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arXiv:physics/0009093v1 [physics.optics] 28 Sep 2000Slow light in moving media U. Leonhardt School of Physics and Astronomy University of St Andrews North Haugh St Andrews, Fife, KY16 9SS Scotland P. Piwnicki Physics Department Royal Institute of Technology (KTH) Lindstedtsv¨ agen 24 S-10044 Stockholm Sweden February 2, 2008 Abstract We review the theory of light propagation in moving media wit h extremely low group velocity. We intend to clarify the most e lemen- tary features of monochromatic slow light in a moving medium and, whenever possible, to give an instructive simpliﬁed pictur e. 11 Introduction Waves experience moving media as eﬀective gravitational ﬁe lds. To under- stand why, imagine a deﬁnite example — light traveling in a mo ving trans- parent ﬂuid such as ﬂowing water [1, 2]. In each drop of the liq uid, light propagates along a straight line and so each drop distinguis hes a particular inertial frame. Now imagine light passing from one drop to a n ext one which happens to move with a diﬀerent velocity vector. Again, the n ew drop in the way of the light distinguishes an inertial frame, but this ne w frame will diﬀer from the previous one. Light traveling in a non–uniformly mo ving medium is constantly forced to adapt to new inertial frames. An anal ogous situation occurs in curved space–time when light, and all other matter , experiences space and time as consisting of connected inertial frames. S pace–time can be thought as being locally ﬂat, even in the vicinity of the most violent gravitat- ing objects, yet each local frame is non-trivially connecte d to the neighboring frames, a connection mathematically described in terms of t he curvature ten- sor [3]. In a moving medium, the local co-moving frames of the medium play the role of the local pieces of ﬂat space–time. Because these s pieces diﬀer in non–uniform ﬂows, light propagation in a moving medium re sembles light propagation in curved space–time, i.e.the medium appears to light as an eﬀective gravitational ﬁeld [4]-[6]. One could conceive of employing moving media for creating ar tiﬁcial as- tronomical objects in the laboratory. For example, water go ing down the drain of a bathtub appears to light as a rotating black hole. H owever, in order to observe spectacular eﬀects of general relativity i n laboratory–based analogues, truly astronomical ﬂow velocities are required . For establishing a black hole, the ﬂuid should move faster than the speed of lig ht in the medium ( cdivided by the refractive index n). The chances of creating anala- goes of black holes are much better when one employs sound ins tead of light in appropriate supersonic ﬂows [7]-[16]. One of the most fas cinating eﬀects of black holes is Hawking radiation [17]-[19], the spontane ous generation of photon pairs near the hole’s horizon where one of the photons falls into the hole and the other tunnels out of the attraction zone and beco me visible. The acoustical equivalent of Hawking radiation is quantum soun d in superﬂuids [7]-[16]. However, superﬂuidity tends to break down before the ﬂuid has a chance to move faster than the speed of sound [20]. Furthermo re, quantum sound and ﬂow are just two aspects of the same object, the quan tum liquid. 2The ﬂow is the macroscopic and the sound the microscopic moti on. Under extreme circumstances such as near the horizon of a sonic bla ck hole, a clear distinction between sound and ﬂow might be diﬃcult. An advan tage of light in a moving medium is the clear separation between wave and ﬂo w, and both light and medium can be regarded as separate quantum systems . Recently, light has been slowed down dramatically [21]-[23 ] due to an ef- fect called Electromagnetically Induced Transparency [24 ]-[26]. The use of slow light could open the opportunity for observing the radi ation of quantum– optical black holes. Note, however, that slow light is a more complicated phenomenon than light in a moving non–dispersive dielectri c where the re- fractive index does not depend on the frequency of light. Slo w light is based on a highly dispersive medium created by dressing the atoms o f the medium with an appropriate light beam. Instead of the phase velocit y only the group velocity is reduced. Therefore, it is not always advisable t o conclude directly from the behavior of light in ordinary dielectrics to the mot ional eﬀects of slow–light media. It is the purpose of these notes to clarify the most elemen- tary features of slow light in a moving medium [27] and, whene ver possible, to give an instructive simpliﬁed picture. 2 Dispersion relation Imagine the medium as being decomposed into drops. Each drop should be small enough such that the ﬂow does not vary signiﬁcantly wit hin the size of the drop, but each drop should be suﬃciently large to sustain several optical oscillations. We thus assume that the wave length of light is much shorter than the typical scale of changes in the ﬂow. In this case, we c an describe light by a dispersion relation in the local co–moving frame o f each medium drop denoted with primes k′2−ω′2 c2−χ(ω′)ω′2 c2= 0. (1) The susceptibility χ(ω′) characterizes the medium. In Electromagnetically Induced Transparency [24]-[26] a coupling beam dresses the upper two levels of a three–level atom shown schematically in Fig. 1. The dres sing of the upper two levels inﬂuences strongly the propagation of the p robe light with a frequency that should match the atomic transition frequen cyω0between 3Drive Probea bc Figure 1: Three level system needed for the creation of elect romagnetically induced transparency. A strong drive laser ﬁeld couples the levels a and c, making the medium transparent for the weak probe laser tun ed to the transition a ↔b. the lower and one of the upper levels. Exactly on resonance, t he medium decouples from the probe light and becomes transparent, whe reas ordinary dielectrics are extremely absorbing at atom–light resonan ces. Furthermore, the susceptibility changes rapidly, see Fig. 2, and, near th e resonance, χ(ω′) assumes a linear dependence on the detuning between the ligh t frequency ω′ and the atomic frequency ω0, χ(ω′) =2c ω0vg(ω′−ω0), (2) giving rise to a very low group velocity vg. If the slow–light medium is moving coherently, the motion will slightly detune each atom from e xact resonance due to the Doppler eﬀect. Imagine that the ﬂow of the medium, u, varies only in two dimensions and that the coupling beam propagates orthogonally to the ﬂo w, see Fig 3. In this case we can ignore the ﬁrst–order Doppler eﬀect of t he coupling beam and we should focus only on the Doppler detuning of the pr obe. In the laboratory frame the probe shall be monochromatic at the ato mic frequency ω0. In the co–moving local frames of the medium the probe–light frequency 4∆1χ IMAGINARY PART REAL PART Figure 2: Susceptibility for the probe laser beam in electro magnetically in- duced transparency. The plot shows the dependence of the rea l (whole lines) and imaginary (dashed lines) parts of the susceptibility χon the detuning ∆ of the probe beam. The drive beam is assumed to be on resonance . Arbitrary units are used. is Doppler–shifted with the detuning ω′−ω0=−u·k, (3) to ﬁrst order in u/c. Let us transform the dispersion relation (1) to the laboratory frame. We take advantage of the Lorentz invarian ce ofk2−(ω/c)2 and obtain to ﬁrst order in u/cthe relation k2−k2 0+ 2k0 vgu·k= 0 (4) with k0=ω0 c. (5) In Ref. [27] a more complicated dispersion relation has been derived which is correct up to second order in u/c. Note that the relation (4) describes eﬀects in leading order of u/(vgc)1/2, because the second–order corrections of Ref. [27] are proportional to v−1 g. 5Figure 3: Slow light in moving media. The medium should move o rthogonally to the coupling beam (grey arrow). This beam prepares the med ium such that the group velocity of light is signiﬁcantly reduced. Th e probe beam (lines) experiences the Doppler detuning due to the moving m edium and becomes deﬂected or even trapped. 63 Magnetic model and metric We are going to use the simpliﬁed relation (4) to illuminate t he most char- acteristic features of slow light in moving media. First we n ote that we can write the relation as /parenleftBigg k+k0u vg/parenrightBigg2 −k2 0u2 v2 g=k2 0. (6) Monochromatic light waves with ﬁxed polarization thus obey the wave equa- tion/parenleftBigg −i∇+k0u vg/parenrightBigg2 φ−k2 0u2 v2gφ=k2 0φ . (7) The ﬂow has a two–fold eﬀect: On one hand, the velocity uappears as an eﬀective vector potential, for example the magnetic vector potential acting on an electron wave [28], and, on the other hand, the hydrodyn amic pressure proportional to u2[29] acts as a scalar potential. This resembles the R¨ ontgen eﬀect of static electromagnetic ﬁelds on polarizable atoms [30, 31]. Alternatively, we can regard the ﬂow as generating an eﬀecti ve gravita- tional ﬁeld. To see this, we introduce the four–dimensional wave vector kν= (k0,−k), (8) and, adopting Einstein’s summation convention, write the d ispersion relation (4) as gµνkµkν= 0 (9) with the contravariant metric tensor gµν=/parenleftBigg 1u/vg u/vg−1/parenrightBigg . (10) Four–dimensional wave vectors are null vectors with respec t to the eﬀective metric gµνand, in turn [1], light rays follow zero-geodesics with resp ect to the line element ds2=gµνdxµdxν, dxν= (ct,x). (11) 7The covariant metric tensor gµνis the inverse of the contravariant one, and is given by gµν=1 1 +u2/v2g/parenleftBigg 1u/vg u/vg−1/parenrightBigg . (12) In contrast to sound [14] or light in non–dispersive media [1 ], a genuine event horizon of slow light cannot exist at this level of appr oximation. The potential existence of an event horizon is thus conﬁned to eﬀ ects to second order in u/(vgc)1/2which are excluded from our simpliﬁed model [32]. 4 Non–relativistic analogue To get more insight into the behavior of slow light in moving m edia, let us study another analogue. Consider a non–relativistic free p article attached to moving frames. The particle is supposed to move freely in eac h of the local frames but is bound to adapt from one frame to the next, simila r to light in drops of ﬂowing water. However we use pure non–relativist ic physics to transform from frame to frame. In case the local frames const itute a global frame, for example a rotating solid body, the particle would move along a straight line with respect to this frame. In the laboratory f rame, the parti- cle’s trajectory appears to be bent, attributed to the eﬀect of Coriolis and centrifugal forces. The Lagrange function of such a ﬁctitio us non–relativistic particle is L=1 2v′2=1 2(v−w)2, (13) up to a constant mass factor that we can set to unity, because w e are inter- ested in inertial eﬀects only. The velocity of the local inte rtial frames is w and we have used the non–relativistic addition theorem of ve locities v′=v−w. (14) Note that we could apply Einstein’s relativistic addition t heorem of velocities to describe light in moving non–dispersive media of refract ive index n, but we should use an eﬀective speed of light of c/nin the theorem. The momentum of the ﬁctitious non–relativistic particle is p≡∂L ∂v=v−w (15) 8and the Hamiltonian is H≡v·p−L=1 2p2−w·p=E . (16) The important point is that we can translate the Hamiltonian (16) into the dispersion relation (4) by setting p=ck k0,w=c vgu, E =1 2c2. (17) with the enhanced eﬀective frame velocity w[33]. Consequently, slow light experiences the moving medium in the same way as a non–relati vistic particle experiences moving inertial frames. Excused by its ﬁctitio us nature, the “non–relativistic” particle would move at an extraordinar y speed reaching c in regions where the medium is at rest, because the energy is 1 2c2=E=H=1 2v2−1 2w2. (18) We obtain from Hamilton’s equations, v=˙r=∂H ∂p,˙p=−∂H ∂r, (19) the equation of motion of the ﬁctitious particle ˙v= (∇ ×w)×v+1 2∇w2. (20) The ﬁrst term describes the Coriolis and the second the centr ifugal force. 5 Light rays around a vortex Consider a speciﬁc example of a non–uniform ﬂow, a vortex. Qu antum vor- tices in alkali Bose–Einstein condensates have been recent ly made [34, 35]. They give rise to intriguing slow-light phenomena. However , vortices do not posses genuine event horizons [32] and they will not radiate spontaneously. The ﬂow of an ideal vortex of vorticity Wis u=W reϕ=iW z∗, (21) 9written in polar coordinates or using the complex notation z=x+iy,z∗= x−iyof the planar Cartesian coordinates x, y. The curl of the vortex ﬂow vanishes, ∇ ×u=0, (22) and hence we obtain the equation of motion of a particle in an a ttractive r−2 potential [36]-[38], ˙v=1 2∇w2=−c2W2 v2gr r4, (23) or, in complex notation, ¨z+c2W2 v2 gz \|z\|4= 0. (24) The trajectories of the fall into a r−2singularity are well known in polar coordinates [36] but here we suggest a more elegant descript ion using the complex notation of the planar Cartesian coordinates, z=/parenleftBigg −ct+ib µ/parenrightBigg(1+µ)/2/parenleftBigg −ct−ib µ/parenrightBigg(1−µ)/2 (25) with the real parameter band µ=/parenleftBigg 1−W2 v2gb2/parenrightBigg−1/2 . (26) One can easily verify that this solution satisﬁes the equati on of motion (24). In the inﬁnite past, t→ −∞ , the trajectories approach z∼ −ct+ib . (27) From this asymptotics we see that the light is incident from t he right to the left with initial velocity cand impact parameter b. Depending on the impact parameter, the coeﬃcient µis real or purely imaginary. In the case of a real µwhenW2< v2 gb2the incident light particle will be able to escape, because the modulus squared of z,c2t2+b2/µ2, will never reach zero, the vortex core. In the inﬁnite future, t→+∞, the particle will approach z∼ct(−1)µ+ib=cteiπµ+ib (28) 100.5 10.40.8 Figure 4: Light rays spiraling towards the vortex core. The p lot shows the limiting spiral of described in Eq. (29). and, consequently, leaves at the angle πµ. In the case of a purely imaginary coeﬃcient µwhenW2> v2 gb2the light is sucked into the vortex, because the modulus squared of z,c2t2+b2/µ2, reaches zero at the time t=−\|b/(µc)\|. In practice, the light will hit the vortex core and will bounce b ack. In theory, the ﬁctitious light particle disappears from this world and rea ppears on another Riemann sheet. In the borderline case when W2=v2 gb2the parameter µ tends to inﬁnity and the trajectory approaches z=−ctlimµ→∞/parenleftBigg 1−1 µib ct/parenrightBiggµ =−ctexp/parenleftBigg −ib ct/parenrightBigg = exp/parenleftBigg ±iW vgct/parenrightBigg .(29) Light falling into the vortex core describes distinctive sp irals illustrated in Fig. 4. Light waves around a vortex Near the vortex core, rays of slow light feel the presence of t he centrifugal r−2attraction but no Coriolis force, because the curl of the ﬂow vanishes. 11Light waves, however, will show a distinctive interference pattern, an optical Aharonov–Bohm eﬀect [39]-[42] due to the long–range nature of a vortex ﬂow. Far outside the vortex core, light rays are hardly deﬂected, yet rays passing the vortex in ﬂow direction will be dragged and those swimmin g against the current lag behind. This does not aﬀect the ray trajectories but it produces a phase shift between the dragged and the lagging rays, and, i n turn, creates a typical interference pattern [41, 42]. Note that the Aharo nov–Bohm eﬀect of waves in moving media is not restricted to light. Indeed, B erryet al. [43] have reported and analyzed a beautiful bathtub experim ent with water waves in a tank, visualizing clearly the interference struc ture generated by the Aharonov-Bohm eﬀect. A further experiment has been repo rted [44] showing spiral waves. Acoustical analogues of the eﬀect hav e been observed in moving classical media [45] and are predicted for superﬂu ids [46]. The acoustical eﬀect might even lead to a friction felt by travel ing vortices due to the so-called Iordanskii force [47]-[49]. The optical Ahar onov–Bohm eﬀect of slow light [27] can be applied to observe in situ the ﬂow of quantum vortices in alkali Bose–Einstein condensates [34, 35] using phase–c ontrast imaging [50]. Consider the propagation of slow–light waves through a vort ex ﬂow. In polar coordinates the wave equation (7) is  ∂2 ∂r2+1 r∂ ∂r+1 r2/parenleftBigg∂ ∂ϕ+iνAB/parenrightBigg2 +ν2 AB r2 φ=k2 0φ (30) with the optical Aharonov–Bohm ﬂux quantum νAB=k0W vg. (31) We could simply eliminate the optical analogue of the vector potential by representing φas φ=φ0exp(−iνABϕ) (32) where φ0feels only the local r−2attraction due to the centrifugal force. The modulation exp( −iνABϕ) describes the long–range Aharonov–Bohm phase pattern [41]. Strictly speaking, however, the ansatz (32) i s only justiﬁed when the function exp( −iνABϕ) is single–valued after a complete cycle of ϕ,i.e. when νABis an integer. For non–integer ﬂux quanta the interference p attern 12is slightly more complicated, showing, most prominently, a line of zeros of the wave function after passing the vortex (which resolves the p roblem of a multi– valued phase, because a phase is not deﬁned when the amplitud e is zero). Even in the case of non–integer optical ﬂux quanta νAB, the modulation exp(−iνABϕ) accounts for the dominant phase pattern. The correct positive–frequency component of a slow–light w ave incident from the right is an appropriate superposition [31] φ=+∞/summationdisplay m=−∞φm (33) of the partial waves φm=um(r) exp( imϕ−iω0t), u m= (−i)νJν(k0r) (34) characterized by the Bessel functions Jνwith the index ν=/radicalBig (m+νAB)2−ν2 AB. (35) Figure 5 shows plots of slow–light waves around a vortex to il lustrate the long–range phase shift brought about by the optical Aharono v–Bohm eﬀect. Close to the vortex core, a light ray is doomed to fall into the singularity when the square of the impact parameters, b2, does not exceed W2/v2 g. In this case the angular momentum mlies in the interval −2νAB≤m≤0, (36) because, according to Eqs. (15,17) and the pattern (21) of th e vortex ﬂow, r×k=r×k0(v−w)/c= (k0b−νAB)ez. (37) The fall of light rays into the vortex core corresponds to a dr astic change of the behavior of the corresponding partial waves φm. In the case (36) the index (35) of the Bessel function in a radial wave (34) is pure ly imaginary. Using the ﬁrst term in the power–series expansion of the Bess el functions [51] we see that umis rapidly oscillating near the core, um∼(−i)ν Γ(ν+ 1)/parenleftBiggk0r 2/parenrightBiggν =(−i)ν Γ(ν+ 1)exp/bracketleftBigg iImνln/parenleftBiggk0r 2/parenrightBigg/bracketrightBigg .(38) 13-15-10-5051015 x-15-10-5051015 y Figure 5: Slow-light waves traveling through a vortex. The d ragging eﬀect of the vortex shifts the phase of the incident light, dependi ng on whether the light propagates with or against the ﬂow. 14This describes a rapid ﬂow of light towards the centre of attr action, if Im ν is negative and an outgoing wave for the positive branch. Far outside the vortex core, r→ ∞, the radial waves approach um∼1√2πk0r/bracketleftbigg exp/parenleftbigg ik0r−iπν−iπ 4/parenrightbigg + exp/parenleftbigg −ik0r+iπ 4/parenrightbigg/bracketrightbigg ,(39) as we see from the asymptotics of the Bessel functions [51]. F or a negative imaginary νthe second term in Eq. (39) dominates and so most of the light is incident, yet a small fraction with weight exp( −2iπν) is able to tunnel out of the attraction zone. Where is the borderline between pure inﬂux and partial tunne ling out of the attraction zone? Consider the geometrical optics of a ra dial wave. We make the eikonal ansatz um=c+ mu+ m+c− mu− mu± m=\|um\|exp(±iR), (40) regard \|um\|as a slowly varying envelope, and obtain from the wave equati on (30) the radial eikonal R=iν/bracketleftBigg/radicalBig ρ2+ 1−arsinh/parenleftBigg1 ρ/parenrightBigg +iπ 2/bracketrightBigg , ρ=k0r iν. (41) Quite typically, valuable insight into the behavior of wave s can be gained by analytical continuation to the complex plane of the wave’ s argument. Consider the line in the complex plane of the radius rwhere the eikonal is purely imaginary, i.e.where Re/bracketleftBigg/radicalBig ρ2+ 1−arsinh/parenleftBigg1 ρ/parenrightBigg/bracketrightBigg = 0, (42) see Fig. 6. This line is called a Stokes line [52, 53]. When pas sing a Stokes line, one component of the superposition (40) becomes expon entially small and hence irrelevant [52]. The asymptotics in the limit of ge ometrical optics switches from a two–component regime, for example in–and–o ut ﬂux, to a single component behavior, such as pure inﬂux [52]. The Stok es line in Fig. 6 thus marks the boundary where light is able to tunnel out of t he optical black hole. 15-0.5 0.5 1 -0.5 i0.5 iIm r Re r Figure 6: Stokes lines of an optical black hole. Stokes lines [52] are lines in the complex plane of the radius rwhere the radial eikonal of light waves is purely imaginary. The lines connect the turning points of li ght rays which are complex in the general case of trapping (and purely imagi nary for an optical black hole). The Stokes lines are calculated from th e radial part of the eikonal (41) and are displayed in the scaled units employ ed there. The arrows indicate the light ﬂux in the regions bounded by a Stok es line. 166 Summary We analyzed the propagation of slow light in moving media in t he case when the light is monochromatic in the laboratory frame. Slow lig ht is generated by the electromagnetically induced transparency of an atom ic transition. The Doppler detuning due to the moving medium generates ﬂow- dependent eﬀects. We considered analogies to magnetism and general re lativity and studied light propagation around a vortex in some detail. Th e subject is far from being exhausted — light detuned from the atomic resonan ce and non– monochromatic light will behave diﬀerently, opening perha ps new exciting analogies and opportunities for laboratory tests of astron omical eﬀects on Earth. 7 Acknowledgements U.L. acknowledges the support of the Alexander von Humboldt Foundation and of the G¨ oran Gustafsson Stiftelse during his stay at the Royal Insti- tute of Technology. P.P. was partially supported by the rese arch consortium Quantum Gases of the Deutsche Forschungsgemeinschaft. References [1] U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301 (1999). [2] U. Leonhardt and P. Piwnicki, Light in moving media , Contemp. Phys. (in press). [3] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Perg- amon, Oxford, 1975). [4] W. Gordon, Ann. Phys. (Leipzig) 72, 421 (1923). [5] Pham Mau Quan, C. R. Acad. Sci. (Paris) 242, 465 (1956). [6] Pham Mau Quan, Archive for Rational Mechanics and Analys is1, 54 (1957/58). [7] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). 17[8] W. G. Unruh, Phys. Rev. D 51, 2827 (1995). [9] T. A. Jacobson, Phys. Rev. D 44, 1731 (1991). [10] T. A. Jacobson and G. E. Volovik, Phys. Rev. D 58, 064021 (1998). [11] N. B. Kopnin and G. E. Volovik, JETP Lett. 67, 140 (1998). [12] T. A. Jacobson and G. E. Volovik, JETP Lett. 68, 874 (1998). [13] G. E. Volovik, JETP Lett., 69, 705, (1999). [14] M. Visser, Class. Quantum Grav. 15, 1767 (1998). [15] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, arXi v:gr-qc/0002015. [16] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, arXi v:gr-qc/0005131. [17] S. M. Hawking, Nature 248, 30 (1974). [18] S. M. Hawking, Commun. Math. Phys. 43, 199 (1975). [19] N. D. Birrell and P. C. W. Davies, Quantum ﬁeld in curved space (Cam- bridge University Press, Cambridge, 1982). [20] L. D. Landau and E. M. Lifshitz, Statistical Physics. Theory of the Con- densed State (Pergamon, Oxford, 1980). [21] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nat ure397, 594 (1999). [22] M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovsev, E. S. Fry, and M. O. Scully, Phys. Re v. Lett.82, 5229 (1999). [23] D. Budiker, D. F. Kimball, S. M. Rochester, and V. V. Yash chuk, Phys. Rev. Lett. 83, 1767 (1999). [24] P. L. Knight, B. Stoicheﬀ, and D. Walls (eds.), Phil. Tra ns. R. Soc. Lond. A 355, 2215 (1997). [25] S. E. Harris, Phys. Today 50(7), 36 (1997). 18[26] M. O. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997). [27] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000). [28] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon, Oxford, 1958). [29] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1987). [30] Haiqing Wei and Rushan Han, and Xiuqing Wei, Phys. Rev. L ett.75, 2071 (1995). [31] U. Leonhardt and M. Wilkens, Europhys. Lett. 42, 365 (1998). [32] M. Visser, arXiv:gr-qc/0002011. [33] U. Leonhardt and P. Piwnicki, arXiv:physics/0003092, Phys. Rev. A (in press). [34] M. R. Matthews, B. F. Anderson, P. C. Haljan, D. S. Hall, C . E Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999). [35] K. W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Phys . Rev. Lett. 84, 806 (2000). [36] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, Oxford, 1976). [37] L. V. Hau, M. M. Burns, and J. A. Golovchenko, Phys. Rev. A 45, 6468 (1992). [38] J. Denschlag, G. Umshaus, and J. Schmiedmayer, Phys. Re v. Lett. 81, 737 (1998). [39] J. H. Hannay, Cambridge University Hamilton prize essa y 1976 (unpub- lished). [40] R. J. Cook, H. Fearn, and P. W. Milonni, Am J. Phys. 63, 705 (1995). [41] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). 19[42] M. Peshkin and A. Tonomura, The Aharonov–Bohm Eﬀect , (Springer, Berlin, 1989). [43] M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J . C. Walm- sley, Eur. J. Phys. 1, 154 (1980). [44] F. Vivanco and F. Melo, Phys. Rev. Lett. 85, 2116 (2000). [45] P. Roax, J. de Rosny, M. Tanter, and M. Fink, Phys. Rev. Le tt.79, 3170 (1997). [46] H. Davidowitz and V. Steinberg, Europhys. Lett. 38, 297 (1997). [47] S. V. Iordanskii, Sov. Phys. JETP 22, 160 (1966). [48] G. E. Volovik, JETP Lett. 67, 881 (1998). [49] M. Stone, arXiv:/cond-mat/9909313. [50] M. R. Andrews, M.–O. Mewes, N. J. van Druten, D. S. Durfee , D. M. Kurn, and W. Ketterle, Science 273, 84 (1996). [51] A. Erd´ elyi, W. Magnus, F. Oberhettinger, and F. G. Tric omi,Higher Transcendental Functions , (McGraw-Hill, New York, 1953). [52] W. H. Furry, Phys. Rev. 71, 360 (1947). [53] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications (Cambridge University Press, Cambridge, 1997). 20
1 COMMENT ON MIE SCATTERING FROM A SONOLUMINESCING BUBBLE WITH HIGH SPATIAL AND TEMPORAL RESOLUTION Physical Review E 61, 5253 (2000) K.R. Weninger, P.G. Evans, S.J. Putterman Physics Department, University of California, Los Angeles CA 90095 Division of Engineering and Applied Science Harvard University, Cambridge MA 02138 ABSTRACT: A key parameter underlying the existence of sonoluminescence [or ‘SL’] is the time relative to SL at which acoustic energy is radiated from the collapsed bubble. Light scattering is one route to this quantity. We disagree with the statement of Gompf and Pecha that -highly compressed water causes the minimum in scattered light to occur 700ps before SL- and that this effect leads to an overestimate of the bubble wall velocity. We discuss potential artifacts in their experimental arrangement and correct their description of previous experiments on Mie scattering. In sonoluminescence the first stage of energy focusing is provided by the collapse of a gas bubble surrounded by water. For example a 40KHz sound field with an amplitude of 1.35atm will cause a helium bubble with an ambient radius of 4microns to expand to a maximum radius of 29 µ. Since the bubble contains 6.7x109 helium atoms and is acted on by about 1atm [=P 0] of ambient external pressure the mechanical energy stored when the bubble is perched at its maximum radius is about 10.eV per helium atom[1]. 2 Sonoluminescence [SL] is due to some fraction of this potential energy being transferred into the thermal degrees of freedom of the relatively few atoms/molecules in the bubble as it implodes under the influence of P 0 . For instance, if all the mechanical energy went into uniformly heating the helium, its temperature at the moment of collapse [when its radius approaches the van der Waals hard core of about .4 µ] would be about 75,000K. A determination of the fraction of potential energy that remains in the bubble at the moment of collapse is a key aspect of SL. Thus the dynamics of the collapse is critical to an understanding of SL and so various techniques [2-10] have been applied to the experimental determination of the bubble radius as a function of time R(t) and the response of the water. Various realizations of Mie scattering [3-10] in particular have proved useful in obtaining bubble parameters. Mie scattering occurs when variations in the index of refraction cause light to be scattered out of the direction of the incident beam. In a recent paper [11] Gompf and Pecha [GP] have used a streak camera to image Mie scattering. In the abstract they claim that “In the last nanoseconds around minimum bubble radius most of the light is scattered at the highly compressed water surrounding the bubble and not at the bubble wall. This leads to a minimum in the scattered light intensity about 700ps before the SL pulse is emitted.” They go on to say that “neglect this changes [sic] leads to a strong overestimation of the bubble wall velocity”. We disagree with a number of aspects of these statements. The 700ps interval which GP quote is specific to their particular experimental arrangement and is unrelated to the physics of a bubble collapsing in highly compressed water. The stated timing resolution of GP is 500ps whereas by 3 the application of time correlated single photon counting to Mie scattering we have achieved a timing resolution of about 50ps. In Figure 1 of reference [8] the flash will be seen to occur 100-200ps before [not 700ps after ] the minimum apparent radius [i.e. ‘y’ axis], which for our experiment is the minimum in total light scattering. We agree [8] that light scattering is due to the index of refraction changes at the wall of the bubble as well as the highly compressed water. But in this case, attribution of the Mie scattering exclusively to the bubble wall would lead to an underestimate , of its velocity, not an overestimate , as quoted above from GP. Perhaps our observation that the flash precedes the minimum in light scattering could be due to this effect. On page 5254 GP discuss our previous experiments on Mie scattering [3,4,5] and state that “In ..former investigations .. the scattered light intensity was assumed to be proportional to the square of the bubble radius which totally neglects the complicated angular distribution of the Mie scattering.” This statement comprises an inaccurate description of past experiments. The complicated angular distribution can be seen in Figure 6 of [4], which was taken from our first paper on Mie scattering from SL [3]. One of the steps that enabled us to obtain quantitative information about bubble radii from Mie scattering was to simplify the scattered intensity as a function of R by collecting light from a large solid angle: such as °−°80 30 [8] or °−°94 46 [3]. In this case the intensity of light scattering is within 20% of R 2 for bubbles bigger than, .6 microns[8], or 1micron[3]. These corrections and their connection to the “complicated angular distribution of Mie scattering” were discussed in these papers. A plot typical of calculations that formed the basis for these corrections was published in ref. [8]. The strong deviations from R 2 Mie scattering displayed in Figure 5 of GP results from their 4 collecting light in a small solid angle [ °−°36 14 ] near the forward direction. On page 5255 GP state that our papers neglected the effect of changes in the refractive indices. This is true; the index of refraction inside the bubble was reckoned to unity for the purpose of deconvolving the scattered intensity. Light scattering techniques have not yet reached the point where changes in the index of refraction, due to say the formation of a plasma, can be extracted. Analysis of our data also neglected the effect of bubble asphericity [see discussion relating to Figure 6 of ref.5]. The time scale of 700ps enters GP in two entirely different contexts: 1) it is a “pronounced minimum in the scattered light intensity .7ns before the SL pulse due to Mie lobe clusters” and 2) “From this time on most of the light is scattered at the highly compressed water around the bubble leading to a strong increase in the scattered light intensity before minimum bubble radius” which is the moment of SL. For the choice of angles over which GP collect scattered light we agree with 1) but emphasize that our choice of angles eliminated this artifactual minimum. Regarding 2) we reiterate our disagreement with GP’s abstract. At more than one location in GP it is claimed that “the bubble wall velocity 1ns before the SL pulse is about 950m/s. This value is much lower than the values found by Weninger et al ”[reference 5 this comment]. First of all, the bubble wall velocity 1ns before collapse, where R is about 1.7 µ, is for our data 900m/sec [5]. So it would appear that GP have confirmed our measurements. Secondly, the 500ps timing resolution of GP means that a 500ps smoothing function has been applied to their rapidly changing data. There can be no question that their value of 950m/s is an underestimate of the bubble wall velocity in their experiment. Furthermore, a statement claiming a significant discrepancy between experiments would have more 5 weight if it was accompanied by a discussion of ‘error bars’. The paper of GP contains no such discussion. We have attempted to give an example of our ‘error’ in Figure 6 of [5]. It has sources in the various processes discussed above, gas concentration, and also run to run variations. The data of GP for R(t) cut off at about 1.7microns up to which point they are largely in agreement with our previously published results[4,5,6]. There remains the issue of whether the bubble wall velocity for some systems [4,5,6] approaches higher velocities [e.g. in excess of 1200m/s] for smaller radii. Systems are characterized by the gas mixture used, acoustic frequency and ambient temperature. [For 1%Xe 99%O dissolved at 150Torr driven at 40KHz at 20C, the maximum velocity was actually found to be less than 950m/s.(8)] In the range µ µ 5. 7.1 ≈>>cRR ,where R c is the collapse or minimum radius, GP provide no data for R. They claim that this is due to the difficulty in subtracting out a large signal due to scattering from highly compressed water. Except for 200ps around the minimum we disagree. Based upon our experience we propose some possible complications that affect their experiment in this range; 1) the GP choice of angles leads to Mie lobes that are sufficiently complicated that the intensity of scattered light is not monotonic with radius, so that deconvolution is difficult, 2) as the GP images are magnified and averaged, small translational motion and concentric pulsation of the bubble can throw its image off the slit, 3) the level of scattered laser light is less than the intensity of SL obscuring dynamics near R c. In Figure 1 we show photos of SL from a bubble undergoing a spontaneous motion on the order of 10 µ. To the eye this particular bubble appears as stable as a star in the sky. When magnified by a factor of 10 this motion is enough to throw the image off the entry slit of the 6 streak camera and introduce errors into the data. For integrated Mie scattering [3,4,5,6] where the width of the incoming light beam [typically 1mm] is larger than this motion and all the scattered light is delivered to a photo-detector the effects of this bubble motion can be minimized. For this reason our investigations of imaged light scattering has been carried out on a shot by shot basis. In Figure 2 one sees a very common example of the bubble falling off the slit during that portion of the cycle that surrounds the minimum by about 1 ns as in GP. In Figure 2 the shrinking bubble is not centered on slit. It is good news that the action of an audible sound field on water has led to a debate about experimental techniques on the scale of 100ps-700ps. As the time scale is narrowed it will eventually be possible to determine the time relative to SL ‘t a-tsl’ at which acoustic energy is radiated by the bubble. This is very important because over 90% of the bubble’s potential energy is radiated as sound[1]. If ‘t a-tsl’ is negative then the average energy per particle at the moment of collapse is less than 1eV which is not high enough to explain the UV spectra [4] for a uniformly heated bubble[12]. Indeed if ‘ta-tsl’ <0 the observation of SL would imply an additional energy focusing mechanism inside the collapsed bubble [13]. Perhaps the nonlinear processes that determine ‘t a-tsl’ are the key to the existence of SL. This research is supported by DARPA. REFERENCES [1] S.J. Putterman and K.R. Weninger Ann. Rev. Fluid Mech. 32, 445 (2000). [2] Delgadino, G.A. and Bonetto, F.J., Phys. Rev. E 56, R6248 (1997). 7 [3] Barber, B.P. and S.J. Putterman, Phys. Rev. Lett. 69, 3839 (1992). [4] B.P. Barber, R.A. Hiller, R. Löfstedt, S.J. Putterman, and K.R. Weninger, Phys. Reports , 281, 67 (1997) [5] Weninger, K., B.P. Barber and S. Putterman, Phys. Rev. Lett. 78, 1799- 1802 (1997). [6] Barber, B.P., K. Weninger, and S.J. Putterman Phil. Trans. R. Soc. Lond. A 355, 641 (1997). [7] W.J. Lenz et al, Appl. Opt. 34, 2648 (1995). [8] Weninger,K.R., Evans,P.G. and Putterman,S.J., Phys. Rev. E , 61, R1020 (2000); [9] G. Vacca, R.D. Morgam, R.B. Laughlin; Phys. Rev E 60,6303 (1999) [10] S.J. Putterman, Scientific American , February 1995 p.32 [11] B. Gompf, R. Pecha, Phys. Rev. E 61, 5253 (2000). [12] S.Hilgenfeldt, S. Grossman, D. Lohse Nature 398, 402 (1999). [13] C.C.Wu and P.H. Roberts, Phys. Rev. Lett. 70, 3424 (1993). 8 FIGURE CAPTIONS 1) A microscope image of light emission from a sonoluminescing bubble moving on the surface of a toroid of 15 µ radius. The bubble is formed from water which has a mixture of 1%He, 99%O 2 dissolved under a pressure of 150Torr. The acoustic frequency is 16.5KHz and the exposure time is .8seconds. In contrast B) shows our most stable image of SL (150Torr 1%Xe in O 2 at 41KHz) accumulated over 700flashes. The 1 µ resolution of the diameter is limited by both the microscope objective and possible bubble motion. A range of motions between those displayed in ‘A’ and ‘B’ may be seen under various circumstances which in general are beyond our experimental control. The intensity of SL is color coded so that red is brightest. 2) Single shot streak camera shadowgraph of a collapsing bubble launching a pulse of sound into the surrounding water [see ref.8 for experimental details]. The image of the bubble is the center line, and the radiated pulse of sound moves at a supersonic velocity relative to the speed of sound in water. The image of this particular bubble is lost during the indicated 1ns time-span. This effect is due to the bubble not being centered on the entrance slit of the streak camera. As the bubble shrinks its shadow falls off the slit. A photo lacking this artifact can be found in [8]. This experiment was carried out at 41KHz with 150Torr 1%Xe, 99%O 2. 4 3 2 1 0micr on 43210 micr on80 60 40 20 0micr on 806040200 micr onA B Figure 1Figure 210 µm 1 ns
arXiv:physics/0009095v1 [physics.acc-ph] 29 Sep 2000EmittanceGrowthfromthe ThermalizationofSpace-Charge N onuniformities∗ StevenM.Lund,John J.Barnard, and EdwardP. Lee LawrenceLivermoreand Berkeley NationalLaboratories,Un iversityofCalifornia, USA Abstract Beams injected into a linear focusing channel typically have some degree of space-charge nonuniformity. In gen- eral, injected particle distributions with systematic cha rge nonuniformities are not equilibria of the focusing channel and launch a broad spectrum of collective modes. These modescanphase-mixandhavenonlinearwave-waveinter- actionswhich, at highspace-chargeintensities, results i n a relaxationtoamorethermal-likedistributioncharacteri zed byauniformdensityproﬁle. Thisthermalizationcantrans- fer self-ﬁeld energy from the initial space-charge nonuni- formitytothelocalparticletemperature,therebyincreas ing beam phase space area (emittance growth). In this paper, we employ a simple kinetic model of a continuous focus- ingchannelandbuildonpreviousworkthatappliedsystem energyandchargeconservation[1,2]toquantifyemittance growth associated with the collective thermalization of an initial azimuthally symmetric, rms matched beam with a radialdensityproﬁlethatishollowedorpeaked. Thisemit- tance growth is shown to be surprisingly modest even for highbeamintensitieswithsigniﬁcantradialstructureint he initial densityproﬁle. 1 INTRODUCTION Experiments with high-current, heavy-ion injectors have observed signiﬁcant space-charge nonuniformities emerg- ing from the source. Sharp density peaks on the radial edge of beam have been measured, but the local incoher- ent thermal spread of particle velocities (i.e., the partic le temperature)acrossthebeamisanticipatedtobefairlyuni - formsincethebeamisemittedfromaconstanttemperature surface. When such a distribution is injected into a linear transportchannel,it will befarfroman equilibriumcondi- tion (i.e., particles out of local radial force balance), an d a broadspectrumofcollectivemodeswill belaunched. The spatial average particle temperature of a heavy ion beam emerging from an injector is typically measured as several times what one would infer from the source ther- maltemperature( ∼0.1eV)andsubsequentbeamenvelope compressions,with ¯Tx∼20eVwhere ¯Tx∼[ǫ2 x/(2R2)]Eb. On the other hand, the radial change in potential energy from beam center to edge is q∆φ∼2.25keV for a beam withline-chargedensity λ∼0.25µC/m(∆φ∼λ/(4πǫ0)). Ifevenasmallfractionofsuchspace-chargeenergyisther- malizedduringcollectiverelaxation,largetemperaturea nd emittanceincreasescanresult. Inthispaper,weemployconservationconstraintstobet- ter estimate emittance increases from collective thermal- ∗This work was performed under the auspices of the U.S.Depart ment of Energy by University of California at Lawrence Livermore National Laboratory and Lawrence Berkeley National Laboratory unde r contract Nos. W-7405-Eng-48 and DE-AC03-76SF00098.ization of normal mode perturbations resulting from ini- tial space-charge nonuniformities characteristic of inte nse beam injectors. Past studies have employed analogous techniques to estimate emittance increases resulting from the thermalization of initial rms mismatches in the beam envelopeandspace-chargenonuniformitiesassociatedwit h combiningmultiplebeamsandotherprocesses[1, 2, 3]. 2 THEORETICAL MODEL Weanalyzeaninﬁnitelylong,unbunched( ∂/∂z= 0)non- relativistic beam composed of a single species of particles of mass mand charge qpropagating with constant axial kinetic energy Eb. Continuous radial focusing is provided by an external force that is proportional to the transverse coordinate x, i.e.,Fext=−2Ebk2 β0x, where kβ0=const is the betatron wavenumber of particle oscillations in the applied focusing ﬁeld. For simplicity, we neglect particle collisions and correlation effects, self-magnetic ﬁelds, and employ an electrostatic model and describe the transverse evolution of the beam as a function of axial propagation distance sintermsofasingle-particledistributionfunction fthat is a function of s, and the transverse position xand anglex′=dx/dsof a single particle. This evolution is describedbytheVlasovequation[2],/braceleftbigg∂ ∂s+∂H ∂x′·∂ ∂x−∂H ∂x·∂ ∂x′/bracerightbigg f(x,x′, s) = 0,(1) where H=x′2/2 +k2 β0x2/2 + (q/2Eb)φis the single- particle Hamiltonian and the self-ﬁeld potential φsatisﬁes the Poissonequation(CGS unitshereandhenceforth) ∇2φ=−4πq/integraldisplay d2x′f (2) subject to the boundary condition φ(r=rp) = 0at the conductingpiperadius r=\|x\|=rp=const. Ifnoparticlesarelostinthebeamevolution,theVlasov- Poissonsystempossessesglobalconstraintscorrespondin g to theconservationofsystemcharge( λ)andscaledenergy (U)perunitaxiallength, λ=q/integraldisplay d2x/integraldisplay d2x′f= const , U=1 2/angb∇acketleftx′2/angb∇acket∇ight+k2 β0 2/angb∇acketleftx2/angb∇acket∇ight+q 2EbλW= const .(3) Here, W≡/integraltextd2x\|∇φ\|2/(8π)is the self-ﬁeld en- ergy of the beam per unit axial length and /angb∇acketleftξ/angb∇acket∇ight ≡ (/integraltext d2x/integraltext d2x′ξ f)/(/integraltext d2x/integraltext d2x′f)is a transversestatis- ticalaverageof ξoverthebeamdistribution f. Notethat U includesbothparticlekineticenergyandtheﬁeldenergyof theappliedandself-ﬁelds. Theseconservationlawsfollow directlyfromEqs.(1)-(2)andprovidepowerfulconstraint s onthenonlinearevolutionofthesystem.Moment descriptions of the beam provide a simpliﬁed understandingof beam transport. For an azimuthallysym- metric beam( ∂/∂θ= 0),a statistical measureofthebeam edge radius R≡2/angb∇acketleftx2/angb∇acket∇ight1/2is employed. Note that Ris the edge radius of a beam with uniformly distributed space- charge. Any axisymmetric solution to the Vlasov-Possion systemwillbeconsistentwiththermsenvelopeequation[1] d2R ds2+k2 β0R−Q R−ǫ2 x R3= 0. (4) Here, Q=qλ/Eb=const is the self-ﬁeld perveance and ǫx= 4[/angb∇acketleftx2/angb∇acket∇ight/angb∇acketleftx′2/angb∇acket∇ight − /angb∇acketleftxx′/angb∇acket∇ight2]1/2is an edge measure of the rmsx-emittanceofthebeamandisastatistical measureof the beamareain x-x′phase-space(i.e.,beamquality). For generaldistributions, ǫxisnotconstantandevolvesaccord- ingto thefullVlasov-Poissonsystem. 3 NONUNIFORMDENSITY PROFILE Weexamineanbeamwithanazimuthallysymmetricradial densityproﬁle n=/integraltext d2x′fgivenby n(r) =/braceleftBigg n0/bracketleftBig 1−1−h h/parenleftBig r rb/parenrightBigp/bracketrightBig ,0≤r≤rb, 0, r b< r≤rp.(5) Here, rbis the physical edge-radius of the beam, n0= n(r= 0)is the on-axis( r= 0) beamdensity, and handp are “hollowing” [ 0≤h≤ ∞,h=n(r=rb)/n(r= 0), p≥0] and radial steepening parameters associated with thedensitynonuniformity.Thisdensityproﬁleisillustra ted inFig.1forthesteepeningindex p= 2andhollowingfac- torsh= 1(uniform), h= 1/2(hollowed), and h= 2 (peaked). Thehollowingparameter hhasrange 0≤h <1 for an on-axis hollowed beam and 0≤1/h < 1for an on-axis peaked beam. The limit h→1corresponds to a uniformdensity beam and h,1/h→0correspondto hol- lowedandpeakedbeamswiththedensityapproachingzero on-axis and at the beam edge ( r=rb), respectively. For large steepening index p≫1, the density gradient will be signiﬁcant only near the radial edge of the beam ( r≃rb), andthedensityisuniformfor h= 1regardlessof p. -1.5 -1 -0.5 0.5 1 1.50.511.52n(x)/n0 x/rb0 <h< 1 h= 1 0 <h<( h= 1/2 shown )hollowed density uniform peaked density∞densityp = 2shown ( h= 2 shown ) Figure1: Uniform,hollowed,andpeakeddensityproﬁles. The beam line-charge density ( λ) and rms edge-radius (R)arerelatedtotheparametersin Eq.(5)by λ=/integraldisplay d2x n=πqn0r2 b/bracketleftbigg(ph+ 2) (p+ 2)h/bracketrightbigg R= 2/angb∇acketleftx2/angb∇acket∇ight1/2=/radicalBigg (p+ 2)(ph+ 4) (p+ 4)(ph+ 2)rb (6)Using these expressions, the Poisson equation (2) can be solvedforthepotential φcorrespondingtothedensitypro- ﬁle (5)andusedto calculatetheself-ﬁeldenergy Was W=λ2/braceleftbigg1 (ph+ 2)2/bracketleftbigg(p+ 2)2h2 4+2(1−h)2 p+ 2+ 4(p+ 2)h(1−h) p+ 4/bracketrightbigg + ln/bracketleftBigg/radicalBigg (p+ 2)(ph+ 4) (p+ 4)(ph+ 2)rp R/bracketrightBigg/bracerightBigg .(7) It is convenient to deﬁnean average phase advance pa- rameter σfor the density proﬁle (5) in terms of an enve- lope matched ( R′= 0 = R′′), rms equivalent beam with uniform density ( h= 1) and the same perveance ( Q) and emittance ( ǫx) as the (possibly mismatched) beam with a nonuniform density proﬁle ( h/negationslash= 1). Denoting the phase advanceper unit axial length of transverse particle oscill a- tions in the matched equivalent beam in the presence and absence of space-charge by σandσ0, we adapt a normal- izedspace-chargeparameter σ/σ0≡/radicalBig k2 β0−Q/R2/kβ0. The limits σ/σ0→0andσ/σ0→1correspondto a cold, space-charge dominated beam and a warm, kinetic domi- nated beam, respectively. Note that this measure applies only in an equivalentbeam sense. In general, distributions fconsistent with the density proﬁle (5) will not be equi- libria ( d/ds/negationslash= 0) of the transport channel and will evolve leaving σill deﬁned. 4 EMITTANCE GROWTH We consider an initial beam distribution fwith a density proﬁlegivenbyEq.(5)andan arbitrary “momentum”dis- tributionin x′. Suchaninitialdistributionisnot,ingeneral, an equilibrium of the focusing channel and a spectrum of collective modes will be launched (depending on the full initialphase-spacestructureof f). Thesemodeswillphase- mix, have nonlinear wave-wave interactions, etc., driving relaxation processes that have been observedin PIC simu- lations to cause the beam space-charge distribution to be- come more uniform for the case of high beam intensities. The conservation constraints (3) are employed to connect the parameters of an initial (subscript i), nonuniform den- sity beam with h/negationslash= 0with those of a ﬁnal (subscript f), azimuthally symmetric and rms envelope matched beam (R′ f= 0 = R′′ f) withuniformdensity( h= 1). Employing Eqs. (4)-(7), conservation of charge ( λi= λf≡λ) and system energy ( Ui=Uf) can be combined intoansingle equationofconstraintexpressibleas (Rf/Ri)2−1 1−(σi/σ0)2+p(1−h)[4 +p+ (3 + p)h] (p+ 2)(p+ 4)(2 + ph)2 −ln/bracketleftBigg/radicalBigg (p+ 2)(ph+ 4) (p+ 4)(ph+ 2)Rf Ri/bracketrightBigg =Eb 2qλ(RiR′ i)′(8) Here,handparethehollowingfactorandindexoftheini- tial density proﬁle, σi/σ0is the initial space-charge inten- sity, and [Eb/(2qλ)](RiR′ i)′is a parameter that measures the initial envelope mismatch of the beam. This nonlinear constraint equation can be solved numerically for ﬁxed h, p,σi/σ0and[Eb/(2qλ)](RiR′ i)′to determine the ratio ofﬁnaltoinitialrmsradiusofthebeam( Rf/Ri). Employing the envelope equation (4), the ratio of ﬁnal to initial beam emittanceisexpressibleas ǫxf ǫxi=Rf Ri/radicalBigg (Rf/Ri)2−[1−(σi/σ0)2] (σi/σ0)2−R′′ i/(k2 β0Ri).(9) Eqs. (8) and (9) allow analysis of emittance growth from the thermalizationofinitialspace-chargenonuniformiti es. We numerically solve Eqs. (8) and (9) to plot (Fig. 2) the growth in rms beam radius ( Rf/Ri) and emittance (ǫxf/ǫxi) due to the relaxation of an initial rms matched beam ( R′ i= 0 = R′′ i) with nonuniform hollowed and peaked density proﬁles to a ﬁnal uniform, matched pro- ﬁle. Final to initial beam ratios are shown for hollowing index of p= 2and are plotted verses the “hollowing fac- tors”h(hollowinitialdensity)and 1/h(peakedinitialden- sity) for families of σi/σ0ranging from σi/σ0→0to σi/σ0→1. Growths are larger for the initially hollowed proﬁle than the peaked proﬁle and increase with stronger space-charge (smaller σi/σ0). However, the change in rms radius ( Rf/Ri) is small in all cases, even for strong space-charge with strong hollowing ( h→0) and peaking (1/h→0) parameters. Moreover, the increases in beam emittance( ǫxf/ǫxi)aresurprisinglymodest(factorof2and less) for intense beam parameters with σi/σ0∼0.1and greater. At ﬁxed σi/σ0and increasing steeping factor p, 0 0.2 0.4 0.6 0.8 1.1.0.9991.1.0011.0021.0031.0041.0051.0061.006 σi/σ0 = 0, 0.1, 0.2, .... 1.0 σi/σ0 = 0 σi/σ0 = 1.0b) p= 2, Peaked On-Axis 0 0.2 0.4 0.6 0.8 1.1.1.1.0051.011.0151.021.02Rf/ Riσi/σ0 = 0, 0.1, 0.2, .... 1.0 σi/σ0 = 0 σi/σ0 = 1.0a) p= 2, Hollowed On-Axis 0 0.2 0.4 0.6 0.8 1.1.1.1.52.εxf/εxi hσi/σ0 = 0.0250.0500.0750.1......σi/σ0 = 0.025, 0.050, 0.075 (dashed) 0.1, (solid) 0.125,0.150,0.175, (dashed), 0.2, (solid) 0 0.2 0.4 0.6 0.8 1.1.1.1.52. 1/hσi/σ0 = 0.025 0.0500.0750.1...... 0.125σi/σ0 = 0.025, 0.050, 0.075 (dashed) 0.1, (solid) 0.125,0.150,0.175, (dashed) 0.2, (solid) Figure 2: Ratio of ﬁnal to initial rms beam size ( Rf/Ri) and emittance ( ǫxf/ǫxi) verses h(a, hollowed beam) and 1/h(b,peakedbeam). similarmodestgrowthfactorsareseenforhollowedbeams forallbutthemostextremehollowingfactors( h∼0.1and less),andasexpected,muchlessgrowthisseenforpeaked beams(closertouniform). 5 DISCUSSION The modest emittance growth at high beam intensities can be understood as general beyond the speciﬁc model em- ployed. Even for signiﬁcant increases in emittance ǫx, the rms matched beam size is given to a good approximation by the envelope equation (4) with the emittance term ne- glected. In this case, R≃√Q/k β0=const during the beam evolution and hence Rf≃Ri. Employing the method of Lagrange multipliers, the free electrostatic en- ergy of the system at ﬁxed rms radius ( R) and line-charge(λ) can be expressed as F=W−/integraltext d2x(µ1r2+µ2)n withµ1,2=const. Takingvariations δφsubjecttothePois- sonequation(2),oneobtainstoarbitraryorderin δφ, δF=/integraldisplay d2x(qφ−µ1x2−µ2)δn+/integraldisplay d2x\|∇δφ\|2 8π.(10) Thus, constrained extrema of Fsatisfy qφ=µ1x2+µ2, correspondingtoauniformdensitybeamcenteredon-axis. Variations about this extremum satisfy δF > 0and are secondorderin δφ. Thus,theavailableelectrostaticenergy forthermalizationinducedemittanceincreaseismodestfo r any smooth density proﬁle. This can be demonstrated for ourspeciﬁcexampleusingequation(7)toplot ∆F=Wi− WfwithRi=Rfverses hand1/hforp= 2,8(Fig.3). 00.250.50.751.1.251.51.752.2.00.0050.010.0150.020.0250.030.03 1/h hΔF /λ2Hollowed Peaked 0.75 0.50 0.25 0p =8p =8 p =2p =2 Figure3: Free energyverseshollowingfactors hand1/h. It has been shown that the rms beam size and emittance undergoverysmall decreases onrelaxationfromauniform density beam to thermal equilibriumover the full range of σi/σ0(Min[ǫxf/ǫxi]≃0.97atσi/σ0≃0.45)[4]. Thusif oneviewstherelaxationasamulti-stepprocedureusingthe conservation constraints to connect the initial nonunifor m proﬁle to a uniform proﬁle and then a thermal proﬁle, any emittancegrowthwillbeexperiencedintheﬁrststep. This result together with the variational argument above show that the emittance growthresults presentedshould be rela- tivelyinsensitivetotheformoftheﬁnal distribution. Finally,caveatsshouldbegivenforvalidityofthetheory. First, the model assumes no generationof halo in the ﬁnal state and that the initial nonuniformbeam can be perfectly rmsenvelopematched. Initialmismatchescan leadto halo productionandprovidealargesourceoffreeenergywhich, if thermalized,canlead to signiﬁcantemittancegrowth[1] . Also,althoughthevelocityspacedistributionisarbitrar yin the present theory, choices that project onto broader spec- trums of modes will more rapidly phase mix and thermal- ize. Small applied nonlinearﬁelds tend to enhancethis re- laxation rate. Initial simulation results in a full AG latti ce are consistent with model predictions presented here and will bepresentedin futurework. 6 REFERENCES [1] MartinReiser, TheoryandDesignofChargedParticleBeams (John Wiley& Sons,Inc., New York,1994). [2] R.C. Davidson, Physics of Nonneutral Plasmas (Addison- Wesley,Reading, MA, 1990). [3] T.P.Wangler,K.R.Crandall,R.S.Mills,andM.Reiser,I EEE Trans. Nucl.Sci., 32, 2196 (1985). [4] S.M. Lund, J.J. Barnard, and J.M. Miller, Proceedings of the 1995 ParticleAccelerator Conference, 3278 (1995).
1A Two Wire Waveguide and Interferometer for Cold Atoms E. A. Hinds, C. J. Vale, and M. G. Boshier Sussex Centre for Optical and Atomic Physics, University of Sussex, Brighton, BN1 9QH, U.K. A versatile miniature de Broglie waveguide is formed by two parallel current- carrying wires in the presence of a uniform bias field. We derive a variety of analytical expressions to describe the guide and present a quantum theory to show that it offers a remarkable range of possibilities for atom manipulation on the sub-micron scale. These include controlled and coherent splitting of the wavefunction as well as cooling, trapping and guiding. In particular we discuss a novel microscopic atom interferometer with the potential to be exceedingly sensitive. An atom whose magnetic moment has projection /G7a/G6d along an external magnetic field of magnitude B experiences a Zeeman interaction potential B U/G7a/G6d/G2d/G3d . The associated force is able to guide weak-field-seeking atoms along a minimum of B. This principle underlies the Stern-Gerlach effect and the magnetic hexapole lens, which have played such important roles in the history of atomic beams. Magnetic forces are now a central feature of atom optics – the subject of manipulating, confining, and guiding cold neutral atom clouds and Bose-condensates [1, 2, 3]. 2Recently there has been great interest in building miniature magnetic structures where small features make a strong field gradient, while a superimposed uniform bias field makes a zero whose position is adjustable [4]. This idea has been realized in several laboratories, using either supported wires [5, 6, 7], printed circuits [8, 9, 10, 11, 12] or microscopic patterns of permanent magnetization [13]. Miniature guides are attractive because they offer the possibility of propagating de Broglie waves in a single transverse mode in 1D [14] or 2D [15]. This is a central goal of many experimental groups because it is required for achieving atom interferometry with guided de Broglie waves. Miniature structures are also promising for studying the physics of quantum gases confined to less than 3D [16]. In this letter we discuss the guide formed by two wires carrying parallel currents I spaced A2 apart in the presence of a bias field, as illustrated in Fig. 1. We point out that the guide is far more adaptable than previously realized [9,10], deriving simple formulae for the various configurations it can produce. We present a quantum theory to show how the guide can be used to manipulate atoms on the sub-micron scale, to make a controlled and coherent splitting of the wavefunction, and to realize a novel atom interferometer. Let us adopt the natural units of A for lengths, A I B /G70 /G6d 2/0 0/Gba for magnetic fields, and 0B/G7a/G6d for energies. Dimensionless quantities based on these units are indicated by lower case letters, e.g. 1/G3dx means A X/G3d. With this scaling the magnetic field produced by the current-carrying wires in Fig. 1 has Cartesian components 32 2 2 22 2 2 2 ) 1(1 ) 1(1) 1( ) 1( y xx y xxby xy y xyb yx /G2b /G2d/G2b/G2d/G2b/G2b /G2b/G2b/G3d/G2b /G2d/G2d/G2b/G2b /G2b/G2d/G3d (1) On the y-axis, 0/G3dyb and the field produced by the wires is xb. This has a single maximum of 1 /G3dxb at height 1/G3dy as shown in Fig. 2. The addition of a bias field 1/G3c/G62 (in normalized units) along the positive x-direction cancels xb at two positions, indicated for 2/1/G3d/G62 by the circles in Fig. 2. For weak-field-seeking atoms, these zeros form guides parallel to the z-axis (a small field may be added along the z- direction to suppress non-adiabatic spin flips, although states without angular momentum around the guide axis can be stable without it [14]). The inset in Fig. 2 shows the dimensionless guiding potential u. The barrier between the two guides is /G62/G2d1 , while the barrier above the upper guide is /G62. As the strength of the bias field is increased, these zeros approach one another until they coalesce to form a single guide at height 1/G3dy when 1/G3d/G62 . Further increase of the bias splits the guide horizontally and the zeros follow the circle 12 2/G3d/G2by x (Fig. 1). In the first two rows of Table 1 we present simple formulae giving the guide centers ) ( 0 0,yx and trap depths 0u in each of the three bias field regimes. Fig. 3a shows the interaction potential u and the field lines for 8.0/G3d/G62 . The trap centered on 2/1/G3dy is smaller and 4 times stronger than the upper one at 2/G3dy (the ratio of gradients is always the inverse of the ratio of heights). Each guide is cylindrically symmetric over a limited region around its center, and has a constant 4potential gradient, a characteristic of the quadrupole symmetry evident in the field lines. We note that the fields are oppositely directed in the two guides. At the two bottom corners, the field can be seen circulating around the current-carrying wires, which are taken in this diagram to be of negligible radius. Figure 3b shows the potential and field lines for the single guide, formed at the critical bias. Here the linear gradient vanishes: the guide is harmonic with curvature 1 / 2 2/G3d/Gb6 /Gb6 /G72u and the corresponding hexapole symmetry can be seen in the field lines. At higher bias field the potential splits laterally into two quadrupole guides as illustrated in Fig. 3c for 5.1/G3d/G62 . Simple formulae for the gradients are given in the last row of Table 1. If we allow the bias field to have a component in the y-direction, many other possibilities open up, one of which is particularly relevant, as we will see below. Let us apply both the critical bias 1/G3d/G62 along x, and an extra bias of magnitude /G62/G44 at angle /G4a to the x-axis. When 1/G3c/G3c/G44/G62 , the potential splits into two guides separated by a distance /G62/G4422 , and the line joining their centers makes an angle 2//G4a with the x-axis. Thus, when the extra bias field is rotated the two guides orbit around the coalescence point at half the frequency. Each of the three regimes of bias field has interesting features to offer for atom optics experiments. To take a concrete example, consider a guide with a 300 /G6dm spacing and 2 A flowing in the wires, for which the characteristic field and gradient are B0 = 2.7 mT and B0/A = 18 T/m (s uch a structure already exists in our laboratory). With a weak bias field of 0.3 mT ( 1.0/G3d/G62 ), the upper guide is centered 3 mm ( /G62/2A ) above the wires and has a field gradient of 0.1 T/m ( AB/02 21/G62 ). This is very well suited to operate as a 5magneto-optical trap (MOT) with two pairs of suitably polarized light beams in the x-y plane, propagating along axes rotated by 45º from the x- and y-axes. For the purpose of initially collecting atoms in the MOT, a pair of auxiliary coils can create a field gradient along the z-direction, allowing a third pair of light beams to produce MOT confinement along the z-axis. The MOT can be lowered and compressed to a maximum gradient of 4.4 T/m ( AB/0 41) by increasing the bias field to 0.23 mT ( 2/30B ). At this point the light can be turned off and the auxiliary coils producing the field gradient along the z-axis can be switched to produce parallel fields. This makes a purely magnetic Ioffe-Pritchard trap, where weak-field-seeking atoms can be left in the ground state by evaporation. Alternatively, if a cold dense sample of atoms is already available in a macroscopic magnetic trap, as for example with a Bose-Einstein condensate, a single mode of the upper guide can be loaded in a “mode-matched” way by suddenly switching off the trap and turning on the guide with I, /G62 and a z-bias field chosen to duplicate the original trap spring constants. In either case, a subsequent adiabatic variation of the field can bring atoms to the coalescence point of the guide, where the potential is harmonic and the transverse frequency 0/G77 is given by 022 0 B A m/G7a/G6d /G77 /G3d . This has the value 1 3s108.2/G2d/Gb4 for the particular guide we are considering here. The ground state size 0 //G77 /G73 m/G68/Gba is 512 nm. Suppose now that the atoms have all been prepared in the transverse ground state of the harmonic guide. An increase /G62/G44 of the bias field deforms the guide potential into a symmetric double well, providing a highly controlled and reproducible 50:50 coherent splitter for the de Broglie wave. If the two halves of the atom cloud are held apart for some length of time they may act as the arms of an interferometer. In order to understand 6the operation of this interferometer in more detail, we have calculated the eigenmodes of the guide numerically by solving the two-dimensional time-dependent Schrödinger equation with the spin degree of freedom adiabatically eliminated. We start with the eigenstates )},{(y xnn of a harmonic potential, then slowly deform the potential to obtain the corresponding eigenstates of the Hamiltonian /G48 for the 2-wire guide at the critical bias 1/G3d/G62 . Next we slowly vary the bias field to find how the three lowest eigenstates and their energies /G48 evolve as a function of /G62. We assume that the atom density is low, although of course the mean field interactions at higher density should produce interesting non-linearities and physics beyond the Gross-Pitaevskii equation. Figures 4 and 5 show the eigenstates and their energies, labeled by the quantum numbers ),(y xnn which count the number of nodes along x and y. At the critical bias, the spectrum is (almost) harmonic with the )0,0( state lying 0/G77/G68 below the nearly degenerate pair of first excited states )0,1( and )1,0( (a small anisotropy of the guide prevents exact degeneracy). An increase /G62/G44 of the bias splits the guide and the single- peaked )0,0( state deforms adiabatically into a double-peaked wavefunction with even reflection symmetry in the yz-plane. As long as it is not perturbed, this of course returns to the harmonic ground state when /G62 is restored to 1. If, however, a differential phase of /G70 is introduced between its two peaks while 1/G3e/G62 , the wavefunction becomes a yz- antisymmetric function, which we recognize in Fig. 4 as the )0,1( state. When 1/Gae/G62 this evolves adiabatically into the first yz-antisymmetric excited state of the harmonic guide. For arbitrary phase shifts the final state of the harmonic guide is a superposition of )0,0( and )0,1( . Thus symmetry dictates that the two output ports of the 7interferometer are two different vibrational states of the harmonic guide. The third state )1,0( in Fig. 4 is even under reflection in the yz-plane, like the ground state, but it has a vibrational excitation in the y-direction. This excitation is preserved as the bias increases and consequently the energy lies approximately 0/G77/G68 above )0,0( . It is also interesting to consider decreasing the bias to 1/G3c/G62 . The interferometer states )0,0( and )0,1( both emerge in the upper guide, with )0,1( having one quantum of excitation along the x-axis. The )1,0( state however, goes into the ground state of the lower guide (Fig. 4). Now we turn to the practical aspects of the interferometer. One can estimate the minimum /G62/G44 needed to achieve splitting, min/G62/G44 , by setting the displacement of the wells equal to the ground state diameter /G732. To lowest order the result can be expressed in the elegant form 0 02 2 min / 2 /2 B A/G7a/G6d/G77 /G73 /G62 /G68/G3d /G3d /G44 , which for the guide in our example amounts to a change of 64 nT in the bias field. Achieving this level of control over the field would require some care but it is not a major technical challenge. When /G62/G44 is increased further, the splitting between the )0,0( and )0,1( levels becomes exponentially small (Fig. 5), being equal to the tunneling frequency between the left and right potential wells. As the bias changes, one wants to avoid non-adiabatic transitions induced by t/Gb6 /Gb6//G48 . This operator, being symmetric under reflections in the yz-plane, cannot excite the yz-antisymmetric state )0,1( , but it does connect the ground state to )1,0( . Since this and the other coupled states are at least 0/G77/G68 away in energy, the adiabatic condition is that /G62/G44 must change slowly in comparison with the period of harmonic oscillation 0/1/G77 (over several milliseconds for the guide in our example). Once the atoms are split, the 8antisymmetric perturbation to be measured is turned on. As is usual in interferometry, this interaction has to be non-adiabatic to mix the states )0,1( and )0,0( (i.e. to introduce a phase shift /G6a2 between the left and right wavepackets) and must therefore be turned on and off in times much less than the tunneling period. This is not a stringent requirement since the tunneling can be made arbitrarily slow by a modest increase in /G62/G44. Finally, reducing /G62/G44 to zero, it remains only to read out the fringe pattern through the population in state )0,0( [ )0,1( ], which is proportional to /G6a2cos [ /G6a2sin ]. In principle the readout can be done by absorption or fluorescence imaging to determine the atom distribution in the guide, although this method requires high spatial resolution. Alternatively, there are at least two methods for separating the )0,0( and )0,1( populations. First, if the guide is operated without any axial bias field and /G62 is reduced below 1 to extract both states in the upper quadrupole guide, the excited state )0,1( will be lost due to spin flips [14], leaving the )0,0( population to be measured by fluorescence. Second, the additional bias field /G62/G44 can be rotated adiabatically from the xˆ direction to xˆ/G2d before reducing its amplitude to zero, thereby transforming state )0,1( into )1,0( . This leaves the output of the interferometer as a superposition of )0,0( and )1,0( in the harmonic guide, which can be read out by reducing /G62 so that the )0,0( component moves into the upper guide while the )1,0( part is transported downward. In an interferometer the statistical signal:noise ratio is proportional to the interaction time /G74and to the square root of the number of atoms N. We can easily imagine 106 atoms trapped in the waveguide with a measurement time of ~10 s, giving 410/G3dN/G74 . In comparison a macroscopic cold atom interferometer of the Kasevich-Chu type [17] has 9108 atoms falling through the apparatus per second with an interaction time of order 30 ms, giving 310/G3dN/G74 over the same 10s. Both interferometers have a separation of ~100 /G6dm between arms. It therefore seems clear that this method of splitting the atoms in time, rather than splitting atoms propagating through space, can significantly advance some kinds of measurement. For example, the new interferometer would be extremely sensitive to electric field gradients and to gravity. Also, it would not suffer from phase shifts due to unwanted rotations because the Sagnac phase is zero. In conclusion, we have shown that two currents and a bias field form an exceedingly versatile structure, producing a waveguide that can be split in a highly controlled way and manipulated on the sub-micron scale. We have shown explicitly how a novel and sensitive atom interferometer could be realized with such a guide. This structure is also ideal for a variety of topical applications including a miniature magneto-optical trap and the study of 1-dimensional quantum gases. Finally, the quantum model of guided atom interferometry presented here will apply to many existing experiments once they reach the level of single-mode operation. We are indebted to David Lau, Stephen Hopkins, and Mark Kasevich for valuable discussions and to Matthew Jones for experimental work on the Sussex microscopic guide. This work was supported by the UK EPSRC and the EU. 10REFERENCES 1 J. P. Dowling and J. Gea-Banacloche, Adv. At. Mol. Opt. Phys. 37, 1 (1996). 2 E. A. Hinds and I. G. Hughes, J. Phys. D 32, R119 (1998). 3 W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, Enrico Fermi Course CXL, M. Inguscio et al. (Eds) IOS Press, Amsterdam (1999). 4 J. D. Weinstein and K. G. Libbrecht, Phys. Rev. A 52, 4004 (1995). 5 J. Fortagh et al. , Phys. Rev. Lett. 81, 5310 (1998). 6 J. Denschlag, D. Cassettari and J. Schmiedmayer, Phys. Rev. Lett. 82, 2014 (1999). 7 M. Key et al. , Phys. Rev. Lett. 84, 1371 (2000). 8 J. Reichel, W. Hänsel and T. W. Hänsch, Phys. Rev. Lett. 83, 3398 (1999). 9 D. Müller et al. , Phys. Rev. Lett. 83, 5194 (1999). 10 D. Müller et al. , arXiv:physics/0003091 (2000). 11 N. H. Dekker et al. , Phys. Rev. Lett. 84, 1124 (2000). 12 R. Folman et al. , Phys. Rev. Lett. 84, 4749 (2000). 13 P. Rosenbusch et al. , Phys Rev. A 61, 31404(R) (2000); Appl. Phys B 61, 709 (2000). 14 E. A. Hinds and Claudia Eberlein, Phys Rev. A 61 33614 (2000) 15 E. A. Hinds, M. G. Boshier and I. G. Hughes, Phys. Rev. Lett. 80, 645 (1998). 16 M. Olshanii, Phys. Rev. Lett. 81, 938 (1998): H. Monien, M. Linn, and N. Elstner, Phys. Rev. A 58, R3395 (1998). 17 M. J. Snadden et al. , Phys. Rev. Lett. 81, 971 (1998). 11Figure Captions Figure 1. Atom guide using two current-carrying wires and a bias field. With increasing bias, two guiding regions move towards each other along Y (dotted line) until they coalesce at Y = A . They then separate along the dashed circle. Figure 2 . Field xb on the x-axis versus height y above the wires. A bias 2/1/G3d/G62 along x makes two zeros (circles). Inset : interaction potential u showing the two linear guides. Figure 3 . Interaction potentials and field lines for (a) 8.0/G3d/G62 (b) 1/G3d/G62 (c) 5.1/G3d/G62 . Figure 4 . Wavefunctions of the lowest three states in the guide for extra bias fields 2,0,2 /min/G2d/G3d /G44/G44 /G62/G62 . The ranges for x and y are 5.0/Gb1 and 72.15.0/G2d respectively. Figure 5 . Energy spectrum for the three lowest eigenstates in the guide versus extra bias field. Table 1 . Center, depth and gradient of the 2-wire guides for each regime of the bias field. Bias /G62 < 1 /G20/G62 = 1 /G62 > 1 (x0, y0) /G28/G2921 1,01/G62/G62/G2d/Gb1 (0,1) /G28/G29 1 ,112/G2d /Gb1/G62/G62 0u /G62 /G62/G2d1, 1 1/G2d/G62 /G72/Gb6/Gb6/u /G28 /G292 21 1 1 /G62 /G62 /G2d/Gb1 /G2d 0 12/G2d/G62/G62 AACoalescence point (0,A) XYBias field Figure 1: Hinds et al.00.20.40.60.811.2 0246 Dimensionless height yDimensionless field bx 00.5 0246y u Figure 2: Hinds et al.0 13 2 1 -10 xy (a) (b) (c)012 1 -10 xy 011 -1 xy 0u y 0 -10 11 2x00.6u y0 -10 11 x01u y 0 -10 12 3 x00.5 1 Figure 3: Hinds et al. Figure 4. Hinds et al -1 0 1 Dimensionless extra bias field Db/Dbmin1234Energy / hw0 (0,0)(0,1)(1,0) Figure 5. Hinds et al. 0
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/D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU/D4/CP/D7/D7/CX/DA /CT /D4/CP/D6/D8/CX /D0/CT/D7/BA /CB/CX/D2 /CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP /CT /CX/D7 /D5/D9/CX/D8/CT /D0/CP/D6/CV/CT/B8 /D8/CW/CT/D4/CW/CX/D0/D3/D7/D3/D4/CW /DD /CX/D7 /CT/D7/D7/CT/D2 /D8/CX/CP/D0/D0/DD /CS/CT/D7 /D6/CX/D4/D8/CX/DA /CT/B8 /CP/D2/CS /D8/D6/CP /CT/D6/D7 /D1/D3/D8/CX/D3/D2/CP/D6/CT /D7/D8/D9/CS/CX/CT/CS /CU/D3/D6 /D3/D2/CT /CP/D6/CQ/CX/D8/D6/CP/D6/DD /CW/D3/D7/CT/D2 /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU/D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/BA /BT/D7 /D8/CW/CT /CU/CP/D6 /AS/CT/D0/CS /D6/CT/CV/CX/D3/D2 /CW/CP/D7 /CQ /CT/CT/D2 /CX/D2 /DA /CT/D7/D8/CX/B9/CV/CP/D8/CT/CS /CJ/BH/BE℄/B8 /DB /CT /CU/D3 /D9/D7 /D3/D2 /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D2/CT/CP/D6 /D3/D6 /CX/D2/D7/CX/CS/CT /D8/CW/CT /D3/D6/CT/D7/B8/DB/CW/CX /CW /DB /CT /CQ /CT/D0/CX/CT/DA /CT /D7/CW/D3/D9/D0/CS /CQ /CT /CV/CT/D2/CT/D6/CX /CT/DA /CT/D2 /CU/D3/D6 /D1/CP/D2 /DD /DA /D3/D6/D8/CT/DC/D7/DD/D7/D8/CT/D1/D7/BA /CF /CT /DB/CX/D0/D0 /D3/D1/CT /D8/D3 /D8/CW/CT /CX/D7/D7/D9/CT /D3/CU /D8/D6/CP/D2/D7/D4 /D3/D6/D8 /D4/D6/D3/D4 /CT/D6/B9/D8/CX/CT/D7 /CP/D2/CS /AS/D2/CX/D8/CT/B9/D8/CX/D1/CT /C4/DD /CP/D4/D9/D2/D3 /DA /CT/DC/D4 /D3/D2/CT/D2 /D8/D7 /CX/D2 /CP /CU/D3/D6/D8/CW /D3/D1/CX/D2/CV/D4/D9/CQ/D0/CX /CP/D8/CX/D3/D2/BA/C1/D2 /CB/CT /BA /BE/B8 /DB /CT /CS/CT/D7 /D6/CX/CQ /CT /CQ/D6/CX/CT/AT/DD /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /CU/D3/D9/D6 /DA /D3/D6/B9/D8/CX /CT/D7/BA /CF /CT /D4/D6/CT/D7/CT/D2 /D8 /CP /C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/B8/DB/CW/CX /CW /D4/D6/D3 /DA/CX/CS/CT/D7 /CP /CV/D3 /D3 /CS /D8/CT/D7/D8 /D8/D3 /D3/D9/D6 /D2 /D9/D1/CT/D6/CX /CP/D0 /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/CP/D2/CS /CP/D0/D0/D3 /DB/D7 /D8/D3 /CW/CP/D6/CP /D8/CT/D6/CX/DE/CT /CT/CP/D7/CX/D0/DD /D8/CW/CT /CW/CP/D3/D8/CX /D3/D6 /D2/D3/D2 /CW/CP/D3/D8/CX /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /D1/D3/D8/CX/D3/D2/BA /BT /D2/CT/DB /D7/CT /D8/CX/D3/D2 /CP/D4/D8/D9/D6/CX/D2/CV /CP/D0/D0 /CT/D5/D9/CX/DA/B9/CP/D0/CT/D2 /D8 /D4/CW /DD/D7/CX /CP/D0 /D6/CT/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /AT/D3 /DB /CX/D7 /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS/BA /CC/CW/CX/D7/D7/CT /D8/CX/D3/D2 /D7/CW/D3 /DB/D7 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /D2/D3/D2/B9/D9/D2/CX/CU/D3/D6/D1/CX/D8 /DD /CX/D2 /D8/CW/CT /D4/CW/CP/D7/CT/D7/D4/CP /CT/B8 /DB/CW/CX /CW /CX/D7 /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2 /D3/CU /DA /D3/D6/D8/CX /CT/D7 /CP/D2/CS /CX/D7/D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /DA /D3/D6/D8/CT/DC/B9/D4/CP/CX/D6/CX/D2/CV/BA /CB/D8/CP/D8/CX/D7/D8/CX /D7 /D3/D2/D4/CP/CX/D6/CX/D2/CV /D8/CX/D1/CT/D7 /CP/D6/CT /D3/D1/D4/D9/D8/CT/CS /CP/D2/CS /CT/DC/CW/CX/CQ/CX/D8 /D4 /D3 /DB /CT/D6/B9/D0/CP /DB /D8/CP/CX/D0/D7/B8 /CX/D1/D4/D0/DD/CX/D2/CV /AS/D2/CX/D8/CT /CP /DA /CT/D6/CP/CV/CT /D4/CP/CX/D6/CX/D2/CV /D8/CX/D1/CT/BA /C1/D2 /CB/CT /BA /BF /D8/CW/CT /D1/D3/D8/CX/D3/D2/D3/CU /D8/D6/CP /CT/D6/D7 /CX/D7 /D7/D8/D9/CS/CX/CT/CS/B8 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /CT /D3/CU /D3/D6/CT/D7 /CX/D7 /D3/D2/AS/D6/D1/CT/CS /CP/D2/CS/D7/D8/CX /CZ/CX/D2/CT/D7/D7 /D8/D3 /D8/CW/CT /DA /D3/D6/D8/CT/DC /D3/D6/CT/D7 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS/BA /CC/CW/CT /CX/D2/AT/D9/CT/D2 /CT/D3/CU /DA /D3/D6/D8/CT/DC /D4/CP/CX/D6/CX/D2/CV /CX/D7 /D7/D8/D9/CS/CX/CT/CS/B8 /DB/CW/CX /CW /D4/D6/D3 /DA /CT/D7 /D8/D3 /CQ /CT /CP /CV/D3 /D3 /CS/D8/D6/CP/D4/D4/CX/D2/CV /B4/D9/D2 /D8/D6/CP/D4/D4/CX/D2/CV/B5 /D1/CT /CW/CP/D2/CX/D7/D1 /D3/CU /D8/D6/CP /CT/D6/D7 /CP/D6/D3/D9/D2/CS /D8/CW/CT /D3/D6/CT/D7/BA /C8 /CP/CX/D6/CX/D2/CV /D3/CU /DA /D3/D6/D8/CX /CT/D7 /CP/D0/D0/D3 /DB/D7 /CP /D7/D4 /CT /CX/CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/D6/CP /B9/CT/D6/D7 /CY/D9/D1/D4/CX/D2/CV /CU/D6/D3/D1 /D3/D2/CT /D3/D6/CT /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /D3/D6/CT/B8 /DB/CW/CX /CW /D3/D4 /CT/D2/D7/D8/CW/CT /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /CX/D2 /D1/CP/D2 /DD /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/D7 /D3/CU /D7/D4 /CT /CX/CP/D0 /D8/D6/CP/D2/D7/D4 /D3/D6/D8/CU/CT/CP/D8/D9/D6/CT/D7 /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1 /CY/D9/D1/D4/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D3/D6/CT/D7/BA /CC/CW/CT /D1/D3/D8/CX/D3/D2/DB/CX/D8/CW/CX/D2 /D8/CW/CT /D3/D6/CT /CX/D7 /D7/D8/D9/CS/CX/CT/CS/B8 /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /CP/D7/CP /CU/D9/D2 /D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /D8/D3 /D8/CW/CT /DA /D3/D6/D8/CT/DC /CP/D6/CT /D3/D1/D4/D9/D8/CT/CS/B8/CP/D2/CS /D2/D3 /D8 /DD/D4/CX /CP/D0 /CS/CX/AR/D9/D7/CX/D3/D2 /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D7 /CU/D3/D9/D2/CS/BA/BE /CE /D3 /D6/D8/CT/DC /D1/D3/D8/CX/D3/D2/BE/BA/BD /BW/CT/AS/D2/CX/D8/CX/D3/D2/D7/CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8 /DB /D3/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /BX/D9/D0/CT/D6 /CT/D5/D9/CP/D8/CX/D3/D2/B8 /CS/CT/B9/D7 /D6/CX/CQ/CX/D2/CV /D8/CW/CT /CS/DD/D2/CP/D1/CX /D7 /D3/CU /CP /D7/CX/D2/CV/D9/D0/CP/D6 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /DA /D3/D6/D8/CX /CX/D8 /DD ω(z) =N/summationdisplay α=1kαδ(z−zα(t)), /B4/BD/B5/DB/CW/CT/D6/CT z /D0/D3 /CP/D8/CT/D7 /CP /D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D3/D1/D4/D0/CT/DC /D4/D0/CP/D2/CT/B8 zα=xα+ iyα /CX/D7 /D8/CW/CT /D3/D1/D4/D0/CT/DC /D3 /D3/D6/CS/CX/D2/CP/D8/CT /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC α /B8 /CP/D2/CSkα /CX/D8/D7/D7/D8/D6/CT/D2/CV/D8/CW/B8 /CX/D2 /CP/D2 /CX/CS/CT/CP/D0 /CX/D2 /D3/D1/D4/D6/CT/D7/D7/CX/CQ/D0/CT /D8 /DB /D3/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /AT/D9/CX/CS /CP/D2 /CQ /CT /CS/CT/D7 /D6/CX/CQ /CT/CS /CQ /DD /CP /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /D7/DD/D7/D8/CT/D1 /D3/CUN /CX/D2 /D8/CT/D6/CP /D8/CX/D2/CV/D4/CP/D6/D8/CX /D0/CT/D7 /B4/D7/CT/CT /CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT /CJ/BH/BF ℄/B5/B8 /D6/CT/CU/CT/D6/D6/CT/CS /D8/D3 /CP/D7 /CP /D7/DD/D7/D8/CT/D1 /D3/CU N /D4 /D3/CX/D2 /D8 /DA /D3/D6/D8/CX /CT/D7/BA /CC/CW/CT /D7/DD/D7/D8/CT/D1/B3/D7 /CT/DA /D3/D0/D9/D8/CX/D3/D2 /DB/D6/CX/D8/CT/D7 kα˙zα=−2i∂H ∂¯zα, ˙¯zα= 2i∂H ∂(kαzα),(α= 1,· · ·, N),/B4/BE/B5/DB/CW/CT/D6/CT /D8/CW/CT /D3/D9/D4/D0/CT (kαzα,¯zα) /CP/D6/CT /D8/CW/CT /D3/D2/CY/D9/CV/CP/D8/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /D3/CU/D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 H /BA /CC/CW/CT /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/D7/D3/D2 /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /CS/D3/D1/CP/CX/D2 /D3 /D9/D4/CX/CT/CS /CQ /DD /D8/CW/CT /AT/D9/CX/CS/B8 /CU/D3/D6/D8/CW/CT /CP/D7/CT /D3/CU /CP/D2 /D9/D2 /CQ /D3/D9/D2/CS/CT/CS /D4/D0/CP/D2/CT/B8 /D8/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /D3/D1/D4/D0/CT/DC/DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CSv(z, t) /CP/D8 /D4 /D3/D7/CX/D8/CX/D3/D2 z /CP/D2/CS /D8/CX/D1/CTt /CV/CX/DA /CT/D2 /CQ /DD /D8/CW/CT/D7/D9/D1 /D3/CU /D8/CW/CT /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0 /DA /D3/D6/D8/CT/DC /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7/B8 /DB/D6/CX/D8/CT/D7/BM v(z, t) =1 2πiN/summationdisplay α=1kα1 ¯z−¯zα(t). /B4/BF/B5/CP/D2/CS /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CQ /CT /D3/D1/CT/D7 H=−1 2π/summationdisplay α>βkαkβln\|zα−zβ\|=−1 4πlnΛ . /B4/BG/B5/CC/CW/CT /D8/D6/CP/D2/D7/D0/CP/D8/CX/D3/D2/CP/D0 /CP/D2/CS /D6/D3/D8/CP/D8/CX/D3/D2/CP/D0 /CX/D2 /DA /CP/D6/CX/CP/D2 /CT /D3/CUH /B8 /D4/D6/D3 /DA/CX/CS/CT/D7/D8/CW/CT /D1/D3/D8/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /D8/CW/D6/CT/CT /D3/D8/CW/CT/D6 /D3/D2/D7/CT/D6/DA /CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/CQ /CT/D7/CX/CS/CT/D7 /D8/CW/CT /CT/D2/CT/D6/CV/DD /B8 Q+iP=N/summationdisplay α=1kαzα, L2=N/summationdisplay α=1kα\|zα\|2. /B4/BH/B5/BT/D1/D3/D2/CV /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D3/D2/D7/D8/CP/D2 /D8/D7 /D3/CU /D1/D3/D8/CX/D3/D2/B8 /D8/CW/CT/D6/CT /CP/D6/CT /D8/CW/D6/CT/CT/CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /AS/D6/D7/D8 /CX/D2 /D8/CT/CV/D6/CP/D0/D7 /CX/D2 /CX/D2 /DA /D3/D0/D9/D8/CX/D3/D2/BM H /B8Q2+P2/CP/D2/CS L2/B8 /D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /D8/CW/D6/CT/CT /DA /D3/D6/D8/CX /CT/D7 /D3/D2 /D8/CW/CT /D4/D0/CP/D2/CT/CX/D7 /CP/D0/DB /CP /DD/D7 /CX/D2 /D8/CT/CV/D6/CP/CQ/D0/CT /CP/D2/CS /CW/CP/D3/D7 /CP/D6/CX/D7/CT/D7 /DB/CW/CT/D2 N≥4 /BA/BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB /BF/BE/BA/BE /BV/CP/D2/D3/D2/CX /CP/D0 /CC /D6/CP/D2/D7/CU/D3 /D6/D1/CP/D8/CX/D3/D2/D7/BW/D9/CT /D8/D3 /D8/CW/CT /CW/CP/D3/D8/CX /D2/CP/D8/D9/D6/CT /D3/CU /BG/B9/D4 /D3/CX/D2 /D8 /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/B8 /D8/CW/CT/D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D3/CU /DA /D3/D6/D8/CT/DC /D1/D3/D8/CX/D3/D2 /D2/CT /CT/D7/D7/CX/D8/CP/D8/CT/D7 /CP /CS/CX/AR/CT/D6/CT/D2 /D8/CP/D4/D4/D6/D3/CP /CW /D8/CW/CP/D2 /CU/D3/D6 /CX/D2 /D8/CT/CV/D6/CP/CQ/D0/CT /BF/B9/DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/D7/BA /CF /CT /CU/D3/D0/D0/D3 /DB/CA/CT/CU/BA /CJ/BG/BL ℄ /CP/D2/CS /D4 /CT/D6/CU/D3/D6/D1 /CP /CP/D2/D3/D2/CX /CP/D0 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/DA /D3/D6/D8/CT/DC /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7/BA /CC/CW/CX/D7 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /CP/D2 /CT/CU/B9/CU/CT /D8/CX/DA /CT /D7/DD/D7/D8/CT/D1 /DB/CX/D8/CW2 /CS/CT/CV/D6/CT/CT/D7 /D3/CU /CU/D6/CT/CT/CS/D3/D1/B8 /D4/D6/D3 /DA/CX/CS/CX/D2/CV /CP /D3/D2/B9 /CT/D4/D8/D9/CP/D0/D0/DD /CT/CP/D7/CX/CT/D6 /CU/D6/CP/D1/CT/DB /D3/D6/CZ/B8 /CQ /CT/D7/D8 /D7/D9/CX/D8/CT/CS /CU/D3/D6 /CP /CS/CT/D8/CP/CX/D0/CT/CS /CP/D2/CP/D0/B9/DD/D7/CX/D7 /D3/CU /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /CU/D3/D9/D6 /CX/CS/CT/D2 /D8/CX /CP/D0 /DA /D3/D6/D8/CX /CT/D7/BA /BY /D3/D6 /CX/D2/D7/D8/CP/D2 /CT/B8/D8/CW/CX/D7 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/D0/D0/D3 /DB/D7 /D8/D3 /D4 /CT/D6/CU/D3/D6/D1 /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D8 /DB /D3/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2/D7/B8 /CU/D6/D3/D1 /DB/CW/CX /CW /D8/CW/CT /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7/D3/CU /D8/CW/CT /D1/D3/D8/CX/D3/D2 /CP/D6/CT /CP/D2/CP/D0/DD/DE/CT/CS/BA /CC/CW/CT /CS/CT/D8/CP/CX/D0/D7 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/B9/D1/CP/D8/CX/D3/D2 /CP/D6/CT /D2/D3/D8 /CV/CX/DA /CT/D2 /CW/CT/D6/CT /CQ/D9/D8 /D8/CW/CT /D3/D9/D8/D0/CX/D2/CT /CX/D7 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/B8/D9/D7/CX/D2/CV zα /B4α= 1,· · ·,4 /B5 /CP/D7 /D8/CW/CT /D3/D1/D4/D0/CT/DC /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /D3/CU /D8/CW/CT/CU/D3/D9/D6 /DA /D3/D6/D8/CX /CT/D7/BA (2Jn)1 2eiθn=N−1 2N/summationdisplay α=1exp(i(2πn/N )(α−1))zα, n= 0,· · ·, N−1,/B4/BI/B5/DB/CX/D8/CW /D2/CT/DB /CP/D2/D3/D2/CX /CP/D0 /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /braceleftbigg φ1=1 2(θ1−θ3), φ2=1 2(θ1+θ3)−θ2, φ 3=θ2 I1=J1−J3, I 2=J1+J2, I 3=J1+J2+J3./B4/BJ/B5/CC/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CX/D7 /D6/CP/D8/CW/CT/D6 /D3/D1/D4/D0/CX /CP/D8/CT/CS /CQ/D9/D8 /CX/D2/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CUφ3 /B8 /D1/CT/CP/D2/CX/D2/CV /D8/CW/CP/D8I3=1 2L2/CX/D7 /CP /D3/D2/D7/D8/CP/D2 /D8/D3/CU /D1/D3/D8/CX/D3/D2/BA /CF /CT /D1/CT/D2 /D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CX/D7 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CQ /DD /D4/D6/CT/B9/D7/CT/D6/DA/CX/D2/CV /CP/D2/CS /D1/CP/CZ/CX/D2/CV /D9/D7/CT /D3/CU /D8/CW/CT /D3/D2/D7/D8/CP/D2 /D8/D7 /D3/CU /D1/D3/D8/CX/D3/D2 /CX/D7 /D8/CP/CZ/B9/CX/D2/CV /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /D7/DD/D1/D1/CT/D8/D6/CX/CT/D7 /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA/C0/D3 /DB /CT/DA /CT/D6/B8 /CQ /CT/D7/CX/CS/CT/D7 /D8/CW/CT/D7/CT /D7/DD/D1/D1/CT/D8/D6/CX/CT/D7/B8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 /CP/D0/D7/D3 /CX/D2/B9/DA /CP/D6/CX/CP/D2 /D8 /D9/D2/CS/CT/D6 /D8/CW/CT /CS/CX/D7 /D6/CT/D8/CT /CV/D6/D3/D9/D4 /D3/CU /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2/D7/BA /CC/CW/CX/D7/D0/CP/D7/D8 /CU/CT/CP/D8/D9/D6/CT /CX/D7 /D4/CP/D6/D8/CX /D9/D0/CP/D6 /D8/D3 /D8/CW/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2 /D3/CU /CU/D3/D9/D6 /CX/CS/CT/D2 /D8/CX /CP/D0/DA /D3/D6/D8/CX /CT/D7/BA /C1/D8 /CW/CP/D7 /CQ /CT/CT/D2 /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /CU/D3/D6 /CP /D7/D9/CQ/CV/D6/D3/D9/D4 /D3/CU /D8/CW/CT/D7/CT/D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CT/AR/CT /D8 /D3/CU /D8/CW/CT/D7/CT /D7/DD/D1/D1/CT/D8/D6/CX/CT/D7 /D3/D2 /D8/CW/CT /D3/D9/D4/D0/CT (I1, φ1) /D0/CT/CP/CS/D7 /D8/D3 /D7/CX/D1/D4/D0/CT /D0/CX/D2/CT/CP/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /B4 /D7/CT/CT /CJ/BG/BL ℄/B8/D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT/D4/D6/D3 /CS/D9 /CT/CS /CX/D2 /CC /CP/CQ/D0/CT /BD/B5/B8 /CQ/D9/D8 /D8/CW/CT /CT/AR/CT /D8 /D3/CU /CU/D3/D6/CX/D2/D7/D8/CP/D2 /CT /D8/CW/CT /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2 (2,1,3,4) /D3/D2 /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/B4/DB/CW/CX /CW /CP/D2 /CQ /CT /D8/CW/D3/D9/CV/CW /D8 /D3/CU /CP/D7 /CP /D6/CT/D0/CP/CQ /CT/D0/CX/D2/CV z′ 1=z2 /B8z′ 2=z1 /B8 z′ 3=z3 /B8z′ 4=z4 /B5/B8 /D0/CT/CP/CS/D7 /D8/D3 /D2/D3 /D7/CX/D1/D4/D0/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/D2 (I1, φ1) /BA /CC/CW/CT /CT/AR/CT /D8 /D3/CU /D8/CW/CT/D7/CT /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2/D7 /D3/D2(I2, φ2) /CS/D3 /CT/D7/D2/CT/CX/D8/CW/CT/D6 /D0/CT/CP/CS /D8/D3 /D7/CX/D1/D4/D0/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/BA /CF /CT /D7/CW/CP/D0/D0 /CS/CX/D7 /D9/D7/D7/D7/D3/D1/CT /D3/D2/D7/CT/D5/D9/CT/D2 /CT/D7 /D3/CU /D8/CW/CT/D7/CT /CX/D7/D7/D9/CT/D7 /CX/D2 /D8/CW/CT /D3/D1/D4/D9/D8/CP/D8/CX/D3/D2 /D3/CU/C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2/D7/BA/D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2 I1 φ1 (1,2,3,4) I1 φ1 (4,1,2,3) I1 φ1−π/2 (2,3,4,1) I1 φ1+π/2 (2,1,4,3) −I1−φ1+π/2 (3,4,1,2) I1 φ1 (4,3,2,1) −I1−φ1−π/2 (3,2,1,4) −I1 −φ1 (1,4,3,2) −I1 −φ1/CC /CP/CQ/D0/CT /BD/BA /BX/AR/CT /D8 /D3/CU /D8/CW/CT /D7/D9/CQ/CV/D6/D3/D9/D4 D4 /D3/D2 /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU(I1, φ1) /BA/CC /D3 /D7/D9/D1/D1/CP/D6/CX/DE/CT /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /CJ/BG/BL ℄/B8 /D8/CW/CT /D1/D3/D8/CX/D3/D2/CX/D7 /CX/D2 /CV/CT/D2/CT/D6/CP/D0 /CW/CP/D3/D8/CX /B8 /CT/DC /CT/D4/D8 /CU/D3/D6 /D7/D3/D1/CT /D7/D4 /CT /CX/CP/D0 /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/B9 /D8/CX/D3/D2/D7/B8 /CU/D3/D6 /CX/D2/D7/D8/CP/D2 /CT /DB/CW/CT/D2 /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /CP/D6/CT /CU/D3/D6/D1/CX/D2/CV /CP /D7/D5/D9/CP/D6/CT/D8/CW/CT /D1/D3/D8/CX/D3/D2 /CX/D7 /D4 /CT/D6/CX/D3 /CS/CX /CP/D2/CS /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /D6/D3/D8/CP/D8/CT /D3/D2 /CP /CX/D6 /D0/CT/B8/D8/CW/CT/D2 /D7/DD/D1/D1/CT/D8/D6/CX /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2 /B4z3=−z1 /CP/D2/CSz4=−z2 /B5/D3/CU /D8/CW/CT /D7/D5/D9/CP/D6/CT /D0/CT/CP/CS /D8/D3 /D5/D9/CP/D7/CX/D4 /CT/D6/CX/D3 /CS/CX /D1/D3/D8/CX/D3/D2 /B4/D4 /CT/D6/CX/D3 /CS/CX /D1/D3/B9/D8/CX/D3/D2 /CX/D2 /CP /CV/CX/DA /CT/D2 /D6/D3/D8/CP/D8/CX/D2/CV /CU/D6/CP/D1/CT/B5/B8 /D7/CT/CT /CJ/BG/BL ℄ /CU/D3/D6 /D8/CW/CT /D3/D1/D4/D0/CT/D8/CT/CS/CT/D8/CP/CX/D0/D7/BA/BE/BA/BF /C8 /D3/CX/D2 /CP /D6/GH /D7/CT /D8/CX/D3/D2/D7/BT/D7 /CP /D4/D6/CT/D6/CT/D5/D9/CX/D7/CX/D8/CT /D8/D3 /D3/D9/D6 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /D3/D2 /D8/CW/CT /D4/CP/D7/D7/CX/DA /CT /D8/D6/CP /CT/D6/B3/D7/D1/D3/D8/CX/D3/D2/B8 /CP /CQ/CP/D7/CX /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/D9/CQ/D7/DD/D7/D8/CT/D1 /CQ /CT/B9/CW/CP /DA/CX/D3/D6 /CX/D7 /D2/CT /CT/D7/D7/CP/D6/DD /BA /BY /D3/D6 /D8/CW/CX/D7 /D1/CP/D8/D8/CT/D6/B8 /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /CX/D7 /CW/D3/D7/CT/D2 /CP/D2/CS /CP /C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC/D7/DD/D7/D8/CT/D1 /CX/D7 /D3/D1/D4/D9/D8/CT/CS/BA /CC/CW/CT /D7/CT /D8/CX/D3/D2 /CX/D7 /CP /D8/D3 /D3/D0 /DB/CW/CX /CW /CX/D2/D7/D9/D6/CT/D7/D8/CW/CP/D8 /CU/D3/D6 /D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/B8 /D8/CW/CT /D8/D6/CP /CY/CT /D8/D3/D6/DD /CW/CP/D7/D8/CW/CT /CS/CT/D7/CX/D6/CT/CS /CV/CT/D2/CT/D6/CX /CW/CP/D3/D8/CX /CQ /CT/CW/CP /DA/CX/D3/D6/BA /CC /D3 /D4 /CT/D6/CU/D3/D6/D1 /C8 /D3/CX/D2 /CP/D6/GH/D7/CT /D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/B8 /DB /CT /D4/D6/D3 /CT/CT/CS /CP/D7 /CX/D2 /CJ/BG/BL℄/B8 /CP/D2/CS /D9/D7/CT /D8/CW/CT/D7/CT/D8 /D3/CU /CP/D2/D3/D2/CX /CP/D0 /D3/D2/CY/D9/CV/CP/D8/CT/CS /DA /CP/D6/CX/CP/CQ/D0/CT/D7/BM /braceleftBigg R1= (I1+I3)1 2cos2φ1R2= (I3−I2)1 2sin 2φ2 P1= (I1+I3)1 2sin 2φ1P2= (I3−I2)1 2cos2φ2, /B4/BK/B5/D8/D3 /D3/D1/D4/D9/D8/CT /D8/CW/CT /D7/CT /D8/CX/D3/D2 R2= 0 /B8˙R2<0 /BA/CC/CW/CT /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CX/D7 /D7/CT /D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CP/D6/CQ/CX/D8/D6/CP/D6/DD /CW/D3/D7/CT/D2 /CX/D2/CX/B9/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /BY/CX/CV/BA /BD/BA /CF /CT /D2/D3/D8/CX /CT /D8/CW/CP/D8 /D8/CW/CT/D7/CT /D8/CX/D3/D2 /D6/CT/D4/D6/D3 /CS/D9 /CT/D7 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /DB /CT/D0/D0 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2/CJ/BH/BG ℄/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D7/CT /D8/CX/D3/D2 /DB /CP/D7 /D3/D1/D4/D9/D8/CT/CS /D9/D7/CX/D2/CV /CP /D2 /D9/D1/CT/D6/CX /CP/D0/CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D6/CT/CS/D9 /CT/CS /D7/DD/D7/D8/CT/D1/BA /CC/CW/CX/D7 /CW/CP/D7 /D8/CW/CT /CP/CS/DA /CP/D2 /D8/CP/CV/CT/D3/CU /D3/AR/CT/D6/CX/D2/CV /CP /D2/CP/D8/D9/D6/CP/D0 /D8/CT/D7/D8 /D3/CU /D8/CW/CT /CP /D9/D6/CP /DD /D3/CU /D3/D9/D6 /D2 /D9/D1/CT/D6/CX /CP/D0/CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/B8 /DB/CW/CX /CW /CX/D7 /D1/CP/CS/CT /CX/D2 /D3/D9/D6 /CP/D7/CT /CX/D2 /D8/CW/CT /D3/D6/CX/CV/CX/D2/CP/D0 /DA /D3/D6/B9/D8/CT/DC /D3/D1/D4/D0/CT/DC /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /D9/D7/CX/D2/CV /CP /AS/CU/D8/CW /D3/D6/CS/CT/D6 /D7/CX/D1/D4/D0/CT /D8/CX /BZ/CP/D9/D7/D7/B9/C4/CT/CV/CT/D2/CS/D6/CT /D7 /CW/CT/D1/CT /CJ/BH/BH ℄/BA /C8/D6/CT/DA/CX/D3/D9/D7 /CP/D2/CP/D0/D3/CV/D3/D9/D7 /D7/CT /D8/CX/D3/D2/D7 /CP/D2 /CQ /CT/D7/CT/CT/D2 /CX/D2 /CJ/BG/BL ℄ /CP/D2/CS /CJ/BH/BG ℄ /CP/D2/CS /CP/D6/CT /CP/D0/D0 /CX/D2 /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW/D3/D9/D6 /D6/CT/D7/D9/D0/D8/BA /CC/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /CX/D7 /CW/CP/D3/D8/CX /BA /CC/CW/CX/D7 /D7/CT/D8/B9/D8/D0/CT/D7 /D8/CW/CT /CW/D3/CX /CT /D3/CU /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/B8/DB/CW/CX /CW /DB /CT /DB/CX/D0/D0 /D9/D7/CT /CU/D6/D3/D1 /D2/D3 /DB /D3/D2/BA/CF /CT /D2/D3/D8/CX /CT /D8/CW/CP/D8 /BY/CX/CV/BA/BD /CW/CP/D7 /D7/DD/D1/D1/CT/D8/D6/CX/CT/D7 /DB/CW/CX /CW /D6/CT/D0/CP/D8/CT /D8/D3/D8/CW/CT /CT/AR/CT /D8 /D3/CU /D8/CW/CT /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2/D7 /CX/D0/D0/D9/D7/D8/D6/CP/D8/CT/CS /CX/D2 /CC /CP/CQ/D0/CT /BD/BA /CC/CW/CX/D7/D7/D3/D1/CT/CW/D3 /DB /D6/CT/AT/CT /D8/D7/B8 /CP/D7 /CP/D2 /D8/CX /CX/D4/CP/D8/CT/CS /CT/CP/D6/D0/CX/CT/D6 /D3/D2/B8 /D8/CW/CP/D8 /D8/D3 /CT/D5/D9/CX/DA/B9/CP/D0/CT/D2 /D8 /D4/CW /DD/D7/CX /CP/D0 /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1 /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS /CS/CX/AR/CT/D6/CT/D2 /D8 /D4 /D3/CX/D2 /D8/D7 /D0/D3 /CP/D8/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/CT/CV/CX/D3/D2/D7 /D3/CU /D8/CW/CT/D7/CT /D8/CX/D3/D2/BA /C5/D3/D6/CT/D3 /DA /CT/D6 /D8/CW/CT /CU/CP /D8 /D8/CW/CP/D8 /D8/CW/CT /D2/CT/DB /CP/D2/D3/D2/CX /CP/D0 /DA /CP/D6/CX/B9/CP/CQ/D0/CT/D7 /CP/D6/CT /D2/D3/D8 /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D9/D2/CS/CT/D6 /CP/D0/D0 /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/B9/D8/CX /CT/D7 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D4/CW /DD/D7/CX /CP/D0 /D6/CT/CP/D0/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT/D7/DD/D7/D8/CT/D1 /CW/CP /DA /CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D0/D3 /CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D6/CT/CS/D9 /CT/CS /D4/CW/CP/D7/CT /D7/D4/CP /CT/B8/CP/D2/CS /CP/D6/CT /D0/CX/CZ /CT/D0/DD /D8/D3 /D2/D3/D8 /CP/D0/D0 /CQ /CT/D0/D3/D2/CV /D8/D3 /D8/CW/CT /D7/CT /D8/CX/D3/D2/BA /BY /D6/D3/D1 /D8/CW/CT/CP/CS/DA /CT /D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB/B8 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D4/CW /DD/D7/CX /CP/D0 /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2/D7/CV/CT/D2/CT/D6/CP/D8/CT /CX/CS/CT/D2 /D8/CX /CP/D0 /AT/D3 /DB/D7/B8 /D7/D3 /D8/CW/CT /D4/D6/D3/D4 /D3/D7/CT/CS /D7/CT /D8/CX/D3/D2 /D1/CP /DD /D2/D3/D8/CQ /CT /CQ /CT/D7/D8 /D7/D9/CX/D8/CT/CS /CU/D3/D6 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D2/CV /CP/CS/DA /CT /D8/CX/D3/D2 /CP/D2/CS /D4 /D3/D7/D7/CX/CQ/D0/CT /D4/CP/D8/B9/D8/CT/D6/D2/D7/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT/DD /D4/D6/D3 /DA /CT /DA /CT/D6/DD /D9/D7/CT/CU/D9/D0 /D8/D3 /CW/CP/D6/CP /D8/CT/D6/CX/DE/CT /D8/CW/CT/D8 /DD/D4 /CT /D3/CU /D1/D3/D8/CX/D3/D2 /B4/D4 /CT/D6/CX/D3 /CS/CX /B8 /D5/D9/CP/D7/CX/D4 /CT/D6/CX/D3 /CS/CX /B8 /CW/CP/D3/D8/CX /B5/BA /C7/D2/CT /D4 /D3/D7/B9/D7/CX/CQ/D0/CT /DB /CP /DD /D8/D3 /CX/D6 /D9/D1 /DA /CT/D2 /D8 /D8/CW/CX/D7 /D0/D3 /CP/D8/CX/D3/D2 /D4/D6/D3/CQ/D0/CT/D1 /CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1/D2/D3/D8/CX /CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 R1= 0 /D3/D6P1= 0 /D6/CT/D1/CP/CX/D2/D7 /D9/D2/B9 /CW/CP/D2/CV/CT/CS /CQ /DD /D8/CW/CT /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2/D7 /D0/CX/D7/D8/CT/CS /CX/D2 /CC /CP/CQ/D0/CT /BD/B8 /CP/D2/CS /D8/D3 /CP/D4/B9/D8/D9/D6/CT /CP/D0/D0 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D7/DD/D7/D8/CT/D1/D7/B8 /DB /CT /CP/D2 /D7/D9/D4 /CT/D6/CX/D1/D4 /D3/D7/CT /D8/CW/CT /D8 /DB /D3/CU/D3/D0/D0/D3 /DB/CX/D2/CV /D7/CT /D8/CX/D3/D2/D7 R1= 0 /B8˙R1<0 /B8 /CP/D2/CSR′ 1= 0 /B8˙R′ 1<0 /B8/DB/CW/CT/D6/CT′/D7/D8/CP/D2/CS/D7 /CU/D3/D6 /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CUR1 /CP/D2/CS˙R1 /D3/CQ/D8/CP/CX/D2/CT/CS /CP/CU/B9/D8/CT/D6 /D8/CW/CT /D6/CT/D0/CP/CQ /CT/D0/CX/D2/CV (2,1,3,4) /CX/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS/BN /B4/D2/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT/BG /BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB/D7/D9/D4 /CT/D6/CX/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /CS/D3 /CT/D7 /D2/D3/D8 /D1/D3 /CS/CX/CU/DD /D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/CT /D8/CX/D3/D2 /CU/D3/D6/D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/B8 /CQ/D9/D8 /CP/D0/D0/D3 /DB/D7 /CU/D3/D6 /D1/D3/D6/CT /D4 /D3/CX/D2 /D8/D7/CP/D2/CS /D6/CT/CS/D9 /CT/D7 /D3/D1/D4/D9/D8/CP/D8/CX/D3/D2 /D8/CX/D1/CT/D7/B5/BA /CC/CW/CT 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/CU/D3/D9/D2/CS /CT/D2/D3/D9/CV/CW /D4/CP/CX/D6/CX/D2/CV/D7 /D8/D3 /D3/CQ/D8/CP/CX/D2 /CP /CS/CX/D7/D8/D6/CX/B9/CQ/D9/D8/CX/D3/D2 /D3/CU /D4/CP/CX/D6/CX/D2/CV/D7 /D8/CW/CP/D8 /D0/CP/D7/D8 /D0/D3/D2/CV/CT/D6 /D8/CW/CP/D2 /CP /D8/CX/D1/CTτ /B8 /D1/CT/CP/D2/CX/D2/CV/DB /CT /D3/D1/D4/D9/D8/CT/CS /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD N(T > τ )∼/integraldisplay∞ τρ(T)dT , /B4/BL/B5/DB/CW/CT/D6/CT ρ(T)dT /CX/D7 /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /CP/D2 /CT/DA /CT/D2 /D8 /D8/D3 /D0/CP/D7/D8 /CP /D8/CX/D1/CT T /BA /CC/CW/CT /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BH/B8 /DB/CW/CT/D6/CT /DB /CT /D2/D3/D8/CX /CT /D8/CW/CP/D8/BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB /BH/D7/D3/D1/CT /D6/CP/D6/CT /CT/DA /CT/D2 /D8/D7 /CP/D6/CT /D0/CP/D7/D8/CX/D2/CV /CP /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D0/D3/D2/CV /D8/CX/D1/CT/B8 /CP/D2/CS /D8/CW/CT/CS/CX/AR/CT/D6/CT/D2 /D8 /D9/D6/DA /CT/D7 /CP/D6/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D4 /D3/D7/D7/CX/CQ/D0/CT /D9/D8/B9/D3/AR /CS/CX/D7/D8/CP/D2 /CT /CU/D3/D6d /B8 /D6/CT/CV/CP/D6/CS/CX/D2/CV /D8/CW/CT /CS/CT/D8/CT /D8/CX/D3/D2 /D3/CU /CT/DA /CT/D2 /D8/D7/BA /CC/CW/CT/CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/B3/D7 /D8/CP/CX/D0 /CX/D7 /CS/D3/D2/CT /CQ /DD /D8/CW/CT /C4/D3/CV/B9/C4/D3/CV/D4/D0/D3/D8 /CX/D2 /BY/CX/CV/BA /BH/BA /BT /D4 /D3 /DB /CT/D6/B9/D0/CP /DB /CS/CT /CP /DD /D3/CU /D8/CW/CT /D8/CP/CX/D0 /D3/CU /D8/CW/CT /D8 /DD/D4 /CT τ−α+1/DB/CX/D8/CW /CT/DC/D4 /D3/D2/CT/D2 /D8 α∼3.66±0.1 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS/B8 /DB/CW/CX /CW/D8/D6/CP/D2/D7/D0/CP/D8/CT/D7 /CX/D2 /D2/D3/D2/B9/DE/CT/D6/D3 /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /DA /CT/D6/DD /D0/D3/D2/CV /D6/CP/D6/CT /CT/DA /CT/D2 /D8/D7/BM/D0/D3/D2/CV /D0/CP/D7/D8/CX/D2/CV /D4/CP/CX/D6/CX/D2/CV /D3 /D9/D6/D7/BA /CC/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/B9/CX/D8 /DD /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D4/CP/CX/D6/CX/D2/CV ρ(τ) /D0/CP/D7/D8/CX/D2/CV /CP /D8/CX/D1/CTτ /B8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS/CU/D6/D3/D1 /BX/D5/BA /B4/BL/B5 ρ(τ) =dN dτ, /B4/BD/BC/B5/DB/CW/CX /CW /D0/CT/CP/CS/D7 /D8/D3 /CP /D4 /D3 /DB /CT/D6 /D0/CP /DB /CS/CT /CP /DD /D3/CU /D8/CW/CT /D8 /DD/D4 /CTρ∼τ−α∼ τ−3.7±0.1/BA /CC/CW/CX/D7 /CQ /CT/CW/CP /DA/CX/D3/D6 /D8/D6/CP/D2/D7/D0/CP/D8/CT/D7 /CX/D2 /AS/D2/CX/D8/CT /CP /DA /CT/D6/CP/CV/CT /D4/CP/CX/D6/B9/CX/D2/CV /D8/CX/D1/CT/B8 /CP/D2/CS /D7/CT /D3/D2/CS /D1/D3/D1/CT/D2 /D8/BA /C0/D3 /DB /CT/DA /CT/D6 /D7/CX/D2 /CT α /CX/D7 /D0/D3/D7/CT/D8/D34 /CP/D2/CS /D8/CW/CT /CP /D9/D6/CP /DD /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6α /CX/D7 /D2/D3/D8 /D4 /CT/D6/CU/CT /D8/B8 /DB /CT /CP/D2/D2/D3/D8 /CQ /CT /D3/D2 /D0/D9/D7/CX/DA /CT /CP/CQ /D3/D9/D8 /D8/CW/CT /CS/CX/DA /CT/D6/CV/CT/D2 /CT /D3/CU /D8/CW/CT /D8/CW/CX/D6/CS/D1/D3/D1/CT/D2 /D8/BA /CF /CT /CX/D2/D7/CX/D7/D8 /D8/CW/CP/D8 /CP/D7 /CT/DC/D4 /CT /D8/CT/CS/B8 /D8/CW/CT /D0/D3/D2/CV/B9/D0/CP/D7/D8/CX/D2/CV /D8/CX/D1/CT/B9 /D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7 /CX/D2/CS/D9 /CT/CS /CQ /DD /CP /D8 /DD/D4/CX /CP/D0 /D7/D8/CX /CZ/CX/D2/CV /CQ /CT/CW/CP /DA/CX/D3/D6/B8 /D6/CT/D7/D9/D0/D8/D7/CX/D2 /CP /D4 /D3 /DB /CT/D6/B9/D0/CP /DB /CS/CT /CP /DD /D3/CU /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/B3/D7 /D8/CP/CX/D0/BA/BE/BA/BI /C5/CX/D2/CX/D1/D9/D1 /CS/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /DA/D3 /D6/D8/CX /CT/D7/CC /D3 /D3/D2 /D0/D9/CS/CT /D3/D2 /DA /D3/D6/D8/CT/DC /D1/D3/D8/CX/D3/D2/B8 /DB /CT /D1/CT/CP/D7/D9/D6/CT /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1/CS/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/BA /C1/D2/CS/CT/CT/CS/B8 /CP/D7 /D7/D9/CV/CV/CT/D7/D8/CT/CS /CX/D2 /CJ/BH/BD ℄/B8/D8/CW/CT /D7/CX/DE/CT /D3/CU /D8/CW/CT /D3/D6/CT/D7 /D7/D9/D6/D6/D3/D9/D2/CS/CX/D2/CV /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /CX/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3/D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CS/CX/D7/D8/CP/D2 /CT /D3/CU /CP/D4/D4/D6/D3/CP /CW /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/CU/D3/D6 /CP /BF/B9/DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1 /CJ/BG/BD℄/BA /CC/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CS/CX/D7/D8/CP/D2 /CT /CX/D7 /D2 /D9/B9/D1/CT/D6/CX /CP/D0/D0/DD /D1/CT/CP/D7/D9/D6/CT/CS /CP/D2/CS /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT/D4 /D3/D6/D8/CT/CS /CX/D2 /CC /CP/CQ/D0/CT /BE/B8/DB/CW/CX /CW /CV/CX/DA /CT /D8/CW/CT /DA /CP/D0/D9/CT dmin≈0.6 /BA /BT/D2 /CP/D2/CP/D0/DD/D8/CX /CP/D0 /CT/D7/D8/CX/D1/CP/B9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CX/D2 /D8/CT/D6/B9/DA /D3/D6/D8/CT/DC /CS/CX/D7/D8/CP/D2 /CT /CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS/B4/D0/D3 /DB /CT/D6 /CQ /D3/D9/D2/CS/B5/B8 /CQ /DD /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D3 /D9/D6/D7 /CS/D9/D6/B9/CX/D2/CV /CP /D4/CP/CX/D6/CX/D2/CV/B8 /CP/D2/CS /D8/CW/CP/D8 /D8/CW/CX/D7 /D1/CX/D2/CX/D1 /D9/D1 /CX/D7 /D7/D1/CP/D0/D0 /D3/D1/D4/CP/D6/CT/CS/D8/D3 /D8/CW/CT /D3/D8/CW/CT/D6 /CX/D2 /D8/CT/D6/B9/DA /D3/D6/D8/CT/DC /CS/CX/D7/D8/CP/D2 /CT/D7 /DB/CW/CX /CW /DB /CT /CP/D2 /CP/D7/D7/D9/D1/CT/D8/D3 /CQ /CT /CP/D0/D0 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /CP /CV/CX/DA /CT/D2 /CS/CX/D7/D8/CP/D2 /CT dav /BA /CD/D2/CS/CT/D6 /D8/CW/CT/D7/CT /D3/D2/B9/CS/CX/D8/CX/D3/D2/D7 /D8/CW/CT /D8/CW/CT /D3/D2/D7/D8/CP/D2 /D8/D7 /D3/CU /D1/D3/D8/CX/D3/D2/D7 /CQ /CT /D3/D1/CT K≡/parenleftBigg4/summationdisplay l=1kl/parenrightBigg L2−(Q2+P2) =d2 min+5d2 av≈5d2 av, /B4/BD/BD/B5/CP/D2/CS Λ= exp( −4πH)≈d2 mind10 av. /B4/BD/BE/B5/CD/D7/CX/D2/CV /CQ /D3/D8/CW /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BD/BD/B5 /CP/D2/CS /B4/BD/BE/B5 /DB /CT /D3/CQ/D8/CP/CX/D2 /CP /D7/CX/D1/D4/D0/CT/CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CX/D2 /D8/CT/D6/B9/DA /D3/D6/D8/CT/DC /CS/CX/D7/D8/CP/D2 /CT dmin=/radicalBigg/parenleftbigg5 K/parenrightbigg5 Λ . /B4/BD/BF/B5/CF /CT /D2/D3 /DB /D9/D7/CT /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BD/BF/B5/B8 /DB/CX/D8/CW /D8/CW/CT /DA /CP/D0/D9/CT/D7 /CU/D3/D6 /D8/CW/CT /D3/D2/D7/D8/CP/D2 /D8 /D3/CU /D1/D3/D8/CX/D3/D2/D7 /CV/CX/DA /CT/D2 /CQ /DD /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT/DA /D3/D6/D8/CX /CT/D7 /D9/D7/CT/CS /CU/D3/D6 /D8/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/D7/BM [(1.747,1.203) ( −√ 2/2,0) (√ 2/2,0) (0,−1)] /BA /CC/CW/CX/D7 /D0/CT/CP/CS/D7 /D8/D3dmin≈0.58 /B8 /CX/D2 /DA /CT/D6/DD /CV/D3 /D3 /CS/CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D6/CT/D4 /D3/D6/D8/CT/CS /CX/D2 /CC /CP/CQ/D0/CT /BE/BA/C0/CP /DA/CX/D2/CV /CP /D6/D3/D9/CV/CW /D4/CX /D8/D9/D6/CT /D3/CU /D8/CW/CT /D9/D2/CS/CT/D6/D0/DD/CX/D2/CV /DA /D3/D6/D8/CT/DC/B9/D1/D3/D8/CX/D3/D2/B8/DB /CT /D2/D3 /DB /CU/D3 /D9/D7 /D3/D2 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/D6/CP /CT/D6/D7/BA /CC/CX/D1/CT /D3/CU /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /C5/CX/D2/CX/D1 /D9/D1 /CS/CX/D7/D8/CP/D2 /CT 10000 0.6000 20000 0.5980 30000 0.5980 50000 0.5960 100000 0.5960/CC /CP/CQ/D0/CT /BE/BA /C5/CT/CP/D7/D9/D6/CT/CS /D1/CX/D2/CX/D1 /D9/D1 /CS/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/B8/DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /D8/CX/D1/CT/BA /CF /CT /CV/CT/D8 /D6/D3/D9/CV/CW/D0/DD min(rij)∼0.6 /B8/DB/CW/CX /CW /D8/D6/CP/D2/D7/D0/CP/D8/CT/D7 /D8/CW/CP/D8 /D8/CW/CT /D3/D6/CT/B3/D7 /D6/CP/CS/CX/D9/D7 r /CX/D7 /D7/D9 /CW /D8/CW/CP/D8r <0.3 /BA/BF /C8 /CP /D6/D8/CX /D0/CT /D1/D3/D8/CX/D3/D2/BF/BA/BD /BW/CT/AS/D2/CX/D8/CX/D3/D2/D7/CC/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /CP /D8/D6/CP /CT/D6 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /D8/CW/CT /CP/CS/DA /CT /D8/CX/D3/D2 /CT/D5/D9/CP/B9/D8/CX/D3/D2 ˙z=v(z, t) /B4/BD/BG/B5/DB/CW/CT/D6/CT z(t) /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D6/CP /CT/D6 /CP/D8 /D8/CX/D1/CTt /B8/CP/D2/CSv(z, t) /CX/D7 /D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CS/BA /BY /D3/D6 /CP /D4 /D3/CX/D2 /D8 /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/B8/D8/CW/CT /DA /CT/D0/D3 /CX/D8 /DD /AS/CT/D0/CS /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /BX/D5/BA /B4/BF/B5/B8 /CP/D2/CS /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/BG/B5 /CP/D2 /CQ /CT /D6/CT/DB/D6/CX/D8/D8/CT/D2 /CX/D2 /CP /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CU/D3/D6/D1/BM ˙z=−2i∂Ψ ∂¯z, ˙¯z= 2i∂Ψ ∂z /B4/BD/BH/B5/DB/CW/CT/D6/CT /D8/CW/CT /D7/D8/D6/CT/CP/D1 /CU/D9/D2 /D8/CX/D3/D2 Ψ(z,¯z, t) =−1 2π4/summationdisplay α=1kαln\|z−zα(t)\| /B4/BD/BI/B5/CP /D8/D7 /CP/D7 /CP /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2/BA /CC/CW/CT /D7/D8/D6/CT/CP/D1 /CU/D9/D2 /D8/CX/D3/D2 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2/D8/CX/D1/CT /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /DA /D3/D6/D8/CT/DC /D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 zα(t) /B8 /CX/D1/D4/D0/DD/CX/D2/CV /CP /D2/D3/D2/B9/CP/D9/D8/D3/D2/D3/D1/D3/D9/D7 /D7/DD/D7/D8/CT/D1/BA/BF/BA/BE /BT /CT/D7/D7/CX/CQ/D0/CT /D4/CW/CP/D7/CT /D7/D4/CP /CT/BT /AS/D6/D7/D8 /D7/D8/CT/D4 /CX/D2 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /D4/CP/D7/D7/CX/DA /CT /D8/D6/CP /B9/CT/D6/D7 /CX/D7 /D8/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT/CX/D6 /CP /CT/D7/D7/CX/CQ/D0/CT /CS/D3/D1/CP/CX/D2 /CX/D2 /D3/D8/CW/CT/D6 /DB /D3/D6/CS/D7/B8/DB/CW/CT/D6/CT /D8/CW/CT/DD /D1/D3 /DA /CT/BA /CB/CX/D2 /CT /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /DB/CW/CX /CW/CP/D6/CT /CS/D6/CX/DA/CX/D2/CV /D8/CW/CT /AT/D3 /DB /CX/D7 /CW/CP/D3/D8/CX /B8 /DB /CT /CP/D2/D2/D3/D8 /D9/D7/CT /C8 /D3/CX/D2 /CP/D6/GH/D1/CP/D4/D7 /D8/D3 /DA/CX/D7/D9/CP/D0/CX/DE/CT /D8/CW/CT /D4/CW/CP/D7/CT /D7/D4/CP /CT /CP/D7 /CX/D2 /D4 /CT/D6/CX/D3 /CS/CX /BF/B9/DA /D3/D6/D8/CT/DC/D7/DD/D7/D8/CT/D1/D7/BA /CC/CW/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /D9/D7/CT /D3/CU /C8 /D3/CX/D2 /CP/D6/GH /D6/CT /D9/D6/D6/CT/D2 /CT/D7 /CP/D7 /CP/D2/CP/D2/CP/D0/D3/CV /D8/D3 /D8/CW/CT /D4 /CT/D6/CX/D3 /CS /D7/CW/D3/D9/D0/CS /CP/D0/D7/D3 /D6/CT/DA /CT/CP/D0 /CX/D8/D7/CT/D0/CU /CW/CP/DE/CP/D6/CS/D3/D9/D7/CP/D2/CS /D2 /D9/D1/CT/D6/CX /CP/D0/D0/DD /D3/D7/D8/D0/DD /B8 /CP/D7 /D4/CP/CX/D6/CX/D2/CV /DB /CP/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 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/D7/DD/D1/B9/D1/CT/D8/D6/DD /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /CS/CX/D7 /D9/D7/D7/CT/CS /CQ /CT/D8 /DB /CT/CT/D2 /DB/CW/CP/D8 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2/D8/CW/CT /CU/CP/D6 /D6/CT/CV/CX/D3/D2 /CJ/BH/BE ℄ /CP/D2/CS /D8/CW/CT /D1/D3/D8/CX/D3/D2 /DB/CX/D8/CW/CX/D2 /D8/CW/CT /D3/D6/CT/BA /C6/CP/D1/CT/D0/DD/D8/CW/CX/D7 /D6/CT/D7/D9/D0/D8/D7 /D8/D6/CP/D2/D7/D0/CP/D8/CT/D7 /CX/D2 /D8/D3 /D8/CW/CT /CU/CP /D8 /D8/CW/CP/D8 /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /D7 /CP/D0/CT/CP/D7∼r6/DB/CW/CT/D6/CT r /CX/D7 /D8/CW/CT /D7/D1/CP/D0/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/B8 /CP/D2/CS /CX/D2 /D8/CW/CT /CU/CP/D6/AS/CT/D0/CS /D6/CT/CV/CX/D3/D2 /CX/D8 /CW/CP/D7 /CQ /CT/CT/D2 /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /D8/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2 /D3 /CTꜶ/B9 /CX/CT/D2 /D8 /CQ /CT/CW/CP /DA /CT/D7 /CP/D7D∼1/R6/B8 /DB/CW/CT/D6/CT R /CX/D7 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /CU/D6/D3/D1/D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU /DA /D3/D6/D8/CX /CX/D8 /DD /CP/D2/CS1/R /CX/D7 /D8/CW/CT /D7/D1/CP/D0/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/BA /C1/D2/CU/CP /D8 /DB /CT /D2/D3/D8/CT /D8/CW/CP/D8 /D0/D3/D7/CT /D8/D3 /D8/CW/CT /DA /D3/D6/D8/CT/DC /DB /CT /CW/CP /DA /CT˙θ≈k/2πr2/B8/D8/CW/CT/D6/CT/CU/D3/D6/CT /D7/CX/D2 /CTσ(r, t)∼r3/B8 /DB /CT /D6/CT/CP/D7/D3/D2/CP/CQ/D0/DD /CT/DC/D4 /CT /D8 /D8/CW/CT /D7/D8/CP/D2/B9/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2 σ(k/2πr2, t)≈σ(˙θ, t) /D8/D3 /CQ /CT /D3/D2/D0/DD /CP /CU/D9/D2 /D8/CX/D3/D2/D3/CUt /BA /CA/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D7/CW/D3 /DB/D2 /D3/D2 /BY/CX/CV/BA /BD/BH/B8 /CP/D2/CS /DB /CT /CT/AR/CT /D8/CX/DA /CT/D0/DD /D2/D3/B9/D8/CX /CT /D8/CW/CP/D8 /CU/D3/D6 /CP/D0/D0 /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/CP/CS/CX/CX/B8 /D8/CW/CT /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D1/D3/D6/CT/D3/D6 /D0/CT/D7/D7 /D3/D2 /CT/D2 /D8/D6/CP/D8/CT/CS /D3/D2 /D3/D2/CT /D9/D6/DA /CT/BA /C1/D8 /CX/D7 /D8/CW/CT/D2 /D7/D9Ꜷ /CX/CT/D2 /D8 /D8/D3 /D3/D2/D7/CX/CS/CT/D6 /D3/D2/D0/DD /D3/D2/CT /D3/D6/CQ/CX/D8 /D8/D3 /D7/D8/D9/CS/DD /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6/BA/C6/D3/D8/CT /CP/D0/D7/D3 /D8/CW/CP/D8 /CU/D3/D6r= 0.18 /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 σ(r)2∼3.4 10−5/B8/D7/D3 /CU/D3/D6 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS /D8/CX/D1/CT /CX/CU /DB /CT /CW/CP/CS /CS/CX/AR/D9/D7/CX/D3/D2 /DB /CT /D7/CW/D3/D9/D0/CS /D3/CQ/B9/D7/CT/D6/DA /CT /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D3/D6/CS/CT/D6 σ2(r, t)∼σ2(r)t≈1 /CU/D3/D6 t= 5.104/BA /C1/D2 /D8/CW/CX/D7 /D0/CX/CV/CW /D8/B8 /D7/CX/D2 /CT /D7/D9 /CW /CV/D6/D3 /DB/D8/CW /D3/CU /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/DB /CP/D7 /D2/D3/D8 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 /BY/CX/CV/BA/BD/BG/B8 /DB /CT /CP/D2 /CT/DC /D0/D9/CS/CT /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD/D3/CU /CS/CX/AR/D9/D7/CX/DA /CT /B4/D3/D6 /D7/D9/D4 /CT/D6/CS/CX/AR/D9/D7/CX/DA /CT/B5 /CQ /CT/CW/CP /DA/CX/D3/D6 /DB/CX/D8/CW/CX/D2 /D8/CW/CT /D3/D6/CT/BA/C0/D3 /DB /CT/DA /CT/D6/B8 /D7/D9/CQ /CS/CX/AR/D9/D7/CX/DA /CT /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D7 /D7/D8/CX/D0/D0 /CP /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /B8 /CP/D7 /CU/D3/D6/CX/D2/D7/D8/CP/D2 /CT (5.104)1/8≈3.87 /D7/D3 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CP/D6/CV/D9/D1/CT/D2 /D8 /CS/D3 /CT/D7/D2/D3/D8 /CW/D3/D0/CS/B8 /CP/D2/CS /DB /CT /D2/CT/CT/CS /DA /CT/D6/DD /D0/D3/D2/CV /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /D8/D3 /D0/CP/D6/CX/CU/DD /D8/CW/CT/D8/CT/D1/D4 /D3/D6/CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7/BA/BG /BV/D3/D2 /D0/D9/D7/CX/D3/D2/C1/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /DB /CT /CW/CP /DA /CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /CP /D4/CP/D7/B9/D7/CX/DA /CT /D8/D6/CP /CT/D6 /CX/D2 /CP /CW/CP/D3/D8/CX /AT/D3 /DB /CV/CT/D2/CT/D6/CP/D8/CT/CS /CQ /DD /CU/D3/D9/D6 /CX/CS/CT/D2 /D8/CX /CP/D0/D4 /D3/CX/D2 /D8 /DA /D3/D6/D8/CX /CT/D7/BA /C1/D2 /D8/CW/CT /D4/D6/D3 /CT/D7/D7 /D3/CU /D8/CW/CX/D7 /D7/D8/D9/CS/DD /B8 /CP /D4/CP/D6/D8/CX /D9/D0/CP/D6/CP/D8/D8/CT/D2 /D8/CX/D3/D2 /CW/CP/D7 /CQ /CT/CT/D2 /D1/CP/CS/CT /D8/D3 /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/D9/CQ/D7/DD/D7/D8/CT/D1/BA /CB/CX/D2 /CT/D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /CP/D6/CT /CX/CS/CT/D2 /D8/CX /CP/D0/B8 /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1 /CX/D7 /D7/D9/CQ /CY/CT /D8 /D8/D3/D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /CS/CX/D7 /D6/CT/D8/CT /D7/DD/D1/D1/CT/D8/D6/DD /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1 /CX/D2 /DA /CP/D6/CX/CP/D2 /CT/D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /CV/D6/D3/D9/D4 /D3/CU /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2/D7/BA /CC/CW/CX/D7 /D0/CT/CP/CS /D8/D3 /CX/D2 /D8/D6/D3 /CS/D9 /CT/CP /D2/CT/DB /C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2/BA /CC/CW/CX/D7 /D7/CT /D8/CX/D3/D2 /CW/CP/D7 /D8/CW/CT /CP/CS/DA /CP/D2 /D8/CP/CV/CT /D8/D3/CT/DC/CW/CX/CQ/CX/D8 /D8/CW/CT /D2/D3/D2 /CW/D3/D1/D3/CV/CT/D2/CT/CX/D8 /DD /D3/CU /D8/CW/CT /D4/CW/CP/D7/CT /D7/D4/CP /CT/B8 /D6/CT/D7/D9/D0/D8/B9/CX/D2/CV /CU/D6/D3/D1 /CP /D7/D4 /CT /CX/CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1 /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D4/CW/CT/D2/D3/D1/CT/D2/D3/D2 /D3/CU /DA /D3/D6/D8/CT/DC/B9/D4/CP/CX/D6/CX/D2/CV/BA /C8 /CP/CX/D6/CX/D2/CV/D6/CT/D7/D9/D0/D8/D7 /CX/D2 /CP /D8/CT/D1/D4 /D3/D6/CP/D6/DD /D0/D3/D7/D7 /D3/CU /CP /CS/CT/CV/D6/CT/CT /D3/CU /CU/D6/CT/CT/CS/D3/D1 /D3/CU /D8/CW/CT/DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1/D7/B8 /CX/D8 /CX/D7 /D8/CW/CT/D6/CT/CU/D3/D6/CT /CP/D7/D7/D3 /CX/CP/D8/CT/CS /DB/CX/D8/CW /CP /D7/D8/CX /CZ/CX/D2/CT/D7/D7/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2 /D8/D3 /CP/D2 /D3/CQ /CY/CT /D8 /D3/CU /D0/CT/D7/D7/CT/D6 /CS/CX/D1/CT/D2/D7/CX/D3/D2 /D8/CW/CP/D2 /D8/CW/CT/CP /CT/D7/D7/CX/CQ/D0/CT /D4/CW/CP/D7/CT /D7/D4/CP /CT/B8 /CP/D2/CS /CX/D7 /D0/CX/D2/CZ /CT/CS /D8/D3 /D0/D3 /CP/D0 /CW/CP/D2/CV/CT/D7 /D3/CU/CS/CT/D2/D7/CX/D8 /DD /CX/D2 /D8/CW/CT /C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2 /CX/D0/D0/D9/D7/D8/D6/CP/D8/CT/CS /CX/D2 /BY/CX/CV/BA /BE/BA /CB/D9 /CW /CQ /CT/CW/CP /DA/CX/D3/D6 /CP/D2 /D8/CW/CT/D2 /CQ /CT /D8/CW/D3/D9/CV/CW /D8 /D3/CU /CP/D7 /CP /CW/CP/D3/D7/B9 /CW/CP/D3/D7 /CX/D2 /D8/CT/D6/B9/D1/CX/D8/D8/CT/D2 /D8 /CQ /CT/CW/CP /DA/CX/D3/D6/BA /CE /D3/D6/D8/CT/DC/B9/D4/CP/CX/D6/CX/D2/CV/B8 /CQ /DD /CX/D2 /DA /D3/D0/DA/CX/D2/CV /D3/D2/D0/DD /D8 /DB /D3/DA /D3/D6/D8/CX /CT/D7/B8 /CX/D7 /D6/CT/D1/CX/D2/CX/D7 /CT/D2 /D8 /D3/CU /CW/CX/CV/CW/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1/D7/DB/CW/CT/D6/CT /D0/D3 /DB/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /DA /D3/D6/D8/CT/DC /CQ /CT/CW/CP /DA/CX/D3/D6 /DB /CP/D7 /D7/CW/D3 /DB/D2 /D8/D3 /CQ /CT/CX/D2/AT/D9/CT/D2 /D8/CX/CP/D0 /CJ/BD/BE ℄/BA /BY /D9/D6/D8/CW/CT/D6 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /CW/CP /DA /CT /D0/CT/CP/CS /D8/D3 /D3/D1/B9/D4/D9/D8/CT /D4/CP/CX/D6/CX/D2/CV/B9/D8/CX/D1/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7/BA /CC/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/D2/D7/CX/D8 /DD/CT/DC/CW/CX/CQ/CX/D8/D7 /CP /D4 /D3 /DB /CT/D6/B9/D0/CP /DB /D8/CP/CX/D0 /D8 /DD/D4/CX /CP/D0 /D8/CW/CT /D3/CU /D7/D8/CX /CZ/CX/D2/CT/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6/BA/CC/CW/CT /D4 /D3 /DB /CT/D6/B9/D0/CP /DB /CT/DC/D4 /D3/D2/CT/D2 /D8 /CX/D7 /CU/D3/D9/D2/CS /D8/D3 /CQ /CT /CP/D6/D3/D9/D2/CS α≈3.7 /B8/CX/D1/D4/D0/DD/CX/D2/CV /AS/D2/CX/D8/CT /D8 /DD/D4/CX /CP/D0 /B4/CP /DA /CT/D6/CP/CV/CT/B5 /D7/D8/CX /CZ/CX/D2/CV /D8/CX/D1/CT/BA /CF /CT /D2/D3/D8/CT/D8/CW/CP/D8 /D7/CX/D2 /CT /D1/D3/D7/D8 /D1/CT/D6/CV/CX/D2/CV /D4/D6/D3 /CT/D7/D7/CT/D7 /CX/D2 /BE/BW /D8/D9/D6/CQ/D9/D0/CT/D2 /CT /D3 /D9/D6/DB/CW/CX/D0/CT /D8 /DB /D3 /D7/CP/D1/CT /D7/CX/CV/D2/B9/DA /D3/D6/D8/CX /CT/D7 /CP/D6/CT /D4/CP/CX/D6/CX/D2/CV/B8 /D8/CW/CT /AS/D2/CX/D8/CT/D2/CT/D7/D7 /D3/CU/D4/CP/CX/D6/CX/D2/CV /D8/CX/D1/CT /CQ /DD /D8/CW/CT /CX/D2 /D8/D6/D3 /CS/D9 /D8/CX/D3/D2 /D3/CU /CP/D2/D3/D8/CW/CT/D6 /D7/D4 /CT /CX/AS /D8/CX/D1/CT/D7 /CP/D0/CT /CQ /CT/D7/CX/CS/CT/D7 /D8/CW/CT /D8 /DD/D4/CX /CP/D0 /D1/CT/D6/CV/CX/D2/CV /D8/CX/D1/CT/B8 /D1/CP /DD /D4/D0/CP /DD /CP/D2 /CX/D1/B9/D4 /D3/D6/D8/CP/D2 /D8 /D6/D3/D0/CT/BA/C8 /CP/D7/D7/CX/DA /CT /D8/D6/CP /CT/D6/D7 /D1/D3/D8/CX/D3/D2 /CP/D6/CT /CP/D2/CP/D0/DD/DE/CT/CS 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/D8/CW/CT /CW/CP/D3/D8/CX /D6/CT/CV/CX/D1/CT /CU/D3/D6 /CP /CV/CX/DA /CT/D2 /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/BA /CC/CW/CT /D7/CT /D8/CX/D3/D2 /CX/D7 /D3/D1/B9/D4/D9/D8/CT/CS /D9/D7/CX/D2/CV /D8/CW/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /D3/CU /D8/CW/CT /D6/CT/CS/D9 /CT/CS /D7/DD/D7/D8/CT/D1 /B4/BK/B5/BA /CC/CW/CT /D3/D2/CS/CX/B9/D8/CX/D3/D2/D7 /CX/D1/D4 /D3/D7/CT/CS /CP/D6/CT /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/BM R1= 0 /B8˙R1<0 /BA /CC/CW/CX/D7 /D7/CT /D8/CX/D3/D2/CW/CP/D7 /D8/CW/CT /CP/CS/DA /CP/D2 /D8/CP/CV/CT /D3/CU /D3/D2 /D8/CP/CX/D2/CX/D2/CV /CP/D0/D0 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D4/CW /DD/D7/CX /CP/D0 /D3/D2/AS/CV/B9/D9/D6/CP/D8/CX/D3/D2/D7 /D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1 /CP/D2 /DD /D4 /CT/D6/D1 /D9/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7/BA /BV/D3/D2/B9/D7/CT/D5/D9/CT/D2 /D8/D0/DD /D7/D8/CX /CZ/CX/D2/CT/D7/D7 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS/B8 /DB/CW/CX/D0/CT /CX/D8 /DB /CP/D7 /D2/D3/D8 /CX/D2 /BY/CX/CV/BA /BD/BA /CC/CW/CT/D0/D3 /DB /CT/D6 /D4/D0/D3/D8 /CX/D7 /CP /DE/D3 /D3/D1 /D3/CU /D8/CW/CT /D9/D4/D4 /CT/D6 /D3/D2/CT/BA /BY/CX/CV/BA /BF/BA /C8 /D3/CX/D2 /CP/D6/GH /D7/CT /D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CU/D3/D9/D6 /DA /D3/D6/D8/CT/DC /D7/DD/D7/D8/CT/D1 /CX/D2/D5/D9/CP/D7/CX/D4 /CT/D6/CX/D3 /CS/CX /D6/CT/CV/CX/D1/CT/D7/BA /CC/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/D6/D6/CT/B9/D7/D4 /D3/D2/CS /D8/D3 /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /CS/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT /D7/D5/D9/CP/D6/CT/BA /CF /CT /D2/D3/D8/CX /CT/D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /CW/CP/D3/D7 /D6/CT/D7/D9/D0/D8/CX/D2/CV /CX/D2 /D0/D3/D7/CT/CS /D9/D6/DA /CT/D7/BA /CC/CW/CT /D7/CT /D8/CX/D3/D2/D7/CP/D6/CT /D3/D1/D4/D9/D8/CT/CS /D9/D7/CX/D2/CV /D8/CW/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7 /D3/CU /D8/CW/CT /D6/CT/CS/D9 /CT/CS /D7/DD/D7/D8/CT/D1 /B4/BK/B5/BA/CC/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CX/D1/D4 /D3/D7/CT/CS /CP/D6/CT /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/BM R1= 0 /B8˙R1<0 /BA 300 320 340 360 380 400 4200.511.522.53 Timer34/BY/CX/CV/BA /BG/BA /CC /DD/D4/CX /CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /CS/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /DA /D3/D6/D8/CX /CT/D7 /DA /CT/D6/D7/D9/D7/D8/CX/D1/CT/BA /CC/CW/CT /CS/CX/D7/D8/CP/D2 /CT r34 /CX/D7 /D4/D0/D3/D8/D8/CT/CS/BA /BT /D4/CP/CX/D6/CX/D2/CV /D3/CU /D8/CW/CT /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/CX/D7 /CX/CS/CT/D2 /D8/CX/AS/CT/CS /CU/D3/D6350< t < 385 /BA /CF /CT /D2/D3/D8/CX /CT /D8/CW/CP/D8 /DB/CW/CX/D0/CT /D8/CW/CT /D4/CP/CX/D6 /CX/D7/CU/D3/D6/D1/CT/CS /D8/CW/CT /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7 /D6/CT/D1/CP/CX/D2 /D0/D3/D7/CT /D8/D3 /CT/CP /CW /D3/D8/CW/CT/D6 /B4r34<1 /B5/B8/CP/D2/CS /D8/CW/CT /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CV/D6/CT/CP/D8/D0/DD /D6/CT/CS/D9 /CT/CS/BA /CC/CW/CX/D7 /CP/D0/D0/D3 /DB/D7 /CP /D7/CX/D1/D4/D0/CT/CS/CX/CP/CV/D2/D3/D7/D8/CX /D8/D3 /D2 /D9/D1/CT/D6/CX /CP/D0/D0/DD /CS/CT/D8/CT /D8 /DA /D3/D6/D8/CT/DC/B9/D4/CP/CX/D6/CX/D2/CV/BD/BE /BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB 1 1.2 1.4 1.6 1.8 200.511.522.53 log(τ)log(N)0 50 1000500100015002000 τN/BY/CX/CV/BA /BH/BA /C6/D9/D1 /CQ /CT/D6 /D3/CU /D4/CP/CX/D6/CX/D2/CV/D7 N /DA /CT/D6/D7/D9/D7 /D8/CX/D1/CT /D0/CT/D2/CV/D8/CW τ /BA /C7/D2/D0/DD /D4/CP/CX/D6/B9/CX/D2/CV/D7 /D0/CP/D7/D8/CX/D2/CV /D0/D3/D2/CV/CT/D6 /D8/CW/CP/D2τ= 11 /CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS/BA /CC/CW/CT /D4/D9/D6/D4 /D3/D7/CT /D3/CU/D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D9/D6/DA /CT/D7 /CX/D7 /D8/D3 /D7/CW/D3 /DB /D8/CW/CT /CT/D6/D6/D3/D6/B9/CQ/CP/D6 /CT/AR/CT /D8/D7 /CX/D2/CS/D9 /CT/CS /CQ /DD/D8/CW/CT /CW/D3/D7/CT/D2 /CX/D2 /D8/CT/D6/B9/DA /D3/D6/D8/CT/DC /CS/CX/D7/D8/CP/D2 /CT /D9/D8/D3/AR/D7 d= 0.8,0.9,1 /DB/CW/CX /CW/D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /CS/CP/D7/CW/CT/CS/B9/CS/D3/D8/D8/CT/CS/B8 /D7/D3/D0/CX/CS /CP/D2/CS /CS/CP/D7/CW/CT/CS/D0/CX/D2/CT/D7/BA /CC/CW/CT /D8/CP/CX/D0 /D3/CU /D8/CW/CT /D9/D6/DA /CT /D7/CW/D3 /DB/D7 /CP /D4 /D3 /DB /CT/D6 /D0/CP /DB /CS/CT /CP /DD /DB/CX/D8/CW /D3/B9/CTꜶ /CX/CT/D2 /D8 (α−1)≈ −2.66 /BA /CC/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /D9/D6/DA /CT /CP/D6/CX/D7/CT/D7/CT/D6/D6/D3/D6/B9/CQ/CP/D6/D7/BA /CC/CW/CT /D6/D9/D2/B9/D8/CX/D1/CT /CX/D7t= 105/BA −5−4−3−2−1012345−5−4−3−2−1012345 XY/BY/CX/CV/BA /BI/BA /BT /CT/D7/D7/CX/CQ/D0/CT /D4/CW/CP/D7/CT /D7/D4/CP /CT /CU/D3/D6 /D8/CW/CT /D4/CP/D6/D8/CX /D0/CT/D7 /B4/CV/D6/CP /DD/B5/B8 /CP/D2/CS /D8/CW/CT/CU/D3/D9/D6 /DA /D3/D6/D8/CX /CT/D7 /B4/CS/CP/D6/CZ/B5/BA /CC/CW/CT /D8/D6/CP /CY/CT /D8/D3/D6/CX/CT/D7 /D3/CU /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /CP/D2/CS /D8/D6/CP /CT/D6/D7/CP/D6/CT /D4/D0/D3/D8/D8/CT/CS /D3/D2 /D8/CW/CT /D4/D0/CP/D2/CT/BA /CF /CT /D2/D3/D8/CX /CT /D8/CW/CP/D8 /D8/CW/CT /D8/D6/CP /CT/D6/D7 /CW/CP /DA /CT /CP /CT/D7/D7/D8/D3 /CP /CQ/D6/D3/CP/CS/CT/D6 /CS/D3/D1/CP/CX/D2 /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D9/D2/CS /CX/D2 /CJ/BH/BE ℄/BA/CC/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D6/CP /CT/D6/D7 /CP/D6/CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/CW/CT /D6/CT/CV/CX/D3/D2 /D3/CU/AG/D7/D8/D6/D3/D2/CV/AH /CW/CP/D3/D7/BA−6 −4 −2 0 2 4 6−6−4−20246 XY −6 −4 −2 0 2 4 6−6−4−20246 XY/BY/CX/CV/BA /BJ/BA /CB/D2/CP/D4/D7/CW/D3/D8/D7 /D3/CU /CP /D7/DD/D7/D8/CT/D1 /DB/CX/D8/CW /BH/BC/BC/BC /D4/CP/D6/D8/CX /D0/CT/D7 /CP/D8 /D8 /DB /D3 /CS/CX/AR/CT/D6/B9/CT/D2 /D8 /D8/CX/D1/CT/D7/BA /CC/CW/CT /DA /D3/D6/D8/CT/DC /D3/D6/CT/D7 /CP/D6/CT /D7/CW/D3 /DB/CX/D2/CV/B8 /CP/D7 /CT/DC/D4 /CT /D8/CT/CS/BA /CC/CW/CT /D3/D6/CT/D7/CX/DE/CT /CX/D7 /CT/D7/D8/CX/D1/CP/D8/CT/CS /CP/D6/D3/D9/D2/CS /BC/BA/BF/BA /C7/D2 /D8/CW/CT /CQ /D3/D8/D8/D3/D1 /D4/D0/D3/D8/B8 /D7/D3/D1/CT /D8/D6/CP /B9/CT/D6/D7 /CP/D6/CT /D7/D8/CX /CZ/CX/D2/CV /D8/D3 /D8/CW/CT /D3/D6/CT/D7/BA /CF /CT /CP/D0/D7/D3 /D2/D3/D8/CX /CT /D8/CW/CT /CP/D0/D1/D3/D7/D8 /D6/CT/CV/D9/D0/CP/D6/D1/D3/D8/CX/D3/D2 /CS/CT/D7 /D6/CX/CQ /CT/CS /CX/D2 /CJ/BH/BE ℄ /CU/D3/D6 /D8/CW/CT /CU/CP/D6 /D6/CT/CV/CX/D3/D2/BA /CC/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2/D3/CU /D8/CW/CT /D8/D6/CP /CT/D6/D7 /CP/D6/CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/CW/CT /D7/D8/D3 /CW/CP/D7/D8/CX /D7/CT/CP/BA/BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB /BD/BF 4.55 4.551 4.552 4.553 4.554 4.555 4.556 4.557 4.558 4.559 4.56 x 1040.511.522.533.5 Timeri4 4.554.5514.5524.5534.5544.5554.5564.5574.5584.559 4.56 x 1040.220.2250.230.2350.240.2450.250.255 Time r/BY/CX/CV/BA /BK/BA /BW/CX/D7/D8/CP/D2 /CT /CQ /CT/D8 /DB /CT/CT/D2 /D3/D2/CT /DA /D3/D6/D8/CT/DC /CP/D2/CS /D8/CW/CT /D3/D8/CW/CT/D6/D7 ri4 /DA /CT/D6/B9/D7/D9/D7 /D8/CX/D1/CT /B4/D9/D4/D4 /CT/D6 /AS/CV/D9/D6/CT/B5/BA /CF /CT /D2/D3/D8/CX /CT /D8/CW/CT /D4/CP/CX/D6/CX/D2/CV /D3/CU /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/CP/D6/D3/D9/D2/CS t= 4.553 104/BA /C1/D8 /CP/AR/CT /D8/D7 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /AG/D8/D6/CP/D4/D4 /CT/CS/AH/D8/D6/CP /CT/D6 /B4/CQ /D3/D8/D8/D3/D1 /AS/CV/D9/D6/CT/B5/BA /CC/CW/CT /CQ /D3/D8/D8/D3/D1 /AS/CV/D9/D6/CT /D7/CW/D3 /DB/D7 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT r /CQ /CT/D8 /DB /CT/CT/D2 /D3/D2/CT /DA /D3/D6/D8/CT/DC /CP/D2/CS /CP /D4/CP/D7/D7/CX/DA /CT /D8/D6/CP /CT/D6 /B3/D8/D6/CP/D4/D4 /CT/CS/B3 /CX/D2 /D8/CW/CT /D3/D6/CT /DA /CT/D6/D7/D9/D7 /D8/CX/D1/CT/B8 /DB/CW/CX/D0/CT /D8/CW/CT /D4/CP/CX/D6/CX/D2/CV /D3 /D9/D6/D7/B8 /AT/D9 /D8/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT/CP/D1/D4/D0/CX/AS/CT/CS/BA /C1/D2/CX/D8/CX/CP/D0 /DA /D3/D6/D8/CT/DC /D4 /D3/D7/CX/D8/CX/D3/D2/D7 /CX/D7[(1.747,1.203) ( −√ 2/2,0) (√ 2/2,0) (0,−1)] /BA /C1/D2/CX/D8/CX/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D6/CP /CT/D6 /CX/D7(0,−1.24)/B4 /D0/D3/D7/CT /D8/D3 /D8/CW/CT /CU/D3/D9/D6/D8/CW /DA /D3/D6/D8/CT/DC/B5/BA 110 112 114 116 118 120 122 124 126 128 13000.511.522.53 Timeri −2 −1 0 1 2−2−1012 XY/BY/CX/CV/BA /BL/BA /CC/CW/CT /D9/D4/D4 /CT/D6 /D4/D0/D3/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT/D7 ri /CQ /CT/D8 /DB /CT/CT/D2 /D3/D2/CT/CP/CS/DA /CT /D8/CT/CS /D4/CP/D6/D8/CX /D0/CT /CP/D2/CS /D8/CW/CT /CU/D3/D9/D6 /DA /D3/D6/D8/CX /CT/D7/BA /CC/CW/CT /D8/D6/CP /CT/D6 /CX/D7 /CX/D2/CX/D8/CX/CP/D0/D0/DD/D4/D0/CP /CT/CS /D0/D3/D7/CT /D8/D3 /D3/D2/CT /DA /D3/D6/D8/CT/DC /CP/D2/CS /D7/D8/CX /CZ/D7 /CP/D6/D3/D9/D2/CS /D8/CW/CT /DA /D3/D6/D8/CT/DC /CU/D3/D6 /CP /CT/D6/D8/CP/CX/D2 /D8/CX/D1/CT /B8 /D8/CW/CT/D2 /CX/D8 /CY/D9/D1/D4/D7 /CP/D2/CS /D7/D8/CX /CZ/D7 /D3/D2 /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /DA /D3/D6/D8/CT/DC /BA/BT/CU/D8/CT/D6 /CP /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D8/CX/D1/CT /CX/D8 /CV/CT/D8/D7 /CQ/CP /CZ /D3/D2 /D8/CW/CT /AS/D6/D7/D8 /DA /D3/D6/D8/CT/DC/B8 /DB /CT /D2/D3/B9/D8/CX /CT /D8/CW/CP/D8 /D8/CW/CT /D8/D6/CP /CT/D6/D7 /CW/CP/D7 /CY/D9/D1/D4 /CT/CS /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /D3/D6/CQ/CX/D8 /CU/D9/D6/D8/CW/CT/D6 /CU/D6/D3/D1/D8/CW/CT /DA /D3/D6/D8/CT/DC/BA /BT /D8 /D8/CW/CT /CT/D2/CS /D3/CU /D8/CW/CT /D4/D0/D3/D8/D7 /D8/CW/CT /D4/CP/D6/D8/CX /D0/CT /AS/D2/CP/D0/D0/DD /CT/D7 /CP/D4 /CT/D7/CU/D6/D3/D1 /D8/CW/CT /D3/D6/CT/D7 /D6/CT/CV/CX/D3/D2/BA /CC/CW/CT /CQ /D3/D8/D8/D3/D1 /D4/D0/D3/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT /D8/D6/CP /CY/CT /D8/D3/D6/CX/CT/D7/CX/D2 /D6/CT/CP/D0 /D7/D4/CP /CT /CU/D3/D6δt= 4 /D3/CU /D8/CX/D1/CT/B9/CX/D2 /D8/CT/D6/DA /CP/D0 /B4 /CU/D6/D3/D1t= 114 /D8/D3118/B5/BA /CC/CW/CT /CQ/D0/CP /CZ /D0/CX/D2/CT /D6/CT/CU/CT/D6/D7 /D8/D3 /D8/CW/CT /D8/D6/CP /CY/CT /D8/D3/D6/DD /D3/CU /D8/CW/CT /D4/CP/D6/D8/CX /D0/CT/B8 /DB/CW/CX/D0/CT/D8/CW/CT /CS/CP/D6/CZ /CP/D2/CS /D0/CX/CV/CW /D8 /CV/D6/CP /DD /D0/CX/D2/CT/D7 /D8/D3 /D8 /DB /D3 /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7 /CX/D2 /DA /D3/D0/DA /CT/CS /CX/D2 /D8/CW/CT/CY/D9/D1/D4/BA /CF /CT /D2/D3/D8/CT /D8/CW/CP/D8 /DB/CW/CX/D0/CT /D8/CW/CT /CY/D9/D1/D4 /D3 /D9/D6/D7 /D8/CW/CT/D6/CT /CX/D7 /CP /AG/D7/CX/D2/CV/D9/B9/D0/CP/D6/CX/D8 /DD/AH /D3/D2 /D8/CW/CT /D8/D6/CP /CY/CT /D8/D3/D6/DD /D3/CU /D8/CW/CT /CP/CS/DA /CT /D8/CT/CS /D4/CP/D6/D8/CX /D0/CT/BA /C1/D2/CX/D8/CX/CP/D0 /DA /D3/D6/D8/CT/DC/D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D7[(1.747,1.203) ( −√ 2/2,0) (√ 2/2,0) (0,−1)] /BA /CC/CW/CT/CX/D2/CX/D8/CX/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D6/CP /CT/D6 /CX/D7(0,−1.27) /BA−1 0 1 2−1−0.500.51 −1 0 1−1.5−1−0.500.511.5 XY −2 0 2−3−2−10123 XY −4−2 024−3−2−10123A B C D /BY/CX/CV/BA /BD/BC/BA /BY /D3/D9/D6 /D3/D2/D7/CT /D9/D8/CX/DA /CT /D7/D2/CP/D4/D7/CW/D3/D8/D7 /CU/D3/D6 /D8/CW/CT /CU/D3/D9/D6 /DA /D3/D6/D8/CT/DC /D3/CU/CT/D5/D9/CP/D0 /D7/D8/D6/CT/D2/CV/D8/CW/D7 /CP/D2/CS /BD/BC/BC/BC /D4/CP/D6/D8/CX /D0/CT/D7/B8 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /CU/D3/D9/D6 /D3/D2/B9/D7/CT /D9/D8/CX/DA /CT /D4/CP/CX/D6/CX/D2/CV /D3/CU /D8/CW/CT /DA /D3/D6/D8/CX /CT/D7/BA /BX/DA /CT/D2 /D8/CW/D3/D9/CV/CW /D8/CW/CT /D4/CP/D6/D8/CX /D0/CT/D7 /CP/D6/CT/CX/D2/CX/D8/CX/CP/D0/D0/DD /D4/D0/CP /CT/CS /CP/D6/D3/D9/D2/CS /D3/D2/CT /DA /D3/D6/D8/CT/DC/B8 /CP/D7 /D4/CP/CX/D6/CX/D2/CV/D7 /D3 /D9/D6/B8 /D7/D3/D1/CT /D3/CU/D8/CW/CT/D1 /CY/D9/D1/D4 /CU/D6/D3/D1 /CP /DA /D3/D6/D8/CT/DC /D3/D6/CT /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /CP/D2/CS /D6/CT/D1/CP/CX/D2 /D3/D2 /D8/CW/CT/D1/CP/CU/D8/CT/D6 /D8/CW/CT /D4/CP/CX/D6/CX/D2/CV/BA /CF/CW/CX/D0/CT /D8/CW/CT /DA /D3/D6/D8/CT/DC/B9/D4/CP/CX/D6/CX/D2/CV /D3 /D9/D6/D7 /D7/D3/D1/CT /D4/CP/D6/B9/D8/CX /D0/CT/D7 /CT/DA /CT/D2 /D8/D9/CP/D0/D0/DD /CT/D7 /CP/D4 /CT /CU/D6/D3/D1 /D8/CW/CT /D3/D6/CT/D7/BA /CF /CT /D2/D3/D8/CX /CT /CP/D0/D7/D3 /D8/CW/CP/D8/CP/CU/D8/CT/D6 /CU/D3/D9/D6 /D4/CP/CX/D6/CX/D2/CV/D7 /CP/D0/D0 /D3/D6/CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /AG /D3/D2 /D8/CP/D1/CX/D2/CP/D8/CT/CS/AH /CP/D2/CS/CP/D6/CT /D4 /D3/D4/D9/D0/CP/D8/CT/CS /DB/CX/D8/CW /D8/D6/CP /CT/D6/D7 /D3/D6/CX/CV/CX/D2/CP/D8/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /AS/D6/D7/D8 /D3/D6/CT /CP/D2/CS/DB/CW/CX/D0/CT /CP/CQ /D3/D9/D8 /BD/BC/B1 /D3/CU /D8/D6/CP /CT/D6/D7 /CW/CP /DA /CT /CT/D7 /CP/D4 /CT/CS /CU/D6/D3/D1 /D8/CW/CT /D6/CT/CV/CX/D3/D2 /D7/D9/D6/B9/D6/D3/D9/D2/CS/CX/D2/CV /CP/D0/D0 /CU/D3/D9/D6 /D3/D6/CT/D7/BA /C1/D2/CX/D8/CX/CP/D0 /DA /D3/D6/D8/CT/DC /D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D7[(1.747,1.203) (−√ 2/2,0) (√ 2/2,0) (0,−1)] /BA /C8 /CP/D6/D8/CX /D0/CT/D7 /CP/D6/CT /D9/D2/CX/CU/D3/D6/D1/D0/DD /CX/D2/CX/D8/CX/CP/D0/B9/CX/DE/CT/CS /D3/D2 /D8/CW/CT /CX/D6 /D0/CT /D3/CU /D6/CP/CS/CX/D9/D7 r= 0.27 /CP/D6/D3/D9/D2/CS /D8/CW/CT /CU/D3/D9/D6/D8/CW /DA /D3/D6/D8/CT/DC/BA/BD/BG /BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB −0.2 0 0.2 0.4 0.6 0.8 1−1.1−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1 XY/BY/CX/CV/BA /BD/BD/BA /CI/D3 /D3/D1 /D3/CU /BY/CX/CV/BA /BD/BC /CU/D3/D6 /D8/CW/CT /D8 /DB /D3 /DA /D3/D6/D8/CT/DC /CX/D2 /DA /D3/D0/DA /CT/CS /CX/D2 /D8/CW/CT /AS/D6/D7/D8/D4/CP/CX/D6/CX/D2/CV /B4/BD/BC/BC/BC /D4/CP/D6/D8/CX /D0/CT/D7/B5/BA /CC/CW/CT /DA /D3/D6/D8/CT/DC /D3/D2 /D8/CW/CT /D6/CX/CV/CW /D8 /CX/D7 /D8/CW/CT /D0/D3/D7/CT/D7/D8/D8/D3 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/CP/D6/D8/CX /D0/CT/D7/BA /CC/CW/CX/D7 /CZ/CX/D2/CS /D3/CU /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D7 /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /D3/CU /D4/CP/CX/D6/CX/D2/CV /D3/CU /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/BA /BW/D9/D6/CX/D2/CV /D8/CW/CT /D4/CP/CX/D6/CX/D2/CV /D3/CU/D8/CW/CT /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7/B8 /D8/CW/CT /D4 /CT/D6/CX/D4/CW/CT/D6/DD /D3/CU /D8/CW/CT /D3/D6/CT/D7 /AG/D1/CT/D6/CV/CT/AH /CP/D2/CS /CU/D3/D6/D1 /CP/D0/CP/D6/CV/CT/D6 /AG/CX/D7/D0/CP/D2/CS/AH /DB/CW/CT/D6/CT /D4/CP/D6/D8/CX /D0/CT/D7 /CP/D6/CT /D8/D6/CP/D4/D4 /CT/CS /D3/D2 /CP/D2/CS /D1/CP /DD /D8/D6/CP/D2/D7/B9/CU/CT/D6 /CU/D6/D3/D1 /CP /DA /D3/D6/D8/CT/DC /D8/D3 /CP/D2/D3/D8/CW/CT/D6/BA /CB/D3/D1/CT /D4/CP/D6/D8/CX /D0/CT/D7 /D1/CP /DD /CT/DA /CT/D2 /D8/D9/CP/D0/D0/DD/CT/D7 /CP/D4 /CT /CX/D2 /D8/CW/CT /D4/D6/D3 /CT/D7/D7/BA /C1/D2/CX/D8/CX/CP/D0 /DA /D3/D6/D8/CT/DC /D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D7[(1.747,1.203) (−√ 2/2,0) (√ 2/2,0) (0,−1)] /BA /C8 /CP/D6/D8/CX /D0/CT/D7 /CP/D6/CT /D9/D2/CX/CU/D3/D6/D1/D0/DD /CX/D2/CX/D8/CX/CP/D0/B9/CX/DE/CT/CS /D3/D2 /D8/CW/CT /CX/D6 /D0/CT /D3/CU /D6/CP/CS/CX/D9/D7 r= 0.27 /CP/D6/D3/D9/D2/CS /D8/CW/CT /CU/D3/D9/D6/D8/CW /DA /D3/D6/D8/CT/DC/BA 0 0.5 1−1−0.500.5 XY 0 0.5 1−1−0.500.5 XY0 0.5 1−1−0.500.5 XY 0 0.5 1−1−0.500.5 XYA B C D /BY/CX/CV/BA /BD/BE/BA /BW/CX/AR/CT/D6/CT/D2 /D8 /D7/D2/CP/D4/D7/CW/D3/D8/D7 /DB/CX/D8/CW /BD/BC/BC/BC /D4/CP/D6/D8/CX /D0/CT/D7/B8 /CX/D2/CX/D8/CX/CP/D0/CX/DE/CT/CS /CP/D8/CS/CX/AR/CT/D6/CT/D2 /D8 /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D6/CP/CS/CX/D9/D7/BA /CC/CW/CT /D7/D2/CP/D4/D7/CW/D3/D8/D7 /CP/D6/CT /D8/CP/CZ /CT/D2 /DB/CW/CX/D0/CT /D8/CW/CT/AS/D6/D7/D8 /D4/CP/CX/D6/CX/D2/CV /D3 /D9/D6/D7/BA /BT/D2 /CT/CV/CV/B9/D7/CW/CP/D4 /CT /CU/D3/D6/D1 /D3/CU /D8/CW/CT /D0/D9/D7/D8/CT/D6 /CU/D3/D6/D1/CT/CS /CQ /DD/D8/CW/CT /D4/CP/D6/D8/CX /D0/CT/D7 /CP/D4/D4 /CT/CP/D6/D7 /CX/D2 /D4/D0/D3/D8 /BV/BA /CB/D9/CV/CV/CT/D7/D8/CX/D2/CV /CP /D7/D8/D6/D3/D2/CV /CX/D2/AT/D9/CT/D2 /CT/D3/CU /D4/CP/CX/D6/CX/D2/CV /CU/D3/D6r >0.2 /BA /C1/D2/CX/D8/CX/CP/D0 /DA /D3/D6/D8/CT/DC /D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D7[(1.747,1.203) (−√ 2/2,0) (√ 2/2,0) (0,−1)] /BA /CA/CP/CS/CX/CX /CU/D3/D6 /D8/CW/CT /BT/B8/BU/B8/BV/B8/BW /D4/D0/D3/D8/D7 /CP/D6/CT/D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD r= 0.135 /B8r= 0.18 /B8r= 0.225 /B8r= 0.27 /BA−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5 XY/BY/CX/CV/BA /BD/BF/BA /CC /D6/CP /CY/CT /D8/D3/D6/CX/CT/D7 /D3/CU /BE/BH /CP/CS/DA /CT /D8/CT/CS /D4/CP/D6/D8/CX /D0/CT/D7/B8 /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8/CX/D2/CX/D8/CX/CP/D0 /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CC/CW/CT /D4/D0/D3/D8 /CX/D7 /D1/CP/CS/CT /CX/D2 /D8/CW/CT /D3/B9/D6/D3/D8/CP/D8/CX/D2/CV /CU/D6/CP/D1/CT/DB/CX/D8/CW /D8 /DB /D3 /DA /D3/D6/D8/CX /CT/D7 /CS/D9/D6/CX/D2/CV /CP /D4/CP/CX/D6/CX/D2/CV /D0/CP/D7/D8/CX/D2/CV ∆t= 15 /B8 /CP/D2/CS /D4/CP/D6/D8/CX /D0/CT/D4 /D3/D7/CX/D8/CX/D3/D2/D7 /D6/CT /D3/D6/CS/CT/CS /CU/D3/D6 /D3/D2/D7/D8/CP/D2 /D8 /CX/D2 /D8/CT/D6/B9/DA /D3/D6/D8/CT/DC /CS/CX/D7/D8/CP/D2 /CT/BA /C1/D2/CX/D8/CX/CP/D0/DA /D3/D6/D8/CT/DC /D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D7[(1.747,1.203) ( −√ 2/2,0) (√ 2/2,0) (0,−1)] /BA/C8 /CP/D6/D8/CX /D0/CT/D7 /CP /D9/D2/CX/CU/D3/D6/D1/D0/DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /D3/D2 /CX/D6 /D0/CT/D7 /D3/CU /D6/CP/CS/CX/CX/BM r= 0.27 /B8 r=.20 /B8r= 0.135 /BA/BT/BA /C4/CP/CU/D3/D6/CV/CX/CP /CT/D8 /CP/D0/BA/BM /C8 /CP/D7/D7/CX/DA /CT /CC /D6/CP /CT/D6 /BW/DD/D2/CP/D1/CX /D7 /CX/D2 /BG /C8 /D3/CX/D2 /D8/B9/CE /D3/D6/D8/CT/DC /BY/D0/D3 /DB /BD/BH 0 1 2 3 4 5 x 1040.17960.17980.180.18020.18040.18060.1808 Time<r(t)> 0 1 2 3 4 5 x 10411.522.533.544.55x 10−3 Timeσ(r,t)/BY/CX/CV/BA /BD/BG/BA /C7/D2 /D8/CW/CT /D0/CT/CU/D8 /D8/CW/CT /D1/CT/CP/D2 /DA /CP/D0/D9/CT /angbracketleftr(t)/angbracketright /CP/D7 /CU/D9/D2 /D8/CX/D3/D2 /D3/CU/D8/CX/D1/CT /CX/D7 /D4/D0/D3/D8/D8/CT/CS/BA /CC/CW/CT /CP /DA /CT/D6/CP/CV/CT /angbracketleftr(t)/angbracketright /CP/D2 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /D3/D2/D7/D8/CP/D2 /D8/CX/D2 /D8/CX/D1/CT/BA /BT/D2/CS /D3/D2 /D8/CW/CT /D6/CX/CV/CW /D8/B8 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2 σ(r,t) =/radicalbig /angbracketleft(r(t)− /angbracketleftr(t)/angbracketright)2/angbracketright /CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT /CX/D7 /D4/D0/D3/D8/D8/CT/CS/BA /CF /CT /CS/D3 /D2/D3/D8/D3/CQ/D7/CT/D6/DA /CT /CP/D2 /DD /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT /CS/CX/AR/D9/D7/CX/D3/D2 /D4/CW/CT/D2/D3/D1/CT/D2/D3/D2/B8 /CP/D2/CS /D3/D2 /D0/D9/CS/CT/D3/D2 /D8/CW/CT /CP/CQ/D7/CT/D2 /CT /D3/CU /CS/CX/AR/D9/D7/CX/D3/D2 /CU/D3/D6 /D0/CP/D6/CV/CT/D6 /D8/CX/D1/CT/D7 /D7/D4/CP/D2/BA /BW/CP/D8/CP /CX/D7 /D3/D1/B9/D4/D9/D8/CT/CS /D9/D7/CX/D2/CV 200 /D8/D6/CP /CT/D6/D7/BA /BT/D0/D0 /D8/D6/CP /CT/D6/D7 /CP/D6/CT /D9/D2/CX/CU/D3/D6/D1/D0/DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS/CP/D8t= 0 /D3/D2 /D8/CW/CT /CX/D6 /D0/CT /D3/CU /D6/CP/CS/CX/D9/D7 r= 0.18 /BA0.1 0.12 0.14 0.16 0.18 0.20.20.40.60.811.21.41.61.82x 10−3 rσ(r) −1−0.9 −0.8−3.6−3.4−3.2−3−2.8−2.6 Log10(r)Log10(σ) 200 400 600 800 10000.40.50.60.70.80.911.11.21.3 Timeσ(1/r2)/BY/CX/CV/BA /BD/BH/BA /C7/D2 /D8/CW/CT /D9/D4/D4 /CT/D6 /D4/D0/D3/D8 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2 σ(r) =/radicalbig /angbracketleft(r− /angbracketleftr/angbracketright)2/angbracketright /CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2 /CT /CU/D6/D3/D1 /D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU/D8/CW/CT /DA /D3/D6/D8/CT/DC r /CX/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS/BA /CC/CW/CT /D0/D3/CV/B9/D0/D3/CV /D4/D0/D3/D8 /D7/D9/CV/CV/CT/D7/D8/D7 /CP /CQ /CT/CW/CP /DA/CX/D3/D6 σ∼r2.93∼r3/BA /C7/D2 /D8/CW/CT /CQ /D3/D8/D8/D3/D1 /D4/D0/D3/D8 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2 σ(1/r2, t) /CX/D7 /D3/D1/D4/D9/D8/CT/CS /CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT /CU/D3/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/D6/CP/CS/CX/CX /D9/D7/CT/CS /CX/D2 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arXiv:physics/0009098v1 [physics.acc-ph] 29 Sep 2000PULSEDSUPERCONDUCTIVITY ACCELERATION M. Liepe∗fortheTESLACollaboration Deutsches Elektronen-SynchrotronDESY, D-22603Hamburg, Germany Abstract ThedesignoftheproposedlinearcolliderTESLAisbased on9-cell1.3GHz superconductingniobiumcavities, oper- atedin pulsedmode[1]. Withintheframeworkofaninter- national collaboration the TESLA Test Facility (TTF) has been set up at DESY, providing the infrastructure for cav- ity R&D towardshigher gradients[2]. More than 60 nine- cell cavities were tested, accelerating gradients as high a s 30 MV/m were measured. In the second production of TTF-cavitiestheaveragegradientwasmeasuredtobe24.7 MV/m. Twomodules,eachcontainingeightresonators,are presently used in the TTF-linac. These cavities are oper- atedinpulsedmode: 0.8msconstantgradientwithupto10 Hz repetitions rate. We will focus on two aspects: Firstly, the cavity fabrication and treatment is discussed, allowin g to reach high gradients. Latest results of single cell cavi- ties will be shown,going beyond40 MV/m. Secondly,the pulsed mode operation of superconducting cavities is re- viewed. This includes Lorentz force detuning, mechanical vibrations (microphonics) and rf ﬁeld control. Both top- ics meet the upcoming interest in superconducting proton linacs, like the sc driverlinac for SNS (SpallationNeutron Source). 1 INTRODUCTION The aim of the cavity program - launched by the TESLA collaborationin1992-istoexplorethetechnologyofhigh gradient superconducting cavities and to demonstrate the performanceand reliability of a superconductinglinac op- eratedinpulsedmode. Theinfrastructureforthetreatment , assembly andtest ofsuperconductingcavitieshasbeenes- tablishedatDESY.Afullintegratedsystemtest withbeam is done at the TTF-linac [2]. In a ﬁrst step the design goal for the cavities of the TTF-linac was set to the value Eacc=15 MV/m. The aim was to gradually approach the acceleratinggradientof 22MV/mrequiredforTESLA with acentre-of-massenergy Ecm= 500GeV [3]. 2 PERFORMANCEOFTHE SUPERCONDUCTINGTTF-CAVITIES The TTF 9-cell cavities are fabricated from RRR =300 niobium by electron-beam welding of half cells that are deep-drawn from niobium sheet metal, see Fig. 1. The cavitieshavebeenorderedatfourEuropeancompaniesand have been preparedand tested at DESY [4]. The presently used standard cavity preparation before the vertical test ∗liepe@sun52a.desy.de conical head plate stiffening ringreference flange pick up flange Effectivelength 1036mm Aperturediameter 70mm Couplingcell tocell 1.98% 2.0 4.2mT/(MV/m) R/Q per cavity 1036 315kHz/mm Cavity bandwidth( )433Hz Figure 1: Cross section and selected design parameters of the 1.3GHz TTF9-cellcavity. consistsofthefollowingsteps: •removalofthedamagelayerbybufferedchemical polishing(BCP): 80 µm fromtheinnerand30 µm fromtheoutercavitysurface •2hoursheattreatmentat800◦C •4hoursheattreatmentat1400◦Cwith titaniumgetter •removalofthetitaniumlayerby80 µminner and30 µmouterBCP •ﬁeldﬂatnesstuning •ﬁnal20 µmremovalfromtheinnersurfacebyBCP •highpressurerinsing(HPR)with ultrapurewater •dryingbylaminarﬂow ina class10cleanroom •assemblyofall ﬂanges,leak-check •2timesHPR, dryingbylaminarﬂow andassembly oftheinputantennawith highexternal Q. 2.1 VerticalTest Results Uptodatemorethan60nine-cellcavitieshavebeentested with cw rf-excitationin the vertical cryostat. The time de- velopment of the test results is summarized in Fig. 2. The performanceimprovementoverthepastyearsisclearlyvis- ible. Inparticular,duringthelastyearnearlyalloftheca v- ities performedabove22 MV/m. More important, the ma- jority of these cavities reached their good performance al- readyintheﬁrsttest. Thedistributionofthemaximumgra- dientisshowninFig.3forall9-cellcavities. Inthecaviti es with low performance defects in the welds or in the bulk niobium were found [4]. For the second cavity production theweldingtechniquewasimprovedandallniobiumsheets wereeddycurrentscanned. Thesecavitiesreachedanaver- ageacceleratinggradientof24.7MV/m,therebyexceedingFirst an Best CW-Tests 05101520253035 1/1/95 1/1/97 1/1/99 1/1/01 DateEacc [MV/m] First Best Date19951996 1997 1998 1999 200035 30 25 20 15 10 5 0E [MV/m]acc Figure 2: Time development of the maximum gradient achieved in TTF 9-cell cavities. Shown is the cavity per- formance in the ﬁrst test and in the best test. Note that several cavitieshave beentested onlyonce (resultslabele d hereasﬁrst test). the required gradient for TESLA500 on average. All cav- ities of the third production which have been tested up to dateshowedaperformanceclearlyabovetheTESLAspec- iﬁcation. 024681012 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 accelerating gradient E [MV/m]accNumber of cavitiesThird production Second production Average: 24,7 MV/m First production Average: 20,5 MV/m Figure 3: Distribution of maximum gradient for the TTF 9-cellcavities(besttest). 2.2 HorizontalTest Results After the cavities have passed the vertical acceptance test successfully,theheliumvesselisweldedtotheheadplates of the cavity, see Fig. 1. A 20 µm removal from the inner surfacefollows. Inthelastpreparationstepbeforethehor i- zontaltest,themainpowercouplerisassembledtothehigh pressurerinsedcavity. Theexternal Qofthepowercoupler istypically 2·106. Morethan30caviteshavebeentestedin pulsed mode operation (see Fig. 7) in the horizontal cryo- stat (”CHECHIA”). The average gradient achieved in the vertical and the horizontal tests are quite similar as shown in Fig. 4. In a few cases the performance was reduced in the horizontal test due to ﬁeld emission. In other cavities the maximum gradient was improved by the fact that the cavites are operated in pulsed mode instead of the cw op- eration in the vertical test. These results demonstrate tha t thegoodperformanceofacavitycanbepreservedafterthe accemblyoftheheliumvesselandthepowercoupler.0 10 20 30 400510152025303540 Eacc [MV/m] vertical testEacc [MV/m] horizontal test Efirst production second production Figure4: Comparisonofverticalandhorizontaltestresult s of cavities from the frist and the second production. The average accelerating gradient in the vertical tests with cw excitationis 23.6MV/m, in the horizontaltest with pulsed operation24.2MV/m. 2.3 CavityPerformanceintheTTF-Modules Eight 9-cell cavities, each with its own helium vessel and main power coupler, are assebled in a string in the clean- room. The cavity string is than installed in a cryo-module. Up to date four accelerator modules have been built, three of them have been tested in the TTF linac (the third one replaced module #1). The cavities in the modules are op- eratedinpulsedmode: 0.8msconstantgradientwith upto 10 Hz repetitions rate (see Fig. 7). Due to the fact that all cavities of a module are operatedat the same gradient, the performanceof a moduleis limitted by the cavity with the lowest performance. The measured average gradient was 16 MV/m, 20 MV/m and 22.5 MV/m for the modules #1, 2 and 3, respectively. The module #4 has been equipped with cavities, that have shown a performanceof Eacc≥25 MV/m intheverticaltest. 3 CAVITY R&D The single-cell cavity R&D program focuses on possible simpliﬁcations in the cavity production and treatment and on pushing the cavity performance towards higher gradients. For a possible energy upgrade of TESLA to Ecm= 800 GeV the performance of the cavities has to be pushed closer to the physical limit of 50-60 MV/m. This limit isdeterminedbythepropertiesofthesuperconductor niobium. However, in the present 9-cell cavities, the maximum gradient is limited at lower gradients by three maineffects: •thermalinstabilities(quenches) •ﬁeldemissioncurrents •decreaseofthequalityfactorwithoutindicationof ﬁeldemissionat 25-30MV/m. The ﬁrst limitation can be caused by defects in the material or weld. Therefore the sensitivity of the eddy current scanning apparatus was improved signiﬁcantly fora better diagnostic of niobium sheets with tiny defects [5]. In order to raise the onset of ﬁeld emission further, effortsare undertakento improvethe highpressurerinsing system and the particle control during the assembly in the cleanroom [6]. The third limitation is not well understood yet but measurements on electropolished superconducting cavities have demonstrated, that electropolishing (EP) ca n overcomethislimit. 3.1 ElectropolishedCavities In a collaboration between KEK and Saclay it has been shown convincingly that EP raises the maximum acceler- ating ﬁeld in 1-cell cavities by more than 7 MV/m with respect to the standard buffered chemical polishing [7]. Therefore in a joined effort between CERN, DESY, KEK and Saclay a electropolishing system for 1-cell cavities has been set up at CERN for further investigations. Us- ing the KEK-style electropolishing, very high gradients around40 MV/m have been achievedin several 1-cell res- onators, as shown in Fig. 5. Interestingly also this cavitie s Eacc [MV/m]Q0 1,00E+081,00E+091,00E+101,00E+11 0 5 10 15 20 25 30 35 40 45100 µm EP half cells + 100 µm EP + 100 C bake 100 um EP half cells + 50 um BCP +220 um EP + 100 C bake 100 um EP half cells + 50 um BCP + 245 um EP + 100 C bakeTESLA 500 GeV goal TESLA 800 GeV goal Figure 5: Performance of electropolished 1-cell 1.3 GHz cavities after backout at moderate temperature( ∼100◦C). Forcomparisonalsoshownistherequiredacceleratinggra- dient for TESLA with a centre-of-massenergy Ecm= 500 GeV andforanenergyupgradeto Ecm=800GeV. showed a decrease of the quality factor without indication of ﬁeld emission at highgradients,beforebakeoutat mod- erate temperature ( ∼100◦C). However, experiments have shown that in-situ baking at moderate temperature drops RBCSby a factor of 2, signiﬁcantly reduces the Q-slope andimprovesthemaximumgradientsinEP cavities[8,9]. Presently it is unclear, why this bakeout seems to be more efﬁcient on EP cavities thanon BCP cavities. It isremark- able, that the high performance of electropolished 1-cell cavities is achieved without high-temperature (1400◦C) treatment, normally done to improve the niobium quality. PresentlytheEPtechnologyistransferredtotheTTF9-cell resonatorsin collaborationwithKEK. 3.2 HydroformedCavities The development of seamless cavity fabrication was mo- tivated by the potential of cost reduction, but recent tests have shown, that seamless cavities are also promising toreach highest gradients. Presently two methods of cav- ity fabrication without welding are studied: spinning [10] andhydroforming[11]. Thehydroformingofcavitiesfrom seamless niobium tubes is being pursued at DESY. Four 1-cell cavites have been successfully built so far. Af- ter a buffered chemical polishing, one of these resonators reacheda highacceleratinggradientof32.5MV/m ata re- markablehighqualityfactor Q0= 2·1010. By electropol- ishing the innersurface of this cavity the gradientwas fur- therenhancedbeyond40MV/m,againshowingaveryhigh quality factor, see Fig. 6. One should note that this cavity Figure 6: Performance of a hydroformend1-cell 1.3 GHz cavity before and after electorpolishing. The cavity was fabricatedat DESYandtested byP. Kneiselat JLAB. was produced from a low RRRniobium ( RRR = 100). Afera1400◦Cheattreatmentthe RRRraisedto300-400. 3.3 Superstructure Theconceptofthesuperstructurehasbeenproposedforthe TESLA main linac, aiming to reduce the spacing between the cavites and to save in rf components[12]. In this con- cept several multi-cell cavities are coupled by short beam tubes. The whole chain can be fed by one FM coupler at- tached at one end beam tube. The superstructure-layout is extensively studied at DESY since 1999. Computations have been performed for the rf properties of the cavity- chain, the bunch-to-bunch energy spread and multibunch dynamics. A copper model of a 4 x 7-cell superstructure hasbeenbuiltinordertocomparewiththesimulationsand for testing the ﬁeld-proﬁle tuning and the HOM damping scheme [13]. A ”proof of principle” niobium prototype of the 4x 7-cellsuperstructureis nowunderconstructionand willbetestedwithbeamattheTESLATestFacilityin2001 [14]. 4 PULSED OPERATIONOF SUPERCONDUCTING CAVITIES ThesuperconductingcavitiesattheTTF-linacareoperated in pulsed mode. The pulse structure of the rf power and the beam is shown in Fig. 7. The rf pulse length is 1300 µs from which 500 µs are required to ﬁll the cavity. Amplitude and phase control is obviously needed duringtime[µs] 500 1000 1500 20000Accelerating voltage cavity detuningincident power cavity phasebeam currentFill time Flat topamplitude [a.u] Figure7: SchematicviewofthepulsestructureoftheTTF cavityoperation. Theacceleratingﬁeldisincreaseddurin g 0.5msﬁllingtime,followedby0.8msconstantgradient. the ﬂat-top of 800 µs when beam is accelerated, but it is equallydesirabletocontroltheﬁeldduringﬁllingtoensur e proper beam injection conditions. The pulsed structure of the rf ﬁeld and the beam current puts demanding requirementsontherfcontrolsystem. Therearetwomajor sourcesofﬁeldperturbationsin asuperconductingcavity: •beamloadingandﬂuctuationofthebeamcurrent •modulationofthecavityresonancefrequency. The resonance frequency is modulated by deforma- tions of the cavity walls induced by external mechanical vibrations (microphonics) or by the time dependent Lorentz force, see Fig. 7. The resulting amplitude and phase perturbations are in the order of 5 %and20◦, re- spectively. These errorsmust be suppressed by one to two orders of magnitude by the rf control system. Fortunately, the dominating perturbations (Lorentz force detuning and beam loading) are highly repetitive from pulse to pulse. Therefore they can be reduced by use of feedforward compensation signiﬁcantly. The feedforward control is optimized continously by making the feedforward system adaptive [15]. The remaining non repetitive errors can be suppressedbyfeedbackcontrol. 4.1 RFFieldControl The two modules installed in the TTF-linac are supplied withrfpowerbyone10MWmulti-beamklystron. Ampli- tudeandphasecontrolcanbeaccomplishedbymodulation of the incident rf wave which is common to the 16 cavi- ties. Therefore control of an individual cavity ﬁeld is not possible, but the sum of the electric ﬁeld vectors of all 16 cavitiesisregulatedbyadigitalrfcontrolsystem[16]. Th e layoutoftheTTFdigitalrfcontrolsystemisbasedoncon- trolling the real and imaginary part of the complex vector sum, as shown in Fig. 8. The rf control system is realized asacombinationoffeedbackandfeedforwardcontrol. The effectivenessof the feedbacksystem is limited by the loop delay of 5 µs and the unity-gain bandwidth of about 20 kHz. When feedback is applied, the residual control er- ror is dominated by a repetitive component. At the TTFReIm......klystronvector modulator1.3 GHz ...... cryomodule 2 cryomodule 1 1.3 GHz + 250 kHz f = 1 MHzs calibrated vector-sum DSP system+ +digital low pass filter ImRepower transmission lineDACDAC 250 kHz ADC...ADC ADC...ADC setpoint tablegain tablefeed forward table Figure8: Schematicofthedigitalrfcontrolsystem. linac it could be demonstrated, that this error can be re- ducedfurtherbymorethananorderofmagnitudewiththe adaptive feedforward[17]. The high degreeof ﬁeld stabil- ity achievedattheTTF linacismainlyduetothelowlevel offast ﬂuctuations,like microphonics. 4.2 Microphonics Externalmechanicalvibrationscanchangetheshapeofthe cavity and thereby the eigenfrequency of the cavity. This results in an amplitude and phase jitter of the accelerating ﬁeld. Superconductingcavitiesaresensitivetomicrophon - ics due to the thin wall thicknessand the small bandwidth. The rms microphonics frequency spread, measured in 16 cavities of the TTF-linac, is 9.5 ±5.3 Hz and thus surpris- inglysmallforasuperconductingcavitysystem. Thespec- trum of this frequency ﬂuctuations is dominated be a few frequencies. 4.3 Lorentz-ForceDetuning The resonance frequency of pulsed cavities is modulated by the time-varying Lorentz force within the rf pulse, see Fig. 7. The dynamic Lorentz force detuning is correlated with the pulsed rf ﬁeld and is depending on the mechani- cal propertiesof the cavity. The steady state Lorentz force detuning at constant accelerating gradient Eaccis propor- tional to the square of the gradient: ∆f=−K·E2 acc. The TTF cavities have been designed for a steady state detuning constant K=1 Hz/(MV/m)2. This quadratical dependence is also reﬂected in the dynamic Lorentz force detuning during the rf pulse. Figure 9 shows the fast in- rease of the dynamic frequency shift at high ﬂat-top gra- dients. Preliminary measurements at CHECHIA and the TTF-linac indicate, that the dynamic detuning during the ﬂat-top rangesform ±120Hz (total drift 240 Hz) to ±300 Hz at thedesigngradientforTESLA500. The correspond- ing additional rf power required to maintain a constant accelerating ﬁeld during the ﬂat-top amounts to ∼5%to ∼30%. Giventhattheextarfpowerforcavityﬁeldcontrol shouldnotexceed10 %thismeansthatthestiffeningofthe cavitiesforTESLAmustbebettercontrolledorincreased.0 500 1000 1500 2000−300−200−1000100200300detuning [Hz]Lorentz Force Detuning of D39 in Chechia 15 MV/m 20 MV/m 25 MV/m 30 MV/m filltime : 500 µs flat-top: 800 µs time [µs] Figure9: DynamicaldetuningofaTTF9-cellcavityduring pulsedoperationat differentﬂat-topgradients. 4.4 Active Compensation of Lorentz-Force De- tuning In case that the present passive stiffening of the TTF cav- ities does not limit the additional power to 10 %, it will be necessary to increase the stiffening (e.g. by copper spray- ing [18]) or to implement a fast active frequency control scheme. A solution for a fast frequency tuner could be basedonpiezotranslators. Thepiezotranslatorwouldallo w for a fast frequency tuning within the rf pulse to compen- sate the detuning induced by the Lorentz force. A proof of principle experiment of this compensation with a fast piezotunerhasbeenconductedsuccessfullyatthehorizon- tal test stand at DESY [19], see Fig. 10. Such a system appearsespeciallyattractivesince itwill improvetheove r- all efﬁciencyoftheTESLAlinacssingniﬁcantly. Morede- taildstudieswillbedoneinthenearfuturetodeterminethe performancelimitationsofafast piezotuner. 0 500 1000 1500 2000−300−250−200−150−100−50050100detuning [Hz]detuning during RF−pulse at 20MV/m, preliminary! time [ µs ]M. Liepe 05.07.00without compensation with piezo compensation1300 µs pulse Figure10: ActivecompensationofLorentz-forcedetuning duringtherfpulse. Intheﬁrst successfulltestshowhere,a TTF9-cellcavityisoperatedat20MV/mﬂat-topgradient.5 CONCLUSIONS The presented results demonstrate the availability of the superconducting technology for the 500 GeV linear col- lider TESLA. The TTF 9-cell cavities are now routinely reaching the TESLA requirementsof 22 MV/m. The third module in the TTF-linac was successfully operated at the TESLA design gradient. The technology needed for the pulsedoperationathighgradientwasdevelopedandthere- liable operationofa superconductinglinacin pulsedmode wasdemonstratedattheTTF-linac. Recentresultsof1-cell cavities justify the optimism that the accelerating gradie nt in the9-cellcavitiescanbe increasedevenfurther. 6 REFERENCES [1] Conceptual Design Report of a 500 GeV e+e−Linear Col- lider with Integrated X-ray Laser Facility, DESY 1997-048, ECFA1997-182, DESY(1997). [2] TESLA TEST FACILITY LINAC-Design Report, TESLA Report 95-01, editedbyD.A.Edwards, DESY(1995). [3] R.Brinkmann, contributiontothis conference. [4] B. Aune et al., Phys. Rev. ST Accel. Beams, Vol. 3, No. 9 (2000). [5] A. Brinkmann, Proceedings of the 9th Workshop on RF Su- perconductivity, SantaFe,USA (1999) tobe published. [6] D. Reschke, Proceedings of the 9th Workshop on RF Super- conductivity, Santa Fe,USA(1999) tobe published. [7] E. Kako et al., Proceedings of the 9th Workshop on RF Su- perconductivity, SantaFe,USA (1999) tobe published. [8] P. Kneisel, Proceedings of the 9th Workshop on RF Super- conductivity, Santa Fe,USA(1999) tobe published. [9] L.Liljeetal.,Proceedingsofthe9thWorkshoponRFSupe r- conductivity, Santa Fe,USA(1999) tobe published. [10] P. Kneisel and V. Palmieri, Proceedings of PAC 1999, New York, USA(1999), p. 943. [11] W.Singeretal.,ProceedingsofEPAC2000,Vienna,Aust ria (2000) tobe published. [12] J.Sekutowicz,M.FerrarioandCh.Tang,Phys.Rev.STAc - cel.Beams, Vol. 2, No.6(1999). [13] N. Baboi etal.,contribution tothisconference. [14] J.Sekutowiczetal.,Proceedingsofthe9thWorkshopon RF Superconductivity, SantaFe,USA(1999) tobe published. [15] M. Liepe and S. Simrock, Proceedings of EPAC 1998, Stockholm, Sweden(1998), p. 1735. [16] T.Schilcher,TESLA-Report,TESLA98-05, DESY(1998). [17] A. Gamp et al., Proceedings of EPAC 1998, Stockholm, Sweden(1998). [18] S.Bousson, Proceedings of the9thWorkshoponRFSuper- conductivity, Santa Fe,USA(1999) tobe published. [19] M. Liepe et al.,tobe published.
Large Coherence Area Thin-Film Photonic Stop-Band Lasers Victor I. Kopp Department of Physics, Queens College of CUNY, Flushing, New York 11367 Chiral Photonics, Inc., New York, New York 10002 Zhao- Qing Zhang Physics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Azriel Z. Genack Department of Physics, Queens College of CUNY, Flushing, New York 11367 Chiral Photonics, Inc., New York, New York 10002 We demonstrate that the shift of the stop band position with increasing oblique angle in periodic structures results in a wide transverse exponential field distribution corresponding to strong angular confinement of the radiation. The beam expansion follows an effective diffusive equation depending only upon the spectral mode width. In the presence of gain, the beam cross section is limited only by the size of the gain area. As an example of an active periodic photonic medium, we calculate and measure laser emission from a dye-doped cholesteric liquid crystal film. PACS numbers: 42.70.Qs, 42.70.Df Photonic band gap materials hold promise as a platform for a new generation of efficient, compact photonic devices. The initial excitement generated for these materials has been sustained by the discovery of new physical effects leading to new applications [ 1]. In this Letter, we present an unexpected conjunction of strong expansion of the region of phase coherence within the medium and weak divergence of the output beam, which is made possible in a photonic stop band structure. Our calculations show that a diffusion-like spreading of the coherent field leads to wide area lasing at the edge of a photonic stop band or at a localized state in the middle of the stop band. These results are supported by measurements of the spatial profile of laser emission from a promising organic material – a dye-doped cholesteric liquid crystal [ 2, ,3]. Since the maximum excitation energy is proportional to the laser area, large-area thin-film devices provide a new approach for high- power lasers. These enables lightweight optical sources for free-space communications, coherent backlighting for 3-D holographic and projection displays, and therapeutic irradiation of large areas of skin. Periodic dielectric materials in one, two or three dimensions may possess a gap in the photon density of states in the corresponding number of dimensions [1]. In 3-D periodic structures with sufficient modulation in the dielectric function, a full photonic band gap may appear in which spontaneous emission is suppressed.2Disturbing the periodicity of an active crystal introduces long-lived localized modes into the photonic band gap at which lasing is facilitated. Though lasing has not yet been observed in 3-D periodic structures, the enhanced photon dwell times and microcavity effects in localized states have suppressed the laser threshold in lower dimensional periodic systems. Unlike lasers based on Fabry-Perot resonators, the band-edge and defect modes of 1-D structures are significantly separated from adjacent modes and have narrower linewidth than other modes. They may therefore operate as single mode lasers over a significant power range. The shift of the band gap with oblique angle at the band-edge frequency only permits the propagation of light with k-vector components near the normal. Unlike conventional spreading of light between two mirrors, which requires oblique k-vector components (Fig. 1a), photonic band structures lead to enhanced spreading of light with only near-normal k-vector components (Fig.1b). It would appear that the best candidate for a 1-D periodic structure, for large area lasing would be a vertical cavity surface emitting laser (VCSEL) [ 4]. But high order transverse modes arise [ 5] in small-diameter VCSEL, while in large-diameter VCSELs spontaneous filamentation results from structural nonuniformities [ 6]. Another class of periodic one-dimensional structures, which have recently been produced as macroscopic single domain materials, is that of self-organized polymeric cholesteric liquid crystals (CLCs) [ 7]. Lasing at the edge of the stop band of CLCs has recently been demonstrated [2] and here we will specifically consider measurements and simulations of these materials. The molecular director in cholesteric structures lies in a molecular plane and executes a helical rotation perpendicular to the planes. A stop band centered at a wavelength in the medium equal to the helical pitch P is created for waves propagating along the helical axis. Optically pumped lasing in dye-doped CLCs occurs at the first mode at the edges of the stop band at vacuum wavelengths of neP or noP, where ne and no are the extraordinary and ordinary indices of refraction, respectively. The narrowest mode in CLCs is the first mode on the high frequency side of the band, which is composed of two counterpropagating components travelling normal to the layers of the structure and has an electric field everywhere aligned with the lowest index axis of the optical indicatrix. Consequently lasing generally occurs only in this mode slightly above threshold. For off- normal incident radiation, the gap shifts to higher frequency, placing oblique waves at the frequency of the lasing mode inside the stop band. As a result, transmitted or emitted radiation is confined to a narrow range of angles about the normal direction even for an incident beam with a large angular divergence as shown in Fig.1 b. The restricted angular distribution corresponds to a wide beam at the output surface of the thin film and hence to a beam inside the sample with divergence greatly exceeding that of free space diffraction divergence. The CLC film studied was a right-handed structure with n ~ 1.7 and Dn ~ 0.2 and a thickness of 35 µm. The samples were doped with laser dye PM-597 with an absorption peak at 530 nm and an emission peak at 590 nm and pumped with 150 ns pulses at the second harmonic of Q-switched pulses of a Nd :YAG laser at 532 nm. Lasing is produced at the mode closest to the upper band edge. The emitted beam is perpendicular to the CLC film and produces the pattern on a screen 11 .5 cm from the sample shown in the insert in Fig. 3. The CLC medium is modeled as a set of anisotropic amplifying layers with thickness h significantly smaller than the wavelength of the incident light. The direction of the molecular axis is rotated between successive layers by a small angle 2 πP/h. A normally incident circularly polarized 1-D Gaussian beam with the helicity of the CLC structure and frequency of the lasing mode is incident upon the sample. For simplicity of calculation, we consider an incident beam, which is a superposition of waves with an intensity distribution that is Gaussian in the plane of incidence (the x-z-plane) and is homogeneous in the perpendicular direction. The main results obtained for this wave are readily generalized to the case of a Gaussian beam. Transmission is calculated using a 4 × 4 transfer matrix method [ 8, 9, 10] for each Fourier component of the beam which are then superposed to give the transmitted wave.3The transfer-matrix is computed numerically for each plane wave component by taking the product of the transfer matrix for each layer of thickness h with uniform dielectric tensor. The value of h is decreased until the value of the transmittance converges. Transmission is simulated in a CLC film with P = 370 nm, thickness L = 35 µm and extraordinary and ordinary dielectric constants ee and eo which correspond to refractive indices ne = 1.8 and no =1.6. For the background medium, we use =bn1.52, which equals the refractive index of the glass substrates. Gain is introduced by adding a negative imaginary part to both ee and eo. The width of the band-edge mode in the transmission spectrum drops with increasing gain and vanishes at a critical value of the gain at the laser threshold. At this point, both the transmittance and reflectance diverge. The simulations show that the wavefunction at the output surface for a gain coefficient slightly below the critical value is proportional to \|]\|)1( exp[ xi−−a at the frequency of the band edge state for the perpendicularly propagating wave. A typical intensity distribution is shown in linear and logarithmic plots in Fig. 2. The intensity distribution is seen to decay exponentially, \|)\|2exp( x I a−∝ . Except for a small region near the beam center, the phase of the electric field increases linearly with the same coefficient a, i.e., \|\|xaf= . The value of α decreases with increasing gain coefficient and vanishes at the critical gain. The value of a, and hence the wavefunction, is found to be independent of the width of the incident beam. The calculated intensity distribution at the output surface gives rise to an oscillatory structure in the far field. The measured and calculated far-field intensities are shown in Fig. 3 at a distance of 11.5 cm from the sample. The agreement is good outside of the central region, but calculations give a flatter peak than is measured. This is due to the use of a beam with a Gaussian intensity distribution only in one direction and to the finite gain region created by the pump beam. We now demonstrate analytically that spatial distribution found is a direct consequence of the Lorentzian shape of the resonant peak of the transmitted electric field, )()( rkkiBCkT −−≈b b; zyx,,=b , ( 1 ) where bC and B are constants, l p/2bnk= and rb rnk l p/2= are wavevectors in the embedding and resonant medium, respectively. The linewidth of the transmission spectrum is Δλ = Blr2/2pnb. When the gain coefficient g is slightly below its critical value gc, B is proportional to ( gc-g). Equation (1) can be generalized to include oblique incident waves if we replace k by Akcos. At the band-edge frequency rkk=, the transmittance is strongly peaked in the normal direction. In the limit of rkB<<, we can expand 2/1 cos2AA−≅ , and Eq. (1) gives, rr kiABCkkAT222),(+≈=b b. ( 2 ) Using Eq. (2) and superposing all transmitted plane waves we obtain the transmitted electric field: \|]\|)1(exp[),( xBki LzxE rT−−∝=b. ( 3 ) This is precisely the wavefunction indicated in Fig. 2 with rBk=a . The wavefunction obtained in Eq. (3) depends only on rBk and is independent of the spatial extent of the incident beam. Since B is proportional to4(gc-g), the wavefunction becomes an unbounded plane wave as the gain approaches the critical value. When rBk x/1\|\|<< , Eq. (3) becomes invalid and it cannot describe the central region of the beam. Using Eq. (3), the width of the beam is defined as 2 0x, where 0x is the position at which the intensity drops to half of its peak value. This gives a beam width at the output surface of rBk xW /2ln20=≡ . By substituting b rnBpl l / 22=Δ and rb rnk l p/2= into the previous relation, we find a universal relation between Wand the linewidth lΔ2in transmission for normally incident radiation at rl, r br Wn ll p l Δ=2 2ln2. (4) This relation is valid as long as rkB<<. This condition is satisfied when the sample is sufficiently thick or )(g g−c is sufficiently small. Since this condition does not explicitly depend on the sample characteristics, we expect that it holds also for lasing modes in other layered media, such as binary-layered ( BL) media and Fabry- Perot (FP) resonators. This is demonstrated in Fig. 4, where we plot lr/nbW vs. Dl/lr on a log-log scale for three arbitrarily chosen systems: (a) CLC ( ◊), (b) BL (+), (c) FP ( x). We also plot Eq. (4) as a solid line for comparison. In the absence of gain, we also show the data for two CLC samples with different thickness ( 1). The universal relation in Eq. (4) has a surprising interpretation in terms of diffusion. Since the linewidth in the transmission spectrum is inversely proportional to the photon dwell time t, Eq. (4) can be interpreted in terms of the diffusion relation: tDW≈2, since t » lr2/(2pcDl). The effective diffusion constant is rb b rkncn cD / 2/2= ≈ p l . So, the diffusion constant is proportional to the wavelength of the lasing mode in the background medium and is independent of the sample thickness and gain coefficient. Although we have assumed the wave is homogeneous in the y direction, this condition can be relaxed in the analytical approach. If the incident beam is a Gaussian wave in both the x and y directions, the generalization of the 1-D approach leads to an outgoing wave: ] )1(exp[)/1( r r b rTBki E −− ∝ , where 2 2yx+=r and again the condition rBk/1>>r should be satisfied. This is similar to the wavefunction found in Eq. (3). These results may be naturally applied to band-edge or localized modes in periodic 1-D structures since only a singe mode of radiation exists over a wide angular range centered on the normal direction to yield wide area thin-film lasers. Acknowledgements This work was supported by the National Science Foundation under Grant No. DMR9973959, and by the Army Research Office. It was performed in part under the auspices of the City University of New York CAT in Ultrafast Photonic Materials and Applications.5Figure Captions Fig. 1. Coherent spreading of beam inside an amplifying medium. a Two flat mirrors define a cavity, in which spreading of the beam involves oblique k-vector components, which correspond to losses from the laser mode. The resulting beam size is linearly proportional to the photon dwell time. b Layered photonic band-gap medium, in which all oblique components of the k-vector are reflected, since their propagation is forbidden, except for k-vector component close to the normal. The beam size follows a diffusion equation and is proportional to the square root of the photon dwell time. Fig. 2. Computer simulation of spatial distribution of laser emission at output surface of CLC. a Intensity and phase in linear scale; b Intensities of incoming and outgoing beams in semi-log scale. Fig. 3. Spatial distribution of laser emission in far field (11.5 cm from sample). Fig. 4. Universal relation of inverse beam width to relative linewidth for different samples (Eq. 4).6Fig. 1.n1 n2 n1 n2 a b7Fig. 2. -0.15 -0.05 0.05 0.15 Distance (cm)Intensity (a. u.)103 101 10-1 10-3Intensity (a. u.)120 80 40 0Phase (degrees)1200 800 400 0◊◊Intensity - -Phase ◊◊ Outgoing - - Incominga b/G01/G02/G03/G04/G05 /G07/G05 -2 -1 0 1 2 Distance (cm)Intensity (a. u.)100 80604020 0◊◊◊◊Experiment —Simulation 9Fig. 4+x1 ◊ +++xx x xxxxxx ◊ ◊◊ ◊◊ ◊ ◊1 +xWavelength divided by half height beam width10-1010-910-810-710-610-5 Relative line width1CLC samples of differe nt thickness without gain xFabry-Perot resonators with different gain ◊CLC samples with different gain +Layered dielectric media with different gain — Square root analytical dependence10-1 10-2 10-3 10-410 [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, in Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, New Jersey, 1995). [2] V.I. Kopp, B. Fan , , H.K.M. Vithana and A.Z. Genack, Optics Letters 23, 1707 (1998). [3] Chen, S.H., Katsis, D., Schmid, A.W., Mastrangelo, J.C., Tsutsui, T., Blanton, T.N., Nature 397, 506 (1999). [4] Hiroyuki Yokoyama, Kikuo Ujihara, in Spontaneous Emission and Laser Oscillation in Microcavities, (CRC Press, 1995). [5] D. Burak, R. Binder, SPIE Proc. 3286, 220 (1998). [6] M. Orenstein, E. Kapon, N.G. Stoffel, J.P. Harbison, L.T. Florez, and J. Wullert, Appl. Phys. Lett. 58, 804 (1991). [7] J.E. Stockley, G.D. Sharp, K.M. Johnson, Opt. Lett. 24, 55 (1999). [8] S. Teitler and B. Henvis, Opt. Soc. Am. 60, 830 (1970). [9] D.W. Berreman, J. Opt. Soc. Am. 62, 502 (1972). [10] H. Wohler, G. Haas, M. Fritsch, and D.A. Mlynski, J. Opt. Soc. Am. A 5, 1554 (1988).
arXiv:physics/0010001v1 [physics.acc-ph] 30 Sep 2000LARGE-SCALE SIMULATIONOF BEAMDYNAMICS INHIGH INTENSITYIONLINACSUSING PARALLEL SUPERCOMPUTERS∗ Robert D.Ryne andJi Qiang,LANL,Los Alamos,NM87545,USA Abstract Inthispaperwepresentresultsofusingparallelsupercom- puters to simulate beam dynamics in next-generationhigh intensityionlinacs. Ourapproachusesathree-dimensiona l space charge calculation with six types of boundary con- ditions. The simulations use a hybrid approach involving transfer maps to treat externally applied ﬁelds (including rf cavities) and parallel particle-in-cell techniques to t reat the space-charge ﬁelds. The large-scale simulation result s presented here represent a three order of magnitude im- provement in simulation capability, in terms of problem size and speed of execution, compared with typical two- dimensional serial simulations. Speciﬁc examples will be presented, including simulation of the spallation neutron source (SNS) linac andthe Low EnergyDemonstratorAc- celerator(LEDA)beamhaloexperiment. 1 INTRODUCTION The high intensity of future accelerator-driven systems places stringent requirements on the allowed beam loss, since very small fractional losses at high energy can pro- duce unacceptably high levels of radioactivity. Previous studies suggest that the low density, large amplitude halo of the beam is a major issue for these systems [1, 2, 3]. Large-scalesimulationsareanimportanttoolforexplorin g thebeamdynamics,predictingthebeamhalo,andfacilitat- ing design decisions aimed at controlling particle loss and meetingoperationalrequirements. The most widely used model for simulating intense beams in ion rf linacs is represented by the Poisson- Vlasov equations. These equations are often solved us- ing a particle-in-cell(PIC) approach. In this paper we will describe a parallel simulation capability that combines th e PICmethodwith techniquesfrommagneticoptics,andwe willpresentresultsofusingparallelsupercomputerstosi m- ulate beamdynamicsinhighintensityionrf linacs. 2 PHYSICALMODELAND NUMERICALMETHODS In the PIC approach a number of simulation particles, calledmacroparticles,areusedtosolve(indirectly)thee vo- lution equationsand model the chargedparticle dynamics. The motion of individual particles in the absence of radia- ∗Work supported by the DOE Grand Challenge in Computational A c- celerator Physics, Advanced Computing for 21st Century Acc elerator Science and Technology Project, and the Los Alamos Accelera tor Code Group using resources at the Advanced Computing Laboratory and the National Energy Research Scientiﬁc Computing Center.tioncanbedescribedbyHamilton’sequations, d/vector q dt=∂H ∂/vector p,d/vector p dt=−∂H ∂/vector q, (1) where H(/vector q, /vector p, t)denotes the Hamiltonian of the system, and where /vector qand/vector pdenote canonical coordinates and mo- menta,respectively. Inthelanguageof mappings wewould say that there is a (generally nonlinear) map, M, corre- sponding to the Hamiltonian H, which maps initial phase space variables, ζi,intoﬁnal variables, ζf, andwe write ζf=Mζi. (2) The potential in the Hamiltonian includes contributions fromboththeexternalﬁeldsandthespace-chargeﬁelds. In the Poisson-Vlasov approach, discreteness effects are ne- glected and the space charge is represented by a smoothly varyingmeanﬁeld. Typically,theHamiltoniancanbewrit- ten as a sum of two parts, H=Hext+Hsc, which corre- spondtotheexternalandspace-chargecontributions. Such a situation is ideally suited to multi-map symplectic split - operator methods [4]. A second-order-accurate algorithm fora singlestepisgivenby M(τ) =M1(τ/2)M2(τ)M1(τ/2),(3) where τdenotes the step size, M1is the map correspond- ingto HextandM2isthemapcorrespondingto Hsc. This approachcanbeeasilygeneralizedtohigherorderaccuracy usingYoshida’sschemeif desired[5]. The electrostatic scalar potential generated by the chargedparticlesisobtainedbysolvingPoisson’sequatio n ∇2Ψ(r) =−ρ(r)/ǫ0. (4) where ρis the charge density. We have developed a Fourier-based transformation and an eigenfunction expan- sion method to handle six different boundary conditions: (1) openin all three dimensions;(2) opentransverselyand periodic longitudinally;(3,4)roundconductingpipe tran s- verselyandopenorperiodiclongitudinally;(5,6)rectang u- lar conducting pipe transversely and open or periodic lon- gitudinally. A discussion of the numerical algorithms for solving the Poisson’s equation with these different bound- aryconditionscanbefoundin [8]. The charge density ρon the grid is obtained by using a volume-weighted linear interpolation scheme[6, 7]. Af- ter the potential and electric ﬁeld is found on the grid, the same scheme is used to interpolate the ﬁeld at the particle locations. Duringthecourseofthesimulationeachstepin- volves the following: transport of a numerical distributio n ofparticlesthroughahalfstepbasedon M1,solvingPois- son’sequationbasedontheparticlepositionsandperform- ing a space-charge “kick” M2, and performing transport throughthe remaininghalfofthestepbasedon M1.3 APPLICATIONS We haveapplied the above3D parallelPIC approachto an early design of the SNS linac and to the proposed LEDA beam halo experiment. Our simulation of the SNS linac starts atthe beginningofthe DTL.Thecodeadvancespar- ticles through drift spaces, quadrupoleﬁelds and RF gaps. The dynamics inside the gaps is computed using external ﬁelds calculated from the electromagnetic code SUPER- FISH [9]. A schematic plot of the SNS linac conﬁguration used in this study is shown in Figure 1 [10]. It consists of threetypesofRFstructures: aDTL,aCCDTL,andaCCL. Thereare a total of425RF segmentsin the linac. Figure2 shows the rms transverse size ( xrms, yrms) and the maxi- mum transverseextent ( xmax, ymax) of the bunchedbeam in the linac with one set of errors. We see that the maxi- mum particle amplitude is well-below the aperture size of the linac. Thismarginis neededto operatethe linac safely and to avoid beam loss at the high energy end. The jump inrmsbeamsizebetweentheDTLandCCDTLat 20MeV is due to a changeoffocusingperiodfrom 8βλto12βλat 805MHz. Figure1: TheSNS linacconﬁguration 00.511.522.5 0 200 400 600 800 1000(cm) Kinetic Energy (MeV)Bore Radius Xrms Yrms Xmax Ymax Figure2: Transversebeamsize asa functionofkineticen- ergyin theSNS linac In the LEDA beam halo experiment, a mismatched high-intensity proton beam will be propagated through a periodic focusing transport system and measurementswill be made of the beam proﬁle. The goals of the experimentare two-fold: ﬁrst, to study beam halo formation and test ourphysicalunderstandingofthe phenomena,andsecond, to evaluate our computationalmodelsand assess their pre- dictive capability through a comparison of simulation and experiment. Fig. 3 gives a schematic plot of the layout of the experiment [11]. It consists of 52 alternating-focusin g quadrupole magnets with a focusing period of 41.96cm. The gradients of the ﬁrst four quadrupole magnets can be adjusted to create a mismatch that excites the breathing modeorthequadrupolemode. Thetransversebeamproﬁle will be measured using a beam-proﬁle scanner. Fig. 4 and Fig.5presentsimulationresultsofthetransversebeamsiz e for the breathing mode and the quadrupole mode, plotted at the center of the drift spaces between quadrupole mag- nets, as a function of distance. The plots include both the rmsbeamsizeandthemaximumparticleextentinthesim- ulation. The physical parameters for the simulation were I=100mA, E=6 .7MeV, and f= 350MHz. The simula- tion was performedusing 100million macroparticles with a 128x128x256(x-y-z)space-chargegrid. Figure3: LEDAhaloexperimentlayout 00.20.40.60.81 0 2 4 6 8 10 12displacement (cm) distance (m)Yrms Xrms Ymax Xmax Figure 4: Transverse beam size as a function of distance forthebreathingmodeintheLEDAhaloexperiment From Fig. 4, the two transverse components of the breathing mode are in phase, while the quadrupole mode in Fig. 5 has the two components out of phase. Evidently, it will be possible in the experiment to clearly excite ei- ther of the two modes. Furthermore, the debunching of the beam will not signiﬁcanly alter the structure of the os- cillations. Fig. 6 shows the accumulated one-dimensional00.20.40.60.81 0 2 4 6 8 10 12Displacement (cm) Distance (m)Xrms Yrms Xmax Ymax Figure 5: Transverse beam size as a function of distance forthequadrupolemodeintheLEDAhaloexperiment 0100000200000300000400000500000600000700000800000 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 x (cm)Breathing Mode Quadrupole Mode Figure 6: Accumulated density proﬁle along xfor the breathingmodeandthequadrupolemode density proﬁles (along x) for the breathing mode and the quadrupole mode just after the magnet # 49. The breath- ing mode is more peaked and has a larger extent than the quadrupolemode. Measurementswillbetakenatthisloca- tion and will be compared with our simulations. The data inFig.6arewell-resolvedoverarangeofabout 6decades. An important piece of informationfrom a design stand- pointisamountofchargebeyondaspeciﬁedradiusorspa- tial locationasafunctionofdistancealongtheaccelerato r. This is shown graphically in Fig. 7 which shows the com- plement of the horizontal and vertical cumulative density proﬁles, at every step, when the quadrupole mode is ex- cited. In other words, the contoursdescribe the fraction of chargethatwouldbeinterceptedbyascraperplacedatthat transverseposition. The above LEDA simulations used 100 million macroparticlesand a 3D Poisson solver, and requiredonly 2 hours to execute on 256 processors. In contrast, beam dynamicssimulations performedon serial computerstypi- cally use 10,000to 100,000macroparticlesand a 2D Pois- sonsolver. Eveniftheabovelarge-scalecalculationscoul d beperformedonaPC,theywouldrequireontheorderofa monthto complete. Inconclusion,whilesmall-scalesimu- Figure 7: Horizontal and vertical cumulative density pro- ﬁles of a quadrupole mode mismatch in the LEDA halo experiment. lationsonserialcomputersareextremelyvaluableforrapi d design and predicting rms properties, large-scale simula- tions are needed for high-resolutionstudies aimed at mak- ingquantitativepredictionsofthe beamhalo. 4 ACKNOWLEDGMENTS We thank the SNS linac design team and the LEDA beam halo experiment team for helpful discussions. We also thankS.HabibforhelpfuldiscussionsandC.ThomasMot- tersheadforgraphicssupport. 5 REFERENCES [1] R.L.Gluckstern, Phys. Rev. Lett. 73, 1247 (1994). [2] T.P.Wangler,K.R.Crandall,R.Ryne,andT.S.Wang,Phys . Rev. STAccel.Beams 1,084201 (1998). [3] J. Qiang and R. D. Ryne, Phys. Rev. ST. Accel. Beams 3, 064201 (2000). [4] E.Forestet al.,Phys.Lett. A 158, 99(1991). [5] H.Yoshida, Phys. Lett. A150, 262(1990). [6] R. W. Hockney and J. W. Eastwood, Computer Simulation UsingParticles,Adam Hilger,New York, 1988. [7] C. K.Birdsall and A. B. Langdon, Plasma Physics Via Com- puter Simulation, McGraw-HillBookCompany, NY, 1985. [8] J. Qiang and R. Ryne, ”High Performance Particle-In-Cel l SimulationinaProtonLinac,” tobe submittedtoPRST-AB. [9] J. H. Billen and L. M. Young, ”POISSON SUPERFISH”, LANLReport LA-UR-96-1834 (revisedJan. 8, 2000). [10] T. Bhatia et al., ”Beam Dynamics Design for the 1-GeV 2- MW SNSLinac,” LANLReport LA-UR-99-3802, 1999. [11] T. Wangler, ”LEDA Beam Halo Experiment-Physics and Concept of the experiment,” LA-UR-00-3181, 2000.
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/DB/CX/D8/CW/CU/D6/CT/D5/D9/CT/D2/CR/DD /CP/D2/CS /D4/D9/D0/D7/CT /D0/CT/D2/CV/D8/CW/BA/CA/BX/BY/BX/CA/BX/C6/BV/BX/CB/CJ /BD /CL /CA /BA/C4 /BA/BU /D3 /DC/D1/CP/D2/B8 /BW/BA /C5/BA /CB/CP/D2/CS/CT/D6/D7 /CP/D2/CS /C8 /BA /C2/BA /C5/CP/D6/D8/CX/D2/B8 /CT/CS/D7/BA/B8/CK/C0/CP/D2/CS/CQ /D3 /D3/CZ /D3/CU /CE /CP/CR/D9/D9/D1 /BT/D6/CR /CB/CR/CX/CT/D2/CR/CT /CP/D2/CS /CC /CT/CR /CW/D2/D3/D0/D3/CV/DDꜼ/B8/C6/D3 /DD /CT/D7 /C8/D9/CQ/D0/CX/CR/CP/D8/CX/D3/D2/D7/B8 /C6/CT/DB /C2/CT/D6/D7/CT/DD /BC/BJ/BI/BH/BI/B8 /BD/BL/BL/BH/BA/CJ/BE/CL/BY /D6/CP/D2/CR/CX/D7 /BY/BA /BV/CW/CT/D2/B8 /CK/C1/D2 /D8/D6/D3 /CS/D9/CR/D8/CX/D3/D2 /D8/D3 /C8/D0/CP/D7/D1/CP /C8/CW /DD/D7/CX/CR/D7Ꜽ/B8/C8/D0/CT/D2 /D9/D1 /C8/D6/CT/D7/D7/B8 /C6/CT/DB /CH /D3/D6/CZ/B8 /BD/BL/BJ/BG/BN /CB/CT/CR/BA /BK/BA/BE/BA/CJ/BF/CL/BT/BA /C1/BA /BT/CZ/CW/CX/CT/DE/CT/D6/B8 /C1/BA /BT/BA /BT/CZ/CW/CX/CT/DE/CT/D6/B8 /CA/BA /CE/BA /C8 /D3/D0/D3 /DA/CX/D2/B8 /BT/BA /BZ/BA/CB/CX/D8/CT/D2/CZ /D3 /CP/D2/CS /C3/BA /C6/BA /CB/D8/CT/D4/CP/D2/D3 /DA/B8 /CK/C8/D0/CP/D7/D1/CP /BX/D0/CT/CR/D8/D6/D3 /CS/DD/D2/CP/D1/CX/CR/D7Ꜽ/B8/C8 /CT/D6/CV/CP/D1/D3/D2 /C8/D6/CT/D7/D7/B8 /BX/D0/D1/D7/CU/D3/D6/CS/B8 /C6/CT/DB /CH /D3/D6/CZ /BD/BC/BH/BE/BF/B8 /BD/BL/BJ/BH/BN /CB/CT/CR/BA/BI/BA/BD/BA/BG/BA/CJ/BG/CL/C2/CT/D2/D7 /C3/D2/D3/CQ/D0/D3 /CR /CW/B8 /CK/BT/CS/DA /CP/D2/CR/CT/CS /CC/CW/CT/D6/D1/D3/D1/CT/D8/D6/DD /CB/D8/D9/CS/CX/CT/D7 /D3/CU/CB/D9/D4 /CT/D6/CR/D3/D2/CS/D9/CR/D8/CX/D2/CV /CA/CP/CS/CX/D3/B9/BY /D6/CT/D5/D9/CT/D2/CR/DD /BV/CP /DA/CX/D8/CX/CT/D7Ꜽ/B8 /C8/CW/BA/BW/BA /D8/CW/CT/B9/D7/CX/D7/B8 /BV/D3/D6/D2/CT/D0/D0 /CD/D2/CX/DA /CT/D6/D7/CX/D8 /DD /B8 /BT/D9/CV/D9/D7/D8/B8 /BD/BL/BL/BJ/BA
arXiv:physics/0010003v1 [physics.hist-ph] 2 Oct 2000On Wolfgang Pauli’s most important contributions to physic s1 Norbert Straumann Institute for Theoretical Physics, University of Zurich, Winterthurerstrasse 190, 8057 Z¨ urich, Switzerland Abstract After some brief biographical notes, Wolfgang Pauli’s main contributions to general re- lativity, quantum theory, quantum ﬁeld theory, and element ary particle physics are reviewed. A detailed description is given how Pauli descovered – befor e the advent of the new quantum mechanics – his exclusion principle. Meine Damen und Herren, Ich f¨uhle mich sehr geehrt, diesen Vortrag halten zu d ¨urfen. Gleichzeitig habe ich ein etwas beklemmendes Gef ¨uhl, denn meine Aufgabe ist nicht leicht. Am Samstag werden wir manches ¨uber Paulis Pers ¨onlichkeit und seine weitgespannten Inter- essen h ¨oren. In meinem Vortrag werde ich hingegen fast ausschliess lich¨uber Paulis Wissenschaft[1] reden. Auch bei ¨ausserster Konzentration auf seine wesentlichsten Beitr ¨age ist eine Stunde nat ¨urlich zu kurz. Deshalb muss ich die Gewichte sehr ungleichm ¨assig verteilen. Ich sollte zumindest auf folgende Themen eingehen: 1) Zu Beginn seiner Karriere hat sich Pauli in sehr jungen Jah ren mit seinen Beitr ¨agen zur Allgemeinen Relativit ¨atstheorie, insbesondere zur Weylschen Erweiterung, sehr schnell einen Namen als ganz aussergew ¨ohnlicher theoretischer Physiker gemacht. 2) Paulis wichtigster Beitrag zur Physik ist zweifellos sei n Ausschliessungsprinzip. Der wenig bekannten Entdeckungsgeschichte werde ich einen wesentli chen Teil meiner Zeit widmen. Die Schwierigkeit dabei ist, dass man sich in die Situation vorder Quantenmechanik (in unserem heutigen Sinne) zur ¨uckversetzen muss. Nach genauem Studium von Paulis Argumentationskette bin ich zutiefst davon beeindruckt, w ie Pauli — auf der Basis der br¨uchigen Bohr-Sommerfeld-Theorie und dem damaligen spektr oskopischen Wissen — der Natur sein Ausschliessungsprinzip ablauschen konnte. Die Genialit ¨at des jungen Mannes zeigt sich, so denke ich, nirgends deutlicher. 3) Die weiteren wichtigsten Beitr ¨age zur Quantenmechanik, zur Quantenfeldtheorie und zur Elementarteilchenphysik werde ich relativ knapp abhandel n m¨ussen. Dies ist nicht so schlimm, da man w ¨ahrend des Studiums mit diesen Themen konfrontiert wird und Paulis Einsichten zum selbstverst ¨andlichen Besitz der Physik geworden sind. 1Opening talk at the symposium: WOLFGANG PAULI AND MODERN PHY SICS, in honor of the 100th anniversary of Wolfgang Pauli’s birthday, ETH (Zurich), Ma i 4-6 2000.Auf den Zusammenhang von Spin und Statistik muss ich aber unbedingt etwas eingehen, da mit diesem — zur grossen Befriedigung von Pauli — das Aussc hlussprinzip eine tiefere Begr¨undung fand. Ferner m ¨ochte ich etwas zur Neutrinohypothese sagen, vor allem weil die damit verbun- denen anf ¨anglichen Schwierigkeiten meist ungen ¨ugend hervorgehoben werden. Es ging eben nicht nur um die Energieerhaltung, sondern u.a. auch um Spin und Statistik der Atomkerne. * * * * * Werdegang In einem Alter, in welchem heute die meisten ihre Dissertati on aufschreiben, war Pauli bereits eine der wichtigsten Autorit ¨aten und treibenden Kr ¨afte in der Welt der theoretischen Physik. “Was, so jung und schon unbekannt” soll Pauli einmal zu einem jungen Mann gesagt hab en. Paulis Ausstrahlung auf die Physiker seiner Zeit hat Paul Eh renfest anl ¨asslich der Verleihung der Lorentzmedaille an Pauli im Jahre 1931 so zusammengefas st[2]: “Aber vielleicht noch von viel gr ¨osserem Gewicht als seine Publikationen sind die unz¨ahligen, unverfolgbaren Beitr ¨age, die er zur Entwicklung der neueren Physik durch m ¨undliche Diskussionen oder Briefe geliefert hat. Die enorm e Sch¨arfe seiner Kritik, seine ausserordentliche Klarheit und vor allem die r¨ucksichtslose Ehrlichkeit, mit der er stets den Nachdruck auf die ungel ¨osten Schwierigkeiten legt, bewirkt, dass er als unsch ¨atzbare Triebkraft innerhalb der neueren theoretischen Fo rschung gelten muss.” Einen lebendigen Eindruck von Paulis Einﬂuss auf die Physik er seiner Zeit und seine unerh ¨ort kritische F ¨ahigkeit gibt der nie abreissende Briefstrom[3], bei dessen Lekt ¨ure man immer wieder den Atem anh ¨alt. Paulis Kritik war gesucht, gef ¨urchtet und oft beissend, wie die folgende etwas erschreckende Kostprobe aus einem Brief an Einstein[4]aus dem Jahre 1929 zeigt, der gleichzeitig auch den stattgefundenen Generationswechsel ¨uberdeutlich macht (als Hintergrund f ¨ur diesen Brief muss man wissen, dass Einstein bereits in den 20er Jahr en mehrere Versuche machte, Gravitation und Elektrodynamik in einer einheitlichen kla ssischen Feldtheorie zu verschmelzen, wovon Pauli wenig hielt): “Es bleibt [den Kritik ¨ubenden Physikern] nur ¨ubrig, Ihnen dazu zu gratulieren (oder soll ich lieber sagen: zu kondolieren?), dass Sie zu de n reinen Mathematikern ¨ubergegangen sind. Ich bin auch nicht so naiv als dass ich gla uben w ¨urde, Sie w¨urden auf Grund irgend einer Kritik durch Andere Ihre Meinun g¨andern. Aber ich w¨urde jede Wette mit Ihnen eingehen, dass Sie sp ¨atestens nach einem Jahr den ganzen Fernparallelismus aufgegeben haben werden, so wie S ie fr¨uher die Aﬃntheorie aufgegeben haben. Und ich will Sie nicht durch Fortsetzung d ieses Briefes noch weiter zum Widerspruch reizen, um das Herannahen dieses nat ¨urlichen Endes der Fernparallelismustheorie nicht zu verz ¨ogern.” Pauli war, was man ein Wunderkind nennt. Als reifer Mann mein te er dazu: “Ja, das Wunder- kind — das Wunder vergeht und das Kind bleibt .. . .” Es war wohl naheliegend, dass Pauli nach dem Abitur noch im Jahre 1918 nach M ¨unchen zog, wo Sommerfeld ‘eine Pﬂanzst ¨atte der 2theoretischen Physik’ geschaﬀen hatte und die besten Talen te anzog. Sommerfeld ist vor allem als begnadeter Lehrer in die Geschichte der Physik eingegan gen. Seine auch heute noch sehr lesenswerten Lehrb ¨ucher geben uns eine Ahnung von der Begeisterung, mit der er s eine Vorlesun- gen hielt und mit welchem Enthusiasmus er die jugendlichen H ¨orer mitriss. Zu den Massst ¨aben seines Unterrichts sagte er einmal[5]: “Was wir betonen m ¨ussen, ist die ideale Seite der mathe- matischen und naturwissenschaftlichen Studien, ihre Sch ¨onheit und innere Wahrhaftigkeit; ihre Kraft, phrasenloses Denken und r ¨ucksichtsloses Schliessen im Sch ¨uler zu entwickeln.” Sommerfeld erkannte sofort die Begabung von Pauli und betra ute ihn schon in den Anfangsse- mestern mit dem f ¨ur die Enzyklop ¨adie der mathematischen Wissenschaften bestimmten Kapite l ¨uber Relativit ¨atstheorie. Zu diesem Zeitpunkt geh ¨orte Pauli bereits zu den besten Kennern der Allgemeinen Relativit ¨atstheorie. Im Jahre 1919 sind von ihm drei Arbeiten[6]auf diesem Gebiet erschienen, wovon zwei der kurz zuvor entstandenen Weylsch en Theorie von Gravitation und Elektrizit ¨at gewidmet sind. Die Weylsche Theorie war — mathematisch gesehen — ein bestec hender Versuch, die beiden damals einzig bekannten Wechselwirkungen in geometrische r Weise einheitlich zu beschreiben. Weyl verallgemeinerte dabei die Riemannsche Geometrie auf sehr nat ¨urliche Weise, wodurch nicht nur das Gravitationsfeld sondern auch das elektromag netische eine geometrische Bedeutung erlangten[7]. Einstein ¨ausserte mit Recht sofort physikalische Bedenken, aber Pau li war nach Weyl der erste, der die Theorie n ¨aher untersuchte und beobachtbare Konsequenzen ausarbeit ete. So berechnete er die Periheldrehung des Merkur und die Lichtab lenkung in der Weylschen Theorie f¨ur eine damals von Weyl bevorzugte Wirkung2. In diesen ersten Arbeiten von Pauli sieht man bereits den aus gereiften Meister. Jedermann wunderte sich, wie dies so fr ¨uh m¨oglich war[8]. Pauli befand sich erst im dritten Semester als er im Sommer 19 20 auf Veranlassung von Sommerfeld mit der Arbeit an dem f ¨ur die Enzyklop ¨adie der mathematischen Wissenschaften bestimmten Kapitel ¨uber Relativit ¨atstheorie begann. In weniger als einem Jahr bew ¨altigte er — neben seinem Studium — die anspruchsvolle Aufgabe und liefe rte Anfang 1921 das Manuskript von 237 Buchseiten mit gegen 400 verarbeiteten Literaturzi taten ab. Damit etablierte sich Pauli als ein Wissenschaftler von seltener Tiefe und alles ¨uberragenden synthetischen und kritischen F¨ahigkeiten. In seiner Besprechung des Enzyklop ¨adieartikels[9]schrieb Einstein[10]: “Wer dieses reife und gross angelegte Werk studiert, m ¨ochte nicht glauben, dass der Verfasser ein Mann von einundzwanzig Jahren ist. Man wei ss nicht, was man am meisten bewundern soll, das psychologische Verst ¨andnis f ¨ur die Ideenentwick- lung, die Sicherheit der mathematischen Deduktion, den tie fen physikalischen Blick, das Verm ¨ogen¨ubersichtlicher systematischer Darstellung, die Literat urkenntnis, die sachliche Vollst ¨andigkeit, die Sicherheit der Kritik.” Im selben Jahr 1921, als Paulis Artikel erschien, promovier te er bereits nach sechs Semestern an der Universit ¨at M¨unchen mit einer Arbeit ¨uber das Wasserstoﬀ-Molek ¨ulion[11]im Rahmen der alten Quantentheorie von Bohr und Sommerfeld. Die Grenzen d ieser Theorie zeichneten sich bereits bei diesem Problem ab. Vollends klar wurde dies alle rdings erst nachdem Heisenberg auch das Heliumatom untersucht hatte. Nach seiner Promotion trat Pauli im Wintersemester 1921/22 eine Assistentenstelle bei Max Born in G ¨ottingen an. In gemeinsamen Untersuchungen ¨ubertrugen die beiden die astronomische 2Da die zugeh ¨origen Feldgleichungen h ¨oherer Ordnung sind, tritt im zentralsymmetrischen Proble m neben der Masse noch eine zweite willk ¨urliche Integrationskonstante auf. Pauli zeigt, dass nur f ¨ur eine spezielle Wahl dieser Integrationskonstante die Einsteinschen Resultate f ¨ur die Periheldrehung und die Lichtablenkung resultieren. 3St¨orungstheorie auf die Atomphysik. Schon am 29. November 192 1 schrieb Born an Einstein[12]: “Der kleine Pauli ist sehr anregend; einen so guten Assisten ten werde ich nie mehr kriegen.” Jahrzehnte sp ¨ater¨ausserte sich Born ¨uber Pauli folgendermassen[13]: “Denn ich wusste seit der Zeit, da er mein Assistent in G ¨ottingen war, dass er ein Genie war, nur vergleichbar mit Ein stein selbst, ja dass er rein wissenschaftlich vielleicht noch gr ¨osser war als Einstein, wenn auch ein ganz anderer Menschentyp, der in meinen Augen Einsteins Gr ¨osse nicht erreichte.” Entdeckungsgeschichte des Ausschliessungsprinzips Einen lebendigen Eindruck von Paulis Entdeckung des Aussch liessungsprinzips[14]— seinem wichtigsten Beitrag zur Physik — gewinnt man wieder aus dem e rsten Band des Briefwechsels[15]. Das Pauliprinzip lag zum Zeitpunkt seiner allgemeinen Form ulierung gegen Ende 1924 keines- wegs in der Luft, stand man doch damals vor zwei grunds ¨atzlichen Schwierigkeiten: Einen all- gemeinen ¨Ubersetzungsschl ¨ussel eines mechanischen Modells in eine koh ¨arente Quantentheorie gab es noch nicht und der Spin-Freiheitsgrad war unbekannt. Die Entdeckungsgeschichte des Ausschliessungsprinzips b eginnt im Herbst 1922 in Kopen- hagen, wo sich Pauli am Bohrschen Institut mit einer Erkl ¨arung des anomalen Zeeman-Eﬀektes abm¨uhte. Land´ e hatte aus den Spektraldaten bereits die Aufspa ltung der Spektralterme in einem schwachen Magnetfeld abgeleitet und f ¨ur die Beschreibung der Dublettstruktur der Al- kalimetalle halbzahlige magnetische Quantenzahlen einge f¨uhrt. Pauli gelang es in einem ersten Schritt, Land´ es Termanalyse f ¨ur das Paschen-Back-Gebiet zu verallgemeinern. Er fand dab ei das “allgemeine formale Gesetz, welches gestattet, die Auf spaltungsfaktoren gim Falle kleiner Felder aus den Energiewerten bei grossen Feldern abzuleite n”[16]. (Darauf werde ich gleich n ¨aher eingehen.) Diese fr ¨uhe Arbeit war — wie Pauli in seinem Nobelvortrag[17]betont — f ¨ur die Entdeckung des Ausschliessungsprinzips wesentlich. Zum Zeitpunkt ih rer Abfassung war er aber dar ¨uber sehr ungl ¨ucklich, wie z. B. aus einem Brief an Sommerfeld hervorgeht[18]: “Ich habe mich sehr lange mit dem anomalen Zeemaneﬀekt gepla gt, wobei ich oft auf Irrwege geriet und eine Unzahl von Annahmen pr ¨ufte und dann wieder verwarf. Aber es wollte und wollte nicht stimmen! Dies ist mir bis jetz t einmal gr ¨undlich schief gegangen! Ein Zeit lang war ich ganz verzweifelt .. .i ch habe das Ganze mit einer Tr ¨ane im Augenwinkel geschrieben und habe davon wenig Freude. ” Pauli hat selber erz ¨ahlt[19]wie ihm auf einer ziellosen Wanderung durch die Strassen Kop en- hagens Harald Bohr begegnete, der freundlich zu ihm sagte: “ Sie sehen so ungl ¨ucklich aus”, worauf er schroﬀ antwortete: “Wie kann man gl ¨ucklich aussehen, wenn man an den anomalen Zeemaneﬀekt denkt.” 1. Schritt: Zeemaneﬀekt in starken Feldern und Paulis Summe nregel[16] Ich m ¨ochte Ihnen diesen ersten Schritt auf dem Weg zum Ausschlies sungsprinzip im Detail vorf¨uhren, um Ihnen zu zeigen, wie Pauli auf unsicherem Grund zu e inem richtigen — und wie sich zeigen sollte — wichtigen Resultat gelangte. Dabei werde ich heutige Bezeichnungen verwenden, um unn ¨otige Schwierigkeiten zu vermeiden. Wie immer beschreibt Pauli zun ¨achst in konziser Weise, was er an bereits Vorhandenem zugrunde legt. Dies muss ich ebenfalls festhalten: 4•Die Beziehung zwischen Energieniveaus und Spektrum wird du rch die Bohrsche Regel E2−E1=hν bestimmt. (Diese kannte man seit Bohrs Pionierarbeiten.) •Den Spektraltermen hatte man bereits verschiedene Quantenzahlen zugeordnet. Es sind dies3: ⊲L[=k−1], L= 0,1,2,3,... (S,P,D,F,... ) (unsere heutige Bahndrehimpuls-Quantenzahl); ⊲S[=i−1]: jeder Term geh ¨ort zu einem Singlett oder Multiplettsystem mit maximaler Multiplizit ¨at 2S+ 1 (S= 0,1 2,1,...) (unsere heutige Spinquantenzahl); ⊲die verschiedenen Terme eines Multipletts (mit denselben LundS) werden durch die Quantenzahl J[=j] unterschieden; deren Werte sind: J=L+S, L+S−1,... L −Sf¨urL≥S, J=S+L, S+L−1,... S −Lf¨urL<S . (Jist nat ¨urlich unsere heutige totale Drehimpulsquantenzahl.) •Ferner kannte man die folgenden Auswahlregeln , die in den meisten F ¨allen g ¨ultig sind: L− →L±1, S− →S , J− →J+ 1,J,J−1 (0 − →0 verboten) . •Bei gegebener Atomzahl Z(Z−pf¨urp-fach ionisierte Atome) gilt: Zgerade − →S,J: ganz, Zungerade − →S,J: halbganz . •Aufspaltung im Magnetfeld: ⊲Jeder Term spaltet in 2 J+ 1 Niveaus auf, welche durch die Quantenzahl M=J,J− 1,...,−Junterschieden werden. ⊲Land´ e: F¨ur schwache Felder ist die Aufspaltung △EM=M·g(µ0B) (¨aquidistante Reihe); dabei ist µ0=e/planckover2pi1/2mcdas Bohrsche Magneton (von Pauli 1920 eingef ¨uhrt) undgist der Land´ esche g-Faktor: g=3 2+S(S+ 1)−L(L+ 1) 2J(J+ 1); ⊲Auswahlregeln f¨ur Zeeman- ¨Uberg ¨ange: M− →M±1 (σ−Komponente) , M− →M (π−Komponente) . 3In eckigen Klammern sind die historischen Bezeichnungen ge geben. 5Bemerkung: Fallsgundg′im Anfangs- und Endzustand gleich sind gilt:
arXiv:physics/0010004v1 [physics.class-ph] 2 Oct 2000WHAT IS THE EVANS-VIGIER FIELD? VALERI V. DVOEGLAZOV Escuela de F´ ısica, Universidad Aut´ onoma de Zacatecas Apartado Postal C-580, Zacatecas 98068, Zac., M´ exico E-mail: valeri@ahobon.reduaz.mx URL: http://ahobon.reduaz.mx/˜ valeri/valeri.htm Abstract. We explain connections of the Evans-Vigier model with theor ies proposed previously. The Comay’s criticism is proved to be i rrelevant. The content of the present talk is the following: −Evans-Vigier deﬁnitions of the B(3)ﬁeld [1]; −Lorentz transformation properties of the B(3)ﬁeld and the B-Cyclic Theorem[2, 3]; −Clariﬁcations of the Ogievetski˘ ı-Polubarinov, Hayashi a nd Kalb-Ramond papers [4, 5, 6]; −Connections between various formulations of massive/mass lessJ= 1 ﬁeld; −Conclusions of relativistic covariance and relevance of th e Evans-Vigier postulates. In 1994-2000 I presented a set of papers [7] devoted to clariﬁ cations of the Weinberg (and Weinberg-like [8, 9]) theories and the con cept of Ogievet- ski˘ ı-Polubarinov notoph . In 1995-96 I received numerous e-mail communi- cations from Dr. M. Evans, who promoted a new concept of the lo ngitudinal phaseless magnetic ﬁeld associated with plane waves, the B(3)ﬁeld (which is later obtained the name of M. Evans and J.-P. Vigier). Reason s for continu- ing the discussion during 2-3 years were: 1) the problem of ma ssless limits of all relativistic equations does indeed exist; 2) the dynami cal Maxwell equa- tions have indeed additional solutions with energy E= 0 (apart of those2 VALERI V. DVOEGLAZOV withE=±\|κ\|, see [10, 11, 12, 13, 14];13) the B(3)concept met strong non-positive criticism (e. g., ref [15, 16, 17]) and the situ ation became even more controversial in the last years (partially, due to the E vans’ illness). What are misunderstandings of both the authors of the B(3)model and their critics? In Enigmatic Photon (1994), ref. [1], the following deﬁnitions of the longitudinal Evans-Vigier B(3)ﬁeld have been given:2 Deﬁnition 1. [p.3,formula (4a)] B(1)×B(2)=iB(0)B(3)⋆,et cyclic . (3) Deﬁnition 2. [p.6,formula (12)] B(3)=B(3)⋆=−iκ2 B(0)A(1)×A(2), (4) and Deﬁnition 3. [p.16,formula (41)] B(3)=B(0)ˆk. (5) The following notation was used: κis the wave number; φ=ωt−κ·ris the phase; B(1)andB(2)are usual transverse modes of the magnetic ﬁeld; A(1)andA(2)are usual transverse modes of the vector potential. The main experimental prediction of Evans [1a,b] that the ma gnetiza- tion induced during light-matter interaction (for instanc e, in the IFE) M=αI1/2+βI+γI3/2where I=1 2ǫ0cE2 0,E0=c\|Bπ\|(6) hasnotbeen conﬁrmed by the North Caroline group [18]. As one can see from Figure 4 of [18] “the behaviour of the experimental curv e does not match with Evans calculations”. Nevertheless, let us try to deepen understanding of the theo retical con- tent of the Evans-Vigier model. In their papers and books [1] Evans and 1If we put energy to be equal to zero in the dynamical Maxwell eq uations ∇ ×[E−iB] +i(∂/∂t)[E−iB] = 0 , (1) ∇ ×[E+iB]−i(∂/∂t)[E+iB] = 0 . (2) we come to ∇ ×E= 0 and ∇ ×B= 0, i. e. to the conditions of longitudinality. The method of deriving this conclusion has been given in [19]. 2I apologize for not citing all numerous papers of Evans et aland papers of their critics due to page restrictions on the papers of this volume .WHAT IS THE EVANS-VIGIER FIELD? 3 Vigier used the following deﬁnition for the transverse anti symmetric tensor ﬁeld: /parenleftbiggB⊥ E⊥/parenrightbigg = B(0)√ 2 +i 1 0  E(0)√ 2 1 −i 0  e+iφ+ B(0)√ 2 −i 1 0  E(0)√ 2 1 +i 0  e−iφ, (7) IfB(0)=E(0)this formula describes the right-polarized radiation. Of course, a similar formula can be written for the left-polari zed radiation. These transverse solutions can been re-written to the real ﬁ elds. For in- stance, Comay presented them in the following way [16c] in th e reference frame Σ: E⊥= cos[ ω(z−t)]ˆi−sin[ω(z−t)]ˆj, (8) B⊥= sin[ ω(z−t)]ˆi+ cos[ ω(z−t)]ˆj, (9) and analized the addition of B\|\|=√ 2ˆkto (9). Making boost to other frame of reference Σ′he claimed that a) B(3)′isnotparallel to the Poynting vector; b) with the Evans postulates E(3)′has a real part; c) transverse ﬁelds change, whereas B(3)is left unchanged when the boost is done to the frame moving in the zdirection. Comay concludes that these observations disprove the Evans claims on these particular questions. Fu rthermore, he claimed that the B(3)model is inconsistent with the Relativity Theory. According to [20, Eq.(11.149)] the Lorentz transformation rules for elec- tric and magnetic ﬁelds are the following: E′=γ(E+cβ×B)−γ2 γ+ 1β(β·E), (10) B′=γ(B−β×E/c)−γ2 γ+ 1β(β·B), (11) whereβ=v/c,β=\|β\|= tanh φ,γ=1√ 1−β2= cosh φ, with φbeing the parameter of the Lorentz boost. We shall further use the n atural unit system c= ¯h= 1. After introducing the spin matrices ( Si)jk=−iǫijkand deriving relevant relations: (S·β)jkak≡i[β×a]j, βjβk≡[β211−(S·β)2]jk,4 VALERI V. DVOEGLAZOV one can rewrite Eqs. (10,11) to the form Ei′=/parenleftigg γ11 +γ2 γ+ 1/bracketleftig (S·β)2−β2/bracketrightig/parenrightigg ijEj−iγ(S·β)ijBj,(12) Bi′=/parenleftigg γ11 +γ2 γ+ 1/bracketleftig (S·β)2−β2/bracketrightig/parenrightigg ijBj+iγ(S·β)ijEj.(13) Pure Lorentz transformations (without inversions) do not c hange signs of the phase of the ﬁeld functions, so we should consider separa tely properties of the set of B(1)andE(1), which can be regarded as the negative-energy solutions in QFT and of another set of B(2)andE(2), the positive-energy solutions. Thus, in this framework one can deduce from Eqs. ( 12,13) Bi(1)′=/parenleftigg 1 +γ2 γ+ 1(S·β)2/parenrightigg ijBj(1)+iγ(S·β)ijEj(1),(14) Bi(2)′=/parenleftigg 1 +γ2 γ+ 1(S·β)2/parenrightigg ijBj(2)+iγ(S·β)ijEj(2),(15) Ei(1)′=/parenleftigg 1 +γ2 γ+ 1(S·β)2/parenrightigg ijEj(1)−iγ(S·β)ijBj(1),(16) Ei(2)′=/parenleftigg 1 +γ2 γ+ 1(S·β)2/parenrightigg ijEj(2)−iγ(S·β)ijBj(2),(17) and Bi(1)′=/parenleftigg 1 +γ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijBj(1), (18) Bi(2)′=/parenleftigg 1−γ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijBj(2), (19) Ei(1)′=/parenleftigg 1 +γ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijEj(1), (20) Ei(2)′=/parenleftigg 1−γ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijEj(2), (21) (when the deﬁnitions (7) are used). To ﬁnd the transformed 3- vector B(3)′ is just an algebraic exercise. Here it is B(1)′×B(2)′=E(1)′×E(2)′=iγ(B(0))2(1−β·ˆk)/bracketleftigg ˆk−γβ+γ2(β·ˆk)β γ+ 1/bracketrightigg . (22)WHAT IS THE EVANS-VIGIER FIELD? 5 We know that the longitudinal mode in the Evans-Vigier theor y can be con- sidered as obtained from Deﬁnition 3 . Thus, considering that B(0)trans- forms as zero-component of a four-vector and B(3)as space components of a four-vector: [20, Eq.(11.19)] B(0)′=γ(B(0)−β·B(3)), (23) B(3)′=B(3)+γ−1 β2(β·B(3))β−γβB(0), (24) we ﬁnd from (22) that the relation between transverse and lon gitudinal modes preserves its form: B(1)′×B(2)′=iB(0)′B(3)∗ ′, (25) that may be considered as a proof of the relativistic covaria nce of the B(3) model. Moreover, we used that the phase factors in the formula (7) ar e ﬁxed between the vector and axial-vector parts of the antisymemt ric tensor ﬁeld for both positive- and negative- frequency solutions if one wants to have pure real ﬁelds. Namely, B(1)= +iE(1)andB(2)=−iE(2). As we have just seen the B(3)ﬁeld in this case may be regarded as a part of a 4-vector with respect to the pure Lorentz transformations. We are now goin g to take oﬀ the abovementioned requirement and to consider the general case: /parenleftbiggB⊥ E⊥/parenrightbigg′ = Λ/braceleftbigg/parenleftbigg˜B(1) ˜E(1)/parenrightbigg e+iφ+/parenleftbigg˜B(2) ˜E(2)/parenrightbigg e−iφ/bracerightbigg = (26) = Λ/braceleftbigg/parenleftbigg˜B(1) eiα(xµ)˜B(1)/parenrightbigg e+iφ+/parenleftbigg˜B(2) −eiβ(xµ)˜B(2)/parenrightbigg e−iφ/bracerightbigg . Our formula (26) can be re-written to the formulas generaliz ing (6a) and (6b) of ref. [2] (see also above (18,19)): Bi(1)′=/parenleftigg 1 +ieiαγ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijBj(1), (27) Bi(2)′=/parenleftigg 1−ieiβγ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijBj(2). (28) One can then repeat the procedure of ref. [2] (see the short pr esentation above) and ﬁnd out that the B(3)ﬁeld may have various transformation laws when the transverse ﬁelds transform with the matrix Λ wh ich can be extracted from (12,13). Since the Evans-Vigier ﬁeld is de ﬁned by the6 VALERI V. DVOEGLAZOV formula (3) we again search the transformation law for the cr oss product of the transverse modes/bracketleftig B(1)×B(2)/bracketrightig′=? with taking into account (27,28). /bracketleftig B(1)×B(2)/bracketrightigi′= (29) =iγB(0)/braceleftigg/bracketleftigg 1−eiα+eiβ 2(iβ·ˆk)/bracketrightigg (1 +γ2(β2−(S·β)2) γ+ 1)ijBj(3)+ +ieiα−eiβ 2(S·β)ijBj(3)−γB(0)/bracketleftigg ieiα+eiβ 2+ei(α+β)(β·ˆk)/bracketrightigg ijβj  . We used again the Deﬁnition 3 thatB(3)=B(0)ˆk. One can see that we recover the formula (8) of ref. [2] (see (22 ) above) when the phase factors are equal to α=−π/2,β=−π/2. In the case α= +π/2 and β= +π/2, the sign of βis changed to the opposite one.3 But, we are able to obtain the transformation law as for antis ymmetric tensor ﬁeld, for instance when α=−π/2,β= +π/2.4Namely, since under this choice of the phases B(1)′×B(2)′=iγ/bracketleftig B(0)/bracketrightig2/parenleftigg ˆk−γβ(β·ˆk) γ+ 1+ (iˆiβy−iˆjβx)/parenrightigg , (30) the formula (30) and the formula for opposite choice of phase s lead precisely to the transformation laws of the antisymmetric tensor ﬁeld s: /bracketleftig Bi(3)/bracketrightig′=/parenleftigg 1±γ(S·β) +γ2 γ+ 1(S·β)2/parenrightigg ijBj(3). (31) B(0)is a true scalar in such a case. What are reasons that we introduced additional phase factor s in Hel- moltz bivectors? In [21] a similar problem has been consider ed in the (1/2,0)⊕(0,1/2) (cf. also [7, 22]). Ahluwalia identiﬁed additional phase factor(s) with Higgs-like ﬁelds and proposed some relation s with a grav- itational potential. However, the Eﬁeld under deﬁnitions ( α=−π/2, β= +π/2) becomes to be pure imaginary. One can also propose a model with the corresponding introduction of phase factors in suc h a way that 3By the way, in all his papers Evans used the choice of phase fac tors incompatible with the B-Cyclic Theorem in the sense that notall the components are entries of anti- symmetric tensor ﬁelds therein. This is the main one but not t he sole error of the Evans papers and books. 4In the case α= +π/2 and β=−π/2, the sign in the third term in parentheses (formula (30) is changed to the opposite one.WHAT IS THE EVANS-VIGIER FIELD? 7 B⊥to be pure imaginary. Can these transverse ﬁelds be observab le? Can the phase factors be observable? A question of experimental possibility of detection of this class of antisymmetric tensor ﬁelds (in fa ct, of the anti- hermitian modes on using the terminology of quantum optics) is still open. One should still note that several authors discussed recent ly unusual con- ﬁgurations of electromagnetic ﬁelds [23, 24]. Let us now look for relations with old formalisms. The equati ons (10) of [4] is read fµν(p)∼[ǫ(1) µ(p)ǫ(2) ν(p)−ǫ(1) ν(p)ǫ(2) µ(p)] (32) for antisymmetric tensor fµνexpressed through cross product of polariza- tion vectors in the momentum space. This is a generalized cas e comparing with the Evans-Vigier Deﬁnition 2 which is obtained if one restricts oneself by space indices. The dynamical equations in the Ogievetski˘ ı-Polubarinov a pproach are 2fµν−∂µ∂λfλν+∂ν∂λfλµ=Jµν, (33) and the new Kalb-Ramond gauge invariance is deﬁned with resp ect to trans- formations δfµν=∂µλν−∂νλµ. (34) It was proven that the Ogievetski˘ ı-Polubarinov equations are related to the Weinberg 2(2 j+ 1) formalism [25, 26] and [7b-f,i]. Furthermore, they [4] also claimed “In the limit m→0 (orv→c) the helicity becomes a relativistic invariant, and the concept of spin loses its meaning. The system of 2 s+ 1 states is no longer irreducible; it decom- poses and describes a set of diﬀerent particles with zero mas s and helicities ±s,±(s−1),...±1,0 (for integer spin and if parity is conserved; the situ- ation is analogous for half-integer spins)5.” In fact, this hints that actually the Proca-Duﬃn-Kemmer j= 1 theory has twomassless limit, a) the well- known Maxwell theory and b) the notoph theory ( h= 0). The notoph theory has been further developed by Hayashi [5] in the conte xt of dila- ton gravity, by Kalb and Ramond [6] in the string context. Hun dreds (if not thousands) papers exist on the so-called Kalb-Ramond ﬁe ld (which is actually the notoph ), including some speculations on its connection with Yang-Mills ﬁelds. 5Cf. with [27]. I am grateful to an anonymous referee of Physics Essays who sug- gested to look for possible connections. However, the work [ 27] does not cite the previous Ogievetski˘ ı-Polubarinov statement.8 VALERI V. DVOEGLAZOV In [28] I tried to use the Ogievetski˘ ı-Polubarinov deﬁniti ons of fµν(see (32)) to construct the “potentials” fµν. We can obtained for a massive ﬁeld fµν(p) =iN2 m 0 −p2 p1 0 p2 0 m+prpl p0+mp2p3 p0+m −p1−m−prpl p0+m0 −p1p3 p0+m 0 −p2p3 p0+mp1p3 p0+m0 , (35) This tensor coincides with the longitudinal components of t he antisym- metric tensor obtained in refs. [9a,Eqs.(2.14,2.17)] (see also below and [7i, Eqs.(16b,17b)]) within normalizations and diﬀerent forms of the spin basis. The longitudinal states reduce to zero in the massless case u nder appro- priate choice of the normalization and only if a j= 1 particle moves along with the third axis OZ.6Finally, it is also useful to compare Eq. (35) with the formula (B2) in ref. [29] in order to realize the correct p rocedure for taking the massless limit. Thus, the results (at least in a mathematical sense) surpris ingly depend on a) the normalization; b) the choice of the frame of referen ce. In the Lagrangian approach we have LProca=−1 4FµνFµν+m2 2AµAµ=⇒ LMaxwell(m→0), (36) and L=−1 2FµFµ+m2 4fµνfµν=⇒ LNotoph=−1 2FµFµ(m→0),(37) where Fµ=i 2ǫµναβ∂βfνα=∂β˜fµβ(38) (if one applies the duality relations). Thus, we observe tha t a) it is impor- tant to consider the parity matters (the dual tensor has diﬀe rent parity properties); b) we may look for connections with the dual ele ctrodynam- ics [30]. The above surprising conclusions induced me to start form th e ba- sic group-theoretical postulates in order to understand th e origins of the Ogievetski˘ ı-Polubarinov-Evans-Vigier results. The set of Bargmann-Wigner equations, ref [31] for j= 1 is written, e.g., ref. [32] [iγµ∂µ−m]αβΨβγ(x) = 0 , (39) [iγµ∂µ−m]γβΨαβ(x) = 0 , (40) 6There is also another way of thinking: namely, to consider “u nappropriate” normal- ization N= 1 and to remove divergent part (in m→0) by a new gauge transformation.WHAT IS THE EVANS-VIGIER FIELD? 9 where one usually uses Ψ{αβ}=mγµ αδRδβAµ+1 2σµν αδRδβFµν, (41) In order to facilitate an analysis of parity properties of th e corresponding ﬁelds one should introduce also the term ∼(γ5σµνR)αβ˜fµν. In order to understand normalization matters one should put arbitrary (dimensional, in general) coeﬃcients in this expansion or in deﬁnitions of the ﬁelds and 4-potentials [28]. The Rmatrix is R=/parenleftbiggiΘ 0 0−iΘ/parenrightbigg ,Θ =−iσ2=/parenleftbigg0−1 1 0/parenrightbigg . (42) Matrices γµare chosen in the Weyl representation, i.e.,γ5is assumed to be diagonal. The reﬂection operator Rhas the properties RT=−R , R†=R=R−1, (43) R−1γ5R= (γ5)T, (44) R−1γµR=−(γµ)T, (45) R−1σµνR=−(σµν)T. (46) They are necessary for the expansion (41) to be possible in su ch a form, i.e., in order the γµR,σµνRand (if considered) γ5σµνRto besymmetrical matrices. I used the expansion which is similar to (41) Ψ{αβ}=γµ αδRδβFµ+σµν αδRδβFµν, (47) and obtained ∂αFαµ+m 2Fµ= 0, (48) 2mFµν=∂µFν−∂νFµ. (49) If one renormalizes Fµ→2mAµorFµν→1 2mFµνone obtains “textbooks” Proca equations. But, physical contents of the massless limits of these equa- tions may be diﬀerent . Let us track origins of this conclusion in detail. If one advo cates the following deﬁnitions [33, p.209] ǫµ(0,+1) = −1√ 2 0 1 i 0 , ǫµ(0,0) = 0 0 0 1 , ǫµ(0,−1) =1√ 2 0 1 −i 0 (50)10 VALERI V. DVOEGLAZOV and (/hatwidepi=pi/\|p\|,γ=Ep/m), ref. [33, p.68] or ref. [34, p.108], ǫµ(p,h) =Lµ ν(p)ǫν(0,h), (51) L0 0(p) =γ , Li 0(p) =L0 i(p) =/hatwidepi/radicalig γ2−1, (52) Li k(p) =δik+ (γ−1)/hatwidepi/hatwidepk (53) for the ﬁeld operator of the 4-vector potential, ref. [34, p. 109] or ref. [35, p.129]7,8 Aµ(x) =/summationdisplay h=0,±1/integraldisplayd3p (2π)31 2Ep/bracketleftig ǫµ(p,h)a(p,h)e−ip·x+ (ǫµ(p,h))cb†(p,h)e+ip·x/bracketrightig , (54) the normalization of the wave functions in the momentum repr esentation is thus chosen to the unit, ǫ∗ µ(p,h)ǫµ(p,h) =−1.9We observe that in the massless limit all deﬁned polarization vectors of the momen tum space do not have good behaviour; the functions describing spin-1 pa rticles tend to inﬁnity. This is not satisfactory, in my opinion, even thoug h one can still claim that singularities may be removed by rotation and/or c hoice of a gauge parameter. After renormalizing the potentials, e. g.,ǫµ→uµ≡mǫµ we come to the ﬁeld functions in the momentum representation : uµ(p,+1) = −N√ 2m pr m+p1pr Ep+m im+p2pr Ep+m p3pr Ep+m , uµ(p,−1) =N√ 2m pl m+p1pl Ep+m −im+p2pl Ep+m p3pl Ep+m , (55) 7Remember that the invariant integral measure over the Minko wski space for physical particles is/integraldisplay d4pδ(p2−m2)≡/integraldisplay d3p 2Ep, E p=/radicalbig p2+m2. Therefore, we use the ﬁeld operator as in (54). The coeﬃcient (2π)3can be considered at this stage as chosen for convenience. In ref. [33] the fact or 1/(2Ep) was absorbed in creation/annihilation operators and instead of the ﬁeld op erator (54) the operator was used in which the ǫµ(p, h) functions for a massive spin-1 particle were substituted b y uµ(p, h) = (2 Ep)−1/2ǫµ(p, h), which may lead to confusions in searching massless limits m→0 for classical polarization vectors. 8In general, it may be useful to consider front-form heliciti es (or “time-like” polariza- tions) too. But, we leave a presentation of a rigorous theory of this type for subsequent publications. 9The metric used in this paper gµν= diag(1 ,−1,−1,−1) is diﬀerent from that of ref. [33].WHAT IS THE EVANS-VIGIER FIELD? 11 uµ(p,0) =N m p3 p1p3 Ep+m p2p3 Ep+m m+p2 3 Ep+m , (56) (N=mandpr,l=p1±ip2) which do not diverge in the massless limit. Two of the massless functions (with h=±1) are equal to zero when the particle, described by this ﬁeld, is moving along the third a xis (p1=p2= 0, p3/negationslash= 0). The third one ( h= 0) is uµ(p3,0)\|m→0= p3 0 0 p2 3 Ep ≡ Ep 0 0 Ep , (57) and at the rest ( Ep=p3→0) also vanishes. Thus, such a ﬁeld operator describes the “longitudinal photons” which is in complete a ccordance with the Weinberg theorem B−A=hfor massless particles (let us remind that we use the D(1/2,1/2) representation). Thus, the change of the normaliza- tion can lead to the change of physical content described by t he classical ﬁeld (at least, comparing with the well-accepted one). Of co urse, in the quantum case one should somehow ﬁx the form of commutation re lations by some physical principles.1 If one uses the dynamical relations on the basis of the consid eration of polarization vectors one can ﬁnd ﬁelds: B(+)(p,+1) = −iN 2√ 2m −ip3 p3 ipr = +e−iα−1B(−)(p,−1),(58) B(+)(p,0) =iN 2m p2 −p1 0 =−e−iα0B(−)(p,0), (59) B(+)(p,−1) =iN 2√ 2m ip3 p3 −ipl = +e−iα+1B(−)(p,+1), (60) and E(+)(p,+1) = −iN 2√ 2m Ep−p1pr Ep+m iEp−p2pr Ep+m −p3pr E+m = +e−iα′ −1E(−)(p,−1),(61) 1I amverygrateful to the anonymous referee of my previous papers (“Fo undation of Physics”) who suggested to ﬁx them by requirements of the dim ensionless nature of the action (apart from the requirements of the translational an d rotational invariancies).12 VALERI V. DVOEGLAZOV E(+)(p,0) =iN 2m −p1p3 Ep+m −p2p3 Ep+m Ep−p2 3 Ep+m =−e−iα′ 0E(−)(p,0), (62) E(+)(p,−1) =iN 2√ 2m Ep−p1pl Ep+m −iEp−p2pl Ep+m −p3pl Ep+m = +e−iα′ +1E(−)(p,+1),(63) where we denoted, as previously, a normalization factor app earing in the deﬁnitions of the potentials (and/or in the deﬁnitions of th e physical ﬁelds through potentials) as N.E(p,0) and B(p,0) coincide with the strengths obtained before by diﬀerent method [9a,28], see also (35). B±(p,0t) = E±(p,0t) = 0 identically. So, we again see a third component of anti- symmetric tensor ﬁelds in the massless limit which is depend ent on the normalization and rotation of the frame of reference. However, the claim of the pure“longitudinal nature” of the antisym- metric tensor ﬁeld and/or “Kalb-Ramond” ﬁelds after quanti zation still requires further explanations. As one can see in [5] for a the ory with L=−1 8FµFµthe application of the condition ( A(+) ij(x),j)\|Ψ>= 0 (in our notation ∂µfµν= 0), see the formula (18a) therein, leads to the above conclusion. Transverse modes are eliminated by a new “gauge ” transforma- tions. Indeed, the expanded lagrangian is LH=1 4(∂µfνα)(∂µfνα)−1 2(∂µfνα)(∂νfµα) = =−1 4L2(2j+1)+1 2(∂µfαµ)(∂νfαν). (64) Thus, the Ogievetski˘ ı-Polubarinov-Hayashi Lagrangian i s equivalent to the Weinberg’s Lagrangian of the 2(2 j+ 1) theory [36] and [7a-e],2which is constructed as a generalization of the Dirac Lagrangian for spin 1 (instead of bispinors it contains bivectors ). In order to consider a massive theory (we insist on making the massless limit in the end of calculation s, for physical quantities) one should add +1 4m2fµνfµνas in (37). The spin operator of the massive theory, which can be found on the basis of the N¨ other formalism, is Jk=1 2ǫijkJij= (65) =ǫijk/integraldisplay d3x/bracketleftig f0i(∂µfµj) +fj µ(∂0fµi+∂µfi0+∂if0µ)/bracketrightig =m 2/integraldisplay d3x˜E×˜A 2The formal diﬀerence in Lagrangians does not lead to physica l diﬀerence. Hayashi said that this is due to the possiblity of applying the Fermi m ethod mutatis mutandis .WHAT IS THE EVANS-VIGIER FIELD? 13 In the above equations we applied dynamical equations as usu al. Thus, it becomes obvious, why previous authors claimed the purelongitudinal na- ture of massless antisymmetric tensor ﬁeld after quantizat ion, and why the application of the generalized Lorentz condition leads to equating the spin operator to zero.3But, one should take into account the normalization is- sues. An additional mass factor in the denominator may appear a) af ter “re-normalization” L → L /m2(if we want to describe long-range forces an antisymmetric tensor ﬁeld must have dimension [ energy ]2in the c= ¯h= 1 unit system, and potentials, [ energy ]1in order the corresponding action would be dimensionless; b) due to appropriate change of the c ommutation relations for creation/annihilation operators of the high er-spin ﬁelds (in- cluding ∼1/m); c) due to divergent terms in E,B,Ainm→0 under certain choice of N. Thus, one can recover usual quantum electrodynamics even if we use ﬁelds (not potentials) as dynamical variables . The conclusions are: −While ﬁrst experimental veriﬁcations gave negative result s, the B(3) construct is theoretically possible, if one develops it in a mathematically correct way ; −TheB(3)model is a relativistic covariant model. It is compatible wi th the Relativity Theory. The B(3)ﬁeld may be a part of the 4-potential vector, or (if we change connections between parts of Helmol tz bivec- tor) may be even a part of antisymmetric tensor ﬁeld; −TheB(3)model is based on deﬁnitions which are particular cases of the previous considerations of Ogievetski˘ ı and Polubarin ov, Hayashi and Kalb and Ramond; −The Duﬃn-Kemmer-Proca theory has two massless limits that s eems to be in contradictions with the Weinberg theorem ( B−A=h); −Antisymmetric tensor ﬁelds after quantization maydescribe particles of both helicity h= 0 and h=±1 in the massless limit. Surprisingly, the physical content depends on the normalization issues an d on the choice of the frame of reference (in fact, on rotations). Acknowledgments. I am thankful to Profs. A. Chubykalo, E. Co- may, L. Crowell, G. Hunter, Y. S. Kim, organizers and partici pants of the Vigier2K , referees and editors of various journals for valuable disc ussions. I acknowledge many internet communications of Dr. M. Evans ( 1995-96) on the concept of the B(3)ﬁeld, while frequently do notagree with him in many particular questions. I acknowledge discussions (199 3-98) with Dr. D. Ahluwalia (even though I do not accept his methods in science ). 3It is still interesting to note that division of total angula r momentum into orbital part and spin part is notgauge invariant.14 VALERI V. DVOEGLAZOV I am grateful to Zacatecas University for a professorship. T his work has been supported in part by the Mexican Sistema Nacional de Inv estigadores and the Programa de Apoyo a la Carrera Docente. References 1. Evans M. W. (1992) Physica B182, 227; ibid. 237; (1993) Modern Non-linear Optics. [Series Adv. Chem. Phys. Vol. 85(2)], Wiley Interscience, NY; (1994-1999) in Evans M. W., Hunter G., Jeﬀers S., Roy S. and Vigier J.-P. (eds.), The Enigmatic Photon. Vols. 1-5 , Kluwer Academic Publishers, Dordrecht. 2. Dvoeglazov V. V. (1997) Found. Phys. Lett. 10, 383. This paper presents itself a comment on the debates between E. Comay and M. Evans and it cri ticizes both authors. 3. Dvoeglazov V. V. (2000) Found.Phys. Lett. 13, 387. 4. Ogievetski˘ ı V. I. and Polubarinov I. V. (1966) Yad. Fiz. 4, 216 [Translation: (1967) Sov. J. Nucl. Phys. 4, 156]. 5. Hayashi K. (1973) Phys. Lett. B44, 497. 6. Kalb M. and Ramond P. (1974) Phys. Rev. D9, 2273. 7. Dvoeglazov V. V. (1994) Rev. Mex. Fis. Suppl. 40, 352; (1997) Helv. Phys. Acta 70, 677; ibid. 686; ibid. 697; (1998) Ann. Fond. L. de Broglie 23, 116; (1998) Int. J. Theor. Phys. 37, 1915; (1999) ibid. 38, 2259; (1998) Electromagnetic Phenomena 1, 465; (2000) Czech. J. Phys. 50, 225; (2000) in V. V. Dvoeglazov (ed.), Photon:Old Problems in Light of New Ideas , Nova Science Publishers, Huntington. 8. Sankaranarayanan A. and Good R. H., jr. (1965) Nuovo Cim. 36, 1303; (1965) Phys. Rev.140, B509; Sankaranarayanan A. (1965) Nuovo Cim. 38, 889. 9. Ahluwalia D. V. and Ernst D. J. (1993) Int. J. Mod. Phys. E 2, 397; Ahluwalia D. V., Johnson M. B. and Goldman T. (1993) Phys. Lett. B316, 102. 10. Oppenheimer J. R. (1931) Phys. Rev. 38, 725. 11. Good R. H., Jr. (1957) Phys. Rev. 105, 1914 (see p. 1915); (1959) in Lectures in theoretical physics. University of Colorado. Boulder , Interscience, p. 30 (see p. 47); Nelson T. J. and Good R. H., Jr. (1969) Phys. Rev. 179, 1445 (see p. 1446). 12. Gianetto E. (1985) Lett. Nuovo Cim. 44, 140; (1985) ibid. 145. See pp. 142 and 147, respectively. E. Recami et al. were very close to re-discover this solution in the papers (1974) Lett. Nuovo Cim. 11, 568 and (1990) in M. Mijatovic (ed.), Hadronic mechanics and nonpotential interactions , Nova Science Pubs., New York, p. 231 but in the analysis of the determinant of the Maxwell equations the y put additional constraint of transversality p·ψ= 0 on the solutions of Eq. (1,2). 13. Ahluwalia D. V. and Ernst D. J. (1992) Mod. Phys. Lett. A7, 1967; Ahluwalia D. V. (1996) in G. Hunter et al.(eds.), Proceedings of The Present Status of Quantum Theory of Light: A Symposium to Honour Jean-Pierre Vigier. York Uni versity, Toronto, Aug. 27-30, 1995 , Kluwer Academic Publishers, p. 443. 14. Dvoeglazov V. V., Tyukhtyaev Yu. N. and Khudyakov S. V. (1 994)Izv. VUZ: ﬁz. 37, 110 [Translation: (1994) Russ. Phys. J. 37, 898]. This is a version of the Saratov University preprint of 1992, which was revised in order to in clude references to the works appeared in 1991-1994. 15. Barron L. D. (1993) Physica B190, 307; Lakhtakia A. (1993) Physica B191, 362; (1995) Found. Phys. Lett. 8, 183; Grimes D. M. (1993) Physica B191, 367. 16. Comay E. (1996) Chem. Phys. Lett. 261, 601; (1996) Physica B222, 150; (1997) Found. Phys. Lett. 10, 245; (1997) Physica A242, 522; (1999) Apeiron 6, 233. 17. Hunter G. (1999) Chem. Phys. Lett. 242, 331; (2000) Apeiron 7, 17. 18. Akhtar Raja M. Y., Sisk W. N., Yousaf M. and Allen D. (1995) Appl. Phys. Lett. 67, 2123; (1997) Appl. Phys. B64, 79. 19. Dvoeglazov V. V. et al(1997) Apeiron 4, 45.WHAT IS THE EVANS-VIGIER FIELD? 15 20. Jackson J. D. (1980) Electrodin´ amica Cl´ asica. Spanish edition, Alhambra S. A. 21. Ahluwalia D. V. (1998) Mod. Phys. Lett. A13, 3123; see also V. V. Dvoeglazov’s comment on this paper, to be published. 22. Dvoeglazov V. V. (1997) in J. Keller and Z. Oziewicz (eds. ),Advances in Cliﬀord Al- gebras – Proceedings of the Int. Conference on the Theory of t he Electron. Cuautitlan, Mexico, Sept. 27-29, 1995. Vol. 7C, UNAM, M´ exico, pp. 303-319. 23. In (1993) A. Lakhtakia (ed.), Essays on the Formal Aspects of Electromagnetic Theory , World Scientiﬁc, Singapore. 24. In (1995) T. W. Barrett and D. M. Grimes (eds.), Advanced Electromagnetism: Foundations, Theory and Applications , World Scientiﬁc, Singapore. 25. S. Weinberg, Phys. Rev. B 133(1964) 1318; ibid B 134(1964) 882; ibid 181(1969) 1893. 26. Dvoeglazov V. V. (1997) Weinberg Formalism and New Looks at the Electromag- netic Theory, in J.-P. Vigier et al. (eds.), The Enigmatic Photon. Vol. IV , Kluwer Academic Publishers, Dordrecht, Chapter 12, and reference s therein. 27. Kirchbach M. (2000) Rarita-Schwinger Fields without Au xiliary Conditions in Baryon Spectra, in A. Chubykalo, V. Dvoeglazov et al. (eds.), Lorentz Group, CPT and Neutrinos. Proceedings of the International Workshop , World Scientiﬁc, Singa- pore, pp. 212-223. 28. Dvoeglazov V. V. (1998) Photon-Notoph Equations, physi cs/9804010, to be pub- lished. 29. Ahluwalia D. V. and Sawicki M. (1993) Phys. Rev. D47, 5161. 30. Strazhev V. I. and Kruglov S. I. (1977) Acta Phys. Polon. B8, 807; Strazhev V. I. (1978) ibid. 9, 449; (1977) Int. J. Theor. Phys. 16, 111. 31. Bargmann V. and Wigner E. P. (1948) Proc. Nat. Acad. Sci. (USA) 34, 211. 32. Luri´ e D. (1968) Particles and Fields , Interscience Publisher. New York, Chapter 1. 33. Weinberg S. (1995) The Quantum Theory of Fields. Vol. I. Foundations , Cambridge University Press, Cambridge. 34. Novozhilov Yu. V. (1975) Introduction to Elementary Particle Theory , Pergamon Press, Oxford. 35. Itzykson C. and Zuber J.-B. (1980) Quantum Field Theory , McGraw-Hill Book Co., New York. 36. Dvoeglazov V. V. (1993) Hadronic Journal 16, 459.
ddB axial length lfield transition regioncenterline 2hNull B fieldDIPOLE SEPTUM MAGNET IN THE FAST KICKER SYSTEM FOR MULTI-AXIS ADVANCED RADIOGRAPHY* L. Wang, S. M. Lund, B. R. Poole, LLNL, Livermore, CA 94550, USA * The work was performed under the auspices of the U. S. Department of Energy by University of California LawrenceLivermore National Laboratory under contract No.W-7405-Eng-48Abstract Here we present designs for a static septum magnet with two adjacent apertures where ideally one aperture has a uniform dipole field and the other zero field. Two designs are considered. One is a true septum magnet with a thin layer of coils and materials separating the dipole field region from the null field region. During the beam switching process, the intense electron beam will spray across this material septum leading to concerns on beam control, vacuum quality, radiation damage, etc. due to the lost particles. Therefore, another configuration without a material septum is also considered. With this configuration it is more difficult to achieve high field quality near the transition region. Shaped shims are designed to limit the degradation of beam quality (emittance growth). This approach is closely related to a previous septum magnet design with two oppositely oriented dipole field regions presented by the authors [1]. Simulations are performed to obtain the magnetic field profile in both designs. A PIC simulation is used to transport a beam slice consisting of several thousand particles through the magnet to estimate emittance growthin the magnet due to field non-uniformity. 1 INTRODUCTION A linear induction accelerator based X-ray technology can provide time-resolved, 3-D radiography capabilities fora hydrodynamic event. A kicker system, which includes a stripline dipole kicker and a dipole septum magnet, is a key component of this technology [2]. The kicker system cleaves a series of intense electron beam micropulses, and steers the beam into separate beam transport lines to achieve multiple lines of sight on target. The first part ofthis fast kicker system is a high current stripline dipole kicker that is only capable of imparting a small angular bend to the beam centroid. This is followed by a static field dipole septum magnet that increases the angular separation of the centroids and steers them into separate transport lines. The ideal "box" geometry of the septum magnet for our application is shown in Figure 1. There are two adjacent apertures where ideally one aperture has a uniform dipole magnetic field and the other zero field, separated by a thin septum in the field transition region. The two beamsemerging from the kicker are incident on the magnet a radial distance d from the centerline and are further separated from each other by the dipole field as the beams traverse the axial length l of the magnet. Given the beam energy, and the incident and exiting angles of the beam at the dipole septum magnet, the magnetic field needed to provide the desired bend can be calculated. Assuming that the effect of the fringe field is negligible, the required dipole magnetic field B is related to the beam energy Eb and the incident and exiting beam angles θi and θf by Bmc eE mcE mcbb fi l=  +  −[] 2222 sin sinθθ . (1) where m and e are the mass and charge of an electron, and c is the speed of light in vacuum. On traversing the magnet, the beam centroids will move radially a distance ∆=−[] −[]lcos cos sin sinθθ θθif fi . (2) Fig. 1: Ideal box geometry of the dipole septum magnet. 2 SEPTUM MAGNET DESIGN The geometric parameters for the magnet designs are an axial length l=50cm, incident and exiting beam angles of θi=1o andθf=22 5.o. This results in a bending field of B≈500Gauss for the beam energy of Eb≈20MeV. The incident beam radius ( )rb and centroid separation ()2d at the septum magnet are 0.7 cm and 5.6 cm, respectively. The beam centroid will displace a horizontal distance ∆~.104cm while traversing the magnet. The first design considered is a true septum magnet with a thin layer of coils and materials separating the dipolefield region from the null field region. Figure 2 shows the 2-D geometry of this septum magnet. The beam goes through the aperture ( ) 2h of 10 cm. The amp-turns of each coil for the needed B = 500 Gauss field is NI Bh≈/~µ01990 amp-turns, where µ0 is the free space permittivity, and N the number of turns of the coil. Figure 2: Geometry of a dipole septum magnet with a thin coil separating dipole and null field regions. During the beam switching process, the intense electron beam will spray across this material septum leading to particle loss and concerns on beam control, vacuum quality, radiation damage, etc. Therefore, another configuration without a material septum is also considered. The schematic of the second design is shown in Figure 3. It essentially consists of two "C" type dipole magnets brought into close radial proximity. With this configuration it is more difficult to achieve high field quality near the field transition region. The rapidity of thetransition between the dipole field and null field regions depends critically on hd/ , the ratio of magnet half-gap h relative to the incident centroid displacement d. The ratio drb/ , where rb is the beam radius, must also be sufficiently large so that the incident beam does not enter the field transition region. Achievable d is limited by the fast kicker technology and the maximum fast kicker to septum magnet drift distance. Thus we need h to be as small as possible in order to obtain good field quality near the field transition region. However, h has to be large enough to allow beam transport through the magnet aperture with negligible scraping over the full range of machine operating parameters. To enhance field linearity and limit the degradation of beam quality (emittance growth), shims are designed and utilized near the field transition region. 3 MAGNETIC FIELD SIMULATIONS The magnetic field in the dipole septum magnet was simulated using the 2-D magnetostatic Poisson code. Only the upper half of the symmetric structure wassimulated. Effects due to the iron saturation over the full possible range of coil excitation are negligible. Figure 4 displays the field contours for the dipole septum magnet which has a thin layer of coils in the middle. There is ahigh quality dipole field on the left side as indicated by the straight field lines. Figure 3: Schematic of a dipole septum magnet which does not have a material septum separating the dipole and the null field regions. Figure 4: Simulated field lines in the half aperture of the dipole septum magnet with a material septum. The vertical magnetic field on the midplane (y = 0) of the magnet as a function of x is plotted in Figure 5. The predicted magnetic field on the left side is about 500 Gauss as expected. The magnetic field goes from the maximum to a small value when crossing the center septum coil. The electron beam entering the dipole field side of the magnet experiences a uniform field when it traverses the magnet. The simulation result also shows that there is a small leakage field in the null field region. With the use of a thin (3 mm) µ−metal sheet enclosure, the magnitude of the magnetic field leakage is reduced to less than 0.2 Gauss. -20 -15 -10 -5 0 5 10 15 20050100150200250300350400450500550 x (cm)By (Gauss) Figure 5: Vertical magnetic field on the midplane of the material septum magnet . The simulated field contours for the magnet without a material septum separating the dipole field region from the null field region is shown in Figure 6. The semicircles in the figure illustrate the positions of the incident electron beams emerging from the kicker. Thecoils coilscoils shims xyµ−metal enclosureµ−metal enclosurecurved field lines near the field transition region indicate non-uniform fields. With this configuration, both beams experience non-uniform field which can lead to emittance growth. Compared to the magnet with a material septum, it is more difficult to achieve high field quality near the transition region with this design. Figure 6: Simulated field lines in the half aperture of the dipole septum magnet without a material septum. To enhance field linearity near the transition region, shims of trapezoidal shape are designed. From Figure 7, note that the field lines are straighter near the incident beam position in the dipole field region because of the shims. In addition, a µ−metal sheet enclosure can also be utilized to reduce the leakage field in the null field region. The effect of the shim is further illustrated in Figure 8, where the vertical magnetic field By on the midplane ( y=0) of the septum magnet is plotted. The solid curve represents the magnetic field for the case where no shim nor µ−metal sheet enclosure is used. It takes more than 10 cm for the magnetic field to go from the maximum value to a small number. The leakage field in the null field region is about 80 Gauss at the edge of the incident beam. With the use of the µ−metal sheet enclosure (dashed curve), the field drops to less than 0.1 Gauss at that location. The third curve represents the case where both shims and a µ−metal sheet enclosure are utilized. The curve shows that the shim reduces the undesirable variation in field at the beam edge and centroid for the beam in the dipole field region, allowing the electron beam to experience a more uniform field when it traverses the magnet. 4 EMITTANCE GROWTH SIMULATIONS To determine beam quality for the various magnet designs, a PIC code was utilized to examine the emittance growth in the magnet [3]. A 20 MeV beam slice wastransported through the magnetic field region using the2-D field map for the structure. The results show that there is no emittance growth when the electron beams go through the material dipole septum magnet with a thin layer of coils separating the dipole field region from the null field region. For the design with no material septum, there is no emittance growth when the beam traverses through the null field region and 6% emittancegrowth when the beam goes through the dipole field region. This emittance growth can be further reduced by refining the design of the shims. These simulations do not take into account the non-ideal effects (vacuum degradation, plasma, etc.) resulting from the particle loss on septum materials. Figure 7: Simulated field lines in the half aperture of the dipole septum magnet without material septum. Shims are used to make fields more uniform in transition region. -20 -15 -10 -5 0 5 10 15 200100200300400500600By (Gauss)no shims, no mu-metal enclosure no shims, with mu-metal sheet enclosure with shims and mu-metal sheet enclosure Figure 8: Vertical magnetic field By on the midplane ( y=0) of the septum magnet without a material septum. 5 CONCLUSION Two designs of a dipole septum magnet for the use in multi-axis advanced radiography are presented. One design avoids the use of a material septum. To improve the field quality near the transition region in this design, shaped shims are designed. Simulations are performed to estimatethe emittance growth in the magnet for both designs. 6 ACKNOWLEDGEMENT The authors wish to thank G. Caporaso, Y.-J. Chen, and G. Westenskow for helpful technical suggestions. REFERENCES [2] L. Wang, et. al., “A Prototype Dipole Septum Magnet for Fast High Current Kicker Systems”, PAC 99, New York, March 29- April 2, 1999. [2] Y.J. Chen, et. al., “Precision Fast Kicker for Kiloampere Electron Beams”, PAC 99, New York, March 29- April 2, 1999. [3] B.R. Poole, et. al., “Analysis and Modeling of a Stripline Beam Kicker and Septum”, LINAC 98, Chicago, IL, August 23-28, 1998.incident beams xy incident beams xy
arXiv reference: physics/0010006 Relativistic angle -changes and frequency -changes Eric Baird (eric_baird@compuserve.com) We generate a set of "relativistic" predictions for the relationship between viewing angle and apparent frequency, for each of three different non -transverse shift eq uations. We find that a detector aimed transversely (in the lab frame) at a moving object should report a redshift effect with two out of the three equations. 1. Introduction One of the advantages of Einstein’s special theory [1] over other contemporary models was that it provided clear and comparatively straightforward predictions for angle -changes and frequency changes at different angles. These calculations could be rather confusing in the variety of aether models that were available at the time ( see e.g. Lodge, 1894 [2] ). In this paper, we derive the relativistic angle - changes and wavelength -changes that would be associated with each of the three main non - transverse Doppler equations, if used as the basis of a relativistic model. 2. Geometry Wavefront shape If a “stationary” observer emits a unidirectional pulse of light and watches the progress of the spreading wavefront, the signal that they detect is not the outgoing wavefront itself, but a secondary (incoming) signal generated when the outgoing wavefro nt illuminates dust or other objects in the surrounding region. If we assumed that the speed of light was constant in the observer’s own frame, any dust - particles that the observer saw to be illuminated at the same moment would be said to lie on a spherical surface, and these “illumination - events” would be said to be simultaneous. In a different inertial frame, the observer’s spatial coordinates change while these light - signals are in flight. The description of the same light -rays and illuminated surfa ces now involves rays emanating from one spatial position, striking dust -particles, and generating secondary signals which then converge on a different spatial position. In the new frame, our original illuminated spherical surface now has to obey the condition that the round -trip flight -time from A→surface →B is the same for rays sent in any direction, and a map of a cross -section through the surface now gives us an elongated ellipse, with the emission and absorption points being the two ellipse foci. The same set of illumination -events can mark out a spherical spatial surface in one frame and a spheroidal spatial surface in the other. [3][4] Ellipse dimensions Instead of using the “fixed flat aether approach” for calculating distances in our diagram (setting the distance between foci as v and the round - trip distance as 2 c), we will instead construct our map around the wavelength -distances that must be fitted into it, regardless of whether or not these distances are expected to fit nicely into flat spaceti me. The three Doppler formulae used are: freq’ / freq = (c-v) / c … (1) freq’ / freq = ( )( )vcvc +−/ … (2) freq’ / freq = c / (c+v) … (3) , where v is recession velocity Equation (1) is often associated with emitter theories, (3) is usually associated with an absolute aether stationary in the observer’s frame, and (2) is the intermediate prediction used by special relativity (and is the root product of the other two equations). [5] Relativistic aberration and frequency -changes Eric Baird 2 October 2000 page 2 / 4 arXiv reference: physics/0010006 3. Resulting wavefront maps Ellipse proportions Each of our three Doppler equations (1), (2), (3) has the same ratio λ0°: λ180° , so all three ellipses share the same proportions and angles. Each ellipse is related to its immediate neighbours by a Lorentz magnification or reduction. Reading information from the wavefront map Each wavefront map is drawn for a specific velocity, and includes the reference -angles and wavelengths for a “stationary” object (dashed circular outline) and the corresponding angles and wavelength distances for the moving object (solid ellipse). Each ray drawn on our map then has two nominal angles (defined by where the ray hits the perimeter of each wavefront outline), and two nominal wavelengths (defined by the distance from the common focus to each wavefront outline). Thus, for special relativity at v=0.75 c (second diagram), we can immediately see that a light - ray aimed at the moving particle at 90° in the background frame will strike the front of the particle at slightly less than 45° from the particle’s path, and will ha ve a wavelength that is about two thirds of the original wavelength. Conversely, when the particle emits a ray that is received at 90° in the background frame, we can see that the wavelength is magnified about 1½ times, and in the particle’s frame was aimed rearwards at roughly 45° from the perpendicular. Precise calculations are given in section 5. 4. Special relativity Lorentz contraction The elliptical outline of the second map can be converted back into a circle of radius c by applying a Lorentz contraction in the direction of motion. The resulting diagram (above) then shows the non-relativistic Nineteenth -Century predictions for angle -changes and wavelength changes associated with an object moving through a flat stationary aether, viewed by a stationary observer (Lodge [2], fig 4, pp.739). Minkowski diagrams We can also obtain special relativity’s elliptical wavefront diagram by taking a standard Minkowski light -cone diagram, [6] and slici ng it at an angle parallel with the plane of simultaneity of an object in a different frame. Lorentz transforms can then be used to turn the “skewed” lightcone diagram back into the usual “unskewed” version. Relativistic aberration and frequency -changes Eric Baird 2 October 2000 page 3 / 4 arXiv reference: physics/0010006 5. Resulting formulae Angle changes All three diagrams produce the same angle - change relationships presented in Einstein’s 1905 paper [1] as “the law of aberration in its most general form”, )/(Acos1/ Acos'Acos cvcv −−−= Wavelength changes at all angles The wavelength -changes in r ays arriving in the laboratory frame, if angle A is measured in the lab frame (and v is positive for recession when A=0°), are: λ’/λ = 2 2/ 1Acos)/(1 cvcv −+ … (i) λ’/λ = 2 2/ 1Acos)/(1 cvcv −+ … (ii) λ’/λ = Acos)/(1 cv+ … (iii) The wavelengths generated by the second diagram are those given in section 7 of the 1905 electrodynamics paper. Wavelengths generated by (1) and (3) are simply special relativity’s values, multiplied or divided by a Lorentz term. 6. “Transverse” shift predictions Emitter ®®lab If we aim a detector transversely at a moving object (with our “ninety degrees” being measured in the detector’s frame), the predicted wavelength changes seen by the detector are: freq’/freq = 1 – v²/c² … (i) freq’/freq = 2 2/ 1 cv− … (ii) freq’/freq = 1 … (iii) As with the non -transverse predictions (1), (2), (3), special relativity’s “transverse” predictions for observed frequencies and apparent ruler - changes are the root pro duct average of the “non -relativistic” predictions made by assuming a fixed speed of light in each frame. Lab®®emitter If a signal is aimed at the moving object at 90° in the lab frame, the wavelength changes seen by the object should be: freq’/freq = 1 … (i) freq’/freq = 1 / 2 2/ 1 cv− … (ii) freq’/freq = 1 / (1 – v²/c²) … (iii) These predictions di ffer from the earlier set because of our asymmetrical decision to measure all angles in the laboratory frame . 7. Round -trip shift predictions … transverse If we aim a signal at a moving transponder and catch the retransmitted signal, then if both signals paths are transverse to the transponder path in the laboratory frame, multiplying the two previous sets of results together gives us final round -trip predictions of: freq’/freq = 1 – v²/c² … (i) freq’/freq = 1 … (ii) freq’/freq = 1/ (1 – v²/c²) … (iii) Equation (1) predicts a Lorentz -squared redshift after two lab -transverse frame transitions, and special relativity gives a null result. … non -transverse We could also aim our source and detector at the transponder non-transversely. Multiplying the approach and recession shifts gives: cv c cvc freqfreq )( ' −− −×= = 1 – v²/c² … (i) 1)()( '= ×=−+−− +− v cv c vcvc freqfreq … (ii) )(' v cc vcc freqfreq −+ +×= =1/ (1 – v²/c²) … (iii) These round -trip frequency shifts a re the same as in the previous “transverse” calculation. Relativistic aberration and frequency -changes Eric Baird 2 October 2000 page 4 / 4 arXiv reference: physics/0010006 … at any angle If we mount our detector and signal source on a turnt able, the angle that the source →detector line makes with the transponder path has no effect on the final calculated round -trip shift. If we aim two lasers collinearly into an ion beam, and measure the non -transverse frequency shifts by measuring how far the lasers need to be detuned in order to achieve resonance in the beam, [7] the relationship between the product of the shifted frequencies and the original frequency should be unaffected by any misalignment of the lasers or spread of observation angles, [8] provided that the angular error is the same for both observations. 8. Consequences Generality of the E=mc2 result In a previous paper [9] we showed that the E=mc2 mass -energy relationship can be derived from the frame -dependent total momentum of two plane waves emitted directly forwards and backwards along the direction of motion, if the non-transverse shift law is (1), with the equivalent calculation using (2) giving E=mc2 / √ (1 – v2/c2). Einstein’s 1905 “inertia” paper [10] gives a more general calcula tion for an opposing pair of plane waves tilted at any angle in the emitter -frame. Since the angle -dependent information needed to calculate the momentum components of light-rays in the first two elliptical maps differs only by a Lorentz scaling, the angle - independence of the 1905 derivation also applies to our “emitter -theory” derivation. Reworking special relativity Ellipse 3(ii) contains the same key relationships as special relativity, but without explicitly using the special theory’s assumption of flat spacetime, or by dividing the phenomena into separate “propagation” and “Lorentz” components. It is possible that other researchers may be interested in examining a class of model in which local lightspeed constancy is regulated by spacetime distortions associated with the relative motion of physical particles (e.g. [11]), but which generates the same basic shift predictions as special theory. We do not expect to see a theory based on 3(iii). 9. Conclusions The wavefront maps presented in section 3 can be us eful as a visualisation aid for relativistic problems. Investigating the properties of these maps raised three additional points: • Special relativity’s aberration formula can be calculated from general principles without committing to special relativity’s shift equations. • Transverse redshifts can also be derived from a first -order Doppler equation, although the strength of the effect is different under special relativity. • Special relativity’s key relationships can be deduced from the principle of relati vity and the “relativistic Doppler” shift equation without assuming that spacetime is flat, suggesting the possibility of a “non - flat” variation on special relativity. Although these points are probably not new, they can be difficult to derive by other means. REFERENCES [1] Alber t Einstein, "On the Electrodynamics of Moving Bodies" (1905), translated and reprinted in The Principle of Relativity (Dover, NY, 1952) pp.35 -65. [2] Oliver J. Lodge, "Aberration problems," Phil. Trans. R. S. A184 739- (1894). [3] If we assume that lightspeed is fixed in the new frame, we also conclude that different sections of the reflecting surface are illuminated at different times – if we have a moving spherical surface divided up into rings that are concentric to the motion axis, and these rings illuminate sequentially, the result is a spheroidal set of spatial coordinates for the illumination events (see: Moreau). This leads us to the concept of “the relativity of simultaneity”. [4] W. Moreau, "Wave front relativity," Am. J. Phys. 62 426-429 (1994). [5] T.M. Kalotas and A.R. Lee, “A two -line derivation of the relativistic longitudinal Doppler formula,” Am.J.Phys 58 187 -188 (Feb 1990) [6] Albert Einstein, 1921 Princeton lectures, translated and reprinted in The Meaning of Relativity 6th ed (Chapman & Hall, NY, 1952) pp.36. [7] R. Klein et.al. , “Measurement of the transverse Doppler shift using a stored relativistic 7Li+ ion beam,” Z.Phys.A 342 455-461 (1992). [8] Hirsch I. Mandelberg and Louis Witten, “Experimental Verification of the Relativistic Doppler Effect,” J.Opt.Soc.Am. 52 529-536 (1962). [9] Eric Baird, “Two exact derivations of the mass/energy relationship, E=mc2,” arXiv: physics/0009062 (2000) [10] Albert Einstein, “Does the Inertia of a Body depend on its Energy -Content?” (1 905), translated and reprinted in The Principle of Relativity (Dover, NY, 1952) pp.67 -71. [11] Eric Baird, “GR without SR: A gravitational domain description of first -order Doppler effects,” arXiv: gr-qc/9807084 (1998).
MODELING OF AN INDUCTIVE ADDER KICKER PULSER FOR DARHT-II* L. Wang, G. J. Caporaso, E. G. Cook, LLNL, Livermore, CA94550, USA * The work was performed under the auspices of the U. S. Department of Energy by University of California LawrenceLivermore National Laboratory under contract No.W-7405-Eng-48Abstract An all solid-state kicker pulser for a high current induction accelerator (the Dual-Axis Radiographic Hydrodynamic Test facility DARHT-2) has been designed and fabricated. This kicker pulser uses multiple solid state modulators stacked in an inductive-adder configuration. Each modulator is comprised of multiple metal-oxide- semiconductor field-effect transistors (MOSFETs) which quickly switch the energy storage capacitors across a magnetic induction core. Metglas is used as the core material to minimize loss. Voltage from each modulator is inductively added by a voltage summing stalk and delivered to a 50 ohm output cable. A lumped element circuit model of the inductive adder has been developed to optimize the performance of the pulser. Results for several stalk geometries will be compared with experimental data. 1 INTRODUCTION Linear induction accelerator based x-ray technology can provide time-resolved, 3-D radiography capabilities for a hydrodynamic event. A key component of this technology is a kicker system[1]. The kicker system cleaves a series of intense electron beam micropulses, and steers the beam into separate beam transport lines to achieve multiple lines of sight. The first part of this fast kicker system is ahigh current stripline dipole kicker. The original design of the pulser which drives this stripline kicker was based on planar triodes[2]. Although the performance of the pulser based on this design was very good, the availability of thehigh frequency planar triodes in the future has become a concern. This led to the development of an all solid-state kicker pulser design for the Dual-Axis Radiographic Hydrodynamic Test facility DARHT-2. The new pulser design was based on the Advanced Radiograph Machine (ARM) modulator technology [3]. It uses multiple solid- state modulators stacked in an inductive-adder configuration. Each modulator is comprised of multiple metal-oxide-semiconductor field-effect transistors (MOSFETs) which quickly switch the energy storage capacitors across a magnetic induction core. Metglas is used as the core material to minimize loss. Voltage from each modulator is inductively added by a voltage summing stalk and delivered to a 50 ohm output cable. The cross section of this solid-state kicker pulser is shown in Figure1. Only three stacked modules are shown in the figure. A picture of the fabricated solid-state kicker pulser is shownin Figure 2. Fig. 1: Cross section of the solid-state kicker pulser. Figure 2: Inductive adder kicker pulser.Transformer SecondaryMOSFET DriveCircuitMOSFETCapacitor Metglas CoreTransformer Primary2 INDUCTIVE ADDER MODEL A lumped element circuit model of the inductive adder has been developed to optimize the rise time and performance of the pulser. Figure 3 shows the lumped element equivalent circuit model of the inductive adder. Figure 3: A lumped element equivalent circuit model of the inductive adder. The capacitance C_s and the resistance R1 are 24 µF and 50 Ohms, respectively. The load resistance R_load is50 Ohms. The inductance of the transformer primary L_primary is 20.9 µH. The effective leakage inductance L_leakage is 20 nH. 3 SIMULATION RESULTS Output voltage waveforms were simulated using the circuit model for different stalk geometries. These cases were chosen based on the stalks made for the experiment. Figure 4 shows the plots of output voltage versus timewith a 50 Ohm stalk. The capacitors are charged to 650 V Figure 4: Output voltage versus time with a 50 Ω stalk.initially. With thirty modulator stacks, the total voltage should go to 19.5 kilovolts. For the simulation, the switches are closed from 10 ns to 100 ns. With a 50 Ohm stalk, L1 and C1 are 6.5 nH and 2.6 pF, respectively. If a stalk is matched to the load, we will expect the output voltage waveform to be a square wave except during the rise and fall time. The output voltage plot of the 30 th stack in Figure 4 shows that the 50 Ohm stalk doesn't match into the 50 Ohm load. The waveform indicates that the impedance of the stalk should be reduced in order to match into the load impedance. This is due to the presence of the effective leakage inductance. The plots of output voltage versus time with a 12.6 Ohm stalk areshown in Figure 5. With 12.6 Ohm stalk, L1 and C1 are 1.6 nH and 10.3 pF, respectively. There is an overshoot in the output voltage waveform in Figure 5. This indicates that in order to match into the load impedance, the impedance of the stalk should be increased . Figure 5: Output voltage versus time with a 12.6 Ω stalk. Figure 6 shows the plots of output voltage versus time with a 18.9 Ohm stalk. With 18.9 Ohm stalk, L1and C1 are 2.5 nH and 6.9 pF, respectively. The output voltage waveform indicates that this 18.9 Ohm stalk provides a better match into the load impedance compared to other available stalks. This result agrees with the experimental data. The output voltage versus time from experimental data with 50 Ohm stalk is shown in Figure7. For the experiment, the output voltage has a negative value. We can see that the output voltage waveforms are similar (except longer rise time) between the simulation result and experimental data. The output voltage versus time from the experimental data with a 12.5 Ohm stalk isshown in Figure 8. There is an overshoot in the output voltage waveform as we have also seen from the simulation result. R1 L_primary switch C_s C1 L1 L_leakage R1 L_primary switch C_s C1 L1 L_leakage V_out R1 L_primary switch C_s C1 L1 L_leakage R_load Stack 1 Stack 2 Stack 30 0n 40n 80n 120n 160n 200n -50.00 120.00 290.00 460.00 630.00 800.00 v(OUT1) T 0n 40n 80n 120n 160n 200n -0.50K 1.00K 2.50K 4.00K 5.50K 7.00K V(OUT10) T 0n 40n 80n 120n 160n 200n -5.00K -1.00K 3.00K 7.00K 11.00K 15.00K V(OUT20) T 0n 40n 80n 120n 160n 200n 0.00K 5.00K 10.00K 15.00K 20.00K 25.00K V(OUT30) Tadder30_50ohm Temperature= 27 Output voltage of the first stack Output voltage of the 10th stack Output voltage of the 20th stack Output voltage of the 30th stack0n 42n 84n 126n 168n 210n -50.00 120.00 290.00 460.00 630.00 800.00 v(OUT1) T 0n 42n 84n 126n 168n 210n -0.50K 1.00K 2.50K 4.00K 5.50K 7.00K V(OUT10) T 0n 42n 84n 126n 168n 210n -5.00K -1.00K 3.00K 7.00K 11.00K 15.00K V(OUT20) T 0n 42n 84n 126n 168n 210n 0.00K 5.00K 10.00K 15.00K 20.00K 25.00K V(OUT30) Tadder30_12p6ohm Temperature= 27 Output voltage of the first stack Output voltage of the 10th stack Output voltage of the 20th stack Output voltage of the 30th stackFigure 6: Output voltage versus time with a 18.9 Ω stalk. Figure 7: Output voltage versus time from experimental data (with a 50 Ohm stalk). 4 SUMMARY A lumped element circuit model of the inductive adder kicker pulser has been developed. Output voltage waveforms were simulated using the circuit model for different stalk geometries. Based on the estimated effective leakage inductance, the simulation results agree with the experimental data on the choice of stalk.Figure 8: Output voltage versus time from experimental data (with a 12.5 Ohm stalk). REFERENCES [1] Y.J. Chen, et. al., “Precision Fast Kicker for Kiloampere Electron Beams”, PAC 99, New York, NY, March 29- April 2, 1999. [2] W.J. DeHope, et. al., “Recent Advances in Kicker Pulser Technology for Linear Induction Accelerators”, 12th IEEE Intl. Pulsed Power Conf., Monterey, CA, June 27-30, 1999. [3] H. Kirbie, et. al., “MHz Repetition Rate Solid-State Driver for High Current Induction Accelerators”, PAC 99, New York, NY, March 29- April 2, 1999.Output voltage of the first stack Output voltage of the 10th stack Output voltage of the 20th stack Output voltage of the 30th stack0n 42n 84n 126n 168n 210n -50.00 120.00 290.00 460.00 630.00 800.00 v(OUT1) T 0n 42n 84n 126n 168n 210n -0.50K 1.00K 2.50K 4.00K 5.50K 7.00K V(OUT10) T 0n 42n 84n 126n 168n 210n -5.00K -1.00K 3.00K 7.00K 11.00K 15.00K V(OUT20) T 0n 42n 84n 126n 168n 210n 0.00K 5.00K 10.00K 15.00K 20.00K 25.00K V(OUT30) Tadder30_18p9ohm Temperature= 27
arXiv:physics/0010008 3 Oct 2000SLC□Final□Performance□and□Lessons □ Nan□Phinney #□ SLAC,□P.O.□Box□4349,□Stanford□CA□94309□□(USA)□□ Abstract□ The□ Stanford□ Linear□ Collider□ (SLC)□ was□ the□ first□ prototype□ of□ a□ new□ type□ of□ accelerator,□ the□ electro n- positron□linear□collider.□Many□years□of□dedicated□e ffort□ were□ required□ to□ understand□ the□ physics□ of□ this□ new □ technology□ and□ to□ develop□ the□ techniques□ for□ maximizing□ performance.□ Key□ issues□ were□ emittance□ dilution,□ stability,□ final□ beam□ optimization□ and□ background□ control.□ Precision,□ non-invasive□ diagnostics□ were□ required□ to□ measure□ and□ monitor□th e□ beams□ throughout□ the□ machine.□ Beam-based□ feedback□ systems□ were□ needed□ to□ stabilize□ energy,□ trajectory ,□ intensity□ and□ the□ final□ beam□ size□ at□ the□ interactio n□ point.□ A□ variety□ of□ new□ tuning□ techniques□ were□ developed□ to□ correct□ for□ residual□ optical□ or□ alignm ent□ errors.□ The□ final□ focus□ system□ underwent□ a□ series□ o f□ refinements□in□order□to□deliver□sub-micron□size□bea ms.□ It□ also□ took□ many□ iterations□ to□ understand□ the□sour ces□ of□ backgrounds□ and□ develop□ the□ methods□ to□ control□ them.□ The□ benefit□ from□ this□ accumulated□ experience□ was□seen□in□the□performance□of□the□SLC□during□its□f inal□ run□in□1997-98.□The□luminosity□increased□by□a□facto r□of□ three□ to□ 310 30 □ and□ the□ 350,000□ Z□ data□ sample□ delivered□was□nearly□double□that□from□all□previous□ runs□ combined.□ 1□ INTRODUCTION□□ The□concept□of□an□electron-positron□linear□collider □was□ proposed□as□a□way□of□reaching□higher□energy□than□wa s□ feasible□with□conventional□storage□ring□technology. □The□ SLC,□built□upon□the□existing□SLAC□linac,□was□intend ed□ as□an□inexpensive□way□to□explore□the□physics□of□the □Z 0□ boson□ while□ demonstrating□ this□ new□ technology□ [1].□ Both□ goals□ were□ much□ more□ difficult□ to□ achieve□ than □ anticipated,□ with□ the□ SLC□ only□ approaching□ design□ luminosity□ after□ten□years□of□operation.□As□the□fir st□of□ an□entirely□new□type□of□accelerator,□the□SLC□requir ed□a□ long□and□continuing□effort□to□develop□the□understan ding□ and□ techniques□ required□ to□ produce□ a□ working□ linear □ collider.□ Precision□ diagnostics,□ feedback,□ automate d□ control□ and□ improved□ tuning□ algorithms□ were□ key□ elements□ in□ this□ progress.□ In□ parallel,□ there□ was□ a n□ international□ collaborative□ effort□ to□ design□ an□ e +e-□ collider□to□reach□an□energy□of□1□Tev□or□higher□[2]. □Both□ projects□ benefited□ from□ a□ close□ interaction.□ The□ SL C□ drew□on□the□ideas□and□techniques□developed□for□a□fu ture□ machine□ while□ the□ collider□ design□ has□ been□ heavily□ influenced□by□the□experience□gained□with□the□SLC.□ _______________________□ Work□supported□by□the□U.S.□Dept.□of□Energy□under□co ntract□ DE-AC03-76SF00515□ #□Email:□ nan@slac.stanford.edu □ □ Figure□ 1:□ □ SLD□ luminosity□ showing□ the□ performance□ improvement□ from□ 1992-1998.□ The□ bars□ show□ luminosity□ delivered□ per□ week□ and□ the□ lines□ show□ integrated□ luminosity□ for□ each□ run.□ The□ numbers□ giv e□ average□polarization.□□ 2□ SLC□HISTORY□ The□ SLC□ was□ first□ proposed□ in□ the□ late□ 1970s,□ with□ design□ studies□ and□ test□ projects□ starting□ soon□ afte r.□ Construction□ began□ in□ October,□ 1983□ and□ was□ completed□ in□ mid-1987,□ with□ many□ upgrades□ in□ succeeding□ years.□ After□ two□ difficult□ years□ of□ commissioning,□ the□ first□ Z 0□ event□ was□ seen□ by□ the□ MARK□ II□ detector□ in□ April□ 1989.□ The□ MARK□ II□ continued□to□take□data□through□1990.□In□1991□the□SL D□ experiment□ was□ brought□ on□ line□ with□ a□ brief□ engineering□ run.□ SLD□ physics□ data□ taking□ began□ the□ next□ year□ with□ a□ polarized□ electron□ beam.□ More□ than □ 10,000□Z 0s□were□recorded□with□an□average□polarization□ of□22%.□In□1993,□the□SLC□began□to□run□with□‘flat□be am’□ optics□ with□ the□ vertical□ beam□ size□ much□ smaller□ tha n□ the□ horizontal,□ unlike□ the□ original□ design□ where□ th e□ beam□ sizes□ were□ nearly□ equal□ [3].□ This□ provided□ a□ significant□increase□in□luminosity□and□SLD□logged□o ver□ 50,000□ Z 0s.□ The□ polarized□ source□ had□ also□ been□ upgraded□ to□ use□ a□ ‘strained□ lattice’□ cathode□ which□ provided□polarization□of□about□62%□[4].□□ For□ the□ 1994-95□ run,□ a□ new□ vacuum□ chamber□ was□ built□ for□ the□ damping□ rings□ to□ support□ higher□ beam□ intensity□[5]□and□the□final□focus□optics□was□modifi ed□to□ produce□smaller□beams□at□the□Interaction□Point□(IP) □[6].□ A□ thinner□ strained□ lattice□ cathode□ brought□ the□ polarization□up□to□nearly□80%.□Over□100,000□Z 0s□were□ delivered□in□this□long□run.□For□the□next□runs,□the□ SLD□ experiment□ was□ upgraded□ with□ an□ improved□ vertex□ detector□with□better□resolution□and□larger□acceptan ce.□In□ 1996,□operations□were□limited□by□scheduling□constan ts□ and□ 50,000□ Z 0s□ were□ delivered.□ The□ final□ run□ of□ the□ SLC□ began□ in□ 1997□ and□ continued□ through□ mid-1998.□ The□luminosity□increased□by□more□than□a□factor□of□t hree□ and□a□total□of□350,000□Z 0s□were□recorded,□nearly□double□ the□total□sample□of□events□from□all□previous□SLD□ru ns□ [7].□Because□of□the□high□electron□beam□polarization ,□the□ small□and□stable□beam□size□at□the□interaction□point ,□and□ a□ high-precision□ vertex□ detector,□ the□ SLD□ was□ able□ to□ make□ the□ world's□ most□ precise□ measurements□of□many□ key□ electroweak□ parameters□ with□ this□ data□ sample.□ Figure□1□shows□the□SLC□luminosity□history.□ 3□ 1997-98□PERFORMANCE□ During□ the□ 1997-98□ run,□ the□ SLC□ reached□ a□ peak□ luminosity□ of□ 300□ Z 0s□ □ per□ hour□ or□ 310 30 □ /cm 2/sec.□ The□ luminosity□ steadily□ increased□ throughout□ the□ ru n,□ demonstrating□ that□ the□ SLC□ remained□ on□ a□ steep□ learning□ curve.□ A□ major□ contribution□ to□ this□ performance□ came□ from□ a□ significant□ disruption□ enhancement,□ typically□ 50-100%.□ The□ improvement□ was□ due□ to□ changes□ in□ tuning□ procedures□ and□ reconfiguration□ of□ existing□ hardware□ with□ no□ major□ upgrade□ projects.□ Improved□ alignment□ and□ emittance□ tuning□ procedures□ throughout□ the□ accelerator□ result ed□ in□minimal□emittance□growth□from□the□damping□rings□ to□ the□ final□ focus.□ In□ particular,□ a□ revised□ strategy□ for□ wakefield□ cancellation□ using□ precision□ beam□ size□ measurements□at□the□entrance□to□the□final□focus□pro ved□ effective□ for□ optimizing□ emittance.□ The□ final□ focus □ lattice□ was□ modified□ to□ provide□ stronger□ demagnification□ near□ the□ interaction□ point□ and□ to□ remove□residual□higher-order□aberrations.□□ The□□luminosity□□of□□a□□linear□□collider□□ L□□is□□given□□by□ d y xHfNNLσπσ =−+ 4□ (1)□ where□ N±□are□the□number□of□electrons□and□positrons□at□ the□interaction□point□(IP),□ f□is□the□repetition□frequency,□ σx,y □are□the□average□horizontal□( x)□and□vertical□( y)□beam□ sizes,□and□ Hd□is□a□disruption□enhancement□factor□which□ depends□on□the□beam□intensities□and□□the□transverse □and□ longitudinal□ beam□ sizes.□ At□ the□ SLC,□ the□ repetition □ frequency□ was□ 120□ Hz□ and□ the□ beam□ intensity□ was□ limited□ by□ wakefield□ effects□ and□ instabilities□ to□ a bout□ 410 10 □ particles□ per□ bunch.□ The□ only□ route□ to□ higher□ luminosity□ was□ by□ reducing□ the□ effective□ beam□ size. □ Taking□emittance□as□the□product□of□the□beam□size□an d□ angular□ divergence□ ( θx,y ),□ εx,y □ =□ σx,y θx,y ,□ □ the□ basic□ strategy□was□to□decrease□the□emittance□and□increase □the□ angular□ divergence.□ A□ key□ breakthrough□ was□ the□ understanding□that□the□effective□beam□size,□ σx,y □,□must□ be□ evaluated□ from□ the□ integral□ over□ the□ beam□ overla p□ distribution□and□not□the□RMS.□Properly□calculated,□ σx,y □□ decreases□with□larger□ θx,y □as□shown□in□Figure□2.□Further□ reduction□ of□ the□ vertical□ size□ was□ possible□ by□ the□ addition□of□a□permanent□magnet□octupole□on□each□sid e□ of□the□final□focus□as□shown□in□Figure□3.□ □ Figure□2:□□Horizontal□beam□size□vs□angular□divergen ce□ at□the□SLC□IP□showing□the□reduction□in□beam□size□fo r□ larger□ θ.□ The□ upper□ curves□ are□ for□ the□ 1996□ optics,□ calculated□using□the□RMS□beam□size□(solid)□and□corr ect□ luminosity-weighted□effective□beam□size□(dashed).□T he□ lower□curve□(dot-dashed)□is□for□the□1998□optics.□□ □ Figure□3:□□Vertical□beam□size□vs□angular□divergence □at□ the□SLC□IP□showing□the□dependence□of□beam□size□ σy□ on□ divergence□ θy,□ without□ (upper)□ and□ with□ octupoles□ (lower)□to□cancel□higher□order□aberrations.□Both□cu rves□ are□the□luminosity-weighted□effective□beam□size.□□ Beam□ sizes□ as□ small□ as□ 1.5□ by□ 0.65□ microns□ were□ achieved□at□full□beam□intensity□of□4*10 10 □particles□per□ pulse.□ With□ these□ parameters,□ the□ mutual□ focussing□ of□ the□beams□in□collision□becomes□significant,□resulti ng□in□ a□ further□ increase□ in□ luminosity.□ The□ strength□ of□ t he□ effect□ is□ characterized□ by□ the□ disruption□ parameter ,□ Dx,y ,□for□each□plane□which□is□the□inverse□focal□length□ in□ units□of□the□bunch□length,□ σz.□ ) (,2 ,y xyxzeNr yxDσ+σγσ σ =□ (2)□ Recorded□ SLD□ event□ rates□ confirmed□ the□ theoretical□ calculations□ of□ the□ disruption□ enhancement□ which□ wa s□ typically□ 50-100%□ [8].□ Figure□ 4□ shows□ the□ measured□ disruption□enhancement.□The□enhancement□is□calculat ed□ as□ the□ ratio□ of□ the□ measured□ SLD□ event□ rate□ to□ that □ predicted□for□rigid□beams□without□disruption.□□□ Figure□4:□□Measured□disruption□enhancement□factor□a s□a□ function□ of□ luminosity.□ At□ the□ highest□ luminosity,□ the□ enhancement□exceeded□100%.□ 4□ DIAGNOSTICS□ A□ key□ element□ in□ improving□ the□ performance□ of□ the□ SLC□ was□ the□ development□ of□ precision,□ non-invasive□ diagnostics□ to□ characterize□ and□ monitor□ the□ beams.□ Breakthroughs□ in□ understanding□ often□ followed□□ quickly□ on□ the□ heels□ of□ a□ new□ diagnostic□ tool□ which □ allowed□insight□into□the□beam□quality□and□correlati ons.□ One□example□was□the□characterization□of□a□microwave □ instability□in□the□damping□rings.□Beginning□in□1990 ,□the□ maximum□ beam□ intensity□ was□limited□by□errant□pulses □ which□ created□ high□ backgrounds□ in□ the□ detector.□ By□ correlating□ the□ energy□ and□ trajectory□ on□ a□ pulse-to - pulse□ basis,□ the□ problem□ was□ traced□ to□ the□ damping□ rings.□ Only□ when□ a□ diagnostic□ was□ developed□ to□ monitor□the□bunch□length□continuously□while□the□bea m□ was□in□the□rings□was□it□possible□to□identify□the□ca use,□a□ longitudinal□ instability□ due□ to□ the□ interaction□ of□ the□ intense□ bunches□ with□ the□ impedance□ of□ the□ vacuum□ chambers.□The□beam□intensity□could□be□increased□onl y□ after□these□chambers□were□rebuilt□in□1994.□□ Emittance□ preservation□ in□ a□ linear□ collider□ require s□ tight□control□of□the□trajectories□and□optical□match ing□in□ the□ linacs□ and□ transport□ lines.□ If□ the□ beam□ is□ not□ matched□ to□ the□ lattice□at□the□entrance□of□the□linac ,□the□ inherent□energy□spread□of□the□beam□will□cause□slice s□of□ different□energy□to□filament.□Dispersion□from□the□b eam□ passing□off-axis□through□the□quadrupoles□interacts□ with□ the□ correlated□ energy□ spread□ along□the□beam□to□crea te□ an□ x–z□ (or□ y–z)□ correlation□ or□ tilt.□ If□ the□ beam□ passes□ off-axis□ through□ the□ structures,□ wakefields□ from□ th e□ head□of□the□bunch□act□on□the□tail□to□cause□another□ x–z□ correlation.□To□avoid□these□effects,□one□must□be□ab le□to□ accurately□characterize□the□beam□profile□and□optimi ze□it□ as□a□function□of□lattice□and□trajectory□changes.□ □The□ key□ to□ emittance□ control□ at□ the□ SLC□ was□ the□ development□of□wire□scanners□which□allowed□a□precis e,□ rapid,□ non-invasive□ measurement□ of□ the□ beam□ profile .□ The□ first□ scanners□ were□ installed□ at□ the□ beginning□ and□ end□of□the□linac□in□1990□[9].□Four□scanners□separat ed□in□ betatron□ phase□ provide□ a□ measurement□ of□ the□ beam□ emittance□in□a□few□seconds.□The□wires□scan□across□t he□ beam□ during□ a□ sequence□ of□ pulses□ scattering□ a□ small □ fraction□ of□ the□ particles□ on□ each□ pulse.□ Downstream □ detectors□ measure□ the□ number□ of□scattered□particles □at□ each□ step□ to□ map□ out□the□beam□profile.□Wire□scanner s□ were□ absolutely□ essential□ for□ matching□ the□ positron □ beam□ into□ the□ SLC□ linac□ since□ invasive□ monitors□ lik e□ fluorescent□ screens□ would□ interrupt□ the□ electrons□ needed□to□produce□more□positrons.□Over□several□year s,□ more□ than□ 60□ wire□ scanners□ were□ installed□ throughou t□ the□ SLC□ from□ the□ injector□ to□ the□ final□ focus□ to□ characterize□ the□ beam□ transverse□ size□ and□ energy□ distribution.□Many□of□these□were□scanned□routinely□ by□ completely□ automated□ procedures□ to□ provide□ real-tim e□ monitoring□ and□ long□ term□ histories□ of□ the□ beam□ properties.□ In□ 1996,□ a□ novel□ ‘laser□ wire’□ beam□ size □ monitor□ was□ developed□ and□ installed□ near□ the□SLC□IP □ [10]□ to□ measure□ the□ individual□ micron-scale□ beams□ which□ would□ destroy□ any□ conventional□ wire.□ This□ device□ placed□ an□ optical□ scattering□ center□ inside□ t he□ beam□ pipe□ with□ light□ from□ a□ high□ power□ pulsed□ laser □ brought□to□a□focus□of□400-500□nm.□The□e +□or□e -□beam□ was□ scanned□ across□ the□ laser□ spot□ and□ its□ shape□ reconstructed□from□the□number□of□scattered□particle s□at□ each□ step.□ This□ device□ was□ a□ prototype□ for□ the□ beam □ size□monitors□which□will□be□needed□for□the□micron-s ize□ beams□of□a□future□linear□collider.□□ To□ maintain□ the□ optical□ matching□ to□ high□ precision□ required□not□only□the□development□of□the□measuremen t□ devices□ themselves□ but□many□iterations□of□refinemen ts□ in□the□data□processing□algorithms.□Typically□four□w ires□ were□used□to□provide□a□redundant□measurement□of□the □ phase□ space.□ Non-gaussian□ distributions□ required□ different□ fitting□ algorithms□ to□ parameterize□ the□ be am□ shape.□Since□a□single□measurement□required□many□bea m□ pulses,□ it□ was□ essential□ to□ filter□ out□ errant□ data. □ Beam□ position□monitors□near□the□scanners□were□used□to□fi t□the□ trajectory□ on□ each□ pulse□ and□ correct□ the□ expected□ position□ of□ the□ beam□ with□ respect□ to□ the□ wire.□ Automated□procedures□require□robust□fitting□algorit hms□ with□ careful□ error□ analysis.□ The□ accumulated□ SLC□ experience□ underscores□ several□ essential□ requiremen ts□ for□linear□collider□diagnostics.□In□addition□to□pro viding□ sufficient□precision,□the□scans□must□be□non-invasiv e□to□ allow□frequent□measurements□during□normal□operation .□ Automated□procedures□are□needed□so□the□scans□can□be □ regularly□ scheduled□ to□ provide□ long□ term□ history□ an d□ allow□ correlation□ with□ other□ events.□ Future□ collide r□ designs□ have□ incorporated□ these□ lessons□ and□ include d□ precision□diagnostics□and□correction□elements.□□ 5□ FEEDBACK□ Another□lesson□from□the□SLC□experience□is□the□cruci al□ importance□ of□ feedback□ to□ combat□ the□ inherent□ instabilities□ of□ a□ linear□ collider.□ Feedback□ contro lled□ the□beam□energy□and□trajectory,□stabilized□the□pola rized□ source,□ and□ maintained□ and□ optimized□ collisions.□ Several□ generations□ of□ development□ were□ required□ to □ produce□the□flexible□feedback□systems□used□througho ut□ the□ SLC.□ The□ first□ ‘slow’□ SLC□ energy□ and□ trajectory □ feedback□was□implemented□in□1985.□This□was□followed □ by□ prototype□ pulse-to-pulse□ feedback□ using□ dedicate d□ hardware.□Energy□and□trajectory□feedback□at□the□end □of□ the□linac□was□developed□in□1987□and□collision□feedb ack□ in□ 1989.□ A□ generalized□ database-driven□ system□ [11]□ was□implemented□starting□in□1991.□This□feedback□use d□ existing□ hardware,□ making□ it□ relatively□ easy□ to□ add □ a□ new□ system□ anywhere□ needed.□ In□ order□ to□ avoid□ overcorrection,□ the□ sequence□ of□ trajectory□ feedback s□ along□ the□ main□ linac□ were□ connected□ by□ a□ ‘cascade’□ system□ which□ allowed□ each□ feedback□ to□ communicate□ with□ its□ next□ downstream□ neighbor.□ Transfer□ matrice s□ between□ the□ feedbacks□ were□ adaptively□ calculated.□ Online□ diagnostics□ of□ the□ feedback□ performance□ were □ expanded□ over□ several□ years□ to□ provide□ better□ monitoring□ and□ histories.□ A□ luminosity□ optimization □ feedback□ was□ developed□ in□ 1997□ to□ improve□ the□ resolution□of□the□final□optical□tuning□at□the□IP.□B y□1998,□ the□ SLC□ had□ more□ than□ 50□ feedback□ systems□ controlling□over□250□beam□parameters.□ The□optimization□feedback□is□an□interesting□example □ of□ a□ system□ which□ may□ have□ wide□ applicability□ for□ future□machines.□To□achieve□and□maintain□the□minimu m□ beam□ size□ at□ the□ SLC□ IP,□ five□ final□ corrections□ wer e□ routinely□ optimized□ for□ each□ beam.□ These□ included□ centering□of□the□ x□and□ y□beam□waist□positions,□zeroing□ of□the□dispersion□ ηx□and□ ηy,□and□minimization□of□an□ x–y□ coupling□ term.□ Since□ the□ first□ SLC□ collisions,□ an□ automated□procedure□was□used□to□scan□the□beam□size□ as□ a□ function□ of□ each□ parameter□ and□ set□ the□ optimum□ value.□The□beam□size□was□measured□with□a□beam-beam□ deflection□scan□but□this□technique□lacked□the□resol ution□ required□ to□ accurately□ measure□ micron-size,□ disrupt ed□ beams.□It□was□estimated□that□poor□optimization□caus ed□a□ 20-30%□ reduction□ in□ luminosity□ during□ the□ 1996□ run□ [12].□ For□ 1997,□ a□ novel□ ‘dithering’□ feedback□ was□ implemented□ which□ optimized□ a□ direct□ measure□of□the □ luminosity□(i.e.□the□beamstrahlung□signal)□as□a□fun ction□ of□small□changes□in□each□parameter□[13].□By□averagi ng□ over□1000s□of□beam□pulses,□it□was□possible□to□impro ve□ the□ resolution□ by□ a□ factor□ of□ 10.□ A□ similar□ ‘dither ing’□ feedback□ was□ developed□ to□ minimize□ emittance□ at□ the □ end□ of□ the□ linac□ but□ never□ fully□ commissioned.□ Optimization□ feedback□ modeled□ on□ this□ system□ has□ been□incorporated□in□the□designs□of□future□collider s.□ The□SLC□feedback□systems□were□essential□for□reliabl e□ operation□ of□ the□ accelerator□ and□ provided□ several□ l ess□ obvious□ benefits.□ Feedback□ compensated□ for□ slow□ environmental□ changes□ such□ as□ diurnal□ temperature□ drifts□or□decreasing□laser□intensity□and□provided□a □fast□ response□ to□ changes□ such□ as□ klystrons□ cycling.□ It□ facilitated□ a□ smooth□ recovery□from□any□interruption □to□ operation.□ Feedback□ improved□ operating□ efficiency□b y□ providing□ uniform□ performance□ independent□ of□ the□ attention□or□proficiency□of□a□particular□operations □crew.□ An□important□benefit□was□that□the□feedback□decouple d□ different□ systems□ so□ that□ tuning□ could□ proceed□ non- invasively□ in□ different□ parts□ of□ the□ machine□ while□ delivering□ luminosity.□ Feedback□ also□ provided□ a□ ver y□ powerful□ monitor□ of□ many□ aspects□ of□ the□ machine□ performance.□ Much□ of□ the□ SLC□ progress□ came□ from□ using□ feedback□ to□ automate□ as□ many□ routine□ tuning□ operations□as□possible.□ In□ spite□ of□ the□ critical□ importance□ of□ feedback□ for □ SLC□ operation,□ there□ were□ several□ areas□ in□ which□ th e□ feedback□performance□was□less□than□optimal.□Many□of □ the□ problems□ occurred□ in□ the□ sequence□ of□ trajectory □ feedbacks□ along□ the□ main□ linac.□ To□ avoid□ having□ multiple□feedbacks□respond□to□an□incoming□disturban ce,□ a□simple□one-to-one□system□was□used□where□each□loop □ communicated□with□the□next□downstream□feedback.□The □ topology□ of□ this□ ‘cascade’□ system□ was□ limited□ by□ bandwidth□ and□ connectivity□ constraints.□ However,□ in □ the□ presence□ of□ strong□ wakefields,□ the□ beam□ transpo rt□ depends□ on□ the□ origin□ of□ the□ perturbation□ and□ a□ mor e□ complex□ interconnection□ is□ required.□ Each□ feedback□ must□ have□ information□ from□ all□ upstream□ systems□ to□ determine□ the□ ideal□ orbit□ correction.□ Simulations□ indicate□ that□ a□ feedback□ system□ with□ a□ many-to-one□ cascade□ can□ avoid□ overcorrection□ problems□ [14].□ Another□ problem□ which□ has□ been□ studied□ and□ understood□concerns□the□configuration□of□monitors□a nd□ correctors□ in□ each□ system.□ Traditional□ SLC□ feedback s□ were□ distributed□ over□ at□ most□ two□ sectors□ (200□ m),□ constraining□ the□ orbit□ locally□ but□ allowing□ oscilla tions□ to□ grow□ elsewhere.□ Simulations□ have□ shown□ that□ a□ better□ trajectory□ can□ be□ achieved□ if□ the□ devices□ ar e□ distributed□ over□ a□ longer□ region.□ The□ success□ of□ th e□ SLC□feedback□systems□coupled□with□new□understanding □ of□their□limitations□has□produced□a□robust□design□f or□the□ feedback□required□for□the□NLC.□ 6□ TUNING□ALGORITHMS□ A□variety□of□innovative□optical□tuning□techniques□w ere□ developed□for□the□SLC.□These□included□precision□bea m- based□alignment□using□ballistic□data□and□other□meth ods,□ as□ well□ as□ betatron□ and□ dispersion□ matching.□ The□ luminosity□ optimization□ feedback□ provided□ the□ resolution□ required□ to□ align□ the□ final□ focus□ sextup oles□ and□ octupoles.□ In□ the□ non-planar□ SLC□ arcs,□ a□ 4-D□ transfer□ matrix□ reconstruction□ technique□ with□ caref ul□ error□analysis□allowed□minimization□of□coupling□ter ms□ and□ synchrotron□ radiation□ emittance□ growth.□ An□ extension□of□this□method□allowed□an□adjustment□of□t he□ effective□ spin□ tune□ of□ the□ arc□ to□ preserve□ maximum□ polarization.□ A□ two-beam□ dispersion□ free□ steering□ algorithm□ developed□ for□ the□ SLC□ linac□ was□ later□ applied□successfully□at□LEP.□□□An□important□technique□used□for□emittance□control□i n□ the□ SLC□ linac□ was□ to□ introduce□ a□ deliberate□ betatro n□ oscillation□ to□ generate□ wakefield□ tails□ which□ compensated□ for□ those□ due□ to□ alignment□ errors□ [15]. □ Wire□ scanner□ measurements□ of□ the□ beam□ profile□ were□ used□ to□ characterize□ the□ wakefield□ tails,□ and□ then□ an□ oscillation□ was□ created□ by□ one□ of□ the□ linac□ traject ory□ feedbacks□ which□ was□ closed□ by□ the□ next□ feedback.□ Since□ 1991,□ this□ method□ was□ applied□ with□ reasonable □ success□using□wires□in□the□middle□and□near□the□end□ of□ the□ linac.□ One□ problem□ was□ that□ careful□ tuning□ was□ required□to□find□the□optimal□phase□and□amplitude□fo r□the□ oscillation.□The□cancellation□is□also□very□sensitiv e□to□the□ phase□advance□between□the□source□of□the□wakefield□a nd□ the□ compensating□ oscillation□ so□ any□ change□ in□ the□ optics□required□retuning.□Simulations□also□showed□t hat□ significant□emittance□growth□could□occur□in□the□200 □m□ of□linac□downstream□of□the□last□wires□[16].□For□the □1997□ run,□a□different□strategy□was□adopted.□Wire□scanner s□at□ the□entrance□to□the□final□focus□were□used□for□tunin g□out□ wakefield□ tails□ to□ ensure□ that□ the□ entire□ linac□ was □ compensated.□In□addition,□the□induced□oscillations□ were□ made□ nearer□ the□ end□ of□ the□ linac□ where□ the□ higher□ energy□ beam□ was□ less□ sensitive□ to□ optics□ changes,□ making□ the□ tuning□ more□ stable.□ This□ technique□ was□ successful□in□reducing□the□emittance□growth□in□the□ SLC□ linac□ by□ more□ than□ an□ order□ of□ magnitude.□ Figure□ 6□ shows□the□evolution□of□beam□size□over□time.□ 012345678910 1985 1990 1991 1992 1993 1994 1996 1998 Y e a r Y e a r Y e a r Y e a r Beam□Size□(microns) 012345678910 σσσσx∗σσσσy□ (microns 2) S L C □ D e s i g n σσ σσX σσ σσYσσ σσXX XX□□ □□∗∗ ∗∗□□ □□σσ σσyy yy □ Figure□5:□□Evolution□of□beam□size□at□the□SLC□IP□ove r□ time□from□1991□to□1998□showing□ σx□(blue-middle),□ σy□ (green-lower)□ and□ the□ product□ σxσy□ (red-upper).□ The□ final□beam□area□was□1/3□of□the□original□design□valu e.□□ 7□ CONCLUSIONS□ More□ than□ ten□ years□ of□ SLC□ operation□ has□ produced□ much□ valuable□ experience□ for□ future□ linear□ collider s.□ Because□a□linear□collider□lacks□the□inherent□stabil ity□of□ a□ storage□ ring,□ it□ is□ a□ much□ more□ difficult□ machine .□ Significant□progress□was□made□on□precision□diagnost ics□ for□ beam□ characterization□ and□ on□ flexible,□ intellig ent□ feedback□ systems.□ New□ techniques□ for□ optical□ matching,□ beam-based□ alignment,□ and□ wakefield□ control□ were□ developed□ and□ refined.□ The□ beam-beam□ deflection□ was□ shown□ to□ be□ a□ powerful□ tool□ for□ stabilizing□and□optimizing□collisions.□Important□le ssons□ were□ also□ learned□ on□ collimation□ and□ background□ control□ and□ on□ many□ other□ issues□ not□ discussed□ here .□ Both□ the□ SLC□ and□ future□ colliders□ benefited□ from□ an □ intense□ exchange□ of□ ideas□ and□ experiments.□ It□ is□ th e□ experience□ and□ knowledge□ gained□ with□ the□ SLC□ that□ gives□confidence□that□the□NLC□design□contains□the□t ools□ required□to□commission□and□operate□a□linear□collide r.□ The□ most□ enduring□ lesson□ from□ the□ SLC□ is□ undoubtedly□ that□ any□ new□ accelerator□ technology□ wil l□ present□ unanticipated□challenges□and□require□consid er- able□hard□work□to□master.□Once□a□technology□becomes □ routine,□ it□ is□ easy□ to□ forget□ the□ initial□ effort□ th at□ was□ needed.□ At□ the□ SLC□ as□ elsewhere,□ the□ most□ difficult □ problems□ were□ almost□ always□ those□ which□ were□ not□ expected.□It□is□also□clear□that□the□experience□gain ed□on□ an□operating□accelerator□is□complementary□to□that□f rom□ demonstration□ projects.□ The□ discipline□ of□ trying□ to □ produce□ physics□ forces□ one□ to□ confront□ and□ solve□ problems□ which□ are□ not□ relevant□ otherwise.□ The□ SLC□ finally□ reached□ near□ design□ luminosity□ due□ to□ the□ creativity□ and□ dedication□ of□ a□ large□ number□ of□ peop le□ over□many□years.□ REFERENCES□ [1]□□B.□Richter,□"The□SLAC□Linear□Collider",□Geneva □Accel.□ Conf.□(1980).□ [2]□□"Zeroth-order□ design□ report□ for□ the□ Next□ Linea r□ Collider", □SLAC-R-474,□2□vol.□(1996).□ [3]□□C.□ Adolphsen□ et□ al.,□ "Flat□ beams□ in□ the□ SLC", □PAC,□ Washington,□DC□(1993).□ [4]□□T.□ Maruyama□ et□ al.,□ "Electron□ spin□ polarizatio n□ in□ photoemission□ from□ strained□ GaAs□ grown□ on□ GaAsP", □ Phys.Rev.B46:4261-4264□(1992).□ [5]□□K.□Bane,□ et□ al.,□ "High□ intensity□ single□ bunch□ instability□ behavior□ in□ the□ new□ SLC□ damping□ ring□ vacuum□ chamber ",□ PAC,□Dallas,□TX□(1995)□ [6]□□F.□ Zimmermann□ et□ al.,□ "Performance□ of□ the□ 1994 /1995□ SLC□final□focus□system",□PAC,□Dallas,□TX□(1995)□ [7]□P.□Raimondi,□et.al.,□"Recent□luminosity□improve ments□at□ the□SLC",□EPAC,□Stockholm,□Sweden□(1998).□ [8]□ T.□ Barklow,□ et.al.,□ "Experimental□ evidence□ for□ beam- beam□disruption□at□the□SLC",□PAC,□New□York,□NY□(199 9).□ [9]□□M.□ Ross□ et□ al.,□ Wire□ scanners□ for□ beam□ size□ an d□ emittance□measurements□at□the□SLC,□PAC,□San□Francis co,□CA□ (1991).□ [10]□R.□Alley□et□al.,□"A□laser-based□beam□profile□m onitor□for□ the□SLC/SLD□interaction□region",□NUIMA,A379,363□(19 96).□ [11]□ T.□ Himel,□ et□ al.,□ "Adaptive□ cascaded□ beam-base d□ feedback□at□the□SLC,"□PAC,□Washington,□DC□(1993).□ [12]□□P.□ Emma□ et□ al.,□"Limitations□of□interaction-p oint□spot- size□tuning□at□the□SLC",□PAC,□Vancouver,□Canada□(19 97).□ [13]□ L.□ Hendrickson,□ et□ al.,□ "Luminosity□ optimizati on□ feed- back□in□the□SLC,"□ICALEPCS,□Beijing,□China,□□(1997) .□ [14]□ L.□ Hendrickson,□ et□ al.,□ "Beam-based□ feedback□ s imula- tions□for□the□NLC□Linac,"□LINAC,□Monterey,□CA□(2000 ).□ [15]□ J.□ Seeman,□ et□ al.,□ "The□ Introduction□ of□ trajec tory□ oscillations□to□reduce□emittance□growth□in□the□SLC□ LINAC",□ HEACC,□Hamburg,□Germany□(1992).□ [16]□ R.□ Assmann,□ "Beam□ dynamics□ in□ SLC",□ PAC,□ Vancouver,□Canada□(1997).□
arXiv:physics/0010009 3 Oct 2000The Effect of an Under-Dense Plasma Density Gradient on the Backstreaming Ion MechanismGeorge J. Caporaso Lawrence Livermore National Laboratory, Livermore, California 94550 USA Abstract The space charge limited emission of ions from a target in the focus of an intense relativistic electron beam is studied analytically for the case of a spatially varying target density profile. In particular, the emission in the presence of an under-dense plasma shelf in contact with the solid density target dramatically differs from the case of an abrupt solid-vacuum boundary. It is found that an under-dense gradient scale length several times that of the beam radius at the focus reduces the emission by at least an order of magnitude over that to be expected from a solid-vacuum boundary. 1 INTRODUCTION High-resolution x-ray radiography requires the production of a small (≈1 mm diameter) spot on the surface of a Bremsstrahlung converter target by a relativistic electron beam of at least several kiloamperes [1]. A mechanism that might possibly disrupt the focal spot was proposed by D. Welch [2]. Bombardment of the target by a high power electron beam would lead to the rapid formation of a surface plasma. A large axial electric field would appear at the surface due to the charge redistribution on the target arising from cancellation of the beam's radial electric field. This axial field would expel the ions into the beam. These backstreaming ions would acquire energies on the order of the space charge depressed potential of the beam and would propagate upstream at very high speeds where they would act as an electrostatic focusing lens. The focusing due to these moving ions would cause the electron beam to pinch upstream of the target and then rapidly diverge. The result would be a spot size that would rapidly increase in time at the converter target. In the case of multiple pulses striking the same target the situation might be modified by the presence of a plasma left over from the previous pulses. The leading edge of this plasma could be very tenuous. We consider the effects of the leading edge of the plasma, which is of lower density than the beam. In this region the plasma electrons will be expelled and a “bare” ion column will be present. We assume that the axial electric field is low enough that we can neglect the motion of these background ions. We treat emission of ions from the “critical surface” where the beam density is equal to the ion density and impose the space-charge limited flow condition. An analytic model is presented for a "beer can" geometry in which a close fitting conducting tube surrounds the beam right up to the target. The under-dense plasma is modeled as having an exponentially varying density.A model of backstreaming ion emission from a sharp boundary was given previously [3]. That reference derived an analytic solution for the emission of ions from a planar target for the same geometry and discussed the subsequent disruption of the electron beam focal spot. This paper considers the more general case of a target that has a tenuous plasma in the vicinity of the target. The general solution is found as a function of the scale length of the under-dense plasma density gradient. In the limit that the scale length approaches zero the solution of reference [3] is recovered.* Because the source of the electric field responsible for the axial flow of ions results from the rapid variation in radial electric field due to the neutralizing effects of the target, it is of interest to consider the effects of a neutralizing background that varies from zero density up to the density of the beam. By spreading out the region over which the radial electric field is neutralized it is expected that the axial electric field and hence the ion emission will be reduced. 2 TARGET GEOMETRY AND MODEL Consider the geometry shown in figure 1. An electron beam that just fits inside a cylindrical conducting tube impinges normally on a conducting plane (target). We assume that the beam and tube have radius a and extend infinitely in the z-direction. We take the target to have a diffuse boundary with an exponentially varying density. We consider the case of steady-state space-charge-limited emission of ions. beam"beer can" x-ray target Fig. 1. "Beer can" geometry proposed to reduce the space charge depressed potential of the beam, which would reduce the backstreaming ion current. * The value of the neutralization fraction given in equation [20] of reference [3] is in error.We will assume that the target surface is sufficiently rich in ions that the flow will be space-charge-limited. The steady-state emission is determined by Poisson's equation for the electrostatic potential (in c.g.s. units) ∇2Φ = −4πρ= −4π ρ b1−f z( )( ) +ρ i [ ]. [1] Here f(z) represents the neutralization fraction of the beam’s space charge (ρb) due to a static ion background. ρi represents the charge density of the backstreaming ions. The ion velocity can be found from the conservation of energy (since the target and tube are grounded) v i= −2eΦ/M [2] where M is the ion mass and e is the ion charge. The (emitted) ion charge density is given by ρi=J r( )/v i [3] where J(r) is the ion current density. Equation [1] is two dimensional (r and z). A great simplification is made possible by choosing the beam profile to be of the form ρb= −ρoJoαr( ) [4] where Jo is the zeroth order Bessel function and α=x01/a. Here a is the radius of the beer can, x01 is the first root of J o and -ρo is the on-axis charge density of the beam. Let us seek solutions which have the following form: Φr,z( )= −ψz( )Joαr( ) [5] and J r( )= ΛoJo3/2αr( ) [6] where ψ(z) and Λo are to be determined. Substitution of Equations [2] through [6] into Equation [1] yields ψ−d2ψ dζ2= −4πρo1−fζ( )[ ] α2+4πΛo α2M 2e1 ψ [7] where we have defined a dimensionless axial coordinate ζ=αz. We now define a dimensionless variable and a dimensionless constant Ω ≡α2ψ 4πρo [8] and µ≡4πΛo α2M 2eα2 4πρo3/2 . [9] We choose f( ζ) such that the critical surface occurs at z=0. fζ( )=e−ζ/λ. [10]The parameter λ is normalized gradient scale length. We would expect to recover previous results for a sharp boundary as λ→0. With these definitions the differential equation for the dimensionless potential becomes ∂2Ω ∂ζ2− Ω −µ Ω=e−ζ/λ−1 . [11] We use the boundary conditions appropriate for space- charge-limited flow arising from the critical surface (where the neutralization fraction is one): Ω0( )=0 dΩ0( )/dζ=0. [12] As ζ→∞ we have Ω ∞+µ Ω ∞=1 . [13] The asymptotic fractional neutralization of the beam due to the backstreaming ions is then just f n=µ Ω ∞. [14] The differential equation [11] can be solved numerically and reveals that the potential changes very abruptly from zero at the target surface to the space charge depressed potential of the beam (as reduced by the backstreaming ions) over a distance of the order of the beam radius. When there is an under-dense plasma gradient the potential changes more slowly and rises to a higher value indicating less neutralization due to backstreaming ions. The solution of Equation [11] is shown in Figure 2. Note that the potential changes rapidly over a distance of order the beam radius when no plasma is present. 0 5 10 15 2000.250.50.751 zetaPlotter I/O 0 5 10 15 2000.250.50.751 zetaPlotter I/O ζΩλ=4 λ=0 Fig. 2. Solution of Equation [11]. Ω is plotted vs. ζ for a sharp boundary (λ=0) and for one with a tenuous plasma shelf (λ=4). The normalized neutralization fraction (f(λ)/f(0)) as a function of λ is shown in Figure 3a. If the actual under- dense plasma scale length is L then the true neutralization fraction as a function of L/a is shown in Figure 3b. 3 Discussion Figure 3b. shows that the neutralization fraction is decreased by an order of magnitude when the scale length is equal to a beam diameter. For typical parameters of interest for radiography the plasma blow-off speed is onthe order of at least several mm/µsec. Thus, the scale length will be on the order of the beam diameter after several hundred nanoseconds. This time is sufficiently short that even though a plasma exists when a second beam pulse hits the target there may be no significant backstreaming ion emission. 1.0 0.8 0.6 0.4 0.2 0.0Normalized neutralization fraction 4 3 2 1 0 lSimulation results from Extend Fit: f(λ)=1/(1+0.6 λ)2 ff fn nλλ( )( ) ( )≡0 fanL Neutralization fraction L a =−ez L/ Background ion density/ ρb Fig. 3(a). Normalized neutralization fraction as a function of λ. 3 (b). The actual neutralization fraction as a function of L/a where L is the actual under-dense plasma scale length and a is the radius of the beam. 4 CONCLUSIONS We have provided a solution to the problem of the space charge limited flow of ions from the critical surface of a target in the presence of an exponentially varying under-dense ion background. Modest density scale lengths are shown to substantially reduce the axial electric field and hence the emission of backstreaming ions from the target. This result indicates that the backstreaming ion mechanism may not be a serious threat to multiple pulse trains that hit a Bremsstrahlung target. 5 ACKNOWLEDGMENTS This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract W-7405-ENG-48. 6 REFERENCES [1]G. J. Caporaso, "Linear Induction Accelerator Approach to Advanced Radiography", Proceeding of 1997 Particle Accelerator Conference, Vancouver, Canada, May 1997. [2]Dale Welch, Target Workshop, Albuquerque, NM. Feb. 6, 1997 (private communication). [3] G. J. Caporaso and Yu-Jiuan Chen, "Analytic Model Of Ion Emission From The Focus Of An IntenseRelativistic Electron Beam On A Target" in Proceedings of the 1998 LINAC Conf.
arXiv:physics/0010010 3 Oct 2000Ion-Hose Instability in Long Pulse Induction Accelerators George J. Caporaso and Jim. F. McCarrick Lawrence Livermore National Laboratory, Livermore, California 94550 USA Abstract The ion-hose (or fast-ion) instability sets limits on the allowable vacuum in a long-pulse, high current accelerator. Beam-induced ionization of the background gas leads to the formation of an ion channel which couples to the transverse motion of the beam. The instability is studied analytically and numerically for several ion frequency distributions. The effects of beam envelope oscillations on the growth of the instability will be discussed. The saturated non-linear growth of the instability is derived analytically and numerically for two different ion frequency distributions. 1 INTRODUCTION With the advent of DARHT-2 and its 2 µsec pulse concern has surfaced over the ion-hose instability that may arise from the beam's interaction with the channel created from the background gas via collisional ionization [1]. We consider a simple model with a τ-dependent neutralization (τ here is defined as the distance back from the beam head divided by βc). The model we use for the channel is really more appropriate for a channel that has been preformed however a more correct treatment (which is also more complicated) produces the same asymptotic result so we adopt this one for simplicity. Let y represent the centroid position of the beam and ψ represents the centroid position of the ion channel. We consider the case of a smooth external focusing force (the case of solenoidal focusing is well represented in the asymptotic solutions by replacing kß the betatron wavenumber by kc/2, one half of the cyclotron wavenumber). Since we will also follow the non-linear development of the instability we choose as a starting point the equations used by Buchanan to describe the coupling of a beam and channel, each assumed to have a Gaussian spatial profile [2]. The beam has Gaussian radius a while the channel has Gaussian radius b such that the parameter Ro is given by Ro2≡a2+b2. [1] This parameter results from integrating the force due to the beam over the distribution of the channel and vice versa. The model is ∂2ˆy ∂ζ2+ˆy+εxα2ˆy−ˆψ( )=0 [2] ∂2ˆψ ∂x2+α2ˆψ−ˆy( )=0 [3] where α2≡1−e−ˆy−ˆψ( )2 ˆy−ˆψ( )2 [4] with ˆy=y/Ro and ˆψ=ψ/Ro. Here ζ=kβz, x=ωoτ and ε≡k2 kβ2ωoτo [5] where τo is the neutralization time of the background gas and is approximately given by τo≅10−9/Ptorrsec [6] k2 is the coupling strength given by (I is the beam current and Io is ≈ 17 kA) k2=2I γβIoRo2, [7] and ωo is the ion (angular) "sloshing" frequency in the field of the beam ωo2=2qI McRo2. [8] Here M is the ion mass, q the ion charge and c the speed of light. 2 LINEARIZED EQUATIONS If both y and ψ are small compared to Ro equations [2] and [3] may be linearized as ∂2y ∂ς2+y+εx y−ψ( )=0 [9] ∂2ψ ∂x2+ψ−y=0. [10] We will solve these equations for a “tickler” excitation, that is ∂y0,x( ) ∂ς=.01sin(x) [11] y0,x( )=ψ0,x( )=∂ψ ζ,0( ) ∂x=0. [12] These equations are appropriate for a beam and channel system that are characterized by a single betatron and ion “slosh” frequency. We are treating the case of solenoidal focusing that we assume is dominant compared to the focusing provided by the ion channel. Under this conditionit is a good approximation to neglect the spread in betatron frequency that will result due to the non-linearity of the beam-channel force which arises from the non-uniform spatial profile of the channel. However, it is not a good approximation to ignore the spread in ion resonance frequencies which arises from the non-uniform spatial profile of the beam. To account for this spread we use the “spread mass” model [3]. We modify the model by splitting the channel centroid into “filaments” labeled by a subscript λ which characterizes the frequency of a particular filament. Equation [10] is thus modified as ∂2ψλ ∂x2+λ ψλ−y( )=0. [13] The position of the channel centroid is then found by averaging the individual positions of the filaments over a distribution function ψ=fλ( )∫ψλdλ. [14] For numerical work we will use the conventional definition and take a “top hat” distribution where fλ( )=1 θ for 1−θ≤λ≤1. [15] For analytic work we will use a Lorentzian distribution (and equation [13] with λ2 instead of λ): fλ( )=δ/π λ−1( )2+δ2 [16] where δ is the half-width of the distribution and the range of λ is from −∞ to +∞. 3 EFFECTS OF ENVELOPE OSCILLATIONS We now wish to investigate the effects of an envelope mismatch in the accelerator on the growth of the instability. Since the dominant focusing for the beam is provided by the solenoidal field, an envelope mismatch will result in a beam radius that varies as rb=a1+µsin2ς( ) [17] where we have assumed a particular choice of phase for the envelope oscillations without loss of generality. Because the channel is formed by the beam we can expect that there will be a similar variation for the channel radius. Thus the ion resonance frequency will be periodically varying in z. This is analogous to the case of “stagger tuning” the resonant frequency of cavities to detune the beam breakup instability. We will investigate this effect by averaging over the fast ζ and x oscillations of both the channel and beam centroid positions [4].First we write the factor g( ζ) as gς( )=1+µsin2ς. [18] Equations [9] and [13] then become ∂2y ∂ς2+y+εx gζ( )y−ψ( )=0 [19] and ∂2ψλ ∂x2+λ2 gζ( )ψλ−y( )=0. [20] Then we may use the Laplace transform method (transforming in x to s and back again) along with equations [14] and [16] to obtain ψ=dx'yζ,x'( ) gζ( )e−δx−x'( ) gζ( )sinx−x'( ) gζ( )+δcosx−x'( ) gζ( )0x∫ . [21] We now write y( ζ,x) as yζ,x( )=Aζ,x( )eiζ−x( ) [22] where A is regarded as a slowly varying amplitude such that 1 A∂A ∂ζ<<1;1 A∂A ∂x<<1. [23] Treating µ as a small parameter, averaging equations [19] and [21] in ζ over 2π, we find after considerable algebra i∂A ∂ζ+εx 2A−i 2dx'Aζ,x'( )Joµ 2x−x'( )e− δx−x'( ) 0x∫=0 . [24] By Laplace transforming equation [24] in x to s and using the method of steepest descents we find the asymptotic growth rate y∝eΓko,so( ) [25] where Γko,so( )≈ −δx+x−µ2 8+µ4 64+εζ 2 2  1/2 . [26] The exponential growth given by equations [25] and [26] is shown in Fig. 1. µ Fig. 1. Asymptotic growth as a function of µ for δ =.05. We see that a small envelope mismatch can significantly reduce the linear growth, particularly for a large growth rate. 4 NON-LINEAR DEVELOPMENT It is clear from equations [2] through [4] that when the beam and channel displacements become of order Ro the beam-channel force falls off significantly as compared to the linear approximation used in equations [9] and [10]. We now extend equation [13] into the non-linear range (for a Lorentzian distribution) as ∂2ˆψλ ∂x2+λ2αλ2ˆψλ−ˆy( )=0. [27] By takingˆy=Aζ,x( )eiζ−x( )and ˆψλ=Bλζ,x( )eiζ−x( ) with A and B both slowly varying we may average equations [2] and [27] to obtain (assuming B>>A) 2i∂A ∂ζ+εxA−B( ) 1+B2 4=0 [28] −Bλ+λ2Bλ−A( ) 1+Bλ−A2 4≅0. [29] Equation [29] can be solved iteratively and integrated with equation [16] to find B. This result can be used to manipulate [28] into the form ∂ψ ∂ξ−2ψ 1+2δψ ψs+ψ2 ψs2≅0 [30] with B2≅8δψ ψs/ 1+ψ2 ψs2 [31] where ψ≡A2, ψ s≡32δ3, andξ≡εζx/ 4δ, the number of e-folds of linear growth.non-linear ion hose 1.00E-031.00E-021.00E-011.00E+00 050100 150 200 250 300 xpsi y Fig. 2. Channel and beam position vs. x at the end of the accelerator from numerical solution. The blue curve is the channel centroid while the purple curve is the beam centroid. The top hat distribution (equation [15]) was used for θ=0.59. Fig. 3. Channel and beam position vs. ξ from equations [30] and [31] with δ=0.0939. This maximum value of the abscissa corresponds to 10 e-folds of linear growth of the beam centroid position, as is the case for figure 2. 5 CONCLUSIONS We have shown that envelope oscillations that lead to a periodic detuning of the ion resonant frequency significantly reduce the linear growth rate of the instability. In addition, when the amplitude of the ion channel motion becomes of the order of Ro, the beam- channel force falls off significantly from the linear approximation. The betatron motion of the beam/channel causes a periodic modulation of the ion resonant frequency which increases the effective damping of the oscillations. This effect leads to the saturation of the beam centroid displacement at an amplitude that is of the order of 2δψ or about 10 - 30% of the channel amplitude. 6 ACKNOWLEDGMENTS This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract W-7405-ENG-48. 7 REFERENCES [1]R. J. Briggs, private communication. [2]H. L. Buchanan, Phys. Fluids 30, 221 (1987). [3]E. P. Lee, Phys. Fluids 21, 1327 (1978). [4]G. V. Stupakov, et. al., Phys. Rev. E 52, 5499 (1995). eΓvar BeamChannel
PROGRESS IN INDUCTION LINACS George J. Caporaso Lawrence Livermore National Laboratory, Livermore, California 94550 USA Abstract This presentation will be a broad survey of progress in induction technology over the past four years. Much work has been done on accelerators for hydrodynamic test radiography and other applications. Solid-state pulsers have been developed which can provide unprecedented flexibility and precision in pulse format and accelerating voltage for both ion and electron induction machines. Induction linacs can now be built which can operate with MHz repetition rates. Solid-state technology has also made possible the development of fast kickers for precision control of high current beams. New insulator technology has been developed which will improve conventional induction linacs in addition to enabling a new class of high gradient induction linacs. 1 INTRODUCTION The last several years have seen dramatic advances in linear induction accelerator technology. There have been revolutionary advances in pulsed power drivers for both accelerators and fast kickers that now offer unprecedented speed, pulse format flexibility and voltage precision. Some of this new technology will be used in the 2nd axis of DARHT. New compact focusing lenses have been developed for Heavy Ion Fusion drivers and advances in insulators and dielectrics for pulse forming lines offer the possibility to realize a new class of induction accelerators with gradients an order of magnitude higher than the current state of the art. 2 ADVANCED RADIOGRAPHY AS A DRIVER OF INDUCTION TECHNOLOGY A significant impetus for the development of advanced technology has come from the area of flash x-ray radiography for stockpile stewardship. The cessation of underground nuclear testing has placed increased emphasis on flash x-ray radiography. In order to meet the need for acquiring multiple line-of-sight, multiple time frame data during a single hydrodynamic test a concept was developed that provides these features using a single accelerator [1]. By equipping a long pulse induction linac with a pulsed power source capable of running in the MHz range and using fast kickers multiple pulses and many lines of sight could be achieved. This concept is illustrated in Figure 1 which shows the key elements of advanced technology necessary to make the scheme viable. A solid-state modulator powered by Field Effect Transistors (FETs) is shown along with a fast kicker capable of switching kiloampere level beams into different transport lines. hard tube modulator100 ns100 ns50 ns 50 ns 50 ns 200 ns 2.5 µs injector20 - 50 MeV, 2 - 6 kA accelerator Metglas cell induction adder FET-switched modulatorkickers 200 ns - 2 µs 500 ns 2.5 µs variable interpulse spacing down to 500 ns for 10 pulses 4 - 12 lines of sight Fig. 1. Linear induction accelerator concept for multi- axis, multi-time frame flash x-ray radiography. 2. A. SOLID-STATE MODULATOR The basic concept of the solid-state modulator is shown in figure 2. The accelerating voltage is established on a large capacitor bank which is connected (and disconnected) from the load by a fast switch, which is this case is a series-parallel array of FETs. The load in this case is an induction core. S1OpenCCCCInduction cellC1 FETsCharge system Opto- isolator Control+ VcellS1 Close S1timeIcell Reset interval timereset circuit reset on Fig. 2. Basic circuit concept for the solid-state modulator. A fast switch (FET arrays) connect and disconnect a pre- charged capacitor to the induction cell. Note that the output voltage is negative in this illustration. The induction core permits these circuits to be stacked in an inductive voltage adder configuration. A fast resetcircuit is also provided to enable the modulator to operate at very high repetition rates. This circuit concept allows the pulse format to be changed without changing the circuit configuration. Simply by programming the optical trigger sequence the width and inter-pulse time for each pulse can be changed at will. A single layer of the ARM-II modulator is shown in Figure 3. Fig. 3. A single switch board of the ARM-II modulator. The board contains 12 FETs which switch 100 Amps each at roughly 800 volts. These circuit boards are stacked in series to provide 15 kV for the ARM-II modulator. There are four of these stacks in parallel around a Metglas core to provide 4.8 kA. The circuit boards shown in Figure 3. Are stacked in series and parallel to provide a modulator capable of running at 1 MHz with full reset and at 2 MHz without reset. Fig. 4. A single ARM-II modulator capable of generating a programmable burst of variable width pulses at 15 kV (open circuit voltage) and 4.8 kA output current. The ARM-II modulator was designed to be stacked in an inductive voltage adder configuration. A three-stage adder is shown in Figure 5. Fig. 5. The three-stage ARM modulator assembly which produces 45 kV (open circuit) at 4.8 kA and 1 MHz repetition rate. A typical burst which illustrates the pulse format flexibility is shown in Figure 6. Fig. 6. Typical burst from the modulator showing the pulse format agility of the system. The horizontal scale is 2 µsec. per division. This technology will also be used to power the high current kicker for the 2nd axis of DARHT. 2. B. FAST, HIGH-CURRENT KICKER The second major piece of technology inspired by the needs of Advanced Radiography is a precision, fast kicker capable of handling long multi-kiloamp beam pulses. Inorder to obtain switching times of order 10 ns. the source of the fields must be inside the beam pipe. A configuration was chosen which is similar to a stripline configuration similar to that of a stripline beam position monitor was chosen. The concept is shown in Figure 7. drive cable drive plate termination septum magnetdriven plate bias dipole windingsdrift space switched beam positions non-driven plate Fig. 7. Fast, high-current kicker concept. The system consists of four equal size striplines. One opposite pair of electrodes is powered to produce switching in a plane. In this illustration the top electrode is powered negatively while the bottom electrode is powered positively to switch the beam downward in the vertical plane. Since the pulser technology employed is unipolar, a D.C. bias dipole winding wrapped over the kicker vacuum housing pre- steers the beam upwards to obtain a full range of vertical motion. The system that has undergone extensive testing on the ETA-II accelerator at Livermore is shown in Figure 8. Fig. 8. Stripline kicker system used on ETA-II. A system very similar to this will be deployed on the second axis of DARHT to produce a sequence of radiographic pulses from a 2 µsec pulse in the accelerator. The kicker system has been very successful. Beams of up to 2 kA, 50 ns wide at 6 MeV have been steered rapidly with precision. An image from the switched beam intercepting a quartz foil is shown in Figure 9. Thepicture captures a single ETA-II pulse in the act of being switched from one position to another. 4 cm Fig. 9. A single beam pulse from ETA-II caught in the act of switching from an initial position on the right to the final position on the left. The image is created by light striking a quartz foil approximately 60 cm downstream of the end of the kicker. The total centroid shift is some 4 cm with a kicker plate voltage of ± 9 kV. 2. C. DARHT The Dual Axis Radiographic Hydrodynamic Test (DARHT) facility is under construction at Los Alamos. It consists of a single 70 ns pulse, 20 MeV, 4 kA induction linac which is in operation and a second, long pulse machine under construction now. The second axis (DARHT-2) will produce a 2 µsec, 20 MeV pulse at up to 4 kA. A kicker system will be used to extract a sequence of 4 relatively short radiographic pulses out of the long pulse and direct these to the x-ray converter target. Fig. 10. The DARHT facility at LANL. The axes of the two radiographic machines are at right angles. 3 ADVANCES IN HEAVY ION FUSION TECHNOLOGY There have been numerous advances in the technologies required for Heavy Ion Fusion driver development. Very compact superconducting lenses have been developedwhich will lead to increased accelerating gradients. An example of such a lens is shown in Figure 11. Fig. 11. Superconducting quadrupole using a Rutherford cable on a flat support (Martovetsky at LLNL). Compact pulsed magnets suitable for focussing arrays of beamlets have also been developed. A prototype array for the Integrated Research Experiment (IRE) is shown in Figure 12. Fig. 12. Compact, pulsed quadrupole lens array for the IRE. Improved characterization of magnetic core materials such as Metglas, Finemet and Silicon-Iron under conditions comparable to those found in a driver have been completed which will lead to the most efficient andeconomical choices for the several different accelerator cell systems employed in a driver. 4 ADVANCED INSULATORS AND THE DIELECTRIC WALL ACCELERATOR 4. A. HIGH GRADIENT INSULATORS In the past few years a new class of insulators has been developed that has superior performance for short, long and bi-polar pulses. Called simply high gradient insulators (HGI) these are novel configurations of conventional insulating materials [2]. The basic idea of the HGI is to interrupt the normal insulator with finely spaced, floating electrodes. The typical spacing between electrodes can be a few mm down to 0.1 mm. In general, the voltage holding ability of these configurations improves as the period length is shortened. Insulators have been fabricated from dielectrics such as kapton, rexolite, lexan and fused silica. A few samples are shown in Figure 12. These insulators have flashover strengths 2 to 5 times higher than conventional insulators. A startling result is the excellent performance of these insulator configurations in the proximity of high current electron beams. One such test is shown in Figure 13. Wall current monitorHGI Velvet cathodeWire mesh 1 kA beam Fig. 13. High gradient insulator test using the kapton/stainless steel insulator shown on the right side of the figure. The insulator measured 22 cm outer diameter by 2 cm in axial length. A velvet cloth 1 cm in radius was used as a cathode while a highly transparent wire mesh was used as an anode. A 20 ns. FWHM, 440 kV pulse was placed across the outer diameter of the insulator. The cathode produced a 1 kA electron beam repeatedly with no breakdowns. As the voltage was increased signs of insulator breakdown at the end of the pulse could be observed. The breakdown-free accelerating gradient in the presence of this beam was 22 MV/Meter. In another test of the insulator in the presence of beam an ETA-II induction cell was modified to accept a high gradient insulator. The standard insulator, a slanted piece of rexolite with a slant width of 3.75 cm was replaced with a high gradient version with a straight wall and only 1 cm wide. This cell was installed on the end of ETA-II about 10 cm from a graphite beam stop. Voltage was applied tothe cell by coupling the beam return current through load resistors. The cell took the full beam current (2 kA, 50 ns pulses at 6 MeV) at 1 Hz for an entire day. The cell logged over 10,000 shots with no breakdowns. The cell was operated at up to twice it’s normal operating voltage with a straight wall insulator having a direct line of sight to the beam and a width almost one quarter of the standard insulator with no breakdowns at 17.5 MV/Meter. These results suggest the possibility of an accelerator configuration that might have a considerably higher gradient than conventional induction accelerators. 4. B. THE DIELECTRIC WALL ACCELARATOR The basic concept for a Dielectric Wall Accelerator (DWA) is shown in Figure 14. A conventional induction machine has an accelerating field only in the gap, which occupies a relatively small fraction of the axial length of an accelerating cell. If the conducting beam pipe could be replaced by an insulating wall, accelerating fields characteristic of the gaps might be applied uniformly over the entire length of the accelerator yielding a much higher gradient. Continuous E-field State of the Art Induction Accelerator 0.75 MeV/meter GradientE-field in gaps only 1 meter Dielectric Wall Accelerator 20 MeV/meter GradientDielectric Wall Pulse Forming Line Fig. 14. Basic idea of the DWA. In order to supply an accelerating voltage over the entire structure a suitable pulse forming line must be used along with a closing switch to initiate the voltage pulse. One such concept is shown in Figure 15. The Asymmetric Blumlein is configured as two radial transmission lines with different dielectrics. These lines are initially charged to the same voltage but opposite polarities so that there is no net voltage across the pair of lines. If switches on the outer diameter are closed, waves will propagate radially inward leaving zero voltage in their wakes. Because the dielectrics in the lines have different values of permittivity, the waves travel at different speeds. When the faster of the two waves hits the inner boundary of the line there will be a reflection because of an impedance mismatch which will boost the voltage of that wave, causing the polarity of that line to reverse which generates a net accelerating voltage across the line I.D.CLCLCL+-+Initially charged Switches closed + +- +--+Fast wave reflects High Gradient Insulator- "Fast" line"Slow" line + -+- Fig. 15. The concept of the Asymmetric Blumlein one of several schemes for providing an accelerating pulse over the DWA. 5 CONCLUSIONS Many important technological advances in the induction accelerator field have taken place over the past several years. A revolution in pulsed power technology has boosted the maximum repetition rate of induction machines by three orders of magnitude over the previous record providing unprecedented pulse format flexibility and voltage precision. Progress in compact magnetic lenses and lens arrays for Heavy Ion Fusion promise systems with higher average accelerating gradients. Advanced understanding of magnetic core material will permit the construction of more efficient and economical fusion drivers. A new class of high gradient insulators has demonstrated superior performance in a variety of modes and promises to make possible the construction of novel high gradient accelerators. 6 ACKNOWLEDGMENTS It is a pleasure to acknowledge the help of many colleagues at LLNL, LANL, LBNL, Honeywell FM&T and elsewhere: Yu—Jiuan Chen, Judy Chen, Steve Sampayan, Jim Watson, John Weir, Ed Cook, Tim Houck, Glen Westenskow, Hugh Kirbie, Dave Sanders, Mike Burns, Mike Krough, John Barnard and Art Molvik. This paper is dedicated to my friend Dan Birx who was a true genius and an outstanding human being who had a profound influence on this field. This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract W-7405-ENG-48. 7 REFERENCES [1]G. J. Caporaso, “Linear Induction Accelerator Approach for Advanced Radiography” in Proc. 1997 Part. Accel. Conf. [2]G. J. Caporaso, in Frontiers of Accelerator Technology , World Scientific, 1996.
1Design and optimization of tapered structure of near-field fiber probe based on FDTD simulation H. NAKAMURA and T. SATOTheory and Computer Simulation Center, National Institute for Fusion Science, 322-6 Oroshi-cho,Toki, Gifu 509-5292, JapanH. KAMBE and K. SAWADADepartment of Applied Physics, Shinshu University, 500 Wakasato, Nagano 380-8553, JapanT. SAIKIKanagawa Academy of Science and Technology, 3-2-1 Sakado, Takatsu, Kawasaki, Kanagawa213-0012, JapanKey words. finite-difference time-domain (FDTD) method, near-field scanning optical micros- copy, aperture probe, taper structure, collection efficiency. Summary The finite-difference time-domain method was employed to simulate light propagation in tapered near-field fiberprobes with small metal aperture. By conducting large-volume simulations, including tapered metal-claddingwaveguide and connected optical fiber waveguide, we il-lustrated the coupling between these guiding modes as wellas the electric field distribution in the vicinity of the aper-ture. The high collection efficiency of a double-taperedprobe was reproduced and was ascribed to the shorteningof the cutoff region and the efficient coupling to the guid-ing mode of the optical fiber. The dependence of the effi-ciency on the tapered structure parameters was also ex-amined. 1. Introduction Improvement of the optical throughput and collection efficiency of aperture probes is the most im-portant issue to be addressed for the application of near-field scanning optical microscopy (NSOM) in optical re-cording, fabrication, and manipulation as well as spectro-scopic studies. The tapered region of the aperture fiberprobe is considered to be the metal-cladding opticalwaveguide, whose propagation properties are character-ized by the cutoff diameter and the absorption coefficientof the cladding metal. Through systematic experimentalstudies, it has been confirmed that the transmission effi-ciency decreases in the region where the core diameter issmaller than the wavelength of the propagating light. On the basis of this finding, we proposed to shorten the nar-row metal-cladding region with strong optical losses bymaking a double-tapered structure with a large cone angle.This structure is easily realized using a multi-step chemi-cal etching technique. It has been demonstrated that the transmission efficiency is much improved by 1-2 orders of magnitude as compared to the single-tapered probe witha small cone angle (Saiki et al. , 1996). Further optimization of the tapered structure is needed to achieve much higher probe efficiency. How-ever, it is very time-consuming to assess many structureparameters, such as the cone angle and taper length, bytrial and error. Numerical analysis is a more reasonableway to attain an optimized structure efficiently and to un-derstand the electromagnetic field distribution in a taperedwaveguide including the vicinity of the aperture. Compu-tational calculation by the finite-difference time-domain(FDTD) method is the most popular and promising methodavailable for this purpose (Furukawa & Kawata, 1996;Nakamura et al. , 2000), because it can be easily applied to actual three-dimensional problems. Although there havebeen many simulations focusing on the electric field dis-tribution in the vicinity of the aperture to examine the spa-tial resolution of NSOM, no calculations have been re-ported dealing with the light propagation in the tapered2region in terms of the sensitivity of the probe. In this pa- per, using the three-dimensional FDTD method, we dem-onstrate the high collection efficiency of double-taperedprobes including guiding optical fibers, as compared withsingle-tapered probes. We also examined the dependenceof the collection efficiency on the cone angle and taperlength in detail. 2. Model and Calculations Figure 1 illustrates the cross-sectional view of the FDTD geometry of the three-dimensional problem,which reproduces the experimental situation of singlequantum-dot imaging (Saiki & Matsuda, 1999). A fiberprobe with a double- or single-tapered structure collectsluminescence ( λ=1 µm) from a quantum dot buried λ/40 beneath the semiconductor (GaAs; n=3.5) surface. Weassume that the source of luminescence is a point-like di-pole current linearly polarized along the x direction. Theradiation caught by the aperture with a diameter of l/5propagates in the tapered region clad with a perfectly con-ducting metal and then is guided to the optical fiberwaveguide. The refractive indices of the core and clad- ding of the fiber are 1.487 and 1.450, respectively. Theintensity of the collected signal, I coll is evaluated by two- dimensionally integrating the electric field intensity in thecore area of the optical fiber. The simulation box consistsof a 120x120x360 grid in the x, y, and z directions; thespace increment is λ/40. We run the simulation with a time step ofc∆t= λ /(40√3) employing Mur’s boundary condi- tion. 3. Results and Discussions To demonstrate the performance of the double- tapered probe, calculations were performed for three typesof probes as shown in Fig. 2, where the spatial distribu- xz yθD GaAs (n=3.5)Core (n=1.487)Cladding (n=1.450) 2 λ 14˚ 0.2 λOptical fiber waveguide MetalTapered metal-claddingwaveguide Figure 1 Cross-sectional view of the FDTD geometry of the three-dimensional NSOM model.Figure 2 Calculated distribution of the electric field intensity forthree types of probes. (a) single-tapered probe with a coneangle θ=28˚, (b) single-tapered probe with θ=90˚, and (c) double-tapered probe with θ=90˚ and neck diameter D= λ. 2λ 0.2λ3.6λ 28˚ 90˚0.9λ λ 90˚0.4λ2λ(a) (b) (c)3tion of the electric field intensity is shown on a logarith- mic scale. In Fig. 2(a) and 2(b), Icoll is compared for probes with θ=28˚ and θ=90˚. The Icoll ratio is estimated to be 1:32. Such a distinct improvement in Icoll can be attributed to the difference in the length of the cutoff region. Bymaking the cone angle large and shortening the cutoff re-gion, much radiation power can be directed towards thetapered region. Figure 2(c) shows the result of calculationin the case of double-tapered probe whose cone angle isthe same as in Fig. 2(b). The neck diameter D is assumedto be λ, which is twice the cutoff diameter (d c~ λ/2) of the cylindrical waveguide clad with a perfectly conductingmetal. I coll of Fig. 2(c) is found to be three times greater than Icoll in Fig. 2(b). The radiation pattern in Fig. 2(c) Figure 3 Plots of the intensity of collected light as a function of (a)cone angle θ and (b) neck diameter D.02468 80 100 120 140 Cone angle θ [degrees]Collected light intensity [arb. units]θλ 0246 0.5 1.0 1.5 Neck Diameter D/λCollected light intensity [arb. units]D 90˚(a) (b)clearly illustrates that the second tapered region modifies the wavefront of the propagating light to match the guid-ing mode of the optical fiber, while the spherical wave-like propagation in Fig. 2(b) cannot be coupled to the guid-ing mode so efficiently. To summarize, the collection ef-ficiency of the double-tapered probe in Fig. 2(c) is greaterby two orders than that of the conventional single-taperedprobe in Fig. 2(a). Although we have demonstrated the advantage of a double-tapered probe, its performance should be de-pendent on various structure parameters. In Figs. 3(a) and3(b), the values of I coll as a function of cone angle θ and neck diameter D, respectively, are plotted The enhance-ment of I coll with the increase in θ can be understood rea- sonably. Icoll will increase monotonously as θ approaches 180˚. In the case of a realistic metal aperture, however, alarge θ will cause diminished spatial resolution due to the finite skin depth of the metal. The optimum value of θ should be chosen by balancing the collection efficiencywith the spatial resolution. As depicted in Fig. 3(b), thedependence of I coll on D is found to be more complicated and seems to be less essential. One significant result isthat a neck diameter D as small as d c is more preferable, compared with D~2 dc, to attain high efficiency in cou- pling to the guiding mode of the optical fiber. 4. Summary FDTD simulation demonstrated the perfor- mance of a double-tapered probe, whose collection effi-ciency was found to be greater by two orders than that ofa common single-tapered probe. Such high efficiencycould be explained as follows: (1) by shortening the cut-off region of the metal-cladding waveguide, much radia-tion was directed into the probe; (2) by introducing a guid-ing region, smooth coupling to the optical fiber wasachieved. We also examined the collection efficiency as afunction of structure parameters. Dependence on the coneangle was evident as expected; the efficiency increasedmonotonously with the cone angle. On the other hand, therelationship between the efficiency and the neck diameter4was found to be complicated. Further study, focusing on a more realistic situation, introducing a complex dielectricconstant of the cladding metal, is now in progress. Acknowledgement This work was carried out by the Advanced ComputingSystem for Complexity Simulation (NEC SX-4/64M2) atthe National Institute for Fusion Science. References Furukawa, H. & Kawata, S. (1996) Analysis of image for- mation in a near-field scanning optical microscope: ef-fects of multiple scattering. Opt. Commun. 132, 170-178. Nakamura, H., Sawada, K., Kambe, H., Saiki, T. & Sato, T. (2000) Spatial resolution of near-field scanning opticalmicroscopy with sub-wavelength aperture. Prog. Theor. Phys. Suppl. 138, 173-174. Saiki, T., Mononobe, S., Ohtsu, M., Saito, N. & Kusano, J. (1996) Tailoring a high-transmission fiber probe forphoton scanning tunneling microscope. Appl. Phys. Lett. 68, 2612-2614. Saiki, T. & Matsuda, K. (1999) Near-field optical fiber probe optimized for illumination-collection hybrid modeoperation. Appl. Phys. Lett. 74, 2773-2775.
DYNAMICS OF CHEMICAL REACTIONS: OPEN EQUILIBRIUM Do we need coefficients of thermodynamic activity? B. Zilbergleyt Independent Scholar E-mail: LIVENT1@MSN.COM ABSTRACT. Dynamic properties of chemical reactions and appropriate relationships for open chemical equilibrium are discussed in approach of the chemical dynamics. New way to calculate composition of chemical systems in open equilibrium, based on the amended equation for the change of Gibbs free energy, was exemplified using reactions of cobalt double oxides with sulfur. Evaluated this way equilibrium reaction extents had a good match with values found by direct thermodynamic simulation. The developed method allows us to perform analysis of complex chemical systems without coefficients of thermodynamic activity. In cases when the coefficients are in need or just desirable, the method offers a simple way to calculate them. Possible systems to apply method of chemical dynamics are discussed. Previous paper [1] has described new approach to the thermodynamics of open chemical systems. With assigned generic name chemical dynamics, that approach is based on assumption that there is a linear relationship between external thermodynamic force and relevant shift of chemical reaction extent in the vicinity of true equilibrium. Corresponding expression for reduced Gibbs free energy change may be written as ∆ g i = ∆g0 i + f ie∆i+ lnΠaki(∆i, ηki)+ τi∆iδi, (1) where ∆i - symbol for reaction extent ∆ξ, in true equilibrium ∆i = 1; δi − symbol for δξ = (1−∆) - reaction shift from equilibrium; ∆gi =∆Gi/RT reduced change of the reaction Gibbs free energy; f ie∆i reduced (divided by RT) external thermodynamic force multiplied by ∆ thus having dimension of energy; a ki(∆i, ηki) - thermodynamic activity; ηki − amount of moles of k-participant of i-reaction, consumed/arrived in the run of this reaction from initial state to true thermodynamic equilibrium; and τ is a parameter of the theory characterizing system interaction with its environment. We have introduced the τi∆iδi product in [1] as chaotic member due to analogy with the well known chaotic equation [2]. Taking into account that ∆g0 i = -lnK i, or ∆g0 i = lnΠ`i, tick mark and asterisk relate values to isolated (ISEQ) and open equilibrium (OPEQ) correspondingly, equation (1) will turn to ∆gi = f ie∆i ln(Π`i/Πi) + τi∆iδI. (2) One of the most important results described in [1] was a specific Hookes law type of shift-force dependency, which is linear up to a certain value of the force, and then 2 shows a kind of saturation. In many cases the curve is still linear (more correctly, quasi-linear) beyond the bend point but under different slope. In the run of this work about a hundred of reactions with essential negative ∆G0 values were investigated, and in no cases deviations from the chemical Hookes law were found. Typical distribution of reaction coordinates in open equilibrium vs. thermodynamic forces for a large group of reactions (Me 1O+S) at 298K in systems, where primary oxide is bound into double oxide Me 1OMe 2O (MeOR), is shown on Fig.1. Related source data is in the Table1. Zero thermodynamic force corresponds to conditional interaction of an oxide with itself thus representing isolated system, and ∆ * i = 1 and δ* i = 0. 0.000.250.500.751.00 0369 1 2 Fig.1. Correlation between reaction shift δ* (ordinate) and external thermodynamic force (abscissa) at open equilibrium in a group of various metal double oxide reactions with sulfur. Table 1. Thermodynamic force and reaction shift in MeOR-S systems. Double oxides (∆g0)/∆ξ, kJ/m δ MeO∗MeO 0,00 0,00 FeO SiO2 1,50 0,03 FeO Cr2O3 1,86 0,04 NiO SiO2 1,60 0,09 NiO Cr2O3 2,51 0,29 CoO Fe2O3 1,82 0,31 FeO Fe2O3 1,84 0,32 CoO TiO2 2,44 0,38 CoO WO3 2,14 0,39 CoO Cr2O4 2,94 0,39 NiO WO3 2,55 0,49 FeO Al2O3 5.03 0,61 MnOSiO2 5,81 0,75 PbO B2O3 8,28 0,76 MnOFe2O3 5,58 0,78 PbO SiO2 8,07 0,82 PbO WO3 8,07 0,85 3 PbO TiO2 10,81 0,86 Physical nature of the parameter τ wasnt clear at the time when [1] was published. Further investigation has unambiguously showed that this parameter has a unique value for every chemical reaction given the stoichiometric equation and reactant amounts, and is responsible for reaction resistance against external thermodynamic force. It makes very clear why combination of τ and external thermodynamic force defines the reaction shift from true equilibrium. Each chemical reaction can be characterized by several dynamic curves. They are already familiar force-shift curve and two others - reaction extent vs. effective Gibbs free energy change, calculated as a sum of internal and external values, and reaction extent vs. chaotic member τ i∆iδi. Shift-force curve that can be used to determine τ for the simplest reaction A+B=AB is shown on Fig.2. Next picture, Fig.3 shows a joint graph of above mentioned curves. An arbitrary value of 0.5 was assumed for η A, thus unilaterally defining ∆g0 for that reaction. Data used for plotting the graphs are in the Table 2. 0.00.51.0 02 0 4 0 Fig.2. Shift-force curve for reaction A+B=AB with ηA= 0.5. 0.000.250.500.751.00 - 4 - 3 - 2 - 1 0123 Fig.3. Reaction extent ∆ (ordinate) vs. (1) [-ln( Π`/Π(∆))+f ie∆i] (ascending curves), and vs. (2) chaotic term τi∆iδI (the descending curve). Points of intersections of the descending curve with all ascending curves define appropriate reaction extents at ISEQ (f ie∆i = 0, ∆i = 1, the leftmost), and OPEQ (f ie∆I =∆G0 CoOR , ∆i < 1, we have taken arbitrary real numbers of 1, 2, 3 for it). Reaction A+B=AB with ηΑ=0.5. Quotient of the activity products (columns 3 and 4) was calculated as Π`/Π(∆ i)=(1/∆ i )∗[(2−η)/(2−∆ i ∗η)]∗[(1−∆ i ∗η)/(1−η)] 2. (2) 4 Now we will give one practical example. Consider reaction between cobalt double oxide and sulfur at initial amounts of reactants n0 (CoOR) = 1m and n0 (S) = 2m, where R represents one of the non-reacting with sulfur oxides TiO 2, Cr 2O3, WO 3 2CoOR+4S=CoS2+CoS+SO2+2R. (3) T a b l e 2 . Data for plotting Fig2 and 3. ∆ δ ln(Π`/Π(∆ i )) [- ln(Π/Π(∆ i ))/∆i ]τi τi∆iδi 0.00 1.00 -3.91 44.64 0.10 0.90 -3.32 33.24 0.20 0.80 -2.55 12.74 0.30 0.70 -2.06 6.85 10.25 2.15 0.40 0.60 -1.67 4.18 7.18 1.72 0.50 0.50 -1.35 2.70 5.51 1.38 0.60 0.40 -1.06 1.76 4.47 1.07 0.70 0.30 -0.79 1.12 3.77 0.79 0.80 0.20 -0.52 0.65 3.28 0.53 0.90 0.10 -0.26 0.29 2.92 0.26 1.00 0.00 0.00 0.00 2.33 0.00 Thermodynamic equivalent of transformation ηCoOR =0.9073 at 1000K was found using HSC Chemistry thermodynamic simulation of reaction 2CoO+4S+2Y 2O3=CoS2+CoS+SO2+2Y 2O3. (3) with neutral (non-reacting with sulfur at chosen temperature) diluent Y 2O3, ratio CoO:Y 2O3 =1. Shift-force curve for this reaction is shown on Fig.4. 0.000.250.500.751.00 0 100 200 300 400-lnγ/∆ (lnΠ`/Π(∆))/∆ Fig.4. Shift-force graphs for reaction (3). It was plotted with data calculated for the force as (-ln γCoO)/∆ and (-ln( Π`/Π(∆))/∆. The match between both on Fig.4 is remarkable. Values of the parameter τ for 0=<δ=>0.40 found from the graph in the first case was 32.61, in the second 41.27. Joint graph for this reaction with data based on τ = 41.27 is plotted on Fig.4. Points of intersection of ascending curves with the descending curve, corresponding to appropriate open equilibria, are shown on an enlarged part of Fig.6. Comparison of 5 simulated with HSC values of reaction extents with those estimated from Fig.5 is given in Table 3. 0.000.250.500.751.00 -20 -10 0 10 Fig.5. Reaction extent ∆ vs. reduced effective Gibbs free energy changes and vs. chaotic term τ∆δ (the descending curve). Ascending curves correspond to, left to right, CoO and CoOTiO 2, Cr 2O3, WO 3. Once again, the leftmost curve meets chaotic term at ∆=1. Reaction 2CoO+4S+2Y 2O3 =CoS2+CoS+ SO2+2Y 2O3. Table 3. Simulated and graphically evaluated equilibrium values or reaction extents at OPEQ. Values of ∆G0 CoOR were taken from HSC Chemistry database. CoOTiO 2CoOCr 2O3 CoOWO 3 (-∆G0 CoOR /RT) 3.77 6.17 7.20 ∆ sim., HSC 0.92 0.89 0.85 ∆ est., τ = 41.27 0.92 0.85 0.82 ∆ est., τ = 32.61 0.90 0.82 0.77 0.800.850.900.951.00 - 2 02468 Fig.6. Rescaled graph of ∆ (ordinate) vs. chaotic member (abscissa). Simulated with HSC values are shown as points on the descending curve. They follow in the same order as the ascending curves for used double oxides. In the table, values ∆ sim. and ∆ est. were determined using two in principle different methods, the last of them splits in its turn by two also slightly different. For graphic 6 evaluation in a system, taken arbitrarily out of the model set, the match is good and proves ability of the method of chemical dynamics to simulate open equilibria and complex chemical systems correctly. Better match for the values found purely by our method is a pleasant but yet not sufficient for far going conclusions and excessive optimism fact. So far we are investigating the method in general. Results of this work show a possibility to find equilibrium composition of a subsystem without simulation the complex chemical system as a whole. We can do it if we know how to write down external thermodynamic force, through which the complimentary part of the system is acting against the subsystem. Along with results of [1] they prove that with the method of chemical dynamics one can avoid usage of thermodynamic activity coefficients at all. Nevertheless, in cases when the coefficients are still in need or just more habitual, the method of chemical dynamics provides quite simple way to find them using shift-force relationship. This relationship has been found to be unilateral within the entire possible scope of reaction shifts (from 0 up to almost 1) thus expanding the opportunities out of linear (more exactly, quasi-linear) area. In many cases this can eliminate expensive experimental works to define coefficients of thermodynamic activity. Double oxides with strong bounds between components, used in this work as a model set, represent only one example out of a big number of possible applications. Next and probably more practically interesting is application to the systems with reactions having comparable energetic characteristics (supposed next step of this work). Besides that, one interesting application of the method of chemical dynamics might also occur in stationary systems. In this case, using Onzagers relation between flows and thermodynamic forces, one can find equivalent value of external force to consider the stationary state formally related to the open equilibrium with chemical reaction shifted by this force. Most probably, this statement can be easily checked out in electrochemical systems, where electrochemical forces are definitely external to chemical reactions in the electrochemical cell, and relationships between them and flows (that is, electrical currents) are well known. To conclude, it is worthy to mention that ideology of here developed method is totally different from classical way of analysis of complex systems. Conventional method aggregates all parts of the system to analyze the whole on a probability basis. We go opposite direction, dividing the system by parts and replacing probability treatment of subsystems interaction by shift-force relationships. Difference between two approaches was in general well discussed in [3]. Developed in this work approach descends back to dAlembert principle of classical mechanics [4, 5]. REFERENCES. 1. B. Zilbergleyt. LANL Printed Archives, Chemical Physics, http://arXiv.org/abs/physics/0004035, April 19, 2000. 2. R. Devaney. Introduction to Chaotic Dynamic Systems; Benjamin/Cummings: New York, 1986. 3. I. Prigogin. From Being to Becoming.; W.H.Freeman: San Francisco, 1980. 4. J. Leech. Classical Mechanics; Wiley and Sons: New York, 1958. 5. B. Zilbergleyt. Russian Journal of Physical Chemistry, 1983, 864-867.
arXiv:physics/0010014v1 [physics.plasm-ph] 4 Oct 2000Connection between the Dielectric and the Ballistic Treatm ent of Collisional Absorption R. Schneider∗ Theoretical Quantum Electronics, Institute of Applied Phy sics, Darmstadt University of Technology, Hochschulstr. 4a Abstract In this work two important models of treating collisional absorption in a laser driven plasma are compared, the dielectric and the ballistic model. We wil l see that there exists a remarkable connection between these basic approaches which could give a hint how to overcome the inherent limitations. The approximations made in the models are not i dentical and lead to diﬀerent advantages and disadvantages. We notice that the dieletric model is able to handle screenin g in a selfconsistent manner, but is limited to ﬁrst order in the electron-ion interaction. The b allistic model calculates the electron- ion collision exactly in each order of the interaction, but h as to introduce a cut-oﬀ to incorporate screening eﬀects. This means in the context of kinetic theor y that the electron-ion correlation has to be calculated either in random phase or inladder approximation, or, in other words, the linearized Lenard-Balescu orBoltzmann collision term has to be used. 1 Basic Results 1.1 The Ballistic Model 1The momentum loss per unit time along the initial direction o f a electron scattered by an ion reads ˙p=−meνei(v)v=−K v3v, K=Z2e4ni 4πε2 0meln Λ,ln Λ =1 2lnb2 max+b2 ⊥ b2 min+b2 ⊥. This equation deﬁnes the collision frequency νei(v). The Coulomb logarithm ln Λ depends on two cut-oﬀ lengths bmaxandbminwhich describe the dynamical screening of the Coulomb poten tial and the quantummechanical closing of the singularity at the ori gin on the scale of a De Broglie wavelength. So we assume bmax=/radicalbig ˆv2os/2 +v2 th max(ω, ωp), bmin=¯h me/radicalbig ˆv2os/2 +v2 th. Notice that the collision parameter b⊥for perpendicular deﬂection is an inherent quantity for the Coulomb collision and not a cut-oﬀ. Calculating the ensemble average over an isotropic distrib ution function, where the Coulomb loga- rithm is treated as a constant, we could determine the time-dependent collision frequency νei(t) =K mev3os(t)/integraldisplayvos(t) 04πv2 ef(ve)dve. In order to compare this result with the dielectric model, we have to determine the time averaged energy absorption of the plasma in the laser ﬁeld for a harmon ic electron movement. The energy absorption is connected to the time-averaged collision frequency νeiby meνeiv2os=meνeiv2os= 2νeiEkin. Hence, we ﬁnd for the cycle averaged absorped energy density ˙E ˙E= 2neνeiEkin=Zω4 pmeln Λ1 vos(t)/integraldisplayvos(t) 0v2ef(ve)dve. (1) ∗e-mail:Ralf.Schneider@physik.tu-darmstadt.de 1The model is based on the usage of the Coulomb cross section wh ich is the subject of standard text books. For a detailed discussion of the ensemble averaging and the Coulo mb logarithm see ref. [1].1.2 The Dielectric Model In many papers about collisional absorption in plasmas the d ielectric theory was the starting point, refs. [2], [3], [4]. As this theory is well known we only prese nt the result for the cycle averaged absorped energy density ˙E=Zω4 pme π2ˆvos/integraldisplaykmax 0dk kF/parenleftbigg k, ω,ˆvos vth/parenrightbigg (2) F(k, ω,ˆvos vth) =ω2∞/summationdisplay n=1nℑ{ǫ−1 n}/integraldisplaykˆvos ωvth 0dxJ2 n(x) (3) ǫn(k, ω) = 1 +1 k2−√ 2nω k3D/parenleftbiggnω√ 2k/parenrightbigg −i/radicalbiggπ 2nω k3e−n2ω2 2k2 with D(x) =e−x2/integraldisplayx 0et2dt, k →k/kD, kD=ωp vth, ω→ω/ω p. The upper integral limit kmaxin eq. (2) is necessary in the classical case due to the diverg ence of the integral for large k. In the quantum case an additional term exp( −k2/8k2 B) (kBDe Broglie wavenumber) appears inside the integral of eq. (2), which co nﬁrms the assumption that the De Broglie wavelength has to be considered in kmax, refs. [5], [6]. 2 The Connection between the Models When analizing the function F(k, ω,ˆvos vth) we get the remarkable equality lim k→∞F(k, ω,ˆvos vth) =G(ˆvos vth) =π2ˆvos1 vos(t)/integraldisplayvos(t) 0v2efM(ve)dve, (4) which connects eq. (1) and eq. (2) if f(ve) is set Maxwellian, see Fig. 1. The approximation that the k-dependence of F(k, ω,ˆvos vth) is a theta function leads us to the Coulomb logarithm kmax/integraldisplay 0dk kF(k, ω,ˆvos vth)≈G(ˆvos vth)lnkmax kmin=G(ˆvos vth)lnbmax bmin. (5) The lower cut-oﬀ kmin, which is nothing else the inverse screening length, will be determined by comparing the integrals /integraldisplayk0 0dk F(k, ω,ˆvos vth) =/integraldisplayk0 0dk G(ˆvos vth)Θ(k−kmin), where k0is chosen large enough that F(k0, ω,ˆvos vth) and G(ˆvos vth) are equal. Comparing the dielectric inverse screening length kminand the one introduced in the ballistic model, Fig. 2, we come to a good qualitative agreement. Nerverthele ss, a quantitative diﬀerence appears. It must be kept in mind that we handled the Coulomb logarithm a s a constant during the ensemble average and also during the time average, which is not done in the dielectric model. The discrepancy should decrease if we overcome this approximation, which wi ll be the subject of further investigations. 3 Conclusions It was shown in the previous section that there exists a stron g connection between the dielectric and the ballistic model. This results from the fact that the i ntegral kernel F(k, ω,ˆvos vth), eq. (3), only becomes a function of vos/vthand agrees with the integral term of eq. (1). When calculatin g it is essential to include enough orders of Bessel functions for l argek. So, as the integral in eq. (3) runs up to large k, it is never a good approximation to take only a few orders of B essel functions, which was done by many authors to get analytical expressions for the ab sorption. Furthermore, it is much easier to ﬁnd approximations of the term in eq. (1), ref. [1], than of the complicated expression eq. (3).00.20.40.60.811.21.4 012345678910 k/kD Fig. 1: The integral kernel F(k, ω,ˆvos vth) (solid, eq. (3)) andG(ˆvos vth) (dashed, eq. (4)) for ω/ω p= 2 and 0 ≤ ˆvos/vth≤6 (bottom to top).0.511.522.53 00.511.522.533.544.55 ˆvos/vth Fig. 2: kmindetermined by the dielectric model (solid) and the one assumed in the ballistic model b−1 max(dashed) for 1 .8≤ω/ω p≤2.8 (bottom to top). When making the approximation eq. (5) in the dielectric trea tment we could see the diﬀerence be- tween both models. In case of the dielectric model the collis ion parameter b⊥for perpendicular deﬂection is missing. This is exactly the term which leads be side the De Broglie wavelength to the convergence of the collision integral for small collision p arameters which means large kin eq. (2). The disappearance of that length is a consequence of the weak cou pling approximation in the dielectric theory, equivalent to the ﬁrst order Born approximation or s traight orbit assumption. We could expect that the integral kernel F(k, ω,ˆvos vth) should show a decay to zero for k > b−1 ⊥when we go beyond the weak coupling approximation, which leads to a red uced absorption. This is in agreement to stopping power calculations, ref. [7], where the authors found an overestimation of the stopping power in the case of the ﬁrst order Born approximation in the e lectron-ion coupling. Including the static shielded T matrix they found good agreement with nume rical results. References [1]Mulser, P., Cornolti, F., Besuelle, E., Schneider, R. ,Time-dependent electron-ion collision frequency at arbitrary laser intensity-tempera ture ratio , Phys. Rev. E, accepted [2]Oberman, C., Ron, A., Dawson, J. , Phys. Fluids 5(1962)1514 [3]Klimontovich, Yu.L. ,Kinetic Theory of Nonideal Gases and Nonideal Plasmas , (Nauka, Moscow 1975) (russ.), Engl. transl.: Pergamon Press, Oxfor d 1982 [4]Decker, C.D., Mori, W.B., Dawson, J.M., Katsouleas, T. , Phys. Plasmas 1(12) (1994)4043 [5]Silin, V.P., Uryupin, S.A. , Sov. Phys. JETP 54(3) (1981)485 [6]Bornath, Th., Schlanges, M., Hilse, P., Kremp, D., Bonitz, M. ,Quantum Kinetic Theory of Plasmas in Strong LaserFields , Laser & Particle Beams, next issue [7]Gericke, D.O., Schlanges, M. , Phys. Rev. E 60(1) (1999)904
arXiv:physics/0010015v1 [physics.flu-dyn] 4 Oct 2000A path-integral approach to the collisional Boltzmann gas C. Y. Chen Dept. of Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, PRC Email: cychen@public2.east.net.cn Abstract : Collisional eﬀects are included in the path-integral form ulation that was proposed in one of our previous paper for the collisi onless Boltz- mann gas. In calculating the number of molecules entering a s ix-dimensional phase volume element due to collisions, both the colliding m olecules and the scattered molecules are allowed to have distributions; thu s the calculation is done smoothly and no singularities arise. PACS number: 51.10.+y. 11 Introduction In our previous works, we proposed a path-integral approach to the col- lisionless Boltzmann gas[1][2]. It is assumed in the approa ch that there are continuous and discontinuous distribution functions i n realistic Boltz- mann gases: continuous distribution functions are produce d by continuous distribution functions that exist previously and disconti nuous distribution functions are caused by boundary eﬀects. (Boundaries can bl ock and reﬂect molecules in such a way that distribution functions become d iscontinuous in the spatial space as well as in the velocity space.) To treat t hese two kinds of distribution functions at the same time, a diﬀerent type o f distribution function, called the solid-angle-average distribution fu nction, is introduced as f(t,r,v,∆Ω) =1 ∆Ω/integraldisplay f(t,r,v)dΩ, (1) where ∆Ω represents one of the solid angle ranges in the veloc ity space deﬁned by the investigator and f(t,r,v) is the “ordinary” distribution func- tion. By letting each of ∆Ω be adequately small, the newly emp loyed dis- tribution function is capable of describing gas dynamics wi th any desired accuracy. Provided that collisions in a Boltzmann gas can be neglected, the solid-angle-average distribution function is found to be f(t,r,v,∆Ω) =1 ∆Ω/integraldisplay ∆S1fct(t0,r0,v,Ω0)\|cosα\|dS0 \|r−r0\|2Ur0r +1 ∆Ω/integraldisplay ∆S2η(t0,r0,v,Ω0)dS0 \|r−r0\|2v3Ur0r,(2) where, referring to Fig. 1, ∆ S1is an arbitrarily chosen virtual surface within the eﬀective cone deﬁned by −∆Ω at the point r, ∆S2stands for all bound- ary surfaces within the eﬀective cone, ηis the local emission rate of boundary surface (acting like a surface-like molecular source), r0represents the posi- tion of dS0, Ω0is the solid angle of the velocity but takes the direction of (r−r0),t0is the local time deﬁned by t0=t− \|r−r0\|/v,αis the angle between the normal of dS0and the vector r−r0,fctis the contin- uous part of the distribution function existing previously , and Ur0ris the path-clearness step function, which is equal to 1 if the path r0ris free from blocking otherwise it is equal to 0. The objective of this paper is to include collisional eﬀects in the path- integral formalism. 2According to the conventional wisdom collisions can be anal yzed by the method developed by Boltzmann long ago, in which it is unders tood that there is a symmetry between the ways molecules enter and leav e a phase vol- ume element. Peculiarly enough, this well-accepted unders tanding includes actually hidden fallacies[3], which can brieﬂy be summariz ed as follows. In terms of studying collisions in a Boltzmann gas, there are tw o issues that are supposedly important. The ﬁrst one is related to how many col lisions take place within a phase volume element and during a certain time ; the second one is related to how the scattered molecules will, after col lisions, spread out over the velocity space and over the spatial space. These two issues involve diﬀerent physics and have to be formulated diﬀerently. If th e molecules leav- ing a phase volume element is of interest, one needs to take ca re of only the ﬁrst issue; whereas if the molecules entering a volume eleme nt is of inter- est, one needs to concern oneself with both the issues aforem entioned. This imparity simply suggests that the time-reversal symmetry, though indeed exists for a single collision between two molecules, cannot play a decisive role in studying collective eﬀects of collision. In the present paper we formulate the collisional eﬀects par tly in an unconventional way. In deriving how many molecules make col lisions, the standard method is employed without much discussion; but, i n formulat- ing how scattered molecules enter a six-dimensional phase v olume element, which is an absolute must for the purpose of this paper, a rath er diﬀerent and slightly sophisticated approach is introduced. In sec. 2, general considerations concerning basic collisi onal process are given. It is pointed out that only the scattering cross secti on in the center-of- mass frame is well deﬁned and can be employed in our studies. S ec. 3 gives a formula that describes how a molecule, when moving along it s path, will survive from collisions. Sec. 4 investigates how molecular collisions create molecules that enter a speciﬁc phase volume element. In the i nvestigation, both the colliding molecules and the scattered molecules ar e allowed to have distributions. (Otherwise, singularities will arise, as R ef. 3 reveals.) Sec. 5 includes all the collisional eﬀects in a complete path-inte gral formulation. In Sec. 6, approximation methods are introduced to make the new formulation more calculable and an application of the method is demonstr ated. Sec. 7 oﬀers a brief summary. Throughout this paper, to make our discussion as simple as po ssible, it is assumed that molecules of interest are all identical, but distinguishable, perfectly rigid spheres and they move freely when not making collisions. 32 General considerations of collision Firstly, we recall general features of binary collisions in terms of classical mechanics. Consider two molecules: one is called molecule 1 and the other molecule 2. Let v1andv2label their respective velocities before the collision. The center-of-mass velocity and the velocity of molecule 1 r elative to the center-of-mass are before the collision c=1 2(v1+v2) and u=1 2(v1−v2). (3) Similarly, the center-of-mass velocity and the velocity of molecule 1 relative to the center-of-mass are after the collision c′=1 2(v′ 1+v′ 2) and u′=1 2(v′ 1−v′ 2). (4) The conservation laws of classical mechanics tell us that c=c′and \|u\|=\|u′\|=u. (5) Fig. 2 schematically illustrates the geometrical relation ship of these ve- locities. Note that, for the collision deﬁned as above the ﬁn al velocities of the two molecules, such as u′,v′ 1andv′ 2, cannot be completely deter- mined unless the impact parameter of the collision is speciﬁ ed at the very beginning[4]. At this point, mention must be made of one misconcept in that t he usual derivation of the Boltzmann equation gets involved[3 ]. In an attempt to invoke the time-reversal symmetry of molecular collisio n, the standard treatment in textbooks[5] deﬁnes the scattering cross sect ion in the labora- tory frame in such a way that ¯σ(v1,v2→v′ 1,v′ 2)dv′ 1dv′ 2 (6) represents the number of molecules per unit time (per unit ﬂu x of type 1 molecules incident upon a type 2 molecule) emerging after sc attering with respective ﬁnal velocities between v′ 1andv′ 1+dv′ 1and between v′ 2and v′ 2+dv′ 2. If a close look at expression (6) is taken, we may ﬁnd that the v alue of ¯σin it is ill-deﬁned. As Fig. 2b clearly shows, the molecules o f type 1, namely the ones with the velocities v′ 1after the collisions, will spread out over a two-dimensional surface in the velocity space (formi ng a spherical shell 4with diameter 2 u) rather than over a three-dimensional velocity volume as suggested by the deﬁnition. Because of this seemingly small fault, the value of ¯σactually depends on the size and shape of dv′ 1and cannot be treated as a uniquely deﬁned quantity theoretically and experiment ally. Another type of scattering cross section, which is elaborat ed nicely in textbooks of classical mechanics and suﬀers from no diﬃcult y, is in terms of the relative velocities uandu′, as shown in Fig. 3. It is deﬁned in such a way that the area element dS=σ(Ωu′)dΩu′ (7) represents the number of molecules per unit time (per unit ﬂu x of type 1 molecules with the relative velocity uincident upon a type 2 molecule) emerging after scattering with the ﬁnal relative velocity u′pointing in a direction within the solid angle range dΩu′. Note that the deﬁnition (7), in which the center-of-mass velocity corc′becomes irrelevant, makes good sense in the center-of-mass frame rather than in the laborat ory frame. Before ﬁnishing this section we turn to discussing how colli sions can generally aﬀect the solid-angle-average distribution fun ction f(t,r,v,∆Ω) deﬁned by (1). In view of that gas dynamics of the Boltzmann ga s develops along molecular paths, as shown by (2), we believe that colli sional eﬀects should also be investigated and formulated in terms of molec ular paths. Fig. 4 illustrates that there are two types of processes. On o ne hand, a molecule that would reach rwith the velocity vat time tmay suﬀer from a collision and become irrelevant to the distribution f unction; on the other hand, an “irrelevant” molecule may collide with anoth er molecule and then become relevant. It should be stressed again that th ere is no symmetry between the two types of processes. For the ﬁrst typ e of process, we only need to investigate what happens to a single molecule . If a collision takes place with it, we know that the molecule will depart fro m its original path, which is suﬃcient as far as our formulation is concerne d. For the second type of process, we need to know: (i) how many collisio ns take place within the eﬀective cone; (ii) how the scattered molec ules spread out over the phase space. As stressed in the introduction, the se cond issue is particularly essential because of that the distribution fu nction is nothing but the molecular density per unit phase volume, in other wor ds we must concern ourselves with the scattered molecules “around” th e phase point (r,v), rather than the scattered molecules “at” the phase point ( r,v). In the next two sections, we will formulate the two processes respectively. 5Since the collisions are assumed to take place in terms of cla ssical mechanics all the calculations can be done without analytical diﬃcult y. 3 The surviving probability Consider a molecule moving along a spatial path where many ot her molecules make their own motions. If P(τ) denotes the probability that the molecule survives a time τwithout suﬀering a collision and w(τ)dτdenotes the prob- ability that the molecule makes a collision between time τand time τ+dτ, we must have a simple relation P(τ+dτ)−P(τ) =−P(τ)w(τ)dτ, (8) which yields 1 PdP dτ=−w(τ). (9) Therefore, the surviving probability associated with a mol ecule moving from r0torwith the velocity vcan be expressed formally by P(r0,r;v) = exp( −/integraldisplay lw(τ)dτ), (10) where lrepresents the path along that the molecule will move if no co llision takes place. For the Boltzmann gas under consideration, who se molecules are assumed to be free from forces except in collisions, the p ath of a molecule is nothing but the segment of straight line linking up the two points. The surviving probability deﬁned by (10) can be evaluated by the stan- dard approach[5]. To make this paper complete, we include th e evaluation in what follows. Suppose that the molecule encounters a molecu lar beam with the velocity v1at the path element dl. In terms of the molecular beam, the molecule has the speed 2 u= 2\|u\|, in which u= (v−v1)/2, and it occupies the volume with respect to the beam 2uσ(Ωu′)dΩu′dτ, (11) where σ(Ωu′) and Ω u′are deﬁned in (7) and illustrated by Fig. 3. The molecular density of the colliding beam is f(τ,rl,v1)dv1, (12) 6where rlis the position of the path element dl. Thus, the total collision probability can be written as wdτ=dτ/integraldisplay v1/integraldisplay Ωu′2uf(τ,rl,v1)σ(Ωu′)dΩu′dv1. (13) In terms of (13), the surviving probability (10) becomes P(r0,r;v) = exp/bracketleftBigg −/integraldisplay l/integraldisplay v1/integraldisplay Ωu′2uσ(Ωu′)f(τ,rl,v1)dΩu′dv1dτ./bracketrightBigg ,(14) where dτis the time period during that the molecule passes the path el ement dl. The formula (14) describes how collisions make the number of molecules along a certain path decrease. The method employed has nothi ng partic- ularly new in comparison with that employed by the textbook t reatment. Before changing our subject, one thing worth mentioning. In deriving (14), we had luck not to be concerned with how the scattered molecul es spread out over the phase space. It is readily understandable that t he same luck will not be there in the next section. 4 The creation probability We now study the process in which collisions make molecules g ive contribu- tions to the solid-angle-average distribution function f(t,r,v,∆Ω). It should be mentioned that in this section, unlike in the las t sections, v′andv′ 1represent the velocities of colliding molecules while vandv1 represent the velocities of scattered molecules. The see how the scattered molecules spread out over the six-d imensional phase space, we consider a relatively small six-dimensiona l volume element as ∆r·∆v= ∆r·v2∆v∆Ω. (15) In (15) ∆ ris chosen to enclose the point rinf(t,r,v,∆Ω), ∆ vto enclose the speed vinf(t,r,v,∆Ω); and ∆Ω is just the ﬁnite velocity solid-angle-range ∆Ω in f(t,r,v,∆Ω). The discussion below will be focused on molecules that really enter, after collisions, this six-dimensional volu me element. Note that in Fig. 5a each point in the spatial volume ∆ rdeﬁnes an eﬀective cone, within which physical events may make impact on the distri- bution function at the point. This means that the entire eﬀec tive cone, with 7respect to the spatial volume element ∆ r, must be slightly larger than the one deﬁned solely by r, as shown by Fig. 5b. Fortunately, we will, in the end of formulation, let ∆r→0 (16) and thus the actual entire eﬀective cone is only academicall y larger. On this understanding, we will not distinguish between the entire e ﬀective cone and the eﬀective cone deﬁned solely by the single point r. Look at molecular collisions taking place within the eﬀecti ve cone shown in Fig. 6. Note that Fig. 6a, while coming to one’s mind immedi ately, is not an appropriate picture to manifest the collision proc ess aﬀecting the distribution function at the point rsince both the colliding molecules and scattered molecules in it have no true distributions. (Ref. 3 analyzes the situation and brings out that such mental picture will ﬁnall y lead to singu- larities.) In Fig. 6b, the velocity distributions of all col liding molecules and scattered molecules are explicitly illustrated. Our task h ere is to formulate the relationship between the distribution functions of col liding molecules and the scattered molecules (including their velocity dist ributions and spa- tial distributions). We divide the entire eﬀective cone into many individual regi ons, denoted by (∆r0)i. It is obvious that within each of the regions collisions can gen- erate a certain number of molecules that will ﬁnally enter th e phase volume deﬁned by (15). Let ncl idenote the number of such molecules. In what immediately follows, it is assumed that the generated molec ules suﬀer no further collisions. The entire contributions of all collis ions to the distribu- tion function f(t,r,v,∆Ω) can then be expressed by fcl(t,r,v,∆Ω) =1 ∆r·∆v/summationdisplay incl i, (17) in which iruns all the divided regions within the eﬀective cone. For la ter use, we wish to rewrite (17) as fcl(t,r,v,∆Ω) =1 ∆Ω/summationdisplay incl i v2(∆r)i(∆v)i. (18) The advantage of (18) over (17) is that (∆ r)iand (∆ v)iin (18) may be chosen to be diﬀerent for diﬀerent ias long as the molecular number of ncl i is counted up accordingly. (Of course, all velocity directi ons of the molecules have to be within the solid angle range ∆Ω.) 8Now, consider a small, much smaller than ∆Ω, solid angle rang e ∆Ω 0 at a point r0towards the point r, as shown in Fig. 7a ( r0is within the eﬀective cone). It is easy to see that if collisions take plac e atr0, the scattered molecules having velocities within ∆Ω 0will spread out over the spatial volume element ∆r≈ \|r−r0\|2v∆Ω0∆t, (19) as shown in Fig. 7b. Accordingly, they will spread out over th e velocity volume element ∆v≈v2∆v∆Ω0, (20) as shown in Fig. 7c. Since ∆Ω 0is much smaller than ∆Ω (the latter one is ﬁnite), a molecule having a velocity within ∆Ω 0can be regarded as one having a velocity within ∆Ω. Thus, by letting (∆ r)iin (18) be equal to ∆rof (19) and letting (∆ v)iin (18) be equal to ∆ vin (20), expression (18) becomes fcl(t,r,v,∆Ω) =1 ∆Ω/integraldisplay lim (∆Ω0,∆v,∆t)→(0,0,0)ρcl(r0)dr0 v3∆v\|r−r0\|2∆Ω0∆t,(21) where the integral is over the entire eﬀective cone deﬁned by rand−∆Ω andρclis the local density (per unit spatial volume) of the molecul es that are “emitted” from r0due to collisions and ﬁnally enter the speed range ∆ v and the solid-angle range ∆Ω 0during the time ∆ t. To determine the density ρcl, we have two tasks. One is to derive the collision rate at r0and the other is to derive what fraction of the scattered molecules emerge with velocities within the range ∆ v∆Ω0. The ﬁrst task can be accomplished in a well-known way while the second one c annot. As discussed in the last section, a speciﬁc molecule with the initial ve- locityv′that collides with a beam of molecules with the initial veloc ityv′ 1 occupies a volume with respect to the beam 2u∆tσ(Ωu)dΩu, (22) where Ω uis the solid angle of the scattered relative velocity u= (v−v1)/2. The number of “such speciﬁc” molecules within dv′dr0is f(t0,r0,v′)dv′dr0. (23) The density of the molecules with v′ 1is characterized by f(t0,r0,v′ 1)dv′ 1. (24) 9Therefore, the number of all collisions within the spatial v olume dr0within the time ∆ tis dr0/integraldisplay dv′/integraldisplay dv′ 1/integraldisplay dΩuf(v′)f(v′ 1)2uσ(Ωu)∆t. (25) We now to evaluate the probability that the molecules expres sed by (25) enter the velocity range ∆ v∆Ω0. Note that the integration of dv′dv′ 1is carried out in the laboratory frame while the integration of dΩuis in the center-of-mass frame, which makes the evaluation quite diﬃ cult. For this reason, we make the integration conversion as /integraldisplay dv′/integraldisplay dv′ 1· · ·=/integraldisplay dc′/integraldisplay dΩu′/integraldisplay u2du/bardblJ/bardbl · · ·, (26) where u=u′is understood and /bardblJ/bardblrepresents the Jacobian between the center-of-mass frame and the laboratory frame /bardblJ/bardbl=∂(v′,v′ 1) ∂(c′,u′). (27) Equation (4) tells us that the Jacobian is equal to 8. By making use of (25) and (26), the distribution function fclexpressed by (21) becomes fcl(t,r,v,∆Ω)≈1 ∆Ω/integraldisplay −∆Ωdr0/integraldisplay dc′/integraldisplay dΩu′/integraldisplay /integraldisplay ∆v∆Ω0u2dΩudu ·/bardblJ/bardbl v3∆v\|r−r′\|2∆Ω02uσ(Ωu)f(t0,r0,c′−u′)f(t0,r0,c′+u′).(28) In regard to expression (28), some observations are made. As mentioned in Sec. 2, if two molecular beams with deﬁnite velocities v′andv′ 1col- lide with each other the scattered molecules will spread out only over a two-dimensional spherical surface in the velocity space, w hich implies that diﬃculty arises if the velocity distributions of scattered molecules are of con- cern. Whereas, in this expression, all the colliding molecu les are allowed to have distributions, the value of u′=uis allowed to vary and therefore the scattered molecules explicitly spread out over the velocit y space (as well as over the spatial space). Furthermore, by using the notation /integraldisplay /integraldisplay ∆v∆Ω0· · · · · · , (29) 10we have ensured that only the scattered molecules of relevan ce are taken into account. In Fig. 8, which is drawn for scattered molecules in the veloc ity space, we are concerned only with molecules that ﬁnally enter the ra nge ∆ v∆Ω0. Allowing uto vary a little bit, we may let the scattered molecules ﬁll ou t the velocity range. Namely, we have /integraldisplay /integraldisplay ∆Ω0∆vu2dΩudu(· · ·)≈v2∆v∆Ω0(· · ·), (30) where ( · · ·) represents other factors that have been treated as constan ts in terms of the inﬁnitesimally small range of ∆ v∆Ω0. Inserting (30) into (28) and taking the limits ∆Ω 0→0 and ∆ v→0, we ﬁnally arrive at fcl(t,r,v,∆Ω) =1 v∆Ω/integraldisplay −∆Ωdr0/integraldisplay dc′/integraldisplay dΩu′ /bardblJ/bardbl \|r−r0\|22uσ(Ωu)f(t0,r0,c′−u′)f(t0,r0,c′+u′),(31) where t0=t− \|r−r0\|/vand the integration /integraldisplay dΩu′· · · (32) is over the entire solid angle (0 →4π). Note that u,u′anduin the integrand of (31) have to be determined skillfully. First us ec=c′and v=v(r−r0)/\|r−r0\|to determine u, then use u=\|u\|and Ω u′to determine u′, as shown in Fig. 9. We have directly formulated the contribution to the solid-a ngle-average distribution function from collisions. It should be noted t hat the formula- tion can be done only under the condition that the velocity so lid-angle range ∆Ω is ﬁnite: if both ∆Ω and ∆Ω 0in the formulation were assumed to be inﬁnitesimally small, the limiting processes concerning t he two quantities would not be in harmony with each other. This shows again that the intro- duction of the solid-angle-average distribution function is a must to the gas dynamics of Boltzmann gas. 5 Complete formulation The complete formulation for the collisional Boltzmann gas is now in order. In Fig. 10, we have depicted a piece of boundary and some colli sions taking 11place within the eﬀective cone. As said before, all these eve nts can directly contribute to the solid-angle-average distribution funct ion. We then use the following sum to represent the total distribu tion function f(t,r,v,∆Ω) = f(i)+f(ii)+f(iii), (33) where f(i),f(ii) and f(iii)represent the contributions from the existing con- tinuous distribution function, from the piece of boundary a nd from the col- lisions respectively. For simplicity, no other types of dis tribution functions are assumed to exist within the eﬀective cone. As has been illustrated in Sec. 3, a molecule that makes its mo tion towards the point rmay suﬀer a collision with other molecules. The involved surviving probability Phas been deﬁned by (14). By taking the probability into account, the ﬁrst term in (2) becomes f(i)(t,r,v,∆Ω) =1 ∆Ω/integraldisplay ∆S1fct(t0,r0,v,Ω0)\|cosα\|dS0 \|r−r0\|2P(r0,r,v0),(34) where v0inPtake the value of vand points to the direction of ( r−r0). By taking the same eﬀect into account, the second term in (2) c an be expressed by f(ii)(t,r,v,∆Ω) =1 ∆Ω/integraldisplay ∆S2η(t0,r0,v,Ω0)dS0 \|r−r0\|2v3P(r0,r,v0).(35) A rather detailed discussion on the molecular emission rate ηhas been included in Ref. 1. Here, we content ourselves with pointing out that the rateηsatisﬁes the normalization condition at the surface elemen tdS0. If no molecular absorption and production by the surface element are assumed, the following expression holds /integraldisplay η(t,r,v1,Ω1)dΩ1dv1=/integraldisplay v2f(t,r,v2,Ω2)\|cosθ\|v2 2dΩ2dv2, (36) in which θis the angle between the velocity v2and the normal of dS0, Ω1 points to an outward direction of dS0and Ω 2points to an inward direction of dS0. The concrete relation between η(t,r,v) and f(t,r,v) must, of course, be ultimately determined by experimental data[6]. In obtaining (31) for the distribution function created by c ollisions, fur- ther collisions were excluded. To include possible further collisions, the 12contribution expressed by (31) needs to be modiﬁed as f(iii)(t,r,v,∆Ω) =1 v∆Ω/integraldisplay −∆Ωdr0/integraldisplay dc′/integraldisplay dΩu′ /bardblJ/bardbl \|r−r0\|22uσ(Ωu′)f(t0,r0,c′−u′)f(t0,r0,c′+u′)P(r0,r,v0),(37) where the integration of dr0is over the entire eﬀective cone deﬁned by r and−∆Ω,\|v0\| ≡vand takes the direction of ( r−r0),u′is deﬁned by u=\|c−v0\|and Ω u′. In these formulas, the probability Pshould be set to be zero at the very beginning if there is physical blocking along the path r0r. Equations (33)-(37) constitute a complete set of integral e quations that describe the collisional Boltzmann gas deﬁned in this paper . The formulation proves in a theoretical way an obvious intuition that the dis tribution function at a spatial point can directly be aﬀected by physical events taking place at other, even remote, points in view of the fact that a molecu le can freely pass any distance in a certain probability. In this sense, th e picture here is more “kinetic” than that associated with the Boltzmann equa tion, in which physical events have to make their inﬂuence region by region (like what happen in a continuous medium). Another comment is about the famous H-theorem. If the involved distribution function is initially nonuniform in the spati al space and non- Maxwellian in the velocity space, the resultant distributi on function given by the formalism will approach the uniform Maxwellian. Thou gh such ex- plicit proof has not been achieved yet, we believe that this m ust be the case by noticing a general discussion stating that as long as a sta tistical process is a Markoﬃan one the H-theorem must hold true[7]. 6 Approximation and application Although the formulation oﬀered in the previous section is f ormally com- plete, there still exist diﬃculties that hinder one from per forming calculation for a realistic gas. Unlike the solution for the collisionle ss Boltzmann gas, given by (2), the equation system in the last section, namely (33)-(37), is an integral-equation set. Without knowing the entire history of the distribution function f(t), the integrals in the system cannot be evaluated accuratel y. Fortunately, there are situations for which adequate appro ximations can be introduced and meaningful results can be obtained. In wha t follows, we 13ﬁrst deal with weakly collisional gases and then give some di scussion on how the consideration can apply to more general cases. If the density of a Boltzmann gas is relatively low, by which w e imply that the mean free path of molecules is not too short comparing wit h the length scale of the system or that the mean free time is not too short c omparing with the time scale of the phenomena of interest, we may apply the following iterating procedure to calculate the distribution functio n. Firstly, we assume that the system can be treated as a collisi onless Boltz- mann gas and the collisionless formulation can directly app lied. Namely, we have the zeroth-order solution f[0] (i)=1 ∆Ω/integraldisplay ∆S1fct(t0,r0,v,Ω0)\|cosα\|dS0 \|r−r0\|2(38) f[0] (ii)=1 v3∆Ω/integraldisplay ∆S2η(t0,r0,v,Ω0)dS0 \|r−r0\|2. (39) Then, we can construct the ﬁrst-order distribution functio n by inserting the zeroth-order solution into all right sides of the equati ons (33)-(37), which yields f[1] (i)=1 ∆Ω/integraldisplay ∆S1fct(t0,r0,v,Ω0)\|cosα\|dS0 \|r−r0\|2P[0](r0,r,v0) (40) f[1] (ii)=1 v3∆Ω/integraldisplay ∆S2η(t0,r0,v,Ω0)dS0 \|r−r0\|2P[0](r0,r,v0). (41) and f[1] (iii)=1 v∆Ω/integraldisplay −∆Ωdr0/integraldisplay dc′/integraldisplay dΩu′/bardblJ/bardbl \|r−r0\|2 2uσ(Ωu)f[0](t0,r0,c′−u′)f[0](t0,r0,c′+u′)P[0](r0,r,v0).(42) In all the ﬁrst-order formulas, the surviving probability i s deﬁned as P[0](r0,r;v0) = exp/bracketleftBigg −/integraldisplay l/integraldisplay v1/integraldisplay Ωu′2uσ(Ωu′)f[0](τ,rl,v1)dΩu′dv1dτ./bracketrightBigg .(43) In equations (42) and (43), f[0]is the total zeroth-order distribution function, namely f[0]=f[0] (i)+f[0] (ii). Along this line, we can formulate higher-order solutions fo r dilute gases. If the gas of interest is rather dense, the approximation met hod presented 14above may not work eﬀectively. One wishes, however, to point out that for the regions near boundaries, where the distribution functi on suﬀers from most irregularities and collisions between molecules have no enough time to erase such irregularities, the introduced method should still make sense. In view of this, it is expected that a hybrid method will be dev eloped, in which the approach here and other eﬀective approaches, such as the ordinary ﬂuid theory, can be combined into one scheme so that more prac tical gases become treatable. To illustrate the application of our approximation scheme, we investigate a gas leaking out of box through a small hole. For simplicity, we assume, referring Fig. 11, that the zeroth-order solution of the lea king gas is con- ﬁned to a “one-dimensional thin pipe” (shaded in the ﬁgure), which can be expressed by f[0]=/braceleftBigg C0exp/bracketleftbig−mv2 x/(2κT)/bracketrightbig(inside the pipe) 0 (outside the pipe)(44) and then we try to determine the collisional eﬀects of the dis tribution func- tion. Note that the distribution function expressed by (44) is kin d of special so that the formula (42) should be slightly modiﬁed. For this pu rpose, we write the diﬀerential collision probability as (the subindex xofvxis suppressed) [f(v′)∆Sdx0dv′][f(v′ 1)dv′ 1][2uσ(Ωu)dΩu], (45) where ∆ Sis the cross area of the pipe. By making the variable transfor ma- tion, we obtain /integraldisplay∞ 0dv′/integraldisplay∞ 0dv′ 1· · ·=/integraldisplay∞ 0dc′/integraldisplay+∞ −∞du′/bardblJ/bardbl · · · =/integraldisplay∞ 0dc′/integraldisplay∞ 0u2du4(u2)−1· · ·. In a way similar to that has been presented in the last section , we ﬁnally arrive at f[1][(∆θ)i] =∆S v(∆θ)i/integraldisplay −(∆θ)idx0∞/integraldisplay 0dc′8σ(Ωu) u\|r−r0\|2f[0](c′+u′)f[0](c′−u′),(46) where (∆ θ)iis the polar angle range set by the investigator (the azimuth al angle range is irrelevant in the case). 15The formula (46) can be calculated easily with a computer. Re ferring to Fig. 11b, we set v= 1, r⊥= 1,m 2κT= 1, let (∆ θ)ibe the interval /bracketleftbigg 0.4π−0.52 +i 50,0.4π−0.5 +i 50/bracketrightbigg and notice σ(Ωu) is constant[4]. The numerical results are listed as the following: (normalized by f[(∆θ)0]) f[(∆θ)0] = 1.00000e+ 00 f[(∆θ)5] = 5.23910e−01 f[(∆θ)10] = 2.01786e−01 f[(∆θ)15] = 5.03200e−02 f[(∆θ)20] = 4.76340e−03 f[(∆θ)25] = 5.11013e−05.(47) 7 Summary In this paper, we have proposed a complete mathematical sche me to deal with the Boltzmann gas. The scheme has many new features. In a ddition to those given in Ref. 1, some related to treating collisiona l eﬀects are the following. Firstly, collisional eﬀects are investigated in the full ve locity-and-position space. In particular, a six-dimensional volume element is e xplicitly deﬁned and a calculation concerning molecules entering the volume element is di- rectly performed. Secondly, both the colliding molecules and scattered molec ules are al- lowed to have distributions. In other words, we consider the full and collec- tive behavior of collisions, in which the time-reversal sym metry existing for a collision of two molecules plays almost no role. Thirdly, the treatment in this approach is consistent with t he previous approach to the collisionless Boltzmann gas in the sense tha t all the formulas are given in terms of what happen along molecular paths. Finally, the resultant formulas of this approach are, in man y practical situations, calculable by means of today’s computer. It is believed that this approach will be developed further s o that a better understanding of complicated ﬂuid phenomena can be achieve d. 16References [1] C.Y. Chen, A path-integral approach to the collisionless Boltzmann ga s, to be published. [2] C.Y. Chen, Perturbation Methods and Statistical Theories , in English, (International Academic Publishers, Beijing, 1999). [3] C.Y. Chen, Mathematical investigation of the Boltzmann collisional o p- erator , to be published. [4] L.D. Landau and E.M. Lifshitz, Mechanics , 3rd edition, (Pergamon Press, 1976). [5] See, for instance, F. Reif, Fundamentals of Statistical and Thermal Physics , (McGraw-Hill book Company, 1965). [6] M.N. Kogan, Rareﬁed Gas Dynamics , (Plenum Press, New York, 1969). [7] R. Kubo, H-theorems for Markoﬃan Processes inPerspectives in Sta- tistical Mechanics , edited by H.J. Revech´ e, (North-Holland, 1981). 17Figure captions 1. A physical surface and a virtual surface within the eﬀecti ve cone de- ﬁned by rand ∆Ω. 2. A collision between two molecules. (a) The molecular velo cities before the collision. (b) The molecular velocities after the colli sion. 3. The scattering cross section in the center-of-mass frame . (a) Solid an- gles and relative velocities. (b) The relation between the c ross section and solid angle range. 4. Two types of collision processes. 5. Eﬀective cones. (a) For a single spatial point. (b) For a gi ven spatial volume. 6. (a) A mental picture in which two molecular beams collide w ith each other. (b) A mental picture in which both colliding molecule s and scattered molecules have distributions. 7. (a) A solid angle range ∆Ω 0towards the point r. (b) The distribution of scattered molecules in the spatial space. (c) The distrib ution of scattered molecules in the velocity space. 8. The velocity distribution of scattered molecules in the c enter-of-mass frame and in the laboratory frame. 9. Relations between various essential vectors in the formu lation. 10. Contribution to the solid-angle-average distribution function from dif- ferent sources. 11. A gas leaking out of a container through a small hole. 18Figure 1 rpppppppppppppppppppppppp ∆Ω−∆Ω ∆S2∆S1 @@ Figure 2 (a) (b)- 1 ppppppppppppjHHHHHHH- UA A AAPPPPPPPPPPPq pppppppppppp *v′ 1 v′ 2c′u′v1 v2c upppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp Figure 3 (a) (b)
arXiv:physics/0010016v1 [physics.optics] 5 Oct 2000Phase-coherent frequency measurement of the Ca intercombi nation line at 657 nm with a Kerr-lens mode-locked femtosecond laser J. Stenger, T. Binnewies, G. Wilpers, F. Riehle and H. R. Tell e Physikalisch-Technische Bundesanstalt, Bundesallee 100 , 38116 Braunschweig, Germany J. K. Ranka, R. S. Windeler and A. J. Stentz Bell Laboratories, Lucent Technologies, 700 Mountain Ave, Murray Hill, New Jersey 070974, USA (February 2, 2008) The frequency of the Calcium3P1—1S0intercombination line at 657 nm is phase–coherently measured in terms of the output of a primary cesium frequency standard using an opti- cal frequency comb generator comprising a sub–10 fs Kerr-le ns mode-locked Ti:Sapphire laser and an external microstruct ure ﬁber for self–phase–modulation. The measured frequency of νCa= 455 986 240 494 276 Hz agrees within its relative un- certainty of 4 ×10−13with the values previously measured with a conceptually diﬀerent harmonic frequency chain and with the value recommended for the realization of the SI unit of length. So far, the use of optical frequency standards for the realization of basic units, e. g. of time and length, was hampered by the diﬃculties of measuring their high fre- quencies. As a result, the most precise frequency values recommended by the Comit´ e International des Poids et Measures (CIPM) are based on very few frequency mea- surements [1]. Recent progress in femtosecond pulse gen- eration and the advent of microstructure optical ﬁbers [2–4], however, have substantially facilitated such opti- cal frequency measurements. These schemes [5–7] start with the highly periodic pulse train of a Kerr–lens mode– locked laser which corresponds in the to a comb–like fre- quency spectrum of equidistant lines. The spectral span of this comb reﬂects the duration of an individual pulse while the spacing between the lines is determined by the pulse repetition frequency. It has been shown, that the fast, spectrally far–reaching Kerr–lens mode–locking mechanism enforces a tight coupling [8] of the optical phases of the individual lines. As a result, the frequency of any of these lines is given by an integer multiple of the pulse repetition frequency frepand a frequency νceo [9–11] which accounts for the oﬀset of the entire comb with respect to the frequency origin. If these quanti- ties are known, the unknown frequency of any external optical signal can be precisely determined by counting the frequency of the beat–note between this signal and a suitable comb line. In this Letter, we report the ﬁrst ap- plication of this method to phase–coherent measurements of the frequency of an optical Ca frequency standard op- erating on the1S0—3P1intercombination transition at657 nm. The frequency of this state–of–the–art standard has al- ready been measured with a conventional harmonic fre- quency chain [12,13], and the frequency value of νCa= 455 986 240 494 150 Hz has been recommended by the CIPM for the ’Practical realization of the deﬁnition of the metre (1997)’ [1] with a relative standard uncertainty of 6×10−13. The present experiments allow for the ﬁrst time the comparison of two conceptually diﬀerent phase– coherent optical frequency measurements, the conven- tional type operating in absolute–frequency domain and the novel type operating in diﬀerence–frequency domain. This comparison is performed without the need for inter- mediate transportable standards. Two of the three required measurements mentioned above, the pulse repetition rate frepand the multiple in- teger factor, are straightforward to carry out. The mea- surement of the c arrier–e nvelope o ﬀset frequency νceo, which arises from the relative velocity of the carrier phase and the pulse envelope, is a more demanding task. Sev- eral more or less complex schemes, depending on the comb span available, have been proposed for the mea- surement of νceo[10]. The simplest concept requires an octave span of the frequency comb which is not directly available from our laser. Thus, the available span has to be expanded, e. g. by external self–phase–modulation (SPM). Microstructure air–silica ﬁbers, which allow for tailoring the group velocity dispersion (GVD) properties [2] are highly suited for SPM of moderate peak power pulses available from mode–locked laser oscillators since they provide both lateral and temporal conﬁnement of the pulses over long interaction lengths. If the octave span is achieved, then the νceomeasurement can be ac- complished by second–harmonic generation (SHG) of the spectral lines in the low–frequency wing of the comb. Whereas the of the comb lines are shifted by νceowith respect to the origin, their harmonics are shifted by 2νceo. Thus, the beat notes between these harmonics and the corresponding high–frequency–wing lines of the comb show the desired component at νceo. 1FIG. 1. Schematic of the experimental set-up. The fre- quency comb generator consists of the Ti:Sapphire femtosec - ond laser and the microstructure ﬁber. SESAM denotes the semiconductor saturable absorber mirror, and PZT piezo transducers. Whereas the repetition rate is phase locked to the hydrogen maser, fCaandνceoare counted. LBO denotes the LiB 3O5–frequency doubling crystal, PD photo detectors, PM photomultiplier, PLL phase–locked loop tracking oscill a- tor, and FVC a frequency–to–voltage converter, respective ly. As depicted in Fig. 1, our measurement set–up consists of four parts: the frequency comb generator (i), detectors and electronics for servo control and/or data acquisition of the pulse repetition frequency (ii), of the beat note between the laser and Ca–standard radiation (iii) and of the carrier–envelope–oﬀset frequency (iv). i) The Kerr-lens mode-locked (KLM) Ti:Sapphire laser is similar to that described in [14]. The laser gener- ates pulses with a duration of ∼10 fs, a repetition rate frep= 100 MHz, an average power of 120 mW and a spectrum centered at 790 nm. The group velocity disper- sion (GVD) of its cavity is controlled by double–chirped mirrors and an intracavity fused silica prism pair. A similar external prism pair (not shown in Fig. 1) is used for pulse recompression. Approximately 30 mW of the laser output is coupled into a 10 cm long piece of mi- crostructure ﬁber with a core diameter of 1.7 µm and a zero–GVD wavelength of 780 nm [2]. The output spec- trum of the ﬁber was measured to cover the range from 520 nm to 1100 nm, while the comb structure with dis- tinct lines spaced by 100 MHz is expected to be preserved even in the extreme wings of the SPM–broadened spec- trum. This was conﬁrmed by measurement of the fringe– visibility (after appropriate optical ﬁltering) at the out - put of a two beam interferometer with 100 MHz fringe spacing yielding a 100 % visibility throughout the entire comb. ii) The repetition frequency of the laser was detected with a fast Si PIN photo diode yielding a 100 MHz signal with a S/N–ratio of 140 dB (BW = 1 Hz). It was phase– locked to the 100 MHz output signal of a hydrogen maser which was controlled by the PTB primary Cs standards.The actuator of this servo loop is a double piezo device (PZT1A and B) which tilts the output coupler where the spectral components are laterally displaced by the intra– cavity double prism GVD compensator. Tilting of the output coupler predominantally modiﬁes the resonator round–trip group delay [9] which, in turn, determines frep. iii) The frequency fCaof the beat note between the ra- diation from the Ca–standard (at optical frequency νCa) and the nearest comb mode was both measured and long– term stabilized. For this purpose, the output of the Ca standard was combined with the emission from the out- put of the microstructure ﬁber, which was ﬁltered by an interference ﬁlter. The beat note was detected by a Si PIN photo diode. Its output signal with a typi- cal S/N–ratio of 30 dB (BW = 100 kHz) was ﬁltered by a phase–lock–loop tracking oscillator (PLL) and subse- quently counted by a totalizing counter. The beat sig- nal frequency was kept within the hold–in range of the PLL by an additional servo loop controlling the length of the Ti:Sapphire–laser resonator with the help of a piezo transducer (PZT2). The error signal of this slow loop was generated by a frequency–to–voltage–converter (FVC) which monitored the output frequency of the PLL. Un- wanted cross–talk between the frepandfCaservo loops was minimized by adjustment of the lever point of the frep–tilting actuator to the lateral position of the red comb components at 657 nm. This was accomplished by choosing appropriate gain values (of opposite sign) for the piezo transducers PZT1A and B. iv) The frequency νceowas measured, as mentioned above, by external self–phase modulation in the mi- crostructure ﬁber and subsequent second–harmonic gen- eration in a nonlinear optical crystal. The infrared por- tion of the ﬁber output around 1070 nm was frequency doubled with a non–critically phase–matched, 10 mm long LiB 3O5(LBO) crystal, which was heated to about 140◦C in order to fulﬁll the type-I phase–matching con- dition for this wavelength. The beat note between the re- sulting green SHG signal and the green output of the ﬁber was detected by a photo multiplier (PM) after spectral and spatial ﬁltering both ﬁelds with a single mode ﬁber and a 600 l/mm grating, respectively. The PM output signal with a S/N ratio of up to 40 dB(BW = 100 kHz) was ﬁltered with a second tracking oscillator and counted by a second totalizing counter. Both counters were syn- chronously gated by the same 1 Hz clock signal derived from the 100 MHz hydrogen maser signal. The optical Ca frequency standard has been described elsewhere [13,15], and only the relevant details shall be given here. This standard is operated in a diﬀerent build- ing and is linked to the Ti:Sapphire laser by a 150 m long polarization–preserving single–mode ﬁber. Two indepen- dent systems are available, producing clouds of between 106and 107ballistic Ca atoms which are released from magneto–optical traps (MOT 1 and MOT 2). Each trap 2is loaded for 15 ms. Then the trapping lasers at 423 nm and the magnetic quadrupole ﬁelds are switched oﬀ, and a homogeneous magnetic ﬁeld of 0.23 mT is switched on for spectral separation of the Zeeman components. At the same time the cloud starts to expand and to fall down due to its thermal velocity ( T∼=3 mK) and gravity, re- spectively. After a settling time for the magnetic ﬁeld of 0.2 ms the atoms are interrogated by pulses from a frequency–stabilized dye laser. For the frequency mea- surements MOT 1 was employed, in which the low veloc- ity atoms are directly captured from a thermal beam by three mutually perpendicular counterpropagating beams with two diﬀerent frequencies. The atoms were excited in a time–domain analogue of the optical Ramsey excita- tion by two pulses separated by 216.4 µs followed by two pulses with the same separation from a counterpropagat- ing beam [15]. The frequency of the interrogating laser ﬁeld was stabilized to the central fringe (FWHM of the fringes 1.16 kHz) of the interference structure that occurs when the ﬂuorescence of the excited atoms is measured as function of the laser frequency. The uncertainty of the frequency of the Ca optical frequency standard dur- ing both measurement series was estimated to 53 Hz. A more detailed uncertainty budget is given in [15]. Sev- eral tests have been performed to check the validity of this estimate. By reversing the temporal order of the excitation pulses a shift of a few Hertz was determined. The frequency of the laser stabilized to atoms of MOT 1 was compared with its frequency when stabilized to a second MOT 2. Here, the atoms of an eﬀusive beam are decelerated in a Zeeman slower and these slow atoms are deﬂected and directed towards the magneto–optical trap [16]. The frequency diﬀerence measured when the laser was stabilized alternatively to atoms from MOT 1 and MOT 2 by three pulses of a standing wave was 12±15 Hz. An additional laser ﬁeld at 672 nm was used in MOT 1 to repump the atoms which are lost via the1D2 state [17]. The magnetic ﬁeld of 0.23 mT caused a second order Zeeman shift of the Ca frequency of +3.1 Hz. νCa− 1σ[Hz] relative 1 σ νCa(CIPM )[Hz] uncertainty Series A 100 328 7 .2×10−13 Series B 148 132 2 .9×10−13 Series A+B 126 180 4 .0×10−13 Previous Meas. [15] -20 113 2 .5×10−13 CIPM recomm. [1] 0 270 6 .0×10−13 TABLE I. The diﬀerences of the measured Ca3P1—1S0 transition frequencies in respect to the value recommended by CIPM and the absolute and relative uncertainties. Series A and B refer to the frequency comb generator and previous measurement refers to the harmonic chain.Two measurement series were carried out on diﬀerent days with a total measurement time of 3000 and 4200 s, respectively. On these days the frequency deviation of the hydrogen maser and the Cs clock resulted in deviations of the measured Ca frequency of +26 Hz and 24 Hz, respec- tively. The deviations of the corrected measured mean values from the CIPM–recommended value of νCa(CIPM ) = 455 986 240 494 150 Hz are listed in Tab. 1 together with the corresponding relative 1 σuncertainties, which comprise the estimated relative frequency uncertainties of the standards and those of the frequency measure- ments. The values of the previous measurement [13] us- ing the harmonic frequency chain and the CIPM values [1] are shown for comparison. FIG. 2. The Allan standard deviation of measurement se- ries B is plotted versus the averaging time τtogether with the frequency instabilities of the Ca standard and the hydro - gen maser. The inset shows the distribution of the measured values of series B (averaging time = 1 s, bin-width = 1 kHz, σ= 5.75 kHz). Fig. 2 shows the Allan standard deviation of the data of series B together with the corresponding frequency in- stabilities of the Ca standard [18] and those of the hy- drogen maser [19]. The inset of Fig. 2 shows a histogram of the data of series B (1 s averages) with respect to the CIPM–recommended frequency. The center frequency of the ﬁtted Gaussian of νCa(CIPM )+ 46 Hz agrees within the uncertainty with the mean value of series B. The in- stability of the frequency measurement process is about one order of magnitude larger than the frequency insta- bility of the Ca standard and of the hydrogen maser. So far, the measurements are still limited by the non– optimum detection of the frequency ﬂuctuations of the Ti:Sapphire laser. Substantial improvement of the mea- 3surement process can be expected if the instantaneous residual phase error between frepand the hydrogen maser reference signal is simultaneously measured, i.e. at a high harmonic of frep. To conclude, the frequency of an optical frequency standard based on the Calcium3P1—1S0intercombina- tion transition at 657 nm has been measured in terms of the output of a primary cesium frequency standard using a novel type of optical frequency chain. Accu- rate frequency measurements of this optical standard are of particular importance since it represents one of the realizations of the unit of length with lowest uncer- tainty. Furthermore, the optical frequency of the Ca standard is used as a reference for highly accurate fre- quency measurements of other optical or ultraviolet tran- sitions [20]. The frequency value presented in this Letter agrees within its relative uncertainty with the CIPM– recommended value, which is based on a measurement with a harmonic frequency chain. The agreement justiﬁes the conﬁdence in the data obtained from both measure- ment schemes. The advantages of the novel scheme em- ployed here, such as the substantially reduced complexity and the accurately known, dense grid of reference fre- quencies throughout the visible and near infrared range, allows a variety of new applications including the real- ization of optical clocks or ultra–precise measurements in fundamental physics. We gratefully acknowledge ﬁnancial support from the Deutsche Forschungsgemeinschaft through SFB 407, ex- perimental assistance by B. Lipphardt and H. Schnatz, and help for the set–up of the Ti:Sapphire laser by G. Steinmeyer and U. Keller. [1] T. J. Quinn, Metrologia 36, 211 (1999). [2] J. K. Ranka, R. S. Windeler, and A. J. Stentz; Opt. Lett. 25, 25 (2000). [3] J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin; Opt. Lett. 21, 1547 (1996). [4] N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson; Opt. Lett. 24, 1395 (1999). [5] A. Apolonski et al.; Phys. Rev. Lett. 85, 740 (2000). [6] S. A. Diddams et al.; Phys. Rev. Lett. 84, 5102 (2000). [7] M. Niering et al.; Phys. Rev. Lett. 84, 5496 (2000). [8] T. Udem, J. Reichert, R. Holzwarth, and T. W. H¨ ansch; Opt. Lett. 24, 881 (1999). [9] J. Reichert, R. Holzwarth, T. Udem, and T. W. H¨ ansch; Opt. Commun. 172, 59 (1999). [10] H. R. Telle et al.; Appl. Phys. B 69, 327 (1999). [11] D. J. Jones, S. A. Diddams, and J. K. Ranka; Science 288, 635 (2000). [12] H. Schnatz et al.; Phys. Rev. Lett. 76, 18 (1996). [13] F. Riehle et al.; IEEE Trans. Instrum. Meas. IM 48 , 613 (1999).[14] D. H. Sutter et al.; Opt. Lett. 24, 631 (1999). [15] F. Riehle et al.; inProceedings of the 1999 Joint Meet- ing of the European Frequency and Time Forum and The IEEE International Frequency Control Symposium (EFTF co/Soci´ et´ e Francaise des Microtechiques et de Chronom´ etrie (SFMC), Besancon, France) p. 700 (1999). [16] T. Kisters, K. Zeiske, F. Riehle, and J. Helmcke; Appl. Phys. B 59, 89 (1994). [17] C. W. Oates, F. Bondu, R. W. Fox, and L. Hollberg; Eur. Phys. J. D 7, 449 (1999). [18] F. Riehle et al.; inTrapped Charged Particles and Fun- damental Physics , edited by D. H. E. Dubin and D. Schneider (American Institute of Physics, Woodbury, New York, 1999), pp. 348–356. [19] A. Bauch; Time–Unit–Section of the Physikalisch– Technische Bundesanstalt, Braunschweig, Germany, pri- vate communication (2000). [20] K. R. Vogel et al.; submitted to Opt. Lett. (2000). 4
1 Can Zero-Point Phenomena truly be the Origin of Inertia? Charles T. Ridgely charles@ridgely.wsGalilean Electrodynamics, in review (2000) Abstract A current approach to the problem of inertia suggests that the origin of the inertial properties of matter is the interaction between matter and vacuum electromagnetic zero-point radiation. Herein, it is shown that zero-point phenomena can be treated as theorigin of inertia only when one chooses to ignore the mass-energy content of matter. Inthe absence of any physical basis for such a choice, it is concluded that zero-point-induced forces must arise in addition to the intrinsic inertial properties of ordinary matter. 1. Introduction Several recent papers have proposed that the origin of the inertial properties of matter can be explained by expressing the inertial mass m in Newton's second law of motion, ()dmdt=fv ,( 1 ) entirely in terms of the vacuum electromagnetic zero-point-field (ZPF).1-5 According to the ZPF proposal, when a body of ordinary matter is accelerated by an external force, thequarks and electrons constituting the body scatter a portion of the zero-point radiationoccupying the space through which the body moves. This scattering of radiation leads toan electromagnetic drag force on the body, which according to ZPF proponents can beassociated with the inertial resistance of the body. 1-5 The ZPF hypothesis seems2 compelling because it not only provides a local description of the inertial properties of matter, but also provides an electromagnetic basis for inertial and gravitational forcesalike. Supporters of the ZPF hypothesis have performed a great deal of work inattempting to show that the interaction between matter and zero-point radiation gives riseto observable forces. 1-5 Even so, the ZPF hypothesis still seems to suffer from at least a couple of inherent conceptual difficulties. Certainly, a real zero-point radiation field must give rise some level of resistance on accelerating matter. The question is, however, what fraction of the inertial force on anaccelerating object is actually induced by interaction with the ZPF? This seems to beultimately hinged on how one chooses to view E = mc 2, referred to as the “law of inertia of energy” in earlier times.6 Those who support the ZPF hypothesis interpret the energy content of ordinary matter as entirely kinetic energy of quarks and electrons induced byinteraction with zero-point radiation. 1,4 Of course, the problem with this interpretation is that it implies that subatomic particles have no intrinsic rest mass-energy content. Thepresent analysis employs the more traditional interpretation that E = mc 2 is merely a statement that all forms of energy possess inertial properties.7-12 Since subatomic particles are known to possess rest mass-energy, such particles ought to exhibit intrinsicinertial properties in addition to any effects arising due to zero-point radiation. Another problem is that the ZPF hypothesis ascribes the inertial properties of matter entirely to the inertial mass m in Newton's second law of motion, 1-5 while ignoring the underlying participation of space-time.7-12 The behavior of space-time clearly forms the basis necessary for the existence of ZPF-induced forces. Since the vacuumelectromagnetic ZPF is Lorentz-invariant, if space-time were incapable of distortion the3 ZPF would remain undetectable in all systems of reference. As a result, ZPF-induced forces would not occur. Of course, space-time distortion does occur and does give rise toobservable forces, as is suggested by general relativity. 13-18 Based on this, the objective of the present analysis is to demonstrate that resistance forces induced by the ZPF act inaddition to the intrinsic inertial properties of ordinary matter. In the next Section, a block of matter undergoing uniform acceleration is considered. It is pointed out that while the ZPF cannot be detected in an inertial system of reference,this is not so in an accelerating system. Observers residing on the accelerating blockdetect a flux of zero-point radiation passing through the block. The force density due tothis flux of radiation is derived for the case of a small, particle-sized block. Uponexpressing the force in terms of space-time distortion within the accelerating system, 11 it becomes straightforward to see that ZPF-induced forces arise due to the behavior ofspace-time. In Section 3, the expression for the force derived in Section 2 is applied to the case in which a body accelerates uniformly through zero-point radiation. To derive the totalresistance force acting on the body, the expression for the force is modified to include allforms of energy possessed by the accelerating body. 7-12 Carrying this out leads to an expression for the resistance force having two terms. One term arises due to the body’sinteraction with the ZPF, and is identical to the force derived solely on the basis of theZPF hypothesis; 1-4 the other term arises due to the body’s intrinsic rest mass-energy. The form of this expression makes it clear that zero-point phenomena cannot be the origin ofinertia; the intrinsic mass-energy of matter must be taken into account, as well. Withoutgood reason for intentionally disregarding the intrinsic energy content of matter, it is4 concluded that forces induced by interaction with zero-point radiation arise in addition to the intrinsic inertial properties of matter. 2. The Resistance Force on Accelerating Matter due to Zero-Point Radiation As mentioned in the Introduction, zero-point radiation cannot be detected in an inertial system of reference. This is not the case in an accelerating system, however. Consider a block of matter, of volume 0V, undergoing uniform acceleration. Observers residing in the comoving reference frame (CMRF) of the block detect a flux of zero-point radiation entering the block through the front side, which we may call wall A, and passing out the block through the back side, which we call wall B. According to these observers, the radiation within the volume of the block possesses a momentum density of the form ()21 ABc∆= − SS pppp ,( 2 ) where AS and BS are the Poynting vectors corresponding to the flux of zero-point radiation detected at walls A and B, respectively. Expressing the Poynting vectors in terms of the energy density of the ZPF at each wall leads to AAcu=Sn , (3a) BBcu=Sn , (3b) where Au and Bu are the energy densities of the ZPF at walls A and B, and n is a unit vector in the direction of the block's acceleration. Since the block is accelerating, zero- point radiation gains energy as it traverses the length of the block, from wall A to wall B. As a result, radiation detected at wall B appears blue-shifted relative to radiation detected5 at wall A. Taking this into account, the Poynting vector given by Eq. (3b) can be expressed as 2 A BA Bdcudτ τ=Sn ,( 4 ) where Adτ and Bdτ are intervals of proper time experienced by comoving observers situated at walls A and B, respectively. Using Eqs. (3a) and (4) in Eq. (2), the momentum density of zero-point radiation within the block becomes 2 1AA Bud cdτ τ∆= − n pppp .( 5 ) Expressing this in terms of coordinate time in the accelerating system, and simplifying a bit, then leads to 22 2 AA ABu d dt dt cd t d dτ ττ  ∆= −   n pppp ,( 6 ) where dt is an interval of coordinate time in the accelerating system. Equation (6) shows that the momentum density of zero-point radiation within the block is a directmanifestation of space-time structure in the accelerating system. 10-12 Were space-time incapable of distortion, the expression within square parentheses guarantees that the zero-point momentum density would be zero in all systems of reference. The force density possessed by zero-point radiation passing through the block can be derived by using τ=∆ ∆fpfpfpfp , where τ∆ is an interval of proper time in the CMRF of the block. Comoving observers at wall A can express τ∆ in terms of the block's length, x′∆, along the x′−axis of the CMRF, by noticing that the time taken for a light signal to6 complete a round trip across the block is 2xcτ′ ∆=∆ . Using this time interval and Eq. (6), the force density within the block can then be expressed as 22 2 ˆ 2AA ABu d dt dt xd t d dτ ττ ′  =−  ′∆ x ffff .( 7 ) where ˆ′x is a unit vector in the direction of the block’s acceleration along the x′−axis of the CMRF. Equation (7) holds for large volumes in which ABddττ≠ and x′∆ assumes sizeable values. However, when the volume of the block is very small, then ABddττ≈ and x′∆ assumes very small values. The force density within a particle-sized volume can be expressed by taking the limit of Eq. (7) as Bdτ tends to Adτ and x′∆ tends to zero: 0BAdd xLim ττ→ ′∆→′=ffffffff .( 8 ) Carrying out the limit and noting that B dt dτ> A dt dτ, the force density assumes the form11 0 ˆ lndtuxdτ∂ ′′=− ′∂x ffff ,( 9 ) where 0u is the proper energy density of the ZPF according to observers residing in the CMRF of the particle. Equation (9) expresses the force density of zero-point radiation passing through a particle of matter undergoing substantial acceleration along the x′–coordinate axis of the CMRF. When the acceleration is weak, however, the force density simplifies to10-11 0 ˆdtuxdτ∂′′=− ′∂x ffff . (10)7 And when transformed from the accelerating system to flat Minkowski space-time, the force density becomes10 0dtudτ=−ffff∇∇∇∇ , (11) where it is assumed that v << c, and the prime has been dropped from ′ffff for simplicity. According to observers in Minkowski space-time, when an external force f is exerted on a particle, a resistance force ZPFf arises due to interaction between the particle and zero-point radiation. Using Eq. (11), observers in Minkowski space-time can express this resistance force as 00 ZPFdtuVdτ=− f ∇∇∇∇ , (12) where 0V is the volume of the accelerating particle. Equation (12) gives the resistance force acting on an accelerating particle due to the scattering of zero-point radiation. According to Eq. (12), the force on the particle arises not only due to the presence ofzero-point radiation, but also due to the distortion of space-time characterized by dt dτ.10-12 It is interesting to note that when dt dτ assumes a constant value, the force given by Eq. (12) is then zero. This suggests that space-time is intimately involved in the generation of ZPF-induced forces. 3. The Total Resistance Force on a Uniformly Accelerating Body In the preceding Section, it was shown that ZPF-induced forces arise due to the behavior of space-time. In this Section, it is shown that zero-point radiation can be thesole origin of inertia only when the intrinsic energy content of matter is ignored.8 According to the ZPF hypothesis, when a material body undergoes acceleration due to an external force, the quarks and electrons constituting the body scatter a portion of thezero-point radiation passing through the body. The energy density of this portion ofradiation is 1-5 ()3 232ZPFudcωηω ωπ=∫!, (13) in which the fraction of zero-point radiation that actually interacts with the accelerating body is governed by the spectral function ()ηω. The interaction between the body and zero-point radiation imparts a quantity of energy to the body. However, another form of energy that must also be taken into account is the intrinsic rest mass-energy of the accelerating body.7-12 Thus, the total energy of the accelerating body must then be ()3 00 232EEV dcωηω ωπ=+∫!, (14) where 0E is the rest mass-energy of the body. In order to determine an expression for the total resistance force acting on the accelerating body, Eq. (12) must be amended to include all forms of energy possessed by the body.7-12 This calls for replacing 00uV in Eq. (12) with the body’s total energy, E. Carrying this out and then inserting the total energy given by Eq. (14) leads to ()3 00 232dtEV dcdωηω ωπτ =− +  ∫f!∇∇∇∇ . (15) According to stationary observers, this is the resistance force acting on a uniformly accelerating body of rest mass-energy 0E that interacts with zero-point radiation. To9 simplify Eq. (15), we may consider a particular body accelerating uniformly along the x−coordinate axis. For this case, we may put12 21dt ax dcτ≈+ , (16a) 2dt dcτ=a∇∇∇∇ . (16b) Substituting the second of these expressions into Eq. (15), and simplifying a bit, then leads to ()3 0 0 22 32Vmdccωηω ωπ=− −∫fa a!, (17) in which 2 00Em c= was used to simplify the first term. Equation (17) is the total resistance force exerted on a body of mass 0m that is accelerating uniformly through zero-point radiation. With the exclusion of the first term, Eq. (17) is identical to the inertial force derived on the basis of the ZPF hypothesis.1-4 In fact, supporters of ZPF hypothesis claim that the inertial mass of an accelerating bodyshould be expressed entirely by the expression within parentheses, in the second term of Eq. (17). Clearly, this makes sense only when one chooses to set 00 m=. This is nothing more than a choice, of course; there exists no rational basis for purposely dismissing the intrinsic mass-energy of matter. The interaction between matter and zero-point radiationmust certainly lead to forces on accelerating objects, 1-5 but such forces must arise in addition to the inertial forces arising due to the intrinsic energy content of matter.10 4. Discussion and Conclusions The zero-point field (ZPF) approach to the problem of inertia asserts that the origin of inertia rests entirely with the interaction between the subatomic particles constitutingmatter and vacuum electromagnetic zero-point radiation. 1-5 As mentioned in the Introduction, one problem with this approach is that it seems to disregard the intrinsicenergy content of matter. According to the ZPF hypothesis, the rest mass-energy contentof matter is entirely kinetic energy of quarks and electrons induced by interaction withzero-point radiation. 1,4 Of course, the present analysis has adhered to the more traditional interpretation that all forms of matter possess intrinsic rest mass-energy, and that allforms of energy exhibit inertial properties in accordance with the law of inertia of energy, 2Em c= .7-12 Since subatomic particles such as quarks and electrons are each endowed with an intrinsic rest mass-energy, it makes sense that these particles must possessintrinsic inertial properties aside from zero-point phenomena. Another problem with the ZPF hypothesis is that it suggests that the origin of the inertial properties of matter rests solely with the inertial mass appearing in Newton’ssecond law of motion, 1-5 while neglecting the crucial role played by space-time.7-12 Herein, it has been shown that ZPF-induced forces manifest directly due to the behaviorof space-time in accelerating systems, and that such forces act in addition to the inertialforces arising due to the intrinsic energy content of matter. Clearly, the only way zero-point phenomena can be treated as the source of inertia is when the intrinsic energycontent of matter is intentionally ignored. In the absence of any good reason for this, theonly rational conclusion to be had is that ZPF-induced forces are resistance forces actingin addition to the intrinsic inertial properties of matter.11 Notes and References 1B. Haisch and A. Rueda, “On the relation between a zero-point-field-induced inertial effect and the Einstein-de Broglie formula,” Phys. Lett. A 268, 224, (2000). 2A. Rueda and B. Haisch, “Contribution to inertial mass by reaction of the vacuum to accelerated motion,” Found. Phys. 28, 1057 (1998). 3A. Rueda and B. Haisch, “Inertia as reaction of the vacuum to accelerated motion,” Phys. Lett. A 240, 115 (1998). 4B. Haisch, A. Rueda, and H. E. Puthoff, “Advances in the proposed electromagnetic zero-point-field theory of inertia,” 34th AIAA/ASME/SEA/ASEE Joint PropulsionConference, AIAA paper 98-3134, (1998). 5B. Haisch, A. Rueda, and H. E. Puthoff, “Physics of the zero-point field: implications for inertia, gravitation and mass,” Speculations in Science and Technology 20, 99 (1997). 6Max Born, Einstein’s Theory of Relativity (Dover, New York, 1965), p. 283. 7A. Einstein, “Does the inertia of a body depend upon its energy-content?,” in Einstein, The Principle of Relativity (Dover, New York, 1952), pp. 69-71. 8Max Born, Ref. 6, p. 286. 9H. Weyl, Space-Time-Matter (Dover, New York, 1952), 4th ed., p.202. 10C. T. Ridgely, “On the nature of inertia,” Gal. Elect. 11, 11 (2000). 11C. T. Ridgely, “On the origin of inertia,” Gal. Elect. 12, 17 (2001). 12C. T. Ridgely, “Space-time as the source of inertia,” Gal. Elect. 13, 15 (2002). 13A. Einstein, The Meaning of Relativity, Including the Relativistic Theory of the Non- Symmetric Field (Princeton, New Jersey, 1988), 5th ed., pp. 57-58.12 14Max Born, Ref. 6, pp. 313-317. 15P. G. Bergmann, Introduction to the Theory of Relativity (Dover, New York, 1976), p. 156. 16B. F. Schutz, A first course in general relativity (Cambridge, New York, 1990), p.122. 17H. Ohanian and R. Ruffini, Gravitation and Spacetime (Norton, New York, 1994), 2nd ed., pp. 53-54. 18I. R. Kenyon, General Relativity (Oxford, New York, 1990), p. 10.
arXiv:physics/0010019v1 [physics.atom-ph] 6 Oct 2000ATOMIC ANTENNA MECHANISM IN HHG AND ATI M. Yu. Kuchiev(a), V. N. Ostrovsky(b) (a)The University of New South Wales, Sydney, Australia (b)Institute of Physics, The University of St Petersburg, Russ ia 1 Introduction This paper reviews recent development of the atomic antenna , a theoretical framework which describes a number of laser-induced multiphoton phen omena in atoms. The localization of atomic electrons inside an atom drasticall y suppresses their interaction with a laser ﬁeld. For many processes this circumstance favo rs multistep mechanisms when at ﬁrst one of atomic electrons is released from an atom b y the ﬁeld. After that an interaction of the ejected electron with the laser ﬁeld re sults in absorption of energy from the ﬁeld and its accumulation in the form of the electron wiggling energy and ATI energy. It is very essential that the energy absorbed by the e lectron can be transferred to the parent atomic core via an inelastic collision of the pr imarily ejected electron with the atom. The collision may trigger a number of phenomena inc luding high harmonic generation (HHG), enhancement of above-threshold ionizat ion (ATI), production of multiply charged ions. In this physical picture the absorpt ion of energy from the ﬁeld takes place in the region of large separations from an atom, w here the electron-laser interaction dominates over the electron-core potential. T his circumstance results in dramatic enhancement of the probability of multiphoton pro cesses. Such a scenario of photoabsorption was suggested, apparently for the ﬁrst tim e, in Ref. [1]. Later the idea was rediscovered by several authors in diﬀerent contexts [2 , 3, 4]. In current literature the above described sequence of events is often referred to a s the rescattering, the three- step mechanism, or even the simpleman model. The term atomic antenna , suggested in Ref. [1], refers to the fact that the ﬁrstly emitted electr on plays a role similar to an aerial in conventional radio devices, enhancing the absorp tion. It is very important that the physical picture drawn above ca n be implemented not only as a model, but also as a clear and rigorous quantum forma lism. In this paper we outline two convenient ways to implement the atomic antenna idea. The one originating from Ref. [1] is called the factorization technique. It was f ormulated in detail in Ref. [5]and recently applied to HG in Refs. [6, 7]. We discuss also ano ther, complimentary technique, which is referred to below as the method of eﬀecti ve ATI channels. It is close in spirit to the approach developed previously by Lewe nstein et al[8, 9] and can in turn be linked to the Corcum model [2]. From the ﬁrst glance , these two schemes diﬀer very signiﬁcantly. However, we prove their identity b y demonstrating that they describe the same physical idea from diﬀerent points of view . The paper is organized as follows. Section 2 describes the fa ctorization technique which allows one to present the amplitude of the ”complicate d” multiphoton process as a product of the amplitudes of much more simple, ”elementary ” processes. Section 3 is devoted to the concept of eﬀective ATI channels, which pro vides important insights into the physical nature of complicated multiphoton proces ses. We derive the eﬀective ATI channels from the factorization technique revealing cl ose links between the two approaches. Section 4 describes two important examples of ” elementary” processes: one of them is the photoionization, another one is the electr on-atom collision in a laser ﬁeld which results in the generation of the high-energy quan ta. These two ”elementary” processes are vital for description of HG in the framework of the factorization technique. Sections 5, 6 are devoted to several examples illustrating n umerically an accuracy of methods developed for HHG and ATI. A number of results report ed in this paper are derived neglecting the Coulomb ﬁeld of the residual atomic p article which inﬂuence the active atomic electron. This approach is well justiﬁed for n egative ions, but needs to be modiﬁed for the processes with neutral atoms. Section 7 ex poses our recent progress based on the eikonal approach which allows us to take into acc ount the Coulomb ﬁeld. The concluding Section 8 summarizes the results. The atomic units are used throughout the paper unless indicated otherwise. 2 Factorization technique The factorization technique of Ref. [5] can be applied to a nu mber of multiphoton processes. In this section our attention is restricted to an important example of HG which has been recently considered by the present authors [6 , 7]. We concentrate on the linearly polarized laser ﬁeld F(t) =Fcosωt (1) that creates the external potential V(t) =F(t)·racting on atomic electrons. Using the second order time-dependent perturbation theory one ca n present the amplitude of HHG d+ Nin the single-active-electron approximation in the follow ing form d+ N=−i T/integraldisplayT 0dt/integraldisplayt −∞dt′/angbracketleftφa(t)\|exp(iΩt)U G(t, t′)V(t′)\|φa(t′)/angbracketright. (2) Here the brackets /angbracketleft \| \| /angbracketrightimply integration over coordinates, T= 2π/ωis the laser period. The initial-state wave function is φa(r, t) =φa(r) exp(−iEat) (3) where Ea=−κ2/2 is the bound state energy and φa(r) is the corresponding atomic eigenfunction. The potential U=ǫ·raccounts for production of the high harmonic withthe frequency Ω = Nωand the polarization ǫ. The Green function G(t, t′) describes the electron propagation in the intermediate state. The amplit ude (2) takes into account only a sequence of events in which HHG follows the absorption of the necessary energy from the laser ﬁeld. A number of omitted, the so called time-r eversed sequences, is strongly suppressed due to multiphoton nature of the proces s. Neglecting the potential of the atomic core (a possible way t o lift this approximation is considered in Section 7), one can present the Green functi on via a complete set of the Volkov wave functions φp(r, t) that account for the electron dressing by the laser ﬁeld G(r, t;r′, t′) =/integraldisplay φp(r, t)φ∗ p(r′, t′)d3p (2π)3, (4) φp(r, t) = exp  i /parenleftBigg p+F ωsinωt/parenrightBigg ·r−1 2/integraldisplayt/parenleftBigg p+F ωsinωτ/parenrightBigg2 dτ   .(5) The multiphoton nature of the problem makes the phases of the integrand in (2) to vary rapidly with t,t′. This circumstance allows one to use the saddle-point appro ximation for integrations over the time variables. After the accurate integration over the momenta pin (4), see details in [5, 6, 7], it is possible to show that (2) , (4), (5) lead to the following convenient presentation of the HHG amplitude d+ N= 2/summationdisplay md+ Nm, (6) d+ Nm=Am µ0(Km)BN mµ 0(Km). (7) Each term in (6) is written as a product of two amplitudes of physical, fully accomplished and observable processes; no ”oﬀ-shell” entities appear. Therefore Eqs. ( 6), (7) clearly demonstrate the stepwise character of the process. The ﬁrst step is described by the ﬁrst factor Am µ0(Km) which is an amplitude of physical ATI process when after abs orption ofmlaser photons the active electron acquires a translational momentum p=Km. In the Keldysh-type approximation this amplitude can be evalu ated using the saddle-point technique discussed in Section 4. The subscript µ0inAm µ0speciﬁes the contribution of one particular saddle point in t′integration, see more details in Section 4. The other factor, BN mµ 0(Km), is a combined amplitude of the second and third steps , which are the propagation and laser assisted recombination (PLAR). I t can be further factorised into the propagation factor 1 /Rmµ0describing the second step and the amplitude of the third step , which is the laser assisted recombination (LAR), CNm(Km): BN mµ 0(Km)≃1 Rmµ0CNm(Km). (8) Rmµ0is merely an approximate expression for the distance passed by the active electron in course of its laser-induced wiggling motion 1 Rm µ0=−ω2 Fcosωt′mµ0. (9) The amplitude CNm(p) of the physical LAR process describes the laser assisted re com- bination, i.e. transition of an electron with momentum pfrom the continuum to thebound state. Since the continuum state is laser-dressed, th e recombining electron can emit the N-th harmonic photon, gaining necessary extra energy from th e laser ﬁeld. More details on LAR amplitude are given in Section 4. The summation in formula (6) runs over a number of photons mabsorbed on the ﬁrst step of the process when the active electron is released . In other words one can say that the energy conservation constraint selects the discre te set of ATI channels in the laser-dressed continuum. These channels serve as intermed iate states for the three-step HG process. In a given channel the electron has translationa l momentum Kmwith the absolute value deﬁned by Km=/radicalBig 2 (mω−Up+Ea). (10) HereUp≡F2/(4ω2) is the well-known ponderomotive potential. ATI plays a rol e of the ﬁrst stage of HHG process only if the electron momentum ha s speciﬁc direction , namely Kmis directed along F. This ensures eventual electron return to the core that makes the ﬁnal step, LAR, possible as discussed in detail in R efs. [5, 6, 7]. The observable HG rates RNare expressed via the amplitudes as RN≡Ω3 2πc3/vextendsingle/vextendsingle/vextendsingled+ N/vextendsingle/vextendsingle/vextendsingle2, (11) Ω =Nωis the frequency of emitted harmonic, cis the velocity of light. Eq. (6) is the major result of this section. It presents the am plitude of the “com- plicated” HG process in terms of the amplitudes of “elementa ry” processes which are the ionization and LAR. Conceptual signiﬁcance of this resu lt is based on the fact that it supports the three-step interpretation of the HG process which, in turn, stems from the physical picture of the atomic antenna discussed in Sect ion 1. Eq. (6) is also very convenient for numerical applications, since a knowledge o n suﬃciently simple elemen- tary amplitudes enables one to calculate accurately the HG p rocess, as discussed below in Section 5. 3 Eﬀective channels The summation over min Eq. (6) has a clear physical interpretation. After the ﬁrs t step (ionization) the released electron can be found in any A TI channel before colliding with the parent atom. After the third step all intermediate c hannels result in the same ﬁnal electron state. Therefore the contributions of interm ediate ATI level interfere, as shows the summation over min Eq. (6). The interference has several prominent manifestations including the cutoﬀ of the HHG rates for high Nand their oscillations in the plateau domain, as discussed in Section 5. The number o f the ATI channels which give signiﬁcant contribution to the amplitude is usua lly large δm≫1. (12) This circumstance does not pose a problem for numerical appl ications based on Eq. (6), but may in some situations obscure the qualitative analyses . To overcome this diﬃculty it is desirable to carry out summation over min (6) in an analytical form. This is the major task of this Section which follows the approach of Ref. [10].Let us verify ﬁrst that the summation over min (6) can be replaced by integration over the related continuum variable d+ N=/summationdisplay md+ Nm≃/integraldisplay d+ Nmdm . (13) To prove the reliability of the approximation based on (13) w e use the Poisson sum- mation formula which allows one to present the amplitude of H HG in the following form/summationdisplay md+ Nm=/summationdisplay j/integraldisplay dmexp(−2πijm)d+ Nm. (14) The summation index jin (14) may be looked at as a variable which is conjugate to the mvariable. Since mωrefers to the spectrum, 2 πj/ω should be identiﬁed as a time variable or, more accurately, as a number of periods of t ime that elapsed between the ejection of the electron from an atom and its return back. The wave function of the released electron spreads in space the more the larger jis. Therefore one should anticipate that the most important contribution to (14) is g iven by the major term j= 0. Same conclusion can be drawn from Eq. (13). The estimate s hows that the spectral variable mcovers a wide region which causes the time variable jto be localized. This discussion demonstrates that one can safely take the le ading j= 0 term in (14) thus supporting (13). An advantage of integration over the variable min (13) is that it can be carried out in closed analytical form using the saddle-point method. To specify this statement let us return to Eqs. (2), (4), (5). One can deduce from them that t he major dependence of the phase of the integrand of (2) on integration variables is associated with the factor exp(iS) where Sis the classical action S=S(t, t′, m) =/integraldisplayt t′dτ 1 2/parenleftBigg Km+F ωsinωτ/parenrightBigg2 −Ea + Ωt . (15) Since we know from Eq. (7) that the momentum variable arises i n the ﬁnal formulas asKm, we can use this momentum in (15) instead of the integration v ariable pthat originates from (4). It should be noted that Eq. (7) where Kmarises was derived using accurate integration over all momenta p; hence the substitution p→Kmin (15) is not an additional approximation. Eq. (6) was obtained using the saddle points approximation f or integrations over the time variables t,t′. Positions of m-dependent saddle points tm,t′ mare governed by the equations ∂ ∂tS(t, t′, m) = 0 , (16) ∂ ∂t′S(t, t′, m) = 0 , (17) in which mis considered as an integer labeling the physical ATI channe l. As was mentioned above, Eq. (13) opens a possibility for integrati on over mthat can also be carried out using the saddle point approximation. The posit ions of corresponding saddle points are governed by the following equation ∂ ∂mS(t, t′, m) = 0 , (18)where one can write the partial derivative over mbecause the m-dependence of tmand t′ mdoes not contribute due to (16), (17). Eqs. (17), (16), (18) d eﬁne two instants of timetm,t′ mat which, respectively, the electron emerges from an atom an d returns back to it, as well as the number of quanta m=meﬀabsorbed in course of the ionization. All these three variables are, generally speaking, the comp lex-valued functions of the frequency Ω = Nωof the generated harmonic. For a given Nthere can be several solutions of Eqs. (16), (17), (18). Integrating over min Eq. (13) by the saddle method gives the following represen ta- tion for the HHG amplitude d+ N= 2/summationdisplay meﬀ/parenleftBigg2π iS′′ meﬀ/parenrightBigg1/2 d+ Nmeﬀ, (19) which is the major result of this Section. Comparing (19) wit h (6) one ﬁnds, along with clear similarities, several distinctions. The most substa ntial of them originates from diﬀerent physical meaning of the summation index in these fo rmulas. In Eq. (6) it is an integer labeling channels in the physical ATI spectrum. In contrast, in formula (19) summation runs over complex-valued meﬀset. It is natural to say that meﬀare labels of eﬀective channels . In order to ﬁnd the amplitude d+ Nmeﬀone can use representation (7) ford+ Nmand continue the amplitudes Am µ0(Km) and BN mµ 0(Km) into the complex- m plane which can be done if they are known suﬃciently well. One more, though less important diﬀerence, is presented by an additional square r oot factor in (19) that arises from integration over mand depends on the second derivative S′′ mof the action (15) overm. The advantage of eﬀective channel representation stems fro m the fact that only small number of eﬀective channels (actually one or two) contribut es, whereas the number of essential real ATI channels is quite large, as discussed abo ve. Bearing this in mind it is worthwhile to illustrate variation of eﬀective channels labels meﬀwithNby solving numerically set of equations (17), (16), (18) for some parti cular case [10]. Fig. 1 shows two important solutions meﬀ(N) that move along trajectories in the complex- mplane asNvaries. The overall picture comprises a characteristic cro ss-like pattern. For small Nthe trajectories are close to the real axis and are well separ ated. They approach each other as Nincreases and almost ”collide” at some particular critical value N=Nc. For larger N,N > N c, the trajectories start to move almost perpendicular to the real- maxis and rapidly acquire large imaginary parts. It can be dem onstrated that large Im(meﬀ) lead to suppression of the HHG process. Therefore the criti calNcmarks the beginning of the cutoﬀ region for HHG. As shown in Ref. [10], a pproximate analytical solution of Eqs. (17), (16), (18) shows that the critical val ueNcis equal to Ncω=\|Ea\|+ 3.1731Up, (20) in agreement with the well known result of Refs. [3, 4, 8, 22]. In order to ﬁnd simple physical interpretation for the eﬀect ive channels, let us note that Eqs. (16), (17), (18) may be considered as classical equ ations of motion in the laser ﬁeld. They deﬁne the classical trajectories along which the electron ﬁrst goes away from the atom, and then returns back accumulating during this mot ion energy from the laser ﬁeld that is necessary for HG. This physical picture agrees w ith the atomic antenna concept discussed in Section 1. It also comes in line with the Corcum model [2] basedFigure 1: Trajectories of the eﬀective channel labels meﬀ(N) in the complex m-plane for laser frequency ω= 0.0043a.u., intensity I= 1011W/cm2and varying harmonic order N for HHG by H−ion. Positions of two eﬀective channel labels for odd intege rNare denoted respectively by circles and diamonds. entirely on the classical trajectories. Even more close rel ation can be found with the approach of Lewenstein et al[8], which uses the saddle point method to integrate over the momenta pin formula (4). Eqs. (6) and (19) provide two diﬀerent ways to describe the at omic antenna concept, either in terms of real physical channels in the intermediat e ATI spectrum, or by using the eﬀective channels for the intermediate state. Each appr oach has its advantages which can be beneﬁcial for diﬀerent aspects of HHG problem. I mportantly, the above discussion ensures identity of the two formulas since (19) w as derived directly from (6). 4 Photoionization and recombination Eq. (6) shows that the process of HHG is intimately related to ionization and LAR. This fact makes the latter processes very interesting from t he perspective of the atomic antenna concept, in addition to their well known importance as the basic events in the laser-matter interaction. This Section describes the rece nt progress in the theory of these two ”elementary” phenomena. Consider ﬁrst the multip hoton ionization. Adopt- ing the Keldysh-type approach which neglects the ﬁeld of the atomic core in the ﬁnal state one can present the amplitude of the ionization in the f ollowing form Am(pm) =1 T/integraldisplayT 0/angbracketleftφpm(t)\|V(t)\|φa(t)/angbracketright, (21) where φp(t) =φp(r, t) is the Volkov wave function (5) and pmsatisﬁes the energy conservation law Ea+m ω=1 2p2 m+Up. (22) Fast variation of the phase of the integrand in (21) allows on e to use the saddle-point approach for integration over the time t. This approximation, ﬁrst proposed by Keldysh [11], was developed in detail in Refs. [12, 13, 14]. Using thi s scheme one presents theFigure 2: Detachment of H−ion in bichromatic ﬁeld with the frequencies ω= 0.0043 a.u. and 3 ωand intensities I1= 1010W/cm2andI2= 109W/cm2respectively. Diﬀerential detachment rate (in units 10−8a.u.) as a function of the electron emission angle θis shown for various values of the ﬁeld phase diﬀerence ϕas indicated in the plots. Open symbols show results of calculations of [19] (in the ϕ=±1 2πplot the open circles show the results for ϕ=1 2π and open triangles these for ϕ=−1 2π). Solid curves show results of the adiabatic theory [18] (which coincide for ϕ=1 2πandϕ=−1 2π). Left - the ﬁrst ATD peak, corresponding to absorption of n= 8 photons of frequency ω. Right - the third ATD peak, corresponding to absorption of n= 10 photons of frequency ω. photoionization amplitude as Am(pm) =/summationdisplay µAmµ(pm), (23) where summation runs over essential saddle points labeled b y subscript µ. Note that in Eq. (6), relating the ionization problem with HG, it is suﬃ cient to take into account the amplitude which arises from only one saddle point (of two ) labeled as µ0; another saddle point gives the same contribution that produces a fac tor 2. The technique described above was reﬁned in Refs. [15, 16]. I n the pioneer publi- cations [11, 12, 13, 14] the momentum pof the ionized electron was treated as a small quantity, and all essential functions were expanded in powe rs ofp/κ. This approxima- tion restricted the applicability of the approach, since fo r high channels in ATI spectrum the momentum is not small. Refs. [15, 16] demonstrated that t he technique can be mod- iﬁed to include large electron momenta as well. Importantly , this modiﬁcation retains simplicity and clear physical nature of the Keldysh approac h. One can anticipate that the Keldysh-type methods should pro duce reliable results for photodetachment of negative ions, where the detached el ectron is inﬂuenced neg-ligibly by the Coulomb ﬁeld of the residual atomic particle. Due to this reason the photodetachment was in the focus of attention of Refs. [15, 1 6] which compared the re- sults of improved Keldysh approximation with a variety of nu merical and experimental data available for negative ions. Refs. [17, 18] continued t his study and extended it to the case of the two-color laser ﬁeld. Detailed description o f all results obtained in these works would bring us too far away from the main topic of this pa per. However, it is important to mention that the overall accuracy of the modiﬁe d Keldysh approximation proves be very high. It closely reproduces results of other, much more sophisticated methods for total probabilities of detachment as well as for spectral and angular distri- butions of photoelectrons both for the weak and strong ﬁeld r egimes (i.e., for any value of the Keldysh parameter γ=κω/F ). An example of photodetachment of the H−in bichromatic laser ﬁeld with the frequencies ω= 0.0043 a.u. and 3 ωshown in Fig. 2 illustrates this point. Let us now turn our attention to the other relevant problem, l aser-assisted photo recombination (LAR). Consider the electron-atom impact in a laser ﬁeld which results in creation of a negative ion and HG. Since the system can acqu ire energy from the laser ﬁeld due to absorption of several laser quanta, the emitted h armonics should exhibit the equidistant spectral distribution. Strange enough, th is important process was not studied theoretically until two recent almost simultaneou s publications [20, 21]. The amplitude of the photorecombination can be written as Cm(p) =1 T/integraldisplayT 0/angbracketleftφa(t)\|exp(iΩt)U\|φp(t)/angbracketright. (24) HereU=ǫ·rdescribes the potential which is responsible for the harmon ic production, and Ω is the energy of the generated harmonic which satisﬁes t he energy conservation constraint Ω =p2 2+Up+\|Ea\|+m ω , (25) in which mis the number of laser photons absorbed during recombinatio n. The wave function of the electron in the continuum in Eq. (24) can be de scribed by the Volkov wave function, similarly to the Keldysh-type approach to ph otoionization. Comparing the amplitudes (24) and (21) one observes their close simila rity. This fact allows one to develop the theory on the basis of formula (24) along the li nes described above for the ionization problem. In particular, one can use the saddl e-point approximation for integration over the time variable in (24). This approach, suggested in Ref. [20], is supplemented in th e cited paper by sev- eral numerical examples. One of them, shown in Fig. 3, depict s the cross section of recombination on hydrogen atom in a laser ﬁeld with ω= 0.0043 a.u. and the intensity I= 1011W/cm2versus the energy of the emitted high harmonic. Since this is the ﬁrst work in the ﬁeld we could not compare our results with other ca lculations. Bearing in mind that the recombination process has similarities wit h the ionization problem, where similar approach works well one can expect reliable re sults in the recombination problem as well.Figure 3: Cross section σm(p) for laser-assisted recombination of the electron with the energy Eel= 10 eV into the bound state in H−ion [20]. The results are shown for the laser ﬁeld with the frequency ω= 0.0043 a.u. and the intensity I= 1011W/cm2. The symbols are joined by lines to help the eye. 5 Quantitative illustrations for HHG To illustrate the applicability of the two methods, the fact orization technique and the method of eﬀective channels, consider an example of HHG by hy drogen negative ion in a laser ﬁeld with ω= 0.0043 a.u. . Using the factorization procedure one needs ﬁrst to calculate the amplitudes of ATI and LAR, which can be d one by the technique discussed in Sections 4. After that employing (6) one ﬁnds th e amplitude of HHG, and from (11) the HHG rates. The results are presented in Fig. 4. It is important to note a strong interference of contributio ns coming from diﬀer- ent intermediate channels m. In the plateau region it is responsible for an oscillatory pattern, and becomes even more important in the cutoﬀ region , where a contribution of any single channel drastically exceeds the results of an a ccurate summation over a large number of channels. In order to apply the technique ba sed on the eﬀective channels one needs to calculate the ”elementary” amplitude s for the complex-valued number meﬀof quanta absorbed on the ﬁrst step of three-step process. Th is can be done by the approach described in Section 3, because it relie s on analytical methods for calculation of these amplitudes that remain valid for a c omplex-valued m. Taking the eﬀective channels (presented in Fig. 1 for I= 1011W/cm2), calculating for them the ”elementary” amplitudes and applying formula (19), we ﬁ nd the rates presented in Fig. 4. If one takes into account in the summation (19) over meﬀa single eﬀective channel, shown by diamonds in Fig. 1, then the cutoﬀ region is nicely described, as well as the overall pattern in the plateau domain. However such on e-saddle-point approxi- mation does not reproduce structures in the N-dependence of HG rates. Taking into account the two eﬀective channels, shown by diamonds and cir cles in Fig. 1 improves the results for the rates in the plateau domain by producing a ppropriate structures in theN-dependence. Remarkably, this two-saddle-point calculat ion gives correct posi- tions of minima and maxima in the rate N-dependence, albeit the magnitudes of the rate variation is reproduced somewhat worse; for instance t he depth of the minimum atN= 17 is quite strongly overestimated.Figure 4: Harmonic generation rates (11) (in sec−1) for H−ion in the laser ﬁeld with the frequency ω= 0.0043 a.u. and various values of intensity Ias indicated in the plots. Closed circles - results obtained by Becker et al[22], open circles - calculations [6, 7] based on factor- ization procedure and performing numerical summation (6) o ver contributions of diﬀerent ATI channels, open diamonds – calculations of Ref. [20] based on the eﬀective channels approach (19) with a single eﬀective channel meﬀ(N) taken into account (namely, the eﬀective channels shown by diamonds in Fig. 1); open squares – same but taking in to account two saddle points mc(N) (namely, the eﬀective channels shown by diamonds and circl es in Fig. 1). The symbols are joined by lines to help the eye. Fig. 4 shows good agreement of the approaches based on the fac torization technique and on the eﬀective channels, which both are in accord with th e results of Ref. [22]. This agreement holds both for multiphoton regime (left part of Fig. 4), as well as in the tunneling regime (right part of Fig. 4). 6 Above-Threshold Ionization in high channels The methods discussed in Sections 2, 3 can be applied to a numb er of other multiphoton problems. To illustrate this point consider the factorizat ion technique for ATI. The ionization amplitude An(pn) in the Keldysh-type approximation, considered in (21), neglects interaction of the released electron with the core . LetA(1) n(pn) be a correction that takes this interaction into account. In this notation t he total amplitude of ATI with absorption of nphotons is Atot n(pn) =An(pn) +A(1) n(pn). Using the approach of Ref. [5] we ﬁnd the following relation A(1) n(pn) =/summationdisplay m/summationdisplay µ/summationdisplay σ=±1Am(σKm)1 RmµfB(pn, σKm), (26) which presents the suﬃciently complicated correction A(1) n(pn) in terms of two ”elemen- tary” amplitudes Am(Km) and fB(p,K). The later one describes the electron-atom impact in the laser ﬁeld in the Born approximation. This scat tering can be named quasielastic, since the atom remains in the same state, but t he electron momentum is changed K→pboth in direction and absolute value. Summation over µin formula (26) reﬂects the fact that the electron emission into the con tinuum takes place at twoFigure 5: Diﬀerential above-threshold detachment rates wn(θ). Left - the rates versus the number nof absorbed photons for ﬁxed value of electron ejection angl eθ. Right - same as functions of θfor ﬁxed n. An H−ion is irradiated by the laser wave with frequency ω= 0.0043 a.u. and intensity I= 5×1010W/cm2. The minimum number of photons necessary for ionization is nmin= 11. Curves - present calculation with account of rescatter ing; closed circles - Keldysh-type approximation of [15]. The arrow indicates t he plateau cutoﬀ as predicted by classical theory [24]. moments of time tmµ(µ= 1,2) per laser period. The electron return to the atom is ensured only if the electron momentum is parallel (for one va lue of µ) or antiparallel (for another µ) to the ﬁeld, see details in Ref. [5]. This fact is taken into a ccount by a summation index σ=±1 in (26). Eq. (26) is obtained via application of the factorization te chnique to ATI. It has a transparent physical meaning. Ionization with absorptio n ofnphotons needs that ﬁrstmquanta are absorbed by an atom removing the electron from an a tom into the continuum state with momentum ±Km. The collision of this electron with the atom (often referred to as rescattering ) results in absorption of additional n−mquanta and transition of the electron into a state with momentum pn. All intermediate ATI channels labeled by index mcontribute coherently. This physical picture of ATI agrees with the atomic antenna concept, as was ﬁrst discussed in [1] . We applied (26) to calculation of Above Threshold Detachmen t (ATD) from H−ion. Fig. 5 shows the results which clearly indicate that the cont ribution of the process (26) to the angular distributions of ATD spectra is dominating fo r higher ATD channels while for low channels rescattering eﬀects are small . 7 Eikonal approach to the Coulomb ﬁeld The Keldysh-type approximation for laser induced ionizati on, as described in Section 4, discards interaction of receding photoelectrons with th e atomic core. The most signiﬁcant part of this interaction arises due to the Coulom b ﬁeld of the core. That is why a number of numerical applications considered above is c arried out for negative ions, where the core is neutral. However, for the multiphoton proc esses with neutral atoms, the Coulomb forces between the active electron and the resid ual positive ion become operative. This interaction was taken into account ﬁrst in R ef. [12] in the quasistaticlimit of the small Keldysh parameter γ=κω/F ≪1. A very simple relation was obtained between the photoionization rate wCfor the electron bound by a potential with the Coulomb ﬁeld produced by a core charge Zand its counterpart wsrfor the electron with the same binding energy κ2/2 but bound by short range forces: wC=/parenleftBigg2κ2 F/parenrightBigg2Z/κ wsr. (27) This result means that for conventional conditions the pres ence of the Coulomb ﬁeld enhances the rates by several orders of magnitude. It is rema rkable that the relation (27) holds in fact for arbitrary value of the Keldysh paramet er, both in the multiphoton and tunneling region, as was established in a more elaborate theory by Perelomov and Popov [14]. This result agrees well with experimental data f or the total rates [23]. The theory of Perelomov and Popov is restricted to ejection o f low-energy electrons. It is usually anticipated that these electrons give princip al contribution to the total rates summed over all ATI channels as well as over ejection an gles. The current exper- iments, however, are able to select an individual ATI channe l even for a large number of absorbed quanta. Both energy and angular distributions o f these electrons mani- fest some fascinating features which are the object of inter est in modern experiment and theory. This fact prompts to develop a theory which, repr oducing the Perelomov- Popov results for low electron energy, could also describe h igh-energy photoelectrons. The antenna-type phenomena considered in this paper provid e an additional inspira- tion for this study. The ionization amplitude in Eqs. (6), (2 6) should be summed over the number mof quanta absorbed that should be large enough. This makes th e elec- tron momentum in the intermediate ATI channels to be also lar ge. Therefore we need to know how the Coulomb ﬁeld aﬀects the amplitudes for large m omenta. In order to address this issue we develop the eikonal approach for this problem. It presents a simpliﬁed version of the semiclassical approximation, tha t assumes that the Coulomb ﬁeld does not produce signiﬁcant distortions of classical t rajectories describing electron propagation in the continuum (i.e., the electron wiggling m otion in the laser ﬁeld). The Coulomb ﬁeld comes into the picture through its contributio n to the action/integraltextt(Z/r)dt calculated along the classical trajectories discussed. Wi thin the semiclassical approx- imation the action plays a role of the wave function phase. It is important that the Coulomb ﬁeld can produce large contribution to the action wh ile its distortion of the trajectories can remain unsubstantial Calculating the act ion, one is able to construct the semiclassical wave function for the released electron, and ﬁnd with its help the ion- ization amplitude. If we totally neglect in this scheme the C oulomb ﬁeld putting Z= 0, the eikonal wave function simpliﬁes to be the Volkov functio n, and we return to the Keldysh-type approximation. An important veriﬁcation of o ur eikonal approach pro- vides the limit of low photoelectron energy where we reprodu ce the Perelomov-Popov result (27). Fig. 6 presents a quantitative example which il lustrates importance of the Coulomb interaction. 8 Conclusions Atomic antenna gives a clear physical idea how the complicat ed multiphoton processes are operative. Absorption of large number of quanta from the laser ﬁeld needs thatFigure 6: Angle-resolved photoionization rates wn(θ) for four lowest open ATI channels labeled by a number of absorbed photons n(n= 11 corresponds to the lowest open ATI channel). Hydrogen atom is illuminated by the laser wave wit h frequency ω= 2 eV and intensity I= 1014W/cm2; solid curves - eikonal theory, dashed curves - the Perelomo v-Popov theory [14] (27). one of the atomic electrons is at ﬁrst released from an atom. A fter that, propagating in the core vicinity, the electron accumulates high energy f rom the laser ﬁeld and, returning to the atomic core, transfers this energy into oth er channels such as HHG, ATI or others. Importantly, this physical picture is implem ented in a simple and reliable formalism. We discussed two convenient ways to present the t heory. One of them, called the factorization technique, is presented by Eqs. (6 ), (26) for the cases of HHG and ATI. In this approach the amplitude of a complicated proc ess is expressed via the physical amplitudes of more simple, ”elementary” processe s. Another scheme, called the eﬀective channels method, is based on Eq. (19) for HHG. Th e eﬀective channels are closely related to the classical trajectories, that makes t hem convenient for qualitative, as well as numerical studies. We demonstrated an equivalenc e of the two approaches. Applications of both formalisms need calculation of the ”el ementary” amplitudes. This can be achieved by using the modiﬁed Keldysh-type appro ach which very ac- curately reproduces data for photodetachment, and, hopefu lly, for recombination and electron-atom scattering in the laser ﬁeld as well. The Coul omb ﬁeld of the atomic core can be taken into account within eikonal approximation. Rel iability of the theoretical approaches is demonstrated by quantitative applications.References [1] M. Yu. Kuchiev, Pis’ma Zh. Eksp. Teor. Fiz. 45, 319 (1987) [JETP Letters 45, 404 (1987)]. [2] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). [3] J. L. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett.68, 3535 (1992). [4] K. C. Kulander, K. J. Schafer, and J. L. Krause, in Super-Intense Laser-Atom Physics , Vol. 316 of NATO Advanced Study Institute, Series B: Physics , edited by B. Piraux et al(Plenum, New York, 1993), p. 95. [5] M. Yu. Kuchiev, J. Phys. B 28, 5093 (1995). [6] M. Yu. Kuchiev and V. N. Ostrovsky, J. Phys. B 32, L189 (1999). [7] M. Yu. Kuchiev and V. N. Ostrovsky, Phys. Rev. A 60, 3111 (1999). [8] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994). [9] M. Lewenstein, P. Salli` ers, and A. L’Huillier, Phys. Re v. A52, 4747 (1995). [10] M. Yu. Kuchiev and V. N. Ostrovsky, http://xxx.lanl.go v/physics/0007016. [11] L. V. Keldysh, Zh. ´Eksp.Teor.Fiz. 47, 1945 (1964) [Sov.Phys.JETP 20, 1307 (1965)]. [12] A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. ´Eksp. Teor. Fiz. 50, 1393 (1966) [Sov. Phys.-JETP 23, 924 (1966)]. [13] A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. ´Eksp. Teor. Fiz. 51, 309 (1966) [Sov. Phys.-JETP 24, 207 (1967)]. [14] A. M. Perelomov and V. S. Popov, Zh. ´Eksp. Teor. Fiz. 52, 514 (1967) [Sov. Phys.-JETP 25, 336 (1967)]. [15] G. F. Gribakin, M. Yu. Kuchiev, Phys. Rev. A 55, 3760 (1997). [16] G. F. Gribakin, M. Yu. Kuchiev, J. Phys. B 30, L657 (1997). [17] M. Yu. Kuchiev and V. N. Ostrovsky, J. Phys. B 31, 2525 (1998). [18] M. Yu. Kuchiev and V. N. Ostrovsky, Phys. Rev. A 59, 2844 (1999). [19] D. A. Telnov, J. Wang, and S. I. Chu, Phys. Rev. A 51, 4797 (1995). [20] M. Yu. Kuchiev and V. N. Ostrovsky, Phys. Rev. A 61, 033414 (2000). [21] A. Jaro´ n, J. Z. Kami´ nsky, and F. Ehlotzky, Phys. Rev. A 61, 023404 (2000). [22] W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994). [23] S. F. J. Larochelle, A. Talebpour, and S. L. Chin, J. Phys . B31, 1215 (1998). [24] G. G. Paulus, W. Becker, W. Nicklich, and H. Walter, J. Ph ys. B27, L703 (1994).
arXiv:physics/0010020v1 [physics.chem-ph] 6 Oct 2000Spontaneous formation and stability of small GaP fullerene s V. Tozzini Istituto Nazionale per la Fisica della Materia and Scuola No rmale Superiore, Piazza dei Cavalieri, 7 I-56126 Pisa, Italy F. Buda Department of Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands A. Fasolino Research Institute for Materials, Institute of Theoretica l Physics, University of Nijmegen, Toernooiveld, NL-6525ED Nijmegen, The Netherlands Abstract We report the spontaneous formation of a GaP fullerene cage i n ab-initio Molecular Dynamics simulations starting from a bulk fragme nt. A systematic study of the geometric and electronic properties of neutral and ionized GaP clusters suggests the stability of hetero-fullerenes form ed by a compound with zincblend bulk structure. We ﬁnd that GaP fullerenes up to 28 atoms have high symmetry, closed electronic shells, large HOMO-LUMO e nergy gaps and do not dissociate when ionized. We compare our results for Ga P with those obtained by other groups for the corresponding BN clusters. 61.48.+c, 71.15.Pd, 81.05.Tp Typeset using REVT EX 1The discovery of carbon fullerenes and nanotubes [1,2] has o pened a completely new ﬁeld at the borderline between chemistry and physics leadin g to many new phenomena and applications. Up to now, most eﬀorts to identify fullerenes based on other e lements have focused on BN which is the most similar to carbon and exists in nature in t he hexagonal(graphite-like) structure [3–10]. However, the (nested-)cages and wires fo und for this material [3,4] do not resemble any of the small preferred structures of the carbon fullerene family, particularly due to the absence of the characteristic pentagonal rings. B esides, nanotubes based on other layered materials, such as GaSe [11] and black-phosphorus [ 12], have been theoretically pre- dicted to be stable. On the basis of density functional calcu lations it has also been proposed that GaN nanotubes could be synthesised by using carbon nano tubes as a nucleation seed [13]. One intriguing question is whether fullerene cages could be realized in typical semicon- ductors of the III-V family, like GaAs, InSb or GaP, which do n ot possess a graphite-like bulk structure. These materials are not considered as good c andidates for hollow structures sinceπbonding should be less eﬀective in these larger atoms of high er rows of the periodic table than in the ﬁrst one [5]. In this letter we show, by means of ab-initio Car-Parrinello Molecular Dynamics [14], that a small GaP bulk fragment spontaneously organizes in a cage f ormed by a diﬀerent number of atoms of the two elements arranged as in carbon fullerenes . We discuss the geometric and electronic structure of GaP cages with either the same or a diﬀerent number of atoms of the two species. Our results strongly suggest that small G aP fullerenes could be stable, since they have high symmetry, closed electronic shells, la rge HOMO-LUMO energy gaps and do not dissociate when ionized. We give quantitative est imates of the relative stability of cages formed either by hexagons and pentagons as in carbon fullerenes or by hexagons and squares as proposed for BN [5–9]. Our results are obtained by the Car-Parrinello approach [14 ] using a Density Functional in the Generalized Gradient Approximation proposed by Beck e and Perdew [15,16]. This approximation reproduces the experimental cohesive energ y of typical bulk semiconductors within ≈5% and underestimates the valence to conduction band excita tion energies [17]. We use nonlocal norm-conserving ﬁrst-principles pseudopo tentials [18] and expand the single particle wavefunctions on a plane wave basis set with a cut-o ﬀ of 12 Rydberg. We use a periodically repeated cubic simulation box of 24 ˚A side, so that periodic images are at least 14˚A apart. We have veriﬁed that this size is large enough to desc ribe isolated clusters. The electronic optimization and structural relaxation have be en performed using damped second order dynamics with electronic mass preconditioning schem e [19,20]. We use throughout an integration time step of 8 a.u. The symmetry of the equilibri um structure is not biased but it is reached spontaneously during the geometry optimization starting from the corresponding regular polyhedron. The process of formation of the fullerene cage with 28 atoms f rom a larger bulk-like cluster of 41 atoms (Ga 28P13) is shown in Fig.1, with the help of three snapshots taken during the structural energy minimization which leads to th e appearance of the Ga 16P12 fullerene cage [21]. The cage has 12 pentagons and 6 hexagons and T dsymmetry as C 28. An analysis of the charge distribution shows 108 valence elect rons on the bonded cage, exactly the number which corresponds to the neutral Ga 16P12cluster [22]. Finally in Fig.1d we show 2the equilibrium structure of the neutral Ga 16P12cluster alone. The observed spontaneous formation of a Ga 16P12cage with pentagons is surprising since, in the case of BN [5–8], (deformed) squares are found t o be energetically much more favorable. For B 12N12[6], there is an energy diﬀerence of 9 eV between the cage with pentagons and the one with squares in favor of the latter whic h contains only heteropolar bonds and is favored for a material composed by atoms with ver y diﬀerent electronegativity as B and N. Therefore, most studies have considered cages B nNnformed by hexagons closed by square rings [5,6,8,9]. Very recently, Fowler et al. [10] have pointed out that, among the cages with pentagons, those with one species in excess of 4 at oms (B nNn+4) minimize the number of homopolar bonds. It is remarkable that the cage Ga 16P12which spontaneously appear in our simulation falls into this class. We have studied the equilibrium structure and electronic st ates of clusters with 20 and 28 atoms of the type III nVn±4, namely Ga 12P8, Ga 8P12, Ga 16P12and Ga 12P16, and compared them to clusters with the same number of III and V atoms, namel y Ga 10P10and Ga 12P12, the latter in the two isomers [6] with hexagons closed either by ﬁve- or four-sided faces. The minimum energy structures of Ga 16P12and Ga 12P16are found to have T dsymmetry, whereas those of Ga 12P8and Ga 8P12have T hsymmetry. Among the clusters Ga nPn, the Ga12P12with 4-membered rings belongs to T hwhereas those with pentagons present very large distortions around the lower C 3vsymmetry. The structural parameters of the cages belonging either to T hor to T dare given in Table I. Hexagons are found with alternating angles of 880−1050and 1260−1340, pentagons with angles of 850−920, 1140−1260,∼1000, ∼1100and deformed squares with angles 750and 980. Ga-P distances are in general shorter than in bulk compounds due to predominant sp2bonding. The radial distance rfrom the center of the cluster given in Table I indicates a tendency of the anion to occupy positions at larger distances from the center than the cation, as found for ultrasmall cluster [23,24]. As in the case of carbon and BN fullerenes, the GaP clusters wo uld represent metastable states with respect to the bulk equilibrium structure. Ther efore only experimental obser- vation can establish with certainty their existence. Never theless, there are a few quantities which are used in the literature as indicators of stability. We support our prediction for the stability of the examined GaP clusters by using the followin g indicators: i) closed electronic shells and large energy gaps; ii) cohesive energy; iii) ther mal stability; iv) stability of the ionized clusters. The ﬁrst indicator of chemical stability is the energy gap be tween the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO). I n carbon fullerenes a cor- relation between this energy and the observed fullerenes ha s been experimentally veriﬁed [25]. In Table II we give the calculated HOMO-LUMO energy gap and the cohesive energy for all the clusters studied. Among the cages with pentagons the highest energy gaps are for cluster with P in excess of 4, a composition which has been suggested to be favorable also for BN [10]. However, a very large gap is also found for th e Ga 12P12with squares. A comparison of the binding energies per atom between the GaP cages and the zincblend bulk phase of this material is possible only for the clusters with the same number of Ga and P atoms. From the results of Table II, we ﬁnd that the cohesive energies per atom for Ga12P12with squares, Ga 12P12with pentagons, and Ga 10P10are about 10% lower than in the bulk. This result is very close to that found for BN and car bon fullerenes of the same size [5]. 3We have studied the thermal stability of two clusters with ve ry diﬀerent energy gaps, namely Ga 12P8and Ga 8P12(see Table II). For both clusters we have performed two annea ling cycles of about 3 ps, up to 1500 K and up to 2000 K. The system is h eated with a rate of 2×1015K/s, then equilibrated for one ps at the highest temperature , and ﬁnally cooled down with the same temperature change rate. For Ga 8P12no bond breaking or structural rearrangements occur in both cycles and the structure comes back to the same minimum energy conﬁguration when the temperature is lowered. This i s also the case for Ga 12P8in the annealing up to 1500 K, whereas at 2000 K some structural r earrangement takes place leading to a distorted structure with higher energy when coo led down. These results indicate that the thermal stability is correlated with the width of th e energy gap. Mass spectrometry experiments use the diﬀerence in mass-to -charge ratio of ionized atoms or clusters to select them. Therefore one basic requir ement for the possible detection of such clusters is that they remain stable also when ionized . We have investigated the stability of some positively ionized clusters, [Ga 8P12]+, [Ga 12P16]+and [Ga 16P12]+. We have included an uniform charge background in order to have an ove rall neutral system in the supercell calculation. The electronic structure remains a lmost unaﬀected and degeneracies are broken by negligible amounts in the order of hundredths o f eV. Only minor structural distortions occur upon ionization. In particular, the six e quivalent P-P (Ga-Ga) bond lengths split into three diﬀerent classes. Remarkably, during a mol ecular dynamics run for [Ga 8P12]+ we observe a dynamical exchange between these three classes of bond lengths with each other. This eﬀect produces features in the low frequency vibration al spectrum in the range 30 −120 cm−1which might be detected by infrared multiphoton ionization spectra [26]. It is interesting to compare cages closed either by 4- or 5-me mbered rings. As already mentioned, such a comparison has been done for B 12N12in Ref. [6]. However there are no results comparing clusters with equal number of atoms III nVnwith square rings to the more favorable structures with pentagons of the type III nVn+4. The authors who have proposed the latter stoichiometry [10], in fact, do not give a comparison to cages with square rings. Although it is impossible to compare directly the cohesive energy of structures with a diﬀerent number of atoms of each species, we are in a pos ition to give an estimate of the bond energy for the two types of cages. Given the number of each type of bond in all the structures with pentagons studied so far and the values o f the total cohesive energy, we estimate by a best ﬁt the following energies per bond: E Ga−P= -2.593 eV, E Ga−Ga=-1.133 eV, EP−P=-2.349 eV [27]. As shown in Table II, these values yield the c orrect cohesive energy with a relative error of 0.4% at most for all cluster with pent agons, whereas overestimate the cohesive energy of the cluster Ga 12P12with squares. In this cluster, in fact, there are only Ga-P bonds yielding directly E Ga−P=-2.499 eV a smaller value than in the clusters with pentagons. However, as it can be seen in Table II, it is th e isomer with squares which has the lowest energy among the two Ga 12P12. The 1.9 eV energy diﬀerence between them is much less than the 9 eV found for B 12N12[6]. This is most probably due to the less ionic character of the GaP bonds. The diﬀerence in electronegativ ity of Ga and P is in fact ∼0.4 against ∼1 for BN. The observed spontaneous formation in our simulati ons of a cage with the same topology of C 28shows a possible evolution pattern from ionized bulk fragme nts to classical fullerene cages formed by pentagons and hexagons . In summary, we have shown, by means of ab-initio Car-Parrine llo Molecular Dynamics, that small GaP fullerenes have highly symmetric structures , closed electronic shells and large 4HOMO-LUMO gaps and cohesive energy. These clusters are ther mally stable and remain in the same structure also when ionized. These ﬁndings togethe r with the observed spontaneous formation in our simulations of a 28 atoms cage with the same s ymmetry of C 28support the possible existence of GaP fullerenes. We hope that our wo rk will stimulate experimental groups to widen their search for hetero-fullerenes also to I II-V compound semiconductors. Acknowledgments : V.T. acknowledges ﬁnancial support of the Research Instit ute for Materials (RIM) and of the Institute of Theoretical Phys ics of the University of Ni- jmegen where this work was carried out. Calculations have be en performed at the Stichting Academisch Rekencentrum Amsterdam (SARA) with a grant of th e Stichting Nationale Computer Faciliteiten (NCF). We thank P. Giannozzi for prov iding relevant data on the pseudopotentials. We are very grateful to A. Janner, R.A. de Groot and J.C. Maan for critical reading of the manuscript. 5REFERENCES [1] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. S malley, Nature 318, 162 (1985). [2] S. Iijima, Nature 354, 56 (1991). [3] D. Goldberg, Y. Bando, O. St´ ephan, and K. Kurashima, App l. Phys. Lett. 73, 2441 (1998). [4] P. A. Parilla, A. C. Dillon, K. M. Jones, G. Riker, D.L. Sch ulz, D.S. Ginley and M.J. Heben, Nature 397, 114 (1999). [5] G. Seifert, P. W. Fowler, D. Mitchell, D. Porezag, and TH. Frauenheim, Chem Phys. Lett.268, 352 (1997). [6] F. Jensen, and H. Toftlund, Chem Phys. Lett. 201, 89 (1993). [7] X. Blase, A. De Vita, J-.C. Charlier, and R. Car, Phys. Rev . Lett. 80, 1666, (1998). [8] M-L. Sun, Z. Slanina and S-.L. Lee, Chem Phys. Lett. 233, 279 (1995). [9] S. S. Alexandre, M. S. C. Mazzoni, and H. Chacham, Appl. Ph ys. Lett. 75, 61 (1999). [10] P. W. Fowler, K. M. Rogers, G. Seifert, M. Terrones, and H . Terrones, Chem Phys. Lett 299, 359 (1999). [11] M. Cˆ ot´ e, M. L. Cohen and D. J. Chadi, Phys. Rev. B 58, R4277 (1998). [12] G. Seifert and E. Hern´ andez, Chem Phys. Lett. 318, 355 (2000). [13] S. M. Lee, Y. H. Lee, Y. G. Hwang, J. Elsner, D. Porezag and T. Frauenheim, Phys. Rev. B 60, 7788 (1999). [14] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985). [15] A.D. Becke, Phys. Rev. A 38, 3098 (1988). [16] J.P. Perdew, Phys. Rev. B 33, 8822 (1986). [17] G. Ortiz, Phys. Rev. B 45, 11328 (1992). [18] X. Gonze, R. Stumpf, and M. Scheﬄer, Phys. Rev. B 44, 8503 (1991). [19] F. Tassone, F. Mauri, and R. Car, Phys. Rev. B 50, 10561 (1994). [20] F. Buda, and A. Fasolino, Phys. Rev. B 60, 6131 (1999). [21] In Ref. [20] we have noted that also III-V clusters inclu ded in sodalite show the tendency to organize in hollow shells of cations and anions. [22] The neutral Ga 28P13cluster with 149 electrons has one unpaired electron in the u pper- most occupied state. Just below this state, the electronic s tructure presents one 3–fold and one 1–fold degenerate state. Interestingly, the 28 atom s cage forms spontaneously whenever we have completely ﬁlled states, namely for 148, 14 2 and 140 electrons, which would describe an ionized bulk-like fragment. [23] W. Andreoni, Phys. Rev. B 45, 4203 (1992). [24] F. Buda, and A. Fasolino, Phys. Rev. B 52, 5851 (1995). [25] H. Kietzmann, R. Rochow, G. Gantef¨ or, W. Eberhardt, K. Vietze, G. Seifert, and P.W. Fowler, Phys. Rev. Lett. 81, 5378 (1998). [26] D. Van Heijnsbergen, G. von Helden, M. A. Duncan A. J. A. v an Roij and G. Meijer, Phys. Rev. Lett. 83, 4983 (1999). [27] The values of E Ga−Gaand E P−Pare close to the experimental bond energies, 1.17 eV and 2.08 eV for Ga and P, respectively, given in J. Emsley, The elements , Oxford University Press, New York, 1989. 6TABLES Ga16P12(Td) Ga12P16(Td) 4 Ga 3m xxx x=-1.96 r=3.39 4 P 3m xxx x=-2.37 r=4.11 12 Ga m xxz x= 0.83 r=3.50 12 P m xxz x= 0.81 r=3.92 z= 3.30 z= 3.75 12 P m xxz x=2.87 r=4.06 12 Ga m xxz x= 2.33 r=3.30 z=-0.03 z=-0.12 Ga12P8(Th) Ga8P12(Th) Ga12P12square (T h) 8 P 3 xxx x=2.05 r=3.55 8 Ga 3 xxx x=1.54 r=2.67 12 P i 0yz y=3.36 r=3.78 z=1.74 12 Ga i 0yz y=2.61 r=2.87 12 P i 0yz y=3.18 r=3.37 12 Ga i 0yz y=2.81 r=3.13 z=1.19 z=1.13 z=1.39 TABLE I. Point symmetry and atomic positions of the clusters with symmetry T dand T h. Values of the parameters and of the radial distance r from the center of the cluster are in Angstrom. The positions of the minimum energy structure vary less than 0.01˚A from the given values. 7cluster HOMO-LUMO gap E cohesive Efit error (eV) (Hartree) (Hartree) (%) Ga12P8 1.28 -2.5374 -2.5370 -0.01 Ga8P12 2.14 -2.8161 -2.8052 -0.39 Ga16P12 1.09 -3.6876 -3.6806 -0.19 Ga12P16 1.55 -3.9452 -3.9488 0.09 Ga10P10 1.03 -2.6632 -2.6711 0.30 Ga12P12(pentagon) 1.24 -3.2362 -3.2429 0.21 Ga12P12(square) 1.86 -3.3060 -3.4308 3.78 TABLE II. HOMO-LUMO gap and cohesive energy of the clusters s tudied. For reference, the indirect energy gap of bulk GaP within the Local Density Appr oximation is 1.62 eV. The cohesive energy is obtained as the diﬀerence of the total energy minus the energy of the isolated pseudoatoms. Within the same approximations these values are -2.15045 Ha rtree and -6.46295 Hartree for Ga and P, respectively. The cohesive energy per GaP pair in the b ulk is found to be 0.2976 Hartree. In the last two columns we give the values and the relative err or of the cohesive energy estimated from the calculated bond energies E Ga−P= -2.593 eV, E Ga−Ga=-1.133 eV, E P−P=-2.349 eV (see text). 8FIGURES FIG. 1. Ga atoms are represented as large light balls, P atoms as small dark balls. Start- ing from a truncated bulk structure with tetrahedral symmet ry, Fig.1a shows a ﬁrst step in the evolution towards structural energy minimization. The mai n rearrangement is the bonding of the peripheral Ga atoms between themselves, 12 atoms in pairs on the edges of the tetrahedron and four triplets on the vertices. In Fig.1b the central P atom br eaks its bonds, followed by 12 Ga atoms (Fig1.c) leading to the formation of a Ga 16P12fullerene cage. Fig.1d shows the equilibrium structure of the neutral Ga 16P12cluster alone. Notice the non-planarity of the pentagons. T he symmetry of the equilibrium conﬁguration is T d(see Table I). 9This figure "fig.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0010020v1
arXiv:physics/0010021v1 [physics.flu-dyn] 6 Oct 2000Measurements of the instantaneous velocity diﬀerence and l ocal velocity with a ﬁber-optic coupler S. H. Yao Department of Physics, Oklahoma State University, Stillwa ter, OK 74078 V. K. Horv´ ath Department of Physics and Astronomy, University of Pittsbu rgh, Pittsburgh, PA 15260 P. Tong and B. J. Ackerson Department of Physics, Oklahoma State University, Stillwa ter, OK 74078 W. I. Goldburg Department of Physics and Astronomy, University of Pittsbu rgh, Pittsburgh, PA 15260 (February 2, 2008) New optical arrangements with two single-mode input ﬁbers a nd a ﬁber-optic coupler are devised to measure the instantaneous velocity diﬀerence and local v elocity. The ﬁbers and the coupler are polarization-preserving to guarantee a high signal-to-no ise ratio. When the two input ﬁbers are used to collect the scattered light with the same momentum tr ansfer vector but from two spatially separated regions in a ﬂow, the obtained signals interfere w hen combined via the ﬁber-optic coupler. The resultant light received by a photomultiplier tube cont ains a cross-beat frequency proportional to the velocity diﬀerence between the two measuring points. If the two input ﬁbers are used to collect the scattered light from a common scattering region but with two diﬀerent momentum transfer vectors, the resultant light then contains a self-beat freq uency proportional to the local velocity at the measuring point. The experiment shows that both the cros s-beat and self-beat signals are large and the standard laser Doppler signal processor can be used t o measure the velocity diﬀerence and local velocity in real time. The new technique will have vari ous applications in the general area of ﬂuid dynamics. OCIS codes: 120.7250, 060.2420, 060.1810, 030.7060. I. INTRODUCTION Measurements of the velocity diﬀerence or the relative velocity, δv(ℓ) =v(x+ℓ)−v(x), between two spatial points separated by a distance ℓhave important appli- cations in ﬂuid dynamics. For example, in the study of turbulent ﬂows one is interested in the scaling behavior ofδv(ℓ) over varying distance ℓ, when ℓis in the in- ertial range, in which the kinetic energy cascades at a constant rate without dissipation. [1,2] If the separation ℓis smaller than the Kolmogorov dissipation length, [1,2] the measured δv(ℓ) becomes proportional to the velocity gradient ∂v/∂r ≃δv(ℓ)/ℓ(assuming ℓis known), a useful quantity which is needed to determine the energy dissi- pation and ﬂow vorticity. In many experimental studies of ﬂuid turbulence, one measures the local velocity as a function of time and then uses Taylor’s frozen turbu- lence assumption to convert the measured temporal vari- ations into the spatial ﬂuctuations of the velocity ﬁeld. [3] The frozen turbulence assumption is valid only when the mean velocity becomes much larger than the velocity ﬂuctuations. For isotropic turbulent ﬂows with a small mean velocity, direct measurement of δv(ℓ) is needed.Over the past several years, the present authors and their collaborators have exploited the technique of ho- modyne photon correlation spectroscopy (HPCS) to mea- sureδv(ℓ). [4,5] With the HPCS scheme, small particles seeded in a ﬂowing ﬂuid are used to scatter the inci- dent laser light. The scattered light intensity I(t), which ﬂuctuates because of the motion of the seed particles, contains Doppler beat frequencies of all particle pairs in the scattering volume. For each particle pair separated by a distance ℓ(along the beam propagation direction), their beat frequency is ∆ ω2=q·δv(ℓ), where qis the momentum transfer vector. The magnitude of qis given byq= (4πn/λ)sin(θ/2), where θis the scattering angle, nis the refractive index of the ﬂuid, and λis the wave- length of the incident light. Experimentally, the Doppler beat frequency ∆ ω2is measured by the intensity auto- correlation function, [6] g(τ) =/angb∇acketleftI(t+τ)I(t)/angb∇acket∇ight /angb∇acketleftI(t)/angb∇acket∇ight2= 1 + bG(τ), (1) where b(≤1) is an instrumental constant and henceforth we set b= 1. The angle brackets represent a time average overt. 1It has been shown that G(τ) in Eq. (1) has the form [7] G(τ) =/integraldisplayL 0dr h(r)/integraldisplay+∞ −∞dδv P (δv, r)cos(qδvτ),(2) where δvis the component of δvin the direction of q,P(δv, r) is the probability density function (PDF) of δv(r), and h(r)dris the number fraction of particle pairs with separation rin the scattering volume. Equation (2) states that the light scattered by each pair of par- ticles contributes a phase factor cos( qτδv) (because of the Doppler beat) to the correlation function G(τ), and G(τ) is an incoherent sum of these ensemble averaged phase factors over all the particle pairs in the scattering volume. In many previous experiments, [7–10] the length Lof the scattering volume viewed by a photodetector was controlled by the width Sof a slit in the collecting optics. While it is indeed a powerful tool for the study of tur- bulent ﬂows, the HPCS technique has two limitations in its collecting optics and signal processing. First, a weighted average over ris required for G(τ) because the photodetector receives light from particle pairs having a range of separations (0 < r < L ). As a result, the mea- sured G(τ) contains information about δv(ℓ) over various length scales up to L. With the single slit arrangement, the range of Lwhich can be varied in the experiment is limited. The lower cut-oﬀ for Lis controlled by the laser beam radius σ. The upper cut-oﬀ for Lis determined by the coherence distance (or coherence area) at the detect- ing surface of the photo-detector, over which the scat- tered electric ﬁelds are strongly correlated in space. [6] When the slit width Sbecomes too large, the photode- tector sees many temporally ﬂuctuating speckles (or co- herence areas), and consequently ﬂuctuations in the scat- tered intensity I(t) will be averaged out over the range ofq-values spanned by the detecting area. Recently, we made a new optical arrangement for HPCS, with which the weighted average over rin Eq. (2) is no longer needed and the upper limit for Lcan be extended to the coherence length of the laser. In the ex- periment, [5] two single mode, polarization-maintaining (PM) ﬁbers are used to collect light with the same po- larization and momentum transfer vector qbut from two spatially separated regions in a ﬂow. These regions are illuminated by a single coherent laser beam, so that the collected signals interfere when combined using a ﬁber- optic coupler, before being directed to a photodetector. With this arrangement, the measured G(τ) becomes pro- portional to the Fourier cosine transform of the PDF P(δv, r). The second limitation of HPCS is related to signal pro- cessing. The correlation method is very eﬀective in pick- ing up small ﬂuctuating signals, but the resulting corre- lation function G(τ) is a time-averaged quantity. There- fore, the correlation method is not applicable to unstable ﬂows. Furthermore, information about the odd moments ofP(δv, r) is lost, because the measured G(τ) is a Fouriercosine transform of P(δv, r). In this paper, we present a further improvement for HPCS, which is free of the two limitations discussed above. By combining the new ﬁber-optic method with the laser Doppler velocimetry (LDV) electronics, we are able to measure the instantaneous velocity diﬀerence δv(ℓ, t) and local velocity v(x, t) at a high sampling rate. With this technique, the statistics of δv(ℓ, t) andv(x, t) are obtained directly from the time series data. The new method of measuring v(x, t) oﬀers several advan- tages over the standard LDV. The remainder of the paper is organized as follows. In Section 2 we describe the ex- perimental methods and setup. Experimental results are presented and analyzed in Section 3. Finally, the work is summarized in Section 4. II. EXPERIMENTAL METHODS A. Measurement of the velocity diﬀerence Figure 1 shows the optical arrangement and the ﬂow cells used in the experiment. A similar setup has been described elsewhere, [5] and here we mention only some key points. FIG. 1. (a) Scattering geometry for the velocity diﬀerence measurement. ki, incident wave vector; ks, scattered wave vector; and q=ks−ki. (b) Experimental setup for the ve- locity diﬀerence measurement in rigid body rotation. (c) Fl ow cell and optical arrangement for a jet ﬂow. As shown in Fig. 1(b), an incident beam from a Nd:YVO 4laser with a power range of 0.5-2W and a wave- length of λ=532 nm is directed to a ﬂow cell by a lens. With the aid of a large beam splitting cube, two single- mode, polarization-maintaining (PM) ﬁbers collect the scattered light from two diﬀerent spots along the laser beam with a separation L. The two PM ﬁbers are con- nected to a ﬁber-optic coupler (purchased from OZ Op- 2tics [11]), which combines the light from the two input ﬁbers and split the resultant light evenly into two output ﬁbers. A graded index lens is placed at each end of the ﬁber to collimate the light entering (or exiting from) the ﬁber core. Each input ﬁber is mounted on a micrometer- controlled translation stage and the distance Lcan be adjusted in a range of 0-25 mm in steps of 0.01 mm. The output ﬁbers of the optical coupler are connected to two photomultiplier tubes (PMT1 and PMT2). PMT1 is op- erated in the digital mode and its output signal is fed to a digital correlator (ALV-5000). PMT2 is operated in the analogue mode and its output signal is fed to a LDV sig- nal processor (TSI IFA655). An oscilloscope connected to PMT2 directly views the analogue signals. A low-noise preampliﬁer (Stanford Research SR560) further ampliﬁes the analogue output of PMT2 before it goes to the LDV signal processor. As shown in Fig. 1(a), the electric ﬁelds detected by each input ﬁber sum in the coupler and consequently in- terfere. In the experiment, we obtain the beat frequency, ∆ω2=q·v1−q·v2=q·δv(L), in two diﬀerent ways. One way is to measure the intensity auto-correlation function g(τ) in Eq. (1). With the ALV correlator, it takes ∼1 minute to collect the data with an adequate signal-to- noise ratio. The other way is to use the LDV signal pro- cessor to measure the instantaneous beat frequency ∆ ω2, giving velocity diﬀerences in real time. The LDV signal processor is essentially a very fast correlator and thus re- quires the beat signal to be large enough so that no signal averaging is needed. In the experiment to be discussed below, we use both methods to analyze the beat signals and compare the results. It has been shown that the correlation function g(τ) has the form: [5] g(τ) = 1 +I2 1+I2 2 (I1+I2)2Gs(τ) +2I1I2 (I1+I2)2Gc(τ) = 1 + bsGs(τ) +bcGc(τ), (3) where I1andI2are the light intensities from the two input ﬁbers. When I1=I2, one ﬁnds bs=bc= 0.5. If one of the input ﬁbers is blocked (i.e., I2= 0), we haveg(τ) = 1 + Gs(τ), where Gs(τ) is the self-beat cor- relation function for a single ﬁber. When the separation L between the two input ﬁbers is much larger than the spot size viewed by each ﬁber, the cross-beat correlation function Gc(τ) takes the form Gc(τ)≃Gs(τ)/integraldisplay+∞ −∞dδv P (δv)cos(qδv(L)τ).(4) Two ﬂow cells are used in the experiment. The ﬁrst one is a cylindrical cuvette having an inner diameter of 2.45 cm and a height of 5 cm. The cuvette is top mounted on a geared motor, which produces smooth rotation with an angular velocity ω= 2.5rad/s. The cell is ﬁlled with 1,2-propylene glycol, whose viscosity is 40 times larger than that of water. The whole cell is immersed in a largesquare index-matching vat, which is also ﬁlled with 1,2- propylene glycol. The ﬂow ﬁeld inside the cell is a simple rigid body rotation. With the scattering geometry shown in Fig. 1(b), the beat frequency is given by ∆ ω2=ksωL withks= 2πn/λ. The sample cell is seeded with a small amount of polystyrene latex spheres. For the correla- tion measurements, we use small seed particles of 1 .0µm in diameter. By using the small seed particles, one can have more particles in the scattering volume even at low seeding densities. This will reduce the amplitude of the number ﬂuctuations caused by a change in the number of particles in each scattering volume. The particle number ﬂuctuations can produce incoherent amplitude ﬂuctua- tions to the scattered light and thus introduce an extra (additive) decay to g(τ). [16] Large particles 4 .75µmin diameter are used to produce higher scattering intensity for instantaneous Doppler burst detection. Because the densities of the latex particles and the ﬂuid are closely matched, the particles follow the local ﬂow well and they do not settle much. The second ﬂow cell shown in Fig. 1(c) is used to gen- erate a jet ﬂow in a 9 cm×9cmsquare vat ﬁlled with quiescent water. The circular nozzle has an outlet 2 mm in diameter and the tube diameter before the contraction is 5.5 mm. The nozzle is immersed in the square vat and a small pump is used to generate a jet ﬂow at a constant ﬂow rate 0 .39cm3/s. The mean velocity at the nozzle exit is 12 .4cm/s and the corresponding Reynolds num- ber is Re=248. This value of Re is approximately three times larger than the turbulent transition Reynolds num- ber,Rec≃80, for a round jet. [12] Because of the large area contraction (7.6:1), the bulk part of the velocity pro- ﬁle at the nozzle exit is ﬂat. This uniform velocity pro- ﬁle vanishes quickly, and a Gaussian-like velocity proﬁle is developed in the downstream region, 2-20 diameters away from the nozzle exit. [13] As shown in Fig. 1(c), the direction of the momentum transfer vector q(and hence the measured velocity diﬀerence) is parallel to the jet ﬂow direction, but the separation L is at an angle of 45oto that direction. B. Measurement of the local velocity In the measurement of the velocity diﬀerence, we use two input ﬁbers to collect the scattered light with the sameq(i.e. at the same scattering angle) but from two spatially separated regions in the ﬂow. We now show that with a diﬀerent optical arrangement, the ﬁber-optic method can also be used to measure the instantaneous local velocity v(x, t). Instead of collecting light from two spatially separated regions with the same q, we use the two input ﬁbers to collect light from a common scatter- ing volume in the ﬂow but with two diﬀerent momentum transfer vectors q1andq2(i.e. at two diﬀerent scatter- ing angles). Figure 2(a) shows the schematic diagram of the scattering geometry. The collected signals at the 3two scattering angles are combined by a ﬁber-optic cou- pler, and the resultant light is modulated at the Doppler beat frequency: [14] ∆ ω1=q1·v−q2·v= ∆q·v(x), where ∆ q=q1−q2. The magnitude of ∆ qis given by ∆q= (4πn/λ)sin(α/2), with αbeing the acceptance angle between the two ﬁbers. The principle of using the scattered light at two dif- ferent scattering angles to measure the local velocity has been demonstrated many years ago. [14] What is new here is the use of the ﬁber-optic coupler for optical mix- ing. The ﬁber-optic technique simpliﬁes the standard LDV optics considerably. As shown in Fig. 2(b), in the standard LDV arrangement the two incident laser beams form interference fringes at the focal point. When a seed particle traverses the focal region, the light scattered by the particle is modulated by the interference fringes with a frequency, [15] ∆ ω1=q1·v−q2·v= ∆q·v(x). The magnitude of ∆ qhas the same expression ∆ q= (4πn/λ)sin(α/2) as shown in the above, but αnow be- comes the angle between the two incident laser beams. The main diﬀerence between the standard LDV and the new ﬁber-optic method is that the former employs two incident laser beams and a receiving ﬁber [Fig. 2(b)], while the latter uses only one incident laser beam and two optical ﬁbers to measure each velocity component [Fig. 2(a)]. Consequently, the beat frequency ∆ ω1in Fig. 2(a) is independent of the direction ( ki) of the inci- dent laser beam, whereas in Fig. 2(b) it is independent of the direction ( ks) of the receiving ﬁber. FIG. 2. (a) One-beam scattering geometry for the local ve- locity measurement. ki, incident wave vector; ( ks)1and (ks)2, two scattered wave vectors; q1= (ks)1−ki;q2= (ks)2−ki; ∆q=q2−q1. (b) Two-beam scattering geometry for the local velocity measurement. ( ki)1and (ki)2, two incident wave vectors; ks, scattered wave vector; q1=ks−(ki)1; q2=ks−(ki)2; ∆q=q2−q1. (c) Schematic diagram of a one-beam probe for the local velocity measurement. S, mea- suring point; P, lens; M, frequency modulator; C, ﬁber-opti c coupler. With the one-beam scheme, one can design various op- tical probes for the local velocity measurement. Figure2(c) shows an example, which would replace the commer- cial LDV probe by reversing the roles of the transmitter and receiver. The two input ﬁbers aim at the same mea- suring point S through a lens P, which also collects the back-scattered light from S. The frequency modulator M shifts the frequency of the light collected by the input ﬁber 1, before it is combined via the ﬁber-optic coupler C with the light collected by the input ﬁber 2. The re- sultant light from an output ﬁber of the coupler contains the beat frequency ∆ ω1and is fed to a photodetector. Because the measured ∆ ω1is always a positive number, one cannot tell the sign of the local velocity when zero velocity corresponds to a zero beat frequency. The fre- quency shift by the modulator M causes the interference fringes to move in one direction (normal to the fringes) and thus introduces an extra shift frequency to the mea- sured beat frequency. This allows us to measure very small velocities and to determine the sign of the mea- sured local velocity relative to the direction of the fringe motion (which is known). [15] The other output ﬁber of the coupler can be used as an alignment ﬁber, when it is connected to a small He-Ne laser. With the reversed He- Ne light coming out of the input ﬁbers, one can directly observe the scattering volume viewed by each input ﬁber and align the ﬁbers in such a way that only the scattered light from the same measuring point S is collected. The one-beam probe has several advantages over the usual two-beam probes. To measure two orthogonal ve- locity components in the plane perpendicular to the in- cident laser beam, one only needs to add an extra pair of optical ﬁbers and a coupler and arrange them in the plane perpendicular to that shown in Fig. 2(c) (i.e., rotate the two-ﬁber plane shown in Fig. 2(c) by 90o). With this arrangement, the four input ﬁbers collect the scattered light from the same scattering volume but in two orthog- onal scattering planes. Because only one laser beam is needed for optical mixing, a small single-frequency diode laser, rather than a large argon ion laser, is suﬃcient for the coherent light source. In addition, the one-beam ar- rangement does not need any optics for color and beam separations and thus can reduce the manufacturing cost considerably. With the one-beam scheme, one can make small invasive or non-invasive probes consisting of only three thin ﬁbers. One can also make a self-sustained probe containing all necessary optical ﬁbers, couplers, photodetectors, and a signle-frequency diode laser. III. RESULTS AND DISCUSSION A. Velocity diﬀerence measurements We ﬁrst discuss the measurements of the velocity dif- ference in rigid body rotation. Figure 3(a) shows a sam- ple oscilloscope trace of the analogue output from PMT2 when the separation L=1.0 mm. The signal is ampliﬁed 1250 times and band-pass-ﬁltered with a frequency range 4of 1-10 kHz. This oscilloscope trace strongly resembles the burst signals in the standard LDV. The only diﬀer- ence is that the signal shown in Fig. 3(a) results from the beating of two moving particles separated by a distance L. Figure 3(a) thus demonstrates that the beat signal between the two moving particles is large enough that a standard LDV signal processor can be used to measure the instantaneous velocity diﬀerence δv(ℓ, t) in real time. Figure 3(b) shows the measured beat frequency ∆ ω2 as a function of separation L. The circles are obtained from the oscilloscope trace and the triangles are obtained from the intensity correlation function g(τ). The two measurements agree well with each other. The solid line is the linear ﬁt ∆ ω2= 41.78L(103rad/s), which is in good agreement with the theoretical calculation ∆ω2=ksωL= 42.28L(103rad/s). This result also agrees with the previous measurements by Du et al. [5] Because ∆ ω2increases with L, one needs to increase the laser intensity at large values of L in order to resolve ∆ ω2. The average photon count rate should be at least twice the measured beat frequency. FIG. 3. (a) Oscilloscope trace of a typical beat burst be- tween two moving particles separated by a distance L=1.0 mm. The signal is obtained in rigid body rotation. (b) Mea- sured beat frequency ∆ ω2as a function of separation L. The circles are obtained from the oscilloscope trace and the tri an- gles are obtained from the intensity correlation function g(τ). The solid line shows a linear ﬁt to the data points. We now discuss the time series measurements of δv(L, t) in a jet ﬂow using the LDV signal processor. The jet ﬂow has signiﬁcant velocity ﬂuctuations as compared with the laminar rigid body rotation. The measuring point is in the developing region of the jet ﬂow, 3 di- ameters away from the nozzle exit and is slightly oﬀ the centerline of the jet ﬂow. Figure 4 shows the measured histogram P(δv) of the velocity diﬀerence δv(L, t) in the jet ﬂow, when the separation L is ﬁxed at L=0.5 mm (cir- cles) and L=0.8 mm (squares), respectively. It is seen that the measured P(δv) has a dominant peak and its position changes with L. Because δv(L, t) increases with L, the peak position moves to the right for the largervalue of L. The solid curve in Fig. 4 is a Gaussian ﬁt to the data points with L=0.5 mm. The obtained mean value of δv(L, t) is/angb∇acketleftδv/angb∇acket∇ight= 1.87cm/s and the standard deviation σ= 0.171cm/s. At L=0.8 mm, the measured P(δv) peaks at the value /angb∇acketleftδv/angb∇acket∇ight= 2.73cm/s. FIG. 4. Measured histogram P(δv) of the velocity diﬀer- enceδv(L, t) in the jet ﬂow. The values of L are: L=0.5 mm (circles) and 0.8 mm (squares). The solid curve is a Gaussian ﬁt to the circles. In the above discussion, we have assumed that each input ﬁber sees only one particle at a given time and the beat signal comes from two moving particles sepa- rated by a distance L. In fact, when the seeding density is high, each input ﬁber may see more than one parti- cle at a given time. The scattered light from these par- ticles can also beat and generate a self-beat frequency proportional to δv(ℓ0), where ℓ0≃0.15mmis the laser spot size viewed by each input ﬁber. [5] The self-beating gives rise to a small peak on the left side of the measured P(δv). Note that the peak position is independent of L, because δv(ℓ0) is determined only by ℓ0, which is the same for both measurements. It is seen from Fig. 4 that the cross-beating is dominant over the self-beating under the current experimental condition. The intensity correlation function g(τ) is also used to analyze the beat signal. In the experiment, we measure the histogram P(δv) and g(τ) simultaneously, so that Eq. (4) can be examined in details. Figure 5 shows the measured g(τ)−1 (circles) as a function of delay time τ at L=0.5 mm. The squares are the self-beat correlation function Gs(τ) obtained when one of the input ﬁbers is blocked. As shown in Fig. 4, the measured P(δv) has a Gaussian form and thus the integration in Eq. (4) can be carried out. The ﬁnal form of g(τ) becomes g(τ) = 1 + Gs(τ)/bracketleftBig bs+bccos[q/angb∇acketleftδv/angb∇acket∇ightτ]e−(qστ)2/2/bracketrightBig .(5) The solid curve in Fig. 5 is a plot of Eq. (5) with bs= 0.5 andbc= 0.13. The values of /angb∇acketleftδv/angb∇acket∇ightandσused in the plot are obtained from the Gaussian ﬁt shown in Fig. 4. It is seen that the calculation is in good agreement with the 5measured g(τ). The ﬁtted value bs= 0.5 agrees with the expected value at I1=I2. The value of bcwould be 0.5 if the collected signals from the two input ﬁbers were fully co- herent and the ﬁber-optic coupler mixed them perfectly. The fact that the ﬁtted value of bcis smaller than 0.5 indicates that the collected signals are not fully corre- lated. This is caused partially by the fact that in the present experiment the scattered light suﬀers relatively large number ﬂuctuations resulting from a changing num- ber of particles in the scattering volume. These number ﬂuctuations produce incoherent amplitude ﬂuctuations to the scattered light and thus introduce an extra (addi- tive) decay to g(τ). [16] Because the beam crossing time (proportional to the beam diameter) is much longer than the Doppler beat time 1 /∆ω2(proportional to the wave- length of the scattered light), the slow decay due to the number ﬂuctuations can be readily identiﬁed in the mea- sured g(τ). This decay has an amplitude 0.4 and has been subtracted out from the measured g(τ) shown in Fig. 5. FIG. 5. Measured intensity correlation function g(τ)−1 as a function of delay time τat L=0.5 mm (open circles). The squares are obtained when one of the input ﬁbers is blocked. The solid curve is a plot of Eq. (5). It is shown in Eq. (5) that to accurately measure the mean velocity diﬀerence /angb∇acketleftδv/angb∇acket∇ight, the beat frequency ∆ω2=q/angb∇acketleftδv/angb∇acket∇ightmust be larger than the decay rate Γ s≃ qδv(ℓ0) for Gs(τ) and also larger than the decay rate Γc≃qσresulting from the ﬂuctuations of the velocity diﬀerence. From the measurements shown in Figs. 4 and 5, we ﬁnd Γ s≃1.33×105s−1and Γ c≃3.8×104s−1, which are indeed smaller than the beat frequency ∆ ω2≃ 4.15×105s−1. Because g(τ) contains a product of Gs(τ) and exp[ −(qστ)2/2] [see Eq. (5)], its decay is determined by the faster decaying function. It is seen from Fig. 5 that the decay of g(τ) is controlled by Gs(τ), which de- cays faster than exp[ −(qστ)2/2]. It should be noted that in the measurements shown in Fig. 4, the beat signals are analogue ones and we have used a band-pass ﬁlter together with a LDV signal analyzer to resolve the beatfrequency. Consequently, many low-frequency self-beat signals are ﬁltered out. This low-frequency cut-oﬀ is ap- parent in Fig. 4. The measurements of g(τ), on the other hand, are carried out in the photon counting mode, and therefore the measured g(τ) is sensitive to all the self- beat signals as well as the cross beat signals. With a simple counting of particle pairs, we ﬁnd that the proba- bility for cross beating is only twice larger than that for the self-beating. B. Local velocity measurements We now discuss the local velocity measurements using the new optical arrangement shown in Fig. 2(a). The ve- locity measurements are conducted on a freely suspended ﬂowing soap ﬁlm driven by gravity. Details about the ap- paratus has been described elsewhere, [17–19] and here we mention only some key points. 2% solution of deter- gent and water is introduced at a constant rate between two long vertical nylon wires, which are held apart by small hooks. The width of the channel (i.e., the distance between the two nylon wires) is 6.2 cm over a distance of 120 cm. The measuring point is midway between the ver- tical wires. The soap solution is fed, through a valve, onto an apex at the top of the channel. The ﬁlm speed ¯ v, rang- ing from 0.5 to 3 m/s, can be adjusted using the valve. The soap ﬁlm is approximately 2-6 µmin thickness and is seeded with micron-sized latex particles, which scatter light from a collimated laser beam. The light source is an argon-ion laser having a total power of 1W. The incident laser beam is oriented perpendicular to the soap ﬁlm and the scattered light is collected in the forward direction. FIG. 6. Measured intensity autocorrelation function A(τ) as a function of delay time τwith the measuring time T=30 ms. The inset (a) shows an enlarged portion of A(τ) for small values of τup to τ= 20µs. The inset (b) shows the frequency power spectrum P(f) of the measured A(τ). To measure the rapidly changing beat frequency ∆ ω1, we build a fast digital correlator board for PC. [20] With a fast sampling rate fs, the plug-in correlator 6board records the time-varying intensity I(t) (number of TTL pulses from the photomultiplier tube per sam- ple time) over a short period of time T and then calcu- lates the (unnormalized) intensity autocorrelation func- tion,A(τ) =/angb∇acketleftI(t+τ)I(t)/angb∇acket∇ight. Figure 6 shows an example of the measured A(τ) as a function of delay time τwith T = 30 ms and fs= 14.32 MHz. Because the burst sig- nal I(t) is a periodic function of t, the measured A(τ) becomes an oscillatory function of τ. The frequency of the oscillation apparent in the inset (a) is the beat fre- quency ∆ ω1. The amplitude of the oscillation decays at largeτ. The inset (b) shows the power spectrum P(f) of the measured A(τ); it reveals a dominant peak at 755.1 kHz. The power spectrum is obtained using a fast Fourier transform (FFT) program. [21] FIG. 7. Measured intensity autocorrelation function A(τ) as a function of delay time τwith the measuring time T=50 µs. The inset (a) shows the frequency power spectrum P(f) obtained by FFT. The inset (b) shows the frequency spectrum Q(f) obtained by the Scargle-Lomb method. To increase the sampling rate of the velocity measure- ments, one needs to keep the measuring time T for each A(τ) as short as possible. The signal-to-noise ratio for A(τ) decreases with shorter measuring time T and with lower mean photon count rate, which was ∼1 MHz in the present experiment. It is found that the shortest useful measuring time Tcis roughly 50 µs. For this value of T, A(τ) is quite noisy and the corresponding peak in P(f) becomes less pronounced [see Fig. 7 and inset (a)]. It is worth mentioning that if one only wants to know the periodicity of a function, rather than its actual power at diﬀerent frequencies, the Scargle-Lomb method [22] is a better alternative to FFT. This method, which does not require evenly spaced sampling, compares the measured data with known periodic signals using the least-square ﬁtting procedure and determines the relevant frequencies by the goodness of the ﬁt Q(f). It can even utilize the un- even sampling to further increase the Nyquist frequency. As shown in Fig. 7(b), the Scargle-Lomb method can still clearly identify the periodicity of the signal, even when the power spectrum P(f) [Fig. 7(a)] becomes lessreliable. The total time required for the measurement of the characteristic frequency is less than 1 ms. Using the correlator board together with an average speed PC (300 MHz), we are able to conduct accurate measurements of the local velocity with a sampling rate up to 1 kHz. IV. SUMMARY We have developed new optical arrangements with two single-mode input ﬁbers and a ﬁber-optic coupler to mea- sure the local velocity v(x) and the velocity diﬀerence, δv(ℓ) =v(x+ℓ)−v(x), between two spatial points sepa- rated by a distance ℓ. The ﬁbers and the coupler are po- larization preserving to guarantee a high signal-to-noise ratio. To measure the velocity diﬀerence δv(ℓ), the two input ﬁbers are used to collect the scattered light with the same momentum transfer vector qbut from two spa- tially separated regions in a ﬂow. These regions are il- luminated by a single coherent laser beam, so that the collected signals interfere when combined via the ﬁber- optic coupler. The resultant light received by a pho- tomultiplier tube therefore contains the beat frequency ∆ω2=q·δv(ℓ). We analyzed the beat signals using two diﬀerent devices and compared the results. First, the in- tensity auto-correlation function g(τ) was measured us- ing a digital correlator. Secondly, a standard LDV signal processor was used to determine the instantaneous beat frequency ∆ ω2. With this device, δv(ℓ,t) can be ob- tained in real time. The technique can be further devel- oped to measure one component of the local ﬂow vorticity vector /vector ω(x, t) =∇ ×v(x, t). [23] To measure the instantaneous local velocity itself, one needs only to reorient the two ﬁbers so that they point to the same scattering volume. With this optical arrange- ment, we have three alternatives to measure a velocity component. They employ (i) an analog photodetector and a standard LDV signal processor (burst detector), (ii) a commercial photon correlator, such as that made by ALV, and ﬁnally (iii) a home-made digital correlator. This latter device completes a velocity measurement in less than 1 ms and is orders of magnitude cheaper than the other two alternatives. The new ﬁber-optic method has several advantages over the standard LDV and can be used widely in the general area of ﬂuid dynamics. Be- cause only one laser beam is needed to obtain two veloc- ity components, a compact single-frequency diode laser can replace a large multi-frequency argon-ion laser. By eliminating the color and beam separation units in the standard LDV, the one-beam scheme is less costly to im- plement. With more optical ﬁber pairs and couplers, one can carry out multi-point and multi-component velocity measurements in various turbulent ﬂows. 7ACKNOWLEDGMENTS We thank M. Lucas and his team for fabricating the scattering apparatus and J. R. Cressman for his contribu- tions. The work done at Oklahoma State University was supported by the National Aeronautics and Space Ad- ministration (NASA) Grant No. NAG3-1852 and also in part by the National Science Foundation (NSF) Grant No. DMR-9623612. The work done at University of Pittsburgh was supported by NSF Grant No. DMR- 9622699, NASA Grant No. 96-HEDS-01-098, and NATO Grant No. DGE-9804461. VKH acknowledges the sup- port from the Hungarian OTKA F17310. [1] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, UK, 1995). [2] K. R. Sreenivasan, “Fluid turbulence,” Rev. Mod. Phys, 71, S383-395 (1999). [3] G. I. Taylor, “The spectrum of turbulence,”Pro. R. Soc. London A, 164, 476-490 (1938). [4] T. Narayanan, C. Cheung, P. Tong, W. I. Goldburg, and X.-L. Wu, “Measurement of the velocity diﬀer- ence by photon correlation spectroscopy: an improved scheme,”Applied Optics, 36, 7639-7644 (1997). [5] Yixue Du, B. J. Ackerson, and P. Tong, “Velocity dif- ference measurement with a ﬁber-optic coupler,” J. Opt. Soc. Am. A. 15, 2433-2439 (1998). [6] B. J. Berne and R. Pecora, Dynamic light scattering (Wi- ley, New York, 1976). [7] P. Tong, W. I. Goldburg, C. K. Chan, and A. Siri- vat, “Turbulent transition by photon correlation spec- troscopy,” Phys. Rev. A, 37, 2125-2133 (1988). [8] H. K. Pak, W. I. Goldburg, and A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,”Phys. Rev. Lett. 68, 938-941 (1992). [9] P. Tong and Y. Shen, “Relative velocity ﬂuctuations in turbulent Rayleigh-B´ enard convection,” Phys. Rev. Lett.69, 2066-2069 (1992). [10] H. Kellay, X.-l. Wu, and W. I. Goldburg, “Experiments with turbulent soap ﬁlms,”Phys. Rev. Lett. 74, 3975- 3978 (1995). [11] Oz Optics Ltd, 219 Westbrook Road, Carp ON Canada, K0A 1L0 (http://ozoptics.com). [12] J. W. Daily and D. R. F. Harleman, Fluid Dynamics , p.421 (Addison-Wesley, Reading, MA, 1966). [13] F. M. White, Viscous Fluid Flow , p.470 (McGrn-Hill, New York, 1991). [14] F. Durst and J. H. Whitelaw, “Optimization of opti- cal anemometers,” Proc. of Royal Soc. A. 324, 157-181 (1971). [15] L. E. Drain, The laser Doppler technique (John Wiley & Sons, New York, 1980). [16] P. Tong, K. -Q. Xia, and B. J. Ackerson, “Incoherent cross-correlation spectroscopy,”J. Chem. Phys. 98, 9256- 9264 (1993). [17] M. A. Rutgers, X. L. Wu, R. Bagavatula, A. A. Peterson, and W. I. Gouldburg, “Two-dimensional velocity proﬁles and laminar boundary layers in ﬂowing soap ﬁlms,”Phys. Fluids 8, 2847 (1997). [18] W. I. Goldburg, A. Belmonte, X. L. Wu, and I. Zus- man, “Flowing soap ﬁlms: a laboratory for studying two-dimensional hydrodynamics,”Physica A 254, 231- 247 (1998). [19] V. K. Horv´ ath, R. Crassman, W. I. Goldburg, and X. L. Wu, “Hysteresis at low Reynolds number: Onset of two- dimensional vortex shedding”, Phys. Rev. E 61, R4702- 4705 (2000) cond-mat/9903067. [20] For a full description of the correlator board, see http://karman.phyast.pitt.edu/horvath/corr/. The tota l cost for the correlator board is less than $100. The de- vice can be duplicated for non-proﬁt applications without permission. [21] see, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes , 2nd edition (Cambridge University Press, UK, 1992). [22] J. D. Scargle, “Studies in astronomical time series ana l- ysis III.”Astrophy. J. 343, 874-887 (1989). [23] S. H. Yao, P. Tong, and B. J. Ackerson, “Instanta- neous vorticity measurements using ﬁber-optic couplers,” manuscript available from the authors. 8
The Principle of Synergy and Isomorphic UnitsEdgar Paternina Electrical engineer Author of Physics and The Principle of Synergy , published in CD ROM in English and Spanish at Amazon.com, on which this paper is based Contact:epaterni@epm.net.co AbstractA solution to the part and whole problem is presented in this paper by using a complex mathematicalrepresentation that permits to define the Holon concept as a unit that remains itself in spite of complexoperations such as integration and derivation. This can be done because of the remarkable isomorphicproperty of Euler Relation. We can then define a domain independent of the observer and the object, aswithin it, the object is embedded. We will then be able to have a Quantum Mechanics solution withoutthe "observer drawback", as Karl R. Popper tried to find all his life but from the philosophical point ofview and which was Einstein main concern about QM too. A unit that has always similar or identicalstructure or form, despite even complex operations such as integration and derivation, is the ideal unit forthe new sciences of complexity or just the systems sciences too, where structure or form, wholeness,organization, and complexity are main requirements. A table for validating the results obtained ispresented in case of the pendulum formula. Complex Numbers and Isomorphic Units In his General System Theory , Ludwig von Bertalanffy wrote: Reality, in the modern conception, appears as a tremendous hierarchical order of organized entities,leading, in a superposition of many levels, from the physical and chemical to biological and sociologicalsystems. Unity of Science is granted, not by a utopian reduction of all sciences to physics and chemistry,but by the structural uniformities of the different levels of reality.Those structural uniformities or isomorphisms in different levels of reality are the main concern of thispaper, and its main aim will be to present a new way of "seeing" reality by means of some isomorphic units, or co-variant units , so to speak, units in which the form is one important attribute as well. A basic recurrent design or pattern that can be used to interpret and explain those problems where dynamicinteractions or an organized complexity appear.But the problem of form appeared in classical physics too but precisely in those fields where the field concept was unavoidable. The magnetic field problem, whose existence can even be felt by putting two permanent magnets near by, is really one of those problems of nature that after all were hiding a greatmystery. With the magnet we can not only present theoretical examples of those three basic fundamentalattributes that are the basic to an isomorphic unit, but the magnetic field to be well represented, from themathematical point of view, we must also use a complex number mathematical symbolism.Those three basic attributes are: wholeness, oneness and openness.The wholeness attribute can be seen easily in the case of a magnet, where each magnet split from anothermagnet is precisely a whole new magnet. This new "part" that came from an old whole is a whole too .The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 1 of 31To obtain a whole from another whole is like to obtain a son from a father, or an object from theinstanciation of a class and in this same sense Ken Wilber wrote To be a part of a larger whole means that the whole supplies a principle(or some sort of glue) not found in the isolated parts alone, and thisprinciple allows the parts to join, to link together, to have something in common, to be connected, inways that they simply could not be on their own...When it is said that "the whole is greater than the sumof its parts," the "greater" means "hierarchy"...This is why "hierarchy" and "wholeness·" are oftenuttered in the same sentenceBut associated with this wholeness attribute is that binary or dual aspect of reality, where we have alwaystwo "opposites" or more appropriately, complementary or polar entities within a comprehensive whole,or just a unit that transcends duality, the oneness attribute, that can be found physically in a magnet. Thewithin and the without some sort of dynamic structure embedded in a unit that cannot be split in its twocomponents. Structures that perform well in a changing environment must be capable to reflect thewithout, they must have, as it were, a storing capacity to reflect that without. But the magnet has also another remarkable attribute associated with form, with the environment orinterface or some sort of a medium to separate an internal milieu from an external environment, as itwere, a field concept, or the openness attribute. At this point is important to recall that these threeattributes cannot be considered on their own. They are linked together by some sort of glue. This is themain characteristic of this new way of "seeing" reality in which we must always have in mind thatglueing principle, we will name the Principle of Synergy. But at this point of reasoning we are taken tothink in open systems, as those systems capable of exchanging with the environment. Prejudices normally come from ways of "seeing" reality that had generated by themselves some sort ofsecure stance in one of those realms. Facts, data or immediate experiences can be obtained in all thosethree realms, of science, philosophy and ontology and in every one of those realms we find in fact anactual practice, some sort of methodology to obtain facts. But the data obtained in every case is limitedby the language used and with objects of study that are obtained. Those that practice philosophy orontology normally have a great prejudice against a mathematical language, which is some sort oflanguage that permits the practicians to avoid any ambiguity, by obtaining with it precise definitions. Buton the other side those that practice philosophy and ontology were always right in the sense that alanguage whose main aim were to avoid any ambiguity or uncertainty was not a good language tointerpret reality after all. That ambiguity or uncertainty can be found there where we find those threeattributes, that has to do with open systems and in general with dynamism or generation of forms:wholeness, oneness and openness.For representing mathematically such kind of problems, it is necessary to use a language that permits todefine a unit embedded in a comprehensive whole, environment or dynamic structure, and which caninclude also a radical duality or polarity, which can be generated, precisely because that inherentcoincidentia oppositorum or tension, as it were, a field, that makes by definition that unit an open system.Complex numbers have the capacity not only to represent that binary or duality aspect of reality or thechance to have two polarities included in one unit, but also the nondual or wholeness attribute, we haverelated with that capacity to generate a new whole. The openness attribute is related with a capacity togenerate a field that is concomitant with those entities we can named holonic -to use a term coined byKen Wilber- such as magnetic entities, electrons, linguistic signs, cells, life, mind and beings in general.But complex numbers were born when trying to solve the simple algebraic equationThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 2 of 31 x² + 1 = 0 x² = -1 x = Sqr(-1) = J where as a solution we have the square root of a negative one or the radical unit J. From the point of view of "real" number perspective it does not exist a solution for this simple problemand as so it was necessary to make a paradigm extension or paradigm shift, and to define a new type ofmore general numbers to solve the problem. The solution J was named "imaginary" by Descartes for the first time and since then, complex numbers remained as some sort of a strange mathematical tool. It wasLeonard Euler in 1745, the one, that finally found a mathematical symbol for representing that new entitythat included both kinds of numbers:- those named real and which we will term nondual for reasons we will see later, and- those named "imaginary", we will name dual on the other side for apparent reasons too.Problems of growth has been associated from the point of view of a mathematical representation with thenumber epsilon or euler number since a long time ago. And it was by studying infinite series, that Eulerfound that entity, that not only could represent those two kinds of numbers, nondual and duals, but also itincluded those cyclical waveform, sine and cosine, that occur so frequently in nature wherever we havecyclical phenomena. But the most important, the most cogen argument to use this kind of new numbers,that have been used by Electrical Engineering since Oliver Heaviside and Steinmetz introduced them atthe end of the 19th century to solve alternating electrical current circuits, is precisely, its inherentisomorphic property, that permits them to make, as it were, co-variant representations, or to makesimpler complex operations.Evidently such a useful mathematical entity, was adequate to represent dynamic realities, and it was usedfor the first time for representing electromagnetic fields at the end of the 19th century, and after that,vectors were born in physics, but then they are taught, in general, without making any references to thiscomplex number origin. Up to that moment complex numbers had not been used in practical cases, andas so it justifies why the complex plane was delayed 100 years from its real birth at the end of the 18thcentury.A unit that has always similar or identical structure or form, despite even complex operations such asintegration and derivation, is the ideal unit for the new sciences of complexity or just the systemssciences too, where structure or form, wholeness, organization, and complexity are main requirements.But another important point is that it can also be used to define then a Basic Unit System concept inwhich uncertainty is included, and as so open systems.Classical physics had as it main aim to resolve natural phenomena into a play of elementary units, as itwere, to resolve those phenomena into their parts, isolated parts, I mean, so the concept of particle wasalways the starting point of the whole framework. But that part or particle needed to be considered as anisolated entity, that is, as a closed system, in which there were no interactions at all with theenvironment. This ideal model to represent reality was so restrictive that it definitely failed, the way weall know.An adequate framework for representing reality, the whole reality, must be complex, in the sense, that itThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 3 of 31must not only include, the dual-logical-nature of reality, but also it must include, that another aspectrelated with form, wholeness and oneness. The form, the structure must be co-variant, that is, it mustremain the same in spite of a progressive modification of that same structure. In this way adaptation orchangeness as a fundamental process can give us persistent properties for that structure or just despitecomplex operations such as integration and derivation done upon that structure. But there is another moreimportant aspect to recall, and it is the need not to reduce uncertainty, in certain cases, but to be able justto manage it. And here we come across with the fundamental problem of open and closed systems. Whenwe have an open system, in general, that uncertainty is an essential part of the problem, as it were, of itsopenness attribute, and in this sense that uncertainty cannot be reduced unless we close the system, so wedetermine its state completely defining then ideal objects of study, that can fail in real cases. Theinteractions of a closed system are reduced almost to zero and then the system becomes a static systemand not precisely as a steady state system. The so-called-new-sciences-of-complexity, with the seminal work of John H. Holland, Adaptation in Natural and artificial systems has as a main drawback precisely this tendency to eliminate uncertainty, reducing it to unprecisions or just by trying to close thesystem under consideration, so it is not strange to find in that school of science a tendency not abandonthe Second Law of Thermodynamics as the main principle. But open systems and that second law are insome sense "incompatible" if we establish a hierarchical framework in which, that second law is just aspecial case of the behavior of an open system. Euler Relation and Its Isomorphic Properties Up to this moment we have been seeing the emergence of a new concept of unit that includes in itsmathematical representation the dual and nondual nature of reality, as it were, a relationship between thepart and the whole or just a unit that is a whole and a part at the same time or a Holon as Ken Wilbernamed it. And what we aim with this paper is to show that Euler Relation, is precisely the mathematicalsymbol necessary to make an adequate representation of that whole/part entity we have named a Bus: e J( Ø ) = Cos (Ø) + J Sin (Ø) By asigning values to Ø, from Ø = 0 to Ø = 90 degrees, we obtain an horizontal and a vertical linerespectively, as it were, the complex plane, which can be "seen" as a fifth sphere of reality or a totalitythat can be used to represent or contain the four dimensional space-time continuum. It can also be seen asa mathematical representation of the domain of Form, in the same line of that domain imagined by theperennial philosophy with Plato. Karl R. Popper in the intent to transcend dualism envisioned thisdomain as a "third world", independent from mind or the subject and the object. Karl R. Popper main concern in Objective Knowledge was precisely to avoid what he called an essentialist explanation by introducing this "third world", as a world independent both from the object and the subject. But onlythrough a mathematical representation we can avoid any semantic pitfall. In figure we can see twosystems S and S´ in interrelation, and apart each other an angle, being the whole domain of representation the complex plane.The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 4 of 31 And in general we can then envision Ø acquiring all values, from zero to 360 degrees, or just, some sortof a clock pointer, or a vector rotating about an origin at a given frequency, where Ø = wt and w = 2¶f A vector rotating around an origin or a point, is what we term the centerness attribute of Euler Relation, that has to do with its natural cyclical behavior and we could say nature loves the cyclical waveformbehavior, as we find it everywhere, from the motions of the stars and the planets, to the tides on earth,our heartbeats and our psycological states and even that motion that has always exerted such afascination upon human mind, and I mean the pendulum motion. In Euler relation we have two types ofThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 5 of 31cyclical waves separated by the radical J, and what this means is that those two type of cyclical wavesare very different in nature, and this is what we are going to show in the following.In Euler Relation we then have a:Dual, symmetric or binary component that can be represented with the sine function: and aNondual or Oneness component that can be represented with the cosine function: Cos(Ø ) = Cos(-Ø)We must recall that the dual component or that component associated with the sine function changeswith changing the sign of the angle Ø, and that the nondual component or cosine function remains thesame with changing that sign angle, so we have in this unit those two requisites we pointed out at thebeginning of this paper necessary for representing reality. But this mathematical process of changingsign is associated with changing the rotation sense, counterclockwise or clockwise as we will see later, soit has to do with a very general sense of rotation of the whole structure.Historically Euler Relation was associated with the problem of the infinite series: where by replacing X = JØ we obtain finally Euler Relation by separating those terms that are affected by J from the others,obtaining the nondual or cosine expresion and the dual affected by J, or sine expression.This infinite series can be called Euler Relation, where, e = 2.71828. But the main point to notice at thisvery moment, is the cosine and sine nature of the two components separated by J. Cosine or nondual nature of Euler RelationFrom Euler relation we obtain the cosine function represented by Cos (wt) = ( e J( wt ))/ 2+ ( e - J( wt )) / 2 so the cosine function is expressed as the sum of two vectors rotating in opposite directions, one of themin counterclockwise or positive direction at an angular velocity w, and the second one in the clockwise ornegative direction at an equal angular velocity w and as the vector rotate the two dual components canceleach other, and as so the sum is a purely real or nondual vector, nondual because we cannot obtain theopposite by changing the sign of the angle. The axis of the cosine function is in fact that axis not affectedby J, the so called real axis, we have named the nondual axis. At this point it is important to recall theradical duality, expressed in that inherent polarity or tension of those two vectors rotating in oppositeThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 6 of 31directions but in a comprehensive whole, or just the complex plane. Sine or dual nature of Euler RelationFrom that same Euler Relation we obtain the sine function represented by Sin (wt) = [( e - J( wt )) / 2- ( e J( wt ))/ 2] J so the sine function is expressed as the sum of two vectors rotating in opposite directions, one of them incounterclockwise or positive direction at an angular velocity w, and the second one in the clockwise ornegative direction at an equal angular velocity w and as the vector rotate the two nondual componentscancel each other, and as so the sum is a purely, as it were, "imaginary" or a dual vector, dual because byjust changing the sign of the angle we can obtain the opposite.The axis of the sine function is in fact the J axis(see how the sine function is affected by J), the so-called-imaginary-axis, or the symmetry axis, where symmetry is defined as similarity of forms orarrangement on either side of a dividing line, so on one side we have a positive magnitude and on theother we have a negative sign for that magnitude. The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 7 of 3From this point of view it is not perfectly general to say that the sinusoidal wave, as an adjective, is thesame for both sine and cosine functions even though we can obtain the one from the other by the additionof a phase angle of -90°; that addition is not at all a trivial one though. In fact to avoid any ambiguity wemust differentiate clearly the asymetrical or nondual nature of the one, and the symetrical or dual natureof the other. Reality is in fact composed of two components, but in the physical domain, as it were, the "imaginary" domain, the dual nature is easily grasped and as so we can really say nature "loves" thesinusoid, and normally it hides as a mystery, its nondual nature. This nondual nature must always bediscovered. It is some sort of fixed or nonchanging component to which the change of the system as awhole can be referred, but as a point of reference it must be discovered, it must be chosen or requires adecision. In this sense we can say that the foundation of all reality is an ultimate frame of reference inwhich the nondual, the nonchanging is at the background or up from the point of view of hierarchy andthis implies then an "inclusive" attribute that is essential to have always in mind when dealing withreality. That ultimate frame of reference, we have named the domain of Form, permits us to define adomain independent of the observer and the object, as within it the object is embedded. We will then beable to have a Quantum Mechanics solution without the "observer drawback", as Karl R. Popper tried tofind all his life but from the philosophical point of view and which was Einstein main concern about QMtoo.Phase angle and magnitud are the two main state variables of that unit obtained with Euler relation andthat has been named a phasor in EE. We will use interchangeably the terms vectors and phasors and infact we will replace them by the Basic Unit System concept, which in general will be a rotating entity in the complex plane at a given frequency . In most practical problems all vectors are riding with the same frequency, so a sort of "merry-go-round" effect exists, in which vectors are seen as stationary withThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 8 of 31respect to each other, so the ordinary rules of vector geometry can be used to manipulate them as withthat "merry-go-round" effect we obtain some sort of static framework within dynamism. But then we have also the chance to have the phenomenon of resonance. We are acquainted with such a phenomenonspecially in the production of musical sounds and certain type of vibrations but the important point torecall is that through resonance we can explain those cases where small changes can often produce large effects. We must recall additionally again the fact that the phase or the angle sign is associated with the positiveor -negative sense of the vector rotation and as so:- In the cosine case we can interchange both rotation vectors, by changing their sign and nothingchanges, the form remains the same, they cannot be segregated just as in a magnet where we cannotdifferentiate the two polar components as they are seen always as one, as it were, the oneness property.- In the sine case that interchange, or changing of signs of the angle, affects the final-resulting vectorwith respect to its position in relation with the nondual axis.Resuming we have with ER a fundamental structure with two components separated by the radical J: - one nondual in which oneness is an essential attribute and which brings us to mind that self-awarenesscapacity that permits us as humans to think about our own thinking process, that permits us reflectivity orto know that we know, as it were, a new way of "seeing"; a way to be conscious of our Being, that makesus different from the animal world. This nondual component is precisely that one that permits us torepresent, as it were, the within of things. This within attribute was not possible to be considered inclassical physics as it used to see reality just from the without of things. But with this within attribute wecan envision a storing capacity of energy as that we find in magnetic fields, but also an informationstoring capacity in general. It is important to recall from the beginning that this within attribute as a storing information capacity at its highest manifestation is very different from consciousness , being more related with the fifth sphere of reality or just in plain mathematical words with the complex plane.So from here that pretension to reduce everything to consciousness is not our pretension anymore. Agreater within means a greater complexity or a greater without, that can or cannot be necessarily a greaterconsciousness.- one dual in which we can have two parts separated or clearly differentiated. Parts separated, is preciselythat condition necessary for the application of analytical procedures, being the second one, the chance tolinearize. We will see in what follows that ER most important isomorphic property is precisely to reduce complexity to a minus one degree of complexity making linear the representation of non-linear operators without reducing them.But these two basic components are united in that mathematical symbolism called Euler Relation and atthe same time separated in a radical way by J, repeating again at a higher level of representation the same basic structure we have found within that same relation, an inclusiveness attribute. We have then,as it were, a fundamental and basic minimum structural complexity represented by Euler Relation. Is thisbasic minimum structural complexity the one necessary to represent reality at its most profoundstructure? Does this basic minimum structural complexity give us those isomorphic properties needed forrepresenting and deducing the most fundamental laws of physics and reality? The answer to these twoquestions will be yes, and this is what this proposal is all about. Complex Algebra and Isomorphic PropertiesThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 9 of 31Vectors are ideal mathematical entities when relationships are important and as so we have vector sumsand differences, but also multiplication, division, derivative and integrals and every one of theseoperations can be represented in the complex plane as some sort of complex metrics and even theresulting geometrical figures are simpler than those obtained in normal geometry. But the powerfuladvantage of complex number is seen in computations, or when using complex-algebra, where theisomorphic property of Euler Relation manifests all its co-variant power.The main restriction we must pose from the very beginning is that of the same frequency, so that allrotating vectors considered must have the same angular velocity or frequency, so that we can have the"merry-go-round" effect. The frequency in this sense is that variable that can in fact produce large effectsin a system that was previously chaotic before acquiring it. The order of this system depends on thatacquired same frequency for all entities conforming the system. So this "restriction" is in fact the central point for the system acquiring a higher ordered state . In fact that frequency is associated with one of the essential state variables of the system, the angle, being the other the magnitude. The state is in fact thatone in which having a "merry-go-round" effect give us as a result, as it were, an organized complexity.Rewritting ER again e J( Ø ) = Cos (Ø) + J Sin (Ø) we can note that it is a unit vector in which we have a length one, because 1 = Sqr [ Cos ²(Ø) + Sin ²(Ø) ]So we have a unit vector standing at an angle Ø from the nondual or real or main axisThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 10 of 31 and we can have then in general, entities represented as A = Abs (A)* e J( Ø ) that can be represented too in a rectangular form taken from the figure The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 11 of 31as A = a-nondual + J * a-dual whereA-nondual = Abs( A)Cos(Ø) andA-dual = Abs( A)Sin(Ø) With these expressions the complex algebra can be established having in mind that we must not mix, asit were, oranges and apples, we must in fact abandon once and for all the reductionistic tendency, withthis complex representation. We must make in fact a radical distinction between the two basiccomponents that are used to represent reality and according to this radical duality, the nondualcomponents must be used with nonduals, and the dual components with the dual components, noting thatJ * J = -1 in case of multiplication and division of complex vector entities.With this in mind, all laws of normal algebra and arithmetic are preserved as with this radical separationwe preserve homogenety on the one side and heterogenety on the other, and as so we then have anotherrules for the basic operations with complex algebra. We must differentiate the dual nature of reality tointegrate them again, through the basic unit system concept, that transforms itself in a powerfulexplaining and simplier tool, some sort of a truly isomorphic unit.The sum or difference of two Buses . To obtain the sum of two Buses, we must decompose it in its two basic components in rectangular form as:The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 12 of 31A = a-nondual + J * a-dual B = b-nondual + J * b-dual soC = A + B equalsC = ( a-nondual + b-nondual ) + J( a-dual + b-dual ) soc-nondual = a-nondual + b-nondualandc-dual = a-dual + b-dualin the same way for the differenceThe product of two Buses. To multiply two Buses we must use instead of the rectangular form, the corresponding polar form as: A1 = Abs (A1) e J( Ø ) B1 = Abs (B1)* e J( Ø ) so we have A1* B1 = Abs(A1)Abs(B1) e J( Ø1) e J( Ø2) or A1 B1 = Abs(A1)Abs(B1) e J( Ø1+Ø2) a resulting vector C, in which their magnitudes are multiplied, as in normal arithmetical cases, but their angles are summed, so a multiplication or a nonlinear operation is transformed, as it were, in a linearoperation. C1 = Abs (C1)* e J( Ø3 ) where Abs(C1) = Abs(A1)Abs(B1) and Ø3 = Ø1 + Ø2The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 13 of 31By similar mathematical reasoning we can obtain the quotient of two buses, in which that quotienttransforms itself both in a difference of the corresponding angles and a quotient of its magnitudes. Themagnitudes behave themselves in the same way normal mathematical arithmetical quantitativeoperations behave. All its rules are preserved.But if we call qualitative those aspects related with the phase angle, we see then, they are summedinstead of multiplied in case of complex algebra representation. This capacity of complex numbers toreduce a nonlinear operation to a linear one, is in fact the one that makes them such a powerful simplifiertool, and adequate to represent the complex nature of reality we have been presenting from the beginningof this paper. But it is important to recall that linearization does not affect at all the quantitative part, justthe qualitative one, giving us a new capacity to think the whole correctly and the chance to introduce bodly in our intellectual frameworks new categories, so a new world outlook is obtained that includes initself an explanation and development of new things and even solves once and for all the paradoxes ofwave and particle as we will see when deducing the complex Schrödinger Wave Equation, but also whenobtaining the exact mathematical representation of the most intriguing open system mankind has everknown and which had always exerted a fascination for him, I mean, the pendulum movement.Conjugate of two complex numbers. Two buses are conjugate if they have the same magnitud and equal angles, but each angle being the negative of the other. As we have pointed out that negative sign has todo with an inversion of the rotational sense in such cases where the angle is a function of time.Differentiation and integration. We have already pointed out that remarkable property of ER we have qualified as isomorphic and that has to with the fact that the integration and differentiation of ER, arethemselves ER of the same frequency. And in that case in which the angle of ER is not complex, but justa simple function of a variable, the differentiation and integration of that ER function is the same, in bothcases. There is no way to distinguish between them. But that differentiation appears clearly in those casewe have a complete ER.By the rules of elementary Calculus, the derivative of an ER with a constant magnitud, has the samemagnitud as the original vector multiplied by the frequency w, but the angle has been advanced by 90°,or just the expression multiplied by J, which is an ER, where the angle is 90°. The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 14 of 31And the integration is the dual operation as that of differentiation and as so the integration of an ER withconstant magnitud, has the same magnitud as the original vector divided by the frequency w F -1 (INTEGRATION) F ' (DIFFERENTIATION) but the angle has been retarded by 90°, or just the expression multiplied by - J , as were, an opposite sense of rotation. This way of reasoning is exactly the application of what I like to name the dualityprinciple, but which is a way of applying a binary logic or symmetry to obtain a dual reality from one ofits components.The complex vector algebra simplifies mathematical operations by one degree, so in this way it reducescomplexity by one degree too and as so it becomes a real isomorphic tool as even complex operationssuch as differentiation and integration are reduced to simple algebraic operators. This technique ofreducing complexity by minus one degree is the most important characteristic of the ElectricalEngineering mathematics, where phasors are its "bread and butter" tool. This reduction of complexity byminus one degree permits us to comply with those two conditions of analytical procedures:- linearization andThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 15 of 31- to have weak interaction between the parts in those cases in which the general open system is closed,which was what Ludwig Von Bertalanffy pointed out as the central and methodological problem ofsystems theory.But the important thing to recall is that Euler Relation is the ideal isomorphic unit, as the sums,differences, integrals, and derivatives of Euler Relation functions of a given frequency are themselvesEuler Relation functions of the same frequency, they do not change their form with these operations, theyremain invariant or just co-variant. No other mathematical function is preserved in this fashion , as so it is the ideal one for filling that requirement Einstein put in his The foundation of the General Theory of Relativity , 1916, when he wrote "The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutionswhatever(generally co-variant)".We can then define a differential complex geometry in which we have a differential of reality defined asa Basic Unit System or a Holon such that DS = Abs (DS) e J( Ø ) with this complex metrics we can find and deduced all the fundamental equations of physics. It is notstrange to find such a powerful explaining tool after we have found all those isomorphic properties ofcomplex numbers. A Holon has embedded a within and a without, as it were, an analytical and asynthetic capacity, or a partness and a wholeness attributes. Leonard Euler must have felt a specialfeeling of wholeness when he found this relation. The Principle of Synergy The late Abraham Maslow was the one who coined the term "synergy" , an obscure term fromanthropology, but he used it for the first time in business to describe how wealth can be created fromcooperation. Creation of wealth, emergence of new things and structures that modify themselves to givebetter performance are main issues of the so-called-sciences-of-complexity or systems sciences.Structure and environment or as we would say, threeness(Oneness) and openness, they both claim for awhole that is greater than the sum of the parts, or for a basic and fundamental framework in which threefundamental and interdependent entities are put in mutual interaction. They pose the need to have asstarting point not only a minimum structure, but also a minimum system of elements in interaction.In the physical domain synergy can be found whenever we have a real transformation of energy toanother useful way or level which presuposses interchange or transformation of energy betweensystems.This "useful way" is in fact the real source of new applications and in this sense synergy is there,where we have emergence of new realities or just open systems that have exchanging-energy-capacitywith the environment. That whole greater than the sum of its parts has nothing to do with metaphysic, asthat "greater" comes from the very nature of an open system.Electrical energy, or the AC we use at home, is in fact the result of the application of the Principle ofSynergy where energy is transformed from a primary source, hidraulic for example, to electrical energyby moving a threefold magnetic structure, which gives at the output the AC energy we can utilize inmany ways. In these cases the open system concept is concomitant.The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 16 of 31If we consider three-double-complex-vectors or phasors, or just Buses constituted each one of them by where they correspond to three physical counterpart dispositions displaced 120° from one another justlike the figure and where that physical disposition cancels, as it were, its apparent cause so that thecause-effect relationship is subtle or the cause become hidden, or just it depends on the interchanging ofenergy or information with the environment or field, so at last we have A' =Abs (A')* e J( wt ) a rotating phasor or Bus at a given frequency.A given frequency is precisely the one attribute that makes it possible, the "merry-go-around" effect, as it were, that new order or reality that comes from the application of the Principle of Synergy. If thefrequencies are not the same for the three independent entities we wont't have the desired"merry-go-around" effect; it is just obtained when that frequency is reached, some sort of emergencestate, in which one small effect -a small change in frequency- gives us a large effect or a new emergentorder.The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 17 of 31 But those three or six independent entities in interrelationship conform just one entity or a whole picture,or a form represented by a circle.We have already come across with a radical separation of reality, in which we have a without and awithin but embedded in a whole, a whole that is not just the result of an arithmetical sum of elements.The Principle of Synergy or that glueing principle can be just represented by a fundamental sevenfoldstructure as the one of the figure in which we have not only a theoretical framework that has embedded that radical separation, but also apractical way of representing that minimum complex structure of nondual and dual, and in which weThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 18 of 31have a new type, as it were, of constitutive characteristic whereby we must not know only the parts, butalso the relations.But the point to recall is that structure represented by- three inner or nondual relations, or just the 1-1, 2-2 and 3-3 relations- three outer or dual relations, or just the 1-2, 2-3 and 1-3 relationsBut additionally the wholeness attribute is associated as we have pointed out with an emergent property,represented by the 1-2-3 relation. But here we have a clear departure of a holistic framework, as eventhough we have that holistic property embedded too in this framework, that important attribute isembedded in a Basic Unit System or Holon, not in a general abstraction that is just the whole, and wherethe hierarchy-relationship between that part and the whole is not cleat at all.The Bus(Holon) concept is embedded in a formal complex mathematical framework, whose generality isassociated with those isomorphic properties we have already presented and that give us a real internalconsistency which allows to treat steady-state systems by the same general techniques or methodology.Furthermore those six relations are some sort of detailed complexity as opposed to the 1-2-3 relationwhich is the complexity of the whole. The traditional problem between the part or detail and the whole isnot a problem anymore with this sevenfold structure that can be used at different levels of reality withoutreducing the one to the other.The fact we use a complex mathematical symbolism, as a tool, for representing this sevenfold structureavoids that syncretic whole reasoning Galileo used when he wrote There are seven windows given to animals in the domicile of the head, through which the air is admitted to the tabernacle of the body, toenlighten, to warm and to nourish it. What are these parts of the microcosmos? Two nostrils, two eyes,two ears, and a mouth. So in the heavens, as in macrocosmos, there are two favorable starts, twounpropitious, two luminaries, and Mercury undecided and indifferent. From this and many othersimilarities in nature, such as seven metals, ect., which is were tedious to enumerate, we gather that thenumbers of planets is necessarely seven . But it also prevents using as a key guiding principle the primacy of the whole which results in arrogant role of dominance when applied to organizations. The Pendulum The pendulum movement is so remarkable not just for its role as a cornerstone in the birth of modernscience that permitted Galileo a paradigm shift, but also because it is the most natural example of anopen system. When looking at the swinging body he saw a body that almost succeded in repeating thesame motion over and over ad infinitum. Its succes in repeating the same motion over an over lies in itsinterrelationship with the environment or in its openness attribute. By looking at the pendulum Galileo reported that the pendulum's period was independent of amplitud for amplitudes as great as 90°, hisview of the pendulum led him to see far more regularity than we can now discover there . Is this view of seeing more regulaties than there really existed part of the incapacity of normal science to explainpendulum-like regularities in a most documented way, and following a mathematical methodology withno leaps? Or How else are we to account for Galileo's discovery that the bob's period is entirely independent of amplitude, a discovery that the normal science stemming from Galileo had to eradicateand that we are quite unable to document today . The exact simple pendulum solution implies the solution of a first order differential equation whichThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 19 of 31implies too an integration whose solution is an elliptic integral. This means the introduction of anapproximation factor that could only be found by observations of the pendulum real behavior, some sortof trial and error procedure. In fact normal mathematical symbolism, I mean, not-complex-mathematicalsymbolism, cannot give reason of that approximation factor without using some sort of methodologicalleaps, to explain deviations. Normal science works with closed systems, that is, systems that do notexchange with their environment. But the pendulum movement seems to violate even that infamoussecond law of thermodynamics. In fact it gives us a natural sense of eternity just as the poet Jacques Bridaine wrote "Eternity is a pendulum whose balance wheel says unceasingly only the two words, in the silence of a tomb, 'always! never! always!... '" After exploiting the cyclical wave nature of Euler Relation in EE, it is obvious to expect we will be ableto explain with it, all those natural phenomena such as that of the pendulum, in which we have cyclicalor wave movements. But for achieving this, it is necessary to realize a real paradigm shift. With thecomplex plane we have introduced in fact a new sphere of reality in which we have embedded both thenondual and dual nature of reality, and as so a complex metrics, whose main characteristics are:- on the one side, its isomorphic property, as we have extensively pointed out up to this point, and whichgives us a powerful methodological tool to explain those cases in which real dynamism is involved- but also that property we have associated with a "merry-go-round" effect, and that permits us to make amathematical representation of real dynamic entities or of the generation of forms, as is the case with thependulum, in which its form is continually generating itself just as a result of a mere impulse.A new sphere of reality or the sphere of form, in which the four-dimensional space-time continuum isembedded. A new category or totality that contains the physiosphere is also that sphere in which we canhave animated forms, as it were, the biosphere, or the Bergson-Theilhard de Chardin DURATIONconcept.A complex metrics is then a top-down metrics and this also means an axiomatic nature, so from thenotion of the Basic Unit System and those normal regulaties already known from physics, but also fromthose properties obtained from geometry we can obtain the state of our Bus system, through amethodology in which instead of starting with the postulation of a differential equation, we start with theBus concept, as a tool to integrate or obtain that state. And this will be our aim for the pendulum case.From the point of view of the BUS, the pendulum movement is a rotational motion. The pendulum as aBus is then an open steady state system and the earth gravitational field is its context, and whenobserving that cyclical movement we can observe a maximum angle Ø, or Ømax, for a correspondingmaximum displacement Smax. Solving this problem means, integrating , some sort of constitutivecharacteristic, not a summative one.The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 20 of 31 We will use Int [ ] as the integration symbol - so the complex trajectory DS betweenS = 0toS = Smaxwill be but additionally we note that this is a static and potential expression related with space, and as so wemust introduce its dynamic counterpart by multiplying both part of the equation by the basic unit of timegiven by: whereØ = wtThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 21 of 31So reality S of the pendulum is generated and represented by: Intuitively, from geometry we know that in a circle we have as a general dynamic expresion in which the principle of synergy is included. Its manifestation can berecognized in that rotating expresion. Balancing Equality and the Pendulum To obtain the pendulum harmonic motion or just a steady state we must apply a balancing or acompensator equality, between two forces or polatities:- an inner one related with the weighting mass of the bob and- an outer one related with the so-called-inertial mass.This equality played a very important role in Einstein works about gravitational fields. Up to that timethat equality had been considered a mystery in the Newtonian framework. It generated those famousmental examples with elevators, in which bodies are subjected on the one side to an inertial force, and onthe other to the gravitational field.When these two forces or polarities within a comprehensive whole become equal then we have aharmonic motion or a motion that almost succeeded in repeating itself over and over ad infinitum. Abalancing loop which implies a rotational motion or the Synergistic Principle means that the law ofopposites must always be considered in a comprehensive whole, or dynamic context, producing acyclical movement. But if we consider inertial forces then we must apply Newton Second Law, but thistime at the fivefold continuum, or complex plane.The first derivative of S(t) S(t) = L* e J(wt + Ømax) is dS/dt = L Ømaxe J(wt + Ømax)The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 22 of 31and the second derivative: d 2 S/ d t 2 = - (LØmaxww)e J(wt + Ømax) being amax = ( L * Ømax * w w ) the maximum absolute value of acceleration.According to that second law we have at the maximum point Smax the inertia "vector", which tends tomaintain the motion as:Fmax = m ( L Ømax * w * w) if we apply, on the other hand, the vector composition of forces and using tangential components at thatpoint we have the mass "vector" as:Fmax = m * g * Sen(Ømax) so equating both the inertial and the mass "vectors" we've got:LØmaxww = gSen(Ømax)so the angular velocity is:w = sqr( g * Sen(Ømax) /( Ømax * L))but ifw = 2Pi fandT = 1/f, where f is the frecuency and T the period thenT = 2Pi sqr((L/g) ( Ømax /Sen(Ømax))which is the exact pendulum formula.The factor sqr( Ømax /Sen( Ømax) ) tends to one when Ømax is small and it can be omitted as long asthe amplitud does not exceed 10°. In Table 1, in the second row we can see the values K of the elliptic integral, in the third row the corresponding factor according that integral, and in the fourth row thecorresponding factor obtained with the Bus concept. In the fifth row we can see the difference in errorbetween the elliptic factor and the Bus Factor. This difference in errors as the amplitud increases can betaken as a fallibility criterion for those angles greater then 30 degrees where the error between the two isdefinitely inaceptable.Table 1. Aproximation Factor for the Period of a simple Pendulum Ø(angle) 0°10°20° 30°60° 90°120° 150°180° K 1.5711.5741.583 1.5981.686 1.8542.157 2.768Infinite 2K/Pi 1.0001.0021.008 1.0171.073 1.1801.373 1.762Infinite Ø /Sen(Ø) 1.0001.0051.002 1.0471.209 1.5702.418 5.235Infinite Error(%) 0.0000.298-0.595 2.86511.248 24.8443.217 66.34N/AThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 23 of 31The exact solution introduces a nonlinear character in the pendulum motion which gives reason of its realbehavior defined by the well known observed laws. Normally this problem was solved by resolving adifferential equation, that gave finally an elliptic integral, or some kind of solution that must be validatedwith the real observed behavior of the pendulum, as an elliptic integral cannot be expressed in terms of the usual algebraic or trigonometric functions, see Vector Mechanics for Engineers . Galileo did not know this factor, and so he extended the application of the observed pendulum laws far beyond its realrange, even to 90 degrees. But the fact that the time of any amplitude were independent of the mass ofthe body made he think in the falling of bodies and specially in the well-known Pisa Tower experiment,in which both an iron and a wood sphere fall with the same time, which means in all cases, the trajectoryfollowed by those bodies is an invariant.The pendulum is essentially a device for measuring time. It is in fact, as it were, the contrivance of time.Its form unfolds itself in time, it is the dynamic device for excellence. Differently from those movementsof classical Newtonian mechanics, like those of the planets, in which time does noy play really any roleand in which we could talk about the chance to invert time, the pendulum movement in this sense is not aclassical movement, and it answers the question posed by Thomas S. Khun about that movement. So thePrinciple of Synergy applied to the pendulum explains why this principle seems opposed to thatinfamous second law, as with it, forms can be generated and we can obtain in natural way a steady stateopen system. Quantum Mechanics The Quantum Mechanics problem before being a scientific problem is a philosophical problem as fromthe beginning it touches the same nature of reality and the relation of the subject with that reality. Therelation between object and subject, the active factor of the subject in the process of cognition is in factan epistemological problem that has to do with the way the subject defines what he understands byobjects. But this philosophical problem can be found in the idea of the active role of language in theshaping of our worldviews or images of the world, and as so it is primitive and was the main concern notonly of philosophers but also of historians as we can see in Adam Shaff work History and Truth. And weknow also the efforts done by Karl Poper during all his life and specially in the introduction of his Quantum theory and the Schism in physics. From the Postscript to the Logic of Scientific Discovery , to exhorcise consciousness or the observer from physics.Reality is independent of human mind, of the subject perceiving it, and we can distinguish in it threegreat realms:- Reality per se- The subject that perceives that reality- the object perceived by the subject as realityThe structure of reality perceived by the subject is crucial in the determination of the objects of study,that as a matter of fact, are those objects normally studied by science. But science is a product of humanactivity and as so language is crucial in the determination of those objects of study. This problem tookKarl R. Popper to define a third world, as an intermediate world between consciousness and objectivereality.As we have seen we can have two kinds of mathematical languages, a partial one that not necessarilyThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 24 of 31denies complex numbers as it uses them in "convinient ways", and an integral one in which complexnumber are used for building a unit in which the radical J is some sort of operator to distinguish betweentwo different orders of reality, as it were, the nondual and the dual nature of reality embedded in thatsame unit. In this sense this is not a new theory but a way of "seeing" and interpreting reality in whichuncertainty is always concomitant.When we talk about Reality per se we are not meaning a third world as that of Plato, divine, superhuman and eternal, but of reality as defined by Mortimer J. Adler in his Adler's Philosophical Dictionary where he wrote:BEING The word "being" is an understanding of that which in the twentieth century is identified with reality.What does the word "real" mean? The sphere of the real is defined as the sphere of existence that istotally independent of the human mind...Another distinction with which we must deal is that betweenbeing and becoming, between the mutable being of all things subject to change, and the immutable beingof that which is timeless and unchangeable. That is eternal which is beyond time and change. In the realof change and time, past events exist only as objects remembered, and future events exits only as objectsimagined.Historically the prevailing structure of reality has been a dualistic one in which reality as a whole isdivided in just two domains, being each one, as it were, parallel to the other, with no possibility ofintegration in case of gross dualism. Normally there has been some sort of partialness depending on theaccent put either in philosophy as the queens of all science, or in science as the real-objective science, orjust two and only two realms of reality. Philosophy and Ontology are not considered normally asindependent domains, but for our purpose and aim, philosophy must look for an integrated image ofreality and in the searching of methods of generalization so that philosophical theses are kept fromcontradicting science, in other words it must contribute to oneness and not to partialness. On the otherhand Ontology must look for the more-being, and as such realism must be concomitant as a main issue.In our context, in which we use a structure defined by a basic unit system, those objects of study can beclassified as real open systems and as ideal closed systems. An open system can have though a variablecenter and as such its external manifestation or its form cannot be determined, as its field or circle ofinfluence is variable, and this will be the case for the electron. Other open systems in which both itscenter and its radius are not variable will have a more determined state and will be near to an ideal closedsystem. But this issue has to do directly with the Uncertainty Principle, in which we have as inphilosophy no univocal conditions as those we can obtained with closed systems. But the important pointis the correlation we have between closed systems, measuring and open system and the UncertaintyPrinciple. The incapacity to measure the momentum and the position of an electron at the same time, hasbeen associated with a philosophical problem that has to do with an emphasis on the subject doing theexperiment, pointing out the fact that the structures of the subject are crucial in the way he defines realityor its object of study, which is similar to that philosophical problem in which language is considered as afactor that creates reality.We can talk about metaphysics when we have assertions for which we don´t have procedures on the oneside, or on the laws on the other, that can be used to verify those assertions. In the latter case when wehave laws we can built measurement instruments to verify the Data. On the other hand we have knownprocedures that when used they take us to a series of consistent results, that can be describedThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 25 of 31qualitatively, but not necessarily quantitatively. In both cases our main task as scientists and philosophermust be the searching of thruth.In our case we will consider this problem of the Uncertainty Principle, in case of an electron, as theimpossibility to determine the state of the corresponding Basic Unit System because there is not a lawwith which we can interrelate its two components, as it were, its dynamic counterpart and its static one.In this way we can separate the scientific problem of Quantum Mechanics from a philosophical problemwhich I think was the aim of Karl Popper when defining his "third world theory". In our case there areclear differentiation between precision and uncertainty, as there are cases such as the electron case inwhich there is no way to reduce the qualitative to the quantitative, which was Karl R. Popper requisite toincrease the degree of contrastability of certain theories.The trajectory or more appropriately the reality of an electromagnetic entity as that of the electron, whichas a matter of fact behaves as a wave, can be represented by the bus concept as But a general conceptualization of the Bus concept can be seen as a mathematical expression of energyin its most primary definition, but energy "is like a frequency multiplied by Plank's constant h"E = hfand that angle ø can be replaced byø = 2p / h (p* x - E *t)a well-known expresion taken from Quantum Mechanicsand The two state variables of the BUS in this case are x and t, so to able to determine its state, we must findor know a relationship between them. The geometric-like behavior of the pendulum gave us the clue tofind that relationship. In case of the planet movements it is the second Kepler Law associated with acentral force movement the one that permits to determine its state.We know that for the electron we cannot find a relationship between its two state variables, and this isanother way to present the Uncertainty Principle which means from the geometrical point of view wewill not have a known trajectory followed by the electron, as was the case with the pendulum or as it iswith the planets. In both of these cases we have a "real" differential equation associated with. In theelectron case we have the well-known complex Schrödinger wave Equation that was presented by him inThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 26 of 311926 as a postulate, that is, as a way of saying that equation does not have just one solution starting frominitial conditions, so the need to measure was replaced by qualitative methods, where one must focus in abehavioral area and not in finding laws that permit us to measure. In case of the pendulum its formchanges with time, but at the same time, its centerness attribute is localized, which seems not to be thecase for the electron case.But our aim is to deduce that complex wave Equation in the context of the Bus concept. Let us supposean unknown general solution, as a function of space and time, as it were, of its two state variables let us rename Abs(S) by Ψ and S(x, t) by Ψ( x, t) to distinguish magnitude from complex quantities on the one side, and on the other to use the traditionalnotation used when representing Schrödinger Wave Equation in Quantum Mechanics.Expressing E as function of momentum and replacingE = p²/2mwe have S(x, t) as the point here is to follow the well-known-wave procedure by making two partial derivatives of thisexpression with respect to space and one partial derivative with respect to time and by equating them weobtain the complex Schrödinger Wave Equation. But at this point it is important to recall the isomorphicproperty of Euler Relation as the one that makes it possible this result, as it were, the permanence of theunstable.The first partial derivative with respect to space givesThe Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 27 of 31the second partial derivative gives us equating both Ψ" expressions 2 and 3 The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 28 of 31we obtain the well-known Schrödinger Wave Equation, introduced by him in 1926, for a free particlemoving in x's direction. This equation was presented then, as we said previously, as a postulate as therewas no means to deduce it, from more basic principles, none the less, we have just applied the waveprocedure to the BUS concept based on the principle of synergy.There is an emergent conclusion in all this, and it is that the fundamental of physical reality is energy , and not a particle( or the mass concept). A particle is a Bus or Holon rotating at an unknown low frequency but in the complex plane or in reality represented in that plane, so that its state is completely determined, or else the qualitative aspect is reduced to the quantitative one in such a way no possibilityof a field or just a storing capacity is feasible different from its well determined state. It is important toremember at this point that the Bus concept as a mathematical symbolism also has that wholenessattribute that is at the base of growth, each whole has the capacity to generate another whole, but alwayshaving in mind that openness attribute that also permits us to define the Bus in general as a symbol forrepresenting a steady state open system. It is not a metaphysical concept that came from nothing. It is aconcept in which a glueing principle as that of the Principle of Synergy permits us to define that onenessattribute. Conclusions and Suggested applications From all this exposition emerges a conceptual framework that not necessarily reduces everything to themost elementary levels of reality, but as a good engineering conceptual tool opens new avenues forfuture research and development. The important point to recall at the outset is the need to abandon theold dualistic framework that has the natural tendency to put the whole or the part as a "primacy". TheBus concept is a part/whole complex mathematical concept that has embedded, as we have seen, thenondual and the dual nature of reality, but also the Principle of Synergy in which the whole is greaterthan the sum of its parts that gives us a medium to interpret reality, as it were, in organic ways as wehave a minimum threshold of complexity.In general, reality can be analyzed as a web of Buses but the Principle of Synergy implies a same frequency to obtain that "merry-go-around" effect or that emergent steady state or new order or that higher complex state where we have an organized complexity. One of the most important point to recallof this symbolism is that we do not need to make any reference to anthropomorphic concepts such aspsyche or consciousness to explain those emergent states where we find a whole greater than the sum ofthe parts. But from the point of view of the whole reality we obtain a framework in which by theintroduction of a fifth sphere of reality, that of form, we have then a sevenfold structure where thespace-time continuum is embedded in that fifth sphere where life can be defined as an animated form,but then we have mind or the noosphere as the sixth sphere of reality, but also the Being as that one thathas embedded them all.Having found such a powerful explaining tool, it is obvious to feel an imperious need to share with thescientifical community such a framework that can not only illuminates its actual practice by defining theobjects of study in certain cases but also establishing clearly the impossibility to reduce those objects toisolable units in other cases. The whole concern of Ludwing von Bertalanffy in its GST when he wroteWe may state as characteristic of modern science that this scheme of isolable units acting in one-waycausality has proved to be insufficient. Hence the appearance, in all fields of science, of notions likewholeness, holistic, organismic, gestalt, etc., which all signify that, in the last resort, we must think interms of systems of elements in mutual interaction.The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 29 of 31is then the same concern of all this paper, but our main point is the integration of those three big three, Being, Mind and Form or as Ken Wilber wrote in its Sex, Ecology and Spirituality the Spirit of EvolutionWith Kant, each of these spheres is differentiated and set free to develop its own potentials withoutviolence...These three spheres, we have seen, refer in general to the dimensions of "it", of "we", and of"I"...In the realm of "itness" or empiric-scientific truths, we want to know if propositions more or lessaccurately match the facts as disclosed...In the realm of "I-ness", the criterion is sincerity...And in the realm of "we-ness" the criterion is goodness, or justness or relational care and concern...What isrequired, of course, is not a retreat to a predifferentiated state...what is required in the integration of the Big Three. And that, indeed, is what might be called the central problem of postmodernity... how does one integrate them?Science, philosophy and ontology and its integration implies a science that looks for truth but with anopenness criterion, a philosophy than looks for oneness without the reductionism tendency but also for an ontology that looks for wholeness as a fundamental principle. To manage complexity properly it is very essential to have a basic structure and among the possibleconceptual structures we can have- a binary or dual one and- a threeness one that by mathematical inevitability becomes a sevenfold structureThis sevenfold structure has been used succesfully in many fields in a natural way but also recently wecan find its application in many other fields too. For example the three-"tiers" architectural approach toclient/server solutions and which looks for separating the various components of a client/server systeminto three "tiers" of services that must come together to create an application is precisely a solutionwhose main aim is to manage the changing complexity and which requires a basic hierarchy that startswith the service to the client. A tool must be adapted in every case so that we can avoid what can benamed the "Galileo syncretic whole reasoning" about that sevenfold structure. Maybe Galileo powerfulmind "saw" the powerful explaining capacity of this structure but his time was just the beginning of ascience in which the bynary or dual structure would prevail.Structure and environment are the main starting point of the new sciences of complexity or in the studiesof complex adaptive systems in which adaptation to the environment implies always a minimumconceptual structural framework.User services or interface or environment, business services or an adaptive plan, and data services or representation of changing estructures in a general way, are just some of the new applications we canfound in the General Systems Sciences or Sciences of complexity of this sevenfold framework withwhich we definitely transcend the dualistic worldview and which is really very different from the holisticparadigm, and not just from the conceptual point of view but even more important from the practicalpoint of view. 1. Bergson Henry.Creative Evolution. University Press of America.1983.2. Bertalanffy von Ludwig. General System Theory George Braziller.1969.3. Beer and Johnston. Vector Mechanics for Engineers. McGraw-Hill. 1962The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 30 of 314. Chardin Teilhard de. The Phenomenon of Man. Perennial Library.19755. Einstein et All. The Principle Of relativity. Dover.1952 6. Epsilon Pi. Physics and The Principle of Synergy Amazon.com.1999 7. Hazen and Pidd. FISICA. Editorial Norma. 19658. Ken Wilber. Sex, Ecology, Spirituality. Shambhala. 19959. Paternina Edgar. Física y Realidad. 1991.10. Saussure Ferdinand de. Course in General Linguistic. 199711. Stuart Kauffman. At Home in the Universe. Oxford Paperbacks.1995.12. Lao Tse. Tao Te King. Fontana. 199413. The I Ching. Richard Wilhelm Translation. Princenton .University Press.199014. Thomas S. Kuhn. The Structure of Scientific Revolutions. The University of Chicago Press.1996© 2000 by Epsilon Pi. All rights reserved. The Principle of Synergy and Isomorphic Units © 1999 by Epsilon Pi 31 of 31
arXiv:physics/0010023v1 [physics.gen-ph] 7 Oct 2000What Future Expects Humanity After the Demographic Transit ion Time? L.Ya.Kobelev, L.L.Nugaeva Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia E-mail: leonid.kobelev@usu.ru The variant of phenomenological theory of humankind future existence after time of demographic transition based on treating the time of demographic transi tion as a point of phase transition and taking into account an appearing of the new phase of mankind i s proposed. The theory based on physical phenomenological theories of phase transitions a nd classical equations for system predatory- preys for two phases of mankind, take into account assumptio n about a multifractal nature of the set of number of people in temporal axis and contains control parameters. The theory includes scenario of destroying of existent now human population by n ew phase of humanity and scenario of old and new phases co-existence. In particular cases when th e new phase of mankind is absent the equations of theory may be formulated as equations of Kapitz a , Foerster, Hoerner, Kobelev and Nugaeva, Johansen and Sornette phenomenological theories of growth of mankind. 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION The problem of mankind population growth is one of the global problems of the mankind existence and its prosperity in the future. Will the demographic ex- plosion existing now at the mankind population of the whole world stops and what will be after its stopping? Almost all the phenomenological theories of future of mankind ( see [1], [2], [3], [4], [5]) include the time of demographic transition t0. Before this time the growth of mankind population is very large ( hyperbolic or ex- ponential growth), after this time the population growth may practically absent [3] or may be regulated by driving parameters [4] that govern the development of mankind (such parameters are present in non-linear open systems (see [6]). Thus the time point t0(the time of demographic transition) is very essential for the future of mankind. What is biological and physical senses of the demographic transition time? The mankind as the whole is a very large system and its behavior can be described by methods of nonlinear dynamic, probability theory or statistical the- ory of open system using the methods of fractal geometry ( [7], [8], or power laws [5]) . Now put the question: how the demographical transition time may be treated from these points of view? There are many diﬀerent ways to answer on this question. In this paper we consider the demographic transition time as a point of a phase tran- sition, i.e. the point of transition of mankind in a new phase ( in physiological, psychological, behaviour and so on senses). What this phase will be if it may exist? Will be the population of men of the new phase friendly to the old phase of mankind or will they ﬁght with men of the old phase and force out them from their ecological niche? What scenario may exist and be admissible in that case? It seems that if the driving parameters that determine growth of the new phase ( thou it is diﬃcultnow to select the main of them, it is the tusk for many research groups ) will the same as now the new phase may diminish the old phase of mankind very soon. We would remind that all phenomenological theories give for the time t0the time in ﬁrst part of twenty one century. If the treating of time t0as a point of phase transition is correct and has relation to existence of humanity , the embryos of the new phase must appear ( as it well known from theory of phase transitions in physics) before the time of demographic transition , i.e. in very near time. What role they will play in the life of mankind society and what it will be like? Will the new phase consist of international groups of islam terrorist and the ﬁght and wars spread to the whole world? Or it will international groups of corrupted oﬃcials that for own prosperity gov- ern the whole world to its death? May be the groups of sexual minorities will became the main part of mankind and diminish it ? Will the new phase characteristics lay in psychological (i.e. behaviour) domain or it lay in the physiological domain ( may be it is incurable genetics diseases or mental disorders reasoned by increasing radi- ation in our world)? Are the Russian revolution in 1917 year or the Germany fascist state in 1933 year are the examples of appearing of such phases? May be the vic- tory of progressive mankind in the World War IIand collapse of socialist system in Soviet Union are results only the fact that the demographic time transition is not come? It seems, the search of answers on these questions is the one of main tusks of governments and research de- mographical gropes of developed countries in near time. We consider in this paper only one way of mathemat- ical analysis and description of two phases of mankind (the old phase and the new phase) and describe time be- haviour of these phases by system of diﬀerential equa- tions that are Volterra-Lotka equations for system of 1predatory-prey. II. EQUATIONS FOR DESCRIBING TWO PHASES MANKIND TIME BEHAVIOUR There are two simplest ways (physical and biological) for describing time dependence of two phases (the old and the new) of mankind : 1. In physics the phenomenological theory of phase tran- sitions was developed by Landau (see [9]). Theory of phase transitions as appearing of order parameters de- veloped by Ginzburg-Landau [10]. These theory may be used for describing time behaviour of two phases. From physical point of view the transition from the old phase of mankind to the new phase may: a) take very short time (as transition of water to ice); b) take long time and has intermediate state when two phases of mankind will co-existent (as in the transition to superconductive state for superconductors of second order);c) be charac- terized by stretched in time axes phase transition as in some ferroelectrics. The diﬀerences of these cases de- termined by characteristics of the small embryos gropes of new phase and their interaction with the old phase of mankind. If interaction between men of old and new phases will attract the man of old phase to transition to the new phase (we will call such interaction as ”nega- tive” ), the transition time will be small and co-existent of two phases of mankind for the long time period is im- possible. Probably the evidences of such transitions are the above examples of Russian revolution of 1917 year and fascist Germany state of 1933 year. If result of in- teraction of men of the old phase with gropes of men of the new phase will neutral, i.e. men of old phase will not be attracted by embryos of new phase ( ”positive” interaction), amount of new the phase will increase and co-existence of new and old phases is possible. The time of coexistence determined by govern parameters and se- lected mathematical models. 2. The biological point of view on behavior of mankind allows to use many mathematical models for describing of time behaviour of old and new mankind phases. In this paper we see only simple predatory-prey models where the old phase of mankind will be prey and the new phase will be predatory. III. EQUATIONS OF WOLTERRA-LOTKA FOR OLD AND NEW PHASES MANKIND POPULATION Let us designate the old phase of mankind by N1(t) and the new phase of mankind by N2(t) where tis cur- rent time. Equations of Volterra-Lotka type is well re- searched. It is well considered the time behaviour of thepredatory and the prey populations (including time be- haviour populations near the point of demographic tran- sition t0where ﬂuctuations play important role). So we only write these equations and mark the main conse- quences of mathematical theories. Let for time dtthe old phase changes are cNν 1dt−γN1N2dt( the second member is the result of diminishing character of inﬂuences (inter- actions) between the old and the new phases) and new phase changes are −δN2dt+bN1N2dt. The equations for N1(t) and N2(t) may be written in this case as ∂ ∂tN1(t) =cNν 1(t)−γN1(t)N2(t) (1) ∂ ∂tN2(t) =−δN2(t) +bN1(t)N2(t) (2) where c, b, δ, γ are constant and deﬁned by selections of considered models. These equations are Volterra equa- tions for ν= 1 and are Lotka equations for ν= 0. The humanity phases N1andN2in (1),(2) have links and de- termine one another. In the case when populations of the old phase and the new phase are multifractal sets on the time axes (see [4]) equations (1) and (2) read Dβ 0,tN1(t) =cNν 1(t)−γN1(t)N2(t) (3) Dµ 0,tN2(t) =−δN2(t) +bN1(t)N2(t) (4) where Dβ 0,tandDµ 0,tare Riemann-Liouville fractional derivatives or generalized Riemann-Liouville derivative s (see [4])and βandµare fractional numbers or functions of govern parameters. The equations (1) and (2) may be considered as the base equations of rude bioligical models for describing future population of mankind if there are exist two phases of mankind and sets of N1andN2are sets of population of old and new phases . As well known the equations of such type ( type of Volterra-Lotka equations) may be generalized and wrote in more general form ( form of Kolmogorov-Fokker-Plank equations for probability den- sityw(N1, N2, t) to ﬁnd in moment tthe populations N1andN2in the considered system [12], [13], or may be considered the case when inﬂuences N2on behaviors ofN1included only at the bound time intervals and so on). These generalized equations are wide spread and give many known models have used not only in biology ( [14]), but in economics ( [15]), non linear physics, the- oretical chemistry and so on. In the next part of paper we write more detailed description of results of interac- tions between two mankind phases N11 and N2based on well known researches of equations of such type. For case when in above equations there are only one phase N1, andν= 0 and coeﬃcient cis function of time and t0 (c=c(t, t0) it is not diﬃcult to chose c(t, t0) in the sim- ple forms and as special cases receive the base equations of work [3], [4], [5]. 2IV. HOW WILL INTERACT OLD AND NEW MANKIND PHASES? The mathematical research of solutions characteristics of equations (1)-(2) or more general equations taking into account the ﬂuctuations is well worked out (see for ex- ample [13]. As was pointed out these equations may be written for mean values of N1andN2or in the form of govern equations for probability w(N1, N2, t) to ﬁnd the system in the moment twith values N1andN2, or in the form of Kolmogorov- Fokker-Plank equations that are consequences of governor equations. The inﬂuences of noises and ﬂuctuations( depending at c, b, δ, γ ) on the behaviour of system may be taken into account too. More complicated equations may be written for case when the phase N2may part of her time have ”a rest time” and in that rest time do not interact with the phase N1. So we omit the process of receiving of solutions for diﬀerent systems of equations describing interaction between old- new phases and only describe main results: 1. Possibility of co-existence between old and new phases with periodical changing of value of N1andN2; 2. Possibility for diminishing of one phase by other phase that depends at the initial values; 3. Possibility for calculation of mean values of N1and N2in one of system states; 4. Possibility for calculation of time existence texfor old or new phases if take into account by including in the main equations (or changing the main equation by its Fokker-Plank probability analogies equations) exter- nal or internal noises ( additive or multiplicative); 5. Possibility of description of ﬂuctuation characteristi cs of kinetic variables near bifurcation points; 6. Possibility of researching of noises role in losing sta- bility of stationary state or in diﬀerent self-excited osci l- lations states; 7. Possibility of determination of conditions for stochas- tic regimes appearing in system; 8. Possibility of governing by processes of ﬁghting or co- existence between old and new phases; 9. Possibility of using more complicated models and equations including fractal characteristics of mankind population distribution on the time axis. 10. Possibility of selection of diﬀerent variants for gover n parameters based on experimental sociological dates and so on. Thus the predatory-prey models and its equations give rich possibilities for researching of the behavior of two phases of mankind. Has this assumption about appearing of two mankind phases relation to problem of future prop- agation of mankind? The future propagation of mankind will gives answer on this question, but if it assumption is correct it is time now to ﬁnd the new phase (or phases) and determine (or make attempts to determine) is it one or other phases useful for future of mankind or not useful.V. CONCLUSIONS As one of advantages to use the equations (1)-(2) for describing demographic problems (with some of them the mankind already has confronted now and may be these problems are consequences of appearing the embryos of new phase of mankind ) we shall stress an opportunity of insert and account in the theory many factors (such as, incurable illnesses, natural cataclysmic etc.) deﬁnin g future of mankind as a result of inﬂuences the control pa- rameters included in the govern parameters of predatory- prey models and fractional dimensions . There are three main scenario of models based on equa- tions type of (1)-(2): a) Two phases of mankind co-existent during long time with periodically changes its values and with mean val- uesN1=δb−1andN2=cγ−1conserves in bound limits if population ﬂuctuations may be omitted; b) Mankind population lost its stability as the result of ﬂuctuation inﬂuences if the equations of Lotta type used; c)The old mankind phase has bound time of existence and this time may by calculated if parameters of theory are deﬁned . The main purpose of this paper was to analyze possibil- ity of introducing of two mankind’s population phases after demographic transition time. Driving parameters included in the considered models (equations of Landau- Ginzburg time theory or equations of predatory-prey type) of the phenomenological theories of the mankind population in the frame of physical Landau or Ginzburg- Landau models or biological predatory-prey models. The predatory-prey type models was chosen asan example. The change of number of mankind (described in the framework of phenomenological theories of the popula- tion) can be adjusted by such choice of control param- eters with concrete contents of dependencies of them at which the mankind will has stability time and population will grow so slowly, that overpopulation and the problems connected with ﬁghting between two mankind phases will do not arise in the foreseen future. Stress now as was done in [5] that after demographic transition time also the type of economical development of humankind may be changed . [1] Foerster, Von H. et al., Science, v.132, p.1291 (1960) [2] Hoerner, von SJ., British Interplanetary Society, v.28 , 691, (1975) [3] Kapitza S.P., Uspehi Fizicheskih Nauk (Russia), 1996, vol.166, No.1, pp.63-79; Kapitza S.P., How Many People Lived, Live and are to Live in the World. An essay on the theory of growth of humankind , Moscow, Inst. Phys. Problem RAS, 1999,238p.(in Russian) 3[4] Kobelev L.Ya., Nugaeva L.L. Will Humanity population Stabilized in Future? arXiv/physics:0003035 [5] Johansen A., Sornette D. The End of growth Era? arXiv:cond-mat/0002075 [6] Klimontovith Yu.L., Statistical Theory of Open Sys- tems. Vol.1 , Moscow, Yanus, 1995, 686p. (in Russian); Kluwer Academic Publishers, Dordrecht, 1995; Klimon- tovich Yu.L., Statistical Physics of Open systems. Vol.2 , Moscow, Yanus, 1999, 450p. (in Rusian). [7] Mandelbrot B., Fractal Geometry of Nature , W.H.Freeman, San Francisco, 1982 [8] Kobelev L.Ya, Fractal Theory of Time and Space , Ekater- inburg, Konros, 1999, 136p. (in Russian);Kobelev L.Ya., What Dimensions Do the Time and Space Have: Integer or Fractional? arXiv:physics/0001035; [9] L.D.Landau, I.M.Livshiz, Statistical Physics, Moskow, Nauka,1964 (in Russian [10] V.L.Ginzburg, L.D.Landau, GETP, 1948, v.20,P.1064 (i n Russian) [11] Samko S.G, Kilbas A.A., Marithev I.I., Integrals and Derivatives of the Fractional Order and Their Applica- tions, (Gordon and Breach, New York, 1993). [12] Alekseev B.B.,Kostin I.K,Stability of simple biogeoc hi- noses to External Stochastic Inﬂuences,in ”Biologi- cal Systems in Agriculture and Forestry”,-Moskwa, Nauka,1974, p.105-112 ( in Russian) [13] Vasiliev,V.A., Romanovski Yu.M. Yaxno V.G. Avtovave processes, Moscow, Nauka,1987,p.240 (in Russian) [14] Murray J.D. Mathematical Biology, Springer-Verlag, Berlin New York Heidelberg, 1989 [15] Lorenz Nonlinear Dynamical Economics and Chaotic Motion , Springer-Verlag, 1993. 4
arXiv:physics/0010024v1 [physics.atom-ph] 7 Oct 2000Continuous-wave Doppler-cooling of hydrogen atoms with two-photon transitions V´ eronique Zehnl´ e and Jean Claude Garreau Laboratoire de Physique des Lasers, Atomes et Mol´ ecules an d Centre d’Etudes et de Recherches Laser et Applications Universit´ e des Sciences et Technologies de Lille F-59655 Villeneuve d’Ascq Cedex, France Abstract We propose and analyze the possibility of performing two-photon continuous- wave Doppler-cooling of hydrogen atoms using the 1 S−2Stransition. “Quenching” of the 2 Slevel (by coupling with the 2 Pstate) is used to increase the cycling frequency, and to control the equilibrium tempe rature. Theoreti- cal and numerical studies of the heating eﬀect due to Doppler -free two-photon transitions evidence an increase of the temperature by a fac tor of two. The equilibrium temperature decreases with the eﬀective (quenching dependent) width of the excited state and can thus be adjusted up to value s close to the recoil temperature. Typeset using REVT EX 1Laser cooling of neutral atoms has been a most active researc h ﬁeld for many years now, producing a great deal of new physics. Still, the hydrogen at om, whose “simple” structure has lead to fundamental steps in the understanding of quantu m mechanics, has not yet been laser-cooled. The recent experimental demonstration of th e Bose-Einstein condensation of H adds even more interest on laser cooling of hydrogen [1]. On e of the main diﬃculties encountered in doing so is that all transitions starting fro m the ground state of H fall in the vacuum ultraviolet (VUV) range (121 nm for the 1 S−2Ptransition), a spectral domain where coherent radiation is diﬃcult to generate. In 1993, M. Allegrini and E. Arimondo have suggested the laser cooling of hydrogen by two-photon πpulses on the 1 S−3Stransition (wavelength of 200 nm for two-photon transitions) [2]. Sinc e then, methods for generation of CW, VUV, laser radiation have considerably improved, and have been extensively used in metrological experiments [3]. This technical progress a llows one to realistically envisage the two-photon Doppler cooling (TPDC) of hydrogen in the continuous wave regime, in particular for the 1 S−2Stwo-photon transition. Laser cooling relies on the ability of the atom to perform a gr eat number of ﬂuorescence cycles in which momentum is exchanged with the radiation ﬁel d. It is well known that 2 S is a long-lived metastable state, with a lifetime approachi ng one second. From this point of view, the 1 S−2Stwo-photon transition is not suitable for cooling. On the ot her hand, the minimum temperature achieved via Doppler cooling is pro portional to the linewidth of the excited level involved on the process [4], a result that w ill be shown to be also valid for TPDC. From this point of view 2 Sis an interesting state. In order to conciliate these antagonistic properties of the 1S−2Ptransition, we consider in the present work the possibility of using the “quenching” [5] of the 2 Sstate to control the cycling frequency of the TPDC process. For the sake of sim plicity, we work with a one- dimensional model. We write rate equations describing TPDC on the 1 S−2Stransition in presence of quenching. The quenching ratio is considered as a free parameter, allowing control of the equilibrium temperature. The cooling method is then in principle limited only by photon recoil eﬀects. 2We also develop analytical approaches to the problem. A Fokk er-Planck equation is derived, describing the dynamics of the process for tempera tures well above the recoil tem- perature Tr(corresponding to the kinetic energy acquired by an atom in e mitting a photon). A numerical analysis of the dynamics of the cooling process c ompletes our study. Let us consider a hydrogen atom of mass Mand velocity vparallel to the z-axis (Fig. 1) interacting with two counterpropagating waves of angular f requency ωLwith 2 ωL=ω0+δ, where ω0/2π= 2.5×1014Hz is the frequency corresponding to the transition 1 S→2S, and also deﬁne the quantity k≃2kL= 2ωL/c. The shift of velocity corresponding to the absorption of two-photons in the same laser wave is ∆ = ¯ hk/M = 3.1 m/s. We will neglect the frequency separation between 2 Sand 2Pstates (the Lamb shift – which is of order of 1.04 GHz) and consider that the one-photon spontane ous desexcitation from the 2 P states also shifts the atomic velocity of ∆ randomly in the + zor−zdirection. Note that Tr=M∆2/kB≈1.2 mK for the considered transition ( kBis the Boltzmann constant). We neglect the photo-ionization process connecting the excit ed states to the continuum. This is justiﬁed by the 1 /Edecreasing of the continuum density of states as a function o f their energy Eand by the fact that a monochromatic laser couples the excite d levels only to a very small range of continuum levels. The atom is subjected to a controllable quenching process th at couples the 2 Sstate to the 2 Pstate (linewidth Γ 2P= 6.3×108s−1). The adjustable quenching rate is Γ q. Four two-photon absorption process are allowed: i) absorption of two photons from the +z-propagating wave (named wave “+” in what follows), with a ra te Γ1and corresponding to the a total atomic velocity shift of +∆; ii) absorption of two photons from the −z- propagating wave (wave “ −”), with a rate Γ −1and atomic velocity shift of −∆;iii) the absorption of a photon in the wave “+” followed by the absorpt ion of a photon in the wave “−”, with no velocity shift and iv) the absorption of a photon in the wave “ −” followed by the absorption of a photon in the wave “+”, with no velocity sh ift. The two latter process are indistinguishable, and the only relevant transition ra te is that obtained by squaring the sum of the amplitudes of these process (called Γ 0). Also, these process are “Doppler-free” 3(DF) as they are insensitive to the atomic velocity (to the ﬁr st order in v/c) and do not shift the atomic velocity. Thus, they cannot contribute to the coo ling process. As atoms excited by the DF process must eventually spontaneously decay to the ground state, this process heats the atoms. In the limit of low velocities, the transition amp litude for each of the four processes is the same. One thus expects the DF transitio ns to increase the equilibrium temperature by a factor of two. We can easily account for the presence of the quenching by int roducing an eﬀective linewidth of the excited level (which, due to the quenching p rocess, is a mixing of the 2 S and 2Plevels) given by Γe= Γ2PΓq Γq+ Γ2P=gΓ2P (1) withg≡Γq/(Γq+ Γ2P). This approximation is true as far as the quenching ratio is much greater than the width of the 2 Sstate (note that this range is very large, as the width of the 2Sstate is about 10−8times that of the 2 Pstate). The two-photon transition rates [6] are given by: Γn= Γ2Pg 2(1 + 3 δn0)¯I2 (¯δ−nKV)2+g2/4(2) where n={−1,0,1}describes, respectively, the absorption from the “ −” wave, DF transi- tions, and the absorption from the “+” wave. ¯I≡I/Iswhere Isis the two-photon saturation intensity, ¯δis the two-photon detuning divided by Γ 2P,K≡k∆/Γ2P= 0.26 and V≡v/∆. The rate equations describing the evolution of the velocity distribution n(V, t) and n∗(V, t) for, respectively, atoms in the ground and in the excited le vel are thus ∂n(V, t) ∂t=−[Γ−1(V) + Γ 0+ Γ1(V)]n(V, t) +Γe 2[n∗(V−1) +n∗(V+ 1)] (3a) ∂n∗(V, t) ∂t= Γ−1(V−1)n(V−1, t) + Γ 0n(V, t) + Γ 1(V+ 1)n(V+ 1, t)−Γen∗(V, t). (3b) The deduction of the above equations is quite straightforwa rd (cf. Fig 1). The ﬁrst term in the right-hand side of Eq. (3a) describes the depopulatio n of the ground-state velocity 4classVby two-photon transitions, whereas the second term describ es the repopulation of the same velocity class by spontaneous decay from the excite d level. In the same way, the three ﬁrst terms in the right-hand side of Eq. (3b) describe t he repopulation of the excited state velocity class Vby two-photon transition, and the last term the depopulatio n of this velocity class by spontaneous transitions. For each term, w e took into account the velocity shift ( V→V±1) associated with each transition and supposed that sponta neous emission is symmetric under spatial inversion. For moderate laser intensities, one can adiabatically elim inate the population of excited level. This is valid far from the saturation of the two-photo n transitions and reduces the Eqs. (3a-3b) to one equation describing the evolution of the ground-state population: dn(V, t) dt=−/bracketleftBigg Γ0+Γ−1(V) 2+Γ1(V) 2/bracketrightBigg n(V, t) + 1 2{Γ0[n(V−1, t) +n(V+ 1, t)] + Γ −1(V−2)n(V−2, t) + Γ 1(V+ 2)n(V+ 2, t)}(4) Eq.(4) is in fact a set of linear ordinary diﬀerential equati ons coupling the populations of velocity classes separated by an integer: V, V±1, V±2,· · ·. This discretization exists only in the 1-D approach considered here, but it does not signiﬁca ntly aﬀect the conclusions of our study, while greatly simplifying the numerical approac h. Eqs.(4) can be recast as dn/dt=Cn,where Cis a square matrix and nis the vector (· · ·n(−i, t),· · ·n(0, t), n(1, t),· · ·). Numerically, the equilibrium distribution is obtained i n a simple way as the eigenvector neqofCwith zero eigenvalue. In this way, the asymptotic temperature is obtained as : T Tr=/angbracketleftBig V2/angbracketrightBig =∞/summationtext i=−∞i2neq(i) ∞/summationtext i=−∞neq(i)(5) Fig. 2 shows the equilibrium distribution obtained by numer ical simulation for ¯δ=−0.25 andg= 1/3. The dotted curve corresponds to the distribution obtaine d by artiﬁcially suppressing DF transitions (i.e., by setting Γ 0= 0). As we pointed out earlier, the DF transitions lead to a heating eﬀect. Doppler cooling is eﬃci ent mainly for atoms distributed 5on a range of g/Karound the velocity V=±\|¯δ\|/K[4] whereas Doppler-free transitions are independent of the velocity; all velocity classes are thus a re aﬀected by the heating. As a consequence, DF transitions induce a deformation of the vel ocity proﬁle, specially for small values of gand¯δ, superimposing a sharp peak of cold atoms on a wide backgroun d of “hot” atoms. In what follows, all numerically calculated-temper atures are deduced form the width of the thin peak of cold atoms. Eqs. (3a) and (3b) or Eq. (4) have no exact solution. However, using some reasonable hypothesis, it is possible to develop analytical approache s. The most usual of these ap- proaches is to derive from the above equations a Fokker-Plan ck equation (FPE) describing the evolution of the velocity distribution. The derivation of the FPE for two-photon cool- ing follows the standard lines that can be found in the litera ture (see [7]). If \|V\| ≫1 the coeﬃcients in the resulting equation can be expanded up to se cond order in 1 /\|V\|(this is the so-called hypothesis of small jumps ). Moreover, if K\|V\| ≪ \|¯δ\|, gthe resulting expression can be expanded up to the order V. The resulting FPE reads ∂n ∂t= 2¯Γ′∂(V n) ∂V+/parenleftbigg 2¯Γ +Γ0 2/parenrightbigg∂2n ∂V2(6) where ¯Γ≡Γ−1(0) = Γ 0/4 and ¯Γ′is the V-derivative of Γ −1evaluated at V= 0. Multiplying this equation by V2and integrating over Vone easily obtains: d/an}bracketle{tV2/an}bracketri}ht dt=−4¯Γ′/an}bracketle{tV2/an}bracketri}ht+/parenleftBig 4¯Γ + Γ 0/parenrightBig (7) As/an}bracketle{tV2/an}bracketri}ht=T/T r, this equation shows that the characteristic relaxation ti me is (4 ¯Γ′)−1= (gΓ2P¯I2¯δK)/(4¯δ2+g2). The equilibrium temperature is then given by T Tr=2¯Γ + Γ 0/2 2¯Γ′=¯δ2+g2/4 K\|¯δ\|(8) This results conﬁrms that the Doppler-free two-photon tran sitions, corresponding to the contribution Γ 0/2 = 2 ¯Γ in Eq. (8) increase the equilibrium temperature (at least i n the range of validity of the FPE) by a factor 2. This fact can also b e veriﬁed from the numerical 6simulations, as shown in Fig. 3, where the dotted curve corre sponds to the temperature obtained without DF transitions. As in one-photon Doppler- cooling, the equilibrium tem- perature is independent of the laser intensity (but the time need to achieve cooling obviously increases as the laser intensity diminishes). Note that the range of validity of the FPE is \|V\| ≫1. It thus fails when the temperature approaches the recoil temperature (or, in other words, \|V\| ≈1). Fig. 4 shows the dependence of the equilibrium temperature as a function of the detuning for diﬀerent values of parameter g. The minimum temperature is clearly reduced by the decreasi ng of g, up to values close to the recoil temperature Tr. Moreover, the ﬁgure shows that the minimum temperature generally agrees with the theoretical predictions: it is go verned bothby the eﬀective linewidth gof the excited state andby the detuning, the optimum value being ¯δ≈ −g/2 (in the range of validity of the FPE). A reasonably good agreement between numerical data and the FPE prediction within its range of validity is also observed. Let us ﬁnally note that an interesting practical possibilit y is to change the quenching parameter as the cooling process proceeds. One starts with a high value of gin order to rapidly cool the atoms to a few recoil velocities. Then, the q uenching parameter andthe detuning are progressively decreased, achieving temperat ures of order of the recoil temper- ature. A detail study of the procedure optimizing the ﬁnal te mperature is however out of the scope of the present paper. In conclusion, we have suggested and analyzed, both analyti cally and numerically, the using of 1 S−2Stwo-photon transition together with the quenching of the 2 S-state to cool hydrogen atoms to velocities approaching the recoil limit. The quenching ratio gives an additional, dynamically controllable parameter. Laboratoire de Physique des Lasers, Atomes et Mol´ ecules (P hLAM) is UMR 8523 du CNRS et de l’Universit´ e des Sciences et Technologies de Lil le. Centre d’Etudes et Recherches Lasers et Applications (CERLA) is supported by Minist` ere d e la Recherche, R´ egion Nord- Pas de Calais and Fonds Europ´ een de D´ eveloppement Economi que des R´ egions (FEDER). 7FIGURES 1S 2S 2P Γ q Γ 2P ω0 ωL δ z v (+) (-) FIG. 1. Hydrogen levels involved in the two-photon Doppler c ooling in presence of quenching./CE/D2 /B4 /CE /B5/BE/BC /BD/BC /BC /B9/BD/BC /B9/BE/BC /BD/BC/BA/BK/BC/BA/BI/BC/BA/BG/BC/BA/BE/BC FIG. 2. Numerically calculated velocity distributions wit h¯δ=−0.25 and g= 1/3. The dotted curve corresponds to the distribution obtained by suppress ing Doppler-free transitions (cf. text). Typically, the distribution exhibits two structures: a bro ad background due to the atoms heat by Doppler-free transitions and a sharp peak of cold atoms. 8/AMÆ/CC /BP/CC /D6/BC /B9/BD /B9/BE /B9/BF /BD/BC/BC/BD/BC/BD FIG. 3. Dependence of the temperature (log scale) on the detu ning. The full curve takes into account all two-photon transitions, whereas in the dotted c urve the Doppler-free transitions have been suppressed. The plot shows that the eﬀect of the latter i s to increase the temperature by a factor of two, in agreement with the FPE prediction./AMÆ/CC /BP/CC /D6/BC /B9/BD /B9/BE /B9/BF /BD/BC/BC/BD/BC/BD/BC/BA/BD 9FIG. 4. Dependence of the temperature (log scale) on the detu ning for three values of g: 0.9 (full line) 0.5 (dashed line) and 0.09 (dotted line). The tri angles correspond to the calculation based on Eq.(8) for g= 0.5 and show the breaking of the Fokker-Planck approach at temp eratures close to Tr. The curve corresponding to g= 0.09 shows that the minimum temperature is very close to the recoil limit. 10REFERENCES [1] T. J. Greytak, D. Kleppner, and S. C. Moss, Physica B 280, 20 (2000). [2] M. Allegrini and E. Arimondo, Phys. Lett. 172, 271 (1993). [3] A. Huber et al., Phys. Rev. Lett. 80, 468 (1998) and references therein. [4] See for example C. Cohen-Tannoudji in Fundamental systems in quantum optics ,´Ecole d’´ et´ e des Houches, Session LIII 1990, J. Dalibard, J. M. Ra imond, and J. Zinn-Justin eds., North-Holland, Amsterdam, 1992; W. D. Phillips, ibid., for a very good review of both theoretical and experimental aspects of Doppler cooli ng. [5] Quenching of the 2 Sstate can be achieved by mixing the 2 Sand the 2 Pstate. This can be done, e.g., by microwave radiation around 1.04 GHz (the sp acing between the two levels) or by a static electric ﬁeld of a few tenths of volts. F or details, see W. E. Lamb and R. C. Retherford, Phys. Rev. 81, 222 (1951); F. Biraben, J. C. Garreau, L. Julien, and M. Allegrini, Rev. Sci. Instrum. 61, 1468 (1990). [6] B. Cagnac, G. Grynberg, and F. Biraben, J. Phys. France 34, 845 (1973). [7] J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980); see also [4]. 11
arXiv:physics/0010025v1 [physics.chem-ph] 9 Oct 2000Photoabsorption spectra in the continuum of molecules and a tomic clusters Takashi Nakatsukasa∗ RI Beam Science Laboratory, RIKEN, 2-1 Hirosawa, Wako 351-0 198, Japan Kazuhiro Yabana† Institute of Physics, University of Tsukuba, Tennodai 1-1- 1, Tsukuba 305-8571, Japan. We present linear response theories in the continuum capabl e of describing photoionization spec- tra and dynamic polarizabilities of ﬁnite systems with no sp atial symmetry. Our formulations are based on the time-dependent local density approximation wi th uniform grid representation in the three-dimensional Cartesian coordinate. Eﬀects of the con tinuum are taken into account either with a Green’s function method or with a complex absorbing potent ial in a real-time method. The two methods are applied to a negatively charged cluster in the sp herical jellium model and to some small molecules (silane, acetylene and ethylene). PACS number(s): 31.15.Ew, 31.70.Hq, 33.80.Eh I. INTRODUCTION Oscillator strength distribution characterizes the opti- cal response of atoms and molecules. Advances in mea- surements with synchrotron radiation and high resolution electron energy loss spectroscopy have enabled us to ob- tain oscillator strength distribution of a whole spectral region originated from valence electrons [1]. The photon energy dependences of molecular photoelectron spectra have also been measured. They provide useful informa- tion about the dynamics of the photoionization processes. Theoretical analysis of photoabsorption spectra above the ﬁrst ionization threshold requires continuum elec- tronic wave function in a non-spherical multi-center po- tential. Several methods have been developed for this purpose [2], including the continuum multiple scatter- ing method [3], the Schwinger variational method [4,5], ﬁnite-volume variational method [6], the linear algebraic method [7], and the K-matrix method [8]. The Stieltjes moment method does not directly utilize the scattering state but extract continuum spectrum from the spectral moments which are calculated with a square-integrable basis set [9]. The method has been extensively applied to large systems [10,11]. The complex basis method also gives continuum spectra by the calculation with a square- integrable basis set [12]. The purpose of the present article is to present alter- native theoretical methods for the continuum spectra of molecules. Our methods rely upon the linear response theory in the time-dependent local-density approxima- tion (TDLDA) and employ a uniform grid representation in a three-dimensional Cartesian coordinate. The TDLDA (or alternatively called the time- dependent density-functional theory) is an extension ofthe static density-functional theory to describe electron ic dynamics under a time-dependent external ﬁeld [13]. In the practical applications, an adiabatic approximation is usually assumed: the same exchange-correlation po- tential as that in the static case is used for the time- dependent problem. The correlation eﬀect is included through the dynamical screening which is represented by an induced local potential. In the TDLDA, the con- tinuum boundary condition was treated with the radial Green’s function for spherical systems such as atoms [14] and clusters in the spherical jellium model [15,16]. The importance of the dynamical screening eﬀect on the con- tinuum response has been stressed. The method has been extended for molecules with a single-center expan- sion technique [17]. However, the application was limited to small, axially symmetric molecules. Since the Hamiltonian in the Kohn-Sham theory is al- most diagonal in coordinate representation, a grid rep- resentation in the coordinate space provides an econom- ical description. A uniform grid representation in the three-dimensional Cartesian coordinate which has been developed by Chelikowsky et.al [18] provides a conve- nient basis for our purpose. The problem is then how to incorporate the scattering boundary condition in the uniform grid representation. Our ﬁrst method is based on the real-time method which one of the present authors has recently devel- oped to calculate dynamic polarizability of ﬁnite systems [19,20]. In the method, the time-dependent Kohn-Sham equation is solved explicitly in real-time as well as in real - space with a uniform grid. The dynamic polarizability of a whole spectral region is then obtained at once through the time-frequency Fourier transformation. An advan- tage of this method is that we do not need to handle ∗Email: Ntakashi@riken.go.jp †Email: yabana@nucl.ph.tsukuba.ac.jp 1matrices of very large dimensionality, since only a single Slater determinant is evolved in time. A drawback is that the single-particle continuum states cannot be treated ex- actly because the electrons are conﬁned in a limited size of box (model space). A possible way to avoid the diﬃ- culty would be employing a complex absorbing potential at the end of the box, which we shall discuss later. Our second method is based on the modiﬁed Stern- heimer method [21]. The method has been widely used in the linear response calculations [22,23]. It recasts the response problem into solving the static Schr¨ odinger-typ e equation with a source term. The problem is then to solve the equation with an appropriate outgoing boundary con- dition. Our recipe here is to solve iteratively the equa- tion taking into account the boundary condition employ- ing a Green’s function of a spherical potential, separating the self-consistent potential into long range spherical an d short range non-spherical parts. The paper is organized as follows. We present our methods in Sec. II. The real-time method with an absorptive boundary potential and the Green’s function method are explained. In Sec. III, applications of the methods will be demonstrated. First the spherical jel- lium cluster is considered to conﬁrm reliability of our methods. We then show calculations of photoabsorption spectra of some small molecules and compare them with measurements. We give interpretation of the obtained spectra. Finally conclusion are drawn in Sec. IV. II. LINEAR RESPONSE IN THE CONTINUUM OF THREE-DIMENSIONAL REAL SPACE A. Real-time method with an absorbing potential In this section we ﬁrst recapitulate the real-time method in TDLDA [19,24]. For systems with relatively large number of particles, the real-time method is one of the most eﬃcient method to calculate the electronic excitations in molecules and atomic clusters. The TDLDA equations for a spin-independent N- electron system are given in terms of the time-dependent Hamiltonian h(t) which is a functional of the density n(r,t). i∂ ∂tφi(r,t) =h(t)φi(r,t), i= 1,···,N/2 (2.1) h(t) =−1 2m∇2+Vion +e2/integraldisplay d3r′n(r′,t) \|r−r′\|+µxc[n(r,t)],(2.2) n(r,t) = 2N/2/summationdisplay i=1\|φi(r,t)\|2. (2.3) We adopt ¯h= 1 throughout the present paper. Vionis an electron-ion potential for which we employ a norm-conserving pseudopotential [25] with a separable approx- imation [26]. µxcis an exchange-correlation potential. We represent the electronic wave function on a uni- form grid points inside a certain spatial area [18]. All potentials in Eq. (2.2) are diagonal in this representation except for the nonlocal part of Vion. The second order dif- ferential operator in the kinetic energy is approximated employing a nine-point formula. For the time evolution, we use an algorithm developed in Ref. [27] which utilize a predictor-corrector method. The integration of Eq. (2.1) is approximately carried out with a fourth-order Taylor expansion.: φi(t+ ∆t) =e−ih(t+∆t/2)∆tφi(t) ≈4/summationdisplay n=0(−ih(t+ ∆t/2)∆t)n n!φi(t).(2.4) For stable iteration, the time step ∆ tshould be smaller than the inverse of the maximum eigenvalue of the Hamil- tonian. The energy and the particle-number conserva- tions are then well satisﬁed. This method was successful in nuclear physics to investigate the dynamics of nuclear reaction [27,28] and has been proved to be fruitful for electron systems as well [19,24]. We start with the static LDA problem solving the ground-state Kohn-Sham equations and determine the initial occupied orbitals φi(r) and density n0(r). The conjugate gradient method is used to solve the static Kohn-Sham equation. Then, an external ﬁeld is turned on instantaneously at t= 0. In this paper, we shall con- sider only the dipole response, adopting Vext=k0rνδ(t) (ν=x,y,z ) which produces an initial state for the time evolution as φi(r,0+) =e−ik0rνφi(r), ν=x,y,z. (2.5) We calculate time evolution with this initial condition without any external ﬁelds. The dynamical polarizabili- ties in the time representation is then obtained as αν(t) =−e2 k0/integraldisplay d3rrνδn(r,t), ν=x,y,z, (2.6) where the transition density is simply given by the dif- ference from that of the ground-state δn(r,t) =n(r,t)−n0(r). (2.7) Since the dynamical polarizability is usually deﬁned in the energy representation, we take the Fourier transform of Eq. (2.6). αν(ω) =/integraldisplayT 0dtαν(t)eiωt−Γt/2, ν=x,y,z, (2.8) where we introduce a smoothing parameter Γ. In order to obtainα(ω) in energy resolution of Γ, we need to cal- culate the time evolution up to T≥2π/Γ. In the next 2section, we discuss a gas-phase average of the x,yandz direction, as α(ω) = 1/3/summationdisplay ν=x,y,zαν(ω). (2.9) When the energy is in a region where ω+ǫi>0, a part of the electron wave functions φi(r,t) can escape to the continuum. This outgoing part of wave function is eventually reﬂected at the edge of the box and produces a spurious discrete structure in the photoabsorption spec- tra. Therefore, we now introduce an imaginary absorb- ing potential W(r) to suppress the reﬂection and mimic the continuum. Since the imaginary potential should be zero in a region where the ground-state density has a ﬁnite value, we should adopt a spherical box of large ra- dius and the imaginary potential is switched on only in a outer shell of width ∆ r. Although the complex potential at the edge of the box slightly violates normalization and energy conservation, the box must be large enough to preserve the conservations if the time evolution is carried out using an initial state of Eq. (2.5) with k0= 0 (the ground state). We adopt an absorptive potential of a linear depen- dence on coordinate, which has been discussed in the wave-packet method for molecular collisions [29,30]: W(r) =/braceleftbigg0 for 0 <r<R, −iW0r−R ∆rforR<r<R + ∆r.(2.10) The height W0and width ∆ rmust be carefully chosen to prevent the reﬂection. There has been an argument based on the WKB theory to elucidate a condition of no reﬂection [29,30]. For simplicity, let us take a one-dimensional system. The absorbing potential is given by a form similar to Eq. (2.10):W(x) = 0 forx<0 andW(x) =−iW0x/∆xfor x>0. The WKB solution is ψ(x) = exp(ikx) +Rexp(−ikx) forx<0,(2.11) ψ(x) = (1 +R)E1/4/parenleftBig E+iW0x ∆x/parenrightBig−1/4 ×exp/braceleftbigg i/integraldisplayx 0dxk(x)/bracerightbigg forx>0,(2.12) wherek(x) = [2m(E+iW0x/∆x)]1/2andk=k(0). Conditions we require are \|R\|2≪1 (no reﬂection) and \|ψ(∆x)\| ≪1 (complete absorption). A continuity condi- tion for a derivative at x= 0 leads to R≈W0 4√ 8mE3/2∆x, (2.13) assumingW0/E≪1. The absolute value of wave func- tion atx= ∆xis approximately given by \|ψ(∆x)\| ≈exp/braceleftbigg −W0∆x/parenleftBigm 8E/parenrightBig1/2/bracerightbigg . (2.14)If we demand \|R\|2<0.001 and \|ψ(∆x)\|2<0.01, we have 20E1/2 ∆r√ 8m<\|W0\|<1 10∆r√ 8mE3/2, (2.15) where ∆xhas been replaced by ∆ r. This is the condi- tion discussed in Ref. [30] which we shall test with nu- merical calculations. Strictly speaking, the conditions, \|R\|2<0.001 and \|ψ(∆x)\|2<0.01, lead to a factor 18 .4 instead of 20 in the left hand side and 0 .128 instead of 1/10 in the right hand side. In actual real-time calcula- tions, electrons with diﬀerent energies are emitted simul- taneously. Thus, ∆ randW0must be chosen properly ac- cording to the energy spectrum of photoabsorption above the ionization threshold. B. Green’s function method with an outgoing boundary condition Exact treatment of the continuum is possible with the use of a Green’s function. For spherical systems, the Green’s function can easily be constructed by making a multipole expansion and discretizing the radial coor- dinate. The method was ﬁrst applied to nuclear giant resonances [31,32] and then applied to photoabsorption in atoms [33,14]. In this section we present a method to construct a Green’s function in the uniform grid repre- sentation for a system without any spatial symmetry. The linear response theory is formulated most conve- niently using a density-density correlation function [34] . The general formalism with local-density approximation is given in the frequency representation [32,14]. The tran- sition density, which corresponds to the Fourier trans- form of Eq. (2.7), can be expressed with the use of the independent-particle density-density correlation funct ion χ0: δn(r,ω) =/integraldisplay d3r′χ0(r,r′;ω)Vscf(r′,ω), (2.16) whereVscfis the self-consistent ﬁeld which is the sum of the external ﬁeld and the dynamical screening (dielectric) ﬁeld: Vscf(r,ω) =Vext(r,ω) +/integraldisplay d3r′δV[n(r)] δn(r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n=n0δn(r′;ω), (2.17) whereV[n(r)] is a single-particle potential of the time- independent version of Eq. (2.2), deﬁned by h= −∇2/2m+V. Although a part of Vis non-local, the screening ﬁeld arises from density-dependent parts of V, the direct Coulomb and the exchange-correlation poten- tials, which are local. The calculation neglecting the sec- ond term of Eq. (2.17) will be called an ”independent- particle approximation” (IPA) in the next section. Theχ0has a form 3χ0(r,r′;ω) = 2occ/summationdisplay iunocc/summationdisplay m/braceleftbigg φi(r)φ∗ m(r)φm(r′) ǫi−ǫm−ω−iηφ∗ i(r′) +φ∗ i(r)φm(r)φ∗ m(r′) ǫi−ǫm+ω+iηφi(r′)/bracerightbigg ,(2.18) whereηis an inﬁnitesimal positive parameter. The fac- tor two in the right hand side of Eq. (2.18) is the spin degeneracy of each single-particle state. Here the summa- tion with respect to unoccupied states mcontains both discrete and continuum states. Instead of taking the ex- plicit summation, we may use the single-particle Green’s function deﬁned by G(±)(r,r′;E) =/an}b∇acketleftr\|(E−h[n0]±iη)−1\|r′/an}b∇acket∇i}ht =all/summationdisplay kφk(r)φ∗ k(r′) E−ǫk±iη. (2.19) The sign determines a boundary condition of the Green’s function; the (+) for a outgoing wave and the ( −) for an incoming wave. In fact, the summation with respect to m in Eq. (2.18) can be extended to all states because the summation over the occupied states for the ﬁrst term will be canceled by a contribution from the second term. Therefore, using Eq. (2.19), we can rewrite (2.18) as χ0(r,r′;ω) = 2occ/summationdisplay iφi(r)/braceleftBig/parenleftBig G(+)(r,r′; (ǫi−ω)∗)/parenrightBig∗ +G(+)(r,r′;ǫi+ω)/bracerightBig φi(r′), (2.20) where we assume that the occupied states have real wave functions. To calculate the dipole photoresponse of a system, one should take Vext=rν,ν=x,y,z . Then, the dynamic polarizability is given by αν(ω) =−e2/integraldisplay d3rrνδn(r,ω), ν=x,y,z. (2.21) Since the number of spatial grid points is large, it is not convenient and even impossible to construct explic- itly the response function χ0(r,r′;ω) and to perform spa- tial integration by summing up over grid points in solving the self-consistent equations, Eqs. (2.16) and (2.17). We can avoid the diﬃculty by converting the integral into the equivalent diﬀerential equation. We introduce a function ψi(r;E,Vscf)≡/integraldisplay d3r′G(+)(r,r′;E)Vscf(r′)φi(r′).(2.22) The transition density is then expressed as δn(r,ω) = 2occ/summationdisplay iφi(r)/braceleftbig (ψi(r; (ǫi−ω)∗,V∗ scf))∗ +ψi(r;ǫi+ω,Vscf)}.(2.23) When the energy ωis far below the ﬁrst ionization thresh- old, all the outgoing channels are closed and Eq. (2.22)is equivalent to the Schr¨ odinger equation of energy E= ǫi±ωwith a source term, (E−h[n0])\|ψi/an}b∇acket∇i}ht=Vscf\|φi/an}b∇acket∇i}ht, (2.24) assuming that the function /an}b∇acketleftr\|ψi/an}b∇acket∇i}htoutside a box area van- ishes. The integral in Eq. (2.22) is thus converted into a linear algebraic equation (2.24). This procedure is known as the modiﬁed Sternheimer method [21]. However, since this cannot describe a correct asymptotic behavior of the wave function, it is not applicable when the energy Eis close to zero. Furthermore, when E >0, the method is incapable to produce the continuum spectra. Thus, the remaining task is to calculate ψi(r;E,Vscf) deﬁned by Eq. (2.22) under an appropriate outgoing boundary condition. For this purpose, we employ an integral equation for the Green’s function. We start with a single-particle problem with a spherical poten- tialV0(r). For instance, the Green’s function for free particles (V0(r) = 0) has an analytic expression, G(+) 0(r,r′;E) =−m 2πe+ik\|r−r′\| \|r−r′\|, (2.25) wherek=√ 2mE(E >0). For a negative energy E <0, exp(+ik\|r−r′\|) in Eq. (2.25) is replaced by exp( −κ\|r−r′\|) withκ=√−2mE. The single-particle Green’s function for a non-spherical (even non-local) potential V(r,r′) then satisﬁes the following integral equation G(+)(r,r′;E) =G(+) 0(r,r′;E) +/integraldisplay d3r′′d3r′′′ G(+) 0(r,r′′;E)˜V(r′′,r′′′)G(+)(r′′′,r′;E),(2.26) where ˜V(r,r′) =V(r,r′)−V0(r)δ3(r−r′). The bound- ary condition of G(+) 0determines an asymptotic behavior ofG(+). Substituting this into Eq. (2.22), we obtain a multi-linear equations for ψi(r;E,Vscf), ψi(r;E,Vscf) −/integraldisplay d3r′d3r′′G(+) 0(r,r′;E)˜V(r′,r′′)ψi(r′′;E,Vscf) =/integraldisplay d3r′G(+) 0(r,r′;E)Vscf(r′)φi(r′). (2.27) In solving this equation, we need to evaluate the following integral Ψ(r;E) =/integraldisplay d3r′G(+) 0(r,r′;E)f(r′), (2.28) wheref(r) is either/integraltext d3r′˜V(r,r′)ψi(r′;E,Vscf) or Vscf(r)φi(r). We note that both of them are zero out- side the box. We calculate Ψ( r;E) again by recasting Eq. (2.28) into equivalent diﬀerential equation, /braceleftbigg E−/parenleftbigg −1 2m∇2+V0(r)/parenrightbigg/bracerightbigg Ψ(r;E) =f(r).(2.29) 4In the discretization, this is a linear algebraic equation for grid points inside the box area. However, the solu- tion outside the box area is needed to apply the Lapla- cian operator and should be prepared by other method. We prepare it by a multipole expansion method. Noting thatV0(r) is a central potential, G(+) 0can be expressed in terms of the regular and irregular solutions of radial Schr¨ odinger equation in the usual way. Ψ(r;E)\|r≥R=lmax/summationdisplay lmw(+) l(r;E) rYlm(ˆr)Φlm(E),(2.30) Φlm(E)≡2m/integraldisplay r<Rd3r′ul(r′;E) r′Ylm(ˆr′)f(r′),(2.31) whereul(r;E) andw(+) l(r;E) are solutions of the radial diﬀerential equation being normalized as the Wronskian W[ul,w(+) l] is unity. The ulis regular at the origin and w(+) lis an outgoing wave at inﬁnity. The lmax= 16 is chosen in the later calculations. We must choose the V0(r) so as to make the poten- tial˜V(r) =V(r)−V0(r) be negligible outside the box. For neutral molecules, the attractive ionic potential is ap - proximately canceled out by the repulsive direct Coulomb term. However we will later employ a gradient-corrected exchange potential which possesses a correct asymptotic form of −e2/r. Therefore we use a jellium potential, Eq. (3.1), with Z= 1 asV0(r). Thew(+) l(r;E) in Eq. (2.30) is an irregular Coulomb wave function with an outgoing boundary condition. We use a FORTRAN77 program “COULCC” in Ref. [35] to calculate Coulomb functions of complex energies. The ul(r,E) in Eq. (2.31) is calculated by integrating the radial Schr¨ odinger equa- tion with a fourth-order Runge-Kutta method. We now summarize our procedure to solve the response equation. Once an external ﬁeld Vext(r,ω) is given, Equa- tions (2.16) and (2.17) constitute a linear equation for the transition density δn(r,ω). Discretizing on a uni- form grid, we have a linear algebraic equation with di- mensionality equal to the number of grid points. In or- der to solve the equation, we need to calculate actions of theχ0on some functions, which is equivalent to cal- culate actions of the G(+).ψi(r;E,Vscf) is calculated by solving another linear equation (2.27). In order to solve Eq. (2.27), we need to calculate actions of the G(+) 0. This operation is achieved by solving Eq. (2.29), again discretizing on grid points. We note that the outgoing boundary condition is imposed at this stage. The wave function Ψ( r;E) outside the box is prepared by a multi- pole expansion method, Eqs. (2.30) and (2.31). Solutions of linear equations (2.16), (2.27) and (2.29) are obtained by iterative methods. The iterative procedure is summa- rized in Fig. 1. Our algorithm thus requires to solve multiple nested linear algebraic equations. Conjugate gradient (CG) method and its variants oﬀer stable and eﬃcient scheme to solve these equations. If the energy Eis real, Eq.(2.29) is a hermitian problem. Therefore, the CG method is very powerful to solve the equation. However, if the energy is complex, Eq. (2.29) becomes a non-hermitian problem to which the CG method is not applicable. For such cases, we adopt a Bi-conjugate gradient (Bi-CG) method. To solve Eq. (2.27), we have tested diﬀerent iter- ative methods to this non-hermitian problem. We found that a generalized conjugate residual (GCR) method is one of the most eﬀective algorithms for the present prob- lem. For the outer most iteration loop of Eq. (2.23), we use the GCR method again. δn=∑φi( )ψ∗ i(εi−ω)+ψi(εi+ω) i( )1−G(+) 0Vψi=G(+) 0V(n) scfφi ( )E+∇2/2m−V0Ψ=fδn=χ0( )Vext+δV δnδn Self-consistent determination of ^^^^^^^ ·Iteration for solving V(n) scf δn(n)(r) ψi=G(+)V(n) scfφi·Iteration for solving ψ(m) i f=V(n) scfφi,Vψ(m) iΨ(k)Ψ(k+1) ψ(m+1) i·Iteration for solving δn(n+1)(r)with outgoing boundary conditionsδn(r) Ψ=G(+) 0f FIG. 1. Algorithm to determine the transition den- sityδn(r, ω) is presented. There are three nested iterative loops to solve linear equations. III. APPLICATIONS A. Spherical jellium model for Na− 7: Test study First we have carried out a calculation for a negatively- charged Na cluster Na− 7. We use a spherical uniformly- charged jellium potential for Vion(r). The jellium poten- tial of radius RIis given by Vion(r) =  −Ze2 2RI/braceleftbigg 3−/parenleftBig r RI/parenrightBig2/bracerightbigg forr≤RI, −Ze2 rforr>R I,(3.1) whereZeis a total charge of a jellium sphere. For the Na− 7, we haveZ= 7 and eight valence electrons. The main photoabsorption peak is calculated to be above the ionization threshold. Thus, this would be a good test case to study a response in the continuum and to check the applicability of the theories in the previous section. We discretize three-dimensional Cartesian coordinates with spacing ∆ x= ∆y= ∆z= 1.5˚A and employ grid 5points inside a spherical box of radius R= 12 ˚A. This is found enough for the ground-state density to be negli- gible at the edge r=R. The number of mesh points is 2109. The TDLDA Hamiltonian is given by Eq. (2.2) in which the exchange-correlation potential is that of Ref. [36]: µxc[n(r)] =−1.222 rs(r)−0.0666 ln/parenleftbigg 1 +11.4 rs(r)/parenrightbigg ,(3.2) in units of Ry where 4 /3πr3 s=n(r). The radius of jellium sphereRIis 3.93×81/3a.u. Since the jellium potential for Na− 7is spherical, we can use the same technique as that of Ref. [32,14] (a Fortran program in Ref. [16]) and conﬁrm that our methods pro- vide the same results. The Green’s function is expanded in partial waves and is given by G(+) l(r,r′;E) =ul(r<;E)w(+) l(r>;E)/W[ul,w(+) l], (3.3) for thel-th partial wave. Here ulandw(+) lare regular and irregular solutions of the radial diﬀerential equation , respectively. The boundary condition of w(+) lat the edge r=Ris deﬁned as exp( ikr) withk2= 2m(E−e2/R) forE > e2/Rand exp( −κr) withκ2= 2m(e2/R−E) forE < e2/R. Of course, this cause a change of the threshold energy by e2/R, but it is enough to check va- lidity of our methodology. We also add a small imaginary parameter Γ /2 to the frequency ω, which plays exactly the same role as a smoothing parameter of the Fourier transform in real-time method, Eq. (2.8). In this calcu- lation, Γ = 0 .1 eV is used and the mesh size for radial coordinate is as small as 0.1 ˚A. 0 1 2 3 4 5 ω [ eV ]0100200300Re α(ω) [ A3 ]0100200300400Im α(ω) [ A3 ](a) (b) FIG. 2. Imaginary (a) and real (b) part of dy- namic polarizabilities for Na− 7cluster calculated with the Green’s function method. Solid lines are the results of one-dimensional calculation of Ref. [16] of jellium sphere and square symbols are those of the three-dimensional cal- culation without assuming any symmetry. The smoothing parameter Γ = 0 .1 eV is used for both calculations.For the three-dimensional Green’s function calculation, in order to justify Eq. (2.26), a condition that ˜V(r)≈0 forr>R must be satisﬁed. We assume that the asymp- totic behavior is the same as that of the (radial) one- dimensional calculation. Namely, we adopt the Green’s function of free particles, Eq. (2.25), with an energy be- ing shifted as E→E−e2/R. We also use the same smoothing parameter Γ = 0 .1 eV. It turns out that the inclusion of Γ helps convergence of iterative procedure. We show the results in Fig. 2. The three-dimensional calculation has been done for a frequency range 0 <ω< 5 eV with a step ∆ ω= 0.1 eV. The solid lines are the re- sults of one-dimensional calculation in which the Green’s functions are expanded in the partial wave, Eq. (3.3). The squares are the results of the three-dimensional cal- culation which perfectly agree with the solid lines. This means that the Green’s function method on the three- dimensional mesh is able to take account properly of the continuum eﬀects. The static calculation indicates the highest occupied molecular orbital (HOMO) −0.37 eV, and a Coulomb potential has a value e2/R= 1.2 eV at the edge of the boxR= 12˚A. Thus, the ionization threshold is 1.57 eV in this calculation. The ﬁgure indicates that the pho- toabsorption has peaks below and above the threshold. In the IPA calculation, we obtain a single peak at an en- ergy 1.4 eV. The screening potential brings the peak into the continuum region (around 2.35 eV) and splits the sin- gle peak into two parts at the threshold. The summed oscillator strengths for energies up to 5 eV correspond to about 95% of the Thomas-Kuhn-Reiche (TRK) sum rule for valence electrons. 0 1 2 3 4 5 ω [ eV ]02004000200400Im α(ω) [ Α3 ]0200400600 (a) ∆r=6A, W0=2eV (b) ∆r=12A, W0=1eV (c) ∆r=21A, W0=1eV FIG. 3. Imaginary part of dynamic polarizabilities for Na− 7cluster calculated with the real-time method. The absorptive potentials are chosen as (a) ∆ r= 6˚A with W0= 2 eV, (b) ∆ r= 12˚A with W0= 1 eV, (c) ∆ r= 21 ˚A with W0= 1 eV. The smoothing parameter Γ = 0 .1 eV is used on the Fourier transform of Eq. (2.8). 6We now turn to the real-time method. A necessary size of space is much larger than that of the Green’s function method because the absorbing potential is set up in a outer region R < r < R + ∆r. We prepare three kinds of absorbing potentials in a linear form of Eq. (2.10): (a) ∆r= 6˚A withW0= 2 eV, (b) ∆ r= 12˚A withW0= 1 eV and (c) ∆ r= 21˚A withW0= 1 eV. Using a square mesh of ∆x= ∆y= ∆z= 1.5˚A, the numbers of mesh points are (a) 7153, (b) 17077, and (c) 44473, respec- tively. We use a duration of time evolution T= 70 eV−1 with a time step ∆ t= 0.01 eV−1and a smoothing param- eter Γ = 0.1 eV for the Fourier transform. Since we have a complex potential at the edge of the box, the energy is not strictly conserved but the scale of violation turns out to be less than 0.1% for this calculation. The imag- inary part of dynamic polarizability is shown in Fig. 3. Parts (a), (b) and (c) correspond to the three cases de- scribed above. For (a) and (b), the main resonance in the continuum region looks like a superposition of three diﬀerent peaks which are located at ω≈2, 2.7 and 3.5 eV, respectively. However, since we cannot see the same behavior in Fig. 2 (a), this should be a spurious eﬀect of the reﬂection of outgoing waves. This is conﬁrmed by a calculation of (c), for which the structure at ω >2.5 eV is almost disappeared and we see only a smooth tail. Now let us check the criterion of a good absorber, Eq. (2.15). The position of the main peak is calculated at Eres−Ethr= 2.35−1.57≈0.8 eV above the ionization threshold. Using E=0.8 eV and (8 m)−1≈1 eV˚A2for electrons, Eq. (2.15) reduces to 18(∆r)−1<\|W0\|<0.07∆r, (3.4) whereW0in units of eV and ∆ rin˚A. This criterion is satisﬁed for the case (c) but not for (a) and (b). Fig. 3 shows that the real-time method provides us with a cor- rect response in the continuum if no reﬂection occurs at the edge of the box. In order to ﬁnd a suitable absorb- ing potential (of a linear form), we ﬁnd the criterion, Eq. (2.15), very useful even when a condition W0/E≪1 is not satisﬁed. B. Valence shell photoabsorption of silane It is well known that the energy of the highest occupied orbital does not coincide with the ﬁrst ionization poten- tial in the simplest local-density approximation. This fact causes a serious problem for the continuum response calculation that the ionization threshold cannot be ade- quately described by the static Kohn-Sham Hamiltonian. Furthermore, the excited states around the ionization threshold appear in too low excitation energies. To rem- edy this defect, a gradient correction for the exchange- correlation potential has been proposed. We utilize the one constructed by van Leeuven and Baerends [37] which we denote as µ(LB). It is so constructed that the poten- tial has a correct −e2/rtail asymptotically. The energyof the highest occupied orbital also approximately coin- cides with the ionization potential. For small molecules, TDLDA calculations with this gradient correction have shown to give accurate description of discrete excitations in small molecules [38]. In the following sections III B, III C and III D, we em- ploy a sum of the exchange-correlation potentials of µ(PZ) of Ref. [39] for the local-density part and µ(LB)for the gradient correction. We should remark here that an ac- curate calculation of the gradient correction µ(LB)(r) be- comes diﬃcult at far outside the molecule, because the µ(LB)depends on \|∇n(r)\|/n(r)4/3which approaches to ﬁnite but both the numerator and the denominator ap- proaches to zero at r→ ∞. Thus, we use an explicit asymptotic form, µ(LB)(r) =−e2/rforr > R c. In the followings, the Rcis chosen as 6.5 ˚A for silane and 5.5 ˚A for acetylene and ethylene. When we evaluate the screening ﬁeld in Eq. (2.17), we neglect the functional derivative of µ(LB)(r) with re- spect to the density because this should be a small cor- rection. For the real-time calculation, we make the same approximation. Namely the time-dependent exchange- correlation potential is calculated by µxc(r,t)≈µxc[n0(r)] +/integraldisplay d3r′δµxc(r) δn(r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n=n0δn(r′,t) ≈µxc[n0(r)] +/integraldisplay d3r′δµ(PZ)(r) δn(r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n=n0δn(r′,t). In this section, we discuss the application of our tech- niques to a deformed molecule, silane SiH 4. We use the pseudopotentials which are calculated by the prescrip- tions of Refs. [25,26]. For Si atom, we employ pseu- dopotentials for s,p,d waves and take d-wave poten- tial as a local part. For the hydrogen atom, we take sandppotentials with the latter as a local potential. The box is a sphere of radius R= 7˚A discretized in meshes of ∆ x= ∆y= ∆z= 0.4˚A. Position of the Si atom (nucleus) is located at the origin and four hydro- gen atoms are at (1 .209,0.0,0.855), ( −1.209,0.0,0.855), (0.0,1.209,−0.855), and (0 .0,−1.209,−0.855) in units of ˚A. The symmetry of this molecule belongs to the tetra- hedral (Td) point group. The valence electronic orbitals in the ground state are (3a1)2,(2t2)6:1A1. In actual calculations, we do not assume a priori the sym- metry, but the degeneracies of electron orbitals according to Eq. (III B) naturally emerge after the minimization of the total energy. For this molecule, the dipole response does not depend on the direction of an external ﬁeld. A smoothing parameter Γ = 0 .5 eV is used in the follow- ings. Utilizing the exchange-correlation potential of µ(PZ)+ µ(LB), the calculated occupied valence orbitals in the ground state are listed in Table I. If we neglect the gra- dient correction term µ(LB), we obtain −13.5 eV for 3a1 7and−8.3 eV for 2t2, respectively. The photoabsorption oscillator strengths for silane calculated by means of the Green’s function method are shown in Fig. 4. The oscil- lator strength df/dω , the photoabsorption cross section σ(ω), and the imaginary part of dynamic polarizability Imα(ω), are related to each other by σ(ω) =2π2e2 mcdf dω=4πω cImα(ω). (3.5) A sharp peak at 10 eV in the calculation consists of bound discrete peaks overlapped by the width Γ. In the experiment [43,44,1], we observe a peak at 10.7 eV, the width of which is considered to originate from a cou- pling of electronic excitation with ionic motion. Since we ﬁx the ion coordinates and calculate vertical electronic excitations, we cannot describe the width below the ion- ization threshold. On the other hand, a broad peak at 14.6 eV in the ex- periment is above the ionization threshold and the main decay channel is an emission of the electron [43,1]. The calculation well reproduces this peak although the energy shifts slightly to the lower (14 eV). Beyond the peak of 14.6 eV, a group of tiny peaks is observed in the exper- iment, which is assigned the Rydberg states with a hole (3a1)−1embedded in the continuum. This ﬁne structure is smeared out in the calculation because the smoothing parameter Γ is much larger than the width of each Ryd- berg state. The integrated oscillator strengths for ener- gies up to 30 eV is calculated to exhaust 87% of the TRK sum rule for valence electrons. About 60% of the valence shell strengths lie between 10 and 20 eV (Fig. 5 (b)). 0 10 20 30 Photon energy ω [ eV ]012df/dω [ eV−1 ]IPA TDLDA (e,e) exp photon exp 13 14 ω [ eV ]00.20.40.60.8df/dω [ eV−1 ]Photoabsorption oscillator strength SiH4 (3a1)−1(2t2)−1total FIG. 4. Calculated and experimental photoabsorption oscillator strengths of silane. The thick solid line is the calculation compared with synchrotron radiation experi- ment (thin solid line) [43] and high resolution dipole (e,e) experiment (dotted line) [44]. The dashed line is the IPA calculation without dynamical screening. The smoothing parameter Γ = 0 .5 eV is used in the calculation. Inset: An energy region of 13 < ω < 15 eV is magniﬁed and the total oscillator strengths are decomposed into those associated with diﬀerent occupied valence electrons, 3 a1 (solid line) and 2 t2(dashed line).The screening ﬁeld in Eq. (2.17) signiﬁcantly reduces a dipole polarizability at low frequency. The calculated static dipole polarizability is 5.1 ˚A3while it is 7.9 ˚A3in the IPA. The dielectric eﬀects also change the absorption spectra. The screening shifts the resonance energies to higher energies because the electron-electron correlatio n is repulsive. Roughly speaking, the IPA overestimates the absorption spectra at low frequency and underesti- mates at high frequency. As you see in Fig. 4, the di- electric eﬀects are very important to reproduce the mag- nitude and shape of the photoresponse. In order to discuss details of each resonance, it is useful to calculate a partial oscillator strength of each occupied orbital which should be identiﬁed with a photoemission spectra if the neutral dissociation channel without elec- tron emission is negligibly small. As we see in Ref. [14], the photoabsorption cross section can be written as a sum of contributions from all occupied orbitals. Here we deﬁne the partial oscillator strengths ( df/dω )ias df dω= 2occ/summationdisplay i/parenleftbiggdf dω/parenrightbigg i, (3.6) /parenleftbiggdf dω/parenrightbigg i=−2ω πe2/integraldisplay d3rd3r′V∗ scf(r)φi(r)× Im/braceleftBig/parenleftBig ˜G(+)(r,r′; (ǫi−ω)∗)/parenrightBig∗ +˜G(+)(r,r′;ǫi+ω)/bracerightBig ×Vscf(r′)φi(r′), (3.7) where ˜G(+)is deﬁned by Eq. (2.19) except that the sum- mation with respect to kis carried out only for unoccu- pied states. 0 10 20 30 Photon energy ω [ eV ]012df/dω [ eV−1 ]W0=0 W0=4 eV 0 10 20 ω [ eV ]05(a) Real−time calculation (b) f(ω) TRK=8 FIG. 5. (a) Photoabsorption oscillator strengths of silane calculated with the real-time method. The thin line corresponds to the result obtained with the box of R= 10 ˚A without a complex absorbing potential, while the thick line to the one with a complex potential of W0= 4 eV with ∆ r= 10˚A in addition to a spherical box of R= 7˚A. In carrying out the Fourier transform, the smoothing pa- rameter Γ = 0 .5 eV is used. (b) The integrated oscillator strength f(ω). The TRK sum rule indicates f(∞) = 8 for the valence shell photoabsorption. 8TABLE I. Calculated eigenvalues of occupied valence orbita ls and experimental vertical ionization potential (IP) in units of eV. The experimental data are take n from Ref. [40] for silane, from Ref. [41] for acetylene and from Ref. [42] for ethylene. Silane Acetylene Ethylene occ cal exp occ cal exp occ cal exp state IP state IP state IP (3a1)2−17.4 18 .2 (2 σg)2−22.4 23 .5 (2 ag)2−22.8 23 .7 (2t2)6−12.4 12 .8 (2 σu)2−18.4 18 .4 (2 b1u)2−18.6 19 .4 (3σg)2−16.7 16 .4 (1 b2u)2−16.3 16 .3 (1πu)4−12.1 11 .4 (3 ag)2−14.7 14 .9 (1b3g)2−13.2 13 .0 (1b3u)2−11.7 11 .0 Below the ionization threshold, there are three bound transitions in our calculations which we ﬁnd are associ- ated with the excitations of 2 t2valence electrons. The ﬁrst two transitions are considered to correspond to the shoulders in the measurements [44,45] and interpreted as transitions to the Rydberg states. The broad resonance at 14 eV (14.6 eV in experiment) is above the ioniza- tion threshold of the 2 t2orbitals. Therefore, the struc- ture may be more complex. An inset of Fig. 4 shows the partial oscillator strengths of 3 a1and 2t2orbitals in a photon-energy region of 13 to 15 eV. The calcula- tion suggests that the resonance structure is due to the bound-to-bound excitation of 3 a1electrons. The exci- tation couples with the bound-to-continuum excitations of 2t2electrons through the dynamical screening eﬀect. The coupling shifts the excitation energy up by about 1 eV and brings about the width due to the autoionization process. The partial cross section of 2 t2electrons also acquires oscillation as a function of the excitation energy due to the rapid change of the induced ﬁeld. Now let us examine the applicability of the real-time method. The time step is chosen as ∆ t= 0.002 eV−1 and the time evolution is calculated up to T= 12 eV−1. We show the results in Fig. 5 (a). The bound excited states is reasonably described in the calculation. On the other hand, the calculation without an absorbing poten- tial (thin line) apparently provides a wrong response in the continuum region. We must adopt an imaginary po- tential to remove several spurious resonances. However, it is very diﬃcult to treat the ionization energy properly. In order to mimic the continuum, the absorber should be eﬀective only for a outgoing electron with positive en- ergy. The problem is that the static potential has −e2/r behavior at large r. Electrons with negative energies (−e2/R < E < 0) may be absorbed by the imaginary potential. Therefore, the eﬀective ionization potential becomes −ǫHOMO −e2/Rfor this calculation. Taking R= 7˚A, this shift in the ionization potential amounts to about 2 eV. The calculated peak in the continuum is located atω= 14 eV. According to the condition (2.15) withE= 14−12.4+2 = 3.6 eV, we adopt the ∆ r= 10˚A andW0= 4 eV. The number of grid points is 321,781 forthis case. The thick line in Fig. 5 (a) shows the result. The spurious peaks disappear and the result well agrees with that of the Green’s function method. The oscillator strengths near the ionization threshold (11 <ω< 13 eV) indicate some discrepancies, which we naturally expect from the above argument. In Fig. 5 (b), integrated os- cillator strength is plotted against the energy. Seven out of eight (TRK sum rule) unit of strength locates below 30 eV. Finally we should mention the feasibility of computa- tion. In order to calculate the absorption spectra of Fig. 5, the real-time method takes about 10 hours using a single CPU of a Fujitsu VPP700E. On the other hand, the Green’s function method takes about 30 minutes for each energy. Thus, the real-time method is faster than the Green’s function method if the response is calculated for over 20 diﬀerent energies. We have carried out the cal- culations for 125 frequencies to obtain the smooth line of Fig. 4. C. Valence shell photoabsorption of acetylene The acetylene molecule, C 2H2, has a symmetry conﬁg- uration ofD∞h. This high symmetry has made possible a calculation of the Green’s function using a single-center expansion [17]. Even so, only two kinds of molecular orbitals, 3σgand 1πu, which are primarily derived from atomicpstates have been taken into account in Ref. [17], because it was diﬃcult to describe the s-derived states in the single-center formulation. In the present paper, we consider all valence orbitals, including the 2 σgand 2σu in addition to the above p-derived orbitals, to calculate the photoresponses. The spherical box is taken as R= 6˚A with meshes of ∆x= ∆y= ∆z= 0.3˚A. All atoms are located on the z-axis at ±0.601˚A for carbon and ±1.663˚A for hydro- gen. There are ten valence electrons and the calculated energies of occupied orbitals are listed in Table I. Using the Green’s function method, we obtain the photoabsorp- tion oscillator strengths as a function of photon energy, shown in Fig. 6. The calculation indicates a sharp bound 9resonance at ω= 9.6 eV and a broad structure around 15 eV which seems to be a superposition of three reso- nances. The resonance at 9.6 eV strongly responds to a dipole ﬁeld parallel to the molecular axis. The large oscillator strengths in the IPA at ω= 5∼8 eV and at ω= 12.5∼13.5 eV are shifted to higher energies by the dielectric eﬀects. The agreements with the experi- mental data are signiﬁcantly improved by the inclusion of the dynamical screening. The static dipole polarizabil- ity is also aﬀected signiﬁcantly: In the IPA calculation, the polarizabilities parallel ( α/bardbl) and perpendicular ( α⊥) to the molecular axis are α/bardbl= 10.7˚A3andα⊥= 3.87 ˚A3. The dynamic screening reduces these values to 4.79 ˚A3and 2.77 ˚A3, respectively, which well agree with the experimental values, α/bardbl= 4.73 andα⊥= 2.87˚A3[47]. We ﬁnd some disagreements between the calculation and the experiment in Fig. 6. We observe two distinct peaks in the experiments for the broad resonance around 15 eV. However, the calculation indicates three peaks. The lowest (13.2 eV) and the highest ones (15.9 eV) are related with responses to a dipole ﬁeld parallel to the molecular axis, while the middle one (14.3 eV) is a re- sponse to the perpendicular ﬁeld. We plot the partial os- cillator strengths in an inset of Fig. 6. The lowest peak at 13.2 eV consists of the transition of 3 σgvalence orbitals. The middle peak at 14.3 eV turns out to consist of contri- butions of 2 σuand 1πuorbitals. We need an energy shift of these strengths by about 1 eV to reproduce the experi- ments. An accurate conﬁguration interaction calculation with the Schwinger variational method is available for this molecule [48]. This calculation succeeds to reproduce the double peak structure. The assignments are consis- tent to ours: 3 σgfor lower and 2 σufor higher transitions. 0 10 20 30 40 Photon energy ω [ eV ]00.51df/dω [ eV−1 ]IPA TDLDA (e,e) exp photon exp 13 15 17 ω [ eV ]00.20.40.6df/dω [ eV−1 ]C2H2Photoabsorption oscillator strength total 2σu3σg1πu FIG. 6. Calculated and experimental photoabsorp- tion oscillator strengths of acetylene. See the caption of Fig. 4. The experimental data are taken from Ref. [41,46]. Inset: An energy region of 12 .5< ω < 17.5 eV is magniﬁed and the total oscillator strength (thick line) is decomposed into partial oscillator strengths associate d with diﬀerent occupied valence orbital, 2 σu(solid line), 3σg(dashed) and 1 πu(dotted). The contributions of 2 σg electrons are negligible.0 10 20 30 40 Photon energy ω [ eV ]00.51df/dω [ eV−1 ]W0=0 W0=4 eV 0 10 20 3005(a) Real−time calculation (b) f(ω) TRK=10 FIG. 7. (a) Photoabsorption oscillator strengths of acetylene calculated with the real-time method. The thin line corresponds to the result obtained with the box of R= 10 ˚A without a complex absorbing potential, while the thick line to the one with a complex potential of W0= 4 eV and ∆ r= 10 ˚A in addition to a spherical box of R= 6˚A. In carrying out the Fourier transform, the smoothing parameter Γ = 0 .5 eV is used. (b) The integrated oscillator strength f(ω). The TRK sum rule indicates f(∞) = 10 for the valence shell photoabsorp- tion. The experiments also indicate discrete Rydberg series around 10 eV. We do not see this structure in the cal- culation because we have used a smoothing parameter Γ = 0.5 eV which is much larger than the experimental energy resolution. In principle, we can calculate these Rydberg states since the potential has a −e2/rbehavior at large distance. In order to obtain these ﬁne struc- tures, we have to calculate the response with Γ ≈0 and the small frequency mesh ∆ ω(order of meV) must be adopted. This is beyond a scope of this paper. The real-time calculation is carried out with the same imaginary potential as we have used for silane ( W0= 4 eV and ∆r= 10˚A). We have 635,371 grid points in this space. The time evolution is calculated up to T= 12 eV−1with a time step of ∆ t= 0.001 eV−1. The results are shown in Fig. 7 (a). Again the absorbing potential is an essential ingredient to obtain sensible results in the continuum energy region. The calculated photoabsorp- tion spectra well agree with those of the Green’s function calculation except for minor oscillatory behaviors at high energy. This spurious oscillations start to appear at an energyω≈22 eV. In fact, the condition of a good ab- sorber, Eq. (2.15), is not satisﬁed in this energy region. Namely the potential ( W0= 4 eV) is too week to absorb such high energy particles. In Fig. 7 (b), we show the integrated oscillator strengths up to 40 eV. One-fourth of the TRK sum rule value of valence electrons is in an en- ergy region over 40 eV. It is worth noting that the CPU time of the real-time calculation is less than one-ﬁfth of that of the Green’s function calculation. 10D. Valence shell photoabsorption of ethylene The ethylene C 2H4is the simplest organic π-system, which possesses the D2hsymmetry. The two carbon atoms are on the x-axis atx=±0.6695˚A and four hydrogen atoms are in the x−yplane at (x,y) = (1.2342,0.9288), (1.2342,−0.9288), ( −1.2342,0.9288), (−1.2342,−0.9288) in units of ˚A. There are twelve va- lence electrons, and calculated eigenenergies of occupied valence orbitals are listed in Table I. The calculations are carried out using the same box (R= 6˚A) and mesh size (0.3 ˚A) as we have used for the acetylene molecule. The photoabsorption oscillator strengths calculated with the Green’s function method are shown in Fig. 8. The agreement with an experiment [49] is excellent. Almost all the main features of pho- toabsorption spectra are reproduced in the calculation. The observed bound excited states show diﬀerent pho- toresponses according to the direction of the dipole ﬁeld. The lowest peak at ω= 7.6 eV is mainly a response to a dipole ﬁeld parallel to the molecular (C −C) axis. This is associated with the excitations of the HOMO 1 b3uelec- trons. On the other hand, states at 9.8 eV respond almost equally to a dipole ﬁeld of x,y, andzdirection, to which both 1b3gand 1b3uoccupied orbitals contribute. A small peak atω= 11.4 eV is calculated as a resonance of 3 ag orbital, which may correspond to a small shoulder in the experiment. Beyond 11.7 eV, the HOMO electrons are in the continuum. The ﬁrst prominent peak at 12.4 eV is a resonance with respect to a dipole ﬁeld of ydirection which is in the molecular plane and perpendicular to the C−C bond. The excitations of the 1 b2uand 1b3gelec- trons are main components of this resonance. In a region of 13.2< ω < 20 eV, the 1 b3gelectrons can be excited into the continuum and produce the smooth background of oscillator strengths (0 .1∼0.2 eV−1). A peak struc- ture at 14.6 eV is made of the excitation of 1 b2uelectrons. The peak at 16.4 eV is constructed by excitations from the 2b1uoccupied orbitals, while the experiment indicates the resonance at 17.1 eV. The above analysis is consistent with that in the liter- ature [49], except for the 12.4 eV peak (11.9 eV in the experiment). Our analysis suggests transition of 1 b3gand 1b2uelectrons while Ref. [49] indicates 3 ag. Comparing the TDLDA calculation with that of the IPA (dashed line), we see again that the dynamic screen- ing eﬀect is very important to reproduce the experimental data. This feature of the IPA result is consistent with the other calculations without the dynamic screening [50]. The calculated dipole polarizability is α= 4.22˚A3(5.47, 3.97, 3.23 ˚A3forx,y,zdirection, respectively), which is in a good agreement with the experimental value, 4.22 ˚A3[47]. In the IPA calculation, we obtain α= 7.04˚A3 (10.4, 5.96, 4.76). Results of the real-time calculation are shown in Fig. 9 (a). The box and imaginary potential we use are the same as those for acetylene. We can obtain a sensible re-sult if we adopt the absorbing potential. However, again, a spurious oscillatory behavior is seen at high energy re- gion because the absorbing potential is not strong enough to erase those high-energy components. 0 10 20 30 40 Photon energy ω [ eV ]00.51df/dω [ eV−1 ]TDLDA IPA (e,e) expPhotoabsorption oscillator strength C2H4 FIG. 8. Calculated and experimental photoabsorption oscillator strengths of ethylene. The thick solid line is th e calculation compared with high resolution dipole (e,e) ex- periment (dotted line) [49]. The dashed line is the IPA calculation without dynamical screening. The smoothing parameter Γ = 0 .5 eV is used in the calculation. 0 10 20 30 40 Photon energy ω [ eV ]00.51df/dω [ eV−1 ]W0=0 W0=4 eV 0 10 20 30 ω [ eV ]0510(b) f(ω) (a) Real−time calcullation TRK=12 FIG. 9. Photoabsorption oscillator strengths of ethy- lene calculated with the real-time method. See the cap- tion of Fig. 7 (a). (b) The integrated oscillator strength f(ω). The TRK sum rule indicates f(∞) = 12 for the valence shell photoabsorption. IV. CONCLUSIONS We have developed methods based on the TDLDA of investigating the photoresponses in the continuum for systems with no spatial symmetry: (1) The real-time method with an absorbing potential, and (2) the Green’s function method. These methods allow us to treat the photoionization and the dynamical screening eﬀects self- consistently. For the real-time method, we have tested 11imaginary potentials of diﬀerent kinds and found that those satisfying a condition, Eq. (2.15), are good to mimic the continuum eﬀect. However, it is very diﬃ- cult to treat the photoabsorption spectra in vicinity of the ionization threshold. There are two reasons: One is because the condition, Eq. (2.15), requires a large model space to treat such a low-energy emission prop- erly. The second is that the ionization threshold is not correct when the potential has a 1 /rbehavior at large dis- tance. In addition to this, since the condition is energy dependent, it is very diﬃcult to construct a good ab- sorber for low-energy and high-energy outgoing electrons simultaneously. The advantage of the real-time method is a computational feasibility. Utilizing the Fourier trans- form, we can obtain the spectra over the whole energy region at once. The Green’s function method possesses a capability of an exact treatment of the continuum. Using a Green’s function of Coulomb asymptotic waves, it is also possi- ble to investigate the photoresponse near the threshold. The main diﬃculty of the Green’s function treatment is a heavy computational task. In order to reduce the CPU time, we use a complex energy for the Green’s function, G(ω+iΓ/2). We have found that the inclusion of the imaginary part Γ facilitates a convergence of iterative procedures in the calculation. The Γ also plays a role of lowering an energy resolution of the calculations. Thus, we can choose a value of Γ depending on the energy res- olution required in each problem. We would like to em- phasize again that the method is capable of calculating response functions of many-electron systems, below, near and above the ionization threshold in a uniﬁed manner. The numerical calculations have been performed for a spherical Na− 7cluster (test study) and non-spherical molecules, SiH 4, C2H2, and C 2H4. Studies of pho- toresponse for deformed molecules including both the dynamic screening and the continuum eﬀect are very few. An exception is a study of nitrogen and acetylene molecules (with a high symmetry D∞h) by Levine and Soven [17] using the single-center expansion technique. However, only 1 πuand 3σgoccupied orbitals are included in their calculation because of limitation of the single- center expansion. Since our calculation has been carried out on three-dimensional coordinate meshes, we do not need any spatial symmetry including all valence orbitals in the calculation. We present the photoabsorption os- cillator strengths compared with dipole ( e,e) and syn- chrotron radiation experiments. The agreement is gen- erally excellent. The inclusion of the dynamic screening turns out to be essential to reproduce the experiments. The IPA calculation overestimates the strengths at low energy and underestimates at high energy. We are strongly encouraged by the success of our meth- ods applied to simple molecules in the present paper. The applications to more complex systems may become pos- sible in near future.ACKNOWLEDGMENTS We would like to thank Y. Hatano and K. Kameta for useful discussion and providing us with the photoabsorp- tion experimental data. Calculations were performed on a Fujitsu VPP700E Super Computer at RIKEN, a NEC SX-4 Super Computer at Osaka University and a HI- TACHI SR8000 at Institute of Solid State Physics, Uni- versity of Tokyo. [1] Y. Hatano, Phys. Rep. 313, 109 (1999). [2] I. Cacelli, V. Carravetta, A. Rizzo, R. Moccia, Phys. Rep . 205, 283 (1991). [3] D. Dill and J. L. Dehmer, J. Chem. Phys. 61, 692 (1974). [4] R. R. Lucchese, G. Raseev, and V. McKoy, Phys. Rev. A25, 2572 (1982). [5] B. Basden and R. R. Lucchese, Phys. Rev. A37, 89 (1988). [6] H. Le Rouzo and G. Raseev, Phys. Rev. A29, 1214 (1984). [7] L. A. Collins and B. I. Schnieder, Phys. Rev. A29, 1695 (1984). [8] I. Cacelli, V. Carravetta, R. Moccia, J. Chem. Phys. 85, 7038 (1986). [9] P. W. Langoﬀ, in Theory and Application of Moment Methods in Many-Fermion Systems , eds B.J. Dalton, S.M. Grimes, J.P. Vary and S.A. Williams (Plenum, New York, 1980), p191. [10] V. Carravetta, L. Yang, and H. ˚Agren, Phys. Rev. B55, 10044 (1997). [11] J. L. Pascual, L. G. M. Pettersson, and H. ˚Agren, Phys. Rev.B56, 7716 (1997). [12] C. Yu, R. M. Pitzer, and C. W. McCurdy, Phys. Rev. A32, 2134 (1985). [13] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984). [14] A. Zangwill and P. Soven, Phys. Rev. A 21, 1561 (1980). [15] W. Ekardt, Phys. Rev. Lett. 52, 1925 (1984). [16] G. Bertsch, Comp. Phys. Comm. 60, 247 (1990). [17] Z. H. Levine and P. Soven, Phys. Rev. A 29, 625 (1984). [18] J. R. Chelikowsky, N. Troullier, K. Wu, and Y. Saad, Phys. Rev. B50, 11355 (1994). [19] K. Yabana and G. F. Bertsch, Phys. Rev. B54, 4484 (1996). [20] K. Yabana and G. F. Bertsch, Int. J. Quantum Chem. 75, 55 (1999). [21] G. D. Mahan, Phys. Rev. A22, 1780 (1980). [22] Z. H. Levine, D. C. Allan, Phys. Rev. B43, 4187 (1991). [23] J.-I. Iwata, K. Yabana, and G. F. Bertsch, Nonlinear Op- tics,26, 9 (2000). [24] F. Calvayrac, P.-G. Reinhard, E. Suraud, and C. A. Ull- rich, Phys. Rep., in press. [25] N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991). 12[26] L. Kreinman and D. Bylander, Phys. Rev. Lett. 48, 1425 (1982). [27] H. Flocard, S. Koonin, and M. Weiss, Phys. Rev. C 17, 1682 (1978). [28] J. W. Negele, Rev. Mod. Phys. 54, 913 (1982). [29] D. Neuhasuer and M. Baer, J. Chem. Phys. 90, 4351 (1989). [30] M. S. Child, Mol. Phys. 72, 89 (1991). [31] G. Bertsch and S. F. Tsai, Phys. Rep. 18, 125 (1975). [32] S. Shlomo and G. Bertsch, Nucl. Phys. A243 , 507 (1975). [33] M. J. Stott and E. Zaremba, Phys. Rev. A 21, 12 (1980). [34] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Body Systems (McGraw-Hill, New York 1971), Sec. 13. [35] I. J. Thompson and A. R. Barnett, Comp. Phys. Comm. 36, 363 (1985). [36] O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). [37] R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421 (1994). [38] M. E. Casida, C. Jamorski, K. C. Casida, andD. R. Salahub, J. Chem. Phys. 108, 4439 (1998). [39] J. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [40] A. W. Potts and W. C. Price, Proc. R. Soc. London A 326, 165 (1972). [41] M. Ukai et al., J. Chem. Phys. 95, 4142 (1991). [42] D. M. P. Holland et al., Chem. Phys. 219, 91 (1997). [43] K. Kameta et al., J. Chem. Phys. 95, 1456 (1991). [44] G. Cooper, G. R. Burton, W. Fat Chan, and C. E. Brion, Chem. Phys. 196, 293 (1995). [45] C. Y. R. Wu, F. Z. Chen, and D. L. Judge, J. Chem. Phys.99, 1530 (1993). [46] G. Cooper, G. R. Burton, and C. E. Brion, J. Electron Spectroy. Relat. Phenom. 73, 139 (1995). [47] N. J. Bridge and A. D. Buchkingham, Proc. R. Soc. A 295, 334 (1966). [48] M. C. Wells, and R. R. Lucchese, J. Chem. Phys. 111 (1999) 6290. [49] G. Cooper, T. N. Olney, and C. E. Brion, J. Chem. Phys. 194, 175 (1995). [50] F. A. Grimm, Chem. Phys. 81, 315 (1983). 13
arXiv:physics/0010026v1 [physics.gen-ph] 10 Oct 2000QUANTUM SUPERSTRINGS AND QUANTIZED FRACTAL SPACE TIME B.G. Sidharth B.M. Birla Science Centre, Adarshnagar, Hyderabad - 500 063 , India Abstract Though Quantum SuperString Theory has shown promise, there are some puzzling features like the extra dimensions, which are curled up in the Kaluza-Klein sense. On the other hand a recent formu la- tion of what may be called Quantized Fractal Space Time leads to a surprising interface with QSS - we deal with the Planck lengt h, the same number of extra dimensions and an identical non-commut ative geometry. It is shown that this is not accidental. On the cont rary it gives us insight into the otherwise inexplicable features o f QSS. 1 Introduction At the outset it must be pointed out that Quantum Mechanics (Q M) and Quantum Field Theory (QFT) operate in the Minkowski space ti me - this is a diﬀerentiable manifold and space time points, infact po int particles are legitimate. However this contradicts the very spirit of QM - arbitrarily small space time intervals imply arbitrarily large momenta and en ergies. Yet both QM and QFT have co-existed with this unhappy situation[1]. F rom this point of view, two approaches are more satisfactory. One is s tring theory and the other is the discrete space time formulation. The for mer has evolved into the present theory of Quantum SuperStrings (QSS) while the latter leads to a formulation in terms of stochastic, fractal space -time. These two approaches appear very diﬀerent, but as we will now show, the y share several strikingly similar characteristics. This will throw light on some features of 1string theory which appear very puzzling, like the extra dim ensions and the non-commutative geometry aspect. 2 String Theory The origin of String Theory lies in two observed features. Th e ﬁrst is the Regge trajectories, J∝M2, which deﬁne the higher spin resonances, and which implies a ﬁnite size or spread in space for them. The second feature is the dual resonance or s−tchannel scattering, which lead to the Venetziano model. It was realized that if the part icles could be given an extension of the order of the Compton wavelength, th at is treated as strings, then the puzzling s−tfeature could be explained. All this leads to strings which are governed by the equation[ 2] ρ¨y−Ty′′= 0 (1) where the frequency ωis given by ω=π 2/radicalBigg T ρ(2) T=mc2 l;ρ=m l(3) /radicalBig T/ρ=c (4) Tbeing the tension of the string, lits length and ρthe line density. The identiﬁcation (3) gives (4) where cis the velocity of light, and (1) then goes over to the usual d’Alembertian or massless Klein-Gordon eq uation. It is worth noting that as l→0 the potential energy which is ∼/integraltextl 0T/parenleftBig∂y ∂x/parenrightBig2dx rapidly oscillates. Further if the above string is quantized canonically, we get ∝angbracketleft∆x2∝angbracketright ∼l2(5) The string eﬀectively shows up as an inﬁnite collection of Ha rmonic Oscil- lators [2]. It must be mentioned that (5) and (3) both show tha tlis of the order of the Compton wavelength. This has been called one of t he miracles 2of String theory by Veniziano[3]. Infact the minimum length lturns out to be given by T/¯h2=c/l2which from (3) and (4) is seen to give the Compton wavelength. If the relativistic quantized string is given rotation[4], then we get back the equation for the Regge trajectories given above. Here we are dealing with objects of ﬁnite extension rotating with the velocity of lig ht rather like spin- ning Black Holes. It must be pointed out that in Super String t heory, there is an additional term a0 J≤(2πT)−1M2+a0¯hwitha0= +1(+2)for the open (closed) string (6) In equation (6) a0comes from a zero point energy eﬀect. When a0= 1 we have the usual guage bosons and when a0= 2 we have the gravitons. It must be mentioned that in the relativistic case, even in Cl assical Theory it is known that[5, 6] there is an extension of the order of the Compton wavelength lseen in (3) above such that within this extension there are negative energies, reminiscent of a Dirac particle with zit terbewegung, and indicative of the breakdown of the concept of space time poin ts. Finally, if we go over to the full theory of Quantum Superstri ngs[7, 8], we deal with the Planck length, a non-commutative geometry and a total of ten space time dimensions, six of which are curled up in the Kaluz a-Klein sense. 3 Quantized Fractal Space time A starting point for QFST is the observation that the electro n can be con- sidered to be a Kerr-Newman Black Hole, which is otherwise cl assical, but which also describes the purely QM anomalous g= 2 factor[9]. However there is a naked singularity, in other words the radius of the horizon becomes complex and is given by r+=GM c2+ıb,b≡/parenleftBiggG2Q2 c8+a2−G2M2 c4/parenrightBigg1/2 (7) whereais the angular momentum per unit mass. The same circumstance is observed in the purely QM descripti on of the elec- tron by the Dirac equation. This time we have x= (c2p1H−1t+a1) +ı 2c¯h(α1−cp1H−1)H−1, (8) 3The bridge between equations (7) and (8) is the fact that the i maginary term in (8) is the zitterbewegung term which again is symptomatic of the fact that space time points are not meaningful. Infact an average over Compton scales removes the naked singularity. Another way of looking at this is that for the Dirac or Klein-G ordan operator we have a non Hermitian position operator given by /vectorXop=/vector xop−ı¯hc2 2/vector p E2(9) The Hermitianization of (9) leads to[10] a quantization of s pace time at the Compton scale. As with strings, we have in this description a background ZPF [10] corresponding to a collection of oscillation bound ed by the Compton wavelength. Interestingly an equation like (2) holds[6]. Thus we have to deal with space time intervals, which are a sor t of a minimum cut oﬀ. This leads to the non commutative geometry [x,y] = 0(l2),[x,px] =ı¯h[1 +l2],[t,E] =ı¯h[1 +τ2] (10) [11, 12]. Interestingly using (10) we can get back the Dirac r epresentation. Next, from the Dirac equation we can get the Klein-Gordon equ ation [13, 14] or alternatively the massless string equation (1) subject t o (4). (Physically this means that bosons are bound states of Fermions.) Furthe r with bound Fermions we recover the equation of Regge trajectories [15] . Interestingly, in Superstring theory also Dirac spinors are introduced ad hoc , averages over an internal time τare taken, and then the Klein-Gordon equation of the bosonic theory is deduced[16]. At the same time this space time fudge is exactly of the type required to explain the s−tchannel feature. It must be observed that in (10) if l2= 0, then we are back with usual space time and Quantum Mechanics. It is only when l2∝negationslash= 0 that we have a non commutativity, leading to Fermions. Indeed, as also noted by Witten, our usual Loren tzian space time diﬀerentiable manifold is bosonic and leads to the boso nic string or Klein-Gordon equation (1), - but there is a substructure giv en by (10) which leads to the Fermionic description and quantized space time . The former is an approximation when l2= 0 and we have equation (1) and what Witten calls bosonic space time (Cf. [17]). The latter gives the spinoria l representation for the Lorentz group which as noted by Einstein(Cf. ref.[18 ]) is the more fundamental representation and also the Dirac equation, ra ther than (1). 4It is quite signiﬁcant that the above discrete space time fea tures can be understood in terms of the Nelsonian Theory in which the diﬀu sion constant, in the light of the above remarks has been meaningfully ident iﬁed with ¯h/m [19]. Here we are dealing with a double Weiner process[20], a nd the minimum space time cut oﬀs are stochastic in nature and may be called Q FST. Infact, a quick way to Quantum Mechanics from the Nelsonian t heory is by starting with the diﬀusion equation, ∆x·∆x=h m∆t This can be rewritten as the usual Quantum Mechanical relati on, m∆x ∆t·∆x=h= ∆p·∆x To throw further light on this, we observe that it is well know n[6] that from a two state formulation we can recover the Schrodinger equati on by considering the diﬀusion of a particle to neighbouring points on either s ide, separated by a distance of the order of the Compton wavelength. Similarly if we could consider a diﬀusion forwards and backwards in time through a Compton time interval reminiscent of the double Weiner process refe rred to, we can get back the d’Alembertian equation, or the string equation (1). To see this more explicitly, let us consider a point xand its neighbouring points x±l and a time t±τ. Then a typical simple diﬀusion equation would be given by [21], following Feynman, Cı(t−τ) +Cı(t+τ) =/summationdisplay j[δıj−ı ¯hHıj(t)τ]Cj(t) (11) where we denote by, Cı(t)≡< ı\|ψ(t)>,the amplitude for the state \|ψ(t)> to be in the state \|ı>,and <ı\|U\|j >≡Uıj(t+τt,t)≡δıj−ı ¯hHıj(t)τt. andıdenotes the point xand the points x±l. In the limit l=cτ→0 (11) gives the Klein-Gordon equation or alternatively the equat ion (1) subject to the condition (4). This also shows the close relation of the d iﬀusion process and special relativity, as noted in [19]. 5Finally in the above formulation using the spirit of ﬂuctuat ions and the non commutative relation (10), we get, ∆ x∼l3, wherelis the electron Compton wavelength[20]. We are thus lead ﬁrstly to the Planck length and secondly to the fact that the single xdimension has now become three dimensional. The two extra dimensions are ﬂuctuations and curled up in the Kaluza-Klein sense[22] as shown elsewhere[20]. It must be stressed that i n this formulation we use as seen in the Super String case, the zero point ﬂuctuat ions. 4 Conclusion It is not surprising that QSS and QFST reach the same extra dim ensions, non commutative geometry and Planck scale, because the form er starts with extension and then the QM input, while the latter starts with extension and the Nelsonian QM input. From this point of view we get an in sight into the otherwise inexplicable features of QSS. As Venezia no put it [3], ”we thus face a kind of paradoxical situation. On the other ha nd quantum mechanics is essential to the success of the KK idea. At the sa me time, QFT gives meaningless inﬁnities and spoils the nice semiclassi cal results. If the beautiful KK idea is to be saved we need a better quantum theor y than QFT. I will argue below that such a theory already exists: it is cal led (super) string theory.” The key lies in space time intervals which underlie a Fermionic sub structure of space time given by (10), which has a perfectly n atural origin in stochastic process or in the spirit of which lies ”Law withou t Law”[23], and is exempliﬁed by QFST. References [1] F.D. Peat, ”Superstrings”, Abacus, Chicago, 1988, p.21 . [2] G. Fogleman, Am.J.Phys., 55(4), 1987, pp.330-336. [3] G. Veneziano, ”Quantum Geometric Origin of All Forces in String The- ory” in ”The Geometric Universe”, Eds. S.A. Huggett et al., O xford University Press, Oxford, 1998, pp.235-243. [4] J. Scherk, Rev.Mod.Phys., 47 (1), 1975, pp.1-3ﬀ. 6[5] C. Moller, ”The Theory of Relativity”, Clarendon Press, Oxford, 1952, pp.170 ﬀ. [6] B.G. Sidharth, Ind.J. Pure & Appld.Phys., Vol.35, 1997, pp.456-471. [7] W. Witten, Physics Today, April 1996, pp.24-30. [8] Y. Ne’eman, in Proceedings of the First Internatioinal S ymposium, ”Frontiers of Fundmental Physics”, Eds. B.G. Sidharth and A . Burinskii, Universities Press, Hyderabad, 1999, pp.83ﬀ. [9] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitation ”, W.H. Free- man, San Francisco, 1973, pp.819ﬀ. [10] B.G. Sidharth, Int.J.Mod.Phys.A, 13 (15), 1998, p.259 9ﬀ. [11] H.S. Snyder, Physical Review, Vol.72, No.1, July 1 1947 , p.68-71. [12] B.G. Sidharth, Chaos, Solitons and Fractals, 11 (2000) , 1269-1278. [13] V. Heine, ”Group Theory in Quantum Mechanics”, Pergamo n Press, Oxford, 1960, p.364. [14] J.R.Klauder, ”Bosons Without Bosons” in Quantum Theor y and The Structures of Time and Space, Vol.3 Eds by L. Castell, C.F. Va n Wei- izsecker, Carl Hanser Verlag, Munchen 1979. [15] B.G. Sidharth, ”Scaled Universe II”, to appear in Chaos , Solitons and Fractals. [16] P. Ramond, Phys.Rev.D., 3(10), 1971, pp.2415-2418. [17] B.G. Sidharth ”A Brief Note on Analaticity and Causalit y, and thee ’Levels of Physics’”, to appear in Chaos, Solitons and Fract als. [18] M. Sachs, ”General Relativity and Matter”, D. Reidel Pu blishing Com- pany, Holland, 1982, p.45ﬀ. [19] B.G. Sidharth, Chaos, Solitons and Fractals, 12(1), 20 00, 173-178. [20] B.G. Sidharth, ”Uniﬁcation of Electromagnetism in Qua ntized Fractal Space Time”, to appear in Chaos, Solitons and Fractals. 7[21] R.P. Feynman, The Feynman Lectures on Physics, 2, Addis on-Wesley, Mass., 1965. [22] H.C. Lee, ”An Introduction to Kaluza-Klein Theories”, World Scientiﬁc, Singapore, 1984. [23] J.A. Wheeler, Am.J.Phys., 51(5), 1983, p.398. 8
1 A Macroscopic Approach to the Origin of Exotic MatterCharles T. Ridgelycharles@ridgely.wsGalilean Electrodynamics, in review (2000) . AbstractHerein the Casimir effect is used to present a simple macroscopic view of the origin ofexotic matter. The energy arising between two nearly perfectly conducting parallel platesis shown to become increasingly negative as the plate separation is reduced. It isproposed that the Casimir energy appears increasingly negative simply because thevacuum electromagnetic zero-point field performs positive work in pushing the platestogether, transforming field energy into kinetic energy of the plates. Next, the inertialproperties of exotic matter are considered. The parallel plates of the Casimir system arereplaced with an enclosed cavity of identical dimensions that is subjected to an externalforce. It is found that zero-point radiation exerts an inertial force on the cavity inopposition to the external force. This ultimately leads to the conclusion that the inertialproperties of exotic matter are identical to the inertial properties of ordinary matter.1. Introduction According to a recent proposal by Miguel Alcubierre [1], general relativity admits metric solutions that predict the possibility of superluminal space travel. In essence,2 these metrics imply that a spaceship can be propelled through space by expanding space- time behind the ship while, at the same time, contracting space-time in front of the ship.The ship then surfs on this space-time deformation, acquiring very high velocity relativeto the rest of the universe. Even more surprising is that these “warp drive” metricspredict that such travel can be carried out without the usual time dilation effects predictedby special relativity. That is, proper time experienced by a space traveler, using anAlcubierre warp drive, is identical to the time experienced by observers remaining on theEarth. As with worm holes, warp drives require tremendous quantities of negative energy, or exotic matter [1-5]. Although there are several conceptual difficulties associated withwarp drive, the energy requirement alone can easily be taken as sufficient grounds for usto dismiss the feasibility of warp drive. More encouraging, however, is that modifiedforms of the Alcubierre metric have been proposed, requiring more reasonable quantitiesof exotic matter [4-5]. We can only hope that as more research is conducted, warp drivetheory will become increasingly tenable. Of course, there is one serious obstacle to our actually engineering a warp drive system of any size: the need for exotic matter. Before we can set up a physically realizable warp drive, we must have a working knowledge of how to generate exoticmatter. As is well known, exotic matter is forbidden classically, but is permitted on thequantum level under special circumstances. When these special circumstances areapplied to warp drive theory, we are left with a warp bubble requiring massive amounts3 of exotic matter and having walls whose thickness is on the order of the Planck length [2- 3]. But even with such constraints, we are still left with the question of how to obtainexotic matter, regardless of quantity. It is an objective of the present analysis to provide asimple macroscopic view of the origin of exotic matter while leaving the question ofquantity for future development. One well-known example in which negative energy produces observable forces is the Casimir effect [1, 4]. The Casimir effect is an attractive force that arises between twonearly perfectly conducting parallel plates when they are placed in close proximity, asshown in Figure 1. In essence, the plates block out some of the vacuum zero-pointelectromagnetic radiation that would otherwise reside in the volume between the plateswere the plates removed [6-7]. This gives rise to a small energy difference between theplates relative to the region outside the plates: ()22 3720π=−!cLUrr,( 1 ) where 2L is the surface area of each plate, and r is the plate separation. Differentiating Eq. (1) with respect to the plate separation then gives the force on the plates: ()()22 4240π ∂=− =−∂! Ur cLFrrr.( 2 ) This attractive force, tending to push the plates together, has been experimentally verified to within 5% of its theoretical value [7].4 While the Casimir effect is typically reserved for discussions involving the quantum world, there may in fact be a way to picture the negative energy-content of the Casimirsystem from a macroscopic point of view. This can be done by taking a macroscopicview of the vacuum electromagnetic zero-point field (ZPF). With this approach, the ZPFis viewed as a real Lorentz-invariant radiation field that is essentially uniform throughoutall of space. So long as the ZPF is isotropic, no direct observation of the ZPF can bemade; but if the ZPF becomes locally anisotropic, such as with the Casimir system, directobservations can then indeed be carried out [7-8]. The next Section deals with the change in Casimir energy that arises when the plate separation is decreased. It is shown that the ZPF performs positive work in pushing theplates together, transforming electromagnetic field energy into kinetic energy of theplates. The resulting drop in field energy appears increasingly negative with respect tothe uniform energy content outside the plates [8-9]. Based on this, it is proposed thatwhenever the ZPF performs positive work in a localized region of space-time, a drop inzero-point field energy occurs, resulting in a negative energy density relative to the restof space-time. Section 3 is devoted to acquiring greater insight into the inertial properties of exotic matter. Applying the law of inertia of energy [10-11] to the Casimir energy leads to anegative inertial mass. Initially, it seems that this small quantity of exotic matter shouldreduce the overall inertia of the Casimir system. To determine whether this is indeed thecase, the parallel plates of the Casimir system are replaced with a cavity of identical5 dimensions. When the cavity is subjected to an external force, space-time distortion [12] within the cavity gives rise to a Doppler-shift, detectable within the co-moving referenceframe of the cavity. It is shown that zero-point radiation responds to this Doppler-shiftby exerting an inertial resistance force on the cavity in a manner identical to that ofordinary, positive radiation [10]. This leads to the conclusion that the inertial propertiesof exotic matter are the same as the inertial properties of ordinary matter. 2. Negative Energy due to Work Performed by Zero-Point Field Consider the energy between two parallel plates when their separation is first 1r and then later 2r, where 1r > 2r. According to Eq. (1), when the plate separation is 1r and 2r the energy between the plates is, respectively 22 1 3 1 720cLUrπ=−!, (3a) 22 2 3 2 720cLUrπ=−!. (3b) As pointed out in the Introduction, the force between the plates is attractive, tending to push the plates together. Suppose the force pushes the plates from separation 1r to separation 2r. The change in energy between the plates is then 2 2 21 33 2111 720UU U c Lrrπ∆= − = − −  ! .( 4 )6 This implies that the Casimir energy becomes increasingly negative as the plate separation is reduced. The work W performed in order to bring about this change in energy is 2 2 12 33 2111 720Wc Lrrπ=−  ! .( 5 ) This is positive work performed by the ZPF in pushing the plates from 1r to 2r, transforming electromagnetic field energy into kinetic energy of the plates. Since the energy content of the ZPF is initially uniform everywhere [8-9], this drop in field energymanifests as a negative energy density between the plates, relative to the energy densityof the region outside the plates. Of particular interest is the case in which the initial separation of the parallel plates is very large. Assuming that no other forces act on the plates, the ZPF should still exert aforce on the plates, no matter how minute that force may be. The work performed by that force can be expressed simply by taking the limit of Eq. (5) as 1r tends to infinity: () 112rWL i m W∞→∞= .( 6 ) Carrying this out leads directly to 22 3720cLWrπ ∞=!,( 7 ) in which the subscript 2r has been dropped for simplicity. This is the work performed by the ZPF in pushing two parallel plates to a separation r when the plates are initially very far apart. The potential energy associated with Eq. (7) is then7 22 3720cLUrπ ∞=−!,( 8 ) which is identical to the original expression for the Casimir energy, given by Eq. (1). So then, we have come full circle, but with the clear assertion that the negative energydensity of the Casimir system arises because the ZPF performs positive work in alocalized region of space-time. More specifically, we propose that in any instance inwhich the ZPF performs positive work in a localized region of space-time, a drop invacuum electromagnetic zero-point field energy occurs, resulting in a negative energydensity in that region relative to the rest of space-time [9]. 3. Inertial Properties of Exotic Matter In the preceding section, the Casimir system of two conducting parallel plates was used to show that when the ZPF performs positive work in a localized region, the energydensity of that region becomes negative. Taking a strict view of the relationship betweenenergy and mass [10], we might suppose that such a region exhibits an inertial mass in accordance with the law of inertia of energy, 2mU c∞= . Applying this expression to the Casimir energy, given by Eq. (8), leads directly to 22 3720Lmcrπ=−!.( 9 ) This implies that the mass-energy between the plates of the Casimir system is negative with respect to the exterior region. At first sight, we might expect this small quantity of8 exotic matter should reduce the overall inertial mass of the Casimir system. As we shall see, however, we have no good reason to presuppose that exotic matter behaves in anyway differently than does ordinary matter. According to previous analyses [10-11, 13], the inertial properties of energy cannot be adequately explained on the basis of inertial mass alone; the contribution by space-timemust also be taken into account. This was demonstrated by considering an ideal cavity of length r, containing monochromatic radiation of frequency 0ν that is subjected to an external force [10]. It was found that when an external force is applied to such a cavity, radiation pressure exerts an inertial resistance force on the cavity, given by ()ABNhFnrνν=−" ", (10) where N is the average number of photons in the cavity, Aν and Bν are the frequencies of radiation measured by co-moving observers at walls A and B of the cavity, respectively, and ˆn is a unit vector in the direction of the cavity’s acceleration. This expression for the force was used to show that the inertial resistance of the cavity arisesas a direct manifestation of space-time distortion within the cavity [10, 12]. The plan here is to use Eq. (10) to gain greater insight into the inertial properties of exotic matter. To do this, the parallel plates of the Casimir system are replaced with acavity of identical dimensions, namely one with r << L, as shown in Figure 2. Also, the zero-point radiation under consideration here is not monochromatic, and thus the average9 indicated by N must be eliminated in favor of a sum over all frequency modes within the cavity. Carrying this out, Eq. (10) can be recast in the form NN ii ii ABFnrωω =−  ∑∑"! ". (11) where N is the number of modes within the cavity [14]. For the case of a cavity in thermal equilibrium with its surroundings, the number of photons observed at each wallwithin the cavity is roughly the same. Using this to our advantage, Eq. (11) can then bewritten as ()N AB i iFnrωω=−∑"! ". (12) This expression implies that when an external force is applied to the cavity, each frequency mode within the cavity exerts a net force on the cavity, given by ()iA B iFnrωω=−"! ". (13) For the case of a uniformly accelerating cavity, as shown in Figure 2, space-time distortion gives rise to an observable Doppler-shift within the cavity [10]: 0 ωω−≠AB . Observers residing in the co-moving reference frame (CMRF) of the cavity find that zero-point radiation arriving at wall B is blue-shifted relative to radiation observed at wall A. If the frequencies observed at wall A are 0ωω=Ai i , then the frequencies at wall B are 01 1ωω+=−R Bi i RVc Vc, (14)10 where RV is the relative velocity of wall B gained during the time in which photons travel from wall A to wall B. Using these frequencies, observers moving with the cavity recast Eq. (12) in the form 001 1ωω +=−−∑"! "N R i RiVcFnrV c. (15) Rearranging a bit, this expression becomes () 01111ω+=− − ∑" "!N R i i RVcFnrV c. (16) To further simplify this expression, we notice that for the case of a very small cavity undergoing Newtonian acceleration, we can use the approximation [10] 1111 τ+ −≈ − ∇  −  "R RVc d t rV c d, (17) where dt is an interval of coordinate time, τd is an interval of proper time experienced by observers moving with the cavity, and the minus sign arises because the gradient is inthe direction of the cavity’s acceleration [10]. Substituting Eq. (17) into Eq. (16) then gives the force as () 0ωτ=− ∇ ∑""!N i idtFd. (18) As a final consideration, we notice that the summation simply expresses the total zero- point energy within the cavity:11 () 0ω=∑!N i iE . (19) At this point, it must be stressed that although the energy density within the cavity is negative with respect to exterior regions, individual photons inside the cavity have their usual positive energies, iiEω=!. The cavity simply blocks out those photons having wavelengths longer than the distance between the walls of the cavity [7]. Using Eq. (19) in Eq. (18), the force on the cavity then becomes dtFEdτ=− ∇"". (20) According to this expression, zero-point radiation within the cavity exerts an inertial force on the cavity that acts in opposition to the externally applied force, just as doesordinary, positive radiation [10, 15]. Thus, the inertial properties of a region having a negative energy density are identical to those of a region having an equivalent positive energy density. Previous analyses have shown that all forms of positive energy resistchanges to their states of motion, and that the source of such resistance is entirely due tospace-time distortion arising within accelerating systems of reference [10-11, 13].Equation (20) implies that the same holds true for negative forms of energy as well. Wecan conclude, therefore, that the inertial properties of exotic matter are the same as theinertial properties of ordinary matter. Both forms of matter resist changes to their statesof motion when acted upon by an external force.12 Returning to the Casimir system, it is now easy to see that the inertial mass predicted by Eq. (9) will contribute to an overall increase in the inertia of the Casimir system rather than reducing it, as the minus sign initially suggested. While the minus sign in Eq. (9)certainly implies that the mass is exotic, having its origin in negative energy, the minussign tells us nothing about the inertial properties of exotic matter. 4. Discussion Current theory places serious restrictions on the existence of exotic matter. According to recent analyses, such restrictions set the thickness of the walls of an Alcubierre warpbubble on the order of the Planck length while requiring an enormous quantity of exoticmatter [2-3]. In the present analysis, a simple macroscopic interpretation of the origin ofexotic matter and its inertial properties has been suggested while leaving the question ofquantity for future development. Current theory predicts that the energy density between two nearly perfectly conducting parallel plates is negative, resulting in an attractive force between the plates[1, 4]. Herein, it was shown that this Casimir energy is negative simply because thevacuum electromagnetic zero-point field (ZPF) performs positive work in pushing theplates together, leading to a drop in field energy between the plates relative to the regionoutside the plates. Since the energy content outside the plates is uniformly distributed,and thus undetectable [7-8], the energy density between the plates appears negative.13 Applying the law of inertia of energy [10-11] to the Casimir system led to a negative inertial mass. At first sight, it seemed that the presence of this small quantity of exoticmatter should reduce the overall inertia of the Casimir system. To determine whether thisis indeed the case, a cavity of identical dimensions that is subjected to an external forcewas considered. It was found that zero-point radiation within such a cavity responds tospace-time distortion within the cavity in a manner identical to that of ordinary, positiveradiation [10]. This led to the conclusion that the inertial properties of a regionpossessing a negative energy density are identical to the inertial properties of a regionpossessing an equivalent positive energy density. Of course, the validity of the interpretation set forth herein is ultimately hinged on whether or not the ZPF is comprised of real radiation, and on the amount of work that canbe performed thereby. Judging from the literature on the subject, whether or not the ZPFis real seems to be an ongoing debate [7-9]. Assuming that the ZPF is indeed real andcapable of performing electromagnetic work [9], we propose that whenever the ZPFperforms positive work in a localized region of neutral space-time, a net drop in energyoccurs in that region. The magnitude of such a drop in energy is directly proportional tothe quantity of work performed, and expresses itself in the form of exotic matter.Furthermore, we conclude that the inertial properties of exotic matter are identical to theinertial properties of ordinary matter; both forms of matter resist changes to their states ofmotion.14 As a final thought, it is interesting to speculate on how the notions put forth herein might be applied to warp drive theory, mentioned in the Introduction. From ahypothetical standpoint, we can imagine a spaceship, residing in Minkowski space-time,which carries out some sort of internal process that alters the ZPF outside the ship [16].If, through this process, the ship can coax the ZPF into performing substantialelectromagnetic work, then the ship can use the ZPF to generate the exotic matterrequired for warp drive. In essence, we imagine a spaceship that manipulates the ZPF inorder to generate stress-energy outside the ship. With the right configuration, this stress-energy can be used to alter the Minkowski geometry outside the ship, forming anAlcubierre-type geometry, or some more efficient variation thereof [1, 4-5]. Assumingthat all goes as planned, the spaceship should then be propelled by the resulting space-time geometry, acquiring very high velocity relative to the rest of the universe. Onenoteworthy feature of this model is that there is no need for the spaceship to carry aquantity of exotic matter on board for use as a fuel source. Rather, such a spaceshipwould generate exotic matter outside the ship as it moves through space.15 References [1] M. Alcubierre, “The warp drive: hyper-fast travel within general relativity,” Class. Quant. Grav. 11, L73-L77 (1994). [2] M. J. Pfenning and L. H. Ford, “The unphysical nature of ‘Warp Drive’,” Class. Quant. Grav. 14, 1743-1751 (1997). [3] M. J. Pfenning and L. H. Ford, “Quantum inequality restrictions on negative energy densities in curved spacetimes,” Doctoral Dissertation, Tufts University (1998). [4] C. Van Den Broeck, “A ‘warp drive’ with reasonable total energy requirements,” Class. Quant. Grav. 16, 3973-3979 (1999). [5] C. Van Den Broeck, “On the (im)possibility of warp bubbles,” Comment for Class. Quant. Grav., gr-qc/9906050. [6] J. P. Dowling, “The mathematics of the Casimir effect,” Mathematics Magazine, Vol. 62, No. 5, 324-331 (1989). [7] B. Haisch and A. Rueda, “The zero-point field and inertia,” in “Casusality and locality in modern physics,” G. Hunter, S. Jeffers, and J. –P. Vigier (eds.), KluwerAcad. Publ., pp. 171-178 (1998). [8] B. Haisch, A. Rueda, and H. E. Puthoff, “Physics of the zero-point field: implications for inertia, gravitation and mass,” Speculations in Science andTechnology, Vol. 20, 99-114, (1997). [9] A very interesting, and informative, discussion on the feasibility of extracting energy from the zero-point field is given by D. Cole, “Energy and thermodynamic16 considerations involving electromagnetic zero-point radiation,” Amer. Inst. Physics Conf. Proc. No. 458, 960, (1999). [10] C. T. Ridgely, “On the origin of inertia,” Galilean Electrodynamics 12, 17-20 (2001). [11] C. T. Ridgely, “On the nature of inertia,” Galilean Electrodynamics 11, 11-15 (2000). [12] The word “distortion” is used in reference to any instance in which local space-time structure deviates from the flat, Minkowski space-time of special relativity. [13] C. T. Ridgely, “Space-time as the source of inertia,” Galilean Electrodynamics, in press. [14] There are two effective limits on the number of modes within the cavity. Wavelengths greater than the dimensions of the cavity are blocked out of theinterior region, and wavelengths shorter that the atomic size do not contribute to theforce on the walls of the cavity. These limits are generally embodied in a cut-offfunction. See, for example, Refs. [6-8]. [15] Of course, exterior ZPF radiation should contribute to the resistance force on an accelerating cavity as well. This arises because the cavity occupies a volume ofspace, and essentially plows through a sea of zero-point radiation in much the sameway as a submarine moves through water. [16] It seems likely that this process will be cyclical. It is difficult to imagine any static embodiment capable of maintaining a sizable alteration of the ZPF.Figure 1 Charles T. Ridgely Galilean Electrodynamics17 Figure 1. Two nearly perfectly conducting plates of area 2L are placed in parallel. The plates are separated by distance r, where r << L. Figure 2 Charles T. Ridgely Galilean Electrodynamics18 Figure 2. An enclosed cavity of identical dimensions as the Casimir system accelerates uniformly under the action of an external force. Space-time distortion, arising due toacceleration, gives rise to a frequency-shift in the zero-point radiation observed at walls A and B within the cavity: 0 ωω−≠AB . Zero-point radiation pressure exerts a net force F" on the cavity in opposition to the externally applied force.
arXiv:physics/0010028v1 [physics.atom-ph] 12 Oct 2000Breit interaction correction to the hyperﬁne constant of an external s-electron in many-electron atom O.P. Sushkov School of Physics, University of New South Wales, Sydney 2052, Australia Correction to the hyperﬁne constant Aof an external s-electron in many-electron atom caused by the Breit interaction is calculated analytically: δA/A = 0.68Zα2. Physical mechanism for this correction is polarization of the internal electronic shel ls (mainly 1 s2shell) by the magnetic ﬁeld of the external electron. This mechanism is similar to the po larization of vacuum considered by Karplus and Klein [1] long time ago. The similarity is the rea son why in both cases (Dirac sea polarization and internal atomic shells polarization) the corrections have the same dependence on the nuclear charge and ﬁne structure constant. In conclusion we also discuss Zα2corrections to the parity violation eﬀects in atoms. PACS: 31.30.Gs, 31.30.Jv, 32.80.Ys I. INTRODUCTION Atomic hyperﬁne structure is caused by the magnetic interac tion of the unpaired external electrons with the nuclear magnetic moment. There are two types of relativistic correc tions to this eﬀect. The ﬁrst type is a single particle correction caused by the relativistic eﬀects in the wave equ ation of the external electron [2]. This correction is of the order of (Zα)2, whereZis the nuclear charge and αis the ﬁne structure constant. Solving the Dirac equation on e can ﬁnd this correction analytically in all orders in ( Zα)2[3]. An alternative way to ﬁnd this correction is direct nume rical solution of the Dirac equation. Correction of the second typ e has a many-body origin: it is due to polarization of paired electrons by the Breit interaction of the external el ectron. There are two kinds of paired electrons in the problem: a) Dirac sea, b) closed atomic shells. The contribu tion related to the Dirac sea was calculated by Karplus and Klein long time ago [1]: δA/A =α/2π−Zα2(5/2−ln 2). Theα/2πpart is due to usual anomalous magnetic moment and Zα2part comes from a combined eﬀect of the Nuclear Coulomb ﬁeld a nd the Breit interaction. We would like to stress that as soon as Z >12 theZα2part is bigger than α/2π. Eﬀect of the polarization of atomic shells by the Breit interaction has been recently calculate d numerically [4–7]. These calculations were performed for Cs (Z= 55) because they were motivated by the interest to parity no nconservation in this atom. Results of these calculations are somewhat conﬂicting, but nevertheless th ey indicate that the correction for an external s-electron i s of the order of ∼ ±0.4%. In spite of being rather small this correction is compara ble with the present accuracy of atomic many-body calculations and therefore it must be taken into a ccount. The purpose of the present work is to calculate analytically the correction induced by the Breit interacti on. This allows to elucidate the physical origin of the eﬀect and its dependence on the atomic parameters. This also provi des an important lesson for a similar correction to the parity non-conserving amplitude which we discuss in the con clusion. II. CONTRIBUTION OF THE DIRECT ELECTRON-ELECTRON MAGNETIC INTERACTION In the present work we do not consider single particle relati vistic eﬀects (Dirac equation), so we assume that in zero approximation the atom is described by the Schroedinge r equation with Coulomb electron-nucleus and electron- electron interaction. For magnetic interaction throughou t this work we use the Coulomb gauge. Vector potential and magnetic ﬁeld of the nucleus are of the form AN(r) =µN×r r3, (1) HN(r) =∇ ×AN→8π 3µNδ(r), whereµNis the magnetic moment of the nucleus. We keep only the spheri cally symmetric part of the magnetic ﬁeld HNbecause in this work we consider only s-electrons. Interact ion of the magnetic moment µ1of the external electron with nuclear magnetic ﬁeld gives the hyperﬁne structure 1Hhfs=∝angb∇acketleft−µ1· HN∝angb∇acket∇ight=−C(µ1·µN) =A(s·I), (2) C=8π 3ψ2 e(0), A=µ1 sµN IC. Hereψe(r) is the wave function of the external electron, and Ais the hyperﬁne constant. Vector potential of the external s-electron is of the form Ae(r) =/integraldisplayµ1×(r−R) \|r−R\|3ψ2 e(R)d3R=−µ1× ∇ r/integraldisplayψ2 e(R) \|r−R\|d3R= 4πµ1×r r3/integraldisplayr 0ψ2 e(R)R2dR. (3) Hence the magnetic ﬁeld is He(r) =∇ ×Ae→8π 3ψ2 e(r)µ1. (4) Let us repeat once more that we keep only the spherically symm etric part of the magnetic ﬁeld. The Hamiltonian of an internal electron in the magnetic ﬁeld A=AN+Ae,H=HN+Heis given by H=(p−e cA)2 2m−µ2· H+U(r), (5) whereµ2is the magnetic moment of the internal electron. Certainly \|µ1\|=\|µ2\|=\|e\|¯h 2mc, however directions of µ1 andµ2are independent. Having eq. (5) in mind one can easily draw di agrams describing correction to the hyperﬁne structure due to the electron-electron magnetic interacti on. This diagrams are shown in Fig.1 N N N N Nn N a b ce e e e e e n m n n m n n FIG. 1. Diagrams describing the direct Breit interaction correcti on to the hyperﬁne structure. The diagrams a) and b) give the paramagnetic contribution, and the diagram c) gives the diamagnetic contribution. N denotes the nucleus, edenotes the external electron, ndenotes the internal electron, and ﬁnally mdenotes a virtual excitation of the internal electron. The w avy line shows magnetic interaction with the external electron , and the dashed line shows magnetic interaction with the nuc leus. Two equal paramagnetic contributions are given by the diagr ams shown in Fig.1a,b. Corresponding energy correc- tions are δEa=δEb=/summationdisplay n∈filled∝angb∇acketleftn\|(−µ2· He\|δψn∝angb∇acket∇ight, (6) where δψn=/summationdisplay m/negationslash=n∝angb∇acketleftm\|(−µ2· HN)\|n∝angb∇acket∇ight En−Em\|m∝angb∇acket∇ight. (7) The diamagnetic contribution shown in Fig.1c is given by the A2term from the Hamiltonian (5), hence δEc=e2 mc2∝angb∇acketleftn\|Ae·AN\|n∝angb∇acket∇ight. (8) Before proceeding to the accurate calculation of δEit is instructive to estimate a magnitude of the correction. Let us look for example at the diamagnetic correction (8). Accor ding to eqs. (1), (3) the vector potentials are AN∼µN/r2 andAe∼µ1rψ2 e(0). Hence the correction (8) is of the order of δE∼e2µ1µNψ2 e(0)/(mc2r), whereris the radius 2of the internal shell. Since all the interactions are singul ar it is clear that the main contribution comes from the K-shell, sor∼aB/Z(aBis the Bohr radius). Together with eq. (2) this gives the foll owing relative value of the Breit correction to the hyperﬁne constant δC/C ∼Zα2. So we see that this eﬀect has exactly the same dependence on t he atomic parameters as the Dirac sea polarization considered in the paper [1]. Now let us calculate the coeﬃcient in the Zα2correction. We consider explicitly only 1s and 2s closed she lls and we also need to consider the external s-electron. In atom ic units the single particle energies of these states are: E1=−Z2/2,E2=−Z2/8,Ee≈0. At small distances the nuclear Coulomb ﬁeld is practicall y unscreened and hence the wave functions are of the simple form [8] ψ1=1√πe−ρ, ψ2=1√ 8πe−ρ/2(1−ρ/2), (9) ψe=/radicalbigg3 16πρJ1(/radicalbig 8ρ). Hereρ=ZrandJ1(x) is the Bessel function. The functions ψ1,2are normalized in the usual way:/integraltext ψ2 id3ρ= 1. The wave function of the external electron is normalized by t he condition ψ2 e(0) = 3/(8π). With this normalization the leading order hyperﬁne constant (2) is equal to unity, C= 1, and therefore this normalization is convenient for calculation of the relative value of the Breit correction to the hyperﬁne constant. Using eqs. (1),(3),(8) and performi ng summation over spins in the closed shells one ﬁnds the diamag netic correction δEc=Zα2(µ1·µN)/summationdisplay n/parenleftbigg4 3/integraldisplay∞ 0ψ2 n(ρ) ρ4d3ρ/integraldisplayρ 0ψ2 e(ρ′)d3ρ′/parenrightbigg = 0.230Zα2(µ1·µN). (10) The numerical coeﬃcient was found by straightforward numer ical integration. Contributions of the inner shells drop down approximately as 1 /n3, so 0.230=0.207+0.023, where the ﬁrst contribution comes f rom the 1s-shell and the second contribution comes from the 2s-shell. To calculate the paramagnetic contributions (6) we use corr ectionsδψndeﬁned by eq. (7) and calculated in the Appendix. Substitution of (4), (9), and (A9) into formula (6 ) and summation over electron spins in the closed shell gives the following result δEa+δEb=−8 3Zα2(µ1·µN)/parenleftbigg/integraldisplay∞ 0e−2ρJ2 1(/radicalbig 8ρ)w1(ρ)dρ+1 8/integraldisplay∞ 0e−ρ(1−ρ/2)J2 1(/radicalbig 8ρ)w2(ρ)dρ/parenrightbigg (11) =−8 3Zα2(µ1·µN)(0.219 + 0.021) = −0.640Zα2(µ1·µN), where we present explicitly the contributions of 1s- and 2s- shells. The numerical coeﬃcient is found by numerical integration. Similar to the diamagnetic term the contribut ions of the inner shells drop down approximately as 1 /n3. The leading order hyperﬁne structure is given by eq. (2) with constantC= 1 due to the accepted normalization. According to eqs. (10) and (11) the total correction caused b y the direct magnetic interaction is δEa+δEb+δEc= −0.410Zα2(µ1·µN). Comparing this with eq. (2) one ﬁnds the relative value of t he direct correction: δA(dir) A=δC(dir) C= 0.410Zα2. (12) III. CONTRIBUTION OF THE EXCHANGE ELECTRON-ELECTRON MAGNE TIC INTERACTION Exchange diagrams contributing to the correction are shown in Fig2. N N N N Ne N a b ce e e n m n m nn en en 3FIG. 2. Diagrams describing the exchange Breit interaction correc tion to the hyperﬁne structure. The diagrams a) and b) give the paramagnetic contribution, and the diagram c) give s the diamagnetic contribution. N denotes the nucleus, edenotes the external electron, ndenotes the internal electron, and ﬁnally mdenotes a virtual excitation of the internal electron. The w avy line shows magnetic interaction with the external electron , and the dashed line shows magnetic interaction with the nuc leus. The diagrams Fig.2a,b show the “paramagnetic” contributio ns, and the diagram Fig.2c shows the ”diamagnetic” contribution. Note that the contributions of the diagrams F ig.2a,b must be doubled because the opposite order of operators is also possible. Let us begin with the “dimagneti c” term. Comparison of Fig.1c and Fig.2c shows that the direct and the exchange contributions are very similar and t herefore the simplest way to derive the exchange term is just to make appropriate alterations in eq. (10) which giv es the direct contribution. The alterations are obvious: 1)opposite sign, 2) ψ2 n→ψeψn, 3)ψ2 e→ψeψn, 4)there is no summation over the intermediate spins, hence 4/3→2/3. Thus the result is δE(ex) c=−Zα2(µ1·µN)/summationdisplay n/parenleftbigg2 3/integraldisplay∞ 0ψn(ρ)ψe(ρ) ρ4d3ρ/integraldisplayρ 0ψn(ρ)ψe(ρ′)d3ρ′/parenrightbigg =−0.107Zα2(µ1·µN). (13) The coeﬃcient is found by numerical integration: 0.107=0.0 96+0.011, where the ﬁrst contribution comes from the 1s-shell and the second contribution comes from the 2s-shel l. The paramagnetic exchange contribution shown in Fig.2b is s imilar to the direct ones given by Fig.1a,b. The only diﬀerence is in algebra of Pauli matrixes and in additional s ign (-). This consideration shows that the paramagnetic exchange contribution is equal to half of that given by eq. (1 1) δE(ex) b=−4 3Zα2(µ1·µN)/parenleftbigg/integraldisplay∞ 0e−2ρJ2 1(/radicalbig 8ρ)w1(ρ)dρ+1 8/integraldisplay∞ 0e−ρ(1−ρ/2)J2 1(/radicalbig 8ρ)w2(ρ)dρ/parenrightbigg (14) =−0.320Zα2(µ1·µN). Note that the sign of the exchange contribution is the same as the sign of the direct one (11). The diagram shown in Fig.2a does not have analogous direct di agram because it has the hyperﬁne interaction attached to the line of the external electron. Nevertheless the calculation of this diagram is quite similar to the calculation described by eqs. (6) and (7). After substituti on ofδψefrom (A13) and performing summation over the polarizations inside the closed shell one ﬁnds the followin g expression for the diagram shown in Fig.2a δE(ex) a=−8πZα2(µ1·µN)/parenleftbigg/integraldisplay∞ 0e−2ρJ1(/radicalbig 8ρ)N1(/radicalbig 8ρ)ρdρ+1 8/integraldisplay∞ 0e−ρ(1−ρ/2)2J1(/radicalbig 8ρ)N1(/radicalbig 8ρ)ρdρ/parenrightbigg (15) = 0.156Zα2(µ1·µN). The total exchange magnetic correction is δE(ex) a+δE(ex) b+δE(ex) c=−0.271Zα2(µ1·µN). Comparing this with eq. (2) one ﬁnds the relative value of the exchange correctio n: δA(ex) A=δC(ex) C= 0.271Zα2. (16) IV. TOTAL BREIT CORRECTION. Zα2CORRECTION DUE TO ELECTRON-ELECTRON COULOMB INTERACTION Adding the direct (12) and the exchange (16) contributions o ne ﬁnds the total Breit correction δAB A=δCB C= 0.681Zα2. (17) In the calculation we have not used the explicit form of the Br eit interaction, but nevertheless this is the correction generated by the interaction which reads in the relativisti c form and in the Coulomb gauge as (see ref. [2]) HB=−1 2r/parenleftbigg α1·α2+(α1·r)(α2·r) r2/parenrightbigg . (18) Herer=r1−r2is distance between the electrons, and αiis theα-matrix of the corresponding electron. The Breit interaction correction to the hyperﬁne structure of Cs was p reviously calculated numerically in the works [4–7], 4but results of these calculations were somewhat conﬂicting . Our result (17) agrees with that of the most recent computation [7]. Note that eq. (17) gives the leading in Zpart of the Breit correction. There are other parts, say the correction to the energy of the external electron which dire ctly inﬂuence the hyperﬁne constant. However the other parts contain lower powers of Z. The correction (17) does not include all Zα2terms. To realize what is left let us look at the electron-ele ctron interaction Hamiltonian in ( v/c)2approximation [9] HB=α2/braceleftbigg −πδ(r)−1 2r/parenleftBig δαβ+rαrβ r2/parenrightBig p1αp2β+1 2r3(−(s1+2s2)·[r×p1] + (s2+2s1)[r×p2]) (19) +s1·s2 r3−3(s1·r)(s2·r) r5−8π 3s1·s2δ(r)/bracerightbigg . Herepi,sidenote the momentum and the spin of the electron. All the term s containing momenta vanish for s- electrons, the two last terms are already taken into account by the calculation performed above, however the ﬁrst term has not been considered yet. The matter is that in spite o f being a (v/c)2-correction it has a nonmagnetic origin. It comes from the ( v/c)-expansion of the electron-electron Coulomb interaction1 r(u† 1u1)(u† 2u2), whereuiis the Dirac spinor of the corresponding electron. This is why this term i s accounted automatically in the Dirac-Hartree-Fock calculations [10,4–7]. Nevertheless if one wants to separa te the total Zα2correction analytically then the ﬁrst term must be also considered explicitly. Since this term is spin i ndependent, it can contribute only via exchange diagrams. A straightforward calculation very similar to that performe d above gives the following result for the Coulomb correctio n. δECoulomb = 2Zα2(µ1·µN)/parenleftbigg/integraldisplay∞ 0e−2ρJ2 1(/radicalbig 8ρ)w1(ρ)dρ+1 8/integraldisplay∞ 0e−ρ(1−ρ/2)J2 1(/radicalbig 8ρ)w2(ρ)dρ/parenrightbigg −4πZα2(µ1·µN)/parenleftbigg/integraldisplay∞ 0e−2ρJ1(/radicalbig 8ρ)N1(/radicalbig 8ρ)ρdρ+1 8/integraldisplay∞ 0e−ρ(1−ρ/2)2J1(/radicalbig 8ρ)N1(/radicalbig 8ρ)ρdρ/parenrightbigg (20) = 0.558Zα2(µ1·µN). Comparing this with eq. (2) one ﬁnds the relative value of the Coulomb correction: δACoulomb A=δCCoulomb C=−0.558Zα2. (21) Combination of the magnetic (17) and of the Coulomb (21) corr ections give the total Zα2correction to the hyperﬁne constant due to the polarization of the closed electronic sh ells. δA A= 0.123Zα2. (22) V. BREIT INTERACTION CORRECTION TO THE PARITY NONCONSERVAT ION EFFECT The Breit interaction correction exists for both nuclear sp in independent (weak charge) and nuclear spin dependent (anapole moment) weak interactions. The correction to the s pin independent eﬀect is the most interesting one because of the high precision in both atomic theory [11] and experime nt [12,13]. Recent computations [6,7] show that the atomic Breit correction to the nuclear spin-independent pa rity nonconservation (PNC) eﬀect in Cs is about 0.6%. This is enough to inﬂuence interpretation of the experiment al data, see Refs. [13,6]. The Breit correction to the PNC eﬀect can be calculated analytically similar to the hyperﬁn e structure correction. However this calculation is out of the scope of the present work. In the present paper I just want a) to estimate this correction parametrically, b) to comment on the importance of the Dirac sea polarization. Let us ﬁrst estimate the correction. It is given by the diagrams similar to that shown in Fig.2a,b. The only diﬀeren ce is that the electron-nucleus magnetic interaction must be replaces by the electron-nucleus weak interaction [14] HW=G 2√ 2mQW[s·pδ(r) +δ(r)s·p], (23) whereGis the Fermi constant, and QWis the weak charge. The relative value of the Breit correctio n is 5δ∝angb∇acketleftHW∝angb∇acket∇ight/∝angb∇acketleftHW∝angb∇acket∇ight ∼α2/r3 ∆E∼α2Z3 Z2∼Zα2. (24) Hereα2/r3is the magnetic interaction between external and internal e lectrons, and ∆ Eis the excitation energy of the virtual state of the internal electron. For the estimation w e take K-electrons, therefore r∼1/Z, and ∆E∼Z2. Thus the Breit correction to the PNC eﬀect has exactly the same dep endence on atomic parameters as the Breit correction to the hyperﬁne structure. It is known that there is a large rela tivistic enhancement factor for PNC eﬀect [14], therefore one might think that the nonrelativistic expansion used in t he estimate (24) has very poor accuracy. However it is not the case, the matter is that the large relativistic facto r appears from the distances r∼nuclear size ≪aB/Z, therefore this factor is more or less the same for the externa l electron and for the K-electron. So it is canceled out in the ratio (24). By the way a similar argument explains a relat ively high accuracy of the nonrelativistic expansion for the hyperﬁne structure correction (17). I would like also to comment on the importance of the Dirac sea polarization. The eﬀect calculated in refs. [6,7] and estimated in eq. (24) is caused by the polarization of the clo sed atomic shells. This is analogous to the contribution (17) for the hyperﬁne structure. However one has to remember that for the hyperﬁne correction there is also the contribution of the Dirac sea polarization [1], α/2π−Zα2(5/2−ln 2)≈α/2π−1.81Zα2, which is bigger and has the opposite sign. Only account of both eﬀects together (ato mic shells + Dirac sea) has the physical meaning. The radiative correction to the nuclear weak charge is known in t he single loop approximation, see Ref. [15]. This includes α/πandα/πln(MZ/µ) terms (MZis the Z-boson mass and µis some infrared cutoﬀ). However Zα2radiative correction to the weak charge has not been considered yet and it is quite possible that this correction is larger than theα/πcontribution (at least this is so for the hyperﬁne constant) . To calculate Zα2radiative correction one needs to go to the two loop approximation, or to work in the single lo op approximation but with the Green’s functions in the external nuclear Coulomb ﬁeld. Thus: 1) account of the Breit correction calculated numerically in Refs. [6,7] and estimated in eq. (24) without account of the Dirac sea con tribution is not suﬃcient, 2) account of the Dirac sea polarization can inﬂuence agreement between theory and exp eriment. VI. CONCLUSION Correction to the hyperﬁne constant Aof an external s-electron in many-electron atom caused by th e polarization of inner atomic shells by the electron-electron Breit inter action is calculated analytically: δA/A = 0.68Zα2. This correction has the same origin as the Dirac sea polarization eﬀectδA/A =α/2π−Zα2(5/2−ln 2) calculated by Karplus and Klein long time ago [1]. It has been shown that the parametric estimate for the Breit c orrection to the parity nonconservation eﬀects is also Zα2. We stress that to take this correction into account one need s to consider both polarization of the inner atomic shells and polarization of the Dirac sea. ACKNOWLEDGMENTS I am grateful to V. A. Dzuba for very important stimulating di scussions. I am also grateful to V. A. Dzuba and W. R. Johnson for communicating me results of their calculat ions [7] prior to the publication. APPENDIX A: CORRECTIONS TO THE ELECTRONIC WAVE FUNCTIONS DU E THE HYPERFINE INTERACTION Correction δψto the single particle wave function is given by eq. (7). Usin g this formula one can easily prove that δψsatisﬁes the following equation (H0−En)δψn=−/summationdisplay m/negationslash=n\|m∝angb∇acket∇ight∝angb∇acketleftm\|V\|n∝angb∇acket∇ight=−V\|n∝angb∇acket∇ight+∝angb∇acketleftn\|V\|n∝angb∇acket∇ight\|n∝angb∇acket∇ight. (A1) Here V=−µ· HN=−8π 3(µ·µN)δ(r) (A2) is the perturbation and 6H0=p2 2m−Ze2 r→ −1 2∆−Z r(A3) is the Hamiltonian of the Coulomb problem. The screening of t he Coulomb ﬁeld is neglected because we consider only small distances, r∼1/Z. The eq. (A1) has an inﬁnite set of solutions. To ﬁnd the corre ct one we have to remember that there is the additional condition of orthogonality ∝angb∇acketleftδψn\|ψn∝angb∇acket∇ight= 0, (A4) which follows from eq. (7). Having the perturbation (A2) it i s convenient to represent δψnin the form δψn=4 3Z5/2(µ·µN)ψn(0)fn(ρ), (A5) whereρ=Zr. Substitution of (A5) into (A1) shows that the functions f(ρ) obey the following equations 1s:/parenleftbigg ∆ρ+2 ρ−1/parenrightbigg f1(ρ) =−4πδ(ρ) + 4e−ρ, (A6) 2s:/parenleftbigg ∆ρ+2 ρ−1 4/parenrightbigg f2(ρ) =−4πδ(ρ) +1 2e−ρ/2(1−ρ/2). To satisfy the boundary conditions at ρ= 0 and atρ=∞it is convenient to use another substitution f1=1 ρe−ρw1(ρ), (A7) f2=1 ρe−ρ/2w2(ρ), wherewi(0) = 1 and wi(ρ) grows at large ρnot faster than a polynomial. Straightforward solution of e qs. (A6) together with the orthogonality condition (A4) gives w1= 1−2ρ[ln(2ρ)−5/2 +c]−2ρ2, (A8) w2= 1−2ρ[(1−ρ/2)lnρ−3/4 +c]−(13/4−c)2ρ2+ρ3/4, wherec= 0.577215 is the Euler constant. Altogether eqs. (A5), (A7), an d (A8) give δψ1=4 3√πZ5/2(µ·µN)1 ρe−ρw1(ρ), (A9) δψ2=4 3√ 8πZ5/2(µ·µN)1 ρe−ρ/2w2(ρ). We also need to know the hyperﬁne correction to the wave funct ion of the external electron. Basically it is also given by eq. (A1), but there is a special point concerning nor malization. We use an artiﬁcial normalization condition ψ2 e(0) = 3/(8π), see eq. (A8). On the other hand the correct normalization i s/integraltext ψ2 e(r)d3r= 1 and hence ψe(0)∝E3/2, whereEis energy of the electron. There are two terms in the right han d side of eq. (A1), the ﬁrst term is proportional toE3/2and the second one is proportional to E9/2. For an external electron E→0 and hence the second term must be neglected. After that the equation (A1) is getting linear in ψand we can return to the normalization ψ2 e(0) = 3/(8π). Similar to (A5) it is convenient to use the substitution δψe(r) =/radicalbigg 2 3πZ(µ·µN)fe(ρ), (A10) wherefesatisﬁes the equation /parenleftbigg ∆ρ+2 ρ/parenrightbigg fe(ρ) =−4πδ(ρ). (A11) Note that due to the diﬀerent normalization of ψethe coeﬃcient in the right hand side of eq. (A10), including p ower ofZis diﬀerent from that in eq. (A5). Solution of eq. (A11) is 7fe=π/radicalbigg2 ρN1(/radicalbig 8ρ), (A12) whereN1(x) is the singular Bessel function. At small xthis function behaves as N1≈ −2/πx, therefore fe(ρ) has correct behavior at small ρ:fe(ρ)≈1/ρ. Together with (A10) this gives δψe=−/radicalbigg 4π 3Z(µ·µN)N1(√8ρ)√ρ. (A13) To make sure that this is the correct solution one has to prove validity of the orthogonality condition (A4). With eqs. (9) and (A13) one ﬁnds that the overlapping is of the form ∝angb∇acketleftδψn\|ψn∝angb∇acket∇ight ∝/integraldisplay∞ 0J1(/radicalbig 8ρ)N1(/radicalbig 8ρ)ρdρ∝/integraldisplay∞ 0J1(x)N1(x)x3dx. (A14) This integral is not well deﬁned at ∞. The origin for this is clear: we are working with zero energy state. To correct the situation, one has to introduce an exponential factor e−βx2and then to consider the limit β→0. ∝angb∇acketleftδψn\|ψn∝angb∇acket∇ight ∝lim β→0/integraldisplay∞ 0e−βx2J1(x)N1(x)x3dx∝lim β→01 β5e−1/2βW3/2,3/2(1/β)∝lim β→01 β13/2e−1/β= 0. (A15) For the evaluation of the integral we have used ref. [16]. For mula (A15) completes the prove of the orthogonality. [1] R. Karplus and A. Klein, Phys. Rev. 85, 972 (1952). [2]Quantum mechanics of one- and two- electron atoms , H.A.Bethe and E.E.Salpeter, Berlin, Springer, 1957. [3] O. P. Sushkov, V. V. Flambaum, and I. B. Khriplovich, Opt. Spectr. 44, 3 (1978), (Sov. Phys. Opt. Spectr. 44, 2 (1978)) [4] S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Re v. A43, 3407 (1991). [5] M. S. Safronova, W. R. Johnson, and A. Derevianko, Phys. R ev. A60, 4476 (1999). [6] A. Derevianko, Phys. Rev. Lett., 85, 1618 (2000). [7] V. A. Dzuba, W. R. Johnson, private communication. [8]Quantum mechanics: non-relativistic theory , L.D. Landau and E.M. Lifshitz. Oxford, New York; Pergamon P ress; 1965. [9]Relativistic quantum theory , V.B.Berestetskii, E.M.Lifshitz, and L.P.Pitaevskii, Ox ford, Pergamon Press, 1971. [10] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, J. Phys. B 17, 1953 (1984); V. A. Dzuba, V. V. Flambaum, A. Ya. Kraftmakher, and O. P. Sushkov, Phys. Lett. A 142, 373 (1989) [11] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, Phys. Lett . A141, 147 (1989); S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990). [12] C. S. Woods et al., Science 275, 1759 (1997). [13] S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2484 (1999). [14]Parity nonconservation in atomic phenomena , I. B. Khriplovich, Philadelphia, Gordon and Breach, 1991. [15] W. J. Marciano and A. Sirlin, Phys. Rev. D 27, 552 (1983). [16]Table of integrals, series, and products , I.S. Gradshteyn and I.M. Ryzhik; Boston, London; Academic Press, 1994. 8
arXiv:physics/0010029v1 [physics.atom-ph] 12 Oct 2000LETTER TO THE EDITOR A quasi classical approach to fully diﬀerential ionization cross sections Tiham´ er Geyer †and Jan M Rost ‡ †– Theoretical Quantum Dynamics – Fakult¨ at f¨ ur Physik, Uni versit¨ at Freiburg, Hermann–Herder–Str. 3, D–79104 Freiburg, Germany ‡Max–Planck–Institute for the Physics of Complex Systems, N ¨ othnitzer Str. 38, D–01187 Dresden, Germany Abstract. A classical approximation to time dependent quantum mechan ical scattering in the Møller formalism is presented. Numerical ly, our approach is similar to a standard Classical–Trajectory–Monte–Carlo c alculation. Conceptu- ally, however, our formulation allows one to release the res triction to stationary initial distributions. This is achieved by a classical forw ard–backward propagation technique. As a ﬁrst application and for comparison with exp eriment we present fully diﬀerential cross sections for electron impact ioniz ation of atomic hydrogen in the Erhardt geometry. PACS numbers: 34.80D, 03.65.Sq, 34.10+x Classical models and approximations are frequently used fo r atomic and molecular problems, despite their inherent quantum nature. One of the reasons is that our intuitive understanding is mainly based on classical terms and pictures by which we are surrounded in every day life. Another reason for not do ing (fully) quantum mechanical calculations is the complexity of a problem: ful ly diﬀerential cross sections in higher dimensional atomic systems, e.g., often require a numerical eﬀort still beyond present computing power. Only recently the quantum mechani cal Coulomb three– body scattering problem was solved numerically [1, 2]. On the other hand, a remarkably successful classical approa ch to collisional atomic problems has been developed over the years, the so called Cla ssical–Trajectory–Monte– Carlo method (CTMC). It was introduced as a purely classical model based on a “planetary atom” with a major axis of two meters (!) [3]. This model has produced reasonable results for total or energy diﬀerential ionizat ion cross sections on atomic hydrogen [4, 5] and for other few–body Coulomb collision pro cesses [6]. Attempts to reduce the limitation of this classical model ai med at changing the description of the initial state for the hydrogen atom fr om a microcanonical distribution to one, that is closer to the quantum density [7 , 8]. Another idea was to introduce additional ad hoc stabilisation potentials in order to be able to treat multi–electron targets [9]. However, all attempts took as a starting point not the quantum problem but the previously formulated classical mo del. Hence, the proposed amendments were accompanied by inconsistencies or the need of ”ﬁt parameters” determined from cross sections. In the end, one must say that it is up to now not possible to describe higher diﬀerential cross sections or t argets with more than one active electron consistently in a classical collision fram ework.Letter to the Editor 2 To achieve progress in this situation we decided to go one ste p backwards and start with a time dependent quantum mechanical scattering f ormalism. By following the approximations which lead to the classical description , i.e. the CTMC method, we can identify the source and nature of deviations between the classical and the quantum result which serve as a guide to improve the classical descri ption to a quasi–classical approximation. We divide the problem into three logically separate steps: ( 1) preparation of the initial state before the collision, (2) propagation in t ime, and (3), extraction of the cross section. For a consistent quasiclassical picture each of these steps has to be approximated in the same way. To keep the derivation transpa rent, we will concentrate on electron impact ionization of one active target electron in the following. Step (1): The initial wave function for the collision proble m is translated ﬁrst into a quantum phase space distribution by the Wigner transf ormation [10]. The resulting Wigner distribution is reduced to a classical dis tribution w(p, q) which can be propagated classically in phase space by taking the usual li mit ¯h→0. The diﬀerence to an a priori classical approach is the use and interpretati on of negative parts of the distribution: Viewed as the ¯ h→0 limit of a quantum problem they contribute to the observables in the same way as the positive parts since th ey do not need to be interpreted as weights for real paths of classical particle s. Yet, there arise additional problems when using this type of general initial phase space distributions in the usual classical framework: Most of them are not stationary under c lassical propagation, their Poisson bracket with the Hamilton function does not vanish, {H, w} /ne}ationslash= 0. Hence, the initial target distribution will look very diﬀerent at t he time the projectile has approached and the collision actually happens. Step (2): The formulation of the propagation is crucial sinc e it must resolve the problem of the non–stationary classical initial distri bution, as described above. Traditionally, the time dependent scattering is described by calculating the transition amplitude between initial and ﬁnal state through the S–matr ix, which is in turn related to the t–matrix describing directly the cross section, see, e.g., [11]. In a simpliﬁed version where the asymptotic initial and ﬁnal states are eig enstates of the asymptotic Hamiltonians H(i) 0andH(f) 0one normally writes for the transition amplitude Sfi= lim t→∞/an}bracketle{tf\|U(t)\|i/an}bracketri}ht (1) where U(t) = exp[ −iHt] denotes propagation with the full Hamiltonian. By a Wigner transform the quantum time evolution operator U(t) can be directly transformed with the help of the quantum Liouville operator Lq, which reduces to the classical Liouville operator Lcin the limit ¯ h→0 [12]. The latter describes the evolution of a phase space distribution w(r, p, t) according to the Poisson bracket ∂tw={H, w} ≡ − iLcw (2) in analogy to the quantum evolution of the density matrix ρgenerated by the commutator, ∂tρ=−i[H, ρ]. (3) Hence, we could directly use the translation of (1) to classi cal mechanics via the Liouville operator. In connection with the microcanonical initial state distribution this is indeed equivalent to the CTMC formulation [13]. However, using non–stationary initial state distributions is inconsistent with the reduc ed quantum description of (1)Letter to the Editor 3 which relies on the fact that the asymptotic states are eigen states of U0and therefore stationary. Instead we have to go back to the full scattering formulation Sfi=/an}bracketle{tf\|Ω† −Ω+\|i/an}bracketri}ht, (4) where Ω∓= lim t→±∞U†(t)U0(t) (5) are the Møller operators. The meaning of Ω +, e.g., is to propagate backwards with U0(t) using the asymptotic Hamiltonian Hi 0without the projectile–target interaction and then forward again under the full Hamiltonian with U(t). Again, with the help of the Liouville operator we can translate the Møller operator s to their classical analogue, thereby obtaining a prescription how to propagate a non–sta tionary initial phase space distributions wi(γ), where γ= (/vector p1, /vector q1, /vector p2, /vector q2) is a point in the 12-dimensional phase space: wf= lim t→+∞lim t′→−∞e−iLf cteiLc(t−t′)eiLi ct′wi≡ Kwi. (6) The diﬀerence to (1) are the explicit propagations under Lf candLi cin the initial and ﬁnal channel (which need not be the same). The mea ning of (6) becomes very transparent if we insert a discretized distribution, w hich is used in the actual calculations, wi(γ) =/summationtext nwnδ12(γ−γi n). The ﬁnal distribution reads wf(γ) =Kwi=/summationdisplay nwnδ12(γ−γf n) (7) where each phase space point γf nemerges from γi nthrough solving successively Hamilton’s equations, ﬁrst with Hi 0, then with H, and eventually with Hf 0. With this propagation scheme a non–stationary initial distribu tion will spread when being propagated backwards with the asymptotic Li c. However, it will be refocused under the following forward propagation with Lc. Hence, when the actual collision happens for t≈0 the original target distribution is restored, slightly po larized by the approaching projectile. Hence, there is no more need for the initial distribution to b e classically stationary. We are able to use any phase space distribution as a target in o ur quasi classical approach. This also includes unstable multi–electron targ ets, e.g., classical helium. Step (3): Before we come to the actual evaluation we have to fo rmulate the cross section such that it can make full use of the non-statio nary initial phase space distribution wi(/vector p1, /vector q1), where “1” refers to the target electron. Without modiﬁcat ion the total energy Eof the ﬁnal state forces by energy conservation for each clas sical trajectory only those parts of the initial phase space distr ibution to contribute to the cross section which have the same energy E. However, this would bring us essentially back to the microcanonical description. In order to make the entire non-stationary initial state distribution “visible” to the collision proc ess, we use the energy transfer ¯E1=E(f) 1−E(i) 1to the target electron rather than its energy E(f) 1itself as a diﬀerential measure. Of course, as long as the initial state is on the ener gy shell with well deﬁned energy E=E(i) 1+E(i) 2the new deﬁnition coincides with the usual expression for th e cross section, d5σ dΩ1dΩ2dE1/vextendsingle/vextendsingle/vextendsingle/vextendsingle E=d5σ dΩ1dΩ2d¯E1/vextendsingle/vextendsingle/vextendsingle/vextendsingle E, (8) where dΩiare the diﬀerentials for the solid angles of the two electron s, respectively.Letter to the Editor 4 To extract this cross section we have to evaluate the phase sp ace integral d5σ dΩ1dΩ2d¯E1=/integraldisplay dx2dy2d/vector p1d/vector q12/productdisplay i=1δ(Ω(f) i−Ωi)δ(¯E(f) 1−¯E1)wi,(9) where the integration is over the initial state variables, n amely the impact parameter areadx2dy2and the phase space of the (bound) target electron d/vector p1d/vector q1, with initial distribution wi(/vector p1, /vector q1, x2, y2). The propagated angles Ωf iof the electrons as well as the energy transfer ¯Ef 1have to coincide with the desired values Ω iand¯E1to contribute to the cross section (9) which is a generalization of the one der ived in [14], e.g., where the initial bound state was assumed to live on a torus, i.e., wi(/vector p1, /vector q1) =δ(/vectorI(/vector p1, /vector q1)−/vectorI0) with a well deﬁned multidimensional action /vectorI0. Finally, we have to respect the Pauli principle for the two id entical electrons. Formally, this can be done easily in the Wigner transform for the two ionized electrons in the ﬁnal state. In the limit ¯ h= 0 one is left with the usual classical symmetrization, i.e., an interchange of indices. To keep th e notation simple we have omitted symmetrization in the outlined derivation, howeve r, it is included in the actual computation which is carried out by applying standard CTMC t echniques to evaluate (9). 024Ei=250eV Eb=5eV ϑa=8°microcan. 024 -180 -90 0 90 180ϑbWigner Figure 1. Fully diﬀerential cross section in Ehrhardt geometry: Ein= 250eV, Eb= 5eV and θa= 80: comparison of measurements of Erhardt et al to quasiclassical calculations with (a), the microcanonic al distribution (standard CTMC), and (b) with the Wigner distribution. Error bars indi cate the statistical error of the Monte Carlo calculations. The theoretical data has been scaled by about 20% to reproduce the correct total ionization cross se ction at 250 eV. For the negative parts in the Wigner cross section see text. As a ﬁrst application we discuss fully diﬀerential ionizati on cross sections for atomic hydrogen comparing three data sets: a calculatio n with theLetter to the Editor 5 00.40.8microcan.Ei=54.4eV Eb=5eV ϑa=23° 00.40.8 -180 -90 0 90 180ϑbWigner Figure 2. Same as in Figure 1, but for Ein= 54.4eV, Eb= 5eV and θa= 230and measurements (circles) by R¨ oder et al[16]. standard microcanonical distribution (CTMC), one with the non–stationary Wigner distribution in our quasi classical framework and experime ntal data at impact energies of 250eV (ﬁg. 1) and 54.4eV (ﬁg. 2), respectively. For each impact energy about 108trajectories have been calculated. The cross section at 250 eV for the Wigner distribution still exhibits negative parts, indicating that this cross section is not yet fully converged. This is no t surprising if one takes into account that the fraction of phase space of the ﬁnal stat e is so small with the chosen bin sizes for energies and angles, that only between 1 00 and 300 events ﬁnally contribute to the shown cross sections. However, a consider able advantage of the present method is that a sampling of 108trajectories contains the complete scattering information, not just one speciﬁc diﬀerential cross sectio n. The ﬁgures show, that the microcanonical distribution, i.e . standard CTMC, is not able to reproduce the binary peak [17], whereas with the W igner distribution it is reproduced fairly well for 250eV and rather well for 54.4eV i mpact energy. Keeping in mind that in contrast to the microcanonical distribution the Wigner distribution has the correct probability densities in momentum andconﬁguration space, one can conclude, that at least for energies between 50 and 250 eV the diﬀerential cross sections ”image” the initial phase space distribution. The present approach is still a classical approximation and cannot reproduce quantum eﬀects. Theref ore, features in the cross section, for which coherence is crucial are represented pur ely or not at all. To summarize, we have shown, that a consequent classical app roximation to a quantum system can give much better results compared to thos e from an a priori classical model, though both approaches are realized numer ically in almost the same way. However, the main diﬀerence is conceptual: in the usual classical limit eachLetter to the Editor 6 individual trajectory represents that of an electron obeyi ng the classical equations of motion, whereas in our classical approximation only the ent ire phase space distribution is meaningful and individual trajectories are only discret ized points of the distribution evolving in time. Hence, there is no problem to deal with ”neg ative probabilities” in the initial distribution, since we regard them not as probab ilities but only as weights of the integration which, of course, may be negative. However, the use of non-stationary distributions like the W igner distribution as an initial state implies additional diﬃculties for a scatte ring description which we have overcome by using a forward-backward propagation sche me akin to the quantum Møller formalism and by a reformulation of the energy diﬀere ntial cross section in terms of the energy transfer during the ionization process. With these modiﬁcations all the tools of the standard CTMC technique can be applied st raightforwardly. Moreover, our approach can be in principle generalized to mu lti–electron targets since we generate our initial phase space distribution from a quantum wave function and we know how to deal with non–stationary initial distribu tions. Financial support by the Deutsche Forschungsgemeinschaft within the SFB 276 at the University Freiburg is gratefully acknowledged. References [1] I. Bray, J. Phys. B 33(1999) 581 [2] T. N. Rescigno etal., Science 286(1999) 2474 [3] R. Abrines and I. C. Percival, Proc. Phys. Soc. 88(1966) 861 [4] R. Abrines etal., Proc. Phys. Soc. 89(1966) 515 [5] R. E. Olson and A. Salop, Phys. Rev. A 16(1977) 531 [6] D. R. Schultz etal., J. Phys. B 25(1992) 4601 [7] D. J. W. Hardie and R. E. Olson, J. Phys. B 16(1983) 1983 [8] D. Eichenauer etal., J. Phys. B 14(1981) 3929 [9] D. Zajfman and D. Maor, Phys. Rev. Lett. 56(1986) 320 [10] E. Wigner, Phys. Rev. 40(1932) 749 [11] J. R. Taylor, Scattering theory , John Wiley & Sons, Inc., 1972 [12] E. J. Heller, J. Chem. Phys. 65(1976) 1289 [13] S. Keller etal., J. Phys. B 26(1993) L737 [14] J. M. Rost, Phys. Rep. 297(1998) 271 [15] H. Ehrhardt etal., Phys. Lett. 110A (1985) 92 [16] J. R¨ oder etal., Phys. Rev. A 53(1996) 225 [17] J. S. Briggs, Comments At. Mol. Phys. 23(1989) 155
arXiv:physics/0010030v1 [physics.gen-ph] 12 Oct 2000Number counts against K-corrections D.L. Khokhlov Sumy State University, R.-Korsakov St. 2, Sumy 40007, Ukraine E-mail: khokhlov@cafe.sumy.ua Abstract Having based on the analysis of Pietronero and collaborator s the galaxy number counts can be considered as an evidence of that the geometry o f the universe is euclidean and that the K-corrections are absent. In such a universe mea suring the cosmological redshift of the object and measuring the ﬂux from the object a re inconsistent. It is considered the model of the universe which describes the abo ve properties. Authors of [1],[2] claimed that galaxies have a fractal dist ribution with constant D≈2 up to the deepest scales probed until now 1000 h−1Mpc and may be even more. They showed that modiﬁcation of the euclidean geometry and the K-correc tions are not very relevant in the range of the present data. The use of K-corrections leads to the unstable behaviour of the number counts, with fractal dimension Dincreasing systematically to substantially larger values as a function of the depth of the volume limited sample. Quantitatively this behaviour can be explained as the eﬀect of K-corrections app lied to an underlying galaxy distribution with fractal dimension D≈2. The use of the FRW geometry instead of the euclidean geometry is equivalent to an eﬀective K-correcti on. So similar to the use of K- corrections the use of the FRW geometry leads to the unstable behaviour of the number counts as a function of depth. The number counts can be considered as an evidence of that the geometry of the universe is euclidean rather than FRW and that the K-corrections are s purious. Let us study the uni- verse with euclidean geometry. It should be noted that here e uclidean geometry is conceived as a real background of the universe not as an approximation o f the FRW geometry. In the euclidean geometry, the radial distance rand the angular diameter distance rθare the same and are given by r=rθ=c H0z 1 +z. (1) The luminosity distance is given by rL=r(1 +z) =rθ(1 +z) =c H0z. (2) From this it follows that the intrinsic luminosity of the obj ectLand the observed ﬂux Fare related as F∝L r2 L∝L r2(1 +z)2∝L r2 θ(1 +z)2. (3) In the case of FRW geometry, the intrinsic luminosity of the o bjectLand the observed ﬂuxFare related as F∝L r2 L∝L r2 θ(1 +z)4. (4) 1Two factors 1 + zare due to the increase of the scale factor. The third factor 1 +zis due to the decrease of the frequency of photon. The fourth factor 1 +zis due to that the time goes faster. The two of four factors 1 + zare absent in the case of euclidean geometry. This is natural to interprete such that the scale factor grows, th e frequency of photon is constant, and the ﬂow of time is constant. Since the frequency of photon is constant, K-corrections are absent. Thus in the universe with the euclidean geometry the K-corrections are absent. We arrive at the situation where there are two inconsistent t ypes of observations. The ﬁrst, when the scale factor do not grow with time a=const, and the clock goes faster with time ∆ t∝1 +zand correspondingly the frequency of photon decreases with time ω∝(1 +z)−1. The second, when the scale factor grows with time a∝1 +z, and the clock goes persistent with time ∆ t=const and correspondingly the frequency of photon is constant with time ω=const. Below we consider the model of the universe with such properties. In the FRW model [3], the universe is considered in the system accompanying to the matter of the universe. On the contrary, let us consider the u niverse in the laboratory system. It should be stressed that here we consider the model of the un iverse in the laboratory system not as an approximation of the FRW model but as a real one. Let t he universe be close. The total mass of the close universe in the laboratory system is equal to zero. Hence, when considering the close universe in the laboratory system, th e gravity (matter) of the universe do not deﬁne the evolution of the universe. So we consider the close universe in the laboratory system. L et the scale factor of the universe be equal to the size of the horizon at every moment of time a=ct. (5) The law (5) can be treated in two ways. The ﬁrst, the clock goes persistent with time ∆t=const, and the scale factor of the universe grows with time a∝t. The second, the scale factor is constant with time a=const, and the clock goes faster with time ∆ t∝t. Thus it is impossible to deﬁne simultaneously the growth of the sc ale factor and the acceleration of the ﬂow of time. The evolution of the universe can be expres sed either as a growth of the scale factor or as an acceleration of the ﬂow of time. Hence th ere are two inconsistent types of observations. Measuring the cosmological redshift of the object correspo nds to the second type. In this case the scale factor of the receiver and the scale factor of t he emitter are the same ar=ae. The clock of the receiver goes faster than the clock of the emi tter ∆ tr= ∆te(1+z), and the spectral line of the receiver is redshifted from the spectra l line of the emitter ωe=ωr(1+z). Measuring the photon ﬂux from the object through the photome tric band corresponds to the ﬁrst type. In this case the scale factors of the receive r and the scale factor of the emitter are related as ar=ae(1 +z). The clock of the receiver and the clock of the emitter go concurrently, and the frequency of photon for the receive r and for the emitter is the sameωr=ωe. Hence the observed ﬂux is a function of redshift F∝r−2 θ(1 +z)−2, and the K-corrections are absent. Thus describing the universe in the laboratory system we com e to the conclusion that the geometry of the universe is euclidean and that the K-corr ections are absent. This is in agreement with the observed galaxy number counts. It should be noted that there is no 2evidences in favour of the use of the system accompanying to t he matter of the universe for description of the universe. So the number counts can be cons idered as an evidence in favour of the use of the laboratory system for description of the uni verse. The observed behaviour of the number counts is D≈2. Description of the universe in the laboratory system leads to the fractal structure D= 2 [4]. Sketch how the fractal structure arises in this case. So we consider the close unive rse in the laboratory system. For the close universe the mass of the matter is equal to the energ y of selfgravity mc2=Gm2 a. (6) From this substituting the law (5) it follows that the scale o f mass grows with time m=c2 Ga=c3 Gt. (7) Let photon be emitted from the centre of the laboratory syste m. Then the motion of photon deﬁnes the distance R=ct. Since the scale of mass grows with time, it grows with the distance from the centre of the laboratory system m∝t∝R. The number of the particles of the mass Mwithin radius Ris given by N(< R) =ρR3 M∝R2. (8) Thus the growth of the scale of mass with time deﬁnes the fract al structure of the universe. References [1] L. Pietronero, M. Montuori, and F. Sylos Labini, in the Pr oc. of the Conference ”Critical Dialogues in Cosmology” N. Turok Ed. (1997) World Scientiﬁc [2] F. Sylos Labini, M. Montuori, L. Pietronero, Phys.Rep. 293(1998) 66 [3] Ya.B. Zeldovich and I.D. Novikov, Structure and evolution of the universe (Nauka, Moscow, 1975). [4] D.L. Khokhlov, astro-ph/9912277 3
arXiv:physics/0010031v1 [physics.atom-ph] 12 Oct 2000Pathways to double ionization of atoms in strong ﬁelds Krzysztof Sacha1,2and Bruno Eckhardt1 1Fachbereich Physik, Philipps Universit¨ at Marburg, D-350 32 Marburg, Germany 2Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiello´ nski, ul. Reymonta 4, PL-30-059 Krak´ ow, Poland (February 23, 2013) We discuss the ﬁnal stages of double ionization of atoms in a s trong linearly polarized laser ﬁeld within a classical model. We propose that all trajectories l eading to non-sequential double ionization pass close to a saddle in phase space which we identify and cha racterize. The saddle lies in a two degree of freedom subspace of symmetrically escaping elect rons. The distribution of longitudinal momenta of ions as calculated within the subspace shows the d ouble hump structure observed in experiments. Including a symmetric bending mode of the elec trons allows us to reproduce the transverse ion momenta. We discuss also a path to sequential ionization and show that it does not lead to the observed momentum distributions. I. INTRODUCTION Present day lasers are powerful enough to ionize several electrons from an atom. The electrons can be removed one by one in a sequential process or all at once in a non- sequential process. Independent electron models give ion- ization rates that are much smaller than experimentally observed [1], indicating that interactions between elec- trons are important. A series of most recent experiments has added the observation that also the ﬁnal state of the electrons is dominated by the interactions: the to- tal momentum of the electrons is aligned along the ﬁeld axis [2–5] and the joint distribution of parallel momenta for the two electrons, in the double ionization experi- ment [5], has pronounced maxima along the diagonal, showing that the electrons typically come with the same momenta. These observations have been reproduced in numerical simulations with varying approximations and simplifying assumptions [6,7]. But given the complexity of the analysis that is required the essential elements are diﬃcult to identify. As a step towards a better under- standing we discuss here the pathways to double ioniza- tion within a classical model for electrons in a combined Coulomb and external ﬁeld. Our aim is not to describe the full ionization process all the way from the ground state to the ﬁnal, multiply ionized state. According to the currently accepted mod- els [8–13,6,7], the whole process of multiphoton multiple ionization can naturally be divided into two steps: in the ﬁrst step a compound state of highly excited electrons close to the nucleus is formed and in the second step sev- eral electrons can escape from this compound state to produce the multiply ionized ﬁnal state. We focus on the last step, the escape of two or more electrons from the highly excited compound state close to the nucleus. The formation of the intermediate compound state is suggested by the rescattering model [8,9] for strong ﬁeld multiple ionization. According to this model the en- hanced cross section for multiple ionization is due to a rescattering of one electron that is temporarily ionizedand accelerated by the ﬁeld before it returns to the nu- cleus when the ﬁeld reverses. During the collision energy is transfered to other electrons, but all of this happens close to the core, where the dynamics of the electrons is fast and the interactions are strong and non-integrable. As a consequence, details of the initial preparation pro- cess are lost. Moreover, the decay of this state is also quick. The compound state is thus a short lived, highly unstable complex that separates the ﬁrst half of the ex- citation process, whose main contribution is the build up of energy in the complex, from the second half, where the decay mode is determined. The compound state has several decay paths: single ionization when only one electron escapes to inﬁnity, dou- ble or multiple ionization with two or more electrons es- caping to inﬁnity, and the case of a single electron that es- capes from the neighborhood of the nucleus but is rescat- tered to the next ﬁeld reversal: in the latter case the whole process repeats itself, another compound state is formed and the decay path has to be selected anew. We discuss here the further evolution after the forma- tion of the compound state. To be speciﬁc, we will focus on the escape of two electrons in the following, but the arguments can easily be extended to the removal of more than two electrons. The main aim is to identify the chan- nels that lead to double escape and to study their signa- tures in the distribution of electron and ion momenta. Our analysis is purely classical. Given the highly excited complex from which we start and the multiphoton na- ture of the process this seems a reasonable point of entry to the ﬁnal stage of the ionization process. Our analysis is very similar to Wanniers approach [14–16,18] to dou- ble ionization by electron impact. The main diﬀerence is that in the present case one has to take into account also the external ﬁeld. Brief summaries of some aspects of this model have been presented in [17]. The pathways to double ionization are discussed in sec- tion 2. The dynamics in the C2vandCvsubspaces in- cluding the eﬀective potential and sample trajectories is discussed in section 3. A key element of our argument is 1the identiﬁcation of a saddle in phase space near which trajectories leading to non-sequential double ionization have to pass; its properties and stability are discussed in section 4. The distributions of electron momenta within theC2vandCvsubspaces are analyzed in section 5. The dynamics outside these symmetry spaces and sequential ionization process are discussed in section 6. We conclude with a summary of the model in section 7. II. PATHWAYS TO DOUBLE IONIZATION As described in the introduction we assume an ini- tial state of two highly excited electrons near the nucleus which then decays to either single ionization or double ionization. During this process the linearly polarized laser ﬁeld is always on. Therefore, the Hamiltonian con- sists of three parts, H=T+Vi+V12 (1) the kinetic energy of the electrons, T=p2 1 2+p2 2 2(2) the potential energies associated with the interaction with the nucleus and the ﬁeld (polarized along z-axis), Vi=−2 \|r1\|−2 \|r1\|+F(t)z1+F(t)z2 (3) and the repulsion between the electrons, V12=1 \|r2−r1\|. (4) The electric ﬁeld strength F(t) has an oscillatory com- ponent times the envelope from the pulse; the discussion applies to general F(t), and speciﬁc choices have to be made for the numerical simulations only. Once the electrons leave the atom the repulsion pushes them apart and becomes weaker the larger the separa- tion. Thus in the asymptotic state after ionization and after the pulse is turned oﬀ, repulsion is minimized. In order to identify the eﬀects of the electron repulsion on the full process it is instructive to consider the double ionization events without repulsion ﬁrst. This will be presented in the next section. The pathways with elec- tron repulsion included will be presented thereafter. A. Without electron repulsion The Hamiltonian for two electrons without electron re- pulsion splits into two independent Hamiltonians for each electron. In view of the fast motion of the electrons close to the nucleus we will frequently use an adiabatic as- sumption and discuss motion of the electrons in a ﬁeldwith ﬁxed amplitude F. Note, however, that all simula- tions use the full time-dependent ﬁeld and do not make use of this adiabatic assumption. With this assumption each electron moves in a constant electric ﬁeld, one of the few non-trivial integrable problems. The initial energy and the other constants of motion of each electron are ﬁxed by the initial conditions (in the compound state) and do not change. Double ionization can thus occur only if both electrons individually have enough energy to ionize (and have the other constants of motion so as to allow ionization). The threshold for ionization is set by the ﬁeld strength: if the ﬁeld is non-zero a Stark saddle forms and the total energy has to be above the Stark saddle. For a constant ﬁeld strength Fthe saddle lies at \|zF\|=/radicalbig 2/\|F\|and has a potential energy (single electron) VF=−2/radicalbig 2\|F\|. Therefore, double ionization is excluded if the total energy for both electrons is less than 2 VF. For E= 2VFthe only path leading to double ionization has both electrons with the same energy. For E >2VFdouble ionization becomes possible even with slight asymmetries in the energy distribution. In the full system of two electrons without repulsion but in a time-dependent ﬁeld integrability is lost but sep- arability is still preserved. As discussed in the introduc- tion the initial state is a compound state with negative energy for each electron. If that energy is above the Stark saddle and if the motion is directed towards it the elec- trons can cross and run away from the nucleus. Once they cross the saddle they have to gain energy to escape from the Coulomb attraction when the ﬁeld is turned oﬀ. This happens essentially by running down the potential energy slope on the other side of the saddle while the ﬁeld is still on. This mechanism of energy gain is the same as in the interacting electron case. B. With electron repulsion With the electron repulsion included the common Stark saddle at zFis no longer accessible since the elec- trons cannot sit on top of each other. The best that can be achieved is a symmetric arrangement of both elec- trons in the same distance from the nucleus and sym- metric with respect to the ﬁeld axis. If the distances are not the same, electron repulsion will push the elec- tron that is further out away from the nucleus and thus help towards ionization, but the one further in has to face not only Coulomb attraction but also the repulsion from the one further out. Thus repulsion acts so as to amplify diﬀerences in energy in this conﬁguration. The conﬁguration that is singled out is a symmetric one, with both electrons moving at the same distance from the nu- cleus. Deviations from this conﬁguration will be ampli- ﬁed suﬃciently so that non-sequential double ionization is suppressed and only single ionization takes place. A remaining electron can, however, be still ionized if its en- ergy is higher than the saddle energy for a single electron 2atom and, in the adiabatic picture, the other constants of motions allow to do so. In such sequential ionization cor- relations between escaping electrons are strongly weak- ened, that is the ﬁnal momenta of the electrons along the polarization axis can be either parallel or anti-parallel. III. DYNAMICS IN THE C2VAND CV SUBSPACES A two electron atom illuminated by a linearly polar- ized electromagnetic wave possesses some symmetry sub- spaces. The simplest C2vsymmetric conﬁguration corre- sponds to both electrons moving in a plane which con- tains the ﬁeld polarization axis and with positions and momenta symmetric with respect to this axis. The elec- trons put in such a conﬁguration never leave it because there is no force which can pull them out of the subspace. The symmetry subspace can be enlarged. That is, with additional bending motion of the electrons with respect to the ﬁeld axis the symmetry subspace is Cv. The symmetric conﬁgurations become more apparent in suitably chosen coordinates. We apply the canonical transformation R= (ρ1+ρ2)/2, p R=pρ1+pρ2 r= (ρ1−ρ2)/2, p r=pρ1−pρ2 Z= (z1+z2)/2, p Z=pz1+pz2 z= (z1−z2)/2, p z=pz1−pz2, ϕ=ϕ1+ϕ2, L= (pϕ1+pϕ2)/2 φ=ϕ1−ϕ2, pφ= (pϕ1−pϕ2)/2 (5) where ( ρi, zi, ϕi) are cylindrical coordinates of the elec- trons, labeled i= 1 and 2. For double ionization in lin- early polarized laser ﬁeld the total angular momentum projection on the polarization axis is conserved. The ex- periments begin with atoms in the ground state, thus, for the ﬁeld directed along the z-axis we may choose L= 0. Then the Hamiltonian of the system reads H=p2 R+p2 r+p2 Z+p2 z 4+p2 φ 2(R+r)2 +p2 φ 2(R−r)2+V(R, r, Z, z, φ, t ) (6) with the potential energy V=−2/radicalbig (R+r)2+ (Z+z)2−2/radicalbig (R−r)2+ (Z−z)2 +1/radicalbig 2R2−2(R2−r2)cosφ+ 2r2+ 4z2 +2ZF(t), (7) where the ﬁeld is given by F(t) =Ff(t)cos(ωt+θ) with F,ωandθthe peak amplitude, frequency and initial phase of the ﬁeld, respectively, and with f(t) = sin2(πt/T d) (8)the pulse envelope of duration Td. Setting r= 0,pr= 0,z= 0 and pz= 0 we deﬁne the Cvsymmetry subspace. The Hamiltonian is then reduced to H=p2 R+p2 Z 4+p2 φ R2+V(R, Z, φ ) (9) where V=−4√ R2+Z2+1 R/radicalbig 2(1−cosφ)+ 2ZF(t).(10) The potential (7) is symmetric in r(forz= 0) and in z (forr= 0), so that the derivatives with respect to randz vanish: once the electrons are in the symmetry subspace r=z= 0 and pr=pz= 0, they cannot leave it. (a) ΖR 0246 -5 0 5 (b) ΖR 0246 -5 0 5 FIG. 1. Sections through equipotential surfaces of the adi- abatic potential Eq. (10) for ﬁxed time tand for φ=π(a) andφ=π/4 (b). Panel (a) corresponds also to the poten- tial Eq. (12) in the C2vsymmetric subspace; the saddle moves along the dashed line when the electric ﬁeld points in the pos - itiveZ-direction and along a second obtained by reﬂection on Z= 0 during the other half of the ﬁeld cycle. 3The further constraint φ=πandpφ= 0 leads to the C2vsymmetry subspace H=p2 R+p2 Z 4+V(R, Z) (11) with potential V=−4√ R2+Z2+1 2R+ 2ZF(t). (12) Let us begin with an analysis of the motion in the C2v subspace. The electrons move in a plane and their posi- tions ( ρi=R,zi=Z) in that plane and their momenta (pρi=pR/2,pzi=pZ/2) are the same. The adiabatic potential (12) for ﬁxed time correspond- ing to the maximal ﬁeld amplitude F= 0.137 a.u., i.e. an intensity of 6 .6·1014W/cm2, is shown in Fig. 1a. The saddle is located along the line ZS=rScosθSand RS=rSsinθSwithθS=π/6 or 5 π/6 and at a distance r2 S=√ 3/\|F(t)\|. (13) The energy of the saddle is VS=−6/radicalBig \|F(t)\|/√ 3. (14) For the above mentioned ﬁeld the saddle has an energy ofVS=−1.69 a.u.. If we switch oﬀ the repulsion between the electrons the saddle will move onto the Z-axis, i.e. both electrons are allowed to escape symmetrically on top of each other. Slow switching on the repulsion results in a change of the positions of the saddles experienced by each elec- tron. The repulsion separates the saddles which ﬁnally are situated at some distance from the Z-axis. A typical trajectory within the symmetric conﬁgura- tion for ω= 0.057 a.u. (800 nm) is shown in Fig. 2. During the ramping of the ﬁeld the electronic motion is little inﬂuenced by the electric ﬁeld, but during the third half cycle of the ﬁeld the saddle is close enough to the electron orbits and ionization takes place. Once on the other side of the saddle, the electrons rapidly gain en- ergy. The saddle thus provides a kind of transition state [19,20] for the correlated double ionization process: once the electrons cross it, they are accelerated by the ﬁeld and pulled further away, making a return rather unlikely. Moreover, the electrons can acquire the missing energy so that both can escape to inﬁnity even when the ﬁeld van- ishes. Note that before the double ionization occurs the eﬀect of the ﬁeld on the electrons is minimal, supporting the adiabatic assumption. In the experiments [2–5] ion momenta both parallel and transverse to the ﬁeld are measured. For ω= 0.057 a.u. momentum transfer by photons is negligible, so the ion momentum reﬂects the sum of the momenta of the emitted electrons, p1+p2=−pion. Symmetric mo- tion in the C2vsubspace takes place in a plane and conse- quently the total transverse momentum of the electronsis zero. In the Cvsubspace the electrons are permitted to leave the plane, and this is a minimal extension to allow ions to have non-vanishing ﬁnal transverse momenta. 0.0 0.5 1.0 1.5 2.0 t/(2π/ω)−5.0−2.50.02.5E (a.u.)0.05.010.015.020.0r (a.u.)(a) (b) FIG. 2. A typical trajectory in the C2vsymmetric sub- space with E=−1.3 a.u., F= 0.137 a.u. and the pulse duration Td= 4×2π/ω. Panel (a) distance of the electrons to the nucleus. The dashed line indicates the distance of the saddle. Note that before the double ionization occurs the ef - fect of the ﬁeld on the electrons is minimal, supporting the adiabatic assumption. Panel (b) energy of the electrons. No te that the initial state has negative total energy and cannot l ead to double ionization. The energy increases once the electro ns have escaped from the nucleus far enough so that acceleratio n by the electric ﬁeld dominates. The sections through equipotential surfaces of Eq. (10) forφ=πandφ=π/4 are shown in Fig. 1. Increasing or decreasing φfromπresults in greater repulsion energy between the electrons. So the electrons can escape, with the coordinate φ/negationslash=π, only if the energy is greater than the saddle energy, Eq. (14). We discuss here double ionization of atoms but the analysis of a symmetric escape can be easily extended to a multiple ionization process [17]. Especially for Nelec- trons the symmetric subspace corresponding to Nparti- cles symmetrically distributed in the plane perpendicular to the polarization axis is CNv. The analysis of the mul- tiple ionization will be published elsewhere. IV. THE SADDLE Within the adiabatic assumption and for a ﬁxed exter- nal ﬁeld strength non-sequential double ionization hinges on the crossing of the saddle in the C2vsubspace. For the dynamics within the subspace this is obvious from the potential. For the motion in the full six-dimensional con- ﬁguration space this is less clear. One possibility to gain insight locally near the saddle is a harmonic analysis and a determination of the frequencies of small deviations. Within the C2vsymmetric subspace there is one hyper- bolic mode corresponding to motion across the saddle (the ’reaction coordinate’, in chemical physics parlance 4[19,20]) and a stable one to perpendicular variations. In the full space we expect at least one additional unstable one, corresponding to the ampliﬁcation of energy diﬀer- ences mentioned before in section (IIB). The analysis in this section is for ﬁxed ﬁeld strength, for electron dynam- ics in a constant external ﬁeld, justiﬁed by the adiabatic reasoning. The second order variations of the potential (7) around the saddle point results in H≈p2 R+p2 Z 4+9 2r3 S(R−RS)2 −3√ 3 r3 S(R−RS)(Z−ZS)−5 2r3 S(Z−ZS)2 +p2 r+p2 z 4+1 2r3 Sr2−3√ 3 r3 Srz−9 2r3 Sz2 +p2 φ R2 S+1 4RS(φ−π)2+VS. (15) Theφ-degree of freedom corresponds to a bending mo- tion of the electrons against each other and is stable on account of the repulsive nature of the Coulomb force. This degree of freedom is also the one that comes in by going from the C2vsubspace to the Cvsubspace. Diagonalization in the ( R−RS, Z−ZS) space reveals one stable and one unstable mode. The latter corre- sponds to the reaction coordinate and its Lyapunov expo- nent is µ≈2.43F3/4. Similar analysis in the ( r, z) space yields another stable and unstable mode with Lyapunov exponent ν≈3.14F3/4. The direction of the unstable mode is ( wr, wz)≈(0.43w, w) and it corresponds to the situation when one electron escapes and the other one is turned back to the nucleus. That is, with positive and in- creasing wthe ﬁrst electron moves away from the nucleus, i.e.ρ1=RS+wrandz1=ZS+wzgrow [see (5)], while the other one returns to the nucleus, i.e. ρ1=RS−wr andz1=ZS−wzdecrease. All in all there are three stable modes and two un- stable ones. Any energy contained in the stable modes is preserved and cannot be transferred to kinetic energy along the reaction coordinates. Whether single or dou- ble ionization occurs is thus determined solely by the energy and initial conditions in the two hyperbolic sub- spaces. For energy equal to the saddle energy only a trajectory within the C2vsymmetry subspace leads to non-sequential escape – any deviation from the subspace are growing faster than the escape along the reaction co- ordinate since ν > µ . For energy higher than the saddle some deviations from the symmetry subspace are allowed. In particular, following Wanniers lead and Rosts gener- alization [14–16,18] it is possible to estimate the critica l behavior for the double ionization cross section at thresh- old. It is algebraic with the exponent given by the ratio of the positive Lyapunov exponents of the unstable modes. A detailed discussion of this is outside the main line of our arguments here and will be given elsewhere. It is instructive to actually calculate numbers for the Lyapunov exponents in the two directions. For laserswith the maximal ﬁeld strength of F= 0.137 a.u. we ﬁnd 1/µ= 1.8 a.u. and 1 /ν= 1.4 a.u.. Compared to the period of the laser, 2 π/ω= 110 .2 a.u. this is rather fast, indicating that the crossing of the saddle and the separa- tion away from the double ionization manifold take place rather quickly. This justiﬁes also our adiabatic analysis in this section. V. FINAL STATE MOMENTA DISTRIBUTION WITHIN THE SYMMETRIC SUBSPACES So far we have discussed the phase space features in an adiabatic approximation for ﬁxed ﬁeld strengths. Now we will use this to draw conclusions about the experi- mentally observed signatures, speciﬁcally about the dis- tributions of ion momenta in the ﬁnal state. This can be calculated within the Cvsubspace by averaging over all initial conditions and all phases of the ﬁeld. A. Parameters of the model The rescattering of an electron leads to a highly excited complex of total energy Ewhich is one of the parame- ters of our model. The maximal energy a rescattering electron can bring in has been estimated to be about 3.17Up[8,9], where Upis the ponderomotive energy of an electron. For the weakest ﬁeld used in the experiment on double ionization of He atoms [2] this maximal en- ergy barely corresponds to the ionization energy of the He1+ions. We therefore assume in the following that the highly excited complex has a negative initial energy, E <0. The absence of detailed knowledge of the structure of the initial compound nucleus suggests to average over the initial conﬁgurations. However, even for a negative en- ergy and ﬁxed time it is diﬃcult to deﬁne a microcanon- ical distribution of initial conditions for the Hamiltonia n (9) since, for non-zero external ﬁeld, the system is open. Therefore, we choose for the calculations initial condi- tions from the energy shell that also lie in the hypersur- faceZ= 0. The results are not sensitive to a particular choice of the hypersurface but the one for Z= 0 has the advantage that the dipole moment along the ﬁled is zero and the choice of the initial conditions does not depend on the initial ﬁeld phase. The second parameter, in addition to the energy, is the timet0during the pulse Eq. (8) when the highly excited complex is formed. The rescattering event is not possible at the beginning of the pulse, so one has to start simula- tions somewhere in the middle of the pulse. In Fig. 3 and 4 ﬁnal distributions of ion momenta for the initial energy E=−0.1 a.u., ﬁeld strength F= 0.137 a.u. and dif- ferent initial time t0are shown. The distributions of the transverse momenta are almost the same but the ones for 5the parallel momenta diﬀer. The latter reveal a double hump structure with widths sensitive to the initial time. −6 −3 0 3 6 ppar (a.u.)0.00.10.2probability density0.00.10.20.3 −4 −2 0 2 4 ptrans (a.u.)0.00.50.00.51.0 (a) (b) (c) (d) FIG. 3. Final distributions of ion momenta parallel, ppar and transverse, ptransto the ﬁeld polarization axis for the ini- tial energy E=−0.1 a.u., peak ﬁeld amplitude F= 0.137 a.u. and pulse duration Td= 20 ×2π/ω. Panels (a)-(b) cor- respond to the initial time t0= 0.25Tdin the pulse dura- tion with the envelop f(t) = sin2(πt/T d) while panels (c)-(d) tot0= 0.75Td. Dashed lines are related to the estimates ±2Ff(t0)/ω=±2.4 a.u.. Note that the distributions are es- sentially the same independently if one chooses t0before or after the peak ﬁeld value provided f(t0) is the same. The results are based on integrations of about 8 ·104trajectories. −6 −3 0 3 6 ppar (a.u.)0.00.1probability density0.00.10.2 −4 −2 0 2 4 ptrans (a.u.)0.00.50.00.51.0 (a) (b) (c) (d) FIG. 4. The same as in Fig. 3 but for t0= 0.4Td [panels (a)-(b)] and t0= 0.6Td[panels (c)-(d)]. The width of the parallel momentum distribution can be estimated as ±2Ff(t0)/ω=±4.3 a.u.. The maximum energy that can be acquired by a free electron in the ﬁeld is 2 Up. So, for parallel emission of two electrons, the maximal parallel ion momentum can be estimated as ppar= 2/radicalbig 4Up= 2Ff(t)/ω. (16)If we substitute in Eq. (16) t=t0we ﬁnd values which correspond very well to the widths of the distributions in Fig. 3 and 4. The ﬁgures indicate also that the widths are the same independently if t0is chosen before or after the peak of the pulse provided f(t0) is the same. This implies that the dominant ionizations take place in the ﬁrst ﬁeld cycle after the complex is formed. Fitting the width of the calculated distribution to the experimental results allows one to estimate the moment in the pulse when majority of doubly ionized ions are created. −6 −3 0 3 6 ppar (a.u.)0.00.1probability density0.00.10.00.10.2 −4 −2 0 2 4 ptrans (a.u.)0.01.00.00.40.00.61.2 (a) (b) (c) (d) (e) (f) FIG. 5. The same as in Fig. 3-4 but for ﬁxed t0= 0.33Td and diﬀerent initial energy: E=−0.05 a.u. [panels (a)-(b)], E=−0.5 a.u. [panels (c)-(d)] and E=−1.5 a.u. [panels (e)-(f)]. Now we ﬁx t0and change the initial energy E; the results are shown in Fig. 5. For the lowest energy, E=−1.5 a.u., the transverse momentum distribution is narrower than for higher energies. E=−1.5 a.u. is actually very close to the minimal saddle energy for the applied ﬁeld, VS=−1.69 a.u.. Thus the eﬀect is natural as, close to the saddle energy, only trajectories near the C2vsubspace can cross the saddle and those with φ/negationslash=π bounce back from the potential barrier and do not ionize. While the width of the distributions of the parallel momenta do not change signiﬁcantly their shapes do, es- pecially for the initial energy close to the saddle. The electrons that cross the saddle when the energy is near VSare slow and the combined interactions of the external and Coulomb ﬁelds shapes the distributions. This is dif- ferent from the high energy case: then shortly after the electrons cross the saddle the interaction with the elec- tric ﬁeld is stronger than the attraction to the nucleus and the distribution is mostly shaped by the laser ﬁeld. The initial energy of the complex is the higher the higher intensity of the laser is and the larger the energy a rescattered electron can bring in [8,9]. From the depen- 6dence of the distributions on the initial energy we may conclude that in the experiment the shape of the distri- bution of the parallel ion momenta should change char- acter when the laser intensity increases. For the intensity at the threshold for non-sequential double ionization the distribution with single maximum around zero momen- tum is expected; for higher intensities the double hump structure should turn up. −6 −3 0 3 6 ppar (a.u.)0.00.10.2probability density0.00.10.20.3 −4 −2 0 2 4 ptrans (a.u.)0.00.50.00.51.0 (a) (b) (c) (d) FIG. 6. Panels (a)-(b): ﬁnal ion momentum distributions measured in the experiment of double ionization of He atoms in the focus of 800 nm, 220 fs (i.e. about 80 ×2π/ω) laser pulses at peak intensity of 2 .9×1014W/cm2(i.e. for the ﬁeld strength F= 0.091 a.u.) from [2]. Panels (c)-(d): the cor- responding distributions calculated in the Cvsymmetry sub- space for the initial energy E=−0.6 a.u. and t0= 0.33Td where Td/2 = 10 ×2π/ω, see Eq. 8. −6 −3 0 3 6 ppar (a.u.)0.00.1probability density0.00.10.2 −4 −2 0 2 4 ptrans (a.u.)0.00.40.00.40.8 (a) (b) (c) (d) FIG. 7. Panels (a)-(b): the same as in the corresponding panels in Fig. 6 but for the peak intensity of 6 .6×1014W/cm2 (i.e. for the ﬁeld strength F= 0.137 a.u.). Panels (c)-(d): the same as in the corresponding panels in Fig. 6 but for the initi al energy E=−0.4 a.u.. All numerical results have been obtained for initial con- ditions taken from the Cvsymmetry subspace. Our re-sults for the C2vsubspace for parallel momentum dis- tributions are essentially the same. The transverse mo- menta of ions for the C2vsubspace are, however, zero because of the symmetry assumption. After this discussion of the two parameters (initial en- ergy and starting time of the integration) we can turn to comparisons with experimental observations. B. Comparison with experimental results Weber et al. [2] carried out double ionization exper- iments with He atoms and measured the distributions of ion momenta. They applied infrared (800 nm) laser pulses of the duration 220 fs (measured on the half peak value) and with the peak intensities in the range (2.9−6.6)×1014W/cm2. In Fig. 6 and 7 we show the experimental distributions and compare with those cal- culated in the Cvsubspace. The agreement is remarkable except for the parallel momentum distribution in Fig. 7 where the calculated distribution possesses a much more pronounced minimum than in the experiment and the positions of the peaks do not exactly coincide with the experimental values. −10 −5 0 5 10 ppar (a.u.)0.000.09probability density0.000.070.14 −5.0 −2.5 0.0 2.5 5.0 ptrans (a.u.)0.00.40.00.20.4(a) (b) (c) (d) FIG. 8. Panels (a)-(b): ﬁnal ion momentum distributions measured in the experiment of double ionization of Ne atoms in the focus of 795 nm, 30 fs (i.e. about 11 ×2π/ω) laser pulses at peak intensity of 13 ×1014W/cm2(i.e. for the ﬁeld strength F= 0.192 a.u.) from [4]. Panels (c)-(d): the corresponding distributions calculated in the Cvsymmetry subspace for the initial energy E=−0.3 a.u. and t0= 0.4Td where Td/2 = 11 ×2π/ω, see Eq. 8. There are a few possible sources for these discrepancies. First, the pulse duration in the experiment was about 8 times longer than in the simulations. The slow ramp- ing of the ﬁeld in the experiment implies that the initial timet0of the ionization is less well deﬁned, i.e. there are contributions from some range of t0. There are also con- tributions from diﬀerent initial energies. Secondly, real ionizing trajectories do not live exactly in the symmetry 7subspace but close to it, leading to asymmetries and ad- ditional diﬀerences in the ﬁnal momenta. And there are also possible contributions from sequential double ioniza - tion events (see below). Moshammer et al. [4] performed experiments with Ne atoms for much shorter pulses, i.e. 30 fs and for radiation with similar wave length (795 nm) as the previous group. The comparison of our calculations with this experiment are presented in Fig. 8. Interactions of the two excited electrons with the other electrons are neglected in our model and the energy values used in the calculations are measured with respect to the threshold for the two elec- tron continuum (i.e. about 2.3 a.u.). The agreement is even better than for the case of He atoms. This is pre- sumably due to the much shorter pulse duration and the fasted ramping of the ﬁeld, so that the time t0when the majority of the ionization events happen is much better deﬁned. −40 −20 0 20 40z (a.u.)−20−1001020−10 0 10 20−4−2024x (a.u.)−50 0 50 100 150−100−50050100 (a) (b) (c) FIG. 9. Trajectories of electrons outside the symmetric subspace for E=−0.58 a.u. and F= 0.137 a.u.. Initial po- sitions are close to the saddle and marked by heavy dots; the electrons are distinguished by dotted and continuous track s. Panel (a) shows a symmetric escape of the electrons. Panel (b) shows a case where outside the symmetry subspace one electron escapes and the other falls back to the ion. Panel (c ) shows an example of sequential ionization of both electrons in opposite directions. VI. SEQUENTIAL DOUBLE IONIZATION Already from the experiment it is clear that double ionization is a rare process, e.g. outnumbered by singleionization by about 104: 1 for He atoms and ﬁeld inten- sity 2.9·1014W/cm2[2]. Hence arbitrarily chosen ini- tial conditions in the full space will typically not lead to double ionization and numerical simulations of the whole process are rather unattainable. We have discussed the non-sequential double escape of the electrons considering trajectories within the sym- metry subspace. Motion in the symmetry subspace is unstable, that is deviations from the subspace will be ampliﬁed leading to single rather than double ionization. We can illustrate this with trajectories started slightly outside the symmetry plane (Fig. 9). Fig 9a shows ini- tial conditions on the saddle and symmetrically escaping electrons. For some deviation from symmetry, one elec- tron escapes, the other remains trapped to the nucleus (Fig. 9b). It is possible, however, that the second electron returns to the nucleus but picks up enough energy to ionize itself (Fig. 9c). In the adiabatic picture, if the energy of the remaining electron is higher than the saddle for a sin- gle electron atom, VF=−2/radicalbig 2\|F\|, whether the electron stays trapped or escapes depends on the other constants of motion (besides the energy) in this integrable system. The second electron can thus escape during the same half ﬁeld cycle as the ﬁrst one but its ﬁnal parallel momen- tum component need not be related to that of the ﬁrst electron, as is shown in Fig. 9c. −6 −3 0 3 6 ppar (a.u.)0.000.150.30probability density −6 −3 0 3 6 ppar (a.u.)0.00.10.2 (a) (b) FIG. 10. Final parallel ion momenta distributions calcu- lated in the two-dimensional non-interacting electrons mo del for the initial time t0= 0.33Tdin the pulse duration, where Td= 20 ×2π/ω, peak ﬁeld amplitude F= 0.137 a.u. and initial energy E=−0.8 a.u. [panel (a)] and E=−0.1 a.u. [panel (b)]. The initial conditions for the electrons have b een chosen to satisfy E=E1+E2with the restrictions E1<0 andE2<0. The saddle energy for a single electron is VF=−1.05 a.u.. The results in each panel are based on about 1 .5·106trajectories. We are not able to calculate the contribution of sequen- tial events to the double ionization process but we can estimate the distribution of the ﬁnal ion momenta if the sequential process were dominant. To simulate sequen- tial escape one may return to a non-interacting electron model, essentially since the electrons cross the barrier at diﬀerent times. In the model the initial conditions for each electron are chosen independently, constraint only by the requirement that the total initial energy is ﬁxed 8E=E1+E2and that E1<0 and E1<0. Simulta- neous double ionization are not explicitly excluded, but events with delayed ionization are more probable, so that the distributions can still reﬂect the contributions from sequential ionization. In Fig. 10 the distributions of parallel ion momenta forE=−0.8 a.u. and E=−0.1 a.u., calculated in the non-interacting electrons model for F= 0.137 a.u, are plotted. The ﬁgure should be compared with the ﬁgures from the previous section. The conclusion is straightfor- ward: the non-interacting electrons model is not able to reproduce correlations between the electrons observed in the experiments. Moreover, sequential escape can not be a dominant mechanism for double ionization in the range of the ﬁeld intensities considered here. VII. CONCLUSIONS We have considered the process of double ionization of atoms in a strong, linearly polarized ﬁeld for inten- sities below the saturation of single electron ionization. We have developed a model for non-sequential double ionization within classical mechanics. The process has been divided into two stages: in the ﬁrst one a rescat- tering process leads to a highly excited complex of two electrons. In the second stage, an ionization of such com- pound state takes place. We have focused on the latter stage considering diﬀerent pathways to double escape of the electrons. The excited complex can doubly ionize even when its energy is negative because the external ﬁeld opens up saddles for electron escape. The pathway favored by the Coulomb interactions and the ﬁeld is simultaneous symmetric escape of both electrons. Deviations form the symmetric conﬁgurations are ampliﬁed by the repulsion between the electrons which pulls one electron to inﬁnity but the other one is pushed back to the nucleus. There- fore we propose that non-sequential double ionization is dominated by motions of electrons in or near the sym- metric subspace with the saddle. The requirement of the symmetric motion greatly sim- pliﬁes the analysis which then can be carried out for the three- or even two-dimensional eﬀective potential. The trajectory simulations within the symmetric conﬁgura- tions turns out to reproduce the experimentally observed ion momenta distributions very well. We have also con- sidered an alternative mechanism of the ionization, i.e. sequential escape of the electrons. By means of the non- interacting electrons model we show that the sequential ionization is not able to explain the experimentally ob- served electrons correlations. The modeling of the experimental distributions re- quires information on two parameters, the initial energy and the time of formation, which reﬂect a lack of knowl- edge on the compound state and the ramping of the ﬁeld. The dependence of the momentum distributions on theparameters and comparison with the experimental results give insights into the dynamics of double ionization. The analysis in the present paper has been restricted to double ionization but its extension to multiple escape is straightforward and will be given elsewhere. In its cur- rent form the model is applicable for linearly polarized ﬁelds only. For other polarizations the number of rescat- tering events is greatly reduced. However, for some el- liptically polarized ﬁeld, if an electron is driven back to the core and a highly excited complex is formed, in the adiabatic approximation the symmetric conﬁguration of the electrons can be deﬁned with respect to the tempo- rary electric ﬁeld axis. Then one can proceed with the analysis as for the linearly polarized case. Our whole discussion has a more than superﬁcial sim- ilarity with Wanniers analysis of double ionization by electron impact [14–16]. The main diﬀerences are the presence of a ﬁeld and its time dependence, which en- larges phase space and complicates the identiﬁcation of the transition state. In the adiabatic approximation at ﬁxed ﬁeld strength we could identify this saddle in the C2vsubspace. The comparison with experiments is com- plicated furthermore by the necessity to average over initial energy and time of preparation of the compound state. Thus, signatures one might attribute to Wanniers analysis, such as threshold exponents (they follow imme- diately from the stability analysis of the saddle, but are not easy to verify), will be even more diﬃcult to study. But we have no doubt that the observation of the corre- lated escape of the electrons [5] is a clear signature of the existence and dominance of the saddle and the pathways to double ionization which we discuss here. VIII. ACKNOWLEDGEMENTS We would like to thank Harald Giessen for stimulat- ing our interest in this problem and for discussions of the experiments. Financial support by the Alexander von Humboldt Foundation and by KBN under project 2P302B00915 are gratefully acknowledged. [1] A. L’Huiller, L. A. Lompre, G. Mainfray, and C. Manus, Phys. Rev. A 27, 2503 (1983). [2] Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen and R. D¨ orner, Phys. Rev. Lett. 84, 443 (2000). [3] Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen and R. D¨ orner, J. Phys. B: At. Mol. Opt. Phys. 33, L1 (2000). 9[4] R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C.D. Sch¨ oter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoﬀmann and W. Sandner, Phys. Rev. Lett. 84, 447 (2000). [5] Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. D¨ orner, Nature 405, 658 (2000). [6] A. Becker and F.H.M. Faisal, Phys. Rev. Lett. 84, 3546 (2000). [7] R. Kopold, W. Becker, H. Rottke, and W. Sandner, (pri- vate communication). [8] P.B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). [9] K.C. Kulander, J. Cooper, and K.J. Schafer, Phys. Rev. A51, 561 (1995). [10] A. Becker and F.H.M. Faisal, J. Phys. B 29, L197 (1996). [11] A. Becker and F.H.M. Faisal, J. Phys. B 32, L335 (1999). [12] B. Sheehy, R. Lafon, M. Widmer, B. Walker, L.F. Di-Mauro, P.A. Agostini, and K.C. Kulander, Phys. Rev. A 58, 3942 (1998). [13] A. Becker and F.H.M. Faisal, Phys. Rev. A 59, R1742 (1999). [14] G.H. Wannier, Phys. Rev. 90, 817 (1953). [15] A. R. P. Rau , Phys. Rep. 110, 369, (1984). [16] J.M. Rost, Phys. Rep. 297, 271 (1999). [17] B. Eckhardt and K. Sacha, physics/0005079; K. Sacha and B. Eckhardt, physics/0006013; B. Eckhardt and K. Sacha, Ref. to Nobel symposium, Physica Scripta (submitted). [18] J.M. Rost, (preprint). [19] E. P. Wigner, Z. Phys. Chemie B 19, 203 (1932); Trans. Faraday Soc. 3429, (1938). [20] E. Pollak, in Theory of Chemical Reactions , vol III, M. Baer, ed., (CRC Press, Boca Raton, 1985, p. 123. 10
arXiv:physics/0010032v1 [physics.plasm-ph] 12 Oct 2000HEP/123-qed A sandpile model with tokamak-like enhanced conﬁnement phenomenology S. C. Chapman1∗, 1Physics Dept. Univ. of Warwick, Coventry CV4 7AL, UK R. O. Dendy2 2EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom B. Hnat1 (February 18, 2014) Abstract Conﬁnement phenomenology characteristic of magnetically conﬁned plas- mas emerges naturally from a simple sandpile algorithm when the parameter controlling redistribution scalelength is varied. Close a nalogues are found for enhanced conﬁnement, edge pedestals, and edge localised mo des (ELMs), and for the qualitative correlations between them. These resul ts suggest that toka- mak observations of avalanching transport are deeply linke d to the existence ∗sandrac@astro.warwick.ac.uk 1of enhanced conﬁnement and ELMs. 52.55.Dy, 52.55.Fa, 45.70.Ht, 52.35.Ra Typeset using REVT EX 2The introduction of the sandpile paradigm [1]- [3] into magn etized plasma physics (fusion [4]- [10], magnetospheric [11]- [13], and accretion disk [1 4]- [16]: for recent reviews see Ref. [17,18]) has opened new conceptual avenues. It provides a fr amework within which observa- tions of rapid nondiﬀusive nonlocal transport phenomena ca n be studied; recent examples include analyses of auroral energy deposition derived from global imaging [13], and of elec- tron temperature ﬂuctuations in the DIII-D tokamak [10], bo th of which involve avalanching. Insofar as such phenomena resemble those in experimental sa ndpiles or mathematically ide- alized models thereof, they suggest that the conﬁnement phy sics of macroscopic systems (plasma and other) may reﬂect unifying underlying principl es. In this paper we present results suggesting that this unity m ay extend to some of the most distinctive features of toroidal magnetic plasma conﬁneme nt: enhanced conﬁnement regimes (“H-modes”), edge localised modes (“ELMs”), steep edge gra dients (“edge pedestal”), and their observed phenomenological and statistical correlat ions – for recent quantitative studies, see for example Refs. [19–21] and references therein. An imp ortant question is whether the L to H transition necessarily reﬂects a catastrophic bifurc ation of conﬁnement properties, or can be associated with a monotonic change in the character of the turbulence [21]. We show that key elements of the observed phenomenology emerge naturally from a simple one-dimensional sandpile model, that of Chapman [22] (here after CDH), which incorporates other established models [1,23] as limiting cases. The cent rally fueled (at cell n= 1) CDH model’s distinctive algorithmic feature relates to the loc al redistribution of sand at a cell (say at n=k) where the critical gradient zcis exceeded: the sandpile is ﬂattened behind the unstable cell over a “ﬂuidization length” Lf, embracing the cells n=k−(Lf−1), k− (Lf−2), ..., k; and this sand is conservatively relocated to the cell at n=k+ 1. In Ref. [22] the CDH sandpile is explored for all regimes 1 < L f< Nfor both constant and ﬂuctuating critical gradient zc. Here we consider the dynamics of the more realistic case wit h random ﬂuctuations in zc; the system is robust in that once ﬂuctuations are introduce d in the critical gradient, the behavior is essentially insensitive to both t heir level and spectral properties [22], see also Ref. [17]. The limit Lf= 1 is the ﬁxed point corresponding to the centrally 3fueled algorithm of Ref. [1] in one dimension. In the limit Lf=N(where Nis the number of cells in the sandpile) the sandpile is ﬂattened everywhere b ehind an unstable cell as in Refs. [23,17]. A real space renormalization group approach [24] s hows that the robust scale free dynamics for the limiting case Lf=Ncorresponds to a nontrivial (repulsive) ﬁxed point (see e.g. Ref. [3]). The essential result of Ref. [22] is that diﬀerent regimes of avalanche statistics emerge, resembling a transition from regular to intermittent dynamics reminiscent of deterministic chaos. The control parameter is the normal ized redistribution scalelength Lf/Nwhich speciﬁes whether the system is close to the nontrivial Lf=Nﬁxed point. Height proﬁles for a CDH sandpile with 512 cells, time averag ed over many thousands of avalanches, are shown in Fig.1 for three diﬀerent values of t he ﬂuidization length Lfin the range 50 < L f<250. The sandpile proﬁle shape, stored gravitational poten tial energy, and edge structure (smooth decline or pedestal) correlate with each other and with Lf. AsLf is reduced, the edge pedestal steepens and the time averaged stored energy rises; multiple “barriers” (regions of steep gradient) are visible in trace (a) and to some extent trace (b) of Fig.1. Time evolution of the CDH sandpile for Lf= 50,150, and 250 respectively is quantiﬁed in Figs.2-4. The top traces show total stored ener gy; the middle traces show the position of the edge of the sandpile (the last occupied cell) ; and the bottom traces show the magnitude and occurrence times of mass loss events (here after MLEs) in which sand is lost from the system by being transferred beyond the 512th cell. Time is normalized to the mean inter-avalanche time ∆ τ(proportional to the fueling rate). The CDH sandpile is fueled only at the ﬁrst cell, so that the great majority of a valanches terminate before reaching the 512th cell (these are classiﬁed as internal). W hile internal avalanches result in energy dissipation (recorded in the upper traces of Figs.2- 4), and may alter the position of the edge of the sandpile, they do not result in an MLE; there ar e corresponding periods of quiescence in the middle and lower traces of Figs.2-4. Conve rsely the MLEs are associated with sudden inward movement of the sandpile edge, and in this important sense appear to be edge localised. However, MLEs and the associated inward e dge movement are in fact the result of systemwide avalanches triggered at the sandpile c enter (cell n= 1). The character 4of the MLEs changes with Lf. In Fig.2, where the mean and peak stored energy are greatest , the MLEs are similar to each other and occur with some regular ity. The regularity of MLE occurrence in Fig.3 is less marked, the magnitude of the larg est MLEs is greater than in Fig.2, and there is greater spread in MLE size. This trend con tinues in Fig.4, which also has the lowest stored energy. These eﬀects correlate with the un derlying dynamics of the CDH sandpile. Figure 5 plots the relation between average store d energy and Lffor the N= 512 system and much larger N= 4096 and 8192 systems (normalized to the system size N). The curves coincide, demonstrating invariance with respec t to system size, with an inverse power law with slope close to −2 for Lf/N < 1/4, and a break at Lf/N∼1/4. These two regimes yield the quasi-regular and quasi-intermitten t dynamics in Figs.2-4 (see also the plot of avalanche length distribution against Lfin Fig.8 of Ref. [22]). The parameter Lf/Nis a measure of proximity of this high dimensional system to t heLf=Nnontrivial ﬁxed point. This determines both the apparent complexity of the timeseries in Figs.2-4 and the underlying statistical simplicity described below, wh ich is also invariant with respect to system size. There is systematic correlation between time averaged stor ed energy < E > and MLE frequency fMLE, as shown in Fig.6. To obtain these curves, which are again no rmalized to system size, we have derived MLE frequencies using a standar d algorithm previously used [20] to assign frequencies to ELMs observed in tokamak plasm as in the Joint European Torus (JET). Since the CDH sandpile often generates bursts of mass loss with structure down to the smallest timescales, which might not be resolvable under ex perimental conditions, we have followed Ref. [20] in applying a (relatively narrow) measur ement window of width 450∆ τto obtain fMLE. The correlation between < E > andfMLEis a noteworthy emergent property, furthermore Fig.6’s characteristic curve is very similar t o that of Fig.6 of Ref. [19], which relates measured energy conﬁnement to ELM frequency in JET. Energy conﬁnement time τccan be deﬁned for the CDH sandpile by dividing the time averag ed stored energy < E > by the time averaged energy dissipation rate <∆E >(where ∆ Eis the energy dissipated in a single avalanche). The embedded plot of Fig. 6 shows τcagainst MLE frequency fMLE. 5Finally, we explore the situation where there is a secular ch ange in the redistribution algorithm: in Fig.7, Lfdecreases slowly, continuously, and linearly with time, fr om one constant value to another over a period encompassing many te ns of thousands of avalanches. There is a corresponding time evolution of the energy conﬁne ment properties of the sandpile and of the character of the MLEs. Figure 7(top) shows total st ored energy as a function of time as Lfchanges from 250 at t= 4×104to 50 at t= 1.15×105, while ∼105avalanches occur: over a period of time corresponding to a few tens of MLE s, the system smoothly evolves from low to high conﬁnement. This is accompanied by a gradual change in character of the time variation in the sandpile edge (position of last o ccupied cell, Fig.7(middle)) and of the MLEs (Fig.7(lower)), from large amplitude to small and f rom irregular to regular. Figure 7 can perhaps be regarded as the CDH sandpile analogue of, for example, Fig.2 of Ref. [20] or Fig.2 of [21]. The essential point here is that the sandpile a pparently freely explores phase space with changing control parameter Lf/N. Characteristic properties of the dynamics (whether quasi-regular or quasi-intermittent) and corres pondingly, conﬁnement properties (such as stored energy and MLE characteristics) smoothly fo llow changes in this parameter rather than exhibiting a sudden phase transition or catastr ophe. By varying a single control parameter in the CDH sandpile alg orithm, we have shown correlations between: stored energy, conﬁnement times, sa ndpile proﬁle, sandpile edge struc- ture, and the amplitude, frequency, and dynamical characte r of mass loss events. We have also seen how slow secular change in the control parameter pr oduces a smooth evolution in conﬁnement properties. If a single control parameter analo gous to Lf/Nexists for tokamaks, it can in principle be found from experimental data by examin ing scaling with respect to system size as above. The existence of such extensive tokamak-like phenomenolog y, emergent from a very sim- ple system, is a novel discovery. Insofar as the phenomenolo gical resemblance is close, there is more to be learnt. A minimalist interpretation starts fro m the premise that the CDH sandpile algorithm provides a simple one-parameter model f or studying generic nonlocal transport, conditioned by a critical gradient, in a macrosc opic conﬁnement system. Chang- 6ing the value of the single control parameter Lfthen corresponds to altering the range in conﬁguration space over which the transport process operat es. It then follows from the results in the present paper that this may be the minimum requ irement to generate those aspects of tokamak-like conﬁnement phenomenology describ ed. This is a signiﬁcant conclu- sion, but one can consider a more far-reaching one. A possibl e maximalist interpretation attaches greater weight to recent observations [6,9,10] of avalanching transport in tokamaks and in largescale numerical simulations [7,8] thereof, and therefore regards the avalanch- ing transport that is built into sandpile algorithms as an ad ditional point of contact with magnetically conﬁned plasmas. One would then infer from the present results that toka- mak observations of avalanching transport are deeply linke d to the existence of enhanced conﬁnement and ELMs. ACKNOWLEDGMENTS We are grateful to Jack Connor, George Rowlands, David Ward a nd Nick Watkins for comments and suggestions. SCC was supported by a PPARC lectu rer fellowship, ROD by Euratom and the UK DTI, and BH by HEFCE. 7REFERENCES [1] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 50, 381 (1987). [2] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988). [3] H.J. Jensen, Self-Organised Criticality: Emergent Complex Behaviour i n Physical and Biological Systems , Cambridge University Press, 1998. [4] D.E. Newman, B.A. Carreras, P.H. Diamond, and T.S. Hahm, Phys. Plasmas 3, 1858 (1996). [5] R.O. Dendy and P. Helander, Plasma Phys. Control. Fusion 39, 1947 (1997). [6] B.A. Carreras et al., Phys. Rev. Lett. 80, 4438 (1998). [7] X. Garbet and R. Waltz, Phys. Plasmas 5, 2836 (1998). [8] Y. Sarazin and P. Ghendrih, Phys. Plasmas 5, 4214 (1998). [9] T.L. Rhodes et al., Phys. Lett. A 253, 181 (1999). [10] P.A. Politzer, Phys. Rev. Lett. 84, 1192 (2000). [11] T.S. Chang, IEEE Trans. Plasma Sci. 20, 691 (1992). [12] S.C. Chapman, N.W. Watkins, R.O. Dendy, P. Helander, an d G. Rowlands, Geophys. Res. Lett. 25, 2397 (1998). [13] A.T.Y. Lui et al., Geophys. Res. Lett. 27, 2397 (2000). [14] S. Mineshige, M. Takeuchi, and H. Nishimori, Astrophys . J.435, L125 (1994). [15] K.M. Leighly and P.T. O’Brien, Astrophys. J. 481, L15 (1997). [16] R.O. Dendy, P. Helander, and M. Tagger, Astron. Astroph ys.337, 962 (1998). [17] S.C. Chapman, R.O. Dendy, and G. Rowlands, Phys. Plasma s6, 4169 (1999). [18] S.C. Chapman and N.W. Watkins, Space Sci. Rev. accepted (2000). 8[19] G.M. Fishpool, Nucl. Fusion 38, 1373 (1998). [20] W. Zhang, B.J.D. Tubbing, and D. Ward, Plasma Phys. Cont rol. Fusion 40, 335 (1998). [21] J. Hugill Plasma Phys. Control. Fusion 42, R75 (2000). [22] S.C. Chapman, Phys. Rev. E 62, 1905 (2000). [23] R.O. Dendy and P. Helander, Phys. Rev. E 57, 3641 (1998). [24] S.W.Y. Tam, T.S. Chang, S.C. Chapman, and N.W. Watkins, Geophys. Res. Lett. 27, 1367 (2000). Captions FIG.1. Time averaged height proﬁles of 512 cell CDH sandpile forLf= (a)50, (b)150 and (c)250. Inset: edge structure. FIG.2. Time evolution of 512 cell CDH sandpile with Lf= 50: (top) stored energy, (middle) position of last occupied cell, (lower) magnitude and occurence of mass loss events. FIG.3. As Fig.2, for Lf= 150. FIG.4. As Fig.2, for Lf= 250. FIG.5. Average stored energy versus Lf/Nfor sandpiles of N= 512 ,4096,8192. Energy is normalized to the Lf= 1 case (eﬀectively to N2). FIG.6. Average stored energy versus MLE frequency, and (ins et)τcversus MLE fre- quency for sandpiles of N= 512 ,4096,8192. Energy and MLE frequency are normalized as in Fig 6. FIG.7. Time evolution of (top) stored energy, (middle) sand pile edge position and (lower) MLEs, as Lfchanges slowly and linearly from 250 to 50. 90 100 200 300 400 500 600050100150200250300350400 Cell NumberHeight440 460 480 500 520010203040 (a) (b) (c) (b) (a) (c) 6.5 77.5 88.5 99.5 10 x 1044567x 104Energy 6.5 77.5 88.5 99.5 10 x 104460480500520Edge Pos. 6.5 77.5 88.5 99.5 10 x 104050100150Flux Out Time6.5 77.5 88.5 99.5 10 x 104012x 104Energy 6.5 77.5 88.5 99.5 10 x 104300400500600Edge Pos. 6.5 77.5 88.5 99.5 10 x 10405001000Flux Out Time6.5 77.5 88.5 99.5 10 x 104012x 104Energy 6.5 77.5 88.5 99.5 10 x 104200400600Edge Pos. 6.5 77.5 88.5 99.5 10 x 104050010001500Flux Out Time−3 −2.5 −2 −1.5 −1 −0.5101102103 log( Lf / N )log( <E> / N )Slope ~ −1.90.20.250.30.350.40.450.50.550.6050100150200250300350400450500 fMLE × N<E> / N0.20.30.40.50.60100200300 fMLE × Nτc / N2 4 6 8 10 12 14 x 10402468x 104Energy 2 4 6 8 10 12 14 x 104200300400500600Edge Pos 2 4 6 8 10 12 14 x 104050010001500Flux Out Time
Submitted to Nature 10/11/00 1LET Corp., 4431 MacArthur Blvd., Washington, DC 20007 2Icarus Research, Inc., P.O. Box 30780, Bethesda, MD 20824-0780Comments on Superluminal Laser Pulse Propagation P. Sprangle, J.R. Peñano1, and B. Hafizi2 Naval Research Laboratory, Plasma Physics Division, Washington D.C. Researchers claim to have observed superluminal (faster than light) propagation of a laser pulse in a gain medium by a new mechanism in which there is no distortionof the pulse. Our analysis shows that the observed mechanism is due to pulsedistortion arising from a differential gain effect and should not be viewed assuperluminal propagation. In a recent article in Nature [1], titled Gain-assisted superluminal light propagation , experimentalists reported observing a laser pulse propagating faster than the speed of light through a gas cell which served as a amplifying medium. They state thatthe "peak of the pulse appears to leave the cell before entering it". This article generateda great deal of press attention around the world. For example, reports appeared in CNN and in newspapers such as the Los Angeles Times, New York Times, Washington Post, South China Morning Post, India Today , and the Guardian of London [2]. The objective of this paper is to analyze, discuss and comment on the conclusions reached in Ref. 1 and related studies [3]. We show that the theory on which thissuperluminal interpretation was based is inaccurate because of an inconsistency in theordering of the terms kept in the analysis. Our analysis of the experiment shows that themechanism responsible for the so-called "superluminal propagation" can be attributed todifferential gain of the laser pulse. That is, the front of the laser pulse is amplified more2than the back, causing the pulse shape to be tilted towards the front. The leading edge of the distorted pulse, however, propagates at the speed of light, as one would expect. Theauthors of Ref. 1 specifically discount this explanation. They claim that superluminalpropagation is observed "while the shape of the pulse is preserved" and "the argumentthat the probe pulse is advanced by amplification of its front edge does not apply" in thepresent experiment. Our analysis indicates that these claims are incorrect. In the experiment a long laser pulse was passed through an amplifying medium consisting of a specially prepared caesium gas cell of length 6=L cm, as depicted in Fig. 1. The laser pulse of duration 7.3=T µsec was much longer (1.1 km) than the gas cell, so that at any given instant only a small portion of the pulse was inside the cell. By measuring the pulse amplitude at the exit, they claim that both the front and the backedges of the pulse were shifted forward in time by the same amount relative to a pulsethat propagated through vacuum. The following analysis considers a laser pulse propagating in a general dispersive and amplifying medium characterized by a frequency dependent complex refractive index )( ωn . To determine the evolution of the pulse envelope we represent the laser electric field amplitude as () .. ) ( exp),()2/1(),( cc t zki tzA tzEo o + − = ω ,( 1 ) where ),(tzA is complex and denotes the slowly varying laser pulse envelope, c n ko o o /)(ωω= is the complex wavenumber, oωis the carrier frequency and c.c denotes the complex conjugate. The field is polarized in the transverse direction and propagates in the z-direction. Since ok is complex, the factor () zko) Im( exp− represents an overall amplification of the pulse at frequency oω which does not result in pulse3distortion. The deviation of the refractive index from unity, i.e., 1 )( )( −=∆ ωω n n , is small in the reported experiment and it is therefore legitimate to neglect the reflection of the laser pulse at the entrance and exit of the gas cell. The envelope equation describing the evolution of a laser pulse propagating in a general dispersive medium has been derived elsewhere [4]. For a one-dimensional laserpulse the envelope equation is given by ),( ...2),(1 22 2 1 tzA ti ttzAtcz     + ∂∂+∂∂−=   ∂∂+∂∂κ κ ,( 2 ) where ()oωωωωκκ=∂∂=l l l /)(, ...,2,1=l , c n /)( )( ωωωκ ∆= and the laser pulse envelope at the input to the amplifying medium ) ,0( t zA= is assumed given. Equation (2) is derived by substituting the field representation given by Eq. (1) into the linearized wave equation and performing a spectral analysis [4,5] that involves expanding the refractive index about the carrier frequency oω and neglecting reflected waves. These assumptions are valid when the refractive index is close to unity and the spectral width of the pulse is narrow. We limit our analysis to terms of order 2κ, i.e., to lowest order in group velocity dispersion; this is sufficient for the present purpose. In a vacuum, 0)(=∆ωn so that the right hand side of Eq. (2) vanishes and the envelope is given by )/ ,0( ),( cztA tzA −= , indicating that the pulse propagates without distortion with velocity c. To solve for the pulse envelope in a general dispersive medium we Fourier transform Eq. (2) in time and solve the resulting differential equation in z for thetransformed envelope. Inverting the transformed envelope yields the solution4() () 2/ exp)/ ( exp),0(ˆ 21),(2 2 1 z iz i czti Ad tzA νκνκ ν νν π+ ∫ −− =∞ ∞−,( 3 ) where ),0(ˆνA is the Fourier transform of the envelope at z = 0, and ν is the transform variable. In the experiment the values of the parameters are such that the following inequalities hold: 2/ 12 2 1 z zνκνκ>> >> , where T/1≈ν and Lz=. To correctly evaluate the integral in Eq. (3), the exponentials in the small quantities should be expanded to an order of approximation consistent with Eq. (2), otherwise unphysical solutions may result. For example, if the higher order term 2 /2 2zνκ is neglected in Eq. (3), the laser envelope is given by () ( ) z i czti Ad tzA νκ ν νν π1 exp)/ ( exp),0(ˆ 21),( ∫ −− =∞ ∞−.( 4 ) Equation (4) can be integrated exactly to give )/ ,0( ),( pvztAtzA −= .( 5 ) The quantity ()[] ) 1/( /11κ ωω c c cn vp +=∂∂=− defines the group velocity of a pulse in a dispersive medium. However, in addition to cases where pv is abnormal i.e., greater than c or negative, there are other instances in which the concept of group velocity does not represent the pulse velocity. These include situations where the interaction length, L,is less than the phase mixing length associated with the spectral components of the pulse,or when L is much less than the pulse length. These conditions apply in the experimentof Ref. 1.5The exact solution, given by Eq. (5), to the approximate envelope equation can lead to unphysical conclusions since it implies that the pulse propagates undistorted with velocity pv [3]. For example, if 0 1 1<<−κc , the pulse velocity exceeds c. For the parameters in Ref. 1, however, 1κc is essentially real and < -1, giving a negative pulse velocity, 310 /c p−=ν . They have, as well as others, ascribed physical meaning to this by considering the delay time cL L T p / /−=∆ν , defined as the difference in the pulse transit time in the gain medium and in vacuum [1,6]. Since 310 /c p−=ν in Ref. 1, the delay time is negative, implying superluminal propagation. This error is due to retaining terms beyond the order of the approximation . That is, the exponential factor in Eq. (4) contains terms to all orders in zνκ1 while terms of order z2 2νκ and higher are neglected. For example, this is equivalent to keeping terms proportional to 2 1) (zνκ while neglecting terms proportional to z2 2νκ which are of the same order. To avoid this incorrect conclusion it is necessary to solve Eq. (3) by keeping the order of approximation consistent. Expanding the exponential terms in Eq. (3) to secondorder gives () ∫ − ++ =∞ ∞−222 1 2 1 ) )(2/1( 1),0(ˆ )2/1(),( νκκ νκννπ z zi zi Ad tzA () )/ ( exp czti−−×ν .( 6 ) Equation (6) can be integrated to give () )/ ,0( ...211 ),(2222 1 2 1 cztA tz zitz tzA −    + ∂∂− −∂∂−= κκ κ .( 7 ) In Eq. (7) the first term on the right hand side denotes the vacuum solution, the second term represents lowest order differential gain, while the third and higher order terms are6small and denote higher order effects. The result in Eq. (7) shows that the pulse propagates at the speed of light while undergoing differential gain. The quantity 1κ can be negative in the presence of gain or absorption. In the case of gain, when 0 1<κ , the front portion of the pulse is amplified more than the back. Note that the differential gain effect, i.e., the first order t∂∂/term in Eq. (7), can be recovered from Eq. (5) through a Taylor expansion. However, this is simply equivalent to expanding Eq. (5) so that the proper order of approximation is recovered, as was done in deriving Eq. (7). The results of this analysis may be used to interpret the experiment. The susceptibility of the medium used in Ref. 1 has the following form near the resonancefrequencies γ γ πωχiffM iffM nf+−++−=∆≅ 22 11 2)()(, ( 8 ) where 02,1> M are related to the gain coefficients. The susceptibility in Eq. (8) represents a medium with two gain lines of spectral width γ at resonance frequencies f1 and f2. The gain spectrum for ==M M2,1 0.18 Hz, =1f3.5 x 1014 Hz, MHz7.21 2+=f f and =γ 0.46 MHz, is shown in Fig. 2(a) (solid curve). For these parameters the deviation of the refractive index from unity ) (ωn∆ shown in Fig. 2(b) closely approximates that in Fig. 3 of Ref. 1. The input laser pulse envelope, which approximates the experimental pulse, is taken to have the form  <<== , otherwise ,02 0 ),2/(sin),0 (2Tt Tt at zAoπ(9)7where oa is the pulse amplitude and 2 /) ( 2/2 1 0 ff+=πω is the carrier frequency. The spectrum associated with the input pulse is shown by the dashed curve in Fig. 2 and has no significant spectral components at the gain lines. For the parameters of the experiment we find that the first order correction in Eq. (7), i.e. the term proportional to t∂∂/ , is of order 2 1 10 /−≈TLκ while the second order correction is smaller than this by 110−. Hence, the expansion performed to obtain Eq. (7) is valid. The differential gain effect, which is misinterpreted as superluminal propagation, requires that 0 1<κ . Using Eq. (8) we find that 1κ is approximately given by M f ff fc2 1 22 11) () (8 −+−≅πκ , where γ>>−2,1f f . In this case it is clear that a gain medium )0 (>M is required for 1κ to be negative. Note that in the absence of gain ) 0 (<M , 1κ can also be negative provided γ<<−2,1f f . In this case differential absorption occurs in which the back of the pulse is absorbed more than the front. This effect has also been presented as superluminal propagation [7]. The validity of Eq. (7) was verified by numerically solving the envelope equation to higher order. Figure 3 compares the solution given by Eq. (7) at the exit of the gain medium (dotted curves) with the vacuum solution )/ ,0( cLtA− (solid curves). Panel (a) shows the entire pulse profile. Consistent with the experimental measurements, the leading edge is shifted forward in time relative to the vacuum solution by 62 nsec. Panel(b) shows three curves: the solid curve denotes the vacuum solution, the dotted curveshows the result obtained from Eq. (7), and the dashed curve shows the result obtained8from Eq. (5). The dotted curve shows that the front of the pulse propagates with velocity c; the propagation is not superluminal. The unphysical solution, given by the dashed curve, shows the front of the pulse propagating at superluminal velocities. Panel (c) is anexpanded view near the peak of the pulse showing that the front is amplified more thanthe back. In conclusion, we find that to properly describe pulse propagation, a consistent ordering of the approximations is necessary. In addition, the distortion of the pulse formthat the authors of Ref. 1 misinterpret as a newly observed mechanism for superluminalpropagation is actually due to differential gain. That is, the modification of the pulseshape is due to the addition of new photons to the front of the pulse. This phenomenonshould not be viewed as superluminal light propagation. Acknowledgements This work was supported by the Office of Naval Research and the Department of Energy.9References 1. Wang, L.J., Kuzmich, A., Dogariu, A ., Gain-assisted superluminal light propagation . Nature 406, 277 (2000). 2. McFarling, U.L., A pulse of light breaks the ultimate speed limit . Los Angeles Times (20 July, 2000). Glanz, J., Faster than light, maybe, but not back to the future . New York Times (30 May, 2000). Suplee, C., The speed of light is exceeded in lab . Washington Post (20 July, 2000). Ray of light for time travel . South China Morning Post (21 July, 2000). It's confirmed: speed of light can be broken . India Today (21 July, 2000). Davies, P., Light goes backwards in time . The Guardian (20 July, 2000). 3. Mitchell, M.W. & Chiao, R.Y., Causality and negative group delay in a simple bandpass amplifier. Am. J. Phys. 66, 14 (1998). Bolda, E., Garrison, J.C. & Chiao, R.Y., Optical pulse propagation at negative group velocities due to a nearby gain line. Phys. Rev. A 49, 2938 (1994). Chiao, R.Y., Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations . Phys. Rev. A 48, R34 (1993). Crisp, M.D., Concept of group velocity in resonant pulse propagation . Phys. Rev. A. 4, 2104 (1971). 4. Sprangle, P., Hafizi B., Peñano, J.R ., Laser pulse modulation instabilities in plasma channels . Phys. Rev. E 61, 4381 (2000). 5. Agrawal, G., Nonlinear Fiber Optics (Academic Press, San Diego, California).106. Marangos, J., Faster than a speeding photon . Nature 406, 243 (2000). Chiao, R.Y., in Amazing Light, a Volume Dedicated to C.H. Townes on His Eightieth Birthday (ed. Chiao, R.Y.) 91-108 (Springer, New York, 1996). 7. Chu, S., Wong, S., Linear pulse propagation in an absorbing medium . Phys. Rev. Lett. 48, 738 (1982). Garrett, C.G.B. & McCumber, D.E. Propagation of a Gaussian light pulse through an anomalous dispersion medium. Phys. Rev. A 1, 305 (1970).11/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1L z = 0c gain mediumlaser pulse envelope Figure 1. Schematic showing a long laser pulse entering a gain medium.12-4 -2 0 2 400.20.40.60.811.2 pulse spectrum (a.u.)/B4 /AX /A0 /AX/BC /B5 /BP /BE /AP /B4/C5/C0/DE/B5-4 -2 0 2 4-1.5-1-0.500.511.5/CA/CT /B4/A1 /D2 /B5 /A2 /BD/BC/BI /A0 /C1/D1/B4 /AK/C4 /B5 (gain spectrum) (a) (b) Figure 2. Gain spectrum (solid curve) obtained using the susceptibility in Eq. (8) for the parameters 18.0=M Hz, 14 1 105.3×=f Hz, 7.21 2+=f f MHz, and 46.0=γ MHz. The dashed curve shows the spectrum associated with the pulse envelope of Eq. (9) with 7.3=T µsec.1300.1 0.2 0.3 0.40.0050.010.0150.020.0250.03/CC/CX/D1/CT /B4 /AM /D7/B50123456700.20.40.60.81/CY /BT /B4 /C4/BN /D8 /B5 /CY /BP/CP /BC/CY /BT /B4 /C4/BN /D8 /B5 /CY /BP/CP /BC/CY /BT /B4 /C4/BN /D8 /B5 /CY /BP/CP /BC/A1 /CC /BP/BI /BE /D2 /D7 /CT /CR 33.5 44.5 50.960.9811.021.04 (c)(b)(a) Figure 3: Dotted curves show the pulse envelope amplitude \|),(\| tLA at Lz= obtained from Eq. (7). Solid curves denote a pulse that has traveled a distance L through vacuum. The dashed curve in panel (b) is the unphysical solutionobtained from Eq. (5) showing superluminal propagation. Panels (b) and (c) areexpanded views of the front and peak of the pulse, respectively. The parametersfor this figure are the same as in Fig. 2.
arXiv:physics/0010034v1 [physics.flu-dyn] 13 Oct 2000The decay of plane wave pulses with complex structure in a nonlinear dissipative medium Sergei N. Gurbatov, Bengt O. Enﬂo, Galina V. Pasmanik Radiophysics Dept., University of Nizhny Novgorod 23, Gagarin Ave., Nizhny Novgorod 603600, RUSSIA e-mail: gurb@rf.unn.runnet.ru Department of Mechanics, Royal Institute of Technology, S–100 44 Stockholm, SWEDEN e-mail: benﬂo@mech.kth.se, phone: Int+46 8 7907156, fax: Int+46 8 7969850 1Abstract Nonlinear plane acoustic waves propagating through a ﬂuid a re studied using Burgers’ equation with ﬁnite viscosity. The evolution of a s imple N-pulse with regular and random initial amplitude and of pulses with mono chromatic and noise carrier is considered. In the latter case the initial pulses are characterized by two length scales. The length scale of the modulation function i s much greater than the period or the length scale of the carrier. With increasing ti me the initial pulses are deformed and shocks appear. The ﬁnite viscosity leads to a ﬁn ite shock width, which does not depend on the ﬁne structure of the initial pulse and i s fully determined by the shock position in the zero viscosity limit. The other eﬀe ct of nonzero viscosity is the shift of the shock position from the position at zero vi scosity. This shift, as well as the linear time, at which the nonlinear stage of evolu tion changes to the linear stage, depends on the ﬁne structure of the initial pul se. It is also shown that the nonlinearity of the medium leads to generation of a nonze ro mean ﬁeld from an initial random ﬁeld with zero mean value. The relative ﬂuctu ation of the ﬁeld is investigated both at the nonlinear and the linear stage. 21 Introduction The propagation of ﬁnite amplitude sound waves is of fundame ntal interest in nonlinear acoustics. In the simplest model of propagation in ﬂuids the se waves are described by the well-known Burgers’ equation (plane waves) [1,2] or modiﬁc ations of Burgers’ equation, which are called generalized Burgers’ equations (cylindri cal and spherical waves) [3–5]. In studies of nonlinear wave propagation an important probl em is to ﬁnd the waveform of the asymptotic wave at long time after the preparation of t he initial wave or at long distance from the source emitting the wave. In the ﬁrst case t he asymptotic wave is called the old-age wave and is an asymptotic solution of an in itial value problem of some wave equation and in the latter case the asymptotic wave is a s olution of a boundary wave problem. The asymptotic wave is damped both by absorpti on and by shock wave dissipation and the asymptotic wave is therefore, for plane waves, described by the linear diﬀusion equation. Because the linear diﬀusion equation de scribes the attenuation of high frequency waves, the asymptotic behaviour is determined by the spectrum of the waves of low frequencies. This spectrum is the result of the distorti on of the wave at the nonlinear stage. The decay of a wave to the old-age waveform is very diﬀerent fo r periodic signals and pulse perturbations. For the periodic signal the old-ag e waveform is an exponen- tially decaying harmonic wave. The pulse perturbation has c ontinuous spectrum and the amplitude of the pulse decays according to a power law. For sinusoidal and N-wave initial perturbations the old-ag e solutions have been studied in several papers for plane, cylindrical and spherical wave s with use of both analytical and numerical methods [6–15]. The dependence of the amplitu de constant of the old-age waveform on the parameters of the initial perturbations has been found. The aim of the present paper is to investigate the asymptotic behaviour of complex pulses, such as modulated or random waves. Plane waves are st udied, which means that we will use the original form of Burgers’ equation [1]. Burge rs’ equation has an exact solution, which is found by reducing it to the linear diﬀusio n equation by mean of the 3Hopf-Cole transformation [16, 17]. Because of the existenc e of an exact solution it is relatively easy to ﬁnd the old-age waveform developing from simple initial perturbations like periodic signals and N-waves in the plane wave case. How ever, for signals with complex structure it is far from trivial to ﬁnd their evolution even f or plane waves [18–20]. For a regular signal with fractal structure [18,20] an unusual s equence of stages of evolution may appear: the nonlinear stage may succeed the linear stage . At the nonlinear stage this wave may decrease more slowly than a periodic wave or a si mple pulse. Still more complicated behaviour is shown by a random signal . The solution of Burgers’ equation with random initial conditions is often called Bur gers turbulence. Numerous papers are devoted to investigations of this problem [22–27 ]. In the case of vanishing viscosity the continuous random initial ﬁeld is transforme d into a sequence of straight lines with some slope and with random locations of the shocks separ ating them. Due to the merging of the shocks the internal scale of the turbulence in creases and the random wave decreases more slowly than the periodic signal. The decay ra te depends on the behaviour of the initial energy spectrum of low frequency [21,26] and i s also sensitive to the statistics of the potential of the initial velocity ﬁeld [22,26,27]. Th e asymptotic behaviour of the ﬁeld in the case of ﬁnite viscosity also depends strongly on t he statistical properties of the initial ﬁeld. In the case where there are large scale compone nts in the initial spectrum the nonlinear stage never transforms to the linear regime of evo lution [21]. In the opposite case the ﬁnal evolution depends on the tail of the initial pro bability distribution [23]. In the present paper we consider the evolution of complex pul ses which are character- ized by two scales: l0- the inner scale of the carrier, L0- the scale of the modulation, and the condition L0>> l 0. For such signals the generation of a low-frequency com- ponent or a non-zero mean ﬁeld takes place. The case of vanish ing viscosity has been investigated [29,31]. It has been shown that, after multipl e merging of the shocks, the inner structure disappears and ﬁnally a ﬁnite pulse ends up w ith an N-wave. This N-wave is fully described by the positions of its zero and shocks, wh ich are determined by the potentialψ0(x) of the initial velocity v0(x):v′ 0(x) =−ψ0(x). It has also been shown that, for a pulse with random carrier, the ﬂuctuations of the positions of the zero and the 4shocks are relatively small for l0<<L 0. In the present paper we will consider the inﬂuence of ﬁnite vi scosity on the nonlinear stage of the evolution of the pulse and investigate its old-a ge behaviour. The paper is organized as follows. In section 2, on the base of the Hopf-Cole solution, the asymp totic behaviour of the pulse at the nonlinear and the old-age stages is analyzed in g eneral terms without assuming the detailed structure of the initial pulse. In section 3 the evolution of simple N-waves with regular and random amplitude is discussed. Section 4 deals with pulses with monochromatic carrier and s ection 5 with pulses with noise carrier. It is shown that the parameters of the asympto tic waveform depend weakly on the ﬁne structure of the initial pulse, but that the old-ag e behaviour is very sensitive to the properties of the carrier. 2 Solution of Burgers’ equation and its large time asymptotics Our starting-point is the Burgers’ equation ∂v ∂t+v∂v ∂x=ν∂2v ∂x2, (1) which governs the propagation of plane acoustic waves in non linear dissipative media. Herevis the velocity of ﬂuid and νis the ﬂuid viscosity. Burgers’ equation (1) has the Hopf-Cole solution [16,17] v(x,t) =−2νUx U, (2) whereU(x,t) is the solution for the linear diﬀusion equation U(x,t) =+∞/integraldisplay −∞U0(y)G(x,y,t)dy . (3) 5HereG(x,y,t) is the Green function of the linear diﬀusion equation G(x,y,t) =1√ 4πνtexp/bracketleftigg −(x−y)2 4νt/bracketrightigg , (4) andU0(x) is the initial condition for this equation U(x,0) =U0(x) = exp/bracketleftiggψ0(x) 2ν/bracketrightigg , (5) which corresponds to the initial condition v(x,0) =v0(x) =−dψ0(x) dx. (6) The main goal of the present paper is the investigation of the old-age behaviour of the localized pulses. But ﬁrst we discuss shortly the asympt otic evolution of nonlocalized waves. Consider a group of perturbation with the bounded ini tial potential \|ψ0(x)\|<∞ assuming that ψ0(x) is a periodic signal with a period L0or homogeneous noise with rather fast decreasing probability distribution of the potential ψ0. For such a perturbation in U(x,t) (3) we separate a constant component ¯U: U(x,t) =¯U+˜U(x,t). (7) Inserting (7) into (3) we see that ¯Udoes not depend on time, since the y-integral of G(x,y,t) is unity. Here ˜U(x,t) is a ﬁeld with zero mean value (on the period or statisti- cally for noise). As times goes on, the viscous dissipation a nd oscillation (inhomogenity) smoothing causes the amplitude (variance) of the ﬁeld ˜U(x,t) to become less. At times when it amounts to \|˜U\| ≪¯Uthe solution (2) is equal to v(x,t) =−2ν˜Ux(x,t) ¯U. (8) As˜U(x,t) satisﬁes the linear diﬀusion equation, then v(x,t) also at these times fulﬁls the linear equation. This testiﬁes precisely to the fact that th e evolution of the velocity ﬁeld enters the linear stage. The accumulated nonlinear eﬀects a re described in this solution by the nonlinear integral relation between the initial velo city ﬁeldv0(x) and the ﬁelds ˜U(x,0),¯U(5,6), and are characterized by the value \|△ψ0\|/ν∼Re0. 6Here△ψ0is the characteristic change in amplitude of ψ0, andRe0is the initial Reynolds number. From (8) it is easy to get the well known result, that for Re0≫1 the initial harmonic waves asymptotically has also harmonics form, but with the a mplitude not depending on the initial amplitude [2,3]. At large initial Reynolds number the homogeneous Gaussian ﬁ eldv0(x) at the non- linear stage transforms into series of sawtooth waves with s trong nongaussian statistical properties [21,25]. Nevertheless at very large time, when t he relation (8) is valid, the dis- tribution of the random ﬁeld v(x,t) with statistically homogeneous initial potential ψ0(x) converges weakly to the distribution of the homogeneous Gau ssian random ﬁeld with zero mean value, and universal covariance function [23]. But the amplitude of this function is nonlinearly related to the initial covariance function o f the ﬁeldv0(x) and increases proportionally exp( Re2 0) with increasing of initial Reynolds number Re0[21,23]. Let us consider the opposite situation when the initial pert urbation is localized in some region. We assume also that initial potential ψ0(x) vanishes very fast for x→ ∞, so we have ψ0(x) = 0,\|x\|>L∗, (9) whereL∗can be considered as the length of initial pulse. The conditi on (9) is identical to the assumption that the integral over the initial pulse is zero. It should be stressed that if the integral over the initial pu lse is△ψ0/ne}ationslash= 0, then the initial perturbation transforms to the unipolar pulse with area△ψ0asymptotically. For Re0=△ψ0/ν≫1 the form of this pulse is close to the triangular with the wid th of shock δ(t)∼νt/xsmuch smaller than the length of the pulse xs=/radicalig 2\|△ψ0\|t. For such a pulse δ(t)/xs∼Re−1 0=const and does not depend on time. Thus the nonlinear stage of the evolution prevails for all times. Let us now consider the time asymptotic behaviour of initial ly localized pulse which satisﬁes (9). If we rewrite (3) as U(x,t) = 1 ++∞/integraldisplay −∞[U0(y)−1]G(x,y,t)dy , (10) 7then we can see that because of (9) the integrand at right hand side of (10) is zero outside the region \|y\|< L∗. It is seen from (4) that the length scale ldif(t) of Green’s function G(x,y,t) is ldif=√ 2νt . (11) If the length scale of U0(y) is calledLU, and the following condition ldif(t)≫LU (12) is valid, then Green’s function can be considered as constan t in the interval where the integrand at the right hand side of (10) is signiﬁcantly diﬀe rent from zero. Thus we obtain from (10) using (5) U(x,t)≈1 +G(x,y∗(x,t),t)B . (13) Herey∗(x,t),\|y∗\|<L∗, is the value of y, in the neighborhood of which the integrand in (10) gives the essential contribution in the integral. In (1 3) we introduce the notation B=+∞/integraldisplay −∞/bracketleftigg exp/parenleftiggψ0(y) 2ν/parenrightigg −1/bracketrightigg dy . (14) We assume B/ne}ationslash= 0. From (3),(13),(14) an approximate solution is obtained v(x,t) =x−y∗ tBG(x,y∗,t) 1 +BG(x,y∗,t). (15) If the initial pulse ψ0is centered around y= 0, and the large scale of the ﬁeld v(x,t) is much greater then L∗, then we can put y∗≈0 and obtain v(x,t) =x tBG(x,0,t) 1 +BG(x,0,t)(16) instead of (15). For large initial Reynolds number Re0=A/2ν, whereA >0 is the maximum of ψ0(y), the constant B(14) may be rewritten as the product of the maximum of the integrand and some length: Leff B=LeffeA/2ν, Re 0=A/2ν , (17) 8Using the deﬁnition of Green’s function (4) and (17) we can re write (16) in the form v(x,t) =x texp/bracketleftigg −1 2ν/parenleftiggx2 2t−A+νln4πνt L2 eff/parenrightigg/bracketrightigg 1 + exp/bracketleftigg −1 2ν/parenleftiggx2 2t−A+νln4πνt L2 eff/parenrightigg/bracketrightigg. (18) In the limit of vanishing viscosity ( ν→0) we get from (18) the well known solution for N-wave v(x,t) =  x t,\|x\|<xs 0,\|x\|>xs,, (19) wherexsis the position of the shock xs=√ 2At . (20) At this stage the form and the energy of the pulse E(t) =/integraldisplay v2(x,t)dx=25/2A3/2 3t1/2(21) are determined only by the value of maximum of initial potent ialAand does not depend on the ﬁne structure of the pulse. This limiting case ( ν→0) for the pulses with complex inner structure was investigated in the paper [31]. For large but ﬁnite Reynolds number the solution (18), compl eted with a solution valid atxnearxsand exhibiting the shock structure, is valid only at ﬁnite ti me. For ﬁnite t values and suﬃciently small xwe still have BG(x,0,t)≫1. (22) Then for \|x\| ≤xs=/radicaltp/radicalvertex/radicalvertex/radicalbt2t/parenleftigg A−νln4πνt L2 eff/parenrightigg , (23) BG(xs,0,t) = 1 (24) we obtain the increasing part (in x) of the N-wave solution (19). On the other hand, for x→ ∞,tﬁxed, we ﬁnd that v(x,t) fades away exponentionally. 9Fort→ ∞,xarbitrary ﬁnite, we have BG(x,0,t)≪1, (25) and the solution (16) can be approximated as v(x,t) =Bx√ 4πνt3exp/bracketleftigg −x2 4νt/bracketrightigg . (26) Using (4) we can rewrite the condition of the validity of the s olution (26) as √ 4πνt≫B ,/radicaltp/radicalvertex/radicalvertex/radicalbt4πνt L2 eff≫eA/2ν. (27) Because (24) gives a solution of the linear diﬀusion equatio n, the condition (27) deﬁnes the linear stage of the pulse evolution. On this stage the pul se energy is E(t) =B2/radicalbiggν 8πt3. (28) Thus at the linear stage of evolution of the pulse has a univer sal form (26) and is determined only by the constant B, which is deﬁned by the initial perturbation by the relation (14). The case B= 0 can be understood by comparing the deﬁnition of B according to the equations (14) and (5) with the formulas (30 ) - (33) in ref. [19]. By this comparison it is clearthat the case B= 0 corresponds to the absence of the Fourier component with n= 1 in the terminology of ref. [19]. The Fourier component wit hn= 0 is absent here already for B/ne}ationslash= 0. The case B= 0 excludes the N-wave solution (19). Below we will consider three cases of initial perturbation, assuming that the initial potential may be written in the form ψ0(x) =M(x)F(x), (29) whereM(x) has the scale L0≃L∗. First we consider the simplest case when Fis constant, either deterministic or random value. Here we dis cuss the main properties of the wellknown solution (see ref. [30]) for the N-wave. The ca ses of monochromatic and noise carrier will be considered in Sections 4 and 5. In these cases we assume that the scalel0of the carrier F(x) satisﬁes the condition l0≪L0and then v(x,0) =v0(x)≃M(x)f(x), f(x) =−F′(x). (30) 103 The decay of a simple pulse with random initial amplitude 3.1 The evolution of a regular N-wave We discuss the evolution of a one scale pulse, whose potentia l has the structure in (29): ψ0(x) =M(x)A. (31) HereAis a constant and M(x) is a dimensionless function with the scale L0: M(x)≈1−x2 2L2 0+... . (32) In particular we will consider the cases where the initial pe rturbation (31), (32) is exact in the interval \|x\|<√ 2L0andψ0(x)≡0 outside this interval. We ﬁrst assume A >0 and ﬁnd from (6) that the initial velocity pulse is an N-wave: v0(x) =xA L0,\|x\|<√ 2L0, v0(x) = 0,\|x\|>√ 2L0. (33) From (5) we then obtain U0(x) = exp/parenleftiggψ0(x) 2ν/parenrightigg = exp/bracketleftiggA 2ν/parenleftigg 1−x2 2L2 0/parenrightigg/bracketrightigg ,\|x\|<√ 2L0; U0(y) = 1, x>√ 2L0. (34) From (34) we ﬁnd that for this special case the length scale LUofU0(y), ﬁrst introduced in (11), is (the square root of the inverted coeﬃcient of x2in (34)) LU≃L0(2ν A)1 2=L0 Re1 2 0,Re0≡A 2ν. (35) The calculation of U(x,t), from which we obtain v(x,t) by use of (2), can now be done 11using (10) with insertion of (34) and (4): U(x,t) = 1 +1√ 4πνt/integraldisplay√ 2L0 −√ 2L0/parenleftigg exp/bracketleftiggA 2ν/parenleftigg 1−y2 2L2 0/parenrightigg/bracketrightigg −1/parenrightigg exp/bracketleftigg −(x−y)2 4νt/bracketrightigg dy= = 1 +1 21 /parenleftigg 1 +At L2 0/parenrightigg1/2exp A 2ν−A 4νL2 0x2 1 +At L2 0 ·   erf  √ 2L0−x 1 +At L2 0 /radicaltp/radicalvertex/radicalvertex/radicalbt1 4νt/parenleftigg 1 +At L2 0/parenrightigg + erf  √ 2L0+x 1 +At L2 0 /radicaltp/radicalvertex/radicalvertex/radicalbt1 4νt/parenleftigg 1 +At L2 0/parenrightigg    −1 2/braceleftigg erf/bracketleftigg√ 2L0−x√ 4νt/bracketrightigg + erf/bracketleftigg√ 2L0+x√ 4νt/bracketrightigg/bracerightigg (36) The deﬁnition of the error function is: erfz=2√π/integraldisplayz 0exp(−t2)dt. (37) The asymptotic behavior of the error function for large argu ments is erfz=±/parenleftigg 1−1 πe−z2/bracketleftiggΓ(1 2) z−Γ(3 2) z3+.../bracketrightigg/parenrightigg , z→ ±∞. (38) Using (38) we ﬁnd from (36) for \|x\|<√ 2L0,tﬁnite,ν/A≪1: U(x,t)≈1/radicaligg 1 +At L2 0exp A 2ν 1−x2 2L2 0(1 +At L2 0)  . (39) On the other hand, for√ 2L0<\|x\|<√ 2L0/parenleftigg 1 +At L2 0/parenrightigg we have for tﬁnite andν/A≪1, νt/L2 0≪1 andAt/L2 0≫ν/A(the last inequality is necessary for the cancellation of th e last two error functions in (36)): U(x,t)≈1/radicaligg 1 +At L2 0exp A 2ν 1−x2 2L2 0(1 +At L2 0)  + 1. (40) 12For\|x\|</radicaltp/radicalvertex/radicalvertex/radicalbt2A/parenleftigg t+L2 0 A/parenrightigg ,tﬁnite andν/A≪1 we can neglect the second term at the righthand side of (40). Taking ν→0 withtﬁnite in (39) or (40) we obtain using (2): v(x,t)≈x tAt L2 0 1 +At L2 0,\|x\|<xs v(x,t) = 0,\|x\|>xs, (41) wherexsis the shock coordinate xs=/radicaltp/radicalvertex/radicalvertex/radicalbt2A/parenleftigg t+L2 0 A/parenrightigg . (42) However, for growing t-values the sharp discontinuity of the N-wave solution (41) is broadened and we can calculate a shockwidth which depends on a small but ﬁnite value ofν. Using (40) in (2) and assuming that t≫L2 0 A, (43) we obtain v(x,t) =x tL0√ Atexp/bracketleftiggA 2ν−x2 4νt/bracketrightigg L0√ Atexp/bracketleftiggA 2ν−x2 4νt/bracketrightigg + 1. (44) Using (14) and (32) we ﬁnd for the present case in (17) Leff=L0/radicalig 4πν/A and B=/radicaligg 4πL2 0ν Aexp/bracketleftbiggA 2ν/bracketrightbigg =L0/radicaligg 2π Re0eRe0, (45) and thus, using (4), we see that the result (44) is a special ca se of (16). From the expression (44) we will now ﬁnd the width of the shock . If we deﬁne the positionxsof the shock as the coordinate where the wave amplitude has de creased to half its maximal value, we ﬁnd from (23) the following expression forxs: xs=/radicaltp/radicalvertex/radicalvertex/radicalbt2t/parenleftigg A−νlnAt L2 0/parenrightigg . (46) 13Evaluating the expression (44) for xin the neighborhood of the shock, or more precisely, for \|x−xs(t)\|2 νt≪1 (47) we ﬁnd v(x,t) =xs(t) 2t/parenleftigg 1−tanhx−xs(t) δ/parenrightigg , (48) where the shockwidth is given as δ=4νt xs(t). (49) In order to decide which of the two waveforms (41) and (48) is m ost appropriate we compare the expressions for xs(42) and (46). From these formulas we see directly that the shock velocity ( vs=dxs dt) decreases faster for growing twith nonzero viscosity. The zero viscosity expression (42) can be used as long as the corr ection to the N-wave shock positionx=√ 2Atis much greater in (42) than in (46), which means L2 0 At≫ν Aln/parenleftiggAt L2 0/parenrightigg , tlnt tM,nl≪A νtM,nl, (50) where the nonlinear time tM,nlis deﬁned as tM,nl=L2 0 A. (51) In the notation tM,nl,Mstands for the kind of pulse and nlstands for ”nonlinear”. The energy of a pulse is deﬁned as the integral over the pulse l ength of the square of the ﬂuid velocity. We will calculate the energies of some of t he pulses studied above. For the N-wave (41) we obtain E(t) =/integraldisplay v2(x,t)dx=25/2 3A2 L0/parenleftigg 1 +At L2 0/parenrightigg1/2. (52) Thus fort≫tM,nland (50) still valid we see from (52) that E(t) behaves like A3 2/t1 2(21) and thus depends on the initial scale only in the next order of t, i.e./parenleftig L2 0A1/2/t3/2/parenrightig . 14The energy of the wave under the condition t≫tM,nlbut (50) no longer valid is obtained from (44). We obtain E(t) =+∞/integraldisplay −∞ x tL0√ Atexp/parenleftiggA 2ν−x2 4νt/parenrightigg 1 +L0√ Atexp/parenleftiggA 2ν−x2 4νt/parenrightigg 2 dx. (53) After transformation of the integral in (53) we obtain E(t) =2ν1/2B2 πt3/2∞/integraldisplay 0ξ1/2 /parenleftigg eξ+B√ 4πνt/parenrightigg2dξ , (54) whereBis given in (45). We need the formula ( [32], formula (2.3.13. 27)) ∞/integraldisplay 0xα−1e−px (ex−z)2dx= Γ(α) [Φ(z,α,p + 2)−(p+ 1)Φ(z,α,p + 2)], (55) where Φ is given by the power series Φ(z,α,p ) =∞/summationdisplay n=0(p+n)−αzn,\|z\|<1. (56) Using (55), with α= 3/2 andp= 0, in (54) an energy expression, valid for t≫tM,nland (cf. (50))tlnt tM,nl≫A νtM,nl, is obtained: E(t) =2ν1/2B2 πt3/2Γ(3/2)/bracketleftigg Φ(−B√ 4πνt,1 2,2)−Φ(−B√ 4πνt,3 2,2)/bracketrightigg . (57) From (57) the energy of the wave in the linear region is obtain ed by keeping only the power zero in the series for Φ: E(t) =√ 2ν3/2L2 0 AeA/ν t3/2, (58) which is a special case of (28) with Bgiven by (45). After introduction of the ”linear” timetM,lin, the validity condition of (58) is written: B√ 4πνt≪1, t≫tMlin=tMnlexp(A/ν) = exp(A/ν)L2 0 A=tMnle2Re0. (59) 15For the negative Athe pulse decays much faster than for A>0. Forν→0 att>tMnl the initial pulse transforms into so called S-wave [2,31], a nd the energy becomes E(t) =25/2 3L3 0 t2, (60) which is independent of the initial amplitude \|A\|of the pulse. Of course the reason why the energy of the S-wave decreases faster with tthan the energy of the N-wave is that the length of the N-wave increases with growing tin contrast to the S-wave. At large Reynolds number \|A\|/2ν≫1 the constant B(14) which determines the linear stage of evolution is B=−23/2L0 (61) and much smaller than for positive Awith the same Reynolds number. The region of validity of linear regime is obtained from (25) and (61) as t≫L2 0/ν . (62) Thus for the pulses with negative Athe linear stage begins much earlier than for the pulses with positive A, for which the linear time tMlin(59) increases exponentionally with the Reynolds number Re0=A/2ν. Nevertheless, we can see on this simple example the main nont rivial properties of the old-age behavior of the pulses in the case of large Reynolds n umber. In the limit of vanishing viscosity ( ν→0) the energy of the pulse (21),(52) and the shape of the pulse (19),(20),(41) do not depend on the length ofL0asymptotically for t→ ∞. They are determined only by the maximum of initial potentia lψ0(in our case – of the area of triangular pulse of initial pulse A). Nevertheless, at the linear stage the energy depends not only of A, but is also proportional to L2 0(53). We can remark that the energy of initial pulse is proportional to A2/L0. The other property that we stress is that the amplitude B(17),(45) of the wave at the old-age stage depends exponentially on the amplitude Aof the initial pulse. For small wave numbers kthe Fourier component C(k,t) of the pulse depends linearly on k:C(k,t)∼b(t)k. At the linear stage the slope b(t) of the Fourier component does not 16depend on time: b(t)≃B=const. At the nonlinear stage the growing of the slope until the very large linear time (59) leads to an exponentionally l arge value (in 1 /ν) ofb(t) at the linear stage (cf.(45)). 3.2 The decay of an N-wave with random amplitude Let us discuss the evolution of N-wave (33) with random initi al amplitude. We assume that the amplitude Ain (31) has a gaussian probability distribution function wi th a zero mean value W(A) =1/radicalig 2πσ2 ψe−A2/2σ2 ψ. (63) From (33) it is easy to see that mean initial ﬁeld /an}bracketle{tv0(x)/an}bracketri}htis zero, and relative ﬂuctuation of the energy (52) at t= 0 is δE(0) =/an}bracketle{tE2(0)/an}bracketri}ht − /an}bracketle{tE(0)/an}bracketri}ht2 /an}bracketle{tE(0)/an}bracketri}ht2= 2. (64) In the previous section it was shown that there is strong diﬀe rence between the decay of the pulse with positive and negative amplitude A. If we introduce ¯E+(t) – the mean energy of pulse with positive amplitude A(N-wave), and ¯E−(t) – the mean energy of pulse with negative amplitude A(S-wave), then at time t≫tM,nl≃L2 0/σψ (65) from (52), (60) and (63) we have ¯E+(t)/¯E−(t)≃(σψt/L2 0)3/2≃(t/tM,nl)3/2≫1. (66) This fast decrease of pulses with negative Awill lead to the generation of a ﬁeld with positive mean value /an}bracketle{tv(x,t)/an}bracketri}htfrom an initial ﬁeld v0(x) with zero mean value /an}bracketle{tv0(x)/an}bracketri}ht= 0. At timet≫tM,nl(65) we can neglect the inﬂuence of the negative pulse on the m ean ﬁeld and use the expressions (19),(20) for the velocity. In d iﬀerent realizations we will have the N-wave with the same slope v′ x= 1/tand random position of the shocks. 17Let us introduce the cumulative probability of the random am plitude QH(H) =Prob(A<H ) =H/integraldisplay −∞W(A)dA . (67) Then from (19,20)   v(x,t) =x/twith probability 1 −QH(x2/2t) v(x,t) = 0 with probability QH(x2/2t)(68) From (68) we obtain for the mean velocity /an}bracketle{tv(x,t)/an}bracketri}htand its variance σ2 v(x,t) =/an}bracketle{t(v(x,t)− /an}bracketle{tv(x,t)/an}bracketri}ht)2/an}bracketri}htthe following expressions: /an}bracketle{tv(x,t)/an}bracketri}ht=x t/parenleftigg 1−QH/parenleftiggx2 2t/parenrightigg/parenrightigg , (69) σ2 v(x,t) =x2 t2QH/parenleftiggx2 2t/parenrightigg/parenleftigg 1−QH/parenleftiggx2 2t/parenrightigg/parenrightigg . (70) For the Gaussian distribution of A(63) we have from (67), (69) /an}bracketle{tv(x,t)/an}bracketri}ht=x 2t/parenleftigg 1−erf/parenleftiggx2 2√ 2σψt/parenrightigg/parenrightigg , (71) σ2 v(x,t) =x2 4t2/parenleftigg 1−erf2/parenleftiggx2 2√ 2σψt/parenrightigg/parenrightigg , (72) where erf(z) is the error function (37). At smallx <(σψt)1/2the mean ﬁeld /an}bracketle{tv(x,t)/an}bracketri}htis half the regular ﬁeld with A∼σψ, due to the fact that only in that half of realizations, in whic hA>0, we have a relatively slowly decaying N-wave. The mean ﬁeld /an}bracketle{tv(x,t)/an}bracketri}ht(71) has not a clear stressed shock front and has a very fast decaying tail at x≫√σψt: /an}bracketle{tv(x,t)/an}bracketri}ht ≃√ 2√πσψ xexp/bracketleftigg −x4 8σ2 ψt2/bracketrightigg . (73) It is easy to see that both mean ﬁeld /an}bracketle{tv(x,t)/an}bracketri}htand variance σ2 v(x,t) have self-similar behaviour /an}bracketle{tv(x,t)/an}bracketri}ht=/radicalbiggσψ t¯V/parenleftiggx√σψt/parenrightigg , σ2 v(x,t) =σψ t¯σ2 V/parenleftiggx√σψt/parenrightigg , (74) 18where ¯V, ¯σVare given as: ¯V(y) =1 2y/parenleftigg 1−erf/parenleftiggy2 2√ 2/parenrightigg/parenrightigg ,¯σ2 V(y) =1 4y2/parenleftigg 1−erf2/parenleftiggy2 2√ 2/parenrightigg/parenrightigg . (75) Due to the self-similarity the relative integral ﬂuctuatio n of the ﬁeld δv2=+∞/integraltext −∞σ2 v(x,t)dx +∞/integraltext −∞/an}bracketle{tv(x,t)/an}bracketri}ht2dx= 3.356 (76) does not depend on tand is of the order of unity. The situation radically changes on the linear stage when the ﬁeld is described by the equation (26). At this stage we have the self-similar evolut ion of the ﬁeld v(x,t) =B t¯Vlin/parenleftiggx√ 2νt/parenrightigg , (77) where ¯Vlin(y) is deﬁned as ¯Vlin(y) =y√ 2πexp/bracketleftigg −y2 2/bracketrightigg , (78) andBis a random amplitude (45) with nonzero mean value. For the ga ussian distribution ofFin (29) we have from (14) /an}bracketle{tB/an}bracketri}ht=+∞/integraldisplay −∞/parenleftig eM2(x)Re2 0/2−1/parenrightig dx. (79) Here we introduce the eﬀective Reynolds number Re0 Re0=/an}bracketle{tF2(x)/an}bracketri}ht1/2/2ν=σψ/2ν. (80) From (79) we see that mean value /an}bracketle{tB/an}bracketri}htdoes not depend on the ﬁne structure of the carrier f(x) and is always positive. The positive mean value /an}bracketle{tB/an}bracketri}htis a result of generation of mean ﬁeld at the nonlinear stage. Here we consider the case of larg e Reynolds number, where we can use the asymptotic expression for B(45). The n-th moment of Bwill be expressed through the probability distribution function (p.d.f.) of A(63): /an}bracketle{tBn/an}bracketri}ht= (4πL2 0ν)n/2/integraldisplayenA/2ν An/2W(A)dA. (81) 19Using the steepest descent method for Re0≫1 we have from (63), (81) /an}bracketle{tBn/an}bracketri}ht=/parenleftbigg2π n/parenrightbiggn/2 Ln 01 Ren 0en2Re2 0/2. (82) Thus we have very fast growing of the momentum with increasin gn. From (77) one can see that at the linear stage both mean ﬁeld and variance are se lf-similar, and that the relative integral ﬂuctuation δv2does not depend on t. But in contrast to the nonlinear stage (76), in the linear stage the relative integral ﬂuctua tion of the ﬁeld δv2is extremely large for large Reynolds number: δv2=/an}bracketle{tB2/an}bracketri}ht − /an}bracketle{tB/an}bracketri}ht2 /an}bracketle{tB/an}bracketri}ht2≃1 2eRe2 0. (83) We have similar behaviour also for the relative ﬂuctuation o f the energy δE(t) (64). At the nonlinear stage at σψt≫L2 0we have from (21), (63), (64) δE(t) =2√π Γ2(5/4)−1≃3.376. (84) Thus at the nonlinear stage δE(t) does not depend on the initial scale L0and the varianceσψof the p.d.f. of amplitude A. At the linear stage the ﬂuctuation of the energy is determine d by the fourth and the second moment of B, and from (82) we have also a strong enhancement of energy ﬂuctuation (84) δE(t) =1 4e4Re2 0. (85) 4 The evolution of a pulse with monochromatic car- rier In this section we consider the evolution of a pulse with mono chromatic carrier ψ0(x) =M(x)Acosk0x, v 0(x)≃M(x)a0sink0x. (86) HereA=a0/k0, and we assume that the length scale L0of the modulation function M(x) is much greater then the period l0of the carrier ( l0= 2π/k0). Below we consider two 20large parameters Re0=A/2νandL0/l0. For diﬀerent ratios of these parameters we have diﬀerent regimes of wave evolution. The pure monochromatic signal ( M(x)≡1) is characterized by the nonlinear time ts,nl= 1/a0k0(”s” stands for ”signal”) and the linear time ts,lin= 1/νk2 0[21]. Atts,nl≪ t≪ts,lin(Re0≫1) the monochromatic wave transforms into the sawtooth wave with the slopev′ x= 1/t. The shock amplitude △V=l0/2t, as well as the energy of the wave E=l2 0/12t2, does not depend on the initial amplitude. The shockwidth δ= 2ν/△V= 4νt/l0 (87) increases with time, and is, at t∼ts,lin, of the same order as a period. For t≫ts,lin we have the linear regime of evolution, where the wave form is sinusoidal again with exponentionally decaying amplitude v(x,t) = 2νk0exp[−νk0t2]sink0x. For a pulse with monochromatic carrier a large-scale compon entvl(x,t) is generated. The interaction of the high frequency component with the lar ge-scale wave vl(x,t) will change the evolution of the carrier. 4.1 The nonlinear stage of evolution at large Reynolds numbe r Below we give a short summary of pulse evolution at ν→0 based on the paper [31], and discuss the inﬂuence of ﬁnite dissipation on the evolution o f large scale and high frequency carrier. The nonlinear eﬀect leads to the generation of the large-sca le component vl(x,t), and att≫ts,nlthis component obtains the stationary form vl(x,t) =−M′(x)A, (88) which is equal to the form of simple pulse v0(x) =M′(x)Awith the same initial potential ψ0(x) =M(x)A. The evolution of the large-scale component is characteriz ed by the nonlinear time tM,nl=L2 0/A(51), and (88) is valid for t≤tM,nl. At time tint=L0l0 A(89) 21the energy of large-scale and small-scale components are of the same order. At t≫ts,nl the evolution of the large-scale component is equal to the ev olution of a simple pulse with ψ0(x) =M(x)A. ForM(x) = (1 −x2/2L2 0),\|x\|<√ 2L0the evolution of the large-scale component is described by the expressions (41), (42), (52). At t≫ts,nl, in the limit ν→0, the evolution of the large-scale component is determined by onl y one parameter A=a0/k0of the initial perturbation. Att≫ts,nlthe amplitudes △Vof the shocks of the small-scale components do not depend on the initial amplitude: △V=l0/2t, but due to the interaction with large-scale component they have nonzero velocity Vr≃ −AM′(xr) =vl(xr) (88), where xris the initial position of the shock. The distance between the shoc ks increases with time as l(t) =l0(1 +At/L2 0). (90) The collision of the shocks of small-scale components movin g with constant velocity of ﬁne structure with shocks of the large-scale structure (42) leads to decrease of the number of shocks (see ﬁg. 3 from [31]). The last collision occurs at t imet=8L2 0 l2 0tM,nl, and after this time a pure N-wave remains. At ﬁnite Reynolds number the width of the shocks of the large- scale component in- creases with time (87). The linear spreading of small struct ure leads to the increase of the distance between the shocks (90). Comparing (87) with (90) w e ﬁnd that, if the initial Reynolds number satisﬁes the condition A 2ν=Re0≫L2 0 l2 0, (91) then the nonlinear structure of shocks will be conserved. Th is is because the relative shock width is δ(t) l(t)= 4ν AL2 0 l2 0. (92) In the opposite case, under the condition 1≪Re0≪L2 0/l2 0 (93) 22the nonlinear structure will dissipate before the nonlinea r distortion of large-scale com- ponent begins. The evolution of the large-scale component at ﬁnite Reynold s number will be described by the same equations as the evolution of a simple wave (47) - ( 49). Only the position of the shocks xs(t) will be determined by some other equation (23), where Leffdepends on the ﬁne structure of the initial pulse. This dependence le ads to the sensitivity of the old-age behaviour on the ﬁne structure of the initial pulse. 4.2 Old-age linear stage evolution of pulse with monochroma tic carrier The ﬁnal linear stage of evolution is described by the equati on (26), where the constant B(14) is now given by B=+∞/integraldisplay −∞{exp [Re0M(x)cosk0x−1]dx}. (94) At large Reynolds number Re0=A/2νthe constant Bmay be written in the form B=Leffexp[Re0] (17). In fact for the large Reynolds number the main contrib ution in the integral in (94) comes from the neighborhood of points yr=l0r(l0= 2π/k0, r= 0,±1,±2,...). An evaluation of Bby the steepest descent method then gives B=/summationdisplay r√ 2π k0/radicalig Re0M(yr)exp[Re0M(yr)]. (95) From (95) we see that the prefactor Leffin (17) strongly depends on the ratio of two large numbers: the Reynolds number Re0and the number L2 0/l2 0(l0= 2π/k0). When the relation (91) is valid, then only the ﬁrst term r= 0 in (95) is signiﬁcant, and from (17), (95) we have Leff=l0√2πRe 0. (96) In the case of moderate Reynolds number, when the condition ( 93) is valid, we have from (17,95) Leff=L0 Re0. (97) 23The results above should be compared with expression (45) of Bfor the simple pulse for which we have Leff=L0/radicalig 2π/Re 0. One can see that the modulation leads to faster transformation of the wave into the linear regime of evoluti on and consequently to decrease of the amplitude of the wave ( B∼Leffexp[A/2ν]) at the old-age stage. 5 The evolution of a pulse with noise carrier In this section we will study the evolution of a pulse with noi se carrierf(x) =−F′(x) (30). We assume that the potential F(x) is homogeneous gaussian noise with rapidly decreasing covariance function Bψ(ρ) =/an}bracketle{tF(x)F(x+ρ)/an}bracketri}ht=σ2 ψR(ρ) =σ2 ψ/parenleftigg 1−ρ2 2l2 0+ρ4 4l2 1+.../parenrightigg . (98) In the limit of vanishing viscosity the continuous homogene ous ﬁeldv0(x) =f(x) transforms into sequence of sawtooth pulses with equal slop ev′ x= 1/tand with random position of shocks. Due to the collision and merging of the sh ocks, their number decreases and the characteristic scale of random ﬁeld increases. This eﬀect makes all the statistical properties of the ﬁeld self-similar and they are determined by only one scale l(t), which can be interpreted as characteristic distance between the z eroes ofv(x,t) or between the shocks [21,25,26]. The evolution of the integral scale l(t) in time due to merging of the shocks ( [21], see eq. 4.15, p. 170) l(t) = (σψt)1/2ln−1/4/parenleftiggtσ2 v 2πσψ/parenrightigg (99) depends on only two integral characteristics of the initial homogeneous ﬁeld σ2 ψ=/an}bracketle{tF2(x)/an}bracketri}ht, σ2 v=σ2 ψ/l2 0=/an}bracketle{tf2(x)/an}bracketri}ht. (100) Herel0is the correlation length of the initial potential. Due to th e merging of the shocks the energy density of the random wave /an}bracketle{tv2(x,t)/an}bracketri}ht=l2(t)/t2∼t−1decreases slower then the energy of harmonic perturbation. 24In case of the ﬁnite Reynolds number the thickness of the shoc k in the sawtooth wave originating from a monochromatic wave is given in (87). We ha ve the same expression for a random wave as well, where △Vis the random amplitude of the shock. For the estimations we can assume △V∼l(t)/t. The ratio between the integral scale l(t) and internal scale δ(t) is the eﬀective Reynolds numberRe(t): Re(t)≈l(t) δ(t)≈l(t)△V(t) ν∼Re0ln−1/2/parenleftbiggt tnnl/parenrightbigg , (101) whereRe0andtn,nl(”n” stands for ”noise”) are deﬁned as Re0=σψ/2ν, tn,nl=l2 0/σψ. (102) Thus the eﬀective Reynolds number Re(t) logarithmically slowly decreases with time, and the linear stage of evolution starts at very large time tn,lin≃tn,nlexp(Re2 0)≫tn,nl, (Re(tn,lin)∼1). At this time the nonlinear eﬀects become unimportant and the noise enters into the linear mode where its damping is mainly deter mined by linear dissipation. On the base of the solution (8), as it was shown in [21], the ene rgy decays as σ2 v(t) = (νl2 0/t3)1/2Re1/2 0eRe2 0. (103) At the linear stage the distribution of the random ﬁeld conve rges to the distribution of the homogeneous Gaussian ﬁeld with zero mean velocity and variance according to (103) [23]. The evolution of the pulse with noise carrier in the limit of v anishing viscosity ( ν→ 0) was considered in the paper [31]. It was shown that an initi al perturbation v0(x) transforms to an N-wave. In the case, when the scale of the car rierl0is much smaller then scale of modulation function L0, it was shown that the ﬂuctuation of the shock positions is relatively small. Below we consider the properties of the pulse with noise carr ier at the nonlinear stage for ﬁnite Reynolds number and the old-age behaviour of the pu lse. 255.1 Nonlinear stage of evolution of pulse with noise carrier In the case of vanishing viscosity we can introduce two chara cteristic times: the nonlinear time for noise carrier evolution tn,nl=l2 0/σψand nonlinear time for modulation evolution tM,nl=L2 0/σψ. Att≫tn,nlthe initial wave transforms into the sequence of sawtooth pulses, with the integral scale lM(x,t)≃M1/2(x)l0(t/tn,nl)1/2and the energy density ¯E(x,t) =l2 M(x,t)/t2≃M(x)σ2 v(t/tn,nl)−1depending slowly on the coordinate [31]. At this stage the nonlinearity leads to partial depression of m odulation and to generation of the mean ﬁeld vl(x,t) =/an}bracketle{tv(x,t)/an}bracketri}ht=−M′(x)σψ(lnM(x)t/2πtnnl)1/2[31]. Due to the merging of the shocks their number decrease, and at t≫tM,nlthe initial pulse with noise carrier transforms into an N-wave with random positio ns of zero and shocks. The position of the N-wave zero is localized in an narrow region △l≃L0/(lnL2 0/2πl2 0)1/2near the center of the initial pulse [31]. The position of the shoc ks are determined by the equation (20), where A=ψmis the value of absolute maximum of the initial potential ψ0(x) (29). It was shown that the cumulated probability QH(H) = Prob(ψm< H) has the form [31] QH(H) = exp[ −N∞(H)], N∞(H) =/parenleftiggσ2 ψ H2L2 0 2πl2 0/parenrightigg1/2 exp/bracketleftigg −H2 2σ2 ψ/bracketrightigg . (104) HereN∞(H) is the mean number of intersections of level Hby the initial potential ψ0(x) in the interval \|x\|<L∗(9),L0is a characteristic scale of the modulation function M(x) (32) (L0∼L∗),l0=σψ/σvis an integral scale of the Gaussian homogeneous carrier potentialF(x) andσψis its variance (100). The mean velocity /an}bracketle{tv(x,t)/an}bracketri}ht, withv(x,t) given by (68) and its variance σ2 v(x,t) are described by equations (69), (70), where now the cumulative probability QH(H) is deter- mined by equation (104), and not by the error function in (71) . It is easy to see from (74) that both mean ﬁeld and variance have self-similar behaviou r. From (69) , (70), (104) we have (cf. (74)) ¯V(y,N) =y(1−qN(y)),¯σ2 V(y,N) =y2qN(y)(1−qN(y)), (105) 26qN(y) =QH(y2σψ/2) = exp/bracketleftigg −√ 2N y2exp/parenleftigg −y4 4/parenrightigg/bracketrightigg , (106) wherey=x/√σψt, andNis a large parameter, proportional to the number of correlat ion lengths on the whole extension of the pulse: N=L0√ 2πl0. (107) Contrary to the case of the simple initial pulse (75), in the c ase of pulse with noise carrier the mean ﬁeld and its variance have two diﬀerent scal es forN≫1 (see ﬁgs. 1 and 2). The two scales in the ”b” and ”c” curves in the ﬁgures ar e the width of the shock and the length of the pulse. In the ”a” curves only the length o f the pulse remains. For N≫1 the function qN(y)≃0 aty < y ∗≃(4 lnN)1/4. The function qN(y) increases rather fast to 1 in the narrow region ( y−y∗)/y∗≃(4 lnN)−1. Thus forN≫1 the mean ﬁeld has the N-wave similar structure with the dime nsionless shock position ys=y∗= (4 lnN)1/4=/parenleftigg 2 lnL2 0 2πl2 0/parenrightigg1/4 . (108) The relative width of the shock of the mean ﬁeld is △xs xs=△y y∗≃1 4 lnN=1 2 ln(L2 0/2πl2 0). (109) In the neighborhood of the shock position we can introduce a n ew variable z: y=y∗(1 +z/y4 ∗), (110) and from (106) we see that the shape of the front is described b y double exponential distribution qN(y) =qd((y−y∗)y4 ∗), qd(z) =e−e−z. (111) The variance σ2 vis diﬀerent from zero in a narrow region (109) near the shock p osition y∗and for the relative integral ﬂuctuation of the ﬁeld (76) we h ave from (105), (106) δv2≃△y y∗≃1 4 lnN≪1. (112) 27A ﬁnite Reynolds number leads to a ﬁnite width of the shock. Th e shock structure in each realization is described by the expressions (48), (49). The shifting of the shock position xsfrom the zero viscosity limit xs=√2ψmt(A≡ψm) is given in (23). At the nonlinear stage we can neglect this shifting. Then in each realization we have self-similar evolution of the pulse, and the relative width of the shock δ/xs= 2ν/ψmdoes not depend on time. While forN≫1 (L0≫l0) the maximum ψmof the initial potential is localized in the narrow region △H/H 0≃(lnN) nearH0≃σψ(2 lnN)1/2( [31] formulas (107), (113)), we can estimate the relative width of shock front as δ xs≃2ν H0≃1 Re01 (lnN)1/2, Re 0=σψ 2ν. (113) Comparing (113) with (109) we see that the inﬂuence of ﬁnite v iscosity on the mean ﬁeld is unimportant if/parenleftigg lnL2 0 2πl2 0/parenrightigg1/2 ≃(lnN)1/2≪Re0. (114) In the opposite case, for extremely large ratio L0/l0, the width of the mean ﬁeld will be determined by the viscosity. The displacements of shock position xs(23) from the vanishing viscosity position xs=√2ψmtleads ﬁnally to the depressing of nonlinear eﬀects. For t≫tlin∼L2 effeψm/ν/ν (cf.(27)) we have the linear stage of evolution. While for th e random perturbation we have a large ﬂuctuation of tlin, the cumulative action of nonlinearity, which is proportio nal to tlin, leads to strong ﬂuctuation of the ﬁeld at the linear stage. 5.2 Old-age linear stage of evolution of a pulse with noise ca rrier At the old-age stage, when the evolution of the pulse is descr ibed by linear solution (26) all the properties of the wave are determined by the constant B, given in (14). The potential ψ0(x) (29) is a random Gaussian function. The mean value of Bis given by the equation (79), and does not depend on the ﬁne structure of the carrier. ForRe0=σψ/2ν≫1 we have from (82) /an}bracketle{tB/an}bracketri}ht=L0√ 2π1 Re0eRe2 0/2. (115) 28Let us ﬁrst consider the case of relatively large Reynolds nu mber, when the integral (14) may be calculated using the steepest descent method, an d only the contribution of the absolute maximum is important: B=/radicaligg 4πν \|ψ′′ m\|eψm/2ν. (116) Hereψm=ψ0(xm) is the value of absolute maximum of ψ0, andψ′′ m=ψ′′ 0(xm) is the second derivative of the potential at this point. To ﬁnd the s tatistical properties of Bwe need to know the joint probability distribution of ψmandψ′′ m. From (98) it is easy to ﬁnd the correlation coeﬃcient for ψmandψ′′ m:r=−l2 1/l2 0. If we consider the conditional probability distribution function W(ψ′′ m/ψm), we then obtain for the conditional mean value/an}bracketle{tψ′′ m/an}bracketri}htψmand variance/parenleftig σ2 ψ′′m/parenrightig ψm: /an}bracketle{tψ′′ m/an}bracketri}htψm=−ψmrσ2 ψ′′m/σ2 ψm=−ψm/l2 0, /parenleftig σ2 ψ′′m/parenrightig ψm=σ2 ψm(1−l4 1/l4 0)/l4 1. (117) WhileBincreases exponentionally with ψmonly the maximum ψm≫σψgives a signiﬁcant contribution to the average characteristics of B. Thus in (116) we can substitute /an}bracketle{tψ′′ m/an}bracketri}htψm (117) instead of ψ′′ m, and so we have B=l0/radicaligg 4πν ψmeψm/2ν. (118) From (104) we have for the probability distribution functio n of the absolute maximum of the initial potential ψ0(x): WH(H) =∂QH ∂H=−N′ ∞(H)e−N∞(H). (119) This function is localized near H0=σψ(2 lnN)1/2(120) and has a tail for H→ ∞: WH(H) =N σψe−H2/2σ2 ψ. (121) 29For the calculation of the moments of Bwe can use the steepest descent method using the tail of the probability distribution function (121): /an}bracketle{tBn/an}bracketri}ht=L0ln−1 0/parenleftbigg2π n/parenrightbiggn/21 Ren 0en2Re2 0/2. (122) We have the main contribution for the n-th moment (122) at the pointH∗,n≃nσψ/2ν= nRe 0. Comparing H∗,nwithH0given in (120) we see that the inequality (114) is necessary for (122) to be valid. Comparing (122) with the relation (82) for the moment of Bfor a simple pulse we see that for the small-scale noise carrier the moments n≥2 depend on the integral scale l0 of the carrier. The relative integral ﬂuctuation δv2of the ﬁeld (76) on this stage is expressed for two ﬁrst moments of Bas δv2=/an}bracketle{tB2/an}bracketri}ht /an}bracketle{tB/an}bracketri}ht2−1≃l0 L01 2eRe2 0, (123) and when the condition (114) is valid it is very large in contr ast to the nonlinear stage (112), where δv2≪1. For the noise carrier with the scale l0≪L0the relative integral ﬂuctuation is the small factor l0/L0times the ﬂuctuation of the simple pulse with random Gaussian amplitude (83). The calculation of /an}bracketle{tB/an}bracketri}ht,/an}bracketle{t△B2/an}bracketri}htmay be done directly on the base of the integral (14), and for large Reynolds number we have the same equation (123) . For (l0/L0)≪exp[−Re2 0] we have relatively small ﬂuctuation of δv2, and with increasing of l0/L0the probability distribution of Bapproaches slowly the normal distribution (normalization ). This normal- ization is similar to the normalization of the homogeneous ﬁ eld at the old-age stage [23]. But while the moment Bnincreases exponentionally with n, we have weak convergence to the normal distribution of Blike in [23]. 6 Discussion and conclusion We have investigated the evolution of pulses with complex st ructure in nonlinear media, described by Burgers’ equation. The investigation for vani shing viscosity was done in our 30paper [31]. There it was shown that the asymptotic form of an a rbitrary initial pulse with zero area is an N-wave. It was also shown that the shock positi ons of the N-wave are determined by only one parameter of the initial perturbatio n - the value of the absolute maximum of the initial potential ψ0(x) (v0(x) =−∂ψ0(x)/∂x). It was also shown in the paper [31] that for the noise carrier the ﬂuctuations of the s hock positions are relatively small if the carrier length scale l0is much smaller than the modulation length scale L0. In the present paper we have considered the evolution of puls es with complex structure at large but ﬁnite Reynolds number. On the base of the Hopf-Co le solution it is shown that the ﬁnite viscosity leads to a ﬁnite shock width δ, which does not depend on the ﬁne structure of the initial pulse and is fully determined by the shock position in the zero viscosity limit. The other eﬀect is the shift of the shock pos ition from the zero viscosity limit position. This shift depends on the ﬁne structure of th e initial pulse, and as a consequence the time tlin, at which the nonlinear stage of evolution changes to the lin ear stage, is determined not only by the value of the maximum of th e initial potential but also by the ﬁne structure of the pulse. Because the amplitude of the pulse at the linear old-age stage is determined by the time tlin, the old-age amplitude is also sensitive to the inner structure of the pulse. In this paper the evolution of a simple N-pulse with regular a nd random initial ampli- tude and of pulses with monochromatic and noise carrier is co nsidered. It is shown that the nonlinearity of the medium leads to the generation of a no n-zero mean ﬁeld from an initial random ﬁeld with zero mean value. It is also shown tha t, at the nonlinear stage, the relative ﬂuctuation of the ﬁeld and its energy is of unit o rder for simple pulses and small for pulses with complex inner structure ( l0<<L 0). However, at the old-age linear stage this ﬂuctuation increases exponentially with increa sing initial Reynolds number. Acknowledgements This work was supported by a grant from the Royal Swedish Acad emy of Sciences, by the grant INTAS No 97-11134 and by the RFBR grant No 99-02- 18354. Sergey Gurbatov 31thanks the staﬀ of the Department of Mechanics (KTH) and othe r friends in Sweden for their warm hospitality. 32-4 -2 0 2 4-2-1012 a b c Figure 1: Self-similar mean velocity ﬁeld form ¯V(y) for the pulse with random initial amplitude (75) (curve a) and for the pulse with noise carrier (105) for N= 100 (curve b) andN= 1000 (curve c). 33-4 -2 0 2 40.00.40.81.2 abc Figure 2: Self-similar ﬂuctuation form ¯ σ2 V(y) for the pulse with random initial amplitude (75) (curve a) and for the pulse with noise carrier (105) for N= 100 (curve b) and N= 1000 (curve c). 34References [1] J.M. Burgers: The Nonlinear diﬀusion equation. D. Reide l, Dordrecht, 1974. [2] G.B. Whitham: Linear and nonlinear waves. Wiley, New Yor k, 1974. [3] O. Rudenko, S. Soluyan: Theoretical foundations of nonl inear acoustics. Plenum, New-York, 1997. [4] D.G. Crighton: Model equations of nonlinear acoustics. J. Fluid Mech. 11 (1979) 11–33. [5] P.L. Sachdev: Nonlinear Diﬀusive Waves: Cambridge Univ ersity Press 1987. [6] D.G. Crighton, J.F. Scott: Asymptotic solutions of mode l equations in nonlinear acoustics. Phil. Trans. R. Soc. Lond. A292 (1979) 101–134. [7] J.F. Scott: Uniform asymptotics for spherical and cylin drical nonlinear acoustic waves generated by a sinusoidal source: Proc. R. Soc. Lond. A 375 (1981) 211–230. [8] B. O. Enﬂo: Asymptotic behavior of the N-wave solution of Burgers’ generalized equation for cylindrical acoustic waves: J. Acoust. Soc. Am . 70 (1981) 1421–1423. [9] B. O. Enﬂo: Saturation of a nonlinear cylindrical sound w ave generated by a sinu- soidal source. J. Acoust. Soc. Am. 77 (1985) 54–60. [10] P.L. Sachdev, V.G. Tikekar, K.R.C. Nair: Evolution and decay of spherical and cylindrical N-waves. J. Fluid Mech. 172 (1986) 347–371. [11] P.L. Sachdev, K.R.C. Nair: Evolution and decay of spher ical and cylindrical acoustic waves generated by a sinusoidal source. J. Fluid Mech. 204 (1 989) 389–404. [12] P.W. Hammerton, D.G. Crighton: Old-age behaviour of cy lindrical and spherical nonlinear waves: numerical and asymptotical results. Proc . R. Soc. Lond. A422 (1989) 387–405. 35[13] B.O. Enﬂo: Some analytic results on nonlinear acoustic wave propagation in diﬀusive media. Radioﬁzika 36 (1993) 665–686. [14] B.O. Enﬂo: Saturation of nonlinear spherical and cylin drical sound waves. J. Acoust. Soc. Am. 99 (1996) 1960–1964. [15] B.O. Enﬂo: On the connection between the asymptotic wav eform and the fading tail of an initial N-wave in nonlinear acoustics: Acustica - Acta Acustica 84 (1998) 401–413. [16] J.D. Cole: On a quasi-linear parabolic equation occurr ing in aerodynamics. Quart. Appl. Math. 9 (1951) 225–236. [17] E. Hopf: The partial diﬀerential equation ut+uux=uxx. Comm. Pure Appl. Mech. 3 (1950) 201–230. [18] S.N. Gurbatov, D.B. Crighton: The nonlinear decay of co mplex signals in dissipative media. Chaos 5(3) (1995) 524–530. [19] O. Brander, J. Hedenfalk: A new formulation of the gener al solution to Burgers’ equation. Wave Motion 28 (1998) 319–332. [20] J.R. Angilella, J.C. Vassilicos, Spectral, diﬀusive a nd spiral ﬁelds, Physica D (1998), in press. [21] S.N. Gurbatov, A.N. Malakhov, A.I. Saichev: Nonlinear random waves and turbu- lence in nondispersive media: waves, rays, particles. Manc hester University Press, Manchester, 1991. [22] S.E. Esipov: Energy decay in Burgers turbulence and int erface growth: the problem of random initial conditions II: Phys. Rev. E 49 (1994) 2070– 2081. [23] S. Albeverio, A.A. Molchanov, D. Surgailis: Stratiﬁed structure of the universe and Burgers’ equation – a probabilistic approach. Prob. Theory Relat. Fields 100 (1994) 457–484. 36[24] J.P. Bouchaud, M. M´ ezard, G. Parisi: Scaling and inter mittency in Burgers turbu- lence. Phys. Rev. E 52 (1995) 3656–3674. [25] S.A. Molchanov, D. Surgailis, W.A. Woyczynski: Hyperb olic asymptotics in Burgers’ turbulence and extremal processes. Comm. Math. Phys. 168 (1 995) 209–226. [26] S.N. Gurbatov, S.I. Simdyankin, E. Aurell, U. Frisch, G . Toth: On the decay of Burgers turbulence. J. Fluid Mech, 344 (1997) 339–374. [27] G.M. Molchan: Burgers equation with self-similar Gaus sian initial data: tail proba- bilities. J. of Stat. Phys. 88 (1997) 1139–1150. [28] J.P. Bouchaud, M. M´ ezard, Universality classes for ex treme-value statistics. Journal of Physics A - Mathematical and General 30 (1997) 7997–8015. [29] S.N. Gurbatov, I.Y. Demin: Transformation of strong ac oustic noise pulses. Akust. Zh. 28(5) (1982) 634-640. [Eng. translation: Sov. Phys. aco ustics 28(5) 375–379.] [30] P.L. Sachdev, K.T. Joseph, K.R.C. Nair: Exact N-wave so lutions for the non-planar Burgers’ equation. Proc. R. Soc. Lond. A445 (1994) 501-517. [31] S.N. Gurbatov, B.O. Enﬂo, G.V. Pasmanik: The decay of pu lses with complex struc- ture according to Burgers’ equation. Acustica - Acta Acusti ca 85 (1999) 181-196. [32] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev: Integr als and series, Vol. 1: Ele- mentary functions. Gordon and Breach, Glasgow, 1988. 37
arXiv:physics/0010035v1 [physics.class-ph] 14 Oct 2000COMMENT ON “DO ZERO-ENERGY SOLUIIONS OF MAXWELL EQUATIONS HAVE THE PHYSICAL ORI- GIN SUGGESTED BY A. E. CHUBYKALO?” BY V.V. DVOEGLAZOV A. E. Chubykalo Escuela de F´ ısica, Universidad Aut´ onoma de Zacatecas Apartado Postal C-580 Zacatecas 98068, ZAC., M´ exico Received October 6, 1999 Key words: electromagnetic ﬁeld energy. Dvoeglazov’s paper is not trivial because we know the classi cal def- inition of any non-trivial truth: a truth is not trivial if and only if anopposite statement is not false(and vice versa, of course). So I strongly recommend his paper for a reader. But I have a few notes: 1) V. Dvoeglazov writes in his footnote number 7: ”In my opin- ion, equations (20,21) of ref. [1] are just another form of th e Maxwell equations for this particular case, in the sense that there i s no new physical content if one expects that the Maxwell electrodyn amics describes also the Coulomb interaction.” The point is that precisely this (“ another ”) form of the Maxwell equations (namely with totalderivatives!) allows us to describe the Coulomb interaction as an action-at-a-distance (it was shown in [2]). It is well-known that generally accepted Maxwell equa- tions (with partial derivatives only) describe just transverse1elec- tromagnetic waves in vacuum which spread with a limited velo city 1“transverse” in this case means that vectors EandBare per- pendicular with respect to wave vector kin every point. 12 A. E. Chubykalo (i.e. they describe so called short-range interaction only). That is why there is new physical content2in Eqs. (18-21) [1]. Note also that V. Dvoeglazov here erroneously mentioned just Eqs. (20 ,21). The point is that Eqs. (20,21) describe exclusively instant aneous (action-at-a-distance) interaction while the whole syste m of our equations (18-21) (Eqs. (1-4) in this Comment) describe bot h in- stantaneous and short-range. These equations are written a s two uncoupled pairs of diﬀerential equations: ∇ ×H∗=1 c∂E∗ ∂t, (1) ∇ ×E∗=−1 c∂B∗ ∂t(2) and ∇ ×H0=4π cj−1 c/parenleftBigg/summationdisplay iVi· ∇/parenrightBigg E0, (3) ∇ ×E0=1 c/parenleftBigg/summationdisplay iVi· ∇/parenrightBigg H0. (4) This system follows from: ∇ ×H=4π cj+1 cdE dt(5) ∇ ×E=−1 cdH dt(6) where E=E0+E∗=E0(r−rq(t)) +E∗(r, t), (7) H=H0+H∗=H0(r−rq(t)) +H∗(r, t), (8) ris the ﬁxed point of observation, rqis the point of the location of a moving charge qat the instant t, the total time derivative of any vector ﬁeld value E(orH) can be calculated by the following rule: dE dt=∂E∗ ∂t−/parenleftBigg/summationdisplay iVi· ∇/parenrightBigg E0, (9) 2I should to note here that a considerable number of papers hav e recently been published which declare (prove?) an existenc e of so called longitudinal electromagnetic waves in vacuum but these ideas still are not generally accepted in classical electro dynamics (see, e.g., review [3]).Comment on ”Do Zero-Energy ...” 3 hereViare velocities of the particles at the same moment of time of observation.3 2) V. Dvoeglazov’s phrase: ”The main problem with the Chuby- kalo derivation is the following: the integrals are divergent when they extend over all the space. ... But, to the best of my knowl - edge, some persons claimed that such procedures do not have s ound mathematical foundation.” I do not see any mathematical prohibition to use such procedures. See, e.g., [4] (Vol. 2, ch. 17, §1, point 282). Fihtengoltz here considers the following deﬁnition of the integral with inﬁnite limits: ∞/integraldisplay af(x)dx= lim A→∞A/integraldisplay af(x)dx. (10) If this limit ( rhs10) does not exist or(attention!) is inﬁnite, they say that this integral diverges . But then Fihtengoltz adds, however, that in the case when the limit ( rhs10)exists and it is equal to inﬁnity one can consider the inﬁnite limit ( rhs10) as a value of the integral ( lhs10). V. Dvoeglazov asks: ”Furthermore, even if one accepts its va - lidity the total energy resulting from integration of (28) o ver the whole space is to be inﬁnite!?”4 Why not? Many years ago, in Newton’s times, for example, nobody (I mean, physicists) doubted that the Universe had an in- ﬁnite number of stars. It meant that the mass of the Universe w as inﬁnite. And thismass is no more than/integraltext ∞̺dVover the inﬁnite vol- ume, where ̺is alimited mass density of the Universe. Nowmany physicists believe that the mass (and, of course, the total energy) of our Universe is limited but their statements have no any evid ence (however the statement that the Universe has an inﬁnite mass also has no evidence). Unfortunately, it is still a matter of the b elief. At last V. Dvoeglazov cites: ”It was noted by Barut [5, p.105] that in the case of non-vanishing ﬁelds at the spatial inﬁnit y ‘we cannot expect to ﬁnd globally conserved quantities ’.” I have only one question here: Is it possible experimentally to ﬁndglobally conserved quantities? 3) The ﬁnal phrase of V. Dvoeglazov’s paper is: ”... we are not yet convinced in the necessity of correction of the formula f or the 3Note (see [2]) that unlike the ﬁelds {}∗the ﬁelds {}0do not retard. 4Recall that Eq. (28) in [1] is: w=2E∗·E0+ 2H∗·H0+E2 0+H2 0 8π.4 A. E. Chubykalo energy density for radiation ﬁeld because of the absence of ﬁ rm experimental and mathematical bases in [1].” In turn, I explained in this short comment that if we suppose that the radiation ﬁeld exists in inﬁnity ( mathematically it is pos- sible, see above, on the other hand the opposite statement ha s no evidences) we must correct the formula of the energy density for the electromagnetic ﬁeld. REFERENCES 1. A. E. Chubykalo, Mod. Phys. Lett. A13, 2139 (1998). 2. A. Chubykalo and R. Smirnov-Rueda, Phys Rev. E53, 5373 (1996); ibid. E 55, 3793E (1997); Mod. Phys. Lett. A12, 1 (1997). 3. V. V. Dvoeglazov, Hadronic J. Suppl. 12, 241 (1997). 4. G. M. Fihtengoltz, “ Osnovy Matematicheskogo Analiza [Foun- dations of Mathematical Analysis]” (GI TTL, Moscow, 1956) (in Russian). 5. A. O. Barut, “ Electrodynamics and classical theory of ﬁelds and particles ” (Dover Publications, Inc., New York, 1979).
arXiv:physics/0010036v1 [physics.class-ph] 14 Oct 2000Coulomb interaction does not spread instantaneously Rumen I. Tzontchev, Andrew E. Chubykalo and Juan M. Rivera-J u´ arez Escuela de F´ ısica, Universidad Aut´ onoma de Zacatecas Apartado Postal C-580 Zacatecas 98068, ZAC., M´ exico e-mails: rumen@ahobon.reduaz.mx andandrew@ahobon.reduaz.mx (February 9, 2008) Abstract The experiment is described which shows that Coulomb intera ction spreads with a limit velocity and thus this kind of interaction canno t be considered as so called “instantaneous action at a distance” I. INTRODUCTION As shown in previous works by one of the authors of this articl e [1-4], instantaneous action at a distance is the direct consequence of generally a ccepted classical electrody- namics. Particularly in the aforementioned works it was sho wn that the Coulomb ﬁeld (unlike a so called freeﬁeld) is spread instantaneously. On the other hand theories exist which aﬃrm that any kind of electric ﬁeld spreads with a limit ed rate, but these theories require signiﬁcant modiﬁcation within classical electrod ynamics [5]. However, one cannot choose theoretically one theory over another; it must be don e through experiment. Thus, the propose of this present work is to verify experimentally if the Coulomb ﬁeld is really spread instantaneously or not. In the following section we describe an experiment which pro vided us with appropriate framework for verifying the instantaneous spread. II. EXPERIMENTAL EQUIPMENT In order to determine the velocity of the Coulomb interactio n, a Coulomb electric ﬁeld generator, which rapidly changes its magnitude, and antenn as which can register this 1electric ﬁeld, are necessary. A. The generator The generator consists of two hollow metallic spheres aandb, connected electrically by means of a plasma discharger and discharge cable (ﬁg.1): Fig1. Coulomb electric ﬁeld generator (regular conﬁguration) The radius of each sphere is R, the distance between the centres of the spheres is equal tol. Let us investigate the ﬁeld at point A, which is on a line which passes through the centres of both spheres and which is at a distance Lfrom the centre of sphere b. Let us call this line the “experimental axis”. Sphere bis charged electrically with a positive charge as far as the b reak-down voltage of the discharger. Then, a spark ﬂies through the discharger , a spark which is a very brief plasma cord of little electrical resistance. As we can see, b oth spheres are electrically connected during the spark by means of the discharge cable al one. Thus begins a process of discharge, during which a part of the electric charge init ially accumulated on sphere b oscillates between sphere aand sphere bwith a high frequency. The distance between this mobile part of the charge and point Achanges with the same frequency. Consequently, the electric ﬁeld at point Aalso oscillates according to the Coulomb law. This ﬁeld has colinear direction at the experimental axis. It is important to note that because of the cilindrical symme try of the experimental construction at point A, the electric ﬁeld is not perceived due to an electromagenet ic transversal wave, regardless of the values of the distances landL. In our examinations further on we will use the quasi-station ary approximation. This approximation can be correctly applied only if the length of the discharge cable, the radius of the spheres and all distances in our experiment are less th an the wavelength of the pro- posed ﬁeld. All our calculations will be carried out supposi ng that “instantaneous action” exists, that is, the velocity of propagation of the Coulomb e lectric ﬁeld is inﬁnite. In this 2case the wavelength will have an inﬁnite value. In case it tur ns out that the electric ﬁeld is propagated with a ﬁnite velocity and if the use of the quasi -stationary approximation is not correct, our calculations can only be considered as th e ﬁrst aproximation in the description of the process. From an electric point of view, both spheres aandbare condensers. Their capacitance is the sum of each sphere’s own capacitance and the capacitan ce in common with the other sphere: C=Cown+Ca,b The discharge cable can be presented as an induction Ldand a resistance Rdconnected in series. When the discharger is in the state of conduction it c an be substituted by a short circuit. Consequently, the variable ﬁeld generator can be p resented as a dipole (ﬁg.2): Fig.2. Electric diagram equivalent to the electric ﬁeld generator . qa(t) andqb(t) are the electric charges corresponding to spheres aandbandUa(t) andUb(t) are the corresponding potentials. Applying the second law o f Kirchoﬀ for this circuit and taking into consideration that each sphere is in itself an eq uipotential surface, we obtain: Rdi(t) +Lddi(t) dt−1 C[qa(t)−qb(t)] = 0, (1) where i(t) is the current which passes through the discharge cable. The process begins by electrically charging sphere bas far as the break-down voltage of the discharger Ub0. Then, when the spark appears on the surface of sphere ba charge of Q0=CUb0 has accumulated. Just as during the process of discharge there is no charge lea k, and the capacitance of the discharge cable is not appreciable, it is always qa(t) +qb(t) =Q0. (2) 3From equations (1) and (2), for the charge qb(t) we obtain the following diﬀerential equa- tion: d2qb(t) dt+Rd Lddqb(t) dt+2 LdCqb(t) =Q0 LdC(3) with the initial conditions in the instant t= 0: dqb(t) dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=0= 0; qb(t)\|t=0=Q0. The solution to this diﬀerential equation is: qb(t) =qme−βtcos(ωt+α) +Q0 2, (4) where β=Rd 2Ld;ω=/radicalBig ω2 0−β2;ω0=/radicalBigg 2 LdC; tan α=−β ω;qm=Q0 2 cosα. For the electric charge of the sphere awe obtain: qa(t) =Q0−qb(t) =Q0 2−qme−βtcos(ωt+α). (5) The electric current discharge i(t) is given by the expression: i(t) =−qmω0e−βtsinωt. (6) The potential of the electric ﬁeld at point Awill be the sum of the potentials generated by the charges qa(t) and qb(t) ϕe(L, t) =qa(t) 4πε0(L+l)+qb(t) 4πε0L= =Q0 8πε02L+l L(L+l)+qm 4πε0l L(L+l)e−βtcos(ωt+α). (7) As we pointed out earlier, in the case of the existence of a ﬁni te velocity of the propagation of the Coulomb interaction, the application of the quasi-st ationary approximation can be not correct and the processes in the generator may only be d escribed qualitatively. However, the basic conclusion of the consideration made is c orrect: there are rapid spatial oscillations of the electric charge along the axis of the exp eriment that produces a rapid change of electric ﬁeld at the point A. 4B. The antenna The antenna is a metallic hollow sphere of radius r, connected by means of a cable to the earth (ﬁg.3). Fig.3. Electric antenna. It is accepted that the earth has a potential equal to zero. Th us, the potential on the surface of the antenna will always be equal to 0 due to the conn ection to the earth. When there is no external electrical ﬁeld the sphere of the antenn a is not electrically charged. Let us suppose that there is an external electrical ﬁeld whos e source is suﬃciently far away from the sphere so that we can accept that the potential o f the external ﬁeld is not altered considerably in the place of the sphere. Then we c an talk of the potential in the region on the sphere of the antenna ϕe,a(t). In this case, there is charge q(t) of such magnitude on the surface of the sphere that equation (8) is fu lﬁlled: ϕe,a(t) +ϕi,a(t) =ϕe,a(t) +q(t) 4πε0r= 0, (8) where ϕi,a(t) is the potential on the surface of the antenna originated by charge q(t), which comes from the earth. Therefore: q(t) =−ϕe,a(t) 4πε0r. (9) When the electric ﬁeld changes, charge q(t) changes. This means that the electric charge passes through the connecting cable between the antenna and the earth, in other words, electric current passes through the connecting cable: ia(t) =dq(t) dt=−4πε0rdϕe,a(t) dt. (10) The current ia(t) can be recorded. C. Experimental conﬁguration The Coulomb electric ﬁeld generator and two antennas are at a distance Ffrom each other, as shown in ﬁg. 4: 5Fig.4. The main diagram of the experimental conﬁguration. The distance from the centre of antenna 1 to the centre of sphe rebof the generator is L. In the cables, which connect the antennas to earth, small R50magnitude resistors are included, which do not aﬀect the working of the antennas. The voltage fall on these resistors is proportional to the current and is supplied at b oth input points to a rapid digital oscilloscope. In this way, the currents passing thr ough the cables are recorded. The discharge in the discharger gives rise to an oscillatory displacement of electric charge along the experimental axis. For this reason, a chang e in the Coulomb electric ﬁeld will be emitted from the generator through the same axis . This change, reaching antennas 1 and 2 will cause the current to ﬂow through the R50resistors and thus, a UR voltage fall upon them. In the case of “instananeous action” , when the Coulomb electric ﬁeld appears simultaneously in the whole space, this UR1(t) voltage fall for antenna 1 satisﬁes the following diﬀerential equation: dUR,1(t) dt+1 4πε0rR50UR,1(t) =dϕe,1(t) dt, (11) wheredϕe,1(t) dtis the ﬁrst derivate of the potential of the Coulomb electric ﬁeld in the region of antenna 1, originated by the generator. The inicial condi tionUR,1(0) = 0 gives us the following solution of this equation: UR,1(t) =D (β−B)2+ω2/bracketleftBig −ωe−Bt+ωe−βtcosωt+ (β−B)e−βtsinωt/bracketrightBig , (12) where D=qmω0 4πε0l L(L+l);B=1 4πε0rR50. The voltage fall UR,2(t) corresponding to antenna 2 obeys the same law and the dif- ference is only in the value of the coeﬃcient D. For antenna 2: D=qmω0 4πε0l (L+F)(L+F+l). The tension UR,1(t) has been supplied to channel 1 of the oscilloscope by means o f a coaxial cable, and the tension UR,2(t) - at channel 2. Consequently, if “instantaneous action” exists, two impulses of equal form and diﬀerent ampl itude should be 6visible on the display of the oscilloscope, which appear sim ultaneously and develop in a parallel fashion in time . We assume that the air and coaxial cable are linear structures which possess no noticeable dispersion f or the frequencies used. The other possibility is that the change in the Coulomb electric ﬁeld is propagated at a ﬁnite velocity, that is, a wave exists. So, the front of the wave pro duced by the Coulomb electric ﬁeld will arrive at antenna 1 ﬁrst and then at antenna 2 after a certain interval of time ∆t. Consequently, two impulses displaced in time ∆ twill be seen on the display of the oscilloscope, which will obey the equation: ∆t=F v, where vis the velocity of propagation of the wave. The interval can b e measured experi- mentally and the velocity of the front of the wave vcan be obtained, which by its essence, is group velocity. III. PRACTICAL EXPERIMENTAL EQUIPMENT The Coulomb electric ﬁeld generator consists of two standar d Van de Graaf generators with the radius of the balls equal to 10 cm. One of the generato rs becomes electrically charged during the experiment, and we shall call it “active” . The other, which is only used for receiving a part of the charge of the active generato r through the discharger and the discharge cable, we shall call “passive”. The genera tors are elevated on insulating supports until the centre of the balls reaches a height of 1.7 m with reference to the earth’s surface. The distance between the centres of both balls is 3 m . The antennas are spheres of a radius of 9.5 cm, whose centres can be found 1.7 m from the e arth’s surface. The centre of antenna 1 is to be found at a ﬁxed distance of 0.5 m fro m the centre of the ball of the Van de Graaf active generator, and the distance be tween the centres of both antennas varies. Fig. 5 gives an overview of the practical ex perimental equipment: Fig.5. Practical experimental equipment (overview). 7Antenna 1 is slightly displaced from the experimental axis s o as not to obstruct the direct visibility between the Van de Graaf and antenna 2 gene rators. If this does not happen, antenna 1 will behave like a screen and the signal fro m antenna 2 decreases drastically. Both antennas are connected at the input of the oscilloscope by means of two high frequency coaxial cables with the characteristic resi stance of 50 Ohms. Antenna 1 is connected to channel 1 and antenna 2 to channel 2. Each cable i s impedance balanced on both sides. The lengths of the cables are equal, with an uncer tainty of 5 mm. The noise at the input of the oscilloscope has a value of 10 mV p−p. The measurement sensitivity of the temporal intervals is 0.3 ns. The signals which are due to the eﬀects of the apparatus (signal penetration from one of the channels of the oscillos cope to the other, bad earth contact, signal penetration through the power supply cable s, etc.) and inductions in the coaxial cables have already been measured. With this pur pose, an experiment has been carried out, in which antenna 1 has approached to a maxim um the active Van de Graaf generator maintaining constant the other variables o f the experiment; the sphere of antenna 2 has been disconnected from the coaxial cable. Th e signal in channel 2 of the oscilloscope in the case of various distances bewteen th e end of the coaxial cable corresponding to antenna 2 and the active Van de Graaf genera tor has been measured. The result is that in all cases, the signal obtained has a valu e below 0 .5% of the useful signal (with the sphere of antenna 2 connected). An analogou s experiment has been done with antenna 1, commuting the exploration of the oscillosco pe to antenna 2. The result was similar. This allows us to exclude corrections in order t o avoid apparatus eﬀects. The temporal symmetry of the antennas of the experimental eq uipment has been veriﬁed. The defect of the aforementioned symmetry can be ob tained from a diﬀerent length of the coaxial cables or an assymetry of both channels of the oscilloscope. With this end, both antennas are disposed symmetrically side-by -side at a distance equal to that of the active Van de Graaf generator. The delay of the fro nt ﬂank of the impulse of channel 1 with reference to the front ﬂank of the impulse of channel 2 in the case of various combinations of the sensitivity of the vertical amp liﬁers of the oscilloscope has been determined. The result is that the delay in all cases is b elow 0.3 ns. For this reason we have considered that, in the following measurements and c alculations, the assymetry shown in time is equal to 0.3 ns. 8The experiments were carried out in the following manner: An tenna 2 is placed at a certain distance Ffrom antenna 1. The oscilloscope is put to work in the framewo rk of the exploration of single ﬁring with a sinchronization alon g the negative front ﬂank of the impulse of channel 1. One waits until the moment of the sponta neous jump of the spark between the electrodes of the discharger. The information o f the impulses of antennas 1 and 2 is recorded in the memory of the oscilloscope. Then the a nalysis of the information begins. It has been predicted theoretically and demonstrated exper imentally that the ﬁrst ﬂanks of impulses from antennas 1 and 2 are negative and have t he same form. For this reason, by means of an adjustment of the sensitivity of the ve rtical ampliﬁers of both channels of the oscilloscope, its parallelism is reached (ﬁ g.6) and then the delay ∆ tof the impulse ﬂank of antenna 2 with reference to the impulse ﬂank o f antenna 1 is measured. Fig.6. Disposition of the antennas’ impulses on the display of the o scilloscope. This investigation has been repeated for diﬀerent distance s between antennas 1 and 2 in the interval of 0.50 m to 1.50 m spaced uniformly 0.10 m. For each value of distance the measurement has been reiterated 20 times. With the purpose o f proving what inﬂuence the power supply cables and the oscillations (which are grad ually propagated along them) have, a complete cycle of investigations has been carried ou t, with the conﬁguration shown in ﬁg.7: Fig.7. Coulomb electric ﬁeld generator (conﬁguration with a turn o f 180◦). The Coulomb electric ﬁeld generator is turned 180◦on the horizontal plane, and with this the passive and active Van de Graaf generators have chan ged places. At the same time, the disposition of the power supply cables and the bloc k of antennas is unalterable. 9According to the theoretical description, the front ﬂanks o f the impulses of channels 1 and 2 in this case should invert their sign from negative to po sitive and the impulses of both channels should maintain sameness of form. This is obse rved in practice and the re- sults of the investigation for ∆ tdo not diﬀer signiﬁcantly from the results obtained using the basic conﬁguration. A basic parasite factor is the trans versal electromagnetic wave which is emitted from the Coulomb interaction generator and is reﬂected on the earth’s surface. It is slightly diﬀerent from the Coulomb interacti on by the sign of its ﬁrst front and by the sign of the ﬂank of the ﬁrst impulse which can be seen on the oscilloscope’s display respectively. In the case of the basic conﬁguration (ﬁg.5), the ﬁrst front of the Coulomb interaction is always negative, while the ﬁrst fron t of the transversal electromag- netic wave is positive. This can be explained using the Elect rodynamic theory in the case of the construction shown of the Coulomb electric ﬁeld gener ator [6]. In order to separate the Coulomb interaction from the signal originated by the tr ansversal electromagnetic wave, the delay observed from the ﬁrst front of the transvers al electromagnetic wave with respect to the ﬁrst front of the Coulomb interaction is of ass istance. This additional time is that needed by the transversal electromagnetic wave in or der to arrive at the earth’s surface and then to the antenna, while the Coulomb interacti on is propagated along a straight line. With the purpose of decreasing additively th e amplitude of the transversal electromagnetic wave, the discharge cable was electricall y and magnetically screened. The aformentioned experimental facts oblige us, in order to avo id the inﬂuence of the transver- sal electromagnetic wave on the experimental results, to wo rk with a maximum distance of 1.5 m between the antennas. In the absence of ”instantaneo us action”, it is necessary to determine which part of our measuring signal is due to the t ransversal electromagnetic wave and which part is due to the lonitudinal component of the electric ﬁeld. Our anten- nas react to the potential of the electric ﬁeld and for this re ason cannot distinguish both components from the electric ﬁeld. With the purpose of separ ating them, screens have been used which considerably decrease the intensity of the t ransversal electromagnetic wave without altering to a relevant degree the amplitude of t he longitudinal component. Two types of screens have been used: a metallic mesh and a thin layer of aluminium. In order to reject the inﬂuence of the waves reﬂected in the vary ing objects, the antenna was placed in a thick aluminium cylinder with one end sealed and t he other open, connected 10to the earth. The opening of the cylinder was directed toward s the Coulomb interaction generator. The screens were placed covering this opening. T he metallic mesh is made of iron wire with a diameter of 1mm and the mesh measures 5 mm ×5 mm. Its eﬀectiveness as armour for the transversal electromagnetic wave with the frequency obtained by us for 9.2 MHz is greater than 50 dB[7]. This means that the signal du e to this wave decreases considerably and now barely aﬀects the results of the experi ment. In this case, we register a decrease of our signal of just 0.5 dB. Because of this, we can conclude the practically all of our signal comes from the logitudinal component of the electric ﬁeld generated by the Coulomb interaction generator. We repeated the same exp eriment using a 0.02 mm thick aluminium sheet as a screen. Its eﬀectiveness as armou r for the transversal electro- magnetic wave of the aformentioned frequency is greater tha n 80 dB[7]. The decrease in the signal is just 1 dB, and this result conﬁrms the previous c onclusion that our signal from the antenna is only due to the longitudinal component of the electric ﬁeld. IV. EXPERIMENTAL RESULTS The results of the experiment are presented in ﬁg.8. Fig.8. Relation between the delay ∆ tbetween the impulses of both antennas and the distance Fbetween the antennas. The distance Fbetween antenna 1 and antenna 2 has been traced along the hori zontal axis. The delay ∆ tof the signal of antenna 2 with reference to antenna 1 has been traced on the vertical axis. The uncertainties of the measurement r esults have also been marked on the distance axis and the time axis. The straight line, which is presented in the drawing, expres ses the empirical linear relation, which relates distance Fto the delay ∆ t. This has been determined by means of the minimum square method. The line, which corresponds th e the speed of light c= 3.00×108m/s cannot be diﬀerentiated from the line shown by the limite d exactness of the representation on the drawing. 11Following a statistical treatment one obtains, that with a r eliable probability P= 0.95 the group velocity of the propagation of the Coulomb interac tion is found in the interval [8]: v= (3.03±0.07)×108m/s We would like to call this type of ﬁeld propagation “ Coulomb waves ”, which, of course, has no analogy with the usual electromagnetic waves. V. CONCLUSIONS P.A.M. Dirac wrote ([9], p. 32): “ As long as we are dealing only with transverse waves, we cannot bring in the Coulomb interactions between p articles. To bring them in, we have to introduce longitudinal electromagnetic waves an d include them in the potentials Aµ.” It is known, however, that generally accepted Maxwellian e lectrodynamics forbids the spreading of anylongitudinal electro-magnetic perturbation in vacuum. So Dirac writes ([9], p. 32): “ The longitudinal waves can be eliminated by means of a mathem atical transformation... ”. But after this transformation one gets the theory in which a charge isalways accompanied by the Coulomb ﬁeld around it, i.e. ([9], p. 32) “ Whenever an electron is emitted, the Coulomb ﬁeld around it is simultaneously emitted, forming a kind ofdressing for the electron. Similarly, when an electron is absorbed, t he Coulomb ﬁeld around it is simultaneously absorbed ”. In accord with the above it was shown [1-4] that the Coulomb in teraction “works” instantaneously , if one accepts that basic equations of classical electrody namics are right. On the other hand our experiment shows that the Coulomb inter action, at least, does not spread instantaneously. The proper inference from this exp eriment is that the Coulomb interaction cannot be considered as so called “instantaneo us action at a distance” and, in turn, the basic equations of classical electrodynamics can be incomplete and moreover, their application is even limited in classical electrodyna mics. 12REFERENCES [1] A. E. Chubykalo and R. Smirnov-Rueda, Phys. Rev. E53, 5373 (1996); [2] A. E. Chubykalo and R. Smirnov-Rueda, Mod. Phys. Lett. A 12(1), 1 (1997). [3] A. E. Chubykalo and R. Smirnov-Rueda, Phys. Rev. E57(3), 3683 (1998). [4] A. E. Chubykalo and S. J. Vlaev, Int. J. Mod. Phys. A 14(24), 3789 (1999). [5] V. V. Dvoeglazov, it Hadronic J. Suppl. 12, 241 (1997) [6] E.M. Purcell, Electricity and Magnetism, Berkeley Physics Course (2nd ed., McGraw- Hill Book Co, 1985), Vol. 2. [7] J. Barcells, F. Daura, R. Esparza, and R. Pallas, Interferencias electromagn´ eticas en sistemas electr´ onicos (Alfaomega, Mexico, 1992). [8] M. R. L. Soto, A.O. Calzadilla and P.A. Perez, An´ alisis y procesamiento de los datos experimentales (Mes. Ciudad de la Habana, Cuba, 1999) p. 156. [9] P.A.M. Dirac, Directions in Physics (Wiley, New York, 1978). 13
arXiv:physics/0010037v1 [physics.class-ph] 14 Oct 2000Comment on “Dependence of Gravitational Action on Chemical Composition: New Series of Experiments” by M. Nanni in Apeiron vol.7, p. 19 5 (2000) Rumen I. Tzontchev and Andrew E. Chubykalo Escuela de F´ ısica, Universidad Aut´ onoma de Zacatecas Apartado Postal C-580 Zacatecas 98068, ZAC., M´ exico In our comment we show that the application of appropriated s tatistical methods to the results of the author proves that the author in his article ha s not been able to reach his goal. In article [1] the results of a very interesting fundamental experiment are de- scribed. The objective of the experiment is to statisticall y demonstrate that the folowing equation is reliable M=Wi Wk(Torino) −Wi Wk(Plateau Ros´ a) /negationslash= 0 where WiandWkare correspondingly the weights of samples of two materials of diﬀerent chemical compositions, measured in the city of Tor ino (180m above sea level) and in Plateau Ros´ a (3480m above sea level). This wou ld seriously question the validity of the Weak Equivalence Principle (WEP). Our pu rpose is to show that the author has not been able to reach his goal in his article. W ith that purpose a standart statistical processing of the author’s presented results has been completed in [1], using the same symbols. The following relationships have been used [2-4] (the letter “ A” corresponds to Torino, the letter “ B” corresponds to Plateau Ros´ a): ∆W=/radicalBig [SDA ·t(P, n)]2+ ∆2 d; ∆/bracketleftbiggWi Wk/bracketrightbigg =Wk·∆Wi+Wi·∆Wk W2 k; ∆M= ∆/bracketleftbiggWi Wk(A)/bracketrightbigg + ∆/bracketleftbiggWi Wk(B)/bracketrightbigg , 1where ∆ Wis the experimental error of a series of measurements of the w eight of a sample under certain conditions; SDA is Standard Deviation Average for the same weight; t(P;n) it is the Student’s coeﬃcient to conﬁdence probability Pand number measurements n; ∆dis the scale error (∆ d= 3×10−6g);Wis the average of the corresponding sample weight; ∆ Mdetermines the limits of the conﬁdence interval ( M−∆M,M+ ∆M). With probability Pthe exact value of magnitude Mis located within this interval. ¿From the results in [1] the accuracy with which the experime nt should be carried out is seen, it is comparable with the accuracy of a metrologi cal experiment. For following, in the statistical processing of the experiment al data, the requirements of a metrological experiment should be respected. For that r eason, a level of the conﬁdence probability Phas been accepted as 0.999. On the other hand, the noted conﬁdence probability is required for each experiment that aspires to demonstrate invalidity in a fundamental physical principle. If the weig ht of a sample is measured 10 times in an experimental series and P= 0.999, the Student’s coeﬃcient is valued ast(0.999,10) = 4 .78. There are two possibilities: 1. If the digit “0” is outside the conﬁdence interval, it can b e conﬁrmed with a probability of 0.999, that the exact value of magnitude M is d iﬀerent from “0”. 2. If the digit 0 is inside the conﬁdence interval, nothing ca n be deduced. The deviation limit ∆ M, that is only due to the scale error ∆ d, equals 4 × 10−6. Because of this, there is no reason to consider those combin ations of chemical substances, where M≤4×10−6, and only the cases where M >4×10−6will be dealt with. In Table 1 magnitude M(calculated by M. Nanni), ∆ M, the reliable interval and the relative error for several chemical substa nce combinations have been presented: Lead Aluminium Gold Bronze Silver Brass-Sand M(×10−6) 8 8 6 6 6 ∆M(×10−6) 9.07 8.11 9.42 8.79 6.26 Conﬁdence (-1.07; (-0.11; (-3.42; (-2.79; (-0.26; interval 17.07) 16.11) 15.42) 14.79) 12.26) ∆M M·100% 113% 101% 157% 146% 104% Table 1 It can be seen that digit “0” participates in all of the conﬁde nce intervals. This clearly indicates that magnitude M can be diﬀerent or equal t o “0”. In Table 1 it can be seen that the relative error for all of the combination s is bigger than 100%! In this case the standard formulas should not be used for a norma l distribution, instead 2more general statistical formulas should be used. But this w ill considerably increase the width of the conﬁdence interval. Finally, it is possible that some deviations of the WEP exist. Regrettably, the author has not been able to de monstrate this thesis in his article [1]. The results of the article can only justify the realization of a new series of measurements with a more precise scale and/or a higher number of the weight measurements for each sample, and a appropriate s tatistical processing. References 1. M. Nanni, Apeiron 7(3-4), p. 195 (2000). 2. N. Adreev, N. Ilkov, L. Dlognikov, “ Fizika - laboratornyi praktikum ” (“TU-Soﬁa”, Soﬁa, 1997) (in Bulgarian). 3. C. Prokopieva, “ Laboratornyi praktikum po Fizike” (VMEI, Varna, 1993) (in Bul- garian). 4. M. Andreev, V. Liudskanov, “ Laboratornaia Fizika ” (“Nauka i Isskustvo”, Soﬁa, 1975) (in Bulgarian). 3
arXiv:physics/0010038v1 [physics.atom-ph] 15 Oct 2000Resummation in the Doubly–Cut Borel Plane: The LoSurdo–Stark Eﬀect Ulrich D. Jentschura∗ National Institute of Standards and Technology, Mail Stop 8 401, Gaithersburg, Maryland 20899-8401 and Institut f¨ ur Theoretische Physik, Technische Univers it¨ at Dresden, 01062 Dresden, Germany and Laboratoire Kastler-Brossel, Case 74, 4 Place Jussieu, F-75252 Paris Cedex 05, France The divergent perturbation series for the LoSurdo–Stark eﬀ ect has purely real coeﬃcients. By contrast, the energy eigenvalue of the quasistationary sta tes is complex (the imaginary part corre- sponds to the autoionization width). Two resummation presc riptions are compared: (i) a contour- improved resummation method based on a combination of Borel and Pad´ e techniques and (ii) a combination of the Borel method with an analytic continuati on by conformal mapping. With both methods, the complex energy eigenvalue can be reconstructe d from the purely real perturbation series. The performance of both methods is compared, calcul ational diﬃculties at strong ﬁeld are addressed, and the connection to divergent perturbative ex pansions in quantum ﬁeld theory is dis- cussed. 11.15.Bt, 11.10.Jj, 32.60.+i, 32.70.Jz I. INTRODUCTION The Rayleigh–Schr¨ odinger perturbation series for the LoSurdo–Stark eﬀect [1,2] can be formulated to arbitrar- ily high order [3]. The perturbative coeﬃcients grow fac- torially in absolute magnitude [4], and the radius of con- vergence of the perturbation series about the origin is zero. The perturbation series is a divergent, asymptotic expansion in the electric ﬁeld strength F, i.e. about zero electric ﬁeld. This means that the perturbative terms at small coupling ﬁrst decrease in absolute magnitude up to some minimal term. After passing through the minimal term, the perturbative terms increase again in magni- tude, and the series ultimately diverges. By the use of a resummation method, it is possible to assign a ﬁnite value to an otherwise divergent se- ries, and various applications of resummation methods in mathematics and physics have been given, e.g., in [5–9]. For a general introduction to divergent series we refer to [10–19]. When a divergent series is resummed, the su- perﬁcial precision limit set by the minimal term can be overcome, and more accurate results can be obtained as compared to the optimal truncation of the perturbation series [20]. In [21] it was shown that the convergence of Rayleigh- Schr¨ odinger perturbation theory can be improved if (di- agonal) Pad´ e approximants to the divergent perturbation series are evaluated instead of the “raw” partial sums of the divergent perturbation series (see Table I in [21]). However, the Pad´ e approximants cannot reproduce the full physical result. The rational approximants produce real output if the (purely real) perturbative coeﬃcients of the Stark eﬀect are used as input data. The imaginary part of the complex energy pseudoeigenvalue, which cor- responds to the autoionization width, is not reproduced. The divergent perturbation series of the LoSurdo– Stark eﬀect has both alternating and nonalternating com- ponents (as explained in Sec. II below). Rather signiﬁ-cant diﬀerences exist between the resummation of nonal- ternating divergent perturbation series (i.e., series who se terms have the same sign) and alternating divergent per- turbation series (i.e., series whose terms alternate in sign). Typically, nonalternating series describe “unsta- ble” physical situations in which the energy eigenvalues acquire an imaginary component. An example for a non- alternating divergent series is the quartic anharmonic os- cillator at negative coupling (the perturbation is of the formg x4where gis the coupling). The perturbative co- eﬃcients for the anharmonic oscillator grow factorially and alternate in sign [22–29]. The energy levels of the quartic anharmonic oscillator acquire a width when the coupling parameter becomes negative ( g <0). This sce- nario is equivalent to a nonalternating perturbation serie s at positive coupling. The energy eigenvalue of the quar- tic anharmonic oscillator as a function of the coupling parameter has a branch cut along the negative real axis (see Ch. 37 of [30]). The resummation of nonalternat- ing series or of series which have a leading or subleading nonalternating component, therefore corresponds to a re- summation “on the cut” in the complex plane. Histori- cally, the resummation of divergent series with nonalter- nating contributions has been problematic [5,6]. Here, we show that the divergent perturbation series of the Stark eﬀect can be resummed by a combination of Borel [10] and Pad´ e [5,7] techniques, where the Pad´ e approximants are used for the analytic continuation of the partial sums of the Borel transform beyond the circle of convergence. The ﬁnal evaluation of the Laplace–Borel integral is car- ried out along an integration contour introduced in [20]. The LoSurdo–Stark eﬀect and in particular the de- cay width, which is nonperturbative and nonanalytic in the electric ﬁeld, have attracted considerable atten- tion [3,4,21,31–61]. Experiments have been performed in ﬁeld strengths up to a couple of MV/cm [62–65]. A rather mathematically motivated investigation regarding the Borel summability of the divergent perturbation se- 1ries for the LoSurdo–Stark eﬀect was performed in [39]. Mathematical considerations on the Borel summability in general imply a priori that the Borel transform can be analytically continued beyond its ﬁnite radius of con- vergence. The problem of how this analytic continua- tion of the Borel transform can be constructed from a ﬁnite number of perturbative terms, is not considered. Also, it should be noted that the investigation [39] is restricted to nonreal, unphysical electric ﬁeld strengths 0<argF < π . The result for a physically relevant, real ﬁeld strength Frequires an additional analytic con- tinuation. The calculation [52] showed that it is pos- sible to perform the analytic continuation of the Borel transform by employing Pad´ e approximants. In com- parison to the investigation [52], we use here a slightly modiﬁed, but equivalent integration contour for the eval- uation of the generalized Borel integral (see [20] and Sec. III below). This contour exhibits the additional terms which have to be added to the otherwise recom- mended principal-value prescription [66–68]. The inte- gration oﬀ the real axis serves to reproduce the full phys- ical result for the energy level (including the width of the quasistationary state). Our calculation extends to higher ﬁeld strengths than [52], and we also investigate an ex- cited state, whereas the previous work [52] was restricted to the ground state. The alternative resummation method discussed here makes use of a conformal mapping of the Borel plane [67–73] and leads to results consistent with the method indicated above. The application of the confor- mal mapping depends on further information about the perturbation series. As explained in Sec. IV below, we de- termine the radius of convergence of the Borel transform by analyzing the large-order growth of the perturbative coeﬃcients. Moreover, we calculate the branch points in the Borel plane. These results make possible the appli- cation of the conformal mapping for the analytic con- tinuation of the Borel transform. The resummation of the perturbative expansion can then be accomplished by evaluating a generalized Laplace–Borel integral. This paper is organized as follows: In Sec. II, we give a brief outline of the perturbative expansion for the LoSurdo–Stark eﬀect. In Secs. III and IV, we describe the resummation methods which are used to obtain the numerical results presented in Sec. V. We conclude with a summary of the results in Sec. VI. Finally, the connec- tion of the current investigation to quantum ﬁeld theo- retic perturbation series and a number of recent investi- gations in this area is brieﬂy discussed in Appendix VI. II. PERTURBATION SERIES FOR THE LoSURDO–STARK EFFECT In the presence of an electric ﬁeld, the SO(4) symmetry of the hydrogen atom is broken, and parabolic quantum numbers n1,n2andmare used for the classiﬁcation ofthe atomic states [74]. For the Stark eﬀect, the pertur- bative expansion of the energy eigenvalue E(n1, n2, m, F ) reads [see Eq. (59) of [3]], E(n1, n2, m, F )∼∞/summationdisplay N=0E(N) n1n2mFN, (1) where Fis the electric ﬁeld strength. For N→ ∞, the leading large-order factorial asymptotics of the perturba - tive coeﬃcients have been derived in [43] as E(N) n1n2m∼A(N) n1n2m+ (−1)NA(N) n2n1m, N → ∞,(2) where A(N) ninjmis given as an asymptotic series, A(N) ninjm∼K(ni, nj, m, N ) ×∞/summationdisplay k=0aninjm k(2nj+m+N−k)!. (3) where the aninjm kare constants. The K-coeﬃcients in Eq. (3) are given by K(ni, nj, m, N ) =−/bracketleftbig 2πn3nj! (nj+m)!/bracketrightbig−1 ×exp{3 (ni−nj)}62nj+m+1(3n3/2)N. (4) Here, the principal quantum number nas a function of the parabolic quantum numbers n1,n2andmis given by [see Eq. (65) in [3]] n=n1+n2+\|m\|+ 1. (5) According to Eq. (2), the perturbative coeﬃcients E(N) n1n2m, for large order N→ ∞ of perturbation theory, can be written as a sum of a nonalternating factorially di- vergent series [ﬁrst term in Eq. (2)] and of an alternating factorially divergent series [second term in Eq. (2)]. Be- cause the aninjm kin Eq. (3) are multiplied by the factorial (2ni+m+N−k)!, we infer that for large perturbation theory order N, the term related to the aninjm 0 coeﬃcient (k= 0) dominates. Terms with k≥1 are suppressed in relation to the leading term by a relative factor of 1 /Nk according to the asymptotics (2nj+m+N−k)! (2nj+m+N)!∼1 Nk/bracketleftbigg 1 +O/parenleftbigg1 N/parenrightbigg/bracketrightbigg (6) forN→ ∞. The leading ( k= 0)–coeﬃcient has been evaluated in [4] as aninjm 0 = 1. (7) According to Eqs. (2), (3) and (7), for states with n1< n2, the nonalternating component of the perturbation series dominates in large order of perturbation theory, whereas for states with n1> n2, the alternating compo- nent is dominant as N→ ∞. For states with n1=n2, the odd- Nperturbative coeﬃcients vanish [43], and the 2even-Ncoeﬃcients necessarily have the same sign in large order [see Eq. (2)]. According to Eq. (2), there are sub- leading divergent nonalternating contributions for state s withn1> n2, and there exist subleading divergent al- ternating contributions for states with n1< n2. This complicates the resummation of the perturbation series./B9 /BI /CP/D6/CV /D8 /BP /AP/BG/A2 /B9/A2 /B9/A2 /B9/B9 /B9 /B9 /B9/A2/A2 /A2 /A2 /A2/A2 /B9 /B9/B9/CA/CT/B4 /D8 /B5 /C1/D1 /B4 /D8 /B5/C8 /D3/D0/CT/D7 /CS/CX/D7/D4/D0/CP /CT/CS /CU/D6/D3/D1 /D8/CW/CT /D6/CT/CP/D0 /CP/DC/CX/D7/A9 /BR/C6/BI /BI /BI/C8 /D3/D0/CT/D7 /D0/DD/CX/D2/CV /D3/D2 /D8/CW/CT /D6/CT/CP/D0 /CP/DC/CX/D7 FIG. 1. Integration contour C+1for the evaluation of the generalized Borel integral deﬁned in Eq. (11). Poles displaced from the real axis are evaluated as full poles, whereas those poles which lie on the real axis are treated as half poles. III. RESUMMATION BY BOREL–PAD ´E TECHNIQUES The resummation of the perturbation series (1) by a combination of the Borel transformation and Pad´ e ap- proximants proceeds as follows. First we deﬁne the pa- rameter λ= 2 max( n1, n2) +m+ 1. (8) The large-order growth of the perturbative coeﬃcients [see Eqs. (2) and (3)] suggests the deﬁnition of the (gen- eralized) Borel transform [see Eq. (4) in [76]] EB(z)≡EB(n1, n2, m, z) =B(1,λ)[E(n1, n2, m);z] =∞/summationdisplay N=0E(N) n1n2m Γ(N+λ)zN, (9) where we consider the argument zofEB(z) as a complex variable. Because the perturbative coeﬃcients E(N) n1n2m diverge factorially in absolute magnitude, the Borel transform EB(z) has a ﬁnite radius of convergence about the origin. The evaluation of the (generalized) Laplace– Borel integral [see Eq. (11) below] therefore requires an analytic continuation of EB(z) beyond the radius of con- vergence. The “classical” Borel integral is performed in thez-range z∈(0,∞), i.e. along the positive real axis[see e.g. Eqs. (8.2.3) and (8.2.4) of [5]]. It has been sug- gested in [66] that the analytic continuation of (9) into regions where Fretains a nonvanishing, albeit inﬁnitesi- mal, imaginary part can be achieved by evaluating Pad´ e approximants. Using the ﬁrst M+ 1 terms in the power expansion of the Borel transform EB(z), we construct the Pad´ e approximant (we follow the notation of [7]) PM(z) =/bracketleftbigg [[M/2]]/slashbigg [[(M+ 1)/2]]/bracketrightbigg EB(z), (10) where [[ x]] denotes the largest positive integer smaller thanx. We then evaluate the (modiﬁed) Borel integral along the integration contour C+1shown in Fig. 1 in or- der to construct the transform TEM(F) where TEM(F) =/integraldisplay C+1dt tλ−1exp(−t)PM(F t).(11) The successive evaluation of transforms TEM(F) in in- creasing transformation order Mis performed, and the apparent convergence of the transforms is examined. This procedure is illustrated in Tables I and II of [20]. In the current evaluation, a slightly modiﬁed scheme is used for selecting the poles in the upper right quadrant of the complex plane as compared to [20]. The contour C+1is supposed to encircle all poles at t=ziin the upper right quadrant of the complex plane with arg zi< π/4 in the mathematically negative sense. That is to say, the contribution of all poles ziwith Rezi>0, Imzi>0 and Re zi>Imzi, −2π i/summationdisplay iRes t=zitλ−1exp(−t)PM(F t), is added to the principal value (P.V.) of the integral (11) carried out in the range t∈(0,∞). Note the further restriction (Im zi<Rezior equivalently arg zi< π/4) with regard to the selection of poles in comparison to the previous investigation [20]. In practical calculation s, this modiﬁcation is observed not to aﬀect the numerical values of the transforms TEM(F) deﬁned in Eq. (11) in higher transformation order M≥10 [i.e. for large M, see also Eq. (14) below], because the poles are observed to cluster near the real axis in higher transformation order, and so the contribution of poles with π/4<argzi< π/2 gradually vanishes. We have, TEM(F) = (P .V.)/integraldisplay∞ 0dt tλ−1exp(−t)PM(F t) −2π i/summationdisplay iRes t=zitλ−1exp(−t)PM(F t). (12) The principal-value prescription [ﬁrst term in Eq. (12)] for the evaluation of the Laplace–Borel integral has been recommended in [66,75]. This prescription leads to a real (rather than complex) result for the energy shift and cannot account for the width of the quasistationary 3state. The additional pole contributions [second term in Eq. (12)] are responsible for complex-valued (imaginary) corrections which lead, in particular, to the decay width. By contour integration (Cauchy Theorem) and Jor- dan’s Lemma, one can show that the result obtained along C+1is equivalent to an integration along the straight line with arg z=π/4, TEM(F) =cλ/integraldisplay∞ 0dt tλ−1exp(−c t)PM(F c t) (13) where c= exp( i π/4). This contour has been used in [52] (see also p. 815 in [30]). The exponential factor exp( −c t) and the asymptotic behavior of the Pad´ e approximant PM(F c t) ast→ ∞ together ensure that the integrand falls oﬀ suﬃciently rapidly so that the Cauchy Theorem and Jordan’s Lemma can be applied to show the equiva- lence of the representations (12) and (13). The representation (13) illustrates the fact that the integration in the complex plane along C+1analytically continues the resummed result in those cases where the evaluation of the standard Laplace–Borel integral is not feasible due to poles on the real axis. The representations (11) and (12) serve to clarify the role of the additional terms which have to be added to the result obtained by the principal-value prescription in order to obtain the full physical result, including the nonperturbative, non- analytic contributions. Note that, as stressed in [20], the pole contributions in general do not only modify the imaginary, but also the real part of the resummed value for the perturbation series. Formally, the limit of the sequence of the TEM(F) as M→ ∞, provided it exists, yields the nonperturbative result inferred from the perturbative expansion (1), lim M→∞TEM(F) =E(F)≡E(n1, n2, m, F ).(14) Because the contour C+1shown in Fig. 1 extends into the complex plane, the transforms TEM(F) acquire an imaginary part even though the perturbative coeﬃcients in Eq. (1) are real. In the context of numerical analysis, the concept of incredulity [77] may be used for the analysis of the con- vergence of the transforms TEM(F) of increasing order M. If a certain number of subsequent transforms ex- hibit apparent numerical convergence within a speciﬁed relative accuracy, then the calculation of transforms is stopped, and the result of the last calculated transfor- mation is taken as the numerical limit of the series under investigation. It has been observed in [20,66] that for a number of physically relevant perturbation series, the apparent numerical convergence of the transforms (11), with increasing transformation order, leads to the phys- ically correct results. Note that in [66], speciﬁc pertur- bation series were examined for which the second term in (12) vanishes. The resummation method by a combination of Borel and Pad´ e techniques, which has been introduced inthe current Section, will be referred to as “method I” throughout the current paper. IV. RESUMMATION BY CONFORMAL MAPPING According to Eqs. (2) and (3), the perturbative coeﬃ- cientE(N) n1n2m, for large N, can be written as the sum of an alternating and of a nonalternating divergent series. In view of Eqs. (4) and (7), we conclude that the series deﬁned in Eq. (9), EB(z) =∞/summationdisplay N=0E(N) n1n2m Γ(N+λ)zN, has a radius of convergence s=2 3n3(15) about the origin, where nis the principal quantum num- ber [see Eq. (5)]. Therefore, the function EB(w) =∞/summationdisplay N=0E(N) n1n2msN Γ(N+λ)wN, (16) has a unit radius of convergence about the origin. It is not a priori obvious if the points w=−1 and w= +1 repre- sent isolated singularities or branch points. The asymp- totic properties (2) and (3) together with Eq. (6) suggest that the points w=−1 and w= +1 do not constitute poles of ﬁnite order. We observe that the leading facto- rial growth of the perturbative coeﬃcients in large per- turbation order Nis divided out in the Borel transform (16), which is a sum over N. The perturbative coeﬃ- cientE(N) n1n2mcan be written as an asymptotic series over k[see Eq. (3)]. We interchange the order of the summa- tions over Nandk, we use Eq. (6) and take advantage of the identity ∞/summationdisplay N=0wk Nk= Li k(w). The Borel transform EB(w) can then be written as a sum over terms of the form Tk(w) where for k→ ∞, Tk(w)∼C(ni, nj, m)aninjm kLik(w). (17) The coeﬃcient C(ni, nj, m) is given by C(ni, nj, m) =−/bracketleftbig 2πn3nj! (nj+m)!/bracketrightbig−1 ×exp{3 (ni−nj)}62nj+m+1. (18) These considerations suggest that the points w=−1 andw= +1 represent essential singularities (in this case, branch points) of the Borel transform EB(w) deﬁned in 4Eq. (16). For the analytic continuation of EB(w) by con- formal mapping, we write was w=2y 1 +y2(19) (this conformal mapping preserves the origin of the com- plex plane). Here, we refer to was the Borel variable, and we call ythe conformal variable. We then express theMth partial sum of (16) as EM B(w) =M/summationdisplay N=0E(N) n1n2msN Γ(N+λ)wN =M/summationdisplay N=0CNyN+O(yM+1), (20) where the coeﬃcients CNare uniquely determined [see, e.g., Eqs. (36) and (37) of [67]]. We deﬁne the partial sum of the Borel transform, re-expanded in terms of the conformal variable y, as E′M B(y) =M/summationdisplay N=0CNyN. (21) We then evaluate (lower-diagonal) Pad´ e approximants to the function E′M B(y), E′′M B(y) =/bracketleftbigg [[M/2]]/slashbigg [[(M+ 1)/2]]/bracketrightbigg E′M B(y). (22) We deﬁne the following transforms, T′′EM(F) =sλ/integraldisplay C+1dw wλ−1exp/parenleftbig −w/parenrightbig E′′M B/parenleftbig y(w)/parenrightbig . (23) At increasing M, the limit as M→ ∞, provided it ex- ists, is then again assumed to represent the complete, physically relevant solution, E(F) = lim M→∞T′′EM(F). (24) We do not consider the question of the existence of this limit here (for an outline of questions related to these issues we refer to [68]). Inverting Eq. (19) yields [see Eq. (23)] y(w) =√1 +w−√1−w√1 +w+√1−w. (25) The conformal mapping given by Eqs. (19) and (25) maps the doubly cut w-plane with cuts running from w= 1 to w=∞andw=−1 tow=−∞unto the unit circle in the complex y-plane. The cuts themselves are mapped to the edge of the unit circle in the y-plane.In comparison to the investigations [67] and [68], we use here a diﬀerent conformal mapping deﬁned in Eqs. (19) and (25) which reﬂects the diﬀerent singular- ity structure in the complex plane [cf. Eq. (27) in [67]]. We also mention the application of Pad´ e approximants for the numerical improvement of the conformal map- ping performed according to Eq. (22). In comparison to a recent investigation [73], where the additional Pad´ e– improvement in the conformal variable is also used, we perform here the analytic continuation by a mapping whose structure reﬂects the double cuts suggested by the asymptotic properties of the perturbative coeﬃcients given in Eqs. (2), (3) and (6) [cf. Eq. (5) in [73]]. The method introduced in this Section will be referred to as “method II”. It is one of the motivations for the current investigation to contrast and compare the two methods I and II. A comparison of diﬀerent approaches to the resummation problem for series with both alter- nating and nonalternating divergent components appears useful, in part because the conformal mapping (without further Pad´ e improvement) has been recommended for the resummation of quantum chromodynamic perturba- tion series [67,68]. We do not consider order-dependent mappings here [69–72]. For an order-dependent mapping to be con- structed, the conformal mapping in Eq. (19) has to be modiﬁed, and a free parameter ρhas to be introduced. The coeﬃcients CNin the accordingly modiﬁed Eq. (21) then become ρ-dependent. The free parameter ρis cho- sen so that the ρ-dependent coeﬃcient CM(ρ) of order M vanishes. Consequently, the choice of ρdepends on the order Mof perturbation theory, and in this way the map- ping becomes order-dependent. Certain complications arise because ρcannot be chosen arbitrarily, but has to be selected, roughly speaking, as the ﬁrst zero of the ρ- dependent coeﬃcient CM(ρ) for which the absolute mag- nitude of the derivative C′ M(ρ) is small (this is explained in [30], p. 886). It is conceivable that with a judicious choice of ρ, further acceleration of the convergence can be achieved, especially when the order-dependent map- ping is combined with a Pad´ e approximation as it is done here in Eq. (22) with our order- independent mapping. In the current investigation, we restrict the discussion to the conformal order-independent mapping (19) which is nevertheless optimal in the sense discussed in [67,68]. V. NUMERICAL CALCULATIONS In this section, the numerical results based on the re- summation methods introduced in Secs. III and IV are presented. Before we describe the calculation in detail, we should note that relativistic corrections to both the real and the imaginary part of the energy contribute at a relative order of ( Zα)2compared to the leading non- relativistic eﬀect which is treated in the current inves- tigation (and in the previous work on the subject, see 5e.g. [43,52]). Therefore, the theoretical uncertainty due to relativistic eﬀects can be estimated to be, at best, 1 part in 104(for an outline of the relativistic and quan- tum electrodynamic corrections in hydrogen see [78–84]). Measurements in very high ﬁelds are diﬃcult [62]. At the achievable ﬁeld strengths to date (less than 0 .001 a.u.or about 5 MV /cm), the accuracy of the theoretical predic- tion exceeds the experimental precision, and relativistic eﬀects do not need to be taken into account. The perturbative coeﬃcients E(N) n1n2mdeﬁned in Eq. (1)for the energy shift can be inferred, to arbitrarily high or- der, from the Eqs. (9), (13–15), (28–33), (59–67) and (73) in [3]. The atomic unit system is used in the sequel, as is customary for this type of calculation [3,33,36,38]. The unit of energy is α2mec2= 27.211 eV where αis the ﬁne structure constant, and the unit of the electric ﬁeld is the ﬁeld strength felt by an electron at a distance of one Bohr radius aBohrto a nucleus of elementary charge, which is 1 /(4π ǫ0)(e/a2 Bohr) = 5.142×103MV/cm (here, ǫ0is the permittivity of the vacuum). TABLE I. Real and imaginary part of the energy pseudoeigenva lueE000(F) for the ground state of atomic hydrogen (parabolic quantum numbers n1= 0, n2= 0, m= 0). The ﬁeld strength Fis given in atomic units. A comparison is made to the earlier work [36]. Discrepancies at large ﬁeld in compar ison to the investigation [36] have also been found in [46,51 ,54]. Real part of the resonance Re E000(F) Autoionization decay width Γ 000(F) F(a.u.) Ref. [36] Our results Ref. [36] Our results 0.04 −0.503 771 591 −0.503 771 591 01(1) 3 .89×10−63.892 70(1) ×10−6 0.06 −0.509 203 452 (1) 5.150 72(5) ×10−4 0.08 −0.517 495 363 −0.517 560 50(5) 4 .511 10 ×10−34.539 63(5) ×10−3 0.10 −0.527 419 3(5) 1.453 8(5) ×10−2 0.12 −0.535 567 −0.537 334(5) 2 .942 3×10−22.992 7(5) ×10−2 0.16 −0.547 78 −0.555 24(5) 7 .119 5×10−27.131(5) ×10−2 0.20 −0.552 60 −0.570 3(5) 1 .249 3×10−11.212(5) ×10−1 0.24 −0.550 82 −0.582 6(1) 1 .892 7×10−11.767(5) ×10−1 0.28 −0.543 4 −0.591 7(5) 2 .643×10−12.32(3)×10−1 0.32 −0.531 1 −0.600(5) 3 .507×10−12.92(3)×10−1 0.36 −0.514 4 −0.604(5) 4 .497×10−13.46(3)×10−1 0.40 −0.493 8 −0.608(5) 5 .631×10−14.00(5)×10−1 We consider the resummation of the divergent pertur- bative expansion (1) for two states of atomic hydrogen. These are the ground state ( n1=n2=m= 0) and an excited state with parabolic quantum numbers n1= 3, n2= 0,m= 1 [note Eq. (5)]. We list here the ﬁrst few perturbative coeﬃcients for the states under investiga- tion. For the ground state, we have (in atomic units), E000(F) =−1 2−9 4F2−3 555 64F4 −2 512 779 512F6−13 012 777 803 16 384F8+. . . (26)The perturbation series for the state n1= 3, n2= 0, m= 1 is alternating, but has a subleading nonalternat- ing component [see Eq. (2)]. The ﬁrst perturbative terms read E301(F) =−1 50+45 2F−31875 2F2 +54 140 625 4F3−715 751 953 125 16F4+. . . (27) Note that for F= 0, the unperturbed nonrelativistic en- ergy is recovered, which is −1/(2n2) in atomic units. In contrast to the real perturbative coeﬃcients, the energy 6pseudoeigenvalue (resonance) E(n1, n2, m, F ) has a real and an imaginary component, E(n1, n2, m, F ) = Re En1n2m(F)−i 2Γn1n2m(F),(28) where Γ n1n2m(F) is the autoionization width. Using the computer algebra system Mathemat- ica[85,86], the perturbative coeﬃcients in the expansion of the energy (1) are evaluated up to N= 40. This makes possible the evaluation of the transforms TEM(F) de- ﬁned in Eq. (11) up to the transformation order M= 40. The apparent convergence of the transforms in higher or- der is examined. This procedure leads to the numerical results listed in Tables I and II. The numerical error of our results is estimated by determining the numerical variation (highest and lowest value) of the four highest- order transforms M= 37,38,39,40. It is conceivable that more accurate numerical results could be obtained by evaluating higher-order transforms of order M >40. An important result of the comparison of the meth- ods introduced in Secs. III and IV is the following: Both methods appear to accomplish a resummation of the perturbation series to the physically correct result. Method II (Borel+Pad´ e-improved conformal mapping, see Sec. IV) appears to perform marginally better than method I (plain Borel+Pad´ e, see Sec. III). Both methods yield approximately the same number of signiﬁcant ﬁg- ures in the ﬁnal result. For the perturbation series stud- ied in [73], considerable improvement was achieved by employing a variant of method II as compared to method I applied to the very problem studied in [73] (roughly two more signiﬁcant ﬁgures were obtained with method II than with method I). In all cases considered (here and in [73]), both methods I and II lead to results which are in mutual agreement. To date, a rigorous theory of the performance of the resummation methods for divergent series of the type discussed in this work (with alternating and nonalternating components) does not exist. In order to illustrate the above statements, we con- sider, for the atomic state with quantum numbers n1= 3, n2= 0 and m= 1, the evaluation of the transforms TEM(F) deﬁned in Eq. (11) (method I) and of the trans- forms T′′EM(F) deﬁned in Eq. (23) (method II) in trans- formation order M= 38,39,40 for a ﬁeld strength of F= 2.1393×10−4. Method II leads to the following results, T′′E37(F= 2.1393×10−4) =−0.015 860 468 42 −0.529 32 ×10−6, T′′E38(F= 2.1393×10−4) =−0.015 860 468 39 −0.529 11 ×10−6, T′′E39(F= 2.1393×10−4) =−0.015 860 468 37 −0.529 11 ×10−6andT′′E40(F= 2.1393×10−4) =−0.015 860 468 39 −0.529 09 ×10−6. Here, we should note one important point: The addi- tional Pad´ e–improvement of method II signiﬁcantly im- proves the convergence. If the plain conformal mapping is used, i.e. if the evaluation of the Pad´ e approximants in Eq. (22) is left out and the partial sum expressed in terms of the conformal variable (21) is directly plugged into the integral (23), then the numerical accuracy dete- riorates. In this case, the real part of the 39th and the 40th transform read −0.015 860 424 and −0.015 860 616, respectively, which corresponds to 5 less signiﬁcant ﬁg- ures in the ﬁnal result than achievable by method II. Method I yields the following data, TE37(F= 2.1393×10−4) =−0.015 860 468 41 −0.529 25 ×10−6, TE38(F= 2.1393×10−4) =−0.015 860 468 43 −0.529 18 ×10−6, TE39(F= 2.1393×10−4) =−0.015 860 468 28 −0.529 31 ×10−6and TE40(F= 2.1393×10−4) =−0.015 860 468 32 −0.529 13 ×10−6. Numerical results obtained by resummation are pre- sented in Tables I and II for a variety of ﬁeld strengths and for the two atomic states under investigation here. At large ﬁeld, the numerical evaluation of the pseu- doeigenvalues has been historically problematic. Follow- ing early work [32,33,35], the numerical results of [36] ap- pear to have been taken as a reference point for a number of subsequent calculations. In [3], results are obtained by optimal truncation of the perturbative expansion (see Table VII of [3]). These results are real rather than com- plex. This restriction is shared by methods which are based on the direct application of Pad´ e approximants to the perturbation series, as pointed out in Sec. I (see also Ref. [21]). Such an approach can only lead to better ap- proximants for the realpart of the energy. The fact that the numerical calculation [36] is inaccu- rate at large ﬁeld has been indicated explicitly in [51]. The calculation [51] yields improved accuracy for the ground state at large ﬁeld. However, e.g. for the excited state with quantum numbers n1= 3,n2= 0 and m= 1, the numerical precision of the calculation [51] has not been suﬃcient to discern any numerical discrepancy with the previous calculation [36] (see Table III in [51]). Very accurate data for the ground state, even at large ﬁeld, have also been obtained in the investigation [46], which is based on a complex coordinate approach. Our data are consistent with the results of [46]. The calculation [46] is 7restricted to the ground state. Another investigation [50] , based on complex coordinates, has lead to very accurate data for the ground state; ﬁeld strengths of F≤0.1 inatomic units are considered. In [52] (generalized Borel– Pad´ e resummation), numerical data are obtained for the ground state, again for ﬁeld strengths F≤0.1. TABLE II. Real part and imaginary part of the energy pseudoei genvalue E301(F) for the excited state with parabolic quantum numbers n1= 3, n2= 0, m= 1. The ﬁeld strength Fis given in atomic units. The data are compared to [36]. As for the ground state, discrepancies are observed at large ﬁeld. Real part of the resonance Re E301(F) Autoionization decay width Γ 301(F) F(a.u.) Ref. [36] Our results Ref. [36] Our results 1.5560×10−4−0.016 855 237 2 −0.016 855 237 140 8(5) 0 .42×10−90.421 6(2) ×10−9 1.9448×10−4−0.016 179 388 5 −0.016 179 388 25(5) 0 .143 8×10−60.143 8(1) ×10−6 2.1393×10−4−0.015 860 468 −0.015 860 468 4(1) 0 .105 7×10−50.105 8(1) ×10−5 2.5282×10−4−0.015 269 204 −0.015 269 292(3) 0 .175 60 ×10−40.176 3(1) ×10−4 2.9172×10−4−0.014 740 243 −0.014 742 64(5) 0 .976 51 ×10−40.100 1(1) ×10−3 3.3061×10−4−0.014 242 49 −0.014 259 7(5) 0 .278 53 ×10−30.295 4(1) ×10−3 The numerical calculation [36], while leading to inac- curate results for excessively large electric ﬁelds, yield s very accurate data for all experimentally accessible elec- tric ﬁeld strengths to date. In addition, it should be no- ticed that the inaccuracies at excessively large ﬁeld of [36 ] have been pointed out by the same authors in [54]. How- ever, not all atomic states considered in [36] were treated in the later investigation [54]. Our data for the ground state indicated in Table I are consistent with the numer- ical results obtained in [54]. However, it should be noted that the later work [54] leaves out the excited state with quantum numbers n1= 3,n2= 0 and m= 1 for which results are given here here in Table II. To the best of our knowledge, the numerical discrepancy with [36] for the excited state with quantum numbers n1= 3,n2= 0 and m= 1 has not been recorded in the literature. We do not claim here that it would have been impossible to dis- cern this discrepancy with the other methods which have been devised for the theoretical LoSurdo–Stark problem. Notably, it appears likely that the approach from [54] or the method presented in [46] could easily be generalized to the particular excited state considered here, and that such a generalization would lead to very accurate results. We merely include Table II here in order to illustrate the utility of the rather unconventional resummation method for the regime of large coupling, where even rather sophis- ticated numerical methods, which avoid the intricacies of a small-ﬁeld perturbative expansion, have been shown to be problematic [36,54]. We conﬁrm that the numerical data given in [36] are accurate up to a ﬁeld strength of F≈0.1 for the ground state and up to F≈3×10−4for the excited ( n= 5)-state with n1= 3,n2= 0 and m= 1. In order to exemplify the utility of a resummation method in comparison to the pure perturbation series, we brieﬂy consider here the ground state for a relatively large ﬁeld F= 0.40. In this case, the ﬁrst partial sums of the perturbation series (26) read −0.5,−0.86,−2.282,−22.3842, . . . This divergent sequence of numbers illustrates that at large coupling, the perturbation series becomes entirely meaningless unless it is combined with a suitable resum- mation process. VI. CONCLUSION The quasistationary states of hydrogenlike atoms in an electric background ﬁeld have been investigated using the perturbative approach in [3,43]. We would also like to mention a recent investigation where allowance is made for polynomial potentials in contrast to the constant elec- tric ﬁeld which leads to a linear potential [61]. These in- vestigations are complemented by approaches which com- bine the perturbative series with resummation methods, as it is done in [21,52] and in the current investigation. Here, we discuss the resummation of the divergent per- turbation series of the LoSurdo–Stark eﬀect using two methods. Method I, which uses a variant of the contour- improved Borel–Pad´ e technique introduced in [52], is de- scribed in Sec. III. The integration contour is modiﬁed so 8that the additional terms which have to be added to the principal value of the Laplace–Borel integral are clearly identiﬁed [see Eq. (12)]. Method II, which comprises an analytic continuation by conformal mapping with addi- tional improvement by Pad´ e approximants in the confor- mal variable [see Eq. (22)], is discussed in Sec. IV. This method is a variant of the method introduced in [67,68] which has been shown to accelerate convergence of per- turbative quantum chromodynamics (by optimal confor- mal mapping of the Borel plane). Both methods accom- plish a resummation of the divergent perturbation series (1) for the LoSurdo–Stark eﬀect, and the decay width of the quasistationary states is obtained (see Sec. V and the numerical results in Tables I and II). It is a characteristic feature of the quasistationary states in an electric ﬁeld that nonperturbative, nonan- alytic imaginary contributions exist which correspond to the autoionization decay width [see Eq. (28)]. These contributions cannot be accounted for by perturba- tion theory (the coeﬃcients are real rather than com- plex), unless the perturbation series is combined with a resummation process. In quantum electrodynamics, we encounter nonperturbative eﬀects in the electron- positron pair-production amplitude in a background elec- tric ﬁeld [87–90]. The vacuum-to-vacuum amplitude ac- quires an imaginary part, whose magnitude is related to the production rate per space-time volume of fermion- antifermion pairs. This nonperturbative, imaginary con- tribution can be inferred from the perturbative expan- sion of the eﬀective action by contour-improved resum- mation [20]. Nonperturbative eﬀects typically involve a nonanalytic factor of exp( −1/g) where gis an appro- priate coupling parameter for the physical system under investigation (in the case of the LoSurdo–Stark eﬀect, the coupling parameter is the electric ﬁeld strength F). The existence of nonperturbative contributions is inti- mately linked with the failure of the Carleman criterion for a particular perturbation series (see for example [24], Theorems XII.17 and XII.18 and the deﬁnition on p. 43 in [91], p. 410 in [5], or the elucidating discussion in Ref. [92]). The Carleman criterion determines, roughly speaking, if nonanalytic contributions exist for a given ef - fect which is described by a speciﬁed perturbation series. The relevance of the Carleman criterion for the perturba- tive expansions encountered in the current investigation, is discussed in [20]. We also mention the connection of the perturbative expansion for the LoSurdo–Stark eﬀect to perturbative expansions in quantum chromodynamics. The divergent nonalternating and alternating contributions to the per- turbation series [see Eqs. (2) and (3)] correspond in their mathematical structure to infrared and ultraviolet renor- malons [93], and the structure of a doubly cut Borel plane (see Sec. IV) is used in [67,68] for the construc- tion of an optimal conformal mapping, which is devised with the notion of accelerating the convergence of per- turbative quantum chromodynamics. The current investigation illustrates the utility of re-summation methods in those cases where perturbation theory breaks down at large coupling. As explained in Sec. V, even in situations where the perturbation series diverges strongly, it can still be used to obtain meaningful physical results if it is combined with a suitable resum- mation method. In a relatively weak ﬁeld, it is possible to obtain more accurate numerical results by resummation than by optimal truncation of the perturbation series (see also [20]). In a strong ﬁeld, it is possible to obtain phys- ically correct results by resummation even though the perturbation series diverges strongly (see the discussion in Sec. V and the data in Tables I and II). By resumma- tion, the perturbation series which is inherently a weak- coupling expansion can be given a physical interpretation even in situations where the coupling is large. In the application discussed here (LoSurdo–Stark energy shift), the contour-improved resummation methods introduced in Secs. III and IV provide results which are in agreement with theoretical data presented in [33,46,52,54]. ACKNOWLEDGMENTS The author would like to thank Professor G. Soﬀ for many insightful discussions and his continuous encour- agement during the course of the work. Helpful discus- sions with Professor P. J. Mohr, E. J. Weniger, J. Sims, I. N´ andori and S. Roether are also gratefully acknowl- edged. The author would also like to acknowledge sup- port from the Deutscher Akademischer Austauschdienst (DAAD). APPENDIX: DIVERGENT PERTURBATION SERIES IN QUANTUM FIELD THEORY We brieﬂy indicate some aspects of certain divergent perturbation series in quantum ﬁeld theory, in particular the quantum electrodynamic (QED) eﬀective action and the associated pair–production amplitude for electron– positron pairs. We use natural units in which the re- duced Planck’s constant, the permittivity of the vacuum and the speed of light (in ﬁeld-free vacuum) assume the value of unity (¯ h=ǫ0=c= 1). The one-loop QED eﬀec- tive action for an arbitrary electric and magnetic ﬁeld per space-time volume reads [this result can be found e.g. in Eq. (4-123) in [89], upon inclusion of an additional coun- terterm; or [95]] S=−1 8π2/integraldisplayi∞ 0ds s3e−(m2 e−iǫ)s/bracketleftbigg (es)2abcoth(eas) ×cot(ebs)−1 3(es)2(a2−b2)−1/bracketrightbigg , (A1) where aandbare the secular invariants , 9a=/radicalBig/radicalbig F2+G2+F2, b=/radicalBig/radicalbig F2+G2− F2, F=1 4FµνFµν=1 2(B2−E2), G=1 4Fµν(∗F)µν=−E·B. It is perhaps interesting to note that the electric- magnetic duality (invariance under the transformation a→ −ib,b→ia) immediately follows from the compact, one-dimensional integral representation for the eﬀective action given in Eq. (A1). The particular cases of a pure magnetic and a pure electric ﬁeld are of some interest, because they can be used as model series for divergent alternating and nonalternating asymptotic perturbation series [8,9,20,76]. In the case of a pure magnetic ﬁeld B=\|B\|, the result reads [see e.g. Eq. (5) in [9]] SB=−e2B2 8π2∞/integraldisplay 0ds s2/braceleftbigg coths−1 s−s 3/bracerightbigg exp/parenleftbigg −m2 e e Bs/parenrightbigg , (A2) where meis the mass of the electron. This result can be expressed as a divergent asymptotic perturbation series in the coupling parameter gB=e2B2/m4 e. For the pure electric ﬁeld, the result can be inferred by the replace- ment B→ −iEwhere E=\|E\|, and the inclusion of a converging factor; it reads SE=e2E2 8π2∞/integraldisplay 0ds s2/braceleftbigg coths−1 s−s 3/bracerightbigg ×exp/bracketleftbigg −i/parenleftbiggm2 e e E−iǫ/parenrightbigg s/bracketrightbigg =−e2E2 8π2∞+iǫ/integraldisplay 0+iǫds s2/braceleftbigg cots−1 s+s 3/bracerightbigg exp/bracketleftbigg −m2 e e Es/bracketrightbigg (A3) [Eq. (7) of [20] and the expression before Eq. (10) of [9] contain typographical errors]. We take the opportunity to supplement the proportionality factor for the expres- sion in Eq. (7) of [20] to yield the eﬀective action per space-time volume element; it reads e2E2/(8π2). As evi- dent from the Eq. (A3), the integration of the Borel–Pad´ e transform for the electric ﬁeld case should be carried out along the contour C+1shown here in Fig. 1. When this contour is used, then a sign change results for the imag- inary contributions in Table 1 of [20] (the sign change of the imaginary part according to the choice of the in- tegration contour has been discussed at length in [20]). The magnitude of the imaginary part yields the pair- production amplitude. The contour C+1is used in the current investigation (and in the context of the relatedbrief discussion in [20]) for the calculation of nonper- turbative imaginary eﬀects, i.e. the autoionization decay width of atomic states (LoSurdo–Stark eﬀect). The divergent asymptotic perturbation series for the cases of the magnetic and electric ﬁeld, generated by ex- pansion of the results in Eqs. (A2) and (A3), can be found in Eqs. (6) and (7) of [9] ( B–ﬁeld, alternating series cou- pling parameter gB=e2B2/m4 e) and in Eqs. (8) and (9) of [20] ( E–ﬁeld, nonalternating series, coupling param- etergE=e2E2/m4 e). The singularity structure of the Borel transform of the series for the magnetic ﬁeld case has been discussed in some detail in [76]. The pertur- bation series for the LoSurdo–Stark eﬀect contains both nonalternating and alternating components so that its re- summation represents a comparatively more interesting task. The same applies to the more complex perturbation series calculated in [94] for the renormalization group γ function, whose resummation – at strong coupling – has been discussed in [73,94]. In this particular case, there is no imaginary part involved, and the integration of the Borel–Pad´ e transform proceeds along the contour C0in- troduced in [73]. ∗Electronic address: jentschura@physik.tu-dresden.de. [1] A. LoSurdo, Atti R. Accad. Lincei 22, 664 (1913). [2] J. Stark, Berl. Ber. 1913, 932 (1913). [3] H. Silverstone, Phys. Rev. A 18, 1853 (1978). [4] L. Benassi, V. Grecchi, E. Harrell, and B. Simon, Phys. Rev. Lett. 42, 704 (1979). [5] C. M. Bender and S. A. Orszag, Advanced Mathemati- cal Methods for Scientists and Engineers (McGraw-Hill, New York, NY, 1978). [6] E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989). [7] G. A. Baker and P. Graves-Morris, Pad´ e Approximants , 2nd ed. (Cambridge University Press, Cambridge, 1996). [8] G. V. Dunne and T. M. Hall, Phys. Rev. D 60, 065002 (1999). [9] U. Jentschura, J. Becher, E. Weniger, and G. Soﬀ, Los Alamos e-print hep-ph/9911265; Phys. Rev. Lett. 85, 2446 (2000). [10] E. Borel, Le¸ con sur les S´ eries Divergentes , 2nd ed. (Gauthiers-Villars, Paris, 1928). [11] G. H. Hardy, Divergent Series (Clarendon Press, Oxford, UK, 1949). [12] W. Balser, From Divergent Power Series to Analytic Functions (Springer, Berlin, 1994). [13] B. Candelpergher, J. C. Nosmas, and F. Pham, Approche de la R´ esurgence (Hermann, Paris, 1993). [14] J. P. Ramis, S´ eries Divergentes et Th´ eories Asymp- totiques (Soci´ et´ e Math´ ematique de France, Marseille, 1993). [15] B. Shawyer and B. Watson, Borel’s Method of Summa- bility(Oxford University Press, Oxford, 1994). [16] L. B. de Monvel (Ed.), M´ ethodes R´ esurgentes (Hermann, 10Paris, 1994). [17] V. Kowalenko, N. E. Frankel, M. L. Glasser, and T. Taucher, Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders (Cambridge Univer- sity Press, Cambridge, 1995). [18] B. Y. Sternin and V. E. Shatalov, Borel-Laplace Trans- form and Asymptotic Theory (CRC Press, Boca Raton, 1996). [19] W. O. Bray and ˇC. Stanojevi´ c, Analysis of Diver- gence – Control and Management of Divergent Processes (Birkh¨ auser, Boston, 1999). [20] U. D. Jentschura, Los Alamos e-print hep-ph/0001135, Phys. Rev. D 62, 076001 (2000). [21] H. S. Silverstone and P. M. Koch, J. Phys. B 12, L537 (1979). [22] C. M. Bender and T. T. Wu, Phys. Rev. 184, 1231 (1969). [23] B. Simon, Ann. Phys. (N. Y.) 58, 76 (1970). [24] S. Graﬃ, V. Grecchi, and B. Simon, Phys. Lett. B 32, 631 (1970). [25] C. M. Bender and T. T. Wu, Phys. Rev. Lett. 27, 461 (1971). [26] C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973). [27] E. J. Weniger, Ann. Phys. (N. Y.) 246, 133 (1996). [28] E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996). [29] E. J. Weniger, Phys. Rev. A 56, 5165 (1997). [30] J. Zinn-Justin, Quantum Field Theory and Critical Phe- nomena , 3rd ed. (Clarendon Press, Oxford, 1996). [31] J. R. Oppenheimer, Phys. Rev. 31, 66 (1928). [32] M. H. Alexander, Phys. Rev. 178, 34 (1969). [33] M. Hehenberger, H. V. McIntosh, and E. Br¨ andas, Phys. Rev. A 10, 1494 (1974). [34] N. A. Gushina and V. K. Nikulin, Chem. Phys. 10, 23 (1975). [35] P. Froelich and E. Br¨ andas, Phys. Rev. A 12, 1 (1975). [36] R. J. Damburg and V. V. Kolosov, J. Phys. B 9, 3149 (1976). [37] T. Yamabe, A. Tachibana, and H. J. Silverstone, Phys. Rev. A 16, 877 (1977). [38] R. J. Damburg and V. V. Kolosov, J. Phys. B 11, 1921 (1978). [39] S. Graﬃ and V. Grecchi, Commun. Math. Phys. 62, 83 (1978). [40] I. W. Herbst and B. Simon, Phys. Rev. Lett. 41, 67 (1978). [41] L. Benassi, V. Grecchi, E. Harrell, and B. Simon, Phys. Rev. Lett. 42, 1430 (Erratum) (1979). [42] R. J. Damburg and V. V. Kolosov, J. Phys. B 12, 2637 (1979). [43] H. Silverstone, B. G. Adams, J. ˇCiˇ zek, and P. Otto, Phys. Rev. Lett. 43, 1498 (1979). [44] J. E. Avron, B. G. Adams, J. ˇCiˇ zek, M. Clay, M. L. Glasser, P. Otto, J. Paldus, and E. Vrscay, Phys. Rev. Lett.43, 691 (1979). [45] B. G. Adams, J. E. Avron, J. ˇCiˇ zek, P. Otto, J. Paldus, R. K. Moats, and H. J. Silverstone, Phys. Rev. A 21, 1914 (1980). [46] L. Benassi and V. Grecchi, J. Phys. B 13, 911 (1980). [47] E. Luc-Koenig and A. Bachelier, J. Phys. B 13, 1743 (1980).[48] E. Luc-Koenig and A. Bachelier, J. Phys. B 13, 1769 (1980). [49] H. J. Silverstone, E. Harrell, and C. Grot, Phys. Rev. A 24, 1925 (1981). [50] A. Maquet, S. I. Chu, and W. P. Reinhardt, Phys. Rev. A27, 2946 (1983). [51] D. Farrelly and W. P. Reinhardt, J. Phys. B 16, 2103 (1983). [52] V. Franceschini, V. Grecchi, and H. J. Silverstone, Phy s. Rev. A 32, 1338 (1985). [53] W. L. Glab, K. Ng, D. Yao, and M. H. Nayfeh, Phys. Rev. A 31, 3677 (1985). [54] V. V. Kolosov, J. Phys. B 20, 2359 (1987). [55] V. V. Kolosov, J. Phys. B 22, 833 (1989). [56] G. Alvarez and H. J. Silverstone, Phys. Rev. Lett. 63, 1364 (1989). [57] G. Alvarez, R. J. Damburg, and H. J. Silverstone, Phys. Rev. A 44, 3060 (1991). [58] F. M. Fern´ andez, J. Phys. A 25, 495 (1992). [59] C. A. Nicolaides and S. I. Themelis, Phys. Rev. A 45, 349 (1992). [60] G. Alvarez and H. J. Silverstone, Phys. Rev. A 50, 4679 (1994). [61] J. Zamastil, J. ˇCiˇ zek, and L. Sk´ ala, Phys. Rev. Lett. 84, 5683 (2000). [62] P. M. Koch, Phys. Rev. Lett. 41, 99 (1978). [63] T. Bergeman et al., Phys. Rev. Lett. 53, 775 (1984). [64] W. L. Glab and M. N. Nayfeh, Phys. Rev. A 31, 530 (1985). [65] H. Rottke and K. H. Welge, Phys. Rev. A 33, 301 (1986). [66] M. Pindor, Los Alamos e-print hep-th/9903151. [67] I. Caprini and J. Fischer, Phys. Rev. D 60, 054014 (1999). [68] I. Caprini and J. Fischer, Los Alamos e-print hep- ph/0002016; Phys. Rev. D 62, 054007 (2000). [69] R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). [70] J. C. LeGuillou and J. Zinn-Justin, Ann. Phys. (N. Y.) 147, 57 (1983). [71] R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (N. Y.) 241, 152 (1995). [72] R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (N. Y.) 249, 109 (1996). [73] U. D. Jentschura and G. Soﬀ, Los Alamos e-print hep- ph/0006089. [74] L. D. Landau and E. M. Lifshitz, Quantenmechanik (Band III der Lehrbuchreihe ¨ uber Theoretische Physik von L. D. Landau und E. M. Lifshitz) (Akademie-Verlag, Berlin, 1979). [75] P. A. R¸ aczka, Phys. Rev. D 43, R9 (1991). [76] U. D. Jentschura, E. J. Weniger and G. Soﬀ, J. Phys. G 26, 1545 (2000). [77] A. van Wijngaarden, B.I.T. 6, 66 (1966). [78] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii , Quantenelektrodynamik (Band IVb der Lehrbuchreihe ¨ uber Theoretische Physik von L. D. Landau und E. M. Lifshitz) , 1 ed. (Akademie-Verlag, Berlin, 1973). [79] W. R. Johnson and G. Soﬀ, At. Data Nucl. Data Tables 33, 405 (1985). [80] P. J. Mohr, in Atomic, Molecular, and Optical Physics 11Handbook , edited by G. W. F. Drake (A. I. P., Wood- bury, NY, 1996), pp. 341–351. [81] U. D. Jentschura and K. Pachucki, Phys. Rev. A 54, 1853 (1996). [82] U. D. Jentschura, G. Soﬀ, and P. J. Mohr, Phys. Rev. A 56, 1739 (1997). [83] P. J. Mohr, G. Plunien, and G. Soﬀ, Phys. Rep. 293, 227 (1998). [84] M. I. Eides, H. Grotch, and V. A. Shelyuto, Los Alamos e-print hep-ph/0002158. [85] S. Wolfram, Mathematica-A System for Doing Math- ematics by Computer (Addison-Wesley, Reading, MA, 1988). [86] Certain commercial equipment, instruments, or materi - als are identiﬁed in this paper to foster understanding. Such identiﬁcation does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equip- ment identiﬁed are necessarily the best available for thepurpose. [87] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). [88] J. Schwinger, Phys. Rev. 82, 664 (1951). [89] C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, NY, 1980). [90] W. Greiner and J. Reinhardt, Quantum Electrodynamics (Springer, New York, 1992). [91] M. Reed and B. Simon, Methods of Modern Mathemati- cal Physics IV: Analysis of Operators (Academic Press, New York, 1978). [92] J. Fischer, Int. J. Mod. Phys. A 12, 3625 (1997). [93] M. Beneke, Phys. Rep. 317, 1 (1999). [94] D. Broadhurst and D. Kreimer, Los Alamos e-print hep- th/9912093; Phys. Lett. B 475, 63 (2000). [95] W. Dittrich and H. Gies, Probing the Quentum Vacuum – Perturbative Eﬀective Action Approach in Quantum Elec- trodynamics and Its Applications – Springer Tracts in Modern Physics No. 266 (Springer, New York, 2000). 12
arXiv:physics/0010039v1 [physics.data-an] 15 Oct 2000Maximally Informative Statistics Maximally Informative Statistics David R. Wolf PO 8308, Austin, TX 78713-8308, USA, E-mail: drwolf@realti me.net Dr. Wolf is the corresponding author. Edward I. George Department of MSIS, University of Texas, Austin, TX 78712-1 175, Email: egeorge@mail.utexas.edu Revision history: April 1996. Presented Bayesian Statisti cs 6, Valencia, 1998. Invited paper Monograph on Bayesian Methods in the Sciences , Rev. R. Acad. Sci. Exacta. Fisica. Nat. Vol. 93, No. 3, pp. 381–386, 1 999. Arxiv version asserts bold vectors dropped in print. Abstract: In this paper we propose a Bayesian, information theoretic a p- proach to dimensionality reduction. The approach is formul ated as a varia- tional principle on mutual information, and seamlessly add resses the notions of suﬃciency, relevance, and representation. Maximally in formative statistics are shown to minimize a Kullback-Leibler distance between p osterior distri- butions. Illustrating the approach, we derive the maximall y informative one dimensional statistic for a random sample from the Cauchy di stribution. Keywords: Bayesian Inference; Kullback-Leibler distance; Maximall y in- formative statistics; Suﬃcient statistics; Mutual inform ation; Calculus of variation. c∝circleco√yrt1999 by ﬁrst author. Reproduction for noncommercial purpos es permitted.Maximally Informative Statistics 2 1 Introduction Dimensionality reduction is a fundamental goal of statisti cal science. In a modeling context, this is often facilitated by estimating a low dimensional quantity of interest. For example, suppose the quantities o f interest are the labels of a classiﬁcation of photographs of objects; of tree s, children, etc. The data are the photographs, and the goal is to infer which of the several classes have been presented. In this case the data space ofte n has dimension on the order of >106, while the parameter space is a small discrete set of labels each having much lower dimension. A low dimensional s ummary of the photograph is then obtained as the estimate of the classi ﬁcation of the photograph. In this paper, we propose a novel fully Bayesian information theoretic approach to dimensionality reduction, based on maximizing the mutual in- formation between a statistic and a quantity of interest. Th e approach is for- mulated as a variational principle on mutual information, a nd it seamlessly addresses the notions of suﬃciency, relevance, and represe ntation. We refer to statistics which maximize this mutual information as maximally informa- tive(MI) statistics. Such statistics are shown to minimize a Kul lback-Leibler distance between posterior distributions. The mutual information between a statistic and a quantity of interest is deﬁned in section 2. The mutual information based variati onal principle for MI statistics is utilized in section 3 to derive non-vari ational derivative forms of the principle. In section 4 several properties of MI statistics are derived. The important result of this section is that MI stat istics provide a generalization of the notion of suﬃciency, because they ar e sensible both when they are not suﬃcient statistics, and when lower-than- data-dimension suﬃcient statistics do not exist. In section 5 we present the result that in inference the Kullback-Leibler (KL) distance is properl y a functional of posterior distributions. There we ﬁnd MI statistics at func tional minima of a KL distance based on posterior distributions of the parame ter of interest. The arguments made here suggest that the KL distance derived here is to be preferred to a maximum relative entropy distance, a fact w hich is not discussed in, for example, Kullback [3] or Shor [5], and nume rous others. In section 6 the MI static for the location parameter of the univ ariate Gaussian distribution is derived, and shown to be the expected result , since in this case a one-dimensional suﬃcient statistic exists. In secti on 7 we ﬁnd a one-Maximally Informative Statistics 3 dimensional MI statistic for the Cauchy distribution, wher e a suﬃciency reduction does not exist. In section 8 we discuss approximat ing the posterior distribution as a Gaussian and apply this technique to show t hat the MI statistics are then Bayes’ estimators of the mean and standa rd deviation. In that section a contrast of the approximate MI inference appr oach with the Maximum Entropy method is made, and it is shown that although they agree for Gaussian likelihoods, they disagree for other distribu tions, with simplicity arguing in favor of the MI statistics. 2 The Mutual Information Between a Statistic and a Quantity o f Interest Let the data x∈Xbe drawn according to a parameterized distribution P(x\|θ), with θ∈Θ, the parameter space. θitself is distributed according to the prior P(θ). The marginal distribution of xis obtained from P(x) =/integraltextP(x\|θ)P(θ)dθ, and the posterior of θgiven xis obtained from Bayes Theorem as P(θ\|x) =P(x\|θ)P(θ) P(x)(1) The quantity of interest q=ξQ(θ) will be a function of θ, a mapping from the parameter space Θ into some Q,ξQ(·) : Θ→Q. It will be useful to use the Dirac delta-function δ(·) to represent the distribution of qas P(q\|θ) := P({θ:q=ξQ(θ)} \|θ) =δ(q−ξQ(θ)) (2) = Πkq i=1δ(qi−ξQ,i(θ)), (3) where δ(z(·)) = Π iδ(zi(·)). Note that (2) may be seen directly by using Bayes’ theorem to expand P(q,θ) asP(q\|θ)P(θ), integrating that over q, which must produce P(θ), and noting that because the support of P(q\|θ) is the unique qsuch that q=ξQ(θ) (θis speciﬁed), P(q\|θ) must therefore be the Dirac delta function. The distribution of qgiven the data x, may be written using (1) and (3) as P(q\|x) =/integraldisplay P(q\|θ)P(θ\|x)dθ (4)Maximally Informative Statistics 4 A statistic r=ξR(x) will be a function of x, a mapping from the data space Xinto some R,ξR(·) :X→R. Again using the delta notation, the distribution of the statistic given data is P(r\|x) = δ(r−ξR(x)) (5) = Πkr i=1δ(ri−ξR,i(x)) (6) The joint distribution of the statistic rand the quantity of interest q, con- ditioned on the data xis P(r,q\|x) =P(r\|x)P(q\|x) (7) (since r=ξR(x) is speciﬁed once xis known, making P(r\|x,q) =P(r\| x)), and the unconditional joint distribution is P(r,q) =/integraldisplay P(r\|x)P(q\|x)P(x)dx. (8) Finally, we deﬁne the mutual information between a statisti cξR(·) and a quantity of interest ξQ(·) as M(ξR(·),ξQ(·)) =/integraldisplay /integraldisplay P(r,q)log/parenleftBiggP(r,q) P(r)P(q)/parenrightBigg dqdr (9) This mutual information is the Kullback-Leibler distance b etween the joint distribution P(r,q) and the marginal product P(r)P(q) corresponding to independence between randq. Note that this Kullback-Leibler distance is diﬀerent from the Kullback-Leibler distance mentioned in t he introduction (and seen later in section 5). A major contribution of this pa per is the demonstration of how these two Kullback-Leibler distances are related. 3 MI Statistics and the Variational Principle We now deﬁne the maximally informative (MI) statistic. LetS={ξR(·)}be a set of statistics under consideration. A MI statistic for a quantity of interest ξQ(·)is any statistic ξR(·)from Smaximizing the mutual information M(ξR(·),ξQ(·))between the statistic and the quantity of interest.Maximally Informative Statistics 5 The following variational principle can be used to obtain an MI statistic. Letδ δf(·)denote the functional derivative with respect to f(·). Choose ξR(·)fromSsuch thatδM(ξR(·),ξQ(·)) δξR(·)= 0andδ2M(ξR(·),ξQ(·)) δξR(·)2 is negative semideﬁnite, i.e. so that ξR(·)maximizes the informa- tion between itself and ξQ(·), the quantity of interest. If possible, choose the global maximum. Note that MI statistics in Smay occur on the boundary of S. This may be a case of interest, which occurs when constraints are i mposed on the statistics, and may be handled with a trivial modiﬁcation. N ote also that the space Sof statistics may be constrained to contain only low-dimens ional statistics, in order to force a dimesionality reduction of t he data. We now demonstrate the variational principle for MI statist ics. The ar- gument proceeds by varying (see, for example [1] for the vari ational calculus) the mutual information of (9) with respect to the statistic f unction ξR(·) of dimension kr, i.e.ξR(·) = (ξr,1(·), . . ., ξ r,kr(·)). We now proceed to substitute ξR(x) =ξ0 R(x) +ǫη(x) in (9), and take the derivative with respect to ǫ. Assuming appropriate regularity conditions, we have ∂ǫM(ξR(·),ξQ(·)) =/integraldisplay /integraldisplay/bracketleftBigg ∂ǫP(r,q)log/parenleftBiggP(r,q) P(r)P(q)/parenrightBigg +P(r)∂ǫP(q\|r)/bracketrightbigg dqdr (10) =/integraldisplay /integraldisplay ∂ǫP(r,q)log/parenleftBiggP(r,q) P(r)P(q)/parenrightBigg dqdr,(11) where simpliﬁcation from (10) to (11) occurs because probab ility is conserved. Utilizing (7) we ﬁnd P(r,q) =/integraldisplay δ(r−ξR(x))P(q\|x)P(x)dx (12) Taking the derivative of (12) with respect to ǫyields ∂ǫP(r,q) = Σkr j=1/integraldisplay δ′(rj−ξR,j(x))ηj(x)Πi/negationslash=jδ(ri−ξR,i(x))P(q\|x)P(x)dx (13)Maximally Informative Statistics 6 Note that because ηis arbitrary, we may choose it to simplify as needed. We proceed by considering krchoices of η. Label the choices by m∈ {1, . . ., k r}, and on choice mtake the components of ηas follows: ηℓ(x) = δ(x−xc),(ℓ=m) (14) ηℓ(x) = 0 ,(ℓ∝ne}ationslash=m) (15) where xcis any data point we may choose. The condition that the mutual information is extremal then becomes the statement that for allxcandi∈ {1, . . ., k r}. ∂ǫM(ξR(·),ξQ(·))\|ǫ=0= 0 (16) =/integraldisplay /integraldisplay δ′(ri−ξ0 R,i(xc)) Π i/negationslash=jδ(ri−ξ0 R,i(xc)) ×P(q\|xc)log/parenleftBiggP(r,q) P(r)P(q)/parenrightBigg dqdr(17) Integrating (17) by parts with respect to r(dropping both the “0” su- perscript and subscript “ c”, since there is no distinction to be made at this point) yields the condition that for all x /integraldisplay P(q\|x)∂rlog/parenleftBiggP(r,q) P(r)P(q)/parenrightBigg \|r=ξR(x)dq= 0 (18) where derivatives with respect to vectors are gradients (ve ctors of deriva- tives). The form from which the theorems of the next section a re proven, is found by rewriting (18) as /integraldisplayP(q\|x) P(q\|r)∂rP(q\|r)\|r=ξR(x)dq= 0 (19) 4 MI Statistics and Suﬃciency Now we prove several important properties concerning MI sta tistics. The ﬁrst property is the intuitively obvious property that data is a MI statistic . The second property is that any suﬃcient statistic is a MI statistic . Finally, we note that MI statistics are not necessarily suﬃcient statistics .Maximally Informative Statistics 7 Theorem 1 .Any 1–1 function of data is a MI statistic of the quantity of interest Proof: Let ξR(·) be the identity so that ξR(x) =xin (19). The fraction in that equation is then 1, and the derivative integ rates to zero because probability is conserved. Having ξR(x) any in- vertible function changes nothing as any value of it determi nes x. Theorem 2 .Any suﬃcient statistic for the quantity of interest is a MI statistic of the quantity of interest Proof: Note that, using the deﬁnition of ξR(x) being a suﬃcient statistic, the ratio in (19) is one - the posterior distribut ion of the quantity of interest given the data xis the same as the posterior distribution of the quantity of interest given the suﬃcient statistic ξR(x). The derivative then integrates to zero because probabili ty is conserved. (Note that in both Theorems 1 and 2 the Hessian condition of th e MI inference variational principle is easily established sin ce then the extremum of the mutual information is easily seen to be a local maximum . Otherwise, one must check convexity.) Although it is true that any suﬃcient statistic is a MI statis tic, the con- verse is false. In problems (of data dimension greater than o ne) where a lower-than-data dimension suﬃcient statistic does not exi st, there will exist a lower-than-data dimension statistic which is MI but not su ﬃcient. Thus, the class of maximally informative statistics contains the suﬃcient statistics, but is broader. MI statistics need not provide all of the avai lable information about the underlying quantity of interest. For example, as w e show in Sec- tion 7, such a lower-than-data dimension MI statistic can be obtained for the Cauchy distribution where a lower-than-data dimension suﬃ cient statistic is a-priori unavailable. In this manner, MI statistics seamle ssly address rele- vance to the consumer of the information because it is about some relevant quantity of interest that MI statistics are maximally informative.Maximally Informative Statistics 8 5 MI Statistics and the KL Distance Equation (19) may be rewritten as ∂r/bracketleftBigg/integraldisplay P(q\|x)log/parenleftBiggP(q\|x) P(q\|r)/parenrightBigg dq/bracketrightBigg \|r=ξR(x)= 0 (20) which, along with the curvature condition, states that Theorem 3 .The Kullback-Leibler distance between the posterior distribution conditioned on the statistic and the posterio r distri- bution conditioned on the data is minimized by a MI statistic . Again, note that MI statistics for the quantity of interest a re generally not suﬃcient statistics for the quantity of interest. Indeed, r ather than making the Kullback-Leibler distance zero, as in the case of suﬃcie nt statistics, MI statistics are found at local minima of the Kullback-Lieble r distance - viewed as a functional of the statistic. This demonstrates how the a pproach of this paper generalizes that performed by Lindley [4]. 6 MI Statistics for the Gaussian distribution This section details the inference of the one-dimensional M I statistic for the one-dimensional Gaussian distribution. We take the positi on parameter of the Gaussian to be q, and the the goal is to ﬁnd ξR(x) so that (19) holds. From there note that the calculation of P(q\|r) and P(q\|x) is necessary, and by Bayes’ theorem therefore it is necessary to ﬁnd P(r\|q), which may be written as P(r\|q) =/integraldisplay P(r\|q,x)P(x\|q)dx =/integraldisplay P(r\|x)P(x\|q)dx =/integraldisplay δ(r−ξR(x))N/productdisplay i=1e−(xi−q)2/2σ2 √ 2πσdx (21) The ansatz ξR(x) =/summationtextN i=1λixiis useful (and not resrictive since the λ′ isare implicitly only restricted to be functions of x), and making the changes ofMaximally Informative Statistics 9 variables yi=λixifollowed by ui=λiqin (21) yields a form which may immediately be recognized as the convolution of NGaussians with means µi=λiqand standard deviations σi=λiσrespectively, P(r\|q) =/integraldisplay δ(r−N/summationdisplay i=1λiq−N/summationdisplay i=1ui)N/productdisplay i=1e−u2 i/2(σλi)2 √ 2π(λiσ)du. (22) This has the solution P(r\|q) =φ(0, σ′)(r−N/summationdisplay i=1λiq) (23) where σ′=σ/radicalBig/summationtextN i=1λ2 iandφ(·,·)(·) is the Gaussian density φ(µ, σ)(z) =1√ 2πσe−(z−µ)2/2σ2. (24) Finally, inserting this result into Bayes’ theorem with uni form prior to ﬁnd the posterior distribution of qconditioned on ryields P(q\|r) =S φ(0, σ′)(r−N/summationdisplay i=1λiq) (25) where S:=/summationtextN i=1λi. The calculation for P(q\|x) is similar with the result is that P(q\|x) =φ(x,σ√ N)(q) (26) where x:=1 N/summationtextN i=1xi. From the forms of (25) and (26) it is clear that not only will the integrand of (20) (that equation equivalen t to (19)) be minimized, but that it will be zero, if all λi= 1/Nis chosen. This of course is the expected result since ξR(x) =/summationtextN i=1xi/Nis a suﬃcient statistic for q when σis known. Alternatively, to satisfy that the calculation indicated i n (19) is successful at ﬁnding the expected result, continue by taking (25) and (2 6) and substi- tuting them into (19) to ﬁnd after some simpliﬁcation the equ ation which must be satisﬁed by ξR 0 =/bracketleftBigg/integraldisplay (r−qN/summationdisplay i=1λi)e−(x−q)2/2(σ/√ N)2)dq/bracketrightBigg \|r=ξR(x). (27)Maximally Informative Statistics 10 This has the unique solution ξR(x) =xwhen the arbitrary scale of the inferred statistic is ﬁxed by setting 1 =/summationtextN i=1λi. To conclude this section, the procedure culminating in (20) or (19) of ﬁnding MI statis tics has been shown to produce the expected known result for the Gaussian c ase. The next section approaches the Cauchy distribution case for lo wer than data dimension statistics, where there is no suﬃcient statistic available and the result is novel. 7 MI Statistics for the Cauchy Distribution This section outlines the inference of the one-dimensional MI statistic for the one dimensional Cauchy distribution. The detailed step s may be taken similarly to those of the last section but taking the Cauchy d istribution instead of the Gaussian distribution. Take the position par ameter of the Cauchy to be q, and the the goal is to ﬁnd ξR(x) so that (19) holds. As in the last section it is necessary to determine both P(q\|r) and P(q\|x). Assuming the same ansatz that ξR(x) =/summationtext iλixi, the necessary convolutions may be carried out with the use of the Fourier convolution the orem, with the results that P(r\|q) =S π(S2+ (r−qS)2), (28) P(q\|r) =S π(S2+ (r−qS)2), (29) and P(q\|x)∝n/productdisplay i=11 π(1 + (xi−q)2)(30) where S:=/summationtextN i=1λi. Substituting (28), (29), and (30) into (19) yields the equation that must be solved for ξR(x) 0 =/bracketleftBigg/integraldisplay/parenleftBiggn/productdisplay i=11 π(1 + (xi−q)2)/parenrightBiggr/S−q 1 + (r/S−q)2dq/bracketrightBigg \|r=ξR(x).(31) Rewriting this equation in more suggestive terms, while tak ing the scale S= 1, gives the result as an implicit equation for ξR(x), ξR(x) =/integraldisplay q P(q\|(x, ξR(x)) )dq. (32)Maximally Informative Statistics 11 The form of the result (32) says that ξR(x) is the posterior mean of qgiven the data and itself as an additional observation . This form also suggests thatξR(x) could be the posterior mean of qgiven the data. However, this is not the case, as a check using the posterior moment forms de rived in [6] immediately shows. Further, assuming a value for ξR(x) on the right-hand side of (32) allows that to be computed in closed form using th e results of [6]. This ﬁnally yields that the left-hand side is a ration al function of the right-hand side, a ﬁxed point equation which may be solve d by standard iterative methods. Other checks immediately show that the s olution is not the maximum likelihood solution, nor the median. To conclude this section, the one-dimensional MI statistic for the Cauchy distribution position parameter has been found as the poste rior mean of the position parameter of the Cauchy distribution given the dat a and the MI statistic, and this statistic is diﬀerent from the Bayes’ es timator which is the posterior mean given the data only. 8 Approximate MI Inference and Bayes Estimators In many cases of interest, if not in all cases of relevance wit h high dimensional data, the convolutions that appear similarly to those in (28 ) etc. will be quite impossible to do in closed form, and probably in a practical s ense will even be numerically intractable. However, there is an approach t hat may be taken which does some harm to a fully rigorous Bayesian approach, b ut which may be necessary. The idea that is applicable in these cases of di ﬃculty is to directly take P(q\|r) in (20) to be Gaussian with parameters r=ξR(x)= (µ(x), σ(x)). The approximate MI approach just outlined is applied bel ow to ﬁnding the approximate MI statistics ( µ(x), σ(x)). The approximate MI approach is then contrasted with an alternative approach using the KL distance inverted from that of (20), one that resembles Maxi mum Entropy inference. The rusults of this section hold for any likeliho od, as will become apparent. Take an arbitrary one-dimensional parameterized likeliho od parameter- ized by q(i.e. with qthe parameter of interest). Parameterize the inferred distribution P(q\|r) of (20) as (see (24)) P(q\|r= (µ, σ)) =φ(µ, σ)(q). (33)Maximally Informative Statistics 12 Equations (20) and (33) imply that the MI statistic is µ=/integraldisplay q P(q\|x)dq σ2=/integraldisplay (q−µ)2P(q\|x)dq (34) These quantities are the Bayes’ estimators for the mean and s tandard devi- ation of the distribution. If, on the other hand, the inverted form of the KL distance is t aken, as it often is in many of the cases we have observed, the statisti cµis µ=/integraltextq φ(µ, σ)(q)log(P(q\|x))dq/integraltextφ(µ, σ)(q)log(P(q\|x))dq(35) which, along with another non-linear equation for σ, is a complicated non- linear system to be solved for r= (µ, σ). Note that when the likelihood P(x\|q) is Gaussian these two approxi- mate approaches produce the same statistic, the posterior m ean and standard deviation; but for the Cauchy likelihood, for example, this is not the case, with necessity to solve the complicated nonlinear system. I n contrast, the approximate MI inference technique always produces the pos terior Bayes’ moment estimators. The diﬀerence between the forms of the approximate MI statis tics and the inverted-KL statistics appearing in (34) and (35) respe ctively makes it clear that one needs a good ﬁrst-principles approach to the K L distance. 9 Conclusion We have formulated the mutual information based variationa l principle for statistical inference, a fully Bayesian approach to infere nce, deﬁned MI statis- tics for a quantity of interest, shown how the principle may b e reformulated as a minimal KL distance principle based on posterior distri butions, and demonstrated how inference proceeds, when lower-than-dat a dimension suf- ﬁcient statistics are absent, using the Cauchy distributio n. Finally, an ap- proximate approach to the inference of MI statistics was dis cussed, and the relationship of the resulting statistics to Bayes’ estimat ors and the Maximum Entropy version of the same approximation was noted.Maximally Informative Statistics 13 10 Acknowledgements Thanks go to the Data Understanding Group at NASA Ames for the ir lively and interactive critique, friendship, mentoring, and supp ort. This paper was improved by comments from Dr. Jeremy Frank and Hal Duncan, bo th of NASA. Much thanks to Tony O’Hagan for detailed comments. Thi s work was suported by NASA Center for Excellence in Information Te chnology contract NAS-214217. This work was supported by NSF grant DM S-98.03756 and Texas ARP grants 003658.452 and 003658.690. References [1] George Arfken. Mathematical Methods for Physicists . Academic Press, Inc., London, 1985. [2] George E. P. Box and George C. Tiao. Bayesian Inference in Statistical Analysis . Wiley, NY, 1973. [3] Solomon Kullback. Information Theory and Statistics . John Wiley and Sons, Inc., New York, 1959. [4] D. V. Lindley. On a measure of the information provided by an experi- ment. Annals of Mathematical Statistics , 27:986–1005, 1961. [5] John E. Shore and Rodney V. Johnson. Axiomatic derivatio n of the prin- ciple of maximum entropy and the principle of minimum cross e ntropy. IEEE Transactions on Information Theory , 26(1):26–37, January 1979. [6] David R. Wolf. Posterior moments of the cauchy distribut ion. In Max- imum Entropy and Bayesian Methods , eds. W. van der Linden et. al., Kluwer Academic, Dordrecht, Netherlands, 1998.
arXiv:physics/0010040v1 [physics.chem-ph] 16 Oct 2000Hydration of Methanol in Water A DFT-based Molecular Dynamics Study Titus S. van Erp and Evert Jan Meijer Department of Chemical Engineering, Universiteit van Amst erdam, Nieuwe Achtergracht 166, NL-1018 WV AMSTERDAM, The Netherlands We studied the hydration of a single methanol molecule in aqueous solution by ﬁrst-principle DFT-based molecular dynamics simulation. The calculations show that the local structural and short-time dynamical properties of the wa- ter molecules remain almost unchanged by the presence of the methanol, conﬁrming the observation from recent exper- imental structural data for dilute solutions. We also see, i n accordance with this experimental work, a distinct shell of water molecules that consists of about 15 molecules. We found no evidence for a strong tangential ordering of the wat er molecules in the ﬁrst hydration shell. INTRODUCTION The solvation of alcohols in water has been studied extensively. [1] It is of fundamental interest in physics, chemistry and biology, but also of importance in technical applications. The characteristic hydroxyl group allows alcohols to form hydrogen bonds and is responsible for the good solubility of the smaller alcohols. In contrast, the alkyl group is hydrophobic and does not participate in the hydrogen bonding network of water. The presence of both hydrophobic and hydrophilic groups make the microscopic picture of solvation of alcohol in water a non- trivial and therefore interesting matter. Understanding the solvation of methanol in water is a prerequisite for the study of chemistry of alcohols in aqueous solution. Important examples of such reactions are the conversion of ethanol into acetaldehyde in biolog- ical systems or the industrial ethanol production by acid- catalysed hydration of ethylene. An accurate microscopic understanding of the mechanism and kinetics of such re- actions is of fundamental interest. However, presently, this picture is still far from complete. Density Functional Theory (DFT) based Molecular Dynamics simulation has proved to be a promising tool provide such an insight. An accurate calculation of the chemical bonding is incorpo- rated via a DFT-based electronic structure calculations. The eﬀect of temperature and solvent on the reactive events is implicitly accounted for via the Molecular Dy- namics technique. The implementation of DFT-based MD as proposed by Car and Parrinello [2] has proven to be extremely eﬃcient. It has successfully been applied to study of a large variety of condensed-phase systems at ﬁnite temperature. Applications to chemical reactions include the cat-ionic polymerization of 1,2,5-trioxane [3 ], or the acid-catalysed hydration of formaldehyde [4]. As a ﬁrst step towards the study of chemical reac- tions involving alcohols we present in this paper a Car- Parrinello Molecular Dynamics (CPMD) study of the hy- dration of the simplest alcohol (methanol) in aqueoussolution. Recent experimental work [5] has provided de- tailed structural information on the solvation shell. Var- ious molecular simulation studies (e.g. Ref. [6–10] have addressed structure and dynamics of both the solute and the solvent. This experimental and numerical work has revealed that there is a distinct solvation shell around the methanol, and that the water structure is little af- fected by the presence of a methanol molecule. In this paper we will address these structural properties and in addition consider the dynamics of the methanol and the water molecules in the solvation shell. This paper is organized as follows. First we outline the computational approach and its validation. Then we present the results for the structure and dynamics of a single solvated methanol in water. We conclude the paper with a summary and discussion. METHODS AND VALIDATION Electronic structure calculations are performed using the Kohn-Sham formulation [11] of DFT. [12] We em- ployed the BLYP functional, that combines a gradient- corrected term for the correlation energy as proposed by Lee, Yang and Parr [14] with the gradient correction for the exchange energy due to Becke [13]. Among the avail- able functionals, the BLYP functional has proven to give the best description of the structure and dynamics of wa- ter. [15,16] All calculations [17] were performed using the CPMD package. [18] The pseudopotential method is used to restrict the number of electronic states to those of the valence elec- trons. The interaction with the core electrons is taken into account using semi-local norm-conserving Martins- Troullier pseudopotentials. [19] The pseudopotential cut - oﬀ radius for the H was chosen 0.50 au. For O and C the radii are taken 1.11 and 1.23 a.u. for both the l=s and l=p term. The Kohn-Sham states are expanded in a plane-wave basis set matching the periodicity of the peri- odic box with waves up to a kinetic energy of 70 Ry. Test calculations showed that for this structural and energetic properties were converged within 0 .01˚A and 1 kJ/mol, respectively. Frequencies are converged within 1 %, ex- pect for CO and OH stretch modes that are underesti- mated by 3 % and 5 % compared to basis-set limit values. To validate the computational methods outlined above we performed a series of reference calculations of rel- evant gas-phase compounds with the CPMD package. Energetics and geometry were calculated for methanol, water, two mono-hydrate conﬁgurations, and the di- 1hydrate conﬁguration shown in Fig. 1. These calcula- tions were performed using a a large periodic box of size 10x10x10 ˚A3. The interactions among the periodic images were eliminated by a screening technique sim- ilar to that of Ref. [20]. In addition we determined for the methanol molecule both the harmonic vibra- tional frequencies and the frequencies at ﬁnite temper- ature (T= 200 K). The latter includes the anharmonic contributions, and were obtained from the spectrum of the velocity auto correlation function (VACF) of a 3 ps CPMD calculation at E= 200 K. The calculated peak po- sitions can be compared with experimental spectra. Re- sults of the gas-phase calculations were compared with results obtained with a state-of-the-art atomic-orbital based DFT package (ADF [21]), and with results from MP2 calculations of Ref. [22]. In the comparison of the energies zero-point energies were not taken into account. 1.981 108.1104.2 1.096 1.094 1.091107.3107.0106.81.446 1.437 1.415108.5 107.00.9640.979 1.993 1.937104.5 104.9 0.9600.981 0.973173.2 173.7 180.00.986 0.967 0.979 0.971 0.9580.990 0.983172.2 104.6 104.9103.6172.9172.0 1.941 1.915 1.8900.979107.7 ...106.8 0.9600.9721.4281.4511.460 108.0108.5111.0 111.1 111.31.094109.6109.7109.9 1.0991.098 108.2(a) (b) 1.907153.3 0.9580.9720.9781.9411.9571.976 0.9740.9900.997148.6152.5153.8 154.5 151.3 1.9171.866 1.8551.8660.999 0.992 0.973 0.979105.3 105.6 105.01.4431.452 1.420106.9 107.3107.0 0.9690.9860.992109.0 109.1 108.3 0.972 0.9591.898 FIG. 1. Energy-optimized geometries of two water /methanol dimers and a trimer. Distances ( ˚A) and angles (degrees) are shown for three computational methods : CPMD-BLYP (top, present work), ADF-BLYP [21] (middle, present work) and MP2 [22] (bottom). Complexation energies and geometries of the methanol hydrates are given in Tab. I and Fig. 1. Deviations among CPMD and ADF are within 1 kcal/mole for the energies, smaller than 0 .005˚A for the inter-molecular bonds and within 0 .03˚A for the weaker intra-molecular bonds. This indicates a state-of-the art accuracy for electronic struc - ture methods employed in CPMD. Diﬀerences among BLYP and MP2 are within acceptable limits, with BLYP complexation energies smaller by 4 kJ/mole (dimer) and 10 kJ/mole (trimer). These deviations are similar to the comparison of BLYP and MP2 for the water dimer bind- ing energy, [15,23] where BLYP is 4 kJ/mole smaller, with the MP2 energy only 1 kJ/mol below the experi- mental value. Assuming similar diﬀerences for the com-plexation energies bonds in the methanol hydrates would suggest that BLYP underestimates the methanol-water binding energy by approximately 5 kJ/mol. Inter- and intra-molecular BLYP bond lengths are up to 0 .02 and 0.06˚A longer compared to the MP2 results, respectively. TABLE I. Complexation energies (kJ/mol) of methanol hydrates shown in Fig. 1. Numbers are bare values without zero-point energy corrections and entropy contributions. Complex CPMD-BLYP ADF-BLYPaMP2b CH3O + H 2O (a) 20.2 20.2 24.4 CH3O + H 2O (b) 17.1 17.6 21.0 CH3O + 2 H 2O 58.3 59.6 68.8 aRef. [32]. bG2(MP2) method. MP2(full)/6-311+G(d,p) optimized ge- ometries. From Ref. [22]. Vibrational frequencies are listed in Tab. II. Again comparison of CPMD and ADF is excellent, consis- tent with the results for the energetics and geometries. Comparing the calculated ﬁnite-temperature frequencies against the experimental values shows that BLYP tends to underestimate the frequencies of almost all modes by ≈10 %. This trend is a known feature of BLYP. For example similar deviations are observed for BLYP calcu- lation of water. [15] Overall we conclude that the reference calculations of gas-phase provides conﬁdence that DFT-BLYP performs with a suﬃcient accuracy for a quantitative study of methanol hydration. TABLE II. Harmonic and T=200 K vibrational frequen- cies of gas-phase methanol molecule. Harmonic Anharmonic ν(cm−1) ν(cm−1) mode CPMD-BLYP ADF-BLYPaCPMD-BLYP Exp.b (T=200 K) τ(OH) 280 380 280 270 ν(CO) 940 950 880 1034 r(CH3) 1040 1050 980 1075 r(CH3) 1130 1130 1070 1145 δ(OH) 1330 1340 1270 1340 δ(CH3) 1430 1430 1320-1430c1454 δ(CH3) 1460 1460 1320-1430c1465 δ(CH3) 1470 1470 1320-1430c1480 ν(CH3) 2940 2910 2640 2844 ν(CH3) 2990 2950 2740 2970 ν(CH3) 3060 3020 2830 2999 ν(OH) 3550 3590 3310 3682 aRef. [32]. bRef. [33]. cModes not separated. Broad peak with width listed. 2SOLVATION We performed Car-Parrinello Molecular Dynamics simulations of the solvation of a single methanol molecule. We considered two systems: one with 31 water molecules and the other with 63 water molecules, yield- ing methanol-water solutions with mole ratios of 1:31 and 1:63. In the following they are referred to as the small and large system, respectively. For reference we also performed a simulation of a pure water sample of 32 molecules. The molecules are placed in a periodic cu- bic box with edges of 9.98 ˚A (small solvated methanol system), 12.50 ˚A (large solvated methanol system), and 9.86˚A(pure water) corresponding to the experimental densities at ambient conditions. The temperature of the ions is ﬁxed at 300 K using a Nos´ e-Hoover thermostat [24–26]. The ﬁctitious mass associated with the plane- wave coeﬃcients is chosen at 900 a.u., which allowed for a time step in the numerical integration of the equations- of-motion of 0.145 fs. The two systems were equilibrated for 1 ps from an initial conﬁguration obtained by a force- ﬁeld simulation. Subsequently we gathered statistical av- erages from a 10 ps trajectory of the 31+1 molecule sys- tem, from a 7 ps trajectory of the 63+1 molecule system, and from a 10 ps trajectory of the pure water system. Structure In Fig. 2 we have plotted the radial distribution func- tions (RDF) of the water oxygen atoms. The minor vari- ations among the RDF’s of the small methanol system, the large methanol system, and the pure water system is an indication that the local water structure, as mea- sured by this RDF, is at only marginally changed by the solvation of a methanol molecule. Note, in this respect, that for the 32 molecule the ﬁrst solvation shell consti- tutes a signiﬁcant fraction of the total number of water molecules (see below). 0.01.02.03.0gCO31 water + 1 methanol 63 water + 1 methanol 0.01.02.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0gOwOw r( )Ao31 water + 1 methanol 63 water + 1 methanol 32 water FIG. 2. Calculated carbon-oxygen (top) and water oxy- gen-oxygen (bottom) radial distribution functions for the small (solid line) and large (dashed line) methanol system. Fig. 2 also shows the RDF of the methanol carbon and water oxygens for the small and large methanol system. A pronounced ﬁrst peak clearly indicates the existence ofshell of water molecules at a distance of ≈3.7˚A. Com- paring the RDF’s of the small and large system shows a noticeable diﬀerence. This should be attributed to the limited size of the small system. It suggests that a proper description of the solvation structure of a single methanol in a cubic periodic simulation box requires at least 50 water molecules. Integrating the RDF for the large system up to the minimum at r= 5.0˚A yields 16 water molecules in the ﬁrst solvation shell. The deﬁnite solvation shell observed in our simulations is consistent with the neutron diﬀraction data of Soper and Finney [5] who studied a 1:9 molar methanol-water system. Diﬀer- ences in molarity limits a quantitative comparison of the carbon-oxygen RDF, but a qualitative comparison learns that peak positions match with the peak values slightly more pronounced in the simulation results. To analyze the orientational ordering of the water molecules around the methanol we computed the distri- bution function of the angle between the C-O H2Obond vector and the normal to the plane of the water molecules in the ﬁrst solvation shell. The results show that angle distribution is relatively uniform with a small tendency towards the tangential orientation, a feature occurs for al l solvation shell radii in the range of 3.7-5.0 ˚A. Over the range of 0o-90othe distribution gradually decays, with the value at the tangential orientation (0o) about a factor of 2 larger than at the perpendicular orientation (90o). Qualitatively, this seems consistent with data for the ori- entational distribution obtained from neutron-diﬀractio n data [5]. However from this experimental data it is con- cluded that the water molecules prefer to lie tangential and form a cage around the methanol. Our data do not give clear evidence for a cage-like structure. However, this might be a diﬀerent interpretation from similar data. Note, in this respect, also that the experimental data can- not be quantitatively compared to our data, as diﬀerent orientational distribution functions are employed. To analyze the hydrogen bonding we adopted the def- inition of Ref. [7]: two molecules are hydrogen bonded if simultaneously the inter-oxygen distance is less than 3.5˚Aand the OHO angle is smaller than 30o. From the simulation of the large system we found that the methanol hydroxyl group donates and accepts on average 0.9 and 1.5 hydrogen bonds, respectively. For a water molecule these numbers are equal and measured to be 1.7 in the simulation of the pure water sample. These results indicate that the methanol hydroxyl group par- ticipates strongly in the hydrogen bonding network with the a donating behavior similar to water hydrogen and a accepting character somewhat smaller than a water oxy- gen. Dynamics The time scale (7-10 ps) of the present simulations al- lows for a reliable analysis of dynamical properties oc- curring on the picosecond time scale. 3The velocity auto correlation function (VACF) of the hydrogen atoms provides an important measure of hy- drogen bonding. Fig. 3 shows the Fourier spectrum of the calculated VACF of hydrogen atoms of the wa- ter molecules in the small and large methanol sample. The three distinct peaks correspond to the vibrational (3100 cm−1), bending (1600 cm−1), and librational- translational (500 cm−1) modes of the water molecules. The most important observation is that mutual compar- ison of the two methanol samples and the comparison of these with the spectrum of the pure water sample (also plotted) shows no signiﬁcant diﬀerence, not even for the small methanol sample where the solvation shell consti- tutes half of the water molecules in the system. This demonstrates that also the short-time dynamics of the water molecules is hardly aﬀected by the solvation of a methanol molecule. 0.00.20.40.60.81.0F(arbitrary units)isolated methanol methanol in 63 water 0.00.20.40.60.8 05001000150020002500300035004000F(arbitrary units) ν (cm-1)31 water + 1 methanol 63 water + 1 methanol 32 water FIG. 3. Bottom: Calculated Fourier spectrum of the ve- locity auto correlation function of the water hydrogens for the small methanol system (solid line), the large methanol syst em (dashed line), and the pure water sample (dotted line). Top: Calculated Fourier spectrum of the velocity auto correlati on function of the hydrogen atom of the methanol hydroxyl group for the large methanol system (dashed line) and for an isolat ed methanol molecule (solid line). An indication for the average residence time of a water molecule in the ﬁrst solvation shell is obtained by moni- toring the trajectories of the individual water molecules. We found that in the large methanol system over 7 ps 10 water molecules left the region within 5 ˚A from the methanol carbon. From this we estimate the average res- idence time to be of the order of a few picoseconds. Fig. 3 shows the Fourier spectrum of the VACF of the hydroxyl H of methanol obtained from the trajectory of the large system. The spectrum is of limited accuracy due to the relatively short trajectories (7 ps). For com- parison, the calculated spectrum for a single methanol molecule at T= 200 Kis also plotted. In solution the OH stretch ( νOH) peak, with a calculated gas-phase po- sition of about 3300 cm−1, has shifted by ≈200 cm−1to lower frequencies and has a relatively large width. The shift and width are both typical characteristics of a hy- drogen bond and are also observed in the water spec-trum (Fig. 3). In contrast to the OH stretch mode, we see that the OH-bending mode ( δOHat 1300 cm−1) is blue-shifted by an amount of 50-100−1. A comparison with experimental frequency shifts in infrared spectra is limited as, to our knowledge, no experimental data for dilute methanol-water solutions are reported. However, a comparison with measured shifts in liquid methanol [27] shows similar trends for the shift of infrared stretch (-354 cm−1) and bend (+78 cm−1) peaks. The torsional mode ( τOH), expected to be shifted upward to around 600 cm−1, is not visible in our calculated spectra due to the large statistical errors. DISCUSSION We have studied the solvation of a single methanol molecule in water using DFT-based Car-Parrinello molecular dynamics simulation. Validation of the ap- proach showed that energetics, structural, and dynam- ical properties of reference gas-phase compounds were suﬃcient to expect a quantitative accuracy of calculated properties. The calculated solvation structure supports the exper- imental observation [5] that a shell of about 15 water molecules is formed around the methanol. Structural analysis also learns that the hydrogen bonded network of water is only minimally distorted by the presence of the methanol molecule. This conﬁrms the proposition of Soper et al. [5] that speculations that the normal wa- ter structure is signiﬁcantly enhanced by the hydropho- bic alkyl group is groundless. The calculations showed that methanol OH group is strongly involved in hydro- gen bonding, both as acceptor and as donor. Analysis of the dynamics learns that the average residence time of a water molecule in the ﬁrst solvation shell is of the order of a few picoseconds. The vibrational spectrum of the water molecules is hardly changed by the presence of the methanol, indicating that the short-time dynam- ics is hardly aﬀected by the presence of the methanol molecule. Vibrational analysis shows that methanol OH- stretch peak is a broad feature that is signiﬁcantly red- shifted upon solvation, conﬁrming its hydrogen-bonding character. In conclusion, from comparison with available exper- imental data we have shown that ﬁrst-principle DFT- based molecular dynamics simulation provides a reason- able accurate description of the structure and dynamics of a dilute aqueous methanol solution. This opens the way towards the study of chemistry involving methanol and larger alcohols in water. Acknowledgements The Netherlands Organization for Scientiﬁc Research is acknowledged ﬁnancial support. E. J. M. acknowledges the ”Royal Netherlands Academy of Arts and Sciences” for ﬁnancial support. 4[1] F. Franks, in: F. Franks (Ed.), Water: A comprehensive treatised, Vol. 2, Plenum, 1973. [2] R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. [3] A. Curioni, W. Andreoni, J. Hutter, H. Schiﬀer, M. Par- rinello, J. Am. Chem. Soc. 116 (1994) 11251. [4] E. J. Meijer, M. Sprik, J. Am. Chem. Soc. 120 (1998) 6345. [5] A. K. Soper, J. L. Finney, Phys. Rev. Lett. 71 (1993) 4346. [6] W. L. Jorgensen, J. D. Madura, J. Am. Chem. Soc. 105 (1983) 1407. [7] M. Ferrario, M. Haughney, I. R. McDonald, M. L. Klein, J. Chem. Phys. 93 (1990) 5156. [8] G. P´ alink´ as, I. Bak´ o, K. Heinzinger, P. Bopp, Mol. Phy s. 73 (1991) 897. [9] H. Tanaka, K. E. Gubbins, J. Chem. Phys. 97 (1992) 2626. [10] A. Laaksonen, P. G. Kusalik, I. M. Svishchev, J. Phys. Chem. A 101 (1997) 5910. [11] W. Kohn, L. J. Sham, Phys. Rev. 140 (1965) 1133. [12] P. Hohenberg, W. Kohn, Phys. Rev. B, 136 (1964) 864. [13] A. D. Becke, Phys. Rev. A 38 (1988) 3098. [14] C. Lee, W. Yang, R. G. Parr, Phys. Rev. B 37 (1988) 785. [15] M. Sprik, J. Hutter, M. Parrinello, J. Chem. Phys. 105 (1996) 1142. [16] P. L. Silvestrelli, M. Bernasconi, M. Parrinello, Chem . Phys. Lett. 277 (1997) 478. [17] Computational resources consisted of an IBM-SP and a cluster of state-of-the-art PC’s. Calculations were exe- cuted in parallel using MPI and amounted to a total of ≈10000 hours of CPU-time. [18] CPMD, version 3.0f, developed by J. Hutter, P. Bal- lone, M. Bernasconi, P. Focher, E. Fois, S. Goedecker, M. Parrinello, and M. Tuckermann, at MPI f¨ ur Festk¨ orperforschung and IBM Zurich Research Labora- tory (1990-1997). [19] N. Troullier, J. L. Martins, Phys. Rev. B 43 (3) (1991) 1993. [20] R. N. Barnett, U. Landman, Phys. Rev. B 48 (1993) 2081. [21] ADF 2.3, [28–30] Theoretical Chemistry, Vrije Univer- siteit, Amsterdam. [22] L. Gonz´ alez, O. M´ o, M. Y´ a˜ nez, J. Chem. Phys. 109 (1998) 139. [23] MP2 limit estimate. See for example [31]. [24] S. Nos´ e, J. Chem. Phys. 81 (1984) 511. [25] S. Nos´ e, Mol. Phys. 52 (1984) 255. [26] W. G. Hoover, Phys. Rev. A 31 (1985) 1695. [27] T. Shimanouchi, Tables of molecular vibrational frequ en- cies consolidated, Volume I, National Bureau of Stan- dards, 1 (1972). [28] E. J. Baerends, D. E. Ellis, P. Ros, Chem. Phys. 2 (1973) 42. [29] G. te Velde, E. J. Baerends, J. Comput. Phys. 99 (1992) 84. [30] C. F. Guerra, J. G. Snijders, G. te Velde, E. J. Baerends, Theor. Chem. Acc. 99 (1998) 391. [31] M. Sch¨ utz, S. Brdarski, P.-O. Widmark, R. Lindh, G. Karlstrm, J. Chem. Phys. 107 (1997) 4597. [32] Kohn-Sham orbitals are expanded in an uncontracted triple- ζSlater type basis set augmented with 2p and 3dpolarization functions for H and 3d and 4f polarization functions for C and O. The core states are kept frozen. [33] W. A. Benedict, N. Gailer, E. K. Plyler, J. Chem. Phys. 24 (1956) 1139. 5
arXiv:physics/0010041v1 [physics.bio-ph] 16 Oct 2000On Optimality in Auditory Information Processing Mattias F. Karlsson∗(f95-mka@nada.kth.se ) and John W.C. Robinson∗(john@sto.foa.se ) Defence Research Establishment, SE 172 90 Stockholm, Swede n Abstract. We study limits for the detection and estimation of weak sinu soidal signals in the primary part of the mammalian auditory system using a stochastic Fitzhugh-Nagumo (FHN) model and an action-reaction model f or synaptic plasticity. Our overall model covers the chain from a hair cell to a point j ust after the synaptic connection with a cell in the cochlear nucleus. The informat ion processing perfor- mance of the system is evaluated using so called φ-divergences from statistics which quantify a dissimilarity between probability measures and are intimately related to a number of fundamental limits in statistics and informat ion theory (IT). We show that there exists a set of parameters that can optimize s everal important φ- divergences simultaneously and that this set corresponds t o a constant quiescent ﬁring rate (QFR) of the spiral ganglion neuron. The optimal v alue of the QFR is frequency dependent but is essentially independent of the a mplitude of the signal (for small amplitudes). Consequently, optimal processing acco rding to several standard IT criteria can be accomplished for this model if and only if t he parameters are “tuned” to values that correspond to one and the same QFR. Thi s oﬀers a new explanation for the QFR and can provide new insight into the r ole played by several other parameters of the peripheral auditory system. Keywords: Auditory system, Information, Detection, Estimation, Div ergences 1. Introduction When a sensory cell in a mammal is presented with a stimulus, t he information about it must in general be communicated throug h several layers of intermediating nerve cells before it reaches the p arts of the brain where the ﬁnal processing takes place. A logical quest ion, there- fore, is how much of the information is lost in the ﬁrst parts o f this processing chain and how have these parts of the chain have (p ossibly) been optimized by evolution to combat information loss, for diﬀerent types of stimuli. One of the simplest settings of this proble m is the auditory system. The frequency ﬁltering process in the inne r ear makes it suﬃcient in general, at least for weak signals, to restric t attention to a single type of stimuli, a pure tone , when studying the response of the auditory nerve cells and their connections in the cochlear n ucleus. From an information-theoretic perspective it is thus of interes t to determine how well the peripheral parts of the auditory processing cha in preserve ∗This work was supported by FOA project E6022, Nonlinear Dyna mics. neuro_sub.tex; 2/02/2008; 1:30; p.12 information about the two parameters, the amplitude and pha se, that characterize a tone at a given frequency. An even more fundam ental question, however, is how well information about the presen ce of such a tone is preserved, i.e. in what ways this part of the auditor y processing chain imposes limits on achievable detection performance. Mathemati- cally, these two problems belong to the realm of statistical decision and information theory (IT); for weak tones the detection probl em is, more- over, intimately connected with the estimation problem of d etermining the amplitude. Despite the extensive literature on information processin g in neu- rons, a relatively small number of works treat the fundament al statis- tical limits for neural detection and estimation that bound the perfor- mance of sensory systems. One notable exception, however, i s Stemm- ler’s work (Stemmler, 1996) on the detection and estimation capabilities of the Hodgkin-Huxley, McCullough-Pitts and leaky integra te-and-ﬁre model neurons in terms of the Fisher Information. Stemmler s hows that there exists a universal small-signal scaling law whic h relates the optimal detection, estimation, and communication perform ance of these model neurons, and that this scaling law also applies to the ( narrow- band) signal-to-noise ratio (SNR) on the output of a neuron w hich is excited by a sinusoidal signal. Manwani and Koch (Manwani an d Koch, 1999) give a detailed analysis of the noise in dendritic cabl e structures and its eﬀect on fundamental limits for detection and estima tion. In particular, they provide relations for minimum mean-squar e error in linear estimation and minimum probability of error (the lat ter under an assumption of Gaussian noise) based on a stochastic versi on of the linear one-dimensional cable equation. In the majority of o ther infor- mation theoretic analyses of neural information processin g the focus is on the spike train on the output of a neuron though, and a long- standing objective has been to try to break the “neural code” of the spike train. However, there is a fundamental component miss ing in modeling that rests solely on considering information in th e spike train and it is the inﬂuence of the synaptic connections . The importance of this aspect of neural computation has recently been recogni zed and it has even been suggested that the synaptic connections in fac t represent the primary bottleneck that limits information transmissi on in neural circuitry (Zador, 1998). Consequently, when studying info rmation pro- cessing in neurons, in particular detection and estimation capabilities of the auditory system, it seems imperative to consider mode ls and methods that describe not only the individual neurons and th eir spike trains but also the synaptic connections between the neuron s. In the present study we investigate, theoretically, the fun damental limits for detection and estimation of weak signals in the ma mmalian neuro_sub.tex; 2/02/2008; 1:30; p.23 auditory system. We model the neurons in the auditory nerve a nd their synaptic connections using ideas from Tuckwell (Tuckwell, 1988) and Kistler-Van Hemmen (Kistler and Van Hemmen, 1999) that take into account the notion of synaptic plasticity . Incorporation of the synaptic eﬃcacy’s dependence on the prehistory of action potentials arriving to the synapse in the model makes it possible to obtain a more rea listic assessment of the information available to the next step in t he audi- tory processing chain, the processing in the cochlear nucle us. Another feature of our study is the use of more general measures of sig nal-noise separation. To quantify signal-noise separation we use the so-called φ-divergences from statistics and IT (Liese and Vajda, 1987) . The φ- divergences are applicable to virtually any kind of signal a nd system (in a stochastic setting), in particular the highly nonline ar dynamic systems represented by neurons, and are intimately related to a number of fundamental limits in statistics/IT. Our main objective is to deter- mine whether the primary auditory system has a structure whe reby (nontrivial) optimizations of φ-divergences with respect to parameters can occur. Given the signiﬁcance of the φ-divergences as performance measures, an aﬃrmative answer to this question would yield a new view on the role played by various parameters in the neurons of the auditory system, such as the quiescent ﬁring rate (QFR), and would ins pire new experiments relating to the function of the auditory proces sing chain. We show that such optimizations indeed are possible, where s ome of the underlying mechanisms are explained in terms of the model st ructure, and we numerically determine the optimal values. The paper is organized as follows. In Section 2 we describe ou r model of the auditory system, in which the central component are the Fitzhugh-Nagumo equations. This section also includes an i ntroduction toφ-divergences and a review of their properties. The divergen ces are computed in Section 3, and discussed in Section 4. 2. Methods 2.1.Physiological modeling We consider the peripheral part of the mammalian auditory ne rvous system (Geisler, 1998), beginning with the acoustic (ﬂuid) pressure at a point in the inner ear and ending at the soma of a cell in the co chlear nucleus. As a model of the chain from the inner ear, via an inne r hair cell and a spiral ganglion cell, to a point a small distance do wn the ganglion axon we employ a stochastic FitzHugh-Nagumo (FHN) model (FitzHugh, 1961; Scott, 1975). This model, which we hencefo rth (with a neuro_sub.tex; 2/02/2008; 1:30; p.34 slight abuse of language) will call the FHN neuron, represen ts an attrac- tive choice in our study for two reasons: It is analytically/ numerically tractable and has the ability to produce a response that is bo th visually and statistically similar to that observed in real neurons. In particu- lar, it is well-known that even simple (white-noise driven) stochastic FHN models are able to accurately reproduce the interspike i nterval histograms (ISIH) in various forms of nerve ﬁbers, such as th e auditory nerve ﬁbers of squirrel monkeys (Massanes and Vicente, 1999 ). For the terminal boutonic connections of the auditory nerve with th e dendrites (or soma) of the cells in the cochlear nucleus, together with the parts of the dendrites from the boutonic connections to the somas, we employ an action-reaction model combined with a time-varying α-function like transformation with additive noise (Tuckwell, 1988; Kistl er and Van Hemmen, 1999). The conjunction of these two model features m akes it possible to capture both the synaptic plasticity and variab ility observed in real neurons. Furthermore, incorporation of plasticity in the model turns out to be of crucial importance for our results since it removes “false optima” that would otherwise be present. 2.1.1. Stochastic FitzHugh-Nagumo Model. Thestochastic FHN model is given by the following system of stochastic diﬀerential equations (Longtin, 1993)1 εdVt=Vt(Vt−a)(1−Vt)dt−Wtdt+dvt, dWt= (Vt−δWt−(b+st))dt,, t∈[0,T],(1) where ε,a,b,δ > 0 are (nonrandom) parameters, Vis the fast (“voltage like”) variable, Wis the slow (“recovery like”) variable, and stis the signal process representing the stimuli , here the acoustic pressure in the inner ear. The parameter aeﬀectively controls the barrier height between the two potential wells in the potential term (i.e., the ﬁrst term on the RHS of the ﬁrst equation) and the variable bis a bias parameter moderating the eﬀect of the signal input. These tw o param- eters aﬀect the stability properties of the FHN neuron, and so does the relaxation parameter δmultiplying the slow variable. The parameter ε sets the time scale for the motion in the potential described by the ﬁrst equation. Normally, the variable Vis thought to represent membrane voltage in the neuron but since the FHN model can be viewed as obtained by “descent” from the higher dimensional Hodgkin- Huxley model (or other likewise more elaborated models) it is not re asonable to attach a too strict physical meaning to it. To us it will mer ely act 1To guarantee global solutions to (1) we must assume that the m odel for very large \|Vt\|is modiﬁed so that the potential in Vtgrows at most linearly. neuro_sub.tex; 2/02/2008; 1:30; p.45 as a convenient way of modeling the timing information in the action potentials generated by the neuron when the latter are deﬁne d via a simple threshold operation on the fast variable V. The signal stis here chosen to enter on the slow variable W, which controls the refractory periods ofV, in order to facilitate a comparison with the qualitative results for the corresponding deterministic dynamics in (A lexander et al., 1990). However, it is easy to transform the system into a n equivalent one (of the same form) where the signal enters on the fast vari able (Alexander et al., 1990). The stochastic process vtis a noise process accounting for the variability in ﬁring pattern observed in real neurons, which we in order to have control over the correlation time (L ongtin, 1993) take to be an Ornstein-Uhlenbeck (OU) process dvt=−λvtdt+σdξt, t∈[0,T], (2) where λ >0 determines the eﬀective correlation time and σ >0 is the intensity of a standard Wiener process (integrated Gaus sian white noise) ξ. We assume that all the input and intrinsic noise sources can be collectively described by this process. This noise model is also often used with λ= 0, so that vtbecomes a Wiener process, which has proved suﬃcient to reproduce real data, see e.g. (Massanes and Vice nte, 1999). An example of an output to the FHN neuron (1),(2) with sinusoi dal signal and parameter values typical for the simulations is s hown in Fig. 1. 2.1.2. Spike train. An important underlying assumption in our model and, indeed , in most rate-based treatments of neural dynamics, is that the intervals between action potentials, not their particular form, in a given neu ron carry all the information relevant to the subsequent neural processi ng by other connected neurons. Accordingly, in the remaining parts of t he model that describe how the output of the FHN neuron is processed we replace the output of the FHN neuron by an equivalent random point pro cess T={0< τ0< ... < τ k−1< τk< τk+1...≤T}, (the number of points in Tmay be ﬁnite or inﬁnite) where τkis deﬁned by level crossings of the fast variable Vin the FHN model as τk+1= inf{t > τ k:Vt> γandVs< γfor some τk≤s≤t}. In other words, τk+1is the ﬁrst time after τkfor an upcrossing over the level γ(τ0is the ﬁrst time for an upcrossing after t= 0), where γis suitably chosen to represent an action potential level . The point process Tthus contains the timing information in the nerve signals at neuro_sub.tex; 2/02/2008; 1:30; p.56 a point in the auditory nerve immediately after the ganglion cell and will therefore be referred to as the spike train . Since the shapes and relative positions of the action potentials are not appreci ably changed as they propagate through the (myelinated) axons of the audi tory nerve we assume that the process Talso represents the timing information in the action potentials as they reach a terminal connection in the telodendria of the ganglion cell.2 2.1.3. Synaptic connections. The model for synaptic response is made up of two parts; a nomi nal (or average) response and a variability from the nominal due to s ynaptic plasticity (Koch, 1999, ch. 13). For a synapse in a nominal state at an electrotonic distance x0from the soma on a dendrite of some length L≥x0, the impulse response r(“Green’s function”) for the transformation from action po tential applied on the presynaptic side of the synapse to the voltage at the soma can be modeled by an expansion of the form (Tuckwell, 198 8, sec. 6.5) r(t) =β∞/summationdisplay n=0An(x0) 1 +λ2n−α/parenleftBig te−αt−e−αt+e−t(1+λ2 n) 1 +λ2n−α/parenrightBig , t≥0,(3) (with uniform convergence) where r(t) = 0 for t <0. Expressions for the constants An,λnin terms of L, and graphs showing the appearance of (3) for typical values of these constants and α,β, are given in (Tuck- well, 1988). In (3) it is assumed that the impulse response fr om action potential to post-synaptic current at the soma is given by a s o-called α-function of the form h(t) =βte−tαfort≥0 and h(t) = 0 for t <0 (Jack and Redman, 1971). From the deﬁnition of rit is clear that the expression (3) actually describes both the synapse and a par t of the dendrite (the part between the synapse and the soma), but sin ce the response at a point down the dendrite is mainly determined by the response of the synapse we shall, for simplicity, refer to rin (3) as the nominal synaptic response . The synaptic connections in the cochlear nucleus are often m ade by synapses having a fair, or even a large amount, of release s ites, such as the endbulb of Held , which is connected to spherical bushy cells in the anteroventral cochlear nucleus (Webster et al. , 1992). As a consequence, the synaptic transmission will be reliable i n the sense that an incoming action potential will almost always yield a nexcita- 2The time delay incurred by the propagation down along the aud itory nerve will be neglected since it will be approximately 3-4% of the lengt hTof the observation time interval in our examples. neuro_sub.tex; 2/02/2008; 1:30; p.67 tory postsynaptic potential (EPSP). However, the EPSPs will vary in strength depending (primarily) on the prehistory of the act ion poten- tials that have arrived at the synapse. This phenomenon, the synaptic plasticity , has a crucial eﬀect on the overall dynamical behavior of the nerve and needs to be taken into account in conjunction wi th the nominal response in (3). We model the plasticity using a simp le action- recovery scheme developed by Kistler and Van Hemmen (Kistle r and Van Hemmen, 1999) which combines the three state plasticity model of Tsodyks and Markham (Tsodyks and Markham, 1997) and the spik e response model of Gerstner and van Hemmen (Gerstner and Van H em- men, 1992). The action-recovery scheme employs a variable Zand its complement 1 −Zthat correspond to “active” and “inactive” resources , respectively, where the term “resources” can be interprete d as resources on both the pre- and the postsynaptic side, such as the availa bility of neurotransmitter substance or postsynaptic receptors. In addition, resources can also be interpreted as some ionic concentrati on gradient, e.g. the membrane potential on the postsynaptic side. This a pproach, therefore, also compensates for the EPSPs’ dependence on th e voltage of the following neuron’s soma. Quantitatively, the amount of available resources are determined by the recursion (Kistler and Van H emmen, 1999) Zτk+1= 1−[1−(1−R)Zτk]exp[−(τk+1−τk)/τ], (4) where 0 < R≤1 is a constant corresponding to the fraction of resources that gets inactive due to a spike and τ >0 is a decay time parameter. The variable Zτkshould be interpreted as the amount of resources available just before time τkand it is therefore proportional to the strength in an eventual EPSP caused by an action potential ar riving to the synapse at time τk. An approximation to the initial condition Zτ0 can be obtained by forming an average of the available resour ces for a number of spike trains, generated by the unforced FHN model f or the studied system, for a large T. Thus, by using the plasticity model above we can calculate the pristine (or noise-free) postsynaptic response Ras R(t,x) =∞/summationdisplay k=0Zτkr(t−τk), t > 0, where ris the nominal response given in (3). This model is capable of producing results in close agreement with real data (cf. (Ts odyks and Markham, 1997)), provided the appropriate choices of const ants are made. In reality there is always also a certain noise present due to e.g. the inherent unreliability of the ionic channels involved in th e transmission of signals in and between the neurons (Koch, 1999). To take th is eﬀect neuro_sub.tex; 2/02/2008; 1:30; p.78 into account we have added zero mean white Gaussian noise wit h in- tensity σ2to the EPSPs given by our model, which thus represents our total synaptic response . 2.2.Information processing We study information processing performance in terms of gen eral sta- tistical signal-noise separation measures applied to the o utput of our model, the soma of a cell in the cochlear nucleus. The output s ignal- noise separation setting was chosen since it can be applied w ith only minimal assumptions about the input signal. Due to the frequ ency selectivity of the primary parts of the auditory system it is suﬃcient, at least as a good ﬁrst approximation for weak signals (Egu´ ı luz et al., 2000; Camalet et al., 2000), to restrict attention to sinuso idal signals (possibly with slowly varying amplitude and phase). The sim plicity that the output-separation setting oﬀers can be contrasted with that of a communications setting which in general would require cons iderably more assumptions in order to deﬁne quantities like alphabet , message, 3coding and channel capacity (Cover and Thomas, 1991). Of cou rse, one could also select some stochastic signal and consider on ly mutual information between input and output but this too would requ ire some further statistical assumptions. For our study however, it is suﬃcient to restrict attention to the very simple class of signals stin the FHN model of the form st=Asin(ω0t+ϕ), (5) where A,ω≥0 are constant in time and ϕ∈[−π,π) is a phase which is also constant in time. 2.2.1. φ-divergences and generalized SNR. A number of fundamental limits in statistical inference and IT can be expressed as monotonic functions of so-called φ-divergences, which can be though of as “directed distances” between probabilit y measures. For example, the minimal probability of error in (Bayesian) detection, Wald’s inequalities (sequential detection), the bound in S tein’s lemma (cutoﬀ rates in Neyman-Pearson detection) and the Fisher in formation for small parameter deviations (the Cram´ er-Rao bound) can all be written as simple functions of a φ-divergence. In the simplest setting, 3The message set involved (at each given frequency), if one ca n be deﬁned, would depend entirely on the situation; it would be diﬀerent for va rious phrases in human languages and would be diﬀerent for natural sounds in diﬀere nt environments. This makes it reasonable to assume that the primary parts of the au ditory system have been optimized by evolution with respect to criteria that ar e largely invariant, such as the ability to detect and possibly determine the amplitud e of a weak tone. neuro_sub.tex; 2/02/2008; 1:30; p.89 where p0,p1are two probability density functions (PDFs) on the real lineR, theφ-divergence dφ(p0,p1) between p0,p1is deﬁned as (Liese and Vajda, 1987) dφ(p0,p1) =/integraldisplay Rφ/parenleftBigp1(x) p0(x)/parenrightBig p0(x)dx, (6) where φis any continuous convex function φon [0,∞) (we assume p0(x) = 0 if p1(x) = 0). A φ-divergence satisﬁes dφ(p0,p1)≥0, with equality if and only if p0=p1almost everywhere, and thus expresses the “separation” between p0,p1in a relative-entropy like way. Indeed, one prominent member of the family of φ-divergences is the Kullback- Liebler divergence or relative entropy, also known as information di- vergence dI(Cover and Thomas, 1991), obtained for φ(x) =−ln(x). Other important members of this family are the Kolmogorov or error divergence d(q) E, obtained for φ(x) =\|(1−q)x−q\|where q∈[0,1] is a parameter, and the χ2-divergence dχ2, obtained for φ(x) = (1 −x)2. Theχ2-divergence is twice the ﬁrst term in a formal expansion of th e information divergence around 0 (i.e. for p0=p1) and is a (tight) upper bound for a family of generalized SNR measures known as deﬂec tion ratios4that depend only on the means and variances of the observable s. Ifhis some function of data, the deﬂection ratio (DR) D(h) is deﬁned as (Basseville, 1989) D(h) =\|E1(h)−E0(h)\|2 Var0(h), where E1(h),E0(h) is the expectation of hcomputed using p0andp1, respectively, and Var 0(h) is the variance of hcomputed using p0. The DR is upper-bounded as D(h)≤dχ2(p0,p1), (7) with equality if and only if C1(h−E0(h)) =C2(p1/p0−1) with p0- probability one, for two constants C1,C2not both zero. In particular, we have equality in (7) if hequals p1/p0, thelikelihood ratio . It follows that a larger χ2-divergence allows for larger SNR, when expressed in terms of DRs. Theχ2and information divergences determine locally the Cram´ er - Rao bound (CRB) for parameter estimation ((Salicru, 1993; C over and Thomas, 1991)). For example, if θis a parameter with values in some 4Indeed, it can be shown that the (narrow-band) SNR measures u sed in stochas- tic resonance can be expressed as limits of deﬂection ratios (Rung and Robinson, 2000; Robinson et al., 2000). neuro_sub.tex; 2/02/2008; 1:30; p.910 open interval Iandpθ,θ∈ I,is a family of PDFs on Rindexed by θ then, under some regularity conditions, lim θ→θ0dI(pθ0,pθ) (θ−θ0)2= lim θ→θ0dχ2(pθ0,pθ) 2(θ−θ0)2=1 2I(θ0), forθ0∈ I, where I(θ0) is the Fisher Information at θ0. Thus, for estimation of θwhen θis near θ0the CRB (which is the inverse of the Fisher Information), and thereby the achievable accuracy f or unbiased estimation of θ, is locally determined by the growth of the χ2and information divergences as a functions of θ, near θ0. The Kolmogorov divergence is directly related to the minima l achiev- able probability of error in Bayesian hypothesis testing. I fp0andp1 are two possible PDFs for the data observed and qis taken as the a priori probability of p0to be correct, so that p1has probability 1−q, then the minimal achievable probability of error5˜P(q) e(p0,p1) for decision between p0,p1(i.e. which is the correct density) based on a single sample xis given by (cf. e.g. (Ali and Silvey, 1966)) ˜P(q) e(p0,p1) =1 2(1−d(q) E(p0,p1)). A larger Kolmogorov divergence thus gives a smaller minimal proba- bility of error. For later reference we point out that all the deﬁnitions and p roper- ties above have counterparts on much more general probabili ty spaces (Liese and Vajda, 1987; Robinson et al., 2000; Rung and Robin son, 2000), for instance in the inﬁnite dimensional context of pr obability measures on the space of continuous functions on [0 ,T]. 2.2.2. Auditory processing performance. In order to apply φ-divergences to assess performance in our model of the auditory processing chain, we need to specify the sett ing in somewhat greater detail, as well as elaborate some of the fea tures of the model. We have chosen to make the parameters A,ωandϕconstant, which in a detection scenario means that we are considering so-cal led coherent detection (detection of a completely deterministic signal ). At ﬁrst this might seem as an oversimpliﬁcation but we argue that it is not , for the following reason. There are a number of nerve cells in the aud itory nerve “tuned” to any given frequency and each corresponding axon, moreover, exhibits spatial divergence near the end where it splits up i nto diﬀerent branches. Connections are then made between these branches and the 5As is well-known, ˜P(q) e(p0, p1) is achieved with a simple likelihood ratio test. neuro_sub.tex; 2/02/2008; 1:30; p.1011 dendritic tree or soma of the following neurons. Since the de ndrites (from the connective synapse to the soma) have diﬀerent leng ths, the time delays in them will be diﬀerent. For sinusoidal input si gnals this can be exchanged for a phase shift of the signal, at least as a g ood ﬁrst approximation. Thus, for a given frequency, the primary aud itory pro- cessing can be viewed as taking place over a “bank” of paralle l channels, similar in characteristics but corresponding to diﬀerent p hase shifts. In a detection setting this corresponds to a bank of coherent de tectors operating on the outputs of these channels. It is conceivabl e that the subsequent processing can take advantage of this low-level parallelism and that detection is possible based on a logical “or” operat ion where one detector indicating presence of the signal is suﬃcient. Therefore, we use a ﬁxed phase ϕin the signal stin (1),(5) and treat the phase as a (variable) parameter. We assess the auditory processing performance by computing the φ-divergences of the output of our model (the voltage to the so ma of a cell in the cochlear nucleus) at a time point T, where Tis the end point of a long time interval [0 ,T]. The two PDFs p0,p1in the deﬁnition (6) are in the present setting given by the PDF for the output when no signal is present in the FHN model (1),(2) (i.e. st≡0) and when a signal stas in (5) is present, respectively. Since the PDFs in this cas e are densities on the real line they are easy to compute, using numerical simulation, but they are dependent on the phase ϕ, and so are the resulting φ-divergences. In order to overcome this, and obtain overall performance measures of the processing over all the paralle ll channels described above, we have weighted together the φ-divergences as dφ=/integraldisplay2π 0dφ(ϕ)dϕ, (8) based on the assumption that there are enough channels to cov er a suﬃciently dense set of the phase interval [0 ,2π). It turns out that for high frequencies the φ-divergences do not vary appreciably over a period but in the medium and low frequency cases there will typicall y be one or two regions of phase values where the divergences are sign iﬁcantly lower, as illustrated in Fig. 2a. However, since the regions where the φ-divergences deviate signiﬁcantly from their average valu es generally are relatively small we argue that average divergences as in (8) are relevant as measures of system performance. Finally we remark that even though the more general “inﬁnite di- mensional” formulas for divergences mentioned above in pri nciple could be applied if we generalized the problem to the case where out put over a whole time interval [0 ,T] was observed (instead of only its end point T), these formulas are considerably more diﬃcult to handle nu merically neuro_sub.tex; 2/02/2008; 1:30; p.1112 and involve solving a general nonlinear ﬁltering problem (L iptser and Shiryayev, 1977). Since the synaptic connection itself rep resents an averaging over time (and thus “dimensionality reduction” i n the prob- lem) we have chosen the approach above as a reasonable compro mise to reduce computational complexity while retaining releva nce of the model. 2.3.Simulations The stochastic diﬀerential equations were solved using the Euler-Maru- yama scheme (Kloeden and Platen, 1992) and the PDFs of the out put to the model were estimated using a histogram approach based on counting the number of samples falling in a grid of intervals on the real line. For calculation of the Kolmogorov divergence the so obtained “raw” histograms were suﬃcient but they proved insuﬃcient f or the χ2and information divergences (which are sensitive to inaccu racies in the representation of the PDFs). Therefore, smoothing with a kernel of the type e−c\|x\|was applied to the estimated PDFs before the latter two divergences were calculated. In order to reduce the depe ndence on the smoothing parameter c, its values were kept in a region where the results for the Kolmogorov divergence did not vary appre ciably depending on weather smoothing was applied or not. Moreover , in this region, the values of the so computed χ2and information divergences were qualitatively independent of the value of c. All our simulations were done using Matlab on UNIX(Digital)/Linux(i386). 3. Results Our main object of study is the variability of performance, q uantiﬁed viaφ-divergences (cf. Sect. 2.2.2), as a function of parameters . We shall primarily focus on the Kolmogorov divergence, since this is easiest to compute numerically, but we shall also consider performanc e in terms of the information and χ2divergences, and deﬂection ratios. The regimes of values used for the parameters in the FHN part of the model are chosen on the basis of previous studies (Mass anes and Vicente, 1999; Alexander et al., 1990). First, a nominal set of parameters is chosen for which the FHN output resembles real neuron data and then the parameters are varied around this point. At all times, however, the parameters are kept inside the region where the output is spike-train like i.e., all the resulting FHN outputs are vis ually similar to the one shown in Fig. 1. The synaptic constants used for the si mulation are chosen in order to give realistic EPSP:s for the studied s ystems and neuro_sub.tex; 2/02/2008; 1:30; p.1213 the distance x0is set rather small ( x0= 0.25 on a dendrite of length L= 1.5) since many synapses in the auditory system, e.g. the endbu lb of Held, form connections close to the soma. 3.1.Performance with respect to variation of aandb. A basic example of performance expressed as a function of par ameters is shown in Fig. 3a where the Kolmogorov divergence for A= 0.2, ω0= 8,δ= 1 and σ= 100√ 2·10−5is displayed as a function of aand b. Both aandbhave eﬀect on how much excitation that is needed to produce spikes in the FHN output. If ais made smaller the potential barrier height decreases, which gives a larger spike rate. I ncreasing the value of bhas the same eﬀect, since an increase in bcan be interpreted as if a bias was added to the input signal. This is illustrated in Fig. 2b where the FHN neuron’s spontaneous activity is displayed fo r diﬀerent values of aandb. A marked “ridge” is present in the divergence surface in Fig. 3a, indicating that there is a family of values of the potential p arameter a and the bias parameter bthat would optimize the ability of the modeled system to detect a (weak) sinusoidal signal. The FHN neurons corre- sponding to these parameter values have the common property that they ﬁre only sparsely without the signal input but ﬁre with a signiﬁ- cant intensity when the signal is present. For parameter val ues outside the region under the ridge, the Kolmogorov divergence, and a ssociated performance, is uniformly lower. The “plateau” on the left o f the ridge is located above parameter values for which the FHN neurons a re very easily excited. Given that the spike intensities of the FHN n eurons corresponding to these parameter values are roughly indepe ndent of the presence or absence of an input signal, the presence of th e plateau may seem counterintuitive. However, the ﬁring that takes pl ace when an input signal is applied is much more regular (since it is phase locked to the signal) compared to that taking place when the excitat ion is just noise. Thus, the divergences corresponding to the systems f or which the FHN parts are easily excited are rather large but still clear ly smaller than those corresponding to the ridge. In the former region o f parameter values it is also possible that an applied input signal decre ases the ﬁring rate since the noise-induced ﬁring rate can be larger than th e rate given by a phase-locked spike train. Consequently, even though th e region of spontaneous ﬁring yields rather large divergences they are are clearly smaller than the divergences on the ridge. The region of low d ivergences to the right of the ridge is generated by parameter values cor responding to systems of FHN neurons that are very diﬃcult to excite and h ardly ever ﬁre, even in the presence of an input signal. neuro_sub.tex; 2/02/2008; 1:30; p.1314 Performing the same type of analysis on the system, but using theχ2 or information divergence instead, yields qualitatively s imilar results, as seen in Figs. 3b,3c. Due to numerical problems it is hard to ca lculate the exact height of the ridges, however, and we therefore lim it the surfaces’ heights in the ﬁgures by truncating values above a certain threshold to the value of the threshold. Even though this pre vents a precise estimation of the optimal combinations of paramete r values it allows the main objective to be fulﬁlled; to show the exist ence of regions with (considerably) better performance in terms of divergences than others. For deﬂection ratios, on the other hand, the num erical problems are minor, since they can be calculated without exp licitly calculating p0andp1, which makes DRs more robust. In Fig. 3d DRs for the output of the model are displayed. Also for the DRs a ri dge can be seen and the resulting set of optimal values is similar to t hat for the divergences (though small changes in the position of the ridge can be seen). This qualitative behavior seen in all examples so f ar, with a (largely) common region of optimal values, is recurrent in all our simulations described in the following. 3.2.Performance for a lower intensity level. In the previous section we described a simulation which was a imed at investigating optimization of performance as a function of the potential parameter aand the bias parameter b, in an otherwise ﬁxed environ- ment. If we change the environment, new values of the paramet ers will emerge as optimal. For instance, if we lower the intensity le vel of the noise the location of the ridge appearing in Fig. 3a will chan ge, as seen in Fig. 4a. Together, these two ﬁgures illustrate, moreover , that care must be exercised when interpreting results of the stochast ic resonance type (Gammaitoni et al., 1998) for neural processing system s: For a ﬁxed pair of parameters values a,b, such as a= 0.6 and b= 0.12, the divergence can be higher for a larger noise level but the maximally achievable divergence, obtained on the ridge in the two ﬁgures, will be lower. Hence, for a system where adaption to environmental changes is possible, a lower noise intensity is always better in our set ting. 3.3.Performance with respect to variation of aandδ. If we instead of varying the potential parameters a,bvary the relaxation parameter δwe get the result illustrated in Fig. 4b. Also this divergenc e surface displays a marked ridge, similar to the one in Fig. 3a , indicating possible combinations of parameter values for best perform ance. neuro_sub.tex; 2/02/2008; 1:30; p.1415 3.4.Performance for other input signal parameters. The ridges in the divergence surfaces discussed so far are on ly relevant for the given input signal and if we change the input by e.g. al tering the amplitude or the frequency of the signal we get a diﬀerent result. Examples of this are shown in Fig. 4c, where the amplitude Ais set to 0.1, and in Fig. 4d where the angular frequency ω0is set to 2. Even though we still can see ridges in both cases they are diﬀe rent in shape than the ﬁrst one in Fig. 3a. Obviously, the divergence decreases with decreased signal amplitude and the height of the ridge b ecomes lower in Fig. 4c, but the location of the ridge changes only sl ightly and it appears as if only a slight change of optimal parameter values occurs. When varying the frequency however, the ridge clear ly moves to an entirely new position and new parameter values render o ptimal performance. 4. Discussion We have described a method for analyzing the information pro cess- ing capability in the primary part of the mammalian auditory ner- vous system using fundamental statistical and information theoretical performance criteria, quantitatively expressed by φ-divergences. Our premise has been that, since these criteria are highly relev ant for the processing taking place in this system, the non-existence o f well deﬁned global maxima of these criteria occuring in the interior of r egions of feasible system parameters would suggest incompleteness o r incorrect- ness of the overall model. (Loosely speaking, one can argue t hat such global interior maxima must exist for the “right” criteria in a “correct” model since otherwise parameters would have to be set at boun daries in order to achieve optimal behavior. Parameters at boundar ies would favorstructural change by evolution until only interior optima occur, whereby the “drive” for structural change ceases). One inst ance of this point is that without taking into account the synaptic plast icity, it can be shown that the divergence surfaces will have a qualita tively diﬀerent shape, with an additional ridge that, at least part ly, will yield optimal parameter values that are unphysical. However, the observed “ridges” in the divergence and deﬂection surfaces in Figs. 3 ,4 indeed allow for optimization of performance by taking parameter v alues in the interior of the domain of values that have physical signiﬁca nce. Since the model is based on fairly standard and well accepted compo nents (e.g. the FHN model), which we feel capture the essential mec hanisms involved in the information processing considered here, we believe that neuro_sub.tex; 2/02/2008; 1:30; p.1516 the results in fact can be interpreted as a quantitative indi cation of how some of the parameters in the auditory system presumably must be set. In particular this applies to the quiescent ﬁring rate (QFR) which, in real systems under this assumption, must take valu es (as a result of evolution) near those that correspond to the maxi ma of the performance measures considered here. Verifying this i s a topic for further research, however. The conclusion about the QFR is based on the qualitative obse r- vation that all the “ridges” appearing in the divergence and deﬂection surfaces correspond to parameter values that lie a certain “ thin” or “manifold like” set in parameter space. A closer examinatio n of this set shows that the combinations of parameter values that corres pond to e.g. the ridge in Fig. 3a describe systems that have virtually the same ﬁring intensity in the absence of an external signal, i.e. virtual ly the same QFR. Since this speciﬁc QFR also is common for all optimal val ues of parameter combinations corresponding to the ridges in Fi gs. 4a,4b, and in all other simulations that we have tried with the same i nput signal, this strongly suggests a connection between the QFR and the information processing performance of the system. Further evidence supporting this hypothesis can be seen in Figs. 4c,4d which s how that the optimal QFR, and thereby the optimal set of parameters, i s very little aﬀected by a change in our (weak) signal amplitude but changes considerably with the applied frequency. This reﬂects well the frequency division of sound performed in the inner ear, as discussed in Section 2.2. A more detailed investigation of the frequency depende nce also shows that the optimal QFR in our model increases with increa sing frequency. Even though existing real data is inconclusive o n this point, Kiang’s classical data (Kiang, 1965) can be interpreted to s upport the hypothesis that such a frequency dependence exists. Howeve r, experi- ments are needed to resolve the issue. Finally we point out th at even though the location of the ridge in e.g. Figs. 3a,3b,3c is lar gely the same it does vary slightly depending on which divergence or d eﬂection is considered, which is to be expected since these performan ce measures are not identical. In particular, the χ2-divergence in Fig. 3b can, as explained in Sec. 2.2.1, be considered to be a ﬁrst order appr oximation of the information divergence in Fig. 3c. All constants in our model have been chosen in order to produc e as realistic data as possible. The choices are not critical tho ugh, since in most of the simulations where the values of the constants are varied (in a reasonable large interval) the results are qualitativ ely invariant. Our approach therefore oﬀers a new qualitative, and possibl y also a quantitative, explanation of the diﬀerent levels of QFRs ob served in the auditory nerve. neuro_sub.tex; 2/02/2008; 1:30; p.1617 5. Acknowledgements The authors would like to acknowledge fruitful discussions with Prof. A. Longtin of Ottawa Univ. which led to improvement of the res ults in several aspects and to Dr. A. Bulsara of SPAWAR SSC, San Die go, CA, for many insightful suggestions which clariﬁed the pres entation of the material. MFK would also like to thank Prof. Longtin fo r his hospitality during a visit in Ottawa. References J.C. Alexander, E.J. Doedel and H.G. Othmer (1990) On the Res onance Structure in a Forced Excitable System. SIAM J. Appl. Math. 50:1373–1418. S. Ali and D. Silvey (1966) A General Class of Coeﬃcients of Di vergence of One Distribution from Another. J. Roy. Stat. Soc. B28:131–142. M. Basseville (1989) Distance Measures for Signal Processi ng and Pattern Recogni- tion.Signal Processing. 18:349–369. S. Camalet, T. Duke, F. J¨ ulicher and J. Prost (2000) Auditor y Sensitivity Provided by Self-Tuned Critical Oscillation of Hair Cells. Proc. Nat. Acad. Sci. 97:3183– 3188. T.M. Cover and J.A. Thomas (1991) Elements of Information Th eory. Wiley, New York, NY. V.M. Egu´ ıluz, M. Ospek, Y. Choe, A.J. Hudspeth and M.O. Magn asco (2000) Essential Nonlinearities in Hearing. Phys. Rev. Lett. 84:5232–5235. R.A. FitzHugh (1961) Impulses and Physiological States in T heoretical Models of Nerve Membrane. BioPhys. J. 1:445–466. L. Gammaitoni, P. H¨ anggi, P. Jung and F. Marchesoni (1998) S tochastic Resonance. Rev. Mod. Phys. 70:223–287. C.D. Geisler (1998) From Sound to Synapse: Physiology of the Mammalian Ear, Oxford University Press, New York, NY. W. Gerstner and J.L. Van Hemmen (1992) Associative Memory in a Network of “Spiking” Neurons. Network. 3:139–164. J.J.B. Jack and S.J. Redman (1971) The Propagation of Transi ent Potentials in some Linear Cable Structures. J. Physiol. 215:283–320. N. Kiang (1965) Discharge Patterns of Single Fibers in the Ca t’s Auditory Nerve. M.I.T. Press, Cambridge, Massachusetts. W.M. Kistler and J.L. Van Hemmen (1999) Short-Term Plastici ty and Network Behavior. Neural Comp. 11:1579–1594. P.E. Kloeden and E. Platen (1992) Numerical Solution of Stoc hastic Diﬀerential Equations. Springer, Berlin. C. Koch (1999) Biophysics of Computation: Information Proc essing in Single Neurons, Oxford University Press, New York. F. Liese and I. Vajda (1987) Convex Statistical Distances. T eubner, Leipzig. R.S. Liptser and A.N. Shiryayev (1977) Statistics of Random Processes I General Theory: Springer, New York. A. Longtin (1993) Stochastic Resonance in Neuron Models. J. Stat. Phys. 70:309– 327. neuro_sub.tex; 2/02/2008; 1:30; p.1718 A. Manwani and C. Koch (1999) Detecting and Estimating Signa ls in Noisy Cable Structures, II: Information Theoretical Analysis. Neural Comp. 11:1831–1873. S.R. Massanes and C.J.P. Vicente (1999) Nonadiabatic Reson ances in a Noisy Fitzhugh-Nagumo Neuron Model. Phys. Rev. E. 59:4490–4497. J.S. Nagumo, S. Arimoto and S. Yoshizawa (1962) An Active Pul se Transmission Line Simulating Nerve Axon. Proc. IRE. 50:2061–2070. J.W.C. Robinson, J. Rung, A.R. Bulsara and M.E. Inchiosa (20 00) General Measures for Signal-Noise Separation in Nonlinear Systems, to appea r inPhys. Rev. E . J. Rung and J.W.C. Robinson (2000) A Statistical Framework f or the Description of Stochastic Resonance Phenomena. In: D.S. Broomhead, E.A . Luchinskaya, P.V.E. McClintock, and T. Mullin, eds. STOCHAOS, Stochasti c and Chaotic Dynamics in the Lakes. American Institute of Physics, Melvi lle, NY. M. Salicr´ u (1993) Connections of Generalized Divergence M easures with Fisher Information Matrix. Information Sciences. 72:251–269. A.C. Scott (1975) The Electrophysics of a Nerve Fiber. Rev. Mod. Phys. 47:487–533. M. Stemmler (1996) A Single Spike Suﬃces: the Simplest Form o f Stochas- tic Resonance in Model Neurons. Network: Computation in Neural Systems. 7:687–716. M.V. Tsodyks and H. Markham (1997) The Neural Code Between Ne ocortical Pramidal Neurons Depends on Neurotransmitter Release Prob ability. Proc. Natl. Acad. Sci. 94:719–723. (See also correction at http://www.pnas.org) H.C. Tuckwell (1988) Introduction to Theoretical Neurobio logy. Cambridge Univer- sity Press, Cambridge. D.B. Webster, A.N. Popper and R.R. Fay, eds. (1992) The Mamma lian Auditory Pathway: Neuroanatomy, Springer, Ney York, NY. A. Zador (1998) Impact of the Unreliability on the Informati on Transmitted by Spiking Neurons. J. Neurophysiol. 79:1219–1229. 6. Figures neuro_sub.tex; 2/02/2008; 1:30; p.1819 0 5 10 15 20 25 30 35 40−0.4−0.200.20.40.60.811.2voltage / (0.10 V) t / (ms) Figure 1. A typical example of the output from the model (1),(2) with pa rameters a= 0.55,b= 0.12,δ= 1,ε= 0.005,σ= 100√ 2·10−5andλ= 100, when an input signal stas in (5) with parameters A= 0.1 and ω0= 8 is applied. neuro_sub.tex; 2/02/2008; 1:30; p.1920 00.10.20.30.40.50.60.70.80.9 10.10.20.30.40.50.60.70.80.91 ω=0.5 ω=4 ω=7 Kolmogorov divergence Fraction of periodtime 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.20.40.60.81Spike intensity b a Figure 2. (a) Top: Kolmogorov divergence variation for q= 0.5 on the output of the model over one period of the signal stin (5) for three diﬀerent frequencies and amplitude A= 0.2. The parameters in the FHN model (1) are a= 0.3,b= 0.12, δ= 1 and ε= 0.005 and the other parameters are σ= 100√ 2·10−5,λ= 100, α= 10, β= 100, x0= 0.25,L= 1.5,R= 0.2,τ= 50, γ= 0.5. (b) Bottom: Spontaneous activity (no input signal applied) for the FHN- model, for diﬀerent values of the parameters aandband with the other parameters set to δ= 1, ε= 0.005,σ= 100√ 2·10−5andλ= 100. neuro_sub.tex; 2/02/2008; 1:30; p.2021 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.10.20.30.40.50.60.70.8 b a d ε (0.5) 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.250510152025303540 b a χ 2 d 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.511.52 b a d Ι 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.511.522.5x 10−4D b a Figure 3. (a) Top left: The Kolmogorov divergence for δ= 1,ε= 0.005,A= 0.2, ω0= 8,σ= 100√ 2·10−5,λ= 100, α= 10, β= 100, x0= 0.25,L= 1.5,R= 0.2, τ= 50, γ= 0.5,q= 0.5 and diﬀerent values of the potential parameter aand the bias parameter b. (b) Top right: The χ2divergence for diﬀerent values of the potential parameter aand the bias parameter bwhen the other parameter values are the same as in Fig 3a. Due to the unreliability for high values of t he divergence no value above 40 has been plotted. (Eventually the χ2-divergence decreases to zero, when abecomes suﬃciently large, since then there will be almost no spikes generated.) (c) Bottom left: The information divergence for diﬀerent va lues of the potential parameter aand the bias parameter bwhen the other parameter values are the same as in Fig 3a. Due to the unreliability for high values of t he divergence no value above 3 has been plotted. (d) Bottom right: The deﬂection rat io for diﬀerent values of the potential parameter aand the bias parameter bwhen the other parameter values are the same as in Fig 3a. neuro_sub.tex; 2/02/2008; 1:30; p.2122 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.20.40.60.81 b a d ε (0.5) 00.10.20.30.40.50.60.70.80.9 00.511.522.500.10.20.30.40.50.60.70.8 d a d ε (0.5) 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.10.20.30.40.5 b a dε (0.5) 00.10.20.30.40.50.60.70.80.9 0.050.10.150.20.2500.20.40.60.81 b a dε (0.5) Figure 4. (a) Top left: The Kolmogorov divergence for diﬀerent values of the poten- tial parameter aand the bias parameter b, when the other parameter values are the same as in Fig 3a except for the noise intensity, which is lowe r (σ= 100√ 2·10−6). (b) Top right: The Kolmogorov divergence for diﬀerent value s of the potential pa- rameter aand the relaxation parameter δforb= 0.12 and with the other parameter values as in Fig 3a. (c) Bottom left: The Kolmogorov divergen ce for diﬀerent values of the potential parameter aand the bias parameter bwhen the other parameter values are the same as in Fig 3a except for the signal amplitud e, which is lower (A= 0.1). (d) Bottom right: The Kolmogorov divergence for diﬀeren t values of the potential parameter aand the bias parameter bwhen the other parameter values are the same as in Fig 3a except for the frequency, which is low er (ω0= 2). neuro_sub.tex; 2/02/2008; 1:30; p.22
arXiv:physics/0010042v1 [physics.flu-dyn] 16 Oct 2000(A Particle Field Theorist’s) Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanics (and d-Branes) R. Jackiw Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA 02139-4307 Text transcribed by M.-A. Lewis, typeset in L ATEX by M. Stock MIT-CTP#3000 Abstract This monograph is derived from a series of six lectures I gave at the Centre de Recherches Math´ ematiques in Montr´ eal, in March and June 2000, while t itulary of the Aisenstadt Chair.Contents Pr´ ecis 5 1 Introduction 8 2 Classical Equations 9 2.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 A word on canonical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 The irrotational case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Nonvanishing vorticity and the Clebsch parameterizati on . . . . . . . . . . . . . 16 2.5 Some further remarks on the Clebsch parameterization . . . . . . . . . . . . . . 18 3 Speciﬁc Models 24 3.1 Galileo-invariant nonrelativistic model . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Lorentz-invariant relativistic model . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Some remarks on relativistic ﬂuid mechanics . . . . . . . . . . . . . . . . . . . . 33 4 Common Ancestry: The Nambu-Goto Action 36 4.1 Light-cone parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Cartesian parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Hodographic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4 Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Supersymmetric Generalization 41 5.1 Chaplygin gas with Grassmann variables . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3 Supermembrane Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 Hodographic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.5 Light-cone parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.6 Further consequences of the supermembrane connection . . . . . . . . . . . . . 48 6 One-dimensional Case 49 6.1 Speciﬁc solutions for the Chaplygin gas on a line . . . . . . . . . . . . . . . . . 50 6.2 Aside on the integrability of the cubic potential in one d imension . . . . . . . . 51 6.3 General solution for the Chaplygin gas on a line . . . . . . . . . . . . . . . . . . 51 6.4 Born-Infeld model on a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.5 General solution of the Nambu-Goto theory for a (d=1)-br ane (string) in two spatial dimensions (on a plane) . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Towards a Non-Abelian Fluid Mechanics 59 7.1 Proposal for non-Abelian ﬂuid mechanics . . . . . . . . . . . . . . . . . . . . . 59 7.2 Non-Abelian Clebsch parameterization . . . . . . . . . . . . . . . . . . . . . . . 60 7.3 Proposal for non-Abelian magnetohydrodynamics . . . . . . . . . . . . . . . . . 64 Solutions to Problems 66 References 68Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 5 Pr´ ecis During the March 2000 meeting of the Workshop on Strings, Dua lity, and Geometry in Montr´ eal, Canada, I delivered three lectures on topics in ﬂ uid mechanics, while titulary of the Aisenstadt Chair. Three more lectures were presented in June 2000, during the Montr´ eal Workshop on Integrable Models in Condensed Matter and Non-E quilibrium Physics. Here are brief descriptive remarks on the content of the lectures. 1. Introduction – The motivation for the research is explain ed. 2. Classical Equations – The classical theory is reviewed, b ut in a manner diﬀerent from textbook discussions. (a) Equations of motion – Summary of conservation and Euler e quations. (b) A word on canonical formulations – An advertisement of th e method for ﬁnding the canonical structure for the above (developed with L.D. Fadd eev). (c) The irrotational case – C. Eckart’s Lagrangian and a rela tivistic generalization for vortex-free motion. (d) Nonvanishing vorticity and the Clebsch parameterizati on – In the presence of vor- ticity, the velocity Chern-Simons term (kinetic helicity) provides an obstruction to the construction of a Lagrangian for the motion. C.C. Lin’s m ethod overcomes the obstruction, and leads to the Clebsch parameterization for the velocity vector. (e) Some further remarks on the Clebsch parameterization – P roperties and peculiarities of this presentation for a 3-vector. 3. Speciﬁc Models – Nonrelativistic and relativistic ﬂuid m echanics in spatial dimensions greater than one. (a) Galileo-invariant nonrelativistic model – The Chaplyg in gas [negative pressure, in- versely proportional to density] is studied, selected solu tions are presented, unex- pected symmetries are identiﬁed. (b) Lorentz-invariant relativistic model – The scalar Born -Infeld model is found to be the relativistic generalization of the Chaplygin gas, and s hares with it unexpected symmetries. (c) Some remarks on relativistic ﬂuid mechanics – Dynamics f or isentropic relativistic ﬂuids is given a Lagrangian formulation, and the Born-Infel d model is ﬁtted into that framework. 4. Common Ancestry: The Nambu-Goto Action – Both the Chaplyg in gas and the Born- Infeld model devolve from the parameterization-invariant Nambu-Goto action, when spe- ciﬁc parameterization is made. (a) Light-cone parameterization – Chaplygin gas is derived . (b) Cartesian parameterization – Born-Infeld model is deri ved.6 R. Jackiw —(A Particle Field Theorist’s) (c) Hodographic transformation – Chaplygin gas is derived ( again). (d) Interrelations – The Chaplygin gas and Born-Infeld are r elated because (1) the former is the nonrelativistic limit of the latter; (2) both descend from the same Nambu-Goto action. 5. Supersymmetric Generalization – Fluid mechanics enhanc ed by supersymmetry. (a) Chaplygin gas with Grassmann variables – Vorticity is pa rameterized by Grassmann variables, which act like Gaussian potentials of the Clebsc h parameterization. (b) Supersymmetry – Supercharges, transformations genera ted by them, and their algebra. (c) Supermembrane connection – Supermembrane Lagrangian i n three spatial dimen- sions. (d) Hodographic transformation – Supersymmetric Chaplygi n gas in two spatial dimen- sions is derived. (e) Light-cone parameterization – Supersymmetric Chaplyg in gas in two spatial dimen- sions is derived (again). (f) Further consequences of the supermembrane connection – Hidden symmetries of the supersymmetric model. 6. One-dimensional Case – The previous models in one spatial dimension are completely integrable. (a) Solutions for the Chaplygin gas on a line – Some special so lutions are presented; inﬁnite number of constants of motion is identiﬁed; Riemann coordinates are intro- duced and the ﬂuid equations as well as constants of motion ar e expressed in terms of them. (b) Aside on the integrability of the cubic potential in one d imension – The one-dimen- sional problem with pressure ∝(density)3possesses the SO(2,1) “Schr¨ odinger sym- metry” and the equations of motion, in Riemann form, become f ree. (c) General solution of the Chaplygin gas on a line – Solution obtained by linearization. (d) Born-Infeld model on a line – When formulated in terms of i ts Riemann coordinates, it becomes trivially equivalent to the Chaplygin gas. (e) General solution of the Nambu-Goto theory for a ( d= 1)-brane (string) in two spatial dimensions (on a plane) – The explicit string soluti on is transformed by a hodographic transformation to the Chaplygin gas solution, and a relation is estab- lished between this solution and the one found by linearizat ion. 7. Towards a Non-Abelian Fluid Mechanics – Motivation for th is theory is given. (a) Proposal for non-Abelian ﬂuid mechanics – A Lagrangian i s proposed; it involves a non-Abelian auxiliary ﬁeld whose Chern-Simons density sho uld be a total derivative.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 7 (b) Non-Abelian Clebsch parameterization (or, casting the non-Abelian Chern-Simons density into total derivative form) – Total derivative form for the non-Abelian Chern- Simons density is found, thereby generalizing the Abelian C lebsch parameterization, which achieves a total derivative form for the Abelian densi ty. (c) Proposal for non-Abelian magnetohydrodynamics – Our pr oposal, which generalizes the one in Section 7.1 to include a dynamical non-Abelian gau ge ﬁeld, reduces in the Abelian limit to conventional magnetohydrodynamics.8 R. Jackiw —(A Particle Field Theorist’s) 1 Introduction Field theory, as developed by particle physicists in the las t quarter century, has enjoyed a tremendous expansion in concepts and calculational possib ilities. We learned about higher and unexpected symmetries, and disc overed evidence for partial or complete integrability facilitated by these symmetries . We appreciated the relevance of topological ideas and structures, like solitons and instan tons, and introduced new dynamical quantities, like the Chern-Simons terms in odd-dimensiona l gauge theories. We enlarged and uniﬁed numerous degrees of freedom by introducing organizi ng principles such as non-Abelian symmetries and supersymmetries. Indeed, application of ﬁe ld theory to particle physics has now been replaced by the study of fundamentally extended str uctures like strings and mem- branes, which bring with them new mathematically intricate ideas. Thinking about research possibilities, I decided to invest igate whether the novelties that we have introduced into particle physics ﬁeld theory can be u sed in a diﬀerent, non-particle physics, yet still ﬁeld-theoretic context. In these Aisens tadt lectures I shall describe an ap- proach to ﬂuid mechanics, which is an ancient ﬁeld theory, bu t which can be enhanced by the ideas that we gleaned from particle physics. As an introduction, I shall begin with a review of the classic al theory. Mostly, I duplicate what can be found in textbooks, but perhaps the emphasis will be new and diﬀerent. After this I shall describe how some instances of the classical the ory are related to d-branes and how this relation explains some integrability properties o f various models. I shall then show how the degrees of freedom can be enlarged to accomodate supe rsymmetry and non-Abelian structures in ﬂuid mechanics. A few problems are scattered t hroughout; solutions are given at the end of the text, before the references. New work that I shall describe here was done in collaboration with D. Bazeia, V.P. Nair, S.-Y. Pi, and A.P. Polychronakos. Textbooks for the classic al theory, which I recommend, are by Landau and Lifschitz [1] as well as by Arnold and Khesin [2] .Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 9 2 Classical Equations 2.1 Equations of motion We begin with nonrelativistic equations that govern a matte r density ﬁeld ρ(t,r) and a velocity ﬁeld vector v(t,r), taken in any number of dimensions. The equations of motion comprise a continuity equation, ∂ ∂tρ(t,r) +∇·/parenleftbig ρ(t,r)v(t,r)/parenrightbig = 0 (1) which ensures matter conservation, that is, time independe nce, ofN=/integraltext drρ, and Euler’s equation, which is the expression of a nonrelativistic forc e law. ∂ ∂tv(t,r) +v(t,r)·∇v(t,r) =f(t,r) (2) Hereρvis the current jandfis the force. We shall deal with an isentropic ﬂuid, that is, entropy is constant and does not appear in our theory. Also we ignore dissipation and take the force to be given by the pressure P:f=−1 ρ∇P. For isentropic motion Pis a function only ofρ, sofcan also be written as −∇V′(ρ): f=−1 ρ∇P=−∇V′(ρ) (3) with the dash (also known as “prime”) designating the deriva tive with respect to argument. V′(ρ) is the enthalpy, ρV′(ρ)−V(ρ) =P(ρ), and/radicalbig P′(ρ)≡sis the speed of sound. (Those familiar with the subject will recognize that I am using an Eu lerian rather than a Lagrangian description of a ﬂuid [3].) The dynamics summarized in (1) and (2) and the deﬁnition (3) m ay be presented as continuity equations for an energy momentum tensor. The ene rgy density E=Too E=1 2ρv2+V(ρ) =Too(4a) together with the energy ﬂux Tjo=ρvj(1 2v2+V′) (4b) obey ∂ ∂tToo+∂jTjo= 0. (4c) Similarly the momentum density, which in the nonrelativist ic theory coincides with the current, Pi=ρvi=Toi(5a)10 R. Jackiw —(A Particle Field Theorist’s) and the stress tensor Tij Tij=δij(ρV′−V) +ρvivj=δijP+ρvivj(5b) satisfy ∂ ∂tToi+∂jTji= 0. (5c) Note thatToi∝ne}a⊔ionslash=Tiobecause the theory is not Lorentz invariant, but Tij=Tjibecause it is invariant against spatial rotations. [Thus Tµνis not, properly speaking, a “tensor”, but an energy-momentum “complex”.] A simpliﬁcation occurs for the irrotational case when the vo rticity ωij≡∂ivj−∂jvi(6) vanishes. For then the velocity can be given in terms of a velo city potential θ, v=∇θ (7) and equation (2) can be replaced by Bernoulli’s equation. ∂θ ∂t+v2 2=−V′(ρ) (8) The gradient of (8) gives (2), with help of (3) and (7). Problem 1 In the free Schr¨ odinger equation for a unit-mass particle, i¯h∂ψ ∂t=−¯h2 2∇2ψ, set ψ=ρ1/2eiθ/¯h, and separate real and imaginary parts. Show that the result ing equations are like those of ﬂuid mechanics. What is the velocity? Is vortic ity supported? What is the forcef? Our equations can be presented in any dimensionality, but we shall mostly consider the cases of three, two, and one spatial dimensions. In the ﬁrst c ase, the vorticity is a (pseudo-) vector ω=∇×v (9) in the second, it is a (pseudo-)scalar ω=εij∂ivj(10) while the last, lineal case is always simple because there is no vorticity and the velocity can always be written as the derivative (with respect to the sing le spatial variable) of a potential.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 11 Dynamics of any particular system is most economically pres ented when a canonical/action formulation is available. To this end we note that the above e quations of motion can be obtained by (Poisson) bracketing with the Hamiltonian H=/integraldisplay dr/parenleftbig1 2ρv2+V(ρ)/parenrightbig =/integraldisplay drE (11) ∂ρ ∂t={H,ρ} (12) ∂v ∂t={H,v} (13) provided the nonvanishing brackets of the fundamental ( ρ,v) variables are taken to be {vi(r),ρ(r′)}=∂iδ(r−r′) {vi(r),vj(r′)}=−ωij(r) ρ(r)δ(r−r′). (14) (The ﬁelds in curly brackets are at equal times, hence the tim e argument is suppressed.) An equivalent, more transparent version of the algebra (14) is satisﬁed by the ﬁeld momentum density, P=ρv. (15) As a consequence of (14) we have {Pi(r),ρ(r′)}=ρ(r)∂iδ(r−r′) {Pi(r),Pj(r′)}=Pj(r)∂iδ(r−r′) +Pi(r′)∂jδ(r−r′). (16) This is the familiar algebra of momentum densities. One veri ﬁes that the Jacobi identity is satisﬁed [4]. Naturally one asks whether there exists a Lagrangian whose c anonical variables lead to the Poisson brackets (14) or (16) and to the Hamiltonian (11). In more mathematical language, we seek a canonical 1-form and a symplectic 2-form that lead t o the algebra (14) or (16). Problem 2 Second-quantized Schr¨ odinger ﬁelds satisfy equal-time c ommutation (anticom- mutation) relations, when describing bosons (fermions): ⌈ψ(r),ψ∗(r′)⌋±=δ(r−r′). Show that the algebra (16) is reproduced (apart from factors of i¯h) whenρ=ψ∗ψ,P= Im ¯hψ∗∇ψ. Since in the nonrelativistic theory P=j, ﬁnd jin terms of ρandv, with vas determined in Problem 1. 2.2 A word on canonical formulations I shall now describe an approach to canonical formulations o f dynamics, publicized by Faddeev and me [5], which circumvents and simpliﬁes the more elabora te approach of Dirac.12 R. Jackiw —(A Particle Field Theorist’s) We begin with a Lagrangian that is ﬁrst order in time. This ent ails no loss of generality because all second-order Lagrangians can be converted to ﬁr st order by the familiar Legendre transformation, which produces a Hamiltonian: H(p,q) =p˙q−L( ˙q,q), wherep≡∂L/∂ ˙q (the over-dot designates the time derivative). The equatio ns of motion gotten by taking the Euler-Lagrange derivative with respect to pandqof the Lagrangian L( ˙p,p; ˙q,q)≡p˙q− H(p,q) coincide with the “usual” equations of motion obtained by t aking theqEuler-Lagrange derivative of L( ˙q,q). [In factL( ˙p,p; ˙q,q) does not depend on ˙ p.] Moreover, some Lagrangians possess only a ﬁrst-order formulation (for example, Lagran gians for the Schr¨ odinger or Dirac ﬁelds; also the Klein-Gordon Lagrangian in light-cone coor dinates is ﬁrst order in the light-cone “time” derivative). Denoting all variables by the generic symbol ξi, the most general ﬁrst-order Lagrangian is L=ai(ξ)˙ξi−H(ξ). (17) Note that although we shall ultimately be interested in ﬁeld s deﬁned on space-time, for present didactic purposes it suﬃces to consider variables ξi(t) that are functions only of time. The Euler-Lagrange equation that is implied by (17) reads fij(ξ)˙ξj=∂H(ξ) ∂ξi(18) where fij(ξ) =∂aj(ξ) ∂ξi−∂ai(ξ) ∂ξj. (19) The ﬁrst term in (17) determines the canonical 1-form: ai(ξ)˙ξidt=ai(ξ)dξi, whilefijgives the symplectic 2-form: d ai(ξ)dξi=1 2fij(ξ)dξidξj. To set up a canonical formalism, we proceed directly. We do not make the frequently heard statement that “the canonical momenta ∂L/∂ ˙ξi=ai(ξ) are constrained to depend on the coordinates ξ”, and we do not embark on Dirac’s method for constrained systems. In fact, if the matrix fijpossesses the inverse fijthere are no constraints. Then (18) implies ˙ξi=fij(ξ)∂H(ξ) ∂ξj. (20) When one wants to express this equation of motion by bracketi ng with the Hamiltonian ˙ξi={H(ξ),ξi}={ξj,ξi}∂H(ξ) ∂ξj(21) one is directly led to postulating the fundamental bracket a s {ξi,ξj}=−fij(ξ). (22) The Poisson bracket between functions of ξis then deﬁned by {F1(ξ),F2(ξ)}=−∂F1(ξ) ∂ξifij∂F2(ξ) ∂ξj. (23)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 13 One veriﬁes that (22) satisﬁes Jacobi identity by virtue of ( 19). Whenfijis singular and has no inverse, constraints do arise, and the development becomes more complicated (see [5]). Our problem in connection with (14) and (16) is in fact the inv erse of what I have here summarized. From (14) and (16), we know the form of fijand that the Jacobi identity holds. We then wish to determine the inverse fij, and alsoaifrom (19). Since we know the Hamiltonian from (11), construction of the Lagrangian (17) should follow immediately. However, an obstacle may arise: If there exists a quantity C(ξ) whose Poisson bracket with all theξivanishes, then 0 ={ξi,C(ξ)}=−fij∂ ∂ξjC(ξ). (24) That is,fijhas the zero mode∂ ∂ξjC(ξ), and the inverse to fij, namely the symplectic 2-form fij, does not exist. In that case, something more has to be done, a nd we shall come back to this problem. Totally commuting quantities like C(ξ) are called “Casimir invariants”. Since they Poisson- commute with allthe dynamical variables, they commute with the Hamiltonian , and are constants of motion. But these constants do not reﬂect any sy mmetry of the speciﬁc Hamilto- nian, nor do they generate any inﬁnitesimal transformation onξi, since the {C(ξ),ξi}bracket vanishes. As will be demonstrated below, the algebra (14), (16) admits Casimir invariants, which create an obstruction to the construction of a canonical for malism for ﬂuid mechanics; this obstruction must be overcome to make progress. (In the Lagra ngian formulation of ﬂuid mechanics these Casimirs are related to a parameterization -invariance of that formalism [3].) 2.3 The irrotational case We now return to the speciﬁc issue of determining the ﬂuid dyn amical Lagrangian. The problem of constructing a Lagrangian which leads to (14) and (16) can be solved by inspection for the irrotational case, with vanishing vorticity [see (6 )]. For then the velocity commutator in (14) vanishes and (7) shows that the ﬁrst equation in (14) c an be satisﬁed by taking ρand θto be canonically conjugate. {θ(r),ρ(r′)}=δ(r−r′) (25) Thus the Lagrangian reads Lirrotational =/integraldisplay dr/parenleftbig θ˙ρ−Hv=∇θ/parenrightbig (26) whereHis given by (11) with vtaken as in (7). The form of this Lagrangian can be understood by the following argument, due to C. Eckart [6].14 R. Jackiw —(A Particle Field Theorist’s) Consider the Lagragian for Npoint-particles in free nonrelativistic motion. With the m ass mset to unity, the Galileo-invariant, free Lagrangian is jus t the kinetic energy. L0=1 2N/summationdisplay n=1v2 n(t) (27) In a continuum description, the particle-counting index nbecomes the continuous variable r, and the particles are distributed with density ρ, so that/summationtextN n=1v2 n(t) becomes/integraltext drρ(t,r)v2(t,r). But we also need to link the density with the current j=ρv, so that the continuity equation holds. This can be enforced with the help of a Lagrange multip lierθ. We thus arrive at the free, continuum Lagrangian. ¯LGalileo 0 =/integraldisplay dr/parenleftig 1 2ρv2+θ/parenleftbig ˙ρ+∇·(ρv)/parenrightbig/parenrightig (28) Since ¯LGalileo 0 is ﬁrst order in time and the canonical 1-form/integraltext drθ˙ρdoes not contain v, the latter may be varied, evaluated, and eliminated [5]. Doing t his, we ﬁnd ρv−ρ∇θ= 0 (29) and we conclude that ∇θis the velocity or, more precisely, that ∇θis the vderivative of the kinetic energy, that is, the momentum p, which in this nonrelativistic setting coincides with v. ∇θ=∂ ∂v1 2v2≡p=v (30) Substituting this in (28), we obtain LGalileo 0 =/integraldisplay dr/parenleftbig θ˙ρ−1 2ρ(∇θ)2/parenrightbig (31) which reproduces (26) with the interaction V(ρ) in (11) set to zero, and leads to the free version of the Bernoulli equation of motion (8). ˙θ+(∇θ)2 2= 0 (32) Taking the gradient gives ˙v+v·∇v= 0. (33) Problem 3 The Lagrange density for the unit-mass Schr¨ odinger equati on can be taken as LSchr¨ odinger =i¯hψ∗∂ ∂tψ−¯h2 2m∇ψ∗·∇ψ. What form does this take after ψis represented by ρ1/2eiθ/¯h? Compare with (26).Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 15 Remarkably, the same equation (33) emerges for a kinetic ene rgyT(v) that is an arbitrary function of v. This will be useful for us when we study a relativistic gener alization of the theory. If we replace (28) by ¯L0=/integraldisplay dr/parenleftig ρT(v) +θ/parenleftbig ˙ρ+∇·(ρv)/parenrightbig/parenrightig (34) and vary vto eliminate it, we get in a generalization of (30) ∇θ=∂T ∂v≡p. (35) So in the general case, it is the momentum – the vderivative of T(v) – that is irrotational. The Lagrange density becomes L0=/integraldisplay dr/parenleftbig θ˙ρ−ρh(p)p=∇θ/parenrightbig (36) whereh(p) is the Legendre transform of T(v). h(p) =v·p−T(v)∂h ∂p=v (37) Again varying θin (36) gives the continuity equation 0 =δL0 δθ= ˙ρ−/integraldisplay drρ∂h(p) ∂p·δp δθ = ˙ρ−/integraldisplay drρv·δ δθ∇θ = ˙ρ+∇·(ρv). (38) Varyingρgives 0 =δL0 δρ=−˙θ−h(p). (39) Taking the gradient, this implies with the help of (35) ∂i˙θ=−v·∂ ∂rip =−vj∂ ∂rjpi =−vj∂pi ∂vk∂ ∂rjvk. (40) On the other hand, (35) implies that ∂i˙θ=∂pi ∂vk˙vk. (41) The two are consistent, provided the free Euler equation hol ds, that is, ˙vk+vj∂jvk= 0 (42)16 R. Jackiw —(A Particle Field Theorist’s) (as long as ∂pi/∂vk=∂2T/∂vi∂vkhas an inverse). Let me observe that free motion is here governed by a Lagrangi an that is not quadratic and the free equations are not linear. Nevertheless, the equ ations of motion (38) and (42) can be solved in terms of initial data. ρ(t= 0,r)≡ρ0(r) (43) v(t= 0,r)≡v0(r) (44) Upon determining the retarded position q(t,r) from the equation q+tv0(q) =r (45) one veriﬁes that the solution to the free equations reads v(t,r) =v0(q) (46) ρ(t,r) =ρ0(q)/vextendsingle/vextendsingledet∂qi ∂rj/vextendsingle/vextendsingle. (47) A ﬁnal remark: Note that the free Bernoulli equation (8) coin cides with the free Hamilton- Jacobi equation for the action. 2.4 Nonvanishing vorticity and the Clebsch parameterizati on We now return to our original Galileo-invariant problem and enquire about the Lagrangian for velocity ﬁelds that are not irrotational, that is, whose vorticity is nonvanishing. Here we specify the spatial dimensionality to be 3, and observe that the algebra (14) possesses a zero mode, since the quantity C(v)≡/integraldisplay drεijkvi∂jvk=/integraldisplay drv·ω (48) (Poisson) commutes with both ρandv. So the symplectic 2-form does not exist: in the language developed above, fijhas no inverse. (Notice that C, also called the “ﬂuid helic- ity”, coincides with the Abelian Chern-Simons term for v[7].) (In the irrotational case with vanishing ω, the obstacle obviously is absent.) To make progress, one must neuteralize the obstruction. Thi s is achieved in the following manner, as was shown by C.C. Lin [8]. We use the Clebsch parameterization for the vector ﬁeld v. Any three-dimensional vector, which involves three functions, can be presented as v=∇θ+α∇β (49) with three suitably chosen scalar functions θ,α, andβ. This is called the “Clebsch parameteri- zation”, and ( α,β) are called “Gaussian potentials” [9]. In this parameteriz ation, the vorticity reads ω=∇α×∇β (50)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 17 and the Lagrangian is taken as L=−/integraldisplay drρ(˙θ+α˙β)−Hv=∇θ+α∇β (51) withvinHexpressed as in (49). Thus ( ρ,θ) remain canonically conjugate but another canonical pair appears: ( ρα,β). The phase space is 4-dimensional, corresponding to the fo ur observables ρandv, and a straightforward calculation shows that the Poisson b rackets (14) are reproduced, with vconstructed by (49). But how has the obstacle been overcome? Let us observe that in the Clebsch parameteri- zationCis given by C=/integraldisplay drεijk∂iθ∂jα∂kβ (52) which is just a surface integral C=/integraldisplay dS·(θω). (53) In this form, Chas no bulk contribution, and presents no obstacle to constr ucting a symplectic 2-form and a canonical 1-form in terms of θ,αandβ, which are deﬁned in the bulk, that is, for all ﬁnite r. Lin gave an Eckart-type derivation of (51): Return to ¯LGalileo 0 in (28) and add a further constraint, beyond the one enforcing current conservation [8]. ¯LGalileo 0 =/integraldisplay dr/parenleftig 1 2ρv2+θ/parenleftbig ˙ρ+∇·(ρv)/parenrightbig −ρα(˙β+v·∇β)/parenrightig (54) Setting the variation (with respect to v) to zero evaluates vas in (49); eliminating vfrom (54) gives rise to (51). This procedure works in any number of dimensions, producing the same canonical 1-form in any dimension. This means that in two spatial dimensions, on the plane, where the ( ρ,v) space is three-dimensional, the four-dimensional phase sp ace (ρ,θ;ρα,β) is larger. Moreover, the analog to Cin two spatial dimensions, that is, the obstruction to const ructing a symplectic 2-form, is not a single quantity: an inﬁnite number of object s (Poisson) commute with ρandv. These are Cn=/integraldisplay drρ/parenleftigω ρ/parenrightign n= 0,±1,±2,.... (55) (Of course, the Cnvanish in the irrotational case where there is no obstructio n.) One can understand why there is an inﬁnite number of obstructions by observing that phase space must be even dimensional, but ( ρ,v) comprise three quantities on the plane. So a nonsingular symplectic form can be constructed either by increasing the number of canonical variables to four, or decreasing to two. The Lin/Clebsch method increase s the variables. On the other hand, decreasing to two entails suppressing of one continuo us and local degree of freedom, and18 R. Jackiw —(A Particle Field Theorist’s) evidently this is equivalent to neutralizing the inﬁnite nu mber of global obstructions, namely, theCn. But I do not know how to eﬀect such a suppression, so I remain w ith (51). Note that ¯LGalileo 0 in (54), apart from a total time derivative, can also be writt en in any number of dimensions as ¯LGalileo 0 =/integraldisplay dr/parenleftbig ρT(v)−ρ(˙θ+α˙β)−ρv·(∇θ+α∇β)/parenrightbig =/integraldisplay dr/parenleftbig ρT(v)−jµ(∂µθ+α∂µβ)/parenrightbig (56) where we have introduced the current four-vector jµ= (cρ,ρv), (57) employed the four-vector gradient ∂µ= (1 c∂ ∂t,∇), and denoted the kinetic energy by T(v). These expressions will form our starting point for a relativ istic generalization of the theory as well as a non-Abelian generalization. (That is why we have in troduced the velocity of light in the above deﬁnitions; of course cdisappears in the Galilean theory, as it has no role there.) Finally we observed that in one spatial dimension, where vcan always be written as θ′ and the vorticity vanishes, the irrotational canonical 1-f orm/integraltext dxθ˙ρis generally applicable and can equivalently be written as −1 2/integraltext dxdyρ(x)ε(x−y)˙v(y), whereεis a±1-step function, determined by the sign of its argument. Evidently this leads to a spatially nonlocal, but otherwise completely satisfactory canonical formulation of ﬂuid motion on a line. 2.5 Some further remarks on the Clebsch parameterization Let me elaborate on the Clebsch parameterization for a vecto r ﬁeld, which was presented for the velocity vector in (49). Here I shall use the notation of e lectromagnetism and discuss the Clebsch parameterization of a vector potential A, which also leads to the magnetic ﬁeld B=∇×A. (Of course the same observations apply when the vector in qu estion is the velocity ﬁeldv, with ∇×vgiving the vorticity.) The familiar parameterization of a three-component vector employs a scalar function θ(the “gauge” or “longitudinal” part) and a two-component transv erse vector AT:A=∇θ+AT, ∇·AT= 0. This decomposition is unique and invertible (on a space w ith simple topology). In contrast, the Clebsch parameterization uses three scala r functions, θ,α, andβ, A=∇θ+α∇β (58) which are not uniquely determined by A(see below). The associated magnetic ﬁeld reads B=∇α×∇β. (59) Repeating the above in form notation, the 1-form A=Aidriis presented as A= dθ+αdβ (60)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 19 and the 2-form is dA= dαdβ. (61) Darboux’s theorem ensures that the Clebsch parameterizati on is attainable locally in space [in the form (60)] [10]. Additionally, an explicit construc tion ofα,β, andθcan be given by the following [11]. Solve the equations dx Bx=dy By=dz Bz(62a) which may also be presented as εijkdrjBk= 0. (62b) Solutions of these relations deﬁne two surfaces, called “ma gnetic surfaces”, that are given by equations of the form Sn(r) = constant ( n= 1,2). (63) It follows from (62) that these also satisfy B·∇Sn= 0 (n= 1,2) (64) that is, the normals to Snare orthogonal to B, orBis parallel to the tangent of Sn. The intersection of the two surfaces forms the so-called “magne tic lines”, that is, loci that solve the dynamical system dr(τ) dτ=B/parenleftbig r(τ)/parenrightbig (65) whereτis an evolution parameter. Finally, the Gaussian potential sαandβare constructed as functions of ronly through a dependence on the magnetic surfaces, α(r) =α/parenleftbig Sn(r)/parenrightbig β(r) =β/parenleftbig Sn(r)/parenrightbig (66) so that ∇α×∇β= (∇S1×∇S2)εmn∂α ∂Sm∂β ∂Sn. (67) Evidently as a consequence of (64), ∇α×∇βin (67) is parallel to B, and because Bis divergence-free αandβcan be adjusted so that the norm of ∇α×∇βcoincides with \|B\|. Onceαandβhave been ﬁxed in this way, θcan easily be computed from A−α∇β. Neither the individual magnetic surfaces nor the Gauss pote ntials are unique. [By viewing Aas a canonical 1-form, it is clear that the expression (60) re tains its form after a canonical20 R. Jackiw —(A Particle Field Theorist’s) transformation of α,β.] One may therefore require that the Gaussian potentials αandβ simply coincide with the two magnetic surfaces: α=S1,β=S2. Nevertheless, for a given A andBit may not be possible to solve (62) explicitly. The Chern-Simons integrand A·Bbecomes in the Clebsch parameterization A·B=∇θ·(∇α×∇β) =∇·(θB) =B·∇θ. (68) Thus having identiﬁed the Gauss potentials αandβwith the two magnetic surfaces, we deduce from (64) and (68) three equations for the three functions ( θ,α,β) that comprise the Clebsch parameterization. B·∇α=B·∇β= 0 B·∇θ= Chern-Simons density A·B (69) Eq. (68) also shows that in the Clebsch parameterization the Chern-Simons density becomes a total derivative. A·B=∇·(θB) (70) This does notmean that the Clebsch parameterization is unavailable when the Chern-Simons integral over all space is nonzero. Rather for a nonvanishin g integral and well-behaved B ﬁeld, one must conclude that the Clebsch function θis singular either in the ﬁnite volume of the integration region or on the surface at inﬁnity bounding the integration domain. Then the Chern-Simons volume integral over (Ω) becomes a surface integral on the surfaces ( ∂Ω) bounding the singularities. /integraldisplay ΩdrA·B=/integraldisplay ∂ΩdS·(θB) (71) Eq. (71) shows that the Chern-Simons integral measures the m agnetic ﬂux, modulated by θ and passing through the surfaces that surround the singular ities ofθ. The following explicit example illustrates the above point s. Consider the vector potential whose spherical components a re given by Ar= (cos Θ)a′(r) AΘ=−(sin Θ)1 rsina(r) AΦ=−(sin Θ)1 r/parenleftbig 1−cosa(r)/parenrightbig . (72) (r, and Θ, Φ denote the conventional radial coordinate and the p olar, azimuthal angles.) The functiona(r) is taken to vanish at the origin, and to behave as 2 πνat inﬁnity ( νinteger or half-integer). The corresponding magnetic ﬁeld reads Br=−2(cos Θ)1 r2/parenleftbig 1−cosa(r)/parenrightbig BΘ= (sin Θ)1 ra′(r)sina(r) BΦ= (sin Θ)1 ra′(r)/parenleftbig 1−cosa(r)/parenrightbig (73)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 21 and the Chern-Simons integral – also called the “magnetic he licity” in the electrodynamical context – is quantized (in multiples of 16 π2) by the behavior of a(r) at inﬁnity /integraldisplay drA·B=−8π/integraldisplay∞ 0drd dr/parenleftbig a(r)−sina(r)/parenrightbig =−16π2ν. (74) In spite of the nonvanishing magnetic helicity, a Clebsch pa rameterization for (72) is readily constructed. In form notation, it reads A= d(−2Φ)+2/parenleftig 1−/parenleftbig sin2a 2/parenrightbig sin2Θ/parenrightig d/parenleftig Φ + tan−1/bracketleftig/parenleftbig tana 2/parenrightbig cos Θ/bracketrightig/parenrightig (75) The magnetic surfaces can be taken from formula (75) to coinc ide with the Gauss potentials. S1= 2/parenleftig 1−/parenleftbig sin2a 2/parenrightbig sin2Θ/parenrightig = constant S2= Φ + tan−1/bracketleftig/parenleftbig tana 2/parenrightbig cos Θ/bracketrightig = constant (76) The magnetic lines are determined by the intersection of S1andS2. cosa 2=εcos(Φ −ϕ0) sin Θ =/radicaligg 1−ε2 1−ε2cos2(Φ−ϕ0)(77) whereεandϕ0are constants. The potential θ=−2Φ is multivalued. Consequently the “surface” integral determining the Chern-Simons term read s /integraldisplay drA·B=/integraldisplay dr∇·(−2ΦB) =/integraldisplay∞ 0rdr/integraldisplayπ 0dΘBΦ/vextendsingle/vextendsingle/vextendsingle Φ=2π. (78) That is, the magnetic helicity is the ﬂux of the toroidal magn etic ﬁeld through the positive- x (x,z) half-plane. Problem 4 Consider a vector potential A, whose Clebsch parameterization reads Ai=∂iΦ+ cos Θ∂ih(r), where Θ and Φ are the azimuthal and polar angles of the vecto rr, andhis a nonsingular function of the magnitude of r. Show that the Chern-Simons density (magnetic helicity density) is given by εijkAi∂jAk=εijk∂iΦ∂jcos Θ∂kh(r). Consider the integral of the Chern-Simons density over all space. This integral may ﬁrst be evaluated over a spherical ball Ω, and then the radius Rof the ball is taken to inﬁnity. When the integrand is a diverg ence of a vector, Gauss’s theorem casts the volume integral onto a surface integral over the sphere ∂Ω bounding the ball: I=/integraltext Ωd3r∇·V=/integraltext ∂ΩdS·V=/integraltext2π 0dΦ/integraltextπ 0dΘsin Θr2Vr\|r=R, but singularities in Vmay modify the equality. The three derivatives in the above C hern-Simons density may be extracted in three diﬀerent ways. Show that th e result always is 4 π/bracketleftbig h(R)− h(0)/bracketrightbig , but various singularities must be carefully handled.22 R. Jackiw —(A Particle Field Theorist’s) Problem 5 Show that/integraltext d3rB·δAvanishes (apart from surface terms) where δAis a variation and A,B=∇×A, as well as δAare presented in the Clebsch parameterization. When the variational principle is implemented by varying th e components of A, one ﬁnds that1 2/integraltext d3rB2is stationary provided ∇×B= 0. Show that implementing the variation by varying the scalar functions in the Clebsch parameterizati on for Agives the weaker condition ∇×B=µB, whereµcan depend on r. How is this r-dependence constrained? How is the constraint satisﬁed? There is another approach to the construction of (Abelian) v ector potentials for which the (Abelian) Chern-Simons density is a total derivative, and a s a consequence a Clebsch parame- terization for these potentials is readily found. The metho d relies on projecting an Abelian potential from a non-Abelian one, and it can be generalized t o a construction of non-Abelian vectors for which the non-Abelian Chern-Simons density is a gain a total derivative. This will be useful for us when we come to discuss non-Abelian ﬂuid mech anics. Therefore, I shall now explain this method – in its Abelian realization [12]. We consider an SU(2) group element gand a pure gauge SU(2) gauge ﬁeld, whose matrix- valued 1-form is g−1dg=Vaσa 2i(79) whereσaare Pauli matrices. It is known that tr(g−1dg)3=−1 4εabcVaVbVc=−3 2V1V2V3(80) is a total derivative; indeed its spatial integral measures the winding number of the gauge functiong[13]. Since Vais a pure gauge, we have dVa=−1 2εabcVbVc(81) so that if we deﬁne an Abelian gauge potential Aby projecting one SU(2) component of (79) (say the third) A=V3, the Abelian Chern-Simons density for Ais a total derivative, as is seen from the chain of equation that relies on (80) and (81). AdA=V3dV3=−V1V2V3=2 3tr(g−1dg)3(82) Of courseA=V3is not an Abelian pure gauge. Note thatgdepends on three arbitrary functions, the three SU(2) local gauge functions. HenceV3enjoys suﬃcient generality to represent the 3-dimensional vectorA. Moreover, sinceA’s Abelian Chern-Simons density is given by tr( g−1dg)3, which is a total derivative, a Clebsch parameterization for Ais easily constructed. We also observe that when the SU(2) group element ghas nonvanishing winding number, the resultant Abelian vec tor possesses a nonvanishing Chern-Simons integral, that is, nonzero magn etic helicity.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 23 Finally we remark that the example of a Clebsch-parameteriz ed gauge potential A, pre- sented above in (72), is gotten by a projection onto the third isospin direction of a pure gauge SU(2) potential, constructed from group element g= exp/parenleftbig (σa/2i)ˆraa(r)/parenrightbig [12]. Further intricacies arise when the Clebsch parameterizati on is used in variational calcula- tions involving vector ﬁelds; see Ref. [14].24 R. Jackiw —(A Particle Field Theorist’s) 3 Speciﬁc Models We now return to our irrotational models both relativistic a nd nonrelativistic, for which we shall specify an explicit force law and discuss further prop erties. 3.1 Galileo-invariant nonrelativistic model Recall that the nonrelativistic Lagrangian for irrotation al motion reads LGalileo=/integraldisplay dr/parenleftig θ˙ρ−ρ(∇θ)2 2−V(ρ)/parenrightig (83) where ∇θ=v. The Hamiltonian density His composed of the last two terms beyond the canonical 1-form/integraltext drθ˙ρ, H=/integraldisplay dr/parenleftig ρ(∇θ)2 2+V(ρ)/parenrightig =/integraldisplay drH. (84) Various expressions for Vappear in the literature. V(ρ)∝ρnis a popular choice, appropriate for the adiabatic equation of state. We shall be speciﬁcally interested in the “Chaplygin gas”. V(ρ) =λ ρ, λ> 0 (85) According to what we said before, the Chaplygin gas has entha lpyV′=−λ/ρ2, negative pressureP=−2λ/ρ, and speed of sound s=√ 2λ/ρ(henceλ>0). Chaplygin introduced his equation of state as a mathematica l approximation to the physi- cally relevant adiabatic expressions with n>0. (Constants are arranged so that the Chaplygin formula is tangent at one point to the adiabatic proﬁle [15]. ) Also it was realized that certain deformable solids can be described by the Chaplygin equatio n of state [16]. These days nega- tive pressure is recognized as a possible physical eﬀect: ex change forces in atoms give rise to negative pressure; stripe states in the quantum Hall eﬀect m ay be a consequence of negative pressure; the recently discovered cosmological constant m ay be exerting negative pressure on the cosmos, thereby accelerating expansion. For any form of V, the model possesses the Galileo symmetry appropriate to no nrelativistic dynamics. The Galileo transformations comprise the time an d space translations, as well as space rotations. The corresponding constants of motion are the energy E. E=/integraldisplay drH=/integraldisplay dr/parenleftig ρ(∇θ)2 2+V(ρ)/parenrightig (time translation) (86) the momentum P(whose density Pequals the spatial current), P=/integraldisplay drP=/integraldisplay drj=/integraldisplay drρv (space translation) (87) and the angular momentum Mij. Mij=/integraldisplay dr/parenleftbig riPj−rjPi/parenrightbig (rotation) (88)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 25 The action of these transformations on the ﬁelds is straight forward: the time and space argu- ments are shifted or the space argument is rotated. Slightly less trivial is the action of Galileo boosts, which boost the spatial coordinate by a velocity u r→R=r−tu. (89) The density ﬁeld transforms trivially: its spatial argumen t is boosted, ρ(t,r)→ρu(t,r) =ρ(t,R) (90) but the velocity potential acquires an inhomogeneous term. θ(t,r)→θu(t,r) =θ(t,R) +u·r−u2 2t (91) Those of you familiar with ﬁeld theoretic realizations of th e Galileo group will recognize the inhomogeneous term as the well-known Galileo 1-cocyle. It e nsures that the velocity, the gradient of θ, transforms appropriately. v(t,r)→vu(t,r) =v(t,r−tu) +u (92) The associated conserved quantity is the “boost generator” . B=tP−/integraldisplay drρr (Galileo boost) (93) Finally, also conserved is the total number. N=/integraldisplay drρ (particle number) (94) The corresponding transformation shifts θby a constant. These transformations and constants of motion ﬁt into the ge neral theory: the action is invariant against the transformations; Noether’s theorem can be used to derive the constants of motion; their time independence is veriﬁed with the help o f the equations of motion – indeed, the continuity equations (1), (4), and (5) as well as the symmetry of Tijguarantee this. Also, using the basic Poisson brackets (25) for the ( ρ,θ) variables, one can check that each inﬁnitesimal transformation is generated by Poisson brack eting with the appropriate constant; Poisson bracketing the constants with each other reproduce s the Galileo Lie algebra with a central extension given by N, which corresponds to the familiar Galileo 2-cocycle. Ther e are a total of1 2(d+ 1)(d+ 2) Galileo generators in dspace plus one time dimensions. Together with the central term, we have a total of1 2(d+ 1)(d+ 2) + 1 generators. Another useful consequence of the symmetry transformation s is that they map solutions of the equations of motion into new solutions. Of course, “new” solutions produced by Galileo transformations are trivially related to the old ones: they are simply shifted, boosted or rotated. [The free theory as well as the adiabatic theory with n= 1 + 2/dare also invariant against the SO(2,1) group of time translation, dilation, an d conformal transformation [17],26 R. Jackiw —(A Particle Field Theorist’s) which together with the Galileo group form the “Schr¨ odinge r group” of nonrelativistic motion, whenever the energy-momentum “tensor” satisﬁes 2 Too=Tii[18].] But we shall now turn to the speciﬁc Chaplygin gas model, with V(ρ) =λ/ρ, which possesses additional and unexpected symmetries. The Chaplygin gas action and consequent Bernoulli equation for the Chaplygin gas in ( d,1) space-time read IChaplygin λ=/integraldisplay dt/integraldisplay dr/parenleftig θ˙ρ−1 2ρ(∇θ)2−λ ρ/parenrightig (95) ˙θ+(∇θ)2 2=λ ρ2(96) This model possesses further space-time symmetries beyond those of the Galileo group [19]. First of all, there is a one-parameter ( ω) time rescaling transformation t→T=eωt, (97) under which the ﬁelds transform as θ(t,r)→θω(t,r) =eωθ(T,r) (98) ρ(t,r)→ρω(t,r) =e−ωρ(T,r). (99) Second, indspatial dimensions, there is a vectorial, d-parameter ( ω) space-time mixing trans- formation t→T(t,r) =t+ω·r+1 2ω2θ(T,R) (100) r→R(t,r) =r+ωθ(T,R) (101) Note that the transformation law for the coordinates involv es theθﬁeld itself. Under this transformation, the ﬁelds transform according to θ(t,r)→θω(t,r) =θ(T,R) (102) ρ(t,r)→ρω(t,r) =ρ(T,R)1 \|J\|, (103) withJthe Jacobian of the transformation linking ( T,R)→(t,r). J= det ∂T ∂t∂T ∂rj ∂Ri ∂t∂Ri ∂rj =/parenleftig 1−ω·∇θ(T,R)−ω2 2˙θ(T,R)/parenrightig−1 (104) (The time and space derivatives in the last element are with r espect totandr.) One can again tell the complete story for these transformations: The acti on is invariant; Noether’s theorem gives the conserved quantities, which for the time rescalin g is D=tH−/integraldisplay drρθ (time rescaling) (105)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 27 while for the space-time mixing one ﬁnds G=/integraldisplay dr(rH −θP) (space-time mixing). (106) The time independence of DandGcan be veriﬁed with the help of the equations of motion (continuity and Bernoulli). Poisson bracketing the ﬁelds θandρwithDandGgenerates the appropriate inﬁnitesimal transformation on the ﬁelds. So now the total number of generators is the sum of the previou s1 2(d+ 1)(d+ 2) + 1 with 1 +dadditional ones 1 2(d+ 1)(d+ 2) + 1 + 1 + d=1 2(d+ 2)(d+ 3). (107) When one computes the Poisson brackets of all these with each other one ﬁnds the Poincar´ e Lie algebra in one higher spatial dimension, that is, in ( d+1,1)-dimensional space-time, where the Poincar´ e group possesses1 2(d+2)(d+3) generators [20]. Moreover, one veriﬁes that ( t,θ,r) transform linearly as a ( d+ 2) Lorentz vector in light-cone components, with tbeing the “+” component and θthe “−” component [21]. Presently, we shall use these additional symmetries to gene rate new solutions from old ones, but, in contrast with what we saw earlier, the new solutions w ill be nontrivially linked to the former ones. Note that the additional symmetry holds even in the free theory. Before proceeding, let us observe that ρmay be eliminated by using the Bernoulli equation to express it in terms of θ. In this way, one is led to the following ρ-independent action for θ in the Chaplygin gas problem: IChaplygin λ=−2√ λ/integraldisplay dt/integraldisplay dr/radicalbigg ˙θ+(∇θ)2 2. (108) Although this operation is possible only in the interacting case, the interaction strength is seen to disappear from the equations of motion. ∂ ∂t1/radicalig ˙θ+(∇θ)2 2+∇·∇θ/radicalig ˙θ+(∇θ)2 2= 0 (109) λmerely serves as an overall factor in the action. The action (108) looks unfamiliar; yet it is Galileo invaria nt. [The combination ˙θ+1 2(∇θ)2 responds to Galileo transformations without a 1-cocycle; s ee (91).] Also IChaplygin λpossesses the additional symmetries described above, with θtransforming according to the previously recorded equations. Let us discuss some solutions. For example, the free theory i s solved by θ(t,r) =r2 2t(110) which corresponds to the velocity v(t,r) =r t. (111)28 R. Jackiw —(A Particle Field Theorist’s) Galileo transforms generalize this in an obvious manner int o a set of solutions. (The charge densityρis determined by its initial condition. In the free theory, ρis an independent quantity, and I shall not discuss it here.) Performing on the above form ula forθthe new transformations of time-rescaling and space-time mixing, we ﬁnd that the sol ution is invariant. We can ﬁnd a solution similar to (110) in the interacting case , ford>1, which we henceforth assume (the d= 1 case will be separately discussed later). One veriﬁes tha t a solution is θ(t,r) =−r2 2(d−1)tρ(t,r) =/radicalbigg 2λ d(d−1)\|t\| r(112) v(t,r) =−r (d−1)tj(t,r) =−ε(t)/radicalbigg 2λ dˆr. (113) Note that the speed of sound s=√ 2λ ρ=√ dr (d−1)t=√ dv (114) exceedsv. Again this solution can be translated, rotated, and booste d. Moreover, the solution is time-rescaling–invariant. However, the space-time mix ing transformation produces a wholly diﬀerent kind of solution. This is best shown graphically, w here thed= 2 case is exhibited (see ﬁgure ) [22]. Another interesting solution, which is essentially one-di mensional (lineal), even though it exists in arbitrary spatial dimension, is given by θ(t,r) = Θ(ˆn·r) +u·r−1 2t/parenleftbig u2−(ˆn·u)2/parenrightbig . (115) Here ˆnis a spatial unit vector, uis an arbitrary vector with dimension of velocity, while Θ is an arbitrary function with static argument, which can be boost ed by the Galileo transform (91). The corresponding charge density is time-independent. ρ(t,r) =√ 2λ ˆn·u+ Θ′(ˆn·r)(116) The current is static and divergenceless. j(t,r) =√ 2λ/parenleftig ˆn+u−ˆn(ˆn·u) ˆn·r+ Θ′(ˆn·r)/parenrightig (117) The sound speed s=√ 2λ/ρ= Θ′(ˆn·r) + ˆn·uis just the ˆ ncomponent of the velocity v=∇θ= ˆnΘ′(ˆn·r) +u. Finally, we record a planar static solution to (109) θ(r) = Θ(ˆn1·r/ˆn2·r) (118) where ˆn1and ˆn2are two orthogonal unit vectors [23]. Problem 6 Show that the solution for θgiven in (110) is invariant under the time rescaling transformation (97), (98), and under the space-time mixing transformation (100)–(102).Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 29 -10 -5 0 5 10x/t-10 -5 0 5 10y/t0.30.50.70.930 R. Jackiw —(A Particle Field Theorist’s) 3.2 Lorentz-invariant relativistic model We now turn to a Lorentz-invariant generalization of our Gal ileo-invariant Chaplygin model in (d,1)-dimensional space-time. We already know from (34)–(37) how to construct the free Lagrangian by using Eckart’s method with a relativistic kin etic energy. T(v) =−c2/radicalbig 1−v2/c2 (119) Recall that mass has been scaled to unity, and that we retain t he velocity of light cto keep track of the nonrelativistic c→ ∞ limit. Evidently, the momentum is p=∂T(v) ∂v=v/radicalbig 1−v2/c2. (120) Thus the free relativistic Lagrangian, with current conser vation enforced by the Lagrange multiplierθ, reads [compare (34)] ¯LLorentz 0 =/integraldisplay dr/parenleftig −c2ρ/radicalbig 1−v2/c2+θ/parenleftbig ˙ρ+∇·(ρv)/parenrightbig/parenrightig . (121) This may be presented in a Lorentz-covariant form in terms of a current four-vector jµ= (cρ,ρv).¯LLorentz 0 of equation (121) is thus equivalent to [compare (56), (57)] ¯LLorentz 0 =/integraldisplay dr/parenleftig −jµ∂µθ−c/radicalbig jµjµ/parenrightig . (122) Eliminating vin (121), we ﬁnd, as before, that p=∂T ∂v=v√ 1−v2/c2=∇θ, v=∇θ/radicalbig 1 + (∇θ)2/c2, (123) and the free Lorentz-invariant Lagrangian reads [compare ( 36), (37)] LLorentz 0 =/integraldisplay dr/parenleftig θ˙ρ−ρc2/radicalbig 1 + (∇θ)2/c2/parenrightig . (124) To ﬁndLGalileo 0 of (31) as the nonrelativistic limit of LLorentz 0 in (124), a nonrelativistic θ variable must be extracted from its relativistic counterpa rt. Calling the former θNRand the latter, which occurs in (124), θR, we deﬁne θR≡ −c2t+θNR. (125) It then follows that apart from a total time derivative LLorentz 0 − − − − → c→∞LGalileo 0. (126) Next, one wants to include interactions. While there are man y ways to allow for Lorentz- invariant interactions, we seek an expression that reduces to the Chaplygin gas in the nonrel- ativistic limit. Thus, we choose [24] LBorn-Infeld a =/integraldisplay dr/parenleftig θ˙ρ−/radicalbig ρ2c2+a2/radicalbig c2+ (∇θ)2/parenrightig , (127)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 31 whereais the interaction strength. [The reason for the nomenclatu re will emerge presently.] We see that, as c→ ∞, LBorn-Infeld a − − − − → c→∞LChaplygin λ=a2/2. (128) [AgainθNRis extracted from θRas in (125) and a total time derivative is ignored.] Although it perhaps is not obvious, (127) deﬁnes a Poincar´ e-invaria nt theory, and this will be explicitly demonstrated below. Therefore, LBorn-Infeld a possesses Lorentz and Poincar´ e symmetries in (d,1) space-time, with a total of1 2(d+ 1)(d+ 2) + 1 generators, where the last “ + 1” refers to the total number N=/integraltext drρ. Whena= 0, the model is free and elementary. It was demonstrated pre viously [eqs. (34)– (42)] that the free equations of motion are precisely the sam e as in the nonrelativistic free model, so the complete solution (43)–(47) works here as well . Fora∝ne}a⊔ionslash= 0, in the presence of interactions, one can eliminate ρas before, and one is left with a Lagrangian just for the θ ﬁeld. It reads LBorn-Infeld a =−a/integraldisplay dr/radicalig c2−(∂µθ)2. (129) This is a Born-Infeld-type theory for a scalar ﬁeld θ; its Poincar´ e invariance is manifest, and again, the elimination of ρis only possible with nonvanishing a, which however disappears from the dynamics, serving merely to normalize the Lagrangi an. The equations of motion that follow from (127) read ˙ρ+∇·/parenleftigg ∇θ/radicaligg ρ2c2+a2 c2+ (∇θ)2/parenrightigg = 0 (130) ˙θ+ρc2/radicaligg c2+ (∇θ)2 ρ2c2+a2= 0. (131) The density ρcan be evaluated in terms of θfrom (131); then (130) reads ∂α/parenleftig1/radicalbig c2−(∂µθ)2∂αθ/parenrightig = 0 (132) which also follows from (129). After θNRis extracted from θRas in (125) we see that in the nonrelativistic limit LBorn-Infeld a (127) or (129) becomes LChaplygin λof (95) or (108), LBorn-Infeld a − − − − → c→∞LChaplygin λ=a2/2(133) and the equations of motion (130)–(132) reduce to (1), (96), and (109). In view of all the similarities to the nonrelativistic Chapl ygin gas, it comes as no surprise that the relativistic Born-Infeld theory possesses additi onal symmetries. These additional symmetry transformations, which leave (127) or (129) invar iant, involve a one-parameter ( ω) reparameterization of time, and a d-parameter ( ω) vectorial reparameterization of space. Both transformations are ﬁeld dependent.32 R. Jackiw —(A Particle Field Theorist’s) The time transformation is given by an implicit formula invo lving also the ﬁeld θ[25], t→T(t,r) =t coshc2ω+θ(T,r) c2tanhc2ω (134) while the ﬁeld transforms according to θ(t,r)→θω(t,r) =θ(T,r) coshc2ω−c2ttanhc2ω . (135) [We record here only the transformation on θ; howρtransforms can be determined from the (relativistic) Bernoulli equation, obtained by varying ρin (127), which expresses ρin terms ofθ. Moreover, (135) is suﬃcient for discussing the invariance of (129).] The inﬁnitesimal generator, which is time independent by virtue of the equati ons of motion, is [26] D=/integraldisplay dr/parenleftig c4tρ+θ/radicalbig ρ2c2+a2/radicalbig c2+ (∇θ)2/parenrightig =/integraldisplay dr(c4tρ+θH) (time reparameterization). (136) A second class of transformations involving a reparameteri zation of the spatial variables is implicitly deﬁned by [25]. r→R(t,r) =r−ˆωθ(t,R)tancω c+ ˆω(ˆω·r)/parenleftbigg1−coscω coscω/parenrightbigg (137) θ(t,r)→θω(t,r) =θ(t,R)−c(ˆω·r)sincω coscω(138) ˆωis the unit vector ω/ωandω=√ω·ω. The time-independent generator of the inﬁnitesimal transformation reads [26] G=/integraldisplay dr(c2rρ+θρ∇θ) =/integraldisplay dr(c2rρ+θP) (space reparameterization). (139) Of course the Born-Infeld action (127) or (129) is invariant against these transformations, whose inﬁnitesimal form is generated by the constants. With the addition of DandGto the previous generators, the Poincar´ e algebra in ( d+1,1) dimension is reconstructed, and ( t,r,θ) transforms linearly as a ( d+ 2)-dimensional Lorentz vector (in Cartesian components) [21]. Note that this symme try also holds in the free, a= 0, theory. It is easy to exhibit solutions of the relativistic equation (132), which reduce to solutions of the nonrelativistic, Chaplygin gas equation (109) [after −c2thas been removed, as in (125)]. For example θ(t,r) =−c/radicalbigg c2t2+r2 d−1(140)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 33 solves (132) and reduces to (112). The relativistic analog o f the lineal solution (115) is θ(t,r) = Θ(ˆn·r) +n·r−ct/radicalbig c2+u2−(ˆn·u)2. (141) [Note that the above proﬁles continue to solve (132), even wh en the sign of the square root is reversed, but then they no longer possess a nonrelativistic limit.] Additionally there exists an essentially relativistic sol ution, describing massless propaga- tion in one direction: according to (132), θcan satisfy the wave equation ⊓ ⊔θ= 0, provided (∂µθ)2= constant, as for example with plane waves θ(t,r) =f(ˆn·r±ct) (142) where (∂µθ)2vanishes. Then ρreads, from (131), ρ=∓a c2f′. (143) Other solutions are given in ref. [27]. 3.3 Some remarks on relativistic ﬂuid mechanics The Born-Infeld model reduces in the nonrelativistic limit to the Chaplygin gas. Equations governing the latter belong to ﬂuid mechanics, but the Born- Infeld equations do not readily expose their ﬂuid mechanical structure. Nevertheless they do in fact describe a relativistic ﬂuid. In order to demonstrate this, we give a pr´ ecis of relat ivistic ﬂuid mechanics. Usually the dynamics of a relativistic ﬂuid is presented in t erms of the energy-momentum tensor,θµν, and the equations of motion are just the conservation equat ions∂µθµν= 0 [28]. [We denote the relativistic energy-momentum tensor by θµν, to distinguish it from the nonrel- ativisticTµνintroduced in (4) and (5). The limiting relation between the two is given below.] But we prefer to begin with a Lagrange density L=−jµaµ−f(/radicalbig jµjµ). (144) Herejµis the current Lorentz vector jµ= (cρ,j). Theaµcomprise a set of auxilliary variables; in the relativistic analog of irrotational ﬂuids we take aµ=∂µθ, more generally aµ=∂µθ+α∂µβ (145) so that the Chern-Simons density of aiis a total derivative [compare (56), (57)]. The function f depends on the Lorentz invariant jµjµ=c2ρ2−j2and encodes the speciﬁc dyanamics (equation of state). The energy momentum tensor for Lis θµν=−gµνL+jµjν√jαjαf′(/radicalbig jαjα). (146)34 R. Jackiw —(A Particle Field Theorist’s) [One way to derive (146) from (144) is to embed that expressio n in an external metric tensor gµν, which is then varied; in the variation jµandaµare taken to be metric-independent and jµ=gµνjν.] Furthermore, varying jµin (144) shows that aµ=−jµ√jαjαf′(/radicalbig jαjα) (147) so that (146) becomes θµν=−gµν[nf′(n)−f(n)] +uµuνnf′(n) (148) where we have introduced the proper velocity uµby jµ=nuµuµuµ= 1 (149) so thatnis proportional to the proper density and 1 /nis proportional to the speciﬁc volume. Eq. (148) identiﬁes the proper energy density eand the pressure P(which coincides with L) through the conventional formula [28]. θµν=−gµνP+uµuv(P+e) (150) Therefore, in our case e=f(n) (151) P=nf′(n)−f(n). (152) The thermodynamic relation involving entropy Sreads Pd/parenleftig1 n/parenrightig + d/parenleftige n/parenrightig ∝dS (153) where the proportionality constant is determined by the tem perature. With (151) and (152) the left side of (153) vanishes and we verify that entropy is c onstant, that is, we are dealing with an isentropic system, as has been stated in the very begi nning. For the free system, the pressure vanishes, so we choose f(n) =cn L0=−jµaµ−c/radicalbig jµjµ. (154) In the “irrotational” case, aµ=∂µθ, and with j=ρvthis Lagrange density produces the free ¯LLorentz 0 of (121), (122) (apart from a total time derivative). For the Born-Infeld theory, we present the pressure Pin (152) by choosing f(n)=c√ a2+n2, which corresponds to the pressure P=−a2c/√ a2+n2. When the current is written as jµ=a∂µθ/radicalbig c2−(∂µθ)2(155a) so that n=a/radicaligg (∂µθ)2 c2−(∂µθ)2(155b)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 35 the Lagrange density L=P=−a2c√ a2+n2(155c) coincides with that for the Born-Infeld model in eq. (129). Other forms for fgive rise to relativistic ﬂuid mechanics with other equatio ns of state. It is interesting to see how the nonrelativistic limit of θµνin (148) produces Tµνof (4)–(5). It is especially intriguing to notice that θµνis symmetric but Tµνis not. To make the connec- tion we recall that uµ= 1//radicalbig 1−v2/c2(1,v/c), we observe that n=/radicalbig ρ2c2−j2, setj=ρvand conclude that n = ρc/radicalbig 1−v2/c2∼ρc−(ρv2/2c). Alsof(n) is chosen to be cn+V(n/c), and thus P= nf′(n)−f(n) = (n/c)V′(n/c)−V(n/c). It follows that θoo=nc−(v2n/c3)V′ 1−v2/c2+V≈ρc2−ρv2/2 1−v2/c2+V(ρ) ≈ρc2+ρv2 2+V(ρ) =ρc2+Too. (156) Thus, apart from the relativistic “rest energy” ρc2,θoopasses toToo. The relativistic energy ﬂux iscθjo(because∂ ∂xµθµo=1 c˙θoo+∂jθjo) cθjo=vj 1−v2/c2/parenleftig nc+n cV′/parenrightig ≈vjρc2−ρv2/2 +ρV′(ρ) 1−v2/c2 ≈jjc2+ρvj/parenleftbig v2/2 +V′(ρ)/parenrightbig =jjc+Tjo. (157) Again, apart from the O(c2) current, associated with the O(c2) rest energy in θoo,Tjois obtained in the limit. The momentum density is θoi/c(becauseθµνhas dimension of energy density). Thus θoi/c=vi/c2 1−v2/c2/parenleftig nc+n cV′/parenrightig ≈ρvi=Pi. (158) Finally, the momentum ﬂux is obtained directly from θij. θij=δij/parenleftign cV′−V/parenrightig +vivj c2−v2/parenleftig nc+n cV′/parenrightig ≈δij/parenleftbig ρV′(ρ)−V(ρ)/parenrightbig +vivjρ=Tij(159) From the limiting formula n∼ρcwe also see that the pressure in (155c) tends to the Chaplygin expression −a/ρ.36 R. Jackiw —(A Particle Field Theorist’s) 4 Common Ancestry: The Nambu-Goto Action The “hidden” symmetries and the associated transformation laws for the Chaplygin and Born- Infeld models may be given a coherent setting by considering the Nambu-Goto action for a d-brane in ( d+ 1) spatial dimensions, moving on ( d+ 1,1)-dimensional space-time. In our context, a d-brane is simply a d-dimensional extended object: a 1-brane is a string, a 2-bra ne is a membrane and so on. A d-brane in ( d+ 1) space divides that space in two. The Nambu-Goto action reads ING=/integraldisplay dϕ0dϕLNG=/integraldisplay dϕ0dϕ1···dϕd√ G (160) G= (−1)ddet∂Xµ ∂ϕα∂Xµ ∂ϕβ(161) HereXµis a (d+ 1,1) target space-time (d-brane) variable, with µextending over the range µ= 0,1,...,d,d + 1. Theϕαare “world-volume” variables describing the extended obje ct withαrangingα= 0,1,...,d ;ϕα,α= 1,...,d , parameterizes the d-dimensional d-brane that evolves inϕ0. The Nambu-Goto action is parameterization invariant, and w e shall show that two diﬀerent choices of parameterization (“light-cone” and “Cartesian ”) lead to the Chaplygin gas and Born- Infeld actions, respectively. For both parameterizations we choose ( X1,...,Xd) to coincide with (ϕ1,...,ϕd), renaming them as r(ad-dimensional vector). This is usually called the “static parameterization”. (The ability to carry out this p arameterization globally presupposes that the extended object is topologically trivial; in the co ntrary situation, singularities will appear, which are spurious in the sense that they disappear i n diﬀerent parameterizations, and parameterization-invariant quantities are singularity- free.) 4.1 Light-cone parameterization For the light-cone parameterization we deﬁne X±as1√ 2(X0±Xd+1).X+is renamed tand identiﬁed with√ 2λϕ0. This completes the ﬁxing of the parameterization and the re maining variable is X−, which is a function of ϕ0andϕ, or after redeﬁnitions, of tandr.X−is renamed as θ(t,r) and then the Nambu-Goto action in this parameterization co incides with the Chaplygin gas action IChaplygin λin (108) [29]. 4.2 Cartesian parameterization For the second, Cartesian parameterization X0is renamed ctand identiﬁed with cϕ0. The re- maining target space variable Xd+1, a function of ϕ0andϕ, equivalently of tandr, is renamed θ(t,r)/c. Then the Nambu-Goto action reduces to the Born-Infeld acti on/integraltext dtLBorn-Infeld a , (129) [29].Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 37 4.3 Hodographic transformation There is another derivation of the Chaplygin gas from the Nam bu-Goto action that makes use of a hodographic transformation, in which independent and dep endent variables are interchanged. Although the derivation is more involved than the light-con e/static parameterization used in Section 4.1 above, the hodographic approach is instructive in that it gives a natural deﬁnition for the density ρ, which in the above static parameterization approach is det ermined from θ by the Bernoulli equation (96). We again use light-cone combinations:1√ 2(X0+Xd+1) is calledτand is identiﬁed with ϕ0, while1√ 2(X0−Xd+1) is renamed θ. At this stage the dependent, target-space variables areθand the transverse coordinates X:Xi, (i= 1,... ,d ), and all are functions of the world- volume parameters ϕ0=τandϕ:ϕr, (r= 1,... ,d );∂τindicates diﬀerentiation with respect toτ=ϕ0, while∂rdenotes derivatives with respect to ϕr. The induced metric Gαβ=∂Xµ ∂ϕα∂Xµ ∂ϕβ takes the form Gαβ=/parenleftigg GooGos Gro−grs/parenrightigg =/parenleftigg 2∂τθ−(∂τX)2∂sθ−∂τX·∂sX ∂rθ−∂rX·∂τX −∂rX·∂sX/parenrightigg (162) The Nambu-Goto Lagrangian now leads to the canonical moment a ∂LNG ∂∂τX=p (163a) ∂LNG ∂∂τθ= Π (163b) and can be presented in ﬁrst-order form as LNG=p·∂τX+ Π∂τθ+1 2Π(p2+g) +ur(p·∂rX+ Π∂rθ) (164) whereg= detgrsand ur≡∂τX·∂rX−∂rθ (165) acts as a Lagrange multiplier. Evidently the equations of mo tion are ∂τX=−1 Πp−ur∂rX (166a) ∂τθ=1 2Π2(p2+g)−ur∂rθ (166b) ∂τp=−∂r/parenleftig1 Πggrs∂sX/parenrightig −∂r(urp) (166c) ∂τΠ =−∂r(urΠ) (166d) Also there is the constraint p·∂rX+ Π∂rθ= 0 (167)38 R. Jackiw —(A Particle Field Theorist’s) [Thaturis still given by (165) is a consequence of (166a) and (167).] Heregrsis inverse togrs, and the two metrics are used to move the ( r,s) indices. The theory still possesses an invariance against redeﬁning the spatial parameters wit h aτ-dependent function of the parameters; inﬁnitesimally: δϕr=−fr(τ,ϕ),δθ=fr∂rθ,δXi=fr∂rXi. This freedom may be used to set urto zero and Π to −1. Next the hodographic transformation is performed: Rather t han viewing the dependent variables p,θ, and Xas functions of τandϕ,X(τ,ϕ) is inverted so that ϕbecomes a function of τandX(renamedtandr, respectively), and pandθalso become functions of t andr. It then follows from the chain rule that the constraint (167 ) (at Π = −1) becomes 0 =∂Xi ∂ϕr/parenleftig pi−∂ ∂Xiθ/parenrightig (168) and is solved by p=∇θ . (169) Moreover, according to the chain rule and the implicit funct ion theorem, the partial derivative with respect to τat ﬁxed ϕ[this derivative is present in (164)] is related to the parti al derivative with respect to τat ﬁxed X=rby ∂τ=∂ ∂t+∇θ·∇ (170) where we have used the new name “ t” on the right. Thus the Nambu-Goto Lagrangian – the ϕintegral of the Lagrange density (164) (at ur= 0, Π = −1) – reads LNG=/integraldisplay dϕ/braceleftbig p·∇θ−˙θ−∇θ·∇θ−1 2(p2+g)/bracerightbig . (171a) But use of (169) and of the Jacobian relation d ϕ= drdet∂ϕs ∂Xi=dr√gshows that LNG=/integraldisplay dr/braceleftbig −1√g˙θ−1 2√g(∇θ)2−1 2√g/bracerightbig . (171b) With the deﬁnition √g=√ 2λ/ρ (171c) LNGbecomes, apart from a total time derivative LNG=1√ 2λ/integraldisplay dr/braceleftbig θ˙ρ−1 2ρ(∇θ)2−λ ρ/bracerightbig . (171d) Up to an overall factor, this is just the Chaplygin gas Lagran gian in (95). The present derivation has the advantage of relating the den sityρto the Jacobian of theXi→ϕtransformation: ρ=√ 2λdet∂ϕs ∂Xi. (This in turn shows that the hodographic transformation is just exactly the passage from Lagrangian to Eulerian ﬂuid variables – a remark aimed at those who are acquainted with the Lagrange fo rmulation of ﬂuid motion [3].) Finally, let me observe that the expansion of the Galileo sym metry in (d,1) space-time to a Poincar´ e symmetry in ( d+ 1,1) space-time can be understood from a Kaluza-Klein–type framework [30].Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 39 4.4 Interrelations The relation to the Nambu-Goto action explains the origin of the hidden ( d+ 1,1) Poincar´ e group in our two nonlinear models on ( d,1) space-time: Poincar´ e invariance is what remains of the reparameterization invariance of the Nambu-Goto act ion after choosing either the light- cone or Cartesian parameterizations. Also the nonlinear, ﬁ eld dependent form of the transfor- mation laws leading to the additional symmetries is underst ood: it arises from the identiﬁcation of some of the dependent variables ( Xµ) with the independent variables ( ϕα). The complete integrability of the d= 1 Chaplygin gas and Born-Infeld model is a con- sequence of the fact that both descend from a string in 2-spac e; the associated Nambu-Goto theory being completely integrable. We shall discuss this i n Section 6. We observe that in addition to the nonrelativistic descent f rom the Born-Infeld theory to the Chaplygin gas, there exists a mapping of one system on ano ther, and between solutions of one system and the other, because both have the same d-bran e ancestor. The mapping is achieved by passing from the light-cone parameterizatio n to the Cartesian, or vice-versa. Speciﬁcally this is accomplished as follows: Chaplygin gas →Born-Infeld: GivenθNR(t,r), a nonrelativistic solution, determine T(t,r) from the equation T+1 c2θNR(T,r) =√ 2t (172) Then the relativistic solution is θR(t,r) =1√ 2c2T−1√ 2θNR(T,r) =c2(√ 2T−t) (173) Born-Infeld →Chaplygin gas: GivenθR(t,r), a relativistic solution, ﬁnd T(t,r) from T+1 c2θR(T,r) =√ 2t (174) Then the nonrelativistic solution is θNR(t,r) =1√ 2c2T−1√ 2θR(T,r) =c2(√ 2T−t) (175) The relation between the diﬀerent models is depicted in the ﬁ gure below. As a ﬁnal comment, I recall that the elimination of ρ, both in the nonrelativistic (Chap- lygin) and relativistic (Born-Infeld) models is possible o nly in the presence of interactions. Nevertheless, the θ-dependent ( ρ-independent) resultant Lagrangians contain the interact ion strengths only as overall factors; see (108) and (129). It is theseθ-dependent Lagrangians that correspond to the Nambu-Goto action in various parameteriz ations. Let us further recall the the Nambu-Goto action also carries an overall multiplicati ve factor: the d-brane “tension”, which has been suppressed in (160). Correspondingly, for a “ tensionless” d-brane, the Nambu- Goto expression vanishes, and cannot generate dynamics. Th is suggests that an action for40 R. Jackiw —(A Particle Field Theorist’s) Dualities and other relations between nonlinear equations . “tensionless” d-branes could be the noninteracting ﬂuid me chanical expressions (95), (127), with vanishing coupling strengths λanda, respectively. Furthermore, we recall that the non- interacting models retain the higher, dynamical symmetrie s, appropriate to a d-brane in one higher dimension.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 41 5 Supersymmetric Generalization Once proven the fact that a bosonic Nambu-Goto theory gives r ise to and links together the Chaplygin gas and Born-Infeld models, which are irrotation al in that the velocity of the former and the momentum of the latter are given by a gradient of a pote ntial, one can ask whether there is a d-brane that produces a ﬂuid model with nonvanishi ng vorticity. We shall show that this indeed can be achieved if one starts wi th a super–d-brane, and moreover the resulting ﬂuid model possesses supersymmetry . However, since theories of ex- tended “super” objects cannot be formulated in arbitrary di mensions, we shall consider the ﬂuid in two spatial dimensions, namely, on the plane [31]. 5.1 Chaplygin gas with Grassmann variables We begin by positing the ﬂuid model. The Chaplygin gas Lagran gian in (95) is supplemented by Grassmann variables ψathat are Majorana spinors [real, two-component: ψ∗ a=ψa,a= 1,2, (ψ1ψ2)∗=ψ∗ 1ψ∗ 2]. The associated Lagrange density reads L=−ρ(˙θ−1 2ψ˙ψ)−1 2ρ(∇θ−1 2ψ∇ψ)2−λ ρ−√ 2λ 2ψα·∇ψ . (176) Hereαiare two (i= 1,2), 2×2, real symmetric Dirac “alpha” matrices; in terms of Pauli matrices we can take α1=σ1,α2=σ3. Note that the matrices satisfy the following relations, which are needed to verify subsequent formulas εabαi bc=εijαj ac αi abαj bc=δijδac−εijεac αi abαi cd=δacδbd−δabδcd+δadδbc; (177) εabis the 2 ×2 antisymmetric matrix ε≡iσ2. In equation (176) λis a coupling strength which is assumed to be positive. The Grassmann term enters with cou pling√ 2λ, which is correlated with the strength of the Chaplygin potential V(ρ) =λ/ρin order to ensure supersymmetry, as we shall show below. It is evident that the velocity should be deﬁned as v=∇θ−1 2ψ∇ψ . (178) The Grassmann variables directly give rise to a Clebsch form ula for v, and provide the Gauss potentials. The two-dimensional vorticity reads ω=εij∂ivj=−1 2εij∂iψ∂jψ=−1 2∇ψ× ∇ψ. The variables {θ,ρ}remain a canonical pair, while the canonical 1-form in (176) indicates that the canonically independent Grassmann variables are√ρψso that the antibracket of the ψ’s is {ψa(r),ψb(r′)}=−δab ρ(r)δ(r−r′). (179)42 R. Jackiw —(A Particle Field Theorist’s) One veriﬁes that the algebra (14) or (16) is satisﬁed, and fur ther, one has {θ(r),ψ(r)}=−1 2ρ(r)ψ(r)δ(r−r′) (180) {v(r),ψ(r′)}=−∇ψ(r) ρ(r)δ(r−r′) (181) {P(r),ψ(r′)}=−∇ψ(r)δ(r−r′). (182) The momentum density Pis given by the bosonic formula P=ρv, but the Grassmann variables are hidden in v, by virtue of (178). The equations of motion read ˙ρ+∇·(ρv) = 0 (183) ˙θ+v·∇θ=1 2v2+λ ρ2+√ 2λ 2ρψα·∇ψ (184) ˙ψ+v·∇ψ=√ 2λ ρα·∇ψ (185) and together with (178) they imply ˙v+v·∇v=∇λ ρ2+√ 2λ ρ(∇ψ)α·∇ψ . (186) All these equations may be obtained by bracketing with the Ha miltonian H=/integraldisplay d2r/parenleftig 1 2ρv2+λ ρ+√ 2λ 2ψα·∇ψ/parenrightig =/integraldisplay d2rH (187) when (14), (16) as well as (179)–(181) are used. We record the components of the energy-momentum “tensor”, a nd the continuity equations they satisfy. The energy density E=Too, given by E=1 2ρv2+λ ρ+√ 2λ 2ψα·∇ψ=Too(188) satisﬁes a continuity equation with the energy ﬂux Tjo. Tjo=ρvj/parenleftbig1 2v2−λ ρ2/parenrightbig +√ 2λ 2ψαjv·∇ψ−λ ρψ∂jψ+λ ρεjkψε∂kψ (189) ˙Too+∂jTjo= 0 (190) This ensures that the total energy, that is, the Hamiltonian , is time-independent. Conservation of the total momentum P=/integraldisplay d2rP (191)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 43 follows from the continuity equation satisﬁed by the moment um density Pi=Toiand the momentum ﬂux, that is, the stress tensor Tij. Tji=ρvivj−δij/parenleftig2λ ρ+√ 2λ 2ψα·∇ψ/parenrightig +√ 2λ 2ψαj∂iψ (192) ˙Toi+∂jTji= 0 (193) ButTijis not symmetric in its spatial indices, owing to the presenc e of spin in the problem. However, rotational symmetry makes it possible to eﬀect an “ improvement”, which modiﬁes the momentum density by a total derivative term, leaving the integrated total momentum unchanged (provided surface terms can be ignored) and rende ring the stress tensor symmetric. The improved quantities are Pi I=Toi I=ρvi+1 8εij∂j(ρψεψ) (194) Tij I=ρvivj−δij/parenleftig2λ ρ+√ 2λ 2ψα·∇ψ/parenrightig +√ 2λ 4/parenleftig ψαi∂jψ+ψαj∂iψ/parenrightig −1 8∂k/bracketleftig (εkivj+εkjvi)ρψεψ/bracketrightig (195) ˙Toi I+∂jTij I= 0. (196) It immediately follows from the symmetry of Tij Ithat the angular momentum M=/integraldisplay d2rεijriPj I=/integraldisplay d2rρεijrivj+1 4/integraldisplay d2rρψεψ (197) is conserved. The ﬁrst term is clearly the orbital part (whic h still receives a Grassmann contribution through v), whereas the second, coming from the improvement, is the sp in part. Indeed, sincei 2ε=1 2σ2≡Σ, we recognize this as the spin matrix in (2+1) dimensions. The extra term in the improved momentum density (194),1 8εij∂j(ρψεψ), can then be readily interpreted as an additional localized momentum density, g enerated by the nonhomogeneity of the spin density. This is analogous to the magnetostatics fo rmula giving the localized current density jmin a magnet in terms of its magnetization m:jm=∇×m. All in all, we are describing a ﬂuid with spin. Also the total number N=/integraldisplay d2rρ (198) is conserved by virtue of the continuity equation (183) sati sﬁed byρ. Finally, the theory is Galileo invariant, as is seen from the conservation of the Ga lileo boost, B=tP−/integraldisplay d2rrρ (199) which follows from (183) and (191). The generators H,P,M,BandNclose on the (extended) Galileo group. [The theory is not Lorentz invariant in (2 + 1) -dimensional space-time, hence the energy ﬂux Tjodoes not coincide with the momentum density, improved or not .]44 R. Jackiw —(A Particle Field Theorist’s) We observe that ρcan be eliminated from (176) so that Linvolves only θandψ. From (184) and (185) it follows that ρ=√ λ/parenleftbig˙θ−1 2ψ˙ψ+1 2v2/parenrightbig−1/2. (200) Substituting into (176) produces the supersymmetric gener alization of the Chaplygin gas La- grange density in (108). L=−2√ λ/braceleftig/radicalig 2˙θ−ψ˙ψ+ (∇θ−1 2ψ∇ψ)2+1 2ψα·∇ψ/bracerightig (201) Note that the coupling strength has disappeared from the dyn amical equations, remaining only as a normalization factor for the Lagrangian. Conseque ntly the above elimination of ρ cannot be carried out in the free case, λ= 0. 5.2 Supersymmetry As we said earlier, this theory possesses supersymmetry. Th is can be established, ﬁrst of all, by verifying that the following two-component supercharge s are time-independent Grassmann quantities. Qa=/integraldisplay d2r/bracketleftig ρv·(αabψb) +√ 2λψa/bracketrightig . (202) Taking a time derivative and using the evolution equations ( 183)–(186) establishes that ˙Qa= 0. Next, the supersymmetric transformation rule for the dynam ical variables is found by constructing a bosonic symmetry generator Q, obtained by contracting the Grassmann charge with a constant Grassmann parameter ηa,Q=ηaQa, and commuting with the dynamical variables. Using the canonical brackets one veriﬁes the fol lowing ﬁeld transformation rules: δρ={Q,ρ}=−∇·ρ(ηαψ) (203) δθ={Q,θ}=−1 2(ηαψ)·∇θ−1 4(ηαψ)·ψ∇ψ+√ 2λ 2ρηψ (204) δψ={Q,ψ}=−(ηαψ)·∇ψ−v·αη−√ 2λ ρη (205) δv={Q,v}=−(ηαψ)·∇v+√ 2λ ρη∇ψ . (206) Supersymmetry is reestablished by determining the variati on of the action/integraltext dtd2rLconse- quent to the above ﬁeld variations: the action is invariant. One then reconstructs the super- charges (202) by Noether’s theorem. Finally, upon computin g the bracket of two supercharges, one ﬁnds {ηa 1Qa,ηb 2Qb}= 2(η1η2)H (207)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 45 which again conﬁrms that the charges are time-independent: {H,Qa}= 0. (208) Additionally a further, kinematical, supersymmetry can be identiﬁed. According to the equations of motion the following two supercharges are also time-independent: ¯Qa=/integraldisplay d2rρψa. (209) ¯Q= ¯ηa¯Qaeﬀects a shift of the Grassmann ﬁeld: ¯δρ={¯Q,ρ}= 0 (210) ¯δθ={¯Q,θ}=−1 2(¯ηψ) (211) ¯δψ={¯Q,ψ}=−¯η (212) ¯δv={¯Q,v}= 0. (213) This transformation leaves the Lagrangian invariant, and N oether’s theorem reproduces (209). The algebra of these charges closes on the total number N. {¯ηa 1¯Qa,¯ηb 2¯Qb}= (¯η1¯η2)N (214) while the algebra with the generators (202), closes on the to tal momentum, together with a central extension, proportional to volume of space Ω =/integraltext d2r {¯ηa¯Qa,ηbQb}= (¯ηαη)·P+√ 2λ(¯ηεη)Ω. (215) The supercharges Qa,¯Qa, together with the Galileo generators ( H,P,M, and B), withN form a superextended Galileo algebra. The additional, nonv anishing brackets are {M,Qa}=1 2εabQb (216) {M,¯Qa}=1 2εab¯Qb (217) {B,Qa}=αab¯Qb. (218) 5.3 Supermembrane Connection The equations for the supersymmetric Chaplygin ﬂuid devolv e from a supermembrane La- grangian,LM. We shall give two diﬀerent derivations of this result, whic h make use of two diﬀerent parameterizations for the parameterization-inv ariant membrane action and give rise, respectively, to (176) and (201). The two derivations follo w what has been done in the bosonic case in Sections 4.1 and 4.3. We work in a light-cone gauge-ﬁxed theory: The supermembran e in 4-dimensional space- time is described by coordinates Xµ(µ= 0,1,2,3), which are decomposed into light-cone components X±=1√ 2(X0±X3) and transverse components Xi{i= 1,2}. These depend on46 R. Jackiw —(A Particle Field Theorist’s) an evolution parameter ϕ0≡τand two space-like parameters ϕr{r= 1,2}. Additionally there are two-component, real Grassmann spinors ψ, which also depend on τandϕr. In the light- cone gauge, X+is identiﬁed with τ,X−is renamed θ, and the supermembrane Lagrangian is [32] LM=/integraldisplay d2ϕLM=−/integraldisplay d2ϕ{√ G−1 2εrs∂rψα∂sψ·X} (219) whereG= detGαβ; Gαβ=/parenleftigg GooGos Gro−grs/parenrightigg =/parenleftigg 2∂τθ−(∂τX)2−ψ∂τψ us ur −grs/parenrightigg (220) G=gΓ Γ≡2∂τθ−(∂τX)2−ψ∂τψ+grsurus grs≡∂rX·∂sX, g= detgrs us≡∂sθ−1 2ψ∂sψ−∂τX·∂sX. (221) Here∂τsigniﬁes diﬀerentiation with respect to the evolution para meterτ, while∂rdiﬀeren- tiates with respect to the space-like parameters ϕr;grsis the inverse of grs, and the two are used to move the ( r,s) indices. Note that the dimensionality of the transverse co ordinatesXi is the same as of the parameters ϕr, namely two. 5.4 Hodographic transformation To give our ﬁrst derivation following the procedure in Secti on 4.3, we rewrite the Lagrangian in canonical, ﬁrst-order form, with the help of bosonic cano nical momenta deﬁned by ∂LM ∂∂τX=p=−Π∂τX−Πur∂rX (222a) ∂LM ∂∂τθ= Π =/radicalbig g/Γ. (222b) (The Grassmann variables already enter with ﬁrst-order der ivatives.) The supersymmetric extension of (164) then reads LM=p·∂τX+ Π∂τθ−1 2Πψ∂τψ+1 2Π(p2+g) +1 2εrs∂rψα∂sψ·X +ur/parenleftig p·∂rX+ Π∂rθ−1 2Πψ∂rψ/parenrightig . (223) In (223)urserves as a Lagrange multiplier enforcing a subsidiary cond ition on the canonical variables, and g= detgrs. The equations that follow from (223) coincide with the Eule r- Lagrange equations for (219). The theory still possesses an invariance against redeﬁning the spatial parameters with a τ-dependent function of the parameters. This freedom may be u sedLectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 47 to setuτto zero and ﬁx Π at −1. Next we introduce the hodographic transformation, as in Section 4.3, whereby independent-dependent variables are interchanged, namely we view the ϕrto be functions of Xi. It then follows that the constraint on (223), which with Π = −1 reads p·∂rX−∂rθ+1 2ψ∂rψ= 0 (224) becomes ∂rX·/parenleftig p−∇θ+1 2ψ∇ψ/parenrightig = 0. (225) Herep,θandψare viewed as functions of X, renamed r, with respect to which acts the gradient ∇. Also we rename pasv, which according to (225) is v=∇θ−1 2ψ∇ψ . (226) As in Section 4.3, from the chain rule and the implicit functi on theorem it follows that ∂τ=∂t+∂τX·∇ (227) and according to (222a) (at Π = −1,ur= 0)∂τX=p=v. Finally, the measure transforms according to d2ϕ→d2r1√g. Thus the Lagrangian for (223) becomes, after setting urto zero and Π to −1, LM=/integraldisplayd2r√g/parenleftig v2−˙θ−v·∇θ+1 2ψ(˙ψ+v·∇ψ)−1 2(v2+g) −1 2εrsψαi∂jψ∂sxj∂rxi/parenrightig . (228a) Butεrs∂sxj∂rxi=εijdet∂rxi=εij√g. After√gis renamed√ 2λ/ρ, (228a) ﬁnally reads LM=1√ 2λ/integraldisplay d2r/parenleftig −ρ(˙θ−1 2ψ˙ψ)−1 2ρ(∇θ−1 2ψ∇ψ)2−λ ρ−√ 2λ 2ψα×∇ψ/parenrightig .(228b) Upon replacing ψby1√ 2(1−ε)ψ, this is seen to reproduce the Lagrange density (176), apart from an overall factor. 5.5 Light-cone parameterization For our second derivation, we return to (219)–(221) and use t he remaining reparameterization freedom to equate the two Xivariables with the two ϕrvariables, renaming both as ri. Also τis renamed as t. This parallels the method in Section 4.1. Now in (219)–(221 )grs=δrs, and ∂τX= 0, so that (221) becomes simply G= Γ = 2 ˙θ−ψ˙ψ+u2(229) u=∇θ−1 2ψ∇ψ . (230)48 R. Jackiw —(A Particle Field Theorist’s) Therefore the supermembrane Lagrangian (219) reads LM=−/integraldisplay d2r/braceleftbigg/radicalig 2˙θ−ψ˙ψ+/parenleftbig ∇θ−1 2ψ∇ψ/parenrightbig2+1 2ψα×∇ψ/bracerightbigg . (231) Again a replacement of ψby1√ 2(1−ε)ψdemonstrates that the integrand coincides with the Lagrange density in (201) (apart from a normalization facto r). 5.6 Further consequences of the supermembrane connection Supermembrane dynamics is Poincar´ e invariant in (3+1)-di mensional space-time. This invari- ance is hidden by the choice of light-cone parameterization : only the light-cone subgroup of the Poincar´ e group is left as a manifest invariance. This is jus t the (2+1) Galileo group generated byH,P,M,B, andN. (The light-cone subgroup of the Poincar´ e group is isomorp hic to the Galileo group in one lower dimension [33].) The Poincar´ e generators not included in the above list correspond to Lorentz transformations in the “ −” direction. We expect therefore that these generators are “dynamical”, that is, hidden and u nexpected conserved quantities of our supersymmetric Chaplygin gas, similar to the situation with the purely bosonic model. One veriﬁes that the following quantities D=tH−/integraldisplay d2r ρθ (232) G=/integraldisplay d2r(rH −θPI−1 8ψαα·PIψ) =/integraldisplay d2r(rH −θP−1 4ψαα·Pψ) (233) are time-independent by virtue of the equations of motion (1 83)–(186), and they supplement the Galileo generators to form the full (3 + 1) Poincar´ e alge bra, which becomes the super- Poincar´ e algebra once the supersymmetry is taken into acco unt. Evidently (232), (233) are the supersymmetric generalizations of (105), (106). We see that ﬂuid dynamics can be extended to include Grassman n variables, which also enter in a supersymmetry-preserving interaction. Since ou r construction is based on a super- membrane in (3+1)-dimensional space-time, the ﬂuid model i s necessarily a planar Chaplygin gas. It remains for the future to show how this construction c ould be generalized to arbitrary dimensions and to diﬀerent interactions. Note that Grassma nn Gauss potentials ψcan be used even in the absence of supersymmetry. For example, our t heory (176), with the last term omitted, posseses a conventional, bosonic Hamiltonia n without supersymmetry, while the Grassmann variables are hidden in vand occur only in the canonical 1-form.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 49 6 One-dimensional Case In this section, I shall discuss the nonrelativistic/relat ivistic models in one spatial dimension. Complete integrability has been established for both the Ch aplygin gas [34] and the Born-Infeld theory [35]. We can now understand this to be a consequence of the complete integrability of the Nambu-Goto 1-brane (string) moving on 2-space (plane ), which is the antecedent of both models. [Therefore, it suﬃces to discuss only the Chapl ygin gas since solutions of the Born-Infeld model can then be obtained by the mapping (172)– (173).] As remarked previously, in one dimension there is no vortici ty, and the nonrelativistic velocityvcan be presented as a derivative with respect to the single sp atial variable of a potentialθ. Similarly, the relativistic momentum p=v//radicalbig 1−v2/c2is a derivative of a potentialθ. In both cases the potential is canonically conjugate to the densityρgoverned by the canonical 1-form/integraltext dxθ˙ρ. Moreover, it is evident that at the expense of a spatial nonlocality, one may replace θby its antiderivative, which is pboth nonrelativistically and relativistically (nonrelativistically p=v), so that in both cases the Lagrangian reads L=−1 2/integraldisplay dxdyρ(x)ε(x−y) ˙p(y)−H . (234) For the Chaplygin gas and the Born-Infeld models, His given respectively by HChaplygin=/integraldisplay dx/parenleftig 1 2ρp2+λ ρ/parenrightig (235) HBorn-Infeld=/integraldisplay dx/parenleftbig/radicalbig ρ2c2+a2/radicalbig c2+p2/parenrightbig . (236) The equations of motion are, respectively Chaplygin gas: ˙ρ+∂ ∂x(pρ) = 0 (237) ˙p+∂ ∂x/parenleftigp2 2−λ ρ2/parenrightig = 0 (238) or∂ ∂t1/radicalig ˙θ+p2 2+∂ ∂xp/radicalig ˙θ+p2 2= 0 (239) Born-Infeld model: ˙ ρ+∂ ∂x/parenleftbigg p/radicaligg ρ2c2+a2 c2+p2/parenrightbigg = 0 (240) ˙p+∂ ∂x/parenleftbigg ρc2/radicaligg c2+p2 ρ2c2+a2/parenrightbigg = 0 (241) or∂ c2∂t/parenleftbigg˙θ/radicalig c2−1 c2˙θ2+p2/parenrightbigg −∂ ∂x/parenleftbiggp/radicalig c2−1 c2˙θ2+p2/parenrightbigg = 0 (242) In the above, eqs. (239) and (242) result by determining ρin terms of θ(p=∂ ∂xθ) from (238) and (241), and using that expression for ρin (237) and (240).50 R. Jackiw —(A Particle Field Theorist’s) 6.1 Speciﬁc solutions for the Chaplygin gas on a line Classes of solutions for a Chaplygin gas in one dimension can be given in closed form. For example, to obtain general, time-rescaling–invariant sol utions, we make the Ansatz thatθ∝ 1/t. Then (109) or (239) leads to a second-order nonlinear diﬀer ential equation for the x- dependence of θ. Therefore solutions involve two arbitrary constants, one of which ﬁxes the origin ofx(we suppress it); the other we call k, and take it to be real. The solutions then read θ(t,x) =−1 2k2tcosh2kx . (243) [Other solutions can be obtained by relaxing the reality con dition onkand/or shifting the argumentkxby a complex number. In this way one ﬁnds that θcan also be1 2k2tsinh2kx, 1 2k2tsin2kx,1 2k2tcos2kx; but these lead to singular or unphysical forms for ρ.] The density corresponding to (243) is found from (96) or (238) to be ρ(t,x) =√ 2λk\|t\| cosh2kx. (244) The velocity/momentum v=p=∂ ∂xθis v(t,x) =p(t,x) =−1 ktsinhkxcoshkx (245) while the sound speed s(t,x) =cosh2kx k\|t\|(246) is always larger than \|v\|. Finally, the current j=ρ∂θ ∂xexhibits a kink proﬁle, j(t,x) =−ε(t)√ 2λtanhkx (247) which is suggestive of complete integrability. Another particular solution is the Galileo boost of the stat ic proﬁles (115), (116): p(t,x) =p(x−ut) (248) ρ(t,x) =√ 2λ \|p−u\|. (249) Hereuis the boosting velocity and p(x−ut) is an arbitrary function of its argument (provided p∝ne}a⊔ionslash=u). Clearly this is a constant proﬁle solution, in linear moti on with velocity u. Further evidence for complete integrability is found by ide ntifying an inﬁnite number of constants of motion. One veriﬁes that the following quantit ies I± n=/integraldisplay dxρ/parenleftig p±√ 2λ ρ/parenrightign , n= 0,±1,... (250) are conserved.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 51 The combinations p±√ 2λ ρare just the velocity ( ±) the sound speed, and they are known as Riemann coordinates. R±=p±√ 2λ ρ(251) The equations of motion for this system [continuity (237) an d Euler (238)] can be succinctly presented in terms of R±: ˙R±=−R∓∂ ∂xR±. (252) 6.2 Aside on the integrability of the cubic potential in one dimension Although it does not belong to the models that we have discuss ed, the cubic potential for 1-dimensional motion, V(ρ) =ℓρ3/3, is especially interesting because it is secretly free – a fact that is exposed when Riemann coordinates are employed. For this problem these read R±=p±√ 2ℓρand again they are just the velocity ( ±) the sound speed. In contrast to (252) the Euler and continuity equations for this system decouple :˙R±=−R±∂ ∂xR±. Indeed, it is seen thatR±satisfy essentially the free Euler equation [compare with ( 42) and identify R± withv]. Consequently, the solution (44)–(46) works here as well. Recall the previous remark in Section 3.1 on the Schr¨ odinge r group [Galileo ⊕SO(2,1)]: in one dimension the cubic potential is invariant against th is group of transformations, and in all dimensions the free theory is invariant [17], [18]. Ther efore a natural speculation is that the secretly noninteracting nature of the cubic potential i n one dimension is a consequence of Schr¨ odinger group invariance. Another interesting fact about a one-dimensional nonrelat ivistic ﬂuid with cubic potential is that it also arises in a collective, semiclassical descri ption of nonrelativistic free fermions in one dimension, where the cubic potential reproduces fermio n repulsion [36]. In spite of the nonlinearity of the ﬂuid model’s equations of motion, there is no interaction in the underlying fermion dynamics. Thus, the presence of the Schr¨ odinger gr oup and the equivalence to free equations for this ﬂuid system is an understandable consequ ence. 6.3 General solution for the Chaplygin gas on a line The general solution to the Chaplygin gas can be found by line arizing the governing equations (continuity and Euler) with the help of a Legendre transform , which also eﬀects a hodographic transformation that exchanges the independent variables ( t,x) with the dependent ones ( ρ,θ); actually instead of ρwe use the sound speed s=√ 2λ/ρand instead of θwe use the momentum p=∂ ∂xθ. Deﬁne ψ(p,s) =θ(t,x)−t˙θ(t,x)−x∂ ∂xθ(t,x). (253)52 R. Jackiw —(A Particle Field Theorist’s) From the Bernoulli equation we know that ˙θ=−1 2p2+1 2s2. (254) Thus ψ(p,s) =θ(t,x) +t 2(p2−s2)−xp (255) and the usual Legendre transform rules govern the derivativ es. ∂ψ ∂p=tp−x (256a) ∂ψ ∂s=−ts (256b) It remains to incorporate the continuity equation (237) who se content must be recast by the hodographic transformation. This is achieved by rewrit ing equation (237) in terms of s=√ 2λ/ρ: ∂s ∂t+p∂s ∂x−s∂p ∂x= 0. (257) Next (257) is presented as a relation between Jacobians: ∂(s,x) ∂(t,x)+p∂(t,s) ∂(t,x)−s∂(t,p) ∂(t,x)= 0 (258a) which is true because here ∂x/∂t =∂t/∂x = 0. Eq. (258a) implies, after multiplication by ∂(t,x)/∂(s,p) 0 =∂(s,x) ∂(s,p)+p∂(t,s) ∂(s,p)−s∂(t,p) ∂(s,p) =∂x ∂p−p∂t ∂p−s∂t ∂s. (258b) The second equality holds because now we take ∂s/∂p =∂p/∂s = 0. Finally, from (255), (256) it follows that (258b) is equivalent to ∂2ψ ∂p2−∂2ψ ∂s2+2 s∂ψ ∂s= 0. (258c) This linear equation is solved by two arbitrary functions of p±s(p±sbeing just the Riemann coordinates) ψ(p,s) =F(p+s)−sF′(p+s) +G(p−s) +sG′(p−s). (259) In summary, to solve the Chaplygin gas equations, we choose t wo functions FandG, constructψas in (259), and regain s(=√ 2λ/ρ),p(=∂ ∂xθ), andθfrom (255), (256). In particular, the solution (243), (244) corresponds to F(z) =G(−z) =±z 2klnz (260) where the sign is correlated with the sign of t.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 53 6.4 Born-Infeld model on a line Since the Born-Infeld system is related to Chaplygin gas by t he transformation described in Section 4.4, there is no need to discuss separately Born-Inf eld solutions. Nevertheless, the formulation in terms of Riemann coordinates is especially s uccinct and gives another view on the Chaplygin/Born-Infeld relation. The Riemann coordinates R±for the Born-Infeld model are contructed by ﬁrst deﬁning 1 c∂ ∂xθ=p/c= tanϕp a/ρc= tanϕρ (261) and R±=ϕp±ϕρ. (262) The 1-dimensional version of the equations of motion (130), (131), that is, (240), (241) can be presented as ˙R±=−c(sinR∓)∂ ∂xR±. (263) The relation to the Riemann description of the Chaplygin gas can now be seen in two ways: a nonrelativistic limit and an exact transformation. For th e former, we note that at large c, ϕp≈p/c,ϕρ≈a/ρcso that RBorn-Infeld ± ≈1 c/parenleftig p±a ρ/parenrightig =1 cRChaplygin ±/vextendsingle/vextendsingle/vextendsingle λ=a2/2. (264) Moreover, the equation (263) becomes, in view of (264), 1 c˙RChaplygin ± =−RChaplygin ∓1 c∂ ∂xRChaplygin ± (265) so that (252) is regained. On the other hand, for the exact tra nsformation we deﬁne new Riemann coordinates in the relativistic, Born-Infeld case by R±=csinR±. (266) Evidently (263) implies that R±satisﬁes the nonrelativistic equations (252), (265) when R± solves the relativistic equation (263). Expressing R±andR±in terms of the corresponding nonrelativistic and relativistic variables produces a map ping between the two sets. Calling pNR,ρNRandpR,ρRthe momentum and density of the nonrelativistic and of the re lativistic theory, respectively, the mapping implied by (266) is pNR=c2ρRpR/radicalig (p2 R+c2)(ρ2 Rc2+a2) ρNR=1 c2/radicalig (p2 R+c2)(ρ2 Rc2+a2). (267)54 R. Jackiw —(A Particle Field Theorist’s) As can be checked, this maps the Chaplygin equations into the Born-Infeld equations. But the mapping is not canonical. We record the inﬁnite number of constants of motion, which pu t into evidence the (by now obvious) complete integrability of the Born-Infeld equati ons on a line. The following quantities are time-independent: I± n=acn−1/integraldisplay dx(ϕp±ϕρ)n sinϕρcosϕp, n= 0,±1,... (268) The nonrelativistic limit takes the above into (250), while expressing I± nin terms of R± according to (266) shows that the integrals in (268) are expr essible as series in terms of the integrals in (250). In the relativistic model ρneed not be constrained to be positive (negative ρcould be interpreted as antiparticle density). The transformation p→ −p,ρ→ −ρis a symmetry and can be interpreted as charge conjuguation. Further, pandρappear in an equivalent way. As a result, this theory enjoys a duality transformation: ρ→ ±a c2p p → ±c2 aρ . (269) Under the above, both the canonical structure and the Hamilt onian remain invariant. Solutions are mapped in general to new solutions. Note that the nonrela tivistic limit is mapped to the ultra-relativistic one under the above duality. Self-dual solutions, with ρ=±a c2p, satisfy ˙ρ=∓c∂ ∂xρ (270) and are, therefore, the chiral relativistic solutions that were presented at the end of Section 3.2. In the self-dual case, when pis eliminated from the canonical 1-form and from the Hamilto nian with the help of (269), one arrives at an action for ρ, which coincides (apart from irrelevant constants) with the self-dual action, constructed some tim e ago [37] /braceleftbigg 1 2/integraldisplay dtdxdy˙ρ(x)ε(x−y)p(y)−/integraldisplay dtdx/radicalbig ρ2c2+a2/radicalbig c2+p2dt/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle p=c2 aρ =2c2 a/braceleftbigg 1 4/integraldisplay dtdxdy˙ρ(x)ε(x−y)ρ(y)−c 2/integraldisplay dtdx/parenleftig ρ2(x) +a2 c2/parenrightig/bracerightbigg (271) 6.5 General solution of the Nambu-Goto theory for a (d=1)- brane (string) in two spatial dimensions (on a plane) The complete integrability of the Chaplygin gas and of the Bo rn-Infeld theory, as well as the relationships between the two, derives from the fact that th e diﬀerent models descend by ﬁxing in diﬀerent ways the parameterization invariance of the Nam bu-Goto theory for string on a plane. At the same time, the equations governing the planar m otion of a string can be solved completely. Therefore it is instructive to see how the strin g solution produces this Chaplygin solution [21].Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 55 We follow the development in Section 4.3. The Nambu-Goto act ion reads ING=/integraldisplay dϕ0LNG (272a) LNG=/integraldisplay dϕ1LNG (272b) LNG=/bracketleftig −det∂Xµ ∂ϕα∂Xµ ∂ϕβ/bracketrightig1/2 . (272c) HereXµ,µ= 0,1,2, are string variables and ( ϕ0,ϕ1) are its parameters. As in Section 4.3, we deﬁne light-cone combinations X±=1√ 2(X0±X2), renameX−asθ, and choose the parameterization X+=ϕ0≡τ. After suppressing the superscripts on ϕ0andX1, we construct the Nambu-Goto Lagrange density as LNG= det1/2/parenleftigg 2∂τθ−(∂τX)2u u −(∂ϕX)2/parenrightigg (273) u=∂ϕθ−∂τX∂ϕX (274) Equations of motion are presented in Hamiltonian form: p≡∂LNG ∂∂τXΠ≡∂LNG ∂∂τθ(275) ∂τX=−1 Πp−u∂ϕX (276a) ∂τθ=1 2Π2/parenleftbig p2+ (∂ϕX)2/parenrightbig −u∂ϕθ (276b) ∂τp=−∂ϕ/parenleftig1 Π∂ϕX/parenrightig −∂ϕ(up) (276c) ∂τΠ =−∂ϕ(uΠ) (276d) and there is the constraint p∂ϕX+ Π∂ϕθ= 0. (277) There still remains the reparameterization freedom of repl acingϕby an arbitrary function ofτandϕ; this freedom may be used to set u= 0, Π = −1. Consequently, in the fully parameterized equations of motion Eq. (276d) disappears; i nstead of (276a) and (276c), we have∂τX=p,∂τp=∂2 ϕX, which imply (∂2 τ−∂2 ϕ)X= 0 (278a) (276b) reduces to ∂τθ=1 2/bracketleftbig (∂τX)2+ (∂ϕX)2/bracketrightbig (278b)56 R. Jackiw —(A Particle Field Theorist’s) and the constraint (277) requires ∂ϕθ=∂τX∂ϕX . (278c) Solution to (278a) is immediate in terms of two functions F±, x(τ,ϕ) =F+(τ+ϕ) +F−(τ−ϕ) (279) and then (278b), (278c) ﬁx θ: θ(τ,ϕ) =/integraldisplayτ+ϕ dz/bracketleftbig F′ +(z)/bracketrightbig2+/integraldisplayτ−ϕ dz/bracketleftbig F′ −(z)/bracketrightbig2. (280) This completes the description of a string moving on a plane. But we need to convert this information into a solution of the Chaplygin gas, and we know from Section 4.3 that this can be accomplished by a hodographic transformation: instead o fXandθas a function of τand ϕ, we seekϕas a function of τandX, and this renders θto be a function of τandXas well. The density ρis determined by the Jacobian \|∂X/∂ϕ \|. ReplaceτbytandXbyxand deﬁneϕto bef(t,x). Then from (279) it follows that x=F+/parenleftbig t+f(t,x)/parenrightbig +F−/parenleftbig t−f(t,x)/parenrightbig . (281) This equation may be diﬀerentiated with respect to tandx, whereupon one ﬁnds ∂f ∂t=−F′ +(t+f) +F′ −(t−f) F′ +(t+f)−F′ −(t−f)(282a) ∂f ∂x=1 F′ +(t+f)−F′ −(t−f). (282b) Thus the procedure for constructing a Chaplygin gas solutio n is to choose two functions F±, solve the diﬀerential equations (282) for f, and then the ﬂuid variables are θ(t,x) =/integraldisplayt+f(t,x)/bracketleftbig F′ +(z)/bracketrightbig2dz+/integraldisplayt−f(t,x)/bracketleftbig F′ −(z)/bracketrightbig2dz (283) √ 2λ ρ=/vextendsingle/vextendsingleF′ +(t+ϕ)−F′ −(t−ϕ)/vextendsingle/vextendsingle. (284) One may verify directly that (283) and (284) satisfy the requ ired equations: Upon diﬀer- entiating (283) with respect to tandx, we ﬁnd ∂θ ∂t= (F′ +)2/parenleftig 1 +∂f ∂t/parenrightig + (F′ −)2/parenleftig 1−∂f ∂t/parenrightig =−2F′ +F′ − (285a) ∂θ ∂x= (F′ +)2/parenleftig∂f ∂x/parenrightig −(F′ −)2/parenleftig∂f ∂x/parenrightig =F′ ++F′ − (285b)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 57 The second equalities follow with the help of (282). From (28 5) one sees that ∂θ ∂t+1 2/parenleftig∂θ ∂x/parenrightig2 =1 2(F′ +−F′ −)2=λ ρ2(286) the last equality being the deﬁnition (284). Thus the Bernou lli (Euler) equation holds. For the continuity equation, we ﬁrst ﬁnd from (284) and (285) ∂ρ ∂t=±∂ ∂t√ 2λ F′ +−F′ − =∓√ 2λ (F′ +−F′ −)2/bracketleftig F′′ +/parenleftig 1 +∂f ∂t/parenrightig −F′′ −/parenleftig 1−∂f ∂t/parenrightig/bracketrightig =±2√ 2λ (F′ +−F′ −)3/parenleftbig F′′ +F′ −+F′′ −F′ +/parenrightbig (287a) ∂ ∂x/parenleftig ρ∂θ ∂x/parenrightig =∂ ∂x/parenleftig ±√ 2λF′ ++F′ − F′ +−F′ −/parenrightig =∓√ 2λ (F′ +−F′ −)2/parenleftbig F′′ +F′ −+F′′ −F′ +/parenrightbig∂f ∂x =∓2√ 2λ (F′ +−F′ −)3/parenleftbig F′′ +F′ −+F′′ −F′ +/parenrightbig (287b) The last equalities follow from (282); since (287a) and (287 b) sum to zero, the continuity equation holds. We observe that the diﬀerentiated functions F′ ±are just the Riemann coordinates: from (285b) and (284) [with the absolute value ignored] we have p±√ 2λ ρ≡R±= 2F′ ±. (288) Also it is seen with the help of (282) that the Riemann formula tion (252) of the Chaplygin equations is satisﬁed by 2 F′ ±. The constants of motion (250) become proportional to I± n∝/integraldisplay dx1 F′ +−F′ −/bracketleftbig F′ ±/bracketrightbign =/integraldisplay dx∂f ∂x/bracketleftbig F′ ±(t±f)/bracketrightbign ∝/integraldisplay dz/bracketleftbig F′ ±(z)/bracketrightbign. (289) Finally we remark that the solution (243), (244) correspond s to F+(z) =−F−(z) =±lnz 2k. (290)58 R. Jackiw —(A Particle Field Theorist’s) There exists a relation between the two functions FandGin (259), which encode the Chaplygin gas solution in the linearization approach of Section 6.3, a nd the above two functions F±, which do the same job in the Nambu-Goto approach. The relation is th at 2F′ +is inverse to 2 F′′and 2F′ −is inverse to 2 G′′, that is, 2F′′[2F′ +(z)] =z 2G′′[2F′ −(z)] =z (291) Problem 7 Derive (291). Verify this relation with (260) and (290).Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 59 7 Towards a Non-Abelian Fluid Mechanics Fluid mechanics and ﬂuid magnetohydrodynamics may very wel l describe the long-wavelength degrees of freedom in a quark-gluon plasma. Moreover it is pl ausible that the group (color) degrees of freedom remain distinct in that regime, so that on e should incorporate them in the ﬂuid approximation. In this way one is led to think about c onstructing non-Abelian ﬂuid mechanics and (color) magnetohydrodynamics. In this secti on we describe an approach to this task [38]. In the course of development, we encounter and sol ve an interesting mathematical problem: how to parameterize a non-Abelian vector potentia l so that the non-Abelian Chern- Simons density becomes a total derivative, and the volume-i ntegrated Chern-Simons term is given by a surface integral. Obviously this is the non-Abeli an generalization of the similar Abelian problem, which is solved by presenting the Abelian v ector potential in Clebsch form. So we shall determine the non-Abelian version of the Clebsch parameterization. 7.1 Proposal for non-Abelian ﬂuid mechanics We review our Lagrange density for relativistic Abelian ﬂui d mechanics (144): L=−jµaµ−f(/radicalbig jµjµ). (292) The equation of state is encoded in the function f. For free ﬂuid motion f(/radicalbigjµjµ) =c/radicalbigjµjµ. Herejµis the matter current and aµis an auxilliary 4-vector, which is presented in the form aµ=∂µθ+α∂µβ . (293) The time component a0, involving time derivatives, determines the canonical 1-f orm; the spatial components aare in the Clebsch parameterization, as is needed for overco ming the obstacle created by a Casimir invariant of the ﬂuid in the alg ebra (14), (16). Another way of characterizing the parameterization of the vector ais that it casts the Chern-Simons density ofa, namely, a·∇×ainto total derivative form: ∇θ·(∇α×∇β) =∇·(θ∇×a). For a non-Abelian generalization, it is plausible to suppos e that the current 4-vector ac- quires an internal symmetry index: Jµ a; correspondingly, the auxiliary 4-vector must also acquire an internal symmetry index: Aa µ. It remains to give a rule for parameterizing Aa µ, which generalizes the Abelian rule (293). Our proposal – and it is a speculative one, since at this stage we have no derivation from microscopic considerations – is that Aa µshould be written in a form so that its non- Abelian Chern-Simons density, CS( A) =AadAa+1 3fabcAaAbAc, is a total derivative ( fabcare the structure constants of the group). This leads us to the pu rely mathematical problem of constructing a parameterization for a non-Abelian vector t hat ensures this property.60 R. Jackiw —(A Particle Field Theorist’s) 7.2 Non-Abelian Clebsch parameterization (or, casting the non-Abelian Chern-Simons density into tot al derivative form) We enquire whether it is possible to parameterize the non-Ab elian 1-form, Aa, such that the Chern-Simons 3-form is a total derivative (is exact): CS(A) =AadAa+1 3fabcAaAbAc= dΩ. (294) That this should be possible follows from the observation th at the left side of (294) is a 3-form on 3-space; hence it is closed, because a 4-form does not exis t in 3-space. [Of course on a 4- dimensional space the exterior derivative of (294) is propo rtional to the non-Abelian anomaly (Chern-Pontryagin density) [13].] But a closed form is also exact, at least locally; this justiﬁes the right side of (294). How this works in the Abelian case has already been explored i n Sections 2.4 and 2.5: the Clebsch parameterization (58), (60) for Aleads to the desired result. But the generalization of (58), (60) for a non-Abelian 1-form is not evident. Howeve r, at the end of Section 2.5, an alternative approach is presented, wherein the Abelian 1-f orm is projected from a non-Abelian pure gauge 1-form. This construction can be generalized to t he non-Abelian case and yields the sought-for parameterization. The mathematical problem can therefore be formulated in the following way: For a given group H, how can one construct a potential Aa µ= (Aa 0,Aa i)such that the non-Abelian Chern- Simons integrand CS(A)is a total derivative? Here we shall only sketch the solution to the problem, referring those interested to Ref. [38] for a detai led discussion. In the solution that we present, the “total derivative” form for the Chern-Simons density of Aais achieved in two steps. The parameterization, which we ﬁnd , directly leads to an Abelian form of the Chern-Simons density: AadAa+1 3fabcAaAbAc=γdγ (295) for someγ. Then Darboux’s theorem [10] (or usual ﬂuid dynamical theor y [11]) ensures that γcan be presented in Clebsch form, so that γdγis explicitly a total derivative. We begin with a pure gauge g−1dgin some non-Abelian group G(called the Ur-group) whose Chern-Simons integral coincides with the winding num ber ofg. W(g) =1 16π2/integraldisplay d3rCS(g−1dg) =1 24π2/integraldisplay tr(g−1dg)3(296) We consider a normal subgroup H⊂G, with generators Ia, and construct a non-Abelian gauge potential for Hby projection: Aa∝tr(Iag−1dg). (297) WithinH, this is not a pure gauge. We determine the group structure th at ensures the Chern-Simons 3-form of Aato be proportional to tr( g−1dg)3. Consequently, the constructedLectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 61 non-Abelian gauge ﬁelds, belonging to the group H, carry quantized Chern-Simons number. Moreover, we describe the properties of the Ur-group Gthat guarantee that the projected potentialAaenjoys suﬃcient generality to represent an arbitrary poten tial inH. Since tr(g−1dg)3is a total derivative for an arbitrary group (although this f act cannot in general be expressed in ﬁnite terms [39]) our construction e nsures that the form of Aa, which is achieved through the projection (297), produces a “total de rivative” expression (in the limited sense indicated above) for its Chern-Simons density. Conditions on the Ur-group G, which we take to be compact and semi-simple, are the following. First of all Ghas to be so chosen that it has suﬃcient number of parameters t o make tr(Iag−1dg) a generic potential for H. Since we are in three dimensions, an H-potential Aa ihas 3×dimHindependent functions; so a minimal requirement will be dimG≥3 dimH . (298) Secondly we require that the H-Chern-Simons form for Aashould coincide with that of g−1dg. As we shall show in a moment, this is achieved if G/H is a symmetric space. In this case, if we split the Lie algebra of Ginto theH-subalgebra spanned by Ia,a= 1,...,dimH, and the orthogonal complement spanned by SA,A= 1,...,(dimG−dimH), the commutation rules are of the form [Ia,Ib] =fabcIc(299a) [Ia,SA] =haABSB(299b) [SA,SB] =N haABIa. (299c) (ha)ABform a (possibly reducible) representation of the H-generators Ia. The constant N depends on normalizations. More explicitly, if the structu re constants for the Ur-group G are named ¯fabc, a,b,c = 1,...,dimG, then the conditions (299a–c) require that ¯fabcvanishes whenever an odd number of indices belongs to the orthogonal c omplement labeled by A,B,.. . Moreover,fabcare taken to be the conventional structure constants for Hand this may render them proportional to (rather than equal to) ¯fabc. We deﬁne the traces of the generators by tr(IaIb) =−N1δab,tr(SASB) =−N2δAB tr(IaSA) = 0. (300) We can evaluate the quantity tr[ SA,SB]Ia= trSA[SB,Ia] using the commutation rules. This immediately gives the relation N1N=N2. Expanding g−1dgin terms of generators, we write g−1dg= (IaAa+SAαA) (301) which deﬁnes the H-potentialAa. Equivalently Aa=−1 N1tr(Iag−1dg) (302)62 R. Jackiw —(A Particle Field Theorist’s) From d(g−1dg) =−g−1dgg−1dg, we get the Maurer-Cartan relations Fα≡dAa+1 2fabcAbAc=−N 2haABαAαB dαA+hαBAAaαB= 0. (303) Using these results, the following chain of equations shows that the Chern-Simons 3-form for theH-gauge group is proportional to tr( g−1dg)3: 1 16π2(AadAa+1 3fabcAaAbAc) =1 48π2(AadAa+2AaFa) =1 48π2(AadAa−NhaABAaαAαB) =1 48π2(AadAa+NdαAαA) =−1 48π2/bracketleftig1 N1tr(AdA) +N N2tr(dαα)/bracketrightig =−1 48π2N1tr(AdA+αdα) =−1 48π2N1trg−1dgd(g−1dg) =1 48π2N1tr(g−1dg)3. (304) In the above sequence of manipulations, we have used the Maur er-Cartan relations (303), which rely on the symmetric space structure of (299a–c), and the trace relations (300), along withN1N=N2. We thus see that/integraltext CS(A) is indeed the winding number of the conﬁguration g∈G. Since tr(g−1dg)3is a total derivative locally on G, the potential (302), with the symmetric space structure of (299a–c), does indeed fulﬁll the requirement o f making CS( A) a total derivative. It is therefore appropriate to call our construction (302) a “non-Abelian Clebsch parameteri- zation”. In explicit realizations, given a gauge group of interest H, we need to choose a group G such that the conditions (298), (299a–c) hold. In general th is is not possible. However, one can proceed recursively. Let us suppose that the desired result has been established for a group, which we call H2. Then we form H⊂Gobeying (299a–c) as H=H1×H2, whereH1is the gauge group of interest, satisfying dim G≥3 dimH1. For this choice of H, the result (304) becomes CS(H1) + CS(H2) =1 48π2N1tr(g−1dg)3(305) But since CS( H2) is already known to be a total derivative, (305) shows the de sired result: CS(H1) is a total derivative. To see explicitly how this works we work out the representati on for a SU(2) ≈O(3) potential Aa i, which possesses nine independent functions.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 63 We takeG=O(5),H=O(3)×O(2). We consider the 4-dimensional spinorial representa- tion ofO(5). With the generators normalized by tr( tatb) =−δab, the Lie algebra generators ofO(5) are given by Ia=1 2i/parenleftigg σa0 0σa/parenrightigg I0=1 2i/parenleftigg −1 0 0 1/parenrightigg (306) SA=1 i√ 2/parenleftigg 0 0 σA0/parenrightigg ˜SA=1 i√ 2/parenleftigg 0σA 0 0/parenrightigg σ’s are the 2 ×2 Pauli matrices. IagenerateO(3), with the conventional structure constants εabc, andI0is the generator of O(2).S,˜Sare the coset generators. A general group element in O(5) can be written in the form g=Mhk whereh∈O(3), k∈O(2), and M=1/radicalig 1 +¯ w·w−1 4(w×¯ w)2 1−i 2(w×¯ w)·σ −w·σ ¯ w·σ 1 +i 2(w×¯ w)·σ  (307) wais a complex 3-dimensional vector, with the bar denoting com plex conjugation. w·¯ w= wa¯waand (w×¯ w)a=εabcwb¯wc. The general O(5) group element contains ten independent real functions. These are collected as six from M(in the three complex functions wa), three inh, and one in k. TheO(3) gauge potential given by −tr(Iag−1dg) reads Aa=Rab(h)ab+ (h−1dh)a aa=1 1 +w·¯ w−1 4(w×¯ w)2/braceleftigg wad¯ w·(w×¯ w) + ¯wadw·(¯ w×w) 2(308) +εabc(dwb¯wc−wbd ¯wc)/bracerightigg whereRab(h) is deﬁned by hIah−1=Rabhbandkdoes not contribute. Aais theh-gauge transform of aa, which depends on six real parameters ( wa). The three gauge parameters of h∈O(3), along with the six, give the nine functions needed to par ameterize a general O(3)- [orSU(2)-] potential in three dimensions. The Chern-Simons form is CS(A) =1 16π2(AadAa+1 3εabcAaAbAc) =1 16π2(aadaa+1 3εabcaaabac)−d/bracketleftbigg1 16π2(dhh−1)aaa)/bracketrightbigg +1 24π2tr(h−1dh)3(309)64 R. Jackiw —(A Particle Field Theorist’s) The second equality reﬂects the usual response of the Chern- Simons density to gauge trans- formations. Using the explicit form of aaas given in (308), we can further reduce this. Indeed we ﬁnd that aadaa+1 3εabcaaabac= (−2)(¯ w×d¯ w)·ρ+ (w×dw)·¯ρ [1 +w·¯ w−1 4(w×¯ w)2]2(310) ρk=1 2εijkd ¯wid ¯wj Deﬁning an Abelian potential a=w·d¯ w−¯ w·dw 1 +w·¯ w−1 4(w×¯ w)2(311) we can easily check that adareproduces (310). In other words CS(A) =1 16π2ada+ d/bracketleftbigg(dhh−1)aaa) 16π2/bracketrightbigg +1 48π2tr(h−1dh)3(312) If desired, the Abelian potential acan now be written in the Clebsch form making adainto a total derivative, while the remaining two terms already ar e total derivatives, though in a “hidden” form for the last expression. This completes our co nstruction. 7.3 Proposal for non-Abelian magnetohydrodynamics We return to our construction of a Lagrange density for non-A belian kinetic theory. As explained in Section 7.1, a plausible non-Abelian generali zation for (292) is L=−Jµa1 N1trIag−1∂µg−c/radicalbig JµaJaµ (313) where for simplicity we have taken the “free” form for f. When the desired group is SU(2), g is anO(5) group element, as detailed in Section 7.2. Magnetohydrodynamics is achieved by introducing a further interaction with a dynamical gauge potential Aa µ. This is accomplished by promoting the derivative of gto a gauge-covariant derivative, gauged on the right Lmagnetohydrodynamics =−1 N1Jµatr(Iag−1Dµg)−c/radicalig JµaJaµ−1 4FaµνFa µν (314) with Dµg=∂µg+egAµ. (315) Aµ=Aa µIaare independent, dynamical gauge potentials (not given by g) leading to the ﬁeld strengths Fa µν. The gauge transformation properties by the gauge function hare g′=gh A′=h−1Ah+1 eh−1dh J′a µIa=h−1Ja µIah . (316)Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 65 We expect that the Lagrangian (314) will describe non-Abeli an magnetohydrodynamics, namely the dynamics of a ﬂuid with non-Abelian charge couple d to non-Abelian ﬁelds. [The Abelian version of (314) does indeed describe ordinary magn etohydrodynamics.] This gluon hydrodynamics can be useful for non-Abelian plasmas such as the quark-gluon plasma. Details of (314) and possible applications are under further study.66 R. Jackiw —(A Particle Field Theorist’s) Solutions to Problems Problem 1 The imaginary part of the Schr¨ odinger equation gives the co ntinuity equa- tion in the form ˙ ρ+∇·(ρ∇θ) = 0. This identiﬁes the velocity vas∇θ, that is, vis irrotational and there is no vorticity. The real part become s the Bernoulli equation ˙θ+ 1 2(∇θ)2=¯h2 2ρ−1/2∇2ρ1/2, whose gradient gives the Euler equation and identiﬁes the f orcef as∇(¯h2 2ρ−1/2∇2ρ1/2). Problem 2 j=ρvwithv=∇θ. Problem 3 LSchr¨ odinger =θ˙ρ−1 2ρ(∇θ)2−¯h2 8∇ρ·∇ρ ρwhere the time derivative of i¯hρ 2−ρθ has been dropped. The results in the solutions to Problems 1 and 3 are called the M¨ adelung formulation of quantum mechanics [40]. Problem 4 CS(A) =εijk∂iΦ∂jcos Θ∂kh(r) (a) Extracting the ﬁrst derivative leaves CS( A) =∂iVi a,Vi a=εijkΦ∂jcos Θ∂kh(r). (This is true because εijk∂i∂jcos Θ = 0 = εijk∂i∂kh(r), since cos Θ and h(r) are nonsingular.) Note thatVi a=εiΘrΦ(−sinΘ r)h′(r) =δiΦΦ/parenleftbig1 rsin Θ/parenrightbig h(r). SinceVr a= 0, the surface integral does not contribute. However, since Φ is multivalu ed, there is a contribution from the Φ integral:/integraltext d3rCS(A) =/integraltextR 0r2dr/integraltextπ 0sinΘ dΘ/integraltext2π 0dΦ/parenleftbig1 rsinΘ∂ ∂ΦΦ/parenrightbig ×/parenleftbig1 rsin Θ/parenrightbig h′(r) = 4π/bracketleftbig h(R)−h(0)/bracketrightbig . (b) Extracting the second derivative leaves CS( A) =∂jVj b−εijk(∂j∂iΦ)cos Θ∂kh(r). The last term is present, owing to the singularity of Φ at the orig in, which gives rise to εkij∂i∂jΦ =δk32πδ(x)δ(y). (See [41].) Also we have Vi b=εijk∂iΦ cos Θ∂kh(r) = εΦjr/parenleftbig1 rsinθ/parenrightbig cos Θh′(r) =−1 rδjθcot Θh′(r). Again there is no r-component to contribute to the surface integral, but the second, singular term leave s /integraldisplay d3rCS(A) =/integraldisplay d3r(2π)δ(x)δ(y)cos Θ∂ ∂zh(r) = 2π/integraldisplayR −Rdzz \|z\|∂ ∂zh(\|z\|) = 4π/integraldisplayR 0dz∂ ∂zh(z) = 4π/bracketleftbig h(R)−h(0)/bracketrightbig . (c) Extracting the last derivative leaves CS(A) =∂kVk c−εijk(∂k∂iΦ)∂jcos Θh(r), Vk c=εijk∂iΦ∂jcos Θh(r) =εΦθk/parenleftig1 rsin Θ/parenrightig/parenleftig −1 rsinΘ/parenrightig h(r) =δkrh(r) r2.Lectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 67 Here the surface integral contributes 4 πh(R). The singular term is −δj3(2π)δ(x)δ(y)(∂jcos Θ)h(r) =−(2π)δ(x)δ(y)/parenleftig∂ ∂zz \|z\|/parenrightig h(\|z\|) =−(4π)δ3(r)h(0). Hence this contribution to the spatial integral is −4πh(0), for a total of 4 π/bracketleftbig h(R)−h(0)/bracketrightbig . Problem 5 In the Clebsch parameterization, B=∇α×∇β, andδA=∇δγ+δα∇β+ α∇δβ. Therefore B·δA=B·∇δγ+B·α∇δβ+ (B·∇β)δα =∇·(Bδγ) +∇·(Bαδβ)−(B·∇α)δβ+ (B·∇β)δα . The last two terms vanish, so/integraltext d3rA·Bis the surface term/integraltext dS·B(δγ+αδβ) with no contribution from the bulk (ﬁnite rspace). This of course is consistent with the Chern- Simons integral being a surface term, since δ1 2/integraltext d3rA·B=/integraltext d3rB·δA. When demanding the variation of1 2/integraltext d3rB2to vanish, we ﬁrst ﬁnd/integraltext d3r(∇×B)·δA= 0. When δAis arbitrary, this condition implies the vanishing of ∇×B. However, in the Clebsch parameterization, all we can conclude is that ∇×Bis proportional to B, with a position-dependent proportionality factor ∇×B=µB. Taking the divergence shows that B·∇µ= 0, that is, µcan be a function of the magnetic surfaces; see (64), (69). Problem 6 θ(t,r) =r2/2t. Time rescaling: θω(t,r) =eωθ(T,r),T=eωt;eωθ(T,r) = eωr2/2eωt=r2/2t=θ(t,r). Space-time mixing: θω(t,r) =θ(T,R),T=t+ω·r+1 2ω2θ(T,R) =t+ω·r+ω2R2/4T, R=r+ωθ(T,R) =r+ωR2/2T. Squaring the second equation gives R2=r2+ω·rR2/T+ ω2R4/4T2. Multiplying the ﬁrst equation by R2/TgivesR2=tR2/T+ω·rR2/T+ω2R4/4T2. Comparing the two shows that R2/T=r2/torθ(T,R) =θ(t,r). Problem 7 From (256) and (259) we learn that t=F′′+G′′,x= (p+s)F′′−F′+ (p−s)G′′−G′, whereFis a function of p+s=R+andGis a function of p−s=R−. Diﬀerentiating these equations with respect to tandx, it follows that 1 = F′′′˙R++G′′′˙R−, 0 =F′′′∂R+ ∂x+G′′′∂R− ∂x, 0 =R+F′′′˙R++R−G′′′˙R−, 1 =R+F′′′∂R+ ∂x+R−G′′′∂R− ∂x, which in turn imply∂R+ ∂x= 1/(R+−R−)F′′′,∂R− ∂x=−1/(R+−R−)G′′′, and ˙R±=−R∓∂R± ∂x[the Riemann equation (252) again]. On the other hand, the functions F±(t±f) describing the Chaplygin gas solution from the Nambu-Goto equation are related to R±by (288):R±= 2F′ ±. Hence∂R± ∂x=±2F′′ ±∂f ∂x= ±2F′′ ±/(F′ +−F′ −) =±4F′′ ±/(R+−R−), where (282b) is used. It follows that 4 F′′ +(z)F′′′/parenleftbig 2F′ +(z)/parenrightbig = d dz/parenleftig 2F′′/parenleftbig 2F′ +(z)/parenrightbig/parenrightig = 1 and 4F′′ −(z)G′′′/parenleftbig 2F′ −(z)/parenrightbig =d dz/parenleftig 2G′′/parenleftbig 2F′ −(z)/parenrightbig/parenrightig = 1, or 2F′′/parenleftbig F′ +(z)/parenrightbig =z and 2G′′/parenleftbig 2F′ −(z)/parenrightbig =z. WhenF+(z) = lnz/2k, 2F′ +(z) = 1/zk; withF(z) =z 2klnz, 2F′′(z) = 1/kzand 2F′′/parenleftbig 2F′ +(z)/parenrightbig =z; similarly for F−(z) andG(z).68 R. Jackiw —(A Particle Field Theorist’s) References [1] L. Landau and E. Lifshitz, Fluid Mechanics (2nd ed., Pergamon, Oxford UK 1987). [2] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics (Springer- Verlag, Berlin 1998). [3] The Lagrangian description of a ﬂuid and its relation to a n Eulerian description is ex- plained, for example, in R. Salmon, Ann. Rev. Fluid Mech. 20, 225 (1988). [4] In the ﬂuid mechanical context, these brackets were put f orward by P.J. Morrison and J.M. Greene, Phys. Rev. Lett. 45, 790 (1980); (E) 48, 569 (1982). [5] L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988). For a detailed exposition, see R. Jackiw in Constraint Theory and Quantization Methods , F. Colomo, L. Lusanna, and G. Marmo, eds. (World Scientiﬁc, Singapore 1994), repri nted in R. Jackiw, Diverse Topics in Theoretical and Mathematical Physics (World Scientiﬁc, Singapore 1995). [6] C. Eckart, Phys. Rev. 54, 920 (1938). [7] For an introduction to the properties of the Abelian and n on-Abelian Chern-Simons terms, see S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (NY)140, 372 (1982), (E) 185, 406 (1985). In ﬂuid mechanics or in magnetohydrodynamics th e Abelian Chern-Simons term is known as the ﬂuid or magnetic helicity, and was introd uced by L. Woltier, Proc. Nat. Acad. Sci. 44, 489 (1958), and further studied in M. Berger and G. Field, J. Fluid Mech. 147, 133 (1984); H. Moﬀatt and A. Tsinober, Ann. Rev. Fluid Mech. 24, 281 (1992). [8] C.C. Lin, International School of Physics E. Fermi (XXI) , G. Careri, ed. (Academic Press, New York NY 1963). [9] A. Clebsch, J. Reine Angew. Math. 56, 1 (1859). [10] A constructive discussion of the Darboux theorem can be found it the second work of Ref. [5]. [11] H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge UK 1932), p. 248. [12] R. Jackiw and S.-Y. Pi, Phys. Rev. D 61, 105015 (2000). [13] S. Treiman, R. Jackiw, B. Zumino, and E. Witten, Current Algebra and Anomalies (Princeton University Press/World Scientiﬁc, Princeton N J/Singapore 1985). [14] S. Deser, R. Jackiw, and A.P. Polychronakos, physics/0 006056. [15] S. Chaplygin, Sci. Mem. Moscow Univ. Math. Phys. 21, 1 (1904). [Chaplygin was a col- league of fellow USSR Academician N. Luzin. Although accuse d by Stalinist authorities of succumbing excessively to foreign inﬂuences, unaccountab ly both managed to escape the fatal consequences of their alleged actions; see N. Krement sov,Stalinist Science (Prince- ton University Press, Princeton NJ 1997).] The same model (8 5) was later put forwardLectures on (Supersymmetric, Non-Abelian) Fluid Mechanic s (and d-Branes) 69 by H.-S. Tsien, J. Aeron. Sci. 6, 399 (1939) and T. von Karman, J. Aeron. Sci. 8, 337 (1941). [16] K. Stanyukovich, Unsteady Motion of Continuous Media (Pergamon, Oxford UK 1960), p. 128. [17] M. Hassa¨ ıne and P. Horvathy, Ann. Phys. (NY)282, 218 (2000); L. O’Raifeartaigh and V. Sreedhar, htp-th/0007199. [18] R. Jackiw, Physics Today 25(1), 23 (1972); U. Niederer, Helv. Phys. Acta 45, 802 (1972); C.R. Hagen, Phys. Rev. D 5, 377 (1972). [19] These symmetry transformations were identiﬁed by D. Ba zeia and R. Jackiw, Ann. Phys. (NY)270, 246 (1998), on the basis of constants of motion that were fou nd previously; see Ref. [20]. [20] M. Bordemann and J. Hoppe, Phys. Lett. B317 , 315 (1993); A. Jevicki, Phys. Rev. D 57, 5955 (1988). These authors consider only the planar, d= 2, case. [21] D. Bazeia, Phys. Rev. D 59, 085007 (1999). [22] Analytic details of the implementation of the space-ti me mixing transformation on (112) are presented by Bazeia and Jackiw [19]. [23] N. Ogawa, Phys. Rev. D 62, 085023 (2000), who also presents other solutions, as does Bazeia, Ref. [21]. [24] R. Jackiw and A.P. Polychronakos, Comm. Math. Phys. 207, 107 (1999). [25] The additional transformation rules were derived in Re f. [24], on the basis of constants of motion identiﬁed previously in the d= 2 case; see Ref. [26]. [26] M. Bordemann and J. Hoppe, Phys. Lett. B325 , 359 (1994). [27] G. Gibbons, Nucl. Phys. B514 , 603 (1998). [28] See, for example, Landau and Lifshitz, Ref. [1], or S. We inberg, Gravitation and Cosmology (Wiley, New York NY 1972). [29] That the theory of a membrane [( d= 2)-brane] in three spatial dimensions is equivalent to planar ﬂuid mechanics was known to J. Goldstone (unpublis hed) and worked out by his student Hoppe (sometimes in collaboration with Bordemann) . The method described in Secs. 4.1 and 4.2 was presented for d= 2 in J. Hoppe, Phys. Lett. B329 , 10 (1994), while a version of the argument in Sec. 4.3 specialized to d= 2 is found in Bordemann and Hoppe, Ref. [20]. Generalization to arbitrary dis given in R. Jackiw and A.P. Polychronakos, Proc. Steklov Inst. Math. 226, 193 (1999) and Ref. [24]. [30] Hassa¨ ıne and Horvathy, Ref. [17]. [31] R. Jackiw and A.P. Polychronakos, Phys. Rev. D 62, 085019 (2000). Some of these results are described in unpublished papers by J. Hoppe, Karlsruhe p reprint KA-THEP-6-93, Karlsruhe preprint KA-THEP-9-93, hep-th/9311059.70 R. Jackiw —(A Particle Field Theorist’s) [32] B. de Wit, J. Hoppe, and H. Nicolai, Nucl. Phys. B305 , [FS23] 525 (1988). [33] L. Susskind, Phys. Rev. 165, 1535 (1968). [34] Landau and Lifshitz, Ref. [1]; Y. Nutku, J. Math. Phys. 28, 2579 (1987); P. Olver and Y. Nutku, J. Math. Phys. 29, 1610 (1988); M. Arik, F. Neyzi, Y. Nutku, P. Olver, and J. Verosky, J. Math. Phys. 30, 1338 (1989); J. Brunelli and A. Das, Phys. Lett. A235 , 597 (1997). [35] B. Barbishov and N. Chernikov, Zh. Eksp. Theor. Fis. 51, 658 (1966) [English translation: Sov. Phys. JETP 24, 437 (1967)]. [36] A. Jevicki and B. Sakita, Phys. Rev. D 22, 467 (1980) and Nucl. Phys. B165 , 511 (1980); J. Polchinski, Nucl. Phys. B362 , 25 (1991); J. Avan and A. Jevicki, Phys. Lett. B266 , 35 (1991) and B272 , 17 (1991). [37] R. Floreanini and R. Jackiw, Phys. Rev. Lett. 59, 1873 (1987). [38] R. Jackiw, V.P. Nair, and S.-Y. Pi, Phys. Rev. D 62, 085018 (2000). [39] C. Cr¨ onstrom and J. Mickelsson, J. Math. Phys. 24, 2528 (1983), (E) 27, 419 (1986). [40] E. M¨ adelung, Z. Phys. 40, 322 (1926); see also E. Merzbacher, Quantum Mechanics (3rd ed., Wiley, New York NY 1998). [41] R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 66, 2682 (1991).
arXiv:physics/0010043v1 [physics.acc-ph] 17 Oct 2000SLAC-AAS-97 KEK-ATF-11 October 2000 Bunch Length Measurements at the ATF Damping Ring in April 20001 K.L.F. Bane, Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA T. Naito, T. Okugi, and J. Urakawa High Energy Research Organization (KEK), Oho 1-1, Tsukuba, Ibaraki, Japan 1Work supported by Department of Energy contract DE–AC03–76 SF00515. 1Bunch Length Measurements at the ATF Damping Ring in April 2000 K.L.F. Bane, T. Naito, T. Okugi, and J. Urakawa 1 Introduction We want to accurately know the energy spread and bunch length depen- dence on current in the ATF damping ring. One reason is to know the strength of the impedance: From the energy spread measureme nts we know whether or not we are above the threshold to the microwave ins tability, and from the energy spread and bunch length measurements we ﬁnd o ut the ex- tent of potential-well bunch lengthening (PWBL). Another r eason for these measurements is to help in our understanding of the intra-be am scattering (IBS) eﬀect in the ATF. The ATF as it is now, running below desi gn energy and with the wigglers turned oﬀ, is strongly aﬀected by IBS. T o check for consistency with IBS theory of, for example, the measured ve rtical beam size, we need to know all dimensions of the beam, including th e longitudinal one. But beyond this practical reason for studying IBS, IBS i s currently a hot research topic at many accelerators around the world (se ee.g.Ref. [1]), and the eﬀect in actual machines is not well understood. Typi cally, when comparing theory with measurements fudge factors are neede d to get agree- ment (see e.g.Ref. [1]). With its strong IBS eﬀect, the ATF is an ideal machine for studying IBS, and an indispensable ingredient f or this study is a knowledge of the longitudinal phase space of the beam. The results of earlier bunch lengthening measurements in th e ATF can be found in Refs. [2]-[4]. Measurements of current dependen t eﬀects, espe- cially bunch length measurements using a streak camera, can be diﬃcult to perform accurately. For example, space charge in the came ra itself can lead to systematic errors in the measurement results. It is i mportant the results be accurate and reproducible. In the measurements o f both Decem- ber 1998[3] and December 1999[4], by using light ﬁlters, the authors ﬁrst checked that space charge in the streak camera was not signiﬁ cant. And then the Dec 99 authors show that their results agree with tho se Dec 98, i.e.on the dates of the two measurements the results were reprodu cible. 2Since IBS is so strong in the ATF, in the Dec 99 measurements an attempt was made to estimate the impedance eﬀect using the fo llowing method: First, from the form of the energy spread vs.current measure- ments it was concluded that the threshold to the microwave in stability was beyond 2 mA. Then, by dividing the bunch length vs.current curve by the energy spread vs.current curve the eﬀect of IBS was divided out, and PWBL was approximated. The assumption is that PWBL can be tre ated as a perturbation on top of IBS. The result was that this compone nt of bunch lengthening was found to grow by 7-15% (depending on the rf vo ltage) be- tween the currents of .5 mA and 2 mA, about a factor of 3 less tha n the total bunch length growth. The conclusion was that the inductive c omponent of the impedance was small, in fact much smaller than had been co ncluded earlier in Ref. [2]. Electron machines generally run in a parameter regime where IBS is an insigniﬁcant eﬀect, and impedance measurements and calcul ations have also normally been performed for machines where IBS is unimporta nt. To sim- plify the interpretation of the impedance from bunch length measurements, in April 2000 the energy spread and bunch length measurement s of Dec 99 were repeated, but now with the beam on a linear (diﬀerence ) coupling resonance, where the horizontal and vertical emittances we re approximately equal. For this case the eﬀect of IBS was expected to be very sm all. An energy spread vs.current measurement under such conditions will also al- low us to more clearly see whether we reach the threshold to th e microwave instability. As part of the April data taking we, in addition , repeated the earlier oﬀ-coupling measurements, in order to check the rep roducibility of the earlier results. In this report we present and analyze th is recent set of data, and compare it with the results of the earlier measurem ents, particu- larly those of Dec 99. The measurements and analysis of data in this report follow e ssentially the same procedure as was used in Ref. [4]. In the present repo rt we will try to be relatively brief. The comparison of our results wit h IBS theory will be given in a following report. For more details about th e measurement and analysis techniques presented in this report, the reade r should consult Ref. [4]. 2 Energy Spread Measurements Energy spread vs.current measurements were performed on March 21-22, 2000, with the beam both on and oﬀ the coupling resonance, for rf volt- 3ages of Vc= 150 kV and 300 kV. As usual the beam width is measured after extraction on a thin screen in a dispersive region. The data is ﬁt to a Gaussian, yielding the rms energy spread σδ(see Fig. 1). (For these mea- surements the energy spread was not calibrated, but from ear lier experience we expect the zero value results to be near 5 .5×10−4, which allows us to estimate the scaling factor.) In the ﬁgure the points in the p lots are ﬁt to y= 5.5(1 + a·xb). Figure 1: Energy spread as a function of current for various v alues of cavity voltage. The measurements were performed on 3/21-22/00. Th e curves give a simple ﬁt to the data. We note from Fig. 1 that, as expected, on the coupling resonan ce the energy spread is almost independent of current, only increa sing by 3-4% by 2.5 mA. With the vertical beam size enlarged, the eﬀect of i ntra-beam scattering becomes small. Oﬀ the coupling resonance the cha nge by 2.5 mA is 35-40%. We also note in (a) and (b) that there is no evidence of the threshold to the microwave instability, whose signature wo uld be a kink in the data. If there is no microwave instability on resonance, then almost certainly there is no microwave instability oﬀ resonance, s ince in the latter case the longitudinal phase space volume is increased, whic h tends to sta- bilize the beam. Finally, note that the 150 kV and 300 kV on-re sonance results are statistically identical, as would be expected i n the case of no mi- 4crowave instability and negligible intra-beam scattering . Note also that for the oﬀ-resonance results the energy spread is slightly larg er for the higher voltage. This would be expected, since a higher voltage mean s a smaller bunch length, which increases intra-beam scattering, whic h, in turn, more increases the energy spread. The measurements were repeated for the nominal (oﬀ-resonan ce,Vc= 300 kV) settings on April 13 and 14 (see Fig. 2), just before an d after the bunch length measurements to be presented in the next sectio n. We note that the results of the two measurements are essentially the same, and they are also essentially identical to the March results with the beam oﬀ-resonance andVc= 300 kV (Fig. 1c). This indicates that the machine condition s, as concerns intra-beam scattering (the horizontal and vertic al beam sizes; the lattice), are the same during all these measurements, as wel l as during the oﬀ-coupling, bunch length measurements presented below. Figure 2: Energy spread as a function of current oﬀ-resonanc e, atVc= 150 and 300 kV. Measurements were performed April 13 (a) and 14 (b ). Finally, in Fig. 3 we compare the ﬁts to the Vc= 150 and 300 kV, oﬀ-coupling measurements with those to measurements—unde r the same conditions—taken in Dec 99 and presented in Ref. [4]. We note that the corresponding curves for the two dates are signiﬁcantly diﬀ erent. For ex- ample, for Vc= 300 kV and I= 1 mA the new results are 8% higher than the earlier results. This suggests that (when oﬀ coupling) t he machine, con- cerning intra-beam scattering, was signiﬁcantly diﬀerent for the two dates. 5Figure 3: Fits to the March, oﬀ-resonance energy spread meas urements, forVc= 150 and 300 kV, compared to results of Dec 99 (the dashed curv es). 3 Bunch Length Measurements For the streak camera measurements ﬁrst, to make sure that sp ace charge in the camera itself was not a problem, tests were made at the hig hest current (2.5 mA) with diﬀerent light ﬁlters, and an appropriate ﬁlte r was chosen. Then, the main data taking process consisted of storing a hig h current beam and measuring the longitudinal bunch proﬁle ∼50−70 times at ﬁxed time intervals, while the current naturally decreased. Each tra ce, along with its DCCT current reading, was automatically saved to disk. To le t in more light at the lower currents the light ﬁlter was automaticall y switched half- way through each data taking sequence. Measurements were pe rformed with the beam on and oﬀ the coupling resonance, and with 4 diﬀerent settings of rf voltage. As before the streak camera traces were ﬁt to an asymmetric Ga ussian, given by λz=A√ 2πσ0exp/bracketleftbigg −1 2(z−¯z)2 σ2 0(1±t)2/bracketrightbigg z≷¯z , (1) with the convention that more negative values of zare more to the front of the bunch. The ﬁtting constants are A,σ0, ¯z, and the asymmetry factor t (a constant, platform oﬀset is also included in the ﬁt). Note that the full- width-at-half-maximum (FWHM) of the ﬁt is zfwhm = 2√ 2ln 2σ0and the rms bunch length σz=σ0/radicalbig 1 + (3 −8/π)t2. For small t,σz≈σ0. The skew moment, deﬁned by s=/angbracketleft(z− /angbracketleftz/angbracketright)3/angbracketright/σ3 z, is given by s≈4t/√ 2π, for small t. Note also that from physical considerations, we generally expect t >0, i.e.the leading edge of the distribution to be steeper than the tr ailing edge. 6Four example scans, with their ﬁts, are shown in Fig. 4. We not e that the ﬁts are reasonably good, though we see some, what are likely a nomalous, deviations from the ﬁts in the data. We have lots of data and we will use the statistical Method of Maximum Likelihood to do the error analysis[5]. Figure 4: Four example scans, and their asymmetric Gaussian ﬁts, for the beam on the coupling resonance with Vc= 250 kV. The horizontal axis has been shifted so that all peaks are at z= 0. The results for the beam on the coupling resonance and for rf v oltages Vc= 150, 200, 250, 300 kV are given in Figs. 5-8, and for the beam o ﬀ the coupling resonance in Figs. 9-12. Shown are the paramete rs of the asymmetric Gaussian ﬁt to the measured proﬁles: the area (a) , the rms σ0(b), the asymmetry factor t(c), and an estimate of the relative rms error in the ﬁt (d). Plots (a)-(c) give the zeroth, the second , and the third moments of the charge distribution. The line in (b) is a strai ght line ﬁt to the bunch length, where each data point has been weighted i nversely by the variance in the ﬁt to the asymmetric Gaussian [note that f or the ﬁt y=x·mfit+bfit]. The line in (c) is a straight line ﬁt to y=x·mfit. In the (a) part of the ﬁgures we note that the area is roughly pr opor- tional to current—with the constant of proportionality diﬀ erent for the low and the high current results due to the two diﬀerent light ﬁlt ers used— though with some scatter in the data. In (b) we note that the sc atter in the measured bunch lengths is quite signiﬁcant, especially at t he lower currents, 7Figure 5: Bunch length as function of current for the beam on t he coupling resonance and Vc= 150 kV. Given are the parameters of the asymmetric Gaussian ﬁt to the measured proﬁles. The lines in the plots ar e straight line ﬁts to the results (with zero oﬀset in the case of the asymmetr y parameter). Figure 6: Bunch length as function of current for the beam on t he coupling resonance and Vc= 200 kV. 8Figure 7: Bunch length as function of current for the beam on t he coupling resonance and Vc= 250 kV. Figure 8: Bunch length as function of current for the beam on t he coupling resonance and Vc= 300 kV. 9Figure 9: Bunch length as function of current for the beam oﬀ t he coupling resonance and Vc= 150 kV. Figure 10: Bunch length as function of current for the beam oﬀ the coupling resonance and Vc= 200 kV. 10Figure 11: Bunch length as function of current for the beam oﬀ the coupling resonance and Vc= 250 kV. Figure 12: Bunch length as function of current for the beam oﬀ the coupling resonance and Vc= 300 kV. 11with an rms deviation from the straight line ﬁt varying from . 35-.80 mm. In (c) we see even more scatter in the asymmetry factor. We exp ect this parameter to start at zero, and to increase with increasing c urrent, and the data does seem to support an increasingly positive asymmetr y factor with current. The results are near .1/mA, and slightly decrease w ith voltage. We will show in a future ATF report that this parameter gives us i nformation about the higher mode losses and the real part of the impedanc e. Note that the measured asymmetry factors oﬀ-resonance are about half of those on-resonance, qualitatively consistent with the bunch bei ng longer, and the higher mode losses being less, when oﬀ-resonance. Finally, in (d) we note that the relative rms error in the ﬁt to the asymmetric Gaussi an is∼4-10%. There is more noise than in the measurements of Dec 99, reﬂect ed in the fact that the ﬁts then were twice as good. 3.1 Comparisons and Discussion In Fig. 13 we reproduce the linear ﬁts to the bunch length para meter for the measurements with the beam oﬀ the coupling resonance (the so lid lines). Note that these ﬁts are not valid below I/lessorsimilar.4 mA, where there is no data; in fact, the bunch lengths are expected to drop below these li nes for lower currents. In (a) the results are compared with the on-coupli ng results; in (b) with oﬀ-coupling results of Dec 99. In all cases the bunch len gth curves move up as the voltage moves down, as expected. Also, in (a) the bun ch length is larger oﬀ the coupling resonance than on, due to the eﬀect o f intra-beam scattering, as expected. We expect σz(0), for Vc= 200 kV, to be 6.2 mm. For the case of on- resonance, we ﬁnd the value at the origin of the curves to be 3. 7%, 10.4%, 4.6%, 5.0% larger than the zero value calculation for Vc= 150, 200, 250, 300 kV, respectively. For the case of oﬀ-resonance the value s are 4.5%, 6.4%, 15.3%, 27.1% larger than the zero value calculations. We exp ect the real bunch lengths to follow curves with negative curvature, so a n overshoot is not unexpected. Note that these overshoots are signiﬁcantl y less than the ∼35% for the Dec 99 results (see Fig. 13b), which, at the time, m ade us suspect a scaling error in the streak camera results. In Sec. 2 we saw that, for the on-coupling case, the energy spr ead was almost independent of current. For example, by I= 2 mA the energy spread has grown by only 3%. Here we see that at I= 2 mA the bunch length has grown by 33-39%, depending on the rf voltage. Sinc e IBS on the coupling resonance is very weak, this growth must be almo st entirely due to PWBL. Thus, it appears that, after all, PWBL is a big fac tor in the 12Figure 13: Linear ﬁts to measurements with the beam oﬀ the cou pling resonance (solid lines). The curves represent results for Vc= 150 kV, 200 kV, 250 kV, and 300 kV. The dashes in (a) are the on-coupling resul ts; those in (b) are results of Dec 99. Note that for each case the bunch l ength monotonically decreases as the voltage is increased. ATF, and the machine must have a large inductive component of impedance. Note that this conclusion is diﬀerent from that reached in De c 99, where PWBL was estimated by dividing the bunch lengthening oﬀ-res onance by the energy spread growth. There it was found, for example, th at the PWBL contribution to the bunch length at 2 mA, divided by the contr ibution at .5 mA, ranged from 7-15% depending on voltage. Here the equiv alent (on- coupling) results range from 18-26%, a marked diﬀerence. Or , if we try to compare the present on and oﬀ-coupling results using the sam e prescription, we ﬁnd that oﬀ-coupling the results are increases of 21.4% an d 6.1%, for respectively the case Vc= 150 and 300 kV; on-coupling the results are 23.5% and 20.1%. These also don’t agree. In making the earlier appr oximate calculation of PWBL we had assumed that PWBL can be treated as a perturbation on top of IBS. That is, we assumed that (1) PWBL d oesn’t aﬀect IBS signiﬁcantly and (2) the bunch length increase due to PWBL can be added on top of that due to IBS, and the result would be simil ar whether on or oﬀ resonance. It appears that these assumptions taken t ogether are wrong and the interaction of IBS and PWBL is, in fact, more com plicated. In Fig. 13b we note that the new oﬀ-coupling results are very d iﬀerent from those of Dec 99. We saw in Sec. 2 above that the energy spre advs. current was diﬀerent for the two measurement days, presumab ly due to a diﬀerence in the IBS eﬀect, so we would expect diﬀerences als o in bunch length, since it is also aﬀected by IBS. In addition the bunch length (below 13threshold) is aﬀected by the impedance through PWBL. To say m ore, quan- tatively, about such measurements we would need a better und erstanding of how IBS and PWBL interact and, in addition, a knowledge of the transverse beam sizes and optics on the days of the measurements. 4 Conclusions We have performed energy spread and bunch length measuremen ts with the beam on and oﬀ the coupling resonance. Our energy spread r esults show that, with the beam on-resonance, the eﬀect of intra-be am scattering is small, and that the threshold to the micro-wave instabili ty is beyond 2.5 mA. Our on-resonance bunch length results show that pote ntial well bunch lengthening is large—by 2 mA the bunch has lengthened b y 33-39%— indicating that the impedance has a large inductive compone nt. Our energy spread and bunch length measurements with the bea m oﬀ- resonance give results that are very diﬀerent from those of D ec 99. This suggests that the horizontal and vertical emittances and/o r the lattice were diﬀerent than during the earlier measurements, resulting i n a diﬀerent intra- beam scattering eﬀect. These measurements should be repeat ed both on and oﬀ the coupling resonance, under the same conditions—ve riﬁable by ﬁnding the same energy spread vs.current curves—to see if the bunch length measurements are reproducible. We have assumed here that the streak camera measurements are suﬃciently accurate for our purposes. The results appear to be consistent, but we have no independent w ay of checking their accuracy. (We might suggest installing in the ATF, som etime in the future, a bunch length measuring apparatus that works in a co mpletely diﬀerent way, such as the spectrum approach of Ref. [6], as an independent check on the streak camera results.) In the future, when doing any such current-dependent measur ements the energy spread vs.current measurement should always be performed, as a speciﬁer of the intra-beam scattering machine conditio ns, and since it is a rather quick and simple measurement. To study speciﬁcal ly impedance questions, measurements should be performed with the beam o n the coupling resonance, where intra-beam scattering is very weak. With t he beam oﬀ- resonance, bunch length and energy spread measurements are important ingredients, along with the horizontal and vertical emitta nces and optics, for attempting a full understanding of intra-beam scattering a nd its interaction with the impedance in the ATF damping ring. 145 Acknowledgements One of the authors (K.B.) thanks the ATF scientists and staﬀ f or their hospitality and help during his visits to the ATF, and M. Ross for his en- couragement and support to make such visits. References [1] C. Kim, “A Three-Dimensional Touschek Scattering Theor y,” LBL- 42305, September 1998. [2] H. Hayano, et al, “Impedance Measurement of ATF DR,” Proc. of EPAC 1998, Stockholm, Sweden, May 1998, p. 481. [3] K. Bane, et al, “Bunch Lengthening and Current-Dependent Energy Spread at ATF,” ATF Report 98-38, December 1998. [4] K. Bane, T. Naito, T. Okugi, “Bunch Length Measurements a t the ATF Damping Ring in 1999,” ATF Report 00-05, May 2000. [5] See, for example, P. Bevington and K. Robinson, Data Reduction and Error Analysis for the Physical Sciences , Second Edition, (McGraw- Hill, Inc., New York, 1992). [6] T. Ieiri, “Measurement of Bunch Length Based on Beam Spec trum at KEKB Rings,” Proc. of EPAC 2000, Vienna, Austria, June 200 0, p. 1735. 15
arXiv:physics/0010044v1 [physics.atom-ph] 17 Oct 2000Coordinate-space approach to the bound-electron self-ene rgy: Self-Energy screening calculation P. Indelicato Laboratoire Kastler-Brossel, Unit´ e Mixte de Recherche du CNRS n◦C8552, ´Ecole Normale Sup´ erieure et Universit´ e Pierre et Marie Cu rie, Case 74, 4 place Jussieu, F-75252 Paris CEDEX 05, France Peter J. Mohr National Institute of Standards and Technology, Gaithersb urg, Maryland 20899-8401 (Time-stamp: ¡Tuesday, October 17, 2000, 10:21:29 dft paul ¿) Abstract The self-energy screening correction is evaluated in a mode l in which the eﬀect of the screening electron is represented as a ﬁrst-ord er perturbation of the self energy by an eﬀective potential. The eﬀective poten tial is the Coulomb potential of the spherically averaged charge density of the screening electron. We evaluate the energy shift due to a 1 s1/2, 2s1/2, 2p1/2, or 2 p3/2electron screening a 1 s1/2, 2s1/2, 2p1/2, or 2p3/2electron, for nuclear charge Z in the range 5 ≤Z≤92. A detailed comparison with other calculations is made. 31.30.Jv Typeset using REVT EX 1I. INTRODUCTION The self-energy correction to the electron-electron inter action is one of the many con- tributions of order α2to an atomic binding energy. These corrections, shown as Fey nman diagrams in Fig. 1, are often called self-energy screening c orrections, and for inner shells they are the largest of all fourth-order radiative corrections. They give rise to three terms, which are represented in Fig. 2, when one distinguishes between th e reducible and irreducible part of the diagram in Fig. 1 (A). A ﬁrst attempt to evaluate the contribution of such diagrams from bound state quantum electrodynamics (BSQED) was made in 1991 [1] in an approxima tion in which the electrons not associated with the self-energy loop were represented a s a perturbing potential (Fig. 3). The potential was obtained by taking a spherical average of t he electron wave function, and calculating the potential associated with the resultin g charge density. More recently, direct evaluations of the diagrams of Fig. 1 for the ground st ate of two-electron ions have been made [2–7], and the method of [1] was used to provide the s elf-energy correction to several interactions [8]. In this paper we report on a comple te calculation of the self-energy screening in the approximation of [1], for all combinations of pairs of the states 1 s1/2, 2s1/2, 2p1/2, or 2p3/2in the range 5 ≤Z≤92. Terms corresponding to the diagram in Fig. 3(A) are obtained by varying the self-energy expression with respect to the external potential of the bou nd-state wave function, leading to the introduction of the ﬁrst-order correction to the wave function in the potential δV. Terms corresponding to the diagram in Fig. 3(B) are obtained by varying the expression for the self-energy with respect to the external potential o f the Green’s function and while those in Fig. 3(A′) correspond to variation with respect to the energy of the bo und state. The expression for the self energy in a large class of potenti alsV(x) can be written as the sum ESE=EL+EHof a low-energy part ELand a high-energy part EHgiven (in units in which ¯ h=c=me= 1) by [9] 2EL=α πEn−α πP/integraldisplayEn 0dz/integraldisplay dx2/integraldisplay dx1ϕ† n(x2)αlG(x2,x1, z)αmϕn(x1)(δlm∇2·∇1− ∇l 2∇m 1)sin[(En−z)x21] (En−z)2x21 (1) and EH=α 2πi/integraldisplay CHdz/integraldisplay dx2/integraldisplay dx1ϕ† n(x2)αµG(x2,x1, z)αµϕn(x1)e−bx21 x21−δm/integraldisplay dxϕ† n(x)βϕn(x), (2) where b=−i[(En−z)2+iδ]1/2,Re(b)>0, and x21=x2−x1. In these expressions, ϕnand Enare the eigenfunction and eigenvalue of the Dirac equation f or the bound state n, andG is the Green’s function for the Dirac equation correspondin g to the operator G= (H−z)−1, where H=α·p+V+βis the Dirac Hamiltonian. The indices landmare summed from 1 to 3, and the index µis summed from 0 to 3. The contour C Hextends from −i∞to 0−iǫand from 0 + iǫto +i∞, with the appropriate branch of bchosen in each case. For the present calculation, we assume that the potential V(x) is close to a pure Coulomb potential, except for a small correction δV(x), which is not necessarily spherically symmetric. Indeed s ome applications of this method have been made with non-spheric ally-symmetric perturbations [8]. We obtain the screening correction to the self-energy b y making the replacements V(x)→V(x) +δV(x), (3a) ϕn(x)→φn(x) +δφn(x), (3b) G(x2,x1, z)→G(x2,x1, z) +δG(x2,x1, z) (3c) En→En+δEn (3d) in Eqs. (1) and (2) and retaining only the ﬁrst-order correct ion terms. In the Eqs. (1) and (2), and (3a) to (3d) above, we use the symbols G,En,ϕnfor the exact quantities in the potential V(x), while the symbols G,En,φnrepresent the corresponding exact quantities in a pure Coulomb potential V(x). The same conventions are employed throughout the paper. We denote the operations on the unperturbed self energy that lead to these three corrections byδφ,δG, and δE, respectively. In particular, we employ the notation 3δφ=δφn∂ ∂ϕn(4) δG=δG∂ ∂G(5) δE=δEn∂ ∂En, (6) and the total correction is the sum δφn∂ ∂ϕn+δG∂ ∂G+δEn∂ ∂En(7) in which the partial diﬀerentiation symbol denotes formal d iﬀerentiation with respect to the indicated variable, with the result evaluated with the unpe rturbed functions. In Sec. II we write expressions for the ﬁrst-order perturbat ion corrections to the energy, the wave function and the Green’s function. In Secs. III, IV, and V we derive the expressions for the various contributions to the screened self-energy c orresponding to the three diagrams of Fig. 3. In a series of three earlier papers [10–12], we have derived and tested a method of analytically isolating divergent contributions to the s elf-energy diagram in coordinate space. In Secs. IV and V we derive from this earlier work the ge neralizations of the analytic subtraction terms which are necessary to make all contribut ions to the self-energy screening ﬁnite. The numerical results are presented in Sec. VI, and Se c. VII is the conclusion. II. PERTURBATION EXPANSION Replacing the potential V(x) byV(x) +δV(x), where δV(x) is spherically symmetric, changes the wave functions, the energy, and the Green’s func tion, which appear in Eqs. (1) and (2). From standard perturbation theory the ﬁrst-order e nergy correction is δEn=/integraldisplay dxφ† n(x)δV(x)φn(x) =/integraldisplay∞ 0dxx22/summationdisplay i=1fn,i(x)δV(x)fn,i(x), (8) where the radial wave function fn,iis deﬁned by writing 4φn(x) = fn,1(x)χµ κ(ˆx) ifn,2(x)χµ −κ(ˆx) , (9) and where χµ κ(ˆx) is the Dirac angular momentum eigenfunction. The ﬁrst-order correction to the wave function is given with the aid of the reduced Green’s function GR(x2,x1, En), deﬁned by (see for example [13]) GR(x2,x1, En) =/summationdisplay m Em/negationslash=Enφm(x2)φ† m(x1) Em−En = lim z→En/bracketleftBigg G(x2,x1, z)−φn(x2)φ† n(x1) En−z/bracketrightBigg , (10) as δφn(x2) =−/integraldisplay dx1GR(x2,x1, En)δV(x1)φn(x1). (11) For a spherically symmetric potential we have δfn,i(x2) =−/integraldisplay∞ 0dx1x2 12/summationdisplay j=1GR,ij κ(x2, x1, En)δV(x1)fn,j(x1). (12) In Eq. (12), the components of the radial reduced Green’s fun ction GR,ij κ(x2, x1, En) are deﬁned in analogy with the components of the full Green’s fun ctionGij κ(x2, x1, z) as given in Ref. [9], Eq. (A.14). To evaluate the ﬁrst-order correction to the Green’s functi on we use the well-known expansion G(z) =1 H+δV−z =1 H−z−1 H−zδV1 H−z +1 H−zδV1 H−zδV1 H−z+···, (13) and the term of ﬁrst order in δVis δG(z) =−1 H−zδV1 H−z, (14) 5which has second-order poles at the eigenvalues of the Dirac equation. In coordinate space, the ﬁrst-order correction is δG(x2,x1, z) = −/integraldisplay dx3G(x2,x3, z)δV(x3)G(x3,x1, z), (15) and for δV(x3) spherically symmetric, we have δGij κ(x2, x1, z) =−/integraldisplay∞ 0dx3x2 3 ×2/summationdisplay k=1Gik κ(x2, x3, z)δV(x3)Gkj κ(x3, x1, z). (16) III. LOW-ENERGY PART The low-energy part, for an arbitrary external spherically symmetric potential, when integrated over the spherical angles of the vectors x2andx1, yields EL=α πEn+α πP/integraldisplayEn 0dzU(z) (17) with U(z) =−/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1 ×/summationdisplay κ2/summationdisplay i,j=1Fn,¯ı(x2)Gij κ(x2, x1, z)Fn,¯(x1)Aij κ(x2, x1, z), (18) where the summation over κruns over all nonzero integers, and where ¯ ı= 3−iand ¯= 3−j. We are concerned with the ﬁrst-order perturbation in this ex pression that arises from variation of the external potential. The self energy depend s on the potential through three quantities that appear in Eq. (17) and (18), the wave functio nFn,i(x), the energy eigenvalue En, and the Green’s function Gij κ(x2, x1, z). The three corrections are denoted by δφEL,δEEL, andδGEL, respectively, with the total δEL=δφEL+δEEL+δGEL. (19) 6A. Lower-order terms The expression (1) contains spurious parts of lower order in Zαthan the complete result. The physically signiﬁcant part is isolated in a function FL(Zα) deﬁned by [10] EL=α π/bracketleftbigg5 6En+2 3/angb∇acketleftϕn\|β\|ϕn/angb∇acket∇ight+7 6/angb∇acketleftϕn\|V\|ϕn/angb∇acket∇ight +(Zα)4 n3FL(Zα)/bracketrightbigg (20) Here the linear perturbation of this function with respect t o variation of the external poten- tial is of interest. A corresponding function δFL(Zα) is deﬁned by δEL=α π/bracketleftbigg 2δEn+4 3/angb∇acketleftφn\|β\|δφn/angb∇acket∇ight+7 3/angb∇acketleftφn\|V\|δφn/angb∇acket∇ight +α(Zα)3 n3δFL(Zα)/bracketrightbigg (21) where the fact that δ/angb∇acketleftφn\|V\|φn/angb∇acket∇ight=/angb∇acketleftφn\|δV\|φn/angb∇acket∇ight+ 2/angb∇acketleftφn\|V\|δφn/angb∇acket∇ight =δEn+ 2/angb∇acketleftφn\|V\|δφn/angb∇acket∇ight (22) has been taken into account. B. Low-order matrix elements The lower-order expectation values involving the ﬁrst-ord er correction to the wave func- tionδφncan be evaluated by direct numerical integration. However, to get an independent check of the precision of the calculation, particularly whe n strong cancellation occurs, we derive a number of useful expressions in which δφndoes not occur. The energy perturbation is given by the conventional expression δEn=/angb∇acketleftφn\|δV\|φn/angb∇acket∇ight =/integraldisplay∞ 0dx x2/bracketleftBig f2 1(x) +f2 2(x)/bracketrightBig δV(x). (23) 7The wave-function perturbation terms are /angb∇acketleftφn\|β\|δφn/angb∇acket∇ight=/integraldisplay dx2/integraldisplay dx1φ† n(x2)β ×/summationdisplay Ei/negationslash=Enφi(x2)φ† i(x1) En−EiδV(x1)φn(x1) (24) and /angb∇acketleftφn\|V\|δφn/angb∇acket∇ight=/integraldisplay dx2/integraldisplay dx1φ† n(x2)V(x2) ×/summationdisplay Ei/negationslash=Enφi(x2)φ† i(x1) En−EiδV(x1)φn(x1). (25) Since we are considering here the case where the unperturbed potential is the Coulomb po- tential with known wave functions, we can simplify the calcu lation of these matrix elements. In particular, we interpret the expressions in Eqs. (24) and (25) as perturbations of the wave function on the left-hand side to give /angb∇acketleftφn\|β\|δφn/angb∇acket∇ight=/angb∇acketleftδβφn\|δV\|φn/angb∇acket∇ight (26) and /angb∇acketleftφn\|V\|δφn/angb∇acket∇ight=/angb∇acketleftδVφn\|δV\|φn/angb∇acket∇ight, (27) where δβφnandδVφnare ﬁrst-order corrections to the wave function due to pertu rbations β andV, respectively. These are simply calculated as the coeﬃcien ts ofδin the power series expansions in δof the wave functions for the appropriately modiﬁed Hamilto nians Hβ=α·p+ (1 + δ)β+V =α·p+ (1 + δ)βm+V (28) and HV=α·p+β+ (1 + δ)V . (29) In (28) the second line restores the mass dependence of the Ha miltonian in order to exhibit the dependence of the modiﬁed Coulomb wave functions on δ. 8The wave function correction δβφnis obtained by replacing mby (1 + δ)minφnand calculating the coeﬃcient of the term linear in δ, which is equivalent to writing δβφn=m∂ ∂mφn (30) and leads to /angb∇acketleftφn\|β\|δφn/angb∇acket∇ight=m 2∂ ∂m/angb∇acketleftφn\|δV\|φn/angb∇acket∇ight, (31) where the derivative acts only on the wave function. A furthe r simpliﬁcation is possible based on the mass dependence of the wave function in (31). If xis replaced by x/min the second line of (28), then the mass factors out of the Hamilton ian, and the wave function is independent of m. As a result, we have /angb∇acketleftφn\|β\|δφn/angb∇acket∇ight=m 2∂ ∂m/integraldisplay∞ 0dx x2/bracketleftBig f2 1(x) +f2 2(x)/bracketrightBig δV(x) =1 m3/integraldisplay∞ 0dx x2 ×/bracketleftBig f2 1(x/m) +f2 2(x/m)/bracketrightBigm 2∂ ∂mδV(x/m) =−1 2/integraldisplay∞ 0dx x3/bracketleftBig f2 1(x) +f2 2(x)/bracketrightBig∂ ∂xδV(x) =−1 2/angb∇acketleftφn\|x(δV)′\|φn/angb∇acket∇ight, (32) where the convention m= 1 has been restored in the last line, and ( δV)′represents the operator corresponding to∂ ∂xδV(x). The correction δVφnis obtained by replacing Zby (1 + δ)Zinφnand calculating the coeﬃcient of the term linear in δwhich is equivalent to writing δVφn=Z∂ ∂Zφn (33) which leads to /angb∇acketleftφn\|V\|δφn/angb∇acket∇ight=Z 2∂ ∂Z/angb∇acketleftφn\|δV\|φn/angb∇acket∇ight (34) where it is understood that the derivative acts only on the wa ve function, or 9/angb∇acketleftφn\|V\|δφn/angb∇acket∇ight=Z/integraldisplay∞ 0dx x2 ×/bracketleftBigg f1(x)∂f1(x) ∂Z+f2(x)∂f2(x) ∂Z/bracketrightBigg δV(x). (35) The derivative of the potential in (32) is calculated analyt ically, and the derivatives in (35) are calculated numerically by evaluating the wave function with 32 ﬁgure precision and using a symmetric derivative formula with δ(Zα) = 10−12. The matrix elements listed above are evaluated by Gaussian q uadrature, and the code was tested in the Coulomb case and compared to the analytic resul ts, as described in Appendix A. C. Low-Energy energy-level and wave function correction The correction from the energy-level perturbation of the lo w-energy part of the self energy (1) is δEEL=δEn∂EL ∂En, (36) where ∂EL ∂En=α π−α π/bracketleftBigg/integraldisplay dx2/integraldisplay dx1φ† n(x2)αlG(x2,x1, z)αmφn(x1)(δlm∇2·∇1− ∇l 2∇m 1)sin[(En−z)x21] (En−z)2x21/bracketrightBigg z=En −α πP/integraldisplayEn 0dz/integraldisplay dx2/integraldisplay dx1φ† n(x2)αlG(x2,x1, z)αmφn(x1)(δlm∇2·∇1− ∇l 2∇m 1)∂ ∂Ensin[(En−z)x21] (En−z)2x21 The second term on the right-hand side of (37) makes no contri bution because (δlm∇2·∇1− ∇l 2∇m 1)sin[(En−z)x21] (En−z)2x21=2 3δlm(En−z) +O(En−z)3, (38) and /integraldisplay dx2/integraldisplay dx1φ† n(x2)αlG(x2,x1, z)αmφn(x1) =/integraldisplay dx2/integraldisplay dx1φ† n(x2)αl/summationdisplay Ej=Enφj(x2)φ† j(x1) En−zαmφn(x1) +O(1) =O(1). (39) 10The ﬁrst term on the right-hand side of Eq. (39) vanishes by vi rtue of the identity /integraldisplay dxφ† n(x)αlφj(x) = i/integraldisplay dxφ† n(x)[H, xl]φj(x) = i ( En−Ej)/integraldisplay dxφ† n(x)xlφj(x).(40) The correction due to variation of the bound state eigenvalu e is thus δEEL=α πδEn+α πP/integraldisplayEn 0dz δEU(z), (41) where δEU(z) =−δEn/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1/summationdisplay κ2/summationdisplay i,j=1 f¯ı(x2)Gij κ(x2, x1, z)f¯(x1)∂ ∂EnAij κ(x2, x1, z). (42) From Eq. (1) one can also easily obtain the correction due to t he variation of the bound state wave function: δφEL=−2α πP/integraldisplayEn 0dz/integraldisplay dx2/integraldisplay dx1φ† n(x2)αlG(x2,x1, z)αmδφn(x1)(δlm∇2·∇1− ∇l 2∇m 1)sin[(En−z)x21] (En−z)2x21, (43) since the dependence on the bound-state wave function is exp licit. D. Low-Energy Green’s function correction The correction due to the variation of the Green’s function i n Eq. (17) is given by δGEL=α πRe/integraldisplay C+dz δGU(z) (44) where ﬁrst-order change in U(z) in Eq. (18), due to variation of Gis δGU(z) =−/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1 ×/summationdisplay κ2/summationdisplay i,j=1fn,¯ı(x2)δGij κ(x2, x1, z)fn,¯(x1)Aij κ(x2, x1, z), (45) and the integration contour C +is deﬁned subsequently. The Green’s function Gij κ(x2, x1, z) in Eq. (18) has poles along the real axis in the range of integr ation over z, and in the 11perturbation expansion of Gij κ(x2, x1, z), in powers of a perturbing potential δVand the unperturbed Green’s function Gij κ(x2, x1, z), higher order poles are introduced as noted in Sec. II. Here, our method of isolating those poles and evalua ting their contribution is described. In terms of the spectral resolution of the unpert urbed radial Green’s function Gij κ(x2, x1, z) =/summationdisplay mfm,i(x2)fm,j(x1) Em−z, (46) Eq. (16) reads δGij κ(x2, x1, z) =−/integraldisplay∞ 0dx3x2 32/summationdisplay k=1/summationdisplay m2,m1 fm2,i(x2)fm2,k(x3) Em2−zδV(x3)fm1,k(x3)fm1,j(x1) Em1−z, (47) which explicitly shows the second and ﬁrst-order poles. In ( 47), only states m2, m1with spin-angular momentum quantum κcontribute. The principal parts of δGij κ(x2, x1, z) are identiﬁed by expanding the functions in Eq. (45) in Laurent s eries about each pole. For z≈Em, Gij κ(x2, x1, z) =fm,i(x2)fm,j(x1) Em−z+GR,ij κ(x2, x1, Em) +O(z−Em), (48) where GR,ij κ(x2, x1, Em) are the radial components of the reduced Green’s function g iven in Eq. (10) GR,ij κ(x2, x1, Em) = lim z→Em/bracketleftbigg Gij κ(x2, x1, z) −fm,i(x2)fm,j(x1) Em−z/bracketrightbigg =/summationdisplay l El/negationslash=Emfl,i(x2)fl,j(x1) El−Em. (49) Hence δGij κ(x2, x1, z) =−/integraldisplay∞ 0dx3x2 32/summationdisplay k=1/bracketleftbigg fm,i(x2)fm,k(x3)δV(x3)fm,k(x3)fm,j(x1)1 (Em−z)2 +GR,ik κ(x2, x3, Em)δV(x3)fm,k(x3)fm,j(x1)1 Em−z +fm,i(x2)fm,k(x3)δV(x3)GR,kj κ(x3, x1, Em)1 Em−z/bracketrightbigg +O(1),(50) 12or, in view of Eqs. (8) and (12), δGij κ(x2, x1, z) =−fm,i(x2)fm,j(x1)δEm (Em−z)2+/bracketleftBig δfm,i(x2)fm,j(x1) +fm,i(x2)δfm,j(x1)/bracketrightBig1 Em−z+O(1).(51) The same result may be obtained by expanding the pole contrib ution to the full Green’s function Gκ(x2, x1, z) =Fm,i(x2)Fm,j(x1) Em−z+O(1) (52) in powers of δV(x), with Em=Em+δEm+··· Fm,i(x) =fm,i(x) +δfm,i(x) +···, (53) and retaining only the ﬁrst-order correction −fm,i(x2)fm,j(x1)δEm (Em−z)2 +/bracketleftBig δfm,i(x2)fm,j(x1) +fm,i(x2)δfm,j(x1)/bracketrightBig1 Em−z. In addition to the expansion of the Green’s function correct ion, we have Aij κ(x2, x1, z) =Aij κ(x2, x1, Em) +(z−Em)Bij κ(x2, x1, Em) +O/parenleftBig (z−Em)2/parenrightBig , (54) where Bij κ(x2, x1, Em) =d dz/bracketleftBig Aij κ(x2, x1, z)/bracketrightBig z=Em. (55) The complete expansion for z≈Emis /integraldisplay∞ 0dx3x2 32/summationdisplay k=1Gik κ(x2, x3, z)δV(x3)Gkj κ(x3, x1, z)Aij κ(x2, x1, z) =fm,i(x2)fm,j(x1)δEm/bracketleftbigg Aij κ(x2, x1, Em)1 (Em−z)2−Bij κ(x2, x1, Em)1 Em−z/bracketrightbigg −/bracketleftBig δfm,i(x2)fm,j(x1) +fm,i(x2)δfm,j(x1)/bracketrightBig Aij κ(x2, x1, Em)1 Em−z+O(1).(56) 13Our strategy for dealing with poles in the low-energy part is to calculate the line integral overzof the diﬀerence between the complete integrand and the pole terms, and add the pole terms integrated analytically. In the unperturbed self-en ergy calculation, the singularities along the real axis in the interval (0 ,1) are poles, and the appropriate prescription for integration over zyields the principal value integral in Eq. (17). In the prese nt context, there are double poles as well, so it is necessary to reexamin e the original derivation to obtain the correct prescription. It follows from the discus sion in Ref. [9] that the integration overzcan be written as Re/integraldisplay C+dzU(z), (57) where C +is a contour that extends from z= 0 to z=Enabove the real axis in the complex z plane. Here we use a method based on an analytic evaluation of the pole terms, as described in Ref. [14]. With the notation Upfor the pole terms of U, we have Re/integraldisplay C+dzU(z) = Re/integraldisplay C+dz[U(z)− Up(z) +Up(z)], =/integraldisplayEn 0dz/bracketleftBigg U(z)− Up(z) + 1 EnRe/integraldisplay C+dz′Up(z′)/bracketrightBigg . (58) The pole at z=En, the endpoint of the integral over z, does not cause any problem, as follows from the discussion in Sec. IIIC [see Eqs. (37) to (40 )]. The relevant integrals for the analytic evaluation of the pole terms are Re/integraldisplay C+dz1 (Em−z)2=−En Em(En−Em), (59) Re/integraldisplay C+dz1 Em−z= ln/bracketleftbiggEm En−Em/bracketrightbigg , (60) where 0< E m< E n. (61) Applying these results to Eq. (44), we have 14δGEL=α πRe/integraldisplay C+dz δGU(z) =α π/integraldisplayEn 0  δGU(z)−/summationdisplay m Em<En/bracketleftBiggR(2) n,m (Em−z)2+R(1) n,m (Em−z)/bracketrightBigg   +α π/summationdisplay m Em<En/bracketleftBigg −R(2) n,mEn Em(En−Em)+R(1) n,mln/parenleftbiggEm En−Em/parenrightbigg/bracketrightBigg (62) where R(2) n,m=δEm/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 12/summationdisplay i,j=1fn,¯ı(x2)fm,i(x2)fm,j(x1)fn,¯(x1)Aij κ(x2, x1, Em) (63) and R(1) n,m=−δEm/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 12/summationdisplay i,j=1fn,¯ı(x2)fm,i(x2)fm,j(x1)fn,¯(x1)Bij κ(x2, x1, Em) −/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 12/summationdisplay i,j=1fn,¯ı(x2)/bracketleftBig δfm,i(x2)fm,j(x1) +fm,i(x2)δfm,j(x1)/bracketrightBig fn,¯(x1)Aij κ(x2, x1, Em) (64) E. Numerical evaluation of the ﬁrst-order correction to the Green’s function for the low-energy part Numerical evaluation of the Coulomb Green’s functions in th is paper is based on the explicit formulas given, for example, in Eqs. (A.16) and (A. 17) of Ref. [9], together with the numerical algorithms described in Ref. [15]. The ﬁrst-o rder correction to the Green’s function in Eq. (16), for the range of arguments relevant to t he low-energy part, is evaluated by numerical integration over x3, where the interval of integration is divided into four subi n- tervals: (0 , y1), (y1, y2), (y2, y3), and ( y3,∞). Deﬁning p= 2√ 1−z2,y0= 3, and assuming x2≤x1, we choose y1= min( y0, px2),y2= min(max( y0, px2), px1), and y3= max( y0, px1). In the ﬁrst interval we make the substitution x3=ξ3, and integrate over ξby Gauss-Legendre quadrature with 12 to 26 integration points. In the second an d third interval we also do Gauss-Legendre integration, with 17 to 33 and 11 to 21 points , respectively. We use 6 to 18 point Gauss-Laguerre integration for the remaining inte rval. The integrations over the 15second and fourth intervals are the least accurate. For a 2 selectron at Z= 20 the integral over (y1, y2) has an error of a few parts in 106in the worst case. F. Reduced Green’s Function In the preliminary version of this calculation [10], a purel y numerical method of evalu- ating the reduced Green’s function was employed. However, w hile that method is adequate at high Z, it gives unsatisfactory results for the 2s state when Z≤20, so a new method was developed that yields better precision. As a check of the coding of the later method, the results of the two methods were compared and are in agreem ent within a relative diﬀer- ence of 10−6over a wide range of the variables. Both methods are brieﬂy de scribed in the following subsections. 1. Evaluation of the Reduced Green’s Function by numerical p ole removal Eq. (48) can be written as Gij κ(x2, x1, En+ǫ) =−fn,i(x2)fn,j(x1) ǫ +GR,ij κ(x2, x1, En) +O(ǫ). (65) As an immediate consequence of this relation, we have 1 2[Gij κ(x2, x1, En+ǫ) +Gij κ(x2, x1, En−ǫ)] =GR,ij κ(x2, x1, En) +O(ǫ2), (66) so the reduced Green’s function can be easily obtained from t he full Green’s function by symmetric interpolation of the energy variable. We form a li near combination of two such interpolations in order to obtain a result with an error of or derǫ4rather than ǫ2. In particular, we have q 2[Gij κ(x2, x1, En+ǫ) +Gij κ(x2, x1, En−ǫ)] 16+(1−q) 2[Gij κ(x2, x1, En+ǫ′) +Gij κ(x2, x1, En−ǫ′)] =GR,ij κ(x2, x1, En) +C2/bracketleftBig q ǫ2+ (1−q)ǫ′2/bracketrightBig +C4/bracketleftBig q ǫ4+ (1−q)ǫ′4/bracketrightBig +O(ǫ6) +O(ǫ′6), (67) where the choice ǫ′=/parenleftBiggq q−1/parenrightBigg1 2 ǫ; q >1 (68) provides q ǫ2+ (1−q)ǫ′2= 0 (69) with the parameter qfree to vary in the range (1 ,∞). The choice q=4 3gives the Lagrange interpolation formula with equally spaced evaluation poin ts:ǫ′= 2ǫ. With the choice in Eq. (68), the fourth-order term is proportional to /vextendsingle/vextendsingle/vextendsingleq ǫ4+ (1−q)ǫ′4/vextendsingle/vextendsingle/vextendsingle=q q−1ǫ4. (70) The coeﬃcient of ǫ4is 4 for equally spaced points and approaches 1 as q→ ∞ . As a compromise between a large coeﬃcient for qnear 1 and the minimum coeﬃcient as q→ ∞, with a correspondingly larger roundoﬀ error, we employ the v alueq= 10. The interpolation interval is taken to be ǫ= 0.02 (En+1−En) where EnandEn+1are the Dirac eigenvalues for principal quantum number nandn+1 for the same κ. This interval avoids overlap with the nearest pole of the Green’s function. 2. Direct evaluation of the Reduced Green’s Function The method developed for the present work is similar to that o f Hylton [13,16], but diﬀers in the details of its implementation. From Eq. (65), i t is evident that we obtain the reduced Green’s function by expanding the various componen ts in the explicit expression for the radial Green’s function in powers of ǫ=z−Enand keeping only the ﬁnal combinations of terms that are of order 1. To implement this, at each step in the numerical evaluation 17of the reduced Green’s function we calculate only the coeﬃci ents of the leading two terms in the power series in ǫand discard the higher-order terms. In certain cases, it is n ecessary to begin with three terms in the expansions, because the lead ing term either vanishes or cancels an equal leading term in forming a diﬀerence. As sugg ested by the form of the following equations, many of the coeﬃcients follow from com binations of coeﬃcients that appear earlier in the calculation. The code for the numerical calculation was written by modify ing the existing code for the radial Green’s function described in Refs. [15] and [17], so only a few details that illustrate the approach are given here. We deﬁne the radial quantum numb ernr=n−\|κ\|. Expansions are needed for z: z=z0+z1ǫ+z2ǫ2+··· z0=En;z1= 1;z2= 0, (71) forc=√ 1−z2: c=c0+c1ǫ+c2ǫ2+··· c0=γEn nr+λ;c1=−nr+λ γ;c2=−1 2c3 0, (72) where γ=Zαandλ=√κ2−γ2, forw=γ/c: w=w0+w1ǫ+w2ǫ2+··· w0=γ c0;w1=−γc1 c2 0;w2=γ(c2 1−c0c2) c3 0, (73) and for ν=γz/c: ν=ν0+ν1ǫ+ν2ǫ2+··· ν0=z0w0;ν1=z0w1+z1w0; ν2=z0w2+z1w1. (74) Expressions that appear in the deﬁnitions of the radial Gree n’s functions include λ−ν: 18λ−ν=−nr−ν1ǫ−ν2ǫ2+··· =/braceleftBig−ν1ǫ−ν2ǫ2+···ifnr= 0 −nr−ν1ǫ+···ifnr/negationslash= 0(75) andΓ(λ−ν): Γ(λ−ν) =Γ(nr) −1ǫ−1+Γ(nr) 0+··· Γ(0) −1=−1 ν1;Γ(0) 0=ν2 ν2 1−γE Γ(nr) −1=−1 nrΓ(nr−1) −1 Γ(nr) 0=−1 nrΓ(nr−1) 0 +ν1 n2rΓ(nr−1) −1, (76) where γE= 0.577. . .is Euler’s constant. The recursion relations used to calcul ate the power series for the Whittaker functions are treated in a similar m anner. For example, for the power series evaluation of Mν+(1/2),λ(2cx) by means of the recursion relations in Eq. (D.2) of [15], we write T(m) =T0(m) +T1(m)ǫ+··· T0(0) = 1; T1(0) = 0 T0(m+ 1) =(m−nr)2c0x (m+ 2λ+ 1)(m+ 1)T0(m) T1(m+ 1) =(m−nr)2c0x (m+ 2λ+ 1)(m+ 1)T1(m) +[(m−nr)c1−ν1c0]2x (m+ 2λ+ 1)(m+ 1)T0(m) (77) The termination of the power series for the leading term in Eq . (77) at m=nrcorresponds to fact that the leading term in Eq. (65) is proportional to th e bound-state wave function. The calculation of the reduced Green’s function in this work is based on the application of two-term expansions in ǫ, as described above, to the numerical evaluation of the comp lete Green’s function (see Appendix D of Ref. [15]). The numerica l value of the reduced Green’s function is just the collection of terms with combined order 1 inǫ. 19G. Numerical evaluation of the ﬁrst-order correction to the wave function The ﬁrst-order correction to the wave function, given by Eq. (12), is evaluated with the aid of the reduced Green’s function as described in Sec. IIIF . The numerical integration in Eq. (12) is divided into 3 segments: δfn,i(x) =1 pn/bracketleftBigg 5/integraldisplay1 0du x 1u4/parenleftBig x1u5/pn/parenrightBig22/summationdisplay j=1GR,ij κ(x, x1u5/pn, En)δV(x1u5/pn)fn,j(x1u5/pn) +/integraldisplayx2 x1dy(y/pn)22/summationdisplay j=1GR,ij κ(x, y/p n, En)δV(y/pn)fn,j(y/pn) +/integraldisplay∞ x2dy(y/pn)22/summationdisplay j=1GR,ij κ(x, y/p n, En)δV(y/pn)fn,j(y/pn)/bracketrightBigg , (78) where pn= 2/radicalBig 1−E2nis the coeﬃcient of the argument in the exponent that governs the behavior of the integrand for large values of the argument, x1= min ( xp,3), and x2= max (xp,3). The ﬁrst and second integrals are evaluated by means of 30 point Gauss- Legendre quadrature and the third is evaluated with 15 point Gauss-Laguerre quadrature. The accuracy of the perturbed wave function calculated acco rding to Eq. (78) has been tested in the Coulomb case by comparison to the result obtained by nu merical diﬀerentiation of the Coulomb wave function. For values of the argument of the ﬁrst-order correction to th e wave function near the origin, we found that greater numerical accuracy and speed c ould be obtained with a nu- merical evaluation based on the expansion in powers of x. This expansion is described in Appendix B. IV. HIGH ENERGY PART The high-energy part, given by the integral in Eq. (2), must b e regularized, since it is formally inﬁnite. We employ the Pauli-Villars regularizat ion scheme, following the method of Refs. [10,12] to isolate and remove the divergent contribut ions. In Ref. [12] we demonstrated that suitable numerical convergence can be achieved throug h the use of a term-by-term 20subtraction method. This particular method has the advanta ge that it does not require a mix of coordinate-space and momentum-space calculations a s do earlier methods, but works entirely within coordinate space. In this method, the high- energy part EH, given by Eq. (2), is separated into two parts: EHAandEHB. The divergences are all contained in EHAand can be calculated completely analytically, while EHBis ﬁnite and is treated numerically. In this section we describe the method used to evaluate δEHB, while the method to compute δEHA is discussed in Sec. V; the total is δEH=δEHA+δEHB. (79) . The high-energy remainder with term-by-term subtraction, from Eq. (32) in [12], is written as EHB=EH− EHA =α 2πi/integraldisplay CHdz/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1 ×/braceleftbigg∞/summationdisplay \|κ\|=1/bracketleftbigg Kκ(x2, x1, z)− K(0,0) κ(x2, x1, z)− K(0,1) κ(x2, x1, z)− K(1,0) κ(x2, x1, z)/bracketrightbigg − K(0,2) D(x2, x1, z)/bracerightbigg ,(80) with Kκ(x2, x1, z) =2/summationdisplay i,j=1[Fn,i(x2)Gij κ(x2, x1, z)Fn,j(x1)Aκ(x2, x1)− F n,¯ı(x2)Gij κ(x2, x1, z)Fn,¯(x1)Aij κ(x2, x1)],(81) K(0,0) κ(x2, x1, z) =Aκ/braceleftBig F11 κ(x2, x1, z)/bracketleftBig F2 n,1(x2)−3F2 n,2(x2)/bracketrightBig +F22 −κ(x2, x1, z)/bracketleftBig F2 n,2(x2)−3F2 n,1(x2)/bracketrightBig/bracerightBig ,(82) K(0,1) κ(x2, x1, z) =/braceleftBigg F11 κ(x2, x1, z)/bracketleftBigg Fn,1(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1 +κn x2/parenrightBigg Fn,1(x2)− F n,2(x2)/parenleftBigg 3Bκ∂ ∂x2−Cκ1−κn x2/parenrightBigg Fn,2(x2)/bracketrightBigg +F22 −κ(x2, x1, z)/bracketleftBigg Fn,2(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1−κn x2/parenrightBigg Fn,2(x2)− F n,1(x2)/parenleftBigg 3Bκ∂ ∂x2−Cκ1 +κn x2/parenrightBigg Fn,1(x2)/bracketrightBigg +2F12 −κ(x2, x1, z)Fn,1(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1−κn x2/parenrightBigg Fn,2(x2) +2F21 κ(x2, x1, z)Fn,2(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1 +κn x2/parenrightBigg Fn,1(x2)/bracerightBigg , (83) 21and K(1,0) κ(x2, x1, z) =−V(x2)Aκ/braceleftBiggd dzF11 κ(x2, x1, z)/bracketleftBig F2 n,1(x2)−3F2 n,2(x2)/bracketrightBig +d dzF22 −κ(x2, x1, z)/bracketleftBig F2 n,2(x2)−3F2 n,1(x2)/bracketrightBig/bracerightBigg . (84) In Eqs. (81) to (84) Aij κ,Aκ,BκandCκare integrals over coordinate directions deﬁned in Refs. [9,12,14,18], Fn,i(x) are radial components of the bound-state wave function as b efore, andκnis the spin-angular momentum quantum number of the bound sta ten. The ex- pressions for the free Green’s function radial components Fij κ(x2, x1, z) and their derivatives can be found in [12] and those of the Coulomb Green’s function can be found in [9]. The methods we used for summation over angular momentum κand for numerical integrations are identical to those described in [12] and will not be repea ted here. We also found that the convergence of the numerical integration was much bette r than in the case of the un- perturbed self-energy. We thus did not use the extra subtrac tion term K(0,2) D(x2, x1, z) that was necessary in [12] to obtain good convergence at low Z. The high-energy remainder for the self-energy screening is obtained from Eq. (80) as de scribed in Sec. II as the sum δφEHB+δGEHB+δEEHB. In the three following subsections we derive the expressio ns that are used to obtain δφEHB,δGEHB, and δEEHBfrom Eqs. (80) to (84). A. Wave function correction To obtain the expression for the high-energy remainder for t he wave function correction, we need the functional derivatives of Eqs. (81) to (84) with r espect to the radial wave functions Fn,i(x). For the full expression (81) we have Kφ,κ(x2, x1, z) =2/summationdisplay i,j=1/braceleftBig/bracketleftBig fn,i(x2)Gij κ(x2, x1, z)δfn,j(x1) +δfn,i(x2)Gij κ(x2, x1, z)fn,j(x1)/bracketrightBig Aκ(x2, x1) −/bracketleftBig fn,¯ı(x2)Gij κ(x2, x1, z)δfn,¯(x1) +δfn,¯ı(x2)Gij κ(x2, x1, z)fn,¯(x1)/bracketrightBig Aij κ(x2, x1)/bracerightBig ,(85) and for the subtraction terms, we obtain 22K(0,0) φ,κ(x2, x1, z) = 2Aκ/braceleftBig F11 κ(x2, x1, z) [fn,1(x2)δfn,1(x2)−3fn,2(x2)δfn,2(x2)] +F22 −κ(x2, x1, z) [fn,2(x2)δfn,2(x2)−3fn,1(x2)δfn,1(x2)]/bracerightBig ,(86) K(0,1) φ,κ(x2, x1, z) =/braceleftbigg F11 κ(x2, x1, z)/bracketleftbigg fn,1(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1 +κn x2/parenrightBigg δfn,1(x2)−fn,2(x2)/parenleftBigg 3Bκ∂ ∂x2−Cκ1−κn x2/parenrightBigg δfn,2(x2) +δfn,1(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1 +κn x2/parenrightBigg fn,1(x2)−δfn,2(x2)/parenleftBigg 3Bκ∂ ∂x2−Cκ1−κn x2/parenrightBigg fn,2(x2)/bracketrightbigg +F22 −κ(x2, x1, z)/bracketleftbigg fn,2(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1−κn x2/parenrightBigg δfn,2(x2)−fn,1(x2)/parenleftBigg 3Bκ∂ ∂x2−Cκ1 +κn x2/parenrightBigg δfn,1(x2) +δfn,2(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1−κn x2/parenrightBigg fn,2(x2)−δfn,1(x2)/parenleftBigg 3Bκ∂ ∂x2−Cκ1 +κn x2/parenrightBigg fn,1(x2)/bracketrightbigg +2F12 −κ(x2, x1, z)/bracketleftbigg fn,1(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1−κn x2/parenrightBigg δfn,2(x2) +δfn,1(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1−κn x2/parenrightBigg fn,2(x2)/bracketrightbigg +2F21 κ(x2, x1, z)/bracketleftbigg fn,2(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1 +κn x2/parenrightBigg δfn,1(x2) +δfn,2(x2)/parenleftBigg Bκ∂ ∂x2+Cκ1 +κn x2/parenrightBigg fn,1(x2)/bracketrightbigg/bracerightbigg and K(1,0) φ,κ(x2, x1, z) =−2V(x2)Aκ/braceleftBiggd dzF11 κ(x2, x1, z) [fn,1(x2)δfn,1(x2)−3fn,2(x2)δfn,2(x2)] +d dzF22 −κ(x2, x1, z) [fn,2(x2)δfn,2(x2)−3fn,1(x2)δfn,1(x2)]/bracerightBigg .(88) In terms of the expressions in Eqs. (85) to (88), the ﬁrst-ord er wave function correction to EHBis δφEHB=α 2πi/integraldisplay CHdz/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1 ×∞/summationdisplay \|κ\|=1/bracketleftbigg Kφ,κ(x2, x1, z)−K(0,0) φ,κ(x2, x1, z)−K(0,1) φ,κ(x2, x1, z)−K(1,0) φ,κ(x2, x1, z)/bracketrightbigg .(89) In order to evaluate the expression in (87), we need the deriv ative of the bound-state Dirac wave function and the derivative of its ﬁrst-order cor rection in the potential δV. The diﬀerential equations for the large and small components of the unperturbed wave function are (see, e.g., [10], Appendix A) d dxfn,1(x) =−1 +κn xfn,1(x) + [1 + En−V(x)]fn,2(x) 23d dxfn,2(x) = [1 −En+V(x)]fn,1(x)−1−κn xfn,2(x), (90) which yield the wave function derivatives from the analytic expressions for the wave function. We obtain analogous expressions for the perturbation of the wave-function components in the potential V(x) +δV(x). Retaining only ﬁrst-order terms in δV(x), we obtain d dxδfn,1(x) =−1 +κn xδfn,1(x) + [1 + En−V(x)]δfn,2(x) + [δEn−δV(x)]fn,2(x) d dxδfn,2(x) = [1 −En+V(x)]δfn,1(x)−1−κn xδfn,2(x) + [δV(x)−δEn]fn,1(x).(91) B. Green’s function correction We do the corresponding calculation for the high-energy ter m to account for variation of the Coulomb Green’s function under a change of the potential . Since the Coulomb Green’s function has no poles on the high energy integration contour (which lies on the imaginary axis), we may directly apply Eq. (16) to obtain KG,κ(x2, x1, z) =−2/summationdisplay i,j,k=1/integraldisplay∞ 0dx3x2 3δV(x3)/bracketleftbigg fn,i(x2)Gik κ(x2, x3, z)Gkj κ(x3, x1, z)fn,j(x1)Aκ(x2, x1) −fn,¯ı(x2)Gik κ(x2, x3, z)Gkj κ(x3, x1, z)fn,¯(x1)Aij κ(x2, x1)/bracketrightbigg .(92) OnlyK(1,0)contributes to the subtraction term, and we thus obtain from Eq. (84) K(1,0) G,κ(x2, x1, z) =−δV(x2)Aκ/braceleftbiggd dzF11 κ(x2, x1, z)/bracketleftBig f2 n,1(x2)−3f2 n,2(x2)/bracketrightBig +d dzF22 −κ(x2, x1, z)/bracketleftBig f2 n,2(x2)−3f2 n,1(x2)/bracketrightBig/bracerightbigg .(93) The Green’s function correction to EHBis thus δGEHB=α 2πi/integraldisplay CHdz/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1∞/summationdisplay \|κ\|=1/bracketleftbigg KG,κ(x2, x1, z)−K(1,0) G,κ(x2, x1, z)/bracketrightbigg (94) 24C. Numerical evaluation of the ﬁrst-order correction to the Green’s function for the high-energy part The evaluation of the the Green’s function correction to the high-energy part is the most diﬃcult and time consuming part of the present calculat ion because of the additional non-trivial integration over x3and the necessity of evaluating integrals over a product of two Coulomb Green’s functions in Eq. (92). The calculation i s facilitated somewhat by subtracting and adding an additional term that closely appr oximates the term in Eq. (92). We write /integraldisplay∞ 0dx3x2 3δV(x3)Gik κ(x2, x3, z)Gkj κ(x3, x1, z) =/integraldisplay∞ 0dx3x2 3/bracketleftbigg δV(x3)−1 2δV(x1)−1 2δV(x2)/bracketrightbigg Gik κ(x2, x3, z)Gkj κ(x3, x1, z) +1 2[δV(x1) +δV(x2)]/integraldisplay∞ 0dx3x2 3Gik κ(x2, x3, z)Gkj κ(x3, x1, z), (95) where (see Eq. (35) from Ref. [10]) 2/summationdisplay k=1/integraldisplay∞ 0dx3x2 3Gik κ(x2, x3, z)Gkj κ(x3, x1, z) =∂ ∂zGij κ(x2, x1, z). (96) We calculate separately the two terms in the sum δGEHB=δGE(1) HB+δGE(2) HB, where δGE(1) HB=α 2πi/integraldisplay CHdz/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1∞/summationdisplay \|κ\|=1/braceleftbigg −1 22/summationdisplay i,j=1[δV(x1) +δV(x2)] ×/bracketleftBigg fn,i(x2)∂ ∂zGij κ(x2, x1, z)fn,j(x1)Aκ(x2, x1)−fn,¯ı(x2)∂ ∂zGij κ(x2, x1, z)fn,¯(x1)Aij κ(x2, x1)/bracketrightBigg −K(1,0) G,κ(x2, x1, z)/bracerightbigg (97) and δGE(2) HB=α 2πi/integraldisplay CHdz/integraldisplay∞ 0dx2x2 2/integraldisplay∞ 0dx1x2 1/integraldisplay∞ 0dx3x2 3/bracketleftbigg −δV(x3) +1 2δV(x1) +1 2δV(x2)/bracketrightbigg ×∞/summationdisplay \|κ\|=12/summationdisplay i,j,k=1[fn,i(x2)Gik κ(x2, x3, z)Gkj κ(x3, x1, z)fn,j(x1)Aκ(x2, x1) −fn,¯ı(x2)Gik κ(x2, x3, z)Gkj κ(x3, x1, z)fn,¯(x1)Aij κ(x2, x1)]. (98) The derivative of the Green’s function in Eq. (97) is evaluat ed numerically with a two-point formula: 25∂ ∂zGij κ(x2, x1, z)≈Gij κ(x2, x1, z(1 +ǫ))−Gij κ(x2, x1, z(1−ǫ)) 2zǫ. (99) D. Energy correction We evaluate δEEHB, given by δEEHB=δEn∂ ∂EnEHB, (100) by numerical diﬀerentiation of EHB. We employ an eﬀective inﬁnitesimal displacement of the energy variable obtained by making the replacement of En(Zα) byEn[(Z±ǫ)α], with ǫ= 0.002. The symmetric derivative formula with this displaceme nt has an uncertainty of a few parts in 106. One expects that numerical integration errors, as discuss ed in [12], are slowly varying functions of Zand largely cancel in forming the derivative, leading to an accurate result for the derivative. Evidently, however, th e error in terminating the sum over κdoes not cancel as strongly, which may lead to signiﬁcant unc ertainties for some Zandnℓ. We have thus found it preferable to diﬀerentiate directly th e running term in the sum over angular momenta in Eq. (80). To study this eﬀect, we compare n umerically∂ ∂EnSA(r, y, u) evaluated by diﬀerentiation of the sum ∂ ∂EnSA(r, y, u) =∂ ∂En∞/summationdisplay \|κ\|=1/bracketleftbigg Kφ,κ(ry, y, iu ) −K(0,0) φ,κ(ry, y, iu )−K(0,1) φ,κ(ry, y, iu ) −K(1,0) φ,κ(ry, y, iu )/bracketrightbigg (101) to a sum of derivatives (term-by-term diﬀerentiation) ∂ ∂EnSA(r, y, u) =∞/summationdisplay \|κ\|=1∂ ∂En/bracketleftbigg Kφ,κ(ry, y, iu ) −K(0,0) φ,κ(ry, y, iu )−K(0,1) φ,κ(ry, y, iu ) −K(1,0) φ,κ(ry, y, iu )/bracketrightbigg (102) where u= (1/2)(1/t−t) and K(i,j) φ,κare deﬁned in Eqs. (85) to (88). The summation is terminated when the remainder estimate as described in Ref. [12] is smaller than a predeﬁned 26cutoﬀ value. Although the two methods of calculation conver ge to the same value for a very small cutoﬀ (10−10), the results with a larger cutoﬀ can disagree by a few parts i n 10−4. Since very small cutoﬀ values would lead to other problems wh en r is close to 1, we evaluate the energy derivative by the term-by-term method. Other ill ustrations of this problem are discussed in Sec. VIA. V. ANALYTIC TERMS In this section we evaluate analytically the terms that are s ubtracted in the numerical calculation described in Sec. IV for each of the three contri butions to the screened self-energy. In Refs. [10,12], with this application in mind, we were care ful not to obscure the origins ofVandEnthrough the use of the Dirac equation. The individual terms c orresponding to the three diagrams of Fig. 3 are thus obtained by direct diﬀer entiation. A. Wave-function correction The analytic portion of the wave-function correction follo ws from Refs. [10,12]. In each of the terms E(i,j) H, we calculate the variation with respect to a change in the wa ve function based on the explicit dependence on the wave function, and ob tain δφE(0,0) H=2α π/angb∇acketleftφn\|β\|δφn/angb∇acket∇ight/bracketleftBigg ln(Λ2)−1 +1−E2 n E2nln/parenleftBig 1 +E2 n/parenrightBig +O(Λ−1)/bracketrightBigg , (103) δφE(0,1) H=2α π/angb∇acketleftφn\|α·p\|δφn/angb∇acket∇ight/bracketleftbigg1 4ln(Λ2)−6−3E2 n+ 7E4 n 24E2 n(1 +E2 n)+1−E4 n 4E4 nln/parenleftBig 1 +E2 n/parenrightBig +O(Λ−1)/bracketrightbigg , (104) and δφE(1,0) H=2α π/braceleftBigg /angb∇acketleftφn\|V\|δφn/angb∇acket∇ight/bracketleftBigg1 4ln(Λ2) +6−E2 n 8E2 n−3 +E4 n 4E4 nln/parenleftBig 1 +E2 n/parenrightBig/bracketrightBigg − /angb∇acketleftφn\|βV\|δφn/angb∇acket∇ight/bracketleftBigg2 En−2 E3 nln/parenleftBig 1 +E2 n/parenrightBig/bracketrightBigg +O(Λ−1)/bracerightBigg . (105) 27In analogy with [10,12], we deﬁne δφEHA= lim Λ→∞/bracketleftbigg δφE(0,0) H+δφE(0,1) H +δφE(1,0) H−2δm(Λ)/angb∇acketleftφn\|β\|δφn/angb∇acket∇ight/bracketrightbigg , (106) where the last term is the renormalization term and δm(Λ) =α π/bracketleftbigg3 4ln(Λ2) +3 8/bracketrightbigg . (107) If we combine the coeﬃcients of ln( Λ2) in Eq. (106), we obtain α 2π/angb∇acketleftφn\|β+α·p+V\|δφn/angb∇acket∇ight=α 2πEn/angb∇acketleftφn\|δφn/angb∇acket∇ight= 0 (108) from the diﬀerential equation for φnand the fact that the ﬁrst-order correction to the wave function is orthogonal to the unperturbed wave functio n. Hence, the wave function correction is separately ﬁnite. B. Energy correction This correction is obtained by diﬀerentiating all terms wit h respect to En. The three terms are δEE(0,0) H=α πδEn/bracketleftBigg /angb∇acketleftφn\|β\|φn/angb∇acket∇ight/parenleftBigg2 En1−E2 n 1 +E2 n−2 E3 nln/parenleftBig 1 +E2 n/parenrightBig/parenrightBigg −1 4ln(Λ2)−6−E2 n 8E2n+3 +E4 n 4E4nln/parenleftBig 1 +E2 n/parenrightBig +O(Λ−1)/bracketrightBigg , (109) δEE(0,1) H=α πδEn/angb∇acketleftφn\|α·p\|φn/angb∇acket∇ight/bracketleftbigg6 + 9E2 n−8E4 n−3E6 n 6E3n(1 +E2n)2−1 E5 nlog(1 + E2 n) +O(Λ−1)/bracketrightbigg ,(110) and δEE(1,0) H=α πδEn/braceleftBigg /angb∇acketleftφn\|V\|φn/angb∇acket∇ight/bracketleftBigg −6 + 3E2 n+E4 n 2E3n(1 +E2n)+3 E5nlog(1 + E2 n)/bracketrightBigg +/angb∇acketleftφn\|βV\|φn/angb∇acket∇ight/bracketleftBigg6 + 2E2 n E2 n(1 +E2 n)−6 E4 nlog(1 + E2 n)/bracketrightBigg +O(Λ−1)/bracerightBigg . (111) 28The total is δEEHA=δEE(0,0) H+δEE(0,1) H+δEE(1,0) H, (112) which contains a divergent term given by δEEHA=−α δE n 4πln(Λ2) +O(1). (113) C. Green’s function correction This term arises entirely from E(1,0) H, which is linear in the potential in the Green’s function. Taking into account the fact that /angb∇acketleftφn\|δV\|φn/angb∇acket∇ight=δEn, we obtain δGEHA=δGE(1,0) H=α π/braceleftbigg δEn/bracketleftbigg1 4ln(Λ2) +6−E2 n 8E2n−3 +E4 n 4E4nln/parenleftBig 1 +E2 n/parenrightBig/bracketrightbigg −/angb∇acketleftφn\|βδV\|φn/angb∇acket∇ight/bracketleftBigg2 En−2 E3 nln/parenleftBig 1 +E2 n/parenrightBig/bracketrightBigg +O(Λ−1)/bracerightbigg/bracketrightbigg . (114) This expression also contains a divergent contribution δGEHA=α δE n 4πln(Λ2) +O(1), (115) which cancels the corresponding term in Eq. (113). The total expression δEHA=δφEHA+δEEHA+δGEHA (116) is thus ﬁnite as expected. VI. RESULTS AND DISCUSSION A. Coulomb tests In order to check the accuracy of the numerical calculation, the equations derived in the previous sections, and the parallelized code, we compar e the calculations obtained by numerical diﬀerentiation of the one-electron self-energy function F(Zα) to the results of 29the method presented in this paper using δV(r) =−ǫ α/r as a perturbing potential. This method, proposed in Ref. [1], is very eﬃcient, as each indivi dual contribution to the screened self-energy can be checked independently. The overall agre ement between the results of these two methods of calculation is good, although diﬀerences bet ween some contributions can be several times the combined uncertainties based only on the a pparent convergence of the numerical integration. These additional errors come from t he numerical problems described in Sec. IVD. An illustrative example is the wave function cor rection. One can compare the term-by term derivative with respect to Zin the wave function to the derivative obtained from two converged sums, as is done in Eqs. (101) and (102). Th ere is also an independent calculation based on the ﬁrst-order correction to the wave f unction from Eq. (9). The evolution of this sum for smaller and smaller values of the cu toﬀ error are displayed in Fig. 4. One can see that although the two calculations conver ge to the same limit when the cutoﬀ is as small as 10−15, the results follow very diﬀerent paths. Only the term-by-t erm diﬀerentiation method follows the result obtained from eva luation of the ﬁrst-order correction to the wave function independently of the cutoﬀ. This consti tutes a very demanding test of our numerical evaluation of the ﬁrst-order correction to th e wave function. The diﬀerence between the two calculations is never smaller than 8 ×108, which is the error from numerical uncertainties of the full Green’s function and the error in t he numerical derivative. To improve the numerical precision, we have employed a subdi vision of the integration overrinto regions with 0 < r < 0.4 and with 0 .4< r < 1, as described in Ref. [12]. This division provides an accurate evaluation that does not requ ire functional evaluations with values of rtoo close to 1. In this way we where able to obtain an accurate c omparison of all contributions in the high-energy part. A few such problems remain in the test calculations in the low -energy part at low Z. We did not attempt to improve the accuracy, because we have enou gh accurate cases to check the code and numerical procedures. From the tests we have per formed, it is clear that the calculation based on numerical diﬀerentiation is the less a ccurate. However since numerical diﬀerentiation is used in the ﬁnal result to obtain the reduc ible correction we have increased 30the total uncertainty of both the pure Coulomb test and spher ically-averaged potential of the next section accordingly. B. Results with spherically-averaged one-electron potent ial The calculations of interest for physical applications are based on realistic potentials obtained from the spherically-averaged potential of Eq. (C 2) of Appendix C. All results presented here are given in terms of the scaled function deﬁn ed by Enℓj,n′ℓ′j′=α2(Zα)3 πn3Fnℓj,n′ℓ′j′(Zα)mc2(117) All 16 possible total scaled functions F(Zα) for the self-energy screening of nℓelectron by an′ℓ′electron, 1 ≤n, n′≤2, 0≤ℓ, ℓ′≤1 are given in Table II and in Figs. 5 to 8. It can be seen that the uncertainty at low Zcan be as high as 30% for the screening of 2 pelectrons atZ= 10, or as low as 10−6. These functions can be used to evaluate the self-energy screening correction to any atom with two to ten electrons, i n a shell n≤2. As an example we treat the case of lithiumlike uranium. With the results pr esented here we can compute the self-energy screening correction for all three states 1 s22s, 1s22p1/2and 1s22p3/2. A ﬁrst approximation is obtained for transition energies by negle cting the core relaxation. The self-energy screening correction to 2 pj→2stransition energy is evaluated using Eq. (117) as ∆Ej= 2E2pj,1s−2E2s,1s, where the factor of two accounts for the fact that there are t wo 1selectrons screening n= 2 electron. A better approximation, which takes into accou nt the relaxation of the core electrons, and provides a value for th e total binding energy is given byE1s22s= 2(E2s,1s+E1s,2s+E1s,1s) and E1s22pj= 2(E2pj,1s+E1s,2pj+E1s,1s). The results of these calculations are presented in Table IV together with a ll other calculations known to date. It should be noted that the present method is equivalent to th e Coulomb approximation in certain cases. If one considers only the Coulomb contribution in the interaction between the two electrons in Fig. 2, the present method provides the e quivalent contribution to the 31ground state of two-electron ions. This follows from the fac t that between sstates, only the monopole part of the 1 /r12operator contributes. This radial contribution of the mono pole part is exactly given by the potential in Eq. (B10). Moreover the retarded part of the Coulomb interaction vanishes in this case. Finally, the exc hange correction only involves the spin of the two electrons and corresponds to a multiplicatio n of the function F1s,1s(Zα) by two. The above arguments can be extended for all cases where an ele ctron of arbitrary quan- tum number interacts with a selectron. The selectron couples only to the monopole term in the angular expansion of the Coulomb interaction. If, how ever, the electrons are not identical, then there is an exchange term with additional mu ltipole terms. In this case, the retardation contribution to the Coulomb interaction is also non-zero. Obviously in the relativistic case the magnetic part of the electron-electr on interaction should be considered. Because of these considerations we can compare our results t o the Coulomb part of the calculation done by the G¨ oteborg Group. In Fig. 5 we plot also the function F(Zα) from Refs. [3,19]. The diﬀerence between the two calculatio ns is displayed on Fig. 9. The agreement is very good for medium- Z, while the diﬀerence between the two calculations increases with increasing Z, which is due to the inclusion of ﬁnite nuclear size in Refs. [ 3,19], while the present results are for a point nucleus. This compa rison thus provides the ﬁnite nuclear size eﬀect on the two-electron self-energy. The diﬀ erence at low Z(5 and 10) are due to numerical inaccuracies. Since no uncertainties are g iven in Ref. [19], which contains more accurate values than Ref. [3], we assume an uncertainty of 1 in the last digit (note thatF(Zα) in Ref. [19] is two times ours.) The low- Zbehavior of the self-energy correction for to the Coulomb in teraction to F1s,1s(Zα) is known from the work of Araki [22] and Sucher [23], and 1 /Zexpansions from Drake [25] to be FCoul 1s,1s(Zα) =1 2/bracketleftbigg 2.588819 +/parenleftbigg7 2−2 log 2/parenrightbigg log/parenleftBig 2 (Zα)2/parenrightBig/bracketrightbigg , (118) 32while the magnetic part is Fmag 1s,1s(Zα) =1 8(119) We cannot directly compare our value for F1s,1s(Zα) with those of Yerokhin et al. [2,4], because their results also include the magnetic and retarda tion contribution to the self energy and our model does not. This correction contributes e ven at very low Zsince it contains the free-electron anomalous magnetic moment from the vertex correction. The diﬀerence between the total contribution (for point nucleu s) from Ref. [4] and the present work is plotted on Fig. 10 for 20 ≥Z. Evidently, the magnetic interaction contribution to the two-electron self-energy is much larger than the contri bution of the ﬁnite nuclear size. A simple ﬁt of the diﬀerence between the present result and the one in Ref. [4] with a second- order polynomial yields 0 .124 for the contribution of the anomalous magnetic moment, i n good agreement with the value in Eq. (119). From the ﬁgure, it is evident that for Zas low as 5 the higher-order terms still make a signiﬁcant contr ibution. On the same ﬁgure we also plot the Breit contribution from Ref. [19], which is i n agreement with the diﬀerence between Yerokhin et al. and the present work for Z≥20, and matches reasonably well the extrapolated values even down to Z= 1. VII. CONCLUSION In this paper we describe a method of approximately evaluati ng two-electron radiative corrections that can easily be generalized to the direct eva luation of the correction repre- sented by the diagrams in Fig. 2. Accuracy and correctness of the method and programs is assessed by extensive comparisons with numerical deriva tives of well-known one-electron self-energy results for a Coulomb perturbation. It is demon strated that the method can work down to Z= 5 in some cases with reasonable accuracy. With the use of a mo re accurate Green’s function evaluation and convergence acceleration techniques, following Refs. [20,21], it is likely that calculation can be performed for He. The res ults presented in the present 33paper also provides approximate self-energy screening cor rections in any ion with less than 10 electrons, thus providing a valuable, QED-based replace ment for methods based on the Welton approximation [26,27] or other, less eﬃcient, scree ning schemes as used in atomic structure codes. It is also equivalent to the direct Coulomb contribution for some states of helium-like ions. This method could also be used with numerical Dirac-Fock pot entials and wave functions from one of the codes in Refs. [28,29]. Preliminary tests sho w that good numerical accu- racy can be achieved. Such an approach would provide more acc urate self-energy screening corrections for the outer shells of very heavy transuranic e lements or for inner hole binding energies [30]. ACKNOWLEDGMENTS The numerical calculations presented here have been made po ssible by a generous com- puter time allocation on the IBM SP2 at the Centre National Un iversitaire Sud de Calcul (Montpellier, France). Some of the calculations and comput er program development were done on the NIST SP2. APPENDIX A: NUMERICAL TEST As a consistency check on the computer code, we carry out a tes t calculation in which the correction terms are generated by numerical diﬀerentia tion of the unperturbed Coulomb self energy with respect to the nuclear charge Z. This should give the same result as the screening calculation where both the unperturbed potentia l and the perturbing potential are the Coulomb potential, with an appropriate normalizati on factor. In other words, we consider the potential V(x) =−(Z+ ∆Z)α x(A1) and let 34V(x) =V0(x) +δV(x) (A2) where V0(x) =−Zα x δV(x) =−∆Zα x(A3) If the level shift E(Z) is known as a function of Zfor the Coulomb potential, then the exact correction due to δV(x) isE(Z+ ∆Z)−E(Z) and the ﬁrst-order correction in ∆ Zis δE(Z) = ∆ Zlim δZ→0E(Z+δZ)−E(Z) δZ = ∆Z∂ ∂ZE(Z)≡∆ZE′(Z) (A4) Thus the ﬁrst-order perturbation due to the potential δV(x), with unit charge shift ∆ Z= 1 should be exactly equal to the derivative with respect to Zof the Coulomb level shift E′(Z). APPENDIX B: ORIGIN EXPANSION OF THE FIRST-ORDER CORRECTION TO THE WAVE FUNCTION This origin expansion is made with the use of the diﬀerential equation for the ﬁrst-order correction to the wave function in Eq. (91). However this exp ansion has a diﬀerent form depending on whether one uses a Coulomb perturbing potentia l or a potential created by an other electron. 1. Coulomb perturbation potential In the case of a Coulomb perturbing potential, the correctio n to the wave function must have a logarithmic contribution, so as to cancel the α/xterm in the lowest order of the development. We write δfi(x) =αxω/bracketleftbigg∞/summationdisplay j=0ζ(j) ixj j!+ log( x)∞/summationdisplay j=0λ(j) ixj j!/bracketrightbigg , (B1) 35and replace in Eq. (91), together with a series expansion of t he unperturbed wave function, which behaves as ωnear the origin. We then extract coeﬃcients of xjlog(x) and of xjand solve for the ζ(j) iandλ(j) icoeﬃcients. The coeﬃcient of the log( x)/xterm for an unperturbed wave function of angular symmetry κnis   (−κn−λ)λ(1) 0+γλ(2) 0= 0 −γλ(1) 0+ (κn−λ)λ(2) 0= 0(B2) where the unperturbed wave function origin behavior is give n byω=λ−1,λ=/radicalBig κ2n−γ2, γ=Zα. The determinant of this equation is zero and thus we can writ e λ(1) 0=κn−λ γλ(2) 0. (B3) The general equation for the term of order iis   (−κn−λ−i)λ(1) i+γλ(2) i= (1 + En)λ(2) i−1 −γλ(1) i+ (κn−λ−i)λ(2) i= (1−En)λ(1) i−1(B4) The determinant of the linear system in Eq. (B4) is given by (2 λ+i)iand is nonzero for i >0. By solving order after order, all higher-order terms can b e expressed as a function of λ(2) 0,γandκn. The expressions are all relatively simple since the unpert urbed wave function does not have a logarithmic contribution. The non-logarith mic terms are obtained by ﬁrst deﬁning a series expansion for the unperturbed wave functio n fi(x) = (−1)i−1Nixω/bracketleftbigg∞/summationdisplay j=0ϕ(j) ixj j!/bracketrightbigg , (B5) where i= 1, 2 and Niis a normalization factor. The equation derived from the ter m of order 1 /xis given by   (−κn−λ)ζ(1) 0+γζ(2) 0=N2ϕ(0) 2+λ−κn γλ(2) 0 −γζ(1) 0+ (κn−λ)ζ(2) 0=N1ϕ(0) 1−λ(2) 0, (B6) where we have used Eq. (B3), and which again has a zero determi nant. Explicit expressions of the n= 1 and n= 2 wave functions can be found in Ref. [18,15] (Note that in Re f. [18] 36the norm N3of the 2 p3/2wave function in Eq. (A3) should be N3=γ5−2δ′/[2Γ(5−2δ′)]). Requiring the compatibility of the two equations enables to calculate λ(2) 0as λ(2) 0=(κn+λ)N1ϕ(0) 1−γN2ϕ(0) 2 2λ. (B7) One can then obtain a relation between ζ(2) 0andζ(1) 0using one of the two equations in (B6) ζ(1) 0=γ2ζ(1) 0+ (κn−λ)λ(0) 2−γN2ϕ(0) 2 γ(κn+λ). (B8) Allζcoeﬃcients can thus be expressed as a function of ζ(2) 0. These coeﬃcients must be determined from the normalization condition of the perturb ed wave function. This obliges to explicitly write ζ(i) n,n >0, as a function of ζ(2) 0rather than keeping them as function of ζ(i) n−1. The latter expressions are simpler, but the former are very large. We use Mathematica to build the equations obeyed by the ζ(i) nanλ(i) ncoeﬃcients, evaluate the explicit expressions ofζ(i) n, extracting the part which depends on ζ(i) 0and the one which doesn’t, and generating FORTRAN code. The code can have hundreds of lines for each pie ce of ζ(i) 4. The ﬁnal expression of dfcan ﬁnally be recast as   δf1(x) =αxω/bracketleftbigg ζ(2) 0/summationtext∞ j=0ζ(j) 1,axj j!+/summationtext∞ j=0ζ(j) 1,bxj j!+ log( x)/summationtext∞ j=0λ(j) 1xj j!/bracketrightbigg δf2(x) =αxω/bracketleftbigg ζ(2) 0/parenleftBig 1 +/summationtext∞ j=1ζ(j) 2,axj j!/parenrightBig +/summationtext∞ j=0ζ(j) i,b+ log( x)/summationtext∞ j=0λ(j) 2xj j!/bracketrightbigg (B9) A comparison for a small value of xof the expansion and of the value obtained by the use of the reduced Green’s function as described in Sec. IIIG yield two values of ζ(2) 0, one for each component of the wave function. A comparison of the two value s provide a good check of the algebra. We compute ζ(j) iandλ(j) icoeﬃcients up to j= 3. The value of ζ(2) 0obtained from each component of the wave function at x= 0.0005 agree with an accuracy of 13 signiﬁcant ﬁgures, for n= 1 and n= 2,κn=−1, 1,−2. 2. Electron screening potential The screening potential described in Eq. (C2) leads to an ori gin expansion rather diﬀerent than the one described in the preceding section. In order to e valuate the origin expansion of 37the ﬁrst-order wave function correction we ﬁrst evaluate th e screening potential expansion. Using the origin expansion of the wave function and Eq. (C2), one can easily show that δV(j)(x) =δV(j)(0) +x2λ(j)/parenleftBigg∞/summationdisplay n=0δV(j) nxn n!/parenrightBigg (B10) where λ(j)=/radicalBig κ2 (j)−γ2(the origin behavior of the screening wave function is λ(j)−1) and δV(j)(0) = α/integraldisplay∞ 0dxx/bracketleftBig f2 1(x) +f2 2(x)/bracketrightBig . (B11) The asymptotic expansion is obtained by substituting Eq. (B 10) in the Poisson equation obeyed by the potential d dx2(xδV(j)(x)) +αx/bracketleftBig f2 1(x) +f2 2(x)/bracketrightBig = 0 (B12) and expanding the two radial component of the wave function i n powers of x. With such an expansion the shape of the origin expansion of th e ﬁrst order correction to the wave function is δfi(x) =αxλ/bracketleftbiggζ(−1) i x+∞/summationdisplay j=0ζ(j) ixj j! +x2λ(j)∞/summationdisplay j=0λ(j) ixj j!/bracketrightbigg . (B13) We obtain the equation for λ(j) 0by looking at the coeﬃcients of x2λ(j)+λ. We get for the equation of order 1 /x:   (−1−2λ(j)−κn−λ)λ(1) 0+γλ(2) 0=−N2ϕ(0) 2δV(j) 0 −γλ(1) 0+ (−1−2λ(j)+κn−λ)λ(2) 0=−N1ϕ(0) 1δV(j) 0(B14) We note that in this case this equation in inhomogeneous and h as a non-zero determinant. The equation for ζ(i) 0is obtained from the term of order 1 /x2as   (−κn−λ)ζ(1) 0+γζ(2) 0= 0 −γζ(1) 0+ (κn−λ)ζ(2) 0= 0, (B15) 38APPENDIX C: MODEL POTENTIALS One of the models considered here for a screening potential i s the spherically averaged potential that arises from the charge distribution of anoth er electron in state jin the atom: δV(j)(x2) =1 4π/integraldisplay dΩ2/integraldisplay dx1α \|x2−x1\|\|φj(x1)\|2 =α/integraldisplay∞ 0dx1x2 1 max(x2, x1) ×/bracketleftBig f2 1,(j)(x1) +f2 2,(j)(x1)/bracketrightBig (C1) To facilitate numerical integration, this equation is writ ten as δV(j)(x2) =/angbracketleftbiggα x/angbracketrightbigg −α/integraldisplayx2 0dx1x2 1/parenleftbigg1 x1−1 x2/parenrightbigg ×/bracketleftBig f2 1,(j)(x1) +f2 2,(j)(x1)/bracketrightBig (C2) forx2< x0, or as δV(j)(x2) =α x2−α/integraldisplay∞ x2dx1x2 1/parenleftbigg1 x2−1 x1/parenrightbigg ×/bracketleftBig f2 1,(j)(x1) +f2 2,(j)(x1)/bracketrightBig (C3) forx2> x0, for a suitable value of x0. The expectation value in (C2) is evaluated with the aid of the identity /angbracketleftbiggα x/angbracketrightbigg =−∂En ∂Z(C4) where Enis the energy eigenvalue of the screening wave function. The crossover point is taken to be x0= 2//radicalBig 1−E2 n. The integral in (C2) is evaluated by 20 point Gauss Legendre quadrature with a new integration va riable tover the range (0 ,1) deﬁned by x1=x2t4, and the integral in (C3) is evaluated by 25 Gauss Laguerre qu adrature with a new integration variable sover the range (0 ,∞) where x1=x2+s/(2/radicalBig 1−E2n). This prescription gives a precision of better than one part in 1012for the range 1 ≤Z≤100, as determined by comparing results of the two methods of integr ation in (C2) and (C3) . The corresponding derivatives are 39d dx2δV(j)(x2) =−α x2 2/integraldisplayx2 0dx1x2 1 ×/bracketleftBig f2 1,(j)(x1) +f2 2,(j)(x1)/bracketrightBig (C5) forx2< x0, or d dx2δV(j)(x2) =−α x2 2 ×/parenleftbigg 1−/integraldisplay∞ x2dx1x2 1/bracketleftBig f2 1,(j)(x1) +f2 2,(j)(x1)/bracketrightBig/parenrightbigg (C6) forx2> x0. The derivatives are calculated with the same integration me thods as the described above for the function δV(j)(x). A simple additional model potential, useful for testing cod e, is generated by employing an exponential charge distribution, which corresponds to t he replacement /bracketleftBig f2 1,(j)(x1) +f2 2,(j)(x1)/bracketrightBig →4γ3e−2γx, (C7) in Eqs. (C2), (C3), (C5), and (C6) and leads to the analytic po tential δVexp(x) =α x−α/parenleftbigg1 x+γ/parenrightbigg e−2γx(C8) with /angbracketleftbiggα x/angbracketrightbigg =αγ (C9) and the derivative d dxδVexp(x) =−α x2+α/parenleftbigg1 x2+2γ x+ 2γ2/parenrightbigg e−2γx(C10) 40TABLES TABLE I. Comparison between the present direct calculation (Dir.) with δV(r) =α/rand a calculation using numerical derivative of the one-electro n self-energy (Num. Der.). Comparison is done for the scaled function F(Zα) . 1s Z Num. Der. Dir. 20 -11.1383 (3) -11.1384 (8) 50 -6.35896 (5) -6.3589 (2) 90 -6.05030 (3) -6.0502 (5) 2s Z Num. Der. Dir. 20 -12.2267 (8) -12.227 (1) 50 -8.0243 (1) -8.0243 (3) 90 -10.03340 (6) -10.0334 (6) 2p1/2 Z Num. Der. Dir. 20 0.3190 (8) 0.319 (6) 50 -0.2444 (1) -0.2445 (8) 90 -2.30603 (4) -2.30607 (7) 2p3/2 Z Num. Der. Dir. 20 -0.606 (1) -0.61 (1) 50 -0.9056 (1) -0.906 (2) 90 -1.37474 (3) -1.3748 (3) 41TABLE II. Self-energy screening function f(Zα) for 1 s, 2s, 2p1/2and 2p3/2electrons F(Zα) for 1 sscreened by Z 1s 2s 2p1/2 2p3/2 5 -5.171 (5) -0.491 (2) -0.243 (2) -0.242 (2) 10 -3.882 (1) -0.3681 (4) -0.1832 (5) -0.1809 (5) 18 -2.901 (1) -0.27670 (1) -0.14115 (1) -0.13555 (1) 20 -2.7386 (3) -0.26184 (10) -0.1345 (1) -0.1279 (1) 24 -2.4726 (3) -0.23772 (7) -0.12394 (8) -0.11536 (8) 30 -2.1726 (1) -0.21111 (4) -0.11286 (5) -0.10092 (5) 32 -2.0917 (1) -0.20410 (4) -0.11010 (5) -0.09696 (5) 40 -1.83507 (8) -0.18267 (3) -0.10245 (3) -0.08403 (3) 44 -1.73770 (6) -0.17506 (2) -0.10028 (3) -0.07891 (2) 50 -1.62045 (5) -0.16663 (2) -0.09870 (2) -0.07243 (2) 54 -1.55825 (4) -0.16272 (1) -0.09867 (2) -0.06875 (2) 60 -1.48502 (4) -0.15912 (1) -0.10011 (1) -0.06402 (1) 66 -1.43296 (4) -0.15809 (1) -0.10334 (1) -0.06003 (1) 70 -1.40906 (3) -0.158821 (9) -0.10661 (1) -0.057727 (10) 74 -1.3935 (2) -0.160737 (9) -0.11085 (1) -0.055657 (9) 80 -1.38620 (2) -0.166065 (8) -0.119409 (9) -0.052952 (8) 83 -1.39010 (7) -0.16998 (1) -0.124865 (10) -0.051754 (7) 90 -1.4214 (2) -0.18313 (4) -0.14160 (4) -0.04933 (2) 92 -1.43690 (2) -0.188136 (6) -0.147674 (8) -0.048726 (6) F(Zα) for 2 sscreened by 1s 2s 2p1/2 2p3/2 10 -7.951 (9) -3.162 (6) -4.180 (6) -4.171 (6) 20 -5.770 (1) -2.2806 (8) -3.0141 (9) -2.9876 (9) 30 -4.740 (1) -1.8624 (4) -2.4615 (5) -2.4120 (5) 4240 -4.1691 (4) -1.6260 (2) -2.1503 (2) -2.0717 (2) 50 -3.8535 (3) -1.4896 (1) -1.9720 (1) -1.8563 (1) 60 -3.7137 (3) -1.4208 (1) -1.88388 (9) -1.71942 (8) 70 -3.7207 (2) -1.4072 (2) -1.86978 (8) -1.63864 (9) 80 -3.87690 (9) -1.44860 (6) -1.92989 (6) -1.60346 (6) 90 -4.2184 (2) -1.55689 (8) -2.08125 (9) -1.61098 (8) 92 -4.3155 (2) -1.58889 (8) -2.12572 (10) -1.61786 (8) F(Zα) for 2 p1/2screened by 1s 2s 2p1/2 2p3/2 10 0.10 (6) -0.03 (2) -0.06 (2) -0.06 (2) 20 0.064 (5) -0.027 (2) -0.049 (2) -0.049 (2) 30 -0.012 (1) -0.0373 (9) -0.0620 (6) -0.0611 (6) 40 -0.119 (5) -0.056 (3) -0.088 (4) -0.084 (4) 50 -0.2566 (5) -0.0817 (3) -0.1250 (2) -0.1158 (2) 60 -0.4316 (2) -0.1164 (2) -0.1757 (1) -0.1547 (1) 70 -0.6607 (9) -0.1636 (7) -0.245 (7) -0.202 (3) 80 -0.9751 (5) -0.2300 (6) -0.3446 (9) -0.2604 (5) 90 -1.4367 (1) -0.33002 (7) -0.49508 (7) -0.33416 (6) 92 -1.5564 (2) -0.3563 (2) -0.5348 (1) -0.35143 (9) F(Zα) for 2 p3/2screened by 1s 2s 2p1/2 2p3/2 10 -0.7 (2) -0.21 (9) -0.3 (1) -0.3 (1) 20 -0.663 (3) -0.185 (3) -0.285 (3) -0.281 (3) 30 -0.691 (4) -0.189 (2) -0.287 (2) -0.279 (2) 40 -0.737 (1) -0.1992 (3) -0.3004 (4) -0.2849 (4) 50 -0.794 (1) -0.2154 (4) -0.3223 (4) -0.2962 (4) 60 -0.8599 (4) -0.2369 (2) -0.3524 (2) -0.3111 (2) 4370 -0.9330 (3) -0.2642 (2) -0.3912 (5) -0.3286 (3) 80 -1.0123 (3) -0.299 (1) -0.440 (1) -0.3486 (8) 90 -1.0972 (2) -0.3419 (2) -0.5022 (1) -0.3704 (1) 92 -1.114 (9) -0.351 (6) -0.516 (6) -0.374 (6) TABLE III. Comparison between F(Zα) using the partial-wave renormalization method [3] and this work. More accurate numerical values from Ref. [19] are used. Rrmsis the nuclear mean-spherical charge radius used (in Fm). Since no error es timate is provided with Ref. [19], in which 2 F(Zα) is tabulated we use an error of 1 on the last displayed ﬁgure. Z R rms Ref. [19] This work Diﬀ. 5 -5.1745 (5) -5.171 (5) 0.004 (5) 10 -3.8820 (5) -3.882 (1) 0.000 (1) 18 3.423 -2.89995 (5) -2.901 (1) -0.001 (1) 24 3.643 -2.47240 (5) -2.4726 (3) -0.0002 (3) 32 4.07 -2.09145 (5) -2.0917 (1) -0.0002 (1) 44 4.480 -1.73720 (5) -1.73770 (6) -0.00050 (8) 54 4.78 -1.55720 (5) -1.55825 (4) -0.00105 (7) 66 5.21 -1.43035 (5) -1.43296 (4) -0.00261 (6) 74 5.37 -1.38890 (5) -1.3935 (2) -0.0046 (3) 83 5.519 -1.38080 (5) -1.39010 (7) -0.00930 (8) 92 5.860 -1.41630 (5) -1.43690 (2) -0.02060 (5) 44TABLE IV. Comparison of the present result for lithium-like ions with earlier work (eV). Results from Refs. [5–8,31–34] all include ﬁnite nuclear si ze correction, not included in the present work. We use results without exchange. The diﬀerence with Re f. [1] on the 2 p1/2screening is due to a programming error in the code used in 1991. Orbital screened by ∆ E Ref. [1] Diﬀ. Refs. [31,32] Ref. [8] Ref. [6] Ref. [33] Ref. [3 4] 1s 1s -3.76607 (5) -3.79 0.02 2s -0.49310 (2) -0.48 -0.01 2p1/2 -0.38705 (2) -0.33 -0.06 2p3/2 -0.12771 (2) 2s 1s -1.41385 (8) -1.44 0.03 -1.375 (5) -1.389 -1.39 -1.375 (30) - 1.385 (15) 2p1/2 1s -0.50991 (5) -0.32 -0.19 -0.485 (5) -0.506 -0.505 -0.475 (30 ) -0.495 (15) 2p3/2 1s -0.365 (3) -0.356 1s22s -11.3460 (1) -11.42 0.08 1s22p1/2 -9.32606 (8) -8.88 -0.45 1s22p3/2 -8.518 (4) Transitions (Valence+Core) 2p1/2→2s 2.0200 (2) 2.55 -0.53 2.01 (1) 2p3/2→2s 2.828 (4) Transitions (Valence) 2p1/2→2s 1.8079 (1) 1.78 (1) 1.766 1.77 1.80 (6) 1.78 (3) 2p3/2→2s 2.098 (4) 2.065 45FIGURES/A1/B4 /BT /B5 /A2/B4 /BU /B5 FIG. 1. Feynman diagrams for radiative corrections to the el ectron-electron interaction. (A) represents the wave function correction and (B) is the verte x correction./A1/B4 /BT /B5 /A2 Æ /BX /BX/B4 /BT /BC/B5 /A3/B4 /BU /B5 FIG. 2. Feynman diagrams for radiative corrections to the el ectron-electron interaction after extraction of reducible (A) and irreducible (A’) parts for t he wave function correction (A) of Fig. 1./A1Æ /CE /A2Æ /BX /A3Æ /CE FIG. 3. Feynman diagrams for the self-energy perturbed by an external potential. In the case of the spherically-averaged screening potential used in th is work, the diagram in the left originates in the diagram (A) in Fig. 2, and represents the wave function correction, while the diagram in the center comes from diagram (A’) in Fig. 2, using the (symbolic ) relation∂ ∂EG(E) =G(E)·G(E). The diagram on the right comes from the vertex correction. 46-0.1496085-0.1496080-0.1496075-0.1496070-0.1496065 10-1510-1310-1110-910-7Derivative Direct Calc. T. by T.Sum Cut-off FIG. 4. Comparison between the convergence of the sum calcul ated using numerical derivation of the unperturbed self-energy sum as in Eq. (80) to the direc t evaluation using perturbed wave function and reduced Green’s function as deﬁned in Eq. (101) , and to an evaluation in which terms in Eqs. (81) to (84) are derived for each κvalue in the sum (T.-by-t deriv.). Evaluation is done for a 2p3/2state and r= 0.9992, y= 1,t= 0.99,Z= 20. 47 -6.0-5.0-4.0-3.0-2.0-1.00.0 5 25 45 65 85 ZF(Zα)1s Sunnergren 1s 2s 2p1/2 2p3/2 FIG. 5. Screening of a 1 selectron by a 1 s, 2s, 2p1/2or 2p3/2electron. Sunnergren: Correction to the Coulomb part of the electron-electron interaction from Ref. [19], improved calculation following Ref. [3]. -9.0-8.0-7.0-6.0-5.0-4.0-3.0-2.0-1.0 10 30 50 70 90 ZF(Zα)1s 2s 2p1/2 2p3/2 FIG. 6. Screening of a 2 selectron by a 1 s, 2s, 2p1/2or 2p3/2electron. 48-2.0-1.5-1.0-0.50.00.5 5 25 45 65 85 ZF(Zα)1s 2s 2p1/2 2p3/2 FIG. 7. Screening of a 2 p1/2electron by a 1 s, 2s, 2p1/2or 2p3/2electron. -1.2-1.0-0.8-0.6-0.4-0.20.0 5 25 45 65 85 ZF(Zα)1s 2s 2p1/2 2p3/2 FIG. 8. Screening of a 2 p3/2electron by a 1 s, 2s, 2p1/2or 2p3/2electron. The large error bar forZ= 10 are only due to the low-energy part of the vertex correcti on 49-0.025-0.020-0.015-0.010-0.0050.0000.0050.0100.015 0 20 40 60 80 100 ZΔF(Zα) Sunn. FIG. 9. Comparison between calculation using the partial-w ave renormalization method [3] and this work. More accurate numerical values from Ref. [19] are used. -0.5-0.4-0.3-0.2-0.100.10.2 0 20 40 60 80 100 ZΔF(Zα) Yer.-pres. Sunner. Breit Pol. extrap. to 0 50FIG. 10. Comparison between calculation for point nucleus ( including Coulomb and magnetic part of the electron-electron interaction) from Ref. [4] an d the Coulomb value from this work. The solid line represents a second-order polynomial ﬁt to the di ﬀerence and extrapolates to −0.124 for Z= 0, as expected from Eq. (119). Also plotted are the Breit val ues from Ref. [19], which are in reasonable agreement with the extrapolation. 51REFERENCES [1] P. Indelicato and P. J. Mohr, Theor. Chem. Acta. 80, 207 (1991). [2] V. A. Yerokhin and V. M. Shabaev, Phys. Lett. A 207, 274 (1995). [3] H. Persson, S. Salomonson, P. Sunnergren, and I. Lindgre n, Phys. Rev. Lett. 76, 204 (1996). [4] V. A. Yerokhin, A. N. Artemyev, and V. M. Shabaev, Phys. Le tt. A234, 361 (1997). [5] V. A. Yerokhin, A. N. Artemyev, T. Beier, V. M. Shabaev, an d G. Soﬀ, J. Phys. B: At. Mol Phys. 31, L691 (1998). [6] V. A. Yerokhin, A. N. Artemyev, T. Beier, V. M. Shabaev, an d G. Soﬀ, Phys. Scr. 59, in press (1998). [7] V. A. Yerokhin, A. N. Artemyev, T. Beier, G. Plunien, V. M. Shabaev, and G. Soﬀ, Phys. Rev. A 60, 3522 (1999). [8] S. A. Blundell, K. T. Cheng, and J. Sapirstein, Phys. Rev. A55, 1857 (1997). [9] P. J. Mohr, Ann. Phys. (N.Y.) 88, 26 (1974). [10] P. Indelicato and P. J. Mohr, Phys. Rev. A 46, 172 (1992). [11] P. Indelicato and P. J. Mohr, J. Math. Phys. 36, 714 (1995). [12] P. Indelicato and P. J. Mohr, Phys. Rev. A 58, 165 (1998). [13] D. J. Hylton, J. Math. Phys. 25, 1125 (1984). [14] P. J. Mohr and Y.-K. Kim, Phys. Rev. A 45, 2727 (1992). [15] P. J. Mohr, Ann. Phys. (N.Y.) 88, 52 (1974). [16] D. J. Hylton, Phys. Rev. A 32, 1303 (1985). [17] P. J. Mohr, G. Plunien, and G. Soﬀ, Phy. Rep. 293, 227 (1998). 52[18] P. J. Mohr, Phys. Rev. A 26, 2338 (1982). [19] P. Sunnergren, Ph.D. thesis, Chalmers University of Te chnology, 1998. [20] U. D. Jentschura, P. J. Mohr, and G. Soﬀ, Phys. Rev. Lett. 82, 53 (1999). [21] U. D. Jentschura, P. J. Mohr, G. Soﬀ, and E. J. Weniger, Co mp. Phys. Communi. 116, 28 (1999). [22] H. Araki, Prog. Theo. Phys 17, 619 (1957). [23] J. Sucher, Phys. Rev. 109, 1010 (1958). [24] G. W. F. Drake and W. C. Martin, Can. J. Phys. 76, 679 (1998). [25] G. W. F. Drake, in Atomic, Molecular and Optical Physics Handbook , edited by G. W. F. Drake (AIP Press, Woodbury, New York, 1996). [26] P. Indelicato, O. Gorceix, and J. P. Desclaux, J. Phys. B : At. Mol Phys. 20, 651 (1987). [27] P. Indelicato and J. P. Desclaux, Phys. Rev. A 42, 5139 (1990). [28] J. P. Desclaux, in Methods and Techniques in Computational Chemistry , edited by E. Clementi (STEF, Cagliary, 1993), Vol. A. [29] K. G. Dyall, I. P. Grant, C. T. Johnson, F. A. Parpia, and E . P. Plummer, Comp. Phys. Communi. 55, 425 (1989). [30] P. Indelicato, S. Boucard, and E. Lindroth, Eur. Phys. J . D3, 29 (1998). [31] S. A. Blundell, Phys. Rev. A 46, 3762 (1992). [32] S. A. Blundell, Phys. Rev. A 47, 1790 (1993). [33] K. T. Cheng, W. R. Johnson, and J. Sapirstein, Phys. Rev. A47, 1817 (1993). [34] H. Persson, I. Lindgren, S. Salomonson, and P. Sunnergr en, Phys. Rev. A 48, 2772 (1993). 53[35] P. J. Mohr, Phys. Rev. A 46, 4421 (1992). 54
arXiv:physics/0010045v1 [physics.bio-ph] 17 Oct 2000Dynamic ﬁtness landscapes: Expansions for small mutation rates Claus O. Wilke Digital Life Laboratory Mail Code 136-93, Caltech Pasadena, CA 91125 wilke@caltech.edu Christopher Ronnewinkel Institut f¨ ur Neuro- und Bioinformatik Medizinische Universit¨ at L¨ ubeck Seelandstraße 1a D-23569 L¨ ubeck, Germany ronne@inb.mu-luebeck.de Abstract We study the evolution of asexual microorganisms with small mutation rate in ﬂuc- tuating environments, and develop techniques that allow us to expand the formal solution of the evolution equations to ﬁrst order in the muta tion rate. Our method can be applied to both discrete time and continuous time syst ems. While the be- havior of continuous time systems is dominated by the averag e ﬁtness landscape for small mutation rates, in discrete time systems it is inst ead the geometric mean ﬁtness that determines the system’s properties. In both cas es, we ﬁnd that in sit- uations in which the arithmetic (resp. geometric) mean of th e ﬁtness landscape is degenerate, regions in which the ﬁtness ﬂuctuates around th e mean value present a selective advantage over regions in which the ﬁtness stays a t the mean. This eﬀect is caused by the vanishing genetic diﬀusion at low mutation r ates. In the absence of strong diﬀusion, a population can stay close to a ﬂuctuating peak when the peak’s height is below average, and take advantage of the peak when i ts height is above average. Key words: dynamic ﬁtness landscape, quasispecies, error threshold, molecular evolution, ﬂuctuating environment PACS: 87.23.Kg Preprint submitted to Elsevier Preprint 2 February 20081 Introduction A major part of all living creatures on Earth consists of prok aryotes and phages. These organisms replicate mainly without sexual re combination [1], and typically produce oﬀspring on a time-scale of hours. Bec ause of their short gestation times, microbes experience ubiquitous environm ental changes such as seasons on an evolutionary time scale. Most of the DNA base d microbes have developed error correction mechanisms, such limiting the amount of dele- terious mutations they experience. In a changing environme nt, however, small mutation rates might severely curtail a species’ ability to react to new sit- uations. The observed genomic mutation rates of asexual org anisms such as bacteria and DNA viruses lie typically around 2 −4×10−3[2], implying that a few out of every thousand oﬀspring get mutated at all. It has b een proposed [3] that even lower genomic mutation rates are not observed simp ly because they would stiﬂe a species’ adaptability in a changing environme nt. While this is certainly a reasonable assumption, we do not currently have a deep under- standing of what types of ﬁtness landscapes require what mut ation rates, and whether a small mutation rate is always disadvantageous in a changing envi- ronment. In this paper, we address the eﬀects of a changing en vironment on a population evolving in a small mutation rate. Our main obje ctive is to de- velop an expansion to ﬁrst order in the mutation rate which en ables us to ﬁnd approximate solutions for inﬁnite asexual populations evo lving in arbitrary dynamic landscapes. Due to the nature of the expansions that we use, we are led to a c omparison between discrete time and continuous time systems. Our main result from the comparison is that in dynamic ﬁtness landscapes, contin uous and discrete time systems have qualitative diﬀerences in the low mutatio n rate regime. This diﬀerence can manifest itself, for example, in populat ions that replicate either continuously or synchronized in discrete generatio ns. Given that all other factors are equal, the continuously replicating stra ins will have a selective advantage. As a generic result for both continuous and discr ete time, we ﬁnd that a low mutation rate can enable a population to draw a sele ctive advantage from ﬂuctuations in the landscape. Our analysis is based on the quasispecies model [4–6]. The qu asispecies lit- erature was for a long time focused on static ﬁtness landscap es, but recently more emphasis has been put on the aspect of changing environm ents [3,7–14]. Here, we mainly use methods developed in Ref. [12]. The paper is structured as follows. In Sec. 2, we demonstrate how systems with discre te as well as con- tinuous time can be treated to ﬁrst order in the mutation rate . In Sec. 3, we discuss the expansions we have found in Sec. 2. We treat the ca se of a vanish- ing mutation rate in Sec. 3.1, and that of a very small but posi tive mutation rate in Sec. 3.2. In Sec. 3.3, we study the localization of a po pulation around a 2oscillating peak, and in Sec. 3.4, we discuss the problems we encounter when approximating a continuous time system with a discrete time system. We close our paper with concluding remarks in Sec. 4. 2 Analysis 2.1 The model Consider a system of evolving bitstrings. The diﬀerent bits trings ireplicate with rates Ai, and they mutate into each other with probabilities Qij. Through- out this paper, we assume that the probability of an incorrec tly copied bit is uniform over all strings, and denote this probability by R. The mutation ma- trixQ= (Qij) is then given by Qij= (1−R)l/parenleftbiggR 1−R/parenrightbiggd(i,j) , (1) where d(i, j) is the Hamming distance between two sequences iandj. The matrix Qis a 2l×2lmatrix, and it is in general diﬃcult to handle numeri- cally. Therefore, in the following we impose the additional assumption that all sequences with equal Hamming distance from a given referenc e sequence have the same ﬁtness. This is the so-called error class assumptio n [15]. The matrix Qis then an ( l+ 1)×(l+ 1) matrix, Qij=min{i,j}/summationdisplay k=max {i+j−l,0}/parenleftiggj k/parenrightigg/parenleftiggl−j i−k/parenrightigg (1−R)l/parenleftbiggR 1−R/parenrightbiggi+j−2k . (2) The generality of our results is not aﬀected by this choice, b ecause the calcu- lations we present in the following can be performed with eit her of the two matrices Q, and they lead to very similar expressions. Let us write down the quasispecies equations for sequences e volving in con- tinuous or discrete time and in a static ﬁtness landscape. We introduce the replication matrix A= diag( A0, A1, . . .). The continuous diﬀerential equation of the (unnormalized) concentration variables y= (y0, y1, . . .) then reads ˙y(t) =QAy(t). (3) The discrete diﬀerence equation, on the other hand, can be wr itten as y(t+ ∆t) = [∆ tQA+λ]y(t), (4) where ∆ tis the duration of one generation, and λgives the proportion of parents that survive one generation and enter the next one to gether with 3their oﬀspring. Both Eq. (3) and Eq. (4) converge for t→ ∞ towards a se- quence distribution given by the Perron eigenvector of the m atrixQA. Hence, for a static landscape the discrete time and the continuous t ime quasispecies equations are equivalent, as far as the asymptotic state is c oncerned. The dis- tinction between discrete and continuous time, however, is important when the ﬁtness landscape changes over time. Consider the situat ion of a dynamic ﬁtness landscape, represented by a time dependent matrix A(t). Equation (3) becomes ˙y(t) =QA(t)y(t). (5) The time-dependent diﬀerence equation, on the other hand, r eads y(t+ ∆t) = [∆ tQA(t) +λ]y(t), (6) The dynamic attractors of both Eqs. (5) and (6) are not immedi ately obvious, and therefore we cannot know to what extent the two systems di ﬀer unless we perform a more elaborate analysis. Moreover, in a static lan dscape, a nonzero λ does not aﬀect the asymptotic state of the system, which is wh y it normally is set to zero in Eq. (4) [16,17]. The situation is diﬀerent in a d ynamic landscape, and we have to allow for a non-zero λin general. 2.2 Discrete time Let us begin our analysis with the discrete system. We set λ= 0, because that leads to the simplest equation describing a discrete time ev olutionary system in a dynamic ﬁtness landscape. The more complicated cases wi thλ >0 can be constructed from the equation for λ= 0, as we will see later on. We address the equation y(t+ ∆t) = ∆ tQA(t)y(t). (7) The solution to this equation is formally given by the time-o rdered matrix product [12] [using n=t/∆tandA′(ν) = ∆ tA(ν∆t)] y(n) =T/braceleftiggn−1/productdisplay ν=0QA′(ν)/bracerightigg y(0) =:Ydisc(n)y(0). (8) In the second line, we have introduced the notation Ydisc(n) for this matrix product. We will occasionally refer to Ydisc(n) as a propagator , since Ydisc(n) fully determines the state of the system at time t=n∆t, given an initial state at time t= 0. 4Ydisc(n) can be evaluated to ﬁrst order in R. The only dependency of Ydisc(n) onRis the one in Q. When we expand Q[Eq. (2)] in R, we ﬁnd Qij=min{i,j}/summationdisplay k=max {i+j−l,0}/parenleftiggj k/parenrightigg/parenleftiggl−j i−k/parenrightigg [δβ,0+R(δβ,1−αδβ,0) +. . .], (9) withα=l−βandβ=i+j−2k. As usual, δi,jdenotes the Kronecker symbol. The sum collapses into a single term, and we ﬁnd to ﬁrst order i nR Qij= (1−lR)δi,j+ (l−j)Rδi,j+1+jRδi,j−1. (10) After some algebra, we obtain from that for the matrix Ydisc(n) /parenleftbigg Ydisc(n)/parenrightbigg ij=/bracketleftigg (1−lRn)n−1/productdisplay ν=0A′ j(ν)/bracketrightigg δi,j + (l−j)Rt−1/summationdisplay µ=0µ/productdisplay ν1=0A′ j(ν1)t−1/productdisplay ν2=µ+1A′ j+1(ν2) δi,j+1 + jRt−1/summationdisplay µ=0µ/productdisplay ν1=0A′ j(ν1)t−1/productdisplay ν2=µ+1A′ j−1(ν2) δi,j−1, (11) This expression fully describes to ﬁrst order in Rthe state of the system after ntime steps. 2.3 Continuous time Let us now turn to the continuous system. We can use the expans ion ofYdisc(n) to ﬁnd an expansion for the propagator of the static continuo us case. If the ﬁtness landscape is static, the solution to Eq. (3) is given b y y(t) = exp( QAt)y(0). (12) It is useful to recall that the exponential operator of a matr ix is deﬁned as exp(QAt) =1+QAt+1 2!(QAt)2+. . . . (13) We can expand the single terms in that sum separately. Equati on (11) allows us to write ( QA)kas /parenleftbigg (QA)k/parenrightbigg ij= (1−klR)Ak jδi,j+ (l−j)Rk/summationdisplay µ=1Aµ jAk−µ j+1 δi,j+1 + jRk/summationdisplay µ=1Aµ jAk−µ j−1 δi,j−1+O(R2). (14) 5Both the sums in the expression for ( QA)kand the remaining sum in Eq. (13) can then be taken analytically. We ﬁnd /parenleftbigg exp(QAt)/parenrightbigg ij= (1−lRA jt)eAjtδi,j+ (l−j)RK[Ajt, Aj+1t]δi,j+1 +jRK[Ajt, Aj−1t]δi,j−1, (15) where the function K[a, b] is of the form K[a, b] =a a−b/parenleftig ea−eb/parenrightig . (16) With Eq. (15), we have an expansion of the propagator of the co ntinuous system in a static landscape to ﬁrst order in R. Similarly, we can treat piece- wise constant landscapes. Under a piecewise constant lands cape we under- stand a landscape for which we can deﬁne intervals I1= [0, t1),I2= [t1, t2), I3= [t2, t3),. . ., such that the landscape does not change within any of these intervals. Any dynamic ﬁtness landscape can be approximate d in that way. The solution to the diﬀerential equation for that type of lan dscapes is given by y(t) = exp[ QA(tn)(t−tn)] exp[QA(tn−1)(tn−tn−1)]··· ···exp[QA(0)t1]y(0). (17) With the two simplifying assumptions that all intervals hav e the same length τand that we are observing the system only at the end of an inter val, Eq. (17) becomes (for n=t/τ) y(t) =T/braceleftiggn−1/productdisplay ν=0exp[QA(ντ)τ]/bracerightigg y(0) =:Ycont(t)y(0). (18) The similarity to Eq. (8) is evident. Hence, in analogy to the calculation that leads from Eq. (10) to Eq. (11), we ﬁnd in the piecewise consta nt, continuous case /parenleftbigg Ycont(t)/parenrightbigg ij=/bracketleftigg/parenleftbigg 1−lRn−1/summationdisplay ν=0Ac j(ν)/parenrightbigg exp/parenleftbiggn−1/summationdisplay ν=0Ac j(ν)/parenrightbigg/bracketrightigg δi,j + (l−j)Rn−1/summationdisplay µ=0K(µ) j,j+1µ−1/productdisplay ν1=0exp[Ac j(ν1)]n−1/productdisplay ν2=µ+1exp[Ac j+1(ν2)] δi,j+1 + jRn−1/summationdisplay µ=0K(µ) j,j−1µ−1/productdisplay ν1=0exp[Ac j(ν1)]n−1/productdisplay ν2=µ+1exp[Ac j−1(ν2)] δi,j−1.(19) We have used the abbreviations Ac(ν) =τA(ντ) and K(ν) i,j=K[Ac i(ν), Ac j(ν)]. (20) 6Equation (19) fully determines to ﬁrst order in Rthe state of the continuous system after tunits of time have passed. 3 Discussion With Eqs. (11) and (19), we have expansions for the propagato rs of discrete- time and continuous time evolving systems in a dynamic ﬁtnes s landscape. In this section, we will examine these expansions and discuss t heir properties. 3.1 A vanishing mutation rate ForR= 0, both Ydisc(t) andYcont(t) turn into diagonal matrices. We ﬁnd (choosing τ= ∆tandn=t/τ) Ydisc(t) = exp/bracketleftign−1/summationdisplay ν=0log/parenleftig A′(ν)/parenrightig/bracketrightig =/parenleftbigg n/radicalig A′(n−1)···A′(0)/parenrightbiggn , (21) Ycont(t) = exp/bracketleftign−1/summationdisplay ν=0Ac(ν)/bracketrightig . (22) This result shows that in a dynamic ﬁtness landscape the disc rete and the continuous model have not only quantitative, but also impor tant qualitative diﬀerences. While in the continuous case the state of the sys tem at time tis determined by the exponential of the arithmetic mean of the ﬁ tness landscape until time t, in the discrete case it is determined by the exponential of t he geometric mean of the ﬁtness landscape, which can be written as arithmetic mean of the logarithm of the ﬁtness landscape. The latter cor responds to results from population genetics [18]. Since arithmetic an d geometric mean are in general diﬀerent, the same ﬁtness landscape can have v ery diﬀerent eﬀects in a continuous or discrete system for R= 0. Consider a landscape, for example, with an oscillating sharp peak, A0(t) =  σ(1−a) for 0 ≤t < T/ 2 σ(1 +a) for T/2≤t < T(23a) Ai(t) = 1 for 0 < i≤l , (23b) with 0 ≤a <1 and σ >0. In the continuous system without mutations, the master sequ ence grows with the rate A0=σif time is measured in integer multiples of T. Hence, if σ > 1, the peak sequence will always supersede all other sequenc es for t→ ∞. Contrasting to that, the geometric mean is/tildewideA0=σ√ 1−a2. Even for σ >1 it 7is possible to have/tildewideA0<1 ifais large enough, in which case in the discrete system the master sequence grows slower than all others. Con sequently, it will be expelled from the population for t→ ∞ . The special case of σ= 1 is depicted in Fig. 1. There, the ﬁtness landscape becomes ﬂat i n continuous time, but acquires a hole in discrete time. 3.2 Small non-zero mutation rates Let us now turn to the case of a small but non-zero R. From the above, we can expect that there is a qualitative diﬀerence between dis crete and con- tinuous time even for ﬁnite R. In order to see this diﬀerence, we take the oscillating sharp peak landscape as a generic example. A two concentration approximation has proven useful to describe situations wit hσ≫1 [14] but is not applicable here, since we are particularly interested i n the case σ= 1, for which the average landscape is ﬂat in continuous time and acq uires a hole in discrete time. The analysis of the landscape Eq. (23) is facilitated by its p eriodicity in time (with period length T). For periodic landscapes, it has been shown in Ref. [12] that a periodic attractor with period length Texists. Its state at phase φ= 0 (the phase is deﬁned as φ:=tmodT) is given by the principal eigenvector of themonodromy matrix X(0) =Y(T), (24) where Y(t) is the propagator of the system. Equation (24) holds regard less of continuous or discrete time. The attractor’s state at oth er phases φcan be calculated in a similar fashion. In Figure 2, we have displayed the order parameter ms[19,20] in the sharp peak landscape as a function of Rfor the discrete time and the continuous time system. The order parameter is given by ms(t) =1 ll/summationdisplay i=0xi(t)(l−2i), (25) where the xi(t) represent the total (normalized) concentration of all seq uences in error class iat time t. We have calculated the order parameter both from the full monodromy matrix and from the expansions to ﬁrst ord er inR. We ﬁnd that the expansions give reliable results for small muta tion rates, but start deviating from the true value as Rapproaches 1 /lT. Note that both expansions must break down beyond 1 /lT, as both the discrete and the con- tinuous propagator assume unphysical negative values on th e diagonal when Rexceeds 1 /lT[Eqs. (11) and (19)]. 8From Fig. 2, it is evident that there exists a qualitative diﬀ erence between the discrete and the continuous time system. In the system with c ontinuous time, the sequences stay centered around the currently active pea k for arbitrarily small but non-zero mutation rates, whereas in the system wit h discrete time, the sequence distribution becomes ever more homogeneous as R→0. The behavior of the discrete system is easily explained. In t he geometric mean of the landscape, the peak position is actually disadvantag eous, and hence the population is driven into the remaining genotype space, which it occupies homogeneously due to the lack of selective diﬀerences. Form ally, the popula- tion feels the geometric mean only for a vanishing mutation r ate. However, by continuity, the disadvantage at the peak position will rema in for some small but non-zero R, which leads to the continuous decay of the order parameter a s R→0. Interestingly, the order parameter does not decay exactl y to zero, but to a value slightly below zero. This happens because the popu lation becomes homogeneously distributed over the whole sequence space ex cept for the po- sition of the oscillating peak. The resulting small imbalan ce in the sequence distribution towards the opposite end of the boolean hyperc ube then leads to a negative order parameter. The inset in Fig. 2 shows that our approximation predicts this behavior accurately for small R. Now consider the continuous system. For an inﬁnitesimal R >0, the depen- dence of the asymptotic state on the initial condition is los t, as we know from the Frobenius-Perron theorem. Since for R= 0 the evolution of the popula- tion in time steps of size Tis guided by the ﬂat average landscape, one might suspect that for inﬁnitesimal R >0 a homogeneous distribution is found as the unique asymptotic state. This is what we observe for a pop ulation evolv- ing in a ﬂat static landscape with little mutation. However, the situation in a dynamic landscape may be diﬀerent, because the dynamics of t he landscape has a signiﬁcant inﬂuence on the asymptotic sequence distri bution. In fact, it is possible that a ﬂat average landscape leads to an ordered a symptotic state for ﬁnite mutation rates R >0. In the next subsection, we will demonstrate this eﬀect for the oscillating sharp peak. 3.3 Localization around an oscillating peak We will now have a closer look at the oscillating sharp peak la ndscape, Eq. (23). We are interested in the case σ= 1, which leads to a degenerate average in continuous time. First we note some general properties of th e monodromy ma- trixX(φ) for a periodic landscape with ﬂat arithmetic mean. If X(φ) is given to ﬁrst order in R, it reads (assuming the average ﬁtness is 1) X(φ) = (1−TlR)1+R˜X(φ), (26) 9where ˜X(φ) is independent of the mutation rate Rand contains only the oﬀ-diagonal entries from Eq. (19). Since ˜X(φ) diﬀers from X(φ) only by a scalar factor and an additional constant on the diagonal, th e eigenvectors of the former matrix are identical to the ones of the latter ma trix, while the eigenvalues are related through λi= (1−TlR)+R˜λi. As a consequence, we ﬁnd that the asymptotic species distribution is given by the pri ncipal eigenvector of the oﬀ-diagonal matrix ˜X(φ), which is independent of R. If we took terms up to the kth order of Rinto account in Eq. (19), we would ﬁnd the higher order contributions to the eigenvectors up to ( k−1)th order of R. However, with our ﬁrst-order approximation, we are only able to calculate the asymptotic sequence concentrations to 0th order in R. For small mutation rates R, we can restrict our analysis to the ﬁrst three error classes. For the oscillating peak, we ﬁnd with help of Eq. (19 ) the following expressions ˜X(φ)≈Texp(T) 0 αφ(aT) 0 lβφ(aT)αφ(aT) 0 2 0 l−1 0 , (27) where αφ(ξ) = (2 /ξ)[1−e−ξ/2]eξ\|φ−1/2\|(28a) βφ(ξ) =eξ/2e−2ξ\|φ−1/2\|(28b) ξ=aT . (28c) The (unnormalized) asymptotic state follows as  y0 y1 y2 (φ, ξ) = αφ(ξ) /radicalig 2(l−1) +α2 φ(ξ)βφ(ξ)l l−1 . (29) Now, if the third error class concentration is negligibly sm all compared to the other two concentrations, the concentrations of the higher error classes can be neglected as well, and the asymptotic state is approximat ely given by the concentrations of the ﬁrst two error classes only. From Eq. ( 29), we can derive the following criterion for this approximation to be valid, exp(ξ/2)≫lξ2/4. (30) Hence, if ξ=aTis large, which means that the ﬁtness ﬂuctuations are large and slow, the population is exclusively distributed over th e peak and the ﬁrst error class. For this case, we ﬁnd the following simpliﬁed de scription of the 10population: x0(φ, ξ) = 1/slashbigg/bracketleftbigg 1 +/radicalig lβφ(ξ)/bracketrightbigg , (31a) x1(φ, ξ) =/radicalig lβφ(ξ)/slashbigg/bracketleftbigg 1 +/radicalig lβφ(ξ)/bracketrightbigg , (31b) ms(φ, ξ) = 1−2x1(φ, ξ)/l . (31c) The last equation implies that in the limit R→0, the order parameter is always larger than 1 −2/l. This means that although the peak does not have an average selective advantage, the evolving sequences are attracted to the peak nonetheless. As long as l >1, the order parameter averaged over one oscillation cycle is positive , which means that a population can draw a selective advantage from being close to the peak in comparison to being far away from it. In Figure 3A, we display the predicted behavior of the system in a very small mutation rate, as given by Eq. (31). The observed change in th e sequence concentrations is explained as follows. During the times at which the peak has above-average ﬁtness, the sequences on the peak replicate f aster than all others and hence grow exponentially until the peak’s concentratio n saturates around one, while all oﬀ-peak sequences assume vanishing concentr ations. Similarly, during the times at which the peak has below-average ﬁtness, the peak’s rela- tive concentration decays, while the population moves onto the nearest advan- tageous sequences, which can be found in the ﬁrst error class . The sequences in all other error classes are adaptively neutral compared to t he ﬁrst error class. Hence, the amount of sequences that move into higher error cl asses is solely determined by the mutation rate. If the mutation rate is smal l enough, the diﬀusion among these neutral sequences becomes negligibly small on the time scale of the peak oscillations T. Therefore, the population stays mainly within the ﬁrst error class until the peak ﬁtness switches back to th e above-average value. Thus, we ﬁnd the qualitative behavior of Eq. (31): In a landscape with a large and slowly oscillating sharp peak and a small mutatio n rate, the pop- ulation oscillates between the peak sequence and the ﬁrst er ror class in the asymptotic state. In short, the population becomes localiz ed close to the peak. For extremely small mutation rates, Eq. (31) agrees perfect ly with the full numerical solution. For somewhat larger mutation rates, th e main discrepancy that arises is a phase shift between the full solution and the approximation (Figure 3B). The phase shift moves the concentration curves towards earlier times, i.e., the system becomes more responsive to the chang ing peak as the mutation rate increases. This is intuitively clear. With a h igher mutation rate, the ﬁrst error class will already be occupied to a larger exte nt when the peak switches to the below-average value, so that the concentrat ion of the one- mutants can grow faster towards their equilibrium value. Si milarly, when the peak switches back to the above-average value, the peak sequ ences have a more 11favorable initial concentration, which makes them grow fas ter in comparison to a lower mutation rate. Let us shortly extend the above argumentation to broader pea ks, like peaks with linear ﬂanks of width 1 ≤w≤l: Ai(t) = max/braceleftbigg 1,w−i w[A0(t)−1]/bracerightbigg for all 0 < i≤l. (32) The sharp peak from above corresponds to a peak of width w= 1. For ar- bitrary chosen width 1 ≤w≤l, the population gets transported to the wth error class due to the selection pressure during the below-a verage peak ﬁtness phases. The wth error class is in that case the boundary of the advantageou s region. Again, if the mutation rate is suﬃciently small, diﬀ usion can be ne- glected and the population will stay in the wth error class until the peak ﬁtness switches back to the above-average value. This implies that for peaks of width w≥l/2, it is possible to have ms(φ)≤0 for some intermediate oscillation phases. In particular for the maximum width w=l, the order parameter ms(φ) will oscillate symmetrically around zero. In this subsection, we have only considered continuous time systems. We have established that in a dynamic ﬁtness landscape with ﬂat aver age, a population can draw a selective advantage from peaks that ﬂuctuate arou nd the average ﬁtness value. The same eﬀect will occur in a discrete time sys tem if we con- sider the geometric mean of the ﬁtness landscape instead. In other words, in a landscape with ﬂat geometric mean, a population with a small mutation rate will draw a selective advantage from a peak that ﬂuctuates ar ound that geo- metric mean. The origin of that eﬀect is again the vanishing d iﬀusion, which causes the population to remain close to the peak when the pea k has a height below the mean. 3.4 Discrete systems with overlapping generations When discussing the discrete system in Sec. 3.1 and 3.2, we ha ve set λ= 0, i.e., we have made the assumption that every sequence can gen erate oﬀspring only once, and dies before the next generation starts to repl icate. The opposite extreme is λ= 1, for which no sequence ever dies. With λ= 1, a sequence can theoretically stay inﬁnitely long in the system (in prac tice, the growth of new sequences is compensated through an out-ﬂux of old seq uences, but that is not our concern here. The details of the out-ﬂux do not inﬂuence the unnormalized concentration variables y(t) in Eqs. (3)–(6) [12]). For λ= 1, Eq. (6) converges to Eq. (5) for ∆ t→0. In other words, for λ= 1 and a small ∆t, Eq. (6) is an approximation to Eq. (5). This fact has been exp loited in Ref. [12] in order to calculate the continuous system numeri cally. However, it 12has not been evaluated in Ref. [12] to what extend the discret e approximation behaves qualitatively diﬀerent from the continuous system . Let us brieﬂy examine how the discrete equation with λ= 1 ﬁts into the concepts we have developed so far. For λ= 1, the propagator Ydisc(t) assumes the form Ydisc(t) =T/braceleftiggn−1/productdisplay ν=0[∆tQA(ν∆t) + 1]/bracerightigg , (33) which can be rewritten into Ydisc(t) =T/braceleftigg 1+ ∆tn−1/summationdisplay ν=0QA(ν∆t) + ∆t2n−1/summationdisplay ν=0ν−1/summationdisplay ν′=0QA(ν∆t)QA(ν′∆t) +···+ ∆tnn−1/productdisplay ν=0QA(ν∆t)/bracerightigg . (34) With the formulae given in Section 2, it is possible to expand this expression to ﬁrst order in R. Since the corresponding calculation is tedious, and the result does not give any new insights, we omit this expansion here. Let us just consider the zeroth order term, Ydisc(t) =1+ ∆tn−1/summationdisplay ν=0A(ν∆t) + ∆t2n−1/summationdisplay ν=0ν−1/summationdisplay ν′=0A(ν∆t)A(ν′∆t) +···+ ∆tnn−1/productdisplay ν=0A(ν∆t) +O(R). (35) Compare this expression to Eqs. (21) and (22). For λ= 1, we neither have the exponential of the averaged landscape, nor do we have an e xpression that depends solely on the geometric mean of the landscape. We obt ain a mix- ture between the two cases, and the size of ∆ tdetermines which case we are closer to. Consequently, we obtain qualitatively wrong res ults from the dis- crete approximation if the arithmetic and geometric mean of the landscape diﬀer signiﬁcantly. Nevertheless, the discrepancies betw een the results can be restricted to arbitrary small values of the mutation rate if we choose ∆ tsmall enough. As an example, consider Fig. 4. There we display the order par ameter in the oscillating sharp peak landscape obtained from the full con tinuous propaga- tor, and compare it to the result from the discrete approxima tion for various values of ∆ t. For a relatively large ∆ t= 2 (n= 50), Eq. (33) gives a poor approximation of the continuous system. Throughout the who le range of R there are signiﬁcant deviations from the full solution. As w e decrease ∆ t(in- crease n), the approximation moves much closer to the true value of th e order parameter. Yet, for very small R, the order parameter always decays to zero in 13the approximation, whereas it stays close to one in the full s olution. However small we choose ∆ t, there will always be some contribution from the geometric mean at R= 0. That causes the order parameter in the discrete approxim ation to vanish for this particular landscape. Contrasting to above situation, however, the diﬀerences be tween approxima- tion and full solution are hardly noticeable in landscapes w here the arithmetic and the geometric mean have a comparable structure (a peak in the averaged landscape is also a peak in the geometric mean of the geometri c mean of the landscape, only with a slightly diﬀerent height). 4 Conclusions We have studied time-dependent ﬁtness landscapes in the qua sispecies model for the particular regime of small mutation rates. We have sh own that the discrete time formulation and the continuous time formulat ion yield qualita- tively diﬀerent outcomes in that regime. If time is updated c ontinuously, an evolving population adapts for R→0 to the exponential of the average ﬁtness landscape, whereas in discrete time, the population adapts to the geometric mean of the landscape. If the arithmetic or the geometric mean of the ﬁtness landsca pe have degen- eracies, then the behavior of the respective continuous tim e or discrete time system for R→0 is determined by the eﬀect of the landscape on the popu- lation for some small but ﬁnite R, which can be very diﬀerent from its eﬀect forR= 0. In particular, for the case of a slowly oscillating peak, the growth of the population onto the peak when the peak is high is much fa ster than the diﬀusion away from the peak when the peak is low, which imp lies that a population can draw a selective advantage from that peak evenif the average (resp. geometric mean) height of the peak does not exceed the surrounding landscape. From that observation, the following picture em erges: If the average height of a slowly oscillating peak is larger than or equal to the surrounding ﬁtness landscape, than in a small mutation rate environment a population will draw a selective advantage from being close to the peak posit ion. Only if the average height is truly smaller than the surrounding ﬁtness , the peak position will be necessarily disadvantageous. The diﬀerences that we have found between continuous time an d discrete time systems are not only interesting from a modeling perspectiv e. They also have implications for the evolution of organisms that have the ab ility to inﬂuence their replication cycle. In a ﬂuctuating environment, a str ain that feels the average of the landscape will have a selective advantage ove r a strain that feels the geometric mean, as the latter is generally smaller . Hence, if two 14strains are identical apart from the fact that one replicate s in a synchronized manner (all individuals generate their oﬀspring at the same time, every ∆ t units of time), whereas the other one replicates unsynchron ized (at any point in time, some individuals may generate oﬀspring), then the u nsynchronized strain will out-compete the synchronized strain. Throughout this paper, we have exclusively considered inﬁn ite populations. It is quite likely that ﬁnite populations experience the arith metic or geometric mean ﬁtness just as inﬁnite populations do. However, since ﬁ nite population sampling occurs at every time step, the sampling might well i nterfere with the averaging, such that ﬁnite populations could experience a s omewhat diﬀerent landscape. Nevertheless, the eﬀect that a ﬂuctuating peak c an lead to a se- lective advantage will also exist in a ﬁnite population. Wit h a small mutation rate, the ﬁnite population will not drift away from the peak w hen it is below average, and hence the population will most likely rediscov er the peak when it rises again to above average. We thank Erik van Nimwegen for useful comments and suggestio ns, and Chris Adami for carefully reading the manuscript. This work was su pported in part by the National Science Foundation under contract No. DEB-9 981397, and by the BMBF under F¨ orderkennzeichen 01IB802C4. References [1] Bruce R. Levin and Carl T. Bergstrom. Bacteria are diﬀere nt: Observations, interpretations, speculations, and opinions about the mec hanisms of adaptive evolution in prokaryotes. Proc. Natl. Acad. Sci. USA , 97:6981–6985, 2000. [2] J. W. Drake, B. Charlesworth, D. Charlesworth, and J. F. C row. Rates of spontaneous mutations. Genetics , 148:1667–1686, 1998. [3] Martin Nilsson and Nigel Snoad. Optimal mutation rates i n dynamic environments. eprint physics/0004042, April 2000. [4] M. Eigen. Selforganization of matter and the evolution o f biological macromolecules. Naturwissenschaften , 58:465–523, 1971. [5] Manfred Eigen, John McCaskill, and Peter Schuster. Mole cular quasi-species. J. Phys. Chem. , 92:6881–6891, 1988. [6] Manfred Eigen, John McCaskill, and Peter Schuster. The m olecular quasi- species. Adv. Chem. Phys. , 75:149–263, 1989. [7] B. L. Jones. Selection in systems of self-reproducing ma cromolecules under the constraint of controlled energy ﬂuxes. Bull. Math. Biol. , 41:761–766, 1979. [8] B. L. Jones. Some models for selection of biological macr omolecules with time varying constants. Bull. Math. Biol. , 41:849–859, 1979. 15[9] Claus O. Wilke. Evolutionary Dynamics in Time-Dependent Environments . Shaker Verlag, Aachen, 1999. PhD thesis Ruhr-Universit¨ at Bochum. [10] Claus O. Wilke, Christopher Ronnewinkel, and Thomas Ma rtinetz. Molecular evolution in time dependent environments. In Dario Florean o, Jean- Daniel Nicoud, and Francesco Mondada, editors, Advances in Artiﬁcial Life, Proceedings of ECAL’99, Lausanne, Switzerland , Lecture Notes in Artiﬁcial Intelligence, pages 417–421, New York, 1999. Springer-Ver lag. [11] Martin Nilsson and Nigel Snoad. Error thresholds on dyn amic ﬁtness landscapes. Phys. Rev. Lett. , 84:191–194, 2000. [12] Claus O. Wilke, Christopher Ronnewinkel, and Thomas Ma rtinetz. Dynamic ﬁtness landscapes in molecular evolution. Phys. Rep. , in press. (eprint physics/9912012). [13] C. Ronnewinkel, C. O. Wilke, and T. Martinetz. Genetic a lgorithms in time- dependent environments. In L. Kallel, B. Naudts, and A. Roge rs, editors, Theoretical Aspects of Evolutionary Computing , New York, 2000. Springer- Verlag. [14] Martin Nilsson and Nigel Snoad. Quasispecies evolutio n on a ﬁtness landscape with a ﬂuctuating peak. eprint physics/0004039, April 2000 . [15] J¨ org Swetina and Peter Schuster. Self-replication wi th errors—A model for polynucleotide replication. Biophys. Chem. , 16:329–345, 1982. [16] Lloyd Demetrius, Peter Schuster, and Karl Sigmund. Pol ynucleotide evolution and branching processes. Bull. Math. Biol. , 47:239–262, 1985. [17] Ellen Baake and Wilfried Gabriel. Biological evolutio n through mutation, selection and drift: An introductory review. Ann. Rev. Comp. Phys. , 7, 1999. in press. [18] Jin Yoshimura and Vincent A. A. Jansen. Evolution and po pulation dynamics in stochastic environments. Res. Popul. Ecol. , 38:165–182, 1996. [19] Ira Leuth¨ ausser. Statistical mechanics of Eigen’s ev olution model. J. Stat. Phys., 48:343–360, 1987. [20] P. Tarazona. Error thresholds for molecular quasispec ies as phase transitions: From simple landscapes to spin-glass models. Phys. Rev. E , 45:6038–6050, 1992. 16/D3/D7 /CX/D0/D0/CP/D8/CX/D2/CV /D4 /CT/CP/CZ /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /D8/CX/D1/CT/B8 /CS/CX/D7 /D6/CT/D8/CT /D8/CX/D1/CT/B8/CA /BP /BC /CA /BP /BC Fig. 1. A landscape with an oscillating peak whose average he ight coincides with the ﬁtness of all other sequences. In continuous time, the lands cape becomes completely ﬂat for R= 0. In discrete time, however, the population feels the geom etric mean of the ﬁtness landscape for R= 0, which has a hole at the position of the peak./CY /D1/D7 /CY/BD/BC /A0 /BG /BD/BC /A0 /BE /BD/BC /A0 /BC/BD/BC /A0 /BJ/BD/BC /A0 /BI/BD/BC /A0 /BH/BD/BC /A0 /BG/CT/D6/D6/D3/D6 /D6/CP/D8/CT /CA/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7/CS/CX/D7 /D6/CT/D8/CT /D8/CX/D1/CT/B8 /A1 /D8 /BP /BD /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /D8/CX/D1/CT/B9/BC/BA/BE /BC/BA/BC/BC/BA/BE/BC/BA/BG/BC/BA/BI/BC/BA/BK/BD/BA/BC/BD/BA/BE/BD/BC /A0 /BK/BD/BC /A0 /BI/BD/BC /A0 /BG/BD/BC /A0 /BE/BD/BC /A0 /BC/CT/D6/D6/D3/D6 /D6/CP/D8/CT /CA/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7 Fig. 2. Order parameter msin a dynamic ﬁtness landscape with a single oscillating peak in a continuous time system and a discrete time system. T he solid lines have been obtained from diagonalizing the full monodromy matrix X, the dotted lines represent the approximation to ﬁrst order in R. We have used the ﬁtness landscape deﬁned in Eq. (23), with a= 8/10,T= 30, and l= 10. The graph shows a snapshot of the order parameter at phase φ= 0 of its limit cycle. The inset shows the same data, but in a log-log plot. There, we have plotted the absolu te value of msfor the discrete-time system, because msassumes a value slightly below 0 in that case. 17/BC /BD/BA/BC /BC/BA/BK /BC/BA/BI /BC/BA/BG /BC/BA/BE/BC /BD/BA/BC /BC/BA/BK /BC/BA/BI /BC/BA/BG /BC/BA/BE /DC/BD /B4 Ꜷ /B5/DC/BE /B4 Ꜷ /B5 /DC/BC /B4 Ꜷ /B5 /D1/D7 /B4 Ꜷ /B5 /BD /A0 /BE /BP/D0/BD /A0 /BE /BP/D0 /D1/D7 /B4 Ꜷ /B5/DC/BC /B4 Ꜷ /B5/D4/CW/CP/D7/CT Ꜷ/BP/CC /BT/BU /DC/BD /B4 Ꜷ /B5/BC /BC/BA/BE/BC/BA/BG/BC/BA/BI/BC/BA/BK/BC /BC/BA/BE/BC/BA/BG/BC/BA/BI/BC/BA/BK/BD/BA/BC/BD/BA/BCFig. 3. Sequence concentrations and order parameter of the s teady state versus the relative phase φ/T. The upper plot (A) shows the predictions for R→0 from the three-concentration model [Eq. (31)], with T= 100, a= 0.4 and l= 10. The line for x2(φ) is indistinguishable from the abscissa. For suﬃciently sm all mutation rates and given Eq. (30), the full numeric solution is in perf ect agreement with the three-concentration model. For larger mutation rates, the main discrepancy arises as a phase shift. For R= 10−4(B), the prediction is still in good agreement with the full numeric solution (shown as circles) if we phase-shi ft our prediction by an amount of ∆ φ/T= 0.088. 18/A1 /D8 /BP /BC /BM /BI/BJ/A1 /D8 /BP /BD /BM /BC/A1 /D8 /BP /BE /BM /BC/BD/BC /A0 /BK/BD/BC /A0 /BI/BD/BC /A0 /BG/BD/BC /A0 /BE/BD/BC /A0 /BC /BC/BA/BC/BC/BA/BE/BC/BA/BG/BC/BA/BI/BC/BA/BK/BD/BA/BC/CT/D6/D6/D3/D6 /D6/CP/D8/CT /CA/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7 Fig. 4. Order parameter in a ﬁtness landscape similar to that of Fig. 2, but with T= 100. The solid line stems from the full continuous time prop agator, and the dots have been calculated from the discrete approximation E q. (33). The number of discretization time steps nis deﬁned as T/∆t. The graph shows a snapshot of the order parameter at phase φ= 0 of its limit cycle. 19
arXiv:physics/0010047v1 [physics.flu-dyn] 19 Oct 2000Magnetic Field Saturation in the Riga Dynamo Experiment Agris Gailitis, Olgerts Lielausis, Ernests Platacis, Serg ej Dement’ev, Arnis Cifersons Institute of Physics, Latvian University LV-2169 Salaspils 1, Riga, Latvia Gunter Gerbeth, Thomas Gundrum, Frank Stefani Forschungszentrum Rossendorf P.O. Box 510119, D-01314 Dresden, Germany Michael Christen, Gotthard Will Dresden University of Technology, Dept. Mech. Eng. P.O. Box 01062, Dresden, Germany (January 9, 2014) After the dynamo experiment in November 1999 [1] had shown magnetic ﬁeld self-excitation in a spiraling liquid m etal ﬂow, in a second series of experiments emphasis was placed on the magnetic ﬁeld saturation regime as the next principal step in the dynamo process. The dependence of the strength of the magnetic ﬁeld on the rotation rate is studied. Vari- ous features of the saturated magnetic ﬁeld are outlined and possible saturation mechanisms are discussed. PACS numbers: 47.65.+a, 52.65.Kj, 91.25.Cw In the last decades, the theory of homogeneous dy- namos has turned out unrivaled in explaining magnetic ﬁelds of planets, stars and galaxies. Enormous progress has been made in the numerical treatment of the so-called kinematic dynamo problem where the velocity vof the ﬂuid with electrical conductivity σis assumed to be given and the behaviour of the magnetic ﬁeld Bis governed by the induction equation ∂B ∂t=∇ ×(v×B) +1 µ0σ∆B. (1) Above a critical value of the magnetic Reynolds num- berRm=µ0σLvwhere Landvdenote a typical length and velocity scale of the ﬂuid, respectively, the obvious solution B= 0 of Eq. 1 may become unstable and self- excitation of a magnetic ﬁeld may occur. The investi- gation of the saturation mechanism which limits the ex- ponential growth of the magnetic ﬁeld is much more in- volved as it requires the simultaneous solution of Eq. 1 and the Navier-Stokes equation ∂v ∂t+ (v· ∇)v=−∇p ρ+1 µ0ρ(∇ ×B)×B+ν∆v(2) where the back-reaction of the magnetic ﬁeld on the ve- locity is included. In Eq. 2, ρandνdenote the density and the kinematic viscosity of the ﬂuid, respectively. A number of recent computer simulations of the Earth’s core has led to impressive similarities betweenthe computed and the observed magnetic ﬁeld behaviour including, most remarkably, ﬁeld reversals [2]. The ac- tual relevance of these simulations for real dynamos is, however, not completely clear as they are either working in parameter regions far from that of the Earth or have resort to numerical trickery as, e.g., the use of anisotropi c hyperdiﬀusivities (for a recent overview, see [3]). Until very recently, a serious problem of the science of hydromagnetic dynamos was the lack of any possibility to verify numerical results by experimental work. This situation changed on 11 November 1999 when in the ﬁrst Riga dynamo experiment self-excitation of a slowly grow- ing magnetic ﬁeld eigenmode was observed for the ﬁrst time in an experimental liquid metal facility [1]. On the background of an ampliﬁed external signal, the growth rate and the rotation frequency of the self-excited ﬁeld were identiﬁed in good agreement with the numerical predictions based on kinematic dynamo theory. Due to some technical problems, the saturation regime was not reached in this ﬁrst experiment. Soon after, Stieglitz and M¨ uller studied self-excitation and the saturation regime in another dynamo facility in Karlsruhe [4]. In the following we will represent the results of a second series of experiments at the Riga facility which were car- ried out during 22-25 July 2000. It was possible to work at considerably lower sodium temperatures and thus at higher electrical conductivities than in the November ex- periment. This fact allowed to study in great detail the dynamo behaviour in the kinematic as well as in the sat- uration regime. Here, we will mainly focus on the results in the saturation regime as they go essentially beyond the results of the November experiment. The Riga dynamo facility consists of three concentric tubes of approximately 3 m length (Fig. 1). In the in- nermost tube of 25 cm diameter a spiraling ﬂow of liquid sodium with a velocity up to 15 m/s is produced by a pro- peller. The sodium ﬂows back in a second coaxial tube and stays at rest in a third outermost tube. For more details of the experimental design see [1] and references therein. 1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 F1 3 4 52 L D1 D2 D3D1=0.25 m D3=0.80 mD2=0.43 m L=2.91 mH1 H2 H3 H4 H5 H6 FIG. 1. The main part of the Riga dynamo facility: 1 - Propeller, 2 - Helical ﬂow region, 3 - Back-ﬂow region, 4 - Sodium at rest, 5 - Thermal insulation F - Position of the ﬂux-gate sensor and the induction coil, H1...H6 - Positions of Hall sensors. As in November 1999, the r-components of the mag- netic ﬁeld outside the dynamo were measured by Hall sensors at eight diﬀerent places, six of which were aligned parallel to the axis. Within the dynamo, close to the in- nermost wall, the z-component of the magnetic ﬁeld was measured by a sensitive magnetometer to study the start- ing phase of ﬁeld generation and the r-component by a less sensitive induction coil recording over all the gener- ation time. -0.2-0.100.10.2 12012513013514014515017501800185019001950Bz [mT] Rotation rate [1/min] t [s]Fluxgate sensor Rotation rate FIG. 2. The signal from the inner ﬂux gate sensor (F in Fig. 1) at the beginning of magnetic ﬁeld self-excitation togeth er with the rotation rate. A typical run is documented in Figs. 2 and 3. Without any external excitation, a rotating ﬁeld starts to grow exponentially around t= 130 s where the rotation rate has reached 1920 min−1(Fig. 2). Starting with thismoment the rotation rate was kept practically constant for 80 s. The whole magnetic ﬁeld pattern rotates around the vertical axis in direction of the propeller rotation, bu t much slower. Hence, each sensor records an AC signal with a frequency of the order 1 Hz. At around t= 240 s the exponential growth goes over into saturation where the magnetic ﬁeld continues to ro- tate with a certain frequency but stays essentially con- stant in its amplitude (Fig 3). Between t= 250 s and t= 350 s a number of diﬀerent rotation rates has been studied, showing a clear dependence of the saturation ﬁeld level on the rotation rate. -1-0.500.51 0100 200 300 400 50002004006008001000120014001600180020002200Br [mT] Rotation rate [1/min] t [s](a)Hall sensor Rotation rate -12-10-8-6-4-2024681012 0 100 200 300 400 50002004006008001000120014001600180020002200dBr/dt [mT/s] Rotation rate [1/min] t [s](b)Induction coil Rotation rate FIG. 3. One of the experimental runs with self-excitation and saturation: Signals measured at the Hall sensor H4 (a), and in the induction coil at position F (b). From in total four experimental runs similar to that shown in Fig. 3, we compiled a number of data con- cerning the dependence of the growth rates (Fig 4a), the frequencies (Fig. 4b), the power consumption of the mo- tors (Fig. 5), and the saturation levels (Fig. 6) on the rotation rate. The ﬁrst observation to be made in Fig. 4 is a sat- isfactory conﬁrmation of the numerical predictions, in particular with respect to the slope of the curves and the temperature dependence. There is a parallel shift of ap- proximately 10 per cent with respect to the growth rate and of approximately 5 per cent with respect to the fre- quency. The shift of the growth rates might be partly ex- 2plained by the fact that the steel walls (with their lower electrical conductivity) were not taken into account in the 2D code underlying the given numerical predictions. From one-dimensional calculations it is known that the general eﬀect of these walls consists in an increase of the critical Rmby about 8 per cent which is also the order of the said parallel shift. In total, there is now a quite coher- ent picture of the kinematic regime in which the results from the November experiment and the recent results ﬁt together. -2.5-2-1.5-1-0.500.51 1200 1400 1600 1800 2000 2200Growth rate [1/s] Rotation rate [1/min](a)Pred. at 150 °C 200°C 250°C Meas. at 155 °C 172°C 210°C 0.250.50.7511.251.5 1200 1400 1600 1800 2000 2200Frequency [1/s] Rotation rate [1/min](b)Pred. at 150 °C 200°C 250°C Meas. at 155 °C 172°C 210°C Sat. at 160 °C 166°C 174°C FIG. 4. Growth rates (a) and frequencies (b) for diﬀerent rotation rates and temperatures in comparison with the nu- merical prediction. For the sake of clarity, data measured a t close temperatures are grouped together. The shown tem- peratures are to be understood as ±3◦C. The two rightmost points at 210◦C are taken over from the November experi- ment. The larger error bars at low rotation rates are due to the fact that for a quickly decaying signal the frequency and the decay rate are much harder to determine than for a slow decaying or an increasing signal. In (b) the frequencies in t he saturation regime are also given, whereas the correspondin g growth rates are (by deﬁnition) zero. As for the saturation regime there is an interesting ob- servation to be made in Fig. 4b. Whereas in this regime the growth rates are by deﬁnition zero, the frequencies continue to increase with the rotation rate. Obviously, the frequency values are even a bit higher than what would be expected from a simple extrapolation of thedata in the kinematic regime. This fact indicates that there is not only an overall brake of the sodium ﬂow by the Lorentz forces but that the spatial structure of the ﬂow is changed, too. We will come back to this point later on. Concerning the motor power increase in the satura- tion regime in Fig. 5 we show for one experimental run the motor power consumption as a function of the ro- tation rate Ω. The points in Fig 5 correspond to those time intervals where the rotation rate was kept constant. The shown data in the ﬁeld-free regime are best ﬁtted with an Ω3curve which is also expected from simple hydraulic arguments. As for the four rightmost points in the saturation regime, there is evidently an increase of the power consumption which must be explained in terms of the back-reaction of the magnetic ﬁeld. In or- der to estimate this additional power the Ohmic losses POhm=/integraltext j2/σ dV have to be computed. Using the magnetic ﬁeld structure as it results from the kinematic code and ﬁxing its strength to the measured ﬁeld strength we get Ohmic losses of approximately 10 kW which is a good approximation of the observed power consumption increase. 6080100120140 1600 1700 1800 1900 2000 2100Mechanical motor power [kW] Rotation rate [1/min]Field free With saturation FIG. 5. Motor Power below and in the saturation regime However, as already mentioned in connection with the observed frequency behaviour, the power increase due to magnetic forces seems to be not the whole story. An- other feature of the ﬁeld might help to identify a fur- ther saturation mechanism. In Fig. 6 we have plotted four saturation levels of the radial magnetic ﬁeld com- ponent measured at diﬀerent positions in dependence of the temperature corrected rotation rate Ω c. Three of the measurements where carried out at the Hall sensors H2, H4, H6 (see Fig. 1). The fourth ﬁeld is inferred from the induced voltage (divided by the frequency) in the in- duction coil at position F. The magnitude of the radial component reaches 7 mT. From numerical simulations of the kinematic regime it is known that the axial compo- nent at the given position is by a factor ﬁve larger than the radial component. Hence a value of about 30 mT 3seems to be a reasonable estimate for this z-component inside the dynamo. The correction of the rotation rates has been done the following way: at ﬁrst note that at a temperature of 157◦C and at a rotation rate of 1840 min−1an extremely slowly decaying mode was observed in one run which will serve as a deﬁnition of a marginal state. Saturation was observed, however, at quite diﬀer- ent temperatures. Therefore, the measured rotation rate Ω at the temperature Twas ﬁrst corrected correspond- ing to Ω c(T) =σ(T)/σ(157◦C)Ω(T). By virtue of this deﬁnition, the corrected rotation rate Ω cis proportional to the magnetic Reynolds number Rm. The four diﬀer- ent saturation levels in Fig. 6 are ﬁtted by some curves proportional to (Ω c- 1840 min−1)β(this is no pre-justice for a corresponding behaviour close to Ω =1840 min−1; a slightly sub-critical behaviour close to the marginal poin t cannot be excluded by our data). The diﬀerent exponents give clear evidence for a remarkable redistribution of the magnetic energy towards the propeller region. In trying to explain the two features of the magnetic ﬁeld which go beyond an overall braking of the ﬂow by the magnetic pressure, namely, the relatively high fre- quency and the redistribution of magnetic energy towards the propeller region, one can utilize numerical results for those ﬂows with an azimuthal velocity component which is decaying along the z-axis. To begin with, note that the self-excited magnetic ﬁeld with its azimuthal depen- dence exp( imφ) with m= 1 gives rise to Lorentz forces withm= 0 and m= 2. Of the former, which will be the only one of interest in the following, there are com- ponents which can be absorbed into pressure derivatives inzandrdirection. The ϕcomponent, however, re- mains. The perturbation δvφof the azimuthal velocity component arising from the Lorentz forces can be esti- mated from Eq. 2 to be in the order of 1 m/s, if one takes into account the measured magnetic ﬁeld. It is in- teresting to observe that the numerical simulation of the inﬂuence of the azimuthal velocity changes along the z- axis for the kinematic dynamo [5] results in similar mag- netic ﬁeld redistributions as shown for the saturation in Fig. 6. Hence, this decay might indeed be relevant and it is therefore the favorite candidate to account for the observed features of the magnetic ﬁeld in the saturation regime.00.20.40.60.81 1800185019001950200020502100215001234567BHall [mT] BInduction [mT] Corrected rotation rate [1/min]Hall sensor 2 Hall sensor 4 Hall sensor 6 Induction coil FIG. 6. Measured magnetic ﬁeld levels in the saturation regime. The corrected rotation rate Ω cis related to the mea- sured rotation rate Ω via Ω c(T) =σ(T)/σ(157◦C) Ω. The lines are interpolating curves of the form (Ω c−1840 min−1)β withβ= 0.417 for H2, β= 0.275 for H4, β= 0.216 for H6, β= 0.205 for the induction coil. The study of the saturation regime will be continued by means of numerical simulations in tight connection with a sort of inverse dynamo approach based on the measured data and, hopefully, with direct velocity measurements inside the dynamo facility. We thank the Latvian Science Council for support un- der grant 96.0276, the Latvian Government and Interna- tional Science Foundation for support under joint grant LJD100, the International Science Foundation for sup- port under grant LFD000 and Deutsche Forschungsge- meinschaft for support under INK 18/B1-1. [1] A. Gailitis et al., Phys. Rev. Lett. 84, 4365 (2000) [2] G. A. Glatzmaier and P. H. Roberts, Nature 377, 203 (1995) [3] F. H. Busse, Annu. Rev. Fluid Mech. 31, 383 (2000) [4] R. Stieglitz and U. M¨ uller, in Proceedings of the inter- national conference ”Magnetohydrodynamic at Dawn of third Millenium” , Giens, France, September 17-22, 2000 , edited by A. Alemany, F. Plunian, (Grenoble, 2000), 175 [5] F. Stefani, G. Gerbeth, and A. Gailitis, in Proceedings of the International Workshop on Laboratory Experiments on Dynamo Action, Riga, June 14-16, 1998 , edited by O. Lielausis, A. Gailitis, G. Gerbeth, F. Stefani, (FZ Rossendorf, 1998) 4
Computer simulation approach to reliability and accuracy in EXAFS structural determinations Paolo Ghigna, Melissa Di Muri, and Giorgio Spinolo Dipartimento di Chimica Fisica, INCM and C.S.T.E./CNR, Università di Pavia I 27100 - Pavia (Italy) Synopsis The fit of a set of simulated noisy EXAFS spectra samples the frequency distribution of structural parameters and gives their statistical estimators (mean, dispersion, correlations). Abstract The frequency distribution of different parameters of an EXAFS spect rum can be directly sampled by analysing a population of simulated spectra produced by adding computer -generated noise to a reference pattern. The procedure gives statistical estimators of the parameters obtained with different data processing strategies t o test the performance of a strategy, to evaluate the bias introduced by random noise, and to clarify the amount of information actually contained in an experimental spectrum. Results are given for the two simple local structures of an Ag atom surrounded by two oxygens or by six iodines. 2 1. Introduction EXAFS (Extended X -Ray Absorption Fine Structure) is well recognised as a powerful and now popular tool for investigating local structural features in different kinds of materials. Strong points of the technique are its well assessed theoretical background [see, for instance, Rehr 2000], the ability to provide information on the environment of a chemically defined scattering centre, and the possibility to deal with different kinds of materials, fr om liquids or disordered solids to well crystallised samples. The popularity of EXAFS determinations has also greatly increased in recent years because of the rapid diffusion of specialised software [Filipponi DiCicco & Natoli 1995, Filipponi & Di Cicco 19 95, Binsted 1998, Rehr 1994 and recent versions of the FEFF programme]. Aim of this paper is to give a first contribution to the problems concerning accuracy and reliability of these interesting determinations. Indeed, these aspects have received so far on ly little attention and, to the knowledge of the present Authors, only a few papers [ Incoccia and Mobilio, 1984, Lytle et al. , 1989, Filipponi, 1995, Curis and Benazeth, 2000, Krappe and Rossner, 2000, ] are available on this regard in the literature. Indee d, there are many particular aspects which make not so straightforward a conventional error analysis of EXAFS data processing. To list just the most obvious ones, there is first the effect of preliminary processing steps such as pre -edge and post - edge back ground removal, which heavily rely on empirical models, and on some amount of subjective evaluation. Another source of error is possibly related to the frequent practice of combining k-space fit with Fourier -space analysis and windowing, and also to the ne ed of introducing some threshold value above which the coordination shells are not taken into account. Finally, different k-space weighting schemes are typically 3 applied not only for the purpose of data presentation and inspection, but are also used in data processing, a practice which does not seem based on sound principles. Other difficulties, which are indeed typical of many full -spectrum structure -based data fit procedures, are due to the strong non -linear relation between experimental data and model p arameters, and to the high degree of correlation between model parameters. This can reasonably produce bias and non -parabolic errors. The approach that is used in this paper has been applied two years ago by the same group to the problem of accuracy and re liability of powder pattern structure refinement [Dapiaggi et al. 1998]. Only the most general ideas and the specific details will be discussed here, while reference is made to the previous work for a deeper discussion of the method. In general, aim of the approach is to know what amount of information is actually contained in an EXAFS spectrum and can be easily retrieved by routine work and, on the contrary, what information is deeply masked by noise (and by the particular structural model) and therefore r equires a specialised strategy in data acquisition and analysis, or cannot be retrieved by whatever kind of data treatment. This task is achieved, as suggested in the well known book by Press et al. (1988), with computer simulated experiments. According to this approach, a ‘large’ number of synthetic data sets is prepared: each set of data differs from the others because of random errors added by computer simulation (according to assumed statistical distribution laws) to the same reference spectrum built fr om an assumed structural model and assumed (‘true’) parameters. Fitting all data sets to the model gives a population of parameters, and the frequency distribution over the computer generated population is used as a numerical sampling of the underlying pr obability distribution. 4 2. Outline As a general rule, we kept things as simple as possible. Consequently, some problems of a real EXAFS local structure determination have not been investigated in this preliminary approach. To quote just two wel l-known problems, we have considered only very simple models containing a single coordination shell, and we kept simple both the pre -edge and post -edge backgrounds. Broadly speaking, the present paper investigates only the effect of random noise. 2.1. Reference XAS spectra The reference spectra have been deterministically produced from assumed models and from a given set of ‘true’ parameters. Two different models are discussed in the present work. In the first one, the photoabsorber (Ag) is surrou nded by a small number (two) of light neighbours (O), in the second one the same photoabsorber is surrounded by a larger number (six) of heavy neighbours (I). As stated before, an unique shell around the photoabsorber is considered in both models. The GNXA S package (Filipponi, DiCicco & Natoli 1995, Filipponi & Di Cicco 1995) has been used to calculate the reference EXAFS of both models. The reference XAS spectrum was built using a Debye -Waller factor equal to exp( -2·a2·k2), with a2 = 0.01Å2 and adding: a) a pre -edge background modelled with a straight line, b) an edge step modelled with an arctg function broadened for the finite core -hole lifetime corresponding to a FWHM of 8.123 eV and with an edge jump set to J = 1 and centred at E0=25516.5 eV; c) a post edge background modelled using the linearized hydrogenic model for the atomic background absorption: J·(1-8v/3) with v = (E-E0)/E0. The 5 presence of XANES structures on and near the edge was deliberately ignored for the sake of simplicity. Each reference XA S spectrum so obtained will be referred to in the following as muref(E). 2.2. Synthetic ‘experimental’ spectra Starting from each reference spectrum, a set of 50 noisy spectra were simulated using a random number generator for adding to each point of the reference spectrum a Gaussian noise in the range muref(E)(1+/-10-4) as an approximation of a Poissonian noise for a counting rate of 108 counts/s. We assume that this is a realistic noise level for a “good” experimental spectrum that can be routinely obtained at a third generation synchrotron radiation source. 2.3. Fitting procedure Each synthetic XAS spectrum has been fitted using again the GNXAS package. In particular, the program performs the following preliminary operations (not repeated successively): 1) linear fit of the pre -edge region; 2) calculation of the first derivative of the absorption signal and determination of the edge position from the maximum; 3) preliminary post edge background extraction, fitting the post edge signal with beta-splines and without considering the structural parameters. The program then compares the post edge absorption with a model which is composed by a smooth post -edge background part ( beta-splines) plus a structural oscillating part which is actually the EXAFS . 6 2.4. Evaluation of structural parameters A total number of twenty four different sets of fit parameters have been produced in correspondence to each different combination concerning: 1. Kind of model (AgO 2 or AgI 6), 2. Freely adjustable parameters: a) n (coordination number), a2 (Gaussian part of the Debye -Waller factor), r (shell radius), E0,T (k-space origin), with J (edge jump) fixed at its theoretical value of 1, or b) the previous set of adjustable parameters and J. 3. Sampling range in k space: a) 2 – 12 Å-1, or b) 3.5 – 12 Å-1, or c) 5 – 12 Å-1, for the AgO 2 model; a) 2 – 16 Å-1, or b) 3.5 – 16 Å-1, or c) 5 – 16 Å-1, for the AgI 6 model. 4. Weighting exponent in k space: a) fit of chi (k) [which should be the correct choice in the case of a constant nois e], or b) fit of k3.chi(k) [which corresponds to what is more usually made in the EXAFS community]; i.e. fit of km. chi (k) with m = 0 (case a) or m = 3 (case b). In the followings, a fit according for a particular case among all these options will be concisely referenced, for instance, with “ J/2-12/3” to indicate that the pertinent model is fitted including the edge jump among the freely adjustable parameters, sampling the k space between 2 and 12 Å-1, and using m = 3 weights, or with “ J=1/5- 16/0” to indicate th at the pertinent model is fitted without including the edge jump among the freely adjustable parameters, sampling the k space between 5 and 16 Å-1, and using m = 0 weights. 7 3. Results and discussion We start our discussion with the AgI 6 model. For this model, the upper part of Fig. 1 gives an example of a single simulated EXAFS (dots) and its fit (continuous line) obtained with the J/2-16/3 procedure. The corresponding Fourier transforms are shown in the lower part of the same figure. By conside ring the whole set of 50 simulated EXAFS and the same procedure, one obtains a population of 50 { J, r, n, E0T, a2} or { r, n, E0T, a2} parameters. From this population, several statistical estimators (mean values, standard deviations, …) are obtained. Table 1 summarises the results concerning mean values and standard deviations for the J/2-16/3 fit procedure. It might be interesting to note that all parameters (with the notable exception of E0,T) are biased by amounts that are much larger than the sample standard deviations. The bias, however, is around 6% for the coordination number, well below 0.01 Å for the coordination distance, and below 2% for the Debye -Waller factor. All these biases are nicely smaller than what is usually believed to be a typical EXAF S error on these parameters. Moreover, the bias is around 10% for the edge jump. Deeper insights can be obtained by exploring the scatter plots (Fig. 2 -6) where the values of couples of parameters obtained from the fit of a particular reference model and according to a fixed fit strategy (again: J/2-16/3) are plotted against each other for the whole set of simulated experiments. These figures show, for instance, that edge jump and Debye -Waller factor appear as only weakly correlated variables (fig. 2), whil e edge jump and coordination number are much more heavily correlated to each other (see Fig. 3). Concerning the latter couple, it is also interesting to note that the fit results are heavily displaced on the scatter plot from the ‘true’ values and that the ir joint correlation 8 line, when extrapolated to J = 1, gives a much more accurate n value. This suggests that taking the edge jump as a fixed parameter produces a much better value for the coordination number. Fig. 7 - 11 compare mean values and standard er rors for different fit procedures and for the simulated experiments on AgI 6. On these plots, the error bars are the standard errors calculated over the sample of computer simulated experiments, and the horizontal lines are the true values. The x axis is si mply a label of the fit procedure and the correspondence is as follows: 1 : J=1/2-16/3; 2 : J=1/3.5 -16/3; 3 : J=1/5-16/3; 4 : J=1/2-16/0; 5 : J=1/3.5 -16/0; 6 : J=1/5-16/0; 7 : J/2-16/3; 8 : J/3.5-16/3; 9 : J/5-16/3; 10 : J/2-16/0 11 : J/3.5-16/0; 12 : J/5-16/0. In some detail, Figs. 7 and 8 show the results obtained for J and n. When J is fixed to its true value ( x-labels 1 to 6), the coordination numbers obtained are quite close to the real value, with a bias that is close to the statistical error and well below 20 % (which is commonly believed to be a good estimate of the error in the coordination numbers as determined by EXAFS). In addition, the statistical error is almost independent on the weighting scheme and on the range of k space used in the fit. On the contrary, allowing J to float (cases 7 - 12) introduces a bias in the determination of coordination numbers which is much greater than the statistical spread (but in any case below 20 %). The correlation between n and J is well known and is due to th e fact that they both contribute to the EXAFS as normalising factors. In particular, it is well apparent from Figs. 7 and 8 that, when J is allowed to float, the n values obtained from the fit are almost always greater than the true value, while the J valu es are always lower. Moreover, J·n (which is the actual normalizing factor of the EXAFS) is regularly underestimated (from the mean values of Table 1, < J·n> = 5.85 instead of 6). In our 9 opinion, this last fact is due to broadening of the edge caused by the finite lifetime of the core hole. The broadened edge jump is then accounted by the beta-splines used for fitting the post -edge background in form of a lower trend of the average signal just after the edge. Then, the extrapolation to the edge energy of this ‘wrong’ trend produces an underestimation of J (and J·n) which is calculated as the d ifference between post -edge and pre -edge backgrounds at the edge. An indirect proof of the above understanding has been obtained by considering a restricted number of simulated experiment based on a reference EXAFS with a stepped edge jump (instead of the rounded arctg shape). In this case, the fit gives much more accurate parameters: < J> = 0.98, < n> = 6.07 and < J·n> = 5.95. The a2 parameter (see Fig. 9) is almost independent on the actual fit conditions and is always quite well determined, both the bias an d the statistical error being of the order of few percents. This despite the fact that there are correlations of a2 both with J and n. Seemingly, what is important here is the total correlation of all the three parameters a2, J and n; J and n may well be h eavily biased, but their combination is accurate enough to allow an accurate determination of a2. The E0T parameter is seemingly well recovered (Fig. 10) for all fit procedures. Concerning the r parameter (Fig. 11), the bias is well above the statistical e rror. E0T and r are strongly correlated, but this correlation is not enough to explain the bias on r. We think that this is (at least partially) an indirect effect of the disorder ( a2), because in calculating the mean geometrical configuration the fitting program uses other factors in addition to the simple Debye -Waller weight coming from the a2 factor: this produces small phase and amplitude shifts which possibly are at the origin of the bias experimentally observed. Of course, this second order effect is expected to vanish in the limit a2 -> 0. Indeed, using a reference model with a ten times lower a2 we actually 10 obtained a six times lower bias on r. It should be noted, however, that the (systematic) error affecting r is of the order of a few thousandths o f Å, that is well below the value (1-2 hundredths of Å) which is usually assumed as a reasonable estimate of the error in the determination of the bond distances by EXAFS. Finally, for what concerns more generally the correlations between different paramet ers, it can be said that the case of the J/2-16/3 procedure is well representative of the results for the different procedures. The results for the AgO 2 model are shown in Fig.s 12 -16. The results for this model are qualitatively similar to those of the Ag I6 model, but in general correlations, biases and the statistical errors are greater. For example, the systematic error which affects r can be of the order of 1 -2 hundredths of Å, and that which affects n can reach the value of 20 % if J is allowed to floa t. This result is expected and is due to the much lower level of the EXAFS oscillations for this model. 11 4. Conclusions The most important result of the present simulation approach is that random noise plays a minor role with respect to systemat ic errors in determining accuracy, precision and correlation of the structural parameters obtained from analysis of an EXAFS spectrum. This inference is well apparent when one considers, on one side, that the simulated experiments only account for random n oise and, on the other side, that for most parameters the statistical spread here obtained is typically much lower than what is reported in the scientific literature as the typical error of a real EXAFS analysis. The conclusion is also supported by the sim ulation results alone, by considering the significant difference (for some parameters) between bias and statistical spread. If we now remind that the background is here modelled only in a drastically simplified way, while a real EXAFS spectrum is much more difficult to analyse on this regard, it seems reasonable to infer that the most important source of error in the analysis of an EXAFS spectrum is related to the procedure of (pre -edge and/or post - edge) background subtraction. On this regard, the computer simulation also makes clear the importance of the a priori knowledge of the correct value of the edge jump ( J), which is essential for an accurate determination of n. It is therefore suggested to start an EXAFS analysis only after having obtained an indepe ndent and reasonably accurate value of the edge jump, for instance by comparison with the spectra of proper standards of precisely known coordination number or, more directly, by accurately determining the amount of photo -absorber atoms in the sample. Finally, the present simulation approach show that the accuracy of the fitted parameters is practically independent both of the weighting scheme and of the fitted range in k space. The latter aspect practically affects only the determination of a correct 12 value of J if – contrary to the suggested procedure – one wants to keep this parameter free to change during fitting. Acknowledgements The Authors want to thank Adriano Filipponi (Università dell’Aquila, Italy) for his deep insight into the details of the GNXAS programme and for stimulating discussions on aim and results of the present work. This work has been partially supported by the Department of University and Scientific and Technological Research of the Italian Government (MURST -40%). 13 References Binsted, N ., Gurman, S. J., Campbell, T. C, and Stephenson, P. C.(1998): EXCURV98: SERC - Daresbury Laboratory. Curis, E. and Benazeth, S. (2000). J. Synchrotron Rad . 7, 262. Dapiaggi, M., Anselmi -Tamburini, U., and Spinolo, G. (1998). J.Appl.Cryst. 31,379- 387. Filipponi, A., Di Cicco, A., and Natoli, C.R. (1995). Phys.Rev. B 52, 15122. Filipponi, A. and Di Cicco, A. (1995). Phys.Rev. B52, 15135. Filipponi, A. (1995). J.Phys.Condensed Matter 7,9343. Incoccia, L. and Mobilio, S. (1984). Il Nuovo Cimento , 3 D (5), 867. James, F. & Roos, M., CERN Computer Centre Program Library, Program D 506 Krappe, H. J. and Rossner, H. H. (2000). Phys Rev. B 61 , 6596. Lytle, F.W., Sayers, D.E. , and Stern, E.A. (1989). Physica B 158, 701. Press W. H., Flannery B. P., Teukolsky S. A. & Vetterling W. T. (1988). Numerical recipes in C, Cambridge University Press. Rehr, J.J., Booth, C.H., Bridges, F., and Zabinsky, S.I. (1994). Phys.Rev. B49, 12347. Rehr, J. J. (2000). Rev. Mod. Phys . 72, 621. 14 Table 1 Mean values and standard errors over the sample of simulated experiments obtained on the AgI 6 model with the J/2-16/3 fit procedure. Parameter True value Mean value from simulation Std. error n 6. 6.43 0.02 r (Å) 3.1 3.1059 2·10-4 E0,T (eV) 25516.5 25516.49 0.01 J 1 0.9082 0.8·10-4 a2(Å2) 0.01 9.86·10-3 4·10-5 15 Fig. 1 - Example of a single simulated EXAFS (dots) and its fit (continuos line) obtained as described in the text (upper panel). The lower panel shows the modulus of the corresponding Fourier Transform. r (Å)0 2 4 6\|FT[k3?(k)\| (a. u.) 01020k (Å-1)0 2 4 6 8 10 12 14 16 18k3?(k) (a. u.) -5-4-3-2-1012345 16 [a/a(true)]20.97 0.98 0.99 1.00J/J(true) 0.9060.9070.9080.9090.9100.9110.912 Fig. 2 - Correlation between the single pattern evaluations of two parameter values [Gaussian part of the Debye -Waller factor ( a2) and edge jump ( J)]. Each parameter is normalized using its true value. 17 n/n(true)1.05 1.06 1.07 1.08 1.09J/J(true) 0.9060.9070.9080.9090.9100.9110.912 Fig. 3 - Correlation between the single pattern evaluations of two parameter values [coordination number ( n) and edge jump ( J)]. Each parameter is normalized using its true value. 18 n/n(true)1.05 1.06 1.07 1.08 1.09[a/a(true)]2 0.9700.9750.9800.9850.9900.995 Fig. 4 - Correlation between the single pattern evaluations of two parameter values [coordination number ( n) and Gaussian part of the Debye -Waller factor ( a2)]. Each parameter is normalized using its true value. 19 r/r(true)1.0017 1.0018 1.0019 1.0020 1.0021E0T/E0T(true) 0.9999980.9999991.0000001.0000011.000002 Fig. 5 - Correlation between the sin gle pattern evaluations of two parameter values [shell radius ( r) and origin of the calculated EXAFS ( E0T)]. Each parameter is normalized using its true value. 20 r/r(true)1.0017 1.0018 1.0019 1.0020 1.0021J/J(true) 0.9060.9070.9080.9090.9100.9110.912 Fig. 6 - Correlation between the single pattern evalua tions of two parameter values [shell radius ( r) and edge jump ( J)]. Each parameter is normalized using its true value. 21 Fig. 7 - Edge jump J as determined from the different fitting procedures ( x-labels, see text), for the AgI 6 model. The horizontal line marks the true value, and the error bars are the statistical errors calculated over the sample of 50 synthetic experiments. The J parameter has been kept fixed at its true value = 1 in the first six fitting procedures. 123456789101112J 0.880.920.961.00 22 Fig. 8 - Same plot as in Fig. 7, b ut for the coordination number n. 123456789101112n 5.76.06.36.6 23 Fig. 9 - Same plot as in Fig. 7, but for the Gaussian part of the Debye -Waller factor a2. 123456789101112a2 (Å2) 0.00960.00980.01000.01020.0104 24 Fig. 10 - Same plot as in Fig. 7, but for the origin of the calculated EXAFS ( E0T). 123456789101112E0,T (eV) 25516.025516.525517.0 25 Fig. 11 - Same plot as in Fig. 7, but f or the shell radius r. 123456789101112r (Å) 3.0983.1003.1023.1043.1063.108 26 Fig. 12 - Same plot as in Fig. 7, but for the AgO 2 model. 123456789101112J 0.880.920.961.00 27 Fig. 13 - Same plot as in Fig.8, but for the AgO 2 model. 123456789101112n 1.92.02.12.22.32.4 28 Fig. 14 - Same plot as in Fig. 9, but for the AgO 2 model. 123456789101112a2 (Å2) 9.0e-31.0e-21.1e-21.2e-2 29 Fig. 15 - Same plot as in Fig. 10, but for the AgO 2 model. 123456789101112E0,T (eV) 2551425515255162551725518 30 Fig. 16 - Same plot as in Fig. 11, but for the AgO 2 model. 123456789101112r (Å) 2.3032.3102.3172.324
arXiv:physics/0010049v1 [physics.atom-ph] 20 Oct 2000Generalized Radial Equations in a Quantum N-Body Problem Zhong-Qi Ma∗, Bing Duan, and Xiao-Yan Gu Institute of High Energy Physics, Beijing 100039, The Peopl e’s Republic of China We demonstrate how to separate the rotational degrees of fre edom in a quantum N-body problem completely from the internal ones. It is shown that any common eigenfunction of the total orbital angular m omentum ( ℓ) and the parity in the system can be expanded with respect to (2 ℓ+ 1) base-functions, where the coeﬃcients are the functions of t he internal vari- ables. We establish explicitly the equations for those func tions, called the generalized radial equations, which are (2 ℓ+ 1) coupled partial diﬀerential equations containing only (3 N−6) internal variables. PACS number(s): 11.30.-j, 03.65.Ge, and 03.65.Fd Symmetry is an important property of a physical system. The s ymmetry of a quantum system can simplify its Schr¨ odinger equation and remove so me variables in the equation. The simplest example is the hydrogen atom problem, where, du e to the spherical sym- metry, the wavefunction is expressed as a product of a radial function and a spherical harmonic function, Ψℓ m(r) =φ(r)Yℓ m(θ,ϕ), (1) and the Schr¨ odinger equation reduces to a radial equation w ith only one radial variable. For a quantum N-body problem with a pair potential, the Schr¨ odinger equat ion is in- variant under the spatial translation, rotation, and inver sion. From those symmetries, one should be able to separate the motion of center-of-mass a nd the global rotation of the system from the internal motions so as to reduce the Schr¨ odinger equation to the generalized ”radial” equation that contains only internal variables. However, this prob- lem has not been solved. In this letter we will solve this prob lem completely. Using the appropriately chosen (3 N−6) internal variables and the (2 ℓ+ 1) base-functions for the total orbital angular momentum ℓ, we establish explicitly the generalized radial equations ∗Electronic address:MAZQ@IHEP.AC.CN 1without any approximation. Only (3 N−6) internal variables are involved in both the generalized radial functions and the equations. Denote by rjthe position vectors of N-particles with masses mjin the laboratory frame (LF), respectively. The Schr¨ odinger equation for th eN-body problem is −/parenleftBig ¯h2/2/parenrightBigN/summationdisplay j=1m−1 j△rjΨ +VΨ =EΨ, (2) whereVis assumed to be a pair potential, depending on the distances of each pair of particles. Therefore, the potential Vis a function of only the internal variables. It is well known that, due to the translation symmetry of the system, th e motion of center-of-mass can be separated completely from others by making use of the J acobi coordinate vectors in the center-of-mass frame (CF) [1-3], Rk=/parenleftbiggmkWk+1 Wk/parenrightbigg1/2 rk−N/summationdisplay j=k+1mjrj Wk+1 ,1≤k≤(N−1). (3) whereWj=/summationtextN t=jmt. In CF, the Laplace operator and the total orbital angular mo men- tum operator Lcan be directly expressed with respect to Rk: △=N/summationdisplay j=1m−1 j△rj=N−1/summationdisplay k=1△Rk, L=−i¯hN/summationdisplay j=1rj× ▽rj=−i¯hN−1/summationdisplay k=1Rk× ▽Rk, (4) The Laplace operator obviously has the symmetry of O(3N−3) group with respect to (3N−3) variables. The O(3N−3) group contains a subgroup SO(3)×O(N−1), where SO(3) is the usual rotational group. The space inversion and th e diﬀerent deﬁnitions for the Jacobi vectors in the so-called ”Jacobi tree” [4] can be obtained by O(N−1) transformations. For the system of identical particles, th e permutation group among particles is also a subgroup of O(N−1) group. Because of the spherical symmetry, the angular momentum is c onserved. The hydrogen atom problem is a typical quantum two-body problem, where th ere is only one Jacobi coordinate vector, usually called the relative position ve ctorr. For a quantum N-body problem, equation (1) should be generalized in three aspect s. The ﬁrst is how to deﬁne the internal variables, which describe the internal motion s completely. The second is 2how to ﬁnd the complete set of the independent base-function s with the given angular momentum. The total wavefunction is expanded with respect t o the base-functions, where the coeﬃcients are the generalized radial functions which o nly depend on the internal variables. The last is how to derive the generalized radial e quations that only contain (3N−6) internal variables. As a matter of fact, these three aspec ts are connected. The parity should also be considered in the generalization. Due to the spherical symmetry, one only needs to study the eigenfunctions of angular momentum w ith the largest eigenvalue ofL3(m=ℓ), which are simply called the wavefunctions with the angula r momentum ℓin this letter for simplicity. Their partners with the small er eigenvalues of L3can be calculated from them by the lowering operator L−. Denote byR=R(α,β,γ ) a spatial rotation, transforming CF to the body-ﬁxed frame (BF), and by ξall the internal variables in a quantum N-body problem for simplicity. Although Wigner did not separate the motion of center-of-ma ss by the Jacobi vectors, he proved from the group theory that any wavefunction with the a ngular momentum ℓin the system can be expressed as follows (see Eq. (19.6) in [5]) : Ψℓ ℓ(α,β,γ,ξ ) =ℓ/summationdisplay q=−ℓDℓ ℓq(α,β,γ )∗ψq(ξ), (5) where we adopt the commonly used form of the D-function [6]. In Eq. (5) Dℓ ℓq(α,β,γ )∗ plays the role of the base-function with the angular momentu mℓ, andψq(ξ) is the gener- alized radial function. What Wigner proved is that there are only (2ℓ+ 1) independent base-functions with the angular momentum ℓ. Unfortunately, due to the singularity of the Euler angles, the generalized radial equations are very diﬃcult to derive based on Eq. (5). Wigner did not discuss the generalized radial equat ions, and to our knowledge, those equations have not yet been established in the literat ure. It is obvious that the generalized radial equations are very easy to obtain for the Swave [7]. However, it seems quite diﬃcult to obtain even for Pwave in a three-body problem [8,9]. Recently, a coupled angular momentum basis was used to predi agonalize the kinetic energy operator [10], where some oﬀ-diagonal elements rema in non-vanishing. In their calculation, the function with a given angular momentum was combined from the partial angular momentum states by the Clebsch-Gordan coeﬃcients. Since the partial angular momenta are not conserved, one has to deal with, in principle , an inﬁnite number of 3the partial angular momentum states. This problem also occu rs in the hyperspherical harmonic function method and its improved ones [2,4,11,12] . However, as Wigner proved, only (2ℓ+ 1) partial angular momentum states are involved in constru cting the base- functions with the angular momentum ℓ. Arbitrarily choose two Jacobi coordinate vectors, say R1andR2. LetR1be parallel with theZ-axis of BF, and R2be located in the XZplane with a non-negative X- component in BF. The rotational degrees of freedom of the sys tem are described by a rotationR(α,β,γ ), transforming CF to BF. Deﬁne (3 N−6) internal variables, which should be invariant in the global rotation R(α,β,γ ): ξj=Rj·R1, η j=Rj·R2, ζ j=Rj·(R1∧R2),1≤j≤(N−1),(6) whereη1=ξ2andζ1=ζ2= 0. It is worth mentioning that ξjandηjhave even parity, butζjhas odd parity. From them we have Ωj= (R1∧Rj)·(R1∧R2) =ξ1ηj−ξ2ξj, ωj= (R2∧Rj)·(R1∧R2) =ξ2ηj−η2ξj, Rj·Rk= Ω−1 2(Ωjηk−ωjξk+ζjζk), (7) where Ω 1=ω2= 0, and Ω 2=−ω1= (R1∧R2)2. Recall that two Jacobi vectors R1andR2completely determine BF and three Euler angles. The base-functions with the angular momentum ℓshould be combined from the products of two spherical harmonic functions Yq m(R1) andYp m′(R2) by the Clebsch-Gordan coeﬃcients /an}bracketle{tq,m,p,m′\|ℓ,(m+m′)/an}bracketri}ht. Deﬁne [3,11] Qℓτ q(R1,R2) =(R11+iR12)q−τ(R21+iR22)ℓ−q (q−τ)!(ℓ−q)!{(R11+iR12)R23−R13(R21+iR22)}τ, τ≤q≤ℓ, τ = 0,1. (8) whereRjais theath component of the Jacobi vector Rj.Qℓτ q(R1,R2) is the common eigenfunction of L2,L3,△Rk, and the parity with the eigenvalues ℓ(ℓ+ 1),ℓ, 0, and (−1)ℓ+τ, respectively. As a matter of fact, the following combinati on of products of two spherical harmonic functions is proportional to Qℓτ q(R1,R2) /summationdisplay mξq/2 1Yq m(R1)η(ℓ−q+τ)/2 2Yℓ−q+τ ℓ−m(R2)/an}bracketle{tq,m,(ℓ−q+τ),(ℓ−m)\|ℓ,ℓ/an}bracketri}ht=CQℓτ q(R1,R2),(9) 4whereCis a normalization factor. Now, we come to the theorem. Theorem . Any function Ψℓλ ℓ(R1,· · ·,RN−1) with the angular momentum ℓand the parity (−1)ℓ+λin a quantum N-body problem can be expanded with respect to Qℓτ q(R1,R2) with the coeﬃcients ψℓλ qτ(ξ,η,ζ), which depend on (3 N−6) internal variables: Ψℓλ ℓ(R1,· · ·,RN−1) =1/summationdisplay τ=0ℓ/summationdisplay q=τψℓλ qτ(ξ,η,ζ)Qℓτ q(R1,R2), (10) ψℓλ qτ(ξ,η,ζ) =ψℓλ qτ(ξ1,· · ·,ξN−1,η2,· · ·,ηN−1,ζ3,· · ·,ζN−1), where the parity of ψℓλ qτ(ξ,η,ζ) is (−1)λ−τ. Equation (5) coincides with Eq. (10), because either of the s et ofDℓ ℓq(α,β,γ )∗and the set ofQℓτ q(R1,R2) is a complete set of base-functions of the angular momentum . However, equation (10) has three important characteristics, which m ake it possible to derive the generalized radial equations. The ﬁrst is that the Euler ang les do not appear explicitly in the base-functions Qℓτ q(R1,R2). The second is the well chosen internal variables (6). The third is that the internal variables ζjhave odd parity. It is due to the existence ofζjthat the base-functions Qℓ0 q(R1,R2) andQℓ1 q(R1,R2) appear together in one total wavefunction. By comparison, all the internal variables in a quantum three-body problem have even parity ( ζj= 0) so that in a total wavefunction with a given parity, only t he base-functions with the same parity appear [5,12]. Now, substituting Eq. (10) into the Schr¨ odinger equation ( 2) with the Laplace operator (4), we obtain the generalized radial equations by a straigh tforward calculation: △ψℓλ q0+ 4{q∂ξ1+ (ℓ−q)∂η2}ψℓλ q0+ 2q∂ξ2ψℓλ (q−1)0+ 2(ℓ−q)∂ξ2ψℓλ (q+1)0 +N−1/summationdisplay j=32Ω−1 2/braceleftBig/bracketleftBig −ωjq∂ξj+ Ω j(ℓ−q)∂ηj+η2ζjq∂ζj+ξ1ζj(ℓ−q)∂ζj/bracketrightBig ψℓλ q0 −q/bracketleftBig ωj∂ηj+ξ2ζj∂ζj/bracketrightBig ψℓλ (q−1)0+ (ℓ−q)/bracketleftBig Ωj∂ξj−ξ2ζj∂ζj/bracketrightBig ψℓλ (q+1)0 −iη2q(q−1)/bracketleftBig ζj∂ηj−Ωj∂ζj/bracketrightBig ψℓλ (q−1)1 −iq/bracketleftBig η2ζjq∂ξj−ξ2ζj(2ℓ−2q+ 1)∂ηj+η2ωjq∂ζj+ξ2Ωj(2ℓ−2q+ 1)∂ζj/bracketrightBig ψℓλ q1 +i(ℓ−q)/bracketleftBig ξ2ζj(2q+ 1)∂ξj−ξ1ζj(ℓ−q)∂ηj+ξ2ωj(2q+ 1)∂ζj+ξ1Ωj(ℓ−q)∂ζj/bracketrightBig ψℓλ (q+1)1 −iξ1(ℓ−q)(ℓ−q−1)/bracketleftBig ζj∂ξj+ωj∂ζj/bracketrightBig ψℓλ (q+2)1/bracerightBig =−/parenleftBig 2/¯h2/parenrightBig [E−V]ψℓλ q0, (11a) 5△ψℓλ q1+ 4{q∂ξ1+ (ℓ−q+ 1)∂η2}ψℓλ q1+ 2(q−1)∂ξ2ψℓλ (q−1)1+ 2(ℓ−q)∂ξ2ψℓλ (q+1)1 +N−1/summationdisplay j=32Ω−1 2/braceleftBig/bracketleftBig −ωjq∂ξj+ Ω j(ℓ−q+ 1)∂ηj+η2ζjq∂ζj+ξ1ζj(ℓ−q+ 1)∂ζj/bracketrightBig ψℓλ q1 −(q−1)/bracketleftBig ωj∂ηj+ξ2ζj∂ζj/bracketrightBig ψℓλ (q−1)1+ (ℓ−q)/bracketleftBig Ωj∂ξj−ξ2ζj∂ζj/bracketrightBig ψℓλ (q+1)1 −i/bracketleftBig ζj∂ηj−Ωj∂ζj/bracketrightBig ψℓλ (q−1)0−i/bracketleftBig ζj∂ξj+ωj∂ζj/bracketrightBig ψℓλ q0/bracerightBig =−/parenleftBig 2/¯h2/parenrightBig [E−V]ψℓλ q1, (11b) △ψℓλ qτ(ξ,η,ζ) =/braceleftBig 4ξ1∂2 ξ1+ 4η2∂2 η2+ (ξ1+η2)∂2 ξ2+ 4ξ2(∂ξ1+∂η2)∂ξ2+ 6 (∂ξ1+∂η2) +N−1/summationdisplay j=3/bracketleftBig ξ1∂2 ξj+η2∂2 ηj+ Ω−1 2/parenleftBig ηjΩj−ξjωj+ζ2 j/parenrightBig /parenleftBig ∂2 ξj+∂2 ηj/parenrightBig +Ω−1 2/parenleftBig Ω2 2+ Ω2 j+ω2 j+ξ1ζ2 j+η2ζ2 j/parenrightBig ∂2 ζj+ 4/parenleftBig ξj∂ξj+ζj∂ζj/parenrightBig ∂ξ1 +4/parenleftBig ηj∂ηj+ζj∂ζj/parenrightBig ∂η2+ 2/parenleftBig ηj∂ξj+ξj∂ηj/parenrightBig ∂ξ2+ 2ξ2∂ξj∂ηj/bracketrightBig/bracerightBig ψℓλ qτ(ξ,η,ζ). (11c) Due to the limited size of a letter, we have to leave the proof o f the theorem and the detailed calculation elsewhere. When establishing BF we ar bitrarily choose two Jacobi coordinate vectors R1andR2. Those two vectors may be replaced with any other two Jacobi vectors. One may change the choice according to the ch aracteristics of the physical problem under study, such as some or all particles in the quan tumN-body problem are the identical particles. In deriving the generalized radial equations, the key is to d iscover the base-functions Qℓτ q(R1,R2) of the angular momentum and to choose the right internal var iables, some of which have odd parity. From Eq. (9) we see that only ﬁnite nu mber of partial angular momentum states are involved in constructing the base-func tionsQℓτ q(R1,R2). Namely, the contributions from the remaining partial angular momen tum states have been incor- porated into those from the generalized radial functions. The two features in this method, that the numbers of both func tionsψℓλ qτ(ξ,η,ζ) and equations are ﬁnite, and they depend only on (3 N−6) internal variables, are important for calculating the energy levels and wavefunctions in a quantu mN-body problem. In fact, in the numerical experiment for the quantum three-body prob lem by the series expansion, much fewer terms have to be taken to achieve the same precisio n of energy than with other methods. The calculation error will be less in comparison wi th the method to truncate the series on the partial angular momentum states. As the num ber of the particles in the 6system increases, we believe, to remove three independent v ariables will greatly decrease the calculation capacity requirement. ACKNOWLEDGMENTS . The authors would like to thank Prof. Hua-Tung Nieh and Prof. Wu-Yi Hsiang for drawing their attention to the quantu m few-body problems. This work was supported by the National Natural Science Foundati on of China and Grant No. LWTZ-1298 of the Chinese Academy of Sciences. [1] M. Viviani, Few-Body Systems, 25, 177 (1998). [2] R. Krivec, Few-Body Systems, 25, 199 (1998), and references therein. [3] Zhong-Qi Ma, Science in China, A43, 1093 (2000). [4] U. Fano, D. Green, J. L. Bohm, and T. A. Heim, J. Phys. B32, R1 (1999), and references therein. [5] E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra , 1959, Academic Press, New York. [6] For example, A. R. Edmonds, Angular Momentum in Quantum Mechanics , Princeton University Press, Princeton, 1957. [7] M. I. Haftle, V. B. Mandelzweig, Ann. Phys. (NY) 189, 29 (1989). [8] N. Barnea, V. B. Mandelzweig, Phys. Rev. A41, 5209. [9] N. Barnea, V. B. Mandelzweig, Phys. Rev. A44, 7053. [10] F. Gatti, C. Iung, M. Menou, and X. Chapuisat, J. Chem. Ph ys.108, 8821 (1998). [11] R. Krivec, V. B. Mandelzweig, Phys. Rev. A42, 3779 (1990). [12] J. Z. Tang, S. Watanabe, M. Matsuzawa, Phys. Rev. A55, 988 (1997). [13] W. T. Hsiang and W. Y. Hsiang, On the reduction of the Schr ¨ odinger’s equation of three- body problem to a system of linear algebraic equations, prep rint. [14] A. V. Matveenko, Phys. Rev. A59, 1034 (1999). 7
1THE PHYSICAL ORIGIN OF MASS AND CHARGE II Erik A. Haeffner Eskadervägen 48, 183 58 TÄBY, SWEDEN, haeffner@algonet.se This article is forthcoming in Galilean Electrodynamics. This article is an alternative version of a document which already can be found on the net. (http://www.aalgonet.se/~haeffner . The scientific message is, however, now presented on a somewhat lower abstraction level, with the ambition that wellknown and generally accepted experimental facts may be identified more convenient and obvious as direct consequences of a new physical concept here disclosed and discussed. The new concept CER (Condensed Electromagnetic Radiation), proposed in this article, indicates anelectromagnetic origin of mass particles, in fact, an overwhelming amount of experimental evidence confirmsthat the CER concept is fundamental for the physical explanation of mass particle properties.Two well established physical phenomena may, among others indicated in the manuscript, be pointed out asconclusive for the new theory: –Parametric down conversion is a commonly used expression to describe the phenomenon when a light wave passing an optically active crystal is converted into two new oppositely polarized light waves. According to thenew theory it should in consequence be possible to produce two new CER particles if the initial electromagneticradiation, EMR, has a high enough energy (frequency) equivalent with the sum of the energies (masses) of thenew particles. The pair production phenomenon (electron and positron) is thus explained by the CER concept.–The second apparent proof is the existence of the first generation leptons and quarks. These may be predicted as CER particles, each produced by the superposition of two plane polarized EMR at right angle with a quantum phase difference of π/6 as described and illustrated in the manuscript. A consequence of the new physical interpretation of mass is the conclusion that all material in the universe is of electromagnetic origin, either in the form of EMR : E = h υ or as CER : E = h υ= m c2 (mass particles) Introduction The concept of mass has a central and fundamental position in the science of physics. Quantities such as charge and energy are correlated to mass. Electrical charge does not exist without mass, in fact, it might be called aproperty of mass which in itself is a form of energy, as discovered by Einstein.The three most important theories of modern physics; Quantum Mechanics, Special Relativity and the StandardModel cannot, however, give a straightforward explanation of the physical nature or the origin of mass. In Quantum Mechanics mass is described both in the form of wave equations and as point particles where the energy content is concentrated to a mathematical point. The waves are then interpreted to express the probabilityto find a particle within a certain space-time region. As pointed out in a study by Marmet ( 2 ) it is not easy tofind a logical definition of Quantum Mechanics as a theory of physics and in this rather diffuse situation todetermine if the concept of mass has a physical meaning at all. The negative opinion of Einstein in consideringthe physical reality of Quantum Mechanics is wellknown but to a large extent ignored. He has, however, given a good reason for his opinion. ”I am, in fact, firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this theory operates with an incompletedescription of physical systems ” ( 3 )There has been many other efforts in the history of physics to introduce mathematical models of wave structuresintended to decribe mass particles. Among those with a wide scientific background is a paper by M. Wolff. (6)Difficulties arise often, however, when the mathematical models are compared with experimental facts or when the nature and origin of waves should be explained . The Standard Model is a classification system in which all elementary particles are ordered in families, generations and classes according to which forces keep particles together. The internal physical nature andstructure of most fundamental particles - leptons and quarks - is so far not known and not needed for theclassification as such but of course much wanted for the understanding of nature. Special Relativity is a rational and logical description of particle physics. Masses exist independently of an observer. Relativistic corrections are used to give a description independent of the observers velocity. Thisrational mathematical description of nature conforms with experimental facts but cannot provide a physicalexplanation of the existence of mass particles.Summarizing: None of these mentioned three theories can, severally or in combination, explain the nature or the origin of mass. For that purpose an additional physical theory is consequently needed. In this article a new physical concept Condensed Electromagnetic Radiation, CER, is introduced with the aim to fill this gap and to constitute a physical explanation of the nature and the origin of mass and charge. The CER2concept will be compared with physical realities, such as the existence of fundamental mass particles and scientifically accepted experimental facts . The CER Concept Following wellknown physics we find that in a linear or plane polarized electromagnetic radiation, EMR, the electric and magnetic field strength vectors oscillate in planes perpendicular to each other and to the direction ofpropagation. If we superimpose two independent electrical oscillations we can produce in different ways polarized EMR. If the two oscillations are in phase and at right angle we find as a result a new plane polarized electromagnetic radiation. Should the two initial EMR:s still have the same amplitude but there exists a phasedifference of 90 degrees, circular polarized electromagnetic radiations will appear. With a phase difference of 30degrees or 150 degrees the radiation is said to be elliptically polarized with the electric vectors rotatingrespectively to the right and to the left when looked upon in the direction of radiation propagation.( fig 1., fig 2.) Ey=1 Ex=0 Ey=1 Ex=1 Ey=0 Ex=1 Ey=1 Ex=−1 Fig 1. Superposition of X-vibrations and Y-vibrations in phase. If the initial vibrations are: Ey=A sin ωτ+φ() , where ω = frequency in radians/s τ = time in seconds φ = phase difference, and Ex=B sin ωτ() ; assu min g A = B = the amplitude of vibrations, we find figures of polarization as shown on fig 2. a) b) c) Fig 2. Superposition at right angle of oscillations Ex and Ey with the same amplitude at phase differences: a) φ =π 6 b) φ =π 2 c) φ = 5π 6 The endpoints of electrical vectors in a circularly polarized EMR moving in space can be illustrated by a spiral spring, fig 3. Circular Polarization C 3Elliptical Polarization Fig 3 Formation (a,d), propagation (b,e) and condensation (c,f) of circulary and elliptically polarized wave packets, CER. If the frequency is high the distance between subsequent threads of the spring is small. We now foresee –the hypothesis– that at a certain small distance the individual electromagnetic wave units are squeezed into a closely packed wave packet, which we call a Condensed Electromagnetic Radiation, CER, and identify as a CER mass particle. In the same way we can produce elliptically polarized CER mass particles. (It should be noted that the expression ”condensed” here means a condensation in space-time, not to be confused with ordinary gas condensation. It should also be noted that we here interpret Planck´s constant h as the energy content of one EMR wave unit. If we multiply h with the frequency υ , we will get an amount of energy hυ, which has been called a photon or elderly a "light quanta" (energy quantum). These EMR energy quantities are not massparticles. The CER concept, on the other hand, describes how EMR wave units after circular or ellipticalpolarization are transformed into mass particles. We can calculate the number of wave units in an electron bydividing the rest energy of the electron with Planck´s constant, and find that the electron as a CER particle contains 1.2356 x 10 20 wave units. ) The CER concept of circularly or elliptically polarized electromagnetic radiation in a condensed state as a physical identification of mass, is discussed in the following sections and investigated as how this concept agreeswith wellknown and scientifically accepted physical phenomena. Leptons and quarks as CER particles Through the theoretical and experimental development of the standard model, the plurality of elementary particles have slowly found their place in a framework of families, classes and generations. Leptons and quarks of the first generation, fig 4, are now assumed to be the real fundamental particles from which all the other”elementary” particles originate either as excitation states or as composite mass particles.It is then obviously important to investigate how the CER hypothesis will conform with the symmetry andproperties of these mentioned most fundamental particles. Fig 4 illustrates polarized EMR waves, each composed of two linear polarized EMR’s with phase differences in 4 steps, 3π 6 ;2π 6;π 6; 0 quadrant I, followed by the 4 symmetrically opposite phase difference angles in quadrant III. We might describe the phase difference going from 0°, in steps of π 630°() to 3π 690°() and from 6π 6180°() to 9π 6270°()as increasing the degree of polarization of the resultant EMR. In the CER concept it is assumed that at a certain small distance between adjacent electrical vector spiral units in circular and elliptical EMR (fig 3) these superimpose to compress the EMR into CER mass particles. It would thus be possible to identify the existing most fundamental particles with the illustrated species of polarized CER.The two bottom lines of fig 4 indicate the first generation of leptons and quarks with their electrical charges. Wefind that these most fundamental particles of the standard model may easily be identified as CER particles by their symmetry and charge which increases in polarization steps of π 630o()4Quadrant I Quadrant III Phase difference φ 90o 3π 6 60o 2π 6 30o π 6 0o 0 180o 6π 6 210o 7π 6 240o 8π 6 270o 9π 6 Leptons an d quarks e u d νe νe d u e Ch arg e -1 -2 3 -1 3 0 0 +1 3 +2 3 +1 Particle name electron up quark antiparticle down quark antineutrino electron type neutrino electron type down quarkantiparticle up quark positron Fig 4 Superposition at right angle of two plane polarized EMR waves with the same amplitude but with phase difference of n π 6 where n = 0,1,2,3 (quadrant I) and 6,7,8,9 (qadrant III). The resultant CER wave packets are identified in symmetry and in electric charge with leptons and quarks of the first generation. This identification will also explain the charge unit of the electron and the positron as corresponding to the maximum polarization possible. The quarks appear as elliptically polarized CER wave packets with different polarization degrees. The electron neutrinos are each formed of two superimposed plane polarized EMR at right angle and in phase. They consequently have a mass according to the fundamental equation hν = m c2 m = hν c2 The explanation of the old problem why the electron has the experimentally well determined charge e, and nothing else, higher or lower, is then simply that it has the maximum polarization degree possible as a CERparticle. Confirmation of the new theory by experimental facts : Pair Production - Annihilation If an EMR travels into a crystal that has nonlinear optical properties, two new EMR:s with opposite angular momentum are created, that is one clockwise and one anticlockwise circularly polarized electromagnetic radiations. The sum of the energies of the two new radiations equals exactly the energy of the parent EMR. Thephenomenon is called spontaneous parametric down conversion.According to the CER concept as described above the secondary polarized radiations should at a certainfrequency be converted to CER mass particles. That is also the case in physical reality, when a high energy EMR > 1.02 MeV, in the vicinity of a nucleus is transformed into one electron (mass energy 0.51 MeV) and its antiparticle the positron with the same mass. The reaction is wellknown as pair production and may physicallybe interpreted as a spontaneous down conversion of a high energy electromagnetic radiation. The pair production phenomenon is an experimental confirmation of how CER mass particles are produced by the condensation of high energy circular polarized electromagnetic radiation.5Relativistic mass effects "A real world", or in other words, a physical realism implies or has a necessary prerequisite that mass and electromagnetic radiation exist independently of an observer. Einsteins discovery or deduction that the velocity of EMR is a constant c in vacuum and independent of any observers own velocity is, in fact, the physical foundation of all relativistic phenomena. A CER mass particle is physically a compressed or condensed wave packet of circularly or elliptically polarized electromagnetic radiation which in itself has the velocity c. If we assume that the wave packet is moving with a velocity v relative to the constant velocity c we find that when calculating the frequency ω (radians per s) of the polarized electromagnetic radiation constituting the CER wave packet we must consider the Doppler effect in order to determine the shift of frequency when the wave packet with a velocity v is moving relative to c. When the velocity v is increasing the frequency ω is also increasing according to the formula: ω1=ωo1+v c        1−v2 c2 As the mass is directly proportional to the frequency according to hωo = m oc2 where h=h 2π we find that the mass of a CER particle follows the formula for relativistic increase with velocity m1=mo1+v c        1−v2 c2 The increase of mass when a particle is moving relative to the fixed EMR velocity c may then be interpreted as a Doppler effect increasing the frequency of the EMR which constitutes the CER wave packet. No observer is needed for this deduction as only internal physical relations are used in the c frame of a reference. Waves associated to mass particles A wellknown mathematical formula used to indicate wave properties of mass particles is the de Brolie wave length λ= h m⋅ v h = Planck´s constant, m = particle mass and v = particle velocity as measured by an external observer. This equation is physically supported by diffraction experiments where a stream of electrons pass through slites in a screen and then produce a diffration pattern on a parallell target screen. There is so far no physicalexplanation of the origin or nature of these mass particle waves producing the diffraction patterns. Textbook formulations are that the de Broglie waves somehow are "associated " with mass. It is also often pointed out that matter waves are not of electromagnetic origin. However, by introducing the new concept CER, CondensedElectromagnetic Radiation, we have established that mass particles really are of electromagnetic origin. We can therefore look on the de Broglie phenomenon with diffraction patterns created by mass particles in the light of the CER concept. Starting with the fundamental equation hυ=m c2 where h = Planck´s constant υ = frequency of electromagnetic radiation, EMR m = the mass of a CER wave packet c = EMR velocity in vacuum6If we exchange υ with c/λ, where λ = wave length, we get λ = h m⋅c It has been shown that the electron and the other leptons and quarks of the first generation confirm with respect to charge and symmetri their identity as CER particles. We can then treat the electron as a CER mass particle with the experimentally determined mass m e and use the formula above to calculate the wavelength λe will decrease, also found by diffraction experiments. It has been shown that the electron and the other leptons and quarks of the first generation confirm with respectto charge and symmetry their identity as CER particles. We can then treat the electron as a CER mass particle with the experimentally determined mass m e and use the formula above to calculate the wavelength λe of an electron at rest. Using the known constants h and c we find λe = 2.42 x 10 -10 cm What happens if the electron moves towards, as an example, a diffraction chamber with a velocity ν? A velocity of a CER particle is always a movement relative to the constant velocity c. As has already been shown the electron mass will then increase according to the relativity equation m1=mo1+v c        1−v2 c2 From the equation λ=h m⋅c we realize that when the electron mass increases its wavelength λe will decrease, as also found by diffraction experiments The CER concept can consequently provide a physical explanation of what has been called " matter waves" or"associated waves". It should be noted that there is no reference to an "observer" or the relative velocity to such an "observer" when calculating the wavelength of a mass particle. An alternative way to estimate the wavelength of the electron is by using the experimentally found Compton scattering relation, which describes the relation of wave lengths when an incoming EMR λ0()is scattered by an electron at an angle θ from its initial direction. The scattered EMR will then get the new wavelenght λs and the scattered electron is found to have the dimension of a wavelenght λeaccording to the equation λs - λ0 = λe 1- cos θ () When θ = 90 θ = 90 θ = 90 θ = 90 ° it is found that the ”Compton wavelenght” λe = h me⋅c which is the same value as already found if we concider the electron as a CER mass particle. This means that we have found another experimental confirmation of the CER concept. Uncertainty relations The basic postulate of Quantum Mechanics is the Heisenberg uncertainty principle. If a complete mathematical description of fundamental physics can be built on this postulate is a question that has been discussed both byphysicists and philosophers for decades. Steven Weinberg ( 4 ) and many others confirm that QuantumMechanics provides a precise framework for calculating energies, trasition rates and probabilities. Marmet ( 2 ) and others have, on the other hand, found many absurdities when comparing the QM mathematical descriptions with a physical reality. These scientific controversies have apparently their root in a problem that already Gödel (5 ) discussed. The mathematics of a system ensures that there are no internal mathematical contradictions.Mathematics cannot, however deduce results from relations that are external to the mathematical system.With the introduction of the CER concept we can end this controversy and give the Heisenberg deduction of theuncertainty principles a physical meaning. The starting point is the same relationship E =hυ as Heisenberg used for the mathematical formation of a Fourier wave packet. The difference is that according to the CER concept the wave packet is identified physically as compressed electromagnetic radiation. In that way we can give thefoundation of Quantum Mechanics a physical reality. This is shown (ref.1) by a simple deduction of theHeisenberg uncertainty relations using the CER wave packet concept . The question, however, to what extent further quantum theory mathematical transformations correspond to physical realities will not be discussed here.7Conclusions The new concept of condensed electromagnetic radiation CER indicates a physical structure of mass particles. It has been demonstrated that the fundamental particles leptons and quarks of the first generation, which in theStandard Model are thought as building blocks of composite particles, coincide in charge and symmetry with aseries of CER wave packets where the phase difference between originally plane polarized EMR waves is integer multiples of π 6 radians. Besides this successful demonstration of the physical origin of mass and charge the CER concept is supported by the explanation it provides of, so far mysterious, phenomena of nature such as mass particle production, mass increase with velocity, elementary particle symmetries, charge quantization, wave structure of matter, etc. The CER concept is consequently of interest for further physical studies and experiments. What comes to mind, as ahigh priority, is the synthesizing of existing or new particles through the building up, or expressed in physicalterminology, the superposition of high energy EMR:s which might differ in phase, frequency and amplitude. Aconsequence of the new physical interpretation of mass is the conclusion that all materia in the universe is of electromagnetic origin, either in the form of EMR : E = h υ or as CER : E = h υ= m c2 The Planck - Einstein energy formulas above appears then as very fundamental equations of the universe. References [ 1 ] E.A. Haeffner, The Physical Origin of Mass and Charge. http://www.algonet.se/~haeffner, (Febr. 1, 1997) (Officially registered June 2, 1995) [ 2 ] P. Marmet, Absurdities in Modern Physics. A Solution. (1993), Ed. Les Editions du Nordir, c/o R. Yergeau, 165 Waller Simard Hall Ottawa, On, Canada K1N6N5. http://www.newtonphysics.on.ca/Heisenberg/contents.html (1997) [ 3 ] P.A. Schilp, ed. Albert Einstein: Philosopher - Scientist, pp 666,672 Evanston,IL: Library of Living Philosophers, (1949) [ 4 ] S. Weinberg, Dreams of a Final Theory. Hutchinson Radius, London, (1993) [ 5 ] S.C. Kleene, The Work of Kurt Gödel. The Journal of Symbolic Logic, 41 No 4, (1976) [ 6 ] M. Wolff, "Beyond the Point Particle - A Wave Structure for the Electron", Galilean Electrodynamics 6 (5), 83-91 (1995) Acknowledgement. The author sincerely thanks the Editor Dr Cynthia K. Whitney for her kind advice and support during the publishing procedure.
1Introduction The 1st law of thermodynamics for heat exchange has the following form: TdS=dU+PdV. (1) There are substances which contract with the temperature: water at T<277 K, liquid He4, Si in a certain temperature interval, honey, Te, Se (monoclinic), quartz glass etc.1,2 Water contracts with the temperature at 273<T<277 K. Let's carry out the process (1) with dU=0 in this temperature interval: dQ>0 and dV<0. Throughout this paper pressure is supposed to be the atmospheric one. Then dQ=PdV. It is a contradiction. Both sides of the equation have different signs. In this paper this paradox is explained. Theory It is possible to prove that at 273<T<277 K and dU=0, water contracts with heating. From thermodynamics1,3 α=(1/V)( ∂V/∂T)P=-(1/V)( ∂V/∂P)T(∂P/∂T)V=kT(∂S/∂V)T. (2) where kT is the isothermal compressibility. If α<0 then ( ∂S/∂V)T<0 and (∂S/∂V)U<0 because ( ∂S/∂V)U is a special case of ( ∂S/∂V)T. From Eq. (1) (∂S/∂V)U=P/T>0. (3)2Really, let's suppose that water expands at U=const, 273<T=T 0<277 K. Then, due to continuity of the function V, there will be an interval T0-ΔT<T<T0+ΔT where water expands at T ≠const. But it contradicts to experiment because limΔT→0(1/V)[(V(T 0+ΔT)-V(T0))/ΔT]P=α<0. To explain this paradox, in4 it has been supposed that P in Eq. (1) is negative for substances with negative α. The author thinks that this opinion is wrong. According to foundations of thermodynamics, pressure P in Eq. (1) is the external pressure3, here the positive atmospheric one. According to thermodynamics, P=-(∂U/∂V)S. (4) and P is negative for metastable states.3,4 In4 it is supposed that substances with negative thermal expansion coefficient are metastable. It is doubtful. The metastable states correspond to local minima, the stable one corresponds to the absolute minimum.3 Water at T<277 K has only one state. For water in weightlessness P in (4) is greater than 0, regardless of is T<277 K or T>277 K. For water in cylinder under piston under pressure P= -(∂U/∂V)S,+0<0, P=-( ∂U/∂V)S,-0>0, regardless of the temperature. Therefore, water at T<277 K does not differ from that at T>277 K in that sense. For liquid with negative pressure, P in Eq. (4) is less than zero4 (regardless of is the derivative forward one or backward one).3In solids P +0 in Eq. (4) is negative, however, it does not mean that the sign before P in Eq. (1) must be negative. For almost all solids α>0. Backward derivative P -0>0. One has to arrive at a conclusion that for substances with negative α the sign before P in Eq. (1) must be minus not plus: dQ=dU-PdV. (5) Whence dQ=dQ 1+dQ2, dQ1=dU, dQ 2=−+PdV. If dQ>0, then dQ 1>0 and dQ 2>0. Almost nobody checked experimentally validity of Eq. (1) for substances with α<0. There are relations using which one can verify the 1st law3,5: the Mayer's relation cV-cP=-Tα2V/kT. (6) The Reech's relation cV=cPkS/kT. (7) kS=-(1/V)( ∂V/∂P)S is the adiabatic compressibility, and the relation (∂cP/∂P)T= -TV(( ∂α/∂T)P+α2). (8) If to derive these relations using Eq. (5) instead of Eq. (1), they will be the following ones cV-cP=Tα2V/kT. (9) cV=cPkT/kS. (10) (∂cP/∂P)T=TV((∂α/∂T)P+α2). (11) For water at 273K cV-cP≈-2,5 J/(kg ⋅K), cP=4217,6 J/(kg ⋅K).5 Using Eq. (6) we can not verify the 1st law because we do not know cV. We can do it using4Eq. (7): for T=273K k T=5,0885 ⋅10-10 Pa-1=kS.5 These values can prove neither Eq. (1) nor Eq. (5). The authors of5 used table dependence of enthalpy H(P, T) from a reference book to obtain ( ∂cP/∂P)T=∂2H/∂P∂T. However, in reference books6,7 enthalpy is obtained from the relations H=ΔHf0+cdTP∫ , (12) (∂H/∂P)T=V-T(∂V/∂T)P (13) where ΔHf0 is the enthalpy of formation of substance and Eq. (8) is a sequence of Eq. (13). Hence, one may not agree with the result from5 that (∂cP/∂P)T>0 in the whole temperature range. Conclusions The following conclusion can be drawn. The author thinks he succeeded to prove that Eq. (5) must be used instead of Eq. (1) for substances with negative thermal expansion coefficient. The final proof can be done by measuring precisely cv, kT and kS. References 1 John S. O. Evans, J. Chem. Soc., Dalton Trans., 1999, (19), 3317.52 C. A. Angell, R. D. Bressel, M. Hemmati, E. J. Sare and J. C. Tucker, PCCP, 2000, 2,1559. 3 Physical Encyclopaedia, Sovetskaja Entsiklopedia, Moscow, 1990. 4 A. Imre, K. Martinas and L. P. N. Rebelo, J. Non- Equilib. Thermod., 1988, 23, 351. 5 J. Guemez, C. Fiolhais and M. Fiolhais, Am. J. Phys., 1999, 67, 1100. 6 L. V. Gurvich and I. V. Veitz, Thermodynamic Properties of Individual Substances, Hemisphere Pub Co., NY, L., 1989, etc. 4th ed. 7 S. L. Rivkin, A. A. Aleksandrov and E. A. Kremenevskaja, Thermodynamic derivatives for water and water vapour, Energija, Moscow, 1977, pp. 3-6.
1The Law of Conservation of Energy in Chemical Reactions I. A. Stepanov Latvian University, Rainis bulv. 19, Riga, LV-1586, Latvia Abstract Earlier it has been supposed that the law of conservation of energy in chemical reactions has the following form: ΔU=ΔQ-PΔV+ i∑µiΔNi In the present paper it has been easy proved by means of the theory of ordinary differential equations that in the biggest part of the chemical reactions it must have the following form: ΔU=ΔQ+PΔV+ i∑µiΔNi The result obtained allows to explain a paradox in chemical thermodynamics: the heat of chemical processes measured by calorimetry and by the Van’t-Hoff equation differ s very much from each other. The result is confirmed by many experiments. I. Introduction The law of conservation of energy for heat exchange is the following one: δQ=ΔU+δA (1) Here δQ is the heat introduced to the system, ΔU is the change in the internal energy and δA is work done by the system. Further it will be assumed that δA=PΔV.2For chemical processes this law is written in the following form [1]: ΔU=ΔQ-PΔV+ i∑µiΔNi (2) where µi are chemical potentials and ΔNi are the changes in the number of moles. It is due to the fact that the motive force of heat exchange is the heat introduced to the system but a motive force of a chemical reaction is the change in the internal energy. In this paper it is assumed that ΔQ is negative for exothermic reactions. Using the theory of differential equations of physical processes i t is possible easy to prove that the energy balance in the form of (2) for the biggest part of the chemical reactions is not correct. In the biggest part of the chemical reactions the law of conservation of energy must have the following form: ΔU=ΔQ+PΔV+ i∑µiΔNi (3) II. Theory Pay attention that if P=const and A is the work of expansion then dQ and dA=PdV are exact differentials. Let's consider heat exchange, one introduces the quantity of heat ΔQ in the system ( ΔV=0): ΔQ=ΔU. Now let's suppose ΔV>0. Let's find out, is it necessary to write ΔQ=ΔU-PΔV or ΔQ=ΔU+PΔV. It is necessary to add the term P ΔV to the right side of the equation. If ΔQ>0 then ΔU>0 and P ΔV>0. Then, according to the theory of differential equations of physical processes one gets (1) with ΔA=PΔV. Here a traditional method from differential equations was used [2]. If there is an equality dfdxdydz =−+−+−+αβγ (α, β and γ>0) and one does not know what signs must there be in3front of α, β and γ one proceeds as following. Let dy and dz=0. Let df<0, if dx<0 then +α. The signs before β and γ are found analogically. Let's consider an exothermic and an isothermic reaction. Let 's suppose that ΔV=0. In this occasion the 1st law of thermodynamics will be the following one: ΔU=ΔQ+ i∑µiΔNi (4) Now let 's suppose that ΔV≠0. For the biggest part of the exothermic reactions with ΔV≠0 the difference in the volume ΔV is less than zero (in isothermal case), for example, the following reaction: 2H2+O2=2H2O (5) The term PΔV must be added to the right side of (4). Let's analyze what sign must it have. ΔU is less than 0, PΔV is less than 0. Hence the 1st law of thermodynamics for a chemical reaction will have the following form: ΔU=ΔQ+PΔV+ i∑µiΔNi (6) Really, dU=dU 1+dU2+dU3 where dU 1=dQ, dU 2=−+PdVand dU3=∑µidNi. If dU>0 then dU 1, dU2 and dU3>0 and dV>0 whence the sign before P is plus. It is important to stress that for heat exchange ( ∂U/∂V)S<0 but for an endothermic reaction ( ∂U/∂V)S,N>0. In an endothermic reaction one heats the reactants, dU increases and dV increases. Equation (6) describes this process correct but according to (2) the volume must decrease. If an open system with no chemical reactions expands on its own then its internal energy is exausted and ( ∂U/∂V)S,N<0. III. Discussion and Experimental Checking4The 1st law of thermodynamics in the form of (6) was obtained not strictly in [3]. In the present work a strict derivation is given. Using the result obtained in the present paper it is possible to explain a paradox in chemical thermodynamics: the heat of chemical reactions, that of dilution of liquids and that of micelle formation measured by calorimetry and by the Van’t-Hoff equation differ significantly [4-6]. The difference is far beyond the error limits. According to thermodynamics the Van’t- Hoff equation must give the same results as calorimetry because the Van’t-Hoff equation is derived from the first and the second law of thermodynamics without simplifications. Nobody succeeded to explain this paradox. Most probably, the reason is that in the derivation of the Van’t-Hoff equation it is necessary to take into account the law of conservation in the form of (6), not of (2). If to derive the Van’t-Hoff equation taking into account (6) the result will be the following one: d/dT ln K=ΔH0/RT2 ( 7) where K is the reaction equilibrium constant and ΔH0=ΔQ0+PΔV0. Let's check experimentally the result obtained. In [7] the dependence of N 2O4=2NO2 reaction equilibrium constant on the temperature was given which was found experimentally (Table1). From Table 1 one sees that the equilibrium constant at 293, 303 and 323 K obeys the equation ln K = 22,151-59677/RT (8) with high accuracy. Values calculated by (8) differ from experimental ones at 0,1; 0,2 and 0,03%, respectively. The equilibrium constant at 273 K deflects from this relationship. Dependence of K on temperature is a broken line. The points 293, 303 and 323 K are at one segment, 273 K is at the other one. From Eqs. (7) and (8) it is possible to calculate ΔH0 and ΔQ0=ΔH0- PΔV0. We suppose that gas is ideal. These values are given in Table 1. There also5the values of the heat of reaction ΔQ are given calculated according to [8]. One sees a good agreement with the present theory. Pay attention that K in (8) is the true reaction equilibrium constant which depends on the activities. In [9] the following reaction was considered: 2Zn(gas)+Se 2(gas)=2ZnSe(solid) (9) ΔGf0=-723,614+382,97T kJ/ mol,1260<T<1410 K (10) As ln K=-ΔG0/RT, from (10) one may find ΔH0 and ΔQ0=ΔH0-PΔV0. The heat of reaction ΔQ is taken from [10] for T=1300 (Table 2). For T=1400 the data in [10] are absent. In [11] the following reaction was given: 2HgTe(solid)=2Hg(gas)+Te 2(gas) (11) lnK=-348496/RT+13,77; 778<T<943 K (12) The heat of this reaction is given in Table 2. In [12] the following reaction was given: 2ZnS(solid)=2Zn(gas)+S 2(gas) (13) lnK=-774142/RT+21,976; 1095<T<1435 K (14) The heat of this reaction is given in Table 2. The equilibrium constant K in Eqs. (12), (14) depends on pressure, not on fugacity. But at a temperature many hundreds grades Kelvin and at the atmospheric pressure one can neglect intermolecular interactions and pressure is very close to fugacity. In [13] the following reaction was considered: 2Te(liquid)=Te 2(gas) (15) lnP=-119,755/RT+11,672469, 722,65<T ≤800 K (16) -116,706/RT+11,21345, 800<T<921,6 K -114,045/RT+10,8661, 921,6<T<1142 K.6Assuming K=P one can build Table 3. One sees that experiment is in a good agreement with the theory developed i n the present paper. References 1. F. Daniels and R. Alberty, Physical Chemistry (John Wiley & Sons, New York, 4th edition, 1975) p. 64. 2. A. M. Samojlenko, S. A. Krivosheja and N. A. Perestjuk, Differential Equations. Examples and Problems (Vysshaja Shkola, Moscow 1989) p. 35. 3. I. A. Stepanov , DEP VINITI, No 37-B96 , 1996. Available from VINITI , Moscow. 4. P. R. Brown, Proc. 12th IUPAC Conf. on Chem.Thermodynamics (Snowbird, Utah, 1992) pp. 238-239. 5. D. S. Reid and M. A. Quickenden , F. Franks, Nature 224 (1969) 1293. 6. J. C. Burrows , D. J. Flynn, S. M. Kutay, T. G. Leriche and D. G. Marangoni, Langmuir 11 (1995) 3388. 7. H. Blend, J. Chem. Phys. 53 (1970) 4497. 8. L. V. Gurvic and I. V. Veitz, Thermodynamic Properties of Individual Substances (Hemisphere Pub Co., NY, L. 4th edition, 1989 etc.). 9. R. Brebrick and H. Liu, High. Temp. Mater. Sci. 35 (1996) 215. 10. I. Barin, Thermochemical Data of Pure Substances (VCH, Weinheim, 1989). 11. R. F. Brebrick and A. J. Strauss, J. Phys. Chem. Solids 26 (1965) 989. 12. W. Hirschwald, G. Neumann and I. N. Stranski, Z. Physik. Chem. (NF) 45 (1965) 170. 13. R. F. Brebrick, High Temp. Science 25 (1988) 187.7Table 1 Dependence of the heat of N 2O4=2NO2 reaction on the temperature [7] T,°K K (atm) [7]ΔΔΗΗ0 (kJ/mol)ΔΔΗΗ0-PΔΔV0 (kJ/mol)ΔΔQ (kJ/mol) [8] 273,15 0,01436 - - - 293,15 0,09600 59,68 57,24 57,30 303,15 0,2140 59,68 57,16 57,25 323,15 0,9302 59,68 56,99 57,158Table 2 The heat of some chemical reactions measured by the Van’t-Hoff equation and by calorimetry 2Zn(gas)+Se 2(gas)=2ZnSe(solid) [9] T,°K ΔΔΗΗ0 (kJ/mol) (10)ΔΔΗΗ0-PΔΔV0 (kJ/mol)ΔΔQ (kJ/mol) [10] 1300 -723,614 -691,20 -690,80 2HgTe(solid)=2Hg(gas)+Te 2(gas) [11] T,°K ΔΔΗΗ0 (kJ/mol) (12)ΔΔΗΗ0-PΔΔV0 (kJ/mol)ΔΔQ (kJ/mol) [10] 800 348,496 328,56 329,50 900 348,496 326,06 325,72 2ZnS(solid)=2Zn(gas)+S 2(gas) [12] T,°K ΔΔΗΗ0 (kJ/mol) (14)ΔΔΗΗ0-PΔΔV0 (kJ/mol)ΔΔQ (kJ/mol) [10] 1100 774,142 746,72 753,00 1200 774,142 744,22 750,00 1300 774,142 741,74 746,92 1400 774,142 739,24 743,769Table 3 The heat of 2Te(liquid)=Te 2(gas) reaction measured by the Van’t-Hoff equation [13] and by calorimetry T,°K ΔΔΗΗ0 (kJ/mol) (16)ΔΔΗΗ0-PΔΔV0 (kJ/mol)ΔΔQ (kJ/mol) [10] 800 119,76 113,11 112,92 900 116,71 109,27 109,41 1000 114,05 105,74 105,97 1100 114,05 104,91 102,59
arXiv:physics/0010053v1 [physics.ed-ph] 22 Oct 2000Maple procedures in teaching the canonical formalism of general relativity Dumitru N. Vulcanovaand Gabriela Ciobanub a) The West University of Timi¸ soara Theoretical and Computational Physics Department V. Pˆ arvan no. 4 Ave., 1900 Timi¸ soara, Romania and b) “Al.I Cuza” University of Ia¸ si Theoretical Physics Department Copou no. 11 Ave., 6600 Ia¸ si, Romania Abstract We present some Maple procedures using the GrTensorII packa ge for teaching purposes in the study of the canonical version o f the general relativity based on the ADM formalism 1 Introduction The use of computer facilities cam be an important tool for te aching general relativity. We have experienced several packages o f procedures, (in REDUCE + EXCALC for algebraic programming and in Math- ematica for graphic visualizations) which fulﬁll this purp ose ([10]). In this article we shall present some new procedures in Maple V using GrTensorII package ([11]) adapted for the canonical versio n of the gen- eral relativity (in the so called ADM formalism based on the 3 +1 split of spacetime). This formalism is widely used ([8],[9]) in th e last years as a major tool in numerical relativity for calculating viol ent processes as, for example the head-on collisions of black holes, massi ve stars or other astrophysical objects. Thus we used these computer pr ocedures 1in the process of teaching the canonical formalism as an intr oductory part of a series of lectures on numerical relativity for grad uated stu- dents. The next section of the article presents shortly the n otations and the main features of the canonical version of the general relativity. Early attemps in using computer algebra (in REDUCE) for the A DM formalism can be detected in the literature ([3], [6],[7]). Obviously we used these programs in producing our new procedures for Mapl e + GrTensorII package, but because there are many speciﬁc feat ures we shall present in some detail these procedures in the section 3 of the article. The last section of the article is dedicated to the c onclusions pointed out by running the Maple procedures presented here a nd some future prospectives on their usage toward the numerical rea lativity. 2 Review of the canonical formalism of general relativity Here we shall use the speciﬁc notations for the ADM formalism [1],[2]; for example latin indices will run from 1 to 3 and greek indice s from 0 to 3. The starting point of the canonical formulation of the general relativity is the (3+1)-dimensional split of the space-tim e produced by the split of the metric tensor : (4)gαβ= (4)goo(4)goj (4)gio(4)gij = NkNk−N2Nj Ni gij (1) wheregijis the riemannian metric tensor of the three-dimensional spacelike hypersurfaces at t = const. which realize the spac etime foliation. Here Nis the ”lapse” function and Niare the components of the ”shift” vector [2]. The Einstein vacuum ﬁeld equations now are (denoting by ” ·” the time derivatives) : ˙gij= 2Ng−1/2[πij−1 2gijπkk] +Ni/j+Nj/i (2) ˙πij=−Ng1/2[Rij−1 2gijR] +1 2Ng−1/2gij[πklπkl−1 2(πkk)2] 2−2Ng−1/2[πimπjm−1 2πijπkk] +g1/2[N/ij−gijN/m /m] + [πijNm]/m−Ni /mπmj−Nj /mπmi(3) whereπijare the components of the momenta canonically conjugate to thegij’s. In the above formulas we denoted by ”/” the three-dimensiona l covariant derivative deﬁned with gijusing the components of the three- dimensional connection [2] : Γi jk=1 2gim(gmj,k+gmk,j−gjk,m) (4) The Ricci tensor components are given by Rij= Γk ij,k−Γk ik,j+ Γk ijΓm km−Γk imΓm jk (5) The initial data on the t = const. hypersurface are not indepe ndent because they must satisfy the constraint equations, which c omplete the Einstein equations H=−√g{R+g−1[1 2(πkk)2−πijπij]}= 0 (6) Hi=−2πij /j= 0 (7) where His the super-hamiltonian, Hithe super-momentum and gis the determinant of the three-dimensional metric tensor gij. The action functional in Hamiltonian form for a vacuum space -time can thus be written as ([1],[2]) : S=/integraldisplay dt/integraldisplay (πij˙gij−NH −NiHi)ω1ω2ω3(8) where theωi’s are the basis one-forms. Thus the dynamic equations (2) and (3) are obtained by diﬀerentiating Swith respect to the canon- ical conjugate pair of variables ( πij,gkm). 3 Maple + GrTensorII procedures Here we shall describe brieﬂy the structure and the main feat ures of the Maple procedures for the canonical formalism of the ge neral 3relativity as described in the previous section. Two major p arts of the programs can be detected : one before introducing the met ric of the spacetime used (consisting in several deﬁnitions of ten sor objects which are common to all spacetimes) and the second one, havin g line- commands speciﬁc to each version. The ﬁrst part of the program starts after initalisation of th e GrTen- sorII package ( grtw(); ) and has mainly the next lines : > grdef(‘tr := pi{^i i}‘); > grdef(‘ha0:=-sqrt(detg)(Ricciscalar+ (1/detg)((1/2)(tr)^2-pi{i j}pi{ ^i ^j }))‘); > grdef(‘ha{ ^i }:=-2(pi{ ^i ^j ;j}-pi{ ^i ^j }Chr{ p j ^p })‘ ); > grdef(‘derge{ i j }:=2N(x,t)(detg)^(-1/2)(pi{ i j } - (1/2)g{ i j}tr)+Ni{ i ;j } + Ni{ j ;i }‘); > grdef(‘Ndd{ ^m j }:= Nd{ ^m ;j }‘); > grdef(‘bum{ ^i ^j ^m}:=pi{ ^i ^j }Ni{ ^m }‘); > grdef(‘bla{ ^i ^j }:=bum{ ^i ^j ^m ;m }‘); > grdef(‘derpi{ ^i ^j }:= -N(x,t)(detg)^(1/2)(R{ ^i ^j }-(1/2)g{ ^i ^j }Riccisc alar)+ (1/2)N(x,t)(detg)^(-1/2)g{ ^i ^j }(pi{ ^k ^l }pi{ k l } - (1/2)(tr)^2)-2N(x,t)(detg)^(-1/2)(pi{ ^i ^m }pi{ ^ j m }- (1/2)pi{ ^i ^j }tr)+ (detg)^(1/2)(Ndd{ ^i ^j }-g{ ^i ^j } Ndd{ ^m m }) + bla{ ^i ^j } - Ni{ ^i ;m }pi{ ^m ^j }- Ni{ ^j ;m }pi{ ^m ^i }‘); Hereha0andha{ˆi}represents the superhamiltonian and the supermomentum as deﬁned in eqs. (6) and (7) respectively and tr is the trace of momentum tensor density πij- which will be deﬁned in the next lines of the program. Here N(x,t) represents the lapse functionN. Also, derge {i j}represents the time derivatives of the components of the metric tensor, as deﬁned in eq. (2) and derpi {ˆiˆ j}the time derivatives of the components of the momentum tenso r πijas deﬁned in eq. (3). The next line of the program is a speciﬁc GrTensorII command for loading the spacetime metric. Here Maple loads a ﬁle (pre viously generated) for introducing the components of the metric ten sor as functions of the coordinates. We also reproduced here the ou tput of the Maple session showing the metric structure of the spacet ime we introduced. > qload(‘Cyl_din‘); 4Default spacetime = Cyl_din For the Cyl_din spacetime: Coordinates x(up) a x = [x, y, z] Line element 2 2 ds = exp(gamma(x, t) - psi(x, t)) d x 2 2 2 + R(x, t) exp(-psi(x, t)) d y + exp(psi(x, t)) d z As is obvious we introduced above the metric for a spacetime w ith cylindrical symmetry, an example we used for teaching purpo ses being a well known example in the literature ([5]). In natural outp ut this metric has the form : gij= eγ−ψ0 0 0R2e−ψ0 0 0 eψ  (9) in cylindrical coordinates x,y,z withx∈[0,∞),y∈[0,2π),z∈ (−∞,+∞) whereR,ψandγare functions of xandtonly. After the metric of the spacetime is established the next seq uence of the programm just introduce the components of the momentu m tensorπijas > grdef(‘Nd{ ^ m } := [diff(N(x,t),x), 0, 0]‘); > grdef(‘Ni{ ^i } := [N1(x,t), N2(x,t), N3(x,t)]‘); > grdef(‘vi1{^i}:=[pig(x,t)exp(psi(x,t)-gamma(x,t)) ,0,0]‘); > grdef(‘vi3{^i} :=[0,0,exp(-psi(x,t))(pig(x,t)+(1/2)R(x,t)pir(x, t)+ pip(x,t))]‘); > grdef(‘vi2{^i}:=[0,(2R(x,t))^(-1)pir(x,t)exp(ps i(x,t)),0]‘); > grdef(‘pi{ ^i ^j } := vi1{ ^i }kdelta{^j $x}+vi2{ ^i }kdelta{ ^j$y }+ vi3{ ^i }kdelta{^j $z}‘); > grcalc(pi(up,up)); > grdisplay(pi(up,up)); 5HereNi{ˆi}represents the shift vector Niand the other objects (Nd,vi1,vi2andvi3) represent intermediate vectors deﬁned in order to introduce the momenum pi{ˆiˆj}having the form : πij= πγeψ−γ0 0 01 2RπReψ0 0 0 e−ψ(πγ+1 2RπR+πψ)  (10) In the program we denoted πγ,πRandπψwithpig,pirandpip, respectively. The momentum components are introduced in or der that the dynamic part of the action of the theory be in canonical fo rm, that is : ˙gijπij=πγ˙γ+πψ˙ψ+πR˙R. The next lines of the programm check if this condition is fullﬁled : > grdef(‘de1{ i }:=[diff(grcomponent(g(dn,dn),[x,x]),t ),0,0]‘); > grdef(‘de2{ i }:=[0,diff(grcomponent(g(dn,dn),[y,y]) ,t),0]‘); > grdef(‘de3{ i }:=[0,0,diff(grcomponent(g(dn,dn),[z,z ]),t)]‘); > grdef(‘ddgt({ i j }:= de1{ i }kdelta{j $x}+de2{ i }kdelta{ j$y }+ de3{ i }kdelta{ j $z}‘); > grcalc(ddgt(dn,dn)); > grdef(‘act:=pi{ ^i ^j }ddgt{ i j }‘); > grcalc(act); gralter(act,simplify); grdisplay(act); By inspecting this last output from the Maple worksheet, the user can decide if it is necessary to rediﬁne the components of the mom entum tensor or to go further. Here the components of the momentum t ensor were calculated by hand but, of course a more experienced use r can try to introduce here a sequence of commands for automatic calcu lation of the momentum tensor components using the above condition, t hrough an intensive use of solve Maple command. Now comes the must important part of the routine, dedicated t o calculations of diﬀerent objects previously deﬁned : > grcalc(ha0); gralter(ha0,simplify); > grdisplay(ha0); > grcalc(ha(up)); gralter(ha(up),simplify); > grdisplay(ha(up)); > grcalc(derge(dn,dn)); gralter(derge(dn,dn),simplify ); > grdisplay(derge(dn,dn)); > d1:=exp(-psi(x,t))grcomponent(derge(dn,dn),[z,z]) +exp(psi(x,t)- 6gamma(x,t))grcomponent(derge(dn,dn),[x,x]); > simplify(d1); > d2:=(1/(2R(x,t)))exp(psi(x,t))grcomponent(derge (dn,dn),[y,y])+ (1/2)R(x,t)exp(-psi(x,t))grcomponent(derge(dn,dn ),[z,z]); > simplify(d2); > d3:=exp(-psi(x,t))grcomponent(derge(dn,dn),[z,z]) ; > simplify(d3); > grcalc(derpi(up,up)); gralter(derpi(up,up),simplify ); > grdisplay(derpi(up,up)); > f1 := exp(gamma(x,t)-psi(x,t))grcomponent(derpi(up, up),[x,x])- pig(x,t)(d3-d1); > simplify(f1); > f2:= 2R(x,t)exp(-psi(x,t))grcomponent(derpi(up,u p),[y,y])+ (1/R(x,t))d2pir(x,t)-pir(x,t)d3; > simplify(f2); > f3 := exp(psi(x,t))grcomponent(derpi(up,up),[z,z])+ d3(pig(x,t)+ (1/2)R(x,t)pir(x,t)+pip(x,t))-f1-(1/2)R(x,t)f2- (1/2)pir(x,t)d2; > simplify(f3); This is a simple series of alternation of grcalc ,gralter andgrdisplay commands for obtainig the superhamiltonian, supermomentu m and the dynamic equations for the theory. d1 ... d3 andf1 ... f3 are the time derivatives of the dynamic variables, ˙ γ,˙R,˙ψand ˙πγ, ˙πR, ˙πψrespectively. Denoting with ” ′” the derivatives with respect to r we display here the results for the example used above (cylin drical gravitational waves) : H0=eψ−γ 2(2R′′−R′γ′+1 2(ψ′)2R−πγπR+1 2R(πψ)2) = 0 H1=Hr=eψ−γ(−2π′ γ+γ′πγ+R′πR+ψ′πψ) = 0 ; H2=H3= 0 ˙γ=N1γ′+ 2N1′−eψ−γ 2NπR;˙R=N1R′−eψ−γ 2Nπγ ˙ψ=N1ψ′+1 Reψ−γ 2Nπψ ; ˙πγ=N1π′ γ+N1′πγ−eψ−γ 2(R′N′+1 2R′ψ′N−1 4ψ′2RN+1 2NπγπR−1 4RNπ2 ψ) 7˙πR=N1π′ R+N1′πR+eψ−γ 2(γ′N′−2N′′−2N′ψ′+1 2γ′ψ′N−ψ′′N−ψ′2+1 2RNπ2 ψ) ˙πψ=N1π′ ψ+N1′πψ+eψ−γ 2(RN′ψ′−R′′N+1 2NR′γ′+R′ψ′N−1 2γ′ψ′NR +ψ′′RN+1 4ψ′2RN+1 2NπRπγ−1 4RNπ2 ψ) These are the well-known results reported in ([5]) or ([6]). One of the important goals of the canonical formalism of the g en- eral relativity (which constitutes the “kernel” of the ADM f ormalism) is the reductional formalism. Here we obtain the true dynami cal sta- tus of the theory, by reducing the number of the variables thr ough solving the constraint equations. This formalism is applic able only to a restricted number of space-time models, one of them bein g the above cylindrical gravitational waves model. Unfortunate ly only a speciﬁc strategy can be used in every model. Thus the next lin es of our program must be rewritten speciﬁcally in every case. Her e, for teaching purposes we present our example of cylindrical gra vitational wave space-time model. Of course we enccourage the student t o apply his own strategy for other examples he dares to calculate. In our example of cylindrical gravitational waves, the redu ctional strategy as described in ([5]) starts with the usual rescali ng of Hand Hito¯Hand¯Hiby ¯H=eγ−ψ 2H;¯N=eψ−γ 2N;¯H1=eγ−ψH1;¯N1=eψ−γN1 wich produce the next sequence of Maple+GrTensorII command s : > grdef(‘aha0:=sqrt(exp(gamma(x,t)-psi(x,t)))ha0‘); > grdef(‘aha{ ^j } := exp(gamma(x,t)-psi(x,t))ha{ ^j }‘); > grdef(‘an:=sqrt(exp(psi(x,t)-gamma(x,t)))n(x,t)‘) ; > grdef(‘ani{ ^i } := exp(psi(x,t)-gamma(x,t))ni{ ^i }‘); The canonical transformation to the new variables, includi ng Kuchar’s ”extrinsic time”, deﬁned by : T=T(∞) +/integraldisplayr ∞(−πγ)dr, ΠT=−γ′+ [ln ((R′)2−(T′)2)]′ R=R, ΠR=πR+ [ln (R′+T′ R′−T′)]′ are introduced with : 8> pig(x,t):=-diff(T(x,t),x); > pir(x,t):=piR(x,t) - diff(ln((diff(R(x,t),x)+diff(T( x,t),x))/ (diff(R(x,t),x)-diff(T(x,t),x))),x); and speciﬁc substitutions in the dynamic objects of the theo ry : > grmap(ha0, subs , diff(gamma(x,t),x)= diff( ln( (diff(R(x,t),x))^2- (diff(T(x,t),x))^2 ),x)-p iT(x,t),‘x‘); > grcalc(ha0); gralter(ha0,simplify); > grdisplay(ha0); > grmap(ha(up), subs , diff(gamma(x,t),x)=diff( ln( (diff (R(x,t),x))^2- (diff(T(x,t),x))^2 ),x)-piT(x,t),‘x‘); > gralter(ha(up),simplify); > grdisplay(ha(up)); > grcalc(aha0); > grmap(aha0, subs , diff(gamma(x,t),x)=diff( ln( (diff(R (x,t),x))^2- (diff(T(x,t),x))^2 ),x)-piT(x,t),‘x‘); > gralter(aha0,simplify,sqrt); > grdisplay(aha0); > grcalc(aha(up)); > grmap(aha(up), subs , diff(gamma(x,t),x)=diff( ln( (dif f(R(x,t),x))^2- (diff(T(x,t),x))^2 ),x)-piT(x,t),‘x‘); > gralter(aha(up),simplify); > grdisplay(aha(up)); > grmap(act, subs , diff(gamma(x,t),x)=diff( ln( (diff(R( x,t),x))^2- (diff(T(x,t),x))^2 ),x)-piT(x,t),‘x‘); > grcalc(act); grdisplay(act); Thus the action yields (modulo divergences) : S= 2π/integraldisplay∞ −∞dt/integraldisplay∞ 0dr(ΠT˙T+ ΠR˙R+πψ˙ψ+πχ˙χ−¯N¯H −¯N1¯H1) where : ¯H=R′ΠT+T′ΠR+1 2R−1π2 ψ+1 2Rψ′2+1 4R−1π2 χ+Rχ′2 ¯H1=T′ΠT+R′ΠR+ψ′πψ+χ′πχ 9Solving the constraint equations ¯H= 0 and ¯H1= 0 for Π Tand ΠR and imposing the coordinate conditions T=tandR=rwe obtain ﬁnally : S= 2π/integraldisplay+∞ −∞dT/integraldisplay+∞ 0dR[πψψ,T+πχχ,T−1 2(R−1π2 ψ+Rψ2 ,R+Rπ2 χ+R−1χ′2)] from the next sequence of programm lines : > R(x,t):=x; T(x,t):=t; grdisplay(aha0); > solve(grcomponent(aha0),piT(x,t)); > piT(x,t):= -1/2(x^2diff(psi(x,t),x)^2+pip(x,t)^2) /x; > eval(piR(x,t)); > piR(x,t):=-diff(psi(x,t),x)pip(x,t); piR(x,t); > grdisplay(aha0); grdisplay(aha(up)); > piT(x,t); 2 /d \2 2 x \|-- psi(x, t)\| + pip(x, t) \dx / - 1/2 ------------------------------- x > piR(x,t); /d \ -\|-- psi(x, t)\| pip(x, t) \dx / > grcalc(act); grdisplay(act); For the Cyl_din spacetime: act /d \ act = \|-- psi(x, t)\| pip(x, t) \dt / > grdef(‘Action:=act+piT(x,t)diff(T(x,t),t)+piR(x,t )diff(R(x,t),t)‘); > grcalc(Action);gralter(Action,factor,normal,sort,e xpand); 10> grdisplay(Action); For the Cyl_din spacetime: Action /d \2 /d \ Action = - 1/2 x \|-- psi(x, t)\| + \|-- psi(x, t)\| pip(x, t) \dx / \dt / 2 pip(x, t) - 1/2 ---------- x > grdef(‘Ham:=piT(x,t)diff(T(x,t),t)+piR(x,t)diff( R(x,t),t)‘); > grcalc(Ham); gralter(Ham,expand); > grdisplay(Ham); For the Cyl_din spacetime: Ham 2 /d \2 pip(x, t) Ham = - 1/2 x \|-- psi(x, t)\| - 1/2 ---------- \dx / x 4 Conclusions. Further improuvements We used the programms presented above in the computer room wi th the students from the graduate course on Numerical Relativi ty. The main purpose was to introduce faster the elements of the cano nical version of relativity with the declared objective to skip th e long and not very straitforward hand calculations necessary to proc ess an entire example of spacetime model. We encouraged the students to tr y to modify the procedures in order to compute new examples. The major conclusion is that this method is indeed usefull fo r an attractive and fast teaching of the methods involved in the A DM for- 11malism. On the other hand we can use and modify these programs for obtaining the equations necessary for the numerical relati vity. In fact we intend to expand our Maple worksheets for the case of axisy mmet- ric model (used in the numerical treatement of the head-on co llision of black-holes). Of course, for numerical solving of the dyn amic equa- tions obtained here we need more improuvements of the codes f or paralel computing and more sophisticated numerical method s. But this will be the object of another series of articles. References [1] Arnowitt R., Deser S., Misner C.W., Gravitation - an introduc- tion to current research ed. by L. Witten, New York, 1962 [2] Misner C.W., Thorne K.S., Wheeler J.A., Gravitation , Freeman, San Francisco, 1973 [3] Moussiaux A., Tombal P., Demaret J., G.R.G., Vol. 15, No. 3, p. 209, 1983 [4] Ryan M., Hamiltonian Cosmology , Lectures Notes in Physics, Vol. 13, Springer, Berlin, 1972 [5] Kuchar K., Phys. Rev.D, Vol. 4, No. 4, p. 955, 1971 [6] Vulcanov D., Intern.J. of Modern Phys., Vol. 5, No. 6, 199 4, p. 973 [7] Vulcanov D., Intern.J. of Modern Phys., Vol. 6, No. 3, 199 5, p. 317 [8] Hehl F.W., Puntingam R.A., Ruder H. (editors) - Relativity and Scientiﬁc Computing , Springer Verlag, Berlin, 1996 (see also the references cited here) [9] Laguna P., The Grand Chalange in Computational Gravitation : Collision of Black-Holes , in Vulcanov D.N., Cot˘ aescu I.I (editors), -Gravity, Theoretical Physics and Computers , Proceedings of the VI-th conference in General Relativity and Gravitation, Bi strit ¸a, Romania, 1996, Mirton Publ.House, Timi¸ soara, 1997 (see al so the references cited here) [10] Vulcanov D.,Ghergu F., The use of algebraic programming in teaching general relativity , submitted to Computing in Science and Engineering, 1999 12[11] Musgrave P., Pollney D., Lake K,. GrTensorII Release 1. 50, Queen’s Univ. of Kingston, Ontario, Canada, july 1996, http://astro.queensu.ca ∼grtensor/ 13
1The Heats of Reactions. Calorimetry and Van't-Hoff . 1 I. A. Stepanov Latvian University, Rainis bulv. 19, Riga, LV-1586, Latvia Abstract Earlier it has been supposed that the law of conservation of energy in chemical reactions has the following form: dU=dQ-PdV+ i∑µidNi In [1-4] it has been shown that for the biggest part of reactions it must have the following form: dU=dQ+PdV+ i∑µidNi In the present paper this result is confirmed by other experiments. 1. Introduction For chemical processes the law of conservation of energy is written in the following form: dU=dQ-PdV+ i∑µidNi (1)2where dQ is the heat of reaction, dU is the change in the internal energy, µi are chemical potentials and dNi are the changes in the number of moles. In [1-4] it has been shown that the energy balance in the form of (1) for the biggest part of the chemical reactions is not correct. In the biggest part of the chemical reactions the law of conservation of energy must have the following form: dU=dQ+PdV+ i∑µidNi (2) The Van’t-Hoff equation is the following one : d/dTlnK= ΔH0/RT2(3) where K is the reaction equilibrium constant and ΔH0 is the enthalpy. According to thermodynamics , the Van’t-Hoff equation must give the same results as calorimetry because it is derived from the 1st and the 2nd law of thermodynamics without simplifications. However, there is a paradox: the heat of chemical reactions, that of dilution of liquids and that of other chemical processes measured by calorimetry and by the Van’t-Hoff equation differ significantly [1-4]. The difference is far beyond the error limits. T he reason is that in the derivation of the Van’t-Hoff equation it is necessary to take into account the law of conservation in the form of (2), not of (1) [1-4]. If to derive the Van’t-Hoff equation using (2) the result will be the following one: d/dTlnK= ΔH0/RT2 (4) where ΔH0=ΔQ0+PΔV0. In the present paper the following processes were checked: Si(liquid)= Si(gas), 2Si(liquid)=Si 2(gas), Si(solid)+SiO 2(solid)=2SiO(gas). . 2. Experiments3In [6] the following processes were considered: Si(liquid)= Si(gas) (5) logP=logK=-2,08 ⋅104/T+10,84 1762≤T≤1998K (6) 2Si(liquid)= Si2(gas) (7) logP=logK=-2,46 ⋅104/T+10,93 1842≤T≤1992K (8) The 1st process is evaporation , the 2nd one is a chemical reaction. Their heats are given in Table 1 . One sees that the heat of evaporation measured by the Van’t-Hoff equation is much closer to the experiment than the heat of the reaction. Evaporation is a physical process , it obeys the traditional thermodynamics. The second process obeys the present theory. In [7] the following reaction has been studied: Si(solid)+SiO 2(solid)=2SiO(gas) (9) logP=13,613-1,785 ⋅104/T, 1300 ≤T≤1580K (10) The heat of this reaction is given in Table 1 . It obeys the present theory, not the traditional one.4References 1) I.A. Stepanov: DEP VINITI , No 37-B96 , (1996). Available from VINITI , Moscow. 2) I.A. Stepanov : DEP VINITI , No 3387-B98. (1998). Available from VINITI , Moscow. 3) I.A. Stepanov : 7th European Symposium on Thermal Analysis and Calorimetry. Aug. 30 - Sept. 4. Balatonfuered , Hungary. 1998. Book of Abstracts. P. 402-403. 4) I.A. Stepanov : The Law of Conservation of Energy in Chemical Reactions .- http://ArXiv.org/abs/physics/0010052 5) I. Barin: Thermochemical Data of Pure Substances , VCH, Weinheim, (1989). 6) T. Tomooka, Y. Shoji and T. Matsui: J. Mass Spectrom. Soc. Jpn., 47 (1999), 49. 7) O. Kubaschewski and T.G. Chart: J. Chem. Thermodynamics, 6 (1974), 467. 8) L.V. Gurvich and I.V. Veitz: Thermodynamic Properties of Individual Substances , 4th edn., Hemisphere Pub Co, NY, L., (1989 etc.). 9) L.V. Gurvich and V.S. Iorish: IVTANTHERMO - A Thermodynamic Database and Software System for the Personal Computer. User's Guide , CRC Press, Inc., Boca Raton, (1993).5Table 1 The heat of evaporation and of some chemical reactions measured by the Van’t-Hoff equation and by calorimetry Si(liquid)= Si(gas)[6]1 T,°K ΔΔΗΗ0, kJ/mol ΔΔΗΗ0-PΔΔV0, kJ/mol ΔQ0, kJ/mol [5, 8] 1800 397,55 - 395,78 1900 397,55 - 395,23 2Si(liquid)=Si 2(gas)[6]1 T,°K ΔΔΗΗ0, kJ/mol ΔΔΗΗ0-PΔΔV0, kJ/molΔQ0, kJ/mol [5, 8] 1850 470,19 454,82 455,57 1900 470,19 454,40 455,05 1990 470,19 453,65 454,11 Si(solid)+SiO 2(solid)=2SiO(gas)[7] T,°K ΔΔΗΗ0, kJ/mol ΔΔΗΗ0-PΔΔV0, kJ/molΔQ0, kJ/mol[9] 1300 683,10 661,50 661,81 1400 683,10 659,83 657,90 1500 683,10 658,17 652,43 1 Thermodynamic data for Si(liquid) is taken from [5], these for Si(gas) and Si 2(gas) is from [9].
arXiv:physics/0010055v1 [physics.acc-ph] 23 Oct 2000SLAC–PUB–8676 October, 2000 Nonlinear δfMethod for Beam-Beam Simulation∗ Yunhai Cai, Alexander W. Chao, Stephan I. Tzenov Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 and Toshi Tajima University of Texas at Austin, Austin, TX 78712 and Lawrence Livermore National Laboratory, Livermore, CA 945 51 Abstract We have developed an eﬃcacious algorithm for simulation of t he beam-beam in- teraction in synchrotron colliders based on the nonlinear δfmethod, where δf is the much smaller deviation of the beam distribution from t he slowly evolving main distribution f0. In the presence of damping and quantum ﬂuctuations of synchrotron radiation it has been shown that the slowly evol ving part of the distribution function satisﬁes a Fokker-Planck equation. Its solution has been obtained in terms of a beam envelope function and an amplitud e of the dis- tribution, which satisfy a coupled system of ordinary diﬀer ential equations. A numerical algorithm suited for direct code implementation of the evolving dis- tributions for both δfandf0has been developed. Explicit expressions for the dynamical weights of macro-particles for δfas well as an expression for the slowly changing f0have been obtained. Submitted to Physical Review Special Topics: Accelerators and Beams ∗Work supported by Department of Energy contract DE–AC03–76 SF00515.1 Introduction The eﬀects of the beam-beam interaction on particle dynamic s in a synchrotron collider are the key element that determines the performance of the colli der such as luminosity [1] - [3]. In order to accurately understand these eﬀects, it is necess ary to incorporate not only the overall collisional eﬀects of the beam-beam interaction, b ut also the collective interaction among individual parts of the beam in each beam and its feedba ck on the beam distribution. The particle-in-cell (PIC) approach [4], [5] has been adopt ed to address such a study need [6], [7], [8]. Particle-in-cell codes typically use macro-particles to r epresent the entire distribution of particles. In the beam-beam interaction for the PEP-II [9 ] (for example), the beams consist of 1010particles each. Simulating this many particles with the PIC technique is computationally prohibitive. With the conventional PIC co de 1010particles are represented by only 103−104macro-particles allowing simulation of the beam-beam inte raction in a reasonable computation time. However, the statistical ﬂuc tuation level of various quantities such as the beam density ρin the code is much higher than that of the real beam. The ﬂuctuation level δρgoes as approximately δρ ρ≈√ N N, (1.1) where Nis the number of particles. Therefore, the ﬂuctuation level of the PIC code is about 103times higher than that of the real beam. Although this probab ility is not signiﬁcant for beam blowup near resonances, the higher ﬂuctuation level ha s a large eﬀect on more subtle phenomenon such as particle diﬀusion. The purpose of the δfalgorithm is to facilitate the study of subtle eﬀects and has been introduced in [10], [11], [12]. Theδfmethod follows only the ﬂuctuating part of the distribution instead of the entire distribution. This is essentially modeling the numerator o f the right-hand side of equation (1.1). So the 103−104macro particles are used to represent√ 1010or 105real ﬂuctuation particles in PEP-II beams. This is only one or two orders of ma gnitude beyond the number of macro particles. Such a modest gap between the number of ma cro particles and the real ﬂuctuating particles maybe ameliorated by the standard tec hniques of the PIC approach, such as the method of ﬁnite-sized macro-particles [4], [5]. PIC strong-strong codes use a ﬁnite number of particles to re present the Klimontovich equation for the microscopic phase space density (MPSD) [13 ]. In the particular case of one-dimensional beam-beam interaction, ∂f ∂s+p∂f ∂x−(K(s)x−F(x;s))∂f ∂p= 0, (1.2) where K(s)xis the usual magnetic guiding force and F(x;s) is the beam-beam force F(x;s) =2eEx(x) mγv2δp(s). (1.3) 2The electric ﬁeld Ex(x) is calculated from the distribution of the particles of the on-coming beam and δp(s) is the periodic δ-function with a periodicity of the accelerator circumfere nce. The distribution function f(x, p;s) is represented by a ﬁnite number of macro-particles by f(x, p;s) =1 NN/summationdisplay n=1δ(x−xn(s))δ(p−pn(s)), (1.4) where Nis the number of macro-particles. The strategy of the δfmethod is that only the perturbative part of the distributio n is followed. The total distribution function f(x, p;s) is decomposed into f(x, p;s) =f0(x, p;s) +δf(x, p;s), (1.5) where f0(x, p;s) is the steady or slowly varying part of the distribution and δf(x, p;s) is the perturbative part. The key to this method is ﬁnding a distrib ution f0(x, p;s) which is close to the total distribution f(x, p;s). The perturbative part δf(x, p;s) is then small, causes only small changes to the distribution, and thus represents only the ﬂuctuation levels. If a distribution f0(x, p;s) close to the total distribution is not found or found poorly , then δf(x, p;s) represents more than the ﬂuctuation part of the total distr ibution; defeating the purpose of the method. The ideal situation is having an analy tic solution for f0(x, p;s). In this case any numerical truncation errors which result from the necessary derivatives of this function are eliminated. If an analytic solution cannot be f ound, then a numerical solution needs to be found which is close to the total distribution f(x, p;s) and is slowly varying. A frequent numerical update of f0(x, p;s) would also defeat the purpose of the δfmethod, since the PIC technique essentially does this also. The beam-beam interaction can lead to beam instabilities th at disrupt or severely distort the beam or gradual beam spreading. The higher the beam curre nt, and thus the beam- beam interaction, the stronger these eﬀects become. Theref ore, when one wants to maximize the luminosity of a collider, one needs to confront the beam- beam interaction eﬀects. The operation of PEP-II, for example, is critically dependent o n the beam-beam interaction and optimal parameters to minimize the related beam instabilit ies are under intense study. The paper is organized as follows. In the next Section we pres ent a brief formulation of the problem of beam-beam interaction in synchrotron collid ers. In Section 3 we develop the nonlinear δfmethod for solving the equation for the microscopic phase sp ace density in the presence of random external forces. The equation for the ﬂuc tuating part δfis being derived and its solution is found explicitly in terms of dynamical we ight functions, prescribed to each macro-particle. In Section 4 we solve the Fokker-Planck equ ation for the averaged slowly evolving part of the distribution. We show that the solution is an exponential of a bilinear form in coordinates and momenta with coeﬃcients that can be r egarded as generalized Courant-Snyder parameters. In Section 5 we outline numeric al algorithms to alternatively solve the Fokker-Planck equation and the macro particle dis tribution with dynamical weight. Finally, Section 6 is dedicated to our summary and conclusio ns. 32 Description of the beam-Beam Interaction In order to describe the beam dynamics in an electron positro n storage ring, we introduce the equations of motion in the following manner. The beam pro pagation in a reference frame attached to the particle orbit is usually described in terms of the canonical conjugate pairs /hatwideu(k)=u(k)−D(k) u/hatwideη(k);/hatwidep(k) u=p(k) u p(k) 0−/hatwideη(k)dD(k) u ds, (2.1) /hatwideσ(k)=/tildewideσ(k)+/summationdisplay u=x,z/parenleftBigg u(k)dD(k) u ds−D(k) up(k) u p(k) 0/parenrightBigg ;/hatwideη(k)=1 β2 k0E(k)−Ek0 Ek0, (2.2) where u= (x, z),sis the path length along the particle orbit, and the index krefers to either beam ( k= 1,2). In equations (2.1) and (2.2) the quantity u(k)is the actual particle displacement from the reference orbit in the plane transver sal to the orbit, p(k) uis the actual particle momentum, and E(k)is the particle energy. Furthermore, p(k) 0andEk0are the total momentum and energy of the synchronous particle, respectiv ely, and D(k) uis the well-known dispersion function. The quantity /tildewideσ(k)=s−ω(k) 0Rt (2.3) is the longitudinal coordinate of a particle from the k-th beam with respect to the syn- chronous particle, where ω(k) 0is the angular frequency of the synchronous particle and Ris the mean machine radius. It is known that the dynamics of an individual particle is gov erned by the Langevin equations of motion: d/hatwideu(k) ds=∂/hatwiderH(k) ∂/hatwidep(k) u−D(k) u/tildewideF(k) η ;d/hatwidep(k) u ds=−∂/hatwiderH(k) ∂/hatwideu(k)+/tildewideF(k) u−/tildewideF(k) ηdD(k) u ds, (2.4) d/hatwideσ(k) ds=∂/hatwiderH(k) ∂/hatwideη(k)−/summationdisplay u=x,zD(k) u/tildewideF(k) u ;d/hatwideη(k) ds=−∂/hatwiderH(k) ∂/hatwideσ(k)+/tildewideF(k) η, (2.5) where /tildewideF(k) u=−p(k) 0Ak/parenleftBigg /hatwidep(k) u+/hatwideη(k)dD(k) u ds/parenrightBigg , (2.6) /tildewideF(k) η=−p(k) 0Ak/bracketleftBigg 1 +/parenleftBig 3−β2 k0+α(k) M/parenrightBig /hatwideη(k)+/summationdisplay u=x,zK(k) u/hatwideu(k)/bracketrightBigg , (2.7) 4Ak=C1\|Bk\|2+/radicalBig C2\|Bk\|3/2ξk(s), (2.8) C1=2ree2 3(mec)3 ; C2=55 24√ 3re¯he3 (mec)6 ;re=e2 4πǫ0mec2. (2.9) Hereα(k) Mis the momentum compaction factor, K(k) u(s) is the local curvature of the reference orbit, and Bk=/parenleftBig B(k) x, B(k) z, B(k) s/parenrightBig is the magnetic ﬁeld. The variable ξk(s) is a Gaussian random variable with formal properties: /an}b∇acketle{tξk(s)/an}b∇acket∇i}ht= 0 ; /an}b∇acketle{tξk(s)ξk(s′)/an}b∇acket∇i}ht=δ(s−s′). (2.10) The hamiltonian part in equations (2.4) and (2.5) consists o f three terms: /hatwiderH(k)=/hatwiderH(k) 0+/hatwiderH(k) 2+/hatwiderH(k) BB, (2.11) where /hatwiderH(k) 0=−K(k) 2/hatwideη(k)2+1 2πβ2 k0∆Ek0 Ek0cos/parenleftBigghk/tildewideσk R+ Φ k0/parenrightBigg , (2.12) /hatwiderH(k) 2=1 2/parenleftBig /hatwidep(k)2 x+/hatwidep(k)2 z/parenrightBig +1 2R2/parenleftBig G(k) x/hatwidex(k)2+G(k) z/hatwidez(k)2/parenrightBig , (2.13) /hatwiderH(k) BB=λkδp(s)Vk/parenleftBig x(k), z(k),/tildewideσ(k);s/parenrightBig . (2.14) The parameter K(k)is the so called slip phase coeﬃcient, hkis the harmonic number of the RF ﬁeld and ∆ Ek0is the energy gain per turn. The coeﬃcients G(k) x,z(s) represent the focusing strength of the linear machine lattice, δp(s) is the periodic delta-function, while λkand Vk/parenleftBig x(k), z(k),/tildewideσ(k);s/parenrightBig are the beam-beam coupling coeﬃcient and the beam-beam pote ntial, respectively. The latter are given by the expressions: λk=reN3−k γk01 +βk0β(3−k)0 β2 k0, (2.15) Vk/parenleftBig x(k), z(k),/tildewideσ(k);s/parenrightBig =/integraldisplay dxdzd/tildewideσGk/parenleftBig u(k)−u,/tildewideσ(k)−/tildewideσ;s/parenrightBig ρ3−k(u,/tildewideσ;s), (2.16) 5where Nkis the number of particles in the k-th beam and the Green’s function Gk(u,/tildewideσ;s) for the Poisson equation in the fully 3D case, in the ultra-re lativistic 2D case and in the 1D case can be written respectively as: Gk/parenleftBig u(k)−u,/tildewideσ(k)−/tildewideσ;s/parenrightBig =  −/bracketleftbigg/parenleftBig x(k)−x/parenrightBig2+/parenleftBig z(k)−z/parenrightBig2+/parenleftBig /tildewideσ(k)−/tildewideσ+ 2s/parenrightBig2/bracketrightbigg−1/2 , δ/parenleftBig /tildewideσ(k)−/tildewideσ+ 2s/parenrightBig ln/bracketleftbigg/parenleftBig x(k)−x/parenrightBig2+/parenleftBig z(k)−z/parenrightBig2/bracketrightbigg , 2πδ/parenleftBig /tildewideσ(k)−/tildewideσ+ 2s/parenrightBig δ/parenleftBig z(k)−z/parenrightBig/vextendsingle/vextendsingle/vextendsinglex(k)−x/vextendsingle/vextendsingle/vextendsingle.(2.17) In what follows we focus on the two-dimensional case, entire ly neglecting the longitudinal dynamics. Let us write down the Langevin equations of motion (2.4) and (2.5) once again in the following form: dx(k) ds=p(k), (2.18) dp(k) ds=F(k) L+F(k) B+F(k) R, (2.19) x(k)=/parenleftBig /hatwidex(k),/hatwidez(k)/parenrightBig ;p(k)=/parenleftBig /hatwidep(k) x,/hatwidep(k) z/parenrightBig , (2.20) where F(k) L=/parenleftBigg −G(k) x R2/hatwidex(k),−G(k) z R2/hatwidez(k)/parenrightBigg (2.21) is the (external) force acting on particles from the k-th beam, that is due to the linear focusing properties of the corresponding conﬁning lattice . Furthermore, F(k) B=λkδp(s)/parenleftBigg −∂Vk ∂/hatwidex(k),−∂Vk ∂/hatwidez(k)/parenrightBigg (2.22) is the beam-beam force and F(k) R=−pk0Ak/parenleftBigg /hatwidep(k) x−dD(k) x ds,/hatwidep(k) z−dD(k) z ds/parenrightBigg (2.23) is the synchrotron radiation friction force with a stochast ic component due to the quantum ﬂuctuations of synchrotron radiation [cf expression (2.8) ]. 63 The Nonlinear δfMethod It can be checked in a straightforward manner that the Klimon tovich microscopic phase space density fk(x,p;s) =1 NkNk/summationdisplay n=1δ/bracketleftBig x−x(k) n(s)/bracketrightBig δ/bracketleftBig p−p(k) n(s)/bracketrightBig (3.1) satisﬁes the following evolution equation: ∂fk ∂s+p· ∇xfk+/parenleftBig F(k) L+F(k) B/parenrightBig · ∇pfk+∇p·/parenleftBig F(k) Rfk/parenrightBig = 0, (3.2) where/braceleftBig x(k) n(s),p(k) n(s)/bracerightBig is the trajectory of the n-th particle from the k-th beam. Next we split the MPSD fkinto two parts according to the relation: fk(x,p;s) =fk0(x,p;s) +δfk(x,p;s), (3.3) where fk0is a solution to the equation ∂fk0 ∂s+p· ∇xfk0+/parenleftBig F(k) L+F(k) L0/parenrightBig · ∇pfk0+∇p·/parenleftBig F(k) Rfk0/parenrightBig = 0. (3.4) The quantity F(k) L0in Eq. (3.4) is the linear part of the beam-beam force F(k) B. The beam- beam force should be calculated with the on-coming beam dist ribution f(3−k)0. In what follows it will prove convenient to cast the beam-beam force into the form: F(k) B=F(k) L0+F(k) N0+δF(k) B, (3.5) where F(k) N0is the nonlinear (in the transverse coordinates) contribut ion calculated with f(3−k)0, while δF(k) Bdenotes the part of the beam-beam force due to δf3−k. It is worthwhile to note here that the representation (3.3) i s unique, embedding the basic idea of the δfmethod. However, one is completely free to ﬁx the f0part, which usually describes those features of the evolution of the system one c an solve easily (and preferably in explicit form). In the next Section we show that fk0, averaged over the statistical realizations of the process ξk(s) satisﬁes a Fokker-Planck equation and ﬁnd its solution. Subtract now the two equations (3.2) and (3.4) to obtain an eq uation for the δfk ∂δfk ∂s+p· ∇xδfk+/parenleftBig F(k) L+F(k) B/parenrightBig · ∇pδfk+∇p·/parenleftBig F(k) Rδfk/parenrightBig = 7=−/parenleftBig δF(k) B+F(k) N0/parenrightBig · ∇pfk0. (3.6) The next step consists in deﬁning the weight function that is relative to the total distribution as Wk(x,p;s) =δfk(x,p;s) fk(x,p;s). (3.7) Substituting δfk=Wkfk ; fk=fk0 1−Wk(3.8) into (3.6) and taking into account (3.2) we ﬁnally arrive at t he evolution equation for the weights: ∂Wk ∂s+p· ∇xWk+/parenleftBig F(k) L+F(k) B+F(k) R/parenrightBig · ∇pWk= =−1 fk/parenleftBig δF(k) B+F(k) N0/parenrightBig · ∇pfk0= =Wk−1 fk0/parenleftBig δF(k) B+F(k) N0/parenrightBig · ∇pfk0. (3.9) Equation (3.9) can be solved formally by the method of charac teristics. The ﬁrst couple of equations for the characteristics are precisely the equa tions of motion (2.18) and (2.19). Suppose their solution (particle’s trajectory in phase spa ce){x(s),p(s)}is known, and let us write down the last one of the equations for the characteri stics 1 Wk−1dWk ds=1 fk0/parenleftBig δF(k) B+F(k) N0/parenrightBig · ∇pfk0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x,p−→trajectory. (3.10) Note that its right-hand-side is a function of sonly, provided xandpare replaced by par- ticle’s trajectory in phase space {x(s),p(s)}. Therefore equation (3.10) can be integrated readily to give: Wk(s) = 1 + [ Wk(s0)−1]exp  s/integraldisplay s0dσ fk0(σ)/bracketleftBig δF(k) B(σ) +F(k) N0(σ)/bracketrightBig · ∇pfk0(σ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x(σ),p(σ)  .(3.11) 84 The Fokker-Planck Equation To derive the desired equation let us deﬁne the distribution function Fk0(x,p;s) and the ﬂuctuation δfk0(x,p;s) according to the relations: Fk0(x,p;s) =/an}b∇acketle{tfk0(x,p;s)/an}b∇acket∇i}ht;δfk0(x,p;s) =fk0(x,p;s)− F k0(x,p;s),(4.1) where /an}b∇acketle{t· · ·/an}b∇acket∇i}htimplies statistical average. Neglecting second order term s and correlators in δfk0 andδf(3−k)0that generally give rise to collision integrals, we write do wn the equations for Fk0andδfk0 ∂Fk0 ∂s+p· ∇xFk0+/parenleftBig F(k) L+F(k) L0/parenrightBig · ∇pFk0+∇p·/parenleftBig¯F(k) RFk0/parenrightBig = =−∇p·/angbracketleftBig/tildewideF(k) Rξk(s)δfk0/angbracketrightBig , (4.2) ∂δfk0 ∂s=−∇p·/parenleftBig/tildewideF(k) Rξk(s)Fk0/parenrightBig +O(δfk0), (4.3) where ¯F(k) Rand/tildewideF(k) Rdenote the deterministic and the stochastic parts of the rad iation fric- tion force F(k) Rrespectively. Moreover, the force F(k) L0should be calculated now with the distribution function Fk0. Equation (4.3) has a trivial solution δfk0(s) =−∇p·∞/integraldisplay 0dσ/tildewideF(k) R(s−σ)ξk(s−σ)Fk0(s−σ), (4.4) which is substituted into equation (4.2) yielding the Fokke r-Planck equation: ∂Fk0 ∂s+p· ∇xFk0+/parenleftBig F(k) L+F(k) L0/parenrightBig · ∇pFk0+∇p·/parenleftBig¯F(k) RFk0/parenrightBig = =∇p·/bracketleftBig/tildewideF(k) R∇p·/parenleftBig/tildewideF(k) RFk0/parenrightBig/bracketrightBig . (4.5) In order to carry out the δfmethod eﬀectively, it is important to ﬁnd an equilibrium solution of f0(or very slowly varying solution) so that the evolution of δfis separate in time scale from that of f0. In the following we discuss the equation and the solution of thef0 distribution. For the sake of simplicity, in what follows bellow in this Sec tion, we consider one dimen- sion only (say x), since the results can be easily generalized to the multidi mensional case, 9provided the x-z coupling is neglected. Let us write down the Fokker-Planck equation (4.5) in the simpliﬁed form: ∂Fk0 ∂s+p∂Fk0 ∂x−Fk(s)x∂Fk0 ∂p= Γk∂ ∂p(pFk0) +Dk∂2Fk0 ∂p2, (4.6) where Γk=pk0C1 2πR2πR/integraldisplay 0ds\|Bk(s)\|2; Dk=p2 k0C2 4πR2πR/integraldisplay 0ds\|Bk(s)\|3/angbracketleftBig p2 kx(s)/angbracketrightBig , (4.7) Fk(s) =G(k) x(s) R2+λkδp(s)A(k) x(s) ; A(k) x(s)x=∂Vk ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle linear part. (4.8) Let us seek for a solution of the Fokker-Planck equation (4.6 ) in the general form: Fk0(x, p;s) =ak(s) exp/bracketleftBigg −/tildewideγk(s)x2+ 2/tildewideαk(s)xp+/tildewideβk(s)p2 2ǫ(k) x0/bracketrightBigg , (4.9) where ǫ(k) x0is a scaling factor with dimensionality and meaning of emitt ance. Direct substi- tution of (4.9) into (4.6) and equating similar powers (up to second order) in xandpyield the following equations for the unknown coeﬃcients: dak ds= Γkak/parenleftBigg 1−/tildewideβk β(eq) k/parenrightBigg , (4.10) d/tildewideαk ds=Fk/tildewideβk−/tildewideγk+ Γk/tildewideαk/parenleftBigg 1−2/tildewideβk β(eq) k/parenrightBigg , (4.11) d/tildewideβk ds=−2/tildewideαk+ 2Γ k/tildewideβk/parenleftBigg 1−/tildewideβk β(eq) k/parenrightBigg , (4.12) d/tildewideγk ds= 2Fk/tildewideαk−2Γk/tildewideα2 k β(eq) k, (4.13) where 10β(eq) k=Γkǫ(k) x0 Dk(4.14) is the equilibrium β-function. It is important to note that when the damping vanishes (Γ k= 0) the above equations are exactly the same as the well-known diﬀerential equations fo r the Courant-Snyder parameters. In this sense the functions /tildewideαk,/tildewideβkand/tildewideγkcan be regarded as a generalization of the Courant- Snyder parameters in the case when radiation damping and qua ntum excitation are present. The well-known quantity /tildewideIk= det/parenleftBigg /tildewideγk/tildewideαk /tildewideαk/tildewideβk/parenrightBigg =/tildewideβk/tildewideγk−/tildewideα2 k (4.15) is no longer invariant. It is easy to check that its dynamics i s governed by the equation d/tildewideIk ds= 2Γ k/tildewideIk/parenleftBigg 1−/tildewideβk β(eq) k/parenrightBigg . (4.16) Comparison between equations (4.10) and (4.16) shows that ak(s) =Ck0/radicalBig /tildewideIk(s) (4.17) withCk0an arbitrary constant as it should be. Therefore the solutio n (4.9) takes its ﬁnal form Fk0(x, p;s) =/radicalBig /tildewideIk(s) 2πǫ(k) x0exp/bracketleftBigg −/tildewideγk(s)x2+ 2/tildewideαk(s)xp+/tildewideβk(s)p2 2ǫ(k) x0/bracketrightBigg , (4.18) Let us deﬁne now the dimensionless envelope function σkaccording to the relations σk=√βke ak; βke=/tildewideβk β(eq) k. (4.19) Manipulating equations (4.11), (4.12) and (4.13) for the ge neralized Courant-Snyder param- eters one can eliminate /tildewideαkand/tildewideγkand obtain a single equation for the envelope σk, which combined with equation (4.10) comprises a complete set: d2σk ds2+ Γkdσk ds+Fkσk=1 β(eq)2 ka2 kσ3 k, (4.20) 11dak ds= Γkak/parenleftBig 1−a2 kσ2 k/parenrightBig . (4.21) By solving equations (4.20) and (4.21) one can obtain a compl ete information about the evolution of the Fk0part of the distribution function. However, solving the abo ve system of equations for the beam envelopes and amplitudes of the dis tributions is not an easy task. For that purpose we develop in the next Section a numerical sc heme which is more suited for direct code implementation. 5 Numerical Algorithm In the previous Sections, we have established the theoretic al foundation of the nonlinear δf method for the beam-beam interaction. In this Section we wil l apply those results to outline numerical algorithms suitable for computer simulation. Starting with Eq. (3.4), because the forces in the equation b oth from lattice and the on-coming beam are linear, its solution is well known Gaussi an distribution (for example as shown in the previous Section in the one-dimensional case) Fk0(z;s) =1 /bracketleftBig 2πdet/parenleftBig/hatwideΣk/parenrightBig/bracketrightBig3 2exp/parenleftbigg −1 2zT·/hatwideΣ−1 k·z/parenrightbigg , (5.1) where/hatwideΣkis the matrix of the second moments for the distribution and zis a vector in the six- dimensional phase space. Based on the method of the beam-env elope [14], the propagation ofFk0can be represented as the iteration of the/hatwideΣkmatrix, /hatwideΣ(i+1) k=/hatwiderMk·/hatwideΣ(i) k·/hatwiderMT k+/hatwiderDk, (5.2) where/hatwiderMkis the one-turn matrix including the linear beam-beam force of the on-coming beam, and the radiation damping and/hatwiderDkis the one-turn quantum diﬀusion matrix. Both /hatwiderMkand/hatwiderDkcan be extracted from the lattice using for example the LEGO c ode [15], [16]. However, there is a diﬀerence compared to the situation of a s ingle storage ring, namely, we have to simultaneously iterate the Gaussian distributio n for both beams, since the linear map/hatwiderMkdepends on the beam size of the other beam. Combining Eqs. (3.1) and (3.7), the perturbative part of the beam distribution δfkhas a representation in terms of macro-particles δfk(x,p;s) =1 NkNk/summationdisplay n=1W(n) k(s)δ/bracketleftBig x−x(k) n(s)/bracketrightBig δ/bracketleftBig p−p(k) n(s)/bracketrightBig , (5.3) where W(n) k(s) is the dynamical weight of the n-th particle from the k-th beam. As a part of the solution for Eq. (3.9), the propagation of the particle coordinates in phase space is the same as the conventional PIC code [8] provi ded that the beam-beam force is the sum of the two parts from both Fk0andδfk. 12For the Fk0part, we can apply the well known Erskine-Bassetti formula [ 17] for a Gaus- sian beam. The force due to the δfkis obtained by solving the two-dimensional Poisson equation. In addition to the change of the coordinate, the we ight of the particle should be propagated according to Eq. (3.11). The weight should be upd ated after the change of the coordinate since the change of the weight depends on the traj ectory of the particle. 6 Summary We have developed an eﬃcacious algorithm for simulating the beam-beam interaction in a synchrotron collider with (or without) synchrotron radiat ion. The nonlinear δfmethod has been introduced into the evolutionary description of subtl e changes of the counter stream- ing distribution of the colliding beams over many revolutio ns. The overall equation that describes this evolution is the Fokker-Planck equation (wi th the radiative process and quan- tum ﬂuctuations). In order to isolate the δfdistribution from the average distribution, we analyze the solution of the Fokker-Planck equation. Obtain ed is a form of solution in which the time dependence is parameterized through a slow evoluti on (slow compared with the changes in the δfdistribution due to the individual beam-beam interaction) in the Courant- Snyder parameters and the emittance of the beam. This algori thm will enhance the analysis capability to scrutinize greater details and subtle eﬀects in the beam-beam interaction than the PIC version which has been widely deployed [8]. The current algorithm as well as the previous one [8] have bee n developed with an immediate application to the PEP-II B-factory collider. Th e code [8] has already been applied to describe the beam-beam interaction in the PEP-II with unprecedented accuracy and reproduction faithfulness, and will be suﬃcient to stud y the overall dynamics such as the analysis of resonance instabilities and associated lum inosity functions. It is anticipated, however, that the numerical noise associated with the PIC wi ll require either an inordinate amount of macro-particle deployment or a level of noise high enough to mask some minute phase space structure that may manifest in subtle but import ant long-time evolution of the beam such as particle diﬀusion. It is here that the current al gorithm will cope with the problem. Acknowledgments We would like to thank John Irwin and Ron Ruth for their contin uous support and encour- agement. It is our pleasure to thank Sam Heifets and Robert Wa rnock for many stimulating discussions. One of the authors (T.T.) is supported in part b y DOE contract W-7405-Eng.48 and DOE grant DE-FG03-96ER40954. References [1] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Acc elerators , Wiley, New York, 1993. 13[2] D. Neuﬀer, A. Riddiford and A. Ruggiero, IEEE Trans. Nucl . Sci.,NS-30 , 2430 (1983). [3] M. Month and J.C. Herrera eds., Nonlinear Dynamics and the Beam-Beam Interaction , AIP, New York, 1979. [4] C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulation , McGraw– Hill, New York, 1983. [5] T. Tajima, Computational Plasma Physics , Addison–Wesley, Reading, Mass., 1989. [6] S. Krishnagopal and R. Siemann, “Coherent Beam-Beam Int eraction 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] Y. Cai, A.W. Chao, S.I. Tzenov and T. Tajima, “Simulation of the Beam-Beam Eﬀects in e+e−Storage Rings with a Method of Reducing the Region of Mesh”, S LAC-PUB-8589, August 2000. [9] “PEP-II: An Asymmetric B Factory”, Conceptual Design Re port, SLAC-418, June 1993. [10] T. Tajima and F.W. Perkins, in Proc. of 1983 Sherwood The ory Meeting, Univ. of Maryland, Arlington, VA, 1983. [11] M. Kotschenreuther, Bull. Am. Phys. Soc., 33, 2109 (1988). [12] J.K. Koga and T. Tajima, J. Comput. Phys., 116, 314 (1995). [13] Yu.L. Klimontovich, The Statistical Theory of Non-equilibrium Processes in a Pl asma, MIT Press, Cambridge, MA, 1967. [14] K. Ohmi, K. Hirata, and K. Oide, “From the Beam-Envelope Matrix to Synchrotron- Radiation Integrals,” Phys. Rev. E 49751 (1994). [15] Y. Cai, M. Donald, J. Irwin and Y. Yan, “LEGO: A Modular Ac celerator Design Code,” SLAC-PUB-7642, August 1997. [16] Y. Cai, “Simulation of Synchrotron Radiation in an Elec tron Storage Ring,” Proceeding of Advanced ICFA Beam Dynamics Workshop on Quantum Aspects o f Beam Physics, Edited by Pisin Chen (1998). [17] M. Bassetti and G. Erskine, CERN ISR TH/80-06 (1980). 14
arXiv:physics/0010056v1 [physics.atom-ph] 23 Oct 2000External-ﬁeld shifts of the199Hg+optical frequency standard[] Wayne M. Itano Time and Frequency Division, National Institute of Standar ds and Technology, Boulder, CO 80305 (Dated: 23 October 2000) Frequency shifts of the199Hg+5d106s2S1/2(F= 0, MF= 0) to 5 d96s2 2D5/2(F= 2, MF= 0) electric-quadrupole transition at 282 nm due to external ﬁe lds are calculated, based on a com- bination of measured atomic parameters and ab initio calculations. This transition is under investigation as an optical frequency standard. The pertur bations calculated are the quadratic Zeeman shift, the scalar and tensor quadratic Stark shifts, and the interaction between an ex- ternal electric ﬁeld gradient and the atomic quadrupole mom ent. The quadrupole shift is likely to be the most diﬃcult to evaluate in a frequency standard and may have a magnitude of about 1 Hz for a single ion in a Paul trap. Key words: atomic polarizabilities; electric quadrupole interactio n; mercury ion; optical fre- quency standards; Stark shift; Zeeman shift. 1. INTRODUCTION It has long been recognized that a frequency stan- dard could be based on the 282 nm transition between the ground 5 d106s2S1/2level and the metastable 5 d96s2 2D5/2level of Hg+[1]. The lifetime of the upper level is 86(3) ms [2], so the ratio of the natural linewidth ∆ ν to the transition frequency ν0is 2×10−15. (Unless oth- erwise noted, all uncertainties given in this paper are standard uncertainties, i.e., one standard deviation esti - mates.) Doppler broadening can be avoided if the tran- sition is excited with two counter-propagating photons, as originally proposed by Bender et al. [1] and subse- quently demonstrated by Bergquist et al. [3]. However, optical Stark shifts are greatly reduced if the transition is driven instead with a single photon by the electric- quadrupole interaction. In this case, Doppler broaden- ing can be eliminated if the ion is conﬁned to dimen- sions much less than the optical wavelength, as was ﬁrst demonstrated by Bergquist et al. [4]. Recently, the ( F= 0,MF= 0) to (F= 2,MF= 0) hy- perﬁne component of the199Hg+5d106s2S1/2to 5d96s2 2D5/2single-photon transition has been observed with a linewidth of only 6.7 Hz by Rafac et al. [5]. A laser servo- locked to this transition is an extremely stable and repro- ducible frequency reference. New developments in opti- cal frequency metrology [6, 7] may soon make this system practical as an atomic frequency standard or clock. While the ( F= 0,MF= 0) to (F= 2,MF= 0) hy- perﬁne component has no linear Zeeman shift, it does have a quadratic Zeeman shift that must be accounted for. In addition, there is a second-order Stark shift and a shift due to the interaction between the electric-ﬁeld gra- dient and the atomic electric-quadrupole moment. None of these shifts has yet been measured accurately, so it is useful to have calculated values, even if they are not veryprecise. Also, it is useful to know the functional form of the perturbation, even if the magnitude is uncertain. For example, the quadrupole shift can be eliminated by av- eraging the transition frequency over three mutually or- thogonal magnetic-ﬁeld orientations, independent of the orientation of the electric-ﬁeld gradient. 2. METHODS AND NOTATION The quadratic Zeeman shift can be calculated if the hy- perﬁne constants and electronic and nuclear g-factors are known. Similarly, the quadratic Stark eﬀect can be cal- culated from a knowledge of the electric-dipole oscillator strengths. The quadrupole shift depends on the atomic wavefunctions. Some of these parameters have been mea- sured, such as the hyperﬁne constants and some of the oscillator strengths. There are also published calcula- tions for some of the oscillator strengths. Here, we estimate, by the use of the Cowan atomic- structure codes, values for parameters for which there are neither measured values nor published calculations. The Cowan codes are based on the Hartree-Fock ap- proximation with some relativistic corrections [8]. The odd-parity conﬁgurations included in the calculation were 5d10np(n= 6,7,8,9), 5d105f, 5d96s6p, 5d96s7p, 5d96s5f, and 5d86s26p. The even-parity conﬁgurations were 5d10ns(n= 6,7,8,9,10), 5d10nd(n= 6,7,8,9), 5d96s2, 5d96s7s, 5d96s6d, and 5d96p2. Recently, San- sonetti and Reader have made new measurements of the spectrum of Hg+and classiﬁed many new lines [9]. They also carried out a least-squares adjustment of the energy parameters that enter the Cowan-code calculations in or- der to match the observed energy levels. We use these adjusted parameters in our Cowan-code calculations. As one test of this method of calculation, we estimated2 the weakly allowed 10.7 µm 5d106p2P1/2to 5d96s2 2D3/2 electric-dipole decay rate. This decay is allowed only be- cause of conﬁguration mixing, since it requires two elec- trons to change orbitals. The calculation shows the de- cay to be due mostly to mixing between the 5 d106pand 5d96s6pconﬁgurations. The calculated rate is 55.6 s−1; the measured rate is 52(16) s−1[2]. Another test is the electric-quadrupole decay rate of the 5 d96s2 2D5/2level to the ground level. The calculated rate is 12.6 s−1, and the measured rate is 11.6(0.4) s−1. Similar calculations have been carried out by Wilson [10]. LetH0be the atomic Hamiltonian, exclusive of the hy- perﬁne and external ﬁeld eﬀects, which are treated as per- turbations. For convenience, we denote the eigenstates ofH0corresponding to the electronic levels 5 d106s2S1/2 and 5d96s2 2D5/2havingJzeigenvalueMJby\|S 1/2MJ/an}b∇acket∇i}ht and\|D 5/2MJ/an}b∇acket∇i}ht, respectively. The corresponding eigenvalues of H0are denoted W(S,1/2) andW(D,5/2). An arbitrary eigenstate of H0with eigenvalue W(γ,J) and electronic angular mo- mentumJis denoted \|γ J M J/an}b∇acket∇i}ht. Since199Hg+has in addition a nuclear angular momentum I, whereI= 1/2, the complete state designation is \|γJFM F/an}b∇acket∇i}ht, whereFis the total angular momentum, and MFis the eigenvalue ofFz. 3. QUADRATIC ZEEMAN SHIFT In order to calculate the energy shifts due to the hy- perﬁne interaction and to an external magnetic ﬁeld B≡Bˆz, we deﬁne eﬀective Hamiltonian operators H′ S andH′ Dthat operate within the subspaces of hyper- ﬁne sublevels associated with the electronic levels 5 d106s 2S1/2and 5d96s2 2D5/2respectively: H′ S=hASI·J+gJ(S)µBJ·B+g′ IµBI·B,(1) H′ D=hADI·J+gJ(D)µBJ·B+g′ IµBI·B,(2) whereASandADare the dipole hyperﬁne constants, gJ(S) andgJ(D) are the electronic g-factors,g′ Iis the nuclearg-factor,his the Planck constant, and µBis the Bohr magneton. All of the parameters entering H′ S andH′ Dare known from experiments, although a more accurate measurement of gJ(D) would be useful. The ground-state hyperﬁne constant AShas been measured in a199Hg+microwave frequency standard to be 40 507.347 996 841 59 (43) MHz [11]. The excited-state hyperﬁne constantADhas been measured recently by an extension to the work described in Ref. [5], in which the diﬀer- ence in the frequencies of the \|S 1/2 0 0/an}b∇acket∇i}htto\|D 5/2 2 0/an}b∇acket∇i}ht and the \|S 1/2 0 0 /an}b∇acket∇i}htto\|D 5/2 3 0 /an}b∇acket∇i}httransition frequen- cies was determined to be 3 AD=2 958.57(12) MHz [12],in good agreement with an earlier, less precise measure- ment by Fabry-P´ erot spectroscopy [13]. The ground- state electronic g-factorgJ(S) was measured in198Hg+ by rf-optical double resonance to be 2.003 174 5(74) [14]. The excited-state electronic g-factorgJ(D) was measured in198Hg+by conventional grating spectroscopy of the 398 nm 5d106p2P3/2to 5d96s2 2D5/2line to be 1.198 0(7) [15]. The diﬀerence in gJ(S) orgJ(D) between 198Hg+and199Hg+is estimated to be much less than the experimental uncertainties. The nuclear g-factorg′ I is−5.422 967(9) ×10−4[16]. The measurement was made with neutral ground-state199Hg atoms, so the diamag- netic shielding factor will be slightly diﬀerent from that in the ion. However, this is eﬀect is negligible, since the magnitude of g′ Iis so small compared to gJ(S) orgJ(D). The determination of gJ(D) could be improved by mea- suring the optical-frequency diﬀerence between two com- ponents of the 282 nm line and the frequency of a ground- state microwave transition at the same magnetic ﬁeld. Since the uncertainty in the quadratic Zeeman shift is due mainly to the uncertainty in gJ(D), it is useful to see how accurately it can be estimated theoretically. The Land´ eg-factor for a2D5/2state, including the correction for the anomalous magnetic moment of the electron, is 1.200 464. The Cowan-code calculation shows that the conﬁguration mixing does not change this value by more than about 10−6, i.e., 1 in the last place. There are sev- eral relativistic and diamagnetic corrections that modify gJ(D), one of which, called the Breit-Margenau correc- tion by Abragam and Van Vleck [17], is proportional to the electron mean kinetic energy. The other corrections are more diﬃcult to calculate. The Cowan-code result for the mean kinetic energy of an electron in the 5 dorbital of the 5d96s2conﬁguration is T= 19.32hcR∞, whereR∞ is the Rydberg constant. Using this value, we obtain a theoretical value of gJ(D), including the Breit-Margenau correction, of 1.199 85, which disagrees with the the ex- perimental value by 1 .85×10−3, which is 2.6 times the estimated experimental uncertainty of Ref. [15]. If we calculategJ(D) for neutral gold, which is isoelectronic to Hg+, by the same method, we obtain a value which diﬀers from the accurately measured experimental one [18] by (7 ±2)×10−5. Thus, the error in the calculated value forgJ(D) of199Hg+might be less than 1 ×10−4, but it is impossible to be certain of this, since there are uncalculated terms. Measurements of the199Hg+optical clock frequency at diﬀerent values of the magnetic ﬁeld should result in a better experimental value for gJ(D) in the near future. For low magnetic ﬁelds ( Bless than 1 mT), it is suﬃ- cient to calculate the energy levels to second order in B. To this order in B, the energies of the hyperﬁne-Zeeman sublevels for the ground electronic level are3 W(S,1/2,0,0,B) =W(S,1/2)−3hAS 4−[gJ(S)−g′ I]2µ2 BB2 4hAS, (3) W(S,1/2,1,0,B) =W(S,1/2) +hAS 4+[gJ(S)−g′ I]2µ2 BB2 4hAS, (4) W(S,1/2,1,±1,B) =W(S,1/2) +hAS 4±[gJ(S) +g′ I]µBB 2. (5) For the 5d96s2 2D5/2level we have W(D,5/2,2,0,B) =W(D,5/2)−7hAD 4−[gJ(D)−g′ I]2µ2 BB2 12hAD, (6) W(D,5/2,2,±1,B) =W(D,5/2)−7hAD 4±[7gJ(D)−g′ I]µBB 6−2[gJ(D)−g′ I]2µ2 BB2 27hAD, (7) W(D,5/2,2,±2,B) =W(D,5/2)−7hAD 4±[7gJ(D)−g′ I]µBB 3−5[gJ(D)−g′ I]2µ2 BB2 108hAD, (8) W(D,5/2,3,0,B) =W(D,5/2) +5hAD 4+[gJ(D)−g′ I]2µ2 BB2 12hAD, (9) W(D,5/2,3,±1,B) =W(D,5/2) +5hAD 4±[5gJ(D) +g′ I]µBB 6+2[gJ(D)−g′ I]2µ2 BB2 27hAD, (10) W(D,5/2,3,±2,B) =W(D,5/2) +5hAD 4±[5gJ(D) +g′ I]µBB 3+5[gJ(D)−g′ I]2µ2 BB2 108hAD, W(D,5/2,3,±3,B) =W(D,5/2) +5hAD 4±[5gJ(D) +g′ I]W(S,B 2. (11) Here,W(γ,J,F,M F,B) denotes the energy of the state \|γJFM F/an}b∇acket∇i}ht, including the eﬀects of the hyperﬁne interac- tion and the magnetic ﬁeld. At a value of Bof 0.1 mT, the quadratic shift of the \|S 1/2 0 0/an}b∇acket∇i}htto\|D 5/2 2 0/an}b∇acket∇i}httransition (optical clock tran- sition)is −189.25(28) Hz, where the uncertainty stems mainly from the uncertainty in the experimental value ofgJ(D). In practice, the error may be less than this if the magnetic ﬁeld is determined from the Zeeman split- tings within the \|D 5/2F M F/an}b∇acket∇i}htsublevels. The reason is that an error in gJ(D) leads to an error in the value ofBinferred from the Zeeman splittings, which partly compensates for the gJ(D) error. If instead we use the calculated value of gJ(D), the quadratic shift for B= 0.1 mT is −189.98 Hz, where the uncertainty is diﬃcult to estimate. 4. QUADRATIC STARK SHIFT The theory of the quadratic Stark shift in free atoms has been described in detail by Angel and Sandars [19]. The Stark Hamiltonian is HE=−µ·E, (12)where µis the electric-dipole moment operator, µ=−e/summationdisplay iri, (13) andEis the applied external electric ﬁeld. In Eq. (13), riis the position operator of the ith electron, measured relative to the nucleus, and the summation is over all electrons. First consider an atom with zero nuclear spin, such as 198Hg+. To second order in the electric ﬁeld, the Stark shifts of the set of sublevels \|γJM J/an}b∇acket∇i}htdepend on two pa- rameters,αscalar(γ,J) andαtensor(γ,J), called the scalar and tensor polarizabilities. In principle, when both mag- netic and electric ﬁelds are present but are not parallel, the energy levels are obtained by simultaneously diago- nalizing the hyperﬁne, Zeeman, and Stark Hamiltonians. In practice, the Zeeman shifts are normally much larger than the Stark shifts, so that HEdoes not aﬀect the di- agonalization. In that case, the energy shift of the state \|γJM J/an}b∇acket∇i}htdue toHEis ∆WE(γ,J,M J,E) =−1 2αscalar(γ,J)E2−1 4αtensor(γ,J)[3M2 J−J(J+ 1)] J(2J−1)(3E2 z−E2). (14)4 TreatingHEby second-order perturbation theory leads to the following expressions for the polarizabilities [19]: αscalar(γ,J) =8πǫ0 3(2J+ 1)/summationdisplay γ′J′\|(γJ/ba∇dblµ(1)/ba∇dblγ′J′)\|2 W(γ′,J′)−W(γ,J), (15) αtensor(γ,J) = 8πǫ0/bracketleftbigg10J(2J−1) 3(2J+ 3)(J+ 1)(2J+ 1)/bracketrightbigg1/2/summationdisplay γ′J′(−1)J−J′/braceleftbigg 1 1 2 J J J′/bracerightbigg\|(γJ/ba∇dblµ(1)/ba∇dblγ′J′)\|2 W(γ′,J′)−W(γ,J).(16) The summations are over all levels other than \|γJ/an}b∇acket∇i}ht. Equations (15) and(16) can be rewritten in terms of the osci llator strengthsfγJ,γ′J′: αscalar(γ,J) =4πǫ0e2¯h2 me/summationdisplay γ′J′fγJ,γ′J′ [W(γ′,J′)−W(γ,J)]2, (17) αtensor(γ,J) =4πǫ0e2¯h2 me/bracketleftbigg30J(2J−1)(2J+ 1) (2J+ 3)(J+ 1)/bracketrightbigg1/2/summationdisplay γ′J′(−1)J−J′/braceleftbigg 1 1 2 J J J′/bracerightbiggfγJ,γ′J′ [W(γ′,J′)−W(γ,J)]2,(18) wheremeis the electron mass. The tensor polarizability is zero for l evels withJ <1, such as the Hg+5d106s2S1/2 level. For an atom with nonzero nuclear spin I, the quadratic Stark shift of the state \|γJFM F/an}b∇acket∇i}htis ∆WE(γ,J,F,M F,E) =−1 2αscalar(γ,J,F )E2−1 4αtensor(γ,J,F )[3M2 F−F(F+ 1)] F(2F−1)(3E2 z−E2). (19) We make the approximation that hyperﬁne interac- tion does not modify the electronic part of the atomic wavefunctions (the IJ-coupling approximation of Angel and Sandars [19]). This approximation is adequate for the present purpose, which is to evaluate the Stark shift of the199Hg+optical clock transition. Obtaining thediﬀerential Stark shift between the hyperﬁne levels of the ground state, which is signiﬁcant for the199Hg+ microwave frequency standard [11], requires going to a higher order of perturbation theory [20]. In the IJ- coupling approximation [19], αscalar(γ,J,F ) =αscalar(γ,J), (20) αtensor(γ,J,F ) = (−1)I+J+F/bracketleftbiggF(2F−1)(2F+ 1)(2J+ 3)(2J+ 1)(J+ 1) (2F+ 3)(F+ 1)J(2J−1)/bracketrightbigg1/2/braceleftbigg F J I J F 2/bracerightbigg αtensor(γ,J).(21) Equations (17) and (18) were used to evaluate the po- larizabilities for the Hg+5d106s2S1/2and 5d96s2 2D5/2 levels. For the calculation of αscalar(S,1/2), the os- cillator strengths for all electric-dipole transitions co n- necting the 5 d106sconﬁguration to the 5 d10np(n= 6,7,8) and 5d96s6pconﬁgurations were included. These were taken from the theoretical work of Brage et al. [21]. The ﬁnal result is αscalar(S,1/2)/(4πǫ0) = 2.41× 10−24cm3, which compares very well with the value of 2.22×10−24cm3obtained by Henderson et al. from a combination of experimental and calculated oscilla- tor strengths [22]. For the calculations of αscalar(D,5/2) andαtensor(D,5/2), the oscillator strengths for electric- dipole transitions to the 5 d10np(n= 6,7,8), 5d105f, and 5d96s6pconﬁgurations were taken from Brage et al.[21]. The oscillator strengths for electric-dipole transitions to the 5 d96s7pand 5d86s26pconﬁgurations were taken from the Cowan-code calculations. The re- sults wereαscalar(D,5/2)/(4πǫ0) = 3.77×10−24cm3and αtensor(D,5/2)/(4πǫ0) =−0.263×10−24cm3. Evalu- ating Eq. (21) for F=2 andF=3 in the 5 d96s2 2D5/2level, we obtain αtensor(D,5/2,2) =4 5αtensor(D,5/2) and αtensor(D,5/2,3) =αtensor(D,5/2). The tensor polarizability is much smaller than the scalar polarizabilities and in any case does not contribute if the external electric ﬁeld is isotropic, as is the case for the blackbody radiation ﬁeld. The net shift of the op- tical clock transition due to the scalar polarizabilities i s 1 2[αscalar(S,1/2)−αscalar(D,5/2)]E2. In frequency units, the shift is −1.14×10−3E2Hz, whereEis expressed in V/cm. The error in the coeﬃcient is diﬃcult to esti- mate, particularly since it is a diﬀerence of two quanti- ties of about the same size. However, the total shifts are small for typical experimental conditions. If the electric ﬁeld is time-dependent, as for the blackbody ﬁeld, the mean-squared value /an}b∇acketle{tE2/an}b∇acket∇i}htis taken. At a temperature of 300 K, the shift of the optical clock transition due to the blackbody electric ﬁeld is −0.079 Hz. The mean-squared blackbody ﬁeld is proportional to the fourth power of the temperature. For a single, laser-cooled ion in a Paul trap, the mean-squared trapping electric ﬁelds can be made small enough that the Stark shifts are not likely to5 be observable [23]. 5. ELECTRIC QUADRUPOLE SHIFT The atomic quadrupole moment is due to a depar- ture of the electronic charge distribution of an atom from spherical symmetry. Atomic quadrupole moments were ﬁrst measured by the shift in energy levels due to an applied electric-ﬁeld gradient in atomic-beam resonance experiments [24, 25]. The interaction of the atomic quadrupole moment with external electric-ﬁeld gradients, for example those generated by the electrodes of an ion trap, is analo- gous to the interaction of a nuclear quadrupole moment with the electric ﬁeld gradients due to the atomic elec- trons. Hence, we can adapt the treatment used for the electric-quadrupole hyperﬁne interaction of an atom [26]. The Hamiltonian describing the interaction of external electric-ﬁeld gradients with the atomic quadrupole mo- ment is HQ=∇E(2)·Θ(2)=2/summationdisplay q=−2(−1)q∇E(2) qΘ(2) −q,(22) where ∇E(2)is a tensor describing the gradients of the external electric ﬁeld at the position of the atom, and Θ(2)is the electric-quadrupole operator for the atom. Following Ref. [26], we deﬁne the components of ∇E(2) as ∇E(2) 0=−1 2∂Ez ∂z, (23) ∇E(2) ±1=±√ 6 6∂E± ∂z=±√ 6 6∂±Ez, (24) ∇E(2) ±2=−√ 6 12∂±E±, (25) whereE±≡Ex±iEyand∂±≡∂ ∂x±i∂ ∂y. The operator components Θ(2) qare deﬁned in terms of the electronic coordinate operators as Θ(2) 0=−e 2/summationdisplay j(3z2 j−r2 j), (26) Θ(2) ±1=−e/radicalbigg 3 2/summationdisplay jzj(xj±iyj), (27) Θ(2) ±2=−e/radicalbigg 3 8/summationdisplay j(xj±iyj)2, (28) where the sums are taken over all the electrons. The quadrupole moment Θ( γ,J) of an atomic level \|γJ/an}b∇acket∇i}htis deﬁned by the diagonal matrix element in the state with maximumMJ: Θ(γ,J) =/an}b∇acketle{tγJJ\|Θ(2) 0\|γJJ/an}b∇acket∇i}ht. (29) This is the deﬁnition used by Angel et al. [24]. In order to simplify the form of ∇E(2), we make a principal-axis transformation as in Ref. [27]. That is, weexpress the electric potential in the neighborhood of the atom as Φ(x′,y′,z′) =A[(x′2+y′2−2z′2) +ǫ(x′2−y′2)].(30) The principal-axis (primed) frame ( x′,y′,z′) is the one in which Φ has the simple form of Eq. (30), while the laboratory (unprimed) frame ( x,y,z) is the in which the magnetic ﬁeld is oriented along the zaxis. The tensor components of ∇E(2)in the principal-axis frame are obtained by taking derivatives of Φ( x′,y′,z′): ∇E(2) 0′=−2A, (31) ∇E(2) ±1′ = 0, (32) ∇E(2) ±2′=/radicalbigg 2 3ǫA. (33) In the principal-axis frame, HQhas the simple form HQ=−2AΘ(2) 0′+/radicalbigg 2 3ǫA/parenleftBig Θ(2) 2′+ Θ(2) −2′/parenrightBig .(34) As long as the energy shifts due to HQare small rel- ative to the Zeeman shifts, which is the usual case in practice,HQcan be treated as a perturbation. In that case, it is necessary only to evaluate the matrix elements ofHQthat are diagonal in the basis of states \|γJFM F/an}b∇acket∇i}ht, where Fis the total atomic angular momentum, includ- ing nuclear spin I, andMFis the eigenvalue of Fzwith respect to the laboratory (not principal-axis) frame. Let ωdenote the set of Euler angles {α,β,γ }that takes the principal-axis frame to the laboratory frame. To be ex- plicit, starting from the principal-axis frame, we rotate the coordinate system about the zaxis byα, then about the newyaxis byβ, and then about the new zaxis byγ so that the rotated coordinate system coincides with the laboratory coordinate system. We can set γ= 0, since the ﬁnal rotation about the laboratory zaxis, which is parallel to B, has no eﬀect. The states \|γJFm /an}b∇acket∇i}ht′deﬁned in the principal-axis frame and the states \|γJFµ /an}b∇acket∇i}htdeﬁned in the laboratory frame are related by \|γJFm /an}b∇acket∇i}ht′=/summationdisplay µD(F) µm(ω)\|γJFµ /an}b∇acket∇i}ht, (35) whereD(F) µm(ω) is a rotation matrix element deﬁned in the passive representation [28, 29]. The inverse relation is \|γJFµ /an}b∇acket∇i}ht=/summationdisplay mD(F) µm∗(ω)\|γJFm /an}b∇acket∇i}ht′. (36) In order to evaluate the diagonal matrix elements of HQin the laboratory frame, it is necessary to evaluate matrix elements of the operators Θ(2) q′, deﬁned in the principal-axis frame. These matrix elements are of the form6 /an}b∇acketle{tγJFµ \|Θ(2) q′\|γJFµ /an}b∇acket∇i}ht=/summationdisplay m′mD(F) µm′(ω)D(F) µm∗(ω)′/an}b∇acketle{tγJFm′\|Θ(2) q′\|γJFm /an}b∇acket∇i}ht′, (37) = (γJF/ba∇dblΘ(2)/ba∇dblγJF)/summationdisplay m′m(−1)F−m′/parenleftbigg F2F −m′q m/parenrightbigg D(F) µm′(ω)D(F) µm∗(ω), (38) = (−1)F−µ−q(γJF/ba∇dblΘ(2)/ba∇dblγJF)/summationdisplay m′m/parenleftbigg F2F −m′q m/parenrightbigg D(F) µm′(ω)D(F) −µ−m(ω), (39) = (−1)F−µ−q(γJF/ba∇dblΘ(2)/ba∇dblγJF)/summationdisplay K m m′n n′(2K+ 1)/parenleftbigg F2F −m′q m/parenrightbigg /parenleftbigg F F K µ−µ n′/parenrightbigg /parenleftbigg F F K m′−m n/parenrightbigg D(K) n′n∗ (ω),(40) = (−1)F−µ−q(γJF/ba∇dblΘ(2)/ba∇dblγJF)/parenleftbigg F2F −µ0µ/parenrightbigg D(2) 0−q∗(ω), (41) where Eq. (38) follows from the Wigner-Eckart theorem, and E qs. (39), (40), and (41) follow from Eqs. (4.2.7), (4.3.2), and (3.7.8) of Ref. [28], respectively. The required rotati on matrix elements are, from Eq. (4.1.25) of Ref. [28] (with correction of a typographical error), D(2) 00∗(ω) =1 2(3 cos2β−1), (42) D(2) 0±2∗(ω) =/radicalBig 3 8sin2β(cos2α∓i sin2α). (43) The 3-jsymbol in Eq. (41) is /parenleftbigg F2F −µ0µ/parenrightbigg = (−1)F−µ 2[3µ2−F(F+ 1)] [(2F+ 3)(2F+ 2)(2F+ 1)2F(2F−1)]1/2. (44) The diagonal matrix elements of HQin the laboratory frame are /an}b∇acketle{tγJFM F\|HQ\|γJFM F/an}b∇acket∇i}ht= −2[3M2 F−F(F+ 1)]A(γJF/ba∇dblΘ(2)/ba∇dblγJF) [(2F+ 3)(2F+ 2)(2F+ 1)2F(2F−1)]1/2[3 cos2β−1)−ǫsin2β(cos2α−sin2α)]. (45) It is simple to show, by directly integrating the angu- lar factor in square brackets in Eq. (45), that the average value of the diagonal matrix elements of HQ, taken over all possible orientations of the laboratory frame with re- spect to the principal-axis frame, is zero. This also fol- lows directly from the fact that the quantity in square brackets is a linear combination of spherical harmonics. It is less obvious that the average, taken over any three mutually perpendicular orientations of the laboratory z quantization axis, is also zero. This result is proven in the Appendix. This provides a method for eliminating thequadrupole shift from the observed transition frequency. The magnetic ﬁeld must be oriented in three mutually perpendicular directions with respect to the trap elec- trodes, which are the source of the external quadrupole ﬁeld, but with the same magnitude of the magnetic ﬁeld. The average of the transition frequencies taken under these three conditions does not contain the quadrupole shift. The reduced matrix element in Eq. (45) is, in the IJ- coupling approximation, (γ(IJ)F/ba∇dblΘ(2)/ba∇dblγ(IJ)F) = (−1)I+J+F(2F+ 1)/braceleftbigg J2J F I F/bracerightbigg/parenleftbigg J2J −J0J/parenrightbigg−1 Θ(γ,J), (46) whereIis included in the state notation in order to spec- ify the order of coupling of IandJ. For the particular case of the199Hg+5d96s2 2D5/2level, the reduced matrix elements are (D 5/2 2/ba∇dblΘ(2)/ba∇dblD 5/2 2) = 2/radicalbigg 14 5Θ(D,5/2),(47)(D 5/2 3/ba∇dblΘ(2)/ba∇dblD 5/2 3) = 2/radicalbigg 21 5Θ(D,5/2).(48) Since the Cowan-code calculation shows that there is very little conﬁguration mixing in the199Hg+5d96s2 2D5/2level, Θ( D,5/2) can be reduced to a matrix ele- ment involving only the 5 dorbital:7 Θ(D,5/2) =e 2/an}b∇acketle{t5d2d5/2, mj= 5/2\|3z2−r2\|5d2d5/2, mj= 5/2/an}b∇acket∇i}ht, (49) =e 2/an}b∇acketle{t5d, m l= 2\|3z2−r2\|5d, m l= 2/an}b∇acket∇i}ht, (50) =e/radicalbigg 4π 5/an}b∇acketle{t5d, m l= 2\|Y20(θ,φ)\|5d, m l= 2/an}b∇acket∇i}ht, (51) =e/radicalbigg 4π 5/an}b∇acketle{t5d\|r2\|5d/an}b∇acket∇i}ht/integraldisplay2π 0/integraldisplayπ 0Y∗ 2 2(θ,φ)Y2 0(θ,φ)Y2 2(θ,φ)sinθdθdφ, (52) = 5e/an}b∇acketle{t5d\|r2\|5d/an}b∇acket∇i}ht/parenleftbigg 2 2 2 −2 0 2/parenrightbigg /parenleftbigg 2 2 2 0 0 0/parenrightbigg , (53) =−2e 7/an}b∇acketle{t5d\|r2\|5d/an}b∇acket∇i}ht. (54) The apparent sign reversal in Eq. (49) relative to Eqs. (26) a nd (29) is due to the fact that the quadrupole moment is due to a single holein the otherwise ﬁlled 5 dshell rather than to a single electron . According to the Cowan-code calculation, /an}b∇acketle{t5d\|r2\|5d/an}b∇acket∇i}ht= 2.324a2 0= 6.509×10−17cm2, (55) wherea0is the Bohr radius. Since the quadrupole shifts are zero in the 5 d106s2S1/2level, the quadrupole shift of the199Hg+optical clock transition is due entirely to the shift of the \|D 5/2 2 0/an}b∇acket∇i}htstate, and is given by /an}b∇acketle{tD 5/2 2 0\|HQ\|D 5/2 2 0/an}b∇acket∇i}ht=4 5AΘ(D,5/2)[(3 cos2β−1)−ǫsin2β(cos2α−sin2α)], (56) =−8 35Ae/an}b∇acketle{t5d\|r2\|5d/an}b∇acket∇i}ht[(3 cos2β−1)−ǫsin2β(cos2α−sin2α)], (57) ≈ −3.6×10−3hA[(3 cos2β−1)−ǫsin2β(cos2α−sin2α)] Hz, (58) whereAis expressed in units of V/cm2. Thus, for typical values A≈103V/cm2and\|ǫ\|<∼1, the quadrupole shift is on the order of 1 Hz. ACKNOWLEDGMENTS We thank Dr. C. J. Sansonetti for making available the result s of Ref. [9] prior to publication. We acknowledge ﬁnancial support from the U.S. Oﬃce of Naval Research. 6. APPENDIX. ANGULAR AVERAGING OF THE QUADRUPOLE SHIFT For the purpose of describing the quadrupole shift, the orie ntation of the laboratory (quantization) axis with respect to the principal-axis frame is deﬁned by the angles βandα. In the principal-axis coordinate system, a unit vector along the laboratory zaxis is deﬁned in terms of βandαby ˆz= (sinβcosα,sinβsinα,cosβ). (59) We wish to show that the angular dependence of the quadrupole shift is such that the diagonal matrix elements given by Eq. (45) average to zero, for ˆzalong any three mutually perpendicular directions. An arbitrary set of three mutually perpendicular unit vecto rse1,e2, and e3can be parameterized by the set of anglesθ,φ, andψin the following way: e1= (sinθcosφ,sinθsinφ,cosθ), (60) e2= (cosφcosθcosψ−sinφsinψ,sinφcosθcosψ+ cosφsinψ,−sinθcosψ), (61) e3= (−cosφcosθsinψ−sinφcosψ,−sinφcosθsinψ+ cosφcosψ,sinθsinψ). (62) It can be veriﬁed by direct computation that ei·ej=δij.8 The quadrupole shift can be evaluated for each of these three unit vectors substituted for ˆz[Eq. (59)] and the average taken. First consider the average of the quantity (3 cos2β−1) that appears in Eq. (45): We use the fact that cosβis the third component of ˆz, so the average is: /an}b∇acketle{t3 cos2β−1/an}b∇acket∇i}ht= cos2θ+ sin2θcos2ψ+ sin2θsin2ψ−1, (63) = cos2θ+ sin2θ−1, (64) = 0, (65) for arbitrary θ,φ, andψ. Similarly, the average of the other angle-dependent term i n Eq. (45), sin2β(cos2α−sin2α), is calculated by making use of the fact that sin βcosαis the ﬁrst component of ˆz, and sinβsinαis the second: /an}b∇acketle{tsin2β(cos2α−sin2α)/an}b∇acket∇i}ht=1 3[sin2θcos2φ−sin2θsin2φ +(cosφcosθcosψ−sinφsinψ)2−(sinφcosθcosψ+ cosφsinψ)2 +(cosφcosθsinψ+ sinφcosψ)2−(sinφcosθsinψ−cosφcosψ)2], (66) = 0, (67) for arbitrary θ,φ, andψ. Hence, the matrix elements of HQgiven by Eq. (45) average to zero for any three mu- tually perpendicular orientations of the laboratory quan- tization axis. REFERENCES [] Work of the U.S. N.I.S.T. Not subject to U.S. copy- right. [1] P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger, and J. B. West, Bull. Am. Phys. Soc. 21, 599 (1976). [2] W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland, Phys. Rev. Lett. 59, 2732 (1987). [3] J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, and G. Leuchs, Phys. Rev. Lett.55, 1567 (1985). [4] J. C. Bergquist, W. M. Itano, and D. Wineland, Phys. Rev. A 36, 428 (1987). [5] R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett.85, 2462 (2000). [6] J. Reichert, M. Niering, R. Holzwarth, M. Weitz, Th. Udem, and T. W. H¨ ansch, Phys. Rev. Lett. 84, 3232 (2000). [7] S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiﬀ, J. L. Hall, J. K. Ranka, R. S. Windeler, R. T. Udem, and T. W. H¨ ansch, Phys. Rev. Lett. 84, 5102 (2000). [8] R. D. Cowan, The Theory of Atomic Structure and Spectra (Univ. California Press, Berkeley, CA, 1981). [9] C. J. Sansonetti and J. Reader, Phys. Scripta (in press). [10] M. Wilson, Atomic Spectra and Oscillator Strengths for Astrophysics and Fusion Research , edited by J. E. Hansen (North-Holland, Amsterdam,1990). [11] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 80, 2089 (1998). [12] J. C. Bergquist, unpublished experimental results. [13] O. Loebich and A. Steudel, Z. Phys. 166, 56 (1961).[14] W. M. Itano, J. C. Bergquist, and D. J. Wineland, J. Opt. Soc. Am. B 2, 1392 (1985). [15] Th. A. M. Van Kleef and M. Fred, Physica 29, 389 (1963). [16] B. Cagnac, Ann. Phys. (Paris) 6, 467 (1961). [17] A. Abragam and J. H. Van Vleck, Phys. Rev. 92, 1448 (1953). [18] W. J. Childs and L. S. Goodman, Phys. Rev. 141, 176 (1966). [19] J. R. P. Angel and P. G. H. Sandars, Proc. Roy. Soc. A305, 125 (1968). [20] W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233 (1982). [21] T. Brage, C. Proﬃtt, and D. S. Leckrone, Ap. J. 513, 524 (1999). [Updated tables of os- cillator strengths were taken from the website http://aniara.gsfc.nasa.gov/sam/sam.html .] [22] M. Henderson, L. J. Curtis, R. Matulioniene, D. G. Ellis, and C. E. Theodosiu, Phys. Rev. A 56, 1872 (1997). [23] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland, J. Appl. Phys. 83, 5025 (1998). [24] J. R. P. Angel, P. G. H. Sandars, and G. K. Woodgate, J. Chem. Phys. 47, 1552 (1967). [25] P. G. H. Sandars and A. J. Stewart, Atomic Physics 3, edited by S. J. Smith and G. K. Walters (Plenum Press, New York, 1973), p. 429. [26] N. F. Ramsey, Molecular Beams (Oxford Univ. Press, London, 1956), Ch. III. [27] L. S. Brown and G. Gabrielse, Phys. Rev. A 25, 2423 (1982). [28] A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, 1974). (Earlier printings include some incorrect equations involving the rotation operators. See Ref. [29].) [29] A. A. Wolf, Am. J. Phys. 37, 531 (1969).9 ABOUT THE AUTHOR: Wayne M. Itano is a physicist in the Time and Fre- quency Division, Physics Laboratory, National Instituteof Standards and Technology. The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce.
arXiv:physics/0010057v1 [physics.flu-dyn] 24 Oct 2000Vortex tubes in velocity ﬁelds of laboratory isotropic turb ulence Hideaki Mouri1, Akihiro Hori2, Yoshihide Kawashima3 Meteorological Research Institute, Nagamine 1-1, Tsukuba 305-0052, Japan To appear in Physics Letters A Abstract The streamwise and transverse velocities are measured simu ltaneously in grid turbu- lence at Reynolds numbers 100–300. We extract typical inter mittency patterns, which are consistent with velocity proﬁles of Burgers and Lamb–Oseen vortices. The radii of these vortex tubes are estimated to be several of the Kolmogorov le ngth. PACS : 47.27.Gs; 07.05.Kf Keywords : isotropic turbulence; data analyses 1Corresponding author. E-mail: hmouri@mri-jma.go.jp 2Aﬃliated with Japan Weather Association 3Aﬃliated with Tsukuba Technology Research 11 Introduction By using direct numerical simulations [1] and bubble-visua lization experiments [2], it has been estab- lished that turbulence contains vortex tubes. Regions of in tense vorticity are organized into tubes. Their radii and lengths are, respectively, of the orders of t he Kolmogorov length ηand the integral length L. Their lifetimes are several turnover times of the largest e ddies. Vortex tubes occupy a small fraction of the volume, and are embedded in a background ﬂow w hich is random and of large scales (see also reviews [3, 4, 5]). The most familiar model for the tubes is the Burgers vortex [6 ]. This is an axisymmetric ﬂow in a strain ﬁeld. In cylindrical coordinates, they are written as uθ∝2ν ar/parenleftBigg 1−exp/parenleftBigg −ar2 4ν/parenrightBigg/parenrightBigg (a >0), (1) and (ur,uθ,uz) =/parenleftbigg −1 2ar,0,az/parenrightbigg . (2) Hereνis the kinetic viscosity. The above equation (1) describes a rigid-body rotation for small radii, and a circulation decaying in radius for large radii. The vel ocity is maximal at r=r0= 2.24(ν/a)1 2. Thus r0is regarded as the tube radius. The same circulation pattern is expected for the Lamb–Oseen vortex, i.e., an unstretched diﬀusing tube [7]. The eﬀect of vortex tubes on the velocity ﬁeld is of great inte rest. This is especially the case at high Reynolds numbers, Reλ≫100. They are achieved only in experiments, where a measurem ent is made with a probe suspended in the ﬂow, and merely a one-dim ensional cut of the velocity ﬁeld is obtained. The velocity signal at small scales is enhanced in a fraction of the volume [8]. This small-scale intermittency is attributable to vortex tubes . However, measurements often deal with the velocity component in the mean-ﬂow direction alone (her eafter, the streamwise velocity u). The transverse velocity vis more suited to detecting rotational ﬂows such as those ass ociated with vortex tubes [9, 10, 11]. This Letter reports simultaneous measurements of the strea mwise and transverse velocities in high-Reynolds-number ﬂows behind a grid. Grid turbulence i s almost isotropic and free from external inﬂuences such as mean shear, and is hence suited to studying universal properties of turbulence [8, 12]. With a conditional averaging technique, we extract typical intermittency patterns, which turn out to agree with velocity proﬁles of Burgers and Lamb–Oseen vorti ces. Then energetic importance of the vortex tubes is studied as a function of the scale. 22 Experiments The experiments were done in two wind tunnels of Meteorologi cal Research Institute. Their test sections are of 3 ×2×18 and 0.8 ×0.8×3 m in size. Turbulence was produced by placing a grid across the entrance to the test section. The grid consists of two layers of uniformly spaced rods, the axes of which are perpendicular to each other. We used three g rids. The separation of the axes of their adjacent rods are M= 10, 20, and 40 cm. The cross sections of the rods are, respect ively, 2 ×2, 4×4, and 6 ×6 cm. The mean wind was set to be U≃4, 8, or 12 m s−1. We consequently conducted 8 sets of experiments as summarized in Table 1. The streamwise ( U+u) and transverse ( v) velocities were measured simultaneously with a hot- wire anemometer. The anemometer is composed of a crossed-wi re probe and a constant temperature system. The probe was positioned on the tunnel axis at d= 2, 3, or 6 m downstream of the grid. The hot wires are 5 µm in diameter, 1.25 mm in eﬀective length, 1.25 mm in separati on, and oriented at ±45◦to the mean-ﬂow direction. Pitch-angle calibration was don e before each of the measurements. The signal was low-pass ﬁltered at fc= 4–12 kHz with 24 dB/octave, and sampled digitally at fs= 8–24 kHz with 16-bit resolution. To avoid aliasing, the samp ling frequency was set to be twice of the ﬁlter cutoﬀ frequency. The entire length of the signal was as long as 2 ×107points, since we have to minimize statistical uncertainties. The turbulence level, i.e., the ratio of the root-mean-squa re value of the streamwise ﬂuctuation ∝angbracketleftu2∝angbracketright1 2to the mean streamwise velocity U, was always less than 10% (Table 1). This good characteristi c of grid turbulence allows us to rely on the frozen-eddy hypot hesis of Taylor, ∂/∂t=−U∂/∂x , which converts temporal variations into spatial variations in th e mean-wind direction [13]. To obtain the maximum spatial resolution, we set fcin such a way that U/fcis comparable to the probe size, ∼1 mm. Since the probe is larger than the Kolmogorov length (se e below), the smallest- scale motions of the ﬂow were ﬁltered out. The present spatia l resolution is nevertheless typical of hot-wire anemometry [10, 14]. Though the RELIEF technique a chieves a greater resolution [9], it is applicable only to the transverse velocity. The ﬂow parameters such as the integral length L, the Kolmogorov length η, the Taylor microscale λ, and the Reynolds number Reλare given in Table 1. The Reλvalues range from 100 to 300. They exceed those in almost all the past studies of grid turbulenc e [4, 8, 12]. This is because our wind tunnels as well as our grids are large. The present data provi de for the ﬁrst time an opportunity to study vortex tubes in isotropic turbulence at high Reynolds numbers. 33 Conditional average To examine the presence or absence of vortex tubes, we extrac t typical intermittency patterns in the streamwise ( u) and transverse velocities ( v). The patterns are obtained by averaging the signals centered at the position where the transverse-velocity inc rement δv(s) =v(x+s)−v(x) is enhanced over its root-mean-square value by more than a factor of 3, i. e.,\|δv(s)\|>3∝angbracketleftδv(s)2∝angbracketright1 2. Here the scale sis set to be the spatial resolution U/fc. Such an event is detected 2–5 times per the integral length. When the increment is negative, we inverse the sign of the vsignal. The results for experiments 1, 3, 7, and 8 are shown in Fig. 1 (solid lines). We applied the con ditional averaging also for the other experiments and with other threshold values. Their results are consistent with those in Fig. 1. For reference, we show velocity proﬁles of Burgers vortices (dotted lines). The transverse compo- nent is from the circulation ﬂow (1), while the streamwise co mponent is from the radial inﬂow −1 2ar of the strain ﬁeld (2). Here it is assumed that the tube center passes through the probe position and the tube axis is perpendicular to the streamwise and transve rse directions. The tube radius r0was determined so as to reproduce the vpattern for each of the experiments. Grid turbulence exhibits nearly the same vpattern as that of a Burgers vortex. Tubes are surely ubiquitous at Reλ≃100–300. Their radii are estimated to be several of the Kolmo gorov length ( r0≃ 6–8η). Since these estimates are well above the svalues, they should represent typical radii of vortex tubes. Similar tube radii have been obtained from direct num erical simulations [1]. Theupattern of grid turbulence exhibits a trace of the strain ﬁel d. Thus at least some of the vortex tubes are of Burgers type. However, the observed stra in is less signiﬁcant than those of Burgers vortices. There could exist unstretched tubes such as Lamb– Oseen vortices. In numerical turbulence [1], vortex tubes are not always oriented to the stretching d irection. Once a tube is stretched to a high amplitude, it decouples from the original strain ﬁeld. This could be the case in grid turbulence [12]. We applied the conditional averaging to larger values of s(not shown here). The velocity patterns fors < L have nearly the same shapes as in Fig. 1. However, with increa sing the scale s, the corresponding tube radius r0becomes large, while the amplitude of the upattern becomes small as compared with that of the vpattern. Velocity signals at large scales reﬂect tubes whic h are inclined to the mean-wind direction. The inclination does not change th e shape of the velocity pattern, but the tube radius along the mean-wind direction rmwis greater than its true radius [11, 15]. In addition, there could exist large-radius unstretched tubes which are not coupled with the strain ﬁeld. Velocity patterns are extracted also for enhancements of th e streamwise-velocity increment δu(s) 4=u(x+s)−u(x). The results for δu(s)>+3∝angbracketleftδu(s)2∝angbracketright1 2are shown in Fig. 2, while those for δu(s)< −3∝angbracketleftδu(s)2∝angbracketright1 2are shown in Fig. 3. The vpattern, which was obtained by conditioning over the sign of the local slope, is close to a velocity proﬁle of a Burgers or L amb–Oseen vortex. Thus an enhancement ofδu(s) is caused by a vortex tube. The upattern is explained by the circulation ﬂow (1) crossing the probe at some distance [15], together with the strain ﬁel d (2). The event δu(s)<−3∝angbracketleftδu(s)2∝angbracketright1 2is detected two times more frequently than the event δu(s)>+3∝angbracketleftδu(s)2∝angbracketright1 2. This fact is consistent with theupattern in Fig. 1, which exhibits a negative slope at the tube position. We comment on controversial results in Figs. 1–3. First, the vpattern is somewhat extended as compared with the velocity proﬁle of a Burgers vortex, espec ially in experiments 7 and 8, where the Reynolds number is high. Vortex tubes in grid turbulence mig ht not be exactly the same as Burgers or Lamb–Oseen vortices. There might exist additional contr ibutions from inclined tubes. Second, the upattern is embedded in a one-sided excursion, u≃a few tenths of ∝angbracketleftu2∝angbracketright1 2>0, which extends to x≃ ±L. This might be due to an artifact of the frozen-eddy hypothes is. The streamwise velocity has a ﬂuctuation of the order of ∝angbracketleftu2∝angbracketright1 2over the scale L. Thus the speed at which a vortex tube crosses the probe is higher or lower than the mean wind speed. The observe d radius of the tube is accordingly smaller or larger than its rmwvalue. Since the scale sis at around the typical radius of vortex tubes, inclined tubes with rmw> sare more numerous than tubes with rmw< s. Fast-moving tubes with rmw> scould be captured in our averaging process for the scale sand cause the observed uexcursion. We admit that these interpretations are tentative. It is imp ortant to conduct detailed analyses of data obtained under wider ranges of conditions. 4 Flatness factor To study energetic importance of vortex tubes as a function o f the scale, we compute the ﬂatness factor of the transverse-velocity increment δv(s) =v(x+s)−v(x): ﬂatness factor =∝angbracketleftδv(s)4∝angbracketright ∝angbracketleftδv(s)2∝angbracketright2. (3) The ﬂatness factor serves as a measure of the relative import ance between vortex tubes and the background ﬂow at each of the scales. If the velocity increme nts follow a Gaussian distribution, the ﬂatness factor is equal to 3. If the tail of the probability di stribution of the velocity increments is more pronounced than that of a Gaussian, the ﬂatness factor i s higher than 3. The velocity-increment distribution observed at s≥L, where the background ﬂow is predominant, is close to a Gauss ian [3, 5, 9]. On the other hand, the vpattern for an enhancement of δv(s) ats < L is close to a velocity 5proﬁle of a Burgers or Lamb–Oseen vortex, i.e., a model for th e vortex tubes ( §3). Fig. 4 shows the ﬂatness factor for the transverse velocity vin experiments 1, 3, 7, and 8. The abscissa is the scale snormalized by the Kolmogorov length η. We indicate the integral length Land the Taylor microscale λ(arrows). At large scales, the ﬂatness factor is equal to the Gaussian value of 3. This is because the velocity increments reﬂect the backgr ound ﬂow. As the scale is decreased from the integral length, the ﬂatness factor begins to increase. This is due to a progressive contribution of vortex tubes. The tubes aﬀect velocity increments at scales around their observed sizes, which could be as large as their lengths ( ≃L). Around the Taylor microscale, the increase becomes signi ﬁcant. Fig. 4 also shows the ﬂatness factor for the streamwise veloc ityu, which is less enhanced than that for the transverse velocity v. This ﬁnding is consistent with past experimental and numer ical studies [1, 14]. From our results in Figs. 2 and 3, the streamw ise ﬂatness factor is expected to reﬂect the energetic importance of vortex tubes, as in the case of th e transverse ﬂatness factor. However, the upattern for a vortex tube is less pronounced than the vpattern. The circulation ﬂow of a vortex tube contributes to the streamwise velocity only if its axis passes at some distance from the probe. Moreover, the streamwise-velocity increment suﬀers from t he strain ﬁeld. The ﬂatness factors in Fig. 4 are displayed in order of increa sing the Reynolds number (from top to bottom). There appears to exist a trend of increasing the s mall-scale ﬂatness. This appearance is conﬁrmed in Fig. 5, which shows the ﬂatness factors at the ﬁxe d scales s= 10η(a) and λ(b). Those for the streamwise and transverse velocities increase in a s imilar manner with the Reλvalue. In both the velocity components, vortex tubes are increasingly pre dominant over the background ﬂow at a higher Reynolds number. The increase of the ﬂatness factors is more signiﬁcant at the smaller scale s = 10η, where the background ﬂow is less important. We also studied the ﬂatness factors at s≃L/2 (not shown here), which appear to depend on the grid size Mrather than the Reλvalue. 5 Conclusion To summarize, vortex tubes are important to the intermitten cy pattern of the transverse velocity at Reλ≃100–300. The pattern is well approximated as a velocity proﬁ le of a Burgers or Lamb–Oseen vortex, if its radius is several of the Kolmogorov length (Fi g. 1). The strain ﬁeld associated with the tube is present in the streamwise velocity. With increasing the Reynolds number, vortex tubes become more predominant over the background ﬂow (Figs. 4 and 5). We b elieve that these ﬁndings are new and convincing, owing to long records of both the streamwise and transverse velocities in isotropic turbulence at high Reynolds numbers. Similar experiments o f grid turbulence at much higher Reynolds 6numbers, Reλ≫300, would be crucial to proceed further [4, 5]. This work was ﬁnancially supported in part as ‘basic atomic e nergy research’ by the Science and Technology Agency, Japan. The authors are grateful to the re feree for useful comments and M. Takaoka for interesting discussion. References [1] J. Jim´ enez, A.A. Wray, P.G. Saﬀman, R.S. Rogallo, J. Flu id Mech. 255 (1993) 65. [2] S. Douady, Y. Couder, M.E. Brachet, Phys. Rev. Lett. 67 (1 991) 983. [3] U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov, Ca mbridge Univ. Press, 1995. [4] K.R. Sreenivasan, R.A. Antonia, Annu. Rev. Fluid Mech. 2 9 (1997) 435. [5] K.R. Sreenivasan, Rev. Mod. Phys. 71 (1999) S383. [6] J.M. Burgers, Adv. Appl. Mech. 1 (1948) 171. [7] H. Lamb, Hydrodynamics, 6th ed., Cambridge Univ. Press, 1932. [8] G.K. Batchelor, A.A. Townsend, Proc. R. Soc. London A 199 (1949) 238. [9] A. Noullez, G. Wallace, W. Lempert, R.B. Miles, U. Frisch , J. Fluid Mech. 339 (1997) 287. [10] R. Camussi, G. Guj, Phys. Fluids 11 (1999) 423. [11] H. Mouri, M. Takaoka, H. Kubotani, Phys. Lett. A 261 (199 9) 82. [12] A. Tsinober, E. Kit, T. Dracos, J. Fluid Mech. 242 (1992) 169. [13] G.I. Taylor, Proc. R. Soc. London A 164 (1938) 476. [14] B. Dhruva, Y. Tsuji, K.R. Sreenivasan, Phys. Rev. E 56 (1 997) R4928. [15] F. Belin, J. Maurer, P. Tabeling, H. Willaime, J. Phys. I I France 6 (1996) 573. 7Table 1. Summary of experimental conditions and turbulence characteristics: grid size M, distance from the grid d, sampling frequency fs= 2fc, mean wind speed U, root-mean-square values of the streamwise and transverse ﬂuctuations ∝angbracketleftu2∝angbracketright1 2and∝angbracketleftv2∝angbracketright1 2, kinematic viscosity ν, mean energy dissipation rate ∝angbracketleftε∝angbracketright= 15ν∝angbracketleft(∂u/∂x )2∝angbracketright, integral length L=/integraltext∝angbracketleftu(x+s)u(x)∝angbracketright/∝angbracketleftu2∝angbracketrightds, Taylor microscale λ= (∝angbracketleftu2∝angbracketright/∝angbracketleft(∂u/∂x )2∝angbracketright)1 2, Kolmogorov length η= (ν3/∝angbracketleftε∝angbracketright)1 4, and Reynolds number Reλ=∝angbracketleftu2∝angbracketright1 2λ/ν. Velocity derivatives were obtained as ∂u/∂x = (8u(x+∆)−8u(x−∆)−u(x+2∆)+u(x−2∆))/12∆, where ∆is the sampling interval U/fs. The experiments 1 and 2 were made in the 0.8 ×0.8× 3 m tunnel, while the other experiments were made in the 3 ×2×18 m tunnel. To check the isotropy, the transverse velocity was also used to compute t heL,λ, and ηvalues, via the relations L= 2/integraltext∝angbracketleftv(x+s)v(x)∝angbracketright/∝angbracketleftv2∝angbracketrightdsand∝angbracketleft(∂u/∂x )2∝angbracketright=∝angbracketleft(∂v/∂x )2∝angbracketright/2. The Lvalue derived from the vdata is smaller by a factor of 0.5–0.6 than that derived from the udata. Thus our grid turbulence is anisotropic at largest scales. On the other hand, the λandηvalues derived from the vdata agree with those derived from the udata within ≤1%. The exceptions are the results for experiments 1 and 2, where the discrepancy is 10–20%. This anisotropic cha racter is consistent with the fact that their measured ratios of ∝angbracketleftu2∝angbracketright1 2/∝angbracketleftv2∝angbracketright1 2are not very close to unity. exp. M d f s U ∝angbracketleftu2∝angbracketright1 2∝angbracketleftv2∝angbracketright1 2 ν ∝angbracketleftε∝angbracketright L λ η Re λ cm m kHz m s−1m s−1m s−1cm2s−1m2s−3cm cm cm 1 10 2 8 4.76 0.234 0.193 0.145 0.282 9.75 0.650 0.0322 105 2 10 2 16 9.60 0.475 0.406 0.145 2.01 9.97 0.494 0.0197 162 3 20 3 8 4.27 0.257 0.255 0.147 0.182 16.1 0.897 0.0364 157 4 20 3 16 8.38 0.506 0.500 0.147 1.45 17.1 0.624 0.0216 215 5 20 3 24 12.7 0.767 0.761 0.149 4.90 17.6 0.518 0.0161 267 6 40 6 8 4.39 0.220 0.213 0.147 0.0718 17.2 1.22 0.0459 182 7 40 6 16 8.70 0.446 0.427 0.147 0.595 17.2 0.858 0.0270 260 8 40 6 24 13.0 0.665 0.642 0.149 2.09 15.6 0.688 0.0199 307 8Figure captions Fig. 1 Conditional averages of the streamwise ( u) and transverse ( v) velocities for \|δv(s)\|>3∝angbracketleftδv(s)2∝angbracketright1 2 withs=U/fc. The abscissa is the spatial position xnormalized by the Kolmogorov length η. The velocities are normalized by the ∝angbracketleftv2∝angbracketright1 2value. For experiments 1, 3, 7, 8, respec- tively, the peak amplitudes correspond to 0.179, 0.198, 0.2 76, and 0.443 m s−1. The values of 3∝angbracketleftδv(s)2∝angbracketright1 2/∝angbracketleftv2∝angbracketright1 2are 0.778, 0.498, 0.527, and 0.647. The detection rates per t he integral length are 1.99, 4.00, 4.51, and 4.00. The proﬁles of Burgers vortices are shown with dotted lines. Their radii r0are 5.54η, 7.35η, 6.03η, and 8 .15η. Fig. 2 Conditional averages of the streamwise ( u) and transverse ( v) velocities for δu(s)>+3∝angbracketleftδu(s)2∝angbracketright1 2 withs=U/fc. The abscissa is the spatial position xnormalized by the Kolmogorov length η. The velocities are normalized by the ∝angbracketleftu2∝angbracketright1 2value. For experiments 1, 3, 7, 8, respec- tively, the peak amplitudes correspond to 0.256, 0.260, 0.3 54, and 0.538 m s−1. The values of 3∝angbracketleftδu(s)2∝angbracketright1 2/∝angbracketleftu2∝angbracketright1 2are 0.529, 0.346, 0.359, and 0.445. The detection rates per t he integral length are 0.464, 0.938, 1.16, and 1.16. Fig. 3 Conditional averages of the streamwise ( u) and transverse ( v) velocities for δu(s)<−3∝angbracketleftδu(s)2∝angbracketright1 2 withs=U/fc. The abscissa is the spatial position xnormalized by the Kolmogorov length η. The velocities are normalized by the ∝angbracketleftu2∝angbracketright1 2value. For experiments 1, 3, 7, 8, respectively, the peak amplitudes correspond to 0.246, 0.248, 0.338, and 0.51 8 m s−1. The detection rates per the integral length are 1.08, 2.18, 2.37, and 2.04. Fig. 4 Flatness factor of the increments of the streamwise ( u) and transverse ( v) velocities. The abscissa is the scale snormalized by the Kolmogorov length η. Arrows denote the integral length Land the Taylor microscale λ. Dotted lines denote the Gaussian value of 3. Fig. 5 Flatness factor of the velocity increments at s= 10η(a) and s=λ(b). The abscissa is the Reynolds number Reλ. Open circles denote the streamwise velocity while ﬁlled ci rcles denote the transverse velocity. We also show the experiment number s. The scale s= 10ηis, roughly speaking, the minimum scale resolved in all the experiments . 9This figure "fig1.png" is available in "png" format from: http://arxiv.org/ps/physics/0010057v1This figure "fig2.png" is available in "png" format from: http://arxiv.org/ps/physics/0010057v1This figure "fig3.png" is available in "png" format from: http://arxiv.org/ps/physics/0010057v1This figure "fig4.png" is available in "png" format from: http://arxiv.org/ps/physics/0010057v1This figure "fig5.png" is available in "png" format from: http://arxiv.org/ps/physics/0010057v1
arXiv:physics/0010058v1 [physics.plasm-ph] 24 Oct 2000Electromagnetic Energy Penetration in the Self-Induced Transparency Regime of Relativistic Laser-Plasma Interac tions M. Tushentsov,1F. Cattani,2A. Kim,1D. Anderson2and M. Lisak2 1Institute of Applied Physics, Russian Academy of Sciences, 603600 Nizhny Novgorod, Russia 2Department of Electromagnetics, Chalmers University of Te chnology, S-412 96 G¨ oteborg, Sweden Two scenarios for the penetration of relativistically inte nse laser radiation into an overdense plasma, accessible by self-induced trans parency, are presented. For supercritical densities less than 1.5 times the critica l one, penetration of laser energy occurs by soliton-like structures moving into the pl asma. At higher back- ground densities laser light penetrates over a ﬁnite length only, that increases with the incident intensity. In this regime plasma-ﬁeld str uctures represent alter- nating electron layers separated by about half a wavelength by depleted regions. Recent developments of laser technology have opened possib ilities to explore laser-matter interactions in regimes previously not achievable, [1]. Th is has meant a strong impulse to the theoretical investigation of phenomena occurring in such e xtreme conditions, when electrons quiver with relativistic velocities and new regimes may app ear. In particular, penetration of ultra-intense laser radiation into sharp boundary, overde nse plasmas is playing a fundamental role in the development of the fast ignitor fusion concept as well as of x-ray lasers, [2,3]. In this regime the optical properties of the plasma are substan tially modiﬁed by the relativistic increase of the inertial electron mass and the consequent lo wering of the natural plasma fre- quency. In the Seventies it was shown that this relativistic eﬀect en ables super-intense electromagnetic radiation to propagate through classically overdense plas mas, the so called induced transparency eﬀect, [4–7]. Recent numerical simulations based on relati vistic PIC codes [3,8,9,19], multiﬂuid plasma codes [10] and Vlasov simulations [11], as well as rec ent experiments [12,13], have re- vealed a number of new features of the interaction dynamics, such as laser hole boring, enhancedincident energy absorption, multi-MeV electron beam, as we ll as ion beam production and gen- eration of strong magnetic ﬁeld. An exact analytical study of the stationary stage of the pene tration of relativistically strong radiation into a sharp boundary, semi-inﬁnite, overdense p lasma, taking into account both the relativistic and striction nonlinearity, has recently led to the determination of an eﬀective threshold intensity for penetration [14]. It is known that, for incident intensities lower than the penetration threshold, an overdense plasma totally reﬂ ects the radiation with the forma- tion of a nonlinear skin-layer structure close to the plasma -vacuum boundary, [6]. For higher intensities the radiation was found to propagate in the form of nonlinear traveling plane waves [4,7], or solitary waves [15]. Further analysis has shown th at other scenarios are possible for incident intensities exceeding the threshold, depending o n the supercritical plasma parameter, [16]. Namely, if no>1.5 (nois the supercritical parameter deﬁned as no=ω2 p/ω2, where ωis the carrier frequency of the laser, ωp= (4πe2No/m)1/2is the plasma frequency of the unperturbed plasma), a quasi-stationary state can be reali zed and, even if still in a regime of full reﬂection, the laser energy penetrates into the overde nse plasma over a ﬁnite length which depends on the incident intensity. The subsequent plasma-ﬁ eld structure consists of alternat- ing electron layers, separated by depleted regions with an e xtension of about half a wavelength which acts as a distributed Bragg reﬂector. How do these stru ctures emerge as a consequence of relativistic laser- overdense plasma interactions? Wha t kind of scenarios are realized? These are the questions we will try to answer in this Letter. Our model is based on relativistic ﬂuid equations for the ele ctrons, in order to avoid plasma kinetic eﬀects which may shade or complicate the problem (se e, for example, [10,17]). Ions are considered as a ﬁxed neutralizing background due to the very short time scales involved, and the slowly varying envelope approximation in time is assume d to be valid. The governing set of self-consistent equations for the 1D case of interest in t he Coulomb gauge reads ∂p/bardbl ∂t=∂φ ∂x−∂γ ∂x, (1) ∂n ∂t+∂ ∂x(np/bardbl γ) = 0, (2)∂2ϕ ∂x2=no(n−1), (3) 2i∂a ∂t+∂2a ∂x2+ (1−no γna) = 0. (4) Variables are normalized as: ωt→t,ωx/c→x, the longitudinal momentum of the electrons p/bardbl/mc→p/bardbl, the scalar potential eϕ/mc2→ϕ, electron density N/N o=n,γ= (1+ p2 /bardbl+a2)1/2 is the Lorentz factor, mandeare the electron rest mass and charge, cis the speed of light in vacuum and we consider circularly polarized laser radiatio n with the amplitude of the vector potential normalized as eA/mc2= (a(x, t)/√ 2)Re[(y+iz) exp(iωt)]. Eqs. (1)-(4) have been numerically integrated for the probl em of normally incident laser ra- diation from vacuum ( x <0) onto a semi-inﬁnite overdense plasma ( x≥0), the numerical interval consisting of two parts: a short vacuum region to th e left of the plasma boundary and a semi-inﬁnite plasma region to the right. As for the boundary conditions, at inﬁnity in the plasma regi on the ﬁeld must vanish, electrons are immobile and the electron density unperturbed, conditi ons that are valid until this right boundary is reached by ﬁeld perturbations. At the vacuum-pl asma boundary the radiation boundary condition reads a−i∂a ∂x= 2ai(t), (5) where ai(t) is the incident laser wave, which means that in the vacuum re gion the total ﬁeld is the sum of the incident and reﬂected wave. At the initial time electrons are in equilibrium with ions, i.e., p/bardbl= 0, n= 1, ϕ= 0. Two diﬀerent cases have been considered for the incident laser pulse: a semi-inﬁnite envelope turning on as ai(t) =ao(tanh t+ 1) and a Gaussian envelope. Finally, the analysis has been performed for overdense plas mas (no>1) and for a quite wide range of incident intensities both higher and lower than the penetration threshold. For maximum incident intensities lower than the threshold, after a transient stage, a station- ary regime with the formation of nonlinear skin-layers is re ached, which is in perfect agreement with previous analytical solutions [6,14]. Furthermore, g ood agreement is found with the cal- culated threshold for laser penetration, [14], for intensi ties above which the nonlinear skin-layer regime is broken and the interaction leads to the penetratio n of laser energy into the overdenseplasma. Above this threshold, interactions drastically co me into play and the analysis of this dynamical process, object of a second set of numerical studi es, has revealed two qualitatively diﬀerent scenarios of laser penetration into overdense pla smas, depending on the supercritical parameter no. Ultimately, the qualitative behavior of the system occurs over a wide range of incident intensities and thus it does not sensitively depen d on the speciﬁc values. If no≤1.5 we have only a dynamical regime where laser radiation slowly pe netrates into the overdense plasma by moving soliton-like structures. In Fig. (1), the tempora l evolution of the semi-inﬁnite tanh- shaped laser radiation interacting with a plasma with no= 1.3 is depicted. Solitary waves are generated near the left boundary and then slowly propagate a s quasi-stationary plasma-ﬁeld structures with a velocity much lower than the speed of light . The contribution to the nonlin- ear dielectric permittivity due to electron density pertur bations is weaker than the one due to the relativistic nonlinearity, therefore we may consider t hese solutions as the extension of pure low-relativistic soliton solutions, [15], to a regime of sl ightly higher amplitudes. Furthermore, the excitation dynamics of such structures is similar to tha t of structures described by the nonlinear Schr¨ odinger equation with a cubic nonlinearity for a slightly overdense plasma in the low relativistic limit, (see, i.g., [18] and references the rein). The generation of similar structures can be inferred from the results of PIC simulations, such as t hose presented in [19]. Thus, if this were the case, i.e., if no−1≪1, solitary structures excited by incident intensities sli ghtly above the threshold may be considered as exact solutions. When the incident pulse has a Gaussian shape, penetration is seen to occur by a ﬁnite number of soliton-like structures. As shown in Fig. 2(a) a Ga ussian pulse with amplitude ao= 0.74 and pulse duration τ= 200, for the same plasma parameters as in Fig. 1, generates two propagating solitary structures instead of a continuou s train. The corresponding spectral analysis, see Fig. 2(b), shows that the spectrum of the trans mitted radiation is on average redshifted, while that of the reﬂected radiation presents a n unshifted and a blueshifted part which can be accounted for in terms of Doppler shift due to the moving real vacuum-plasma boundary. It should be underlined that in the limit of strongly relativ istic intensities, when localized solutions have the form of few-cycle pulses as in [15], our mo del cannot be applied since theslowly varying envelope approximation will break down, and the question of what happens at intensities largely exceeding the threshold is still open. At higher background densities, no>1.5, the dynamic regime of interaction is completely diﬀerent, as shown in Fig. 3, where a tanh-like pulse with ao= 1.3 that is an intensity of 3.6×1018W/cm2for a wavelength of 1 µminteracts with a plasma with No= 1.6Ncr(Ncr= mω2/4πe2is the critical density). The earliest stage of the spatial evolution presents the cha racteristic distribution of a nonlin- ear skin-layer, but the ponderomotive force acting at the va cuum-plasma boundary is pushing electrons into the plasma, thus shifting the real boundary t o a new position. When the ﬁeld amplitude on the real boundary exceeds the threshold calcul ated in [14], the interaction leads to the creation of a deep electron density cavity whose size i s about half a wavelength and which acts as a resonator. The whole plasma-ﬁeld structure t hen starts to slowly penetrate into the plasma and the same process is repeated at the bounda ry, where now the perturbed plasma has diﬀerent parameters. What is interesting is that, after a transient stage during w hich deep intensity cavities are produced, the plasma settles down into a quasi-stationary p lasma-ﬁeld distribution, allowing for penetration of the laser energy over a ﬁnite length only, whi ch increases with increasing incident intensities. The electron density distribution becomes st ructured as a sequence of electron layers over the ion background, separated by about half a wavelengt h wide depleted regions. The peak electron density increases from layer to layer reaching an a bsolute maximum in the closest layer to the vacuum boundary. At the same time the width of the layer s becomes more and more narrow. Such nonlinear plasma structures can act as a distri buted Bragg reﬂector and they are very close to those described analytically in [16]. If the incident laser pulse has a ﬁnite duration, the electro magnetic energy penetrates into the plasma over a ﬁxed ﬁnite length but, after the laser drive has vanished, the energy localized inside the plasma is reﬂected back towards the vacuum space, as in some sort of ”boomerang” eﬀect. The transient regime is obviously more complicated a s the depleted regions surrounded by electron layers act like resonators, with the electromag netic energy being excited by the incident pulse. Fig. 4 shows how these structures excited by a pulse 400 fslong ( λ= 1µm)bounce back. Clearly, these excited localized plasma-ﬁeld structures may live much longer than the duration time of the drive pulse. However, it should be un derlined that, on a longer time scale, the dynamics can be rather unpredictable. For instan ce, in a run with a laser drive 200 fs long, the interaction between two structures has resulted i n one long-lived cavity, whereas a second run evolved into a moving localized structure simila r to those presented in Fig. (1). It is obvious that, when dealing with long-time dynamics, ab sorption processes acting on the electromagnetic energy in the cavities should be taken into account. In conclusion, we have shown that there are two qualitativel y diﬀerent scenarios of laser energy penetration into overdense plasmas in the regime of r elativistic self-induced trans- parency, depending on the background supercritical densit y. For slightly supercritical densities No<1.5Ncr, the penetration of laser energy occurs in the form of long-l ived soliton-like struc- tures which are generated at the vacuum-plasma boundary pla sma and then propagate into the plasma with low velocity. At higher plasma densities No>1.5Ncr, the interaction results in the generation of plasma-ﬁeld structures consisting of a lternating electron and electron dis- placement regions, with the electromagnetic energy penetr ating into the overdense plasma over a ﬁnite length only, as determined by the incident intensity . The work of M.T. and A.K. was supported in part by the Russian F oundation for Basic Re- search (grants No. 98-02-17015 and No. 98-02-17013). One of the authors (F.C.) acknowledges support from the European Community under the contract ERBF MBICT972428.FIG. 1. Electron density (solid lines) and ﬁeld amplitude (d ashed lines) distributions at various moments, for no= 1.3 and semi-inﬁnite pulse with the maximum incident intensit y ao= 0.74. The ion ensity distribution is in dotted line. All the qua ntities are dimensionless. FIG. 2. Temporal distribution (a) of the ﬁeld structures gen erated in a plasma with no= 1.3 andath= 0.62 by a Gaussian incident pulse with amplitude ao= 0.74 and width τ= 200 and relative spectra (b). All the quantities are dimensionless . FIG. 3. Snapshots of the time evolution of the electron densi ty (continuous line) and the solitary structures (dashed line) generated by a semi-inﬁn ite pulse ao(tanh t+1 )with ao= 1.3, propagating into a plasma with no= 1.6 and ath= 0.99. All the quantities are dimensionless. FIG. 4. Temporal distribution of the ﬁeld structures genera ted in a plasma with no= 1.6 andath= 0.99 by a Gaussian incident pulse with amplitude ao= 1.5 and width τ= 800. All the quantities are dimensionless.[1] S.C. Wilks and W. Kruer, IEEE Trans. QE 33 , 154 (1997); see also in Superstrong Fields in Plasmas , AIP Conf. Proc. 426(1998). [2] M. Tabak et al., Phys. Plasmas 1, 1626 (1994). [3] S.C. Wilks et al., Phys. Rev. Lett. 69, 1383 (1992). [4] A.I. Akhiezer and R.V. Polovin, Sov. Phys. JETP 3, 696 (1956). [5] P. Kaw and J. Dawson, Phys. Fluids 13, 472 (1970); C. Max and F. Perkins, Phys. Rev. Lett. 27, 1342 (1971). [6] J.H. Marburger and R.F. Tooper, Phys. Rev. Lett. 35, 1001 (1975). [7] C.S. Lai, Phys. Rev. Lett. 36, 966 (1976); F.S. Felber and J.H. Marburger, Phys. Rev. Lett .36, 1176 (1976). [8] E. Lefebvre and G. Bonnaud, Phys. Rev. Lett. 74, 2002 (1995); H. Sakagami and K. Mima, Phys. Rev. E 54, 1870 (1996); S. Guerin et al., Phys. Plasmas 3, 2693 (1996); B.F. Lasinski et al., Phys. Plasmas 6, 2041 (1999). [9] A. Pukhov and J. Meyer-ter-Vehn, Phys. Rev. Lett. 79, 2686 (1997); J.C. Adam et al., Phys. Rev. Lett. 78, 4765 (1997). [10] J. Mason and M. Tabak, Phys. Rev. Lett. 80, 524 (1998). [11] H. Ruhl et al., Phys. Rev. Lett. 82, 2095 (1999). [12] J. Fuchs et al., Phys. Rev. Lett. 80, 2326 (1998). [13] M. Tatarakis et al., Phys. Rev. Lett. 81, 999 (1998). [14] F. Cattani et al., Phys. Rev. E 62, 1234 (2000). [15] V.A. Kozlov, A.G. Litvak and E.V. Suvorov, Sov. Phys. JE TP49, 75, (1979); P.K. Kaw, A. Sen and T. Katsouleas, Phys. Rev. Lett. 68, 3172 (1992).[16] A. Kim et al., JETP Lett. 72, 241 (2000). [17] X.L. Chen and R.N. Sudan, Phys. Fluids 5, 1336 (1993). [18] A.V. Kochetov, Sov. J. Plasma Phys. 12, 821 (1986). [19] Y. Sentoku et al., Phys. Rev. Lett. 83, 3434 (1999); S.V. Bulanov et al., JETP Lett. 71, 407 (2000).
arXiv:physics/0010060v1 [physics.class-ph] 25 Oct 2000Wannier-Stark states of a quantum particle in 2D lattices M. Gl¨ uck, F. Keck, A. R. Kolovsky[] and H. J. Korsch Fachbereich Physik, Universit¨ at Kaiserslautern, D-6765 3 Kaiserslautern, Germany (Dated: February 20, 2014) A simple method of calculating the Wannier-Stark resonance s in 2D lattices is suggested. Using this method we calculate the complex Wannier-Stark spectru m for a non-separable 2D potential realized in optical lattices and analyze its general struct ure. The dependence of the lifetime of Wannier-Stark states on the direction of the static ﬁeld (re lative to the crystallographic axis of the lattice) is brieﬂy discussed. I. INTRODUCTION The quantum states of a particle in a periodic po- tential plus homogeneous ﬁeld (known nowadays as the Wannier-Stark states, WS-states in what follows) are one of the long-standing problems of single-particle quantum mechanics. The beginning of the study of this problem dates back to the paper by Bloch of 1929, followed by contributions of Zener, Landau, Wannier, Zak and many others [1]. In the late eighties the problem got a new impact by the invention of semiconductor superlattices. The unambiguous observation of the WS-spectrum in a semiconductor superlattice [2] ended a long theoretical debate about the nature of WS-states, and now it is com- monly accepted that they are the resonance states of the system. Besides, WS-states were recently studied in a system of cold atoms in an optical lattice [3] and some other (quasi) one-dimensional systems. Although WS-states are resonances, i.e. metastable states, in the theoretical analysis of related problems they were usually approximated by stationary states (one-band, tight-binding, and similar approximations). Beyond the one-band approximation, WS-states in the semiconductor and optical lattices were studied in recent papers [4] and [5] by using the scattering matrix approach of Ref. [6] (see also Ref. [7] for details). This approach ac- tually solves the one-dimensional Wannier-Stark problem and supplies exhaustive information about 1D WS-states. In the present letter we extend the method of Ref. [6, 7] to the case of two-dimensional lattices. For the ﬁrst time we ﬁnd the complex spectrum of 2D WS-states and ana- lyze its general structure. To be concrete, we choose the following system: H=p2/2 +V(r) +F·r,r= (x,y),(1) V(r) = cosx+ cosy−ǫcosxcosy, (2) where 0 ≤ǫ≤1 [8]. Two limiting cases ǫ= 0 and ǫ= 1 correspond to an ‘egg crate’ potential, for which the system is separable, and a ‘quantum well’ potential, where the coupling between two degrees of freedom is maximal (see Fig. 1). Let us also note that the choice ǫ= 1 corresponds to a 2D optical potential created by two standing laser waves crossing at right angle. Thus the results presented below can be directly applied to the system of cold atoms in a 2D optical lattice. FIG. 1: Potential energy (2) for ǫ= 0 (a) and ǫ= 1 (b). II. 2D WANNIER-BLOCH SPECTRUM We brieﬂy recall the key points of the 1D theory. The spectrum of the Bloch particle in the presence of a static ﬁeld consists of several sets of equidistant levels Eα,l=Eα+ 2πFl−iΓα/2, (3) known as Wannier-Stark ladders of resonances. In Eq. (3), 2πstands for the lattice period, Fis the am- plitude of the static force, l= 0,±1,...is the site index and the index α= 0,1,...labels diﬀerent ladders. The lifetime of WS-states Ψ α,l(x) is deﬁned by the resonance width Γ αasτα= ¯h/Γα. Typically, the lifetime ταrapidly decreases with increasing index α. Because of this only the ﬁrst few WS-ladders are of physical importance. Along with the WS-states Ψ α,l(x), one can also intro- duce Wannier-Bloch states (WB-states) by ψα,k(x) =/summationdisplay lΨα,l(x)exp(i2πkl). (4) As follows from the deﬁnition (4), the continuous evo- lution of WB-states obeys the equation ψα,k(x,t) = exp(−iEαt/¯h)ψα,k−Ft/¯h(x), where Eα=Eα−iΓα/2. Thus, WB-states can be alternatively deﬁned as the eigenfunction of the evolution operator over the Bloch pe- riodTB= ¯h/F. (Note that the eigenvalues of the evolu- tion operator form degenerate bands Eα(k) =Eα). Addi- tionally, to ensure that ψα,k(x) are resonance states of the system, the eigenvalue equation for the evolution opera- tor should be accomplished by the speciﬁc non-hermitian boundary condition. It was proven in Ref. [7] that the required boundary conditions are imposed by the trunca- tion of the evolution operator matrix in the momentum representation. We proceed with the two-dimensional case. As men- tioned above, WB-states in a 1D lattice can be deﬁned2 FIG. 2: Illustration of the transformation (8). The sub- lattices and the reduced Brillouin zone are shown for the cas e q=r= 1. as the non-hermitian eigenstates of the evolution opera- tor over one Bloch period. In the 2D problem there are two diﬀerent Bloch periods associated with the two com- ponents of the static ﬁeld. Therefore the notion of the WB-states can be introduced only in the case of com- mensurate periods, i.e., in the case of ‘rational’ directio n of the ﬁeld ( q,rare coprime integers): Fx=qF (r2+q2)1/2, F y=rF (r2+q2)1/2.(5) Provided condition (5) is satisﬁed, we deﬁne 2D WB- states as the non-hermitian eigenfunctions of the sys- tem evolution operator over the common Bloch period TB= (r2+q2)1/2¯h/F. Using the Kramers-Henneberger transformation, the evolution operator can be presented in the form /hatwideU(TB) = e−iqxe−iry/hatwidestexp/parenleftBigg −i ¯h/integraldisplayTB 0dt˜H(t)/parenrightBigg ,(6) ˜H(t) =(ˆpx−Fxt)2 2+(ˆpy−Fyt)2 2+V(x,y),(7) which reveals its translational invariance (the hat over the exponent sign denotes time ordering). Alternatively, we can rotate the coordinates so that the direction of the ﬁeld coincides with the x′-axis: x′=qx+ry (r2+q2)1/2, y′=qy−rx (r2+q2)1/2. (8) Transformation (8) introduces a new lattice period a= 2π(r2+q2)1/2and reduces the size of the original Bril- louin zone s=r2+q2times (see Fig. 2). Associ- ated with the new lattice period is a new Bloch time Ta= (r2+q2)−1/2¯h/F, which isstimes shorter than the original Bloch time TB. Using ˆp′ x=−i¯h∂/∂x′and ˆp′ y=−i¯h∂/∂y′, the time evolution operator over the new Bloch time Tain the rotated coordinates has the form /hatwideU′(Ta) = e−i2πx′/a/hatwidestexp/parenleftBigg −i ¯h/integraldisplayTa 0dt˜H′(t)/parenrightBigg ,(9) FIG. 3: Position of the ground WB-band repeated by the subband energy interval 2 πF(r2+q2)−1/2as a function of the ﬁeld direction θ= arctan( r/q) (parameters ¯ h= 2,F= 0.08√ 2,ǫ= 0, integers q, r≤21). ˜H′(t) =(ˆp′ x−Ft)2 2+ˆp′ y 2+V(x′,y′). (10) Then, presenting the wave function as ψ(r′) = eik′r′/summationdisplay n′cn′/angbracketleftr′\|n′/angbracketright,/angbracketleftr′\|n′/angbracketright=1 aei2πn′·r′/a, (11) we get the matrix equation /summationdisplay m′U′(k′) n′m′cm′= e−iETa/¯hcn′, (12) whereU′(k′) n′m′denotes the k′-dependent matrix elements of the operator (9): U′(k′) n′m′=/angbracketleftn′\|e−ik′·r′/hatwideU′(Ta)eik′·r′\|m′/angbracketright. (13) Similar to the 1D case, the truncation of the inﬁnite uni- tary matrix (13), \|n′ x\|,\|m′ x\| ≤N→ ∞,\|n′ y\|,\|m′ y\| ≤M→ ∞,(14) which is presumed in the numerical calculations, auto- matically imposes the non-hermitian boundary condition along thex′-direction. (Truncation of the matrix over the indexn′ y,m′ ydoes not change the hermitian boundary condition along the y′-direction.) Then the eigenvalues Eobtained by numerical diagonalization of the truncated matrix correspond to the quantum resonances. In the transformed coordinates, the unit cell with area a2= (2π)2scontainssdiﬀerent sublattices (see Fig. 2), and each of them supports its own WB-states. The sublattices are related by primitive translations of the unrotated lattice, and correspondingly the energies of3 their WB-states diﬀer by multiples of aF/s. Further- more, as function of the quasimomentum, the energies E=E(i) β(k′ x,k′ y) (hereβ= 0,1,...is the ‘Bloch band’ index andi= 1,...,s is the sublattice index) do not de- pend onk′ x. This follows from the fact that a change of k′ xin Eq. (13) can be compensated by shifting the time origin in Eq. (9). For the y′-degree of freedom the Bloch theorem can be applied, and therefore E(i) β(k′ x,k′ y) is a periodic function of k′ ywith generally nonzero amplitude ∆Eβ. Thus, assuming a rational direction of the ﬁeld, in each fundamental energy interval aF, the static ﬁeld inducess=r2+q2identical sub-bands, separated by the energy interval aF/s. Simultaneously, the size of the Brillouin zone is reduced by a factor s. This result re- sembles the one obtained for 1D Wannier-Stark system aﬀected by a time-periodic perturbation. In the latter case – provided the condition of comensurability between the Bloch period and the period of the driving force is fulﬁlled – the quasienergy spectrum of the system has a similar structure [9]. We conclude this section with a remark concerning the numerical procedure. Although the reduced Brillouin zone approach described above is the most consistent, we found it more convenient to diagonalize the evolu- tion operator without preliminary rotation of the coordi- nate. In other words, in order to ﬁnd the WB-spectrum, we solve the eigenvalue equation (12) with the truncated matrix constructed on the basis of the operator (6). As a result of the diagonalization, one obtains eigenvalues Eβ(kx,ky) with quasimomentum k= (kx,ky) deﬁned in the original Brillouin zone. Because the WB-bands are uniform along the direction of the ﬁeld, Eβ(kx,ky) is a periodic function of both kxandkywith periods 1/rand 1/qrespectively. The energies obtained in this way can then be used to construct the complete WB- spectrum E(i) β(k′ x,k′ y),i= 1,...,s . In the next section we present results of a numerical calculation of the dis- persion relation Eβ(kx,ky) for the periodic potential (2) and moderate values of the static ﬁeld F= (Fx,Fy), \|F\|=F=const. III. NUMERICAL RESULTS It is instructive to begin with the separable case ǫ= 0. In this case, 2D WB-states are given by the product of 1D states and 2D WB-energies are just the sum of 1D energies. In what follows we restrict ourselves to analyz- ing only the ground band. First we consider the real part of the spectrum E0= Re( E0). It was shown in the previous section that for rational directions of the ﬁeld the ground WB-subbands repeat with energy splitting aF/s. As an example, Fig. 4 shows the relative positions of these subbands as a function of the angleθ= arctan(r/q) for ¯h= 2 andF= 0.08√ 2. We recall that in the considered case of a separable potential the bands have zero width for any θ/negationslash= 0,π/2. FIG. 4: Real (left) and imaginary (right) parts of the dis- persion relation Eβ(kx, ky) for the ground WB-states and dif- ferent values of the potential parameter ǫ= 0, 0 .1,0.5, and 1 (from top to bottom). The system parameters are ¯ h= 2, Fx=Fy= 0.08, and ky= 0. The main diﬀerence between separable and non- separable potentials is that the subbands E(i) 0(k) have a ﬁnite width in the latter case. This is illustrated by Fig. 4(a) which shows the dispersion relation E0(kx,ky= 0) for the potential (2) with (from top to bottom) ǫ= 0, 0.1, 0.5, and 1. The direction of the ﬁeld is θ=π/4, i.e.r=q= 1. The amplitude of the static ﬁeld and the value of the scaled Planck constant are the same as in Fig. 4. It is seen in Fig. 4(a) that the WB-bands gain a ﬁnite width as ǫis increased. We also calculated the dispersion relation E0(kx,ky= 0) for diﬀerent an- glesθ= arctan(r/q), withr,q≤6. It was found that the band widths ∆ E0= ∆E0(r,q) are typically much smaller than the mean energy separation between the subbands. Thus, for practical purpose, one can neglect the band width for the real part of the spectrum. (An exception is the caseθ= 0,π/2 where the width of the WB-bands ap- proximately coincides with the width of the Bloch band in the absence of the static ﬁeld.) Neglecting the width of the bands they were found to form a structure similar to that shown in Fig. 4. We proceed with the analysis of the decay rate of the WB-states, which is determined by the imaginary part of the complex energy, Γ 0=−2Im(E0). In the case of a separable potential the dependence Γ 0= Γ0(F,θ) is obviously given by the equation Γ0(F,θ) = Γ′ 0(Fcosθ) + Γ′ 0(Fsinθ), (15) where Γ′ 0(F′) stands for the width of 1D WS-resonances. For the parameters used (¯ h= 2 andF= 0.08√ 2) the dependence (15) is shown in Fig. 5 by a solid line. The maximum around θ=π/2 originates from a peak-like behavior of Γ′ 0(F′) and is explained by the phenomenon of 1D resonant tunneling [7]. For a non-separable potential and rational direction of the ﬁeld the decay rate depends on the quasimomen-4 tum. For the particular case θ=π/4 this dependence is depicted in Fig. 4(b). We would like to note the compli- cated behavior of Γ 0(k). The oscillating character of the decay rate is an open problem for the present day. Be- cause the decay rate depends on the quasimomentum it might be convenient to introduce the notion of ¯Γ0, where the average is taken over the reduced Brillouin zone. The dots in Fig. 1 show the values of ¯Γ0for some rational di- rection of the ﬁeld and two diﬀerent values of ǫ. It is seen that for a small ǫ= 0.1 the ratio ∆Γ 0/¯Γ0is small and the obtained dependence ¯Γ0=¯Γ0(r,q) essentially reproduces that of the separable case. However, this is not valid for ǫ= 1, where the decay rate varies wildly. Thus, in the case of strong coupling between two degrees of freedom the description of WS-state by a mean decay rate is insuﬃcient. IV. CONCLUSION We studied Wannier resonances in a 2D system, mainly discussing the complex energy spectrum of the Wannier-Bloch states. However, because the latter are related to the Wannier-Stark states by a Fourier transformation, the obtained results can be easily reformulated in terms of the Wannier-Stark resonances. Then the following is valid. (i) Neglecting the asymptotic tail, WS-states are localized functions along the direction of the ﬁeld. (This follows from the degeneracy of WB-bands along the ﬁeld direction.) (ii) For any rational direction of the ﬁeld [see Eq. (5)] WS-states are Bloch waves in the transverse direction. (iii) For a non-separable potential the corre- sponding energy bands have a ﬁnite width. (iv) For the realpart of the spectrum, the band widths are small and can be well neglected for r,q> 1. We also found a nontrivial dependence of the resonance width (inverse lifetime of WS-states) on the direction of the ﬁeld. Because the value of the resonance width de- ﬁnes the decay of the probability, a complicated behavior of the survival probability is expected when the direction of the ﬁeld is varied. The detailed study of the probabil- ity dynamics is reserved for future publication. [] Also at L. V. Kirensky Institute of Physics, 660036 Kras- noyarsk, Russia. [1] F. Bloch, Z. Phys. 52, (1929) 555; G. Zener, Proc. R. Soc. London, Ser. A 137, 523 (1934); L. D. Landau, Phys. Z. Sov.1, 46 (1932); G. H. Wannier, Phys. Rev. 117, 432 (1960); A. Rabinovitch and J. Zak, Phys. Rev. B 4, 2358 (1971). [2] E. E. Mendez, F. Agullo-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988); E. E. Mendez and G. Bastard, Phys. Today 46, 34 (1993). [3] M. Raizen, C. Solomon, Qian Niu, Physics Today, July 1997, p.30. [4] M. Gl¨ uck, A. R. Kolovsky, H. J. Korsch and F. Zimmer (unpublished) [5] M. Gl¨ uck, A. R. Kolovsky, H. J. Korsch, Phys. Rev. Lett.83, 891 (1999); Phys. Rev. A 61, 061402(R) (2000); J. Opt. B: Quantum Semiclass. Opt. 2, 612 (2000). [6] M. Gl¨ uck, A. R. Kolovsky, H. J. Korsch, Phys. Rev. Lett. 82, 1534 (1999); Phys. Rev. E 60, 247 (1999). [7] M. Gl¨ uck, A. R. Kolovsky, H. J. Korsch, J. Opt. B: Quan- tum Semiclass. Opt. 2, 694 (2000). [8] Dimensionless variables are used where the amplitude of the static ﬁeld and the scaled Planck constant are the inde- pendent parameters of the system. Alternatively, one can set ¯h= 1 and introduce the notion of the scaled amplitude V0/negationslash= 1 for the potential (2). [9] X.-G. Zhao, R. Jahnke, and Q. Niu, Phys. Lett. A 202, 297 (1995).5 FIG. 5: Decay rate of the ground WB-states as a function of the ﬁeld direction θin the case of separable potential ( ǫ= 0, solid curve). The dashed and dashed-dotted lines are an interpolation to arbitrary θof the mean decay rate calculated for some rational directions of the ﬁeld (dots) for ǫ= 0.1 and ǫ= 1, respectively. The maximum and minimum values of the decay rate for these angles are indicated by the ’error’ bars.
arXiv:physics/0010061v1 [physics.atom-ph] 25 Oct 2000A variational Monte Carlo calculation of dynamic multipole polarizabilities and van der Waals coeﬃcients of the PsH system Massimo Mella Dipartimento di Chimica Fisica ed Elettrochimica, Universita’ degli Studi di Milano, via Golgi 19, 20133 Milan o, Italy Electronic mail: Massimo.Mella@unimi.it Dario Bressaninia, and Gabriele Morosib Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Universita’ dell’Insubria, via Lucini 3, 22100 Como, Italy aElectronic mail: Dario.Bressanini@uninsubria.it bElectronic mail: Gabriele.Morosi@uninsubria.it February 2, 2008 Abstract The ﬁrst three dynamic multipole polarizabilities for the g round state of hydrogen, helium, hydride ion, and positronium hydride PsH have been computed using the variational Monte Carlo (VMC) method and explicitly correlated wave function s. Results for the static dipole polarizability by means of the diﬀusion Monte Carlo (DMC) me thod and the ﬁnite ﬁeld ap- proach show the VMC results to be quite accurate. From these d ynamic polarizabilities van der Waals dispersion coeﬃcients for the interaction of PsH with ordinary electronic systems can be computed, allowing one to predict the dispersion energy for the interaction between PsH and less exotic atoms and molecules. PACS number(s): 36.10.-k, 02.70.Lq 1While experimentalists relay everyday on positrons and pos itronium atoms (Ps) to collect informa- tion about microscopic features of macroscopic systems lik e solutions, polymers and crystals, much less eﬀort has been devoted to the theoretical understandin g of the complex interactions that take place between ordinary matter and positrons. Among the expl ored avenues of this ﬁeld, we mention the interest in predicting the stability of classes of compo unds like e+M and MPs [1-9], where M represents an atomic or molecular system, and the calculati on of the cross sections in the scattering process of e+and Ps on a molecule or an atom [10-16]. On the contrary, the evaluation of the interaction energy be tween e+M or MPs and a molecule or atom is an almost unexplored issue [17]. We believe this fa ct is primarily due to the need of a very accurate trial wave function to describe correctly th e correlated motions of electrons and positrons. So far, only variational calculations with expl icitly correlated Gaussians (ECG) [1, 5] or Hylleraas-type functions [7-9], and the DMC method [2-4] ha ve shown to be able to adequately recover the correlation energy in positron-containing sys tems. Related to the calculation of the interaction energies is th e calculation of second order properties of positron-containing systems, a problem whose surface ha s been barely scratched in the past [6]. These properties, speciﬁcally the dynamic polarizabi lities, are strictly related to the van der Waals coeﬃcients that describe the long range interaction b etween systems [18], representing a way to tackle the problem of the asymptotic intermolecular inte ractions. Recently, Caﬀarel and Hess showed that these properties can be computed by means of quan tum Monte Carlo simulations [19] connecting the imaginary-time-dependent dynamics of the u nperturbed system with the transition probabilities of a reference diﬀusion process. In this work we apply a modiﬁed version of their method to compute dynamic multipole polarizabilities for PsH, H, H e, and H−as a way to understand the behaviour of these systems when interacting with an externa l ﬁeld, and as a ﬁrst step towards the deﬁnition of the interaction potential between PsH and the o rdinary matter. As far as we know, the work by Le Sech and Silvi [6] is the only on e reporting calculations on the eﬀect of a constant electric ﬁeld on PsH. In that work they computed both the static dipole polarizability, 123 a.u., and the behaviour of the annihila tion rate Γ 2γversus the intensity of the ﬁeld employing explicitly correlated wave functions, numerica l integration, and a variation-perturbation approach. As by-product of our calculations of the potentia l energy curve of the e+LiH system [20], we obtained an estimation of the static dipole polarizabili ty of 49(2) a.u., a value quite diﬀerent from the one computed by Le Sech and Silvi. Since we believe this di ﬀerence to be too large to admit an explanation based on the diﬀerent accuracy of the methods us ed to compute this value, we plan to solve this puzzle in this work. In the method by Caﬀarel and Hess [19] the frequency-depende nt second order correction to the ground state energy is written as a sum of the two time-center ed autocorrelation functions of the perturbing potential V E(2) ±(ω) =−/integraldisplay∞ 0e±tωCV V(t)dt (1) where the autocorrelation function CV V(t) is given by CV V(t) =/angbracketleftV(0)V(t)/angbracketrightΨ2 0− /angbracketleftV(0)2/angbracketrightΨ2 0(2) Here, /angbracketleft.../angbracketrightΨ2 0indicates that the average has to be taken using the Langevin dynamics that samples the square of the exact ground state wave function of the unpe rturbed system. Caﬀarel and Hess [19] showed that it is possible to compute CV V(t) employing an optimized trial wave function and the pure-diﬀusion Monte Carlo (PDMC ) method, an alternative algorithm to the commonly used DMC with branching, where each walker ex plicitly carries its own weight along all the simulation [21]. In their work on He and H 2, Caﬀarel, R´ erat, and Pouchan [22] reported that the autoco rrelation function CV V(t) becomes dominated by the noise at large times, and this fact might be due to the ﬂuctuations of the walker weights that increase during a PDM C simulation, while the value of the autocorrelation function itself becomes smaller. While th e second eﬀect is intrinsic to the stochastic 2method, the ﬁrst can be reduced employing a more accurate tri al wave function that is able to reduce the weight ﬂuctuations. Another possibility, givin g up the exactness of the method (i.e. not sampling the exact Ψ2 0), is represented by the sampling of a quite accurate trial wa ve function without carrying around the weight for each walker, a method we call P erturbation Theory variational Monte Carlo (PT-VMC). This algorithm can be useful for those syste ms whose autocorrelation function has a large decaying time, as the case of H−and PsH. This large decaying time will increase the ﬂuctuations of the carried weights, and hence the statistic al noise in the autocorrelation functions in the long-time region. As a test of the correctness of our computer program and of the accuracy of the method, we computed the ﬁrst three autocorrelation functions, and hen ce the dynamic polarizabilities up to the octupolar one, for the two systems H and He. The analytical fo rms of the perturbing potentials were taken from Ref. [22]. While for H we employed the exact ground state wave function and compared with the analytic values of the multipole polarizability [1 8], for the He case we used a 25 term Hylleraas-type wave function optimized by means of the stan dard energy minimization [23]. We ﬁtted the numerical CV V(t) results of our simulations with a linear combination of thr ee exponential functions CV V(t)≃3/summationdisplay i=1aie−λit(3) in order to have an analytical representation of the autocor relation functions at all the times. Since it is important to reproduce accurately the long time behavi our of CV V, the smallest λiin Eq. 3 was independently calculated ﬁtting ln[CV V] in the long time region with a ﬁrst order polynomial. This choice was found to improve sensibly the goodness of the total ﬁtting in this time range. These analytical representations of CV Vallow us to compute easily the integrals in Eq. 1 and to obtain simple expressions of α(ω). The parameters obtained by the ﬁtting procedure are avail able from the authors upon requests. For both systems we found excellent agreement of the static p olarizabilities (H αdip=4.495 au, αquad=15.034 au, αoct=133.105 au; He αdip=1.382 au, αquad=2.401 au, αoct=10.367 au) with the exact results for H [18], with PDMC results by Caﬀarel, R´ erat, and Pouchan [22], with Glover and Weinhold upper and lower bounds for He [24], and with the a ccurate results by Yan et al. [25]. At this point we would like to stress that, although in the PT- VMC method the walkers carry always a unitary weight because the branching process is abs ent, similarly to the PDMC method the time step has to be chosen short enough to produce only a sm all time step bias. For these two systems we found the time step of 0.01 hartree−1to be adequate to compute statistically exact results. As a check of the ability of the PT-VMC method to compute polar izabilities also for highly polarizable systems whose exact wave function is more diﬀus e than the one of He and H, we selected the hydride ion as test case. For this system we optimized a 5 t erm Hylleraas-type wave function whose average properties are shown in Table 1 together with t he accurate results obtained in Ref. [26]. Table 1 contains also the multipole static polarizabi lities computed in this work employing a time step of 0.01 hartree−1, and the static polarizability computed by Glover and Weinh old [24]. Comparing the mean values in Table 1, one can notice that our 5 term wave function gives lower values than the ones obtained in Ref. [26] except for /angbracketleftr−/angbracketright. This fact may explain the underestimation of the αdipby PT-VMC, that recovers 92(2) percent of the accurate value . Nevertheless, this result represents a fairly good estimation of the static dipole pol arizability for H−, a quantity that appears diﬃcult to compute even with more complex approaches [27]. As far as PsH is concerned, we computed the autocorrelation f unctions using two diﬀerent trial wave functions, including one (Ψ1 T) and 28 (Ψ28 T) terms [3]. The choice of two trial wave functions to guide the Langevin dynamics was aimed at testing the depen dency of CV V(t) on the quality of the wave function itself. Employing the PT-VMC method and our wave functions for PsH, w e computed the autocorre- lation functions for three perturbation potentials: 3V1=x1+x2−xp (4) V2=3(x2 1+x2 2−x2 p)−(r2 1+r2 2−r2 p) 2(5) V3=x3 1+x3 2−x3 p−3[x1(y2 1+z2 1) +x2(y2 2+z2 2)−xp(y2 p+z2 p)] 2(6) where the subscripts 1 and 2 indicate the two electrons, whil e the subscript pindicates the positron. These potentials are the cartesian forms of the dipole, quad rupole, and octupole moment operators for the PsH system. Figure 1 shows the averaged correlation f unctions for V1,V2, andV3as obtained by the VMC method employing the 28 term trial wave function. E ach value of the correlation functions was computed employing roughly 1010conﬁgurations. From Figure 1 one can note the eﬀect at large evolution times of the dispersion of the “trajector ies” used to compute the autocorrelation function. This eﬀect makes diﬃcult the reproduction of the l ong-time regime of these functions due to the exponential decay and the roughly constant statistical error introduced by the method. Moreover, the statistical error strongly depends on the perturbation potential whose autocorrelation function is computed, i.e., more speciﬁcally on the dispersion of its mean value over the Ψ2 Tdistribution. The results for the static multipole polarizabilities, i.e . for ω= 0, computed with both trial wave functions, are shown in Table 2. While for the dipole pol arizabilities there is a good agreement between the two values, larger diﬀerences are present for th e higher multipole polarizabilities. This fact is an indication of the diﬀerent accuracy of the two func tions in approximating the exact wave function at large distances from the nucleus. In fact it can b e shown that if one approximates the autocorrelation functions taking care only of the excit ation to the ﬁrst state of the appropriate symmetry, the autocorrelation function is proportional to /angbracketleftV2 i/angbracketright − /angbracketleftVi/angbracketright2, where Viis the perturbing potential. Comparing the dipole results with the value obta ined by Le Sech and Silvi [6], again the large diﬀerence of the computed polarizabilities is app arent. As a ﬁnal check for this problem, we computed the energy of the PsH when immersed in a weak stati c electric ﬁeld Fby means of standard DMC simulations adding the linear potential F(x1+x2−xp). To make our simulations stable, i.e. to avoid the dissociation of the PsH, we truncat ed the eﬀect of the linear potential at \|xi\|= 15 bohr. We ﬁtted the DMC results by means of the simple polyn omial a−αdipF2/2, where αdipis the static dipole polarizability, obtaining αdip= 42.3(8) a.u. We believe that this result, statistically indistinguishable from the αdipobtained by the PT-VMC method, gives the deﬁnitive answer to the problem of the PsH polarizability. Neverthele ss, the discrepancy between our PT- VMC and DMC αdipand the one computed by Le Sech and Silvi [6] remains puzzling . In our experience [3], to compute the matrix elements they needed, millions of conﬁgurations must be used even for systems like PsH to avoid to be fooled by a false c onvergence. Unfortunately, Le Sech and Silvi did not report any information about the number of c onﬁgurations they used to compute the integrals, so we cannot judge the numerical accuracy of t heir results. An attempt to estimate the total accuracy of our αresults can be made comparing the polariz- ability values obtained by the two wave functions. These diﬀ er by 10 percent at most, a value that we feel might give a conservative estimate of the relative er rors for the higher multipolar ﬁelds. As stated previously, although dynamical polarizabilitie s are interesting on their own, they repre- sent the basis to compute van der Waals dispersion coeﬃcient s for the interaction between diﬀerent systems. Therefore, following Ref. [25], we present the cal culation of the C6,C8, andC10dispersion coeﬃcients between H, He, and PsH as a ﬁrst eﬀort to obtain acc urate information on the interac- tion between positronic systems and ordinary matter in the f ramework of the Born-Oppenheimer approximation and second-order perturbation theory. Using the ﬁtted parameters for H, He, and PsH we computed the c oeﬃcients for the interaction between the ordinary systems and between these and PsH. The v alues are reported in Table 3. Since the values for the H-H, H-He, and He-He coeﬃcients are accura tely known [25], we use them as a test of the accuracy of our approach: all the values diﬀer fro m the accurate results by Yan et al. [25] at most by one part over hundreds. 4Comparing the Cn’s for the ordinary systems with the ones for the interaction with PsH, it strikes that these last are more than an order of magnitude larger tha n the formers. These features, due to the larger PsH polarizability, indicates that positroni c systems strongly interact with ordinary matter even at large distances. Unfortunately, nothing can be said about location and depth of the total potential minimum. This strongly depends also on t he eﬀect of the repulsion between the positron cloud and the H and He nuclei, so that we believe a sup ermolecule approach is needed. In a previous work [17] we computed the interaction energy be tween H and PsH, showing that this system could have a metastable state. Although the disp ersion coeﬃcients for the interaction between He and PsH are smaller than those for PsH and H, they mi ght be large enough to give rise to a potential well that could support at least a stable state . If this turns out to be the case, the He-PsH system could be the lightest van der Waals (i.e. bound by means of dispersion forces) stable dimer. ACKNOWLEDGMENTS Financial support by the Universita’ degli Studi di Milano i s gratefully acknowledged. The authors are indebted to the Centro CNR per lo Studio delle Relazioni t ra Struttura e Reattivita’ Chimica for grants of computer time. 5Figure captions: Figure 1. Logarithm of the correlation functions of the pert urbing potentials. 6References [1] G. G. Ryzhikh, J. Mitroy, and K. Varga, J. Phys. B: At. Mol. Opt. Phys. 31, 3965 (1998). [2] T. Yoshida, G. Miyako, N. Jiang, and D. M. Schrader, Phys. Rev. A 54, 964 (1996). [3] M. Mella, G. Morosi, and D. Bressanini J. Chem. Phys. 111, 108 (1999). [4] N. Jiang and D. M. Schrader, J. Chem. Phys. 109, 9430 (1998), Phys. Rev. Lett. 81, 5113 (1998). [5] K. Strasburger, J. Chem. Phys. 111, 10555 (1999). [6] C. Le Sech and B. Silvi, Chem. Phys. 236, 77 (1998). [7] D. C. Clary, J. Phys. B At. Mol. Opt. Phys. 9, 3115 (1976). [8] Y. K. Ho, Phys. Rev A 34, 609 (1986). [9] A. M. Frolov, and V. H. Smith, Jr. Phys. Rev. A 56, 2417 (1997). [10] M. T. McAlinden, F. G. R. S. MacDonald, and H. R. J. Walter s, Can. J. Phys. 74, 434 (1996). [11] S. Sur, S. K. Adhikari, and A. S. Ghosh, Phys. Rev. A 53, 3340 (1996). [12] M. Comi, G. M. Prosperi, and A. Zecca, Nuovo Cimento 2D, 1347 (1983). [13] P. K. Biswas and A. S. Ghosh, Phys. Lett. A 223, 173 (1996). [14] P. K. Biswas and S. K. Adhikari, J. Phys. B: At. Mol. Opt. P hys.31, L315 (1998). [15] P. Chaudhuri, S. K. Adhikari,B. Talukdar, S. Bhattacha ryya, Eur. Phys. Journal 5, 217 (1999). [16] B. Nath, C. Sinha, Eur. Phys. Journal 6, 295 (1999). [17] M. Mella, G. Morosi, and D. Bressanini, J. Chem. Phys. 112, 1063 (2000). [18] G. Lamm and A. Szabo, J. Chem. Phys. 72, 3354 (1980). [19] M. Caﬀarel and O. Hess, Phys. Rev. A 43, 2139 (1991). [20] M. Mella, G. Morosi, D. Bressanini, and S. Elli, J. Chem. Phys.113, 6154 (2000). [21] B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry , 1st ed., (World Scientiﬁc, Singapore, 1994). [22] M. Caﬀarel, M. R´ erat, and C. Pouchan, Phys. Rev. A 47, 3704 (1993). [23] G. Migliavacca, unpublished results (Thesis, Univers ity of Milan 1998). [24] R. M. Glover and F. Weinhold, J. Chem. Phys. 65, 4913 (1976). [25] Z. Yan, J. F. Babb, A. Dalgarno, and G. W. F. Drake, Phys. R ev. A54, 2824 (1996). [26] T. Koga, Y. Kasai, and A. J. Thakkar, Int. J. Quantum. Che m.46, 689 (1993). [27] D. Spelsberg, T. Lorenz, and W. Meyer, J. Chem. Phys. 99, 7845 (1993). 7VMCaHylleraasb /angbracketleftE/angbracketright -0.52701(2) -0.52775b /angbracketleftV/angbracketright -1.0448(2) -1.0555b /angbracketleftr−/angbracketright 2.7262 2.7102b /angbracketleftr2 −/angbracketright 11.844 11.915b /angbracketleftr−−/angbracketright 4.4119 4.4127b /angbracketleftr2 −−/angbracketright 24.957 25.20b αdip 189.30 206(3)c αquad 5761.5 αoct 450758 Table 1: Mean values for observables of the ground state1Sof H−. All values are in atomic units. aThis work (5 term wave function) bRef. [26] cRef. [24] 8ΨT αdip αquad αoct Ψ1 T43.66(3) 972.7(2) 39178(32) Ψ28 T42.99(4) 876.9(3) 34848(71) Table 2: Static multipole polarizabilities for the ground s tate2,1Sof the PsH computed with one term (Ψ1 T) and 28 term (Ψ28 T) wave functions. All values are in atomic units. 9C6 C8 C10 H-H 6.480 125.23 3318.2 6.499a124.39a3285.8a H-He 2.813 41.671 866.33 2.821a41.836a871.54a He-He 1.454 13.880 177.01 1.461a14.117a183.69a H-PsH 40.30 2596.1 86292 He-PsH 15.718 950.80 23490 Table 3: Computed dispersion coeﬃcients. All values are in a tomic units. aRef. [25] 10-4-20246810 0 510 15 20 25 30ln[C(t)] t (a.u.)Dipole Quadrupole Octupole
arXiv:physics/0010062v1 [physics.atom-ph] 25 Oct 2000Stability and production of positron-diatomic molecule complexes Massimo Mella Dipartimento di Chimica Fisica ed Elettrochimica, Universita’ degli Studi di Milano, via Golgi 19, 20133 Milan o, Italy Electronic mail: Massimo.Mella@unimi.it Dario Bressaniniaand Gabriele Morosib Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Universita’ dell’Insubria, via Lucini 3, 22100 Como, Italy aElectronic mail: Dario.Bressanini@uninsubria.it bElectronic mail: Gabriele.Morosi@uninsubria.it January 25, 2014 Abstract The energies at geometries close to the equilibrium for the e+BeO and e+LiF ground states were computed by means of diﬀusion Monte Carlo simulations. These results allow us to predict the equilibrium geometries and the vibrational frequencie s for these exotic systems, and to discuss their stability with respect to the various dissoci ation channels. Since the adiabatic positron aﬃnities were found to be smaller than the dissocia tion energies for both complexes, we propose these two molecules as possible candidates in the challenge to produce and detect stable positron-molecule systems. PACS number(s): 36.10.-k, 02.70.Lq 1Despite the wide diﬀusion of positron and positronium (Ps) b ased analytical techniques to study solids [1], polymers [2], solutions [3], and organic molecu les in the gas phase [4, 5], a direct ob- servation of the compounds between the positron and an atom o r a molecule is still lacking. In fact Γ2γannihilation rate from positron annihilation life-time sp ectroscopy and angular correlation anni- hilation radiation are the only standard measurements carr ied out during the interaction positron- matter. The prediction of these observables is required to i nfer the formation of the positronic compounds, a task that appears complex, especially for heav y atoms and ions or large molecules, due to the high accuracy that is needed for the wave function t hat describes the complexes. The theoretical work on positron containing systems is scar ce, and in our opinion this is due to the diﬃculty in describing accurately the electron-posi tron correlation using standard quantum chemistry methods like Self Consistent Field (SCF), Conﬁgu ration Interaction, and Coupled Cluster methods [6]. Two more approaches have been pursued during the last years, namely Density Functional The- ory (DFT) [7] and variational calculations based on Explici tly Correlated Gaussian (ECG) trial wave functions [8, 9]: they also suﬀer from practical drawbacks. Although DFT methods have a conve- nient scaling of the computational cost versus the system co mplexity, the exact exchange-correlation potential between electrons and the correlation potential between electrons and positron are only approximately known. As far as ECG wave functions are concer ned, two groups [8, 9] showed that accurate results can be obtained even for positron containi ng systems. Unfortunately, the ECG wave functions suﬀer from the fast increase of the computati onal cost with the number of particles, therefore preventing their use for medium and large systems . Nevertheless, accurate results can be obtained employing the frozen-core approximation for atom s and molecules [8]. In our ongoing project to study positronic compounds as a way to understand matter-antimatter interactions and to predict the existence of a bound state fo r positron-atom or positron-molecule complexes [10-15], we employ the ﬁxed node diﬀusion Monte Ca rlo (FN-DMC) method [16]. This technique is known to be able to recover most of the correlati on energy between electrons and between electrons and a positron [10-15,17-19]. Although F N-DMC is a powerful technique, it is not easy to reduce the nodal error introduced by the ﬁxed node approximation. This result might be achieved in principle by employing more accurate trial wa ve functions or resorting to the nodal release technique, but both approaches do not easily apply t o large systems, i.e. more than ten electrons, due to their computational cost. Nevertheless, the FN-DMC method has given accurate positron aﬃnities, as well as electron aﬃnities [20], for sy stems up to twelve electrons, both atoms and molecules, exploiting the cancellation of nodal errors [12]. In the quest for stable positronic complexes, we studied the potential surface for e+LiH by FN- DMC calculations [21] and found that the equilibrium distan ce and the vibrational transitions are diﬀerent from those of LiH, opening the possibility for a spe ctroscopic detection of this compound. However, the LiH adiabatic positron aﬃnity (APA) is larger t han the dissociation energy (DE), and a third body would be required to dissipate the excess energy . We suggested to start from a van der Waals complex of LiH with a rare gas, and to attach the positro n to this so that the rare gas should dissipate the excess energy. Similar consideration can be e xtracted from the work by Mitroy and Ryzhikh [22], where they employed a full non-adiabatic appr oach and ECG functions to establish the stability of e+LiH. To avoid this complex mechanism, in this Letter we investiga te other systems to see if we can ﬁnd a molecule whose APA is smaller than the DE, allowing the p ositron to be attached and to form the complex without the intervention of a third body. If the spectroscopic properties of this compound diﬀer from those of the parent molecule, it could be a good candidate for experimental observation. We have performed accurate calculations of the total energy for e+BeO and e+LiF systems at various internuclear distances by means of FN-DMC. These re sults allow us to obtain the equilibrium distances for both molecules and to compute the vibrational frequencies. In the FN-DMC algorithm we sample a distribution of conﬁgura tions in 3N dimensional space that represents Ψ 0ΨT, where Ψ 0is the ground state wave function having the same nodal surfa ces 2of the trial wave function Ψ T. Using this distribution we obtain a MC estimate of the ﬁxed n ode energy E0using the mixed estimator E0=1 NN/summationdisplay i=1Eeloc(Ri) =1 NN/summationdisplay i=1HΨT(Ri) ΨT(Ri)(1) In our calculations the trial wave function Ψ Tis ΨT=Det \|φα\|Det \|φβ\|eU(rµν)Ω (rp, rpν) (2) φα,βare orbitals and eU(rµν)is the electronic correlation factor used by Schmidt and Mos kowitz in their works on atoms and ions [23, 24]. In Eq. 2 Ω (rp, rpν) =Nterms/summationdisplay i=1ciΦi(rp, rpν) (3) where [25, 26] Φi(rp, rpν) =fi(rp)exp/bracketleftBigg ki,1N/summationdisplay ν=1rpν−Nnuc/summationdisplay n=1ki,n+1rp,n/bracketrightBigg (4) In this equation fi(ra) is a function that contains explicitly the dependence on th e spatial coordinates of the positron and kiis a vector of parameters for the i–th term of the linear expan sion. While the φα,βorbitals were obtained by means of standard SCF calculation s on the parent neutral molecule, the other parameters of Ψ Twere optimized minimizing the variance of the local energy using a ﬁxed sample of conﬁgurations. Although this m ethod produces wave functions whose properties are generally less accurate than those obtained by minimizing the energy [27], it is much faster. Moreover, the FN-DMC energy value depends only on th e location of the nodal surfaces of the electronic part of the wave function, so that it is not e xtremely important to have the best possible description of its positronic part. Nevertheless , if one is interested in properties diﬀerent from the energy, whose accuracy is strongly dependent on the quality of the trial wave function (for example the δ(r+−)), a re-optimization of all the wave function parameters is needed [14]. All the FN-DMC simulations were carried out using a target po pulation of 5000 conﬁgurations and a time step of 0.001 hartree−1. Few more simulations employing a time step of 0.0005 hartre e−1 were run to check for the absence of the time step bias in the me an energy values. The FN-DMC energy results for various internuclear distances of e+LiF and e+BeO are shown in Table 1. We ﬁtted these energy values by means of a second order polyno mial and computed equilibrium geometrical parameters and the fundamental vibrational wa venumber ωefor the two complexes e+7Li19F and e+9Be16O. All the results are collected in Table 2. Comparing the results in Table 2 with the experimental value s [28] for7Li19F (Re= 2.955 bohr, ωe= 910.34 cm−1) and9Be16O (Re= 2.515 bohr, ωe= 1487.32 cm−1), we note that after the addition of the positron both molecules have larger equilib rium distances and vibrational frequencies. While the increase of Reis similar to the one we found for e+LiH [21] and can be rationalized invoking the repulsive interaction of the positron with the nuclei, the increase of stiﬀness of the two bonds is an unexpected result. However, it must be pointed ou t that the computed frequencies have an estimated statistical accuracy of the order of 10%, and th is means that care must be taken in discussing the change of this property. In a previous work [12] we computed the total energies for LiF (-107.4069(9) hartree) and BeO (-89.7854(13) hartree) at their equilibrium distances by m eans of FN-DMC. Together with the Emin values shown in Table 2, these energies allow us to compute th e adiabatic positron aﬃnity (APA) for these two systems, namely 0.022(1) hartree for e+LiF, and 0.025(2) hartree for e+BeO. These two values are smaller than the APA for the e+LiH (0.0366(1) hartree). This result was already 3observed for the vertical PA [12], and is in contrast with the fact that the dipole moment of LiF (µ=6.33 Debye) and BeO ( µ=6.26 Debye) are larger than the one of LiH ( µ=5.88 Debye) [29]. This indicates that the dipole moment is not suﬃcient to pred ict a qualitative trend in the PA, and that this value strongly depends on the speciﬁc features of e ach molecule. As far as the dissociation of these complexes is concerned, c are must be taken in choosing balanced values for the energies of the fragments for the possible dis sociation channels. For a positron-diatomic molecule complex e+MX, where M=Li or Be and X=O or F, the possible fragmentations are e+M + X, M++ PsX, M + e+X, and PsM + X+. Although not all the energy values of the fragments are known, one can safely assume that the PsM + X+dissociation pattern has the highest energy with respect to the other possibilities. This is due ﬁrstly to the large ionization potential of X (0.5005 hartree for O, and 0.6403 hartree for F) [30], at least twice a s large as the positronium (Ps) ground state energy (-0.25 hartree); secondly, to the usually smal l binding energy of Ps to metal atoms (for instance, the binding energy of Ps to Li in the PsLi complex is just 0.01158 hartree [8]). Moreover, we believe it is reasonable to discard also the M + e+X channel, since the possibility of obtaining binding between e+and X is hindered by the small polarizability of X. To support this conclusion, we stress that even for HF and H 2O, that are both polar molecules, DMC did not show binding wit h the positron [12]. Although this is not a proof, it strongly s uggests that e+O and e+F probably are not bound. Accepting these conclusions, we are left only with e+M + X and M++ PsX as possible fragmentations. To compute the total energy for bo th channels we use the ECG results for e+Li (-7.532323 hartree), e+Be (-14.669042 hartree), Li+(-7.279913 hartree), and Be (-14.667355 hartree) [8], and the FN-DMC results for O (-75.0518(4) hart ree), F (-99.7176(3) hartree) [31], PsO (-75.3177(5) hartree), and PsF (-100.0719(8) hartree) [11]. Moreover, we estimate the Be+ energy (-14.3248 hartree) subtracting the ionization pote ntial (0.3426 hartree) [30] to the total energy of Be. Using these results, we end up with an energy of - 107.2499(3) hartree for e+Li and F, and an energy of -107.3518(8) hartree for Li+and PsF. This fragmentation, similar to the one found for e+LiH (i.e. Li+and PsH), is primarily driven by the small value of the Li ioni zation potential. Diﬀerently, for e+BeO we obtain -89.6642(5) hartree for Be+and PsO, and -89.7208(4) hartree for e+Be and O, so that the most stable dissociation fragments pres ent a positron-atom bound state. Using the lowest energy dissociation threshol d for the two systems one gets a DE of 0.080(1) hartree for e+LiF, and 0.090(2) hartree for e+BeO. Both these values are larger than the APA, and this fact means that the two positron-molecule c omplexes do not dissociate after positron addition to the parent molecules. This outcome is d iﬀerent from what we found for the addition of e+to LiH, where the e+LiH complex breaks up due to the excess of the APA with respect to the DE [21]. Therefore, it does not appear necessa ry for LiF and BeO to use a third body, and a simple positron addition will give birth to stabl e complexes in rotovibrational excited states. As previously stated, the possibility to produce th ese stable species could give the chance to experimentally detect stable positron complexes. Roughly speaking, a mean lifetime on the order of 10−9seconds is expected for these systems, and this may be large e nough to allow a spectroscopical analysis in the reaction chamber by means of Fourier Transfo rm Infrared Spectroscopy if a suﬃcient concentration of e+MX can be produced, and if the frequency shift after positron attachment is large enough that the vibrational spectrum of the complex does not overlap with the neutral molecule one. Unfortunately the large uncertainty in ωedoes not allow a quantitative prediction of this frequency shift. Moreover, positrons having kinetic energy larger th an the diﬀerence DE-APA can open the various fragmentation channels depending on the excess of t heir relative energies. For instance, the collision between positron and BeO can produce e+Be and O as fragments, so that the annihilation of e+with the electronic cloud of Be can be directly recorded from the 2 γphotons. Moreover, it might be possible to detect the stable state of PsF, a system t hat, diﬀerently from PsCl and PsBr, has not been prepared in solution [3]. In conclusion, we have presented accurate APA and DE for e+LiF and e+BeO systems computed by means of FN-DMC. These results allow us to discuss possibl e mechanisms of formation for positron-molecule complexes by direct attachment of e+to the molecules, and the possibility to produce e+M and PsX systems. It should be now interesting to compute the Γ2γannihilation rate 4for these complexes, in order to predict their mean lifetime after e+addition. Unfortunately, more technical work on the method appears to be necessary before t hese calculations can be carried out for these large systems. ACKNOWLEDGMENTS Financial support by the Universita’ degli Studi di Milano i s gratefully acknowledged. The authors are indebted to the Centro CNR per lo Studio delle Relazioni t ra Struttura e Reattivita’ Chimica for grants of computer time. 5References [1] M. H. Weber, S. Tang, S. Berko, B. L. Brown, K. F. Canter, K. G. Lynn, A. P. Mills, Jr., L. O. Roellig, and A. J. Viescas, Phys. Rev. Lett. 61, 2542 (1988). [2] G. Consolati, R. Rurali, and M. Stefanetti, Chem. Phys. 273, 493 (1998). [3] P. Castellaz, J. Major, C.Mujica, H. Schneider, A. Seege r, A. Siegle, H. Stroll, and I. Billard, J. Radioanal. Nucl. Chem. 210,457 (1996). [4] K. Iwata, G. F. Gribakin, R. G. Greaves, C. Kurz, and C. M. S urko. Phys. Rev. A 61, 022719 (2000). [5] D. M. Schrader, F. M. Jacobsen, N. Frandsen, and U. Mikkel sen, Phys. Rev. Lett. 69, 57 (1992). [6] K. Strasburger, Chem. Phys. Lett. 253, 49 (1996). [7] R. M. Nieminen, E. Boronski, and L. Lantto, Phys. Rev. B 32, 1377 (1985). [8] G. G. Ryzhikh, J. Mitroy, and K. Varga, J. Phys. B: At. Mol. Opt. Phys. 31, 3965 (1998). [9] K. Strasburger, J. Chem. Phys. 111, 10555 (1999). [10] D. Bressanini, M. Mella, and G. Morosi, Phys. Rev. A 57, 1678 (1998). [11] D. Bressanini, M. Mella, and G. Morosi, J. Chem. Phys. 108, 4756 (1998). [12] D. Bressanini, M. Mella, and G. Morosi, J. Chem. Phys. 109, 1716 (1998). [13] D. Bressanini, M. Mella, and G. Morosi, J. Chem. Phys. 109, 5931 (1998). [14] M. Mella, G. Morosi, and D. Bressanini J. Chem. Phys. 111, 108 (1999). [15] M. Mella, G. Morosi, and D. Bressanini, J. Chem. Phys. 112, 1063 (2000). [16] B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry , 1st ed., (World Scientiﬁc, Singapore, 1994). [17] T. Yoshida, G. Miyako, N. Jiang, and D. M. Schrader, Phys . Rev. A 54, 964 (1996). [18] T. Yoshida and G. Miyako, Phys. Rev. A 54, 4571 (1996). [19] N. Jiang and D. M. Schrader, J. Chem. Phys. 109, 9430 (1998), Phys. Rev. Lett. 81, 5113 (1998). [20] G. Morosi, M. Mella, and D. Bressanini, J, Chem. Phys. 111, 6755 (1999). [21] M. Mella, G. Morosi, D. Bressanini, and S. Elli, to be pub lished on J. Chem. Phys. [22] J. Mitroy and G. G. Ryzhikh, submitted to J. Phys. B. [23] K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. 93, 4172 (1990). [24] J. W. Moskowitz and K. E. Schmidt, J. Chem. Phys. 97, 3382 (1992). [25] D. Bressanini, M. Mella, and G. Morosi, Chem. Phys. Lett .272, 370 (1997). [26] D. Bressanini, M. Mella, and G. Morosi, Phys. Rev. A 55, 200 (1997). [27] M. Snajdr and S. M. Rothstein, J. Chem. Phys. 112, 4935 (2000). [28] K. P. Huber and G. Herzberg, Molecular spectra and Molecular Structure. Constants of di atomic Molecules (Van Nostrand Reinhold, New York, 1979). 6[29] G. L. Gutsev, M. Nooijen, and R. J. Bartlett, Chem. Phys. Lett.276, 13 (1997). [30] C. E. Moore, Ionisation Potentials and Ionization Limits Derived from t he Analysis of Optical Spectra , National Bureau of Standard, Report NSRDS-NBS34 (1970). [31] A. L¨ uchow and J. B. Anderson, J. Chem. Phys. 105, 7573 (1996). 7R /angbracketleftE/angbracketright e+LiF 2.955 -107.4243(8) 3.200 -107.4291(8) 3.400 -107.4249(10) 3.500 -107.4176(8) e+BeO 2.30 -89.7975(13) 2.40 -89.8089(15) 2.51 -89.8108(18) 2.75 -89.7998(14) Table 1: Total energy at various internuclear distances. Al l values are in atomic units. 8e+7Li19F e+9Be16O Emin(hartree) -107.429(1) -89.8108(16) Re(bohr) 3.18 2.53 ωe(cm−1) 1073 1537 R0(bohr) 3.20 2.55 Table 2: Equilibrium properties for e+7Li19F and e+9Be16O 9
arXiv:physics/0010063v1 [physics.data-an] 25 Oct 2000Optimal Recovery of Local Truth Carlos C. Rodr´ ıguez Department of Mathematics and Statistics University at Albany, SUNY Albany NY 12222, USA carlos@math.albany.edu http://omega.albany.edu:8008/CONTENTS 1 Contents 1 Introduction 1 1.1 Nonparametrics with the World in Mind . . . . . . . . . . . . 2 2 Estimating Densities from Data 3 2.1 The knn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Double Smoothing Estimators . . . . . . . . . . . . . . . . . . 5 3 The Truth as n→ ∞? 6 3.1 The Natural Invariant Loss Function and Why the MSE is not that Bad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Some Classic Asymptotic Results 9 4.1 Asymptotic Mean Square Errors . . . . . . . . . . . . . . . . . 12 5 Choosing the Optimal Norm 14 5.1 Generalized knn Case with Uniform Kernel . . . . . . . . . . . 17 5.2 Yet Another Proof When The Hessian is Deﬁnite . . . . . . . 22 5.3 Best Norm With General Kernels . . . . . . . . . . . . . . . . 24 6 Asymptotic Relative Eﬃciencies 26 7 An Example: Radially Symmetric Distributions 27 8 Conclusions 30 9 Acknowledgments 31Abstract Probability mass curves the data space with horizons!. Let fbe a mul- tivariate probability density function with continuous se cond order partial derivatives. Consider the problem of estimating the true va lue off(z)>0 at a single point z, fromnindependent observations. It is shown that, the fastest possible estimators (like the k-nearest neighb or and kernel) have minimum asymptotic mean square errors when the space of obse rvations is thought as conformally curved. The optimal metric is shown t o be generated by the Hessian of fin the regions where the Hessian is deﬁnite. Thus, the peaks and valleys of fare surrounded by singular horizons when the Hessian changes signature from Riemannian to pseudo-Riemannian. A daptive esti- mators based on the optimal variable metric show considerab le theoretical and practical improvements over traditional methods. The f ormulas simplify dramatically when the dimension of the data space is 4. The si milarities with General Relativity are striking but possibly illusory at this point. How- ever, these results suggest that nonparametric density est imation may have something new to say about current physical theory.1 INTRODUCTION 1 1 Introduction During the past thirty years the theory of Nonparametrics ha s been domi- nating the scene in mathematical statistics. Parallel to th e accelerating dis- covery of new technical results, a consensus has been growin g among some researchers in the area, that we may be witnessing a promisin g solid road towards the elusive Universal Learning Machine (see e.g. [1 , 2]). The queen of nonparametrics is density estimation. All the f undamen- tal ideas for solving the new problems of statistical estima tion in functional spaces (smoothing, generalization, optimal minimax rates , etc.) already ap- pear in the problem of estimating the probability density (i .e. the model) from the observed data. More over, it is now well known that a s olution for the density estimation problem automatically implies s olutions for the problems of pattern recognition and nonparametric regress ion as well as for most problems that can be expressed as a functional of the den sity. In this paper I present a technical result, about optimal non parametric density estimation, that shows at least at a formal level, a s urprising simi- larity between nonparametrics and General Relativity. Sim ply put, probability mass curves the data space with horizons. What exactly it is meant by this is the subject of this paper bu t before proceeding further a few comments are in order. First of all, let us assume that we have a set {x1,...,x n}of data. Each observation xjconsisting of pmeasurements that are thought as the pcoordinates of a vector in I Rp. To make the data space into a probability space we endow I Rpwith the ﬁeld of Borelians but nothing beyond that. In particular no a priori metric structure on the data space is assumed. The nobservations are assumed to benindependent realizations of a given probability measure Pon I Rp. By the Lebesgue decomposition theorem, for every Borel set Bwe can write, P(B) =/integraldisplay Bf(x)λ(dx) +ν(B) (1) whereνis the singular part of Pthat assigns positive probability mass to Borel sets of zero Lebesgue volume. Due to the existence of pa thologies like the Cantor set in one dimension and its analogies in higher di mensions, the singular part νcannot be empirically estimated (see e.g. [3]). Practicall y all of the papers on density estimation rule out the singular par t ofPa priori. The elimination of singularities by ﬁat has permitted the co nstruction of a1 INTRODUCTION 2 rich mathematical theory for density estimation, but it has also ruled out a priori the study of models of mixed dimensionality (see [4]) that may be nec- essary for understanding point masses and spacetime singul arities coexisting with absolutely continuous distributions. We assume further that in the regions where f(x)>0 the density fis of classC2i.e., it has continuous second order partial derivatives. 1.1 Nonparametrics with the World in Mind The road from Classical Newtonian Physics to the physics of t oday can be seen as a path paved by an increasing use of fundamental conce pts that are statistical in nature. This is obvious for statistical m echanics, becoming clearer for quantum theory, and appearing almost as a shock i n General Relativity. Not surprisingly there have been several attem pts to take this trend further (see e.g. [5, 6, 7, 8]) in the direction of Physics as Inference . Now suppose for a moment that in fact some kind of restatement of the foundations of physics in terms of information and statisti cal inference will eventually end up providing a way to advance our current unde rstanding of nature. As of today, that is either already a solid fact or rem ains a wild spec- ulation, depending on who you ask. In any case, for the trend t o take over, it will have to be able to reproduce all the successes of current science and make new correct predictions. In particular it would have to repr oduce General Relativity. Recall that the main lesson of General Relativi ty is that space and time are not just a passive stage on top of which the univer se evolves. General Relativity is the theory that tells (through the ﬁel d equation) how to build the stage (left hand side of the equation) from the data (right hand side of the equation). The statistical theory that tells how to bu ild the stage of inference (the probabilistic model) from the observed data is:Nonparametric Density Estimation . It is therefore reassuring to ﬁnd typical signatures of General Relativity in density estimation as this paper does . Perhaps Physics is not just a special case of statistical inference and all th ese are only coin- cidences of no more relevance than for example the fact that m ultiplication or the logarithmic function appear everywhere all the time. That may be so, but even in that case I believe it is worth noticing the con nection for undoubtedly GR and density estimation have a common goal: The dynamic building of the stage . More formally. Let fbe a multivariate probability density function with continuous second order partial derivatives. Consider the problem of esti-2 ESTIMATING DENSITIES FROM DATA 3 mating the true value of f(z)>0 at a single point z, fromnindependent observations. It is shown that, fastest possible estimator s (including the k- nearest neighbor and kernel as well as the rich class of estim ators in [9, the- orem3.1]) have minimum asymptotic mean square errors when t he space of observations is thought as conformally curved. The optimal metric is shown to be generated by the Hessian of fin the regions where the Hessian is deﬁ- nite. Thus, the peaks and valleys of fare surrounded by horizons where the Hessian changes signature from Riemannian to pseudo-Riema nnian. The result for the case of generalized k-nearest neighbor es timators [9] has circulated since 1988 in the form of a technical report [1 0]. Recently I found that a special case of this theorem has been known sinc e 1972 [11] and undergone continuous development in the Pattern Recogn ition literature, (see e.g. [12, 13, 14, 15]). 2 Estimating Densities from Data The canonical problem of density estimation at a point z∈I Rpcan be stated as follows: Estimatef(z)>0fromnindependent observations of a random variable with density f. The most successful estimators of f(z) attempt to approximate the den- sity of probability at zby using the observations x1,...,x nto build both, a small volume around zand, a weight for this volume in terms of probability mass. The density is then computed as the ratio of the estimat ed mass over the estimated volume. The two classical examples are the k-n earest neighbor (knn) and the kernel estimators. 2.1 The knn The simplest and historically the ﬁrst example of a nonparam etric density estimator is [16] the knn. The knn estimator of f(z) is deﬁned for k∈ {1,2,...,n }as, hn(z) =k/n λk(2) whereλkis the volume of the sphere centered at the point z∈I Rpof radius R(k) given by the distance from zto the kth-nearest neighbor observation. Ifλdenotes the Lebesgue measure on I Rpwe have,2 ESTIMATING DENSITIES FROM DATA 4 λk=λ(S(R(k))) (3) where, S(r) ={x∈I Rp:/ba∇dblx−z/ba∇dbl ≤r} (4) The sphere S(r) and the radius R(k) are deﬁned relative to a given norm, /ba∇dbl · /ba∇dblin I Rp. The stochastic behavior of the knn depends on the speciﬁc va lue of the integer kchosen in (2). Clearly, in order to achieve consistency (e.g . stochastic convergence of hn(z) asn→ ∞ towards the true value of f(z)>0) it is necessary to choose k=k(n) as a function of n. The volumes λkmust shrink, to control the bias, and consequently we must have k/n→0 forhn(z) to be able to approach a strictly positive number. On the othe r hand, we must havek→ ∞ to make the estimator dependent on an increasing number kof observations and in this way to control its variance. Thus , for the knn to be consistent, we need kto increase with nbut at a rate slower than n itself. The knn estimator depends not only on kbut also on a choice of norm. The main result of this paper follows from the characterizat ion of the /ba∇dbl · /ba∇dbl that, under some regularity conditions, produces the best a symptotic (as n→ ∞) performance for density estimators. 2.2 The kernel If we consider only regular norms /ba∇dbl · /ba∇dbl, in the sense that for all suﬃciently small values of r>0, λ(S(r)) =λ(S(1))rp≡βrp(5) then, the classical kernel density estimator can be written as: gn(z) =Mµ λ(S(µ))(6) where, Mµ=1 n /summationdisplay xj∈S(µ)Kµ−1(xj−z)  (7) The smoothing parameter µ=µ(n) is such that k= [nµp] satisﬁes the conditions for consistency of the knn, Kµ−1(x) =K(µ−1x) where the kernel2 ESTIMATING DENSITIES FROM DATA 5 functionKis a non negative bounded function with support on the unit sphere (i.e. K(x) = 0 for /ba∇dblx/ba∇dbl>1) and satisfying, /integraldisplay /bardblx/bardbl≤1K(x)dx=β (8) Notice that for the constant kernel (i.e. K(x) = 1 for /ba∇dblx/ba∇dbl ≤1) the estimator (6) approximates f(z) by the proportion of observations inside S(µ) over the volume ofS(µ). The general kernel function Kacts as a weight function allocating diﬀerent weights Kµ−1(xj−z) to thexj’s insideS(µ). To control bias (see (32) below) the kernel Kis usually taken as a decreasing radially symmetric function in the metric generated by the norm /ba∇dbl·/ba∇dbl. Thus,Kµ−1(xj− z) assigns a weight to xjthat decreases with its distance to z. This has intuitive appeal, for the observations that lie closer to zare less likely to fall oﬀ the sphere S(µ), under repeated sampling, than the observations that are close to the boundary of S(µ). The performance of the kernel as an estimator for f(z) depends ﬁrst and foremost on the value of the smoothness parameter µ. The numerator and the denominator of gn(z) depend not only on µbut also on the norm /ba∇dbl · /ba∇dbl chosen and the form of the kernel function K. As it is shown in theorem (8) these three parameters are inter-related. 2.3 Double Smoothing Estimators The knn (2) and the kernel (6) methods are two extremes of a con tinuum. Both,hn(z) andgn(z) estimate f(z) asprobability-mass-per-unit-volume . The knn ﬁxes the mass to the deterministic value k/nand lets the volume λk to be stochastic, while the kernel method ﬁxes the volume λ(S(µ)) and lets the massMµto be random. The continuum gap between (2) and (6) is ﬁlled up by estimators that stochastically estimate mass and volu me by smoothing the contribution of each sample point with diﬀerent smoothi ng functions for the numerator and denominator (see [9]). Letb≥1 and assume, without loss of generality that bkis an integer. The double smoothing estimators with deterministic weight s are deﬁned as, fn(z) =1 n/summationtextn i=1K/parenleftig z−xi R(k)/parenrightig 1 cb/summationtextbk i=1ωiλi(9)3 THE TRUTH AS N→ ∞ ? 6 where, ωi=/integraldisplayi/bk (i−1)/bkω(u)du (10) andω(·) is a probability density on [0 ,1] with mean c. 3 The Truth as n→ ∞? In nonparametric statistics, in order to assess the quality of an estimator fn(z) as an estimate for f(z), it is necessary to choose a criterion for judg- ing how far away is the estimator from what it tries to estimat e. This is sometimes regarded as revolting and morally wrong by some Ba yesian Fun- damentalists. For once you choose a loss function and a prior , logic alone provides you with the Bayes estimator and the criterion for j udging its qual- ity. That is desirable, but there is a problem in high dimensi onal spaces. In inﬁnite dimensional hypothesis spaces (i.e. in nonparam etric problems) almost all priors will convince you of the wrong thing! (see e .g. [17, 18] for a non-regular way out see [19]). These kind of Bayesian no nparametric results provide a mathematical proof that: almost all fundamental religions are wrong, (more data can only make the believers more sure that the wron g thing is true!). An immediate corollary is that: Subjective Bayesians can’t go to Heaven . Besides, the choice of goodness of ﬁt criterion is as ad-hoc (an equivalent) to the choice of a loss function. 3.1 The Natural Invariant Loss Function and Why the MSE is not that Bad The most widely studied goodness of ﬁt criterion is the Mean S quare Error (MSE) deﬁned by, (MSE) =E\|fn(z)−f(z)\|2(11) where the expectation is over the joint distribution of the s amplex1,...,x n. By adding and subtracting T=Efn(z) and expanding the square, we can express the MSE in the computationally convenient form, (MSE) = E\|fn(z)−T\|2+\|T−f(z)\|2 = (variance) +(bias)2(12)3 THE TRUTH AS N→ ∞ ? 7 By integrating (11) over the z∈I Rpand interchanging Eand/integraltext(OK by Fubbini’s theorem since the integrand ≥0) we obtain, (MISE) =E/integraldisplay \|fn(z)−f(z)\|2dz (13) The Mean Integrated Square Error (MISE) is just the expected L2distance offnfromf. Goodness of ﬁt measures based on the ( MSE ) have two main advantages: They are often easy to compute and they enab le the rich Hilbertian geometry of L2. On the other hand the ( MSE ) is unnatural and undesirable for two reasons: Firstly, the ( MSE ) is only deﬁned for densities inL2and this rules out a priori all the densities in L1\L2which is unaccept- able. Secondly, even when the ( MISE ) exists, it is diﬃcult to interpret (as a measure of distance between densities) due to its lack of in variance under relabels of the data space. Many researchers see the expecte dL1distance between densities as the natural loss function in density es timation. The L1 distance does in fact exist for all densities and it is easy to interpret but it lacks the rich geometry generated by the availability of the inner product in L2. A clean way out is to use the expected L2distance between the wave functionsψn=√fnandψ=√f. Theorem 1 TheL2norm of wave functions is invariant under relabels of the data space, i.e., /integraldisplay \|ψn(z)−ψ(z)\|2dz=/integraldisplay \|˜ψn(z′)−˜ψ(z′)\|2dz′(14) wherez=h(z′)withhany one-to-one smooth function. Proof: Just change the variables. From, the change of variables the orem the pdf ofz′is, ˜f(z′) =f(h(z′))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(h) ∂(z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(15) from where the wave function of z′is given by, ˜ψ(z′) =ψ(h(z′))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(h) ∂(z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2 (16) Thus, making the substitution z=h(z′) we get,3 THE TRUTH AS N→ ∞ ? 8 /integraldisplay \|ψn−ψ\|2dz=/integraldisplay \|ψn(h(z′))−ψ(h(z′))\|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(h) ∂(z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledz′ =/integraldisplay \|ψn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(h) ∂(z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2 −ψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(h) ∂(z′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2 \|2dz′ =/integraldisplay \|˜ψn−˜ψ\|2dz′(17) Q.E.D. The following theorem shows that a transformation of the MSE of a con- sistent estimator provides an estimate for the expected L2norm between wave functions. Theorem 2 Letfn(z)be a consistent estimator of f(z). Then, E/integraldisplay \|ψn−ψ\|2dz=1 4/integraldisplayE\|fn(z)−f(z)\|2 f(z)dz+(smaller order terms) (18) Proof: A ﬁrst order Taylor expansion of√xaboutx0gives, √x−√x0=1 2(x−x0)√x0+o((x−x0)2) (19) Substituting x=fn(z),x0=f(z) into (19) squaring both sides and taking expectations we obtain, E\|ψn(z)−ψ(z)\|2=1 4E\|fn(z)−f(z)\|2 f(z)+o(E\|fn(z)−f(z)\|2) (20) integrating over zand interchanging Eand/integraltextwe arrive at (18). Q.E.D. Proceeding as in the proof of theorem 1 we can show that /integraldisplay\|fn−f\|2 fdz=/integraldisplay\|˜fn−˜f\|2 ˜fdz′(21) where, as before, z↔z′is any one-to-one smooth transformation of the data space and ˜fis the density of z′. Thus, it follows from (21) that the leading term on the right hand side of (18) is also invariant under rel abels of the data space. The nice thing about the L2norm of wave functions, unlike (21), is that it is deﬁned even when f(z) = 0.4 SOME CLASSIC ASYMPTOTIC RESULTS 9 4 Some Classic Asymptotic Results We collect here the well known Central Limit Theorems (CLT) f or the knn and kernel estimators together with some remarks about nonp arametric den- sity estimation in general. The notation and the formulas in troduced here will be needed for computing the main result about optimal no rms in the next section. Assumption 1 Letfbe a pdf on I Rpof class C2with non singular Hessian, H(z)atz∈I Rp, and withf(z)>0, i.e., the matrix of second order partial derivatives of fatzexists, it is non singular and its entries are continuous atz. Assumption 2 LetKbe a bounded non negative function deﬁned on the unit sphere, S0={x∈I Rp:/ba∇dblx/ba∇dbl ≤1}and satisfying, /integraldisplay /bardblx/bardbl≤1K(x)dx=λ(S0)≡β (22) /integraldisplay /bardblx/bardbl≤1xK(x)dx= 0∈I Rp(23) Theorem 3 (CLT for knn) Under assumption 1, if k=k(n)is taken in the deﬁnition of the knn (2) in such a way that for some a>0 limn→∞n−4/(p+4)k=a (24) then, if we let Zn=√ k(hn(z)−f(z))we have, limn→∞P(Zn≤t) =/integraldisplayt −∞1√ 2πexp/parenleftigg −(y−B(z))2 2V(z)/parenrightigg dy (25) where, B(z) = ap+4 2p 2f2/p(z) /braceleftigg β−1−2/p/integraldisplay /bardblx/bardbl≤1xTH(z)xdx/bracerightigg (26) and, V(z) =f2(z) (27)4 SOME CLASSIC ASYMPTOTIC RESULTS 10 Proof: This is a special case of [9, theorem3.1]. Theorem 4 (CLT for kernel) Under assumptions 1, and 2 if µ=µ(n)is taken in the deﬁnition of the kernel (6) in such a way that for s omea>0, k= [nµp]satisﬁes (24) then, if we let Zn=√ k(gn(z)−f(z))we have (25) where now, B(z) = ap+4 2p 2 /braceleftigg β−1/integraldisplay /bardblx/bardbl≤1xTH(z)xK(x)dx/bracerightigg (28) and, V(z) =f(z)/braceleftigg β−2/integraldisplay /bardblx/bardbl≤1K2(x)dx/bracerightigg (29) Proof: The sample x1,...,x nis assumed to be iid fand therefore the kernel estimatorgn(z) given by (6) and (7) is a sum of iid random variables. Thus, the classic CLT applies and we only need to verify the rate (24 ) and the asymptotic expressions for the bias (28) and variance (29). We have, E[gn(z)] =1 βµp1 nn/summationdisplay j=1/integraldisplay K/parenleftiggxj−z µ/parenrightigg f(xj)dxj (30) =1 βµp/integraldisplay K(y)f(z+µy)µpdy (31) =/integraldisplayK(y) β/braceleftigg f(z) +µ∇f(z)·y+µ2 2yTH(z)y+o(µ2)/bracerightigg dy(32) =f(z) +µ2 2β/integraldisplay yTH(z)yK(y)dy+o(µ2) (33) where we have changed the variables of integration to get (31 ), used assump- tion 1 and Taylor’s theorem to get (32) and used assumption 2 t o obtain (33). For the variance we have, var(gn(z)) =1 nβ2µ2pvar(K((X−z)/µ)) (34) =1 nβ2µ2p/braceleftigg/integraldisplay /bardbly/bardbl≤1f(z+µy)K2(y)µpdy4 SOME CLASSIC ASYMPTOTIC RESULTS 11 −/parenleftigg/integraldisplay /bardbly/bardbl≤1f(z+µy)K(y)µpdy/parenrightigg2  (35) =f(z) nβ2µp/integraldisplay /bardbly/bardbl≤1K2(y)dy+o/parenleftigg1 nµp/parenrightigg (36) where we have used var( K) =EK2−(EK)2and changed the variables of integration to get (35), used assumption 1 and (0th order) Ta ylor’s theorem to get (36). Hence, the theorem follows from (33) and (36) aft er noticing that (24) and k=nµpimply, √ kµ2=k4+p 2pn−2/p= (n−4 p+4k)p+4 2p− →ap+4 2p (37) k nµp=k k= 1 (38) Q.E.D. Theorem 5 (CLT for double smoothers) Consider the estimator fn(z) deﬁned in (9). Under assumptions 1, 2, and (24) if we let Zn=√ k(fn(z)− f(z))we have (25) where now, B(z) = ap+4 2p 2[βf(z)]2/p β−1/braceleftigg/integraldisplay /bardblx/bardbl≤1xTH(z)x[K(x) +λ0]dx/bracerightigg (39) and, V(z) =f2(z)/braceleftigg β−1/integraldisplay /bardblx/bardbl≤1K2(x)dx−λ1/bracerightigg (40) with, λ0=b2/p c/integraldisplay1 0u1+2 pω(u)du−1 (41) λ1= 1−c−2b−1/integraldisplay1 0/braceleftbigg/integraldisplay1 yω(x)dx/bracerightbigg2 dy (42) Proof: See [9, theorem3.1]. Remember to substitute Kbyβ−1Ksince in the reference the Kernels are probability densities and in h ere we take them as weight functions that integrate to β.4 SOME CLASSIC ASYMPTOTIC RESULTS 12 4.1 Asymptotic Mean Square Errors Letfnbe an arbitrary density estimator and let Zn=√ k(fn(z)−f(z)). Now suppose that fn(z) is asymptotically normal, in the sense that when k=k(n) satisﬁes (24) for some a >0, we have (25) true. Then, all the moments of Znwill converge to the moments of the asymptotic Gaussian. In p articular the mean and the variance of Znwill approach B(z) andV(z) respectively. Using, (12) and (24) we can write, limn→∞n4/(p+4)E\|fn(z)−f(z)\|2=V(z) a+B2(z) a(43) We call the right hand side of (43) the asymptotic mean square error (AMSE) of the estimator fn(z). The value of acan be optimized to obtain a global minimum for the (AMSE) but it is well known in nonparametrics that the raten−4/(p+4)is best possible (in a minimax sense) under the smoothness asumption 1 (see e.g. [20]). We can take care of the knn, the ke rnel, and the double smoothing estimators simultaneously by noticing th at in all cases, (AMSE) = α1a−1+α2a4/p(44) has a global minimum of, (AMSE)∗=/braceleftigg (1 + 4/p)/parenleftbiggp 4/parenrightbigg4 p+4/bracerightigg α4 p+4 1αp p+4 2 (45) achieved at, a∗=/parenleftbiggpα1 4α2/parenrightbiggp p+4(46) Replacing the corresponding values for α1andα2for the knn, for the kernel, and for the double smoothing estimators, we obtain that in al l cases, (AMSE)∗= (const. indep. of f)  f(z)/parenleftigg∆2 f(z)/parenrightiggp p+4  (47) where, ∆ =/integraldisplay /bardblx/bardbl≤1xTH(z)xG(x)dx (48)4 SOME CLASSIC ASYMPTOTIC RESULTS 13 =p/summationdisplay j=1ρj∂2f ∂z2 j(z) (49) withG(x) = 1 for the knn, G(x) =K(x) for the kernel, G(x) =K(x)+λ0for the double smoothers (see (39) and (41)) and, if ejdenotes the jth canonical basis vector (all zeroes except a 1 at position j), ρj=/integraldisplay /bardblx/bardbl≤1(x·ej)2G(x)dx (50) Notice that (49) follows from (48), (23) and the fact that H(z) is the Hes- sian offatz. The generality of this result shows that (47) is typical for density estimation. Thus, when fnis either the knn, the kernel, or one of the estimators in ([9, theorem3.1]), we have: limn→∞n4/(p+4)E\|fn(z)−f(z)\|2≥cf(z)/parenleftigg∆2 f(z)/parenrightiggp p+4 (51) The positive constant cmay depend on the particular estimator but it is independent of f. Dividing both sides of (51) by f(z), integrating over z, using theorem 2 and interchanging Eand/integraltextwe obtain, limn→∞n4/(p+4)E/integraldisplay \|ψn(z)−ψ(z)\|2dz≥4c/integraldisplay/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆ ψ(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2p p+4 dz (52) The worst case scenario is obtained by the model f=ψ2that maximizes the action given by the right hand side of (52), L=/integraldisplay/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 ψ(z)p/summationdisplay j=1ρj∂2ψ2 ∂z2 j(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2p p+4 dz (53) This is a hard variational problem. However, it is worth noti cing that the simplest case is obtained when the exponent is 1, i.e. when th e dimension of the data space is p= 4. Assuming we were able to ﬁnd a solution, this solution would still depend on the pparameters ρ1,...,ρ p. A choice of ρj’s is equivalent to the choice of a global metric for the data spa ce. Notice also, that the exponent becomes 2 for p=∞and that for p≥3 (but not for p= 1 or 2) there is the possibility of non trivial (i.e. diﬀeren t from uniform) super-eﬃcient models for which estimation can be done at rat es higher than n−4/(p+4). These super-eﬃcient models are characterized as the non ne gative5 CHOOSING THE OPTIMAL NORM 14 solutions of the Laplace equation in the metric generated by theρj’s, i.e., non negative ( f(z)≥0) solutions of, p/summationdisplay j=1ρj∂2f ∂z2 j(z) = 0 (54) Recall that there are no non trivial (diﬀerent from constant ) non negative super-harmonic functions in dimensions one or two but there are plenty of solutions in dimension three and higher. For example the New tonian poten- tials, f(z) =c/ba∇dblz/ba∇dbl−(p−2) ρ (55) with the norm, /ba∇dblz/ba∇dbl2 ρ=p/summationdisplay j=1/parenleftiggzj√ρj/parenrightigg2 (56) will do, provided the data space is compact. The existence of (hand picked) super-eﬃcient models is what made necessary to consider bes t rates only in the minimax sense. Even though we can estimate a Newtonian po tential model at faster than usual nonparametric rates, in any neigh borhood of the Newtonian model the worst case scenario is at best estimated at raten−4/(p+4) under second order smoothness conditions. 5 Choosing the Optimal Norm All ﬁnite (p <∞) dimensional Banach spaces are isomorphic (as Banach spaces) to I Rpwith the euclidian norm. This means, among other things, that in ﬁnite dimensional vector spaces all norms generate t he same topology. Hence, the spheres {x∈I Rp:/ba∇dblx/ba∇dbl ≤r}are Borelians so they are Lebesgue measurable and thus, estimators like the knn (2) are well deﬁ ned for arbitrary norms. It is possible, in principle, to consider norms that a re not coming from inner products but in this paper we look only at Hilbert norms /ba∇dbl · /ba∇dbl Aof the form, /ba∇dblz/ba∇dbl2 A=zTATAz (57) whereA∈Λ with Λ deﬁned as the open set of real non-singular p×p matrices. For each A∈Λ deﬁne the unit sphere,5 CHOOSING THE OPTIMAL NORM 15 SA={x∈I Rp:xTATAx≤1} (58) its volume, βA=λ(SA) =/integraldisplay SAλ(dx) (59) and theA-symmetric (i.e. /ba∇dbl · /ba∇dbl Aradially symmetric) kernel, KA, KA(x) = (K◦A)(x) =K(Ax) (60) whereKsatisﬁes assumption 2 and it is I-symmetric, i.e., radially symmetric in the euclidian norm. This means that K(y) depends on yonly through the euclidian length of y, i.e. there exists a function Fsuch that, K(y) =F(yTy) (61) The following simple theorem shows that all A-symmetric functions are really of the form (60). Theorem 6 For anyA∈Λ,˜KisA-symmetric if and only if we can write ˜K(x) =K(Ax)for allx∈I Rp(62) for someI-symmetric K. Proof: ˜K(x) isA-symmetric iﬀ ˜K(x) =F(/ba∇dblx/ba∇dbl2 A) for some function F. ChooseK(x) =˜K(A−1x). ThisKisI-symmetric since K(x) =F/parenleftig (AA−1x)T(AA−1x)/parenrightig = F(xTx). More over, ˜K(x) =˜K(A−1(Ax)) =K(Ax). Thus, (62) is necessary forA-symmetry. It is also obviously suﬃcient since the assumed I-symmetry ofKin (62) implies that ˜K(x) =F((Ax)T(Ax)) =F(/ba∇dblx/ba∇dbl2 A) so it isA- symmetric. Q.E.D. An important corollary of theorem 6 is, Theorem 7 LetA,B∈Λ. Then, ˜KisAB-symmetric if and only if ˜KB−1 isA-symmetric.5 CHOOSING THE OPTIMAL NORM 16 Proof: By the ﬁrst part of theorem 6 we have that ˜K=K◦A◦BwithK someI-symmetric. Thus, ˜K◦B−1=K◦AisA-symmetric by the second part of theorem 6. Q.E.D. Let us denote by β(A,K) the total volume that a kernel Kassigns to the unitA-sphereSA, i.e., β(A,K) =/integraldisplay SAK(x)dx (63) In order to understand the eﬀect of changing the metric on a de nsity esti- mator like the kernel (6), it is convenient to write gnexplicitly as a function of everything it depends on; The point z, the metric A, theA-symmetric kernel function ˜K, the positive smoothness parameter µand, the data set {x1,...,x n}. Hence, we deﬁne, gn(z;A,˜K,µ,{x1,...,x n}) =1 n/summationtextn j=1˜K/parenleftigxj−z µ/parenrightig β(A,˜K)µp(64) The following result shows that kernel estimation with metr icABis equiv- alent to estimation of a transformed problem with metric A. The explicit form of the transformed problem indicates that a perturbati on of the met- ric should be regarded as composed of two parts: Shape and vol ume. The shape is confounded with the form of the kernel and the change of volume is equivalent to a change of the smoothness parameter. Theorem 8 LetA,B∈Λ,µ >0, and ˜KanAB-symmetric kernel. Then, for allz∈I Rpand all data sets {x1...,x n}we have, gn(z;AB,˜K,µ,{x1,...,x n}) =gn(ˆBz;A,˜K◦B−1,\|B\|−1/pµ,{ˆBx1,...,ˆBxn}) (65) where \|B\|denotes the determinant of BandˆB=\|B\|−1/pBis the matrix B re-scaled to have unit determinant. Proof: To simplify the notation let us denote, µB=µ \|B\|1/p(66) Substituting ABforAin (64) and using theorem 6 we can write the left hand side of (65) as,5 CHOOSING THE OPTIMAL NORM 17 1 n/summationtextn j=1K/parenleftig AB/parenleftigxj−z µ/parenrightig/parenrightig β(AB,˜K)µp=1 n/summationtextn j=1(K◦A)/parenleftbigg ˆBxj−ˆBz µB/parenrightbigg β(A,K◦A)(µB)p whereKis someI-symmetric kernel and we have made the change of vari- ablesx=B−1yinβ(AB,˜K) (see (63) ). The last expression is the right hand side of (65). Notice that, K◦A=˜KB−1isA-symmetric. Q.E.D. 5.1 Generalized knn Case with Uniform Kernel In this section we ﬁnd the norm of type (57) that minimizes (47 ) for the estimators of the knn type with uniform kernel which include the double smoothers with K(x) = 1. As it is shown in theorem 8 a change in the determinant of the matrix that deﬁnes the norm is equivalent to changing the smoothness parameter. The quantity (57) to be minimized was obtained by ﬁxing the value of the smoothness parameter to the best pos sible, i.e. the one that minimizes the expression of the (AMSE) (43). Thus, t o search for the best norm at a ﬁx value of the smoothness parameter we need to keep the determinant of the matrix that deﬁnes the norm constant. We have, Theorem 9 Consider the problem, min \|A\|=1/parenleftbigg/integraldisplay SAxTH(z)xdx/parenrightbigg2 (67) where the minimum is taken over all p×pmatrices with determinant one, SAis the unit A-ball andH(z)is the Hessian of the density f∈ C2atz which is assumed to be nonsingular. WhenH(z)is indeﬁnite, i.e. when H(z)has both positive and negative eigenvalues, the objective function in (67) achieves its ab solute minimum value of zero when Ais taken as, A=c−1diag(/radicaligg ξ1 p−m,...,/radicaligg ξm p−m,/radicaligg ξm+1 m,...,/radicaligg ξp m)M (68) where theξjare the absolute value of the eigenvalues of H(z),mis the number of positive eigenvalues, Mis the matrix of eigenvectors and cis a5 CHOOSING THE OPTIMAL NORM 18 normalization constant to get detA= 1(see the proof for more detailed deﬁnitions). IfH(z)is deﬁnite, i.e. when H(z)is either positive or negative deﬁnite, then for all p×preal matrices AwithdetA= 1we have, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay SAxTH(z)xdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥2pπ p+ 3p\|detH(z)\|1/p(69) with equality if and only if, A=H1/2 +(z) \|H1/2 +(z)\|1/p(70) whereH1/2 +(z) denotes the positive deﬁnite square-root of H(z) (see the proof below for explicit deﬁnitions). Proof: Sincef∈ C2the Hessian is a real symmetric matrix and we can therefore ﬁnd an orthogonal matrix Mthat diagonalizes H(z), i.e. such that, H(z) =MTDM withMTM=I (71) where, D= diag (ξ1,ξ2,...,ξ m,−ξm+1,...,−ξp) (72) where all the ξj>0 and we have assumed that the columns of Mhave been ordered so that all the mpositive eigenvalues appear ﬁrst and all the negative eigenvalues −ξm+1,...,−ξpappear last so that (71) agrees with (72). Split Das, D= diag (ξ1,...,ξ m,0,...,0)−diag(0,...,0,ξm+1,...,ξ p) =D+−D− (73) and use (71) and (73) to write, H(z) =MTD+M−MTD−M =/parenleftig D1/2 +M/parenrightigT/parenleftig D1/2 +M/parenrightig −/parenleftig D1/2 −M/parenrightigT/parenleftig D1/2 −M/parenrightig = ΣT +Σ+−ΣT −Σ− (74) Using (74) and the substitution y=Axwe obtain, /integraldisplay SAxTH(z)xdx =/integraldisplay yTy≤1yT/parenleftig A−1/parenrightigT/parenleftig ΣT +Σ+−ΣT −Σ−/parenrightig A−1ydy =/integraldisplay yTy≤1/angbracketleftig ΣA−1y,ΣA−1y/angbracketrightig dy (75)5 CHOOSING THE OPTIMAL NORM 19 where, Σ = Σ ++ Σ −= (D++D−)1/2M (76) and<.,.> denotes the pseudo-Riemannian inner product, /a\}b∇acketle{tx,y/a\}b∇acket∇i}ht=m/summationdisplay i=1xiyi−p/summationdisplay i=m+1xiyi (77) By letting G= diag(1,...,1,−1,...,−1) (i.e.mones followed by p−m negative ones) be the metric with the signature of H(z) we can also write (77) as, /a\}b∇acketle{tx,y/a\}b∇acket∇i}ht=xTGy (78) Let, B= [b1\|b2\|...\|bp] = ΣA−1(79) whereb1,...,b pdenote the columns of B. Substituting (79) into (75), using the linearity of the inner product and the symmetry of the uni t euclidian sphere we obtain, /integraldisplay SAxTH(z)xdx =/integraldisplay yTy≤1/a\}b∇acketle{tBy,By /a\}b∇acket∇i}htdy =/summationdisplay j/summationdisplay k/a\}b∇acketle{tbj,bk/a\}b∇acket∇i}ht/integraldisplay SIyjykdy (80) =/summationdisplay j/summationdisplay k/a\}b∇acketle{tbj,bk/a\}b∇acket∇i}htδjkρ =ρp/summationdisplay j=1/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht (81) whereρstands for, ρ=/integraldisplay SI/parenleftig y1/parenrightig2dy=2pπ p+ 3(82) From (79) and (81) it follows that problem (67) is equivalent to, min \|B\|=\|Σ\| p/summationdisplay j=1/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht 2 (83) WhenH(z) is indeﬁnite, i.e. when m /∈ {0,p}it is possible to choose the columns of Bso that/summationtext j/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht= 0 achieving the global minimum. There are many possible choices but the simplest one is, B=c·diag(√p−m,√p−m,...,√p−m/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright m,√m,√m,...,√m/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright p−m) (84)5 CHOOSING THE OPTIMAL NORM 20 since, p/summationdisplay j=1/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht=c2m(√p−m)2−c2(p−m)(√m)2= 0. (85) The scalar constant cneeds to be adjusted to satisfy the constraint that detB= det Σ. From (79), (84) and (76) we obtain that the matrix for t he optimal metric can be written as, A=B−1Σ =c−1 √p−mΣ++c−1 √mΣ− (86) From (86) we get, ATA=c−2 p−mΣT +Σ++c−2 mΣT −Σ− (87) Finally from (74) we can rewrite (87) as, ATA=c−2MT/parenleftigg1 p−mD++1 mD−/parenrightigg M (88) =c−2MTdiag(ξ1 p−m,...,ξm p−m,ξm+1 m,...,ξp m)M (89) Notice that when p−m=m(i.e. when the number of positive equals the number of negative eigenvalues of H(z)) the factor 1 /mcan be factorized out from the diagonal matrix in (89) and in this case the optim alAcan be expressed as, A=H1/2 +(z) \|H1/2 +(z)\|1/p(90) where we have used the positive square-root of H(z) deﬁned as, H1/2 +(z) = diag(/radicalig ξ1,...,/radicalig ξp)M (91) In all the other cases for which H(z) is indeﬁnite, i.e. when m /∈ {0,p/2,p} we have, A=c−1diag(/radicaligg ξ1 p−m,...,/radicaligg ξm p−m,/radicaligg ξm+1 m,...,/radicaligg ξp m)M (92)5 CHOOSING THE OPTIMAL NORM 21 The normalization constant cis ﬁxed by the constraint that det A= 1 as, c= (p−m)−m 2pm−(p−m) 2p\|detH(z)\|1 2p (93) This shows (68). Let us now consider the only other remaining case when H(z) is deﬁnite, i.e. either positive deﬁnite ( m=p) or negative deﬁnite ( m= 0). Introducing λ0as the Lagrange multiplier associated to the constraint det B= det Σ we obtain that the problem to be solved is, min b1,...bp,λ0L(b1,b2,...,b p,λ0) (94) where the Lagrangian Lis written as a function of the columns of Bas, L(b1,b2,...,b p,λ0) = p/summationdisplay j=1/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht 2 −4λ0(det(b1,...,b p)−det Σ) (95) The−4λ0instead of just λ0is chosen to simplify the optimality equations below. The optimality conditions are, ∂L ∂bj= 0 forj= 1,...,p and∂L ∂λ0= 0 (96) where the functional partial derivatives are taken in the Fr ´ echet sense with respect to the column vectors bj. The Fr´ echet derivatives of quadratic and multi linear forms are standard text-book exercises. Writi ng the derivatives as linear functions of the vector parameter hwe have, ∂ ∂bj/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht(h) = 2 /a\}b∇acketle{tbj,h/a\}b∇acket∇i}ht (97) ∂ ∂bjdet(b1,...,b p)(h) = det(b1,..., h/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright j-th col.,...,b p) (98) Thus, using (97) and (98) to compute the derivative of (95) we obtain that for allhand allj= 1,...,p we must have, ∂L ∂bj(h) = 2/braceleftiggp/summationdisplay k=1/a\}b∇acketle{tbk,bk/a\}b∇acket∇i}ht/bracerightigg 2/a\}b∇acketle{tbj,h/a\}b∇acket∇i}ht −4λ0det(b1,...,h,...,b p) = 0 (99)5 CHOOSING THE OPTIMAL NORM 22 When/summationtext k<bk,bk>/\e}atio\slash= 0 we can rewrite (99) as, /a\}b∇acketle{tbj,h/a\}b∇acket∇i}ht=c−1 0det(b1,...,h,...,b p) (100) But now we can substitute h=biwithi/\e}atio\slash=jinto (100) and use the fact that the determinant of a matrix with two equal columns is zero, to obtain, /a\}b∇acketle{tbj,bi/a\}b∇acket∇i}ht= 0 for all i/\e}atio\slash=j. (101) In a similar way, replacing h=bjinto (100), we get /a\}b∇acketle{tbj,bj/a\}b∇acket∇i}ht=c−1 0detB=c (102) wherecis a constant that needs to be ﬁxed in order to satisfy the cons traint that detB= det Σ. We have shown that the optimal matrix Bmust have or- thogonal columns of the same length for the G-metric. This can be expressed with a single matrix equation as, BTGB=cI (103) Substituting (79) into (103) and re-arranging terms we obta in, ATA=c−1ΣTGΣ (104) =c−1(ΣT ++ ΣT −)G(Σ++ Σ −) =c−1(ΣT +Σ+−ΣT −Σ−) ATA=c−1H(z) (105) From (105), (103), (82) and (81) we obtain, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay SAxTH(z)x dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ρp\|c\| (106) and replacing the values of ρandcwe obtain (69). Q.E.D. 5.2 Yet Another Proof When The Hessian is Deﬁnite Consider the following lemma. Lemma 1 LetA,Bbe twop×pnon-singular matrices with the same deter- minant. Then /integraldisplay SA/ba∇dblx/ba∇dbl2 Bdx≥/integraldisplay SB/ba∇dbly/ba∇dbl2 Bdy (107)5 CHOOSING THE OPTIMAL NORM 23 Proof: Just splitSAandSBas, SA= (SASB)∪(SASc B) (108) SB= (SBSA)∪(SBSc A) (109) and write, /integraldisplay SA/ba∇dblx/ba∇dbl2 Bdx=/integraldisplay SB/ba∇dblx/ba∇dbl2 Bdx−/integraldisplay Sc ASB/ba∇dblx/ba∇dbl2 Bdx+/integraldisplay SASc B/ba∇dblx/ba∇dbl2 Bdx (110) Now clearly, min x∈SASc B/ba∇dblx/ba∇dbl2 B≥1≥max y∈Sc ASB/ba∇dbly/ba∇dbl2 B (111) from where it follows that, /integraldisplay SASc B/ba∇dblx/ba∇dbl2 Bdx≥min x∈SASc B/ba∇dblx/ba∇dbl2 B/integraldisplay SASc Bdx (112) ≥max y∈Sc ASB/ba∇dbly/ba∇dbl2 B/integraldisplay Sc ASBdy (113) ≥/integraldisplay Sc ASB/ba∇dbly/ba∇dbl2 Bdy (114) where (112) and (114) follow from (111). To justify the middl e inequality (113) notice that from (108), (109) and the hypothesis that \|A\|=\|B\|we can write,/integraldisplay SASc Bdx+/integraldisplay SASBdx=/integraldisplay Sc ASBdy+/integraldisplay SASBdy (115) The conclusion (107) follows from inequality (114) since th at makes the last two terms in (110) non-negative. Q.E.D. IfBis a nonsingular matrix we deﬁne, ˆB=B \|detB\|1/p(116) An immediate consequence of lemma 1 is, Theorem 10 IfH(z)is deﬁnite, then for all p×pmatrices with \|A\|= 1we have, ∆ =/integraldisplay SA/ba∇dblx/ba∇dbl2 H1/2(z)dx≥/integraldisplay SˆH1/2(z)/ba∇dblx/ba∇dbl2 H1/2(z)dx (117)5 CHOOSING THE OPTIMAL NORM 24 Proof: ∆ = \|H(z)\|1/p/integraldisplay SA/ba∇dblx/ba∇dbl2 ˆH1/2(z)dx (118) ≥ \|H(z)\|1/p/integraldisplay SˆH1/2(z)/ba∇dblx/ba∇dbl2 ˆH1/2(z)dx (119) =/integraldisplay SˆH1/2(z)/ba∇dblx/ba∇dbl2 H1/2(z)dx (120) where we have used lemma 1 to deduce the middle inequality (11 9). Q.E.D. 5.3 Best Norm With General Kernels In this section we solve the problem of ﬁnding the optimal nor m in the general class of estimators (9). Before we optimize the norm we need to state explicitly what i t means to do estimation with diﬀerent kernels and diﬀerent norms. Fir st of all a general kernel function is a nonnegative bounded function deﬁned on the unit sphere generated by a given norm. Hence, the kernel only makes sense relative to the given norm. To indicate this dependence on the norm we wri teKAfor the kernel associated to the norm generated by the matrix A. We let KA=K◦A (121) whereK=KIis a ﬁx mother kernel deﬁned on the euclidian unit sphere. Equation (121) provides meaning to the notion of changing th e norm without changing the kernel. What this means is not that the kernel is invariant under changes ofAbut rather equivariant in the form speciﬁed by (121). Recall also that a proper kernel must satisfy (22). To control bias we mus t also require the kernels to satisfy (23). It is easy to see (just change the variables) that if the mother kernel Khas these properties so do all its children KAwith the only proviso that \|A\|= 1 in order to get (22). Notice also that radial symmetry of Kis a suﬃcient but not a necessary condition for (23). The optimization of the norm with general kernels looks more complicated than when the kernel is uniform since the best ( AMSE )∗also depends on/integraltext SAK2 A(x)dx. Consider the double smoothing estimators, which are the mo st general case treated in this paper. From, (39), (40) and (45) we have, (AMSE )∗= (const.)/braceleftbigg β−1/integraldisplay SAK2 A(x)dx−λ1/bracerightbigg4 p+4f(z)/parenleftigg∆2 f(z)/parenrightiggp p+4 (122)5 CHOOSING THE OPTIMAL NORM 25 where the constant depends only on the dimension of the space . Even though the dependence of (122) on Alooks much more complicated than (47) this is only apparently so. In fact the two expressions deﬁne very si milar optimiza- tion problems as we now show. First notice that the search for best Amust be done within the class of matrices with a ﬁx determinant. For otherwise we will be ch anging the value of the smoothness parameter that was ﬁxed to the best po ssible value in order to obtain (122). If we let \|A\|= 1 we have, /integraldisplay SAK(x)dx=β=/integraldisplay SAdx=λ(SI) (123) We also have that, /integraldisplay SAK2 A(y)dy=/integraldisplay SAK2(Ay)dy=/integraldisplay SIK2(y)dx (124) From (123) and (124) we deduce that the term in (122) within cu rsive brack- ets is the same for all matrices Aand it depends only on the ﬁx kernel K. Finally notice that the value of ∆ in (122) is given by ∆ =/integraldisplay SAxTH(z)xG(Ax)dx (125) whereG(x) =K(x) +λ0in the general case. By retracing again the steps that led to (81) we can write, ∆ =/summationdisplay j/summationdisplay k/a\}b∇acketle{tbj,bk/a\}b∇acket∇i}ht/integraldisplay SIyjykG(y)dy (126) =/summationdisplay j/summationdisplay k/a\}b∇acketle{tbj,bk/a\}b∇acket∇i}htδjkρk(G) =p/summationdisplay j=1/a\}b∇acketle{tbj,bj/a\}b∇acket∇i}htρj(G) (127) where now, ρj(G) =/integraldisplay SI/parenleftig xj/parenrightig2G(x)dx (128) There are three cases to be considered. 1. All theρj(G) =ρforj= 1,...,p . The optimization problem reduces to the case when the kernel is uniform and therefore it has the same solution.6 ASYMPTOTIC RELATIVE EFFICIENCIES 26 2. All theρj(G) have the same sign, i.e. they are all positive or all nega- tive. If e.g. all ρj>0 just replace bjwith√ρjbjand use the formulas obtained for the uniform kernel case. 3. Some of the ρj(G) are positive and some are negative. This case can be handled like the previous one after taking care of the signs f or diﬀerent indicesj. The ﬁrst case is the most important for it is the one implied wh en the kernels are radially symmetric. The other two cases are only include d for complete- ness. Clearly if we do estimation with a non radially symmetr ic kernel the optimal norm would have to correct for this arbitrary builti n asymmetry, eﬀectively achieving at the end the same performance as when radially sym- metric kernels are used. The following theorem enunciates t he main result. Theorem 11 In the general class of estimators (9) with radially symmetr ic (mother) kernels, best possible asymptotic performance (u nder second order smoothness conditions) is achieved when distances are meas ured with the best metrics obtained when the kernel is uniform. 6 Asymptotic Relative Eﬃciencies The practical advantage of using density estimators that ad apt to the form of the optimal metrics can be measured by computing the Asymp totic Rel- ative Eﬃciency (ARE) of the optimal metric to the euclidian m etric. Let us denote byAMSE (I) andAMSE (H(z)) the expressions obtained from (122) when using the euclidian norm and the optimal norm respectiv ely. For the Euclidean norm we get, AMSE (I) = (const.)/braceleftbigg β−1/integraldisplay SIK2(x)dx−λ1/bracerightbigg4 p+4f(z)/parenleftigg(ρtrH(z))2 f(z)/parenrightiggp p+4 (129) where tr stands for the trace since, ∆ =/integraldisplay SIxTH(z)xG(x)dx=/summationdisplay i,jhij(z)/integraldisplay SIxixjG(x)dx=ρtrH(z) (130)7 AN EXAMPLE: RADIALLY SYMMETRIC DISTRIBUTIONS 27 Using (123), (124) and (69) we obtain that when H(z) is deﬁnite, AMSE (H(z)) = (131) (const.)/braceleftbigg β−1/integraldisplay SIK2(x)dx−λ1/bracerightbigg4 p+4f(z)/parenleftigg(ρp\|detH(z)\|1/p)2 f(z)/parenrightiggp p+4 Hence, when H(z) is deﬁnite the ARE is, ARE =AMSE (I) AMSE (H(z))=/parenleftiggtrH(z) p\|detH(z)\|1/p/parenrightigg2p p+4 (132) Ifξ1,...,ξ pare the absolute value of the eigenvalues of H(z) then we can write, ARE = 1 p/summationtext jξj /parenleftig/producttext jξj/parenrightig1/p 2p p+4 =/parenleftiggarith. mean of {ξj} geom. mean of {ξj}/parenrightigg2p p+4 (133) It can be easily shown that the arithmetic mean is always grea ter or equal than the geometric mean (take logs, use the strict concavity of the logarithm and Jensen’s inequality) with equality if and only if all the ξj’s are equal. Thus, it follows from (133) that the only case in which the use of the optimal metric will not increase the eﬃciency of the estimation of th e density at a point where the Hessian is deﬁnite is when all the eigenvalu es ofH(z) are equal. It is also worth noticing that the eﬃciency increa ses withp, the dimension of the data space. There is of course inﬁnite relat ive eﬃciency in the regions where the H(z) is indeﬁnite. 7 An Example: Radially Symmetric Distri- butions When the true density f(z) has radial symmetry it is possible to ﬁnd the regions where the Hessian H(z) is positive and negative deﬁnite. These models have horizons deﬁned by the boundary between the regi ons where H(z) is deﬁnite. We show also that when and only when the density i s linear in the radius of symmetry, the Hessian is singular in the inte rior of a solid sphere. Thus, at the interior of these spheres it is impossib le to do estimation with the best metric.7 AN EXAMPLE: RADIALLY SYMMETRIC DISTRIBUTIONS 28 Let us denote simply by Lthe log likelihood, i.e., f(z) = exp(L) (134) If we also denote simply by Ljthe partial derivative of Lwith respect to zj then, ∂f ∂zj=f(z)Lj (135) and also, ∂2f ∂zi∂zj=∂f ∂ziLj+f(z)Lij=f(z){LiLj+Lij} (136) where we have used (135) and the deﬁnition Lij=∂Lj ∂zi. It is worth notic- ing, by passing, that (136) implies a notable connection wit h the so called nonparametric Fisher information I(f) matrix, /integraldisplay H(z)dz=I(f)− I(f) = 0 (137) our main interest here however, is the computation of the Hes sian when the density is radially symmetric. Radial symmetry about a ﬁx po intµ∈I Rpis obtained when L(and thusfas well) depends on zonly through the norm /ba∇dblz−µ/ba∇dblV−1for some symmetric positive deﬁnite p×pmatrixV. Therefore we assume that, L=L(−1 2(z−µ)TV−1(z−µ)) (138) from where we obtain, Li=/parenleftig −vi·(z−µ)/parenrightig L′(139) Lij=L′′vi·(z−µ)vj·(z−µ)−L′vij(140) wherevi·andvijdenote the i-th row and ij-th entries of V−1respectively. Replacing (139) and (140) into (136), using the fact that V−1is symmetric and thatvj·(z−µ) is a scalar and thus, equal to its own transpose ( z−µ)Tv·j, we obtain H(z) =f(z)L′/braceleftigg/parenleftigg L′+L′′ L′/parenrightigg V−1(z−µ)(z−µ)T−I/bracerightigg V−1(141) We have also assumed that L′is never zero. With the help of (141) we can now ﬁnd the conditions for H(z) to be deﬁnite and singular. Clearly H(z)7 AN EXAMPLE: RADIALLY SYMMETRIC DISTRIBUTIONS 29 will be singular when the determinant of the matrix within cu rly brackets in (141) is zero. But that determinant being zero means that λ= 1 is an eigenvalue of (L′+L′′/L′)V−1(z−µ)(z−µ)T(142) and since this last matrix has rank one its only nonzero eigen value must be equal to its own trace. Using the cyclical property of the tra ce and letting y=−1 2(z−µ)TV−1(z−µ) we can write, Theorem 12 The Hessian of a radially symmetric density is singular when and only when either L′= 0or L′+d dylogL′=−1 2y(143) Notice that theorem 12 provides an equation in yafter replacing a par- ticular function L=L(y). Theorem 12 can also be used to ﬁnd the functions L(y) that will make the Hessian singular. Integrating (143) we o btain, L(y) + logL′(y) =−1 2log(\|y\|) +c (144) and solving for L′, separating the variables and integrating we get, L(y) = log/parenleftbigg a/radicalig \|y\|+b/parenrightbigg (145) whereaandbare constants of integration. In terms of the density equati on (145) translates to, f(z) =a/ba∇dblz−µ/ba∇dblV−1+b (146) Hence, in the regions where the density is a straight line as a function of r=/ba∇dblz−µ/ba∇dblV−1the Hessian is singular and estimation with best metrics is not possible. Moreover, from (141) we can also obtain the reg ions of space where the Hessian is positive and where it is negative deﬁnit e. WhenL′>0, H(z) will be negative deﬁnite provided that the matrix, I−(L′+L′′/L′)V−1(z−µ)(z−µ)T(147)8 CONCLUSIONS 30 is positive deﬁnite. But a matrix is positive deﬁnite when an d only when all its eigenvalues are positive. It is immediate to verify that ξis an eigenvalue for the matrix (147) if and only if (1 −ξ) is an eigenvalue of the matrix (142). The matrix (142) has rank one and therefore its only nonzero e igenvalue is its trace so we arrive to, Theorem 13 When, L′+d dylogL′<−1 2y(148) H(z)is negative deﬁnite when L′>0and positive deﬁnite when L′<0. When, L′+d dylogL′>−1 2y(149) H(z)is indeﬁnite. For example when f(z) is multivariate Gaussian L(y) =y+cso that L′= 1 and the horizon is the surface of the V−1-sphere of radius one i.e., (z−µ)TV−1(z−µ) = 1. Inside this sphere the Hessian is negative deﬁnite and outside the sphere the Hessian is indeﬁnite. The results in this section can be applied to any other class of radially symmetric distr ibutions, e.g. multivariate Twhich includes the Cauchy. 8 Conclusions We have shown the existence of optimal metrics in nonparamet ric density estimation. The metrics are generated by the Hessian of the u nderlying density and they are unique in the regions where the Hessian i s deﬁnite. The optimal metric can be expressed as a continuous function of t he Hessian in the regions where it is indeﬁnite. The Hessian varies contin uously from point to point thus, associated to the general class of density est imators (9) there is a Riemannian manifold with the property that if the estimato rs are computed based on its metric the best asymptotic mean square error is m inimized. The results are suﬃciently general to show that these are absolu te bounds on the quality of statistical inference from data. The similarities with General Relativity are evident but so are the diﬀer- ences. For example, since the Hessian of the underlying dens ity is negative deﬁnite at local maxima, it follows that there will be a horiz on boundary9 ACKNOWLEDGMENTS 31 where the Hessian becomes singular. The cross of the boundar y corresponds to a change of signature in the metric. These horizons almost always are null sets and therefore irrelevant from a probabilistic point of view. However, when the density is radially symmetric changing linearly wi th the radius we get solid spots of singularity. There is a qualitative chang e in the quality of inference that can be achieved within these dark spots. But u nlike GR, not only around local maxima but also around local minima of the d ensity we ﬁnd horizons. Besides, it is not necessary for the density to reach a certain threshold for these horizons to appear. Nevertheless, I bel ieve that the in- fusion of new statistical ideas into the foundations of Phys ics, specially at this point in history, should be embraced with optimism. Onl y new data will (help to) tell. There are many unexplored promising avenues along the lines of the sub- ject of this paper but one that is obvious from a GR point of vie w. What is missing is the connection between curvature and probabilit y density, i.e. the ﬁeld equation. I hope to be able to work on this in the near futu re. The existence of optimal metrics in density estimation is no t only of the- oretical importance but of signiﬁcant practical value as we ll. By estimating the Hessian (e.g. with kernels that can take positive and neg ative values, see [21]) we can build estimators that adapt to the form of the opt imal norm with eﬃciency gains that increase with the number of dimensions. The antidote to the curse of dimensionality! 9 Acknowledgments I would like to thank my friends in the Maximum Entropy commun ity spe- cially Gary Erickson for providing a stimulating environme nt for this meeting. I am also in debt to Ariel Caticha for many interesting conver sations about life, the universe, and these things. References [1] V. N. Vapnik, Statistical Learning Theory , John Wiley & Sons, Inc., 1998. [2] L. G. L. Devroye and G. Lugosi, A Probabilistic Theory of Pattern Recog- nition , Springer, New York, 1996.REFERENCES 32 [3] L. Devroye and L. Gy¨ orﬁ, “No empirical probability meas ure can con- verge in the total variation sense for all distributions,” Annals of Statis- tics,18, (3), pp. 1496–1499, 1990. [4] A. R´ enyi, “On the dimension and entropy of probability d istributions,” Acta Math. Acad. Sci. Hungar. ,10, pp. 193–215, 1959. [5] E. Jaynes, “Information theory and statistical mechani cs,”Phys. Rev. , 106, p. 620, 1957. Part II; ibid, vol 108,171. [6] B. R. Frieden, Physics from Fisher Information, a Uniﬁcation , Cam- bridge University Press, 1998. [7] C. C. Rodr´ ıguez, “Are we cruising a hypothesis space?,” inMaximum Entropy and Bayesian Methods , R. F. W. von der Linden, V. Dose and R. Preuss, eds., vol. 18, (Netherlands), pp. 131–140, Kluwe r Academic Publishers, 1998. Also at xxx.lanl.gov/abs/physics/9808 009. [8] A. Caticha, “Change, time and information geometry,” in Maxi- mum Entropy and Bayesian Methods , A. Mohammad-Djafari, ed., vol. 19, Kluwer Academic Publishers, 2000. too appear. Also at math- ph/0008018. [9] C. C. Rodriguez, “On a new class of multivariate density e stimators,” tech. rep., Dept. of Mathematics and Statistics, The Univer sity at Al- bany, 1986. (http://omega.albany.edu:8008/npde.ps). [10] C. C. Rodriguez, “The riemannian manifold induced by a d ensity esti- mator,” tech. rep., Dept. of Mathematics and Statistics, Th e University at Albany, 1988. (http://omega.albany.edu:8008/rmide.h tml). [11] K. Fukunaga and L. D. Hostetler, “Optimization of k-nea rest-neighbor density estimates,” IEEE Trans. on Information Theory ,IT-19 , pp. 320–326, May 1972. [12] R. D. Short and K. Fukunaga, “The optimal distance measu re for nearest neighbor classiﬁcation,” IEEE Trans. on Information Theory ,IT-27 , pp. 622–637, September 1981. [13] K. Fukunaga and T. Flick, “An optimal global nearest nei gbor metric,” IEEE Trans. on Pattern Analysis and Machine Intelligence ,PAMI-6 , pp. 314–318, May 1984.REFERENCES 33 [14] K. Fukunaga and D. M. Hummels, “Bayes error estimation u sing parzen and k-nn procedures,” IEEE Trans. on Pattern Analysis and Machine Intelligence ,PAMI-9 , pp. 634–643, September 1987. [15] J. P. Myles and D. J. Hand, “The multi-class metric probl em in near- est neighbour discrimination rules,” Pattern Recognition ,23, (11), pp. 1291–1297, 1990. [16] E. Fix and J. L. Hodges, “Discriminatory analysis. nonp arametric dis- crimination: Consistency properties,” Tech. Rep. 4 Projec t number 21- 49-004, USAF School of Aviation Medicine, Randolph Field, T x., 1951. [17] L. M. Le-Cam and G. Lo-Yang, Asymptotics in Statistics: Some Basic Concepts , Springer series in statistics, Springer-Verlag, 1990. [18] P. Diaconnis and D. Freedman, “On the consistency of bay esian esti- mates (with discussions),” Ann. Stat. ,14, (1), pp. 1–67, 1986. [19] C. C. Rodr´ ıguez, “Cv-np bayesianism by mcmc,” in Maximum Entropy and Bayesian Methods , G. J. Erickson, ed., vol. 17, Kluwer Academic Publishers, 1997. (physics/9712041). [20] I. Ibragimov and R. Has’minskii, Statistical Estimation , vol. 16 of Ap- plications of Mathematics , Springer-Verlag, 1981. [21] R. S. Singh, “Nonparametric estimation of mixed partia l derivatives of a multivariate density,” Journal of Multivariate Analysis ,6, pp. 111–122, 1976.
arXiv:physics/0010064v1 [physics.data-an] 26 Oct 2000ROLE AND MEANING OF SUBJECTIVE PROBABILITY SOME COMMENTS ON COMMON MISCONCEPTIONS G. D’AGOSTINI Dipartimento di Fisica dell’Universit` a “La Sapienza” Piazzale Aldo Moro 2, I-00185 Roma (Italy)† Abstract. Criticisms of so called ‘subjective probability’ come on th e one hand from those who maintain that probability in physics has only a frequentistic inter- pretation, and, on the other, from those who tend to ‘objecti vise’ Bayesian theory, arguing, e.g., that subjective probabilities are indeed ba sed ‘only on private in- trospection’. Some of the common misconceptions on subject ive probability will be commented upon in support of the thesis that coherence is t he most crucial, universal and ‘objective’ way to assess our conﬁdence on eve nts of any kind. Key words: Subjective Bayesian Theory, Measurement Uncertainty 1. Introduction The role of scientists is, generally speaking, to understan d Nature, in order to forecast as yet unobserved (‘future’) events, independent ly of whether or not these events can be inﬂuenced. In laboratory experiments and all t echnological applica- tions, observations depend on our intentional manipulatio n of the external world. However, other scientiﬁc activities, like astrophysics, a re only observational. Nev- ertheless, to claim that cosmology, climatology or geophys ics are not Science, be- cause “experiments cannot be repeated” - as pedantic interp reters of Galileo’s scientiﬁc method do - is, in my opinion, short-sighted (for a recent defence of this strict Galilean point of view, advocating ‘consequently’ f requentistic methods, see Ref. [1]). The link between past observations and future obs ervations is provided by theory (or model). It is accepted that quantitative (and, often, also qualitat ive) forecasting of future observations is invariably uncertain, from the mome nt that we deﬁne suﬃ- ciently precisely the details of the future events. The unce rtainty may arise because we are not certain about the parameters of the theory (or of th e theory itself), and/or about the initial state and boundary conditions of th e phenomenon we want †Email: giulio.dagostini@roma1.infn.it. URL: http://www -zeus.roma1.infn.it/∼agostini. Contribution at the XX International Workshop on Bayesian I nference and Maximum Entropy Methods in Science and Engineering, Gif sur Yvette, France, July 8–13, 2000.2 G. D’AGOSTINI to describe. But it may also be due to the stochastic nature of the theory itself, which would produce uncertain predictions even if all param eters and boundary conditions wereprecisely known. Nevertheless, the constant state of uncer tainty does not prevent us from doing science. As Feynman wrote, “it is scientiﬁc only to say what is more likely and what is less likely”.[2] This obse rvation holds not only for observations, but also for the values of physical quanti ties (i.e. parameters of the theory which have eﬀect on the real observations). And in deed, physicists ﬁnd probabilistic statements about, for example, top quark mas s or gravitational con- stant very natural,[3] and several equivalent expressions are currently used, such as “to be more or less conﬁdent ”, “to consider something more or less probable , or more or less likely”, “to believe more or less something”. However, the subjective deﬁnition of probability, the only one consistent with the a bove expressions, is usually rejected because of educational bias according to w hich “the only scien- tiﬁc deﬁnition of probability is the frequentistic one,” “q uantum mechanics only allows the frequency based deﬁnition of probability,” “pro bability is an objective property of the physical world,” etc. In this paper I will com ment on these and other objections against the so called ‘subjective Bayesia n’ point of view. Indeed, some criticisms come from ‘objective Bayesians’, who have b een, traditionally, in a clear majority during this workshop series. I don’t expect to solve these debates in this short contribut ion, especially con- sidering that many aspects of the debate are of a psychologic al and sociological nature. Neither will I be able to analyse in detail every obje ction or to cite all the counter-arguments. I prefer, therefore, to focus here only on a few points, referring to other papers [4–6] and references therein for points alre ady discussed elsewhere. 2. Subjective Probability and Role of Coherence The main aim of subjective probability is to recover the intu itive concept of prob- ability as degree of belief. Probability is then related to u ncertainty and not (only) to the outcomes of repeated experiments. Since uncertainty is related to knowl- edge, probability is only meaningful as long as there are hum an beings interested in knowing (or forecasting) something, no matter if “the eve nts considered are in some sense determined , or known by other people.”[7] Since - fortunately! - we do not share identical states of information, we are in diﬀ erent conditions of uncertainty. Probability is therefore only and always cond itional probability, and depends on the diﬀerent subjects interested in it (and hence the name subjective ). This point of view about probability is not related to a singl e evaluation rule. In particular, symmetry arguments and past frequencies, as we ll as their combination properly weighted by means of Bayes’ theorem, can be used. Since beliefs can be expressed in terms of betting odds, as is well known and done in practice, betting odds can be seen as the most general way of making relative beliefs explicit, independently of the kind of eve nts one is dealing with, or of the method used to deﬁne the odds. For example, everybod y understands Laplace’s statement concerning Saturn’s mass, that “it is a bet of 10000 to 1 that the error of this result is not 1/100th of its value.”[8] I wis h all experimental results to be provided in these terms, instead of the mislead ing [9] “such and suchCOMMENTS ON SUBJECTIVE PROBABILITY 3 percent CL’s.” What matters is that the bet must be reversibl e and that no bet can be arranged in such a way that one wins or loses with certai nty. The second condition is a general condition concerning bets. The ﬁrst c ondition forces the subject to assess the odds consistently with his/her belief s and also to accept the second condition: once he/she has ﬁxed the odds, he/she must be ready to bet in either direction. Coherence has two important roles: the ﬁr st is, so to speak, moral, and forces people to be honest; the second is formal, allowin g the basic rules of probability to be derived as theorems, including the formul a relating conditional probability to probability of the conditionand and their jo int probability (note that, consistent with the use of probability in practice and with the fact that in a theory where only conditional probabilities matter, it ma kes no sense to have a formula that deﬁnes conditional probability, see e.g. Section 8.3 of Ref. [4] fo r further comments and examples). Once coherence is included in the subjective Bayesian theor y, it becomes ev- ident that ‘subjective’ cannot be confused with ‘arbitrary ’, since all ingredients for assessing probability must be taken into account, inclu ding the knowledge that somebody else might assess diﬀerent odds for the same events . Indeed, the coherent subjectivist is far more responsible (and more “objective” , in the sense that ordi- nary parlance gives to this word) than those who blindly use s tandard ‘objective’ methods (see examples in Ref. [4]). Another source of object ions is the confusion between ‘belief’ and ‘imagination’, for which I refer to Ref . [5]. 3. Subjective Probability, Objective Probability, Physic al Probability To those who insist on objective probabilities I like to pose practical questions, such as how they would evaluate probability in speciﬁc cases , instead of letting them pursue mathematical games. Then it becomes clear that, at most, probability evaluations can be intersubjective, if we all share the same education and the same real or conventional state of information. The probabi lity that a molecule ofN2at a certain temperature has a velocity in a certain range seems objective: take the Maxwell velocity p.d.f., make an integral and get a n umber, say p= 0.23184 . . .This mathematical game gets immediately complicated if one thinks about a real vessel, containing real gas, and the molecule ve locity measured in a real experiment. The precise ‘objective’ number obtained from t he above integral might no longer correspond to our conﬁdence that the velocity is re ally in that interval. The idealized “physical probability” pcan easily be a misleading “metaphysical” concept which does not correspond to the conﬁdence of real si tuations. In most cases, in fact, pis a number that one gets from a model, or a free parameter of a model. Calling Etherealevent and P(E) the probability we attribute to it, the idealized situation corresponds to the following conditio nal probability: P(E\|Model →p) =p . (1) But, indeed, our conﬁdence on Erelies on our conﬁdence on the model: P(E\|I) =/summationdisplay ModelsP(E\|I,Model →p)·P(Model →p\|I), (2)4 G. D’AGOSTINI where Istands for a background state of information which is usuall y implicit in all probability assessments. Describing our uncertainty o n the parameter pby a p.d.f. f(p) (continuity is assumed for simplicity), the above formula can be turned into P(E\|I) =/integraldisplay1 0P(E\|p, I)·f(p\|I)dp . (3) The results of Eqs. (2) and (3) really express the meaning of p robability, describing our beliefs, and upon which (virtual) bets can be set (‘virtu al’ because it is well known that real bets are delicate decision problems of which beliefs are only one of the ingredients). For those who still insist that probability is a property of t he world, I like to give the following example, readapted from Ref. [10]. Six ex ternally indistinguish- able boxes each contain ﬁve balls, but with diﬀering numbers of black and white balls (see Ref. [6] for details and for a short introduction o f Bayesian inference based on this example). One box is chosen at random. What will be its white ball content? If we extract a ball, what is the probability that it will be white? Then a ball is extracted and turns out to be white. The ball is reintr oduced into the box, and the above two questions are asked again. As a simple appli cation of Bayesian inference, the probability of extracting a white ball in the second extraction be- comes P(E2=W) = 73%, while it wasP(E1=W) = 50% for the ﬁrst extraction. One does not need to be a Bayesian to solve this simple text boo k example, and everybody will agree on the two values of probability (we hav e got “objective” results, so to say). But it is easy to realize that these proba bilities do not rep- resent a ‘physical property of the box’, but rather a ‘state o f our mind’, which changes as the extractions proceed. In particular, ‘measur ing’, or ‘verifying’, that P(E2=W) = 73% using the relative frequency makes no sense. We could i magine a large number of extractions. It is easy to understand, give n our prior knowledge of the box contents, that the relative frequencies “will ten d” (in a probabilistic sense) to ≈20%,≈40%,≈60%,≈80%, or ≈100%, but ‘never’ 73%1. This certainly appears to be a paradox to those who agree that P(E2=W) = 73% is the ‘correct’ probability, but still maintain that proba bility as degree of belief is a useless concept. In this simple case the six a priori prob abilities pi=i/5, withi= 0,1, . . .,5, can be seen as the possible “physical probabilities”, but the “real” probability which determines our conﬁdence on the ou tcome is given by a discretized version of Eq. (3), with f(pi) changing from one extraction to the next. I imagine that at this point some readers might react by sayin g that the above example proves that only the frequentistic deﬁnition of pro bability is sensible, be- cause the relative frequency will tend for n→ ∞ to the ‘physical probability’, identiﬁed in this case by the white ball ratio in the box. But t his reaction is quite na¨ ıve. First, a deﬁnition valid for n→ ∞ is of little use for practical applications (“In the long run we are all dead”[11]). Second, it is easy to s how [6] that, for the cases in which the ‘physical probability’ can be checked and the number of ex- traction is ﬁnite, though large, the convergence behaviour of the frequency based 1Does this violate Bernoulli’s theorem? I leave the solution to this apparent paradox as amus- ing problem to the reader.COMMENTS ON SUBJECTIVE PROBABILITY 5 evaluation is far poorer than the Bayesian solution and can a lso be in paradoxical contradiction with the available status of information. Mo reover, only the Bayesian theory answers consistently, in unambiguous probabilisti c terms, the legitimate question “what is the box content?”, since the very concept o f probability of hy- potheses is banished in the frequentistic approach. Simila rly, only in the Bayesian approach does it make sense to express in a logical, consiste nt way the conﬁdence on the diﬀerent causes of observed events and on the possible values of physics quantities (which are unobserved entities). To state that “ in high energy physics, where experiments are repeatable (at least in principle) th e deﬁnition of probabil- ity normally used”[12] is to ignore the fact that the purpose of experiments is not to predict which electronic signal will come out next from th e detector (following the analogy of the six box example), but rather to narrow the r ange in which we have high conﬁdence that the physics quantities lie.2 4. Observed Frequencies, Expected Frequencies, Frequenti stic Approach and Quantum Mechanics Many scientists think they are frequentists because they ar e used to assessing their beliefs in terms of expected frequencies, without being awa re of the implications for a sane person of sticking strictly to frequentistic ideas. C ertainly, past frequencies can be a part of the information upon which probabilities can be assessed [6,9]. Similarly, probability theory teaches us how to predict fut ure frequencies from the assessment of beliefs, under well deﬁned conditions. But id entifying probability with frequency is like confusing a table with the English wor d ‘table’. This confu- sion leads some authors, because they lack other arguments t o save the manifestly sinking boat of the frequentistic collection of adhoc-eries , to argue3that “probabil- ity in quantum mechanics is frequentistic probability, and is deﬁned as long-term frequency. Bayesians will have to explain how they handle th at problem, and they are warned in advance.”[14] Probability deals with the belief that an event may happen, g iven a particular state of information. It does not matter if the fundamental l aws are ‘intrinsically probabilistic’, or if it is just a limitation of our present i gnorance. The impact on our minds remains the same. If we think of two possible events resulting from a quantum mechanics experiment, and say (after computing no m atter how compli- cated calculations to also take into account detector eﬀect s) that P(E1)≫P(E2), this simply means that we feel more conﬁdent inE1than in E2, or that we will be surprised if E2happens instead of E1. If we have the opportunity to repeat the experiment we believe that events of the ‘class’ E1will happen more frequently 2To be more rigorous, simple laboratory experiments can be pe rformed in conditions of re- peatability[13], but thinking of repeating very complex pa rticle physics experiments run for a decade make no sense (even in principle!). Perhaps the remar k “in principle” in the above quo- tation from Ref. [12] is to justify Monte Carlo simulation of the experiments. But one has to be aware that a Monte Carlo program is nothing but a collectio n of our best beliefs about the behaviour of the studied reaction, background reactions an d apparatus. 3A similar desperate attempt is try to throw a bad shadow over B ayesian theory, saying that in this theory “frequency and probability are completely di sconnected”[14], using as argument an ambiguous sentence picked up from the large Bayesian litera ture, and severed from its context.6 G. D’AGOSTINI than events of class E2. This is what we ﬁnd carefully reviewing the relevant liter- ature and discussing with theorists: probability is ‘proba bility’, although it might be expressed in terms of expected frequencies, as discussed above. Take for exam- ple Hawking’s A Brief History of Time ,[15] (which a statistician[16] said should be called ‘a brief history of beliefs’, so frequently do the w ords belief, believe and synonyms appear in it). For example: “In general, quantum me chanics does not predict a single deﬁnite result for an observation. Instead , it predicts a number of diﬀerent possible outcomes and tells us how likely each of th ese is” [15]. Looking further to the past, it is worth noting the concept of “degree of truth” introduced by von Weizs¨ acker, as reported by Heisenberg [17]. It is diﬃ cult to ﬁnd any diﬀer- ence between this concept and the usual degree of belief, esp ecially because both Heisenberg and von Weizs¨ acker were fully aware that “natur e is earlier than man, but man is earlier than natural science” [17], in the sense th at science is done by our brains, mediated by our senses. It is true that, reading s ome text books on quantum mechanics one gets the idea that “probability is fre quentistic probabil- ity”[14], but one should remember the remarks at the beginni ng of this section, and the fact that many authors have used, uncritically, the d ominant ideas on probability in the past decades. But some authors also try to account for proba- bility of single events, instead of ‘repeated events’, and have to admit that this is possible if probability is meant as degree of belief. In conclusion, invoking intriguing fundamental aspects of quantum mechanics in the discussion of inferential frameworks shows little aw areness of the real issues involved in the two classes of problems. First, the debate ab out the interpretations of quantum mechanics is far from being settled [18]. Second, as far as ‘natural science’ is concerned, it doesn’t really matter if nature is deeply deterministic or probabilistic, as eloquently said by Hume: “Though there is no such thing as Chance in the world; our ignorance of the real cause of any eve nt has the same inﬂuence on the understanding, and begets a like species of b elief or opinion.”[19] 5. Who is Afraid of Subjective Bayesian Theory? “It is curious that, even when diﬀerent workers are in substa ntially complete agree- ment on what calculations should be done, they may have radic ally diﬀerent views as to what we are actually doing and why we are doing it.” [20] I t is indeed surprising that strong criticisms of subjective probabili ty come from people who essentially agree that probability represents “our degree of conﬁdence” [21] and that Bayes’ theorem is the proper inferential tool. I would l ike to comment here on criticisms (and invitations to convert.. . ) which I have r eceived from objective Bayesian friends and colleagues, and which can be traced bac k essentially to the same source. [20] The main issue in the debate is the choice of theprior to enter in the Bayesian inference. I prefer subjective priors becau se they seem to me to correspond more closely to the spirit of the Bayesian theory and the results of the methods based on them are more reliable and never paradoxica l [5]. Nevertheless, I agree, in principle, that a “concept of a ‘minimal informat ive’ prior speciﬁcation – appropriately deﬁned!”[22] can be useful in particular ap plications. The prob- lem is that those who are not fully aware of the intentions and limits of the soCOMMENTS ON SUBJECTIVE PROBABILITY 7 called reference priors tend to perceive the Bayesian approach as dogmatic. Let us analyse, then, some of the criticisms. - “Subjective Bayesians have settled into a position interm ediate between or- thodox statistics and the theory expounded here.”[20] I thi nk exactly the opposite. Now, it is obvious that frequentistic methods are conceptually a mess, a collection of arbitrary prescriptions. But those wh o stick too strictly to the theory expounded in Ref. [20] tend to give up the real (u navoidably subjective!) knowledge of the problem in favour of mathemat ical convenience or blindly following the stance taken by the leading ﬁgures i n their school of thought. This is exactly what happens with practitioners us ing blessed ‘ob- jective’ frequentistic ‘procedures’ (for example, see Ref . [5] for a discussion on the misuse of Jeﬀreys’ priors). - “While perceiving that probabilities cannot represent on ly frequencies, they [subjective Bayesians] still regard sampling probabiliti es as representing fre- quencies of ‘random variables”’.[20] The name‘random vari able’ is avoided by the most authoritative subjective Bayesians [7] and the t erms ‘uncertain (aleatoric) numbers’ and ‘aleatoric vectors’ (form multi- dimensional cases) are currently used. Even the idea of ‘repeated events’ is rej ected [7], as ev- ery event is unique, though one might think of classes of anal ogous events to which we can attribute the same conditional probability, but these events are usually stochastically dependent (like the outcomes black and white in the six box example of Ref. [6]). In this way the ideas of uncertainty and probabil- ity are completely disconnected from that of randomness ` a la von Mises. [23] Nevertheless, I admit that there are authors, including mys elf [4], who mix the terms ‘uncertain numbers’ and ‘random variables’, to ma ke life easier for those who are not accustomed to the concept of uncertain numb ers, since the formal properties (like p.d.f., expected value, variance, etc) are the same for the two objects. - “Subjective Bayesians face an awkward ambiguity at the beg inning of a prob- lem, when one assigns prior probabilities. If these represe nt merely prior opin- ions, then they are basically arbitrary and undeﬁned”. [20] Here the confusion between subjective and arbitrary, discussed above, is obvi ous. - “It seems that only private introspection could assign the m and diﬀerent peo- ple will make diﬀerent assignments”. No knowledge, no scien ce, and there- fore no probability, is conceivable if there is no brain to an alyse the external world. Fortunately there are no two identical brains (yet), and therefore no two identical states of knowledge are conceivable, though i ntersubjectivity can be achieved in many cases. - “Our goal is that inferences are to be completely ‘objectiv e’ in the sense that two persons with the prior information must assign the same p rior probabil- ity.” [20] This is a very na¨ ıve idealistic statement of litt le practical relevance. - “The natural starting point in translating a number of piec es of prior infor- mation is the state of complete ignorance.” [20] When should we deﬁne the state of complete ignorance? At conception or at birth? How m uch is learned and how much was already coded in the DNA?8 G. D’AGOSTINI 6. Conclusions To conclude, I think that none of the above criticisms is real ly justiﬁed. As for criticisms put forward by frequentists or by self-designed frequentistic practition- ers (who are more Bayesian than they think they are [3]) there is little more to comment in the context of this workshop. I am much more intere sted in making some ﬁnal comments addressed to fellow Bayesians who do not s hare some of the ideas expounded here. I think that users and promoters of Bay esian methods of the diﬀerent schools should make an eﬀort to smooth the tones of the debate, because the points we have in common are without doubt many mo re, and more relevant, than those on which there is disagreement. Workin g on similar problems and exchanging ideas will certainly help us to understand ea ch other. There is no denying that Maximum Entropy methods are very useful in solv ing many compli- cated practical problems, as this successful series of work shops has demonstrated. But I don’t see any real contradiction with coherence: I am re ady to take seri- ously the result of any method, if the person responsible for the result is honest and is ready to make any combination of reversible bets based on the declared probabilities. References 1. C. Giunti, Proc. Workshop on Conﬁdence Limits, CERN, Gene va, Switzerland, January 2000. CERN Report 2000–005, May 2000, pp. 63–72, e-print arX iv: hep-ex/0002042. 2. R. Feynman, The character of physical law , MIT Press, 1967. 3. G. D’Agostini, Proc. of the XVIII International Workshop on Maximum Entropy and Bayesian Methods, Garching (Germany), July 1998. (Kluwer A cademic Publishers, Dor- drecht, 1999), pp. 157–170, e-print arXiv: physics/981104 6. 4. G. D’Agostini, CERN Report 99–03, July 1999, electronic v ersion available at the author’s URL. 5. G. D’Agostini, Rev. R. Acad. Cienc. Exact. Fis. Nat., Vol. 93, Nr. 3, 1999, pp. 311–319, e-print arXiv: physics/9906048. 6. G. D’Agostini, Am. J. Phys. 67(1999) 1260-1268, e-print arXiv: physics/9908014. 7. B. de Finetti, Theory of Probability (J. Wiley & Sons, 1974). 8. D.S. Sivia, Data analysis – a Bayesian tutorial , Oxford University Press, 1997. 9. G. D’Agostini, Proc. Workshop on Conﬁdence Limits, CERN, Geneva, Switzerland, January 2000. CERN Report 2000–005, May 2000, pp. 3–23. 10. R. Scozzafava, Pure Math. and Appl., Series C 2, 1991, pp. 223-235. 11. J.M. Keynes, A tract on Monetary Reform (Macmillan, London, 1923). 12. Particle Data Group, D.E. Groom et al., Eur. Phys. J. C15(2000) 1 (Section 28). 13. DIN Deutsches Institut f¨ ur Normung, “Grundbegriﬀe der Messtechnick – Behandlung von Unsicheratine bei der Auswertung von Messungen” , (DIN 1319 Teile 1–4), Beuth Verlag GmBH, Berlin, Germany, 1985. ISO International Organization for Standardization, International vocabulary of basic and genaral terms in metrology ,” Geneva, Switzerland, 1993. 14. F. James, Proc. Workshop on Conﬁdence Limits, CERN, Gene va, Switzerland, January 2000. CERN Report 2000–005, May 2000, pp. 1–2. 15. S.W. Hawking, A Brief history of time , Bantam Doubleday Dell, 1988. 16. D.A. Berry, Talk at the Annual Meeting of American Statis tical Association, Chicago, August 1996. 17. W. Heisenberg, Physics and philosophy. The revolution in modern science , 1958 (Harper & Row Publishers, New York, 1962). 18. See e.g. G. ’t Hooft, In search of the ultimate building blocks , 1992 (Cambridge University Press, 1996).COMMENTS ON SUBJECTIVE PROBABILITY 9 19. D. Hume, Enquiry concerning human understanding ”, 1748. 20. E.T. Jaynes, Probability Theory: The logic of Science , Chapter 12 on Ignorance priors and transformation groups , (book available at http://bayes.wustl.edu/etj/prob.ht ml and http://omega.albany.edu:8008/JaynesBook, but Chapter 1 2 is missing since beginning of 2000). 21. H. Jeﬀreys, Theory of probability , 1939 (Clarendon Press, Oxford, 1998). 22. J.M. Bernardo and A.F.M. Smith, Bayesian Theory , John Wiley & Sons, 1994. 23. R. von Mises, Probability, Statistics and Truth , 1928, (George Allen & Unwin, 1957), second edition.
arXiv:physics/0010065v1 [physics.atom-ph] 26 Oct 2000An optical trap for collisional studies on cold fermionic po tassium G. Roati‡, W. Jastrzebski∗, A. Simoni, G. Modugno†, and M. Inguscio INFM, LENS, and Dipartimento di Fisica, Universit` a di Fire nze, Largo E. Fermi 2, I-50125 Firenze, Italy. ‡Dipartimento di Fisica, Universit` a di Trento, I-38050 Pov o (Tn), Italy (January 10, 2014) We report on trapping of fermionic40K atoms in a red- detuned standing-wave optical trap, loaded from a magneto- optical trap. Typically, 106atoms are loaded at a density of 1012cm−3and a temperature of 65 µK, and trapped for more than 1 s. The optical trap appears to be the proper environment for performing collisional measurements on th e cold atomic sample. In particular we measure the elastic col - lisional rate by detecting the rethermalization following an intentional parametric heating of the atomic sample. We als o measure the inelastic two-body collisional rates for unpol ar- ized atoms in the ground hyperﬁne states, through detection of trap losses. 32.80.Pj, 34.50.-s, 34.50.Pi, 05.30.Fk I. INTRODUCTION The fermionic isotope of potassium40K is among the most interesting atomic species for the experimental study of quantum degenerate gases, as early discussed in [1]. Recently, the possibility of cooling a sample of fermionic40K atoms to the regime of quantum degener- acy has been demonstrated [2], by means of evaporative cooling in a magnetic trap. This is the ﬁrst step towards further studies on a dilute Fermi system, including the possibility of Cooper pairing. As for this prospect, the presence of Feshbach resonances in the scattering length of selected Zeeman substates of40K [3], induced by a magnetic ﬁeld, could be exploited to bring the required temperatures for a BCS-like phase transition to a range experimentally achievable. Optical traps appear to be very useful tools for inves- tigating such collisional properties of atoms at low tem- perature, since they can hold high-density samples in any spin state, and an homogeneous magnetic ﬁeld can be ap- plied without modiﬁcation of the trapping potential [4]. For the particular case of40K, the most interesting Fes- hbach resonance has been predicted for two spin states which are not magnetically trappable, and therefore an optical trap would be necessary [3]. In this paper we report for the ﬁrst time on optical ∗also Institute of Physics, Polish Academy of Science, 02 668 Waszawa, Poland †modugno@lens.uniﬁ.it; http://www.lens.uniﬁ.ittrapping of a sample of40K, precooled in a magneto- optical trap to sub-Doppler temperatures [5]. By using a compression phase followed by optical molasses cooling we obtain trapped samples at relatively low temperature (65µK) and high density (1012cm−3). In this tempera- ture and density regime, our optical trap appears to be an interesting environment to investigate the collisional properties of40K, possibly including the predicted Fesh- bach resonances. We are indeed able to perform elastic collisional measurements on unpolarized sample of atoms in the ground F=9/2 state, and inelastic measurements on samples in the F=9/2 and F=7/2 hyperﬁne substates, which are both held in the optical trap. A comparison with theoretical predictions is made in both cases, and a good agreement is found. II. LOADING AND CHARACTERIZATION OF THE OPTICAL TRAP The optical trap is realized with linearly polarized light from a single-mode Ti:Sa laser, detuned to the red of both the D 1and D 2transitions of potassium, respectively at 769.9 nm, and 766.7 nm. The laser radiation is arranged in a vertical standing-wave conﬁguration, by retroreﬂect- ing the beam, achieving a 1D optical lattice with the strongest conﬁnement against gravity. The laser beam is weakly focused within a two-lens telescope to a waist-size w0=90µm, with a Rayleigh length zR=3 cm; the eﬀective running wave power at the waist position is P=600 mW. A schematic of the experimental apparatus is shown in Fig. 1. The use of a tunable Ti:Sa laser allows us to change in a continuous way the trap detuning from a few nanome- ters to about 40 nm, allowing a variety of trap condi- tions. For a typical wavelength of the trapping laser λt=787 nm, the calculated trap depth [6] for the ground state is U0=300 µK (6.2 MHz) and the scattering rate is Γsc=30 s−1. Since the trapping light is very far detuned from any transition starting from the excited 4P states, the light shift for both the excited states is positive, and it is approximately half of the trap depth. The vibrational frequencies of each lattice site are given, in the harmonic approximation, by ωA= 2π/radicalBig 2U0/Mλ2 t (2.1) ωR=/radicalBig 4U0/Mw2 0 1in the axial and radial directions, respectively. Here Mis the mass of potassium. The expected values of the radial and axial trap frequencies for the conditions above are respectively 2 π×870 Hz and 2 π×450 kHz. The power of the trapping laser can be controlled with a fast AOM (with a fall time of 300 ns); a polarizer cube after the AOM provides the required linear polarization. The pump laser is a low-noise doubled Nd:YVO, and no active stabilization or tuning of the Ti:Sa laser frequency is used, to avoid amplitude noise which can possibly heat up the trapped atoms [7]. We load the optical trap directly from a vapor cell magneto-optical trap (MOT) of40K as described in [5]. We note the typical cooling and repumping light inten- sities of 36 mW/cm2and 4 mW/cm2respectively, and the detuning of the cooling light of -20 MHz from the cy- cling F=9/2 →F′=11/2 transition (the natural linewidth and saturation intensities of40K are Γ=2 π×6.2 MHz and Is=1.8 mW/cm2). In the MOT we collect in 4 s about 5×107atoms at a peak density of 1010cm−3and at a temperature of 60 µK. The dipole trap beam, which is passing approximately through the center of the MOT, is left on during the MOT loading, with no detectable eﬀect on the atom number or temperature. To load the optical lattice from the MOT we use the following procedure: the trapped atoms are compressed for 30 ms by ramping the quadrupole gradient of the MOT from the normal value of 18 Gauss/cm up to 44 Gauss/cm, while the intensity of the cooling and re- pumping light are abruptly reduced to 20 mW/cm2and 0.06 mW/cm2respectively. Moreover, the detuning of the cooling light is reduced to -6 MHz to improve the compression eﬀect. The MOT quadrupole ﬁeld is then switched oﬀ in 1 ms, and the detuning of the cooling light is increased to -20 MHz for other 2 ms, providing the usual optical molasses cooling. Then the MOT beams are turned oﬀ by switching oﬀ the AOMs in a few µs. The repumping beams are turned oﬀ 1 ms later than the cooling ones, to pump the atoms in the lower hyperﬁne ground state, F=9/2. For a complete shielding of the optical trap from the resonant stray light of the MOT beams, which could cause undesired optical excitation and pumping, we also block the light with a mechanical shutter within 3 ms from the switch-oﬀ of the AOMs. Us- ing this procedure, about 5% of the atoms in the MOT can be loaded in the optical trap, corresponding to a few 106atoms. The atoms in the optical trap are probed by turning on again the MOT beams at half the normal intensity and in resonance with the cycling transition, while the repumper is in resonance with the F=7/2 →F′=9/2. The ﬂuorescence is detected by a photomultiplier with the help of a lens and of an optical ﬁber with a core diameter of approximately 0.5 mm, which provides good spatial selectivity to discard the light scattered by the windows of the vacuum cell, and allows detection of as few as 103atoms. A typical measurement of the lifetime of the optical trap is shown in Fig. 2. After about 0.3 sof storage, the decay curve of the atoms pumped in the ground F=9/2 state is a single exponential, with a 1/e lifetime of τ=1.4(3) s. We expect the main source of loss in this regime to be collisions with background gas, the trap lifetime being roughly one third of the MOT loading time. The faster decay at shorter times could possibly be due to losses connected to an evaporative cooling process, which we will consider below. Alternatively, the density and the absolute number of trapped atoms can be measured by absorption imaging, using a pulse of light derived from the MOT laser lasting 200µs, at half the saturation intensity and in resonance with the cycling transition. To compensate for the light shift due to the optical trap, the frequency is blue shifted by approximately 1.5 U0from the unperturbed resonance. The beam is passing through the optical trap in the hor- izontal plane and is imaged on a CCD camera with a two-lens system. The distribution of atoms in the op- tical trap, as shown in Fig. 3, is gaussian in the radial direction, with a typical FWHM of 70 µm. Also the axial distribution is generally gaussian, and the typical FWHM of 500 µm indicates that approximately 1200 lat- tice sites, spaced by 395 nm, are occupied. The mean number density is determined by averaging the optical density over the trap section and scaling by the ratio of the lattice spacing λtto the the mean axial extension dA of an individual lattice site. Such mean axial extension is estimated from the temperature and axial trap frequency as dA=ω−1 A/radicalbig 2πkBT/M , (2.2) and is of the order of 100 nm. The typical mean density of atoms after 100 ms of storage is 7 ×1011cm−3, and de- cays proportionally to the number of atoms for increasing storage times. The radial temperature is measured by detecting the radial extension of the atomic cloud after 1 ms of expan- sion, following a sudden switch-oﬀ of the optical trap. Al- ternatively it is possible to estimate the temperature di- rectly from the radial extensions of the atoms in the trap; this estimation is aﬀected by a larger uncertainty, since it relies on the knowledge of the radial trap frequency. The axial temperature is extracted from the temporal width of a time-of-ﬂight (TOF) signal, as detected by the ab- sorption of a 1-cm wide sheet of light placed 1.5 cm be- low the trapping region, in resonance with the cycling transition. The TOF beam, generated by an extended- cavity diode laser, is doubly passing through the vacuum cell, and the absorption signal is detected with a fre- quency modulation technique. The typical TOF signal, as shown in Fig. 4, indicates the presence of two diﬀer- ent velocity distributions, corresponding to temperature s around 70 µK for the broad peak and around 4 µK for the narrow one. We think that the colder component is due to a small fraction of the trapped atoms which are adiabatically cooled during the release from the optical trap. Indeed, the AOM which controls the optical trap 2is capable of cutting only 95% of the light in less than 300 ns, while the residual tail is extinguished in about 5µs. Since the latter switch-oﬀ time is longer than ω−1 A, the atoms in the bottom of the trap are further cooled down. We have veriﬁed this conjecture by intentionally slowing the main switch-oﬀ time of the AOM down to 5µs, and observing an adiabatic cooling of hotter com- ponent to 8 µK. We expect to be possible to eliminate such residual adiabatic cooling eﬀect by using the AOM in a double-pass conﬁguration, but since this would re- duce by about 30% the available laser power, we prefer to ﬁt the TOF signal with a two-component curve to ac- curately extract the temperature of the majority of the atoms. The temperature measurement described above can be performed only after a minimum storage time of 80 ms, to allow for the separation of the TOF signal due to the atoms in the optical trap from the signal due to the atoms in the MOT which have not been loaded in the trap. We have veriﬁed that after this time interval the axial and radial temperature are identical within the uncertainty. To look for any relationship between the temperatures of the atoms in the MOT and of those in the optical trap, we have performed a series of temperature measurements after 100 ms of storage in optical traps of diﬀerent depths. The result is shown in Fig. 5: the temperature appears to scale almost linearly with the trap depth, and for deep traps we measure a ratio U0/kBT∼4.5. Note that for very deep traps such a temperature can be much higher than the MOT one, which is about 70 µK after the com- pression phase. This behavior, which has been reported also in [8], seems to indicate that the temperature of the atoms in the optical trap is determined by the process of loading and optical trapping itself, and not by the origi- nal temperature in the MOT. We note also that the number of atoms loaded in the trap is observed to decrease almost linearly with the trap depth, with the density decreasing even faster due to the accompanying reduction in the spring constant of the trap. As a result, the optimal density conditions for per- forming collisional measurements are found only at large trap depths, where the temperature of the trapped atoms is comparable to that of the MOT. A detailed descrip- tion of these observations goes beyond the scope of this paper, and will therefore be presented elsewhere. We have also observed that the temperature of the atoms in the optical trap remains almost constant with time, although the calculated scattering rate would lead to a constant heating with a rate as large as 100 µKs−1for the tightest trap investigated. Also this result is suggest - ing the presence of an evaporative cooling eﬀect, which counteracts the heating due to photon scattering. The eﬀective trap depth and spring constants are mea- sured with the help of a parametric heating technique [9]. If the power of the trap light is sinusoidally modulated by the AOM, a strong heating of the atoms, accompanied by trap losses, arises whenever the modulation frequency matches twice the trap oscillation frequency or a subhar-monic thereof. By monitoring the trap population after ∼100 ms of parametric heating it is therefore possible to measure both radial and axial trap frequency with an un- certainty of the order of 10%. A typical measurement of the trap frequencies is shown in Fig. 6, for a trap depth of 180 µK. Note the presence of a strong resonance also at four times the axial frequency, which can be explained only by considering the anharmonicity of the optical trap in the axial direction, due to its sinusoidal shape [10]. III. COLLISIONAL MEASUREMENTS A. Elastic collisions To characterize the elastic collisions we use a technique which has already proved useful in this kind of optical trap [4], which is based on the detection of the rether- malization of the trapped atoms after the excitation of one of the degrees of freedom. In particular, we paramet- rically excite the axial vibrational mode of the trap by applying a 2% modulation to the laser power for about 5 ms, and measure the subsequent rapid increase of the axial temperature, followed by an exponential decay due to thermalization of the axial mode with the radial ones, as mediated by elastic collisions. In order to excite uni- formly the atoms distributed over the vibrational levels in the anharmonic trap, we sweep the modulation fre- quency roughly between ωAand 2 ωAduring the 5 ms heating phase. Indeed, if only a single frequency is used, we observe a large loss of atoms from the trap, with no larger increase of the temperature, as if only the popula- tion of a few levels were excited. Moreover, modulation amplitudes larger than 2% cause quite a large loss of atoms, which are detected by the TOF beam together with those released from the trap, therefore altering the temperature measurements. The decay curve for the con- ditions T= 65µK,n=7.5×1011cm−3, is shown in Fig. 7; each data point is the average of two series of about 20 TOF measurements, and the original temperature was 67µK. The ﬁt to an exponential decay yelds τth=10(2) s. We have checked that the decay time is halved, within the uncertainty, when the atomic density in the optical trap is halved, to verify that the rethermalization is ac- tually mediated by elastic collisions, and not simply by cross-dimensional anharmonic eﬀects. We compare the observed collision rates with the re- sults of a full close-coupling numerical calculation with the standard Hamiltonian including hyperﬁne structure. The model is parametrized by the long range dispersion coeﬃcients of the atom-atom interaction potential and by single channel singlet and triplet scattering lengths. We adopt the high precision value C6=3897au from Ref. [11] for the van der Waals coeﬃcient, and choose the range of possible scattering lengths by scaling to40K the results obtained for39K from an analysis of the 0− g[12] molecular state photoassociation spectroscopy. Similar 3but less restricted values are also presented in [13] from an analysis of the 1 gspectrum. We obtain in this way the limits 100 < a s<110a 0and 150 < a t<250a 0. It is consistent with the experimental errors in the present measurements to ﬁx the singlet scattering length to say as= 105 a0and leave only the triplet as a free parameter. From our calculation and from the results of an anal- ogous experiment performed in a magnetic trap [14], we expect that both s- and p-wave collisions are contribut- ing to the cross sections at T=65µK for40K. Indeed, a p-wave shape resonance close to threshold enhances the p-wave contribution at low temperatures [14,15]. Since our sample is unpolarized we calculate spherical sandp cross sections σ, and estimate the thermalization rate as 1 τth=/parenleftbigσs αs+σp αp/parenrightbig nv (3.1) where αis the average number of collisions necessary for the rethermalization, nis the atomic density, and vis the rms relative velocity between two colliding atoms. Note that we have implicitly assumed an identical dis- tribution of all possible spin states in F=9/2 over the optical trap, since the trap potential is independent of the magnetic moment of the atoms. We adopt for the parameters αthe values αs= 2.5 and αp= 4.1 quoted in [14]. We ﬁnd the best agreement with the measured rate at the lowest allowed value of the triplet scattering length (at=150a0),i.e.when the p-wave resonance is closer to threshold. In this situation the average thermal rates areσsv=6.2×10−11cm3/s and σpv=1.75×10−10cm3/s and the characteristic time for rethermalization for the given density in the trap is then about τth=20 ms. It is interesting to notice that a comparison between the pho- toassociation results in [12] and [13] seems to privilege lower values of atin agreement with what we ﬁnd here. From the calculated elastic collision rate 1/ τel≡nσv, withσthe total cross section, is also possible to estimate the typical speed of the thermalization due to evaporative cooling processes. Using the formalism presented in [16], the ratio of the evaporation to the elastic collision time constant is τev/τel≈√ 2eηη−1(3.2) where η=U0/kBT. From the measured value of η∼4.5 (see Fig. 5), the timescale for the thermalization due to evaporation is therefore expected to be a few hun- dreds of ms. Although this process can be neglected with regard to the cross-dimensional collisional processes re- ported above, which are proceeding at a much faster rate, we speculate that it could play an important role in the observed long term stability of the temperature of the atoms in the optical trap. B. Inelastic collisions The inelastic two-body collisions between atoms in the F=7/2 state, and between atoms in the F=9/2and F=7/2 states, can be detected in a relatively easy way through trap losses, since the hyperﬁne energy of 1.285 GHz released under the transition F=7/2 →9/2 is much larger than the trap depth. We can vary the pop- ulation of the upper hyperﬁne state F=7/2 by means of an optical pumping beam, derived from the MOT laser, in resonance with the F=9/2 →F′=7/2 transition, and controllable in intensity with an AOM. By shining the beam on the trapped atoms with an intensity of about 10µW/cm2, and for a time interval variable between 0.1 and 2 ms, we transfer up to 95% of the atoms from the ground to the upper state. The relative populations of the two hyperﬁne states after the pumping pulse are mea- sured by monitoring the TOF signal, to which only the atoms in F=9/2 contribute. We have veriﬁed that the typical time scale for hyperﬁne optical pumping due to inelastic Raman scattering of the trap photons is much longer than 1 s, and therefore such eﬀect is negligible. The optical pumping pulse is applied after about 100 ms from the loading of the dipole trap, to let the trapped atoms equilibrate, and then the decay of the number of atoms in the trap is measured with the usual procedure. In Fig. 8 the decay of the trapped atoms is shown for the cases of 95% and 85% of the total population pumped into the F=7/2 state. In this par- ticular measurement the temperature of the atoms was T=50µK. We ﬁt the experimental data by assuming a constant volume regime, which in an harmonic trap is equivalent to the assumption of a constant temperature. In this regime, the density is proportional to the num- ber of atoms, and the decay curves are the solutions of the coupled diﬀerential equations for the densities of the populations in both hyperﬁne states ˙n1=−γn1−G1,2n1n2 (3.3) ˙n2=−γn1−2G2,2n2 2−G1,2n1n2. (3.4) Here the suﬃxes 1 and 2 refer to the levels F=9/2 and F=7/2, respectively, γis the rate of linear trap loss, and Gi,jis the spherical rate of two-body inelastic collisions between atoms in states iandjat temperature T. To account for the loss of both atoms in the pair for each collision event, factors of 2 are added for collisions of atoms belonging to the same hyperﬁne level. The constant γcan be obtained from the tails of the decay curve, where the binary collision rate is negligible. The ﬁt yields γ=(0.90±0.05) s−1. We keep this value ﬁxed for the ﬁts where we have 95%, 83% and 66% of the atoms in F=7/2. We assume a 30% uncertainty in the atomic density and obtain best estimates of the decay rates from the individual curves. Next we combine the three measurements to obtain G7/2,7/2=3.2×10−11cm3/s andG9/2,7/2=3.8×10−12cm3/s with an uncertainty of the order of 50%. From numerical calculations we ﬁnd thermally averaged spherical rates of G7/2,7/2=4.4×10−11cm3/s andG9/2,7/2=3.5×10−12cm3/s for T=50µK, in agree- ment with the measured rate. We obtain these values at 4the lower edge a3=150a0of the initial conﬁdence range. The agreement holds until the value a3≃180a0, where the loss coeﬃcient G9/2,7/2drops to values of the order of G9/2,7/2≃10−12cm3/s. In all this range p-wave losses provide a signiﬁcant contribution to both G7/2,7/2and G9/2,7/2, and the agreement with the measured values represents therefore further evidence of the presence of a low energy p-wave resonance. It is interesting to no- tice that the rate coeﬃcient G(9/2,7/2) rapidly increases above a3=180a0, reaching values larger than 10−11cm3/s fora3>190a0, in disagreement with the measured value. This seem to rule out the higher range of a3values, as already indicated by the large observed rethermalization rate. We also note that inelastic collisions cannot be re- sponsible for the faster decay of the atom number at short trapping times (see Fig. 2), even assuming a residual 5% on the F=7/2 state after the normal loading of the optical trap, due to the relatively small value of G9/2,7/2. IV. CONCLUSIONS We have reported on loading of fermionic40K atoms in a 1D optical lattice from a single MOT. The 1.4-s lifetime of the optical trap is large enough to allow the observation of elastic and inelastic collisional processe s, due to the high density of the atomic sample in the trap. Indeed, the typical time scale for collisional processes is a few tens of ms, at a density of 7 ×1011cm−3and at a temperature of 65 µK. Furthermore, optical pumping and heating eﬀects arising from the scattering of trap photons do not seem to play a signiﬁcant role on such time scale. The experimental collisional rates for unpolarized sam- ples we determine are quite consistent with the theoreti- cal expectations. Further studies of inelastic and elastic collisions in spin-polarized samples in our optical trap could lead to a characterization of the collisional prop- erties of fermionic potassium at low temperature. As already noted, particularly interesting is the possibilit y of searching for the occurrence of s-wave Feshbach reso- nances in the collisions between Zeeman sublevels of the ground state. For such purpose, a temperature lower than the present 65 µK would be preferred, to reduce the p-wave contribution, and to avoid temperature broaden- ing of the resonance. Although the temperature can be lowered by loading the atoms in very shallow traps, the achieved density is not large enough to perform accurate collisional measurements, and therefore methods for re- ducing the temperature of a dense sample are needed. In particular, we note that the axial degree of freedom of our trap is in the Lamb-Dicke regime ( dA≪λ), allowing for the application of fast degenerate Raman sideband cooling [17], which would allow for an eﬃcient reduction of the temperature without atom losses. We thank D. Lau for his partecipation to the early phases of the experiment, and Laboratorio di Fisica dellaScuola Normale Superiore for loaning the Ti:Sa laser for the optical trap. We also acknowledge contributions by P. Hannaford and N. Poli, and useful discussions with V. Vuletic and F. S. Cataliotti. One of us (A. S.) gratefully thanks E. Tiesinga for helpful discussions, and the NIST group for the ground state collision code. This work was supported by the European Community Council (ECC) under the Contract HPRICT1999-00111, and by MURST under a PRIN Program (Progetto di Ricerca di Interesse Nazionale). W. J. was funded by the NATO-CNR Senior Guest Fellowship Programme 1998. [1] F. S. Cataliotti, E. A. Cornell, C. Fort, M. Inguscio, F. Marin, M. Prevedelli, L. Ricci, and G. M. Tino, Phys. Rev. A 57, 1136 (1998). [2] B. DeMarco and D. S. Jin, Science 258, 1703 (1999). [3] J. Bohn, Phys. Rev. A 61, 053409 (2000). [4] V. Vuletic, C. Ching, A.J. Kerman, and S. Chu, Phys. Rev. Lett. 82, 1406 (1999). [5] G. Modugno, C. Benko, P. Hannaford, G. Roati, and M. Inguscio, Phys. Rev. A 60, R3373 (1999). [6] See for example: R. Grimm, M. Weidem¨ uller, and Y. B. Ovchinikov, Adv. At. Mol. Opt. Phys. 4295 (2000). [7] For a discussion of the heating of the trapped atoms due to noise in the trap laser, see T. A. Savard, K. M. O’Hara, and J. E. Thomas Phys. Rev. A 56, R1095 (1997). [8] S. J. M. Kuppens, K. L. Corwin, K. W. Miller, T. E. Chupp, and C. E. Wieman, Phys. Rev. A 62013496-1 (2000). [9] S. Friebel, C. D’Andrea, J. Walz, M. Weitz, and T. W. H¨ ansch, Phys. Rev. A 57, R20 (1998). [10] For a discussion of the classical case of parametric hea ting of an anharmonic oscillator see L. D. Landau and E. M. Lifschitz, Mechanics (Pergamon, Oxford , 1976), pp. 80- 84. [11] A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb, Phys. Rev. Lett 82, 3589 (1999). [12] J. P. Burke,Jr., C. H. Greene , J. L. Bohn, H. Wang, P. L. Gould, and W. C. Stwalley, Phys. Rev. A 60, 4417 (1999). [13] C. J. Williams, E. Tiesinga, P. S. Julienne, H.Wang, W.C. Stwalley, and P. L. Gould, Phys. Rev. A 60, 4427 (1999). [14] B. DeMarco et al., Phys. Rev. Lett. 82, 4208 (1999). [15] J. Bohn et. al. Phys. Rev. A 593660 (1999) [16] W. Ketterle and N. J. van Druten, Adv. At. Mol. Opt. Phys.37181 (1996). [17] V. Vuletic, C. Chin, A. J. Kerman, and S. Chu, Phys. Rev. Lett. 81, 5768 (1998). 5PD/c108/c47/c52CCD AOM Trap□beamTOF□beamMOTPM FIG. 1. Schematics of the experimental apparatus. PD: photodiode; PM: photomultiplier. 0 0,51,01,52,02,5105106 Trapped atoms Storage time (s) FIG. 2. Decay of the number of atoms in the optical trap, in the ground state F=9/2. The ﬁt with an exponential decay yields a time constant of 1.4 s. FIG. 3. Image of the atoms in the optical trap acquired in absorption with a CCD camera.The radial (horizontal) FWHM is about 70 µm, while about 1200 individual traps are piled vertically within 500 µm. 0 20 40 60 80 10069 µK4 µKTOF signal Time (ms) FIG. 4. Typical TOF signal from the atoms in the opti- cal trap. The narrow peak is due to atoms which have been adiabatically cooled during the release from the trap. The continuos line is the best ﬁt with a two-component TOF dis- tribution, which gives the reported temperatures. 60100 200 300 400 500 600020406080100120140 Temperature (µK) Trap Depth (µK) FIG. 5. Temperature of the trapped atoms after 100 ms of storage as a function of the trap depth. 0,1 1 10000,00,20,40,60,81,01,2Trapped atoms fraction Modulation frequency (kHz) FIG. 6. Spectrum of the resonances in the parametric heat- ing of the atoms in the optical trap. The low-frequency resonance is at twice the radial trap frequency, while the high-frequency ones are at the fundamental, second and fourth harmonic of the axial trap frequency, respectively.01020304050606065707580859095100 Axial temperature (µK) Storage Time (ms) FIG. 7. Decay of the axial atomic temperature as a con- sequence of the rethermalization with the radial degrees of freedom of the trap, mediated by elastic collision. The best ﬁt to an exponential decay yields a time constant τ=10(2) ms. 0,0 0,5 1,0 1,5 2,0 2,50.11Total density (10 11 cm-3) Storage Time (s) FIG. 8. Decay of the number of atoms in the optical trap for diﬀerent populations of the upper hyperﬁne level F=7/2: open triangles 87%; solid triangles 95%. The lines represen t the best ﬁt with the solution of the coupled diﬀerential equa - tion described in the text. The third set of points (circles) shows for comparison the decay for the F=9/2 state. 7
arXiv:physics/0010066v1 [physics.comp-ph] 26 Oct 2000Optimization of ground and excited state wavefunctions and van der Waals clusters M. P. Nightingale and Vilen Melik-Alaverdian Department of Physics, University of Rhode Island, Kingsto n RI 02881, USA (September 24, 2013) A quantum Monte Carlo method is introduced to optimize excit ed state trial wave functions. The method is applied in a correlation function Monte Carlo c alculation to compute ground and excited state energies of bosonic van der Waals clusters of u pto seven particles. The calculations are performed using trial wavefunctions with general three -body correlations. PACS codes: 03.65 02.50.N, 02.70.L 36.40.M 34.30 Solving the Schr¨ odinger equation for systems in which the z ero-point energy is relatively large and gives rise to stron g anharmonicity poses a computational challenge, even for fe w-particle systems. With discrete variable representatio n methods (DVR)1one can obtain many states for systems with upto six vibratio nal degrees of freedom and intermediate zero-point energy. Projector Monte Carlo methods such as th e correlation function method (CFMC)2,3and the projector operator imaginary time spectral evolution (POI TSE)4method, are applicable to more degrees of freedom, but both methods are restricted to a smaller number of states . A diﬀerence between the correlation function and projector operator Monte Carlo methods is the way in which th ey extract energies from correlation functions, but both require ancillary quantities to boost the spectral wei ght of the excited states. The method presented in this Letter makes it possible to construct these quantities in a s ystematic and eﬃcient way, which has been used to enhance the CFMC method signiﬁcantly. In fact, without optimized wa vefunctions the CFMC methods does not produce any meaningful results for most of the clusters for which we repo rt results. We also expect that results of the POITSE method can be improved substantially, by using spectral enh ancement operators based on optimized wavefunctions. A variant of the method described here was applied previousl y to study critical dynamics of lattice systems.5,6 The application in this Letter is to bosonic van der Waals clu sters, which we treat in the position representation. We denote by Rthe Cartesian coordinates of a cluster consisting of Ncatoms. As a preliminary step in the parameter optimization, we generate a set of typically thousands of co nﬁgurations Rσ,σ= 1,...,Σ sampled from a guiding functionψ2 gwith relative probabilities ψg(Rσ)2≡w−2 σ; to ensure that the sample has suﬃcient overlap with all stat es in our computations, we choose a power pwith 2<∼p<∼3 andψp g=˜ψ(1), where ˜ψ(1)is the optimized ground state wavefunction, which is obtained after a few initial optimiz ation iterations for a simple ground state wave function. The trial functions are linear combinations of about a hundr ed elementary basis functions, each of which depends on non-linear optimization parameters. The trial function s are constructed one at a time from the ground state up, as follows. Suppose we ﬁx the non-linear parameters of the elementary ba sis functions, denoted by βi,i= 1,...,n , at initial values. First, we consider the ideal case in which these func tions span an n-dimensional invariant subspace of the Hamiltonian H, i.e., we assume that there exists an n×nmatrix Eso that Hβi(Rσ) =/summationdisplay jβj(Rσ)Eji. (1) In this ideal case ˜ψ(k)(R) =/summationdisplay iβi(R)d(k) i (2) is an eigenvector of Hwith an eigenvalue ˜Ekwhich equals the exact energy Ek, ifd(k)is a right eigenvector of Ewith the same eigenvalue. In practice, the subspace spanned by the basis functions is n ot invariant, and Eq. (1) yields an overdetermined set of Σ×nequations for the n2unknowns Eij. The equations can be solved approximately by minimizing th e sum of squared residuals with respect to the Eji. This gives the normal equations /summationdisplay σw2 σ Hβi(Rσ)−/summationdisplay jβj(Rσ)Eji βk(Rσ) = 0, (3) such that E=N−1H, (4) whereNij=/summationdisplay σβi(Rσ)w2 σβj(Rσ)≡/summationdisplay σBσiBσj (5) Hij=/summationdisplay σβi(Rσ)w2 σHβj(Rσ)≡/summationdisplay σBσiB′ σj. (6) The weights w2 σwere chosen in the orthogonality relation Eq. (3) to reprodu ce the standard quantum mechanical overlap integrals and matrix elements in the limit of an inﬁn ite Monte Carlo sample. In this case each variational energy ˜Ekis an upper bound to the exact energy Ek. In all but the simplest cases, the elementary basis function s are quasi-dependent, in the sense that some of the eigenvalues of Ndiﬀer from zero less than the roundoﬀ error. As a consequence , direct application of Eq. (4) yields numerically unstable results. The standard solution to thi s problem is to use a singular value decomposition to write7 B=USrVT, (7) whereUandVare square orthogonal matrices of order Σ and n, whileSris a rectangular Σ ×nmatrix with zeroes everywhere except for its leading diagonal elements σ1≥σ2≥σr>0;ris chosen such that the remaining singular values are close to zero.8From Eq. (7) one obtains E=VrS−1 rUT rB′, (8) whereUris the Σ ×rmatrix consisting of the ﬁrst rcolumns of U; andVris then×rmatrix likewise obtained from V. With this choice of the linear variational parameters, we ca n now —following Umrigar et al.,9— optimize the non-linear parameters in the elementary basis functions by minimization of χ2=/summationdisplay σw2 σ/parenleftBig H˜ψ(k)−˜Ek˜ψ(k))/parenrightBig2 //summationdisplay σw2 σ, (9) the variance of the energy of the wavefunction given in Eq. (2 ). We consider clusters of atoms of mass µ, interacting pairwise via a Lennard-Jones potential. In di mensionless form, the pair potential can be written as v(r) =r−12−2r−6and the Hamiltonian as H=−P2/2m+V, wherePandVare the total kinetic and potential energy operators, while the only parameter is the dimensionless massm−1= ¯h2/µσ2ǫ, which is proportional to the square of the de Boer parameter.10 The trial functions11,12are deﬁned in terms of the interparticle distances rστand scaled variables ˆ rστ=f(rστ); fmaps the interparticle distances monotonically onto the in terval ( −1,1) in the region in which the wavefunction diﬀers appreciably from zero, but the explicit form of fis not important for the current discussion. The elementary basis functions used for the trial wavefunction of energy le velkhave non-linear variational parameters a(k) j, and are of the form βi(R) =si(R) exp /summationdisplay ja(k) jsj(R)−/summationdisplay σ<τ/parenleftbigg κkrστ+1 5√mr5στ/parenrightbigg  (10) whereσ,τ,υ = 1,...,N care particle indices. The polynomial siis characterized by three non-negative integers nil: si(R) =/summationdisplay σ<τ<υ3/productdisplay l=1(ˆrl στ+ ˆrl τυ+ ˆrl υσ)nil. (11) The prefactor polynomial sihas bosonic symmetry, and contains general three-body corr elations, since all polynomials symmetric in x,y, andzcan be written as polynomials in the three invariants Il=xl+yl+zl, withl= 1,2,3 and v.v..11The number of elementary basis functions is limited by an upp er bound on the total degree/summationtext llnil; the polynomials sjin the exponent are of the same form as those in the prefactor, and their number in Eq. (10) is limited similarly. The constant κkis determined after a few iterations so that the wavefunctio n has the correct exponential decay in the limit that a single particle goes oﬀ to inﬁnity. Assuming —as is plausible for the small clusters studied here— that the energy of a cluster is roughly proportional to the nu mber of particle pairs13, we ﬁndκk=2 Nc−1/radicalBigg −m˜Ek Nc. (12) Ther−5 στterm in Eq. (10) ensures that Hβi/βihas a weaker divergence than with r−12 στin the limit rστ→0.11 States of higher energy are found with the same optimization scheme by selecting the appropriate eigenvector d(k) of the matrix Eand inserting it into Eq. (2). We use the same scaling functio nffor all states, but diﬀerent non-linear parameters a(k) jandκk. This scheme works as long as the variational freedom of the t rial functions can accurately represent the true eigenstates. Otherwise, for the Monte Ca rlo samples of the size we are using, states can be skipped and spurious ones may be introduced. In this context, noting that the exponential factors diﬀer for each state, we found it useful to check consistency of eigensystems obtain ed with this basis set and one that includes the variational wavefunctions of the previously determined, lower-lying s tates. The trial wavefunctions constructed in this fashion are use d as basis functions in a correlation function Monte Carlo calculation.2,3. Formally, this means that the nelementary basis functions βiare replaced by a small number of functions exp( −1 2tH)˜ψ(k). For this part of the computation the analog of Eq. (4) is used to compute eigenvalues, rather than Eq. (8). The reason for this is that, except during optim ization, too many conﬁgurations Rαare sampled to store the full matrices BandB′, deﬁned in Eqs. (6). These matrices are required for the sing ular value decomposition in Eq. (8). However, this is not a serious problem since the ﬁn al basis functions ˜ψ(k)are few in number and roughly orthonormal, at least for small projection times t. It may appear that we used stationarity of the energy to deter mine the linear variational coeﬃcients d(k) i, but a subtlety arrises because the integrals in matrix elements w ere approximated by weighted sums. Consider, therefore, in more detail the expectation value ˜Eof the energy in state ψand its functional derivative δ˜E δψ(R)∝(H −˜E)ψ(R). (13) For an energy eigenstate, the derivative vanishes both poin twise and in norm, which respectively corresponds to stationarity of the energy and minimality of its variance. A lthough these two criteria are equivalent for arbitrary functional variations, they yield diﬀerent approximation s, for the restricted variations of trial functions used in p ractice, as is well known. Orthogonality of the functional derivativ e and any wavefunction, suggests yet a third approximation: orthogonality —in the ﬁnite sample sense of Eq. (3)— of the fu nctional derivative and the elementary basis functions, or more generally the derivatives of the trial function with respect to the variational parameters. For a ﬁnite sample this orthogonality is not equivalent to stationarity of the approximate, sample average of the energy with respect to the variational parameters, which yields Eq. (4) with Hreplaced by its symmetrized analog. An important advantage of the orthogonality and variance cr iteria is that they yield zero variance for the energy and the variational parameters, because the right-hand sid e of Eq. (13) vanishes for an exact eigenstate even on a ﬁnite sample. On the other hand, stationarity of ﬁnite-sample est imate of the energy yields estimates of the variational parameters and the energies that diﬀer from sample to sample . Instead of this hybrid method which treats linear and non-li near variational parameters diﬀerently, we ﬁrst at- tempted to use minimization of the variance of the energy by a pplication of the Levenberg-Marquardt algorithm to all parameters simultaneously. Although that method works for statistical mechanical applications,6it fails for our current application, unless one starts out with good initia l guesses for all parameters.14 Next we present results for excited state energies for clust ers with up to seven atoms. First, we computed energies for trimers of Ne, Ar, Kr, and Xe ( m= 7.092×10−3,6.9635×10−4,1.9128×10−4, and 7.8508×10−5). Since our variational functions contain general three-body corr elations, the accuracy of the wavefunctions and energies fo r the trimers can be improved without apparent limit other tha n the machine precision. During optimization of the wavefunctions for the trimers we typically start with the gr ound state wavefunction which has prefactor degree of 5 or 6. For the trimers we chose not to vary the polynomial coeﬃ cients in the exponent and simply used the ﬁxed terms to satisfy the boundary conditions. The quality of the wave functions may be improved by varying polynomial coeﬃcients in the exponent, and for larger clusters it becom es important to include them. For the optimization we used a sample consisting of 4000 conﬁ gurations and gradually increased the prefacor degree to improve quality of the trial functions. For Ne clusters we performed diﬀusion Monte Carlo15calculations using optimized wavefunctions with prefactor degrees up to 14. We found that the projected energies have converged when the prefactor degree of 12 and higher is used. The result s presented in Table I were obtained using variational wavefunctions with degree of 12 in the prefactor. Before we present detailed estimates of the energy we mentio n the sources of errors of this method.2In addition to the usual statistical errors of Monte Carlo computations, t here are two systematic errors. For any ﬁnite projection timetand in the limit of vanishing statistical errors, the energi es computed by this method are upper bounds to theTABLE I. Energy levels Ekof the rare gas trimers; the errors are estimated to be a few un its in the least signiﬁcant decimal. k Ne3 Ar3 Kr3 Xe3 1 -1.719 560 -2.553 289 43 -2.760 555 34 -2.845 241 50 2 -1.222 83 -2.250 185 5 -2.581 239 0 -2.724 955 8 3 -1.142 0 -2.126 361 -2.506 946 8 -2.675 064 8 4 -1.038 -1.996 43 -2.412 444 -2.608 615 5 -0.890 -1.946 7 -2.387 973 -2.592 226 TABLE II. Energy levels of Ar clusters of up to seven atoms; th e errors are estimated to be a few units in the least signiﬁcan t decimal. k Ar4 Ar5 Ar6 Ar7 1 -5.118 11 -7.785 1 -10.887 9 -14.191 2 -4.785 -7.567 -10.561 -13.969 3 -4.674 -7.501 -10.51 -13.80 4 -4.530 -7.39 -10.46 -13.74 5 -4.39 -7.36 -10.35 -13.71 exact energies.16In practice, since the errors increase with projection time one should choose the smallest projection time such that the corresponding error and statistical erro rs are of the same order of magnitude. A troublesome detail is that at that point the results tend to have a non-Gaussian d istribution,17which makes it diﬃcult to produce error bars with a sharply deﬁned statistical meaning. In addition , there is the time-step error, which arrises because the imaginary-time evolution operator exp( −tH) has to be evaluated as the limit τ→0 of [exp( −τH) +O(τ2)]t/τ, but this error is much more easy to control. In order to eliminate the time-step error from the diﬀusion M onte Carlo calculation we veriﬁed that there was no statistically signiﬁcant diﬀerence between time steps τ=0.4, 0.2 and 0.1. In the diﬀusion Monte Carlo calculations we use 1.3 million Monte Carlo steps (16 blocks with 80 000 dat a per block). For Ar, Kr, and Xe trimers we found that the quality of the wavefunctions does not improve beyon d the degree of 10 in the prefactor. The results in Table I for the three more massive noble gas atoms were obtained usi ng trial wavefunctions with polynomials of degree 10. Independence of the time-step error within the statistical error was established by comparing τ=0.8, 0.6, and 0.4. The number of Monte Carlo steps is the same as for Ne. The resul ts in Table I agree with, and in some cases improve upon, those of Leitner et al.13. In Table II we present results for the energies of the ﬁrst ﬁve levels of Ar clusters of sizes 4 through 7. Our method allows one to go beyond 7 atom clusters, but as one can see from Table II the statistical error increase with system size. To obtain more accurate results for larger clusters it would probably be helpful to include higher order correlations in the wavefunction. In the calculations for 4 through 7 atom clusters we used a 10 degree prefactor and an exponent of degree three. Again, 1.3 million step diﬀusion Monte Carl o results were compared for τ=0.8, 0.6, and 0.4. Finally, Fig. 1 contains three energy levels as a function of mass for a four particle cluster. The results are plotted using variables chosen so that there is linear dependence bo th for large masses and for energies close to zero.12 As the energy of the levels approaches zero, both the optimiz ation and the projection methods begin to fail, and correspondingly data points are missing. Again, the use of t rial wavefunctions with four-body correlations are likely to allow one continue to smaller masses. This research was supported by the (US) National Science Fou ndation (NSF) through Grant DMR-9725080. It is our pleasure to thank David Freeman and Cyrus Umrigar for val uable disscusions and suggestions for improvements of the manuscript. 1D.H. Zhang, Q. Wu, J.Z.H. Zhang, M. von Dirke, and Z. Ba˘ ci´ c, J. Chem. Phys. 102, 2315 (1995). 2D.M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988). 3B. Bernu, D.M. Ceperley, and W.A. Lester, Jr., J. Chem. Phys. 93, 552 (1990); W.R. Brown, W.A. Glauser, and W.A. Lester, Jr., J. Chem. Phys. 103, 9721 (1995). 4D. Blume, M. Lewerenz, P. Niyaz, and K.B. Whaley, Phys. Rev. E 55, 3664 (1997). D. Blume and K. B. Whaley, J. Chem. Phys.112, 2218 (2000) and references therein.-2.5-2-1.5-1-0.50 00.05 0.10.15 0.20.25 0.30.35−\|Ek\|1 2 m−1 2 KrAr Ne He✛ ✻❄ ❄ ❄k= 1 : ✸ ✸✸✸✸✸✸✸✸✸✸✸✸k= 2 : + +++++++++++ k= 3 : ✷ ✷✷✷✷✷✷✷✷✷ FIG. 1. −√−Ekof lowest three levels ( k= 1,2,3) for four particle clusters vs m−1 2. The estimated errors for most energies are smaller, than the plot symbols. The vertical arrows indi cate Kr, Ar, Ne, and He; the horizontal arrow indicates the cl assical value -√ 6. 5M.P. Nightingale and H.W.J. Bl¨ ote, Phys. Rev. Lett. 80, 1007 (1998). 6M.P. Nightingale and H.W.J. Bl¨ ote, Phys. Rev. B 62, 1089 (2000). 7G.H. Golub and C.F. van Loan, Matrix computations (Second Edition), (Johns Hopkins University Press, 1989) Chapter 5.5. 8We ignored all singular values σkwithσk<103σ1ǫdbl, where ǫdblis the double precision machine accuracy. 9C.J. Umrigar, K.G. Wilson, and J.W. Wilkins, Phys. Rev. Lett .60, 1719 (1988); C.J. Umrigar, K.G. Wilson, and J.W. Wilkins, in Computer Simulation Studies in Condensed Matter Physics, R ecent Developments, edited by D.P. Landau K.K. Mon and H.B. Sch¨ uttler, Springer Proceedings in Physcs (Sp ringer, Berlin, 1988). 10J. de Boer, Physica ,14, 139 (1948). 11Andrei Mushinski and M. P. Nightingale, J. Chem. Phys. 101, 8831, (1994). 12M. Meierovich, A. Mushinski, and M.P. Nightingale, J. Chem. Phys.105, 6498 (1996). 13D.M. Leitner, J.D. Doll, and R.M. Whitnell, J. Chem. Phys. 94, 6644 (1991). 14V. Melik-Alaverdian and M.P. Nightingale, Int. J. Mod. Phys . C10, 1409, (2000). 15C.J. Umrigar, M.P. Nightingale, and K.J. Runge, J. Chem. Phy s.99, 2865 (1993) and references therein; we used a single, weighted walker, and an accept reject step with τeﬀ= 1 for accepted moves and τeﬀ= 0 for rejected ones. 16J.K.L. MacDonald, Phys. Rev. 43, 830 (1933). 17H. Hetherington, Phys. Rev. A 30, 2713 (1984).
arXiv:physics/0010067v1 [physics.gen-ph] 26 Oct 2000The second ”Einstein paradox”. V.A.Kuz’menko Troitsk Institute of Innovation and Fusion Research, Troitsk, Moscow region, 142190, Russian Federation. Abstract Time invariance problem of a photon absorption process in at oms and molecules is discussed. PACS number: 42.50.-p The Dirac equation, in the case of electromagnetic interact ion, is not invariant under unitary time inversion [1]. However, a comm on opinion exists that the electromagnetic interaction in nature must be time invariant [2]. The attempts to ﬁnd what is a proof basis of such point of view in a s pecial case of a photon absorption process in atoms and molecules give on ly indistinct references point at Einstein’s works [3]. If one keeps in mind the Einstein coeﬃcients of absorption an d stimulated emission, this basis is erroneous. The Einstein coeﬃcients characterize inte- gral cross-section of optical transition. Time invariance preserving demands equality not only the integral cross-section, but also it de mands equality of spectral width of forward and backward optical transitions . Einstein nothing writes about the width of optical transitions. So, Einstein coeﬃcients have no direct connection to T-invariance of photon absorption p rocess. There is also exists the Einstein’s opinion ”that physics co uld be restricted to the time-symmetric case for which retarded and advanced ﬁ elds are equiv- alent” [4]. Obviously there is a main basis of existing point of view, because of any experimental result in proof of T-invariance preserv ing in a photon absorption process is absent. In contrast, for the opposite point of view we have one direct and com- plete experimental proof and a number of indirect evidences . The direct and complete experimental proof is connected with the experime ntal study of the so-called line wings [5]. The experiments clearly show a very strong T- invariance violation in a photon absorption process in mole cules. Although the integral cross-section of forward and backward optical transitions are ob- viously the same, the spectral width and cross-section for s uch transitions diﬀer on several order of magnitude [6]. The concept of T-invariance violation of a photon absorptio n process is a good basis for explanation of most eﬀects in nonlinear opti cs from a pure quantum position without using any semiclassical approxim ation [7]. There 1are indirect experimental proofs. The most striking exampl e is the popula- tion transfer eﬀect in the case of sweeping a resonance condi tions in a two level system. On the basis of T-invariance violation of abso rption process this eﬀect has a simple and natural explanation [8]. In contr ast, on the basis of semiclassical wave approximation theory the explanatio n of this eﬀect looks like as the ravings of a madman [9]. On the whole such situation may be called as the second ”Einst ein paradox”a. When great authority and delusion of one scientist delay on d ecades progress of physical theory in some ﬁelds of scientiﬁc research. So, d o we have ”Einstein paradox” in quantum optics? aThe ﬁrst paradox is connected with quantum statistics [10,1 1]. References [1] W.M.Jin, eprint, quant-ph/0001029 [2] V.B.Berestetsky, E.M.Lifshits, L.P.Pitaevsky, ”Relativistic Quantum Theory” , part 1, Nauka, Moscow, p.66 (1968) in Russian. [3] A.K.Popov, eprint, quant-ph/0005118 [4] A.Bohm, N.L.Harshman, eprint, quant-ph/9805063, p.2 [5] V.A.Kuz’menko, eprint, aps1998dec29 002, http://publish.aps.org/eprint/ [6] V.A.Kuz’menko, eprint, hep-ph/0002084 [7] V.A.Kuz’menko, eprint, physics/0007076 [8] V.A.Kuz’menko, eprint, aps1999sep30 002, http://publish.aps.org/eprint/ [9] R.L.Shoemaker, in ”Laser and Coherence Spectroscopy” , Ed. J.I.Steinfeld, N.Y., Plenum, p.197 (1978). [10] Ya.M.Gel’fer, V.L.Lyuboshits, M.I.Podgoretsky, ”Gibbs Paradox and Identity of Particles in Quantum Mechanics” , Nauka, Moscow (1975), in Russian. [11] P.Ehrenfest, G.E.Uhlenbeck, ”Paul Ehrenfest” , Collected Scientiﬁc Pa- pers, North-Holland Publishing Company, Amsterdam, p.539 , (1959). 2
arXiv:physics/0010068v1 [physics.atom-ph] 27 Oct 2000The no-sticking eﬀect in ultra-cold collisions Areez Mody, Michael Haggerty and Eric J. Heller Department of Physics, Harvard University, Cambridge, MA 0 2138 (August 2000) We provide the theoretical basis for understanding the phen omenon in which an ultra cold atom incident on a possibly warm target wi ll not stick, even in the large nlimit where nis the number of internal degrees of freedom of the target. Our treatment is non-perturbative in which th e full many-body problem is viewed as a scattering event purely within the con text of scatter- ing theory. The question of sticking is then simply and natur ally identiﬁed with the formation of a long lived resonance. One crucial phy sical insight that emerges is that the many internal degrees of freedom ser ve to decohere the incident one body wavefunction, thus upsetting the deli cate interference process necessary to form a resonance in the ﬁrst place. This is the physical reason for not sticking. I. INTRODUCTION The problem of low energy sticking to surfaces has attracted much attention over the years [1–5]. The controversial question has been the ultral ow energy limit of the incoming species, for either warm or cold surfaces. A battle has ensue d between two countervailing eﬀects, which we will call classical sticking and quantum re ﬂection. The concept of quantum reﬂection is intimately tied into threshold laws, and was re cognized in the 1930’s by Lennard- Jones [1]. Essentially, ﬂux is reﬂected from a purely attrac tive potential with a probability which goes as 1 −α√ǫ, asǫ→0, whereαis a constant and ǫis the translational energy of the particle incident on the surface. Classically the tra nsmission probability is unity. Reﬂection at long range prevents inelastic processes from o ccurring, but if the incoming particle should penetrate into the strongly attractive reg ion, the ensuing acceleration and hard collision with the repulsive short range part of the pot ential leads to a high probability of inelastic processes and sticking. The blame for the quantum reﬂection can be laid at the feet of t he WKB approximation, which breaks down in the long range attractive part of the pot ential at low energy. Very far out, the WKB is good even for low energy, because the potentia l is so nearly ﬂat. Close in, the kinetic energy is high, because of the attractive potent ial, even if the asymptotic energy is very low, and again WKB is accurate. But in between there is a breakdown, which has been recognized and exploited by several groups [6–11]. We s how in the paper folling this one that the breakdown occurs in a region around \|V\| ≈ǫ; i.e. aproximately where the kinetic and potential energies are equal. It would seem that quantum reﬂection would settle the issues of sticking, since if the particle doesn’t make it in close to the surface there is no st icking. (Fig 1) There is one caveat, however, which must be considered: quantum reﬂecti on can be defeated by the existence of a resonance in the internal region, i.e. a thres hold resonance. (Fig 2) The situation is very analogous to a high Q Fabry-Perot cavit y, where using nearly 100% reﬂective, parallel mirrors gives near 100% reﬂection exce pt at very speciﬁc wavelengths. 1At these speciﬁc energies a resonace buildup occurs in the in terior of the cavity, permiting near 100% transmission. Such resonances are rare in a one dim ensional world, but the huge number of degrees of freedom in a macroscopic solid particle makes resonance ubiquitous. Indeed, the act of colliding with the surface, creating a pho non and dropping into a local bound state of the attractive potential describes a Feshbac h resonance. Thus, the resonances are just the sticking we are investigating, and we must not tr eat them lightly! Perhaps it is not obvious after all whether sticking occurs. After the considerable burst of activity surrounding the st icking issue on the surface of liquid Helium [12,13], and after a very well executed theo retical study by Clougherty and Kohn [4], the controversy has settled down, and the commo n wisdom has grown that sticking does not occur at suﬃciently low energy. While we ag ree with this conclusion, we believe the theoretical foundation for it is not complete , nor stated in a wide enough domain of physical situations. For example, Ref. [4] treats only a harmonic slab with one or two phonon excitation. It is not clear whether the results apply to a warm surface. On the experimental side, even though quantum reﬂection was ob served from a liquid Helium surface, that surface has a very low density of available sta tes (essentially only the ripplons) which could be a special case with respect to sticking. Thus, the need for more rigorous and clear proof of non-sticking in general circumstances is evident. This paper gives such an analysis. In a following paper, application is made to spe ciﬁc atom-surface and slab combinations, and the rollover to the sticking regime as ene rgy is increased (which can be treated essentially analytically) is given. The strategy we use puts a very general and exact scattering f ormalism to work, providing a template into which to insert the properties of our target a nd scatterer. Then very general results emerge, such as the non-sticking theorem at zero ene rgy. The usual procedure of deﬁning model potentials and considering one phonon proces ses etc. is not necessary. All such model potentials and Hamiltonians wind up as parameter s in the R-matrix formalism. The details of a particular potential are of course importan t for quantitative results, but the range of possible results can be much more easily examined by inserting various parameters into the R-matrix formalism. All the possible choices of R-m atrix parameters give the correct threshold laws. Certain trends are built into the R-matrix f ormalism which are essentially independent of the details of the potentials. Before commencing with the R matrix treatment, we brieﬂy con sider the problem per- turbatively in order to better elucidate the role played by q uantum reﬂection. We emphasize that none of the perturbation section is actually necessary for our ﬁnal conclusions. In a perturbative treatment for our slab geometry, quantum r eﬂection simply results in the entrance channels’ wave function (at threshold) having its amplitude in the interaction region go to zero as ke∼√ǫwhen normalized to have a ﬁxed incoming ﬂux. ( keis the magnitude \|/vectorke\|of the incident wavevector of the incoming atom). The inelas tic transition probabilities are proportional to the potential weighted o verlap of the channel wavefunctions and this immediately leads to the conclusion that the inelas tic probability itself vanishes as ke∼√ǫ. As mentioned, this conclusion is shown to rigorously remai n true using the R matrix. We show in this paper that in spite of the inherently m any-body nature of the problem, in the ultra-cold limit we can correctly obtain the long-range form of the entrance channel’s wavefunction by solving for the one-dimensional motion in the long-range surface- atom attraction (i.e. the diagonal element of the many-chan nel potential matrix). This 2FIG. 1. The stationary state one body wavefunction of the inc ident atom moving in the y-independent mean potential felt by it. The amplitude insid e the interaction region is supressed byke∼√ǫ. This is tantamount to the reﬂection of the atom. energytarget absorbs bound stateBefore: After: FIG. 2. A schematic view of a Feshbach resonance wherein the i ncident atom forms a long lived quasi-bound state with the target. The many body wavefuncti on in this situation (not shown) has a large amplitude in the ‘interior’ region near the slab. allows quantitative predictions of the sticking probabili ty, which we do in the following paper. There, we further exploit the perturbative point of v iew together with an analysis of WKB to predict a ‘post-threshold’ behavior as quantum reﬂec tion abates, when the incoming energy is increased. II. GEOMETRY AND NOTATION The incident atom is treated as a point particle at position ( x,y). To keep the notation simple we leave out the z-coordinate and conﬁne our discussion to two spatial dimens ions. Thus a cross-section will have dimensions of length etc. It w ill be quite obvious how and wherezmay be inserted in all that follows. Let urepresent all the bound degrees of freedom of the scattering target, which we take to be a slab of crystalline or amorphous material. Let Ω c(u),c= 1,2,···, be the manybody target wave functions in the absence of interactions with the incident particle, and having energy Etarget c. These are normalized as/integraltext all udu\|Ωc(u)\|2= 1.xis the distance of the scatterer (atom) from the face of the sl ab which 3is approximately (because the wall is rough) along the line x= 0. The internal constituents of the slab lie to the left of x= 0 and the scatterer is incident from the right with kinetic energyǫ= ¯h2k2 e/2m. The total energy Eof the system is E=ǫ+Etarget e (1) wherec=eis the index of the ‘entrance channel’ i.e. the initial inter nal state of the slab before the collision is Ω e(u). Notice that we say nothing about the value of Etarget eitself. In particular the slab need not be cold. kcis the magnitude of the wave vector /vectorkcof the particle when it leaves the target in the state Ω c(u) after the collision. Our interest focusses onke→0.keis the magnitude of the wavevector of the incoming particle. For the open channelsc= 1,···n(this deﬁnes n) for which E >Etarget c kc≡/radicalBigg 2m(E−Etarget c) ¯h2 (c≤n) ; (2) whereas for the closed channels ( c>n),E <Etarget cand kc≡i/radicalBigg 2m(Etarget c−E) ¯h2≡iκc(c>n). (3) κc>0. We will use ( kcx,kcy) as thex,ycomponents of /vectorkc. LetUint(x,y,u ) = (2m/¯h2)Vint(x,y,u ), whereVint(x,y,u ) describes quite generally the interaction potential between the incident atom and all the internal degrees of fre edom of the slab. For simplicity we assume for the moment that there is no interaction between slab and atom for x>a. III. PRELIMINARIES: PERTURBATION As stated above, we excercise the perturbative treatment fo r insight only; our ﬁnal conclusions are based on nonperturbative arguments. We treat the interaction Uint(x,y,u ) between slab and atom by separating out a ‘mean’ potential felt by the atom that is independent of yandu; call itU(0)(x). The remainder U(1)(x,y,u )≡Uint(x,y,u )−U(0)(x) is treated as a perturbation. Now the incident beam is scattered by the entire length (say f romy=−LtoL= 2L) of wall which it illuminates. If all measurements are made clos e to the wall so that its length 2Lis the largest scale in the problem, then it is appropriate to speak of a cross-section per unit length of wall, a dimensionless probability. More spec iﬁcally, we will assume that the matrix elements U(1) cc′(x,y)≡/integraltext all uduΩ∗ c(u)U(1)(x,y,u )Ωc′(u) of the perturbation U(1)(x,y,u ) in the Ω c(u) basis are given by the simple form U(1) cc′(x,y) =U(1) cc′(x)f(y) fory∈[−L,L] and 0 elsewhere. f(y) is a random persistent (does not die to 0 as \|L\| → ∞ ) function that models the random roughness of the slab and is character ized by its so-called spectral density function S, a smooth positive-valued non-random function, such that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL/integraldisplay −Ldyeikyf(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≡2LS(k)∀k (4) 4asL→ ∞. Now, applying either time-independent perturbation (equi valently the Born approxima- tion for this geometry) or time-dependent perturbation the ory via the Golden Rule, gives that the cross-section per unit length of wall for inelastic scattering to a ﬁnal channel cis Pin c←e(θ) =2π ke a/integraldisplay −∞dx′φ(x′;kcx)U(1) ce(x′)φ(x′;kex) 2 S(kcy−key) (5) whereφ(x;kx) is the solution of the o.d.e. /parenleftBiggd2 dx2−U(0)(x) +k2 x/parenrightBigg φ(x;kx) = 0 (6) which is regular or goes to zero as x→ −∞ inside the slab and is normalized as φ(x;kx)∼sin(kxx+δ) asx→ ∞ (7) Accepting for the moment that as ke→0 the amplitude of φ(x;kex) in the internal region x < a goes to zero as ke∼√ǫ, then the square of the overlap integral in Eq. (5) behaves ask2 e, because by our proposition the amplitude of φ(x′;kex)∼kex∼ke. Together with the 1/keprefactor we get an overall behavior of kefor the inelastic probability as claimed. To show that indeed as ke→0 the amplitude of φ(x;kex) in the internal region x<a goes to zero aske∼√ǫ, we temporarily disregard the required normalization of φ(x;kx) of Eq. (7) and ﬁx its initial conditions (slope and value) at some point inside the interaction region x < a such that the regularity condition is ensured. We then integ rate out to x=a. Let us denote this unnormalized solution with a prime, as φ′(x;kx). The point is for kxvarying near 0, both v(the value) and s(the slope) that the solution emerges with at x=a, are independent of kxand in fact the interior solution thus obtained is itself ind ependent of kx. This is because the local wave vector k(x) =/radicalBig 2m(ǫ−U(x))/¯h2essentially stays the same function of xfor allǫnear 0. Therefore for x>a φ (x;kx) continues onto vcos[kx(x−a)] +s kxsin[kx(x−a)]x>a (8) This is a phase-shifted sine wave of amplitude ∼1/kx. We must enforce the normalization of Eq. (7) and get φ(x;kx)∼kxφ′(x;kx). As a result, the interior solution gets multiplied by kxand we thereby have our result. φ(x;kx) is the solution of a one-dimensional Schrodinger equation for the incoming particle in the one-dimensional l ong-range potential created by the slab. The suppression of its amplitude by√ǫnear the slab is due to the reﬂection it suﬀers where the interaction turns on. Within the perturb ative set-up the non-sticking conclusion is then already foregone [1]. The problem is whether we can really accept this verdict of th e one-dimensional un- perturbed solution, when in fact we know that the turning on o f the perturbation (many body interactions) causes a multitude of resonances to be cr eated, internal resonances being exactly the situation in which the Proposition above is know n to badly fail. It appears that the perturbation is in no sense a small physical eﬀect. There fore a nonpeturbative approach is needed. Here we use R-matrix theory in its general form to a ccomplish the task. 5IV. S-MATRIX AND R-MATRIX One point that the preceding section has made clear is that it is the energies (both initial and ﬁnal) in the x-direction, perpendicular to the slab that are most relevan t. In fact as regards the ﬁnal form of our answers the motion of the ydegree of freedom may as well have been the motion of another internal degree of freedom of the slab. In other words, mathematically speaking, the ydegree of freedom may be subsumed by incorporating it as just another u. For example, we may imagine the incident atom being conﬁned in the y-direction by the walls of a wave-guide at y=−LandLthat is large enough so that it could not possibly change the physics of sticking. Then we quite ri gorously have a bound internal state of the form Ωc,n(y,u) = Ωc(u) sinnπy L(9) xis now the only scattering degree of freedom. There will be no necessity in carrying along the extra index nand variable yas in Eq. (9), and we will simply continue to write Ω c(u) instead. Thus with this understanding, the problem is essen tially one-dimensional in the scattering degree of freedom. We proceed to derive the expression for the Smatrix in terms of the so-called Rmatrix, and derive the structure of the Rmatrix. For simplicity we continue to assume for the moment that there is no interaction for x>a. Then forx>a, the scattering wavefunction of the interacting system corresponding to the scattering p article coming in on one entrance channel, say c=e, with energy ǫ= ¯h2k2 e/(2m) is ψ(x,u) =∞/summationdisplay c=1/parenleftBigge−ikex √keδce−eikcx √kcSce/parenrightBigg Ωc(u)x>a (10) where the sum must include all channels, even though the open channels are ﬁnite in number. The factors of k−1/2 cin Eq. (10) mean that the ﬂux in each channel is proportional o nly to the square of the coeﬃcient and hence ensure the unitarity of S. With this convention, the open-open part of the S-matrix—the n×nsubmatrixScc′withc,c′= 1,2,...,n —is unitary.√kc≡eiπ/4√κcmay be arbitrarily chosen since it cannot aﬀect the open-ope n part of S. Sis found in analogy to the one-dimensional case by introduci ng the matrix version of the inverse logarithmic derivative at x=acalledR(E) the Wigner R-matrix deﬁned by /vector v=R(E)/vector s (11) where the components of /vector vand/vector sare the expansion coeﬃcients of ψ(x=a,u) and∂ψ(x=a,u) ∂x respectively in the Ω c(u) basis. Supposing∂ψ(x=a,u) ∂xto be known, we will (like in electro- statics) use the Neumann Green’s function GN(x,u;x′,u′) to construct ψ(x,u) everywhere in the interior x<a.ψ(x,u) satisﬁes the full Schr¨ odinger equation with energy E. We need χλ(x,u)λ= 1,2,···,the normalized eigenfunctions of the full Schr¨ odinger equ ation in the interiorx<a with energies Eλ, satisfying Neumann boundary conditions∂χ(x=a,u) ∂x= 0. So /parenleftBigg−¯h2 2m∇2+Vint(x,u)−E/parenrightBigg ψ(x,u) = 0 (12) 6/parenleftBigg−¯h2 2m∇2+Vint(x,u)−Eλ/parenrightBigg χλ(x,u) = 0 (13) /parenleftBigg−¯h2 2m∇2+Vint(x,u)−E/parenrightBigg GN(x,u;x′,u′) =δ(x−x′)δ(u−u′) (14) where ∇2≡∂2 ∂x2+∂2 ∂u2and ∂GN(x=a,u;x′,u′) ∂x= 0 and∂χ(x=a,u) ∂x= 0 (15) ⇒GN(x,u;x′,u′) =∞/summationdisplay λ=1χλ(x,u)χλ(x′,u′) Eλ−E(16) GNis symmetric in the primed and unprimed variables. By Stokes ’ Theorem, (−¯h2/2m)/integraldisplay x′<adx′/integraldisplay all u’du′/parenleftBig φ1∇′2φ2−φ2∇′2φ1/parenrightBig = (−¯h2/2m)/integraldisplay x’=a, all u’du′(φ1∇′ ˆnφ2−φ2∇′ ˆnφ1) (17) where ∇′ ˆn(·)≡ˆx′(·)· ∇′withφ1=ψ(x′,u′) andφ2=GN(x,u;x′,u′) gives ψ(x,u) =¯h2 2m/integraldisplay all u′du′GN(x,u;x′,u′)∂ψ(x′=a,u′) ∂x′x<a (18) Putx=aand it is deduced using Eqs. (11) and (18) together that Rcc′(E) =∞/summationdisplay λ=1γλcγλc′ Eλ−E(19) whereγλc=/radicalBig ¯h2 2m/integraltext all uduχλ(a,u)Ωc(u). A. The S matrix Now shifting attention to the outside ( x>a), we see that we can compute both ∇ˆnψ(a,u) andψ(a,u) on the surface x=ausing the asymptotic form of Eq. (10) which automatically gives these expanded in the Ω c(u) basis. Writing the matrix Eq. (11) is now simple. It is best to do it all in matrix notation, and thus be able to trea t all possible independent asymptotic boundary conditions simultaneously. Leteikx,√ kand 1/√ kbe diagonal matrices with diagonal elements eikcx,√kcand 1/√kc. Then Eq. (11) reads e−ika √ k−eika √ kS=iRk/parenleftBigg−e−ika √ k−eika √ kS/parenrightBigg . (20) Each column c= 1,...,n of the matrix equation above is just Eq. (11) for the solution corresponding to an incoming wave only in channel c(Forc>n the wavefunctions blow up asx→ ∞). Remembering that non-diagonal matrices don’t commute, w e solve for Sto get 7S=e−ika√ k1 1−iRk(1 +iRk)1√ ke−ika(21) or, with some simple matrix manipulation, S=e−ika 1 1−i√ kR√ k(1 +i√ kR√ k)e−ika. (22) V. S MATRIX NEAR A RESONANCE As discussed in the introduction, the resonances are a key to the sticking issue. Sticking is essentially a long lived Feshbach resonance in which ener gy has been supplied to surface and bulk degrees of freedom, temporarily dropping the scatt ering particle into a bound state of the attractive potential. Thus we must study resona nces in various circumstances in the low incident translational energy regime. We derive t he approximation for S(E) near E=E0, a resonant energy of the compound system. E0is the total energy of the joined (resonant) system. Within the R-matrix approach, the χλ(x,u) of section IV are bound, compound states with Neuman boundary conditions at x=a.R-matrix theory properly couples these bound state to the continuum, but some of the ei genstates are nonetheless weakly coupled to the continuum, as evidenced by small value s of theγλc’s of section IV; these are the measure of the strength of the continuum coupli ngs. While every one of the R−matrix bound states will result in a pole Eλin theRmatrix expansion, only the weakly coupled ones are the true long lived Feshbach resonances of p hysical interest. It is also helpful to know that the values of these ‘truly’ resonant pol es atEλare the most stable to changes in the position x=aof the box. This in fact provides one unambiguous way to identify them. Our purpose here is to derive the resonant app roximation to the Smatrix in the vicinity of one of these Feshbach resonances. We do so usi ng the form of the R-matrix in Eq. (19). Note that the energy density ρ(E) = 1/D(E) of these Feshbach resonances will be large because of the large number of degrees of freedom of t he target.D(E) is the level spacing of the quasibound, resonant states. A. Isolated Resonance As mentioned, the point of view we will take is to identify a re sonant energy with a particular pole Eλin the Rmatrix expansion of Eq. (19). Those Eλcorresponding to resonances are a subsequence of the Eλappearing in the expansion in Eq. (19). For Enear a well isolated resonance at Eλwe separate the sum-over-poles expansion of the R-matrix into a single matrix term having elementsγλcγλc′ Eλ−E, plus a sum over all the remaining terms, call itN. If the energy interval between Eλand all the other poles is large compared to the open-open residue at Eλthen we may expect that the n×nopen-open block of Nwill have all its elements to be small. Then rewriting the inverse in Eq . (22) 1 1−i√ kR√ k≡1 1−i/parenleftBig M+V Eλ−E/parenrightBig (23) 8whereM≡√ kN√ kandVcc′≡(√kcγλc)(√kc′γλc′), and setting M= 0 allows us to simplify the central term in Eq. (22) exactly. (We will return to the ca seM∝negationslash=0.) 1 1−i√ kR√ k(1 +i√ kR√ k) (24) = 1 +1 1−i√ kR√ k2i√ kR√ k (25) = 1 +1 1−iV Eλ−E2iV Eλ−E(withM= 0) (26) = 1 +1 Eλ−E−iV2iV (27) = 1 +1 Eλ−E−i(Γλ/2 +i∆E)2iVk (28) where we used V2=/parenleftBig (γ2 λ1k1+···+γ2 λnkn) + (γ2 λ(n+1)κn+1+···)/parenrightBig V (29) ≡/parenleftbigg/parenleftbiggΓλ1 2+···+Γλn 2/parenrightbigg +i(γ2 λ(n+1)κn+1+···)/parenrightbigg V (30) ≡/parenleftbiggΓλ 2+i∆Eλ/parenrightbigg V (31) to get the identities [Eλ−E−iV]V= [Eλ−E−i(Γλ/2 +i∆E)]V (32) ⇒1 Eλ−E−i(Γλ/2 +i∆E)V=1 Eλ−E−iVV (33) Also deﬁne (Γ λc/2)1/2≡γλc√kc,c= 1,2,···,n. This deﬁnes the sign of the square-root on the lhs. to be the sign of γλcand allows the convenience of expressing things in terms of t he Γλc’s and their square-roots, and not having to use the γλc’s themselves. Thus we arrive at Scc′=e−ikca δcc′+iΓ1/2 λcΓ1/2 λc′ E(r) λ−E−iΓλ/2 e−ikc′a(34) whereE(r) λ≡Eλ+ ∆Eλ, for then×nopen-open unitary block of Sin the neighbourhood of a single isolated resonance after neglecting the contrib ution of the background matrix M. For us the essential point is that Γλc= 2kc(E)γ2 λc, (35) that the partial widths Γ λcdepend on the energy E, through the kinematic factor kc(E). Mostly this energy dependence is small and irrelevant excep t where the kc’s and hence Γ λc’s are varying near 0. These are the partial widths of the open ch annels near threshold. Hence \|Sce\|2(c∝negationslash=e) an inelastic probability behaves like ke∼√ǫwhen the entrance channel is at threshold. Including the background term ( M∝negationslash= 0) does not change this. To see this we may 9perform the inverse in Eq. (22) to ﬁrst order in Mand then get an additional contribution of the terms e−ika 2i 1−iV Eλ−EM+1 1−iV Eλ−E+1 1−iV Eλ−EiM1 1−iV Eλ−E2iV e−ika(36) to theS-matrix. Now, both MandVhave a factor of√kcmultiplying their cth columns (and rows) from their deﬁnitions and so a matrix element bcc′of the matrix in parentheses in Eq. (36) will have a√kcand√kc′dependence. An inelastic element of S(c∝negationslash=c′) would now take the form Scc′=e−ikca bcc′+iΓ1/2 λcΓ1/2 λc′ E(r) λ−E−iΓλ/2 e−ikc′a, (37) As mentioned our interest is in the case when the entrance cha nnel is at threshold so that this dependence is√ke, making the inelastic probability \|Sce\|2still continue to behave as ke∼√ǫ. B. Overlapping Resonances Here we require the form of the Smatrix near an energy Ewhere many of the quasi- bound states may be simultaneously excited, i.e. the resona nces overlap. Again, neglecting background for the moment, the Smatrix is simply taken to be a sum over the various resonances. S= 1−/summationdisplay λiAλ E−E(r) λ+iΓλ/2(38) whereAλis an×nrank 1 matrix with the cc′th component as Γ1/2 λcΓ1/2 λc′. There is no entirely direct justiﬁcation of this form, but one can see that there i s much which it gets correct. TheAλare symmetric, hence Sis symmetric. Obviously it has the poles in the right places allowing the existence of decaying states with a pure ly outgoing wave at the resonant energies. A crucial additional assumption that also makes Sapproximately unitary is that the signs of the Γ1/2 λcare random and uncorrelated both in the index λas well asc, regardless of how close the energy intervals involved may be. One simple consequence is that we approximately have that AλAλ′=δλλ′ΓλAλ (39) in the sense that the l.h.s. is negligible for λ∝negationslash=λ′in comparison to the value for λ=λ′. With Eq. (39) it is easy to verify the approximate unitarity o fS. We investigate now the onset of the overlapping regime as Eincreases.D(E), the level spacing of the resonant E(r) λ, is a rapidly decreasing function of its argument. On the oth er hand, Γλ= Γλ1+ Γλ2+···+ Γλn, and since more channels are open at higher energy, Γ λis increasing with the energy of the resonance. The widths must therefore eventually overlap, 10and Γλ≫D/parenleftBig E(r) λ/parenrightBig for the larger members of the sequence of E(r) λ’s. In this regard there is a useful estimate due to Bohr and Wheeler [15], that for nlarge Γλ D(E(r) λ)≃n . (40) Appendix A derives this using a phase space argument. Here we point out that this is entirely consistent with the assumption of the random signs , indeed requiring it to be true. Take for example a typical inelastic amplitude Scc′=−i/summationdisplay λΓ1/2 λcΓ1/2 λc′ E(r) λ−E−iΓλ/2(c∝negationslash=c′) (41) First let us note that the Γ λbeing the sum of many random variables (the partial widths Γ λc) do not ﬂuctuate much. Let Γ denote their typical value over th enoverlapping resonances. Also since Γ = nDit follows that the typical size of a partial width Γ λcisD. Therefore the typical size of the product Γ1/2 λcΓ1/2 λc′isDbut these random variables ﬂuctuate randomly over the indexλ, and moreover the sign is random. Thus for energies in the ove rlapping domain Scc′is a sum of ncomplex numbers each of typical size D/Γ = 1/n, but random in sign. This makes for a sum of order 1 /√n. Clearly this is as required to make the n×nmatrix Sunitary. Note that the above argument fails (as is should) if c∝negationslash=c′because then the signs of Γ1/2 λcΓ1/2 λc= Γλ>0 are of course not random. Unlike the case of the isolated resonance, the S-matrix elem ents here are smoothly varying in E. Addition of a background term Bcc′ Scc′=Bcc′−i/summationdisplay λΓ1/2 λcΓ1/2 λc′ E(r) λ−E−iΓλ/2. (42) just shifts this smooth variation by a constant. If Bcc′is also thought of as arising from a sum over the individual backgrounds then for the same reason s as discussed at the end of the preceding section \|Bce\|2∼ke∼√ǫfor an entrance channel near threshold. For simplicity we will continue to take Bcc′to be 0 and look at the case with background in the appendix. VI. Q-MATRIX AND STICKING From the viewpoint of scattering theory, the sticking of the incident particle to the target is just a long-lived resonance. It is natural then to investi gate the time-delay for the collision. Smith [14] introduced the collision lifetime or Q-matrix Q≡i¯hS∂S† ∂E(43) which encapsulates such information. We review some of the r elevant properties of Q. The rhs of Eq. (43) involves the ‘open-open’ upper left block of Sso that Qis also ann×n energy-dependent matrix, having dimensions of time. For 1- dimensional elastic potential scattering S=eiφ(e)andQreduces to the familiar time delay i¯h∂φ(E) ∂E. If/vector vis a vector 11whose entries are the coeﬃcients of the incoming wave in each channel then /vector vtrQ(E)/vector vis the average delay time experienced by such an incoming wave. Bec ause physically the particle is incident on only one channel, /vector vconsists of all 0’s except for a 1 in the eth slot so that the relevant quantity is just the matrix element Qee(E). Smith shows that this delay time is the surplus probability of being in a neighborhood of the t arget (measured relative to the probability if no target were present) divided by the ﬂux arr iving in channel e. This matches our intuition that when the delay time is long, there is a high er probability that the particle will be found near the target. Furthermore, as a Hermitian matrix, Q(E), can be resolved into its eigenstates /vector v(1)···/vector v(n) with eigenvalues q1···qn. The components of /vector v(1)are the incoming coeﬃcients of a quasi- bound state with lifetime q1and so on. Then /vector vtrQ(E)/vector v=n/summationdisplay j=1qj\|/vector v(j)·/vector v\|2. (44) As can be seen from this expression, the average time delay re sults, in general, from the excitation of multiple quasi-stuck states each with its lif etimeqjand probability of formation \|/vector v(j)·/vector v\|2. However, we will ﬁnd that using our resonant approximation to the Smatrix near a resonant energy E(r) λthe time delay will consist of only one term from the sum on the rhs of Eq. (44), all the other eigenvalues being identically 0. Using equation Eq. (43), Q(E) =i¯h/parenleftBigg/summationdisplay λ′−iAλ′ /bracketleftBig E−E(r) λ′−iΓλ′/2/bracketrightBig2−/summationdisplay λλ′AλAλ′ /bracketleftBig E−E(r) λ+iΓλ/2/bracketrightBig/bracketleftBig E−E(r) λ′−iΓλ′/2/bracketrightBig2/parenrightBigg (45) which using Eq. (39) simpliﬁes to =/summationdisplay λ¯h (E−E(r) λ)2+ (Γλ/2)2Aλ, (46) a remarkably simple answer. We need Qee(E), whereeis the entrance channel. Qee(E) =/summationdisplay λ¯hΓλe (E−E(r) λ)2+ (Γλ/2)2(47) =/summationdisplay λ/parenleftBigg¯hΓλ (E−E(r) λ)2+ (Γλ/2)2×Γλe Γλ/parenrightBigg (48) where the second equation has the interpretation (for each t erm) as the life-time of the mode, multiplied by the probability of its formation. Note how for each resonance E(r) λthere is only one term corresponding to the decomposition of Eq. (44) . The actual measured lifetime is the average of Qee(E) averaged over the energy spectrum \|g(E)\|2of the collision process. A. Energy averaging over spectrum With the target in state Ω e(u) wherec=eis the entrance channel, the energy of the target is ﬁxed, and the time-dependent solution will look li ke 12ψ(x,u,t) =/integraldisplay dE g(E)∞/summationdisplay c=1 e−ikc(E)x /radicalBig ke(E)δce−eikc(E)x /radicalBig kc(E)S(E)ce Ωc(u) . (49) Recall,Eis the total energy of the system. We are interested in the thr eshold situation where the incident kinetic energy of the incoming particle ǫ→0. This can be arranged ifg(E) is peaked at E0with a spread ∆ Esuch that i) E0is barely above Etarget eand ii) ∆E=δǫis some small fraction of ǫ, the mean energy of the incoming particle. The second condition ensures that we may speak unambiguously of the inc oming particle’s mean energy. So, ∝angb∇acketleftQee(E)∝angb∇acket∇ight ≡/integraldisplay dE\|g(E)\|2Qee(E) (50) ≃1 ∆E/integraldisplay dEQee(E) (51) ∝angb∇acketleft∝angb∇acket∇ightdenotes the average over the ∆ Einterval. Now Qee(E) is just a sum of Lorentzians centred at the E(r) λ’s with width Γ λand Eq. (51) is just a measure of their mean value over the ∆Einterval. So long as the ∆ Einterval around which we are averaging, is broad enough to st raddle many of these Lorentzians, the mean height is just 1 ∆E×ρ(E)∆E×¯hπΓλe Γλ(52) where the second factor is the number of Lorentzians in the ∆E interval and the third factor is the area under the ‘ λth’ Lorentzian. This is true regardless of whether or not the y are overlapping. It will be convenient to write Γ λas Γλ=n×2¯kλvar(γλ) (53) where var(γλ) is the variance of the set of γλc′sover thenopen channels and ¯kλis a mean or eﬀective wavenumber kcover the open channels, which for a particular realization λwe take to be deﬁned by Eq. (53) itself. Let ∝angb∇acketleft ∝angb∇acket∇ightdenote the average over the occurrences of the quantity in the ∆ Einterval. Γ ≡ ∝angb∇acketleftΓλ∝angb∇acket∇ight,¯k≡ ∝angb∇acketleft¯kλ∝angb∇acket∇ight. Then Eq. (52) simpliﬁes ∝angb∇acketleftQee(E)∝angb∇acket∇ight ≃¯h1 Dke∝angb∇acketleftγ2 λe∝angb∇acket∇ight n¯k∝angb∇acketleftvar(γλ)∝angb∇acket∇ight(54) ≃¯h Γke ¯k(55) which tends to 0 as ke∼√ǫ. The form of Eq. (55) and all the steps leading up to it remain valid whether the Lorentzians are overlapping or not, as lon g as the ∆E= ∆ǫinterval which we are averaging over includes many of them. B. On an isolated resonance If the target is cold enough that the resonances are isolated , then as the incident particle’s energyǫ→0, adhering to the condition ∆ ǫ < ǫ will eventually result in ∆ ǫbecoming 13narrower than the resonance widths. It becomes possible the n for ∆ǫto be centered right around a single isolated resonance at E(r) λ. In this case ∝angb∇acketleftQee(E)∝angb∇acket∇ightis found simply by putting E=E(r) λ, because the spectrum \|g(E)\|2is well approximated by δ(E−E(r) λ). So ∝angb∇acketleftQee(E)∝angb∇acket∇ight=¯hΓλe Γ2 λ=¯h ΓλΓλe Γλ=¯h Γλke n¯k. (56) Even in this case there is the√ǫbehavior as ǫ→0 and there is no sticking. In the extreme case that there are no other open channels at al l (n= 1),∝angb∇acketleftQee(E)∝angb∇acket∇ight ≃ ¯hΓλe Γ2 λ=¯h Γλebecause Γ λ= Γλe. In fact,e= 1, and ∝angb∇acketleftQee(E)∝angb∇acket∇ightdiverges, implying in this case that it is possible to have the particle stick. This is an exce ption to all the cases above but is experimentally not so relevant because we may always expe ct to ﬁnd some exothermic channels open for a target with many degrees of freedom. VII. INELASTIC CROSS SECTIONS AND STICKING Another physically motivated measure of the sticking proba bility may be obtained by studying the total inelastic cross-section of the collisio n. The idea is that any long lived “sticking” is overwhelmingly likely to result in an inelast ic colision process; i.e. that the scattering particle will leave in a diﬀerent channel than it entered with. Using the original Wigner approach it is possible to show that for our case where we have only one scattering degree of freedom, the inelastic probability for an exother mic and endothermic collision vanishes like ke. The only possible exception to this is a measure zero chance of a resonance exactly at the threshold energy, Etarget e. In the event that there is a resonance E(r) λclose to but above this threshold energy, it is only necessary that Eis belowE(r) λ(by an energy of at least ∆E, the spread in energy) in order to observe the usual Wigner th reshold behavior: Pinelastic →0 likeke∝√ǫ (57) for the inelastic probability. However our problem is unusu al in the sense that because of the large number of degrees of freedom of the target, we will a lways ﬁnd resonances between Etarget eandEno matter how small E−Etarget e=ǫis. Thus the Wigner regime is not accessible. Still the surprise is that a simple computation reveals the same behavior holds for largen: Pinelastic (E) =/summationdisplay c/negationslash=ePc←e(E) (58) =/summationdisplay c/negationslash=e\|Sce(E)\|2(59) =/summationdisplay c/negationslash=e/summationdisplay λ/summationdisplay λ′Γλc1/2Γλe1/2 E−E(r) λ−iΓλ/2Γλ′c1/2Γλ′e1/2 E−E(r) λ+iΓλ/2(60) ⇒Pinelastic (E) =/summationdisplay λΓλ (E−E(r) λ)2+ (Γλ/2)2Γλe (61) where we used the random sign property of the Γ1/2 λc’s and the understanding that/summationtext c/negationslash=eΓλc≃ /summationtext all cΓλc= Γλ. Since the sum/summationtext c/negationslash=eis over the n≫1 open channels, omission of a single term 14can hardly matter. Apart from the factor ¯ h/Γλ, the rhs of the above equation is identical to the expression for Qee(E) in Eq. (48). Averaging Pinelastic (E) over many resonances E(r) λ (overlapping or not) we may use the same algebraic simpliﬁca tions as before to show ∝angb∇acketleftPinelastic∝angb∇acket∇ight=ke ¯k(62) Asketends to 0, this gives the√ǫWigner behavior showing that there is no sticking. The above argument fails when there is only one open channel. There are no inelastic channels to speak of. In this case, if the energy Ecoincides with a resonant energy E(r) λwe will have the exceptional case of sticking, as discussed at t he end of the previous section. But as pointed out there, this is primarily of theoretical in terest only. VIII. CHANNEL DECOHERENCE The only case for which we stick is seen to be the case of when we are sitting right on top of a resonance with the incoming energy so well resolved t hat we are completely within the resonance width, AND there are no exothermic channels op en. Having no such channels open amounts to an inﬁntesimally low energy for a large targe t. Otherwise, the sticking probability tends to 0 as√ǫin every case. A. Time dependent picture From the time independent point of view, the physical reason for the absence of low energy sticking is contained in the factorΓλe Γλof Eq. (48). This is the formation probability for the compound state. We will explain physically why it is s mall forn≫1. The resonance state is a many-body entangled state. If we imagine the decay of this compound state (already prepared by some other means say) each open channel carries away some fraction of the outgoing ﬂux, with no preference for any one particula r channel. Running this whole process in reverse it becomes evident that the optimum way to formthe compound state is to have each channel carry an incoming ﬂux with exactly the right amplitude and phase. This corresponds to however an entangled initial state. Wit h all the incoming ﬂux instead constrained to be in only one channel it becomes clear that we are not exciting the resonance in the optimal way and the buildup of amplitude inside is not s o large; i.e., the compound state has a small probability of forming. The time dependent view is even more revealing. Imagine a wav e packet incident on the system. For a single open channel Feshbach resonance, th e build-up of amplitude in the interior region can be decomposed as follows. As the lead ing edge of the wavepacket approaches the region of attraction, most is turned away due to the quantum reﬂection phenomena. (It is a useful model to think of the quantum reﬂec tion as due to a barrier located some distance away from the interaction region.) Th e wavefunction in the interaction region constructively interferes with new amplitude enter ing the region. At the same time, the amplitude leaving the region is out of phase with the reﬂe cted wave, cancelling it and assisting more amplitude to enter. 15Now suppose many channels are open. All the ﬂux entering the i nterior must of course return, but it does so fragmented into all the other open chan nels. Only the fraction that makes it back into the entrance channel has the opportunity t o interfere (constructively) with the rest of the entering wavepacket. The constructive inter ference is no longer eﬃcient and is in fact almost negligible for n≫1, thereby ruining the delicate process that was responsibl e for the buildup of the wave function inside. The orthogonali ty of the other channels prevents interference in the scattering dimension. If we trace over t he target coordinates, leaving only the scattering coordinate, most of the coherence and the con structive interference is lost, and no resonant buildup occurs. Therefore, one way to unders tand the non-sticking is to say that decoherence is to blame. B. Fabry-Perot and Measurement Analogy Suppose we have a resonant quantum mechanical Fabry-Perot c avity, where the particle has a high probability of being found in between the two reﬂec ting barriers. Now, during the time it takes for the probability to build up in the interior, suppose we continually measure the position of the particle inside. In doing so we decohere t he wave function and in fact never ﬁnd it there at all. Alternatively, imagine simply til ting one barrier (mirror) to make it non-parallel to the ﬁrst and redirecting the ﬂux into an orth ogonal direction, again spoiling the resonance. Measurement entangles other (orthogonal) d egrees of freedom with the one of interest, resulting in ﬂux being eﬀectively re-directed in to orthogonal states. Thus the states of the target (if potentially excitable) are in eﬀect contin ually monitoring (measuring) to see if the incoming particle has made it in inside, ironically th en preventing it from ever doing so. The buildup process of constructive interference in the interaction region, described in the preceding paragraph, is slower than linear in t. Therefore, the constant measurement of the particle’s presence (and resultant prevention of sti cking) is an example of the Zeno “paradox” in measurement theory. IX. CONCLUSION We have presented a general approach to the low energy sticki ng problem, in the form ofR-matrix theory. This theory is well suited for the task, sinc e it highlights the essential features of multichannel scattering at low incident transl ational energy. We did not need to make a harmonic or other approximate assumptions about th e solid target, which is characterized by its long range interaction with the incomi ng particle and its density of states. “Warm” surfaces are included in the formalism, and d o not change the non-sticking conclusion. Several supporting arguments for the non-sticking conclus ion were given. Perhaps most valuable is the physical decoherence picture associated wi th the conclusion that there is no sticking in the zero translational energy limit. Reviewing the observations leading up to the non-sticking c onclusion, we start with the near 100% sticking in the zero translational energy limit cl assically (sticking probability 1). We then invoke the phenomenon of quantum reﬂection (Fig. 1), which keeps the incident particle far from the surface (sticking probability 0). Thi rd, we note that quantum reﬂection 16can be overcome by resonances (Fig. 2), and since resonances are ubiquitous in a many body target, being the Feshbach states by which a partice could st ick to the surface, perhaps sticking approaches 1 after all. Fourth, we suggest that dec oherence (from the perspective of the incoming channel, with elestic scattering deﬁnded as coherent) ruins the resonance eﬀect, reinstating the quantum reﬂection as the determinin g eﬀect. Finally, then, there is no sticking, and the short answer as to why is: quantum reﬂe ction and many channel decoherence. The ultrashort explanation is simply quantum reﬂection, but this is dangerous and non-rigorous, as we have tried to show. All this does not tell us much about how sticking turns on as in cident translational energy is raised. This is the subject of the following paper, where a WKB analysis proves very useful. Quantum reﬂection is a physical phenomenon lik ed directly to the failure of the WKB approximation. ACKNOWLEDGMENTS This work was supported by the National Science Foundation t hrough a grant for the In- stitute for Theoretical Atomic and Molecular Physics at Har vard University and Smithsonian Astrophysical Observatory:National Science Foundation A ward Number CHE-0073544. APPENDIX A: Γ≃ND With the large number of degrees of freedom involved and assu ming thorough phase space mixing associated with the resonance we may reasonabl y describe the compound state wavefunction by a classical ensemble of points ( x,px,u,pu) in the combined phase space of the joint system given by the normalized distribution 1 ρC(E)δ(E−H(x,px,u,pu)). (A1) It is understood in the above that the system is restricted to be in the region x<a . This makes all accessible states of energy Ewithx < a equally likely. Then the rate of escape Γ/¯hthrough the hypersurface x=aof the members of this ensemble is Γ ¯h=1 ρC(E)/integraldisplay x=adudpu/integraldisplay px∈[0,∞]dpxpx mδ(E−H(x,px,u,pu)). (A2) px/mis just the velocity in phase space of a point at x=ain the ˆxdirection. At x=awe have supposed no interaction. Hence the Hamiltonian separa tes in Eq. (A2). Therefore Γ ¯h=1 ρC(E)/integraldisplay dudpu∞/integraldisplay 0d/parenleftBiggp2 x 2m/parenrightBigg δ/parenleftBigg E−/parenleftBiggp2 x 2m+Htarget(u,pu)/parenrightBigg/parenrightBigg (A3) =1 ρC/integraldisplay Htarget(u,pu)<Edudpu (A4) =1 ρCΩC≃1 2π¯hρQΩQ=1 2π¯hnD. (A5) 17ThereforeΓ D≃n.ρQ(ρC) is the quantum (classical) density of states (phase space v olume) of the joint system at energy E. Ω Q(ΩC) is the quantum (classical) total number of states (total phase space volume) of only the target below energy E. We have used the correspondence between the Classical and Quantum density of states. 1 /ρQis identiﬁed with D, and the number of states of the target having energy less that Eis justn, the number of open channels. APPENDIX B: INELASTIC PROBABILITY WITH BACKGROUND We show here that the inelastic probabilities remain essent ially unaﬀected in magnitude with the presence of a background term in the S-matrix. In the isolated case the addition ofbcc′to an inelastic element Scc′simply changes the Lorentzian proﬁle of \|Scc′\|2. In the more important overlapping case, the energy variation of Scc′is smooth in any case without background and \|Scc′\|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleBcc′−i/summationdisplay λΓ1/2 λcΓ1/2 λc′ E(r) λ−E−iΓλ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (B1) =\|Bcc′\|2+/summationdisplay λΓλcΓλc′ (E(r) λ−E)2+ Γλ2/4(B2) where we have used the random sign property of the products Γ1/2 λcΓλc1/2to neglect the 2nd cross-term in comparison to the last one where again the same property is used to simplify the double sum to a single one. Summing over all the inelastic channels then leads to the same result of Eq. ( 61) with an added term of/summationtext c/negationslash=e\|Bcc′\|2which itself is proportional to ke as discussed at the end of Section VB. [1] J. E. Lennard-Jones et. al. ,Proc. R. Soc. London, Ser. A 1566, (1936); Ser. A 15636, (1936). [2] T.W. Hijmans, J.T.M. Walraven, and G.V. Shlyapnikov, Phys. Rev. B45, 2561 (1992). [3] W. Brenig, Z. Phys. B36, 227 (1980). [4] D.P. Clougherty and W. Kohn, Phys. Rev. B,464921 (1992). [5] E. R. Bittner, J. Chem. Phys. 100, 5314 (1993). [6] P. S. Julienne and F. H. Mies, J. Opt. Soc. Am. B 6, 2257 (1989). [7] P. S. Julienne, A.M. Smith, and K. Burnett, Adv. At. Mol. O p. Phys. 30, 141 (1992). [8] L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Non-relativistic Theory) (Pergamon Press, Oxford (UK) 1981). [9] C.J. Joachain, Quantum Collision Theory (North-Holland, Amsterdam 1975). [10] G.F. Gribakin and V. V. Flambaum, Phys. Rev. A48546 (1993). [11] R. Cˆ ote´, E. J. Heller, and A. Dalgarno, “Quantum suppression of cold atom collisions” Phys. Rev. A 53, 234-41 (1996). 18[12] I. A. Yu, J. Doyle, J. C. Sandberg, C. L. Cesar, D. Kleppne r, and T. J. Greytak, Phys. Rev. Lett711589 (1993). [13] J. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppne r, and T. J. Greytak, Phys. Rev. Lett67603 (1991); C. Carraro and M.W. Cole, Phys. Rev. B45, 12931 (1992); T.W. Hijmans, J.T.M. Walraven, and G.V. Shlyapnikov, Phys. Rev. B45, 2561 (1992). [14] F. T. Smith, Phys. Rev. 118, 349 (1960). [15] N. Bohr and J. Wheeler, Phys. Rev. 56(5),416-450 (1939). see p. 426 Sec. III. 19
arXiv:physics/0010069v1 [physics.atom-ph] 27 Oct 2000Transition from reﬂection to sticking in ultracold atom-su rface scattering Areez Mody, John Doyle and Eric J. Heller Department of Physics, Harvard University, Cambridge, MA 0 2138 (August 2000) In paper I [1] we showed that under very general circum- stances, atoms approaching a surface will not stick as its in - coming energy approaches zero. This is true of either warm or cold surfaces. Here we explore the transition region from non sticking to sticking as energy is increased. The key to un - derstanding the transition region is the WKB approximation and the nature of its breakdown. Simple rules for understand - ing the rollover to higher energy, post-threshold behavior , in- cluding analytical formulae for some asymptotic forms of th e attractive potential are presented. We discuss a practical ex- ample of atom-surface pair in various substrate geometries . We also discuss the case of low energy scattering from clus- ters. I. INTRODUCTION The problem of sticking of atoms to surfaces at very low collision velocities has a long history and has met with some controversy. The issue goes back to the early distorted wave Born approximation results of Lennard- Jones [3], who obtained the threshold law sticking prob- ability going as kin the limit of low velocities. This paper is a companion to paper I [1], wherein we put the problem on a ﬁrmer theoretical foundation. We showed (non-perturbatively) that in an ultracold collision a sim- plistic one-body view of things is essentially correct even if the number of internal degrees of freedom is very large. We concluded that approaching atoms will not stick to surfaces if the approach velocity is low enough, even if the surface is warm. From the methods used, it is clear that the non-sticking rule would apply to clusters as well as semi-inﬁnite surfaces, and would also apply to projectiles more complex than atoms. From an experimental perspective atom-surface stick- ing could impact the area of guiding and trapping atoms in material wires and containers. In those applications it is necessary to predict the velocities needed for quantum reﬂection, sticking, and the transition regime between them. We do so in this paper. Above a certain temperature or kinetic energy, but still well below the attractive well depth of the atom surface potential, atoms will stick to surfaces with near 100 per- cent certainty. The reason for this is simple: Classical trajectory simulations of atom-surface collisions at low collision velocities indicate sticking with near certaint y because the acceleration in the attractive regime is fol- lowed by a hard collision with the wall. This almost always leads to suﬃcient energy loss from the particle to the surface that immediate escape is not possible. Thisis true so long as the approach energy is signiﬁcantly less than the well depth, which is itself greater than the temperature of the surface. Therefore, the onset of quan- tum reﬂection is heralded by a break down in the WKB approximation - an approximation based purely on the (sticking) classical trajectories. Thus, there must exist a transition region between the non-sticking regime for very low collision velocities, and the sticking regime at higher velocities. The key to un- derstanding the transition region is to understand the validity of classical mechanics (WKB) as applied to the sticking problem. The correctness of the simplistic one- body physics of quantum reﬂection from the surface, fo- cusses our study on the WKB approximation to the co- ordinate normal to the surface. The entrance channel wavefunction thus obtained may also be used as input into the Golden Rule to study the threshold behaviour of the inelastic cross-sections. We do this in Section VI. The nonsticking threshold behavior we established in pa- per I is interpreted as an extension of the validity of the so-called Wigner threshold behavior. We are also able to make deﬁnite predictions about the nature of the post threshold behavior of sticking in terms of inelastic cross- sections. (Section VI). II. QUANTUM REFLECTION AND WKB We consider the typical case of an attractive potential arising out of the cumulative eﬀect of Van der Waals at- tractions between target and incident atoms. A classical atom would proceed straight into the interaction region showing no sign of reﬂection, but the quantum mechan- ical probability of being found inside is suppressed by a factor of k(ask→0) as compared to the classical proba- bility (Section VI A1 ), where kis the wave vector of the incoming atom. This is tantamount to saying that quan- tum mechanically the amplitude is reﬂected back without penetrating the interaction region, analogous to the ele- mentary case of reﬂection from the edge of a step-down potential in one dimension while attempting to go over the edge. A useful way to view this is to attribute the reﬂection to the failure of the WKB approximation. To be speciﬁc, we keep the geometry of paper I in mind: an atom is incident from the right ( x >0) upon the face of a slab ( x= 0) that lies to the left of x= 0. For a low incoming energy ǫ≡¯h2k2/2m, a left-moving WKB solution begun well inside the interaction region will fail to match onto a purely left-going WKB solution as we integrate out to large distances because the WKB 1criterion \|λ′(x)\|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯hp′ p2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1 (1) for the local accuracy of the wavefunction will in general fail to be valid in some intermediate region. For bounded potentials that turn on abruptly for example at x=a, it is obvious that WKB will fail near x∼a. For long-range potentials such as a power law V(x) =−cn/xnit is not immediately obvious where this region of WKB failure lies, if it exists at all. It turns out that even in this case it is possible to identify (for small enough ǫ) a distance (dependent on ǫ) at which the potential ‘turns on’ and where WKB will fail. We will show below that WKB is at its worst ( \|λ′(x)\|is maximized) at a distance xwhere the kinetic and potential energies are approximately equal, i.e where \|V(x)\| ∼ǫ. The distance away from the slab at which the particle is turned around - or quantum re- ﬂected - is precisely this distance. Furthermore, one may heuristically expect that the greater the failure of WKB, the greater the reﬂection. Fig. 1 shows a plot of the error term in Eq. (1) for three diﬀerent values of the incoming energy of neon on a semi-inﬁnite slab of SiN. The essential points to notice are: 1) There is a greater error incurred in attempting to apply the WKB (classical mechanics) approximation to colder atoms than to warmer ones. Consequently, we ex- pect that the slower the atom, the more non-classical its behavior. In particular, slow enough atoms will be ‘quan- tum reﬂected’ and will not stick. 2) As the incoming velocity is decreased the atom is reﬂected at distances progressively further and further away from the slab. This is because the interval in x around which the WKB error is large, may be identiﬁed as the region from which the atom is reﬂected. A useful qualitative rule of thumb obtained in Section III below is that the region of WKB error reaches all the way out to those regions where the potential energy is still roughly the same order of magnitude as the incom- ing energy (Eq. (5)). This means that as ǫ→0 the error is still large where the potential energy graph looks essen- tially ﬂat. In fact as ǫ→0 it is easily shown that a plot of the WKB error will show a non-uniform convergence to a polynomial proportional to xn 2−1for all n >0 (2) Fig. 1 shows the case for n= 3. III. WKB FAILURE Diﬀerentiating p2/2m+V(x) =ǫw.r.t. x, we have p′=−mV′ p(3)1060 2120 3180 42401357 50 100 150V(x) = -2.55 K nm3 x3WKB error (nm)x Energy (nK)200 nK2 nK0.02 nK FIG. 1. The WKB error of Eq. (1) for three diﬀerent values of the incoming energy 200, 2 and 0.02 nK, vs. the dis- tance xnm from the slab (SiN). The long range form of the potential −c3/x3(c3=220 meV ˚A3) is also shown for which the negative ‘y-axis’ is calibrated in the diﬀerent units of en- ergy. The sticking probabilities for the three cases are ap- proximately 1, 0.6, 0.1. which when used repeatedly to eliminate p′shows that \|p′/p2\|in Eq. (1) is maximized when p2 3m=V′2 V′′. (4) ForV(x) =−cn/xn, this is exactly when \|V(x)\|=ǫ/parenleftbigg2(n+ 1) n−2/parenrightbigg . (5) We discover that for n >2 only, we have a point where WKB is at its worst at a distance xwhere \|V(x)\| ∼ǫ, and moreover, that this maximum behaves like max/vextendsingle/vextendsingle/vextendsingle/vextendsinglep′ p2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼1 c1/n nǫ1 2−1 n∼1 c1/n nk1−2 n(6) which for n >2 diverges as k→0. Note how a weaker potential (smaller cn) isbetter at reﬂecting a particle at the same energy, but allows the atom to approach closer. Heuristically a sketch of V(x) =−cn/xnreveals why: the weaker potential is seen to turn on more abruptly at a point closer to x= 0, promoting an greater breakdown of WKB there. Alternatively a simple scaling argument with Schrodinger’s equation reveals the same trend. The above conclusions are valid only for n >2. For n≤2 the error term of Eq. (1) looks qualitatively dif- ferent from that in Fig. 1. It is small at all distances except near x= 0 where it diverges to inﬁnity, as is ev- ident from Eq.(2). If the physical parameters are such that this region where WKB fails very close to the slab is never actually manifest in the long-range part of the potential then the ‘no-reﬂection’ classical behaviour wil l be valid all the way up to distances near the slab where the atom will begin to feel the short range forces and loose energy to the internal degrees of freedom. For such 2a case then with n <2 we believe one will notobserve quantum reﬂection. IV. STICKING PROBABILITY Having established that the reﬂection is caused by a well-deﬁned localized region, we solve the one- dimensional Schrodinger equation around this region to accurately compute the reﬂection probability. For an at- tractive power law potential V(x) =−cn/xn, the rele- vant one dimensional equation is /parenleftbiggd2 dx2+an−2 n xn+k2/parenrightbigg φe(x) = 0. (7) φe(x) is the entrance channel wavefunction. The length scale an≡(2mcn/¯h2)1 n−2, (8) contains all the qualitative information about the reﬂec- tion. Its relevance is twofold. Firstly, the sticking prob- ability for small k, behaves as Psticking ∼Nnk an (9) where Nnis a pure numeric constant (roughly of order 10 for n= 3, and of order 1 for n= 4,5), see Ref. [6].Psticking may be computed numerically for any k, and Fig.2 shows Psticking vs.kanforn= 3,4, and 5. Secondly, the distance at which the particle is turned around is estimated by solving /parenleftBigan x/parenrightBign = (kan)2(10) for x, which is just the requirement that \|V(x)\|=ǫ. Equation (9) together with equation (8) makes plain that a smaller cnis more conducive to making quantum reﬂec- tion happen, while Eq. (10) indicates that the turnaround point is then necessarily closer to the surface. With these eﬀects in mind, we look at some speciﬁc cases. V. EXAMPLES We examine the case of incidence on a slab which may be treated as semi-inﬁnite, and also the case when it is a thin ﬁlm. It is useful to ﬁrst look at these cases pretend- ing there is no Casimir interaction, and assuming that the short range form of the potential is everywhere valid. Afterwards we put in the Casimir interaction. For clar- ity we will pick a speciﬁc example of target and incident atoms for most of our discussions, by specifying the nu- meric values for the short range potential between the atom and semi-inﬁnite slab, since these are most com- prehensively tabulated in reference [4].0.2 0.4 0.6 0.8 11.2 1.40.20.40.60.81 k anvs. k a 3 vs. k aP 4 vs. k a 5sticking FIG. 2. Sticking probabilities for an atom incident on surface providing a long range interaction of the form V(x) =−cn/xnfor the cases n=3,4,5. Note that the length scale anused to compute the dimensionless kancoordinate on the ‘x-axis’ vs. which we plot the sticking probabilities , is diﬀerent for each n. Fig. 3 shows the sticking probability vs. the temper- ature of an incoming Ne atom in units of 10−9Kelvin. The slab is silicon-nitride (SiN). The various curves are for the diﬀerent cases depending on whether we are we are considering a thick or thin slab, and whether the Casimir eﬀect is included or not. We will discuss these cases be- low, pointing out the relevant length and energy scales involved in deciding to label the slab as semi-inﬁnite or thin. The mapping from the mathematically natural kan (with n= 3 and c3= 220 meV ˚A3) scale of Fig. 2 to the more physical temperature scale of Fig. 3 is made using T≃[69.08 Kelvin]/parenleftbiggmH matom/parenrightbigg3/parenleftBigg meV˚A3 c3/parenrightBigg2 (ka3)2 (11) where we used ∝angbracketleftǫ∝angbracketright= (3/2)kBTto compute the tem- perature by setting ∝angbracketleftǫ∝angbracketrightequal to the incoming energy. mH= mass of hydrogen atom, and for our example mNe= 20.03mH. All the graphs in Fig. 3 have an initial slope of 0.5 indicating the√ǫbehaviour of the sticking probabilities once the energies are low enough to be in the Quantum Reﬂection regime. A particular temperature at which there is a transition to the post-threshold sticking regime , we arbitrarily (but intuitively) deﬁne as the temperature where the slope becomes 0.4. For the thin ﬁlm case of 10 nmin our example this temperature is 10 nK. While the parameters in our example are fairly typi- cal, it is clear that the cubic dependence on mass and quadratic dependence on the c3coeﬃcient in Eq.(11), will make this temperature range over quite a few or- ders of magnitude. The c3’s in Ref [4], listed in units of meV−˚A3for a variety of surface atom pairs, range in values from 100 to 3000. 30.1 1 10. 100. 1000.1 .1 .01thin slab (10 nm) thin slab (10 nm)(C)stickingP thin slab (1 nm) 0.01Temperature (nK)semi-infinite slabsemi-infinite slab(C) FIG. 3. Sticking probabilities vs temperature of incident Ne atoms on SiN. The broken line indicate the inclusion of the very long range Casimir forces (see text). The large dot demarcates the regions of threshold and post-threshold, us ing the criterion suggested at the end of Section V A. Semi-Inﬁnite Slab (without Casimir) Even though c3coeﬃcients are known both theoret- ically and experimentally for many surface-atom pairs, for completeness we take a moment to look at a quick way of estimating them. This is provided by the London formula Vatom−atom(r) =−3 2IAIB IA+IBαAαB r6≡−c6 r6,(12) which estimates the Van der Waals interaction between any two atoms. Iis the ionization potential and αthe polarizability of each atom. Then summing over all the atoms in the semi-inﬁnite slab (thick) we get Vslab−atom(x) =−πc6ρatoms 6×1 x3≡−c3 x3(13) where ρatoms = the density of slab atoms. These esti- mates are not very accurate, but correctly indicate the physical quantities on which the answer depends. Ref- erence [4] provides a useful compendium of these coef- ﬁcients. We have used c3= 220 ±4 meV ˚A3for neon atoms incident on silicon nitride from work of [5]. This choice of c3makes a3≃212 nm (14) Thus the ‘semi-inﬁnite slab’ curve of Fig. 3 is the n= 3 curve of Fig. 2 scaled to temperature units using Eq. (11). B. Thin Slab (without Casimir) From far enough away any slab will appear thin. The surface-atom interaction will behave like −c3 x3−−c3 (x+d)3≃−3dc3 x4(15)forx≫d, where d is the thickness of the slab. The resulting c4coeﬃcient equal to 3 dc3gives an a4coeﬃcient that can be written as a4=/radicalbigg 2m ¯h23dc3=a3/bracketleftbigg3d a3/bracketrightbigg(1/2) . (16) For macroscopic values of d(≫a3) then, it is only for vanishingly small incident energies that the ﬁniteness of the slab becomes apparent. For any macroscopic dthis will be physically irrelevant. For microscopic d(≪a3) however, this window in energy over which the thinness of the slab makes an appreciable diﬀerence can be larger and even prevail for all energies. To continue our illustrative example we pick the microscopic value of d= 10 nm. This makes a4≃800nm. The ‘thin slab’ curve of Figure 3 shows that the sticking probabilities are substantially reduced and the onset of Quantum Reﬂection occurs at a much higher energy. As a benchmark case, we also include what will likely be the physically limiting case for a continuous ﬁlm of d= 1 nm. This further reduces the sticking prob- abilities for a ﬁxed temperature by a factor of√ 10, because the important quantity a4is reduced by this much. (Eq.(16)). The transition temperature appears to have increased by 3 orders of magnitude versus the semi-inﬁnite case. C. Semi-inﬁnite slab (Casimir Regime) As the incoming energy ǫtends to 0, we have seen that the turn-around region from which the atom ‘quantum reﬂects’ moves progressively further away from the slab. But at large distances, however, it is well known that the interaction potential itself takes on a diﬀerent form due to Casimir eﬀects. In particular, a semi-inﬁnite dielectri c slab (dielectric constant ǫs) has an interaction potential with an atom of polarizability αgiven by Vslab−atom(x) =−3 8π¯hcα x4ǫs−1 ǫs+ 37/23x→ ∞ (17) =−235(eV−˚A)α x4ǫs−1 ǫs+ 37/23x→ ∞ (18) Even for suﬃciently large x, the form above is not exact but a good approximation found in Ref. [7]. Our purpose here is only to estimate the various numbers to see their relevance. It will suﬃce to put αNe= 0.39˚A3and the last factor involving ǫsis replaced by 1 since most solids and liquids have ǫssubstantially greater than 1. This gives ac(C) 4coeﬃcient of 9 ×104meV˚A4and hence a resulting a(C) 4= 93 nm. The superscript ‘C’ reminds us it is due to the Casimir interaction which is valid only for large enough x. To estimate the distance beyond which the Casimir form itself is valid, we use the statement from Ref. 4[8]: ‘Within a factor of 2, the van der Waals poten- tial is correct at distances less than 0.12 λtr, while the Casimir potential is correct at distances at longer range.’ λtr= [1,240 nm](eV ∆E) here is the wavelength associated with the transition between the ground and excited state that gives the atom its polarizability. ∆ Eis the tran- sition energy. Knowing this much we may deduce the qualitative features of the sticking probability curve the arguments being similar to the cases above. For this Casimir case and the one below, however, there is a caveat to all this. The exact manner in which the potential changes its near range form to its long-range Casimir form can certainly aﬀect the sticking probabili- ties at the intermediate energy where it makes this tran- sition. Some numerical experimentation choosing arbi- trary forms of the potential having the correct short range and long-range behavior, conﬁrms this. Therefore the curves in ﬁgure 3 involving Casimir forces are only quantitatively and notquantitatively correct. D. Thin Slab (Casimir Regime) Even for a thin slab we expect that the distance at which the Casimir interaction is valid remains the same as for a semi-inﬁnite slab made of the same material. At these distances if x≫dis also valid, then one may expect the surface atom interaction to behave like −c(C) 4 x4−−c(C) 4 (x+d)4≃−4dc(C) 4 x5(19) The length scale a(C) 5=a(C) 4/bracketleftBigg 4d a(C) 4/bracketrightBigg1/3 (20) associated with this c5= 4dc(C) 4coeﬃcient makes a(C) 5= 717nm .Figure 3 shows a slight decrease in the sticking probabilities, the eﬀect being evidently less here than in the case of the thick slab. E. Hydrogen on ‘thick’ Helium Rather atypical, but extremely favourable parameters (c3= 18 meV ˚A3) are found in the case of Hydrogen atoms incident on bulk liquid Helium. Evidence for quan- tum reﬂection was experimentally seen in this system. [9] A comparison with the parameters used in our example of Ne on SiN: mNe/mH= 20.03 and c(Ne−SiN) 3 /c(H−He) 3 = 220 /18. (21) With the use of Eq. (11), we see that the sticking proba- bilities for this case are in fact the same curves as in Fig- ure 3 except shifted to the right in temperature by about6.1 orders of magnitude. This puts it exactly in the milli- Kelvin regime where sticking probabilities of about 0.01 to 0.03 were observed as temperatures ranged from about 0.3 mK to 5 mK. [9] However, the sticking probabilities predicted by the ‘semi-inﬁnite slab(C)’ curve of Fig. 3 are about a factor 2.5 too large, but we feel there is good reason for this. We already mentioned the qualitative manner in which the Casimir forces were included but it seems that a greater error is caused for another reason. The length scale a3= 17 nm for H-He is so small that the WKB error is close in (see Fig. 1) where the interac- tion potential is not exactly of the form ∼1/x3. Practi- cally speaking this means that the region over which we must integrate Eq.(7) must include points close to the slab to get some convergence and thus we are violating the assumption that the potential is ∼1/x3there. This problem would not plague the Ne-SiN case too much, be- cause the length scale there is substantially bigger. For H-He we must include some short-range information to get an improvement. Still it is the long-range forces that are mostly responsible. Ref [10,11] and others have mod- eled this close range behaviour and obtained better agree- ment; the improvement coming from explicit considera- tion of the bound states supported by the close range potential. These appear in the potential matrix elements of perturbation theory. VI. RELATION TO THRESHOLD BEHAVIOR We now wish to take a broader view of the quantum re- ﬂection behaviour at threshold ( k→0), and the sticking that sets in as the energy is increased - a Post Thresh- old behavior. In particular we want to make connection to, and extend the well-known threshold behaviors of in- elastic rates which were ﬁrst stated most generally by Wigner in Ref [12]. For example, Wigner showed that the exothermic excitation rates for collisions between two bodies with bound internal degrees of freedom tend to a constant value as their relative translational energy tend s to 0, provided there is no resonance at the 0 translational energy threshold. Equivalently, the exothermic inelastic cross-section diverges as 1 /v, a fact known in the still older literature as the ‘1 /vlaw’. vis the relative ve- locity of the collision. Notice especially the proviso in the statement above, that there be no resonance at the threshold energy; suggesting that the many resonances between 0 and ǫprovided by a many body target could make the law inoperative. But the entire thrust of Part 1 was to establish quite generally that this many-resonance regime was precisely the one for which the old ‘1 /vlaw’ is reinstated. Here we re-examine the Wigner behaviour from a dif- ferent point of view using our understanding of quan- tum reﬂection. In addition to furthering an intuitive un- derstanding of the Wigner behaviour, viewing things in this way will lead naturally to predicting a generic post 5threshold behavior (e.g. the 1/v law is replaced by a 1 /v2 law) and an understanding of when the sticking sets back in asǫis increased. The reader will have noted that we have shifted our attention to a three dimensional geometry of incidence on a localized cluster instead of the one dimensional case of incidence on a slab. So long as the target dimensions are dwarfed by the incidence wavelength we will ﬁnd that both problems are eﬀectively one dimensional due to the fact that it is only the s-wave which can penetrate the interaction region. For clarity we will deal with both cases separately. A. Threshold and Post-Threshold Inelastic Cross-sections The starting point is the template provided by the golden rule dσe→c∝1 kρ(Ec)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay all/vectorrd3r φ(−) c,/vectorkc(/vector r)Uce(/vector r)φ(+) e,/vectork(/vector r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (22) for the diﬀerential cross-section for inelastic transitio ns from internal state Ω e(u) to Ω c(u) where /vectorkand/vectorkcare the incoming and outgoing directions of the incident atom. We describe brieﬂy how Eq. (22) is arrived at. For each internal state Ω c(u) (c= 1,2,···n) that we may imagine freezing the target in ( uincorporates all the target degrees of freedom), there is some eﬀective poten- tial felt by the incoming atom. These potentials are just the diagonal elements of the complete interaction poten- tialU(x, u) in the Ω c(u) basis, which if present all by themselves (oﬀ-diagonal elements 0) could only cause an elastic collision to occur. It is the oﬀ-diagonal elements that may be thought of as causing the inelastic transi- tions. Treating them as a perturbation on the elastic scattering wavefunctions we use the Golden Rule to ob- tain Eq.(22). ρ(E) is the energy density of states of the free atom. φ(+) e,/vectork(/vector r) is the entrance channel wavefunction andφ(−) c,/vectorkc(/vector r) is the ﬁnal channel wavefunction. They are both exact elastic scattering wavefunctions in the poten- tialsUee(/vector r) and Ucc(/vector r) respectively. The factor of 1 /k divides the Golden Rule rate by the ﬂux to get the prob- ability. Now all the kdependence of dσe→cand hence σe→cis due to 1) the factor 1 /kand 2) The sensitive k-dependence of the amplitude of the en- trance channel wavefunction inside the interaction region over which the overlap integral of Eq.(22) takes place. This is simply because the incoming amplitude is more reﬂected away by the potential as k→0 resulting in the interior amplitude being suppressed by a factor of kas compared to what one would expect classically.1. Incidence on a Slab For this one-dimensional situation we speak of an in- elastic probability instead of a cross-section, but other- wise Eq.(22) remains entirely valid here also with the obvious modiﬁcations. Fork→0, when WKB is invalid, we established quite generally [2] that the entrance channel wavefunc- tionφe(x) when normalized to have a ﬁxed incoming ﬂux, had its amplitude inside the interaction region behaving like φe(xinside)∼k Threshold (23) Now the change from quantum reﬂection at threshold to sticking at post threshold (see Fig.2) begins to a oc- cur at those energies at which the WKB wavefunctions - which show no quantum reﬂection - may be increas- ingly trusted. At these energies where WKB is valid we may simply use the well-known WKB amplitude factor 1//radicalbig k(x), to conclude that φe(xinside)∼√ k Post−Threshold .(24) The probability density of being found inside then be- haves like k2at threshold (quantum reﬂection) and like kat post threshold (no quantum reﬂection) respectively. It is quite natural that the probability density inside the interaction region is smaller compared to the out- side by a factor of k, even when there is no quantum reﬂection. This is simply a kinematical eﬀect: where the particle is moving faster, it is less likely to be found by a factor inversely proportional to its velocity there. Class i- cally what is unexpected is that for small enough knear threshold, the probabilities inside are further suppressed by a factor of k. Quantum reﬂection of the amplitude from the region around \|V(x)\|=ǫ(section II), goes hand in hand with the quantum suppression of the amplitude within this region. So ﬁnally including this k-dependence of the amplitude of φe(x) found in equations (23) and (24) we get Pe→c∝k Threshold Pe→c∝const. Post−Threshold (25) 2. Incidence on a cluster Since for large wavelengths only the s-wave interacts with the cluster it is clear that the problem may be re- duced in the usual manner to a one dimensional problem again. Therefore for a unit s-wave ﬂux the inelastic prob- abilities will behave as before as in equations (25), but what is really relevant is a unit plain wave ﬂux which provides a s-wave ﬂux of π/k2. i.e. Even though the problem is one-dimensional in the radial co-ordinate, the required normalization for the incoming ﬂux is not ﬁxed 6to be a constant as before, but is now required to grow as ∼1/k2, in order to correctly account for the increasing (ask→0) range of impact parameters that all ‘count as’ s-wave. Thus we have simply to multiply the one- dimensional probabilities of equations (25) by this factor of 1/k2, and conclude that the inelastic cross-sections for this cluster geometry behave like σe→c∝1 kThreshold σe→c∝1 k2Post−Threshold (26) The Threshold result of Eq. (26) is just the Wigner ‘1/v law’ we spoke of in section VI. But now we can say more. As the incoming wavelength λincreases, we ﬁrst witness for large enough λa quadratic dependence to the exother- mic cross-section ( σ∝λ2). It is only at still larger wave- lengths that this dependence eventually changes over to a linear one ( σ∝λ). This happens when the sticking yields to the quantum reﬂection. This energy is mostly determined by the long range form of the potential, and has nothing to do with the bound state energies or any other details involving the interaction potential. VII. CONCLUSION Examining the WKB error term provided a quick and easy way to estimate the threshold temperatures required to observe quantum reﬂection. It became transparent that only power laws dying faster than −1/x2were ca- pable of acting as quantum reﬂectors. The validity of WKB at higher temperatures heralded a post-threshold behavior in which the atom sticks. Even for other geome- tries such as incidence on a localised three dimensional cluster, a WKB analysis together with the Fermi Golden Rule provided a simple understanding of this threshold and post-threshold behavior in terms of inelastic pro- cesses being shut oﬀ due to a reﬂection of the incoming amplitude. The extremely long incoming wavelength is invariably impedance mismatched (for potentials shorter ranged than 1 /r2) by the abrupt change of wavelength in the interaction region, and is therefore reﬂected. It should be clear that even a repulsive interaction will ob- viously provide such a mismatch so that the Wigner be- havior, or quantum reﬂection, is quite general; though of course most dramatic if the potential is attractive as we have been considering throughout. This eﬀect of quantum reﬂection/suppression, which is ultimately responsible for the threshold behaviour, is dynamical in that it is caused by the presence of the interaction potential. We feel that the original derivatio n by Wigner that focuses on the k→0 behaviour of the free space wave functions tends to obscure this physical origin of threshold behaviour. The golden rule approach makes it more explicit and especially paves the way for predicting the Post-Threshold behaviour.ACKNOWLEDGMENTS A.M. is most grateful to Michael Haggerty for his kind help and advice and for always being available to discus things with. A.M. thanks Alex Barnett for pointing him to the references on the Casimir interactions. This work was supported by the National Science Foundation through a grant for the Institute for Theoret- ical Atomic and Molecular Physics at Harvard University and Smithsonian Astrophysical Observatory:National Science Foundation Award Number CHE-0073544. This work was also supported by the National Science Foundation by grants PHY-0071311 and PHY-9876927. [1] paper1 [2] We gave the heuristic elementary reason for the sup- pressed behaviour of Eq. (23) in Sec III of paper I,where we then attempted to rigorously justify it. [3] J. E. Lennard-Jones et. al. ,Proc. R. Soc. London, Ser. A1566, (1936); Ser. A 15636, (1936). [4] G. Vidali, G. Ihm, Hye-Young Kim, and Milton Cole, Surface Science Reports 12, 133-181 (1991) [5] R. E. Grisenti, W. Schoelkopf, J. P. Toennies, G. C. Hegerfeldt, and T. Khler Phys. Rev. Lett. 83, 1755 (1999) [6] R.Cote et. al. Phys. Rev. A56, 1781 (1997) [7] L. Spruch, Y. Tikochinsky Phys. Rev. A48, 4213 (1997) [8] E.A. Hinds V. Sandoghdar, Phys. Rev. A43, 398 (1991) in particular section II. [9] Ite A. Yu, John M. Doyle, et. al. Physical Review Letters Vol. 71(10), 1589 (September 1993). [10] C. Carraro and M. Cole Physical Review B. Vol. 45(22), 1589 (June 1992). [11] T.W. Hijmans and J.T.M. Walraven and G.V. Shlyap- nikov Physical Review B. Vol. 45(5), 2561 (February 1992). [12] E. P. Wigner, Physical Review , Vol. 73, 1002 (1948) 7
arXiv:physics/0010070v1 [physics.gen-ph] 27 Oct 2000Quantum Fluctuations of the Gravitational Field and Propagation of Light: a Heuristic Approach1. Stefano Ansoldi∗and Edoardo Milotti∗∗ (∗)Dipartimento di Fisica Teorica dell’Universit` a di Triest e and I.N.F.N. - Sezione di Trieste, Strada Costiera, 11 - I-34014 Miranare - Trieste, Italy e-mail: ansoldi@trieste.infn.it (∗∗)Dipartimento di Fisica dell’Universit` a di Udine and I.N.F.N. - Sezione di Trieste, Via delle Scienze, 208 - I-33100 Udine, Italy e-mail: milotti@fisica.uniud.it Abstract Quantum Gravity is quite elusive at the experimental level; thus a lot of interest has been raised by recent searches for quantu m gravity eﬀects in the propagation of light from distant sources, lik e gamma ray bursters and active galactic nuclei, and also in earth-b ased inter- ferometers, like those used for gravitational wave detecti on. Here we describe a simple heuristic picture of the quantum ﬂuctuati ons of the gravitational ﬁeld that we have proposed recently, and show how to use it to estimate quantum gravity eﬀects in interferometer s. 1 Introduction The propagation of light in random or ﬂuctuating media has lo ng been used as a probe of their statistical properties [1]. This holds tr ue also for exotic media like the background of gravitational waves in the spac e between the earth and some faraway light source [2, 3, 4] (see ﬁgure 1). Recently several authors have proposed searches for quantu m gravity ef- fects in the propagation of light over cosmological distanc es or in earth-based interferometers; for instance, according to Ellis, Mavrom atos and Nanopou- los [5], the vacuum of quantum gravity should be dispersive a nd this should 1Talk presented at Qed 2000, 2nd Workshop on Frontier Tests of Quantum Electrody- namics and Physics of the Vacuum. 1Figure 1: In addition to atmospheric scintillation we expect scintil lation eﬀects due to the background of gravitational waves that aﬀect star light as it travels to the earth. The light emitted by a star does not travel in a stra ight line, but is scattered by this background: therefore it displays angula r displacement and the time of arrival ﬂuctuates randomly. show up as an energy-dependent spread in the arrival times of energetic pho- tons from distant sources like gamma-ray bursters, active g alactic nuclei, and gamma-ray pulsars. This opens up interesting observationa l opportunities for experimentalists (see, e.g. [6, 7]), but requires either a v iolation of Lorentz invariance or of the equivalence principle [8]. The proposal [5] is based on the latest developments of D-brane theory, and it is yet another picture of the “space-time foam” ﬁrst pr oposed by Wheeler in 1957 [9], and later considered by other authors su ch as Hawking [10]. Here we propose a heuristic treatment of these background ﬂu ctuations, which is based on the direct use of the uncertainty principle for energy and time, with additional assumptions on the dynamic and statis tical behaviour of the resulting quantum ﬂuctuations. Afterwards we use thi s model to evaluate the spectral properties of the ﬂuctuations of the e nergy density and of the gravitational potential, and to derive the behaviour of light in a generic two-arm interferometer. P. Bergmann [2] originally suggested to search for low-freq uency com- ponents of gravitational waves by attempting to detect ﬂuct uations in the intensity of light, but actually most searches for this eﬀec t have been carried 2out looking for ﬂuctuations in the arrival times of pulses fr om millisecond pulsars and other such sources. See [3] for the details of how one such search is performed, and [4] for a recent review. 2 Fluctuations of the gravitational energy density We consider now a single quantum ﬂuctuation of the vacuum ene rgy density and assume that it is uniformly spread over a spherical regio n, then the total energy associated with the ﬂuctuation is E=4πR3 0 3ρ, (1) where R0is the radius of the bubble. We assume that the bubble expands at the speed of light, so that R0=ct, where tis the time of creation of the bubble, and that it satisﬁes the time-energy uncertaint y principle at all times2, so that E≈¯ht, and therefore we ﬁnd ρ≈3¯h 4πc3t4. (2) It is also important to note that the expanding bubbles are ju st a represen- tation of the light cones in 3-space, therefore they are Lore ntz invariant, i.e., they would look the same in any other Lorentz-boosted refere nce frame. A rough form of Lorentz invariance is thus present in our model , whose causal structure is compatible with a relativistic model of spacet ime, and these ﬂuc- tuations satisfy the requirement, ﬁrst discussed by Zeldov ich [12], that the quantum vacuum must indeed be Lorentz invariant. Now let n(/vector x, t) be the number density of the ﬂuctuations that occur in spacetime, i.e., n(/vector x, t)dV dt ﬂuctuations are created in a small space-time volume dV dt at position /vector xand time t, and assume that n(/vector x, t)dV dt is a Poisson variate with average n0dV dt and variance n0dV dt (see ﬁg. 2); then the average density observed at ( /vector x0, t) and due to the ﬂuctuations created at 2Quantum mechanics enters this simple model only by way of the uncertainty principle, but it does so “peacefully”, and coexists with relativistic invariance in a rather natural way. 3Figure 2: a.We picture the ﬂuctuations of the gravitational energy dens ity as “bubbles” that expand at the speed of light, and are unifor mly ﬁlled with a decreasing energy density that satisﬁes at all times the tim e-energy uncertainty principle; b.an observer sees the whole of space seething with bubbles, wh ose number density in a given space-time volume follows a simple Poisson statistics. More pictures and simulations of these ﬂuctuations can be fo und on the web [11]. a distance rand at an earlier time t′is3 d2ρ=3¯hn0 c3(t−t0)r2drdt. (3) Then the total energy density observed at ( /vector x0, t) and due to all the prior ﬂuctuations in the light-cone of the observer is given by ρtot(t)≈/integraldisplayrmax rmindr/integraldisplayt−r/c −T0dt03¯hn0 c3(t−t0)r2≈¯hn0lnrmax rmin, (4) where T0is the age of the Universe, and rmin,rmaxare the minimum and the maximum distance from the observer. Furthermore in (4) w e have dropped the /vector xdependence, because we assume translational invariance. T he maximum distance can be taken to be the present radius of the U niverse rmax≈c/H, while the minimum radius corresponds to the smallest bubbl e that can possibly be observed, that is that bubble that emerg es from a “mini 3Here we assume that the energy densities are suﬃciently smal l so that they can indeed be added linearly. 4black hole” stage and turns into a “normal” ﬂuctuation, so th atrminis just the Schwartzschild radius of the ﬂuctuation: rmin=2G c2·E c2≈2G c2·¯h c2t=2G¯h c3rmin. (5) We solve eq. (5) and ﬁnd rmin=/radicalBigg 2G¯h c3, (6) which is just the Planck length. Thus eq. (4) gives an average density ρtot≈1 2¯hn0lnc5 2G¯hH2. (7) It is worthwhile to notice that a minimum and a maximum scale s how up nat- urally in the calculations, and that the total energy densit y (7) is seamlessly related to both the micro and the macrostructure of the Unive rse. The same formalism can be used to estimate the variance of the energy density ﬂuctuations, and proceeding as before one ﬁnds σ2 ρ=9¯h2cn0 28π/integraldisplayrmax rmindr r5=9¯h2cn0 112π/parenleftBigg1 r4 min−1 r4 max/parenrightBigg ≈9c7n0 448G2π. (8) It is also important to notice that the average energy densit y (7) contributes to the eﬀective cosmological constant [13]: ∆Λ (t) =8π c4Gρtot≈8π c4G¯hn0lnc ℓPH(t); (9) recent observations favour a nonzero and positive cosmolog ical constant [14, 15] such that Ω Λ≈0.7 and Λ = 3Ω ΛH2/c2; this means that we can use (9) to estimate the order of magnitude of the free parameter n0, if we assume that most of the eﬀective cosmological constant is due to the ﬂuctuations considered here: n0≈3ΩΛH2c2 4πG¯hln (c5/2G¯hH2)≈3·1022m−3s−1. (10) 53 Fluctuations of the gravitational potential Since we assume that ﬂuctuations spread with the speed of lig ht, there can be no gravitational potential outside the bubble, because f or an external observer there is no way to know that it exists, while inside t he bubble we assume that the gravitational potential is just the (Newton ian) potential of a sphere of uniform energy density ρ: Φ (r) =  −4πGρ 3c2/parenleftbigg3 2R2 0−1 2r2/parenrightbigg ifr < R 0 0 if r > R 0. (11) Integrating as in the case of the energy density one ﬁnds: Φtot≈ −16πG¯hn0 3c2r2 max≈ −16πG¯hn0 3cH(12) and σ2 Φ=68πG2¯h2n0 35c3lnrmax rmin. (13) 4 Correlation functions A somewhat more complicated integration yields the correla tion function of the gravitational potential: /angbracketleft∆Φ (/vector x1, t1) ∆Φ ( /vector x2, t2)/angbracketright=/parenleftBiggG¯hc 2c2/parenrightBigg2 n0/integraldisplay Sd3x/integraldisplay t<t⋆dt/parenleftBigg 3−r2 1 c2(t1−t)2/parenrightBigg × ×/parenleftBigg1 c2(t1−t)2/parenrightBigg/parenleftBigg 3−r2 2 c2(t2−t)2/parenrightBigg/parenleftBigg1 c2(t2−t)2/parenrightBigg , (14) where the shorthand notation r1,2=\|/vector x1,2−/vector x\|has been used, t⋆= min ( t1−r1/c, t2−r2/c) andSis the range of acceptable values of /vector x1,2— i.e., values such that rmin< r1,2< rmax. The expression (14) can be calculated for diﬀerent special c ases, e.g., the time correlation function at a given space point is R(τ) =/angbracketleft∆Φ (/vector x,0)∆Φ ( /vector x, τ)/angbracketright ≈68 35G2¯h2π c3n0lnrmax+cτ rmin+cτ(15) 6and then, using the Wiener-Kintchine theorem, one ﬁnds /angbracketleftBig \|∆Φ (f)\|2/angbracketrightBig =/integraldisplay−∞ −∞R(τ) e−2πifτdτ≈34 35G2¯h2n0π c3·1 f. (16) 5 Irradiance ﬂuctuations in a two-arm interferometer In the weak ﬁeld approximation the frequency of light is line arly related to the gravitational potential ν(/vector x, t)≈ν0/parenleftBigg 1−Φ (/vector x, t) c2/parenrightBigg ; (17) therefore, using the correlation function (14), one ﬁnds th at the time cor- relation function for the irradiance ﬂuctuations in a two-a rm interferometer is RI(τ) = /angbracketleft∆I(t) ∆I(t+τ)/angbracketright =4I1I2ν2 0sin2ϕ0 c6/integraldisplay /integraldisplay L1,L2dℓ1dℓ2× ×/angbracketleftBigg Φ/bracketleftBigg /vector x/parenleftBiggℓ1 c/parenrightBigg ,ℓ1 c/bracketrightBigg Φ/bracketleftBigg /vector x/parenleftBiggℓ2 c+τ/parenrightBigg ,ℓ2 c+τ/bracketrightBigg/angbracketrightBigg ,(18) where ℓ1,2is the position along the path L1,2,ν0is the frequency of light in a zero-potential region, I1,2is the irradiance of light along each path, and ϕ0 is the average phase diﬀerence. We plan to complete the calculation of the time correlation f unction (18) for speciﬁc interferometer designs in the near future. References [1] Sheng P (ed.) Scattering and Localization of Classical Waves in Random Me - dia, Singapore, World Scientiﬁc, 1990 [2] Bergmann P (1971) Phys. Rev. Lett. 261398 [3] Romani R W and Taylor J H (1983) Astrophys. J. 265L35 7[4] Giovannini M “Stochastic GW Backgrounds and Ground Base d Detectors” inCAPP-2000 , AIP Conference Proceedings (to appear) [5] Ellis J, Mavromatos N E and Nanopoulos D V (2000) Gen. Rel. Grav. 32127 [6] Biller S et al. (1999) Phys. Rev. Lett. 832108 [7] Schaefer B E (1999) Phys. Rev. Lett. 824964 [8] Mavromatos N E gr-qc/0009045 [9] Wheeler J A (1957) Ann. Phys. 2604 [10] Hawking S W (1996) Phys. Rev. D533099 [11] http://www.ﬁsica.uniud.it/ ∼milotti/stf/index.html [12] Zeldovich Ya B (1968) Sov. Phys. Uspekhi 11381 [13] Weinberg S (1989) Rev. Mod. Phys. 611 [14] Schmidt B et al. (1998) Astrophys. J. 50746 [15] Perlmutter S et al. (1999) Astrophys. J. 517565 8
arXiv:physics/0010071v1 [physics.atom-ph] 27 Oct 2000Coherent time evolution of highly excited Rydberg states in pulsed electric ﬁeld: Opening a stringent way to selectively ﬁeld-ionize the high ly excited states M. Tada, Y. Kishimoto, I. Ogawa∗, H. Funahashi1, K. Yamamoto2, and S. Matsuki Nuclear Science Division, Institute for Chemical Research , Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan 1Physics Department, Kyoto University, Kyoto 606-8503, Jap an 2Department of Nuclear Engineering, Kyoto University, Kyot o 606-8501, Japan (February 20, 2014) Coherent time evolution of highly excited Rydberg states in Rb (98 ≤n≤150) under pulsed electric ﬁeld in high slew- rate regime was investigated with the ﬁeld ionization detec - tion. The electric ﬁeld necessary to ionize the Rydberg stat es was found to take discrete values successively depending on the slew rate. Speciﬁcally the slew-rate dependence of the ionization ﬁeld varies with the quantum defect value of the states, i.e. with the energy position of the states relative to the adjacent manifold. This discrete transitional behavio r of the ionization ﬁeld observed for the ﬁrst time is considered to be a manifestation of the strong coherence eﬀect in the time evolution of the Rydberg states in pulsed electric ﬁeld and opens a new eﬀective way to stringently select a low- ℓstate from the nearby states by ﬁeld ionization. PACS numbers: 32.60.+i, 31.70.Hq, 32.80.Bx Highly excited Rydberg states [1] in the ramped elec- tric ﬁeld is one of the most interesting systems which provide ideal and versatile situations for investigating t he coherence eﬀects in the time evolution of a quantum sys- tem with many potential-energy curves crossing one an- other [2]. In spite of this interesting feature and also of the potential applicability to the wide area of fundamen- tal physics including cavity QED and quantum compu- tation [3], the Rydberg states with high principal quan- tum number n>∼80 have not been investigated in detail, partly because of the diﬃculty in selectively detecting a particular state from many close-lying states; in such highly excited states, the ﬁeld ionization process gener- ally occurs both through the non-adiabatic and adiabatic transitions, resulting multiple ionization thresholds an d the selective detection of a particular state becomes in- creasingly more diﬃcult. The purpose of this Letter is to present the experimen- tal results on the ﬁeld ionization of the Rydberg states withn= 98−150. It was observed for the ﬁrst time that in high slew rate regime in the applied pulsed elec- tric ﬁeld, the ﬁeld ionization process has single threshold value: Speciﬁcally the ionization electric ﬁeld takes dis- crete values successively with increasing slew rate, and this dependence varies with the position of the states rel- ative to the adjacent manifold. This transitional behavior in the ﬁeld ionization shows regular dependence on the principal quantum number n, thus indicating that this behavior is quite general and applicable to a wide range of higher-lying Rydberg states. Since the diﬀerences inthe ﬁeld ionization values were found to be large enough, i.e. 300 % for the 111 p3/2and 111 s1/2states, it is possi- ble to stringently select a low- ℓstate from the close-lying states by ﬁeld ionization. From these characteristic be- haviors, it is strongly suggested that the coherence in the time evolution under the pulsed electric ﬁeld plays decisive role to the behavior of the ﬁeld ionization. The experimental setup is shown in Fig. 1. Thermal Rb atoms in the ground-state atomic beam are passed through a laser excitation region and then the ﬁeld ion- ization region, which are about 40 mm apart each other. The whole volume of the excitation and the ﬁeld ion- ization regions is surrounded with three pairs of planer copper electrodes to compensate the stray ﬁeld in three axes and also to apply the pulsed electric ﬁeld for the ionization. The selective ﬁeld ionization (sﬁ) electrodes consist of two parallel plates of 120 mm length, in one of which a ﬁne copper-mesh grid was incorporated into the area of 20 ×20 mm2, thus allowing to pass and detect the ﬁeld ionized electrons with a channel electron multiplier. The sﬁ electrodes and the laser interaction region were attached to a cold ﬁnger in a cryostat, thus the tempera- ture can be varied from room temperature down to lower temperature with liquid N 2and He. Two-step cw-laser excitation was adopted to excite the Rydberg njstates from the 5 s1/2ground state of85Rb through the 5 p3/2second excited state. A diode laser (780 nm for the ﬁrst step) and a dye laser of coumarin 102 excited by a Kr ion-laser (479 nm for the second step) were used. The main reason to use the cw lasers to ex- cite the Rydberg states instead of the usually adopted pulsed lasers is that we want to use this ﬁeld ionization scheme for the selective ionization of highly excited Ry- dberg states in a continuous mode as discussed later. The pulse shape applied for the ﬁeld ionization is shown also in Fig. 1. The pulse sequence was produced with a waveform generator NI5411 and the ﬁeld ioniza- tion signals were detected and analyzed with the Lab- VIEW data acquisition system on a pc computer. Rep- etition rate of the pulse was kept to 5 kHz so that the detection eﬃciency of the Rydberg states is optimum for the atoms with velocity of 350 m/s [4]. The ionization mainly occurs at the steep rise of the pulse during the timetf, but the peak ﬁeld can be kept for a time th (holding time) to ionize also the states with longer life time than tfunder the electric ﬁeld. This point will be discussed later. 1atomic beamlaser laser interaction point field ionization electrodeelectron channel electron multiplierElectric fieldPulse shape of the applied electric field Time t0tstfth vsvf12 cmionization pointF t FIG. 1. Experimental setup for investigating the time evo- lution and the ﬁeld ionization process under the pulsed elec - tric ﬁeld in high slew rate regime. Also shown is the pulse shape of the applied electric ﬁeld. During the course of the present experiment, stray ﬁeld of∼80 mV/cm was found to appear in the interaction region. By applying suitable compensation potentials in three axes, the stray ﬁeld was reduced to less than 10 mV/cm. In this stray ﬁeld, the pstate with nless than 120 is well separated from the adjacent manifold levels. The eﬀect of the mixing of the low- ℓstates to the neigh- boring levels on the ﬁeld ionization behavior is discussed later. In Fig. 2 shown are ﬁeld ionization spectra of 111 s1/2 and 111 p3/2states [5] as a function of the applied electric ﬁeldF=vf/l(lis the distance of the sﬁ electrode), which were measured by varying the slew rate S=F/tf. Here the slow component of the pulsed ﬁeld vswas set to zero. In these spectra, only one prominent peak was found as the threshold electric ﬁeld. Remarkably the sﬁ ﬁeld value at the peak changes to a smaller value with increasing slew rate at a particular slew rate. Moreover the sﬁ ﬁelds for the sandpstates are quite diﬀerent from each other at the same slew rate. For example, the electric ﬁeld necessary to ionize the 111 p3/2state at the slew rate of 11 V/(cm ·µs) is 1.7 V/cm, while the value is 5.2 V/cm for the 111 s1/2state, more than 300 % diﬀerence (see the spectrum din Fig.2). It should also be noted here that the transitional be- havior in the 109 dstate (not shown in Fig. 2 to avoid complexity) is the same as in the s1/2state. This means that the transitional behavior depends on the position of the states relative to the adjacent manifold: As seen from the relevant Stark energy-ﬁeld diagramin in Fig. 3, the upper −positioned [6]pstate shows diﬀerent transitional behavior from the lower −positioned sanddstates. In Fig.4 shown is the eﬀective principal-quantum- number n∗dependence of the critical slew rate Scand the ionization electric ﬁelds Fcfor the sandpstates. Herethe critical slew rate Scis deﬁned as the value at which the transition of the sﬁ ﬁeld just starts to a new value. Also the sﬁ ﬁeld corresponds to the peak position of the ionization signal. These values vary quite regularly with n∗ranging from 95 to 147; approximately Fc∝(n∗)−4.0 for both the pandsstates, while Sc∝(n∗)−4.0for the pstate and Sc∝(n∗)−2.8for the sstate. These results indicate that this transitional behavior is quite general for a wide range of nand thus can be applicable to selectively ionize the sandpstates for a wide range of higher excited states. The diﬀerence in the ionization ﬁeld is ∼300 % for the sandpstates, quite large compared to the case of adiabatic transition in which the diﬀerence would be only 5 % for the states atn∼110. Electric field [ V / cm ]s - state p - state 20.0 16.7 14.3 5.7 2.0a b c d e f FCounts [ arb. unit ] 0 1 2 3 4 5 6 7g11.1 6.7 FIG. 2. Typical ﬁeld ionization spectra for the 111 s1/2and 111p3/2states measured by varying the slew rate, values of which are shown at the right side of each spectra. The spectra for the repspective states ( sandp) taken separately were su- perposed upon each other in these spectra. Solid and dashed lines are the ﬁtted results with a Gaussian plus linear back- ground. Related to the selectivity in the Rydberg states, it should be noted that there observed some sﬁ signals at the lower ﬁeld region (1.0 ∼2.4 V/cm) in the sﬁ spec- trum of sstate: The signal peak at this portion changed discretely with slew rate as in the same manner as of thepstate. This part of signal counts is due to the ef- fect of blackbody radiations which induce the transition from the initial sstate to the pstate. In fact the counts at this portion measured by varying the temperature of the excitation-detection region from 120 K to 40 K was found to depend linearly on the temperature as expected. The observed transition rate to the pstate is in roughly agreement in its absolute values and in good agreement in its temperature dependence with the theoretical predic- tions. This agreement indicates also that the selectivity of the excited states with the ﬁeld ionization method in the pulsed electric ﬁeld regime is quite good even at such 2highly excited region. The detailed discussion on the sto ptransitions and the eﬀect of blackbody radiations will be reported elsewhere. The above results were all obtained without the slow component vsof the pulsed ﬁeld. Switching on this value, the transitional behavior of the sstate remains the same, while that of the pstate changes drastically as in the fol- lowing: When the applied slow component ﬁeld with its slew rate less than 1 mV/(cm ·µs) exceeds the ﬁrst anti- crossing ﬁeld ( ∼75 mV/cm for the 111 p3/2state, see Fig. 3), the transitional behavior changes abruptly, be- coming the same as of the sstate. This means that once the ﬁrst anti-crossing is traversed adiabatically and the state is mixed with the adjacent manifold levels, then the following sﬁ behavior for the pstate under the high slew rate regime changes completely, resulting no diﬀerence in their behavior between the states of opposite positions to the adjacent manifold. These experimental results, especially the transitional behavior, are in general not in agreement with simple predictions from the incoherent contributions of adia- batic and non-adiabatic transition processes: The exper- imental sﬁ spectra have only one prominent peak and this peak ﬁeld does neither correspond exactly to the ex- pected position from the purely adiabatic (paths 2 or 3 in Fig. 3), nor diabatic transitions (paths 1 or 4) lead- ing to the reddest or bluest trajectory in the adjacent manifold; the expected ﬁelds for these paths (1 to 4) inn= 108 (corresponding to the adjacent manifold of 111sand 111 pstates from the quantum defect values in Rb) are 1.8, 2.4, 2.4, and 4.2 V/cm respectively, which should be compared to the observed ﬁeld values of 1.7 and 5.2 V/cm for the pandsstates, respectively. More importantly, these spectra show very clear slew rate de- pendence as described above which can not be explained from a simple incoherent process. 109 reddest1 2 4 Electric field [ mV / cm ]Energy [ cm ]-1111p 111s108 manifoldbluest reddest 107 bluest9.25 9.3 9.35 9.4 9.45 9.5 9.55 0 50 100 150 200110d 109d3 0 50 100 150 200-9.55-9.50-9.45-9.40-9.35-9.30-9.25 E F7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 Electric field [ V / cm ]1 2 , 3 4classical ionization threshold 0 1 2 3 4 5 6-7 -8 -9 -10 -11 -12 -13 -14 F FIG. 3. Stark energy diagram near the 108 manifold in Rb together with the classical ﬁeld-ionization threshold- line (E=−6.12√ F), where all the avoided crossings are not ex- plicitely shown for simplicity. The arrows 1 to 4 indicate th e possible trajectories for the adiabatic (2 and 3) and the ex- treme non-adiabatic (1 and 4) transitions. Recently Harmin [2] examined coherent time evolu- tion on a grid of Landau-Zener anti-crossing under a linear ramped electric ﬁeld in which each manifold lev-els are treated as linear in time, parallel, and equally spaced and inﬁnite number. The time development of an initially populated state is then governed by two level Landau-Zener (LZ) transitions at avoided crossings and adiabatic evolution between them. The key parameters in this model study are 1)the LZ transition probabilities Dfor making a non-adiabatic transition process at the anti-crossing traversals and 2) the dynamical phase unit ϕ. The phase unit ϕis the area covered by the pair of the adjacent up- and down-going levels [2]. The overall dif- ference in phase advance ∆Φ for two paths from zero to the ionization ﬁeld is equal to the sum of the phase units ϕin the whole area covered by the two paths. In the actual Stark energy-ﬁeld grid system, the phase units ϕi are not a constant but vary with respective anti-crossings. These parameters Dandϕare estimated approximately byD= [exp( −π˜µ2ϕ)]2, ϕ∼(3˙Fn10)−1, where ˜ µis an average low- ℓquantum defect and ˙Fis the slew rate of the applied electric ﬁeld F. The probability Dincreases monotonically with in- creasing slew rate, while the overall diﬀerence in phase advance ∆Φ of the state wave function through the grid is eﬀective in modulo 2 πand strongly aﬀects the coherent nature of the process through the interference between many states populated along the way of traversals in the applied electric ﬁeld [2]. In the present experimental setup with slew rate ∼20 V/(cm ·µs) atn∼100, the probability of non-adiabatic transition Dis quite high, reaching ∼99.8% from the above estimation so that our case corresponds to the non- adiabatic limit in Harmin’s treatment in a good approx- imation. F [ V / cm ] S [ V /(cm sec) ] n*-3.9 -4.0-3.9 -4.2 -2.8 -2.7-3.9 -3.9ionization threshold critical slew rateµ 90 100 110 120 130 140 15020 1098 7 6 5 4s sp ps sp p 1234567810 9 FIG. 4. Dependence of the selective ﬁeld ionization (sﬁ) value and the critical slew rate on the eﬀective princi- pal-quantum-number n∗in Rb Rydberg states. Discrete two values of the sﬁ ﬁeld observed and their correspond- ing slew rates are plotted together with the ﬁtted lines of n-dependence. The number to each plot is the coeﬃcient α in the ﬁtting of ( n∗)−αdependence with the estimated error of±0.2. Taking into account the quantum defect values of 3lowℓstates, estimated phase unit ϕvaries from 10−2 to 10−3with increasing electric ﬁeld. The number of avoided crossings traversed between zero ﬁeld and the ﬁeld ionization region in Rb is estimated to be Nblue∼ 900/parenleftbign 100/parenrightbig2, Nred∼1150/parenleftbign 100/parenrightbig2for the up- (bluest) and down- (reddest) going trajectories, respectively. At the region of n∼110, this number of anti-crossing traver- sals suggests that the phase advance is over 2 πat some ﬁeld value. When this resonance condition is fulﬁlled, the constructive interference between the many number of Stark states along the advance of the pulsed electric ﬁeld may result in one prominent peak in the sﬁ ﬁeld. In Harmin’s analysis [2], the population of the states under the electric ﬁeld was found to make a series of resonances in a form of lanes along the up- or down-going directions of the initially excited state at zero ﬁeld. This general feature in the non-adiabatic limit seems to be satisﬁed experimentally, since a discrete sequence of sﬁ threshold ﬁeld was found, depending on the slew rate. However there is a signiﬁcant diﬀerence in the ex- perimental observations compared to the model calcula- tion in that the observed sﬁ ﬁeld values for both of the s andpstates decrease with increasing slew rate. Contrary to this observation, the model calculation predicts that with increasing slew rate, the most populated state ap- proaches to the limiting trajectory in the manifold, i.e. to the bluest state for the up-going initial state or the reddest state for the down-going initial state, depending on the direction (up- or down-going) of the initial state. This suggests thus that the sﬁ threshold ﬁeld for the s anddstates in Rb should increase with increasing slew rate, in disagreement with the experimental results. Finally we note that the sﬁ spectra observed with the holding time thextended up to 500 µs showed no dis- tinguishable diﬀerence from those taken with the short holding time of 1 µs, indicating all the states ionized have decay lifetime shorter than 1 µs in the electric ﬁeld. Also we observed no signiﬁcant sﬁ-signals over 10 V/cm in the spectra measured. The decay rate of the blue states in the electric ﬁeld by the tunneling process, estimated from the hydrogenic approximation, is much higher than 2×103s−1. Therefore the ionization process for the states along the blue lines is not due to the tunneling pro- cess but of autoionization-like one due to their mixing to the red continuum, even though such coupling becom- ing weaker as nincreases [1]. In conclusion, we observed for the ﬁrst time a dis- crete transition of the threshold sﬁ ﬁeld with slew rate in the highly excited Rydberg states in Rb, the behavior of which depends also on the position of the low ℓstates rel- ative to the adjacent manifold. The experimental results strongly suggest that the coherent interference eﬀect in the time evolution on the grid of anti-crossings under the pulsed electric ﬁeld plays decisive role for the occurrence of such transitional behavior. The transitional behavior observed here brings us a new powerful method to selectively ﬁeld-ionize the lowℓstates from the many close-lying states, thus opening a new way to apply the highly excited Rydberg states to fundamental physics research. One of the example of such applications is to search for dark matter axions with a Rydberg-atom cavity detector [7]. In this kind of search experiment, it is essential to do the experiment in a continuous way so as to keep the detection eﬃciency as high as possible. It is thus inevitable to use cw lasers to excite the Rydberg atoms continuously which is the main reason for developing the present experimental setup. The authors would like to thank Akira Masaike for his continuous encouragement throughout this work. This research was partly supported by a Grant-in-Aid for Spe- cially Promoted Research (No.09102010) by the Ministry of Education, Science, Sports, and Culture, Japan. ∗Present address: Department of Physics, Osaka Univer- sity, Toyonaka, Osaka 560-0043, Japan [1] T. M. Gallagher, Rydberg Atoms (Cambridge University Press, Cambridge, England, 1994) and references cited therein. [2] D. A. Harmin, Phys. Rev. A 56232 (1997); see also D. A. Harmin and P. N. Price, Phys. Rev. A 491933 (1994). [3] See for example, CavityQuantumElectrodynamics , edited by P. Berman (Academic Press, Boston, 1994); A. Rauschenbeutel etal., Phys. Rev. Lett. 835166 (1999), and references cited therein. [4] Taking into account the velocity distribution of atoms a nd the detection eﬃciency in the pulsed ﬁeld ionization, the overall detection eﬃciency is estimated to be 75 %. [5] Note that the sﬁ detector used is of integral type in a sens e that the sﬁ signals are integrated with increasing sﬁ ﬁeld vfand thus the derivative of the spectra with respect to vfis the true counts at each bin which is shown in Fig. 2. [6] Here the upper −positioned levels are referred to the states with the quantum defect value \|δ\|(mod 1) ≥0.5. [7] S. Matsuki and K. Yamamoto, Phys. Lett. B 263, 523 (1991); I. Ogawa, S. Matsuki and K. Yamamoto, Phys. Rev. D 53, R1740 (1996); K. Yamamoto and S. Matsuki, Nucl. Phys. B (Proc. Suppl.) 72, 132 (1999); M. Tada, et al., Nucl. Phys. B (Proc. Suppl.) 72, 164 (1999); A. Kitagawa, K. Yamamoto and S. Matsuki, LANL e-print archive, hep-ph/9908445. 4
arXiv:physics/0010072v1 [physics.gen-ph] 27 Oct 2000On the Four Dimensional Conformal Anomaly, Fractal Spaceti me and the Fine Structure Constant Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia, 30314 October 2000 Abstract Antoniadis, Mazur and Mottola ( AMM) two years ago computed t he intrinsic Hausdorﬀ dimension of spacetime at the infrared ﬁxed point of the quantum conforma l factor in 4 DGravity. The fractal dimen- sion was determined by the coeﬃcient of the Gauss-Bonnet top ological term associated with the conformal gravitational anomaly and was found to be greater than 4. It i s explicitly shown how one can relate the value of the Hausdorﬀ dimension computed by AMM to the univer sal dimensional ﬂuctuation of spacetime ǫgiven by φ3/2, where φis the Golden Mean 0 .618... Based on the infrared scaling limit of the theory and using recent Renormalization Group arguments by El Naschie , we conjecture that the unknown coeﬃcient Q2, associated with the four dimensional gravitational confo rmal anomaly, could be precisely equal to the inverse ﬁne structure constant values ranging between 137 .036 and 137 .081. Our results generate decimal digits up to any arbitrary number. The Conformal Anomaly and Fractal Spacetime at Large Scales Antoniadis, Mazur and Mottola [1] more than two years ago com puted the intrinsic Hausdorﬀ dimension of spacetime at the infrared ﬁxed point of the quantum confor mal factor in 4 DGravity. The fractal dimension was determined by the coeﬃcient of the Gauss-Bonnet topological term associated with the conformal anomaly : trace anomaly and was found to be greater than 4. They also discussed a plaus ible physical mechanism for the screening of the cosmological constant at very large distances in full agreeemnt with Nottale’s work [7] and other results obtained by the present author and collaborators [6]. The Hausdorﬀ dimension for spacetime is related to the geode sic distance l(x, x′) between points x, x′ and the volume Vlenclosed by the spherical surface radius equal to l. The scaling relation between the two isVl∼ldH. For large l, this scaling relation deﬁnes the intrinsic dimension dHof the space. In 2Dquantum gravity [2,3] the most appropriate way to calculate thedHis by the heat kernel methods associated with the Laplacian operator DµDµand the proper time s: K2(x, x′;s, g) =< x\|e−sDµDµ\|x′> . (1) The heat kernel K2has a short distance expansion whose anomalous scaling beha viour can be calculated based on the standard techniques pioneered by B. de Witt in th e sixties. The average geodesic length-squared that a scalar particle is able to diﬀuse after a proper time sis given by : l2 s≡1 V</integraldisplay d2x√g/integraldisplay d2x′/radicalbig g′l2K2(x, x′;s, g)>V. (2) where the average is taken with respect to the ﬁxed volume Lio uville ﬁeld theory partition function. By expanding the heat Kernel K2(x, x′;s;g) in a power series sone can see that l2 s∼s as s →0. (3a) which is the standard result undergoing Brownian motion. The relevant scaling behaviour ( with area/volume ) is the ex pectation value of the following quantity appearing in the heat kernel expansion, under the conformal scalings of the metric gab=exp(2σ)¯gab: s </integraldisplay d2x√g DµDµfǫ(x, xo)>V=s </integraldisplay d2x√¯g¯Dµ¯Dµ¯fǫ(x, xo)>V∼sVα−1 α1 (3b) 1where fǫ(x, x′) is any smooth function with support only for distances of l(x, x′)∼ \|x−x′\|< ǫ. Asǫgoes to zero it approaches a delta function : (1 /√g)δ2(x−x′) The ﬁnite area/volume scaling behaviour of the last proport ionality factor follows by a constant shift in the Liouville ﬁeld σ→σ+σo. The αnare the anomalous scaling dimensions associated with the ﬁe lds of the Liouville ﬁeld theory and are given in terms of the weight snby the formulae : αn=n+α2 n Q2=2n 1 +/radicalBig 1−4n Q2. (4a) where the charge Q2is determined in terms of the matter central charge ( anomaly coeﬃcient ) cmby : Q2=25−cm 6. D= 2. (4b) We refer to the references [1,2,3 ] for further details. The m ain result is that the scaling behaviour of s under a global area/volume scaling s→λ−α−1 α1s. (5) will determine the Hausdorﬀ Dimension dHfrom the relation : l2 s∼s∼V−α−1 α1 l∼l−dHα−1 α1. (6) giving ﬁnally : dH=−2α1 α−1= 2√25−cm+√49−cm√25−cm+√1−cm≥2. D= 2 (7) The authors [1] repeated this analyis associated with the co nformaly anomaly in D= 4 where the charge Q2is now the coeﬃcient of the Gauss-Bonnet curvature squared t erms present in the four dimensional conformal anomaly : Q2=1 180(NS+11 2NWF+ 62NV−28) + Q2 grav. (8) theunknown charge Q2is given in terms of the number of free scalars NS, Weyl fermions NWFand vector ﬁelds NV F. While −28 and the unknown value of the gravitational charge Q2 gravare the contributions of the spin-0 conformal factor and spin-2 graviton ﬁelds of the metric itself. The scaling behaviour of the proper time under a global scali ng of the volume V→λVinD= 4 is : s→λ−β8 βos. (9) where βo, β8are the conformal scaling exponents corresponding to the vo lume operator and the higher quartic derivative operator ( square of the Laplacian plus curvatur e and other derivative terms). The Hausdorﬀ dimension is given, in the long scale limit, by the relation : l4 s∼s∼V−β8 βo l∼l−dHβ8 βos (10) then the Haussdorﬀ dimension of fractal spacetime is explic itly given by : dH=−4β0 β8= 41 +/radicalBig 1 +8 Q2 1 +/radicalBig 1−8 Q2≥4. (11) This ﬁnal expression for the spacetime fractal dimension is allwe need to show that the unknown charge Q2in eq-(8) can be equated with the inverse of the ﬁne structure constant 137 .036....( given by the Particle data booklet ). The authors [1] emhasized that the value of Q2was uncertain , principally because of the unknown infrared contributions of gravitons to the value of Q2 gravwhich appears in the r.h.s of eq-(8). 2To calculate what the value of Q2may be , which in turn will yield the value of Q2 gravpresent in the r.h.s of (8), we conjecture that this value can be related to t he inverse of the ﬁne structure constant 137 .036.. based on the recent papers by the author, Granik and El Naschi e [4,6] . The main results of [4] and [6] is that there is a universal dimensional ﬂuctuation in Nature given in terms of the Golde n Mean by : ∆Dfluctuation =ǫ=φ3 2=φ−1 2= (0.618−1 2) = 0.118..... where φ +1 =1 φ⇒φ=√ 5−1 2= 0.618....(12) and that the inverse of the ﬁne structure constant [4] , rangi ng between 137 .036...and 137 .081.., can be thought of as an internal dimension or a central charge, using the language of Irrational Conformal ﬁeld Theory, as Eddington envision long ago. This line of reasoni ng is nothing but following the path chartered by Einstein himself on the geometrization of allphysics, with the new ingredient that we believe that Nature isfractal at its core. A simple numerical calculation shows that by simply setting Q2= 137 .036....inside the basic equation (11) yields automatically nothing more, nothing else but 4+ ǫfor the Hausdorﬀ dimension of fractal spacetime , where ǫis the universal dimensional ﬂuctuation given by φ3/2 : 41 +/radicalBig 1 +8 137.036 1 +/radicalBig 1−8 137.036= 4.11856∼4 +φ3 2= 4.118..... !!!!!!. (13) Is this a numerical concidence ordesign ???? El Naschie [4] has presented very convincing arguments rega rding the unﬁcation of gravity with the electroweak and strong forces, based on Renormalization Gr oup Arguments, that there is a very deep and explicit connection bewteen Nottale’s Scale Relativity [7 ] , Irrational Conformal Field theory, El Naschie’s Cantorian-Fractal spacetime and the New Extended Scale Rel ativity formulated by the author [5], and that the inverse ﬁne structure constant ( an internal dimens ion or charge ) plays a fundamental role in determining the scaling regimes of the electroweak and stro ng interation , and the hierarchy of the 16 ,6 internal dimensions, present in the Heterotic string theor y and its compactiﬁcations , from 26 →10→4 . Argyris et al [8] have recently shown how a fractalization of spacetime may be an intrinsic property of all processes in Nature, from the microworld to cosmos, having w ell deﬁned signatures in Cosmic Strings and in the phenomenon of Spontaneous Symmetry breaking . Acknowledegements The author thanks M. S. El Naschie for his hospitality and for the series of discussions which led to this work. Also we wish to thank S.Ansoldi for his assitance and to E. Spallucci, E. Gozzi, M. Pavsic, A. Granik, T. Smith, C. Handy. A. Schoeller, A. Boedo, J.Mahecha, J. Gir aldo, L. Baquero for their support. References 1. Antoniadis, P. Mazur and E. Mottola : “ Fractal Geometry of Quantum Spacetime at Large Scales “ hep-th/9808070 . Antoniadis, P. Mazur and E. Mottola : Nuc. Phys. B 388 (1992 ) 627 2- V. Knizhnik, A. Polyakov and A. Zamolodchikov : Mod. Phys. LettA 3 (1988 ) 819. F. David, Mod. Phys. Lett A 3 (1988) 1651. F. David : Nuc. Phys. B 257 (1985) 45 3- J. Distler, H. Kawai : Nuc. Phys. B 257 (1985) 509 V. Kazakov, Phys. Lett B 150 ( 1985) 282 L. Ambjorn, B. Durhuus, J. Frohlich and P. Orland : Nuc. Phys. B 270 (1986) 457 . 4- M. S. El Naschie : “ Coupled oscillations and mode locking o f Quantum Gravity ﬁelds, Scale Relativity andE(∞)space “. Chaos, Solitons and Fractals 12(2001) 179-192. 5-C. Castro : ‘’ Hints of a New Relativity Principle from p-Brane Quantum Mechanics “ Chaos, Solitons and Fractals 11(2000) 1721 C. Castro : “ Noncommutative Geometry, Negative Probabliti es and Cantorian Fractal Spacetime “ Chaos, Solitons and Fractals 12(2001) 101-104 36-C. Castro, A. Granik : “ Scale Relativity in E(∞)space and the Average Dimension of the World ’ To appear in Chaos, Solitons and Fractals. hep-th/0004152 7- L. Nottale : “ Fractal Spacetime and Microphysics : Toward s a theory of Scale Relativity. World Scientiﬁc , 1993. L. Nottale : “ La Relativite dans touss ses Etats “ Hachette Li terature, Paris 1998. 8- J. Argyris, C. Ciubotarium H. Matuttis : “ Fractal space, c osmic strings and spontaneous symmetry breaking “ Chaos, Solitons and Fractals 12(2001) 1-48 4
1Is the Basic Unit System a String? Edgar Paternina. Electrical engineer Author of Physics and The Principle of Synergy, published in CD ROM in English and Spanish at Amazon.com. Contact: epaterni@epm.net.co Abstract The main aim of this paper is to present an overview of the need of a new way of coping the fundamental equations of physics, see the references. Our mainaim is to build a new metrics in which both time and space are included, but insome sort of minkowskian union of the two and in such a way that that unionpreserves an independent reality, and where energy, can be taken as the real fundamental issue of that metrics, instead of the particle concept. A departure from General Theory of Relativity is clear, but then the BUS concept behavesitself as a fundamental string. Introduction We are not interested from the beginning in the motion of a particular point in space. Our main aim is to build a new metrics in which both time and space are included, but in some sort of minkowskian union of the two and in such away that that union preserves an independent reality, and where energy can betaken as the real fundamental issue of that metrics. The particle point of view in physics is true but partial. And it is this partialness the one we are trying to avoid from the beginning and as so gravitation will not occupy an exceptional position with regard to the other forces, particularly the electromagnetic forces , but energy as a conceptual starting point will be the one that will occupy an exceptional epistemologicalposition in our constructs. Being our starting point an integral one we will relate that problem with the need to frame in a unified conceptual scheme the radical duality of theUniverse expressed, in general, in the following dyads:2- the particle and the wave problem expressed dramatically in physics as the momentum and position electron problem of Quantum Mechanics. - the relation between the whole or form and the part, the so-called generalization-specialization problem - the qualitative aspects of reality and its quantitative ones - the relation between rotational movement and linear movement being the latter a special case of the former - the relation between time and space and the fact they are always in a unified framework or sphere of reality, the so-called space time continuum. - The relation between two poles in a magnetic field and the fact each one of them cannot be isolated makes the magnetic field a very fundamental onewhere oneness, openness and wholeness are main features. - And finally the relation of the signifier and the signified in case of the linguistic sign and as an example that transcends physics but that has the same nature or dynamic pattern. A Symbol for differentiating These two different orders of reality must be clearly differentiated, that is, separated, so to speak, in different boxes, just as apples and oranges. And for this we need a special symbol of differentiation, but also of another one that integrates them both in a unified framework, having then a Basic UnitSystem in which both components are integrated. This Bus concept is then awhole/part entity or a holon as defined by Ken Wilber[6], and as so with it anew holonic metrics emerges. The symbol for separation was discovered byCardano in 1545, when trying to solve the simple algebraic equation x² +1 = 0 and the symbol for its integration or the interdependence of the two state variables was found by Leonard Euler in 1745 when studying infinite series. J(θ) e = Cos( θ) + JSin( θ) (1)3Apparently the cos and sin functions seem to be the same mathematical functions, except by the fact they differ by 90 degrees. But they do indeed aredifferent from each other not only by that fact -which is some way to escapefrom unidimensionality- but also by the way they change their sign by changing their angle θ, so Cos(θ) = Cos(-θ) Sin(θ) = -Sin(θ) having this second one two solutions, the plus and minus sign of the square root of minus one problem. We can say then that the first component of ER isnondual, and the second one is dual. The first one has to do with the whole,the second one with the part or a binary logic. The first one with a geometricor just a graphical representation and the second one with its correspondingalgebraic or mathematical representation. The first one with the dynamic nature or time and the second one with space. From the physical point of view we need a metrics where we have two state variables, the well-known S and T variables, one related with space, and theother one, the dynamic part of reality, related with time. We will replace themthough by E and R, because we will use S for representing the whole physicalreality or what Minkowski named the quadratic differential expression, which is an invariant or else it represents the same nature of reality, physical reality in this case. In this sense reality is independent from the observer or its frameof references and space by itself, and time by itself, are doomed to fade away into mere shadows as Minkowski put it in his classical paper Space and Time[7]. We have then a fifth sphere of reality as an intermediate domainbetween the observer and the object that we have named the sphere of Form, where reality can be represented not as something relative but as a new metrics in which the laws of nature hold good independently of the system ofreference as the holon concept is a Basic Unit System that preserves theisomorphic properties of all systems. A New Sphere of Reality and the Complex Plane The introduction of this sphere of reality is necessary both from the point of view of Euler Relation where we have the complex plane by just assigning θ a value of 90 degrees, and from the point of view of the need of a metrics with4which we can represent the space time continuum so we will have a new differential quadratic element defined as: J( θ) DS = Abs(DS) e (2) that can be represented in rectangular coordinates as DS = d E + J* d R (2) that gives us according to the Pythagorean Theorem DS² = d E² - d R² (3) where d R² = d x² +d y² + d z²and it is this integral space time mathematical representation the one that permits us to find again all the fundamental equations of physics, including the well-known Schrödinger Wave Equation[1,2,3,4,5]. With this new concept of unit in which the part and the whole are included, we have defined a newmetrics with which reality can be represented without the reductionisticdrawback, because the Uncertainty Principle is included. But on the other sidethe concept of dimension acquires its real connotation associated with the fourdimensional space time continuum, where for having closed system andobjects we need to solve (2) or (3), that is, we need to find laws or relations between the two main component of the BUS concept. And for this we need to consider not only the observed manifestation of that entity represented by the 5BUS, associated in some cases with a clear graphical representation, but also with the chance it has a corresponding and almost exact mathematicalrepresentation, such as in a planet, where on the one side we have an ellipseand on the other we have a corresponding equation for it. In this case we havea closed system with its state completely determined, and when this is the casewe can make predictions, we can make measurements. The main departure of this new approach in regard to the well-known approach due to Einstein where (3) or the “linear element” was generalized as DS ² = Σ Gij dXidXj is precisely to reduce the whole problem to find 10 functions Gij according to the rules of Tensor Analysis, so to speak, to just an analytical problem, not a geometric problem anymore, as geometry was then reduced to algebra. So Einstein wrote “Thus, according to the general theory of relativity, gravitationoccupies an exceptional position with regard to other forces, particularly theelectromagnetic forces, since the ten functions represent the gravitational fieldat the same time define the metrical properties of the space measured..” If weassimilate each Gij to a dimension, mathematical dimension, we will need toposit 10 dimensions to solve the whole problem, and in spite those additional dimensions have no physical meaning at all. Expression (2) is an infinitesimal rotating vector or pointer or else a tiny loop or string vibrating at certain frequency, but it can also vibrate at other modesor frequencies, but in the complex plane. The fact that with it we can deducethe Schrödinger Wave Equation takes us to think that the main and fundamental concept behind everything physical is energy, and defined like a frequency multiplied by Planck’s constant h, and not precisely the particleconcept, even though the latter is a derived one, and an electron can be seenjust as a whirlpool of energy. The fact that with (2) and (3) we can find all the fundamental equations of physics as is shown in the references, makes it possible for us to have ways to express the same general laws of nature or dynamic patterns from a unified conceptual point of view, which was the central claim of the systemssciences, so the unity of science is granted not by the reduction of all sciencesto physics but by those isomorphic regularities[5] of the different levels ofreality, including the physiosphere or the so-called four dimensional spacetime continuum and the biosphere too as in that new domain we have named6the domain of form and represented by the complex plane, life can be defined as an animated form. The problem of two arrows of time and the Second Law of Thermodynamics is associated with the definition of open systems, and in this case that problemis overcome because with (2) we have found the equations of the pendulum, a truly open system. The Principle of Synergy The production of electrical energy or Alternating Current is a real example of what I have called the application of the principle of synergy. In thatproduction we have a threefold physical magnetic structure rotating because of the hydraulic turbines and at the end, what we have is a rotating magnetic field that by Maxwell Laws induces in each one of the terminals of that threefoldstructure the three phase AC we use daily. Mathematically we can representthis situation by considering three dyads of space and time vectors, that is, J( θ) J(wt) A = Abs(A) e + Abs(A)* e J( θ+120) J(wt) A = Abs(A)* e + Abs(A)* e J( θ+240) J(wt) A = Abs(A)* e + Abs(A)* e by using ER we decompose each one of the above equations in sin and cos, and after summing them up we finally obtain that the three space componentsbecome null, so we have just a dynamic expression: J(wt) A = Abs(A’)* e By mathematical inevitability we can represent that sum as in the figure7What we have here is just a sum 1-2-3 that is greater than the sum of its parts and what this means is that we have an open system that is interchangingenergy with the environment, so in this sense this is something practical, asthe extra that can be in the whole that is not in the parts comes from the environment, from the field concept as a rotating entity and which is a complete whole by itself just in case of the magnetic field, a reason why itseems so fundamental in the universe. Synergy is today a fashionable catchword, a word we hear almost everyday in the business environment and it means precisely that sum that is greater than the sum of its parts. In this sense it is synergy with the help of the field concept and the Basic Unit System or Holon concept, the one that permits usto define Entropy = f (-Synergy)so that problem of the two time arrows is not our problem anymore, and Entropy is just the result of the not application of that principle of synergy. If we apply this principle we can have emergent states, or movements as that ofthe pendulum that seemed to violate the infamous Second Law of 8Thermodynamics. Before finishing we want to point out some of the main features of this new metrics - We do not need to use Tensor Analysis whose main disadvantage being its abstraction from the point of view of physical representation - We do not need then to introduce additional strange dimensions, with no physical meaning at all - A Basic Unit System by definition is a rotating entity but in the complex plane and as such it behaves like a string - The complex representation of the BUS concept reduces complexity by minus one degree[5] - We have with the BUS concept a new way of presenting the Uncertainty Principle [1,5] - We have a new way of finding the fundamental equations of physics in the line of reasoning of that claim put by the systems sciences - We have then a new holonic worldview by introducing an intermediate level of reality between the observer and the object that definitely permitsus to exorcise the ghost of consciousness from physics. References 1. Epsilon Pi. Physics and The Principle of Synergy Amazon.com.1999 2. The Basic Unit System Concept and the Principle of Synergy ( http://xxx.lanl.gov/html/physics/9908040 ) 3. Electromagnetism, Relativity and the Basic Unit System ( http://xxx.lanl.gov/html/physics/9908042) 4. Gravitational Fields and The Basic Unit System concept ( http://xxx.lanl.gov/html/physics/9908045 ) 5. The Principe of Synergy and Isomorphic Units ( http://xxx.lanl.gov/html/physics/0010022 ) 6. Ken Wilber . Sex, Ecology and Spirituality. Shambhala Boston & London. 1995 7. Einstein at all . The Principle of Relativity. Dover Publications, INC.1952 © 2000 by Epsilon Pi. All rights reserved.
ArXiv reference: http://xxx.lanl.gov/abs/ physics/0010074 Transverse redshift effects without special relativity Eric Baird (eric_baird@compuserve.com) Transverse redshift effects are sometimes presented as being unique to special relativity (the "transverse Doppler effect"). We argue that if the detector is aim ed at 90 degrees in the laboratory frame, most theories will predict a redshifted frequency at the detector, although these predictions can be concealed by specifying that angles should be defined in a frame other than the laboratory frame. These redshifts are often stronger than special relativity's predictions. We list some of the situations in which lab -transverse redshifts would be expected. 1. Introduction According to Einstein’s special theory, light - signals coming from an object moving in the laborato ry frame should have an increased wavelength when they arrive at a transversely - aimed lab -frame detector [1]. These “transverse redshifts” are sometimes presented as being a unique feature of special relativity, e.g. Rosser 1964: “… According to the theory of special relativity, if a beam of atoms which is emitting light is observed in a direction which according to the observer is at right angles to the direction of relative motion, then the frequency of the light should differ from the frequency the lig ht would have if the source were at rest relative to the observer. This is the transverse Doppler effect. According to the classical ether theories there should be no change in frequency in this case. ” [2] Other reference texts agree that transverse redshifts should not occur in classical theory [3][4], but are less specific about how the word “transverse” should be interpreted. We show that Rosser’s statement is incorrect, and that not only are “laboratory -transverse” redshift predictions common to a ra nge of models, but that many of these predicted redshifts are stronger than their “special relativity” counterparts. In this paper, we briefly look at and list the lab - transverse predictions of a number of different models. 2. “Stationary” and “moving” aether predictions “Aberration shift” If we assume that light travels throughout space at c relative to the observed object, aberration effects cause an observer aiming their detector at 90° degrees in their own frame to see more of the “back -side” of the movi ng object, and can lead to the observer expecting to see a partial recession redshift [5] (e.g.: Lodge “… Doppler effect caused by motion of the observer is … a case of common aberration. ” 1893 [6] “ .. a spurious or apparent Doppler effect. ” 1909 [7] ). We can rederive these effects by starting with special relativity (which “relativises” the stationary -aether and moving -aether calculations) and working backwards to find the original moving -aether predictions. Non-transverse shift tests Special relat ivity’s transverse predictions are sometimes tested experimentally by measuring the non -transverse (“radial”) frequency shift relationships, and then analysing the data to find a residual Lorentz component after first - order propagation effects have been accounted for [8]-[14]. [9] [10] [11] [12] [13] [14] The three main Doppler equations for the apparent frequency f ’ and apparent front -back depth d’ of a receding or approaching radiating object [15][16][17], with v as recession velocity , are: d’/ d = f ’/ f = ( c-v) / c … (1) d’/ d = f ’/ f = ( )( )vcvc +−/ … (2) d’/ d = f ’/ f = c / (c+v) … (3) Special relativity’s “relativistic Doppler” predictions (2) are the root -product average of the predictions associated with “absolute aether”s that are i) stationary in the emitter’s frame (1) and ii) stationary in the observer’s frame (3) [18]. Any model that generates the first -order Doppler equation (1) should give a residual Lorentz -squared redshift when stationary - aether propagation effects (3) are divided out, a stronger result than special relativity’s single residual Lorentz redshift. In the case of Ives -Stilwell 1938 [8], the mean position of approach - and recession shifted spectral lines gives a central position that is not affected by velocity with (3), and that has velocity -dependent positional offsets with (1) and (2). Transverse redshift effects … Eric Baird Sunday , 29 October 2000 5:47 AM page 2 / 2 ArXiv reference: http://xxx.lanl.gov/abs/ physics/0010074 Transverse motion The same relationships should hold for data taken at other angles in the laboratory frame – any “special relativity” result should be interpretable either as a stationary -aether propagation effect supplemented by a Lorentz redshift (time -dilation of the moving emitter), or as a moving -aether propagation shift supplemented by a Lorentz blueshift (time - dilation of the moving observer’s reference - clocks). Where the special theory predicts a lab - transverse Lorentz redshift, an unmodified “moving -aether” model should (again) predict a Lorentz -squared la b-transverse redshift (Note: all viewing angles must be specified in a particular frame to avoid aberration issues, otherwise this approach will fail [18] ). 3. Emitter -theory If we are only observing a single object, the simplest predictions for a “ballistic light - corpuscle” model superimposed on flat spacetime should coincide with the predictions for an absolute aether moving with the object (lab-transverse Lorentz -squared redshift). 4. Dragged -light models If light is completely dragged by a particle - cloud or object, we should again expect the most extreme scenario (where dragging is effectively absolute over extended regions of space), to be equivalent to a “moving aether” model, giving us a Lorentz -squared lab - transverse redshift. Dragged -light models producing weaker dragging effects (or with more “democratic” dragging characteristics) should produce correspondingly weaker lab -transverse frequency changes. 5. Relativistic calculations using the emitter -theory shift equation In another paper, we have derived the relativistic aberration and wavelength -changes associated with (1), (2) and (3) [19]. In that exercise, the relativistic application of the emitter -theory equation is once again associated with a Lorentz -squared “lab - transverse” redshift prediction. In a round -trip version of the experiment (where a signal is aimed and the reflection received at 90° in the same frame), (1) gives a double Lorentz redshift and special relativity gives a null result [19]. 6. Gravitational redshifts Verifications of general relativity’s gravity - shift predictions are sometimes used as indirect supporting evidence in favour of the special theory. The prediction that light from high -gravity stars should be seen to be spectrally shifted was made by John Michell in 1783, and again by Einstein in 1910 [20]. If we calculate the strength of the effect by dropping an object across a gravitational gradient and using Doppler equation (1) to calcu late its final motion shift (Einstein [21], MTW [22] §7.2), we get a one -way gravity -shift prediction of ΔE=~gh/c2 (good Earth -surface approximation), and ΔE=2gh/c2 for round -trip shifts (exact relationship) [23]. Verifications of these relationships are often considered to be verifications of general relativity [24][25], although they do not depend on general relativity’s mathematics, special relativity’s frequency -shift relationships, or the principle of relativity. 7. Centrifugal redshifts The equivalence pri nciple requires that centrifugal redshifts must be calculable from gravitational principles [26], because of the apparent outward gravitational field seen in the rotating frame (the “Coriolis field” [27]). If we attach two clocks to the centre and to the rim of a rotating disc, observers in the disc’s rotating frame are entitled to claim that the disc is immersed in a effective gravitational field that pulls objects away from the rotation axis. We can then apply the general arguments given in Einstein’s 1911 gravity -shift paper for signals passed through this field [21] to argue that the perimeter clock must run more slowly than the central clock. These calculations do not require special relativity. Huyghens’ principle and gravitation If two light -clocks do have a genuine measurable difference in clock -rate, we can apply Huyghens’ principle to the apparent lightspeed differential between the two regions and predict a deflection of lightrays towards the slower cloc k [21][28]. By this argument, an effective gravitational field should be present in any experiment producing physical clock -rate differences. Transverse redshift effects … Eric Baird Sunday , 29 October 2000 5:47 AM page 3 / 3 ArXiv reference: http://xxx.lanl.gov/abs/ physics/0010074 8. Other rotating -body problems Similar considerations apply to the Hafele - Keating experiment [29][30] and other experiments involving the comparison of rates of clocks orbiting with and against the earth’s rotation ( e.g. GPS and other satellite -based systems [31] §3 pp.54 -64). If a clock -rate difference is large enough to be deemed “significant”, then the geometrical deviation from flat spacetime should be considered to be equally ”significant” (since the former should be calculable from the latter). Einstein’s equatorial clocks The issue of gravitational -equivalence is nicely illustrated by the example in section 4 of the “electrodynamics” paper, in which Einstein suggests that a clock at the earth’s equator should tick more slowly than one at the pole. If we are using sea -level clocks, these gravitational effects conspire to make the effect disappear [32] – if a sea -level clock -rate differential is associated with a gravitational gradient, the earth’s oceans should flow “downhill” across this time -dilation gradient towards the equator, only reaching equilibrium when all parts of the ocean surface have the same clock rate (the resulting equatorial bulge should, of course, also be calculable from more conventional “centrifugal force” arguments). More complex problems Although it is useful to be able to calculate clock -lags by as suming flat spacetime and applying a Lorentz correction, the success of this approach over small regions does not mean that these are intrinsically flat -space problems (cartographers once used similar equations to compensate for the Earth’s curvature, but their success did not prove that the Earth’s surface “really was” flat). We would suggest that where “flat” and “gravitational” arguments disagree, the second approach may have greater validity. The calculation of route -dependent gravitational effects fr om apparent clock rate differences is a much more complex subject, [33][34] and is beyond the scope of this paper. 9. Thermal redshifts Similar arguments can be applied to the case of the thermal second order Doppler effect in Fe57 [25][35]. If the Fe57 atoms have “significant” velocities while locked into a “stationary” crystal lattice, then they must also be continually undergoing “significant” accelerations. 10. Muon lifetimes “Muon -decay” experiments are widely cited in textbooks as supporting evidence of special relativity’s time -dilation predictions [36]. C.M. Will [31] Appendix: pp.245 -257: “But the [upper atmospheric] muon is so unstable that it would decay long before reaching sea level … if it weren’t for the time dilation of special relativity, which increases its lifetime as a consequence of its high velocity. ” This statement about the time -dilated muon depends on the assumption that the speed of light is “really” fixed in the observer’s frame – but since the special theory ought to predict the same outcome when we assume that lightspeed is fixed in the object’s frame, we are also entitled to claim, with equal validity, that the muon’s ageing rate is anomalously fast, and that (with a fixed lightspeed in the muon frame) the muon would actually penetrate further , if is was not for the time -compaction effect of special relativity! The “muon” statement obviously involves a certain amount of interpretation being applied to the experimental data. If we return for a moment to Newtonian mechanics, the muon’s decay position x for a given Newtonian rest mass m, particle lifetime t and momentum p, is (with v=p/m) x = vt = pt/m. Calculating the equivalent decay point under special relativity with x’ = vSRt’, we have a smaller velocity value vSR =p/mγSR [37] and a larger (time -dilated) decay time t’=t γSR , with the two Lorentz factors cancelling ( x’=x). In this particular calculation, the effect of the time-dilation path -lengthe ning effect is to compensate for the path -contraction due to special relativity’s reduced nominal velocity values, so that the muon’s decay position is as it would have been under Newtonian mechanics. 11. “SR-similar” aether models Although this paper is intended to be about recognisably “non -SR” models, we should also mention that there are a range of “Lorentzian” aether models that also incorporate time - dilation effects (see e.g. [38]-[40] and many articles in dissident journals). Many of these models only predict small or non -existent deviations from special relativity. Where they agree exactly, the special theory is usually assumed to be preferable because of its reduced number of physical assumptions. 39 40 41 42 Transverse redshift effects … Eric Baird Sunday , 29 October 2000 5:47 AM page 4 / 4 ArXiv reference: http://xxx.lanl.gov/abs/ physics/0010074 12. “Physical” and “interpreted” time dilations The distinction between “physically -verifiable” time dilation effects and “interpreted” time dilation effects is not always obvious. In the case of “moving aether” calculations, the “lab-transverse red shift” result is usually overlooked, possibly because it seems unreasonable that a transverse redshift could be detected if the emitter was not “really” ageing more slowly. Since a rectilinearly -moving point -particle only has transverse motion with respect to a point -observer for a vanishingly short period of time, these shifts do not have to be “sustainable”, and do not have to be associated with “real” clock -rate differences. In the “muon” case, time -dilation seems to be an “interpreted” property, whose reality depends on the statement that the speed of light is “really” locked to the observer’s own frame – this statement cannot be physically verified without breaking the principle of relativity. In the case of a relativistic model based on the moving -aether equations (e.g. this author [33][34]), time dilation is more difficult to pin down, as this class of model seems to require a non-Euclidean spacetime in which relative ageing rates can be route -dependent. General Summary: In general: a) We can test the correctness of the special theory’s shift equations, but cannot isolate an unambiguous physical difference in clock rates unless the experiment involves gravitation or acceleration. In the absence of these effects, the “time - dilation” results are interpretative. b) If two physical light -clocks do have a verifiable difference in clock rate, then Huyghens’ principle applied to this apparent lightspeed differential shoul d give us a description of a gravitational gradient between the two clocks, and the problem can be treated as a gravitational exercise without involving special relativity. Exception - “twins” problem The one possible exception to this rule seems to be the original version of the infamous “twin,” “astronaut,” or “clock” problem ( [43] [44], [37] §4.6 pp.125 -126), where a traveller coasts away from their twin at a constant v m/s, experiences a sudden abrupt accele ration that reverses their course, and then coasts back to their twin’s position at a constant speed of -v m/s. In special relativity’s analysis of the problem, the returning twin shows a final clock -lag equal to the total nominal time -dilation effect accumulated during the constant -velocity stages of its journey. It is difficult to model this outcome gravitationally, since the application of gravitational effects to signals belonging to the slow astronaut’s coasting stages can undermine the special the ory’s calculations [45][46]. If we assume that the sudden acceleration of the traveller produces a shift - inducing gravitational field effect, the characteristics of this abruptly -introduced field are not straightforward [47]. The situation is also difficult to test experimentally. The favoured GR approach to the problem seems to be to amend the experiment so that the traveller does not coast, but experiences a constant acceleration throughout the journey (MTW [22] §6.2-6.6 pp.166 -176.). This, of course, brings us back to a situation where all of the final measurable clock - difference is accumulated while the object is accelerating. 13. Checklist Inertial motion: Model freq’/freq @90° LAB Stationary absolute aether 1 Moving absolute aether 1-vv/cc special relativity (1-vv/cc)1/2 emitter -theory in flat space 1-vv/cc dragging (extreme) 1-vv/cc Dragging (intermediate) 1 to 1-vv/cc “relativistic Doppler” equation, applied relativistically [19], no assumption of flat spacetime (1-vv/cc)1/2 “emitter -theory” Doppler equation, applied relativistically [19], no assumption of flat spacetime 1-vv/cc aether models incorporating time dilation (1-vxvy/cc) n Aether models incorporating Lorentz time dilation typically ~(1-vv/cc)1/2 Transverse redshift effects … Eric Baird Sunday , 29 October 2000 5:47 AM page 5 / 5 ArXiv reference: http://xxx.lanl.gov/abs/ physics/0010074 Inertial motion – timeflow: SR Result Calculable without SR time dilation? Muon track length Yes “Original” twins problem (combination of inertial an d non-inertial motion) Problematic Non-inertial motion: Test shift expected without SR? Gravity -shift Yes Centrifuge test Yes Haefe -Keating Yes Orbiting atomic clocks Yes Rotating object Yes Thermal atoms Yes “Constant -g” twins problem Yes 14. CONCLUSIONS A small amount of investigation shows that transverse redshifts (where “transverse” means “transverse in the laboratory frame”) do seem to appear in most models – of those considered here, only one (flat absolute aether stationary in the observer f rame) is not immediately associated with a lab -transverse redshift prediction. Although it might be considered convenient to dismiss many of these redshift predictions by specifying that the “transverse” detector should be aimed at an angle other than 90° in the laboratory frame, this introduces an additional level of interpretation and theory -dependence into our experiments, and invites confusion about which sets of predictions apply to which experiments. Since some of these redshift predictions belong to models that predate special relativity and produce stronger lab-transverse redshifts than Einstein’s special theory, casual statements that “transverse redshifts only appear under special relativity” need to be treated with a certain amount of trepidation. REFERENCES [1] A. Einstein, "On the Electrodynamics of Moving Bodies" (1905), translated in The Principle of Relativity (Dover, NY, 1952) Section 7 pp.35 -65. [2] W.G.V. Rosser, An Introduction to the Theory of Relativity (Butterworths, London, 1964) section 4.4.7 pp.160. [3] “… transverse Doppler effect . This is a relativistic effect, for classically one would not expect a frequency shift from a source that moves by right angles. ” Richard A. Mould, Basic Re lativity , (Springer -Verlag, NY, 1994) pp.80. [4] “… transverse Doppler shift … this is a purely relativistic effect …” Ray d’Inverno Introducing Einstein’s Relativity (OUP, Oxford, 1992) pp.40. [5] A.I.A. Adey, “A Note on Transverse Doppler Effects,” Galilean Electrodynamics 7 99-100 (Sept/Oct 1996). [6] Oliver Lodge, “Aberration Problems,” Phil.Trans.Roy.Soc. (1893) sections 56 -57. [7] Oliver Lodge, The Ether of Space (Harper & Brothers London 1909) chapter X. [8] Herbert E. Ives and G.R. Stilwel l, “An Experimental Study of the Rate of a Moving Atomic Clock,” J.Opt.Soc.Am. 28 215-226 (1938). [9] Hirsch I. Mandelburg and Louis Witten, “Experimental Verification of the Relativistic Doppler Effect,” J.Opt.Soc.Am. 52 529-536 (1962). [10] Kaivola, Poulson, Riis & Lee “Measurement of the Relativistic Doppler Shift in Neon,” Phys.Rev.Lett. 54 255-258 (1985) [11] P. Juncar et.al., “New Method to Measure the Relativistic Doppler Shift: First Results and a Proposal,” Phys.Rev.Lett. 54 11-13 (1985) [12] D.W. MacArthur “Special relativity: Understanding experimental tests and formulations,” Phys.Rev. A33 1-5 (1986). [13] R.Klein, R Grieser et. al ., “Measurement of the transverse Doppler shift using a stored relativistic 7Li+ ion beam,” Z.Phys.A 342 455-461 (1992) [14] Roger W. McGowan et. al. “New Measurement of the Relativistic Doppler Shift in Neon,” Phys.Rev.Lett. 70 251-254 (1993). [15] Roy Weinstein, "Observation of Length by a Single Observer," Am. J. Phys. 28 607-610 (1960) Transverse redshift effects … Eric Baird Sunday , 29 October 2000 5:47 AM page 6 / 6 ArXiv reference: http://xxx.lanl.gov/abs/ physics/0010074 [16] Eric Baird “Rul er-changes and relative velocity” arXiv ref: physics/9807015 (1998) [17] This is sometimes referred to as “the spatial analogue of the Doppler effect”. [18] T.M. Kalotas and A.R. Lee, “A two -line derivation of the relativistic longitudinal Doppler formula,” Am.J.Phys 58 187-188 (1990) [19] Eric Baird “Relativistic angle -changes and frequency -changes” arXiv ref: physics/0010006 (2000) [20] John Michell, “On the Means of discovering the Distance, Magnitude, &c. of the Fixed Stars … ,” Phil.Trans.Royal Soc. (1784) pp. 35 -57 & Tab III, sections 30 -32. [21] A. Einstein, "On the Influence of Gravitation on the Propagation of Light" (1911), translated in The Principle of Relativity (Dover, NY, 1952) pp.97 -108. [22] Misner, Thorne and Wheeler (MTW) Gravitation (Freeman NY 1971). [23] For a photon round -trip, the emitter -theory shift calculation gives ΔE = 1 - [(c-v)(c+v)/c2] = v2/c2 = 2gh/c2 The “Newtonian” mgh one-way result referred to in MTW §7.2 is a first -order approximation. [24] R.V. Pound and J.L. Snider, "Effect of Gravity on Gamma Radiation," Phys.Rev. 140 B 788-803 (1965). [25] Robert V. Pound, “Weighing photons,” Class.Quantum Grav. 17 2303 -2311 (2000) [26] H.J.Hay, J.P. Schiffer et al, “Measurement of the red shift in an accelerated system using the Mössbauer effect in Fe57,” Phys.Rev.Lett. 4 165-166 (1960). [27] A. Einstein The Meaning of Relativity (Chapman & Hall NY 1967) pp.95 -98. [28] Einstein’s 1911 paper describes a gravitational field as a change in refractive index. Newton’s previous attempt at this approach was a failure. [29] J.C. Hafele and R. E. Keating, "Around -the- world atomic clocks: Observed relativistic time gains," Science 177 168-170 (1972). [30] A. G. Kelly, "Reliability of Relativistic Effect Tests on Airborne Clocks" Inst. Engineers.Ireland Monograph No. 3 (February 1996). ISBN 1 -898012 -22-9 [31] Clifford M. Will Was Einstein Right?: Putting General Relativity to the Test (Basic Books NY 1986). [32] Jeremy Bernstein, Cranks, Quarks and the Cosmos (Basic Books 1993) “How can we be sure that Einstein was not a crank?” pp.21 -22. [33] Eric Baird , “GR without SR: A gravitational - domain description of Doppler shifts” arXiv ref: gr-qc/9807084 (1998). [34] The 1998 version of the “GR without SR” paper has some known problems. A revised version should be available during 2000/2001 [35] R.V.Pound and G.A.Rebka Jr., “Variation with temperature of the energy of recoil -free gamma rays from solids,” Phys.Rev.Lett. 4 274-275 (1960). [36] “[muons] … the discrepancy … ca n only be removed by means of the time -dilatation factor. … agreement [with SR] is remarkably good…” J.G. Taylor Special Relativity (Clarendon Oxford 1975) §2.3 pp.16 -19. [37] Edwin F. Taylor and John Archibald Wheeler Spacetime Physics: Introduction to Special relativity; second edition (W.H. Freeman NY 1992) §7.7 pp.211 -213 [38] Herman Erlichson, “The Rod Contraction - Clock Retardation Ether Theory and the Special Theory of Relativity,” Am.J.Phys. 41 1068 -1077 (1973). [39] J.P. Cedarholm and C.H. Towne s, “A new experimental test of special relativity,” Nature 184 1350 -1351 (1959) [40] Reza Mansouri and Roman U. Sexl, “A Test Theory of Special Relativity: II. First Order Tests,” Gen.Rel. and Gravitation 8 515-524 (1977) [41] A.K.A. Maciel and J. Tiomno, “Experiments to Detect Possible Weak Violations of Special Relativity,” Phys.Rev.Lett. 55 143 -146 (1985). [42] Herbert E. Ives, “The Doppler Effect Considered in relation to the Michelson -Morley Experiment,” J.Opt.Soc.Am. 27 389-392 (1937). [43] Paul J . Nahin Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction (AIP Press NY 1993), Tech Note 5 pp.317 -321. [44] William Moreau, “Nonlocality in frequency measurements of uniformly accelerating observers,” Am.J.Phys. 60 561-564 (1992). [45] C.B. Leffert and T.M. Donahue “Clock Paradox and the Physics of Discontinuous Gravitational Fields,” Am.J.Phys. 26 515-523 (1958). [46] C. Møller “Motion of Free Particles in Discontinuous Gravitational Fields” [1959?] [47] Eric Baird, “Warp drives, w avefronts and superluminality” arXiv ref: gr-qc/9904019 (1999).
The Heats of Dilution. Calorimetry and Van't-Hoff. I. A. Stepanov Latvian University, Rainis bulv. 19, Riga, LV-1586, Latvia e-mail istepanov@email.com Abstract Earlier it has been found that the re is a big difference between heats of dilution measured by calorimetry and by the Van't-Hoff equation. In the present paper a reason for that is proposed. Experimental data for dilution of benzene and n-hexane in water were used. 1. Introduction There is a big difference between heats of dilution measured by calorimetry and by the Van't-Hoff equation. Let’s cite [1]: “Historically, the Van’t-Hoff isochore has frequently been used as a means of deriving enthalpy change values for reactions of interest from experimental data gathered at a number of temperatures. ... Detailed review of the data in articles for reactive systems where the enthalpy change has been determined both by calorimetric measurement and other temperature-change techniques show that there is often a sizeable discrepancy between the two values. ... It is possible to show that improvements ininstrumentation and computing power have not overcome earlier difficulties in the estimation of enthalpy values from data taken at several temperatures. Perusal of literature over the last three decades shows that many authors have been aware that obtaining enthalpy values by these techniques is inherently unsatisfactory”. In private communication to the author the author of [1] wrote: “There are a number of comprehensive collections of enthalpy data, particularly those authored by R. M. Izatt and J. J. Christensen, which contain all the thermodynamic data for many systems. Further, many of these systems contain data collected by calorimetric determination and data from 1/T estimation methods in the same table. As I examined those it seemed that there was always a consistent difference between the two methods”. In the present paper an attempt is made to explain this papadox. 2. Theory For chemical processes the law of conservation of energy is written in the following form: dU=dQ-PdV+ i∑µidNi (1) where dQ is the heat of reaction, dU is the change in the internal energy, µi are chemical potentials and dN i are the changes in the number of moles. In [2-6] it has been shown that the energy balance in the form of (1) for the biggest part of the chemical reactions is not correct. In the biggest part of the chemical reactions the law of conservation of energy must have the following form: dU=dQ+PdV+ i∑µidNi (2) The Van’t-Hoff equation is the following one :d/dTlnK= ΔH0/RT2(3) where K is the reaction equilibrium constant and ΔH0 is the enthalpy. According to thermodynamics , the Van’t-Hoff equation must give the same results as calorimetry because it is derived from the 1st and the 2nd law of thermodynamics without simplifications. However, there is a paradox: the heat of chemical reactions, that of dilution of liquids and that of other chemical processes measured by calorimetry and by the Van’t-Hoff equation differ significantly [1-6]. The difference is far beyond the error limits. T he reason is that in the derivation of the Van’t-Hoff equation it is necessary to take into account the law of conservation in the form of (2), not of (1) [2-6]. If to derive the Van’t-Hoff equation using (2) the result will be the following one: d/dTlnK= ΔH0/RT2 (4) where ΔH0=ΔQ0+PΔV0. 3. Experimental Check and Discussion In [7] heats of dilution of benzene and n-hexane in water were given (Table 1). One sees that ΔH is a few times bigger than the experimental value ΔQ. It is impossible to explain that by non-ideality of solution. In [7-9] they use K c and Ka in (3) (the constants depending on the concentrations and on the activities, respectively). It confirms the result [2-6] that in chemical processes ΔQ is not equal to ΔH. From Table 1 it is possible to find the change in the volume ΔV for dilution. For benzene ΔV≈10-2 and 2⋅10-2 m3/mol, for n-hexane ΔV≈2⋅10-2 m3/mol. ΔV=V2-V1≈V2 because for both substances V 1<<ΔV. Pay attention that for both substances V 2 is close to the volume of the ideal gas (2,5 ⋅10-2 m3/mol).Let's analyze a phase transition which is not a chemical reaction [10] Zn(liq)=Zn(gas) 700<T<1000 K. (5) lnP=-118366/RT+12,049 (6) From (6) ΔH0=118,366 kJ/mol. From [11] for T=900 ΔQ=118,31 kJ/mol, for T=800 ΔQ=119,37 kJ/mol. One sees that in this case the usual Van’t-Hoff equation is valid. The Van’t-Hoff equation (3) was often used for determination of the heat of reaction. It gives correct results for reactions with ΔV→0. Therefore, it was assumed that for chemical reactions ΔQ=ΔH=ΔU+PΔV (7) However from (1) and (7) it follows ΔQ=ΔQ+∑µiΔNi (8) This conclusion is an absurd: one can not neglect the last term in (8). Therefore, the Van’t- Hoff equation must not give correct results neither for ΔV→0 nor for ΔV≠0. For reactions with ΔV≠0 this equation gives wrong results [1-6]. The present theory explains this paradox: in chemical processes (4) must be used.References 1. P. R. Brown, in Proc. 12th IUPAC Conf. Chem. Thermodynamics , Snowbird, Utah, August, 1992, 238-239. 2. I. A. Stepanov, DEP VINITI , No 37-B96 , (1996). Available from VINITI , Moscow. 3. I. A. Stepanov , DEP VINITI , No 3387-B98. (1998). Available from VINITI , Moscow. 4. I. A. Stepanov , 7th European Symposium on Thermal Analysis and Calorimetry. Aug. 30 - Sept. 4. Balatonfuered, Hungary. 1998. Book of Abstracts. P. 402-403. 5. I. A. Stepanov , The Law of Conservation of Energy in Chemical Reactions.- http://ArXiv.org/abs/physics/0010052. 6. I. A. Stepanov , The Heats of Reactions. Calorimetry and Van't-Hoff. 1 . - http://ArXiv.org/abs/physics/0010054. 7. D. S. Reid, M. A. Quickenden , F. Franks, Nature, 224 (1969) 1293. 8. C. V. Krishnan, H. L. Friedman, J. Phys. Chem. , 73 (1969) 1572. 9. R. L. Bohon, W. F. Claussen, J. Amer. Chem. Soc. , 73 (1951) 1571. 10. R. Hultgren, P. Desai, D. Hawkins, M. Gleiser, K. Kelly, Selected Values of the Thermodynamic Properties of the Elements , (American Society for Metals, Metals Park, OH) 1973. 11. I. Barin, Thermochemical Data of Pure Substances (VCH, Weinheim, Germany) 1989.Table 1 Heats of dilution of benzene and n-hexane in water [7] ΔΔQ/J/mol, CalorimetryΔΔH/J/mol, Van't-Hoff Benzene 800 460 [8]1820 2430 [9] n- hexane470 2500
arXiv:physics/0010076v1 [physics.optics] 30 Oct 2000Optical interpretation ofspecial relativity and quantummechanics Jos´eB. Almeida UniversidadedoMinho,PhysicsDepartment,4710-057Braga ,Portugal Tel:+351-253604390,e-mail:bda@ﬁsica.uminho.pt Abstract: The present work shows that through a suitable change of vari ables rela- tivistic dynamics can be mapped to light propagation in a non -homogeneous medium. A particle’s trajectory through themodiﬁed space-timeis t hus formallyequivalentto a light ray and can be derived from a mechanical equivalent of F ermat’s principle. The similaritiesbetweenlightpropagationandmechanicsaret henextendedtoquantumme- chanics,showingthatrelativisticquantummechanicscanb ederivedfromawaveequa- tion in modiﬁed space-time. Non-relativistic results, suc h as de Broglie’s wavelength, Schr¨ odingerequationanduncertaintyprincipleareshown tobedirectconsequencesof thetheoryand itis argued thatrelativisticconclusionsar e alsopossible. 1 Introduction This paper is presented in a rather crude state; the text is im perfect and some conclusions are deferred toulteriorpublications;neverthelesstheautho rfeelsthatthisworkmustbediffusedeven in a preliminary stage due to its signiﬁcance. The author’s p resence at the OSA’s annual meeting providedan opportunityforthepresentationofhiswork tha thecouldnotdespise. The similarities between light propagation and wave mechan ics have been pointed out by numer- ous authors, although a perfect mapping from one system to th e other has never been achieved. Almeida et al.1showed that near-ﬁeld light diffraction could be calculate d using the Wigner Dis- tributionFunction(WDF) andobtainedresultsprovingthee xistenceofsuper-resolutionincertain circumstances. Thestudyofwideanglelightpropagationmakesuseofatrans formationwhichbringstomindthe Lorentz transformationof special relativity.It was then n atural to try an associationof Newtonian mechanics to paraxial optics and special relativity to wide angle propagation. This process pro- moted the deﬁnition of a coordinate transformation to rende r the relativisticspace homologousto the optical space. The introduction of a variational princi ple allowed the derivation of relativistic dynamics in the modiﬁed space-time in a process similar to th e derivation of optical propagation fromFermat’sprinciple.Oneimportantconsequenceisthat each particletravelsthroughmodiﬁed space-timewith thespeed oflight. The similarity could be carried further to diffraction phen omena and quantum mechanics. It was postulated that a particle has an intrinsicfrequency relat ed to its mass and many important results were derived directly from this statement. More general res ults will probably be feasible in the future.Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 2 2 Notes onHamiltonianoptics Thepropagationofan opticalray is governedby Fermat’spri nciple,which can bestated:2 δ/integraldisplayP2 P1nds= 0. (1) The integral quantity is called point characteristic and measures the optical path length between points P1andP2. V(x1, y1, x2, y2) =/integraldisplayP2 P1nds . (2) Thequantity dsisthelengthmeasuredalong aray pathand can bereplaced by: ds=dz (1−u2x−u2y)1/2=dz uz, (3) where ux,uyanduzare theray directioncosineswithrespect tothe x,yandzaxes. InsertingEq. (3)intoEq.(1)wehave: δ/integraldisplayz2 z1ndz (1−u2 x−u2 y)1/2= 0, δ/integraldisplayz2 z1ndz uz= 0. (4) Ofcoursewecan also write: ds= (1 + ˙ x2+ ˙y2)1/2dz , (5) with ˙x=dx dzand ˙y=dy dz. (6) It iseasy torelate ˙xand˙ytouxanduy: ux=˙x (1 + ˙x2+ ˙y2)1/2, uy=˙y (1 + ˙x2+ ˙y2)1/2, (7) whereonlythepositiveroot isconsidered. InsertingEq. (5)intoEq.(1)weget: δ/integraldisplayz2 z1n(1 + ˙x2+ ˙y2)1/2dz= 0. (8) We can usethe positioncoordinates xandyas generalized coordinates and zfor time, in order to deﬁnetheLagrangian. Wehave:3,4 L=n(1 + ˙x2+ ˙y2)1/2(9)Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 3 EulerLagrange’spropagationequationsare: d dz∂L ∂˙x−∂L ∂x= 0, (10) d dz∂L ∂˙y−∂L ∂y= 0 ; (11) We can go a step further ifwe deﬁne a systemHamiltonianand wr itethe canonical equations; we start by ﬁnding the components of the conjugate momentum ( p) from the Lagrangian. Knowing thatnis afunctionof x,yandz, theconjugatemomentumcomponentscan bewrittenas: px=∂L ∂˙x=n˙x (1 + ˙x2+ ˙y2)1/2, py=∂L ∂˙y=n˙y (1 + ˙x2+ ˙y2)1/2. (12) IfweconsiderEq.(7), theresultis: px=nux, py=nuy. (13) ThesystemHamiltonianis: H=px˙x+py˙y−L =n( ˙x2+ ˙y2) (1 + ˙x2+ ˙y2)1/2−L =−n (1 + ˙x2+ ˙y2)1/2 =−n(1−u2 x−u2 y)1/2, =−nuz, (14) The Hamiltonian has the interesting property of having the d ependence on the generalized co- ordinates and time, separated from the dependence on the con jugate momentum. The canonical equationsare: ˙x=∂H ∂px=ux (1−u2x−u2y)1/2=ux uz, ˙y=∂H ∂py=uy (1−u2x−u2y)1/2=uy uz, n˙ux+ ˙nux=−∂H ∂x= (1−u2 x−u2 y)1/2∂n ∂x=uz∂n ∂x, n˙uy+ ˙nuy=−∂H ∂y= (1−u2 x−u2 y)1/2∂n ∂y=uz∂n ∂y. (15) Obviouslytheﬁrst twocanonical equationsrepresent justa trigonometricrelationship.Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 4 Itisinterestingtonotethatiftherefractiveindexvaries onlywith z,thentheconjugatemomentum will stay unaltered; the direction cosines will vary accord ingly to keep constant the products nux andnuy. We will now consider an non-homogeneous medium with a direct ion dependent refractive index and willaddthisdependenceas acorrection to anominalinde x. n=n0−nc (1 + ˙x2+ ˙y2)1/2, (16) where n0isthenominalindexand ncisacorrection parameter. Eq.(9)becomes L=n0/parenleftBig 1 + ˙x2+ ˙y2/parenrightBig −nc. (17) We will follow the procedure for establishing the canonical equations in this new situation. It is clearthat themomentumis stillgivenby Eq.(13)if nis replaced by n0. ThenewHamiltonianis givenby H=−n0(1−u2 x−u2 y) +nc, (18) and thecanonical equationsbecome ˙x=∂H ∂px=ux (1−u2x−u2y)1/2=ux uz, ˙y=∂H ∂py=uy (1−u2 x−u2 y)1/2=uy uz, n0˙ux+ ˙n0ux=−∂H ∂x=uz∂n0 ∂x−∂nc ∂x, n0˙uy+ ˙n0uy=−∂H ∂y=uz∂n0 ∂y−∂nc ∂y. (19) The present discussion of non-homogeneous media is not comp letely general but is adequate for highlighting similarities with special relativity and qua ntum mechanics, as is the purpose of this work. 3 Diffraction andWignerdistribution function Almeida et al.1have shown that the high spatial frequencies in the diffract ed spectrum cannot be propagated and this can even, in some cases, lead to a diffr action limit much lower than the wavelength;here wedetailthosearguments. TheWignerdistributionfunction(WDF)ofascalar,timehar monic,andcoherentﬁelddistribution ϕ(q, z)can be deﬁned at a z= const .plane in terms of either the ﬁeld distribution or its Fourier transform ϕ(p) =/integraltextϕ(q) exp(−ikqTp)dq:5–7 W(q,p) =/integraldisplay ϕ/parenleftBigg q+q′ 2/parenrightBigg ϕ∗/parenleftBigg q−q′ 2/parenrightBigg exp/parenleftBig −ikq′Tp/parenrightBig dq′(20) =k2 4π2/integraldisplay ϕ/parenleftBigg p+p′ 2/parenrightBigg ϕ∗/parenleftBigg p−p′ 2/parenrightBigg exp/parenleftBig ikqTp′/parenrightBig dp′, (21)Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 5 where k= 2π/λ,∗indicatescomplexconjugateand q= (x, y), (22) p= (nux, nuy). (23) In theparaxial approximation,propagationin ahomogeneou smediumofrefractiveindex ntrans- formstheWDFaccording totherelation W(q,p, z) =W(q−z np,p,0). (24) AftertheWDFhas been propagatedoveradistance,theﬁeld di stributioncan berecovered by6,7 ϕ(q, z)ϕ∗(0, z) =1 4π2/integraldisplay W(q/2,p, z) exp(iqp)dp. (25) Theﬁeld intensitydistributioncan alsobefoundby \|ϕ(q, z)\|2=4π2 k2/integraldisplay W(q,p, z)dp. (26) Eqs. (25) and (26) are all that is needed for the evaluation of Fresnel diffraction ﬁelds. Consider the diffraction pattern for a rectangular aperture in one di mension illuminatedby a monocromatic wave propagating in the zdirection. The ﬁeld distribution immediately after the ape rture is given by ϕ(x,0) = 1 → \| x\|< l/2, ϕ(x,0) = 0 → \| x\| ≥l/2, (27) withlbeingtheaperturewidth. Consideringthat ϕ(x,0)isreal wecan write ϕ/parenleftBigg x+x′ 2/parenrightBigg ϕ∗/parenleftBigg x−x′ 2/parenrightBigg = H/parenleftBiggl 2+x′ 2− \|x\|/parenrightBigg H/parenleftBiggl 2−x′ 2− \|x\|/parenrightBigg . (28) Wethen applyEq.(28)to theWDF deﬁnitionEq.(20)toﬁnd W(x, px) = 0 → \| x\| ≥l/2, W(x, px) =2 sin[kpx(l−2x)] kpx→0≤x < l/ 2, W(x, px) =2 sin[kpx(l+ 2x)] kpx→ − l/2≤x≤0, (29) Afterpropagationwe obtainthefollowingintegralﬁeld dis tribution \|ϕ(x, z)\|2=4π2 k2/braceleftBigg/integraldisplaynx/z n(2x−l)/(2z)2 sin[kpx(l−2zpx/n−2x)] kpxdpx +/integraldisplayn(2x+l)/(2z) nx/z2 sin[kpx(l+ 2zpx/n+ 2x)] kpxdpx/bracerightBigg . (30)Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 6 01234 z(mm)-0.15-0.1-0.0500.050.10.15q(mm) Fig.1.Fresneldiffractionpatternfora one-dimensionala pertureofwidth 0.1 mmwithk= 107. Fig. 1showsatypicaldiffractionpattern obtainedbynumer icalintegrationofEq. (30). Forwideanglesparaxialapproximationnolongerappliesan dtheappropriateWDFtransformation isnowgivenby W(q,p, z) =W(q−zp/radicalBig n2− \|p\|2,p,0)→ \|p\|< n, W(q,p, z) = 0 →otherwise . (31) Eq. (31) shows that only the moments such that \|p\|< ncan be propagated.8In fact, if \|p\|/n= sinα, with αtheangletheray makes withthe zaxis, It is obviousthat thehighermomentswould correspond to values of \|sinα\|>1; these moments don’t propagate and originate evanescent waves instead,Fig. 2. The net effect on near-ﬁeld diffracti on is thatthehigh-frequency detail near theapertureisquicklyreduced. Theﬁeld intensitycan nowbeevaluatedbytheexpression k2\|ϕ(x, z)\|2 4π2=/integraldisplayp0 p11 kpxsin/braceleftbigg kpx/bracketleftbigg l−2/parenleftbigg x−zpx//radicalBig n2−p2x/parenrightbigg/bracketrightbigg/bracerightbigg dpx +/integraldisplayp2 p01 kpxsin/braceleftbigg kpx/bracketleftbigg l+ 2/parenleftbigg x−zpx//radicalBig n2−p2 x/parenrightbigg/bracketrightbigg/bracerightbigg dpx,(32) with p0=nx√ x2+z2, p1=n(2x−l)/radicalBig (2x−l)2+ 4z2,Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 7 -10-50510 q-1-0.500.51p a)-10-50510 q-1-0.500.51p b) Fig. 2. Propagationof the WDF in wide angle condition( k= 106m−1, horizontalscale in µm). a) Originaldistribution,b)afterpropagationover 3µm. p2=n(2x+l)/radicalBig (2x+l)2+ 4z2. (33) 0246810 z-4-2024q Fig. 3. Near-ﬁeld diffraction pattern when the aperture wid th is exactly one wavelength; ( k= 106m−1, bothscalesin µm). Fig. 3 shows the near-ﬁeld diffraction pattern when the aper ture is exactly one wavelength wide. The situationis such that all the high frequencies appear at values of \|px\|>1and are evanescent, resultinginaﬁeldpatternwithonesmallminimumimmediate lyaftertheaperture,afterwhichthe beam takes aquasi-gaussianshape, withoutfurtherminima. Thewidthofthesharp peak justafter the aperture is considerably smaller than one wavelength de termining a super-resolution on this region.Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 8 4 Special relativity Special relativitydealswitha 4-dimensionalspace-timee ndowedwitha pseudo-Euclideanmetric whichcan bewrittenas ds2=dx2+dy2+dz2−dt2, (34) where the space-timeis referred to the coordinates (t, x, y, z ). Here theunits were chosen so as to makec= 1,cbeing thespeed oflight. For a more adequate optical interpretation one can use coord inates (x, y, z, τ )withτthe proper time:9 τ=t/parenleftBigg 1−v2 c2/parenrightBigg1/2 =t γ. (35) Herev2=\|v\|2, withv v=/parenleftBiggdx dt,dy dt,dz dt/parenrightBigg , (36) theusual 3-velocity vector.Fig. 4 showsthe coordinates of an event Ewhen just two spatial coor- dinatesare usedtogetherwithcoordinate τ. Fig. 4. The relativistic frame in two dimensions. The τcoordinate is the proper time while the distancetotheoriginis thetimemeasuredinthe frameat res t. ConsideringEqs.(35, 36)andthenewcoordinates themetric deﬁned byEq. (34)becomes ds2=v2dt2−dt2=dτ2. (37) The trajectory of an event in 4-space is known as its world lin e. In Fig. 5 we represent the world line of an event Ewith coordinates (x, τ). At each position the derivative of the world line with respect to τis ˙x=dx dtdt dτ=γvx. (38) Moregenerallywecan write − →ν=γv, (39)Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 9 Fig. 5. The curved line represents the word line of event E. The speed is represented by the vector v, whichistheside ofarectangulartrianglewhosehypotenus ehasamagnitude c. with − →ν= ( ˙x,˙y,˙z). (40) It follows from Eq. (40) that, at each point, the components o fvare the direction cosines of the tangentvectortotheworldlinethroughthat point. We must now state a basic principle equivalentto Fermat’s pr inciple for ray propagation in optics given in Eq. (1). Taking into consideration the relativisti c Lagrangian we can state the following variationalprinciple:3,4 δ/integraldisplay (mγ−V)ds= 0, (41) where mis the rest mass of the particle and Vis the local potential energy. It can be shown that mγrepresentsthekineticenergyoftheparticle.UsingEq.(37 )astraightforwardcalculationshows thatds=dτso thatwecan also write δ/integraldisplay (mγ−V)dτ= 0. (42) Thenewprinciplemeansthattheparticlewillchooseatraje ctorymovingawayfromhighpotential energiesmuchastheopticalraymovesawayfromlowrefracti veindex.Fromthisprinciplewecan derivealltheequationsofrelativisticmechanics,inasim ilarmannerasfrom Fermat’s principleit ispossibletoderivelightpropagation. ComparingthisequationwithEq. (9), wecan deﬁnetheLagran gian ofthemechanical systemas L= (mγ−V). (43) We now have a 4-dimensional space, while in the Hamiltonian f ormulation of optics we used 3 dimensions. The following list shows the relationship betw een the two systems, where we refer ﬁrst totherelativisticsystemandthen totheopticalsyste m. x↔x, y↔y, z↔no equivalent , τ↔z. In the mapping from optical propagation to special relativi ty the optical axis becomes the proper timeaxis and the ray direction cosines correspond to the compone nts of the speed vector v. TheAlmeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 10 refractive index has no direct homologous; it will be shown t hat in special relativity we must consider a non-homogeneous medium with different refracti ve indices in the spatial and proper timedirections. We can derive the conjugate momentum from the La grangian using the standard procedure: p=/parenleftBiggm− →ν γ/parenrightBigg =mv. (44) Comparing with Eq. (13) it is clear that mis the analogous of the position dependent refractive index. ThesystemHamiltoniancan becalculated H=p·v−L =mν2 γ−L =−m γ+V. (45) Thecanonicalequationsfollowdirectlyfrom Eq.(15) − →ν=γv, dp dτ=−gradH, (46) wherethegradientis takenoverthespatialcoordinates,as usual. Theﬁrst oftheequationsaboveisthesameasEq. (40), whilet hesecondonecan bedevelopedas d dτ/parenleftBiggm γ− →ν/parenrightBigg = grad/parenleftBiggm γ−V/parenrightBigg . (47) Considering that from the quantities inside the gradient on ly the potential energy should be a functionofthespatialcoordinates,wecan simplifythesec ondmember: d dτ/parenleftBiggm γ− →ν/parenrightBigg =−gradV (48) d dτ(mv) = −gradV. (49) Eq. (49)is formallyequivalentto thelast twoEqs. (19), con ﬁrmingthat theconjugatemomentum componentsareproportionaltoworldline’sdirectioncosi nesin4-space.Thetotalrefractiveindex analoguecannowbefoundtobe m−V/γ.Wecancheck thevalidityofEq.(49)byreplacingthe τderivativebyaderivativewithrespect to t. d dt(mγv) =−gradV, (50) where mγistherelativisticmassandtheproduct mγvis therelativisticmomentum. If the mass is allowed to be coordinate dependent, as a mass di stribution through the Universe, the passage between Eqs. (47) and (48) is illegitimate and we are led to equations similar to Eq. (19). The consideration of a coordinate dependent mass allo ws the prediction of worm tubes, the analogues of optical waveguides, and black holes, which wou ld have optical analogues in high refractiveindexglass beadswithgradual transitionto vac uumrefractiveindex.Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 11 5 De Broglie’s wavelength The formal equivalence between light propagation and speci al relativity in the x, y, z, τ frame suggeststhatdeBroglie’swavelengthmaybetheformalequi valentoflightwavelength.Wewould liketoassociateanevent’sworldlinetoalightrayandsimi larlywewanttosaythat,intheabsence of a potential, the event’s world line is the normal to the wav efront at every point in 4-space. We muststartfromabasicprinciple,statingthateach particl ehasan intrinsicfrequencyrelated toit’s mass in a similar way as the wavelength of light is related to t he refractive index; we state this principleby theequation f=m h, (51) where his Planck’s constant. If we remember that everything is norm alized to the speed of light byc= 1,Eq. (51)isequivalenttoa photon’senergy equation E=hf. (52) So we have extended an existing principle to state that any pa rticle has an intrinsicfrequency that istheresultofdividingit’sequivalentenergy E=mc2byPlanck’s constant. In the 4-space x, y, z, τ frame a material particle travels in a trajectory with direc tion cosines givenby thecomponentsof vand consistentlya photontravelsalong aspatial direction with zero component in the τdirection.The intrinsicfrequency deﬁned by Eq. (51) origi natesa wavelength alongtheworldlinegivenby λw=c f=h mc, (53) wherewehavetemporarilyremovedthe cnormalizationforclarity reasons. Fig. 6. The moving particle has a world wavelength λw=h/(mc)and a spatial wavelength λ= h/(mv). As shownin Fig.6, when projectedonto3-space λwdeﬁnes aspatialwavelengthsuchthat λ=λwc v=h p. (54) The previous equation deﬁnes a spatial wavelength which is t he same as was originally proposed by de Broglie in the non-relativistic limit. When the speed o f light is approached Eq. (54) will produce a wavelength λ→λwwhile de Broglie’s predictions use the relativistic moment um and soλ→ ∞whenv→c.Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 12 6 Wavepropagationand Schr ¨odinger equation The arguments in the previous paragraphs lead us to state the general principle that a particle has an associated frequency given by Eq. (51) and travels on a wor ld line through 4-space with the speed oflight.In ageneralization to wavepackets,we willr espect theformal similaritywithlight propagation and state that all waves travel in 4-space with t he speed of light. A particle about which we know mass and speed but know nothing about its positi on will be represented by a monocromaticwaveand amovingparticleingeneral willbere presented by awavepacket. Accordingto general opticalpracticewewillsay thattheﬁe ld mustverifythewaveequation /parenleftBigg ∇2−∂2 ∂t2/parenrightBigg ϕ(P, t) = 0, (55) wherePisapointin4-spaceand ∇2isan extendedlaplacian operator ∇2=∂2 ∂x2+∂2 ∂y2+∂2 ∂z2+∂2 ∂τ2. (56) In Eq. (55) we have returned to the c= 1normalization in order to treat all the coordinates on an equal footing.Dueto thespecial 4-spacemetricwe willassu methat ϕ(P, t)isoftheform ϕ(P, t) = Φ( P)ei2πft, (57) withfgivenbyEq.(51).Noticethatweusedaplussignintheexpone ntinsteadoftheminussign used inopticalpropagation;thisisduetothespecial 4-spa cemetric. Notsurprisinglywewillﬁndthat,intheabsenceofapotenti al,Eq.(55)canbewrittenintheform ofHelmoltzequation/parenleftBig ∇2+k2/parenrightBig Φ(P) = 0, (58) with k=2π λw. (59) IfwetakeintoconsiderationEq.(35), thelaplacianbecome s ∇2=∂2 ∂x2+∂2 ∂y2+∂2 ∂z2+γ2∂2 ∂t2 =∇2 3+γ2∂2 ∂t2, (60) where∇2 3representstheusuallaplacian operatorin3-space. In ordertoderiveSchr¨ odingerequationwere-writeEq. (55 ) ∇2ϕ(P, t) +ik∂ϕ(P, t) ∂t= 0, (61) and usingEq.(60) ∇2 3ϕ(P, t) +ik(γ2+ 1)∂ϕ(P, t) ∂t= 0. (62)Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 13 In a non-relativistic situation γ→1. Considering Eq. (53) we can write Eq. (62) in the form of Schr¨ odingerequation10 i¯h∂ϕ(P, t) ∂t=−¯h2 2m∇2 3ϕ(P, t), (63) where ¯h=h/(2π). Eq. (63) retains the symbol Prepresenting a point in 4-space. It must be noted, though, th at in a non-relativisticsituation τ→tand we can say that ϕ(P, t)→ϕ(P, t)withPhaving the 3-space coordinatesof P. Inthepresenceofapotentialwehavetoconsiderthatmoment umisnolongerpreserved,asshown by Eq. (49). This can be taken into account when we evaluate th e laplacian in Eq. (62) by the inclusion of an extra term −V ϕ(P, t). We will the end up with the Schr¨ odinger equation in a potential. 7 Heisenberg’s uncertainty principle Forthepairofassociatedvariables xandpx,Heisenberg’suncertaintyprinciplestatesthatthereis an uncertaintygovernedbytherelation ∆x∆px≥¯h 2. (64) The interpretation of the uncertainty principle is that the best we know a particle’s position, the least we know about its momentum and vice-versa; the product of the position and momentum distributionwidthsisasmallnumbermultipliedbyPlanck’ sconstant.Anapplicationoftheuncer- taintyrelationshipis usuallyfound inthediffractionofa particleby an aperture. Ifweassumethatthelocalizationofaparticlewithmomentu mpxcanbedonewithanapertureof width ∆x,wecanuseFraunhoferdiffractiontheorytosaythatfurthe ralongthewaytheprobability ofﬁndingtheparticlewithanyparticularvalueofit’smome ntumwillbegivenbythesquareofthe aperture Fouriertransform, consideringdeBroglie’s rela tionshipfor thetranslationofmomentum intowavelength. A rectangularapertureofwidth ∆xhasaFouriertransformgivenby A(fx) = ∆ xsinc(∆ xfx), (65) where sincis theusual sin(x)/xfunction. Considering de Broglie’s relationship given by Eq. (54), ma kingfx= 1/λand the fact that the Fouriertransformmustbesquaredwecan write P(px) = ∆ x2sinc2/parenleftbigg∆xpx h/parenrightbigg . (66) The second member on the previous equation has its ﬁrst minim um for ∆xpx/h=πand so we can say thatthespread inmomentumisgovernedby ∆x∆px=πhandEq. (64)isveriﬁed.Almeida,Optical interpretation . . . OSA Annualmeeting/2008 Page 14 If we accept that wave packets propagate in 4-space at the spe ed of light and that the momentum is given by Eq. (44), there is a upper limit to the modulus of th e momentum \|p\| ≤mc. In the propagationoflightrayswefoundasimilarlimitationas \|p\| ≤nwiththeresultsinlightdiffraction exempliﬁed by Eq. (32) and Fig. 3. It is expected that the same effects will be present in particle diffraction and in fact Figs. 2 and 3 could also represent the diffraction of a stationary particle by an aperture with width equal to λw. The strong peak about half one wavelength in front of the aperture shows that the particle is localized in a region con siderably smaller than its wavelength and, aboveall, showstheinexistenceofhigherorderpeaks. 8 Conclusions Special relativitywasshowntobeformallyequivalenttoli ghtpropagation,providedthetimeaxis is replaced by the proper time . In this coordinate set all particles follow w world line at t he speed oflightandcanbeassumedtohaveanintrinsicfrequencygiv enbymc2/h.Quantummechanicsis then a projection of 4-space wave propagation into 3-space. Important conclusions were possible throughtheanalogy withlightpropagationand diffraction . It was possible to derive Schr¨ odinger equation and it was sh own that Heisenberg’s uncertainty principle may be violated in special cases in the very close r ange, similarly to what had already been showntohappen inlightdiffraction.1 Future work will probably allow the derivation of relativis tic quantum mechanics conclusions, throughtheuseoftheWignerDistributionFunctionforthep redictionofwavepacket propagation in4-space. 9 Acknowledgements The author acknowledges the many fruitful discussions with Estelita Vaz from the Mathematics DepartmentofUniversidadedo Minho,especially onthesubj ectofrelativity. References 1. J. B. Almeida and V. Lakshminarayanan, ”Wide Angle Near-F ield Diffraction and Wigner Distribution”, Sub- mittedtoOpt.Lett.(unpublished). 2. M.BornandE.Wolf, Principlesof Optics ,6th.ed.(CambridgeUniversityPress, Cambridge,U.K.,19 97). 3. H.Goldstein, ClassicalMechanics ,2nd.ed.(AddisonWesley,Reading,MA,1980). 4. V. J. Jos´ e and E. J. Saletan, Classical Mechanics – A Contemporary Aproach , 1st. ed. (Cambridge University Press, Cambridge,U.K.,1998). 5. M. J. Bastiaans, “TheWignerDistributionFunctionand Ha milton’sCharacteristicsof a Geometric-OpticalSys- tem,”Opt.Commun. 30,321–326(1979). 6. D.Dragoman,“TheWignerDistributionFunctioninOptics andOptoelectronics,”in ProgressinOptics ,E.Wolf, ed.,(Elsevier,Amsterdam,1997),Vol. 37,Chap.1, pp.1–56 . 7. M. J. Bastiaans, “Application of the Wigner Distribution Functionin Optics,” In The WignerDistribution - The- ory and Applications in Signal Processing , W. Mecklenbr¨ auker and F. Hlawatsch, eds., pp. 375–426 (El sevier Science,Amsterdam,Netherlands,1997). 8. J. W. Goodman, IntroductiontoFourierOptics (McGraw-Hill,NewYork,1968). 9. R. D’Inverno, IntroducingEinstein’sRelativity (ClarendonPress, Oxford,1996). 10. S.Gasiorowicz, QuantumPhysics ,2nded.(J.WileyandSons,NewYork,1996).
ProgressintheNextLinearColliderDesign
arXiv:physics/0010078v1 [physics.acc-ph] 30 Oct 2000STUDIESOF BEAM OPTICSAND SCATTERING INTHE NEXT LINEAR COLLIDERPOST-LINACCOLLIMATIONSYSTEM∗ P. Tenenbaum,R. Helm,L. Keller,T.O.Raubenheimer, SLAC, S tanford, CA, USA Abstract We present a new conceptual and optical design for the Next Linear Collider post-linac collimation system. En- ergy collimation and passive protection against off-energ y beams are achieved in a system with large horizontal dis- persion and vertical betatron functions. Betatron collima - tion is performedin a relativelylow-beta(FODO-like) lat- tice in which only thin spoilers intercept particles near th e beam core, while thick absorbers maintain a large stay- clear from the beam. Two possible schemes for the spoil- ersareconsidered: oneinwhichthespoilersarecapableof tolerating a certain number of damaging interceptions per colliderrun(”consumable”spoilers),andone in whichthe spoilers are potentially damaged on every machine pulse and are self-repairing (”renewable” spoilers). The colli- mation efﬁciency of the system is evaluated, considering bothhaloparticleswhicharerescatteredintothebeamand muon secondaries which are passed to the interaction re- gion. We concludethat the new designis a promisingcan- didatefortheNLCpost-linacsystem. 1 INTRODUCTION The experience of the Stanford Linear Collider (SLC) in- dicates that collimation of the beam halo at the end of the mainlinacsoftheNextLinearCollider(NLC)willbeane- cessity. The principal requirementson the NLC post-linac collimationsystem areasfollows: •The system should stop particles which would gen- erate unacceptable backgrounds in the detector from enteringthe ﬁnalfocus •The collimationefﬁciencyshould be sufﬁciently high thatthenumberofhaloparticleswhicharetransmitted to the ﬁnal focus is comparable to the number gener- ated bybeam-gasand thermal-photonscatteringfrom the collimationregionandtheﬁnal focus •Thenumberofmuonsecondariesfromthecollimation system whichreachthedetectormust beminimized •The opticalandwakeﬁeld dilutionsofthe beamemit- tancesdueto thecollimationsystemmust besmall •Thesystemmustprotecttheﬁnalfocusandthedetec- tor from beams which have large energy or betatron excursionswithoutbeingdestroyedin theprocess. The 1996 NLC design included a post-linac collima- tion system shown in Figure 1 [1]. The system design ∗Work supported by U.S. Department of Energy, Contract DE-AC 03- 76SF005150.0 250. 500. 750. 1000. 1250. 1500. 1750. 2000. 2250. 2500. s (m) δE/ p0c = 0. Table name = twissSample MAD command file for NLC BDS Windows NT 4.0 version 8.23/acc 30/06/99 18.43.45 0.022.545.067.590.0112.5135.0157.5180.0202.5225.0β1/2(m1/2) -0.04-0.03-0.02-0.010.00.010.020.030.04 Dx(m) βx1/ 2βy1/ 2Dx Figure 1: Optical functions of 1996 NLC post-linac colli- mationsystem. was driven primarily by the machine protection require- mentthatasinglebunchtrain(80kJat500GeVperbeam) at nominal emittances ( γǫx,y= (4 ×0.06)mm.mrad) should not be able to damage the collimators. This re- quired a scheme of optically-thin spoilers and thick ab- sorbers in each plane, large betatron functions, and strong optics, which in turn introduced difﬁculties due to nonlin- earitiesandwakeﬁelds. The difﬁculties envisioned in the operation of the colli- mationsystemledtoreconsiderationofthedesignassump- tionsanda newconceptualdesign. 2 DESIGN ASSUMPTIONS The design of the post-linac collimation system is most strongly governed by the expected properties of large ex- cursions which can impact the collimators. Previously it had been assumed that neither energy nor betatron ex- cursions could be trapped actively in the NLC due to its low repetition rate (120 linac pulses per second). A re- examinationofthe SLCoperationalhistory,aswell as that of other accelerators, indicated that failures which could causeafast(inter-pulse)betatronoscillationoftherequ ired magnitude were either rare or could be eliminated by de- sign,whilepulse-to-pulseenergyvariationsoftherequir ed magnitudecannotberuledoutforalinac. The expected charge of the beam halo was originally 1010particles per linac pulse (1% of the beam), based on early SLC experience. Later SLC experience showed that the halo could be reduced substantially through care- ful tuning of the injection (damping ring and compressor) systems. In the present NLC design a collimation system downstream of the damping ring and ﬁrst bunch compres-sor is expected to dramatically reduce the halo intensity at the end of the main linac. The present estimate of the halo is107particles per pulse; we have chosen to design for a safety factor of 100 over this estimate. This reduction eliminatestherequirementsforwatercoolinginthespoile r elementsandeasesthetolerancesonmuongeneration. 3 NEW COLLIMATIONSYSTEM OPTICS Figure 2 shows the optical functions of the new post-linac collimationlattice. Theenergyandbetatroncollimatorsa re separated,with theformerprecedingthelatter. 0.0 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. 1100. 1200. s (m) δE/ p0c = 0. Table name = TWISSEnergy and Betatron Collimation Plus All Matching Regions Windows NT 4.0 version 8.23/acc 28/03/00 13.24.43 0.020.40.60.80.100.120.140.160.β1/2(m1/2) 0.00.0250.0500.0750.1000.1250.1500.1750.2000.2250.250 Dx(m) βx1/ 2βy1/ 2Dx Figure 2: Optical functions of proposed new NLC post- linac collimationsystem. 3.1 EnergyCollimation Theenergycollimationsectionachievespassiveprotectio n against off-energy pulses through a 0.5 radiation length (R.L.) spoileranda 20 R.L. absorberseparatedbyapprox- imately 30 meters. The ﬁrst few R.L. of the absorber are titanium, for which the RMS beam size σr≡√σxσy must be larger than 560 µm to ensure survival [2]. Beams which pass through the spoiler will develop RMS scat- tering angles of 19 µradians in horizontal and vertical; combined with the dispersive beam size at the absorber (ησδ= 500 µm), the expected size of a beam at the ab- sorberwhichﬁrst passesthroughthespoileris660 µm. Survival of the 0.5 R.L. spoiler is also a consideration. At the spoiler location in the energy collimation region, σr= 89µm. For the NLC bunch train at 500 GeV per beam, the minimum beam size for survival of a 0.5 R.L. beryllium spoiler is approximately 50 µm, thus we have chosenberylliumasthematerialforthespoilers[3]. Thecollimationdepthinenergyshouldbenarrowerthan the bandpassoverwhich beamsare well-behavedin the ﬁ- nal focus. The present system is designed to remove ±1% off-energy particles, which requires a half-gap of 1.3 mm forthespoilersand2.0mmfortheabsorbers.The jitter ampliﬁcation effect of collimator wakeﬁelds must be minimized at all points in the collimation system. In the energy collimation region, the ratio ηx/βxis large andthusthecollimatorwakeﬁeldsprimarilycoupleenergy jitter into horizontal position jitter. This aberration is can- celled by placing a second spoiler-absorber pair at a loca- tion which is −Iin betatron optics from the ﬁrst pair but with equal dispersion functions. The cancellation is only exactforon-energyparticles,buttheexpectedenergyjitt er of 0.22% only causes a horizontal jitter of 0.5% of σx. A similareffectiscausedbyhigh-orderdispersion,butthee f- fect is approximately1/3 aslarge asthe residualwakeﬁeld jitter contribution. 3.2 BetatronCollimation Because large betatron oscillations are not expected to de- velop during one inter-pulse period, it is expected that the betatron collimators will rarely be hit by the beam core. The baseline design for the betatron collimation system, which is the system pictured in Figure 2, utilizes “con- sumable” spoilers, in which the spoilers can be moved to present a fresh surface to the beam after every incident of beam-core interception; we assume that 1,000 such inci- dents can occur per year of operation. An alternative de- sign would permit damage on every pulse and require that the collimators be self-repairing, “renewable” collimato rs. While more techincally challenging, the renewable colli- matorswouldpermitsmalleraperturestobeused,whichin turnwouldpermitsmallerbetatronfunctions. The system in Figure 2 is based on a triplet lattice with phase advancesof π/2and3π/2percell in horizontaland vertical, respectively. Thus the system collimates in two phases, two planes, two iterations per phase/plane. Each high-beta region in the system contains 2 adjustable spoil- ers (xandy) and 2 ﬁxed cylindrical absorbers. Multiple coulomb scattering in the spoilers gives the halo a large angular divergence, which causes particles to hit the ab- sorbersinthenextcell. The required collimation aperture is set by acceptable limits on synchrotronradiation in the ﬁnal doublet. Based on studies of the 1996 ﬁnal focus [4], the nominal spoiler half-gaps are approximately 200 µm for 500 GeV beams. The ﬁxed absorbers have a round aperture with a radius of 1 mm. Spoilers and absorbers are 0.5 and 20.0 R.L., respectively. The vertical jitter ampliﬁcation factor for the betatron collimation system is 46%, smaller than the 66% expected forthe1996design. Fortheexpectedincomingjiter(0.375 σy), the collimators contribute 0.17 σyjitter in quadrature withtheincomingjitter. Theseestimatesarebasedonana- lytic modelsfor collimatorswith a ztaper and a large x/y aspect ratio [5]; however, recent experiments indicate tha t the actual wakeﬁeld effect may be smaller than this [6]. The horizontal jitter ampliﬁcation is expected to be about halfthat ofthe vertical.4 SCATTERING STUDIES The efﬁciencyof primary-particlecollimationandthe pro- ductionofmuonswhicharetransmittedtotheIPwerestud- ied using a combination of TURTLE, EGS, and MUCUS (MUltipleCoUlombScatteringprogram). 4.1 PrimaryParticles Figure 3 shows the halo attenuation based on tracking of 2 million halo particles which originate at a point on one collimator. Figure 3 (a) shows the attenuationfor particle s 240µm from the beam axis at each of the ﬁrst 4 spoil- ers (2 vertical, 2 horizontal); the attenuation is shown for cases in which off-energyprimary particles are collimated downstream of the collimation system (eliminating parti- cles which are more than 2% off-energy), and cases with- outdownstreamenergyattenuation. Theattenuationistyp- icallybetween 0.6×10−5and8×10−5,whilethedesired value is 0.1×10−5. Figure 3 (b) shows the attenuationas a functionofsourceoffsetfortheﬁrst spoiler. 0 1 2 3 4 510−710−610−510−410−3 Spoiler NumberAttenuation(a) 200 250 300 350 400 450 5000246x 10−5 Source Offset, micrometersAttenuation(b) Figure 3: Collimation efﬁciency of betatron collimation system. (a): attenuation for ﬁrst y, ﬁrstx, second y, sec- ondxspoiler, respectively,both with (circles) and without (crosses) ﬁnal energycollimation; solid line shows the de- siredattenuation. (b): Attenuationasafunctionofprimar y particleoffsetfrombeamaxis,ﬁrst yspoiler. Note that these estimates are preliminary, and recent studies have indicated that a substantial improvementmay be achieved by optimizing the zpositions of the spoilers. Also,increasingthespoilerthicknessto1.0R.L.wouldim- proveattenuationby an order of magnitude,but the result- ingenergydepositioninthespoilerswouldhavetobestud- ied. 4.2 MuonSecondaries The problemof muon secondariesfrom the post-linac col- limatorsenteringthedetectorismoreseverethanitwasfor the 1996 NLC design, primarily because the collimation systemandﬁnalfocusareshorterinthepresentdesign(2.5km per side compared to 5.2 km per side), which puts all sourcesofmuonsclosertotheIP.Inthepresentdesign,we includetwo largemagnetizedtoroidsformuonattenuation on each side of the IP; despite this, we expecton the order ofseveralhundredmuonsperlinacpulsetoenterthemuon endcap of a detector similar to the “LCD Large” design [7], as well as tens of muons per linac pulse in the elec- tromagneticcalorimeter. Thesestudieswereperformedfor 500 GeV beams; for 250 GeV beams, 2 orders of magni- tude improvement are expected. The muon rate can also bereducedbyaddingadditionalspoilers,reducingthehalo intensity,orconstructinga smallerdetector(suchas“LCD Small”). Since the 500 GeV center-of-mass (CM) results are quite acceptable, a reasonable approach to the muon situation mightbe to buildthe system with 2 muontoroids and spaces allocated for additional toroids, to be added in later yearsifrequired. 5 CONCLUSIONSAND FUTURE DIRECTIONS We have presented an optics design for the Next Linear Colliderpost-linaccollimationsystemwhichaddressesth e difﬁcultiesin the previoussystem design. Thenew system hasweakeroptics,loosertolerances,largerbandwidth,an d better wakeﬁeld propertiesthan the original. The new sys- tem is somewhat poorer than desired in the areas of halo attenuation and muonproduction;futurework will seek to addressthisweakness.. The present energy collimator includes a 5 milliradian arc,whichchangestheanglebetweenthelinacandtheﬁnal focus. Recent developmentsin ﬁnal focus design have ex- pandedthe potentialenergyreachof the NLC [8]; in order to take full advantage of this change, we plan to redesign thecollimationsystemtoeitheradoglegorachicane,such thatthepost-linacsystemandthelinacareco-linearandth e former can be expanded by “pushing back” into the latter whenthelinacgradientandenergyareincreased. 6 ACKNOWLEDGEMENTS This work would not have been possible without the ideas and assistance of G. Bowden, J. Frisch, J. Irwin, T. Markiewicz,N. PhinneyandF. Zimmermann. 7 REFERENCES [1] NLCDesignGroup, Zeroth-OrderDesignReportfortheNext Linear Collider, 555-641 (1996). [2] NLCDesignGroup, Zeroth-OrderDesignReportfortheNext Linear Collider, 573-574 (1996). [3] W.RalphNelson, private communication. [4] S.Hertzbach, private communication. [5] G.Stupakov, SLAC-PUB-7167(1996). [6] P.Tenenbaum etal, these proceedings. [7] http://hepwww.physics.yale.edu/lc/graphics.html. [8] P.Raimondi and A.Seryi,these proceedings.
arXiv:physics/0010079v1 [physics.flu-dyn] 30 Oct 2000Description of Vorticity by Grassmann Variables and an Extension to Supersymmetry R. Jackiw Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA 02139-4307 Typeset in L ATEX by M. Stock MIT-CTP#3036 Abstract Hagen Kleinert’s early interest in particle physics quantu m ﬁeld theory served him well for his subsequent researches on statistical physics and colle ctive phenomena. Therefore, on the occasion of a signiﬁcant birthday, I oﬀer him this essay, in w hich particle physics concepts are blended into a ﬁeld theory for macroscopic phenomena: Fl uid mechanics is enhanced by anticommuting Grassmann variables to describe vorticity, while an additional interaction for the Grassmann variables leads to supersymmetric ﬂuid mecha nics. 1 Pr´ ecis of Fluid Mechanics (With No Vorticity) Let me begin with a pr´ ecis of ﬂuid mechanical equations [1]. An isentropic ﬂuid is described by a matter density ﬁeld ρand a velocity ﬁeld v, which satisfy a continuity equation involving the current j=ρv: ˙ρ+∇·(ρv) = 0 (1) and a force equation involving the pressure P: ˙v+v·∇v=−1 ρ∇P . (2) (Over-dot denotes diﬀerentiation with respect to time.) Fo r isentropic ﬁelds, the pressure P is a function only of the density, and the right side of (2) may also be written as −∇V′(ρ), whereV′(ρ) is the enthalpy, P(ρ) =ρV′(ρ)−V(ρ), and/radicalbig ρV′′(ρ) =/radicalbig P′(ρ) is the sound speed (prime denotes diﬀerentiation with respect to argume nt).2 R. Jackiw Equations (1) and (2) can be obtained by bracketing the dynam ical variables ρandvwith the Hamiltonian H(ρ,v) H(ρ,v) =/integraldisplay dr/parenleftbig1 2ρv2+V(ρ)/parenrightbig (3) ˙ρ={H,ρ} (4a) ˙v={H,v} (4b) provided the nonvanishing brackets of the fundamental vari ables (ρ,v) are taken to be [2] {vi(r),ρ(r′)}=∂iδ(r−r′) (5a) {vi(r),vj(r′)}=−ωij(r) ρ(r)δ(r−r′). (5b) (The ﬁelds in the brackets are at equal times, hence the time a rgument is suppressed.) Here ωijis the vorticity, deﬁned as the curl of vi: ωij=∂ivj−∂jvi. (6) One naturally asks whether there is a canonical 1-form that l eads to the symplectic struc- ture (5); that is, one seeks a Lagrangian whose canonical var iables can be used to derive (5) from canonical brackets. When the velocity is irrotational , the vorticity vanishes, vcan be written as the gradient of a velocity potential θ,v=∇θ, and (5) is satisﬁed by postulating that {θ(r),ρ(r′)}=δ(r−r′) (7) that is, the velocity potential is conjugate to the density, so that the Lagrangian can be taken as L/vextendsingle/vextendsingle irrotational=/integraldisplay dr θ˙ρ−H/vextendsingle/vextendsingle v=∇θ(8) whereHis given by (3) with v=∇θ. 2 Extending the Formalism to Include Vorticity The traditional method of including vorticity in a Lagrangi an formalism [3] involves writing the velocity in a more elaborate potential representation, the so-called Clebsch parameteriza- tion [4], v=∇θ+α∇β (9)Description of Vorticity by Grassmann Variables and an Exte nsion to Supersymmetry 3 which supports nonvanishing vorticity ωij=∂iα∂jβ−∂jα∂iβ . (10) The Lagrangian L=−/integraldisplay drρ(˙θ+α˙β)−H/vextendsingle/vextendsingle v=∇θ+α∇β(11) identiﬁes canonical pairs to be {θ,ρ}(as in the irrotational case) and also {β,αρ}. It then follows that the algebra (5) is satisﬁed, provided vis given by (9). The quantities ( α,β) are called the “Gauss potentials”. The situation here is similar to the electromagnetic force l aw: The Lorentz equation can be presented in terms of the electric and magnetic ﬁeld stren gths, but a Lagrangian for the motion requires describing the ﬁelds in terms of potentials . 3 Some Further Observations on the Clebsch Decomposition of the Vector Field v In three dimensions, (9) involves the same number of functio ns on the left and right sides of the equality: three. The total number of dynamical variable s (ρ,v) is even – four – so an appropriate phase space can be constructed from the four pot entials (ρ,θ,α,β ). Nevertheless the Gauss potentials are not uniquely determined by v. The following is the reason why a canonical formulation of (5) requires using the Clebsch de composition (9). Although the algebra (5) is consistent in that the Jacobi identity is sati sﬁed, it is degenerate in that the kinematic helicity h h≡1 2/integraldisplay d3rv·(∇×v) =1 2/integraldisplay d3rv·ω (12) (ωi=1 2εijkωjk) has vanishing bracket with ρandv. (Note that his just the Abelian Chern- Simons term of v[5].) Consequently, a canonical formulation requires elim inating the kernel of the algebra, that is, neutralizing h. This is achieved by the Clebsch decomposition: v= ∇θ+α∇β,ω=∇α×∇β,v·ω=∇θ·(∇α×∇β) =∇·(θ∇α×∇β). Thus in the Clebsch parameterization the helicity is given by a surface integralh=1 2/integraltext dS·θ(∇α×∇β) – it possesses no bulk contribution, and the obstruction to a canonical realization of (5) is removed [6]. In two spatial dimensions, the Clebsch parameterization is redundant, involving three func- tions to express the two velocity components. Moreover, the kernel of (5) in two dimensions comprises an inﬁnite number of quantities kn=/integraldisplay d2rρ/parenleftBigω ρ/parenrightBign (13)4 R. Jackiw for which the Clebsch parameterization oﬀers no simpliﬁcat ion. (Hereωis the two-dimensional vorticityωij=εijω.) Nevertheless, a canonical formulation in two dimensions also uses Clebsch variables to obtain an even-dimensional phase space. 4 Kinematical Grassmann Variables for Vorticity Rather than using the Gauss potentials ( α,β) of the Clebsch parameterization (9) in the de- scription of vorticity (10), we propose an alternative that makes use of Grassmann variables [7]. We write v=∇θ−1 2ψa∇ψa (14) whereψais a multicomponent, real Grassmann spinor ψ∗ a=ψa, (ψaψb)∗=ψ∗ aψ∗ b. (The number of components depends on spatial dimensionality.) Evident ly the nonvanishing vorticity is ωij=−∂iψa∂jψa. (15) Moreover, the canonical 1-form in the Lagrangian that repla ces (11) reads L=−/integraldisplay drρ(˙θ−1 2ψa˙ψa)−H/vextendsingle/vextendsingle v=∇θ−1 2ψ∇ψ(16) The Hamiltonian retains its (bosonic) form (3), but the Gras smann variables are hidden in the formula for the velocity. From the canonical 1-form, we dedu ce that (θ,ρ) remain a conjugate pair [see (7)] and that the canonically independent Grassma nn variables are√ρψ. Thus we postulate, in addition to the Poisson bracket (7) satisﬁed b y (θ,ρ), a Poisson antibracket for the Grassmann variables {ψa(r),ψb(r′)}=−δab ρ(r)δ(r−r′) (17) and this, together with (7), has the further consequence tha t the following brackets hold: {θ(r),ψ(r′)}=−1 2ρ(r)δ(r−r′) (18) {v(r),ψ(r′)}=−∇ψ(r) ρ(r)δ(r−r′). (19) The algebra (5) follows. One may state that it is natural to describe vorticity by Gras smann variables: vortex motion is associated with spin, and the Grassmann descripti on of spin within classical physics is well known. In the model as developed thus far the Grassman n variables have no role beyond the kinematical one of parameterizing vorticity (15 ) and providing the correct bracket structure. They do not contribute to the equations of motion forρandv, (1) and (2) [even though they are hidden in the formula (14) for v]. Moreover, they satisfy a free equation: from (16) it follows that ˙ψ+v·∇ψ= 0. (20)Description of Vorticity by Grassmann Variables and an Exte nsion to Supersymmetry 5 5 Dynamical Grassmann Variables for Supersymmetry Thus far the Grassmann variables’ only role has been to param eterize the velocity/vorticity (14), (15) and to provide canonical variables for the symple ctic structure (5). The equations for the ﬂuid (1), (2) are not polluted by them and they do not ap pear in the Hamiltonian, beyond their hidden contribution to v. Thus the equation for the Grassmann ﬁelds is free (20). But now we enquire whether we can add a Grassmann term to the Ha miltonian so that the Grassmann variables enter the dynamics and the entire mo del enjoys supersymmetry. We have succeeded for a speciﬁc form of the potential V(ρ): V(ρ) =λ ρ(21) and for the speciﬁc dimensionalities of space-time: (2+1) a nd (1+1). The reason for these speciﬁcities will be explained in the next Section. The potential (21), with λ>0, leads to negative pressure P(ρ) =ρV′(ρ)−V(ρ) =−2λ/ρ (22) and sound speed s=/radicalbig P′(ρ) =√ 2λ/ρ (23) (henceλ>0). This model is called the “Chaplygin gas”. Chaplygin introduced his equation of state as a mathematica l approximation to the physi- cally relevant adiabatic expressions V(ρ)∝ρnwithn>0 [8]. (Constants are arranged so that the Chaplygin formula is tangent at one point to the adiabati c proﬁle.) Also it was realized that certain deformable solids can be described by the Chapl ygin equation of state [9]. These days negative pressure is recognized as a possible physical eﬀect: exchange forces in atoms give rise to negative pressure; stripe states in the quantum Hall eﬀect may be a consequence of negative pressure; the recently discovered cosmologica l constant may be exerting negative pressure on the cosmos, thereby accelerating expansion. 5.1 Planar model In (2+1) dimensions the Grassmann variables possess 2-comp onents and two real 2 ×2 Dirac “α”-matrices act on them: α1=σ1,α2=σ3. The supersymmetric Hamiltonian is H=/integraldisplay d2r/braceleftBig 1 2ρv2+λ ρ+√ 2λ 2ψα·∇ψ/bracerightBig (24) where it is understood that v=∇θ−1 2ψ∇ψ[7, 10]. While the continuity equation retains its form (1), the force equation acquires a contribution from th e Grassmann variables ˙v+v·∇v=∇λ ρ2+√ 2λ ρ(∇ψ)α·∇ψ (25)6 R. Jackiw andψis no longer free: ˙ψ+v·∇ψ=√ 2λ ρα·∇ψ . (26) These equations of motion, together with (1), ensure that th e following supercharges are time independent: Q=/integraldisplay d2r/braceleftbig ρv·(αψ) +√ 2λψ/bracerightbig (27a) ˜Q=/integraldisplay d2rρψ . (27b) They generate the following transformations: δρ=−∇·/bracketleftbig ρ(ηαψ)/bracketrightbig˜δρ= 0 (28a) δψ=−(ηαψ)·∇ψ−v·αη−√ 2λ ρη ˜δψ=−η (28b) δv=−(ηαψ)·∇v+√ 2λ ρ(η∇ψ) ˜δv= 0 (28c) whereηis a two-component constant Grassmann spinor. The antibrac kets of the supercharges produce other conserved quantities: {Qa,Qb}=−2δabH (29a) {˜Qa,˜Qb}=−δabN (29b) {˜Qa,Qb}=αab·P+√ 2λδabΩ. (29c) HereNis the conserved number/integraltext d2rρ,Pis the conserved momentum/integraltext d2rρv, and Ω is a center given by the volume of space/integraltext d2r. 5.2 Lineal model In (1+1) dimensions the Chaplygin gas equation can be writte n in compact form in terms of the Riemann coordinates R±=v±√ 2λ/ρ . (30) Both eqs. (1) and (2) are equivalent to ˙R±=−R∓∂ ∂xR±. (31) It is known that this system is completely integrable [11]. O ne hint for this is the existence of an inﬁnite number of constants of motion: In ±=/integraldisplay dxρ(R±)n(32)Description of Vorticity by Grassmann Variables and an Exte nsion to Supersymmetry 7 are time-independent by virtue of (31). The supersymmetric Hamiltonian makes use of a real, 1-compo nent Grassmann ﬁeld ψ[12]: H=/integraldisplay dx/parenleftBig 1 2ρv2+λ ρ+√ 2λ 2ψ∂ ∂xψ/parenrightBig . (33) The velocity is given by v=∂ ∂xθ−1 2ψ∂ ∂xψand the equations of motion for the bosonic variables retain the same form as the absence of ψ, that is, (1), (2) continue to hold. The Grassmann ﬁeld satisﬁes ˙ψ+R−∂ ∂xψ= 0 (34) and a general solution follows immediately with the help of ( 31):ψis an arbitrary function ofR+. ψ= Ψ(R+) (35) Thus the system remains completely integrable. The supersymmetry charges and transformation laws are obvi ous dimensional reductions of (27)–(28): Q=/integraldisplay dxρR +ψ (36a) ˜Q=/integraldisplay dxρψ (36b) δρ=−η∂ ∂x(ρψ) ˜δρ= 0 (37a) δψ=−ηψψ′−ηR+˜δψ=−η (37b) δv=−η(ψv)′+ηR+ψ′˜δv= 0. (37c) The algebra of these is {Q,Q}=−2H (38a) {˜Q,˜Q}=−N (38b) {˜Q,Q}=P+√ 2λΩ. (38c) In view of (35), we see that evaluating the supercharges Qand˜Qon the solution gives expressions of the same form as the bosonic conserved charge s (33). Indeed, we recognize that two charges in (36) are the ﬁrst two in an inﬁnite tower of con- served supercharges, which generalizes the inﬁnite number of bosonic conserved quantities (32): Qn=/integraldisplay dxρRn +ψ . (39)8 R. Jackiw 6 The Origins of Our Models We have succeeded in supersymmetrizing a speciﬁc model – the Chaplygin gas – in speciﬁc dimensionalities – the 2-dimensional plane and the 1-dimen sional line – leading to nonrelativis- tic, supersymmetric ﬂuid mechanics in (2+1)- and (1+1)-dim ensional space-time. The reason for these speciﬁcities is that both models descend from Namb u-Goto models for extended sys- tems in a target space of one dimension higher than the world v olume of the extended object. Speciﬁcally, a membrane in three spatial dimensions and a st ring in two spatial dimensions, when gauge-ﬁxed in a light-cone gauge, can be shown to devolv e to a bosonic Chaplygin gas in two and one spatial dimensions, respectively [13]. The ﬂu id velocity potential arises from the single dynamical variable in the gauge-ﬁxed Nambu-Goto theory, namely, the transverse direction variable for the membrane in space and the string o n a plane. Although purely bosonic Chaplygin gas models in other dimensions can devolv e from appropriate Nambu-Goto models for extended objects, for the supersymmetric case we need a superextended object, and these exist only in speciﬁc dimensionalities. In our case it is the light-cone parameterized su- permembrane in (3+1)-dimensional space-time [14] and the s uperstring in (2+1)-dimensional space-time [15] that give rise to our planar and lineal super symmetric ﬂuid models. One naturally wonders whether an arbitrary bosonic potenti alV(ρ) has a supersymmetric partner in arbitrary dimensions, and this problem is under f urther investigation. One promising approach is to consider parameterizations of extended obje cts other than the light-cone one. It is known that in the purely bosonic case, other parameteri zations of the Nambu-Goto actions lead to other ﬂuid mechanical models, and this shoul d carry over to a supersymmetric generalization. Incidentally, the existence of Nambu-Goto antecedents of t he ﬂuid models that we have discussed allows one to understand some of their remarkable properties: complete integrability in the lineal case; existence of further symmetries (which w e have not discussed here) and relation to other models (which devolve from the same extend ed system, but are parameterized diﬀerently from the light-cone method) [16]. References [1] See, for example, L. Landau and E. Lifshitz, Fluid Mechanics (2nd ed., Pergamon, Oxford UK 1987). [2] This algebra implies the familiar translation algebra f or the momentum density P=ρv (= current density). In the ﬂuid mechanical context, the bra ckets (5) were posited by P.J. Morrison and J.M. Greene, Phys. Rev. Lett. 45, 790 (1980), (E) 48, 569 (1982). [3] C.C. Lin, International School of Physics E. Fermi (XXI) , G. Careri, ed. (Academic Press, New York NY 1963).Description of Vorticity by Grassmann Variables and an Exte nsion to Supersymmetry 9 [4] A. Clebsch, J. Reine Angew. Math. 56, 1 (1859); H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge UK 1932), p. 248. [5] For a discussion of Abelian and non-Abelian Chern-Simon s terms, see S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (NY)140, 372 (1982), (E) 185, 406 (1985). In ﬂuid mechanics and magnetohydrodynamics the Abelian Chern-Sim ons term is known as the ﬂuid or magnetic helicity, and was introduced by L. Woltier, Proc. Nat. Acad. Sci. 44, 489 (1958). [6] Some further peculiarities of the Clebsch parameteriza tion and the Chern-Simons term are discussed by S. Deser, R. Jackiw, and A.P. Polychronakos , physics/0006056. A non- Abelian generalization of the Clebsch parameterization is in R. Jackiw, V.P. Nair, and S.-Y. Pi, Phys. Rev. D 62, 085018 (2000). [7] R. Jackiw and A.P. Polychronakos, Phys. Rev. D 62, 085019 (2000). [8] S. Chaplygin, Sci. Mem. Moscow Univ. Math. Phys. 21, 1 (1904). [Chaplygin was a col- league of fellow USSR Academician N. Luzin. Although accuse d by Stalinist authorities of succumbing excessively to foreign inﬂuences, unaccountab ly both managed to escape the fatal consequences of their alleged actions; see N. Krement sov,Stalinist Science (Prince- ton University Press, Princeton NJ 1997).] The same model (2 1) was later put forward by H.-S. Tsien, J. Aeron. Sci. 6, 399 (1939) and T. von Karman, J. Aeron. Sci. 8, 337 (1941). [9] K. Stanyukovich, Unsteady Motion of Continuous Media (Pergamon, Oxford UK 1960), p. 128. [10] Some of these results are described in unpublished pape rs by J. Hoppe, Karlsruhe preprint KA-THEP-6-93, Karlsruhe preprint KA-THEP-9-93, hep-th/9 311059. [11] Landau and Lifshitz, Ref. [1]; Y. Nutku, J. Math. Phys. 28, 2579 (1987); P. Olver and Y. Nutku, J. Math. Phys. 29, 1610 (1988); M. Arik, F. Neyzi, Y. Nutku, P. Olver, and J. Verosky, J. Math. Phys. 30, 1338 (1989); J. Brunelli and A. Das, Phys. Lett. A235 , 597 (1997). [12] Y. Bergner ( in preparation ). [13] J. Goldstone (unpublished); M. Bordemann and J. Hoppe, Phys. Lett. B317 , 315 (1993), B329 , 10 (1994); R. Jackiw and A.P. Polychronakos, Proc. Steklov Inst. Math. 226, 193 (1999); Comm. Math. Phys. 207, 107 (1999). [14] B. de Wit, J. Hoppe, and H. Nicolai, Nucl. Phys. B305 , [FS23] 525 (1988). [15] J. Gauntlett, Phys. Lett. B228 , 188 (1989). [16] For more details on these and related topics, see R. Jack iw, physics/0010042.
arXiv:physics/0010080v1 [physics.gen-ph] 31 Oct 2000Comment on the Electron Self-Energy Calculation Yong-Gwan Yi∗ February 21, 2014 Abstract A quantum-electrodynamical description of the self-energ y in which the interaction is propagated with the velocity of light instea d of instantaneously gives rise to a modiﬁcation of the photon propagator identic al with the conver- gence factor. The self-energy is discussed in relation to th e static polarization induced in the vacuum due to the presence of an electron itsel f. The self-energy of an electron is an old problem of electrody namics. The quan- tum theory has put the problem in a critical state. In 1939, We isskopf calculated the self-energy of an electron according to the positron the ory of Dirac, in which the self-energy diverges only logarithmically [1]. In 1949 , Feynman put forward an intuitive and intelligible method of calculation in atta cking this problem [2]. The fact that the integral is not ﬁnite is unavoidable, but th e convergence factor that is adopted for computational purposes deﬁnes the mathe matics [3]. From the discussion of “action-at-a-distance” in classical electr odynamics, meanwhile, I have attempted to evaluate the self-energy in which the interact ion is propagated with the velocity of light instead of instantaneously. The neces sity of such an evaluation occurred to me on the assumption that action of a source shoul d be explicable solely on the ﬁnite propagation velocity. I shall comment on the phy sical signiﬁcance of the convergence factor. Looking at the Feynman diagram for the self-energy, one sees at once that it describes steady-state interaction phenomena. The diagra m represents both the instantaneous interaction of the electron with the Coulomb ﬁeld created by the electron itself and the self-interaction due to the emissio n and reabsorption of a virtual transverse photon. We know, however, in classical e lectrodynamics that the Coulomb potential does not act instantaneously, but is dela yed by the time of prop- agation. We also know that the velocity of an electron cannot exceed the velocity of light, and hence any electron cannot reabsorb a photon it e mitted at a retarded time. From these points of view the Feynman graph illustrati ng the self-energy is unnatural. We are thus led to suspect the physical descripti on of the Feynman di- agram. A physical description of time-dependent interacti on necessitates a further eﬀect. What further eﬀect can be assumed in the Feynman diagram with out violating the established thought? Perhaps the vacuum polarization i s an illustration. To be ∗Geo-Sung Furume Apt. 101-401, Gebong-1-dong Guro-ku, Seou l, Korea 1consistent with time-dependent interaction, it is necessa ry to regard an electron- positron pair as existing part of the time in the form of a virt ual photon. If the photon virtually disintegrates into an electron-pos itron pair for a certain fraction of the time, the electron loop gives an additional e2correction to the photon propagator through which an electron interacts with itself . The modiﬁcation of the photon propagator is then the replacement 1 q2−→1 q2/bracketleftbig −Πµν(q)/bracketrightbig1 q2+1 q2/bracketleftbig −Πµν(q)/bracketrightbig1 q2/bracketleftbig −Πµν(q)/bracketrightbig1 q2+···, (1) where Π µν(q) is the vacuum polarization tensor and the iǫprescription is implicit. The polarization tensor Π µν(q) is written as the sum of a constant term Π(0) for q= 0 and terms proportional to ( qµqν−δµνq2). The leading term Π(0) is quadrat- ically divergent constant. In the limit q2→0, the q2term is absorbed into the renormalization constant. Keeping only the leading term, w e see that (1) can be written as 1 q2−→1 q2+ Π(0)−1 q2,or1 q2−1 q2−Π(0)by letting q2→q2−Π(0),(2) where we have used the operator relation ( A−B)−1=A−1+A−1BA−1+···. Let us recall that the convergence factor for a light quantum introduced by Feynman is 1 q2−→1 q2−Λ2 q2−Λ2,that is ,1 q2−1 q2−Λ2, (3) where Λ2is the cut-oﬀ constant for q2and the iǫprescription is implicit. It becomes evident that the additional electron loop gives rise to a mod iﬁcation of the photon propagator identical with the convergence factor consider ed in connection with the divergent self-energy integral. This fact throws light on t he physical implication of the convergence factor. Physically the modiﬁcation (3) i s now looked upon as the result of superposition of the eﬀects of vacuum polariza tion rather than that of quanta of various masses. From a minus sign associated with t he closed loop, we can then understand why we must associate the minus sign with the convergence factor which has not been explained so far from the latter point of vi ew. By interpreting Λ2as Π(0), we can clarify the mathematical content of Feynman’ s approach and emphasize the physical meaning of Weisskopf’s result that t he self-energy is due to the static polarization induced in the vacuum due to the pres ence of an electron itself rather than to the reaction back on the electron of its own radiation ﬁelds. There have been many arguments which say that the quadratica lly divergent constant Π(0) must be discarded [4]. This is because it modiﬁ es the photon propa- gator into a propagator for a neutral-vector meson of a mass/radicalbig Π(0). Even though any “honest” calculation gives Π(0) /negationslash= 0, at present, the way we compute the vac- uum polarization is consistent with assigning a null value t o Π(0) which leads to a nonvanishing photon mass. When viewed from the present poin t, however, one sees that the modiﬁcation (3) amounts to the substitution of the v alue Π(0). Whenever the photon propagator is supplied with the convergence fact or, it amounts to taking account of the closed loop contribution to the photon propag ator. The assumption of Π(0) has been removed from the discussion, but the appeara nce of the value is explicit in the practical calculation. 2The polarization tensor Π µνdiverges severely. A method of making Π µνconver- gent without spoiling the gauge invariance has been found by Bethe and by Pauli [5]. The method states that if we subtract from the integrand of Π µνa similar expression with mreplaced by a very large mass ( m2+Λ2)1/2, we get a much more reasonable result. Even though such a procedure has no meani ng in terms of phys- ically realizable particles, it characterizes a possible c alculation of the closed loop path of virtual particles. It should be noted that the closed loop is a virtual process in which an electron-positron pair is physically imaginabl e particles. The energy of radiation ﬁelds of an electron is, as usually understood, far less than 2 mc2, so the virtual electron-positron pair is negative in energy co mpared to the rest mass. The virtual pair is nothing more than a pair-like concept. Th e problem is how to calculate the closed loop path of such a virtual pair. One ﬁnd s that the conver- gence procedure, which suggests taking the diﬀerence of the result for electrons of p2=m2andp2=m2+ Λ2, is to be expected also from the physical standpoint. The convergence procedure shows itself in a deﬁnite way Πµν−→/integraldisplay/parenleftBig/bracketleftbig Πµν/bracketrightbig p2=m2−/bracketleftbig Πµν/bracketrightbig p2=m2+λ2/parenrightBig dλ. (4) The formulation suggests an obvious modiﬁcation Πµν−→/bracketleftbig Πµν/bracketrightbig p2<m2via/bracketleftbig Πµν/bracketrightbig p2=m2−/bracketleftbig Πµν/bracketrightbig p2>m2, (5) which ﬁts in completely with the physical description of suc h a virtual electron- positron pair. The further eﬀect which we have assumed for computing the ele ctron self-energy is the vacuum polarization. In radiative corrections, a “fu ndamental” eﬀect which gives rise to the modiﬁcation of the photon propagator would be the vacuum po- larization that describes corrections to a virtual photon. References [1] V. F. Weisskopf, Phys. Rev. 56, 72 (1939); Physics Today (November 1981) p.69. [2] R. P. Feynman, Phys. Rev. 76, 749; 769 (1949); Quantum Electrodynamics (W. A. Benjamin, 1961) p.128. [3] W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949); J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964) p.147. [4] J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967) p.267. [5] R. P. Feynman’s remark in his 1949 paper [2]. 3
arXiv:physics/0010081v1 [physics.data-an] 31 Oct 2000Capacity of multivariate channels with multiplicative noise: I.Random matrix techniques and large-N expansions for full transfer matrices Anirvan Mayukh Sengupta Partha Pratim Mitra Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07 974 February 2, 2008 Abstract We study memoryless, discrete time, matrix channels with ad ditive white Gaussian noise and input power constraints of the form Yi=/summationtext jHijXj+Zi, whereYi,XjandZi are complex, i= 1..m,j= 1..n, andHis a complex m×nmatrix with some degree of randomness in its entries. The additive Gaussian noise vect or is assumed to have uncor- related entries. Let Hbe a full matrix (non-sparse) with pairwise correlations be tween matrix entries of the form E[HikH∗ jl] =1 nCijDkl, whereC,Dare positive deﬁnite Hermi- tian matrices. Simplicities arise in the limit of large matr ix sizes (the so called large- n limit) which allow us to obtain several exact expressions re lating to the channel capacity. We study the probability distribution of the quantity f(H) = log det(1 + PH†SH).Sis non-negative deﬁnite and hermitian, with TrS=n. Note that the expectation E[f(H)], maximised over S, gives the capacity of the above channel with an input power c onstraint in the caseHis known at the receiver but not at the transmitter. For arbit raryC,Dexact expressions are obtained for the expectation and variance o ff(H) in the large matrix size limit. ForC=D=I, whereIis the identity matrix, expressions are in addition obtaine d for the full moment generating function for arbitrary (ﬁnit e) matrix size in the large signal to noise limit. Finally, we obtain the channel capacity wher e the channel matrix is partly known and partly unknown and of the form αI+βH,α,βbeing known constants and entries ofHi.i.d. Gaussian with variance 1 /n. Channels of the form described above are of interest for wireless transmission with multiple antenn ae and receivers. 11 Introduction Channels with multiplicative noise are in general diﬃcult t o treat and not many analytical results are known for the channel capacity and optimal input distributions. We borrow techniques from random matrix theory [1] and associated sad dle point integration meth- ods in the large matrix size limit to obtain several analytic al results for the memoryless discrete-time matrix channel with additive Gaussian noise . Apart from the intrinsic in- terest in multiplicative noise, these results are relevant to the study of wireless channels with multiple antennae and/or receivers [2, 3, 4]. The channel input-output relationship is deﬁned as Yi=n/summationdisplay j=1HijXj+Zi (1) where all the quantities are in general complex, and i= 1...m,j= 1...n.Ziare Gaussian distributed with zero mean and a unity covariance matrix, E[ZiZ∗ j] =δij. Note that this ﬁxes the units for measuring signal power. For most of the pap er we employ an overall power constraint n/summationdisplay j=1E[\|Xj\|2] =nP (2) except in one case where we are able to employ an amplitude (or peak power) constraint. The entries of the matrix Hijare assumed to be chosen from a zero mean Gaussian distribution with covariance matrix E[HikH∗ jl] =1 nCijDkl (3) HereC,Dare positive deﬁnite Hermitian matrices. Note that althoug h we assume the distribution of Hto be Gaussian, this assumption can be somewhat relaxed with out substantially aﬀecting some of the large nresults. This kind of universality is expected from known results in random matrix theory [1]. However, for simplicity we do not enter into the related arguments. We consider the case where C,Dare arbitrary positive deﬁnite hermitian matrices, as well as the special case where C,Dare identity matrices. In either case, one needs to consider the scale of H. SinceHmultipliesX, we absorb the scale of HintoP. The formulae derived in the paper can be converted into more expl icit ones exhibiting the scale ofH(sayh) and the noise variance σby the simple substitution P→Ph2/σ2. A note about our choice of convention regarding scaling with n: We chose to scale the elements of the matrix Hijto be order 1 /√nand let each signal element Xjbe order 1. In the multi-antenna wireless literature, it is common to do the scaling the other way round. In these papers [2, 3], Xj’s are scaled as 1 /√nbut keeping Hij’s are kept order 1 so that the average totalpower isP. Our choice of convention is motivated by the fact that we want to treat the systems with channel known at receiv er and those with partially unknown channel within the same framework. For reasons that will become clear later, it is convenient for us to keep the scaling of the input space and the output space to be the same, i. e. to keep Yi,XjandZiall to be order 1 and to scale down Hijto be order 1 /√n. 2The advantage of this is that the singular values of Hhappens to be order 1. For the results in the last section, it is convenient that the ﬂuctua ting part of the matrix scales this way, in order to have a meaningful result . The ﬁnal answe r for capacity is obviously the same in either convention. While using our results in the context of multiantenna wireless, we just have to remember that the total power, in ph ysical units, is P, and not nP. In this paper, we discuss two classes of problems. The ﬁrst cl ass consists of cases whereHis known to the receiver but not to the transmitter. Hbeing known to neither corresponds to problems of the second class. The case where His known to both could be solved by a combination of random matrix techniques used i n this paper and the water-ﬁlling solution [2]. As for the ﬁrst class of problems, we need to maximise the mutu al information I(X,(H,Y)) over the probability distribution of Xsubject to the power constraint. Following Telatar’s argument [2], one can show that it is enough to maximise over G aussian distributions of X, withE(X) = 0. Let E(X∗ iXj) =PSij.TrS=nso that the power constraint is satisﬁed.Shas to be chosen so that E(I(X,Y\|H)), i. e. mutual information of X,Yfor givenH, averaged over diﬀerent realisations of H, is maximum. Most of the paper deals with the statistical properties of th e quantity f(H) = log det(1 + PH†SH) =rank(H)/summationdisplay i=1log(1 +Pµi) (4) whereµiare the squares of the singular values of the matrix S1 2H. The conditions for optimisation over Sare as follows: Let E(H(1 +PH†SH)−1H†) = Λ (5) Λ is a nonnegative deﬁnite matrix. Then •Sand Λ are simultaneously diagonalizable. •In the simultaneously diagonalizing basis, let the diagona l elements Sii=siand Λii=λi. Then for all i, such that si>0,λi=λ. •Forisuch thatsi= 0,λi<λ. The derivation of these conditions are provided in Appendix A. 2 Channel known at the receiver: arbitrary ma- trix size, uncorrelated entries We start with the simplest case, in which the matrix entries a re i.i.d. Gaussian, corre- sponding to C=I,D=I. In this case, one obtains S=Ifor the capacity achieving distribution [2]. In this case, the joint probability densi ty of the singular values of His explicitly known to be given by [1] P(µ1,...,µ min(m,n)) =1 Z/productdisplay i<j(µi−µj)2/productdisplay iµ\|m−n\| ie−n/summationtext iµi(6) 3where the normalisation constant can be obtained as a conseq uence of the Selberg integral formula ([1], Pg.354, Eq.17.6.5) Z=min(n,m)/productdisplay j=1Γ(j)Γ(\|m−n\|+j) (7) In the following, we assume (without loss of generality) min(n,m) =n. This form has been utilised before to obtain the expectation off(H) in terms of inte- grals over Laguerre polynomials [2]. However, it is also fai rly straightforward to obtain the full moment generating function (and hence the probability density) off(H), particularly at largeP. Consider the moment generating function F(α) of the random variable f(H), given by F(α) =E[exp(αf(H))] =E[/productdisplay i(1 +Pµi)α] (8) 2.1 Large P limit In the limit of large P, the expectation can be simply computed as an application of the integral formula stated above. Note that the large Plimit is obtained when Pis much larger than the inverse of the typical smallest eigenvalue. For the case m=n, this would require that P >>n , whereas if m/n=β >1, then we require P >> (√β−1)−1. Taking the largePlimit, we obtain F(α)≈(P)αnE[/productdisplay iµα i] (9) E[/productdisplay iµα i] =n/productdisplay j=1Γ(α+\|m−n\|+j) Γ(\|m−n\|+j)(10) In this limit, it follows that E[f(H)]≈nlog(P) +n/summationdisplay j=1ψ(m−n+j)−nlog(n) (11) V[f(H)]≈n/summationdisplay j=1ψ′(\|m−n\|+j) (12) whereψ(j) = Γ′(j)/Γ(j). Settingm/n=βand for large n, we get E[f(H)]≈nlog(βP/e) (13) Forβ>1 and large n, V[f(H)]≈log(m m−n) = log(β β−1) (14) Forβ= 1 and large m(=n), V[f(H)]≈log(m) + 1 +γ (15) 4whereγis the Euler-Mascheroni constant. Laplace transforming the moment generating function, one o btains the probability density of C=f(H). In the large Plimit, the probability density is therefore given by p(C −nlog(P/e)) wherep(x) is given by p(x) =1 2π/integraldisplay∞ −∞dαe−iαn(log(n)−1)−ixαn/productdisplay j=1Γ(iα+\|m−n\|+j) Γ(\|m−n\|+j)(16) An example of p(x) is presented in Fig.1 for m=n= 4. 2.2 Arbitrary P For arbitrary P,F(α) does not simplify as above, but can nevertheless be written in terms of ann×ndeterminant as follows: F(α) =detM(α) detM(0)(17) where the entries of the complex matrix Mare given by ( i,j= 1...n) Mij(α) =/integraldisplay∞ 0dµ(1 +Pµ)αµi+j+\|m−n\|−2e−nµ(18) To obtain this expression for F(α), one has to simply express the quantity/producttext i/negationslash=j(µi−µj) as a Vandermonde determinant and perform the integrals in th e resultant sum. The in- tegral can be expressed in terms of a Whittaker function (rel ated to degenerate Hyper- geometric functions), and can be evaluated rapidly, so that for small values of m,nthis provides a reasonable procedure for numerical evaluation o f the probability distribution off(H). 3 Channel known at the receiver: large matrix size, correlated entries. For the more general case of correlations between matrix ent ries as in Eq.3, the matrix ensemble is no longer invariant under rotations of H, so that the eigenvalue distribution used in the earlier section is no longer valid. However, by us ing saddle point integration [5], it is still possible to compute the expectation and vari ance off(H) in the limit of large matrix sizes. In this section, we simply state the results fo r the expectation and variance, and explore the consequences of the formulae obtained. The s addle point method used to obtain these results was used in an earlier paper to obtain th e singular value density of random matrices [5] and is described in Appendix B . The expectation and variance of f(H) are given in terms of the following equations: E[f(H)] =m/summationdisplay i=1log(w+ξir) +n/summationdisplay j=1log(w+ηjq)−nqr−(m+n)log(w) (19) V[f(H)] =−2log\|1−g(r,q)\| (20) 5where w2=1 P(21) g(r,q) = [1 nn/summationdisplay j=1(ηj w+ηjq)2][1 nm/summationdisplay j=1(ξj w+ξjr)2] (22) In the above equations, ξ,ηdenote the eigenvalues of the matrices ˜C=S1 2CS1 2,D respectively. The numbers r,qare determined by the equations r=1 nn/summationdisplay j=1ηj w+ηjq(23) q=1 nm/summationdisplay j=1ξj w+ξjr(24) These equations are expected to be valid in the limit of large m,nassuming that a suﬃcient number of the eigenvalues ξ,ηremain nonzero. These equations could be used to design optimal multi-antenna systems [6]. 4 Calculating Capacity In this section we provide the step by step procedure for calc ulating capacity using the results from the previous sections. We found that the optima l covariance matrix Sand the matrixCcould be diagonalized together. Let us work in the diagonali zing basis. Deﬁne ˜C as before. This is a diagonal matrix in this basis, with diago nal elements ξi=cisi, where ci,siare the diagonal elements of C,Srespectively. We assume that ci’s are sorted in decreasing order. That is, c1>c2>· · ·>cm. The optimality condition, Eq.5, becomes: cir w+cisir=λ,fori= 1,...,p. (25) pis the number for nonzero si’s. One way to see this is as follows: Take the expression in Eq.19, replace ξbycisiand take its derivative with respect to non-zero si’s. Note that q,r changes aξichanges. However, this expression is evaluated at a point wh ich is stationary with respect to variation in qandr. Hence, to ﬁrst order, changes of q,rdue to changes inξdo not have a contribution. We just change ξkeepingq,rﬁxed. Since ∂ξi/∂si=ci, we got the expression in Eq.25. Eq.25, along with Eq.23 and Eq.24, provide p+ 2 equations for p+ 3 unknowns, namelyr,qandsi,i= 1,..,p. The additional condition comes from total power constrain t/summationtext isi=P. Once we ﬁnd such a solution, we could check whether the condi tionssi>0 andλi=cir/w < λ is satisﬁed for all i > p. If any of them is not satisﬁed, we need to changep, the number of non-zero eigenvalues of S. After getting a consistent set of solutions we use Eq.19 to calculate capacity. Schematically, the algorithm is as follows: 1. Diagonalize Cand arrange eigenvalues in the decreasing order along the di agonal. 62. Start with p=1. 3. Solve equations 25,23,24 along with the power constraint . 4. Check whether si>0 fori= 1,..,p, and,cp+1r/w<λ . 5. If any of the previous conditions are not satisﬁed, go back to step 3 with pincre- mented by 1. Otherwise, proceed to next step. 6. Calculate capacity using Eq.19. 5 Channel known at the receiver: large matrix size, uncorrelated entries The results of the previous section simplify if we assume tha t the matrix entries are uncorrelated with unit variance. In this case, the equation s become E[f(H)] =mlog(w+r) +nlog(w+q)−nqr−(m+n)log(w) (26) V[f(H)] =−2log\|1−1 (w+q)2β (w+r)2\| (27) r=1 w+q(28) q=β w+r(29) First, consider the special case where m=n. In this case, we obtain E[f(H)] =n[log/parenleftBigP e/parenrightBig + log/parenleftBig 1 +1 x/parenrightBig +x P] (30) V[f(H)] = 2log/parenleftBig(1 +x)2 (2x+ 1)/parenrightBig (31) wherex2+x=P(x positive). For large P, the expectation and variance tend to nlog(P/e) and log(P) respectively. Note that the variance grows logarithmical ly with power, but does not depend on the number of channels. Form,nnot equal, one obtains expressions which are analogous by so lving the simul- taneous equations above for qandr(which lead to quadratic equations for either qorr by elimination of the other variable): r(w) =−(w2+m−n) + ∆ 2w(32) q(w) =−(w2−m+n) + ∆ 2w(33) ∆ =/radicalBig (w2+m+n)2−4mn (34) Substituting these formulae in Eq.26 and Eq.27 gives the des ired expressions for the expectation and variance of the capacity f(H). 76 H unknown at both receiver, transmitter: large matrix size, uncorrelated entries The case where His unknown both to the transmitter and receiver is in general hard [4]. For example, analytical formulae for the capacity are not av ailable even in the scalar case. However, in the case that the matrix entries are uncorrelate d, the problem reduces to an eﬀective scalar problem which exhibits simple behaviour at large m. To proceed, one ﬁrst obtains the conditional distribution p(/vectorY\|/vectorX). This can be done by noting that for ﬁxed /vectorX,/vectorYis a linear superposition of zero mean Gaussian variables an d is itself Gaussian with zero mean and variance given by E[YiY∗ j] = (1 +1 n/summationdisplay k\|Xk\|2)δij (35) Note that only the magnitude of the vector /vectorXenters into the equation, and the dis- tribution of /vectorYis isotropic. Eﬀectively, since the transfer matrix is unkn own both at the transmitter and receiver, only magnitude information and n o angular information can be transmitted. Since we are free to choose the input distribut ion ofx=\|/vectorX\|/√n, we can henceforth regard xas a positive scalar variable. As for y=\|/vectorY\|/√m(√mis just to ar- range the right scaling),we still have to keep track of the ph ase space factor y2m−1which comes from transforming to 2 mdimensional polar coordinates. Note that we need 2 m dimensions since /vectorYis a complex vector. Thus, the problem can be treated as if it w ere a scalar channel, keeping track only of the magnitudes yandx, except that the measure for integration over yshould bedµ(y) = Ω 2my2m−1dywhere Ω 2mis from the angular integral. The conditional probability p(y\|x) is given by p(y\|x) =/bracketleftbiggm π(1 +x2)/bracketrightbiggm exp(−my2 2(1 +x2)) (36) The conditional entropy of ygivenxis easy to compute from the original obervation that the conditional distribution is Gaussian, and is given by H(y\|x) =mEx/bracketleftbigg log/parenleftbiggπe m(1 +x2)/parenrightbigg/bracketrightbigg (37) The entropy of the output is H(y) =−Ex/integraldisplay dµ(y)p(y\|x)log(Ex′p(y\|x′)) (38) Thus, the mutual information between input and output is giv en by subtracting the two expressions above and rearranging terms: I=−Ex/integraldisplay dµ(y)p(y\|x)log(Ex′[(1 +x2 1 +x′2)mexp(−my2 (1 +x′2)+m)]) (39) Theyintegral contains the factor y2m−1exp(−my2 (1 +x2)) (40) 8which is sharply peaked around y2= (1 +x2) formlarge. Thus, the yintegral can be evaluated using Laplace’s method to obtain (for mlarge) I≈ −ExlogEx′[(1 +x2 1 +x′2)mexp(−m(1 +x2) (1 +x′2)+m)] (41) Applying Laplace’s method again to perform the integral ins ide the logarithm, as- suming that the distribution over xis given by a continuous function p(x), we ﬁnally obtain I=1 2log(2m π) +/integraldisplay dxp(x)log[x 1 +x21 p(x)] (42) The capacity and optimal input distribution is straightfor wardly obtained by max- imising the above. It is easier to treat the case where a peak p ower constraint is used, namelyx≤√ P. In this case, the optimal input distribution is ( x∈[0,√ P]) p(x) =1 log(1 +P)2x 1 +x2(43) and the channel capacity is C=1 2log(m 2π) + log(log(1 + P)) (44) Notice that the capacity still grows with m, which is somewhat surprising, but this growth is only logarithmic. Secondly, the dependence on the peak power is through a double logarithm. With an average power constraint/integraltextx2dxp(x) =Pthe optimal input distribution is given by p(x) =a2x 1 +x2e−x2 a(1+P) (45) whereais a constraint determined by the normalisation condition, which yields the equa- tion a=/integraldisplay∞ 0dy 1 +ye−y a(1+P) (46) The capacity is given by C=1 2log(m 2π) + log(a) +P 1 +P1 a(47) For largeP,a≈log(1 +P), thus recovering the double logarithm behaviour. 7 Information loss due to multiplicative noise We could generalize the calculation in the previous section to a problem which interpolates smoothly between usual additive noise channel and the case c onsidered above. This is a 9problem with same number of transmitters and receivers ( m=n) and is deﬁned by Yi=n/summationdisplay j=1(αδij+βHij)Xj+Zi (48) β= 0 is the usual channel with additive gaussian noise. α= 0 corresponds the problem we have just discussed. In the ﬁrst case, capacity increases logarithmically with input power, whereas in the second case it has a much slower (double logarithmic) dependence on input power. Apart from the theoretical interest in study ing the crossover between these two kinds of behavior, this problem has much practical importance [7]. The easy thing to calculate is c= lim n→∞C/n. Notice that this quantity is zero in the limitα→0, capacity being logarithmic in nin that limit. For simplicity, we choose the input power constraint/summationtext i\|Xi\|2≤nP. We relegate the details of the saddle point calculation to Appendix C. The result is c= log/bracketleftBigg 1 +α2P 1 +β2P/bracketrightBigg (49) The result tells us that, in the large Nlimit, the eﬀect of multiplicative noise is similar to that if an additive noise whose strength increases with th e input power. It is of particular interest to note that there exists a lower bound to the channel capacity, which is given by the capacity of a ﬁctitious addit ive gaussian channel with the same covariance matrix for ( /vectorX,/vectorY) as the channel in question. Remarkably, this bound coincides with the saddle point answer. 8 Appendix A The condition of optimality with respect to Sis E[Tr{(1 +PH†SH)−1H†δSH}] =Tr(ΛδS)≤0 (50) for all allowed small δS.δShas to satisfy two conditions: that S+δSis non-negative deﬁnite and that Tr(δS) = 0. The matrix Λ has been deﬁned in the ﬁrst section. It is a non-negative deﬁnite hermitian matrix. IfShas only positive eigenvalues then adding a small enough her mitianδSto it does not make any of the eigenvalues zero or negative. Then on ly way the optimisation condition can be satisﬁed is by choosing Λ to be proportional to the Identity matrix. This can be seen as follows: for Λ = λI,Tr ΛδS=λTrδS = 0. If Λ /ne}ationslash=λI, then, in general, TrΛδS/ne}ationslash= 0 even though δS= 0, and can therefore be chosen to be positive. What ifShas few zero eigenvalues? Let us choose a basis so that Sis diagonal. The eigenvalue of S siare ordered so that s1,...,s kare positive and si= 0 fori > k. We could choose δSijto be non zero only for 1 ≤i,j≤kand repeating the argument of the last paragraph, Λ ij=λδij, for 1 ≤i,j≤k. In fact, even if we choose δSijto be nonzero fori≤k < j , andj≤k < i we do not violate, to ﬁrst order in δS, non negativity of eigenvalues of S+δS. This would give us Λ ij= 0 fori≤k<j andj≤k<i. Hence Λ is of block-diagonal form. The k×kblock is already constrained to be proportional to Identity matrix. We would now constrain the other block of Λ which is of size (n−k)×(n−k). 10Since the last n−keigenvectors of Scorrespond to zero eigenvalues, we are free to rotate them among each other. Using this freedom, we diagona lise the lower ( n−k)×(n−k) block of Λ. Choosing diagonal δSijwith with negative values for i=j≤kbut positive valuesi=j >k, and satisfying Tr(δS) = 0, we can show that the last n−keigenvalues of Λ are smaller than or equal to λ. 9 Appendix B In this section, it is assumed without loss of generality tha tm≥n. We consider ﬁrst the caseS=I, but derive the results for arbitrary C,D. It is easy to recover the results for generalSby making the transformation H→S1 2HandC→S1 2CS1 2. We start from the identity det([wiH;−iH†w])−α=/integraldisplay dµ(X)dµ(Y)exp( −1 2α/summationdisplay a=1[w(Y† aYa+X† aXa)+i(Y† aHXa−X† aH†Ya)]) (51) where dµ(X) =n/productdisplay i=1α/productdisplay a=1dXR iadXI ia 2π(52) withR,Idenoting real and imaginary parts respectively. dµ(Y) is deﬁned analogously. The introduction of multiple copies of the Gaussian integra tion is the well known ‘replica trick’. This allows us to compute f(H), since it is easily veriﬁed that det([w iH ;−iH†w])−α=w−(m+n)αe−αf(H)(53) where we have set w2=n/P. The moment generating function of f(H) can be obtained by studying the expectation of the determinant above with re spect to the probability distribution of H. We therefore obtain for the moment generating function F(−α) =w(m+n)α/integraldisplay dµ(X)dµ(Y)exp( −1 2[wα/summationdisplay a=1(Y† aYa+X† aXa)+1 2nα/summationdisplay a,b=1(Y† aCYbX† bDXa)]) (54) The last term in the exponent can be decoupled by introducing theα×αcomplex matricesP,Qwith contour integrals over the matrix entries in the comple x plane to obtain F(−α) =w(m+n)α/integraldisplay dµ(X)dµ(Y)dµ(R)dµ(Q)exp( −1 2S) (55) where S=wα/summationdisplay a=1(Y† aY+X† aX) +α/summationdisplay a,b=1(Y† aCYbRab+QabX† aDXb−nRabQba) (56) dµ(R)dµ(Q) =α/productdisplay a,b=1dRabdQab 2π(57) 11TheR,Qintegrals, in contrast with the X,Yintegrals, are complex integrals along appropriate contours in the complex plain. For example, if t heQijintegrals are along the imaginary axis, so that the Qintegrals give rise to delta functions which can then be integrated over Rto obtain the above equation. The integrals over X,Ycan now be performed to obtain F(−α) =w(m+n)α/integraldisplay dµ(R)dµ(Q)exp( −log(det(w+CR))−log(det(w+DQ))+nTr(RQ)) (58) whereCRandDQare understood to be outer products of the matrices. Introdu cing the eigenvalues ξ,ηofC,Dthe exponent may be written as m/summationdisplay i=1log(det(w+ξiR)) +n/summationdisplay j=1log(det(w+ηjQ))−nTr(RQ) (59) Ifm,nbecome large and the number of non-zero ξi,ηigrow linearly with m,n, then we can perform the R,Qintegrals using saddle point methods. If we assume that at th e saddle point the matrices R,Qdo not break the replica symmetry , i.e R=rI,Q=qI whereIis the identity matrix, then the saddle point equations are ∂C/∂r=∂C/∂q= 0, where Cis deﬁned below, leading to r=1 nn/summationdisplay j=1ηj w+ηjq(60) q=1 nm/summationdisplay j=1ξj w+ξjr(61) Expanding the exponent upto quadratic order around the sadd le point and performing the resulting Gaussian integral, we obtain F(α) = exp(αC(r,q) +α2 2V(r,q)) (62) C(r,q) =m/summationdisplay i=1log(w+ξir) +n/summationdisplay j=1log(w+ηjq)−nqr−(m+n)log(w) (63) V(r,q) =−2log\|1−g(r,q)\| (64) g(r,q) = [1 nn/summationdisplay j=1(ηj w+ηjq)2][1 nm/summationdisplay j=1(ξj w+ξjr)2] (65) SinceF(α) is the moment generating function for f(H), the expressions for C,Vgive the expressions for the expectation and variance of f(H), as presented in section (3). 1210 Appendix C In this case, P(/vectorY\|/vectorX) =1 [π(1 +β2\|X\|2)]ne−\|/vectorY−α/vectorX\|2 (1+β2\|X\|2/n) (66) Let us redeﬁne /vector x=/vectorXand/vector y=/vectorY/√n. The optimal probability distribution of /vector xdepends only on its norm x=\|/vector x\|/√n. Letq(x) to be the probability distribution of x. Once more, H(/vector y\|/vector x) =E/vector x/bracketleftBig nlog/parenleftBig πe(1 +β2x2)/n/parenrightBig/bracketrightBig =n/integraldisplay dxq(x)log/bracketleftbiggπe n(1 +β2x2)/bracketrightbigg (67) However, p(/vector y) =E/vector x[p(/vector y\|/vector x)]≈/integraldisplay dxq(x)nn [π(1 +β2x2)]ne−n(y2+α2x2) (1+β2x2)+2nφ/parenleftBig αxy 1+β2x2/parenrightBig (68) where φ(a) = lim d→∞1 dlog/bracketleftBigg/integraltextπ 0dθsind−2(θ)edacos(θ) /integraltextπ 0dθsind−2(θ)/bracketrightBigg (69) Saddle point evaluation of φ(a) (which is equivalent to doing an expansion of the Bessel functionsIν(z) with large order νand large argument z, but the ratio z/νheld ﬁxed) gives φ(a) =acosθ(a) + log sinθ(a) (70) cosθ(a) =asin2θ(a) (71) In fact we would need dφ(a)/da. dφ(a) da= cosθ(a) =√ 1 + 4a2−1 2a(72) Variation of H(/vector y) =/integraltextd/vector yp(/vector y)log1 p(/vector y)with respect to q(x) produces δH(/vector y) δq(x)=−/integraldisplay d/vector yp(/vector y\|x)(1 + logp(/vector y)) (73) where p(/vector y\|x) =/bracketleftbiggn π(1 +β2x2)/bracketrightbiggn exp(−nf(x,y)) =p(y\|x) (74) and f(y,x) =y2+α2x2 (1 +β2x2)−2φ(αxy 1 +β2x2) (75) Now we can do the /vector yintegral in Eq.73 by the saddle point method. After going ove r to polar coordinates and doing some straightforward calculat ions, we ﬁnd that the integral peaks aty=y(x) given by y(x)2= (1 + (α2+β2)x2) (76) 13This is expected, as variance of /vector ygiven a uniform angular distribution of /vector xwith a ﬁxed normxis the right hand side of (76). On the other hand, the variance isy(x)2in the saddle point approximation. Thus ﬁnally, we have the condition for the stationarity of th e mutual information, − C= log/integraldisplay dx′q(x′)p(y(x)\|x′) +nlog/bracketleftbiggπe n(1 +β2x2)/bracketrightbigg (77) where Cis a constant, which turns out to be the channel capacity. The constant is ﬁxed by the condition that q(x) is a normalised probability distribution. This condition , along with the fact/integraltextd/vector yp(y\|x) = Ω 2n/integraltextdyy2n−1p(y\|x) = 1, Ω 2n= 2πn/Γ(n), can be used to determineC. 1 = Ω 2n/integraldisplay dxy′(x)y(x)2n−1/integraldisplay dx′q(x′)p(y(x)\|x′) (78) =e−CΩ2n/integraldisplay√ P 0dx/bracketleftbiggn πe(1 +β2x2)/bracketrightbiggny′(x) y(x)y(x)2n(79) ≈e−C/radicalbigg2n π/integraldisplay√ P 0dxy′(x) y(x)/bracketleftBigg y(x)2 (1 +β2x2)/bracketrightBiggn (80) For anyα>0, f(x) = log/bracketleftBigg y(x)2 (1 +β2x2)/bracketrightBigg = log/bracketleftBigg 1 + (α2+β2)x2 1 +β2x2/bracketrightBigg (81) is a monotonically increasing function of x, for positive x. Hence the last integral is dominated by the contribution from the region near the upper limit. For a monotonically increasing function f(x), /integraldisplayz 0g(x)exp(nf(x))≈g(z)exp(nf(z)) nf′(z). (82) Using this, we get c= limn→∞C/n= log/bracketleftBigg 1 + (α2+β2)P 1 +β2P/bracketrightBigg (83) Acknowledgements: The authors would like to thank Emre Telatar for many useful and inspiring discussions. 14Figure Captions Figure 1. The probability density function of f(H) is given for m=n= 4 in the limit of largeP. The origin is shifted to the value 4log( P/e). References [1] M.L.Mehta, “Random Matrices”, Academic Press, New York , 1991. [2] I.E.Telatar, “Capacity of multi-antenna Gaussian chan nels”, to appear in Eur. Trans. Telecommun. [3] G.J.Foschini “Layered space-time architecture for wir eless communications in a fad- ing environment when using multi-element antennas”, Bell L abs. Tech. J., V1, N2, P41-59, 1996. [4] T.L.Marzetta and B.M.Hochwald, “Capacity of a mobile mu ltiple-antenna commu- nication link in a Raleigh ﬂat-fading environment”, to appe ar in IEEE Trans. Info. Theory. [5] A.M.Sengupta and P.P.Mitra, “Distribution of Singular Values of some random ma- trices”, Phys. Rev. E60, P3389-3392, 1999. [6] A.L.Moustakas et. al. , “Communication through a diﬀusi ve medium: Coherence and capacity”, Science 287, P287-290, 2000. [7] P. P. Mitra and J. Stark, Bell Labs Tech Rep. 11111-990318 -05TM; J. Stark, P. P. Mitra and A. M. Sengupta, Bell Labs Tech Rep. to appear. 15−5−4−3−2−101234500.050.10.150.20.250.3 xp(x)Capacity distribution for m=n=4
arXiv:physics/0010082v1 [physics.class-ph] 31 Oct 2000Thomas precession angle and spinor algebra Shao-Hsuan Chiu1,2∗and T. K. Kuo2† 1Department of Physics, Rochester Institute of Technology, NY 14623 2Department of Physics, Purdue University, IN 47907 Abstract We present an alternative derivation of the Thamos precessi on angle. Us- ing Pauli matrices, the ﬁnite precession angle can be comput ed exactly. We also discuss a new physical interpretation of the precessio n angle. ∗Email address: sxcsps@rit.edu †Email address: tkkuo@physics.purdue.edu 1In the theory of relativity, it is well known that the order of successive Lorentz transfor- mations is crucial in relating two inertial reference frame s [1]: [Ji, Jj] =iǫijkJk, [Ji, Kj]=iǫijkKk, [Ki, Kj]=−iǫijkJk, (1) where Ji,j,kandKi,j,kare the inﬁnitesimal generators of rotations and boosts, re spectively. The phenomenon of Thomas precession is known to originate fr om this non-commutativity, in which the eﬀect of two successive Lorentz transformation is equivalent to the product of a single Lorentz transformation and a rotation. The prece ssion formula can be derived approximately through the inﬁnitesimal Lorentz transform ation consisting of rotations and boosts [2–4]. In this note we intend to present an alternativ e derivation, showing that the ﬁnite precession angle can be computed exactly through appl ications of the spinor algebra [5]. In the two-component spinor algebra, the Pauli matrices sat isfy the commutator: [σi, σj] = 2iǫijkσk (2) It is clear that Ji=σi/2 (rotations) and Ki=iσi/2 (boosts) represent the two-dimensional Lorentz group. A ﬁnite rotation about an axis ˆ nthrough an angle θis written as R= exp(iθ/vector σ·ˆ n 2), while a pure boost in an arbitrary direction ˆ nbecomes B= exp(−ζ/vector σ·ˆ n 2), where ζrepresents the rapidity parameter, /vectorζ=ζˆn. This simple properties of σienable one to manipulate explicitly these ﬁnite rotations. We will explo it this property to calculate the result of the combination of ﬁnite Lorentz transformations . We start with the combination of two pure boosts by choosing o ne boost with rapidity parameter 2 ηalong the direction ˆ nα=−(sin 2αˆx+ cos 2 αˆz), and the other with rapidity 2ξˆz: K=e−/vectorξ·/vector σe−/vector η·/vector σ=e−ξσ3eη(cos 2 ασ3+sin2 ασ1). (3) The combination of two pure boosts is equivalent to the combi nation of a third boost and a rotation: K=eΛ(cos 2Θ σ3+sin2Θ σ1)eiΨσ2 =e−iΘσ2eΛσ3ei(Θ+Ψ) σ2. (4) Here 2Λ is the third rapidity in the direction ˆ nΘ=−(sin 2Θˆ x+ cos 2Θˆ z), while 2Ψ is the Thomas precession angle rotated about ˆ y. Given ξ,η, and α, we are interested in how Λ, Θ, and the angle Ψ in particular, can be derived exactly. It tu rns out that KKTandKTK provide a simple approach to this problem. We may write KKTfrom eq.(3): KKT=e−ξσ3e2η(cos 2 ασ3+sin2 ασ1)e−ξσ3. (5) But from eq.(4), KKT=e2Λ(cos 2Θ σ3+sinΘ σ1)=e−iΘσ2e2Λσ3eiΘσ2 = cosh 2Λ + (cos 2Θ σ3+ sin Θ σ1) sinh 2Λ . (6) 2We may then cast the above two equations into [6]: KKT=e−iΘσ2e2Λσ3eiΘσ2 =e−ξσ3e2η(cos 2 ασ3+sin2 ασ1)e−ξσ3 = cosh 2 ξcosh 2 η−cos 2αsinh 2ξsinh 2η+ [−sinh 2ξcosh 2 η+ cos 2 αcosh 2 ξsinh 2η]σ3 +[sin 2 αsinh 2η]σ1. (7) Comparing eq.(7) and the second line of eq.(6), it follows th at tan 2Θ =sin 2αsinh 2η −cosh 2 ηsinh 2ξ+ cos 2 αsinh 2ηcosh 2 ξ, (8) cosh 2Λ = cosh 2 ξcosh 2 η−cos 2αsinh 2ξsinh 2η. (9) Note that Ψ is canceled out in KKT. Eqs.(8) and (9) are equivalent to eq.(11.32) in Ref.2. To compute Ψ, we now turn to KTK. From eq.(3), we ﬁnd KTK=eη(cos 2 ασ3+sin 2 ασ1)e−2ξσ3eη(cos 2 ασ3+sin2 ασ1), (10) while eq.(4) gives KTK=e−i(Θ+Ψ) σ2e2Λσ3ei(Θ+Ψ) σ2. (11) We may further simplify eq.(10) by noting that e−iασ2eησ3eiασ2=eη(cos 2 ασ3+sin2 ασ1). (12) This leads us to KTK=e−iασ2(eησ3e−2ξσ(−α)eησ3)eiασ2, (13) where σ(−α)≡cos(−2α)σ3+ sin(−2α)σ1. We therefore arrive at a simple relation from eq.(11) and eq.(13): e−i(Θ+Ψ −α)σ2e2Λσ3ei(Θ+Ψ −α)σ2=eησ3e−2ξσ(−α)eησ3. (14) It is interesting to observe that, with the substitutions η→ −ξ,ξ→ −η, and α→ −α, the right-hand side of eq.(14) is nothing but KKTin eq.(5). For this reason, we can readily write down tan2(Θ + Ψ −α) directly from eq.(8): tan 2(Θ + Ψ −α)≡tan2Φ =sin 2αsinh 2ξ cosh 2 ξsinh 2η−cos 2αsinh 2ξcosh 2 η. (15) The ﬁnite Thomas precession angle 2Ψ can therefore be expres sed in terms of ξ,η, and α: tan2Ψ =tan 2Φ −tan2Θ + (1 + tan2Φ tan 2Θ) tan2 α 1 + tan 2Φ tan2Θ −(tan 2Φ −tan2Θ) tan2 α, (16) 3where tan2Θ and tan 2Φ are given in eq.(8) and eq.(15), respec tively. We next check our result under inﬁnitesimal Lorentz transfo rmations. Let −2/vector η= 2/vectorξ+δ/vectorξ andα≪1 for simplicity, where \|δ/vectorξ\| ≃ \|2/vectorξ\|\|2α\| ≪ \|2/vectorξ\|. Note that Θ, Φ, and Ψ, become inﬁnitesimal as well. To ﬁrst order in δξ, sinh 2 η≃sinh 2ξ+δξcosh 2 ξand cosh 2 η≃ cosh 2 ξ+δξsinh 2ξ. Eq.(8) then becomes tan 2Θ ≃2Θ≃2α δξsinh 2η≃2α δξ(sinh 2 ξ+δξcosh 2 ξ), (17) while eq.(15) yields tan2Φ ≃2Φ≃2α δξsinh 2ξ. (18) The inﬁnitesimal Thomas precession angle about ˆ yfollows immediately: 2Ψ≃2(α−Θ + Φ) ≃2α(1−cosh 2 ξ). (19) To compare eq.(19) with the result in Ref.2, we note that /vectorβandδ/vectorβin Ref.2 are simply 2/vectorξandδ/vectorξ, respectively. We then write cosh 2 ξ≡γand sinh 2 ξ≡γ(2ξ). From Ref.2, ∆/vectorΩ =−γ2 γ+ 1(/vectorβ×δ/vectorβ)≃−γ2 γ+ 1\|2/vectorξ\|\|δ/vectorξ\|ˆy = (−2α)γ2(2ξ)2 1 +γˆy= 2α(cosh 2 ξ−1)ˆy (20) Therefore, to the ﬁrst order in δξ, eq.(16) reduces to 2Ψ≃ −∆Ω. (21) where ∆Ω is the rotated angle in Ref.2. The minus sign in the ab ove equation arises from the fact that 2Ψ in our formulation is the rotated angle assoc iated with the third boost, while the angle ∆Ω in Ref.2 associates with the combined boos ts. It is also worthwhile to examine the role played by Thomas pre cession angle as two boosts are combined in reverse order. One notes from eq.(3) and eq.( 4), e−ξσ3eησα=eΛσΘeiΨσ2, (22) where σα≡cos 2ασ3+ sin 2 ασ1, and σΘ≡cos 2Θ σ3+ sin 2Θ σ1. It follows that e−ξσ3eησαe−iΨσ2=eΛσΘ. (23) Note that σ1andσ3are symmetric while σ2is antisymmetric: (eΛσΘ)T=eΛσΘ=e−ξσ3eησαe−iΨσ2. (24) After taking the transpose, the left-hand side of eq.(23) be comes 4eiΨσ2eησαe−ξσ3. We thus reach the relation: eiΨσ2eησαe−ξσ3eiΨσ2=e−ξσ3eησα. (25) The combination of two boosts, exp( −ξσ3) exp(ησα), is related to its reverse, exp(ησα) exp(−ξσ3), by two identical rotations, eiΨσ2. In other words, operating the same rotations on a reference frame before and after two boosts ca n bring this frame to another ref- erence frame which is reached by the same two boosts in revers e order. The angle associates with this particular rotation is the Thomas precession angl e. In this note, we showed how the use of Pauli matrices can provi de a general and eﬀective way of solving the problem of combining ﬁnite Lorentz transf ormations. This advantage is illustrated through the derivation and a physical interpre ting of the ﬁnite Thomas precession angle. ACKNOWLEDGMENTS We thank G.-H Wu for useful discussions. S. H. C. was supporte d in part by the Pur- due Research Foundation. T. K. K. is supported in part by DOE g rant No. DE-FG02- 91ER40681. 5REFERENCES [1] See, e.g., L. H. Ryder, Quantum Field Theory , Cambridge University Press (1985). [2] J. D. Jackson, Classical Electrodynamics (2nd Edition), John Wiley & Sons (1975), Sec- tion 11.8. [3] D. Shelupsky, Am. J. Phys. 35, 650 (1967). [4] C. Misner, K. Thorne, and J. wheeler, Gravitation , W. H. Freeman and Company (1975), Section 41.4. [5] For a review of related spinor algebra, see, e.g., Chapte r 41 of Ref.[4] and J. J. Sakurai, Modern Quantum Mechanics , The Benjamin/Cummings Publishing Company (1985), Chapter 3. [6] T. K. Kuo, G.-H. Wu, and S.-H. Chiu , Phys. Rev. D62, 051301 (2000). 6
arXiv:physics/0010083v1 [physics.geo-ph] 31 Oct 20001 P-V-T equation of state of MgSiO 3perovskite from molecular dynamics and constraints on lower mantle composition Frederic C. Marton, Joel Ita,1and Ronald E. Cohen Geophysical Laboratory and Center for High Pressure Resear ch, Carnegie Institution of Washington, Washington, DC 20015-1305 Abstract The composition of the lower mantle can be investigated by ex amining densities and seismic velocities of compositional models as functions of depth. I n order to do this, it is necessary to know the volumes and thermoelastic properties of the compos itional constituents under lower mantle conditions. We determined the thermal equation of st ate (EoS) of MgSiO 3perovskite using the non-empirical VIB interatomic potential with mol ecular dynamics simulations at pressures and temperatures of the lower mantle. We ﬁt our P-V -T results to a thermal EoS of the form P(V, T) =P0(V, T0) + ∆Pth(T), where T0= 300 K and P0is the isothermal Universal EoS. The thermal pressure ∆ Pthcan be represented by a linear relationship ∆ Pth=a+bT. We ﬁndV0= 165.40 ˚A3,KT0= 273 GPa, K′ T0= 3.86, a= -1.99 GPa, and b= 0.00664 GPa/K for pressures of 0-140 GPa and temperatures of 300-3000 K. By ﬁxi ngV0to the experimentally determined value of 162.49 ˚A3and calculating density and bulk sound velocity proﬁles alo ng a lower mantle geotherm, we ﬁnd that the lower mantle cannot co nsist solely of (Mg,Fe)SiO 3 perovskite with X Mgranging from 0.9-1.0. Using pyrolitic compositions of 67 mo l% perovskite (XMg= 0.93-0.96) and 33 mol% magnesiow¨ ustite (X Mg= 0.82-0.86), however, we obtained density and velocity proﬁles that are in excellent agreemen t with seismological models for a reasonable geotherm. Submitted to: J. Geophys. Res., 29 March 2000 Revised: 28 September 20002 Introduction By comparing density and seismic velocity proﬁles of compositional with seismological models, it is pos- sible to investigate the composition of the Earth’s lower mantle. Therefore, an understanding of the high pressure, high temperature properties and equa- tion of state (EoS) of (Mg,Fe)SiO 3perovskite is vi- tal to our understanding of the lower mantle, as this mineral accounts for perhaps two-thirds of the miner- alogy of this region [ Bina, 1998], and one-third of the volume of the entire planet. Experiments have been performed up to pressures approaching those found at the base of the mantle, but direct coverage of the lower mantle geotherm has been limited to perhaps the uppermost one-third (Figure 1). Experiments in- Figure 1 clude resistively heated multi-anvil apparati [ Wang et al., 1994;Kato et al., 1995;Utsumi et al., 1995;Fu- namori et al., 1996], diamond anvil cells (DACs) at ambient temperature [ Kudoh et al., 1987;Ross and Hazen, 1990], resistively heated DACs [ Mao et al., 1991;Saxena et al., 1999], laser heated DACs [ Knittle and Jeanloz 1987;Fiquet et al., 1998, 2000], and shock wave experiments [ Duﬀy and Ahrens, 1993; Akins and Ahrens, 1999]. Theoretical methods, such as those used here, per- mit the investigation of thermodynamic and ther- moelastic properties of minerals at high pressures and temperatures. Previous theoretical work on or- thorhombic MgSiO 3perovskite (Mg-pv) has been done using molecular dynamics (MD) using the semi- empirical potential of Matsui [1988] at 0-1000 K and 0-10 GPa (at 300 K) [ Matsui, 1988] and 300-5500 K and 0-400 GPa [ Belonoshko, 1994; Belonoshko and Dubrovinsky, 1996], and combined with lattice dy- namics at 500-3000 K and 0-100 GPa [ Patel et al., 1996]. Wolf and Bukowinski [1987] used a rigid ion Modiﬁed Electron Gas (MEG) model at 0-800 K and 0-150 GPa, while Hemley et al. [1987] used a MEG model combined with shell stabilized ion charge den- sities at 0-2500 K and 0-200 GPa. Cohen [1987] did quasiharmonic lattice dynamics calculations from 0- 3000 K and 0-150 GPa using the Potential Induced Breathing (PIB) model. First principles static lattice calculations (T = 0 K) have also been done up to pressures of 150 GPa, using plane-wave pseudopoten- tial (PWPP) MD [ Wentzcovitch et al., 1995] and the Linearized Augmented Plane Wave (LAPW) method [Stixrude and Cohen, 1993]. Karki et al. [1997] also used the PWPP method to examine the athermal elastic moduli of Mg-pv.In addition to the equation of state, the thermo- dynamic stability of perovskite in the lower mantle is an open question and has been studied with experi- mental and theoretical methods. Work by Meade et al.[1995], Saxena et al. [1996, 1998] and Dubrovin- sky et al. [1998] indicate that perovskite will break down to oxides under lower mantle conditions. The thermodynamic analysis of Stixrude and Bukowinski [1992] and the experimental work of Serghiou et al. [1998], however, suggest that it will not. We used MD simulations using the non-empirical Variational Induced Breathing (VIB) model, similar to the model of Wolf and Bukowinski [1988], to inves- tigate the properties and EoS of Mg-pv at pressures (0-140 GPa) and temperatures (300-3000 K) that cover the bulk of the conditions found in the lower mantle. Newton’s equations of motion are solved as functions of time, and equilibrium properties are ob- tained as time averages over a suﬃciently long inter- val. This accounts for all orders of anharmonicity, but not for quantum lattice dynamics eﬀects. Thus our results are more appropriate at high tempera- tures above the Debye temperature (1076 K [ An- derson, 1998]), and entirely suitable for the Earth’s mantle. We then used our results, combining them with data for other components and phases where appropriate, to calculate density and seismic veloc- ity proﬁles along a geothermal gradient for diﬀerent compositional models. These proﬁles were then com- pared with proﬁles from a reference Earth model in order to test the compositional models’ ﬁtness for the lower mantle. Method MD simulations were performed on supercells of 540-2500 atoms of orthorhombic ( Pbnm ) Mg-pv using the non-empirical VIB potential. VIB is a Gordon- Kim type model [ Gordon and Kim, 1972] where the potential is obtained by overlapping ionic charge densities which are computed using the local den- sity approximation (LDA) [ Hedin and Lundqvist, 1971]. The total energy is a sum of (i) the self- energy of each atom, (ii) the long-range electrostatic energy computed using the Ewald method, and (iii) the short-range interaction energy, i.e.,the sum of the kinetic, short-range electrostatic, and exchange- correlation overlap energies from the LDA. Free oxy- gen ions are not stable, and are stabilized in VIB by surrounding them with Watson spheres with charges equal in magnitude but opposite in sign to the ions,3 e.g.,2+ spheres for O2−. Interactions are obtained for overlapping ion pairs at diﬀerent distances with diﬀerent Watson sphere radii for each oxygen. The interactions are ﬁt with a 23 parameter analytical expression as functions of the interatomic distance r andUi=zi/Ri, where Ui,zi, and Riare the Wat- son sphere potential, charge, and radius for atom i. For each oxygen i,Riis adjusted to minimize the to- tal energy at each time step. This gives an eﬀective many-body potential. The oxygen ions respond to changes in their environment. For example, they are compressed at high pressures relative to low pressures. Previous work on Mg-pv using the related PIB model gave an equation of state in excellent agreement with experiment [ Cohen, 1987]. Also, work using VIB on MgO has shown that this model accurately predicts EoS and thermal properties [ Inbar and Cohen, 1995]. Nominally charged ions give semi-quantitative, but less accurate, results than desired for Mg-pv. This is due to a small degree of covalent relative to ionic bonding as revealed by electronic structure calcula- tions of cubic ( Pm3m ) Mg-pv performed using the ﬁrst-principles LAPW method [ Cohen et al., 1989]. These calculations show that while the Mg is nearly a perfectly spherical Mg2+ion, there is some charge transfer from O to Si and a small covalent bond charge (involving ≪0.1 electrons) between Si and O. By varying the ionic charges on Si (3.1+ to 3.55+) and O (1.70- to 1.85-) to account for the covalency, but otherwise using the same methods as before, we com- pared the resulting zero temperature pressure-volume data to the LAPW results of Stixrude and Cohen [1993]. The best agreement was found with Si3.4+ and O1.8−. Using these charges, the resulting poten- tial gives excellent agreement with the octahedral ro- tation energetics obtained using the LAPW method [Stixrude and Cohen, 1993;Hemley and Cohen, 1996] (Figure 2). Figure 2 MD runs were performed for 20 ps with a 1 fs time step using a sixth order Gear predictor-corrector scheme [ Gear, 1971] in the constant pressure-tempera- ture ensemble using the thermostat and barostat of Martyna et al. [1994]. Initial atomic positions for the genesis run were the same as those for the unro- tated octahedra of Stixrude and Cohen, [1993]. Sub- sequent runs at higher pressures or temperatures used the positions generated by a previous run. Statistical ensembles were obtained in ∼2000 iterations for the genesis run and in <1000 for subsequent ones. We ﬁtted P-V-T data obtained from the MD sim-ulations to a thermal EoS of the form P(V, T) =P0(V, T0) + ∆Pth(T) (1) with a reference temperature T0= 300 K and the thermal pressure ∆ Pthrelative to the 300 K isotherm. We analyzed our results using the third-order Birch- Murnaghan isothermal EoS [ Birch, 1952] P0= 3KT0f(1 + 2 f)5/2(1−ξf), (2) with the Eulerian strain variable fand the coeﬃcient ξgiven by f≡1 2/bracketleftBigg/parenleftbiggV0 V/parenrightbigg2/3 −1/bracketrightBigg , ξ≡ −3 2(K′ T0−4),(3) and the Universal EoS [ Vinet et al., 1987] P0= 3KT0(1−x)x−2exp/bracketleftbigg3 2(K′ T0−1)(1−x)/bracketrightbigg ,(4) where x=/parenleftBig V V0/parenrightBig1/3 ,KTis the isothermal bulk mod- ulus, and K′ Tis its pressure derivative. ∆Pthis given by ∆Pth=/integraldisplayT T0/parenleftbigg∂P ∂T/parenrightbigg VdˆT=/integraldisplayT T0(αKT)dˆT,(5) where αis the volume coeﬃcient of thermal expan- sion. If αKTis independent of T, then the thermal pressure can be represented by a linear relationship ∆Pth=a+bT (6) [Anderson, 1980, 1984; Anderson and Suzuki, 1983], which applies to a wide range of solids at high T, in- cluding minerals, alkali metals, and noble gas solids [Anderson, 1984]. The linearity of ∆ Pthin T starts at 300 K [ Masuda and Anderson, 1995] in miner- als. There should be a small anharmonic correction at high T, which results in an additional c T2term in (6), which is often suﬃciently small so that it can be ne- glected [ Anderson, 1984]. Indeed, including the c T2 term did not statistically improve our ﬁts. Likewise, neither did including volumetric compression terms (e.g., Jackson and Rigden, [1996]). Once the P-V-T data were ﬁt, other parameters, such as α, α=/parenleftbigg∂lnV ∂T/parenrightbigg P, (7)4 its volume dependence, the Anderson-Gr¨ uneisen pa- rameter, δT=/parenleftbigg∂lnα ∂lnV/parenrightbigg T, (8) the Gr¨ uneisen parameter, γ=αKTV CV, (9) and its volume dependence, q=/parenleftbigg∂lnγ ∂lnV/parenrightbigg T, (10) were obtained by numerical diﬀerentiation. CPwas found by calculating enthalpy ( U + PV ) at each MD run point and diﬀerentiating with respect to T. Using the relation CP=CV(1 +γαT), equation (9) can be rewritten as γ=/parenleftbiggCP αKTV−Tα/parenrightbigg−1 . (11) Results MD calculations were performed at 46 pressure- temperature points, ranging from 0-140 GPa and 300- 3000 K, thus covering the P-T conditions of the lower mantle (Table 1). Axial ratios b/a and c/a follow 1 smooth trends at lower mantle pressures (Figure 3a-Figure 3b), with deviations away from the Pbnm structure seen at P ≤12 GPa and T ≥2400 K. At 3000 K and 12 GPa, the structure can be observed to move towards cubic lattice parameters (b/a →1, c/a →√ 2 ). This point in particular is close to the melting curve of Mg-pv [ Heinz and Jeanloz, 1987;Knittle and Jeanloz 1989; Poirier, 1989] and such temperature- induced phase transitions have been predicted for Mg- pv by MD [ Wolf and Bukowinski, 1987] and lattice dynamics [ Matsui and Price, 1991]. In addition, it has been found for RbCaF 3and KMnF 3perovskites by MD simulations [ Lewis and L´ epine, 1989; Nos´ e and Klein, 1989], and observed experimentally in the ﬂuoride perovskite neighborite (NaMgF 3) [Chao et al.,1961]. Regardless of these changes, the volumes are well-behaved as functions of pressure and temper- ature (Figure 3c) and no other structural anomalies or melting were encountered during runs at other P,T points. The resulting P-V-T EoS ﬁts and their reduced χ2values are given in Table 2. Both the Birch- 2 Murnaghan and Universal EoSs are in excellent agree- ment with experimental results. As for the accuracyof our thermal pressure expression, RMS diﬀerences between volumes found using equation (1) for either EoS and those found using isothermal EoSs are on the order of 10−3˚A3. As for the diﬀerences between the two equation of state forms, Jeanloz [1988] com- pared them over moderate compressions and found that they are quite similar. More recent work by Co- hen et al. [2000] supports this, but they found that for large ( >30%) strains and for determining parameters such as V0, KT0,andK′ T0, the Universal EoS works best, and so, for the rest of our analyses, we use that here. However, given that under the pressure range studied, compressions will be no greater that 25%, so we can conﬁdently compare our results with those of others that were found with the Birch-Murnaghan EoS. Once we determined the EoS parameters, we de- termined volumes for Mg-pv over a wide P-T range (-10-150 GPa, 0-3300 K). Using this extended data set, we were determined the isothermal bulk modu- lus/bracketleftbig KT=−V/parenleftbig∂P ∂V/parenrightbig T/bracketrightbig at those points, and α,δT,γ, andqusing equations 7, 8, 11, and 10. We found/parenleftbig∂K ∂T/parenrightbig P=0= -0.0251 GPa/K, close to those found ex- perimentally, -0.023 GPa/K [ Wang et al., 1994] and -0.027 GPa/K [ Fiquet et al., 1998], as well as one found by the inversion of multiple experimental data sets, -0.021 GPa/K [ Jackson and Rigden, 1996]. Figure 4 shows αat pressures from 0 to 140 GPa. Figure 4 Experimental results shown have higher T slopes, as well as higher extrapolated values for most tempera- tures, though an average value found by Jackson and Rigden [1996] at zero GPa of 2 .6×10−5K−1over 300- 1660 K matches our value over the same T range ex- actly. Kato et al. [1995] found an average value of 2.0±0.4×10−5K−1over 298-1473 K and 25 GPa, close to our average value of 1 .89×10−5K−1at the same conditions. The lack of quantum eﬀects in our MD results can be seen at low temperatures; instead of tending towards zero at zero temperature, αre- mains large. δTincreases with T, but decreases as P increases (Figure 5). The dependency of δTon T Figure 5 decreases as a function of P, with convergence to a value of 2.87 at ∼130 GPa, similar to the high pres- sure behavior in MgO found by the same MD method [Inbar and Cohen, 1995]. Our value of δT= 3.79 at ambient conditions is in excellent agreement with the experimentally derived value of Wang et al. [1994] of 3.73, though it is lower than the value of 4.5, found by Masuda and Anderson [1995] from the experimental data of Utsumi et al. [1995]. Other theoretical deter- minations of δTare much higher, however. Hama and5 Suito [1998a] found a value of 5.21, based on calcula- tions using the LAPW results of Stixrude and Cohen [1993] and the Debye approximation for lattice vibra- tion. Using MD and semi-empirical potentials [ Mat- sui,1988], Belonoshko and Dubrovinsky [1996] found δT= 5.80, and Patel et al. [1996] found δT= 7.0 us- ing the same potentials, combining MD with lattice dynamics. Anderson [1998], using Debye theory, cal- culated zero pressure values ranging from 4.98 at 400 K to∼4 at 1000 K ≤T≤1800 K, coming close to our values at those temperatures. The Gr¨ uneisen parameter also increases as a func- tion of T and decreases as a function of P, and ap- proaches a value of 1 at pressures of 130-140 GPa (Fig- ure 6). Our value of 1.33 at zero P and 300 K matches Figure 6 well with Wang et al.’s [1994] value of 1.3 and Utsumi et al.’s [1995]/ Masuda and Anderson’s [1995] value of 1.45. Stixrude et al. [1992], using the experimental data of Mao et al. [1991], found a higher value of γ = 1.96. Values determined by the inversion of multi- ple experimentally determined P-V-T data sets match well also: 1.5 [ Bina, 1995], 1.33 [ Jackson and Rigden, 1996], and 1.42 [ Shim and Duﬀy, 2000]. Two shock wave studies ﬁnd a value of 1.5 [ Duﬀy and Ahrens, 1993;Akins and Ahrens, 1999] based on limited data sets: four for the former (with qﬁxed equal to 1) and two for the latter. Values of γfrom theoretical studies range from very close to ours, 1.279 [ Hama and Suito, 1998a] and 1.44 [ Hemley et al., 1987], to 1.97 [Belonoshko and Dubrovinsky, 1996;Patel et al., 1996]. Anderson’s [1998] zero pressure, 300 K value of 1.52 is somewhat higher than ours, but at 400-1800 K his values of ∼1.4 are very close to ours. The volume dependence of γ,q, is 1.03 at 300 K (Figure 7). Our equation of state form, with ∆ Pth Figure 7 linear in T and independent of volume, constrains q = 1 if the isochoric heat capacity CV= 3nR, the clas- sical harmonic value. As T increases, we ﬁnd that q increases to a value of 1.07 at 3000 K due to anhar- monicity. Values from inverted experimental data sets range from 1.0 [ Bina, 1995] to 2.0 [ Shim and Duﬀy, 2000]. Stixrude et al. [1992] found a high qvalue of 2.5 to go along with their value for γ.Akins and Ahrens’s [1999] value was even higher, q= 4.0±1.0 withCV= 5nR, but these are preliminary conclu- sions based on limited data. Patel et al. [1996] found a value of 3.0 at 0 GPa, decreasing to 1.7 at 100 GPa using a combination of molecular and lattice dynam- ics.Anderson’s [1998] Debye calculations gave val- ues close to unity at T ≥1000 K, with qdecreasing slightly with increasing T, to 0.82 at 1800 K.Discussion and Conclusions Taking sets of experimental P-V-T data, we ﬁt the high temperature data [ Fiquet et al., 1998, 2000; Sax- ena et al., 1999] and available experimental MgSiO 3 data to our thermal Universal EoS form (Table 2) in order to have consistent bases for comparison. The resulting ﬁts have higher statistical uncertainties but compare well with our results. Volumes calculated along the lower mantle geothermal gradient of Brown and Shankland [1981], which has a starting T of 1873 K at 670 km (Figure 1), show that those calculated from our EoS are 3 to 4 ˚A3per unit cell larger, about 2.5%, than those calculated from the inverted experi- mental data sets (Figure 8). This corresponds to den- Figure 8 sity diﬀerences of ∼0.1 g/cm3. This is the opposite of the typical error of the local density approximation to density functional theory, on which our method is based. The larger volume is due to the choice of ionic charges and other model assumptions. If we ﬁx V 0 to the experimentally determined value of 162.49 ˚A3 [Mao et al., 1991], but otherwise use our EoS param- eters, our calculated volumes are very close, within 1%, to the volumes derived from the inverted exper- imental data sets. The comparisons also improve as pressure increases. To compare our model with other zero temperature theoretically-derived EoSs, we cal- culated volumes at 0 K from 0-140 GPa and ﬁt an isothermal Universal EoS to them (Table 3). Table 3 Compositional models of the lower mantle can be tested against seismological models by examin- ing densities and seismic wave velocities as func- tions of depth. Candidates include pyrolite [ Ring- wood, 1975] and chondritic or pure perovskite models. Studies support both the former [ Bina and Silver, 1990; Stacey, 1996] and latter [ Butler and Ander- son,1978;Stixrude et al., 1992], though uncertainties in the thermodynamic parameters of the constituent minerals make it diﬃcult to resolve this question with certainty. Indeed, several studies have been able to support both models depending on whether high (pure perovskite) or low (pyrolite) values of αandδT orγandqare used [ Zhao and Anderson, 1994;An- derson et al., 1995;Karki and Stixrude, 1999]. Calculating the densities of Mg-pv along a geother- mal gradient [ Brown and Shankland, 1981] (Fig- ure 9a), we see that they are ∼0.25 g/cm3(∼5%) too Figure 9 low compared with the ak135-f seismological model [Kennett et al., 1995;Montagner and Kennett, 1996]. The calculated bulk sound velocities, Vϕ=/radicalbig KS/ρ, where KS=KT(1 +γα T), are, in turn, ∼0.6-0.76 km/s (7-8%) too high (Figure 9b). Using the experi- mentally determined value of V0= 162.49 ˚A3[Mao et al.,1991] in place of our calculated value does increase the densities somewhat, but not enough to match the ak135-f values. As the perovskite found in the lower mantle should be a solid solution of the Mg and Fe end-members, we added iron by adjusting density by ρ(XFe) =ρ(0)(1 + 0 .269XFe) (12) [Jeanloz and Thompson, 1983], but did not change the bulk moduli [ Mao et al., 1991]. Including 10 mol% Fe does cause densities to agree with the seis- mological values (Figure 10a), but the bulk sound ve- Figure 10 locities are still much too high (Figure 10b). Conse- quently, we ﬁnd that the lower mantle cannot consist solely of (Mg,Fe)SiO 3perovskite. We also tested pyrolitic compositions consisting of mixtures of perovskite and magnesiow¨ ustite (mw). Densities of mw were calculated using the thermody- namic data set of Fei et al. [1991], as well as KTs for the Mg end-member. KSs were then calculated using the relation α(P, T) =α(P0, T)/bracketleftbiggV(P, T) V(P0, T)/bracketrightbiggδT , (13) withδTandγfromInbar and Cohen [1995]. Iron was accounted for in KSvia the relationship KS(XFe) =KS(0) + 17 XFe (14) [Jeanloz and Thompson, 1983]. X Mg= 0.93-0.96 for pv and 0.82-0.86 for mw [ Katsura and Ito, 1996; Martinez et al., 1997] were used. Densities and bulk sound velocities were calculated using Voigt-Reuss- Hill averaging for two simpliﬁed pyrolite models (high [1] and low [2] Fe content) consisting of 67 mol% pv and 33 mol% mw and were found to be in excellent agreement with the seismological model. (Figure 11). Figure 11 The two pyrolite models have partitioning coeﬃcients Kpv−mw Fe−Mgof 0.34 and 0.26, where Kpv−mw Fe−Mg=Xpv Fe/Xpv Mg Xmw Fe/Xmw Mg. (15) Experimental evidence suggests that for such a bulk composition, Kpv−mw Fe−Mgshould increase in the mantle from∼0.20 (660 km) to ∼0.35 (1500 km), with Xpv Mg decreasing and Xmw Mgincreasing with depth [ Mao et al.,1997]. Compositional models with high Fe con- tent of pv and low Fe content of mw (and vice-versa)fall in-between the results of those shown in Figure 11. Given the range of Fe-Mg partitioning between perovskite and magnesiow¨ ustite under the appropri- ate pressure and temperature conditions, we ﬁnd py- rolite to be the most likely compositional model for the lower mantle. Given that other components (Al, Fe3+) and phases (CaSiO 3perovskite) should be present in the lower mantle, we do not expect exact agreement of a sim- pliﬁed pyrolite model with any seismological model. It is believed that, under lower mantle conditions, Al2O3is mainly incorporated into the Mg-pv struc- ture [Irifune, 1994;Wood, 2000]. Generally, the eﬀect of the incorporation of Al into Mg-pv on its physical properties has been considered small, e.g., Weidner and Wang [1998]. However, Al, unlike Fe, causes signiﬁcant distortion in the Mg-pv lattice [ O’Neill and Jeanloz, 1994], which may aﬀect the compress- ibility. Recent experimental work by Zhang and Wei- dner[1999], at pressure of up to 10 GPa and tempera- tures of up to 1073 K, indicates that, compared with MgSiO 3, Mg-pv with 5 mol% Al has a smaller K T value (232-236 GPa) and a ( ∂KT/∂T)Pvalue more than double that in magnitude. The values for α0, (∂α/∂T )P, andδTare also larger. Smaller bulk mod- uli and the higher density (0.2% at 300 K and 0 GPa) would cause seismic velocities to decrease. In addi- tion, the presence of Al tends to equalize the par- titioning of Fe into perovskite and magnesiow¨ ustite and may allow garnet to coexist with perovskite in the uppermost ∼100 km of the lower mantle [ Wood and Rubie, 1996; Wood, 2000]. These partitioning experiments, however, were done at high fO2, so it is uncertain how applicable they are to the mantle. The eﬀect of the presence of Fe3+in Mg-pv on the EoS and elasticity, while known for defect concentra- tions and electrostatic charge balance [ McCammon, 1997, 1998], is unknown. As for Ca-pv, experimental data suggest that its elastic properties are in excel- lent agreement with seismological models, and thus would be invisible in the lower mantle [ Wang et al., 1996;Hama and Suito, 1998b]. Performing MD calculations over the range pres- sures and temperatures found in the Earth’s mantle, we ﬁnd a thermal EoS that is in excellent agreement with experimental results. Thermodynamic parame- ters can be derived that agree well with experimen- tally determined values and that can be conﬁdently interpolated to conditions found in the lower mantle. Moreover, no solid-solid phase transitions or melting were found during the runs under lower mantle condi-7 tions, so orthorhombic MgSiO 3perovskite should be stable to the core-mantle boundary. Using these re- sults, we ﬁnd that pyrolite with Kpv−mw Fe−Mg= 0.26-0.34 is the most likely compositional model for the lower mantle. Acknowledgments. We thank Iris Inbar, Russell Hemley, Stephen Gramsch, Joe Akins, Mark Woods, Sheila Coleman, and Lars Stixrude for helpful discussions and comments. We acknowledge the support of the Na- tional Science Foundation through grant EAR-9870328. The computations were performed on the Cray SV1 at the Geophysical Laboratory, with the support of NSF grant EAR-9975753 and the W. M. Keck Foundation. References Akins, J. A., and T. J. Ahrens, Shock wave equa- tion of state measurements on MgSiO 3perovskite at lower mantle pressures indicate γ0= 1.5 and q = ∂lnγ/∂lnV >3,Eos Trans. AGU, 80, F757, 1999. Anderson, O. L., An experimental high-temperature ther- mal equation of state bypassing the Gr¨ uneisen param- eter,Phys. Earth Planet. Inter., 1, 173-183, 1980. Anderson, O. L., A universal thermal equation-of-state, J. Geodyn., 1, 185-214, 1984. Anderson, O. L., Thermoelastic properties of MgSiO 3per- ovskite using the Debye approach, Am. Mineral., 83, 23-35, 1998. Anderson, O. L., K. Masuda, and D. Guo, Pure silicate perovskite and the PREM lower mantle model: a ther- modynamic analysis, Phys. Earth Planet. Inter., 89, 35-49, 1995. Anderson, O. L., and I. Suzuki, Anharmonicity of three minerals at high temperature: Forsterite, fayalite, and periclase, J. Geophys. Res., 88, 3549-3356, 1983. Belonoshko, A. B., Molecular dynamics of MgSiO 3per- ovskite at high pressures: Equation of state, structure, and melting transition, Geochim. Cosmochim. Acta, 58,4039-4047, 1994. Belonoshko, A. B., and L. S. Dubrovinsky, Equations of state of MgSiO 3-perovskite and MgO (periclase) from computer simulations, Phys. Earth Planet. Inter., 98, 47-54, 1996. Bina, C. R., Conﬁdence limits for silicate perovskite equa- tions of state, Phys. Chem. Minerals, 22, 375-382, 1995. Bina, C. R., Lower mantle mineralogy and the geophysical perspective, Rev. Mineral., 37, 205-239, 1998. Bina, C. R., and P. G. Silver, Constraints on lower mantle composition and temperature from density and bulk sound velocity proﬁles, Geophys. Res. Lett., 17, 1153- 1156, 1990. Birch, F., Elasticity and constitution of the Earth’s inte- rior,J. Geophys. Res., 57, 227-286, 1952.Brown, J. M., and T. J. Shankland, Thermodynamic pa- rameters in the Earth as determined from seismic pro- ﬁles,Geophys. J. R. Astron. Soc., 66, 579-596, 1981. Butler, R., and D. L. Anderson, Equations of state ﬁts to the lower mantle and outer core, Phys. Earth Planet. Inter., 17, 147-162, 1978. Chao, E. C. T., H. T. Evans, B. J. Skinner, and C. Milton, Neighborite, NaMgF 3, a new mineral from the Green River Formation, South Ouray, Utah, Am. Mineral., 46,379-393, 1961. Cohen, R. E., Elasticity and equation of state of MgSiO 3 perovskite, Geophys. Res. Lett., 14, 1053-1056, 1987. Cohen, R. E., L. L. Boyer, M. J. Mehl, W. E. Pickett, and H. Krakauer, Electronic structure and total energy calculations for oxide perovskites and superconductors, inPerovskite: A Structure of Great Interest to Geo- physics and Materials Science, edited by A. Navrotsky and D. J. Weidner, pp. 55-66, American Geophysical Union, Washington, D.C., 1989. Cohen, R. E., O. G¨ ulseren, and R. J. Hemley, Accuracy of equation-of-state formulations, Am. Mineral., 85, 338- 344, 2000. Dubrovinsky, L., S. K. Saxena, R. Ahuja, and B. Jo- hansson, Theoretical study of the stability of MgSiO 3- perovskite in the deep mantle, Geophys. Res. Lett., 25, 4253-4256, 1998. Duﬀy, T. S., and T. J. Ahrens, Thermal properties of perovskite above 100 GPa from shock and static com- pression data, Eos Trans. AGU, 74, 571, 1993. Fei, Y., H. K. Mao, and B. O. Mysen, Experimental de- termination of element partitioning and calculation of phase relations in the MgO-FeO-SiO 2system at high pressure and high temperature, J. Geophys. Res., 96, 2157-2169, 1991. Fiquet, G., D. Andrault, A. Dewaele, T. Charpin, M. Kunz, and D. Ha¨ usermann, P-V-T equation of state of MgSiO 3perovskite, Phys. Earth Planet. Inter., 105, 21-31, 1998. Fiquet, G., A. Dewaele, D. Andrault, M. Kunz, and T. Le Bihan, Thermoelastic properties and crystal struc- ture of MgSiO 3perovskite at lower mantle pressure and temperature conditions, Geophys. Res. Lett., 27, 21-24, 2000. Funamori, N., T. Yagi, W. Utsumi, T. Kondo, and T. Uchida, Thermoelastic properties of MgSiO 3perovskite determined by in situ x ray observations up to 30 GPa and 2000 K, J. Geophys. Res., 101, 8257-8269, 1996. Gear, C. W., Numerical Initial Value Problems in Or- dinary Diﬀerential Equations, 253 pp., Prentice-Hall, Englewood Cliﬀs, N.J., 1971. Gordon, R. G., and Y. S. Kim, Theory for the forces between closed-shell atoms and molecules, J. Chem. Phys., 56, 3122-3133, 1972.8 Hama, J., and K. Suito, Equation of state of MgSiO 3per- ovskite and its thermoelastic properties under lower mantle conditions, J. Geophys. Res., 103, 7443-7462, 1998a. Hama, J., and K. Suito, High-temperature equation of state of CaSiO 3perovskite and its implications for the lower mantle conditions, Phys. Earth Planet. Inter., 105,33-46, 1998b. Hedin, L., and B. I. Lundqvist, Explicit local-exchange correlation potentials, J. Phys. C., 4, 2064-2083, 1971. Heinz, D. L., and R. Jeanloz, Measurement of the melt- ing curve of Mg 0.9Fe0.1SiO3at lower mantle conditions and its geophysical implications, J. Geophys. Res., 92, 11,437-11,444, 1987. Hemley, R. J., and R. E. Cohen, Structure and bonding in the deep mantle and core, Phil. Trans. R. Soc. Lond. A, 354, 1461-1479, 1996. Hemley, R. J., M. D. Jackson, and G. G. Gordon, Theoret- ical study of the structure, lattice dynamics, and equa- tions of state of perovskite-type MgSiO 3and CaSiO 3, Phys. Chem. Minerals, 14, 2-12, 1987. Inbar, I., and R. E. Cohen, High pressure eﬀects on ther- mal properties of MgO, Geophys. Res. Lett., 22, 1533- 1536, 1995. Irifune, T., Absence of an aluminous phase in the upper part of the Earth’s lower mantle, Nature, 370, 131-133, 1994. Jackson, I., and S. M. Rigden, Analysis of P-V-T data: constraints on the thermoelastic properties of high- pressure minerals, Phys. Earth Planet. Inter., 96, 85- 112, 1996. Jeanloz, R., Universal equation of state, Phys. Rev. B Condens. Matter ,38,805-807, 1988. Jeanloz, R., and A. B. Thompson, Phase transitions and mantle discontinuities, Rev. Geophys. Space Phys., 21, 51-74, 1983. Karki, B. B., and L. Stixrude, Seismic velocities of major silicate and oxide phases of the lower mantle, J. Geo- phys. Res., 104, 13,025-13,033, 1999. Karki, B. B., L. Stixrude, S. J. Clark, M. C. Warren, G. J. Ackland, and J. Crain, Elastic properties of orthorhom- bic MgSiO 3perovskite at lower mantle pressures, Am. Mineral., 82, 635-638, 1997. Kato, T., E. Ohtani, H. Morishima, D. Yamazaki, A. Suzuki, M. Suto, T. Kubo, T. Kikegawa, and O. Shi- momura, In situ x ray observation of high-pressure phase transitions of MgSiO 3and thermal expansion of MgSiO 3perovskite at 25 GPa by double-stage mul- tianvil system, J. Geophys. Res., 100, 20,475-20,481, 1995. Katsura, T., and E. Ito, Determination of Fe-Mg parti- tioning between perovskite and magnesiow¨ ustite, Geo- phys. Res. Lett., 23, 2005-2008, 1996.Kennett, B. L. N., E. R. Engdahl, and R. Buland, Con- straints on seismic velocities in the Earth from travel- times, Geophys. J. Int. ,122,108-124, 1995. Knittle, E., and R. Jeanloz, Synthesis and equation of state of (Mg,Fe)SiO3 perovskite to over 100 gigapas- cals,Science, 235, 668-670, 1987. Knittle, E., and R. Jeanloz, Melting curve of (Mg,Fe)SiO3 perovskite to 96 GPa: Evidence for a structural tran- sition in lower mantle melts, Geophys. Res. Lett., 16, 421-424, 1989. Kudoh, Y., E. Ito, and H. Takeda, Eﬀect of pressure on the crystal structure of perovskite-type MgSiO 3,Phys. Chem. Minerals, 14, 350-354, 1987. Lewis, L. J., and Y. L´ epine, Structural transitions in ﬂuorine-based perovskites: A molecular dynamics study of KMnF 3,Phys. Rev. B, 40, 3319-3322, 1989. Mao, H. K., R. J. Hemley, Y. Fei, J. F. Shu, L. C. Chen, A. P. Jephcoat, and Y. Wu, Eﬀect of pressure, tem- perature, and composition on lattice parameters and density of (Fe,Mg)SiO 3-perovskites to 30 GPa, J. Geo- phys. Res., 96, 8069-8079, 1991. Mao, H. K., G. Shen, and R. J. Hemley, Multivariable dependence of Fe-Mg partitioning in the lower mantle, Science, 278, 2098-2100, 1997. Martinez, I., Y. Wang, F. Guyot, R. C. Liebermann, and J. C. Doukhan, Microstructures and iron partitioning in (Mg,Fe)SiO 3perovskite - (Mg,Fe)O magnesiow¨ ustite assemblages: An analytical transmission electron mi- croscopy study, J. Geophys. Res., 102, 5265-5280, 1997. Martyna, G. J., D. J. Tobias, and M. L. Klein, Constant pressure molecular dynamics algorithms, J. Chem. Phys., 101, 4177-4189, 1994. Masuda, K., and O. L. Anderson, The isentropic density proﬁle of silicate perovskite computed by thermal pres- sure,Geophys. Res. Lett., 22, 2211-2214, 1995. Matsui, M., Molecular dynamics study of MgSiO 3per- ovskite, Phys. Chem. Minerals, 16, 234-238, 1988. Matsui, M., and G. D. Price, Simulation of the pre-melting behaviour of MgSiO 3perovskite at high pressures and temperatures, Nature, 351, 735-737, 1991. McCammon, C., Perovskite as a possible sink for ferric iron in the lower mantle, Nature, 387, 694-696, 1997. McCammon, C., The crystal chemistry of ferric iron in Fe 0.05Mg0.95SiO3perovskite as determined by M¨ ossbauer spectroscopy in the temperature range 80- 293 K, Phys. Chem. Minerals, 25, 292-300, 1998. Meade, C., H. K. Mao, and J. Hu, High-temperature phase transition and dissociation of (Mg,Fe)SiO 3per- ovskite at lower mantle pressures, Science, 268, 1743- 1745, 1995. Montagner, J.-P., and B. L. N. Kennett, How to reconcile body-wave and normal-mode reference earth models, Geophys. J. Int. ,125,229-248, 1996.9 Nos´ e, S., and M. L. Klein, Structure and dynamics of the ﬂuorperovskite, RbCaF 3,J. Chem. Phys., 90, 5005- 5010, 1989. O’Neill, B., and R. Jeanloz, MgSiO 3-FeSiO 3-Al2O3in the Earth’s lower mantle: Perovskite and garnet at 1200 km depth, J. Geophys. Res., 99, 19,901-19,915, 1994. Patel, A., G. D. Price, M. Matsui, J. P. Brodholt, and R. J. Howarth, A computer simulation approach to the high pressure thermoelasticity of MgSiO 3perovskite, Phys. Earth Planet. Inter., 98, 55-63, 1996. Poirier, J. P., Lindemann law and the melting temperature of perovskites, Phys. Earth Planet. Inter., 54, 364-369, 1989. Ringwood, A. E., Composition and Petrology of the Earth’s Mantle, 618 pp., McGraw-Hill, New York, 1975. Ross, N. L., and R. M. Hazen, High-pressure crystal chem- istry of MgSiO 3perovskite, Phys. Chem. Minerals, 17, 228-237, 1990. Saxena, S. K., L. S. Dubrovinsky, P. Lazor, Y. Cerenius, P. H¨ aggkvist, M. Hanﬂand, and J. Hu, Stability of per- ovskite (MgSiO 3) in the Earth’s mantle, Science, 274, 1357-1359, 1996. Saxena, S. K., L. S. Dubrovinsky, P. Lazor,, and J. Hu, In situ X-ray study of perovskite (MgSiO 3): Phase tran- sition and dissociation at mantle conditions, Eur. J. Mineral., 10, 1275-1281, 1998. Saxena, S. K., L. S. Dubrovinsky, F. Tutti, and T. Le Bi- han, Equation of state of MgSiO 3with the perovskite structure based on experimental measurement, Am. Mineral., 84, 226-232, 1999. Serghiou, G., A. Zerr, and R. Boehler, (Mg,Fe)SiO 3- perovskite stability under lower mantle conditions, Sci- ence, 280, 2093-2095, 1998. Shim, S.-H., and T. S. Duﬀy, Constraints on the P-V-T equation of state of MgSiO 3perovskite, Am. Mineral., 85,354-363, 2000. Stacey, F. D., Thermoelasticity of (Mg,Fe)SiO 3perovskite and a comparison with the lower mantle, Phys. Earth Planet. Inter., 98, 65-77, 1996. Stixrude, L., and M. S. T. Bukowinski, Stability of (Mg,Fe)SiO 3perovskite and the structure of the lower- most mantle, Geophys. Res. Lett., 19, 1057-1060, 1992. Stixrude, L., and R. E. Cohen, Stability of orthorhombic MgSiO 3perovskite in the Earth’s lower mantle, Nature, 364,613-616, 1993. Stixrude, L., R. J. Hemley, Y. Fei, and H. K. Mao, Ther- moelasticity of silicate perovskite and magnesiow¨ ustite and stratiﬁcation of the Earth’s mantle, Science, 257, 1099-1101, 1992. Utsumi, W., N. Funamori, T. Yagi, E. Ito, T. Kikegawa, and O. Shimomura, Thermal expansivity of MgSiO 3 perovskite under high pressures up to 20 GPa, Geophys. Res. Lett., 22, 1005-1008, 1995.Vinet, P., J. Ferrante, J. H. Rose, and J. R. Smith, Com- pressibility of solids, J. Geophys. Res., 92, 9319-9325, 1987. Wang, Y., D. J. Weidner, and F. Guyot, Thermal equation of state of CaSiO 3perovskite, J. Geophys. Res., 101, 661-672, 1996. Wang, Y., D. J. Weidner, R. C. Liebermann, and Y. Zhao, P-V-T equation of state of (Mg,Fe)SiO 3perovskite: Constraints on composition of the lower mantle, Phys. Earth Planet. Inter., 83, 13-40, 1994. Weidner, D. J., and Y. Wang, Chemical- and Clapeyron- induced buoyancy at the 660 km discontinuity, J. Geo- phys. Res., 103, 7431-7441, 1998. Wentzcovitch, R. M., N. L. Ross, and G. D. Price, Ab ini- tio study of MgSiO 3and CaSiO 3perovskites at lower- mantle pressures, Phys. Earth Planet. Inter., 90, 101- 112, 1995. Wolf, G. H., and M. S. T. Bukowinski, Theoretical study of the structural properties and equations of state of MgSiO 3and CaSiO 3perovskites: Implications for lower mantle composition, in High-Pressure Research in Mineral Physics, edited by M. H. Manghnani and Y. Syono, pp. 313-331, American Geophysical Union, Washington, D.C., 1987. Wolf, G. H., and M. S. T. Bukowinski, Variational sta- bilization of the ionic charge densities in the electron- gas theory of crystals: Applications to MgO and CaO, Phys. Chem. Minerals, 15, 209-220, 1988. Wood, B. J., Phase transformations and partitioning rela- tions in peridotite under lower mantle conditions, Earth Planet. Sci. Lett., 174, 341-354, 2000. Wood, B. J., and D. C. Rubie, The eﬀect of alumina on phase transformations at the 660 kilometer discontinu- ity from Fe-Mg partitioning eﬀects, Science, 273, 1522- 1524, 1996. Zhang, J., and D. J. Weidner, Thermal equation of state of aluminum-enriched silicate perovskite, Science, 284, 782-784, 1999. Zhao, Y., and D. L. Anderson, Mineral physics constraints on the chemical composition of the Earth’s lower man- tle,Phys. Earth Planet. Inter., 85, 273-292, 1994. R. E. Cohen, J. Ita, and F. C. Marton, Geophysical Laboratory and Center for High Pressure Research, Carnegie Institution of Washington, 5251 Broad Branch Road, NW, Washington, DC 20015-1305. (e- mail: cohen@gl.ciw.edu; j.j.ita@siep.shell.com; mar- ton@gl.ciw.edu) 1Now at Shell International E&P, Rijswijk, The Netherlands. This preprint was prepared with AGU’s L ATEX macros v5.01. File pv formatted September 26, 2013.10 0 20 40 60 80 100 120 14005001000150020002500300035004000 Fig. 1 T (K) P (GPa) Figure 1. Experimental pressure-temperature coverage of (Mg,Fe)Si O3perovskite. Open symbols are for static experiments on X Mg= 1.0, ﬁlled are for X Mg= 0.9, and open symbols with crosses are shock wave data. Hexa gons: Kudoh et al., [1987]; right-facing triangles: Ross and Hazen, [1990]; left-facing triangles: Wang et al., [1994]; upward-pointing triangles: Fiquet et al., [1998]; downward-pointing triangles: Fiquet et al., [2000]; diamonds: Saxena et al., [1999]; open circles: Utsumi et al., [1995]; ﬁlled circles: Knittle and Jeanloz, [1987]; squares: Funamori et al., [1996]; stars: Mao et al., [1991]; pentagons: Kato et al., [1995]; circle with cross: Duﬀy and Ahrens, [1993]; square with cross: Akins and Ahrens, [1999]. Solid curve is the lower mantle geothermal gradient ofBrown and Shankland [1981]. Static experimental work has direct coverage of app roximately the upper one-third. The shock wave data point that falls near the geotherm is perovskite + m agnesiow¨ ustite.11 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15-0.35-0.30-0.25-0.20-0.15-0.10-0.050.000.050.10 Fig. 2 Energy (Ry) VIBLAPW118130142154166 Å3 δMδR Figure 2. Energetics of octahedral rotations. Calculated total ener gies relative to the cubic structure ( Pm3m ) using VIB (lines) and LAPW [ Stixrude and Cohen, 1993;Hemley and Cohen, 1996] (symbols) as a function of R- (left) and M-point (right) rotation angles as represented b y the fractional change in the oxygen coordinate ( δRand δM). The orthorhombic structure ( Pbnm ) occurs at δR= 0.0912 and δM= 0.0766 at P = 0 GPa.12 0 20 40 60 80 100 120 1401.0051.0101.0151.0201.0251.0301.035Fig. 3a 3000 K300 K b/a P (GPa) 0 20 40 60 80 100 120 1401.4201.4251.4301.4351.4401.4451.450 300 K 3000 KFig. 3b c/a P (GPa)13 -20 0 20 40 60 80 100 120 140 160120130140150160170180 300 K3000 KFig. 3c V (Å3) P (GPa) Figure 3. Structural parameters and volumes of MD results: (a) b/a axi s ratio and (b) c/a axis ratio. At low pressures and high temperatures the structure begins to dev iate from orthorhombic Pbnm , but no phase transitions, with the possibility of the structure heading towards cubic at 12 GPa and 3000 K. (c) volume. Symbols are MD results and curves are Universal EoS ﬁts every 300 K.14 500 1000 1500 2000 2500 30001.0x10-51.5x10-52.0x10-52.5x10-53.0x10-53.5x10-5Fig. 4 0 GPa (exp) 50 GPa12 GPa0 GPa 140 GPa25 GPa 25 GPa (exp) α (K-1) T (K) Figure 4. Volume coeﬃcient of thermal expansion. Solid lines are this work. Experimental work: Wang et al. [1994]: dashed line (0 GPa); Fiquet et al. [1998]: circle (0 GPa); Funamori et al. [1996]: square (0 GPa) and dashed-dotted line (25 GPa).15 0 500 1000 1500 2000 2500 3000 3500 40003.03.54.04.55.0 140 GPa50 GPa25 GPa12 GPa0 GPa Fig. 5δΤ T (K) Figure 5. Anderson-Gr¨ uneisen parameter as a function of temperatur e. The circle is the experimentally determined value from Wang et al. [1994]. The square is the value determined by Masuda and Anderson [1995] from the experimental data of Utsumi et al. [1995]. Open circles are δTs determined from Debye theory by Anderson [1998].16 0 500 1000 1500 2000 2500 3000 3500 40001.01.21.41.6 12 GPa0 GPa 25 GPa 50 GPa 140 GPa Fig. 6γ T (K) Figure 6. Gr¨ uneisen parameter as a function of temperature. The soli d circle is the experimentally determined value from Wang et al. [1994] and the square is the value determined by Masuda and Anderson [1995] from the experimental data of Utsumi et al. [1995]. Other symbols are from inversion of multiple experi mentally determined data sets: Bina[1995] (triangle) Jackson and Rigden [1996]) inverted triangle, and Shim and Duﬀy 2000] (diamond). Open circles are γs determined from Debye theory by Anderson [1998].17 4.8 4.9 5.0 5.1 5.20.00.10.20.30.4Fig. 7 ln γ ln V (Å3) 300 K q = 1.03 1200 K q = 0.98 2100 K q = 1.02 3000 K q = 1.07 Figure 7. lnγversus ln V. The slope of each line gives q. Solid line and squares, 300 K; dashed line and circles, 1200 K; dotted line and triangles, 2100 K; dash-dotted line a nd diamonds, 3000 K. The lower temperatures are all close to 1, with a slight rise to 1.07 at 3000 K.18 20 40 60 80 100 120 140120130140150160Fig. 8 V (Å3) P (GPa)1000 1500 2000 2500 All ExpHigh T ExpExpMD w/ corrected VMD Depth (km) Figure 8. Volumes of MgSiO 3perovskite along the geothermal gradient of Brown and Shankland [1981]. Dashed- dotted curve is volume from our Universal EoS ﬁt. Dashed curv e is from our Universal ﬁt with V 0ﬁxed to 162.49 ˚A3[Mao et al., 1991]. Dotted curve is for experimental data sets with V 0ﬁxed to 162.49 ˚A3(see Table 2). Both high temperature (F98+F00+S99) and all MgSiO 3data data sets fall on the same line. The shaded areas indicat e the uncertainties in the ﬁts of the experimental data sets.19 1000 1500 2000 25004.04.55.05.56.0Fig. 9a V uncorrectedV correctedak135-f ρ (g/cm3) 1000 1500 2000 25008.08.59.09.510.010.511.011.512.0 Fig. 9bVϕ (km/s) Depth (km) Figure 9. (a) Densities and (b) bulk sound velocities of MgSiO 3perovskite calculated along the lower mantle geotherm of Brown and Shankland [1981] using the thermal Universal EoS. Dashed line is for V0set to the ex- perimental value of Mao et al., [1991]. Circles are from the ak135-f seismological model [ Kennett et al., 1995; Montagner and Kennett, 1996].20 1000 1500 2000 25004.04.55.05.56.0 Xpv Fe = 0.05 Xpv Fe = 0.00Xpv Fe = 0.10ak135-fFig. 10a ρ (g/cm3) 1000 1500 2000 25008.08.59.09.510.010.511.011.512.0Fig. 10b Vϕ (km/s) Depth (km) Figure 10. (a) Densities and (b) bulk sound velocities of (Mg,Fe)SiO 3perovskite calculated along the geotherm ofBrown and Shankland [1981] using the experimental value of V0Mao et al., [1991] for Xpv Fe= 0.00 (solid), 0.05 (dashed), and 0.10 (dotted). Circles are from the ak135-f se ismological model [ Kennett et al., 1995;Montagner and Kennett, 1996].21 1000 1500 2000 25004.04.55.05.56.0 ak135-fpyrolite1 pyrolite2 Fig. 11a ρ (g/cm3) 1000 1500 2000 25008.08.59.09.510.010.511.011.512.0 Fig. 11b Vϕ (km/s) Depth (km) Figure 11. (a) Densities and (b) bulk sound velocities of two simpliﬁed pyrolite models (67 mol% pv, 33 mol% mw) along the geotherm of Brown and Shankland [1981]. Solid: Xpv Mg= 0.93, Xmw Mg= 0.82. Dashed: Xpv Mg= 0.96, Xmw Mg= 0.86. Circles are from the ak135-f seismological model [ Kennett et al., 1995;Montagner and Kennett, 1996].22 Table 1. MD P-V-T Data Used in EoS Fits P (GPa) T (K) a ( ˚A) b ( ˚A) c ( ˚A) V ( ˚A3) 0 300 4.8166 4.9283 6.9649 165.33 0 600 4.8299 4.9397 6.9787 166.50 0 900 4.8439 4.9519 6.9944 167.77 0 1200 4.8586 4.9648 7.0107 169.11 0 1500 4.8732 4.9769 7.0289 170.48 0 1800 4.8886 4.9901 7.0455 171.87 0 2100 4.9109 5.0145 7.0578 173.80 0 2400 4.9311 5.0354 7.0688 175.52 0 2700 4.9525 5.0538 7.0831 177.28 0 3000 4.9672 5.0633 7.1076 178.76 12 900 4.7790 4.8856 6.8895 160.86 12 1200 4.7917 4.8951 6.9024 161.90 12 1500 4.8036 4.9051 6.9165 162.97 12 1800 4.8169 4.9157 6.9288 164.06 12 2100 4.8295 4.9256 6.9441 165.18 12 2400 4.8429 4.9367 6.9586 166.36 12 2700 4.8628 4.9539 6.9690 167.88 12 3000 4.9035 4.9531 6.9695 169.27 25 300 4.6971 4.8093 6.7728 152.99 25 600 4.7073 4.8169 6.7830 153.80 25 900 4.7181 4.8250 6.7948 154.68 25 1200 4.7285 4.8330 6.8055 155.53 25 1500 4.7390 4.8409 6.8166 156.38 25 1800 4.7497 4.8487 6.8291 157.27 25 2100 4.7609 4.8572 6.8408 158.19 25 2400 4.7728 4.8658 6.8514 159.11 25 2700 4.7850 4.8736 6.8645 160.08 25 3000 4.8015 4.8859 6.8773 161.34 50 900 4.6194 4.7307 6.6445 145.20 50 1200 4.6280 4.7366 6.6531 145.84 50 1500 4.6366 4.7414 6.6632 146.48 50 1800 4.6450 4.7474 6.6726 147.14 50 2100 4.6545 4.7524 6.6821 147.81 50 2400 4.6627 4.7589 6.6920 148.49 50 2700 4.6728 4.7644 6.7014 149.20 50 3000 4.6814 4.7700 6.7129 149.90 140 300 4.3572 4.5023 6.2698 123.00 140 600 4.3627 4.5046 6.2768 123.35 140 900 4.3687 4.5076 6.2839 123.74 140 1200 4.3747 4.5097 6.2903 124.10 140 1500 4.3806 4.5121 6.2967 124.46 140 1800 4.3867 4.5146 6.3027 124.82 140 2100 4.3923 4.5164 6.3104 125.18 140 2400 4.3986 4.5186 6.3170 125.5523 Table 1. (continued) P (GPa) T (K) a ( ˚A) b ( ˚A) c ( ˚A) V ( ˚A3) 140 2700 4.4048 4.5209 6.3235 125.92 140 3000 4.4109 4.5230 6.3309 126.3124 Table 2. Equation of State Parameters V0(˚A3) K T0(GPa) K′ T0a (GPa) b (GPa/K) χ2(GPa) This work B-M 165.40(6) 274(1) 3.73(2) -2.00(2) 0.00667(5) 0.08561 Universal 165.40(5) 273(1) 3.86(2) -1.99(1) 0.00664(5) 0. 06712 Exp M91 162.49(7) 261(4) 4† F98 162.65(15) 262(6) 3.41(22) -1.79(21) 0.00597(70) 1.47 438 162.49∗267(3) 3.25(10) -1.79(21) 0.00597(70) 1.41566 F00 162.74(56) 239(8) 4.41(18) -1.85(6) 0.00618(20) 0.610 52 162.49∗245(4) 4.29(11) -1.85(6) 0.00618(20) 0.59369 S99 163.25(63) 242(6) 4.45(10) -1.72(1) 0.00574(2) 0.2671 9 162.49∗256(5) 4.20(10) -1.72(1) 0.00572(2) 0.26227 F98 + F00 + S99 162.93(33) 235(6) 4.72(13) -1.85(12) 0.00618 (38) 1.56307 162.49∗244(4) 4.52(13) -1.86(12) 0.00621(40) 1.55478 All 162.31(10) 249(6) 4.40(20) -1.92(10) 0.00639(32) 0.60 337 162.49∗243(8) 4.55(28) -1.90(10) 0.00635(34) 0.61628 Note: MD are best ﬁt EoS parameters to molecular dynamics P-V-T da ta. Exp are best ﬁt Universal EoS parameters to experimental data sets, except data taken from Mao et al. [1991] (M91). References: F98: Fiquet et al. [1998] (27 data points), F00: Fiquet et al. [2000] (38 data points), S99: Saxena et al. [1999], (37 data points) All: all X Mg= 1.0 data (363 data points; see Figure 1).∗Denotes ﬁxed to V 0fromMao et al. [1991].†K′ T0ﬁxed equal to 4.25 Table 3. Zero Temperature Equation of State Parameters V (˚A3) K (GPa) K′ This work∗164.22 280 3.84 PIB∗∗164.78 252 4.05 LAPW†160.74 266 4.2 PWPP‡157.50 259 3.9 Note:∗Universal ﬁt. Other theoretical ﬁts are third order Birch-Murnaghan ﬁts.∗∗Cohen [1987],†Stixrude and Cohen [1993],‡Wentzcovitch et al. [1995].
arXiv:physics/0011001v1 [physics.chem-ph] 1 Nov 2000Application of time-dependent density functional theory to electron-vibration coupling in benzene A. Schnell, G. F. Bertsch∗ Institute for Nuclear Theory, University of Washington, Se attle, WA 98125, USA K. Yabana Institute of Physics, University of Tsukuba, Tsukuba 305-8 571, Japan February 2, 2008 Abstract Optical properties of symmetry-forbidden π-π∗transitions in benzene are calculated with the time-dependent density functional theory (TDDFT) , using an adiabatic LDA functional. Quantities calculated are the envelopes of the Franck-Condon factors of the vibrationally promoted transitions and the associated oscillator strengths. The strengths, which span three orders of magnitude, are reprod uced to better than a factor of two by the theory. Comparable agreement is found for the Fr anck-Condon widths. We conclude that rather detailed information can be obtaine d with the TDDFT and it may be worthwhile to explore other density functionals. The time-dependent density functional theory (TDDFT) has p roven to be a surprisingly successful theory of excitations and particularly the opti cal absorption strength function. The theory is now being widely applied in both chemistry and i n condensed matter physics. The literature in quantum chemistry is cited in a recent stud y on the electronic excitations in benzene [1]. Benzene is an interesting molecule for testi ng approximations because its spectra have been very well characterized, both electronic and vibrational. In this note we will apply the TDDFT to coupling between vibrational and ele ctronic excitations. In our previous studies, we have investigated many diﬀerent elect ronic structure questions using a rather simple version of the density functional theory, the local density approximation (LDA). Our emphasis has been to study the overall predictive power o f a ﬁxed functional rather than to try to ﬁnd the best functional for each properties. The app roximation scheme we consider is straightforward and uses the same computer programs as fo r calculating purely electronic excitations. We treat the electronic dynamics in the adiaba tic approximation, taking the same energy function for the dynamic equation as is used in th e static structure calculation. In our view, this is the only consistent scheme available tha t guarantees conservation of the oscillator sum rule. The electron-vibration coupling is tr eated in a vertical approximation, so only information at frozen nuclear coordinates is requir ed. ∗E-mail: bertsch@phys.washington.edu 10 2 4 6 excitation energy [eV] 1A1g1B2u1B1u1E1uExp. TDLDA CASSCF-CCI Heinze et al. Figure 1: Electronic excitations of benzene in the π-π∗manifold. Besides the experimental data and the present TDDFT, we show the TDDFT of ref. [1] and th e CI calculation of ref. [2] We consider only spin-singlet states in this work and drop th e spin designation in labeling the states. Empirically, the lowest states derive from the π-π∗manifold, exciting an electron from the two-fold degenerate e1gHOMO orbital to the two-fold degenerate e2uorbital. The four states consist of a strongly absorbing two-fold degene rateE1uexcitation and two other states, B1uandB2u, for which symmetry forbids any transition strength. This b asic spectrum is shown in Fig. 1, comparing also with our TDDFT calculation , the TDDFT calculation of ref. [1], and the CI theory of ref. [2]. It is seen that the TDDFT gives an excellent account of the ene rgies. In fact the TDDFT gives a good description of the higher frequency absorption including σ-σ∗transitions as well [3]. The detailed optical properties of the three trans itions have been studied gas phase absorption [4, 5]. The strong transition is the E1uwithf= 0.9−0.95. The B1umode is seen as a shoulder on the strong E1upeak. Its total transition strength is about a factor of 10 lower than the strong state; ref. [4] quotes a value f= 0.09. The B2utransition is very weak and is seen as a partially resolved set of vibrational tr ansitions with a total strength about f≈1.3×10−3[4]. The strength associated with the most prominent resolv ed states is 0.6×10−3[5]. The vibrational couplings of the Bstates has been recently studied using the CASSCF method and analytic expressions for the linear coupling to v ibrations[6], and we shall compare with their results. The TDDFT includes correlation eﬀects i n a diﬀerent way, and has some well-known advantages such as the automatic conservation o f required sum rules. Also, as mentioned earlier, the present method does not require any r eprogramming. For our treatment of the vibrational motion, we assume that t he the vibrations are har- 2monic in the electronic ground state. The Hamiltonian may be deﬁned H=−3N/summationdisplay i¯h2 2Mim∂2 ∂q2 i+1 23N/summationdisplay ijFijqiqj. (1) where qiare the 36 Cartesian displacement coordinates of the 12 atom ic centers, mis the atomic mass unit, Miis the mass of the atom in daltons, and Fijis the matrix of force con- stants. The matrix M−1/2FM−1/2(Mis the diagonal matrix of masses Mi) is diagonalized by an orthogonal transformation Uto obtain the normal modes Qkand the eigenfrequencies ωk= 2πνk. The Cartesian displacements are obtained directly from th e rows of the trans- formation matrix U,qi=M−1/2 i/summationtext kUikQk. The translational and rotational motions will also be contained in the transformation matrix Uas zero frequency modes. The probability distribution of the zero point motion is then given simply by P(/vectorQ)∼exp(−/summationdisplay kQ2 k/2Q2 0k). (2) where Q0k=/radicalBigg ¯h 2mωk=4.1[˚A]√ncm is the r.m.s. amplitude of the zero-point motion1. The last equality expresses the formula in common units with ncm=c/νthe energy of the vibration in wavenumbers [cm−1]. The optical absorption strength function in the presence of the zero point motion is determined by the convolution the probability distributio n of displacements with the strength calculated as a function of displacement, f=/integraldisplay dNQkP(/vectorQ)f(/vectorQ). (3) We thus need the absorption strength as a function of the norm al mode coordinates Qk. In the case of a forbidden transition promoted by the vibration k, the coupling is linear for small displacements and the transition strength will be quadrati c inQk, f(Qk) =f0kQ2 k Q2 0k+... (4) We verify below that this functional dependence is satisﬁed for the couplings of interest in benzene. Then the convolution over the ground state probabi lity distribution gives simply f=f0k. We also consider widths of the transitions due to the Franck- Condon factors of multiply excited vibrations. This is calculated by replacing fwith the strength function S(E, Q) = f(Q)δ(E−E(Q)) in eq. 3. Assuming that the excitation energy is linear in Q, E(Qk)≈E0+KkQk/Q0k+... the Gaussian probability distribution Pgives a Gaussian envelope for the Franck-Condon factors, P(E)∼exp(−(E−E0)2/2K2 k). (5) 1At ﬁnite temperature the r.m.s. amplitude is increased by a f actor 1 //radicalbig tanh(¯hωk/2kBT). 3Figure 2: Cartesian displacements of the vibrational modes 6a (left) and 8a (right). These modes have symmetry E2gand give the most important couplings for our purposes. The r.m.s. displacement of the atoms are magniﬁed by a factor 40, i.e.Qk= 40Q0kwith respect to the scale for the equilibrium positions. For the numerical studies reported here, we constructed the transformation matrix U using the empirical force ﬁeld of Goodman and Ozkabak [7], wh ich ﬁts the observed frequen- cies extremely well. Ab initio calculations of the force ﬁeld have also reached a high level of accuracy [9]. However, as mentioned earlier, we do not make o ur own DFT calculations of the force constants because our goal is the dynamic behavior of t he electrons. The frequencies and symmetries of the normal modes are listed in Table 1, take n from ref. [7]. The most important modes for the induced transition strengths are th eE2gand the B2gmodes2. The E2gvibrations couple the strong electronic excitation to the o ther states in the π-π∗mani- fold. The B1gcan induce out-of-plane dipole strength for these excitati ons. The theoretical widths of the excitations are largely due to mode 1, which is a nA1gradial oscillation mode that favors carbon displacements. In Fig. 2 we show the Carte sian displacements associated with the two strongest E2gmodes with respect to carbon displacements. The present TDDFT calculations were performed making use of the same representation of the Kohn-Sham operator as in our previous study of the full energy distribution in optical absorption [3]. The wave functions are represented on a coor dinate-space mesh as has been introduced in condensed matter physics [ ?]. However, the algorithm in the present program is a new one [10] that uses the conjugate gradient method to ex tract individual states rather that the direct real-time propagation of the wave function. While the real-time method is very eﬃcient for calculating the global strength function, it is less suited for locating individual eigenstates when they are weakly excited by the dipole opera tor. In both methods, the electronic ground state for a given nuclear geometry is ﬁrst computed with the Kohn-Sham 2TheB1gsymmetry would also give couplings between the electronic s tates, but there are no vibrations of that symmetry. 4-3 -2 -1 0 1 2 3 Q/Q000.050.10.150.20.25f Figure 3: Dependence of the oscillator strength of the1B2u←1A1gtransition on the vibrational coordinate for the 8a mode.-0.2 -0.1 0 0.1 0.2 Q/Q04.955.15.25.35.45.5E [eV] Figure 4: Dependence of the1B1u←1A1g transition energy on the vibrational coor- dinate for the mode 1. equation, −∇2 2mφi+δV δnφi=ǫiφi. We use a simple LDA energy density functional [12] for the ele ctron-electron interaction in V and a pseudopotential approximation [13, 14] to treat the in teraction of the valence electrons with the ions. The important numerical parameters in the cal culation are the the mesh spacing, taken as ∆ x= 0.3˚A, and the volume in which the wave functions are calculated, which we take as a sphere of radius 7 ˚A. With these parameters, orbital energies are converged to better than 0.05 eV. Next the TDDFT equations are solved in an representation similar to the RPA equations, −∇2 2mφ± i+δV δnφ± i−ǫiφi+δ2V δn2δnφi= (ǫi±ω)φ± i. Here the transition density δnand normalization are given by δn=/summationdisplay iφi(φ+ i+φ− i),/angb∇acketleftφ+ i\|φ+ i/angb∇acket∇ight−/angb∇acketleftφ− i\|φ− i/angb∇acket∇ight= 1. The equations are solved by the conjugate gradient method fo r the generalized eigenvalue problem [11]. In Fig. 4 we show the dependence of transition s trengths fand excitation energies ωon the coordinates of two of the normal modes. We see that the c onditions for applying eq. (4) and (5) are reasonably well satisﬁed. We may then extract the transition strength f0kand the width Kkby ﬁtting the Qk-dependence of these quantities. The results for the symmetry-allowed vibrations are shown in shown in Ta ble 2. We ﬁrst discuss the widths. The empirical values were obtain ed by making a three-term Gaussian ﬁt to the absorption data of ref. [15]. The only vibr ations that contribute in lowest 5order are the two A1gbreathing modes. The vibrations aﬀect all three transition s identically; mode 1 has the larger amplitude of displacement of the carbon atoms and gives the greater contribution. The results agree rather well with the empiri cal widths. The magnitude of the widths and its independence of the electronic state can be un derstood in very simple terms with the Hueckel model. This is to be expected, since the exci tation energy of the electronic states is mainly due to the orbital energy diﬀerence, and tha t is describe quite well by the Hueckel model. For benzene, the energy diﬀerence is related to the hopping matrix element β byELUMO−EHOMO = 2β.. Allowing changes in the nuclear coordinates, the hopping m atrix element will depend on the distance between neighboring ato msd; this may be parameterized by the form β(d) =β0/parenleftBiggd0 d/parenrightBiggα . Then the HOMO-LUMO gap ﬂuctuates due to the breathing mode vi brations with widths given by ∆E= 2β0α∆r r where ris the radial distance of the carbons from the center and ∆ ris atQk=Q0kin anA1gmode. From ﬁtting orbital energies in various conjugated ca rbon systems one may extract values α≈2.7 and β0= 2.5 eV[3]. Inserting these values in the above equation, one obtains 0.145 eV for the widths associated with mode 1, quite close to the values obtained by TDDFT. We have included in the table also the r.m.s. widths of the Franck-Condon factors obtained by the CASSCF theory, which gives quite similar res ults. One thing should be remarked on the comparison with experiment. While the theor y gives practically identical widths for all three states, the experimental strength is si gniﬁcantly narrower for the the E1u excitation, and this seems to not be understandable in the TD DFT. Next we examine the transition strengths of the B-transitions induced by the zero-point vibational motion. In the middle table of Table 2 we show the c ontributions by the six active vibrational modes. The main contribution for the B2utransition comes from mode 6. This is also found in the CASSCF theory, and is how the observed spe ctrum was interpreted in [5]. In the case of the B1uexcitation, the TDDFT predicts that the coupling of mode 8 is dominant. Experimentally, the situation is unclear becaus e the vibrational spectrum of the excited state is strongly perturbed. Ref. [5] assigns both m ode 6 and mode 8 vibrational involvement. Irrespective of the spectrum of the vibration al modes in the excited state, the total transition strength is given by the same convolution o f the ground state vibrational wave function. As in the case of the widths, the induced B1utransition strength can be understood roughly with the tight-binding model. The charge densities are displaced in the vibration, giving the B1uconﬁguration an induced dipole moment just from the atomic g eometry. The Hueckel Hamiltonian of the orbital energy is also aﬀected by the changed separations between carbons, and that cause a violation of the B1usymmetry. Finally, the Coulomb interaction, which is mainly responsible for the splitting of the three el ectronic states, is aﬀected by the changed separations. Of these three mechanisms, only the eﬀ ect of the symmetry-violation in the Hueckel Hamiltonian is important, and mode 8 crries the l argest ﬂutuation in d. Taking the same d-dependence as before, the strength obtained in the tight-b inding model is 0.05, rather close to the TDDFT result. The tight-binding model ca nnot be used to estimate the very weak B2utransition because the charge density on the atoms is identi cally zero. 6The lower table gives the empirical transition strengths [4 ] and comparison to theory. The agreement between theory and experiment is quite good fo r all states. For the weakest transition, the B1u, the TDDFT gives a transition strength 25% higher than the em pirical For the case of the B1utransition, the TDDFT prediction is within 35% of the measur ed value. We also show the previously reported value for the E1uwhich is within 20%. We consider this remarkable success of the TDDFT considering t hat the strengths that range over three orders of magnitude. In conclusion, we have shown that the TDDFT gives a semiquant itative account of the eﬀect of zero-point vibrational motion on the optical absor ption spectrum in benzene. In this respect this extends the possible domain of utility from the region of infrared absorption, where it is known that the TDDFT gives a description of transi tion strengths accurate to a factor of two or so[16]. We are encouraged by these results to apply the TDDFT to other problems involving the electron-vibrational coupling. Pe rhaps it should be mentioned that not all excitation properties are reproduced so well in the T DDFT. In particular, one can not expect accurate numbers for HOMO-LUMO gap of insulators [18 ] and the optical rotatory power of chiral molecules [19]. Of course, there may be bette r energy functionals for studying particular properties, and it might be interesting to exami ne theories including gradient terms in the functional. We acknowledge stimulating discussions with G. Roepke. Thi s work was supported by the Department of Energy under Grant DE-FG06-90ER40561. References [1] H. Heinze, A. Goerling, and N. Roesch, J. Chem. Phys. 113 2 088 (2000). [2] J. Mauricio O. Matos, B. O. Roos, and P- ˚A Malmqvist, J. Chem. Phys. 86, 3 (1987). [3] K. Yabana, and G. F. Bertsch, Int. J. Quant. Chem. 75, 55 (1999). [4] E. Pantos, J. Philis, A Bolovinos, Jour. Mol. Spectro. 7236 (1978). [5] A. Hiraya and K. Shobatake, J. Chem. Phys. 947700 (1991). [6] A. Bernhardsson, et al., J. Chem. Phys. 112 2798 (2000). [7] L. Goodman, A. G. Ozkabak, and S. N. Thakur, J. Phys. Chem. 95, 9044 (1991). [8] J. Chelikowsky, et al., Phys. Rev. 5011355 (1994). [9] J. M.L. Martin, P. R. Taylor, and T. J. Lee, Chem. Phys. Let t.275, 414 (1997) (and refs. therein). [10] K. Yabana, to be published. [11] W.W. Bradbury and R. Fletcher, Num. Math. 9259 (1966). [12] J. Perdew and A. Zunger, Phys. Rev. B23 5048 (1981). 7mode species ¯νobs[cm−1]mode species ¯νobs[cm−1] 1 A1g 993.1 9 E2g 1177.8 2 A1g 3073.9 18 E1u 1038.3 3 A2g 1350 19 E1u 1484.0 12 B1u 1010 20 E1u 3064.4 13 B1u 3057 11 A2u 674.0 14 B2u 1309.4 4 B2g 707 15 B2u 1149.7 5 B2g 990 6 E2g 608.1 10 E1g 847.1 7 E2g 3056.7 16 E2u 398 8 E2g 1601.0 17 E2u 967 Table 1: The 20 normal modes of vibration of benzene, numbere d according to Wilson [17], with their symmetry and observed frequency. The data of this table are taken from [7] where other references can also be found. [13] N. Troullier and J.L. Martins, Phys. Rev. B43 1993 (1991 ). [14] L. Kleinman and D. Bylander, Phys. Rev. Lett. 48 1425 (19 82). [15] H.-H. Perkampus, UV Atlas of organic compounds , (Vol. 1, Butterworth Verlag Chemie, 1968). [16] G.F. Bertsch, A. Smith, and K. Yabana, Phys. Rev. B52 787 6 (1995). [17] E. B. Wilson Jr., J. C. Decius, and Paul C. Cross, Molecular Vibrations , (McGraw-Hill, New York, 1955). [18] L. Hedin, J. Phys. Condens. Matter 11 R489 (1999). [19] K. Yabana and G.F. Bertsch, Phys. Rev. A60 1271 (1999). 8Width A1gvibrations CASSCF Exp. Kk(ev) 1 2 Tot. 1B2u0.12 0.03 0.15 0.14 0.18 1B1u0.12 0.03 0.15 0.14 0.17 1E1u0.12 0.03 0.15 0.125 f0k/10−3B2gvib. E2gvibrations TDDFT 4 5 6 7 8 9 Total 1B2u- - 1.4 0.2 - - 1.6 1B1u- 1.6 0.4 - 44. 13. 59 f/10−3TDDFT CASSCF Exp. 1B2u 1.6 0.5 1.3 1B1u 59 75 90 1E1u 1100 900-950 Table 2: Vibrational coupling properties in benzene molecu le. The upper table shows the predicted r.m.s. widths associated with the breathing mode vibrations. The total is compared to the CASSCF calculation of ref. [6] and to experiment (see t ext). In the middle table, the predicted transition strength associated with the various vibrations are given, with blank entries having values smaller than 10−4. In the lower table, the predicted total transition strength is compared with the CASSCF theory and to experimen t [4]. 9
arXiv:physics/0011002v1 [physics.bio-ph] 1 Nov 2000physics/0011002 A HISTORICAL PERSPECTIVE ON CANCER * Rafael D. Sorkin Department of Physics, Syracuse University, Syracuse, NY 1 3244-1130, U.S.A. internet address: sorkin@physics.syr.edu Abstract It is proposed that cancer results from the breakdown of univ ersal con- trol mechanisms which developed in mutual association as pa rt of the historical process that brought individual cells together into multi-cellular communities. By systematically comparing the genomes of un i-celled with multi-celled organisms, one might be able to identify the mo st promising sites for intervention aimed at restoring the damaged contr ol mechanisms and thereby arresting the cancer. More or less by deﬁnition, a cancerous cell is one that grows a nd reproduces uncon- trollably. Of course, this characterization presupposes t hat the cell in question lives within a multi-celled organism — a community of cells. The identica l behavior would appear perfectly normal for a cell existing in isolation. But there is more to cancer than uncontrolled growth; for thi s behavior occurs in association with another trait which — in itself — would not s eem to be related to growth at all: loss of diﬀerentiation. Why should a “histologic” ch aracteristic like (lack of) dif- ferentiation be correlated in this way with a “dynamical” ch aracteristic like growth rate? I believe that, far from being necessary for reasons of bioch emistry or cellular dynamics, this association is historical in origin. Let us try to imagine the transformation that took place when the ﬁrst multicellular organisms coalesced from collections of cells living in rel ative isolation from each other. * The main idea exposed herein occurred to me some 20 years ago . I tried once to get it published, but failed. After some hesitation, I am postin g it here in the hope that the historical viewpoint it utilizes may still be of some use to p eople trying to understand cancer. 1Aside from the mutual “solidarity” of its members, such a com munity of cells has the advantage that diﬀerent cells can specialize in diﬀerent ta sks and collectively accomplish all these tasks more eﬀectively than any individual cell cou ld by itself. In other words, a division of labor becomes possible (specialized tissues, specialized organs, etc.) But for such an arrangement to function, the relative multiplic ities of the cells with diﬀerent specializations must be suitable, which in practice means c ontrolling their absolute numbers as well. To participate in such a community, then, a cell need s (at least ) the ability to specialize itself for a range of functions and the ability to regulate its rate of reproduction as needed: precisely the features whose absence typiﬁes the disease of cancer. It seems clear that, in order to provide these twin abilities , new biological “machinery” would have been required, either at the cellular level, or th e supercellular level, or both. Furthermore, since the required mechanisms must have arise n in association with one another as part of the same historical process (a process pre sumably occupying an extended period of time), it would be natural for them to overlap and sh are components to a great extent. That is, there should have developed, in some measur e, only a single (and universal) mechanism responsible on one hand for regulating growth and on the other hand for producing functional diﬀerentiation. Cancer would then re sult from the breakdown of this mechanism. From this perspective, cancer would not be a dise ase like measles, caused by the presence of some pathogen or some actively harmful abnormality, but a disease of deﬁciency like scurvy, caused by the absence or the failure of some mechanism that is present in normal cells (or normal tissue). It would be the co nsequent regression , on the part of the tumor cells, to an earlier, pre-social form of beh avior. The focus for ﬁnding a cure (or for prevention) would thus shi ft from removing the abnormality to restoring the missing or damaged machinery, and a strategy of trying to destroy the defective cells would be less likely to succeed t han one of providing them with “replacement parts” for their damaged control mechani sms. How this could be done would naturally depend on the nature of the defect, but one mi ght imagine, for example, delivering the replacement parts by means of retro-viruses or similar agents. With luck, such replacement parts would not harm healthy tissue, and he nce they (unlike traditional chemotherapeutic agents) could be introduced in numbers su ﬃcient to reach all the cells they needed to reach. 2Before it could be repaired, however, the relevant “machine ry” would have to be identiﬁed. Here a plausible concomitant of the postulated h istorical transition to communal life becomes relevant, namely the universality of the control mechanisms that evolved then. Assuming that the transition to multi-celled organisms was a prolonged process, one would expect the control mechanisms that developed in its course t o be shared by all, or almost all, multi-celled forms. Conversely, uni-celled organisms sho uld lack these control mechanisms. To test this idea, one could compare the genomes of one-celle d organisms with those of multi-celled organisms susceptible to cancer, searching f or genes, or sequences thereof, that were universally present in the latter and absent in the form er. These genes (or sequences) would then be the ones responsible for the postulated contro l mechanisms. Moreover, the oldest such genes (or sequences), would furnish better candidates than the newer ones, assuming that age could be identiﬁed reliably. Any attempt t o develop a “gene therapy” or diagnostic test for cancer could then focus on the compone nts of the genome identiﬁed in this way. Of course, it might be overly optimistic to expect that the po stulated control mech- anisms could be identiﬁed simply with some well delineated p ortion of a cell’s genome. They might involve non-genetic components whose failure wa s not occasioned by genetic mutations. On the other hand, one knows that genetic change i s involved in cancer and that, in particular, there is a close correlation between mu tagenicity and carcinogenicity. Given these facts, it does not seem too much to hope that the re quired “social skills” are coded into universal portions of the genome which could b e identiﬁed and catalogued. Conversely, the failure to discover such portions would con stitute evidence against the historical explanation of cancer put forward in this articl e. This research was partly supported by NSF grant PHY-9600620 and by a grant from the Oﬃce of Research and Computing of Syracuse University. 3
arXiv reference: http://xxx.lanl.gov/abs/ physics/0011003 Newton’s aether model Eric Baird (eric_baird@compuserve.com) Isaac Newton is usually associated with the idea of absolute space and time, and with ballistic light -corpuscle arguments. However, Newton was also a proponent of wave/particle duality, and pub lished a “new” variable -density aether model in which light and matter trajectories were either bent by gravitational fields, or deflected by an aether density gradient. Newton’s (flawed) aether model can be considered as an early attempt at a curved -space model of gravity. 1. Introduction Modern textbooks typically say that Newton believed that space and time were absolute and inviolable. However, a reading of Newton’s “Principia” [ 1] and “Opticks” [ 2] reveals a rather different picture, with Optiks in particular documenting Newton’s attempt to produce a model of gravity in which a gravitational field could be represented as a series of light -distance differentials, or as a variation in lightspeed or refractive index. This can be compared to Einstein’s “ref ractive” approach to gravitational light -bending in 1911 ( [3] §4 ) and to his description of general relativity as a (nonparticulate!) gravitational aether model in 1920 [ 4][5]. We briefly look at some of the features of Newton’s model, the mistake that doomed it to obscurity [ 6], and some of the consequences of this mistake on the subsequent development of physics. 2. Absolute space? In “Principia”, Newton was careful to distinguish between relative space and time , which were to be defined by observations a nd instrument readings, and absolute space and time, which were to relate to more abstract (and possibly arbitrary) quantities that might or might not have an identifiable grounding in physical reality. Finite -lightspeed effects had already been seen in th e timing offsets in the orbits of Jupiter’s moons, so this distinction was important. Newton insisted that the words “space” and “time” should by default refer to “absolute” (deduced, mathematical) quantities rather than their “apparent” counterparts, but statements from Newton regarding “absolute space” and “absolute time” do not automatically mean that Newton believed that directly - measurable distances and physical clock -rates were also absolute – (Principia, Definitions: “… the natural days are truly une qual, though they are commonly considered as equal … it may well be that there is no such thing as an equable motion, whereby time may be accurately measured. ”). In “Opticks”, Newton’s idealised absolute space is occupied by a “new” form of medium whose density depends on gravitational properties, with variations in aether density producing the effects that would otherwise be described as the results of a gravitational field. The resulting metric associates a gravitational field with signal flight -time di fferences ( see: Shapiro effect) that deflect light, leading to a normalised lightbeam -geometry that is not Euclidean. Since these effects are described in modern theory as the effects of curved space, it seems reasonable to interpret Newton’s “absolute spa ce” as an absolute Euclidean embedding -space that acts as a container for non -Euclidean geometry, rather than as an indication that Newton believed that gravity had no effect on measured or perceived distances, times, or “effective” geometrical relationshi ps. 3. Lightspeed problems Newton and Huyghens had opposing ideas on how a lightspeed differential deflected light: Newton view: ”A gravitational gradient is associated with a change in speed of freely falling particles, with the speed being higher where the gravitational field is stronger. The bending of light at an air -glass boundary and the falling of light -corpuscles in a gravitational field can be described as the deflection of light towards regions of higher lightspeed.” Huyghens view: ”If a region has a slower speed of light, it will tend to collect light from the surrounding region. If a light -signal wavefront encounters a lightspeed gradient across its surface, with a faster natural speed on one side and a slower speed on the other, the retardatio n of the wavefront’s “slower” side will steer the wavefront towards the slower -speed region.” Measurements of relative lightspeeds in different media in the 19th Century showed that it was Huyghens’ explanation that was correct. “Newton’s aether model” Eric Baird 1 November 2000 page 2 / 2 arXiv reference: http://xxx.lanl.gov/abs/ physics/0011003 4. Huyghens’ principle Huygh ens’ principle is illustrated in the following diagrams of a light plane -wave hitting a glass block:. (a) (b) Figure (a) shows the progress of a wavefront hitting a lightspeed transition boundary, (b) shows the resulting change in direction of the w ave normal. At the start of the experiment, the leftmost edge of the advancing wavefront hits the air/glass boundary at A. A short time later, the rightmost edge of the wavefront has advanced by a distance c1 and the rightmost edge of the wavefront has re ached the boundary at B. By this time, the leftmost edge has penetrated the glass a smaller distance c2 (because lightspeed in glass is slower) and the new wavefront lies along a line between B and a tangent centred on A with radius c2 . The wavefront nor mal is deflected to point more towards the region of slower lightspeed, with the exact relationship being 21 sinsin cc ANGLE REFRACTIONANGLE INCIDENCE = 5. The “corrected” gravitational aether We can make Newton’s description compatible with Huyghens’ principle by inverting hi s lightspeed and aether -density relationships. After these substitutions, the relevant queries in Opticks read: Qu 19x (rewritten) Doth not the Refraction of Light proceed from the different density of this Aetherial Medium in different places, the Ligh t receding always from the [rarer] parts of the Medium? And is not the density thereof [less] in free and open spaces void of Air and other grosser Bodies, than within the Pores of Water, Glass, Crystal, Gems, and other compact Bodies? … Qu. 21x (rewri tten) Is not this Medium much [denser] within the dense Bodies of the Sun, Stars, Planets and Comets, than in the empty Celestial spaces between them? And in passing from them to great distances, doth it not grow [rarer] and [rarer] perpetually and there by cause the gravity of those great Bodies towards one another, and of their parts toward the Bodies; every Body endeavouring to go from the [rarer] parts of the Medium towards the [denser] ? … … And though this [Decrease] of density may at great distan ces be exceeding slow, yet is the elastick force of this Medium be exceeding great, it may suffice to impel Bodies from the [rarer] parts of the medium towards the [denser] , with all that power which we call Gravity. … 6. Some other issues Opticks’ Query 1 , on light -bending effects: Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action strongest at the least distance? This query does not make a distinction between gravitational light bending (action of gravity on unspecified “corpuscles”, mentioned in Principia), and more conventional lensing effects. Query 4 makes conventional optical effects more “gravitational” by proposing that “… rays of Light … reflected of refracted, begin to bend before they arrive at th e Bodies … ” Query 17 : Total internal reflection at a glass -air boundary introduced the philosophical problem of how the behaviour of light in glass could be affected by properties of a region that the light did not actually reach. How does light “know ” what is beyond the glass, if it never actually passes beyond the glass? Newton’s answer – that there must also be a hyper -fast wave -effect whose interference patterns then steer the subsequent light signal – predates the “pilot wave” description of the t wo-slit problem in quantum mechanics. Query 21 introduces the supposition that the aether might be particulate, but includes a slight qualification: “… (for I do not know what this Aether is) … ”. Query 28 recognises that lightwaves are not compression waves in the gravitational medium, since a compression -wave would have a tendency to spread out into less compressed regions (a lightbeam would then be deflected towards “dark” regions). Newton’s perplexity at how a single medium could then support both gravitational signals and electromagnetic signals can be compared to Einstein’s similar musings in 1920 (“ … two realities that are completely separated from each other conceptually, although connected causally, namely, gravitational ether and electromagnet ic field … ”. [4] Quest. 30 (sic) asks whether light and matter are not interconvertible, and Quest. 31 touches on the idea of stronger short -range forces being at the heart of chemical reactions. “Newton’s aether model” Eric Baird 1 November 2000 page 3 / 3 arXiv reference: http://xxx.lanl.gov/abs/ physics/0011003 7. Effective curvature In the d iagrams below, (a) shows genuinely flat space, (b) shows the aether -density gradient associated with the “corrected” version of Newton’s variable -density aether model, and (c) shows the same map, extruded so that light - distances can be measured directly fr om map distances in the map: (a) :truly flat space, (b): variable -density aether on flat background, (c): normalised light -distance map The physics of maps (b) (“aether -density gradient”) and (c) (“curved space”) can be equivalent ( see: Thorne [ 7], Ch apter 11 “What is Reality?”). Gravitational field approximated as a series of shells of increasing refractive index Although Cavendish did not calculate the sun’s light-bending effect until the late 18th Century, Newton had already made similar calculat ions for the effects of a variable -density medium, in order to predict optical effects caused by the Earth’s (variable density) atmosphere [ 6] ). 8. Effective closure Our “corrected” version of Newton’s aether model (with the tim e taken by light to cross a distance increasing or decreasing with gravitational field strength), allows a nominally -infinite universe containing a “central” concentration of matter to appear to its inhabitants as a closed hyperspherical universe with an e ven distribution of matter and no distinguishable centre [ 8] – figure 2 of Einstein’s ”Geometry and Experience” lecture [9] gives a method of mapping between these two equivalent descriptions. 9. Historical consequences The advantage of expressing a gravitati onal field as a variation in density of an underlying medium was that wavefronts and particles would then be deflected by the same amount (w/p equivalence, gravity as a “spatial density” effect). John Michell’s 1783 letter to Henry Cavendish was able to bu ild on the arguments in Principia and Opticks and conclude that light climbing out of a gravitational well should lose energy, with the image of a high -gravity star viewed through a prism being offset towards the weaker end of the spectrum [ 10]. Michell’s paper also calculated the R=2M event horizon radius, and discussed the “modern” method of finding non -radiating stars from the motions of their “normal” companions. Michell did not describe distance -dependent signal flight -time differences in his double -star scenario (such as those expected in simple ballistic -photon theory superimposed on flat space, discounted by deSitter in 1913 [ 11][12][13]), possibly because of uncertainty as to whether Newton’s aether should support multiple superimposed lightspeeds. LaPl ace also derived the R=2M relationship, and Cavendish and Soldner both calculated “Newtonian” values for the Sun’s bending of light ( see: [14][15][16] and Thorne [7] pp.122 -123 & 132 -133). The disagreement betwe en Newton’s and Huyghens’ arguments (Section 3), and the subsequent disproof of Newton’s lightspeed predictions (Foucault, 1850) had serious consequences for the idea of wave -particle duality, with Optiks going out of print until 1931, and its contents app arently unknown to Einstein as late as 1921[ 4]. Other related work suffered a similar fate – laPlace removed his reference to the r=2M radius in later editions of his book (Thorne [ 7] p.122 -123), Cavendish’s calculation of solar light -deflection did not find its way into print until 1921, and Michell’s paper dropped out of the citation chain and was only “rediscovered” in about 1979 [ 17][18]. After Laplace and Soldner’s published “light - corpuscle” pi eces in 1799 and 1801, there seems to be a “gap” in the reference chain until Einstein’s 1911 paper. Even after Riemann’s groundbreaking work on non -Euclidean geometry [ 19], attempts to construct curved - space models were not always taken seriously (e.g. Cle rk Maxwell on W.K. Clifford’s work, ~1869, “the work of a space -crumpler” [ 20]). Spatial -curvature models remained problematical until after Einstein had repeated Michell’s gravity -shift exercise (apparently oblivious to most or all of these earlier wo rks!) and concluded that the situation could not be resolved unless an increased gravitational field was also associated with a reduction in the rate of timeflow. By arguing that gravity distorted maps of timeflow across a region, Einstein then opened the door to models of space time curvature based on Riemann’s geometry, the most famous being his own general theory. “Newton’s aether model” Eric Baird 1 November 2000 page 4 / 4 arXiv reference: http://xxx.lanl.gov/abs/ physics/0011003 10. Conclusions Newton’s aether model arguably represents one of the most serious missed opportunities in the history of gravitational physics. Newton’s repeated attempts to unify various branches of physics led him to the concept of wave/particle duality and to a model of gravity in which the gravitational field could be described as a density gradient, and in which the deflection of light or mat ter by the field was modelled as the effect of a variation in refractive index. In singly -connected space, this approach can be topologically equivalent to a curved -space model of gravity [ 7] (by contrast, general relativity i s a curved space time model of gravity). However, Newton’s model inverted a key lightspeed relationship. Instead of being a description in which the gravitational field itself was the medium (to misquote Marshall McLuhan, “The medium is the metric.”, see also: Einstein: “ the aether of general relativity ” [4] “If we imagine the gravitational field … to be removed … no ‘topological space’. ” [21]), Newton’s model produced a description of a gravitational medium that was displaced by the gravitational field, and this led to the model and its associated principles and predictions (such as wave/particle duality, gravitational light - bending and gravitational shifts) being largely forgotten until Einstein’s 20th Century work on quantum mechanics and general relativity. Rindler has already pointed out that the mathematical machinery for general relativity was available in the Eighteenth Century [ 22]. Given that Michell’s gravity -shift prediction was tantalisingly close to being a predict ion of gravitational time dilation (the effect missing from 18th Century curved -space models), it seems that the loss of Newton’s aether model may have significantly held back the progress of gravitational physics. REFERENCES [1] Isaac Newton Principia Vol s. I & II [2] Isaac Newton Opticks (Bell London 1931). [3] A. Einstein, "On the Influence of Gravitation on the Propagation of Light" (1911), translated in The Principle of Relativity (Dover, NY, 1952) pp.97 -108. [4] A. Einstein, "Aether and the Theory of Relativity" (1920), translated in Sidelights on Relativity (Dover, NY, 1983) pp.1 -24. [5] Under general relativity, spacetime is a stressable, draggable thing that interacts with matter and can support a range of p hysical parameters – the gravitational field parameters (under general relativity, spacetime is a stressable, draggable thing that interacts with matter and can support a range of physical gravitational field parameters (the g mn ). [6] D.T.Whiteside (edi tor) The Mathematical Papers of Isaac Newton, Volume VI ,1684 -1691 (C.U.P. Cambridge 1974), Appendix 3,4 footnotes pp.422 -437. [7] Kip S. Thorne, Black holes and timewarps: Einstein’s outrageous legacy (W.W. Norton, NY, 1994) [8] Ronald Gautreau. “ Cosmological Schwarzchild radii and Newtonian gravitational theory,” Am.J.Phys. 64 1457 -1467 (1996) [9] A. Einstein, "Geometry and Experience” (1921), translated in Sidelights on Relativity (Dover, NY, 1983) pp.25 -56. [10] John Michell, “On the Means of discovering … of the Fixed Stars … ,” Phil.Trans.Royal Soc. (1784) pp. 35 -57 & Tab III, sections 30 -32. [11] W. deSitter, "A proof of the constancy of the velocity of light," Kon.Acad. Weten. 15 1297 -1298 (1913). [12] W. deSitter, "On the constancy o f the velocity of light," Kon.Acad.Weten. 16 395-396 (1913). [13] Kenneth Brecher, "Is the Speed of Light Independent of the Velocity of the Source?," Phys. Rev. Letters 39 1051 -1054 (1977). [14] Clifford M. Will, "Henry Cavendish, Johann von Soldner, and the deflection of light," Am. J. Phys. 56 413 -415 (1988). [15] Pierre -Simon laPlace Exposition du System du Monde , page reproduced in Misner, Thorne and Wheeler Gravitation (W.H. Freeman NY 1971) fig 24.1, pp.623. [16] Peter Simon Laplace, essay (1799), translated and reprinted in The Large -Scale Structure of Spacetime Stephen W. Hawking and G.F. Ellis (Cambridge 1973) Appendix A, pp. 365 -368 [17] Simon Schaffer, "John Michell and Black Holes," Journ. Hist. Astronomy 10, 42-43 (1979). [18] Gary Gibbon s, "The man who invented black holes," New Scientist 28 June 1979, 1101 (1979) [19] Wolfgang Rindler “General Relativity before special relativity: An unconventional overview of relativity theory,” Am.J.Phys. 62 (10) 887 -893 (1994) [20] Paul Nahin, Time Machines: Time Travel in Physics, Metaphysics and Science Fiction (AIP Press, NY, 1993) Tech Note 4:5 pp. 315 -316 [21] A. Einstein Relativity: The Special and the General Theory, 15th ed. (1954) Appendix V: "Relativity and the Problem of Space" pp.135 -157. [22] Wolfgang Rindler, “General relativity before special relativity: An unconventional view of relativity theory,” Am.J.Phys. 62 887- 893 (1994).
arXiv:physics/0011004v1 [physics.class-ph] 1 Nov 2000Ideal Constraints - A Warning Note Antonio S de Castro UNESP - Campus de Guaratinguet´ a - Caixa Postal 205 - 1250000 0 Guaratinguet´ a - SP - Brasil Abstract. A criticism against the conception adopted by some textbook ’s authors concerning ideal constraints is presented. The criticism i s strengthen with two traditional examples. Portuguese version published in Revista Brasileira de Ensi no de F´ ısica 22, 444 (2000): V´ ınculos Ideais: Uma Nota de Esclarecimento.Ideal Constraints - A Warning Note 2 Constraints are restrictions that limit the motion of the pa rticles of a system. The forces necessary to constrain the motion are said to be fo rces of constraint. The constraints whose unknown forces of constraint can be elimi nated are called ideal constraints. The principle of virtual work and D’Alambert’ s principle are of paramount importance in pedagogical presentations of analytical mec hanics because they are readily derived from Newton’s laws and because they enable us to get r id of the unknown forces of constraint from the equations of motion as well. Furtherm ore, these principles are starting points for obtaining Lagrange’s equations of moti on. To obtain the principle of virtual work and D’Alambert’s principle textbook’s auth ors quite generally consider the virtual work done on a system of Nparticles: δW=N/summationdisplay i=1/vectorF(e) i·δ/vector ri+N/summationdisplay i=1/vectorfi·δ/vector ri where the force on each particle is written as the externally applied force /vectorF(e) iplus the force of constraint /vectorfi. The principle of virtual work is applied only to static problems whereas D’Alambert’s principle is applied to dyna mical situations. The diﬀerence between these principles is not highlighted in th is note neither is relevant to the discussion which follows because the attention is focus ed on the desembarrassment of the unwanted forces of constraint. Hauser [1] argues that “If the δ/vector ri’s are chosen so that any constraints which exist between the coordinates of the particles are satisﬁed, the constraint forces /vectorfiacting on the particles will be perpendicular to the displac ements δ/vector ri.” Similarly Lanczos [2] argues that “ The vanishing of this scalar product means that the force /vectorfiis perpendicular to any possible virtual displacement. ” Taylor [3] also argues that “One thing known about otherwise unknown forces of constrain t and that is that they always act at right angles to any conceivable displacement c onsistent with the constraint under the condition of “stopped time”, i.e., to any virtual d isplacement.” In the same line of reasoning Chow [4] claims that “ Most of the constraints that commonly occur, such as sliding motion on a frictionless surface and rolling contact without slipping, do no work under a virtual displacement, that is /vectorfi·δ/vector ri= 0 (4 .8) This is practically the only possible situation we can imagi ne where the forces of constraint must be perpendicular to δ/vector ri; otherwise, the system could be spontaneously accelerated by the forces of constraint alone, and we know th at this does not occur... ”. Indeed, the statements cited above hold for a system consist ing of just one particle. Nevertheless, there is no compelling reason to believe that they are true for a system with more than one particle, and they are not indeed. Let us il lustrate this point with two very instructive and traditional problems: the rigid bo dy and the Atwood machine. A rigid body is a system of particles connected in such a way th at the distance between any two particles is invariable. Newton’s third law implies that for any pair of particles the forces of constraint are equals and opposites and besides they are parallel to the relative position vector, whatever the virtual displ acements. These facts ensure that the net virtual work of the forces of constraint vanishe s.Ideal Constraints - A Warning Note 3 In the Atwood machine two particles are connected by an inext ensible string passing over a pulley. If the string and the pulley are massless and th e motion is frictionless then the forces of constraint will reduce to the tension in the str ing. The virtual displacements of the particles compatible with the constraint will be in th e vertical direction and so will the forces of constraint. The virtual works of the forces of c onstraint on each particle are the same, unless a sign, ensuring that the net virtual wor k done by the forces of constraint vanishes. It is worthwhile to observe that the conclusions obtained th rough the former examples do not depend of the state of movement of the particl es,i.e., if it is a static or a dynamical problem, in this way such conclusions are suit able to the principle of virtual work as well as to D’Alambert’s principle. From the previous two examples one can drawn the lesson that i n order to eliminate the forces of constraint is solely required that the net virt ual work vanishes: N/summationdisplay i=1/vectorfi·δ/vector ri= 0 This less restrictive condition allows forces of constrain t not perpendicular to δ/vector ri, for systems with more than one particle, without implying in spo ntaneous accelerated motion. In short, there is absolutely no need for resorting to errone ous restrictions on the forces of constraint, as those ones presented by Hauser, Lan czos, Taylor and Chow, in or- der to eliminate them from the analytical formulations of th e classical mechanics. Ideal constraints are those which the net virtual work on the entir e system is zero whatever the relative orientation among the forces of constraint and the virtual displacements. This proviso is suﬃcient enough to ensure that the system is n ot spontaneously accel- erated. [1] Hauser W 1965 Introduction to the Principles of Mechanics (Reading: Addison-Wesley) p 313 [2] Lanczos C 1970 The Variational Principles of Mechanics 4th ed (Toronto: Toronto) p 76 [3] Taylor T T 1976 Mechanics: Classical and Quantum (Oxford: Pergamon) p 31 [4] Chow T L 1995 Classical Mechanics (New York: Wiley) p 104
arXiv:physics/0011005v1 [physics.atom-ph] 1 Nov 2000Elastic and Inelastic Evanescent-Wave Mirrors for Cold Ato ms D. Voigt, B.T. Wolschrijn, R.A. Cornelussen, R. Jansen, N. B hattacharya, H.B. van Linden van den Heuvell, and R.J.C. Spreeuw Van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, the Netherlands http://www.science.uva.nl/research/aplp/ (February 2, 2008) We report on experiments on an evanescent-wave mirror for cold87Rb atoms. Measurements of the bouncing fraction show the importance of the Van der Waals attraction to the surface. We have directly observed radiation pressure para llel to the surface, exerted on the atoms by the evanescent-wave mirror. We analyze the radiation pressure by imaging the motion of the atom cloud after the bounce. The number of photon recoils ranges from 2 to 31. This is independent of laser power, inversely proportional to the detuning and pro - portional to the evanescent-wave decay length. By operatin g the mirror on an open transition, we have also observed atoms that bounce inelastically due to a spontaneous Raman tran- sition. The observed distributions consist of a dense peak a t the minimum velocity and a long tail of faster atoms, show- ing that the transition is a stochastic process with a strong preference to occur near the turning point of the bounce. 32.80.Lg, 42.50.Vk, 03.75.-b I. INTRODUCTION The use of evanescent waves (EW) as a tool to manip- ulate the motion of neutral atoms has been proposed by Cook and Hill [1]. Since then, EW mirrors have become an important tool in atom optics [2]. They have been demonstrated for atomic beams at grazing incidence [3] and for ultracold atoms at normal incidence [4]. In many experiments the scattering of EW photons was undesir- able because it makes the mirror incoherent. However, an EW mirror is also a promising tool for eﬃ- cient loading of low-dimensional optical atom traps in the vicinity of the dielectric surface [5–8]. In these schemes, spontaneous optical transitions play a crucial role in pro- viding dissipation [9–11]. Since inelastic bouncing may increase the atomic phase-space density, this may open a route towards quantum degenerate gases, which does not use evaporative cooling. Thus one may hope to achieve “atom lasers” [12] which are open, driven systems out of thermal equilibrium, similar to optical lasers [13]. It is this application of EW mirrors which drives our inter- est in experimental control of the photon scattering of bouncing atoms. In a ﬁrst experiment involving the scattering of evanes- cent photons we have observed directly for the ﬁrst time the radiation pressure exerted by evanescent waves oncold atoms [14]. In a second experiment we directly observe clouds of atoms which bounce inelastically by changing their hyperﬁne ground state. II. EXPERIMENTAL SETUP FIG. 1. Experimental setup. (a) Cold87Rb atoms (MOT), are released 6.6 mm above a right-angle prism. An evanescent wave is created by beam EW. Fluorescence from probe beam P is imaged onto a CCD camera. (b) Confocal relay telescope for adjusting the angle of incidence θ. The lenses L1,2have equal focal length, f= 75 mm. A translation of L1by a distance ∆ achanges the angle of incidence by ∆ θ= ∆a/fn. The position of the EW spot remains ﬁxed. M is a mirror. Our experiments are performed in a vapor cell. Ap- proximately 107atoms of87Rb are loaded out of the background vapor into a magneto-optical trap (MOT) and are subsequently cooled to 10 µK using polarization gradient cooling (optical molasses). The cold atom cloud is released in the F= 2 ground state. After a free fall of 6.6 mm the atoms reach the horizontal surface of a right-angle BK7 prism (refractive index n= 1.51), see Fig. 1(a). The EW beam emerges from a single-mode optical 1
arXiv:physics/0011006v1 [physics.bio-ph] 2 Nov 2000Spherical harmonic decomposition applied to spatial-temp oral analysis of human high-density EEG Brett. M. Wingeier∗ Brain Sciences Institute, Swinburne University of Technol ogy, 400 Burwood Road, Hawthorn, Victoria 3122, Australia. Paul. L. Nunez Department of Biomedical Engineering, Boggs Center, Tulan e University, New Orleans, Louisiana, 70118. Richard. B. Silberstein Brain Sciences Institute, Swinburne University of Technol ogy, 400 Burwood Road, Hawthorn, Victoria 3122, Australia. (February 2, 2008) Abstract We demonstrate an application of spherical harmonic decomp osition to anal- ysis of the human electroencephalogram (EEG). We implement two methods and discuss issues speciﬁc to analysis of hemispherical, ir regularly sampled data. Performance of the methods and spatial sampling requi rements are quantiﬁed using simulated data. The analysis is applied to e xperimental EEG data, conﬁrming earlier reports of an approximate frequenc y-wavenumber re- lationship in some bands. Typeset using REVT EX 1I. INTRODUCTION The human electroencephalogram (EEG), as measured at the sc alp, represents a super- position of electric ﬁelds resulting from post-synaptic po tentials in neocortex, the thin (2 to 5 mm) surface layer of human brains. Several models of neocor tical dynamics treat EEG as a mixed global/local phenomenon [1–3], and a better underst anding of its spatial-temporal dynamics is necessary for evaluation and reﬁnement of these models. Its temporal behavior has been studied at length, both by clinical observation [4] and with such tools as power spectra [5], coherence [6], the Hilbert transform [7], and m any others. However, until re- cently poor spatial resolution (due to minimal electrode sa mpling and under-use of head models) has limited spatial analysis of EEG [1,2,8]. As a potential ﬁeld on a near-hemispherical surface, EEG is a menable to analysis by spherical harmonic decomposition. In this paper, we apply t wo methods of decomposition (one described by Cadusch [9] and one adapted from Misner [10 ]) to 131-channel EEG data. Using simulated data, we discuss issues and pitfalls releva nt to such an analysis, speciﬁcally the eﬀects of limited and irregular sampling density, integ ration over a hemisphere, and deviations from a spherical surface. From the experimental data, we then draw conclusions regarding the frequency-wavenumber relation of neocortic al activity. II. METHOD We use the real spherical harmonics [11], deﬁned on the spher e Ω and described by the orthogonality integral /angbracketleftYlm\|Yl′m′/angbracketright=2π/integraldisplay 0π/integraldisplay 0Ylm(θ, φ)Yl′m′(θ, φ) sinθdθdφ =δl,l′δm,m′. (1) In theory, a potential ﬁeld Φ(Ω) may be decomposed into spher ical harmonic amplitudes Φlmdeﬁned by Φlm=/integraldisplay ΩYlm(θ, φ)Φ(θ, φ)d2Ω. (2) In the example of EEG and similar data we encounter three majo r and two minor issues. A. Sampling First, when attempting to decompose experimental data, we s ample Φ(Ω) at speciﬁc locations Γ. Assuming near-regularly spaced electrodes, o ur maximum resolvable lis deter- mined by a spherical analog of the familiar Nyquist limit [12 ]fmax= 1/(2∆T). With mean angular inter-electrode distance γ, we initially adopt a conservative limit of lmax=/bracketleftBiggπ 2γ/bracketrightBigg , (3) orlmax= 6 for our 131-channel electrode cap. Analog pre-ﬁltering t o avoid spatial aliasing is not required here due to the low-pass characteristics of t he head volume conductor [13]. 2B. Regularization With sampled data F(Γ), the discretized version of the decomposition in Eq. 2 is unstable in higher l-indices. (An apparently accurate reconstruction of the si gnal may be generated, with large artifacts in the higher spatial frequencies.) We must invoke constraint or regular- ization techniques to address this issue. Cadusch et al[9] approached the problem as a side issue of spherical spline interpolation. The estimate ˆΦlmis constrained by the spline con- straints, and the problem for a given sampling grid and lmaxis reduced to a multiplication by a matrix of µlmcoeﬃcients: ˆΦlm=/summationdisplay x∈Γµlm(x)F(x). (4) Recently, Misner [10] introduced a more complex method for d ecomposition on a rect- angular three-dimensional grid; that is, generalized to us eYnlm(r, θ, φ). We implement here the special case of sampling on a spherical surface, more rel evant to EEG analysis, in which ris constant. In Misner’s method, a correction matrix of GABaccounts for discretization and limited sampling: GAB=/summationdisplay x∈ΓYA(x)YB(x)wx, (5) where A and B refer to index groups ( lm). Here we use the real harmonics and replace Misner’s weight function wxwith the eﬀective area of each electrode. GAB, the matrix inverse of the GAB, is used to calculate “adjoint spherical harmonics” YA. Finally, a set of coeﬃcients Rlm, analogous to Cadusch’ µlm, is generated and used as in Eq. 4 to estimate ˆΦlm. C. Hemispherical sampling Particularly relevant to EEG analysis is the error introduc ed by sampling over only half of the sphere. This sampling corresponds to only1 2of a spatial cycle of the l= 1,m= 0 function, suggesting potential accuracy problems for func tions involving l= 0 and 1. It is also clear that the functions Ylmwill no longer be orthonormal for 0 ≤θ≤π/2 only; rather, we replace the δin Eq. 1 with an error quantity ǫ: /angbracketleftYA\|YB/angbracketright=2π/integraldisplay 0π 2/integraldisplay 0YA(θ, φ)YB(θ, φ) sinθdθdφ =ǫAB. (6) In general, our hemispherical estimates ˆΦ′Awill be related to the hypothetical full-sphere result by the matrix of ǫAB’s: ˆΦ′=ǫˆΦ (7) and our ˆΦ′Abecome somewhat ambiguous between certain sets of ( l, m). Although it may seem appropriate to invert ǫand calculate a more accurate result, the matrix is ill- conditioned ( R >108, where the 2-norm condition number Ris the ratio of the largest singular value of ǫto the smallest) and thus the inversion is problematic. 3In addition, if we use the hemispherical region to calculate Misner’s GABin Eq. 5, the resulting matrix of GABis ill-conditioned ( R >108) and thus the GABcannot be reliably found. Rather, we created a mirrored set of electrodes Γ′, calculated the matrix RofRlm for sample set (Γ ∪Γ′), and discarded the antipodal rows of R. Cadusch’ spline method, while still subject to Eq. 7, is native to the hemispherical s urface and requires no further manipulation. D. Coordinate orientation In many problems, the sphere has no preferred direction. The m-indices are usually col- lapsed [14] to produce an angular power spectrum estimate ˆG(l) as a function of wavenumber l: ˆG(l) =l/summationdisplay m=−l/parenleftBigˆΦ′lm/parenrightBig2(8) which is independent of coordinate system orientation. As w ell, we found the “hemispherical error” in l-spectrum to be independent of orientation. In some EEG stud ies, of course, the orientation of the underlying cerebral hemispheres may be r elevant. In such cases, local spatial Fourier analysis [1] should adequately complement our decomposition without the complication of distinguishing m-modes. E. Non-spherical media We assume that our medium Ω is a sphere, whereas biological da ta is often sampled on an irregular surface. The upper surface of the “average” human head [15] may be represented as a hemiellipsoid with axes a= 10.52 cm, b= 7.66 cm, and c= 8.41 cm, or alternatively 25%, -9%, and 0% elongation from a perfect sphere. Although p rolate spheroidal harmonics have been applied to biophysical ﬁeld problems [16–18], the technique is often unwieldy. In comparison to error from ǫAB, especially for low l, we assume the error due to approximating the ellipsoidal surface with a spherical surface is negligi ble. III. APPLICATION TO SIMULATED DATA We generated evenly tessellated, hemispherical electrode maps of 74, 187, 282, and 559 electrodes, in addition to common experimental maps of 20, 6 4, and 131 [8] electrodes. Five hundred potential maps were simulated for each electrode co nﬁguration. Each potential map was randomly generated with harmonics of degree l= 6, such that the Φ lmvaried with uniform distribution between 0 and 1. Power spectrum estima tesˆG(l) were then calculated for each map, using both methods. Figure 1 shows Pearson’s co rrelation coeﬃcients rl, calculated between G(l) and ˆG(l) over the 500 trials for each electrode map. We have noted that the error due to ǫABcauses power from one ( l, m)-component to be misinterpreted as power in another, often of diﬀerent l. Therefore, we might expect either method’s performance to depend on the l-spectrum being analyzed. Using preliminary 4experimental data, we constructed an approximate power spe ctrum Gnorm(l) for average- referenced scalp EEG, peaked at l= 1 and l= 2, and decaying with l−1thereafter. Another ﬁve hundred potential maps were generated, with Φ lmuniformly distributed between 0<Φlm<Gnorm(l) 2l+ 1(9) to simulate a physiologically realistic distribution of l-spectra. Power spectrum estimates ˆG(l) were calculated for each map using both methods. Results ar e shown in Fig. 2. In general, results for the spline method — though often quit e accurate — were dependent on the distribution of l-spectra being measured, exact electrode positions, and el ectrode numbers. Results for the adjoint harmonic method seemed mor e robust, even for sparse (n= 64) sampling, although accuracy was somewhat less in the hi gher harmonics. In both methods, for l= 6 we observed minimal improvement for more than 131 elec- trodes. We thus believe that our 131-channel sampling is an a ppropriate tool for further study. Furthermore, given the limit in Eq. 3 and the known vol ume-conductor attenuation of higher modes [13], we suggest that study of spatial freque ncies higher than approximately l= 8 will be better served by intracranial EEG than by denser el ectrode maps. In general, for low lthe adjoint harmonic method seemed more consistent. We exam ined typical 131-channel decompositions (Fig. 3) to investigat e further. Both methods accurately reproduced the potential maps ( r >0.9 for 131 channels). The spline method, however, was slightly unstable for low l, and the erroneous negative ˆΦlmare reﬂected in the power spectrum. IV. REFINEMENTS AND ANALYSIS OF ERROR Any application of the spherical harmonic decomposition sh ould take into account the estimated relative contribution of various error sources. Aside from measurement and ex- perimental error, these may be divided into three categorie s: sampling error, orientation error, and hemispherical error. A. Sampling error Figures 1 and 2 indicate minimal improvement for l≤6 with more than 131 electrodes. We can thus deduce that the Nyquist-like limit in Eq. 3 is an ap propriate guideline. When using coarser sampling, we expect some decrease in performa nce for higher l. Decreased accuracy for 20, 64, and 74 channels (seen in Figs. 1 and 2, par ticularly for 20 channels) may be attributed to sampling error. B. Orientation error For a given l-spectrum, results will vary if power is randomly distribut ed across the m’s; that is, various m-components interact diﬀerently with our hemispherical sa mple grid. Five 5hundred random l-spectra, with realistic distribution of G(l), were generated. Thirty 131- channel maps, randomly varying in m-power, were generated for each original l-spectrum. Figure 4 displays the resulting accuracy, again shown as cor relation coeﬃcient rlbetween G(l) and ˆG(l) over the 100 trials. Very little is gained in this simulation by decomposition of multiple epochs. As de- scribed in the following section, error due to m-distribution of power is largely swamped by hemispherical error. In practice, however, we must empha size (in the presence of ran- dom measurement noise) the importance of averaging decompo sitions across many epochs. Orientation error will also become signiﬁcant if our sampli ng grid is severely non-uniform. C. Hemispherical error In Sec. II above, we have discussed the hemispherical error ǫAB. Although it is impos- sible to improve our decomposition results by inverting the matrix ǫ, we may generate a corresponding matrix for the power spectrum result and use i t to estimate the contribution of hemispherical error. Power in a single harmonic Ylm(θ, φ) is blurred by the hemispherical decomposition into surrounding harmonics. Using the 131-channel sampling map , we generated ﬁve hundred potential maps for each of l= 0. . .6, each with one unit power distributed randomly among the available m’s. By averaging over the ﬁve hundred resulting power spectr a, for each l, we obtained an empirical “averaged blurring matrix” Efor power spectra obtained from hemispherical decomposition. That is, ˆG(l)≈EG(l). (10) The typical Efor both methods is a blurred identity matrix; that is, error in power spectra is largely between adjacent l. It is again tempting to invert E, de-blur our spectra, and cal- culate a more accurate result, but although most Eare invertable we found that for realistic spectra the beneﬁt was marginal at best. Instead, Emay be used to better understand the implications of hemispherical error. We calculated correlation coeﬃcients as in Fig. 4, between EG(l) and ˆG(l) over the 500 trials of 30 epochs for 131 electrodes. The resulting hig her correlations (although not applicable to a decomposition of real data) are plotted in Fi g. 5. By comparison with Figs. 2a and 4, the result indicates the importance of hemispheric al error. In particular, after examination of typical Eandǫmatrices, we may interpret the decreased performance at lowlas blurring between adjacent wavenumbers. Furthermore, th e increased eﬀect, seen in Fig. 5, of averaging across various m-distributions indicates that some abrupt changes in performance may be attributed to sensitive interactions between E-blurring and random m-distribution. Practically, the near-identity character of Eis extremely useful. Hemispherical error manifests as blurring between adjacent l. Thus, we may expect composite measures such as the sum of power in l=0,1 to be substantially more accurate than individual esti mates. Figure 6 displays the accuracy of l=0,1 and l=2...6 adjoint harmonic power estimates (used below in our experimental trials), for 500 epochs, rea listicl-distribution, and various sampling densities. 6V. APPLICATION TO EXPERIMENTAL DATA Nunez in 1974 [19,20] and Shaw in 1991 [21], using Fourier ana lysis along linear electrode EEG arrays, observed a relationship between increasing spa tial and increasing temporal frequency in the 8-13 Hz band, roughly consistent with simpl e wave dispersion relations. We attempted to duplicate this result in order to test the adj oint harmonic method under experimental conditions. We analyzed 131-channel EEG (res ting, eyes closed) in ﬁve human subjects. Temporal Fourier coeﬃcients were determined for 300 to 600 one-second epochs (depending on available data), and l-spectra averaged over these epochs. Results are summarized in Fig. 7 as the ratio of power in low ( l= 0,1) to power in high (l= 2,3,4,5,6) spatial frequencies. Above approximately f= 8 Hz, with increasing fwe observed a general trend towards power in higher l. We also observed high-wavenumber power in the delta band ( f≤3 Hz). The alpha band (c. 8–13 Hz) was characterized by the highest power in low spatial frequencies. In order to rule out methodological artifact, we generated a nd analyzed 300 seconds of simulated EEG using 3602 uncorrelated sources, each genera ting 1/f noise through a head volume conductor model as described in [13]. As expected, th e EEG-like noise (labeled RND in the ﬁgure) showed no relation between spatial and tempora l frequencies. VI. CONFIDENCE INTERVAL ESTIMATION Estimates of the temporal power spectrum are known to vary in chi-square distribution [12], assuming normally distributed estimates of the under lying Fourier coeﬃcients. Error distribution for spatial spectrum estimates, on the other h and, is complicated by the de- pendence of hemispherical and orientation error on the enti rel-spectrum. For composite measures of both spatial and temporal spectra, such as shown in Fig. 7, the situation be- comes even more problematic. We propose an empirical test fo r estimation of such conﬁdence levels, analogous to the randomization tests commonly appl ied in nonparametric statistical analysis [22]. Let A=G01 G2...6, (11) where G01is the total power in harmonics l=0 and l=1, and G2...6is the total power in harmonics l=2 through l=6. Let ˆG01,ˆG2...6, and ˆArepresent estimates of the same. Above, we calculated ˆAffor various temporal frequencies. Here, we calculate an app roximate 95% conﬁdence interval for single-epoch estimates of the actua lAf. The conﬁdence interval will apply only to the spatial spectrum composite measure, negle cting error (or nonstationarity) in temporal frequency spectra, which for many applications may be as important. Note, though, that for 300 epochs the normalized standard error of a temporal power spectrum estimate is less than 6%. To determine an empirical conﬁdence interval, we would typi cally examine the distri- bution of random re-samples. In this application, we create d many random l-spectra from an estimated distribution of l-power, simulated many decompositions, and examined the resulting distribution as follows. 7Since hemispherical error is dependent on l-spectrum, the result will be inﬂuenced by the distribution of the random spectra. Srinivasan et al [13] analytically estimated the spatial frequency domain transfer function for volume-con duction blurring of scalp potential as proportional to (2 l+1)−1. This “spatial smearing” is due mainly to the poorly conduct ing skull and physical separation between cortical current sou rces and scalp electrodes. In our calculation, we assumed that underlying l-spectra vary in uniform distribution in proportion to (2l+ 1)−1, and that with average-referenced data the contribution of l=0 is negligible. A large number (20,000) of l-spectra were generated, randomly selecting for each l-bin a value from the appropriate distribution. The decomposition was p erformed, and the composite measure Acalculated, for each randomized spectrum. By examining the distribution of known surrogate Arandwhich produce a certain estimate ˆArand, we can estimate an empirical conﬁdence interval for our spe ctral estimate. In Fig. 8, we show the scatter plot of Arandagainst ˆArandwith 95% conﬁdence intervals. For a given estimate ˆAand the assumptions discussed above, 95% of the time, the act ualAwill fall between the two lines shown. A similar procedure may be used to calculate conﬁdence inter vals for other measures, whether the actual Glor other composite measures. Careful judgment must be appli ed when estimating conﬁdence intervals for multiple-epoch me asures such as shown in Fig. 6. As demonstrated earlier in this paper, variation in the m-component of an l, m-spectrum only allows us to “average out” the minimal orientation erro r. Variation in hemispherical error (dependent on l-spectrum), without gross violation of the stationarity as sumption, is necessary for the average of estimates ˆAover multiple epochs to converge to A. VII. DISCUSSION Our simulations provide a ﬁrm basis for application of spher ical harmonic decomposition to irregularly sampled, hemispherical data such as EEG. Our hemispherical modiﬁcation of Misner’s adjoint harmonic method [10] proved most consis tent. However, for physiologi- cal data of known power distribution, the spline method [9] i s complementary and may be slightly more accurate with high-density sampling. It seem s that, within the conservative band-limit of equation [3] and the known spatial ﬁlter prope rties of the head [13], decom- position accuracy will not be materially improved by more th an 131 electrodes for scalp EEG. We suggest that conﬁdence intervals for such decomposi tions, or for decomposition- derived measures, be determined empirically using randomi zed data. Furthermore, while single-decomposition errors are relatively large, with mu ltiple epochs the experimental ac- curacy may be increased substantially. For this averaging t o be both valid and eﬀective, we must assume a quasi-stationary wavenumber spectrum across our epochs, but with suﬃcient random variation in hemispherical error for our estimates t o converge upon the mean. In addition, especially in EEG applications, we must remain aw are of the limitations inherent in collapse across m’s (we assume the orientation of the underlying cerebral hem ispheres is irrelevant) and the use of spherical harmonics on a hemisphe roidal surface. The dynamical properties of human EEG rhythms are quite comp licated, varying sub- stantially between individuals and brain states. Furtherm ore, physiologically-based theo- retical models point to substantial nonlinear eﬀects and in teractions across spatial scales 8[2,23–26]. Despite all the obvious complications, results from the spherical harmonic de- composition of experimental EEG agreed qualitatively with crude linear electrode array results [1,21]. These results were seen in all subjects and a re consistent with a mixed global/local model of cortical dynamics, in which lower glo bal mode oscillations produce alpha rhythm, superimposed on local (spatially uncorrelat ed) activity in various frequency bands [2]. Further study of spatiotemporal EEG dynamics, us ing spherical harmonic de- composition, should shed more light on these issues. ACKNOWLEDGMENTS This work was supported by the Australian Research Council g rant #A10019013 and by the U.S. National Science Foundation (B.M.W.). 9REFERENCES ∗Also at Department of Biomedical Engineering, Tulane Unive rsity; e-mail wingeier@bsi.swin.edu.au. [1] P. L. Nunez, Electric Fields of the Brain: The Neurophysics of EEG (Oxford University Press, New York, 1981). [2] P. L. Nunez, Neocortical Dynamics and Human EEG Rhythms (Oxford University Press, New York, 1995). [3] F. H. Lopes da Silva, in Neocortical Dynamics and Human EEG Rhythms , by P. L. Nunez (Oxford University Press, New York, 1995). [4] K. E. Misulis, Essentials of Clinical Neurophysiology (Butterworth-Heinemann, Boston, 1997). [5] W. Klimesch, Brain. Res. Rev. 29, 169 (1999). [6] P. L. Nunez et al., Electroen. Clin. Neuro. 103, 516 (1997). [7] Tass P et al., Phys. Rev. Lett. 81, 3291 (1998). [8] P. L. Nunez, B. M. Wingeier, and R. B. Silberstein (unpubl ished). [9] P. J. Cadusch, W. Breckon, and R. B. Silberstein, Brain To pogr.5, 59 (1992). [10] C. W. Misner, http://xxx.lanl.gov/abs/gr-qc/991004 4, 1999. [11] M. Boas, Mathematical Methods in the Physical Sciences (J. Wiley & Sons, New York, 1983), p. 568. [12] J. S. Bendat and A. G. Piersol, Random Data (J. Wiley & Sons, New York, 1986). [13] R. Srinivasan, P. L. Nunez, D. M. Tucker, R. B. Silberste in, and P. J. Cadusch, Brain Topogr. 8, 355 (1996). [14] P. J. E. Peebles, Astrophys. J. 185, 413 (1973). [15] S. K. Law and P. L. Nunez, Brain Topogr. 3, 365 (1991). [16] G. C. K. Yeh and J. Martinek, Ann. NY. Acad. Sci. 67, 1003 (1957). [17] B. N. Cuﬃn and D. Cohen, IEEE Trans. Biomed. Eng. 24, 372 (1977). [18] J. C. de Munck, J. Appl. Phys. 64, 464 (1988). [19] P. L. Nunez, Math. Biosci. 21, 279 (1974). [20] P. L. Nunez, IEEE Trans. Biomed. Eng. 21, 473 (1974). [21] G. R. Shaw, Ph.D. thesis, University of Alberta, 1991. [22] E. S. Edgington, Randomization Tests (Marcel Dekker, New York, 1987). [23] V. K. Jirsa and H. Haken, Physica D 99, 503 (1997). [24] V. K. Jirsa, R. Friedrich, H. Haken, and J. A. S. Kelso, Bi ol. Cybern. 71, 27 (1995). [25] L. Ingber, Physica D 5, 83 (1982). [26] L. Ingber, in Neocortical Dynamics and Human EEG Rhythms , by P. L. Nunez (Oxford University Press, New York, 1995). 10FIGURES FIG. 1. Correlations between actual and estimated l-power for uniformly distributed random spectra. Five hundred potential maps were generated from kn own wavenumber spectra, with random power in each l-component, uniformly distributed between 0 and 1. For each of seven electrode densities, wavenumber spectra were estimated by spherical harmonic decomposition of the 500 sampled maps. Correlations between actual power and estimated power were calculated over the 500 trials for each l-component. Shown here for (a) adjoint harmonic and (b) spli ne methods, these correlations are a measure of the quality of a single decomposed power spectrum. FIG. 2. Correlations between actual and estimated l-power for more realistically distributed random spectra based on genuine EEG data. As in Fig. 1, but in o riginal spectra random power in each l-component is uniformly distributed between 0 and (2 l+ 1)−1. FIG. 3. Topography (left column), l,m-spectra (center column), and l-power (right column) for a typical 131-channel spherical harmonic decompositio n. The original map is shown in (a). The adjoint harmonic method (b) reconstructs topography and gi ves an approximation of l-spectrum. Although the spline method (c) also reconstructs potential topography, we observe irregularities in the lower lamplitude estimates that contribute to decreased performa nce for these wavenumbers, and a less accurate l-spectrum estimate. FIG. 4. Correlations between actual and estimated l-power for multiple-epoch, ad- joint-harmonic estimates of the same l-spectrum, with epochs varying only in m-component. For a reasonably isotropic and dense sample array, such as the 13 1-channel EEG grid used here, there is little orientation error and thus little improvement in r esults. FIG. 5. Sampling a full sphere with 262 channels and the adjoi nt harmonic method, correlations between actual and estimated l-power are shown for multiple-epoch (varying only in m-component) estimates of the same l-spectrum. By sampling over the full sphere, we eliminate he mispherical errors illustrated in Fig. 4. Remaining errors are due to ori entation (note improvement with multiple epochs) and imperfect sampling. FIG. 6. Correlation coeﬃcients, using the adjoint harmonic method, obtained by comparisons of estimated to actual summed power measures. The solid line represents power in l=0 and l=1 modes, and the broken line represents power summed over mode sl=2 through l=6. Increased accuracy (as compared to part A of Fig. 2) is because most hemi spherical error manifests as blurring between power in adjacent l’s. 11FIG. 7. Five to ten minutes of resting, eyes-closed EEG were c ollected with 131 channels from each of six subjects (one duplicated). Complex temporal Fou rier coeﬃcients were calculated for one-second epochs and subjected to spherical harmonic spat ial decomposition using the adjoint harmonic method. Resulting wavenumber spectra were averag ed for each 1-Hz band over the 300 to 600 epochs. The ratio of power in l=0,1 to l=2,3,4,5,6 is plotted as a simple indicator of a bias toward higher spatial frequencies at higher temporal frequ encies (greater than about 10 Hz). This result is qualitatively consistent with the postulated exi stence of an approximate EEG dispersion relation, perhaps with alpha rhythm (8–13 Hz) representing the fundamental and lower overtones. A surrogate signal (dotted line), composed of random EEG-li ke noise and subjected to the same analysis, showed no such relation. FIG. 8. 95% conﬁdence intervals for single-epoch estimates of the power ratio A=Gl=0,1/Gl=2...6. Twenty thousand potential maps were generated from known, random, real- istically distributed (based on genuine EEG data) l-spectra, and decomposed using 131 channels and the adjoint harmonic method. Here, known Aare plotted against the resulting estimated A. Solid lines indicate the empirical 95% conﬁdence interval for a given estimate of A. Multi- ple-epoch estimates will result in much smaller intervals, depending on the variation in l-spectra being decomposed. 1220 64 74 131 187 282 559Spline□method,□uniform l,□correlation between□actual□and□estimated□power 0 1 2 3 4 5 6-0.500.51 l-indexCorrelation□coefficient 0 1 2 3 4 5 6-0.500.51 l-indexAdjoint harmonic□method,□uniform l, correlation□between□actual□and□estimated□power 20 64 74 131 187 282 559Correlation□coefficient Figure□1□-□Wingeier(a) (b) two-column□repro□requestedCorrelation□coefficientCorrelation□coefficient(a) (b) 0 1 2 3 4 5 6-0.500.51 l-indexAdjoint harmonic□method,□realistic l, correlation□between□actual□and□estimated□power 20 64 74 131 187 282 559Correlation□coefficient 0 1 2 3 4 5 6-0.500.51 l-index20 64 74 131 187 282 559Spline□method,□realistic l,□correlation between□actual□and□estimated□power Figure□2□-□Wingeier two-column□repro□requestedOriginal l-power Y0 Y1 Y2 Y3 Y4 Y5 Y6 Spherical HarmonicAmplitudeOriginal l,m-spectrum 0123456 l-indexPower Original potential Estimated l-power Y0 Y1 Y2 Y3 Y4 Y5 Y6 Spherical HarmonicAmplitudeSpline method l,m-spectrum 0123456 l-indexPowerSpline reconstructionEstimated l-power Y0 Y1 Y2 Y3 Y4 Y5 Y6 Spherical HarmonicAmplitudeAdjoint harmonic method l,m-spectrum 0123456 l-indexPower Adj. harm. reconstruction two-column repro requested(a) (b) (c) Figure 3 - Wingeierl-indexCorrelation coefficientAdjoint harmonic method, multiple epochs, correlation between actual and estimated power 01234560.20.30.40.50.60.70.80.91 1 epoch 3 10 30 50 Figure 4 - Wingeierl-indexCorrelation coefficientAdjoint harmonic method, multiple epochs, simulation of no hemispherical error 01234560.20.30.40.50.60.70.80.91 1 epoch 3 10 30 50 Figure 5 - WingeierElectrode DensityCorrelation coefficientAdjoint harmonic method, accuracy of composite measures 2064741311872825590.20.30.40.50.60.70.80.91 l = 0,1 l = 2...6 Figure 6 - Wingeier0510152025303540450.20.250.30.350.40.450.50.55Ratio of power in l=0,1 to l=2,3,4,5,6 Frequency (Hz)CV1 RS1 BW1 KR1 PE1 BW2 RND Figure 7 - Wingeier00.20.40.60.811.21.41.61.8200.20.40.60.811.21.41.61.82 estimate of Aknown AKnown v. estimated A - scatterplotand empirical 95% confidence interval Figure 8 - Wingeier
arXiv:physics/0011007v1 [physics.ins-det] 3 Nov 2000A monitor of beam polarization proﬁles for the TRIUMF parity experiment1 A.R. Berdozf, J. Birchalla, J.B. Blanda, J.D. Bowmanb, J.R. Campbella, G.H. Coombesc, C.A. Davisd,a,2, P.W. Greenc, A.A. Hamiana, Y. Kuznetsove,3, L. Leea, C.D.P. Levyd, R.E. Mischkeb, S.A. Pagea, W.D. Ramsaya, S.D. Reitznera, T. Riesd, G. Royc, A.M. Sekulovicha, J. Soukupc, T. Stockic, V. Suma, N.A. Titove, W.T.H. van Oersa, R.J. Wooa, A.N. Zelenskie aUniversity of Manitoba, Winnipeg, MB bLos Alamos National Laboratory, Los Alamos, NM cUniversity of Alberta, Edmonton, AB dTRIUMF, Vancouver, BC eInstitute for Nuclear Research, Academy of Sciences, Mosco w, Russia fCarnegie Mellon University, Pittsburgh, PA Abstract TRIUMF experiment E497 is a study of parity violation in ppscattering at an energy where the leading term in the analyzing power is expec ted to vanish, thus measuring a unique combination of weak-interaction ﬂavour conserving terms. It is desired to reach a level of sensitivity of 2 ×10−8in both statistical and systematic errors. The leading systematic errors depend on transverse polarization components and, at least, the ﬁrst moment of transverse polarization. A novel polarimeter that measures proﬁles of both transverse components of polariza tion as a function of position is described. Key words: Beams; Polarimeter; Parity Violation PACS: 24.80+y; 29.27.Hj; 13.75.Cs Preprint submitted to Elsevier Preprint 2 February 20081 Introduction Several measurements of the parity violating component in t he nucleon-nucleon interaction have been reported over the years [1] [2] [3] [4] [5], achieving greater precision over time. Such an experiment that aims to measure longitudinal an- alyzing power, Az, to a precision of ±2×10−8(in statistics and systematics) is underway at TRIUMF [6]. Other experimenters have measure d the same quantity with protons incident on light nuclei [7] [8]. The T RIUMF experi- ment is unique in that it seeks to measure the parity-violati ng eﬀect at an energy, 221 MeV, where the leading term (which dominates bel ow 100 MeV), Az(1S0−3P0), is zero (averaged over the acceptance of the detector), th us observing the Az(3P2−1D2) term as the dominant component [9]. The diﬀer- ence in these terms is that they are dependent on diﬀerent com binations of the weak meson couplings [9]. In addition, another experime nt is planned at 450 MeV at TRIUMF [10]. Initially, it was recognized that residual transverse comp onents of polarization which changed sign with the longitudinal component of polar ization gave rise to a systematic error if the detection system was asymmetric or if the incident proton beam were oﬀ the symmetry axis [11]. Later, it was reco gnized that even if the transverse polarization components of a ﬁnite-s ized beam averaged to zero, an inhomogeneous distribution of the transverse po larization over the beam proﬁle could result in a signiﬁcant contribution to the measured Az[12] [13]. The ETH-SIN-Z¨ urich-Karlsruhe-Wisconsin group [1] [2] de scribe a beam in- tensity/polarization proﬁle monitor [14] that operates wi th two wheels ( xand y) each driving two graphite targets through their 50 MeV prot on beam. Pro- tons scattered at 51◦, near the maximum of the12C(p, p)12Canalyzing power, were observed in four scintillators left, right, bottom and top, and timed with respect to a reference on the wheel, i.e. the position of the t arget for that scat- tered proton. The data, along with information about the spi n state, were read into a series of spectra from which intensity and polarizati on proﬁles could be deduced. They used two such devices in their beamline. This p olarimeter is also described modiﬁed for use at lower energies [16]. The ta rgets used were much thinner and they adapted a multi-channel scaler and mul ti-channel an- alyzer to record and store the proﬁles. 1Work supported in part by a grant from the Natural Sciences an d Engineering Research Council of Canada 2Corresponding author. TRIUMF, 4004 Wesbrook Mall, Vancouv er, B.C., Canada V6T 2A3; Tel.:1-604-222-1047, loc. 6316; fax: 1-604-222-1 074 E-mail address: cymru@triumf.ca 3Deceased 2The Bonn group [3] describe a beam proﬁle scanner that measur es polarization [15] by physically moving a polarimeter (one for vertical an d one for horizontal proﬁle) with a thin graphite target through the beam. The tar get is optimized to allow passage of one target at a time through the beam while data collection is enabled. They detected protons scattered at 48◦. An elastically scattered proton in any of four Si detectors generated a sampling of an A DC that read a voltage picked up from a linear potentiometer related to the device’s position. Again, position dependent spectra were generated from whic h intensity and polarization proﬁles can be deduced. They also used two such devices in their beamline. The Los Alamos-Illinois group [4] [7] describe a simple scan ning target used in a conventional polarimeter. 2 Speciﬁcations and design Requirements of the TRIUMF Parity experiment are that it be a ble to inti- mately (i.e., within the data collection cycles of the exper iment, each cycle being eight periods of 25 ms duration) monitor the proﬁle of t he transverse polarization components as a function of position ( Py(x) and Px(y)), and that one be able to determine the corrections derived therefrom t oAzto a level at or below ±6×10−9over the whole data collection period (several hundred hours not counting calibrations and other overhead). These quantities are the average transverse components of polarization which given an eﬀect propor- tional to displacement from the detector symmetry axis, < x >< P y>and < y >< P x>, and the intrinsic ﬁrst moments of polarization, < xP y>and < yP x>, which will contribute even if the proton beam is perfectly a ligned with the apparatus. Higher order terms and in-plane terms (s uch as < xP x> and< yP y>) should be negligible [12] [2]. Typical run time conditions keep transverse components of p olarization under ±1% per data run (typically one hour). Intrinsic ﬁrst moments of transverse polarization are typically within ±25µm per run. As mentioned above, some researchers [14] [15] [16] use grap hite targets which have relatively high counting rates and high analyzing powe rs. However, the angular dependence of both cross section and analyzing powe r and the con- tribution of inelastic scattering, especially at higher en ergies, make this unde- sirable in the present case. By moving the polarimeter detec tors rigidly with the target, Chlebek et al. [15] avoid position correlated ac ceptance problems. Such a device had initially been considered [17] but abandon ed when it be- came apparent that the higher energy and larger beam size wou ld make such a scheme too unwieldy. 3Fig. 1. General schematic view of a PPM. The forward and recoi l paths for one arm are shown with the scintillators indicated as dark volum es. The paths originate from the plane in which the blades lie. The mechanics of the present detector have been described in [19], though there have been modiﬁcations since then which will be explai ned below. The device is a four-branch polarimeter whose target consists o f two wheels that can drive strips (‘blades’) of CH2(two on each wheel) through the beam at a speed locked to the experiment cycle time. It is shown in Figu re 1. Two blades per wheel were chosen as an optimal compromise between polar ization mea- suring time and Az(i.e., experimental determination of the helicity depende nt asymmetry of the beam transmission through an LH2target) measuring time. Two of these detectors are mounted in the experimental beaml ine to allow for extrapolation of the polarization proﬁles to the target location. They are shown in Figure 2. They are upstream of the Transverse ﬁeld Io nization Cham- bers (TRIC’s) that sandwich the LH2target. The TRIC signals (proportional to the beam current) determine the parity violating longitu dinal analyzing power that is the observable of interest. 4Fig. 2. A section of the E497 experiment along TRIUMF beamlin e 4A/2 showing the whole of the upstream PPM on the right, the rear of the down stream PPM on the left, and, between them, a beam position monitor [19]. Several scintillators, light guides, and PMT housings can clearly be seen on the upst ream PPM. 2.1 PPM Detectors Each branch consists of a forward arm of two scintillators at 17.5◦from the axis and a recoil arm. The angle of 17 .5◦was chosen as a reasonable compro- mise near the p−panalyzing power maximum over the energy range at which parity violation may be investigated at TRIUMF, see Figure 3 . The ﬁgure of merit for a polarimeter can be deﬁned as: A2 t(θ)dσ dΩ(θ). (1) This is also shown in Figure 3. The forward arm consists of two scintillators. The solid-an gle of acceptance for scattered protons is deﬁned by a rotated counter (Ω) whose an gle of rotation along its axis perpendicular to the scattering plane is chos en to cancel the eﬀect of the change in p−pscattering cross-section and detector geometry with target blade position [21]. At 223 MeV, this rotation an gle is determined to be 49◦with respect to the plane perpendicular to the nominal 17 .5◦center- 5Fig. 3. Analyzing power (solid line) and ﬁgure of merit (dash ed line) as a function of lab angle for p+pscattering as determined from Ref. [18]. The peak in ﬁgure of merit is ideal for optimal statistical error and the peak i n analyzing power is optimal (ﬂattest response as a function of target position) for systematic errors. line of the scattered protons, in the direction as shown in Fi gure 4. Between the Ω-counter and the target plane is a counter ( C) whose function is to determine that the scattered protons are collinear with the target. Th e recoil arm is at 70.6◦at which the recoil protons from p−pscattering will be stopped in the front ( R) counter or in a 1.6 mm thick aluminum shield immediately beh ind it. Protons from other sources that are too penetrating will pass through and hit the veto ( V) counter immediately behind. The location and dimensions o f each counter are recorded in Table 1. A schematic of a single b ranch lay-out is given in Figure 4. Each scintillator3was attached to a light-pipe viewed by a two-inch RCA 85754photomultiplier tube. The TRIUMF-built bases were equippe d with zener-diodes on the ﬁrst three dynodes and the voltages had t o be carefully adjusted due to the high rates. The front arm counters, Cand Ω, of each branch were mounted externally with the protons passing thr ough a 3.2 mm (at 17 .5◦) thick spun-aluminum shell, at a distance of 470 mm from the t arget, into air. The recoil arm counters, RandV, were mounted internally with a vacuum seal along each scintillator light pipe. The externa l counters and light 3BC-404; Bicron; 12345 Kinsman Rd., Newbury, OH, U.S.A., 440 65 4RCA Corp. 6Table 1 PPM scintillator counter dimensions. Distances are from th e center of the target plane. Counter Height Width Thickness Distance Arm (mm) (mm) (mm) (mm) C 37.5 37.5 6.4 600.2 Forward Ω 28.5 46.0∗6.4 900.0 Forward R 120.2 22.5 6.4 104.4 Recoil V 156.9 30.0 6.4 151.7 Recoil ∗Counter is rotated at 49◦with suitably beveled edges. Fig. 4. Schematic diagram of a single branch of one of the PPM’ s. pipes were wrapped with aluminized mylar and tape to keep out the ambient lighting; the internal counters were wrapped with a light-t ight aluminum foil only and their external sections of light pipe were wrapped a s the external counters. 7Fig. 5. A view of the rear of a PPM showing the external belt dri ve, the stepping motor at bottom, and the shaft encoder shielding at top. 2.2 PPM Targets The mounting of the target blades and the drive arrangement i s shown in Figure 5. The wheel pivots are 215 mm from the beam centre. Eac h arm holds two targets to better balance the statistics of the PPM with t he experiment statistics. This gives two xscans and two yscans per PPM, a total of eight. Each scan occurs during one spin-state of an eight state cycl e. The direction of the spin in each state is deﬁned by the eight state cycle whi ch can be (+− − +−+ +−) or its complement. The initial state of each cycle is chosen according to the same (+ − − +−+ +−) pattern, making up a 64-state ‘supercycle’. The initial spin direction of each s upercycle is chosen randomly. This timing sequence is shown in Figure 7. Each blade target is 1.6 mm wide, 5 mm along the beam, and 85 mm h igh (past its holder-clamp) and is machine cut from sheets of high dens ity polyethylene. As each blade passes through the beam, a proton scattered in a plane contain- ing the direction of motion of the blade is observed in one of t he two forward arms, left-right (horizontal motion) or bottom-top (verti cal motion), and the corresponding recoil proton from free p−pscattering is observed in the recoil arm on the opposite side. Protons scattered in a plane perpen dicular to the direction of motion of the blade (i.e., those that would give Px(x) and Py(y)) 8are not recorded as their recoil protons would in many cases b e stopped or severely multiple-scattered in the target. The target blades are driven through the beam by a D.C. servo- motor/tacho- meter unit5salvaged from an old reel-to-reel tape drive. The two wheels are connected by a timing belt that is mounted external to the PPM housing. This was done because it was proven necessary to ensure prope r cooling. The power to the wheels is transmitted through ferroﬂuidically -sealed shafts6. The read-out of the shaft position was done through a shaft encod er7. It was found necessary to shield this encoder and switch to a rad-hard ver sion as radiation damage caused failure after the ﬁrst few weeks of running. Th is has not been a problem since. In addition, with the blades turned oﬀ and parked out of the be am, it is possible to insert a ﬁxed target of CH2some 0.2 mm thick. This target has a very thin ﬁlm of aluminum evaporated on the surface to preve nt charging, and is mounted in a circular aluminum frame 100 mm in diameter . This allows a rapid determination of PyandPxin the parity beamline, useful for initial tuning of the solenoids that provide longitudinal polariza tion. 2.3 Synchronization and Control The PPM’s rotate at ﬁve revolutions per second and are adjust ed for 180◦an- gular mismatch. A full 200 ms cycle compromises eight blade p assages with 25 ms between passages. The synchronization of the PPM’s, as we ll as the main- tenance of the rotation speed, is accomplished by an applica tion of electronic gearing. Each PPM is equipped with a 2500 line incremental sh aft encoder and DC brushed servo motor. The motors are controlled by a Galil DMC103083-axis PC ISA bus based digital servo motion control card. A functional block diagr am of the control system is shown in Figure 6. A reference 60 Hz square wave sign al is generated from the 60 Hz AC line which has a frequency regulation of 0.06 %. This signal is frequency multiplied by a factor of 125 and phase locked to the 60 Hz line via a voltage controlled oscillator (VCO) feedback regulator c ircuit. The resulting 7500 Hz is phase shifted to produce a double phase quadrature signal. This is directed to the x-axis encoder input of the DMC and represents the master 5Electro-Craft Corporation; 1600 Second St. So., Hopkins, M N, U.S.A. 55343 6Ferroﬂuidics Corp.; 40 Simon St., Nashua, NH 03061, U.S.A. 7Type H25D; BEI Sensors and Motion Systems Co., Industrial En coder Division; 7230 Hollister Ave., Goleta, CA, U.S.A. 93117-2891 8Galil Motion Control, Inc.; 203 Ravendale Drive, Mountain V iew, CA, U.S.A. 94043-5216 9Fig. 6. Schematic of the PPM control system. The YandZreferences and positions are input in quadrature counts, + and −respectively to the Up/Down counter, whose output is fed to the PID ﬁlter. The PID ﬁlters’ outputs a re fed to Pulse Width Modulated (PWM) switching 20 kHz ampliﬁers in current /torque mode. The ampliﬁer output runs the DC servo motor as discussed in se ction 2.2. The optical encoders provide the position signals. The referen ce circuit, adjusted by φ60 is used to provide the reference signals. axis signal which the slave axes, yandz, are commanded to follow through the gear function ratios, FyandFz. The phase slip function factor, φ60, utilizes machine round oﬀ error which comes from the fact that 2500 is n ot evenly divisible by 60, so that at 5 Hz the right amount of phase slip r elative to 60 Hz is obtained. This means that rotator speeds of 3 Hz, 6 Hz, 9 H z, etc., can be set precisely to zero phase slip, while speeds in between c annot (unless the encoder resolution were changed to 3000 lines per turn). Nor mally, the phase slip is set to one 60 Hz cycle in 20 minutes. At 5 Hz, the encoder frequency is perfectly suited for this application. However, the fact orsφ60,Fy, and Fz can be conﬁgured interactively by the user from the windows g raphical user interface (GUI) at any time. In normal operation the gearing is set for 1:1 on both PPM’s an d the phase diﬀerence between the two PPM’s is set for 180◦. To compensate for small mechanical misalignment in the mechanisms, a ﬁne phase adju stment is made so that the actual blade passages through the beam (between t he two PPM’s) are exactly 180◦apart. The gear ratios FyandFzmodify the output signal, θ′ cfrom the x-axis phase slip function and produces command frequency re f- 10erences, θycandθzc, so the PPM speed is correctly calibrated, as required by the user, based on the 60 Hz line signal. During standard use Fy=Fz, which means that the two PPM’s are phase locked to run at the same spe ed with zero relative phase slip. Measurements with a digital oscilloscope showed that durin g rotation at 5 Hz the servo loops kept the two PPM’s within ±1 encoder tick (i.e, ±0.003 radians) of each other. The two reference signals, θycandθzc, are treated by the microprocessor as quadrature counts which are the Basic Length Unit at machine hardware level. This means that the phase synchroni zation and posi- tion accuracy of the servo loop is four times greater than the line frequency of the encoder. These signals are compared in an up/down counte r against the encoder feedback signal and the diﬀerence is used to produce an analog com- mand voltage signal for the servo ampliﬁer via a PID ﬁlter and DAC running at a sampling rate of 1 kHz. This produces the current to drive the motors. The PID ﬁlter parameters, KP,KD, and KI, are the same for both PPM’s due to their similar plant dynamics and shaft torque resista nces. However, the stable operating region is very narrow due to the ﬂexible couplings and the large inertia mis-match between the motor armature and t he blade rotor mechanism (required due to space constraints). The aim is to increase KP in order to minimize the phase lock and position error, but no t high enough to make the static gain loop unstable. To help stabilize the l atter, KDis in- creased high enough to damp out the low frequency instabilit ies, but not high enough to destabilize the derivative loop gain. KIis set to zero in order not to induce low frequency oscillations into the loop due to the hi gh load inertia. An interesting aspect to Figure 6 is that the distance between t he PPM/Servo- Amp units at beam line level and the DMC controller is over 150 meters. This is very unusual in servo control applications due to the dest abilizing eﬀects of phase delay in the encoder and command signal cables; but was required due to the radiation environment. For control measurements, it is possible to run a single PPM o r to park a PPM’s blades at a speciﬁc angle. 3 Signal Processing and Data Acquisition The PPM data collection is an integral part of the experiment al data collec- tion cycle. The PPM blade scans are carefully synchronized n ot only to each other, but are used to drive the polarized source spin ﬂip cyc le and the signal integration gates on the two transverse ion-chambers that b racket the target and whose helicity-dependent output constitutes the parit y-violating signal of the experiment. Each shaft encoder pulse forms the time-bas e for the experi- 11Fig. 7. The data collection cycle. Shown are the ﬁrst two stat es of the eight-state cycle, which can be (+ − −+−+ +−) or its complement. During each state of the cycle, one of the eight PPM blades passes through the be am. For the two states shown here, the two vertically scanning blades of PPM 1 would scan during the ‘Vert. 1 gate’ intervals. While the blade is passing through the beam the appropriate electronics and MCS channels are gated/enabled and the MCS c hannel is advanced by a signal from the shaft encoder (see Fig. 8). The PPM data is read out while the other equipment (TRIC/IPM) data collection is enabled. The spin state selection and sequencer start are controlled by the front-end process or. ment as an input to a timing and sequence module9. A schematic diagram of the data collection cycle is shown in Figure 7. 3.1 Electronics A schematic lay-out of the PPM electronics is given in Figure 8. The signals from the phototube bases were fed through an ampl iﬁer, thus allowing the tubes to be run at lower voltages, important due to the high singles rates, and into individual linear discriminators. Each pair of forward 9Model 221; Jorway Corp.; 27 Bond St., Westbury, NY, U.S.A. 11 590 12Fig. 8. Schematic of the PPM electronics, two branches of one four-branch PPM are shown. The discriminated signals from each counter (10 ns wi dth) are timed in to form the logic coincidence for each branch, Ln, Rn, Bn, Tn, and their corresponding delayed coincidence (accidental), Lnacc, Rn acc, Bn acc, Tn acc;n= 1,2 labelling either one of the two PPM’s. The four signals for any PPM plane (horizontal [Ln, Rn, Ln acc, Rn acc]or vertical [Bn, Tn, Bn acc, Tn acc]) are OR’ed together with the other planes and presented to the four MCS units, suitably ga ted for the appropriate blade as explained in the caption to Figure 6. arm signals, Cand Ω, were formed into a logical coincidence, C·Ω, and each recoil arm was formed into an anticoincidence, R·¯V. These were timed together to form ( C·Ω)·(R·¯V) and ( C·Ω)·(R·¯V)del(delindicating that the signal has been delayed by one cyclotron RF period — 43 ns), where the ﬁrst is the coincident signals, L,R,B, andT, and the latter are their corresponding accidentals, La,Ra,Ba, andTa. These signals are grouped together in common modules for L−RandB−Tand for the two PPMs, which can be inhibited by the timing sequence. This allows a fan-in of the signals, for example L1,B1, L2, and B2, together and they are then presented to the same scaler inp ut, as their respective blades are never in the beam at the same ti me. The scanning scalers and memory modules10read in the data in synch with a clock signal. As each blade moves through the beam the scale r advances through a sequence of channels that are related to the positi on of the blade. 103521A, MM8206A; LeCroy Research Systems; 700 Chestnut Ridg e Road, Chest- nut Ridge, NY, U.S.A. 10977-6499 13The data is then read out through a routine running in a dedica ted processor11 that stores the results in memory according to the timing seq uence, e.g.,LI, BI,LII, andBII, and the spin-state. Thus there is a requirement for only fou r such scalers and memory modules to record two true and two acc idental signals per blade. This allows many of the more crucial experimental modules to reside in a single crate, important for the in-crate control throug h the Starburst and timing sequences. The status of each spin state (+ or −helicity) is nowhere introduced as a gating signal to any of the hardware; thus avo iding undesirable cross-talk which might lead to a helicity-dependent electr onics eﬀect. Rather, the spin state status of the initial state (state 1, see Fig. 7 ) in a pattern of eight states is separately reported, as a frequency modulated/en crypted signal, to the computer. 3.2 Data Acquisition The PPM information was read out of the front-end processor a s a separate event, there being separate events for the TRIC and other mon itor informa- tion. This allowed the PPM information to be transferred to t he main data acquisition computer12while other data was being collected, and vice versa . The data was then written to tape and made available to other p rocessors for on-line analysis and monitoring. The last was especiall y important for the PPM data as it allowed us to monitor both transverse polariza tion compo- nents and the ﬁrst moments of polarization on a run-by-run (a pproximately hourly) basis. If these observables became excessively lar ge, then the beam- line solenoids or other cyclotron parameters were tuned to r educe them. As PPM information was available in each data buﬀer, each buﬀer (200 ms of data) could be analyzed separately and bundled as seemed app ropriate for a regression analysis. A ﬁrst analysis of such a kind was done i n a semi-online manner so that more sophisticated monitoring of the experim ent could be carried out. 4 Results The PPM counter’s were run at a comparatively high rate. Tabl e 2 shows the peak rates for both singles in each individual counter and th e coincidence rate at a beam current of 200 nA and a size of 5 mm. At this current, ﬁr st order 11Starburst J11; Creative Electronic Systems; 70 route de Pon t Butin, 1213 Petit- Lancy 1, Switzerland 12VAXstation 3200; Digital Equipment Corp.; Maynard, MA, U.S .A. 14Table 2 Singles and coincidence rates in PPM detectors. Detector Peak Singles Rate (MHz) C 3.1 Ω 0.8 R 3.0 V 1.5 Coincidence Peak Rate (kHz) (C·Ω)·(R·¯V) 110. (C·Ω)·(R·¯V+ 43ns) 38. accidentals (forward arm accidentally in coincidence with the recoils arm) were typically 35% of the ( C·Ω)·(R·¯V) coincidence rate. Two higher order accidentals were examined: (1) C·(R·¯V) with an accidental hit in the Ω counter; and (2) ( C·Ω)·Rwith an accidental hit in the Vcounter. The ﬁrst are ‘near’ events in the sense that they are close to t he acceptance of the PPM with a similar (very slightly lower, see Fig. 3) analy zing power; and the second are ‘stolen’ events in that they would have been ac cepted as true events but for the accidental veto. Both might result in erro rs in the measured intrinsic ﬁrst moment of polarization coupled to a helicity correlated change in the beam intensity (otherwise, they just tend to pull down the average analyzing power slightly). The eﬀects were measured by taki ng data with a 43 ns (one RF period for the TRIUMF cyclotron) delay in the Ω coun ter for case (1), and a 43 ns delay in the Vcounter for case (2). For an assumed helicity dependent variation of current,∆I I= 10−5, the change in the ﬁrst moment due to case (1) was 1 .2±0.2×10−3µm, and for case (2) was 0 .0±0.2×10−3µm. As these would result in false terms to Azof the order of 10−11, they were inconsequential for the experiment. Tests were also done with carbon blades replacing the usual CH2blades in the polarimeter. These indicated that 1% of the true events i n the PPM came from the carbon in the CH2blades (12C(p,2p)X, etc.). This had a very small contribution to the eﬀective analyzing power. As the data collection involved spin oﬀ periods intersperse d with the polarized beam, it was possible to monitor the PPM’s response (instrum ental asymme- try) to zero polarization (ideally what we would like to see i n the experiment with a perfect longitudinally polarized beam). An instrume ntal asymmetry as a function of blade position plot is presented in Figure 9. It was found that the slope of the instrumental asymmetry was strongly depend ent on the di- vergence or convergence of the beam, as then the angle of inci dence is position 15Fig. 9. Instrumental asymmetry as a function of target (blad e) position. dependent and the scattering angle dependence on position i s diﬀerent from the assumption of a parallel beam. Note that the requirement for the exper- iment is for as parallel a beam (i.e., very weakly focussed at the target) as reasonably achievable. Figure 10 shows a helicity-correlated polarization proﬁle measured by the up- stream PPM with 200 nA beam and a beam size ( σ) of 5.0 mm. Under those conditions, each PPM measures an average < P x>and< P y>to±0.002 and< xP y>and< yP x>to±7µm in one hour. The eﬀective analyzing power as a function of blade position is determined by moving the beam across the range of the blade sweep with the be am trans- versely polarized. Absolute calibration was done by compar ing the integrated result to the existing IBP. 5 Conclusions The PPM rotation control system has worked very well. It is co nvenient to use and normally maintains PPM synchronization to ±one shaft encoder line. The PPM’s have been successfully used throughout the TRIUMF Parity ex- periment (E497). For three runs (not the full data set) of dat a taken in 1997, 16Fig. 10. A beam proﬁle (top) and polarization proﬁle (bottom ) for a longitudinally polarized beam. In this example < Px>is obviously non-zero (about 1%). 171998 and 1999 (about four months) consisting of about 240 hou rs of TRIC data (the actual parity violation measurement) collection : The ‘false’ parity violating analyzing power ( Az) derived from transverse components of polar- ization coupled with a displacement from the ideal instrume ntal symmetry axis has been measured as (0 .02±0.01)×10−7. The false Azderived from the ﬁrst moments of polarization has been measured as (0 .72±0.19)×10−7. This conﬁrms the expectation that the latter is a large (inde ed, so far, the largest) correction. It is also the largest contribution to the E497 error. Im- proved PPM error to total error could be achieved by changing the number of targets (and thus the ratio of PPM data collection time to TRI C data collec- tion time) or seeking some other means of rapidly and accurat ely measuring the polarization proﬁles. References [1] S. Kistryn et al., Phys. Rev. Lett. 58, (1987) 1616. [2] R. Balzer et al., Phys. Rev. C 30, (1984) 1409. [3] P.D. Eversheim et al., Phys. Lett. B 256, (1991) 11. [4] V. Yuan, H. Frauenfelder, R.W. Harper, J.D. Bowman, R. Ca rlini, D.W. MacArthur, R.E. Mischke, D.E. Nagle, R.L. Talaga, and A .B. McDonald, Phys. Rev. Lett. 57, (1986) 1680. [5] J.M. Potter, J.D. Bowman, C.F. Hwang, J.L. McKibben, R.E . Mischke, D.E. Nagle, P.G. Debrunner, H. Frauenfelder, and L.B. Soren sen, Phys. Rev. Lett.33, (1974) 1307. [6] A.R. Berdoz et al., in Proc. of the Int’l. Conf. on Quark Lepton Nuclear Physics (QULEN‘97) , H. Ejiri, T. Kishimoto, Y. Mizuno, T. Nakano, and H. Toki, ed s., Nucl. Phys. A629 , (1998) 433c; S.A. Page et al., in Proc. of the Int’l. Symposium on Weak and Electromagnetic Interactions in Nuclei (WEIN-8 9), edited by P. Depommier, Editions Frontieres (1989) 557. [7] R.W. Harper, V. Yuan, H. Frauenfelder, J.D. Bowman, R. Ca rlini, R.E. Mischke, D.E. Nagle, R.L. Talaga, and A.B. McDonald, Ph ys. Rev. D 31, (1985) 1151. [8] J.D. Bowman et al., Phys. Rev. Lett. 34, (1975) 1184. 18[9] J. Birchall et al., inPolarization Phenomena in Nuclear Physics , AIP Conf. Proc.339, (1995) 136. [10] J. Birchall et al., TRIUMF Experiment proposal E761, (1995) unpublished. [11] M. Simonius, Phys. Lett. 41B, (1972) 415. [12] M. Simonius, R. Henneck, Ch. Jacquemart, J. Lang, W. Hae berli, and Ch. Weddigen, Nucl. Instr. and Meth. 177, (1980) 471. [13] D.E. Nagle, J.D. Bowman, C. Hoﬀman, J. McKibben, R. Misc hke, J.M. Potter, H. Frauenfelder, and L. Sorenson, in High Energy Physics with Polarized Beams and Targets , ed. by G.H. Thomas, AIP Conf. Proc. 51, (1978) 224. [14] W. Haeberli, R. Henneck, Ch. Jacquemart, J. Lang, R. M¨ u ller, M. Simonius, W. Reichart, and Ch. Weddigen, Nucl. Instr. and Meth. 163, (1979) 403. [15] J. Chlebek, S. Kuhn, P.D. Eversheim, and F. Hinterberge r, Nucl. Instr. and Meth. in Phys. Res. A256 , (1987) 98. [16] B. Vuaridel, K. Elsener, W. Gr¨ uebler, V. K¨ onig, and P. A. Schmelzbach, Nucl. Instr. and Meth. in Phys. Res. A244 , (1986) 335. [17] J. Birchall, Can. J. Phys. 66, (1988) 530. [18] R.A. Arndt, Interactive dial-in program SAID, 1994; R. A. Arndt, I.I. Strakovsky, and R.L. Workman, Phys. Rev. C 50, 2731 (1994). [19] J. Soukup et al.,Beam Instrumentation Workshop , AIP Conf. Proc. 333, (1994) 492. [20] A.R. Berdoz et al., Nucl. Instr. and Meth. A307 , (1991) 26. [21] L.G. Greeniaus and J. Soukup, TRIUMF Report No. TR-DN-8 1-1, (1987) unpublished. 19
arXiv:physics/0011008v1 [physics.atom-ph] 3 Nov 2000Higher-order binding corrections to the Lamb shift of 2Pstates U. D. Jentschura †and K. Pachucki ‡ †Max–Planck–Institut f¨ ur Quantenoptik, Hans-Kopfermann -Straße 1, 85748 Garching, Germany∗ ‡Institute of Theoretical Physics, Warsaw University, Ho˙ z a 69, 00-681 Warsaw, Poland∗ Abstract We present an improved calculation of higher order correcti ons to the one- loop self energy of 2 Pstates in hydrogen-like systems with small nuclear charge Z. The method is based on a division of the integration with res pect to the photon energy into a high and a low energy part. The high energy part is calculated by an expansion of the electron propagato r in powers of the Coulomb ﬁeld. The low energy part is simpliﬁed by the applica tion of a Foldy- Wouthuysen transformation. This transformation leads to a clear separation of the leading contribution from the relativistic correcti ons and removes higher order terms. The method is applied to the 2 P1/2and 2P3/2states in atomic hydrogen. The results lead to new theoretical values for the Lamb shifts and the ﬁne structure splitting. PACS numbers 12.20.Ds, 31.30Jv, 06.20 Jr Typeset using REVT EX 1I. INTRODUCTION The evaluation of the one-loop self-energy of a bound electr on is a long standing problem in Quantum Electrodynamics. There are mainly two approache s. The ﬁrst, developed by P. Mohr [1], relies on a multidimensional numerical integral i nvolving a partial wave expansion of the electron propagator in the Coulomb ﬁeld. This approac h is particularly useful for heavy hydrogen-like ions. The second approach is based on an expansion of the electron self-energy in powers of Zα, δESE=α π(Zα)4mF, (1) where F=A40+A41ln/bracketleftBig (Zα)−2/bracketrightBig + (Zα)A50+ (Zα)2/parenleftBig A60+A61ln/bracketleftBig (Zα)−2/bracketrightBig +A62ln2/bracketleftBig (Zα)−2/bracketrightBig +o(Zα)/parenrightBig . (2) The leading contribution as given by A41has been originally calculated by Bethe in [2]. Many others have contributed to the evaluation of higher ord ers corrections, for details see an excellent review by Sapirstein and Yennie in [3]. A very ge neral analytical method has been introduced by Erickson and Yennie in [4]. Erickson and Y ennie were able to calculate all the coeﬃcients in (2) except for A60. The calculation of corrections of ( Zα)2relative order is a highly nontrivial task because the binding Coulomb ﬁeld en ters in a nonperturbative way, and there is no closed form expression for the Dirac-Coulomb propagator. Additionally, one- loop electron self-energy contributes to all orders in Zα, and the separation of the ( Zα)2 relative contribution involves hundreds of terms. A very eﬃ cient scheme of the calculation has been introduced in [5]. It was calculated there the A60coeﬃcient for the 1 Sand 2S states in hydrogen atom. The method was based on the division of the whole expression into two parts, ELandEH, by introducing an artiﬁcial parameter ǫwhich is a cutoﬀ in the photon frequency. In the high energy part EHone expands the electron propagator in powers of the Coulomb ﬁeld and uses Feynman gauge. In the low energy p art one uses Coulomb gauge and applies a multipole expansion. The most important ingredient of this method is the expansion in the parameter ǫafter the expansion in Zαis performed (for details see the next section). The calculation presented in this paper is a further develop ment of this original method. In the low energy part we use a Foldy-Wouthuysen transformat ion. The transformation clearly identiﬁes the leading order contribution and separ ates out all higher order terms. An additional advantage is that the nonrelativistic Schr¨ o dinger-Coulomb propagator can be used here. A closed-form expression of this propagator is known in coordinate and in momentum space (for details see [6]). This method is applied to the 2P1/2and 2P3/2states. All coeﬃcients including A60are obtained. We recover all the previously known results, a nd the new results for A60are in agreement with those obtained from the extrapolation of P. Mohr’s data. Our results are relevant for single electron , smallZsystems (for example atomic hydrogen and He+), which are currently investigated with very high precisio n. New theoretical values for the Lamb shift of the 2 P1/2and 2P3/2states and the ﬁne structure summarize our calculations. 2II. THE ǫfw-METHOD The self-interaction of the electron leads to a shift of the h ydrogen energy levels. This shift at the one-loop level is given by δESE=ie2/integraldisplayd4k (2π)4Dµν(k)/angbracketleft¯ψ\|γµ 1 /negationslashp− /negationslashk−m−γ0Vγν\|ψ/angbracketright − /angbracketleft¯ψ\|δm\|ψ/angbracketright, (3) whereδmrefers to the mass counter term, and it is understood that the photon propagator Dµνhas to be regularized to prevent ultraviolet divergences. ¯ψis the Dirac adjoint ¯ψ=ψ+γ0. For theω-integration ( k0≡ω), the lower part of the Feynman integration contour CFis bent into the “right” half plane with ℜ(ω)>0 and divided into two parts, the low energy contourCLand the high energy contour CH, see Fig. 1. The ǫparameter corresponds to the cut-oﬀKwhich was introduced by H. Bethe in his original evaluation o f the low energy part of the electromagnetic shift of energy levels [2] (speciﬁca lly,K=ǫm). The two contours are separated along the line ℜ(ω) =ǫm, whereǫis some arbitrary dimensionless parameter, which we assume to be smaller than unity. This method of ω-integration has been described in detail in [5]. The two integrations lead to the high and low energy parts ELandEH, which are functions of the ﬁne structure constant αand of the free parameter ǫ. Their sum, however, δESE(α) =EL(α,ǫ) +EH(α,ǫ), (4) does not depend on ǫ. The most important step is the expansion in ǫafter the expansion in α. It eliminates, without actual calculations, many terms th at vanish in the limit ǫ→0. To be more speciﬁc, in expanding ELandEHinǫwe keep only ﬁnite terms (the ǫ0-coeﬃcients) and the terms which diverge as ǫ→0. The divergent terms cancel out in the sum, the ﬁnite terms contribute to the Lamb shift. This cancelation of the d ivergent terms is an important cross-check of the calculation. One may use diﬀerent gauges of the photon propagator for the two parts, because the gauge-dependent term vanishes in the limitǫ→0. For convenience, we use the Feynman gauge for the high and the Coulomb gauge for the low energy part. In this work, the treatment of the low energy part is largely s impliﬁed by the introduc- tion of a Foldy-Wouthuysen (fw) transformation. It enables one to clearly separate out the leading (nonrelativistic dipole) term, which gives the α(Zα)4-contribution, from the rela- tivistic corrections, which give terms in α(Zα)6. An additional advantage is the fact that all contributions to the low energy part can be evaluated using t he nonrelativistic Schr¨ odinger- Coulomb-Green’s function, whose closed-form solution is w ell known [6]. Terms which con- tribute to the Lamb shift up to α(Zα)6can be readily identiﬁed, and each of these can be calculated independently. In the low energy part we may expa nd in the photon momentum k. The terms which contribute to the Lamb shift in the order of α(Zα)6correspond to the “non-relativistic dipole” term (involving the non-relati vistic propagator and wave function), the “non-relativistic quadrupole” term and the “relativis tic dipole” term (which involves the relativistic corrections to the wave function and the Dirac -Coulomb propagator). The terms of higher order in kvanishes in the limit ǫ→0. Calculations of the high-energy part are performed almost e ntirely with the computer algebra system Mathematica [7]. Because of the presence of an infrared cut-oﬀ, one can expand the Dirac-Coulomb propagator in powers of the Coulom b potential. A subsequent 3expansion of the propagator in electron momenta is also perf ormed. This leads ﬁnally to the calculation of matrix elements of operators containing Vandpon the P-states. Because P-wave functions vanish at the origin, all of the relevant mat rix elements are ﬁnite up to the order of ( Zα)6. III. THE HIGH-ENERGY PART In this part we use the Feynman gauge ( Dµν(k) =−gµν/k2). and the Pauli-Villars regularization for the photon propagator 1 k2→1 k2−1 k2−M2, (5) so that the following expression remains to be evaluated: EH=−ie2/integraldisplay CHd4k (2π)4/bracketleftbigg1 k2−1 k2−M2/bracketrightbigg /angbracketleft¯ψ\|γµ 1 /negationslashp− /negationslashk−m−γ0Vγµ\|ψ/angbracketright − /angbracketleft¯ψ\|δm\|ψ/angbracketright(6) We start by calculating the matrix element ˜P=/angbracketleft¯ψ\|γµ 1 /negationslashp− /negationslashk−m−γ0Vγµ\|ψ/angbracketright (7) up to the order of ( Zα)6. The ﬁrst step in the evaluation of ˜Pis the expansion of the matrix M=γµ 1 /negationslashp− /negationslashk−m−γ0Vγµ in powers of the binding ﬁeld. We denote the denominator of th e free electron propagator byD(D=/negationslashp− /negationslashk−m). Realizing that the binding ﬁeld V=−(Zα)2m/ρcarries two powers of (Zα) (withρ=r/aBohr), we expand the matrix Mup toV3, which leads in turn to four matrices, denoted Mi, M0=γµ1 Dγµ, M 1=γµ1 Dγ0V1 Dγµ, M 2=γµ1 Dγ0V1 Dγ0V1 Dγµ, (8) M3=γµ1 Dγ0V1 Dγ0V1 Dγ0V1 Dγµ. withM=M0+M1+M2+M3+O((Zα)7). Deﬁning ˜Pi=/angbracketleft¯ψ\|Mi\|ψ/angbracketright, we write the element ˜Pas the sum ˜P=˜P0+˜P1+˜P2+˜P3+O((Zα)7). (9) This expansion corresponds to a division of the initial expr ession into 0-,1-,2- and 3-vertex parts. We then expand each of the matrices Miinto the standard 16 Γ matrices, which form a basis set of 4 ×4 matrices. Mi=15/summationdisplay β=0ci,βΓβwhereci,β=1 4Tr(ΓβMi). (10) 4The expansion coeﬃcients ci,βare rational functions of the binding ﬁeld, the electron and photon energy and momenta. They can therefore be expanded in powers ofα, leaving none of the electron momentum operators in the denominator. Next , we evaluate the matrix elements of these operators with the relativistic (Dirac) w ave function ψ. It is a property ofPstates, which vanish at the origin, that up to order ( Zα)6, all of the desired matrix elements are ﬁnite. As an example, we describe here the evaluation of the three-v ertex matrix element ˜P3= /angbracketleft¯ψ\|M3\|ψ/angbracketright. It takes on the same values for both 2 Pstates. Expanding M3into the 16 Γ- matrices, we ﬁnd that up to order ( Zα)6, all expansion coeﬃcients vanish except for the identity Id and γ0-matrices. The expansion coeﬃcients are explicitly c3,Id= 16V3k2−k2ω−4ω+ 3ω2+ 2 (k2+ 2ω−ω2)4≡b3,IdV3, (11) wherek=\|k\|and for simplicity m= 1, and c3,γ0= 2V3k4−8k2+ 12k2ω+ 16ω−6k2ω2−12ω2+ 4ω3−ω4−8 (k2+ 2ω−ω2)4 ≡b3,γ0V3.(12) So up to order ( Zα)6, the twoc-expansion coeﬃcients are (except for their dependence on kandω) functions of the binding ﬁeld only. Thus, the matrix elemen t˜P3is given by ˜P3=b3,Id/angbracketleft¯ψ\|V3\|ψ/angbracketright+b3,γ0/angbracketleft¯ψ\|γ0V3\|ψ/angbracketright. (13) The relevant matrix element of the wave function is /angbracketleft¯ψ\|V3\|ψ/angbracketright=/angbracketleft¯ψ\|γ0V3\|ψ/angbracketright=−1 24(Zα)6m3+O((Zα)7). (14) where the ﬁrst equality holds only in the order of ( Zα)6. The above matrix elements take on the same values for the 2 P1/2and 2P3/2states because the radial parts of both 2 Pstates are the same in the non-relativistic limit. For the other vertex parts, many more terms appear, and the ma trix elements contribute in the lower order also. We give one example here, to be evalua ted for the 1-vertex part, /angbracketleft¯ψ\|γ0p·(Vp)\|ψ/angbracketright=−5 48(Zα)4m3−283 1152(Zα)6m3for 2P1/2. (15) and /angbracketleft¯ψ\|γ0p·(Vp)\|ψ/angbracketright=−5 48(Zα)4m3−71 1152(Zα)6m3for 2P3/2. (16) For a more detailed review of the calculations see [8]. Havin g calculated ˜P, we subtract the mass-counter-term before integrating with respect to kandω. The ﬁnalkandωintegration is performed in the following way. Those terms which appear t o be ultraviolet divergent are regularized and integrated covariantly using Feynman para meter approach. The remaining terms are integrated with respect to kby residual integration and with respect to ωby changing the integration variable to 5u=√ 2mω−ω2+iω√ 2mω−ω2−iω. (17) This integration procedure is described in details in [5]. T he ﬁnal results for the high-energy- part are (for the deﬁnition of Fsee Eq. (1)) FH(2P1/2) =−1 6+ (Zα)2/bracketleftbigg4177 21600−103 180ln(2) −103 180ln (ǫ)−2 9ǫ/bracketrightbigg (18) and FH(2P3/2) =1 12+ (Zα)2/bracketleftbigg6577 21600−29 90ln(2)−29 90ln (ǫ)−2 9ǫ/bracketrightbigg . (19) IV. THE LOW ENERGY PART In this part we are dealing with low energy virtual photons, t herefore we treat the binding ﬁeld non-pertubatively. Choosing the Coulomb gauge for the photon propagator, one ﬁnds that only the spatial elements of this propagator contribut e. Theω-integration along CLis performed ﬁrst, which leads to the following expression for EL, EL=−e2/integraldisplay \|k\|<ǫd3k (2π)32\|k\|δT,ij/angbracketleftψ\|αieik·r 1 HD−(Eψ−ω)αje−ik·r\|ψ/angbracketright(ω≡ \|k\|).(20) HDdenotes the Dirac-Coulomb-Hamiltonian, δTis the transverse delta function, and αi refers to the Dirac α-matrices. In the matrix element Pij=/angbracketleftψ\|αieik·r 1 HD−(Eψ−ω)αje−ik·r\|ψ/angbracketright (21) we introduce a unitary Foldy-Wouthuysen transformation U, Pij=/angbracketleftUψ\|(Uαieik·rU+)1 U(HD−(Eψ−ω))U+(Uαje−ik·rU+)\|Uψ/angbracketright. (22) The lower components of the Foldy-Wouthuysen transformed D irac wave function ψvanish up to (Zα)2, so that we may approximate \|Uψ/angbracketrightby \|Uψ/angbracketright=\|φ/angbracketright+\|δφ/angbracketrightwith /angbracketleftφ\|δφ/angbracketright= 0, (23) where \|φ/angbracketrightis the nonrelativistic (Schr¨ odinger-Pauli) wave functio n, and \|δφ/angbracketrightis the relativistic correction. We deﬁne an operator acting on the spinors as even if it does no t mix upper and lower components of spinors, and we call the odd operator odd if it m ixes upper and lower com- ponents. The Foldy-Wouthuysen Hamiltonian consists of eve n operators only. For the upper left 2 ×2 submatrix of this Hamiltonian, we have the result [9] HFW=U(HD−(Eψ−ω))U+=m+HS+δH, (24) 6whereHSrefers to the Schr¨ odinger Hamiltonian, and δHis is the relativistic correction, δH=−(p)4 8m3+πα 2m2δ(r) +α 4m2r3σ·L (25) Now we turn to the calculation of the Foldy-Wouhuysen transf orm of the operators αiexp (k·r). The expression Uαiexp (ik·r)U+is to be calculated. Assuming that ω=\|k\| is of the order O((Zα)2), we may expand the expression Uαieik·rU+in powers of ( Zα). The result of the calculation is Uαieik·rU+=αi/parenleftbigg 1 +i(k·r)−1 2(k·r)2/parenrightbigg −1 2m2pi(α·p) (26) +γ0pi m/parenleftbigg 1 +i(k·r)−1 2(k·r)2/parenrightbigg −γ01 2m3pip2−1 2m2α r3(r×Σ)i +1 2mγ0(k·r)(k×Σ)i−i 2mγ0(k×Σ)i. In the limit ǫ→0 the odd operators in the above expression do not contribute to the self energy in (Zα)2relative order, so one can neglect the odd operators. It can b e shown easily that also the last term in the above expression (proportiona l tok×Σ) does not contribute to the Lamb shift in ( Zα)2relative order for ǫ→0. Because we can ignore odd operators, and because the lower co mponents of the Foldy- Wouthuysen transformed wave function vanish, we keep only t he upper left 2 ×2 submatrix of Eq. (26), and we write Uαieik·rU+as Uαieik·rU+≃pi m/parenleftbigg 1 +i(k·r)−1 2(k·r)2/parenrightbigg (27) −1 2m3pip2−1 2m2α r3(r×σ)i +1 2m(k·r)(k×σ)i, This can be rewritten as Uαieik·rU+=pi meik·r+δyi, (28) whereδyiis of order ( Zα)3. It is understood that the termpi meik·ris also expanded up to the order (Zα)3. Denoting by Ethe Schr¨ odinger energy ( E=−(Zα)2m/8 for 2P states) and byδEthe ﬁrst relativistic correction to E, we can thus write the matrix element Pijas Pij=/angbracketleftφ+δφ\|/bracketleftBiggpi meik·r+δyi/bracketrightBigg1 HS−(E−ω) +δH−δE/bracketleftBiggpj me−ik·r+δyj/bracketrightBigg \|φ+δφ/angbracketright.(29) In this expression, the leading term and the (ﬁrst) relativi stic corrections can be readily identiﬁed. Spurious lower order terms are not present in Eq. (29). By expansion of the denominator HS−(E−ω) +δH−δEin powers of α, the whole expression can be written in a form which involves only the Schr¨ odinger-Coulomb-Gre en’s function 7G(E−ω) =1 HS−(E−ω), (30) whose closed-form expression in coordinate space is given i n Eq. (33). We now deﬁne the dimensionless quantity P=m 2δT,ijPij. (31) Using the symmetry of the P-wave functions and Eq. (29), we easily see that Pcan be written, up to ( Zα)2, as the sum of the contributions (32, 39, 40, 41, 42, 43). The l eading contribution (the “non-relativistic dipole”) is given by Pnd=1 3m/angbracketleftφ\|pi 1 HS−(E−ω)pi\|φ/angbracketright. (32) The evaluation of this matrix element is described here as an example. For the Schr¨ odinger- Coulomb propagator, we use the following coordinate-space representation [6], G(r1,r2,E−ω) =/summationdisplay l,mgl(r1,r2,ν)Yl,m(ˆr1)Y∗ l,m(ˆr2), (33) withE−ω≡ −α2m/(2ν2). gl(r1,r2,ν) =4m aν/parenleftbigg2r1 aν/parenrightbiggl/parenleftbigg2r1 aν/parenrightbiggl e−(r1+r2)/(aν)∞/summationdisplay k=0L2l+1 k/parenleftBig 2r1 aν/parenrightBig L2l+1 k/parenleftBig 2r2 aν/parenrightBig (k+ 1) 2l+1(l+ 1 +k−ν), (34) wherea=aBohr= 1/(αm), and (k)cis the Pochhammer symbol. The evaluation of eq. (32) proceeds in the following steps: The angular integration is performed ﬁrst. Secondly, the remaining integrals over r1andr2are evaluated using the formula (see e.g. [10]), /integraldisplay∞ 0dte−sttγ−1Lµ n(t) =Γ(γ)Γ(n+µ+ 1) n! Γ(µ+ 1)s−γ 2F1/parenleftBig −n,γ,1 +µ;1 s/parenrightBig . (35) The following formula is useful for carrying out the summati on with respect to k[11], ∞/summationdisplay n=0Γ(n+λ) n!sn 2F1(−n,b;c;z) = Γ(λ) (1−s)−λ 2F1/parenleftBig λ,b;c;−sz 1−s/parenrightBig . (36) The summations lead to hypergeometric functions in the resu lt, Pnd(t) =2t2(3−6t−3t2+ 12t3+ 29t4+ 122t5−413t6) 9 (1−t)5(1 +t)3 + 256t7(−3 + 11t2) 9 (1−t)5(1 +t)52F1/parenleftBigg 1,−2t; 1−2t;/parenleftbigg1−t 1 +t/parenrightbigg2/parenrightBigg (37) where t≡√ −2mE/radicalBig −2m(E−ω)=1 2ν. (38) In this expression the terms that gives divergent in ǫterms are separated out of the hyper- geometric function, so the could be easily integrated out. T he other contributions to P(for deﬁnition fo Psee eq. (31)) are 8•the non-relativistic quadrupole, Pnq=1 3m/angbracketleftφ\|pieik·r 1 HS−(E−ω)pie−ik·r\|φ/angbracketright −Pnd, (39) •the corrections to the current αifrom the Foldy-Wouthuysen transformation, Pδy=δT,ij/angbracketleftφ\|δyi 1 HS−(E−ω)pje−ik·r\|φ/angbracketright, (40) •the contribution due to the relativistic Hamiltonian, PδH=−1 3m/angbracketleftφ\|pi 1 HS−(E−ω)δH1 HS−(E−ω)pi\|φ/angbracketright, (41) •the contribution due to the relativistic correction to the e nergy, PδE=1 3m/angbracketleftφ\|pi 1 HS−(E−ω)δE1 HS−(E−ω)pi\|φ/angbracketright, (42) •and due to the relativistic correction to the wave function, Pδφ=2 3m/angbracketleftδφ\|pi 1 HS−(E−ω)pi\|φ/angbracketright. (43) For almost all of the matrix elements we use the coordinate-s pace representation of the Schr¨ odinger-Coulomb propagator given in Eq. (33). There a re two exceptions: For the non- relativististic quadrupole, we use Schwinger’s momentum s pace representation and carry out the calculation in momentum space. A rather involved contri bution is PδH=−1 3m/angbracketleftφ\|piG(E−ω)/bracketleftBigg −(p)4 8m3+πα 2m2δ(r) +α 4m2r3σ·L/bracketrightBigg G(E−ω)pi\|φ/angbracketright.(44) whereG(E−ω) = 1/(HS−(E−ω)). The form of δHimplies a natural separation of PδH into three terms, PδH=Pp4+Pδ+PL·S. (45) ForPδ, Pδ=−1 3m/angbracketleftφ\|piG(E−ω)/bracketleftbiggπα 2m2δ(r)/bracketrightbigg G(E−ω)pi\|φ/angbracketright, (46) which involves the zitterbewegungs-term (proportional to theδ-function), we use a coordinate-space representation of the Schr¨ odinger-Cou lomb propagator involving Whit- taker functions (this representation is also to be found in [ 6]). The result for Pδ(t) is 9Pδ(t) =−α2 27t4(−3 + 4t+ 7t2−8tF2(t))2 (t2−1)4 (47) where F2(t) =2F1/parenleftbigg 1,−2t,1−2t,t−1 t+ 1/parenrightbigg . (48) Both terms Pp4andPL·S, Pp4=−1 3m/angbracketleftφ\|piG(E−ω)/bracketleftBigg −(p)4 8m3/bracketrightBigg G(E−ω)pi\|φ/angbracketright (49) PL·S=−1 3m/angbracketleftφ\|piG(E−ω)/bracketleftbiggα 4m2r3σ·L/bracketrightbigg G(E−ω)pi\|φ/angbracketright, (50) involve two propagators G(E−ω). We use the Schr¨ odinger equation and the identity [HS−(E−ω),1 r∂ ∂rr] =L2 mr3−Zα r2. (51) to rewrite them to the form that contain only one propagator w ith modiﬁed parameters. Namely, to the desired order in ( Zα), the expression with two propagators can be replaced by an expression with just one propagator, in which an ( Zα)2-correction is added to the angular momentum parameter lor to the ﬁne structure constant αin the radial part of the Schr¨ odinger-Coulomb propagator as given in Eq. (33). For t hePp4andPL·Scontributions, many more terms appear in the calculation, and derivatives o f the hypergeometric functions with respect to parameters have to be evaluated. The result c onsists of terms involving elementary functions and hypergeometric functions only, a nd other terms which involve slightly more complex functions. Some of the summations giv e rise to the Lerch transcendent Φ. Summations of the form ∞/summationdisplay k=0knξk∂ ∂b2F1(−k,b,c,z ). (52) can be evaluated with the help of Eq. (36), for more details se e [8]. Although we do not describe the calculations in detail, we stress that the summ ation with respect to the k-index is the decisive point in the calculation. In general, a sensi ble use of contiguous relations is necessary to simplify the result of any of the summations. Symbolic procedures were written to accomplish this. Through the compartmentalizat ion of the calculation achieved by the Foldy-Wouthuysen transformation, it has been possib le to keep the average length of intermediate expressions below 1000 terms. The contribution to ELdue to theδESEis given by EL=−2α πm/integraldisplayǫ 0dωωP (ω). (53) Changing the integration variable to t, we have F=−1 2/integraldisplay1 tǫdt1−t2 t5P(t). (54) 10TheP-terms are integrated with respect to tby the following procedure. Terms which give a divergence for ǫ→0 are extracted from the integrand. The extraction can be ach ieved by a suitable expansion in the argument of the hypergeometri c function(s) which appear in P(t). The extracted terms consist of elementary functions of tonly, so they can be integrated analytically. After integration, the terms are ﬁrst expand ed in (Zα) up to (Zα)2, then inǫup toǫ0. The remaining part, which involves hypergeometric functi ons, is integrated numerically with respect to tby the Gaussian method. Thet-integration leads to F-terms which we name according to the P-termsFnd,Fnq, Fδy,FδH,FδEandFδφ. TheFnd-term, which is the same for both 2 P-states, is given by Fnd=−4 3lnk0(2P) +2 9(Zα)2 ǫ. (55) We have recovered the ﬁrst 9 digits of the Bethe logarithm wit h our (Gaussian) integration procedure (the value for the Bethe logarithm given in [3] is l nk0(2P) =−0.0300167089(3)). TheFnd-term has, for ǫ→0, a divergence of +2 /9(Zα)2/ǫ, which cancels the corresponding divergence in the high energy part. All other F-terms produce logarithmic divergences in (Zα)2ln(ǫ) (see Table I). The results for the low-energy parts of the 2 P-states are FL(2P1/2) =−4 3lnk0(2P) + (Zα)2/bracketleftbigg −0.79565(1) +103 180ln/parenleftBig (Zα)−2/parenrightBig +103 180ln (ǫ) +2 9ǫ/bracketrightbigg (56) and FL(2P3/2) =−4 3lnk0(2P) + (Zα)2/bracketleftbigg −0.58452(1) +29 90ln/parenleftBig (Zα)−2/parenrightBig +29 90ln (ǫ) +2 9ǫ/bracketrightbigg .(57) The divergence in 1 /ǫand in ln(ǫ) cancels out when the low- and high-energy-parts are added. The results for the F-factors (sum of low-energy-part and high-energy-part) ar e: F(2P1/2) =−1 12−4 3lnk0(2P) + (Zα)2/bracketleftbigg −0.99891(1) +103 180ln/parenleftBig (Zα)−2/parenrightBig/bracketrightbigg (58) for the 2P1/2-state and F(2P3/2) =1 6−4 3lnk0(2P) + (Zα)2/bracketleftbigg −0.50337(1) +29 90ln/parenleftBig (Zα)−2/parenrightBig/bracketrightbigg (59) for the 2P3/2-state. The A60coeﬃcients are given by A60(2P1/2) =−0.99891(1) (60) and A60(2P3/2) =−0.50337(1). (61) The last digit is the cumulated inaccuracy of the numerical i ntegrations. The values for the A40andA61coeﬃcients are in agreement with known results [3]. 11These results can be compared to those obtained by P. Mohr [13 ] by extrapolation of his numerical data for higher Z, GSE(2) = −0.96(4), GSE(1) = −0.98(4) for 2 P1/2, (62) and GSE(2) = −0.46(2), GSE(1) = −0.48(2) for 2 P3/2, (63) where the function GSE(Z) for 2P-states is deﬁned by F=A40+ (Zα)2/bracketleftBig A61ln/parenleftBig (Zα)−2/parenrightBig +GSE(Z)/bracketrightBig . (64) BecauseGSE(Z= 0) =A60, these values are clearly in very good agreement with the res ults of our analytical calculation. Using P. Mohr’s numerical da ta [12], we have obtained the following estimates for higher order terms summarized by GSE,7 F=A40+ (Zα)2/bracketleftBig A60+A61ln/parenleftBig (Zα)−2/parenrightBig + (Zα)GSE,7(Z)/bracketrightBig , (65) GSE,7(2P1/2,Z= 1) = 3.1(5) and GSE,7(2P3/2,Z= 1) = 2.3(5). (66) One of the most important aspects of rather lengthy calculat ions such as those presented here is to avoid errors. The result has been checked in many wa ys. Except for checking the values of the terms divergent in ǫ, it was also checked the value of each P-contribution as ω→0. It can be shown easily that the sum of all contributions to t he matrix element P in the low-energy part must vanish in the limit ω→0. Care must be taken when checking the sum, because after the Foldy-Wouthuysen transformatio n, hidden terms are introduced which do not contribute to the Lamb shift, but contribute in t he limitω→0. The hidden terms originate from the odd operators in Eq. (26). Taking in to account these terms, the sum vanishes for both states. V. OTHER CONTRIBUTIONS TO THE LAMB SHIFT For the Lamb shift L, we use the implicit deﬁnition E=mr[f(n,j)−1]−m2 r 2(m+mN)[f(n,j)−1]2+L+Ehfs, (67) whereEis the energy level of the two-body-system and f(n,j) is the dimensionless Dirac energy,mis the electron mass, mris the reduced mass of the system and mNis the nuclear mass. For the ﬁnal evaluation of the Lamb shift the following contr ibutions are added: 1. One-loop self energy. The coeﬃcients are presented in thi s work. For the determination of the Lamb shift the reduced mass dependence of the terms has to be restored. The relevant formulae are given in [3]. For example, the A60have a reduced mass dependence of ( mr/m)3. We use Eq. (66) to estimate the theoretical uncertainty fro m the one–loop contribution. 122. Vacuum polarization correction. It enters for P-states in higher order (for the formulae see [3], p. 570). 3. Two-loop contributions due to the anomalous magnetic mom ent [15]. It is given in analogy to the one-loop contribution as δE2−loop=/parenleftbiggα π/parenrightbigg2 m(Zα)4 n3[B40+...] (68) where theB-coeﬃcients are labeled in analogy to the A-coeﬃcients for the one-loop self energy. The B40coeﬃcient is due to the anomalous magnetic moment of the elec tron. It is given as B40=Cjl 2(2l+ 1)/bracketleftBigg197 72+π2 6−π2ln 2 +3 2ζ(3)/bracketrightBigg/parenleftbiggmr m/parenrightbigg2 , (69) whereCjl= 2(j−l)/(j+ 1/2). 4. Two loop contributions in higher order. Recently, the log arithmic term B62=/bracketleftBigg4 27n2−1 n2ln2/parenleftBig (Zα)−2/parenrightBig/bracketrightBigg/parenleftbiggmr m/parenrightbigg3 , (70) has been calculated in [14]. The B62term, which is enlarged by the logarithm, probably dominates the contributions to the two-loop self energy in h igher order. So the result may also be used to estimate the theoretical uncertainty of t he two–loop contribution, coming mainly from the unknown B61coeﬃcient. It is taken to be half the contribution fromB62. 5. Three-loop self energy as given by the anomalous magnetic moment [15]. δE3−loop=/parenleftbiggα π/parenrightbigg3 m(Zα)4 n3[C40+...] (71) where C40=/bracketleftBigg 2Cjl 2(2l+ 1)1.17611(1)/bracketrightBigg/parenleftbiggmr m/parenrightbigg2 . (72) 6. The additional reduced mass dependence of order ( mr/mN)2(Zα)4[3], which we will refer to as the ( Zα)4recoil contribution, δErec,4=(Zα)4 2n3m3 r m2 N/parenleftBigg1 j+ 1/2−1 l+ 1/2/parenrightBigg (1−δl0), (73) 137. The Salpeter correction (relativistic recoil) in order ( Zα)5as given in [3]. The formula is forP-states δErec,5=m3 r mmN(Zα)5 πn3/parenleftBigg −8 3lnk0(n)−7 31 l(l+ 1)(2l+ 1)/parenrightBigg . (74) 8. Relativistic recoil corrections in the order of ( Zα)6mr/mN, δErec,6=m2 mN(Zα)6/bracketleftBigg1 2/angbracketleftφ\|L2 r4\|φ/angbracketright/bracketrightBigg . (75) The formula for P-states has been ﬁrst calculated in [16]. Th is general form has been obtained by us. The above contributions are listed in table II for the 2 Pstates. VI. RESULTS AND CONCLUSIONS The new theoretical values for the Lamb shifts of the 2 P1/2and 2P3/2states are L(2P1/2) =−12835.99(8) kHz (76) and L(2P3/2) = 12517.46(8) kHz. (77) From the values of the 2 PLamb shifts, the ﬁne structure can be determined. It turns ou t that the limiting factor in the uncertatinty is the experime ntal value of the ﬁne structure constantα. Using a value of [17] (1987) α−1= 137.0359895(61) (44 ppb) , (78) the ﬁne structure can be determined as E(2P3/2)−E(2P1/2) = 10969043(1) kHz . (79) With the most recent and most precise value of αavailable [18] (1995), α−1= 137.03599944(57) (4.2 ppb) , (80) we obtain a value of E(2P3/2)−E(2P1/2) = 10969041 .52(9)(8) kHz , (81) where the ﬁrst error originates from the uncertainty in αand the second from the uncertainty in the Lamb shift diﬀerence. Our result for the ﬁne structure disagrees with that used by Hagley and Pipkin in [19] for the determination of L(2S−2P1/2). Therefore their result of L(2S−2P1/2) = 1057839(12) is to be modiﬁed and according to our calculat ion it should be 14L(2S−2P1/2) = 1057842(12) kHz . (82) Precise theoretical predictions for P-states could be used to compare two diﬀerent kind of measurements of Lamb shifts in the hydrogen. One is the cla ssic 2S1/2-2P1/2Lamb shift measured by several groups [20], [21], [19], and the second i s the combined Lamb shift L(4S−2S)−1 4L(2S−1S) as measured by the H¨ ansch group (for a review see [22]). The experimental value of 2S Lamb shift can be extracted from E(2 S-2P1/2) having the precise value for 2 P1/2Lamb shift, and can also be determined from the combined Lamb shift through the formula L(2S) =8 7/bracketleftBigg/parenleftbigg L(4S)−5 4L(2S) +L(1S)/parenrightbigg exp−/parenleftbigg L(4S)−17 8L(2S) +L(1S)/parenrightbigg theo/bracketrightBigg ,(83) where the subscript expdenotes experimental, and the subscript theodenotes theoretical values. This theocombination has the property that terms scaling 1 /n3cancel out, which means that almost all QED eﬀects do not contribute, and there fore the quantity can be precisely determined. Such a comparison of completely diﬀe rent experimental techniques is an interesting and valuable test of high precision experime nts. The method of calculation presented in this paper could be di rectly applied for the evalu- ation of Lamb shifts and the ﬁne structure in two electron sys tems, for example in helium or positronium. It was a purpose of this method to use only a Schr ¨ odinger-Coulomb propaga- tor, and relativistic eﬀects are incorporated through the F oldy-Wouthuysen transformation. This method clearly separates out the lower and the higher or der terms, and expresses the energy shift through the matrix elements of nonrelativisti c operators. ACKNOWLEDGMENTS This work was done while one of us (K. P.) was a guest scientist at the Max-Planck- Institute for Quantum Optics. The authors would like to than k T. W. H¨ ansch for hospitaliy, encouragement and stimulation. We are very grateful to P. Mo hr for supplying extrapola- tion data of his 1992 calculations, and to M. Weitz and A. Weis for carefully reading the manuscript. (U. J.) would also like to thank H. Kalf for helpf ul discussions with respect to the treatment of hypergeometric functions. Note added (2000): The analytic results for higher-order bi nding corrections to the Lamb shift of2P1/2and2P3/2–states (in particular, the A60–coeﬃcient) have recently been conﬁrmed by an improved numerical calculation in the range of low nuclea r charge numbers Z= 1–5. For details see the e-print physics/0009090 . 15REFERENCES ∗email adresses of authors: (U.D.J.) jentschura@physik.tu-dresden.de, (K.P.) krp@f uw.edu.pl. [1] P. J. Mohr, Ann. Phys. (N.Y) 88, 26, 52 (1974). [2] H. A. Bethe, Phys. Rev. 72(1947), p. 339. [3] J. Sapirstein and D. Yennie, in “Quantum Electrodynamic s”, edited by T. Kinoshita, World Scientiﬁc, Singapore, 1990. [4] G. W. Erickson and D. R. Yennie, Ann. Phys. (N.Y) 35, 271, 447 (1965). [5] K. Pachucki, Phys. Rev. A, 46648, (1992), Ann. Phys. (N.Y.), 2261, (1993). [6] R. A. Swainson and G. W. F. Drake, J. Phys. A Math. Gen. 24(1991), 79. [7] S. Wolfram, “Mathematica-A System for Doing Mathematic s by Computer”, Addison- Wesley, Reading (MA), 1988. [8] U. Jentschura, “Theorie der Lamb-Verschiebung in wasse rstoﬀartigen Systemen”, mas- ter thesis, University of Munich, 1996. [9] C. Itzykson and J. Zuber, Quantum Field Theory, Mc Graw-H ill, New York, 1980. [10] H. Buchholz, “The Conﬂuent Hypergeometric Function”, Springer Verlag, New York/Berlin, 1969. [11] H. Bateman, “Higher Transcendental Functions”, McGra w-Hill, New York, 1953. [12] P. Mohr, Phys. Rev. A, 46(1992), p. 4421. [13] P. Mohr, in Atomic, Molecular, and Optical Physics Handbook , ed. by G. W. F. Drake, AIP N. Y., 1996 (to be published). [14] S. Karshenboim, J. Phys. B 29, 29 (1996). [15] T. Kinoshita in “Quantum Electrodynamics”, edited by T . Kinoshita, World Scientiﬁc, Singapore, 1990. [16] E. Golosov, A. S. Elkhovskii, A. I. Milshtein, I. B. Khri plovich, JETP 80 (2), 208. [17] E. R. Cohen and B. N. Taylor, Rev. Mod. Phys. 59(1987), 1121. [18] T. Kinoshita, Phys. Rev. Lett. 75(1995), 4728. [19] E. W. Hagley and F. M. Pipkin, Phys. Rev. Lett. 72(1994), 1172. [20] S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 46(1981), 232. [21] V. G. Pal’chikov, Yu. L. Sokolov, and V. D. Yakovlev, JET P lett. 38(1983), 418. [22] K. Pachucki, D. Leibfried, M. Weitz, A. Huber, W. K¨ onig , and T. W. H¨ ansch, J. Phys. B29(1996), 177. 16TABLES contribution 2P1/2 2P3/2 Fnq −1.201150(1) + 49 /90ln/parenleftbigǫ/(Zα)2/parenrightbig−1.201150(1) + 49 /90ln/parenleftbigǫ/(Zα)2/parenrightbig Fδy 0.791493(1) −2/9ln/parenleftbigǫ/(Zα)2/parenrightbig0.531475(1) −2/9ln/parenleftbigǫ/(Zα)2/parenrightbig FδH 0.322389(1) −47/288ln/parenleftbigǫ/(Zα)2/parenrightbig0.293749(1) −35/288ln/parenleftbigǫ/(Zα)2/parenrightbig FδE 0.040095(1) + 5 /96ln/parenleftbigǫ/(Zα)2/parenrightbig0.008019(1) + 1 /96ln/parenleftbigǫ/(Zα)2/parenrightbig Fδφ −0.748478(1) + 13 /36ln/parenleftbigǫ/(Zα)2/parenrightbig−0.216612(1) + 1 /96ln/parenleftbigǫ/(Zα)2/parenrightbig sum −0.79565(1) + 103 /180ln/parenleftbigǫ/(Zα)2/parenrightbig−0.58452(1) + 29 /90ln/parenleftbigǫ/(Zα)2/parenrightbig TABLE I. Contributions of relative order ( Zα)2to the low energy part FLfor the 2 P1/2and 2P3/2states contribution 2P1/2in kHz 2P3/2in kHz one-loop self-energy −12846.92(2) 12547.95(2) two-loop self-energy 25.98(7) −12.79(7) three-loop self-energy −0.21 0.10 vacuum polarization −0.35 −0.08 (Zα)4recoil 2.16 −1.08 (Zα)5recoil −17.08 −17.08 (Zα)6recoil 0.42 0.42 sum for 2 P1/2 −12835.99(8) 12517.46(8) TABLE II. Contributions to the Lamb shift in kHz for the 2 P1/2and 2P3/2states. Estimates of the contributions of uncalculated higher order terms are gi ven in the text. Where no uncertainties are speciﬁed, they are negligible at the current level of pre cision. 17FIGURES -6 # !' &- - -××××× ℜ(ω)ℑ(ω) CF CFCL CLCH CHǫ 2m FIG. 1. The ω-integration contour used in the calculation. Bending the F eynman contour CF in the speciﬁed way leads to the high and low energy parts CHandCL. Lines directly below and above the real axis denote branch cuts from the photon and ele ctron propagator. Crosses denote poles originating from the discrete spectrum of the electro n propagator. 18
arXiv:physics/0011009 3 Nov 2000 1 Abstract-- Accurate phase-locked 3:1 division of an optical frequency was achieved, by using a continuous-wave (cw) doubly resonant optical parametric oscillator. A fr actional frequency stability of 2 ×10-17 of the division process has been achieved for 100 s integration time. The technique developed in this work can be generalized to the accurate pha se and frequency control of any cw optical parametric osci llators. Index Terms—Optical parametric oscillators, high pre cision measurements, phase-locking loops. I. INTRODUCTION Novel compact optical frequency chains rely on visi ble to near IR broad frequency combs (spanning one octave) provided by a femtosecond laser that is spectrally broadened by self-phase modulation in photonic band gap fibers [1-2]. The traceability of the frequency mea surement to the microwave primary standard is provided by th e phase-locking of the RF repetition rate of the femt osecond laser, which defines the comb mode spacing. Provide d that we are able to synthesize a frequency difference eq ual to ν/N within the span of the comb ( ν is the frequency to be determined and N an integer divider), the frequenci es ν, and the related subharmonics ν/Ν and (1-N-1)ν can be deduced in a straightforward step. Thus, the use of an opti cal division by N can considerably reduce the required span of the comb generator, and at the same time, reduce by N2 the φ noise requirement for the phase-locking (or beatno te counting) of two of the frequency markers to the ne arest comb modes. Continuous-wave optical parametric oscillators (OPO ) are promising optical frequency dividers. Divide-by -two (N=2) frequency degenerate OPOs have been demonstra ted [3]. For a non-degenerate 3:1 OPO (3 ω→2 ω, ω), the idler wave must be frequency doubled so that the RF beat between the signal and the doubled-idler waves can be used to phase lock the device to zero or to an RF refere nce frequency [4]. The advantage of these OPO-based div iders is that they require only a single pump laser sourc e to generate subharmonics outside the comb spectrum. In a free running OPO, the phase coherence of the subharmonic Laboratoire Primaire du Temps et des Fréquences (BN M- LPTF), Bureau National de Métrologie / Observatoire de Paris, 61, Avenue de l’observatoire, F-75014, Paris (France) waves is ultimately limited by the phase diffusion noise stemming from the spontaneous parametric fluorescen ce. Once these waves are phase-locked, the coherent nat ure of the parametric division process leads to subharmoni c phase- noise reduction by N2 relative to the pump laser phase noise. We report here on an accurate phase-locked o ptical frequency divider of a diode laser operating in the range 840-850 nm. We show that even with a weak beatnote signal-to-noise ratio, and without a fast electro-o ptic cavity length actuator, it is possible to achieve a very h igh fractional stability and long-term operation of the divider. II. THE DRO DIVIDER SETUP A. Experimental The doubly resonant optical parametric oscillator ( DRO) is pumped by a master-oscillator power amplifier (MOPA ) AlGaAs diode laser which is optically injected by a AlGaAs extended-cavity diode laser (Fig.1). The pum p wavelength is λp=843.06 nm and its short-term linewidth 100 kHz. The available power at the DRO input is 0. 4 W. The nearly spherical DRO cavity consists of two hig hly reflective (at the signal and idler wavelengths) Zn Se mirrors spaced by L=106 mm. Their radius of curvature is R= 50 mm. The cavity length can be tuned using a 20 mm lo ng piezoelectric transducer (PZT). A multi-grating, 19 mm long periodically poled lithium niobate (PPLN, poli ng period Λ=29.2 µm) is temperature phase-matched (T= 100°C) for the 3 ω→(2ω±δ)+(ω/G109δ) interaction. The δ quantity ( δ<<ω) represents the small radio-frequency mismatch from perfect division ratio. An intra-cavi ty CaF 2 Brewster plate was inserted in order to couple out as much as 3 mW of idler power (without the plate only 200 µW exits the rear mirror), at the expense of an increa sed oscillation threshold (65 mW compared with 15 mW). The available signal power is 6 mW. We perform the seco nd- harmonic generation (SHG) of the idler wave in a PP LN sample (Λ=35 µm, T=68°C). About 3 mW of signal (2 ω±δ) and 5 nW of useful doubled-idler (2 ω/G1092δ) waves are mixed on a 5 GHz bandwidth avalanche photodiode. Th e resulting 3 δ frequency beat signal is used to control the division ratio of the DRO. Further details of the experimental setup can be found elsewhere [5]. A phase-locked frequency divide-by-3 optical parametric oscillator A. Douillet, J.-J. Zondy, G. Santarelli, A. Makdiss i, and A. Clairon 2 Fig. 1. The DRO, idler SHG and 3 δ beanote detection set- up. FI: Faraday isolator; PBS: polarizing beamsplit ter; LWP: long-wave pass filter; CFP: confocal Fabry-Per ot; PZT: piezo-electric transducer.. Two wavemeters (no t shown) enable to tune the DRO close to the 3:1 divi sion. B. Single mode pair operation and beatnote detectio n Due to the weak dispersion of the extraordinary ind ex of refraction of lithium niobate from 2.53 µm to 1.26 µm, the signal and idler free spectral ranges (FSR ≈ 1.15 GHz) differ by only 1 or 2 MHz. Consequently more than 7 00 nearly resonant signal-idler mode pairs can experie nce similar gain within a mode cluster, giving rise to permanent axial mode hops (a few gigahertz away) or cluster h ops (a few tens of gigahertz) under free running DRO opera tion. The DRO natural tendency to mode-hop requires a permanent and effective sub-nanometric servo contro l of its cavity length in order to detect a stable beatnote prior to the phase-locking step. Single mode pair operation is a chieved with a standard side-of-fringe ( sidelock) servo. The error signal of this servo loop is built by comparing the signal wave power detected by an InAs photodiode to a refe rence voltage that sets the frequency detuning s cssω−ω=∆ of the signal wave (sω) from the cavity eigenmode frequency ss cLqc2 /=ω (sL is the signal wave optical path length of the resonator and q the mode number). This error signal is then integrated and fed back to the PZT transducer. The bandwidth of the sidelock integral servo is limited to 1 kHz by mode-hops. Let us note that this sidelock servo is not a pure frequency servo. While it maintains constant b oth relative detunings, it does not correct for the dri ft of the cavity eigenmode frequencies. Furthermore it conver ts any power fluctuation into a frequency correction. The resulting beat at 3δ under pure sidelock servo is hence expected to be noisy. Under sidelock servo, the DRO can oscillate on a single mode pair during 5-15 minutes, which leaves enough time to proceed to the phase-locking step. The selection of the appropriate mode pairs (e.g. t hose yielding a beanote falling within the 5 GHz electro nic detection bandwidth) is the first experimental diff iculty we had to overcome. Because of the high mode pair dens ity, the particular pair captured by the sidelock is usu ally random. Fine tuning of the OPO-PPLN temperature and of the sidelock reference voltage have to be repetitiv ely processed until a 3 δ beat signal in the frequency range 200 MHz-5 GHz, with a typical power level of –50dBm and a signal to noise (S/N) ration of /G9730 dB in a 100 kHz bandwidth, is detected on the spectrum analyzer. Th e beat frequency drift is rather low (less than 1 MHz per minute). Unfortunately the large short-term frequency jitter (/G97300 kHz peak-to-peak excursion) under sidelock servo combined with the limited bandwidth of the unique P ZT transducer (<10 kHz) prevented us from performing a direct phase locking, by summing (after a proper filtering ) the phase error derived from a stable RF source with th e sidelock error signal. With a wide enough control bandwidth provided for instance by an electro-optic phase modulator [3-4], such a procedure would readily wor k. In the present case, since the sidelock servo (which m aintains the pre-requisite single mode operation) has to be maintained during the phase locking step, various n ested servo loops that we shall detail in the following s ection have to be implemented simultaneously in order to a chieve a phase locked divider. III. THE PHASE-COHERENT FREQUENCY DIVIDER The ways we found to circumvent these difficulties are multiple. Given the noisy feature of the free runni ng beatnote, the procedure must comply to the followin g preliminary steps. At first we reduced the large be at frequency fluctuations by replacing (in a continuou s way to prevent a mode hop) the sidelock servo by a much qu ieter pure frequency lock servo. This consequently reduces th e required dynamical range of the phase servo by mini mizing the beat frequency jitter and drift. Secondly we in creased further the dynamic range of the phase error signal by dividing the beanote frequency (by 128) in order to reduce the control bandwidth required to achieve a stable (free of phase jumps) phase locked loop. Finally, to increas e the long-term operation of the phase-locked divider (li mited by the occurrence of a mode hop), we developed a metho d allowing the coexistence of the frequency (FL) and the phase locked (PL) loops. Fig.2 sketches the various nested loops that we have implemented to achieve a long-te rm phase locked, mode-hop-free operation during more t han one hour. Let us describe now in detail the phase locking ste ps. Once a “free running” beatnote is captured accordin g to the procedure described in section II.B, its frequency is down- converted to a fixed RF frequency of 100 MHz by mix ing it with a microwave synthesizer synchronized to a high stability 10 MHz reference from an H-maser. To avoi d false triggering of the digital by-128 frequency divider, due to additive AM noise, we have improved the S/N ratio o f the down-converted beatnote by phase-locking a 100 MHz voltage-controlled oscillator (VCO) to it. This ser vo loop is denoted the tracking loop in fig.2. The bandwidth of this phase loop is wide enough (1 MHz) to track almost 3 perfectly the phase/frequency fluctuations of the o ptical beat. BS idler SHG Avalanche photodiode Microwave synthesizer Loop filter ~ VCO Tracking Loop Loop filter Frequency Loop RF Synthesizer Phase Loop Sidelock Loop amplifier DBM Phase detector PZT Loop filter Loop filter OPO cavity signalPD H-MASER ÷128 Σ DBM PPLN Fig. 2. The phase/frequency nested loops block diag ram, comprising. The sidelock loop is disabled once the frequency loop is effective. DBM: double-balanced m ixer; PD: InAs photodiode. The 20 dB S/N ratio achieved in the loop bandwidth makes very improbable noise-induced cycle slips. The high er S/N ratio at the VCO output also allows convenient coun ting of the beat frequency with a RF counter referenced to the H- maser, for further diagnosis of the stability of th e DRO divider. When the tracking loop is closed, the correction voltage applied to the VCO varicap is proportional to the frequency difference between the down-converted instantaneous frequency and the free-running VCO frequency. This voltage provides the error signal o f the frequency loop (FL) of fig.2. By feeding back (with an appropriate sign) this correction voltage to the PZ T, one slaves in turn the jittered 3 δ beat frequency to the natural VCO frequency. Tuning the VCO frequency results hen ce in the tuning of the 3 δ beat frequency. To insure a permanent and stable control of the DRO cavity the FL and sidelock error signals are summed before integratio n and both loops are used simultaneously for a while. The stable coexistence of both loops is possible because both error signals have a large dynamic range and provide simi lar physical information. When the gain of the FL servo is sufficiently large the sidelock loop can be disable d without inducing a mode hop. Operating the divider under pu re FL provides a strong reduction of the beatnote fluctua tions down to a few kiloHertz level compared with 300 kHz when both loops are coexisting, meaning that the si delock acts as a noise source for the FL. As a consequence , under pure FL operation the DRO tendency to mode hop is t hen notably reduced. In Fig. 3 we have plotted the Alla n standard deviation σy(τ) of the DRO, under sidelock (upper curve) and frequency lock (lower curve). The data w ere derived from 3 δ frequency measurements using 1s counting gate time. The quantity σy , where y is defined as y=<δ(t)>/iν and νi=118.6 THz is the idler frequency, measures the stability of the optical frequency div ision. Fig 3 illustrates the additional frequency noise brough t by the sidelock servo, as compared with the FL servo. The origin of this additional noise stems from the hybrid natu re of the information carried by the side-of-fringe discrimin ator signal, as discussed in section II. In particular w e suspect a specific thermally induced AM to FM noise conversio n processes [5] to be the cause of the large frequenc y jitter of the 3δ beat under pure sidelock servo. Indeed, the 300 kH z jitter cannot be reasonably imputed either to the s mall pump laser amplitude noise (1% intensity fluctuation wou ld correspond to 10 kHz frequency excursion given an estimated cavity linewidth of ~5 MHz) or to the aco ustical cavity perturbations since they are efficiently cor rected by the FL loop. 10010110210-1210-1110-1010-9 Allan Deviation, σy(τ) Averaging Time, τ(s) Fig. 3. Allan standard deviation of the 3:1 divider frequency noise, relative to the idler frequency under sidelo ck (circles) and frequency lock (triangles). The frequency locki ng of the OPO beatnote to the VCO decreases the beat nois e by almost two decades at τ=1s. To this point, the beat fluctuations are low enough to allow the final phase-locking step. The 100 MHz VCO output is digitally divided by 128 and the resultin g output signal mixed in a digital phase detector with a 781 kHz synthesizer, synchronized to the H-maser. The outpu t phase error signal has a 128 π peak-to-peak dynamic range, because of the division stage, and hence allows a r educed phase locking loop bandwidth. The most natural way to control the beatnote frequency when the FL loop is closed is to change the VCO varicap voltage. We took advan tage of this entry point to phase lock the DRO divider. In this way we achieved in fact a cascaded phase locking of the beatnote. This loop is denoted the phase loop (PL) in fig.2. Hence the VCO itself acts as a second fast transduc er of the DRO divider. Furthermore, summing on the VCO varica p the tracking and phase lock correction signals lead s to a simultaneous operation of both the FL and the PL lo ops. Under FL and PL action, the down-converted beat frequency fluctuations dramatically dropped to the millihertz level for τ=100 s counting gate time. The residual phase error at the output of the phase detector was displayed on an FFT spectrum analyzer (fig.4) and used to adjust the relative gains of the FL and PL by minimizing the residual phase error variance dffS)(2 φ∫=σφ . The small peak at 22 kHz is due to a PZT 4 resonance frequency which ultimately limits the ove rall gain of the system. The sharp low-frequency peaks a t ~ 50 Hz and ~100Hz are due to the electrical power suppl y pickups and the remaining broader peaks (20, 250 a nd 1000 Hz) probably due to the acoustical resonance of the DRO mechanical structure. The major contribution to 2 φσ arises from the broad noise feature between 103 and 104Hz in fig.4, which we identified as possibly originati ng from thermo-optically induced AM to FM noise conversion (a weak thermal self-stabilization of the optical path length of the DRO was observed as in ref. [5]). The total pha se noise variance deduced from the spectrum of fig.4 is 2 . 12=σφ rad2 corresponding to only =σ−φ)exp(230% of the beatnote energy in a narrow coherent peak. Indeed, the measu red residual 3 δ beatnote linewidth was 3 kHz, and no coherent peak was clearly visible. The DRO could remain phas e- locked for more than one hour, without any mode hop , which highlights the robustness of the combined FL/ PL loops. The numerical modeling of the action of the nested loops reveals that such a coexisting FL/PL loop sys tem can tolerate very large frequency jumps (hence the redu ced mode hop events). In particular, when the phase goe s out of lock the FL brings back the frequency error within the capture range of the phase lock loop, shortening it s recapture delay by a factor of 100. Fig. 4. Residual phase noise spectrum of the phase locked divider (relative to the offset δ quantity scaled to the idler frequency), measured from the phase error signal, w hen all loops in fig.2 (except the sidelock loop) are close d. The Allan standard deviation of the phase-locked divider is reported in fig.5. The improvement of th e short- term (τ=1s) stability of the division process is nearly th ree orders of magnitude compared with the stability und er FL action only (see fig.3). For longer integration tim e (up to τ=100s gate time), the stability decreases as τ-1, meaning that the residual phase excursion is bounded to les s than 2π in the long term. This also demonstrates that no cycle slips occurred during the measurement. This is a clear si gnature of the phase locking of the nearly subharmonic wave s. IV. CONCLUSION We have achieved a phase-coherent optical frequency division by 3 of 843 nm light, using a doubly reson ant optical parametric oscillator. A relative optical d ivider frequency stability of 2 ×10-17 for 100 s integration time, at the idler wave is achieved, which highlights the po tential resolution of parametric dividers in a frequency measurement setup. This PPLN-DRO also generates a comb of phase-locked frequencies ( ωp/3, 2ωp/3, ωp, 4/3ωp, 5ωp/3, 2ωp) from various non-phase matched conversion mixings that occur within the chip. Measuring any i nterval between these sidebands leads to a simple determina tion of all frequencies of the comb. Combined with a femtos econd Ti:Sa laser clock [2], such a phase-locked OPO subharmonic comb can provide a direct microwave to deep mid-IR link or reduce by one third the required spa n of repetition rate stabilized femtosecond lasers. An a lternative implementation of 3:1 DRO dividers would consist in patterning both OPO and idler SHG interactions on a single PPLN chip. The intra-cavity cascaded nonlinearities would then induce a self phase locking of the signal and idler waves by mutual injection locking. Such an all-opti cal phase locking would avoid the use of the complex electronic phase-lock loops. 1 001 011 02 1 0- 1 71 0- 1 61 0- 1 5 σ(τ) (Hz)σ y(τ) τ ( s )1 0-21 0-1 Fig. 5. Allan standard deviation of the 3:1 divider frequency noise, relative to the idler frequency (left axis) and absolute variance of δ, as a function of the integration counter gate time (τ =1, 10, 100s). V. REFERENCES [1] J.Reichert, R.Holzwarth, Th.Udem, and T.W. Häns ch, “Measuring the frequency of light with mode-locked lasers” Opt . Commun., Vol. 172, pp. 59-68 (1999) [2] S.A. Diddams, D.J. Jones, J. Ye,nS.T. Cundiff, J. L. Hall, J.K. Ranka, R.S. Windeler, R. Holzwarth, T. Udem and T.W . Hänsch “Direct link between microwave and optical frequenc ies with a 300 THz femtosecond laserg comb,” Phys. Rev. Lett., Vol . 25, pp. 186- 188 (2000) [3] D. Lee and N.C. Wong, “ Tunable optical frequen cy division using a phase-locked optical parametric oscillator” , Opt. Lett., Vol. 17, pp. 13-15 (1992 ) [4] S. Slyusarev, T. Ikegami, S. Ohshima, “Phase-co herent optical frequency division by three of 532-nm laser light w ith a continuous- wave optical parametric oscillator”, Opt. Lett., Vo l. 24, pp. 1856- 1858 (1999). [5] A. Douillet, J.- J. Zondy, A. Yelisseyev,S. Lob anov and L. Isaenko, “Stability and frequency tuning of thermal ly loaded continuous-wave AgGaS 2 optical parametric oscillators”, J. Opt. Soc. Am. B, Vol. 16, pp. 1481-1498, (1999)
arXiv:physics/0011010v1 [physics.plasm-ph] 3 Nov 2000Diﬀusion eﬀects on the breakdown of a linear ampliﬁer model driven by the square of a Gaussian ﬁeld A. Asselaha, P. Dai Prab, J. L. Lebowitzc, and Ph. Mounaixd aLATP, UMR 6632 du CNRS, Centre de Math´ ematique et Informati que, Universit´ e de Provence, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France. bDipartimento di Matematica, Politecnico di Milano Piazza L eonardo da Vinci 32, I-20133 Milano, Italy. cDepartments of Mathematics and Physics, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854-8019. dCentre de Physique Th´ eorique, UMR 7644 du CNRS, Ecole Polyt echnique, 91128 Palaiseau Cedex, France. (February 2, 2008) Abstract We investigate solutions to the equation ∂tE − D ∆E=λS2E, where S(x,t) is a Gaussian stochastic ﬁeld with covariance C(x−x′,t,t′), and x∈Rd. It is shown that the coupling λcN(t) at which the N-th moment /an}b∇acketle{tEN(x,t)/an}b∇acket∇i}ht diverges at time t, is always less or equal for D>0 than for D= 0. Equality holds under some reasonable assumptions on Cand, in this case, λcN(t) = Nλc(t) where λc(t) is the value of λat which /an}b∇acketle{texp[λ/integraltextt 0S2(0,s)ds]/an}b∇acket∇i}htdiverges. TheD= 0 case is solved for a class of S. The dependence of λcN(t) ond is analyzed. Similar behavior is conjectured when diﬀusion is replaced by diﬀraction, D → iD, the case of interest for backscattering instabilities in laser-plasma interaction. PACS numbers: 05.10.Gg, 02.50.Ey, 52.40.Nk Typeset using REVT EX 1I. INTRODUCTION We investigate the development of a linear ampliﬁcation in a system driven by the square of a Gaussian noise. This problem arose and continues to be of interest in modeling the backscattering of an incoherent high intensity laser light by a plasma. There is a large litterature on this topic, and we refer the interested reade r to Ref. [1] for background. Our starting point here is the work by Rose and Dubois [2] who i nvestigated the following equation for the complex amplitude E(x,z) of the scattered electric ﬁeld   ∂zE(x,z)−iD∆E(x,z) =λ\|S(x,z)\|2E(x,z), z∈[0,L], x∈Λ⊂R2,andE(x,0) =E0(x).(1) In Eq. (1), zandxcorrespond to the axial and transverse directions in a plasm a of length Land cross-sectional domain Λ. The input at z= 0,E0(x), is a given function of xand Λ will be generally taken to be a torus (e.g. in numerical solut ions of Eq. (1) using spectral methods). The coupling constant λ >0 is proportional to the average laser intensity and Dis a constant parameter introduced for convenience. The com plex amplitude of the laser electric ﬁeld S(x,z) is a homogeneous Gaussian stochastic ﬁeld deﬁned by /an}b∇acketle{tS(x,z)/an}b∇acket∇i}ht=/an}b∇acketle{tS(x,z)S(x′,z′)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tS(x,z)S(x′,z′)∗/an}b∇acket∇i}ht=C(x−x′,z−z′), where the correlation function C(x,z) is the solution to   ∂zC(x,z) +i 2∆C(x,z) = 0, z∈[0,L], x∈Λ,andC(x,0) =C(x),(2) withC(x) a given function of x[3], normalized so that C(0)≡ /an}b∇acketle{t\|S(x,z)\|2/an}b∇acket∇i}ht= 1. Using heuristic arguments and numerical simulations, Rose and DuBois found that the expected value of the energy density of the scattered ﬁeld /an}b∇acketle{t\|E(x,L)\|2/an}b∇acket∇i}htdiverged for every L>0 asλincreased to some critical value λc(L). The average /an}b∇acketle{t\|E\|2/an}b∇acket∇i}htis over the realizations of the Gaussian ﬁeld S. This divergence indicates a breakdown in the assumptions m ade in 2deriving Eq. (1), which neglects both nonlinear saturation and transient time evolution [2,4]. Physically, it can be interpreted as indicating a change in t he nature of the ampliﬁcation caused by the plasma. To see the origin of this divergence in its simplest form, con sider the case where Dis set equal to zero in Eq. (1), and neglect all dependence of Sonxandz. We are then led to the equation dE(z) dz=λS2E(z), (3) which yields E(z) =E(0)eλS2z, z> 0. HereS2=S2 1+S2 2andS1,S2are two independent real Gaussian random variables with zer o mean and unit variance. It is easily seen that the probabilit y distribution of E(z), setting E(0) = 1, has the density W(E,z) = (2λz)−1E−[1+(2 λz)−1]forE ≥1, z> 0. (4) If we now take moments of Eat some value Lofz, we ﬁnd that /an}b∇acketle{tEN(L)/an}b∇acket∇i}htwill diverge whenever 2 NλL ≥1. At the critical coupling λcN(L) = (2NL)−1, there is a qualitative transition of the ampliﬁcation of /an}b∇acketle{tEN(L)/an}b∇acket∇i}htfrom a regime where it is dominated by the bulk of the order-one-ﬂuctuations of Sto a regime where it is dominated by the large ﬂuctuations ofS. This toy model can be thought of as an idealization in which t he size of the plasma is very small compared to the correlation length of the laser ﬁeld. This is certainly not a reasonable physical approximation and we shall later consi der situations in which Sin Eq. (3) isz-dependent with a covariance C(z,z′). The equation is then still solvable more or less explicitly, depending on the form of C, at least as far as the dependence of the divergence of the moments of EonλandLis concerned. The main diﬀerence from Eq. (4) is that for small enough values of λ, the ﬁrst few moments need not diverge for any L. In this paper, we extend these results to the x-dependent case where iDin Eq. (1) is replaced by D, i.e. we consider a diﬀusive process in xrather than a diﬀractive one. Some- what surprisingly the diﬀusion does not suppress the onset o f divergences in moments of the 3ﬁeld. This suggests a similar behavior for the diﬀractive ca se – in accord with the numerical results of [2] – but we are unable to prove this at the present t ime. Before going on to the formulation and presentation of resul ts for the diﬀusive case, we make some remarks about the relation between expectations o ver diﬀerent realizations of the Gaussian driving term \|S\|2and the outcome of a given experiment. Accepting the idealizations inherent in assuming Gaussian statistics an d neglect of nonlinear terms, the physically relevant question relating to the solution of th e stochastic PDE (1) appears to be the following: What is the probability that for given Λ and Lthere will be small regions in Λ through which a signiﬁcant fraction of the total incoming p ower is backscattered, (here ”total” means through the whole domain Λ). Put more physical ly, imagine Λ to be divided up intoM≫1 cells of equal area \|Λ\|/Mand letR≫1/Mbe a speciﬁed number. We want to compute the probability Pthat in at least one of the cells the integral of \|E\|2over that cell exceeds R\|Λ\|. In the case where Dis set equal to zero, this can be answered by taking for the cell size the transverse correlation length of \|S\|2and assuming the ﬁeld inside each cell to be transversally constant and evolving along zunder Eq. (3) with a z-dependent S. One ﬁnds that Pgreatly increases as λpasses its critical value for the divergence of the second moment, from P≪1 forλ<λ c2(L) toP≃1 forλ>λ c2(L). We expect that this probability will behave similarly in real systems. The outline of the rest of this paper is as follows. In Sec. II w e introduce our diﬀusion- ampliﬁcation model. In Sec. III we prove that the value of the critical coupling obtained without the diﬀusion term cannot be less than the one obtaine d with the diﬀusion term. In Sec. IV we prove that for a large class of Gaussian ﬁelds Sthe values of the critical coupling obtained with or without the diﬀusion term are the same. Sect ion V is devoted to the explicit solution of the diﬀusion-free problem in the parti cular case where the on-axis ﬁeld S(0,z) is a linear functional of a Gauss-Markov process. Finally, in Sec. VI we study the dependence of the critical coupling on the space dimensiona lity in the case of a factorable correlation function C. 4II. MODEL AND DEFINITIONS As explained in the introduction, we consider a modiﬁed vers ion of the linear convective ampliﬁer model obtained by replacing iDbyDon the left-hand side of Eq. (1). Taking D= 1/2 without loss of generality, one is thus led to the problem   ∂tE(x,t)−1 2∆E(x,t) =λS(x,t)2E(x,t), t∈[0,T], x∈Rd,andE(x,0) =E0(x),(5) where, following the usual notation used in diﬀusion proble ms, the time variable t(resp.T) plays the role of the axial space variable z(resp.L). In Eq. (5), we restrict ourselves to the cases where S(x,t) is a real homogeneous Gaussian ﬁeld deﬁned by /an}b∇acketle{tS(x,t)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tS(x,t)S(x′,t′)/an}b∇acket∇i}ht=C(x−x′,t,t′), with the normalization C(0,0,0)≡ /an}b∇acketle{tS(x,0)2/an}b∇acket∇i}ht= 1, and we will take E0(x)≡1 as an initial condition. Note that S(x,t) is not assumed to be stationary in t, and that the rest of our analysis is essentially unaﬀected if we replace Rdby ad-dimensional torus. The critical coupling λcN(T) and its diﬀusion-free counterpart λcN(T) are deﬁned by λcN(T) = inf {λ>0 :/an}b∇acketle{tE(0,T)N/an}b∇acket∇i}ht= +∞}, (6a) λcN(T) = inf {λ>0 :/an}b∇acketle{teNλ/integraltextT 0S(0,t)2dt/an}b∇acket∇i}ht= +∞}, (6b) where /an}b∇acketle{t./an}b∇acket∇i}htdenotes the average over the realizations of S. For a given T >0, Eqs. (6) gives the value of λat which /an}b∇acketle{tE(x,T)N/an}b∇acket∇i}htblows up with and without diﬀusion respectively. Finally, in order not to make the calculations too cumbersom e, we will use in the following the compact notation 5t≡(n,t), /integraldisplay dt≡N/summationdisplay n=1/integraldisplayT 0dt, S(t)≡S(xn(t),t), C(s,t)≡ /an}b∇acketle{tS(s)S(t)/an}b∇acket∇i}ht=C(xn(s)−xm(t),s,t), C0(s,t)≡C(0,s,t), (ϕ,ψ) =/integraldisplay ϕ(t)ψ(t)dt, withs,t∈[0,T],n,m∈N(1≤n,m≤N), and where the xn(·) are given continuous paths on Rd. The covariance operators ˆTCandˆtC0, respectively acting on ϕ(t)∈L2(dt) and ϕ(t)∈L2(dt), are deﬁned by (ˆTCϕ)(s) =/integraldisplay C(s,t)ϕ(t)dt, (ˆtC0ϕ)(s) =/integraldisplayT 0C(0,s,t)ϕ(t)dt. III. COMPARISON OF λCN(T)AND λCN(T) In this section we prove that λcN(T)≤λcN(T). We begin with two technical lemmas that will be useful in the following. Lemma 1 : Suppose the covariance function C(x,t,t′) is continuous. Let µx(t) 1≥µx(t) 2≥ ...≥0 be the eigenvalues of the covariance operator ˆTC. Here, the superscript x(t) denotes theNcontinuous paths xn(t), 1≤n≤N. Then /an}b∇acketle{texpλ/integraltextS(t)2dt/an}b∇acket∇i}ht<+∞if and only if λ<(2µx(t) 1)−1, and in this case one has log/an}b∇acketle{teλ/integraltext S(t)2dt/an}b∇acket∇i}ht=−1 2/summationdisplay i≥1log/parenleftig 1−2λµx(t) i/parenrightig ≤Nλ/integraltextT 0C(0,t,t)dt 1−2λµx(t) 1. (7) To show (7), consider the Hilbert space of the L2(dt) functions ϕ(n,t)≡ϕ(t) with the scalar product ( ϕ,ψ). SinceC(s,t) is continuous in ( s,t), and therefore bounded in compact sets, we have that/integraltext/integraltextC(s,t)2dsdt<+∞. By [5], Theorem VI.23, it follows 6that the covariance operator is compact (and self-adjoint) inL2(dt). Therefore there is an orthonormal basis {ϕj}j≥1such that ˆTCϕj=µx(t) jϕj. Consider now the sequence of random variables Xj= (S,ϕ j). As linear functionals of the Gaussian ﬁeld S, theXj’s form a Gaussian sequence with /an}b∇acketle{tXi/an}b∇acket∇i}ht= 0 and /an}b∇acketle{tXiXj/an}b∇acket∇i}ht= (ϕi,ˆTCϕj) =µx(t) jδij. The equality in Eq. (7) is then obtained straightforwardly from/integraltextS2(t)dt=/summationtext+∞ j=1X2 jand the simple Gaussian identity/angbracketleftig eλX2 i/angbracketrightig =/parenleftig 1−2λµx(t) i/parenrightig−1/2, for 2λµx(t) i<1. The inequality in Eq. (7) follows from −log(1−x)≤x/(1−x) and the fact that/summationtext iµx(t) i=/integraltextC(t,t)dt=N/integraltextT 0C(0,t,t)dt. In the following subsection, ϕ(t)≡ϕ(n,t) will denote a set of Ntest functions normalized such that (ϕ,ϕ) =/summationtextN n=1/integraltextT 0ϕ(n,t)2dt= 1. Lemma 2 : Assume that for every T >0 one has lim x→0sups,t∈[0,T]\|C(x,s,t)−C(0,s,t)\|= 0. Then, ∀ε>0,∃δ >0 such that /vextendsingle/vextendsingle/vextendsingle(ϕ,ˆTCϕ)−(ϕ,ˆTC0ϕ)/vextendsingle/vextendsingle/vextendsingle<ε for everyxn(·)∈Bδ,T, 1≤n≤N, whereBδ,Tis the set of continuous paths x(·) such that \|x(t)\|<δfor everyt∈[0,T]. The proof of this lemma is straightforward: from the uniform convergence condition on C(x,s,t) it follows that ∀ε >0,∃δ >0 such that \|C(s,t)−C0(s,t)\|< εfor everyxn(·)∈ Bδ,T, 1≤n≤N. Thus, ∀ε′>0,∃δ >0 such that /vextendsingle/vextendsingle/vextendsingle(ϕ,ˆTCϕ)−(ϕ,ˆTC0ϕ)/vextendsingle/vextendsingle/vextendsingle≤(\|ϕ\|,ˆT\|C−C0\|\|ϕ\|) <ε′/parenleftbigg/integraldisplay \|ϕ(s)\|ds/parenrightbigg2 ≤ε′NT, for everyxn(·)∈Bδ,T, 1≤n≤N. It remains to take ε′=ε/(NT), which proves Lemma 2. We can now state the main result of this section. Namely, that one of the diﬀusion eﬀects on the divergence of the moments of E(x,T) is a lowering (or, more exactly, a non-increasing) of the critical coupling. The rigorous formulation of this r esult can be stated as the following proposition. 7Proposition 1 : For every T >0, if lim x→0sups,t∈[0,T]\|C(x,s,t)−C(0,s,t)\|= 0, then λcN(T)≤λcN(T). In order to prove this proposition, one writes the moments of Ein terms of the Feynman- Kac formula /an}b∇acketle{tE(0,T)N/an}b∇acket∇i}ht=/angbracketleftbigg/angbracketleftbigg exp/bracketleftbigg λ/integraldisplay S(t)2dt/bracketrightbigg/angbracketrightbigg/angbracketrightbigg x(t), (8) where /an}b∇acketle{t·/an}b∇acket∇i}htx(t)denotes aN-fold Wiener integral over NBrownian paths xn(t), 1≤n≤N, each arriving at x= 0. Letλ>λcN(T), i.e.µ1>(2Nλ)−1, whereµ1is the largest eigenvalue of the covariance operator ˆtC0. Letφ1(t) be the normalized eigenfunction associated with µ1, andφ(t)≡φ(n,t) =N−1/2φ1(t) for every 1 ≤n≤N. [N.B. : the factor N−1/2ensures the normalization ( φ,φ) = 1]. By deﬁnition of µx(t) 1, one has µx(t) 1≥(φ,ˆTCφ). (9) By Lemma 2, ∀ε>0,∃δ >0 such that (φ,ˆTCφ)≥(φ,ˆTC0φ)−ε=Nµ1−ε (10) for everyxn(·)∈Bδ,T, 1≤n≤N. If one now takes ε<Nµ 1−1 2λ, it follows from Eqs. (9) and (10) that µx(t) 1>1/2λand so, by Lemma 1, /angbracketleftbigg exp/bracketleftbigg λ/integraldisplay S(t)2dt/bracketrightbigg/angbracketrightbigg = +∞ for everyxn(·)∈Bδ,T, 1≤n≤N. Finally, since the set of the Brownian paths xn(t) that are inBδ,Thas a strictly positive Wiener measure, one ﬁnds from Eq. (8) that/an}b∇acketle{tE(0,T)N/an}b∇acket∇i}ht= +∞, soλ≥λcN(T) which proves the proposition 1. Note that imposing the uniform convergence of C(x,s,t) toC(0,s,t) is not a very restric- tive condition. As far as we know, it seems to be fulﬁlled by an y nonpathological stochastic ﬁeldSof physical interest. 8IV. EQUALITY OF λCN(T)AND λCN(T)FOR A CLASS OF S For a large class of Gaussian ﬁelds Sit is possible to prove that diﬀusion has no eﬀect on the onset of the divergence of /an}b∇acketle{tE(x,T)N/an}b∇acket∇i}ht, i.e.λCN(T) =λcN(T). Proposition 2 : Assume that for every T > 0 one has lim x→0sups,t∈[0,T]\|C(x,s,t)− C(0,s,t)\|= 0, and that \|C(x,s,t)\| ≤C(0,s,t) for every x∈Rdands,t∈[0,T]. Then λcN(T) =λcN(T). The proof of this proposition is as follows: By the uniform co nvergence condition on C(x,s,t) and Proposition 1 one already has λcN(T)≤λcN(T). It remains to show that λcN(T)≤λcN(T). Letµ1be the largest eigenvalue of the covariance operator ˆtC0. Letφ1(t) be a principal (normalized) eigenvector for the covariance operator ˆTC. One has µx(t) 1= (φ1,ˆTCφ1)≤(\|φ1\|,\|ˆTCφ1\|)≤(\|φ1\|,ˆTC0\|φ1\|)≤Nµ1, where the second inequality follows from the condition \|C(x,s,t)\| ≤C(0,s,t). Suppose now λ<λcN(T), i.e.λ<(2Nµ1)−1. Thenλ<(2µx(t) 1)−1and by Lemma 1 /angbracketleftbigg exp/bracketleftbigg λ/integraldisplay S(t)2dt/bracketrightbigg/angbracketrightbigg ≤exp/bracketleftiggNλ/integraltextT 0C(0,t,t)dt 1−2λµx(t) 1/bracketrightigg ≤exp/parenleftiggNλ/integraltextT 0C(0,t,t)dt 1−2Nλµ 1/parenrightigg . Since this inequality is uniform over all Brownian paths, we ﬁnally have /an}b∇acketle{tE(0,T)N/an}b∇acket∇i}ht ≤exp/parenleftiggNλ/integraltextT 0C(0,t,t)dt 1−2Nλµ 1/parenrightigg <+∞, and therefore λ<λ cN(T), which proves the proposition 2. This result shows that for Gaussian ﬁelds Sfulﬁlling the not so restrictive conditions of Proposition 2, it is suﬃcient to solve the diﬀusion-free pro blem to determine the onset of the divergence of /an}b∇acketle{tE(x,T)N/an}b∇acket∇i}ht. It is therefore interesting to show how such ﬁelds can be act ually obtained. To this end, the remaining of this section will be d evoted to explicitely construct two typical examples of stochastic ﬁelds Swhich fulﬁll the conditions of Proposition 2. 9A. an example of nonstationary S The ﬁrst example is the diﬀusive counterpart of the Gaussian ﬁeld deﬁned by Eq. (2). LetS(x,t) be the solution to   ∂tS(x,t)−1 2∆S(x,t) = 0, t∈[0,T], x∈Rd,andS(x,0) =S(x),(11) where S(x) is a real homogeneous Gaussian ﬁeld deﬁned by /an}b∇acketle{tS(x)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tS(x)S(x′)/an}b∇acket∇i}ht=C(x−x′),(12) withC(x) a given [3] function of xnormalized such that C(0)≡ /an}b∇acketle{tS(x,0)2/an}b∇acket∇i}ht= 1. One has S(x,t) =/integraldisplay S(k)eikx−1 2k2tddk, (13) where S(k) is the Fourier transform of S(x), and from Eqs. (12) and (13) it follows that S(x,t) is a real homogeneous nonstationary Gaussian ﬁeld with /an}b∇acketle{tS(x,t)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tS(x,t)S(x′,t′)/an}b∇acket∇i}ht=/integraltextC(k)eik(x−x′)−1 2k2(t+t′)ddk,(14) where C(k) is the Fourier transform of C(x). Since C(k) is real and positive [3], one has \|C(x,s,t)\| ≡ \|/an}b∇acketle{tS(x,s)S(0,t)/an}b∇acket∇i}ht\| =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay C(k)eikx−1 2k2(s+t)ddk/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/integraldisplay C(k)e−1 2k2(s+t)ddk=C(0,s,t), for everyx∈Rdands,t∈[0,T], soS(x,t) fulﬁlls the conditions of Proposition 2. B. an example of stationary S The second example is provided by a modiﬁed version of Eq. (11 ) obtained by adding a source term ` a la Langevin on its right-hand side. Namely, le tS(x,t) be the solution to   ∂tS(x,t)−1 2∆S(x,t) =L(x,t), t∈]− ∞,T], x∈Rd,andS(x,−∞) = 0,(15) 10where the Langevin source term L(x,t) is a homogeneous Gaussian white noise deﬁned by /an}b∇acketle{tL(x,t)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tL(x,t)L(x′,t′)/an}b∇acket∇i}ht=−δ(t−t′)∆xC(x−x′),(16) withC(x) a given [3] function of xnormalized such that C(0) = 1. The solution to Eq. (15) reads S(x,t) =/integraldisplay ddk/bracketleftbigg eikx/integraldisplayt −∞e−1 2k2(t−s)L(k,s)ds/bracketrightbigg , (17) whereL(k,t) is the Fourier transform of L(x,t). From Eqs. (16) and (17) it can be shown thatS(x,t) is a real homogeneous stationary Gaussian ﬁeld with /an}b∇acketle{tS(x,t)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tS(x,t)S(x′,t′)/an}b∇acket∇i}ht=/integraltextC(k)eik(x−x′)−1 2k2\|t−t′\|ddk,(18) where C(k) is the Fourier transform of C(x). As previously, since C(k) is real and positive [3], one has \|C(x,s,t)\| ≡ \|/an}b∇acketle{tS(x,s)S(0,t)/an}b∇acket∇i}ht\| =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay C(k)eikx−1 2k2\|s−t\|ddk/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/integraldisplay C(k)e−1 2k2\|s−t\|ddk=C(0,s,t), for everyx∈Rdands,t∈[0,T], and soS(x,t) fulﬁlls the conditions of Proposition 2. More generally, it can be checked that any real homogeneous G aussian ﬁeld S(x,t) deﬁned by /an}b∇acketle{tS(x,t)/an}b∇acket∇i}ht= 0, /an}b∇acketle{tS(x,t)S(x′,t′)/an}b∇acket∇i}ht=/integraltextC(k,t,t′)eik(x−x′)ddk, where C(k,t,t′) is real and positive, fulﬁlls the conditions of Propositio n 2. V. EXPLICIT SOLUTION OF THE DIFFUSION-FREE PROBLEM FOR A CLASS OF S In this section we show that an explicit computation of the di ﬀusion-free ampliﬁcation factor /an}b∇acketle{texp(Nλ/integraltextT 0S(0,t)2dt)/an}b∇acket∇i}htcan be achieved if S(0,t) is a linear functional of a Gauss- 11Markov process. Note that determining λcN(T) amounts to ﬁnding the largest eigenvalue of the covariance operator ˆtC0, which in principle can always be achieved, at least numeric ally. As shown above, λcN(T)≥λcN(T) with equality holding when Proposition 2 is applicable. SinceλcN(T) =N−1λc1(T) in the diﬀusion free case, we will take N= 1 in the remaining of this section without loss of generality. A. Solution of the diﬀusion-free problem using the Feynman- Kac formula We consider the case where the Gaussian process S(0,t) can be written as S(0,t) =/an}b∇acketle{tc,Y(t)/an}b∇acket∇i}ht, (19) where /an}b∇acketle{tx,y/an}b∇acket∇i}ht ≡x†y=/summationtext ixiyi,cis a given n-dimensional vector, and Y(t) is a n-dimensional Gauss-Markov process deﬁned as the solution to the linear st ochastic diﬀerential equation   dY(t) +AY(t)dt=GdB(t), Y(0) Gaussian with /an}b∇acketle{tY(0)/an}b∇acket∇i}ht= 0.(20) Here,AandGare constant n×nmatrices, and B(t) is an-dimensional Brownian motion. From Eqs. (19) and (20), it follows that one can write the diﬀu sion-free ampliﬁcation factor as a Feynman-Kac formula /angbracketleftbigg eλ/integraltextT 0S(0,t)2dt/angbracketrightbigg =/angbracketleftbigg eλ/integraltextT 0/angbracketleftY(t),CY(t)/angbracketrightdt/angbracketrightbigg =/integraldisplay v(y,T)dny, (21) whereCdenotes the symmetrical n×nmatrixc⊗c, andv(y,t) is the solution to the parabolic equation   ∂v ∂t= (TrA+λ/an}b∇acketle{ty,Cy/an}b∇acket∇i}ht)v+/an}b∇acketle{tAy,∇/an}b∇acket∇i}htv+1 2/an}b∇acketle{tG†∇,G†∇/an}b∇acket∇i}htv, v(y,0) =/parenleftig 1 2π/parenrightign/21√ \|U\|exp/bracketleftig −1 2/an}b∇acketle{ty,U−1y/an}b∇acket∇i}ht/bracketrightig ,(22) withU= Cov[Y(0),Y(0)]. The solution to Eq. (22) has the form v(y,t) =/parenleftbigg1 2π/parenrightbiggn/21/radicalig \|K(t)\|exp/bracketleftbigg −1 2/an}b∇acketle{ty,K(t)−1y/an}b∇acket∇i}ht+λ/integraldisplayt 0TrCK(s)ds/bracketrightbigg , (23) 12whereK(t) is a symmetrical n×nmatrix which is the solution to   dK(t) dt=GG†−[AK(t) +K(t)A†] + 2λK(t)CK(t), K(0) =U.(24) Thus, from Eqs. (21) and (23) one has /angbracketleftbigg eλ/integraltextT 0S(0,t)2dt/angbracketrightbigg = eλ/integraltextT 0TrCK(t)dt. (25) withK(t) given by the Riccati equation (24). The solution to Eq. (24) is known to explode in ﬁnite time for l arge enough λ. Forn= 1, in which case S(0,t) is itself Markovian, Eq. (24) is solved straightforwardly (see Sec. VB). Forn≥2, the solution to Eq. (24) can be obtained by the so-called Ha miltonian method: we deﬁne the 2 n×2nmatrix H= A†−2λC GG†−A  and solve the linear diﬀerential equation d dt Q(t) P(t) =H Q(t) P(t) , (26) with the initial condition  Q(0) P(0) = I U . The solution K(t) to the Riccati equation (24) is easily checked to be given by K(t) =P(t)Q(t)−1, (27) which explodes if and only if Q(t) becomes singular [6]. Since Eq. (26) is a linear equation, it can in principle be solved by a symbolic computation progr am. B. Application to the n =1 case As an example, let us consider the simplest case n= 1 withC(0,t,t′) = e−\|t−t′\|. In this limit, the diﬀusion-free ampliﬁcation factor reads /angbracketleftbigg eλ/integraltextT 0S(0,t)2dt/angbracketrightbigg =/angbracketleftbigg eλ/integraltextT 0Y(t)2dt/angbracketrightbigg = eλ/integraltextT 0K(t)dt, (28) 13whereY(t) is the Ornstein-Uhlenbeck process   dY(t) +Y(t)dt=√ 2dB(t), /an}b∇acketle{tY(0)/an}b∇acket∇i}ht= 0,/an}b∇acketle{tY(0)2/an}b∇acket∇i}ht= 1,(29) andK(t) is the solution to the Riccati equation   1 2dK(t) dt= 1−K(t) +λK(t)2, K(0) = 1.(30) Equation (30) can be easily solved by means of the substituti on 2λK(t) =−dlogu(t)/dt. Inserting the result into Eq. (28), one obtains /angbracketleftbigg eλ/integraltextT 0S(0,t)2dt/angbracketrightbigg =eT/2 /radicalig cosh(αT) +α−1(1−2λ) sinh(αT), λ< 1/4, (31) /angbracketleftbigg eλ/integraltextT 0S(0,t)2dt/angbracketrightbigg =eT/2 /radicalig 1 +T/2, λ= 1/4, (32) /angbracketleftbigg eλ/integraltextT 0S(0,t)2dt/angbracketrightbigg =eT/2 /radicalig cos(αT) +α−1(1−2λ) sin(αT), λ> 1/4, (33) whereα=\|1−4λ\|1/2. It can be seen from Eq. (33) that, for λ>1/4,/an}b∇acketle{texp(λ/integraltextT 0S(0,t)2dt)/an}b∇acket∇i}ht diverges as Ttends (from below) to the critical time Tc(λ) given by Tc(λ) =1√ 4λ−1tan−1/parenleftigg√ 4λ−1 2λ−1/parenrightigg , (34) where the determination of tan−1is such that 0 <tan−1≤π. Inverting Eq. (34) and using λcN(T) =N−1λc1(T) gives the diﬀusion-free critical coupling λcN(T) in the cases where C(0,t,t′) = e−\|t−t′\|. VI. DEPENDENCE OF THE CRITICAL COUPLING ON SPACE DIMENSIONALITY In this section we study the dependence of λcN(T) on the space dimensionality D. We will restrict ourselves to the cases where the correlation f unctionCcan be written out as CD(x,t,t′) =Cd(x\|\|,t,t′)CD−d(x⊥,t,t′), (35) 14withCD,CdandCD−dcontinuous, symmetric, and positive deﬁnite, and where x\|\|is the projection of xonto ad-dimensional subspace ( d<D ) andx⊥=x−x\|\|. In the following, a correlation function of this type will be called a factorabl e correlation function. It is worth noting that such a correlation function can be very easily ob tained, e.g. when the Gaussian ﬁeldSis deﬁned by either Eq. (14) or Eq. (18) in the cases where C(k) is factorable as C(k) =Cd(k\|\|)CD−d(k⊥). We prove that as λincreases, the divergence of /an}b∇acketle{tE(x,T)N/an}b∇acket∇i}htobtained in the original D- dimensional problem cannot occur before the one obtained in the projected d-dimensional problem whenever 0 ≤CD−d(0,t,t)≤1. Since many stochastic ﬁelds Sof physical interest, e.g. in optics, do have a factorable correlation function, w e expect this result to be useful for the comparison of two-dimensional numerical simulatio ns with experiments and three- dimensional numerical simulations. Before expressing thi s result in a more rigorous way, we begin with two technical lemmas that will be needed in the fol lowing. Lemma 3 : Consider a D-dimensional problem and let µx(t) 1be the largest eigenvalue of the covariance operator ˆTCDand N given continuous paths x(t). ThenλcN(T,D) = [2 supx(t)µx(t) 1]−1. This lemma can be proven straightforwardly by successively considering the inequalities λ>[2 supx(t)µx(t) 1]−1andλ<[2 supx(t)µx(t) 1]−1, and by following the same lines of reasoning as for the proofs of Propositions 1 and 2 respectively, where one replaces the N paths x(t) = 0 corresponding to λcN(T) = [2µx(t)=0 1]−1by N paths maximizing µx(t) 1[7]. Lemma 4 : LetK0(s,t),K1(s,t), andK2(s,t) be three symmetric kernels such that: (i)K0(s,t) =K1(s,t)K2(s,t); (ii)K2is a positive deﬁnite continuous symmetric kernel; (iii) 0 ≤K2(t,t)<1 and the largest eigenvalue of K1is positive, or K2(t,t) = 1 and no condition on the sign of the largest eigenvalue of K1. Thenµ1(K0)≤µ1(K1), whereµ1(Kα) denotes the largest eigenvalue of Kα. 15The proof of this lemma is as follows: since K2is a positive deﬁnite continuous symmetric kernel, Mercer’s theorem holds [8] and this kernel admits th e expansion K2(s,t) =/summationdisplay iaifi(s)fi(t), (36) whereai≥0 andfi(t) respectively denote the itheigenvalue of the operator ˆTK2and the associated normalized eigenfunction. Let φ1(t) be a principal (normalized) eigenfunction of the operator ˆTK0andµ1(K0) the corresponding largest eigenvalue. From the condition (i) and Eq. (36), one has µ1(K0) = (φ1,ˆTK0φ1) =/summationdisplay iai(fiφ1,ˆTK1fiφ1) =/summationdisplay iaiMi(ηi,ˆTK1ηi), (37) whereMiandηi(t) are given by Mi= (fiφ1,fiφ1), and ηi(t) =M−1/2 ifi(t)φ1(t), such that (ηi,ηi) = 1. By the deﬁnition of µ1(K1) and from K2(t,t)≤1, condition (iii), one has respectively µ1(K1)≥(ηi,ˆTK1ηi), (38) and /summationdisplay iaiMi=/integraldisplay/bracketleftigg/summationdisplay iaifi(t)2/bracketrightigg φ1(t)2dt =/integraldisplay K2(t,t)φ1(t)2dt≤/integraldisplay φ1(t)2dt= 1. (39) So, from Eqs. (37), (38), (39) and the condition (iii), it fol lows thatµ1(K0)≤µ1(K1), which proves Lemma 4. We can now proceed to rigorously express and prove the result stated at the beginning of this section. Let λcN(T,D) be the critical coupling associated with a D-dimensional problem in which the correlation function of the Gaussian ﬁe ldSis given by CD. One has the following proposition: 16Proposition 3 : for every T >0, ifCD(x,t,t′) is a factorable correlation function such that 0 ≤CD−d(0,t,t)≤1 for 0 ≤t≤T, thenλcN(T,D)≥λcN(T,d). The proof of this proposition is straightforward. By the deﬁ nition of a factorable corre- lation function one has CD(s,t) =Cd(s,t)CD−d(s,t), where both Cd(s,t) andCD−d(s,t) are continuous, symmetric, and positive deﬁnite. Since CD−d(t,t)≡CD−d(0,t,t) and 0≤CD−d(0,t,t)≤1 by assumption, one can apply the lemma 4 with K0=CD,K1=Cd, andK2=CD−d. It follows immediately that µx(t) 1≤˜µx\|\|(t) 1, where ˜µx\|\|(t) 1denotes the largest eigenvalue of the operator ˆTCd. Letxmax(t) be N paths maximizing µx(t) 1[7]. One has supx(t)µx(t) 1=µxmax(t) 1 ≤˜µxmax \|\|(t) 1 , from which it follows that supx(t)µx(t) 1≤supx\|\|(t)˜µx\|\|(t) 1 and, by Lemma 3, λcN(T,D)≥λcN(T,d), which proves the proposition 3. VII. SUMMARY AND PERSPECTIVES In this paper, we have studied the eﬀects of diﬀusion on the di vergence of the moments of the solution to a linear ampliﬁer driven by the square of a G aussian ﬁeld. We ﬁrst proved that the divergence yielded by a diﬀusion-free calculation cannot occur at a smaller coupling constant than the one obtained from the full calculation (i. e. with diﬀusion). Then we have shown that, in the case where the absolute value of the (unifo rmly continuous) pump ﬁeld correlation function is bounded from above by its one-point value, there is no diﬀusion eﬀect on the onset of the divergence which is therefore given by a di ﬀusion-free calculation. In this context, we have solved the diﬀusion-free problem expl icitly when the pump ﬁeld is a linear functional of a Gauss-Markov process. Finally, we h ave studied the dependence of the critical coupling on the space dimensionality in the c ase of a factorable correlation function. In particular, we have proved that the divergence obtained in a D-dimensional problem cannot occur at a smaller coupling constant than the one obtained in the projected d-dimensional problem ( d<D ). As mentioned in the introduction, we would like to extend our results for the diﬀusion- ampliﬁcation model (5) to the more diﬃcult diﬀraction-ampl iﬁcation problem (1). According 17to Eq. (5), the results obtained in this paper also apply, bes ide some minor technical modiﬁ- cations, if the pump ﬁeld is a complex Gaussian ﬁeld as in Eq. (1). The remaining diﬃculty in extending our results to Eq. (1) lies in controlling the co mplex Feynman path-integral, compared to that of the Feynman-Kac formula for the diﬀusive case. Expressing E(x,t) as a Feynman path-integral and averaging over the realizations ofS, one cannot a priori exclude the possibility that destructive interference eﬀects betw een diﬀerent path contributions make the sum of the divergent contributions ﬁnite. Thus one canno t deduce the divergence of the moments of E(x,L) from that of the ampliﬁcation along paths arriving at the po int (x,L). It is however not unreasonable to expect that Propositions 1 , 2, and 3 also apply to the diﬀraction-ampliﬁcation problem (1). Proving this conjec ture is another matter and is the subject of a future work. Note that in the case of Proposition 2, the on-axis correlation function of the pump ﬁeld must be real and positive, which is q uite restrictive if the pump ﬁeld is complex. From a practical point of view (e.g. in optic s), it would therefore be very interesting to ﬁnd out whether there exists an enlarged vers ion of this proposition applying to complex on-axis correlation functions as well. VIII. ACKNOWLEDGMENTS We thank Harvey Rose for introducing us to this problem and fo r providing many valuable insights. The work of J.L.L. was supported in part by AFOSR Gr ant F49620-98-1-0207 and NSF Grant DMR-9813268. A.A., P.D.P. and J.L.L. acknowledge the hospitality of the IHES at Bures-sur-Yvette, France, where part of this work was don e. 18REFERENCES [1] G. Laval et al., Phys. Fluids 20, 2049 (1977) ; E. A. Williams, J. R. Albriton, and M. N. Rosenbluth, Phys. Fluids 22, 139 (1979) ; and references therein. [2] H. A. Rose and D. F. DuBois, Phys. Rev. Lett. 72, 2883 (1994). [3] Since C(x) is a covariance, it must be chosen in such a way that C(x−x′) is a positive deﬁnite kernel, i.e. its Fourier transform must be real, eve n, and positive. Note that C(x,z) withz∈[−L,0[ can be obtained straightforwardly from (2) and the Hermit ian symmetryC(x,−z) =C(−x,z)∗. [4] Ph. Mounaix, Phys. Rev. E 52(2), 1306 (1995). [5] M. Reed and B. Simon, Methods in Mathematical Physics. I - Functional Analysis (Aca- demic Press, San Diego, 1980). [6] P. Crouch and M. Pavon, Syst. Control Lett. 9, 203 (1987). [7] In the cases where there is no such a set of paths, one shoul d consider N paths that realize the supremum up to a arbitrarily small constant. ¡ [8] R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989), Vol. 1, p. 138. 19
arXiv:physics/0011011v1 [physics.acc-ph] 4 Nov 2000Wake and Impedance G. V. Stupakov Stanford Linear Accelerator Center Stanford University, S tanford, CA 94309 I INTRODUCTION In this lecture we will develop a concept of wakes and impedan ces for relativistic beams interacting with the surrounding environment. Among the numerous publi- cations and reviews on this subject, we refer here to recent b ooks [1–3], where the reader can ﬁnd a more detailed treatment and further referen ces. We will use the CGS system of units throughout this paper. II INTERACTION OF MOVING CHARGES IN FREE SPACE We begin with interactions of particles that moving with con stant velocity in free space. If the material walls are far from the particles, their eﬀect in the ﬁrst approximation can be neglected. Let us consider a leading particle of charge qmoving with velocity v, and a trailing particle of unitcharge moving behind the leading one on a parallel path at a distance swith an oﬀset x, as shown in Fig. 1. We want to ﬁnd the force which the leading particle exerts on the trailing one. We will use the following expressions for the electric and ma gnetic ﬁelds of a particle moving with a constant velocity (see, e.g., [4]): E=qR γ2R∗3,H=1 cv×E, (1) where Ris the vector drawn from point 1 to point 2, R∗2=s2+x2/γ2, and γ= (1−v2/c2)−1/2. From Eq. (1) we ﬁnd that the longitudinal force acting on the t railing charge is Fl=Ez=−qs γ2(s2+x2/γ2)3/2, (2) and the transverse force is Ft=Ex−v cBy=qx γ4(s2+x2/γ2)3/2. (3)1vv2E zx sx FIGURE 1. A leading particle 1 and a trailing particle 2 traveling in fr ee space with parallel velocities v. Shown also is the coordinate system x, z. In accelerator physics, the force Fis often called the space-charge force . It is easy to see that for any position given by sandx, the longitudinal force decreases with the growth of γasγ−2. For the transverse force, if s≫x/γ, Ft∼γ−4, but for s= 0,Ft∼γ−1. Hence, in the limit of ultrarelativistic particles moving parallel to each other, γ→ ∞, the electromagnetic interaction in free space vanishes. In this lecture, we will focus on the case of ultrarelativist ic charges, where v→c. The space-charge eﬀects discussed above disappear in this l imit, and the interaction between the particles is due only to the presence of material walls. Note that, taking the limit v→cin Eq. (1) and recalling that s=vt−z, we can write the electromagnetic ﬁeld of an ultrarelitivistic charge in free space as E=2qr r2δ(z−ct),H=ˆz×E, (4) where r=ˆxx+ˆyyis a two-dimensional radius vector in a cylindrical coordin ate system ( ˆxandˆyare the unit vectors in the directions of xandy, respectively). III PARTICLES MOVING IN A PERFECTLY CONDUCTING PIPE If particles from the above example move parallel to the axis in a perfectly conducting cylindrical pipe of arbitrary cross section, th ey induce image charges, on the surface of the wall, that screen the metal from the elec tromagnetic ﬁeld of the particles. The image charges travel with the same velo cityv(see Fig. 2). Since both the particles and the image charges move on parall el paths, in the limit v=c, according to the results in Section II, they do not interact with each other, no matter how close to the wall the particles are. Interaction between the particles in the ultrarelativisti c limit can occur if 1) the wall is not perfectly conducting, or 2) the pipe is not cylind rical (which is usually due to the presence of RF cavities, ﬂanges, bellows, beam pos ition monitors, slots, etc., in the vacuum chamber).12/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr /BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr image charges FIGURE 2. Particles traveling inside a perfectly conducting pipe of a rbitrary cross section. Shown are the image charges on the wall generated by the leadi ng charge. IV CAUSALITY AND THE “CATCH-UP” DISTANCE If a beam particle moves along a straight line with the speed o f light, the elec- tromagnetic ﬁeld of this particle scattered oﬀ the boundary discontinuities will not overtake it and, furthermore, will not aﬀect the charges tha t travel ahead of it. The ﬁeld can interact only with the trailing charges in the beam t hat move behind it. This constitutes the principle of causality in the theory of wakeﬁelds, according to which the interaction of a point charge moving with the speed of light propagates only downstream and never reaches the upstream part of the be am. /BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr 0 z-s zb FIGURE 3. A wall discontinuity located at z= 0 scatters the electromagnetic ﬁeld of an ultrarelativistic particle. When the particle moves to loc ation z, the scattered ﬁeld arrives to point z−s. We can estimate the distance at which the electromagnetic ﬁe ld produced by a leading charge reaches a trailing particles traveling at a d istance sbehind. Let us assume that a discontinuity located at the surface of a pipe o f radius bat coordinate z= 0 is passed by the leading particle at time t= 0, see Fig. 3. If the scattered ﬁeld reaches point sat time t, then ct=/radicalbig (z−s)2+b2, where zis a coordinate of the leading particle at time t,z=ct. Assuming that s≪b, from these twoequations we ﬁnd z≈b2 2s. (5) The distance zgiven by this equation is often called the catch-up distance . Only after the leading charge has traveled this distance away fro m the discontinuity, can a particle at point sbehind it feel the wakeﬁeld generated by the discontinuity. V ROUND PIPE WITH RESISTIVE WALLS Consider a round pipe of radius b, with ﬁnite wall conductivity σ. A point charge moves along the zaxis of the pipe with the speed of light, and a trailing partic le follows the leading one at a distance s. Both particles are assumed to be on the axis of the pipe. Because of the symmetry of the problem, the only n on-zero component of the electromagnetic ﬁeld on the axis is Ez, which, depending on the sign, either accelerates or decelerates the trailing charge. Our goal no w is to ﬁnd the ﬁeld Ez as a function of s. If the conductivity of the pipe is large enough, we can use per turbation theory to ﬁnd the eﬀect of the wall resistivity. In the ﬁrst approxim ation, we consider the pipe as a perfectly conducting one. In this case, the electro magnetic ﬁeld of the charge is the same as in free space and is given by Eqs. (4). For what follows, we will need only the magnetic ﬁeld Hθ, Hθ=2q rδ(z−ct). (6) Using the mathematical identity δ(z−ct) =1 2πc/integraldisplay∞ −∞dωe−iω(t−z/c), (7) we will decompose Hθinto a Fourier integral, Hθ(r, z, t) =/integraldisplay∞ −∞dωH θω(r)e−iωt+iωz/c, (8) where Hθω(r) =q πrc. (9) In the limit where the skin depth δcorresponding to the frequency ω,δ= c/√ 2πσω, is much smaller than the pipe radius , δ≪b, we can use the Leontovich boundary condition [5] that relates the tangential electri c ﬁeld Eton the wall with the magnetic one, Et=ζH×n, (10)where nis the unit vector normal to the surface and directed toward t he metal, and ζ(ω) = (1 −i)/radicalbiggω 8πσ. (11) Combining Eqs.(10), (11) and (9), we ﬁnd Ezω\|r=b=−(1−i)/radicalbiggω 8πσq πbc. (12) Equation (12) gives us the longitudinal electric ﬁeld on the wall, but we need the ﬁeld on the axis of the pipe. To ﬁnd the radial dependence of Ezω, we use Maxwell’s equations, from which it follows that the electric ﬁeld in a v acuum satisﬁes the wave equation. In the cylindrical coordinate system the wave equ ation for Ezis 1 c2∂2Ez(r, z, t) ∂2t−∆Ez(r, z, t) =1 c2∂2Ez(r, z, t) ∂2t−∂2Ez(r, z, t) ∂2z−1 r∂ ∂rr∂Ez(r, z, t) ∂r= 0. (13) Substituting the Fourier component Ezω(r)e−iω(t−z/c)into this equation, we ﬁnd 1 r∂ ∂rr∂Ezω ∂r= 0. (14) This equation has a general solution Ezω=A+Blnr, where AandBare arbitrary constants. Since we do not expect Ezto have a singularity on the axis, B= 0. Hence the electric ﬁeld does not depend on r,Ezω= const, and Ezω\|r=0=Ezω\|r=b, (15) implying that Ezω\|r=0is given by the same Eq. (12). Note that we have shown here that in the ultrarelativistic case the longitudinal el ectric ﬁeld inside the pipe is constant throughout the pipe cross section. To ﬁnd Ez(z, t) we make the inverse Fourier transformation, Ez(z, t) =/integraldisplay∞ −∞dωE zωe−iω(t−z/c), (16) which gives Ez(z, t) = (i−1)q πcb/radicalbigg 1 8πσ/integraldisplay∞ −∞dω√ωe−iω(t−z/c). (17) The last integral can be taken analytically in the complex pl ane (see the Appendix), with the result Ez(z, t) =q 2πb/radicalbiggc σs3, (18)fors >0. For the points where s <0, located in front of the charge, Ez= 0 in agreement with the causality principle. The positive sign o fEzindicates that the trailing charge (if it has the same sign as q) will be accelerated in the wake. In our derivation we assumed that the magnetic ﬁeld on the wal l is the same as in the case of perfect conductivity. However, the magnetic ﬁ eld is generated not only by the beam current, but also by a displacement current, jdisp z=1 4π∂Ez ∂t. (19) that vanishes in the limit of perfect conductivity. To be abl e to neglect the correc- tions to Hθdue to jdisp z, we must require the total displacement current to be much less then the beam current. In the Fourier representation, t he time derivative ∂/∂t reduces to multiplication by −iω, and the requirement is πb21 4πωEzω≪Iω=q 2π, (20) or /parenleftBigω c/parenrightBig3/2 ≪/radicalbigg 4πσ cb2. (21) In the space-time domain, the inverse wavenumber c/ωcorresponds to the distance s, and the condition of applicability of Eq. (18) is, s≫s0=/parenleftbiggcb2 4πσ/parenrightbigg1/3 . (22) The behavior of Ezfor very small values of s,s < s 0, can be found in Ref. 6.Here we note only that the singularity in Eq. (18) saturates at sma lls, and the electric ﬁeld changes sign and becomes negative at s= 0. This ﬁeld decelerates the leading charge, as expected from the energy balance consideration. VI WAKE DEFINITION The electromagnetic interaction of charged particles in ac celerators with the sur- rounding environment is usually a relatively small eﬀect th at can be considered as a perturbation. In the zeroth approximation, we can assum e that the beam moves with a constant velocity along a straight line. We solv e Maxwell’s equation, ﬁnd the ﬁelds, and then take into account the eﬀect of these ﬁe lds on a particle’s motion. In this approach we neglect the second-order eﬀects because the motion along the perturbed orbit can only slightly change the ﬁelds computed in the zeroth approximation. Those corrections are usually small, espec ially for ultrarelativistic particles.Another important feature of the interaction between the ge nerated electromag- netic ﬁeld and the particles is that in many cases of practica l importance it is localized in a region small compared with the length of the be am orbit. It also occurs on a time scale much smaller than the characteristic o scillation times of the beam in the accelerator (such as betatron and synchrotron pe riods). This allows us to consider this interaction in the impulse approximatio n and characterize it by the amount of momentum transferred to the particle. Taking into account the above considerations, we will intro duce the notion of the wake in the following way. Consider a leading particle 1 of charge qmoving along 1vv2 zx sΡ yR FIGURE 4. A leading particle 1 and a trailing particle 2 move parallel t o each other in a vacuum chamber. axiszwith a velocity close to the speed of light, v≈c, so that z=ct(see Fig. 4). A trailing particle 2 of unit charge moves parallel to the leading one, with the same velocity, at a distance swith oﬀset ρrelative to the z-axis. The vector ρ is a two-dimensional vector perpendicular to the z-axis, ρ= (x, y). Although the two particles move in a vacuum, there are material boundarie s in the problem that scatter the electromagnetic ﬁeld and result in interaction between the particles. Let us assume that we solved Maxwell’s equation and found the electromagnetic ﬁeld generated by the ﬁrst particle. We calculate the change of the momentum ∆pof the second particle caused by this ﬁeld as a function of the oﬀset ρand the distance s, ∆p(ρ, s) =/integraldisplay∞ −∞dt[E(ρ, z, t) +ˆz×B(ρ, z, t)]z=ct−s. (23) Note that we integrate here along a straight line — the unpert urbed orbit of the second particle. The integration limits are extended from m inus to plus inﬁnity, assuming that the integral converges. Since the beam dynamics is diﬀerent in the longitudinal and t ransverse direc- tions, it is useful to separate the longitudinal momentum ∆ pzfrom the transverse component ∆ p⊥. With the proper sign and the normalization factor c/q, these twocomponents are called the longitudinal andtransverse wake functions (or simply wakes ), wl(ρ, s) =−c q∆pz=−c q/integraldisplay dtEz\|z=ct−s, wt(ρ, s) =c q∆p⊥=c q/integraldisplay dt[E⊥+ˆz×B]z=ct−s. (24) Note the minus sign in the deﬁnition of wl— it is introduced so that the positive longitudinal wake corresponds to the energy loss of the trai ling particle (if both the leading and trailing particles have the same sign of char ge). The deﬁned wakes have dimension cm−1in CGS units and V/C in SI units.1 Because of the causality principle, the wakeﬁeld does not pr opagate in front of the leading charge, hence wl(ρ, s)≡0,wt(ρ, s)≡0,for s<0. (25) It was assumed above that the electromagnetic ﬁeld is locali zed in space and time and the integral in Eq. (23) converges. There are cases, however, where this is not true and the source of the wake is distributed uniforml y along an extended path, such as the resistive wall wake of a long pipe, consider ed in Section V. In this case it is more convenient to introduce the wake per unit length of the path by dropping the integration in Eq. (23): wl(ρ, s) =−1 qEz\|z=ct−s, wt(ρ, s) =1 q[E⊥+ˆz×B]z=ct−s. (26) In this deﬁnition, the wakes acquire an additional dimensio n of inverse length, and has the dimension cm−2in CGS and V/C/m in SI. VII PANOFSKY-WENZEL THEOREM Several general relations between longitudinal and transv erse wakes can be ob- tained from Maxwell’s equation without specifying the boun dary condition for the ﬁelds. Let us introduce the vector R= (ρ,−s) (the negative sign in front of sis due to measuring sin the negative direction of z, see Fig. 4) and consider momentum ∆pin Eq. (23) as a function of R. Let us assume that the electric and magnetic ﬁelds are speciﬁed through the vector potential A(r, t) and the scalar potential 1)A useful relation between the units is 1 V /pC = 1 .11 cm−1.φ(r, t), and compute ∆ pfor the given ﬁelds. It is convenient to use the Lagrangian formulation of the equations of motion,2 d dt∂L ∂v=∂L ∂r≡ ∇L, (27) with the Lagrangian for the trailing unitcharge in the electromagnetic ﬁeld L=−mc2/radicalbigg 1−v2 c2+1 cAv−φ. (28) Putting Eq. (28) into Eq. (27) yields ( p=mγv) d dt/parenleftbigg p+1 cA/parenrightbigg =∇/parenleftbigg1 cAv−φ/parenrightbigg . (29) Now, integrating this equation along the orbit of the traili ng particle, x= const, y= const and z=ct−s, and assuming that the ﬁelds Aandφvanish at inﬁnity, we ﬁnd ∆p(R) =/integraldisplay dt∇/parenleftbigg1 cAv−φ/parenrightbigg =q c∇RW(R), (30) where we introduced the wake potential W, W(R) =c q/integraldisplay dt/parenleftbigg1 cAv−φ/parenrightbigg =c q/integraldisplay dt(Az−φ). (31) In the last equation we used v≈cˆz. We just proved an important relation that states that all thr ee components of the vector ∆ pcan be obtained by diﬀerentiation of a single scalar functio nW. Recalling now the relation between the components of ∆ pand the wakes, Eq. (24), we ﬁnd that wl=−∂W ∂(−s)=∂W ∂s,wt=∇ρW, (32) and hence ∂wt ∂s=∇ρwl. (33) 2)This approach to the derivation of the Panofsky-Wenzel theo rem is due to A. Chao.This relation is usually referred to as the Panofsky-Wenzel theorem. Note that ∇ρ is a two-dimensional gradient with respect to coordinates xandy. One of the most important computational applications of the Panofsky-Wenzel theorem is that knowledge of the longitudinal wake function wlallows us to ﬁnd the transverse wake wtby means of a simple integration of Eq. (33). We now prove another important property of W: it is a harmonic function of variables xandy, ∆⊥W≡∂2W ∂x2+∂2W ∂y2= 0. (34) To prove this, we will use the fact that both Aandφsatisfy the wave equation in free space, ( ∂2/∂t2−c2∆)A= (∂2/∂t2−c2∆)φ= 0. Hence 0 =c q/integraldisplay dt/parenleftbigg∂2 ∂t2−c2∆/parenrightbigg (Az−φ) =−c q/integraldisplay dt/parenleftbigg∂2 ∂x2+∂2 ∂y2/parenrightbigg (Az−φ) +c q/integraldisplay dt/parenleftbigg∂2 ∂t2−c2∂2 ∂z2/parenrightbigg (Az−φ) =−∂2W ∂x2−∂2W ∂y2+c q/integraldisplay dt/parenleftbigg∂ ∂t+c∂ ∂z/parenrightbigg /parenleftbigg∂ ∂t−c∂ ∂z/parenrightbigg (Az−φ). (35) The last integral in this equation vanishes because ∂ ∂t+c∂ ∂z≈∂ ∂t+v∇=d dt(36) and /integraldisplay dt/parenleftbigg∂ ∂t+c∂ ∂z/parenrightbigg /parenleftbigg∂ ∂t−c∂ ∂z/parenrightbigg (Az−φ) =/integraldisplay dtd dt/parenleftbigg∂ ∂t−c∂ ∂z/parenrightbigg (Az−φ) = 0. (37) It is interesting that the wake potential Wturns out to be a relativistic invariant. A covariant expression for it can be written as W=−1 q/integraldisplay∞ −∞Akukdτ, (38) where Ak= (φ,−A) is the 4-vector potential, uk= (cγ, cγv) is the 4-vector velocity, andτis the proper time for the particle.VIII SYSTEMS WITH A SYMMETRY AXIS In Section VI we deﬁned the wake as a function of the trailing p article oﬀset relative to the path of the leading particle. In practical ap plications we are also interested in how the wake depends on the trajectory of the le ading particle. We will assume that the system under consideration has a symmet ry axis, and choose it as the z-axis of the coordinate system (see Fig. 5). Now the leading p article 1 1vv2 zxsΡ yΡ'R FIGURE 5. Both the leading particle 1 and the trailing particle 2 are oﬀ set relative to the axis of the chamber. moves in the zdirection with an oﬀset given by vector ρ′, and the trailing particle travels parallel to the leading one, with the same velocity, at a distance sbehind the leading one, and with oﬀset ρrelative to the axis. The vectors ρ′andρare the two-dimensional vectors perpendicular to the z-axis. The wake is still deﬁned by Eq. (24), but now it will be considered as a function of ρ′,ρ, and s wl=wl(ρ,ρ′, s), wt=wt(ρ,ρ′, s). (39) Usually the vacuum chamber is designed so that the system axi s serves as an ideal orbit for the beam. Deviations from it are relatively s mall, and both vectors ρandρ′are typically much smaller than the size of the vacuum chambe r. We can neglect them in wland introduce the longitudinal wake function that depends o nly ons, wl(s) =wl(0,0, s). (40) If the vacuum chamber also has some symmetry elements (e.g., it has either circular, elliptical or rectangular cross section), the tr ansverse wake on the axis, where ρ,ρ′= 0, vanishes, wt(0,0, s) = 0. For small values of ρ,ρ′we can expand wt(ρ,ρ′, s) keeping only the lowest-order linear terms. That gives a te nsor relation between the transverse wake and the oﬀsets, wt(ρ,ρ′, s) =↔ W1(s)ρ+↔ W2(s)ρ′, (41)where↔ W1and↔ W2are the two-dimensional tensors of rank 2. An example of the wake calculation for elliptical and rectangular cross s ections of the pipe can be found in Ref. 7. IX AXISYMMETRIC SYSTEMS In an axisymmetric system Wdepends only on the absolute values of ρ,ρ′, and the angle θbetween them. We can always chose a coordinate system such th at the vector ρ′lies in the x-zplane (see Fig. 6), so that the potential function Wwill be a periodic even function of angle θin a cylindrical coordinate system. Decomposing 12 ρ ρ'/j113 FIGURE 6. Vectors ρandρ′in an axisymmetric system. Win Fourier series in θyields W(ρ, ρ′, θ, s) =∞/summationdisplay m=0Wm(ρ, ρ′, s) cosmθ. (42) Putting this equation into Eq. (34) gives ∞/summationdisplay m=0/parenleftbigg1 ρ∂ ∂ρρ∂Wm ∂ρ−m2 ρ2Wm/parenrightbigg cosmθ= 0, (43) from which we can ﬁnd an explicit dependence of Wmofρ, Wm(ρ, ρ′, s) =Am(ρ′, s)ρm. (44) In Eq. (44) we discarded a singular solution of Eq. (43) Wm∝ρ−m. It is also possible to ﬁnd the dependence of Wmversus ρ′(see [8]), which turns out to be Am(ρ′, s) =Fm(s)(ρ′)m. (45) Using Eq. (32) we now ﬁnd for the longitudinal and transverse wake functions wl=/summationdisplay w(m) l,wt=/summationdisplay w(m) t (46)where w(m) l= (ρ′)mρmF′ m(s) cosmθ, w(m) t=m(ρ′)mρm−1Fm(s)/bracketleftBig ˆrcosmθ−ˆθsinmθ/bracketrightBig , (47) where ˆrandˆθare the unit vectors in the radial and azimuthal directions i n the cylindrical coordinate system. Remember that in this equat ion we assume that the leading particle is in the plane θ= 0. Equations (47) are valid for arbitrary values of ρandρ′. Near the axis, where the oﬀsets are small, the higher-order terms with large values o fmin these equations also become small. In this case, we can keep only the lower-or der terms with m= 0 (monopole ) and m= 1 (dipole ) wakes. For the monopole wake we ﬁnd wl≡w(0) l=F′ 0(s), (48) which shows that the longitudinal wake does not depend on the radius in an axisym- metric system. We also see that the monopole transverse wake vanishes, w(0) t= 0. Since w(0) ldoes not depend on ρ′, sometimes it is more convenient to compute the monopole wake for an oﬀset orbit, ρ′∝negationslash= 0, rather than on the axis. ρ ρ'/j113r /j113/j94 /j94 r cos /j113/j45/j113/j32sin /j113/j94 /j94 FIGURE 7. Vectors ρandρ′and unit vectors in cylindrical coordinate system. For the dipole wake ( m= 1), the vector ˆrcosθ−ˆθsinθlies in the direction of thexaxis, that is in the direction of ρ′, see Fig. 7. Hence, w(1) t=ρ′F(s). (49) The wake given by Eq. (49) is usually normalized by the absolu te value of the oﬀset ρ′, and the scalar function w(1) t/ρ′is called the transverse dipole wake wt, wt(s) =F(s). (50) Such a transverse wake has the dimension cm−2or V/C/m.3In this deﬁnition, a positive transverse wake means a kick in the direction of the oﬀset of the driving particle (if both particles have the same charge). 3)If the original wake is deﬁned per unit length, as in Eq. (26), thenwtwill have the dimension V/C/m2or cm−3.X RESISTIVE WALL WAKE FUNCTIONS We are now in a position to ﬁnd the wakes generated by a particl e in a circular pipe with resistive walls. Using Eq. (18) and the deﬁnition E q. (24) gives the longitudinal wake wl(s) =−1 2πb/radicalbiggc σs3. (51) The minus sign here means that the trailing charge is acceler ated in the wakeﬁeld. Let us now calculate the dipole transverse wake due to the res istive wall. First, we need to solve for the electromagnetic ﬁeld of the leading c harge qmoving with an oﬀset ρ′in a circular pipe. From the point of view of excitation of dip ole modes, this charge can be considered as having a dipole moment d=qρ′. In the zeroth approximation of perturbation theory, the electromagneti c ﬁeld of a dipole moving with the speed of light in a perfectly conducting pipe is E= 2δ(z−ct)/bracketleftbigg2(d·r)r−dr2 r4+d b2/bracketrightbigg ,H=ˆz×E. (52) The ﬁrst term in the expression for Eis a vacuum ﬁeld of a relativistic dipole, and the second one is due to the image charges, which are generate d in order to satisfy the boundary condition on the metal surface. Following the derivation in Section V, we ﬁnd the magnetic ﬁe ld on the wall, Hθ=4d b2cosθδ(z−ct), (53) and take its Fourier transform, Hθω=2d πcb2cosθ , (54) where the angle θis measured from the direction of d. Then using the Leontovich boundary condition, Eq. (12), for the electric ﬁeld Ezω, Ezω\|r=b=−(1−i)/radicalbiggω 8πσ2d πcb2cosθ, (55) and making the inverse Fourier transformation, we obtain Ezon the wall Ez(z, ρ=b, ρ′, t) =qρ′ πb2/radicalbiggc σs3cosθ , (56) where s=ct−z. Recalling that, according to Eq. (47) the dipole wake is a li near function of ρ, we conclude that Ez(z, ρ, ρ′, t) =qρρ′ πb3/radicalbiggc σs3cosθ , (57)and the function F′ 1(s) is F′ 1(s) =−1 πb3/radicalbiggc σs3, (58) which gives the following result for the transverse wake deﬁ ned by Eq. (50): wt(s) =2 πb3/radicalbiggc σs. (59) Analogous to the longitudinal wake, Eq. (18), this formula i s valid only for s≫s0 (see Eq. (22)). XI WAKEFIELD IN A BUNCH OF PARTICLES Up to now we have studied the interaction of two point charges traveling some distance sapart. If a beam consists of Nparticles with the distribution function λ(s) (deﬁned so that λ(s)dsgives the probability of ﬁnding a particle near point s), a given particle will interact with all other particles of the beam. To ﬁnd the change of the longitudinal momentum of the particle at point swe need to sum the wakes from all other particles in the bunch, ∆pz(s) =Ne2 c/integraldisplay∞ sds′λ(s′)wl(s′−s). (60) Here we use the causality principle and integrate only over t he part of the bunch ahead of point s. In the ultrarelativistic limit the energy change ∆ E(s) caused by the wakeﬁeld is equal to c∆pz, so Eq. (60) can also be rewritten as ∆E(s) =Ne2/integraldisplay∞ sds′λ(s′)wl(s′−s). (61) Two integral characteristics of the strength of the wake are the average value of the energy loss ∆ Eav, and the rms spread in energy generated by the wake ∆ Erms. These two quantities are deﬁned by the following equations: ∆Eav=/integraldisplay∞ −∞ds∆E(s)λ(s), (62) and ∆Erms=/bracketleftbigg/integraldisplay∞ −∞ds(∆E(s)−∆Eav)2λ(s)/bracketrightbigg1/2 . (63)As an example, let us calculate ∆ Eavand ∆ Ermsfor the resistive wall wake given by Eq. (51) and a Gaussian distribution function, ρ(s) =1√ 2πσsexp/parenleftbigg −s2 2σ2s/parenrightbigg , (64) where σsis the rms bunch length. Note that, since win Eq. (51) is the wake per unit length of the pipe, we need to multiply the ﬁnal answer by the pipe length L. A direction substitution of the wake Eq. (51) into Eq. (61) gi ves a divergent integral when s′→s.4To remove the divergence, we need to recall that according to Eq. (48) the longitudinal wake is equal to the derivative o f the longitudinal wake potential, wl=F′ 0(s) with F0= (πb)−1/radicalbig c/σsfors >0, and F0= 0 for s <0. We then rewrite Eq. (61) as ∆E(s) =Ne2L/integraldisplay∞ −∞ds′λ(s′)dF0(s′−s) ds =−Ne2L/integraldisplay∞ sds′dλ(s′) dsF0(s′−s) =Ne2Lc1/2 bσ3/2 zσ1/2G/parenleftbiggs σz/parenrightbigg , (65) where the function G(x) is G(x) =1 21/2π3/2/integraldisplay∞ xye−y2/2dy√y−x. (66) The plot of the function G(s/σz) is shown in Fig. 8, where the positive values ofscorrespond to the head of the bunch. We see that in the resisti ve wake the particles lose energy in the head of the bunch and get acceler ated in the tail. On the average, of course, the losses overcome the gain. For the average energy loss one can ﬁnd an analytical result: ∆Eav=Γ(3 4) 23/2π3/2Ne2c1/2 bσ3/2 zσ1/2. (67) Numerical integration of Eq. (63) shows that the energy spre ad generated by the resistive wake is approximately equal to ∆ Eav: ∆Erms= 1.06 ∆Eav. (68) If the beam is traveling in the pipe with an oﬀset yrelative to the axis, it will be deﬂected in the direction of the oﬀset, by the transverse wak eﬁelds. To calculate the deﬂection angle θwe use the relation 4)The integral diverges because Eq. (51) is not valid for very s mall values of s, see Eq. (22).-5-2.5 0 2.5 5 s□sz00.1G FIGURE 8. Plot of the function G(s/σz). θ(s) =∆p⊥(s) p=yLNe2 cp/integraldisplay∞ sds′λ(s′)wt(s′−s) =NLr eyc1/2 γb3σ1/2 zσ1/2H/parenleftbiggs σz/parenrightbigg , (69) where the function H(x) is H(x) =21/2 π3/2/integraldisplay∞ xe−y2/2dy√y−x. (70) The plot of the function H(s/σz) is shown in Fig. 9. The deﬂection angle averaged over the distribution functio n is θav=Γ(1 4) 21/2π3/2NLr eyc1/2 γb3σ1/2 zσ1/2, (71) and the rms spread is ∝angb∇acketleft(θ−θav)2∝angb∇acket∇ight1/2=A1/2NLr eyc1/2 γb3σ1/2 zσ1/2, (72) where A=2 π5/2/bracketleftbigg K/parenleftbigg3 4/parenrightbigg −Γ2(1/4) 4√π/bracketrightbigg (73) andKis the complete elliptic integral. The numerical value of A1/2is 0.186.-5-2.5 0 2.5 5 s□sz00.20.40.6H FIGURE 9. Plot of the function H(s/σz). XII DEFINITION OF IMPEDANCE AND RELATION BETWEEN IMPEDANCE AND WAKE Knowledge of the longitudinal and transverse wake function s gives complete in- formation about the electromagnetic interaction of the bea m with its environment. However, in many cases, especially in the study of beam insta bilities, it is more con- venient to use the Fourier transform of the wake functions, o rimpedances . Also, it is often easier to calculate the impedance for a given geomet ry of the beam pipe, rather than the wake function. Recall, that in Section V we ac tually ﬁrst com- puted the Fourier components of the wakes, and then, using th e inverse Fourier transformation, found the wakes. For historical reasons the longitudinal Zland transverse Ztimpedances are de- ﬁned as Fourier transforms of wakes with diﬀerent factors, Zl(ω) =1 c/integraldisplay∞ 0dsw l(s)eiωs/c, Zt(ω) =−i c/integraldisplay∞ 0dsw t(s)eiωs/c. (74) Note that the integration in Eqs. (74) can actually be extend ed into the region of negative values of s, because wlandwtare equal to zero in that region. Impedance can also be deﬁned for complex values of ωsuch that Im ω >0 and the integrals, Eq. (74), converge. So deﬁned, the impedance is an analytic function in the upper half-plane of the complex variable ω. We must keep in mind that other authors sometimes introduce d eﬁnitions of the impedance that diﬀer from the one given above. In Refs. 2 and 9 the longitudinalimpedance is deﬁned as a complex conjugate to the one given by Eq. (74). Here we follow the deﬁnitions of Refs. 1 and 10. From the deﬁnitions in Eq. (74) it follows that the impedance satisﬁes the following symmetry conditions: ReZl(ω) = Re Zl(−ω), ImZl(ω) =−ImZl(−ω), ReZt(ω) =−ReZt(−ω),ImZt(ω) = Im Zt(−ω). (75) The inverse Fourier transform relates the wakes to the imped ances: wl(s) =1 2π/integraldisplay∞ −∞dωZ l(ω)e−iωs/c, wt(s) =i 2π/integraldisplay∞ ∞dωZ t(ω)e−iωs/c. (76) It turns out that the wakeﬁeld can actually be found if only th e real part of the impedance is known. Indeed, we can rewrite Eq. (76) for wlas wl(s) =1 2π/integraldisplay∞ ∞dω/bracketleftBig ReZl(ω) cosωs c−ImZl(ω) sinωs c/bracketrightBig . (77) For negative values of sthis formula should give wl= 0, hence 0 =/integraldisplay∞ ∞dω/bracketleftbigg ReZl(ω) cosωs c+ ImZl(ω) sinω\|s\| c/bracketrightbigg , (78) from which it follows that wl(s) =1 π/integraldisplay∞ ∞dωReZl(ω) cosωs c=2 π/integraldisplay∞ 0dωReZl(ω) cosωs c. (79) A similar derivation for the transverse wake gives wt(s) =2 π/integraldisplay∞ 0dωReZt(ω) sinωs c. (80) XIII ENERGY LOSS AND ReZL We can relate the energy loss by the bunch to the real part of th e longitudinal impedance. Indeed, ∆Eav=N2e2/integraldisplay∞ −∞dsλ(s)/integraldisplay∞ −∞ds′λ(s′)wl(s′−s) =N2e2/integraldisplay∞ −∞dsλ(s)/integraldisplay∞ −∞ds′λ(s′)1 2π/integraldisplay∞ −∞dωZ l(ω)e−iω(s′−s)/c =N2e2 2π/integraldisplay∞ −∞dωZ l(ω)\|ˆλ(ω)\|2, (81)where ˆλ(ω) =/integraltext∞ −∞dsλ(s)eiωs/c. Since ˆλ(−ω) =ˆλ∗(ω),\|ˆλ(ω)\|2is an even function ofω, and ∆Eav=Q2 π/integraldisplay∞ 0dωReZl(ω)\|ˆλ(ω)\|2, (82) where Q=Ne. For a point charge, λ(s) =δ(s),ˆλ(ω) = 1, and the energy loss is ∆Eav=e2 π/integraldisplay∞ 0dωReZl(ω). (83) XIV KRAMERS-KRONIG RELATIONS Equations (79) and (80) relate Re Z(ω) to the wake function. Since Z(ω) is given by Fourier transformation of w(s), the knowledge of Re Z(ω) allows us to ﬁnd Z(ω), and hence Im Z(ω). This means that Im Z(ω) and Re Z(ω) are functionally related to each other. Mathematically this relation is manifested i n the Kramers-Kronig dispersion relation, which can be written as Z(ω) =−i πP.V./integraldisplay∞ −∞Z(ω′) ω′−ωdω′, (84) where P.V. stands for the principal value of the integral. Ta king the real and imaginary parts of this equation gives explicit relations b etween Re Zand Im Z: ReZ(ω) =1 πP.V./integraldisplay∞ −∞ImZ(ω′) ω′−ωdω′, ImZ(ω) =−1 πP.V./integraldisplay∞ −∞ReZ(ω′) ω′−ωdω′. (85) XV USEFUL FORMULA FOR IMPEDANCE CALCULATION Assume that we have a solution of an electromagnetic problem corresponding to the current on the axis of a pipe with the time and space depe ndence given by e−iωt+iωz/c. Speciﬁcally, we know the electric ﬁeld on the axis, Ezω(z)e−iωt. How can longitudinal impedance be found in terms of Ezω(z)? The longitudinal wake is equal to the integrated ﬁeld Ezgenerated by a point charge moving with the speed of light. The current correspon ding to this point charge can be decomposed into a Fourier integral: cqδ(z−ct) =/integraldisplay∞ −∞dωq 2πe−iω(t−z/c). (86)Since we know the electric ﬁeld generated by each harmonic, w e can ﬁnd the ﬁeld due to the point charge as a superposition of Ezω: Ez(z, t) =q 2π/integraldisplay∞ −∞dωE zω(z)e−iωt. (87) Forwlwe then have wl(s) =−1 2π/integraldisplay∞ −∞dωd(ct)Ezω(ct−s)e−iωt =−1 2π/integraldisplay∞ −∞dωdzE zω(z)e−iω(z+s)/c. (88) Comparing Eq. (88) with Eq. (74) we ﬁnd Zl(ω) =−/integraldisplay∞ −∞dzE ω(z)e−iωz/c. (89) Hence the longitudinal impedance can be obtained simply by m aking Fourier trans- formation of Ezω(z). XVI SMALL PILLBOX CAVITY IN ROUND PIPE As an example of using Eq. (89) we will derive here the longitu dinal impedance Zl(ω) for a small axisymmetric cavity (pillbox) in a round perfec tly conducting pipe, see Fig. 10. We assume that the wavelength associated w ith the frequency ω is much larger than the dimension of the pillbox, c/ω≫g, h. hg b FIGURE 10. Small pillbox cavity in a round pipe. The dashed line near the wall shows the integration path in Eq. (92). First, we need to ﬁnd the solution of Maxwell’s equations cor responding to a unit current Iω=e−iωt+iωz/con the axis. Since the cavity is small, the magnitude of themagnetic ﬁeld at the location of the cavity ( z= 0) is approximately equal to Hθ on the wall of the pipe Hθ=2 bc. (90) Because Ezωdoes not depend on r(see Section V) we can choose the integration path in Eq. (89) close to the wall, as shown in Fig. 10, rather t han the pipe axis. Along this path Ezωis not equal to zero only in the cavity gap, where \|z\| ∼h, and e−iωz≈1. We have Zl(ω) =−/integraldisplay∞ −∞dzE zω(z)e−iωz/c ≈ −/integraldisplay∞ −∞dzE zω(z) =1 cdΦ dt=−iω cΦ, (91) where Φ is the magnetic ﬂux in the cross section of the cavity, Φ =Hθhg. As a result, Zl(ω) =−iω2gh bc2=−iωZ0gh 2πbc, (92) where Z0= 4π/c= 377 Ohm. What we obtained is a purely inductive longitudinal impedance, which can be rewritten as Zl(ω) =−iω c2L, (93) where the inductance L= 2gh/b. In CGS units the inductance has a dimension of cm, 1 cm = 1 nH. A more detailed calculation [11] shows that our method gives only an approximate solution of the problem. In addition to the solenoidal elect ric ﬁeld generated by the time-varying magnetic ﬂux in the cavity, there is also a c ontribution due to the potential component of the electric ﬁeld. This results i n a diﬀerent numerical coeﬃcient in Eq. (92) which depends on the ratio g/h. For example, for g=hthe correction factor is 0.84. XVII INDUCTIVE IMPEDANCE We saw in the previous section that a small pillbox is charact erized by inductive impedance if the frequency is not very large. This is a common feature of many small perturbations whose size is much smaller than the pipe radius (e.g., small holes, shallow obstacles on the wall, etc.) — for not very lar ge frequencies their impedance is purely inductive.The longitudinal wake corresponding to the inductive imped ance can be found by using Eq. (76):5 wl(s) =Lδ′(s). (94) Because of the inductive wake, slices of the beam can change t heir energy, al- though the net energy loss for the bunch is zero because the re al part of the impedance vanishes. We can ﬁnd the energy change as a functio n of position within the bunch by using Eq. (61). Integration gives ∆E(s) =−e2NLλ′(s). (95) For a Gaussian distribution function this reduces to ∆E(s) =e2NL σ2zξe−ξ2/2 √ 2π, (96) where ξ=s/σz. For the rms energy spread we ﬁnd ∝angb∇acketleft∆E2∝angb∇acket∇ight1/2= 3−3/4(2π)−1/2e2NL σ2z. (97) XVIII CAVITY IMPEDANCE In the more general case of a large cavity (Fig. 11) the beam ex cites cavity eigenmodes and the longitudinal wake in the cavity is compos ed of contributions FIGURE 11. An RF cavity with beam pipes. from single modes, wl(s) =/summationdisplay nw(n)(s). (98) 5)Since the integral involved in the calculation of wl(s) actually diverges at ω→ ∞, it should be treated as a generalized function. It is easier to verify E q. (94) by putting it into Eq. (74) and checking that the resulting impedance is given by Eq. (93 ).For perfectly conducting walls, assuming that the modes do n ot propagate into the beam pipes,6the mode frequencies ωnare real. It should be no surprise that each partial wakeﬁeld oscillates with the frequency of the mode ωn, w(n)(s) = 2kncos/parenleftBigωns c/parenrightBig , (99) where knis the loss factor , which depends on the geometry of the cavity and the mode number. As an example, for a cylindrical cavity with b=land TM 010mode, k010= 4.5/l. For a more rigorous derivation of the wake for a cavity, see R ef. 12. The cavity impedance for this wake can be calculated by using Eqs. (74). They give ReZl=πkn[δ(ω+ωn) +δ(ω−ωn)], ImZl=kn/bracketleftbigg1 ω+ωn+1 ω−ωn/bracketrightbigg . (100) It is also easy to generalize the above wake for a cavity with l ossy walls when the frequency of the mode has a small imaginary part γn,γn≪ωn. The wake now decays with time as wl(s) = 2kne−γs/ccos/parenleftBigωns c/parenrightBig . (101) Again using Eq. (74), we can calculate the impedance. It has t wo peaks: one in the vicinity of ω=ωnand the other in the vicinity of ω=−ωn. Assuming that ω is close to ωn, we ﬁnd Zl=kn γ−i(ω−ωn). (102) REFERENCES 1. A. W. Chao, Physics of Collective Beam Instabilities in High Energy Acc elerators (Wiley, New York, 1993). 2. B. W. Zotter and S. A. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientiﬁc, Singapore, 1998). 3. A. W. Chao and M. Tigner, Handbook of Accelerator Physics and Engineering (World Scientiﬁc, Singapore, 1999). 4. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields , vol. 2 of Course of Theoretical Physics , 4th ed. (Pergamon, London, 1979) (Translated from the Rus- sian). 5. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media , vol. 8 of Course of Theoretical Physics , 2nd ed. (Pergamon, London, 1960) (Translated from the Russian). 6)This is true if the frequency of the mode is above the cutoﬀ fre quency for the pipe.6. K. L. F. Bane and M. Sands, The Short-Range Resistive Wall Wakeﬁelds , Tech. Rep. SLAC-PUB-95-7074, SLAC (December 1995). 7. R. L. Gluckstern, J. van Zeijts, and B. Zotter, Phys. Rev. E47, 656 (1993). 8. K. L. Bane, P. B. Wilson, and T. Weiland, in Proc. US Particle Accelerator School: Physics of Particle Accelerators, Upton, N.Y., 1983 (American Institute of Physics, New York, 1985), no. 127 in AIP Conference Proceedings, pp. 8 75–928. 9. P. B. Wilson, in M. Month and M. Dienes, eds., Proc. US Particle Accelerator School: Physics of Particle Accelerators, Batavia, 1987 (American Institute of Physics, New York, 1989), no. 184 in AIP Conference Proceedings, pp. 5 25–564. 10. S. A. Heifets and S. A. Kheifets, Review of Modern Physics 63, 631 (1991). 11. S. S. Kurennoy and G. V. Stupakov, Particle Accelerators 45, 95 (1994). 12. P. Wilson, High Energy Electron Linacs: Applications to Storage Ring R F Systems and Linear Colliders , Tech. Rep. SLAC-AP-2884 (Rev.), SLAC (November 1991). APPENDIX We show here how to calculate the integral in Eq. (17), I(ξ) =/integraldisplay∞ −∞dω√ωe−iωξ, (A1) where ξ=t−z/c=s/c. First, we change the integration variable ωtoζ=ωξ, I(ξ) =1 ξ3/2/integraldisplay∞ −∞dζ/radicalbig ζe−iζ. (A2) We then consider ζas a complex variable. In order to make the integrand a single - valued function in the complex plane, we make a cut along the n egative imaginary half-axis and deform the integration path from a straight li ne to a contour, shown in Fig. 12. The integral I(ξ) can now be split into two parts, I1(ξ) and I2(ξ), corresponding to the left and right branches of the integrat ion contour. In the ﬁrst integral I1(ξ) we change the complex integration variable ζto the real positive variable τ,ζ=e3πi/2τ=−iτ, so that√ζ=e3πi/4√τ= (i−1)√τ/√ 2. This gives for I1 I1(ξ) =1 ξ3/2/integraldisplay0 ∞(−idτ)(i−1)√τ√ 2e−τ =(i+ 1)√ 2ξ3/2/integraldisplay0 ∞dτ√τe−τ =−√π(i+ 1) 23/2ξ3/2. (A3) For the second integral I2(ξ) we choose the real positive variable τsuch that ζ= e−πi/2τ=−iτ, which means that√ζ=e−πi/4√τ= (1−i)√τ/√ 2 and/K49/K6D/K20ζ /K52/K65/K20ζ /K43 /K32/K43 /K31 FIGURE 12. Complex plane ζ. The wavy line indicates the cut. The deformed integration contour consists of two paths, C1andC2, corresponding to the integrals I1andI2, respectively. I2(ξ) =1 ξ3/2/integraldisplay∞ 0(−idτ)(1−i)√τ√ 2e−τ =I1(ξ). (A4) Hence I(ξ) =I1(ξ) +I2(ξ) =−√π(i+ 1) 21/2ξ3/2. (A5) Substituting this equation into Eq. (17) gives Eq. (18).
arXiv:physics/0011012v1 [physics.chem-ph] 5 Nov 2000Coherent control of enrichment and conversion of molecular spin isomers P.L. Chapovsky∗ Institute of Automation and Electrometry, Russian Academy of Sciences, 630090 Novosibirsk, Russia (February 2, 2008) Abstract A theoretical model of nuclear spin conversion in molecules controlled by an external electromagnetic radiation resonant to rotationa l transition has been developed. It has been shown that one can produce an enrichme nt of spin isomers and inﬂuence their conversion rates in two ways, thr ough coherences and through level population change induced by radiation. I nﬂuence of con- version is ranged from signiﬁcant speed up to almost complet e inhibition of the process by proper choice of frequency and intensity of th e external ﬁeld. 03.65.-w; 32.80.Bx; 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]. Spin isomers are important for f undamental science and have various applications. They can serve as spin labels, inﬂuen ce chemical reactions [2,3], or tremendously enhance NMR signals [4,5]. Progress in the spi n isomers study depends heavily on available methods for isomer enrichment. Although this ﬁ eld has been signiﬁcantly advanced recently (see the review in [6]), one needs more eﬃc ient enrichment methods. Recently two enrichment methods based on optically induced change of molecular level populations have been proposed [7,8]. The purpose of the pre sent paper is to investigate the isomer enrichment caused by optically induced coherences i n the molecule. II. QUANTUM RELAXATION Nuclear spin isomers of molecules were discovered in the lat e 1920s. The most known example is the isomers of H 2. These isomers have diﬀerent total spin of the two hydrogen nuclei, I= 1 for ortho molecules and I= 0 for para molecules. Symmetrical polyatomic molecules have nuclear spin isomers too. For example, CH 3F can be in ortho, or in para state depending on the total spin of three hydrogen nuclei equal 3/ 2, or 1/2, respectively (see, e.g., [1]). Diﬀerent spin isomers are often distinguished a lso by their rotational quantum numbers. Consequently, all rotational states of the molecu le are separated into nondegen- erate subspaces of diﬀerent spin states. Schematically, th ese subspaces for the case of two spin states, ortho and para, are presented in Fig. 1. (For a mo ment, an optical excitation shown in the ortho space has to be omitted.) Nuclear spin conversion can be produced by collisions with m agnetic particles. This is a well-known mechanism for the conversion of hydrogen imbed ded in paramagnetic oxygen [9]. If a molecule is surrounded by nonmagnetic particles, c ollisions alone cannot change the molecular spin state. In this case the spin conversion is gov erned by quantum relaxation which can be qualitatively described as follows. Let us spli t the molecular Hamiltonian into two parts, ˆH=ˆH0+ ¯hˆV , (1) 2where ˆH0is the main term which has the ortho and para states as the eige n states (states in Fig. 1); ˆVis a small intramolecular perturbation able to mix the ortho and para states. Suppose that the test molecule was placed initially in the or tho subspace. Due to colli- sions the molecule starts undergo fast rotational relaxati oninside the ortho subspace. This running up and down along the ortho ladder proceeds until the molecule reaches the ortho statemwhich is mixed with the para state kby the intramolecular perturbation ˆV. Then, during the free ﬂight just after this collision, the perturb ation ˆVmixes the para state kwith the ortho state m. Consequently, the next collision can move the molecule to o ther para states and thus localize it inside the para subspace. Such me chanism of spin conversion was proposed in the theoretical paper [10] (see also [11]). Relevance of the described mechanism to actual spin convers ion in molecules is not at all obvious. The problem is that the intramolecular perturb ations, ˆV, able to mix ortho and para states are very weak. They have the order of 10 −100 kHz (hyperﬁne interactions) which should be compared with other much stronger interacti ons in molecules, or with gas collisions. Nevertheless, the experimental and theoretic al proves have been obtained that spin conversion in molecules is indeed governed by quantum r elaxation [6,12–14]. Although, only three molecules (CH 3F, H 2CO and C 2H4) have been studied in this context so far, it is very probable that spin conversion in other polyatomic mo lecules of similar complexity is governed by quantum relaxation too. It is useful for the foll owing to give a few examples of spin conversion rates. Most studied is the spin conversion i n13CH3F which has the rate, γ13/P= (12.2±0.6)·10−3s−1/Torr, (2) in case of pure CH 3F gas [6]. Spin conversion in another isotope modiﬁcation,12CH3F, is by almost two orders of magnitude slower [6]. Similar slow conv ersion was observed in ethylene, 13CCH 4, [13], γeth/P= (5.2±0.8)·10−4s−1/Torr. (3) These data show that spin conversion by quantum relaxation i s on 9−11 orders of magnitude slower than the rotational relaxation, ν∼106−107s−1/Torr. 3III. OPTICAL EXCITATION In this section we will analyze the spin conversion by quantu m relaxation in the presence of a resonant electromagnetic radiation. The level scheme i s shown in Fig. 1. In order to reveal the main features of the phenomenon we will conside r the process in a simplest arrangement. First of all, we assume that molecular states a re nondegenerate and that only one ortho-para level pair, m−k, is mixed by the intramolecular perturbation, ˆV. Monochromatic radiation is chosen to be in resonance with th e rotational transition m−n. In this arrangement the molecular Hamiltonian reads, ˆH=ˆH0+ ¯hˆG+ ¯hˆV . (4) New term, ˆG, describes the molecular interaction with the radiation, ˆG=−(E0ˆd/¯h) cosωLt, (5) where E0andωLare the amplitude and frequency of the electromagnetic wave ;ˆdis the operator of the molecular electric dipole moment. We have ne glected molecular motion in the operator ˆGbecause homogeneous linewidth of pure rotational transiti on is usually larger than the Doppler width. Kinetic equation for the density matrix, ˆ ρ, in the representation of the eigen states of the operator ˆH0has standard form, dˆρ/dt=ˆS−i[ˆG+ˆV ,ˆρ], (6) where ˆSis the collision integral. Molecules in states ( m, n, k ) interacting with the perturbations ˆGandˆVconstitute only small fraction of the total concentration of the test molecu les. Thus, one can neglect collisions between molecules in these states in comparison with collis ion with molecules in other states. The latter molecules remain almost at equilibrium. Consequ ently, the collision integral, ˆS, depends linearly on the density matrix for the disturbed sta tesm,n, and keven in one component gas. Further, we will assume model of strong colli sions for the collision integral. The oﬀ-diagonal elements of ˆSare Sjj′=−Γρjj′;j, j′∈m, n, k ;j/negationslash=j′. (7) 4Here, jandj′indicate rotational states of the molecule. The decoherenc e rates, Γ, were taken equal for all oﬀ-diagonal elements of ˆS. Collisions cannot alter molecular spin state in our model. I t implies that the diagonal elements of ˆShave to be determined separately for ortho molecules, Sjj=−νρo(j) +νwo(j)ρo;ρo=/summationdisplay jρo(j);j∈ortho, (8) and for para molecules, Sjj=−νρp(j) +νwp(j)ρp;ρp=/summationdisplay jρp(j);j∈para. (9) Hereρo,ρpandwo(j),wp(j) are the total concentrations and Boltzmann distributions of ortho and para molecules. The rotational relaxation rate, ν, was taken equal for ortho and para molecules, because kinetic properties of diﬀerent spi n species are almost identical. One can obtain from Eq. (6) an equation of change of the total c oncentration in each spin space [11]. For example, for ortho molecules one has, dρo/dt= 2Re(iρmkVkm). (10) In fact, this result is valid for any model of collision integ ral as long as collisions do not change the total concentration of molecules in each subspac e, i.e.,/summationtext jSjj= 0 if j∈ortho, orj∈para. Let us approximate the density matrix by the sum of zero and ﬁr st order terms over perturbation ˆV, ˆρ= ˆρ(0)+ ˆρ(1). (11) In zero order perturbation theory the ortho and para subspac es are independent, i.e., ρ(0) mk= 0. Consequently, equation of change (10) is reduced to, dρo/dt= 2Re(iρ(1) mkVkm). (12) Note that the spin conversion appears in the second order of ˆV. Kinetic equations for zero and ﬁrst order terms of the densit y matrix are given as, dˆρ(0)/dt=ˆS(0)−i[ˆG,ˆρ(0)]. (13) dˆρ(1)/dt=ˆS(1)−i[ˆG,ˆρ(1)]−i[ˆV ,ˆρ(0)]. (14) 5We start with zero order perturbation theory. In this approx imation, the para subspace remain at equilibrium, ρ(0) p(j) =ρ(0) pwp(j);j∈para. (15) Hereρ(0) pis the total concentration of para molecules. Density matrix for ortho molecules can be determined from th e two equations which follows from Eqs. (7),(8),(13): dρ(0) o(j)/dt=−νρ(0) o(j) +νwo(j)ρ(0) o+ 2Re(iρ(0) o(m\|n)Gnm)[δjm−δjn]; dρ(0) o(m\|n)/dt=−Γρ(0) o(m\|n)−iGmn[ρ(0) o(n)−ρ(0) o(m)]. (16) We will assume further the rotational wave approximation, Gmn=−Ge−iΩt;G≡E0dmn/2¯h; Ω = ωL−ωmn, (17) where the line over symbol indicate a time-independent fact or. Rabi frequency, G, is assumed to be real. Rotational relaxation, ν, and decoherence, Γ, are on many orders of magnitude faster than the spin conversion. It allows to assume station ary regime for ortho molecules, thus having, dρ(0) o(j)/dt= 0. The substitution, ρ(0) o(m\|n) =ρ(0) o(m\|n) exp (−iΩt), transforms Eqs. (16) to algebraic equations which can easily be solved. Thus one has, ρ(0) o(j) =ρ(0) o/bracketleftBigg wo(j) +2Γ νG2∆w Γ2 B+ Ω2(δjm−δjn)/bracketrightBigg ; ρ(0) o(m\|n) =ρ(0) oiG∆wΓ +iΩ Γ2 B+ Ω2; Γ2 B= Γ2+ 4ΓG2/ν; ∆w≡wo(n)−wo(m). (18) We turn now to the calculation of the ﬁrst order term, ρ(1) mk, which has to be substituted into the equation of change (12). The density matrix element ,ρ(1) mk, can be found from the two equations which are derived from Eqs. (7),(14), dρ(1) mk/dt+ Γρ(1) mk+iGmnρ(1) nk=−iVmk[ρ(0) o(k)−ρ(0) o(m)]; dρ(1) nk/dt+ Γρ(1) nk+iGnmρ(1) mk=iVmkρ(0) o(n\|m). (19) Substitution, Vmk=V eiωt,(ω≡ωmk);ρ(1) mk=ρ(1) mkeiωt;ρ(1) nk=ρ(1) nkei(ω−ωkn)t, (20) 6transforms Eq. (19) to algebraic equations from which one ﬁn dsρ(1) mk. Then, Eq. (12) gives the following equation for the total concentration of ortho molecules, dρo dt= 2\|V\|2Re[Γ +i(Ω +ω)][ρ(0) p(k)−ρ(0) o(m)]−iGρ(0) o(n\|m) F(Ω); F(Ω)≡(Γ +iω)[Γ +i(Ω +ω)] +G2. (21) After an appropriate change of notations, the right hand sid e of Eq. (21), coincides formally with the solution [15] for the work of weak optical ﬁ eld in the presence of strong optical ﬁeld. Strong ﬁeld splits the upper state mon two, which appears as two roots of the denominator of Eq. (21) being the second order polynomial on ω. It results in two ortho-para level pairs mixed by the perturbation ˆVinstead of one pair in the absence of an external ﬁeld. In analogy with the optical case, one can distinguish t he isomer conversion caused by population eﬀects (terms proportional to the level populat ionsρ(0) p(k) and ρ(0) o(m)) and by coherences (term proportional to the oﬀ-diagonal density m atrix element, ρ(0) o(n\|m)). Eq. (21) describes time dependence of the concentration of o rtho molecules, ρo, in the second order of ˆV. One can neglect at this approximation small diﬀerence betw eenρoand ρ(0) o. The density of para molecules can be expressed through the d ensity of ortho molecules as,ρ(0) p=n−ρ(0) o, where nis the total concentration of the test molecules. Using thes e points and zero order solution given by Eqs. (15),(18), one can obta in ﬁnal equation of change for ortho molecules, dρo/dt=nγop−ρoγ;γ≡γop+γpo+γn+γcoh, (22) where partial conversion rates were introduced, γop= 2\|V\|2wp(k)f(Ω); f(Ω)≡ReΓ +i(Ω +ω) F(Ω); γpo= 2\|V\|2wo(m)f(Ω); γn= 2\|V\|2G2∆w Γ2 B+ Ω22Γ νf(Ω); γcoh= 2\|V\|2G2∆w Γ2 B+ Ω2ReΓ−iΩ F(Ω). (23) Here the rates γop,γpo, and γnare due to molecular level populations and the rate γcohis due to coherences. The terms γopandγpoare the only ones which remain in the absence of an external ﬁeld. In the radiation free case ( G= 0) the result (23) becomes identical with the solution given in [11]. 7IV. ENRICHMENT Solution to Eq. (22) can be presented as, ρo=ρo+δρoexp(−γt), where time-independent part is given by ρo=nγop γ. (24) Without an external radiation (at the instant t= 0), the equilibrium concentration of para molecules is equal to, ρp(0) = n−ρo(0) = nwo(m) wp(k) +wo(m)=n 2. (25) For simplicity, the Boltzmann factors in Eq. (25) were assum ed to be equal, wp(k) = wo(m)≡w, which implies, γop=γpo. External ﬁeld produces a stationary enrichment of para molecules. One can derive from Eqs. (24),(25), β≡ρp ρp(0)−1 = 1−2γop γ. (26) An enrichment coeﬃcient, β, is deﬁned here in such a way that β= 0 if there is no external electromagnetic ﬁeld. Enrichment of ortho molecules is equ al to−β. Note, that the en- richment, β, does not depend on the magnitude of intramolecular perturb ation ˆV. It is the consequence of the assumption that only one ortho-para leve l pair is mixed. Enrichment, β, depends on the ratio of Boltzmann factors, wo(n)/wo(m), but does not depend on the mag- nitude of wo(n) itself. In further numerical examples relative diﬀerence of the Boltzmann factors will be chosen as wo(n) = 1.2w. One needs to specify a few other parameters in order to invest igate properties of the optically induced enrichment. We will use, where it is possi ble, parameters relevant to the spin conversion in13CH3F. Thus the decoherence rate, Γ, will be chosen equal 6 MHz, wh ich corresponds to the gas pressure of pure CH 3F equal 0.2 Torr [6]. Rotational relaxation, ν, will be chosen by one order of magnitude slower than the decoh erence rate, ν= 0.1Γ. Expressions for the enrichment, β, are given in the Appendix. If ortho-para mixing is performed for a degenerate pair of states m−k(ω= 0), the enrichment of para states, β, has one peak at Ω = 0. This peak is determined mainly by populat ion eﬀects. More interesting is the case of nondegenerate states ( ω/negationslash= 0). In this case one has two peaks, at Ω = −ωand at Ω = 0 (Fig. 2). Peak 1 (Ω = −ω) is due to the coherent eﬀects 8determined by γcoh. Peak 2 (Ω = 0) is mainly due to the optically induced level pop ulation changes determined by γn. In the case of well-separated peaks (Γ B≪ω), the amplitudes of the peak 1 and peak 2 read, A1=∆w wG2 Γ2+G2/bracketleftBigg1 4+D Γ2 B+ω2/bracketrightBigg ;D≡Γ2+ 5G2/4−ΓG2/ν+ω2/4; A2=∆w wG2 Γ2 B/bracketleftBiggΓ ν−3 4+D Γ2 B+G2+ω2/bracketrightBigg . (27) Amplitude of the peak 2 grows rapidly with Gup to β≃4.5%. Amplitude of the peak 1 grows with Gto even bigger value β≃5.5%. These data are shown in Fig. 3 (upper panel) where points are obtained by ﬁtting an exact expression (26) by two Lorentzians and solid curves are given by Eqs. (27). ωwas chosen equal 130 MHz which corresponds to the ortho-para level gap in13CH3F [6]. Note, that there is no optically induced enrichment if ∆w= 0. Widths of the enrichment peaks are given by the expressions, W1= 2√ Γ2+G2;W2= 2Γ B. (28) The two enrichment peaks experience completely diﬀerent po wer broadening which are shown in Fig. 3 (low panel). Peak 1 has the width much smaller t han the Peak 2. Solid curves in Fig. 3 (low panel) are given by Eqs. (28). Points are obtained by ﬁtting the exact expression for enrichment, β, by two Lorentzians. V. CONVERSION Conversion rate in the presence of an external electromagne tic ﬁeld has complicated dependence on radiation frequency detuning, Ω, and Rabi fre quency, G. It is convenient to characterize the conversion rate in relative units, γrel=γ γfree−1, (29) where γfreeis the ﬁeld free conversion rate. Similar to the enrichment, β, this parameter does not depend on magnitude of the perturbation ˆVand on absolute values of the Boltzmann factors. Expressions for the conversion rate, γrel, are given in the Appendix. In the case of degenerate ortho-para level pair m−k(ω= 0),γrelhas narrow negative structure at Ω = 0 9(Fig. 4). Amplitude of this dip grows rapidly with increasin gG. IfG≫Γ≫ν, and all Boltzmann factors have the same order of magnitude, the conv ersion rate at Ω = 0 is given by γrel∼Γ2 G2−1. (30) IfGis large, the relative conversion rate, γrel≃ −1, which corresponds to γ≃0. Thus the spin conversion can be inhibited by radiation having large Gand Ω = 0. In the nondegenerate case ( ω/negationslash= 0) conversion rate, γrel, has two peaks (Fig. 5). If these peaks are well resolved, Γ B≪ω′, conversion rate is given by, γrel=Γ′Γ−1G2 Γ′2+ (Ω + ω′)2+∆w wG2 Γ2 B+ Ω2/parenleftbiggΓ ν−1 2/parenrightbigg , (31) where new parameters are determined as, Γ′≡Γ/parenleftBigg 1 +G2 Γ2+ω2/parenrightBigg ;ω′≡ω/parenleftBigg 1−G2 Γ2+ω2/parenrightBigg . (32) The two peaks in the conversion have Lorentzian proﬁles and a mplitudes determined by the expressions, A3=G2 Γ′Γ;A4=∆w wG2 Γ2 B/parenleftbiggΓ ν−1 2/parenrightbigg . (33) These amplitudes are shown in Fig. 6 (upper panel) by solid cu rves. Points are obtained by ﬁtting the exact solution by two Lorentzians. In this exampl es,ωwas chosen equal 130 MHz. Peak 4 at Ω = 0 is proportional to the ratio of Boltzmann factor s. This peak does not grow signiﬁcantly with G. The peak 3 at Ω = −ω′almost does not depend on Boltzmann factors and at large Ggrows up to γrel= (ω/Γ)2. Thus strong electromagnetic ﬁeld can speed up the conversion signiﬁcantly, viz., by two orders of magnitu de in our numerical example. Widths of the two peaks in the conversion rate are determined by the equations, W3= 2Γ′;W4= 2Γ B, (34) and have very diﬀerent ﬁeld dependences. In fact, the peak 4 a t Ω = 0 is broadened even faster than 2Γ B. Peak 3 at Ω = −ωhas almost no power broadening, if G≪ω(Fig. 6, low panel). Solid curves in Fig. 6, low panel, corresponds to the Eqs. (34). Points are obtained by ﬁtting the exact solution by two Lorentzians. 10VI. CONCLUSIONS We have shown that an external resonant radiation can inﬂuen ce spin isomer conversion in two ways, through level populations and through opticall y induced coherences. The coher- ences introduce in the process new and interesting features . In many cases, the coherences play more important role than the level populations. This an alysis have been performed in a simplest arrangement in order to reveal the main features o f the phenomenon. Optically induced coherences introduce extra resonances b oth in enrichment and in con- version frequency dependences. These new resonances are im portant for future experimental realizations of the optical control of isomer conversion. F irst, they give convenient oppor- tunity to ﬁnd coincidences between molecular transitions a nd available sources of powerful radiation. Second, an observation of the phenomenon will be easier also because electro- magnetic radiation can signiﬁcantly speed up the conversio n. Thus steady state enrichment can be achieved much faster than without ﬁeld. It allows to wo rk at low gas pressures where described above eﬀects can be achieved at smaller radiation intensity. Another advantage is that one can use resonance at which ther e is no large radiation absorption and, consequently, no signiﬁcant level populat ion change by radiation. It should decrease some spurious eﬀects, like molecular resonance ex change. The latter eﬀect can cause serious problems in realization of the optically indu ced enrichment by population eﬀects [7,8]. VII. APPENDIX Here we give expressions for the enrichment, β, and conversion rate, γrel. Enrichment of para molecules is given by Eq. (26). For equal Boltzmann fact ors,wo(m) =wp(k)≡w, one has β= 1−/parenleftBigg 1 +γn+γcoh 2γop/parenrightBigg−1 , (35) and an approximate expression in case of small enrichment, β≃γn+γcoh 2γop. (36) Using Eqs. (23), this expression can be reduced to, 11β≃∆w w/bracketleftBigg1/4 Γ2+G2+ (Ω + ω)2+Γ/ν−3/4 Γ2 B+ Ω2+D (Γ2 B+ Ω2)[Γ2+G2+ (Ω + ω)2]/bracketrightBigg ; D≡Γ2+ 5G2/4−ΓG2/ν+ω2/4. (37) There are two peaks in enrichment, at Ω = −ωand at Ω = 0. These peaks have Lorentzian shape if ω≫ΓB. In the degenerate case ( ω= 0) there is one peak of complicated form at Ω = 0. The spin conversion rate, γrel, is deﬁned as, γrel=γ γfree−1. (38) Using Eqs. (23) one can obtain after straightforward calcul ations the following expression, γrel=g(1 +g)(ω2−Γ2) + 2ω(Ω +ω′) Γ′2+ (Ω + ω′)2+ ∆w wG2 Γ2 B+ Ω2/parenleftbiggΓ ν−1 2/parenrightbigg/bracketleftBigg 1 +g(1 +g)(ω2−Γ2) + 2ω(Ω +ω′) Γ′2+ (Ω + ω′)2/bracketrightBigg + ∆w 2wG2 Γ2 B+ Ω2(1 +g)(2Γ2+ω2)−ω(Ω +ω′) Γ′2+ (Ω + ω′)2; g≡G2 Γ2+ω2; Γ′≡Γ(1 + g);ω′≡ω(1−g). (39) Thus the conversion rate has two peaks situated at Ω = −ω′and at Ω = 0. Position of the peak at Ω = ω′depends on the ﬁeld intensity. In the case of degenerate ortho-para level pair ( ω= 0), frequency dependence of γrel has complicated shape with a dip in the center at Ω = 0 (Fig. (4) ). The whole structure is described by the expression, γrel=−(1 +g)G2 Γ′2+ Ω2+∆w wG2 Γ2 B+ Ω2/bracketleftBigg/parenleftbiggΓ ν−1 2/parenrightbigg/parenleftBigg 1−(1 +g)G2 Γ′2+ Ω2/parenrightBigg +Γ′Γ Γ′2+ Ω2/bracketrightBigg . (40) ACKNOWLEDGMENTS This work was made possible by ﬁnancial support from the Russ ian Foundation for Basic Research (RFBR), grant No. 98-03-33124a 12REFERENCES [1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon Press, Oxford, 1981). [2] M. Quack, Mol. Phys. 34, 477 (1977). [3] D. Uy, M. Cordonnier, and T. Oka, Phys. Rev. Lett. 78, 3844 (1997). [4] C. R. Bowers and D. P. Weitekamp, Phys. Rev. Lett. 57, 2645 (1986). [5] J. Natterer and J. Bargon, Prog. NUCL. Magn. Reson. Spect r.31, 293 (1997). [6] P. L. Chapovsky and L. J. F. Hermans, Annu. Rev. Phys. Chem .50, 315 (1999). [7] L. V. Il’ichov, L. J. F. Hermans, A. M. Shalagin, and P. L. C hapovsky, Chem. Phys. Lett.297, 439 (1998). [8] A. M. Shalagin and L. V. Il’ichov, Pis’ma Zh. Eksp. Teor. F iz.70, 498 (1999). [9] E. Wigner, Z. f. Physikal Chemie 23, 28 (1933). [10] R. F. Curl, Jr., J. V. V. Kasper, and K. S. Pitzer, J. Chem. Phys.46, 3220 (1967). [11] P. L. Chapovsky, Phys. Rev. A 43, 3624 (1991). [12] G. Peters and B. Schramm, Chem. Phys. Lett. 302, 181 (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 and E. Ilisca, (2000), http://arXiv.or g/abs/physics/0008083. [15] S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear resonances in atom and molecular spectra (Nauka, Siberian Branch, Novosibirsk, Russia, 1979), p. 31 0. 13ortho paranm kVmk FIG. 1. Molecular ortho and para states. Bent lines indicate the rot ational relaxation. Vertical line shows an optical excitation. Vmkgives the ortho-para state mixing by an intramolecular perturbation. 14-200 -100 0 100 200012345 peak 2peak 1 G=1 G=6 Enrichment, β (%) Detuning, Ω (MHz) FIG. 2. Enrichment of para molecules, β, as a function of radiation frequency detuning, Ω, at ω= 130 MHz and two Rabi frequencies, G= 1MHz and G= 6MHz. 150 10 20 30 4001234567 Peak 1 Peak 2 Enrichment amplitudes, β (%) 0 10 20 30 400100200300400500 Peak 1 Peak 2 Rabi frequency, G (MHz) Width of the peaks, β (MHz) FIG. 3. Amplitudes of the peaks in enrichment, β(upper panel) and width (FWHM) of these peaks (low panel) for the case of ω= 130 MHz. 16-50 0 50-0.02-0.010.000.01 G=1 G=1.5 Relative conversion rate, γrel Detuning, Ω (MHz) FIG. 4. Relative conversion rate, γrel, for the case of degenerate ortho-para states, ω= 0and the two values of Rabi frequency, G= 1MHz and G= 1.5MHz. 17-200 -100 0 100 2000.00.20.40.60.81.0 peak 4peak 3 x10 G=1 G=6 Relative conversion rate, γrel Detuning, Ω (MHz) FIG. 5. Relative conversion rate, γrel, for the case of nondegenerate ortho-para states (ω= 130 MHz) at two Rabi frequencies, G= 1MHz and G= 6MHz. 180 20 40 60 801E-41E-30.010.11101001000 Peak 3 Peak 4 Amplitudes of conv. peaks, γrel 0 20 40 60 80020406080100120140 Peak 3 Peak 4 Rabi frequency, G (MHz) Width of conv. peaks, γrel (MHz) FIG. 6. Amplitudes of the conversion peaks, γrel, (upper panel) and widths (FWHM) of these peaks (low panel) for the nondegenerate case, ω= 130 MHz. 19
arXiv:physics/0011013v1 [physics.gen-ph] 6 Nov 2000Neutrino ﬂux in the rotating reference frame D.L. Khokhlov Sumy State University, R.-Korsakov St. 2, Sumy 40007, Ukraine E-mail: khokhlov@cafe.sumy.ua Abstract It is considered neutrino ﬂux in the rotating reference fram e. Due to the rotation of the frame, neutrino is observed as a superposition of two sta tes P-transformed one from another. Since P-transformation is forbidden for neutrino , in the rotating reference frame one can detect a half of neutrino ﬂux. Due to the rotatio n of the earth, the detector of neutrinos can measure a half of the solar neutrin o ﬂux predicted by the SSM that may provide a solution for the solar neutrino puzzle . Weak interactions violate P-invariance and C-invariance b ut conserve CP-invariance [1]. The operations of C-transformation and of P-transformatio n are forbidden for neutrinos, but the operation of CP-transformation is allowed for neutrino s. Take the rotating reference frame. Turnover of the frame at t he angleπcorresponds to P-transformation of the frame. Under rotation, the positio ns of the frame in the range from πto 2πare P-transformed from the positions of the frame in the rang e from 0 to π. Neutrino in the rotating frame is observed in the superpositional sta te \|ψ>=1√ 2(\|ν >+P\|ν >). (1) However P-transformation is forbidden for neutrino theref orer it is forbidden to detect neu- trino in the state P \|ν >. Hence when measuring neutrino ﬂux in the rotating frame it i s forbidden to detect a half of neutrino ﬂux. As known [2], [3] experiments measured solar neutrino ﬂuxes signiﬁcantly smaller than those predicted by the standard solar model (SSM), e.g. [4], [5]. Discrepancy between the theoretical and experimental neutrino ﬂux from the sun pres ents the solar neutrino puzzle. The Homestake experiment measured [6] an average37Ar production rate by solar neu- trinos in37Cl of Rate (Chlorine) = 2 .55±0.17(stat) ±0.18(syst)SNU (2) which is 32 ±5% of the 8 .1+1.0 −1.2SNU predicted by the SSM [5]. The Kamiokande II and III experiment reported [7] the data co nsistent with8B solar neutrino ﬂux of φν⊙= [2.9±0.2(stat)±0.3(syst)]×106cm−2s−1(3) which is 51% ±9% of that predicted by the SSM [5]. The GALLEX experiment measured [8] a71Ge production rate by solar neutrinos in71Ga of Rate (Gallium) = 79 ±10(stat) ±6(syst)SNU (4) 1which is 60% ±10% of the 131 .5+7 −6SNU predicted by the SSM [5]. The SAGE experiment measured [9] an average71Ge production rate by solar neutrinos in71Ga of Rate (Gallium) = 74+13 −12(stat)+5 −7(sys)SNU (5) which is 56% ±11% of the 131 .5+7 −6SNU predicted by the SSM [5]. Due to the rotation of the earth, the detector of neutrinos ca n measure a half of the solar neutrino ﬂux predicted by the SSM. This is consistent with th e data of the Kamiokande, GALLEX, SAGE experiments and signiﬁcantly reduces the disc repancy between the theoret- ical and experimental data for the Homestake experiment. So the eﬀect under consideration may provide a solution for the solar neutrino puzzle. References [1] E.D. Commins, P.H. Bucksbaum, Weak interactions of leptons and quarks (Cambridge University Press, 1983) [2] J.N. Bahcall, Nucl. Phys. B (Proc. Suppl.) 38(1995) 98 [3] N. Hata et al., Phys. Rev. D 52(1995) 3622 [4] S. Turck-Chieze et al., Apj. 335(1988) 415 [5] J.N. Bahcall, & M. Pinsonneault, Rev. Mod. Phys. 64(1992) 885 [6] B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38(1995) 47 [7] Y. Suzuki, Nucl. Phys. B (Proc. Suppl.) 38(1995) 54 [8] P. Anselmann et al., Nucl. Phys. B (Proc. Suppl.) 38(1995) 68 [9] J.N. Abdurashitov et al., Nucl. Phys. B (Proc. Suppl.) 38(1995) 60 2

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