abstract It is considered the procedure of deﬁnition of the elements o f length and time by means of photon. In order to deﬁne the element of length one ne eds an a priori given element of time. In order to deﬁne the element of time one need s an a priori given element of length. It is taken the laboratory time and length as a priori given time and length respectively. It is deﬁned the elements of length and time for the reference frame moving with the velocity vand for the reference frame with the gravitational potential Φ. The elements of length and time can be deﬁned by means of photo n. Given the element of time t, one can deﬁne the element of length as a path covered by photo n for the time t l=ct. (1) Given the element of length l, one can deﬁne the element of time as a duration required for photon to cover the distance l t=l c. (2) From this it follows that in order to deﬁne the element of leng th by means of photon one needs an a priori given element of time. It is natural to take t he laboratory time as an a priori given time. In order to deﬁne the element of time by mea ns of photon one needs an a priori given element of length. It is natural to take the labo ratory element of length as an a priori given element of length. Take two reference frames and deﬁne the elements of length an d time by means of photon moving from the second frame to the ﬁrst frame. Let the ﬁrst fr ame be the laboratory one. Take the time of the ﬁrst frame as an a priori given time. Consi der the case when the velocities and gravity in the second frame are negligible v≪c, Φ≪c2. Photon moving from the second frame to the ﬁrst frame deﬁnes an euclidean el ement of length l=ct. (3) Consider the case when the second frame moves with the veloci tyvin the direction perpendicular to the direction towards the ﬁrst frame. Phot on emitted by the second frame possesses the component of velocity vin the direction of the motion of the second frame. In the direction towards the ﬁrst frame, photon possesses the c omponent of velocity c′=√ c2−v2=c/radicalBigg 1−v2 c2. (4) 1Photon moving from the second frame to the ﬁrst frame deﬁnes t he element of length l′=c′t=l/radicalBigg 1−v2 c2. (5) Take the distance between the reference frames given by eq. ( 3) as an a priori given element of length. Then photon moving from the second frame t o the ﬁrst frame deﬁnes the element of time t′=l c′=t/radicalBigg 1−v2 c2. (6) Formulas (5), (6) resemble the Lorentz reduction of length a nd retardation of time as in the special theory of relativity [1]. However in the appro ach under consideration these formulas have another interpretation. In the special theor y of relativity tandlare the proper elements of time and length in the moving frame, t′andl′are the laboratory elements of time and length in the rest frame. In the approach under considera tiontandlare the laboratory elements of time and length in the rest frame, t′andl′are the elements of time and length in the moving frame from the point of view of the rest frame. He re the notions of the proper elements of time and length are absent. The time and length fr om the point of view of the laboratory frame have the physical meaning. The reduction of length (5) and the retardation of time (6) is a consequence of that the velocity of photon in the moving frame decreases from the poi nt of view of the rest frame. From this it follows that one cannot simultaneously deﬁne th e elements t′andl′. While using the laboratory time t, one can deﬁne the length l′, and vice versa while using the laboratory length l, one can deﬁne the time t′. In the special theory of relativity it is used either the laboratory time in the rest frame and the proper le ngth in the moving frame or the laboratory length in the rest frame and the proper time in the moving frame. Therefore in spite of the diﬀerent interpretation the approach under con sideration yields the same results as the special theory of relativity. Let the second reference frame possess the gravitational po tential Φ. This case is equiva- lent to that when the second frame moves with the velocity vin the direction perpendicular to the direction towards the ﬁrst frame. Photon emitted by th e second frame possesses the component of eﬀective velocity vin the direction perpendicular to the direction towards the ﬁrst frame v=√ 2Φ. (7) In the direction towards the ﬁrst frame, photon possesses th e component of eﬀective velocity c′=√ c2−v2=c/radicalBigg 1−2Φ c2. (8) Photon moving from the second frame to the ﬁrst frame deﬁnes t he element of length l′=c′t=l/radicalBigg 1−2Φ c2. (9) 2Take the distance between the reference frames given by eq. ( 3) as an a priori given element of length. Then photon moving from the second frame t o the ﬁrst frame deﬁnes the element of time t′=l c′=t/radicalBigg 1−2Φ c2. (10) The reduction of length (9) and the retardation of time (10) c orrespond to those in the general theory of relativity [1] given by the Schwarzschild metric for the spherical symmetry gravitational ﬁeld where lcorresponds to the radial coordinate r, the gravitational potential is given by Φ = GM/r . However in the approach under consideration these formula s have another interpretation. In the general theory of relativit ytandlare the proper elements of time and length in the frame with potential Φ, t′andl′are the laboratory elements of time and length in the remote frame. In the approach under con sideration tandlare the laboratory elements of time and length in the remote frame, t′andl′are the elements of time and length in the frame with potential Φ from the point of view of the remote frame. Similar to the case with the moving frame the notions of the pr oper elements of time and length are absent. The reduction of length (9) and the retardation of time (10) i s a consequence of that the velocity of photon in the frame with potential Φ decrease s from the point of view of the remote frame. From this it follows that one cannot simultane ously deﬁne the elements t′and l′. While using the laboratory time t, one can deﬁne the length l′, and vice versa while using the laboratory length l, one can deﬁne the time t′. In the linear general theory of relativity it is used either the laboratory time in the remote frame and t he proper length in the frame with potential Φ or the laboratory length in the remote frame and the proper time in the frame with potential Φ. Therefore in spite of the diﬀerent in terpretation the approach under consideration yields the same results as the linear general theory of relativity. The results yielded in the approach under consideration are diﬀerent from those in the non-linear general theory of relativity. In particular in t he non-linear general theory of rela- tivity the physical velocity of photon in the radial directi on is deﬁned by the proper elements of length and time and is equal to the constant c. In the approach under consideration the physical velocity of photon in the radial direction is deﬁne d from the point of view of the laboratory frame (remote frame) via the laboratory time tand the reduced length r′or via the laboratory length rand the retarded time t′ c′=r′ t=r t′=c/radicalBigg 1−2Φ c2. (11) When the radial coordinate reachs the Schwarzschild radius r→rg= 2GM/c2, the veloc- ity of photon in the radial direction tends to zero c′→0. From this it follows that the Schwarzschild radius is the limiting one. References 3[1] L. Landau and E.M. Lifshitz, The classical theory of ﬁeld s, 4th Ed. (Pergamon, Oxford, 1976). 4
arXiv:physics/0009075v1 [physics.acc-ph] 26 Sep 2000 /CB/C4/BT /BV/AL/C8/CD/BU/AL/BK/BH/BH/BL/BT/D9/CV/D9/D7/D8 /BE/BC/BC/BC/C6/D9/D1/CT/D6/CX /CP/D0 /BV/CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /BV/D3/CW/CT/D6/CT/D2 /D8 /CB/DD/D2 /CW/D6/D3/D8/D6/D3/D2 /CA/CP/CS/CX/CP/D8/CX/D3/D2 /BX/AR/CT /D8/D7/CD/D7/CX/D2/CV /CC/D6/CP/BY/CX/BV /BG∗/BT/BA /C3/CP/CQ /CT/D0/CB/D8/CP/D2/CU/D3/D6/CS /C4/CX/D2/CT/CP/D6 /BT /CT/D0/CT/D6/CP/D8/D3/D6 /BV/CT/D2 /D8/CT/D6/BE/BH/BJ/BH /CB/CP/D2/CS /C0/CX/D0/D0 /CA/CS/BA/B8 /C5/CT/D2/D0/D3 /C8 /CP/D6/CZ/B8 /BV/BT /BL/BG/BC/BE/BH/C5/BA /BW/D3/CW/D0/D9/D7/B8 /CC/BA /C4/CX/D1 /CQ /CT/D6/CV/BW/CT/D9/D8/D7 /CW/CT/D7 /BX/D0/CT/CZ/D8/D6/D3/D2/CT/D2/B9/CB/DD/D2 /CW/D6/D3/D8/D6/D3/D2/C6/D3/D8/CZ /CT/D7/D8/D6/BA /BK/BF/B8 /BE/BE/BI/BC/BJ /C0/CP/D1 /CQ/D9/D6/CV/B8 /BZ/CT/D6/D1/CP/D2 /DD/BT/CQ/D7/D8/D6/CP /D8/BV/D3/CW/CT/D6/CT/D2 /D8 /D7/DD/D2 /CW/D6/D3/D8/D6/D3/D2 /D6/CP/CS/CX/CP/D8/CX/D3/D2 /B4/BV/CB/CA/B5 /D3 /D9/D6/D7 /DB/CW/CT/D2 /D7/CW/D3/D6/D8 /CQ/D9/D2 /CW/CT/D7 /D8/D6/CP 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/D3/CU /CQ /CT/CP/D1 /D4 /D3/D4/D9/D0/CP/D8/CX/D3/D2 /CP/D6/CT /CX/D1/D4/D0/CT/D1/CT/D2 /D8/CT/CS /CX/D2 /CC/D6/CP/BY/CX/BV /BG/BM /CC/CW/CT /CQ/D9/D2 /CW/D0/CT/D8/D7 /CP/D2/CT/CX/D8/CW/CT/D6 /CQ /CT /D7/CT/D8 /D3/D2 /CP /D6/CT/CV/D9/D0/CP/D6 /D0/CP/D8/D8/CX /CT /CX/D2 /D7/CX/DC/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /D4/CW/CP/D7/CT/D7/D4/CP /CT/B8 /D8/CW/CT/DD /CP/D2 /CQ /CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS/D5/D9/CP/D7/CX/B9/D6/CP/D2/CS/D3/D1/D0/DD /D3/D6 /D4/D7/CT/D9/CS/D3/B9/D6/CP/D2/CS/D3/D1/D0/DD /BA /BT/D0/D0 /D8/CW/D6/CT/CT /D1/CT/D8/CW/D3 /CS/D7 /CP/D6/CT /CQ/CP/D7/CT/CS /D3/D2 /CP /CV/CT/D2/CT/D6/CP/D0 /D1/D3 /CS/D9/D0/CT /CU/D3/D6/CV/CT/D2/CT/D6/CP/D8/CX/D2/CV /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /DB/CX/D8/CW /CP /CV/CX/DA /CT/D2 /D3/D6/D6/CT/D0/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC σik=/angbracketleftxixk/angbracketright /BA/CC/CW/CT /BV/CW/D3/D0/CT/D7/CZ/DD /CS/CT /D3/D1/B9/D4 /D3/D7/CX/D8/CX/D3/D2 C /B8 /DB/CW/CT/D6/CT CCT= Σ /B8 /CX/D7 /D9/D7/CT/CS /D8/D3 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /CP /D2/D3/D6/D1/CP/D0/B9/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /DA /CT /D8/D3/D6 /D7/CT/D5/D9/CT/D2 /CT /CX/D2 /D8/D3/D3/D2/CT /DB/CX/D8/CW /CV/CX/DA /CT/D2 M /BA /CC/CW/CT /DA /CT /D8/D3/D6 /D7/CT/D5/D9/CT/D2 /CT /CP/D2 /CQ /CT /CV/CT/D2/CT/D6/CP/D8/CT/CS /CQ /DD/BD/BA /CT/D2 /D9/D1/CT/D6/CP/D8/CX/D2/CV /CPd /B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /D4/CP/D6/CP/D0/D0/CT/D0/CT/D4/CX/D4 /CT/CS /D0/CP/D8/D8/CX /CT/BE/BA /CT/D2 /D9/D1/CT/D6/CP/D8/CX/D2/CV /CPd /B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /CB/D3/CQ /D3/D0 /D7/CT/D5/D9/CT/D2 /CT/BF/BA /CV/CT/D2/CT/D6/CP/D8/CX/D2/CV d /B9/D8/D9/D4/D0/CT/D7 /D3/CU /BZ/CP/D9/D7/D7/CX/CP/D2 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /D4/D7/CT/D9/CS/D3/B9/D6/CP/D2/CS/D3/D1 /D2 /D9/D1 /CQ /CT/D6/D7/C5/CT/D8/CW/D3 /CS /B4/BD/B5 /CX/D7 /D9/D7/CT/CS /D8/D3 /CV/CT/D2/CT/D6/CP/D8/CT /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/CX/D2/CV /CQ/D9/D2 /CW/B8 /DB/CW/CX/D0/CT /B4/BE/B5 /D3/D6 /B4/BF/B5 /CP/D6/CT /D9/D7/CT/CS /CU/D3/D6 /D8/CW/CT/D7/CP/D1/D4/D0/CX/D2/CV /CQ/D9/D2 /CW/BA /C1/D2 /D8/CW/CT /CC/D6/CP/BY/CX/BV /BG/CX/D2/D4/D9/D8 /AS/D0/CT/B8 /D8/CW/CT /D3/D6/D6/CT/D0/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC /CX/D7 /D2/D3/D8 /CV/CX/DA /CT/D2 /CS/CX/D6/CT /D8/D0/DD /B8/CQ/D9/D8 /CX/D2 /CX/D8/D7 /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CU/D3/D6/D1 /B4/CP/D7/D7/D9/D1/CX/D2/CV /DA /CP/D2/CX/D7/CW/CX/D2/CV /D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/D6/CP/D2/D7/DA /CT/D6/D7/CP/D0 /D4/D0/CP/D2/CT/D7 /CP/D2/CS/D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D4/D0/CP/D2/CT/B5/B8 /D2/CP/D1/CT/D0/DD /CQ /DD /D7/D4 /CT /CX/CU/DD/CX/D2/CV /CC /DB/CX/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 α, β, ǫ /CU/D3/D6 /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CP/D0 /D4/D0/CP/D2/CT/CP/D2/CSδincoherent , δcoherent , σz /CU/D3/D6 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D4/D0/CP/D2/CT/BA/BE/BA/BJ /C7/D9/D8/D4/D9/D8 /BW/CP/D8/CP/CC/D6/CP/BY/CX/BV /BG /CP/D2 /D4/D6/D3 /DA/CX/CS/CT /D8/CW/CT /D3/D1/D4/D0/CT/D8/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CV/CP/D8/CW/CT/D6/CT/CS /CX/D2 /CP /D6/D9/D2 /CU/D3/D6 /CU/D9/D6/D8/CW/CT/D6 /D4/D6/D3 /CT/D7/D7/CX/D2/CV/BM /D8/CW/CT/AS/CT/D0/CS/D7 /CP /D8/CX/D2/CV /D3/D2 /CT/CP /CW /D4 /D3/CX/D2 /D8 /CP/D2/CS /D8/CW/CT /D4/CW/CP/D7/CT/D7/D4/CP /CT /D3/B9/D3/D6/CS/CX/D2/CP/D8/CT/D7 /D3/CU /CT/CP /CW /D4/CP/D6/D8/CX /D0/CT/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CX/D7/CW /D9/CV/CT /CP/D1/D3/D9/D2 /D8 /D3/CU /CS/CP/D8/CP /CP/D2 /CQ /CT /D7/D9/D4/D4/D6/CT/D7/D7/CT/CS/BA /CC/D6/CP/BY/CX/BV /BG/CP/D0/D7/D3 /CP/D0 /D9/D0/CP/D8/CT/D7 /D8/CW/CT /D1/D3/D6/CT /D9/D7/CT/CU/D9/D0 /D3/D0/D0/CT /D8/CX/DA /CT/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /B4/D6/D1/D7 /DA /CP/D0/D9/CT/D7/B8 /CP /DA /CT/D6/CP/CV/CT /DA /CP/D0/D9/CT/D7/B8 /CC /DB/CX/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/B8 /D8/D6/CP/D2/D7/CU/CT/D6 /D1/CP/D8/D6/CX /CT/D7/B8 /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6 /D8/D6/CP/D2/D7/B9/CU/CT/D6 /D1/CP/D8/D6/CX /CT/D7 /D8/D3 /BE/D2/CS /D3/D6/CS/CT/D6/B8 /CW/D6/D3/D1/CP/D8/CX /CX/D8/CX/CT/D7/B5/BA /CC/CW/CT /CT/D1/CX/D8/D8/CP/D2 /CT /CX/D7 /D8/CW/CT /D9/D7/D9/CP/D0 /D7/D8/CP/D8/CX/D7/D8/CX /CP/D0 /CT/D1/CX/D8/D8/CP/D2 /CT/BN/CP/D0/D7/D3/B8 /D8/CW/CT /CP/D6/CT/CP /D3/CU /D8/CW/CT /D3/D2 /DA /CT/DC /CW /D9/D0/D0 /D3/CU /D8/CW/CT /D2/B9σ /D4/CP/D6/D8/CX /D0/CT/D7 /CX/D7 /CP/D0 /D9/D0/CP/D8/CT/CS/BA /CC/CW/CT /CS/CP/D8/CP /CX/D7 /DB/D6/CX/D8/D8/CT/D2/CX/D2 /BT/CB/BV/C1 /C1 /CU/D3/D6/D1/CP/D8 /CX/D2 /D8/D3 /CP /D7/CX/D2/CV/D0/CT /D3/D9/D8/D4/D9/D8 /AS/D0/CT/B8 /CU/D6/D3/D1 /DB/CW/CX /CW /CX/D8/D7 /D4/CP/D6/D8/D7 /B4/CT/BA/CV/BA/B8 /CC /DB/CX/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /DA/D7/BA/CQ /CT/CP/D1/D0/CX/D2/CT /D4 /D3/D7/CX/D8/CX/D3/D2/B5 /CP/D2 /CQ /CT /CT/DC/D8/D6/CP /D8/CT/CS /DB/CX/D8/CW /CP/D2 /CX/D2 /D0/D9/CS/CT/CS /D8/D3 /D3/D0 /CU/D3/D6 /CT/CP/D7/DD /D4 /D3/D7/D8/B9/D4/D6/D3 /CT/D7/D7/CX/D2/CV/BA /BY /D3/D6/CX/CS/CT/D2 /D8/CX/AS /CP/D8/CX/D3/D2 /CP/D2/CS /CS/CT/CQ/D9/CV/CV/CX/D2/CV /D4/D9/D6/D4 /D3/D7/CT/D7/B8 /D8/CW/CT /D3/D1/D4/D0/CT/D8/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /D6/D9/D2/B8 /CX/D2 /D0/D9/CS/CX/D2/CV /DA /CT/D6/B9/D7/CX/D3/D2 /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/D2 /CP/D0/D0 /D1/D3 /CS/D9/D0/CT/D7/B8 /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /D8/D3 /D8/CW/CT /D3/D9/D8/D4/D9/D8 /AS/D0/CT/BA/BE/BA/BK /CC/CW/CT /BV/D3 /CS/CT/CC/D6/CP/BY/CX/BV /BG/CX/D7 /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /BY /C7/CA /CC/CA/BT/C6/BJ/BJ /B4/AS/CT/D0/CS /CP/D0 /D9/D0/CP/D8/CX/D3/D2/B5 /CP/D2/CS /BT/C6/CB/C1 /BV/B7/B7 /B4/D8/D6/CP /CZ/CX/D2/CV/B8 /D7/CT/D8/D9/D4 /CP/D2/CS/CT/DA /CP/D0/D9/CP/D8/CX/D3/D2/B5/BA /C1/D8 /D9/D6/D6/CT/D2 /D8/D0/DD /D3/D1/D4/D6/CX/D7/CT/D7 /CP/CQ /D3/D9/D8 10 000 /D0/CX/D2/CT/D7 /D3/CU /D7/D3/D9/D6 /CT/D7 /D8/CT/DC/D8/BA /C1/D8/D7 /D3/CQ /CY/CT /D8/B9/D3/D6/CX/CT/D2 /D8/CT/CS/CP/D4/D4/D6/D3/CP /CW /CP/D0/D0/D3 /DB/D7 /CU/D3/D6 /CT/CP/D7/DD /CP/D9/CV/D1/CT/D2 /D8/CP/D8/CX/D3/D2 /CQ /DD /D2/CT/DB /D8 /DD/D4 /CT/D7 /D3/CU /CT/D0/CT/D1/CT/D2 /D8/D7/BA /BT /D7/DD/D1 /CQ /D3/D0/CX /CX/D2/D4/D9/D8 /D0/CP/D2/CV/D9/CP/CV/CT/DB/CX/D8/CW /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /D8/D3 /CS/CT/AS/D2/CT /CQ /CT/CP/D1/D0/CX/D2/CT/B9/DA /CP/D0/D9/CT/CS /CU/D9/D2 /D8/CX/D3/D2/D7 /D1/CP/CZ /CT/D7 /CX/D8 /CT/CP/D7/DD /D8/D3 /CW/CT /CZ /CQ /CT/CP/D1/D0/CX/D2/CT/CS/CT/D7/CX/CV/D2 /CP/D0/D8/CT/D6/CP/D8/CX/D3/D2/D7/BA/BHφ150 [deg]γε [mm mrad ]Τbunch,fwhm [ps] 2468101214Exp. `97, q=6 nC γεx (Exp. CTF II) γεy (Exp. CTF II) Τb (Exp. CTF II)γεx (TraFiC4) γεy (TraFiC4) Τb (TraFiC4)20406080100120140 6 7 8 9 10 11 12/BY/CX/CV/D9/D6/CT /BD/BM /BX/D1/CX/D8/D8/CP/D2 /CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D2/CS /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D8/CW/CT /BV/C4/C1/BV /CC /CT/D7/D8 /BY /CP /CX/D0/CX/D8 /DD/BF /BT/D4/D4/D0/CX /CP/D8/CX/D3/D2 /BX/DC/CP/D1/D4/D0/CT/D7/BF/BA/BD /CC/CW/CT /BV/CC/BY/BE /BU/D9/D2 /CW /BV/D3/D1/D4/D6/CT/D7/D7/D3/D6/CC/D6/CP/BY/CX/BV /BG/DB /CP/D7 /D9/D7/CT/CS /D8/D3 /D7/CX/D1 /D9/D0/CP/D8/CT /D8/CW/CT /CT/D1/CX/D8/D8/CP/D2 /CT /CS/CX/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /BV/C4/C1/BV /CC /CT/D7/D8 /BY /CP /CX/D0/CX/D8 /DD /BU/D9/D2 /CW/BV/D3/D1/D4/D6/CT/D7/D7/D3/D6/CJ/BK℄/B8 /DB/CW/CX /CW /CW/CP/D7 /CQ /CT/CT/D2 /D1/CT/CP/D7/D9/D6/CT/CS/CJ/BL℄/BA /BY/CX/CV/D9/D6/CT /BD/CJ/BL /B8 /BD/BC℄ /D7/CW/D3 /DB/D7 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6ǫx,y /CP/D2/CSσz /BA/CC/CW/CT/D6/CT /CX/D7 /D7/D3/D1/CT /D9/D2/CS/CT/D6/CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D1/CX/D8/D8/CP/D2 /CT /CV/D6/D3 /DB/D8/CW /CU/D3/D6 /CQ /CT/D2/CS/CX/D2/CV /CP/D2/CV/D0/CT/D7 /D0/CT/CU/D8 /D3/CU /D8/CW/CT /D4 /CT/CP/CZ/BA/BT /CP/D9/D7/CT /D3/CU /D8/CW/CX/D7 /D1/CX/CV/CW /D8 /CQ /CT /D8/CW/CT /D3 /DA /CT/D6/CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CQ/D9/D2 /CW /D0/CT/D2/CV/D8/CW/B8 /DB/CW/CX /CW /D1/CX/CV/CW /D8 /D7/D8/CT/D1 /CU/D6/D3/D1/CP /D2/D3/D2/B9/BZ/CP/D9/D7/D7/CX/CP/D2 /CQ/D9/D2 /CW /CX/D2 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /D7/CX/CS/CT /D3/CU /D8/CW/CT /D4 /CT/CP/CZ/B8 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/CW/D3 /DB/D7 /CP /D7/D8/D6/D3/D2/CV/D0/DD /CS/CT/DA/CX/CP/D8/CX/D2/CV /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2ǫx /CP/D2/CSǫy /BA /BT/D7ǫy /CP/D0/D7/D3 /CX/D2 /D6/CT/CP/D7/CT/D7/B8 /D3/D2/CT /D1/CP /DD /D3/D2 /D0/D9/CS/CT /D8/CW/CP/D8/D3/D8/CW/CT/D6 /D7/D3/D9/D6 /CT/D7 /D8/CW/CP/D2 /BV/CB/CA/B9/CX/D2/CS/D9 /CT/CS /CS/CX/D7/D4 /CT/D6/D7/CX/DA /CT /D1/CX/D7/D1/CP/D8 /CW /D3/CU /CT/D1/CX/D8/D8/CP/D2 /CT /D1/CX/CV/CW /D8 /CQ /CT /D6/CT/D7/D4 /D3/D2/D7/CX/CQ/D0/CT /CU/D3/D6/D8/CW/CP/D8/B8 /D7/D9 /CW /CP/D7 /D8/CW/CT /D4/D6/D3 /DC/CX/D1/CX/D8 /DD /D3/CU /D8/CW/CT /DA /CP /D9/D9/D1 /CW/CP/D1 /CQ /CT/D6 /DB /CP/D0/D0/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CX/D7 /D5/D9/CX/D8/CT/D6/CT/CP/D7/D3/D2/CP/CQ/D0/CT/B8 /CP/D2/CS /D8/CW/CT /D7/CX/CV/D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8 /CX/D7 /D0/CT/CP/D6/D0/DD /D6/CT/D4/D6/D3 /CS/D9 /CT/CS/BA/BF/BA/BE /C4/BV/C4/CB /BW/D3/CV/D0/CT/CV/BX/D1/CX/D8/D8/CP/D2 /CT /CV/D6/D3 /DB/D8/CW /CX/D2/CS/D9 /CT/CS /CQ /DD /BV/CB/CA /CP/D2/CS /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D1/CX/D7/D1/CP/D8 /CW /CX/D7 /CP /CW/CX/CV/CW/D0/DD /D3/D6/D6/CT/D0/CP/D8/CT/CS /D4/D6/D3 /CT/D7/D7/BA/CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /CX/D8 /CP/D2/B8 /CX/D2 /D4/D6/CX/D2 /CX/D4/D0/CT/B8 /CQ /CT /D9/D2/CS/D3/D2/CT/BM /D3/D2/CT /CP/D2 /D9/D2 /D8/CP/D2/CV/D0/CT /D8/CW/CT /CS/CX/D7/D8/D9/D6/CQ /CT/CS /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT/D4/CW/CP/D7/CT/D7/D4/CP /CT /CQ /DD /CP/D4/D4/D0/DD/CX/D2/CV /D8/D3 /CT/CP /CW /D4/CP/D6/D8/CX /D0/CT /D8/CW/CT /D3/D4/D4 /D3/D7/CX/D8/CT /CS/CX/D7/D4 /CT/D6/D7/CX/DA /CT /CZ/CX /CZ /CX/D8 /D7/D9/AR/CT/D6/CT/CS /CP/D8 /CP/D2 /CT/CP/D6/D0/CX/CT/D6/D7/D8/CP/CV/CT/CJ/BD/BD℄/BA /BY /D3/D6 /CP/D2 /CT/DC/CP/D1/D4/D0/CT/B8 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D3/D4/D8/CX /D7 /CU/D3/D6 /CP /D4/D6/D3/D4 /D3/D7/CT/CS /CS/D3/CV/D0/CT/CV /CX/D2/CY/CT /D8/D3/D6 /D0/CP /DD /D3/D9/D8 /CU/D3/D6 /D8/CW/CT/C4/BV/C4/CB/CJ/BD/BE℄/BA /C1/D8 /CX/D7 /D9/D7/CT/CS /D8/D3 /D8/D6/CP/D2/D7/D4 /D3/D6/D8 /CP /BD/BH/BC/C5/CT /CE /CQ /CT/CP/D1 /D8/D3 /CP /D4/CP/D6/CP/D0/D0/CT/D0 /D3/AR/D7/CT/D8 /D8/D9/D2/D2/CT/D0 /D8/CW/D6/D3/D9/CV/CW /D8 /DB /D3/CQ /CT/D2/CS/D7 /D3/CU±38 deg /BA /CC/CW/CT /CQ /CT/D2/CS/D7 /CP/D6/CT /D6/CT/CP/D0/CX/D7/CT/CS /CQ /DD /CU/D3/D9/D6 /CQ /CT/D2/CS/CX/D2/CV /D1/CP/CV/D2/CT/D8/D7/B8 /DB/CW/CX /CW /CW/CP /DA /CT /D7/CX/CV/D2/CP/D8/D9/D6/CT +1+−2− /B8 /CP/D2/CS /D7/D3/D1/CT /D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT /CW/D3/D7/CT/D2 /D7/D9 /CW /D8/CW/CP/D8 /D8/CW/CT /D8/D6/CP/D2/D7/CU/CT/D6 /D1/CP/D8/D6/CX/DC Tx,12= 1 /BA /BE /D7/CW/D3 /DB/D7/BI8.6e-078.8e-079e-079.2e-079.4e-079.6e-079.8e-07 0 2 4 6 8 10 12-0.15-0.1-0.0500.050.10.15Normalized Emittance [m] Dispersion [m] Beamline Position [m]Emittance Dispersion/BY/CX/CV/D9/D6/CT /BE/BM /BX/D1/CX/D8/D8/CP/D2 /CT /CV/D6/D3 /DB/D8/CW /D3/D1/D4 /CT/D2/D7/CP/D8/CX/D3/D2 /CP/D0/D3/D2/CV /D8/CW/CT /D4/D6/D3/D4 /D3/D7/CT/CS /C4/BV/C4/CB /CX/D2/CY/CT /D8/D3/D6 /CS/D3/CV/D0/CT/CV/D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /D4/D6/D3 /CY/CT /D8/CT/CS /CT/D1/CX/D8/D8/CP/D2 /CT /CV/D6/D3 /DB/D8/CW /CU/D3/D6 /D8/CW/CX/D7 /D7/CT/D8/D9/D4 /CP/D7 /CP/D0 /D9/D0/CP/D8/CT/CS /CQ /DD /CC/D6/CP/BY/CX/BV /BG/BN /D8/CW/CT8%/CV/D6/D3 /DB/D8/CW /CP/CU/D8/CT/D6 /D8/CW/CT /AS/D6/D7/D8 /CQ /CT/D2/CS/D7 /CX/D7 /CP/D0/D1/D3/D7/D8 /D3/D1/D4/D0/CT/D8/CT/D0/DD /CP/D2 /CT/D0/CT/CS /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D0/CP/D7/D8 /D3/D2/CT/D7/BA /CC/CW/CX/D7 /CX/D7/D4 /D3/D7/D7/CX/CQ/D0/CT /CQ /CT /CP/D9/D7/CT /D8/CW/CT /CQ/D9/D2 /CW /D6/CT/D8/CP/CX/D2/D7 /CX/D8/D7 /D0/CT/D2/CV/D8/CW/B8 /D7/D3 /D8/CW/CT /AS/CT/D0/CS/D7 /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2 /CT/CP /CW/CQ /CT/D2/CS/BA /C1/D2 /CQ/D9/D2 /CW /D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CW/CX /CP/D2/CT/D7/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D3/D2/CT /CW/CP/D7 /D8/D3 /CW/D3/D7/CT /CP /CS/CX/AR/CT/D6/CT/D2 /D8 /D7 /CW/CT/D1/CT/BA /C7/D2/CT/D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /CX/D7 /D8/D3 /D9/D7/CT /D7/CT/DA /CT/D6/CP/D0 /CW/CX /CP/D2/CT/D7 /DB/CX/D8/CW /CP/D21 /D3/D6−1 /D8/D6/CP/D2/D7/CU/CT/D6 /D1/CP/D8/D6/CX/DC /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D1 /CP/D2/CS /D8/D3/D7 /CP/D0/CT /D1/CP/CV/D2/CT/D8 /D0/CT/D2/CV/D8/CW/D7 /CP/D2/CS /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CP /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /CT/DC/D4 /CT /D8/CT/CS /AS/CT/D0/CS /D7/D8/D6/CT/D2/CV/D8/CW/D7/BA/BF/BA/BF /CC/CC/BY /BU/D9/D2 /CW /BV/D3/D1/D4/D6/CT/D7/D7/D3/D6/BX/DA /CT/D2 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /CP/CS/CY/D9/D7/D8/CP/CQ/D0/CT /D3/D4/D8/CX /CP/D0 /CT/D0/CT/D1/CT/D2 /D8/D7 /D8/CW/CT/D6/CT /CX/D7 /D6/D3 /D3/D1 /CU/D3/D6 /D3/D4/D8/CX/D1/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /D8/CT/D6/D1/D7/D3/CU /CT/D1/CX/D8/D8/CP/D2 /CT/BA /CC/CW/CT /CC/BX/CB/C4/BT /CC /CT/D7/D8 /BY /CP /CX/D0/CX/D8 /DD /BU/D9/D2 /CW /BV/D3/D1/D4/D6/CT/D7/D7/D3/D6 /C1 /C1/CJ/BD/BF ℄ /D3/D1/D4/D6/CX/D7/CT/D7 /CU/D3/D9/D6 /CQ /CT/D2/CS/CX/D2/CV/D1/CP/CV/D2/CT/D8/D7 /CP/D2/CS /D2/D3 /D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT/D7 /DB/CX/D8/CW/CX/D2 /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/DA /CT /D7/CT /D8/CX/D3/D2/BA /BV/D3/CW/CT/D6/CT/D2 /D8 /D7/DD/D2 /CW/D6/D3/D8/D6/D3/D2 /CA/CP/CS/CX/CP/D8/CX/D3/D2/CX/D7 /CP /D7/CT/D6/CX/D3/D9/D7 /CX/D7/D7/D9/CT /CV/CX/DA /CT/D2 /CX/D8/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/CT/D8 /CJ/BD/BG/B8 /BD/BH/B8 /BD/BI ℄ /C0/D3 /DB /CT/DA /CT/D6/B8 /D3/D2/CT /CP/D2 /D9/D7/CT /D8/CW/D6/CT/CT /D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT/D7/D9/D4/D7/D8/D6/CT/CP/D1 /D3/CU /D8/CW/CT /CW/CX /CP/D2/CT /D8/D3 /CP/CS/CY/D9/D7/D8 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /CC /DB/CX/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /BT/D7 /D8/CW/CT /CQ /CT/CP/D1 /CS/CX/D7/D8/D3/D6/D8/CX/D3/D2/CT/DC/CW/CX/CQ/CX/D8/D7 /CP /D2/D3/D2/B9/D0/CX/D2/CT/CP/D6 /CQ /CT/CW/CP /DA/CX/D3/D6/B8 /D8/CW/CX/D7 /CP/D2 /CQ /CT /D9/D8/CX/D0/CX/DE/CT/CS /D8/D3 /CP/D2 /CT/D0 /D7/D3/D1/CT /D3/CU /D8/CW/CT /CX/D2/CS/D9 /CT/CS /CZ/CX /CZ/D7 /CQ /DD/CP/D2 /CP/D4/D4/D6/D3/D4/D6/CX/CP/D8/CT /CW/D3/CX /CT /D3/CU /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /CQ /CT/CP/D1 /D7/CX/DE/CT/D7 /CP/D2/CS /CS/CX/DA /CT/D6/CV/CT/D2 /CT/D7/BA /B4/BT /D4/D9/D6/CT/D0/DD /D0/CX/D2/CT/CP/D6 /CQ /CT/CW/CP /DA/CX/D3/D6/DB /D3/D9/D0/CS /D0/CT/CP /DA /CT /D8/CW/CT /CT/D1/CX/D8/D8/CP/D2 /CT /CV/D6/D3 /DB/D8/CW /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D9/D2/CS/CT/D6 /D7/DD/D1/D4/D0/CT /D8/CX /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0/D4/CW/CP/D7/CT/D7/D4/CP /CT/B5/BA /BY/CX/CV/D9/D6/CT /BF /D7/CW/D3 /DB/D7 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /CP /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7 /CP/D2 /D3/CU /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /CC /DB/CX/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BN/D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /CT/D7 /CX/D2 /D7/D0/CX /CT /CT/D1/CX/D8/D8/CP/D2 /CT /D7/D8/D6/D3/D2/CV/D0/DD /D7/D9/CV/CV/CT/D7/D8 /D3/D4 /CT/D6/CP/D8/CX/D2/CV /D8/CW/CT /D3/D1/D4/D6/CT/D7/D7/D3/D6 /D2/CT/CP/D6 /D8/CW/CT /AG/D7/DB /CT/CT/D8/D7/D4 /D3/D8/AH /CP/D6/D3/D9/D2/CS α= 1.2 /B8β= 15 /D1/BJεfinal 2e-06 1.5e-06 1.2e-06 1e-06 9e-07 051015202530βinitial [m]00.511.522.53 αinitial []01e-062e-063e-064e-065e-066e-06εfinal [m]/BY/CX/CV/D9/D6/CT /BF/BM /CB/D0/CX /CT /CT/D1/CX/D8/D8/CP/D2 /CT /CV/D6/D3 /DB/D8/CW /CX/D2 /D8/CW/CT /CC/BX/CB/C4/BT /CC /CT/D7/D8 /BY /CP /CX/D0/CX/D8 /DD /BU/BV/C1 /C1 /CW/CX /CP/D2/CT /CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 /D3/CU/CX/D2/CX/D8/CX/CP/D0 /CC /DB/CX/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /B4/CU/D6/D3/D1 /CJ/BD/BJ℄/B5/BK/CA/CT/CU/CT/D6/CT/D2 /CT/D7/CJ/BD℄ /CH/BA /CB/BA /BW/CT/D6/CQ 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1Analysis of Causality Issue in Near-field Superluminally Propagating Electromagnetic and Gravitational Fields William D. Walker Royal Institute of Technology, KTH-Visby Department of Electrical Engineering Cramérgatan 3, S-621 57 Visby, Sweden bill@visby.kth.se 1 Introduction A simple relativistic analysis is presented which shows that near-field superluminal longitudinal electric signals generated by an electric dipole cannot beused to violate Einstein causality by using the relativistic “sync shift” effect a. The analysis shows that because a signal has some time extent (i.e. after a signal isinitiated one must wait at least one period for the frequency and amplitudeinformation to be determined), an electric dipole can be used to transmit a signalbackward in time, but not before the same signal was initiated, thereby prohibiting auser from changing the signal that was transmitted. Although the analysis is presentedspecifically for superluminal near-field longitudinal electric fields, the result wouldalso apply for any type of signal that is superluminal in a spatial region less than onewavelength. Therefore the superluminal near-field transverse electric and magneticfields, which are also generated by an electric dipole, cannot be used to violatecausality using the “sync shift” effect, since they are known to be superluminal in aregion less than one wavelength near the source. In addition, the gravitational fieldsgenerated by an oscillating mass, which are known to be superluminal in a region lessthan one wavelength near the source, cannot be used to violate causality using the“sync shift” effect. 2 Theoretical analysis The following relativistic analysis looks at the consequences of transmitting a superluminal near-field longitudinal electric field from an amplitude-modulatedelectric dipole to a moving electron which then reflects the signal back to the source.The result is analysed to see if it is possible to transmit a signal backward in time sothat the information can be used to change the same information that was transmitted,thereby violating causality. a Sync shift effect – Term used to refer to the relativistic effect that enables superluminal signals to be transmitted backward in time. The Lorentz transformation for time change: /G01t´= /G02 [ /G01t - (v/c2) /G01x], where /G01t = L/w and /G01x = L (transmission distance), can become negative when the signal velocity (w) is much greater than the velocity (v) of a moving observer: w > (c2)/v [3].2A stationary electric dipole transmits a longitudinal electric field toward an electron which moves away at constant velocity (v) (ref. Figure 1). Figure 1: Stationary electric dipole (A) transmitting longitudinal signal to moving electron (B) at a distance (L 1) away from the stationary electric dipole. The dipole and the moving electron are separated by distance (L 1). From the analysis presented in several papers by the author, the longitudinal electric field generated byan oscillating electric dipole is known to propagate (group speed) nearlyinstantaneously provided the propagation distance (L 1) is much less than one wavelength (~ T/10) [1, 2, 3]. It is assumed that the signal to be transmitted is an amplitude-modulated signal with a modulation wavelength ( T) (ref. Figure 2). The beginning of the signal starts at time (B T) and the end of the signal ends at time (E T). Figure 2: Diagram of an amplitude-modulated signal with modulation wavelength ( T). The modulated signal starts at time (B T) and ends at time (E T). This process can be represented graphically using a Minkowski space-time diagram (ref. Figure 3).wtStationary electric dipoleMoving electronTransmitted longitudinalsi gnal v L1AB Signal Time BT ET T3Figure 3 : Space-time diagram of an instantaneous signal ( T) propagating from stationary point (A) to moving point (B), and then instantaneously reflecting back to point (A), resulting in signal ( R). The diagram shows the space-time co-ordinates (x, ct) of the stationary electric dipole (A) superimposed with the moving electron (B) space-time co-ordinates (x´, ct´) [4].The stationary electric dipole is timelike and is represented by the (ct) axis, and themoving electron (travelling with velocity v) is represented by the (ct´) axis. The signal (wavelength = T) is transmitted nearly instantaneously (w t) to the moving electron and is seen as a horizontal line intersecting the (ct) and (ct´) axes. Since the signal has some time extent (i.e. the signal has a period cT = c/f = T), the end of the signal (E T) will be transmitted some time ( T) after the beginning of the signal (B T). The angle () between the space-time co-ordinates is known to have the following relationship [Tan( ) = v/c]. This follows from the fact that the moving electron (ct´ axis) travelling with velocity (v) is related to the graph co-ordinates: v = x/t, therefore v/c = x/(ct) = Tan( ). If the longitudinal signal were then reflected (w r) by the moving electron (B) toward the stationary electric dipole (A) (ref. Figure 4), it would propagate nearly instantaneously according to its co-ordinate system (i.e. parallel tothe x´ axis). Figure 4: Moving electron (B) reflecting longitudinal signal back to stationary electric dipole (A) located a distance (L 2) away.ct xx´ct´ BTET BRER/G03T /G03R /G01/G01c /G03T/10 ABwt wtwr wr wrStationary electric dipoleMoving electronReflected longitudinal signal v L2A B4The beginning of the signal arrives at the electric dipole (A) at time (B R) and the end of the signal arrives at the electric dipole (A) at time (E R). Note that the largest separation distance between the dipole and the moving electron is ( T/10), thereby enabling the longitudinal signals to propagate nearly instantaneously. The time difference (c t) between the transmission of the end of the signal (E T) and the return of the end of the signal (E R) can then be calculated using simple trigonometric relationships (ref. Figure 5). Figure 5 : Trigonometric diagram relating the time difference (c t) between the transmission of the end of the signal (E T) and the return of the end of the signal (E R) to parameters ( T, ). From Figure 5 it can be seen that Tan( ) = (c t)/(T/10) and it is known that Tan() = v/c. Equating these two relations and solving for c t yields: ct = (v T)/(10 c). This result can also be obtained by using Lorentz transformations. From the moving electron’s perspective, the end of the signal (E T) propagates backward in time: t´= [t - (v/c2)x], where t = 0, x = L1, and = 1/Sqrt[1-(v/c)2]. When the moving electron reflects the signal, from its perspective, the electric dipole moves with velocity (v) and sees a contracted distance x = L2 = L1/. The end of the signal (E R) therefore arrives at the electric dipole (A) at time: t = [0 - (v/c2)L1/], where L 1 = T/10. Therefore one obtains the same solution as was obtained from the Minkowski space-time diagram: ct = -(v T)/(10 c). Since the velocity of the moving electron (v) can be at most (c), then: c t < T/10 < T. This result indicates that although the signal can be transported backward in time ( R arrives before T), it cannot be transported before the same signal was initiated (E R > BT), thus preserving causality. The signal transmitted backward in time cannot be used to change the same signal that was sent. 3 Conclusion A simple relativistic analysis has been presented which shows that near-field superluminal longitudinal electric signals generated by an electric dipole cannot beused to violate Einstein causality with the relativistic “sync shift” effect. The analysishas shown that because a signal has some time extent (i.e. after a signal is initiatedone must wait at least one period for the frequency and amplitude information to bedetermined), an electric dipole can be used to transmit a signal backward in time, butnot before the same signal was initiated, thereby prohibiting a user from changing thesignal that was transmitted. It should be noted that in the model presented it was assumed that the signal information was contained in one period ( T) of an amplitude-modulated signal. In aT/10 ctET ER5real physical system several wavelengths may be required to encode and decode the information. One consequence of this is that it would reduce the spatial region overwhich the information travels superluminally [3]. In addition, it would also increase the effective signal wavelength ( T) used in the modelling of the problem, thus making it even more difficult for the signal to be transported before the same signal was initiated (E R >>BT), and thus making it more difficult to violate causality. Although the analysis is presented specifically for superluminal near-field longitudinal electric fields, the result would also apply for any type of signal that issuperluminal in a region less than one wavelength. In order to violate causality asignal must be superluminal in a spatial region greater than at least one wavelength.Therefore, the superluminal near-field transverse electric and magnetic fieldsgenerated by an electric dipole cannot be used to violate causality using the “syncshift” effect, since these fields are also known to be superluminal in a region less thanone wavelength near the source. In addition, in reference [1] the gravitational fields generated by an oscillating mass are also found to be superluminal in a region less than one wavelength near thesource. Analogous to the electric dipole and moving electron system, if a movingmass were to reflect the superluminal near-field gravitational fields generated by anamplitude-modulated mass, then the signal would arrive back in time at thetransmitter mass, but not before the signal was initiated, thereby prohibiting a userfrom changing the same signal that was transmitted. It is therefore concluded thatsuperluminal near-field gravitational fields cannot be used to violate causality usingthe “sync shift” effect. In conclusion, contrary to Einstein’s hypothesis that superluminal signals are incompatible with relativity theory [5], this paper has shown that superluminalsignals, such as the superluminal near-field electromagnetic fields generated by anelectric dipole or the superluminal near-field gravitational fields generated by avibrating mass, are compatible with relativity because the spatial region in which theyare superluminal is less that one wavelength. Although these superluminal fields canbe used to transmit signals backward in time using the “sync shift” effect, the signalscannot be decoded before the same signals were initiated, thereby preservingcausality. References 1 W. D. Walker, (1998) Superluminal propagation speed of longitudinally oscillating electrical fields , Conference on causality and locality in modern physics , Kluwer Acad. Expanded paper found at electronic archive: http://xxx.lanl.gov/abs/gr-qc/9706082 2 W. D. Walker, (1999) Superluminal near-field dipole electromagnetic fields, International Workshop “Lorentz Group, CPT and Neutrinos” , Zacatecas, Mexico, 23-26 June 1999, to be published in conference proceedings, World Scientific. Expanded paper found at electronic archive: http://xxx.lanl.gov/abs/physics/0001063 3 W. D. Walker, (2000) Experimental evidence of near-field superluminally propagating electromagnetic fields, Vigier III Symposium “Gravitation and Cosmology”, Berkeley, California, USA, 21-25 August 2000, to be published in conference proceedings, Kluwer Acad. Expanded paper found at electronic archive: http://xxx.lanl.gov/abs/physics/0009023 4 R. Resnick, (1968) Introduction to special relativity , John Wiley pub., Appendix A, Also reference J. W. Hinson, relativity website: http://www.physics.purdue.edu/~hinson/ftl/index.html 5 A. Einstein, (1907) Die vom relativitätsprinzipgeforderte trägheit der energie, Ann. Phys., 23, 371-384, English translation in A. Miller, (1998) Albert Einstein’s special theory of relativity , Springer-Verlag New York, 223-225
1A ‘New Theory’ based on the Charge Density Mass (CDM) Ali Riza Akcay TUBITAK-UEKAE P.O. Box 21 41470-Gebze Kocaeli-TURKEY aakcay@yunus.mam.gov.tr ABSTRACT: This paper describe a ‘New Theory’ based on a new concept titled as CDM (Charge Density Mass) towards the evolution of the TOE (Theory of Everything). Acccording to the CDM, there is an indirect proportion between charge density and mass density of any particle. The ‘New Theory’ (‘NT’) can predict and describe the states of the universe at the beginning and the end of the Big-Bang and Big-Crunch. The ‘NT’ proves that the speed of light is relative to the charge density (or current density), and is relatively constant (not always equal to 300,000 Km/sec). The ‘NT’ can predict and describes the space-time singularities without the distribution of mass and energy. The ‘NT’ (or CDM) can provide the unification of gravity, strong, weak and electromagnetic forces into the electromagnetic force for the unification all the fundamental interactions. As known, the main aim of the TOE is to unify all the fundamental interactions. 1. THE ‘NEW THEORY’ BASED ON CDM Description: I have developed a ‘New Theory’ based on the CDM (Charge Density Mass) towards the evolution of the Theory of Everything (TOE). The CDM proves that there is an indirect proportion between the charge density ( DJ) and mass density (Dm) of any particle. This means that, the Dm approaches to zero ( 0→Dm ) when DJ approaches to its maximum value or critical value (maxD DJJ→ ). This relation may be shown as     − = max01 DD D D JJmm . (1) Here, 0Dm denotes the rest mass density of the particles, DJ denotes the charge density of particles and maxDJ the maximum charge density that can be reached by the particles. In case of the relative mass density the same relation is shown as 22 11 max0 cvJJm mDD D D −    − = . (2)2The Big-Bang, Big-Crunch and the Classification of Mass in the Universe: The ‘New Theory’ (‘NT’) can predict and describe the states of the universe at the beginning and the end of Big-Bang and Big-Crunch. According to the ‘NT’ there are basically two types of mass in the universe: Charged Mass (CM) and Uncharged Mass (UCM). Both CM and UCM can be splitted as rest and relative mass (rest CM, relative CM, rest UCM and relative UCM). The impact of CM is electromagnetic force, and the impact of UCM is gravity force. The mass density of CM is low, and can reach to a minimum value (approximatelly to zero) depending to the levels of the charge density (or current density). The charge density CM is high, and can reach to a maximum (or critical) value depending to the levels of charge density. The mass density of UCM is high but, the charge density of UCM is low. The total mass of the universe ( ∑Um) in both cases (rest and relative mass) is equal to the total value of the CM ∑CMm( ) and UCM ( ∑UCMm) in the universe ∑ ∑ ∑ + = UCM CM Um m m . (3) The total mass of the universe cannot be considered as fully CM (FCM) or fully UCM (FUCM) in the normal conditions. In case of FCM, 0= ∑UCMm , and ∑ ∑=CM Um m . In case of FUCM, 0= ∑CMm , and ∑ ∑=UCM Um m . If the total mass of the universe ( ∑Um) is FCM ( ∑ ∑=CM Um m ); in this state the electromagnetic force is equal to its maximum value but, the gravity force is equal to its minimum value (or zero) in the universe. This state will cause of the Big-Bang, and describes the state of the beginning of the Big-Bang. If the total mass of the universe is FUCM ( ∑ ∑=UCM Um m ); in this state the gravity force is equal to its maximum value but, the electromagnetic force is equal to its minimum value (or zero) in the universe. This state will cause of the Big-Crunch, and describes the state of the beginning of the Big-Crunch.3The Theory of Everything: Having built a model of elementary particles and forces, particle physicists and cosmologists are today embarked on a difficult search for a so- called Theory of Everything-a theory that unifies all the fundamental interactions. As essential ingrediant in all major candidate theories is the concept of symmetry breaking . Experiments have determined that there are four physical forces in nature: gravity, strong, weak and electromagnetic forces. At the very instant of the Big-Bang, when energies were at their highest, it is believed that these forces were unified in a single, all-encompassing interaction. As the universe expanded and cooled down, first the gravitational interactions, then the strong interaction, and lastly the weak and the electromagnetic forces would have broken out of the unified scheme and adopted their present distinct identities in a series of symmetry breakings. The Theory of Everything (TOE) intends to unify the gravity, strong, weak and electromagnetic forces into a unique force. But, the theoretical physicists are still struggling to understand how the gravity can be united with other interactions. The ‘New Theory’ (‘NT’) based on the Charge Density Mass (CDM) can easily solve this problem by unifying the gravity, strong and weak forces into the electromagnetic force in case of fully charged mass (FCM); ∑ =0UCMm , and ∑ ∑=CM Um m . The Speed of Light: According to the ‘New Theory’ (‘NT’) the speed of ( c) is not constant, is relative to the charge density ( DJ) of the mediums in which the light travels, and is only constant in the homogenous mediums (the mediums which have fixed or invariable charge or current density. e.g. vacuum since, the charge density in vacuum is fixed and approximately equal to zero). The charge density of vacuum is approximately equal to zero ( 0≈DvacuumJ ). Therefore, the speed of light in vacuum is always constant and equal to 300,000 Km/s. This does not mean that the speed of light is constant in all mediums including the mediums which have high charge densities. The 300,000 Km/s is the speed of light in the vacuum, is a constant, and is the minimum value of the speed of light. The ‘NT’ titled this minimum value of the speed of light as 0c and ≅0c300,000 Km/s. The ‘NT’ proves that there is a direct proportion between the charge density ( DJ) of mediums in which the light travels and the relative speed of light (the ‘NT’ titled the relative speed of light as c). The relative speed of light ( c) can exceed the minimum value of the speed of light ( skm cc /000,3000=≥ ) and can reach to an infinite value ( ∞≈c) depending the levels of charge density of the mediums (e.g. high-level charge density, medium-level charge density and low-level charge density). This means that, the relative speed of light approaches to infinite ( ∞→c) when the charge density (or current density) of mediums in which the light travels approaches to its maximum or critical value (maxD DJJ→ ). The relation between the c and DJ is shown as4max10 DD JJcc −= . (4) This formula can be cosidered three conditions: Condition 1 (0=DJ): sKm cc / 000,3000== . Condition 2 (max0 D DJJ<< ): ∞<<cc0. Condition 3 (maxD DJJ=): ∞=c. Travel Faster Than Light: As known, according to the Special Theory of Relativity (STR), any particle cannot travel faster than 0c=300,000 Km/s which is described as the minimum value of the speed of light by the ‘New Theory’ (‘NT’). But, according to the ‘NT’, any particle can travel faster than 0c (0c is described by the ‘NT’ as the minimum value of the speed of light which is approximately equal to 300,000 Km/s.), and any particle cannot travel faster than c (c is described by the ‘NT’ as the relative speed of light which can reach to an infinite speed). The Space-Time Singularities: The fact that Einstein’s general theory of relativity turned out to predict singularities led to a crisis in physics. The equations of general relativity, which relate to curvature of space-time with the distribution of mass and energy, cannot be defined as a singularity. This means that general relativity cannot predict what comes out of a singularity [1] . Whereas, The ‘New Theory’ can predict and describes the Space-Time Singularities without the distribution of mass and energy. If we consider all conditions of Formula (1):     − = max01 DD D D JJmm . (1) Condition 1 (0=DJ): 0D Dmm=. Condition 2 (max0 D DJJ<< ): 00>>D Dmm . Condition 3 ( maxD DJJ=): 0=Dm .5It is quite clear that the mass density of any particle ( Dm) is equal to zero in the Condition 3 of the Formula (1). This means that the Condition 3 of Formula (1) can predict and describe the Space-Time Singularities without the distribution of mass and energy REFERENCES: [[1]] S.W. Hawking, Black Holes and Baby Universes and other Essays (Bantam Books, 1994).
arXiv:physics/0009078v1 [physics.atom-ph] 26 Sep 2000Spin-axis relaxation in spin-exchange collisions of alkal i atoms S. Kadlecek and T. Walker Department of Physics, University of Wisconsin–Madison, M adison, Wisconsin 53706 D.K. Walter, C. Erickson and W. Happer Department of Physics, Princeton University, Princeton, N ew Jersey 08544 (February 15, 2014) We present calculations of spin-relaxation rates of alkali -metal atoms due to the spin-axis in- teraction acting in binary collisions between the atoms. We show that for the high-temperature conditions of interest here, the spin relaxation rates calc ulated with classical-path trajectories are nearly the same as those calculated with the distorted-wave Born approximation. We compare these calculations to recent experiments that used magnetic deco upling to isolate spin relaxation due to binary collisions from that due to the formation of triplet v an-der-Waals molecules. The values of the spin-axis coupling coeﬃcients deduced from measuremen ts of binary collision rates are consistent with those deduced from molecular decoupling experiments. All the experimental data is consistent with a simple and physically plausible scaling law for the sp in-axis coupling coeﬃcients. I. INTRODUCTION Spin-exchange optical pumping [1,2] of3He uses spin-exchange collisions between3He atoms and optically pumped Rb atoms to produce large quantities of highly spin-polariz ed3He for a variety of applications, including medical imaging [3] and spin-polarized targets [4]. The eﬃciency of polarized3He production is determined by two fundamental rates: the Rb-He spin-exchange rate and the Rb spin-relaxat ion rate. The measured spin-exchange rates [5] are in fairly good agreement with theory [6]. At the high temperatu res needed for the3He spin-exchange rates to exceed the wall relaxation rates, both Rb-Rb and Rb-He relaxation l imit the eﬃciency for spin-exchange. While collisions between alkali atoms nominally conserve the spin-polariza tion, Bhaskar et al.[7] discovered that rapid spin-relaxation occurs in high-density optically pumped Cs, with a surprisi ngly large inferred spin-relaxation cross section in exces s of 1˚A2. The corresponding cross section for Rb-Rb relaxation [5,8 –10], while smaller, still limits the eﬃciency of 3He production. The yet smaller cross section for K-K [8,11] r elaxation suggests that, if technical diﬃculties are surmounted, K may be the optimum partner for spin-exchange w ith3He [5]. A few years ago, we discovered that 1/2 to 2/3 of the alkali-al kali relaxation decouples in magnetic ﬁelds of a few kG [10], making implausible the interpretation of the relax ation exclusively in terms of binary collisions. Recently, careful magnetic decoupling studies in low pressure, isoto pically pure Rb and Cs samples deﬁnitively identiﬁed the source of the ﬁeld-dependent relaxation as the spin-axis in teraction in triplet molecules [12]. The remaining alkali- alkali relaxation mechanism at high pressure is then presum ably from binary collisions. It is the purpose of this paper to show that the deduced values of the spin-axis interaction from binary collisions are consistent with the values recently obtained from magnetic decoupling studies of trip let molecules [12]. We hypothesize a simple scaling law for the second-order spin-orbit interaction that explains the relative magnitudes of the observed spin-axis interact ion strengths. Nevertheless, the measured cross sections are i n every case at least a factor of 10 larger than would be expected from ab initio calculations [13]. Table I contains a summary of the existin g data on alkali-alkali spin relaxation. The spin-axis interaction between two alkali-metal atoms i s V1=2λ 3S·(3ζζ−1)·S. (1) Here the total electron spin of the valence-electron pair is S,Ris the internuclear separation, and ζis a unit vector lying along the direction of the internuclear axis. The coeﬃ cientλ=λ(R), a rapidly decreasing function of R, is currently believed to arise from both the direct spin-dipol ar coupling (averaged over the electron charge distributio n) and the spin-orbit interaction in second order [13]. Accumu lating evidence, using both high temperature [12,14] and low temperature experiments [15], suggests that the predic ted spin-axis coupling (presumably arising almost entirel y from second-order, spin-orbit interactions) is too small i n Cs by a factor of 3 to 4, and in Rb by a factor of more than 10 [12]. Put another way, the theoretical spin-relaxat ion cross-sections for Rb are smaller than experiment by a factor of more than 50. This paper carefully documents how the collisional averagi ng of the interaction (1) leads to a spin relaxation rate. The somewhat complicated averaging can be done exactly with in the limitations of the classical path approximation, 1TABLE I. Spin-relaxation cross sections /angbracketleftvσ/angbracketright/¯v(in cm2) for alkali atoms. Zero-ﬁeld “cross sections” include cont ributions from both molecular formation and binary collisions. High- ﬁeld cross sections are assumed to arise solely from binary c ollisions. The theoretical values are calculated, at a temperature of 4 00 K, using estimates of the spin-axis coupling strength eit her from a simple scaling law or from Ref. [13]. Also shown is the ratio ofab initio predictions to experiment. Zero-ﬁeld High-ﬁeld Scaling ab initio Atom Ref. Experiment Experiment Theory Theory Ratio K [8] 2 .4×10−18 K [16] 1 .0×10−186.2×10−194.4×10−206.7×10−209.2 Rb [8] 1 .6×10−17 Rb [9] 1 .8×10−17 Rb [5] 0 .92×10−17 Rb [10] 0 .93×10−173.4×10−181.4×10−186.1×10−2056 Rb [17] 1 .5×10−175.6×10−181.4×10−186.1×10−2092 Cs [7] 2 .03×10−16 Cs [14] 2 .3×10−161.1×10−161.1×10−160.11×10−1610 so the origin of the discrepancy cannot be due to any inadequa cies of the averaging but must lie with the potential (1), or with the spin-independent interatomic potential which d escribes the classical paths, or with the neglect of some unknown collisional relaxation mechanisms other than sudd en binary collisions. For very cold collisions, where not so many partial waves are involved, one could use the distorted -wave Born approximation (DWBA) [18] to account for any limitations of the classical-path method, but at the tem peratures of interest here, as we will show, the results of the classical path approximation are within a few percent of those of the DWBA. Our recent experiments have studied the spin-relaxation ra tes as a function of magnetic ﬁeld for the three alkali atoms K, Rb, and Cs, as described in Refs. [10,12,16,17]. We a ssume here that the remaining alkali density dependent contributions to the relaxation rate at 12 kG arise entirely from binary collisions. The data are summarized in Table I. II. CLASSICAL PATHS In a binary collision at temperatures of a few hundred Kelvin many partial waves contribute to spin-exchange relaxation, so a classical-path treatment should be adequa te. Methods for averaging over all classical-path cross sections to obtain spin relaxation rates were given in an ear lier paper by Walter et al.[19] (referred to in the following as WHW) for collisions between alkali-metal atoms and noble -gas atoms. In particular, the anistropic magnetic hyperﬁne interaction between the spin of a noble gas nucleus and the electron spin of the alkali-metal atom, Eq. (3) of WHW has the same tensor symmetry as the spin-axis interact ion (1), and the detailed calculations are so closely analogous to those of WHW that we will simply summarize the re sults here. As outlined in WHW and illustrated by Fig. 1 of that paper, we c an assume that the orbit of the colliding pair follows a trajectory governed by the triplet potential V0=V0(R). The time-dependence of the internuclear separation Rcan be found from the equation describing conservation of en ergy and angular momentum, dR dt=±w/radicalbigg 1−b2 R2−2V0 Mw2. (2) HereMis the reduced mass of the pair of colliding alkali-metal ato ms. For the relatively high-temperature exper- imental conditions of interest here, possible changes in di rection of the electron spin Scause such small changes in the energy or angular momentum of the orbital motion that we c an neglect them and parametrize the orbital energy with the initial relative velocity wand the angular momentum by the impact parameter b. It is convenient to let the time of closest approach of the pai r bet= 0, so the orbital angle at time tis ψ=ψ(t) = cos−1ζ(t)·ζ(0). (3) The time dependence of the orbital angle can be found by numer ical quadrature of the equation for conservation of angular momentum dψ dt=wb R2. (4) 2When averaged over a thermal distribution of trajectories, collisions in an alkali-metal vapor of atomic number densitynwill cause the mean longitudinal electron spin polarizatio n/angbracketleftSz/angbracketrightof the atoms to relax at the rate d dt/angbracketleftSz/angbracketright=−n/angbracketleftvσ/angbracketright/angbracketleftSz/angbracketright. (5) The rate coeﬃcient can be readily calculated by methods anal ogous to those used in WHW for relaxation due to the anisotropic magnetic dipole hyperﬁne interaction. The average over all angles of the collisions can be carried out analytically and we ﬁnd in analogy to (33) of WHW, /angbracketleftvσ/angbracketright=/integraldisplay∞ 0dw p(w)w/integraldisplay∞ 0db b8π 32/summationdisplay m=−2\|ϕ2m\|2. (6) The probability p(w)dwof ﬁnding the magnitude wof the relative velocity of the colliding pair between wandw+dw is p(w)dw= 4πw2/parenleftbiggM 2πkBT/parenrightbigg3/2 e−Mw2/2kBTdw, (7) whereTis the absolute temperature and kBis Boltzmann’s constant. The tensor phases accumulated dur ing the collision are ϕ2m=1 ¯h/integraldisplay∞ −∞dtλ d2 0m(ψ) (8) Hered2 0m(ψ) is a Wigner dfunction, for example, d2 00(ψ) = (3 cos2ψ−1)/2 [20]. In the integrand of Eq. (8), both the spin-axis coupling coeﬃcient λand the orbital angle ψare functions of time t, obtained from numerical integration of Equations (2) and (4). Since ψ(−t) =−ψ(t),d2m(ψ) is an even function of tifmis even and odd if mis odd. Alsoλis an even function of t(measured from the time of closest approach), so φ2mis identically zero if m=±1. In practice, the rapid decrease of λwith increasing internuclear separation rmeans that φ2,±2is so small as to be negligible in practice. The variable of integration for (8) is more conveniently chosen to be ψrather than t, since the range of integration is then between ﬁnite rather than inﬁni te limits. III. PARTIAL WAVES We can also calculate the spin relaxation due to the spin-axi s interaction (1) by the distorted-wave Born approxi- mation (DWBA), outlined by Newbury et al. [18], which we refer to as Newbury in this section. According to (47) of Newbury, the rate coeﬃcient corresponding to (6) is /angbracketleftvσ/angbracketright=/integraldisplay∞ 0dw p(w)w64πM2 3¯h4k6∞/summationdisplay l=0(2l+ 1)λ2 ll (9) For the high-temperature conditions of interest here the th ree matrix elements λll′for spin-ﬂip scattering from an initial wave of angular momentum lto a ﬁnal wave of angular momentum l′=l,l±2 are nearly equal. We have assumed exact equality to reduce the double sum on landl′to a single sum on lin (9). The classical initial relative velocity wis related to the asymptotic spatial frequency kof the scattered wave by w=¯hk M(10) In (9), the integral over impact parameters bwhich occurs in the classical-path expression (6) is replac ed by a sum over partial waves l. The matrix element of the spin-axis coupling coeﬃcient bet ween partial waves landl′is λll′=k/integraldisplay∞ 0gl(r)λ(r)gl′(r)dr (11) The wave functions gl(r) are solutions of the Schr¨ odinger equation 3/parenleftbigg −d2 dr2+l(l+ 1) r2+2M ¯h2V0−k2/parenrightbigg gl= 0, (12) with the boundary condition as r→0 gl→0 (13) and with the asymptotic boundary condition for kr/l→ ∞ gl→sin/parenleftbigg kr−πl 2+δl/parenrightbigg . (14) From comparison of (6) with (9) we conclude the classical-pa th and partial wave treatment will give practically the same answers if 2/summationdisplay m=−2\|ϕ2m\|2=/parenleftbigg2λll E/parenrightbigg2 . (15) In (15) we have assumed that l=kb. The initial relative energy of the colliding pair is E=¯h2k2 2M(16) Numerical solutions to the diﬀerential equations (2) and (1 2) are readily obtained. We have conﬁrmed that relation (15) is indeed true, establishing the equivalence of the cla ssical-path and partial-wave methods of calculating the sp in- relaxation rate coeﬃcients. Although the two methods give t he same results, the classical-path approach is much less numerically intensive, requiring only the solution of the s imple ﬁrst-order diﬀerential equations (2) and (4), wherea s the partial-wave analysis requires solving the second-ord er equation (12), with rapidly oscillating solutions. IV. COMPARISON OF AB INITIO CALCULATIONS TO EXPERIMENT In the context of ultracold collisions, Mies et al. [13] recently published ab initio calculations of λ, as the sum of two contributions λ=λSO+λDD. (17) For the heavier alkali-metal atoms, and for small R, they found that second-order spin-orbit interactions, re presented in (17) by λSO, were much larger than the term λDD, which describes the direct interaction between the magnet ic moments of the two valence electrons. Second-order contrib utions analogous to those responsible for λSOare well known from the theoretical literature on molecular spectro scopy [21,22]. Mies et al. parameterize their calculations ofλas follows (we have converted their results from atomic unit s): λ=3g2 Sµ2 B 4a3 B/bracketleftbigg Ce−β(R−RS)−/parenleftBigaB R/parenrightBig3/bracketrightbigg . (18) HereaBis the Bohr radius, µBis the Bohr magneton and gS= 2.00232 is the electronic g-factor. The results (18) of theab initio calculations are parametrized as follows: for Rb, RS= 5.292˚A,C=.001252 and β= 1.84˚A−1; for Cs, RS= 5.292˚A,C=.02249 andβ= 1.568˚A−1. The ﬁrst term in (18) represents λSO. The second term, λDD=−3g2 Sµ2 B 4R3, (19) represents the magnetic interaction of electrons, taken as point particles separated by a distance R. This is an excellent approximation at very large R, but as we show below, it is not a very good approximation at sm aller values of R where the most important contributions to spin relaxation o ccur. Table I shows cross-sections calculated as described in Sec tions II and III. For K, we assume only the classical spin-dipolar term, because the spin-orbit contribution es timated by scaling from Mies et al. [13] is negligible. As can be seen from Table I, the theoretical estimates are smaller t han experiment by about a factor of 10 for Cs and K, and a factor of almost 60 for Rb where the ab initio calculations predict that λgoes to zero at R= 5.5˚A. In order to describe the eﬀects of spin-relaxation in a numbe r of experiments on Cs, where (18) predicts relaxation rates that are much too small, the NIST group has chosen to mul tiplyλSO(the computed contribution to λfrom second-order spin-orbit interactions) by a constant value . This assumes that the R-dependence of the calculation is correct [14,15]. Guided by new experimental data, we will di scuss similar scaling arguments in Section VI. 410-410-310-210-1λ (cm-1 ) 10 8 6 4 R (Å) Cs scaled Cs ab initio Rb scaled Rb ab initio FIG. 1. Assumed spin-axis coupling parameters λas a function of internuclear separation R. V. EFFECTS OF WAVEFUNCTION OVERLAP ON λDD The expression (19) of λDDby Mies et al. neglects the spatial distribution of the electron charge. I n this section we present a simple estimate of the eﬀect of the spatial distr ibution. We ﬁnd that neglect of the spatial distribution cannot be responsible for the discrepancy between experime nt and theory. A simple estimate of the wavefunction \|Ψ/angbracketright=ψ(r1,r2)\|χ/angbracketrightof the triplet state of an alkali dimer is \|Ψ/angbracketright=N[ϕA(1)ϕB(2)−ϕB(1)ϕA(2)]\|χ/angbracketright (20) where, for example, ϕA(1) is a spatial orbital for electron 1 centered at nucleus A, Na normalizing factor, and \|χ/angbracketrightis a three-component spinor representing the triplet spin sta te. The matrix element between a ﬁnal triplet state \|Ψf/angbracketright and an initial triplet state \|Ψi/angbracketrightof the electronic magnetic dipole interaction (at ﬁxed R), is /angbracketleftΨf\|g2 Sµ2 B r5 12S1·(r2 121−3r12r12)·S2\|Ψi/angbracketright=4λDD 3/angbracketleftχf\|S1·(3ζζ−1)·S2\|χi/angbracketright (21) where λDD=3g2 Sµ2 B 4/integraldisplay d3r1d3r2r2 12−3z2 12 2r5 12\|ψ(r1,r2)\|2(22) We make the simplifying assumption that ϕAandϕBcan be approximated by the ground-state wavefunction of the valence electron of an isolated alkali-metal atom. The results are shown in Fig. 2. The principle eﬀect of the wav efunction overlap is to reduce the value of λDD as compared to the point dipole approximation, and to reduce the predicted ab initio cross section for Rb-Rb to 5.3×10−20cm2, increasing the experiment/theory discrepency. Obviousl y, it would be good to use better electron wave functions than the simple form (20) used for this estima te, but we doubt that the results would change by the orders of magnitude needed to obtain agreement with experim ent. VI. SCALING RELATION FOR λSO In this section we show that the spin-axis coupling coeﬃcien ts deduced from triplet molecules are consistent with a simple and plausible scaling relation that, in turn, accur ately predicts the relative binary spin-relaxation cross 530 20 10 0B'(R) (GHz) 14 12 108 6 4 R (Å)Point dipole Spatially averaged dipole Spin-orbit (✕ -1) FIG. 2. Ab initio calculations of λDDfrom Ref. [13], and the modiﬁed version obtained by spatiall y averaging the magnetic dipole-dipole contribution as described in the text. The ve rtical solid line is at the classical turning point for a zero -impact parameter collision at kTcollision energy. sections. Clearly, the contribution λDDfrom the direct interaction of the magnetic dipole moments o f the electrons is much too small to account for observed relaxation rates in th e heavier alkali-metal atoms. The additional contribution λSOcalculated by Mies et al. is also too small. In the absence of any fundamental understa nding of why the ab initio calculations work so poorly, we shall assume that Mies et al. have correctly identiﬁed the second-order spin-orbit interaction as a major contributor to λ, which implies that λSOshould be proportional to the square of the P1/2– P3/2ﬁne structure splitting ∆ ν, and inversely proportional to the valence-electron bindi ng energyE, as predicted by perturbation theory. To obtain the radial dependence of λSO, we shall make the physically plausible assumption that it scales as \|φ(r)\|2, the valence electron density of an unperturbed alkali-met al atom at a distance rfrom the nucleus. (The radial dependence of the values of λSOcalculated by Mies et al. is very nearly that of \|φ(r)\|2.) Thus our scaling law is λSO= Ω(hc∆ν)2 E\|φ(r)\|2. (23) The ﬁne structure splittings ∆ νfor Cs, Rb, and K are 554 cm−1, 237.6 cm−1, and 57.7 cm−1, respectively; the binding energiesEare 3.89 eV, 4.18 eV, and 4 .34 eV. For the required wavefunctions φ(r), we use the asymptotic expansion of the Coulomb wave functi on [25], namely φ(r) =N r/parenleftbigg2r n∗/parenrightbiggn∗ e−r/n∗, (24) where the radius ris measured in Bohr radii aB. The eﬀective principal quantum number n∗of the valence electron in its ground state is related to the ionization energy E(in eV) and the Rydberg R∞= 13.61 eV byn∗= (R∞/E)1/2, and the normalization factor Nis given by N= [(4π)1/2(n∗)3/2Γ(n∗)]−1, where Γ(x) is the Euler gamma function. The universal constant Ω of (23), which has units of volume, i s deduced from experiments as follows. Recent experimental studies [12] of the Cs spin relaxation in tripl et dimers yield a spin-axis coupling \|λCs/h\|= 2.79 GHz. This value reﬂects a thermal average over the rovibrational states of the triplet molecules, but for simplicity we will take it to be the value of λ/hat the Cs triplet dimer equilibrium internuclear separatio n ofR= 12aB, whence, by Eqs. (17) and (19), λCs SO= 2.94 GHz. Thus, using \|φ\|2= 3.48×10−6a−3 BatR= 12aB, we obtain Ω = 2876a3 B, (25) roughly an atomic volume. In Rb, the equilibrium internuclear separation for triplet molecules is R= 11.5aB, where \|φ\|2= 3.23×10−6a−3 B, whence Eqs. (23) and (19) yield λRb/h= 294 MHz, in close agreement with the value \|λRb/h\|= 290 MHz deduced from the observation of spin-relaxation due to Rb triplet di mers [12]. Using the results of Sections II and III and the spin-axis cou pling coeﬃcients deduced from the above simple arguments, we may readily compute the spin relaxation due to the spin-axis interaction in binary collisions between alkali-metal atoms. The calculated cross sections for K-K, Rb-Rb and Cs-Cs at 400 K, using the well-known ab initio interatomic potentials of Krauss and Stevens [23] to comput e the needed trajectories, are respectively 4 .4×10−20cm2, 1.4×10−18cm2and 1.1×10−16cm2. As can be seen from Table I, these values agree well with expe riment. We note that for potassium—unlike rubidium and cesium—it is the dip ole-dipole interaction that dominates the contribution toλ, and so the larger discrepancy with experiment for K is proba bly due to the simpliﬁed estimate (19) of λDD. 6VII. CONCLUSIONS We have shown here that the strength of the spin-axis couplin g deduced from measurements of binary collisions between alkali atoms is consistent with the values deduced f rom magnetic decoupling of relaxation due to formation of weakly bound triplet molecules. The deduced spin-axis co upling strengths are much larger than predicted by ab initio theory. Clearly more reliable theoretical ways to estimate the spin-axis coupling are needed. VIII. ACKNOWLEDGEMENTS Support for this work came from the National Science Foundat ion, AFOSR, and DARPA. D.K.W. is supported by the Hertz Foundation. We appreciate helpful discussions wi th P. Leo, P. Julienne, and C. Williams. [1] M. A. Bouchiat, T. R. Carver, and C.M. Varnum, Phys. Rev. L ett.5, 373 (1960). [2] T. G. Walker and W. Happer, Rev. Mod. Phys. 60, 629 (1997). [3] J. MacFall et al., Radiology 200, 553 (1996); M. S. Albert et al., Nature 370, 199 (1994). [4] P. Anthony et al. (E124 Collaboration), Phys. Rev. Lett. 71, 959 (1993). [5] A. Baranga, S. Appelt, M. Romalis, C. Erickson, A. Young, G. Cates, and W. Happer, Phys. Rev. Lett. 80, 2801 (1998). [6] T. Walker, Phys. Rev. A 40, 4959 (1989). [7] N. D. Bhaskar, J. Pietras, J. Camparo, W. Happer, and J. Li ran, Phys. Rev. Lett. 44, 930 (1980). [8] R. Knize, Phys. Rev. A 40, 6219 (1989). [9] M. Wagshul and T. Chupp, Phys. Rev. A 49, 3854 (1994). [10] S. Kadlecek, L. Anderson, and T. Walker, Phys. Rev. Lett .80, 5512 (1998). [11] S. Kadlecek, L. Anderson, and T. Walker, Nucl. Inst. Met h. Sci. Res. A 402, 208 (1998). [12] C. J. Erickson, D. Levron, W. Happer, S. Kadlecek, B. Cha nn, and T. G. Walker, to be published. [13] F. Mies, C. Williams, P. Julienne, and M. Krauss, J. Res. Natl. Inst. Stand. Technol. 101, 521 (1996). [14] P. Leo, E. Tiesinga, P. Julienne, D.K. Walter, S. Kadlec ek, and T. Walker, Phys. Rev. Lett. 81, 1389 (1998). [15] P. Leo, C. Williams, and P. Julienne, preprint. [16] S. Kadlecek, Ph. D. thesis, University of Wisconsin (19 99), unpublished. [17] C. Erickson, Ph. D. thesis, Princeton University (2000 ), unpublished. [18] N. Newbury, A. Barton, G. Cates, W. Happer, and H. Middle ton, Phys. Rev. A 48, 4411 (1993). [19] D.K. Walter, W. Happer, and T.G. Walker, Phys. Rev. A. 58, 3642 (1998). [20] D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonski i,Quantum Theory of Angular Momentum (World Scientiﬁc, Teaneck, NJ, 1988). [21] M. Tinkham and M. Strandberg, Phys. Rev. 97, 937 (1955). [22] P. Julienne, J. Mol. Spectrosc. 63, 60 (1976). [23] M. Krauss and W. Stevens, J. Chem. Phys. 93, 4236 (1990). [24] J. S¨ oding et al., Phys. Rev. Lett. 80, 1869 (1998). [25] D.R. Bates and A. Damgaard, Phil. Trans. R. Soc. London 242, 101 (1949). The Coulomb wavefunctions used in the present work are, in atomic units, Ψ K(r) = 0 .16099 r0.77e−r/1.77, ΨRb(r) = 0 .15161 r0.80e−r/1.80, and Ψ Cs(r) = 0.13142 r0.87e−r/1.87. 7
arXiv:physics/0009079v1 [physics.gen-ph] 26 Sep 2000Relativistic Quantum Physics with Hyperbolic Numbers S. Ulrych Bombachsteig 16, CH-8049 Z¨ urich, Switzerland Summary. — The hyperbolic number system is rarely used though it is a he lpful tool for the description of relativistic physics. It allows a rewriting of the quadratic Dirac equation in terms of a 2 ×2-matrix equation. In addition, the relativistic description of spin and orbital angular momentum is simpliﬁ ed with the hyperbolic number system. Beside of other applications this new develo ped formalism can be used for the solution of relativistic single particle equat ions with non-static poten- tials. PACS 12.20.-m – 03.65.Pm, 11.15.-q, 32.30.-r. 1. – Introduction The present work wants to show basic concepts for the applica tion of hyperbolic numbers in relativistic quantum physics. The hyperbolic nu mber system has a long history [1, 2], but is rarely used in physics. In analogy to the complex numbers, which provide a relations hip between the sine, the cosine and the exponential function, the hyperbolic number s give a relationship between the hyperbolic sine, the hyperbolic cosine and the exponent ial function. Therefore they are a good tool for the representation of relativistic coord inate transformations and considerations of the spin and the orbital angular momentum within the Lorentz group. The quadratic Dirac equation can be rewritten, using the hyp erbolic numbers, in terms of a 2 ×2-matrix equation, which has the consequence that also rela tivistic spinors for massive particles are represented as two component spinors . The quadratic Dirac equa- tion, understood as a single particle particle equation, is the starting point of calculations with non-static potentials. 2. – Hyperbolic numbers In the present investigation hyperbolic numbers are used fo r the mathematical for- mulation of the electron wave equation. Since these numbers are rarely used in physical applications a brief introduction of the number system is gi ven. 12 S. ULRYCH The hyperbolic numbers x∈Hare deﬁned as x=x0+jx1, x 0, x1∈R, (1) where the hyperbolic unit jhas the property j2= 1. (2) This leads to the following rules for the multiplication and addition of two hyperbolic numbersx=x0+jx1andy=y0+jy1 x+y= (x0+y0) +j(x1+y1), xy = (x0y0+x1y1) +j(x0y1+x1y0). (3) Since there are non-zero elements which have no inverse elem ent these numbers form a commutative ring. The hyperbolic unit jprovides a relation between the hyperbolic sine, cosine and the exponential function coshφ+jsinhφ=ejφ, (4) which can be derived in the same way as the corresponding rela tion for the complex numbers. In addition to the complex conjugation, a hyperbolic conjug ation will be used in the following which changes only the sign of the hyperbolic unit x−=x0−jx1. (5) 3. – Relativistic vectors and the spin group 3.1.Relativistic vectors . – With respect to the investigation of the spin angular mo- mentum in one of the following chapters a new relativistic al gebra is introduced, which can not be expressed in terms of the Dirac or the quaternion al gebra. A contravariant Lorentz vector with the coordinates xµ= (x0,xi) is represented as X=xµσµ. (6) In contrast to the quaternion formalism, the basis vectors σµare made up of the unity and the elements of the Pauli algebra multiplicated by the hy perbolic unit j σµ= (1,jσi). (7) Separating time and space coordinates one therefore ﬁnds X=x0+jx·σ. (8) The Pauli algebra is characterized by its multiplication ru les, which can be written as σiσj=δij1 +iǫijkσk. (9)RELATIVISTIC QUANTUM PHYSICS WITH HYPERBOLIC NUMBERS 3 Using the Pauli matrices as the explicit representation of σi, the vector Xcan be ex- pressed in terms of a 2 ×2 matrix according to X=/parenleftbigg x0+jx3jx1−ijx2 jx1+ijx2x0−jx3/parenrightbigg . (10) A scalar product between two vectors can be deﬁned using the t race of the matrix ¯XY /an}b∇acketle{tX\|Y/an}b∇acket∇i}ht=1 2Tr(¯XY). (11) The symbol ¯X=X† −denotes transposition, complex and hyperbolic conjugatio n of the matrix. ¯XYcorresponds to a matrix multiplication of the two 2 ×2 matrices ¯Xand Y. As stated above, the Pauli matrices can be considered as the basis vectors of the relativistic vector space σµ≡eµ. These basis vectors form a non-cartesian orthogonal basis with respect to the scalar product deﬁned in Eq. (11) /an}b∇acketle{teµ\|eν/an}b∇acket∇i}ht=gµν, (12) where the metric tensor gµνis the diagonal 4 ×4 matrix with the matrix elements gµν= 1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1 . (13) As an example the energy-momentum vector of a free classical pointlike particle, moving with the velocity vrelative to the observer, is expressed in terms of the matrix algebra. The relativistic momentum vector for this particl e can be written as P=E c+jp·σ=mcexp(jξ·σ), (14) withcdenoting the velocity of light, ξthe rapidity, Ethe energy and pthe momentum of the particle. The rapidity is deﬁned as tanhξ =v c=pc E, (15) whereξ=\|ξ\|andp=\|p\|. Rapidity and momentum point into the same direction n=v/\|v\|as the velocity. In quantum mechanics energy and momentum are substituted by diﬀerential opera- tors. With ∇=∂µσµthe momentum operator is then given by P=i¯h∇. (16) This operator forms the basis of the wave equation with spin, which will be introduced in Section 4. In the following cand ¯hwill be set equal to one.4 S. ULRYCH 3.2.Lorentz Transformations . – In analogy to the relation between SU(2) andSO(3) the transformation properties of the vectors deﬁned in the l ast subsection give a relation betweenSO(3,1) and a spin group deﬁned as an extension of the unitary group SU(2). In the following rotations and boosts will be investigated. The rotation of a vector has the form X=⇒X′=RXR†, R= exp( −iθ·σ/2), (17) whereX=xµσµ. In addition, a vector can be boosted to a diﬀerent system. Th e boost parameters ξare chosen to make the considered vector describe an object m oving into the positive direction for positive values of ξ. In many investigations a diﬀerent sign convention is used. For the boosts one ﬁnds the transformati on rule X=⇒X′=BXB†, B= exp(jξ·σ/2). (18) The dagger in the above equations includes only a hermitian c onjugation and nota hyperbolic conjugation. For the boost transformation one ﬁ nds the relation B†=B, whereas the inverse of the boost operator corresponds to B−1=¯B. The explicit matrix representations of the boost matrices Bare B1=/parenleftbigg coshξ1/2jsinhξ1/2 jsinhξ1/2 coshξ1/2/parenrightbigg (19) for a boost in the direction of the x-axis and B2=/parenleftbigg coshξ2/2−ijsinhξ2/2 ijsinhξ2/2 cosh ξ2/2/parenrightbigg , B 3=/parenleftbigg ejξ3/20 0e−jξ3/2/parenrightbigg (20) for boosts along the y- and thez-axis. To proof that these transformation matrices are a represent ation of the Lorentz group the corresponding Lie algebra has to be investigated. If boosts and rotations are combined as follows X=⇒X′=LXL†, L= exp ( −i(J·θ+K·ξ)), (21) the inﬁnitesimal generators of these transformations can b e identiﬁed with Ji=σi 2, K i=ijσi 2. (22) With the commutation relations of the Pauli matrices one can derive that the generators satisfy the Lie algebra of the Lorentz group SO(3,1) [Ji, Jj] =iεijkJk, [Ki, Jj] =iεijkKk, (23) [Ki,Kj] =−iεijkJk.RELATIVISTIC QUANTUM PHYSICS WITH HYPERBOLIC NUMBERS 5 Therefore, the matrices RandBgiven in Eqs. (17) and (18) can be identiﬁed as the transformation matrices of the spin group of SO(3,1). Alternatively, the Lorentz transformations can be express ed with relativistic second rank tensors. Using the generators of the spin s=1 2representation given in Eq. (22) the relativistic tensor of the spin angular momentum operator i s Sij=ǫijkJk, S 0i=−Si0=Ki (24) and the Lorentz transformations given in Eq. (21) can be form ulated according to L= exp( −i 2Sµνωµν). (25) The boost parameters ωµνareωij=ǫijkθkandωi0=ξi. 4. – The fermion wave equation In the present work a relativistic wave equation is used, whi ch is closely related to the classical wave equation. The diﬀerential operator of this w ave equation is formed by the momentum operator Pmultiplied by the dual operator ¯P. P¯Pψ(x) =m2ψ(x), (26) whereP=i∇. The diﬀerential operator P¯Pcan be replaced by PP−since the mo- mentum operator is hermitian. In the following investigati ons there are no diﬀerences between these two choices even if interactions are introduc ed. The wave function ψ(x) has the general structure ψ(x) =ϕ(x) +jχ(x), (27) whereϕ(x) andχ(x) are two-component spinor functions. They depend on the fou r space-time coordinates xµ. The transformation properties of the operator ¯Pcan be deduced by a hermitian and hyperbolic conjugation of the cor responding equations given in the last section. In order to clarify the structure of the equation above, some explicit details are given. The Pauli matrices in Eq. (26) can be seen if one inserts Def. ( 6) σµ¯σνPµPνψ(x) =m2ψ(x). (28) The tensor σµ¯σνrepresents the spin structure which is acting on the spinor f unction. The explicit form is obtained by a matrix multiplication of t he 2×2 basis matrices. The tensor can be separated into a symmetric and an antisymmetri c contribution σµ¯σν=gµν−iσµν, (29) wheregµνcorresponds to the metric tensor and the antisymmetric part is given by σµν= 0 −ijσ1−ijσ2−ijσ3 ijσ1 0σ3 −σ2 ijσ2−σ3 0σ1 ijσ3σ2 −σ1 0 . (30)6 S. ULRYCH The antisymmetric contribution σµνis directly related to the relativistic generalisation of the spin angular momentum operator. Using the generators of the spin s=1 2repre- sentation given in Eq. (24) one ﬁnds Sµν=σµν 2. (31) SincePµPνis symmetric and σµν=−σνµthe operator P¯Pis equivalent to P¯P=PµPµ. At this point the particular form of the diﬀerential operato r seems to be without any eﬀect. However, the spin information which is included i n the diﬀerential operator becomes essential if the momentum operators are replaced by covariant derivates. The inﬂuence of this spin structure can be illustrated by the fol lowing example. Coordinate and momentum vector satisfy the relation X¯P=XµPµ−iSµνLµν, (32) whereLµν=XµPν−XνPµcorresponds to the relativistic orbital angular momentum. 5. – Plane wave states In this section the solutions of the wave equation will be stu died in the free non- interacting case. Due to the simpliﬁcation of the diﬀerenti al operator P¯P=PµPµ, which is identical with the mass operator of the Poincar´ e group, t he solutions will be expressed in terms of the corresponding plane wave representations. T he section is separated into two parts. The ﬁrst part describes how the plane wave states a re generated with the induced representation method. The second section investi gates the connection between spin and Pauli-Lubanski vector. 5.1.Induced representation method . – The irreducible representations of the Poin- car´ e group are labelled by the mass mand the spin s[3]. In the present work these states will be generated with the induced representation me thod, where a state vector is deﬁned within the little group of the Poincar´ e group, i.e . the subgroup that leaves a particular standard vector invariant. An arbitrary state is then generated with the remaining transformations. In the following the transform ation rules of Section 3.2 will be applied. For massive fermions one can choose the standard vector pµ t= (m,0,0,0). The little group of this standard vector is SO(3). The explicit representation of the spin s=1 2 states is given by the Pauli spinor, which will be denoted by \|σt/an}b∇acket∇i}ht=χσ. The polarisation is chosen along the z-axis. For the description of the mass qu antum number mthe ket \|pµ t/an}b∇acket∇i}htis introduced. One therefore starts with the following state \|pµ t/an}b∇acket∇i}ht ⊗ \|σt/an}b∇acket∇i}ht=\|pµ tσt/an}b∇acket∇i}ht. (33) Now, the boosts are acting on this state according to D(B)\|pµ tσt/an}b∇acket∇i}ht=\|pµ/an}b∇acket∇i}htBχσ, (34) whereBhas been deﬁned in Eq. (18) and pµ= (B)µ νpν t. (B)µ νcan be derived from Eq. (63). Since the boost transforms from the rest frame to a p articular frame, in whichRELATIVISTIC QUANTUM PHYSICS WITH HYPERBOLIC NUMBERS 7 the state is described by the momentum pµ, the boost parameters can be identiﬁed with the rapidity. With this information it is possible to calcul ate the explicit form of the relativistic spinor. In analogy to the Dirac formalism one c an introduce the notation u(p,σ) =Bχσ. (35) Explicitly the boost matrix can be written as B= exp(jξ·σ/2) = coshξ/2 +jn·σsinhξ/2, (36) where the rapidity ξsatisﬁes the following relations coshξ/2 =/parenleftbiggp0+m 2m/parenrightbigg1/2 , sinhξ/2 =/parenleftbiggp0−m 2m/parenrightbigg1/2 . (37) Inserting these results into Eq. (35) the spinor is given by u(p,σ) =/radicalbigg p0+m 2m/parenleftbigg 1 +jp·σ p0+m/parenrightbigg χσ. (38) The antiparticle spinor is constructed in analogy to the Dir ac theory, where upper and lower components are interchanged compared to the particle spinor. In the formalism presented here, this can be achieved by multiplying the part icle spinor by the hyperbolic unitj, i.e.v(p,σ) =ju(p,σ). Therefore, one can write v(p,σ) =/radicalbigg p0+m 2m/parenleftbiggp·σ p0+m+j/parenrightbigg χσ. (39) 5.2.The Pauli-Lubanski vector and spin operators . – To show that the plane wave states, derived in the last part of this section, correspond to an irreducible representation of the Poincar´ e group, the connection of these states with t he second Casimir operator, the Pauli-Lubanski vector, will be investigated. Since the spins=1 2representation is considered, the spin angular momentum operators of Eq. (2 4) will be used for the deﬁnition of the Pauli-Lubanski vector Wµ=1 2ǫµρσνSρσPν=˜SµνPν, (40) where ˜Sµνis the dual tensor of the relativistic angular momentum tens or. Explicitly the Pauli-Lubanski vector has the form W0=−J·P, W=−JP0−K×P (41) With the Pauli-Lubanski vector the relativistic spin opera tors can be deﬁned, where the investigation follows the methods given in [4, 5]. One ch ooses a set of four orthogonal vectorsn(ν)satisfying the relation n(µ) ρnρ(ν)=gµν. (42)8 S. ULRYCH Using these vectors the spin operators are deﬁned according to S=1 mWµnµ, (43) where nµ=nµ(i). If the plane wave states derived in the ﬁrst part of this sect ion shall be eigenstates of the spin operators, a particular set of orthogonal vectors nµ(ν)= (n0(ν),nk(ν)) has to be introduced nµ(0)=/parenleftbiggP0 m,Pk m/parenrightbigg , nµ(i)=/parenleftbiggPi m,δki+PkPi m(P0+m)/parenrightbigg . (44) Using these vectors the three spin operators can be written e xplicitly as S=1 m/parenleftbigg JP0+K×P−(J·P)P P0+m/parenrightbigg . (45) For the spin operators one ﬁnds S2=s(s+ 1) = −WµWµ/m2. The operators satisfy the commutation relations of the little group SO(3). The third component of the spin vector can be used to characterize the polarisation. The spin operators were constructed in that way, that they co incide with a vector of boosted generators J, where the boost matrices are acting on the coordinates S=BJ¯B . (46) In other words, Srefers to the same operator as J. As well as in the derivation of the plane wave states, the boost parameters in Bhave to be identiﬁed with the rapid- ity of the state vector. From the above equation follows that under arbitrary Lorentz transformations the spin operators have to transform accor ding to S=⇒S′=LS¯L, (47) whereLcorresponds to the Lorentz tranformation matrix given in Eq . (21). From Eq. (46) one can deduce that it is suﬃcient to deﬁne the spin in the rest frame of the state, according to the non-relativistic description with the non-relativistic spin operators Ji=σi/2, whereas the boost operators Ki=ijJiare not used to characterize the single- particle state for positive energies. Vector products of the form A¯B, with two arbitrary vectors AandB, transform in the same way as the spin operators (A¯B) =⇒(A¯B)′=L(A¯B)¯L. (48) Therefore, P¯Pwas chosen as the diﬀerential operator of the wave equation w ith spin. This guarantees that the operator shows the correct transfo rmation property. Now, the properties of the positive energy states can be summ arized. The plane wave states for positive energies are eigenstates of the four ope ratorsRELATIVISTIC QUANTUM PHYSICS WITH HYPERBOLIC NUMBERS 9 {PµPµ,Pµ,WµWµ,S3}and satisfy the relations PµPµ\|pµσ/an}b∇acket∇i}ht=m2\|pµσ/an}b∇acket∇i}ht, Pµ\|pµσ/an}b∇acket∇i}ht=pµ\|pµσ/an}b∇acket∇i}ht, WµWµ\|pµσ/an}b∇acket∇i}ht=−m2s(s+ 1)\|pµσ/an}b∇acket∇i}ht, (49) S3\|pµσ/an}b∇acket∇i}ht=σ\|pµσ/an}b∇acket∇i}ht. With these states and the contributions from the negative en ergies the solution of the wave equation ψ(x) can be expressed as the following plane wave expansion ψ(x) =/summationdisplay σ/integraldisplayd3p (2π)32p0/parenleftig u(p,σ)e−ipµxµb(p,σ) +v(p,σ)eipµxµ¯d(p,σ)/parenrightig . (50) 6. – Wave equation and Dirac equation The fermion wave equation should be invariant under local ga uge transformations. This has the consequence that a gauge ﬁeld has to be introduce d by a substitution of the momentum operators. In the following only electromagnetic interactions are con sidered. The gauge ﬁeld is then introduced with the minimal substitution of the moment um operators Pµ=⇒Pµ−eAµ(x), (51) where the charge e <0 corresponds to the negative charge of the electron. Using t he above substitution the electron wave equation transforms i nto (P−eA(x))(¯P−e¯A(x))ψ(x) =m2ψ(x). (52) Remember that A(x) =A0(x) +jA(x)·σand¯A(x) =A0(x)−jA(x)·σ. The Dirac equation is able to explain experimental data with highest accuracy. The electron wave equation should therefore be in agreement wit h the Dirac equation. One can show that the wave equation is equivalent to the quadrati c form of the Dirac equation from which one knows that the energy spectrum of hydrogen lik e systems is exactly the same as for the Dirac equation. To show this relationship, the wave equation (52) will be con sidered in detail. A short calculation is leading to /parenleftbig (P0−eA0)2−((P−eA)·σ)2−j[P0−eA0,P−eA]·σ−m2/parenrightbig ψ(x) = 0. (53) One can evaluate the second term of the equation according to ((P−eA)·σ)2= (P−eA)·(P−eA)−eB·σ, (54) where Bcorresponds to the magnetic ﬁeld. The commutator can be calc ulated according to [P0−eA0,P−eA] =ieE, (55)10 S. ULRYCH with the electric ﬁeld E. Inserting these results into Eq. (53) gives /parenleftbig (P−eA)µ(P−eA)µ−eijE·σ+eB·σ−m2/parenrightbig ψ(x) = 0. (56) It is possible to express Eq. (56) completely in the relativi stic tensor formalism if Pauli matrices and electromagnetic ﬁelds are expressed with the a ntisymmetric tensor σµν given in Eq. (30) and Fµν=∂µAν−∂νAµ /parenleftig (P−eA)µ(P−eA)µ−e 2σµνFµν−m2/parenrightig ψ(x) = 0. (57) This equation is formal identical to the quadratic form of th e Dirac equation, which can be derived from the Dirac formalism. The Dirac equation i s given by (γµPµ−eγµAµ(x)−m)ψ(x) = 0 (58) with the Dirac matrices γµ. The quadratic form can be found if one multiplies the Dirac equation by the operator γµPµ−eγµAµ(x) +m. This yields /parenleftbig (γµPµ−eγµAµ)2−m2/parenrightbig ψ(x) =/parenleftbigg (P−eA)µ(P−eA)µ−i 2σµν[Pµ−eAµ,Pν−eAν]−m2/parenrightbigg ψ(x) =/parenleftig (P−eA)µ(P−eA)µ−e 2σµνFµν−m2/parenrightig ψ(x) = 0. (59) The two wave equations (57) and (59) have the same form. Howev er, there are two diﬀerences: The ﬁrst diﬀerence is given in the structure of t he spinorsψ(x). In the case of the wave equation ψ(x) has a two-component structure, whereas in the Dirac equati on ψ(x) corresponds to a four-component spinor Wave equation : ψ(x) =ϕ(x) +jχ(x), Dirac equation : ψ(x) =/parenleftbigg ϕ(x) χ(x)/parenrightbigg . (60) The other diﬀerence is the spin tensor σµν. In the Dirac theory this term is deﬁned according to σµν=i/2 [γµ,γν]. With this tensor one is able to express Eq. (59) according to /parenleftbig (P−eA)µ(P−eA)µ−eiE·α+eB·σ−m2/parenrightbig ψ(x) = 0. (61) Comparing this equation with Eq. (56) one observes that in bo th cases the term including the electric ﬁeld is the only term which couples th e upper and the lower com- ponent of the spinor. In the wave equation the coupling term i s proportional to jσ, in the quadratic Dirac equation the term corresponds to α=γ0γ=γ5σ. One can show, using the Dirac representation of γ5, thatjandγ5have the same eﬀect on the spinor, an interchange between upper and lower components Wave equation : jψ(x) =χ(x) +jϕ(x), Dirac equation : γ5ψ(x) =/parenleftbigg 0 1 1 0/parenrightbigg /parenleftbigg ϕ(x) χ(x)/parenrightbigg =/parenleftbigg χ(x) ϕ(x)/parenrightbigg . (62)RELATIVISTIC QUANTUM PHYSICS WITH HYPERBOLIC NUMBERS 11 One therefore ﬁnds in both cases the same two coupled diﬀeren tial equations. In the wave equation the terms proportional to the hyperbolic unit belong to one diﬀerential equation, the other terms to the second equation. In the quad ratic Dirac equation the diﬀerential equations are separated by the component struc ture. 7. – Orbital angular momentum and single particle potential s As well as in the description of the relativistic spin, the hy perbolic numbers provide a simpliﬁed description of the orbital angular momentum. Th e transformation of the vector components xµcan be performed with 4 ×4 transformation matrices xµ=⇒xµ′= (L)µ νxν, L= exp ( −i(J·θ+K·ξ)). (63) For the generators JiandKionly the third components are displayed (J3)µ ν= 0 0 0 0 0 0 −i0 0i0 0 0 0 0 0 ,(K3)µ ν= 0 0 0 i 0 0 0 0 0 0 0 0 i0 0 0 . (64) In the following Eq. (56) will be considered, where Aµ(x) is interpretated as a single particle potential. The relative coordinates xµcan be restricted to spacelike coordinates, i.e.xµxµ=−ρ2<0. The parametrisation of the free space coordinates suitab le for the description of a single particle moving in a relativistic po tential is then xµ= x0 x1 x2 x3 = ρsinhξ ρcoshξsinθcosφ ρcoshξsinθsinφ ρcoshξcosθ . (65) The relative time between the particles is parametrized as ρsinhξ, whereρ>0. In the limit ofξ→0 the vector reduces to a non-relativistic vector in spheric al coordinates. Using the representation of the Lorentz transformations gi ven in Eq. (63) the above vector can be obtained from a standard vector xµ t= (0,0,0,1) with the following trans- formation L(θ,φ,ξ) = exp ( −iJ3φ) exp( −iJ2θ) exp ( −iK3ξ). (66) Now, one can adopt the results of the considerations of the sp in angular momentum. Changing from the matrix representation given above to the i rreducible group represen- tation one can write K=ijJ (67) and the transformation of Eq. (66) can be written as L(θ,φ,ξ) = exp ( −iJ3φ) exp ( −iJ2θ) exp (jJ3ξ). (68)12 S. ULRYCH As stated in Section 5.2, the boost operators of the transformations are not used to characterize the positive energy states. One can specify ag ainL=Jand deﬁne the following relations for the irreducible states L2\|lm/an}b∇acket∇i}ht=l(l+ 1)\|lm/an}b∇acket∇i}ht, (69) L3\|lm/an}b∇acket∇i}ht=m\|lm/an}b∇acket∇i}ht. Now the transformation given above can be represented expli citly in matrix form accord- ing to Dl(θ,φ,ξ)mlmr=/an}b∇acketle{tlml\|e−iL3φe−iL2θejL3ξ\|lmr/an}b∇acket∇i}ht =e−imlφdl(θ)mlmrejmrξ, (70) with dl(θ)mlmr=/an}b∇acketle{tlml\|e−iL2θ\|lmr/an}b∇acket∇i}ht. (71) The only diﬀerence of the relativistic rotation matrices co mpared to the nonrelativistic rotation matrices is absorbed in the phase ejmrξ. Since these rotation matrices are related to the spherical harmonics, it is possible to deﬁne r elativistic spherical harmonics according to Ylmlmr(Ω) =/parenleftbigg2l+ 1 4πf(Λ)/parenrightbigg1/2 [Dl(θ,φ,ξ)mlmr]∗−, (72) where Ω ≡(θ,φ,ξ). The normalisation is a function of the cut-oﬀ parameter Λ. To deﬁne a compact group the parameter space of ξ∈[−Λ,Λ] is restricted. The limit Λ → ∞ can be taken at the end of explicit calculations, if this is neede d. The additional magnetic quantum number mrreﬂects the fact that, compared to non-relativistic physic s, there is the additional free parameter ξ. The above description implies a range of mrbetweenl and−lwith integer steps. With the relativistic spherical harmonics the dependence o f thePµPµ=−∂µ∂µterm in Eq. (56) on the rotation and boost parameters θ,φ, andξcan be eliminated and absorbed in the dependence on the angular momentum land the seperation constants mlandmr. It is non-trivial task to derive relativistic single partic le potentials from ﬁrst principles. However, one can perform calculations with appropriate mod el potentials. One possibility would be a relativistic generalisation of the 1 /\|x\|central potential describing an electron moving in the potential of a nucleus. eAµ(x) =−Zα ρǫµ(x), (73) where the polarisation vector corrsponds to the unit vector of theξ-coordinate ǫµ(x) =1 ρ∂ ∂ξxµ(ρ,θ,φ,ξ ) = coshξ sinhξsinθcosφ sinhξsinθsinφ sinhξcosθ . (74)RELATIVISTIC QUANTUM PHYSICS WITH HYPERBOLIC NUMBERS 13 In the static limit ξ→0 this potential reduces to the 1 /\|x\|potential. Equation (56) has not been solved yet with the above potentia l. Therefore, only some general remarks on the solution procedure are given here. Th eσ-terms in Eq. (56) imply a coupling of the orbital angular momentum with the spin Jc=L+S. In addition, there is a term proportional to the hyperbolic unit jwhich has a diﬀerent parity. To obtain a solution the general ansatz for the wave function ψ(x) must be ψ(x) =φl(x) +jχl′(x), (75) with l′=/braceleftbiggl−1 l+ 1forl=jc+ 1/2 l=jc−1/2. (76) A relativistic single particle potential depends on the rel ative time. Therefore, one can not ask for the energy spectrum. The spectrum to be consid ered here is the mass spectrum. One can introduce a mass operator M2= (P−eA(x))µ(P−eA(x))µ−e 2σµνFµν(x), (77) where the solutions of Eq. (56), ψ(x)≡ψλ(x) are eigenstates of this mass operator M2ψλ(x) =m2 λψλ(x), (78) with a suitable set of quantum numbers λ. Solving Eq. (56) with the model potential of Eq. (73) should lead to results close to the spectrum of the Dirac equation. For the ground state it is expected to ﬁnd approximately m2 GS≈1−Z2α2. (79) 8. – Summary and Conclusions The preceding investigations have shown that the hyperboli c numbers are a helpful tool for the investigation of relativistic quantum physics . In particular, they oﬀer sim- pliﬁcations in the description of the relativistic angular momentum. Boost generators are constructed with a multiplication of the original SO(3) generators by the complex and the hyperbolic unit. Adding them to the SO(3) generators the Lorentz algebra is satisﬁed. The quadratic Dirac equation can be expressed in terms of a 2 ×2-matrix equation using the hyperbolic numbers. This equation is the starting point for calculations with non-static single particle potentials. The eigenvalue of t he relativistic single particle equation is the (squared) mass. In addition, a new quantum number arises compared to the non- relativistic description or the quasi-relativistic Dirac approximations often used . This quantum number is due to the consideration of the relative time. The additional quan tum number does not indicate new physics. The three quantum numbers of the orbital angula r momentum correspond to the three quantum numbers of the momentum in the plane wave representation and are therefore needed for a consistent description of the sin gle particle states.14 S. ULRYCH REFERENCES [1]A. Dura ˜nona Vedia andJ. C. Vignaux ,Publ. de Facultad Ciencias Fisicio-matematicas Contrib. (Universidad Nacional de La Plata - Argentina) ,104(1935) 139. [2]P. Capelli ,Bull. of American Mathematical Society ,47(1941) 585. [3]E. P. Wigner ,Annals of Mathematics ,40(1939) 149. [4]L. Michel ,Il Nouvo Cimento Supplemento ,14(1959) 95. [5]A. S. Wightman , inRelations de Dispersions et Particules ´El´ ementaires , edited by C. DeWitt andM. Jacob (Hermann and John Wiley, New York) 1960. 1
arXiv:physics/0009080v1 [physics.gen-ph] 27 Sep 2000The Energy of the Universe, the Scaling-Dependent Coupling Constant β and the Pion Mass H. C. G. Caldas†‡∗, and P. R. Silva‡ †Departamento de Ciˆ encias Naturais, DCNAT Funda¸ c˜ ao de Ensino Superior de S˜ ao Jo˜ ao del Rei, FUNREI, Pra¸ ca Dom Helv´ ecio, 74, CEP:36300-000, S˜ ao Jo˜ ao del Rei , MG, Brazil ‡Departamento de F´ ısica, Universidade Federal de Minas Ger ais, CP 702,CEP:30.161-970,Belo Horizonte, MG, Brazil (September, 2000) From the hypothesis of a quadratic dependence of the binding energy of the quarks on the length scale and from the idea of a universal running coupling const ant, we get the closure mass of the Universe and the pion mass. Besides this, we propose a free en ergy for the pions which stationarity condition leads to a closed form for the pion mass as a functio n of the Planck mass and of the Hubble constant. PACS numbers: 04.20.Cv, 04.70.-s In a recent paper [1], was proposed a universal interaction b etween the quarks, running from the Universe gauge factor ato the Planck length LP. In this way, the model of Skalsk´ y and S´ uken´ ık as describe d in [1] comprises both the asymptotic freedom and the quark conﬁnement. As pointed out by Gross in Ref. [5] the forces between the quarks vary with the distance between them: the dynamics of the vacu um enhances the force at large distances, while at short distances, the interaction grows weaker. The assumption that the binding energy of conﬁned quarks inc reases quadratically [1] was a “ad hoc” hypothesis in order to be consistent with the experimental ﬁndings. In t his letter we intend to pursue further on this subject, through somewhat more formal way. Let us represent the globa l behavior of the Universe as a bound system. In terms of an average separation rbetween the particles with average momentum pwe can write the classical Hamiltonian as H=p2 2µ−β¯hc r(1) In (1), µis the reduced mass of the Universe of mass M, and βis a scaling-dependent coupling constant, to be deﬁned latter. From the uncertainty relation, we have: r=¯h p(2) Putting (2) in (1) and minimizing the obtained relation rela tive to p, we obtain the energy: E=−1 2β2µc2=−1 4β2Mc2(3) Although we are not working on the basis of a non-Abelian ﬁeld theory we can think in terms of a scaling-dependent coupling constant which could exhibit both asymptotic free dom and conﬁnement. It seems that a simple achievement of these requirements can be obtained by deﬁning β(r) =βor a(4) where ris the scale of the interaction. Relation (4) into (3) leads to: ∗e-mail: hcaldas@funrei.br 1Er≡ \|E(r)\|=1 4Mc2β2 o(r a)2(5) We see that the binding energy given by (5) has the desired dep endence on the scaling of the interactions, namely: the quadratic dependence on r. Let us now write Erin a more convenient way multiplying and dividing the r.h.s. of equation (5) by L2 p. Thus, Er=1 4Mc2(Lp a)2β2 o(r Lp)2(6) Now we write the well known relations: M=1 2c3 GHo(7) L2 p=¯hG c3(8) and Ho=c a(9) where Mis the closure mass of the Universe, HoandGare respectively the Hubble and the Gravitational constant s, and by substituting relations (7), (8) and (9) into (6) we obt ain: Er=β2 o 16π/parenleftbigghc a/parenrightbigg /parenleftbiggr Lp/parenrightbigg2 (10) Despite our scaling-dependent coupling constant could lea d to the sought dependence of the “binding” energy with the scaling parameter r, we must permit a slight dependence of the βoparameter on r. So, by putting β2 o= 16πinto (10), we reproduce relation (3) of Skalsk´ y and S´ uken´ ık, which works well in the neighborhood of r=¯h mπc, and also gives Er(r=Lp)≡Emin, where Eminis given by relation (1) of Skalsk´ y and S´ uken´ ık. On the other hand the relation (2) of Skalsk´ y and S´ uken´ ık, namely Emax=Mc2, could be obtained from the relation (6) of this paper only with the choice β2 o= 4;r=a. This choice also reproduces the result found in [4] for the minimal energy, Er(r=Lp) =1 2¯hH0, due the interaction between a particle and the rest of the Un iverse. Now, let us suppose that we did not know any previous results a nd we want to ﬁx the parameter β2 oof (10) in the neighborhood of r=¯h mπc. So, as a means to ﬁx βo, we can make the following considerations: Recently one of the present authors [6] have found that to a pa rticle of mass mwe can associate a zero-point energy given by1 2¯hωdB=1 2mc2, where ωdBis the de Broglie frequency. Then we can associate the diﬀere nce between the rest energy of a particle and this zero-point energy as an eﬀe ct of the thermal excitations. Let us suppose yet the pion as a particle of surface of radius equal to Lπ=¯h mπcbehaving as an event-horizon surface to wich we can associat e a Bekenstein entropy [2,3]. We also assume that the pion is in thermal equilibrium with the matter of the Universe with a temperature Tπ, given by: kBTπ=¯hc a= ¯hHo (11) where kBis the Boltzmann constant. Seeking for the condition of a sta tionary free energy for the pion in thermal equilibrium with the Universe, we can write: ∆F= ∆U−Tπ∆S= 0 where F,UandSare respectively the free and the internal energies and the e ntropy of the pion. Writing: ∆S=kB 44π/parenleftbiggLπ Lp/parenrightbigg2 =πkB/parenleftbiggmp mπ/parenrightbigg2 (12) 2and ∆U=1 2mπc2(13) where Lpandmpare respectively the Planck radius and mass, we obtain: mπc2=hc a/parenleftbiggmp mπ/parenrightbigg2 (14) The above relation can also be obtained through relation (3) of [1] upon identifying E(r=Lπ) with the pion rest energy. Naturally, in order to relation (10) of this pap er be able to reproduce (14), we must make the choice β2 o= 16π;r=Lπ. It is intriguing to verify that by considering 1026m as the radius of the Universe, we get ∆ SU=KBπ10122as its entropy, calculated according to the Bekenstein recipe, wh ere we have considered the Universe as a black hole. On the other hand, the entropy of the pion is given by ∆ Sπ=KBπ1040. Therefore the ratio between these entropies is equal to 1082, which is of the same order of magnitude of the number of hadro ns of the Universe. In this way, the entropy of the Universe, ∆ SU(orSMAX [7]), turns out to be the (present) bound from above on entrop y, as is the case when we think on the grounds of Quantum Gravity [8]. It would be interesting to comment that in the case of (14) wil l prove to be an exact relation, it would be a way of obtaining the accurate value of the radius of the Universe (o r the Hubble constant), once the other quantities which appear there are known with relatively great accuracy. Acknowledgements CNPq-Brazil is acknowledged for invaluable ﬁnancial help. [1] V. Skalsk´ y and M. S´ uken´ ık, Astrophys. Space Sci. 236,295 (1996). [2] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973). [3] J. D. Bekenstein, Phys. Rev. D 9, 3292 (1974). [4] H. C. G. Caldas and P. R. Silva, gr-qc/9809007; to be publi shed in Apeiron. [5] D. J. Gross, Phys. Today 40, 39 (1987). [6] P. R. Silva, Phys. Essays. 10,628 (1997). [7] P. A. Zizzi, gr-qc/0008049. [8] J. D. Bekenstein, hep-th/0003058. 3
arXiv:physics/0009081v1 [physics.plasm-ph] 27 Sep 2000On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with ﬂow G. N. Throumoulopoulos1and H. Tasso Max-Planck-Institut f¨ ur Plasmaphysik, EURATOM Associat ion D-85748 Garching, Germany October 1999 Abstract It is shown that the magnetohydrodynamic equilibrium state s of an axisym- metric toroidal plasma with ﬁnite resistivity and ﬂows para llel to the magnetic ﬁeld are governed by a second-order partial diﬀerential equ ation for the poloidal magnetic ﬂux function ψcoupled with a Bernoulli type equation for the plasma density (which are identical in form to the corresponding id eal MHD equilibrium equations) along with the relation ∆⋆ψ=Vcσ. (Here, ∆⋆is the Grad-Schl¨ uter- Shafranov operator, σis the conductivity and Vcis the constant toroidal-loop volt- age divided by 2 π). In particular, for incompressible ﬂows the above mention ed partial diﬀerential equation becomes elliptic and decoupl es from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasma s5, 2378 (1998)]. For a conductivity of the form σ=σ(R,ψ) (Ris the distance from the axis of symmetry) several classes of analytic equilibria with inco mpressible ﬂows can be constructed having qualitatively plausible σproﬁles, i.e. proﬁles with σtaking a maximum value close to the magnetic axis and a minimum value o n the plasma surface. For σ=σ(ψ) consideration of the relation ∆⋆ψ=Vcσ(ψ) in the vicinity of the magnetic axis leads therein to a proof of the non-exist ence of either com- pressible or incompressible equilibria. This result can be extended to the more general case of non-parallel ﬂows lying within the magnetic surfaces. PACS numbers: 52.30.Bt, 52.55.-s 1Permanent address: University of Ioannina, Association EU RATOM / Hellenic Republic, Physics Department, Section of Theoretical Physics, GR 451 10 Ioannina, GreeceI. Introduction In addition to the case of the long living astrophysical plas mas, understanding the equilibrium properties of resistive fusion plasmas is i mportant, particularly in view of the next step devices which will possibly demand pu lse lengths of the order of 103secs (or more for an ITER size machine) ([1] and Refs. cited th erein). Theoretically, however, it was proved by Tasso [2] that resi stive equilibria with σ=σ(ψ) are not compatible with the Grad-Schl¨ uter-Shafranov equ ation. (Here, σis the conductivity and ψis the poloidal magnetic ﬂux function.) The non- existence of static axisymmetric resistive equilibria wit h a uniform conductivity was also suggested recently [3, 4, 5]. Also, in the collision al regime Pﬁrsch and Schl¨ uter showed that the toroidal curvature gives rise to a n enhanced diﬀusion, which is related to the conductivity parallel to the magneti c ﬁeld. In the above mentioned studies the inertial-force ﬂow term ρ(v· ∇)vis neglected in the equa- tion of momentum conservation. For ion ﬂow velocities of the order of 100 Km/m, which have been observed in neutral-beam-heating experime nts [6, 7, 8] the term ρ(v· ∇)vcan not be considered negligible. Therefore, it is worthwhi le to inves- tigate the nonlinear resistive equilibrium, in particular to address the following issues: (a) the impact of the non-linear ﬂow in the Pﬁrsch-Sc hl¨ uter diﬀusion, and (b) the existence of resistive equilibria, in particular eq uilibria with σ=σ(ψ). Since the magnetohydrodynamic (MHD) equilibrium with arbi trary ﬂows and ﬁ- nite conductivity is a very diﬃcult problem, in a recent stud y [9] we considered an axisymmetric toroidal plasma with purely toroidal ﬂow in cluding the term ρ(v· ∇)vin the momentum-conservation equation. It was shown that th e non- linear ﬂow does not aﬀect the static-equilibrium situation , i.eσ=σ(ψ) equilibria are not possible. A way of constructing more plausible equilibria from the phy sical point of view could be by considering ﬂows less restricted in directi on. Taking also into account the fact that the poloidal ﬂow in the edge region of ma gnetic-conﬁnement systems plays a role in the transition from the low-conﬁneme nt mode to the high- conﬁnement mode, in the present report we extend our previou s studies to the case of ﬂows having non-vanishing poloidal components in additi on to toroidal ones. Because of the diﬃculty of the problem we consider ﬂows paral lel to the magnetic ﬁeld. Some of the conclusions, however, can be extended to no n-parallel ﬂows lying within the magnetic surfaces. It is also noted that pos sible equilibria with 2parallel ﬂows would be free of Pﬁrsch-Schl¨ uter diﬀusion be cause the convective termv×Bin the Ohm’s low vanishes. The main conclusion is that for the system under consideration the existence of equilibria dep ends crucially on the spatial dependence of conductivity. The report is organize d as follows. The equilibrium equations for an axisymmetric toroidal resist ive plasma with parallel ﬂows surrounded by a conductor are derived in Sec. II. The exi stence of solutions is then examined in Sec. III for the cases σ=σ(R,ψ) (Ris the distance from the axis of symmetry), and σ=σ(ψ). Sec. IV summarizes our conclusions. II. Equilibrium equations The MHD equilibrium states of a plasma with scalar conductiv ity are governed by the following set of equations, written in standard notat ions and convenient units: ∇ ·(ρv) = 0, (1) ρ(v· ∇)v=j×B− ∇P, (2) ∇ ×E= 0, (3) ∇ ×B=j, (4) ∇ ·B= 0, (5) E+v×B=j σ. (6) It is pointed out that, unlike to the usual procedure followe d in equilibrium studies with ﬂow [10, 11, 12, 13, 14, 15] in the present work an equatio n of state is not included in the above set of equations from the outset and the refore the equation of state independent Eqs. (15) and (16) below are ﬁrst derive d. This alternative procedure is convenient because the equilibrium problem is then further reduced for speciﬁc cases associated with several equations of stat e. The system under consideration is a toroidal axisymmetric m agnetically con- ﬁned plasma, which is surrounded by a conductor (see Fig. 1 of Ref. [9]). With the use of cylindrical coordinates R,φ,z the position of the surface of the con- ductor is speciﬁed by some boundary curve in the ( R,z) plane. The equilibrium quantities do not depend on the azimuthal coordinate φ. Consequently, the di- vergence free magnetic ﬁeld Band current density jcan be expressed, with the 3aid of Ampere’s low (4), in terms of the stream functions ψ(R,z) andI(R,z) as B=I∇φ+∇φ× ∇ψ, (7) and j= ∆⋆ψ∇φ− ∇φ× ∇I. (8) Here, ∆⋆is the elliptic operator deﬁned by ∆⋆=R2∇ ·(∇/R2) and constant ψ surfaces are magnetic surfaces. Also, it is assumed that the plasma elements ﬂow solely along B: ρv=KB, (9) whereKis a function of Randz. Acting the divergence operator on Eq. (9) and taking into account Eq. (1) one obtains ∇K·B= 0. Therefore, the function K is a surface quantity: K=K(ψ). (10) Another surface quantity is identiﬁed from the toroidal com ponent of the mo- mentum conservation equation (2): /parenleftBigg 1−K2 ρ/parenrightBigg I=X(ψ). (11) From Eq. (11) it follows that, unlike the case in static equil ibria, I is not (in general) a surface quantity. Furthermore, expressing the t ime independent electric ﬁeld by E=−∇Φ +Vc∇φ, (12) whereVcis the constant toroidal-loop voltage divided by 2 π, the poloidal and toroidal components of Ohm’s law (6), respectively, yield ∇Φ =∇φ× ∇I σ(13) and ∆⋆ψ=Vcσ=EφRσ. (14) Here,Eφis the toroidal component of E. Eq. (14) has an impact on the boundary conditions, i.e. the component of Etangential to the plasma-conductor interface does not vanish. Therefore, the container can not be conside red perfectly con- ducting. Accordingly, Ohm’s law with ﬁnite conductivity ap plied in the vicinity 4of the plasma-conductor interface does not permit the exist ence of a surface layer of current [16]. It is now assumed that the position of the con ductor is such that its surface coincides with the outermost of the closed magne tic surfaces. Thus, the condition B·n= 0, where nis the outward unit vector normal to the plasma surface, holds in the plasma-conductor interface and there fore the pressure P must vanish on the boundary. It is noticed that this is possib le only in equilib- rium, because in the framework of resistive MHD time depende nt equations, the magnetic ﬂux is not conserved. With the aid of equations (7)- (11) the compo- nents of Eq. (2) along Band perpendicular to a magnetic surface are put in the respective forms B·/bracketleftBigg ∇/parenleftBiggK2B2 2ρ2/parenrightBigg +∇P ρ/bracketrightBigg = 0 (15) and /braceleftBigg ∇ ·/bracketleftBigg/parenleftBigg 1−K2 ρ/parenrightBigg∇ψ R2/bracketrightBigg +K ρ∇K· ∇ψ R2/bracerightBigg \|∇ψ\|2 +  ρ∇/parenleftBiggK2B2 2ρ2/parenrightBigg +∇I2 2R2−ρ 2R2∇/parenleftBiggIK ρ/parenrightBigg2 +∇P  · ∇ψ= 0. (16) Eq. (16) has a singularity when K2 ρ= 1. (17) On the basis of Eq. (9) for ρvand the deﬁnitions v2 Ap≡\|∇ψ\|2 ρfor the Alfv´ en velocity associated with the poloidal magnetic ﬁeld and the Mach number M2≡v2 p v2 Ap=K2 ρ, (18) Eq. (17) can be written as M2= 1. Summarizing, the resistive MHD equilibrium of an axisymmet ric toroidal plasma with parallel ﬂow is governed by the set of Eqs. (14), ( 15) and (16). Owing to the direction of the ﬂow parallel to B, Eqs. (15) and (16) do not con- tain the conductivity and are identical in form to the corresponding equations governing ideal equilibria. Therefore, on the one hand, sev eral properties of the ideal equilibria, e.g. the Shafranov shift of the magnetic s urfaces and the de- tachment of the isobaric surfaces from the magnetic surface s (see the discussion 5following Eq. (26) in Sec IIC) remain valid. On the other hand , as will be shown in Sec. III, the conductivity σin Eq. (14) plays an important role on the existence of equilibria. To reduce further equations (15) and (16), the starting set o f equations (1)-(6) must be supplemented by an equation of state, e.g. P=P(ρ,T), along with an equation determining the transport of internal energy. Suc h a rigorous treatment, however, makes the equilibrium problem very cumbersome. Al ternatively, one can assume additional properties for the magnetic surfaces associated with either isentropic processes, or isothermal processes, or incompr essible ﬂows. These three cases are separately examined in the remainder of this secti on. A. Isentropic magnetic surfaces We consider a plasma with large but ﬁnite conductivity such t hat for times short compared with the diﬀusion time scale, the dissipativ e term ≈j2/σcan be neglected. This permits one to assume conservation of the en tropy: v· ∇S=0, which on account of Eq. (9) leads to S=S(ψ) (Sis the speciﬁc entropy). It is noted that the case S=S(ψ) was considered in investigations on ideal equilibria with arbitrary ﬂows [11, 12] and purely toroidal ﬂows [17, 18], as well as on resistive equilibria with purely toroidal ﬂows [9]. In addition, the plasma is assumed to being a perfect gas whose internal energy densi tyWis simply proportional to the temperature. Then, the equations for th e thermodynamic potentials lead to [17] P=A(S)ργ(19) and W=A(S) γ−1ργ−1=H γ. (20) Here,A=A(S) is an arbitrary function of S,H=W+P/ρis the speciﬁc enthalpy and γis the ratio of speciﬁc heats. For simplicity and without los s of generality we choose the function Ato be identical with S. Consequently, integration of Eq. (15) yields K2B2 2ρ2+γ γ−1Sργ−1=H(ψ). (21) Eq. (16) reduces then to ∇ ·/bracketleftBigg/parenleftBigg 1−K2 ρ/parenrightBigg∇ψ R2/bracketrightBigg + (v·B)K′+Bφ RX′+ρH′−ργS′= 0, (22) 6where the prime denotes diﬀerentiation with respect to ψ. Apart from a factor 1/(γ−1) in the last term of the right-hand side ([1 /(γ−1)]ργS′instead ofργS′) Eq. (22) is identical in form with the corresponding ideal MHD eq uation obtained by Hameiri [12] (Eq. (7) therein). It should be noted that Eq. (2 2) remains regular for the case of isothermal plasmas ( γ= 1) while Hameiri’s result would make the equilibrium equation strangely singular. In particula r, forS=S(ψ) and T= const. Eq. (19) leads to ρ=ρ(ψ) and consequently the incompressibility equation ∇ ·v= 0 follows from Eq. (1). Incompressible ﬂows, however, are described by Eq. (27) below which is free of the above mention ed singularity. Unlike the case of static equilibria, Eq. (22) is not always e lliptic; there are three critical values of the poloidal-ﬂow Mach-number M2at which the type of this equation changes, i.e. it becomes alternatively ell iptic and hyperbolic [10, 12]. The toroidal ﬂow is not involved in these transitio ns because this is incompressible by axisymmetry and, therefore, does not rel ate to hyperbolicity (see also the discussion in the beginning of Sec. IIC). B. Isothermal magnetic surfaces Since for fusion plasmas the thermal conduction along Bis expected to be fast in relation to the heat transport perpendicular to a magneti c surface, equilibria with isothermal magnetic surfaces are a reasonable approxi mation [17, 18, 19, 20, 21, 22]. In particular, the even simpler case of isothermal r esistive equilibria has also been considered [23]. ForT=T(ψ) integration of Eq. (15) leads to K2B2 2ρ2+λTlnρ=H(ψ), (23) whereλis the proportionality constant in the ideal gas law P=λρT. Conse- quently, Eq. (16) reduces to ∇ ·/bracketleftBigg/parenleftBigg 1−K2 ρ/parenrightBigg∇ψ R2/bracketrightBigg + (v·B)K′+Bφ RX′+ρH′−λρ(1−logρ)T′= 0.(24) We remark that apart from the fact that the S terms have been re placed by T terms, Eqs. (23) and (24) are identical with the respective E qs. (21) and (22). C. Incompressible ﬂows 7The existence of hyperbolic regimes may be dangerous for pla sma conﬁnement because they are associated with shock waves which can cause equilibrium degra- dation. In this respect incompressible ﬂows are of particul ar interest because, as is well known from gas dynamics, it is the compressibility that can give rise to shock waves; thus for incompressible ﬂows the equilibriu m equation becomes always elliptic. For ∇ ·v= 0 it follows from Eqs. (1) and (9) that the density is a surface quantity ρ=ρ(ψ), (25) consistent with the fact that in fusion experiments equilib rium density gradients parallel to Bhave not been observed. With the aid of Eq. (25), integration of Eq. (15) yields an exp ression for the pressure: P=Ps(ψ)−v2 2=Ps−K2B2 2ρ. (26) We note here that, unlike in static equilibria, in the presen ce of ﬂow magnetic surfaces in general do not coincide with isobaric surfaces b ecause Eq. (2) implies thatB· ∇Pin general diﬀers from zero. In this respect, the term Ps(ψ) is the static part of the pressure which does not vanish when v=0. If it is now assumed thatK2 ρ∝negationslash= 1 and Eq. (26) is inserted into Eq. (16), the latter reduces t o the elliptic diﬀerential equation (1−M2)∆⋆ψ−1 2(M2)′\|∇ψ\|2+1 2/parenleftBiggX2 1−M2/parenrightBigg′ +R2P′ s= 0. (27) Eq. (27) is identical in form to the corresponding ideal equi librium equation (Eq. (22) of Ref. [22]). It is also noted that special cases of incompressible ideal equilibria have been investigated in Refs. [24] and [2 5]. Unlike to the corresponding sets of compressible S=S(ψ) equations (21) and (22), and T= T(ψ) equations (23) and (24), Eq. (27) is decoupled from Eq. (26) . Once the solutions of Eq. (27) are known, Eq. (26) only determines the pressure. III. The existence of solutions in relation to the conductivity proﬁle We shall show that the compatibility of Eq. (14) containing t he conductivity σwith the “ideal” equations (15) and (16) depends crucially o n the spatial de- pendence of σ. In this respect the cases σ=σ(R,ψ), andσ=σ(ψ) are examined below. 8A.σ=σ(R,ψ) An explicit spatial dependence of σ, in addition to that of ψ, is interesting because it makes the equilibrium problem well posed, i.e. in this case Eq. (14) can be decoupled from the other Eqs. (15) and (16). A possible explicit spatial dependence of σcan be justiﬁed by the following arguments: (a) Even in Spitz er conductivity, σ=αT3/2 e, the quantity αhas a (weak) spatial dependence and (b) cylindrically symmetric resistive σ=σ(ψ) equilibria are possible [9] and therefore the non-existence of axisymmetric static toroidal σ=σ(ψ) equilibria is related to the toroidicity involving through the scale factor \|∇φ\|= 1/R; this could also imply an explicit dependence of σonR. In addition, we may remark that the neoclassical conductivity depends on the aspect ratio Abecause the fraction of trapped particles relates to A(see [26] and Refs. cited therein). It should be noted, however, that a knowledge of the σ-proﬁle in the various collisionality regimes of magnetic conﬁnement has not been obtained to date . For us the main advantage in allowing σ=σ(R,ψ) lies in the fact that Eq. (14) can then be considered as a formula determining the cond uctivity σ=∆⋆ψ Vc, (28) providedψis known. Also, the poloidal electric ﬁeld can then be obtain ed by Eq. (13). To determine ψin the case of compressible ﬂows with isentropic magnetic surfaces the set of Eqs. (21) and (22), which are coupled thro ugh the density ρ, should be solved numerically under appropriate boundary co nditions. This can be accomplished by the existing ideal MHD equilibrium codes [13, 14, 15]. The problem of compressible ﬂows with isothermal magnetic surf aces [Eqs. (23) and (24)] can be solved in a similar way. For incompressible ﬂows ψcan be determined by Eq. (27) alone, which is amendable to several classes of analytic solutions. In part icular, sheared- poloidal- ﬂow equilibria associated with “radial” (poloidal) electr ic ﬁelds which play a role in the L-H transition can be constructed by means of the trans formation [27, 28] U(ψ) =/integraldisplayψ 0[1−M2(ψ′)1/2]dψ, M2<1, (29) 9Under this transformation Eq. (27) reduces (after dividing by (1−M2)1/2) to ∆⋆U+1 2d dU/parenleftBiggX2 1−M2/parenrightBigg +R2dPs dU= 0. (30) It is noted here that the requirement M2<1 in transformation (29) implies that v2 p< v2 s, wherevs= (γP/ρ)1/2is the sound speed. This follows from Eqs. (18) and (in Gaussian units) /parenleftBiggvs vAp/parenrightBigg2 = (γ/2)8πP h2\|∇ψ\|2≈1. Since, according to experimental evidence in tokamaks [29] , the (maximum) value of the ion poloidal velocity in the edge region during the L-H transition is of the order of 10 Km/sec and the ion temperature is of the order of 1 K eV, the scaling vp≪vsis satisﬁed in this region. Therefore, the restriction M2<1 is of non- operational relevance. The simplest solution of Eq. (27) co rresponding to M2= const.,X2= const. and Ps∝ψ, is given by ψ=ψc/parenleftbiggR Rc/parenrightbigg2/bracketleftBigg 2−/parenleftbiggR Rc/parenrightbigg2 −d2/parenleftbiggz Rc/parenrightbigg2/bracketrightBigg , (31) whereψcis theψvalue on the magnetic axis located at ( z= 0,R=Rc) andd is a parameter related to the shape of ﬂux surfaces. Equation (31) describes the Hill’s vortex conﬁguration [30]. The conductivity then fol lows from Eq. (28): σ=σc/parenleftbiggR Rc/parenrightbigg4/bracketleftBigg 2−/parenleftbiggR Rc/parenrightbigg2 −d2/parenleftbiggz Rc/parenrightbigg2/bracketrightBigg , (32) whereσcis the value of σon the magnetic axis. The conductivity proﬁle in the middle-plane z= 0 is illustrated in Fig. 1. We remark the outward displace- ment of the maximum-conductivity position Rmaxwith respect to Rc(Rmax/Rc= 2/√ 3) and the asymetry of the inner part of the proﬁle as compared with the outer part due to the explicit Rdependence of σ. B.σ=σ(ψ) For this case we consider Eq. (14) in the vicinity of the magne tic axis by trans- forming the coordinates from ( R,z,φ ) to (x,y,φ ) (Fig. 2). The transformation is given by R=Rc+x=Rc+rcosθ z=y=−rsinθ. (33) 10The quantities ψ(x,y) andσ(ψ) are then expanded to second-order in xandy: ψ(x,ψ) =ψc+c1x2 2+c2y2 2+c3xy+... (34) and σ=σc+σ1(ψ−ψc) +...=σc+σ1(c1x2 2+c2y2 2+c3xy+...) +.... (35) Here,c1= (∂2ψ/∂x2)c,c2= (∂2ψ/∂y2)c,c3= (∂2ψ/∂x∂y )c,σcis the conductivity on the magnetic axis and σ1= const. On the basis of Eqs. (34) and (35) Eq. ∆⋆ψ=Vcσ(ψ) becomes a polynomial in xandywhich should vanish identically. This requirement leads to c1=c3= 0 and, therefore, it follows from Eq. (34) that the magnetic surfaces in the vicinity of the magnetic ax is are not closed surfaces. The non-existence of σ(ψ) equilibria with closed magnetic surfaces can be extended to the case of non-parallel ﬂows lying within the ma gnetic surfaces. Indeed, if the relation v· ∇ψ= 0 is assumed instead of v∝ba∇dblB, the toroidal component of Eq. (6) leads again to Eq. (14). A possible proof of the non-existence of η=η(ψ) equilibria far from the magnetic axis has not been obtained to date. It may be noted, h owever, that forσ=σ(ψ), Eq. (16) becomes parabolic . This follows by considering in this equation the determinant Dof the symmetric matrix of coeﬃcients. On account of ∆⋆ψ=Vcσ(ψ), andρ=ρ(R,ψ,\|∇ψ\|) by Eq. (15), the second derivatives of equation (16) are contained only in the term K2 ρ∂ρ ∂\|∇ψ\|2∇\|∇ψ\|2· ∇ψ, which comes from the term ∇ ·[(1−K2/ρ)∇ψ/R2]. Subsequent evaluation of Dleads to D= 0. Therefore, the function ψis (over)restricted everywhere to satisfy a parabolic equation and the elliptic equation ∆⋆ψ=Vcσ(ψ). IV. Conclusions The equilibrium of an axisymmetric plasma with ﬂow parallel to the magnetic ﬁeld has been investigated within the framework of the resistive magnetohydrodynamic (MHD) theory. For the system under consideration the equili brium equations 11reduce to a set of a second-order diﬀerential equation for th e poloidal magnetic ﬂux function ψcoupled through the density with an algebraic Bernoulli equ ation, which are identical in form with the corresponding ideal MHD equations, and the equation ∆⋆ψ=Vcσ. (∆⋆,Vcandσare the Grad-Schl¨ uter-Shafranov elliptic operator, the constant toroidal loop voltage and the conduc tivity, respectively. The existence of solutions of the above mentioned set of equa tions is sensitive to the spatial dependence of σ. For a conductivity of the form σ=σ(R,ψ), Eq. ∆⋆ψ=Vcσcan be considered uncoupled to the other two equations, thus determining only the conductivity. For compressible ﬂows and isentopic magnetic surfaces the d iﬀerential equation forψ[(Eq. (22)], pending on the value of the poloidal ﬂow, can be e ither elliptic or hyperbolic. Solutions of the set of this equation and the c oupled Bernoulli equation [Eq. (21)] can be obtained numerically. The proble m of compressible equilibria with isothermal magnetic surfaces [Eqs. (23) an d (24)] can be solved in a similar way. For incompressible equilibria ψobeys an elliptic diﬀerential equation [(Eq. (27)], uncoupled to the associated Bernoull i equation [Eq. (26)] which just determines the pressure. Several classes of anal ytic equilibria with incompressible ﬂows having qualitatively plausible σproﬁles, i.e, proﬁles with σ taking a maximum value close to the magnetic axis and a minimu m value on the plasma surface, can be constructed. In particular, sheared -poloidal-ﬂow equilibria can be derived by means of the transformation (29) for ψ. Forσ=σ(ψ) appreciation of ∆⋆ψ=Vcσin the vicinity of the magnetic axis proves therein, irrespective of plasma compressibili ty, the non-existence of closed magnetic surfaces. This result can be extended to the case of non-parallel ﬂows lying within the magnetic surfaces. In addition, for pa rallel ﬂows ψis (over)restricted to satisfy throughout the plasma an ellip tic and a parabolic dif- ferential equations. According to the results of the present investigation, the e xistence of resistive equilibria is sensitive to the spatial dependence of conduc tivity. Thus, the task of obtaining this dependence in the various conﬁnement regime s of fusion plasmas may deserve further experimental and theoretical investig ations. A conductivity with a spatial dependence in addition to that of ψ, on the one hand, would open up the possibility of the existence of several classes of res istive equilibria free of Pﬁrsch-Schl¨ uter diﬀusion. On the other hand, a strict Spit zer-like conductivity, 12σ=σ(ψ), should imply the persistence of a Pﬁrsch-Schl¨ uter-like diﬀusion also in the non-linear ﬂow regime. Acknowledgments Part of this work was conducted during a visit by one of the aut hors (G.N.T.) to the Max-Planck Institut f¨ ur Plasmaphysik, Garching. Th e hospitality of that Institute is greatly appreciated. References [1] D. Moreau and I. Voitsekhovitch, Nucl. Fusion 39, 685 (1999). [2] H. Tasso, Lectures on Plasma Physics , Report IFUSP/P-181, LFP-8, Uni- versidade de S˜ ao Paulo, Instituto de F ´isica, S˜ ao Paulo (1979). [3] D. Montgomery, and X. Shan, Comments Plasma Phys. Contol led Fusion 15, 315 (1994). [4] J. W. Bates and H. R. Lewis, Phys. Plasmas 32395 (1996). [5] D. Montgomery, J. W. Bates, and H. R. Lewis, Phys. Plasmas 4, 1080 (1997). [6] S. Suckewer, H. P. Eubank, G. J. Goldston E. Hinnov and N. R . Sauthoﬀ, Phys. Rev. Lett. 43, 207 (1979). [7] K. Brau, M. Bitter, R. J. Goldston, D. Manos K. McGuire, S. Suckewer, Nucl. Fusion 23, 1643 (1983). [8] H. F. Tammen, A. J. H. Donn´ e, H. Euringer and T. Oyevaar, P hys. Rev. Lett.72, 356 (1994). [9] G. N. Throumoulopoulos, J. Plasma Physics 59, 303 (1998). [10] H. P. Zehrfeld and B. J. Green, Nucl. Fusion 12, 569 (1972). [11] A. I. Morozov and L. S. Solov´ ev, Reviews of Plasma Physics 8, 1 (1980), edited by M. A. Leontovich (Consultants Bureau, New York). [12] E. Hameiri, Phys. Fluids 26, 230 (1983). 13[13] S. Semenzato, R. Gruber and H. P. Zehrfeld, Comput. Phys . Rep. 1, 389 (1984). [14] W. Kerner, and S. Tokuda, Z. Naturforsch. 42a, 1154 (1987) [15] R. ˙Zelazny, R. Stankiewicz, A. Galkowski and S. Potempski et al., Plasma Phys. Contr. Fusion 35, 1215 (1993). [16] J. D. Jackson Classical Electrodynamics , Second Edition (John Wiley & Sons, New York, 1975) p. 335. [17] E. K. Maschke and H. Perrin, Plasma Phys. 22, 579 (1980). [18] G. N. Throumoulopoulos and G. Pantis, Phys. Plasmas B 1, 1827 (1989). [19] R. A. Clemente and R. Farengo, Phys. Fluids 27, 776 (1984). [20] H. Tasso, Phys. Lett. A 222, 97 (1996). [21] G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 4, 1492 (1997). [22] H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas 5, 2378 (1998). [23] H. Grad and J. Hogan, Phys. Rev. Lett. 24, 1337 (1970). [24] K. Avinash, S. N. Bhattacharyya and B. J. Green, Plasma P hys. Control. Fusion 34, 465 (1992). [25] Zh. N. Andruschenko, O. K. Cheremnykh and J. W. Edenstra sser, J. Plasma Physics 58, 421 (1997). [26] O. Sauter, C. Angioni and Y. R. Lin-Liu, Phys. Plasmas 6, 2834 (1999). [27] R. A. Clemente, Nucl. Fusion 33, 963 (1993). [28] P. J. Morrison, Private communication; transformatio n (29) was discussed in the invited talk entitled “A generalized energy principle” which was delivered in the Plasma-Physics APS Conference, Baltimore 1986. [29] K. H. Burrell, Phys. Plasmas 4, 1499 (1997). [30] W. B. Thompson, An introduction to Plasma Physics (Addison-Wesley, Reading, Massachusetts, 1964), p. 55. 14Figure captions FIG. 1. The conductivity proﬁle on the middle-plane z= 0 described by Eq. (32) FIG. 2. The system of coordinates ( x,y,φ ). 15R Rc 2/√ 3√ 232/27σ σc FIG. 1. The conductivity proﬁle on the middle-plane z= 0 described by Eq. (32)r Rcθ yx RZ φ FIG. 2. The system of coordinates ( x,y,φ ).
arXiv:physics/0009082v1 [physics.atom-ph] 27 Sep 2000Approximation properties of basis functions in variational tree body problem∗ Vladimir S. Vanyashin† Dnepropetrovsk State University, Dnepropetrovsk 049050, Ukraine Abstract A new variational basis with well-behaved local approximat ion properties and multiple output is proposed for Coulomb syst ems. The trial function has proper behaviour at all Coulomb centres. Nonlin- ear asymptotic parameters are introduced softly: they do no t destroy the self-optimized local behaviour of the wave function at v anishing interparticle distances. The diagonalization of the Hamil tonian on a ﬁnite Hilbert subspace gives a number of meaningful eigenv alues. Thus together with the ground state some excited states are a lso re- liably approximated. For three-body systems all matrix ele ments are analytically obtainable up to rational functions of asympt otic param- eters. The feasibility of the new basis usage has been proved by a pilot computer algebra calculation. The negative sign of an elect ron pair local energy at their Coulomb centre has been revealed. PACS number: 31.15.Pf 1 Introduction In variational methods the required energy eigenvalue is ob tained by aver- aging the Hamiltonian, or, in other terms, by averaging the l ocal energy, deﬁned by action of the Hamiltonian operator on a trial wave f unction. The ∗Talk given at the International Conference ”Quantization, Gauge Theory, and Strings” dedicated to the memory of Professor Eﬁm Fradkin. Moscow. 5 – 10 May 2000. To be published in the Conference Proceedings. †e-mail: vvanyash@ﬀ.dsu.dp.ua vanyashv@heron.itep.ru 1very variability of the local energy is a direct consequence of unavoidable approximate character of any chosen variational wave funct ion. As a result, the global (averaged) quantities are much better reproduce d in variational calculations than the local ones. Moreover, in practical ca lculations, the lo- cal reliability may be sacriﬁced in favour of faster converg ence to the global values sought for. The known notion of the eﬀective charge Z∗=Z−5/16 in a two-electron atom (ion) is a good example of such sacriﬁce, since the local behaviour of an exact wave function in the Coulomb centre is d etermined by Zunscreened but not by Z∗. The good local approximation quality of a variational wave f unction is a highly desirable goal since the problem onset ( Criteria of goodness for ap- proximate wave functions [1]) up until the present ( Quality of variational trial states [2]). For any Coulomb system this quality is extremely vulne rable at the Coulomb singularity points. Without special care the lo cal energy inﬁni- ties take place at the Coulomb singularities, destroying th e desired picture of the uniform approximation [3]. Luckily, the average Hamilt onian values are rather insensitive to these local inﬁnities due to: (1) much higher degree of smallness of the neighbouring integrated volume ( ∼r3) in comparison with the degree of the Coulomb singularity ( ∼r−1) and (2) neutralizing of contri- butions with diﬀerent signs at diﬀerent points, as in custom ary cases of an eﬀective charge or scale factor introduction. In spite of great achievements of modern variational calcul ations in ﬁnd- ing high precision energy values [4, 5, 6], the problem of the reliable local approximation of the wave functions persists. From now onwa rds, for brevity, the attribute “local” will be related only to the Coulomb cen tres vicinities. The Kato cusp conditions [7] can be imposed as a supplementar y condition for mean energy minimum, thus introducing conditional extr emum technique, usually more laborious. The method not aﬀecting variationa l freedom and, nevertheless, avoiding local energy inﬁnities was propose d earlier in [8]. The development of this method, presented below, adds the possi bility to repro- duce both local and asymptotic properties of an exact soluti on in the basis functions, thus striving for better uniformity of the wave f unction approxi- mation. 22 Coulomb Variational Basis with Both Local and Asymptotic Proper Behaviour The mentioned local behaviour of a many-body wave function i s, in essence, that of some superposition of Coulomb solutions for the corr esponding pair of particles. In the case of an isolated Coulomb pair, the wav e function is a well-known product of a normalization factor, radial, and a ngular functions: ψ=N R(ρ)Y(n);ρ=−Z1Z2e2m1m2 ¯h2(m1+m2)\|r1−r2\|,n=r1−r2 \|r1−r2\|, R(ρ) = exp( −ρ/n)ρlΦ(1 +l−n,2 + 2l; 2ρ/n).(1) The standard angular functions Y(n) can be equivalently represented as sym- metric irreducible tensors of the rank l, composed from the Cartesian pro- jections of the unit vector nand Kronecker,s deltas: l= 0, Y= 1;l= 1, Yi=ni;l= 2, Yij= 3ninj−δij; l= 3, Yijk= 5ninjnk−δijnk−δiknj−δjkni;... (2) The written Coulomb solution will be used not only for attrac ting pairs, when it leads to the discrete spectrum of bound states, but al so for repulsing pairs. In the latter case the sequence of integral principal quantum numbers ngives the corresponding sequence of the Hamiltonian discre te ”eigenvalues”. They are not physically meaningful for isolated pairs: the ” eigenfunctions” grow exponentionally and are not normalizable. For repulsi ve pairs, which are embedded in a bound system, the negative Hamiltonian ”ei genvalues” aquire the meaning of their local energies near the Coulomb c entre, as the exponentional growth of the ”eigenfunctions” will be dampe d by the envi- ronment. All products of the Coulomb wave functions of all pairs, attr acting and repulsing as well, can constitute the variational basis [8] . A necessary con- traction on dumb indices and selection of admissible asympt otics are implied. The permutation symmetry of identical particles should be i mposed on the ﬁnal form of the basis. Such a basis, put in order by integer principal and orbital qu antum num- bers of diﬀerent pairs, is full enough to approximate any ana lytical many- body wave function. The Kato cusp conditions are rigorously satisﬁed by the basis functions themselves. We stress the point that any approximate 3fulﬁllment of the Kato cusp conditions leaves the diﬃculty o f local energy inﬁnities unsettled, no matter what precision of averaged q uantities has been achieved. For any pair of particles from a many-body system only a densi ty matrix can be attributed and not a wave function. In the density matr ix, unlike in the wave function of an isolated pair, the local and asymptot ic parameters cannot remain identical. In order to reproduce in the basis f unctions this den- sity matrix property we modify the Coulomb radial functions so as to allow independent adjustment of local and asymptotic variationa l parameters. The modiﬁed radial function R(a, ρ) is deﬁned as the product of the exponential factor exp( −aρ) and a ﬁnite segment of the Maclaurin expansion of the ratio of the unmodiﬁed function R(ρ) to the same exponential factor. The modi- ﬁed function has two adjustable parameters: afor the asymptotic behaviour, andn, real or imaginary, for the local behaviour near the Coulomb centre. This local behaviour is not aﬀected by the performed soft int roduction of the independent asymptotic behaviour, which is not connect ed with that of a conﬂuent hypergeometric function. The Maclaurin series o f exp(aρ)R(ρ) up toρ1+l+2kappears to be the Laurent series in inverse even powers of nup ton−2k. Just so the modiﬁed function has also the rearranged form: R(a, ρ) =kmax/summationdisplay k=0ck(a,ρ) n2k. (3) The Laurent coeﬃcients ck(a,ρ) are proposed as the new two-body con- stituents of the many-body variational basis. They are inde pendent of n polynomials in both ρandawith the common exponential factor exp( −aρ). Along with stretching the basis set, the usage of ck(a,ρ) instead of R(a, ρ) will absorb the nonlinear parameters nin easily obtainable coeﬃcients of a linear superposition. The proposed basis has inseparable c luster structure and the variational wave function should terminate only at t he end of a clus- ter. In case of three-body Coulomb systems both the Hamilton ian and unity matrix elements can be computed analytically up to rational functions of all asymptotic parameters a. The squares of eﬀective principal quantum numbers of all att ractive and repulsive Coulomb pairs, being included in the superpositi on coeﬃcients, are tuned automatically with the latters. So the proposed ba sis produces the multiple output: not only the lowest root of the secular e quation has the physical meaning, but some higher roots are also meaning ful. Still a 4majority of higher roots remains a mathematical artefact, h ence, from the physical point of view, the Hamiltonian diagonalizes only p artially. 3 Computer Algebra Feasibility of the New Method The pilot computation conﬁrms the possibility of cooperati ve treatment of several lowest states. With a relatively short wave functio n containing 54 terms (49 terms in the antisymmetric case) the helium para-S and ortho-S energy levels have been calculated in one run as: para-S levels: -2.902900, -2.145871, -2.055637; ortho-S levels: -2.175026, -2.068634, -2.036463, while the results of high precision calculations given in [4 ] are: para-S levels: -2.903724377033982, -2.145974046054634, -2.06127198974091; ortho-S levels: -2.175229378237014, -2.068689067472468 , -2.03651208309830. Though these pilot numerical results are far from the record accuracy, they have deﬁnitely established the negative sign of the electro n pair local energy in the Helium atom. This phenomenon can be tested experiment ally in, e. g., the Helium double ionization. The universal Mathematica program vSlevels is available upon request from the author. In our approach interparticle (Hylleraas) variables are us ed for the ba- sis formation, that is natural for elimination of the local e nergy inﬁnities, and perimetric (Heron) variables — on the stage of analytica l evaluation of integrals, that simpliﬁes calculations. In hyperspherica l variables the local energy inﬁnities,problem appears more intricate. It has been solved princi- pally through frame transformations in the recent work [9]. Acknowledgments The author would like to thank V. B. Belyaev, B. V. Geshkenbei n, L. B. Okun,, V. S. Popov, and Yu. A. Simonov for constructive discussion s and valuable comments. References 5[1] N. M. James and A. S. Coolidge Phys. Rev. 51(1937) 860 [2] W. Lucha and F. F. Sh¨ oberl hep-phys/9904391 [3] H. A. Bethe and E. E. Salpeter Quantum Mechanics of One- and Two- Electron Atoms (New York: Plenum Press) 1977 [4] G. W. F. Drake and W. C. Martin Can. J. Phys. 76(1998) 679 [5] S. P. Goldman Phys. Rev. A 57(1998) R677 [6] A. M. Frolov Phys. Rev. A 58(1998) 4479 [7] T. Kato Communic. Pure Appl. Math. 10(1957) 151 [8] V. S. Vanyashin Heron Variables in 3-body Coulomb Problem physics/9905042 Proc. XI Int. Workshop on High Energy Physics and Quantum Fie ld Theory, 12-18 September 1996, St.-Petersburg ed B. B. Levtchenko (Moscow: MSU Press) 1997 p 403 [9] T. A. Heim and D. Green J. Math. Phys. 40(1999) 2162 physics/9905053 6
arXiv:physics/0009083v1 [physics.gen-ph] 27 Sep 2000QUANTIZED FRACTAL SPACE TIME AND STOCHASTIC HOLISM B.G. Sidharth∗ B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract The space time that is used in relativistic Quantum Mechanic s and Quantum Field Theory is the Minkowski space time. Yet, as poi nted out by several scholars this classical space time is incompa tible with the Heisenberg Uncertainity Principle: We cannot go down to arbi- trarily small space time intervals, let alone space time poi nts. Infact this classical space time is at best an approximation, and th is has been criticised by several scholars. We investigate, what e xactly this approximation entails. Over the past several decades, time has been studied by sever al scholars from diﬀerent perspectives[1]-[30]. As pointed out in Chapter 3 , Newtonian space time was purely geometrical, as compared to Einstein’s phys ical study of it, whether it be the Minkowski space time of Special Relativity or the Rieman- nian space time of General Relativity. In almost all these st udies, as also in Quantum Field Theory, we still speak of space time points a nd deal with rigid scales even though the Quantum Mechanical Uncertaint y Principle con- tradicts these notions as discussed in preceding Chapters l ike Chapters 2, 3 and 6. In the preceding Chapters we have highlighted these shortco mings and have referred to the concept of discrete space time. There is a nua nce, though. Discrete space time could still be thought of in the context o f rigid minimum 0∗E-mail:birlasc@hd1.vsnl.net.in 1units, for example the Planck length, or more generally the C ompton wave- length of the most massive elementary particle[31]. However, as highlighted in Chapter 6 we have considered the C ompton wave- length, ﬁrstly without restricting it to the most massive pa rticle, and secondly in the context of a stochastic underpinning: We were lead to Q uantized Frac- tal Space Time (QFST). Using QFST, we then saw in Chapter 6 that (/vector x,t)→(/vector x,t)γ (1) where theγ’s are matrices. However the constraints imposed by the comm u- tation relations [y,z] = (ıa2/¯h)Lx,[t,y] = (ıa2/¯hc)My, [x,py] = [y,px] =ı¯h(a/¯h)2pxpy; require the γ’s to be atleast, 4 ×4 Dirac matrices. At the same time these commutation relations underlie the double connectivity or spin half, which is closely connected with non-commutativity, as we saw (Cf. also ref.[32]). From equation (1) it would appear that there are extra dimens ions - indeed we encountered this in the previous Chapter. As pointed out, the situation is similar to that in Quantum Superstrings. These extra dime nsions are curled up or supressed, in the unphysical Compton region. In the Kaluza- Klein theory, on the other hand, the curling up takes place wi thin the Planck length. Still, we could reconcile the two. We also saw in Chapter 6 that the double Weiner process within the Compton scale gives rise to Special Relativity and Quantum Mechanics, and indeed time itself, through the equation x→x+ıx′wherex′=ct and the complex velocity potential V−ıU. (We also saw in Chapter 3 that the so called non-relativistic Quantum Theory is not really Galilean invari- ant, as a true non-relativistic theory should be. This is bor ne out by the Sagnac eﬀect [33].) On the other hand if there were no double Weiner process (or zi tterbewe- gung), then the diﬀusion constant νwould vanish and there would be nei- ther Special Relativity nor Quantum Mechanics (nor time!). This would also 2amount to the disappearance of the quantized vortices in the hydrodynami- cal formulation seen in Chapter 3. These situations expose the mismatch of the classical ideas of space time and Quantum Theory. Indeed as Wheeler put it [34], ”No predictio n of space time, therefore no meaning for space time is the verdict of th e Quantum Principle. That object which is central to all of Classical G eneral Relativ- ity, the four dimensional space time geometry, simply does n ot exist, except in a Classical approximation.” Inspite of this, to understa nd the nature of the non- Quantum Mechanical space time of Classical Theory, let us take the diﬀusion constant of Chapter 6, ν(or the Planck constant h) to be very small, but non-vanishing. That is we consider the semi class ical case. This is because, a purely classical description, does not provid e any insight. It is well known that in this situation we can use the WKB appro ximation[35]. In this case the right hand side of the equation ψ=√ρeı/¯hS goes over to, in the one dimensional case, for simplicity, (px)−1 2eı ¯h/integraltext p(x)dx so that we have, on comparison, ρ=1 px(2) In this case the condition U= 0 implies ν· ∇ln(√ρ) = 0 (3) (Cf. Chapter 6). This semi classical analysis suggests that√ρis a slowly varying function of x, infact each of the factors on the left side of (3) would be ∼0(h), so that the left side is ∼0(h2) (which is being neglected). Then from (2) we conclude that pxis independent of x, or is a slowly varying function ofx. The equation of continuity now gives ∂ρ ∂t+/vector∇(ρ/vector v) =∂ρ ∂t= 0 3That is the probability density ρis independent or nearly so, not only of x, but also of t. We are thus in a stationary and homogenous scenario. This is strictly speaking, possible only in a single particl e universe, or for a completely isolated particle, without any eﬀect of the envi ronment. Under these circumstances we have the various conservation laws a nd the time re- versible theory, all this taken over into Quantum Mechanics as well. With Wheeler’s turn of phrase, though with a slightly diﬀerent co nnotation, this is ”time without time” or ”change without change”, an approx imation valid for small, incremental changes, as indeed is implicit in the concept of a dif- ferentiable space time manifold. All this should not be surp rising. In the preceding Chapter, section 2, we have indeed noted the limit ations of the Field approach, a framework otherwise necessary for studyi ng a multitude of particles[36]. We noted in Chapter 2, Prigogine’s statement that ”Our physi cal world is no longer symbolised by the stable and periodic planetary moti ons that are at the heart of classical mechanics”. The moment we consider th e simplest of cases, viz., the three body problem, even in Newtonian Mecha nics, it amounts to bringing in instabilities due to the environmental or hol istic feature. In our considerations in the preceding Chapters, we encount ered exactly this holistic feature in a stochastic setting, what may be called stochastic holism. For example this is embodied not only in the various Large Num ber relations deduced in Chapter 7 or the ﬂuctuations underlying interact ions in Chapter 8, but also in the fact that as we saw in the last Chapter, the nu mber of par- ticles in the universe (and a maximal universal velocity) ca n be considered to be the only free parameter - the other microphysical constan ts are dependent on this. If Classical Theory can be compared to a strucutre co nstructed with rigid building blocks, with ideas like local realism thrown in[37], we are here talking about a picture where the building blocks themselve s depend on the overall structure stochastically. In this case the puzzle of the irreversibility of time, as dis cussed by several scholars[38]-[46] also disappears. Irreversibility is a c onsequence of the sta- tistical, or in a manner of speaking Thermodynamic nature of space time. Indeed we saw in Chapter 6 and Chapter 7 that the equation T=√ Nτ provides an immediate arrow of time, while in Chapter 10 we sa w how QFST could explain the Kaon decay. 4It was shown in Chapter 7 that this picture of ﬂuctuations in t he context of QFST leads automatically to the Large Number Relations. O fcourse it is possible to trivialise these Large Number relations as coin cidences with an anthropic type argument [47]. Our approach on the other hand has been not so much an explanation for cosmic numerology, as it has been a search for minimum underlying simple principles, in the spirit of Occa m’s razor, that would explain disparate phenomena. That is what science is a ll about. References [1] S. Odenwald, Sky & Telescope, February 1996, p.24ﬀ. [2] H.F. Harmuth, ”Information Theory Applied to Space-Tim e Physics”, World Scientiﬁc, Singapore, 1992, p.18ﬀ. [3] Y.S. Vladimirov, Gravitation & Cosmology, Vol.1, No.3, 1995, p.184- 190. [4] D. Finkelstein, Graham Frye and Leonard Susskind, Physi cal Review D, Vol.9, No.8, April 1974, p.2231-2236. [5] D. Finkelstein, Physical Review D, Vol.9, No.8, April 19 74, p.2219-2230. [6] A.F. Ionov, Gravitation & Cosmology, Vol.1, No.3, 1995, p.258-259. [7] W.K. Wootters, Int.J.Th.Phys., Vol.23, No.8, 1984, p.7 01-711. [8] K.G. Denbigh, ”Three Concepts of Time”, Springer-Verla g, New York, 1981, p.50ﬀ. [9] D. Finkelstein and W.H. Hallidy, Int.J.Th.Phys., Vol.3 0, No.4, 1991, p.463-486. [10] A.R. Marlow, Int.J.Th.Phys., Vol.34, No.8, 1995, p.15 59-1565. [11] V. Kreinovich, Int.J.Th.Phys., Vol.35, No.3, 1996, p. 693-695. [12] D.N. Page and W.K. Wootters, Physical Review D, Vol.27, No.12, 15 June 1983, p.2885-2892. 5[13] S. Sonego, Phys.Lett.A., 208, 1995, p.1-7. [14] D. Park, in ”Fundamental Questions in Quantum Mechanic s”, Eds. L.M., Roth and A. Inomata, Gordan & Breach, New York, 1986, pp.263ﬀ. [15] W.K. Wootters, in ”Fundamental Questions in Quantum Me chanics”, Eds. L.M. Ruth and A. Inomata, Gordan & Breach, New York, 1986 , pp.279-290. [16] J. Audretsch, Physical Review D, Vol.27, No.12, 15 June 1983, p.2872ﬀ. [17] J. Audretsch and C. Lammerzahl, J.Math.Phys., 32 (8), A ugust 1991, p.2099ﬀ. [18] D. Finkelstein, Physical Review D, Vol.5, No.12, 15 Jun e 1972, p.2922- 2931. [19] E.H. Kronheimer and R. Penrose, Proc. Camb. Phil. Soc., 63, 1967, p.481-501. [20] H. Kitada, Il Nuovo Cimento, Vol.109 B, No.3, 1994, p.28 1ﬀ. [21] T. Gold, ”The Nature of Time”, Cornell University Press , New York, 1967. [22] V. Weizsacker in ”Quantum Theory and Beyond”, Ed. T. Bas tin, Cam- bridge University Press, Cambridge, 1971, pp.129-134. [23] E.C. Zeeman, J.Math.Phys., Vol.5, No.4, April 1964, p. 490-493. [24] D. Finkelstein, Physical Review D, Vol.5, No.2, 15 Janu ary 1972, p.320- 328. [25] P.T. Landsberg, ”The Enigma of Time”, Adam Hilger Ltd., Bristol, 1982, pp.91-177. Also P.T. Landsberg, in ”The Study of Time I II”, Eds. J.T. Fraser, N. Lawrence and D. Park, Springer-Verlag, New Y ork, 1978, pp.115-140. [26] E. Schrodinger, ”Space-Time Structure”, Cambridge Un iversity Press, Cambridge, 1963, pp.1-119ﬀ. 6[27] T. Tati, Supplement of the Progress of Theoretical Phys ics, No.29, 1964, pp.1-96. [28] S.S. Schweber, Rev.Mod.Phys., 58(2), 1986, pp.449-50 8. [29] D.V. Nanopoulos, Rivista Del Nuovo Cimento, Vol.17, No .10, 1994, pp.1-53. [30] D. Hestenes, ”Space-Time Algebra”, Gordon and Breach, New York, 1966, pp.1-93. [31] D. Finkelstein, private communication to the author. [32] S. Zakrewski, ”Quantization, Coherent States and Comp lex Structures”, Ed. by J.P. Antoine et al, Plenum Press, New York, 1995, p.249 ﬀ. [33] J. Anandan, Phys. Rev., D 24, 1981, pp.338-346. [34] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitatio n”, W.H. Free- man, San Francisco, 1973, p.1183. [35] B.H. Bransden and C.J. Joachin, ”Introduction to Quant um Mechan- ics”, Longman, Essex, 1989, pp.388ﬀ. [36] V. Heine, ”Group Theory in Quantum Mechanics”, Pergamo n Press, Oxford, 1960, p.364. [37] E. Bitsakis, ”Open Questions in Quantum Physics”, Eds. Tarozzi, G., and Van der Merwe, A., D. Reidel Publishing Company, 1985, p. 18-59. [38] F.H.M. Faisal and U. Schwengelbeck, Pramana Journal of Physics, Vol.51, No.5, November 1998, p.585-595. [39] M. Abolhasani and M. Golshani, Foundations of Phys.Let t., Vol.12, No. 3, 1999, p.299-306. [40] H. Price, Nature, Vol.348, 22 November 1990, p.356. [41] F. Rohrlich, Found.of Phys., Vol.28, No.7, 1998, p.104 5-1055. [42] E. Schrodinger, PROC. R.I.A., Vol.53, SECT.A., August 1950, p.46ﬀ. 7[43] P.V. Coveney, Nature, 333, 1988, p.409-415. [44] J.B. Barbour, Int.J.Th.Phys., Vol.36, No.11, 1997, p. 2459-2460. [45] S.W. Hawking, Physical Review D, Vol.32, No.10, 15 Nove mber 1985, p.308-314. [46] P.V. Coveney and R. Highﬁeld, ”The Arrow of Time”, Fauce tt Columbine, New York, 1990. [47] B. Carter, ”Large Number Coincidences and the Anthropi c Principle in Cosmology”, Proc. of IAU, Ed. M.S. Longair, 1974, p.291-298 . 8
arXiv:physics/0009084v1 [physics.gen-ph] 27 Sep 2000ISSUES AND RAMIFICATIONS IN QUANTIZED FRACTAL SPACE TIME: AN INTERFACE WITH QUANTUM SUPERSTRINGS B.G. Sidharth∗ B.M. Birla Science Centre, Hyderabad 500 063 (India) Abstract Recently a stochastic underpinning for space time has been c onsid- ered, what may be called Quantized Fractal Space Time. This l eads us to a number of very interesting consequences which are testa ble, and also provides a rationale for several otherwise inexplicab le features in Particle Physics and Cosmology. These matters are investig ated in the present paper. 1 The Background ZPF We observe that the Bohm formulation discussed in detail in C hapter 3 con- verges to Nelson’s stochastic formulation in the context of the QMKNBH. Indeed Bohm’s non local potential as also Nelson’s three con ditions merely describe the QMKNBH as a vortex, the mass being given by the se lf interac- tion, the radius of the vortex being the Compton wavelength. [1]. We can get a clue to the origin of Quantum Mechanical ﬂuctuations: Foll owing Smolin we observe that the non local stochastic theory becomes the c lassical local theory in the thermodynamic limit, in which Nthe number of particles in the universe becomes inﬁnitely large. However if Nis ﬁnite but large, these 0∗E-mail:birlasc@hd1.vsnl.net.in 1ﬂuctuations are of the order 1 /√ Nof the dimensions of the system, the uni- verse in this case. Indeed as we will see in the nexr Section th is provides a holistic rationale for the ”spooky” non-locality of Quantu m Theory. We now remark that in the above formulation elementary parti cles, typically electrons, can be thought of as ’twisted bits’ of the electro magnetic ﬁeld. Indeed it was pointed out by Barut and co-workers that wave pa cket solu- tions of the mass less scalar ﬁelds appear as massive particl es, while such solutions for the electromagnetic ﬁeld would provide a form ulation of the wave mechanics without assuming the Planck constant[2]. Fo r example this givesE=lωrather than E= ¯hω,lbeing the angular momentum and Ethe energy and ωthe frequency. Boudet[3], also questions the necessity of t he Planck Constant. These theories do not give a value to Planck ’s constant which merely appears as a proportionality factor, because a ll the equations considered are linear. All this as also the zitterbewegung f ormulation of Barut and Bracken and Hestenes described in Chapter 3, is sup erseded by the QMKNBH theory and electrons appear as twisted bits of the ZPF given by the relation, E= ¯hω, instead of Barut’s, E=lω, wherelis the angular momentum. The question, whether this characterizes the Pla nck constant, will be answered at the end of Section 5. 2 Stochastic Conservation Laws Conservation Laws, as is universally known, play an importa nt role in Physics, starting with the simplest such laws relating to momentum an d energy. These laws provide rigid guidelines or constraints within which p hysical processes take place. These laws are observational, though a theoretical facade c an be given by relating them to underpinning symmetries[4]. Quantum Theory, including Quantum Field Theory is in confor mity with the above picture. On the other hand the laws of Thermodynamics h ave a dif- ferent connotation: They are not rigid in the sense that they are a statement about what is most likely to occur, or is an averaged out state ment. In our formulation, the Compton wavelength represents a sta tistical uncer- tainty (Cf. Chapter 6), given by l∼R√ N(1) 2By now (1) is a familiar relation. Given the above background we consider the following simpliﬁed EPR experiment, discussed elsewhe re[5]. Two structureless and spinless particles which are initial ly together, for ex- ample in a bound state get separated and move in opposite dire ctions along the same straight line. A measurement of the momentum of one o f the par- ticles, sayAgives us immediately the momentum of the other particle B. The latter is equal and opposite to the former owing to the con servation law of linear momentum. It is surprising that this statement sho uld be true in Quantum Theory also because the momentum of particle Bdoes not have an apriori value, but can only be determined by a separate acaus al experiment performed on it. This is the well known non locality inherent in Quantum Theor y. It ceases to be mysterious if we recognize the fact that the conservati on of momen- tum is itself a non local statement because it is a direct cons equence of the homogeneity of space as we will see again in the next Chapter: Infact the displacement operatord dxis, given the homogeneity of space, independent ofxand this leads to the conservation of momentum in Quantum The ory (cf.ref.[6]). The displacement δxwhich gives rise to the above displacement operator is an instantaneous shift of origin corresponding to an inﬁnite ve- locity and is compatible with a closed system. It is valid if t he instantaneous displacement can also be considered to be an actual displace ment in real timeδt. This happens for stationary states, when the overall energ y remains constant. It must be borne in mind that the space and time displacement o perators are on the same footing only in this case[7]. Indeed in relati vistic Quantum Mechanics, xandtare put on the same footing - but special relativity itself deals with inertial, that is relatively unaccelerated fram es. Any ﬁeld theory deals with diﬀerent points at the same instan t of time. But if we are to have information about diﬀerent points, then giv en the ﬁnite velocity of light, we will get this information at diﬀerent t imes. All this in- formation can refer to the same instant of time only in a stati onary situation. We will return to this point. Further the ﬁeld equations are o btained by a suitable variational principle, δI= 0 (2) In deducing these equations, the δoperator which corresponds to an arbi- trary variation, commutes with the space and time derivativ es, that is the 3momentum and energy operators which in our picture constitu te a complete set of observables. As such the apparently arbitrary operat orδin (2) is con- strained to be a function of these (stationary) variables[8 ]. All this underscores two facts: First we implicitly conside r an apriori ho- mogenous space, that is physical space. Secondly though we c onsider in the relativisitic picture the space and time coordinates to be o n the same foot- ing, infact they are not as pointed out by Wheeler[9]. Our und erstanding or perception of the universe is based on ”all space (or as much o f it as possible) at one instant of time”. However, in conventional theory this is at best an approxima tion. Moreover in our formulation, the particles are ﬂuctuationally creat ed out of a back- ground ZPF, and, it is these Nparticles that deﬁne physical space, which is no longer apriori as in the Newtonian formulation. It is only in the thermo- dynamic limit in which N→ ∞ andl→0, in (1), that we recover the above classical picture of a rigid homogenous space, with the cons ervation laws. In other words the above conservation laws are strictly vali d in the thermo- dynamic limit, but are otherwise approximate, though very n early correct becauseNis so large. Our formulation leads to a cosmology in which√ Nparticles are ﬂuctuation- ally created from the background ZPF (Cf. Chapter 7), so that the violation of energy conservation is proportional to1√ N. From (1) also we could sim- ilarly infer that the violation of momentum conservation is proportional to 1√ N(per particle). All this implies that there is a small but non-zero probabili ty that the mea- surement of the particle Ain the above experiment will not give information about the particle B. This last conclusion has also been drawn by Gaeta[10] who con siders a back- ground Brownian or Nelson-Garbaczewski-Vigier noise(the ZPF referred to above) as sustaining Nelson’s Stochastic Mechanics (and th e Schrodinger equation). In conclusion, the conservation laws of Physics are conserv ation laws in the thermodynamic sense. 43 Quantized Space Time, Time’s Arrow and Parity Breakdown The arrow of time has been a puzzle for a long time. As is well kn own, the laws of Newtonian Mechanics, Electromagnetism or Quantum T heory do not provide an arrow of time - they are equally valid under time re versal, with only one exception. This is in the well known problem of Kaon d ecay. On the other hand it is in Thermodynamics and Cosmology that we ﬁnd a n arrow of time [11]. Indeed it has been shown that stochastic proces ses are needed for irreversibility[12]. It is also true that there has been no theoretical rationale f or the Kaon puzzle which we will touch upon shortly. We will try to ﬁnd such a theo retical understanding in the context of our quantized space-time, ∼¯h/(energy), that is the Compton time[13]. Let us start with one of the simplest quantum mechanical syst ems, one which can be in either of two sates separated by a small energy[14]. The system ﬂips from one state to another unpredictably and this ”life time” and the energy spread satisfy the Uncertainity Principle, so that the form er is a Compton time. We have: ı¯hdψı dt≈ı¯h[ψı(t+nτ)−ψı(t) nτ] =2/summationdisplay ı=1Hıjψı (3) ψı=eı ¯hEtφı where,H11=H22(which we set = 0 as only relative energies of the two levels are being considered) and H12=H21=E, by symmetry. Unlike in the usual theory where δt=nτ→0, in the case of quantized space-time nis a positive integer. So the second term of (3) reduces to [E+ıE2τ ¯h]ψı= [E(1 +ı)]ψı,asτ= ¯h/E (4) Interestingly, in the above analysis, in (4), the fact that t he real and imagi- nary parts are of the same order is infact borne out by experim ent. From (3) we see that the Hamiltonian is not Hermitian that is i t admits complex Eigen values indicative of decay, if the life times o f the states are ∼τ. 5In general this would imply the exotic fact that if a state sta rts out asψ1and decays, then there would be a non zero probability of seeing i n addition the decay products of the state ψ2. In the process it is possible that some sym- metries which are preserved in the decay of ψ1orψ2separately, are voilated. In this context we will now consider the Kaon puzzle. As is wel l known from the original work of Gellmann and Pais, the two state ana lysis above is applicable here[15, 16]. In the words of Penrose[17], ”th e tiny fact of an almost completely hidden time-asymmetry seems genuinely t o be present in theK0-decay. It is hard to believe that nature is not, so to speak, t rying to tell something through the results of this delicate and beau tiful experiment.” On the other hand as Feynman put it[18], ”if there is any place where we have a chance to test the main principles of quantum mechanic s in the purest way.....this is it.” What happens in this well known problem is, that given CPinvariance, a beam ofK0masons can be considered to be in a two state system as above, one being the short lived component KSwhich decays into two pions and the other being the long lived state KLwhich decays into three pions. In this caseE∼1010¯h[15], so that τ∼10−10sec. After a lapse of time greater than the typical decay period, no two pion decays should be se en in a beam consisting initially of the K0particle. Otherwise there would be violation of CPinvariance and therefore also Tinvariance. However exactly this viola- tion was observed as early as 1964[19]. This violation of tim e reversal has now been conﬁrmed directly by experiments at Fermilab and CE RN[20]. We would like to point out that the Kaon puzzle has a natural ex planation in the quantized time scenario discussed above. Further, we have shown that the discreteness leads to the non commutative geometry [x,y] = 0(l2),[x,px] =ı¯h[1 +l2] (5) and similar equations. If terms ∼l2are neglected we get back the usual Quantum Theory. However retaining these terms, we deduced i n Chapter 6 the Dirac equation. Moreover it can be seen that given (5) spa ce rerlection symmetry no longer holds. This violation is an O(l2) eﬀect. This is not surprising. It has already been pointed out that t he space time divide viz., x+ıctarises due to the zitterbewegung or double Weiner process in the Compton wavelength - and in this derivation terms ∼(ct)2∼l2were neglected. However if these terms are retained, then we get a correction to the usual theory including special relativity. (We will com e back to this point 6shortly.) To see this more clearly let us as in Chapter 3 as a ﬁrst approxi mation treat the continuum as a series of discrete points separated by a di stancel, which then leads to Ea(xn) =E0a(xn)−Aa(xn+l)−Aa(xn−l) (6) Whenlis made to tend to zero, it was shown that from (6) we recover th e Schrodinger equation, and further, we have, E=E0−2Acoskl. (7) The zero of energy was chosen such that E= 2A=mc2, the rest energy of the particle, in the limit l→0. However if we retain terms ∼l2, then from (7) we will have instead/vextendsingle/vextendsingle/vextendsingle/vextendsingleE mc2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼0(l2) (8) Equation (8) shows the correction to the energy mass formula , where again we recover the usual formula in the limit O(l2)≈0. It must be mentioned that all this would be true in principle f or discrete space time, even if the minimum cut oﬀ was not at the Compton sc ale. Intuitively this should be obvious: Space time reﬂection sy mmetries are based on a space time continuum picture. Let us now consider some further imprints of discrete space t ime[21]. First we consider the case of the neutral pion. As we saw in Cha pter 4, this pion decays into an electron and a positron. Could we thi nk of it as an electron-positron bound state also? In this case we have, mv2 r=e2 r2(9) Consistently with the above formulation, if we take v=cfrom (9) we get the correct Compton wavelength lπ=rof the pion. However this appears to go against the fact that there would b e pair anni- hilation with the release of two photons. Nevertheless if we consider discrete space time, the situation would be diﬀerent. In this case the Schrodinger equation Hψ=Eψ (10) 7whereHcontains the above Coulumb interaction could be written, in terms of the space and time separated wave function components as, Hψ=EφT=φı¯h[T(t−τ)−T τ] (11) whereτis the minimum time cut oﬀ which in the above work has been take n to be the Compton time. If, as usual we let T=exp(irt) we get E=−2¯h τsinτr 2(12) (12) shows that if, \|E\|<2¯h τ(13) holds then there are stable bound states. Indeed inequality (13) holds good whenτis the Compton time and Eis the total energy mc2. Even if inequality (13) is reversed, there are decaying states which are relati vely stable around the cut oﬀ energy2¯h τ. This is the explanation for treating the pion as a bound state of an electron and a positron, as indeed is borne out by its decay mode. The si tuation is similar to the case of Bohr orbits– there also the electrons w ould according to classical ideas have collapsed into the nucleus and the at oms would have disappeared. In this case it is the discrete nature of space t ime which enables the pion to be a bound state as described by (9). 4 Magnetic Eﬀects If as discussed in Chapter 3 and subsequently, the electron i s indeed a Kerr- Newman type charged black hole, it can be approximated by a so lenoid and we could expect an Aharonov-Bohm type of eﬀect, due to the vec tor potential /vectorAwhich would give rise to shift in the phase in a two slit experi ment for example[22]. This shift is given by ∆δˆB=e ¯h/contintegraldisplay /vectorA./vectords (14) while the shift due to the electric charge would be ∆δˆE=−e ¯h/integraldisplay A0dt (15) 8whereA0is the electrostatic potential. In the above formulation we would have /vectorA∼1 cA0 (16) Substitution of (16) in (14) and (15) shows that the magnetic eﬀect∼v c times the electric eﬀect. Further, the magnetic component of a Kerr-Newman black hole , as we saw in Chapter 3 is given by Bˆr=2ea r3cosΘ + 0(1 r4),BˆΘ=easinΘ r3+ 0(1 r4),Bˆφ= 0, (17) while the electrical part is Eˆr=e r2+ 0(1 r3),EˆΘ= 0(1 r4),Eˆφ= 0, (18) Equations (17) and (18) show that in addition to the usual dip ole magnetic ﬁeld, there is a shorter range magnetic ﬁeld given by terms ∼1 r4. In this context it is interesting to note that an extra so called B(3)magnetic ﬁeld of shorter range and probably mediated by massive photons ha s indeed been observed and studied over the past few years[23]. 5 Stochastic Holism and the Number of Ar- bitrary Parameters The discrete space time or zitterbewegung has an underpinni ng that is stochas- tic. The picture leads to the goal of Wheeler’s ’Law without L aw’ as we saw in Chapter 6. Furthermore the picture that emerges is Machia n. This is evident from equations like (2), (3) and (6) of Chapter 7– the micro depends on the macro. So the ﬁnal picture that emerges is one of stocha stic holism. Another way of expressing the above point is by observing tha t the interac- tions are relational. For example, in the equation leading t o (7), of Chapter 7, if the number of particles in the universe tends to 1, then a s we saw in Chapter 4, the gravitational and electromagnetic interact ions would be equal, this happening at the Planck scale, where the Compton wavele ngth equals the Schwarzchild radius[24]. Infact as was shown in Chapter 6, when Nthe number of particles in the 9universe is 1 we have a Planck particle with a short life time ∼10−42secs due to the Hawking radiation but with N∼1080particles as in the present universe we have the pion as the typical particle with a stabl e life time ∼of the age of the universe due to the Hagedorn on radiation. Let us now consider the following aspect[25]: It is well know n that there are 18 arbitrary parameters in contemporary physics. We on t he other hand have been working with the micro physical constants referre d to earlier viz., the electron (or pion) mass or Compton wavelength, the Planc k constant, the fundamental unit of charge and the velocity of light. The se along with the number of particles Nas the only free parameter can generate the mass, radius and age of the universe as also the Hubble constant. If we closely look at the equations (11) of Chapter 4 or (7) of C hapter 7, giving the gravitational and electromagnetic strength rat ios, we can actually deduce the relation, l=e2 mc2(19) In other words we have deduced the pion mass in terms of the ele ctron mass, or, given the pion mass and the electron mass, we have de duced the ﬁne structure constant. From the point of view of the order of magnitude theory in which the distinction between the electron, pion a nd proton gets blurred, what equation (19) means is, that the Planck consta nt itself depends oneandc(andm). Further in the Kerr-Newman type characterisation of the electron, in Chapter 3 the charge eis really equivalent to the spinorial tensor density ( n= 1). In this sense ealso is pre determined and we are left with a minimum length viz. the Compton length and a minum um time viz. the Compton time (or a maximal velocity c) as the only fundamental microphysical constants. Let us try to further reﬁne this line of thought. We observe th at a discrete space time picture leads to the non commutative geometry all uded to earlier (5). Infact we would have in this case, more fully, [x,y] = 0(l2),[x,px] =ı¯h[1 +l2],[t,E] =ı¯h[1 +τ2] (20) What (20) means is that there is a higher order correction to t he Heisenberg Uncertainity Principle. Infact from (20) we can easily conc lude that there is 10an extra energy E′given by E′ mc2∼τ2∼l4∼1√ N(21) In (21), the appearance of1√ NwhereNis the number of particles in the universe appears at ﬁrst sight to be purely accidental: We ha ve not deduced it. However this is not so. Infact from the picture of the ﬂuct uational creation of particles alluded to in section 2, we get E′ mc2∼1√ N(22) It can be seen that (22) and (21), deduced from two totally diﬀ erent stand- points, are infact the same. A consequence is the following f act: We have just seen that the micro physical constants namely an elementary particle mass, for example the electron mass m(or Compton wavelength), a universal max- imal velocity ctogether with Nthe number of particles in the universe were the only free parameters or arbitrary constants. From (21) w e can see that there is a further narrowing down to just two arbitrary param eters, for ex- ample the maximal velocity candN. Given these two, the microphysical constants, including the Planck constant can be characteri ized, thus answer- ing the question at the end of section 1. It must be emphasized that what is required is a universal maximum velocity in principle - its e xact value is not important. Then, Nbecomes the only parameter! All this is very much in the spirit of Feynman’s quotation in Chapter 1 as also the ancien t Upanishadic tradition of seeing nature as diﬀerent aspects of one phenom enon. 6 The Origin of a Metric We ﬁrst make a few preliminary remarks. When we talk of a metri c or the dis- tance between two ”points” or ”particles”, a concept that is implicit is that of topological ”nearness” - we require an underpinning of a s uitably large number of ”open” sets[26]. Let us now abandon the absolute or background space time and consider, for simplicity, a universe (or set) that consists solely of two particles. The question of the distance between these particles (quite apart from the question of the observer) becomes meaningles s. Indeed, this is 11so for a universe consisting of a ﬁnite number of particles. F or, we could iso- late any two of them, and the distance between them would have no meaning. We can intuitively appreciate that we would infact need dist ances of inter- mediate points. So for a meaningful distance, the concepts o f open sets, connectedness and the like reenter in which case such an isol ation would not be possible. More formally let us deﬁne a neighbourhood of a particle (or p oint) A of a set of particles as a subset which contains A and atleast one o ther distinct element. Now, given two particles (or points) A and B, let us c onsider a neighbourhood containing both of them, n(A,B) say. We require a non- empty set containing atleast one of A and B and atleast one oth er particle C, such that n(A,B)⊃n(A,C), and so on. Strictly, this ”nested” sequence should not terminate. For, if it does, then we end up with a set n(A,P) consisting of two isolated ”particles” or points, and the ”d istance”d(A,P) is meaningless. For practical purposes, in the spirit of Whe eler’s approxima- tion, this sequence has to be very large. Such an approximation has an immediate application. Our uni verse consists of someN∼1080particles (or points), each point being ”deﬁned” within the Compton wavelength l. Insidel, space time in the usual sense breaks down - we have the unphysical zitterbewegung eﬀects. Indeed lfor a Planck particle of mass ∼10−5gmis precisely the Planck scale. We now assume the following property[27]: Given two distinc t elements (or even subsets) AandB, there is a neighbourhood NA1such thatAbelongs to NA1,Bdoes not belong to NA1and also given any NA1, there exists a neigh- bourhoodNA1 2such thatA⊂NA1 2⊂NA1, that is there exists an inﬁnite sequence of neighbourhoods between AandB. In other words we introduce topological closeness. From here, as in the derivation of Urysohn’s lemma[26], we co uld deﬁne a mappingfsuch thatf(A) = 0 andf(B) = 1 and which takes on all inter- mediate values. We could now deﬁne a metric, d(A,B) =\|f(A)−f(B)\|.We could easily verify that this satisﬁes the properties of a me tric. It must be remarked that the metric turns out to be again, a res ult of a global or a series of larger sets, unlike the usual local pict ure in which it is the other way round. 127 Kaluza-Klein Theories and Quantized Su- per Strings In Chapter 1, we brieﬂy alluded to string theory. Though our s ubsequent considerations were in a diﬀerent class, there is a surprisi ng interface, as we will now see. Our starting point is the fact encountered in Ch apter 6 that the fractal dimension of a Brownian quantum path is 2. This wa s further analysed and it was explained that this is symptomatic of Qua ntized Fractal space time and it was shown that infact the coordinate xbecomesx+ıct. The complex coordinates or equivalently non-Hermitian positi on operators are symptomatic of the unphysical zitterbewegung which is elim inated after an averaging over the Compton scale. In this picture the ﬂuctua tional creation of particles was taken into account in a consistent cosmolog ical scheme in Chapter 7. It is well known that the generalization of the complex xcoordinate to three dimensions leads to quarternions[28], and the Pauli spin ma trices. We next return to the model of an electron as a Quantum Mechani cal Kerr- Newman Black Hole. Infact in Chapter 3, we deduced electroma gnetism in two ways. The ﬁrst was by considering an imaginary shift, xµ→xµ+ıaµ,(aµ∼Compton scale) (23) in a Quantum Mechanical context. This lead to ı¯h∂ ∂xµ→ı¯h∂ ∂xµ+¯h aµ(24) and the second term on the right side of (24) was shown to be the electro- magnetic vector potential Aµ, Aµ= ¯h/aµ(25) The second was by taking into account the fact that at the Comp ton scale, it is the so called negative energy two spinors χof the Dirac bispinor that dominate where, χ→ −χ under reﬂections. This lead to the tensor density property, ∂ ∂xµto∂ ∂xµ−Γµν ν (26) 13the second term on the right side of (26) being identiﬁed with Aµ, Aµ= ¯hΓµν ν (27) It was pointed out that (27) is formally and mathematically i dentical to Weyl’s original formulation, except that here it arises due to the purely Quan- tum Mechanical spinorial behaviour whereas Weyl had put it b y hand. Another early scheme for the uniﬁcation of gravitation and e lectromagnetism as referred to earlier was that put forward by Kaluza and Klei n[29, 30, 31] in which an extra dimension was introduced and taken to be cur led up. This idea has resurfaced in recent years in String Theory. We will ﬁrst show that the characterization of Aµin (25) is identical to a Kaluza Klein formulation. Then we will show that equations ( 26) and (27) really denote the fact that the geometry around an electron i s non-integrable. Finally we will show that infact both (24) or (25) and (26) or ( 27) are the same formulations (as can be guessed heuristically by compa ring (24) and (26)). We ﬁrst observe that the transformation (23) can be written a s, xı→xı+αı5x5 (28) whereαı5in (28) will represent a small shift from the Minkowski metri c gıj, andı,j= 1,2,3,4,5,x5being a ﬁfth coordinate introduced for purely mathematical conversion. Owing to (28), we will have, gıjdxıdxj→gıjdxıdxj+ (gıjαj5)dxıdx5 (29) In Kaluza’s formulation, Aµ∝gµ5 (30) Comparison of (30), (28) and (29) with (23) and (25) shows tha t indeed this is the case. That is, the formulation given in (23) and (24) co uld be thought of as introducing a ﬁfth curled up dimension, as in the Kaluza -Klein theory. To see why the Quantum Mechanical formulation (26) and (27) c orresponds to Weyl’s theory, we start with the eﬀect of an inﬁnitesimal p arallel displace- ment of a vector[32]. δaσ=−Γσ µνaµdxν(31) 14As is well known, (31) represents the extra eﬀect in displace ments, due to the curvature of space - in a ﬂat space, the right side would vanis h. Considering partial derivatives with respect to the µthcoordinate, this would mean that, due to (31) ∂aσ ∂xµ→∂aσ ∂xµ−Γσ µνaν(32) The second term on the right side of (32) can be written as: −Γλ µνgν λaσ=−Γν µνaσ where we have utilized the property that in the above formula tion as seen in Chapter 3, gµν=ηµν+hµν, ηµνbeing the Minkowski metric and hµνa small correction whose square is neglected. That is, (32) becomes, ∂ ∂xµ→∂ ∂xµ−Γν µν (33) The relation (33) is the same as the relation (26). We will next show the correspondence between (33) or (27) or ( 26) and (25) or (24). To see this simply we note that the geodesic equation is, ˙uµ≡duµ ds= Γµ νσuνuσ(34) We also use the fact that in the Quantum Mechanical Kerr-Newm an Black Hole model referred to, we have as in Chapter 3 uµ=cforµ= 1,2 and3, while, \|˙uµ\|=\|uµ\|mc2 ¯h So, from (34) we get, Γµ νµ=1 aν,\|aν\|=¯h mc This establishes the required identity. We now come to the interface with Quantum Super Strings. We ha ve al- ready seen that the Quantized Fractal space time referred to really leads to 15a non-commutative geometry, given by (20) It was also seen th at these re- lations directly lead to the Dirac equation: Quantized Frac tal space time is the underpinning for Quantum Mechanical spin or the Quantum Mechanical Kerr-Newman Black Hole, that is ultimately equations like ( 24) or (25) and (26) or (27). It is also true that both the Kaluza Klein formulation and the non commu- tative geometry (20) hold in the theory of Quantum Superstri ngs (QSS). Infact we get from here a clue to the mysterious six extra curl ed up dimen- sions of Quantum Superstring Theory. For this we observe tha t (20) gives an additional contribution to the Heisenberg Uncertainity Principle and we can easily deduce ∆p∆x∼¯hl2 Remembering that at this Compton scale ∆p∼mc It follows that ∆x∼l3(35) asl∼10−11cmsfor the electron we recover from (35) the Planck Scale, as well as a rationale for the peculiar fact that the Planck Scal e is the cube of the electron Compton scale. More importantly, what (35) shows is, that at this level, the single dimension along thexaxis shows up as being three dimensional. That is there are tw o extra dimensions, in the unphysical region below the Compto n scale. As this is true for the yandzcoordinates also, there are a total of six curled up or unphysical or inaccessible dimensions in the context of the preceding section. If we start with equations (23) to (25) which are related to QF ST (Quan- tized Fractal space time) and the non-commutative relation (20) we obtain a uniﬁcation of electromagnetism and gravitation. On the ot her hand if we consider the spinorial behaviour of the Dirac wave function , we get (26) or (27). The former has been seen to be the same as the Kaluza form ulation while the latter is formally similar to the Weyl formulation - but in this case (27) is not put in by hand. Rather it is a Quantum Mechanical co nsequence. We have thus shown that these two approaches are the same. The extra dimensions are thus seen to be conﬁned to the unphysical Comp ton scale - classically speaking they are curled up or inaccessible. 16In a sense this is not surprising. The bridge between the two a pproaches was the Kerr-Newman metric which uses, though without a clea r physical meaning in classical theory, the transformation (23). The r eason why an imaginary shift is associated with spin is to be found in the Q uantum Me- chanical zitterbewegung and the consequent QFST. Wheeler remarked as quoted in Chapter 4 [9], ”the most eviden t shortcom- ing of the geometrodynamic model as it stands is this, that it fails to supply any completely natural place for spin 1 /2 in general and for the neutrino, in particular”, while ”it is impossible to accept any descript ion of elementary particles that does not have a place for spin half.” Infact th e bridge between the two is the transformation (23). It introduces spin half i nto general rela- tivity and curvature to the electron theory, via the equatio n (27) or (32). In this context it is interesting to note that El Naschie has g iven the fractal formulation of gravitation[33]. Thus apparently disparate concepts like the Kaluza Klein an d Weyl formula- tions, Quantum Mechanical Black Holes, Quantized Fractal s pace time and QSS are seen to have a harmonius overlap, in the context of QFS T with its roots in the ﬂuctuational creation of particles[34, 35]. It is worth pointing out some of the similarities between Str ing theories and our formulation. The former started oﬀ, by considering one d imensional extended objects or strings, the extension being of the orde r of the proton Compton wavelength, vibrating and rotating with the speed o f light (Cf. refs. given in Chapter 1). Not only could space time points and sing ularities be fudged, but further the angular momenta were proportional t o the squares of the masses, deﬁning the well known Regge trajectories, as also in our for- mulation (Cf. Chapter 12, Equation (14)). All this is not sur prising. In particular, QSS deals with Planck length phenomena, the K aluza-Klein curled up extra dimensions and leads to the non commutative g eometry (20). QFST on the other hand, deals with phenomena at the Compton sc ale, space time being unphysical below this scale. Yet it leads us back t o the Planck scale, the same number of extra, curled up, Kaluza-Klein dim ensions and the same non- commutative geometry (20), once the meaning of (35) (or the modiﬁcation of the Heisenberg Uncertainity Principle) is r ecognized. In this interpretation, the situation is similar to the fractal one dimensioinal Brown- ian path becoming two (or three) dimensional. The key is the t ransformation (23), which we ﬁrst encountered right at the beginning, in Ch apter 3 itself. It conceals zitterbewegung, leads to the Kerr-Newman metri c, QFST and 17what not! Finally it is worth emphasizing that both in Strings and in ou r formulation the Compton wavelength extension provides a rationale for t he dual reso- nance model, which originated from the Regge trajectories a nd then gave the initial motivation for String theory. 8 Resolution, Uniﬁcation and the Core of the Electron El Naschie[36] has referred to the fact that there is no aprio ri ﬁxed length scale (the Biedenharn conjecture). Indeed it has been argue d in the above context that depending on our scale of resolution, we encoun ter electro- magnetism well outside the Compton wavelength, strong inte ractions at the Compton wavelength or slightly below it and only gravitatio n at the Planck scale. The diﬀerences between the various interactions are a manifestation of the resolution. In this connection it may be noted that we can refer to the core of the elec- tron∼10−20cms, as indeed has been experimentally noticed by Dehmelt and co-workers[37]. It is interesting that this can be deduced i n the context of the electron as a Quantum Mechanical Kerr-Newman Black Hole . It was shown in Chapter 3 that for distances of the order of the Compton wavelength the potential is given in its QCD form V≈ −βM r+ 8βM(Mc2 ¯h)2.r (36) For small values of rthe potential (36) can be written as V≈A re−µ2r2, µ=Mc2 ¯h(37) It follows from (37) that r∼1 µ∼10−21cm. (38) Curiously enough in (37), rappears as a time, which is to be expected be- cause at the horizon of a black hole randtinterchange roles. 18One could reach the same conclusion, as given in equation (38 ) from a dif- ferent angle. In the Schrodinger equation which is used in QC D, with the potential given by (36), one could verify that the wave funct ion is of the type f(r).e−µr 2, where the same µappears in (37). Thus, once again we have a wave packet which is negligible outside the distance given b y (38). It may be noted that Brodsky and Drell[38] had suggested from a very dif- ferent viewpoint viz., the anomalous magnetic moment of the electron, that its size would be limited by 10−20cm. The result (38) as pointed out, was experimentally conﬁrmed by Dehmelt and co-workers. 9 Levels of Physics We now return to the relation (5) or (20) which expresses the u nderlying non-commutative geometry of space-time. What we would like to point out is that we are seeing here diﬀerent levels of physics. Indeed , rewriting (5) or (20) as, [x,ux] =ı[l+l3], we can see that if l= 0, we have classical physics, while if 0( l3) = 0,we have Quantum Mechanics and ﬁnally if 0( l3)∝negationslash= 0 we have the above discussed fractal picture, and from another point of view, the superst ring picture. Interestingly, in our case the electron Compton wavelength l∼10−11cm,so that 0(l3)∼10−33as in string theory. The expansion in terms of lgiven above can be continued[39], and thus one could in principle go into deeper levels as well. 10 Gravitation and Black Holes In our formulation we have not invoked the full non linear The ory of Gen- eral Relativity. General Relativity itself comes up as an ap proximation, in its linear version and also through the fact that while Gthe gravitational constant, varies with time, over intervals small compared t o the age of the universe, it is approximately constant. (Dirac however rec onciles the vari- ation ofGwith General Relativity by invoking the so called gravitati onal units of measurement[40], the units of our common usage bein g the atomic units). The question arises, is it possible to accommodate B lack Holes within 19such a non General Relativistic formulation? We will now sho w that Black Holes could also be understood without invoking General Rel ativity at all. We start by deﬁning a Black Hole as an object at the surface of w hich, the escape velocity equals the maximum possible velocity in the universe viz., the velocity of light. We next use the well known equation of K eplerian orbits[41], 1 r=GM L2(1 +ecosθ) (39) whereL, the so called impact parameter is given by, Rc, whereRis the point of closest approach, in our case a point on the surface of the o bject andcis the velocity of approach, in our case the velocity of light. Choosingθ= 0 ande≈1, we can deduce from (39) R=2GM c2(40) Equation (40) gives the Schwarzchild radius for a Black Hole and can be deduced from the full General Relativistic theeory. We will now use (40) to exhibit Black Holes at three diﬀerent s cales, the micro, the macro and the cosmic scales. Our starting point is the observation that a Planck mass, 10−5gmsat the Planck length, 10−33cmssatisﬁes (40) and, as such is a Schwarzchild Black Hole. As pointed out Rosen has used non-relativistic Quantu m Theory to show that such a particle is a mini universe. We next come to stellar scales. It is well known that for an ele ctron gas in a highly dense mass we have[42] K/parenleftBigg¯M4/3 ¯R4−¯M2/3 ¯R2/parenrightBigg =K′¯M2 ¯R4(41) where/parenleftbiggK K′/parenrightbigg =/parenleftbigg27π 64α/parenrightbigg/parenleftBigg¯hc γm2 P/parenrightBigg ≈1040(42) and ¯M=9π 8M mP¯R=R (¯h/m ec), Mis the mass, Rthe radius of the body, mPandmeare the proton and electron masses and ¯ his the reduced Planck Constant. From (41) and (42) 20it is easy to see that for ¯M < 1060, there are highly condensed planet sized stars. (Infact these considerations lead to the Chandrasek har limit in stellar theory). We can also verify that for ¯Mapproaching 1060corresponding to a mass∼1036gms, or roughly a hundred to a thousand times the solar mass, the radius Rgets smaller and smaller and would be ∼108cms, so as to satisfy (40) and give a Black Hole in broad agreement with the ory. (On 13th Septembe, 2000, NASA announced the discovery of exactl y such Black Holes.) Finally for the universe as a whole, using only the theory of N ewtonian gravitation, it is well known that we can deduce, as we saw in C hapter 7, R∼GM c2(43) where this time R∼1028cmsis the radius of the universe and M∼1055gms is the mass of the universe. Equation (43) is the same as (40) and suggests that the univer se itself is a Black Hole. It is remarkable that if we consider the univers e to be a Schwarzchild Black Hole as suggested by (43), the time taken by a ray of light to traverse the universe equals the age of the universe ∼1017secsas shown elsewhere [43]. 11 Dimensionality and the Field and Particle Approach In a recent paper, Castro, Granik and El Naschie have given a r ationale for the three dimensionality of our physical space within the fr amework of a Cantorian fractal space time and El Naschie’s earlier work t hereon[44]. An ensemble is used and the value for the average dimension invo lving the golden mean is deduced close to the value of our 3 + 1 dimensions. We no w make a few remarks based on an approach which is in the spirit of the above con- siderations. Our starting point is the fact that the fractal dimension of a quantum path is two, which, it has been argued in Chapter 6 is described by t he coordi- nates (x,ict). Infact this lead to the Dirac equation of the spin half elec tron. Given the spin half, it was pointed out that it is then possibl e to deduce the dimensionality of an ensemble of such particles, which turn s out to be three. 21There is another way of looking at this. If we generalise from the one space dimensional case and the complex ( x,ıt) plane to three dimensions, we in- fact obtain the four dimensional case and the Theory of Quart ernions, which are based on the Pauli Spin Matrices[28]. As has been noted by Sachs, had Hamilton identiﬁed the fourth coordinate in the above gener alisation with time, then he would have anticipated Special Relativity its elf. It must be ob- served that the Pauli Spin Matrices which denote the Quantum Mechanical spin half form, again, a non commutative structure. Curiously enough the above consideration in the complex pla ne can have an interesting connection with an unproven nearly hundred yea r old conjecture of Poincare. Poincare had conjectured that the fact that closed loops cou ld be shrunk to points on a two dimensional surface topologically equivale nt to the surface of a sphere can be generalised to three dimensions also[45]. After all these years the conjecture has remained unproven. We will now see w hy the three dimensional generalisation is not possible. We ﬁrstly observe that a two dimensional surface on which clo sed smooth loops can be shrunk continuously to arbitrarily small sizes is simply con- nected. On such a surface we can deﬁne complex coordinates fo llowing the hydrodynamical route exploiting the well known connection between the two. If we consider laminar motion of an incompressible ﬂuid we wi ll have[46] /vector∇ ·/vectorV= 0 (44) Equation (44) deﬁnes, as is well known, the stream function ψsuch that /vectorV=/vector∇ ×ψ/vector ez (45) where/vector ezis the unit vector in the zdirection. Further, as the ﬂow is irrotational, as well, we have /vector∇ ×/vectorV= 0 (46) Equation (46) implies that there is a velocity potential φsuch that, /vectorV=/vector∇φ (47) The equations (45) and (47) show that the functions ψandφsatisfy the Cauchy-Reimann equations of complex analysis[47]. 22So it is possible to characterise the ﬂuid elements by a compl ex variable z=x+ıy (48) The question is can we generalise equation (48) to three dime nsions? Infact as we saw a generalisation leads not to three but to four dimen sions, with the three Pauli spin matrices /vector σreplacingı. Further these Pauli spin matrices do not commute, and characterise spin or vorticity. This close connection can be established by other arguments as well[48]. This is not surprising - the reason lies in equation (45) or eq uivalently in the multiplication law of complex numbers. (Infact, there is a g eneral tendency to loverlook this fact and this leads to the mistaken impress ion that complex numbers are just an ordered pair of numbers, which latter are usually asso- ciated with vectors.) The above considerations give an explanation for the 3 + 1 dim ensionality of space time[49]. Moreover equations like (45) and (48) re- emphasize the hydrodynamical model discussed earlier. Incidentally as B arrow [50] puts it, ”Interestingly, the number of dimensions of space which we experience in the large plays an important role.... It also ensures that wa ve phenomena behave in a coherent fashion. Were there four dimensions of s pace, then sim- ple waves would not travel at one speed in free space, and henc e we would simultaneously receive waves that were emitted at diﬀerent times. Moreover, in any world but one having three large dimensions of space, w aves would become distorted as they travelled. Such reverberation and distortion would render any high-ﬁdelity signalling impossible. Since so mu ch of the physical universe, from brain waves to quantum waves, relies upon tra velling waves we appreciate the key role played by the dimensionality of ou r space in ren- dering its contents intelligible to us.” We make a ﬁnal remark. We saw in Chapters 1 and 2 that while the c ontem- porary Field approach is based on guage interactions and spi n 1 Bosons, these Bosons, as seen in Chapter 9 are not the Quantuzed Vortices, b ut rather their bound states - they can be thought of as, approxmately steaml ines. On the other hand, our approach has been based on Fermions, spin hal f particles, which are like the Quantized Vortices encountered in Chapte r 3. References 23[1] L. Smolin, in ”Quantum Concepts in Space and Time”, Eds., R. Penrose and C.J. Isham, OUP, Oxford, 1986, pp.147-181. [2] A.O. Barut, Phys.Lett.A., 171, 1992, pp.1-2. [3] R. 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arXiv:physics/0009086v1 [physics.plasm-ph] 27 Sep 2000Formation and Primary Heating of The Solar Corona — Theory and Simulation S.M. Mahajana) Institute for Fusion Studies, The Univ. of Texas, Austin, TX 78712 R. Miklaszewskib) Institute of Plasma Physics and Laser Microfusion, 00–908 W arsaw, Str. Henry 23, P.O. Box 49, Poland K.I. Nikol’skayac) Institute of Terrestrial Magnetism, Ionosphere and Radio W ave Propagation(IZMIRAN), Troitsk of Moscow Region, 142092 Russia N.L. Shatashvilid) Plasma Physics Department, Tbilisi State University, 3800 28, Tbilisi, Georgia, International Centre for Theoretical Physics, Trieste, It aly An integrated Magneto–Fluid model, that accords full treat ment to the Velocity ﬁelds as- sociated with the directed plasma motion, is developed to in vestigate the dynamics of coronal structures. It is suggested that the interaction of the ﬂuid and the magnetic aspects of plasma may be a crucial element in creating so much diversity in the s olar atmosphere. It is shown that the structures which comprise the solar corona can be cr eated by particle (plasma) ﬂows observed near the Sun’s surface — the primary heating of thes e structures is caused by the viscous dissipation of the ﬂow kinetic energy. PACS No.: 47.65.+a; 52.30.-q; 50.30.Bt; 96.60.Pb; 96.60.N a a)Electronic mail: mahajan@mail.utexas.edu b)Electronic mail: rysiek@ifpilm.waw.pl c)Electronic mail: hellab@izmiran.troitsk.ru d)Electronic mail: nanas@iberiapac.ge shatash@ictp.tries te.it 11 INTRODUCTION The TRACE observations [1–3] reveal that the solar corona is comprised of lots of thin loops that are intrinsically dynamic, and that continually evolve. These very thin strings, the observations indicate, are heated for a few to tens of min utes, after which the heating ceases, or at least changes signiﬁcantly in magnitude [1]. I n this paper we examine a class of mechanisms, which, through the viscous–dissipation of t he plasma kinetic energy, pro- vide the primary and basic heating of the coronal structures during their very formation. The basic input of the theory is the reasonable assumption th at the coronal structures are created from the evolution and re–organization of a rela tively cold plasma ﬂow [1–16] emerging from the sub–coronal region (between the solar sur face and the visible corona) and interacting with the ambient magnetic ﬁeld anchored ins ide the solar surface. During the process of trapping and accumulation, a part of the kinet ic energy of the ﬂow is con- verted to heat by viscous dissipation and the coronal struct ure is born hot and bright. For this to happen, we must ﬁnd alternative fast and eﬃcient heat ing mechanisms because, for the conditions prevalent in the coronal structures, the standard viscous dissipation is neither eﬃcient nor fast. The rates of viscous dissipation c an be considerably increased by processes which either enhance the local viscosity coeﬃc ient, or induce short scale structures in the velocity ﬁeld. At present we do not know of a ny convincing mechanism for the former possibility. This paper, therefore, is limit ed to an examination of processes of the latter kind. We ﬁnd that as long as the ﬂow–velocity ﬁel d is treated as an essential and integral part of the plasma dynamics, fast and desirable viscous dissipation does, indeed, result. Consequently, during its very formation, t he coronal structure can become hot and bright. Of the several possible mechanisms by which the ﬂow kinetic e nergy may be converted into heat we emphasize the following two: The ﬁrst is the abil ity of supersonic ﬂows to create nonlinear perturbations which steepen to produce sh ort scale structures which can dissipate by ordinary viscosity. The second stems from the r ecently established property of the magnetoﬂuid equilibria for extreme sub–Alfv´ enic ﬂows (most of the observed coronal ﬂows fall in this category) – such ﬂows can have a substantial , fastly varying (spatially) velocity ﬁeld component even when the magnetic ﬁeld is mostl y smooth. Viscous damp- 2ing associated with this varying component could be a major p art of the primary heating needed to create and maintain the bright Corona. From a gener al framework describing a plasma with ﬂows, we have been able to “derive” several of th e essential characteristics of the coronal structures. Theoretical basis for both these mechanisms will be discussed. Our simulation (for which we developed a dissipative two–ﬂu id code), however, concen- trates only on the ﬁrst mechanism, and preliminary results r eproduce many of the salient observational features. There is clear cut evidence of nonl inearly steepened velocity ﬁelds which eﬀectively dissipate and heat the coronal structure r ight through the process of formation. The numerical investigation of the second mecha nism, which will require a much higher spatial resolution, will be undertaken soon. Naturally all these processes require the existence of part icle ﬂows with reasonable amounts of kinetic energy. There are several recent publica tions [1–11] cataloguing enough observational evidence for such ﬂows in the regions between the sun and the corona to warrant a serious investigation in this direction. It must b e admitted that we still have little understanding of the nature of the processes by which the relatively cool material (no hotter than about 20000K) moves upward from low altitude s (as low as a few thou- sand kilometers) to the outer atmosphere. For this paper, we shall simply exploit the observation that the ﬂows exist, and work out their conseque nces. We believe that the ﬂows might prove to be a crucial element in solving the riddle of coronal heating. The model for the solar atmosphere that we propose and invest igate is obtained by injecting an essential new feature into several extant noti ons — the plasma ﬂows are allowed to play their appropriate role in determining the ev olution and the equilibrium properties of the structures under investigation. We reite rate that the distinguishing ingredient of our model is the assumption (observationally suggested) that relatively cold particles spanning an entire range of velocity spectrum — sl ow as well as fast, continually ﬂow from the sub–coronal to the coronal regions. It is the int eraction of these cold primary ﬂows with the solar magnetic ﬁelds, and the strong coupling b etween the ﬂuid and the magnetic aspects of the plasma that will deﬁne the character istics of a typical coronal structure (including Coronal Holes). In this paper we limit ourselves to the formation and primary heating aspects; we do not deal with instabilities, their nonlinear eﬀects, ﬂaring 3etc. These are the problems that we will confront at the next s tage of the development of the model. In Sec.2, we desribe in relative detail our basic model for th e upper solar atmosphere, a time–dependent, two–ﬂuid system of currents and ﬂows. The ﬂows are treated at par with other determining dynamical quantities, the curre nts and the solar magnetic ﬁelds. Section 3 is devoted to the derivation of the characte ristics of typical coronal structures from the basic model. Following a general discus sion, we numerically simulate the evolution of a cold plasma ﬂow as it interacts with the sol ar magnetic ﬁeld and gravity in Sec.3.1. We follow the fate of an initial cold supersonic ﬂ ow as the particles get trapped by the magnetic ﬁeld. By the time a sizeable density is built u p we also ﬁnd a considerable rise in temperature. In a very short time the velocity ﬁeld de velopes a shocklike structure which dissipates with ordinary viscosity to convert the ﬂow kinetic energy to heat. In Sec. 3.2 we take a diﬀerent approach, and describe elements o f the recently investigated magneto–ﬂuid theory (see Mahajan and Yoshida, 1998, 2000) w hich allows the existence of equilibrium solutions missing in the ﬂowless MHD. We ﬁnd t hat a short–scale velocity component is predicted to be an essential aspect of a class of magnetoﬂuid states in terms of which a typical coronal structure could be modelled. The m agnetoﬂuid states are the equilibrium states created by the strong interaction of the magnetic and the ﬂuid character of a plasma, and are derived from the normal two–ﬂuid equatio ns when the velocity ﬁeld is treated at par with the magnetic ﬁeld. In a somewhat detail ed discussion, we argue for the relevance of these states for the solar corona. These sta tes could be seen as a set of quasi–equilibria evolving to an eventual hot coronal struc ture; the dissipation of the small scale velocity component provides the necessary source of h eating. Since the numerical simulation of these states requires a much ﬁner resolution t han we have in our code, their time dependent simulation is deferred to a future work. 42 THE BASIC MODEL — GENERAL EQUATIONS FOR THE QUIESCENT SOLAR ATMOSPHERE In this section we will develop a general theoretical framew ork from which the typical solar coronal structure will be “derived.” In our model, the plasma ﬂows from the Sun’s surface provide the basic source of matter and energy for the myriad of coronal structures (including Coronal Holes). Although the magnetic ﬁeld is, n aturally, the primary culprit behind the structural diversity of the corona, the ﬂows (and their interactions with the magnetic ﬁeld) are expected to add substantially to that ric hness. The primary objective of this paper is to investigate how the se ﬂows are trapped and heated in the closed magnetic ﬁeld regions, and create one of the typical shining coronal elements. We shall, however, make a small digression to sugg est a possible fate of the fast ﬂows making their way through the regions where the magn etic ﬁeld is weak, or has open ﬁeld lines. The faster particles could readily esca pe the solar atmosphere in the open ﬁeld-line regions. They could also do so by punching temporary channels in the neighboring closed ﬁeld–line structures. The ﬂows esca ping through these existing or “created” coronal holes (the coronal holes (CH) are highl y dynamical structures with open and “nearly open” magnetic ﬁeld regions, see e.g. [17]) may eventually appear as the fast solar wind. In the closed ﬁeld–line, the magnetic ﬁelds will trap the ﬂow s, and the trapping will lead to an accumulation of particles and energy creating the coronal elements with high temperature and density. We shall not consider the solar act ivity processes, since the activity regions (AR) and ﬂares, though an additional sourc e of particles and energy, cannot account for the continuous supply needed to maintain the corona. Moreover, in the theory we suggest, the ﬂare is understood to be a secondar y event and not the primary source for the creation of the hot corona. To describe the entire atmosphere of the quiescent, non–ﬂar ing Sun we use the two– ﬂuid equations where we keep the ﬂow vorticity and viscosity eﬀects (Hall MHD). The general equations will apply in both the open and the closed ﬁ eld regions. The diﬀerence between various sub–units of the atmosphere will come from t he initial, and the boundary 5conditions. LetVdenotes the ﬂow velocity ﬁeld of the plasma in a region where t he primary solar magnetic ﬁeld is Bs. It is, of course, understood that the processes which gener ate the primary ﬂows and the primary solar magnetic ﬁelds are indepe ndent (say at t= 0 time). The total current j=jf+js(herejfis the self–current that generates the magnetic ﬁeld Bfandjsis the source of the solar ﬁeld Bs) is related to the total (that can be observed) magnetic ﬁeld B=Bs+Bfby Amp´ ere’s law: j=c 4π∇ ×B. (1) Notice that in the framework we are developing (assumption o f the existence of primary ﬂows), the boundaries between the photosphere, the chromos phere and the corona become rather artiﬁcial; the diﬀerent regions of each coronal stru cture are distinguished by just the parameters like the temperature and the density. In fact , these parameters should not show any discontinuities; they must change smoothly alo ng the structure. At some distance from the Sun’s surface, the plasma may become so hot and dense that it becomes visible (the bright, visible corona), and this altitude cou ld be viewed as the base of the corona. But to study the creation and dynamics of bright coro nal structures (loops, arches, arcades etc.) we must begin from the photosphere, an d determine the plasma behavior in the closed ﬁeld regions. Assuming that the primary ﬂows provide, on a continuous basi s, the entire material for coronal structures, the solar ﬂow with density nwill obey the Continuity equation: ∂ ∂tn+∇ ·(nV) = 0. (2) We must add a word of caution: in the closed ﬁeld regions, the t rapped particle density may become too high for the conﬁning ﬁeld, resulting in insta bilities of all kinds. In this paper we shall not deal with instabilities and their consequ ences; it will constitute the next stage of development of the model. Since the corona as well as the SW are known to be mostly hydrog en plasmas (with a small fraction of Helium, and neutrons, and an insigniﬁcan t amount of highly ionized 6metallic atoms) with nearly equal electron and proton densi ties:ne≃ni=n, we expect the quasineutrality condition ∇ ·j= 0 to hold. In what follows, we shall assume that the electron and the pro ton ﬂow velocities are diﬀerent (two–ﬂuid approximation was used e.g. in Sturr ock and Hartly, (1966). Neglecting electron inertia, these are Vi=V, andVe= (V−j/en), respectively. We assign equal temperatures to the electron and the protons fo r processes associated with the quiescent Sun. For the creation processes of a typical coron al structure, this assumption is quite good. For the fast SW, however, we know from recent obse rvations (Banaszkiewicz et al. 1997 and references therein), that the species temperature s are found to be diﬀerent: Ti∼2·105K and Te∼1·105K. Since the fast SW is not the principal interest of this paper, we shall persist with the equal temperatures assumpt ion; the kinetic pressure pis given by: p=pi+pe≃2nT, T =Ti≃Te. (3) With this expression for p, and by neglecting electron inertia, the two–ﬂuid equation s are obtained by combining the proton and the electron equations of motion: ∂ ∂tVk+ (V· ∇)Vk= =1 en(j×b)k−2 nmi∇k(nT) +∇k/parenleftbiggM⊙G r/parenrightbigg −1 nmi∇lΠi,kl, (4) and ∂ ∂tb− ∇ ×/bracketleftBigg/parenleftBigg V−j en/parenrightBigg ×b/bracketrightBigg =2 mi∇/parenleftbigg1 n/parenrightbigg × ∇(nT), (5) where b=eB/mic,miis the proton mass, Gis the gravitational constant, M⊙is the solar mass, ris the radial distance, and Π i,lkis the ion viscosity tensor. For ﬂows with large spatial variation, the viscous term will end up playin g an important part. To obtain an equation for the evolution of the ﬂow temperature T, we begin with the energy balance equations for a magnetized, neutral, isothermal electron– proton plasma: ∂ ∂tεα+∇k(εαVα,k+Pα,klVα,l) +∇qα=nαfα·Vα, (6) where αis the species index. The ﬂuid energy εα(thermal energy + kinetic energy) and the total pressure tensor Pα,klare given by εα=nα/parenleftBigg3 2Tα+mαV2 α 2/parenrightBigg ,Pα,kl=nαTαδkl+ Π α,kl, (7) 7and fα=eαE+eα cVα×B+mα∇GM⊙ r, (8) is the volume force experienced by the ﬂuids ( Eis the electric ﬁeld). In Eq. (6), qαis the heat ﬂux density for the species α. After standard manipulations we arrive at the temperature evolution equation 3 2nd dt(2T) +∇(qi+qe) =−2nT∇ ·V+minνi 1 2/parenleftBigg∂Vk ∂xl+∂Vl ∂xk/parenrightBigg2 −2 3(∇ ·V)2 + +5 2n/parenleftBiggj en· ∇T/parenrightBigg −j en∇(nT) +EH+ER (9) where ERis the total radiative loss, EHis the local mechanical heating function, and νiis the ion kinematic viscosity. Note that we have retained visc ous dissipation in this system. If primary ﬂows are ignored in the theory, various anomalous heating mechanisms need to be invoked, and a corresponding term EHhas to be added. The full viscosity tensor relevant to a magnetized plasma is rather cumbersome, and we do not display it here. However just to have a feel for the importance of spatial vari ation in viscous dissipation, we display its relatively simple symmetric form. It is to be c learly understood that this version is meant only for theoretical elucidation and not fo r detailed simulation. We notice that even for incompressible and currentless ﬂows, h eat can be generated from the viscous dissipation of the ﬂow vorticity. For such a simple s ystem, the rate of kinetic energy dissipation turns out to be /bracketleftBiggd dt/parenleftBiggmiV2 2/parenrightBigg/bracketrightBigg visc=−minνi/parenleftbigg1 2(∇ ×V)2+2 3(∇ ·V)2/parenrightbigg . (10) revealing that for an incompressible plasma, the greater th e vorticity of the ﬂow, the greater the rate of dissipation. Let us now introduce the following dimensionless variables : r→rR⊙;t→tR⊙ VA;b→bb⊙;T→T T⊙;n→n n⊙; V→VVA;j→jVAen⊙;qα→qαn⊙T⊙VA;νi→νiR⊙VA (11a), and parameters: b⊙=eB(R⊙) mic;λi⊙=c ωi⊙;c2 s=2T⊙ mi;ω2 i⊙=4πe2n⊙ mi;VA=b⊙λi⊙; 8rA=GM⊙ V2 AR⊙= 2β rc;rc=GM⊙ 2c2sR⊙;α=λi⊙ R⊙;β=c2 s V2 A, (11b) where R⊙is the solar radius. Note that in general νiis a function of density and temper- ature: νi= (Vi,thT2/12πne4). In terms of these variables, our equations read: ∂ ∂tV+ (V· ∇)V= =1 n∇ ×b×b−β1 n∇(nT) +∇/parenleftbiggrA r/parenrightbigg +νi/parenleftbigg ∇2V+1 3∇(∇ ·V)/parenrightbigg , (12) ∂ ∂tb− ∇ ×/parenleftbigg V−α n∇ ×b/parenrightbigg ×b=αβ∇/parenleftbigg1 n/parenrightbigg × ∇(nT), (13) ∇ ·b= 0, (14) ∂ ∂tn+∇ ·nV= 0, (15) 3 2nd dt(2T) +∇(qi+qe) =−2nT∇ ·V+ 2β−1νin 1 2/parenleftBigg∂Vk ∂xl+∂Vl ∂xk/parenrightBigg2 −2 3(∇ ·V)2 + +5 2α(∇ ×b)· ∇T−α n(∇ ×b)∇(nT) +EH+ER. (16) This set of equations will now be studied for diﬀerent types o f magnetic ﬁeld regions, in particular the regions with closed ﬁeld lines. Before we embark on a detailed theory of the formation and hea ting of the corona, we would like to give a short list of heating mechanisms which ha ve been invoked to deal with this rather fundamental and still unresolved problem of Sol ar physics : Alfv´ en waves [20– 27], Magnetic reconnection in Current sheets [28–36], and M HD Turbulence [37–39]. For all these schemes, the predicted temperature proﬁles in the coronal structures come out to be highly sensitive to the form of the heating mechanism [40, 41]. Parker (1988) suggested that the solar corona could be heated by dissipation of many t angential discontinuities arising spontaneously in the coronal magnetic ﬁeld that is s tirred by random photospheric footpoint motions. This theory stimulated numerous search es for observational signatures of nanoﬂares. Unfortunately, all of these attempts fall sho rt of providing a continuous (both in space and time) energy supply that is required to ﬁrs t create in a few minutes, and then support for longer periods the observed bright coro nal structures (see e.g. [1,2]). 9Our attempt to solve this problem makes a clean break with the conventional approach. We do not look for the energy source within the corona but plac e it squarely in the primary ﬂows emerging from the Sun (see the results of [1–3]). We prop ose (and will test) the hypothesis that the energy and particles associated with th e primary ﬂows, in interaction with the magnetic ﬁeld, do not only create the variety of conﬁ gurations which constitute the corona, but also provide the primary heating. The ﬂows ca n give energy and particle supply to these regions on a continuous basis — we will show th at the primary heating takes place simultaneously with the accumulation of the cor ona and a major aspect of the ﬂow–magnetic ﬁeld interaction, for our system, is to pro vide a pathway for this to happen. A mathematical modeling of the coronal structure (for its cr eation and primary heat- ing) will require the solution of Eqs. (12)–(16) with approp riate initial and boundary conditions. We will use a mixture of analytical and numerica l methods to extract, what we believe, is a reasonable picture of the salient aspects of a typical coronal structure. 3 CONSTRUCTION OF A TYPICAL CORONAL STRUCTURE Though the solar atmosphere is highly structured, it seems t hat most of the constituent elements have something common in their creation and heatin g. In order to construct a uniﬁed theory for the entire corona, one would have to confr ont large variations in plasma density and temperature. It seems, however, that bey ond the coronal base, the equilibrium temperature tends to be nearly constant on each one of these structures; the temperature of a speciﬁc structure increases insigniﬁcant ly (about 20 p.c.) from its value at the base to its maximum reached at the top of the structure. This change is much less than the temperature change (about 2 orders of magnitud e) that occurs between the solar surface and the coronal base. This observation is an ou tcome of the investigation of several authors (see, for example, [1,2,41–45]). Their res ults show that the bright elements of the corona are composed of quasi–isothermic and ultra–th in arcs (loops) of diﬀerent 10temperature and density, situated (located) close to one ot her. This state is, perhaps, brought about by the isolating inﬂuence of magnetic ﬁelds wh ich prevent the particle and energy transfer between neighboring structures. It is safe to assume, then, that in the quasi–equilibrium sta te, each coronal structure has a nearly constant temperature, but diﬀerent structures have diﬀerent characteristic temperatures, i.e., the bright corona seen as a single entit y will have considerable temper- ature variation. Observations tell us that the coronal temp eratures are much higher than those of the primary ﬂows (which we are proposing as the mothe r of the corona). For the consistency of the model, therefore, it is essential that th e primary “heating” must take place during the process of accumulation of a given coronal e ntity. This apparent problem, in fact, can be converted to a theoret ical advantage. We distinguish two important eras in the life of a coronal struc ture; a hectic period when it acquires particles and energy (accumulation and heating ), and the relatively calmer period when it ”shines” as a bright, high temperature object . In the ﬁrst era, the most important issue is that of heating wh ile particle accumulation (trapping) takes place in a curved magnetic ﬁeld. This is, in fact, the essential new ingredient of the current approach. We plan to show: 1) that the kinetic energy contained in the primary ﬂows can b e dissipated by viscosity to heat the plasma, and 2) that this dissipation can be large e nough to produce the observed temperatures. Naturally, a time dependent treatment will be needed to desc ribe this era. Any additional heating mechanisms, operative after the eme rgence of the coronal structure, will not be discussed in this paper. For an essent ial energy inventory of the quasi-equilibrium coronal structure, we also ignore the co ntributions of ﬂares and other “activities” on the solar surface because they do not provid e a continuous and suﬃcient energy supply [2]. The second era is that of the quasi-equilibrium of a coronal s tructure of given density and temperature - neither of which has to be strictly constan t. The primary heating has already been performed, and in the equilibrium state, we can neglect viscosity, resistivity and other collisional eﬀects in addition to neglecting the t ime dependence. The calcu- 11lations in this regime will be limited to the determination o f the magnetic ﬁeld and the velocity–ﬁeld structures that the collisionless magnetoﬂ uids can generate and we will also examine if these structures can conﬁne plasma pressure. 3.1 Creation and heating of coronal structure In this subsection we will concentrate on numerical methods to test our basic conjecture that the primary solar ﬂows are responsible for the creation and heating of a typical bright coronal structure. The numerical results (obtained by modeling Eqs. (12)–(16) with viscosity tensor relevant to magnetized plasma) are ex tremely preliminary, but they clearly indicate that the proposed mechanism has considera ble promise. Let us ﬁrst make order of magnitude estimates on the requirem ents that must be met for this scheme to be meaningful. It is well known that (se e e.g. [46]) the rate of energy losses Ffrom the solar corona by radiation, thermal conduction, and advection is approximately 5 ·105erg/cm2s. For the brightest loops the rate loss could even reach 5·106erg/cm2s. If the conversion of the kinetic energy in the primary ﬂows were to compensate for these losses, we would require a radial energ y ﬂux 1 2min0V2 0V0≥F, (17) where V0is the initial ﬂow speed. For V0∼300 km/s this implies an initial density in the range: (3 ·107−4·108)cm−3. For slower ( ∼100 km/s) velocity primary ﬂows the starting density has to b e higher (≥109cm−3). These values seem reasonable according to the latest obse rvational data [1-3]. The normal viscous dissipation of the ﬂow takes place on a tim e (using Eq. (10)): tvisc∼L2 νi, (18) where Lis the length of the coronal structure. For a primary ﬂow with T0= 3 eV = 3.5·104K and n0= 4·108cm−3creating a quiet coronal structure of size L= (2·109− 7·1010) cm, the dissipation time can be estimated to be of the order o f (2·108−1010) s. The shorter the structure and hotter the ﬂow, the faster is th e rate of dissipation. This 12estimated time is much longer than what is actually found by t he latest observations by TRACE [1]. Mechanisms much faster than the one embodied in (1 8), therefore, will be needed for the model to work. In the absence of “anamalous vis cosity”, the only way to enhance the dissipation rates (to the observed values) is to create spatial gradients of the velocity ﬁeld that are on a scale much much shorter than that of the structure length (deﬁned by the smooth part of the magnetic ﬁeld). Thus , the viability of the model depends wholly on the existence of mechanisms that ind uce short–scale velocity ﬁelds. Numerical simulations show that the short–scale vel ocity ﬁelds are, indeed, self– consistently generated in the two–ﬂuid system. For numerical work (to illustrate the bright coronal struct ure formation), we model the initial solar magnetic ﬁeld as a 2D arcade with circular ﬁ eld lines in the x–zplane (see Fig. 1 for the contours of the vector potential, or the ﬂu x function). The ﬁeld attains its maximum value Bmax(xo, z= 0) at x0at the center of the arcade, and is a decreasing function of the height z(radial direction). The set of model equations (12-16) was solved in 2D ﬂat geometry (x,z) using the 2D version of Lax –Wendroﬀ numerical scheme (Richtmyer and Morton 1967) alongwith applying the F lux–Corrected–Transport procedure [48]. Equation (13) was replaced with its equival ent for the y–component of the vector potential which automatically ensures the div ergence-free property of the magnetic ﬁeld. The equation of heat conduction was treated s eparately by Alternate Direction Implicit method with iterations [49]. Transport coeﬃcients for heat conduction and viscosity were taken from Braginski, 1965. A numerical m esh of 200 ×150 points was used for computation. To illustrate the formation and heating of a general coronal structure, we have modeled several cases with diﬀerent initial and boundary condition s for cold primary ﬂows. The dynamical picture is strongly dependent on the relation of t he initial ﬂow pressure and the magnetic ﬁeld strength. Two limiting cases are interesting : 1) the initial magnetic ﬁeld is weak, and the ﬂow signiﬁcantly deforms (and in speciﬁc cases , drags) the magnetic ﬁeld lines, 2) the initial magnetic ﬁeld is strong, and the ﬂow lea ves the ﬁeld lines practically unchanged. For sub–Alfv´ enic ﬂows, we present in Figs. 2-5 the salient f eatures of our preliminary 13results. We have plotted (as functions of xandz) four relevant physical quantities: the ﬂux function A, the density n, the temperature T, and the magnitude of the velocity ﬁeld\|V\|(for speciﬁc cases, when needed, we give the radial componen t of velocity ﬁeld Vzalso). The plots correspond to two (in some cases to three) diﬀerent time frames. The results are described under three separate headings, covering resp ectively, the fully uniform, the spatially non–uniform, and the time–dependent as well as sp atially non–uniform initial ﬂows. A.Initially uniform primary ﬂow and an Arcade-like magnetic ﬁ eld struc- ture— Fig. 2. This case is highly idealized but illustrates the main aspec ts of the creation of the hot coronal structures, and of the basic heating process. When discussing the temporally uniform initial ﬂows, we cho ose the parameters to satisfy the observational constraint that, over a period of some tens of minutes, the lo- cation of the heating must have a relatively smooth evolutio n [1]. The ﬁnal shape and location of the coronal structure (of the associated B(r, t), for example) will be naturally deﬁned by its material source, by the heating dynamics, and b y the initial ﬁeld B0(r, t). For these studies, the initial ﬂow velocity ﬁeld is taken to b e uniform at the surface and has only a radial component, Vz= 300 km/s. Other parameters are: Maximum value of the magnetic ﬁeld Bmax(xo, z= 0) = 7 G, initial density of the ﬂow 4 ·108cm−3and the initial temperature 3 eV. Simulations yield the following results: 1) The ﬂow particles begin to accumulate at the footpoints ne ar the solar surface (Fig. 2, see density at t= 750 s). The accumulation goes on with time, and gradually the entire volume under the arcade (starting from the centra l short loops) is ﬁlled with particles (Fig. 2, density at t= 1400 s). First the shorter loops are ﬁlled, and then the larger ones. 2) The heating of the particles goes hand in hand with the accu mulation (Fig. 2, plots for density and temperature). 3) The regions of stronger magnetic ﬁelds are denser in popul ation (Fig. 2, plots for Aandn). In earlier stages of the formation of a coronal structure, the regions near the 14base (where the ﬁeld is stronger) are denser and hotter than t he distant regions (Fig. 2, t= 750 s plots for nandT); for shorter loops, the density increases (as a function of height z) from the bottom of the structure, and then falls — ﬁrst rapid ly and later insigniﬁcantly; the maximum density is much greater than th e initial density of the ﬂows. 4) The dissipation of the ﬂow kinetic energy is faster in the ﬁ rst stage of formation (Fig. 2, t= 750 s plot for \|V\|). The plot \|V\|versus zshows steep (shock–like) gradients near the base. Thus the bright base is created in the very ﬁrst stage in the stronger magnetic ﬁeld regions (shorter loops). For given parameter s, the initial ﬂow is strongly supersonic. Thus the shocks are generated with eﬃcient tran sfer of kinetic energy into heat. As the mean free path of ions in the plasma is of the order of (106−107)cm(in the direction parallel to the magnetic ﬁeld) and the dimensi on of the structure is much greater – of the order of 1010cm– eﬃcient conditions for the kinetic energy dissipation exist. The plots for the velocity, temperature and density r eveal that with increasing z, and in the regions away from the arcade center, we ﬁrst ﬁnd an u ndisturbed ﬂow with low temperature, then see a transient area with high density and temperature, and ﬁnally a shock consistent with Hugoniot conditions. The short scale represented by the width of the shock-layer (determined by viscosity) is the main enhan cer of viscous dissipation. 5) For later times, the brightening process spreads over wid e regions (Fig. 2, t= 1400 s plot for temperature). 6) In the very ﬁrst stage, the shorter loops are a bit overheat ed, but they cool down somewhat at later times when the longer loops begin to get hea ted (Fig. 2, plots for temperature). 7) The base ( T≥100 eV) of the bright region is at about 1 .4·109cm∼0.02R⊙(Fig. 2, t= 1400 splots for nandT) from the solar surface. This number is in a very good agreement with the latest TRACE results [1]. Outwards from t he base, the accumulated layer has somewhat lower, but more or less uniform, insigniﬁ cantly decreasing density. In the accumulated layer the kinetic energy of the ﬂow is essent ially uniform (again, decreases insigniﬁcantly); the dissipation has practically stopped (Fig. 2, t= 1400 s= 23min, plot for \|V\|versus z). The temperature is practically uniform in the longer loop s and increases insigniﬁcantly in shorter loops (for some specia l conditions these conclusions 15may be somewhat modiﬁed in speciﬁc regions of the arcade; see point 8)). Outwards from the hottest region of the arcade, the temperature decre ases gradually and at some radial distance the outer boundary of the bright part is reac hed (Fig. 2, t= 1400 splot for temperature). Thus, in a very short time a dense and brigh t “coronal structure” is created — this object survives for a time much longer than was needed for its creation. The simulations show that the heating process may continue duri ng this so–called equilibrium stage, but at a rate much slower than the earlier primary heat ing. This heating seems just additional and supporting to the heat content of the nas cent hot structure. At this time, however, the velocity ﬁeld is already much smaller in m agnitude as compared to the initial values; the ﬂows in the hot coronal structure are already subsonic. This is a possible explanation why supersonic ﬂows may not be seen in t he hot observable coronal structures. 8) When relatively dense primary ﬂows interact with weak arc ade-like magnetic ﬁelds (Bmax(x0, z0= 0)≤10Gfor our initial ﬂow with given above parameters), the ﬁeld lines begin to deform (soon after the creation of the solar ba se) in the central region of the arcade but far from the base (see t= 1400 splots for density and temperature in Fig. 2). The particle accumulation is still strong, and the d issipation, though quite fast, stops rather rapidly. Consequently, the temperature ﬁrst r eaches a maximum (up to the deformed ﬁeld–line region this maximum is reached at the sum mit for each short loop) and later falls rapidly. Gradually one can see signs for the c reation of a local gravitational potential well behind the shortest loops (see t= 1400 s plot for Ain Fig. 2). This well supports a relatively dense and cold plasma in the central ar ea of the arcade ( t= 1400 s fornandTof Fig. 2). The density of this structure is considerably gre ater than that of the surrounding areas, and the temperature is considerably less than that of the rest of the accumulated regions at the same height of the arcade. Our preliminary simulations show that for the same paramete rs of the primary outﬂow, such cold and dense plasma objects (conﬁned in the so–called potential well) will not form in the regions where the initial magnetic ﬁelds are stronger (Bmax(x0, z0= 0)≥20G). B.Spatially non–uniform primary ﬂow interacting with an arca de–like mag- netic ﬁeld structure — Figs. 3, 4. 16The latest observations support the idea that the coronal ma terial is injected disconti- nously (in pulses or bunches, for example) from lower altitu des into the regions of interest (e.g. spicules, jet–like structures [6,7,12,13,1,2]). A r ealistic simulation, then, requires a study of the interaction of spatially non–uniform initial ﬂ ows with arcade–like magnetic ﬁeld structures. These “close to the actual” cases represen t more vividly the dynamics of the hot coronal formation. 1) When the spatially symmetric initial ﬂow (plot for Vzatt= 0 in Fig. 3a) inter- acts with the arcade (plots in Fig. 3), and the initial magnet ic ﬁeld is rather strong (Bmax(x0, z0= 0) = 20 G), the primary heating is completed in a very short time (∼(2−3)min) on distances ( ∼10000 km) shorter than the uniform–ﬂow case when the initial magnetic ﬁeld was weaker. This is also consistent wi th observations. The heating is very symmetric and the resulting hot structure is uniform ly heated to 1 .6·106K. 2) Observations reveal the existence of cool material and do wnﬂows, right within the hot coronal structures; they also show an imbalance in the pr imary heating on the two sides of the loops (see [12, 1]). To reproduce these characte ristics, we have modeled the coronal structure formation process using an asymmetric, s patially non–uniform initial ﬂow interacting with a strong magnetic ﬁeld (see Fig. 4). For both of the discussed cases, the downﬂows can be clearly s een for the velocity ﬁeld component Vz. In Fig. 4, the downﬂow is created simply by changing the init ial character of the ﬂow (initially we had only the right pulse from the velo city ﬁeld distribution given in Fig. 3a), while in Fig. 3a (plot at t= 297 s), the downﬂows are the result of more complicated events (see explanation below, in the next para graph). The ﬁnal parameters of the downﬂows are strongly dependent on the initial and bou ndary conditions. In the pictures, the imbalance in the primary heating process is al so revealed. When two identical pulses (Fig. 3a, plot at t= 0s) enter in succession into our standard, arcade–like initial magnetic ﬁeld, we simulate t he equivalent of two colliding ﬂows on the top of a structure. Shocks, though not very strong , are generated in a very short time ( t= 30s). Such shocks, on both sides of the arcade–center, have hot f ronts and cold tails. Soon ( t= 42s) these shocks become ”visible”, a hot and dense area is created on top of the structure where these shocks (at this mo ment they have become 17stronger) collide. After the collision (and ”reﬂection”), the entire area within the arcade becomes gradually hot. At some moment, a practically unifor mly heated structure is created, and the primary heating stops. This process is acco mpanied by downﬂows much slower than the primary ﬂows; much of the primary ﬂow kinetic energy has been converted to heat via shock generation (the shock and downﬂow velociti es diﬀer signiﬁcantly). It is clear that in the case of spatially assymetric initial ﬂow s, the downﬂows on diﬀerent sides of the arcade–center will have diﬀerent characterist ics. Due to the high pressure prevalent in the nascent hot structure (loop), there is no mo re inﬂow of the plasma and the ﬂow deposits its energy at the base; the base becomes over heated. Later this energy can be again transferred upwards via thermal conduction (th is mechanism can work in all the discussed cases), but at that moment the ﬂow could be a lso changed (see initially time–dependent ﬂow cases below). Plots for the temperature and velocity ﬁeld in Figs. 3b,4 als o indicate that some cold particles still remain in the body of the newly created hot st ructure. These particles are perhaps from the slower aggregates (our initial ﬂow was not u niform) which did not have suﬃcient energy to be converted to heat. C.Time dependent non–uniform initial ﬂows interacting with a rcade–like magnetic ﬁeld structures. – Fig. 5 To simulate reality further we introduce time dependence in the initial primary ﬂow velocity ﬁeld. We discuss two distinct cases: 1) Initially, the velocity ﬁeld has a pulse–like distributi on with a time–period nearly half of the “formation time” of the quasi-equilibrium struc ture corresponding to the case with time–independent initial conditions. The results dis played in Fig. 5 show that the emerging coronal structure has a rather uniform distributi on of temperature along the magnetic ﬁeld, and the latter is practically undeformed dur ing formation and heating. We see that when the basic heating ceases, the hot structure s urvives for the time of computation which happens to be shorter than the time necess ary for losses that destroy the structure. 2) The velocity ﬁeld has a fast amplitude modulation near its maximum value ( for these simulations the maximum radial velocity was taken to b e 300 km/s). We ﬁnd 18that the dynamics of the hot coronal structure creation is qu ite similar to the initially time–independent, spatially symmetric case. Because of th is, we don’t give here the corresponding plots. We only note that for this case, the str ucture tends to become even hotter (by a factor 1 .2 for the same parameters) and when quasi–equilibrium is established (time for this to happen is longer than for the ti me–independent initial ﬂows) the base of the structure is hotter than the top although at an earlier time the top was hotter, i.e, there is a temperature oscillation with a time– period longer compared to the creation time of the hot structure. The main message of numerical simulation is that the dynamic al interaction of an initial ﬂow with the ambient solar magnetic ﬁeld leads to a re –organization of the plasma such that the regions in the close vicinity of the solar surfa ce are characterized by strongly varying (in space and time) density and temperature, and eve n faster varying velocity ﬁeld, while the regions farther out from the bright base are n early uniform in these physical parameters. This phenomenon pertains generally, and not for just a set of speciﬁc structures. The creation and primary heating of the coronal structures are simultaneous, accompanied by strong shocks. These are fast processes (few tens of minutes) taking place at very short radial distances from the Sun ( ∼10000 km) in the strong magnetic ﬁeld regions with signiﬁcant curvature. The ﬁnal character istics of the created coronal structures are deﬁned by the boundary conditions for the cou pled primary ﬂow–solar magnetic ﬁeld system. The stronger the magnetic ﬁeld, the fa ster is the process of creation of the hot coronal structure with its base nearer the solar su rface. To investigate the near surface region one must use general time–dependent 3D equat ions. Quasi–stationary (equilibrium) equations, on the other hand, will suﬃce to de scribe the hot and bright layers — the already existing visible coronal structures. 3.2 Construction of quasi–equilibrium coronal structure The familiar MHD theory (single–ﬂuid) is a reduced case of th e more general two–ﬂuid theory discussed in this paper. Constrained minimizaion of the magnetic energy in MHD leads to force–free static equilibrium conﬁgurations [50, 51]. The range of two–ﬂuid relaxed 19states, however, is considerably larger because the veloci ty ﬁeld, now, begins to play an independent fundamental role. The presence of the velocity ﬁeld not only leads to new pressure conﬁning states [52,53], but also to the possibili ty of heating the equilibrium structures by the dissipation of kinetic energy. The latter feature is highly desirable if these equilibria were to be somehow related to the bright cor onal structures. We begin investigating the two–ﬂuid states by ﬁrst studying the simplest, almost analytically tractable, equilibria. This happens when the pressure term in the equation of motion (12) becomes a full gradient, i.e, whenever an equa tion of state relating the pressure and density can be invoked. For our present purpose , we limit ourselves to the constant temperature states allowing n−1∇p→2T∇lnn. Normalizing nto some constant coronal base density n0(reminding the reader that n0is diﬀerent for diﬀerent structures!), and using our other s tandard normalizations (λi0=c/ωi0is deﬁned with n0), our system of equations reduces to: 1 n∇ ×b×b+∇/parenleftBiggrA0 r−β0lnn−V2 2/parenrightBigg +V×(∇ ×V) = 0, (19) ∇ ×/parenleftbigg V−α0 n∇ ×b/parenrightbigg ×b= 0, (20) ∇ ·(nV) = 0, (21) where rA0, α0, β0are deﬁned with n0, T0, B0. This is a complete system of seven equations in seven variables. Following Mahajan and Yoshida (1998) and [54], we seek equil ibrium solutions of the simplest kind. Straightforward algebra leads us to the foll owing system of linear equations: b+α0∇ ×V=d nV (22) and b=a n/bracketleftbigg V−α0 n∇ ×b/bracketrightbigg , (23) where aanddare dimensionless constants related to the two invariants: the magnetic helicity/integraltext(A·B)d3xand the generalized helicity/integraltext(A+V)·(B+∇ ×V)d3x(or /integraltext(V·B+A·∇×V+V·∇×V)d3x) of the system. We will discuss aanddlater. The equilibrium solutions (22), (23) encapsulate the simple ph ysics: 1) the electrons follow 20the ﬁeld lines, 2) while the ions, due to their inertia, follo w the ﬁeld lines modiﬁed by the ﬂuid vorticity. These equations, when substituted in (19), (20), lead to ∇/parenleftBiggrA0 r−β0lnn−V2 2/parenrightBigg = 0, (24) giving the Bernoulli condition which will determine the den sity of the structure in terms of the ﬂow kinetic energy, and solar gravity. Equations (22) an d (23) are readily manipulated to yield α2 0 n∇ × ∇ × V+α0∇ ×/parenleftbigg1 a−d n/parenrightbigg V+/parenleftBigg 1−d a/parenrightBigg V= 0. (25) which must be solved with (24) for nandV; the magnetic ﬁeld can, then, be determined from (22). Equation (24) is solved to obtain n= exp/parenleftBigg −/bracketleftBigg 2g0−V2 0 2β0−2g+V2 2β0/bracketrightBigg/parenrightBigg , (26) where g(r) =rc0/r. This relation is rather interesting; it tells us that the va riation in density can be quite large for a low β0plasma (coronal plasmas tend to be low β0; the latter is in the range 0 .004−0.05) if the gravity and the ﬂow kinetic energy vary on length scales comparable to the extent of the coronal structure. In this system of equations, as we mentioned above, the temperature (which deﬁnes β0) has to be ﬁxed by initial and boundary conditions at the base of the structure. Substitut ing (26) into (25) will yield a single equation for velocity which is quite nontrivially no nlinear. Numerical solutions of the equations are tedious but straightforward. For analytical progress, essential to revealing the nature of the self–consistent ﬁelds and ﬂows, we will now make the additional simplifying assump tion of constant density. This is a rather drastic step (in numerical work, we take the d ensity to be a proper dynamical variable) but it can help us a great deal in unravel ing the underlying physics. There are two entirely diﬀerent situations where this assum ption may be justiﬁed: 1) the primary heating of corona has already been performed, i.e., a substantial part of ﬂow initial kinetic energy has been converted to heat. The re st of the kinetic energy, i.e., the kinetic energy of the equilibrium coronal structure is n ot expected to change much within the span of a given structure. Note that the ratio of ve locity components will have a large spatial variation, but the variation in V2is expected to be small. It is also 21easy to estimate that within a typical structure, gravity va ries quite insigniﬁcantly. There will be exceptional cases like the neighborhood of the Coron al holes and the streamer belts, where signiﬁcant heating could still be going on, and the temperature and density variations could not be ignored. Such regions are extremely hard to model; 2) if the rates of kinetic energy dissipation are not very lar ge, we can imagine the plasma to be going through a series of quasi–equilibria befo re it settles into a particular coronal structure. At each stage we need the velocity ﬁelds i n order to know if an appro- priate amount of heating can take place. The density variati on, though a factor, is not crucial in an approximate estimation of the desired quantit ies. The constant density assumption n= 1 will be used only in Eq. (25) to solve for the velocity ﬁeld (or the bﬁeld which will now obey the same equation). These solutions , when substituted in Eq. (26), would determine the density pr oﬁle (slowly varying) of a given structure. In the rest of this sub–section we will present several class es of the solutions of the following linear equation: α2 0∇ × ∇ × Q+α0/parenleftbigg1 a−d/parenrightbigg ∇ ×Q+/parenleftBigg 1−d a/parenrightBigg Q= 0, (27) whereQis either Vorb. To make contact with existing literature, we would use bas our basic ﬁeld to be determined by Eq. (25); the velocity ﬁeld Vwill follow from Eqs. (22) and (23), which for n= 1, become b+α0∇ ×V=dV (22′) and b=a[V−α0∇ ×b]. (23′) It is worth remarking that in order to derive the preceding se t of equations, all we need is the constant density assumption; the temperature ca n have gradients and, these are determined from the Bernoulli condition (20) with β0(T) replacing β0lnn. 223.3 Analysis of the Curl Curl Equation, Typical Coronal Equi- libria The Double Curl equation (27) was derived only recently [53] (Mahajan and Yo shida 1998); its potential, is still, largely unexplored(see [53 ], [54]). The extra double curl(the very ﬁrst) term distinguishes it from the standard force-fr ee equation [55,50,56] (Woltjer 1958; Taylor 1974, 1986; Priest 1994 and references therein ) used in the solar context. Since aanddare constants, Eq. (25), without the double curlterm, reproduces what has been called the “relaxed state” [50,56]. We will see that thi s term contains quantitative as well as qualitative new physics. In an ideal magnetoﬂuid, the parameters aanddare ﬁxed by the initial conditions; these are the measures of the constants of motion, the magnet ic helicity, and the ﬂuid plus cross helicity or some linear combination thereof [53, 52,57,38]. In our calculations, aanddwill be considered as given quantities. The existence of two , rather than one (as in the standard relaxed equilibria) parameter in this theor y is an indication that we may have, already, found an extra clue to answer the extremely im portant question: why do the coronal structures have a variety of length scales, and w hat are the determinants of these scales? We also have the parameter α0, the ratio of the ion skin depth to the solar radius. For typical densities of interest ( ∼(107−109)cm−3), its value ranges from ( ∼10−7−10−8); a very small number, indeed. Let us also remind ourselves tha t the\|∇\|is normalized to the inverse solar radius. Thus \|∇\|of order unity will imply a structure whose extension is of the order of a solar radius. To make further discussion a little more concrete, let us suppose that we are interested in investigating a structure that has a span ǫR⊙, where ǫ is a number much less than unity. For a structure of order 1000 km,ǫ∼10−3. The ratio of the orders of various terms in Eq. (25) are ( \|∇\| ∼ L−1) α2 0 ǫ2:α0 ǫ/parenleftBig 1 a−d/parenrightBig :/parenleftBig 1−d a/parenrightBig (1) (2) (3). (28) Of the possible principal balances, the following two are re presentative: 23(a) The last two terms are of the same order, and the ﬁrst ≪them. Then ǫ∼α01/a−d 1−d/a. (29) For our desired structure to exist ( α0∼10−8forn0∼109cm−3), we must have 1/a−d 1−d/a∼105, (30) which is possible if d/atends to be extremely close to unity. For the ﬁrst term to be negligible, we would further need α0 ǫ≪1 a−d⇒ǫ≫10−8 1/a−d, (31) which is easy to satisfy as long as neither of a≃dis close to unity. This is, in fact, the standard relaxed state, where the ﬂows are not supposed to pl ay an important part for the basic structure. For extreme sub–Alfv´ enic ﬂows, both aanddare large and very close to one another. Is the new term, then, just as unimportant as i t appears to be? The answer is no; the new term, in fact, introduces a qualitative ly new phenomenon: Since ∇ ×(∇ ×b) is second order in \|∇\|, it constitutes a singular perturbation of the system; its eﬀect on the standard root (2) ∼(3)≫(1) will be small, but it introduces a new root for which the \|∇\|must be large corresponding to a much shorter length scale (l arge\|∇\|). Foraanddso chosen to generate a 1000 km structure for the normal root, a possible solution would be d/a∼1 + 10−4,d≃a=−10 , then the value for \|∇\|for the new root will be (the balance will be from the ﬁrst two terms) \|∇\|−1∼102cm, that is, an equilibrium root with variation on the scale of 10 0 cm will be automatically introduced by the ﬂows. The crucial lesson is that even if the ﬂows are relatively weak (a≃d≃10), the departure from ∇ ∇∇×B=αB, brought about by the double curlterm can be essential because it introduces a totally diﬀerent and sm all scale solution. The small scale solution could be of fundamental importance in unders tanding the eﬀects of viscosity on the dynamics of these structures; the dissipation of thes e short scale structures may be the source of primary plasma heating. 24We do understand that to properly explain the parallel (to th e ﬁeld–line) motion one must use kinetic theory since the mean free path along Blines can become of the order of (106−107)cmfor the hot plasma (100 eV). But since the dissipation acts on the perpendicular energy of the ﬂow, we expect the two–ﬂuid theo ry to give qualitatively (and even quantitatively) correct results. We would like to remind the reader that by manipulating the fo rce free state ∇×B= α(x)B, Parker has built a mechanism for creating discontinuities (short scales) (Parker 1972, 1988, 1994). It is important to note that short length s cales are automatically there if plasma ﬂows are properly treated. (b) The other representative balance arises when we have a co mplete departure from the one–parameter, conventional relaxed state. In this cas e, all three terms are of the same order. In the language of the previous section, this bal ance would demand ǫ∼α01 1/a−d∼α01/a−d 1−d/a(32) which translates as:/parenleftbigg1 a−d/parenrightbigg2 ∼1−d a(33) and 1 a−d∼α01 ǫ. (34) For our example of a 1000 km structure, α0·1/ǫ∼10−5, both aanddnot only have to be awfully close to one another, they have to be awfully close to unity. To enact such a scenario, we would need the ﬂows to be almost perfectly Alfv´ enic. However, let us think of structures which are on the km or 10 km size. In that case α0·1/ǫ∼10−2or 10−3, and then the requirements will become less stringent, altho ugh the ﬂows needed are again Alfv´ enic. At a density of (1 −4)·108cm−3, and a speed ∼(200−300) km/s, the ﬂow becomes Alfv´ enic for B0∼(1−3) G. It is possible that the conditions required for such ﬂows may pertain only in the weak magnetic ﬁeld regions. Following are the obvious characteristics of this class of ﬂ ows: (1) Alfv´ enic ﬂows are capable of creating entirely new kind s of structures, which are quite diﬀerent from the ones that we normally deal with. Noti ce that here we use the term ﬂow to denote not the primary emanations but the plasmas that constitute the existing coronal structures, or the structures in the making. 25(2) Though they also have two length scales, these length sca les are quite comparable to one another: This is very diﬀerent from the extreme sub–Al fv´ enic ﬂows where the spatial length–scales are very disparate. (3) In the Alfv´ enic ﬂows, the two length scales can become co mplex conjugate, i.e., which will give rise to fundamentally diﬀerent structures i nbandV. Deﬁning p= (1/a−d) and q= (1−d/a), Eq. (27) can be factorized as (α0∇ × − λ)(α0∇ × − µ)b= 0 (35) where λ(λ+) and µ(λ−) are the solutions of the quadratic equation α0λ±=−p 2±/radicalBigg p2 4−q. (36) IfGλis the solution of the equation ∇ ×G(λ) =λG(λ), (37) then it is straightforward to see that b=aλG(λ) +aµG(µ), (38) where aλandaµare constants, is the general solution of the double curlequation. Using Eqs. (23’), (37), and (38), we ﬁnd for the velocity ﬁeld V=b a+α0∇ ∇∇×b=/parenleftbigg1 a+α0λ/parenrightbigg aλG(λ) +/parenleftbigg1 a+α0µ/parenrightbigg aµG(µ). (39) Thus a complete solution of the double curlequation is known if we know the solution of Eq. (37). This equation, also known as the ‘relaxed–state ’, or the constant λBeltrami equation, has been thoroughly investigated in literature ( in the context of solar astro- physics see for example Parker (1994); Priest (1994)). We sh all, however, go ahead and construct a class of solutions for our current interest. The most important issue is to be able to apply boundary conditions in a meaningful manner. We shall limit ourselves to constructing only two–dimensio nal solutions. For the Carte- sian two–dimensional case ( zrepresenting the radial coordinate and xrepresenting the direction tangential to the surface, ∂/∂y = 0) we shall deal with sub–Alfv´ enic solutions only. This is being done for two reasons: 1) The ﬂows in a major ity of coronal structures 26are likely to be sub–Alfv´ enic, and 2) this will mark a kind of continuity with the literature. The treatment of Alfv´ enic ﬂows will be left for a future publ ication. We recall from earlier discussion that extreme sub–Alfv´ en ic ﬂows are characterized by a∼d≫1. In this limit, the slow scale λ∼(d−a)/α0d a, and the fast scale µ=d/α0, and the velocity ﬁeld becomes V=1 aaλGλ+daµG(µ) (40) revealing that, while, the slowly varying component of velo city is smaller by a factor (a−1≃d−1) as compared to the similar part of the magnetic ﬁeld, the fas t varying component is a factor of dlarger than the fast varying component of the magnetic ﬁeld! In a magnetoﬂuid equilibrium, the magnetic ﬁeld may be rathe r smooth with a small jittery (in space) component, but the concomitant velocity ﬁeld ends up having a greatly enhanced jittery component for extreme sub–Alfv´ enic ﬂows (Alfv´ en speed is deﬁned w.r. to the magnitude of the magnetic ﬁeld, which is primarily smo oth, and for consistency we will insure that even the jittery part of the velocity ﬁeld re mains quite sub–Alfv´ enic). We shall come back to elaborate this point after deriving expre ssions for the magnetic ﬁelds. Equation (37) can also be written as ∇2G(λ) +λ2G(λ) = 0, (41) and solving for one component of G(λ) determines all other components up to an inte- gration. For the boundary value problem, we will be interest ed in explicitly solving for thez(radial) component. The simplest illustrative problem we solve is the boundary v alue problem in which we specify the radial magnetic ﬁeld bz(x, z= 0) = f(x), and the radial component of the velocity ﬁeld Vz(x, z= 0) = v0g(x), where v0(≃d−1≪1) is explicitly introduced to show that the ﬂow is quite sub–Alfv´ enic. A formal solution o f (Gz(λ) =Qλ) ∂2Qλ ∂x2+∂2Qλ ∂z2+λ2 α2 0Qλ= 0 (42) may be written as Qλ=/integraldisplay∞ λ/α0dk e−κλzCkeikx+/integraldisplayλ/α0 0dkcosqλz Akeikx+ c.c. (43) 27where κλ= (k2−λ2/α2 0)1/2,qλ= (λ2/α2 0−k2)1/2, and CkandAkare the expansion coeﬃcients. The equivalent quantities for Qµareκµ,qµ,Dk, and Ek. The boundary conditions at z= 0 yield (we absorb an overall constant in the magnitude of bz, and aµ/aλis absorbed in DkandEk): f(x) =Qλ(z= 0) + Qµ(z= 0), (44a) v0g(x) =1 aQλ(z= 0) + d Qµ(z= 0). (44b) Taking Fourier transform (in x) of Eq. (44), we ﬁnd, after some manipulation, that ( v0∼ d−1,\|/tildewidef(k)\| ≃ \|/tildewideg(k)\|) Ck≃/tildewidef(k), (45) Dk≃ −/tildewidef(k) d2+v0 d/tildewideg(k)≃d−2/tildewidef(k), (46) and functionally (in their own domain of validity) Ck=AkandDk=Ek. With the expansion coeﬃcients evaluated in terms of the known functi ons (their Fourier transforms, in fact), we have completed the solution for bz, Vzand hence of all other ﬁeld components. The most remarkable result of this calculation can be arrive d at even without a nu- merical evaluation of the integrals. Although/tildewidef(k) and/tildewideg(k) are functions, we would assume that they are of the same order \|/tildewidef(k)\|=\|/tildewideg(k)\|. Then for an extreme sub–Alfv´ enic ﬂow (\|V\| ∼d−1∼0.1, for example), the fastly varying part of bz(Qµ) is negligible (∼d−2= 0.01) compared to the smooth part ( Qλ). However, for these very parameters, the ratio/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleVz(µ) Vz(λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≃\|Ck/a\| \|d Dk\|≃\|Ck/a\| \|Ck/d\|≃1; (47) the velocity ﬁeld is equally divided between the slow and the fast scales. We believe that this realization may prove to be of extreme importance t o Coronal physics. Viscous damping of this substantially large as well as fastly varyin g ﬂow component may provide the bulk of primary heating needed to create and maintain the bright, visible Corona. The preceding analysis warns us that neglecting viscous ter ms in the equation of motion may not be a good approximation until a large part of th e kinetic energy has been dissipated. It also appears that the solution of the basic he ating problem may have to be sought in the pre–formation rather than the post–formati on era. Our time dependent 28numerical simulation to study the formation of coronal stru ctures was strongly guided by these considerations. It is evident that for extreme sub–Alfv´ enic ﬂows, the magne tic ﬁeld, unlike the velocity ﬁeld, is primarily smooth. But for strong ﬂows, the magnetic ﬁelds may also develop a substantial fastly varying component. In that case the res istive dissipation can also become a factor to deal with. We shall not deal with this probl em in this paper. Depending upon the choice of f(x) (from which/tildewidef(k) follows) we can construct loops, arcades and other structures seen in the corona. 4 Conclusions and discussions In this paper we have investigated the conjecture that the st ructures which comprise the solar corona (for the quiescent Sun) owe their origin to part icle (plasma) ﬂows which enter the “coronal regions” from lower altitudes. These primary e manations (whose eventual source is likely to be the sun itself) provide, on a continuou s basis, much of the required material and energy which constitutes the corona. From a gen eral framework describing a plasma with ﬂows, we have been able to “derive” several of th e essential characteristics of the typical coronal structures. The principal distinguishing component of the investigate d model is the full treatment accorded to the velocity ﬁelds associated with the directed plasma motion. It is the interaction of the ﬂuid and the magnetic aspects of the plasm a that ends up creating so much diversity in the solar atmosphere. This study has led to the following preliminary results: 1. By using diﬀerent sets of boundary conditions, it is possi ble to construct various kind of 2D loop and arcade conﬁgurations. 2. In the closed magnetic ﬁeld regions of the solar atmospher e, the primary ﬂows can accumulate, in periods of a few minutes, suﬃcient material t o build a coronal structure. The ability of the supersonic ﬂows to generate shocks, and th e viscous dissipation of these shocks can provide an eﬃcient and suﬃcient source for the pri mary plasma heating which may take place simultaneously with the accumulation. The st ronger the spatial gradients 29of the ﬂow, the greater is the rate of dissipation of the kinet ic energy into heat. The hot base of the structures is reached at typical distances of a ∼10000 km from the origin of simulation. 3. A theoretical study of the magnetoﬂuid equilibria reveal that for extreme sub– Alfv´ enic ﬂows (most of the created corona ﬂows) the velocit y ﬁeld can have a substantial, fastly varying (spatially) component even when the magneti c ﬁeld may be mostly smooth. Viscous damping associated with this fast component could b e a major part of the pri- mary heating needed to create and maintain the bright, visib le coronal structure. The far–reaching message of the equilibrium analysis is that ne glecting viscous terms in the equation of motion may not be a good approximation until a lar ge part of the kinetic energy in the primary ﬂow has been dissipated. 4. The qualitative statements on plasma heating, made in poi nts 1 and 2, were tested by a numerical solution of the time–dependent two-ﬂuid syst em. For sub–Alfv´ enic pri- mary ﬂows we ﬁnd that the particle-accumulation begins in th e strong magnetic ﬁeld regions (near the solar surface), and soon spans the entire v olume of the closed magnetic ﬁeld region. It is also shown that, along with accumulation, the viscous dissipation of the kinetic energy contained in the primary ﬂows heats up the acc umulated material to the observed temperatures, i.e., in the very ﬁrst (and fast, ∼(2−10)min) stage of accumu- lation, much of the ﬂow kinetic energy is converted to heat. T his happens within a very short distance (transition region) of the solar surface ∼0.03R⊙. In the transition region, the ﬂow velocity has very steep gradients. Outside the trans ition layer the dissipation is insigniﬁcant, and in a very short time a nearly uniform (with insigniﬁcantly decreasing density and temperature on the radial distance), hot and bri ght quasi-equilibrium coronal structure is created. In this newborn structure, one ﬁnds ra ther weak ﬂows. One also ﬁnds downﬂows with their parameters determined by the initi al and boundary conditions. The transition region from the solar surface to this equilib rium coronal structure is also characterized by strongly varying (both radial and acr oss) temperature and density. Depending on the initial magnetic ﬁeld , the base of the hot re gion (of the bright part) of a given structure acquires its appropriate density and te mperature. 5) The details of the ensuing dynamics are strongly dependen t on the relative values of 30the pressure of the initial ﬂow, and of the ambient solar magn etic ﬁeld in the region. Two limiting cases were studied with the expected results: 1) Th e ﬂow entering a relatively weak initial magnetic ﬁeld strongly deforms (and in speciﬁc cases drags) the magnetic ﬁeld lines, and 2) the ﬂow interacting with a relatively stro ng magnetic ﬁeld leaves it virtually unchanged. We end this paper with several qualifying remarks: 1) This study, in particular the numerical work, is prelimin ary. We hope to be able to extend the numerical work to make it considerably more quant itative, and to cover a much greater variety of the initial and boundary conditions to si mulate the immense coronal diversity. Then a thorough comparison with observations ca n be undertaken. To show the dissipation of small scale velociy component just like t he dissipation of shock–like structures is postponed for future since it requires much hi gher resolution. 2) This paper is limited to the problem of the origin, the crea tion and the primary heating of the coronal structures. The processes which may g o on in the already existing bright equilibrium corona (secondary or supporting heatin g, instabilities, reconnection) etc., for example, are not considered. Because of this lack o f overlap between our model and the conventional coronal heating models, we do not ﬁnd it meaningful to compare our work with any in the vast literature on this subject. Led b y observations alone, we have constructed and investigated the present model. 3) We do not know much about the primary solar emanations on wh ich this entire study is based. The merit of this study, however, is that as lo ng as they are present (see e.g. [1–3]), the details about their origin are not crucial. 4) We are just beginning to derive the consequences of accord ing a co–primacy (with the magnetic ﬁeld) to the ﬂows in determining overall plasma dynamics. The addition of the velocity ﬁelds (even when they are small) brings in ess ential new physics, and will surely help us greatly in understanding the richness of the p lasma behavior found in the solar atmosphere. 31Acknowledgments The work was supported in part by the U.S. Dept. of Energy Cont ract No. DE-FG03- 96ER-54346. The study of K.I.N. was supported in part by Russ ian Fund of Fundamental Research (RFFR) within a grant No. 99–02–18346. The work of N .L.S. was supported in part by the Joint INTAS–Georgia call–97 grant No. 52. S.M. M and N.L.S are also thankful to the Abdus Salam International Center for Theore tical Physics at Trieste, Italy. 32References 1. C.J. Schrijver, A.M. Title, T.E. Berger, L. Fletcher, N.E . Hurlburt, R.W. Nightin- gale, R.A. Shine, T.D. Tarbell, J. Wolfson, L. Golub, J.A. Bo okbinder, E.E. Deluca, R.A. McMullen, H.P. Warren, C.C. Kankelborg, B.N. Handy and B. DePontieu, So- lar Phys., 187, 261 (1999). 2. M.J. Aschwanden, T.D. Tarbell, R.W. Nightingale, C.J. Sc hrijver, A. Title, C.C. Kankelborg, P. Martens and H. P. Warren, Astrophys. J., 535, 1047 (2000). 3. L. Golub, J. Bookbinder, E. DeLuca, M. Karovska, H. Warren , C.J. Schrijver, R. Shine, T. Tarbell, A. Title, J. Wolfson, B. Handy and C. Kanke lborg, Phys. Plas- mas,6, 2205 (1999). 4. J.M. Beckers, Ann. Rev. 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Fluids, 26, 3540 (1983). 37Figure Captions Fig. 1 Contour plots for the vector potential A(ﬂux function) in the x−zplane for a typical arcade–like solar magnetic ﬁeld (initial distribu tion). The ﬁeld has a maximum Bmax(x0= 0, z0= 0) = 7 G. Fig. 2 Hot coronal structure formation by the interaction of the sp atially homogenuous pri- mary ﬂows with 2D arcade–like structure given in Fig. 1. The i nitial parametrs are: Vz0= 300 km/s, the temperature and density of the ﬂow, T0= 3eVandn0= 4·108cm−3 respectively, and the background density = 108cm−3. The vector potential A, the ﬂow density n(normalized to n0), the ﬂow temperature T(ineV) and the magnitude of the ﬂow velocity \|V\|(incm/s) are plotted for t= 750 sandt= 1400 s. The base of the hot structure is created at a radial distance ∼14000 km. The distnace scale on the plots is 1 = 4 ·1010cm. The primary heating (and brightening) of the structure is p ractically stopped in about 23 minutes. Fig. 3a The distribution of the radial component Vz(with a maximum of 300 km/s att= 0) for the symmetric, spatially non–uniform velocity ﬁeld . Th e plot scale is 1 = 5 ·109cm. The process of interaction of such primary ﬂows with the arca de–like magnetic ﬁelds (given in Fig. 1 with Bmax= 20G) is accompanied by downﬂows much slower than the primary ﬂows (plot for Vzatt= 297 s). The ﬁnal parameters of downﬂows are strongly dependent on the initial and boundary conditions. Fig. 3b Hot coronal structure formation by the interaction of the in itially symmetric spatially non–uniform primary ﬂows (see plot for Vz(x, z) in Fig. 3a) with the 2D arcade–like structure given in Fig. 1. Initial parameters are: the tempe rature and density of the ﬂow, T0= 3eVandn0= 4·108cm−3respectively, the initial background density = 2 ·108cm−3, and the ﬁeld maximum Bmax(x0, z0= 0) = 20 G. The plot scale is 1 = 5 ·109cm. The primary heating is completed in a very short time ∼(2−3 )minon distances ( ∼10000 km) 38shorter than the uniform–ﬂow case when magnetic ﬁeld was wea ker. The heating is symmetric and the resulting hot structure is uniformly heat ed to 1 .6·106K. Much of the primary ﬂow kinetic energy has been converted to heat via sho ck generation. Fig. 4 The interaction of an initially asymmetric, spatially non- uniform primary ﬂow (just the right pulse from the distribution given in Fig. 3a) with a strong arcade–like magnetic ﬁeld ( Bmax(x0, z0= 0) = 20 G). Downﬂows, and the imbalance in primary heating are revealed. Fig. 5 The interaction of the time–dependent non–uniform initial ﬂow (see plot for the time– distribution of Vzin this Figure; the spatial distribution of the pulse is the s ame as in Fig. 3a) with the arcade–like magnetic ﬁeld structure (pl ot in Fig.1 with Bmax= 20G). The emerging coronal structure has uniform distribution of temperature along the magnetic ﬁeld (plot for Tatt= 371 s) and the latter is practically undeformed during the formation and heating. 39This figure "FIG1.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1This figure "FIG5_1.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1This figure "FIG2.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1This figure "FIG5_2.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1This figure "FIG3A.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1This figure "FIG3B.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1This figure "FIG4.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0009086v1
arXiv:physics/0009087v1 [physics.acc-ph] 27 Sep 2000MEASURING AND CONTROLLING ENERGYSPREAD INCEBAF∗ G.A. Krafft, J.-C. Denard, R. W. Dickson,R. Kazimi, V. A. Lebedev,and M.G. Tiefenback TJNAF,NewportNews,VA23606,USA Abstract Ascomparedtoelectronstoragerings,oneadvantageofre- circulatinglinacsisthatthebeampropertiesattargetare no longer dominatedby the equilibriumbetweenquantumra- diativediffusionandradiationdampingbecausenewbeam is continuallyinjected into the accelerator. Thisallows t he energyspreadfromaCEBAF-typemachinetoberelatively small; the measuredenergyspread from CEBAF at 4 GeV is less than 100 parts per million accumulated over times oforderseveraldays. Inthispaper,thevarioussubsystems contributing to the energy spread of a CEBAF-type accel- eratorarereviewed,aswellasthemachinediagnosticsand controls that are used in CEBAF to ensure that a small energy spread is provided during routine running. Exam- ples of relevant developments are (1) stable short bunches emerging from the injector, (2) precision timing and phas- ing of the linacs with respect to the centroid of the beam bunches on all passes, (3) implementing 2 kHz sampling rate feedback systems for ﬁnal energy stabilization, and (4) continuous beam energy spread monitoring with opti- cal transition radiation devices. We present measurement resultsshowingthatsmallenergyspreadsareachievedover extendedperiods. Figure1: SchematicoftheCEBAF Accelerator ∗Work supported by the United States Department of Energy und er Contract No.DE-AC05-84ER40150.1 INTRODUCTION In this paper we summarize the present status on energy spread measurement and control in the Jefferson Lab nu- clear physics accelerator called CEBAF. A schematic dia- gram of CEBAF appears in Fig. 1 and a summary of rele- vant beam parameters is given in Table 1, where all sizes arermsquantities. CW beam, originating in the injector, is recirculated up to ﬁve times through each linac. The beam may be directed into up to three experimental halls simultaneously, the beam current in the halls being at the thirdsubharmonicoftheacceleratoroperatingfrequencyo f 1497 MHz. Because of the low charge-per-bunch at even the highest operating current, collective effects are not a n important source of energy spread in CEBAF. Fig. 1 lo- cates some of the feedback system hardware, discussed in Section6below. Table1: CEBAF AcceleratorParameters Item Value Unit Beam Energy 0.8-6 GeV Beam Current 180 µA/Hall Normalized rmsEmittance 1mmmrad RepetitionRate 499MHz/Hall ChargeperBunch <0.4 pC Extracted rmsEnergySpread <10−4 Transverse rmsBeam Size <100 µm Longitudinal rmsBeam Size 60(200) µm(fsec) BeamrmsAngularSpread <0.1/γ 2 SOURCESAND TYPES OFENERGY SPREADS INRECIRCULATING LINACS Because the electron beam remains in CEBAF for times that are short compared to the usual radiation damping times for the recirculation rings, the energy spread of the recirculating beam is not determined by the equilibrium deﬁned by the quantum character of the emission of syn- chrotron radiation. What effects do determine the energy spread? The sources of energy spread will be grouped into two broad categories: the single bunch energy spread which is the same for all bunchesin the train, and ﬂuctua- tionenergyspreadwhichisderivedfromﬂuctuationsinthe beamcentroidenergy. Sources of single bunch energy spread are: (1) the in- jected single bunch energy spread, (2) energy spread gen- erated by the ﬁnite phase extent of the bunchesinteractingwiththetime-dependentacceleratingﬁeld,(3)synchrotro n emission in the arcs, (4) average phase errors in the syn- chronization of the cavity RF to the beam, (5) summed phaseerrorsfromwholelinacsectionsthatarenotproperly balanced, and (6) interactions of the beam energy spread with non-zero M56in the arcs, which might cause the in- jectedbunchlengthtogrow. Sources of energy ﬂuctuationsare: (1) RF phase ﬂuctu- ationsinindividualRF cavities, (2)RF amplitudeerrorsin the individual cavities, (3) master oscillator noise, and ( 4) magnetic ﬁeld ﬂuctuationsin dipole magnets that are used for energymeasurementsby the feedbacksystem. We will address each of these potential sources of energy spread. The generalphilosophyusedat CEBAF is to use measure- ments to ensure the machine setup minimizes the single bunch energy spread, and to use feedback systems to cor- rect energyﬂuctuations. Our point-of-deliverydiagnosti cs allow us to ensure that the energy spread is under control throughoutthedurationofphysicsrunning. 3 LONGITUDINALMANIPULATIONSIN INJECTOR Ingeneralterms,thefunctionoftheinjectoristoaccelera te the electron beam to an energyhigh enoughthat the phase slip caused by different passes being at different energies is small, andto manipulatethe longitudinalphasespace of the beam in a way that minimizesthe overallextracteden- ergy spread. To solve the ﬁrst problem, 45 MeV injection energy is sufﬁcient for 4 GeV total acceleration. A simple calculationgivesguidanceoninjectionconditionsthatpr o- duce the optimal energy spread. Assume for the moment that one could phase each linac cavity for exactly maxi- mum energy gain. Then the energy of a bunch electron after leaving the accelerator is E=Einj+Egaincos(Φ) where EinjandEare the initial and ﬁnal energy, respec- tively, and Φis the phase of the electron with respect to the bunch centroid (assumed on crest at Φ = 0). Utilizing the single particle distribution functionfor the electron sat injection to performthe properstatistical averagesone ob - tains σE/E=/radicalBig σ2 E,inj/E2+σ4 Φ/2, where σEandσE,injare thermsenergy spreads after ac- celeration and at injection, respectively, and σΦis therms phase spread at injection. The ﬁrst term damps as energy is increased because the initial spread becomes a smaller part of the total, whereas the ﬁnal term does not depend on the energy because both energy and energy spread ac- cumulate at the same rate due to a non-zero bunch length. Given a certain longitudinal emittance from the source ǫl and the ﬁnal energy, there is an optimum energyspread of σE/E=/radicalbig 3/2(ǫl/E)2/3at the optimal injected bunch length of σΦ,opt= (ǫl/E)1/3. A typical measured value forǫlis 6.7 keV◦, yielding an optimal energy spread of 1.16×10−5at4GeV,withabunchlengthof σΦ= 0.18◦= 320fsec. A primaryfunctionoftheinjectoristo providealongitudinalphasespace“matched”tothisbunchlength. A way of providing this match has been developed and documented in various conference proceedings and work- shops [1, 2]. Here we concentrate on the measurements done routinely to ensure that proper bunching has been achieved. A main diagnosticused at CEBAF is to perform phasetransferfunctionmeasurements[3,4]. Thebasicidea is to phase modulate the beam at the beam chopper, with the rest of the RF phases in the accelerator held constant. By analyzing the longitudinal transfer function for its lin - ear and non-linear behavior, one has a way to ensure that thebeamlongitudinalphasespaceisbunchedinawaythat minimizesdistortioninthebunchingprocess,includingth e non-linearitiesdue to RF curvatureand higher order terms [2]. Such measurements are routinely used to restore the proper operation of the injector after machine downs, or whencertaintypesofoperationalproblemsarise. Nextwepresentasummaryviewgraphfromanothertalk at this conference, which shows that the bunch length is properly adjusted [5]. In Fig. 2, we present the bunch length as measured by the zero-phasing method [6], as a functionofcurrentoverthefulloperatingrangeoftheCE- BAF accelerator. One observes a roughly constant bunch length between 150 and 200 fsec (45 and 60 µm). This value is matched well enough that the extracted single bunchenergyspreadislessthan 1.5×10−5. 050100150200250 0 50 100 150 Beam current ( µµ µµA)bunch length (fs ) Figure2: BunchLengthvs.Beam CurrentoutofInjector What are the effects associated with breaking the as- sumption of ideal phasing? In an analysis that was used to set tolerances for the RF controls [7, 8, 9], it was demon- stratedthataslongas: (1)theuncorrelatedamplitudeerro rs in the cavities were under 2×10−4, (2) the uncorrelated phase ﬂuctuationsin the cavities were underseveral tenths of a degree, (3) the phasor sum of the gradients obtained from each cavity is purely real, and (4) the thermal drifts along the linac were stabilized to an error less than 2.6◦, thentheresultingenergyﬂuctuationsinthebeamwouldbe less than 2.5×10−5for an assumed bunchlengthof 0.3◦. Another way of stating condition (3) is that for each passthroughtheaccelerator,onewouldliketoarriveatthetime thatprovidesthecrestenergyforthewholelinac. Next,we discusshowthisconditionisachievedin practice. 4 PATHLENGTH AND M56 Supposeforthemomentthatthephaseofonepassthrough one linac was off crest by Φeradians. Then the relative energyspreadgeneratedbythiserroris σe/E=σΦΦe/10, the factor of ten appearing because we have assumed one linac pass is not phased properly out of ten linac passes total. To have the resulting energy spread at 10 ppm, one needs the phase error to be less than 35mrad = 2◦for a bunchlengthof300fsec. Likewise, supposethatwe requirelessthan10%growth in the bunch length going througheach arc of CEBAF. By a statistical argument, there will be less than 30% bunch length growthafter going throughthe nine arcs of the CE- BAF accelerator. Given a beam energy spread less than 10−4,theM56ofthearcsshouldnotexceed10cm,afairly weaklimitation. Presently, the apparatus in routine use to perform this measurement is based on measuring the time-of-arrival of each separate beam pass with a longitudinal pickup cavity tunedtothebeamfundamental,whoseoutputismixedwith the master oscillator in a phase detector arrangement[10]. The development of this device from ﬁrst experiments to ﬁnal instrumentis documentedin severalParticle Acceler- ator Conference contributions [10, 11, 12]. Because only relative times-of-arrival are required, the precision of t he method is very high. With 4.2 µsec 4µA beam pulses, a precision of 0.1◦= 185 fsec is routinely achieved. Such precision is clearly sufﬁcient for setting the path length, andallows M56ofthearcstobedeterminedtounder3cm by an energy modulation experiment where the energy is changedby 2×10−3. 5 MASTER OSCILLATOR MODULATIONSYSTEM There is a signiﬁcant limitation in the present system used to set the path length. Path lengthchecksmust be donein- vasively to normal beam delivery, by going into a pulsed beam mode. It would be far better to have a method to monitor the linac phases, including higher passes, contin- uously and accurately. During the last few years a system has been developed that will allow continuous monitoring and cresting of the linacs on all passes [13]. This system had its origin in an automatic beam-based linac cresting routine [14], and it is already used routinelyto set the ﬁrst passthrougheachlinacclose tocrest. The system takes advantage of the CW electron beam delivered by CEBAF and standard lockin techniques. It is based on phase modulating the master reference going to each of the linacs, at 383Hz for measurementsof the ﬁrst, so-called north linac, and at 397 Hz for the south linac. Simultaneously and coherently with the modulations, oneobserves the position motion on a beam position monitor (BPM) downstream of both linacs at a point of non-zero dispersion. Linaccrestingcorrespondsto zerooutputfrom the BPM at the modulation frequency. Long integration times permit cresting to be performedwith high precision. Therequiredphasemodulationissmallenoughthattheen- ergyspreadgeneratedbytheditherremainssmall. Table 2 summarizes the system parameters and perfor- manceoftheMasterOscillatorModulationsystem. Itsper- formance, especially in the next step in setting the higher passbeamsclosetocrest,shouldallowustoreducetheen- ergy spread of the extracted beam by roughly a factor of two. Table2: Master OscillatorModulationSystem Item Value Unit ModulationAmplitude 0.05 1497MHz◦ ModulationFrequencies 383,397 Hz Sensitivity >6000 µV/◦ OperatingCurrent >2 µA Dispersionat BPM 1.4 m MeasurementPrecision <0.11497MHz◦ 6 FAST FEEDBACK SYSTEM As mentionedin the introduction,sourcesofbunchenergy centroidﬂuctuationsarecorrectedbyafastdigitalfeedba ck system[15]. Thesystemcorrectsbeampositionandenergy near the targets of the nuclear physics experiments utiliz- ing energymeasurementsobtainedfrom the bendmagnets whichdeliverthebeamtothevarioushalls,seeFig.1. The system is capable of suppressing beam motion in the fre- quency band from 0 to 80 Hz and also performs narrow band suppression at the ﬁrst twelve power line harmon- ics. The system operateswith a 2.1 kHz samplingrate and utilizes two VME board computersto computethe correc- tions. Energycorrectionsare fed back as analogue signals tothegradientsetpointsintheRFcontrolsofafewcavities in thelinaccalledverniercavities. ForthestandardopticsinCEBAF, thehorizontaldisper- sion is maximumin the middleof the bend magnetsdeliv- eringthebeamtothehalls. Itsvalueisapproximately4m, meaning position ﬂuctuations at 10 ppm correspond to 40 µm of beam motion. The feedback system suppresses the ﬂuctuations to around 20 µm, limited by BPM noise [15]. The beam noise to be corrected is primarily at frequencies of60Hz anditsﬁrst few harmonics. Because the energy information is so closely tied to the magnetic ﬁelds in the beam delivery lines to the halls, a question arises about the stability of the magnetic ﬁelds themselves at the 10−5level. The total magnetic ﬁeld at several points within the magnets have been veriﬁed to be stable to 10 ppm, and the power supplies deliver current having similarly small ﬂuctuations. Recently, we have in- stalled a magnetic ﬂux loop monitor through the dipolestrings to the halls. This monitorwill providebetter quan- titative information than we currently possess on residual ﬂuctuationsin the magnets,and will be able to addressthe issue ofmagnetstability directly. 7 ENERGY SPREAD DIAGNOSTICS The accelerator is equipped with slow wire scanners us- ing 22 µm diameter tungsten wires. Beam proﬁles for currents in the 2 to 5 µA range can be accurately mea- sured once a minute with such scanners. More recently, wehavedevelopedaproﬁlemonitorthatcanmeasureeven the mostintensebeamsusingforwardopticaltransitionra- diation (OTR) [16, 17]. A very thin (1/4 µm) carbon foil insertedintothebeampathisnotinvasivetophysicsexper- iments for most CEBAF energies and currents. Presently, OTR monitors are installed in each of the Experimental Hall A and C beam transport lines at the high dispersion points of the beam optics. These monitors provide the ex- perimentsandtheacceleratorwith5Hzmeasurementrates for each instrument by using a common image process- ing hardware. A dedicated software, developed under the EPICS [18] control system, multiplexes up to four video inputchannelsconnectedtoasingleMaxVideoimagepro- cessing board [19]. The global processing speed is 10 Hz, 5 Hz for each of the two OTR monitors. The Hall A OTR measures beams in the 1 to 180 µA operational range; in Hall C, the dynamic range extends down to 0.1 µA. The resolutionof these monitorsis limited by the CCD camera toabout2pixels. Thisamountstoapproximately70 µmof rmsbeamsize. 050100150200 0 20 40 60 80 100 120 Beam Current in µArms Beam Sizes in µmHorizontal beam size Vertical Beam Size Figure 3: Beam Size vs. Beam Current at high dispersion pointinbeamdeliveryline. Fig. 3 shows that the energy spread is relatively stable and below 4×10−5for a wide range of beam currents. The horizontal size, measured at the 4 m dispersion point, is mostly due to the energy spread. Neglecting the beta- tron beam size, 40 µm, and the camera resolution, 70 µm,overestimatestheactualenergyspreadbylessthan25%. Continuous small energy spread became an operational requirement at CEBAF in Dec. 1999, for a hypernuclear experiment housed in Hall A, and continued until May 2000 with a similar experiment in Hall C. Both experi- ments ran simultaneously during one month last March, with 2-pass beam for Hall A and a 4-pass beam for Hall C. Delivering two beams with tight energy spread and en- ergy stability requirementsinstead of one proveddemand- ing. The energy requirements for each experiment were similar: dp/p≤5×10−5,withenergystabilitybetterthan 1×10−4. In addition, Hall A needed the transverse beam sizes at the target to be less than 200 µm but greater than 100µmandabeampositionstable within250 µm. The energy spread requirements have been routinely achieved for the hall under feedback control. Because the feedbacksystemcancorrecttheenergyﬂuctuationsonlyin a single hall as presentlyconﬁgured,there were uncertain- ties that the spread in the other hall would remain small. Fig. 4 shows energy spreads and relative energies in the Hall C beam recorded over a 2-week period, with Hall C transport line providing the energy corrections to the en- ergy feedback system. Small energy spreads were deliv- ered throughout the period to Hall C, however drifts led to energy spread increases in the Hall A beam, as seen in Fig. 5. Hall C Beam 00.40.81.21.6 23-Mar 25-Mar 27-Mar 29-Mar 31-Mar 2-Apr 4-Apr TimeX and sigma X in mmX Position => relative energy rms X width => Energy Spread dE/E = 1E-4 Figure4: HorizontalpositionandsizeofHall Cbeamdur- ing delivery period. Note that the Hall C beam line pro- videdtheenergylockingdata. Throughout the experiments in either hall, the energy spread and stability of both beams were continuously recorded. TheOTRmonitorshavebeencriticalinthistask. Theywereinitiallytoocumbersometobeeasilyusedbyall operatorcrews. Theimplementationofscriptsthatperiodi - callycheckandadjustthecameraillumination,thatinitia l- ize the imageprocessingboardaccordingto the beam,and thatseta“datavalid”ﬂagquicklyimprovedtheinstrument availability to 95% [19]. After these improvements, theHall A Energy spread and Energy Stability for 2 Weeks 00.40.81.21.6 23-Mar 25-Mar 27-Mar 29-Mar 31-Mar 2-Apr 4-Apr TimeX and sigma X in mmX position => relative Energy rms X Width => Energy spread dE/E = 5E-5dE/E = 1 E-4 Figure5: HorizontalpositionandsizeofHallA beamdur- ingdeliveryperiod. Notethedegradationofthespreadwith time due touncorrecteddrifts. Evenwithdrifts, the spread is remarkablysmall. machine crews were able to correct quickly unacceptable energyspreads,usuallywithoutinterruptingbeamdeliver y. We are planning to improve the energy spread monitor- ing for two reasons: 1) At lower energies( <1.2 GeV), the beamcurrenthadtobeloweredtounder50 µAtohaveac- ceptableradiationlevelsonsensitivebeam-lineequipmen t. 2) Experiments scheduled in 2002 require monitoring an energy spread of 2×10−5. As an alternate to OTR moni- toring,weareplanningtousesynchrotronlightbeammon- itoring, which is less invasive to the experimenters. How- ever,theresolutionofsuchadeviceislimitedtoabout100 µm in the visible using the bending magnets of the hall transport lines. We are starting a developmenteffort in or- der to reach about 30 µm resolution utilizing the UV syn- chrotronemission. 8 CONCLUSIONS We have demonstrated the ability of a CEBAF-type accel- erator to produce beams with small energy spreads over long periodsof time. We ensure that the energyspread re- mains small by: (1) ensuring the bunch length out of the injector is small, (2) ensuring that the beam remains close tothecrestphaseoneachseparatepass(sooncontinuously andautomatically!),and(3)providingcontinuousfast cor - rection of 60 cycle harmonic noise on the beam. We have developed beam diagnostic devices to continuously moni- tor and record beam conditionswith 5 Hz update rates us- ingdigitizationofmultiplevideomonitors. 9 REFERENCES [1] R. Abbott, et. al., “Design, Commissioning, and Operation of the Upgraded CEBAF Injector”, Proc. of the 1994 Linac Conf., 777(1994)[2] G. A. Krafft, “Correcting M56andT566to obtain very short bunches at CEBAF”, Proc. of the Microbunches Workshop, AIPConference Proceedings 367, 46 (1996) [3] C.G.Yao,“ANewSchemeforMeasuringtheLengthofVery Short Bunches at CEBAF”, Proc. of the 1990 Beam Instru- mentation Workshop, AIP Conference Proceedings 229, 254 (1990) [4] G. A. Krafft, “Status of the Continuous Electron Beam Ac- celerator Facility”,Proc.of the 1994 Linac Conf.,9(1994) [5] R.Kazimi,C.K.Sinclair,andG.A.Krafft,“SettingandM ea- suring the Longitudinal Optics in CEBAF Injector”, these proceedings [6] D. X. Wang, G. A. Krafft, and C. K. Sinclair, “Measurement of Femtosecond Electron Bunches Using a RF Zero-phasing Method”, Phys.Rev. E, 57, 2283 (1998) [7] A detailed calculation is performed in an internal note, G.A.Krafft,J.J.Bisognano,andR.Miller,“RevisedEnergy SpreadEstimate”,CEBAF-TN-0050(1987).Asimilarcalcu- lationisperformedinL.MermingaandG.A.Krafft,“Energy Spread from RF Amplitude and Phase Errors”, Proc. of the 1996 European Part.Acc.Conf., 756 (1996) [8] G. A. Krafft, et. al., “Energy Vernier System for CEBAF”, Proc. of the 1993 Part. Acc. Conf., 2364 (1993) summarizes the results inthe internal note [7] above. [9] S. N. Simrock, et. al., “Operation of the RF Controls in the CEBAF Injector”, Proc. of the 2nd European Part.Acc.Conf., 824 (1990) [10] Y.Chao, et.al.,“CommissioningandOperationExperience withtheCEBAFRecirculationArcBeamTransportSystem”, Proc.of the 1993 Part.Acc. Conf.,587 (1993) [11] G. A. Krafft, et. al., “Measuring and Adjusting the Path Length at CEBAF”, Proc.of the 1995 Part.Acc. Conf., 2429 (1995) [12] D. Hardy, et. al., “Automated Measurement of Path Length andM56” Proc.of the 1997 Part.Acc.Conf., 2265 (1997) [13] V. A. Lebedev, J. Musson, and M. G. Tiefenback, “High- precision Beam-based RF Phase Stabilization at Jefferson Lab”, Proc.of the 1999 Part.Acc.Conf., 1183 (1999) [14] M. G.Tiefenback andK. Brown,“Beam-based Phase Mon- itoring and Gradient Calibration of Jefferson Laboratory R F Systems”, Proc.of the 1997 Part.Acc.Conf., 2271 (1997) [15] R. Dickson andV. A.Lebedev, “Fast DigitalFeedback Sys - temforEnergyandBeamPositionStabilization”,Proc.ofth e 1999 Part.Acc.Conf., 646 (1999) [16] P. Piot, et. al., “High Current CW Beam Proﬁle Monitors Using Transition Radiation at CEBAF”, Proc. of the 1996 Beam Instrumentation Workshop, AIP Conference Proceed- ings 390, 298-305 (1997) [17] J.-C. Denard, et. al., “High Power Beam Proﬁle Moni- tor with Optical Transition Radiation”, Proc. of the 1997 Part.Acc.Conf., 2198 (1997) Vancouver, BC. [18] L. Dalesio, et. al., Proc. Int. Conf. on Accelerators and Large Experimental Physics Control Systems (ICALEPS), 278 (1992) [19] D. Hardy, et. al., “Multivideo Source Image Processing for Beam Proﬁle Monitoring System”, Proc. of the 2000 Beam InstrumentationWorkshop, AIPproceedings tobepublished .
arXiv:physics/0009088v1 [physics.gen-ph] 27 Sep 2000On the Quantum Aspects of the Logarithmic Corrections to the Black Hole Entropy Carlos Castro∗and Alex Granik† Abstract An extension of the conventional space-time to noncommutat ive Cliﬀord manifolds where all p-branes are treated on equal fo oting al- lowed authors to write a master action functional. The respe ctive functional equation is simpliﬁed and applied to the p-loop o scillator on Cliﬀord manifolds. Its solution represents a generaliza tion of the conventional quantum point oscillator which also extends t o the region of the Planck scales. In the latter the solution yields in an e lemen- tary fashion the basic relations of string theory including string tension quantization. In addition, it is shown that the degeneracy o f thefirst collective excited state of the p-loop oscillator yields no t only the well- known Bekenstein-Hawking area-entropy linear relation bu t also the logarithmic corrections therein. 1 The p-Loop Harmonic Oscillator in Clif- ford Manifolds A new relativity theory [4]-[11] introduced an extension of the ordinary space- time by considering non-commutative C-spaces where all p-branes are uniﬁed on the basis of Cliﬀord multivectors. This resulted in a full poly-dimensional covariance of the quantum mechanical loop wave equations de rived from a master action functional motivated by earlier work [1],[2] . ∗Center for Theoretical Studies of Physical Systems,Clark A tlanta University,Atlanta, GA. 30314; E-mail:castro@ts.infn.it †Department of Physics, University of the Paciﬁc, Stockton, CA.95211; E- mail:galois4@home.com 1The respective master action functional S{Ψ[X(Σ)]}of quantum ﬁeld theory in C-space [5]-[7] is S{Ψ[X(Σ)]}=/integraltext[DX(Σ)] [1 2(δΨ δX∗δΨ δX+E2Ψ∗Ψ) +g3 3!Ψ∗Ψ∗Ψ+ g4 4!Ψ∗Ψ∗Ψ∗Ψ].(1) where Σ is an invariant evolution parameter (a generalizati on of the proper time in special relativity) such that (dΣ)2= (dΩp+1)2+ Λ2p(dxµdxµ) + Λ2(p−1)dσµνdσµν+... +(dσµ1µ2...µp+1dσµ1µ2...µp+1),(2) X(Σ) = Ω p+1I+ Λpxµγµ+ Λp−1σµνγµγν+... (3) is a Cliﬀord algebra-valued line ”living” on the Cliﬀord man ifold outside space-time, Λ is the Planck scale that allows to combine obje cts of diﬀer- ent dimensionality in Eqs.(2,3) and the multivector XEq.(3) incorporates both a point history given by the ordinary ( vector) coordina tesxµand the holographic projections of the nested family of all p-loop histories onto the embedding coordinate spacetime hyperplanes ,that is σµν, ....σ µ1µ2...µp+1. The scalar ( from the point of view of ordinary Lorentz transform ations but not from the C-space point of view ) Ω p+1is the invariant proper p+ 1 = D- volume associated with a motion of a maximum dimension p-loop across the p+1 = D-dim target spacetime. The vector Xhas 2Dcomponents. In what follows we concentrate our attention on a truncated version of the theory by applying it to a linear p-loop oscillator. The truncated version follows from 3 simpliﬁcations. First , in (1) the cubic ( corresponding to vertices) and the quartic (braided scatt ering) terms are dropped. Secondly, all the holographic modes are frozen and only the zero modes are kept. This would yield conventional diﬀerential e quations instead of functional ones. Thirdly, we assume that the metric in C-s pace is ﬂat. We proceed along by properly deﬁning what we mean by ”relativ istic” based on the new relativity. The complete theory is the master ﬁeld theory whose action functional admits a noncommutative braided quantum Cliﬀord al- gebra. As a result of the postulated simpliﬁcations, we are p erforming a reduction of such a ﬁeld theory to an ordinary quantum mechan ical the- ory. It must be kept in mind that ﬁelds are notquantized wave functions. 2For this reason the wave equations that we will be working wit h refer to a nonrelativistic theory in C-spaces. Employing the above simplifying assumptions, we obtain fro m the action (1) the following wave equation describing a p-loop linear osci llator in C-space {−1 21 Λp−1[∂2 ∂xµ2+ Λ2∂2 (∂σµν)2+ Λ4∂2 (∂σµνρ)2+...+ Λ2p∂2 (∂Ωp+1)2]+ mp+1 21 L2[Λ2pxµ2+ Λ2p−2σµν2+...+ Ω2 p+1]}Ψ =TΨ(4) where in the case of a ﬂat C -space metric one can write : ∂2 (∂xµ)2=gµν∂ ∂xµ∂ ∂xν,∂2 (∂σµν)2=Gµνρτ∂ ∂σµν∂ ∂σρτ, etc, Gµνρτis some suitably symmetrized product of the two ordinary met ric- tensors gµνgρτ. It should be noted that in general this is not the case. In C-space a metric is a collection of components G={Gµν;Gµνρτ, ...} this set of generalized higher rank tensors is similar to the ones found in W-geometry [13] [22] that bear a resemblance to the maximal ac celeration of the Finsler type ( jet bundles ) geometries [3] as discusse d by C. Hull. For references see [13] . The Einsteinian gravity is an eﬀect ive ﬁeld theory representing a long range limit of a more fundamental Finsle rian ,Wgeom- etry. The latter ﬁts nicely into the framework of the extende d relativity in C-spaces due to the fact that the Planck scale is the lowest lim iting scale. In Eq.4 Tis tension of the spacetime-ﬁlling p-brane, D=p+ 1,mp+1is the parameter of dimension ( mass)p+1, the parameter L(to be deﬁned later) has dimension ( length )p+1and we use units ¯ h= 1, c= 1. We rewrite Eq.(4) in the dimensionless form as follows {∂2 ∂˜x2 µ+∂2 ∂˜σ2 µν+...−(˜Ω2+ ˜x2 µ+ ˜σ2 µν+...) + 2T }Ψ = 0 (5) where T=T mp+1√ Ais the dimensionless tension which is related (due to the linearization procedure) to the “ mass-parameter-like ” qu antity Eappearing in the action functional (1) by E2=mp+1T+ (m2 p+1Vharmonic /L2). The latter term is the harmonic potential for the p-loop oscillator. Ais a scaling parameter that will be determined below. 3˜xµ=A1/4Λp Lxµ,˜σµν=A1/4σµνΛp−1 L, ...,˜Ωp+1=A1/4Ωp+1 L are the dimensionless arguments, ˜ xµhasCD1≡Dcomponents, ˜ σµνhas CD2≡[D!]/[(D−2)!2!] components, etc. Inserting the usual Gaussian solution for the ground state i nto the wave equation (4) we get the value of A: A ≡mp+1L2/Λp+1 Without any loss of generality we can set A= 1 by absorbing it into L. This will give the following geometric mean relation betwee n the parameters L, m p+1, and Λ L2= Λp+1/mp+1⇒Λp+1< L <1 mp+1 meaning that there are three scaling regimes. The scale repr esented by gen- eralized Compton wavelength (1 /mp+1)(1/p+1)will signal a transition from a smooth continuum to a fractal ( but continuous ) geometry. The scale L will signal both a discrete andfractal world, like El Naschie’s Cantorian- fractal spacetime models [14] and p-adic quantum mechanics [21] and Λ the quantum gravitational regime. The dimensionless coordinates then become ˜xµ=/radicalBig Λp+1mp+1xµ/Λ,˜σµν=/radicalBig Λp+1mp+1σµν/Λ2, ..., ˜Ωp+1=/radicalBig Λp+1mp+1Ωp+1/Λp+1 The dimensionless combination Λp+1mp+1(which indicates existence of two separate scales , Λ and (1 /mp+1)1 p+1) obeys the following double inequality: /radicalBig mp+1Λp+1<1</radicalBigg 1 mp+1Λp+1(6) In turn, relations (6) deﬁne two asymptotic regions: 1)the ”discrete-fractal” region characterized by mp+1Λp+1∼1, or the Planck scale regime, and 2)the ”fractal/smooth phase transition ”, or the low energy region charac- terized by mp+1Λp+1<<1. 4Since the wave equation (5) is diagonal in its arguments (tha t is separable) we represent its solution as a product of separate functions of each of the dimensionless arguments ˜ xµ,˜σµν, etc. Ψ =/productdisplay iFi(˜xi)/productdisplay j<kFjk(˜σjk)... (7) Inserting (7) into (5) we get for each of these functions the W hittaker equa- tion: Z′′−(2T −˜y2)Z= 0 (8) where Zis any function Fi, Fij, ..., ˜yis the respective dimensionless variable ˜xµ,˜σµν, ..., and there are all in all 2Dsuch equations. The bounded solution of (8) is expressed in terms of the Hermite polynomials Hn(˜y) Z∼e−˜y2/2Hn(˜y) (9) Therefore the solution to Eq.(5) is Ψ∼exp[−(˜x2 µ+ ˜σ2 µν+...+˜Ω2 p+1)]/productdisplay iHni(˜xi)/productdisplay jkHnjk(˜σjk)... (10) where there are Dterms corresponding to n1, n2, ..., n D. There are D(D− 1)/2 terms corresponding to holographic area excitation modes n01, n02, ..., etc. Thus the total number of terms corresponding to the N-th excited state (N=nx1+nx2+...+nσ01+nσ02+...) is given by the degree of the Cliﬀord algebra in Ddimensions, that is 2D. The respective value of the tension of the N-th excited state is TN= (N+1 22D)mp+1 (11) yielding quantization of tension. Expression (11) is the analog of the respective value of the N-th energy state for a point oscillator. The analogy however is not complete. We point out one substantial diﬀerence. Since according to a new relativ ity principle [4], [5] all the dimensions are treated on equal footing (there ar e no preferred dimensions) all the modes of the p-loop oscillator( center of mass xµ, holo- graphic modes, p+ 1 volume) are to be excited collectively. This behavior is in full compliance with the principle of polydimensional in variance by Pez- zaglia [20]. As a result, the ﬁrst excited state is not N= 1 ( as could be naively expected) but rather N= 2D. Therefore 5T1→T2D=3 2(2Dmp+1) instead of the familiar (3 /2)m. Having obtained the solution to Eq.(5), we consider in more d etail the two limiting cases corresponding to the above deﬁned 1) fractal and 2) smooth regions. The latter (according to the correspondence princ iple) should be described by the expressions for a point oscillator. In part icular, this means that the analog of the zero slope limit in string theory, the ﬁ eld theory limit, is a collapse of the p-loop histories to a point history : Λ→0, m p+1→ ∞, T→ ∞, σµν, σµνρ, ......→0, L→0. ... and these limits are taken in such a way that the following com bination reproduces the standard results of a point-particle oscill ator : ˜xµ=xµ Λ/radicalBig mp+1Λp+1→xµ/a (12) where the nonzero parameter a >Λ is a ﬁnite quantity and is nothing but the amplitude of the usual point-particle oscillator. In string theory, there are two scales, the Planck scale Λ and the string scale ls>Λ. Without loss of generality we can assign a∼ls. A large value of a >> Λ would correspond to a “macroscopic” string. We shall retur n to this point when we address the black-hole entropy. For further de tails we refer to [12] To consider the black hole entropy we ﬁnd the degeneracy asso ciated with theN-th excited level of the p-loop oscillator. The degeneracy dg(N) is equal to the number of partitions of the number Ninto a set of 2Dnumbers N={nx1+nx2+...+nxD+nσµν+nσµνρ+...+nΩp+1}. This means that there is a collective center of mass excitations, holographic area excitations , holographic volume excitations,....et c. And the latter are given by the set of quantum numbers nxD;nσµν, ..., n Ωp+1respectively. These collective extended excitations are the truequanta of a background indepen- dent quantum gravity. Thus spacetime becomes a “ process “ in the making [2] emerging from the C-space. The degeneracy dg(N) is dg(N) =Γ(2D+N) Γ(N+ 1)Γ(2D)(13) 6where Γ is the gamma function. Based on these results it is tempting to propose that the exac t analytical expression for the entropy associated with the first collective excitation of thep-loop oscillator is given by the logarithm of the degeneracy : S=ln{Γ(2D+N) Γ(N+ 1)Γ(2D)}. N= 2D(14) where 2Dis the degree of Cliﬀord algebra. In [12] in order to make cont act with the entropy of a black hole in ddimensions, and its relation to the logarithm of the degeneracy of the highly exited states of a s tring living in ddimensions, we imposed the following relation between N= 2Dand the black hole ( d−2) dimensional horizon area N= 2D=Ad−2 cdGd. G d= Λd−2 where cdis in general a dimension-dependent coeﬃcient. For example , in d= 4;cd= 4. In essence, this coeﬃcient indicates that a surface area of a two-dim sphere is 4 times the area of the holographic disk. In turn, the bits are counted in terms of holographic area, or disk units. The N ewton constant inddimensions is given by Λd−2in units ¯ h=c= 1. For black holes, the areaAis the area transverse to the radial and temporal directions, i.e. it is a (d−2)-dimensional area which precisely matches the power of th e Planck scale entering the deﬁnition of the Newton constant in ddimensions. The number N, the degree of the Cliﬀord algebra in Ddimensions, associated with the p-loop oscillator living in Ddimensions, is nothing but the number of (d−2)-dimensional horizon area bits of the black hole. Both the area and the tension quantization follow quite naturally from this theory! If we consider large values of Nand 2Dthen using Stirling formula in (14) we get the following expression for the entropy: S= 2Nln2−1 2ln N−1 2ln2−ln√ 2π−O(1/N). (15) In a particular case of d= 4 one has then N=A/4G=A/4Λ2, and we will recover not only the logarithmic corrections recently reviewed in the literature [18] but also allthe higher order corrections to the Bekenstein- Hawking entropy in d= 4 dimensions, up to numerical coeﬃcients: S= 2A 4Λ2ln2−1 2ln[A 4Λ2]−1 2ln2−ln√ 2π−(A 4Λ2)−1+... (16) 7As could be expected, Einstein’s gravity ( an eﬀective theor y) is recovered in thelongrange limit, i.e. for N >> 1, eq.(16) yields a linear dependence be- tween SandA. At smaller scales a departure from this behavior is observe d, since at Planck scales the new (extended) scale relativisti c eﬀects (and as- sociated with it Cantorian-fractal spacetime, braided Hop f quantum Cliﬀord algebras, p-adic quantum mechanics, irrational conformal ﬁeld theory , non- commutative geometry, etc.) begin to take over leading to deviations from a linear dependence S=S(A). From the entropy-area expression (16) we ﬁnd that if entropy cannot de- crease, the entropy of a black hole of an area A3(expressed in terms of N3 bits) must be not less than the sum of the entropies of two blac k holes of areas A1( expressed in terms of N1bits ) and A2(expressed in terms of N2 bits ) whose ”fusion” forms a black hole of area A3: (2ln2)(N3−N2−N1) +1 2lnN1N2 N3≥0. The respective necessary condition looks like a relation between N3, a squared mean geometric, and a double arithmetic values of N1andN2: N1N2≥N3≥N1+N2. (17) If one would deﬁne the exponential of the entropy as the number =Nof micro-physical states then we get: N=eS=e2Nln2−1 2lnN−...∼22NN−1/2=22D+1 2D/2(18) Notice the double exponents ( a “googolplexus“ ) appearing in the number of micro-physical states . For a k-th excited level of the p-loop oscillator we obtain the following ex- pression: Nk=kN1=k2D=kAd−2 cdGd=kAd−2 cdΛd−2. (19) This number may correspond to the entropy of a bound state of m any black holes. In [12] we equated the degeneracy dg(N, D) of the ﬁrst collective excited of ap-loop oscillator (14) in Ddimensions with the asymptotic quantum degeneracy of a massive (super) string state dg(n) inddimensions where n >> 1 is the very large string’s spectral level number . Li and Yoneya [16] 8have shown explicitly that the logarithm of the asymptotic ( large spectral levelnlimit ) quantum degeneracy of the massive string states coin cided precisely with the Bekentsein-Hawking Black Hole Entropy SBH∼A/G. For example, in this fashion we were able to show by ﬁxing d= 26 (bosonic string dimension ) that indeed : SBH∼Ad−2 Gd∼ln[dg(n)]∼√n∼N= 2D(17) This equality between dg(d, n) =dg(N, D) was justiﬁed by the following physical reasoning. It is possible to introduce diﬀerent fr ames in a new rel- ativity: one frame where an observer sees only the strings in d-dimensions (with a given degeneracy) and another frame where the same ob server sees a collective excitations of points, strings, membranes,... ,p-loops in D-dimensions etc. The observed degeneracy (given by a number) should be in variant in any frame. The string’s spectrum Regge behaviour also follo ws naturally in this theory [12]. This represented a rather amazing result: the Shannon entro py of a p-loop oscillator in D-dimensional space ( for a suﬃciently large D), that is the number N= 2D, the total number of bits representing allthe holographic coordinates, the degree of the Cliﬀord Algebra in Ddimensions, is propor- tional to the Bekenstein-Hawking entropy of a Schwarzschil d black hole in d dimensions. The latter is just the number of d−2 dimensional area bits in Planck units. For a detailed derivation of (17) we refer a reader to [12]. A m ore rigor- ous study of the connection between Shannon’s information e ntropy and the quantum-statistical (thermodynamical) entropy is given b y Fujikawa [15]. Because light is trapped inside, the Black Hole horizon is al so an information horizon. 2 Conclusion Application of a simpliﬁed linearized equation derived fro m the master action functional of a new ( extended) relativity to a p-loop oscillator has allowed us to elementary obtain rather interesting results. First o f all, the solution explicitly indicates existence of 2 extreme regions charac terized by the values of the dimensionless combination mp+1Λp+1: 91) the fractal region where mp+1Λp+1∼1 and 2 scales collapse to one, namely Planck scale Λ and 2) the smooth region where mp+1Λp+1<<1 and we recover the description of the conventional point oscillator. Here 2 scales are present , a characteristic ”length” athat we identiﬁed with the string scale lsand the ubiquitous Planck scale Λ ( a >Λ) thus demon- strating explicitly the implied validity of the quantum mec hanical solution in the region where a/Λ>1. For a speciﬁc case of p= 1 (a string) the solution yields ( once again in an elementary fashion) a quan tization of the sting tension Tand one of the basic relations of string theory T= 1/α′. A comparison of the degeneracy of the first collective state of the p-loop oscillator, living in a very large number of dimensions D, with the highly excited level number nof the degeneracy of the massive ( super) string theory, given by Li and Yoneya [16], demonstrated that the Shannon en tropy ( which is in agreement with the logarithm of the degeneracy of state s ) of a p-loop oscillator in D-dimensional space , that is the number N= 2D: the number of bits representing allthe holographic coordinates , is proportional to the Bekenstein-Hawking entropy of the Schwarzschild black hol e inddimensions. In the linear regime of entropy versus area, the latter is jus t the number of (d−2) dimensional area bits in Planck units. Moreover, the solution allowed us to ﬁnd ( up to numerical coe ﬃcients ) the logarithmic , and higher order corrections to the entropy-area linear la w ( in full agreement with [18]) by retaining the ﬁrst order term s in Stirling’s formula. It is also suggestive to propose that the exact anal ytical expression for the black hole entropy is associated with the first collective excitation of the p-loop oscillator and is given by Eq.(14) S=Ln{Γ(2D+N) Γ(N+ 1)Γ(2D)}, N= 2D=Ad−2 cdGd, Gd= Λd−2 where cdis a dimension-dependent coeﬃcient which for d= 4 is c4= 4 Thus a study of a simpliﬁed model ( or ”toy”) problem of a linea rizedp-loop oscillator gives us ( with the help of elementary calculatio ns) not only a set of the well-known relations of string theory but also the log arithmic correc- tion to the well-known black hole entropy-area relation ( ob tained earlier on the basis of a much more complicated mathematical technique ). This indi- cates that the approach advocated by a new relativity might b e very fruitful, 10especially if it will be possible to obtain analogous analyt ic results on the ba- sis of the full master action functional and the resulting fu nctional nonlinear equations whose study will involve Braided Hopf Quantum Cli ﬀord Algebras. Acknowledgements The authors would like to thank E.Spallucci and S.Ansoldi fo r many valuable discussions and comments. Special thanks to A. Quiroz for hi s valuable comments. References [1] S. Ansoldi, C. Castro, E. Spallucci , ” String Representa tion of Quantum Loops ” Class. Quant. Gravity 16(1999) 1833;hep-th/9809182 [2] A. Aurilia, S. Ansoldi , E. Spallucci, J. Chaos, Solitons and Fractals. 10(2-3) (1999) 197 . [3] H. Brandt : Journal of Chaos, Solitons and Fractals 10( 2-3 ) ( 1999 ) 267. [4] C. Castro , ” Hints of a New Relativity Principle from p-Brane Quantum Mechanics ” J. Chaos, Solitons and Fractals 11(11)(2000) 1721 [5] C. Castro, ”An Elementary Derivation of the Black-Hole A rea-Entropy Relation in Any Dimension ” hep-th/0004018 [6] C. Castro , ” The Search for the Origins of MTheory : Loop Quantum Mechanics and Bulk/Boundary Duality ” hep-th/9809102 [7] C.Castro, ” Is Quantum Spacetime Inﬁnite Dimensional?” J. Chaos, Solitons and Fractals 11(11)(2000) 1663 [8] C. Castro, ” The String Uncertainty Relations follow fro m the New Relativity Theory ” hep-th/0001023; Foundations of Physic s , to be published [9] C. Castro, A.Granik, ”On MTheory, Quantum Paradoxes and the New Relativity ” physics/ 0002019; 11[10] C. Castro, A.Granik, ”How a New Scale Relativity Resolv es Some Quan- tum Paradoxes”, J.Chaos, Solitons, and Fractals 11(11) (2000) 2167. [11] C.Castro, A.Granik, ”Scale Relativity in Cantorian E∞Space and Av- erage Dimensions of the World”, J.Chaos,Solitons and Fract als (2000) [12] C.Castro and A.Granik, ”P-Loop Oscillator on Cliﬀord M anifolds and Black Hole Entropy”, physics/0008222 [13] C. Castro : Journal of Geometry and Physics 33( 2000 ) 173. [14] M. S. El Naschie : Journal of Chaos, Solitons and Fractal s10( 2-3 ) ( 1999 ) 567. [15] K.Fujikawa, ”Shannon’s Statistical Entropy and the H- theorem in Quan- tum Statistical Mechanics”, cond-mat/0005496 [16] M.Li and T.Yoneya, ”Short Distance Space-Time Structu re and Black Holes”,J. Chaos, Solitons and Fractals. 10(2-3) (1999) 429 [17] L. Nottale, ”Fractal Spacetime and Microphysics :Towa rds a Theory of Scale Relativity ” World Scientiﬁc, 1993; [18] P.Majumdar, ”Quantum Aspects of Black Hole Entropy”, h ep- th/0009008 [19] L. Nottale, ”La Relativite dans tous ses Etats ” Hachett e Literature, Paris, 1998. [20] W.Pezzaglia, ”Dimensionally Democratic Calculus and Principles of Polydimensional Physics”, gr-qc/9912025 [21] M. Pitkannen : “ Topological Geometrydynamics : TGD, Mathematical Ideas “ hep-th/9506097. Book available at http://www.blues.helsinki.ﬁ/ ∼matpitka V. Vladimorov, I. Volovich and E. Zelenov : “ p-Adic Numbers i n Math- ematical Physics “ World Scientiﬁc, Singapore 1994. L. Brekke. P. Freund : Physics Reports 231(1993) 1-66 12[22] P. Prokushin , M. Vasiliev :” 3 DHigher Spin Gauge theories with Mat- ter “ hep-th/9812242. E. Sezgin, P. Sundell : “ Higher Spin N= 8 Supegravity “ hep- th/9805125. 13
arXiv:physics/0009089v1 [physics.chem-ph] 28 Sep 2000Extracting molecular Hamiltonian structure from time-dependent ﬂuorescence intensity data Constantin Brif and Herschel Rabitz Department of Chemistry, Princeton University, Princeton , New Jersey 08544 Abstract We propose a formalism for extracting molecular Hamiltonia n structure from inversion of time-dependent ﬂuorescence intensity data. The propose d method requires a minimum of a priori knowledge about the system and allows for extracting a compl ete set of information about the Hamiltonian for a pair of molecular electronic sur faces. 1 Introduction A long standing objective is the extraction of molecular Ham iltonian information from laboratory data. The traditional approaches to this problem attempt to make use of time-independent (spectroscopic and scattering) data [1, 2, 3]. Another appr oach aims to use ultrafast temporal data, with information on molecular potentials and dipole m oments obtained for spatial regions sampled by evolving wave packets. Research in this directio n has been especially intense during the last few years [4, 5, 6, 7, 8, 9]. This activity is inspired by recent progress in the technology of ultrafast laser pulses [10, 11], which makes possible obs ervations of molecular dynamics with increasingly higher spatial and temporal resolution. Due to the diﬃculty of the Hamiltonian inversion problem, it is common to assume that somea priori knowledge of the system is available. For example, one techn ique [6] proposes to extract time-evolving wave functions and excited-state potentials using time-resolved and frequency-resolved ﬂuorescence data and knowledge of the g round-state potential, the transition frequencies, and the transition dipole moment. The inverse tracking method [7], proposed for recovering the potential energy and dipole moment of a mo lecular electronic surface by monitoring the temporal evolution of wave packets, explici tly assumes knowledge of the initial excited wave functions. Clearly, such assumptions impair s elf-consistency and at least partially undermine the inversion objectives. Although the desire to simplify the inversion algorithm by making a priori assumptions about what is known and unknown is understandab le, it has remained an open question about whether these assumptions a re actually necessary. This letter addresses the latter point by proposing an inver sion formalism that makes use of minimal a priori knowledge about the system. The formalism is designed to ope rate between two electronic surfaces, with electronic and vibrational t ransitions driven by two fast laser pulses, which allows for extracting the potential energies and dipo le moments for both surfaces as well as the electronic transition dipole moment. The extraction is based on the inversion of the time-dependent ﬂuorescence intensity data obtained from t he detection of spontaneous emission in transitions between the electronic surfaces. The propos ed formalism lays the ground work for extracting a complete set of information about a pair of e lectronic surfaces in a closed way, with a minimum of a priori assumptions about the molecular Hamiltonian. This letter p resents 1the conceptual foundation of this novel approach, and a deta iled numerical algorithm with simulations will be presented elsewhere. 2 The physical picture Consider the ground and excited electronic molecular surfa ces with potential energies Vg(x) and Ve(x) and dipole moments µg(x) and µe(x), respectively. The dipole moment for the electronic transition between the two surfaces is M(x). For the sake of conceptual clarity, we consider a one-dimensional problem; the generalization for the multi dimensional case is straightforward. The setup includes two time-dependent locked laser ﬁelds: ǫ0(t) drives transitions between the two electronic surfaces (the carrier frequency of this l aser will be typically in the visible or ultraviolet part of the spectrum), and ǫ1(t) drives transitions between vibrational levels within each of the two surfaces (the carrier frequency of this laser will be typically in the infrared). The role of the driving ﬁelds is to excite the molecular wave p acket and guide its motion on the surfaces. It is physically reasonable that the potentia ls and dipole moments may be reliably extracted only in the region sampled by the evolving wave pac ket. We assume that interactions with other electronic surfaces and incoherent processes (e .g., thermal excitation and collisional relaxation) are negligible. Ultrafast laser technology ha s made great advances recently, but preparation of the infrared pulse of a desired shape is still a challenging technical problem. We will consider the general situation, with two locked drivin g ﬁelds and ﬁve unknown functions (two potentials and three dipole moments), but taking ǫ1= 0 the problem is easily reduced to a simpler one, with only one driving ﬁeld ǫ0and three unknown functions ( Vg,VeandM). The Hamiltonian of the system in the Born-Oppenheimer, elec tric-dipole and rotating-wave approximations takes the form: H=Hg(x,p,t)σgg+He(x,p,t)σee−M(x)ǫ0(t)(σeg+σge), (1) where σij=\|i/angbracketright/angbracketleftj\|(with i,j=g,e) are transition-projection operators for the electronic s tates \|g/angbracketrightand\|e/angbracketright. Here, xandpare the canonical position and momentum for the vibrational degree of freedom, HgandHeare the vibrational Hamiltonians in the ground and excited e lectronic states, Hi(x,p,t) =T+Vi(x)−µi(x)ǫ1(t), i =e,g, (2) andT=p2/2mis the kinetic energy of the vibrational motion. We assume that the initial state of the system is \|Ψ(0)/angbracketright=\|u0/angbracketright\|g/angbracketright, where u0(x) is the vibrational ground state localized in the known harmonic pa rt of the potential Vg(x). The state of the system at any time twill be of the form \|Ψ(t)/angbracketright=\|u(t)/angbracketright\|g/angbracketright+\|v(t)/angbracketright\|e/angbracketright, (3) with the normalization condition/integraltext dx/parenleftbig \|u(x,t)\|2+\|v(x,t)\|2/parenrightbig = 1. The Schr¨ odinger equation, i/planckover2pi1∂t\|Ψ(t)/angbracketright=H\|Ψ(t)/angbracketright, then takes the form i/planckover2pi1∂tu(x,t) =−/planckover2pi12 2m∂2 xu(x,t) + [Vg(x)−µg(x)ǫ1(t)]u(x,t)−M(x)ǫ0(t)v(x,t),(4) i/planckover2pi1∂tv(x,t) =−/planckover2pi12 2m∂2 xv(x,t) + [Ve(x)−µe(x)ǫ1(t)]v(x,t)−M(x)ǫ0(t)u(x,t), (5) with the initial conditions u(x,0) =u0(x),v(x,0) = 0. 2The radiation emitted spontaneously by the molecule via tra nsitions between the excited and ground electronic surfaces contains information about the wave packet. This fact has been used to reconstruct unknown vibrational wave packets in the meth od of emission tomography [13]. Our aim is diﬀerent: we assume that the initial state of the sy stem is known and want to extract the unknown potentials ( VgandVe) and dipole moments ( µg,µe, and M) from information contained in the time-dependent ﬂuorescence. We choose the time-dependent intensity of the emitted radiation, I(t), as the observable. This intensity is I(t) =E(+)(t)E(−)(t), where E(+)(t) is the negative-frequency part of the electric ﬁeld operato r of the emitted radiation. E(+)is proportional to Mσeg, so the measured quantity is /angbracketleftI(t)/angbracketright=κ/angbracketleftΨ(t)\|M2σee\|Ψ(t)/angbracketright. (6) where κis a proportionality constant. 3 Extraction of the Hamiltonian structure The physical picture above leads to the the following mathem atical problem: extract the po- tentials and dipole moments from the measured intensity /angbracketleftI(t)/angbracketright, assuming that the initial state and the two driving ﬁelds are known (note that a number of adva nced experimental techniques have been recently developed for characterization of ultra short optical pulses [14, 15, 16]). We start from the Heisenberg equation of motion, i/planckover2pi1dI/dt = [I,H], to obtain i/planckover2pi1 κd/angbracketleftI/angbracketright dt=/angbracketleftΨ(t)\|[M2,T]σee−ǫ0(t)M3(σeg−σge)\|Ψ(t)/angbracketright. (7) Using form (3) of the wave function, we rewrite (7) as an integ ral equation for M(x): /integraldisplay dx[M2(x)F(x,t) +M3(x)G(x,t)] =/planckover2pi1 2κd/angbracketleftI/angbracketright dt, (8) where F(x,t) =/planckover2pi12 2mIm[v(x,t)∂2 xv∗(x,t)], G (x,t) =ǫ0(t)Im[u∗(x,t)v(x,t)]. (9) In order to obtain equations for the other unknown functions (two potentials and two dipole moments), we consider the second time derivative of /angbracketleftI(t)/angbracketright. Then, using (3), we derive the following integral equation: /planckover2pi12 κd2/angbracketleftI/angbracketright dt2+TM(t) =/integraldisplay dx[Ve(x)−Vg(x)−ǫ1(t)µe(x) +ǫ1(t)µg(x)]SM(x,t) +/integraldisplay dx[Ve(x)−ǫ1(t)µe(x)]RM(x,t), (10) where RM(x,t) =/planckover2pi12 mRe/braceleftbig v∗(x,t)[∂2 x,M2(x)]v(x,t)/bracerightbig , (11) SM(x,t) =−2ǫ0(t)M3(x)Re [u∗(x,t)v∗(x,t)], (12) 3TM(t) =/planckover2pi14 4m2/integraldisplay dxv∗(x,t)[M2(x)∂4 x−2∂2 xM2(x)∂2 x+∂4 xM2(x)]v(x,t) +/planckover2pi12 mǫ0(t)Re/braceleftbigg/integraldisplay dxv∗(x,t)[M3(x)∂2 x+M2(x)∂2 xM(x)−2∂2 xM3(x)]u(x,t)/bracerightbigg +2ǫ2 0(t)/integraldisplay dxM4(x)/parenleftbig \|v(x,t)\|2− \|u(x,t)\|2/parenrightbig . (13) It is convenient to formally enumerate the unknown function s, f1(x) =Vg(x), f 2(x) =Ve(x), f 3(x) =d0µg(x), f 4(x) =d0µe(x), (14) where d0= 1 V/m, so all the functions fr(x) have the dimension of energy. Then the integral equation (10) takes the form /integraldisplay dx4/summationdisplay r=1Kr(x,t)fr(x) =g(t), (15) where K1(x,t) =−SM(x,t), K 2(x,t) =RM(x,t) +SM(x,t), (16) K3(x,t) =−˜ǫ1(t)K1(x,t), K 4(x,t) =−˜ǫ1(t)K2(x,t), (17) g(t) =/planckover2pi12 κd2/angbracketleftI/angbracketright dt2+TM(t), (18) and ˜ǫ1=d−1 0ǫ1is the scaled (dimensionless) ﬁeld. It is important to emphasize that in fact equations (8) and (1 5) represent an inﬁnite number (or, in practice, a large number) of equations correspondin g to diﬀerent times. We will use this fact in the regularization procedure below. Of course, equa tion (15) is nonlinear because the wave function depends on the potentials and dipole moments. Similarly, a solution Mof (8) depends on the wave function and thereby depends on other unk nown functions. Consequently, the problem at hand, including the integral equations and th e Schr¨ odinger equation, is highly nonlinear. More importantly, the solution for such a system of integral equations is generally not unique and the problem is ill-posed (i.e., the solution is no t stable against small changes of the data). These characteristics are common to virtually all in verse problems and arise because the data used for the inversion are inevitably incomplete. Cons equently, we need to regularize the problem by imposing physically motivated constraints on th e unknown functions. For example, we may use the fact that physically acceptable potentials an d dipoles should be smooth functions and tend to zero asymptotically as x→ ∞ (in the case of the dipole, the atoms are assumed to separate as neutrals). By taking into account this informat ion, some constraints are imposed on the solutions, singling out the functions with desirable ph ysical properties. This regularization procedure will stabilize the solution. The regularized solution of equation (15) is achieved by min imizing the functional J=/integraldisplayt 0dt′/bracketleftBigg/integraldisplay dx4/summationdisplay r=1Kr(x,t′)fr(x)−g(t′)/bracketrightBigg2 +4/summationdisplay r=1αr/integraldisplay dxf2 r(x). (19) Here, αrare standard regularization parameters which denote the tr adeoﬀ between reproducing the laboratory data and obtaining the solution with smooth a nd regular functions. The time 4integration in (19) has a simple physical meaning: the measu red intensity brings in information about the potentials and dipoles at each instance of time and we want to use all the laboratory information which has been accumulated during the period fr om time zero until t. The choice of the functional (19) is not unique, and other forms of regular ization may be considered as well. Taking the variation of the functional Jwith respect to the unknown functions fr(x) involves a subtlety related to the nonlinearity of the problem: the ke rnelsKrand the free term gdepend on the wave function and on M(x) and thereby depend on fr(x). The practical (numerical) solution of any nonlinear problem includes some kind of line arization. Here, the point at which we make the linearization is determined in the regularizati on procedure. We choose to take the variation of the functional Jin equation (19) only with respect to the explicit dependence on fr(x). Then we obtain the set of regularized equations: /integraldisplay dx′4/summationdisplay r=1Kpr(x,x′,t)fr(x′) +αpfp(x) =gp(x,t), (20) where Kpr(x,x′,t) =/integraldisplayt 0dt′Kp(x,t′)Kr(x′,t′), (21) gp(x,t) =/integraldisplayt 0dt′Kp(x,t′)g(t′). (22) With p,r= 1,2,3,4, we have the system of four integral equations with four unk nown functions (two potentials and two moments). Now we want to regularize equation (8) for the electronic tra nsition dipole M. This equation is highly nonlinear: in addition to the dependence on Min the wave function, it also involves second and third powers of M. Once again, we may choose at which point to make the lineariz a- tion. We prefer to linearize at an early stage, in order to obt ain an equation of a simple form. Thus we deﬁne FM(x,t) =M2(x)F(x,t), G M(x,t) =M2(x)G(x,t), (23) and write the functional JM=/integraldisplayt 0dt′/bracketleftbigg/integraldisplay dxM(x)GM(x,t′) +/integraldisplay dxFM(x,t′)−gM(t′)/bracketrightbigg2 +αM/integraldisplay dxM2(x).(24) Here, gM(t)) is the right-hand side of equation (8). The regularized so lution of equation (8) is achieved by minimizing this functional. And we choose the li nearization procedure by taking the variation of JMin equation (24) only with respect to the explicit dependenc e onM(that is, we treat FMandGMas independent of M). Then we obtain /integraldisplay dx′KM(x,x′,t)M(x′) +αMM(x) =gM(x,t), (25) where KM(x,x′,t) =/integraldisplayt 0dt′GM(x,t′)GM(x′,t′), (26) gM(x,t) =/integraldisplayt 0dt′GM(x,t′)/bracketleftbigg gM(t′)−/integraldisplay dx′FM(x′,t′)/bracketrightbigg . (27) 5Finally, the integral equations (20) and (25) for the potent ials and dipole moments and the Schr¨ odinger equations (4) and (5) for the components of the wave function form the full set of coupled equations for the unknown functions, with /angbracketleftI(t)/angbracketright,ǫ0(t), and ǫ1(t) as input data. We conclude the presentation of the formalism with a schemat ic outline of the inversion algorithm which will be numerically implemented in a forthc oming work. First, the algorithm will start with trial functions for the potentials and dipol es to propagate the wave function from t= 0 to t= ∆t. Using trial functions at the ﬁrst step is not an excessive de mand for two reasons: (i) For a suﬃciently small time increment ∆ t, the evolution of the wave function is mainly aﬀected by the values of the potentials and dipoles in the region where u0(x) is localized, i.e., in the harmonic region of the ground potential surface ; such information is usually known with reasonable accuracy. (ii) As more data becomes availab le, the initial trial functions will be replaced by those which match the measured ﬂuorescence in tensity. The second step will use the measured ﬂuorescence intensity, the wave function c omponents u(∆t) and v(∆t), and the initial trial functions for solving equations (25) and ( 20) to obtain the next evaluation of the potentials and dipoles. These functions will be once aga in substituted into the Schr¨ odinger equation to propagate the wave packet from t= ∆ttot= 2∆t. The procedure will be repeated many times with new laboratory data incorporated at each tim e step. The recorded ﬂuorescence intensity, /angbracketleftI(t)/angbracketright, contains information about the potentials and dipoles in t he region where the wave packet is localized at moment tas well as where it was prior to that time. The sequential marching forward in time over the data track acts to reﬁne the potentials and dipoles at each time step. 4 Discussion This letter sets forth the formalism of a novel comprehensiv e approach to the inversion of molecular dynamics from time-dependent laboratory data. O ne of the main features of the proposed inversion method is initiation in the well-known g round state and use of external driving ﬁelds to excite the wave packet and guide its motion o n the ground and excited potential surfaces. Diﬀerent driving ﬁelds will induce diﬀerent dyna mics and may be more or less helpful for the inversion procedure. Consequently, we are left with the attractive prospect of choosing the driving ﬁelds to be optimally suited for assisting the ex traction of unknown potentials and dipoles from laboratory data. This choice may be facilitate d by a closed learning loop [17] in the laboratory, starting with a number of diﬀerent trial ﬁelds. According to the inversion objectives, a learning algorithm will determine the best candidates and direct the ﬁelds to shapes which are best suited to produce these objectives. Natural object ives are to maximize the spatial region where the potentials and dipoles are reliably extrac ted. Physical intuition suggests that one may learn more about the Hamiltonian at a speciﬁc spatial point if the wave packet is not spread over the whole potential surface but is essentially l ocalized in a narrow region around this point. Consequently, the driving ﬁelds best suited for the i nversion will control the dispersion of the wave packet and guide its motion in a desired large spatia l region. A numerical simulation of the algorithm, including its closed-loop learning featu res, will be the next step towards its laboratory implementation. 6Acknowledgments This work was supported by the U.S. Department of Defense and the National Science Founda- tion. References [1] von Geramb H V (Ed) 1994 Quantum Inversion Theory and Applications (New York: Springer) [2] Ho T-S and Rabitz H 1993 J. Phys. Chem. 9713447 Ho T-S, Rabitz H, Choi S E and Lester M I 1996 J. Chem. Phys. 1041187 [3] Zhang D H and Light J C 1995 J. Chem. Phys. 1039713 [4] Bernstein R B and Zewail A H 1990 Chem. Phys. Lett. 170321 Gruebele M and Zewail A H 1993 J. Chem. Phys. 98883 [5] Baer R and Kosloﬀ R 1995 J. Phys. Chem. 992534 [6] Shapiro M 1995 J. Chem. Phys. 1031748 Shapiro M 1996 J. Phys. Chem. 1007859 [7] Lu Z-M and Rabitz H 1995 J. Phys. 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arXiv:physics/0009090v1 [physics.atom-ph] 28 Sep 2000Electron Self Energy for the K and L Shell at Low Nuclear Charg e Ulrich D. Jentschura,1,2,∗, Peter J. Mohr,1,‡, and Gerhard Soﬀ2,§ 1National Institute of Standards and Technology, Mail Stop 8 401, Gaithersburg, MD 20899-8401, USA 2Institut f¨ ur Theoretische Physik, TU Dresden, Mommsenstr aße 13, 01062 Dresden, Germany A nonperturbative numerical evaluation of the one-photon e lectron self energy for the K- and L-shell states of hydrogenlike ions with nuclear charge num bersZ= 1 to 5 is described. Our calculation for the 1S 1/2state has a numerical uncertainty of 0.8 Hz in atomic hydroge n, and for the L-shell states (2S 1/2, 2P1/2, and 2P 3/2) the numerical uncertainty is 1.0 Hz. The method of evaluation for the ground state and for the excited states is described in detail. The numerical results are compared to results based on known terms in the ex pansion of the self energy in powers ofZα. PACS numbers 12.20.Ds, 31.30.Jv, 06.20.Jr, 31.15.-p Contents I Introduction 1 II Method of Evaluation 2 A Status of Analytic Calculations . . . . 2 B Formulation of the Numerical Problem 3 C Treatment of the divergent terms . . . 5 III The Low-Energy Part 6 A The Infrared Part . . . . . . . . . . . . 6 B The Middle-Energy Subtraction Term . 9 C The Middle-Energy Remainder . . . . . 10 IV The High-Energy Part 12 A The High-Energy Subtraction Term . . 12 B The High-Energy Remainder . . . . . . 14 C Results for the High-Energy Part . . . 16 V Comparison to Analytic Calculations 17 VI Conclusion 19 I. INTRODUCTION The nonperturbative numerical evaluation of radiative corrections to bound-state energy levels is interesting fo r two reasons. First, the recent dramatic increase in the accuracy of experiments that measure the transition fre- quencies in hydrogen and deuterium [1–3] necessitates a numerical evaluation (nonperturbative in the binding Coulomb ﬁeld) of the radiative corrections to the spec- trum of atomic systems with low nuclear charge Z. Sec- ond, the numerical calculation serves as an independent test of analytic evaluations which are based on an expan- sion in the binding ﬁeld with an expansion parameter Zα. In order to address both issues, a high-precision nu- merical evaluation of the self energy of an electron in the ground state in hydrogenlike ions has been per- formed [4,5]. The approach outlined in [4] is generalized here to the L shell, and numerical results are obtainedfor the (n= 2) states 2S 1/2, 2P1/2and 2P 3/2. Results are provided for atomic hydrogen, He+, Li2+, Be3+, and B4+. It has been pointed out in [4, 5] that the nonpertur- bative eﬀects (in Zα) can be large even for low nuclear charge and exceed the current experimental accuracy for atomic transitions. For example, the diﬀerence between the sum of the analytically evaluated terms up to the order ofα(Zα)6and the ﬁnal numerical result for the ground state is roughly 27 kHz for atomic hydrogen and about 3200 kHz for He+. For the 2S state the diﬀerence is 3.5 kHz for atomic hydrogen and 412 kHz for He+. The large diﬀerence between the result obtained by an expansion in Zαpersists even after the inclusion of a re- sult recently obtained in [6] for the logarithmic term of orderα(Zα)7ln(Zα)−2. For the ground state, the dif- ference between the all-order numerical result and the sum of the perturbative terms is still 13 kHz for atomic hydrogen and 1600 kHz for He+. For the 2 S state, the diﬀerence amounts to 1 .6 kHz for atomic hydrogen and to 213 kHz for He+. These ﬁgures should be compared to the current exper- imental precision. The most accurately measured tran- sition to date is the 1S–2S frequency in hydrogen; it has been measured with a relative uncertainty of 1 .8 parts in 1014or 46 Hz [3]. This experimental progress is due in part to the use of frequency chains that bridge the range between optical frequencies and the microwave cesium time standard. The uncertainty of the measurement is likely to be reduced by an order of magnitude in the near future [3, 7]. With trapped hydrogen atoms, it should be feasible to observe the 1S–2S frequency with an ex- perimental linewidth that approaches the 1 .3 Hz natural width of the 2S level [8,9]. The perturbation series in Zαis slowly convergent. The all-order numerical calculation presented in this pa- per essentially eliminates the uncertainty from unevalu- ated higher-order analytic terms, and we obtain results for the self-energy remainder function GSEwith a preci- sion of roughly 0 .8×Z4Hz for the ground state of atomic hydrogen and 1 .0×Z4Hz for the 2S state. 1In the evaluation, we take advantage of resummation and convergence acceleration techniques. The resumma- tion techniques provide an eﬃcient method of evaluation of the Dirac-Coulomb Green function to a relative un- certainty of 10−24over a wide parameter range [5]. The convergence acceleration techniques remove the princi- pal numerical diﬃculties associated with the singularity of the relativistic propagators for nearly equal radial ar- guments [10]. The one-photon self energy treated in the current in- vestigation is about two orders of magnitude larger than the other contributions to the Lamb shift in atomic hy- drogen. A comprehensive review of the various contri- butions to the Lamb shift in hydrogenlike atoms in the full range of nuclear charge numbers Z= 1–110 has been given in [11–14]. This paper is organized as follows. The method of eval- uation is discussed in Sec. II. The calculation is divided into a low-energy part and a high-energy contribution. The low-energy part is treated in Sec. III, and the high- energy part is discussed in Sec. IV. Numerical results are compiled in Sec. V. Also in Sec. V, we compare numeri- cal and analytic results for the Lamb shift in the region of low nuclear charge numbers. Of special importance is the consistency check with available analytic results [15,16] for higher-order binding corrections to the Lamb shift. We make concluding remarks in Sec. VI. II. METHOD OF EVALUATION A. Status of Analytic Calculations The (real part of the) energy shift ∆ ESEdue to the electron self-energy radiative correction is usually writ - ten as ∆ESE=α π(Zα)4 n3F(nlj,Zα)mec2(2.1) whereFis a dimensionless quantity. In the following, the natural unit system with ¯ h=c=me= 1 and e2= 4παis employed. Note that F(nlj,Zα) is a dimen- sionless function which depends for a given atomic state with quantum numbers n,landjon only one argument (the coupling Zα). For excited states, the (nonvanishing) imaginary part of the self energy is proportional to the (spontaneous) decay width of the state. We will denote here the realpart of the self energy by ∆ ESE, exclusively. The semi-analytic expansion of F(nlj,Zα) aboutZα= 0 for a general atomic state with quantum numbers n,land jgives rise to the semi-analytic expansion, F(nlj,Zα) =A41(nlj) ln(Zα)−2 +A40(nlj) + (Zα)A50(nlj) + (Zα)2/bracketleftbig A62(nlj) ln2(Zα)−2+A61(nlj) ln(Zα)−2+GSE(nlj,Zα)/bracketrightbig . (2.2) For particular states, some of the coeﬃcients may van- ish. Notably, this is the case for P states, which are less singular than S states at the origin [see Eq. (2.4) below]. For thenS1/2state (l= 0,j= 1/2), none of the terms in Eq. (2.2) vanishes, and we have, F(nS1/2,Zα) =A41(nS1/2) ln(Zα)−2 +A40(nS1/2) + (Zα)A50(nS1/2) + (Zα)2/bracketleftbig A62(nS1/2) ln2(Zα)−2 +A61(nS1/2) ln(Zα)−2+GSE(nS1/2,Zα)/bracketrightbig .(2.3) TheAcoeﬃcients have two indices, the ﬁrst of which de- notes the power of Zα[including those powers implicitly contained in Eq. (2.1)], while the second index denotes the power of the logarithm ln( Zα)−2. For P states, the coeﬃcients A41,A50andA62vanish, and we have F(nPj,Zα) =A40(nPj) +(Zα)2/bracketleftbig A61(nPj) ln(Zα)−2+GSE(nPj,Zα)/bracketrightbig .(2.4) For S states, the self-energy remainder function GSEcan be expanded semi-analytically as GSE(nS1/2,Zα) =A60(nS1/2) +(Zα)/bracketleftbig A71(nS1/2) ln(Zα)−2 +A70(nS1/2) + o(Zα)/bracketrightbig (2.5) (for the “order” symbols o and O we follow the usual con- vention, see e.g. [17,18]). For P states, the semi-analytic expansion of GSEreads GSE(nPj,Zα) =A60(nPj) +(Zα) [A70(nPj) + o(Zα)]. (2.6) The fact that A71(nPj) vanishes has been pointed out in [6]. We list below the analytic coeﬃcients and the Bethe logarithms relevant to the atomic states under in- vestigation. For the ground state, the coeﬃcients A41 andA40were obtained in [19–25], the correction term A50was found in [26–28], and the higher-order binding corrections A62andA61were evaluated in [15, 29–37]. The results are, A41(1S1/2) =4 3, A40(1S1/2) =10 9−4 3lnk0(1S), A50(1S1/2) = 4π/bracketleftbigg139 128−1 2ln 2/bracketrightbigg , 2A62(1S1/2) =−1, A61(1S1/2) =28 3ln 2−21 20. (2.7) The Bethe logarithm ln k0(1S) has been evaluated in [38] and [39–43] as lnk0(1S) = 2.984 128 555 8(3) . (2.8) For the 2S state, we have A41(2S1/2) =4 3, A40(2S1/2) =10 9−4 3lnk0(2S), A50(2S1/2) = 4π/bracketleftbigg139 128−1 2ln 2/bracketrightbigg , A62(2S1/2) =−1, A61(2S1/2) =16 3ln 2 +67 30. (2.9) The Bethe logarithm ln k0(2S) has been evaluated (see [38–43], the results exhibit varying accuracy) as lnk0(2S) = 2.811 769 893(3) . (2.10) It might be worth noting that the value for ln k0(2S) given in [44] evidently contains a typographical error. Our independent re-evaluation conﬁrms the result given in Eq. (2.10), which was originally obtained in [38] to the required precision. For the 2P 1/2state we have A40(2P1/2) =−1 6−4 3lnk0(2P), A61(2P1/2) =103 108. (2.11) Note that a general analytic result for the logarithmic correctionA61as a function of the bound state quan- tum numbers n,landjcan be inferred from Eq. (4.4a) of [34, 35] upon subtraction of the vacuum polarization contribution implicitly contained in the quoted equation. The Bethe logarithm for the 2P states reads [38,45] lnk0(2P) = −0.030 016 708 9(3) . (2.12) Because the Bethe logarithm is an inherently nonrela- tivistic quantity, it is spin-independent and therefore in - dependent of the total angular momentum jfor a given orbital angular momentum l. For the 2P 3/2state the analytic coeﬃcients are A40(2P3/2) =1 12−4 3lnk0(2P), A61(2P3/2) =29 90. (2.13)We now consider the limit of the function GSE(Zα) as Zα→0. The higher-order terms in the potential expan- sion (see Fig. 3 below) and relativistic corrections to the wavefunction both generate terms of higher order in Zα which are manifest in Eq. (2.2) in the form of the nonva- nishing function GSE(Zα) which summarizes the eﬀects of the relativistic corrections to the bound electron wave function and of higher-order terms in the potential ex- pansion. For very soft virtual photons, the potential ex- pansion fails and generates an infrared divergence which is cut oﬀ by the atomic momentum scale, Zα. This cut- oﬀ for the infrared divergence is one of the mechanisms which lead to the logarithmic terms in Eq. (2.2). Some of the nonlogarithmic terms of relative order ( Zα)2in Eq. (2.2) are generated by the relativistic corrections to the wave function. The function GSEdoes not vanish, but approaches a constant in the limit Zα→0. This constant can be determined by analytic or semi-analytic calculations; it is referred to as the A60coeﬃcient, i.e. A60(nlj) =GSE(nlj,0). (2.14) The evaluation of the coeﬃcient A60(1S1/2) has been his- torically problematic [15,34–37]. For the 2S state, there is currently only one precise analytic result available, A60(2S1/2) = −31.840 47(1) [15]. (2.15) For the 2P 1/2state, the analytically obtained result is A60(2P1/2) = −0.998 91(1) [16], (2.16) and for the 2P 3/2state, we have A60(2P3/2) = −0.503 37(1) [16]. (2.17) The analytic evaluations essentially rely on an expansion of the relativistic Dirac-Coulomb propagator in powers of the binding ﬁeld, i.e. in powers of Coulomb interactions of the electron with the nucleus. In numerical evalua- tions, the binding ﬁeld is treated nonperturbatively, and no expansion is performed. B. Formulation of the Numerical Problem Numerical cancellations are severe for small nuclear charges. In order to understand the origin of the numer- ical cancellations it is necessary to consider the renor- malization of the self energy. The renormalization pro- cedure postulates that the self energy is essentially the eﬀect on the bound electron due to the self interaction with its own radiation ﬁeld, minus the same eﬀect on a free electron which is absorbed in the mass of the elec- tron and therefore not observable. The self energy of the bound electron is the residual eﬀect obtained after the subtraction of two large quantities. Terms associ- ated with renormalization counterterms are of order 1 in 3theZα-expansion, whereas the residual eﬀect is of or- der (Zα)4[see Eq. (2.1)]. This corresponds to a loss of roughly 9 signiﬁcant digits at Z= 1. Consequently, even the precise evaluation of the one-photon self energy in a Coulomb ﬁeld presented in [46] extends only down toZ= 5. Among the self-energy corrections in one- loop and higher-loop order, numerical cancellations in absolute terms are most severe for the one-loop problem because of the large size of the eﬀect of the one-loop self- energy correction on the spectrum./B9 /BI/AJ/AI /BX/D2 /BE /D1 /B9/AR /BI/BI /BV/C4/BV/C4 /BV/C0/BV/C0 /A2 /A2 /A2 /A2 /A2 /CA/CT/B4 /AX /B5 /C1/D1/B4 /AX /B5 FIG. 1. Integration contour Cfor the integration over the energy ω=En−zof the virtual photon. The contour C consists of the low-energy contour CLand the high-energy contour CH. Lines shown displaced directly below and above the real axis denote branch cuts from the photon and elec- tron propagator. Crosses denote poles originating from the discrete spectrum of the electron propagator. The contour used in this work corresponds to the one used in [47]. For our high-precision numerical evaluation, we start from the regularized and renormalized expression for the one-loop self energy of a bound electron, ∆ESE= lim Λ→∞/braceleftbigg ie2Re/integraldisplay CFdω 2π/integraldisplayd3k (2π)3Dµν(k2,Λ) ×/angbracketleftbigg ¯ψ/vextendsingle/vextendsingle/vextendsingle/vextendsingleγµ 1 ∝ne}ationslashp− ∝ne}ationslashk−1−γ0Vγν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ/angbracketrightbigg −∆m/bracerightbigg = lim Λ→∞/braceleftbigg −ie2Re/integraldisplay Cdω 2π/integraldisplayd3k (2π)3Dµν(k2,Λ) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xG(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig −∆m/bracerightbigg ,(2.18) whereGdenotes the Dirac-Coulomb propagator, G(z) =1 α·p+β+V−z, (2.19)and ∆mis the Λ-dependent (cutoﬀ-dependent) one-loop mass-counter term, ∆m=α π/parenleftbigg3 4ln Λ2+3 8/parenrightbigg ∝an}bracketle{tβ∝an}bracketri}ht. (2.20) The photon propagator Dµν(k2,Λ) in Eq. (2.18) in Feyn- man gauge reads Dµν(k2,Λ) =−/parenleftbigggµν k2+ iǫ−gµν k2−Λ2+ iǫ/parenrightbigg .(2.21) The contour CFin Eq. (2.18) is the Feynman contour, whereas the contour Cis depicted in Fig. 1. The contour Cis employed for the ω-integration in the current evalu- ation [see the last line of Eq. (2.18)]. The energy variable zin Eq. (2.19) therefore assumes the value z=En−ω, (2.22) whereEnis the Dirac energy of the atomic state, and ω denotes the complex-valued energy of the virtual photon. It is understood that the limit Λ → ∞ is taken afterall integrals in Eq. (2.18) are evaluated./B9 /BI/AJ/AI /B9/AR /B9/AR /BI/BI /BV/C1/CA/BV/C1/CA /BV/C5/BV/C5 /A2 /A2 /A2 /A2 /A2 /CA/CT/B4 /AX /B5 /C1/D1/B4 /AX /B5 FIG. 2. Separation of the low-energy contour CLinto the infrared part CIRand the middle-energy part CM. As in Fig. 1, the lines directly above and below the real axis denote branch cuts from the photon and electron propaga- tor. Strictly speaking, the ﬁgure is valid only for the groun d state. For excited states, some of the crosses, which denote poles originating from the discrete spectrum of the electro n propagator, are positioned to the right of the line Re ω= 0. These poles are subtracted in the numerical evaluation. The integration contour for the complex-valued energy of the virtual photon ωin this calculation is the contour Cemployed in [46–49] and depicted in Fig. 1. The inte- grations along the low-energy contour CLand the high- energy contour CHin Fig. 1 give rise to the low- and the high-energy contributions ∆ ELand ∆EHto the self en- ergy, respectively. Here, we employ a further separation of the low-energy integration contour CLinto an infrared contourCIRand a middle-energy contour CMshown in Fig. 2. This separation gives rise to a separation of the low-energy part ∆ ELinto the infrared part ∆ EIRand the middle-energy part ∆ EM, ∆EL= ∆EIR+ ∆EM. (2.23) 4For the low- Zsystems discussed here, all complications which arise for excited states due to the decay into the ground state are relevant only for the infrared part. Ex- cept for the further separation into the infrared and the middle-energy part, the same basic formulation of the self-energy problem as in [47] is used. This leads to the following separation: ω∈(0,1 10En)±iδ; infrared part ∆ EIR, ω∈(1 10En,En)±iδ; middle-energy part ∆ EM, ω∈En+ i (−∞,+∞) ; high-energy part ∆ EH. Integration along these contours gives rise to the infrared , the middle-energy, and the high-energy contributions to the energy shift. For all of these contributions, lower- order terms are subtracted in order to obtain the contri- bution to the self energy of order ( Zα)4. We obtain for the infrared part, ∆EIR=α π/bracketleftbigg21 200∝an}bracketle{tβ∝an}bracketri}ht+43 600∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3FIR(nlj,Zα)/bracketrightbigg , (2.24) whereFIR(nlj,Zα) is a dimensionless function of order one. The middle-energy part is recovered as ∆EM=α π/bracketleftbigg279 200∝an}bracketle{tβ∝an}bracketri}ht+219 200∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3FM(nlj,Zα)/bracketrightbigg , (2.25) and the high-energy part reads [47,48] ∆EH= ∆m+α π/bracketleftbigg −3 2∝an}bracketle{tβ∝an}bracketri}ht −7 6∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3FH(nlj,Zα)/bracketrightbigg .(2.26) The infrared part is discussed in Sec. III A. The middle- energy part is divided into a middle-energy subtraction termFMAand a middle-energy remainder FMB. The subtraction term FMAis discussed in Sec. III B, the re- mainder term FMBis treated in Sec. III C. We recover the middle-energy term as the sum FM(nlj,Zα) =FMA(nlj,Zα) +FMB(nlj,Zα).(2.27) A similar separation is employed for the high-energy part. The high-energy part is divided into a subtraction term FHA, which is evaluated in Sec. IVA, and the high-energy remainderFHB, which is discussed in Sec. IV B. The sum of the subtraction term and the remainder is FH(nlj,Zα) =FHA(nlj,Zα) +FHB(nlj,Zα).(2.28) The total energy shift is given as∆ESE= ∆EIR+ ∆EM+EH−∆m =α π(Zα)4 n3[FIR(nlj,Zα) +FM(nlj,Zα) +FH(nlj,Zα)].(2.29) The scaled self-energy function Fdeﬁned in Eq. (2.1) is therefore obtained as F(nlj,Zα) =FIR(nlj,Zα) +FM(nlj,Zα) +FH(nlj,Zα).(2.30) In analogy to the approach described in [46, 47,49], we deﬁne the low-energy part as the sum of the infrared part and the middle-energy part, ∆EL= ∆EIR+ ∆EM =α π/bracketleftbigg3 2∝an}bracketle{tβ∝an}bracketri}ht+7 6∝an}bracketle{tV∝an}bracketri}ht+(Zα)4 n3FL(nlj,Zα)/bracketrightbigg ,(2.31) where FL(nlj,Zα) =FIR(nlj,Zα) +FM(nlj,Zα).(2.32) The limits for the functions FL(nlj,Zα) andFH(nlj,Zα) asZα→0 were obtained in [5,48,50]. C. Treatment of the divergent terms The free electron propagator F=1 α·p+β−z(2.33) and the full electron propagator Gdeﬁned in Eq. (2.19) fulﬁll the following identity, which is of particular impor - tance for the validity of the method used in the numerical evaluation of the all-order binding correction to the Lamb shift: G=F−FV F +FV GV F. (2.34) This identity leads naturally to a separation of the one- photon self energy into a zero-vertex, a single-vertex, and a many-vertex term. This is represented diagrammati- cally in Fig. 3. 5/A1 /BP/A2/B7/A3 /B7/A4 FIG. 3. The exact expansion of the bound electron prop- agator in powers of the binding ﬁeld leads to a zero-potentia l, a one-potential, and a many-potential term. The dashed line s denote Coulomb photons, the crosses denote the interaction with the (external) binding ﬁeld. All ultraviolet divergences which occur in the one- photon problem (mass counter term and vertex diver- gence) are generated by the zero-vertex and the single- vertex terms. The many-vertex term is ultraviolet safe. Of crucial importance is the observation that one may additionally simplify the problem by replacing the one- potential term with an approximate expression in which the potential is “commuted to the outside.” The ap- proximate expression generates all divergences and all terms of lower order than α(Zα)4present in the one- vertex term. Unlike the raw one-potential term, it is amenable to signiﬁcant further simpliﬁcation and can be reduced to one-dimensional numerical integrals that can be evaluated easily (a straightforward formulation of the self-energy problem requires a three-dimensional numer- ical integration). Without this signiﬁcant improvement, an all-order calculation would be much more diﬃcult at low nuclear charge, because the lower-order terms would introduce signiﬁcant further numerical cancellations. In addition, the special approximate resolvent can be used eﬀectively for an eﬃcient subtraction scheme in the middle-energy part of the calculation. In the infrared part, such a subtraction is not used because it would in- troduce infrared divergences. We now turn to the construction of the special ap- proximate resolvent, which will be referred to as GAand will be used in this calculation to isolate the ultraviolet divergences in the high-energy part (and to provide sub- traction terms in the middle-energy part). It is based on an approximation to the ﬁrst two terms on the right-hand side of Eq. (2.34). The so-called one-potential term FVF in Eq. (2.34) is approximated by an expression in which the potential terms Vare commuted to the outside: −FVF ≈ −1 2/braceleftbig V,F2/bracerightbig . (2.35) Furthermore, the following identity is used:F2=/parenleftbigg1 α·p+β−z/parenrightbigg2 =1 p2+ 1−z2+2z(β+z) (p2+ 1−z2)2 +2z(α·p) (p2+ 1−z2)2.(2.36) In 2×2 spinor space, this expression may be divided into a diagonal and a non-diagonal part. The diagonal part is diag(F2) =1 p2+ 1−z2+2z(β+z) (p2+ 1−z2)2.(2.37) The oﬀ-diagonal part is given by F2−diag(F2) =2z(α·p) (p2+ 1−z2)2. We deﬁne the resolvent GAas GA=F−1 2/braceleftbig V,diag/parenleftbig F2/parenrightbig/bracerightbig . (2.38) All divergences which occur in the self energy are gen- erated by the simpliﬁed propagator GA. We deﬁne the propagator GBas the diﬀerence of GandGA, GB=G−GA =1 2/braceleftbig V,diag(F2)/bracerightbig −FV F +FV G V F. (2.39) GBdoes not generate any divergences and leads to the middle-energy remainder discussed in Sec. III C and the high-energy remainder (Sec. IVB). III. THE LOW-ENERGY PART A. The Infrared Part The infrared part is given by ∆EIR=−ie2Re/integraldisplay CIRdω 2π/integraldisplayd3k (2π)3Dµν(k2) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xG(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig , (3.1) where relevant deﬁnitions of the symbols can be found in Eqs. (2.18–2.21), the contour CIRis as shown in Fig. 2, and the unregularized version of the photon propagator Dµν(k2) =−gµν k2+ iǫ(3.2) may be used. The infrared part consists of the following integration region for the virtual photon: 6ω∈/parenleftbig 0,1 10En/parenrightbig ±iδ z∈/parenleftbig9 10En,En/parenrightbig ±iδ/bracerightBigg infrared part ∆ EIR.(3.3) Following Secs. 2 and 3 of [47], we write ∆ EIRas a three-dimensional integral [see, e.g., Eqs. (3.4), (3.11) , and (3.14) of [47]] ∆EIR=α πEn 10−α π(P.V.)/integraldisplayEn 9 10Endz /integraldisplay∞ 0dx1x2 1/integraldisplay∞ 0dx2x2 2MIR(x2,x1,z),(3.4) where MIR(x2,x1,z) =/summationdisplay κ2/summationdisplay i,j=1 f¯ı(x2)Gij κ(x2,x1,z)f¯(x1)Aij κ(x2,x1). (3.5) Here, the quantum number κis the Dirac angular quan- tum number of the intermediate state, κ= 2 (l−j)(j+ 1/2), (3.6) wherelis the orbital angular momentum quantum num- ber andjis the total angular momentum of the bound electron. The functions fi(x2) (i= 1,2) are the radial wave functions deﬁned in Eq. (A.4) in [47] for an arbi- trary bound state (and in Eq. (A.8) in [47] for the 1S state). We deﬁne ¯ ı= 3−i. The functions Gij κ(x2,x1,z) (i,j= 1,2) are the radial Green functions, which re- sult from a decomposition of the electron Green function deﬁned in Eq. (2.19) into partial waves. The explicit for- mulas are given in Eq. (A.16) in [47]. The photon angular functions Aij κ(i,j= 1,2) are de- ﬁned in Eq. (3.15) of Ref. [47] for an arbitrary bound state. In Eq. (3.17) in [47], speciﬁc formulas are given for the 1S state. In Eqs. (2.2), (2.3) and (2.4) of [49], the special cases of S 1/2, P1/2and P 3/2states are con- sidered. Further relevant formulas for excited states can be found in [51]. The photon angular functions depend on the energy argument z, but this dependence is usually suppressed. The summation over κin Eq. (3.5) extends over all negative and all positive integers, excluding zero . We observe that the integral is symmetric under the in- terchange of the radial coordinates x2andx1, so that ∆EIR=α πEn 10−2α π(P.V.)/integraldisplayEn 9 10Endz /integraldisplay∞ 0dx1x2 1/integraldisplayx1 0dx2x2 2MIR(x2,x1,z).(3.7) The following variable substitution, r=x2/x1, y=ax1, (3.8)is made, so that r∈(0,1) andy∈(0,∞). The scaling variableais deﬁned as a= 2/radicalbig 1−E2n. (3.9) The Jacobian is /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(x2,x1) ∂(r,y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂x2 ∂r∂x1 ∂r ∂x2 ∂y∂x1 ∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=y a2. (3.10) The function SIRis given by SIR(r,y,z) =−2r2y5 a6MIR/parenleftBigry a,y a,z/parenrightBig =−2r2y5 a6∞/summationdisplay \|κ\|=1/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1f¯ı/parenleftBigry a/parenrightBig ×Gij κ/parenleftBigry a,y a,z/parenrightBig f¯/parenleftBigy a/parenrightBig Aij κ/parenleftBigry a,y a/parenrightBig =−2r2y5 a6∞/summationdisplay \|κ\|=1TIR,\|κ\|(r,y,z), (3.11) where in the last line we deﬁne implicitly the terms TIR,\|κ\| for\|κ\|= 1,...,∞as TIR,\|κ\|(r,y,z) =/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1 f¯ı/parenleftBigry a/parenrightBig Gij κ/parenleftBigry a,y a,z/parenrightBig f¯/parenleftBigy a/parenrightBig Aij κ/parenleftBigry a,y a/parenrightBig .(3.12) Using the deﬁnition (3.11), we obtain for ∆ EIR, ∆EIR=α πEn 10+α π(P.V.)/integraldisplayEn 9 10Endz /integraldisplay1 0dr/integraldisplay∞ 0dySIR(r,y,z). (3.13) The speciﬁcation of the principal value (P.V.) is neces- sary for the excited states of the L shell, because of the poles along the integration contour which correspond to the spontaneous decay into the ground state. Here we are exclusively concerned with the real part of the energy shift, as speciﬁed in Eq. (3.1), which is equivalent to the speciﬁcation of the principal value in (3.13). Evaluation of the integral over zis facilitated by the subtraction of those terms which generate the singularities along the in- tegration contour (for higher excited states, there can be numerous bound state poles, as pointed out in [51,52]). For the 2S and 2P 1/2states, only the pole contribution from the ground state must be subtracted. For the 2P 3/2 state, pole contributions originating from the 1S, the 2S and the 2P 1/2states must be taken into account. The nu- merical evaluation of the subtracted integrand proceeds 7along ideas outlined in [49,51] and is not discussed here in any further detail. The scaling parameter afor the integration over yis chosen to simplify the exponential dependence of the functionSdeﬁned in Eq. (3.11). The main exponen- tial dependence is given by the relativistic radial wave functions (upper and lower components). Both compo- nents [f1(x) andf2(x)] vary approximately as (neglecting relatively slowly varying factors) exp (−ax/2) (for large x). The scaling variable a, expanded in powers of Zα, is a= 2/radicalbig 1−E2n = 2/radicalBigg 1−/parenleftbigg 1−(Zα)2 2n2+ O [(Zα)4]/parenrightbigg2= 2Zα n+ O/bracketleftbig (Zα)3/bracketrightbig . (3.14) Therefore,ais just twice the inverse of the Bohr radius n/(Zα) in the nonrelativistic limit. The product f¯ı/parenleftBigry a/parenrightBig ×f¯/parenleftBigy a/parenrightBig for arbitrary ¯ ı,¯∈ {1,2} [which occurs in Eq. (3.11)] depends on the radial argu- ments approximately as e−y×exp/bracketleftBig1 2(1−r)y/bracketrightBig (for largey). Note that the main dependence as given by the term exp(−y) is exactly the weight factor of the Gauß- Laguerre integration quadrature formula. The deviation from the exact exp( −y)–type behavior becomes smaller asr→1. This is favorable because the region near r= 1 gives a large contribution to the integral in (3.13). TABLE I. Infrared part for the K and L shell states, FIR(1S1/2, Zα),FIR(2S1/2, Zα),FIR(2P1/2, Zα), and FIR(2P3/2, Zα), evaluated for low- Zhydrogenlike ions. The calculations were performed with th e numerical value ofα−1= 137 .036 for the ﬁne-structure constant. Z F IR(1S1/2, Zα) FIR(2S1/2, Zα) FIR(2P1/2, Zα) FIR(2P3/2, Zα) 1 7.236 623 736 8(1) 7.479 764 180(1) 0.085 327 852(1) 0.082 73 6 497(1) 2 5.539 002 119 1(1) 5.782 025 637(1) 0.086 073 669(1) 0.083 27 9 461(1) 3 4.598 155 821 8(1) 4.840 923 962(1) 0.087 162 510(1) 0.084 09 1 830(1) 4 3.963 124 140 6(1) 4.205 501 798(1) 0.088 543 188(1) 0.085 14 0 788(1) 5 3.493 253 319 4(1) 3.735 114 958(1) 0.090 180 835(1) 0.086 40 3 178(1) The sum over \|κ\|in Eq. (3.11) is carried out locally, i.e., for each set of arguments r,y,z. The sum over \|κ\|is absolutely convergent. For \|κ\| → ∞ , the convergence of the sum is governed by the asymptotic behavior of the Bessel functions which occur in the photon functions Aij κ (i,j= 1,2) [see Eqs. (3.15) and (3.16) in [47]]. The pho- ton functions contain products of two Bessel functions of the form Jl(ρ2/1) where Jlstands for either jlorj′ l, and the indexlis in the range l∈ {\|κ\|−1,\|κ\|,\|κ\|+1}. The argument is either ρ2= (En−z)x2orρ1= (En−z)x1. The asymptotic behavior of the two relevant Bessel func- tions for large l(and therefore large \|κ\|) is j′ l(x) =l xxl (2l+ 1)!!/bracketleftbigg 1 + O/parenleftbigg1 l/parenrightbigg/bracketrightbigg and (3.15) jl(x) =xl (2l+ 1)!!/bracketleftbigg 1 + O/parenleftbigg1 l/parenrightbigg/bracketrightbigg . (3.16) This implies that when min {ρ2,ρ1}=ρ2< l, the func- tionJl(ρ2) vanishes with increasing lapproximately as (eρ2/2l)l. This rapidly converging asymptotic behavior sets in as soon as l≈ \|κ\|>ρ2=rωy/a [see Eqs. (2.22) and (3.12)]. Due to the rapid convergence for \|κ\|> ρ2,the maximum angular momentum quantum number \|κ\| in the numerical calculation of the infrared part is less than 3 000. Note that because z∈(9 10En,En) in the infrared part, ω<1 10En. The integration scheme is based on a crude estimate of the dependence of the integrand SIR(r,y,z) deﬁned in Eq. (3.11) on the integration variables r,yandz. The main contribution to the integral is given by the region where the arguments of the Whittaker functions as they occur in the Green function [see Eq. (A.16) in [47]] are much larger than the Dirac angular momentum, 2cy a≫ \|κ\| (see also p. 56 of [48]). We assume the asymptotic form of the Green function given in Eq. (A.3) in [48] applies and attribute a factor exp[−(1−r)cy/a] to the radial Green functions Gij κas they occur in Eq. (3.11). Note that relatively slowly varying factors are replaced by unity. The products of the radial wave 8functionsf¯ıandf¯, according to the discussion following Eq. (3.14), behave as e−yexp/bracketleftBig1 2(1−r)y/bracketrightBig for largey. The photon functions Aij κin Eq. (3.11) give rise to an approximate factor sin[(1 −r)(En−z)y/a] (1−r). (3.17) Therefore [see also Eq. (2.12) in [48]], we base our choice of the integration routine on the approximation e−yexp/bracketleftbigg −/parenleftbiggc a−1 2/parenrightbigg (1−r)y/bracketrightbigg ×sin[(1 −r)(En−z)y/a] (1−r)(3.18) forSIR. The three-dimensional integral in (3.13) is eval- uated by successive Gaussian quadrature. Details of the integration procedure can be found in [5]. In order to check the numerical stability of the results, the calculations are repeated with three diﬀerent values of the ﬁne-structure constant α: α<= 1/137.036 000 5, α0= 1/137.036 000 0 and, α>= 1/137.035 999 5.(3.19) These values are close to the 1998 CODATA recom- mended value of α−1= 137.035 999 76(50) [53]. The calculation was parallelized using the Message Passing Interface (MPI) and carried out on a cluster of Silicon Graphics workstations and on an IBM 9276 SP/2 multi- processor system [54]. The results for the infrared part FIR, deﬁned in Eq. (2.24), are given in Table I for a value ofα−1=α−1 0= 137.036. This value of αwill be used exclusively in the numerical evaluations presented here. For numerical results obtained by employing the values ofα<andα>[see Eq. (3.19)] we refer to [5]. B. The Middle-Energy Subtraction Term The middle-energy part is given by ∆EM=−ie2/integraldisplay CMdω 2π/integraldisplayd3k (2π)3Dµν(k2) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xG(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig , (3.20) where relevant deﬁnitions of the symbols can be found in Eqs. (2.18)–(2.21) and Eq. (3.2), and the contour CMis as shown in Fig. 2. The middle-energy part consists of the following integration region for the virtual photon:ω∈/parenleftbig1 10En,En/parenrightbig ±iδ z∈/parenleftbig 0,9 10En/parenrightbig ±iδ/bracerightBigg middle-energy part ∆ EM. (3.21) The numerical evaluation of the middle-energy part is simpliﬁed considerably by the decomposition of the rela- tivistic Dirac-Coulomb Green function Gas G=GA+GB, (3.22) whereGAis deﬁned in (2.38) and represents the sum of an approximation to the so-called zero- and one-potential terms generated by the expansion of the Dirac-Coulomb Green function Gin powers of the binding ﬁeld V. We deﬁne the middle-energy subtraction term FMAas the expression obtained upon substitution of the propagator GAforGin Eq. (3.20). The propagator GBis simply calculated as the diﬀerence of GandGA[see Eq. (2.39)]. A substitution of the propagator GBforGin Eq. (3.20) leads to the middle-energy remainder FMBwhich is dis- cussed in Sec. III C. We provide here the explicit expres- sions ∆EMA=−ie2/integraldisplay CMdω 2π/integraldisplayd3k (2π)3Dµν(k2) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xGA(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig (3.23) and ∆EMB=−ie2/integraldisplay CMdω 2π/integraldisplayd3k (2π)3Dµν(k2) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xGB(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig .(3.24) Note that the decomposition of the Dirac-Coulomb Green function as in (3.22) is not applicable in the infrared part, because of numerical problems for ultra-soft photons (in- frared divergences). Rewriting (3.23) appropriately into a three-dimensional integral [5,47,48], we have ∆EMA=α π9 10En−2α π/integraldisplay9 10En 0dz /integraldisplay∞ 0dx1x2 1/integraldisplayx1 0dx2x2 2MMA(x2,x1,z). (3.25) The function MMA(x2,x1,z) is deﬁned in analogy to the function MIR(x2,x1,z) deﬁned in Eq. (3.5) for the in- frared part. Also, we deﬁne a function SMA(x2,x1,z) in analogy to the function SIR(x2,x1,z) given in Eq. (3.11) for the infrared part, which will be used in Eq. (3.28) below. We have, 9SMA(r,y,z) =−2r2y5 a6MMA/parenleftBigry a,y a,z/parenrightBig =−2r2y5 a6∞/summationdisplay \|κ\|=1/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1f¯ı/parenleftBigry a/parenrightBig ×Gij A,κ/parenleftBigry a,y a,z/parenrightBig f¯/parenleftBigy a/parenrightBig Aij κ/parenleftBigry a,y a/parenrightBig =−2r2y5 a6∞/summationdisplay \|κ\|=1TMA,\|κ\|(r,y,z). (3.26) The expansion of the propagator GAinto partial waves is given in Eqs. (5.4) and (A.20) in [47] and in Eqs. (D.37) and (D.42) in [5]. This expansion leads to the compo- nent functions Gij A,κ. The terms TMA,\|κ\|in the last line of Eq. (3.26) read TMA,\|κ\|(r,y,z) =/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1 f¯ı/parenleftBigry a/parenrightBig Gij A,κ/parenleftBigry a,y a,z/parenrightBig f¯/parenleftBigy a/parenrightBig Aij κ/parenleftBigry a,y a/parenrightBig .(3.27)With these deﬁnitions, the middle-energy subtraction term ∆EMAcan be written as ∆EMA=α π9 10En+α π/integraldisplay9 10En 0dz /integraldisplay∞ 0dy/integraldisplay1 0dr SMA(r,y,z). (3.28) The subtracted lower-order terms yield, ∆EMA=α π/bracketleftbigg279 200∝an}bracketle{tβ∝an}bracketri}ht+219 200∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3FMA(nlj,Zα)/bracketrightbigg .(3.29) The three-dimensional integral in (3.28) is evaluated by successive Gaussian quadrature. Details of the integra- tion procedure can be found in [5]. The numerical results are summarized in Table II. TABLE II. Numerical results for the middle-energy subtract ion term FMA, the middle-energy remainder term FMB, and the middle-energy term FM. The middle-energy term FMis given as the sumFM(nlj, Zα) =FMA(nlj, Zα) +FMB(nlj, Zα) [see also Eqs.(2.25), (3.29), and (3.33)]. Z F MA(1S1/2, Zα) FMA(2S1/2, Zα) FMA(2P1/2, Zα) FMA(2P3/2, Zα) 1 2.699 379 904 5(1) 2.720 878 318(1) 0.083 207 314(1) 0.701 70 5 240(1) 2 2.659 561 381 1(1) 2.681 820 660(1) 0.084 208 832(1) 0.701 85 0 024(1) 3 2.623 779 453 0(1) 2.647 262 568(1) 0.085 831 658(1) 0.702 09 1 147(1) 4 2.591 151 010 1(1) 2.616 290 432(1) 0.088 040 763(1) 0.702 42 6 850(1) 5 2.561 096 522 1(1) 2.588 297 638(1) 0.090 803 408(1) 0.702 85 4 461(1) Z F MB(1S1/2, Zα) FMB(2S1/2, Zα) FMB(2P1/2, Zα) FMB(2P3/2, Zα) 1 1.685 993 923 2(1) 1.784 756 705(2) 0.771 787 771(2) −0.094 272 681(2) 2 1.626 842 294 5(1) 1.725 583 798(2) 0.770 778 394(2) −0.094 612 071(2) 3 1.571 406 090 7(1) 1.670 086 996(2) 0.769 153 314(2) −0.095 165 248(2) 4 1.519 082 768 6(1) 1.617 650 004(2) 0.766 954 435(2) −0.095 922 506(2) 5 1.469 482 409 0(1) 1.567 873 140(2) 0.764 220 149(2) −0.096 874 556(2) Z F M(1S1/2, Zα) FM(2S1/2, Zα) FM(2P1/2, Zα) FM(2P3/2, Zα) 1 4.385 373 827 7(1) 4.505 635 023(2) 0.854 995 085(2) 0.607 43 2 559(2) 2 4.286 403 675 7(1) 4.407 404 458(2) 0.854 987 226(2) 0.607 23 7 953(2) 3 4.195 185 543 6(1) 4.317 349 564(2) 0.854 984 972(2) 0.606 92 5 899(2) 4 4.110 233 778 8(1) 4.233 940 436(2) 0.854 995 198(2) 0.606 50 4 344(2) 5 4.030 578 931 1(1) 4.156 170 778(2) 0.855 023 557(2) 0.605 97 9 905(2) C. The Middle-Energy Remainder The remainder term in the middle-energy part involves the propagator GBdeﬁned in Eq. (2.39), GB=G−GA, whereGis deﬁned in (2.19) and GAis given in (2.38).In analogy to the middle-energy subtraction term, the middle-energy remainder can be rewritten as a three- dimensional integral, 10∆EMB=α π/integraldisplay9 10En 0dz /integraldisplay1 0dr/integraldisplay∞ 0dySMB(r,y,z), (3.30) where SMB(r,y,z) =−2r2y5 a6∞/summationdisplay \|κ\|=1/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1f¯ı/parenleftBigry a/parenrightBig ×Gij B,κ/parenleftBigry a,y a,z/parenrightBig f¯/parenleftBigy a/parenrightBig Aij κ/parenleftBigry a,y a/parenrightBig .(3.31) The functions Gij B,κare obtained as the diﬀerence of the expansion of the full propagator Gand the simpliﬁed propagator GAinto angular momenta, Gij B,κ=Gij κ−Gij A,κ, (3.32) where the Gij κare listed in Eq. (A.16) in [47] and in Eq. (D.43) in [5], and the Gij A,κhave already been de- ﬁned in Eqs. (5.4) and (A.20) in [47] and in Eqs. (D.37) and (D.42) in [5]. There are no lower-order terms to sub- tract, and therefore ∆EMB=α π(Zα)4 n3FMB(nlj,Zα). (3.33) The three-dimensional integral (3.30) is evaluated by suc- cessive Gaussian quadrature. Details of the integration procedure are provided in [5]. Numerical results for the middle-energy remainder FMBare summarized in Ta- ble II for the K- and L-shell states. For the middle-energy part, the separation into a sub- traction and a remainder term has considerable compu- tational advantages which become obvious upon inspec- tion of Eqs. (3.29) and (3.33). The subtraction involves a propagator whose angular components can be evalu- ated by recursion [5,48], which is computationally time- consuming. Because the subtraction term involves lower- order components [see Eq. (2.25)], it has to be evaluated to high precision numerically (in a typical case, a rela- tive uncertainty of 10−19is required). This high preci- sion requires in turn a large number of integration points for the Gaussian quadratures, which is possible only if the numerical evaluation of the integrand is not compu- tationally time-consuming. For the remainder term, no lower-order terms have to be subtracted, and the rela- tive precision required of the integrals is in the range of 10−11...10−9. A numerical evaluation to this lower level of precision is feasible, although the calculation of the Green function GBis computationally more time con- suming than that of GA[5,47,48]. The separation of the high-energy part into a subtraction term and a remain- der term, which is discussed in Sec. IV, is motivated byanalogous considerations as for the middle-energy part. In the high-energy part, this separation is even more im- portant than in the middle-energy part, because of the occurrence of inﬁnite terms which need to be subtracted analytically before a numerical evaluation can proceed [see Eq. (4.8) below]. We now summarize the results for the middle-energy part. The middle-energy part is the sum of the middle- energy subtraction term FMAand the middle-energy re- mainderFMB[see also Eq. (2.27)]. Numerical results are summarized in Table II for the K- and L-shell states. The low-energy part FLis deﬁned as the sum of the infrared contribution FIRand the middle-energy contribution FM [see Eq. (2.32)]. The results for FLare provided in the Table III for the K- and L-shell states. The limits for the low-energy part as a function of the bound state quantum numbers can be found in Eq. (7.80) of [5]: FL(nlj,Zα) =4 3δl,0ln(Zα)−2 −4 3lnk0(n,l) +/parenleftbigg ln 2−11 10/parenrightbigg1 n +/parenleftbigg 2 ln 2−16 15/parenrightbigg1 2l+ 1+/parenleftbigg3 2ln 2−7 4/parenrightbigg1 κ(2l+ 1) +/parenleftbigg −3 2ln 2 +9 4/parenrightbigg1 \|κ\|+/parenleftbigg4 3ln 2−1 3/parenrightbigg δl,0 +/parenleftbigg ln 2−5 6/parenrightbiggn−2l−1 n(2l+ 1)+ O(Zα). (3.34) The limits for the states under investigation in this paper are FL(1S1/2,Zα) = (4/3) ln(Zα)−2−1.554 642 + O( Zα), FL(2S1/2,Zα) = (4/3) ln(Zα)−2−1.191 497 + O( Zα), FL(2P1/2,Zα) = 0.940 023 + O( Zα), FL(2P3/2,Zα) = 0.690 023 + O( Zα). (3.35) These limits are consistent with the numerical data in Table III. For S states, the low-energy contribution FL diverges logarithmically as Zα→0, whereas for P states, FLapproaches a constant as Zα→0. The leading loga- rithm is a consequence of an infrared divergence cut oﬀ by the atomic momentum scale. It is a nonrelativistic eﬀect which is generated by the nonvanishing probabil- ity density of S waves at the origin in the nonrelativistic limit. The presence of the logarithmic behavior for S states [nonvanishing A41-coeﬃcient, see Eqs. (2.2) and (2.3)] and its absence for P states is reproduced consis- tently by the data in Table III. 11TABLE III. Low-energy part FLfor the K- and L-shell states FL(1S1/2, Zα),FL(2S1/2, Zα), FL(2P1/2, Zα), and FL(2P3/2, Zα), evaluated for low- Zhydrogenlike ions. Z F L(1S1/2, Zα) FL(2S1/2, Zα) FL(2P1/2, Zα) FL(2P3/2, Zα) 1 11.621 997 564 5(1) 11.985 399 203(2) 0.940 322 937(2) 0.690 169 056(2) 2 9.825 405 794 7(1) 10.189 430 095(2) 0.941 060 895(2) 0.690 5 17 414(2) 3 8.793 341 365 4(1) 9.158 273 526(2) 0.942 147 482(2) 0.691 01 7 729(2) 4 8.073 357 919 4(1) 8.439 442 234(2) 0.943 538 386(2) 0.691 64 5 132(2) 5 7.523 832 250 6(1) 7.891 285 736(2) 0.945 204 392(2) 0.692 38 3 083(2) IV. THE HIGH-ENERGY PART A. The High-Energy Subtraction Term The high-energy part is given by ∆EH=−lim Λ→∞ie2/integraldisplay CHdω 2π/integraldisplayd3k (2π)3Dµν(k2,Λ) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xG(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig , (4.1) where relevant deﬁnitions of the symbols can be found in Eqs. (2.18)–(2.21), and the contour CHis as shown in Fig. 1. The high-energy part consists of the following integration region for the virtual photon, ω∈(En−i∞,En+ i∞) z∈(−i∞,i∞)/bracerightBigg high-energy part ∆ EH. (4.2) The separation of the high-energy part into a subtraction term and a remainder is accomplished as in the middle- energy part [see Eq. (3.22)] by writing the full Dirac- Coulomb Green function G[Eq. (2.19)] as G=GA+GB. We deﬁne the high-energy subtraction term FHAas the expression obtained upon substitution of the propagator GAforGin Eq. (4.1), and a substitution of the prop- agatorGBforGin Eq. (4.1) leads to the high-energy remainderFHBwhich is discussed in Sec. IVB. The sub- traction term (including all divergent contributions) is generated by GA, the high-energy remainder term corre- sponds toGB. We have ∆EHA=−lim Λ→∞ie2/integraldisplay CHdω 2π/integraldisplayd3k (2π)3Dµν(k2,Λ) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xGA(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig (4.3) and ∆EHB=−ie2/integraldisplay CHdω 2π/integraldisplayd3k (2π)3Dµν(k2) ×/angbracketleftbig ψ/vextendsingle/vextendsingleαµeik·xGB(En−ω)ανe−ik·x/vextendsingle/vextendsingleψ/angbracketrightbig . (4.4)The contribution ∆ EHAcorresponding to GAcan be sep- arated further into a term ∆ E(1) HA, which contains all di- vergent contributions, and a term ∆ E(2) HA, which contains contributions of lower order than ( Zα)4, but is conver- gent as Λ → ∞. This separation is described in detail in [47,50]. We have ∆EHA= ∆E(1) HA+ ∆E(2) HA. (4.5) We obtain for ∆ E(1) HA, which contains a logarithmic di- vergence as Λ → ∞, ∆E(1) HA=α π/bracketleftbigg/parenleftbigg3 4ln Λ2−9 8/parenrightbigg ∝an}bracketle{tβ∝an}bracketri}ht+/parenleftbigg1 2ln 2−17 12/parenrightbigg ∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3F(1) HA(nlj,Zα)/bracketrightbigg . (4.6) For the contribution F(1) HA, an explicit analytic result is given in Eq. (4.15) in [47]. This contribution is therefore not discussed in any further detail here. The contribution ∆E(2) HAcontains lower-order terms: ∆E(2) HA=α π/bracketleftbigg/parenleftbigg −1 2ln 2 +1 4/parenrightbigg ∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3F(2) HA(nlj,Zα)/bracketrightbigg .(4.7) Altogether we have ∆EHA= ∆E(1) HA+ ∆E(2) HA =α π/bracketleftbigg/parenleftbigg3 4ln Λ2−9 8/parenrightbigg ∝an}bracketle{tβ∝an}bracketri}ht −7 6∝an}bracketle{tV∝an}bracketri}ht +(Zα)4 n3FHA(nlj,Zα)/bracketrightbigg . (4.8) The scaled function FHA(nlj,Zα) is given by FHA(nlj,Zα) =F(1) HA(nlj,Zα) +F(2) HA(nlj,Zα).(4.9) The term ∆ E(2) HAfalls naturally into a sum of four con- tributions [47], 12∆E(2) HA=T1+T2+T3+T4, (4.10) where T1=−1 10∝an}bracketle{tV∝an}bracketri}ht+(Zα)4 n3h1(nlj,Zα), T2=/parenleftbigg7 20−1 2ln 2/parenrightbigg ∝an}bracketle{tV∝an}bracketri}ht+(Zα)4 n3h2(nlj,Zα), T3=(Zα)4 n3h3(nlj,Zα), T4=(Zα)4 n3h4(nlj,Zα). (4.11) The functions hi(i= 1,2,3,4) are deﬁned in Eqs. (4.18), (4.19) and (4.21) in [47] (see also Eq. (3.6) in [49]). The evaluation of the high-energy subtraction term proceeds as outlined in [47–49], albeit with an increased accu- racy and improved calculational methods in intermediate steps of the calculation in order to overcome the severe numerical cancellations in the low- Zregion. We recover F(2) HAas the sum F(2) HA(nlj,Zα) =h1(nlj,Zα) +h2(nlj,Zα) +h3(nlj,Zα) +h4(nlj,Zα). (4.12) The scaled function FHA(nlj,Zα) [see also Eqs. (2.26) and (2.28)] is given by FHA(nlj,Zα) =F(1) HA(nlj,Zα) +F(2) HA(nlj,Zα).(4.13)The limits of the contributions F(1) HA(nlj,Zα) and F(2) HA(nlj,Zα) as (Zα)→0 have been investigated in [47, 49,50]. For the contribution F(1) HA(nlj,0), the re- sult can be found in Eq. (3.5) in [49]. The limits of the functionshi(nlj,Zα) (i= 1,2,3,4) asZα→0 are given as a function of the atomic state quantum numbers in Eq. (3.8) in [49]. For the scaled high-energy subtraction termFHA, the limits read (see Eq. (3.9) in [49]) FHA(nlj,Zα) =/parenleftbigg11 10−ln 2/parenrightbigg1 n +/parenleftbigg16 15−2 ln 2/parenrightbigg1 2l+ 1+/parenleftbigg1 2ln 2−1 4/parenrightbigg1 κ(2l+ 1) +/parenleftbigg3 2ln 2−9 4/parenrightbigg1 \|κ\|+ O(Zα). (4.14) Therefore, the explicit forms of the limits for the states under investigation in this paper are FHA(1S1/2,Zα) =−1.219 627 + O( Zα), FHA(2S1/2,Zα) =−1.423 054 + O( Zα), FHA(2P1/2,Zα) =−1.081 204 + O( Zα), FHA(2P3/2,Zα) =−0.524 351 + O( Zα). (4.15) Numerical results for FHA, which are presented in Ta- ble IV, exhibit consistency with the limits in Eq. (4.15). TABLE IV. Numerical results for the high-energy subtractio n term FHAand the high-energy remainder term FHB. The high-energy term FHis the sum FH(nlj, Zα) =FHA(nlj, Zα) +FHB(nlj, Zα). Z F HA(1S1/2, Zα) FHA(2S1/2, Zα) FHA(2P1/2, Zα) FHA(2P3/2, Zα) 1 −1.216 846 660 6(1) −1.420 293 291(1) −1.081 265 954(1) −0.524 359 802(1) 2 −1.214 322 536 9(1) −1.417 829 864(1) −1.081 451 269(1) −0.524 385 053(1) 3 −1.212 026 714 1(1) −1.415 635 310(1) −1.081 760 224(1) −0.524 427 051(1) 4 −1.209 942 847 4(1) −1.413 693 422(1) −1.082 192 995(1) −0.524 485 727(1) 5 −1.208 059 033 6(1) −1.411 992 480(1) −1.082 749 845(1) −0.524 561 017(1) Z F HB(1S1/2, Zα) FHB(2S1/2, Zα) FHB(2P1/2, Zα) FHB(2P3/2, Zα) 1 −0.088 357 254(1) −0.018 280 727(5)a0.014 546 64(1) −0.042 310 69(1) 2 −0.082 758 206(1) −0.012 729 99(1) 0.014 574 21(1) −0.042 296 81(1) 3 −0.076 811 229(1) −0.006 861 02(1) 0.014 620 51(1) −0.042 273 58(1) 4 −0.070 590 991(1) −0.000 746 40(1) 0.014 685 82(1) −0.042 240 92(1) 5 −0.064 146 139(1) 0.005 567 16(1) 0.014 770 52(1) −0.042 198 76(1) Z F H(1S1/2, Zα) FH(2S1/2, Zα) FH(2P1/2, Zα) FH(2P3/2, Zα) 1 −1.305 203 915(1) −1.438 574 018(5) −1.066 719 31(1) −0.566 670 50(1) 2 −1.297 080 743(1) −1.430 559 85(1) −1.066 877 06(1) −0.566 681 86(1) 3 −1.288 837 943(1) −1.422 496 33(1) −1.067 139 72(1) −0.566 700 63(1) 4 −1.280 533 839(1) −1.414 439 82(1) −1.067 507 18(1) −0.566 726 65(1) 5 −1.272 205 173(1) −1.406 425 32(1) −1.067 979 33(1) −0.566 759 78(1) aResult obtained with a greater number of integration nodes t han are used for the higher- Zresults. 13B. The High-Energy Remainder The remainder term in the high-energy part involves the propagator GBdeﬁned in Eq. (2.39), GB=G−GA, whereGis deﬁned in (2.19) and GAis given in (2.38). The energy shift is ∆EHB=−iα π/integraldisplayi∞ 0dz/integraldisplay∞ 0dx1x2 1 /integraldisplayx1 0dx2x2 2{MHB(x2,x1,z) + c.c.},(4.16) where c.c. denotes the complex conjugate. The photon energy integration is evaluated with the aid of the sub- stitution z→iu whereu=1 2/parenleftbigg1 t−t/parenrightbigg . (4.17) In analogy with the middle-energy subtraction and remainder terms discussed in Secs. III B and III C [see especially Eqs. (3.26) and (3.31)], the functions MHB(x2,x1,z) andSHB(r,y,z) and the terms THB,\|κ\|are deﬁned implicitly in the following: SHB(r,y,t) = =/parenleftbigg 1 +1 t2/parenrightbiggr2y5 a6Re/bracketleftBig MHB/parenleftBigry a,y a,iu/parenrightBig/bracketrightBig =/parenleftbigg 1 +1 t2/parenrightbiggr2y5 a6∞/summationdisplay \|κ\|=1/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1 Re/bracketleftBig fi/parenleftBigry a/parenrightBig Gij B,κ/parenleftBigry a,y a,iu/parenrightBig fj/parenleftBigy a/parenrightBig Aκ/parenleftBigry a,y a/parenrightBig −f¯ı/parenleftBigry a/parenrightBig Gij B,κ/parenleftBigry a,y a,iu/parenrightBig f¯/parenleftBigy a/parenrightBig Aij κ/parenleftBigry a,y a/parenrightBig/bracketrightBig =/parenleftbigg 1 +1 t2/parenrightbiggr2y5 a6∞/summationdisplay \|κ\|=1THB,\|κ\|(r,y,t). (4.18) The only substantial diﬀerence from the treatment of the middle-energy remainder lies in the prefactor generated by the parameterization of the complex photon energy given in Eq. (4.17). The photon angular functions Aκ andAij κ(i,j= 1,2)for the high-energy part are deﬁned in Eq. (5.8) of Ref. [47] and in Eq. (4.3) in [49] for an arbitrary bound state. Special formulas for the ground state can be found in Eq. (5.9) of Ref. [47]. The func- tionsAκandAij κarenotidentical to the photon angular functions for the infrared and middle-energy parts Aij κ (i,j= 1,2) which are used for the low-energy part of the calculation in Sec. III. It might be worth mentioning that in [46–49], both the functions Aij κandAij κare denoted by the symbol Aij κ. It is clear from the context which of the functions is employed in each case.In the last line of Eq. (4.18), we implicitly deﬁne the termsTHB,\|κ\|as THB,\|κ\|(r,y,t) =/summationdisplay κ=±\|κ\|2/summationdisplay i,j=1 Re/bracketleftBig fi/parenleftBigry a/parenrightBig Gij B,κ/parenleftBigry a,y a,iu/parenrightBig fj/parenleftBigy a/parenrightBig × A κ/parenleftBigry a,y a/parenrightBig −f¯ı/parenleftBigry a/parenrightBig Gij B,κ/parenleftBigry a,y a,iu/parenrightBig f¯/parenleftBigy a/parenrightBig × Aij κ/parenleftBigry a,y a/parenrightBig/bracketrightBig . (4.19) With these deﬁnitions, the high-energy remainder can be rewritten as ∆EHB=α π/integraldisplay1 0dt/integraldisplay1 0dr/integraldisplay∞ 0dySHB(r,y,t).(4.20) There are no lower-order terms to subtract, and therefore ∆EHB=α π(Zα)4 n3FHB(nlj,Zα). (4.21) For the high-energy remainder FHB, the limits as Zα→0 read [see Eq. (4.15) in [49]] FHB(nlj,Zα) =1 2l+ 1/bracketleftbigg/parenleftbigg17 18−4 3ln 2/parenrightbigg δl,0 +/parenleftbigg3 2−2 ln 2/parenrightbigg1 κ +/parenleftbigg5 6−ln 2/parenrightbiggn−2l−1 n/bracketrightbigg + O(Zα). (4.22) For the atomic states under investigation, this leads to FHB(1S1/2,Zα) =−0.093 457 + O( Zα), FHB(2S1/2,Zα) =−0.023 364 + O( Zα), FHB(2P1/2,Zα) = 0.014 538 + O( Zα), FHB(2P3/2,Zα) =−0.042 315 + O( Zα). (4.23) The integration procedure for the high-energy part is adapted to the problem at hand. To this end, a crude estimate is found for the dependence of the function SHB deﬁned in Eq. (4.18) on its arguments. The consider- ations leading to this estimate are analogous to those outlined in Sec. III A for the infrared part. The result is the approximate expression e−yexp/bracketleftbigg −/parenleftbigg1 at−1 2/parenrightbigg (1−r)y/bracketrightbigg (4.24) forSHB. This leads naturally to the deﬁnition 14qHB= 1 +/parenleftbigg1 at−1 2/parenrightbigg (1−r), (4.25) so that the (approximate) dependence of SHBon the ra- dial variable at large yis exp ( −qHBy). Note that qHB may assume large values ( ≫1) ast→0; this is unlike the analogous quantity 1 +/parenleftbiggc a−1 2/parenrightbigg (1−r) in the infrared and the middle-energy part, where \|c\|= \|√ 1−z2\|<1 becausez∈(0,En). Having identiﬁed the leading exponential asymptotic behavior of the integrand SHB, it is rather straightforward to evaluate the three- dimensional integral in Eq. (4.20) by Gauss-Laguerre and Gauss-Legendre quadrature [5] [the scaling parameter a is deﬁned in Eqs. (3.9) and (3.14)]. The numerical results for the high-energy remainder function FHBare found in Table IV. These results are consistent with the limits in Eq. (4.23). We now turn to a brief discussion of the convergence acceleration techniques used in the evaluation of the func- tionSHBdeﬁned in Eq. (4.18). The angular momen- tum decomposition of SHBgives rise to a sum over the termsTHB,\|κ\|[see the last line of Eq. (4.18)], where \|κ\| represents the modulus of the Dirac angular momentum quantum number of the virtual intermediate state. In shorthand notation, and suppressing the arguments, we have SHB∝∞/summationdisplay \|κ\|=1THB,\|κ\|. (4.26) The radial Green function GB=GB(ry/a,y/a,z ) in co- ordinate space needs to be evaluated at the radial argu- mentsry/aandy/a(where 0<r< 1), and at the energy argumentz=En−ω= i/2 (t−1−t) [see Eq. (4.18)]. A crucial role is played by the ratio rof the two radial ar- guments. Indeed, for \|κ\| → ∞ , we have [see Eq. (4.7) in [48]] THB,\|κ\|=r2\|κ\| \|κ\|/bracketleftbigg const.+ O/parenleftbigg1 \|κ\|/parenrightbigg/bracketrightbigg , (4.27) where “const .” is independent of \|κ\|and depends only on r,yandt. The series in Eq. (4.26) is slowly convergent forrclose to one, and the region near r= 1 is known to be problematic in numerical evaluations. Additionally, note that the region at r= 1 is more important at low Zthan at high Z. This is because the function SHB, for constanty, depends on rroughly as exp [ −y(1−r)/(at)] [see Eq. (4.24)], where a= 2 (Zα)/n+ O[(Zα)3]. For smallZ, the Bohr radius 1 /(Zα) of the hydrogenlike system is large compared to high- Zsystems, which em- phasizes the region near r= 1. In this region, the se- ries in (4.26) is very slowly convergent. We have foundthat the convergence of this series near r= 1 can be ac- celerated very eﬃciently using the combined nonlinear- condensation transformation [10] applied to the series/summationtext∞ k=0tkwheretk=THB,k+1[see Eqs. (4.26) and (4.27)]. We ﬁrst transform this series into an alternating series by a condensation transformation due to Van Wijngaar- den [55,56], ∞/summationdisplay k=0tk=∞/summationdisplay j=0(−1)jAj, (4.28) where Aj=∞/summationdisplay k=02kt2k(j+1)−1. (4.29) We then accelerate the convergence of the alternating se- ries/summationtext∞ j=0(−1)jAjby applying the nonlinear delta trans- formδ(0) n(1,S0), which is discussed extensively in [57]. The explicit formula for this transformation is given by deﬁning Sn=n/summationdisplay j=0(−1)jAj (4.30) as thenth partial sum of the Van Wijngaarden trans- formed input series. The delta transform reads [see Eq. (8.4-4) of [57]], δ(0) n(1,S0) =n/summationdisplay j=0(−1)j/parenleftbiggn j/parenrightbigg(1 +j)n−1 (1 +n)n−1Sj Bj+1 n/summationdisplay j=0(−1)j/parenleftbiggn j/parenrightbigg(1 +j)n−1 (1 +n)n−11 Bj+1,(4.31) where Bj= (−1)jAj. (4.32) The convergence acceleration proceeds by calculating a sequence of transforms δ(0) nin increasing transformation ordern. It is observed that the transforms converge much faster than the partial sums Sndeﬁned in Eq. (4.30). The upper index zero in Eq. (4.31) indicates that the trans- formation is started with the ﬁrst term A0. The combined transformation (combination of the con- densation transformation and the Weniger transforma- tion) was found to be applicable to a wide range of slowly convergent monotone series (series whose terms have the same sign), and many examples for its application were given in Ref. [10]. For the numerical treatment of radia- tive corrections in low- Zsystems, the transformation has the advantage of removing the principal numerical diﬃ- culties associated with the slow convergence of angular momentum decompositions of the propagators near their singularity for equal radial arguments. In a typical case, suﬃcient precision (10−11) in the convergence of the sum in Eq. (4.26) is reached in a 15transformation order n <100 for the nonlinear trans- formationδ(0) n(1,S0), a region in which the nonlinear se- quence transformation δis numerically stable. Although the delta transformation exhibits considerable numeri- cal stability in higher transformation orders [10,57], in- evitable round-oﬀ errors start to accumulate signiﬁcantly in an excessively high transformation order of n≈500 in a typical case [5], and this situation is avoided in the current evaluation because the transforms exhibit appar- ent convergence to the required accuracy before numer- ical round-oﬀ errors accumulate. Note that evaluation of the condensed series Ajin Eq. (4.29) entails sam- pling of terms THB,\|κ\|for rather large \|κ\|, while elimi- nating the necessity of evaluating alltermsTHB,\|κ\|up to the maximum index. The highest angular momentum \|κ\|encountered in the present calculation is in excess of 4 000 000. However, even in extreme cases less than 3 000 evaluations of particular terms of the original se- ries are required. The computer time for the evaluation of the slowly convergent angular momentum expansion near the singularity is reduced by roughly three orders of magnitude by the use of the convergence acceleration methods. In certain parameter regions (e.g. for large energy of the virtual photon), a number of terms of the input series tkhave to be skipped before the convergence acceleration algorithm deﬁned in Eqs. (4.28)–(4.32) can be applied (in order to avoid transient behavior of the ﬁrst few terms in the sum over κ). In this case, the input data for the combined nonlinear-condensation transformation are the termstk=THB,k+1+κs, whereκsdenotes the number of terms which are directly summed before the transforma- tion is applied. These issues and further details regarding the application of the convergence acceleration method to QED calculations can be found in Appendix H.2 of [5].C. Results for the High-Energy Part The limit of the function FHasZα→0 can be derived easily from Eqs. (4.14), (4.22) as a function of the bound state quantum numbers. For FHthe limit is FH(nlj,Zα) = =/parenleftbigg11 10−ln 2/parenrightbigg1 n+/parenleftbigg16 15−2 ln 2/parenrightbigg1 2l+ 1 +/parenleftbigg −3 2ln 2 +5 4/parenrightbigg1 κ(2l+ 1)+/parenleftbigg3 2ln 2−9 4/parenrightbigg1 \|κ\| +/parenleftbigg17 18−4 3ln 2/parenrightbigg δl,0+/parenleftbigg5 6−ln 2/parenrightbiggn−2l−1 n(2l+ 1) +O(Zα). (4.33) For the atomic states investigated here, this expression yields the numerical values FH(1S1/2,Zα) =−1.313 085 + O( Zα), FH(2S1/2,Zα) =−1.446 418 + O( Zα), FH(2P1/2,Zα) =−1.066 667 + O( Zα), FH(2P3/2,Zα) =−0.566 667 + O( Zα). (4.34) Numerical results for the high-energy part FH(nlj,Zα) =FHA(nlj,Zα) +FHB(nlj,Zα) (4.35) are also summarized in Table IV. Note the apparent con- sistency of the numerical results in Table IV with their analytically obtained low- Zlimits in Eq. (4.34). TABLE V. Numerical results for the scaled self-energy funct ionFand the self-energy remainder function GSE. Z F (1S1/2, Zα) F(2S1/2, Zα) F(2P1/2, Zα) F(2P3/2, Zα) 1 10.316 793 659(1) 10.546 825 185(5) −0.126 396 37(1) 0.123 498 56(1) 2 8.528 325 061(1) 8.758 870 25(1) −0.125 816 16(1) 0.123 835 55(1) 3 7.504 503 432(1) 7.735 777 20(1) −0.124 992 24(1) 0.124 317 10(1) 4 6.792 824 089(1) 7.025 002 41(1) −0.123 968 79(1) 0.124 918 48(1) 5 6.251 627 086(1) 6.484 860 42(1) −0.122 774 94(1) 0.125 623 30(1) Z G SE(1S1/2, Zα) GSE(2S1/2, Zα) GSE(2P1/2, Zα) GSE(2P3/2, Zα) 1 −30.290 24(2) −31.185 15(9) −0.973 5(2) −0.486 5(2) 2 −29.770 967(5) −30.644 66(5) −0.949 40(5) −0.470 94(5) 3 −29.299 169(2) −30.151 93(2) −0.926 37(2) −0.456 65(2) 4 −28.859 223(1) −29.691 27(1) −0.904 12(1) −0.443 13(1) 5 −28.443 372 3(8)a−29.255 033(8) −0.882 478(8) −0.430 244(8) aThe result for this entry given in [4] contains a typographic al error. 16V. COMPARISON TO ANALYTIC CALCULATIONS The numerical results for the scaled self-energy func- tionF(nlj,Zα) deﬁned in Eq. (2.1) are given in Table V, together with the results for the nonperturbative self- energy remainder function GSE(nlj,Zα), which is im- plicitly deﬁned in Eq. (2.2). Results are provided for K- and L-shell states. The results here at Z= 5 are consistent with and much more precise than the best previous calculation [46]. The numerical results for the self-energy remainder GSEare obtained by subtracting the analytic lower-order terms listed in Eq. (2.2) from the complete numerical result for the scaled self-energy functionF(nlj,Zα). No additional ﬁtting is performed. Analytic and numerical results at low Zcan be com- pared by considering the self-energy remainder function GSE. Note that an inconsistency in any of the ana- lytically obtained lower-order terms would be likely to manifest itself in a grossly inconsistent dependence of GSE(nlj,Zα) on its argument Zα; this is not observed. For S states, the following analytic model for GSEis commonly assumed, which is motivated in part by a renormalization-group analysis [58] and is constructed in analogy with the pattern of the analytic coeﬃcients Aij in Eq. (2.2) and (2.3) GSE(nS1/2,Zα) =A60(nS1/2) +(Zα)/bracketleftbig A71(nS1/2) ln(Zα)−2+A70(nS1/2)/bracketrightbig +(Zα)2/bracketleftbig A83(nS1/2) ln3(Zα)−2 +A82(nS1/2) ln2(Zα)−2 +A81(nS1/2) ln(Zα)−2+A80(nS1/2)/bracketrightbig .(5.1) The (probably nonvanishing) A83coeﬃcient, which intro- duces a triple logarithmic singularity at Zα= 0, hinders an accurate comparison of numerical and analytic data forGSE. A somewhat less singular behavior is expected of the diﬀerence ∆GSE(Zα) =GSE(2S1/2,Zα)−GSE(1S1/2,Zα),(5.2) because the leading logarithmic coeﬃcients in any given order ofZαare generally assumed to be equal for all S states, which would mean in particular A71(1S1/2) =A71(2S1/2) and A83(1S1/2) =A83(2S1/2). (5.3) Now we deﬁne ∆ Aklas the diﬀerence of the values of the analytic coeﬃcients for the two lowest S states: ∆Akl=Akl(2S1/2)−Akl(1S1/2). (5.4) The function ∆ GSEdeﬁned in Eq. (5.2) can be as- sumed to have the following semi-analytic expansion aboutZα= 0:∆GSE(Zα) = ∆A60+ (Zα)∆A70 +(Zα)2/bracketleftbig ∆A82ln2(Zα)−2 +∆A81ln(Zα)−2+ ∆A80+ o(Zα)/bracketrightbig . (5.5) In order to detect possible inconsistencies in the numer- ical and analytic data for GSE, we diﬀerence the data for ∆GSE, i.e., we consider the following ﬁnite diﬀerence approximation to the derivative of the function ∆ GSE: g(Z) = ∆GSE((Z+ 1)α)−∆GSE(Zα). (5.6) We denote the analytic and numerical limits of ∆GSE(Zα) asZα→0 as ∆A(an) 60and ∆A(nu) 60, respec- tively, and leave open the possibility of an inconsistency between numerical and analytic data by keeping ∆ A(nu) 60 and ∆A(an) 60as distinct variables. In order to illustrate how a discrepancy could be detected by investigating the functiong(Z), we consider special cases of the function ∆GSE(Zα) andg(Z). We have for Z= 0, which is de- termined exclusively by analytic results, ∆GSE(0) = ∆A(an) 60, (5.7) whereas for Z= 1, which is determined by numerical data, ∆GSE(α) = ∆A(nu) 60+α[∆A70+ o(α)], (5.8) and forZ= 2, ∆GSE(2α) = ∆A(nu) 60+α[2 ∆A70+ o(α)],(5.9) etc. Hence for Z= 0, we have g(0) = ∆GSE(α)−∆GSE(0) = ∆A(nu) 60−∆A(an) 60+α[∆A70+ o(Zα)].(5.10) ForZ= 1, the value of gis determined solely by numer- ical data, g(1) = ∆GSE(2α)−∆GSE(α) =α[∆A70+ o(Zα)], (5.11) and forZ= 2, we have g(2) = ∆GSE(3α)−∆GSE(2α) =α[∆A70+ o(Zα)]. (5.12) Analogous equations hold for Z >2. The analytic and the numerical data from Table V lead to the ﬁve values g(0),g(1),g(2),g(3), andg(4). A plot of the function g(Z) serves two purposes: First, the values g(1),...,g (4) should exhibit apparent convergence to some limiting valueα∆A70asZ→0, and this can be veriﬁed by in- spection of the plot. Secondly, a discrepancy between the analytic and numerical approaches would result in a non- vanishing value for ∆ A(nu) 60−∆A(an) 60which would appear as an inconsistency between the trend in the values of g(1),...,andg(4) and the value of g(0) [see Eq. (5.10)]. 170 1 2 3 40 Z0.02000.02050.02100.02150.0220g(Z) FIG. 4. Plot of the function g(Z) deﬁned in Eq. (5.6) in the region of low nuclear charge. For the evaluation of the data point at Z= 0, a value of A60(1S1/2) =−30.924 15(1) is employed [4,15,59]. Among the separate evaluations of A60for the ground state, the result in [15] has the smallest quoted uncer- tainty. In Fig. 4 we display a plot of g(Z) for low nu- clear charge Z. A value of A60(1S1/2) =A(an) 60(1S1/2) = −30.92415(1) [4,15,59] is used in Fig. 4. The results in- dicate very good agreement between the numerical and analytic approaches to the Lamb shift in the low- Zre- gion up to the level of a few Hz in frequency units for the low-lying atomic states (where nis the principal quan- tum number). The error bars represent the numerical uncertainty of the values in Table V, which correspond to an uncertainty on the level of 1 .0×Z4Hz in frequency units. Analytic work on the correction A60has extended over three decades [15, 34–37]. The complication arises that although the calculations are in general analytic, some re- maining one-dimensional integrations could not be evalu- ated analytically because of the nature of the integrands [see e.g. Eq. (6.96) in [15]]. Therefore a step-by-step com- parison of the analytic calculations is diﬃcult. An addi- tional diﬃculty is the isolation of those analytic terms which contribute in a given order in Zα, i.e., the isola- tion of only those terms which contribute to A60. The apparent consistency of the numerical and analytic data in Fig. 4 represents an independent consistency check on the rather involved analytic calculations.0 1 2 3 40 Z0.0200.0210.0220.0230.0240.0250.0260.0270.028g1/2(Z) FIG. 5. Comparison of numerical data and analyti- cally evaluated higher-order binding corrections for the 2 P1/2 state. We plot the function g1/2(Z) deﬁned in Eq. (5.15) in the region of low Z. The numerical data obtained in the cur- rent investigation appear to be consistent with the analyti c result of A60(2P1/2) =−0.998 91(1) obtained in [16]. Our numerical results are not inconsistent with the an- alytic result [6] for a higher-order logarithm, A71=π/parenleftbigg139 64−ln 2/parenrightbigg = 4.65, (5.13) although they do not necessarily conﬁrm it. As in [4], we obtain as an estimate A71= 5.5(1.0) (from the ﬁt to the numerical data for both S states). Logarithmic terms corresponding to the (probably) nonvanishing A83coef- ﬁcient should be taken into account for a consistent ﬁt of the corrections to GSE. These highly singular terms are diﬃcult to handle with a numerical ﬁtting procedure. The termsA83,A82andA81furnish three more free pa- rameters for the numerical ﬁt, where only ﬁve data points are available (in addition to the quantities A60,A71and A70, which may also be regarded as free parameters for the ﬁtting procedure). The determination of A60by a ﬁt from the numerical data is much more stable than the de- termination of the logarithmic correction A71. We brieﬂy note that our all-order evaluation essentially eliminates the uncertainty due to the unknown higher-order ana- lytic terms. Also, it is interesting to note that the same numerical methods are employed for both the S and P states in our all-order (in Zα) calculation, whereas the analytic treatment of S and P states diﬀers [15,16]. The comparison of numerical and analytic results is much less problematic for P states, because the function GSEis less singular [see Eqs. (2.4) and (2.6)]. For the 2P states, we observe that the function GSE(2Pj,Zα) has the same semi-analytic expansion about Zα= 0 as the function ∆GSE(Zα) deﬁned for S states in Eq. (5.2). We have 18GSE(2Pj,Zα) =A60(2Pj) + (Zα)A70(2Pj) +(Zα)2/bracketleftbig A82(2Pj) ln2(Zα)−2 +A81(2Pj) ln(Zα)−2+A80(2Pj) + o(Zα)/bracketrightbig .(5.14) Hence, we plot the function gj(Z) =GSE(2Pj,(Z+ 1)α)−GSE(2Pj,Zα)(5.15) forj= 1/2 andj= 3/2 in the region of low Z, with the notion that an inconsistent analytic result for A60(2Pj) would lead to irregularity at Z= 0, in analogy with the S states. The numerical data shown in Figs. 5, and 6 appear to be consistent with the analytic results of A60(2P1/2) =−0.998 91(1) and A60(2P3/2) =−0.503 37(1) (5.16) obtained in [16]. In this context it may be interesting to note that analytic results obtained in [16,52] for the higher-order binding corrections to 2P, 3P, and 4P states have recently been conﬁrmed indirectly [60]. Finally, al- though it may be possible to obtain more accurate esti- mates of some higher-order analytic corrections, notably theA70coeﬃcient for P states and ∆ A70for the two lowest-lying S states, we have not made such an analysis in the current work; we have restricted the discussion to a check of the consistency with the available results for A60. 0 1 2 3 40 Z0.0100.0120.0140.0160.0180.020g3/2(Z) FIG. 6. For the 2P 3/2state, we plot the function g3/2(Z) deﬁned in Eq. (5.15) in the region of low Z. The numerical data obtained in the current investigation appear to be con- sistent with the analytic result of A60(2P3/2) =−0.503 37(1) from [16]. VI. CONCLUSION There has recently been a rather broad interest in the numerical calculation of relativistic, QED self energy, an dtwo-body corrections at low Zand the comparison of an- alytic and numerical results [58,61–72]. Traditionally, t he self-energy correction for hydrogenlike systems has posed a computational challenge. Here we have described a nonperturbative evaluation of the one-photon self-energy correction in hydrogenlike ions with low nuclear charge numbersZ= 1 to 5. The general outline of our approach is discussed in Sec. II. In Sec. III, the numerical evalua- tion of the low-energy part (generated by virtual photons of low energy) is described. In Sec. IV, we discuss the nu- merical evaluation of the high-energy part, which is gen- erated by high-energy virtual photons and contains the formally inﬁnite contributions, which are removed by the renormalization. Sec. IV also contains a brief discussion of the convergence acceleration methods as employed in the current evaluation. We discuss in Sec. V the com- parison of analytic and numerical data for K- and L-shell states in the region of low Z. The main results of this paper are contained in Table V: numerical data, non- perturbative in Zα, for the scaled self-energy function F and the self-energy remainder function GSEfor K- and L-shell states at low nuclear charge. The numerical ac- curacy of our data is 1 Hz or better in frequency units for 1S, 2S and both 2P states in atomic hydrogen. The comparison of analytic and numerical results to the level of accuracy of the numerical data, which is dis- cussed in Sec. V, indicates that there is very good agree- ment for the K- and L-shell states. The analytic and numerical data are shown in Figs. 4, 5, and 6. Our all- order evaluation eliminates any uncertainty due to the unknown higher-order analytic terms; the current numer- ical uncertainty in the self energy is at the level of 1 Hz for atomic hydrogen. ACKNOWLEDGMENTS U. D. J. thanks the National Institute of Standards and Technology for kind hospitality during a number of extended research appointments. He would also like to acknowledge support from the Deutscher Akademis- cher Austauschdienst (DAAD). The authors would like to acknowledge helpful discussions with K. Pachucki, S. Karshenboim and J. Sims. P. J. M. acknowledges the Alexander von Humboldt Foundation for continued sup- port. The authors wish to acknowledge support from BMBF, DFG and from GSI. ∗Electronic address: jentschura@physik.tu-dresden.de. ‡Electronic address: mohr@nist.gov. §Electronic address: soﬀ@physik.tu-dresden.de. 19[1] B. de Beauvoir, F. Nez, L. Julien, B. Cagnac, F. Biraben, D. Touahri, L. Hilico, O. Acef, A. Clairon, and J. J. Zondy, Phys. Rev. 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arXiv:physics/0009091v1 [physics.atom-ph] 28 Sep 2000Electromagnetically induced absorption in magneto-optic ally trapped atoms. A. Lipsich, S. Barreiro, P. Valente and A. Lezama∗ Instituto de F ´isica, Facultad de Ingenier ´ia. Casilla de correo 30. 11000, Montevideo, Uruguay. (February 2, 2008) Electromagnetically induced absorption (EIA) was ob- served on a sample of85Rbin a magneto-optical trap using low intensity cw copropagating pump and probe optical ﬁelds. At moderate trapping ﬁeld intensity, the EIA spectrum is deter - mined by the Zeeman eﬀect produced on the atomic ground- state by the trapping quadrupolar magnetic ﬁeld. The use of EIA spectroscopy for the magnetic ﬁeld mapping of cold atomic samples is illustrated. 42.50.Gy, 32.80.Qk, 32.80.Pj, 32.60.+i. I. INTRODUCTION. The interaction of radiation with a coherently prepared atomic medium has attracted considerable attention in recent years [1]. While most coherent processes have been studied using three or more atomic energy levels, it was recently demonstrated that degenerate two-level systems constitute an interesting host for coherent spec- troscopical eﬀects [2–4]. A new coherent eﬀect observed in degenerate two-level systems, is the increase of the atomic absorption occurring when the lower atomic en- ergy level has a smaller (angular momentum) degeneracy than the upper level and the frequencies of two mutually coherent optical ﬁelds match the condition for Raman resonance between ground-state Zeeman sublevels. This phenomenon, the change in sign taken apart, share sev- eral properties with electromagnetically induced trans- parency (EIT) [5] and was designated electromagnetically induced absorption (EIA) [2,3]. The spectral properties of EIA depending on ﬁeld intensity, magnetic ﬁeld and optical ﬁelds polarizations were recently analyzed [6]. In particular, the sensitive dependence of the EIA spectrum on magnetic ﬁeld makes it a potentially useful tool for the characterization of the magnetic atomic environment [7]. To date, the reported observations of EIA were carried in vapor samples. It is worth noticing that the atomic tran- sitions normally used for magneto-optical trapping and cooling verify the conditions for EIA [3]: they are closed transitions with a larger angular momentum in the ex- cited state than in the ground state. This paper presents the ﬁrst observation of EIA on a cold atomic sample in a ∗E-mail: alezama@ﬁng.edu.uymagneto optical trap (MOT) using the trapping transi- tion. It is shown that EIA constitute a simple and direct tool for the inspection of the atomic density distribution with respect to the trapping quadrupolar magnetic ﬁeld [8]. We have performed EIA spectroscopy on a sample of magneto-optically cooled85Rbatoms in the presence of the trapping MOT ﬁeld. In addition to the ﬁelds neces- sary for the MOT operation, the sample was submitted to a ﬁxed frequency pump ﬁeld and a tunable probe ﬁeld with linear and mutually orthogonal polarizations. Both ﬁelds were quasiresonant with the closed 5 S1/2(F= 3)→ 5P3/2(F′= 4) transition of85Rb(trapping transition). A copropagating geometry was used for the pump and the probe waves. This geometry has the advantage of avoiding a spatial dependence of the relative phase be- tween the pump and the probe ﬁeld that would exist, for instance, in a counter-propagating geometry and allow more direct comparison with theory. However, the co- propagating geometry has the disadvantage of requiring very low intensity in both the pump and probe waves to avoid “blowing” the trapped atoms by the radiative force exerted by the quasiresonant ﬁelds. Consequently, EIA was observed only in a regime of pump and probe intensities many orders of magnitude below saturation for which the perturbation of the MOT by the pump and probe ﬁeld was observed to be negligible. Since small ﬁeld intensities were used, the corresponding EIA resonances were small compared to the linear absorption. This re- quired the use of a highly sensitive detection technique in order to distinguish the EIA resonances from the large linear absorption background. A double-frequency lock- in detection was used. II. SETUP. The experiments were done on a standard magneto op- tical trap for85Rbwith three pairs of counterpropagating trapping beams along orthogonal directions. The max- imum available total trapping ﬁeld power was 20 mW and the trapping beam cross section 1 cm. A repump- ing ﬁeld of 1 mWgenerated by an independent laser was superimposed to the trapping beams. The MOT used a quadrupolar magnetic ﬁeld distribution with a magnetic ﬁeld gradient of 1 G/mm along the symmetry axis (vertical). The MOT produced a cold atom cloud of typical 0 .7mmtransverse dimension containing (at 1maximum trapping ﬁeld power) 107atoms. The trap- ping and pump ﬁelds were obtained from the same (fre- quency stabilized) extended cavity diode laser (master laser) using in each case the following procedure: the output of the master laser was frequency shifted by an acousto optic modulator (AOM) to a ﬁxed frequency po- sition relative to the 5 S1/2(F= 3)→5P3/2(F′= 4) atomic transition and used to injection-lock a separate laser diode for power increase. As in [2], the probe ﬁeld was obtained from the pump by the use of two addi- tional consecutive AOMs, one of them driven by a tun- able RF generator. In this way, all three trapping, pump and probe ﬁelds were highly correlated. The pump and probe waves with orthogonal linear polarizations are su- perimposed on a 50 cmlong polarization-preserving sin- gle mode optical ﬁber and subsequently focussed on the cold atoms sample (0 .3mmbeam waist). Intensity was controlled with neutral density ﬁlters in order to have approximately the same power in the pump and probe waves. At the sample, both waves had a central inten- sity around 0 .1µW/cm2. The pump and the probe ﬁelds were mechanically chopped at frequencies f1andf2(1 and 1 .2kHzrespectively). After the sample, a photo- diode detected the transmitted light. The photodiode output was analyzed with a lock-in ampliﬁer using as reference the sum frequency f=f1+f2. With this tech- nique, only the nonlinear component of the absorption, proportional to the product of the pump and probe ﬁeld intensities was detected. The spectra were recorded by scanning the frequency oﬀset between the tunable probe and the ﬁxed-frequency pump. III. RESULTS AND DISCUSSION. An example of a low resolution nonlinear absorption spectrum as a function of the frequency diﬀerence be- tween the probe and the pump ﬁelds is presented in Fig. 1. In this spectrum, the pump frequency is kept ﬁxed at the center of the 5 S1/2(F= 3)→5P3/2(F′= 4) transi- tion of85Rb. The spectrum presents three distinct fea- tures. Two of these are centered around δ= 0. There is a broad resonance (dip) whose width is given by the ex- cited state width Γ /2π= 6MHz . At the bottom of this resonance there is a much narrower one with opposite sign. The broad resonance corresponds to an increase in the transmission due to the saturation of the sample by the pump ﬁeld while the narrow feature represents an in- crease of the absorption. The latter corresponds to EIA. The sub-natural width of this resonance is an indication of its coherent nature. The inset in Fig. 1 shows the cal- culated signal around δ= 0 using the model described in [3,6]. An interesting feature in the spectrum in Fig. 1 is the large dip appearing around δ≃ −10MHz . This resonance occurs when the probe frequency equals the frequency of the trapping beams (red-detuned 10 MHzwith respect to the atomic transition). Since, as a conse- quence of the technique used for the ﬁelds preparation, the probe, pump and trapping beams are highly corre- lated, it is rather natural to observe a coherent resonance involving the probe and the trapping ﬁelds. However, keeping in mind the two-frequencies detection technique used, the pump ﬁeld must also participate in the gen- eration of any detected signal. Consequently the peak observed around δ≃ −10MHz corresponds to a non- linear process involving at least one photon from each of the tree ﬁelds (pump, probe and trapping). The study of this multiphoton eﬀect is beyond the scope of this paper. We focus our attention on the EIA structure observed around δ= 0. Fig. 2(a) presents a detailed view of this structure observed in a MOT with a total trapping beams power attenuated to approximately 4 m W/cm2. Under this condition the linear absorption by the cold atoms was maximum (40%) at the trapping transition. A triplet structure is clearly visible which is reminiscent of the EIA spectrum obtained in a homogeneous sample for perpen- dicular pump and probe polarizations in the presence of a magnetic ﬁeld orthogonal to both polarizations [6]. The observed value of the frequency separation between the center and the two sidebands would correspond in that case to a magnetic ﬁeld of B≈0.3G. This ﬁgure is of the order of the typical magnetic ﬁeld magnitude within the cold atoms volume due to the quadrupolar distribution. Since in our experiment the pump and probe beams over- lap a signiﬁcant fraction (20%) of the cold atomic sample, the EIA spectrum is due to the average contribution from atoms in diﬀerent magnetic environments. The hypothesis that the observed structure of the EIA spectrum is determined by the magnetic ﬁeld present at the MOT can be tested by adding a small bias magnetic ﬁeld at the sample. This was achieved by varying the current in the Helmholtz coils surrounding the trap that are normally used to compensate the earth ﬁeld. The eﬀect of the bias ﬁeld is to change the magnetic ﬁeld distribution at the position explored by the pump and probe waves. It also produces a translation of the cloud that results in a variation in the total number of atoms interacting with the pump and probe waves. Signiﬁcant changes in the relative weight and positions of the res- onances occur for diﬀerent bias magnetic ﬁelds. Since in these measurements the pump, probe and trapping ﬁelds are kept unchanged, we conclude that the observed spectral structure is mainly determined by the magnetic ﬁeld. When the pump-probe beam was carefully aligned through the atomic cloud for maximum linear absorp- tion, the observed structure of the EIA spectrum was a triplet [Fig. 2(a)]. This structure represents the total response of the magnetically inhomogeneous sample. We have compared this observation with the numerical cal- culation of the response arising from 1000 atoms in diﬀer- ent positions in an ideal MOT. We used a Monte Carlo procedure where the position of the atom is randomly 2chosen (uniform distribution) within an oblate ellipsoida l volume, representing the trapped cloud, centered at the zero of a quadrupolar magnetic ﬁeld. The vertical dimen- sion (along the symmetry axis) of the ellipsoid is shorter by a factor of two than the horizontal one. The abso- lute value of the magnetic ﬁeld at the points where the principal axis intercept the ellipsoid surface is BMAX. The pump ﬁeld polarization is vertical and the probe po- larization horizontal. For each atomic position the local magnetic ﬁeld is evaluated [8] and the corresponding non- linear response is calculated using the model described in [6]. Then the total absorption is calculated as the sum of the individual atomic contributions. The result of the simulation is shown in Fig. 2(b) where a realistic value ofBMAX= 0.8Ghas been chosen. The total absorption spectrum is a triplet in agreement with our observation. Unlike, the ideal ellipsoidal sample assumed for the cal- culation, the actual cloud of cold atoms is not symmetric around the zero of the magnetic ﬁeld distribution. This is due to power imbalance and wavefront irregularities in the counterpropagating trapping beams. This produces a cold cloud displaced from the zero of the magnetic ﬁeld with an asymmetric shape and an irregular atomic den- sity distribution. This explains the fact that varying the position of the pump-probe beam across the atomic sam- ple we were able to observe signiﬁcant variations of the nonlinear absorption shape ranging from a single-peak spectrum, a doublet, a triplet or even a ﬁve-peaks struc- ture. Considering that the EIA spectra were recorded with the trapping ﬁeld on, it is somehow surprising to ob- serve spectral structures determined by the Zeeman shift of ground-state sublevels by the quadrupolar magnetic ﬁeld. In the trapping region, the resonant interaction of the atoms with the trapping ﬁeld results in signiﬁcant displacement of the atomic levels due to (AC Stark) light shift. The magnitude of the light shift is diﬀerent for dif- ferent ground-state sublevels and depends on the local trapping ﬁeld intensity and polarization experienced by the atom. The six trapping beams are responsible for a rather complicate interference pattern resulting in three dimensional modulation of the trapping ﬁeld intensity and polarization with a spatial period of half wavelength [9]. The importance of the ground-state sublevels light shifts in a MOT is well illustrated by the peculiar shape of the absorption spectra of the trapped atoms in the MOT [10] analyzed by Grison and coworkers [11]. The inﬂu- ence of the ground-state Zeeman sublevels light shifts on the absorption spectra of strongly driven degenerate two- level systems was studied in [12]. Since the light shifts are uncorrelated with the Zeeman shift produced by the quadrupolar magnetic ﬁeld, one should expect that the former will result in an inhomogeneous dispersion of the energies of the ground-state sublevels producing a smear- ing of the EIA resonance frequencies. We have observed the variation of the EIA spectrumwith the intensity of the trapping ﬁeld (Fig. 3). For low trapping ﬁeld intensities a well resolved triplet was ob- tained. The width of the central peak of the upper trace in Fig. 3 is 30 kHz. This is to our knowledge the narrow- est coherent resonance observed to date in an operating MOT [10,11,13–15]. As the trapping ﬁeld intensity is in- creased, the EIA resonances broaden. For large enough trapping ﬁeld intensity, the triplet structure transforms into a unique broad peak. It is worth noticing that a regime can be found where the trapped cloud is rather dense (linear absorption around 40%) while at the same time the magnetic-ﬁeld-dependent structure of the EIA spectrum remains clearly resolved. In this regime the modiﬁcation of the ground-state sublevels by the light shift produced by the trapping ﬁeld is small compared to the Zeeman shift. The fact that the EIA spectrum reﬂects the magnetic- ﬁeld distribution within the trapped atoms cloud may be used for mapping the magnetic-ﬁeld variations inside the cold atoms sample. This possibility is illustrated in ﬁgure 4 showing the modiﬁcation of the EIA spectrum as the pump-probe beam is translated in the equatorial plane of the cold atom cloud. Between two consecutive traces of Fig. 4 the pump-probe beam was translated by 100 µm. From bottom to top, the spectra in ﬁgure 4 evolve from a well resolved tripled into a broad unre- solved peak. The series in Fig. 4 may be interpreted as a mapping of the magnetic ﬁeld inside the cold atoms cloud averaged over the volume explored by the pump-probe beam. In these measurements the pump-probe beam di- ameter at the sample is 0 .3mma ﬁgure which is only a factor of 2 .3 smaller than the estimated cold atoms cloud diameter. This limits in our case the spatial res- olution of the magnetic ﬁeld mapping. A two orders of magnitude improvement in the resolution is in principle possible with a tight focussed pump-probe beam. The spectra presented in Fig. 4 clearly indicate the lack of symmetry of our MOT. As a matter of fact, the zero of the magnetic ﬁeld, around which symmetric variations of the EIA spectrum should take place, is located outside the region of large atomic density. The mapping of the magnetic ﬁeld using EIA spec- troscopy, appear as a rather simple and straightforward technique. It can be achieved in a cw regime without the need for turning oﬀ the MOT trapping beams and with very low-power pump and probe ﬁelds. As demonstrated here, these ﬁelds can be easily obtained from the trapping laser. As in the example shown, qualitative information about the atomic density distribution with respect to the quadrupolar magnetic ﬁeld is directly inferred from the spectra. The ﬁtting of the experimental EIA spectra to theoretical predictions (based on consistent assumptions for the trapped cloud density distribution) can in princi- ple be used as a tomographic technique for the character- ization of the atoms in the MOT [8]. This approach was not developed in the present case in view of the irregular 3atomic density distribution observed in our MOT. IV. CONCLUSIONS. We have observed EIA resonances on a sample of cold atoms in a MOT in the presence of the trapping light beams. An operating regime of the MOT (with moder- ate trapping beams intensity) could be found where the atomic density is quite large while at the same time the atomic ground state is weakly perturbed by the trapping ﬁeld. In this regime the energy shift of the ground state sublevels is essentially governed by the Zeeman eﬀect. Coherent resonances as narrow as 30 kHzwere observed for the ﬁrst time in an operating MOT. EIA absorption spectroscopy, performed in a cw regime with very low optical intensities, appears as a suitable tool for the in- spection of the Zeeman shift produced for the quadrupo- lar magnetic ﬁeld at diﬀerent positions of the cold atom sample. V. ACKNOWLEDGMENTS. The authors are thankful to A. Akulshin for his contri- bution to the early stages of this study and to H. Failache for helpful discussions. This work was supported by the Uruguayan agencies: CONICYT, CSIC and PEDECIBA and by the ICTP/CLAF. [1] See M.O. Scully and M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge (1997) and ref- erences therein. [2] A.M. Akulshin, S. Barreiro and A. Lezama, Phys. Rev. A57, 2996 (1998). [3] A. Lezama, S. Barreiro and A.M. Akulshin, Phys. Rev. A.59, 4732 (1999). [4] A. Akulshin, S. Barreiro and A. Lezama, Phys. Rev. Lett. 83, 4277 (1999). [5] see S. E. Harris, Physics Today, 50(7) 36 (1997) and references therein. [6] A. Lezama, S. Barreiro,. A. Lipsich and A.M. Akulshin, Phys. Rev. A 61, 013801 (2000). [7] M.O. Scully and M. Fleishhauer, Phys. Rev. Lett. 69, 1360 (1992). H. Lee, M. Fleishhauer and M.O. Scully, Phys. Rev. A 58, 2587 (1998). [8] T.A. Savard, S.R.. Granade, K.M. O’Hara, M.E. Gehm and J.E. Thomas, Phys. Rev. A 60, 4788 (1999). [9] S.A. Hopkins and A.V. Durrant, Phys. Rev. A 56, 4012 (1997). [10] J.W.R. Tabosa, G. Chen, Z. Hu, R.B. Lee and H.J. Kim- ble, Phys. Rev. Lett. 66, 3245 (1991).[11] D. Grison, B. Lounis, C. Salomon, J.Y. Courtois and G. Grynberg, Europhys. Lett. 15, 149 (1991). [12] A. Lipsich, S. Barreiro, A.M. Akulshin and A. Lezama, Phys. Rev. A 61, 053803 (2000). [13] T. van der Veldt, J.F. Roch, P. Grelu and P. Grangier, Opt. Commun. 137, 420 (1997). [14] S.A. Hopkins, E. Usadi, H.X. Chen and A.V. Durrant, Opt. Commun. 138, 185 (1997). [15] A.V. Durrant, H.X. Chen, S.A. Hopkins and J.A. Vac- caro, Opt. Commun. 151, 136 (1998). -10 -5 0 5-101 Non-linear absorption (Arb.units) Probe frequency offset (MHz)-5 0 50 FIG. 1. Nonlinear absorption signal as a function of the pump to probe frequency oﬀset. Total trapping beam power 4mW(see text). Pump frequency ﬁxed at the center of the 5S1/2(F= 3)→5P3/2(F′= 4) transition of85Rb. Linear and orthogonal pump and probe polarizations. Inset: calcu- lated signal using the model in [6] for parameters B= 0 and Ω1= 10−3Γ. a -0.4 -0.2 0.0 0.2 0.4bNon-linear absorption signal Probe frequency offset (MHz) 4FIG. 2. a) Nonlinear absorption signal observed with total trapping beam power 4 mW(see text). The pump frequency is ﬁxed at the center of the 5 S1/2(F= 3)→5P3/2(F′= 4) transition of85Rb. b) Calculated nonlinear absorption spectra for a sample 1000 atoms randomly distributed in an oblate ellipsoidal volume centered in a quadrupolar magnetic ﬁeld with maximum value BMAX = 0.8G. The calculation used the model in [6] with parameters: Ω 1= 10−3Γ, γ= 5×10−3Γ. -0.4-0.20.00.20.40.1 0.13 0.16 0.2 0.25 0.4 Probe frequency offset (MHz)Non-linear absorption signal FIG. 3. Nonlinear absorption spectra for diﬀerent intensi- ties of the trapping ﬁeld. The relative trapping ﬁeld power i s indicated above each trace.-1.0 -0.5 0.0 0.5 1.0 Probe frequency offset (MHz)Non-linear absorption signal FIG. 4. Nonlinear absorption spectra obtained for diﬀerent positions of the pump-probe beam. Consecutive traces corre - spond to a 100 µmhorizontal translation of the pump-probe beam. 5
1 Microtorus: a High Finesse Microcavity with Whispering-Gallery Modes Vladimir S. Ilchenko, Michael L. Gorodetsky†, X. Steve Yao, and Lute Maleki Jet Propulsion Laboratory, California Institute of Technology, MS 298-100, 4800 Oak Grove Dr., Pasadena, CA 91109-8099 † Physics Department, Moscow State University, Moscow 119899 Russia ABSTRACT We have demonstrated a 165 micron oblate spheroidal microcavity with free spectral range 383.7 GHz (3.06nm), resonance bandwidth 25 MHz (Q ≈107) at 1550nm, and finesse F ≥ 104. The highly oblate spheroidal dielectric microcavity combines very high Q-factor, typical of microspheres, with vastly reduced number of excited whispering-gallery (WG) modes (by two orders of magnitude). The very large free spectral range in the novel microcavity - few hundred instead of few Gigahertz in typical microspheres – is desirable for applications in spectral analysis, narrow-linewidth optical and RF oscillators, and cavity QED. OCIS terms: 060.2340 Fiber optics components; 140.4780 Optical resonators 2 Microspherical resonators supporting optical whispering gallery (WG) modes have attracted considerable attention in various fields of research and technology. Combination of very high Q-factor (108-109 and greater) and sub-millimeter dimensions (typical diameters ranging from few tens to several hundred microns), makes these resonators attractive new components for a number of applications, including basic physics research, molecular spectroscopy, narrow-linewidth lasers, optoelectronic oscillators (OEO), and sensors [1-4]. Effective methods of coupling light in and out of WG modes in microspheres are currently being developed, including single mode fiber couplers and integrated waveguides [5,6]. Whispering-gallery modes are essentially closed circular waves trapped by total internal reflection (TIR) inside an axially symmetric dielectric body. The very high Q of microspheres results from 1) ultra- low optical loss in the material (typically, fiber-grade fused silica), 2) fire-polished surface with Angstrom- scale inhomogeneities, 3) high index contrast for steep reduction of radiative and scattering losses with increasing radius, and 4) two-dimensional curvature providing for grazing reflection of all wave vector components. Grazing incidence is essential for minimizing surface scattering that would otherwise limit the Q far below that imposed by attenuation in the material [7]. For example, in the integrated optical micro- ring and micro-disk cavities [8] based on planar waveguide technology (the light in planar devices is effectively bouncing from flat surfaces under finite angle), typical Q-factor is only 10 4 to 105. The substantially higher Q in the spheres, as compared to micro-disks and micro-rings, comes at the price of a relatively dense spectrum of modes. In ideal spheres, the spectrum consists of TE(TM) lmq modes separated by “large” free spectral range (FSR) defined by the circumference of the sphere, and related to consecutive values of index l. In silica spheres of diameter 150 to 400 micron, the “large” FSR should be in the range of 437 to 165GHz, or in the wavelength scale, 3.5 to 1.3nm near the center wavelength 1550nm. Each of TE(TM) lmq modes is (2 l+1)-fold degenerate with respect to the index m. Residual nonsphericity lifts this degeneracy and leads to a series of observable TE(TM) lmq modes separated by “small” FSR in the range 6.8 - 2.5GHz, for the same sphere dimensions, center wavelength, and the eccentricity ε 2 = 3×10-2 , typical of current fabrication methods. 3 A relatively dense spectrum complicates important applications of microsphere resonators, such as spectral analysis and laser stabilization, and necessitates using intermediate filtering. In this Letter, we demonstrate a microcavity with a novel geometry that retains two-dimensional curvature confinement, low scattering loss, and very high Q typical of microspheres, and yet approaches spectral properties of the single mode waveguide ring: a highly-oblate spheroidal microcavity, or microtorus. Calculation of the spectrum of the dielectric spheroid (i.e. ellipsoid of revolution) is not a trivial task, even numerically. In contrast to the case of small eccentricity (see, for example, Ref.[9]), analysis in highly-eccentric spheroids cannot be based on simple approximations. Unlike the case of cylindrical or spherical coordinates, orthogonal spheroidal vector harmonics that satisfy boundary conditions may not be derived via a single scalar wave function [10]. Furthermore, the calculation of spheroidal angular and radial functions is also a nontrivial task. To analyze the properties of eigenfrequencies, we shall rely on physical considerations; however the same approximations can be derived from the asymptotic forms of the spheroidal functions. Very good approximation of WG mode eigenfrequencies in dielectric sphere with radius a much larger than the wavelength can be found as follows: 12−−≈ nta nklq lmqχ, ( 1 ) where tlq is the q-th zero of the spherical Bessel function of the order l and χ = n for TE-mode and χ = 1/n for TM-mode. For large l, t lq ~ l + O( l1/3 ), and it may be calculated either directly or approximated via the zeroes of the Airy function ([11]). The meaning of small second term on the right hand side of Eq.(1) can be understood if we remember that 1) a WG mode is quasiclassically a closed circular beam supported by TIR, 2) optical field tunnels outside at the depth 1 /12−nk , and 3) the tangential component of E (TE- mode), or normal of D (TM-mode) is continuous at the boundary. Eigenfrequencies of high-order WG modes ( ml l≈>>;1 ) in dielectric spheres can be approximated via solutions of the scalar wave equation with zero boundary conditions, because most of the energy is concentrated in one component of the field (Eθ for TE-mode and Er for TM-mode). 4 Based on above considerations, let us estimate WG mode eigenfrequencies in oblate spheroids of large semiaxis a, small semiaxis b, and eccentricity 2 21 ab−=ε . Since WG modes are localized in the “equatorial” plane, we shall approximate the radial distribution by cylindrical Bessel function )~( rknJmq m with mq lmq mq T k knaakn ≈− =⊥2 2 ~, where 0)(=mq mTJ and ⊥kis the wavenumber for quasiclassical solution for angular spheroidal functions. For our purposes a rough approximation is enough: m amlk 2 22 11) (2 ε−+−≈⊥; more rigorous considerations can follow the approach given in [12]. Taking into account that Tmq ≈ tlq – (l – m + 1/2), we finally obtain the following approximation: )1 11(21) (2 2~ 1222 2 2 2− −+−+≈+≈+ ≈ −−⊥ ⊥εχ mltTakT k kna na nklq mqmq mq lmq (2) For very small ε, Eq.(2) yields the same value for “small” FSR (frequency splitting between modes with successive m ≈ l) as earlier obtained by the more rigorous perturbation methods in [9]. In addition, we have compared the prediction of the approximation (2), l = 100, with numerically calculated zeroes of the radial spheroidal functions. Even with ε = 0.8, the error is no more than 5% in the estimate of m, m+1 mode splitting and 0.1% in the absolute mode frequencies. For larger l and smaller ε, the error will evidently be smaller. As follows from (2), with increasing eccentricity, “small” FSR – frequency interval between modes with successive m – becomes compatible with the “large” FSR: l k k k k klmq lmq q lm lmq mql / ~) (~) (1 1 − −+ +. In addition, excitation conditions for modes with different m become more selective: e.g. optimal angles for prism coupling vary with ε as 4/1 2) 1(− ⊥−∝ε lmqkk. In our experiment , we prepared a spheroidal (microtorus) cavity by compressing a small sphere of low-melting silica glass between cleaved fiber tips. The combined action of surface tension and axial compression resulted in the desired geometry (see typical microcavity in Fig.1). One of the fiber “stems” was then cut and the whole structure was installed next to a standard prism coupler. The WG mode spectrum was observed using a DFB laser near the wavelength of 1550nm. The laser was continuously 5 frequency-scannable within the range of ~80GHz (by modulating the current), and temperature tunable between 1545.1 to 1552.4 nm. A high-resolution spectrum over 900GHz was compiled from 15 individual scans with 60GHz increments obtained by changing the temperature. The frequency reference for “stitching” the spectral fragments was obtained by recording the fringes of high-finesse ( F ~ 120) Fabry- Perot ethalon (FSR=30GHz) and additional frequency marks provided by 3.75GHz amplitude modulation (Fig.2). Total drift of the FP was less than 400MHz over the full 15-minute measurement time. The compiled spectrum is presented in Fig.3. The spectrum is reduced to only two whispering-gallery modes of selected polarization excited within the “large” free spectral range of the cavity FSR = 383GHz, or 3.06nm in the wavelength domain. The transmission of “parasitic” modes is at least 6dB smaller than that of the principal ones. With individual mode bandwidth of 23MHz, the finesse F = 1.7×10 4 is therefore demonstrated with this micro-resonator. We can compare this result with the predictions of our approximate expression (2). For dimensions of our cavity a = 82.5µm, b = 42.5 µm (ε = 0.86) and the refraction index of the material n = 1.453, the principal mode number for TE-modes at the wavelength 1550nm should be l ≈ 473 and the “large” FSR GHz lO lnacttnac ql lq mql lmq 402)) ( 617.01(2) (23/5 3/2 ,1 ,1 ≈ + + =− =−− − − −π πνν . The “small” free spectral range should be GHznac q ml lmq 382)1 11(22,1, ≈− −=−−ε πνν In the experimental spectrum, frequency separation between the largest peaks 1 and 2 in Fig.3 is equal to 383.7 ± 0.5GHz. It may therefore be attributed as corresponding to the small FSR, in good agreement with the above estimate, if we take into account an approximately 2% uncertainty in the measurement (limited mainly by the precision of geometrical evaluation of cavity dimensions). The separation between the peaks 3 and 4 in Fig.3, is equal to 400.3 ± 0.5GHz, and is in turn close to the estimate of “large” free spectral range. It may be argued that despite the large splitting between the modes with adjacent values of index m (of the order of “large” FSR), one may still expect a dense spectrum resulting from the overlap of many mode families with different principal number l. In practice, however, it is exactly the coincidence in the 6 frequency domain of WG modes with different main index l, and rapidly increasing difference l – m, that should be responsible for effective dephasing of the “idle” modes from the evanescent coupler, resulting in the reduction of modes in the observed spectrum. A complete analytic interpretation is beyond the format of this Letter and should include a wider range spectral mapping, a study of microcavities of different eccentricity, the calculations of field distribution, and analysis of phase-matched excitation in the evanescent coupler. In conclusion , experimental results indicate a substantial reduction (up to 2 orders of magnitude) in the number of excited WG modes in highly oblate spheroids compared to typical microspheres. This reduction in the mode density is obtained without sacrificing the high Q associated with these structures. The novel type of optical microcavity demonstrates a true finesse on the order of 10 4, a free spectral range of the order of few nanometers, and a quality-factor Q ~ 1 ×107. Based on these results, we conclude that a complete elimination of “transverse” WG modes maybe expected in spheroidal cavities of higher eccentricity. Further increase of the Q and the finesse may also be expected with a refinement of the fabrication technique. The decrease in the density of mode spectrum of ultra-high-Q microcavities offers new applications in laser stabilization, microlasers, various passive photonics devices, and in fundamental physics experiments. The original concept and the experimental research described in this paper was conceived and performed at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with NASA. V. S. Ilchenko’s e-mail address is ilchenko@jpl.nasa.gov. 7 REFERENCES 1. V.B.Braginsky, M.L.Gorodetsky, V.S.Ilchenko, Phys.Lett.A 137, 397-397 (1989) 2. R.K.Chang, A.J.Campillo, eds., Optical Processes in Microcavities, World Scientific, Singapore, 1996; H.Mabuchi, H.J.Kimble, Opt.Lett. 19, 749-751 (1994) 3. V.V.Vassiliev, V. L.Velichansky, V.S.Ilchenko, M.L.Gorodetsky, L.Hollberg, A.V.Yarovitsky, Opt. Commun. 158, 305-312 (1998) 4. V.S.Ilchenko, X.S.Yao, L.Maleki, Proc. SPIE 3611 , 190-198 (1999) 5. V.S.Ilchenko, X.S.Yao, L.Maleki, Opt.Lett. 24, 723-725 (1999) 6. B.E.Little, J.-P.Laine, D.R.Lim, H.A.Haus, L.C.Kimerling, S.T.Chu, Opt.Lett. 25, 73-75 (2000) 7. M.L.Gorodetsky, A.D.Pryamikov, V.S.Ilchenko, J. Opt. Soc. Am. B 17, 1051-1057 (2000) 8. B.E.Little, J.S.Foresi, G.Steinmeyer, E.R.Thoen, S.T.Chu, H.A.Haus, E.P.Ippen, L.C.Kimmerling, W.Greene, IEEE Photon. Technol. Lett. 10, 549-551 (1998) 9. H.M.Lai, P.T.Leung, K.Young, P.W.Barber, S.C.Hill, Phys. Rev. A 41, 5187-5198 (1990) 10. C.Flammer, Spheroidal wave functions, Stanford, Calif., Stanford University Press (1957 ) 11. M.Abramowitz, I.A.Stegun, eds., Han dbook on mathematical functions, New-York, Dover Publ. (1970) 12. I.V.Komarov, L.I.Ponomarev, S.Yu.Slavyanov, Spheroidal and Coulomb Spheroidal Functions, Moscow, Nauka (1976) (in Russian) 8 FIGURE CAPTIONS Fig.1. Photograph and the schematic of the microcavity geometry. Near the symmetry plane (at the location of WG modes), toroidal surface of outer diameter D and cross-section diameter d coincides with that of the osculating oblate spheroid with large semiaxis a = D / 2 and small semiaxis Dd b21= Fig.2. Schematic of the experimental setup to obtain wide range (~900GHz, or 7.2nm) high-resolution spectra of WG modes in microcavity Fig.3. Spectrum of TE whispering-gallery modes in spheroidal dielectric microcavity ( D = 2a = 165µm; d = 42µm; 2b = 83µm). Free spectral range (frequency separation between largest peaks 1 and 2) 383.7GHz (3.06nm) near central wavelength 1550nm. Individual resonance bandwidth 23MHz (loaded Q = 8.5 ×10 6). Finesse F = 1.7×104 9 d ba D Fig.1. V.S.Ilchenko et al., “Microtorus: a High Finesse…” 10 DFB laser 3.75GHz oscillatorEOM 30GHz FSR FP ethalon Temper ature control Ramp source Microcavity Prism cou pler for WG modes Fig.2. V.S.Ilchenko et al., “Microtorus: a High Finesse…” 11 0 200 400 600 8004 3 2 1Cavity transmission Laser frequency tuning, GHz-200 -100 0 100 200 300Individual resonance Lorentzian fit FWHM 22.6MHz Frequency offset from resonance, MHz Fig.3. V.S.Ilchenko et al., “Microtorus: a High Finesse…”

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